Continuous Process Dynamics, Stability, Control and Automation 163463635X, 9781634636353

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CHEMICAL ENGINEERING METHODS AND TECHNOLOGY

CONTINUOUS PROCESS DYNAMICS, STABILITY, CONTROL AND AUTOMATION

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CHEMICAL ENGINEERING METHODS AND TECHNOLOGY

CONTINUOUS PROCESS DYNAMICS, STABILITY, CONTROL AND AUTOMATION

KAL RENGANATHAN SHARMA

New York

Copyright © 2015 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: [email protected]

NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers‟ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the e-book version of this book. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Continuous process dynamics, stability, control and automation / editors, Kal Renganathan Sharma. pages cm. -- (Chemical engineering methods and technology) Includes index. ISBN:  (eBook) 1. Chemical process control. I. Sharma, Kal Renganathan. TP155.75.C665 2014 660'.2815--dc23 2014044403

Published by Nova Science Publishers, Inc. † New York

To my eldest son, R. Hari Subrahmanyan Sharma (alias Ramkishan) who turned thirteen on August 13 th 2014

CONTENTS Preface

ix

About the Author

xvii

Book Description

xix

Chapter 1

Introduction

Chapter 2

Continuous Polymerization Process Technology

33

Chapter 3

Mathematical Process Models

59

Chapter 4

Continuous Process Dynamics

249

Chapter 5

Proportional, Proportional Integral, Proportional Derivative Feedback Control

323

Chapter 6

Frequency Response Analysis

369

Chapter 7

Advanced Control Methods

393

Chapter 8

Pharmacokinetic Analysis

487

Chapter 9

Instruments

527

Chapter 10

Nanorobots in Nanomedicine

545

Index

1

577

PREFACE Automation came about after mechanization . Chemical process control is a subset of the field of automation theory and practice. AIChE, American Institute of Chemical Engineers, New York, NY recently celebrated their centennial or 100th anniversary in 2008. Both the CPI and the industrial controls market size is increasing. Process Control as a required course in the chemical engineering degree plan in a number of US universities came out in the late 60s or 70s. I instruct CHEG 4033 Process Dynamics and Control, MCEG 3073 Automatic Controls and MCEG 3193 Introduction to Robotics, CHEG 3043 Equilibrium Staged Separations, ELMT 2043 Industrial Electronics etc. I have served as the chairman of Continuous Mass Club: Technical Community of Monstanto Minichapter between 19901993. This book is a natural outgrowth of these scholarly and research activities. A modern textbook is proposed that keeps pace with the progress in personal computing resources available to the student these days. Emphasis is placed on industrial application over rosy predictions. For instance, ideal PID control, proportional integral derivative control is not physically realizable. The competition of this book has a entire chapter devoted to PID control. Filters have to be added and then it is longer PID, it becomes FPID, filtered PID control! Discussions on overshoot is in excess of its occurrence in real applications. A careful scrutiny of overshoot can lead to the conclusion that it is more a mathematical artifact and less a engineering possibility It can be shown that overshoot per se violates second law of thermodynamics.. This book is more focused on the applications rather than on tuning and elloborate algebra. Calculus, Laplace transforms and other mathematical methods are used when necessary. A entire chapter is devoted to pharmacokinetic studies. Discussions on life systems draws the student attention more than hypothetical mathematical exercises. A entire chapter is devoted to control of robots. Advances made in nanorobots are also discussed in a separate chapter. Separate Chapters are devoted to mathematical models and process dynamics. Lessons in control are drawn from Trommsdorf effect, thermal wear design, 2 arm manipulator, Fukushima earthquake and deepwater horizon oil spill. Mathematical models can be used where pilot plant data is not available for scale-up. trouble shooting and in globalization. Process models can be classified as: A. Simulations from the Computer; B. Semi-Empirical Models; C. Mechanistic Models; D. Shell Balance Models; E. State Space Models; F. Dimensionless groups; G. Stochastic Models; H. Thermodynamic Analysis; I. Optimization Studies; J. Engineering Analysis and; K. Molecular Basis for Consitutive Laws. State space models can be used to describe a set of

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variables in vector form in terms of matrix equations. The stability of the system can be studied using Eigenvalues and Eigenvector analysis. A scheme of 7 simultaneous simple irreversible reactions were considered. The state space model to describe this system is given by Eq. (2.90). This system can be viewed as an integrating system since all but Eigenvalues are negative with 4 Eigenvalues 0 The composition of a copolymer as a function of comonomer composition, reactivity ratios and reactor choice were derived from the kinetics of free radical initiation, propagation and termination reactions. The copolymerization equation is obtained using the QSSA quasi-steady-state approximation. As an example, copolymer composition with 4 monomers as a function of monomer composition made in CSTR is illustrated 1 The copolymer composition is sensitive to reactivity ratios to a considerable extent. The copolymer composition equation for a multi-component copolymer with n monomers were derived. This methodology was generalized for n monomers. The general form of the equation was represented in the matrix form using linear algebra. The rate matrix and rate equation are given. The QSSA can be written in the vector notation. When the Eigen values of the rate equation become imaginary the monomer concentration can be expected to undergo subcritical damped oscillations as a function of time. The occurrence of multiplicity in model solutions was illustrated by calculation of launch angle of stream of water during firefighting and elbow-up and elbow down solutions in the solution of inverse kinematics of a 3 arm manipulator with end effector (robot) and for a copolymer for certain values of reactivity ratios. Use and significance of dimensionless groups such as Reynolds number, Prandtl number, Biot number, Nusselt number, Mach number, Fourier number, Fick number, Newton number, Maxwell number (mass), Venrnotte number, Sharma number (mass), Maxwell number (momentum), Sherwood number, Shcmidt number, storage number/Sharma number (heat), Peclect number were discussed. Weiner-Hopf integral equation can be used to estimate the effects of mixing in a CSTR, contiuous stirred tank reactor. This can be done by estimating the degree of mixedness from the variance in SPC, statistical process control charts. The compositional distribution of AN in copolymer can be used as input. J is minimized with respect to  and the Weiner-Hopf integral equation is obtained. Groot and Warren developed a mesoscopic simulation tool. Molecules are treated as spherical objects that can obey the Newton‟s laws of motion. Physical properties and EOS of polymeric substances were obtained from this analysis. Conservative, dissipative and random forces were taken into account. In the canonical ensemble, the Gibbs-Boltzmann distribution can seen to be the solution to the Fokker Planck equation. The transient conversion in a PFR, plug flow reactor was derived. For a reaction of first order assuming that the conversion is analytic in space and time the governing equation can be shown to be of the hyperbolic type. For the zeroth order reaction the governing equation was found to be a wave equation! The hyperbolic PDEs can be solved for using the methods of relativistic transformation of coordinates leading to modified Bessel composite function solution, method of separation of variables and method of Laplace transforms. Numerical solution procedures can be used when the equations become nonlinear such as in the case of free radical polymerization reactions. Transient analysis of Denbeigh scheme of reactions, first order reversible reactions, autocatalytic reactions, second order reversible reactions, reactions in series, Michaelis and Menten kinetics, reactions in circle kinetics are discussed in detail. The transient concentration of oxygen during diffusion and reaction in islets of Langerhaans are also

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discussed in detail. The conditions where the concentration can undergo underdamped oscillations were derived. The use of final time condition and physically reasonable energy balance considerations can lead to solutions that are bounded and within the scope of the second law of thermodynamics. The temperature profile in a PFR is also obtained. Energy balance under transient conditions are used in the analysis Transient dynamics of concentration, temperature separately and concentration and temperature both in a CSTR were discussed. The dimensionless group called Damkohler number in addition to conversion, dimensionless time, residence time were used to describe the transient output response from a CSTR. The hydrolysis reaction of ethylene oxide to ethylene glycol was considered in a CSTR. For incompressible flow and for a constant volume system the steady state and transient conversion were derived. The transient model solution is given. The conversion of ethylene oxide as a function of dimensionless time contoured at various values of Damkohler number are displayed CSTR with recycle was considered. The transient temperature in a mixing tank that is heated is given. A fourth order RungeKutta method was used for numerical integration of the ODE, ordinary differential equation. Temperature vs. Time in reactor as response to a step change in input is obtained and displayed. A state space model was developed to describe a jacketed CSTR. The output variables of interest are dimensionless conversion, dimensionless temperature and is give. If all of the Eigenvalues of the state space A matrix are negative then the system is expected to be stable. A system with one Eigenvalue zero and one Eigenvalue negative is called an integrating system. If the Eigenvalues are complex conjugates the system is considerd oscillatory. Sometimes the oscillatory output response may be subcritical and damped, underdamped or critically damped. The stability type and characterization of stability for each of these cases are given in Table 4.3. The transient concentration of initiator and monomer during free radical polymerization in a CSTR was obtained. DaI, Damkohler number (initiator) and DaM, Damkohler number (monomer) were introduced. The steady state conversions were obtained as a function of the Damkohler number. Equation used to describe the transient conversion of monomer is nonlinear. The conversion of initiator and monomer as a function of time is displayed in Figure 4.9. Multiplicity was found in model solution of conversion of monomer. Conversion of initiator was monotonic. It can be seen from Figure 4.9 for the parameters used in the simulation study the transient conversion of monomers undergoes a maximum value. The general form of prototypical first order and prototypical second order system were provided. The first order process is characterized by process gain constant, kp and process time constant, p. The second order process is characterized by the process gain, kp, damping coefficient,  and process time constant, p. When the damping coefficient,  > 1 the system is overdamped, when  = 1 the system is said to be critically damped and when  < 1 the system is expected to undergo a overshoot and is said to be underdamped oscillatory. The use of on-off controller in CHP, combined heat and power system was discussed in detail. On-off controllers are used to shut down the system when output variable is greater than the desired limit. The system is turned back on when the output variable is lower than desired limits. Control action that is proportional to the error measured is called P only, proportional only controllers. Off-set is seen in P only controllers. PI, proportional integral

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control is devised to get rid of the offset. The control action is proportional to the integral of the error generated from initial time to a given instant in time. In Example 5.1, the PI control action for prototypical first order process was discussed. The output function to a step change in input was obtained by inversion of Laplace transform expressions. A closed loop transfer function GCL(s) was derived for the combined system. The conditions when the system will exhibit an overshoot are derived. In Example 5.2 the closed loop transfer function of a hybridized feed forward and feedback control loop was obtained. How the blocks in a control block diagram can be simplified was illustrated. In Example 5.3, PI control of voltage supplied to automatic washing machine was discussed. A Toshiba patent discussed such a machine. Although the model equation predicts the relation between the rotor speeds with the applied torque, there is no model available for applied voltage on the motor and the applied torque. The controller will have to increase the voltage or decrease the voltage, check the angular speed against set point sp and adjust the voltage again till that set point angular speed is attained. PI controller was used. The closed loop transfer function GCL(s) was derived. Conditions for instability was when Gp(s)Gc(s) >> -1. The conditions for underdamped oscillatory conditions to arise during PI control of prototypical first order process were when; Where I is the time constant of PI controller, p is the time constant of prototypical first order process, kc is the controller gain constant and kp is the process gain constant. During PD control, proportional derivative control, the action taken is proportional to the derivative of the error between the measurement and set point. Ideal PID control is not realizable due to the numerator dynamics. The order of the polynomial P(s) is greater than the order of the polynomial in the denominator Q(s) in the controller transfer function. As will be discussed later, a filter of nth order can be added to make the PID controller physically realizable. The PI control of concentration during hydrolysis of ethylene oxide was discussed in Example 5.4. The output transfer function y(s) is given by Eq. (5.73). The conditions for underdamped response can be seen to be; The output transfer function for a system with PD control of prototypical first order process is given by Eq. (5.79). The output function, y(t) is given by Eq. (5.84). For any value of the tie constants and gains of the process and controller the output is stable and has a monotonic exponential decay. In Example 5.5 the PD control of concentration during the formation of ethylene glycol in a CSTR by hydrolysis process was discussed. The conditions for underdamped response are given by Eq. (4.93). PI control of concentration during formation of intermediate during reactions in series was discussed in Example 5.6. The stability of this system of PI control of intermediate product yield of chloroform, CHCl3 can be determined from the poles of Eq. (5.74). As the order of the polynomial in the denominator is increased beyond quadratic, cubic closed form analytical solutions are not possible. Numerical solutions are needed to obtain a solution of the polynomial of higher degrees. Routh stability criteria can be used to analyze the stability of systems such as the one encountered here. Conditions for instability were derived and displayed. Tyreus-Luyben oscillation based tuning was found to make the system less oscillatory for the parameters chosen. There are three important tuning parameters in PID control action. These are: (i) the control gain constant, kc; (ii) integral time constant, I; (iii) derivative time

Preface

xiii

constant, D. An algorithmic or trial and error approach can be used for obtaining optimal results from tuning the controller. The tuning of PID control was developed by Ziegler and Nichols [5]. This method is not widely used in the industry because the closed loop behavior tends to be oscillatory and sensitive to errors. The Tyreus-Luyben tuning parameters for PI and PID control are given in Table 5.1. Closed loop stability analysis is discussed for a number of systems such as: (i) PI control of prototypical first order process; (ii) PI control of prototypical second order underdamped and overdamped processes; (iii) PI control of prototypical third order process; (iv) PI control of Integrating Stable Systems; (v) PI control of marginally stable systems; (vi) PI control of systems that make a center and/or saddle pointl (vii) Best control strategy for systems with inverse response; (viii) P, PD, PI control for jacketed exothermic CSTR; (ix) P, PD, PI control for exothermic PFR; (x) P control of polymerization kettle; (xi) P control of CSTR with recyle and PFR with recyle; (xii) Best Control Strategy for Systems with Dead Time; (xiii) Feedback control during semiconductor processing; (xiii) Systems with periodic disturbance; (xiv) Optimization and control of intermediate product in reactions in series with first first order and second zeroth order, first with zeroth order and second with first order, both first and second first order; (xv) PI control of intermediate species in Denbeigh scheme of reactions; (xvi) Best control strategy for PCR, polymerase chain reactions; (xvii) Best control strategy for systems that obey Michaelis and Menten kinetics; (xviii) Best control strategy for systems that obey reactions in circle kinetics. The mathematical models, experimental trials and computer simulations that are undertaken to study the change of concentration of drug or other compounds of interest with time in the human physiology is called Pharmacokinetics. Application of pharmacokinetics allows for the processes of liberation, absorption, distribution, metabolism and excretion to be characterized mathematically. The absorption of drug can be affected by 14 different methods. The change with time of the concentration of the drug can be by three different types as shown in Figure 8.1: i) Slow absorption (A); ii) maxima and rapud bolus (B); iii) constant rate delivery (C). Pharmacokinetics studies can be performed by 5 different methods including compartment methods. The five different methods are non-compartment method, compartment method, bioanalytical method, mass spectrometry and population pharmacokinetic methods. The factors that affect how a particular drug is distributed throughout the anatomy are: i) rate of blood perfusion; ii) permeability of capillary; iii) biological affinity of drug; iv) rate of metabolism of drug and; v) rate of renal extraction. Drugs may bind to proteins sometimes. Single compartment models were developed for: i) first-order absorption with elimination; ii) fourth order absorption with elimination; (iii) second order absorption with elimination; iv) Michaelis-Menten absorption with elimination and; v) Reactions in circle Absorption with elimination. A system of n simple reactions in circle was considered. The conditions when subcritical damped oscillations can be expected are derived. A model was developed for cases when absorption kinetics exhibit subcritical damped oscillations can be expected. The solution was developed by the method of Laplace transforms. The solution for dimensionless concentration of the drug in single compartment for different values of rate constants and dimensionless frequency are shown in Figures 8.10 – Figures 8.14. The drug profile reaches a maximum and drops to zero concentration after a said time. The fluctuations in concentration depends on the dimensionless frequency resulting from the subcritical damped oscillations during absorption.

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At low frequencies the fluctuations are absent. As the frequency is increased the fluctuations in concentration are pronounced. The frequency of fluctuations were found to increase with increase in frequency of oscillations during absorption. A two-compartment model for absortion with elimination is shown in Figure 7.15. The concentration that has diffused to the tissue region in the human anatomy is also accounted for in addition to the concentration of drug in the blood plasma. The model equation for concentration of drug in the tissue is found to be a ODE of the second order with constant coefficients (8.79). The model solution is given in Eq. (8.81) and obtained by the method of complementary function and particular integral. Software has been developed for rhe implementation of the pharmacokinetic models on the personal computer. Ratio control is used in keeping the reactor adiabatic during the manufacture of flue gas. Heat of reactions from Boudard reaction and oxidation of carbon reactions are made to cancel out each other by control of the ratio of the CO2/air mixture. SPC, statistical process control methods are discussed including Deming‟s quality principles, QIT, quality improvement teams. IMC, internal model control uses the mathematical model developed for the process. Perfect control is achieved when model transfer function, M(s) is the reciprocal of process transfer function, Gp(s). A filter has to be added to make the system realizable. This keeps the numerator of the transfer function of the combined system at a lower degree of polynomial compared with the degree of the polynomial of the denominator of the transfer function of the system. Inversion of a process model alone may not be sufficient for good control. In order for the controller to be stable and realizable the process transfer function must be factorized. Examples are given. In order to make M(s) proper the order of the filter in some cases is increased. A lead/lag controller was considered by use of a model based transfer function as given. Inverse response is expected for the process. A first order filter was added to make the system more „proper‟. The order of the filter was increased to two. Feedforward control is different from feedback control. During feedforward control the load or disturbance is measured or gauged from other considerations and control action taken accordingly. Carrier corporation has patented [11] a feedforward control system for absorption chillers. The feedforward control method comprise of the following steps; (i) determine the disturbance transfer function; (ii) determine the capacity valve transfer function; (iii) measure the occurred disturbance; (iv) implement the feedforward control function. The block diagram for feedforward control of absorption chillers is shown in Figure 7.11. An example of feedforward controller for a resulation of fuel supply to a furnace was discussed. Estimation and control of polymerization reactors was discussed. Control of polymerization of reactors is a difficult task because of the exothermic nature of the polymerization reactions and higher viscosities of monomer/polymer syrups encountered in the reactor. The objective of the reactor control is to obtain superior product quality. Quality is characterized by different parameters. Good structure-property relations are needed for devising control strategies based on measurement of end-use property values. Technical hurdles during the development of mathematical models for polymerization reactors are the nonlinear equations encountered, sensitivity to impurities etc. Feedback control of a variable is not possible if the parameter cannot be measured or estimated. In Figure 8.13 is shown the scheme for possible feedback control configurations. The scheme comprises of state-model, subsystem 1, subsystem 2 and the measurement equations. State estimation techniques have

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xv

been developed to provide estimates of the state variables even in cases where they cannot be obtained by direct measurement. Estimation techniques such as Kalman filter, Weiner filter and extended Kalman filter may be used. In non-linear estimation problems the objective may be to minimize the average squared errors in the initial estimate in the process model and in measurement device. Neural networks can be used to approximate any reasonable function to any degree of required precision. Three different architectures for ANNs, artificial neural networks are possible. The behavior of each unit in time can be described using either differential equations or discrete update equations. The use of ANNs to control a distillation column is discussed. Robotics is introduced. Mathematics used to describe positions and orientations in 3 dimensional space is reviewed. Kinematics of mechanical manipulators are discussed. Forces, velocities and moments are discussed. Inverse kinematics of the manipulator with both the geometric and algebraic methods of solution is outlines. Trajectory analysis of end effector and control strategy for robots are outlined. Submarine nanorobots are being developed for use in branchy therapy, spinal surgery, cancer therapy, etc. Nanoparticles have been developed for use in drug delivery systems and for cure in eye disorders and for use in early diagnosis. Research in nanomedicine is under way in development of diagnostics for rapid monitoring, targeted cancer therapies, localized drug delivery, and improved cell material interactions, scaffolds for tissue engineering and gene delivery systems. Novel therapeutic formulations have been developed using PLGA based nanoparticles. Nanorobots can be used in targeted therapy and in repair work of DNA. Drexler and Smalley debated whether „molecular assemblers‟ that are devices capable of positioning atoms and molecules for precisely defined reactions in any environment is possible or not. Feynman‟s vision of miniaturization is being realized. Smalley sought agreement that precision picking and placing of individual atoms through the use of „Smalleyfingers‟ is an impossibility. Fullerenes, C60, are the third allotropic form of carbon. Soccer ball structured, C60, with a surface filled with hexagons and pentagons satisfy the Euler‟s law. Fullerenes can be prepared by different methods such as: (i) first and second generation combustion synthesis; (ii) chemical route by synthesis of corannulene from naphthalene. Rings are fused and the sheet that is formed is rolled into hemisphere and stitched together; (iii) electric arc method. Different nanostructuring methods are discussed. These include: (1) sputtering of molecular ions;(2) gas evaporation; (3) process to make ultrafine magnetic magnetic powder; (4) triangulation and formation of nanoprisms by light irradiation ; (5) nanorod production using condensed phase synthesis method; subtractive methods such as; (6) lithography; (7) etching; (8) galvanic fabrication; (9) lift-off process for IC circuit fabrication; (10) nanotips and nanorods formation by CMOS process; (11) patterning Iridium Oxide nanostructures; (12) dip pen lithography; (13) SAM, self assembled monolayers; (14) hot embossing; (15) nanoimprint lithography; (16) electron beam lithography; (17) dry etching; (18) reactive ion etching; (19) quantum dots and thin films generation by; (20) sol gel; (21) solid state precipitation; (22) molecular beam epitaxy; (23) chemical vapor deposition; (24) CVD; (25) lithography; (26) nucleation and growth; (27) thin film formation from surface instabilities; (28) thin film formation by spin coating;(29) cryogenic milling for preparation of 100-300 nm sized titanium; (30) atomic lithography methods to generated structures less than 50 nm; (31) electrode position method to prepare nanocomposite; (32) plasma compaction methods; (33) direct write lithography; (34) nanofluids by dispersion. Thermodynamic miscibility of nanocomposites can be calculated from the free energy of

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mixing. The four thermodynamically stable forms of Carbon are diamond, graphite, C60, Buckminster Fullerene and Carbon Nanotube. 5 different methods of preparation of CNTs, carbon nanotubes were discussed. Thermodynamically stable dispersion of nanoparticles into a polymeric liquid is enhanced for systems where the radius of gyration of the linear polymer is greater than the radius of the nanoparticle. Tiny magnetically-driven spinning screws were developed. Molecular machines are molecules that can with an appropriate stimulus be temporarily lifted out of equilibrium and can return to equilibrium in the observable macroscopic properties of the system. Molecular shuttle, molecular switches, molecular muscle, molecular rotors, molecular nanovalves are discussed. Supramolecular materials offers alternative to top-down miniaturization and bottom-up fabrication. Self-organization principles holds the key. Gene expression studies can be carried out in biochips. CNRs are a new generation of self-organizing collectives of intelligent nanorobots. This new technology includes procedures for interactions between objects with their environment resulting in solutions of critical problems at the nanoscopic level. Biomimetic materials are designed to mimic a natural biological material. Characterization methods of nanostructures include SAXS, small angle X-ray scattering, TEM, transmission electron microscopy, SEM, scanning electron microscopy, SPM, scanning probe microscope, Raman microscope, AFM atomic force miscroscopy, HeIM helium ion microscopy.

ABOUT THE AUTHOR Dr. Kal Renganathan Sharma received his B.Tech. in chemical engineering from Indian Institute of Technology, Chennai, India in1985 and MS and PhD in chemical engineering from West Virginia University, Morgantown, WV in 1987 and 1990 respectively. He underwent post doctoral research training at the lifesciences and microgravity space lab, Clarkson University, Potsdam, NY under the guidance of former chair and Prof. R. Shankar Subramanian. He completed an industrial post doctoral research training with Monsanto Plastics Technology, Indian Orchad, MA. There he was told that he was on their “fast track” to become a research fellow. He has obtained cash prizes for exemplary work from Monsanto Plastics, Indian Orchad, MA, SASTRA University, Thanjavur, India and Prairie View A & M University, Prairie View TX. As a naturalized US citizen he obtained a ministry of human resources and development permit to serve as Professor at deemed universities in India. He is sole author of 15 books and 7 book-chapters. Author of 53 Refereed Journal Articles, 553 Conference papers and 113 other presentations. He has instructed 2930 students in 109 courses. Service at HBCUs in this country, United States and deemed universities in India. Editorial Board Member of 7 journals. Fellow of Indian Chemical Society. Who‟s Who in America. Paper cited more than 300 times searcable by Google Scholar. Reviewed more than 60 journal articles. Developed new course and books in emerging areas. One article on “reactions in circle” in the open domain has been downloaded more than 1031 times and another article on nanorobots has been downloaded more than 980 times. In recent years, he has been instructing courses in the Houston, TX area at Texas Southern University, Houston, TX, Lone Star College, North Harris, Houston, TX, Prairie View A & M University, Prairie View, TX. Prior to that he served at SASTRA University, Thanjavur, India, Vellore Institute of Technology, Vellore, India, George Mason University, Faifax, VA and West Virginia University, Morgantown, WV. Titles held include Adjunct Professor, Professor, Head, etc. He has a linked in profile at http://www.linkedin. com/pub/kal-sharma/48/b69/466.

BOOK DESCRIPTION Continuous Process Dynamics, Stability, Control and Automation is a modern first course on process control, instruments, process dynamics and stability. MS Excel spreadsheets are used in order to obtain solutions to non-linear equations when needed and closed form analytical solutions where possible are obtained using Laplace transforms and other methods. The solutions are presented in 210 figures and the book has 1319 equations. With a industrial controls market size of about 150 billion dollars and a chemical process industry market size of 3 trillion dollars the practioners can use this book to master techniques of P, proportional, PI, Proportional Integral, PD, Proportional Derivative feedback control, feedforword control, hybrid control, adaptive control, internal model control, ratio control, filtered real proportional integral derivative control, ANNs, artificial neural networks, SPC, statistical process control. Control block diagrams are developed using MS Paint. Flavor for what is a continuous process is given using 18 process flow diagrams. Be it a feedback control of temperature in a mixing tank or a neural network design for a distillation column, the details and the big picture are both given. Pioneers who made this area possible such as Maxwell, Galileo, Sherwood, Levenspiel, Kalman, Laplace, Fermat, Damkholer, Newton, Fourier, Fick, Michaelis, Menten, Monod, Staudinger, Ziegler, Natta, Flory, Peclect, Bode, Nyquist, Biot, Bessel, Bernoulli (both father and son!), Euler, Stokes, Mach, Reynolds, Prandtl, Nusselt, Weiner, Hopf, Clapeyron, Clausius, Lorenz, Krebs mentioned where their theories were used in the analysis. Coverage Includes:   



Separate Chapters are devoted to Continuous Polymerization Process Tech nology, Mathematical Models, Continuous Process Dynamics, Pharmacokinetics, Instruments and Nanorobots Multiplicity in Model solutions discussed with Examples Eleven different modeling approaches are discussed: Computer Simulations, SemiEmpirical; Shell Balance; State Space Models; Dimensionless Groups; Stochastic; Thermodynamic Analysis; Optimization Routines; Engineering Analysis and Derivation of Constitutive Laws Copolymer Composition with n monomers derived using State Space Models

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                      

Control Lessons are drawn from Tormsdorff Effect, Regulation of Human Anatomical Temperature, 3 arm mechanical manipulator, centralized heating and cooling, Fukushima Earthquake, Deepwater Horizon Oil Spill Side by Side Comparison of Reactor Dynamics of Initiatiated and Thermal Polymerization Side by Side Comparison of Reactor Dynamics of CSTRs in Series and CSTR Dynamics of PFR and Wave Equation Continuous Mass Polymerization Process for Polystyrene, HIPS, High Impact Polystyrene, SAN, Styrene Acrylonitrile Copolymers, ABS, Acrylonirile, Butadiene and Styrene, PMMA, Poly Methyl Methacrylate, SMA, Styrene Maleic Anhyride, Condensation Processes for Nylon 6,6, Polyamide, Polyesters Gas Phase Catalytic Processes for Polyolefins and Ethylene Propylene Copolymer described as background Selectivity in Yield of Biodiesel over Glycerol in Consecutive-Competitive Reactions Computer Simulation of Centrifugal Separation of High Volume Oil and Water Control of Intermediate Yield of Chloroform during Chlorination Reactions in Series Supply Chain Robotics, Camera Drones Mathematical Artefact vs. Physically Reasonable Solution Overshoot Phenomena Frequency Analysis of Damped Wave Condution and Fourier Condution Equations Reactions in Circle Kinetics Michaelis and Menten Kinetics Nanorobots in Photodynamic Therapy of Alzheimers Disease Microwave Temperature Sensors, MEMS 9 different Viscometers Pharmacokinetics of Styrene in Rats, Alchohol in Brain, Paclitaxel in cancer patients, Single Compartment and Two Compartment Models Subcritical Damped Oscillations Taylor Series Solution, Runge Kutta Method, Closed Form Solutions Intraocular Pressure Control Example in Toshiba Washing Machine Solar Aided Combined Cycle Power Plant Continuous Pharmaceutical Production Annular Plug flow Reactor for Single Layer Graphene Sheet Production Kalman Estimation and Control of Polymerization Reactor Bioartifical Pancreas 20 Different Specifications in Binary Distillation and Desktop Computer

Chapter 1

INTRODUCTION 1.1. INDUSTRIAL CONTROLS - MARKETS The size of the worldwide market for industrial controls is expected to reach 150 billion dollars by the year 2019. Continuous processes have been selected over batch operations. Good reasons are lower cost, better product uniformity and sometimes result in less pollution to the environment. Process dynamics can be studied using desktop computers. Proactive process solutions can be offered. Stability analysis using a pencil and paper can lead to better perspectives on the range of operation of the process variables. Control and automation have grown tremendously with the advent of computers and programmable logic controllers, PLCs. More attention is paid by the lead engineer to the Plant start-up and Plant shut-down operations compared with steady-state operations of the plant. Transient analysis is not well studied. Transient behavior of chemical reactors, distillation columns, absorption towers, adsorption beds, extraction units and other unit operations needs to be better studied. Collegiate education methods have to keep pace with the developments in the field of industrial controls. Moore‟s law states that computing speed of microprocessors double every 18 months. Biological databanks double in size every 10 months. Mathematical methods for model development have been refined over centuries. The methods and means available to the engineer need be better utilized. Computer simulation and model development can be an integral part of an engineer‟s endeavors. The days when the effect of professionals who do mathematical modeling and computer simulation on the bottom line of the enterprise is only indirect are over. In the coming era the PW, Present Worth of chemical plants shall be higher because of the value added by an army of engineers and Ph.D. scholars who perform process dynamics studies and develop process models. They also develop control block diagrams, instrument the chemical plant with sensors and connect the sensor output using data acquisition to the desktop computer.

1.1.2. Chemical Process Industry Over 70,000 different products are manufactured in the chemical process industry. Be it batch or continuous process for production the CPI is in constant need for control engineers, specialists who understand the dynamics, stability of different unit operations and processes.

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Automation is an identified goal in recent years. The size of CPI is estimated at 3 trillion US dollars. Depending on the boom and bust cycles of the economy the growth rate of CPI is a fraction of the GDP. The fraction is about 0.87. The first product made since industrial revolution was sulfuric acid. The lead chambers process was the industrial standard for two centuries. The largest volume product made today is polyethylene. The CPI is a science based industry. The performance of the CPI in a given year depends on a number of factors. In 2013, cheap shale gas and a robust economy gave chemical producers a lift in terms of raw materials costs, utility costs, labor costs, tax and tariffs and interest on debt payments. President has urged Congress to increase the minimum wage rate for labor to $10.10 per hour. In Table 1.1 is listed the top 20 chemical producers in United States ranked by sales. Their assets are also provided. The products from the CPI include inorganic chemicals, plastics and petrochemicals, drugs and pharmaceuticals, soaps and detergents, paints and allied products, organic chemicals and phosphates, agricultural chemicals and miscellaneous chemicals.

1.2. EXAMPLES OF CONTROL APPLICATIONS Here are some examples of why continuous process dynamics, stability, control and automation are important.

1.2.1. Supply Chain Robots By the year 2019, the industrial controls market and industrial robotics market is expected to reach $147.7 billion worldwide. According to the market report prepared by Transparency market research in 2012 the size of the market is $102.2 billion. This would mean a compound annual growth rate of the industry, CGAR of 5.6% between 2013-2019. The largest end user is the automotive followed by the semi-conductor industry. The shipments of industrial robots in North America in the year 2000 were close to $1.02 billion. The discrepancy in the amount from $1.02 billion to $102.2 billion may be because of; (i) rapid growth rate and; (ii) differences in which machines that counts as robots amongst different regions of the world. Japan for example counts some machines as robots that in other parts of the world they do not. Industrial robot was recognized as a unique device in the 1960s. CAD. Computer-aided design, computer aided manufacturing, CAM systems, sensors, controllers hooked to computers using data acquisition are emerging trends in industrial automation. The definition of an industrial robot according to Information Standards Organization (ISO) in standard ISO/TR/8373-2.3 is the following: A robot is an automatically controlled, reprogrammable, multipurpose, manipulating machine with several programmable axes, which may be either fixed in place or mobile for use in industrial automation applications.

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Introduction Table 1.1. Top 20 by Chemical Sales in this Country, United States

Rank

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Company

Dow Chemical, Midland, MI ExxonMobil, Irving, TX DuPont, Wilmington, DE PPG Industries, Pittsburgh, PA Chevron Phillips, The Woodlands, TX Praxair, Danbury, CT Huntsman, Salt Lake City, UT Mosaic, Plymouth, MN Air Products, Allentown, PA Eastman Chemical, Kingsport, TN Honeywell, Morristown, NJ Celanese, Irving, TX Ecolab, St. Paul, MN Lubrizol, Wickliffe, OH Ashland, Covington, KY Dow Corning, Midland, MI CF Industries, Deerfield, IL Trinseo, Berwyn, PA Momentive Specialty Chemicals, Columbus, OH Occidental Petroleum, Los Angeles, CA

Chemical Sales (Billions of US $) 57.1 39.0 31.0 14.0 13.1 11.9 11.1 10.0 9.7 9.4 6.8 6.5 6.5 6.4 5.8 5.7 5.5 5.3 4.9 4.6

Assets of the Company (Billions of US $) 69.5 27.5 18.1 11.9 10.5 20.3 9.2 18.1 16.2 11.8 6.8 9.0 19.6 10.0 9.3 12.3 10.7 2.6 2.9 3.9

Per the Robotic Institute Association an industrial robot system is identified as the following: An industrial robot system includes the robot(s) (hardware and software) consisting of the manipulator, power supply, and controller; the end-effector(s); any equipment, devices, and sensors required for the robot to perform its task; and any communications interface that is operating and monitoring the robot, equipment and sensors.

Robots can be programmed to move objects through the entire 3D, three dimensional workspace. The mechanical manipulator has a certain number of links, joints and end effector. The pitch, roll and yaw of the wrist can be controlled. Robots can be used to cause changes similar to what can be done using the motor skills of the human arm. The study of this field of study is called robotics. Stanford University developed a book used as textbook in collegiate course on Introduction to Robotics: Mechanics & Control, J. J. Craig since 1983. Robotics comprises of four major areas:

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Kal Renganathan Sharma (i) Mechanical Manipulation; (ii) Locomotion; (iii) Computer Vision and; (iv) Artificial Intelligence.

Artificial intelligence development in the form of software is an attempt to synthesize human intelligence by computer programming and execution. Mechanical manipulation is an amalgamation of engineering mechanics, control theory and computer science. Progress made in robot programming languages can lead to more user friendly robots. Often times the interface between human and robot is the programming language. This is an important consideration in the design and operation of industrial robots. Library of robot-specific subroutines can be added. JARS, written in Pascal at NASA‟s Jet Propulsion Laboratory is an example. RAPID is a general-purpose language and software developed by ABB Robotics. The user is allowed to command specific sub-tasks of the goal directly. Manipulator programming takes some trial and error. Objects can be moved through 3D, three dimensional space using programming. VAL was written by Unimation in order to control robots. IBM developed AML. KAREl was developed by GMF robotics. Cartesian and interpolated motions can be affected. Vision systems can be used to specify the coordinates of an object of interest. Interaction with sensors is important. Some difficulties in programming robots are the congruence of model predictions and real time experience. This can lead to poor grasping of objects and collisions. Accuracy of manipulator is another worry. Force strategies needs to be developed for constrained motion. Manipulator programs are sensitive to initial conditions. Trajectory and velocity of arm depends on the initial conditions. Program segments need be tuned in a bottom-up programming approach. Changes in configuration can cause large arm motions. Errors in object location can also be causative in problems. Singularities can be identified by writing the Jacobian that can be used to relate the joint velocities to Cartesian velocities of the tip of the arm. For example, say the end effector is required to move with a certain velocity in Cartesian space. Inverse Jacobian can be used to calculate the joint rates from the Cartesian rates. For some values of the vector of joint angles, the jacobian becomes singular. Singularities in the boundaries can be found for most manipulators and have a loci of singularities inside the workspace.

1.2.2. Robotics as Collegiate Course Industrial robots came about in the 1960s. The adoption of robotic equipment came about in the 1980s. In the late 1980s there was a pullback in the use of robots. Use of industrial robots has been found to be cheaper than manual labor. More sophisticated robots are emerging. Nanorobot drug delivery systems can be one such robot. 78% of the robots installed in the year 2000 were welding and material-handling robots. The Adept 6 manipulator had 6 rotational joints. The most important form of the industrial robot is the mechanical manipulator. The mechanics and control of the mechanical manipulator is described in detail in introductory robotics courses [Craig 2005]. The programmability of the device is a salient consideration. The math needed to describe the spatial motions and other

Introduction

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attributes of the manipulators is provided in the course. The tools needed for design and evaluation of algorithms to realize desired motions or force applications are provided by control theory. Design of sensors and interfaces for industrial robots is also an important task. Robotics is also concerned with the location of objects in three dimensional space such as its: (i) position and orientation; (ii) coordinate system and frames of reference such as tool frame and base frames; (iii) transformation from one coordinate system to another by rotations and translations. Kinematics is the science of motion that treats motion without regard to the forces that cause it. In particular, attention is paid to velocity, acceleration of joints and acceleration of the end effector. The geometrical and time based properties of the motion are studied. Manipulators consist of rigid links that are connected by joints that allow relative motion of neighboring links. Joints are instrumented with position sensors which allow the relative motion of neighboring links to be measured. In case of rotary or revolute joints these developments are called joint offset. The number of independent position variables that would have to be specified in order to locate all parts of the mechanism is the number of degrees of freedom. End effector is at the end of the chain of links that make up of the manipulator. This can be a gripper, a welding torch, an electromagnet, etc. Inverse kinematics is the calculation of all possible sets of joint angles that could be used to attain the given position and orientation of the end-effector of the manipulator. For industrial robots the inverse kinematic algorithm equations are non-linear. Solution to these equations is not possible in closed form. The analysis of manipulators in motion in the workspace of a given manipulator includes the development of the Jacobean matrix of the manipulator. Mapping from velocities in joint space to velocities in Cartesian space is specified by the Jacobean. The nature of mapping changes with configuration. The mapping is not invertible at points called singularities. Dynamics is the study of actuator torque functions of motion of manipulator. State space form of the Newton-Euler equations can be used. Simulation is used to reformulate the dynamic equations such that the acceleration is computed as a function of actuator torque. One way to effect manipulator motion from here to there in a specified smooth fashion is to cause each joint to move as specified by a smooth function of time. To ensure proper coordination, each and every joint starts and stops motion at the same time. The computation of these functions is the problem of trajectory generation. A spline is a smooth function that passes through a set or via points. End effector can be made to travel in a rectilinear manner. This is called Cartesian trajectory generation. The issues that ought to be considered during the mechanical design of manipulator are cost, size, speed, load capability, number of joints, geometric arrangement, transmission systems, choice and location of actuators, internal position and sensors. The more joints a robot arm contain the more dexterous and capable it will be. But it will also be harder to build and be more expensive. Specialized robots are developed to perform specified tasks and universal robots are capable of performing wide variety of tasks. Three joints allow for the hand to reach any position in three dimensional spaces. Stepper motors or other actuators can be used to execute a desired trajectory directly. This is called the linear positional control. Kinetic energy of the manipulator links can be calculated using the Langrangian formulation. Both the linear kinetic energy and rotational kinetic energies can be tracked.

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Kal Renganathan Sharma Table 1.2. State of Art in Robots over a 40 Year Period Year 1974 1975 1977 1979 1982 1983 1986

1990/ 1991

1994 1996 1998

2000 2001 2002

2004 2005

Product World‟s First Microcomputer controlled Electric Industrial Robot, IRB6 from ASEA was delivered to a small mechanical engineering company in Southern Sweden. IRB6 – First Robot for Arc Welding First Robots installed in France and Italy IRB 60 – First Electrical Robot for Spot Welding; First Robot installed in Spain Robots introduced in Japan S2, New Control System. Outstanding, HMI, Menu Programming, TCP (Tool Center Point) and the Joy Stick introduced. Allows control of several axes. ASEA bought Trallfa Robot operations, Bryne, Norway. Trallfa launched the world‟s First Painting Robot in 1969. Sales boomed in the mid 1980s when the automotive industry started to paint bumpers and other plastic parts. IRB2000 – 10 Kg Robot. First to be driven by AC motors. Large working range. Great accuracy. ABB acquired Cincinnati Milacron, USA, Graco, USA (robotic painting), Rarsburg Automotive (electrostatic painting atomizers). IRB 6000 – 200 kg Robot introduced. First Modular Robot that is the fastest and most accurate spot welding robot on the market. Unique hollow wrist introduced on Painting Robots. Allows faster and more agile motion. S4 – Breakthrough in user friendliness, dynamic models, gives outstanding performance. Flexible rapid language. Integrated Arc Welding Power Source in Robot Cabinet. Launch of Flex Picker Robot, the World‟s fastest Pick and Place Robot. Robot Studio – First simulation tool based on virtual controller identical to the real one revolutionize off-line programming. Pick and Place Software Pick Master introduced. IRB7000 – First Industrial Robot to handle 500 kg. ABB – First company in World to sell 100,000 Robots Virtual Arc – True arc welding simulation tool that gives robot welding engineers full „offline‟ control of the MIG/MAG process. IRB 6000 – Power Robot with bend over backwards flexibility. IRC5- Robot Controller. Windows interface unit. Launch of 55 new products and robot functions included with 4 new robots: IRB 660, IRB 4450, IRB1600, and IRB260.

The evolution of robots can be studied from the market experience of one leading manufacturer of robots for example, ABB robotics, Zurich, Switzerland. This shall be used later to place in perspective the increased interest in use of nanorobots in the hospital for drug delivery. The different products from ABB Robotics during a 30 year period are given below in Table 1.2.

1.2.3. Infeasibility of “Molecular Assemblers” R.E. Smalley and K.E. Drexler debated on whether “molecular assemblers,” which are devices capable of positioning atoms and molecules for precisely defined reactions in any environment, are possible. In his book Engines of Creation: The Coming Era of

Introduction

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Nanotechnology, Drexler envisioned a world ubiquitous with molecular assemblers. These would provide immortality and lead to the colonization of the solar system. He received a Ph.D. from Massachusetts Institute of Technology (MIT) in 1991. He is also the CEO of Foresight Institute, Palo Alto, California. Smalley, a recipient of the Nobel Prize in chemistry in 1996 for his work on fullerenes, outlined his objections based on science to the molecular assembler idea and called it the “fat fingers” problem” or the “sticky fingers problem.” He was also worried about funding for nanotechnology due to the portrayed darker side of it. In Chemical and Engineering News, “Point–Counterpoint” column, an open letter from Drexler to Smalley was posted challenging Smalley to clarify the “fat fingers problem,” with a response from Smalley and three letters with Drexler countering and Smalley concluding the exchange. Drexler sought clarification from Smalley on the fat fingers problem. He felt that like enzymes and ribosomes, the proposed molecular assemblers neither have nor need the “Smalley Fingers.” Drexler alluded to the long-term goal of molecular manufacturing and its consequences, which can pose opportunities and dangers to long-term security of the United States and the world. Theoretical studies and implementation capabilities are akin to the preSputnik studies of spaceflight or the pre-Manhattan project calculations regarding nuclear chain reactions and are of more than academic interest. He referred to his 20-year history of technical publications in the area of chemical synthesis of complex structures by mechanically positioning reactive molecules and not by manipulating individual atoms. The proposal was successfully defended in the doctoral thesis on Nanosystems: Molecular Machinery, Manufacturing, and Computation and is based on well-established physical principles. Smalley responded with an apology should he have offended Drexler in his article in Scientific American in 2001. His call for using “Smalley Fingers” is impossible. A “Smalley Finger” type of molecular assembler tool will never work. Smalley pointed out the infeasibility of tiny fingers placing one atom at a time. This is also applicable to placing larger, more complex building blocks. As each incoming reactive molecule building block has multiple atoms to control during the reaction, more fingers are needed to ensure they do not go astray. Computer-controlled fingers will be too fat and too sticky for providing the control needed. Fingers cannot perform the chemistry necessary. He called attention to the mention of enzymes and ribosomes needed in the reaction medium. He quarreled with the vision of selfreplicating nanobot. Is there a living cell inside the nanobot that cranks these out? Water is needed inside the nanobot with the necessary nutrients for life. How do the nanobot pick the right enzyme and join in the right fashion? How do the nanobot perform error detection and error correction? He worried about the scope of the chemistry that the nanobot could perform. Enzymes and ribosomes need water to be effective. He mentioned that although biology is wondrous, a crystal of silicon, steel, copper, aluminum, titanium, and other key materials of technology could not be produced by biology. Therefore, without these materials how could a nanobot manufacture a laser, ultrafast memory and other salient components of modern society? Drexler applauded Smalley‟s goal of debunking nonsense in nanotechnology. He sketched the fundamental concepts of molecular manufacturing. He referred to Feynman‟s after-dinner visionary talk in 1959, discussed in Sec. 1.2, and nanomachines building atomically precise products. Feynman‟s nanomachines were largely mechanical and not biological. In order to understand how a nanofactory system could work, he considered a conventional factory system. Some of Smalley‟s questions reach beyond chemistry to systems

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engineering and to problems of control, transport, error rates, and component failure and to answers from computers, conveyors, noise margins, and failure-tolerant redundancy. Nanofactories contain no enzymes, no living cells, and no replicating nanobots, but they do use computers for precise control, conveyors for parts transport, and positioning devices of assorted sizes to assemble small parts into large parts when building macroscopic products. The smallest devices position molecular parts to assemble structures through mechanosynthesis or machine-based chemistry. Conveyors and positioners bring reactants together unlike solvents and thermal motion. Positional control enables a strong catalytic effect by aligning reactants for repeated collisions in optimal geometries at vibrational frequencies greater than terahertz. Positional control can lead to voiding unwanted side reactions. From transition state theory, for suitably chosen reactants, positional control will enable synthetic steps at megahertz frequencies with reliability approaching that of digital switching operations in a computer. When molecules come together and react, their atoms, being sticky, stay bonded to neighbors and thus do not need sticky fingers to hold them. Direct positional control of reactants is revolutionary and achievable. Mechanosynthetic reactions and its field have parallels in the field of computational chemistry. The flourishing of nanotechnology in 2003 suggests a bottom-up strategy using self-assembly. It used to be scaling down microscopic machines in 1959. Progress toward molecular manufacturing is achieved by research in computational chemistry, organic synthesis, protein engineering, supramolecular chemistry, and scanning probe manipulation of atoms and molecules. Scaling down moving parts by a factor of one million results in multiplication of their frequency of operation by the same factor. Progress in the United States on molecular manufacturing has been impeded because of the illusion that it is infeasible. He called for augmentation of nanoscale research with systems engineering effort and achievement of the grand vision articulated by Richard Feynman. Smalley concluded by observing that Drexler left the talk about real chemistry and went to the mechanical world. He felt that precise chemistry could not be made to happen as desired between two molecular objects with simple mechanical motion along a few degrees of freedom in the assembler fixed frame of reference. It was agreed that a reaction would be obtained when a robot arm pushes the molecules together but it may not be the reaction desired. More control is needed than mentioned about molecular assemblers. A molecular chaperone is needed that serves as catalyst. Some agent such as an enzyme is needed. A liquid medium such as water is needed to complete the desired chemical reactions. Smalley recalled a talk on nanotechnology he gave to 700 middle and high school students. The students were asked to write an essay on “Why I am a Nanogeek.” Smalley read the top 30 essays and he picked his favorite five. Half assumed that self-replicating nanobots were possible. What if the self-replicating nanobots fill the world, was the worry of some. However, they have been misinformed.

1.2.4. Trends in Industrial Robots Industrial robots are increasingly used. According to Prof. D. Rus, director of MIT‟s computer science and Artificial Intelligence Laboratory a tipping point in the field of robotics has been reached. There is a paradigm shift from robots as job terminators to robots as job creators. Companies report more productivity per worker with more usage of robots.

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Universal robot, specialized robots, numerically controlled milling machines are increasingly used. Electronics industry is reaching the trillion dollar mark rapidly. Automotive and semiconductor industries increasingly use the robots. Robots can be used in miniaturization operations. Nanotechnology products are expected to have a market size of 3 trillion dollars by the year 2015. Microarrayer and nanoarrayers are expected to be increasingly used in genome sequencing and making biochips. Mechanical micro spotting and ink jet printing use principles of robotics and automation. Apple is planning on investing $10.5 billion in supply chain robotics and lasers and new technology ranging from assembly robots to milling machines. These machines are used for mass production of iphones, ipads, and smart phones. Amazon has recently announced the development of drones for rapid door delivery of ordered items. With a service called „Prime Air‟, by 2015 drones are expected to be used in order to deliver packages less than 30 minutes. Pizza deliveries are also expected. Internet access is expected to be delivered by drones to remote areas. Robots have been made with a spectrum of capabilities. Researcher at University of California at Berkeley has developed a robot for drying and folding clothes from the laundry. In 2000, 78% of installed robots were used for welding or material handling. Samsung, Hewlett Packard, Apple and other competitors use robots for spray painting. Robotic Equipment is used to polish new iPhone and carve plastic MacBook‟s aluminum body. Other consumer electronics investments are $ 22 billion by Samsung, $3.7 billion by Hewlett Packard and $3.95 billion by Sony Corp. Apple has hired robot experts and engineers to oversee operation of high-end manufacturing equipment. When the baby-boomers reach the retirement age in the near future there is a shortage expected in skilled workers. Developments like Baxter that can be used to perform tasks of two workers can increase the usage of robots. Robots can be used to unassemble pipe from conveyor. Marlin steel use robots in order to produce wire baskets and sell it to car makers and pharmaceutical firms. Lear a major auto-parts maker near Detroit, MI, with nearly $15 billion in annual revenue uses robots developed by Universal Robotics to help screw together seats and put together electronics dashboards. Sensors, instrumentation and industrial automation are expected to be a tidy addition to modern chemical plants. Key Competitors identified in the market report by Transparency Market Research are: (i) Denso Wave Inc; (ii) FANUC Corp.; (iii) KUKA AG; (iv) Yaskawa Electric Corp; (v) Toshiba Machine Corp; (vi) Yogakoaw Electric Corp; (vi) ABB Ltd. (vii) Honeywell Intl.; (viii) Emerson Electric; (ix) General Electric; (x) Invensys PLS; (xi) Mitsubishi Electric; (xii) Rockwell Automation; (xiii) Siemens; (xiv) Omron Corp; (xv) Schneider Electric; (xvi) Kawasaki Robotics.

1.2.5. Drones Robo-roo is a robot built by engineers in Germany that can be made to hop like a kangaroo. The kangaroo's jump is unique and contains a mechanism where the jump speed can be controlled in an efficient manner. The Achilles tendon stretches from the kangaroo's head to its calf. Festo scientists in Germany have devised an elastic spring that can be used in order to enable the robo-roo's functionality much like the real Achilles tendon. The kinetic energy of the kangaroo is absorbed by the elastic spring when the robo-roo lands. The next jump is powered by the energy released from the spring. Electronic circuitry are being developed to store and release energy in a more efficient and controlled manner. The robo-roo

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is powered using rechargeable batteries. Bionics is an emerging field of using things found in nature as vision for designing objects with similar functionality. With two feet and tail when the robo-roo touches the floor there is provided a stable three point contact. Two motors are used to control the robots legs. The motors are placed in between the "hip" and "thigh" of the robo-roo. The spring is made of rubber. The jump is cushioned by the elastic spring which is intended to provide the same functionality as provided by the Achilles tendon. The robo-roo is built to provide more bounce. Drones are unmanned aerial vehicles that can be used for various purposes. It can be used to snap high-definition photos of apple orchad from up above from the skies. The acquired photos are processed using software and the health of the crops is analyzed. Corners of the field where insects are attacking the apples are identified. Drone turns its path and heads to that area. Pesticides are dispensed from the wings of the drones. The insects that harming the trees are killed. Drone is back on its patrol. This is but one example in an emerging field of precision agriculture. The farms can be as large as 10,000 acres. The farmers can use this technology in order to keep an eye on their crops. For now, the satellite pictures are used. Resolution of satellite pictures to the 6th coordinate is tedious! Drones may be used by the farmers every day and lead to use of less water, less fertilizer and optimal use of pesticides. Drones can be made to perform 'door-delivery'. The customer calls the apple warehouse and orders the apple for door delivery. One half hour later the drone lands at your doorstep with the apple. The buzz of the drone can act as a door-bell. The apple is left at the doorstep and the drone levitates away into the sky. Drones can be used in conjunction with other police activity. Crimes as they are committed can be spotted and if possible changed into acceptable activities. Parking tickets can be issued using drones. Life saving medicines can be supplied to hospitals using drones during medical emergencies. Drones may be used to fight fires. Drones can be used in warfare amongst nations. One in three American military aircraft are drones. They come in various sizes. The MQ-9 Reaper can be used in order to carry about 3000 pounds of weaponry. In this country Unites States, drones are used to combat terrorism. Al Qaeda and the Taliban in Pakisthan and Afghanisthan are targets. The perpetrator of the 9/11 bombings Osama Bin Laden is now dead by a commando mission viewed by the White House. Drones are operated from remote locations. Hence it is a case in automation! Drones seldom appear in man vs. machine controversies. Drones were first built in the early 1900s. Hollywood finds some use for drones. Filmmakers like Steve Spielberg, Richard Attenborough for example capture footage using “camera drones” that would otherwise be dangerous and expensive to shoot. Action scenes that were shot using helicopters in films were Gemini Ganesan was the hero in India are now taken by camera drones. Helicopters are expensive and the pilots have to perform maneuvers in order to fly low to get a better shot. The costs can be as high as $10,000 per day. Camera drones can be less expensive and can be sent close to the action without causing danger. The cost of a camera drone is $25,000 and can be used over and over again. Sporting events can be filmed using camera drones. Flight control is affected using software to determine if the motor needs more or less power in order to keep the drone steady. Radio control is used to let controllers say to the drone what to do such as „fly faster‟, „begin descent‟, etc. Navigation is accomplished using GPS and a barometer. The atmospheric pressure is measured using the barometer. Flight control software receives as input the location of the drone and path it can take in order to return home.

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Camera drones is a solution that came from a $ 5 million research grant study conducted by WWF, world wildlife fund to the wildlife crime problem in Africa and Asia. $5 million grant was received from Google. The University of Minnesota Uninhabited Aerial Vehicle Laboratory estimates that most farmers will use camera drones in the next five years. Animal conservation group uses camera drones to “animal-sit” rhinos, elephants and tigers. Scientists can track down poachers who kill the endangered animals and without impunity sell their horns, tusks and eyes to be used in Chinese medicines.

1.2.6. Runaway Reactions in Polymerization Kettles An interesting challenge in modeling and control of mass polymerization reactors is the runaway polymerization under certain conditions. This is called the Trommsdorff effect. Manufacturing plants that use suspension polymerization plant has encountered a problem during operation where the reactor was “Set-up”. During free-radical polymerizations there are three important sets of reactions: (i) Initiation; (ii) Propagation and; (iii) Termination. As the polymerization reaction proceeds in suspension kettles, the viscosity of the polymer mass increases. The termination reactions are hindered. The polymer chain grows unbounded in size. This has “set-up” the reactor. This effect is also called auto-acceleration effect. It can be considered as a runaway reaction. The action taken to prevent such occurrences during continuous polymerization of styrene or copolymer of styrene and acrylonitrile a solvent is added. A viscometer may be added to measure the viscosity of the reaction mass. The control action taken can be to decrease the flow of the monomer supply or decreasing the reactor temperature or decreasing the reactor pressure, should the viscosity of the mass remain high. The termination reactions during the gel effect do not take place on account of diffusion limitations. The growing chains have to diffuse and meet in order for the chain recombination mechanism for termination to take place. When the monomer conversion is high in the reactor and there is no solvent or diluent, the viscosity of the reaction mass will be high. Studies have shown an exponential increase in viscosity with conversion. The diffusion process in higher viscous paths will take a lot longer. Without termination the molecular weight grows in an unchecked manner. This causes the gel effect. Gharaghani et al. [2012] developed a control strategy for a two-stage bulk styrene polymerization. They conducted dynamic simulation in order to predict the performance of auto-refrigerated CSTR and tubular reactors. The PFR has temperature controlled jacket zones. In one control strategy they used the jacket temperature and in another strategy the reactor temperature at the exit of each section was used as the controlled variable. GA, genetic algorithm was used in order to obtain the set points for polymer grade transition after optimization. Optimization goals were; (i) higher monomer conversion; (ii) final number averaged molecular weight of the polymer and; (iii) narrower molecular weight distribution. High heats of reaction are associated with mass polymerization of styrene. High viscosity and hence laminar flow are also other features of mass polymerization of styrene. PFR, plug flow reactor with recycle or CSTRS, continuous stirred tank reactors are used for industrial styrene homopolymerization and copolymerization. Dynamic simulation studies of a twostage continuous bulk polymerization process was undertaken. The first stage was auto-

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refrigerated CSTR and the second stage was a tubular reactor. The tubular reactor was subdivided into three temperature-control jacket zones. Monomer conversions, X, number and weight averaged molecular weights and PDI poly dispersity index can be predicted using the mathematical model. The simulation of Chen [1994] is for a basis of production of polystyrene at 5682 kg/h. Conversion in the auto-refrigerated CSTR was 65-75% and that in the second tubular reactor was at 90%. The gel effect/auto acceleration phenomena were modeled by Hui and Hamielec [1972]. The model can be written as follows, neglecting chain transfer effects; kt = kt0exp[-2(A1X + A2X2 + A3X3)]

(1)

kt0 = 1.225 x 106 exp(-844/T)

(2)

A1 = 2.57 – 5.05 x 10-3T

(3)

A2 = 9.56 – 1.76 x 10-2T

(4)

A3 = -3.03 + 7.85 x 10-3T

(5)

The termination rate constant as a function of conversion is given by Eq. (1) during the auto acceleration period. It can be seen to decrease exponentially with conversion. The temperature dependence of the preexponential factor is given by Eq. (2). Eqs. (3-5) can be used to obtain the temperature dependence of the conversion dependence of the termination rate constant. The thermal initiation rate constant is given by; kh = 0.219exp(-13910/T)

(6)

The propagation rate constant is given by; kp = 1.051x104exp(-3557/T)

(7)

The density of monomer as a function of temperature is given by; 

m = 924 – 0.918(T-273.1)

(8)

The number average molecular weight is later derived in Chapter 4.0 for thermal polymerization of styrene as; Mn = kp/SQRT(ktkhCM)

(9)

The number averaged molecular weight was obtained at different conversions and plotted in Figure 1.0. Microsoft Excel 2010 was used for the simulations. The temperature of the reactor chosen was 100 0C. The results are shown in Table 1.3 below;

13

Introduction Table 1.3. Molecular Weight Building during „Runaway Condition‟

T

C 100 0 C

K 373 0K

A1 A2 A3 kt0

0.6677 2.9952 -0.0833 1.27E+05

m3/mol/s

kh kp Density of Styrene CA0

1.3952E-17 0.759 832.3 8.00

m6/mol2/s m3/mol/s kg/cu.m mol/lit

X 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95

C0 m3/mol/s kt 1.275E+05 1.175E+05 1.051E+05 9.123E+04 7.691E+04 6.295E+04 5.003E+04 3.862E+04 2.896E+04 2.110E+04 1.493E+04 1.027E+04 6.863E+03 4.458E+03 2.815E+03 1.728E+03 1.032E+03 5.988E+02 3.380E+02 1.856E+02

Mn 2.011E+05 2.149E+05 2.335E+05 2.578E+05 2.895E+05 3.305E+05 3.837E+05 4.532E+05 5.447E+05 6.666E+05 8.310E+05 1.056E+06 1.370E+06 1.818E+06 2.471E+06 3.454E+06 4.999E+06 7.577E+06 1.235E+07 2.357E+07

mol/lit CM 8.003 7.603 7.203 6.802 6.402 6.002 5.602 5.202 4.802 4.402 4.001 3.601 3.201 2.801 2.401 2.001 1.601 1.200 0.800 0.400

Development of control strategy for a process such as shown in Figure 1.0 is a challenge. Solvent can be used to keep the diffusion paths of growing free radicals short, allowing for termination reactions to proceed with recombination. Online viscometers can be used to check the viscosity of the batch. When the viscosity is very high jacket cooling of cooling by vaporization of solvent/monomer can be called for.

1.2.7. Electric Motor Controls In United States, $50 billion is spent every year on electricity for electric-motor driven systems. Department of Energy estimates an energy savings of about $250 million per year by using motor drives in the past decade. Motors are made in a manner that the motor speed and torque developed needs to be controlled depending on the objectives at a given time. Motor direction, speed and acceleration or negative acceleration times can be controlled. Motors can be powered by AC alternating current or DC direct current. In AC motors work is produced in order to drive a load using a rotating shaft. The amount of work that is produced varies as a function of the amount of torque that can be developed using the designed shaft and the

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Kal Renganathan Sharma

angular speed in RPM, revolutions per minute of the shaft. Motor drives can be used to control the speed and torque of a motor. Motor drive is built using solid-state components and is an electronic unit. It can be used to control the speed of a motor. These drives are used in HVAC, heating ventilation and air conditioning systems. Motor control functions include speed control, acceleration, negative acceleration, motor-start boosting, fault tolerating, programming speeds, different pause methods and several other control functions. Due to the complex dependence of the operation of motors for different loads under different conditions, no one mathematical model can be used to describe any possible conceivable scenario of operation. Therefore reliable measurements are needed to achieve the control objectives. Motor current, motor voltage, drive temperature and other operating conditions are parameters that needs to be monitored. Safety considerations leads to a mechanism of power shut-off when there is an apparent problem. Conditions and faults are allowed in some designs to be displayed. A PLC, personal logic controller or PC personal computer can be used to include additional specialty functions to be sent to the drive through on-board communication.

Figure 1.0. Gel Effect as Seen in Molecular Weight Increase with Conversion.

Introduction

15

The speed of an AC motor is controlled by tweaking the frequency in Hz input into the motor. The frequency range can be from 0 – 100s of hertz. They are programmed to operate at a minimum and a maximum operating speed in order to proactively avoid damage to the load or internal contents of the motor. Operation at more than 10% of the rated capacity in the name plate can lead to damage. Volts per hertz (V/Hz) is a ratio of the voltage and frequency found in a motor during its operations [Rockis and Mazur, 2009]. The motor torque can be controlled using the V/Hz ratio that is applied to the motor. The motor is allowed to develop the rated torque when the V/Hz ratio is maintained. During the acceleration phase, constant torque is delivered as voltage is increased at the same rate as the frequency is increased. After a certain frequency the voltage remains a constant for further increased in frequency. The normal operating speed zone lies at frequencies less than the frequency cut-off. At frequencies greater than this cut-off value the motor is operating at an above normal rating. At the cut-off frequency the motor is said to deliver the full motor voltage. Torque is constant in the linear range of operation. At frequencies higher than the cut-off the motor drive is in a regime of constant output horse power. PWM, pule width modulation can be used in order to control the amount of voltage sent to the motor. The amount of voltage produced has to be controlled in order to control the speed and torque of the motor. Motors can be turned ON or OFF using magnetic starters. They can be used to provide overload protection. Voltage control can be accomplished by using resistances in series to the motor windings. Current control can be accomplished by shunting resistance in parallel to the motor windings. The intensity of the magnetic field can be checked for peak operation. The cost of control is roughly the same when the starter or drive is used. It is more a function of the hp rating of the motor. A potentiometer can be used to control the speed of a motor. The resistance of the potentiometer can be from 5 k- 10 k. Most drives allow a voltage or current input in order to control the speed of the motor. 0 – 10 Volt of direct current supply voltage can be connected to the input terminals of the motor. At 5 V, the motor runs at 50% of the rated speed, and at 7.5 V, the motor runs at 75% speed. 4 mA – 20 mA direct current supply can be connected to the input terminals of the motor drive. The motor runs at 50% speed at 8 mA and 75% speed at 12 mA. Standard control-circuit power is supplied in the 0 – 10 V and 4mA – 20 mA range. Switches are placed in certain positions and the motor control and drive functions are set using programming. Keypad is located on the drive and may be removable. Motor drives can contain up to 30 – 100 parameters. Some of the parameters that can be displayed are; (i) drive output frequency; (ii) motor current draw; (iii) drive output voltage; (iv) drive DC bus voltage; (v) drive internal temperature; (vi) motor fault; (vii) drive elapsed operating time; and (viii) motor operating speed.

1.2.8. Centralized Heating and Cooling The modern homes are equipped with a thermostat to set the desired temperature of the residence (Tsp). Valves to the furnace and air conditioner can be controlled by using measurement of temperature in the residence (Tm ). When the measured temperature, Tm < Tsp the furnace is turned on and the cooling is shut-off. When Tm > Tsp the furnace is shut-off and the air conditioner is turned on. Such a control method is called feedback control. This shall be discussed in more detail later. The block diagram for such control action is shown below in Figure 2.0. Air from the blower is heated by the furnace or cooled by the air-conditioner as

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Kal Renganathan Sharma

the case may be. The heating or cooling action depends on the error signal and whether the measured temperature, Tm is lower or higher compared with the set point temperature, Tsp. Thermocouples are used to measure the temperature in the room. Comparison of measured room temperature to set-point is performed by a comparator. The feedback step is the measurement of actual room temperature using temperature sensor (output) with the input (set-point). Figure 2.0 contains the essential elements of a block flow diagram. These are;          

Comparators Blocks Reference sensor Output sensor Controller Input/reference signals Plant operating system Feedback loop Disturbance signal Output signal

1.2.9. Regulation of Human Anatomical Temperature The body temperature in humans can be seen to be held in dynamic balance by; (i) the generation of heat by metabolic activities within the human anatomy and (ii) by transfer of heat outside the human anatomy to the surroundings. (iii) The heat gain, heat storage and heat transfer mechanism coexist in human anatomy. As can be seen from Figure 3.0 there are five layers between the surroundings and the blood [Sharma, 2010]. The thermophysical properties of the blood, skin, fat and bone are different from each other. The modes of heat transfer can be molecular heat conduction, heat convection, heat radiation and by a fourth mode of heat transfer called damped wave conduction . Metabolism in one word includes all the chemical reactions taking place within the human anatomy. Energy is liberated from chemical reactions that are exothermic. This is used to sustain life and to perform the various functions, basic and chosen. Work is done by human anatomy. The minimal rate of metabolism needed to sustain life is referred to as basic rate of metabolism. This rate is obtained while the patient is awake and resting and is at a stress-less state. Digestive activities should cease, the external hot weather does not cause any heat exchange or thermoregulation. There is enough energy generated at this state for the heart to pump blood throughout the human anatomy, retain normal electrical activity in the nervous system, and generate calories of energy. The basic rate of metabolism can be measured using the rate at which oxygen is consumed and the energy generated from metabolism of oxygen. Some work done by human anatomy is allowed. The energy needed for metabolic activity is obtained from chemical reactions that get coupled resulting in a net decrease in free energy. The basic rate of metabolism in a average patient is roughly, 75 watts. The major organs such

Introduction

17

as brain, skeletal muscle, liver, heart, gastrointestinal tract, kidneys, lungs, etc, participate in the base metabolism. The muscles in the human skeleton requires less energy at the rest state compared with the state of exercise. When the patient is asleep the metabolic rate falls below the basic rate of metabolism. The metabolic rate at all other activities such as walking, sitting, mating, eating, cooking, growing, etc are higher than the basic rate of metabolism. The rate of metabolic rate can exceed the basic rate of metabolism by a factor of 10-20 during strenuous exercise.

Figure 2.0. Centralized Heating and Cooling System in Homes using Feedback Control.

Figure 3.0. Heat Conduction through Skin, Fat, Muscle and Bone Layers from Surroundings to Blood Flow in Human Anatomy.

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The basic rate of metabolism in homo sapiens varies with the body mass m, as m0.75 . The relationship between the basic rate of metabolism and human anatomical parameters can be expressed in terms of the surface to volume ratio of the patient as

S V

1.5

.

Within the human anatomy, mechanisms are in place that will take effect to cool the anatomy when the average temperature reaches 350 C. The average temperature within the human anatomy is usually, 370 C. Au contrairie, when the skin temperature reaches below 22.5 0 C, cellular mechanisms will take effect that will result in generation of heat. The core human anatomical temperature is maintained within a narrow range by use of insulation and heat production. Two mechanisms that can cause cooling within the human anatomy are: i) vasodilation and; ii) evaporative cooling affected by sweat. After sternous exercise, on account of vasodilation, the skin exterior appears a bit reddish. The blood near the skin surface gets refrigerated and flows back to the veins and arteries thereby affecting energy transfer. The human anatomy reduces heat loss in the temperature range of 24 – 32 0 C by reduced blood flow to the dermis. Below 24 0 C, vasoconstriction mechanism is not sufficient and the heat production is by shivering or physical activity. There appears a set point in thermoregulation. Regulatory process is a bit more complicated than a first order feedforward control process. Transient receptor potential ion channels are sensitive to hot and cold temperatures. TRP channels get activated upon control action from the hypothalamus and stimulate the nerves. Nerve signals and harmone signals result in vasodilation/vasoconstriction or blood flow regulation and changes in metabolism and heat generation. Block flow diagram of regulation of human anatomical temperature is shown in Figure 4.0. The thermophysical properties of the biological tissues and other materials are provided in Sharma (2010). The role of fat under the skin in the human anatomy as an insulator can be evaluated using the thermophysical properties provided. Consider thicknesses of skin, fat, muscle and bone of 2.5 mm, 10 mm, 20 mm and 7.5 mm respectively. The thermal conductivity of the muscle and bone are 0.5 W/m/K and 0.6 W/m/K respectively. The effect of the layer of fat on the heat flux from the human anatomy can be evaluated as follows. The governing equation for steady state temperature in the composite assembly of skin, fat, muscle and bone can be written as follows; d 2T dz2

(10)

0

The temperature profile can be seen to be linear with respect to the space coordinate. The heat flux can be seen to be a constant throughout the composite assembly. The effective thermal conductivity of the composite assembly can be written as; L keff

Lskin

L fat

Lmuscle

Lbone

kskin

k fat

kmuscle

kbone

(11)

19

Introduction

Figure 4.0. Regulation of Human Anatomical Temperature seen as Feedforward Control.

Where kskin, kfat, kmuscle and kbone are the thermal conductivities of the skin, fat, muscle and bone respectively. Examples of insulators of heat used to cover the human anatomy are fur, hair and sweat. The effective thermal conductivity of layer of hair on the human skull can becalculated from the idealized model where the hair was reduced to a composite of cylindrical fibers aligned parallel to the axis parallel to the flow of air. Let the thermal conductivity of fiber and air be taken as kfiber, kair respectively. The effective thermal conductivity of composite assembly of hair and air was shown to be (Sharma, 2010); keff , zz kair keff , xx kair

k fiber

1

kair

(12)

kair

2

1 k fiber

kair

k fiber

kair

k fiber

kair

k fiber

kair

(13) 0.30584

4

0.013363

8

.....

Where  is the volume fraction of the fibers. The above expressions tend to capture the role of fat and hair on the heat insulation process.

1.2.10. Feedback Control of 3 Arm Robotic Manipular with End Effector Consider a 3 arm robotic manipulator with end effector. This can be programmed to perform tasks such as to pick a bolt from the table as shown in Figure 5.0. The manipulator is instrumented with sensors at each joint to measure the joint angle. The joints are revolute [Craig, 2005]. Each joint has an actuator that can be made to apply a torque on the neighbouring link. Each joint has a position sensor. Velocity sensors or tachometers are also

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Kal Renganathan Sharma

present at these joints. The manipulator joints follow a prescribed position trajectory. The actuators are commanded in terms of torque. A feedback control system can be deployed. The appropriate actuator commands that will realize the desired motion can be given. The feedback from the sensors in the joints is used to accomplish this task. The torque required is computed. This feedback control loop is shown in Figure 5.0 A vector of joint torques, , is given as input into the robot from the control system. The sensors from the joints measure the joint angles, , and joint velocities. ‟. This is sent as signal to the control box. The number of parameters sent as a vector depends on the number of joints, N in the system. The torque is calculated in the control box from the information input from the trajectory generator and by comparison with the measurements received from the sensors in the joints. A dynamic model can be used if necessary. The feedback is used to detect any servo error . The control action taken has to be in such a fashion that the robotic system is stable.

1.2.11. Lessons in Modeling and Control from Fukushima Earthquake On March 11th 2011, the earth shook for more than 2 minutues in Iwaki, Japan. Skyscrapers began to oscillate and buildings collapsed. Tsunami came in. Earthquake of the order of 8.9 on the Richter scale is the most severe earthquake Japan has suffered in seismic history. The ocean floors heaved and the water came all the way into the living areas. Floods and fires came about. Four tectonic plates are near the island nation of Japan. This tsunami was even of a bigger magnitude compared with the one in Indian Ocean in 2004. Electric power was completely cut off. The Fukushima nuclear power plant (Figure 7.0) was damaged during the earthquake. Seawater was directed onto the fuel during the meltdown of the nuclear power plants. There was no better control action in place. Called The Fukushima 50 the firefighters were risking death and were braving 250 millisieverts of radioactivity. This is five times more than the permissible dose. The damaged nuclear reactors spewed radiation. There are no controls in place although that geographical area is prone to earthquakes.

Figure 5.0. 3 Arm Manipulator with End Effector Picking a Bolt from Table.

Introduction

21

Figure 6.0. Block Diagram of Feedback Control of Robot Picking a Bolt.

Figure 7.0. Fukushima Nuclear Power Plant.

The nuclear energy may be the way to go for power plant needs in the future when oil reserves are depleted. Less pollution is caused on account of nuclear power plants. Safety issues remain after the 3 mile island, Chernobyl and Fukushima accidents. The Fukushima disaster included damage of three units of BWRs, boiling water reactors. The heat transfer from core to outside was lost. Nuclear plants are equipped with earthquake recognition

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Kal Renganathan Sharma

systems. Chain reaction shut down or emergency shut down procedures are outlined in cases of earth quake. The reactor is switched off and the runaway reactions are stopped. In Fukushima plant only three of the six reactors were functioning at the time of the earthquake. The generators were flooded. The cooling system lost power supply. The heat generated increased in the reactor core. The zirconium fuel rod cladding reacted with the steam and oxygen and hydrogen were produced. This may have caused the explosion with release of steam polluted with radioactive isotopes into the atmosphere. A massive volume of smoke spread the radioactive particles throughout the European Union. People evacuated within a 30 km radius of the power plants. The evening television news showed people traveling overseas on account of the tsunami and with the yen being strong. 133 Cs, 137Cs and 131I radioisotopes dosage received by the animals close to the accident were high. The implications of the accident in future design, operation and control of similar power plants are several. First a better project site can be selected with water, land and other requirements needed for a viable and profitable operation of a project. The structures needs to be strengthened against natural disasters. The environmental damage from the accident was infected clouds and leakage of radioactive particles on soil. Entire Xeon gases, 60% of Iodine, 40% of Caesium, 10% of Tellurium and 5% of reminder of the radioactive particles were released to the atmosphere. The amount of radiation released was 8 x 1015 Bq.

1.2.12. Deepwater Horizon Oil Spill The disaster that changed the price at the gasoline pumps in summer of 2010 was in part beacause they did not have better control and sensors in place than they did. The explosion of the Deepwater Horizon drilling rig claimed 11 lives. The Gulf region lost its economic engine and left thousands with mental and physical problems due to stress and pollution. Five million barrels of oil gushed into the ocean for 86 days. Tourism dropped in the Gulf of Mexico. Offshore drilling can be better instrumented and have alarms in place and better controlled. Deepwater Horizon was a semi-submersible, ultra-deepwater offshore oil drilling rig built in 2001 by Hyandai Heavy Industries, South Korea. The rig was used to drill the deepest oil well in history at a vertical depth of 10,683 m in Gulf of Mexico 250 miles southeast of Houston, TX. An explotion on the rig caused by a blowout ignited a fireball visible from a distance of 56 km. The fire was inextinguishable and the Deepwater Horizon sank. The oil well was gushing at the sea floor and caused the largest offshore oil spill in the history of United States. The oil spill causes a lot of damage to the environment and fisheries industry. Transocean received an early partial settlement of $401 million for the loss of Deepwater Horizon. The gushing of oil was at about 62,000 barrels per day on the average and 162,000 barrels a day on the worst case. The spewing went on for three months. The gush rate decreased as the oil reservoir was depleted. The amount of Lousiana shoreline affected by oil grew to 510 km by Nov‟ 2010. The oil spill amounted to about 4.9 million barrels. The blowout preventer valves were not closed. Remotely operated underwater vehicles were used to attempt to close the blowout preventer valves on the well head. A containment dome that was 125 tons in weight was tried and failed when gas leaking from the pipe combined with cold water formed methane hydrate crystals that blocked the opening at the top of the dome. The “top kill” was a procedure where heavy drilling fluids were pumped into the blowout

Introduction

23

preventer in order to restrict the gushing of oil before sealing it for ever with cement. This also failed. The gushing could have been averted by better control, alarm and shut down systems. “Fail-closed” valves could have averted the gushing even after the sinking of the oil rig.

1.3. CONTROL STRATEGY The main components of a control system are as follows; (i) Process or Physico-chemical System that needs to be Controlled (ii) Control Objectives (iii) Set Points and Error Bands (iv) Sensors and Measuring Elements (v) Input Variables, Manipulatable (vi) Disturbance or Load Variables (vii) Output Variables (viii) Constraints (ix) Operational Characteristics such as Batch, Continuous, Semi-batch (x) Safety, Environmental and Economic Considerations (xi) Control Type: Feedback, Feedforward, Adaptive, Multivariable, IMC, Filter Cascaded etc As can be seen from the examples in section 1.2 the control objectives can be clearly identified for each process. In the polymer kettle in Example 1.2.6 in order to avoid the geleffect the viscosity of the reactor mass has to be set as the control objective. Control action can be taken upon finding out should the viscosity be too high or too low for operation of the kettle. Input variables can be initial concentration of the reactant, inlet flow rate to the reactor, jacket temperature, inlet fluid temperature into the reactor, etc. Disturbance variable is one that gets perturbed in an unexpected manner in the reactor for example. Output variables are for example the unconverted monomer conversion in the outlet of reactor. The temperature of the reactor in the case of CSTR may be an output variable. The reactant and product concentrations in a CSTR may be treated as output variables. Constraints can arise from product quality considerations. For example the color of a SAN copolymer cannot be more than certain degree of yellowness. This imposes an upper limit on the AN composition in the copolymer and/or an upper limit on the time-temperature history of the product. In this example, with sufficient time and temperature exposure the product can turn yellow and blue black specifications may arise in the product. Better automation and control leads to operations closer to “optimal” conditions. Optimization of the process can result in millions of dollars of cost savings for the manufacturing division. Materials and energy can be better used. What a dollar can buy can be stretched a bit. Safer and less hazardous a process needs to be more expensive the process can become. “Fail-safe” concept is a salient consideration in the development of instrumentation. In order to change the flow rate using control valves energy needs to be

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Kal Renganathan Sharma

supplied. The stem of the valve is moved upon receipt of a signal. When the signal is lost the stem will go to the position with the lowest value of pressure range of operation. Should the valve be made air-to-open the loss of air would lead to closure of the valve? Such a valve is called as failed-safe valve. The valve can be air-to-close. In this case when instrument air is lost the valve will be at its open state. This is referred to as fail-safe-open valve.

1.4. CONTROL TYPES There can be different types of control action sought. These are as follows;

1.4.1. Feedback Control Examples of feedback control was shown in Example 1-2.9, Example 1.2.11. During feedback control the output variable is measured. The measured signal is compared against a set point using comparators. The error signal forms the input into the controller. Control action depends on the air. For example, as will be discussed in the later chapter, a P only controller is made in a manner that the control action is directly proportional to the error. In Chapter 5.0, feedback control of single-input or manipulated variable and single output or measured variable is discussed in greater detail. The feedback control structure is arrived at by selection of manipulated variable and increased or decreased to control the measured output variable. The desired value of the measured process output is also called as setpoint. Later control gain and control time constant would be decreased. In a similar manner, the process also can be characterized with a process gain and process time constant. The gain is the rate of change of the output to the rate of change of input. This can provide a measure of the sensitivity of the process or controller. The gain can be positive or negative. Negative gains leads to inverse response. This is one kind of instability as will be discussed in chapter 3.0 and chapter 4.0. A control algorithm can be constructed and control parameters can be identified. These control parameters can be tuned in order to obtain better performance. The feedback control can further be of different types. These are as follows; a. b. c. d. e.

On-off Controller P, Proportional only Controller PI, Proportional Integral Controller PD, Proportional Derivative Controller PID, Proportional, Integral and Derivative Controller

1.4.2. Feedforward Control An example of feedforward control is discussed in Example 1-2.9. Vasodilation and evaporative cooling/sweat formation happens immediately after a change in surroundings. This kind of control action is different from the control action described in feedback control

Introduction

25

above. The disturbance variable is measured and control action is taken proactively. An estimate is made on the effect of the control action on the output variable. Output variable is not measured directly. Therefore the control action can work well when there are reliable models that can relate the process output to the manipulated variables. Further the measured disturbance need to be the only aberration in the scheme of things.

1.4.3. Hybrid Feed forward and Feedback Control The feedback and feedforward schemes can be hybridized. In such cases the engineer has more things on his hands to adjust. (see Example 5.2).

1.4.4. Internal Model Control, IMC IMC, internal model control action uses the knowledge about the process that is being controlled. This is different from the black-box approach of feedback control of single output variable by tweaking the inpuit variable based on measurements of output variable. The model developed for the mixing tank heated by the hot fluid in the jacket in Figure 7.4 can be used to design an internal model controller. A good knowledge of when the process is stable and when the process is underdamped oscillatory unstable can lead to better control action. Control action of unstable systems may result in unsatisfactory results. In general the modelbased controller can be added as shown in Figure 7.8. Filters can be added to make the controller more realizable. The transfer function of the output has to be proper. The Laplace transform expression of the transfer function can be represented by; P (s ) Q (s)

(14)

When the order of the polynomial of Q(s) in the denominator of Eq. (14) is greater than the order of polynomial P(s) in the numerator then the transfer function is said to be proper. When the order of the polynomial in the numerator P(s) is the same as the order of the polynomial in the denominator Q(s) then the transfer function is said to be semi-proper.

1.4.5. Ratio Control Ratio controllers are used where ratio of the reactant mixture is of increased significance. For example when flue gas is needed the ratio of air to CO2, is controlled in such a manner that the heat of reactions from the Boudard reaction and heat of reaction from the oxidation reaction “cancel” each other rendering the reactor at a adiabatic state. One or more valves are used in a split range control. More on ratio control is discussed in section 7.1. Ratio control can be used in distillation columns where the reflux ratio is an important design variable.

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Kal Renganathan Sharma

1.4.6. Statistical Process Control, SPC SPC is discussed in greater detail in section 7.2. SPC is used to control the output variable within certain specifications. Control charts are used to better understand the errors between measured and set point values. Attributable causes are assigned to variations. Noise is delineated from a disturbance. Control action is taken according to a 3 sigma or a 6 sigma limit. The statistical measures of mean and variance are only used. This approach falls intermediate between feedback control of a process in a black-box and an IMC, internal model control where the process models are clearly known.

1.4.7. Estimation and Control of Polymerization Reactors The high viscosities and exothermic nature of the polymerization reactions make the control of polymerization reactors an ardous task. The control objectives need to be specified. The process dynamics is non linear. Some of the control strategies that were discussed when the process can conform to a prototypical first order or prototypical second order process for linear systems cannot be applied to systems that are governed by equations that are nonlinear. Measurement of polymer structure is by no means a done deal. Estimation techniques such as Kalman filter and Weiner filter are needed to obtain parameters that are difficult to measure.

1.4.8. Neural Networks ANNs, artificial neural networks may be used in control of distillation columns where the number of output variables is in the hundreds and the relation to the input variables are governed by equations that are nonlinear. ANNs comprise of a number of computing elements which resemble neurons and synapses of a human brain in a network. NNs can be used to approximate any mathematical function to a desired level of accuracy. In supervised learning, a network is given an input along with its desired output. On the other hand, a network in unsupervised learning is given only an input. After each presentation of an input, the performance is measured to tell how the network is doing. A network is expected to selforganize information by using the performance measure as guidance.

1.4.9. Multivariable Process Control Most control schemes discussed is single variable input and single variable output, SISO systems. MIMO systems are multiple variable input and multiple variable output systems. Interaction effects of variables in MIMO systems are often important. A disturbance at any input causes a response in some or all of the outputs. Control and stability analysis of MIMO systems are more tedious compared with analysis of SISO systems. The control system can be decoupled so that control of some output variables may be affected. Cross controllers [Marlin, 2000] in addition to principal controllers may be used to accomplish the control objectives. The number of controllers needed for the operation increases ponentially with the number of

Introduction

27

inputs and outputs. 4 controllers are needed for a system of two inputs and two outputs. 9 controllers are needed for systems of 3 inputs and 3 outputs. The characteristic equation for a MIMI system can be examined for stability from its roots.

1.4.10. Adaptive Control Rather than a black box approach to control action taken or use of a process model adaptive control is based on observations made at discrete intervals of time, updation of model based on the observations and then control action taken.

1.12. GLOSSARY Batch Process – Product is made from raw materials batch by batch. The raw materials and catalyst are charged into the reaction vessel. The reaction vessel is operated at the selected temperature and pressure. The product is removed after the reaction and separated from the unreacted raw materials. The vessel may be stirred at the set RPM, revolutions per minute. Continuous Process – Product is recovered at certain lb/hr from the process plant. Raw materials are fed continuously into the reactors. Reactors are operated at the set temperature, pressure, degree of intensity (agitator RPM). The product is separated from the unreacted raw materials in a suitable unit operation such as devolatilizer, distillation column, centrifuge, etc. Industrial Controls Market – This includes all the elements of the control process: (i) Measurement; (ii) Comparisons; (iii) Computations and; (iv) Corrective Actions. Sensors, comparators, desktop computers dedicated to control action and actuators, signal transmitters form the industrial controls market. Detectors, transducers, transmitters and controllers are all part of the industrial controls market. Process Dynamics – Study of the response of the different unit operations in the process to a step change or some other perturbation given to the system. PLC – Programmable Logic Controller. A computer dedicated to automation of electromechanical processes. Automation – Operation of a device or process with minimal human input. Runaway Reaction – Conversion of the reactant increases in an uncontrolled manner. Heat is given out in exothermic reactions. The heat generated, if not removed, increases the reaction temperature. Often reaction rate doubles with every 10 0C increase in temperature. This further increases the reaction rate followed by more heat generation. Lack of heat removal causes a runaway condition. Transient Study – Study of physical quantities that have not reached steady state yet. Block Diagrams - Boxes are used to denote functions and lines are used to show the relations between the functions. Data Acquisition – Enables measurements at different time intervals from the equipment to be converted to digital format to be stored in the computer. A/D analog to digital converter can be used to convert transducer voltages to digital values in the desktop computer. Robot – Programmable, automatic machine that is multi-purpose and manipulatable.

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Free Radical Polymerization – Method of polymerization where the initiation, propagation and termination reactions take place by the free radical mechanism. Free radicals are formed from initiators and monomer and propagating species is a free radical. Supply Chain Robotics – Use of robots in the distribution and supply of goods PW, Present Worth – Money value of a stream of cash payments of receipts and costs projected in the future in terms of today‟s dollars include capital, interest, tax and tariff, labor, utilities, materials, overhead, revenue from sales, salvage value over the life of the plant. Artificial Intelligence – Synthesize intelligence using computer software programming and execution. Kinematics – Science of motion without worry about the forces that causes it. 3 Arm Mechanical Manipulator Device with three rigid lengths and two joints. The third arm is usually the end effector. Inverse Kinematics The science of understanding the set of joint angles that should correspond to the position and orientation of the end effector Dynamics study of torque functions of the motion of the manipulator Molecular Assemblers – hypothetical device that be used to assemble atoms/molecules at will Trajectory Generation – Set of mathematical functions that specifies where the joints needs to be present in order for the manipulator to go from here to there Spline smooth mathematical function that passes through a set of given points Cartesian Trajectory when the end effector is specified to move in a straight line

1.13. SUMMARY Simulation, modeling, instrumentation and control of chemical processes are important in the manufacture of products at the lowest cost, good quality in an environmentally safe manner. The industrial controls market and chemical process industry market size are discussed. Supply chain robots are increasingly used. Camera drones are expected to be used in the future for interesting applications ranging from fighting fires to precision agriculture. Infeasibility of molecular assemblers is discussed. Robots as a collegiate course are outlined. Nanorobots are possible as envisioned in the movie fantastic voyage. Transient analysis of unit operations and reactors are used in start-up and shut down operations. Real industrial world happenings were cited to better motivate the discipline: (i) Trommsdorff effect during free radical polymerization is when the reactor gets “setup” due to runaway polymerization. Causes are attributed to viscosity increase in the monomer/polymer mass and the termination reactions cease on account of difficulty with diffusion; (ii) centralized heating and cooling in modern homes using feedback control of air from the furnance or air conditioner using a thermostat; (iii) feedforward control of human body temperature. Heat conduction through five layers, i.e., skin, fat, muscle, bone and blood are considered. Vasodilation and evaporative cooling are the two mechanisms of cooling effect within the human anatomy; (iv) feedback control of 3 arm robotic manipulator with end effector. Sensors are used to measure the joint angles and joint velocities. Torques needed is calculated from trajectory generator. Feedback is used to detect any servo error. Control action is taken based on deviations; (v)

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nuclear meltdown in Fukushima nuclear power plant after the tsunami and earthquake in 2011; (vi) deepwater horizon oil spill of 2010. 11 main components of a control system was discussed including control ojectives, set points, measuring elements, input, disturbance and output variables and control types. Better automation and control leads to closer to optimal operation of the plant. Process is operated in a less hazardous and safer manner without increasing the cost. Feedback control action is taken based on the error generated during operation of a process. The output variable is measured and compared against the set point. The difference is the error. Different feedback controllers used are on-off, P, proportional only control, PI, proportional integral control, PD, proportional derivative control and PID, proportional integral and derivative control. Feed forward control action is taken upon direct measurement of the disturbance variable. The control action is in anticipation of the changes expected in the output variable. Often times success of this type control action depends on availability of reliable mathematical models between the process output variables and control variables and effect of perturbations. Internal Model Control, IMC IMC, internal model control action uses the knowledge about the process that is being controlled. Filters can be added to make the controller more realizable. The transfer function of the output has to be proper. Ratio control is used to control ratios of reactant mixtures, reflux rate in a distillation column etc. Statistical process control, SPC is used to control the output variable within certain limits. Control charts are used to better understand the errors between measured and set point values. Attributable causes are assigned to variations. Noise is delineated from a disturbance. Control action is taken according to a 3 sigma or 6 sigma control limit. Estimation and control of polymerization reactors is an ardous task due to exothermic nature of the reaction, high viscosity etc. Some of the desirable parameters are not easty to measure. Estimation techniques such as Kalman filter and Weiner filter are beginning to be used in order to obtain parameters that are difficult to measure. The dynamic equations in polymerization reactors are non-linear. They are linearized and made similar to other existing systems. ANNs, artificial neural networks may be used in control of distillation columns where the number of output variables is in the hundreds and the relation to the input variables are governed by equations that are nonlinear. NNs can be used to approximate any mathematical function to a desired level of accuracy. Most control schemes discussed is single variable input and single variable output, SISO systems. MIMO systems are multiple variable input and multiple variable output systems. Interaction effects of variables in MIMO systems are often important. Cross controllers in addition to principal controllers may be used to accomplish the control objectives. During adaptive control observations made at discrete intervals of time are used to update the mathematical model for the process. This is used in the control action taken.

1.14. FURTHER READING Chen, C. C. (1994). “A Continuous Bulk Polymerization Process for Crystal Polystyrene”, Polym. Plast. Technology Eng., Vol 33, 55-58. Chen, C. (2000). “Continuous Production of Solid Polystyrene in Back-Mixed and LinearFlow Reactors”, Polym. Eng. Sci., Vol. 40, 441-464.

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Coughanowr, D. R. & LeBlanc, S. E. (2009). Process Systems Analysis and Control, McGraw Hill Professional, New York, NY. Sources Craig, J. (2005). An Introduction to Robotics and Control-Mechanics and Control, Prentice Hall, Upper Saddle River, NJ. http://en.wikipedia.org/wiki/Deepwater_Horizon. Gharaghani, M., Adedini, H. & Parvazinia, M. (2012). “Dynamic Simulation and Control of Auto- Refrigerated CSTR and Tubular Reactor for Bulk Styrene Polymerization”, Chemical Engineering Research and Design, Vol. 90, 1540-1552. Hui, A. W. & Hamielec, A. E. (1972). “Thermal Polymerization of Styrene at High Conversions and Temperatures: An Experimental Study”, J. Appl. Polym. Sci., Vol 16, 749-769. Internet – Market Report, “Industrial Controls and Robotics Market – Global Industry, Analysis, Size, Share, Growth, Trends and Forecast 2013-2019”, Transparency Market Research, Kang, C. (March 7th 2013). “New Robots in the Workplace: Job Creators or Job Terminators?” Washington Post. Kruse, R. L. (1990). Tormsdorff effect, Private Communication, Monsanto Plastics Technology, Indian Orchad, MA. Marlin, T. E. (2000). Process Control:Designing Processes and Control Systems for Dynamic Performance, McGraw Hill, 345-349. Newsweek, Issue Dated March 28th & April 4th 2011, with cover Story Apocalypse Now Rehg, J. A. & Sartori, (J. G. J.). Industrial Electronics, Prentice Hall. Rockis, G. J. & Mazur, G. A. (2009). Electrical Motor Controls for Integrated Systems, American Technical Publishers, Inc., Homewood, IL. Satariano, A. (November 24th 2013). “Apple Fortifies its Own Supply Chain – Building Robots and Lasers”, Washington Post. Sharma, K. R. (2010). Transport Phenomena in Biomedical Engineering: Artificial Organ Design and Development and Tissue Design, McGraw Hill Professional, New York, NY. Sharma, K. R. (2010). Nanostructuring Operations in Nanoscale Science and Engineering, McGraw Hill Professional, New York. Sharma, K. R. (2013). “On Photodynamic Therapy of Alzheimer‟s Disease Using Intrathecal Nanorobot Drug Delivery of Curcuma Longa for Enhanced Bioavailability”, Journal of Scientific Research and Reports, Vol. 2, 1, 206-227. Yokota, T. Newsweek, Issue Dated March 21st 2011, with cover Story Earthquake 8.9.

1.15. EXERCISES 1.0. Why is dynamics an important consideration during plant start-up and plant shut down? 2.0. What are the differences between computer simulation and mathematical model? 3.0. What causes the Trommsdorf effect? 4.0. What is meant by reaction run away? 5.0. What are the differences between feedback control and feedforward control? 6.0. What are the differences in function between comparator and controller? 7.0. In which type of control is measurement of the output variable part of the operation?

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8.0. In which type of control is measurement of the disturbance part of the operation? 9.0. Airline pilot is sent a signal from the airport about the weather conditions in the arrival location. On account of snowstorm he delays arrival at Minneapolis, MN. What kind can this control action are classified as. Classify as Feedforward or Feedback Control (Exercises 10-30); 10.0. Steering Action in Cars. The direction of the wheels is the output variable. Direction of the steering wheel is the signal. 11.0. Control of Speed of Car by pushing the gas pedal or brake pedal. 12.0. Marathon athlete drinking lemonade to cool off. 13.0. Sun Tracker in a solar power plant. 14.0. Agitator speed control in a CSTR at a continuous polymerization plant. 15.0. Temperature control of a devolatilizer at a continous polymerization plant. 16.0. Level control of a devolatilizer at a continuous polymerization plant. 17.0. Reactor pressure control at a continuous polymerization plant. 18.0. Homogeneity control in rubber dissolver in monomers at a continuous polymerization Plant. 19.0. Time Watering the Lawn in modern house based on lawncare conditions. 20.0. Control of the trajectory of the end effector of a 2 arm manipulator.. 21.0. Inlet Temperature of Gas turbine in Combined Cycle Power Plant. 22.0. Vibration s of a Nuclear Reactor during Earth Quake. 23.0. Temperature of a Kettle in Continuous Pharmaceutical Produ ction. 24.0. Solar Cell Temperature in Solar Aided Combined Cycle Power Plant. 25.0. Temperature of Catalytic Oxidation Reactor of Sulfur Dioxide. 26.0. Bubble Size in a Plate Pulsed Column. 27.0. Volume Fraction in a Fiber Reinforced Composite. 28.0. Vicosity in a Blender of Polyethylene and Nylon. 29.0. Temperature of a Polymerization Reactor for PMMA, Poly Methyl Methacrylate. 30.0. Pressure of a Devolatilizer during Terpolymer Manufacture. 31.0What is the difference between “fail-safe” and “fail-open” valves? 32.0. What are the differences between adaptive control and ratio control? 33.0. What are the differences between IMC, internal model control and control action using neural networks? 34.0. What are the differences between SPC, statistical process control and use of Kalman filter during control in polymerization reactors? 34a.0. What are the nuances to multivariate process control? 35.0. What is the radioactive damage to the environment on account of Fukushima earth quake? 36.0. What was the role of the cooling break-down in the Fukushima disaster? 37.0. Can motor speed and motor torque be predicted using a mathematical model? 38.0. What is meant by “Constant torque” mode of operation? 39.0. When is the output hp of a motor constant? 40.0. Why is the work done by the fluid in the motor not predictable? 41.0. What happens to the termination rate constant when the temperature of the reactor is increased? Why?

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Kal Renganathan Sharma 42.0. At which temperature, higher or lower is the reactor more prone to the “gel effect”? 43.0. What is the role of the addition of solvent to the “gel effect”? 44.0. Can the molecular weight that is high in Figure 1.0 be relied on for scale-up purposes? 45.0. How does the “gel effect” confirm the termination by combination mechanism and not the other mechanisms such as disproportination?

Chapter 2

CONTINUOUS POLYMERIZATION PROCESS TECHNOLOGY 2.1. INTRODUCTION Often times the choice of technology with continuous mode of operation in order to manufacture commodity and engineering polymer products in large scale, is made compared with batch mode. Continuous mode is usually the lowest cost, safer and environmentally friendlier, better product quality with less batch to batch variability. Batch processes require several hours, in some cases greater than eight hours, to feed the reactants, including monomer or monomers into the reactor, conduct the polymerization reaction, cool the resulting polymer, remove the polymer, and clean the reactor. The equipment required for batch processes typically includes reactors which can hold up to 75,000 liters and may cost more than $1,000,000 per reactor. To improve the deficiencies of the batch processes, continuous polymerization processes have been developed. Continuous polymerization processes are potentially more efficient than a batch process. In a continuous process, monomer and other reactants are continuously fed into and through the reactor while, at the same time, polymer is continuously removed from the reactor. The unreacted monomers are separated from the polymer product and recycled back to the reactors. A continuous process may produce more products per day with a typical plant operated at 15,000 pph, utilizing smaller, less expensive reactors. Continuous processes utilizing continuous stirred tank reactors or tubular reactors are two types of continuous processes. The three methods of making polymers are emulsion, suspension and solution or bulk polymerization. A good number of the emulsion and suspension processes are operated in the batch mode. The solution polymerization or continuous mass polymerization has been the choice of the manufacturers of; (i) (ii) (iii) (iv) (v) (vi)

Polystyrene HIPS, rubber modified polystyrene SAN, styrene acrylonitrile copolymer ABS, Acrylonitrile Butadiene and Styrene engineering thermoplastic Polymethyl methacrylate Formaldehyde dioxolane semi-batch copolymerization

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Kal Renganathan Sharma (vii) Continuous process for high quality anhydrosugar alcohols etc, polycarbonate. Other than the free radical mechanism of polymerization other mechanisms such Zeigler Natta coordination covalent process for making polypropylene and polyethylene can be used in the continuous mode of plant operation. Examples of processes with continuous mode of operation are; (viii) The condensation and step growth polymerization can be performed continuously. (ix) A variety of compounding processes such as polymer polymer miscible blends that mix at a molecular level and compatible blends with product property improvement such as the making of the extra tough polyacetal resin by compounding with polyurethane and viscosity and color, colorability modifications of polymers etc are operated in the continuous mode. (x) Some emulsion processes are evaluated to be operated in the continuous fashion. (xi) Waste tires can be devulcanized and polybutadiene reclaimed in a continuous manner

2.2. OVERVIEW OF MONSANTO 2 CSTR CONTINUOUS MASS POLYMERIZATION PROCESS FOR ABS A need for a continuous mass polymerization process for ABS polymers with high conversion rates and low energy requirements was identified by leading companies such as Monsanto Plastics (currently Bayer), Indian Orchad, MA, Dow Chemical Company, Midland, MI, and General Electric, Pittsfield, MA. High polymerization efficiency for ABS polymers having superior properties was sought and the process needed to be robust enough to handle high conversion on a large scale. ABS polymers comprise of a matrix phase copolymer of styrene and acrylonitrile and a dispersed phase of a conjugated polybutadiene rubber grafted with the SAN chains. Various processes have been utilized for the manufacture of such polymers including emulsion, suspension and mass polymerization techniques and combinations of the three. Although mass polymerized products exhibit desirable properties, this technique has a practical limitation upon the maximum degree of conversion of monomers to polymer which can be effected because of the high viscosities and accompanying power and equipment requirements, which are encountered when the reactions are carried at high conversion [Kruse, 1983]. In a typical ABS mass polymerization process styrene and acrylonitrile are copolymerized in the presence of a diene-based rubber [Sharma, 1997]. Initially the rubber is dissolved in the monomers and a continuous homogeneous phase prevails. When polymerization begins, the monomers are simultaneously copolymerized alone and also as a graft on the rubber backbone. As the monomers polymerize two phases appear: the polymer dissolved in monomer and the rubber dissolved in monomer. Initially the latter phase predominates and the smaller "polymer in monomer" phase is dispersed in the larger "rubber in monomer" phase. However as polymerization progresses the "polymer in monomer" phase becomes greater in volume. At this point the phenomenon of phase inversion occurs and the "rubber in monomer" phase becomes dispersed as discrete particles in a matrix of the "polymer in monomer" phase. Usually in a mass polymerization process, the rubber will

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contain occlusions of polymer/monomer which serve to swell the volume of the rubber particle. As polymerization progresses, monomer is converted to polymer, the viscosity of the mixture increases and greater power is needed to maintain temperature and compositional uniformity throughout the polymerization. Another limitation of previous attempts at the process, is that production of polymers with a high proportion of acrylonitrile as well as high rubber volume fraction is not possible. This is because although rubber dissolves readily in styrene, its solubility in a mixture of styrene and acrylonitrile monomers decreases with the increase in concentration of acrylonitrile. It is found for example that styrene monomer can dissolve about 20% of its weight of a diene rubber whereas a monomer mixture containing 58% styrene and 42% acrylonitrile can dissolve less than 10% of its weight of the same rubber. Thus the amount of rubber that can be added in solution in the monomer mixture is restricted by the proportion of acrylonitrile monomer. However for many purposes such as solvent resistance and toughness it is desirable to have a proportion of acrylonitrile as high as 40% or more by weight. Kruse [1983] described a continuous mass polymerization process for preparing an ABS polymer having a matrix phase comprising a copolymer of SAN and a dispersed phase comprising rubber particles having a weight average particle size of about 0.1-10 microns. The process consists of feeding a solution of polybutadiene rubber dissolved in styrene to the first CSTR. The composition of the dissolved polybutadiene was from 3 - 33% by weight. By a second feed stream, acrylonitrile was charged simultaneously and continuously to the stirred reactor. The conversion and solids level in the first reactor was maintained at a point above that at which phase inversion is expected to occur, i.e., up to 70% by weight based on the weight of the polymerization mixture. The mixture is continuously polymerized while maintaining stirring such that the polymerizing mixture has a substantially uniform composition and such that the rubber is dispersed in the polymerizing mixture as rubber particles. The ABS polymer is continuously separated from the partially polymerized mixture. Operation at such a polymer solids content ensures that upon addition, the rubber immediately forms small particles containing a monomer component, dispersed in the partially polymerized reaction mixture. It was found that, using this technique the monomer content (all species) remaining is less than 0.5 % and substantially all monomer is removed within 30 seconds such that polymerization is not significantly advanced during the polymer separation. What remains is polymeric and the percentage of the sample weight that this represents is the polymer solids of the polymerizing mixture at that time. Because acrylonitrile is separately but simultaneously fed and because the point of phase inversion for the system has been passed such that the rubber disperses as particles as it enters the reaction mixture, the process has the capability of employing high rubber concentrations while still realizing a high acrylonitrile concentration in the final ABS composition. Preferred ABS molding compositions have high gloss and an average rubber particle size less than about 500 nm and most preferably 200 – 400 nm. Conventional ABS polymers having rubber particles this small however lack toughness. Raising the acrylonitrile content of such ABS polymers from the conventional 24% to the range of about 27 to 40% permits these small rubber particles effectively to toughen such an ABS polymer to an unexpected degree. The polymer conversion at which the reaction is conducted is limited by two practical considerations. At the lower end of the range, (as has been indicated above), it is important that the polymer solids level in the reactor, (or the initial reactor where a series of reactors is

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used), to which the monomer streams are added be such that the polymer/monomer phase has a greater volume than the rubber/monomer phase such that the rubber/monomer immediately forms a dispersed phase. In practice this implies a monomer to polymer conversion level of about 35%. At the upper level, the practical constraints of power requirements for the reactor agitator place a limit of about 70% solids. This does not necessarily imply a similar conversion level since up to 50% and from 10 to 30% by weight of a suifigure solvent, (based on the weight of the monomers fed to the reaction), can be used to dilute the reaction mixture to a point at which, even with up to 99% monomer to polymer conversion, the power requirements are not excessive. Some or all of the diluent can be introduced with the rubber in styrene stream either as an added component or by the use of a rubber which is already dissolved in a suitable solvent such as hexane or cyclohexane. The diluents used are ethyl benzene or methyl ethyl ketone. Those ABS polymers having a rubber particle size, of from about 100 -500 nm; acrylonitrile monomer content of about 27 to 40; a rubber content of about 14 to 25%; and a graft level of 150 to 200% generally require a lower molecular weight matrix phase copolymer to ensure proper flow properties, e.g., a molecular weight of about 3,000-60,000 (Mn) or approximately 5,000-50,000 (Mv). After polymerization has progressed to the desired conversion level, the residual monomer is stripped from the polymer. This operation, which is the same whether a single reactor or a series of reactors is employed in the polymerization stage, is conventionally done in a separate device such as a wiped film devolatilizer or a falling strand devolatilizer. Wu and Virkler [2001] have patented a continuous mass polymerization method in order to manufacture ABS for refrigerator liners. Polybutadiene with Mw, weight averaged molecular weight ~ 80,000-250,000 was used in the two reactor and devolatilizer process. Peroxy initiator was used in the first reactor and no initiator was used in the second reactor. The matrix SAN molecular weight was ~ 65,000 – 70,000 gm.mole-1. The first reactor feed comprises of PBd rubber dissolved in styrene and mixed with acrylonitrile, diluent. Chemical initiator was also used. Reactor 1 was operated at 20-30% steady state monomer conversion to polymer forming 20-30% solids. As polymerization takes place the initiator radicals form grafting sites at the vinyl bonds of PBd. Graft chains of SAN grow to a larger size. The rubber phase is formed with grafted SAN chains. It precipitates into a solid phase. This step is similar to “crystallization” kinetics from a supersaturated solution with swing in solvent composition. The rubber phase was found to have a weight averaged particle size of Dw ~0.3 – 0.7 m. As the graft molecular weight is greater than PBd molecular weight the grafted chain provides the surface coverage of formed rubber particles. SAN chains lower in length than a critical length leads to the formation of occlusions within the particle. The PBd chains can cross-link during the consecutive-competitive reactions. Cross-linking reactions may be responsible for formation of particles with cell morphology. Some cross-linking offers better gloss of the final product. It was found,serendipitously, that higher the volume of occlusion within the rubber particle the more efficiently the rubber phase is used in toughening the polymer. The grafted chains stabilize the formed rubber particles. The second reactor is well stirred and conversions of 50-90% are achieved here. The polymer solids formed in the second reactor is ~ 75%. The temperature in the first reactor was held constant for isothermal operation at ~ 75-85 0C. The temperature in the second reactor was held constant for isothermal operation at 130 – 155 0C. No chemical initiator was added to the second reactor. The heat released from the exothermic polymerization reactions is cooled by vaporization of

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some monomer from the reacting mass. SAN chains that form by thermal initiation form the matrix of the ABS blend. Jacket cooling is also available. The devolatilization is conducted in WFD, wiped film devolatilizer, FSD, falling strand devolatilizer or Extruder. The DV temperatures are ~ 200 – 280 0C, operating pressures of 0.01 – 700 mm Hg. Residual monomer and oligomer levels are about 2000 ppm. The particle size distribution of rubber phase was characterized by a Dw/Dn = 2.5. The rubber level in the ABS product is ~10-16%. The impact strength, notched Izod of the sample was 25 KJ.m-2 and tensile modulus was 2.2 GPa.

2.3. THE DOW PROCESS TO PREPARE MAGNUM ABS Polybutadiene was dissolved in a feed stream of styrene, acrylonitrile, and ethyl benzene to form a mixture. The mixture was polymerized in a continuous process under agitation. The polymerization occurred in a three stage reactor system over an increasing temperature profile [Bredewig, 1980]. During the polymerization process, some of the forming copolymer grafts to the rubber particles while some of it does not graft, but, instead, forms matrix copolymer. The resulting polymerization product was then devolatilized, extruded, and pelletized. Different rubber particle sizes in the final polymer product are achieved by changing certain process parameters. These process parameters and how they must be changed to produce rubber particles of a desired size include the degree of agitation, temperature, initiator level and type, chain transfer agents and amounts, and diluent type and concentration. Immediately after the polymerization reaction commences, the rubbery material in the monomer mixture separates into two phases, of which the former, consisting of a solution of the rubber in the monomer mixture, initially forms the continuous phase, whereas the latter, consisting of a solution of the resultant copolymer in the monomer mixture, remains dispersed in form of droplets in the continuous phase. As polymerization and hence conversion proceed the quantity of the latter phase increases. As soon as the volume of the SAN matrix phase equals that of the rubber dispersed phase with grafted SAN a phase change occurs, generally known as phase inversion. When this phase inversion takes place, droplets of rubber solution form in the polymer solution. These rubber solution droplets incorporate by themselves small droplets of what has now become the continuous polymer phase. During the process, grafting of the polymer chains on the rubber takes place, too. Generally, the polymerization is carried out in several stages. In the first polymerization stage, known as prepolymerization, the solution of the rubber in the monomer mixture is polymerized until phase inversion is reached. Polymerization is then continued up to the desired conversion. Mass polymerization offers rubber-modified SAN or ABS copolymers with a good balance of physical and mechanical properties, however the surface gloss of such copolymers is not always quite satisfactory. The polymerization is conducted in one or more substantially linear, stratified flow or so-called "plug-flow" type reactor which may or may not be recirculated as shown in Figure 1.0. The temperatures at which polymerization is most advantageously conducted are dependent on a variety of factors including the specific initiator and type and concentration of rubber, comonomers and reaction diluent, if any, employed. In general, polymerization temperatures from 60-160 0C are employed prior to phase inversion

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with temperatures from 100-190 0C being employed subsequent to phase inversion. Mass polymerization at such elevated temperatures is continued until the desired conversion of monomers to polymer is obtained. Generally, conversion of from 65-90% of the monomers added to the polymerization system (i.e., monomer added in the feed and any additional stream, including any recycle stream) to polymer is desired. Following conversion of a desired amount of monomer to polymer, the polymerization mixture is then subjected to conditions sufficient to cross-link the rubber and remove any unreacted monomer. Such cross-linking and removal of unreacted monomer, as well as reaction of diluent, if employed, and other volatile materials is advantageously conducted employing conventional devolatilization techniques, such as introducing the polymerization mixture into a devolatilizing chamber, flashing off the monomer and other volatiles at elevated temperatures, e.g., from 200-300 0C., under vacuum and removing them from the chamber. The rubber-modified aromatic copolymer composition is thermoplastic. When softened or melted by the application of heat, the compositions of this invention can be formed or molded using conventional techniques such as compression molding, injection molding, gas assisted injection molding, calendaring, vacuum forming, thermoforming, extrusion and/or blow molding. Surface gloss is very dependent upon molding conditions, i.e., for injection molding, machine parameters such as barrel and mold temperatures, injection and holding speed/pressure/times, etc., and method of testing gloss can dramatically affect the gloss value for a given material. The gloss value for materials molded under unfavorable conditions is referred to as intrinsic gloss. Unfavorable conditions generally are those which limit or reduce the flow of the material during molding. For example, it is well known that surface gloss is reduced when during injection molding lower melt temperature, lower mold temperature, lower injection pressure, slower injection speed, lower holding pressure, etc., are applied. Conversely, conditions which enhance the flow of the material will improve gloss. Materials with an intrinsic gloss less than 70 percent can, when molded under favorable or ideal conditions, demonstrate higher gloss (e.g., 70 percent or higher). The mass polymerized rubber-modified ABS copolymer compositions can also be formed, spun, or drawn into films, fibers, multi-layer laminates or extruded sheets, or can be compounded with one or more organic or inorganic substances, on any machine suitable for such purpose. Some of the fabricated articles include household appliances, toys, automotive parts, extruded pipe, profiles and sheet for sanitary applications. These compositions can even find use in instrument housings such as for power tools or information technology equipment such as telephones, computers, copiers, etc.

Figure 1.0. Reactor Configuration in the Dow Process for Magnum ABS.

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2.4. THE PLUG FLOW MULTIZONE PROCESS Multizone plug flow bulk processes include a series of polymerization towers, consecutively connected to each other, providing multiple reaction zones. Stereospecific polybutadiene rubber is dissolved in styrene or in styrene/acrylonitrile monomers mixture, and the rubber solution is then fed into the reaction system. The polymerization can be thermally or chemically initiated, and viscosity of the reaction mixture will gradually increase. During the reaction course, the rubber will be grafted with grafted SAN copolymer and, in the rubber solution, matrix phase SAN is also being formed. At a point where the free, ungrafted SAN cannot be "held" in one single, continuous "phase" of rubber solution, it begins to form domains of SAN phase. The polymerization mixture now is a two-phase system. As polymerization proceeds, more and freer SAN is formed, and the rubber phase starts to disperse itself as particles in the matrix of the ever-growing free SAN. Eventually, the free SAN becomes a continuous phase. This is actually a formation of an oil-in-oil emulsion system. Some matrix SAN is occluded inside the rubber particles as well. This stage is usually given a name of phase inversion. Pre-phase inversion means that the rubber is a continuous phase and that no rubber particles are formed, and post phase inversion means that substantially all of the rubber phase has converted to rubber particles and there is a continuous SAN phase. Following the phase inversion, more matrix SAN (free SAN) is formed and, possibly, the rubber particles gain more grafted SAN. When a desirable monomer conversion level and a matrix SAN of desired molecular weight distribution is obtained, the reaction mixture is "cooked" at a higher temperature than that of previous polymerization. Finally, bulk ABS pellets are obtained from a pelletizer, after devolatilization where volatile residuals are removed. A continuous bulk ABS process was described by GE Plastics [Sue et al. 1996] that provides controllable molecular weight distribution and micro gel particle size using a "threestage" reactor system, for extrusion grade ABS polymers. In the first reactor, the rubber solution is charged into the reaction mixture under high agitation to precipitate discrete rubber particle uniformly throughout the reactor mass before appreciable cross-linking can occur. Solids levels of the first, the second, and the third reactor are carefully controlled to a desired molecular weight. The process flow diagram is shown in Figure 2.0. Different processes of ABS manufacturing give different properties to the final ABS products. One of these properties is the surface gloss of the end products, and technology development to produce ABS materials that could meet with different gloss requirements is still an on-going task for the ABS industry. The gloss of an ABS product is partially the result of molding conditions under which the product is manufactured. However, for a given molding condition, the rubber particle size (diameter) of the ABS material is a major contributing factor to the gloss. In general but not always, ABS materials from emulsion processes produce rubber particles of small sizes (from about 0.05 to about 0.3 microns). Therefore, high gloss products are often made from emulsion ABS materials. On the other hand, ABS materials from mass processes usually form rubber particles of large sizes (from about 0.5 to 5 microns). Therefore, low gloss products are often made using the bulk ABS materials. Although it is possible to produce small particles using bulk processes, the gloss and impact resistance balance will be difficult to reach. Overall, current technology described

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above has not been able to produce, by a bulk process alone, ABS materials of rubber particles of "cell" morphology with monomodal particle size distributions and with average particle sizes less than 0.3 microns of number average diameter, without compromising impact resistance properties. To synthesize ABS polymers with high performance by bulk processes, three aspects are important;. These three aspects; 1) grafting of the rubber substrate prior to phase inversion, 2) particle formation during phase inversion, 3) and cross-linking of the rubber particle at the completion point of the bulk ABS polymerization. However bulk ABS processes discussed above, are somehow deficient by different degrees in controlling and in adjusting the grafting, the phase inversion, and the crosslinking. Accordingly, technologists attempted to develop the continuous mass polymerization process which yields the desired rubber morphology and maximizes grafting thereby allowing a minimization of rubber use for a given level of property performance. Additionally, there is a desire to provide a bulk process capable for producing ABS resins of low gloss as well as high gloss. The products can have the particle sizes that are corresponding to a low gloss characteristic or can have small particle sizes that are corresponding to a high gloss characteristic. One of the preferred products has a number average particle size of less than 0.3 microns, a monomodal particle size distribution, and particles of "cell" morphology, and exhibits high gloss, high impact resistance properties. The "cell" morphology may also be described as a rubber membrane network of spherical surface with the occluded rigid polymer (SAN) filled in the interior spaces. Furthermore, with the "cell" morphology, the grafted rigid polymer (SAN) is grafted on both sides of the rubber membranes, i.e., exterior or interior of the rubber particle. The materials produced are generally not transparent in nature, but rather are generally opaque. However, the opacity of the material is, in most cases, relatively lower than that of emulsion ABS. One of the preferred products of this process can provide high gloss and high impact resistance ABS by producing rubber particles of "cell" morphology with small particle sizes of less than 0.3 microns number average diameter and monomodal size distributions. That is, the present invention offers technology to produce rubber particles with sizes close to those of emulsion particles and with sufficient grafted and occluded vinyl aromatic-unsaturated nitrile (SAN) polymers by a bulk process, leading to high surface gloss and good impact resistance for the bulk vinyl aromatic-unsaturated nitrile-alkadiene (ABS) materials.

2.5. PROCESS FOR HIGH IMPACT POLYSTYRENE (HIPS) High impact polystyrene polyblends (HIPS) comprises of polystyrene matrix having a rubber phase dispersed within the continuous phase, as cross-linked rubber particles. These have been manufactured and marketed. Historically, mechanical blends were prepared by melt blending polystyrene with raw rubber which was incompatible and dispersed as cross-

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linked rubber particles to reinforce and toughen the polymeric polyblend. More recently, HIPS polyblends have been prepared by mass polymerizing solutions of diene rubber dissolved in styrene monomer in batch reactors wherein the rubber molecules were grafted with styrene monomer forming polystyrene polymer grafts on the rubber along with polystyrene polymer in situ in the monomer. As the polystyrene-monomer phase increases during polymerization the grafted rubber phase inverts readily as rubber particles comprising grafted rubber and occluded polystyrene contained therein with said particles cross-linked to maintain the rubber particles as discrete particles dispersed in the polystyrene which forms a matrix phase of the HIPS polyblend. Analysis reveals that the novel polyblends comprise rubber particles structured in new morphological forms comprising rubber fibers or rubber sheets and mixtures. The novel rubber particle structure has been found to provide the HIPS polyblends with a more efficient rubber particle providing improved physical properties for the polyblend such as gloss, impact strength, melt flow and falling dart impact strength. By contrast, HIPS polyblends previously attempted have rubber particles structured with relatively large amounts of occluded polystyrene contained in a network of continuous rubber membranes as the only morphological structure. HIPS polyblends containing such particles only, have lower reinforcing ability for the polyblend and are relatively deficient in flow and gloss. Hitachi [1983] developed a continuous process to produce high impact polystyrene, HIPS. Rubber and styrene form the feed to a multi-stage, horizontal dissolving tank where the polybutadiene is dissolved in styrene monomer by stirring. Heat released is allowed to increase the temperature of the solution. Grafting reactions are facilitated in the first reactor and the phase inversion from rubber as continuous phase to rubber as dispersed phase is affected. The subsequent reactors in the process are used to complete the bulk polymerization of polystyrene. Heat generated from exothermic reactions is removed. The unreacted monomers are separated from the polymer product in a separator and recycled back to the reactor. The process flow diagram is shown in Figure 3.0.

Figure 2.0. GE Continuous Mass Three Stage Polymerization Process with a Prepolymerizer and Finisher to Manufacture ABS.

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Figure 3.0. Hitachi Process for Continuous Polymerization Process for HIPS.

During the production of HIPS, it is important to remove heat generated from the exothermic polymerization reactions and heat of agitation. The temperature of the reactor has to be controlled as specified. The viscosity of the polymerized syrup increases as the reaction proceeds. The heat removal is considered difficult. Addition of solvent or diluent such as ethyl benzene can soak up the heat released. The presence of diluent will obviate the runaway polymerization condition, i.e., the Tormsdorff effect that was discussed in Chapter 1.0. Partially filled reactor kettles can allow vaporization of monomers and diluent. During the vaporization of the liquid heat is consumed. This provides for ease of temperature control of the reactor. Foaming can be avoided. Polybutadiene and styrene are brought in contact with each other in the dissolving tank that is agitated. The dissolving step is stage wise and the tank is operated at atmospheric pressure. The temperatures are raised stepwise from room temperature for the first stage up to the polymerization temperature for the final stage. The temperature policy is 20 – 40 0C for the first stage, 40 – 60 0C for the second stage and 80 – 110 0C for the third stage. The residence time for each stage is about 1- 2 hrs. Four or 5 stages may be used for this step. The first reactor is operated at temperatures 100 – 130 0C under atmospheric pressure. The initiator can be added as desired. The phase inversion of the rubber phase from continuous to disperse is allowed to happen in this reactor. The outlet of the first reactor is about 25 – 40% solids. Heat removal from the reactor is by external jacket. The second reactor has a rotary shaft with number of disk blades. The reactor operating temperatures are ~ 100-150 0C under reduced pressure. The reaction conversion is allowed to go as high as 60% at the outlet of the second reactor. The finisher has mechanism for heating and cooling. The finisher can be hotter than the second reactor and the conversion can be 70-85%. Heating type monomer separator is used in order to separate the polymerized product from the unreacted monomers which can be returned to the rubber tank or feed line. Shearing elements, flash heater, rotary shaft, close clearance helical blades are present in the separator. Vacuum

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pressures of 0.5 – 200 torr can be achieved in the separator. Temperature of the separator is about 230 0C. The product with 4% rubber content can be made. The residence time in the reactor is 4.3 hrs. and that in the finisher is 3.8 hrs. Vents and traps are provided for rapid volatiles to be captured. Condensers, vacuum pumps, monomer tanks, pumps can be seen in Figure 3.0. Once separated the tanks are used to store the monomers. Die head, cooling bath, chip cutter are used to receive the polymer strands and make them into pellets.

2.6. BASELL GAS PHASE POLYMERIZATION PROCESS FOR POLYOLEFIN Basell polyolefin practices a process for the gas-phase fluidized bed catalytic polymerization of olefins. Two or more interconnected polymerization zones are used to feed the monomers and collect the product polymer. The development of olefin polymerization catalysts with high activity and selectivity, particularly of the Ziegler-Natta type and, more recently, of the metallocene type, has led to the widespread use on an industrial scale of processes in which the polymerization of olefins is carried out in a gaseous medium in the presence of a solid catalyst. Basell was formed in October 2000. It is owned equally by BASF and Shell. It develops, produces, and markets polypropylene, polyethylene, advanced polyolefin materials and polyolefin catalysts and also develops and licenses polyolefin processes. It serves customers in more than 120 countries with materials produced in eighteen countries. A widely used technology for gas-phase polymerization processes is the fluidized-bed technology. In fluidized-bed gas-phase processes, the polymer is confined in a vertical cylindrical zone. The reaction gases exiting the reactor are taken up by a compressor, cooled and sent back, together with make-up monomers and appropriate quantities of hydrogen, to the bottom of the bed through a distributor. Entrainment of solid in the gas is limited by an appropriate dimensioning of the upper part of the reactor (freeboard, i.e. the space between the bed surface and the gas exit point), where the gas velocity is reduced, and, in some designs, by the interposition of cyclones in the exit gas line. The flow rate of the circulating gas is set so as to assure a velocity within an adequate range above the minimum fluidization velocity and below the "transport velocity". The heat of reaction is removed exclusively by cooling the circulating gas. The catalyst components may be fed in continuously into the polymerization vessel. The composition of the gas-phase controls the composition of the polymer. The reactor is operated at constant pressure, normally in the range 1-3 MPa. The reaction kinetics is controlled by the addition of inert gases. A significant contribution to the reliability of the fluidized-bed reactor technology in the polymerization of olefins was made by the introduction of suitably pre-treated spheroidal catalyst of controlled dimensions and by the use of propane as diluent. Since fluidized-bed reactors approximate very closely the ideal behavior of a "continuous stirred-tank reactor" (CSTR), it is very difficult to obtain products which are a homogeneous mixture of different types of polymeric chains. In fact, the composition of the gaseous mixture that is in contact with the growing polymer particle is essentially the same for all the residence time of the particle in the reactor. As an example, one of the major limits of fluidized-bed processes is the difficulty of broadening the molecular weight distribution of the obtained polymers. It is

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generally known that, in the continuous polymerization of olefins in a single stirred stage (which also involves steady composition of the monomers and of the chain transfer agent, normally hydrogen) with Ti-based catalysts of the Ziegler-Natta type; polyolefin having a relatively narrow molecular weight distribution are obtained. This characteristic is even more emphasized when metallocene catalysts are used. The breadth of the molecular weight distribution has an influence both on the rheological behavior of the polymer (and hence the processability of the melt) and on the final mechanical properties of the product, and is a characteristic which is particularly important for the (co)polymers of ethylene. It is possible to broaden the molecular weight distribution of polymers without affecting their homogeneity by means of a gas-phase process performed in a loop reactor. The gas-phase polymerization is carried out in two interconnected polymerization zones to which one or more monomers are fed in the presence of a catalyst under reaction conditions and from which the polymer produced is discharged. The process is characterized in that the growing polymer particles flow through the first of the polymerization zones under fast fluidization conditions, leave the first polymerization zone and enter the second polymerization zone, through which they flow in a densified form under the action of gravity, leave the second polymerization zone arid are reintroduced into the first polymerization zone, thus establishing a circulation of polymer between the two polymerization zones. It is possible to broaden the molecular weight distribution of the polymers simply by properly balancing the gas-phase compositions and the residence times in the two polymerization zones of the gas-phase loop reactor. This is due to the fact that, while the polymer moves forward in the second polymerization zone flowing downward in a plugflow mode, owing to the monomer consumption, it finds gas-phase compositions richer in molecular weight regulator. Consequently, the molecular weights of the forming polymer decrease along the axis of this polymerization zone. This effect is also enhanced by the temperature increase due to the polymerization reaction. Only a limited control of the molecular weight distribution is possible. In fact, even if hindered by the packed polymer, the diffusion of the gas within the polymerization zone in which the polymer particles flow in a densified form makes it difficult to establish substantial differences in the gas compositions at different heights of that zone. Moreover, it is not easy to achieve an effective balance of the residence times in the two different polymerization zones of the reactor. Sometimes means are provided which are capable of totally or partially preventing the gas mixture present in the riser from entering the down comer, and a gas and/or liquid mixture having a composition different from the gas mixture present in the riser is introduced into the down comer. The introduction into the down comer of the gas and/or liquid mixture having a composition different from the gas mixture present in the riser is effective in preventing the latter mixture from entering the down comer. The state of fast fluidization is obtained when the velocity of the fluidizing gas is higher than the transport velocity, and it is characterized in that the pressure gradient along the direction of transport is a monotonic function of the quantity of injected solid, for equal flow rate and density of the fluidizing gas. Generally, in the down comer the growing polymer particles flow downward in a more or less densified form. Thus, high values of density of the solid can be reached (density of the solid =kg of polymer per cu.m of reactor occupied by polymer), which can approach the bulk density of the polymer. A positive gain in pressure can thus be obtained along the direction of flow, so that it becomes possible to reintroduce the polymer into the riser without the help of special

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mechanical means. In this way, a "loop" circulation is set up, which is defined by the balance of pressures between the two polymerization zones and by the head losses introduced into the system. The gas mixtures involved in the process of the invention can contain entrained droplets of liquid composed of liquefied gas, as it is customary when operating in the socalled "condensing mode". The introduction of the gas and/or liquid mixture of different composition into the down comer are such to establish a net gas flow upward at the upper limit of the down comer. The established flow of gas upward has the effect of preventing the gas mixture present in the riser from entering the down comer. Basell‟s Spheripol process is the most licensed technology ever developed for the production of polypropylene. Since 1982, proof of its enduring success is the number of leading polypropylene producers choosing this technology. This includes Exxon, Dow, Borealis, Showa Denko, Hyundai and Sinopec to name a few. Spheripol process is the result of 40 years of quality improvement. In the 1960s, polypropylene processes employed first generation low yield catalysts (< 1000 Kg PP/kg catalyst) in mechanically stirred reactors filled with an inert hydrocarbon diluent. Polymer produced with these catalysts had unacceptably high residual metals, and contained 10% a tactic polypropylene which required separation. Removal of catalyst residues and a tactic PP involved treatment of the polymer with alcohol, multiple organic and/or water washings, multistage drying and elaborate solvent, amorphous and catalyst separation systems. These processes were costly and difficult to operate and also required extensive water treatment facilities and catalyst residue disposal systems. In the 1970s the discovery of second generation high yield catalysts (6000 kg PP/kg catalyst) eliminated the need for catalyst residue removal but a tactic was still unacceptably high. This simplified the washing but did not eliminate the tactic recovery steps. In the 1980s the third generation high yields, high selectivity catalysts (30,000 kg PP/kg catalyst) eliminated the need for catalyst and a tactic removal. This further simplified the process and improved product quality. Other breakthroughs occurred in the process design, through the refinement of gas-phase and bulk polymerization reactors that lead to the development of Spheripol technology in 1982. Current catalyst generation has the ability to produce new families of reactor-based products with improved properties. They offer even greater control over morphology, isotacticity and molecular weight. Polypropylene is the world‟s fastest growing thermoplastic. The safety record of Basel technologies is among the best in the world. They achieved nearly 7 million operating hours without any major incident. Spheripol technology includes features that reduce both resource consumption and emissions from the process. These include use of high yield highly stereospecific catalysts the absence of solvents in the process to suspend the polymer (the suspension agent is the monomer itself, recovery and recycling of unreacted monomers and the absence of undesired by-products from the reaction. At the end of 2002 a year-on-year analysis of operating records from over 80 spheripol process plants worldwide showed the average overall operability rate is about 98%. Of an average 2% downtime, less than 1% is due to process features. Spheripol process is versatile and offers a wide range of homopolymers, random copolymers and terpolymers as well as heterophasic impact and specially impact copolymers covering all PP application fields. The continuous process offers good quality product with minimum property variation due to excellent process stability and consistency of Basel‟s catalysts performance. A range of single line capacities from 40 – 450 kilotons/annum are available for homopolymer, random copolymer, either using polymer or chemical grade monomer. They have a modular plant installation whereby

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easy adaptation to new business opportunities is effected. Capital costs for spheripol process are competitive with other used PP processes. It offers the lowest operating costs and excellent plant reliability and transition efficiency. The sheripol process using high yield and high selectivity catalysts supplied by Basel has unique ability to produce polymer spheres directly in the reactor. Spherical PP differs considerably from the small irregularly shaped granular particles produced with some other technologies and provides significant advantages in terms of process reliability. It consists of the following unit operations (Figure 4.0); a) Catalyst Feeding b) Polymerization  bulk polymerization (homopolymer, random copolymer and terpolymer)  Gas-phase polymerization (heterophasic impact and specialty copolymer. Can be added at a later stage without affecting initial plant configuration) c) Finishing Tubular loop reactors are used to conduct the bulk polymerization. These are filled with liquid propylene to produce homopolymer or random copolymer or terpolymer. The catalyst, liquid polypropylene and hydrogen for molecular weight control are continuously fed into the loop reactor. Residence time in the reactor is lower compared with other technologies. This is due to the high monomer density and increased catalyst activity. The loop reactor is used because it is low cost, exhibits good heat transfer characteristics and maintains uniform temperature, pressure and catalyst distribution. The low residence time also results in short transitions during grade changes. The complete filling of the reactors results in low contamination between different grades.

Figure 4.0. Spheripol Process Schematic to Manufacture Spherical Polypropylene Homopolymer, Random Copolymer and Terpolymer.

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A homogeneous mixture of polypropylene spheres is circulated inside the reactor loop. If the production of random copolymer or terpolymer is desired, ethylene and/or butene-1 are introduced in small quantities into the loop reactor. This process achieves very high solid concentration (>50% by weight), excellent heat removal (by water circulation in the reactor jacket) and temperature control (no hot spots). The resulting polymer is continuously discharged from the reactor through a flash heater into a first-stage de-gassing cyclone. Unreacted propylene from the cyclone is recovered, condensed and pumped back into the loop reactor. For the production of impact and specialty impact copolymers, polymer from the first reactor is fed to a gas-phase fluidized bed reactor that operates in series with the loop reactor (this gas-phase reactor is bypassed when homopolymer or random copolymer is produced). In this reactor, an elastomer (ethylene/propylene rubber) formed by the introduction of ethylene is allowed to polymerize within the homopolymer matrix that resulted from the first reaction stage. The carefully developed pores inside the polymer particle allow the rubber phase to develop without the sticky nature of the rubber to disrupt the operation by forming agglomerates. Fluidization is maintained by adequate recirculation of reacting gas: reaction heat is removed from the recycled gas by a cooler, before the cooled gas is recycled back to the bottom of the gas-phase reactor for fluidization. This type of gas-phase reactor is efficient because it maintains a high degree of turbulence in order to enhance monomer diffusion and reaction rates, and offers an efficient heat removal system. Some specialty products, incorporating two different ethylene content copolymers, require a second gas phase reactor in series. In impact copolymer production, at least 60% of the final product is produced in the first-stage loop reactor. In addition, since ethylene is more reactive than propylene, the gasphase reactors are smaller than would be required if this design were to be used for homopolymer production. Spherical morphology ensures high reliability and elimination of fouling phenomena, which frequently disrupt other gas-phase systems. Polymer discharged from the reactors flows to a low-pressure separator and subsequently to a steam treatment vessel where catalyst residues are neutralized and the dissolved monomer is removed, recovered and recycled back to the reactor system. From the steamer, polymer is discharged into a small fluidized-bed dryer with a hot nitrogen closed loop system to remove the moisture. The final product is conveyed to an extrusion unit, where it is mixed with additives and extruded to pellets. Spheripol technology is designed to lower environmental impacts. The design of each Spheripol process plant includes a number of safety features, such as:      

Proprietary Catalyst Deactivation System, which immediately stops all reaction Computer controlled emergency shutdown systems Uninterruptible Power Supply (UPS) for computer control and critical instrumentation control Instrument air emergency buffer Emergency Blowdown System to empty the plant quickly, in the event of an emergency Gas detectors which instantly determine and highlight (on a graphic easy-to-read board) the source of any hydrocarbons in the event of leakage into the atmosphere

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Automatic fire protection systems

Depending upon the severity of the situation, the plant can be shut down manually in a step-by-step, controlled fashion, more rapidly by both manual and computer control, or by instant automatic shutdown. Spheripol process units are configured so that unreacted monomers are recovered and recycled. If necessary, other discontinuous hydrocarbon purges can be sent to "off-gas recovery” for use as a fuel supply or to a flare system. The Spheripol process does not use hydrocarbon diluents or contaminant chemicals and the only wastewater is released from the steaming/ drying section of the plant which contains steam condensate and a small amount of inert polymer fines, which are recovered by a separator. Polypropylene development has for many years been focused on the use of bimodality as well as increasing operating temperature to improve the product properties. Bimodality had always been reached by operating two reactors in series at different operating conditions, resulting in a polymer with the two different qualities mixed on "intra-particle” level. In the Spherizone process, bimodality is created within one single reactor operating at different conditions between the various zones inside the reactor, resulting in an intimate mixing of the various propertydetermining phases at "macro-molecular” level. Spherizone technology was brought from lab scale through two scale-ups of pilot plants to commercial size in 2002, when the very first Spheripol process plant in Brindisi was upgraded by the installation of the Multi-Zone Circulating Reactor, replacing the two slurry loop reactors and the flash line with the new reactor module. The plant has been running well as of the start-up, and is delivering products with excellent quality. The development of the manufacturing platform and the catalysts used will further continue, and with new investments in Spherizone process plants planned, Spherizone technology will become the new standard in PP industry. In the spherizone process (Figure 5.0) catalyst is continuously fed to the multi-zone circulating reactor. In this specially designed loop-reactor consisting of two reaction zones, the growing polymeric granules are circulated between the two different zones. In the so-called "riser” the polymer particles are entrained upward in a fast fluidization regime by the monomer gas flow from a blower. Then, in the top of the reactor the polymer particles enter the so-called "down comer”. Then there is a downward dense-phase plug-flow regime under gravity. At the bottom of the reactor the polymer particles are again fed to the "riser” section. The reactor can be operated in different conditions with regard to hydrogen (as chain transfer agent) and comonomer concentration in the two sections, allowing for the development of a bimodal (MFR, comonomer concentration/type) polymer structure at macro-molecular level. This split between the reaction conditions is achieved by injection of a monomer stream with different composition than in the riser-section at the transition (barrier) section between the riser- and down comer-section. The reactor can also yield monomodal homopolymer and random copolymer products by operating the sections in equal conditions. From the top of the reactor, unreacted monomer is withdrawn and then enters a monomer recovery section. Product is continuously withdrawn from the reactor and solid polymer is separated from the unreacted monomer gas at intermediate pressure. The gas is recycled back to the MZCR. As an option, the polymer can be fed to a fluidized bed gas-phase reactor that is operated in series to the MZCR; here additional copolymerization can take place to yield high-impact copolymer PP. This gas-phase reactor can be bypassed when homopolymer or random

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copolymer is produced. In the reactor, an elastomer (ethylene/propylene rubber) formed by the introduction of ethylene is allowed to polymerize within the homopolymer matrix that resulted from the first reaction stage. The carefully developed pores inside the polymer particle allow the rubber phase to develop without the sticky nature of the rubber to upset operation by forming agglomerates. Fluidization is maintained through adequate recirculation of reacting gas; reaction heat is removed from the recycle gas by a cooler before the cooled gas is recycled back to the bottom of the gas-phase reactor for fluidization. This type of gas-phase reactor is efficient because it maintains a high degree of turbulence in order to enhance monomer diffusion and reaction rates while also offering an efficient heat removal system. Depending on the configuration, from the intermediate separator or the fluidized bed reactor the product is discharged to a receiver; from there the unreacted monomer gas is recovered and the polymer is sent to a vessel for monomer removal and neutralization of residual active catalyst by steam stripping. The removed residual hydrocarbons are recovered and can be sent back to the reactor system, while the polymer is dried by a closed-loop nitrogen system in small fluidized bed drier.

2.7. CONTINUOUS MASS POLYMERIZATION ROUTE FOR PMMA Mitsubishi Gas Chemical Co. has patented a process [Hieda et al., 1998] for preparation of high quality PMMA, poly methyl methacrylate at high product rates. Initiated, solution polymerization in the presence of solvent was used in a continuous manner. The polymer product was discharged into a vented extruder. PMMA has interesting properties such as transparency, weather resistance, and surface appearance. Batch suspension methods have usually been deployed for making PMMA. Casting material and low molecular PMMA as coating has been reported using solution polymerization. Gel effect has been seen when higher conversions were attempted during preparation of PMMA. At low conversions the energy needed at the devolatilizer is high in order to separate the unreacted monomers from the product. Addition of solvent prevents the runaway polymerization condition. The thermal decomposition resistance deteriorates during solution polymerization of PMMA. Zipper decomposition begins at a C-C single bond that is found adjacent to a terminal double bond in the temperature range of 230 0C – 270 0C. At polymerization temperatures less than 100 0C thermally extremely weak head-head bond that is found to cleave at 200 0C remains. The deterioration of decomposition resistance can be decreased by decreasing the zipper decomposition and the head-head bond cleavage. This is accomplished by decreasing the concentration of terminal double bonds in the polymer and the reaction temperature is increased above 100 0C. Methanol solvent, MMA, methyl methacrylate monomer, free radical initiator, CTA, chain transfer agent is added continuously into two agitated reactors connected in series. The conversion achieved is ~ 55-93 % by mole fraction at polymerization temperatures of 100 0C – 180 0C. The polymerized mass is discharged into screw extruder that is vented in a continuous manner. The devolatilizer is operated at temperatures of 170 0C – 270 0C. The volatiles are collected at the vents. Vacuum of 1 – 400 mm Hg downstream is used to reduce the residuals in the polymer further. PMMA with Mw, weight averaged molecular weight of 80,000 – 200,000 were prepared. The thermal decomposition is only 3 wt%.

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Figure 5.0. Spherizone Process to Manufacture Spherical Polypropylene.

Some more examples of continuous polymerization technology can be found in the patent literature. The flow diagrams with labels for the important unit operations are given in the Figures below. The reactor configuration for solution polymerization of polyglycolide is shown in Figure 8.0. SAN styrene and acrylonitrile can be prepared in a continuous manner. The devolatilizer used will depend on the composition of AN in the polymer. In Figure 6.0 is shown two CSTRs in series and wiped film devolatilizer process in order to make SAN copolymer. The kinetics of SAN copolymerization is obtained from experimental study. The reaction rate reaches a maxima as the AN concentration is increased. The sequence distribution of SAN copolymerization is discussed in Sharma (2011). More monomers can be added as termonomer or tetra monomer as the product need arises. The process equipment can be used as retrofit. Thermal polymerization of styrene can be used. In chapter 3.0 is discussed a method to calculate the polymer composition from monomer composition using state space representation during free radical multi-component copolymerization in CSTR. Numerical solutions are needed if made in a PFR. BASF uses a tube bundle reactor to make SAN (Figure 7.0). At azeotropic composition of AN = 25 wt % the monomer and polymer composition will be the same. PFR can be used for more productivity. When operated at the azeotrope monomer-polymer composition in the copolymer is readily calculable. In Figure 9.0 is shown a continuous mass polymerization process for polystyrene. SMA, styrene maleic anhydride copolymer can be prepared in a continuous manner as shown in Figures 10.0 and 11.0. Examples of condensation polymerization process for production of polyamide and nylon 6,6 are shown in Figures 12.0 and 13.0. Gas phase polymerization process is shown in Figure 14.0. Solution copolymerization process developed by Exxon Mobil Corp. is shown in Figure 15.0. Esterification processes are shown in Figures 16.0 – 17.0. Poly tri methylene terephthalate can be prepared as shown by du Pont. Shell has a process for preparation of polyester as shown in Figure 18.0.

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1 Monomer Storage Tank; 2 Flow Control Valve 3,4 - 2 CSTRs in Series 6,9 - Decanters 8 – Pump; 5 – Static Mixer Figure 6.0. LG Continuous Mass Polymerization Process for Preparing Styrene-Acrylonitrile Copolymer.

1 – Tube bundle reactor 2,3 – intermeshing static mixers 4 – Circulating pump; 5 – feed mixture inlet; 6 – outlet 7 – Degassing extruder 8 – Fresh monomer; 9 – volatile content Figure 7.0. BASF Continuous Mass Polymerization Process for SAN.

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Figure 8.0. Reactor Configurations in the Continuous Solution Process for Biodegradable Polyglycolide.

I – Circulating Line; II – Main Polymerization Line 1- Plunger pump; 5,10,13 – Gear pump 2,3,4,6,7,8,9 – Tubular Reactors in Series 11 – Preheater; 12 – Devolatilization Chamber Figure 9.0. Continuous Bulk Polymerization of Styrene.

1 – Maleic Anhydride Feed; 2 – Styrene Feed

Continuous Polymerization Process Technology 3 – First Stage Reactor; 4 – Pump; 7 - shaft 5 – 1 Shaft Horizontal Reactor 6 – 2 Shaft Horizontal Reactor Figure 10.0. Continuous Mass Polymerization of Styrene Maleic Anhydride Copolymer.

Figure 11.0. Bayer Continuous Mass Polymerization Process for Styrene Copolymer.

Figure 12.0. Continuous Preparation of Polyamide.

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Figure 13.0. Continuous Process for Nylon 6,6 from Hexamethylene Diamine and Adipic Acid.

1,2 – Fluidized Bed Reactor; 3 – Distributor 5, 21 – Separator; 6,7, 22 – Discharge Pipe 8 – heat exchanger; 9 – 10, 23 – Compressor; 11 – Feed Pipe Figure 14.0. Continuous Gas Phase Copolymerization Process with a Fluidized Bed Reactor.

Continuous Polymerization Process Technology

8 – CSTR; 2,4,58 – pressurized feed 14,34,40 – separation of solvent, unreacted monomer 20 – lean phase; 22 – concentrated 3 – high capacity, low viscosity pump 18 – pressure reducing means; 12 – heating stage 10 – catalyst feed; 24 – final cooler 32 – drier; 3 – pump Figure 15.0. ExxonMobil Continuous Solution Copolymerization of Ethylene and Propylene.

12 – Prepolymerizer; 20 – preheater; 22 – tray; 24 - dome 14 – Finisher 10- Direct Esterification Reactor Figure 16.0. Du Pont Continuous Polymerization Process to Manufacture Polytrimethylene Terepthalate.

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2 - Esterification Reactor 6 – Prepolymerization Column 10 – Finisher 20 - Glycol Recovery System 26 - Adsorption Bed 32 - Horizontal Agitated Cylindrical Vessel Figure 17.0. Du Pont Continuous Production Process for Polyesters.

2.8. SUMMARY 18 process flow diagrams are reviewed to discuss continuous processes to manufacture ABS, Acrylonitrile Butadiene Styrene engineering thermoplastic, SAN, Styrene – Acrylonitrile Copolymer, PS, Polystyrene, Nylon, PMMA, Poly Methyl Methacrylate, SMA, Styrene Maleic Anhydride and Polyolefins. The Dow process, GE process and Bayer process are compared side by side. The tower process of Dow and the three stage process with a finisher used by GE and the two CSTR process of Bayer are reviewed from a process performance and cost view point. The spheripol process and spherizone process commercialized by Basell polyolefins is discussed. The use of the circulating fluidized bed and the combination of the liquid pressurized process and gas phase Ziegler Natta catalyzed process is reviewed. The spherical polypropylene is manufactured in one of them. The continuous mass polymerization process to manufacture PMMA by the Rohm and Haas

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process is discussed. The azeotropic distillation separation method for separation of the unreacted monomer from the polymerized product and the helical ribbon agitated polymerization reactor is evaluated. The use of the decanter for separation of the unreacted monomers from the product is highlighted in the LG 2 CSTR process to manufacture SAN. The use of tube bundle reactor and degasing extruder to prepare SAN by BASF is compared with the LG process. The reactor configuration to prepare polyglycolide is elloborated. The use of tubular reactor series for the manufacture of polystyrene is discussed. The use of a multistage rubber dissolver and a multistage reactor can be seen in the Hitachi process to prepare HIPS. Shaft horizontal reactors are used in the preparation of SMA copolymer in a continuous fashion. The continuous kettles for the preparation of nylon from hexamethylene diamine and adipic acid and directly from aminonitrile are reviewed. The ethylene propylene copolymer preparation by continuous solution process developed by ExxonMobil is sketched. Some issues in developing a styrenic copolymer using the Bayer continuous process is discussed.

Figure 18.0. Shell Continuous Reactors in Series Polymerization Process to Manufacture Polyester.

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2.9. FURTHER READING Aliberti, V. A., Kruse, R. L., Valcarce, E. M. (1983). “Mass Polymerization Process for ABS Polyblends”, US Patent 4, 417.030, Monsanto Co., St. Louis, Mo. Bredeweg, C. J. (1980). “Process for the Polymerization of Acrylonitrile-Butadiene-Styrene Resins”, US Patent 4, 239,863, The Dow Chemical Company, Midland, MI. Fukumoto, C., Furukawa, T. & Oda, C. (1983). “Process for Continuous Production of High Impact Polystyrene”, US Patent 4,419,488, Hitachi, Tokyo, Japan. Hieda, S., Kurokawa, M., Higuchi, Y. & Kawahara, S. (1998). “Process for Preparing Polymer”, US Patent 5,804,676, Mitsubishi Gas Chemical Co., Tokyo, Japan. Sharma, K. R. (1997). “A Statistical Design to Design the Process Capability of Continuous Mass Polymerization of ABS at the Pilot Plant”, 214th American Chemical Society National Meeting, Las Vegas, NV, September. Sharma, K. R. (2011). “Polymer Thermodynamics: Blends, Copolymers and Reversible Polymerization”, CRC Press, Boca Raton, FL. Sue, C. Y., Koch, R., Pace, J. E. & Prince, G. R. (1996). “Grafting, Phase-Inversion and Cross-Linking Controlled Multi-Stage Bulk Process for Making ABS Graft Copolymers”, US Patent 5,569,709, General Electric Company, Pittsfield, MA. Virkler, T. L. & Wu, W. C. (2001). “Process for Preparing Extrusion Grade ABS Polymer”, US Patent 2001/0031827A1, Bayer Corp., Pittsburgh, PA.

Chapter 3

MATHEMATICAL PROCESS MODELS 3.0. OVERVIEW Mathematical models are used for scale-up of novel chemistry recently invented in the laboratory to the manufacturing plants. Pilot plant studies often times are understaffed and have a heavy backlog of experimental trials that needs to be performed. Mathematical models can be used where pilot plant data is not available for scale-up. Sir Albert Einstein said: "How can it be that mathematics, a product of human thought independent of experience, is so admirably adapted to the objects of reality?"

Chemical process models are developed for a variety of reasons. The overall objective is to gain better understanding of the process. This in turn would help the team of engineers to manufacture new products at the lowest cost, in large scale, with high quality. Better process understanding of the process can also lead to more safe operations. The recent BP Americas spill called Deep Water Horizon [Telegraph, 2010] and the nuclear power plant disasters in Japan that followed the earthquake with a magnitude of 8.9 on the Richter scale in March 2011 are cases in point that poor project planning can lead to unsafe operations. Proactive solutions are needed. It is better to be safe than sorry. The choice of the location of the nuclear power plants near the “ring of fire” that is prone to earthquakes was poor. Good process models can lead to more efficient process operations with less pollution to the environment. The ecosystem needs to be preserved. Dynamic process models can be used to train operators, to design processes, in safety analysis, in process control, in project trouble shooting and in globalization of an enterprise, etc. There are different kinds of process models. These are as follows; (i) Empirical Models (ii) Semi-Empirical Models (iii) Mechanistic Models (iv) Models from Shell Balance and Applications of Equations of Continuity, Energy, Momentum, Mass and Charge. (v) Supercomputer based models such as the one used for weather prediction (vi) Simulation on the Computer

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Kal Renganathan Sharma (vii) Mesoscopic and Stochastic Models (viii) Monte Carlo Trials

A. SIMULATIONS ON COMPUTER 3.1. CONSECUTIVE-COMPETITIVE REACTIONS DURING BIODIESEL PRODUCTION 3.1.1. Selectivity of Biodiesel over Glycerol during Transesterification Oil reserves are expected to be depleted by the year 2050 at the current levels of production. The crude oil reserves are estimated at 4.16 trillion liters worldwide [Podolski et al., 2008]. Global consumption is 84.6 million barrels/day. Earth‟s entire oil reserves according to one estimate are 1.2 trillion barrels without oil sands and 3.74 trillion barrels with oil sands. At the present rate of consumption the oil reserves will be depleted in the next 38.8 – 122.2 years. Search is on for alternative oil finds. Per geological survey 3-4.5 billion barrels was found in Montana and North Dakota. If oil shale can be used as source of oil the reserves can last for 110 more years. Oil finds have been found in Russia, Columbia and Africa. According to the big rollover theory global oil production is already past its peak production. M. K. Hubert, Shell Oil Co., Houston, TX, studied the exhaustion of oil fields. Initial oil find, exploitation and exhaustion phases were identified. This followed the bell curve. He concluded that United States would peak in its oil production in 1970 [Congressional Record, 2005]. The curve is called the Hubert curve. The peak is also called the rollover. Oil experts note that the peak production has been reached. Every year since 1970, we have found less oil and pumped less oil than we consume. Largest known reserves of crude oil are located in the Middle East, along the equator and in Russia and its neighbors. Transportation of crude oil has not been without spills. In 1989 the Exxon Valdez dumped 11 million gallons of oil into the waters and onto the shores of Alaska. 85 million gallons of oil onto previously pristine Arctic tundra was spewed because of a rupture in 1994 in a pipeline in Russia. In 1994 Exxon was ordered to pay $5 billion for Alaskan oil spill. 206 million gallons of oil was spilled in the recent deep water horizon oil spill in the Gulf of Mexico by BP Americas in 2010. This was the worst environmental disaster ever in the history of technology. This was attributed to a rig explosion. The gusher was from the ocean floor. Wars have been fought to prevent monopoly of oil supply. Analogous to how in the history of mankind Stone Age gave way to Iron Age, oil age may be eclipsed by sustainable energy. Man dropped the use of stones when he learnt the use of iron. Air pollution has been found as a result of continued and increased use of petroleum. Global warming has been concluded as a problem because of significant increase in concentration of CO2 in the earth‟s atmosphere. The principles of Sustainable Engineering were developed at the Sandestin Conference of 2003 [Abraham and Nguyen, 2005]. This ought to set the direction of engineers who work on developing sustainable alternatives to current engineering practices. Energy is considered a primary component of sustainable engineering. This is because of;

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61

(i) the pollution problems that arise from current methods of consumption of energy such as acid rain formation from emission of oxides of sulfur and nitrogen and; (ii) the depleting reserves of fossil fuel such as bituminous coal, petroleum, peat, lignite, Anthracite, oil shale, natural gas, etc. In most formulations of sustainability the use of renewable energy resources is desirable. The Sandestin principle states that, “Minimize depletion of natural resources”. Renewable energy resources may be viewed as fuels that are produced at rates equal to or greater than the rates at which they are consumed. Thus there is no “net depletion” of the resources with the passage of time. For example, biomass production rates may jeopardize food supplies and hence have biomass fall out of the column of renewable resources and into the column of non-renewable energy resource. Sandestin principle of life cycle analysis may be applicable for evaluation of biomass as energy resource.

3.1.2. Biodiesel Biodiesel is an interesting choice for use as alternate energy. Biodiesel is a mixture of FAME, fatty acid methyl esters. It is an EPA, Environmental Protection Agency designated advanced biofuel. It can be derived from vegetable oil or animal fats. It can be used from a spectrum of resources that include waste fats, greases and agricultural oils. Biodiesel can save money for the consumer. For example, Ray Mabus, secretary of the Navy testified in Capitol Hill that by purchasing 20% biodiesel blend, savings of 13 cents a gallon and ~ $30,000 total in winter in heating oil costs were realized. Renewable Fuel standard makes good policy as it preserves the ecosystem from the advantages of life cycle use. Biodiesel is nontoxic. It has low emission profiles and is environmentally benign [Krawczyk, 1996]. A century ago R. Diesel successfully used vegetable oil as fuel for his engine. Prior to WW II vegetable oils were blended with diesel fuels time and again. It is recommended for use as a substitute for petroleum-based diesel because it is renewable and biodegradable. One of the methods of preparation of biodiesel is the trans esterification of triacylglycerides in vegetable oil or animal fat with an alcohol such as methanol in the presence of an alkali or acid catalyst. The products are FAME (s) and are called biodiesel. Glycerin is formed as byproduct. Alkali catalysts used are NaOH or KOH. Triglyceride + CH3OH  3R-COOCH3 + C3H5(OH)3 (methanol) (FAME) (Glycerin)

(3.1)

Three synthesis methods are reported for commercial manufacture of biodiesel. These are: Type I: Trans esterification of Vegetable Oil Raw oil was mixed with methanol in the presence of catalyst such as sodium methylate in order to produce the FAME. Glycerin is formed as a by-product. A two or three stage process of reaction and centrifugal separation is used. Centrifugal separation is used to separate the glycerin and biodiesel layers by gravity differences. More degree of separation can be

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Kal Renganathan Sharma

achieved by increased torque of the rotor [Sharma, 2012a]. A trade-off is seen between utility cost for rotor speed and purity level. Optimal operation of rotor can be derived at for maximum revenue. At end of the second stage [Connemann, 1994] with 99.2 – 99.6% conversion the mixture is passed through a vacuum distillation tower in order to separate the unreacted methanol, recover the sodium methylate catalyst and recycle the unreacted oil. B & P Process patented a process [Martin et al., 2011] to make biodiesel with less equipment, more yield and at a higher purity. They use a higher temperature than the boiling point of methanol and increased pressure of the reactor in order to keep the methanol from boiling. The centrifugal separator was made with perforated concentric cylinders. The separation process was affected in a counter-current manner. This makes the throughput higher and use less floor space. The glycerin passes through the rims and the biodiesel separates out through the axial region of the separator. The reaction is between the triglycerides present in the oil and methanol. Spectrum of different feedstock types have been used for the vegetable oil. This ranges from waste cooking oil to jatorpa crop cultivated with a targeted purpose of generating fuel. Feedstock types used are: (i) Soybean oil; (ii) Rapseed Oil; (iii) Sun flower oil; (iv) Coconut Oil; (v) Palm Oil ; (vi) Tung Oil. The catalyst used can be alkali, acid or enzyme. When the FFA, free fatty acid content is greater than 1% the acid catalyst would be better [Kulkrani and Dalai, 2006]. Alkaline catalyst is used in commercial plants. Alkaline catalysts are preferred when the FFA, free fatty acid content in the feedstock is less than 0.5 wt %. Process is sensitive to water and FFA. Saponification of ester may occur in presence of water. Type 2: Pyrolysis/Thermal Cracking of Vegetable Oil Since WWI, World War I, many investigators have studied the pyrolysis of vegetable oils in order to obtain biodiesel. Thermal cracking of Tung oil calcium soaps were reported in 1947. Tung oil was sapponified with lime and then thermally cracked in order to yield a crude oil. Pyrolysis methods have been found to result in more bio gasoline compared with biodiesel [Ma, 1999]. Pyrolysis usually involves heating in the absence of oxygen. Soybean oil was thermally decomposed and separated by distillation. About 75% of the distillate was hydrocarbons such as alkanes and alkenes. Vegetable oils can be catalytically cracked into useful fuels. Catalyst used are silica/alumina and palm and copra oils were used. Condensed organic phase can be fractionated into bio gasoline and biodiesel. Type 3: Physical Blending and Emulsification Process The alternate to use fuel for food has been discussed [Ma, 1999]. During the oil embargo Caterpillar Brazil in South Africa used pre-combustion chamber engines with a blend of 10% vegetable oil in order to maintain total power without any alterations to the engine. 20% vegetable oil and 80% diesel fuel blends were successfully tested. 50/50 blends were also tested. Diesel fleet was powered with filtered frying oil at 95/5 blend with diesel. Polymerization of polyunsaturated vegetable oil lead to viscosity increase and was problematic. Vegetable oil has 80% of calorific value of diesel fuel. After prolonged operation of direct-injection engines problems such as coking, trumpet formation on injectors, carbon deposits, oil ring sticking and thickening and gelling were found. 1:2 and 1:1 blends of degummed soybean oil and diesel fuel were tested for engine performance in a 6 cylinder, 6.6 liter displacement, direct-injection, turbocharged prime mover made from John Deere for 60 hours. Incomplete combustion, unwanted polymerization and gum formation were noted

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63

when using vegetable oil directly as fuel. Micro emulsions of vegetable oil with solvents such as methanol were studied. Micro emulsion was defined as a dispersion of optically isotropic fluid microstructures in 1-150 nm size range in colloidal equilibrium when two normally immiscible liquids and amphiphiles are mixed. Micelle formation leads to improvements in spraying.

3.1.3. Forecast of Biodiesel Production Over the past decade the world production of biodiesel has gone up from 15,200 barrels per day in the year 2000 to 300,000 barrels per day in 2010. In terms of volume this is about 5 billion gallons in 2010. In the past two years, biodiesel production has exceeded targets. Production plants are present in nearly every state. 1000s of jobs are created. The sigmoidal growth of the biodiesel volume is seen from 2006. Biodiesel is designated by ASTM D 6751. New laws and mandates on biodiesel came about in Brazil, China, United States, and Argentina. Germany and Brazil are the world's leading biodiesel producers. Federal excise tax credits are provided for producers and distributors of agro-biodiesel at $1 for every gallon of biodiesel they blend with regular diesel. Forecasts of global dynamics of biodiesel production are available for 2015-2020 by feedstock used such as vegetable oil feed stocks, jatorpa oil, algae biodiesel and cellulose. The expected growth rate of biodiesel production in the world is about 6% between 2009-2018 according to OECD, Organization for Economic Cooperation and Development. By 2017 biodiesel production is expected at 25 billion liters. European biodiesel board estimated that the production of biodiesel in EU is about 9.6 million tons in 2010. By the year 2022, biofuel production is projected to consume a significant amount of total world production of sugar cane (28%), vegetable oils (15%) and coarse grains (12%) [Sharma, 2011]. In India, the former President of India, ABJ Abdul Kalam during his address to the nation on National Science Day, Feb' 28th 2006 [The Hindu, 2006] called for an increase in output of biodiesel from jatorphha crop from current levels of 2 tons per hectare to 4-6 tons per hectare. The oil content of most jatorpha varieties range from 25-35 %. Research in selection, intra-specific, inter-specific hybridization and mutation breeding is needed to develop varieties with more than 45% oil content so that a recovery of 35% under mechanical expelling. Stebbins investigated the technical and economic feasibility of producing biodiesel and livestock feed from Vermont oilseeds at a farm scale and commercial scale. Commercial scale biodiesel facility in Vermont was found to be more profitable as oil prices rose in the simulation models used. India has 60 million hectares of wasteland of which 30 million hectares are available for energy crops such as jatorpha. Cars that can run on biodiesel need be developed and encouraged. The Indian Railways runs passenger trains with diesel engine with 5% blend of biodiesel. 15 million jatorpha saplings are planted in Railways' land. President B. Obama as a senator endorsed the budding biodiesel industry at a new biodiesel plant in Cairo, IL in 2006 [Moran, 2006]. The Renewable Energy group announced that it would build a 60 million gallon per year refinery and had raised $100 million in financing. Bunge Ltd., a major food processor and other venture capital firms were the contributors. About 76 biodiesel plants were in production in 2006, up from 22 in 2004. A biodiesel plant on an average costs up to $20 million to build and yields 30 million gallons per year of fuel.

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Biodiesel serves an important need of meeting the energy security of United States and the developing countries in the world.

3.1.4. Process Analysis The electrical utility costs increases in a non-linear manner with increase in rotor speed in centrifuge used to separate glycerol from FAME. More separation is achieved with more utility costs but with more revenue. A point of optimal operation can be arrived at for maximum revenue. This may vary depending on the price of gasoline at the pump at a given time. The reactions in the reactor during biodiesel production may be modeled as scheme of multiple reactions of the consecutive-competitive/series-parallel type [Levenspiel, 1999]. The methanol can be assumed to be in excess. Hence the reactions shown below can be assumed to obey the pseudo first order kinetics. The concentration of methanol can be lumped with the intrinsic second order reaction rate constant to give a pseudo first order lumped rate constant. The catalytic effect is also captured here. The reactions are modeled as follows; k1

A B

P

R

P

S

P

T

k2

R

B

(3.2)

k3

S

B

Where A - triglyceride, B - Methanol (CH3OH), R-1,2 and 1,3 diglyceride, S – mono glyceride, P – (FAME), T – Glycerol. It may be assumed that once the product P is formed it does not participate in the reaction any further. The FAME is harvested from the kettle. As glycerol (T) can be sold for profit this scheme is of more interest. This reaction set is applicable for successive attacks of a compound by a reactive material. In this case the reactive material is methanol and the compound is triglyceride. The kinetics of the reactions can be written as follows; dC A dt dC R dt dC S dt dCT

k1C A k1C A

k2C R

k2C R

k3C S

k3C S dt C P C A0 C R

CS

(3.3)

CT

CA

It is assumed that the initial concentration of triglycerides is C A0 and that of diglycerides, monoglycerides, glycerol is zero at time zero. The four ordinary differential equations in Eq. (2) can be written in the state space form as follows;

Mathematical Process Models CA d CR dt C S CT

k1 k1

0 0

0 k2

0 0

0 0

CA

k2

k3

CS

0

k3

0 0

CR

65

(3.4)

CT

The eigenvalues of the rate matrix in Eq. (3.4) can be seen to be –k1, -k2, -k3 and 0. Since three Eigen values are negative and one is zero the system can be seen to be of the integrating type [Sharma, 2012c]. This is an example of three-step reaction where the final T, glycerol and P (FAME) are desired. The addition policy of methanol along the length in case of PFR, plug flow reactor and the timings in the case of CSTR can influence the product mix. The method of mixing the reactants such as slow mixing of A to B, slow mixing of B to A and rapid mixing of A and B may be important design criteria. In order to obtain more yield of P the points where R and S will reach a maxima need be avoided as operating points

3.1.5. Economic Analysis The cost of raw materials is a critical factor in the profitability of biodiesel manufacture. Twelve reports were reviewed [Bender, 1999;Weber, 1993] on economic feasibility of biodiesel production using different feed stocks and scales of operation. The production costs for conversion to biodiesel from different feed stocks are given in Table 3.1. At the time of the study in 1999 the projected costs for biodiesel from oilseed or animal fats have a range from 30 cents – 69 cents per liter. The estimates include the soymeal and glycerin credits expected. Crushing and esterification facility is by retrofit of existing tallow facility. Cost of biodiesel from vegetable oil and waste grease are 54 – 62 cents per liter. When the pre-tax price of diesel is 18 cents per liter the biodiesel venture is not profitable. Significant factors that contribute to the bottom line of the biodiesel production were identified in [Nelson and Shrock, 1993]. These include the cost of raw materials, plant size, credit received for glycerin as by-product sales. When waste cooking oil was used the material costs went down. Restaurant greases cost less than food-grade canola and soybean oils. The first factory that produced biodiesel at 300 tons per year from waste cooking oil was started in Chiayi county of Taiwan in October of 2004. About 700 garbage trucks were fueled from biodiesel in 2005 in Taiwan. Table 3.1. Review of 12 Different Routes to Biodiesel

1 2 3 4

Feedstock (Type) Soybean Oil Animal Fats Canola Oil, Sunflower Oil Rapeseed Oil

Per Liter Production Cost 30 cents 32-37 cents 40-63 cents 69 cents

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3.1.6. Annual Worth, AW of a Bio-Diesel Plant in Taiwan [You et al., 2008 ] Biodiesel is produced by Trans esterification reactions. Triglycerides present in virgin soybean oil are reacted with anhydrous alcohol such as methanol, ethanol, proponal etc. to form FAME, fattyacid methyl esters, or biodiesel and glycerol. The alkali catalyst used was sodium hydroxide, NaOH. It can potentially be used as an alternate fuel to the ones currently in vogue that tend to cause pollution problems such as global warming, acid rain, greenhouse gas emissions, etc. Biodiesel is an attractive fuel due to the environmental benefits that comes with its operation. It is prepared from renewable energy resources such as vegetable oil. Using the information provided in Table 3.2 annualized worth AW of the biodiesel plant is calculated. The plant is expected to last for 20 years. The interest rate for equivalence calculations can be taken as 3.0%. The main cost [Turton et al., 1998] in these plants are the raw material costs. The AW analysis is completed using a MS Excel spreadsheet as shown in Table 3.2. The capital cost is obtained by using the SUM command from cells D4 to D10. These are the costs of Reactors, Washing Column, Distillation Column, Heat Exchangers, Pumps, Vacuum Systems. The capital cost of $765,000 is amortized over 20 year period by using the capital recovery factor. The capital recovery factor is obtained from Table A-7 in Appendix A in [Sharma, 2011] for (A/P,3%,20). This was found to be 0.067216. The materials costs include the soybean oil, methanol, catalyst and solvent. The Annual costs are obtained by adding the Materials Cost, Labor Costs, Utility Costs and Overhead Costs. The annual revenue is obtained by adding the sales of biodiesel and glycerin. The AW was calculated. The AW is about $1.915 million. Thus it is profitable to operate the biodiesel plant as described in Taiwan.

3.1.7. Economic Evaluation of Biodiesel Production from Waste Cooking Oil The study of alternate fuel sources to gasoline and coal is of national importance given the supply and demand characteristics of fuels that are used extensively. Biodiesel is derived from vegetable oil or animal fats. It is recommended for use as a substitute for petroleumbased diesel because it is renewable and biodegradable. The common method of preparation of biodiesel is the Trans esterification of triacylglycerol in vegetable oil or animal fat with an alcohol such as methanol in the presence of an alkali or acid catalyst. The products are FAME, s and are called biodiesel. Glycerin is formed as a byproduct. Sodium hydroxide or potassium hydroxide is used as alkali catalyst. A student for his masters thesis, [Weber, 1993] evaluated the economic feasibility of a manufacturing plant producing approximately 22 million pounds per year of biodiesel. Tallow was Trans esterified with methanol in the presence of an alkali catalyst. A second plant is based on canola seed used as the raw material. A by-product credit can be awarded for glycerin produced from seed crushing. A summary of the capital investment, process cost, revenue accrued of the three different plants are shown in Table 3.0. The capital cost can be assumed to paid off over a 30 year period at an interest charge of 6% per year.

67

Mathematical Process Models Table 3.2. Cost and Revenue Data for Biodiesel Production in Taiwan

A. A1 A2 B.

C. D. E.

Description Material Costs (~6.8 million) Soybean Feedstock Methanol, Catalyst and Solvent Capital Equipment (per year) Reactors, Distillation Column, Heat Exchangers, Separators, Pumps Labor Costs Utilities Overhead Revenue from Biodiesel Glycerin Credit

Cost $6,234,000 $564,000 $765,000

$564,000 $124,000 $431,000 $6,845,000 $3,038,000

Table 3.3. Economic Evaluations for Biodiesel Production Plants

Plant Capacity Raw Material Used Total Capital Cost Total Manufacturing Cost Glycerin Credit Price

Plant I Alkali Catalyzed Continuous Process 22 million lbs./year Beef Tallow $ 12 million $34 million

1.7 million lbs./year Canola Oilseed $1 million $5.95 million

Plant III AlkaliCatalyzed Continuous Process 2.2 million lbs./year Animal Fats $3.12 million $3.4 million

$6 million $ 2 /lb.

$0.9 million $4/lb.

$1.2 million $3/lb.

Plant II AlkaliCatalyzed Batch Process

Which process is the most profitable ? It is assumed that the capital cost is amortized at an interest rate of 6% per year for 30 years.

Plant I (A/P, 6%, 30) from Table A-10 in [Sharma, 2011] the capital recovery factor can be seen to be 0.072649 Profit = 2*22 – 12*(A/P, 6%,30) – 34 + 6 = 15.128 million Revenue – capital recovery – cost of production + by-product credit = profit

(3.5)

Plant II (A/P, 6%, 30) from Table A-10 in [15] the capital recovery factor can be seen to be 0.072649

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Kal Renganathan Sharma Profit = 4*1.7 – 1*(A/P, 6%,30) – 5.95 + 0.9 = 1.68 million Revenue – capital recovery – cost of production + by-product credit = profit

(3.6)

Plant III (A/P, 6%, 30) from Table A-10 in [15] the capital recovery factor can be seen to be 0.072649 Profit = 3*2.2 – 3.12*(A/P, 6%,30) – 3.4 + 1.2 = 4.17 million Revenue – capital recovery – cost of production + by-product credit = profit (3.7) So Plant I is the most profitable followed by Plant III and then Plant II.

3.1.8. Selectivity Improvement It can be seen from the economic analysis the yield of biodiesel compared with other byproducts such as glycerin can be a critical design criteria in making these plants more profitable. The reaction sequence for formation of FAME, fatty acid methyl ester from triglycerides found in palm oil and other feedstock involve the formation of diglycerides, monoglycerides and glycerin in sequence with FAME produced in each intermediate step (Darnoko and Cheryan, 2000]. The reaction scheme can be represented as shown in Figure 1.0. The reactions are catalyzed. The catalyst type depends on the FFA content in the feedstock. The triglycerides species is represented with symbol A, diglycerides with R, monoglycerides with S and glycerin with T. The product FAME formed in each step is given by P and the methanol used is given as B.

Figure 1.0. Trans esterification Catalyzed Reactions from Triglycerides to Glycerin and FAME.

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69

The scheme of reactions can be modeled as shown in Eq. (3.2) as a consecutivecompetitive type [Levenspiel, 1999]. The reaction rate expressions in Eq. (3.3) or Eq. (3.4) can be written in dimensionless form as follows after making the following substitutions; C A0

XA

CA

C A0 CR

XR

C A0 CS

XS

C A0 CT

XT

(3.8)

C A0 CP

XP

C A0 k1t k2 k1 k3 k1

In dimensionless form the rate expressions given in Eq. (2) can be seen to become; dX A

1 XA

d dXR

1 XA

d

dXS

XR

d dXT d

(3.9)

XR

(3.10)

XS

(3.11)

(3.12)

XS

The rate expression for the product, FAME can be obtained by adding the contributions from the methanolysis of triglyceride, diglyceride and monoglyceride steps and can be seen to be; dXP d

1 XA

XR

XS

(3.13)

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In order to evaluate the selectivity of the FAME product P over the by-product, glycerin, T solution to Eq. (3.9-3-12) were obtained by the method of Laplace transforms. The solutions are as follows; 1 s s 1

XA s

1 e

XA

1

XR s

XS

XT s XT

s

s 1 s

1

1 1

1

(3.15)

e

1

e

(3.16) 1

1

e

e

e

s 1 s

s s

1

s 1

s

XR

XS s

(3.14)

1

(3.17) 1

(1 e

)

1

(1 e

)

(1 e

)

The product yield can be found by difference as; Xp

XA

XR

XS

XT

(3.18)

Model solutions given by Eqs. (13-17) were plotted in Microsoft Excel 2010 for Windows 7.0 on a Hewlett Packard Compaq Elite 8300 desktop computer with Intel Core i7 processor with 3.9 GHz speed. The results for the product distribution is shown in Figures 2.0 – Figures 5.0. The simulations were conducted for values of reaction rate constant ratios  < 1 and  < 1 and further for  < . It can be seen from the Figures 2-5.0 that the conversion of species A, XA increases in in a monotonic manner as predicted in Eq. (13). The monoglyceride and diglyceride yields go through a maxima. A change in curvature from convex to concave can be seen in the product yields of FAME and glycerin. There is a rate increase later in time in the formation of glycerin. The selectivity of FAME can be poor compared with glycerin formation as can be seen in Figure 4.0. FAME yield can be high as shown in Figure 2.0, 3.0. There can also be cross-over from higher selectivity of FAME to lower selectivity of FAME compared with glycerin as can be seen in Figure 5.0. In such cases, CSTR can be used with residence times less than the cross-over point in order to obtain higher yield of FAME. The convexo-concave

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71

curvature in the product yields is consistent with experimental studies such as that reported by [Jaya and Ethirajulu, 2011].

3.1.9. Conclusions Alternate energy sources are sought after in order to preserve the energy security of the nation and that of the world. This is because the oil reserves are being depleted. Further the increased use of petroleum causes global warming and increased pollution. Biodiesel is a renewable fuel. One of the methods of preparation of biodiesel is the Trans esterification of triacylglycerol in vegetable oil or animal fat with an alcohol such as methanol in the presence of an alkali or acid catalyst. The products are FAME (s) and are called biodiesel. Glycerin is formed as byproduct. Alkali catalysts used are NaOH or KOH.

Figure 2.0. Triglyceride (A), Diglyceride (R), Monoglyceride (S), Glycerin (T) and FAME (P) Product Distribution in Progressive Methanolysis at  = 0.75 and  = 0.4.

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Figure 3.0. Trigylceride (A), Diglyceride (R), Monoglyceride (S), Glycerin (T) and FAME (P) Product Distribution in Progressive Methanolysis at  = 0.75 and  = 0.6.

Three synthesis methods are reported for commercial manufacture of biodiesel. These are: (i) trans esterification of vegetable oil; (ii) Pyrolysis of vegetable oil and; (iii) Physical blending and emulsification. Spectrum of different feedstock types have been used for the vegetable oil. This ranges from waste cooking oil to jatorpa crop cultivated with a targeted purpose of generating fuel. Feedstock types used are: (i) Soybean oil; (ii) Rapeseed Oil; (iii) Sun flower oil; (iv) Coconut Oil; (v) Palm Oil; (vi) Tung Oil. FAME, are produced upon Trans esterification of vegetable oil. A two or three stage process of reaction and centrifugal separation is used. Centrifugal separation is used to separate the glycerin and biodiesel layers by gravity differences. More degree of separation can be achieved by increased torque of the rotor (Sharma 2012a).

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73

Figure 4.0. Trigylceride (A), Diglyceride (R), Monoglyceride (S), Glycerin (T) and FAME (P) Product Distribution in Progressive Methanolysis at  = 0.75 and  = 0.25.

Pyrolysis usually involves heating in the absence of oxygen. Soybean oil was thermally decomposed and separated by distillation. About 75% of the distillate were hydrocarbons such as alkanes and alkenes. Vegetable oils can be catalytically cracked into useful fuels. Catalyst used are silica/alumina and palm and copra oils were used. Condensed organic phase can be fractionated into bio gasoline and biodiesel. Micro emulsions of vegetable oil with solvents such as methanol were studied. Micro emulsion was defined as a dispersion of optically isotropic fluid microstructures in 1-150 nm size range in colloidal equilibrium when two normally immiscible liquids and amphiphiles are mixed.

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Figure 5.0. Trigylceride (A), Diglyceride (R), Monoglyceride (S), Glycerin (T) and FAME (P) Product Distribution in Progressive Methanolysis at  = 0.75 and  = 0.35.

The expected growth rate of biodiesel production in the world is about 6% between 20092018. This is according to OECD, Organization for Economic Cooperation and Development. By 2017, the biodiesel production is expected to reach 25 billion liters. European biodiesel board estimated that the production of biodiesel in EU is about 9.6 million tons in 2010. President of India has called for crops with increased yield of biodiesel. President B. Obama has endorsed the biodiesel fuel industry by offering tax credits to the farmers and suppliers. The reactions in the reactor during biodiesel production may be modeled as scheme of multiple reactions of the consecutive-competitive/series-parallel type [Levenspiel, 1999]. The methanol can be assumed to be in excess. The scheme of reactions are shown in Figure 1.0. The four ordinary differential equations in Eq. (3.2) can be written in the state space form (Eq. (3.3)). The system can be seen to be of the integrating type given the values of the eigenvalues. This is an example of three-step reaction where the product P for (FAME) and by-product, T for glycerol are desired. The addition policy of methanol along the length in case of PFR and the timings in the case of CSTR, continuous stirred tank reactor can influence the product mix. The method of mixing the reactants such as slow mixing of A to B,

Mathematical Process Models

75

slow mixing of B to A and rapid mixing of A and B may be important design criteria. In order to obtain more yield of P the points where R and S will reach a maxima need be avoided as operating points. The cost of raw materials is a critical factor in the profitability of biodiesel manufacture. The AW analysis of a biodiesel factory in Taiwan operated in 2004 is completed using a MS Excel spreadsheet as shown in Table 2.0. It can be seen that the material cost and by-product credit are critical to the profitability of biodiesel. A summary of the capital investment, process cost, revenue accrued of the three different plants are shown in Table 3.0. These plants have been operated in the batch and continuous mode. Careful evaluation of batch vs. continuous mode has not been done. Plant I has been found to be more profitable compared with Plant II and Plant III. It I s not clear whether this is because of the larger size and continuous production [Shiva Prasad and Sharma, 2012b]. The selectivity of FAME over glycerin was obtained from model solutions by method of Laplace Transforms. Product distribution curves for triglycerides, diglycerides, monoglycerides, glycerin and FAME are presented in Table 2-5.0. Formation of FAME is after competition for feed A with glycerin formation. Reactor choice, residence time, temperature can be arrived at in order to obtain higher yield of FAME compared with glycerin. This may increase the profitability of biodiesel production.

3.2. TRANSIENT DRAG EFFECTS ON PROJECTILE MOTION OF SPHERICAL OBJECTS IN AIR 3.2.1. Introduction Projectile motion was discussed by Galileo in his essay de motu in 1592. Later he identified that the trajectory of a projectile as parabolic. Galileo asserted differently from the observations of Aristotle around 343 BC that among falling objects the more earthy one will reach the earth sooner than the less heavier object. Galileo stated that both objects will fall at the same acceleration and when air resistance is not a factor both objects will fall in the same duration to ground from a higher location such as the Leaning tower of Pisa. Astronaut David Scott (1971) demonstrated this in moon during the Apollo 15 Mission to the moon. The heavier hammer and lighter feather reached the ground at the same time when Scott dropped them at the same time. Air resistance is an important factor depending on the material of the object and the shape of the object in motion. Halloun and Hestenes (1985) surveyed common sense beliefs of college students about motion and its causes. According to their study many students have some notion of parabolic motion. Few of them recognized it as caused by a constant force. 66% of pretested students were able to identify the correct parabolic path for the projectile of assigned task. Some students believe that a projectile‟s motion is not only determined by its initial velocity but also how that velocity was imparted. Some students said the ball will move in horizontal path when launched in air from hand, airplane and table. The ball will begin to fall and follow a curvilinear path which is parabolic when there is no air resistance.

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This study is to further explore the effects of transient drag on projectile motion. Parker (1977) considered two dimensional motion of a projectile experiencing a constant gravitational force and a constant fluid drag force. The quadratic force said to vary as the square of the velocity of the object. The drag coefficient was considered to be constant throughout the entire trajectory of the object and taken as approximately (CD = 0.5). The drag force in the horizontal and vertical directions were given in terms of the absolute velocity. d p2 C D vx vx2

Fx

v2y

8 d p2 C D vy vx2

Fy

8

v2y

(3.19)

(3.20)

The governing equations were coupled non-linear equations. They presented short and long time solutions. In this study, it is realized that the drag force can be resolved into vertical and horizontal components. Resolution of drag forces in horizontal and vertical directions results in decoupling of the non-linear equations. The drag coefficient variation with Reynolds number is also taken into account. The drag forces in horizontal and vertical directions can be written as follows;

Fx

Fy

d p2 C D vx2

8 d p2 C D v2y

8

(3.21)

(3.22)

The use of Eqs. (3.21,3.22) in place of Eqs. (3.19,3.20) is based on the realization that the vertical and horizontal components of the velocity and acceleration of the object in motion is independent of each other. This is captured mathematically in Eq. (3.21,3.22). The form of drag force seen in Eqs. (3.21,3.22) was first introduced by Lord Raleigh. The area of circle is used for the projected area for a sphere. The projection is orthographic. The drag force can be realized using stream function representation of flow past a sphere. Form and friction drag can be realized from viscous flow past a sphere (Bird, Stewart and Lightfoot, 2007, Bachelor, 1967, Landau and Lifshitz, 1959). The drag coefficient varies as a function of velocity. Deakin and Troup (1998) suggested that trajectories of projectile motion with air resistance can be approximated by cubic curves. Shipman et al. (2013) discuss projectile motion in their textbook for students in their introductory physical science course. The equations that can be used to projectile motion are not discussed. Symmetry of path, same range at complementary angles are discussed. Illustrations with and without air resistance are shown. Examples from Olympic track and field events such as the event Javelin Throw, Basketball, Football are discussed. Air resistance is suspected as causing asymmetry in the trajectory of the projectile. Longer range appears to be associated with a launch angle of 38 0 compared with a launch angle of 45 0 for a projectile when air resistance was taken into account.

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77

In this study the variation of drag coefficient with velocity is taken into account. The 5 constant expression suggested for steady state drag coefficient at terminal settling velocity as a function of Reynolds number by Turton and Levenspiel (1989) is used in this study. The equation used is given below as; C Dt

24 1 0.173Ret0.657 Ret

0.413 1 16300 Ret 1.09

(3.23)

Wake separation effects are taken into account in Eq. (3.23) at large Reynolds‟ number. It spans a wide range of Reynolds number from 1 at Stokes regime at low Reynolds‟ numbers through transition flow to turbulent flow at large Reynolds number up to 200,000. Eq. (3.23) is shown in Figure 6.0. Renganathan et al. (1989) calculated the vertical distance travelled by geometric and spherical particles in a fluid up to 90% of its terminal settling velocity using computer simulations. This study was extended to two dimensions for design of sedimentation tanks, (Sharma, 2012c). The variation of drag coefficient as given in Eq. (5) was taken into account in this study. Chabra (1993) presented an analytical solution for acceleration motion of a spherical particle in an unbounded Newtonian fluid. The drag coefficient used was of the following form; 2

A

CD

(3.24)

B

Re

They use a Re = p2 substitution. Eq. (6) is good for Re < 6000. Should the system under consideration fall within a Reynolds number of 6000 during the entire trajectory then Eq. (6) may be selected. Eq. (5) spans a wider range. During the launch and final phases of examples considered below such as spin less golf ball the Reynolds number of surrounding air that spans the example is 136-122,690.

3.2.2. Theory - Equations of Motion Consider a spherical or geometric object in projectile motion in an unbounded fluid. The object will traverse a trajectory in two dimensions. The equation of motion in the vertical and horizontal directions can be written as follows (Clift et al.(1978), Renganathan et al. (1989); s

d 3p dvy

6

s

d 3p dvx

6

d p2 C D v2y

6

dt

s

d 3p g

dt

8 d p2 C D vx2

8

(3.25)

(3.26)

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Figure 6.0. Drag Coefficient, CD as a function of Reynolds‟ Number for Range 1 ≤ Re ≤ 200,000.

Subramanian et al. (2001) have studied motion of bubbles and drops in microgravity conditions. In this study the earth‟s gravitational field was considered. The terminal settling velocity of the object in the vertical direction can be obtained by setting the RHS, right hand side of Eq. (3.25) to 0. The terminal settling velocity of the object can be seen to be; 4 sd p g

vyt

3 C Dt

(3.27)

The equations of motion in the horizontal and vertical direction can be written as follows; dvy gdt dvx d

1

1

3C D v2y 4 gd p

(3.28)

3C D vx2 4 dp

Eqs. (3.28) was integrated numerically using the 5th order Runge-Kutta method. The simulations were performed in MS Excel 2007 for Windows. The Butcher‟s method (1964) as described in Chapra and Canale (2006) was used. The recurrence relations used is given in Sharma (2012b). The variation of drag coefficient as a function of velocity in horizontal and vertical directions was assumed to be of the same form as in Eq. (3.23) developed for particles in terminal settling spheres in fluids. The recurrence relations used for the solution of

dy dx

f ( x, y) is given below as;

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Mathematical Process Models

k1

h 7k1 90 f ( xi , yi )

k2

f xi

0.25h, yi

0.125k1h

k3

f xi

0.25h, yi

0.125h (k1

k4

f xi

0.5h, yi

0.5k2 h k3 h

k5

f xi

0.75h, yi

k6

f x i h, yi

yi

1

32k3 12k4

yi

3k1h 16 3k1 7

h

32k5

7k6

(3.29)

k3 )

9k4 h 16 2k2 7

12k3 7

12k4 7

8k5 7

3.2.3. Results The terminal settling velocity of the object is first obtained by trial and error between Eq. (3.27) and Eq. (3.23). From the launch velocity, an determination can be made as to whether during motion will the velocity of the object in either direction exceed that of the terminal settling velocity ? If that is the case then the trajectory of the object would become rectilinear in the direction where it reaches terminal settling velocity. Accelerating particles can be expected to have a curvilinear trajectory. Example 1.0 A golf ball is made of nanocomposite with a density of 2600 kg.m-3. The diameter of the ball is 6.7 cm. The viscosity of the air at a temperature of 20 0 C is 1.8 E-5 Pa.s. The density of air assuming ideal gas was calculated to be 1.307 kg.m-3. The terminal settling velocity of the golf ball in air was calculated by trial and error between Eq. (3.27) and the 5 constant correlation for drag coefficient given by Eq. (3.23) for spheres. The values found were;

Vt

61.4 m.s-1

CDt

0.461

(3.30) (3.31)

The Reynolds number at terminal settling velocity was found to be 298,812. Equations for vx and vy given in Eqs. (3.28) was integrated numerically on MS Excel spreadsheet for Windows using the 5th order Runge and Kutte method as shown by Butcher in Chapra and Canale. The step size used for 0.01 s. The variation of drag coefficient as a function of velocity in horizontal and vertical directions was assumed to be of the same form as in Eq. (3.23) developed for particles in terminal settling spheres in fluids. The case without air resistance for a launch angle of 30 0 and launch velocity of 32 m.s-1 was calculated and plotted in Figure 2.0. The point where maximum height was recognized in the spreadsheet. The

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Kal Renganathan Sharma

descent portion of the projectile had a different equation of motion in the vertical direction. This is given below; dvy gdt dvx d

1

1

3C D vy2 4 gd p

(3.32)

3C D vx2 4 dp

In both the ascend and descend phases of the trajectory of the golf ball the air drag force is in opposition to the motion. The vertical and horizontal displacements of the gold ball (x,y) at a given instant t was recovered from the integrated instantaneous vx and vy values. Average values in two cells were multiplied by the step size and added with the previous displacement. The resulting trajectory of the golf ball is shown side-side by side in Figure 7.0 along with the case without the air drag holding everything else the same. It can be seen that the air drag has caused a reduction in the range and maximum height reached by the golf ball. The trajectory of the golf ball with the air drag can be seen to be asymmetrical. The horizontal displacement of the golf ball during its ascend is greater than the horizontal displacement of the golf ball during its descend for the case when the air drag was accounted for. For the case without the air drag the horizontal displacement of the ball during the ascend and descend were equal to each other. The equations that describe the vy and vx for the case without the air drag are given as follows; vx

v0 cos

vy

v0 sin

gt

x

v0 cos

t

y

v0 sin

t

(3.33) gt 2 2

The golf ball was found to reach the ground after 3.21 seconds for the case with the air drag. The case without the air resistance took 3.26 seconds. The time to descend was found to be 1.6 s for the case with the air resistance and 1.63 s for the case without the air drag. The maximum height of the gold ball was reached after 1.6 s for the case with the air drag and 1.63 s for the case without the air drag. The trajectories of the golf with a launch angle of 45 0 and launch velocity of 32 m.s-1 are shown in Figure 8.0. The range can be seen to be reduced from 104.3 m for the case without the air resistance to 89.8 m for the case with the air drag. This is a reduction of 16.1%. The time to ascend to maximum height of the golf ball with the air resistance was 2.29 s and the time of descent to ground was 2.20 s. The case without the air resistance was symmetrical at 2.3 s. The effect of air resistance was pronounced during the descend and not during ascend. The numerical integration was completed successfully. At some cells when the ball was at the maximum height a number error was found at low Reynolds number.

Mathematical Process Models

Figure 7.0. Effect of Air Drag during Projectile Motion of Golf Ball at a Launch Angle of 300 and Launch Velocity of 32 m.s-1.

Figure 8.0. Effect of Air Drag during Projectile Motion of Golf Ball at a Launch Angle of 45 0 and Launch Velocity of 32 m.s-1.

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Kal Renganathan Sharma 0

Example 2.0 A launch angle of 38 was used in this example. All other parameters were the same as used in Example 1.0. The variation of drag coefficient as a function of velocity in horizontal and vertical directions was assumed to be of the same form as in Eq. (3.23) developed for particles in terminal settling spheres in fluids. A separate case where the drag coefficient was taken as constant in the equations of motion given in Eq. (3.32) was calculated. The drag coefficient was taken as 0.2. The results are plotted in Figure 9.0. Fifth Order Runge Kutte method were used in both cases. The step size used in the case of the use of Eq. (3.23) for CD vs. Re was t = 0.01 s and the step size used for the case of constant drag coefficient, CD = 0.2 was t = 0.05s. It can be seen that the trajectory of the projectile when air resistance was taken into account was asymmetrical. The displacements in x and y direction for the case of varying CD is closer to the case without air resistance than the displacements in x and y direction for the case of constant CD compared with the case without the air resistance.

Figure 9.0. Effect of Air Drag during Projectile Motion of Golf Ball at a Launch Angle of 38 0 and Launch Velocity of 32 m.s-1.

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83

3.2.4. Conclusions Projectile motion is an interesting topic for discussion in introductory physics courses. Air resistance effects on projectile motion was studied. A 5 constant expression for variation of CD vs. Reynolds‟ number was used and the variation of air resistance with velocity was taken into account. The air resistance force was the most (about 11.9%) in the initial and final stages of the projectile. Closer to the maximum height the drag force is near zero. The projectile motion with the air resistance was asymmetrical. The effect of drag coefficient variation with velocity was found as shown in Figure 4.0. Greater range (97 m) was found for the case of launch angle of 38 0 for the example considered compared with launch angles of 45 0 and 30 0 with air resistance. Maximum range of projectile without the air resistance can be expected at a launch angle of 45 0. The use of Eq. (3.23) may be better than the use of a constant value for CD during the entire trajectory. Studies are underway to include the Magnus effect when there is circulation around the object. The case of geometric particles are also being studied.

3.3. TRAJECTORY OF ACCELERATING PARTICLES IN SEDIMENTATION TANKS Symbols Used in Section 3.3 Ap As CD CDt dp g h H kj L mp ∆P Q

Projected Area of Particle (m2) Cross-Sectional Area of Rectangular Tank (m2) Instantaneous Drag Coefficient Drag Coefficient at Terminal Settling Velocity of Particle Particle Size (m) Acceleration Due to Gravity (m.s-2) Step Size used in Numerical Integration in Independent Variable Y Height of Sedimentation Tank (m) Weighting Factors used in Fifth Order Runge-Kutta Method Length of Sedimentation Tank (m) Mass of Particle (kg) Pressure Drop (N.m-2) Discharge Rate (m3.s-1)

 vd p     

Re

Reynolds‟ Number, Re  

t

Time (s)

v

t px

Terminal Settling Velocity of Particle (Vertical) (m.s-1)

v tp y

Terminal Settling Velocity of Particle(Horizontal) (m.s-1)

vy

Vp

Velocity of the Fluid in direction as a function of x (m.s-1) Average Velocity of the Fluid in y Direction (m.s-1) Volume of Particle (m3)

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Kal Renganathan Sharma W

Width of Sedimentation Tank (m)

x dp

X

Dimensionless Distance (Vertical) X 

Y

 v px  Dimensionless Velocitry of Particle (Vertical)  Y  t 

Z

 

 v py Dimensionless Velocity of Particle (Horizontal)  X  t  v py 

Greek



 s



Density Ratio

  s xy 

Viscosity of Water (kg.m-1s-1) Density of Fluid (kg.m-1s-1) Density of Particle (kg.m-3) Tangential Shear Stress (N.m-2) Residence Time (s)



v px  

Dimensionless Distance (Horizontal) X 

   

y dp

3.3.1. Overview of Sedimentation Tank Design Type I sedimentation is characterized by particles that settle discretely and do so at a constant terminal settling velocity. Individual particles are assumed not to flocculate. Sedimentation is a gravity settling method to remove suspended solids (Reynolds and Richards, 1996). Sedimentation is used in waste water and water treatment for removal of sludge. In manufacturing plants this method is used for removal of iron and manganese. Coe and Clavenger (1916) came up with the four types of settling. In Type I the settling particles are assumed to sink relative to fluid in motion free of interference from other particles. Attempts have been made to describe sedimentation of particles using Mason-Weaver equation (1924). This attempt also takes into account the diffusion down a concentration gradient using Fick‟s law of diffusion (1855). Sharma (2005) has pointed out eight reasons to seek a generalized Fick‟s law of diffusion for use in transient applications. Camp‟s theory (1936) (in Davis, 2011) was based on the principle that in order for a particle to be removed from the flowing stream of fluid the particle must have a settling velocity, vpxt sufficiently large so that it can reach the bottom of the tank during the time available, i.e., during the residence time, , of the fluid in the tank. The residence time of the fluid in the tank can be estimated from the discharge rate Q and volume of the tank:

Mathematical Process Models WHL Q

85 (3.34)

Where W, H and L are the width, height and length of the reactangular sedimentation tank and Q is the discharge rate in (m3.s-1). In terms of the terminal settling velocity of the particle; vtpx

Q WL

(3.35)

The overflow rate v0 needs to be set at the terminal settling velicity of the particle as given by Eq. (3.35) as given by Camp (1946). The flow is assumed to be of plug flow type. Most real applications the flow is of the laminar type and a parabolic velocity profile can be expected. In this method the particle removal efficiency is independent of the depth of the tank H and the residence time of the fluid . Stokes law (1822) is assumed for estimation of the terminal settling velocity of the particle in Camp‟s method. Stokes law is applicable at low Reynolds number of fluid around the particle. i.e., Re < 200. In this study the flow of fluid in transition and turbulent flow is also included. The effect of height of the tank is examined by introduction of the viscous effects of the fluid around the particle. A parabolic velocity profile with maximum velocity at the top of the tank (when lid is open) and zero velocity at the sludge zone is derived from the Hagen (1839) and Poiseulle (1841) law for circular conduits applied to a rectangular chamber. The results from an earlier study (Renganathan et al., 1989) for acceleration motion of geometric and spherical particles in a stationary Newtonian fluid is extended to two dimensions in this study. In the earlier study, the vertical distance travelled by a spherical particle prior to reaching 90% of its terminal settling velocity was found from the results of computer experiments to be; X

x dp

1.27

(3.36)

0.93 C Dt

Based on Eq. (3.36) a iron particle with a particle size of 2.5 cm would travel 49.9 cm prior to reaching 90% of its terminal settling velocity. This falls in the range of 25-50% of the depths used in current standard designs of settling tanks. By trial and error, the Reynolds number and drag coefficient at terminal settling velocity of the particle can be calcualted from the 5 constant expression proposed by Turton and Levenspiel (1989); C Dt

24 1 0.173Ret0.657 Ret

0.413 1 16300 Ret 1.09

(3.37)

Eq. (3.37) is capable of exhibiting a minimum (Figure 6.0) unlike other predictions in the literature. This can be used for predictions spanning a spectrum of fluid flow conditions ranging from laminar to transition to turbulent flow. This expression was found to correlate

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Kal Renganathan Sharma

well with experimental data over a wide range of Reynolds number from 1 – 200,000. This expression reverts to the Stokes‟ expression at low Reynolds number. For the iron particle under consideration, the Reynolds number turns out to be 54,250 and the corresponding drag coefficient turns out to be 0.47. Stokes regime at this Reynolds number is not a reasonable assumption. For such cases Camp‟s assumption that the overflow velocity of the sedimentation tank is independent of the height is a poor one. Since the work of Camp (1936, 1946) a number of advances have been made in the prediction of the drag coefficient at steady state. Raleigh (1883) was a pioneer in expressing drag coefficient as a function of Reynolds number (1883). Clift et al. (1978) have provided a review of available empirical correlations for drag coefficient of the particle at terminal settling velocity of the particle, CDt vs. Ret, Reynolds number of the fluid at terminal settling velocity of the particle. Chabra (1993) presented an analytical solution for acceleration motion of a spherical particle in a unbounded Newtonian fluid. The drag coefficient used was of the following form;

CD

A

Re

2

B

(3.38)

They suggest a Re = p2 substitution. In this study, Eq. (3.37) was used for the drag coefficient of the particle in transient state. Numerical simulations were conducted on the desktop computer using the 5th order RungeKutta method using MS Excel 2007 Spreadsheet for Windows software. The results from these computer experiments are used to recommend modifications to the design procedure laid out in Camp (1936). In Camp‟s procedure the sedimenting particles were assumed to be spherical. In this study particle of other shapes are also included. Sphericities of geometric particles such as disks, isometric particles are used in Eq. (3.37) in a manner explained ny Haider and Levenspiel (1989). There are little reports in the literature on two dimensional trajectory of sedimenting particles relative to the moving fluid. Lapple and Shepherd (1940) wrote equations for calculating paths taken by bodies in accelerated motion taking into account fluid friction. Since the work of Lapple and Shepherd advances have been made in description of drag coefficient as well as in numerical integration techniques using the computer. A continuous, rectangular, gravity sedimentation tank with viscous, Newtonian fluid such as water is considered in this study.

3.3.2. Theory and Simulation Consider continuous flow of Newtonian fluid in a prototypical horizontal ideal sedimentation tank with a sectional view as shown in Figure 10.0. W, H and L are width, height and length of the rectangular tank. A sludge zone is allowed to form at the bottom of the tank where the particles are captured during their residence time, . Flow of fluid in vertical direction is neglected. Flow is assumed to be in the horizontal direction only. This is a reasonable assumption for large and deep tanks and for inlet and outlet flow through weirs near the top of the tank. The fluid in the „bulk‟ portion of the tank tends to be pseudostationary.

87

Mathematical Process Models

Figure 10.0. Horizontal Ideal Sedimentation Tank Fluid.

A momentum balance on a slice W∆x∆y of fluid (Bird, Stewart and Lightfoot, 2007) at steady state can be written as follows; W x Py

Py

y

W y

xy x

xy x

x

(3.39)

Momentum transfer from flowing layers of fluid happens from the pressure drop in the fluid into kinetic energy of layers of fluid below the flowing layer via tangential shear stress. In the limits of ∆x and ∆y tend to zero, Eq. (3.39) becomes; P y

xy

(3.40)

x

It is assumed that the pressure drop in the fluid is linear in the horizontal direction; P y

xy

x

P L

(3.41)

Eq. (3.41) can be integrated to yield;

xy

P x c1 L

(3.42)

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Kal Renganathan Sharma

c1 can be seen to be 0 by noting that the velocity profile is maximum at x = 0. The Newton‟s law of viscosity may in order to express the shear stress as a function of velocity gradient and viscosity of the fluid. Eq. (3.42) can be rewritten as; vy

P x L

x

(3.43)

The velocity vy can be obtained by integrating Eq. (10) as; P x2 2 L

vy

(3.44)

c2

c2 can be evaluated by noting that the velocity vy at the sludge zone is 0. Thus;

vy

PH 2 2 L

1

x H

2

(3.45)

The average velocity of the fluid can be obtained by integration of Eq. (3.45) between the defenite limits of 0 and H and can be seen to be;

vy

PH 2 3 L

(3.46)

In terms of residence time; HWL vy WH

L vy

3 L2 PH 2

(3.47)

Eq. (3.47) may be used to include viscous effects in design considerations of sedimentation tanks.

3.3.3. Single Particle – Vertical Direction Consider the relative motion of a spherical particle that is settling towards the sludge zone in the vertical direction. A force balance on the particle can be written taking into account the gravity, Archimedes‟ buoyancy and drag forces as follows;

mp

vpx t

s

Vp g

C D Ap v2fx

2

(3.48)

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Mathematical Process Models

The fluid velocity, vxf surrounding the particle with a vertical velocity of vxp will be equal to each other. Defining the following dimensionless variables; ;Y s

vpx vtpx

x dp

;X

(3.49)

It was shown in Renganathan et al. (1989) that Eq. (3.48) can be written in dimensionless form. By rendering time variable implicit by use of “adx = vxdvx”Eq. (3.48) can be written as follows; YdY

dX

0.75 C Dt

(3.50)

Y2 C D

The expression for the drag coefficient of the particle at terminal settling velocity, CDt can be seen to be;

C Dt

4 gd p 3

s t2 v px

(3.51)

Eq. (3.50) was integrated using numerical integration on the computer. The form of Eq. (17) used was found to be less error prone compared to the second order form of the equation using explicit expression of time variable. The drag coefficient CD in the accelerating phase of the particle was assumed to be given by Eq. (3.37) as given by Turton and Levenspiel (1989) for drag coefficient at terminal settling velocity phase. The equations that are explicit in time are a orderh higher than the one given by Eq. (17). The numerical integration method used was the 5th order Runge-Kutte method using MS Excel 2007 Spreadsheet for Windows. The Butcher‟s method (1964) as described in Chapra and Canale (2006) was used. The recurrence relations used are as follows;

k1

h 7k1 90 f xi , yi

k2

f xi

0.25h, yi

0.125k1h

k3

f xi

0.25h, yi

0.125k1h 0.125k2 h

k4

f xi

0.5h, yi

k5

f xi

0.75h, yi

3k1h 16

k6

f xi

h, yi

3k1 7

yi

1

32k3 12k4

yi

32k5

7k6

(3.52)

0.5k2 h k3 h

h

9k4 h 16 2k2 7

12k3 7

12k4 7

8k5 7

90

Kal Renganathan Sharma

Where h is the increment in the independant varibale, Y. The simulations were run till the particle reached 95% of its terminal settling velocity.

3.3.4. Single Particle – Horizontal Direction The governing equations for relative motion of the particle in the horizontal direction can be written as follows;

mp

vpy

PxAp

C D Ap v2fy

t

L

2

(3.53)

The fluid velocity, vyf surrounding the particle with a horizontal velocity vyp will become equal to each other at the terminal settling stage. The following dimensionless variables are defined as follows;

;Z s

vpy vtpy

y dp

;

(3.54)

Where vpyt is the terminal settling velocity of the particle in the horizontal direction. Eq. (3.53) is made dimensionless using the substitutions in Eq. (3.54). Further the time variable is made implicit by use of the expression for acceleration in terms of velocity of the particle, i.e., “ady = vydvy” and Eq. (3.53) can be written as follows; ZdZ

d

0.75 C Dt

Z 2C D

(3.55)

The expression for the drag coefficient of the particle at terminal settling velocity, CDt can be seen to be; y C Dt

2 Px 3 s Lvtpy2

(3.56)

Eq. (3.55) was integrated using numerical integration on the computer. The form of Eq. (3.55) used was found to be less error prone compared to the second order form of the equation using explicit expression of time variable. The drag coefficient CD in the accelerating phase of the particle was assumed to be given by Eq. (3.37) as given by Turton and Levenspiel (1989) for drag coefficient at terminal settling velocity phase. The equations that are explicit in time are a order higher than the one given by Eq. (3.52). The numerical integration method used was the 5th order Runge-Kutte method using MS Excel 2007 Spreadsheet for Windows. The Butcher‟s method (1964) as described in Chapra and Canale (2006) was used.

Mathematical Process Models

91

3.3.5. Results A 2.5 cm iron particle was considered. The water rate was million gallons per day delivered from the sedimentation tank. The weir area was 0.001 m2 and the pressure drop allowed was 50 N.m-2. The trajectory of the iron particle as obtained from the computer simulations is shown in Figure 11.0. It can be seen that 66% of the terminal settling velocity of the particle is reached when the iron particle reaches the sludge zone. Thus the particle is still accelerating. It has travelled 75 cm in the vertical direction and 29 cm in the horizontal direction. The acceleration zone ought to be a small fraction of the depth of the tank in a modern design of sedimentation tank. The sedimentation tank has to be made deeper. The current procedure overestimates the velocity of the sedimenting particle. Only in deeper tanks the design overflow velocity based upon the settling velocity of the particle will result in the estimated separation efficiency. Other wise the separation efficiency realized would be lower. It can be seen from Eq. (3.48) that during the trajectory of the particle, it can be expected to face an “increasing force”. This is because of the velocity profile as shown in Figure 11.0. The drag force will oppose the motion of the particle in the y direction as described in Eq. (3.48) and rise to meet the applied kinetic force at the terminal settling stage of the particle in the horizontal direction. Recent designs of horizontal gravity sedimentation tanks provide a slope or taper to minimize motion of particles in horizontal direction in the sludge zone.

Figure 11.0. Trajectory of 2.5 cm Iron Particle in Horizontal Sedimentation Tank.

92

Kal Renganathan Sharma

Trajectory of the iron particle at higher pressure drop (100 N.m-2) and deeper tank, H = 2 m is shown in Figure 12.0. At the sludge zone the particle was found to have reached 93% of the terminal settling velocity of the particle.

Figure 12.0. Trajectory of 2.5 cm Iron Particle at a Pressure drop of 100 N.m-2.

Figure 13.0. Trajectories of Accelerating and Settled 2.5 cm Sand Particles.

Mathematical Process Models

93

A sand particle with a particle size of 2.5 cm was considered. The trajectory of the accelerating sand particle was compared with a trajectory of the sand particle if it was at terminal settling velocity of the particle from the entrance of the particle to the tank. The trajectory of the accelerating particle was found to be curvilinear and the trajectory of the steady particle was found to be rectilinear (Figure 13.0).

3.3.6. Conclusion The following conclusions were drawn from the study; 1. Trajectory of sedimenting particles can be simulated using desktop computers and numerical integration. 2. Viscous effects of the fluid can be accounted for the in the design considerations of rectangular sedimentation tanks. 3. Drag correlations with Reynolds number for spherical and geometric particles under laminar to transition to turbulent flow were used in the computer simulations. 4. By writing the acceleration term in terms of velocity the time variabe is rendered implicit and an order reduction is achieved in governing equation. 5. Fifth order Runge Kutta method was used with success. 6. Iron and sand particles were simulated. Trajectories are shown as Figures. 7. Sedimentation tanks needs to be made deeper to allow for accelerating phase. 8. Overflow rate is no longer independent of the tank height. 9. Trade-offs between pressure drop (pumping costs), separation efficiency achived and throughput (capital costs) can be seen. Total cost can be optimized. 10. Shape of the trajectory is plateau in the initial stages and a sharper fall later. 11. Trajectory of accelerating particles are found to be curvilinear as opposed to linear for settled particles. 12. Expression for terminal settling velocity of the partilce in horizontal direction is now given by Eq. (3.56). 13. Acceleration zone was found to be shorter for geometric particles.

B. SEMI-EMPIRICAL MODELS 3.4. MESOSCOPIC CORRELATION Semi-empirical models are used to correlate experimental data using dimensionless groups that stem from first principle based models. Sharma and Turton [1998] presented mesoscopic correlations for heat transfer coefficients between gas-solid fluidized beds and immersed surfaces with a dimensionless group called frequency number. 756 sets of pressure signals were measured simultaneously along with measurements of local, time-averaged bedto-surface heat transfer coefficients. 13 different powders with average particle size in the range of 20 m – 1.789 mm were tested in a 5 cm transparent walled fluidized bed over a wide range of fluidization velocities ( < 2.4 m/s). Fast Fourier transform analysis were

94

Kal Renganathan Sharma

performed and the dominant frequency of pressure fluctuations was noted. This frequency in the dimensionless form is the frequency number . As can be seen in Figure 13.0 the experimental measurements correlated well with the frequency number. The lines in Figure 13.0 were obtained by scaling first principle mechanistic models into the correlation form of the chart. Figure 13.0 is an example of semi-empirical correlations. Mickley and Fairbanks [1955] proposed a packet theory to predict heat transfer coefficients between gas-solid fluidized beds to immersed surfaces. Packets of solids with interstitial gas resides in the surface and the surface gets renewed by fresh packets. The contact times of the packets of various sizes form an exponential distribution. During the time the packet was in contact with the surface the heat transfer between the surface and packet is unsteady. The temperature profile in the packet can be derived by using a Boltzmann transformation [Bird et al., 2007] and can be shown to be given by an error function. The packet is assumed to be a semi-infinite medium. Let the surface temperature of the fresh packet arriving at a hot surface and that of the temperature of the surface be Ts. The packet is assumed to be at an uniform initial temperature of T0. The Fourier conduction based governing equation for temperature in the packet can be written as follows; 2

T t

T

e

x2

(3.57)

Where, e is the thermal diffusivity of the emulsion packet residing at the surface. The mathematical problem is one of a partial differential equation, PDE, in two variables, space and time. The PDE is second order in space and first order in time. One time condition and two space conditions are needed to completely describe the problem. These conditions are provided from the initial time condition and two space conditions and are as follows; t

0, T

T0

x

0, T

Ts

x

,T

(3.58)

T0

The temperature is made dimensionless by defining a u such that;

u

T

T0

Ts

T0

(3.59)

Eq. (3.57) in terms of u can be written now as; u t

Eq. (3.58) in terms of u can be written as;

2 e

u

x2

(3.60)

95

Mathematical Process Models 0, u 1

(3.61) ,u

0

The Boltzmann transformation can be used to convert the PDE in two variables given in Eq. (3.60) into an ODE, ordinary differential equation in one variable, i.e., the transformation variable, . The transformation variable, , can be defined based on similarity observations at infinite space and initial time. At infinite space, x =  and at initial time, t = 0 the temperature u is the same at u = 0. This can be tapped into.



x

4 e t

(3.62)

Eq. (3.60) in terms of the transformation variable now can be written as;



   2u   u  e  2  2 t 4 e t     u   2u  2    0 2    

Or,

Eq. (3.64) can be integrated as follows. Let, p 

(3.64)

u . Then Eq. (3.58) can be written in 

p  2 p 

terms of p as follows;

(3.63)

(3.65)

Separating the variables p and  and integration of both sides would yield;

ln  p   

2 2

c

(3.66)

Where c is the integration constant.  u  c' e 2 

2

Or,

(3.67)

Eq. (3.67) can be integrated to give the solution for dimensionless transient temperature as;

u c1 c2erf

(3.68)

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Kal Renganathan Sharma

The integration constants c1 and c2 can be solved for from the boundary conditions given by Eq. (2.5). The temperature profile in the packet can be seen to be;

u

T T0 T s T0

x

1 erf

2

1

4 t

2

e

d

(3.69)

The wall heat flux can be calculated from;

qx

ke Ts

T x

ke

T0 et

x 0

(3.70)

The instantaneous, local heat transfer coefficient between the bed and the gas-solid fluidized bed can be seen to be; ke

hi

e C pe

(3.71)

t

Eq. (3.71) can be integrated with an exponential distribution for the packet contact time at the surface and the time averaged local heat transfer coefficient between the bed and surface can be seen to be; t _

hbs

hi

t

e

0

_

dt

t

ke

e C pe _

(3.72)

t

Eq. (3.72) is the model solution for the Mickley and Fairbanks‟ packet theory. It is an example of an mechanistic model as discussed below in section 2.3. Sharma and Turton [1998] scaled the model solution given by Eq. (3.72) into a dimensionless correlation of the form of Nusselt number, Nu, vs. frequency number, Nf. The frequency number, Nf is defined as;

Nf

fd p2

(3.73)

e

Where f is the dominant frequency of pressure fluctuations, dp is the average size of the particle that is fluidized. Eq. (3.72) scales to; Nu

Nf

(3.74)

Eq. (3.74) in the log-log plot appears as a straight line as shown in Figure 13.0. In a similar manner, Sharma and Turton [1998] scaled mathematical models presented in the

97

Mathematical Process Models

literature into correlations. These models are due to finite length of packet with variable thermal properties by Chandran and Chen [1982], gas gap between the packet and wall by Baskakov [1973] and finite speed conduction by Renganathan and Turton [1989]. These correlations are also shown in Figure 13.0.

C. MECHANISTIC MODELS 3.5. WHICH IS THE DOMINANT MECHANISM – FOURIER CONDUCTION OR GAS CONVECTION? Consider the combined effects of particle conduction and gas convection during transient heat transfer between gas solid flow past a surface. The packet is assumed to possess constant thermo physical properties, ke, (sCps(1-), thermal conductivity, thermal mass, i.e., the product of density and heat capacity. The governing equation that includes conduction into the packet in the x direction and convection in the azimuthal, z direction can be written using the Fourier parabolic model as follows. The azimuthal velocity of the gas phase is assumed to be constant and equal to U. T t

U

T z

2

T

x2

(3.75)

Figure 13.0. Theoretical Correlation Between Nusselt Number and Frequency Number (Sharma and Turton [1998])

98

Kal Renganathan Sharma The dimensionless form of the equation can be obtained as follows; u

Where,

u

(T T0 ) ; (Ts T0 )

t

x

;X

2

u Z

Pe

r

u

(3.76)

X2

z

;Z

U

; Pe

r

/

r

(3.77) r

It is known that Boltzmann transformation can lead to an error function solution for the parabolic heat conduction equations. Hence the convective term is also lumped along with the accumulation term as follows. Let,  = Z+Pe Eq. (3.76) becomes, 2

u

2 Pe

u

(3.78)

X2

The Boltzmann transformation can be used to convert Eq. (3.78) from a second order PDE to an ODE in one variable of the second order as follows. Let, X

2 2Pe

u

2

X

2

2

u 1 or 2 2

2Pe

u

2

u 2

(3.79)

The solution to the ODE in  can be seen to be; u

Pe1/2 X

1 erf (

2(Z

Pe

)

(3.80)

The conductive portion of the heat flux and hence the conductive component of the heat transfer coefficient between the gas-solid fluidized beds and immersed surfaces can be seen to be from q*=-u/X;

hc*

ke ( s C ps )(1 r

)

U /

1 r

(Z

Pe )

(3.81)

The azimuthal distance Z can be set to zero for the conductive component. It does vary with Z!

Mathematical Process Models

99

The convective component can be obtained by differentiating Eq. (6) wrt. Z and evaluated at X = ndp.

hconv

kg

g C pg rg

(

X 2 Pe Pe X )exp( ) 2 2(Z Pe ) (2( Z Pe )3/2 )

(3.82)

The maximum convective heat transfer coefficient can be obtained when  =0 and

hconv

kg

g C pg rg

(

X 2 Pe Pe X )exp( ) 2 2( Z ) 2( Z )3/2

(3.83)

3.5.2. Evaluation of Dominant Mechanism – Non-Fourier Conduction or Convection during Heat Transfer in Circulating Fluidized Beds Circulating Fluidized Bed, CFB technology was developed by Foster Wheeler, Babcock & Wilcox, Pyro power, Keeler Dorr-Oliver, Tennessee Valley Authority, Westinghouse and Combustion Engineering. Heat transfer between gas-solid fluidized beds and surfaces have been reviewed by Saxena, Grewal et al., [1978]. Very little has been reported on convective heat transfer in CFB to a horizontal tube. During short time scales the system may be far from equilibrium and the high rate unsteady state transient processes cannot be fully described using Fourier‟s heat conduction law. The Cattaneo and Vernote non-Fourier equation, for heat conduction which was shown to be the only mathematical modification (Boley [1964]) that can remove the singularity in surface flux is: q

k

T x

r

q t

(3.84)

Combined with the energy equation this was solved for the semi-infinite medium case by the method of Laplace Transforms and method of relativistic coordinate transformation presented earlier. The velocity of heat is given by  (/ r ). A study was conducted with the Cattaneo and Vernotte non-Fourier heat conduction equation used in the energy balance in 1 dimension along with the convection term. Consider the flow of gas solid suspension past a horizontal tube in a CFB. Two cases can be identified; Case 1: Flow Away from Tube of Gas – Solid Suspension The governing equation from an energy balance may be written after obtaining the dimensionless variables as; 2

u 2

2

Pe

u Y

u Y

2

u

Y2

(3.85)

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Kal Renganathan Sharma

Case 2: Flow of Gas – Solid Suspension Toward the Horizontal Tube The governing equation for flow toward the horizontal tube from an energy balance after obtaining the dimensionless variables may be written as; 2

2

u

Pe

2

u Y

2

u Y

u

Y2

(3.86)

The minus sign indicates the relative velocity of the suspension with respect to the direction of heat conduction path. For the bottom of the tube the y axis is selected as bottom direction is positive. Where y is the direction of flow of gas and solid, T the temperature, t the time and q the flux.

u

T

T0

Y

t

;

T1 T0

r

y

; Pe

(3.87)

Uy

r

r

The Peclet number, Pe represents the ratio of the gas velocity to the velocity of heat by conduction. When Pe is 0, Eqs. (3.86,3.85) reduce to the hyperbolic heat wave propagative equation by conduction alone which was solved for and presented earlier for the semi-infinite medium case under CWT. Obtaining the Laplace transform of Eqs.(3.85,3.86) the second order ODEs result;

u (s ) s s 1

d 2u ( s ) 2

dY

Pe s 1

du ( s ) dY

(3.88)

Pe s 1

du ( s ) dY

(3.89)

(top of tube)

u (s ) s s 1

d 2u ( s ) 2

dY

(bottom of tube) Solving for the ordinary differential equation given by Eq.(3.88), Eq. (3.89);

Pe s 1 1 r1 , r2

1

4s Pe s 1 2

2

u (s )  c1e r1Y  c 2 e  r3Y At Y = , u = 0 and Y = 0, u = 1

(3.90)

(3.91)

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Mathematical Process Models u YY

s s 1

2

0

4s s 1

Pe 2

Pe

(3.92)

The heat transfer coefficient obtained in the Laplace domain may be written as;

h*y

L1

Pe 2s

1

1

4s

(3.93)

s 1 Pe 2

(top of tube) Where, h*y

hy k Cp r

For the case of bottom of tube, the axes is drawn in a manner that y is positive in the bottom direction and;

h*y

L1

Pe 2s

1

1

4s

(3.94)

s 1 Pe 2

(bottom of tube) Inverting the Laplace domain after expanding the square root using the binomial theorem we have;

h

e Pe

k Cp

2

e

1

Pe

2 1 2

3

e

2! Pe5

......

(3.95)

r

(top of tube)

h

Pe

k Cp

e Pe

1

2

e Pe

3

2 1 2

e

2! Pe5

......

(3.96)

r

(bottom of tube) This expansion is good for large Peclet numbers. The terms in the series were inverted by the residue theorem by Heaviside expansion as shown by Mickley, Sherwood and Reed

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Kal Renganathan Sharma

[1957] and by expressing the term as P(s)/Q(s) and finding the poles and writing the inversion. The results are presented in Table 2.1. The results are shown in Figure 5.1 for three different Peclect numbers. It can be seen that at short contact times, the micro scale conduction effects become a dominant contribution to the heat transfer coefficient. At large contact times, the heat transfer coefficient becomes independent of the contact time and is found to be function of the Peclect number. The difference in values between top and bottom of the tube gives the twice the convective contribution. The driving force for heat transfer as the suspension moves towards the tube is away from the tube leading to a minus sign in the convective contribution in the governing equation. As the suspension moves away from the tube the heat transfer driving force and the velocity of the suspension are in the same direction. Hence the conduction and convection have to be added to obtain the total heat transfer. In the case of the bottom of the tube the convection contribution has to be subtracted from the conduction contribution to obtain the net heat transfer coefficient. Thus when the difference between the values from the top and bottom of the tube is obtained the contribution to convection alone will result and when the values from top and bottom are added the twice the conduction contribution will result. Hence Eq. [3.98-3.97] and normalizing for direction the convective contribution may be seen to be Pe/2. The conductive contribution is the addition of Eqs. [3.97,3.98] and dividing by 2 to give after converting heat transfer coefficient to a dimensionless Nusselt number,

1 Maxh

Nu

A1

A2

Pe

A3

Pe

3

....

Pe5

(3.97)

(top of tube)

Nu

Pe 1 Maxh 2

A1

A2

Pe

Pe

A3 3

Pe5

....

(3.98)

(bottom of tube) As   0, in the short time limit the transient solution for top of the tube becomes; Nu

1 1 Maxh Pe

Where Maxwell Number (heat), Maxh =

1 Pe

2 3

Pe5

....

(3.99)

r

D2

For very large Peclet numbers, the transient portion of the solution drops out and the portion that results is invariant with time. In the absence of convection, conduction alone is the mechanism of heat transfer. Here the open interval solution developed earlier can be used. (Uy = 0) and is given for both the bottom and top of the tube by; hy* = exp(-/2) I0 (/2)

(3.100)

103

Mathematical Process Models Table 3.4. Term by Term Inverse Laplace Transform of the Infinite Series

  m1   Am    e   A1  A2  ....  A m m  1!  

Term # 1 2

3

4

5

6

7

8

9

10

11

L1

-

e

e  1   



2e  1  2  0.5 2







 3   5e  1  3  1.5 2   3 



 2 3  4  14e  1  4  3 2    3 4!  



 5 3 5 4  5   42e  1  5  5 2     3 4! 5!  



 15 20 3 15 4 6 5  6      132e  1  6   2  2! 3! 4! 5! 6!  



 21 35 3 35 4 21 5 7 6  7   429e  1  7   2       2! 3! 4! 5! 6! 7!  



 56 3 70 4 56 5 28 6 8 7  8  1430e  1  8  14 2        3! 4! 5! 6! 7! 8!  

 36 2 84 3 126 4 126 5 84 6  1  9       2! 3! 4! 5! 6!     4862e  36 7 9 8  9      7! 8! 9!   2 3 4 5 6  45 120 210 126 210  1  10      

2! 3! 4! 16796e    120 7 45 8 10 9  10     7! 8! 9! 10! 

5!

6!

   

 

s  1k 1 cs k

1 s  1

s  12 s

s  13 2s 2

s  14 5s 3

s  15 14s 4

s  13 42s 5

s  17 132s 6

s  18 429s 7

s  19

1430s 8

s  110 4862s 9

16796s10 s 1

11

When the relaxation time becomes infinite as in the insulator of heat, the D‟Alambert‟s wave solution results. A aliter would be to invert Eq. [3.101] the case for bottom of tube, from the Laplace domain as follows;

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Kal Renganathan Sharma

h*y

L1

Pe 2s

4s

1

1

A

4 Pe2 ; B

s 1 Pe

L1

2

Pe 2s

A s

B

(3.101)

2s s 1

Where,

h*y

Pe 2

B 1 2

Ae

2

I0

1 B 2

AB (e B 1

Pe 2

(3.102)

4 Pe 2 B 1

2

(I 0

1 B 2

I1

1 B 2

1) (3.103)

The inversions can be read from the tables provided in Mickley, Sherwood and Reed [1957]. Eq. [3.104] is the exact solution and it includes modified Bessel function of the first kind and zeroth order. For the special case when Pe = 1, h*y

0.5 1.118e

0.6

I 0 0.4

0.373e

0.6

( I 0 0.4

I1 0.4

1)

(3.104)

3.5.3. Representative Experimental Data The key parameter in using the model is the contact time of the suspension with the tube wall and the dimensionless Peclet. This may be a function of the operating parameters. In addition to this the physical properties of the suspension is required and the relaxation time of the suspension is an added intrinsic parameter of the system. Some experimental data was available for comparison from Sundaresan and Kolar [2002]. They used silica at 3 mean sizes between 143 – 263 m, air velocity 4 – 8 m/s, suspension density of 9-25 kg/m3, 0.5 – 1.0 m high U horizontal tubes, 5.5 m CFB, solid circulation flux, 13-80 kg/m2/s and reported heat transfer coefficients between 30 – 105 W/m2/K in their core heat transfer studies in a CFB. A uniform distribution of the suspension packets with the wall tube is assumed as a first approximation and the Nusselt number calculated as (Nu = (h D/k) = 105 * .05/0.04 = 131). Where the diameter of the tube is taken as 2 inches and the effective thermal conductivity of the suspension calculated from a simple average with the voidage as the weight from the values of air and silica as 0.04 w/m K. Taking the effective thermal diffusivity of the suspension as 3 E-6 m2/s. (Pe = Uy/  (/r ). = 8/(3 E-6/1E-3) = 146; el = (r / D2 ) = 3E-6 * 1 E-3/.052; Short Time Limit: Nu = (1/Rel) ( 1/Pe + …) = 6.25). The relaxation time can be estimated from an analogy to the speed of molecules. The Maxwellian distribution for the speed of the molecules was used to get the root mean square velocity of the molecule and by analogy to heat transfer the velocity of heat transfer was

105

Mathematical Process Models

estimated. The relaxation time was obtained from this estimate given the thermal diffusivity of the suspension. The convective contribution is thus Pe/2 = 146/2 = 73. The conductive contribution to the Nusselt number is then given a typical contact time of the suspension with the tube as 5 milliseconds (Nu = 73 + exp(-5)/73 - exp(-5)*(-4)/733 … = 73). Experimental validation of the model will require contact time of the solids in the wall information, preferably the statistical distribution of their residence times at the surfaces. The relaxation time is another intrinsic property of matter and needs measurement prior to an elaborate study of experimental data. Thus, the Cattaneo and Vernotte non-Fourier heat conduction equation is used to model the mesoscale conduction effects in convection dominated heat transfer in CFB to a horizontal tube. Hence the exact solution is obtained for the dimensionless heat transfer coefficient by conduction and convection and found to be

h*y

Pe 2

B 1 2

Ae

2

1 B

I0

2

B 1

AB (e B 1

2

(I 0

1 B

I1

2

1 B 2

1) (3.105)

for the case of bottom of tube. This can be used especially for Peclet numbers close to 1. Two cases were identified, i.e., one for suspension flow toward the tube and the other for flow of suspension away from the tube. The Peclet number that gives the ratio of the convective fluid velocity to the velocity of heat conduction and the relaxation number that is the inverse of the Fourier number with the relaxation time substituted for time are important parameters for predicting the Nusselt number based upon the tube diameter. The convective contribution is given by Pe/2 = Uy/(/r)/2. A more usable form is obtained by expansion of the Laplace domain using binomial infinite series and upon obtaining the inversion by the method of residues. 13 terms in the binomial series for the case of bottom of tube is taken and results of the inversion presented in Table 1. This is useful at large Peclet numbers.

Nu

Pe 1 Maxh 2

A1

A2

Pe

Pe

A3 3

Pe5

....

(3.106)

The representative data available was used to show how the model can be applied to data from the experimental measurements from the CFB. In the short time limit the transient solution for top of the tube becomes;

Nu

1 1 Maxh Pe

1 Pe

2 3

Pe5

....

(3.107)

For very large Peclet numbers the transient portion of the solution drops out and the portion that results is invariant with time. In the absence of convection, conduction alone is the mechanism of heat transfer. Here the open interval solution developed earlier can be used.

106

Kal Renganathan Sharma

(Uy = 0). When the relaxation time becomes zero as in the insulator of heat, the D‟Alambert‟s wave solution was developed as presented earlier.

3.6. EVAPORATIVE COOLER There are some humidification operations where heat exchange is necessary from an external source. For example, consider an evaporative cooler. Air is blown past the water on the outside of the tube. The heat is carried away from the tube-side fluid. The fluid is cooled when it flows through the tube. Water is sprayed on the outside of the tube and flows as a film. A large heat is released when the sprayed water evaporates into the air stream. The schematic of an evaporative cooler is shown in Figure 14.0 The tube banks are shown in Figure 14.0. The temperature and enthalpy profiles are shown in Figure 14.0. The bulk water is at the temperature, tL corresponds to a saturation gas enthalpy, H1‟*. The overall heat transfer coefficient U0 based on the outside tube surface from tube fluid to bulk water is then given by [Treybal, 1980];

1 U0

d0

d 0 zm

1

d i hT

d av km

hL'

(3.108)

The overall coefficient KY for use with gas enthalpies, from H1‟* to H‟, is;

1 KY

1 kY

m hL0

(3.109)

which m is the slope of the equilibrium line near the operating region in Figure H '* H i'

m

tL

ti

(3.110)

Strictly speaking, m is the slope of the enthalpy and temperature diagram, m

dH '* dt L

(3.111)

If A0 is the outside surface of all the tubes, A0x is the area from the bottom to the level where the bulk-water temperature is tL. Here x is the fraction of the heat transfer surface to that level. For a differential portion of the exchanger the heat loss by the tube-side fluid is; wTCTdtL = U0A0dx (tT - tL)

(3.112)

Mathematical Process Models

Figure 14.0. Schematic of an Evaporative Cooler.

Figure 15.0. Schematic of Tube-Side Fluid Flow of an Evaporative Cooler.

107

108

Kal Renganathan Sharma dtT

U 0 A0

dx

wT CT

tT

(3.113)

tL

The heat lost by the cooling water is;





WALCALdtL = KYA0dx(H‟* - H‟) - U0A0dx(tT - tL)

 dt L   KY A0      dx   wAL C AL

 '*  U A  H  H '   0 0   wAL C AL

 t T  t L  

(3.114)

Heat gained in in the air is written as;



WsdH‟ = KYA0dx(H‟* - H‟)

 dH   dx

'



  KY A0  '*  H  H '      ws 

(3.115)

Subtracting Eq. (3.114) from Eq. (3.111); d tT

tL 1

dx

1

tT

tL

1

H '*

U 0 A0

U O A0

wT CT

wALC AL

mU 0 A0 2

2

0

(3.116)

(3.117)

(3.118)

wT CT

KY A0 1

H'

(3.119)

WALC AL mKY A0

KY A0

wALC AL

ws

(3.120)

Eqs. (3.116) and Eq.(3.115) can be combined into a second order differential equation with constant coefficients. The auxiliary equation which is a quadratic can be examined to realize that the roots of the equation are real and hence the solution will not undergo any pulsations. The solution is given in terms of the integration constants. The enthalpy is related to temperature difference by Eq. (3.110);

tTL

c1er1x

c2e r2 x

(3.121)

Mathematical Process Models

109

Figure 15.0. Temperature and Enthalpy Profile and Resistances in Evaporative Cooler.

H '*

H ' d1er1x

d 2e r2 x

(3.122)

H‟* - H‟ = d1exp(r1x) + d2exp(r2x) Where,

dj

c j rj

1

(3.123)

1

The design of the evaporative cooler needs values of the transfer coefficients. Some of this data availability is scarce. Correlations for the Nusselt number and Sherwood for the given geometry needs to be looked up [Treybal, 1980].

3.7. TRANSIENT DIFFUSION OF OXYGEN IN ISLETS OF LANGERHAANS: PARABOLIC AND HYPERBOLIC MODELS 3.7.1. Overview of Gas Transport Diffusion of oxygen in pancreatic islets in an important consideration during the development of models for the functioning of the islets of Langerhans (Figure 16.0). Islets of Langerhans are spheroidal aggregate of cells located in pancreas. They secrete hormones such as insulin during glucose metabolism. These islets are transplanted in order to effect cure for Type I diabetes and are often encapsulated in devices. Islets that are removed from pancreas lose their internal vascularization. The cure is depends on diffusion of oxygen from external environment.

110

Kal Renganathan Sharma

Figure 16.0. Islets of Langerhans.

Avgoustiniatos et al. [2007] and Colton et al. [2011] developed a method for estimation of maximum oxygen consumption rate and the oxygen permeability in the tissue in a suspension of spherical aggregates from measurements of partial pressure of oxygen in batch experiments. A simultaneous reaction and diffusion model was used. Williams et al. [2012] discusses a dramatic diffusion barrier that leads to core cell death during islet transplantation as cure for Type I diabetes. They attempted to measure the diffusion barrier in intact human islets and determine its role in cessation of insulin secretion. They monitored impeded diffusion into islets using fluorescent dextran beads. They linked the poor diffusion properties with necrosis and not apoptosis. The Fick diffusion model was used in the analysis of [Sharma, 2005, 2012]. Sharma [2005, 2010] has shown that non-Fick damped wave diffusion effects can become significant at short time scales. Bounded solutions from hyperbolic models can be obtained when final time condition and physically reasonable initial time rate of change conditions are used. In this study, damped wave diffusion effects during the diffusion of oxygen in islets of Langerhans is accounted for in detail The metabolic process becomes diffusion limited. Oxygen availability becomes limited in some regions of the tissue. The metabolic rate in the cells and demand for oxygen is greater than the oxygen that has diffused to that region. Growth of multicellular systems over 100 m cease to happen. A condition called hypoxia has been observed in Brockman bodies in fish. During islet transplantation, lack of oxygen supply may restrict graft function especially when encapsulated tissue is used. Oxygen partial pressure was measured in the islet organs of Osphronemus gorami (Brackman bodies) placed in culture. A microelectrode was used to detect oxygen partial pressure in the surrounding region of the islet organ that is 800 m in diameter and within the cells. This was achieved in a radial track. Within a distance of 100 m for the case of no convection (Schrezenmeir et al. [1994]), pO2 is close to zero. A condition called necrosis is reached where the cells begin to die without sufficient oxygen supply. The experiments with

111

Mathematical Process Models

convection showed increased pO2 at the surface and core regions of the islet. The experiments were conducted in a thermostatically controlled measuring chamber at 37 0C. Oxygen supply in addition to diffusion also comes about by the circulatory system and by hemoglobin molecule. Oxygen is carried in the blood by convection to capillaries by the circulatory system. Islets of Langerhans (Figure 16.0) are spheroidal aggregates of cells that are located in the pancreas. The metabolic requirement of the cells requires oxygen to diffuse from the external environment and through the oxygen-consuming islet tissue. The oxygen supply is a critical limiting factor for the functionality and feasibility of islets that are encapsulated, placed in devices for implantation, cultured, used in anaerobic conditions. Theoretical models are needed to describe the oxygen diffusion. The parameters of the model require knowledge of the consumption rate of oxygen, oxygen solubility, effective diffusion coefficient to oxygen in the tissue.

3.7.2. Theory Avgoustiniatos et al. [2007] developed an oxygen reaction and diffusion model. They assumed that the islet preparation is a suspension of tissue spheres that can be divided into m groups. Each sphere in group i has the same equivalent radius, Ri that varies from group to group. The tissue is assumed to be uniform with constant physical properties that is invariant in space. The governing equation for oxygen diffusion and reaction in spherical coordinates with azimuthal symmetry accounting for Fick‟s diffusion can be written as; CO2 t

DT

1 r

2

r

CO2

r2

C E 0CO2

r

CM

CO2

(3.124)

Where DT is the diffusion coefficient in the tissue, CE0 is the total enzyme or complexation species concentration including the rate constant kE0 and CM is the Michaelis constant. The oxygen consumption rate is assumed to obey the Michaelis-Menten kinetics. The interplay of transient diffusion and metabolic consumption of oxygen in the tissue in spherical coordinates is given by Eq. (3.124). The concentration of oxygen CO2 can be expressed in terms of its partial pressure, pO2 This is obtained by using the Bunsen solubility coefficient, t such that;

CO2

t

(3.125)

pO2

Substituting Eq. (3.125) in Eq. (1), Eq. (3.124) becomes; pO2 t

t

t DT

1 r

2

r

r2

pO2 r

C E 0 pO2 ' CM

pO2

(3.126)

112

Kal Renganathan Sharma

The product tDT can be seen to the product of solubility and diffusivity and hence is the permeability of oxygen in the tissue. The Michaelis constant CM, is also modified, C‟M expressed in units of mm Hg. The initial condition can be written as; pO20 , t = 0

pO2

(3.127)

From symmetry at the center of the sphere; pO2 r

(3.128)

0

At the surface the oxygen diffusive transport from within the sphere must be equal to the oxygen transport by convection across the boundary layer surrounding each sphere;

Ji

ki

m ( pO2 m

pO2 ( Ri ))

pO2

tD

r

T

(3.129)

Where the partial pressure of oxygen at the surface is given by pO2 (Ri), the mass transfer coefficient between the surrounding space and the surface of the sphere is given by ki and the oxygen solubility in the surrounding space is given by m. Total rate of oxygen transfer N from the surrounding space to all of the spheres can be summed up as; m

J i (4 Ri2 )ns fi

N

Vm

pm m

(3.130)

t

i 1

Where the volume of the surrounding space is given by Vm, the total number of spheres is ns and the fraction of spheres in group i is given by fi. The initial condition for surrounding space is; pO2 m

pO2 m (0) , t = 0

(3.131)

The mass transfer coefficient can be obtained from suitable Sherwood number correlations. For instance the mass transfer coefficient for spherical particles in an agitated tank in the islet size range of 100-300 m can be written as;

Shi

ki d i Dm

1/3

2

Dm

d imp d tan k

0.15

f

3 4 di 3

(3.132)

Where  is the power input per unit fluid mass, f is the function that has to be obtained from experimental data. Numerical methods are needed to obtain the solution to Eq. (3.126). This is because of the nonlinearity of Michaelis-Meten kinetics.

113

Mathematical Process Models

3.7.3. Asymptotic Limits of Michaelis-Menten Kinetics Closed formed analytical solutions to Eq. (3.126) can be obtained in the asymptotic limits of; i)

High concentration of oxygen, the rate is independent of pO2 (zeroth order)

ii) Low concentration of oxygen, the rate is first order with respect to pO2 The reasons for choosing the asymptotic limits is elucidated in Figure 17.0. It can be seen that at low reactant concentrations the rate is linear [Levenspiel, 1999]. At high enzyme or complexing agent concentration the rate is invariant with respect to concentration. Hence a zeroth order can be assumed at high concentrations and first order at low reactant concentrations. Thus, at high reactant concentrations Eq. (3.126) becomes, pO2 t

t DT

t

1 r

2

r2

r

pO2 r

(3.133)

rmax

Eq. (3.133) can be non-dimensionalized as follows; Let, DT t Ri2

;u

pO2

pO2 m

p O20 pO2 m

;X

r * ; rmax Ri

rmax Ri2 pO20

pO 2 m DT

(3.134) t

Eq. (3.135) becomes; u

1 X

2

X

X2

u X

* rmax

(3.135)

3.7.4. Parabolic Fick Diffusion and Reaction Model The zeroth order reaction at high concentrations of oxygen is a heterogeneity in the partial differential equation. Systems such as this can be solved for by assuming that the solution consists of a steady state part and a transient part; Let, u = uss + ut

(3.136)

114

Kal Renganathan Sharma

Figure 17.0. Rate-Concentration Curve Obeying Michaelis-Menten Kinetics.

Substituting Eq. (3.136) in Eq. (3.135), Eq. (3.135) becomes; ut

1 X

2

(u t

X2

X

u ss ) X

* rmax

(3.137)

Eq. (3.136) holds good when, * rmax

and

ut

1 X2 X

1 X2 X

X2

u ss X

(3.138)

X2

ut X

(3.139)

Eq. (3.138) can be integrated twice and the boundary condition given by Eq. (3.128) applied to yield; u ss

* X 2 rmax

6

d

(3.140)

In order to obtain the solution of the integration constant d in Eq. (3.140), the boundary condition given by Eq. (3.127, 3.129) needs to be modified. Assuming that after attainment of steady state, the surface concentration of the sphere would have reached the surrounding space concentration, d can be solved for the solution for the pO2 at steady state written as;

115

Mathematical Process Models

u

Ri2

ss

* r 2 rmax

(3.141)

6 Ri2

The solution to Eq. (3.139) may be obtained by separation of variables as follows; Let ut = V()g(X). Then Eq. (3.139) becomes; V'

1 g(X)X2 X

V

V

Hence,

ce

X 2 g "( X ) 2 Xg '( X )

X 2 g '( X )

2 n

2 n

(3.142)

(3.143) X2

2 n g (X)

0

(3.144)

Comparing Eq. (3.144) with the generalized Bessel function [Varma and Morbidelli, 1997]; a=2; c=0; s= 1; d =

2 n ;

p = 1/2;

The solution to Eq. (3.144) can be seen to be; g(X)

c1

J 1/2 (

n X)

X

d1

J

1/2 ( n X )

(3.145)

X

From the boundary condition given by Eq. (3.129,3.128), it can be seen that d1 can be set to 0 and, g(X)

c1

J 1/2 (

n X)

.

(3.146)

X

The eigenvalues n can be solved for from the boundary condition given by Eq. (3.129); In the dimensionless form Eq. (3.129) may be written as;

Where

ki R i DT

m

ki Ri

t

DT

u

u , r = Ri X

(3.147)

Bi m , the Biot number (mass). This represents the ratio of mass transfer

from the surrounding space and the diffusion within the sphere. To simplify matters from a mathematical standpoint, Eq. (3.146) can be written in terms of elementary trigonometric functions as;

116

Kal Renganathan Sharma

g(X)

2 Sin( n X ) X n

c1

(3.148)

The Eigenvalues can be obtained from the solution of the following transcendental equation; tan(

nX

n X)

m

1

(3.149)

Bim X

t

The general solution for the dimensionless pO2 can be written as;

u

2 n

cn e

J 1/2

nX

(3.150)

X

0

The Eigen values are given by Eq. (3.149). The cn can be solved for from the initial condition given by Eq. (3.127) using the principle of orthogonality and; Ri

J 1/2 (

0

cn

X

Ri

0

n X)

dX

(3.151)

2 J 1/2 ( n X) dX X

Thus the oxygen concentration profile at high oxygen concentration is obtained. At low concentration of oxygen the rate of consumption of oxygen is first order. The governing equation, Eq. (3.126) can be written as; pO2 t

t DT

t

1 r2

pO2

r2

r

k p pO2

r

(3.152)

Obtaining the dimensionless form of Eq. (29); u

1 X

2

X

X2

u X

2

(3.153)

u

Where,

u

pO2 pO2 m

;X

r ; Ri

DT t Ri2

;

2

k p Ri2 t DT

(3.154)

117

Mathematical Process Models

It can be recognized that  is the Thiele modulus. Eq. (3.153) can be solved for by the method of separation of variables. Let u =V()g(X). V' V

1 gX

2

g X

X2

X

(

2

2 n)

(3.155)

The solution in time domain can be seen to be;

V

ce

2 n

(

2

)

(3.156)

The solution in space domain can be seen to be; X 2 g "( X ) 2 Xg '( X )

X2(

2

2 n ) g ( X)

(3.157)

0

Comparing Eq. (3.157) with the generalized Bessel function [Varma and Morbidelli, 1997]; a=2; c=0; s= 1; d = (2+

2 n );

p = 1/2;

The solution to Eq. (34) can be seen to be;

g( X)

c1

J 1/2 (

2 n

2

X)

d1

X

2 n

1/2 (

J

X

2

)

(3.158)

From the boundary condition given by Eq. (3.128), it can be seen that d1 can be set to 0 and,

g(X)

c1

J 1/2 (

2 n

2

X

X)

.

(3.159)

The Eigenvalues n can be solved for from the boundary condition given by Eq. (3.129); In the dimensionless form Eq. (3.147) may be written as;

Where

ki R i DT

m

ki Ri

t

DT

u

u , r = Ri X

(3.160)

Bi m , the Biot number (mass). This represents the ratio of mass transfer

from the surrounding space of the diffusion within the sphere. To simplify matters from a mathematical standpoint, Eq. (3.158) can be written in terms of elementary trigonometric functions as;

118

Kal Renganathan Sharma

g(X)

c1

2 n

Sin(

2 2 n

2

X)

X

2

(3.161)

The eigenvalues can be obtained from the solution of the following transcendental equation;

tan(

2 n

2

2 n

X)

2 m

1

X

(3.162)

Bim X

t

The general solution for the dimensionless pO2 can be written as;

u

(

cn e

2

2 n)

2 n

J 1/2

2

X

(3.163)

X

0

The Eigen values are given by Eq. (39). The cn can be solved for from the initial condition given by Eq. (3.127) using the principle of orthogonality and; Ri

cn

J 1/2 (

0

2 n

2

X)

X Ri

0

2 ( J 1/2

2 n

X

2

dX

(3.164) X)

dX

Thus the oxygen concentration profile at low oxygen concentration is obtained.

3.7.5. Damped Wave Diffusion and Reaction Hyperbolic Model The times associated with oxygen consumption are low. Oxidation reactions are fast. Recent experimental reports of relaxation times in biological materials by Mitra et al. [1995] is in the order of 16 seconds. Hence in times associated with the oxygen consumption (~10-3 – 1s) the finite speed of diffusion effects cannot be ignored. The damped wave diffusion and relaxation effects may be included in the following manner. At low oxygen concentration a first order rate of reaction can be assumed. A semi-infinite medium of tissue is considered. A step change in concentration is given at the surface. At times zero the concentration of oxygen is at an initial value. At infinite distances the concentration of oxygen would be unchanged at the initial value. The mass balance equation for oxygen can be written as;

119

Mathematical Process Models J O2

CO2

kCO2

x

(3.165)

t

where k is the lumped first order reaction rate constant. Combining Eq. (3.165) with the damped wave diffusion and relaxation equation given below in Eq. (3.166);

J O2

CO2

DT

J O2 mr

x

(3.166)

t

the governing equation is obtained.mr is the mass relaxation time. When it is zero Eq. (3.146) reverts to the Fick‟s law of diffusion. When the rate of mass flux is greater than an exponential rise the wave regime would be more dominant mechanism of transport compared with the Fick regime. Eq. (3.166) is differentiated by x and Eq. (3.165) is differentiated by t 2

and the cross term

J eliminated between the two equations and the governing equation t x

obtained as follows; 2

DT

2

CO2

CO2

mr

2

x

t

1 k

2

CO2 mr

kCO2

t

(3.167)

The governing equation for oxygen concentration in the tissue is obtained in the dimensionless form by the following substitutions;

k* k

mr ; u

CO2 CO2 m

;

t

;X

mr

x DT

(3.168) mr

The governing equation is a partial differential equation of the hyperbolic type. It is second order with respect to time and second order with respect to space. 2

u

X

2

2

u 2

1 k*

u

k *u

(3.169)

The Eq. (3.169) can be solved by a recently developed method given in Sharma [2005] called relativistic transformation of coordinates. The expression for the transient temperature during damped wave conduction and relaxation developed by Baumeister and Hamill [1971] by the method of Laplace transforms was further integrated in Sharma [2009]. A Chebyshev polynomial approximation was used for the integrand with a modified Bessel composite function in space and time. A telescoping power series leads to a more useful expression for the transient temperature. By the method of relativistic transformation, the transient temperature during damped wave conduction and relaxation was developed. There are four regimes to the solution. These include:

120

Kal Renganathan Sharma (i) A regime comprising a Bessel composite function in space and time, (ii) Another regime comprising a modified Bessel composite function in space and time; (iii) The temperature solution at the wave front was also developed separately, and; (iv) The fourth regime at a given location X in the medium is at times less than the inertial thermal lag time. In this regime, the temperature was found to be unchanged at the initial condition.

The solution for the transient temperature from the method of relativistic transformation is compared side by side with the solution for the transient temperature from the method of Chebyshev economization. Both solutions are within 12% of each other. For conditions close to the wave front, the solution from the Chebyshev economization is expected to be close to the exact solution and was found to be within 2% of the solution from the method of relativistic transformation. Far from the wave front, i.e., close to the surface, the numerical error from the method of Chebyshev economization is expected to be significant and verified by a specific example. The solution for transient surface heat flux from the parabolic Fourier heart conduction model and the hyperbolic damped wave conduction and relaxation models are compared with each other. For τ > 1/2 the parabolic and hyperbolic solutions are within 10% of each other. The parabolic model has a “blow-up” as τ → 0, and the hyperbolic model is devoid of singularities. The transient temperature from the Chebyshev economization is within an average of 25% of the error function solution for the parabolic Fourier heat conduction model. A penetration distance beyond which there is no effect of the step change in the boundary is predicted using the relativistic transformation model. The method of relativistic transformation of coordinated has been shown [Sharma, 2009] to bounded solutions close to the integrated expression of exact solution presented by other investigators. This method of analysis is used in this study, The damping term is first 1 k* removed by multiplying Eq. (3.169) by en. Choosing n and let W = uen Eq. (3.169) 2 becomes;

 2W  2W W 1  k *   4 X 2  2

2

(3.170)

The significance of W is it that this can be recognized as the wave concentration. During the transformation of Eq. (3.169) to Eq. (3.170) the damping term has vanished. Now let us define a spatio-temporal symmetric substitution (relativistic transformation);  = 2 – X2 for  > X

(3.171)

Eq. (3.170) becomes; 2 2

W 2

W

1 k* 16

2

0

(3.172)

121

Mathematical Process Models

Comparing Eq. (3.172) with the generalized Bessel equation [Varma and Morbidelli, 1997] 1 k* 16

a = 1; b = 0; c = 0; s = ½; d

2

. The order p = 0.

d = ½ i(1 – k*) and is imaginary. Hence the solution is; s

 1 k *  2  X2 W c1I 0   2 

2 2     c K  1 k *   X  2 0 2  

   

(3.173)

c2 can be seen to be zero from the condition that at  = 0 W is finite.

W

2

1 k*

c 1 I0

X2

2

(3.174)

From the boundary condition at X = 0,

1e

(1 k*) 2

c 1 I0

1 k* 2

(3.175)

c1 can be eliminated between Eqs. (51-52) in order to yield; I0 u

1 2

2

X2 1 k *

(3.176) I0

2

1 k*

This is valid for  > X, k*  1. For X > , J0 u

1 X2 2 I0

At the wave front,  = X,

2

2

1 k*

(3.177) 1 k*

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Kal Renganathan Sharma

u

(1 k*) 2

e

X

e

(1 k*) 2

(3.178)

3.7.6. Results The mass inertia can be calculated from the first zero of the Bessel function at 2.4048. Thus, X 2p inertia

23.1323

1 k*

(3.179)

2

A de no vo dimensionless group called Sharma Number (Mass) was introduced [Sharma, 2010]. This can be written as follows; ki

Sha

r

(3.180)

a

It is the ratio of mass transfer rate to the transport taking place by damped wave diffusion. This can be seen to a product of Maxwell number and Sherwood number. Di

Max

a

r

(3.181)

2

Maxwell number (mass) as given by Eq. (3.181) can be seen to be the Fick number with the relaxation time substituted for time, t. This gives the ratio of the square of the speed of mass (vm =

Di

) by damped wave diffusion divided by the square of a characteristic r

speed (a/r). The Sherwood number, Sh can be written as; ki a

Sh

(3.182)

Di

The product of Maxwell number (mass) and Sherwood number can be seen to yield the Sharma number (mass) as follows;

Sha

ki a

r

Di a

r

2

ki a Di

Max * Sh

(3.183)

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123

Sharma number (mass) may be used in evaluating the importance of the damped wave diffusion process in relation to other processes such as convection, Fick steady diffusion in the given application. The concentration at an interior point in the semi-infinite medium is shown in Figure 3.0 Four regimes can be identified. These are; i)

Zero Transfer Inertial Regime, 0 0

inertia

ii) Times greater than inertial regime and less than at the wave front, Xp> iii) Wave front,  = Xp iv) Open Interval, of times Greater than at the wave front,  > Xp During the first regime of mass inertia there is no transfer of mass up to a certain threshold time at the interior point Xp = 10. The second regime is given by Eq. (3.177) represented by a Bessel composite function of the first kind and zeroth order. The rise in dimensionless concentration proceeds from the dimensionless time 2.733 up to the wave front at Xp = 10.0. The third regime is at the wave front. The dimensionless concentration is described by Eq. (3.178). The fourth regime is described by Eq. (3.176) and represents the decay in time of the dimensionless concentration. It is given by the modified Bessel composite function of the first kind and zeroth order. In Figure 4.0 is shown the three regimes of the concentration when k* = 2.0. It can be seen from Figure 4.0 that the mass inertia time has increased to 8.767. The rise is nearly a jump in concentration at the interior point Xp = 10.0. When k* = 0.25 as shown in Figure 5.0 the inertia time is 7.673. In Figure 6.0, the three regimes for the case when k* = 0.0 is plotted. In Table 1.0 the mass inertia time for various values of k* for the interior point Xp = 10.0 is shown. k* needs to be sufficiently far from 1 to keep the inertia time positive. Table 3.5. Mass Inertia Time vs k* for Interior Point Xp = 10.0 S.No. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

k* (k”‟mr) 0.01 0.1 0.25 0.3 0.4 0.5 1.75 2.0 4.0 8.0 25.0 10.0

Mass Inertia Time (t/mr) 8.741 8.452 7.673 7.266 5.979 2.733 7.673 8.767 9.871 9.976 9.998 10.0

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Kal Renganathan Sharma

Dimensionless Concentration (C/Cs)

Xp = 10

"k* = 0.5

0.4 0.3 0.2 0.1 0 0

5

10

15

20

Dimensionless Time

Figure 18.0. Dimensionless Concentration at an Interior Point Xp = 10 in a Semi-infinite Medium during Simultaneous Reaction and Diffusion. k* = 0.5.

Figure 19.0.

Figure 20.0. Dimensionless Concentration at an Interior Point Xp = 10 in a Semi-infinite Medium during Simultaneous Reaction and Diffusion. k* = 0.25.

Mathematical Process Models

125

Figure 21.0. Dimensionless Concentration in a Semi-infinite Medium during Simultaneous Reaction and Diffusion. k* = 0.0.

The steady state solution for Eq. (46) can be written as; uss = exp(-(k*)1/2 X)

(3.183a)

3.7.7. Conclusion 1) Diffusion and Reaction in Islets of Langerhans was studied using Parabolic and Hyperbolic Models. This is needed to better treat Type I diabetes by the transplantation method. 2) Sharma Number can be used to evaluate when the wave term becomes important in the application. It represents the ratio of Mass Transfer in Bulk to Relaxation Transfer. 3) Sharma number (mass) may be used in evaluating the importance of the damped wave diffusion process in relation to other processes such as convection, Fick steady diffusion in the given application. 4) Four regimes can be identified in the solution of hyperbolic damped wave diffusion model. These are; i) Zero Transfer Inertial Regime, 0 0 inertia ; (ii) Rising Regime during times greater than inertial regime and less than at the wave front, Xp > ; (iii) at Wave front,  = Xp; (iv) Falling Regime in Open Interval, of times greater than at the wave front,  > Xp. 5) Method of superposition of steady state concentration and transient concentration used in both solutions of parabolic and hyperbolic models. 6) Expression for steady state concentration developed. 7) Closed form analytic model solutions developed in asymptotic limits of Michaelis and Menten kinetic at zeroth order and first order.

126

Kal Renganathan Sharma 8) 9) 10) 11)

Expression for Penetration Length Derived – Hypoxia Explained Expression for Inertial Lag Time Derived Solution within bounds of Second Law of Thermodynamics No Overshoot Phenomena Observed.

D. MATHEMATICAL MODELS FROM SHELL BALANCE AND EQUATIONS OF MOMENTUM, ENERGY AND CONTINUITY 3.8. HIGH VOLUME CENTRIFUGAL SEPARATION OF OIL/WATER 3.8.1. Background Oil spills such as the recent one in the Gulf of Mexico called the Deepwater Horizon [Telegraph, 2010] needs to be cleaned-up. During development of technology and advancement of human mankind, it is not sufficient to discover a new gadget. The technological device has to be used in a safe manner and benignant to the environment. The psyche of the general public would be boosted when appropriate methods for handling these spills are in place. Five oil spills are listed in Table 3.6. The Deepwater Horizon in 2010, and the Exxon Valdez in 1989 were at a monetary loss of close to $ 0.5 billion in lost revenue from the crude oil. The largest one is the one in California in 1911. The other two spill are at the Persian Gulf and Mexico. When more than one country is involved oil spills may cause friction among otherwise friendly neighbors. One method for cleaning the oil from the sea water is the use of a centrifuge. A typical CINC centrifugal liquid-liquid separator can be obtained from the CINC Processing Equipment, Inc. The CINC Liquid-Liquid Centrifugal Separator utilizes the force generated by rotating an object about a central axis. By spinning two fluids of different densities within a rotating container or rotor the heavier fluid is forced to the wall at the inside of the rotor while the lighter fluid is forced toward the center of the rotor. A cut-away view of such a centrifugal separator may be viewed at the internet webpage http:///www. cincmfg.com/How_our_Centrifuges_Work_s/108.htm. Table 3.6. Some Illustrative Oil Spills # 1.0 2.0 3.0 4.0 5.0

Oil Spill Deepwater Horizon, Gulf of Mexico Lakeview Gusher, Kern County, CA Gulf War Oil Spill (Iraq, Persian Gulf and Kuwait) Ixtoc I, Gulf of Mexico, Mexico Exxon Valdez, Alaska

Date 04-10-10 to 07-15-10 05-14-10 to 09-1911 01-19-1991 – 01-281991 06-1979 – 03-1980 March 24th 1989

Amount 175 million gallons 378 million gallons 275 million gallons 150 million gallons 30 million gallons

Mathematical Process Models

127

The theory for separation used currently is the Stokes settling of oil droplets. For high volume separation such as the oil that can be recovered from the oil spills a centrifuge such as the one described in this study may be used. Here layers form with one layer that is oil rich and another layer that is water rich. The peripheral layer is water rich and may be collected from a port at the outer centrifugal bowl (Figure 22.0) and the oil rich layer may be collected from the inner rotor wall that is rotating. There is not much discussion in the literature for the theory of centrifugal separation of layered flow. In this study the velocity profiles of the oil rich layer and water rich layer are derived from the equations of continuity and motion. The thickness of the interface of the oil and water is calculated from a component mass balance of the oil in the inlet and outlet streams of the continuous centrifuge. Numerical simulations are run on a desktop computer for a given angular speed of rotor,  (RPM) and density ratio of the fluids and viscosities of the fluids. A set of four simultaneous equations and simultaneous unknowns are solved for using the MINVERSE command in Microsoft Excel for Windows 2007. These constants are used to obtain the power draw at the rotor from the torque required. A log-log plot is developed form the simulations for the power draw at the rotor that may be used in the design of such systems.

3.8.2. Theory Consider a centrifuge with an outer bowl radius of R (m) and an inner rotor radius of R (m). The inner rotor is allowed to rotate at an angular velocity of  RPM. The feed has high concentration of oil about 33% mass fraction oil (xF). It is desired to achieve a separation efficiency of 97.9%. The outlet oil stream is from the inner rotor and the outlet water stream is from the periphery of the bowl. The density ratio of the oil and water is . Viscous flows are considered at steady state. Consider a thin shell of fluid with thickness r and at a distance r from the center of the centrifuge as shown in Figure 1.0. It is assumed that the momentum transfer is predominantly in the radial direction. The tangential velocity assumes a profile that varies with the distance r from the center of the centrifuge. It is assumed that for high volume oil and water feeds such as the one that can be expected from the clean-up of the recent Oil Spill of BP Americas [Telegraph, 2010] two layers are formed, i.e., one rich in oil and the second layer rich in water. As the tangential force from the rotor is increased the species with the higher specific gravity will gain more momentum and move to the periphery of the centrifuge. The species with the lower specific gravity will remain in the inner layer close to the rotor. The density of the crude oil was assumed to be “heavy” and was taken as 900 kg. m3 and the density of the water was taken as 1000 kg.m3. For such a pair, the peripheral layer would be water rich and the inner layer would be oil rich. Earlier discussions in the literature have been largely on droplet formation of oil and layer formation or “slick” formation is not discussed much. Let the radius of the outer centrifugal bowl that is held stationary be R (m) and that of the inner rotor be R (m). The inner rotor is allowed to rotate at an angular velocity of  RPM (revolutions per minute). The water is collected by a port at the periphery of the bowl and the oil is collected through the port in the inner rotor. The feed is introduced from the top of the centrifuge.

128

Kal Renganathan Sharma

Figure 22.0. Cross-Sectional View of Centrifugal Separator of Oil and Water.

The feed location has not been optimized in the study. The equation of continuity and motion for v and the equation of motion for shear stress, r can be written from the Appendix in Bird, Stewart and Lightfoot [2007] as follows;

1 r2 r

r2

0

r

(3.184)

Integrating Eq. (2.64); c1 r

(3.185)

r2

The Newton‟s law of viscosity for the shear rate is given by;

r

r

r

v r

r

(3.186)

For the oil rich inner layer (Figure 22.0) combining Eq. (3.186) and Eq. (3.184);

oil

r

v

c1

r

r3

(3.187)

129

Mathematical Process Models Integrating Eq. (3.187) twice; v

c1

r

2

oil r

(3.188)

c2

2

Eq. (3.188) is valid for, R  r  R. For the water rich peripheral layer (Figure 22.0), in a similar manner the tangential velocity of the fluid can be written as follows; v

c3

r

2

water r

(3.189)

c4

2

Eq. (3.189) is valid for, R  r  R. The boundary conditions can be seen to be; r = R, v = 0

at the outer stationary wall,

(3.190)

Substituting Eq. (3.190) in Eq. (3.189); c1

0

at the inner rotor wall,

c2

(3.191)

r = R, v = R

(3.192)

2

oil R

2

Substituting Eq. (3.192) in Eq. (3.189); v

c1

R

2

2 oil

R2

(3.193)

c2

at the interface of oil rich and water rich layer, Interface is assumed to be without any accumulation of forces; (oil )

r

c1 2

R

r

( water )

(3.194)

c2 2

2

R

2

The velocity across the interface of oil rich and water rich layer is assumed to be continuous; v

c1 R

2

oil

R

2

c2

c3

2

2 water

R2

c4

(3.195)

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Kal Renganathan Sharma

In this study, Eqs. (3.193-3.195) were used to solve for the integration constants, c1, c2, c3 and c4 using the MINVERSE function in Microsoft Excel for Windows 2007. The set of equations Eq. (3.193-3.195) that are needed to obtain the integration constants are given in the matrix form as follows; 0 1 2

2 oil

2

1

R2

1 2 oil

1

2

0 0

1 2

2 water

0

c1

1

1

R2

water R

0

0

1 2

1

0

c2 c3

0

c4

0

(3.196)

1

R2

Eq. (3.196) is a set of four simultaneous equations and four unknowns. The vector of constants can be obtained as follows;

0 c1 c2 c3

0

1 2

2 oil

R2

1

c4

2

2 oil

2

1

R2

water R

2

0

1

0

1

1

2

1 0

0

0

1

1

1

2 water

R2

0

(3.197)

0

1

The layer thickness ratio,  can be estimated as follows. A component balance on the oil in the feed stream, peripheral water stream and inner rotor oil stream would yield; xFv = (v - vrot)xper + vrotxrot

or,

vrot

xF

xper

v

xrot

xper

(3.198)

(3.199)

Let the residence time of the fluid in the continuous centrifuge be  (hr). Then;

and,

vrot = R2(2-2)H

(3.200)

v = R2(1 -2)H

(3.201)

Dividing Eq. (3.200) by Eq. (3.201) and Equating with Eq. (3.199);

Mathematical Process Models

1

2

xF

xper

xrot

xper

2

131

(3.202)

3.8.3. Results Simulations were run on the desktop computer using Microsoft Excel for Windows, 2007. An example calculation is shown below; The calculations were performed for a rotor speed of 1000 RPM. The separation efficiency is about 97.9%. The values in bold face are obtained by using the MINVERSE command in Microsoft Excel for Windows 2007. The results are the inverse of the matrix as described in Eq. (3.197). Simulations were repeated for 29 different values of angular speeds of rotor. Each of the torque values were recorded in another column in the spreadsheet. The torque is calculated from the shear stress the rotor wall multiplied with the surface area of the rotor and the moment arm distance, R, and multiplied with the angular speed  in RPM. The results of these simulations are shown in Figure 23.0 on a log-log plot. The relationship is found to be linear in the log-log plot. For the example run as shown in Table 3.3 the separation efficiency is about 37%. In order to achieve more separation more stages need be considered. The set of simulations were repeated for a higher viscosity of oil, oil (5000 Pa.s). The power draw at the rotor is also shown in Figure 23.0 in the log log plot. The increase in power draw corresponding to an increase in viscosity of oil was not high.

Figure 23.0. Power Draw as a Function of Rotor Speed for  = 0.74.

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Kal Renganathan Sharma

Table 3.7. Calculations for a Given Set of Oil and Water Viscosities and  = 10 RPM Oil Water Separation by Centrifugation

w oil    R oil water res v  

c1 c2 c3 c4 T 199

Feed

Outlet (Oil) 0.99 0.01

Outlet (Water)

0.001

Pa.S

xoil xw

0.33 0.67

1000

Pa.S

vrotor

13.3877551

0.74

vper

27.6122449

0.4524

Separation

0.979591837

0.326531

Ratio

Gallons Ft

V H

0.522028

m

0.9

0

0

0.00002

1

0.83386

3.6523E-05

1

0

0

1 2.87636E-05 -0.034766137 1.26976E-06 -0.034766137 1.000000695

0 1 0.034766137 0.99999873 0.034766137 -6.9532E-07

-1 -28.7636 1 -3.7E-05 -2.7E-07 5.4E-12

0 -1 -0.034766137 1.26976E-06 -0.034766137 6.95323E-07

0.83386 5 900

Efficiency 10000 kg.m3 3

1000

kg.m

1

hr

41

m3.h-1

34.76614 999.9987 34.76614 -0.0007 RPM  1000

0.01 0.99

 0.83386

0 1000 0 0

Torque 42.32546536

3.8.4. Discussions It has been reported in the literature that when the angular speed reaches a critical value, crit secondary flow will develop. The secondary flow is periodic in axial direction and gets superimposed on the tangential flow. Toroidal vortices named Taylor vortices will form. When the angular velocity is increased further travelling waves form and then the turbulent regime is reached. The Rayleigh regime [Taylor, 1923], doubly periodic flow regimes, etc. have been developed for a single fluid. The system under consideration has two components, i.e., oil and water. The Taylor vortex flow, Rayleigh flow regimes can be avoided during the operation of the centrifuge. By running the centrifuge shallow or deep the velocity can be kept close to the laminar regime. The requirements for layer formation are not clear. The transition from a drop regime to a layer regime can be expected to depend on the surface tension of the oil and water. The Marangoni instability [Coles, 1965] may also be an issue. The centrifuge may be the solution to high volume oil and water mixtures that needs to be separated. A computer solution procedure was developed to evaluate centrifugal separation of oil and water. At high volumes, oil rich and water rich layers may be expected to form. Expressions for the interlayer thickness ratio,  was developed from component material

133

Mathematical Process Models

balance on the inlet and outlet streams of the continuous centrifuge. The equations of motion and continuity for the tangential velocity, v were derived for the oil rich and water rich layers. The water is recovered from the outlet at the periphery of the centrifugal bowl that is held stationary and the oil is recovered from the inner rotor that is rotating. The four integration constants in the velocity profile is solved for by use of the MINVERSE operation in Microsoft Excel for Windows 2007. For each RPM and set of parameters the power drawn at the rotor is calculated. The power draw would to be linear with the angular speed of the rotor on a log-log plot. When the density ratios of the two fluid streams,  approach 1 this method may not be efficient. In a similar manner when the feed contains only small quantity of oil other methods such as molecular sieve adsorption may be considered. Centrifugal separation of high volume oil and water may be designed using the methods shown in this study.

3.9. WARM/COOL FEELING AND THERMAL WEAR The effect of damped wave conduction and relaxation on warm/cool feeling of the human skin was studied by Sharma [2010]. In this study a two layer mathematical model was developed to study the transient heat conduction of the human skin and thermal fabric layer during use to protect the human body from cold winter outdoors. A schematic of the relevant aspects from a cross-section of human skin near a finger pad is shown in Figure 24.0. Many kinds of receptors in human skin are known to transmit information about the surroundings to the central nervous system. The role of these receptors in generating sensations caused by stimulus from the surroundings is analyzed. The response in the receptors is physico-chemical in nature. The correlation between mechanical stimulus and sensation of touch from a neurophysiological standpoint was studied experimentally by earlier investigators. They discuss the relation between the surface roughness of fabric and human sensory feel. “Krause‟s end bulb” is the receptor attributed with the cool feeling and “Ruffini‟s ending” is the receptor that is responsible for warm feeling. The transient heat conduction in the neighborhood of these receptors as the outside temperature plummets to low levels as typical of winter in the northern part of USA needs to be modeled. The skin layer and the thermal layer assembly are approximated as shown in Figure 24-0. The blood flow in the vessels results in a constant temperature environment at x = b at T = Tbl, where Tbl is the blood temperature. The origin is taken at the interface between the winter surroundings and outer surface of the thermal wear used to protect the skin from the winter cold weather. The thickness of the thermal fabric is „a‟ and the interface of the thermal wear and human skin occurs at x = a. Let the ambient temperature be Tamb. In winter this can be expected to be much lower than the blood temperature, i.e., Tamb 1 the flow is said to be supersonic. As can be seen from Figure 36.0, an oblique shock wave is formed and attached to the sharp nose of the wedge. Across this shock wave, the streamline direction changes in a discontinuous manner. Ahead of the shock, streamlines are straight and parallel and horizontal. These are not forewarned about the presence of the body. Theoretical analysis about such flow regimes will ensue in a later chapter. When M becomes greater than 5 the flow is said to be hypersonic. The temperature, T, pressure, p and molar volume, v increase almost explosively across the shock wave. The oblique shock that is formed moves closer to the surface. The shock layer becomes very hot. It reaches temperatures hot enough to become dissociated from the wedge. Some ionization of the gas may also occur. Chemically reacting flows pose a complex flow problem.

3.13.5. Fourier, Fick and Newton Numbers During transient heat conduction, transient mass diffusion and transfer momentum transfer the temperature profile, concentration profile and velocity profile varies with time and space. These profiles for different geometries and different boundary conditions with the governing equation obtained from energy balance, mass balance and momentum balance and Fourier‟s law of heat conduction, Fick‟s law of mass diffusion and Newton‟s law of viscosity are given in graduate textbooks such as Bird, Stewart and Lightfoot [2006]. The transient temperature profile, concentration profile and velocity profile for standard geometries and

Mathematical Process Models

167

boundary conditions derived from the governing equations from energy balance, mass balance and momentum balance and damped wave conduction and relaxation equation, damped wave diffusion and relaxation equation and damped wave kinematic diffusion and relaxation are given in Sharma [2005]. The transient temperature, concentration and velocity profiles from Fourier‟s law of heat conduction, Fick‟s law of mass diffusion and Newton‟s law of viscosity are described in dimensionless form by use of Fourier number, Fick number and Newton numbers. These can be calculated by; t

Fo d

2

kt Cpd 2

(3.331)

where Fo is the Fourier number,  is the thermal diffusivity (m2.s-1), d the characteristic length (m), k the thermal conductivity (W.m-1.K-1),  the density (kg.m-3), Cp the heat capacity, (J.kg-1K-1). The physical significance of Fourier number is the ratio of the square of penetration distance of heat disturbance at the surface (t)0.5 to the square of the characteristic length of the object or medium considered. Fi

D ABt d2

(3.332)

Where Fi is the Fick number, DAB is the binary diffusivity (m2.s-1). The physical significance of Fick number is the ratio of the square of penetration distance of mass disturbance at the surface (DABt)0.5 to the square of the characteristic length of the object or medium considered. t Ne (3.333) d2 Where Ne is the Newton number,  is the kinematic viscosity (m2.s-1). The physical significance of Newton is the ratio of the square of penetration distance of velocity disturbance at the surface (t)0.5 to the square of the characteristic length of the object or medium considered. In addition to the Fourier number, Fick number and Newton number, Vernotte number or Maxwell number (heat), Maxwell number (mass), Maxwell number (momentum) are needed to characterize the transient temperature, concentration and momentum profiles obtained from damped wave transport and relaxation equations. These equations generalize the Fourier‟s law of heat conduction, Fick‟s law of mass diffusion and Newton‟s law of viscosity aimed for wider applicability. These are defined as follows; Ve

r

d2

(3.334)

Where Ve is the Vernotte number (Cattaneo and Vernotte independently in 1948 and 1958 postulated the constitutive relation for heat conduction that may be used to generalize

168

Kal Renganathan Sharma

Fourier‟s law of heat conduction with wider applicability), r, is the relaxation time (s). The relaxation time has been shown by Sharma [2006] to be a third of the collision time of electrons in the free electron theory when the acceleration of the electron is accounted for. It is also shown to describe the speed of heat propagation or speed of mass propagation or speed of momentum propagation as follows;

; vm

vh

D AB

r

; ; vmom

r

(3.335) r

Where, vh, vm and vmom are the velocity of heat propagation, velocity of mass propagation and velocity of momentum propagation. The velocity of mass propagation was shown related to the root mean square velocity of molecule from Stokes-Einstein formulation of diffusion and kinetic representation of pressure by Sharma [2007]. The Vernotte number can be rewritten as follows;

Ve

vh2

2 r 2

(3.336)

v2

rd

The physical significance of Vernotte number is that it represents the heat velocity in the dimensionless form. In the asymptotic limit of the heat velocity reaching infinity the relaxation time, r would reach zero and the damped wave conduction and relaxation equation would revert to the Fourier‟s law of heat conduction. The Vernotte number can also be seen as the ratio of the square of the velocity of heat to the square of another characteristic velocity Maxm

D AB

r

(3.337)

d2

Maxm given by Eq. (2.145) is Maxwell number (mass). It is the analogous group of Vernotte number in heat to damped wave diffusion and relaxation. The physical significance of Maxm (mass) is mass diffusion velocity in dimensionless form. Maxmom

D AB d

2

r

(3.338)

Maxmom is given by Eq. (2.146) is Maxwell number (momentum). It is analogous to the Vernotte number in heat to damped wave momentum transfer and relaxation. The physical significance of Maxmom is that it represents the kinematic viscous momentum transfer velocity in dimensionless form. It is the ratio of the diffusional speed to the relaxation/ballistic speed in the process. It can be used to gauge how significant is the relaxation phenomena in a given application.

Mathematical Process Models

169

3.13.6. Sharma Number (MASS) Mathematical theories have been developed to explain the way the mass transfer coefficients correlates with the Reynolds number and Schmidt number. In the film theory, an effective film thickness is assumed over which the concentration changes happen. When a fluid flows past a surface a certain concentration profile will be established. Upon observation of the concentration change with distance it can be inferred that most of the drop in concentration can be attributed to a short region that can be denoted as a “effective film thickness”. The mass transfer coefficient can be calculated as;

N"

k Cli

D AB Cli

Clb

Or,

Clb

D AB

k

(3.339)

(3.340)

The penetration theory was developed by Higbie [1935], 7 decades ago. Higbie pointed out that in most mass transfer applications the contact time is short that during the transfer the process is transient. Consider a semi-infinite medium at a uniform initial concentration of C0. At times greater than zero one of the surfaces is maintained at a constant surface concentration Cs. By the Fick‟s second law of mass diffusion the governing equation can be written as; u t

Where,

u

2

D AB

u

z2

C A C A0 C As

C A0

(3.341)

(3.342)

The governing equation given in Eq. (2.130) can be solved using the Boltzmann z transformation such as . Eq. (2.130) becomes; 4 D ABt u

2

u 2

(3.343)

The dimensionless concentration profile can be shown to be;

u

1 erf

z

4 D ABt

(3.344)

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Kal Renganathan Sharma

The mass transfer coefficient can be calculated from Eq. (2.133). The surface flux can be written as;

Js

DAB C As

C A0

k C As

4 DABt

Thus,

D AB

k

C A0

(3.345)

(3.346)

4 t

Higbie assumed a constant time of exposure for all the eddies of fluid at the interface. Instead of the Fick‟s second law of diffusion when the generalized Fick‟s law of diffusion and relaxation is used, it was shown in Sharma [2007] that for the semi-infinite medium subject to constant wall concentration the surface flux by the method of Laplace transforms can be given by; J*

e

2I

0

(3.347)

2

The mass transfer coefficient can then be seen to be;

k

D AB

e

2I

0

r

Where,

2

(3.348)

tc r

Dankwerts [1955] developed the surface renewal theories. Here he derived the mass transfer coefficient for the general case where the eddies stay at the surface for varying lengths of time and Higbie‟s penetration theory is a particular case where the contact times of the eddies is constant. k = (DABS)0.5

(3.349)

where s is the surface renewal rate. Dobbins [1964] noted that whereas the film theory assumes that the concentration profile has reached steady state in the time of mass transfer and the Higbie and Dankwerts theories account for the transient nature of diffusion during the eddy contact time, they use a semi-infinite boundary condition for zero transfer. Dobbins used a finite length boundary condition and modified Eq. (3.349) as;

k

D AB s coth

szb2 D AB

(3.350)

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Mathematical Process Models

A revisit of Dobbins set of boundary conditions to the governing equation derived by using the generalized Fick‟s law of diffusion for finite width can result in the solution given earlier in Sharma [3007]. The dimensionless concentration when the relaxation times are large can be found as;

u

cn e

2

2 n

cos

0.25 cos

nX

(3.351)

1

From a mass balance on a slice of thickness x in the time increment t, and in the limit of t and x 0, the following equation results; J* X

u

(3.352)

Eq. (3.352) relates the accumulation of concentration to the rate of change of flux in space. u

(

2 n

0.25cn e

2

2 n

sin

0.25 cos

nX

(3.353)

1

0.5cn e

2

2 n

cos(

0.25 ) cos

nX

)

Eq. (3.353) was obtained by differentiating Eq. (3.351) with respect to time. Substituting Eq. (3.353) in Eq. (3.352) and integrating the dimensionless mass flux can be obtained;

J*

c'

(

2 n

0.25

cn

0.5

e

2

2

2 n

sin

0.25 sin

nX

n

1

cn

e

cos(

2 n

(3.354)

0.25 )sin

nX

)

n

cn

Where, From the final condition,

4 1

n 1

2n 1

;

2n 1 n

D AB

r

2a w

c‟ = 0

The mass transfer coefficient, k can be calculated from the flux at the surface, J*;

(3.355)

172

Kal Renganathan Sharma k

J*

2 n

(

D AB

0.25

cn

2

e

2 n

sin

0.25 sin

nX

n

1

(3.356)

r

0.5

cn

e

2

cos(

2 n

0.25 )sin

)

nX

n

Taking only the first term in the infinite series 2

8a we

2

cos

D AB

2 r

4a w2

k

0.25

4a w

r

0.25e

4a w2 2

2

D AB

2

DAB

D AB

2

DAB 2 4aW

sin

r

0.25

r

r

Or, 2

4 e k

r

aw

Sha

2

cos

2

D AB

r

0.25

4a w2

2

D AB

2 r

0.25e

4a w2

2

sin

D AB

2 4aW

r

0.25

(3.357)

2

Where Sha is the Sharma number. In the short time limit at 0, k

r

aw

Sha

4

(3.358)

0.41

2

Thus the dimensionless group Sharma number is arrived at. Sharma Number can be written in terms of Sherwood number, Sh and Maxwell number (mass), Maxm. Sh

Maxm

ka w

(3.359)

D AB D AB

r

(3.360)

2 aW

The Maxwell number (mass) is the Fick number (analogous to Fourier number) evaluated at time t equal to the relaxation time, r. Sh * Max

ka w D AB m

DAB

a w2

r

k

r

aw

Sha

(3.361)

Mathematical Process Models

173

Thus Sharma number is the product of Sherwood number and Maxwell number (mass). The physical significance of Sharma number gives the ratio of the bulk mass transfer rate and the relaxation transfer rate. Maxm is the Maxwell number (mass) and gives the ratio of the diffusion rate and relaxation rate. The Sherwood number gives the ratio of the mass transfer rate in bulk compared with the diffusion rate. At large relaxation times when the above correlation is expected to hold well, it can be seen that the dimensionless mass transfer rate becomes independent of the diffusivity. By analogy between mass diffusion and heat conduction, Sharma number (heat) can be seen to be; h

Sha

r

C paw

h Sa w

(3.362)

Where h is the heat transfer coefficient (W.m-2K-1). S is the storage coefficient. Storage coefficient has been used in analysis of damped wave conduction and relaxation in Sharma [2005]. The storage number is the ratio of thermal mass to that of the relaxation time. It is a measure of how much heat that is accumulated per cycle of the propagation of the heat wave. It would be interesting to see in an analogous manner to mass transfer whether the product of Nusselt number and Vernotte number would give the Sharma Number (heat); ha w

h

r 2 aw

k

r

C paw

h Sha Sa w

Nu Ve

(3.363)

It does! The product of Nusselt number and Vernotte number can be called Sharma Number (heat). The physical significance of the Sharma Number (heat) is that it is a measure of the ratio of the bulk convective heat transfer rate to the relaxation/ballistic transfer rate in the process. It can be seen that storage coefficient S, has units of W/m3/K, when thermal conductivity k has units of W/m/K and heat transfer coefficient, h has units of W/m2/K. S is volumetric, thermal conductivity is linear and heat transfer coefficient areal when heat resistances are calculated from them. The ratio of convection heat transfer to Fourier conduction heat transfer is given by Peclect number (heat).

Pe

a wu

Re Pr

ua w C p k

(3.364)

Peclect number (heat) can be obtained as the product of Reynolds number and Prandtl number. The ratio of convection mass transfer to Fick molecular diffusion is given by Peclect number (mass).

Pe

a wu D AB

Re Sc

ua w DAB

(3.365)

174

Kal Renganathan Sharma

G. STOCHASTIC MODELS 3.14. WEINER HOPF INTEGRAL EQUATION Weiner-Hopf integral equation can be used to estimate the effects of mixing in a CSTR, continuous stirred tank reactor. This can be done by estimating the degree of mixedness from the variance in SPC, statistical process control charts. For example, consider the manufacture of ABS, acrylonitrile butadiene and styrene, engineering thermoplastic using two or three CSTRs operated in series and a devolatilizer. The composition distribution of SAN, styrene acrylonitrile copolymer can be measured and monitored using data acquisition hooked to desktop computers. Often times, the RPPS, rubber phase particle size has to be maintained less than a threshold size. Sometimes bimodal distribution may be needed to obtain am optimal balance of properties: high gloss and high impact strength. Smaller the RPPS higher the gloss. Larger the size of the particle in a certain range higher is the notched Izod impact strength. In the patent literature and product brochures continuous polymerization of ABS engineering thermoplastic has been discussed. ABS with better balance of properties such as surface gloss, tensile strength, impact strength, processability, heat resistance with a bimodal distribution of particle size. Helical ribbon agitators are used in the CSTR to shear and blend well the high viscous, low Reynolds‟ number systems. The monomers and inert diluents used are pumped into the reactors with preformed rubber dissolved in the monomers and diluents along with the initiator, chain transfer agent, antioxidant, mineral oil, etc. The matrix and graft copolymer are formed in the two reactors and the unreacted monomers and diluent are removed from the product in the devolatilizer and recycled back to the reactors. The schematic of the two CSTR and DV, devolatilizer is shown in Sharma [1997]. It was found that the first reactor agitator speed had an effect of lowering the RPPS in product with improved gloss (Sharma, [1997]). The second reactor agitator speed was found to have an effect on the compositional heterogeneity in the second reactor. It is not clear whether the higher agitator speed lowered the mixing times and hence the superior product or by shear effect of anchor agitator resulted in a lower RPPS. Lower mixing times would result in better grafting which in turn has an effect of lowering the particle size in the product. A method is developed to evaluate the mixedness from compositional distribution data. Mixing effects can be expected to manifest as the broader variance in the composition data from SPC charts for SAN copolymer. The Weiner-Hopf integral equation can be used to estimate the degree of mixedness. This can delineate the blend-time mechanism and shear rate mechanisms. This can be done by filtering. This approach is in contrast to Atiqullah-Nauman model (). They obtain the copolymer composition distribution in CSTR with unmixed feed streams using lamellar stretch model that accounts for back mixing. Let the composition of copolymer AN be given by y(t) recorded in SPC charts. X is the derived signal. N is the noise that can be considered to predominantly arise from improper mixing and experimental error. The estimator can be introduced to derive this information.

x

t1

t

A t1 ,

t t0

y

d

(3.366)

175

Mathematical Process Models The rest of the problem is to find A(t1,) that would minimize the least squared error; T

J

E

x t1

x

t1

(3.367)

t

Let A(t1,) = A0(t1,) + (t1,) Minimization of J with respect to ; J

t

J

tr E x t1 xT t1

(3.368)

0

t

E x t1 yT

AT d

t t

AE y

t0

xT t1 d

t0

AE y

yT

AT d d

(3.369)

t 0 t0

J is minimized with respect to  and the Weiner-Hopf integral equation is obtained. t

yT

A0 E y

d

E x t1 yT

(3.370)

yT

} 0

(3.371)

0

(3.372)

t0

E{ x

t1

x t1

t

The error is orthogonal to all data in the set. ~

E x

t1

t

yT

t0    t

(3.373)

The degree of mixedness can be estimated from SPC charts of AN composition.

3.15. MESOSCOPIC MODELS - DISSIPATIVE PARTICLE DYNAMICS Groot and Warren [1997] reviewed dissipative particle dynamics, DPD as a mesocopic simulation tool. They have established useful parameter ranges for simulations. They consider atoms as a set of interacting particles. The time evolution of the ensemble is governed by Newton‟s equation of motion.

176

Kal Renganathan Sharma

vi fi

dri dt dvi

(3.374)

dt

They have developed an EOS, equation of state of DPD fluid applicable for polymeric systems. EOS for polymeric systems are discussed in more detail in Sharma [19]. EOS of DPD fluid is essentially quadratic in density. De Groot and Warren [34] have established useful parameter ranges for simulations. These parameters were linked to  interaction parameters in Flory-Huggins type models. The link opens the ways to perform large scale simulations effectively describing millions of atoms. The molecular fragments are simulated first. All atomistic details are retained to yield the  parameters. These results are input into a DPD simulation. These simulations are used to study the formation of micelles, networks, mesophases and other related phenomena. An illustrative application, they [Groot and Warren, 1997] have calculated the interfacial tension, , between homopolymer melts as a function of interaction parameter,  and N and found a universal scaling collapse when k BT

04

is plotted against N for N >1. The use of DPD to simulate the dynamics of

mesoscopic systems was discussed and a possible problem with the timescale separation between particle diffusion, and momentum diffusion (viscosity) was indicated. An example of soft condensed matter was given. They are neither completely solid nor completely liquid. The relevant length scale is between the atomistic scale and the macroscopic scale. For example, in polymer gels, this length-scale is set by the distance between the cross-links in the gel. Phase diagrams of polymer and its rheology was studied using a simple bead-andspring model. This model can be used to predict linear viscoelastic behavior. No hydrodynamic interactions were considered. For polymer-gels, the nature of the chemistry is not as important as the life-time structure of the polymer connections are. Three methods used to describe polymers confined to lattice conformations: Monte Carlo methods of lattice polymers, self-consistent field theory and dynamic density functionality theory are not well suited for branched polymers. The full connection from atomistic to the macroscopic world cannot be made. DPD, discrete particle dynamics simulation technique was introduced by Hoogerbruge and Koelman. The simulation of soft spheres was performed whose motion is governed by certain collision rules. The force acting on a “particle” consists of three parts each of which is pairwise additive. The three forces are as follows: FC (conservative force)

(3.375)

FD (dissipative force

(3.376)

Fk (random force

(3.377)

177

Mathematical Process Models

They showed that the dissipative force and the random force have to satisfy a certain relation. FijC

fi

FijD

FijR

(3.378)

j i

where the sum runs over all other particles within a certain cut-off radius rc. Here the cut-off radius, rc was taken as 1. The conservative force is a soft repulsion acting along the line of centers and is given by; ^

FijC

a ij 1 rij rij (rij < 1)

(3.379)

0 (rij  1)

FijC

aij is a maximum repulsion between particle i and particle j; rij

ri

rij

rij

^

rij

rj

(3.380)

rij rij

The remaining two forces are a dissipative or drag force and a random force. They are given by; FijD

D

FijR

^

rij

R

^

(3.381)

r ij vij r ij

^

rij

(3.382)

ij r ij

Where D and R are r –dependent weight functions vanishing for r > rc =1. where, vij = vi – vj ij (t) – random fluctuating variable using Gaussian statistics. ij (t ) ij

t

0 kl

t'

ik

jl

il

jk

t

t'

(3.383)

These forces also act along the line of centers and conserve linear and angular momentum. An independent random function for each pair of particles can be associated

178

Kal Renganathan Sharma

with. One weight function can be chosen arbitrarily and the other is then fixed. A relation between the amplitudes and kBT exist. D

R

r

2

2

r

(1 r )2 ,(r

1),0, r

0

(3.384)

2 kBT

Euler-type algorithm was used to advance the set of positions and velocities; ri t

t

ri t

tvi t

vi t

t

vi t

tfi t

fi t

t

(3.385)

t ,v t

fi r t

t

Random force becomes; R ij

R

rij

t

ij

1/2

^

(3.386)

r ij

Where, ij is a random number with zero mean and unit variance, chosen independently for each pair of interacting particles and at each time step. In the context of DPD, a modified version of the velocity-Verlet algorithm is as follows;

ri t

t

ri t

vi t

t

vi t

fi t

t

vi t

t

t 2 fi t

tvi t

2

~

tfi t

(3.387)

~

fi r t vi t

t ,v t t fi t 2

t fi t

t

It can be seen from Eq. (3.387) that the force varies with the velocity. If there is no random or dissipative force present this algorithm would be exact to O

t 2 at  = ½. The

order of algorithm is not clear because of the stochastic nature of the process. The stochastic differential equations can be integrated and the random force can be interpreted as a Weiner process. They considered the motion of a particle in a liquid. On account of collisions with other particles a random force can be seen to be exerted on the particle. The mean force acting on the particle is 0 but the variance of the force is finite. The force uncorrelated between time steps;

Mathematical Process Models 2

1

F

2

179

f t ' dt ' 0 2

N

(3.388) t N

fi i 1

2

2 2

t N

2

t t

The Fokker-Planck equation can be written as; c

D

(3.389)

t

Where, c and D are evolution operators that can be extracted from the time-evolution governing the motion of the particles. The Louisville operator of the Hamiltonian system interacted with conserved forces FC, the second operator D contains the dissipative and noise terms. The dissipative and random forces are set to zero. What is left is the Hamiltonian system. In the canonical ensemble, the Gibbs-Boltzmann distribution can be seen as the solution to the Fokker Planck equation.

eq

ri , Pi

i

e

Pi 2 U 2 mkBT kBT

(3.390)

Eq. (3.390) is the solution of; eq c

t

eq

0

(3.391)

The time step size was chosen as a compromise between fast simulation the equilibrium condition. The repulsion parameter „a‟ in Eq. (3.197) need be chosen. For the thermodynamic state of an arbitrary liquid is to be described correctly by the soft sphere model, the fluctuations in the liquid should be described correctly. The EOS of the polymer is obtained from simulations: Pressure as a function of density for various repulsion parameters. The virial theorem pressure P is obtained.

H. THERMODYNAMIC ANALYSIS 3.16. ANALYSIS OF NTE MATERIALS 3.16.1. Overview The volume expansivity of pure substances is defined as [Smith et al. 2005];

180

Kal Renganathan Sharma

1 V

V T

(3.392) P

It is a parameter that is used to measure the volume expansivity of pure substances and is defined at constant pressure, P. In the field of materials science, the property of linear coefficient of thermal expansion is an important consideration in materials selection and design of products. This property is used to account for the change in volume when the temperature of the material is changed. The linear coefficient of thermal expansion is defined as; lf l0 T f

l0 T0

l l0 T

T

(3.393)

For isotropic materials,  = 3. Instruments such as dilatometers, XRD, X-ray diffraction can be used to measure the thermal expansion coefficient. Typical values of volume expansivity for selected isotropic materials at room temperature is provided in Table 3.9. As can be seen from Table 3.9 the volume expansivity for pure substances are usually positive. Some cases it can be negative. Examples of materials given with negative values for volume expansivity in the literature are water in the temperature range of O – 4 K, honey, mononoclinic Sellenium, Se, Tellerium, Te, quartz glass, faujasite, cubic Zirconium Tungstate, ZrW2O8 [Jakubinek et al. 2008] in the temperature range of 0.3 – 1050 K. ZerodourTM [Askeland and Fulay, 2006] is a glass-ceramic material that can be controlled to have zero or slightly negative thermal coefficient of expansion and was developed by Schott Glass Technologies. It consists of a 70-80 wt % crystalline phase with high-quartz structure. The rest of the material is a glassy phase. The negative thermal expansion coefficient of the glassy phase and the positive thermal expansion coefficient of the crystalline phase are expected to cancel out each other leading to a zero thermal coefficient material. ZerodurTM has been used as the mirror substrate on the Hubble telescope and the Chandra X-ray telescope. A dense, optically transparent and zero-thermal expansion material is necessary in these applications since any changes in dimensions as a result of the changes in the temperature in space will make it difficult to focus the telescopes appropriately. Material scientists have developed ceramic materials based on sodium zirconium phosphate, NZP that have a near-zero-thermal-expansion coefficient. The occurrence of negative values of volume expansivity ab initio, is a violation of second law of thermodynamics according to some investigators such as Stepanov [2000]. They propose that the first law of thermodynamics be changed from dU = dQ – PdV to dU = dQ + PdV in order to work with materials with zero or negative coefficient of thermal expansion. In a famous problem such as the development of the theory of the velocity of sound such a change from defining an isothermal compressibility to isentropic compressibility brought the experimental observations closer to theory [Anderson, 2003]. The proposal from this study is in part motivated by the work of Laplace (see II. Historical Note below).

181

Mathematical Process Models Table 3.9. Volume Expansivity of Selected Materials at Room Temperature # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

, Volume Expansivity (*10-6 K-1) 75.0 49.8 36.0 9.0 36.0 51.9 165.0 240 300.0 210 31.8 27.0 -27.0 -12.0 Negative Negative

Material Aluminum Copper Iron Silicon 1020 Steel Stainless Steel Epoxy Nylon 6,6 Polyethylene Polystyrene Partially Stabilized ZrO2 Soda-Lime Glass Zirconium Tungstate ZrW2O8 Faujasite Water (0 – 4 K) Honey

3.16.2. Historical Note By 17th century it was realized that sound propagates through air at some finite velocity. Artillery tests has indicated that the speed of sound was approximately 1140 ft/s. These tests were performed by standing a known large distance away from a cannon, and noting the time delay between the light flash from the muzzle and the sound of the discharge. In proposition 50, Book II of his Principia Newton [1687] theorized that the speed of sound was related to the elasticity of the air and can be given by the reciprocal of the compressibility. He assumed that the sound wave propagation was an isothermal process. He proposed the following expression for the speed of sound;

1

c

(3.394) T

Where the isothermal compressibility is given by;

T

1 V

V P

(3.395) T

Newton calculated a value of 979 ft/s from this expression to interpret the artillery test results. The value was 15% lower than the gunshot data. He attributed the difference between experiment and theory to existence of dust particles and moisture in the atmosphere. A

182

Kal Renganathan Sharma

century later Laplace [1787] corrected the theory by assuming that the sound wave propagation was isentropic and not isothermal. He derived the expression used to this day to instruct senior level students in Gas Dynamics [Anderson, 2003] for the speed of sound as;

1

c

(3.396) s

1 V

s

V P

(3.397) s

By the demise of Napoleon the great the relation between propagation of sound in gas was better understood.

3.16.3. Theoretical Analysis Clapeyron equation can be derived to obtain the lines of demarcation of solid phase from liquid phase, liquid phase from vapor phase and solid phase from vapor phase in a PressureTemperature diagram for a pure substance. This can be done by considering a point in the demarcation line and a small segment in the demarcation line. At the point the free energy of the solid and liquid phases can be equated to each other at equilibrium. The enthalpy change during melting can be related to the entropy change of melting at a certain temperature of phase change. Along the segment the change in free energy dGs and dGl in the solid and liquid phases at equilibrium can be equated to each other. The combined two laws of thermodynamics are applied and an expression for dP/dT can be obtained. Sometimes when ideal gas can be assumed this expression can be integrated to obtain useful expressions that are used in undergraduate thermodynamics instruction. A similar analysis is considered here using Helmholz free energy. At the point and segment of the phase demarcation line of solid and liquid in a Pressure-Temperature diagram of a pure substance; As

US

AL

dAs

dAL

TS S

UL

T S SL

(3.398)

TS L

(3.399)

U SL

Applying the first and second law of thermodynamics to change in Helmholz free energy; dA dU

For reversible changes at equilibrium;

TdS dA

PdV P S dV sat

(3.400)

SdT S s dT sat

dAL

P L dV sat

S L dT sat

183

Mathematical Process Models dV dT

sat

S SL

U SL

P SL

T P SL

(3.401)

For physical changes Eq. (3.401) can be seen to be always positive. This is because of the lowering of pressure as solid becomes liquid and the internal energy change is positive resulting in a net positive sign in the RHS, right hand side of Eq. (3.401). No ideal gas law was assumed. Only the first two laws of thermodynamics was used and reversible changes were assumed in order to obtain Eq. (3.401). Thus reports of materials with negative thermal expansion coefficient is inconsistent with Eq. (3.401). What may be happening are chemical changes. Strong hydrogen bonded water in 0-4 0K shrinks upon heating due to chemical changes. This cannot be interpreted using laws that are developed to describe physical changes. In the example of faujasite may be lattice structure changes take place upon heating. What do you do ? For ideal gas, it is shown below that the volume expansivity can be related to the reciprocal of absolute temperature. Per the third law of thermodynamics the lowest achievable temperature is 0 0K. Hence volume expansivity is always positive for physical changes. From Eq. (3.392);

1 V

V

V T

P

From Maxwell Relations, V T

S P

P

(3.402) T

Eq. (3.402) can be seen to be the case as follows. The free energy G of pure substances is defined as; G

H

(3.403)

TS

Where, H is the enthalpy (J/mole), S is the entropy (J/mole/K)

dG

d H TS

dH TdS SdT

(3.404)

Combining Eq. (3.404) with the First Law of Thermodynamics, dG TdS VdP TdS SdT VdP

It may be deduced from Eq. (14) that;

SdT

(3.405)

184

Kal Renganathan Sharma G P

T

G T

P

V

(3.406) S

(3.407) G T

(3.408)

S P

The reciprocity relation can be used to obtain the corresponding Maxwell relation. The order of differentiation of the state property does not matter as long as the property is an analytic function of the two variables. Thus, 2

2

G P T

G T P

(3.409)

Combining Eq. (3.409) with Eq. (3.406); V T

S P

P

(3.410) T

Thus Eq. (3.410) can be derived. Combining Eq. (3.392) and Eq. (3.410); S P

V

(3.411) T

For a reversible process, the combined statement of 1st and 2nd laws [Smith et al., 2005] can be written as; dH

(3.412)

TdS VdP

At constant temperature for reversible process for real substances; H P

T T

S P

V

(3.413)

T 1

(3.414)

T

Combining Eq. (21) and Eq. (19);

1 V Or

H P

T

185

Mathematical Process Models

1 T

1 VT

H P

(3.415) T

For ideal gases Eq. (3.415) would revert to the volume expansivity,  would equal the reciprocal of absolute temperature. This would mean that  can never be negative as temperature is always positive as stated by the third law of thermodynamics. So materials with negative values for  ab initio are in violation of the combined statement of the 1st and 2nd laws of thermodynamics. Negative temperatures are not possible for vibrational and rotational degrees of freedom. A freely moving particle or a harmonic oscillator cannot have negative temperatures for there is no upper bound on their energies. Nuclear spin orientation in a magnetic field is needed for negative temperatures [Kittel, 1980]. This is not applicable for engineering applications. Enthalpy variation with pressure is weak and small for real substances. This has to be large to obtain a negative quantity in Eq. (3.415).

3.16.4. Proposed Isentropic Expansivity Along similar lines to the improvement given by Laplace to the theory of the speed of sound as developed by Newton (as discussed in Section II Historical Note) a isentropic volume expansivity is proposed.

1 V

s

V T

(3.416) S

Using the rules of partial differential for three variables, any function f in variables (x,y,z) it can be seen that; f x

z

f x

y

f y

P

V P

x

y x

(3.417) z

Thus, V T

s

V T

T

P T

(3.418) S

Let, P T

Plugging Eq. (27) into Eq. (26);

(3.419) S

186

Kal Renganathan Sharma

1 V

s

V T

P

(3.420)

T

S

At constant pressure, dH

dU

(3.421)

PdV

Eq. (3.421) comes from H = U + PV the definition of specific enthalpy in terms of specific internal energy, U, pressure and volume, P and V respectively. Eq. (3.421) can be written for ideal gas as;

CP

CV dT

PdV

(3.422)

It can be realized from Eq. (3.422) that; V T

V

CP P

CV

(3.423)

P

P

Plugging Eq. (32) into Eq. (30); V T

sV

CP

CV P

S

CP

T

CP

T

(3.424)

Or,

s

1 V

V T

CP

CV

VP

S

V

(3.425)

Eq. (3.425) can be written in another form as follows; 2 PT s

VC p V

2 T

(3.426)

T

The isobaric expansivity gets squared in Eq. (3.426). Even when negative values arise for this quantity the isentropic volume expansivity can be expected to be positive as long as; 2 PT

VC p

2 T

(3.427)

For substances with negative coefficient of thermal expansion under the proposed definition of isentropic volume expansivity, s does not violate the laws of thermodynamics

187

Mathematical Process Models

quid pro quo. Considering the thermal expansion process for pure substances in general and materials with negative coefficient of thermal expansion in particular, the process is not isobaric. Pressure can be shown to be related to the square of the velocity of the molecules.

3.16.5. Measurments of Volume Expansivity Not Isobaric The process of measurement of volume expansivity cannot be isobaric in practice. When materials expand the root mean square velocity of the molecules increases. For the materials with negative coefficient of thermal coefficient the velocity of molecules are expected to decrease. In either case, forcing such a process as isobaric is not a good representation of theory with experiments. Such processes can even be reversible or isentropic. Experiments can be conducted in a reversible manner and the energy may be supplied or may be removed as the case may be. Hence it is proposed to define volume expansivity at constant entropy. This can keep the quantity per se from violation the laws of thermodynamics.

3.16.6. Significance of Treatment of Materials with NTE Recently, Miller et al. [2009] presented a review article on materials that were observed to exhibit negative thermal expansion. Most materials demonstrate an expansion upon heating. Few materials are known to contract. These materials are expected to exhibit a NTE, negative thermal expansion coefficient. These materials include complex metal oxides, polymers and zeolites as shown in Table 3.10. These can be used to design composited with zero coefficient of thermal expansion. When the matrix has a positive thermal expansion coefficient and the filler material has a negative thermal expansion coefficient the net expansion coefficient of the composite can be dialed in to zero. They explore supramolecular mechanisms for exhibition of NTE. Examples of materials where reports indicate NTE with the references are as follows; Table 3.10. Materials that Possess NTE # 1.0

Journal Acta Crystallography US Patent J. Appl. Phys.

5.0 6.0 7.0

Material ZrW2O8, Zirconium Tungstate (cubic lattice) (ZrO)2VP2O7 HfW2O8, Halfnium Tungstate (orthorhombic) ZrMo2O8, Zirconium molybdate, (cubic) Silicalite-1 & Zirconium Silicalite-1 CuScO2 (delafossite structure) Polydiacetylene Crystal

8.0

Graphite Fiber Composites

Proc. of Royal Society

2.0 3.0 4.0

Chem. Mater. Mater. Res. Bulletin Chem. Mater. J of Polym. Sci.

Reference [Evans et al., 1999] [Sleight, 1994] [Jorgensen et al., 2001] [Lind et al., 1998] [Bhange et al. 2005] [Li et al., 2002] [Baughman et al., 1973] [Rupnowski et al., 2005]

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Kal Renganathan Sharma

A careful study of these materials under isentropic heating is suggested. The volume expansivity then will be within the predictions of the laws of thermodynamics. Further studies on chemical changes on heating for these materials are suggested. Strong hydrogen bonded systems may also be considered as strongly interacting systems.

3.16.7. Conclusions For physical changes Eq. (3.401) can be seen to be always positive. This is because of the lowering of pressure as solid becomes liquid and the internal energy change is positive resulting in a net positive sign in the RHS, right hand side of Eq. (3.401). No ideal gas law was assumed. Only the first two laws of thermodynamics were used and reversible changes were assumed in order to obtain Eq. (3.401). Thus reports of materials with negative thermal expansion coefficient are inconsistent with Eq. (3.401). For ideal gases Eq. (3.415) would revert to the volume expansively,  would equal the reciprocal of absolute temperature. This would mean that  can never be negative as temperature is always positive as stated by the third law of thermodynamics. So materials with negative values for  ab initio are in violation of the combined statement of the 1st and 2nd laws of thermodynamics. Along similar lines to the improvement given by Laplace to the theory of the speed of sound as developed by Newton (as discussed in Section II Historical Note) an isentropic volume expansivity is proposed by Eq. (3.416). This can be calculated using Eq. (3.426) from isobaric expansivity, isothermal compressibility and a parameter  that is a measure of isentropic change of pressure with temperature. Eq. (34) can be used to obtain the isentropic expansivity in terms of heat capacities at constant volume and constant pressure and isothermal compressibility at a given pressure and temperature of the material. Chemical changes have to be delineated from physical changes when heating the material.

I. OPTIMIZATION STUDIES 3.17. LIGHT TO ELECTRICITY CONVERSION – OPTIMAL TEMPERATURE The world‟s largest electric power generating system using CSP concentrated solar power technology has been commissioned by NRG solar along with Bright source energy in 2013. The capacity is 392 MW. CSP used a field of mirrors to concentrate sunlight onto thermal receivers mounted on top of towers. Lower cost of production and zero emissions are expected from use of solar power plants. The technology is low risk. The efficiency of conversion of sunlight to useful energy using photovoltaic cell technology hovers around 15%. Photovoltaic efficiency of PV cells is 8% for solar cells made from amorphous silicon. Their efficiency has increased now to 14%. This can be further increased to 20% by use of thin films that contain small amounts of crystals of silicon. Single crystal silicon can be used

Mathematical Process Models

189

to make the “most efficient” solar cells with 30% efficiency. These PV cells are more expensive. A team of researchers from North Carolina State University has fabricated a “super absorbing” design that can be used in order to maximize the light absorption efficiency of the thin film solar cells while minimizing the manufacturing costs, according to EE times – Asia dated March 3rd 2014. Their design can decrease the thickness of the semiconductor materials in thin film solar cells by an order of magnitude without compromising the capability of solar light absorption. 100 nm amorphous silicon layer is a requirement in state-of-the art solar cell design. 90% of incident solar beam can be absorbed using 10 nm thick layer of amorphous silicon. In Figure 32.0 is shown the design of the super absorbing solar cell. The core of the rectangular onion like configuration is made up of a semiconductor layer. This layer is coated by 3 layers of AR, anti-reflection coating that transmits light at a greater efficiency. The researchers first estimated the intrinsic light trapping efficiency of the semiconductor material. Then they devised a structure where the light absorption efficiency is equal to that of the intrinsic efficiency of the semiconductor material. A team of scientists from CIT, California Institute of Technology, Pasadena, CA, developed flexible solar cells that enhance the absorption of sunlight and hence the photovoltaic efficiency using fraction of expensive semi-conductor material. According to Harry Atwater, director of Caltech‟s Resnick Institute that focuses on sustainability research, “these solar cells have, for the first time, surpassed conventional light-trapping limit for absorbing materials” (Anon., 2010). They built their solar cells using silicon wires embedded within a transparent, flexible polymer film. Black paint can absorb light well, but may not generate electricity. These solar cells convert most photons absorbed into electrical energy. These wires have what is called a near perfect internal quantum efficiency. A high-quality solar cell is built for high absorption of light and good conversion of photons to electric current. These wires are painted with anti-reflective coating prior to being embedded into the transparent polymer. Each wire is about 1 mm in diameter and 30-100 mm in length. 2% of the material is silicon and 98% polymer. This brings down the cost of the solar cell. These solar cells are also flexible. Photovoltaic cells respond only to a narrow part of the sun‟s spectrum. In order to circumvent the lower efficiency on account of absorption of narrow part of the spectrum some developers prepare layered materials. The efficiency goes up, but the material becomes expensive as well.

Figure 37.0. Design of Super absorbing Solar Cell Design with Higher Light Absorption Efficiency.

190

Kal Renganathan Sharma

Cloudy days may lower the efficiency. Scientists at Ohio State University developed a doped polymer, oligothiopene. The resulting substance was responsive to wavelengths from 300 nm to 1000 nm. The spectrum of ultraviolet (UV) to near infra-red is spanned in this range. Traditional silicon cells, au contraire, function in the 600 nm to 900 nm range. This narrower range is between orange and red. The doped polymer both fluoresces and phosphoresces. Fluorescence emanates from electrons that get excited by incident rays of sunlight travel from a higher energy state and drop back to a lower energy state. Some light is emitted. The wavelength of the emitted light is in infrared range and not visible. This emitted light is seldom reused. Reuse of emitted light may improve the photovoltaic efficiency. These polymers are cheaper to produce compared with silicon. Hence they can be considered even if their photovoltaic efficiencies are lower. The relaxation time (Sharma, 2005) of these electrons during fluorescence of the doped polymer comes up from a few picoseconds in other solar materials to a few microseconds. A full spectrum solar cell that absorbs the full spectrum of sunlight from the near infrared and far ultraviolet to electric current can be prepared from an alloy of indium, gallium and nitrogen. This was made possible by a serendipitous observation by researchers at Lawrence Berkeley National Laboratory interacting with the crystal-growing research team at Cornell University and Japan‟s Ritsumeikan University. This observation was that the band gap of the semiconductor indium nitride is not 2 eV as previously thought, but instead is a much lower 0.7 eV. Solar cells made from this alloy would be the most efficient and can be lower in cost as well. The efficiency of photovoltaic cells is limited because of a number of factors. Some light energy that gets absorbed is rejected as waste heat. There exists a band gap in semi-conductor materials that the solar cells are made out of. Incoming photons of the right energy knock electrons loose and leave holes and migrate in the np junction to form an electric current. Photons with less energy than the band gap slip right through. Red light photons are not absorbed by high-band-gap semiconductors. Photons such as blue light photons that possess higher energy than the band gap are absorbed. Excess energy is dissipated as heat. There is a maximum efficiency limit for a solar cell made from a single material for converting light into electric power. This is about 30%. In practical applications it is about 25%. Stacks or layers of different materials are attempted in order to increase the efficiency. CIGS, CuInxGa1-xSe2 based photovoltaic thin films can deliver sunlight to electricity conversion performance greater than that of CdTe or silicon based thin films. Nanosolar (Contreras et al., 2006) has developed a process with high-throughput, high-yield printing of nanoparticles onto low-cost substrates and formation of solar cells. CIGS based PV thin films can deliver sunlight to electricity conversion efficiencies of 19.5% (Hegedus, 2006). NREL has certified the solar cell efficiency of 14% achieved by nanosolar with lower cost materials using nanotechnology. CIGS based thin films result in higher efficiencies. They are coated with a homogeneously mixed ink of nanoparticles using wet coating techniques. CIGS rollprinting technology developed by nanosolar uses a combination of high-speed, high-yield, nonvacuum, wet coating of nanoparticles onto low cost per unit area of metal foil substrates with RTP (Rapid Thermal Processing) techniques. Nanosolar‟s rapid thermal processing of nanoparticle-based coatings resulted in solar-cell efficiencies confirmed by NREL (National Renewable Energy Laboratory) to be 14.5% which amounts to a world record for any printable solar cell.

Mathematical Process Models

191

New discoveries such as the CNT (carbon nanotubes) are expected to increase the photovoltaic efficiency of solar cells to over 40%. For now, in areas, where the population density is sparse, and sunlight is abundantly available for most of the year the solar power plants may be profitable. This can be seen by a positive PW (present worth) value. They end up using large area. The solar panels are not protected from birds and other forms of dust that degrade their operations. Lenses and mirrors can be used to concentrate the sunlight and energy storage devices can store the energy in useful chemical forms such as batteries for use at night and during rainy days. Someday technology in solar energy generation will be as technically efficient as that of the combined steam and gas cycle power plant. By use of both the steam and gas turbines the Carnot cycle efficiency has been increased from 30% to 50%. After nearly two centuries since the discovery of the steam engine by James Watt, from the point of view of thermodynamics that defines the limits of machines, what are the issues involved in solar power plant. How can a paradigm shift be effected from: “there is no such thing as a free lunch”. Can solar energy be used to power a robot that can be used for transportation? What happens when this goes out of control? What are the hidden safety hazards in solar technology? Solar energy may be tapped in three different ways; (i) Solar Thermal Power, (ii) Photovoltaic Panels, and (iii) solar heaters. In method (i) steam is generated in large boilers to turn turbines and generate electricity comparable in capacity to coal-fired boiler-based power plants. Photovoltaic panels can be used to convert directly the solar irradiance (w/m2) into electricity. This technology is used to meet peak load needs and distributed power needs. Small power plants of up to 50 MW can be built using panels. The capacity of the recently commissioned De Soto Solar Power Generation Center in Florida is 24 MW and is less than 50 MW. Solar irradiance (w/m2) is used to heat water or air and can be used for residential heating purposes. Solar power plant technology can be used to produce base-load, large-scale power at low technical risk. These can replace coal-fired boiler based power plants. Heat energy storage devices have been invented that can provide for uninterrupted services such as during the night hours, rainy days, etc. Lunar power can also be tapped into. Heat storage elements used are concrete, molten salts and pressurized water. The capital solar plant costs and the plant utilization factor continue to affect the bottom line. Spain has five such plants under development and two that have been already commissioned. Investments have been made in several countries across the globe to advance the design of solar mirrors and lenses. These are used to gather the sunlight and focus on a fuel source to generate of electricity. Full scale commercial operations of such power plants with capacities in the range of 20-200 MW are expected by 2011. The cost per kWh is continuing to be the main concern. Optimization strategies that are being developed at this writing could lead to significant cost reductions and may be expected to improve the overall output efficiency. Cost effective storage devices are also expected. Present worth (pw) of solar power plants: A rapidly declining cost curve is seen in the photovoltaic cell technology (Sharma, 2011). The price of solar modules is expected to fall from around $2/W today to $1 in the near future. The price per watt installed is likely to fall from $6.0-$8.0 to $3.0 per watt. Dominant in the costs are power electronics and installation. With current incentives in the United States and European Union the cost of electricity generation using solar technology is in the range of the IGCC (Integrated Gasification Combined Cycle) power generation plants. In the state of California the cost of electricity

192

Kal Renganathan Sharma

from solar power plants is about 14-15 cents per kWh. This is against the 8-10 cents per kWh cost of electricity from IGCC power plants. By 2013 with some incentives from the federal government the cost of electricity from solar power plant is expected to fall to 10-12 cents per kWh. The per kWh cost is sensitive to the capital cost and the cost of materials of construction of the plant. Sequestration related carbon credits at $30 t-1 of CO2 can affect a per kWh reduction of 3 cents. Government carbon tax, cost of capital are sensitivity parameters on the bottom line of solar power plants. Worldwide, the solar thermal power capacity can grow at least 30% per year from 2010 to 2020 or 2030. 200 GW of new power plant capacity can be added each year. About 1-2 GW per year could be added in 2012. Solar technology is relatively simpler. The NREL, National Renewable Energy Laboratory, has identified potential for 6 TW (terra watts) of solar thermal power in the Southwestern US. The mass of the sun can be calculated from the period of the earth which is 365.25 days. The mass of the sun is ~ 2 x 1033 gm. The surface temperature of the sun is 5500 0C. Few things are known about the light to electricity conversion. The Carnot limit of efficiency of any man made machine is given by; carnot = (1 – T0/TH)

(3.428)

E = h = hc/

(3.429)

From the Planck‟s theorem,

The IV, current-voltage characteristics of a photodiode cannot be described using the Ohm‟s law of electricity. GaAs multijunction devices are the most efficient solar cells reaching a record high of 40.7%. 20-30 different semi-conductors are layered in series. AR, antireflection coatings can be used to minimize reflection from the top surface. The IV characteristics can be given by; I = I0(exp (qV/kBT) – 1) – qAG0(Lp+Ln)

(3.430)

33% of solar insolation is lost as waste heat. The temperature of operation of the photovoltaic module can be determined from energy balance as follows; IT = cIT + UL(Tc – Ta)

(3.431)

The physical significance of each term in Eq. (3.431) is as follows; The fraction of radiation incident on surface of solar cells is given by (IT), where  is the transmittance of any cover; The efficiency of conversion of incident radiation into current is given by (cIT); Radiation and convection losses from top and bottom is given by (U L(TC – Ta). Ta is the ambient temperature. The temperature of the solar cell, Tc can be obtained by rearranging Eq. (3.431); Tc = Ta +(IT/UL)(1- c/)

(3.432)

Mathematical Process Models

193

It can be seen that as there exists an optimal working temperature of the solar cell. As the amount of concentrated solar flux is increased the absorption efficiency first increases and then decreases. As the working temperature increases the heat losses increase by radiation and convection. The Carnot thermal efficiency increases. The optimal temperature of operation for maximum efficiency can be arrived at. The heat losses vary as T4 and the efficiency increases linearly with the working temperature. /T = 0 and the working temperature solved for. The UL in Eq. (3.432), the overall transfer coefficient will vary with temperature.

J. ENGINEERING ANALYSIS 3.18. AMPACITY RISKS IN PCB INTERCONNECTIONS 3.18.1. Introduction IHS, Information handling systems are used to process, compile, store and communicate data for business personnel or for similar purposes. The ampacity risks on PCB, printed circuit board interconnections need be better understood. Depending on the information handled the IHS may differ from one application to the other. They may differ as to what information is handled, how information is processed, stored, communicated and how soon and efficiently the information may be processes, stored or communicated. IHS may be configured for general use or for specific use such as passenger reservation in the airlines, railways, credit card transactions, global communications, enterprise data storage. Variety of hardware and software components may be configured to perform different tasks such as storage, communication, processing. It may include computer, data storage and networking systems. During the design and manufacture of HIS one salient consideration is the detection of areas of the system or circuit that are prone to certain risks. Corrective steps have to be taken in order to minimize these risks. One example is the ampacity risks on circuit boards. The circuit or interconnect may go back to the drawing board stage for rerouting. “Ampacity‟ is defined [Murugam et al., 2009] as the current in amperes that a conductor can carry continuously under the conditions of use without exceeding the temperature rating or fuse point. The simulation, analysis, validation to laboratory data the DC, direct current and short duration AC, alternating current transient pulse effects on the ampacity risks on PCB interconnectivity is needed. This would result in assessment of ampacity risks on PCB interconnections and improvement of the reliability of the operation of IHS. One critical aspect to this aspect is the computation of heat conduction at short times where non-Fourier effects can be expected. The damped wave conduction equation may be applicable here. The damped wave conduction and relaxation equation was sought over Fourier‟s law of heat conduction for eight reasons by Sharma [2005]. The damped wave conduction and relaxation equation was originally suggested by Maxwell [1867], and postulated independently by Cattaneo [1948, 1958] and Vernotte [1958]. The damped wave conduction and relaxation equation in one dimension across constant area may be written as follows;

194

Kal Renganathan Sharma

qx

kA

T x

r

A

qx

(3.433)

t

Where qx is the heat transfer rate in x direction in (watts, w), A is the cross-sectional area across which the heat conduction occurs in (m2), k is the thermal conductivity of the material in (w.m-1.K-1), r is the relaxation time (s). Reviews of the use of this equation have been presented by Joseph and Preziosi [1989] and Ozisik and Tzou [1996]. Extensive theoretical treatment of the equation have been reported by Tzou [1996] and Sharma [2005]. Experimental measurement of relaxation times has been reported by Mitra et al. [1995] recently for biological materials. Taitel [1973] found a overshoot in his transient temperature solution for a finite slab subject to constant wall temperature boundary condition. Bai and Lavine [1995] was concerned about Eq. (3.433) violating the second law of thermodynamics. Zanchini [14], Barletta and Zanchini [2003], calculate a entropy production term and are concerned of a violation of Clausius‟ inequality. Al Nimir et al. [2000] discuss an “overshoot” and equilibrium entropy production. Haji Sheik et al. [2002] point out some anomalies in Eq. (3.433). Tzou [1996] has found Eq. (1) to be admissible within the framework of second law of thermodynamics. Sharma [2003, 2006,2007,2005] have presented closed form analytical solutions for different geometries that are within the bounds of second law of thermodynamics. He has derived the damped wave equation by accounting for acceleration of the molecules in the Stokes-Einstein formulation. He has proved that when physically reasonable initial condition is used the overshoot disappears. Final condition in time was used and bounded solutions without violation of second law of thermodynamics were presented. Antaki [1998] has discussed some analytical solutions for convective boundary condition. In this study the damped wave conduction equation is studied in 3D, three dimensions and analytical solution derived. The solutions are in the form of Bessel composite function of the third order. The derived spatio-temporal temperature profiles can be used to gauge the ampacity risks in PCB interconnections.

3.18.2. Theory The „defect‟region is modeled as a spherical shell with radius R0. An energy balance on the spherical shell at a distance r from the origin can be written and when combined with the damped wave conduction and relaxation equation can be written as; 2

T

r

t

2

T t

r

2

r

r2

T r

(3.434)

Let;

u

T T0 Ts

T0

;

t r

;X

r

(3.435) r

195

Mathematical Process Models

Eq. (3.434) is made dimensionless by using the variables defined in Eq. (3.435). Eq. (3.434) becomes; 2

u

1

u

2

X

2

X

X2

u X

(3.436)

The time and space conditions can be written as;  = 0, u = 0

(3.437)

 = , u = 1

(3.438)

 > 0, X= XR0, u = 1

(3.439)

X = , u = 0

(3.440)

Consider the substitution, V = u/X. Eq. (4) becomes, 2

V

2V

V

2

X

2

4 X

2

V X

V

X2

(3.441)

The damping term can be removed by a u = wexp(-n) substitution. As shown in the preceding sections for n = ½, Eq. (9) becomes; 2

W

2W

W 4

2

X

2

4 X

2

W X

W

X2

(3.442)

Let  = 2 – X2 The term 2W/X2 can be neglected for large X. W is small for large r as u = Wexp(-/2)/r. For large X Eq. (10) can be modified as follows; Now,

4 X

W X

2

W

4

12

2

2 2

W 2

3

8

W

W

W

W 4 W 16

(3.443)

0

(3.444)

0

(3.445)

196

Kal Renganathan Sharma Comparing Eq. (13) with the generalized Bessel equation the solution is; a = 3; b = 0; c = 0; d = -1/16; s = ½ The order p of the solution is then p = 2 sqrt(1) = 2 2

I2 W

c1

X2

2

K2

2 2

2

c2

X2

X2

2

X2

(3.446)

c2 can be seen to be zero as W is finite and not infinitely large at  = 0. 2

I2 V

c1e

X2

2

2 2

(3.447)

X2

An approximate solution can be obtaining by eliminating c1 between the above equation and the equation from the boundary condition. 2

I2

1 X R0

c1e

X R2 0

2

2 2

(3.448)

X R2 0

Thus for  > X 2

V

1 X R0

2

X R2 0

2

X2

I2

X2

2

(3.449) 2

I2

X R2 0

2

For X > ,

u

X X R0

2

X2

X R2 0

J2

2

X2 2 2

I2

2

X R2 0

2

(3.450)

197

Mathematical Process Models

On examining Eq. (3.450) it can be seen that the Bessel function of the second order and first kind will go to zero at some value of . The first root of the Bessel function occurs when ½(X2 - 2 )1/2 = 5.1356

(3.451)

X2 - 2 = 105.498

(3.452)

Or

When an exterior point in the infinite sphere is considered a lag time can be calculated prior to which there is no heat transfer to that point. After the lag time there exists two regimes. One is described by Eq. (3.450) and the third regime is described by Eq.(3.449). Thus, lag = sqrt( Xp2 - 105.498) All the three dimensions of the spherical coordinates are considered. The V = u/r substitution is used and the spatio-temporal temperature in the infinite sphere is derived as follows; The governing equation for the temperature is obtained when the energy balance equation and the constitutive damped wave diffusion and relaxation equation are combined. The equation is made dimensionless by using the substitutions in Eq. (3.435). Then the governing equation in three dimensions in spherical coordinates can be written as; 2

u

u

2

2 X

2

u X

u

X

2

1 X

2

2

1

u 2

X

Xp = 11 (tou > X)"

2

2

Sin 2

u

Cot

2

2

X

u

X > tou"

0.5 0.45

Concentration (C - Co)/(Cs - Co)

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0

10

20

30

40

Dimensionless Time

Figure 38.0. Three Regimes of Dimensionless Temperature at a Exterior Point from the Defect.

(3.453)

198

Kal Renganathan Sharma Consider the substitution, V = u/X. Eq. (21),

2

V

2V

V

2

X

4 X

2

1 2V X X2

V X

2

1 X

2

1

V

2

2

X

2

1

V

Sin 2

2

X

2

V

Cot

2

X2

2

V

(3.454)

The damping term can be removed by a V = wexp(-n) substitution. As shown in the preceding sections for n = ½, Eq. (22) becomes. 2

W 2

2W

W 4

X

4 X

W X

W 4

2W

2

2

X

2

1

W 2

X

2

2

1

W 2

X

2

1

W

Sin 2

2

X

2

W

Cot

2

2

2

X

W

(3.455)

For small , 2

W 2

X

2

4 X

2

W X

X

2

1

W 2

X

2

2

1

W 2

X

2

1

W

Sin 2

2

X

W 2

(3.456)

Let  = X, 2

1 X

2

W

2

W

2

(3.457)

2

 = XSin, 2

1

Then,

2

X

2

W

Sin 2

W

2

(3.458)

2

Eq. (24 )then becomes for large X, 2

W 2

W 4

4 X

2

W X

2

W

X

W

2

2

W

2

1

W

2

(3.459)

Consider the transformation,  = ( 2 – X2 - 2 - 2 ) As shown above in Eq. (3.445) the derivatives in Eq. (3.459) in four independent variables become converted into 1 independent variable,  and Eq. (3.459) becomes; 2

W

4

2

2

Or,

2

W 2

18

9 2

W

W 4

W

W 16

0

(3.460)

0

Comparing Eq. (29) with the generalized Bessel equation the solution is;

(3.461)

199

Mathematical Process Models a = 9/2; b = 0; c = 0; d = -1/16; s = ½ The order p of the solution is then p = 7/2 2

X2

I 7/2 W

c1

2

2

2

I

2 2

X2

2

7/2

2

2

2

c2

2

X2

2

X2

2

2

(3.462)

c2 can be seen to be zero as W is finite and not infinitely large at  = 0. An approximate solution can be obtaining by eliminating c1 between the above equation and the equation from the boundary condition. The equation from the boundary condition can be written as; 2

I 7/2

1

2

2c 1

X R0 e

X R2 0

2

(3.463)

X R2 0

Dividing Eq. (30) by Eq. (31) 2

u

2

X X R0

2

X R2 0

X2

2

X2

I 7/2

2

2

2

(3.464)

2

2

I 7/2

X R2 0

2

For small X,

u

2

X X R0

2

J 7/2

X R2 0

X2

2

X2

2

2 2

2

(3.465)

2

2

I 7/2

X R2 0

2

In the creeping heat transfer limit Eq.(3.459) can be approximated as; 2

W 2

W 4

4 X

W X

2

W

X

2

2

W 2

2

W 2

(3.466)

After the transformation the PDE with 4 variables is converted to a Bessel equation in 1 variable:

200

Kal Renganathan Sharma 2

W

2

W

4

2

W 16

0

(3.467)

The order of the Bessel solution for Eq. (3.467) can be calculated by comparing Eq. (3.467) with the generalized Bessel equation and the solution is; a = 4; b = 0; c = 0; d = -1/16; s = ½. The order p of the solution is then p = 3 2

X2

I3 W

c1

2

2

2

2 2

X2

X2

K3 2

2

2

c2

2

2

2

X2

2

2

(3.468)

c2 can be seen to be zero as W is finite and not infinitely large at  = 0. An approximate solution can be obtaining by eliminating c1 between the above equation and the equation from the boundary condition. The equation from the boundary condition can be written as; 2

I3

1

2c 1

X R0 e

X R2 0

2 2

(3.469)

X R2 0

Dividing Eq. (36) by Eq. (37), 2

u

X X R0

2 2

X2

I3

X R2 0 2

X2

2

2

2

(3.470)

2

2

I3

X R2 0

2

For small X,

u

X X R0

2 2

X2

J3

X R2 0 2

X2

2

2

2

(3.471)

2

2

I3

In the limit of zero radius of the defect;

2

X R2 0

2

201

Mathematical Process Models 2

u

X X R0

I3

2 2

X2

2

X2

2

2

2

(3.472)

2

I3

2

For small X,

u

X X R0

J3

2 2

X2

2

X2

2

2

2

2

(3.473)

2

I3

2

The solution is in terms of a Bessel composite function of the third order and first kind for small X and a modified Bessel composite function of the third order and first kind for times greater than X. The first root of the Bessel function of the third order was calculated by using 17 terms of the series expansion of the Bessel function in a Pentium IV microprocessor using a Microsoft Spreadsheet up to 4 decimal places. The root was found to be 6.3802.

Or

½(X2 + 2 + 2 - 2 )1/2 = 6.3802

(3.474)

X2 + 2 + 2 - 2 = 162.828

(3.475)

When an exterior point in the infinite sphere is considered a lag time can be calculated prior to which there is no heat transfer to that point. After the lag time there exists two regimes. One is described by Eq. (3.473) and the third regime is described by Eq.(3.472). Thus,

lag

X p2

2 p

2 p

162.828

(3.476)

K. MOLECULAR BASIS FOR CONSTITUTIVE LAWS 3.19. NON-FOURIER CONDUCTION EQUATION A CAPITE AD CALCEM TEMPERATURE Not all materials behave in the same manner. Some materials do not behave in the same manner over all the conditions over which they are expected to be applied. For some applications when the experimental observations cannot be explained using theoretical models and for materials that have not been measured before constitutive equations that may be more relevant may have to be derived. For example Newton‟s law of viscosity may not be the best constitutive law for systems such as tomato puree or suspensions of gas and solid. Fourier‟s law of heat conduction may not be the best possible relation between the heat flux

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and temperature gradient for systems where the transfer of heat is transient. Phenomena were short time scales or short space scales or very high heat flux or very high temperature gradient have been found not be adequately represented by Fourier‟s law of heat conduction.

3.19.1. Introduction Fourier‟s law of heat conduction [1822] and their analogs during; (i) mass diffusion Fick‟s law of mass diffusion [1855]; (ii) viscous momentum transfer - Newton‟s law of Viscosity [1687]; (iii) electrical conduction - Ohm‟s law of electricity [1817]; (iv) convection – Netwon‟s law of cooling [1687] are laws derived used in engineering practice for nearly two centuries. These laws have been developed from empirical observations atsteady state. Certain times these laws are not sufficient to predict the experimental observations. There are some reasons to seek an alternate equation to Fourier‟s law of heat conduction (Sharma [2007]); i) the microscopic theory of reversibility of Onsager [1931] is violated; ii) neglects the time needed for the acceleration of heat flow by free electrons(Sharma, [2006]); iii) Singularities were found in a number of important industrial applications of the transient representation of temperature, concentration, velocity [2005]; iv) development of Fourier‟s law was from observations at steady state; v) Over prediction of theory to experiment have been found in a number of industrial applications [Renganathan, K., 1990, Sharma, 2003, Sharma, 2009]; vi) Landau and Lifshitz observed the contradiction of the infinite speed of propagation of heat with Einstein‟s light speed barrier [1987]; vii) Fourier‟s law breaks down at the Casimir limit [1938]. A non-Fourier heat conduction equation was postulated independently by Cattaneo [1948] and Vernotte [1958]. This equation was first suggested by Maxwell from considerations of kinetic theory of gases [1867]. Joseph and Preziosi[1989], Ozisik and Tzou [1996 ] have reviewed the work done investigating this equation. Sharma [2005, 2008a, 2008b, 2009, 2010] discussed several analytical solutions for transient temperature obtained by use of this equation along with the energy balance equation. He considered a semi-infinite medium at an initial temperature of T0 subject to a constant surface temperature boundary condition for times greater than zero. The hyperbolic PDE that forms the governing equation of heat conduction was solved for by a new method called relativistic transformation of coordinates. The hyperbolic PDE is multiplied by e/2and transformed into another PDE in wave temperature. This PDE is converted to an ODE by the transformation variable that is spatiotemporal symmetric. The resulting ODE is seen to be a generalized Bessel differential equation. The solution from this approach is within 12% of the exact solution obtained by Baumeister and Hamill using the method of Laplace transforms. There are no singularities in the solution. There are three regimes to the solution: a) inertial regime; b) regime characterized by Bessel composite function of the zeroth order and first kind and; c) a regime characterized by modified Bessel composite function of the zeroth order and first kind. Expressions for penetration length and

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203

inertial lag time are developed. The comparison between the solution from the method of relativistic transformation of coordinates and the method of Laplace transforms was made by use of Chebyshev polynomial approximation and numerical integration. In a similar manner, the exact solution to the hyperbolic PDE is solved for by the method of relativistic transformation of coordinated for the infinite cylindrical and infinite spherical medium. For the case of heating a finite slab the Taitel paradox problem is revisited. Taitel [1972] found that when the hyperbolic PDE was solved for the interior temperature in the slab was found to exceed the wall temperature of the slab. This is in violation of the second law of thermodynamics [Bai, 1995, Zanchini, 1999, Barletta and Zanchini, 2003]. By use of the final condition in time at steady state the wave temperature was found to be become zero at steady state. This condition when mathematically posed as the 4th condition for the second order PDE leads to well bounded solutions within the bounds of second law of thermodynamics. For systems with large relaxation times, i.e., 2 r

a2

, subcritical damped oscillations can be seen in the temperature. In a

similar manner the transient temperature for a finite sphere and finite cylinder are also derived. Nanoscale effects in time domain [Sharma, 2009] is as important, if not more as nanoscale phenomena in space domain in a number of applications. When the advances in microprocessor speed is approaching the limits of physical laws on gate width and miniaturization, there is incentive to re-examine the physical laws at a level of scrutiny never done before. In this study, the acceleration of the electron is accounted for in the formulation of free electron theory. The non-Fourier conduction equation that results is evaluated for use in prediction of transient temperature in a finite slab.

3.19.2. Theoretical Development 3.19.2.1. Free Electron Theory Ohm‟s law of electric conduction and Fourier‟s law of heat conduction can be derived from the free electron theory. The range of electrical resistivity of materials vary by 30 orders of magnitude. The range of thermal conductivity of materials vary by 5 order of magnitude. No one theory can predict the thermal and electrical conductivity of materials. The free electron model is used. The outermost electrons of the atom are assumed to take part in the conduction process. These electrons are not bound to the atom but are free to move through the entire solid. These electrons have been called free electron cloud, free electron gas, or the Fermi gas. Potential field due to the ion cores is assumed to be uniform throughout the solid. The free electrons possess the same free energy everywhere in the solid. Because of the electrostatic attraction between a free electron and the ion core this potential energy will be a finite negative value. Only energy differences are important and the constant potential can be taken to be zero. Then the only energy that has to be considered is the kinetic energy. The kinetic energy is substantially lower than that of the bound electrons in an isolated atom as the field of motion

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Kal Renganathan Sharma

for the free electron is considerably enlarged in the solid as compared to the field around an isolated atom. The free electron theory can be used to better understand electrical conduction. By Lorenz analogy [1892] the heat conduction can also be predicted in a similar manner. The independent electron assumption was developed by Drude in [1905]. Some of the assumptions in the free electron theory claim that electrons are responsible for all of the conduction. The electrons behave like an ideal gas, occupy negligible volume, undergo collisions, and are perfectly elastic. Electrons are free to move in a constrained flat bottom well. Electron distribution of energy is a continuum.

3.19.3. Derivation of Alternate Non-Fourier Conduction Equation Let the number of electrons per unit volume be given by n. This is the electron density in the material during the conduction process. From the Boltzmann equipartition energy theorem [1876] the microscopic temperature, T in (0K) can be given by; (3.477) Where kBis the Boltzmann constant, m is the mass of the electron in (kg) and ve is the velocity of the free electron in (m.s-1). The heat flux, qz in (W.m-2) is the rate of energy transfer across a cross-sectional area across the heat conduction path. The heat flux in terms of the properties of the moving electron can be written as follows; (3.477a) Eq. (3.477a) is the product of Eq.(3.4777) and nve. Currently, the rate of energy transfer across is estimated this way. This is sort of momentum flux of the electrons as well. The applied temperature gradient acts as a driving force for the motion of the electron. Let the collision time of the electron be given by τ in units of seconds. The applied force and frictional force can be written from free electron theory as follows; (3.478)

(3.479) In this study, the acceleration of the electron is also taken into account. The derivations in the literature for the Fourier‟s law of heat conduction from free electron theory assume that the electron has attained a steady drift velocity. This assumption may be reasonable at steady state. But in transient applications, immediately after the application of the temperature gradient the electron would be in acceleration. This phase of the motion of the electron has not been well studied in the literature. In this study this is taken into account. The Newton‟s second law of motion may be applied to the moving electron(s) as follows;

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205

(3.480) Eliminating the velocity of the electron in Eq. (3.480) by using Eq. (3.477) leads to the damped wave conduction and relaxation equation as shown earlier (Sharma [2006]). Here Eq. (3.477) is differentiated with respect to time as;

(3.481)

Eliminating the acceleration of the electron term between Eq. (3.481) and Eq. (3.480) and plugging Eq. (3.477a) for heat flux in place of the velocity of the electron; (3.482)

Multiplying thoughout Eq. (7)

by Eq. (7) becomes;

(3.483) The expression for thermal conductivity of the material from derivation at steady state can be seen to be; (3.484) The coefficient to the rate of temperature variation with time term can be simplified as follows. The rate of temperature variation with time can henceforth be called as ballistic term. This term is unique to the acceleration of the electron and is expected to become significant in transient applications. The term” ballistic” can be used to denote acceleration effects. (3.485) In an earlier study (Sharma [2006]] the collision time of the electron was shown to be 3rwhere ris the relaxation time of materials as defined by Cattaneo [1948] and Vernotte [1958]. The velocity of heat can be written as follows;

(3.486)

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Kal Renganathan Sharma

In terms of properties of moving electrons Eq. (3.486) can be rewritten as follows. Heat capacity at constant volume, Cv by definition is the energy needed to raise one mole of a substance by 1 0K. For one molecule this would be 1.5kB as a corollary of the equi-partition energy theorem (Eq. (3.477)). The electron cloud is assumed to be ideal gas. It can be shown that for an ideal gas (Sharma [2006]); Cp

Cv

(3.487)

R

Eq. (3.487) is for one mole of the material. For one molecule of the material, Eq. (3.487) can be written as follows; Cp

Cp

mn

(3.488)

2.5kB

Cv kB

5kB n 2mn

(3.489)

Plugging Eq. (3.489), Eq. (9) and Eq. (3.477) into Eq. (3.486), Eq. (3.486) becomes;

vh

54kB2 nT 20mkB n

54kBT 20m

18 ve 20

0.95ve

(3.490)

Thus it can be seen from Eq. (3.490) that the velocity of the electron, ve is approximately equal to the velocity of heat, vh. The velocity of heat is given in terms of the thermo physical properties of materials as given in Eq. (3.486). Eq. (3.485) may be simplified as;

9kB2 nT 4mve

k ve

Cpk

Cp

r

r

(3.491)

Plugging Eq. (3.491) into Eq. (3.483), Eq. (3.483) becomes; qz

k

T z

Cp

r

T t

(3.492)

Eq. (3.491) is an alternate non-Fourier heat conduction equation compared with that postulated by Cattaneo [1948] and Vernotte [1958]. Eq. (3.492) for heat flux comprises of two parts: (i) one part that is the “Fourier part” that gives spatial gradient of temperature, both transient and steady states and; (ii) second part that is the “ballistic part” that gives the contribution of acceleration motion effects of the electron. Another way of viewing the “ballistic term” is in terms of accumulation effects in time of energy at the interfacial area through which the heat is travelling. In the kinetic representation of temperature in terms of square of the velocity of the molecules as given by Eq. (3.477) the heat flux across an interfacial area is defined as the energy of the molecules leaving the

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207

surface less the energy of the molecules that enters the surface per unit time. Accumulation of energy during the said time at the interfacial surface was neglected in the Fourier representation of heat flux with spatial temperature gradient [Sharma, 2005]. Here the accumulation of energy at the surface is also taken into account in the heat flux expression. This added term can also be viewed as a place to capture the acceleration effects of the moving electron under an applied temperature gradient/force. The postulated non-Fourier equation of Cattaneo [1948] and Vernotte [1958] is restated as; q

z

k

qz

T z

r

t

(3.493)

3.19.4. Solution of Transient Temperature in a Finite Slab Using the Alternate Ballistic Transport Equation Consider a semi-infinite medium with homogenous thermo physical properties k, α as shown in Figure 39.0. The initial temperature of the semi-infinite medium is T0 for times less than 0. At time 0 the surface of the semi-infinite medium are raised to a higher temperature Ts(Ts> T0) and maintained constant at Ts for all times t > 0. The initial time condition and the boundary conditions can be written as follows; t = 0, 0 < x < , T = T0

(3.494)

t > 0, x = 0, T = Ts

(3.495)

t > 0, x = , T = T0

(3.496)

The energy balance equation in 1D, one dimension can be written as follows; qx x

Cp

T t

(3.497)

Figure 39.0. Semi-Infinite Medium Subject to CWT, Constant Wall Temperature Boundary Condition.

208

Kal Renganathan Sharma

The governing equation for temperature in 1D is obtained by combining Eq. (17) and Eq. (22) and can be written as follows; 2

T t

2

T

T x t

r

2

x

(3.498)

Eq. (3.498) may be made dimensionless by the following substitutions;

Z

x

t

;

;u

r

r

T

T0

Ts

T0

(3.499)

Eq. (3.499) becomes; 2

u

u

2

2

Z

Z

u

(3.500)

The dimensionless distance Z can be rewritten as; x r

Z

(3.501)

vh r

The physical significance of dimensionless distance can be seen to be the ratio of the relaxation speed calculated for the disturbance to be seen at the considered point to the speed of composite heat transfer in the medium due to both Fourier diffusive and ballistic/relaxation mechanisms. The solution to Eq. (3.500) may be obtained by the method of Laplace transforms. The Laplace transformed Eq. (3.500) may be written as follows;

su ( s )

d 2u ( s )

s

dZ 2

du ( s ) dZ

(3.502)

The solution to the second order ODE given by Eq. (27) can be written as follows;

u (s )

e

sZ 2

sZ 1

c1e

1 s

sZ 1

c2 e

1 s

(3.503)

From the undisturbed temperature, at ad infinitum, as given by BC in Eq. (21), c1 can be seen to be zero.

209

Mathematical Process Models

u (s )

sZ 2

e

1 s

sZ 1

c2 e

(3.504)

c2 is obtained from the constant wall temperature BC as given in Eq. (20) and seen to be given by (1/s). The solution for the dimensionless temperature in Laplace domain may be written as follows; sZ 2

e

u (s )

1 s

sZ 1

e

s

(3.505)

The lag property in Laplace transforms in invoked as follows;

Lf t

s

e

F (s )

(3.505a)

The binomial infinite series expansion is written for the power ponentiation as follows;

1

1 s

1 s

1

1 8s

2

1

5

16s

3

128s 4

.....

(3.506)

Plugging Eq. (3.506) into Eq. (3.505), Eq. (3.506) becomes;

u (s)

e

sZ 2

sZ

e

s

Z Z 2 8s

Z ... 16 s 2

(3.507)

From the inversion tables of Laplace transforms in [Mickley et al., 1957];

I 0 2 kt

L1

k es

s

(3.508)

By Eq. (33); Z

I0

Z 2

L

1

e 8s s

(3.509)

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Kal Renganathan Sharma

L

1

e

Z 8s 2

L1 1

Z

Z2

Z3

8s 2

16s 4

48s

Z 8

... 6

Z2 3 96

Z3 5 5760

.... (3.510)

By invocation of the convolution property the analytical solution may be obtained to varying degrees of accuracy. An approximate solution by truncation of the 4th and higher order terms in Eq. (3.510) leads to the solution for transient temperature. Upon using the lag property as shown in Eq. (3.505a) the solution for transient temperature is given as follows;

u

e

Z 2I

3Z 2 4

Z 2

0

(3.511)

The solution to the transient temperature to with everything else such as the semi-infinite medium used, boundary and time conditions used remaining the same except that the Fourier model was used can be written as (Sharma, 2010) as follows; Z

u 1 erf

(3.512)

4

The solution to the transient temperature to with everything else such as the semi-infinite medium used, boundary and time conditions used remaining the same except that the damped wave conduction and relaxation model was used can be written as (Sharma, 2010]) as follows; 2

X2

I0

0.5

4

u I0

(3.513)

2

Eq. (3.513) is applicable for conditions where  > X. For conditions where  < X the dimensionless temperature is given by;

J0

2 0.5

X2

4

u I0

(3.514)

2

For conditions where  = X, the dimensionless temperature is given by;

u

e

X 2

e2

(3.515)

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Mathematical Process Models

Figure 40.0. Comparison of Transient Temperature from Fourier Model, Damped Wave Transport Model and Ballistic Transport Model.

The predictions for transient temperature in a semi-infinite medium subject to CWT, constant wall temperature boundary condition from the; (i) Fourier model; (ii) Damped Wave Conduction and Relaxation Model and; (iii) Ballistic/Acceleration Model are plotted in Figure 40.0 side-by-side for the same set of parameters at  = 0.5. The theoretical predictions from ballistic model as given by (3.514) were found to be closer in numerical value to the theoretical prediction from Fourier model as given by Eq. (37) compared with that of the theoretical prediction from damped wave conduction and relaxation model given by Eq. (38). The transient temperature can be seen to be convex at short distances and change to concave at later distances in the damped wave conduction and relaxation model. The transient temperature from the ballistic model is also convex at shorter distances and changes to concave later as a function of distance. The Fourier parabolic model for transient temperature is concave as a function of distance. Eq. (3.511) is valid in the open interval,  > 1.5 for values of space and time when  < 1.5Z;

u

e

Z 2

3Z 2 4

J0

Z 2

(3.516)

When  = 1.5Z, the expression for transient temperature can be written as;

u

e

Z 2

e

3

(3.517)

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Kal Renganathan Sharma

The first zero of the Bessel function J0(y) occurs at y = 2.4048. Thus the penetration distance Zpen can be estimated for a given instant of time,  as follows; 2.4048

3Z 2pen 4

Z pen

2

(3.518)

3.19.5. Conclusions An alternate non-Fourier heat conduction equation is derived from consideration of translation motion of spin less electron under a driving force due to an applied temperature gradient. This equation is a capite ad calcem, temperature. Elimination of the rate of change of velocity with respect to time leads to a non-Fourier heat conduction equation with an accumulation of temperature or ballistic term in it. The new constitutive heat conduction equation is combined with the energy balance equation in 1 dimension. The governing equation for transient temperature a partial differential equation (Eq. (3.498)) is solved for by the method of Laplace transforms. The problem considered is the semi-infinite medium with constant thermo physical properties with constant wall temperature boundary condition. A closed form analytical expression for the transient temperature was obtained (Eq. (3.511)) after truncation of higher order terms in the infinite binomial series. This solution is compared with that obtained using the parabolic Fourier model and the damped wave model as presented in an earlier study. The predictions of Eq. (3.511) and Eq. (3.516) are closer to the Fourier model; further work on Eq. (3.492) is underway.

Nomenclature A c CP Cv G H l m P S T U v V

Helmholz free energy (J/mole) Speed of Sound (m/s) Heat Capacity at Constant Pressure (J/kg/K) Heat Capacity at Constant Volume (J/Kg/K) Gibbs Free Energy (J/mole) Specific Enthalpy (J/mole) length of the box (m) mass of molecule (kg) Pressure (Nm-2) Specific Entropy (J/mole/K) Temperature (K) Internal Energy (J/mole) velocity of molecule (m/s) Molar Volume (m3/mole)

Mathematical Process Models

213

Greek  P s  T s 

Linear coefficient of thermal expansion (K-1) Isobaric Volume Expansivity (K-1) Isentropic Volume Expansivity (K-1) Elongational Strain Isothermal Compressibility (ms2/mole) Isentropic Compressibility (ms2/mole) molar density (mole/m3)

Subscripts 0 f T P S

Intial state Final state isothermal isobaric isentropic

3.20. SUMMARY Mathematical models can be used where pilot plant data is not available for scale-up. Model development can lead to better understanding of the process. It can lead to better project planning and decrease pollution to the environment. Dynamic process models can be used to train operators, to design processes, to perform safety analysis, offer better process control strategies, in project trouble shooting and in globalization. Process models can be classified as: (i) Empirical Models; (ii) Semi-Empirical Models; (iii) Mechanistic Models; (iv) Models from shell balance and equations of energy, continuity, mass, momentum and charge; (v) supercomputer based models; (vi) computer simulations; (vii) mesoscopic models and; (viii) Monte Carlo trials. Biodiesel is a renewable fuel that can be made from vegetable oil and waste restaurant greases by catalysed transesterification reactions. Over 5 billion gallons of biodiesel was produced in 2010. The European Union and United States are seeing the sigmoidal portion of the growth curve in biodiesel production.. Economic analysis such as profitability and annualized worth (AW) of a biodiesel plant in Taiwan is presented. With the revenue from glycerine by-product recovery and with lower raw material costs, biodiesel may be profitable especially during days of higher gasoline prices. Multiple reactions of the consecutivecompetive type may be used to model the methonolysis of trigylcerides. The reaction rate constant ratios and residence time in the reactor are important parameters in determining higher selectivity of FAME, fatty acid methyl ester product yield over glycerol by-product production. Illustrations of higher FAME yield, higher glycerol yield and cross-over from

214

Kal Renganathan Sharma

FAME to glycerol are shown for some values of reaction rate constant ratios  and . Reaction scheme from triglycerides, to diglycerides to monoglycerides to glycerol along with formation of FAME in each step by addition of methanol and catalyst are shown. Product distribution curves are presented in Figure 2.0 – 5.0 for different values or reaction rate constant ratios. Projectile motion is an interesting topic for discussion in introductory physics courses. Air resistance effects on projectile motion was studied. A 5 constant expression for variation of CD vs. Reynolds‟ number was used and the variation of air resistance with velocity was taken into account. The air resistance force was the most (about 11.9%) in the initial and final stages of the projectile. Closer to the maximum height the drag force is near zero. The projectile motion with the air resistance was assymetrical. The effect of drag coefficient variation with velocity was found as shown in Figure 4.0. Greater range was found for the case of launch angle of 380 for the example considered compared with launch angles of 450 and 300 with air resistance. Maximum range of projectile without the air resistance can be expected at a launch angle of 450. The use of Eq. (5) may be better than the use of a constant value for CD during the entire trajectory. Studies are underway to include the Magnus effect when there is circulation around the object. The case of geometric particles are also being studied. Trajectory of sedimenting particles can be simulated using desktop computers and numerical integration. Viscous effects of the fluid can be accounted for the in the design considerations of rectangular sedimentation tanks. Drag correlations with Reynolds number for spherical and geometric particles uder laminar to transition to turbulent flow were used in the computer simulations. By writing the acceleration term in terms of velocity the time variabe is rendered implicit and an order reduction is achieved in governing equation. Fifth order Runge Kutta method was used with success. Iron and sand particles were simulated. Trajectories are shown as Figures. Sedimentation tanks needs to be made deeper to allow for accelerating phase. Overflow rate is no longer independent of the tank height. Trade-offs between pressure drop (pumping costs), separation efficiency achived and throughput (capital costs) can be seen. Total cost can be optimized. Shape of the trajectory is plateau in the initial stages and a sharper fall later. Trajectory of accelerating particles are found to be curvilinear as opposed to linear for settled particles. Expression for terminal settling velocity of the partilce in horizontal direction is now given by Eq. (24). Acceleration zone was found to be shorter for geometric particles. Sharma and Turton [1998] presented mesoscopic correlations between heat transfer coefficients between gas-solid fludized beds and immersed surfaces using a de no vo dimensionless group called frequency number. This is an example in semi-empirical models. The correlations were scaled from theories such as Mickley and Fairbanks packet theory [1955] and the theories that modify the packet theory. The experimental data was acquired in a 5 cm fluidized bed over a wide range of powder types and fluidization regimes. The correlations gave reasonable correspondence to experimental data given there was no empirical fit or use of adjustable parameters. Mechanistic models can be used to evaluate different mechanisms such as conduction or convection in industrially important systems such as CFB, circulating fluidized beds. The Peclect number is used to gauge the convective contributions. As the fluid flows away from the surface both convection and conduction act in series and as the fluid flows toward the surface the convection and conduction act with opposite effects. This can be used to obtain

Mathematical Process Models

215

the convective and conductive contributions by adding and subtracting the heat transfer coefficients at the top and bottom of the tube. The convective contribution is given by Eq. (2.23) and conductive contribution is given by Eq. (2.22). The same analysis was repeated for accounting for damped wave conduction and relaxation effects. The method of Laplace transforms was used in order to obtain the solution. A binomial series expansion in the Laplace domain was used. Term by term inversion from Laplace domain to time domain for 13 terms are given in Table 2.1. The model was compared with representative experimental data available in the literature. The use of mathematical model in operations such as evaporative cooler was shown. Humdification and heat exchange operations were modeled. The temperature and enthalpy profiles are calculated and shown in Figure 2.4. Diffusion and Reaction in Islets of Langerhans was studied using Parabolic and Hyperbolic Models. This is needed to better treat Type I diabetes by the transplantation method. Sharma Number can be used to evaluate when the wave term becomes important in the application. It represents the ratio of Mass Transfer in Bulk to Relaxational Transfer. Sharma number (mass) may be used in evaluating the importance of the damped wave diffusion process in relation to other processes such as convection, Fick steady diffusion in the given application. Four regimes can be identified in the solution of hyperbolic damped wave diffusion model. These are; i) Zero Transfer Inertial Regime, 0 0 inertia ; (ii) Rising Regime during times greater than inertial regime and less than at the wave front, X p > ; (iii) at Wave front,  = Xp; (iv) Falling Regime in Open Interval, of times greater than at the wave front,  > Xp. Method of superposition of steady state concentration and transient concentration used in both solutions of parabolic and hyperbolic models. Expression for steady state concentration developed. Closed form analytic model solutions developed in asymptotic limits of Michaelis and Menten kinetic at zeroth order and first order. Expression for Penetration Length Derived – Hypoxia Explained Expression for Inertial Lag Time Derived. Solution within bounds of Second Law of Thermodynamics. No Overshoot Phenomena Observed. Centrifugal separation of oil and water from streams that is concentrated with oil such as the ones from the oil spill disasters is investigated in this study. The existing theories are largely for Stokes settling of oil drops. In this study two layers, one rich in oil and another rich in water are allowed to spin in a centrifuge. The tangential velocity profile is derived from the equations of continuity and motion. The power drawn at the inner rotor is calculated for a set of parameters fo the system and a angular speed  RPM. Each simulation required the solution of four simultaneous equations and simultaneous unknowns. The power draw was found to be log linear with angular rotor speed on a log-log plot. The viscosity of the oil was increased five times to study the effect on the power draw. An expression of the interlayer thickness ratio,  was obtained by use of a component mass balance on oil streams that flow in and out of the continuous centrifuge.  was given in terms of , the ratio of the inner and outer radii of the centrifuge, the mole fraction of oil in feed, peripheral layer and inner layer. State space models can be used to describe a set of variables in vector form in terms of matrix equations. The stability of the system can be studied using Eigenvalues and Eigenvector analysis. A scheme of 7 simultaneous simple irreversible reactions were

216

Kal Renganathan Sharma

considered. The state space model to describe this system is given.. This system can be viewed as an integrating system since all but Eigenvalues are negative with 4 Eigenvalues 0. The composition of a copolymer as a function of comonomer composition, reactivity ratios and reactor choice were derived from the kinetics of free radical initiation, propagation and termination reactions. The copolymerization equation obtained using the QSSA quasisteady-state approximation is given. As an example, copolymer composition with 4 monomers as a function of monomer composition is illustrated in Example 3.1 made in CSTR. The copolymer composition is sensitive to reactivity ratios to a considerable extent. The copolymer composition equation for a multi-component copolymer with n monomers were derived. This methodology was generalized for n monomers. The general form of the equation was represented in the matrix form using linear algebra. The rate matrix and rate equation is given.. The QSSA becomes in the vector notation;

K T (MM *T ) K(M * M

T

)

When the Eigen values of the rate equation become imaginary the monomer concentration can be expected to undergo subcritical damped oscillations as a function of time. The occurrence of multiplicity in model solutions was illustrated by calculation of launch angle of stream of water during firefighting and elbow-up and elbow down solutions in the solution of inverse kinematics of a 3 arm manipulator with end effector (robot). Use and significance of dimensionless groups such as Reynolds number, Prandtl number, Biot number, Nusselt number, Mach number, Fourier number, Fick number, Newton number, Maxwell number (mass), Venrnotte number, Sharma number (mass), Maxwell number (momentum), Sherwood number, Shcmidt number, storage number/Sharma number (heat), Peclect number were discussed. Weiner-Hopf integral equation can be used to estimate the effects of mixing in a CSTR, contiuous stirred tank reactor. This can be done by estimating the degree of mixedness from the variance in SPC, statistical process control charts. The compositional distribution of AN in copolymer can be used as input. J is minimized with respect to  and the Weiner-Hopf integral equation is obtained. Groot and Warren [1997] developed a mesoscopic simulation tool. Molecules are treated as spherical objects that can obey the Newton‟s laws of motion. Physical properties and EOS of polymeric substances were obtained from this analysis. Conservative, dissipative and random forces were taken into account. In the canonical ensemble, the Gibbs-Boltzmann distribution can seen to be the solution to the Fokker Planck equation. Damped wave conduction and relaxation equation is solved for in three dimensions in order to gauge the ampacity risks in PCB interconnections. V = u/r substitution is used and the temperature profile obtained in three dimensions and in 1 dimension. In the creeping transfer limit, the spatio-temporal profile is given as a modified Bessel composite function in space and time of the third order. Three regimes of solution are identified; (i) lag regime; (ii) rising regime given by Bessel composite function in space and time and; (iii) rising regime given by modified Bessel composite function in space and time. In the general case the order of the solution was found to be 7/2 and the order of the solution was found to be two in the case of one dimension.

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Thermodynamic analysis of materials with negative thermal exmapnsion coefficient was performed., For physical changes the expression for internal energy change can be seen to be always positive. This is because of the lowering of pressure as solid becomes liquid and the internal energy change is positive resulting in a net positive sign in the RHS, right hand side of Eq. (3.401). No ideal gas law was assumed. Only the first two laws of thermodynamics was used and reversible changes were assumed in order to obtain Eq. (3.401). Thus reports of materials with negative thermal expansion coefficient is inconsistent with Eq. (3.401). For ideal gases Eq. (3.415) would revert to the volume expansivity,  would equal the reciprocal of absolute temperature. This would meanthat  can never be negative as temperature is always positive as stated by the third law of thermodynamics. So materials with negative values for  ab initio are in violation of the combined statement of the 1st and 2nd laws of thermodynamics. Along similar lines to the improvement given by Laplace to the theory of the speed of sound as developed by Newton (as discussed in Section II Historical Note) a isentropic volume expansivity is proposed by Eq. (3.420). This can be calculated using expression from isobaric expansivity, isothermal compressibility and a parameter  that is a measure of isentropic change of pressure with temperature. Eq. (34) can be used to obtain the isentropic expansivity in terms of heat capacities at constant volume and constant pressure and isothermal compressibility at a given pressure and temperature of the material. Chemical changes have to be delineated from physical changes when heating the material. An alternate non-Fourier heat conduction equation is derived from consideration of translation motion of spinless electron under a driving force due to an applied temperature gradient. Thisequation is a capite ad calcem, temperature. Elimination of the rate of change of velocity with respect to time leads to a non-Fourier heat conduction equation with a accumulation oftemperature or ballistic term in it. The new constitutive heat conduction equation is combinedwith the energy balance equation in 1 dimension. The governing equation for transienttemperature a partial differential equation (Eq. (3.498)) is solved for by the method of Laplacetransforms. The problem considered is the semi-infinite medium with constant thermo physicalproperties with constant wall temperature boundary condition. A closed form analytical expression for the transient temperature was obtained (Eq. (3.511)) after truncation of higher order terms in the infinite binomial series. This solution is compared with that obtained using the parabolic Fourier model and the damped wave model as presented in an earlier study. The predictions of Eq. (3.511) and Eq. (3.516) are closer to the Fourier model, Further work on Eq. (3.492) is underway.

3.21. NOMENCLATURE  ratio of the interface radius with the centrifugal bowl radius R radius of the interface of oil and water c1, c2,c3, c4 integration constants in tangential velocity profile 

density ratio

H

height of centrifuge (m)

oil water

218

Kal Renganathan Sharma  R oil water P r R oil water r T  v vrot vper v V  xF xper xrot

ratio of inner rotor radius with the centrifugal bowl radius radius of the inner rotor (m) viscosity of oil (Pa.S) viscosity of water (Pa.s) power draw at rotor (watts) radial distance radius of the centrifugal bowl (m) density of oil (kg.m-3) density of water (kg.m-3) tangential shear stress (N.m-2) torque (N.m.s-1) residence time of fluid in centrifuge (hr) tangential velocity of fluid (m/s) volumetric flow rate of fluid out through the rotor (m3.hr-1) volumetric flow rate of fluid through the periphery (m3.hr-1) volumetric flow rate of fluid into the centrifuge (m3.hr-1) volume of centrifuge (m3) angular speed of rotor (rad.s-1) weight fraction of oil in feed weight fraction of oil in outlet stream through periphery weight fraction of oil in outlet stream through inner rotor

3.22. FURTHER READING Abraham, M. A. & Nguyen, N. (2005). “Results from the Sandestin Conference: Green Engineering: Defining Principle”, Environmental Progress, Vol. 22, 233-236. Al-Nimr, A. & Naji, M. (2000). “The Hyperbolic Heat Conduction Equation in an Anisotropic Material”, Int. J of Thermophysics, 21, 281-287. Al-Nimr, M. A. & Haddad, O. M. (2003). “The Dual-Phase-Lag Heat Conduction Model in Thin Slab under Fluctuating Thermal Disturbance”, Heat Transfer Engineering , 24, 5, 47-54. Anderson, J. (2003). Modern Compressible Flow with Historical Perspective, McGraw Hill Professional, Third Edition, New York, NY. Antaki, P. J. (1998). Solution for non-Fourier Dual Phase Lag Heat Conduction in a Semiinfinite Slab with Surface Heat Flux, International Journal of Heat and Mass Transfer , Vol. 41, (14), 2253-2258. Askleland, D. R. & Phule, P. P. (2006). The Science and Engineering of Materials, Thomson, Toronto, Ontorio, Canada. Avgoustiniatos, E. S., Dionne, K. E., Wilson, D. F., Yarmush, M. L. and Colton, C. K. (2007). “Measurements of the Effective Diffusion Coefficient of Oxygen in Pancreatic Islets”, IEC , Res., Vol. 46, 6157-6163. Bai, C. & Lavine, A. S. (1995). On Hyperbolic Heat Conduction and Second Law of Thermodynamics, J. Heat Transfer , 117, 2, 256-263.

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Barletta, A. & Zanchini, E. (1997). Thermal-Wave Heat Conduction in a Solid Cylinder which undergoes a change of Boundary Temperature, Heat and Mass Transfer/ Waremaund Stoffuebertragung, 32, 4, 285-291. Baskakov, A. P., Berg, B. V., Vitt, O. K., Filippovsky, N. F., Kirakosyan, V. A., Goldobin, J. M. & Maskaev, V. K. (1973). “Heat Transfer to Objects in Fluidized Beds”, Powder Technology, Vol. 8, 273-282, Vol. 22, 2005, 233-236. Batchelor, G. K. (1967). An Introduction to Fluid Dynamics, Cambridge University Press, 233-234, Cambridge, UK. Baughman, R. H. & Turi, E. A. (1973). “Negative Thermal Expansion of a Polydiacetyelene Single Crystal”, Journal of Polym. Sci., Polymer Physics, Vol. 11, 2453-2466. Baumeister, K. J. & Hamill, T. D. (1971). “Hyperbolic Heat Conduction Equation – A Solution for the Semi-Infinite Body Problem”, ASME Journal of Heat Transfer , Vol. 93, 1, 126-128. Bender, M. (1999). “Economic Feasibility Review for Communit-Scale Farmer Cooperatives for Biodiesel”, Bioresource Technology, Vol. 70. 1, 81-87. Bhange, D. S. & Veda. “Negative Thermal Expansion in Silicalite-1 and Zirconium Silicalite1 having MFI Structure, Materials Research Bulletin, Vol. 41, 1392-1402. Bird, R. B., Stewart, W. E. & Lightfoot, E. N. (2007). Transport Phenomena , John Wiley & Sons, Second Edition, New York. Boley, A. K. (1964). Heat Transfer Structures and M]aterials, Pergamon, New York. Boltzmann, L. (1876). Über die Natur der Gasmoleküle (On the nature of gas molecules). WienerBerichte, 1876, 74, 553–560. Butcher, J. C. (1964). “On Runge-Kutta Processes of Higher Order”, J. Austral. Math. Soc., Vol. 4, 179. Camp, T. R. (1936). “A Study of the Rational Design of Settling Tanks”, Sewage Works Journal, Vol. 8, 9, 116-125. Camp, T. R. (1946). “Sedimentation and the Design of Settling Tanks”, Trans. of American Society of Civil Engineers, Vol. 111, 895. Casimir, H. B. G. (1938). Note on the Conduction of Heat in Crystals, Physica, 5, 495 -500. Cattaneo, C. (1958). A Form of Heat Conduction which Eliminates the Paradox of InstantaneousPropagation, ComptesRendus, 247, 431-433. Cattaneo, C. (1948). Sulla Coduzione del Calone, Atti. Sem. Fis. Univ. Moderna , 3, 83. Chapra, S. C. & Canale R. P. (2006). Numerical Methods for Engineers, Mcgraw Hill Professional, New York. Chhabra, R. P. (1993). Bubbles, Drops and Particles in Non-Newtonian Fluids, CRC Press, Baco Raton, FL. Clift, R., Grace, J. R. & Weber, W. E. (1978). Bubbles, Drops and Particles, Academic Press, New York. Coe, H. S. & Clevenger, G. H. (1916). “Methods for Determining the Capacities of SlimeSettling Tanks”, Trans. of the American Institute of Mininig and Metallurgical Engineers, vol. 55, 356. Chandran, R. & Chen, J. C. (1982). AIChE Journal, Vol. 29, (6), 907-913. Chapra, S. C. & Canale, R. P. (2006). Numerical Methods for Engineers, McGraw Hill Education, New York, NY. Chhabra, R. P. (1993). Bubbles, Drops and Particles in Non-Newtonian Fluids, CRC Press, Baco Raton, FL.

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Jaya, N. & Ethirajulu, K. (2011). “Kinetic Modeling of Transesterification Reaction for Biodiesel Production using Heterogeneous Catalyst”, Int. J Engineering Science and Technology, Vol. 3, 4, 3463-3466. Johnson, A. S., O‟Sullivan, E., D‟Aoust, L. N., Omer, A., Bonner-Weir, S., Fisher, R. J., Weir, G. C. & Colton, C. K. (2011). “Tissue Engineering Part C: Methods”, Vol. 17, 4, 435-449. Joseph, D. D. & Preziosi L. (1989). Heat Waves, Review of Modern Physics, 61, 41-73. Jukes, T. H. & Cantor, C. R. (1969). “Evolution of protein molecules” in H. N. Munro, ed. Mammalian Protein Metabolism., Academic Press, New York. Jorgensen, J. D., Human, Z. & Short, S. (2001). “Pressure-induced Cubic to Orthorhombic Phase Transformation in the Negative Thermal Expansion Material HfW2O8”, J of Appl. Phys., Vol. 89, 6, 3184-3188. Kimura, M. (1980). A Simple Method for Estimating Evolutionary Rate of Base Substitution through Comparative Studies of Nucleotide Sequences”, Journal of Molecular Evolution, Vol. 16, 111-120. Kittel, C. & Kroemer, H. (1980). Thermal Physics, Freeman & Co , Second Edition, New York, NY. Krawczyk, T. (1996). “Biodiesel – Alternative Fuel makes Inroads but Hurdles Remain”, INFORM, 1, Vol. 7, 801-829. Kreith, F., Manglik R. M. & Bohn, M. S. (2010). Principles of Heat Transfer, Brooks/Cole, Pacific Grove, CA. Kulkarni, M. G. & K. Dalai, A. (2006). “Waste Cooking Oil – An Economic Source for Biodiesel: A Review”, Industrial & Engineering Chemistry, Vol. 45, 9, 2901-2913. Landau, L. & Liftshitz, E. M. (1987). Fluid Mechanics, Pergamon, UK. de Laplace, P. S. M. (1816). “Sur la vitesse du son dans l‟aire et dan l‟eau”, Annales de Chimie et de Physique. Lapple C. E. & Shepherd C. B. (1940). “Calculation of Particle Trajectories”, Ind. and Eng. Chemistry, Vol. 32, 5, 605-617. Levenspiel, O. (1999). Chemical Reaction Engineering, John Wiley & Sons, Third Edition, New York, NY. Li, J., Yokochi, A., Amos, T. G. & Sleight, A. W. (2002). “Strong Negative Thermal Expansion along the O-Cu-O Linkage in CuScO2”, Chem. Mater., Vol. 14, 2602-2606. Lind, C., Wilkinson, A. P., Hu, Z., Short, S. & Jorgensen, J. D. (1998). “Synthesis and Properties of the Negative Thernal Expansion Material Cubic ZrMo2O8, Chem. Mater., Vol. 10, 2335-2337. Lord Raleigh. (1883). Phil. Transactions., 316. Lorentz, H. A. (1892a). at the Internet Archive" , La Théorieelectromagnétique de Maxwell et sonapplication aux corps mouvants, Archives néerlandaises des sciences exactes et naturelles, 25, 363–552. Martin, A., Flynn, J. E. & Lange, H. (2011). “Biodiesel Production Method”, US Patent 8,030,505, B & P Process Equipment and Systems, LLC, Saginaw, MI. Ma, F. & Hanna, M. A. (1999). “Biodiesel Production: A Review”, Bioresource Technology, Vol. 70, 1-15. Mason, M. & Weaver, W. (1924). “The Settling of Small Particles in a Fluid”, Physical Review, Vol. 23, 412-426. Maxwell, J. C. (1867). On the Dynamical Theory of Gases, Phil. Trans. Roy. Soc., 157, 49.

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Sharma, K. R. (2009). “Comparison of Solutions from Parabolic and Hyperbolic Models for Transient Heat Conduction in Semi-Infinite Medium”, Int. J of Thermophysics, Vol. 20, 5, 1671-1687. Sharma, K. R. (2010). “On Damped Wave Diffusion of Oxygen in Islets of Langerhans: Part I- Comparison of Parabolic and Hyperbolic Models in a Finite Slab”, 102nd AIChE Annual Meeting, Salt Lake City, UT, November. Sharma, K. R. (2009). Bioinformatics: Sequence Alignment and Markov Models, McGraw Hill Professional, New Yor Sharma, Sharma, K. R. (2009). Comparison of Solutions from Parabolic and Hyperbolic Models for TransientHeat Conduction in Semi-Infinite Medium, In. J. Thermophysics, 30 (5), 16711687. Sharma, K. R. (2008). Analytical Solution of Damped Wave Conduction and Relaxation in a Finite Sphereand Cylinder, J. of Thermophysics and Heat Transfer , 22,( 4), 783-786. Sharma, K. R. (2008). On the Solution of Damped Wave Conduction and Relaxation Equation in a Semi-Infinite Medium Subject to Constant Wall Flux, Int. J. Heat and Mass Transfer , 51, 25-26, 6024-6031. Sharma, K. R. (2008). Analytical Solution of Damped Wave Conduction and Relaxation in a Finite Sphereand Cylinder, J. of Thermophysics and Heat Transfer , 22, (4), 783-786. Sharma, K. R. (2008). On the Solution of Damped Wave Conduction and Relaxation Equation in a Semi-Infinite Medium Subject to Constant Wall Flux, Int. J. Heat and Mass Transfer , 51, 25-26, 6024-6031. Sharma, K. R. (2007). Damped Wave Conduction and Relaxation in Cylindrical and SphericalCoordinates, J. Thermophysics. and Heat Transfer , 21, 4, 688-693. Sharma, K. R. (2007). Principles of Mass Transfer , Prentice Hall of India , N. Delhi, India. Sharma, K. R. (2006). A Fourth Mode of Heat Transfer called Damped Wave Conduction, 42nd Annual Convention of Chemists Meeting , Santiniketan, India, February. Sharma, K. R. (March, 2003). Storage Coefficient of Substrate in a 2 GHz Microprocessor, 225th ACSNational Meeting, New Orleans, LA. Sharma, K. R. (2009). Bioinformatics: Sequence Alignment and Markov Models, McGraw Hill, New York. Sharma, K. R. (March 26th – March 30th, 2006). Errors in Gel Acrylamide Electrophoresis Due to Effect of Charge, 231st ACS National Meeting , Atlanta, GA. Sharma, K. R. (March 2007). On the Derivation of an Expression for Relaxation Time from Stokes-Einstein Relation, 233 rd ACS National Meeting, Chicago, IL, American Chemical Society, Washington, DC. Sharma, K. R. (2006). Manifestation of Acceleration during Transient Heat Conduction, Journal of Thermophysics and Heat Transfer , Vol. 20, 4, 799-808. Sharma, K. R. (2005). Damped Wave Transport and Relaxation, Elsevier , Amsterdam, Netherlands. Sharma, K. R. (November, 2001). Simulation of Instantaneous Local Heat Transfer Coeffficient in Fine Particle Fluidized Beds, IMECE, New York, NY. Sharma, K. R. (November, 2001). Simulation of Instantaneous Local Heat Transfer Coefficient in Slugging Fluidized Beds, IMECE, New York, NY. Sharma, K. R. (November, 2001). Simulation of Instantaneous Local Heat Transfer Coefficient in Intermediate Sized Particle Fluidized Beds, IMECE , New York, NY.

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Sharma, K. R. (2001). Simulation of Local Heat Transfer Coefficient for Coarse Particle Fluidized Beds, IMECE, New York, NY, November. Sharma, K. R. (2000). “Thermal Terpolymerization of Alphamethylstyrene, Acrylonitrile and Styrene, Polymer , 1305-1308 Sharma, K. R. & Turton, R. (1998). “Mesoscopic Approach to Correlate Surface Heat Transfer Coefficients with Pressure Fluctuations in Dense Gas-Solid Fluidized Beds”, Powder Technology, Vol. 99, 109-118. Sharma, K. R. (March, 1998). Mixing in Helical Ribbon Agitatorsin Continuous Mass Polymerization Reactor – Is it Stochastic? , 30th IEEE Southeastern Symposium on System Theory, Morgantown, WV, USA. Sharma, K. R. (September, 1997). A Statistical Design to Define the Process Capability of Continuous Mass Polymerization of ABS at the Pilot Plant, 214th ACS National Meeting, Las Vegas, NV. Shipman, J. T., Wilson, J. D. & Higgins, C. A. (2013). An Introduction to Physical Science, Cengage Learning , Mason, OH. Shiva Prasad, B. G. & Sharma, K. R. (2012). “Alternative Energy for Energy Sustainability”, Annals of Arid Zone, Special Issue, April, Jodhpur, India. Sleight, A. W. (1994). “Negative Thermal Expansion Material”, US Patent 5, 322, 559. Smith, J. M., Van Ness, H. C. & Abbott, M. M. (2005). Introduction to Chemical Engineering Thermodynamics, McGraw Hill Professional, 7th Edition, New York, NY. Stebbins, E. J. (2009). “Technical and Economic Feasibility of Biodiesel Production in Vermont: Evidence from a Farm-Scale Study and a Commercial Scale Simulation Analysis”, Masters Thesis, University of Vermont. Stepanov, L. A. (2000). “Thermodynamics of Substances with Negative Thermal Expansion Coefficient”, Computer Modelling & New Technologies, Vol. 4, 2, 72-74. Stokes, G. G. (1851). “On the Effect of the Internal Friction of Fluids on the Motion of Pendulums”, Trans. of Cambridge Philosophical Society, Vol. 9, 8. Subramanian, R. S. & Balasubramaniam, R. (2001). “The Motion of Bubbles and Drops in Reduced Gravity”, Cambridge University Press, Cambridge, UK. Sundaresan, R. & Kolar, A. (2002). “Core Heat Transfer Studies in a Circulating Fluidized Bed”, Powder Technology, Vol. 124, 1-2, 138-151. Taitel, Y. (1972). On the Parabolic, Hyperbolic and Discrete Formulation of Heat Conduction Equation, Int. J. Heat Mass Transfer , 15, 2, 369-371. Taylor, G. I. (1923). Phil. Trans. A223, 289-343. “BP leak the world‟s accidental oil spill” Telegraph, London, United Kingdon, 2010-08-03. The Hindu, „Science Day‟, February, 2006/ Treybal, R. E. (1980). Mass-Transfer Operations, McGraw Hill Professional, New York, NY. Turton, R., Bailie, R. C., Whiting, W. B. & Shaeiwitz, J. A. (1998). Analysis, Synthesis, and Design of Chemical Processes, Prentive Hall, Upper Saddle River, NJ. Turton, R. & Levenspiel, O. (1989). “A Short Note on the Drag Correlation for Spheres”, Powder Technology, Vol. 47, 83-86. Tzou, D. Y. (1996). Macro to Microscale Heat Transfer: The Lagging Behavior , CRC Press, New York. Varma, A. & Morbidelli, M. (1997). Mathematical Methods in Chemical Engineering, Oxford University Press, New York, NY.

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Vernotte, P. (1958). Les Paradoxes de la Theorie Continue de l‟equation de la Chaleur, C. R. Hebd. Seanc. Acad. Sci. Paris, 246, 22, 3154-3155. Weber, J. A. (1993). “The Economic Feasibiloty of Community-Based Biodiesel Plants”, Master’s Thesis, University of Missouri, Columbia, MO. Williams, S. J., Schwasinger-Schmidt, T., Zamierowski, D. & Stehno-Bittel, L. (2012). “Diffusion into Human Islets in Limited to Molecules Below 10 kDa”, Tissue and Cell, http://dx.doi.org/10.1016/j.tice.2012.05.001 You, Y. D., Shie, J. L., Chang, C. Y., Huang, S. H., Pai, C. Y., Yu, Y. H. & Chang, C. H. (2008). “Economic Cost Analysis of Biodiesel Production: Case in Soybean Oil,” Energy & Fuels, Vol.22, 182-189. Zanchini, E. (1999). Hyperbolic Heat Conduction Theories and non-Decreasing Entropy, Phy. Review B- Condensed Matter and Material Physics, Vol. 60, (2), 991-997.

3.23. EXERCISES Review Questions 1.0. How does the deepwater horizon spill provide impetus to study of dynamics and control ? 2.0. What are the differences between empirical and semi-empirical models ? 3.0. What are the differences between mechanistic models and shell balance models ? 4.0. What are the differences between supercomputer based models and simulation studies on a personal computer. 5.0. What are the differences between mesoscopic models and Monte Carol trials. 6.0. What is th e physical significance of frequency number ? 7.0. What is Brinkman number and what is the physical significance of this dimensionless group. Discuss the use and significance of the following dimensionless numbers (Exercise 8 – 23.0) 8.0. Froude Number 9.0. Fanning Friction Factor 10.0. Drag Coefficient 11.0. Aeration Number 12.0. Graetz Number 13.0. Heat Transfer Factor 14.0. Mass-Transfer Factor 15.0. Sharma Number 16.0. Storage Number 17.0. Power Number 18.0. Flow Number 19.0. Peclect Number 20.0. Condition Number 21.0. Separation Number 22.0. Schmidt Number

227

Mathematical Process Models

23.0. Sherwood Number 24.0. Maxwell Number (heat)/Vernotte Number 25.0. Maxwell Number (mass) 26.0. What is meant by “blow-up” in the model to predict heat transfer coefficient in fluidized beds to immersed surfaces ? 27.0. What are the differences between convective contribution and conductive contribution during the heat transfer between flow of dispersed systems past a surface ? 28.0. Can radiation effects play a role during heat transfer between the dispersed systems and surface in contact with ? 29.0. What is the differences in heat transfer expectations when fluid flow towards a horizontal tube and when the fluid flow away from the tube ? 2

30.0. What is the significane of the cross-derivative term,

u in Eq. (2.26) for Y

temperature near a hot tube in a CFB, circulating fluidized bed. 31.0. Under what range of conditions is the binomial expansion used in Eq. (2.36) a reasonable assumption to make ? 32.0. What are the differences between overall mass transfer coefficient and local mass transfer coefficient as given in Eq. (2.49). 33.0. What are the differences between overall heat transfer coefficient and local heat transfer coefficient as used in Eq. (2.48). 34.0. Can the coupled governing equations for temperature and humidity in a evaporative cooler lead to instability under certain conditions ? Why not ? 35.0. What are the two mechanism of heat removal designed in a evaporative cooler ? 36.0. Rewrite the equations for the oil layer and water later during centrifugal separation under transient conditions. 37.0. Guess the velocity profile of oil layer and water layer during transient conditions in the centrifuge. 38.0. What is the significance of the separation efficiency in centrifugal separation as given by Eq. (2.82) ? 39.0. What is the significance of a log-linear relation between power draw and rotor speed during centrifugal separation ? 40.0. What is the signifance of toroidal vortice formation during centrifugal flow ? 41.0. Rewrite the state space model for a scheme of 7 simultaneous reactions in series/parallel discussed in section 2.5.1 when the reactions are reversible. 42.0. What would happen to the rate matrix when temperature is changes in Eq. (2.100). Assume that all rate constants obey a Arrhenius relationship. 43.0. Can multiplicity occur in calculation of time calculations during fire fighting ? 44.0. What are the differences between “elbow-up” and “elbow-down” solutions of the joint angles of a 3 arm manipulator with end effector ? 45.0. What happens to the physical significance when a third dimensionless group is formed by multiplication of two existing dimensionless groups.

228

Kal Renganathan Sharma

Problems 46.0. Effect of Bubble Residence Time at the Immersed Surface in a Fluidized Bed In section 2.2 the packet model developed by Mickley and Fairbanks [4] was discussed in detail. The model equation that provides an expression for the time averaged heat transfer coefficient between the fluidized bed and immersed surfaces was given by Eq. (2.13). Rather than a exponential distribution of residence times of the packets at the heat transfer surface consider an alternate bathing of solid rich packet and gas rich bubbles at the surface. A saw tooth pattern may be assumed as shown below in Figure 41.0. Assume that the particle contact times as shown in Figure 41.0, tp1, tp2, tp3 etc conform to a exponential distribution and the bubble contact times tb1, tb2, tb3 obey a uniform distribution. What would the time average heat transfer coefficient, ? What happens to the “blow-up” at the asymptotical limit of zero contact time for the instantaneous heat transfer coefficient between the bed and the surface ? 47.0. Simulation of Instantaneous Local Heat Transfer Coeffficient in Fine Particle Fluidized Beds [35] Fine particle fluidized beds in general can be characterized as Geldart A type fluidization [39]. These are bubbling fluidized beds. The mean particle size is small and density os less than 1.4 gm.cm-3. Beds in this group expand appreciably prior to commencement of bubbling. Bed collapse is slow on cut-off of air supply. Gross circulation of powder occurs even when few bubbles are present increasing the mixing process. Bubbles split and recoalesce often. Bubbles rise at a greater velocity compared with the interstitial gas velocity. Regardless of bubble size a maximum bubble velocity was found. The ratio of the bubble cloud volume to the bubble volume id negligible. The hydrodynamics of these beds are such that the bubble size grow to a maximum bubble size. The bed expands uniformly beyond the minimum fluidization bed height. The dominant frequency of pressure fluctuations can be corresponded with the bubble frequency at the surface. This frequency can be taken to be in the range of 1-6 Hz. Simulate a instantaneous heat transfer coefficient signal as a function of time. Use the Mickley and Fairbanks packet theory to estimate the packet heat transfer coefficient and Eq. (2.24) for the convective heat transfer when the bubble stays at the surface. Plot the instantaneous simulated heat transfer coefficient as a function of time. 48.0. Simulation of Instantaneous Local Heat Transfer Coefficient in Slugging Fluidized Beds [36] Slugs are bubbles that are bounded only by the walls of the fluidized bed chamber. Slug frequencies may be taken to be about 2 Hz. During contact of the slugs and the heat transfer surface it may be assumed that the primary mode of heat transfer is by convection. Simulate a instantaneous heat transfer coefficient signal as a function of time. The transient heat transfer as described in Mickley & Fairbanks[4] for Fourier conduction or by Sharma [27] for damped wave conduction and relaxation can be used for obtaining heat transfer coefficients during the time the packets are in contact with the surface. A suitable statistical distribution such as exponential distribution for packet arrival times or Poisson distribution for slug residence times at the surface may be used for purposes of the simulation study.

Mathematical Process Models

229

49.0. Simulation of Instantaneous Local Heat Transfer Coefficient in Intermediate Sized Particle Fluidized Beds [37]

Figure 41.0. Prototypical Trace of Instantaneous Heat Transfer Coefficient between Fluidized Bed and Immersed Surfaces.

Fluidized beds with intermediate size particle size is classified under Geldart B type fluidization. The particle size lies in the range 40 m  dsv  500 m and instrinsic particle density lies in the range of 1.4 g.cm-3  s  4.0 g.cm-3. Bubbling is seen as soon as minimum fluidization velocity is reached. Bed expansion is small and the bed collapse is rapid when the air supply is cut-off. There was found no powder circulation in the absence of bubbles. Bubbles burst at the surface of the bed as discrete entities. Bubbles rise more rapidly compared with the interstitial gas velocity and bubble size increases linearly with both bed height and excess gas velocity (UUmf). Bubble coalsence is pronounced. There was found no maximum stable bubble size. Bubble sizes have been found to be independent of mean particle size or particle size distribution. Assume that the contributions from the particle conduction and gas convection would be equal to each other. Simulate a instantaneous heat transfer coefficient signal as a function of time. The transient heat transfer as described in Mickley & Fairbanks[4] for Fourier conduction or by Sharma [18] for damped wave conduction and relaxation can be used for obtaining heat transfer coefficients during the time the packets are in contact with the surface. A suitable statistical distribution such as exponential distribution for packet arrival times or Poisson distribution for slug residence times at the surface may be used for purposes of the simulation study. The gas convective contribution to the heat transfer coefficient can be given by Eq. (2.24). 50.0. Simulation of Local Heat Transfer Coefficient for Coarse Particle Fluidized Beds [38] In coarse particle fluidized beds the thermal time constant of the particle [3] is larger compared with the residence time of the particle at the surface. In such cases the heat transfer coefficient from Fourier conduction through the particles can be neglected. The gas flow is at a higher rate. This can lead to even turbulent flow. The gas convective heat transfer contribution is more siginificant. Simulate a instantaneous heat transfer coefficient as a function of time for such systems. Allow for bubble contact with the surface. Choose an appropriate statistical distribution for the arrival of packets and bubbles at the surface. The gas velocity in the dense phase is high. 51.0. Archimedes Number

230

Kal Renganathan Sharma The dimensionless group called Archimedes number can be given by the following expression;

Ar

gd 3p

s

s

U

2

f

(3.519)

Where g is the acceleration due to gravity, (m.s-2), dp particle size (m), U is the superficial velocity, m.s-1 and s and f are the densities of the solid particle and fluid respectively. Discuss the significance of Archimedes number. Can it be used to characterize fluidization behavior. Use Ref [39] if needed. 52.0. Grashoff Number Discuss the use and significance of Grashoff number. How is Grashoff number used in free convection problems. Can Raleigh Bernard instability be predicted using Grashoff number analysis. Grashoff number can be calculated as follows;

Gr

Tl 3

g

2

(3.520)

Where l is the characteristic length (m) and  is the kinematic viscosity (m2.s-1),  is the volumetric thermal expansion coefficient, T is the temperature difference between the flat pate and surroundings far from the plate. 53.0. Morton Number Discuss the use and significance of Morton number. How is Morton number used in characterization of bubble shape and bubble velocity. Morton number can be calculated as follows;

M

g

4 3

(3.521)

Where,  is the viscosity of the fluid (Kg.m-1.s-1) and  is the interfacial tension (N.m-1) between the dispersed and continuous phases. 54.0. Eotvos Number Discuss the use and significance of Eotvos number. How is Eatvos number used in characterization of bubble shape and bubble velocity. Eatvos number can be calculated as follows;

E

gd 2

Where d is the volume equivalent diameter. 55.0. Weber Number

(3.522)

Mathematical Process Models

231

Discuss the use and significance of Weber number. How is Weber number used in characterization of bubble shape and bubble velocity. Weber number can be calculated as follows;

W

U2 d

(3.523)

56.0. Anoxic Region in Trickle Bed Filter Develop the concentration profile of Oxygen at steady state in the aerobic region in Figure 42.0. Assume that the concentration of oxygen, at the interface of water and aerobic region is CO2i. The tank is impervious to diffusion of oxygen. The bed diameter if DT. Assume that the Michaelis Menten kinetics obeyed by the oxygen reaction with the microorganisms reaches an asymptotic zeroth order and a maximum reaction rate of Rmax. Show that the steady state concentration of Oxygen, CO2 as a function of x is a quadratic equation. Further calculate the xT is the distance beyond which there is no further diffusion of oxygen. 57.0. Capillary Number Discuss the use and significance of Capillary number. How is capillary number used in description of thermocapillary migration of drops, bubbles and particles. Capillary number can be calculated as follows;

C

T

T d

(3.524)

Where the rate of interfacial tension gradient variation with temperature is given by T

, (N.m-1.K-1) and d is the diameter of the drop, T is the temperature gradient

(K.m-1).

Figure 42.0. Anoxic Region in Trickle Bed Filter.

232

Kal Renganathan Sharma 58.0. Marangoni Number Discuss the use and significane of Marangoni number. How is Marangoni number used in description of thermocapillary migration of drops, bubbles and particles in a vertical temperature gradient. Marangoni number can be calculated as follows;

Mar

T

d 2C p

T k

(3.525)

Where Cp is the heat capacity of the fluid, (J.Kg-1.K-1),  is the viscosity of the fluid, (kg.m-1.s-1) and k is the thermal conductivity of the fluid, (W.m-1.K-1). 59.0. HMM, Hidden Markov Model – First Order HMMs are constructed using concepts of conditional probability. The yare increasingly used in bioinformatics applications (Sharma [40]). The sequence in DNA, deoxy ribonucleic acid molecule can be represented using HMM. Multiple alignments of protein sequence alignments can be achieved using HMMs. It can be protein structure modeling, gene finding, phylogenetic analysis, modeling time series, speech recognition, modeling coding and non-coding regions of DNA, protein subfamilies and machine learning techniques and other areas. A Markov chain is a sequence of random variables whose probabilities at a time interval depends upon the value of the number at the previous time or previous time(s). The controlling parameter in a Markov chain is the transition probability. This is a conditional probability for the system to go to a particular new state, given the current state of the system. In a kth order Markov chain the distribution of Xt depends on the k values immediately preceding it. Transition probability of Xt = P(Xt = X/Xt-k, Xt-k-1,…., Xt-1) The transition probabilities in a first order Markov model for Xt would only depend on one previous value, Xt-1. Given the sequence: T: TATATGCGTAACCGGTT Construct a first order HMM to represent the information in Sequence T. Show the transition probabilities would be as shown in the Figure 43.0 below. 60.0. Second Order HMM Construct a second order HMM to represent the information in sequence S. S: ACGTTGACTGACTGTATACTGGTTAGTGT Show that the dyad probabilities can be given by the values tabulated below; S: ACGTTGACTGACTGACTGTATACTGGTTAGTGT 61.0. Two Balls Crossing Each Other. A ball is thrown with an upward velocity of 3 m.s-1 from the top of a 10 m high building. Half a second later another ball is thrown vertically from the ground with a velocity of 7 m.s-1. Determine the height from the ground where the two balls pass each other. Which of the following is best applicable for the crossing of the two balls: (i) Ascend of A and Ascend of B (ii) Ascend of A and Descend of B (iii) Descend of A and Ascend of B

Mathematical Process Models (iv) Descend of A and Descend of B

Figure 43.0. First Order Markov Model to Represent Sequence T: TATATGCGTAACCGGTT.

Table 3.11. Second Order Markov Model to Represent Sequence

233

234

Kal Renganathan Sharma 62.0. Car’s Stopping Distance A car travels on a rectilinear stretch if road at constant speed vi = 85 mph. At si =0, the driver applies the brakes hard enough to cause the car to skid. Assume that the car keeps sliding until is stops. Assume that throughout this process the car‟s acceleration is given by a = -kg

(3.526)

where the coefficient of kinetic friction, k = 0.7 and g is the acceleration due to gravity. Estimate the car‟s stopping distance and time taken. 63.0. Trajectory of Basketball The basketball is launched at a velocity of 5 m.s-1 at a launch angle of 40 0. What is the height of the basket should the ball fall into the basket after 1.5 secs. The height of the gentleman can be assumed to be 5 ft. 10”. What is the maximum height reached by the ball. The distance between the gentleman and the basket as measured on the ground is 6 m. 64.0. Multiplicity Jack Taylor drove his pick-up truck down a dirt road with variable acceleration. The acceleration varies with displacement as follows; a

2s 12

(3.527)

The initial velocity of the truck is 8 m.s-1 (18 mph). A cheetah from the woods came across the path of Jack. Jack applied his brakes and the truck came to a halt after a certain distance. What is the distance ? Do you obtain more than one answer ? If so, which is the correct one ?

Figure 44.0. Trajectory of Basketball.

Mathematical Process Models

235

Figure 45.0. Launch Angle during Fire-Fighting.

65.0. Launch Angle during Firefighting Calculate the launch angle of the stream of water that emanates from the hose the fireman is holding in Figure 45.0. The height of the building where the fire is rampant is 2.4 m and the distance between the launch location to building is 6 m. Do you obtain more than one answer ? If so, what is the significance of the multiple results ?

Figure 46.0. Belleville Spring Washers.

236

Kal Renganathan Sharma 66.0. Belville Spring Washers The cylinder has a weight of 11 kg and is pushed against a series of Belleville spring Washers so that the compression in the spring is s = 1.8 cm (Figure 46.0). If the force in Newton, of the spring on the cylinder is;

F

646 s

1 4

(3.528)

Where s is given in m, determine the speed of the cylinder just after it moves away from the spring, i.e., at s = 0. 67.0. Projectile Launched from Roof Top A projectile is launched with an initial velocity of 100 m/s from the rooftop of a 100 m tall building (Figure 47.0) at a launch angle of . Determine launch angle  given that the distance from the building to the place the projectile strikes the ground, R‟ is 259.8 m. 68.0. Wedge Action The s, coefficient of static friction for all contact surfaces shown in Figure 48.0 is 0.2. Does the 50 lb force move the block A up, hold it in equilibrium or is it too small to prevent A from coming down and B from moving out ? The 50-lb force is exerted at the midplane of the blocks so that it can be considered as a coplanar problem.

Figure 47.0. Projectile Launched from a Rooftop.

Mathematical Process Models

237

Figure 48.0. Do blocks move ?.

69.0. Stopping Distance of Car during Rain on a Rough Road Heavy rains cause a particular stretch of road to have a coefficient of friction that changes as a function of location. Specifically, measurements indicate that the coefficient of kinetic friction has a 3% decrease per meter. Under these circumstances the acceleration of a car skidding while trying to stop can be approximated by;

a

g

k

c1s

(3.529)

Let k = 0.5, c1 = 0.015 m-1. The initial velocity of the car is 12.5 m/s. Determine the distance to complete stop or stopping distance. Compare the stopping distance in the rain with that obtained during dry conditions under which case the rate of friction decrement, c1 = 0. 70.0. Multiplicity in Launch Angle Calculation The engine room of a freighter is on fire. A fire-fighting tuboat has drawn alongside and is directing a stream of water to enter the stack of freighter as shown in Figure 49.0. If the initial velocity of the jet is 67 ft/sec find the value of the jet launch angle  that will get the job done. 71.0. Spring-Mass System in a Inclined Plane Consider the spring-mass system in a inclined plane as shown in Figure 50.0. When s = 0.3 m the spring is unstretched and the 5 kg block has a speed of 2 m/s down the smooth plane. Determine the distance s when the block stops.

238

Kal Renganathan Sharma

Figure 49.0. Fire in a Freighter and Jet of Water Directed from Fire-Fighting Tug Boat.

Figure 50.0. Spring Mass System in Inclined Plane.

72.0. Tension in Supporting Cables The uniform concrete slab has a weight of 5500 kg. Determine the tension in each of the three parallel supporting cables when the slab is held in the horizontal plane as shown in Figure 51.0. 73.0. Time Taken for Elevator The elevator starts from rest and can accelerate at 3 m/s2 and then decelerate at 1 m/s2. Determine the shortest time taken for the elevator to reach a floor 10 m above the ground. 74.0. Variable Force The force F acting in a constant direction on a 20 kg block has a magnitude that varies with position s of the block as F = 50s. Determine how far the block slides befor its velocity beomes 5 m/s. When s = 0, the block is moving to the right at 2 m/s. The coefficient of kinetic friction between the block and surfaces is 0.3.

Mathematical Process Models

239

Figure 51.0. Concrete Slab Supproted by Three Parallel Supporting Cables.

75.0. Crane Boom Write the state space form of the equations of mechanical equilibrium of translation and rotation. Solve for the reaction forces at Pin A and tension in the cable BC (Figure 52.0).

Figure 52.0. Crane Boom.

240

Kal Renganathan Sharma 76.0. Jukes-Cantor Model [41] A phylogenetic tree is a representation of evolutionary relations among several species with a common ancestor. P(Data/tree) is the chance of the sequence occurring given the tree. P(tree/Data) is change of the tree occurring given the data. During the course of evolution several substitutions of residues and deletions and insertions may have occurred. Mathematical models to track deletions and insertions can be developed. P(b/a, t) is the probability of residue a being substituted by residue b iver a edge length t. P x ,t y

xu

P

yu

(3.530)

,t

x, y are two aligned gapless sequences. u represent sites in the alignment. Different possibilities of P(b/a,t) for residues a and b are examined. A K x K matrix can be written for alphabet of size K. . . P

Ak

.

. . P

Ak

..

.

. .

.

.

.

. .

.

P

A1

P

A1

S t

P

A1

A1 A2

Ak

,t

P

A2

,t

,t

A2

P

A1

,t

Ak

,t

. . P

Ak

A1 A2

Ak

,t ,t

(3.531)

,t

For families of substitution matrices;

S t S s

S t s

(3.532)

i.e., the family is multiplicative for all values of lengths s and t. P a ,t P b , s b c

P a ,s t c

(3.533)

For all a, c, s and t. The substitution process is Markovian and stationary. The probabilities are multiplicative. Jukes and Cantor [41] suggested a model for modeling DNA sequences. A matrix R that contain rate of substitution can be written as follows;

3 3

(3.534)

3 3

241

Mathematical Process Models

The rows and columns are for the nucleotides Adenine, A, Cytosine, C, Guanine, G and thymine, T. Nucleotides undergo transition at a constant rate . The substitution matrix for a small time interval s() is given after approximation as;

s

I

R

(3.535)

Where I is the identity matrix. By multiplicativity (I+R) becomes;

s t

s t s

s t I

R

(3.536)

In the limit of small ; s t

s t

Rs t

(3.537) ds t dt

Rs t

Combining the above two equations; dr dt ds dt

3 r

3 s (3.538)

s

r

Solve for these equations and obtain mathematical expressions for rt and st. Show that at infinite time the nucleotide equilibrium, frequencies can reach a limit of 0.25. 77.0. Kimura’s Modification [42] The Jukes and Cantor model [30] does not capture some important features of nucleotide substitution such as transitions from purine to purine or from pyrimidine to pyrimidine substitutions are common. Transversions where the nucleotide type is changed is rare. Kimura [31] proposed a model with the rate matrix;

2 2

(3.539)

2 2

Obtain the model equations and obtain the solutions. Show that the solutions can be given by; st

0.25 1 e

4 t

242

Kal Renganathan Sharma

0.25 1 e

vt

4 t

2e

2t

(3..540)

rt = 1 – 2st - ut 78.0. Copolymer Composition Discuss the Monomer – Copolymer Composition Curve for the system Methacrylonitrile-Betamethylstyrene formed in a CSTR. Let the monomer 1 be Methacrylontirle, MAN and monomer 2 be BMS, betamethyl styrene. The reactivity ratios r12 and r21 are 1.25 and 0.25 respectively. Compare the monomer and copolymer composition curve for MAN/BMS system the SAN system. The reactivity ratios for styrene and acrylonitrile are r12 = 0.29 and r21 = 0.02. 79.0. Second Order Reaction in a PFR Consider a second order reaction in a PFR, plug-flow reactor as shown in Figure 53.0. At steady state the mass balance across a slice z can be written as follows; (mass rate in) - (mass rate out) – (mass rate reacted) = 0 The intrinsic reaction rate is second order and is; dC A

k2C A2

dt

vC A

vC A

z

z

rA

k2C A2 A z 0

z

(3.541)

(3.542)

Divide Eq. (3.542) throughout by Az and obtain the limit as z  0. v CA A z or

1 CA

dC A

k2 A

C A2

v

1 C A0

C A0 CA or

k2C A2

CA C A0

L

dz o

k2 AL v

k2

(3.543)

1 k2C A0 1 1 k2C A0

Where L is the length of the PFR,  is the residence time in the reactor. Derive a similar performance equation when the rate of the reaction is of the third order ? Number of CSTRs that is Equivalent of PFR 80.0. A graphical procedure can be developed to calculate the number of CSTRs needed to equal the performance of a PFR.

Mathematical Process Models

243

Figure 53.0. Plug-Flow Reactor.

Show that the Steps are as follows: Obtain a graph of the rate of disappearance of species A, (-rA) as a function of the species concentration, CA.  Obtain the outlet concentration of the species A for a PFR for the stated residence time.  Write the steady state mass balance for one CSTR and confirm that; 

rA



C A0

C A1

(3.544)

where CA1 is the exit concentration from reactor 1. The operating line is a straight line with a slope of -1/ as shown in Figure 54.0 in a (-rA) vs. CA graph. Write the steady state mass balance for a train of CSTRs and confirm that; rAi

C Ai

1

C Ai

(3.545)



Note the intersection point between the operating line from the first CSTR to the (-rA) vs. CA curve. This is the exit concentration of species A from reactor 1.  Construct the second operating line with the slope = -1/2 where 2 is the residence time in the second reactor.  Repeat the procedure until the point in the graph representing the PFR performance is reached.  Count the number of CSTRs needed in order to achieve this. 81.0. Model for Site Utilization during Adsorption and Derivation of Langmuir’s Isotherm The Langmuir’s isotherm can be derived from a mass balance of the sites in the adsorbent that are available for adsorption. [Sitestotal] = [Sitesfilled] + [Sitesempty]

(3.546)

244

Kal Renganathan Sharma

Figure 54.0. Graphical Solution for Number of CSTRs needed to Equal Performance of PFR.

The adsorption of bulk solute onto the available sites can be viewed as a reversible reaction. Solutebulk

Keq 

SitesEmpty

 Solutebulk   Sites Empty  [ Sites filled ]



SitesFilled



Keq [ Solutebulk ]  Sitestotal    Sites filled    Sites filled  [ Sites filled ] 1  Keq  Solutebulk   Keq [ Solutebulk ] [ Sitestotal ]





 Sites filled  Keq  Solutebulk   [ Sitestotal ] (1  Keq [ Solutebulk ])

q

q0 y

( K  y)

(3.547)

(3.548)

Mathematical Process Models

245

where q is the amount adsorbed and is the concentration of [filled sites], y is the concentration of the bulk solute in the feed. K is a different constant from Keq the equilibrium constant as defined in Eq. (). Thus the Langmuir isotherm is derived. It relates the amount adsorbed onto solid adsorbents from the bulk solute in the feed. 82.0 Derivation of van’t Hoff’s Law Show that the van‟t Hoff‟s law for osmotic pressure in terms of the solute concentration can be derived from a model of the equilibrium between the permeate and retentate. The phenomena of osmotic flow is the flow of solvent from a region of low solute concentration to a region of higher solute concentration across a semi-permeable membrane. At equilibrium across the semi-permeable membrane the fugacities of the solution and solute free solvent are equal. fw = fs

(3.549)

The fs, the fugacity of the solution can be written in terms of the pure component fugacity, fw using the Poynting correction factor. The Poynting correction factor corrects for the effect of pressure on the pure component fugacity. Let Vw be the molar volume in cc.mole-1.

fw

Vw Pw

w xw fw exp

RT

Ps

(3.550)

Where w is the activity coefficient, xw is the mole fraction of water, T is the temperature of operation. Let  = (Pw – Ps) Eq. (3.250) then becomes; RT ln(

w xw )

Vw

(3.551)

For ideal solution, activity coefficient, w can be taken as 1. By Taylor series representation of ln(1-xs) =-xs RTxs Vw

RTCs

(3.552)

Eq. (3.252) is the van‟t Hoff‟s law that can be used to calculate the osmotic pressure. 83.0. Concentrator for Orange Juice You are hired by New Mexico freshly squeezed orange juice Corp. You have been asked to design a concentrator for orange juice. The concentration operation is based on osmotic flow of water from the orange juice side to a bath of brine solution. The orange juice is filled in transparent containers made of semi-permeable membrane from Toray, Japan. The juice contains 2% by weight sucrose. The containers are dropped in a bath containing brine solution. It can be assumed that the brine solution is 30% NaCl by weight and the rest predominantly water.

246

Kal Renganathan Sharma 84.0. Batch Adsorption One thousand kilograms of a tinted solution must be clarified by adsorbing ninetynine Per cent of the tint on a carbon adsorbent. The tint concentration per kg is 0.96. The equilibrium relation a Freundlich isotherm, for this case is;

q 275 y0.6 (3.553) Where q and y are the adsorbent and solution concentrations respectively. If the carbon initially contains no tint, how much carbon is needed for batch adsorption? 85.0. Rapid Sand Filter The rapid sand filter being designed for Laramie has the characteristics shown below. Determine the head loss for the clean filter bed in a stratified condition. The viscosity of water may be taken as 0.001 Pa.S. The depth of the filter is 75 cm and density of sand grains are 2,800 kg.m-3. The shape factor can be taken as 1.0 and bed porosity is 0.40 The equation developed by Rose in 1945 for head loss of through a clean-stratified sand filter with uniform porosity may be used. The filter loading rate is 250 m3.d-1.m-2.

hL

1.067va2 D g

4

CD f

(3.554)

d

The drag coefficient may be calculated from the 5 constant expression provided by Turton and Levenspiel (1989) applicable for Reynolds number 1 < Re < 200,000. CD

24 1 0.173Re0.657 Re

0.413 1 16300 Re

1.09

86.0. Manufacture of Terpolymer in Two CSTRs in Series Two CSTRs of equal volumes are to be used in series to manufacture a terpolymer . The terpolymer consists of three monomers: (i) alphamethyl styrene, AMS; (ii) acrylonitrile, AN and; (iii) styrene, STY. Acrylonitrile is mildly soluble in water and needs to be separated prior to discharge as AN is a carcinogen at greater than 2 ppm. The product rate is 8000 pounds per hour. The conversion is a total of 55% from both reactors combined. The composition of the product is 50% AMS, 30% AN and 20% styrene (weight fraction). The monomer composition in equilibrium with the product, is 75% AMS, 10% AN and 15% Styrene. The reaction is kinetics were obtained using tube polymerization (K. R. Sharma, [43] in the temperature range 408-418 K can be given by; rM

0.694 0.274 0.835 CM f AN fSty

(3.555)

Mathematical Process Models

247

Table 3.12. Sieve Analysis Sieve No: 16-20 20-30 30-40

d(microns) 1000 m 714 m 505 m

Mass % Retained 10.1 44.5 20.2

Where rM is expressed as mol.lit-1h-1. The molecular weight of AMS = 118 gm.mole-1 is gm.mole-1, AN = 53 gm.mole-1, STY = 104 gm.mole-1 Determine: (a) The mean residence time and volume for each of the two reactors. Express the volume as gallons. (b) The concentration of monomers in each reactor and their composition if applicable in each reactor and its effluent. Allow for addition of monomers into the second reactor to allow for the same composition of monomers in the second reactor. (c) The total mean residence time and total volume for the two reactors. Express volume as gallons. (d) The mean residence time and volume for a single CSTR having the same conversion of monomers to terpolymer. Express volume as gallons. Use graph paper attached if necessary.

Chapter 4

CONTINUOUS PROCESS DYNAMICS 4.1. TRANSIENT ANALYSIS-OVERVIEW Most of the discussions in leading textbooks such as Levenspiel [1999], Froment and Bischoff [2011] on reaction engineering are at steady state. There is very little discussion about events under transient conditions. This can become important during plant start-up and shut down periods. Control scheme design without good understanding of the transient dynamics in the reactor or distillation column can lead to instability and other problems that are avoidable. The overall mass balance can be used to confirm that the velocity of the influx and efflux streams are equal to each other for incompressible flow and constant volume reactor. The component mass balance can be used to obtain the governing equation that can be used to describe the dynamics of the reactant, product and by-product species in the reactor. The reactor type can be CSTR, continuous stirred tank reactor or PFR, plug flow reactor. It can be seen from the worked examples below that in a CSTR the governing equation for a reactant species can be seen to be ODE, ordinary differential equation. They are seen to be linear for the case of irreversible reactions of the reaction order 1. The nature of the governing equation depends on the nature of the kinetics, order of the reaction etc. More than one species such as reactant, product, by-product can be tracked using state space model. The differential equation is seen to be nonlinear for the case when the kinetics of reaction is of the free radical polymerization scheme. The energy balance equation may be used to study the dynamics of the temperature in the reactor. In case of exothermic reactions the reactor may be jacketed with cooling water for removal of heat. The rate of heat removal may be described using an overall heat transfer coefficient, U. The combined dynamics of the reactant species and temperature in the reactor can be solved for using numerical methods such as Runge Kutta fourth order method. The numerical scheme can be implemented in a MS Excel 2007 for Windows on desktop computer. The results are sketched as output response as a function of time. Multiplicities can be looked for. Stability types such as stable, integrating unstable, underdamped oscillatory, inverse response, spiral response, saddle point response, wave form response can be studied from the simulated results. Some idea of the stability types can also be obtained by examining the governing equations. In case of a PFR, the governing equation to describe the reactant species that undergoes first order irreversible reaction can be shown to be a PDE, partial differential equation, of the

250

Kal Renganathan Sharma

hyperbolic type. Transient dynamics of the conversion of reacting species, temperature in the reactor and temperature in the jacket of a CSTR can be studied using numerical solution to differential equations, linear or non-linear. A fourth order Runge-Kutta method can be used for numerical integration. Conversion as a function of time for different values of Damkohler numbers can be obtained as output response. The temperature as a function of time in response to a step change in input temperature can be obtained. A state space model may be used to represent the conversion of the reacting species, yield of the by-product, yield of the product, temperature in the CSTR and temperature of the jacket. The output response to a step change in the input can be obtained by the method of Eigenvalues and Eigenvectors. If all of the Eigen values of the state space A matrix are negative then the system is expected to be stable. A system with one Eigen value zero and one Eigen value negative is called an integrating system. If the Eigen values are complex conjugates the system is considered oscillatory. Sometimes the oscillatory output response may be subcritical and damped, underdamped or critically damped. The stability types can be: (i) improper node characterized as asymptotically stable when the eigenvalues are both less than zero and real; (ii) improper node characterized as unstable when the eigenvalues are both greater than zero and real; (iii) proper or improper node characterized as asymptotically stable when both Eigenvalues are both equal and less than 0, and characterized as unstable when both Eigenvalues are both equal and greater than 0; (iv) saddle point and characterized as unstable when one Eigenvalue is less than zero and the other Eigenvalue is greater than 0 and real; (v) focus or spiral, when the Eigenvalues are complex and characterized as stable when a < 0, and characterized as unstable if a > 0; (vi) center and characterized as marginally stable when the Eigenvalues are complex and a = 0. (Varma and Morbidelli [1997]). The transient concentration of monomer when free radical polymerization reactions are conducted in a CSTR can be obtained. DaI, Damkholer number (initiator) and DaM, Damkohler number (monomer) are introduced. For thermal polymerization binomial infinite series expansion can be used as shown in the examples below to obtain the output response for a step change in input. For the initiated polymerization the equations are non-linear. Numerical solution can be sought to obtain the output response. A maximum in the conversion of the monomer is found. Multiplicity is found in some results as can be seen in the worked examples below. The general form of prototypical first order and prototypical second order system are provided below. The first order process is characterized by process gain constant, kp and process time constant, p. The second order process is characterized by the process gain, kp, damping coefficient,  and process time constant, p. When the damping coefficient,  > 1 the system is over damped, when  = 1 the system is said to be critically damped and when  < 1 the system is expected to undergo an overshoot and is said to be underdamped oscillatory. Reports in the literature about occurrence of temperature overshoot upon use of non-Fourier heat conduction equation is revisited. The Taitel paradox is when the transient temperature in a finite slab heated by isothermal, hot walls, was obtained by the method of separation of variables using the hyperbolic PDE, partial differential equation, for certain values of the thermo physical properties of the finite slab the interior temperature of the slab was found to be greater than the wall temperature. This “overshoot” led investigators to abandon the nonFourier heat conduction equation stating that it violated second law of thermodynamics. A

Continuous Process Dynamics careful side by side study of this problem is presented below. The time condition

251 T was t

assumed to be 0 in Taitel [1972]. Consider a finite slab of width 2a, at an initial temperature of T0 heated by hot isothermal walls brought a temperature, Ts < T0 for times t > 0, greater than 0. As can be seen from Figure 4.1 below only the left hand limit of the rate of change of average temperature of the slab with respect to time is zero. A lumped analysis on the heating process leads to the expression for the rate of change of average temperature of the interior of the slab as shown in the ordinate of Figure 4.1.

T t

t

a2

e

a2

(4.1)

The right hand limit, of the rate of change of average temperature in the finite slab is a maximum! With increase in time this decreases exponentially to an asymptotic zero at large times or upon attainment of steady state. The right hand limit was used in the analysis. For a material with large relaxation time the overshoot was found for the model results when initial accumulation time condition is taken as 0.. For the same set of material and parameters when a physically reasonable time accumulation condition was used the overshoot disappears. The transient temperature was subcritical damped oscillatory. A steady state temperature was attained after a said time. Lumped analysis was further explored. The average temperature in a finite slab subject to convective heating was obtained using: (i) Fourier parabolic model. The model solution rises monotonically to a constant asymptotic value. (ii) Hyperbolic model with the first derivative of temperature with respect to time set to zero as used by Taitel [1972] by the method of Laplace transforms. This model solution appears to have an overshoot; (iii) Hyperbolic model with the initial temperature at T0 and the additional constraint that the average transient temperature should obey the energy balance equation from a lumped analysis. The dimensionless temperature was expressed as a sum of a steady state temperature and a transient temperature. The transient temperature was expressed as a product of wave temperature and decaying exponential in time. The model solution was found to be;

u

S*

t

1

S*

e

2

1

S*

sin

(4.2)

Where, S* is the dimensionless storage number. As shown in Figure 4.16-4.18 the model solution does not exhibit any overshoot. It appears that the damped wave conduction and relaxation equation can be applied to transient heat conduction problems without violation of the second law of thermodynamics. The storage number appears to be an important parameter in determination of the average transient temperature in a finite slab during damped wave conduction and relaxation. The time taken to attain steady state was found to be;

252

Kal Renganathan Sharma

Figure 4.1. Rate of Change of Average Temperature in Interior of Finite Slab as a Function of Time, t.

(4.3)

ss

The maxima in the transient temperature were found to increase with decreasing S* starting with large values such as 10. A cross-over was found after S* became less than about 2.2. Then the maxima in the average transient temperature was found to decrease with decrease in storage number, S*. The average transient temperature becomes zero in the infinite limit of S*.

A. Damkohler Number The governing equation can be made dimensionless by introducing varaibles such as fractional conversion, dimensionless time and Damkohler number . The physical significane of the Damkholer number can be seen to be the the ratio of the residence time of the reactant in the reaction to the intrinsic time of reaction when performed in a test tube. It can also be viewed as the ratio of the reaction speed to the speed of the fluid by convection. The Damkohler number may vary with the reaction kinetics. For example, Da = k, for a first order irreversible reaction, where k is the reaction rate constant for first order and  is the residence time of the flow stream in the reactor. Da

k p2 kD C I kt

for polymerization

reactions of the free radical type, where kp is the propagation rate constant, kD is the rate constant of dissociation of the initiator and kt is the rate constant of the termination steps.

B. Prototypical First Order Process Consider the hydrolysis of oxirane (ethylene oxide) to produce ethylene glycol by the following chemical reaction.

253

Continuous Process Dynamics

Let the reaction be performed in a CSTR such as the one shown in Figure 4.2. For incompressible flow and constant volume reactor the component mass balance for oxirane or species A can be written as follows; v C Ai

CA

VkC A

V

dC A

(4.4)

dt

The fractional conversion of oxirane and other dimensionless quantities are defined as follows;

XA

C Ai

CA

C Ai

;

V ; v

t

; Da

k

(4.5)

The governing equation for the dynamics of fractional conversion of oxirane in a CSTR given by Eq. (4.4) after introduction of the dimensionless variables as given in Eq. (4.5) becomes; XA

Da 1

dX A

XA

d

(4.6)

Eq. (4.6) can be rearranged into a form called the “prototypical first order process”. dX A 1 1 Da d

XA

Da 1 Da

(4.7)

The general form of a prototypical first order process can be written as follows;

p

dY Y dt

kpu

(4.8)

Where, p is the time constant, kp is the process gain and u is the input variable and Y(t) is the output variable. Thus the transient fractional conversion of oxirane is a first order process with a time constant of (1+Da)-1 and process gain of (Da/(1+Da)). Input variables are

254

Kal Renganathan Sharma

those that must be specified to completely define the problem. Without this the problem solution cannot be obtained. A number of processes conform to a a prototypical first order process. The Laplace transform of Eq. (4.8) with a initial condition of Y(0.) = 0 can be written as follows; k p u (s )

Y( s )

s

(4.9)

1

p

Eq. (4.9) can be used in block flow diagrams to show the process transfer function in the box allocated for process.

C. Prototypical Higher Order Processes The general form of the “prototypical second order process” that is linear can be represented as follows (Bequette [2003], Golnaraghi and Kuo [2010]) and Coughanowr and LeBlanc [2009]); 2 p

d 2Y dt

2

2

p

dY dt

Y

k pu

(4.10)

In addition to the process gain, kp, and time constant p there is third parameter  called the damping coefficient. The Laplace transform of prototypical second order process assuming an initial condition of y(0) = 0 and y‟(0) = 0 can be written as follows; Y( s )

u (s )

kp 2 2 ps

2s

1

p

(4.11)

For a step change in the input variable of u, the Laplace transform expression for the output variable can be written as follows; kp u

y( s ) s

2 2 ps

2s

(4.12) p

1

Four cases can be recognized for the output variable y(t) for various values of the damping factor; Case 1: Over damped System For damping coefficient values,  > 1, the output variable in the time domain y(t) is found to be over damped. Eq. (4.12) can be inverted into the time domain. p1 and p2 can be obtained as the roots of the quadratic equation in the denominator of Eq. (4.12). The poles of Eq. (4.12) are 0, -b and –c. The quadratic in the denominator is factorized as;

255

Continuous Process Dynamics y s

Y s

1

kp u

2 p

s

s

(4.13)

2 s

2

1 2 p

p

2

s2

s b s c

Let

1

s

(4.14)

2 p

p

The two factors a and b can be seen to be; 2

c

1

(4.15)

p

2

b

1

(4.16)

p

The inverse of the Laplace transformed expression in Eq. (4.13) can be looked from tables [Mickley, Sherwood and Reid (1957)] as;

Y(t )

bt

1 c1 e 2 p

b1 e

ct

(4.17)

bc c b

The dimensionless output response Y(t) as given by Eq. (4.17) for a typical over damped second order system is shown in Figure 4.2 for different values of the damping coefficient. Case 2: Critically Damped System For damping coefficient value,  = 1, the output variable in the time domain y (t) is said to be critically damped. When the damping coefficient becomes 1 the “b” and “c” values given by Eqs. (4.15,4.16) become equal and found to be; b= c

(4.18) p

Eq. (4.13) becomes;

Y s

y s kp u

1 2 2 p

s

s

1 p

(4.19)

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Kal Renganathan Sharma

The inverse of Laplace transformed expression Y(s) given by Eq. (4.19) can be obtained by invoking convolution theorem; t

Y(t )

1

p

t

2 p 0

pe

p

t

dp

t

p

te

p

1 e

1 e

p

p

1

t

(4.20)

p

The dimensionless output response for a critically damped system as given by Eq. (4.20) is shown in Figure 4.3. Case 3: Undamped System For damping coefficient values,  =0.., the output variable in the time domain y(t) is found to be undamped and oscillatory. Eq. (4.13) becomes;

Y s

y s kp u

1 2 p

s

s

2

(4.21)

1 2 p

Figure 4.2. Output Response to Step Change in Input for an Over damped Second Order Process.

257

Continuous Process Dynamics

Figure 4.3. Output Response to Step Change in Input for a Critically Damped Second Order Process.

Figure 4.4. Output Response to Step Change in Input for a Undamped Second Order Process.

The inverse of Laplace transformed expression Y(s) given by Eq. (4.19) can be obtained by invoking convolution theorem and the dummy variable p;

Y(t ) 

p

 p  sin  p2 0   p t

   dp  1  cos t      p 

   

(4.22)

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Kal Renganathan Sharma

The dimensionless output response for an undamped system as given by Eq. (4.22) is shown in Figure 4.4. As can be seen the oscillations are pronounced and there is a maximal overshoot. There is no return to the set point of 1.0. Case 4: Underdamped System For damping factor values,  < 1, the output variable in the time domain y(t) is found to be underdamped. Some investigators find an “overshoot” for these systems. Upon further scrutiny of systems that are suspected to undergo an “overshoot” it has been found (Sharma (20.12)) that the “overshoot” sometimes can be a mathematical artifact. No real machine or device can undergo an entropic decrease in a spontaneous manner. Entropy always rises for real processes. A spontaneous “overshoot” in a real engineering system would be in violation of Clausius’ inequality. This is further illustrated in Example 4.1. When,  0.. The fluid temperature is at Tf where Tf.< Th. Both sides of the slab are cooled by the fluid. The transient temperature as a function of space and time need be obtained. The two time and two space conditions are given as follows; t = 0, -a  x  +a, T = Th t > 0, x= 0, t  0, x =  a,

k

T x

(4.135) (4.136)

0

T x

h T

Tf

(4.137)

In one dimension the energy balance equation on a thin slice of thickness x can be written as; q x

T t

Cp

(4.138)

Combining Eq. (4.138) with the damped wave heat conduction and relaxation equation the governing equation can be written as; u

2

u

2

2

X2

u

(4.139)

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Kal Renganathan Sharma

where,

u

(T Th ) ; (Tf T h )

t

x

;X r

(4.140) r

The time and space conditions are as follows;  = 0, u = 0 u X

X = 0,

(4.141) 0

a

X

r

(4.142) u X

h* u 1

It can be noted that the heat transfer coefficient is defined with respect to the average temperature in the slab . The energy balance on a thin spherical shell at x with thickness x is written. The governing equation can be obtained after eliminating q between the energy balance equation and the derivative with respect to x of the flux equation and introducing the dimensionless variables; 2

u

u

2

2

X2

u

(4.143)

Eq. (4.143) is integrated between –Xa and +Xa and give; 2

u

u

S* ( u

2

1)

(4.144)

h Sa

Where the storage number S* is defined as S*

A.2. Fourier Parabolic Model Near Steady State Eq. (4.144) in the limit of zero relaxation time reverts to the Fourier parabolic model and the governing equation becomes; u

S*

u

1)

(4.145)

Continuous Process Dynamics

285

Obtaining the Laplace transforms of Eq. (4.144) s

u

or,

( s ) 0 S*

1

(s)

u

(s )

u

(4.146)

(4.147)

S*

s s

1 s

Obtaining the inverse Laplace transform of Eq. (4.147);

u

e

hS p a dp

1 e

ht C pa

(4.148)

0

Eq. (4.148) is the average temperature of the finite slab in response to a step change in convective cooling at both sides of the finite slab. Eq. (4.148) is not a function of the relaxation time and is a function of the heat transfer coefficient, h, density, , and heat capacity, Cp, and half-width of the finite slab, a.

A.3. Initial Accumulation Condition Zero For the hyperbolic model other investigators such as Taitel [1973] use the following initial condition;  = 0,

u

u

0

(4.149)

Obtaining the Laplace transform of Eq. (4.144) realizing the initial condition from Eq. (4.149)) the Laplace domain expression for (s) can be seen to be; u

1

(s ) s

Where, S*

s

2

S*

(4.150) s

S*

1

h Sa

Eq. (4.150) can be compared with the prototypical second order systems with a step input (Eq. (4.12) as discussed above in section 4.1C. For underdamped systems, i.e., when the damping factor,  < 1 the time response to Eq. (4.150) can be seen to exhibit an “overshoot” as shown in Figure 4.5. Without any heat sources the internal temperature greater than the

286

Kal Renganathan Sharma

boundary temperature can be a violation of second law of thermodynamics. When the damping coefficient,

1

2 p

2

(4.151)

S*

1

1

S*

S*

(4.152)

So, the damping factor is;

1 2 S

*

For large values of the dimensionless ratio, S*, the damping coefficient will become less than one and the output response appears to have a overshoot. This can be expected for materials with large relaxation time, Cpa r

(4.153)

h

A.4. Physically Realizable Steady State/Final Time Condition The boundary condition used during convective cooling is a constant value for the heat flux rate from the air flowing past the finite slab as given in Eq. (4.137). The governing equation (6) is the damped wave conduction and relaxation equation taking into account the effect of acceleration of electrons. A boundary condition that includes the wave term, i.e., air r

q would be a better representation of transient events at the air solid interface. Thus. t

Eq. (4.137) need be modified as; u X

h*

u

h*

1

air r

u

(4.154)

rs

Eq. (4.154) is integrated with respect to X and the governing equation written in terms of the average temperature in the slab as follows; u

2

1

S*

u 2

S*

u

S*

(4.155)

287

Continuous Process Dynamics air r

Where,

the ratio of the relaxation time in the air and the relaxation time in

rs

the solid. Let the be expressed as a sum of transient temperature and steady state temperature.  is the ratio of relaxation times of the air and that of the solid. u

u

ss

t

u

(4.156)

Substituting Eq. (4.156) into Eq. (4.155); u

2

t

1

t

u

S*

S*

2

u

t

u

ss

(1 S* ) 0

(4.157)

Eq. (4.157) is valid when the following two equations are valid;

1

2

t

u

S*

t

u

S*

2

S* ( u

ss

u

1) 0

t

0

(4.158)

(4.159)

The steady state temperature can be seen to be; ss

u

1

(4.160)

Eq.(4.159) is a second order differential equation that has been made homogeneous. Eq. (4.158) is multiplied by en throughout. The terms group such that, W

and for n

S*

1 2

t

e

n

u

t

(4.161)

Eq. (4.158) can be seen to become;

2

W

t

W

2

t

S* 1

2

(4.161)

The solution to Eq. (4.161) can be seen to be;

W

t

c1 cos

c2 sin

(4.162)

288

Kal Renganathan Sharma

Where

S* 1

Two time conditions are needed to fully describe the problem. The accumulation condition equal to zero appears to be physically unreasonable. When a slab with an initial temperature is immersed in a hot fluid with convective heat transfer coefficient the temperature in the slab would increase, rapidly in the beginning of the process and can be expected to slow down with the passage of time. At steady state the temperature would be steady and invariant with time. From the initial condition given by Eq. (4.141) the integration constants in Eq. (4.162) can be seen to be; c1 =0

(4.163)

Then Eq.(4.162) becomes; S*

1

u

t

2

e

c2 sin

(4.164)

The transient average temperature t also has to obey the following equation that can be obtained from a lumped analysis of the finite slab from energy balance; u

t

S* )

(1

S* 1

u

t

(4.165)

Eq. (4.165) at zero time, or =0, can be seen to be! S*

t

u

1

(4.166)

S*

The integration constant c2 can be obtained by applying Eq. (4.166) to Eq. (4.167). u

t

S*

c2 1

2

e

2

1

S*

sin

c2 e

2

cos

(4.167)

and c2 seen to be;

c2

S*

1

S*

The average transient temperature in the slab is then written as;

(4.168)

289

Continuous Process Dynamics

u

S*

t

1

S*

e

2

1

S*

sin

(4.169)

Eq. (4.169) is plotted in Figures 4.16 – 4.18, for different values of S* and . The time taken for the slab to reach steady state is finite and can be calculated when (ss ) becomes  in Eq. (4.169). This happens when;

ss

(4.170)

Figure 4.16. Transient Average Temperature in Finite Slab from Damped Wave Conduction and Relaxation ( =0..2).

290

Kal Renganathan Sharma For the special case when S* = 1, Eq. (4.161) becomes; 2

t

W

0

2 t

W

c1

(4.171) c2

1

u

Or,

t

e

2

c2 c1

(4.172)

From the initial condition given by Eq.(8), c2 = 0. c1 can be solved for the energy balance equation given by Eq. (4.166) at  = 0, u

t

c1 1

1 1

2 1

c1

1

.

c1

(4.173)

(4.174)

Thus for the special case when S*=1, 1

u

t

e

2

1

(4.175)

Reports in the literature about appearance of an “overshoot” in the transient temperature for a finite slab were analyzed for the case of convective boundary condition with the wave term. The solution from Fourier parabolic model given by Eq. (17) predicts a monotonic rise in temperature with an asymptote of ss = 1 at steady state. The solution from the hyperbolic damped wave conduction and relaxation for average temperature of the finite slab was solved for by method of Laplace transforms for the time conditions such as the first derivative of temperature in time equal to zero at time zero as used by other investigators in similar problems with constant wall temperature boundary condition. The Laplace domain expression compares from this model compares well with the expression for prototypical second order systems as discussed in a process dynamics and control textbooks. This expression when inverted has an overshoot. The temperature in a finite slab subject to convective boundary condition was assumed to comprise of a transient component and steady state component. The steady state component temperature was obtained and given by Eq. (4.160.). The governing equation for transient temperature is rendered homogeneous upon obtaining the solution of steady state temperature. The transient temperature was expressed as a product of decaying exponential and wave temperature. The governing equation for the wave temperature (Eq. (4.161) was solved for. The integration constants were obtained from the initial temperature condition of t = 0 at

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291

zero time. The second integration constant was calculated by verifying that the model solution obeys the energy balance equation for average transient temperature. This leads to a solution for transient temperature that has a decaying exponential term and a cosinuous term with a phase lag, . Eq. (4.169) does not result in any overshoot as shown in Figures 4.16– 5.18 for various values of frequency and ratio of thermal relaxation times between air and the solid. It h was found that the dimensionless storage number given by S* is a significant Sa parameter in the analysis. Expression for time taken to steady state was found and given by Eq. (4.160).

Figure 4.17. Transient Average Temperature in the Finite Slab Subject to Convective Boundary Condition with Wave Term ( = 0..01).

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Kal Renganathan Sharma

Figure 4.18. Transient Average Temperature in the Finite Slab Subject to Convective Boundary Condition with Wave Term ( = 2.0).

No overshoot was found when the wave term was taken into account in the boundary condition even for small values of , the ratio of the relaxation time of air with that of the solid. This is an indication that the fundamental phenomena such as “acceleration of free electron” as discussed before [Sharma, 2006] can be used to model transient heat conduction at short times and be within the second law of thermodynamics. The maxima in the transient temperature were found to increase with decreasing S* starting with large values such as 10. A cross-over was found after S* became less than about 2.2. Then the maxima in the average transient temperature was found to decrease with decrease in storage number, S*. Eq. (4.169) can be seen to go to zero in the asymptotic limits of infinite storage number, S*  .

293

Continuous Process Dynamics

B. Delineation of Maxima and Steady State Consider the mixing tank in Figure 4.19 with a jacket provided for heating the tank. The governing equations for the temperature in the reactor and the temperature of the coolant in the jacket are obtained from an energy balance on the control volume of the contents of the reactor and can be seen to be;

dT dt

Reactor:

Jacket:

dTj

vj

dt

Vj

v Ti V

T

Tjin

Tj

UA Tj VC p

T

UA Tj jVj C pj

(4.176)

T

At Steady State

0

1 50 125 10

UA 150 125 61.3(10)

UA (3)(61.3) 183.9 Btu.F

Figure 4.19. Mixing Tank Heated by hot Fluid in Jacket.

1

ft

3

(4.177)

294

Kal Renganathan Sharma

0

vj

2.5

200 150

75 1.5 ft 3 .min 50

vj

183.9 150 125 61.3 2.5

(4.178)

1

The model equations are represented in state space form as follows; dT dt dT j

0.4 1.2

0.3 T 1.8 T j

5 120

(4.179)

dt

The model solutions were obtained using fourth Order Runge-Kutta Method for Integration of ODE in MS Excel Spreadsheet 20.07 for Windows. The classical fourth-order RK method was used. The recurrence formula as given in (S. C. Chapra and R. P. Canale, (2006) was used. The recurrence relations used are as follows; h k1 6

Tl

1

Tl

Tjl

1

Tjl

h ' k1 6

2k2

2k2'

2k3

2k3'

k4

k4'

(4.180)

(4.181)

Where,

From Eq. (4.176), f(t,T) =

vi V

k1 = f(ti,Ti)

(4.182)

k2 = f(ti+0.5h, yi+0.5k1h)

(4.183)

k3 = f(ti+0.5h,Ti+0.5k2h)

(4.184)

k4 = f(ti+h, Ti+k3h)

(4.185)

Ti

T

Q

Integration was performed using a MS Excel spreadsheet. The key results are shown in Figure 4.20 and Table 4.4. The step size used was h = 0.01 min. From the last row of Table 4.4 it appears that the steady state temperature values, Tis = 152.2 and Tjs = 199.7 F. The values from solution in (a) are Tis = 150 F and Tjs = 150 F.

Continuous Process Dynamics

Figure 4.20. Transient Reactor Temperature and Jacket Temperature vs. Time.

Table 4.4. Temperature in the Reactor, Jacket vs. Time t 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 3 4 5 6 7 8 9 10

T 50.000 54.263 58.066 58.066 64.567 67.374 69.946 72.316 74.514 76.566 78.492 93.615 105.150 114.938 123.438 130.854 137.329 142.985 147.925 152.240

Tj 200 189.145 180.2475 180.2475 167.0994 162.3572 158.5734 155.5905 153.2766 151.5209 150.2307 150.7255 158.6955 166.9491 174.3797 180.9078 186.6161 191.6031 195.9591 199.7639

295

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Kal Renganathan Sharma

Figure 4.20. Transient Reactor Temperature and Jacket Temperature vs. Time.

(a) Large Step Change in Jacket Flow Rate: 10 times to 15 ft3.min-1

Figure 4.21. Transient Reactor Temperature and Jacket Temperature vs. Time Response to a Big Step Change in Jacket Flow Rate by 10. Times.

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297

(b) Small Step Change in Jacket Flow Rate: 10% Change to 1.65 ft3.min -1

Figure 4.22. Transient Reactor Temperature and Jacket Temperature vs. Time Response to a Small Step Change in Jacket Flow Rate by 10.%.

(b) Larger Vessel v = 10 ft3.min-1; V = 100 ft3 UA 10 50 125 150 125 100 61.3(100) 7.5 UA (4)(61.3) 73.56 Btu.F 1 ft 3 25

(4.186)

10 73.56 T js 125 50 125 100 100(61.3) 7.5 1.5 T js 750 F 0.012

(4.187)

0

0

(c)

298

Kal Renganathan Sharma vj

0

2.5

220 300

vj

dT dt dT j

(d)

73.56 750 125 61.3 2.5

200 750 3

0.733 ft .min

0.012 T 0.779 T j

0.112 0.046

(4.188)

1

5 5.864

(4.189)

dt

Eigenvalues of the A matrix Characteristic second degree polynomial equation

0.112

0.779

2

0.000552

0.891

0.0867 0

(4.190))

Eigenvalues are 1 = -0.78; 2 = -0.11 Both eigenvalues are negative. Hence the system is stable. It can be seen from Figure 4.20 that the transient temperature of the reactor goes through a minima. Both at minima as well as at steady state; dT dt

(4.191)

0

Use of Eq. (4.191) merely is not sufficient to identify the minima or steady steady state value. They may have to be obtained as shown in the above example by other means.

4.4. OTHER KINETIC TYPES A. Denbeigh Scheme of Reactions Consider the Denbigh scheme of reactions performed in the CSTR shown in Figure 4.23. A scheme of reactions as shown in Figure 4.23 was discussed in Levenspiel [1999] as a special case of Denbigh reactions. A state space model that can be developed to describe the dynamics of the 5 species, CA, CR, CT, CB and CS. The assumptions are that the inlet stream contains species A and B at a concentration of CAi and CBi and the initial concentrations of the other species are zero. The kinetics of the simple irreversible reactions shown in Figure 4.23 can be written as follows; dC A

k1

dt dC B dt

k3 C A

k4C B

(4.192)

(4.193)

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Continuous Process Dynamics

Figure 4.23. Denbigh Scheme of Reactions.

dCS

k4C B

dt

dC R dt dCT dt

k3C A

(4.194)

k5CS

k1C A

k2C R

(4.195)

k2C R

k5CS

(4.196)

The reactions are considered to be performed in the CSTR similar to the one shown in Figure 4.6. Component mass balances on each of the species assuming incompressible flow and constant volume reactor can be written as follows; Species A C Ai

Where ; Da1

k1 ; Da 3

C A 1 Da1

t

k3 ;

;

dC A

Da 3

(4.197)

d

V ,  with units of (hr.) is the residence time of v

the species in the reactor, V is the volume of the reactor, (liter) and v is the volumetric flow rate (lit.hr-1) in and out of the reactor. Species B C Bi

C B 1 Da 4

dCB

(4.198)

d

Where Da4 = (k4) Species S CS 1 Da 5

Da 4C B

Da 3C A

dC S d

(4.199)

300

Kal Renganathan Sharma Species R C R 1 Da 2

Da1C A

CT

Da 5C S

dC R

(4.200)

d

Species T Da 2C R

dCT

(4.201)

d

The model equations that can be used to describe the dynamics of the 5 reactant/product species in a CSTR can be written in the state space form as follows; CA

1 Da1

CB

d CS dt CR CT

0

Da 3

0

0

0

CA

C Ai C Bi

0 Da 3

1 Da 4 Da 4

0 1 Da 5

0 0

0 0

CB CS

0

Da1

0

0

1 Da 2

0

CR

0

0

Da 5

Da 2

0 0

1 CT

(4.202)

The stability of the dynamics of the 5 reactants/products in the Denbigh scheme performed a CSTR can be studied by obtaining the eigenvalues of the rate matrix. The characteristic equation for the eigenvalues is obtained by evaluation of the following determinant;

1 Da1 det

Da 3

0 Da 3

0 1 Da 4 Da 4

0

0

0

0 0

0 0

1 Da 5

0

Da1

0

0

0

1 Da 2

Da 5

1 Da1 Da3

1 Da 4

Da 2

1 Da5

1 Da 2

0 1

1

0

(4.203)

The 5 eigenvalues are negative when Damkohler numbers are greater than zero. When eigenvalues are all negative the system is considered to be stable. The Laplace transform of the model equations developed in order to describe the transient dynamics (Eqs. 4.194 – 4.201) for the 5 species in the Denbigh scheme in a CSTR can be written as follows; CA s

C Ai s s 1 Da1

Da 3

(4.204)

.

301

Continuous Process Dynamics C Bi (s ) s 1 Da 4

CB s

(4.205)

Da 3C Ai

CS s

s s 1 Da1

CR s

Da 3

Da 4C Bi s s 1 Da 4 s 1 Da 5

s 1 Da 5

Da1C Ai s 1 Da 2 s s 1 Da1

4.206)

(4.207)

Da 3

  1  Da 5 Da 3C Ai Da 4 Da 5C Bi Da1Da 2C Ai   CT  s      s  s  1                 s Da Da s Da s Da s Da s Da s Da Da 1 1 1 1 1 1            1 3 5 4 5 2 1 3   

(4.208)

The inverse Laplace transform of Eq. (4.204) can be obtained by invocation of the convolution theorem. The expression for transient concentration of species A can be written as; CA t

1

C Ai

1 Da1

Da 3

1 Da1 Da 3

1 e

(4.209)

The inverse Laplace transform of Eq. (4.205) can be obtained by use of the time shift property. The transient concentration of species B can be seen to be; CB t C Bi

1 1 e 1 Da 4

1 Da 4

(4.210)

The inverse Laplace transform of Eq. (4.207) can be obtained by look-up of Laplace inversion Tables in Mickley, Sherwood and Reed [1939]. The transient concentration of species R can be seen to be; CR t C Ai

e

1 Da 2

1 Da 2 Da 2

Da1

e Da 3

1 Da1

1 Da1 Da 3

Da3 Da 2

Da1

Da3

1 1 Da 2 1 Da1

(4.211) Da3

The inverse Laplace transform of Eq. (4.206) can be obtained by look-up of Laplace inversion Tables in Mickley, Sherwood and Reed [1939. The transient concentration of species S can be seen to be;

302

Kal Renganathan Sharma CS t

Da 4C Bi

1

C Ai

C Ai

1 Da 4 1 Da 5

Da 3

1 1 Da 5 1 Da1

1 Da 4

e Da 4

1 Da 5

e

Da 5 1 Da 4

Da 4

Da 3

e 1 Da 5 Da1

(4.212)

Da 5 1 Da 5

1 Da 5

1 Da1 Da 3

Da 3

1 Da1

Da 5

e Da 3 Da1

Da 3

Da 5

The transient concentration of species T can be obtained from the Laplace inversion of Eq. (4.208)

B. Michaelis and Menten Kinetics The production of monosaccharides from starch, aspartame found in Nutrasweet and other sweeteners from phenyl amine, acrylamide from acrylonitrile, penicillin, high fructose corn syrup from starch slurry, fructose from glucose, acetone-butanol from cheese whey, pentose sugar from lignocellulosics etc can be catalyzed using enzymes. Biological processes can be classified as: (i) fermentations; (ii) metabolic processes/functional genomics or metabolomics; (iii) by action of living cells. Fermentations can be further sub-classified into processes as those that are promoted by microorganisms such as yeast, algae, bacteria, protozoa and those that are catalyzed by enzymes. Stem cell research advances also can lead to useful products. The kinetics of the microbe promoted reactions and enzyme catalyzed reactions can be written as follows;

dC A

rA

dt

k

C E 0C A CM

(4.213)

CA

Where the enzyme concentration is CE0, CM is a Michaelis constant. Eq. (4.213) is referred to the equation describing Michaelis and Menten kinetics. At high concentrations, of the reactant, CA the rate expression in Eq. (4.213) becomes; Lt rA

CA

Lt k CA

CE 0 CM CA

kC E 0

1

CM

(4.214)

The reaction rate at the asymptotic limit of infinite concentration of reactant, CA can be seen to become zeroth order rate (Eq. (4.214). At the asymptotic limit of zero reactant concentration Eq. (4.213) can be seen to become a first order reaction rate;

Lt rA

CA

0

k

C E 0C A CM

(4.215)

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Continuous Process Dynamics

The Monod type of kinetics can be used to quantitate the production of cells and can be given by the rate expression;

rc

CC C A

kobs

(4.216)

C A CM '

The reaction scheme for which Eq. (4.216) is applicable is given in Figure 4.24. C

A

C

(4.217)

R

Where C is the microbe acting, CM‟ is the Monod constant and A is the food for the microbe and R is the waste material generated. More microbes are produced when food is supplied to the microbes. The waste product R can inhibit the production of product by phenomena called product poisoning. Activated sludge treatment of waste water is an example of bioprocess that is devoid of product poisoning. When Monod kinetics is in effect the cells reproduce the substrate is built and the waste R that can be poisonous to the product form. An induction period, sigmoidal growth period, stationary period and lysis/death of cell period. The product P can be obtained by cell rupture of cells C using centrifugation and separation of product P from the waste material R and unreacted reactant A and the disrupted cells using downstream processing methods. Consider the reactions described by Michaelis and Menten kinetics performed in the CSTR shown in Figure 4.24. Component mass balances on each of the species assuming incompressible flow and constant volume reactor can be written as follows; Species A

C Ai

CA

DaC E 0C A

dC A

C A CM

dt

(4.218)

Eq. (4.218) is made dimensionless as follows;

Let

XA

C Ai

CA

C Ai

t

;

kC E 0

; Da

C Ai

;

CM M

C Ai

(4.219)

Substituting Eq. (4.219) in Eq. (4.218), Eq. (4.218) can be seen to become; dX A d

Da 1 X A

1

M

XA

XA

(4.220)

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Kal Renganathan Sharma

Figure 4.24. Reaction Scheme that can be represented by Monod Kinetics.

The variables XA and  in Eq. (4.220) can be separated and integrated as follows; XA

dX A 1

2 0 XA

XA

XA

M

(4.221)

d

1 Da

M

Da

0

Eq. (4.221) can be solved for by using a substitution such as; y

XA

q

(4.222)

1 Da 2

M

q

The integral in Eq. (4.221) then becomes;

1

M

Da

2

y

y0

y

dy y

2

Da

q

ydy

2 y0

y

2

Da

(4.223)

q2

The integrals in the LHS can be obtained by a (y2+Da-q2 =z2) substitution and found to be;

ln

Da X A2

q2

2qX A

1 Da

M

4 q2

Da Da

ln

Da

q2

XA

q

q2

Da

Da

q2

XA

q

q2

Da

(4.224)

Eq. (4.224) is the closed form analytical solution to Eq. (4.220). Eq. (4.220) is not explicit in conversion as a function of time. For a given time in order to obtain the corresponding conversion a transcendental equation needs to be solved. There is no guarantee that the solution is monovalued. Multiplicity in solution may arise (Ramkrishna et, al,

305

Continuous Process Dynamics

[1982]). A Taylor series expansion may be written for the conversion as follows near the initial condition at  = 0. 2

XA

X A (0)

X A '(0)

3

2!

X A "(0)

4

X A "'(0) .

3!

4!

X A'"' 0

(4.225)

........

From the initial condition, XA 0

(4.226)

0

From Eq. (4.220) at =0, Da

X A' 0

Da

M

3

1

1

M

X "A 0 X A' 0

0

X "A

(4.229)

M

3Da 3 2

M

4

1

(4.228) M

2 X A' X A'

2q 1

2qDa 2 2

(4.227) M

Da 2 2

X "A 0

X "'A

1

1

M

M 5

(4.230)

M

Truncation of fourth and higher order terms in Eq. (4.226) and substitution of Eqs. (4.226-4.230) in Eq. (4.225) would yield;

XA

2

Da

1

M

2! 1

2

Da

3

1

M

Da 2

M

3

.

2qDa 2 2

3!

M

1

M

4 M

3Da 3 2 1

M 5

........

(4.231)

M

Eq. (4.231) is plotted in Figure 4.25 for dimensionless Michaelis constant of M = 4.0 at three different Damkohler numbers, Da = 1, 4 and 8 respectively. It can be seen from the graph that the transient concentration undergoes a maxima at larger Damkohler numbers. The steady state conversion can be obtained from Eq. (4.220.) as follows;.

0

Da 1 X As

1

M

X As

X As

(4.232)

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Kal Renganathan Sharma

The quadratic equation that needs to be solved for in order to obtain the steady state conversion, XAs can be written as follows; 2

X As

2qX As

X As

q

q2

Da

0

(4.233)

Da

Where q is given by Eq. (4.222). Since Eq. (4.233) is a quadratic there can be multiple values for the steady state conversion. In such cases other considerations may have to go into arriving at the residence time and other operating parameters for the desired output performance.

Figure 4.25. Transient Conversion XA in CSTR for Nutrients that obey Michaelis and Menten Kinetics

Continuous Process Dynamics

307

C. Gel Effect – Runaway Polymerization Reactions Free radical polymerization reactions are used to prepare useful polymers from monomers that can be manufactured by separating crude naphtha in distillation columns and by catalytic conversions. For example catalytic dehydrogenation of ethyl benzene can lead to the formation of styrene. Ammonia and propylene can be contacted in a fluidized bed reactor and in the presence of catalyst acrylonitrile can be made in an economically profitable manner. Switching one of the reactors to iso-propylene as feed material can lead to the manufacture of methacrylonitrile. Isomerization acid catalyst can be used to prepare alpha methyl styrene from ethyl benzene. Beta methyl styrene formation requires a suitable catalyst. Free radical reactions consists of three important phases: (i) initiation by appropriate initiators such as peroxy free radical provider or azoisobutyrontirle, AIBN or by thermal or by photolytic or by other means; (ii) propagation reactions and growth of backbone polymer and; (iii) termination reactions by combination or by chain transfer or by other mechanisms. During high conversion and when little diluent is used some times the growing chains do not diffuse and combine and participate in termination reactions. This can lead to a diffusion limited “gel effect” or runaway reaction condition. Control of such reactions if allowed to take place may be difficult and sometimes needs granite to separate the set mass. One time in Muscatine, IA the suspension kettle was “set-up” due to “gel effect” as touched upon in Chapter 1.0.

4.5. SUMMARY The transient conversion in a PFR, plug flow reactor was derived. For a reaction of first order assuming that the conversion is analytic in space and time the governing equation can be shown to be of the hyperbolic type. For the zeroth order reaction the governing equation was found to be a wave equation! The hyperbolic PDEs can be solved for using the methods of relativistic transformation of coordinates leading to modified Bessel composite function solution, method of separation of variables and method of Laplace transforms. Numerical solution procedures can be used when the equations become nonlinear such as in the case of free radical polymerization reactions. Transient analysis of first order reversible reactions, reactions in series-parallel, reactions that obey the Michaelis and Menten kinetics are discussed in detail. The use of final time condition and physically reasonable energy balance considerations can lead to solutions that are bounded and within the scope of the second law of thermodynamics for problems that can be described using non-Fourier conduction equation. The temperature profile in a PFR is also obtained. Energy balance under transient conditions is used in the analysis. The case of two interacting and non-interacting tanks is discussed as exercises. Transient dynamics of concentration, temperature separately and concentration and temperature both in a PFR and CSTR were discussed. The dimensionless group called Damkohler number in addition to conversion, dimensionless time, and residence time was used to describe the transient output response from a CSTR. The hydrolysis reaction of ethylene oxide to ethylene glycol was considered in a CSTR. For incompressible flow and for a constant volume system the steady state and transient conversion were derived. The

308

Kal Renganathan Sharma

transient model solution is given as equation. CSTR with recycle was considered in Example 4-6. The transient temperature in a mixing tank that is heated is given. A fourth order RungeKutta method was used for numerical integration of the ODE, ordinary differential equation. Temperature vs. Time in reactor as response to a step change in input is obtained and displayed in Figure 4.20. A state space model was developed to describe a jacketed CSTR. The output variables of interest are dimensionless conversion, dimensionless temperature and is given in Eq. (4.41).. If all of the Eigenvalues of the state space A matrix are negative then the system is expected to be stable. A system with one Eigenvalue zero and one Eigenvalue negative is called an integrating system. If the Eigenvalues are complex conjugates the system is considered oscillatory. Sometimes the oscillatory output response may be subcritical and damped, underdamped or critically damped. The stability type and characterization of stability for each of these cases are given in Table 4.3. The transient concentration of initiator and monomer during free radical polymerization in a CSTR was obtained in Example 4.4. DaI, Damkohler number (initiator) and DaM, Damkohler number (monomer) were introduced. The steady state conversions were obtained as a function of the Damkohler number. Eq. (4.58) that is used to describe the transient conversion of monomer is non-linear. The conversion of initiator and monomer as a function of time is displayed in Figure 4.10. Multiplicity was found in model solution of conversion of monomer. Conversion of initiator was monotonic. It can be seen from Figure 4.10 for the parameters used in the simulation study the transient conversion of monomers undergoes a maximum value. The general form of prototypical first order and prototypical second order system were provided. The first order process is characterized by process gain constant, kp and process time constant, tp. The second order process is characterized by the process gain, kp, damping coefficient,  and process time constant, tp. When the damping coefficient, z > 1 the system is over damped, when z = 1 the system is said to be critically damped and when z < 1 the system is expected to undergo a overshoot and is said to be underdamped oscillatory.

4.6. GLOSSARY Damkohler number is the ratio of the residence time of the reactants and other species in the reactor compared with the intrinsic reaction time depending on the chemistry. Denbigh Scheme of Reactions PFR plug flow reactor is a contacting scheme where the velocity of the fluid is constant throughout the tube, the conversion varies along the length of the tube and is one of the best performing reactor type. CSTR continuous stirred tank reactor. CSTRs in Series number of CSTRS are operated in series or other configuration. 3 or more CSTRs are needed to provide an equivalent performance in a PFR. free radical polymerization mechanism of polymerization reactions by free radical propagation of initiation, propagation and termination. thermal initiation polymerization, mechanism of initiation of free radical propagation reactions during polymerization by heat rather than initiator.

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309

polyrate, rate of polymer formation per unit time or rate of monomer consumption per unit time. time constant, p parameter that can be used to characterize exponential rise and exponential decay in transient applications for first and higher order systems. A system with large time constant would give a sluggish response and one with a smaller time constant would provide a quicker response. process gain, kp parameter used to characterize input-output relation in first and higher order systems fractional conversion, Xj amount of species that has been reacted compared with the initial concentration of the species thermal conductivity, k, proportionality constant in Fourier‟s law of heat conduction. Material property. heat capacity, Cp. thermo physical property of material. Amount of energy in kilojoules needed to raise on gram of a material by 1 degree K. relaxation time, r, thermo physical property of material. Measure of velocity of heat. Materials with larger relaxation time will allow slower speeds of heat. residence time,  the time the reacting and product species stays in the reactor.

4.7. FUTHER READINGS Agrawal, P., Lee, C., Lim, H. C. & Ramkrishna, D. (1982). Vol. 37, 3, 453-452, Chemical Engineering Science. Al-Nimr, M., Naji, M. & Al-Wardat, S. (2003). Vol. 42, 8, 5383-5386, Japanese Journal of Applied Physics, Part I: Regular Papers and Short Notes & Review Papers. Antaki, P. J. (1998). Vol. 41, 14, 2253-2258, Int. J. of Heat Mass Transfer . ASME Transactions, Heat Transfer Division, Bai, C. & Lavine, A. S. (1995). Vol. 117, 256-263, J. Heat Transfer . Barletta, A. & Zanchini, E. (2003). Vol. 32, 8, 5383-5386, Heat Mass Transfer/Warema-und Stoffuebertragung. Bird, R. B., Stewart, W. E. & Lightfoot, E. N. (1960). Transport Phenomena, Hoboken, NJ: John Wiley & Sons. Casimir, H. B. G. (1938). Vol. 5, 495-500, Physica. Cattaneo, C. (1948). Sulla conduzione del Calore Syracuse, Societa Tipograpfica Moderne Chapra, S. C. & Canale R. P. (2006). Numerical Methods for Engineers, New York: McGraw Hill Professional. Coughabowr, D. R. & LeBlanc, S. E. (2009). Process Systems Analysis and Control, New York: McGraw Hill Professional. Froment, G. F., Bischoff, K. B. & De Wilde, J. (2011). Chemical Reactor Analysis and Design, Hoboken, NJ: John Wiley & Sons. Golnaraghi, F., Kuo, B. C. (2010). Automatic Control Systems, Hoboken, NJ: John Wiley & Sons.

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Haji-Sheik, A. M., Minkowycz, W. J. & Sparrow, E. M. (2002). Vol. 124, 2, 307-319, J. Heat Transfer . Joseph, D. D. & Preziosi, L. (1989). Vol. 61, 41-73, Review of Modern Physics. Landau, L. & Liftshitz, E. M. (1987). Fluid Mechanics, Pergamon, United Kingdom. Levenspiel, O. (1996). Chemical Reactor Omnibook, Corwallis, OR: 1989. Levenspiel, O. (1999). Chemical Reaction Engineering. Hoboken, NJ: John Wiley & Sons. Maxwell, J. C. (1867). Vol. 157, 49, Phil. Trans. Roy. Michaelis, L. & Menten, M. L. (1913). Vol. 49, 333, Biochem Z. Mitra, K., Kumar, S., Vedavarz A. & Moallemi, M. K. (1995). Vol. 117, 568-573, J. Heat Transfer . Monod, J. (1949). Vol. 3, 371, Ann. Rev. Microbiology. Nernst, W. (1917). Die Theoretician Grundalgen des n Warmestatzes, Knapp Hall, Frankfurt, Germany. Onsager, L. (1931). Vol. 37, 405-426, Phys. Review. Ozisik, M. N. & Tzou, D. Y. (1994). Vol. 116, 526-535, ASME Journal of Heat Transfer. Qiu, T. Q. & Tien, C. L. (1992). Vol. 196, 41-49. Renganathan, K. (1990). “Correlation of Heat Transfer with Pressure Fluctuations”, Ph.D. Dissertation, Morgantown, WV: West Virginia University. Sharma, K. R. (2005). Damped Wave Transport and Relaxation, Amsterdam, Netherlands: Elsevier. Sharma, K. R. (2006). “Errors in Gel Acrylamide Electrophoresis Due to Effect of Charge”, March Atlanta, GA: 231st ACS National Meeting. Sharma, K. R. (2009). Bioinformatics: Sequence Alignment and Markov Models, New York: McGraw Hill Professional. Sharma, K. R. (April, 2006). “Finite Speed Diffusion and Delivery of Drugs and Response to Pulse Decay”, Easton, PA: 32nd Northeast Bioengineering Conference. Sharma, K. R. (February 2006). “A Fourth Mode of Heat Transfer called Damped Wave Conduction”, Santiniketan, India: 42nd Annual Convention of Chemists. Sharma, K. R. (March, 2003). “Storage Coefficient of Substrate in a 2 GHz Microprocessor”, New Orleans, LA: 225th ACS National Meeting, Sharma, K. R. (March/April 2003). “Critical Radii Neither Greater than the Shape Limit nor Less than Cycling Limit”, New Orleans, LA: AIChE Spring National Meeting. Shuler, M. L. & Kargi, F. (2002). Bioprocess Engineering Basic Concepts, Upper Saddle River, NJ: Prentice Hall. Taitel, Y. (1972). Vol. 15, 2, 369-371, Int. J. Heat Mass Transfer. Tzou, D. Y. (1996). Macro to Microscale Heat Transfer: The Lagging Behavior , New York, CRC Press. Varma, A. & Morbidelli, M. (1997). Mathematical Methods in Chemical Engineering, Oxford, UK: Oxford University Press. Vernotte, P. (1958). Paris, Vol. 246, 22, 3154-3155, C. R. Hebd. Seanc. Acad. Sci Wayne Bequette, B. (2003). Process Control: Modeling, Design and Simulation. Upper Saddle River, NJ: Prentice Hall. Xie, Y. S., Yuan, Y. X. & Zhang, X. B. (2006). Vol. 27, 24-28, Bimggmg Xuebao/Acta Aramamentarii. Zanchini, E. (1999). Vol. 60, 2, 991-997, Phys. Review B – Condensed Matter and Material Physics.

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4.8. EXERCISES 1. CSTR with Recycle Consider the operation of a CSTR with provisions for recycle as shown in Figure 4.26. Obtain the model equations for transient conversion as a function of dimensionless time. The r CSTR is operated at constant volume V. The reaction performed in the CSTR obeys the first order reaction kinetics. Part of the effluent (v-w) from the CSTR is recycled back to the feed as shown in Figure 4.26. The feed is v moles/liter at a initial concentration of reactant A at CA0.. The reaction is a first order reaction;. dC A

rA =

(4.234)

k1C A

dt

Where k1 is the first order rate constant with units of hr-1 and CA is the concentration of reactant A (mol.lit-1). The residence time in the CSTR be given by  (hr) and recycle ratio R is given by;

R

v

w

(4.235)

w

Show that the governing equation for conversion of A, X A

C A0

CA

C A0

can be

expressed as a function of the (Da = k1) and recycle ratio, R as follows;

( X A)

Da

R 1 XA 1 R R R 1

XA

dX A

Da 1 X A

1 R 1

Da

d dX A d

(4.236)

(4.238)

Show that Eq. (4.238) can be integrated by separation of variables and the solution can be written as follows;

XA

1 Da 1 Da ( R 1)

R Da

1 e

1 Da R 1

(4.239)

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Kal Renganathan Sharma

Figure 4.26. Dynamics of CSTR with Recycle.

What happens when the reflux ratio R  0, and R , 2.0. Reversible Reaction Performed in a CSTR Consider a CSTR with a single reversible reaction. kf

A

(4.240)

P kb

The reaction rate of A is first order in the forward and reverse direction and is given by; dC A

k f C A kb C P dt C P C A0 C A dC A dt

(k f

kb )C A

(4.241) kb C A0

Where kf, and kb are the forward and reverse reaction rate constants with units of (sec-1). Develop the model equation for conversion of reactant A. Use the dimensionless group of Damkohler number and conversion X if necessary. Obtain the model solution for the dynamics of reactant A. 3.0. Toricelli’s Theorem Applied to Drainage from a Storage Tank The inlet velocity of the fluid to a storage tank as shown in Figure 4.27 is v1 (m3.s-1). g h , where h is the instantaneous height of the fluid in the v1 can be given by v1 tank. The height of the tank, h(t) varies with time, t. (a) Develop an mathematical expression for the efflux velocity v2. (b) Calculate the drainage time of the tank in Figure 4.27. At time t = 0, assume that the initial height of the fluid in tank is H0.

Continuous Process Dynamics

313

Aorifice is the cross-sectional are of the outlet orifice and Atank is the cross-sectional area of the tank. Assume that the fluid is incompressible during flow.

Figure 4.27. Drainage of Tank of Fluid.

4.0. Zeroth Order Reaction in a CSTR. Consider a single reaction performed in a CSTR with a volume of the reactor V liters. AP

(4.242)

The reaction rate is given by; dC A dt

k0

(4.243)

Where k0 is the zeroth order rate constant with units of (mol/(litsec)-1). Develop the model equation for conversion of reactant A as a function of dimensionless time. Assume that the flow is incompressible and the reactor is operated at constant volume throughout the reaction. Use the dimensionless group of Damkohler number if necessary and discuss the transient behavior of conversion from start-up of the reactor to the time it reaches steady state. What happens after steady state is reached ? 5.0. Transient PFR with Zeroth Order Reaction in a PFR Consider a PFR, plug flow reactor as shown in Figure 4.13. The concentration varies along the length of the reactor. The performance of the reactor is expected to be better than that in a CSTR. Assume that the flow is incompressible and the velocity is in a state of “plug flow” i.e., velocity of the fluid across the cross-section of the tube is the same. Develop the model equations for the transient conversion of species A

314

Kal Renganathan Sharma that obeys the zeroth order kinetics. As the flow is incompressible the velocity of the fluid is same throughout the length of the reactor and is v (m3.s-1). Let z be the distance from the entrance of the reactor. Show that the governing equation can be given by; CA

v A

CA

k0

z

t

(4.244)

Where k0 is the zeroth order rate constant with units of (mol/(litsec)-1). The zeroth order reaction rate for species A can be written as follows; dC A

(4.245)

k0

dt

Discuss the transient behavior of conversion from start-up of the reactor to the time it reaches steady state. What happens after steady state is reached ? The dimensionless variables of conversion, XA, dimensionless time,  and Damkohler number Da are defined as follows; C Ai

XA Z

CA

C Ai z ; Da l

t

;

(4.246)

k0 C Ai

Show that the governing equation for concentration of species A becomes; XA

XA

Da

Z

(4.247)

Differentiating the above equation with respect to Z and with respect to  separately and subtracting one from the other the governing equation can be seen to take the form of the wave equation; 2

XA

Z

2

2

XA 2

(4.248)

6.0. Hydrolysis of Ethylene Oxide in Two CSTRs in Series Consider performing the hydrolysis of ethylene oxide in Example 4.1 in two CSTRs in series as shown in Figure 4.28. Assume incompressible flow and first order reaction reaction for species A, i.e., ethylene oxide. Develop the model equations that can be used to describe transient dynamics of hydrolysis of ethylene oxide in Example 4.1 in two CSTRs in series as

315

Continuous Process Dynamics

shown in Figure 4.28. The volumes of the two reactors are V1 and V2 respectively. The volumes of the two reactors may be assumed to be a constant during the course of the reaction. Show that the transfer functions of conversion of species A (ethylene oxide) at the outlet of CSTR 2 can be given by; Da1

g p2 s s

Da 2

Da1 1 1 Da 2

s

2

1 Da 2

1

(4.249) s

2 1

Where Da1 = (k1); Da2 = (k2); 1 is the residence time in CSTR 1 and 2 is the residence time in CSTR 2. Sketch the conversion of species A from CSTR 1 and CSTR 2 as a function of time. From the final value theorem show that at steady state the conversion from the exit of CSTR 2 would be;

X A2 s

Da 2

Da1

1 Da 2

1 Da1

(4.250)

7.0. Underdamped Oscillatory Response in Two CSTRs in Series. 1 For a step input of r s to the system in exercise 6.0 show that the output s

transfer function is given by; Da1

X A2 s s2

s2

2 1

s 1 2 Da 2

Da 2 2 1

1 Da1 1 Da 2

s 2 1 Da 2

(4.251) s

2 1

Figure 4.28. Transient Dynamics during Hydrolysis of Ethylene Oxide in Two CSTRS in Series.

316

Kal Renganathan Sharma Show that the first term in the RHS, right hand side of Eq. (4.194) can be seen to be of the form of prototypical second order process of the form; kp u

y( s ) s

2 2 ps

(4.252)

2s

1

p

Show that the gain, kp, time constant, p and damping coefficient,  of the prototypical second order process can be seen to be; kp 2 p

Da1 1 Da1 1 Da 2

1 1 Da1 1 Da 2

2 1

2

1 2 Da 2

1

1 2

(4.253)

2 1 Da1 1 Da 2

Can the damping coefficient  < 1 ? What are the ramifications of this ? 8.0. Show that the space equation for the two conversions, XA1, XA2 from the two CSTRs in Exercise 16.0 can be written as follows;

d X A1 dt X A2

1 Da1

0 1

1

Da1 Da 2

Da 2

(4.254)

2

Show that the eigenvalues are –(1+Da1) and

Da 2

1

. Confirm that negative

2

eigenvalues indicate a stable system for any value of the Damkohler number and residence times in the reactors. 9.0. Sutro Weir Consider the tank that is filled and depleted at different rates. The height in the tank varies with time and is given by the function h(t). A mass balance on the control volume of the fluid in the tank can be written as follows; vi t

v0 t

A

dh dt

(4.255)

Where A is the cross-section area of the tank. Incompressible flow is assumed. The discharge rate v0(t) in lit.sec-1 can be written in terms of the height in the tank by invoking the linear flow-head relation as found in Sutro weir. This relation is;

Continuous Process Dynamics

317

Figure 4.29. Level in Tank.

v0 t

h(t ) R

(4.256)

Where R is the resistance to flow. Plugging. Eq. (4.256) in Eq. (4.255) and obtaining the Laplace transfer function and assuming that the initial height in the tank is zero show that the transfer function can be given by; h s

R sRA 1

vi s

(4.257)

10.0. Non-Interacting Tanks. Consider two tanks in series as shown in Figure 4.30. These can be considered to be “non-interacting”. The discharge from the first tank is through the atmosphere to the second tank. The height in the first tank does not depend on the happenings in the second tank. From mass balance show that the governing equations for h1(t) and h2(t) can be written as follows; vi t

h1 R1

h1

h2

R1

R2

A1 A2

dh1 dt

dh2 dt

Where A1 and A2 are the cross-sectional areas of the two tanks in Figure 4.30.

(4.258)

318

Kal Renganathan Sharma

Figure 4.30. Two Non-Interacting Tanks.

Show that the transfer function can be written as; h2 s

R2

vi s

sA2 R2 1 sA1R1 1

(4.259)

1 . Confirm that for any positive value of s resistances and cross-sectional areas the output response is monotonically stable. 11.0. Two Interacting Tanks Consider two tanks in series as shown in Figure 4.31. These can be considered to be “interacting”. The discharge from the first tank feeds directly into the second tank. The height in the first tank does depend on the happenings in the second tank. From mass balance show that the governing equations for h1(t) and h2(t) can be written as follows;

Further consider a step change of r ( s )

vi t

h1

h2

R1

R1

h1

h2

R1

R2

A2

A1

dh2

dh1 dt

(4.260)

dt

Where A1 and A2 are the cross-sectional areas of the two tanks in Figure 4.31. Show that the transfer function for the heights in the tanks, h1(s) and h2(s) can be written as follows;

319

Continuous Process Dynamics

Figure 4.31. Two Interacting Tanks.

R1 sA2 R2

h1 s vi s

s

2

A1 A2 R1 R2 1 R2

h2 s h1 s

1

A R A2 R2 s 1 1 1 R2

1 R1 sA2 R2 1

R2

(4.261) 1

(4.262)

12.0. Underdamped Oscillatory System Consider a step change to the height in tank 1 in Figure 4.31 Confirm that the transfer function for level in tank 1, h1(s) conforms to a prototypical second order system. Under what conditions of the cross-sectional areas of the tanks and the resistances to flow in the two tanks would there be a underdamped oscillatory response ? Where does the fluid come from for causing the “overshoot “ ? 13.0. Average Concentration of Oxygen in Islets of Langerhans during Parabolic Fick Diffusion Oxygen availability becomes limited in some regions of the tissue [55]. An oxygen reaction and diffusion model was developed by Colton [56]. The transient diffusion of oxygen in blood plasma and tissue that undergoes simultaneous reaction by Michaelis-Menten kinetics is modeled. At the asymptotic limit of high reactant concentration the reaction rate becomes zero order. The model solution consists of a steady state and transient part. This removes the heterogeneity in the governing

320

Kal Renganathan Sharma equation. The governing equation for oxygen diffusion and reaction in Cartesian coordinates in one dimension accounting for Fick‟s diffusion can be written as; CO2 t

2

DT

CO2

kC E 0CO2

z2

CM

(4.263)

CO2

Where DT is the diffusion coefficient in the tissue, CE0 is the total enzyme or complexation species concentration and CM is the Michaelis constant and k is the rate constant. The oxygen consumption rate is assumed to obey the Michaelis-Menten kinetics. The governing equation describes the interplay of transient diffusion and metabolic consumption of oxygen in the tissue. The concentration of oxygen, CO2 can be expressed in terms of its partial pressure, pO2 . This is obtained by using the Bunsen solubility coefficient, t such that; CO2

t

pO2

(4.264)

Substituting Eq. (4.206) in Eq. (5.195), Eq. (4.205) becomes; pO2 t

t

2 t DT

pO2 z

2

kC E 0 pO2 ' CM

(4.265)

pO2

The product tDT can be seen to the product of solubility and diffusivity and hence is the permeability of oxygen in the tissue. The Michaelis constant CM, is also modified, C‟M expressed in units of mm Hg. The initial condition can be written as; pO2

pO20 , t = 0

(4.266)

From symmetry at the center of the slab; pO2 z

(4.267)

0

At the surface,(z=a) the oxygen diffusive transport from within the tissue must be equal to the oxygen transport by convection across the boundary layer surrounding each tissue;

Ji

ki

m ( pO2 m

pO2 ( Ri ))

pO2

tD T

z

(4.268) z a

An average partial pressure of oxygen in the tissue or slab can be defined as;

321

Continuous Process Dynamics

1 2a

pO2

a

(4.269)

pO2 dz a

Integrating Eq. (4.265) with respect to z between the finite limits of –a and +a and substituting Eq. (4.269) in the resulting equation;

pO2 t

t

ki

t

pO2 m

pO2

kCE 0

(4.270)

Eq. (3.112) is valid at the asymptotic limit of high oxygen concentration. Obtain the transient average partial pressure of oxygen in the finite slab as a function of time. Obtain the transient average concentration for oxygen in a finite slab at the asymptotic limit of low oxygen concentration.

Chapter 5

PROPORTIONAL, PROPORTIONAL INTEGRAL, PROPORTIONAL DERIVATIVE FEEDBACK CONTROL 5.1. CONTROL SYSTEMS The field of controls is an interdisciplinary one. It is a confluence of mathematics and engineering. Electrical, mechanical, chemical and computer are branches of engineering that have courses where students learn control theory. Dynamics of a system is of primary importance in this field. It all began when the scholar J. C. Maxwell investigated the dynamics of the centrifugal governor in an 18th century Boulton and Watt engine. He looked at self-oscillations and how the lags in the system results in overcompensation that leads to an unstable system. The conditions when a system can be expected to undergo underdamped oscillations was discussed in Chapter 4.0. Mathematical modeling of the process is an important task in the analysis and design of control systems. When the dynamics of the system modeled can be converted to Laplace domain they can also be represented in block diagram format. The control system may be linear or non-linear. Analysis of non-linear systems need numerical solutions. Closed form analytical solutions may be obtained for linear systems. Sometimes when multiplicity in solution is seen, another criterion is needed to select the correct solution from the results obtained. Different components such as mechanical, thermal, fluid, chemical/concentration of reacting species, pneumatic and electrical, sensors, robotic manipulators, actuators, and computers can be part of a control system. The governing equations for these systems are represented by differential equations, originating from Newton‟s second law of motion of particle, Kirchoff‟s law of circuits. The results obtained from these models are only good for the assumptions made in deriving the model. A number of problems can be represented by ordinary differential equations.

5.1.1. On-Off Control On-off controllers are used in centralized heating and air conditioning in modern homes, control of level in a tank used as a process vessel, remote control of Christmas tree lighting etc. For example, consider a CHP, combined heat and power system. This technology is also

324

Kal Renganathan Sharma

called as cogeneration. A gas turbine is used to drive an electrical generator and the heat from the exhaust of the turbine is used to produce thermal energy in the form of steam, hot water or hot air. In Figure 5.1 is shown an on-off control strategy for a typical CHP system,[Mackay, 2003]. Gas turbine ratings can vary between 500 kw to 30 kw in cases of micro turbines. Control system is needed to safeguard the system from spikes in loads as various appliances are used. Comfort heating and domestic water heating units cycle ON and OFF. The CHP system is operated using a sub atmospheric Brayton cycle. A heating system is designed to operate in cyclical fashion. For example, a commercial water heater or a furnace operates in the full ON or full OFF mode. When heating system is required and turned ON the system starts and provides the heat using the rejected heat from the gas turbine. Electricity is produced in parallel. When the TT, temperature transmitter in Figure 5.1 detects a temperature when compared with the set point temperature Tsp calls for heating, a signal is sent to the ON/OFF fuel valve enabling it but not opening it. In a similar manner the PT, pressure transmitter detects the pressure and this value is compared with the set point pressure, Psp in the PC, pressure controller. A signal is sent to the ON/OFF fuel valve. Thus On-Off controller is used to shut down the system when no more heat is required. This can be found when Tm > Tsp, i.e., when the measured temperature exceeds the set point temperature by a threshold amount, .

5.2. PROPORTIONAL CONTROL A control action that is taken that is proportional to the error detected is called as Proportional Control. The equations that can be written to capture such control actions are as follows;

Figure 5.1. On-off Control of Gas Turbine Operated Power Plant.

Proportional, Proportional Integral, Proportional Derivative Feedback Control

325

Pv1 = b1 + kc1(Tsp – Tm)

(5.1)

Pv2 = b2 + kc2(Psp – Pm)

(5.2)

Where the error in temperature and pressure can be written as follows; e1 = Tap - Tm

(5.3)

e2 = Psp - Pm

(5.4)

b1 and b2 in Eq. (5.1) and Eq. (5.2) are the bias terms. At steady state when the error e1(t) and e2(t) becomes zero the bias terms are equal to the steady state temperature Tss and steady state pressure, Pss. kc1 and kc2 are the proportional gain of the temperature controller and pressure controller respectively.

5.3. CONTROL BLOCK DIAGRAMS A controller transfer function can be developed for every control action taken. Consider, for example the proportional control action illustrated in section 5.2. The proportional control law for temperature and pressure can be written by substituting the steady state pressure and steady state temperature in the bias terms in Eqs. (5,1. 5.2) as follows; Pv1 - Tss = kc1(Tsp – Tm)

(5.5)

Pv2 -Pss = kc2(Psp – Pm)

(5.6)

The errors e1(t) and e2(t) can be written as; e1(t) = Tap - Tm(t)

(5.7)

e2(t) = Psp - Pm(t)

(5.8)

The controller outputs can be written as c1(s) and c2(s) in the Laplace domain as follows; c1(s) = kc1 e1(s) = gc1(s)e1(s)

(5.9)

c2(s) = kc2 e2(s) = gc2(s)e2(s)

(5.10)

Eq. (5.5) and Eq. (5.6) is the transfer function form for a proportional only controller. This can be shown as a block diagram as shown in Figure 5-2.

326

Kal Renganathan Sharma

Figure 5.2. Block Diagram Relation for Transfer Function for Temperature Controller and Pressure Controller as Shown in Figure 5.1 for CHP Systems.

In a similar manner block diagram relation for valve and the process system under consideration can be constructed. In addition a transfer function may be written for disturbance or load and measurement or sensor. A control block diagram for the temperature and pressure control as shown in Figure 5.1 for a CHP, combined heating and power system can be constructed. This is shown in Figure 5.3.

Figure 5.3. Control Block Diagram for Pressure, Temperature Control for CHP System.

Proportional, Proportional Integral, Proportional Derivative Feedback Control

327

The pressure and temperature in the gas in the compressor are measured using temperature and pressure sensors. The sensor location is such the combined effect of process and the load are captured in the measurement. The error signal is generated in the controller by comparisons with the set point temperature, Tsp and set point pressure. The control action taken is proportional to the error or on-off as the case may be. In any case the control transfer function is shown as a block in Figure 5.3. The valve transfer function also is an important link to the process. The process transfer function can also be shown in the box. It can be a prototypical first order or second order process as discussed in the previous chapter. This kind of control action is called as feedback control. The temperature and pressure measurements were used in the control action taken. Gains can be defined as the ratio of steady state change in output in response to a steady state change in input variable. This can be defined for the valve, process, controller etc. Valve Gain:Valve gain may be defined as; kv

v Pv1

(5.11)

Where v is the change in flow rate and Pv1 is the change in valve top pressure. An example of on-off control action using valve can be seen in end of chapter exercise 2.0. The gain is said to be positive when the flow rate increases with increase in the valve top pressure. The gain is said to be negative when the flow rate decreases with increase in valve top pressure. Process Gain: Process gain may be defined as; V Pv1

k p1

(5.12)

For example in exercise 2.0 at the end of chapter the process gain is the change in the volume of the CSTR in response to a change in valve top pressure. For a positive gain the volume of the reactor contents increases and for a negative gain the volume of the reactor contents decreases when the valve top pressure is increased. The controller gain is kc as defined in Eq. (5.5) and Eq. (5.6).

5.4. OFFSET USING PROPORTIONAL ONLY CONTROLLERS Consider a simplified block flow diagram of a proportional controller for a prototypical first order process. Although Figure 5.4 is for PI controller it can be used for illustration of proportional controller only as well. kp

g p (s) gc (s)

ps

kc

1

(5.13)

328

Kal Renganathan Sharma r(s) = gCL(s) y(s)

(5.14)

A closed loop transfer function gCL(s) is defined as shown in Eq. (5.10). This can be seen from simplification of the transfer functions in the block diagram in Figure 5.4 to be; g p (s ) g c (s )

g CL ( s )

(5.15)

1 g p (s ) g c (s )

Substituting Eq. (5.9) in Eq. (5.11); kc k p ps

g CL ( s )

1

kCL

kc k p kc k p

1 1

kCL

Where,

1

CL s

(5.16)

1

ps

kc k p kc k p

1

(5.17)

p CL

kc k p

1

The closed loop transfer function gCL(s) conforms to the prototypical first order process. Stability considerations require that; CL > 0

(5.18)

kckp > -1

(5.19)

The time output response to a step change in input with magnitude Q can be shown to be; t

y(t )

QkCL 1 e

CL

(5.20)

At steady state or after infinite time, y(t) = QkCL

(5.21)

Only for values of kCL = 1 there is no offset. For other values of kCL 1, an off-set will be realized after enough time has elapsed since the step change was initiated. The process output will not be equal exactly to the set point. The final value theorem can be used to estimate the offset when P only control is used.

Proportional, Proportional Integral, Proportional Derivative Feedback Control Lt e(t ) t

Lt se(s ) s

Lt s r (s )

0

s

s

y( s )

0

Offset = Lt sr ( s ) 1 g CL (s )

329

kCL

Q 1

CL s

0

1

Q 1 kCL

Q 1 kc k p

(5.22)

5.5. PROPORTIONAL-INTEGRAL, PI CONTROL It can be seen from the discussion in sections 5.3-5.5 the use of P only control can lead to an off-set between the desired set point and actual output. The control action can be modified by inclusion of a component that is proportional to the integral of the error. This can be in addition to the proportional to the error component used in P only control. This kind of control is called proportional-integral or PI control. The process history is taken into account. In this method the rectification takes place until the offset vanishes. Say there is an offset for a said period of time. The integral of the error would be finite. Control action is proportional to the integral of the error. This would reduce the error term. This progressively down can lead to zero error. The two tuning parameters are kc and kI. The PI control action law can be written as follows; t

u (t )

kc e(t ) kI e

d

(5.23)

d }

(5.24)

0

Eq. (5.19) can also be written as;

u (t )

kc {e(t )

1

t

e

I 0

I is the integral time and is given by kc/kI, the ratio of the tuning parameters. Obtaining the Laplace transform of Eq. (5.20);

u (s )

kc {e( s )

s I e( s ) } kc e( s ) s I s

1

(5.25)

I

Figure 5.4. Simplified Control Block Diagram for PI Controller for Prototypical First Order Process.

The control transfer function for PI control is thus given by gc(s) where u(s) = gc(s)e(s).

330

Kal Renganathan Sharma

gc (s)

s

kc

1

I

s

(5.26)

I

Example 5.1. PI Control Action for Prototypical First Order Process Consider any prototypical first order process. Consider a PI controller used to control this process. Let the process gains, kp = kc =1. What would be the time response of the process output variable y(t) to a step change in the set point, r(s) = 1/s. The time constant of the controller, I becomes 1% of the time constant of the process, p. The simplified control block diagram is given below. Disturbances are neglected and valve and measurement dynamics are lumped into the process transfer function. Develop the expression for the output variable y(s) in the Laplace domain and time domain y(t). Sketch the response y(t) as a function of time. The transfer functions for prototypical first order process, gp(s) and that for PI control, gc(s) can be seen to be; kp

g p (s)

1

ps

g c (s )

kc

Is

kp

g p (s ) g c (s )

kc

1 ps 1

1

ps

(5.27) 1

Is

1

Is

(5.28)

Is

1 s

r (s )

y(s) = gp(s) u(s) = gp(s) gc (s) e(s) = gp(s) gc (s)(r(s) – y(s))

I

y( s ) p Is

L1

Is 2

0.01 0.01

2 2 ps

1 2s

I

p

s 1

Is

1

0.01

p

1 I s 1 (s )

(5.30)

0.01 p s 1

1 y( s ) 1 (s)

p

0.02s

1

Is

y( s )

0.01

L1

Where (As+B)(Cs+D) = 0.01p2s2 + 0.02sp + 1

(5.29)

2 2 ps

0.02s

1 As

B

L1

p

1 1 (s )

1 CS

D

(5.31)

(5.32)

Proportional, Proportional Integral, Proportional Derivative Feedback Control

331

Eq. (5.12) in the time domain can be seen to be;

0.01

e p

Bt A

e

A

Dt C

(5.33)

B

2t

1

L1 s

2 2

2

s

1 s 1

e

1

2

sin

2

1

(5.34)

2 tan

t

1

1

2

y(t) is the sum of Eq. (5.13) and Eq. (5.14). It can be seen from Figure 4.5 that y(t) can undergo a overshoot. This would happen when the damping factor,  < 1.0. The system described by Eq. (5.11) can be seen to be an underdamped system. The damping factor,  = 0.1 which is less than 1.0. It is not clear as to what is causing the overshoot from the control action of the process. Mathematically the control transfer function and process combine in a form where an underdamped system can occur. Figure 5.4a is an example of a simplified closed loop of a controller and process. A closed-loop transfer function can be defined as follows;

y( s )

g CL ( s )r (s )

g p (s) gc (s)

1 g p (s) g c (s)

The denominator in Eq. (5.15) is called the characteristic equation. For stability, gp(s)gc(s) >>-1. The open loop configuration is shown in Figure 5.4. The denominator in Eq. (5.15) is called the characteristic equation. For stability, gp(s)gc(s) >> -1. The denominator in Eq. (5.15) is called as the characteristic equation. For stability, gp(s)gc(s) >>-1. The open loop configuration is shown in Figure 5.4a.

Figure 5.5. Open Loop Configurations of Controller and Process.

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Kal Renganathan Sharma

Example 5.2. Closed Loop Transfer Function Derive the closed-loop transfer function between y(s) and l(s) and r(s)for the following control block diagram. This is for a feed forward/feedback controller.

y(s) = X(s) + Z(s)

(5.36)

Z(s) = GL(s) L(s)

(5.37)

X(s) = Gp(s) u(s)

(5.38)

u(s) = v(s) + w(s)

(5.39)

V(s) = Gc(s) e(s)

(5.40..)..

w(s) = Gf(s) L(s)

(5.41)

r(s)- ym(s) = e(s)

(5.42)

y(s) = Gp(s) {Gc(s){r(s) – ym(s)}+ Gf(s) L(s)} +GL(s)L(s)

(5.43)

Now, ym(s) = Gm(s) y(s)

(5.44)

y( s )

G p ( s )Gc ( s )r ( s )

L( s ) G f s G p s

GL s

(5.45)

1 G p ( s )Gc ( s )Gm ( s )

Example 5.3. PI Control of Voltage Supplied to Automatic Washing Machine The operation of an automatic washing machine comprises of the following Steps; (i) wash step; (ii) rinse step and; (iii) dehydration step. A brushless DC motor drives an agitator and rotating tub during the wash step. During the rinse step only the rotating tub is used. Voltage is applied to the motor and controls the torque on the motor. The model equations that relate torque to agitator RPM are as follows; d

m

dt

Bm Jm

m

1 Jm

m

L

(5.46)

m is the rotor speed and m is the applied torque to the agitator. Both rotor speed and applied torque vary with time. Although the model equation predicts the relation between the rotor speeds with the applied torque, there is no model available for applied voltage on the motor and the applied torque. The controller will have to increase the voltage or decrease the voltage, check the angular speed against set point sp and adjust the voltage again till that set point angular speed is attained. Use a PI controller and develop a block diagram for the voltage control of the automatic washing machine. Noise and vibration need be minimized. Comment on the closed-loop stability transfer function, gCL(s).

Proportional, Proportional Integral, Proportional Derivative Feedback Control

333

Figure 5.5a. Control Block Diagram of Feedback/Feed forward Control.

Figure 5.6. Block Diagram for PI Control of Automatic Washing Machine.

Toshiba washing machine [2] comprises of an electric motor where the torque needed for wash, rinse and dehydration steps are generated. It also contains a current detector that measures current flowing into the motor. A torque controller is used to optimize the torque developed for each of the wash, rinse and dehydration steps. The current measurement is used as feedback prior to control action being taken. Typical automatic washing machines comprise of a DC motor that is used to drive an agitator and a rotating tub in the wash step and only the rotating tub in the rinse step. A third dehydration step is also completed. According to the driving condition of the motor the voltage applied to the motor is increased or decreased thereby controlling the torque developed by the motor in order to drive the agitator and tub. PI control action is on the rotational speed of the motor. The basis or target speed is a ref, angular speed or RPM at the dehydrating step. A measured speed  can be acquired during the operation of the machine. One of the problems encountered during the operation of such washing machines is as follows. A rotational speed of the motor is proportional to the torque developed. The developed torque, on the other hand, is not proportional to the voltage supplied. The control action is by increasing or decreasing the applied voltage. An unstable operation condition may be encountered. This would happen when the operating speed  and target angular speed

334

Kal Renganathan Sharma

ref will diverge and not be equal to each other. Further, a motor speed variation is increased in the wash operation from 0. – 150 RPM in about 200 milliseconds the PI control action.cannot be applied to the wash step. Motor torque control that is more precise is the goal of the Toshiba technology. Electric current measurements are used as feedback in the torque control. d

Bm

m

dt

m

m

Jm

L

(5.47)

Jm

Obtaining the Laplace transform of Eq. (5.25);

s

m

Bm

0

s

m

Jm m (s)

L

s J ms

L

s J ms

(5.48)

(5.49)

Bm

m (s)

s

g p (s )

In Figure 5.6,

L

s

Jm

s

m (s )

Or,

m (s)

s

(5.50)

Bm

The control transfer function for PI Controller can be written as;

g c (s )

kc

1

Is

s

(5.51)

I

y(s) = gp(s) u(s)

(5.52)

y(s) = gp(s) gc(s) {r(s) – y(s)}

(5.53)

g p ( s ) g c ( s )r ( s )

y( s )

(5.54)

1 g p (s) g c (s)

For closed loop stability, gp(s) gc(s) >> -1. i.e.,

1

s

m (s ) 2

s

sJ m

L

kc Bm

Is I

1

1

(5.55)

Proportional, Proportional Integral, Proportional Derivative Feedback Control

335

5.6. CONDITIONS FOR UNDERDAMPED RESPONSE OF PI CONTROL OF PROTOTYPICAL FIRST ORDER PROCESS Consider a prototypical first order process. Consider PI control of a prototypical first order process. The block diagram in shown in Figure 5.4. The control transfer functions for the process and controller are as follows; kp

g p (s)

1

ps

g c (s )

kc

(5.56) 1

Is Is

The closed loop transfer function, gCL(s) can be written as; g CL ( s )

g p (s ) g c (s )

(5.57)

1 g p (s ) g c (s )

Consider a step change in the input r(s) = 1/s. The following is the analysis of the output response to a step change in input. kc k p

y(s) = r(s)gCL(s) =

Is

Is

1 s 1

Is

or,

1

ps

kc k p

(5.58) Is 1

ps

s

I

p

kc k p

s

2

s

1 kc k p I

1 k p kc

1

y( s )

1

1

kc k p

I

p

s

2

s

I

I

1 kc k p

(5.59) kc k p

Comparison of the first term of RHS of Eq. (5.37) and the Laplace transfer function of prototypical second order process system as given by Eq. (5.59) in chapter 3.0 reveal that for certain conditions the system can become underdamped. This would happen when the damping ratio,  < 1. Thus, 2

I

p

kc k p

2

I

1 kc k p kc k p

(5.60)

336

Kal Renganathan Sharma 1 kc k p

I

2

p kc k p

For damping factor,  < 1, 1 kc k p

I

2

p kc k p

1

4 p kc k p

Or,

I

1 kc k p

2

(5.61)

(5.62)

Thus for small values of integral time, I, as given by Eq. (5.62) the closed loop system of PI control and prototypical first order process would become underdamped!

5.7. PD, PROPORTIONAL DERIVATIVE CONTROL AND PID, PROPORTIONAL INTEGRAL DERIVATIVE CONTROL 5.7.1. Development of Control Action Law The P, proportional only control action can be modified by added a component to the control action that is proportional to the derivative of the error;

u (t )

kc e(t ) kD

de(t ) dt

(5.63)

The control action law depicted in Eq. (5.41) is called as PD, proportional derivative control. It can be used in applications where it is important to preserve the curvature of the process variable with respect to time. For example a linear increasing function or constant function cannot have a steep curvature. Control action taken that is proportional to the derivative of the error can have a smoothing effect on the output response. The control transfer function for Eq. (5.41) can be written as; u (s)

kc e( s )

skD e(s ) 0

u (s)

e( s )kc 1 s

(5.64)

D

Where, D is the time constant of the derivative control. kD D

kc

(5.65)

Proportional, Proportional Integral, Proportional Derivative Feedback Control

337

The P, proportional only control can be modified by addition two components, one proportional to the derivative of the error and the other proportional to the integral of the error. t

u (t )

kc e(t ) kI e

d

kD

0

de(t ) dt

(5.66)

The control action law depicted in Eq. (5.44) is called PID control. Ideal PID control is not physically realizable. No instrument can take a perfect derivative. Real PID control can be affected after suitable modification of the control transfer function. The control transfer function for ideal PID control can be seen to be;

u (s )

kc

I Ds

2

s

s

I

1

(5.67)

I

Example 5.4. PI Control during Desalination of Seawater using Nanoporous Carbon Consider sea water pumped across a packed bed of graphene nanoplatelets, GNP with molecular sieve diameter ~ 5A. Assume a voidage of 0.75 and material density of carbon of 1.8 gm/cc. The interfacial area of ~ 5 million m2/gm. The inlet concentration of sea water can be taken as 3.6% by weight. Potable drinking water is filtered from the column at 1 m/s. The packed bed reached saturation after it has passed through 20 cm of the bed. The concentration of NaCl in potable water that is permitted is 30 ppm. For PI control of the packed bed operation develop the close loop transfer function. Under what conditions would there be instability; The mass balance of salt in the liquid phase pumped across the packed column at transient state can be written as;

(Flow in) - (Flow out) - (Amount Filtered) = (Accumulation) Let L = length of the packed bed column (m), A – cross-sectional area of the packed bed (m2) a-Interfacial area for mass transfer between graphene nanoplatelets, NGP and water m – molar density (mol/m3) v – superficial velocity of the fluid (m/s) xA – weight fraction of salt in water mvA(xA/z) + kx(xA – 0)(a/A) = Am(xA/t) Let Z = z/L;  = vt/L; XA = xA/xA*; Dam = (kxLa)/(vAm), Damkohler number (mass transfer) xA* = saturation concentration of salt in the packed bed column The fluid can be assumed to be incompressible and constant density.

338

Kal Renganathan Sharma The mass balance equation in transient state can be seen to become; (XA/Z) – DamXA = (XA/)

Obtaining the Laplace transforms and for cases where (XA/Z = ) Eq. () can be seen to become; XA(s + Dam) = /s or XA = /s/(s+Dam) Above Eq. can be seen to be of the form XA(s) = u(s) gp(s) Thus, u(s) = /s and gp(s) = 1/(s+Dam) The closed loop transfer function for the PI control of a packed bed adsorption process for desalination of sea water using NGP can be found from the closed loop transfer relation;

g CL ( s )

g p (s ) g c (s )

1 g p (s ) g c (s )

The weight fraction of salt in water can be measured and used for control purposes in the effluent water. Example 5.5. PI Control during Hydrolysis of Ethylene Glycol Consider PI control for hydrolysis of ethylene oxide in a CSTR as described in worked example 4-1. Develop the closed loop transfer function. Under what conditions would there be instability? Ethylene glycol is prepared in a CSTR by hydrolysis of ethylene oxide. The reaction can be written as follows; C2 H 4O

H 2O

C 2 H 4 OH

2

(5.68)

The CSTR is assumed to be operated at constant volume and constant temperature and with excess water. A model was developed to find the concentration of each species as a function of time in Example 4-1. From Eq. (4.6) XA

Da (1

Da

dX A d

XA)

dX A d

X A (1 Da )

(5.69)

Proportional, Proportional Integral, Proportional Derivative Feedback Control

339

Figure 5.7. PI Control during Hydrolysis of Ethylene Oxide in a CSTR.

Obtaining Laplace transform of Eq. (5.47); Da 1 1 Da s s

X A(s )

(5.70.)

Eq. (5.52) can be seen to be of the form; y(s) = u(s).gp(s)

(5.71)

where, u(s) = 1/s and gp(s) = (Da/(1+Da +s)). For PI conrol gc(s) is given by Eq. (5.26). The closed loop transfer function gCL(s) for PI control of hydrolysis of ethylene oxide to ethylene glycol can be written as follows;

g CL ( s )

g p (s ) g c (s )

1 g p (s ) g c (s ) kc Da

y( s )

X A (s )

1 s

1 Da 1

1 Is

Is

s

(5.72)

1 s I s 1 Da

kc Da

Is

Eq. (5.50) can be seen to be;

1

kc

y( s )

2

kI

s kI Da

1 Da (1 kc s kc Da

1

s2 s kI Da

s 1 Da 1 kc kc Da

(5.73)

1

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Kal Renganathan Sharma

The RHS, right hand side of Eq. (5.51) consists of two terms. The second term is the response to a step change of a prototypical second order system as given by Eq. () in chapter 3.0. When the damping factor,  < 1 the system can become underdamped. This can be seen to happen when;

4kc2 Da

kI

1 Da 1 kc

2

(5.74)

The first term in RHS of Eq. (5.51) when inverted from the Laplace domain would contain two exponential in time terms; Ae r1t " Ber2t

(5.75)

When the roots r1, r2 become complex the contribution to the output response can become sinuous! This happens when “b2 – 4ac” < 0 in the quadratic expression in the denominator of the first term in RHS of Eq. (5.51). This happens when;

4kc2 Da

kI

1 Da 1 kc

2

(5.76)

Eq. (5.76) is identical with Eq. (5.74)! Example 5.6. On the Use of PI Proportional Integral Control and Estimation of Static Formation Temperature in Oil Wells The importance of the use of initial condition was discussed in Chapter 4.0 (kindly see "sensitivity" of initial conditions to overshoot"). SFT, static formation temperature of oil wells is used during shut-in time process. Mathematical models can be developed in order to describe the processes involved during the well-bore formation system. The model equations are PDEs, partial differential equations that accounts for the heat transfer processes. Transient convective heat transfer due to circulation losses to the surrounding rocks of the well is an example of a process involved in well-bore formation system. A strategy of PI, proportional integral control can be applied in order to obtain the solution of an IHT, inverse heat transfer problem. This is used in the estimation of the SFT, static formation temperature of the oil well. The error between logged temperature and simulation temperature is the feed-back during shut-in time process. The SFT, static formation temperature, is unknown. This is the initial condition of the governing PDEs. Espinosa-Paredes et al. [2009] tested this method in two oil wells MB-3007 and MB-3009 in Gulf of Mexico. Only one temperature measurement for each fixed depth of estimate was needed. In the Fourier representation of transient temperature the initial accumulation in temperature condition is taken as zero. This can lead to overshoot that is more a mathematical artefact than a heat transfer event as discussed in Chapter 4.0. The damped wave heat conduction and relaxation equation can be used in the mathematical model. The IHT problem of using measured temperatures at a certain depth as a function of time in order to estimate the initial condition can be used with a non-zero

Proportional, Proportional Integral, Proportional Derivative Feedback Control

341

accumulation of temperature initial condition. This may lead to more accurate estimates of the SFT, static formation temperatures.

5.7.2. Time Response of PD Control of Prototypical First Order Process Consider a prototypical first order process. Consider PD control of a prototypical first order process. The block diagram in shown in Figure 5.4. The control transfer functions for the process and controller are as follows; kp

g p (s )

1

ps

g c (s )

kc 1 s

(5.77) D

The closed loop transfer function, gCL(s) can be written as; g p (s ) g c (s )

g CL ( s )

(5.78)

1 g p (s ) g c (s )

Consider a step change in the input r(s) = 1/s. The following is the analysis of the output response to a step change in input. kc k p

y(s) = r(s)gCL(s) =

1

ps

1 s

(5.79)

kc k p

1

Ds

k p kc 1 s

y( s ) s s(

kc k p

p

1

1

ps

or,

1

Ds

D

(5.80)

D ) kc k p

The inverse Laplace transform in Eq. (5.58) can be found by the method of Heaviside expansion. Let,

y( s )

p(s ) q (s )

A1 s

a1

k p kc

The poles of Eq. (5.59), are 0, p

D k p kc

A2 s

a2

respectively.

(5.81)

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Kal Renganathan Sharma

A1

p a1

k p kc

q ' a1

k p kc

p

k p kc

p

(5.83) D

k p kc

t

D k p kc

y(t ) 1 (

(5.82)

D k p kc

1

A2

1

D k p kc

)e

p

D k p kc

(5.84)

Eq. (5.62) is stable decays monotonically. For positive values of kp, kc, p and D the system will not exhibit an overshoot. Thus switching from PI control to PD control the occurrence of “overshoot” or underdamped system has vanished. The conditions when an overshoot can be expected from mathematical analysis for PI control of a prototypical first order system have been derived in section 5.6. Example 5.7 PD Control during Hydrolysis of Ethylene Glycol Consider PD control for hydrolysis of ethylene oxide in a CSTR as described in worked example 4-1. Develop the closed loop transfer function. Under what conditions would there be underdamped response? Ethylene glycol is prepared in a CSTR by hydrolysis of ethylene oxide. The reaction can be written as follows; C2 H 4O

H 2O

C 2 H 4 OH

2

(5.85)

The CSTR is assumed to be operated at constant volume and constant temperature and with excess water. Aa model was developed to find the concentration of each species as a function of time in Example 4-1. From Eq. (4-14) XA

Da (1 X A )

Da

dX A d

dX A d

(5.86)

X A (1 Da )

Obtaining Laplace transform of Eq. (5.86); X A(s )

Da 1 1 Da s s

(5.87)

Eq. (5.87) can be seen to be of the form; y(s) = u(s).gp(s)

(5.88)

Proportional, Proportional Integral, Proportional Derivative Feedback Control

343

where, u(s) = 1/s and gp(s) = (Da/(1+Da +s)). For PD control gc(s) is given by Eq. (5.77). The closed loop transfer function gCL(s) for PD control of hydrolysis of ethylene oxide to ethylene glycol can be written as follows; g p (s ) g c (s )

g CL ( s ) y( s )

1 g p (s ) g c (s ) Dakc 1 s

1 s s2

X A(s )

s 1 Da

Da

D

(5.89)

D kc

Dakc

Eq. (5.89) can be seen to be; Dakc

y( s ) s

2

s 1 Da

Dakc

D

Da

D kc

Dakc

s s

2

s 1 Da

Da

(5.90) D kc

Dakc

There are two terms in the Laplace domain in Eq. (5.90). The first term can be inverted in the time domain to the following form; Dakc

De

at

sin

(5.91)

t

Where 1 Da (1 2

a

Where, 2

D kc )

1 Da 1

Da 2 kc2

D kc

2

(5.92)

4

The second term in Eq. (5.90) when inverted to the time domain can lead to an underdamped response under certain conditions. This happens when the “damping factor”,  < 1. This is when; Dakc

2

1 Da 1

D kc

1

(5.93)

Example 5.8 Control of Intermediate in Reactions in Series Consider the following scheme of reactions performed in a CSTR. k1

A B

R k2

R

B

S

(5.94)

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Kal Renganathan Sharma

The scheme of reactions shown in Eq. (5.94) can be seen during the manufacture of chloroform by chlorination reactions as follows; CH 4

Cl2

CH 3Cl CH 2Cl2 CHCl3

CH 3Cl

Cl2 Cl2 Cl2

HCl

CH 2Cl2

HCl

CHCl3

HCl

CCl4

(5.95)

HCl

Methane can be chlorinated to manufacture chloromethane, dimethylene dichloride, chloroform and carbon tetrachloride by successive chlorination reactions. The 2011 Spring prices for CHCl3, chloroform is 75 cents per pound and CCl4, carbon tetrachloride is 32.5 cents per pound. Hence it is profitable to optimize the intermediate yield of CHCl3. Comparing the scheme of reactions shown in Eq. (5.94) and the chlorination reactions shown in Eq. (5.95) “A” can be CH4, “B” can be Cl2, “R” can be CHCl3 and S can be CCl4. Levenspiel (3) discusses the kinetics of the scheme of reactions given by Eq. (5.94) and gives an expression for CR,max, an optimal intermediate concentration. There is interest in designing a PI control for the intermediate product R as shown in Figure 5.8. Discuss the PI control for the concentration of R. What are the process and control transfer functions? What is the closed loop transfer function, gCL(s). Under what conditions would there be an underdamped response?

Figure 5.8. Feedback Control by PI Controller of Intermediate Product R in a CSTR Component Material Balance.

Proportional, Proportional Integral, Proportional Derivative Feedback Control

345

The material balance of components A, B, R and S can be written over the control volume as follows; (mass rate in) - (mass rate out) – (mass rate reacted) = (mass accumulated)

(5.96)

Species A (Methane) dC A

(5.97) dt The rate of the reaction of chlorination of methane can be given by a pseudo-first order expression as; v C Ai

CA

rA

kC AV

k1C A

V

dC A

(5.98)

dt

Species R (Chloroform) v C Ri

(k1C A k2C R )V V

CR

dC R dt

(5.99)

Species S (Carbon Tetrachloride) v 0 CS

(k2C R )V V

dCS

(5.100)

dt

V , and Eq. (3.8) and Eq. (3.10) v

It can be noted that the residence time in the CSTR, would become; C Ai

C Ri

CA

k CA

(k1 C A

1 k2

CS

k2 C R

dC A

(5.101)

dt

CR )

dCS dt

dCR dt

(5.102)

(5.103)

Let conversion, XA, Damkohler number Da1, Da2 and dimensionless time,  are as follows;

346

Kal Renganathan Sharma C Ai

XA

CA

C Ai t

(5.104)

Da1

k1

Da 2

k2

Obtaining the Laplace transform of Eqs.(5.101-5.103); C Ai

C A (s )

CR (s )

s 1 Da1

(5.105)

s

C Ri

Da1C Ai

s 1 Da 2

s 1 Da1

s

s

1 Da 2

s

Da 2C R s

CS s

(5.106)

(5.107)

CS s 1 s

Eq. (5.106) can be seen to be of the form; y(s) = u(s).gp(s) where, u ( s )

(5.108)

1 s gp s C Ai

C Ri

Da1

C Ai 1 Da 2

s

1 Da1

s

1 Da 2

s

(5.109)

Eq. (5.72) can be seen to comprise of two terms. The first term can be reduced to the general form of a prototypical first order process as given by Eq.() and reproduced below;

y( s )

u (s )

kp s

p

1

(5.110)

The process gain, kp and time constant, p for this term can be seen to be; kp

p

C Ri C Ai 1 Da 2

1 Da 2

(5.111)

Proportional, Proportional Integral, Proportional Derivative Feedback Control

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The second term in Eq.(5.109) can be reduced to the general form of a prototypical second order process as given by Eq.() and reproduced below as;

y( s )

kp

u (s)

2 2 ps

2s

p

(5.112)

1

The second term in Eq. (5.109) can be rearranged as; Da1 s

2 2

2 Da1

s

(5.113) 1 Da1 1 Da 2

Da 2

By comparison of Eq. (5.112) with Eq. (5.113), the process gain kp, time constant, p and damping factor,  can be seen to be; Da1

k

p

1 Da1 1 Da 2

(5.114)

p

1 Da1 1 Da 2 1

Da1

Da 2

2

1 Da1 1 Da 2

For PI control gc(s) is given by Eq. (5.26). The closed loop transfer function gCL(s) for PI control of chlorination of methane to chloroform or carbon tetrachloride can be written as follows;

g CL ( s )

Da1C Ai kc s

gCL s

g CL s

1 Da1 s

s s

C Ri kc 3 3

s

s

2

2 Da1

Da1kc kI C Ai

s2 Da 2

C Ai kc Da1

C Ri kc kI 1 Da1

(5.115)

1 g p (s ) g c (s ) kI

1 Da 2

C Ri kc

g p (s ) g c (s )

C Ri kc s

kI

C Ai kc Da1 s

kI

C Ri kc 1 Da 2

s 1 Da1 1 Da 2

1 Da 2

s

C Ri kc s

C Ri kc kI Da1C Ai kc

kI s

(5.116) 1 Da1

C Ri kc kI 1 Da 2 C Ri kc 1 Da1

C Ri kc

(5.117)

Where kI is the gain during PI control action. The stability of this system of PI control of intermediate product yield of chloroform, CHCl3 can be determined from the poles of Eq. (5.74). As the order of the polynomial in the denominator is increased beyond quadratic, cubic closed form analytical solutions are not possible. Numerical solutions are needed to obtain a solution of the polynomial of higher degrees. Routh (1905) stability criteria can be used to analyze the stability of systems such as the one encountered here. The Routh array of a general nth degree polynomial can be written as follows;

348

Kal Renganathan Sharma an sn

1

a n 1s n

2

an 2sn

.... a 2 s 2

a1s

a0

(5.118)

For all the poles to be negative all the coefficients in the 3rd degree polynomial in the denominator of Eq. (5.75) need be positive. The sufficiency test establishes that the array; Row Coefficients I

an

II III

an

1

an

2

an

4

an

3

an

5

IV

b1

b2

V

c1

c2

(5.119)

Where, a n 1a n

b1

2

an a n 1a n

b2

an an

b1a n

; c1

anan

5

a n 1b2

3

b1

1

4

an

3

b1a n

; c2

(5.120)

a n 1b3

5

b1

1

The number of changes in sign as the first column is traversed down the number gives the number of unstable positive roots. Unstable roots would arise when sign of the coefficients are negative For the system of PI control of the intermediate product yield chloroform from chlorination of methane to chloroform and carbon tetrachloride the conditions for instability would arise when; a 2 a1

(5.121)

a 3a 0

This would happen when

2 Da1

Da 2

CRi kc

1 Da1 1 Da 2

Da1kc kI C Ai

CRi kc kI 1 Da1 (5.122)

Eq. (5.122) can be written in terms of the residence time in the CSTR  and the reaction rate constants k1 and k2 as follows; 3

k1k2 k1

k2

2

2k1k2

CRi kc k1 k2

kI

k1

k2

2

3 k1 k2

C Ri kc kI 1 k1

2 0 (5.123)

For the special case when the further chlorination reaction rate constant k2 = kI the PI control gain Eq. (5.123) reduces to; 3

k1k2 k1

k2

2

2k1k2

k1

k2

2

3 k1 k2

C Ri kc k2 1 k1

2 0 (5.124)

Proportional, Proportional Integral, Proportional Derivative Feedback Control

349

5.8. TYREUS-LUYBEN OSCILLATION BASED TUNING Tyreus-Luyben oscillation based tuning was found to make the system less oscillatory for the parameters chosen. There are three tuning parameters in PID action. These are: (i) the control gain constant, kc; (ii) integral time constant, I; (iii) derivative time constant, D. An algorithmic or trial and error approach can be used for obtaining optimal results from tuning the controller. The tuning of PID control was developed by Ziegler and Nichols [1942]. This method is not widely used in the industry because the closed loop behavior tends to be oscillatory and sensitive to errors. The Tyreus-Luyben tuning parameters for PI and PID control are given below in Table 5.1. Where u is the ultimate time period and ku is the ultimate gain constant. For P only closed-loop control the magnitude of the proportional gain is increased until attainment of continuous oscillation. For larger values of the gain the closed-loop system becomes unstable. The value of the control gain constant where continuous oscillation is found is called the ultimate gain constant and notated by ku. The peak to peak time period that corresponds to this gain is called the ultimate time period, u. The tuning parameters are given in Table 5.1 and vary according to the control action taken.

5.9. SUMMARY The control theory began with the investigations of J. C. Maxwell of the dynamics of the centrifugal governor in a Boulton and Watt engine of the 18th century. Self-oscillations and the lags in the system can draw overcompensation and lead to instability. Mathematical modeling of any process is an important task in the analysis and design of control systems. The model equations when converted into the Laplace domain can be represented using block diagram format. Control systems can be linear or non-linear. Non-linear systems may need numerical solutions for analysis. Closed form analytical solutions can be obtained for linear systems. Sometimes when multiplicity arises in the solutions other criteria is needed to select the feasible solution from amongst the solutions obtained. On-off controllers are used in centralized heating and air conditioning systems in modern homes, Christmas tree lighting, level in water and process tank. An on-off control strategy patented by Mackay (2003) for CHP, combined heat and power systems was discussed. During proportional control, the control action taken is proportional to the error detected between the measured value and set points of the process variables. Control actions taken can be represented using control block diagrams. Figure 5.3 is the control block diagram of the pressure, temperature control for a CHP system. Gains are defined as the ratio of steady state change in output in response to a steady state change in input variable. These are defined for valve, process, controller, etc. Proportional control is used for a prototypical first order process. The closed loop transfer function for the process and control action is derived and shown in Eq. (5.15). The gain and time lag of the closed loop transfer function can be seen in Eqs. (5-16, 5.17). The time output response to a step change in the input can be seen in Eq. (5.20). For values of kCL, closed loop gain other than 1 there is seen an offset at steady state.

350

Kal Renganathan Sharma Table 5.1. Tyreus-Luyben Tuning Parameters Control Action PI PID

Integral Time Constant I 2.2 u 2.2 u

Derivative Time Constant D 0.159 u NA

Control Action Gain kc 0.3125ku 0.4545ku

In order to do better than an offset after implementation of offset PI, proportional integral control can be used. The control action is modified by addition of a component that is proportional to the integral of the error in addition to the action proportional to the error. This can lead to zero offset after rectification. kc and kI are the tuning parameters. The control transfer function for PI control is given in Eq. (5.26). In worked example 5.-1 is given an example of PI control action. The output response is seen to undergo an overshoot. The damping factor,  = 0.1, is less than 1.0. It is not clear as to what is causing the overshoot from the control action of the process. In example 5-2 is the analysis of closed-loop transfer function for feed forward/feedback controller. In example 5-3 is discussed the PI control of voltage supplied to an automatic washing machine patented by Toshiba. Model equations that relate torque to agitator RPM is given in Eq. (5.46). Electric motor is used in order to provide torque needed to perform the wash, rinse and dehydration steps. The control action may lead to instability because the error between target angular speed and operating angular speed in RPM can be rectified indirectly by changing the voltage supplied to the motor. The voltage will affect the torque in a manner that is not completely modeled or understood. The angular speed of the agitator and tub depends on the torque in a manner that is predicted by the mathematical model. Electric current measurements are used as feedback in the torque control of the Toshiba machine. The conditions when underdamped response of PI control of prototypical first order process is derived are section 5.6. Eq. 5.61 gives the criteria for underdamped response to be less than 1.0. This may be expected for small values of integral time. During PD control the control action taken is proportional to the derivative of the error. It may be used in processes where the curvature of the process variable with respect to time needs to be preserved. During PID control, components proportional to the error, proportional to the integral of the error, and proportional to the derivative of the error are introduced. Ideal PID control is not physically realizable! Real PID control can be affected after suitable modification of the control transfer function. In Example 5.4 the PI control during desalination of seawater using nanoporous carbon is discussed. PI Control action during hydrolysis of ethylene glycol is discussed in Example 5.5. For small values of integral gain the system can be expected to be underdamped as shown in Eq. (5.74). The PI control action can be used in the estimation of SFT, static formation temperature of oil wells as discussed in Example 5.6. Fourier parabolic and damped wave hyperbolic models can be used to describe the transient heat conduction processes. The problem is one in IHT, inverse heat transfer as the initial conditions are estimated from logged temperatures at specified locations. The time response to PD control of a prototypical first order process is given by Eq. (5.84). In Example 5.7 is discussed the PD control action during hydrolysis of ethylene glycol. The conditions for underdamped oscillations are given in Eq. (5.93). In Example 5.8

Proportional, Proportional Integral, Proportional Derivative Feedback Control

351

the control action for intermediate in reactions in series is discussed. Industrial applications for these types of reactions are during the production of chloroform and carbon tetrachloride by chlorination of methane and biodiesel and glycerol production by consecutive-competitive reactions of methanolysis of vegetable oil. The stability of the system of PI control of chloroform production by chlorination of methane can be seen from the closed loop transfer function given in Eq., (5.117). The order of the polynomial in the denominator of Eq. (5.117) is cubic. Instability would arise when the criteria in Eq. (5.121) is met. The corresponding expression for PI control gain is given in Eqs. (5.123). Tyreus-Luyben oscillation based tuning was found to make the system less oscillatory for the parameters chosen. There are three tuning parameters in PID action. These are: (i) the control gain constant, kc; (ii) integral time constant, I; (iii) derivative time constant, D. An algorithmic or trial and error approach can be used for obtaining optimal results from tuning the controller.

5.10. GLOSSARY Control Theory: Deals with dynamics of industrial processes. Behavior modification affected by feedback. Control objectives are met by using measurements and set points. Controllability, stability and observability are related issues. Closed form analytical and numerical solutions are obtained when needed. Error: Difference in value between the measurements and set points of the process variable. Feedback Control: Control action based on feedback from the process such as measurements and comparisons against the set points. Mathematical Model: A set of equations that can be used to describe a process or phenomena. Usually difficult to measure parameters can be predicted from readily measurable parameters. Laplace Domain: Often times differential equations become algebraic equations and integrals become algebraic expressions in this domain. The Bromwich integral and FourierMellin integral and Laplace transformation may be used to transform quantities from the Laplace domain and time domain and vice versa. Control Block Diagram: Flow diagram that can be used to show the control action and prototypical first and higher order processes using transfer functions. Valve Gain: Ratio of change in flow rate to the change in valve top pressure (Eq. (5.11) Process Gain: Change in volume of the reactor to the change in valve top pressure (Eq. 5.12). Proportional Gain, kc: Ratio of the change in process variable to the error Proportional Integral Gain, kI: Tuning parameter. Pre „integral of the error‟ constant. Integral Time, I: Ratio of the proportional gain, kc and proportional integral gain, kI. On/Off Control: Shut-down and start-up is used at certain error levels. Used as safeguards. Proportional Control, P Control: Control action taken that is proportional to the error detected.

352

Kal Renganathan Sharma

Proportional Integral Control, PI Control: The control action is modified by addition of a component that is proportional to the integral of the error in addition to the action proportional to the error. This can lead to near zero offset Proportional Derivative Control, PD Control: The control action is modified by addition of a component that is proportional to the derivative of the error in addition to the action proportional to the error. Used in cases where the curvature of the process output with time needs to be preserved. Proportional, Integral, Derivative, PID Control: During PID control, components proportional to the error, proportional to the integral of the error, and proportional to the derivative of the error are introduced. Ideal and Real PID Control: Ideal PID control is not physically realizable! Real PID control can be affected after suitable modification of the control transfer function. Methods to make PID control feasible are discussed in Chapter 7.0 Advanced Controls. Offset: It is the difference between the set point and process variable at steady state.

5.11. REFERENCES Espinosa-Paredes, G., Maralan-Diaz, A., Olea-Gonzalez, U., & Ambiz-Garcia, J. J. (2009). "Application of a Proportional Integral (PI) Control for the Estimation of Static Formation Temperature in Oil Wells", Marine & Petroleum Geology, Vol. 26, 2, 259268. Fujui, A. “Chemical Concentration Control Device for Semiconductor Processing Apparatus”, US Patent 6,921,193 (2005), Kaijo Corp., Hamura, JP. Hosoito, T., Tanaka, T., Okazaki, Y., Kawabata, S., Nagai, K. & Isono, F. (2006). “Washing Machine with Vector Control for Drive Motor”, US Patent 7, 017, 377, Kabushiki Kaisha Toshiba, JP. Levenspiel, O. Chemical Reaction Engineering , (1999). Third Edition, John Wiley, New York, NY. Luyben, M. L. & Luyben, W. L. (1997). Essentials of Process Control, McGraw Hill Professional, New York. MacKay, R. (2007). “Combined Heat and Power System”, US Patent, 7,299,638 Routh, E. J. (1905). Dynamics of a System of Rigid Bodies, Part II, MacMillan, London, United Kingdom (1905). Ziegler, J. G. & Nichols, N. B. (1942). “Optimum Settings for Automatic Controllers”, Trans. ASME, 64, 759-768.

5.12. EXERCISES Review Questions 1.0. What are the differences between “on-off” controller and “P” only control ? 2.0. What are the differences between „PI‟ control and „PD‟ control ? 3.0. How can ideal PID control be made realizable ?

Proportional, Proportional Integral, Proportional Derivative Feedback Control

353

4.0. How is on-off‟ controller used in CHP, combined heat and power systems ? 5.0. What is the significance of controller gain kc during „P‟ only control ? 6.0. What is the significance of constant „b‟ in „P‟ only control action law ? 7.0. What is the significance of valve gain constant kv ? 8.0. What are the differences between process gain constant kp and controller gain constant kc ? 9.0. What is meant by „offset‟ during „P‟ only control ? 10.0. How does PI control action remove „offset‟ found during P only control ? 11.0. What are the differences between control action law during; (i) PI Control (ii) PID Control (ideal) (iii) P only Control 12.0. What are the differences between control action law during; (i) PD Control (ii) PI Control 13.0. How is the final value theorem used to obtain „offset‟ found after control action is taken ? 14.0. What are the transfer functions of control action law of; (i) P only Control (ii) PI only Control (iii) PD only Control (iv) PID Control (ideal) 15.0. Can underdamped oscillatory conditions arise after P only control action of prototypical first order process is taken ? Why not ? 16.0. Can underdamped oscillatory conditions arise after PI only control action of protypical first order process is taken ? Why ? 17.0. What is the best control action for prototypical third order process from closed loop stability considerations ? 18.0. Does the answer tp Q15, Q16 change on account of change in input to periodic signal ? 19.0. Can underdamped oscillatory conditions develop after PD control of prototypical first order process ?Why not ? 20.0. What is the physical significance of Bm in Eq. (5.47). 21.0. Consider a PD control of the voltage in automatic washing machine in Example 5.3. What are the stability issues ? 22.0. What is meant by „smoothing effect‟ on output response ?

Problems 23.0. PI Control of Prototypical First Order Process: Time Response to Periodic Input Consider any prototypical first order process. Consider a PI controller used to control this process. Let the process gains, kp = kc =1. What would be the time response of the process output variable y(t) to a periodic disturbance in the set point,

354

Kal Renganathan Sharma

1

r (s ) s

2

1

The time constant of the controller, I is 1and the time constant of the

process, p is 2.0. The simplified control block diagram is given below. Disturbances are neglected and valve and measurement dynamics are lumped into the process transfer function. Develop the expression for the output variable y(s) in the Laplace domain and time domain y(t). Sketch the response y(t) as a function of time. The transfer functions for prototypical first order process, gp(s) and that for PI control, gc(s) can be seen to be; kp

g p (s) g c (s )

1

ps

kc

(5.125)

Is 1 Is

24.0. PI Control of Prototypical First Order Process: Time Response to Input that undergoes Exponential Decay with Periodicity Consider any prototypical first order process. Consider a PI controller used to control this process. Let the process gains, kp = 3; kc =0.33. What would be the time response of the process output variable y(t) to a change in the set point,

1

r (s )

s 1

(5.126)

2

1

The time constant of the controller, I becomes 10% of the time constant of the process, p. The simplified control block diagram is given below. Disturbances are neglected and valve and measurement dynamics are lumped into the process transfer function. Develop the expression for the output variable y(s) in the Laplace domain and time domain y(t). Sketch the response y(t) as a function of time. The transfer functions for prototypical first order process, gp(s) and that for PI control, gc(s) can seen to be; kp

g p (s) g c (s )

1

ps

kc

Is Is

1

(5.127)

Proportional, Proportional Integral, Proportional Derivative Feedback Control

355

Figure 5.8a. Simplified Control Block Diagram for Q1.0.

Figure 5.9. Control of CSTR Volume.

25.0 On-off Control of Storage Tank A storage tank is used for the monomers prior to feeding them to the reactors during continuous polymerization of HIPS, high impact rubber modified polystyrene. The styrene monomer, solvent such as ethyl benzene and PBd, poly butadiene in solution form are brought from the suppliers as well as from recycle and stored in the storage tank. In order to keep the storage tank from overflowing and from being depleted and ruining the pumps that are starved a on-off controller is devised. The control action suggested is shown in Figure 5.9. Write the transfer function for the on-off control action. The residence time of the reactor depends on the volume of the reactants and products in the reactor. The volume of the reactor contents is measured by the volume transmitter, VT. This information is sent to the VC, volume controller. The set point volume and the measured volume are compared with each other. Desired action is then signaled to the control valve bu a pressure variable. The valves can be operated using pressure transducer values. The valve can be made to open upto a full open position. This action is called for when the tank is found to be depleted. The

356

Kal Renganathan Sharma valve can be made to partially close or close completely until the input reactant streams are shut off. This action is called for when the reactor is found to be overflowing. 26.0. PI Control Action for Prototypical Second Order Overdamped Process. Consider any prototypical second order process. Consider a PI controller used to control this process. Let the process gains, kp = kc =1. Let the damping factor  = 4.0. What would be the time response of the process output variable y(t) to a step change in the set point, r(s) = 1/s. The time constant of the controller, I becomes 5% of the time constant of the process, p. The simplified control block diagram given in Figure 5.4 is applicable. Disturbances are neglected and valve and measurement dynamics are lumped into the process transfer function. Develop the expression for the output variable y(s) in the Laplace domain and time domain y(t). Sketch the response y(t) as a function of time. The transfer functions for prototypical first order process, gp(s) and that for PI control, gc(s) can be seen to be; g p (s) g c (s )

kp 2 2 ps

kc

2

s 1

(5.128)

1

Is Is

27.0. PI Control Action for Prototypical Second Order Underdamped Process Consider any prototypical second order process. Consider a PI controller used to control this process. Let the process gains, kp = kc =1. Let the damping factor  = 0.4. What would be the time response of the process output variable y(t) to a step change in the set point, r(s) = 1/s. The time constant of the controller, I becomes 5% of the time constant of the process, p. The simplified control block diagram given in Figure 5.4 is applicable. Disturbances are neglected and valve and measurement dynamics are lumped into the process transfer function. Develop the expression for the output variable y(s) in the Laplace domain and time domain y(t). Sketch the response y(t) as a function of time. The transfer functions for prototypical first order process, gp(s) and that for PI control, gc(s) can be seen to be; g p (s) g c (s )

kp 2 2 ps

kc

2

Is

s 1

(5.129)

1

Is

28.0. PI Control Action for Prototypical Second Order Undamped Process Consider any prototypical second order process. Consider a PI controller used to control this process. Let the process gains, kp = kc =1. Let the damping factor  = 0.0. What would be the time response of the process output variable y(t) to a step change in the set point, r(s) = 1/s. The time constant of the controller, I becomes 5% of the

Proportional, Proportional Integral, Proportional Derivative Feedback Control

357

time constant of the process, p. The simplified control block diagram given in Figure 5.4 is applicable. Disturbances are neglected and valve and measurement dynamics are lumped into the process transfer function. Develop the expression for the output variable y(s) in the Laplace domain and time domain y(t). Sketch the response y(t) as a function of time. The transfer functions for prototypical first order process, gp(s) and that for PI control, gc(s) can be seen to be; kp 2 2 ps

g p (s) g c (s )

1

(5.130)

Is 1

kc

Is

29.0. Third Order Process with P only Control Consider a third order process whose process function is given by;

1

G p (s)

3s

2

(5.131)

5s 3s 1

What is the maximum proportional gain for P-only control in order for the control action to be closed loop stable. Use Routh stability criterion. 30.0. PI Control of a Process that is Integrating Unstable Consider a process that is integrating unstable and has the following transfer function.

0.5 s 2

Gp s

(5.132)

The output response in the time domain of this process is; 0.5e 2t

y(t )

(5.133)

Show that a PI controller can stabilize this integrating unstable process. Consider the control transfer function;

Gc s

kc

s

1

I

s

(5.134)

I

Show that the closed loop transfer function can be given by;

GCL s

0.5kc s s

2 I

s

I

I

1

0.5kc

2

(5135) 0.5kc

358

Kal Renganathan Sharma Further show that for stability 0 < kc < 4. Show that the system would be underdamped when;

2kc

4 0.25kc2

I

2kc (5.136)

2kc I

0.5kc

2

2

Sketch the output variable as a function time before and after the control action considered. 31.0. Best Control Strategy for Systems with Inverse Response Some systems exhibit a inverse response. The output response to a unit step response in such systems decreases initially and increases to a final steady state later. The transfer function of such a system has positive pole in the numerator; s 3

Gp s

s

0.2 s 2

.u s

(5.137)

What is the best controller to use that would have a stabilizing effect on the process. Consider a PD controller for example with a transfer function given by; Gc s

kc s

D

(5.138)

1

Show that the closed-loop transfer function is given by; Ds

GCL s D

1 2 s kc

2

3 s 3

D

1

0.4 3 s (1 3 kc

(5.139) D

2.2 ) kc

Show that for small D the system can be stabilized. Show that, for stability, kc < 0.733 0.133 and D 0.33 . Is this both possible for positive values of gain kc

constant of controller, kc. and controller time constant D. Under what conditions for the combined process and controller the system becomes underdamped oscillatory. 32.0. Proportional Controller for Jacketed Exothermic CSTR The model equations for dimensionless concentration and dimensionless temperature for the jacketed CSTR described in Figure 4.19 are as follows; dY d

Y1 U*

( X H *) U *

H*

(5.140)

Proportional, Proportional Integral, Proportional Derivative Feedback Control

Da

dX A

(5.141)

X A (1 Da )

d

359

Obtaining Laplace transform of the two model equations the process transfer functions equations for dimensionless concentration and dimensionless temperature for the jacketed CSTR described in Figure 4.19 are as follows; X (s )

Da s s 1 Da

Y( s )

Da H * s s 1 Da s 1 U *

U* s s 1 U*

H* s s 1 U*

(5.142)

Consider a proportional controller with the control transfer function; Gc s

(5.143)

kc

Show that the closed loop transfer function for the dimensionless concentration and dimensionless concentration can be given by; GCL s GCL ( s )

kc Da s2

s 1 Da

kc Da

(5.144) kc s

s3

s 2 Da

U*

H* U*

H * U * 1 Da

s 1 Da 1 U *

H* U*

kc

kc

H * U * 1 Da

Derive the conditions for the system to develop underdamped oscillatory instability in dimensionless concentration and dimensionless temperature. 33.0. PI Controller for Jacketed Exothermic CSTR Consider a PI controller for the jacketed exothermic CSTR in Example 3-3. The controller transfer function is given by;

Gc ( s )

kc

s

1

I

s

(5.145)

I

Derive the closed loop transfer functions for the dimensionless concentration and dimensionless temperature. What are the conditions under which under damped oscillatory instability would develop ? 34.0. Proportional Controller for Polymerization Kettle Consider a proportional controller for the dimensionless initiator concentration and dimensionless monomer concentration discussed in Example 3-4. The model equations for dimensionless initiator concentration is as follows; dY d

Da D

Y 1 Da D

(5.146)

360

Kal Renganathan Sharma Obtaining the Laplace transform of the model equation the transfer function can be written as; Da D

Y( s )

(5.147)

s 1 Da D s

The model equations for dimensionless monomer concentration is as follows; Da M 1 Y

X 1 Da M 1 Y

dX d

(5.148)

Use a binomial approximation for (1-Y)1/2 and render the differential equation amenable for obtaining the Laplace transform;

Da M 1

Y 2

Y 2

X 1 Da M 1

dX d

(5.149)

The equation can me linearized by double differentiation and combination of the other model equation. Obtain the process transfer function for dimensionless monomer concentration. Consider a Proportional controller for both the initiator and monomer concentrations. What are the conditions under which underdamped oscillatory instability may be expected in the dimensionless initiator and monomer concentrations. 35.0. Proportional Integral, PI Controller for Polymerization in CSTR For the problem 13 consider a PI controller in place of the Proportional controller. What are the conditions under which underdamped oscillatory instability may be expected in the dimensionless initiator and monomer concentrations. 36.0. PI Controller Consider a second-order process with the process transfer function with one pole at the origin; Gp s

3 s 2s 1

(5.150)

Show that a PI controller with integral time constant of I = 2 can lead to a pole-zero cancellation. Show that the closed loop transfer function can be given by; 3kc

G CL s I

s

2

(5.151) 3 I

Proportional, Proportional Integral, Proportional Derivative Feedback Control

361

Obtain the output response for a step change in input of magnitude R. Comment on the stability of the system given the order of the system and nature of control action taken. 37.0. Proportional Controller of CSTR with Recycle Consider the process conducted in a CSTR with recycle as shown in Figure 4.26. The model equation for the dimensionless concentration in transient state is given by; R R 1

Da

1

XA

Da

R 1

dX A

(5.152)

d

where R is the recycle ratio. Obtaining the Laplace transform of the model equation the transfer function for the process can be seen to be;

Da

R R 1

XA

Da s s

Da

1

1

dX A

Da

R 1 R R 1

R 1

d

1 s s

Da

1

XA s

(5.153)

R 1

Consider a P only controller for the dimensionless concentration. What is the closed loop transfer function for the system. Under what conditions will there be underdamped oscillations in the dimensionless concentration. 38.0. Proportional Integral, PI Controller of CSTR with Recycle Consider a PI controller in place of P only controller in problem 16.0. Under what conditions will there be underdamped oscillations in the dimensionless concentration. 39.0. Proportional Integral, PI Controller for a First Order Process with Dead Time Consider a PI, proportional integral control for a first order process with a lag time, . The process transfer function with dead time,  can be written as; G p (s )

kpe

s

s

1

(5.154)

p

The controller transfer function is given by;

Gc ( s )

kc

s

1

I

s

(5.155)

I

In order to evaluate the stability of closed loop transfer function Pade‟ approximation can be used to rationalize the process transfer function. The Pade‟ approximation is;

362

Kal Renganathan Sharma

e

2 s 2 s

s

(5.156)

Show that the closed loop transfer function can be written as follows;

G CL s

kc k p s 2 s3

I

p

s2 2

I

s

I

kc k p

p

2

I I

(5.157) s

I

2

Discuss the stability of the closed loop. Under what conditions can underdamped oscillatory conditions be expected ? Comment on the stabilizing effect of the PI controller. 40.0. Proportional P Controller for a First Order Process with Dead Time Consider a P only, proportional control for a first order process with a lag time, . The process transfer function with dead time,  can be written as; G p (s )

kpe

s

s

1

p

(5.158)

The controller transfer function is given by; Gc (s )

(5.159)

kc

In order to evaluate the stability of closed loop transfer function Pade‟ approximation can be used to rationalize the process transfer function. The Pade‟ approximation is; e

s

2 s 2 s

(5.160)

Show that the closed loop transfer function can be written as follows;

G CL s

kc k p 2 s s2

p

s

2

p

k p kc

(5.161) 2 1 k p kc

Discuss the stability of the closed loop. Under what conditions can underdamped oscillatory conditions be expected ? Comment on the stabilizing effect of the P controller on the process with a lag time. 41.0. Proportional Derivative, PD Controller for a First Order Process with Dead Time Consider a PD, proportional derivative controller for a first order process with a lag time, . The process transfer function with dead time,  can be written as;

Proportional, Proportional Integral, Proportional Derivative Feedback Control

G p (s )

kpe

s

s

1

p

363

(5.162)

The controller transfer function is given by; Gc (s )

kc 1 s

D

(5.163)

In order to evaluate the stability of closed loop transfer function Pade‟ approximation can be used to rationalize the process transfer function. The Pade‟ approximation is; e

s

2 s 2 s

(5.164)

Develop the expression for the closed loop transfer function. Show that the system is not physically realizable. 42.0. Feedback Control of Semiconductor Processing Consider the following control instrumentation for a semiconductor wafer processing unit as shown in Figure 5.10. Three replenishing tanks A, B and C are used to refill the chemicals needed for wafer washing and processing. Concentration meter is used [2007] to monitor the concentration of the chemicals used in the wafer processing apparatus. Control action is taken using comparisons of the measured concentration against reference concentrations. Construct a block flow diagram labeling all signals and transfer functions.

Figure 5.10. Feedback Control of Chemical Concentration during Semi-conductor Processing.

364

Kal Renganathan Sharma 43.0. Periodic Disturbance and System Stability Consider a process with the transfer function Gp(s) and a controller with a transfer function Gc(s) as shown in Figure 5.11. The feedback control is based on measurement of the output variable y(t). The load or disturbance transfer function is given by L(s). The set point transfer function is given by r(s). Show that the output transfer function y(s) can be written as follows;

y( s )

G p ( s ) L( s ) Gc ( s )r ( s )

1 G p (s )Gc (s )

(5.165)

Consider a prototypical first order process with the transfer function; kp

G p (s )

s

p

(5.166) 1

Consider PI controller with the transfer function given by;

Gc ( s )

kc

1

Is Is

(5.167)

Consider a disturbance that is periodic in nature with the transfer function; L( s )

s2

2

Figure 5.11. Feedback Control of Chemical Concentration during Semi-conductor Processing.

(5.168)

Proportional, Proportional Integral, Proportional Derivative Feedback Control

365

Consider a step change in the set point and the transfer function r(s) can be given by; R s

r (s)

(5.169)

Show that the closed loop transfer function can be written as;

GCL ( s )

s 3 kc k p s

3 p

s

2

s 2 kc k p

I

2

s

2

s 2

p

s

I

kc k p

I

1 k p kc

kc

I

(5.170).

k p kc

Under what conditions can underdamped oscillations be expected ? What is the final steady state value of the output function y(t). Did the disturbance affect stability ? Why not ? 44.0. PI Control of Intermediate Product in Series with First Order Followed by Zeroth Order Reaction Consider the following scheme of reactions performed in a CSTR given in Levenspiel [1999]. k1

A

k2

R

(5.171)

S

The kinetics of species A and R can be written as follows; dC A dt dC R dt

k1C A

(5.172) k1C A

k2

There is interest in designing a PI control for the intermediate product R as shown in Figure 5.8. Dicuss the PI control for the concentration of R. Show that the model equations for reacting species A and intermediate species R can be written as follows; C Ai

0 CR

CA

k1 C A

( Da1C A

Da 2 )

dC A

(5.173)

dt dC R dt

The process transfer functions can be written as follows;

(5.174)

366

Kal Renganathan Sharma C Ai

C A (s )

CR s

s 1 Da1

Da1C Ai s

1 s s

(5.175)

s

Da 2

1 Da1

s s

1

(5.176)

Where  is the residence time of the reacting species in the reactor and Da1 is the Damkohler number given by (k1) and Da2 is given by (k2). The control transfer function can be written as follows;

Gc ( s )

kc

Is

1

Is

(5.177)

What is the closed loop transfer function, gCL(s). Under what conditions would there be a underdamped response ? 45.0. PI Control of Intermediate Species in Denbigh Scheme of Reactions Denbigh [1999] presented a scheme of general scheme of reactions for series, parallel and series- parallel reactions. The kinetics of these reactions; dC A dt dC R dt dC S dt dCT dt dCU dt

k12C A k1C A k3C R

k34C R

(5.179)

k2C A k4C R

Where k12 = k1 + k2 and k34 = k3 + k4. Show that the process transfer functions for the reacting species in a CSTR during transient operation can be written as follows;

(5.178)

Figure 5.12. Denbigh Scheme of Reactions.

Proportional, Proportional Integral, Proportional Derivative Feedback Control

CA s CR s C

S

s

CT s CU s

367

C Ai s s

1 Da12 s

s s

C Ai Da12 1 Da12 s

s s

C Ai Da 3 1 Da12 s 1 Da 34 s

1 Da 34 1

(5.180)

Da 2C Ai s s

1 s

1 Da12 Da 4 Da12 C Ai

s s

1 s

1 Da12 s

1 Da 34

Consider a PI controller with the transfer function;

Gc ( s )

kc

Is Is

1

(5.181)

What is the closed loop transfer function for each of the processes of species A, R, S, T and U. Under what conditions would underdamped oscillatory concentrations develop ?

Chapter 6

FREQUENCY RESPONSE ANALYSIS 6.1. MOTIVATION The criterion for closed loop stability was discussed in Chapter 5.0. on Feedback control. The poles of the closed loop transfer function as given by Eq. (5.15) have to be stable, i.e., when zero it can become unstable and when negative the output response moves in the opposite direction compared with the set point change. Even when the process and control action devised appear stable according to the stability criterion developed there can be some stability problems. How? There can be uncertainties associated with the parameters that are used to describe the system and the gain constants and time constants for the control system. The uncertainties can become large enough for a reversal of stability criterion arrived at. This raises the question of how robust is the control action considered for a given process. If there are time delays in the process and control action taken the transfer function would contain an e-s. This renders the transform function irrational. The P(s)/Q(s) rational form is not preserved any more. The concern is the robustness of the control system for the considered process. Often times when what the problem is, is realized then dozen solutions can be offered to the problem. An old adage is even before dumping it all into the river, count it before dumping. Measure of robustness can be obtained from the method of frequency response analysis. Transfer functions that are irrational on account of time delays can be rationalized by use of what is called Pade approximation. [Bequette, 2003]

6.2. OUTPUT RESPONSE OF A PROTOTYPICAL FIRST ORDER PROCESS TO A PERIODIC INPUT Consider a prototypical first order system that is linear. The differential equation that can be used to describe such a process is as follows;

p

dy dt

y

kpu

(6.1)

370

Kal Renganathan Sharma

Where p is the time constant, kp is the process gain, u is the input variable and y is the output variable. The Laplace transform of Eq. (6.1) given that the initial value of the output variable is 0 can be written as; k p u (s )

y( s )

s

(6.2)

1

p

Consider a sinuous input, u = Asin(t). The output would then be in the Laplace domain; A kp

y( s ) s

1 s2

p

2

(6.3)

The time domain y(t) can be obtained by using the convolution theorem [Mickley, Sherwood and Reed, 1957] on Eq. (6.3), and;

y(t )

A

kp

t p

t

p

e

sin

p dp

(6.4)

p 0

Integration by parts can be used in order to obtain the integral in Eq. (6.4). An I can be defined as; p

t

I

e

p

sin

t

p dp

(6.5)

0

Let;

sin

u

t

p

(6.6)

p

dv

e

p

dp

Realizing that; t

t

udv uv 0

vdu

(6.7)

0

Another I emerges in RHS, right hand side of Eq. (6.7) and groups with the I on LHS, left hand side of Eq. (6.5) to give;

371

Frequency Response Analysis t

Ak p

y(t )

1

2 2 p

sin wt

p

e

p

(6.8)

 was introduced to simplify the expression for y(t) such that cos() = 1 and sin() = p. or tan() = (p). The amplitude ratio is obtained as the ratio of the amplitude of the output sine wave and the amplitude of the input since wave. This can be obtained at long times after the exponential term in Eq. (6.8) has decayed to a large measure as; kp

AR

1

2 2 p

(6.9)

AR, amplitude ratio is found to be independent of the initial input amplitude A. The phase shift is another measure of this analysis. It is given by the relative shift in phase between the output and input sine waves. This can be seen to be  from Eq. (6.8). The input and output responses are plotted in Figure 6.1.

Figure 6.1. Response y(t) to input u = sin(2t) to a First Order Process with Gain k p = 3 hr-1 and Process Time Constant p = 1.5 hr.

372

Kal Renganathan Sharma

It can be seen from Figure 6.1 that there is damping of the input function. AR, amplitude ratio of 0..3 is realized. There also can be seen a phase lag. The process acts like a low pass.filter.

6.3. BODE AND NYQUIST DIAGRAMS In a Bode diagram, the AR amplitude ratio is plotted against the frequency in a log-log plot and the phase angle,  is plotted against the frequency in a semi-log plot [van Vackenberg, 1984]. A semi-log plot is chosen for the phase angle diagram in anticipation of negative values for phase shift. The Bode plot for the input function u = A sin(t) to a prototypical first order process as discussed in section 6.2 is shown in Figure 6.2 and Figure 6.3. Aliter, another method of obtaining the Bode plot [Golnaraghi and Kuo, 20..09] is by..substitution of (i) for s in the Laplace domain of the transfer function. Then the expression is expressed as a complex number Z = a + ib. Further the steady state process gain can then be obtained from the absolute value of Z, i.e. a2

b2

ATAN

b a

Z

(6.10)

For example for the prototypical first order process given by Eq. (6.1) the transfer function can be seen to be; kp

G (s ) s

p

(6.11)

1

s is replaced with (i) in Eq. (6.11) and; kp

G (i ) i

p

Z

1

kp 2 2 p

p kp 2 2 p 1

i

1

(6.12)

Applying Eq. (6.10) into Eq. (6.12)

Z

a2

ATAN

b2 b a

kp 2 2 p

ATAN (

1

(6.13) p)

Frequency Response Analysis

373

Figure 6.2. Bode AR Plot of Response y(t) to input u = sin(2t) to a First Order Process with Gain k p = 3 hr-1 and Process Time Constant p = 1.5 hr.

Figure 6.3. Bode Phase Angle Plot of Response y(t) to input u = sin(2t) to a First Order Process with Gain kp = 3 hr-1 and Process Time Constant p = 1.5 hr.

In a Nyquist diagram [1932] the imaginary part “b” can be plotted against the real part “a” in Eq. (61.2) for example.

374

Kal Renganathan Sharma

Example 6.1 Construct the Bode plot and Nyquist diagram for a point in a semi-infinite medium subject to constant wall temperature boundary condition. Compare the results between damped wave conduction and relaxation and Fourier conduction. More detailed discussion on damped wave conduction and relaxation can be seen in chapter 4.0 of this book and also in Sharma [2005]. The governing equation for transient temperature for a semi-infinite medium heated by maintaining a constant wall temperature at the surface and assuming that the heat transmission is by damped wave conduction and relaxation mechanism can be written as follows; 2

u

Where, u

(T T0 ) ; (T s T 0 )

t

;X

u

2

2

X2

u

(6.14)

x

r

r

Ts is the surface temperature and T0 is the initial temperature of the semi-infinite medium. Obtaining the Laplace transform of Eq. (6.14) and applying the boundary conditions u = 0 at X = , u = 1 at X = 0; e

u (s )

X s ( s 1)

(6.15)

s

Replacing s with (i) in Eq. (6.15) and obtaining the “a” and “b”;

u (i )

a

ib

i

Let

2

1 X

2

c id

X2 2!

i

2

i

....

(6.16)

(6.17)

i

Taylor series approximation was used to rationalize the Laplace domain expression with exponential term in Eq. (6.15). c and d in Eq. (6.17) can be solved for by squaring both sides of Eq. (6.17) and equating the real and imaginary parts on both sides, LHS and RHS of the resulting equation and;

1 i 2 c

2

d

2 a and b in Eq. (6.16) can then be seen to be;

(6.18) 2

4

2

2

Frequency Response Analysis

a b

Xd

1

X2 2! 2

1 cX

X2 2!

Figure 6.4. Bode Process Gain Plot for X = 0..5 in Semi-infinite Medium Subjected to Constant Wall.Temperature during Damped Wave Conduction and Relaxation.

375

(6.19)

376

Kal Renganathan Sharma

The same analysis is repeated assuming Fourier conduction in the semi-infinite medium. The governing equation for transient temperature would then become; 2

u

u

(6.20)

X2

Obtaining the Laplace transform of Eq. (6.20) and applying the boundary conditions u = 0 at X =, u = 1 at X = 0; e

u (s )

X s)

(6.21)

s

Replacing s with (i) in Eq. (6.21) and obtaining the “a” and “b”; a

u (i )

ib

1 e i

X i

(6.22)

Taylor series approximation is used to rationalize the exponential term in Eq. (6.22);

a

ib

i

2

X2 2!

1 X i

(6.23)

Let i c id . It can be seen by squaring both sides and by equating the real and imaginary parts of both sides of the resulting equation c and d can be obtained and is; c

1

d

(6.24)

2

Combining Eq. (6.24) with Eq. (6.23) it can be seen that; X

a

b

1

(6.25)

2 X

2

X2 2!

2

(6.26)

The Bode plot for X = 0.5 in the semi-infinite medium heated by Fourier conduction from a constant temperature surface is shown in Figure 6.5. The Bode plot for the same point in the medium heated by damped wave conduction and relaxation is also shown in Figure 6.5. A cross-over in the steady-state process gain can be seen at a frequency of approximately  = 2. After the cross-over the process gain is higher for parabolic case compared with the hyperbolic case. Before the cross-over the process gain is lower for the parabolic case

Frequency Response Analysis

377

compared with the hyperbolic case. Before the cross-over the process gain is insensitive to the frequency for the parabolic case. After the cross-over the process gain is more sensitive to frequency for the parabolic case compared with the hyperbolic case. The process gain is equally sensitive to frequency for the hyperbolic case both before and after the cross-over frequency. The Bode phase angle plots for the parabolic and hyperbolic cases at X = 0.5 in the semi-infinite medium is shown in Figure 6.6. S shaped curvatures are found for both parabolic and hyperbolic cases. The Nyquist diagram for the hyperbolic cases at X = 0.5 in the semi-infinite medium is shown in Figure 6.7.

Figure 6.5. Bode Process Gain Plot for X = 0..6 in Semi-infinite Medium Subjected to Constant Wall.Temperature during Damped Wave Conduction and Relaxation and Fourier Conduction (Side by Side Comparison).

378

Kal Renganathan Sharma

Figure 6.6. Bode Phase Angle Plot for X = 0.5 in Semi-infinite Medium Subjected to Constant Wall.Temperature during Damped Wave Conduction and Relaxation and Fourier Conduction (Side by Side Comparison).

Figure 6.7. Nyquist Diagram for X = 0.5 in Semi-infinite Medium Subjected to Constant Temperature Surface during Damped Wave Conduction and Relaxation.

Frequency Response Analysis

379

6.4. FREQUENCY ANALYSIS OF SECOND ORDER PROCESSES The discussions about prototypical second order systems and how over damped, critically damped and underdamped systems can arise for a step input was discussed in chapter 4.0. Here the frequency analysis as discussed in the above sections is applied to prototypical second order systems. The steady state process gain and phase characteristics that can be expected as a function of frequency, for prototypical second order systems are sketched in Figure 6.8. A resonant peak, Rpeak can be expected in the AR or steady process gain as shown in Figure 6.8 at a frequency of R for higher order systems such as prototypical second order systems. For systems with overshoot Rpeak would be very high. It is desirable to maintain Rpeak at values of about 1.1-1.5. R is called the resonant frequency corresponding to the occurrence of resonant peak Rpeak in the gain. A BW, bandwidth can be defined as the frequency corresponding to a value of 0.707 of the zero frequency ABS(Z) or the steady process gain. This is also shown in Figure 6.8. The BW frequency value can be used to better understand the transient properties of the time domain of the system under study. Quicker rise time manifests as larger bandwidth. Time response is slow for systems with lower bandwidth. BW can be used to study the filtering characteristics as well as the robustness of the system. The slope of the ABS(Z) vs.  curve can be representative of a „cut-off‟ rate. This can be used to gauge the signal to noise ratio of the system.

6.5. CLOSED LOOP STABILITY – BODE AND NYQUIST CRITERION The closed loop shown in Figure 5.4 is applicable in general for any control action taken on any considered process. Consider the case when the loop in Figure 5-4 is opened as a hypothetical experiment in thought. A periodic wave such as a cosinuous function may be applied to the set point. The controller settings and tuning parameters are such that the output lags the input by PI radians. The amplitude of the output is the same as that of the input. Consider stoppage of the signal r(t) suddenly and closure of the loop subsequently. The error, e(t) = -y(t) the output variable. The output oscillates at the same frequency and magnitude as before the loop was closed. Such a control loop is said to be nominally stable. The control loop may be considered unstable when upon closure of the loop the output increases with time and the amplitude of the output is greater than the amplitude of the input. In the open loop the output is out of phase with input. The amplitude of the output signal is greater than the set point. Such a stability criterion is called as Bode Criterion. For a stable process, Gp(s) has no poles in the Right Half Plane and has the phase angle cross – PI radians only once in the Bode diagram. The system is considered closed loop stable if and only if; ARc

Gc

Gp

1

(6.27)

ARc is the amplitude ratio at the cross-over frequency, c. This is when the phase angle is –PI radians. The Nyquist stability criterion states that the process is said to be closed loop stable if the Nyquist diagram of Gp(s)Gc(s) does not engulf the critical identified point (-1,0.).

380

Kal Renganathan Sharma

The gain margin can be calculated as the reciprocal of the amplitude of the cross over.frequency. So the Bode stability criterion in terms of the gain margin can be written as; Mgain > 1

(6.28)

The phase margin m is the corresponding phase when the amplitude AR of Gc(s)Gp(s) = 1. The corresponding frequency is m. The stability criterion can be written as; m > 0

(6.29)

It is up to what extent or degree of measure of phase angle that can decrease prior to onset of instability. Example 6.2. Construct the Bode diagram for the process in Example 6.1. The transfer function in worked Example 5.1 can be written as follows; 0.01 p s 1

y( s )

0.01

2 2 ps

0.02s

1 ( 1 s)

p

(6.30)

Let s = i, Then,

0.01

pi

Z i (1 0.01

2 p

1

) 0.02

a

ib

2 p

Figure 6.8. Typical Frequency Response Characteristics of Prototypical Second Order System.

(6.31)

381

Frequency Response Analysis

Figure 6.9. Bode AR Plot for Example 6.1 when p = 0..01.

The “a” and “b” of the analytic function Z in Eq. (6.31) can be found by multiplying the numerator and denominator of Eq. (6.31) by (-0.022p -i(1-0.01(2p2)). 2 p

0.01 10

4

2 2 p

Z

1 10

0.02 4

4 4 p

i 1 0.00981

p

0.0199

2 2 p

2 2 p

a

ib

(6.32)

The AR is given by; AR

a2 ATAN

b2 b a

(6.33)

The Bode AR plot when the process time constant, p = 0.01 is shown in Figure 6.9. The Bode phase angle  plot for the system in Example 5.1 for p = 0.01 is shown in Figure 6.10.

6.6. SUMMARY For a given process of first order or second order suitable control action can be devised as shown in Chapter 5.0. The closed loop transfer function can be calculated and the system can be tuned to be stable. This is achieved by changing the control gain and lag parameters. There can be uncertainties associated with the parameters that are used to describe the system and the gain constants and time constants for the control system. This can cause a reversal in stability to instability. So the system has to be made robust. Robustness of system is measured using methods of frequency analysis described in this chapter. The output response

382

Kal Renganathan Sharma

to a sinuous input to a prototypical first order process was discussed. The amplitude reduction and phase angle development can be seen in Figure 6.1 as well as in the expression given by Eq. (6.8). Aliter is provided for calculation of AR, amplitude ratio and phase angle f. The Laplace transform variable „s‟ is replaced with „(iw)‟ in the transfer function. The real and imaginary part of the resulting complex number, Z = X +iY is calculated. The AR is given by. In a Bode diagram, the AR amplitude ratio is plotted against the frequency in a log-log plot and the phase angle, f is plotted against the frequency in a semi-log plot. A semi-log plot is chosen for the phase angle diagram in anticipation of negative values for phase shift. The AR and phase angle Bode plots for the sinuous input to a first order process was given in Figure 6.2 and Figure 6.3. In Example 6.1, the Bode plots and Nyquist diagram for a point in a semi-infinite medium subject to constant wall temperature boundary condition were constructed. The results between damped wave conduction and relaxation and Fourier conduction models were compared with each other. Taylor series approximation was used to rationalize the Laplace domain expression with exponential term in Eq. (6.15). The results were presented in Figures 6.4 -6.7. Typical frequency response characteristics of prototypical second order system were discussed. Resonant peak can be expected in the frequency domain where there was an „overshoot‟ in the time domain or where there was a damping coefficient of z < 1 in the Laplace domain. Bandwidth is defined as the frequency corresponding to a value of 0.707 of the zero frequency ABS(Z) or the steady process gain. Quicker rise time manifests as larger bandwidth. The closed loop stability considerations discussed in Chapter 5.0 takes shape in the frequency domain as Bode criterion and Nyquist criterion. Bode criterion states that; The system is considered closed loop stable if and only if; The Nyquist stability criterion states that the process is said to be closed loop stable if the Nyquist diagram of Gp(s)Gc(s) does not engulf the critical identified point (-1,0). Gain margin and phase margin are defined. AR and phase angle Bode plots were constructed for the transfer function obtained in Example 6.1.

Figure 6.10. Bode Phase Angle Plot for Example 6.1 when p = 0.01.

Frequency Response Analysis

383

6.7. FURTHER READING E. van Valkenburg, M. (1984). University of Illinois at Urabana-Champaign, “In Memorian: Hendrik W. Bode, (1905-1982)”, IEEE Trans. on Automatic Control, Vol. AC-29, 3, 193194. Quote: “Something should be said about his name. To his colleagues at Bell laboratories and the generations of engineers that have followed, the pronounciation is boh-dee. The Bode family preferred that the original Dutch be used as boh-dah”. Golnaraghi, F. & Kuo, B. C. (2009). Auntomatic Control Systems, John Wiley & Sons, 9th Edition, Hoboken, NJ. Levenspiel, O. (1999). Chemical Reaction Engineering, John Wiley & Sons, Hoboken, NJ. Mickley, H. S., Sherwood, T. E. & Reed, C. E. (1957). Applied Mathematics in Chemical Engineering, McGraw Hill Professional, Second Edition, New York, NY. Nyquist, H. (1932). “Regeneration Theory” Bell System Technical Journal, Vol. 11, 126-147. Sharma, K. R. (2005). Damped Wave Transport and Relaxation, Elsevier, Amsterdam, Netherlands. W. Bequette, B. (2003). Process Control: Modeling, Design, and Simulation, Prentice Hall, Upper Saddle River, NJ. Yeung, K. S. (1985). “A Reformulation of Nyquist‟s Criterion”, IEEE Trans. Educ., Vol. E28, 59-60.

6.8. EXERCISES Review Questions 1.0. How do uncertainities in parameter values affect stability of output response ? 2.0. What is meant by „robustness‟ of control ? 3.0. What is meant by reversal of „stability criterion‟ ? 4.0. What happens to the realizability of the system when a lag term such as e-s appear ? 5.0. Illustrate a use of Pade approximation. 6.0. What are the differences between AR amplitude ratio and phase angle  obtained during frequency analysis ? 7.0. What in Figure 6.1 leads to the inference of a property such as „low pass filter‟ characteristic ? 8.0. Why is a semi-log plot chosen in phase angle Bode plot ? 9.0. Does process gain and time constant affect AR and phase angle ? Why ? 10.0. Derive the amplitude ratio, AR, phase angle  for a given system output response for a given input to a prototypical second order process. 11.0. What are the ordinates of Nyquist diagram ? 12.0. What is meant by resonant peak ? 13.0. What is the significance of bandwidth ? 14.0. Can filtering characteristics be obtained from bandwidth analysis ? 15.0. What is meant by „cut-off‟ rate ?

384

Kal Renganathan Sharma 16.0. When is a loop considered to be „nominally stable‟ ? 17.0. What is meant by „Bode criterion’ ? 18.0. What is meant by „Nyquist criterion’ ? 19.0. What are the differences between „gain margin ‟ and „phase margin‟ ? 20.0. What is the significance of minima and maxima in Bode diagram of Example 4.1 as shown in Figure 6.9 ? 21.0. Does the curve found in Figure 6.10 inflect ? 22.0. What do you expect to see in AR Bode and phase angle Bode of a critically damped system ? 23.0. How does the Nyquist diagram of a overdamped system differ from that of an underdamped system ? 24.0. Can anything peculiar be noted in the Nyquist diagram of a critically damped system ?

Problems 25.0. Bode Diagram of Output Reponse of PI Control of Automatic Washing Machine Consider the output transfer function obtained in Example 4.3. The design is a PI control of voltage supplied to automatic washing machine. s

Gp s

L

s sJ m

Gc s

y s

s

m

kc

s

Bm

1

I

s

I

G p s Gc s

1 Gp s

Gc s

Construct Bode plots for y(s). What can be inferred from AR plot and phase angle plot. 26.0. Construct Nyquist diagram for output transfer function in Exercise 22.0.. 27.0. Bode Diagram of Output Response of PI Control during Hydolysis of Ethylene Oxide The output transfer function of the system with PI control during hydrolysis of ethylene oxide is given by Eq. (4.73). Construct AR Bode and phase angle Bode plots for the system output response. How does the curve change when damping coefficient,  < 1. i.e., what are the differences seen in the AR Bode of underdamped systems and AR Bode of a overdamped system ? What are the differences between phase angle Bode of underdamped system and phase angle Bode of overdamped system? 28.0. Bode Diagram of Output Response of System of PD Control of Prototypical First Order Process

Frequency Response Analysis

385

The closed loop transfer function of system with PD control of prototypical first order process is given by Eq. (4.80). Construct; (i) AR Bode Plot (ii) Phase Angle Bode Plot (iii) Nyquist Diagram 29.0. Bode Diagram of Output Response of System of PD Control during Hydrolysis of Ethylene Oxide The closed loop transfer function response of system with PD control during hydrolysis of ethylene xide to form ethylene glycol is given by Eq. (4.90). Construct; (i) AR Bode Plot (ii) Phase Angle Plot (iii) Nyquist Diagram 30.0. Frequency Analysis of Intermediate Formation in CSTR from Reactions ni Series Transient analysis of intermediate formation from reactions in series was discussed in Example 4.6. An example of such a reaction is chlorination of methane leading to formation of methyl chloride, dimethylene dichloride, choloform and carbon tetrachloride. The closed loop transfer function, GCL(s) of the system of PI control of such reactions to a step input is given in Eq. (4.117). Construct; (i) AR Bode Plot (ii) Phase Angle Bode Plot (iii) Nyquist Diagram (iv) What is the gain margin ? 31.0. Bode Plot of System Response of PI Control of Prototypical First Order Process to Input that undergoes Exponential Decay with Periodicity Consider any prototypical first order process. Consider a PI controller used to control this process. Let the process gains, kp = 3; kc =0.33. The time response of the process output variable y(t) to a change in the set point, r (s ) 

1 (( s  1)2  1)

was discussed in exercise 2.0 at the end of chapter 4.0. The time constant of the controller, I becomes 10% of the time constant of the process, p. Disturbances are neglected and valve and measurement dynamics are lumped into the process transfer function. The transfer functions for prototypical first order process, gp(s) and that for PI control, gc(s) can seen to be; kp

g p (s) g c (s )

1

ps

kc

Is

1

Is

Construct the AR and phase angle Bode plots for the closed loop transfer function GCL(s).

386

Kal Renganathan Sharma 32.0. Bode Plot for System with PI Control Action for Prototypical Second Order Overdamped Process Consider any prototypical second order process. Consider a PI controller used to control this process. Let the process gains, kp = kc =1. Let the damping factor  = 4.0. A step change in the set point, r(s) = 1/s is given. The time constant of the controller, I becomes 5% of the time constant of the process, p. The simplified control block diagram given in Figure 4.4 is applicable. Disturbances are neglected and valve and measurement dynamics are lumped into the process transfer function The transfer functions for prototypical first order process, gp(s) and that for PI control, gc(s) can be seen to be; kp

g p (s) g c (s )

2 2 ps

kc

2 s 1 1

Is Is

Construct the AR Bode and Phase angle Bode for the closed loop transfer function of the system. What happens to the characteristics of the Bode diagram when the process gain constant equals the controller gain constant. 33.0. Bode Plot of System with PI Control for Prototypical Second Order Undamped Process Consider any prototypical second order process as in Exercise 7.0 at end of chapter 4.0. Consider a PI controller used to control this process. Let the process gains, kp = kc =1. Let the damping factor  = 0.0. A step change in the set point, r(s) = 1/s is given. The time constant of the controller, I becomes 5% of the time constant of the process, p. The simplified control block diagram given in Figure 5.4 is applicable. Disturbances are neglected and valve and measurement dynamics are lumped into the process transfer function. Develop the expression for the output variable y(s) in the Laplace domain and time domain y(t). Sketch the response y(t) as a function of time. The transfer functions for prototypical first order process, gp(s) and that for PI control, gc(s) can be seen to be; kp 2 2 ps

g p (s) g c (s )

kc

1

Is

1

Is

Construct the AR Bode and Phase angle Bode for the closed loop transfer function of the system. What happens to the characteristics of the Bode diagram when the system becomes undamped from a system that is underdamped ? 34.0. Bode Plot of System with PI Control of a Process that is Integrating Unstable Consider a process that is integrating unstable and has the following transfer function.

Frequency Response Analysis

G p (s) 

387

0.5 ( s  2)

Show that a PI controller can stabilize this integrating unstable process. Consider the control transfer function; ( s I  1) s I Show that the closed loop transfer function can be given by; Gc ( s )  kc

GCL ( s ) 

0.5kc (s I  1) (s  I  s I (0.5kc  2)  0.5kc ) 2

As discussed in problem 9.0 at end of Chapter 4.0, r stability 0 < kc < 4. The system would be underdamped when;

2kc   I (4  0.25kc2  2kc )

I 

2kc (0.5kc  2)2

Construct the AR and phase angle Bode plots for; (i) Process that is Integrating Stable (ii) Closed Loop Transfer Function that is Underdamped (iii) Closed Loop Transfer Function that is Overdamped How do the characteristics of the Bode diagram change with change in stability from integrating stable to overdamped stable and to underdamped unstable ? 35.0. Bode Plot of Systems with Inverse Response Some systems exhibit a inverse response as discussed in Problem 10 at end of Chapter 4.0. The output response to a unit step response in such systems decreases initially and increases to a final steady state later. The transfer function of such a system has positive pole in the numerator;

G p (s) 

(s  3) .u ( s ) (s  0.2)(s  2)

Consider a PD controller for example with a transfer function given by; Gc s

kc s

D

1

The closed-loop transfer function of the system is given by;

388

Kal Renganathan Sharma Ds

GCL s D

2

1 2 s kc

3 s 3

D

1

0.4 3 s (1 3 kc

2.2 ) kc

D

Show that for small D the system would be stable. For stability, kc < 0.133 and 0.733 . Is this both possible for positive values of gain constant of 0.33 D kc

controller, kc. and controller time constant D. Construct the AR and phase angle Bode diagrams for the closed loop transfer function. What are the differences in characteristics found in the Bode diagram for; (i) Process with Inverse Response (ii) Closed Loop Transfer Function (stable regime) (iii) Closed Loop Transfer Function (underdamped oscillatory regime) 36.0. Bode Plot of System with P only Control of Jacketed CSTR The Lapalce transfer functions of model equations for dimensionless concentration and dimensionless temperature for the jacketed CSTR described in Figure 4.19 are as follows; X (s )

Da s s 1 Da

Y( s )

Da H * s s 1 Da s 1 U *

U* s s 1 U*

H* s s 1 U*

Consider a proportional controller with the control transfer function; Gc s

kc

Show that the closed loop transfer function for the dimensionless concentration and dimensionless concentration can be given by; GCL s GCL ( s )

kc Da s2

s 1 Da

kc Da kc s

s

3

s

2

Da

U*

H* U*

s 1 Da 1 U *

H * U * 1 Da kc

H* U*

kc

H * U * 1 Da

Construct the AR and phase angle Bode plots for; (i) Dimensionless Concentration, X(s) (ii) Dimensionless Temperature, Y(s) (iii) Closed Loop Transfer Function, GCL(s) for P only control of dimensionless concentration (iv) Closed Loop Transfer Function, GCL(s) for P only control of dimensionless temperature

389

Frequency Response Analysis

37.0. Bode Plot of System with PI Controller with Process Transfer Function with One Pole at Origin Consider a second-order process with the process transfer function with one pole at the origin; Gp s

3 s 2s 1

A PI controller with integral time constant of I = 2 can lead to a pole-zero cancellation (see Exercise 15.0 at end of chapter 4.0). Show that the closed loop transfer function can be given by; 3kc

G CL s

3

s2

I

I

Construct the AR and Phase Angl Bode plots. Compare the characteristics of these plots with the AR and phase angle Bode plot of the process transfer function. 38.0. Nyquist Diagarm of Proportional Controller of CSTR with Recycle Consider the process conducted in a CSTR with recycle as shown in Figure 4.26. The transfer function of model equation for the dimensionless concentration in transient state is given by;

Da

R R 1

XA

Da s s

Da

1

1 R 1 R R 1

R 1

dX A

Da

d

1 s s

Da

1

XA s

R 1

where R is the recycle ratio. Consider a P only controller for the dimensionless concentration. What is the closed loop transfer function for the system. Construct a Nyquist diagram for the closed loop transfer function. What are the differences in characteristics between the Nyquist diagram of the process transfer function and the Nyquist diagram of the closed loop transfer function ? How does the charactertistics of the Nyquist diagram change when the combined system changes in regime from overdamped system to underdamped oscillatory systems ? 39.0. Bode Diagram for a Prototypcial First Order Process with Dead Time Consider a first order process with a lag time, . The process transfer function with dead time,  can be written as; G p (s )

kpe

s

s

1

p

390

Kal Renganathan Sharma Construct the AR and phase angle Bode plots for this process transfer function. Can you arrive at a suitable controller for this process from the Bode plots. 40.0. Bode Plots of Systems with Proportional Derivative, PD Controller for a First Order Process with Dead Time Consider a PD, proportional derivative controller for a first order process with a lag time, . The process transfer function with dead time,  can be written as (see Exercise 20 at end of Chapter 5.0);

G p (s )

kpe

s

s

1

p

The controller transfer function is given by; Gc (s )

kc 1 s

D

In order to evaluate the stability of closed loop transfer function Pade‟ approximation can be used to rationalize the process transfer function. The Pade‟ approximation is; e

s

2 s 2 s

Develop the expression for the closed loop transfer function. In order for the system to be realizable add a filter with order n whose transfer function can be written as; 1

F s s

1

n

Construct the AR and phase angle Bode plots for; (i) Process (ii) Nth order filter (iii) Closed loop transfer function Comment on the characteristics of Bode plots in each case. 41.0. Bode Plots of System with PI Control of Intermediate Species in Denbigh Scheme of Reactions Denbigh [Levenspiel, 1999] presented a scheme of general scheme of reactions for series, parallel and series- parallel reactions. The scheme is given in Figure 5.12 (see Exercise 24.0 at end of Chapter 4.0). The kinetics of these reactions are

391

Frequency Response Analysis dC A

k12C A

dt dC R

k1C A

dt dC S

k34C R

k3C R

dt dCT

k2C A

dt dCU

k4C R

dt

Where k12 = k1 + k2 and k34 = k3 + k4. Show that the process transfer functions for the reacting species in a CSTR during transient operation can be written as follows; CA s CR s C

S

s

CT s CU s

C Ai s s

1 Da12 s C Ai Da12

s s

1 Da12 s

1 Da 34

C Ai Da 3 s s

1 Da12 s

1 Da 34 s

s s

Da 2C Ai 1 s 1 Da12

s s

1 s

1

Da 4 Da12 C Ai

1 Da12 s

1 Da 34

Consider a PI controller with the transfer function;

Gc ( s )

kc

Is

1

Is

What is the closed loop transfer function for each of the processes of species A, R, S, T and U. Construct the AR And Phase angle Bode plots for; (i) Transfer Functions of species A, R, S, T and U (ii) Closed Loop Transfer Function of PI Control of Concentration of Species A, R, S, T and U

Chapter 7

ADVANCED CONTROL METHODS 7.1. ESTIMATION AND CONTROL OF POLYMERIZATION REACTORS The control of polymerization reactors is a non-trivial task due to the following reasons; (a) (b) (c) (d) (e)

Specification of control objectives High viscosities and exothermic nature of the reaction Process dynamics is nonlinear Measurement of polymer structure is difficult Estimation techniques such as Kalman filter and Weiner filter are needed to estimate parameters that can be measured

Elicabe and Meira [1988] presented a review of the estimation and control literature in polymerization reactors. Polymers can be prepared in different types of reactors such as; (a) (b) (c) (d)

Batch Reactor, BR CPFR, Continuous Plug Flow Reactor HCSTR, Homogeneous Continuous Stirred-Tank Reactor SCSTR, Segregated Continuous Stirred-Tank Reactor

Polymerization kinetics can be classified into monomer-monomer chain reactions with termination, polymer linkage without termination and polymer linkage by step reactions with monomer or graft chains. Nine different possibilities for the molecular weight distribution according to Elicabe and Meira [1988] from the different combinations of kinetics and reactors. The polymers prepared in BR or CPFR are found to have molecular weight distribution, MWD; (i) wider than Schulz-Flory distribution for monomer linkage with termination; (ii) Poisson or Gold distribution for monomer linkage without termination; (iii) Schulz-Flory distribution for polymer linkage with monomer or graft chains. The objective of the reactor control is to obtain superior product quality. Quality is characterized by parameters such as: (i) Chemical composition of repeat units; (ii) Chain sequence distribution of copolymers;

394

Kal Renganathan Sharma (iii) Alternating, block or random microstructure; (iv) MWD in linear homopolymers; (v) Multivariate distribution of molecular weight, composition; (vi) Degree of chain branching; (vii) Stereo-regularity; (viii) Particle size distribution in lattices obtained from emulsion polymerization; (ix) Partial porosity and surface area.

End-use properties of polymers processed from molding, extrusion are governed by parameters such as; (a) Mechanical strength; (b) Elastomer relaxation; (c) Adhesive tack; (d) Melt viscosity; (e) Brittleness; (f) Impact strength; (g) Drawability; (h) Elastic modulus; (i) Hardness; (j) Softening temperature; (k) Tear strength; (i) Stress-crack resistance and; (j) Adhesive strength. Good structure-property relations are needed for devising control strategies based on measurement of end-use property values. Another difficulty in setting control objectives is that in polymerization reactors a desirable molecular weight distribution, MWD is the objective and a deterministic scalar quantity. The control objective is a mathematical function and not a vector of scalar parameters. Most propagation reactions are exothermic with an energy release of 20 cal/mole for vinyl monomers. The viscosity of the reaction mixture increases in an exponential manner with increases in conversion. Problems associated with heat removal, mixing and transporting the reactor syrups need be solved by control. Suspension and emulsion processes using water as suspending medium in part to avoid these problems. The polymerization process needs to be well understood before it can be controlled. Development of mathematical models for the processes is constrained by the following difficulties; (a) Kinetic equations for monomer are nonlinear. Control theory developed for linear systems is not applicable. (b) Parameters involved in dynamic models are not all known. This is more the case for reversible reactions and heterophase systems. (c) Highly sensitive to impurities at even low concentrations in the raw material streams. Disturbances are not measurable.

Advanced Control Methods

395

Model equations are developed from mass and energy balances. These result in state space equation. The first moments of the MWD yields information about the molecular structure. Model calculations are performed in two phases. In the first phase, the monomer concentrations, temperature in the reactor are obtained from the input variables such as inlet flow rate, inlet reactor temperature. In the second phase, the molecular parameters are derived. The model equations written for BRs or HCSTRs are modeled by sets of non-linear, time-invariant, ordinary first order differential equations of a general form; *

x y

x, u , d

(7.1)

f ( x, u )

Where x is a vector of n states, u is a vector of m controls and d is a vector of k disturbances and y is a vector of p measured outputs. Eq. (7.1) are the state and measurement equations. Eq. (7.1) when modeled in discrete form can be implemented in DDC, direct digital control. Controls are held piecewise constant over each sampling interval. Models of PFR, plug flow reactors result in partial differential equations. Some of this was discussed in Chapter 3.0. Feedback control of a variable is not possible if the parameter cannot be measured or estimated. On-line equipment for polymer characterization has limited availability. Independent variables such as inlet monomer, inlet initiator concentrations and inlet flow rate of the reactor are readily measurable, and intermediate variables such as reactor temperature, monomer concentration in the reactor are easier to measure compared with the structural properties. Indirect measurements such as level in the reactor are measured using nuclear probes. Vapor pressure transmitters are used for temperature detection. Melt index is correlated with viscosity. Viscometers are deployed. The electrical power consumed in the reactors can be used for estimation of viscosity. Structural parameters such as molecular weight, molecular weight distribution may be measured using SEC, size exclusion chromatography. Composition distribution can be obtained from elemental C, H and N analyzer. Sequence distribution can be measured using 13C NMR, nuclear magnetic resonance spectroscopy. Weight average molecular weight, Mw can be measured using light scattering photometers. Mn, the number average molecular weight can be measured using vapor pressure and membrane osmometers. Particle size distribution in lattices can be measured using hydrodynamic chromatography, SEC light scattering, field flow fractionation or electron microscopy. Implementation of on-line SEC requires; (a) a dead time in the reactor sample; (b) sample conditioning equipment for sample injection involving dilution and ultrafiltration; (c) fractionation time; (d) digital system for the acquisition and treatment of data. The nonlinear equations in the models need be linearized. Some control algorithms are designed from input-output measurements alone, without consideration of internal chemistry or local events within the reactor. This approach is also called „black-box‟ approach. The

396

Kal Renganathan Sharma

resulting models are empirical in nature. These are discrete linear transfer functions or difference equations. The output, yt may be found from r previous values of the same output, present value of input and s previous values of the input. A time delay b may also be included in the model structure. A parameter estimation algorithm can be used to estimate the coefficients t and t. Control strategies may be open-loop or closed-loop or self-regulating. An open-loop strategy may be derived from empirical or empirical and theoretical considerations. The control objectives would be met only if the model used is accurate and robust and no unmeasurable disturbances are present. Feedback of controlled variables are necessary. In Figure 7.1 is shown the scheme for possible feedback control configurations. The scheme comprises of state-model, subsystem 1, subsystem 2 and the measurement equations. It would be desirable to use the end use performance measurements for use in feedback control. This is not possible. Structure –property relations developed by in-house experts can be used to correlate the property measurements with the corresponding structure. Structural parameters can be measured directly or estimated from models that relate process variables with the structure. Different paths such as structural variables path, output to ^

subsystem path, feedback the measured variables directly to obtain the state estimates x(t ) from such measures. State estimation techniques have been developed to provide estimates of the state variables even in cases where they cannot be obtained by direct measurement. Sequential estimates can be updated using recursive algorithms. Filtered and predicted estimates are taken up. State estimation is a method of determination of values of the states from the knowledge of the output data and the input data. A unique estimate needs to be made from the input-output data set. The physical significance of these criteria is that the system is said to be observable and reachable. The architecture of the problem is shown in Figure 7.2.

Figure 7.1. Possible Feedback Configurations.

397

Advanced Control Methods

The process model is disturbed by the noise process (t) due to either unknown load/disturbance or model errors. The error in reading of output data is (t). The mean and covariance of the error and noise are known quantities. The symbol “^” is for estimated values in Figure 7.2. K is estimated by different methods. In non-linear estimation problems the objective may be to minimize the average squared errors in the initial estimate in the process model and in measurement device. This has been shown to lead to infinite set of coupled equations that are not tracfigure. Closed form analytical solutions are not possible. One way out is the approximation called extended Kalman filter for manufacturing units that can be modeled using linear equations. One of the benefits of this estimation approach is that when some of the model parameters are not known then under some conditions these unknown model parameters may also be estimated in addition to the states. Jo and Bankoff [1976] developed a Kalman filter in order to obtain estimates of conversion and Mw, weight averaged molecular weight. One of the parameters was augmented resulting in improvement of filter performance. Due to the long residence time in the study, the adaptive, exponential and/or iterative schemes did not improve the filter performance. Quasi-steady state assumption was used. Poor initial estimates of the state variables or of the state covariance matrix could readily be tolerated. Too small estimates of the process noise covariance matrix caused the filter to diverge. The Kalman filter is an on-line computer algorithm which is intended to provide better estimates of the state variables that be obtained by direct solution of measurement equations. If the system is observable, optimal estimates of variables that cannot be measured directly can be made. Consider a linear n-dimensional system in which the random-variable state vector xk, at time tk, evolves by means of a deterministic transition matrix, (k+1,k) to a new value, xk+1 at time tk+1 = tk+t, where t is a fixed sampling interval A zero-mean Gaussian disturbance k is imposed on the transition matrix. A m dimensional vector, yk is obtained, subject to a zero mean Gaussian measurement error, vk. The Kalman filter can be written by the following equations for predictions between observations; ^k

xk Pkk 1

k 1, k Pkk ^k 1 1

xk

Pkk 11

I

(7.2)

T

k 1, k

T

(k)Q k 1

^k

xk

(7.3)

k

^k 1

K (k 1) yk

K k 1 M k 1 Pkk 1 I

K k 1

^k

k 1, k xk

1

1

M k 1 xk

K k 1 M k 1

T

(7.4)

1

K k 1 R k 1 KT k 1

Pkk 1M T k 1 M k 1 Pkk 1M T k 1

R k 1

1

(7.5)

(7.6)

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Kal Renganathan Sharma

Figure 7.2. System with Corresponding Sequential Estimation.

The algorithm for the filter can be processed as follows; ^0 x0

and P00 (i) Obtain (ii) Set k = 0 ^k

(iii) Calculate xk

1

P0 at time t0.. from..prior knowledge of the system or by guess.

and Pkk 1 at tk from Eq. (7.2) and Eq. (7.3). These are a priori estimates

of xk+1 and Pk+1 and tk. (iv) Calculate the Kalman gain matrix at tk+1, K(k+1) from Eq. (7.6) determining M and R from the measurement equations and properties of noise. (v) Once the measurement yk+1 is available at time t = tk+1 calculate the a posteriori ^k 1 1

estimates xk

and Pkk 11 from Eqs. (7.4), (7.5).

(vi) Increment k by 1 and return to step 3 and repeat the loop. The linear filter may be applied to nonlinear systems after linearization of nonlinear equations. The nonlinear dynamic systems is described by the stochastic differential equation for a vector of size n; dx dt

f x, t

G t

t

(7.7)

Measurements are given by; y

h x, t

vt

(7.8)

x, , v are the same as in the linear case and are random variable state vector, zero-mean Gaussian disturbance, and Gaussian measurement error respectively. A deterministic _

reference trajectory x t is generated with a given;

399

Advanced Control Methods _

_

x0

x t0

(7.9)

These equations are linearized and discretized as follows; xk

k 1, k

1

xk

k

k 1

(7.10)

Where, _

xk

x tk

(7.11)

x tk _

yk

y tk

E y tk

h x tk , tk

y tk

(7.12)

_

hx x tk , tk

M k

(7.13)

and  is the transition matrix. This can be obtained from; _

*

t,

f x x, t

t,

(7.14)

With the initial condition;

,

I

(7.15)

The linear filter is then directly applicable to the linearized system when the differences between reference and actual trajectories are small. In order to keep the differences small, the _

x0 can be estimated when new observations are made available. This method is called the

global iteration. This can be repeated till constant reference trajectories. An extended Kalman filter was developed; ^k

xk

^k 1

tk

^k

1

f xt , t dt

xk

(7.16)

tk

^k 1 1

xk

^k

xk K k 1 yk

^k 1

An iterative Kalman filter can also be developed.

h xk

1

(7.17)

400

Kal Renganathan Sharma

Congalidis, Richards and Ray [1989] discuss different approaches to the control of polymerization reactors. Homopolymers and copolymers can be made in CSTR by continuous mass polymerization. Increased competition and emphasis on product quality has rendered method of recipe control obsolete. During recipe control the operating conditions are updated infrequently based on results for testing of polymer samples. Control hardware equipment is available with improvements being made. Advances have been made on development of control theory. Control of polymerization reactors is a tough and complex problem. On-line sensors for product quality monitoring are needed. The steady state values are extremely sensitive to small changes in parameter values or operating conditions. Model equations written to describe reactor dynamics are nonlinear. There are three types of approaches to control of polymerization reactors; A. Development of optimal trajectories for the manipulated variables. The narrowing of the molecular weight distribution, or copolymer composition distribution can be achieved in batch and semi-batch reactions by manipulation of temperature, monomer and initiator during the batch. Manipulated variable trajectories were generated. The control problem becomes one of optimization problem. B. Use of nonlinear state estimation techniques to estimate infrequently measured control variables. Extended Kalman filter was used in order to obtain estimates of conversion and weight average molecular weight in an experimental continuous mass polymerization of vinyl acetate in a small glass CSTR. State estimation using both online and off-line delayed measurements was looked at in order to estimate the molecular weight distribution and branching distribution in a polymerization reactor. C. Evaluation of specific feedback controllers. Multivariable adaptive controller was used to control the reactor monomer concentration and temperature in the continuous mass polymerization of MMA, methyl methacrylate in a CSTR. Simulations were performed and acceptable performance was found in the presence of noise and load changes. Difficulties were encountered in the multiple steady state regions with strong interactions. Additional control problem is posed when there is need for recycle of monomer and solvent. 4 PI controllers were tuned to control the reactor outputs using the selected manipulated variables. The feedback strategy was implemented hierarchy above the feed forward level. A PI controller can be tuned to make it more proper and can be used to manipulate the coolant flow rate in order to control the jacket temperature. Jacket temperature is selected as manipulated variable and not the coolant temperature. The separator and hold tank are described as isothermal first-order lags with constant level and residence time equal to the reactor residence time. The need for level controllers is obviated. Congalidis, Richards and Ray [1989] considered copolymerization of MMA, methyl methacrylate and vinyl acetate in benzene solvent and AIBN, azo-bis-isobutyronitrile initiator. The reactivity ratios are r12 = 26 and r21 = 0.03. The steady state operating point was designed upon consideration of polymer production rate, composition of copolymer, weight average molecular weight and reactor temperature. The inputs are reactor flows of monomers, initiator, chain transfer agent, solvent, and inhibitor, the temperature of the reactor jacket and the temperature of the reactor feed. The reactor, separator and hold tank are preheated. The inputs to the nonlinear model were varied in order to obtain steady state values for output

401

Advanced Control Methods

variables that are acceptable. Viscosity of the medium is held at moderate levels. The equations that govern the dynamics of the reactor assuming that the polymer chemistry is of the free radical type are as follows; Species mole balance can be written for monomer, initiator, solvent, CTA, chain transfer agent and the inhibitor;

C j0

dC j d

(7.18)

C j 1 Da j

Where, Daj = Damkohler number (kj) of species j,  = t.-1 and Cj0 is the concentration of the species j in the inlet of the reactor and Cj is the concentration of the species in the reactor and in the effluent. The energy balance of the reactor can be written as; dTr dt

Tr

Trf r

2

2

Hr j 1 i 1

k C C jk pjk j k

U r A Tr

Tj

r

VC p

Tr 0

(7.19)

r

They used the long chain hypotheses for the polymer reaction rates. By quasi-steady state assumption they derived the expression for concentration of free radicals of all the different dyad types, AA, AB, BB, BA. The rate of change of concentration of radicals can be taken to be zero. The number and weight average molecular weight was calculated by equating the rate of initiation and termination rates. Dead polymer and live polymers are distinguished from each other. The moments of the dead polymer and live polymer were calculated. The separator and hold tanks were modeled as first-order lags on species concentration with constant volume reactor. It can be seen that the governing equations are nonlinear ordinary differential and algebraic equations. They were integrated using a 5th and 6th order Runge Kutta numerical method. The steady state and transient values of the polymerization rate, reactor temperature, copolymer composition and molecular weight. They suggest feed forward control strategy for their recycle streams. The polymer properties are affected by introduction of disturbances in the reactor monomer feed. The fresh feeds are manipulated and a constant feed composition is maintained and allowed to flow into the reactor. The recycle composition is measured by online gas chromatograph. This may introduce some time delays in the circuit. The open-loop output variable response to a purge disturbance was studied. The polymerization rate was found to have an inverse response with the molecular weight. The molecular weight was found to have an inverse response with the residence time in the reactor. The conversion was found to have a maxima as a function of time. It undergoes an integrating phase prior to undergoing a stable phase. The performance of the feed forward controllers was illustrated by examination of the response of the output and manipulated variables to a pulse disturbance of the purge ratio that occurs while the reactor operates at steady state. Feedback controllers were evaluated for polymer properties and production rate. The design procedure used in this study to select a control structure is based on ranking various candidate structures according to a condition number . They discuss use of CONSYD package

402

Kal Renganathan Sharma

of control system design programs in conjunction with the nonlinear model and how it cut the time needed for completion of the design. One of the issues in the control design is the specification of 4 manipulated variables to control previously specified output variables. The selected variables are monomer flow rates of monomers, initiator flow rate, chain transfer agent, reactor jacket temperature. Step tests were conducted using CONSYD package on the nonlinear reactor model. The transfer functions were fitted by least squares to the responses of the output variables for all choices of manipulated variables. The denominator polynomial of the transfer function was restricted to first or second order. The summary of transfer functions obtained from this method is summarized in for of Tables. There are 4! ways that the selected four manipulated variables can be paired with 4 output variables. All pairings have the same zero frequency condition number and minimum singular value. It is assumed that accurate measurements of output variables are available for feedback control. Reactor temperature is readily available. The remaining outputs are not available. Periodic laboratory analysis of monomer concentration, copolymer composition and molecular weight are available. Significant sampling dead time is introduced. The required variables can be obtained by improved online instrumentation such as viscometers, refractometers or by use of state estimator in order to infer polymer properties between the measurements. The estimator can be implemented in the form of an extended Kalman filter as discussed in [Jo and Bankoff, 1976]. Multiple steady states may cause problems during plant start-ups and plant shut-downs. The controllers will not operate efficiently when the assumptions in their derivations are not met. Feed forward control of the recycle stream eliminates recycle disturbances. The control of the reactor is separated from the rest of the process. Ratio controllers on the flows to the reactor partially decouple the multivariable nature of the reactor control problem. Feedback control of polymer rate, composition, molecular weight and reactor temperature is accomplished using PI controllers. The feedback PI and ratio controllers control the system on the arrival of disturbances and the nonlinear behavior of the reactor. These control schemes have been implemented successfully in several polymerization plants.

7.2. IMC, INTERNAL MODEL CONTROL The methods of feedback control discussed in Chapter 5.0 such as P, PI control or Tuyreus-Luben oscillation based tuning does not require a model for the process a priori to control action. Closed loop response and stability considerations were paramount in devising of the control action. Control action law such as P control, PI control, PD control and PID control were obtained without any detailed knowledge of the process. A model based procedure can be developed with the culmination of embedment of the process in the controller. Consider the mixing tank (Figure 7.3) with a jacket provided for heating the tank. Model Equations: Reactor:

dT dt

v Ti V

T

UA Tj VC p

T

(7.21)

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Advanced Control Methods

Jacket:

dTj

vj

dt

Vj

Tjin

UA Tj jVj C pj

Tj

T

Figure 7.3. Mixing Tank Heated by hot Fluid in Jacket.

(a) At Steady State 0

1 50 125 10

UA 150 125 61.3(10)

UA (3)(61.3) 183.9 Btu.F

0 vj

vj

2.5

200 150

75 1.5 ft 3 .min 50

1

ft

(7.22)

3

183.9 150 125 61.3 2.5

(7.23)

1

(b) Matrices in State Space Model dT dt dT j

0.4 1.2

0.3 1.8

T Tj

5 120

(7.24)

dt

(c) Fourth Order Runge-Kutta Method for Integration of ODE in MS Excel Spreadsheet 20.07 for Windows. Classical fourth-order RK method The recurrence formula is given from [Chapra and Canale, 2006]

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Kal Renganathan Sharma

Tl

1

Tl

Tjl

1

Tjl

h k1 6 h ' k1 6

2k2

2k2'

2k3

2k3'

k4

k4'

(7.25)

(7.26)

Where, k1 = f(ti,Ti)

(7.27)

k2 = f(ti+0.5h, yi+0.5k1h)

(7.28)

k3 = f(ti+0.5h,Ti+0.5k2h)

(7.29)

k4 = f(ti+h, Ti+k3h)

(7.30)

vi Ti T Q V Integration was performed using a MS Excel spreadsheet. The key results are shown in Table 7.3 and Table 7.1. The step size used was h = 0.01 min. Tj

From Eq. (7.21), f(t,T) =

Table 7.1. Temperature in the Reactor, Jacket vs. Time T

T

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 3 4 5 6 7 8 9 10

50.000 54.263 58.066 58.066 64.567 67.374 69.946 72.316 74.514 76.566 78.492 93.615 105.150 114.938 123.438 130.854 137.329 142.985 147.925 152.240

Tj 200 189.145 180.2475 180.2475 167.0994 162.3572 158.5734 155.5905 153.2766 151.5209 150.2307 150.7255 158.6955 166.9491 174.3797 180.9078 186.6161 191.6031 195.9591 199.7639

Advanced Control Methods

405

Figure 7.4. Transient Reactor Temperature and Jacket Temperature vs. Time.

(d) From the last row of Table 7.1 it appears that the steady state temperature values, Tis = 152.2 and Tjs = 199.7 F. The values from solution in (a) are Tis = 150 F and Tjs = 150 F. (e1) Large Step Change in Jacket Flow Rate: 10. times to 15 ft3.min-1.

Figure 7.5. Transient Reactor Temperatures and Jacket Temperature vs. Time Response to a Big Step Change in Jacket Flow Rate by 10. Times.

406

Kal Renganathan Sharma (e2) Small Step Change in Jacket Flow Rate: 10% times to 1.65 ft3.min-1

Figure 7.6. Transient Reactor Temperature and Jacket Temperature vs. Time Response to a Small Step Change in Jacket Flow Rate by 10.%.

(f) Larger Vessel v = 10 ft3.min-1; V = 100 ft3 10 UA 0 50 125 150 125 100 61.3(100) 7.5 UA (4)(61.3) 73.56 Btu.F 1 ft 3 25 (g)

10 73.56 T js 125 50 125 100 100(61.3) 7.5 1.5 750 F T js 0.012

(7.31)

0

(7.32)

407

Advanced Control Methods vj

0

2.5

220 300

vj

73.56 750 125 61.3 2.5

200 750 3

0.733 ft .min

(7.33)

1

(h) dT dt dT j

0.112 0.046

0.012 0.779

T

5 5.864

Tj

(7.34)

dt

Eigenvalues of the A matrix Characteristic second degree polynomial equation 0.112

0.779

0.000552

2

0.891

0.0867

0

(7.35)

Eigenvalues are 1 = -0.78; 2 = -0.11 Both eigenvalues are negative. Hence the system is stable. The model developed for the mixing tank heated by the hot fluid in the jacket can be used to design a controller. A good knowledge of then when the process is stable and when the process is underdamped oscillatory unstable can lead to better control action. Control action of unstable systems may result in unsatisfactory results. In general the model-based controller can be added as shown in Figure 7.7. The analysis of the block diagram in Figure 7.7 leads to; y(s) = Gp(s) M(s) r(s)

(7.36)

Consider a prototypical first order process such that;

G p (s )

kp s

p

(7.37) 1

Figure 7.7. Model Based Control in Open Loop Configuration.

408

Kal Renganathan Sharma

When M(s) is a constant, km. For a step change in the set point of magnitude R‟ the output function can be written as; k p km

y( s )

s

p

R' 1 s

(7.38)

From the final value theorem it can be seen that after infinite time; Lts

0 sy( s )

y(t )

Ltt

k p km R '

(7.39)

In order for the output to be free of “offset”, km

1 kp

(7.40).

The output of the model based controller and process in the time domain can be seen to be; t

y(t )

km k p R ' 1 e

p

(7.41)

In order to achieve perfect control; M (s )

1 G p (s )

(7.42)

If this perfect control, can be realized the output would track the input set point. Perfect control is easier written in Eq. (7.42) than can be implemented in practice. In order for a controller to be physically realizable the order of the polynomial equation that is used to determine the poles must be at least one order greater than the polynomial equation that is used to determine the zeros. A filter can be added in order to make the control action physically realizable. A first order filter has the following transfer function; F (s)

1 s

1

(7.43)

Where  is a filter tuning parameter with units of time? The model based control is made “more proper” so to speak. The transfer function is now;

Advanced Control Methods F (s) G p (s )

M (s )

G p 1 (s ) F (s )

409

(7.44)

For the first order transfer process whose transfer function is given by Eq. (7.37) the model based controller transfer function M(s) becomes; s M (s )

1

p

(7.45)

1

kp s

By addition of a first-order filter the first order controller is made more physically realizable. The model based controller transfer function as given by Eq. (8.45) can be viewed as a lead/lag controller. This way the order of the polynomial equation that determines the poles is at least the same as the order of the polynomial equation that determines the zeros. The transfer function of the output variable y(s) is now; y(s) = Gp(s) M(s) r(s) r (s ) y( s ) s 1

(7.46)

The output response is first-order with a time constant of. This controller is dynamic as compared with the static control in Eq. (7.40). The time response of the dynamic control action is faster than that compared with the static controller as long as; (7.47)

p

Inversion of a process model alone may not be sufficient for good control. In order for the controller to be stable and realizable the process transfer function must be factorized. For example, consider the following transfer function for the process; k Gp s

s

p

p1

1

s

1 s

p2

(7.48) 1

When the parameter  is positive and real the zero is positive. This can lead to an inverse response of the process transfer function. A model based controller for this process is attempted as; s M (s)

Consider a first order filter such that;

p1

1 s

kp 1

p2

s

1 F (s)

(7.49)

410

Kal Renganathan Sharma s M (s )

1 s

p1

kp 1

1

The poles of the controller are

1

1

s

1

and

p2

1

s

(7.50)

. The system is unstable. The zero of the

process has become the pole of the control transfer function that can lead to inverse response. The unstable pole in Eq. (8.50) can be removed. The model transfer function then becomes; s M (s )

p1

1 s

p2

1

1

kp

1

s

(7.51)

It can be seen from Eq. (7.51) that M(s) is not proper. In order to make M(s) proper the order of the filter can be increased. Thus when a second order filter is used instead of the first order filter; s M (s )

1 s

p1

kp

p2

1

1 s

1

2

(7.52)

For such control action the output response y(s) can be seen to be; y(s) = Gp(s) M(s) r(s) 1 1 s y( s ) 2 s s 1

(7.53)

The output y(t) can be seen to be; t

y(t ) 1

2

e

te

t

(7.54)

Eq. (7.54) is plotted in Figure 7.8. It can be seen from Figure 7.8 that there is an inverse response for values of  = 1.0 and  = 0.1. Although Figure 7.8 is for the indicated values of  and  in general Eq. (7.54) results in an inverse response. For larger values of  there is a minima present prior to the inverse response. What can be inferred from this analysis is that the inverse response of the process cannot be removed by a stable filter or stable control system. The structure of IMC is shown as a block diagram in Figure 7.9. The components of.IMC structure are (i) load/disturbance GD(s); (ii) process, Gp(s); (iii) Model controller, M(s); (iv) Model, Gm(s); (v) set point, r(s); (vi) process output, y(s). Uncertainties in model can be designed for by improving the robustness of the control action. The ramifications of parameter uncertainties was discussed in Chapter 5.0. This can be achieved by adjustment of filter parameters. In some cases the model transfer function is not invertible readily. For example, in systems with time lag the transfer function of the process would be of the form;

Advanced Control Methods

G p (s )

kpe

s

s

1

p

411

(7.55)

Figure 7.8. Output Response to Model Transfer Function Given by Eq. (7.52) for  = 1.0. and  = 0.1.

Figure 7.9. Block Diagram of IMC Internal Model Control.

412

Kal Renganathan Sharma A first-order Pade approximation can be used as follows; s 2 s 2

1 s

e

1

(7.56)

Plugging Eq. (7.56) in Eq. (7.55); kp 2

G p (s)

s

s

1 2

p

(7.57) s

A model transfer function with an nth order filter can be developed as; 1 2

s

kp 2

s s

1

s

1 2

s

s s

1

s

p

M (s)

n

(7.58)

For a first order filter, n = 1, p

M (s)

kp 2

(7.59)

The output transfer function for a step input is seen to be; 1 s s

y( s )

(7.60)

1

The output transfer function can be written as;

y(t )

1

t

0

p

e

t

dp 1 e

(7.61)

7.3. RATIO CONTROL In some applications, ratio control can be the method of choice. Consider the following example. Consider preparation of flue gas with CO, carbon monoxide in the mixture. A mixture of CO2, carbon dioxide and air is passed through a packed bed of pure carbon. The reactor is operated in an adiabatic manner. There are two reactions that are expected to take place to completion. These are as follows;

413

Advanced Control Methods Reaction A

CO2(g) + C(s)  2CO(g)

(7.62)

2C(s) + O2(g)  2CO(g)

(7.63)

Reaction B

The inlet mixture, CO2/air is at a certain ratio such that the heat of reactions from Reaction A and Reaction B cancel out each other. Reaction B is exothermic and Reaction A is endothermic. The feed is preheated to 600 0C and the bed is operated at 600 0C. The heat of formation at 298 K for CO from Smith, van Ness and Abbott [2005], can be seen to be; Hf298 = -110.53 kJ/mole

(7.64)

The heat of reaction at 600 K can be calculated using the relation; H = dT The temperature variation of heat capacity information can also be used in the analysis as follows;

C ig p

600 298

8.314 600 298

(600 K ) C ig p

(298) C ig p

R

R

A BT

3.507 3.376 302 0.557 *10

CT 2 3

D

(7.65)

T2

6002 2

2982 3

0.031*10 T

5 600 298

(8.314)(7.1163) 59.2 J / mole / K 600

H R0(600 K )

H R0(298)

C p dT

110.5

298

59.2(3.2) 1000

128.4kJ / mole

(7.66)

For Reaction A, 600

H R0(600 K )

H R0(298)

C p dT 298

2

110.5

59.2 * 302 1000

393.51

86.28 * 302 (7.67) 1000

162.8kJ / mole

Let the outlet flue gas stream composition be as follows in mole fraction; yCO = 0.2; yCO2 = 0.01; yO2 = 0.005; yN2 = 0.781

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Kal Renganathan Sharma

For a basis of 1 mole of CO formed x moles are formed by Reaction A from CO 2 and (1x) moles are formed from Reaction B from carbon. The heat balance from the heat generated from Reaction B and heat absorbed by Reaction A can be written as follows; 162.8x – (1-x) 128.4 = 0 x

128.4 162.8 128.4

(7.68) (7.69)

0.44

The corresponding inlet stream ratio between CO2 and air can be calculated as;

0.44 * 0.2 Flue Air 0.56 * 0.2 0.005 0.21

0.088 0.557

1 6.33

(7.70)

The flue to air ratio of 1:6.33 can be seen to be an important process parameter to control in order to maintain the adiabatic reactor status. The ratio control block diagram is shown in Figure 7.10. The set point of 1:6.33 is input into the comparator. The measured flow velocities of CO2 stream and air streams are sent by flow transmitters to the comparator. The control action is taken proportional to the error between the set point and measured ratio. Ratio control can also be effected the flow stream velocity of CO 2 can be maintained at a certain “ratio” of flow rate of air inlet stream. The flow rate ratio is calculated and this information is sent to the comparator. In another strategy the flow rate of CO2 inlet stream is measured and multiplied with a desired ratio in order to determine the set point for the air stream flow rate. This provides a linear input-output relation. The measured output is;

Figure 7.10. Feedback Control of Ratio of CO2 to Air during Manufacture of Flue Gas.

Advanced Control Methods

Ratio

vair vCO2

1 vair vCO2

415

(7.71)

The steady state process can be seen to be; kp

( Ratio) u

1 vCO2

(7.72)

The process gain is a function of the flow rate of CO2 stream. For the air flow stream the process gain would be 1.

7.4. NEURAL NETWORKS 7.4.1. Overview of Neural Networks Neural networks can be used to approximate any reasonable function to any degree of required precision. ANN, artificial neural network is used in control and in pattern recognition and knowledge acquisition. The structure of an ANN consists of a number of computing elements which resemble neurons and synapses of a human brain organized in a network (McClelland, Rumelhart et al., [1986]. Presently most of the implementations of neural networks are software based. Interconnections higher than 2 units may lead to „higherorder‟ or „Sigma-Pi‟ networks. A number of important architectures can be recognized. These are; i) Recurrent ii) Feed-Forward and iii) Layered. A recurrent architecture contains directed loops. An architecture without directed loops is said to be feed forward. Recurrent architectures are less simple. An architecture is layered if the units are partitioned into classes also called layers, and the connectivity patterns are defined between the classes. A feed forward architecture is not necessarily layered. The number of layers is referred to as the depth of the network. In the back-propagation model the network is processed in three distinct steps. The first step is the forward sweep. In the forward sweep the input is given to the input units. The output values of each unit is calculated and moved over the connections to the units in the next layer. The units in the next layer receive the input from units in the previous layer. The output values of the units are then calculated and passed to the units in the next layer and so on. The next step is the error calculation. In this step the values of the output units are compared to the desired output, teaching. If the difference between the actual output and the teaching is within the acceptable error range, then learning is successful. If the difference is not within an acceptable range then an error value is calculated and learning is unsuccessful. The third step is the back-propagation of the error value. In this step, should the learning be unsuccessful

416

Kal Renganathan Sharma

then the error value is propagated backward through the net. The weights of the connections between the units are adjusted to minimize the error value. The main objective of this step is to close the gap between the actual output and the desired output. The above three steps are repeated until learning is successful. The behavior of each unit in time can be described using either differential equations or discrete update equations. Typically a unit i receives a total input Xi from the units connected to it and then produces an output, Yi – f(xi) where f is the transfer function of the unit. In general, all the units in the same layer have the same transfer function, and the total input is a weighted sum of incoming outputs from the previous layer so that,

Xi

Wij y j

Wi

(7.73)

j N (i )

Yi

f ( xi )

f(

WijYj

Wi )

(7.74)

j N (i )

Where Wi is called the bias or threshold of the unit. Wij and Wi are the parameters of the NNs. Other parameters such as time constants, gains and delays are possible. Usually, the total number of parameters is determined by the number of layers, the number of units per layer and the connectivity between layers. It is said to be „fully connected‟ when each unit in one layer is connected to every unit in the following layer. A normalized exponential unit is used to compute the probability of an event with n possible outcomes, such as classification into one of n possible classes. Let j run over a group of n output units, computing the n membership probabilities, and xj denote the total input provided by the rest of the NN into each output unit. Then the final activity yi of each output unit is given by; e

yi

xi

(7.75)

n xk

e k 1

n

yi

(7.76)

1

i 1

When n = 2, the normalized exponential is equivalent to a logistic function via a simple transformation.

yi

e e

x1

x1

e

x2

(7.77)

Any probability distribution Pi (1  j  m) can be represented in normalized exponential from a set of variables xj (1  j  m):

Advanced Control Methods

Pi

e

417

xi

(7.78)

m

e

xk

k 1

As long as m  n. This can be done in infinitely many ways, by fixing a positive constant k and letting Xi = log(pi) + kj for i = 1, n. If m < n there is no exact solution, unless the pi assume only m distinct values at most. The radial basis functions, RBFs, is another type of widely used functions. Here f is a bell shaped function like Gaussian. Each RBF unit i has a reference input xi and f operates in the distance d(xi, xi) measured with respect to some metric yi = f(d). In spatial problem, d is usually the Euclidean distance. Thus some of the important features of ANN model depend on the task at hand. The process of computing approximate weights is called „learning‟ or „training‟ in the ANN paradigm. There are many ANN learning algorithms that employ the principles described above. In general, ANN learning algorithms are classified are classified by either the tasks to be achieved or the methodologies to achieve a task: a) b) c) d)

Auto association Classification Heteroassociation Regularity Detection. There are two classes of ANN learning algorithms; a) Supervised and b) Unsupervised.

In supervised learning, a network is given an input along with its desired output. On the other hand, a network in unsupervised learning is given only an input. After each presentation of an input, the performance, is measured to tell how the network is doing. A network is expected to self-organize information by using the performance measure as guidance. Algorithms in these two categories are further divided into two groups on the basis of the input formats; binary or continuous valued input.

7.4.2. Neural Modeling and Control of a Distillation Column Steck et al. [1991] considered control of a 9 stage, three component distillation column. The distillation column with control valves to the coolant to the condenser and steam flow to the jacket of the reboiler and the reflux to the top of the distillation column is shown in Figure 7.11. Their control objectives were achieved using neural estimator and neural controller. The neural estimator was trained to represent the chemical process as precisely as possible. The neural controller was trained to provide input to the chemical process that will yield desired output. The training of the two neural networks was accomplished using a recursive least squares training algorithm implanted on an Intel iPSC/2 multicomputer (hypercube). ANNs can be used for control purposes. Control of any process system depends on the availability of suitable models that captures all the major phenomena of the system. Mathematical modeling of all the subsystems of a distillation column is an arduous task. Some of the theory such as the McCabe and Thiele method of design of distillation column is discussed in Sharma (2012). 20 different distillation problem types are discussed. Each of the

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Kal Renganathan Sharma

distillation problem type offers a different problem in control. Mathematical model of a distillation column can contain hundreds of state variables. The model equations tend to be nonlinear. The range of operating conditions are wide. The dynamic response is slow. There are significant lag times or dead times. Distillation columns are widely used. A suitable control strategy is needed. ANNs can be trained offline in order to model the nonlinear inverse dynamics of the process with a pre determined level of accuracy. The input space can be large without a priori knowledge of the system equations. The ANNs are made to learn the nonlinear inverse dynamics during online applications. The feed input to the distillation column in Figure 7.11 is at the fourth stage. A side stream by-product is removed at the fifth stage. A heavy product is removed at the bottoms and a light product at the holdup tank at the top. A kettle type reboiler with steam jacket is used and a partial condenser is used. The control objective is to produce bottoms with a specified composition and the distillate with a pre-determined composition. The feed rate and feed composition gets disturbed a bit during the implantation of the process. The control objectives can be met by regulating the heat load on the reboiler and partial condenser. A mathematical model can be obtained by writing the mass and energy balance equations for each of the 9 stages of the column, the reboiler and condenser. The partial condenser can be treated as an extra stage. A computer program can be written that can be used to simulate the operations of the distillation column. The program can be used to simulate hundreds of state variables at discrete time intervals. Training pairs were generated from such simulation results in [Steck et al. 1991] for training the neural controller. A NN is trained as an estimator for the column. As part of the feed forward ANN controller, a neural network is trained as an estimator for the column. The state variables of the distillation column that were selected in the estimator are: flow rate; temperature and compositions of the three components for the bottom and overhead products; and temperature and three component compositions for the by-product as side stream. The data for the estimator comprises of discrete data sampled once per minute. The input to the estimator consists of the reboiler heat valve setting, condenser temperature, current feedrate, and the limited state vector at two previous sample times. The estimator is trained to provide as output a prediction of the limited state vector at the current time. The estimator is a fully connected feed forward network with 31 linear input neurons, 14 linear output neurons, and 157 nonlinear sigmoid neurons distributed over two hidden layers. Training data is generated by simulating the column during startup, from an initial state to a steady operating state, and then varying the feed rate, reboiler heat value setting, and condenser temperature setting sinusoidal about their startup settings. The feedrate is varied at 0.0125 cycle.min-1 with an amplitude of 2.0 moles/min about a mean of 25.0 moles/min. The reboiler valve is varied at 0.0375 cycle.min-1 with an amplitude of 0.05 about a mean of 0.308 both as fractions of the full open position. The condenser temperature is varied at 0.025 cycle.min-1 with an amplitude of 2.0" C about a mean of 75.0" C. These sinusoidal variations are continued for an additional 135 minutes after the 20 minute startup period giving a total of 155 sampled data training pairs for the network. Training was accomplished using a recursive least squares (RLS) training algorithm implemented on an Intel iPSC/2 multicomputer (hypercube) [Steck et al. 1991]. The training with the RLS algorithm is much faster when compared to gradient descent. This is because of the parallel implementation and higher order nature of the RLS training algorithm.

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Figure 7.11. Control of Reboiler Heat Load, Condenser Heat Load, Reflux during Binary Distillation.

Another ANN was trained as a controller for the distillation column. The data for the controller consists of discrete data sampled once per minute, the same sampling rate as the estimator data. The input to the controller consists of the desired compositions of the light component of the bottoms and the heavy component at the overhead tank (the controller reference inputs), the feedrate, and the limited state vector at two previous sample times. The mass fractions of the two components used as reference inputs are generally desired to be small as the goal of the distillation process is to concentrate the heavy component at the bottom and the light component at the top. The controller output consists of a reboiler heat valve setting and condenser temperature at which the distillation column should give the specified reference input compositions. The controller is also a fully connected feed forward network having 31 linear input neurons, 2 linear output neurons, and 157 nonlinear sigmoid neurons distributed over two hidden layers. Training data for the controller is generated by rearranging the data used to train the estimator. The estimator inputs are the reboiler heat valve setting, condenser temperature, feedrate, and the state vector at the two previous sample times and that the training output

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Kal Renganathan Sharma

data is the state vector at the current time. The controller is trained by taking as input the two desired composition values from the state vector at the current time, the current feedrate, and the two delayed samples of the state vector. The controller is trained to give as output the reboiler valve setting and condenser temperature which will produce the state vector at the current time. Training is again accomplished using the RLS training algorithm on the Intel iPSC/2 multicomputer. In order to investigate the capability of the trained controller to control the distillation column, the neural controller is connected to the neural estimator. The controller outputs, the reboiler valve setting and the condenser temperature, are connected to the inputs of the estimator by neural connections with unity weights. The state vector at two previous sample times are provided to the estimator and controller networks by feedback delay lines from the output of the estimator. The state vector for two sample periods prior to the beginning of the simulation is provided as known initial conditions. The current feedrate and the desired compositions at the bottom and top (the controller reference inputs) are known and specified at each simulation sample time. The heavy component (Xl) at the condenser can be manipulated by the controller and is moderately successful at controlling the light component at the bottom product. The moderate success is most likely due to the fact that the light component at the bottom is very small even without control and varies only by a small amount with the changes in reboiler heat used to train the estimator and controller.

7.5. SPC, STATISTICAL PROCESS CONTROL 7.5.1. Overview- Deming and 10. Step Process for Quality Improvement The emergence of Toyota motor corp. as the largest producer of automobiles by sales and by production surpassing the General Motors, corp., is seen as a vindication of the emphasis placed on product quality. One of the fruits of labor from Deming‟s principles [2000] is the SPC, statistical process control methods. Those who have worked with SPC have found the benefits from it as substantial and always in the positive quadrant. With more than 30 million cars sold the corolla brand is one of the more popular and bestselling unit. The reasons attributed to this success are fewer customer complaints, higher quality in terms of fuel efficiency, reliability, warranty etc. The 10 step process of Quality management include: (i) identify a broad area of improvement; (ii) survey the customers; (iii) form QIT, quality improvement team; (iv) Develop cause and effect relations from results of surveys (fish-bone diagrams); (v) develop action plan based on causes identified; (vi) settle for process measures and see whether SPC is applicable; (vii) Continue cycle of Plan-Do-Check; (viii) Continuous Improvement Strategy; (ix) Quality Audit and Documentation; (x) Certification and Management Review. Deming‟s 14 points for management include; Point 1: Create a constancy of purpose in order to improve quality and service, to become competitive and to stay in business. Point 2: Adopt the new philosophy.

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Point 3: Create dependence on mass inspection. Point 4: End the practice of awarding business on the basis of price tag alone. Point 5: Constantly and forever improve the system of production and service. SPC, statistical process control refers to a set of methods and techniques of monitoring and control of a process so that the process is operated close to design premises. SPC is different from APC, automatic process control. The feedback and feed forward control methods discussed in earlier sections provide for correction of the deviation from set point on an as happened basis. No analysis goes into what caused the disturbance. So should there be a source for disturbances the deviation from set point and corrective control action would happen again and again. This would put a strain on the tuning controller elements due to hysteresis effects. In SPC control, the cause and effect analysis would lead to knowledge of source of disturbance among other things. Action on removal of the source of disturbance would result in decreased load on the control elements as compared with APC. Further in SPC, only statistically significant deviation are agreed upon as deviation. In APC what could be noise either random white or even biased ones can trigger corrective action. As with any device repeated use can lead to repair. The long term effect of SPC results in better product quality. The APC method suffers from the quality improvement by changes made upon infrequent and ad hoc inspection method. A more careful study is undertaken in the SPC method.

7.5.2. Example of Application of SPC – Fluidization Quality Analyzer Here is an example of an application of SPC. Fluidization Quality Analyzer [Daw and Hawk, 1995]. The pressure drop in a gas-solid fluidized beds as a function of time is measured. From theory, the pressure drop during fluidization is expected to stay a constant. Experimental measurements indicate the pressure drop in a fluidized bed varies with time. Distinct patterns can be seen from the variation of pressure fluctuations with time [Sharma and Turton, 1998, Sharma, 1998, Renganathan 1990]. The pressure signals can be converted to a frequency domain by use of a FFT, Fast Fourier Transform analysis (Sharma [1998]). Time series analysis such an ACF, autocorrelation function, CCF., cross-correlation function can be deployed to study the hydrodynamic regimes of fluidization. The dominant frequency of slugging beds were found to be lower and distinct such as 2 Hz for a 5 cm bed. For a bubbling bed the dominant frequency of FFT of P measurements were found to be significantly higher than that of that found for slugging beds. Those skilled in the art of fluidization generally agree with the following regimes during gas-solid fluidization:

(i) Minimum fluidization regime (ii) Bubbling Bed (iii) Slugging Bed (iv) Turbulent Bed (v) Phase Inversion and Solid Packets at Dispersed Systems

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Kal Renganathan Sharma (vi) Elutriating Beds

The hydrodynamics of fluidization depends on the particle size and particle size distribution and other characteristics of the powder fluidized. Geldart [1973] has classified the powders according to the quality of fluidization that can be realized. The characteristics of fluidization of Geldart A type powder include a maximum stable bubble size, distinct regimes of minimum fluidization with bed expansion and bubbling regime. The characteristics of Geldart D type powder is for coarse particle systems where onset of bubbling begins upon fluidization. The bubbles grow and coalesce into slugs. Geldart B type powder characteristics have been found to be between that of Geldart A and Geldart D type powders. Geldart C type powders do not fluidize well. The particles adhere to each other. Jetting or gas by-passing is found. At fluidization velocities near the minimum fluidization velocity, U mf the mixing of solids achieved is insufficient to prevent agglomeration. This limits the mass transfer rates. During the slugging regimes the gas by-passes contact with the solid particles resulting in decreased process efficiency. The most efficient contacting condition is that which produces significant agitation in order to prevent solids clumping while minimizing the degree of bypassing. The fluidization quality can be related to the intensity of flow oscillations and degree of gas-solids mixing. A control loop based on P, fluctuations in pressure drop in fluidized beds, measurements with a control valve on the flow rate of the fluidizing gas is shown in Figure 7.12. This was patented by Daw and Hawk [1995]. The P of the fluidized bed is measured using a PT, pressure transducer installed near the distributor plate of the fluidized bed. A signal conditioned circuit was introduced in order to obtain the derivative of signal. During severe slugging regime of fluidization (Bi and Grace [1995]) larger, faster pressure fluctuations compared with slightly slugging conditions can be found. The rate of pressure fluctuations was averaged in order to produce a fluidization intensity signal. Thus P measurements can be directly correlated with fluidization quality. The signal conditioning proposed in [Daw and Hawk, 1995] was based on chaotic time series analyses and in an advance over Fourier analysis methods. During some conditions of fluidization the P signals have been found to be non-periodic [Daw and Hawk, 1995]. Sharma [1998] by use of saddle point analysis in probability density functions of P, measurements for the experimental conditions used in the study found the P measurements in fluidized beds to be periodic. The control loop in Figure 7.12 can be used to control the degree of turbulence or fluidization quality in a fluidized bed. Daw and Hawk [1995] interpreted the P measurements as a function of time as comprising of turbulence induced time-varying component that can be associated with the bubble activity within the fluidized bed. Signal processing to separate the turbulence-induced time-varying component associated with bubble activity in the bed was introduced. The output signal display in the indicator of bubble activity within the fluidized bed. The signal processing circuitry includes obtaining the first derivative of pressure signal, operation amplifier, low pass filter and rectifier. The control action is effected by use of optimal flow rate of fluidizing gas.

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Figure 7.12. Control Loop for Fluidization Quality Analyzer.

One of the goals of SPC is to minimize production costs. This is accomplished with a “make it right the first time” program. Attainment of product consistency and compliance with product specifications and achievement of customer satisfaction are the goals of the SPC. They tend to create opportunities for all members of the organization to contribute towards quality improvement. They help both management and employees make economically sound decisions about actions affecting the process (Smith, [1998]). The basic tools for SPC include: (a) Flowchart; (b) Pareto Chart; (c) Check sheet; (d) Cause and Effect Diagram to identify the root cause of the problem; (e) Histogram; (f) Control Chart and; (g) Scatter Plot; (h) Factorial Design of Experiments. SPC is an important tool and leads to many process improvements and positive results such as;             

uniformity of output reduced rework fewer defective products increased output lower average cost fewer errors higher quality output less scrap less machine downtime less waste in production labor hours increased job satisfaction improved competitive position more jobs

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7.5.3. Control Charts Process control charts are used to achieve and maintain statistical control at each phase of the process. Process control charts are used in process capability studies to assess process capability in relation to product specifications and customer demands. Statistical sampling is part of a self-certification plan for vendors. The capability studies are gauged. Control charts are used in manufacturing units to delineate random noise and variability in process variables that can have Attributable causes. No control action is needed when the variation is due to random noise. Control action can be initiated when the Attributable cause is identified. When the variability is due to random noise it is also referred to as stable system of chance causes. Other kinds of variability can also be classified (Bi and Grace, [1995)] into four categories: (i) improper setting of machines; (ii) operator incompetence and errors; (iii) defects in raw materials, water supply and; (iv) utility malfunction. The variability is larger compared with random noise. These sources of variability are called as assignable causes. A process that is operating in the presence of assignable causes is said to be out of control. One of the goals of SPC is the elimination of batch to batch variability in product quality. A prototypical control chart is shown in Figure 7.13. The control chart can be treated as a graphical display of quality characteristics as a function of time. The samples are measured at a certain predetermined frequency, i.e., once every hour. The data points in Figure 7.13 corresponds to a process in control state. The UCL and LCL are upper control limit and lower control limit. These limits can be obtained given a certain set of objectives and characteristics of the statistical distribution that best corresponds with the raw data from the measurements. Sometimes even if the all the points lie within the UCL and LCL some biases can be detected. This is for example, when 15 out of 18 points measured lie above the center line. This would be indicative of a systematic error. Only random scatter can be concluded as in-control state. In Figure 7.13 is also shown a normal distribution. Often times, the LCL and UCL are calculated as shown in the Figure 7.13. Let the UCL and LCL be at a distance km from the mean value of the measurements, m. m is the square root of the variance of the statistical distribution the measurements conform to. It is also called the standard deviation. Sometimes Bessel’s correction can be used to calculate the sample standard deviation. Here (N-1) is used compared with N, where N is the number of data points in the sample. UCL = m + km

(7.79)

Center Line, CL = m

(7.80)

LCL = m – k m

(7.81)

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Figure 7.13. Prototypical Control Chart.

The units of “k” is in terms of standard deviation units. When k = 3, industrial practioners call it 3 sigma quality. When k = 6, industrial practioners call it 6 sigma quality. The inventor of control charts was Shewhart. These types of charts are also called as Shewhart control charts. The central limit theorem can be invoked for the sample means to conform to a normal distribution. Let the confidence level be given by 100(1-)%. Then, 100(1-)% of the sample means can be expected to fall in the interval of; _

LCL

m

Z

/2

x n

m

Z

/2

UCL n

(7.82)

_

Where x is the sample mean. Two of the parameters that determine the characteristics of the statistical distribution that the population of measurements the samples belong to are the sample mean and the sample standard deviation. The sample standard deviation can be used to estimate the degree of scatter in the information. Two charts may be generated an M, chart or chart that contains the sample means as a function of time. The second chart is an S chart that contains the sample standard deviations as a function of time. The UCL and LCL values as given by Eq. (8.13) can be calculated as long as the population variance,  and mean, m is known. When the parameters m and  are not known then they can be calculated using m preliminary samples. The population mean can be estimated by use of a grand mean;

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Kal Renganathan Sharma

1 m _ xi mi 1

^

x

m

(7.83)

The UCL and LCL of the means of the samples lie in a range that can be expressed as; _

UCL

DU q _

CL

(7.84)

q _

LCL

DL q

_

Where, q is the range of sample means. The constants DU and DL varies with the sample size, n and is given below in Table 8.1. The range the sample means can fall under can be estimated as follows; _

q R

;

(7.85)

m

R

A random variable  is introduced. This is given by the ratio of the range of means with the standard deviation of the population. The statistical distribution that  conforms to for any sample size n has been determined. The mean of this distribution is say m and the standard deviation is say,. These values are given in Table 8.1. The UCL and LCL for the sample mean can be seen to be; UCL

3

x

m

LCL

q n

3

x

m

(7.86) q

n

q is the average range. A range can be attributed to each sample.

On some occasions, the sample variances can be plotted in an S chart. The standard deviation of the population, , can be estimated from sample variance, s. The LCL and UCL for the S chart can be seen to be; LCL c '

3

1 c' 2

CL c '

c'

3

1 c' 2

UCL

(7.87)

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Advanced Control Methods Table 7.2. Factors for Determining UCL and LCL of Sample Means and Sample Variance Charts _

xchart

N

3 m

2 4 6 8 10 12 14 16 18 20 22 24

n

_

xchart m

R chart DL

R Chart DU

S Chart c’

1.13 2.06 2.53 2.85 3.08 3.26 3.41 3.53 3.64 3.74 3.82 3.90

0 0 0 0.136 0.223 0.28 0.33 0.364 0.392 0.414 0.43 0.45

3.27 2.28 2.0 1.86 1.78 1.72 1.67 1.64 1.61 1.59 1.57 1.55

0.80 0.92 0.95 0.97 0.97 0.98 0.98 0.98 0.98 0.99 0.99 0.99

1.13 0.729 0.483 0.373 0.308 0.266 0.235 0.212 0.194 0.180 0.167 0.157

The estimator of  from S chart can be given by; _ ^

s c'

(7.88)

The control chart for sample variances can be whipped up as follows; _

_

_

UCL

s s 3 1 c' 2 c'

CL

_

_

s

s 3

s c

'

1 c' 2

(7.89)

c‟ values for difference sample sizes are given in Table 7.1.

7.6. FEEDFORWARD CONTROL Feed forward control is different from feedback control. During feed forward control the load or disturbance is measured or gauged from other considerations and control action taken accordingly. This is different from feedback control when the process output is measured and the measurement is compared with the set point. The error generated is used in the control action taken. Feed forward control can be recognized in human physiological and anatomical systems. Feed forward control can be used in conjunction with feedback control.

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Carrier Corporation has patented [Kolk et al., 2011] a feed forward control system for absorption chillers. Absorption chillers are different from mechanical vapor compression chillers. Thermochemical processes are used in order to produce refrigeration needed to generate chilled water. Lithium bromide and water are used as fluid pair. In order to generate, for example 44 F of chilled water the shell side of the machine has to be maintained in vacuum such that the water boils at 40 F. The vaporized refrigerant is absorbed by the Lithium bromide solution prior to being pumped into the generator section of the machine where heat is added to concentrate the solution. The boiled off water vapor is then condensed and returned to the evaporator as liquid. Disturbances to the chilled water temperature can arise from cooling water temperature and the entering chilled water temperature. The feed forward control method comprise of the following steps; (i) determine the disturbance transfer function; (ii) determine the capacity valve transfer function; (iii) measure the occurred disturbance; (iv) implement the feed forward control function. The control block diagram for feed forward control in combination with the control action such as PI control that was in place before for chilled absorber is shown in Figure 7.13. The feed forward control action law can be written as; UD

GD ( s )

D

Gu ( s )

(7.90.)

Y is the exit chilled water temperature, U is the capacity valve position and D is the disturbance in wither the entering chilled water temperature or entering cooling water temperature. Change in chilled water temperature can be expressed as; Y Gu (s ) GD (s ) D Y

Gu ( s )

U

UD

Y

Gu ( s )

U

GD ( s )

Y

Gu ( s ) U

Y

Gu ( s ) U

(7.91)

GD ( s ) D

GU ( s ) GD ( s ) D

D

GD ( s ) D

(7.92)

GD (s ) D

As shown in Figure 7.14, the feed forward control strategy in a nutshell comprises of: (i) computing the transfer functions GD(s) and Gu(s); (ii) measuring the disturbance signal D; (iii) implement the feed forward control as shown in Figure 7.13. GD(t) and Gu(t) are measured by application of a prescribed a known amplitude input perturbation and by recording the perturbation in the exit chilled water temperature. The ratio of the perturbation in exit chilled water temperature divided by perturbation of change in the disturbance is reflected in GD(t). The ratio of the perturbation of exit chilled water temperature divided by the perturbation in the capacity valve is captured in Gu(t). The entering cooling water temperature is around 70- 80 F. Fans in the cooling tower are used to maintain this temperature.

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Figure 7.14. Block Diagram for Feed forward Control of Absorption Chillers.

Another example of the use of the feed forward controller is in fuel supply control to a furnace as shown in Figure 7.14. The control strategy shown in Figure 7.14 is different from the feedback control strategy explained in Chapter 4.0. In the feedback control strategy the furnace temperature is measured. When found deviant from the set point temperature control action is taken such as P only control, PI only control, PD control etc. In feed forward control strategy the disturbance is attempted to be measured. One possible source for disturbance in the furnace example is the air flow rate to the furnace. The air gets heated and pumped to the room. Feed forward controller is used to regulate the supply of fuel to the furnace.

Figure 7.15. Feed forward Control of Fuel Supply to Furnace.

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Kal Renganathan Sharma

One of the advantages of the feed forward control strategy is the control action is based on gauging the disturbance as it occurs. But one of the drawbacks is that the control action performance depends on the availability of reliable mathematical models. For example in the example, shown in Figure 7.14 a mathematical model that relates the temperature of the furnace, flow rate of air and fuel gas rate needed for achieving a given set point is needed. The less uncertain the model predictions are the better the performance of the control action taken is. Energy balance on the furnace at steady state can be written as follows; H c.

vC p T i T f

.

Where

fg

. fg

0

(7.93)

is the fuel rate and Hc is the heat of combustion of the fuel gas,  is the fuel

conversion efficiency from chemical energy to thermal energy. .

C p v Ti H

c

Tf

(7.94)

It can be seen from Eq. (8.66) that a certain percentage change in temperature of the furnace will have to be compensated by the same percentage change in the fuel flow rate to the furnace.

7.7. MEASUREMENT AND CONTROL OF BIOCHEMICAL REACTIONS The optimal operation of reactors in the biotechnology industry depends on the catalyst environment. The catalyst environment influences the useful lifetime of an enzyme catalyst of cell population. The state of the catalyst environment need be monitored and control schemes developed according to the observations against set points. There are three important steps towards achieving this goal; (i) Measurement; (ii) Analysis of Measurement Data and; (iii) Control. Advances have been made in the available reactor instrumentation, data acquisition and computer control areas. These methods can also be used on downstream processing operations and preparation of raw materials. Some of the quantities that can be measured and instruments available for biological systems are as follows [Stephanopoulos et al. 1985];        

Torsional Dynamometer Gene Expression Levels Piezoelectric Sensors Tachometer Electromagnetic Flow Meter Additive Drop Count Diaphragm Pressure Gauge Glass Electrode

Shaft Power – from power supply Biochips Internal Force, Moment Realized Tip Speed, Shear Rate, Agitator Speed in RPM Liquid Flow Rate Flow Rate of Additive Pressure pH

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  

Combined Platinum and Reference Electrode Sterilized Electrode Thermometer, Thermistor, Thermocouple Load Cell for Level Control Rota meter, Mass flowmeter Paramagnetic Electrochemical Cell IR Analyzer Spectrophotometer Sensors



Viscometer

    

Redox Dissolved Oxygen Temperature Product Removal Gas Flow rate Partial Pressure of Oxygen, pO2 Partial Pressure of Carbon Dioxide, pCO2 Turbidity, Biomass Measurement for Glucose, Ethanol, DNA, RNA, NADH Ions of ammonium, magnesium, potassium, sodium, copper, phosphate Viscosity

Muscle strength can be measured using dynamometers. Handgrip strength is a measure of sarcopenia. Handgrip strength can be measured using dynamometers. The input shaft power that is used for agitating bioreactors can be measured using torsion dynamometer. Dynamometer is a device used to measure power. Power, torque and speed can be used to characterize the mechanical aspects of rotating machinery. These quantities needs to be measured in order to determine the efficiency of the agitator and identify operating regimes that are safe.

7.8. DESIGN AND CONTROL OF BIOARTIFICIAL PANCREAS Nomura (5) used control theory to characterize the insulin release rate. He studies a step change in glucose concentration. The dynamics of glucose induced secretion of insulin can be expressed as the sum of the proportional response to the step change and a derivative response to the rate of change in the glucose concentration. Each of them have a first order lag time. The Laplace transform of the islet insulin release rate can be expressed as follows:

risl

K prop

1 s

_

Tder

1 s

1

(7.95)

C glu cos e 2

The lag times are 1 and 2 respectively. The Laplace domain expression in Eq. (8.1) can be inverted to give; t

risl

K prop C glu cos e 1

t z

e

1

t

dz

2

dC glu cos e

1

dz

t z

e

2

dz

(7.96)

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Kal Renganathan Sharma

The lag times can be obtained by use of nonlinear regression of islet release-rate experimental data. The ramp function of glucose concentration is written as follows; 0 C glu cos e for t < 0.

C glu cos e

C glu cos e

Where m

(7.97).

mt for 0  t  t0

0 C glu cos e

(7.98)

0 C glu cos e

ss C glu cos e

t0 ss C glu cos e for t  t0

C glu cos e

(7.99)

Eqs. (8.3-8.5) can be substituted into Eq. (8.2) and integrated to yield;

t

risl

0 K prop C glu cos e e

t

t

K prop 1m

1

1 e

t

0 K prop C glu cos e

1

1 e

1

1

(7.100)

t

Tder m(1 e

2

) for 0  t  t0 t

risl

0 K prop C glu cos e e

K prop 1m (

1

(7.101)

t t0

t0

1)e

t t0

t

e

1

1

0 K prop C glu cos e e

1

t

e

1

1 t t0 ss K prop C glu cos e

1 e

1

t t0

Tder m(e

2

t

e

for t  t0

2

)

(7.102)

The insulin release rate from an islet and its dependence on plasma glucose levels needs to be better understood. This is needed for better design of bio artificial pancreas. A step change in glucose concentration is given to islets that have been isolated from a pancreas of mammals. The islet viability and glucose responsiveness are studied from the F curve. Insulin release have been found to be biphasic. Pharmacokinetic models have been developed to describe glucose and insulin metabolism. A model was proposed by Sturis et al. (6) to predict the oscillations of insulin and glucose concentrations with time observed experimentally. Insulin formed in human anatomy have been found to exhibit two kinds of oscillations: i) a rapid oscillation with a time period of 10-15 minutes and small amplitude; ii) longer or ultradian, damped oscillations

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with a period of 100-150 min and larger amplitude. Compartment model was proposed by Sturis et al. to describe glucose and insulin interactions Sturis et al. [1984]. 4 negative feedback loops form their model involving glucose and insulin interactions: i) insulin formation is triggered when glucose levels become more than tolerable limit; ii) increase in insulin level increases the utilization of glucose and hence reduces the glucose levels; iii) rise in glucose level inhibits production of glucose; iv) increase in glucose levels stimulates its utilization. The glucose and insulin never reach stable equilibrium. The model includes two time delays that are critical in describing the observed oscillatory dynamics. The suppression of glucose levels by insulin production is captured by one time delay and the correlation of biological action of insulin with insulin concentration is captured by another time delay in interstitial compartment. Six differential equations describe the system. The plasma plasma variables are C glu cos e , concentration of glucose in plasma, Cinsulin , concentration of insulin

int erstitial in the plasma, Cinsulin , concentration of insulin in the interstitial fluid. Three additional variables used to describe the insulin and glucose system are the delay between the plasma insulin level and its effect on glucose production x1, x2, x3 and time lag, delay. The 6 differential equations can be written as follows:

Vplasma

plasma dCinsulin

dt

plasma risl C glu cos e

plasma dC glu cos e

plasma Cinsulin

kE

dt

dt

int erstitial Cinsulin

plasma Cinsulin Vplasma

(7.103)

plasma

int erstitial dCinsulin

VplasmaG

plasma kE Cinsulin

dt

int erstitial Cinsulin

Vint erstitial

plasma f2C glu cos e

rglu cos e(in )

dx1

int erstitial Cinsulin

3

(7.104)

int erstial

plasma int erstitial f3 f4C glu cos e Cinsulin

plasma Cinsulin Vint erstitial

x1

x3 f5

(7.105)

(7.106)

delay

dx2 dt

dx3 dt

3

x1

x2

(7.107)

delay

3

x2

x3

(7.108)

delay

The volumes of the insulin plasma compartment, insulin interstitial fluid compartment and glucose plasma compartments are denoted by Vplasma, Vinterstitial and VplasmaG respectively. The kE is the rate constant that is used to describe the insulin transport rate into the interstitial fluid compartment. The first order degradation time constants for insulin in plasma

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compartment, insulin in interstitial fluid compartments are given by plasma,interstitial respectively. Utilization functions are given by f. The subscripts 2 and 3 are used to denote the glucose utilization function through the glucose plasma compartment. The 4 subscript is used for dependence on interstitial insulin concentration. The glucose inhibition on account of insulin formation is given by subscript 5. The above pharmacokinetic model developed by Sturis et al. (6), can be combined with an insulin-release model to monitor the glucose control that is achievable using bio artificial pancreas. There are two types of tests used: i) IVGTT, intravenous glucose tolerance test and; ii) OGTT, oral administration of glucose tolerance test. The initial conditions for IVGTT or OGTT can be selected based on the fasting levels of the patient.

7.9. MODELING AND CONTROL OF APFR, ANNULAR PLUG FLOW REACTOR FOR PRODDUCTION OF SINGLE LAYER GRAPHENE Graphene can be made into thin films or thick films. Thin films of graphene can be made on flexible substrates. Other thin films such as Nickel needs rigid substrate. Graphene made by CVD, chemical vapor deposition methods can be rolled into thin films by a transfer process. The process [11] comprises of three steps: (i) Adhesion of polymer supports to the graphene on copper foil. Two rollers are used to get the graphene film grown on a copper foil to be attached to a polymer film coated with adhesive film as it passes through; b; (ii) Etching of copper layers. Electrochemical reaction with aqueous 0.1 M ammonium persulphate solution (NH4)2S2O8 enables the removal of copper layers and; (iii) Release of the graphene layer onto a target substrate. Thermal treatment is used to detach the graphene from the polymer support and reattach the film onto a target substrate. This target substrate could have been placed below the copper foil in order to obviate the third step.

7.9.1. Adhesion, Etching and Transfer Graphene is synthesized in an annular plug flow reactor, APFR. The annular reactor space is generated by an 8 inch outer quartz tube and an inner 7.5 inch copper foil, wrapped quart tube. The use of annular reactor in place of the tubular reactor is to minimize radial temperature gradients. This was found to cause inhomogeneity in the film formation. The inner tube is heated to 1000 0 C. Hydrogen, H2 is allowed to flow at 8 s.c.c.m and 90 mtorr. Annealing process comes next. Annealment for 30 minutes allows for increase in grain sized in copper foil from a few micron to 100 microns. This has been found to increase the graphene growth. Methane, CH4 is allowed to mix with the flowing hydrogen at a flow rate of 24 s.c.c.m at 460 m torr. The sample is rapidly cooled at about 10 0C.s-1. The graphene film grown on copper foil is attached to a thermal release tape by applying pressure on the rollers at 0.2 MPa. Copper foil is etched in a plastic bath filled with etchant. The etched film is washed with deionized water to remove and unused etchant. The graphene film is ready for transfer to a target substrate such as a curved surface. 150- 200 mm.min-1 transfer rates by thermal treatment can be achieved by letting the graphene film pass through the rollers at mild heat of 90-120 0C. Multilayered graphene can be made by repetition of this process. The

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product that comes as a result of this would be different from the bilayer or multilayer material formed during the reaction by other methods. This can be viewed as physical stacking of the formed graphene layers. Screen printing can be used to generate 4 wire touch panels. Continuous production of graphene on large scale is possible. Scalability of the process is high. Processability is good. Carbon has limited solubility in Copper even at 1000 0C. The copper may have a catalytic effect on the graphene formation reaction. Transparent electrodes can be made using graphene in large scale and replace the currently used indium tin oxide, ITO, electrodes. Monolayer graphene was confirmed using Raman spectra. Bilayer and multilayer islands were found from atomic force microscope, AFM and transmission electron microscope, TEM. Stacked layers reduces the optical transmittance by 2.2-2.3% a layer and the conductivity also decreases. Dopants can be added as desired. p junction formation by doping can be achieved by addition of nitric acid, HNO3. Sheet resistance can be increased by chemical doping. Poly methyl methacrylate, PMMA can be used as polymer supports. Some of the challenges in using this method is the formation of polycrystalline graphene due to occurrence of nucleation again to form a second layer. Oxidation of copper has to be avoided. High rates of evaporation of copper from the foil can hinder graphene growth. Copper is not as effective as Ni, nickel to lower the energy barrier to form graphene. Other carbon sources in addition to methane that can be used for the purpose of preparing graphene are carbon monoxide, CO, ethane, C2H6, ethylene, C2H4, ethanol, C6H5OH, acetylene, C2H2, propane, C3H8, butane, C4H10, butadiene, C4H6, pentane, C5H12, pentene, C5H10, cyclopentadiene, C5H6, hexane, C6H14, cyclohexane, C6H12, benzene, C6H6, toluene, C7H8, iso-butane, iso-pentane and hexene. Mixtures of these sources may also be used. Carbon formation from carbon sources are thermodynamically favored only at high temperatures. The temperature that can be used in the reactor is 300 – 2000 0C. The flow rate range can be 5 – 1000 s.c.c m (standard cubic centimeters per minute). Inert gases can prevent undesirable oxidation. Reynolds number in the reactor is in the regime where laminar flow can be assumed. The reactor is operated at low pressure. This would enable the entire vapor and solid system to fall near the sublimation curve of the P-T, pressure-temperature diagram of reactants and products.

7.9.2. Reactor Performance Horizontal low pressure chemical vapor deposition reactor, LPCVD is discussed in Fogler [12]. The reactor is operated at 100 Pa. The APFR discussed in the above paragraph (Figure 7.16) is operated at about 60 Pa. One of the advantages of using LPCVD is the capability of large number of wafers without sacrifice to film uniformity. At low pressures, the diffusion coefficient is expected to increase. Sometimes Knudsen diffusion effects cannot be ignored. The surface reactions are expected to be rate limiting compared with other mass transfer effects. Assume the reaction mechanism for the formation of graphene on copper foil in the APFR above is as follows; Dissociation 2CH4  C2H2 + 3H2

(7.109)

436

Kal Renganathan Sharma 1000 0 C Adsorption C2H2 + 2Cu  2Cu.C + H2

(7.110)

Cu.C  Cu + C

(7.111)

Surface Reaction

Given the monolayer formation of graphene, the Langmuir-Hinshelwood kinetics may be a reasonable assumption for the adsorption kinetics. At high temperatures such as 1000 0 C the dissociation rate can be expected to be rapid. Adsorption is a surface phenomenon. Molecules adsorb on the surface. Other molecules in the interior of the solid substrate are attracted by other surrounding molecules in all directions. The molecules that reside in the surface are in a state of imbalance and are pulled inward. The dissociated products from methane will interact with the molecules in the surface. The amount adsorbed would be proportional to the surface area available for adsorption. The Langmuir isotherm can be derived as follows [13]; [filled sites] + [empty sites]  [filled sites]

(7.112)

Sites are subject to chemical equilibrium; [bulk solute] + [empty site]  [filled site] K‟ = K '

[ filled _ sites ]

[ filled _ sites ] [bul _ solute][empty _ site]

K '[bulk _ solute]

[total _ sites ] 1 K '[bulk _ solute]

Figure 7.16. APFR, Annular Plug Flow Reactor for Dissociation of Methane and Formation of Graphene Deposition on Copper Foil.

(7.113) (7.114)

(7.115)

437

Advanced Control Methods The rate of adsorption can be written as follows; Rate of adsorption r” =

k0 pC2 H 2

(7.116)

1 K ' pC2 H 2

k0 is a measure of total concentration of sites available. Eq. (8) is an alternate form of Langmuir isotherm. The reciprocal of the rate of adsorption varies in a linear manner with the reciprocal of the partial pressure of acetylene. The equilibrium rate constant and the total concentration of the sites can be obtained from the slope and intercept of the straight line. Acetylene decomposition can become autocatalytic [14]. It is a free radical process. Free radicals can combine with the copper metal and become inert. This may keep the reaction from becoming a thermal autocatalytic runaway. The axial flow in the annulus can be assumed to be in laminar flow. This is from the Reynolds‟ number estimated for APFR of less than 50. As the reactant gas flows through the annulus simultaneous reaction and diffusion can be expected. The dissociated products diffuse to the surface of the copper foil and get adsorbed. At high temperatures the surface reactions of carbide formation and graphene formation are found. Diffusion direction is radially inward. The cross-sectional area of the annulus is given by; Ac = 0.25π (Dt2 – Dc2)

(7.117)

Where Dt and Dc are the diameters of the outer tube made out of quartz and inner quartz tube wrapped with copper foil. With the progress of CVD the mole fraction of acetylene in the annulus decreases as the reactant flows down the length of the annulus. An effectiveness factor can be calculated to determine the overall rate of reaction per unit volume of the reactor space. The reactants diffuse radially inward toward the copper foil. Graphene deposits grow on the copper foil by the surface reaction given by Eq. (3). Concentration of the acetylene in the surface of the copper is less than the concentration of acetylene in the bulk. The effectiveness factor is defined as; rate _ of _ reaction(actual ) (7.118) rate _ of _ reaction(if _ entire _ foil _ is _ at _ sup plied _ concentration)

2

Dc [ r "]dD Dc2

r "AA

DC2 H 2 Dc

CC2 H 2 r r "AA

D Dc

(7.119)

Consider a cylindrical shell of length L and thickness r from a distance r from the center of the APFR. The acetylene formed from dissociation of methane undergoes simultaneous diffusion and reaction at the copper surface of the inner tube. The copper carbide becomes graphene upon further reaction. A mass balance on a cylindrical shell of thickness on the

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Kal Renganathan Sharma

dissociated acetylene that undergoes simultaneous diffusion and reaction to the surface of the quartz inner tube can be written as follows;

1  rJ r  C C 2 H 2  k "'C C 2 H 2  0  r r t

(7.120)

The boundary conditions used are as follows; At the center of the reactor the concentration profile has to be symmetric in the annular space. There is no reason for asymmetry. Hence at r = 0,

C C

r

2

H2

0

(7.121)

At the quartz wall of the outer cylinder, the wall is impervious r = R,

C C

r

2

H2

0

(7.122)

The governing equation for the concentration profile that varies with space and time can be obtained by combining Eq. (7.120) with the damped wave diffusion and relaxation equation given in Eq. (7.123);

J r   DC2 H 2

C C 2 H 2 r

 r

J r t

(7.123)

The governing equation for concentration of acetylene species can be seen to be;

r

 2 C C2 H 2 t

2

 1  k" ' r 

C C 2 H 2 t

 DC 2 H 2

 2 C C2 H 2 r

2



DC 2 H 2 C C 2 H 2 r

r

 k" ' C (7.124)

The dimensionless form of Eq. (7.124) can be obtained by defining the following nondimensionalizing variables;

 CC H  C s u   2 r Cs 

 t ;  ; X   r 

r

DC2 H 2  r

; k*  (k" ' r )

(7.125)

The dimensionless governing equation is as follows;

u  2u  2u 1 u        k *u 1 k * 2 2  X X X 

(7.126)

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Advanced Control Methods

The solution is obtained by the method of separation of variables. First the damping term is removed by multiplying the above equation by e n. It can be seen that the terms group as W = uen and the governing equation becomes at n = (1-k*)/2;

 2W  2W 1 W 2 W   1 *    k  16 X 2 X X  2

(7.127)

The method of separation of variables can be used in order to obtain the analytical solution to Eq. (7.127). W = V()  (X)

Let

" ( X ) 

'(X) X

(7.128)

  2 ( X )  0

V"  1  k *  2    V  2  2

(7.129)

 = c1 J0 ( X) + c2 Y0 ( X)

(7.130)

It can be seen that c2 = 0 from the symmetry condition that the derivative of the concentration with respect to r = 0. Now from the BC at the surface,

u  0  c1 X

  R n J 1  DC 2 H 2  r  DC 2 H 2  r



   

(7.131)

The solution for time domain is the sum of two exponentials. The term containing the positive exponential power exponent will drop out as with increasing time the system may be assumed to reach steady state. At steady state or infinite time W = uexp(/2), becomes zero multiplied with infinity. This is an inderdeterminate form of the fourth kind (Piskunov). This can be shown to go to zero. Thus,

V  c4e

 1 k *  2     n  2 

u  1 c n J 0 n X e 

2

 1 k *       2 

(7.132)  1 k *  2    n  2  2

(7.133)

Eq. (7.133) is an infinite series of Fourier Bessel that is modified for inclusion of damped wave diffusion and relaxation effects. The cn can be solved for from the initial condition by

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Kal Renganathan Sharma

using the principle of orthogonally for Bessel functions. At time is zero the LHS and RHS are multiplied by J0 (m X). Integration between the limits of 0 and R is performed. When n is not m the integral is zero from the principle of orthogonally. Thus when n = m, cn = - 0R J0 (n X) / 0R J02 (n X)

(7.134)

It can be noted from Eq.(7.133) that when ( 1 + k*)2/4 < n2

(7.135)

The solution will be periodic with respect to time domain. This can be obtained by using De Movries theorem and obtaining the real part to exp(-i  (n2 - ( 1 + k*)2/4 ) ). For large relaxation times the concentration of graphene product will undergo oscillations.

r

2  1  k * R 2 

58.7 DC 2 H 2

(7.136)

Estimates of relaxation time from Stokes-Einstein formulation from chemical potential (Sharma, [13] ) for graphene at 5 torr this can be as high as 50 minutes.

r 

DC H 2

2

(7.137)

P

The surface to volume ratio needs to be maintained high. It can be seen that there exist a critical value of R above which the rate at which the free radicals are produced in the reaction is larger than the rate at which it is removed by diffusion. This will lead to a runaway condition in autocatalytic reactions. At the critical value,

or

(2R h ) u/ X Cs D/Dr = ( R2 h) (k”‟ C)

(7.138)

Rcrit = 4 D/k”‟ J1 (R (k”‟/D) )/J0(R (k”‟/D)

(7.139)

Considering the average reaction rate instead, Rcrit = 2D/k”‟ J1 (R (k”‟/D)

(7.140)

Due to high temperature and low pressure the primary mode of heat transfer is expected to be due to radiation. Small temperature differences may exist between the copper foil and graphene sheet. There is no need to couple the energy and mass balances at these small temperature gradients. CVD, chemical vapor deposition methods can be used to prepare monolayer graphene sheet. Hydrocarbons are decomposed on transition metals such as Nickel, Ni, Cu, Copper, Co, Cobalt and Ruthenium, Ru. Hydrocarbons used are methane, ethylene, acetylene and benzene.

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Foils made of Ni and Co with thickness of 0.5 mm x 2 mm were used as catalysts [15]. These foils were chopped into five, 5 mm2 pieces and polished mechanically. CVD process was carried out by effecting the decomposition of hydrocarbons around 800 – 1000 0C. Methane is passed at 60 – 70 sccm along with flow of hydrogen at 500 sccm at 1000 0C for 510 minutes over a Nickel foil. Ethylene at 4 – 8 sccm can be used in place of methane at 6070 sccm. Benzene can be used as the carbon source at 1000 0C for 5 minutes. Dilution of benzene is effected using argon and hydrogen. Acetylene at 4 sccm was decomposed on a Co foil at 800 0C. Methane was decomposed at 1000 0C over Co foil. Cooling step of the Metal foils after the decomposition step were performed in a gradual manner.

7.10. COMBINED CYCLE POWER PLANT Integrated Solar Combined Cycle, ISCC, power plants can result in more efficient use of fossil fuels such as coal, natural gas and petroleum. The waste heat from the exhaust of the gas turbine can be used along with the solar energy to raise steam and drive a steam turbine. The world record for obtained power for less fuel was accomplished in April, 2011. Siemens set-up a 580 MW power station with combined cycle operations of steam and gas turbines. The gas turbine blades were made out of crystalline nickel. The inlet gas turbine temperature was 1400 0C – 1500 0C. The boiler weighs 7000 metric tons and the heat exchanger has an area of 510,000 m2. Combined cycle power plant in China operates at efficiencies of about 59.7% recorded at a 430 MW power plant in operation. Thermo economic analysis and 4E analysis – Energy, Exergy, Environment and Economic Analysis, second law costing methods can be used to confirm the feasibility of increases in thermal efficiency from 30% to ~ 60-75%. There is potential for power plant with higher combined cycle efficiencies in this country, United States from the current levels reported at ~ 40%. By use of both the steam and gas turbines the Carnot cycle efficiency has been increased from 30% to 50%. After nearly two centuries since the discovery of the steam engine by James Watt, from the point of view of thermodynamics that defines the limits of machines, what are the issues involved in solar power plant. The power plant size for combined cycle power plant can be high, as high as nuclear power plants. The 2 GW power plant that is being constructed at Pembroke, Wales will be the largest and most efficient combined cycle power plant in the United Kingdom. A 1642 MW combined cycle power plant is expected to be commissioned in Iraq in 2016. Alstom will supply four gas turbines, four HRSGs, heat recovery steam generators, two steam turbines and six air-cooled turbo generators. 2.6 GW is the plant capacity that is expected to come on stream in 2016 or 2017 in Taipei, Taiwan. Mitsubishi Heavy Industries Ltd and CTCI Corp. in Engineering and Procurement have jointly received the order. The gas turbine inlet temperature is slated at 1600 0 C. The largest power plant there is the hydroelectric power plant with 22.5 GW capacity at Three Gorges, China. The suspended nuclear power plant in Kashiwasaki Kariwa in Japan has a capacity of 8.2 GW. The federal government by its Sun Shot Initiative expects to make the abundant solar energy resources in this country, United States more affordable and accessible. Within 10 years the cost of photovoltaic, PV panels, is expected to be reduced by 75%. The solar PV panels can be used to raise saturated steam. Safety of birds and other living creatures near the

442

Kal Renganathan Sharma

concentrated regions of light is an important consideration during the design of the plant. Acreage use can come down from increases in light to electricity conversion. The mass of the sun can be calculated from the period of the earth which is 365.25 days. The mass of the sun is ~ 2 x 1033 gm. The surface temperature of the sun is 5500 0C. Using Boltzmann‟s law if hydrogen molecules were made to travel at the speed of light the micro scale temperature will be 7.22 x 1015 K. If that molecule was polyfullerene then the temperature will be ~ 2.6 x 1018 K. Sun is a hot reservoir from which more heat engines can be made to work from. With improvements in energy storage materials energy from lightning can someday be transformed into useful work in earth surface. The world‟s largest electric power generating system using CSP concentrated solar power technology has been commissioned by NRG solar along with Brightsource energy in 2013. The capacity is 392 MW. CSP used a field of mirrors to concentrate sunlight onto thermal receivers mounted on top of towers. Lower cost of production and zero emissions are expected from use of solar power plants. The technology is low risk. The efficiency of conversion of sunlight to useful energy using photovoltaic cell technology hovers around 15%. Photovoltaic efficiency of PV cells is 8% for solar cells made from amorphous silicon. Their efficiency has increased now to 14%. This can be further increased to 20% by use of thin films that contain small amounts of crystals of silicon. Single crystal silicon can be used to make the “most efficient” solar cells with 30% efficiency. These PV cells are more expensive. By 2021, six Public sector undertakings: BHEL, Bharat Heavy Electricals Ltd., PGCIL, Power Grid Corp. of India Ltd., SECI, Solar Energy Corporation of India, SSL, Sambhar Salt Ltd., REIL, Rajasthan Electronics & Instruments Ltd., and SJVNL, Satluj Jal Vidyut Nigam in India are planning to set-up solar power plant with 4 GW capacity in Rajasthan, India. The solar photo-voltaic power plant will use PV modules based on crystalline silicon technology. CO2, Carbon dioxide emissions can be reduced by 4 million tons per year. The land area proposed to be used is 19,000 acres at Sambhar, Rajasthan. It can be the largest power plant in the world in its kind. In 1955 Bell Labs, created a 6% PV cell for everyday use. Hoffman electronics increased the light to electricity conversion efficiency to 10% in 1959 in commercial applications. The first thin film cell to exceed 10% mark in conversion efficiency was made at University of Delaware, Institute of Energy Conversion in 1980. In 1985, Stanford University, Stanford, CA created a solar cell that is 25% efficiency using 200 X concentration. The 30% mark was shown using Stirling cycle at NREL in 1994 using a gallium indium phosphide/gallium arsenide solar cell. A team of researchers from North Carolina State University has fabricated a “super absorbing” design that can be used in order to maximize the light absorption efficiency of the thin film solar cells while minimizing the manufacturing costs. According to EE times – Asia dated March 3rd 2014. Their design can decrease the thickness of the semiconductor materials in thin film solar cells by an order of magnitude without compromising the capability of solar light absorption. 100 nm amorphous silicon layer is a requirement in state-of-the art solar cell design. 90% of incident solar beam can be absorbed using 10 nm thick layer of amorphous silicon. The core of the super absorbing layer in solar cell comprises of a rectangular onion like configuration that is made up of a semiconductor layer. This layer is coated by 3 layers of AR, anti-reflection coating that transmits light at a greater efficiency. The researchers first

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443

estimated the intrinsic light trapping efficiency of the semiconductor material. Then they devised a structure where the light absorption efficiency is equal to that of the intrinsic efficiency of the semiconductor material. A team of scientists from CIT, California Institute of Technology, Pasadena, CA, developed flexible solar cells that enhance the absorption of sunlight and hence the photovoltaic efficiency using fraction of expensive semi-conductor material. They built their solar cells using silicon wires embedded within a transparent, flexible polymer film. Black paint can absorb light well, but may not generate electricity. These solar cells convert most photons absorbed into electrical energy. These wires have what is called a near perfect internal quantum efficiency. A high-quality solar cell is built for high absorption of light and good conversion of photons to electric current. These wires are painted with anti-reflective coating prior to being embedded into the transparent polymer. Each wire is about 1 mm in diameter and 30-100 mm in length. 2% of the material is silicon and 98% polymer. This brings down the cost of the solar cell. These solar cells are also flexible. Photovoltaic cells respond only to a narrow part of the sun‟s spectrum. In order to circumvent the lower efficiency on account of absorption of narrow part of the spectrum some developers prepare layered materials. The efficiency goes up, but the material becomes expensive as well. Cloudy days may lower the efficiency. Ohio State University developed a doped polymer, oligothiopene. The resulting substance was responsive to wavelengths from 300 nm to 1000 nm. The spectrum of ultraviolet (UV) to near infra-red is spanned in this range. Traditional silicon cells, au contraire, function in the 600 nm to 900 nm range. This narrower range is between orange and red. The doped polymer both fluoresces and phosphoresces. Fluorescence emanates from electrons that get excited by incident rays of sunlight travel from a higher energy state and drop back to a lower energy state. Some light is emitted. The wavelength of the emitted light is in infrared range and not visible. This emitted light is seldom reused. Reuse of emitted light may improve the photovoltaic efficiency. These polymers are cheaper to produce compared with silicon. Hence they can be considered even if their photovoltaic efficiencies are lower. The relaxation time of these electrons during fluorescence of the doped polymer comes up from a few picoseconds in other solar materials to a few microseconds. A full spectrum solar cell that absorbs the full spectrum of sunlight from the near infrared and far ultraviolet to electric current can be prepared from an alloy of indium, gallium and nitrogen. This was made possible by a serendipitous observation by researchers at Lawrence Berkeley National Laboratory interacting with the crystal-growing research team at Cornell University and Japan‟s Ritsumeikan University. This observation was that the band gap of the semiconductor indium nitride is not 2 eV as previously thought, but instead is a much lower 0.7 eV. Solar cells made from this alloy would be the most efficient and can be lower in cost as well. The efficiency of photovoltaic cells is limited because of a number of factors. Some light energy that gets absorbed is rejected as waste heat. There exists a band gap in semi-conductor materials that the solar cells are made out of. Incoming photons of the right energy knock electrons loose and leave holes and migrate in the np junction to form an electric current. Photons with less energy than the band gap slip right through. Red light photons are not absorbed by high-band-gap semiconductors. Photons such as blue light photons that possess higher energy than the band gap are absorbed. Excess energy is dissipated as heat. There is a

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Kal Renganathan Sharma

maximum efficiency limit for a solar cell made from a single material for converting light into electric power. This is about 30%. In practical applications it is about 25%. Stacks or layers of different materials are attempted in order to increase the efficiency. CIGS, CuInxGa1-xSe2 based photovoltaic thin films can deliver sunlight to electricity conversion performance greater than that of CdTe or silicon based thin films. Nanosolar) has developed a process with high-throughput, high-yield printing of nanoparticles onto low-cost substrates and formation of solar cells. CIGS based PV thin films can deliver sunlight to electricity conversion efficiencies of 19.5%. NREL has certified the solar cell efficiency of 14% achieved by nanosolar with lower cost materials using nanotechnology. CIGS based thin films result in higher efficiencies. They are coated with a homogeneously mixed ink of nanoparticles using wet coating techniques. CIGS roll-printing technology developed by nanosolar uses a combination of high-speed, high-yield, no vacuum, wet coating of nanoparticles onto low cost per unit area of metal foil substrates with RTP (Rapid Thermal Processing) techniques. Nanosolar‟s rapid thermal processing of nanoparticle-based coatings resulted in solar-cell efficiencies confirmed by NREL (National Renewable Energy Laboratory) to be 14.5% which amounts to a world record for any printable solar cell. New discoveries such as the CNT (carbon nanotubes) are expected to increase the photovoltaic efficiency of solar cells to over 40%. For now, in areas, where the population density is sparse, and sunlight is abundantly available for most of the year the solar power plants may be profitable. This can be seen by a positive PW (present worth) value. They end up using large area. The solar panels are not protected from birds and other forms of dust that degrade their operations. Lenses and mirrors can be used to concentrate the sunlight and energy storage devices can store the energy in useful chemical forms such as batteries for use at night and during rainy days. Someday technology in solar energy generation will be as technically efficient as that of the combined steam and gas cycle power plant. Solar energy may be tapped in three different ways; (i) Solar thermal power, (ii) Photovoltaic Panels (iii) Solar Heaters. (iv) Concentrated Solar Power (v) Balance of Systems Costs (vi) Systems Integration In method (i) steam is generated in large boilers to turn turbines and generate electricity comparable in capacity to coal-fired boiler-based power plants. Photovoltaic panels can be used to convert directly the solar irradiance (w/m2) into electricity. This technology is used to meet peak load needs and distributed power needs. Small power plants of up to 50 MW can be built using panels. The capacity of the recently commissioned De Soto Solar Power Generation Center in Florida is 24 MW and is less than 50 MW. Solar irradiance (w/m2) is used to heat water or air and can be used for residential heating purposes. Solar power plant technology can be used to produce base-load, large-scale power at low technical risk. These can replace coal-fired boiler based power plants. Heat energy storage devices have been invented that can provide for uninterrupted services such as during the night hours, rainy days, etc. Lunar power can also be tapped into. Heat storage elements used

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are concrete, molten salts and pressurized water. The capital solar plant costs and the plant utilization factor continue to affect the bottom line. Spain has five such plants under development and two that have been already commissioned. Investments have been made in several countries across the globe to advance the design of solar mirrors and lenses. These are used to gather the sunlight and focus on a fuel source to generate of electricity. Full scale commercial operations of such power plants with capacities in the range of 20-200 MW are expected by 2011. The cost per kWh is continuing to be the main concern. Optimization strategies are being developed. Cost effective storage devices are also expected. Present worth (pw) of solar power plants : A rapidly declining cost curve is seen in the photovoltaic cell technology. The price of solar modules is expected to fall from around $2/W today to $1 in the near future. The price per watt installed is likely to fall from $6.0-$8.0 to $3.0 per watt. Dominant in the costs are power electronics and installation. With current incentives in the United States and European Union the cost of electricity generation using solar technology is in the range of the IGCC (Integrated Gasification Combined Cycle) power generation plants. In the state of California the cost of electricity from solar power plants is about 14-15 cents per kWh. This is against the 8-10 cents per kWh cost of electricity from IGCC power plants. By 2013 with some incentives from the federal government the cost of electricity from solar power plant is expected to fall to 10-12 cents per kWh. The per kWh cost is sensitive to the capital cost and the cost of materials of construction of the plant. Sequestration related carbon credits at $30 t-1 of CO2 can affect a per kWh reduction of 3 cents. Government carbon tax, cost of capital are sensitivity parameters on the bottom line of solar power plants. Worldwide, the solar thermal power capacity can grow at least 30% per year from 2010 to 2020 or 2030. 200 GW of new power plant capacity can be added each year. About 1-2 GW per year could be added in 2012. Solar technology is relatively simpler. The NREL, National Renewable Energy Laboratory, has identified potential for 6 TW (terra watts) of solar thermal power in the Southwestern US. Few things are now known about the light to electricity conversion. The Carnot limit of efficiency of any man made machine is given by;

1

carnot

TC TH

By the Planck‟s theorem, E

h

hc

The IV, current-voltage characteristics of a photodiode cannot be described using the Ohm‟s law of electricity. GaAs multijunction devices are the most efficient solar cells reaching a record high of 40.7%. 20-30 different semi-conductors are layered in series. AR, antireflection coatings can be used to minimize reflection from the top surface. The IV characteristics can be given by;

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I

qV kBT

I 0 (e

1) qAG0 Lp

Ln

33% of solar insolation is lost as waste heat. The temperature of operation of the photovoltaic module can be determined from energy balance as follows; IT

c IT

U L Tc

Ta

The physical significance of each term above Eq is as follows; The fraction of radiation incident on surface of solar cells is given by (IT), where  is the transmittance of any cover; The efficiency of conversion of incident radiation into current is given by (cIT); Radiation and convection losses from top and bottom is given by (UL(TC – Ta). Ta is the ambient temperature. The temperature of the solar cell, Tc can be obtained by rearranging the terms and;

Tc

Ta

IT UL

1

c

It can be seen that as there exists an optimal working temperature of the solar cell. As the amount of concentrated solar flux is increased the absorption efficiency first increases and then decreases. As the working temperature increases the heat losses increase by radiation and convection. The optimal temperature of operation for maximum efficiency can be arrived at. The heat losses vary as T4 and the efficiency increases linearly with the working temperature. At /T = 0 the working temperature can be solved for. The UL the overall transfer coefficient will vary with temperature. Solar aided combined cycle power plant may first come about in large numbers before solar along power plants will become ubiquitous. Nuclear power plant can be aided by solar or other cycles starting from the exhaust of the steam turbine. Supercritical steam can be generated from nuclear boilers and from the steam recovered from the steam turbines can be used to drive a bottom second cycle. This second cycle can be aided by solar or coal. This way large power plant size, higher efficiency of energy transformation and lesser land acreage can be used. The solar module can be operated at the optimal temperature from the tradeoffs from radiation losses and temperature dependent heat absorption. A typical scheme for solar aided combined cycle power plant is shown in Figure 7.11. Similar design has been proposed by Kelly et al. (2001). The parabolic trough solar power plant is integrated into a modern combined cycle power plant with gas and steam turbines. A solar collector field can be seen in Figure 7.11. Kelly et al. (2001) found that the most efficient use of solar thermal energy was by the production of high pressure saturated steam for addition to the heat recovery steam generator. This way the solar energy and fuel energy in the inlet of the gas turbine is converted to electricity at higher conversion efficiency. Steam turbine alone power plant or gas turbine alone power plant operate at ~ 30% efficiency and the combined cycle efficiency is ~ 55-60%. The increase in combined cycle efficiency can be explained using the waste heat recovery from the exhaust of the gas turbine and use of solar

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insolation in the input. The parabolic trough collector can be used in order to heat mineral oil. This hot oil is used in a shell and tube heat exchanger in order to make steam. This steam is used to operate steam turbines with the generation of electricity. Optimization of layout and complete integration of solar technology is an ongoing saga. Natural gas may be used as the input fuel to the gas turbine. The combined cycle technology offers interesting technical problems to the control engineer. How much load the gas turbine ought to carry and how much load the steam turbine ought to carry can be provided by a feedforword scheme. Internal model control is another possibility. Feedback control using temperature, pressure measurements as discussed in Chapter 5.0 can be continued to be used. Performance models can be developed. The ambient temperature plays a more significant than expected role in determination of the overall thermal conversion efficiency. Plant designs can be conceived using software such as Gate Cycle from Enter Software.

7.11. CONTINUOUS PHARMACEUTICAL PRODUCTION Chemical engineering principles can be applied to scale-up formulations of drugs into large production in continuous manner. There are number of advantages for preferring continuous processes over batch modes. The quality will be higher, process cost will be lower. Capital investment and plant costs can be depreciated over reasonable periods of time. Centralized control is possible in continuous processes. The raw materials can be procured in whole sale prices. The finishing operations for the product such as formation of capsules, coating, coloring etc. can be sewn into the process flow with benefit. Productivity will be higher and product uniformity will be more. The residence time in the reactors can be lowered and the throughput can be increased. Less manual labor is required. Inventory costs will be on the lower side. Environmental impact is more benign in the case of continuous process compared with batch mode. Depending on the demand optimal strategies can be devised. Sometimes space can be minimized for the same throughput. Some of the hurdles that needs to be overcome in order to have widespread use of continuous processes for pharmaceutical production is the misperceptions of higher risk. A paradigm shift and outside the box thinking is needed for continuous production of drugs. Investors who are happy with current methods need be better informed. They may be used to the conventional pharma units with stepwise reactions, purifications and final product formulations were performed in the batch mode. Change to continuous does not mean that there is need for more engineering and operating skill. Chemical engineers are trained in order to accomplish some of the goals such as crystallization, purification, reaction, separation and heat transfer steps needed in pharmaceutical production. Handling solids need expertise in powder technology, continuous crystallization and particulate processing. Use of continuous crystallization can result in cost savings of about 50%. Start-up and shut down procedures in the continuous processes can be optimized using transient models tested on the computer. FDA approvals for the drug can be used effectively using quality inspections in process implementation. FDA encourage continuous production. Optimization of the total cost of operation with tradeoffs of operating costs and capital costs can result in arrival of certain number of stages for each unit operation used. Expertise in separation, purification and integration of biocatalysts are needed for continuous pharmaceutical production.

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Figure 17. Integrated Solar Combined Cycle Power Plant.

The industrial controls market size is expected to reach ~ 150 billion dollars worldwide. Process dynamics can be studied using desktop computers and proactive process solutions can be sought. Stability analysis using a pencil and paper can lead to a better perspective on the range of operation of the process variables. Control and automation have grown tremendously with the advent of computers and programmable logic controllers, PLCs. Plant start-up, Plant Shut-down operations gets more attention of the lead engineer compared with those during steady-state operation of the plant. Transient analysis is not well studied. Transient behavior of chemical reactors, distillation columns, absorption towers, adsorption beds, extraction units and other unit operations needs to be better studied. Collegiate education methods have to keep pace with the developments in the field of industrial controls. Moore‟s law states that computing speed of microprocessors double every 18 months. Biological databanks double in size every 10 months. Mathematical methods for model development have been refined over centuries. The methods and means available to the engineer need be better utilized. Computer simulation and model development can be an integral part of an engineer‟s endeavors. The days when the effect of professionals who do mathematical modeling and computer simulation on the bottom line of the enterprise is only indirect are over. The coming era is when the PW, Present Worth of chemical plants are better for having an army of engineers and Ph.D. scholars who perform process dynamics studies, develop process control block diagrams, instrument the chemical plant with sensors hooked with data acquisition to the desktop computer.

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GlaxoSmithKline, Novartis, DSM, Lonza are developing continuous process technologies for pharmaceutical production. Over $ 1 billion has been invested in continuous process for pharmaceutical production. Glaxo is building a $ 50 million plant in Singapore in order to manufacture asthma and allergy drug fluticasone propionate. Contract manufacturing can be used in order to perform flow hydrogenations that are catalyzed in packed bed configuration, distillation, extraction, crystallization from the tube to manufacturing. SK Life Sciences sill run 60-70% of the chemistries in batch mode. The operators need to be trained in the PFD, process flow diagram. This can also be shown to the customers. Material and Energy balance can lead to cost savings for the enterprise. Spreadsheets can be used for this purpose. It does not come as a surprise that the high attrition rate in pharmaceutical industry can be decreased when continuous process is adopted. Several unit operations have to be made to work in consonance with each other. It is hard to miss the efficient conversion of raw materials to product using continuous process in large scale. Scale-up principles such as L/D, length over diameter ratio of the reactors, distillation columns, adsorption towers, agitator speed and power draw used, pre-treatment of raw materials can be augmented using reliable mathematical models for mixing, heat transfer, mass transfer for a given pharmaceutical chemistry. Waste heat recovery and energy audit can lead to a more optimal operation. For new products the continuous processes are readily adapted. The cost of over the counter drugs may decrease. Value addition during development of pharmaceutical products are high. It is not hard to break down the value addition along the sequence of unit operations used from raw material pre-treatment to post-treatment of the drug capsule steps. Industrial scale flow processes can be developed for a step identified in the bench scale. One example here is the nitration step developed by DSM. DSM manufactured API, active pharmaceutical ingredient at a capacity of few hundreds of tons per year. They ran hundreds of micro scale reactors in parallel. Usually PFR, plug flow reactor is more efficient. Utility costs will be high in the parallel configuration. The cost of recovery of the unreacted materials and recycling back to the reactors will consume the money budgeted. This is because of the low conversion in the micro reactors and high conversion in the case of the PFR. In cases where the yield of the reaction is low in the bench scale attempts have to made in order to improve the yield. Advances have been made in the downstream processing methods used in pharmaceutical industry. Implementation of these changes can result in higher overall yield of the product. Research in continuous manufacturing has led to the development of a pilot effort to make the API in Tekturna a cardiovascular drug, i.e., Aliskirne Hemifumarate. The number of operations in the batch mode was 21. The unit operations used in the continuous process is 14. Novartis is spending $ 65 million for more R & D over a 10 year period. Some steps used in continuous manufacturing of pharmaceuticals are reactions, separations, purification, crystallization, drying, formulation, tableting. 3 million tablets a year can be made at a reactor output of 41 g/hr. Portable units are possible. Nutritional supplement, NBMI was developed by University of Kentucky in 6 years with the help of contract manufacturing. The drug is intended for use as antioxidant and can be used for reversing ageing. Clinical trials are being conducted. It works by mercury chelation. PCI the contract manufacturer eliminated chloroform used as solvent, increased the yield to 98%, isolated the material in powder form and made into capsules.

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7.12. SPECIFICATIONS LIMITATIONS IN BINARY DISTILLATION Distillation is the oldest and most common refinery process It is based on the principle that liquids with different boiling points can be separated by vaporization and condensation. Crude oil distillation involves feeding crude oil at its boiling point of 400 0C into a large fractionation column. The oil is split up into a series of product fractions depending on desired boiling point ranges. Vaporized oil on its way up through the column encounters condensed liquid on its way down. They are made to come in contact and exchange components at a number of sieve trays in the column. This contact caused light fluid molecules to vaporize and heavy gas molecules to condense. The further up the column, lighter the fluid found at the sieve tray it is. Part of the crude oil that cannot be vaporized is extracted as heavy oil at the bottom of the column. Exxon-Mobil is the world largest company that explores oil and in their refineries manufactures fuel. A schematic of the overall process used by Exxon-Mobil is shown below in Figure 7.12. In India the major oil companies are Indian Oil, Bharat Petroleum, Hindustan Petroleum, Oil and Natural Gas Corporation, Chennai Petroleum, Kochi Refineries, Bangagoan Refinery, Mangalore Refinery and IBP. They are increasing their refinery capacity. The largest refinery in India is owned by Reliance under the management of Mr. Mukesh Ambani with a capacity of 31 million tons. For example, recently the 3 million ton refining capacity expansion and modernization project of Chennai Petroleum Corporation Ltd. CPCL, in Manali near Chennai, in order to improve the overhead distillates yield and produce cleaner fuel came on stream. It is a 9.5 million ton refinery. The expansion was on 120 acres at 2360 crore rupees. It is a first of its kind facility to process 100% hydrocracker bottom and as a result the liquefied petroleum gas production has increased from 1.8 lakh tons to about 4 lakh tons. The petrol production improved from 3.9 lakh tons to 7 lakh tons. The hydrocracker unit is a crucial component in the production of automobile fuels with low sulfur content. Industrial distillation is typically performed in large, vertical cylindrical columns known as "distillation towers" or "distillation columns" with diameters ranging from about 65 cm - 6 m and heights ranging from about 6 m - 60 m. The distillation towers have liquid outlets at intervals up the column which allow for the withdrawal of different fractions or products having different boiling ranges. The "lightest" products (those with the lowest boiling point) exit from the top of the columns and the "heaviest" products (those with the highest boiling point) exit from the bottom of the column. Large-scale industrial towers also use reflux to achieve more complete separation of products. Fractional distillation is also used in air separation, producing liquid oxygen, liquid nitrogen, and high purity argon. Distillation of cholorosilanes also enables the production of high-purity silicon for use as a semiconductor. In industrial uses, sometimes a packing material is used in the column instead of trays, especially when low pressure drops across the column are required, as when operating under vacuum. This packing material can either be random dumped packing (1-3" wide) or structured sheet metal. Typical manufacturers are Koch, Sulzer and other companies. Liquids tend to wet the surface of the packing and the vapors pass across this wetted surface, where mass transfer takes place. Unlike conventional tray distillation in which every tray represents a separate point of vapor liquid equilibrium, the vapor liquid equilibrium curve in a packed column is continuous. However, when modeling packed columns it is useful to compute a number of "theoretical stages" to denote the separation efficiency of the packed

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column with respect to more traditional trays. Differently shaped packing has different surface areas and void space between packing. Both of these factors affect packing performance. Typical number of plates required for a desired level of separation for different aromatic, organic and aqueous systems are given in Table 4.1.

7.12.2. Method of McCabe and Thiele for Multistage Binary Distillation The widely accepted method popular with the practioniers in the chemical process industry is the method of McCabe and Thiele to calculate the number of theoretical stages needed to perform a desired level of separation of a binary system. A schematic of a multistage binary distillation column showing the key components is shown in Figure 7.13. The components are; 1) 2) 3) 4) 5) 6) 7) 8)

Central Feed Vertical Cascade of Stages Enriching Section Overhead Condenser Reflux Distillate Stripping Section Reboiler

In the enriching section the vapor rises above the feed and is washed with the liquid to strip the less volatile component. The washing liquid is prepared in the overhead condenser and part of the distillate is returned to the distillation column as reflux. A reflux ratio is calculated and comes in handy during further design estimates. The distillate is the product that is removed from the condenser and contains the more volatile component in high concentration. The stripping section is that section below the feed where the vapor that boils sweeps the more volatile component with it and moves upward through the column. The vapor is prepared in the reboiler by partial vaporization. The liquid that is removed from the reboiler is the residue and has predominantly the less volatile component. The reboiler can be viewed as an additional stage. The liquid and vapor at each of the n trays are at their dew point and bubble point. The highest temperature is at the bottom and the lowest temperature is at the top. The purity levels achieved for the distillate and residue depend on the liquid/gas ratios used and the number of theoretical stages provided in the two sections of the tower and the operating efficiency resulting from mass transfer diffusion resistances. The cross-sectional area of the distillation column varies with the quality and price of the material of interest. The interrelationship between the purity levels of the distillate, residue, feed and the number of stages required ideally, was outlined by McCabe and Thiele. The most common problem in distillation is designing a distillation column. Given the flow rate and concentration of entering stream, and the flow rate and concentration of the product stream, the size of column needs to be calculated. The industrial columns consist of a series of plates of the sieve tray or bubble cap variety and are used to separate the stages. The gas and liquid phases are made to come in contact with each other until it can reach equilibrium.

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Figure7.12. Exxon Mobil Corporation‟s Refining of Crude Oil into Fractions.

The main steps of the design involve; 1. 2. 3. 4.

Mass Balance - Derive Operating Line Energy Balance - Obtain Equilibrium Curve Transient Mass Transfer Rate Equations - Derive Murphree Efficiency Step off the Ideal and Real Stages required (staircase graph)

The method is less rigorous compared with some other methods such as the Ponchon Savarit method that used the enthalpy information as well. This method was found to give good results for concentrated feeds. The reflux ratio is given as an input parameter in the estimations. Equimolal overflow and vaporization is assumed. At each stage let Ln and xn represent the liquid phase flow rate and the mole fraction of the desired species and Gn and yn be the vapor phase flow rate and mole fraction of the desired species. At the 1st stage y1 is the vapor phase mole fraction and x0 is the liquid phase mole fraction. There is no y0 as the condenser returns will be all liquid.

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Figure 7.13. Schematic of Typical Multistage Distillation Column.

7.12.3. The Mass Balances and Operating Lines From an overall mass balance on the feed, distillate and bottoms streams; F = Dis + W From a component mass balance on the feed, distillate and bottoms stream: xFF = xDDis + xwW

453

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Kal Renganathan Sharma When the column operates at steady state; Ln-1 + Gn+1 = Ln + Gn The component balance would then be; L x0 + Gyn+1 = Lxn + Gy1

The vapor flux G and the liquid flux L are assumed constant between the top of the tower and the nth plate. Rewriting above Eq. yn+1 = (y1 – L/G x0) + (L/G)xn It can be seen that yn+1 varies linearly with xn. The slope can be seen to be (L/G) and the intercept (y1 – L/Gx0). In the special case of the total condenser, (Dis = G + L), xD = x0 = y1 Noting that the vapor phase entering the nth plate is at the composition yn+1 and the liquid exiting the nth stage is at composition xn. From a mass balance on the nth plate and the condenser exit streams; xDDis = -xnL + yn+1G or yn+1 = xn(L/G) + xD(Dis/G) Defining a reflux ratio, R = L/Dis Then, Dis/G = Dis/(L + Dis) = 1/(R + 1) L/G = L/(L + Dis) = R/(R + 1) Combining Eqs. [4.9-4.10] into Eq. [4.4] the operating line of the rectifying section is then; yn+1 = xn(R/(R +1)) + xD(1/(R +1)) The operating line of the rectifying section is essentially a mass balance relationship between the components of the different inlet and outlet streams. The reflux ratio R can be used as a lever to achieve the desired objectives during the operation of the distillation column. In a similar fashion analysis of the reboiler section and a plate in the column leads to the derivation of the operating line of the stripping section: yn+1 = xn(W/G‟ -1) - xW(W/G‟) yn+1 = xn B/(B-1) - xW/(B-1)

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where B is the reboil ratio. The reboil ratio is that ratio of the vapor produced by the reboiler to the residue withdrawn (L‟/W). The nature of feed can be several kinds ranging from all vapors, all liquid or a partial mixture of both liquid and vapor. The feed can be found at the intersection of the two operating lines from the rectifying and stripping sections. For any feed, F + L + G‟ = G + L‟ Where G‟ and L‟ are the vapor phase and liquid phase flow rates in the stripping section. q can be defined as; q =(L‟ – L)/F = LF/F When the feed is all liquid below the bubble point the q value is > 1.0. For a saturated liquid feed q = 1, and for a mixture of liquid and vapor in the feed, the q value lies between 0 and 1.0. For the saturated vapor q is 0 and for superheated vapor the q value is less than 0 and is negative. Later an expression for q in terms of the enthalpies will be shown. From the component mass balances; FzF = Dis xD + W xw The intersection of the two operating lines can be found as; yn+1 = xn(L/G) + xD(Dis/G) yn+1 = xn(L‟/G‟) - xW(W/G‟) Subtracting the equation for the rectifier from the equation for the stripper; yn+1 (G‟ – G) = xn(L‟ - L) - (xWW + Dis xD) Rewriting Eq. [4.17] by using Eq. [4.15), Eq. [4.13) yn+1 = qxn/ (q-1) - zD/(q - 1) Above equation is the feed line. The slope of the line gives the energy required to convert the feed to the vapor phase. The feed line for different feed conditions is shown in Figure 7.14.

7.12.4. Feed Location, Pinch Point, Minimum Reflux and Minimum Stages The information from the VLE, vapor-liquid equilibrium data and the operating lines derived from the mass balances as shown above are combined to calculate the number of theoretical stages needed to achieve a desired level of separation. A step by step illustration of this method is shown in worked example 4.1 in Sharma [2007]. The Pentium 4, 3 GHz PC was used. The graph of the VLE data called the equilibrium line relates the vapor composition

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on a particular stage to the liquid composition in the same stage. For a given reflux ratio, R, the two operating lines from the rectifying and stripping section can be shown to intersect with the feed line for any feed. Each stage can be seen as a staircase graph and stepped off from the graph. This is shown in the worked example 4.1. Thus by a combined graph of the equilibrium line and operating lines the number of stages can be stepped off with ease and is an important procedure. The feed can be located in a stage at the intersection of the operating lines. This is the optimal location of the feed. Other locations are suboptimal. The minimum number of stages required can be calculated when the reflux ratio becomes infinite. At infinite reflux ratio (R = ), the operating lines can be seen to revert to yn+1 = xn. Both the slopes of the rectifying and stripping sections become equal. At infinite reflux, zero distillate is produced. Thus for a lot of energy supplied, in a few stages, that can be stepped off from the graph, only a few drops of pure product is recovered. The minimum reflux ratio can be calculated by letting the operating line from the rectifying section to intersect with the equilibrium line and the q feed line and the number of stages stepped off from the graph. The reflux ratio that is used is often times 1.2 to 1.6 times the minimum reflux ratio needed. This ratio depends on the relative capital costs and operating costs of energy. If capital is cheap, a low reflux ratio with more number of stages can be used and where the energy is cheap a large reflux with fewer stages is preferred. The reboiler load will be low for the former and large for the latter. The condition of minimum reflux ratio also denotes the point of maximum ratio where infinite plates for the desired separation are needed. Above this ratio the desired separation can be effected. The intersection of the equilibrium line and operating line is called the pinch point. Two pinch points can be expected. One from the intersection of the rectifying operating line and the equilibrium curve and another from the stripping operating line and the equilibrium line. It can be seen that infinite stages are required to reach the pinch point from the point of the desired purity level. From this condition of infinite plates the minimum reflux ratio can be calculated by noting the y intercept of the graph. (xD/(Rm+1)).

Figure 7.14. q Line for Different Feed Condition.

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Figure 7.15. Minimum Reflux Ratio and Minimum Number of Stages.

7.12.5. Reboilers The reboiler that is used to supply energy for the fractionation by vaporization of the residue and return to the fractionator may be of several kinds. The reboiler may be built either internal or external to the column. This depends on the heat load. For low capacity of separation, jacketed kettle may be used. Here the vapor capacity and heat transfer area are small. This can usually be found in pilot plants. Sometimes a tubular heat exchanger is built into the bottom of the tower. This is an internal reboiler and the heat transfer surface is larger. It provides a vapor in equilibrium with the residue product so that the last stage represents enrichment due to reboiler. The problem is when cleaning the heat exchanger requires the shutdown of the entire distillation operation. In order to avoid this maintenance routine, external reboilers can be used. They can be arranged with spares for cleaning. This can be used for large installations. The kettle reboiler has the heating medium inside the tubes. The vapor is in equilibrium with the residue product and behaves as an extra theoretical stage. The heating medium in a vertical thermo siphon reboiler is outside the tubes. All of the liquid entering the tubes are vaporized. Fouling is a problem. In horizontal reboilers steam is used for heating. In some vertical thermo siphon reboilers the liquid is received from the traps of the bottom tray. In order to reduce fouling sometimes the boiler is operated in the partially vaporized condition.

7.12.6. Specifications Limitations The parameters of the design of a multistage binary distillation column at the beginning are usually available;

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Kal Renganathan Sharma 1. 2. 3. 4.

Temperature, Pressure, Composition and Flow rate of Feed Pressure of Distillation Optimal Feed Location Heat Losses

It can be shown that only 3 more design specifications after the above is needed to completely specify the problem. (Treybal, 1980). The rest can be calculated from the other parameters. In the above sections it can be seen a lot of the parameters are related to each other by the mass balances and equilibrium line from VLE data. The 6 other parameters are; 1. 2. 3. 4. 5. 6.

Total Number of Trays, N Reflux Ratio, R Reboil ratio, B Composition of product Product Split, Psplit (xDDis/xwW) Ratio of total distillate to total residue (Dis/W)

There are some pitfalls in specifying the problem completely. Sometimes the problem may be over specified or underspecified. For example, it can be seen that say three parameters from the above six are given, i.e., the product split, Psplit, W/Dis and distillate composition are given. Then, from a component balance of the three main streams of feed, distillate and product. FzF = Dis xD + WxW Rewriting he above equation using F = Dis + W (Dis + W)zF = Dis xD + WxW

(4.21)

Dividing throughout by Dis (1 + W/Dis)zF = xD + W/Dis xW From the product split information and W/Dis and the distillate composition the residue composition can be calculated. Now that the residue and distillate compositions are known from the above equation the feed composition can be calculated. It cannot be made available. Otherwise the problem will be over specified. For problems that are underspecified insufficient information is available to complete the problem. There can be 6C3 = 20 problem types. With the advent of the PC it is easier to derive the necessary relationships and work with the information available to achieve the desired objectives. These problem types and the solution approach are as follows; I.

Given Reflux Ratio R, Reboil Ratio B and Dis/W find the number of trays, N p, product split and the bottoms composition. The intersection point (xint, yint) can be seen to obey Eqs. [4.11, 4.12, 4.19]

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yint = xint(R/(R +1)) + xD(1/(R +1)) yint = xint B/(B-1) - xW/(B-1) yint = qxint/ (q-1) - 1/(q - 1) zF Further, (Dis/W + 1)zF = Dis/W xD + xw There are four equations and 4 unknowns, xw, xD, xint, yint which can be solved simultaneously. Once the intersection point is found the same procedure as shown in worked example 4.1 to step of the number of ideal stages required for a given desired level of separation can be stepped from the staircase graph. II. Given W/Dis, product split, R, calculate N p, xw, B Eq. [4.26] can be written in terms of xw only given the information in the product split and W/Dis. Once the xw, xD can be calculated from the product split information. Eqs. [4.25] and (4.23] can be solved simultaneously as they form a set of two equations and two unknowns. From this and Eq. [4.24] the reboil ratio, B can be calculated. III. Given Product Split, P split (xDDis/xwW), Reboil Ratio, B, Residue composition, x W, calculate N p, R, W/Dis. Straightforward solution. Given the residue composition and reboil ratio the operating line for the stripping section can be constructed in the graph. The intersection with the feed line can be identified. Rewriting Eq. [4.22], (Psplit xw/xD + 1)zF = Psplit xw + xw (4.27) Thus the distillate composition can be calculated given the product split and residue mole fraction. Now from Eq. [4.23] the reflux ratio can be calculated. This completes the specification of the problem. IV. Given Product Split, Psplit (xDDis/xwW), Reboil Ratio, B and Reflux Ratio R calculate Residue composition, xW, Np, W/Dis. The four equations can be solved for by simultaneous equations, simultaneous unknowns. V. Given Product Split, P split (xDDis/xwW), Reflux Ratio, R, Residue Composition, xW Calculate N p, W/Dis and Reboil Ratio Straightforward solution. Given product split and residue composition, the distillate composition can be calculated from Eq. [4.27]. Now the W/Dis can be calculated from the product split information. The intersection points of the two operating lines of rectification and stripping can be calculated by solving for Eqs. [4.23, 4.25]. The reboil ratio can then be calculated from Eq. [4.24]. This completes the description of the specifications. The slope of the rectifying line can be calculated and graphically it is straight forward to identify the intersection point. VI. Given Product Split, Residue Composition, and W/Dis calculate the N p, Reflux Ratio R, and Reboil Ratio, B Given the product split, residue composition and W/Dis the distillate composition can be calculated. The feed composition can be calculated and cannot be specified. If stated it would be a overspecification. Between the feed line, operating lines of

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Kal Renganathan Sharma rectification and stripping as given in Eqs. [4.23-4.25) and the feed line, the abscissa and ordinate of the intersection points and the R and B can be calculated by solving a system of 4 equations and 4 unknowns. VII. Given the Product Split, xDDis/xwW, W/Dis and Reboil Ratio, B, calculate N p, R, xW. Given the product split and W/Dis the distillate composition can be written in terms of the residue composition. Using this information in Eq. [4.20] there remains three simultaneous equations (Eqs. [4.23,4.24,4.25] and three unknowns to solve for the intersection point ordinates and the residue composition. This completes the description of the problem VIII. Given W/Dis, Residue composition, x W, Reflux ratio, R calculate N p, Reboil ratio, B and Product Split, x DDis/xwW The distillate composition can be calculated from Eq. [4.26] given the W/Dis and residue composition as the feed is completely specified. Now the product split can be calculated. Graphically the operating line for the rectifying section can be constructed. The intersection point with the feed line is identified in the graph. Given the residue composition the operating line of the stripping section is constructed. From the slope of the line the reboil ration can then be calculated. The number of ideal stages can be stepped off the staircase graph. IX. Given W/Dis., Reboil Ratio, B, Residue composition, x W, calculate N p, Reflux Ratio, R, Product Split, Psplit xDDis/xwW The distillate composition can be calculated from Eq. [4.26] given the W/Dis and residue composition as the feed is completely specified. Now the product split can be calculated. Graphically the operating line for the stripping section can be constructed. The intersection point with the feed line is identified in the graph. As the distillate composition is now available, the operating line of the rectifying section is constructed. From the slope of the line the reflux ratio can then be calculated. The number of ideal stages can be stepped off the staircase graph. X. Given the Reflux Ratio, R, Reboil Ratio, B and Residue Composition, x W, Calculate N p, Product Split, P split xDDis/xwW, W/Dis. Given the residue composition and reboil ration the operating line of the stripping section can be constructed. The intersection point with the feed line can be identified in the combination graph of both the equilibrium and operating lines. Given the reflux ratio the slope of the operating line of the rectifying section is calculated and constructed. The intersection of the operating line of the rectifying section gives the distillate composition. From the information gained the product split is now calculable. The Np the number of ideal stages needed for a desired level of separation can be stepped off from the staircase graph. XI. Given N p, Reboil Ratio, B, Reflux Ratio, R calculate W/Dis, Residue compositiobn, xW, Product Split, Psplit (xDDis/WxW) This problem requires a iterative solution. Here is a suggested procedure: Guess Distillate Composition, xD--------------    Calculate (xint, yint) 

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 check   Calculate xW      Calculate Np -------------------------------------- XII. Given N p, Residue composition xW, P split (xDDis/WxW) calculate Reflux Ratio, R, W/Dis, Reboil Ratio, B Guess B, given residue composition the operating line of the stripping section can be constructed. The intersection point with the feed line can be identified in the graph. Given the product split and residue composition the distillate composition can be calculated from Eq. [4.27]. From the distillate composition and the intersection point the operating line of the rectifying section can be constructed. The Np can be stepped off from the graph. This is compared with the given Np. Then the reboil ratio is changed and the routine repeated until the calculated and given Np are identical within numerical error. XIII. Given N p, Reboil Ratio, B, P split (xDDis/WxW) calculate Reflux Ratio, R calculate W/Dis, Residue composition, xW Guess the residue composition, xW and given the reboil ratio the operating line of the stripping section can be constructed. The intersection point with the feed line can be identified in the graph. Given the product split and residue composition the distillate composition can be calculated from Eq. [4.27]. From the distillate composition and the intersection point the operating line of the rectifying section can be constructed. The Np can be stepped off from the graph. This is compared with the given Np. Then the residue composition is changed and the routine repeated until the calculated and given Np are identical within numerical error. XIV. Given N p, Reboil Ratio, B, W/Dis calculate Residue composition x W, calculate Reflux Ratio, R, Psplit (xDDis/WxW) Guess the residue composition, xW and given the reboil ratio the operating line of the stripping section can be constructed. The intersection point with the feed line can be identified in the graph. Given the W/Dis and residue composition the distillate composition can be calculated from Eq. [4.26]. From the distillate composition and the intersection point the operating line of the rectifying section can be constructed. The Np can be stepped off from the graph. This is compared with the given Np. Then the residue composition is changed and the routine repeated until the calculated and given Np are identical within numerical error. XV. Given N p, Residue composition xW, Reboil Ratio, B, calculate, Reflux Ratio, R, P split (xDDis/WxW), W/Dis Given the residue composition and reboil ratio the operating line of the strippping section can be constructed in the combination graph. The intersection point of the operating lines can be identified by letting the feed line intersect with the stripping line. The three remaining variables are W/Dis, distillate composition. By trial and error by a initial guess of reflux ratio, R, the other parameters are calculated. The Np calculated is compared with the given Np.

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Kal Renganathan Sharma XVI. Given N p, Reflux Ratio, R, W/Dis calculate Reboil Ratio, B, P split (xDDis/WxW), Residue composition xW Guess the distillate composition. Given the reflux ratio the operating line of the rectifying section can be constructed in the graph. The intersection between this line and that of the feed line is identified. Given W/Dis and the distillate composition the residue composition can be calculated from Eq. [4.26]. The operating line for the stripping section is then obtained by connecting the residue point on the y = x line and the intersection operating point. The number of ideal stages can be stepped off and then compared with the given Np. The procedure is repeated till the two values come within numerical error of each other. XVII. Given N p, Residue composition xW, Reflux Ratio, R calculate P split (xDDis/WxW), Reboil Ratio, B, W/Dis Guess the reboil ratio. Given the residue composition and reboil ratio the operating line of the stripping section is constructed. From the feed line intersection with the stripping line the intersection point is obtained. From the intersection point and the reflux ratio the operating line of the rectifying section is constructed. The distillate composition is calculated from the distillate point on the y = x diagonal. The procedure is repeated until the calculated Np is within numerical error of the given N p. XVIII. Given N p, Reflux Ratio, R, P split (xDDis/WxW) calculate Reboil Ratio, B W/Dis, Residue composition xW Guess the distillate composition. Given the reflux ratio the operating line of the rectifying section can be constructed in the graph. The intersection between this line and that of the feed line is identified. Given product split and the distillate composition the residue composition can be calculated from Eq. [4.27]. The operating line for the stripping section is then obtained by connecting the residue point on the y = x line and the intersection operating point. The number of ideal stages can be stepped off and then compared with the given Np. The procedure is repeated till the two values come within numerical error of each other. XIX. Given N p, W/Dis, P split(xDDis/WxW) calculate Reflux Ratio, R calculate, Residue composition xW Reboil Ratio, B Given W/Dis and product split a relation between xD and xW can be established. This can be solved simultaneously for two unknowns with the Eq. [4.27]. Guess the reboil ratio. Construct the operating line of the stripping section. This intersects with the feed line. The line connecting the intersection point and the distillate point is the operating line of the rectifying section. The reflux ratio is calucalated. The Np stepped from the graph is compared with the given value and the guess of the reboil ratio changed until they match. XX. Given N p, W/Dis, Residue composition, xW calculate Reboil Ratio, B, Reflux Ratio, R, Product Split, P split (xDDis/xWW) Given the residue composition and W/Dis from Eq. [4.26] the distillate composition can be calculated. The product split is then calculated. Guess the reboil ratio. Construct the operating line of the stripping section. This intersects with the feed line. The line connecting the intersection point and the distillate point is the operating line of the rectifying section. The reflux ratio is calucalated. The Np stepped from the

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graph is compared with the given value and the guess of the reboil ratio changed until they match.

7.12.7. Equilibrium Line As can be seen from the discussions above and the worked example 4.1, the construct of the equilibrium line is an important aspect to the design procedure due to McCabe and Thiele. Use of stages was made. Here a refresher is provided on the understanding of equilibrium between the vapor and liquid phases, VLE. This would provide the means to obtain the equilibrium line and later the real line limited by diffusion resistances. Every pure liquid exerts an equilibrium pressure, the vapor pressure in the gas phase. The extent of the vapor pressure depends on the temperature. A typical P-T diagram for a pure substance is shown in Figure 4.7. T is the tripe point of the substance where all the three phases co-exist at one temperature and one pressure. The phase rule gives the degrees of freedom in terms of the number of components and phases and is; Ff = Nc –  + 2 The number of degrees of freedom, Ff is given by subtracting the number of phases from the number of chemical species or components present in the system and adding 2. Thus for a 1 component system, the number of degrees of freedom for a two phase system can be seen from Eq. [4.28] to be, 1. Thus if the pressure is specified, the temperature can be calculated. When the number of phases is 3 the degrees of freedom become zero. This means that the temperature and pressure at which the three phases co-exist in equilibrium cannot be varied independently but are determinable values. This is also called the triple point of the pure substance and is shown as T in Figure 7.16. Equilibrium is the state of the system when after sufficient contact the compositions of the system and other parameters of the system such as temperature and pressure do not change with time. It is at steady state. The curve TB in Figure 4.7 can be seen to be the vapor pressure curve of the system. This gives the temperature and pressure relationship of the 1 component, two phase system in equilibrium. The points in the curve TB represent the saturated vapor and saturated liquid. The region sectored by CTB is pure liquid. The region sectored by CTA is solid. This is when the Pressure, P is high and temperature, T is low. The region bordered with ATB is pure vapor and the region beyond the point B is gas. These are regions of high temperature, T and low pressure, P. B is said to be the critical point and the corresponding temperature, Tc the critical temperature and pressure, Pc the critical pressure. Beyond point B the liquid and vapor phase are indistinguishable from each other. State of the system beyond this point is the gas. Gas cannot be liquefied by pressure increase beyond this critical point. The temperature corresponding to 1 atm pressure in the curve TB is called the normal boiling point of the system. The path EF in Figure 4.7 is an isobaric process where the phase changes from liquid to vapor upon supplying heat. The latent heat of vaporization at constant temperature is the amount of heat per mole supplied to effect the phase change. Vapor pressure data is scarce. The vapor pressure temperature curve TB can be established by interpolation or extrapolation of available data using the Clausius Clapeyron equation. The Clausius Clapeyron equation is;

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Kal Renganathan Sharma dP/dT = vap/T(vv – vL)

(4.29)

where vap is the latent heat of vaporization. The liquid molar volume can be neglected and the vapor molar volume can be written assumingideal gas law as; dP/dT = Pvap/RT2

(4.30)

Integrating both sides of the equation, ln(P) = -vapR/T + c1

(4.31)

For a 2 component system with two phases the number of degrees of freedom can be found to be from Eq. (4.28) to be 2. Thus if both the temperature and pressure of the system is known then the compositions of the two phases are fixed and can be determined. This can be represented as a Txy phase diagram for a vapor-liquid mixture. This would be important in the equilibrium calculations during distillation as shown in the above sections. The VLE for a binary system of species A and B where B is the more volatile are shown in Figure 7.18. The Temperature vs y* is given by the DEW line and the temperature vs x is given by the bubble line. The horizontal tie lines shown join the equilibrium mixtures of liquid and vapor phases. tvu is a tie line. The compositions become fixed as the pressure and temperature are known. There can be infinite tie line. The relative amounts of the two phases, liquid and vapor in this example can be calculated as; Moles (Liquid)/Moles(Vapor) = vu/vt

Figure 7.16. Phase Diagram of a Pure Substance.

(4.32)

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The VLE data can be represented as a distribution diagram as shown in the lower part of Figure 7.17. The greater the distance between the equilibrium line and the y = x diagonal greater are the differences in composition between the vapor and liquid phases. A separation factor can be defined or a relative volatility can be defined as follows in order to quantitatively monitor the nature of the equilibrium line:  = (y*(1-x))/(x(1 – y*)) When  is 1 no separation is possible. Some examples of the distribution diagrams are given in Figures 7.17, 7.18. Some of the systems shown exhibit the condition of azeotropy where the two phases behaves like one phase and boils at a constant temperature. Ideal solutions are first considered as a model for the liquid phase in the distillation operations. It is good for several systems but may not be for all systems. Later special provisions can be made for dealing with liquid phase systems that deviates from the ideal solutions. For ideal solutions, the vapor phase partial pressure can be calculated from knowledge of the composition of the liquid phase alone apriori to any experimentation. This can be useful for systems where data is not readily available. Four characteristics that can be mentioned for ideal solutions are;

1. The average intermolecular forces of attraction and repulsion in the solution are unchanged on mixing of the constituents. 2. Volume of solution varies linearly with composition. 3. The mixing is adiabatic. 4. The total vapor pressure of the system varies linearly with the composition expressed as mole fraction. Real systems only approach the ideal solution in the limits. Search can be made from among isomers, or compounds from a series with similar structure differing only with a methyl group. The Raoult's law states that; pA = pA* x The partial pressure of the vapor phase in equilibrium with the ideal solution is given by the product of the liquid phase composition and the pure component vapor pressure of the component at that temperature. For dilute non-ideal solutions, the Henry‟s law states that; y* = pA/P = mx Where m is the Henry‟s law constant. Other models for VLE in addition to the Raoult‟s law are available in the literature. These models are derived using the concept of fugacity and at equilibrium the fugacity of the component in different phases are equal when at equilibrium. Activity coefficients are defined for liquids. Van Laar, Margules, Wilson, NRTL and UNIFAC are some of the other models that can be used to interpolate and extrapolate VLE data. The resulting information will only

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be good for conditions where the data was measured and for those circumstances where the assumptions of the model are reasonably good. When the sum of the partial pressures of the components calculated exceeds the total system pressure deviations from ideality can be seen. Examples of maximum boiling azeotrope and minimum boiling azeotrope are shown in Figures 7.19 and7.29. Large positive or negative deviations from ideality can be seen for some systems. Some systems are partially miscible. An example of such a system is the isobutanol-water system as shown in Figure 7.18. When the sum of the component partial pressures are less than the total system pressure negative deviations from ideality can be seen. In a similar manner when the component pressures exceed the total system pressure positive deviations from ideality can be seen.

Figure 7.17. VLE Phase Diagram of Two Component System.

Advanced Control Methods Point k represents the tie line, tvu. B is the more volatile species.

Figure 7.18. Maximum Boiling Azeotrope.

Figure 7.19. Distribution Diagram of Isobutanol-Water Partially Miscible Systems.

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Figure 7.20. Minimum Boiling Azeotrope in a Constant Pressure Txy Diagram & Distribution Diagram.

7.13. AZEOTROPIC DISTILLATION AND EXTRACTIVE DISTILLATION Azeotropic systems are difficult to separate using binary fractionation methods. Azeotropes as indicated in the above section are constant boiling mixtures where a two component mixture behaves as a one component mixture. Some examples of binary azeotropes are; 1) 2) 3) 4) 5) 6) 7) 8) 9) 10)

Tetrahydrofuran, THF/Carbon Tetrachloride THF/Chloroform Ethanol/Toluene CS2/acetone Acetone/Choloroform Acetic Acid/Water MMA/Methanol Ethanol/Water Acetone/Methanol Acrylic Acid/Water

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Ethyl Acetate/Ethanol Hydrazine/Water Methanol/Ethylacetate n-Heptane/Ethanol Methanol/Benzene Ethylene Glycol/Water Chloroform/Methanol Ethanol/Benzene

In order to separate such systems a third component called the entrainer is added to the system. The entrainer is chosen such that the volatility is in a fashion that a new azeotrope is formed which is lower boiling and can be easily separated. For example, benzene is used as an entrainer in the ethanol/water azeotropic system. The normal boiling points of ethanol is 78.3 0C, benzene 80.1 O C and that of water 100 0 C. Benzene is added as an entrainer to the ethanol/water azeotropic system. At the right pressure the lower boiling binary azeotrope of ethanol/benzene is formed and is boiled off. This system ethanol/water/benzene is reported to form a ternary azeotrope. One method of separating the mixture is shown in Figure 7.20. The water is collected as residue. The reboiler needs to be designed accordingly. The vapor from the first column is condensed and fed into a second column. In the second column the benzene is collected as a residue. The benzene/ethanol azeotrope is boiled off from the top of the column 2. Another method of separating the mixture is by tapping into the formation of ternary azeotrope. This boils at 64.9 0C. As this is lower than the normal boiling point of ethanol, ethanol can be collected as a residue. Table 7.3. Typical Number of Trays used in the Real World of Chemical Plants S.No

System

1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0

Ethylene/Ethane Propylene/Propane Propyne/1,3 – Butadiene 1,3 Butadiene /Vinyl Acetylene Benzene/Toluene Benzene/Ethyl Benzene Toluene /Xylene Ethyl Benzene /Styrene o-Xylene/m-Xylene Methanol/Formaldehyde Dichloroethane/Trichlorethane Acetic Acid/Acetic Anhydride Vinyl Acetate/Ethyl Acetate Ethylene Glycol/Diethylene Glycol Acetic Anhydride/Ethylene Diacetate Cumene/Phenol Phenol/Acetophenone

Typical Number of Trays 73 138 40 130 34,53 20 45 34 130 23 30 50 90 16 32 38 39,54

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Kal Renganathan Sharma Some other examples of entrainers that can be used are as follows; 1. Butyl Acetate - Acetic Acid/Water 2. Hexane - MMA/methanol 3. Toluene - Hydrazine/water

In addition to toluene, aniline, benzene, pyridine, xylene, propylalchohols, cresol, glycol, hexyl amine and glycol ethers can be used as entrainer to separate the hydrazine/water azeotropic system. The normal boiling point of the acetic acid is 118.1 0C and that of water is 100 0C. In the first column glacial acetic acid is collected as the residue and the butlyl acetate and water heteroazeotrope is boiled off at 90.2 0C. In the second column water is collected as residue and the ester-water azeotrope is collected as an overhead product. A decanter can be used in between to let the ester rich and water rich layer separate. Some of the criteria for the choice of the selection of entrainer; 1. Choice of entrainer is most important consideration 2. Added compound should form a low boiling azeotrope with only one of the components of the binary mixture 3. It has to have the lowest composition of the three to minimize energy Requirements 4. New azeotrope should have sufficient volatility to make it readily Separable 5. Residue should be lean in entrainer contents 6. Must be inexpensive and readily available 7. Chemical stability high and does not react with the component to be separated 8. non-Corrosive toward common construction materials 9. non-Toxic 10. Low latent heat of vaporization 11. Low freezing point and can be easily stored outside 12. Low viscosity to produce high tray efficiencies Extractive distillation is a technique where the third component added to the system acts as a solvent for one of the components. The solution is removed as the residue as the other component will be the more volatile species and therefore removed as an overhead product from the top of the first distillation column. The solution is separated in a second distillation column with the solvent as the residue and the dissolved component as the overhead product. The solvent is recycled back to the top of the first distillation column. For example the acetone/methanol system forma binary azeotrope and is difficult to separate. A solvent such as butanol is used and the acetone dissolves in the butanol. The normal boiling point of acetone, methanol and butanol are 56.4 0C, 64.7 0C and 117.8 0C respectively. As shown in Figure 7.21, as the acetone dissolves in butanol the VLE changes such that the azeotrope is no longer present. Methanol although less volatile than acetone is removed as an overhead product from the column A and the butanol is collected as the residue from column B. The residue from column A, the solution of acetone in butanol is feed into column B and the acetone is collected as an overhead distillation product from column B. Thus butanol served as an extractive solvent. Another example of an extractive solvent is acetone or furfural to aid in the separation of buetten-2/n-butane mixture. Phenol is used as a solvent in the separation of toluene/isooctane mixture. The second column is an auxiliary column. The solvent changes

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the relative volatility of the binary system. The solvent itself, a low boiler does not enter into the rectifying section much. The requirements of an extractive distillation solvent are as follows; 1. High Selectivity – ability to change VLE of original mixture in such a fashion that easy separation is permitted 2. High Capacity to dissolve the components of the system 3. Low Volatility – must not enter the overhead condensers 4. Separability – Solvent must be readily separated from the mixture to which it is added 5. Solvent must not form azeotrope 6. Low cost 7. Less Toxic 8. non-Corrosive 9. Good Chemical Stability 10. Freezing Point suitable for storage 11. Viscosity 12. Optimal solvent circulation rate – low rate may indicate many stages and large rates require fewer trays but higher circulation costs

Figure 7.21. Azeotropic Distillation of Ethanol/Water using Benzene as Entrainer.

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In both azeotropic and extractive distillation an extraneous third component is added to the system. This is not desirable. It may surface as impurities in the product. Solvent losses to the tune of 1000 ppm can add to the cost of the process. The required solvent/feed ratios are often 3 or 4 for an effective separation. Materials of construction problems increase with the treatment of an additional chemical. Despite these problems it is the choice of separation in some cases. Generally speaking, extractive distillation is more desirable compared with azeotrpic distillation since; 1. there is a greater choice of added component 2. no need for another azeotrope to form 3. smaller quantities of solvent must be volatilized As an exception is the ethanol/water system using n-pentane as entrainer. This is because n-pentane azeotropes with water. This is less expensive compared with extractive distillation with ethylene glycol. As the volatilized impurity is a minor constituent of feed and azeotrope composition is favorable the cost is less.

Figure 7.22. Extractive Distillation of Acetone/Methanol with Butanol as Solvent.

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7.14. SUMMARY Ratio control is used in keeping the reactor adiabatic during the manufacture of flue gas. Heat of reactions from Boudard reaction and oxidation of carbon reactions are made to cancel out each other by control of the ratio of the CO2/air mixture. The control scheme is shown in Figure 8.1. The energy balance equations are derived in detail. SPC, statistical process control methods are discussed including Deming‟s quality principles, QIT, quality improvement teams. SPC, statistical process control refers to a set of methods and techniques of monitoring and control of a process so that the process is operated close to design premises. SPC is different from APC, automatic process control. In SPC control, the cause and effect analysis would lead to knowledge of source of disturbance among other things. Action on removal of the source of disturbance would result in decreased load on the control elements as compared with APC. The pressure fluctuations from a fluidized can be measured and used to control the expected quality of fluidization. Control charts are used in manufacturing units to delineate random noise and variability in process variables that can have Attributable causes. No control action is needed when the variation is due to random noise. Control action can be initiated when the Attributable cause is identified. Variability found can be classified into 4 different categories. A prototypical control chart with UCL and LCL control limits is displayed in Figure 7.3. Let the UCL and LCL be at a distance km from the mean value of the measurements, m. Calculation of control limits from use of statistical distributions such as normal distribution is given by Eq. (7.13). Sample variances can be plotted in an S chart. The control limits for an S chart is given in Table 8.1. IMC, internal model control uses the mathematical model developed for the process. Perfect control is achieved when model transfer function, M(s) is the reciprocal of process transfer function, Gp(s). A filter has to be added to make the system realizable. This keeps the numerator of the transfer function of the combined system at a lower degree of polynomial compared with the degree of the polynomial of the denominator of the transfer function of the system. Inversion of a process model alone may not be sufficient for good control. In order for the controller to be stable and realizable the process transfer function must be factorized. Examples are given. In order to make M(s) proper the order of the filter in some cases is increased. A lead/lag controller was considered by use of a model based transfer function as given by Eq. (8.45). Inverse response is expected for the process. A first order filter was added to make the system more „proper‟. The order of the filter was increased to two. The output function is given by Eq. (8.54) and plotted in Figure 8.9. It can be seen from Figure 8.9 that there is an inverse response for values of  = 1.0 and  = 0.1. Although Figure 8.9 is for the indicated values of  and  in general Eq. (8.54) results in an inverse response. For larger values of  there is a minima present prior to the inverse response. What can be inferred from this analysis is that the inverse response of the process cannot be removed by a stable filter or stable control system. Feed forward control is different from feedback control. During feed forward control the load or disturbance is measured or gauged from other considerations and control action taken accordingly. Carrier corporation has patented [11] a feed forward control system for

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absorption chillers. The feed forward control method comprise of the following steps; (i) determine the disturbance transfer function; (ii) determine the capacity valve transfer function; (iii) measure the occurred disturbance; (iv) implement the feed forward control function. The block diagram for feed forward control of absorption chillers is shown in Figure 8.11. An example of feed forward controller for a regulation of fuel supply to a furnace was discussed. Estimation and control of polymerization reactors was discussed. Control of polymerization of reactors is a difficult task because of the exothermic nature of the polymerization reactions and higher viscosities of monomer/polymer syrups encountered in the reactor. The objective of the reactor control is to obtain superior product quality. Quality is characterized by different parameters. Good structure-property relations are needed for devising control strategies based on measurement of end-use property values. Technical hurdles during the development of mathematical models for polymerization reactors are the nonlinear equations encountered sensitivity to impurities etc. Feedback control of a variable is not possible if the parameter cannot be measured or estimated. In Figure 8.13 is shown the scheme for possible feedback control configurations. The scheme comprises of state-model, subsystem 1, subsystem 2 and the measurement equations. State estimation techniques have been developed to provide estimates of the state variables even in cases where they cannot be obtained by direct measurement. Estimation techniques such as Kalman filter, Weiner filter and extended Kalman filter may be used. In non-linear estimation problems the objective may be to minimize the average squared errors in the initial estimate in the process model and in measurement device. Jo and Bankoff [13] developed a Kalman filter in order to obtain estimates of conversion and Mw, weight averaged molecular weight. The Kalman filter is an on-line computer algorithm which is intended to provide better estimates of the state variables that be obtained by direction solution of measurement equations. If the system is observable, optimal estimates of variables that cannot be measured directly can be made. Congalidis, Richards and Ray [14] discuss different approaches to the control of polymerization reactors. They write species mole balance equations for initiator, monomer, CTA, etc. They suggest feed forward control strategy for their recycle streams polymer etc. Energy balance equation is also written for the temperature of the reactor. The molecular weight was found to have an inverse response with the residence time in the reactor. Feedback controllers were evaluated for polymer properties and production rate. They discuss use of CONSYD package of control system design programs. The controllers will not operate efficiently when the assumptions in their derivations are not met. Neural networks can be used to approximate any reasonable function to any degree of required precision. Three different architectures for ANNs, artificial neural networks are possible. The behavior of each unit in time can be described using either differential equations or discrete update equations. The use of ANNs to control a distillation column is discussed. A refresher was provided on methods and theory of distillation in Chapter 6.0. Control of any process system depends on the availability of suitable models that captures all the major phenomena of the system. Mathematical model of a distillation column can contain hundreds of state variables. The model equations tend to be nonlinear. The range of operating conditions is wide. ANNs can be trained offline in order to model the nonlinear inverse dynamics of the process with a pre determined level of accuracy. Training pairs were generated from such simulation results in [6] for training the neural controller.. The state variables of the distillation column that were selected in the estimator are: flow rate;

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temperature and compositions of the three components for the bottom and overhead products; and temperature and three component compositions for the by-product as side stream. The input to the estimator consists of the reboiler heat valve setting, condenser temperature, current feedrate, and the limited state vector at two previous sample times. The estimator is trained to provide as output a prediction of the limited state vector at the current time. The estimator is a fully connected feed forward network with 31 linear input neurons, 14 linear output neurons, and 157 nonlinear sigmoid neurons distributed over two hidden layers. In order to investigate the capability of the trained controller to control the distillation column, the neural controller is connected to the neural estimator. Typical solar aided combined cycle power plant is discussed. Continuous pharmaceutical operations are emerging. The azeotropic distillation and 20 different specification problem types distillation needs advanced control methods. Single layer grapheme sheets needs to be manufactured in large scale at lower costs. Some interesting control issues are associated with the roll to roll transfer process. Biochemical reactions and design of bio artificial pancreas needs separate control strategies.

7.15. GLOSSARY Control Objective: The purpose for performing the control action. For example the objective of control of polymer reactor is superior product quality. There can be more than one control objective. Exothermic Reaction: Chemical reactions when there is a spontaneous liberation of of heat. Enthalpy change is negative. Nonlinear Process: The governing equation for the process output variable such as fractional conversion or temperature of the reactor is not a linear differential equation. Polymer Structure: Polymer can be characterized by its structure. Structures can be molecular chain structure, tautomeric structure, microscopic structure, behavior in solution, crystallographic structure, macrostructure etc. Molecular weight distribution is an example of structural parameter. Kalman Filter: linear Quadratic estimate of unknown variable from series of noisy data of measurable variable. The Apollo mission to the moon had four Kalman filters. Estimation Technique: The method used in order to obtain the information from the measured data. For example, Weiner filter or Kalman Filter or Low pass filter Weiner Filter: Linear filter. Estimates obtained by minimization of least square error found in the measured data from random process CPFR Continuous Plug Flow Reactor: Raw materials, catalyst and additives are input into a tubular reactor and the product and unreacted raw materials are collected from the effluent of the reactor in a continuous manner. The flow in the tube is plug flow, i.e. there is no mixing between the late elements and early elements of the fluid. The velocity of all the different layers of the fluid are constant. When the velocity profile is parabolic the reactors are called parabolic flow reactors. HCSTR, Homogeneous Continuous Stirred Tank Reactor: Raw materials, catalyst and additives are input into a tubular reactor and the product and unreacted raw materials are collected from the effluent of the reactor in a continuous manner. All the elements within the

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reactor are homogenized by use of an agitator. The effluent concentration of the important species and the reactor concentration of the same species are the same. SCSTR, Segregated Continuous Stirred-Tank Reactor: A set of batch reactors that operate in parallel with the same age distribution. For example suspension polymerization without coalescence is a said to be a SCSTR. Molecular Weight: The molar mass of a concatenated chain polymer with certain degree of polymerization. Molecular Weight Distribution: The statistical distribution of molar masses found in a given sample. The MWD may be characterized by moments of the distribution such as mean and variance. It can be fitted to the Gaussian normal distribution. Schulz-Flory Distribution: Statistical distribution named after P. Flory and G. V. Schulz used to describe the relative ratios of polymers of different length based on their relative probabilities of occurrence. Poisson distribution: Discrete distribution with the same mean and variance,. For example the number of typos made by a good typist can be described using Poisson distribution. Arrival of guests at a event can be described using Poisson distribution. Probability of the first couple of possibilities will be high and the rest will decrease in probability with increase in number of possibilities. The first error and second error of the typist will have higher probability and the probability of x errors will decrease with increase in x. Polymer Composition: The chemical elemental mix of the polymer. For example SAN copolymer contains 75 wt % styrene and 25 wt % of acrylonitrile. The chemical elements present in the copolymer are carbon, hydrogen, and nitrogen. Chain Sequence Distribution: The probability of dyads, triads, tetrads, pentads, etc. of one repeat unit in a copolymer. More than 2 repeat units can be present in the copolymer. Block Microstructure: Copolymer made out of blocks of repeat unit is said to have block microstructure. When all the present members of two species are found next to each other concatenated the copolymer is said to have a diblock architecture. A triblock copolymer is when the species concatenate as three blocks. AAABBBBBBB is a 30/70 diblock copolymer of repeat units A and B. Triblock copopolymer for example would look like AAABBBBBBBBAAA. Alternating Microstructure: Pronounced in a copolymer with two monomers. Architecture is alternating, i.e., ABABABABABABA…. Multicomponent copolymer with more than 2 repeat units can alternate. For example ABCABCABC….is a lternating terpolymer with monomers A, B and C repeat every third unit from it. Multivariate Distribution: When there are more than independent random variable the distribution is said to be multivariate. Control Theory: Description of dynamical systems with inputs. Controllers are used to manipulate noisy inputs and provide a desired output. State Estimator: Variables introduced that can be estimates of real values underlying observed data Observable System: System that can be measured and estimation techniques can be meaningfully applied Corresponding Sequential Estimation: Sample size is not fixed a priori. As data arrives sampling is stopped according to stopping rules.

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Nonlinear Estimation: A set of m values can be fit to n parameter non-linear model using least squares estimate. Feedback Controllers: Control action based upon the error between the measurements of the process variable and set point. Feedforward Control: Control action taken a priori to any feedback from the process variable. Disturbance measured or gauged and used in control action taken. Controller Tuning: Quantities such as controller gain kc, controller time constant, c and other parameters can be varied in order to achieve a desired output response. This process is called controller tuning. Jacket Temperature: The temperature of the steam in the jacket of the kettle is the jacket temperature Damkohler Number: The ratio of the reaction time to the mass transport time is called the Damkohler number. Different expressions can arise depending on the kinetics of the reactions such as its order, or mechanism such as free radical or autocatalytic, etc. 5th Order Runge Kutta Method: The numerical solution method in order to solve ODE. The increment function has 7 weights. It represents a certain slope of the function. Error falls rapidly with effort to low levels. Inverse Response: The output function decreases with increase in input function. Online Gas Chromotograph: Sensor that is used during the performance of the process that can be used to obtain the composition of the species of interest. Sample is eluted in a column of adsorbent. The elution times are used to determine the species‟ type. Condition Number: Index used to evaluate and select controller schemes. CONSYD Package: Integrated Software for Compuater Aided Control System Design and Analysis IMC, Internal Model Control: Control action arrived at based on a mathematical model of the process Filter Tuning Parameter: Filters are added in order to make the system physically realizable. The parameters of the filter are the tuning parameters “more proper”: By adding a filter an otherwise physically unrealizable system can be made to be physically realizable. For instance, when the degree of the numerator of the system transfer function is greater than the degree of the system transfer function in the denominator the system is said to be physically unrealizable. When a filter is added with a transfer function that makes the degree of the denominator higher the system becomes physically realizable. This is said to make the system “more proper”.

7.16. FUTHER READINGS Bi and Grace, J. R. (1995). “Flow Regime Diagrams for Gas- Solid Fluidization and Upward Transport”, International Journal of Multiphase Flow, Vol. 21, 6, 1229-1236. Chapra, S. C. & Canale, R. P. (2006). Numerical Methods for Engineers, 5th Edition, McGraw Hill Professional, New York, NY. Congalidis, J. P., Richards J. R. & Ray, W. H. (1989). Feedforward and Feedback Control of a Solution Copolymerization Reactor”, AIChE Journal, Vol. 35, 6, 891-907.

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Deming, W. E. (2000). The New Economics for Industry Government and Education , MIT Press Boston, MA. Daw, C. S. & Hawk, J. A. (1995). “Fluidization Quality Analyzer for Fluidized Beds”, US Patent 5, 435, 972. Elicabe G. E. & Meira, G. R. (1988). “Estimation and Control in Polymerization Reactors. A Review”, Polymer Engineering and Sci., Vol. 28, 3, 121-135. Geldart, D. (1973). “Types of Fluidization”, Powder Technology, Vol. 7, 285-292. Jo, J. H. & Bankoff, S. G. (1976). “Digital Monitoring and Estimation of Polymerization Reactors”, AIChE Journal, Vol. 22, 361-369. Kelly, B., Herrmann, U. & Hale, M. J. “Optimization Studies for Integrated Solar Combined Cycle Systems”, Proc. of Solar Forum 2001 Solar Energy: The Power to Choose, April 21st – 25th, Washington, DC. Kolk, R., Martini, D. M., Sheetan, D. & Jenkins, N. (2004). “Feedforward Control for Absorption Chiller”, US Patent 6,742,347, Carrier Corporation, Farmington, CT. Montgomery and Runger, G. C. (2007). Applied Statisticsand Probability for Engineers, Fourth Edition, John Wiley & Sons, Hoboke, NJ. McClellland, J. L., Rumelhart D. E. & Hilton, G. E. (1986). The Appeal of Parallel Distributed Processing – in D. E. Rumelhart & J. C. McCelelland (Eds.) Parallel Distributed Processing Explorations in the Microstructure of Cognition, MIT Press, Cambridge, MA Renganathan, K. (1990). “Correlation of Heat Transfer Coefficients with Pressure Fluctuations in Gas-Solid Fluidized Beds”, Ph.D. Dissertation, Department of Chemical Engineering, West Virginia University, Morgantown, WV. Smith, G. M. (1998). Statistical Process Control and Quality Improvement, Prentice Hall, Upper Saddle River, NJ. Steck, J., Krishnamurthy, K., Mcmillin, B. & Leininger, G, (1991). “Neural Modeling and Control of a Distillation Column”, Proceedings International Joint Conference on Neural Networks, Seattle, WA. Stephanopoulos, G. & Erickson, L. E. (1985). Biological Reactors. Chapter 13, in “Chemical Reaction and Reactor Engineering”, Carberry J. J. and Varma, A., Marcel Dekker, Inc., New York. Sharma, K. R. & Turton, R. (1998). “Mesoscopic Approach to Correlate Surface Heat Transfer Coefficients using Pressure Fluctuations in Dense Gas-Solid Fluidized Beds”, Powder Technology, 99, 2, 109-118. Sharma, K. R. (1998). Periodicity in Pressure Fluctuations in Gas-Solid Fluidized Beds, Fluidization IX, Durango, CO, USA. Sharma, K. R. (2007). Principles of Mass Transfer , Prentice Hall of India, N. Delhi, India. Sharma, K. R. (2010). Nanostructuring Operations in Nanoscale Science and Engineering, McGraw Hill Professional, NY. Sharma, K. R. (2012). Polymer Themodynamics: Blends, Copolymers and Reversible Polymeization, CRC Press, Boca Raton, FL. Smith, J. M., van Ness, H. C. & Abbott, M. M. (2005). Introduction to Chemical Engineering Thermodynamics, 7th Edition, McGraw Hill Professional, New York, NY. Sturis, J., C. Knudsen, N. M. O‟Meara, et al., “Phase-locking regions in a forced model of slow insulin and glucose oscillations,” in J. Belair, L. Glass, U. An der Heiden, and J.

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Milton (eds.), Dynamical Disease: Mathematical Analysis of Human Illness, Woodbury, NY: AIP Press, 1995.

EXERCISES 1.0. Batelle/FERCO Low Pressure Process for Hydrogen Hydrogen is produced by a water-gas shift reaction of syngas. An alternative to fossil fuels is hydrogen. Hydrogen is manufactured from renewable resources such as sunlight, wind, and biomass and is more environmentally friendly. The Batelle/FERCO low-pressure process involves gasification of the biomass followed by steam reforming of the syngas produced. Pressure swing adsorption is used to separate the hydrogen from the unreacted raw materials and by-products. This process is used at the existing McNeil small power plant in Burlington, Vermont. The biomass feed rate for the inlet stream shown in Figure is 314 Mg/day. The effluent from the gasifier is separated and purified using cyclones. Steam reforming is followed by L.T. and H.T. steps. The calorific value of hydrogen is 141.8 MJ/kg. The gasifier is operated at low pressure and high temperature. The steam reformer is operated at high temperature and high pressure. The plant can be assumed to be operating at 80% capacity utilization during the year. What is a good control scheme for this process ? What advanced control concepts discussed in Chapter 7.0 are applicable for this process ? 2.0. Bioethanol from Sugarcane Bagasse There are two processes to use sugarcane bagasse. In Process I as shown in Figure 7.18 the sugarcane bagasse is used as fuel in steam boilers. The steam produced is allowed to operate a turbine and electricity is generated. In Process II the hemicellulose in bagasse is treated with dilute sulfuric acid and steam in a high temperature, short residence time reactor. The cellulose and pentose are hydrolyzed in a second unit. The reactant and product mixture is separated using distillation in order to produce furfurol and ethanol. What are the differences in the control strategy that can be deployed for Process I and Process II ? What are the control variables ? What ought to the control objectives ? What advanced control methods discussed in Chapter 7.0 are applicable for Processes I and II ? 3.0. Combustion Flame Synthesis Process for Production of Fullerene Fullerenes are a distinct allotrope of carbon. C60 is its structural formula. Hexagons and pentagons of carbon atoms for a cage-like structure that looks like a soccer ball. The nobel prize was given in 1996 to H. Kroto, R. Smalley and R. Curl for their discovery of fullerenes. Sharma (2010) discussed the reduction in process cost from $25,000 per kg in the Arizona process, $16,000 per kg in the first generation flame synthesis process down to $200 per kg in the second generation flame synthesis process. The second generation flame synthesis process comprises of two plug flow reactors in series. A oxy-benzene flame that is flat is generated. The feed flow rate determines the temperature in the first reactor. Vacuum conditions are maintained in the chamber. This has been found to be needed in order to generate the fullerenes. The O/C, oxygen/carbon ratio can be changed in order to increase the yield and

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Kal Renganathan Sharma purity of the fullerene production. Turbulence is found to enhance the micromixing and diffusion in the reactor. The flame can be considered as diffusion flame. The fullerene laden gas from the effluent of the reactor can be separated into fullerenes. The yield is sufficient for scale-up to commercial production quantities. Factories in Japan and USA produce fullerenes at a capacity of ~ 40 tons per year. Distance from the burner is variable of choice to look for fullerene production. The residence time in the two reactors are 5 ms and 15 ms respectively. The pressure in the first reactor is 10-400 torr and temperature is in the range of 1500 – 2500 K in both the reactors. What ought to the control strategy for this process ? Which of the advanced control methods discussed in Chapter 7.0 is applicable for this process ? Is feedback control discussed in Chapter 5.0 sufficient ? What ought to be the control variables ?

Figure 7.17. Biomass to Hydrogen: Gasification, Separation, Steam Reforming and Separation by Preesure Swing Adsorption.

Figure 7.18. Process I: Combustion of Sugarcane Bagasse to Generate Steam and then Electricity Process II: Sugarcane Bagasse to Ethanol and Furfural: Acid Hydrolysis and Separation.

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4.0. Hydrogen by Fast Pyrolysis and Steam Reforming Hydrogen can be manufactured by fast pyrolysis of biomass followed by steam reforming. Biomass is dried and then converted to oil by fast exposure to hot particles in a fluidized bed. The thermal energy needed for operation of the reactors may be obtained from combustion of chars and gases that were produced from the biomass. The oil products are condensed and cooled. Bio-oil is shipped in trucks to the hydrogen production unit. Shipment of bio-oil would be less expensive compared with the transport of biomass. This is because of the higher energy density of bio-oil compared with biomass. Bio-oil is fractionated into carbohydrate and lignin fractions using water extraction. Carbohydrate fraction is reformed. Hydrogen generated is separated from other products using PSA, Pressure Swing Adsorption. What ought to the control strategy for this process ? Which of the advanced control methods discussed in Chapter 7.0 is applicable for this process ? Is feedback control discussed in Chapter 5.0 sufficient ? What ought to be the control variables ? 5.0. Rice Bran Oil as Feedstock for Hexane Rice-bran lipids can be extracted using hexane solvent from rice bran at a yield of 26%. Rice is milled in order to produce the white ling-grain rice sold in supermarkets. The outer layers of the rice kernel are removed. These outer layers of the rice kernel are removed. These layers include the hull, the germ and the bran. The bran includes the testa, peripcarp, nucellus and aleurone layers. Rice bran becomes rancid and to be discarded in landfills. Parboiled rice bran has higher lipid levels compared with unstabilized rice bran. Rice-bran oil consists largely of saponifiable compounds. What ought to be the control strategy for the hexane extraction unit ? What are the control variables ? 6.0. Ammoxidation of Glycerol to Acrylonitrile Acrylonitirle can be manufactured from glycerol in a catalytic fluidized bed reactor. Ammonia and Oxygen are added to glycerol to form acrylonitrile and water as products. The reaction is as follows; CH3OH-CH3OH- + NH3 + 0.5 O2  CH3CN + H2O -CH3OH Triglycerol + Ammonia + Oxygen  Acrylonitrile + Water The glycerol is reacted in the vapor phase with ammonia and oxygen in the presence of an acid catalyst. The reaction temperature is about 400-500 0C and reaction pressure is ~ 1 – 5 atm. Pressure. The acid catalyst is an oxide of aluminum, nickel, etc. The ammonia/glycerol molar ratio is ~ 1 – 1.2 and oxygen/glycerol ratio is in the range of 0.5 – 10.0. Which is the best physic-chemical system that can be controlled in this process ? What ought to be the control objectives ? Which process variables needs to be measured using sensors and suitable instrumentation ? Which of the process variables are input variables and which are manipulatble ? What are suitable process constraints ? Which is the best control strategy from among feedback, feedforword and the number of control methods discussed in Chapter 7.0. 7.0. Continous Mass Polymerization of Vinyledene Difluoride

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Kal Renganathan Sharma High melting temperatures of PVDF makes the separation process energy intensive. Centrifugation or atomization may result in less energy cost compared with extrusion. Continuous solution free radical processmay be used in order to make PVDF followed by fluidized bed devolatilizer for separation. The temperatures of the devolatilizer is ~ 120- 220 0C. The reactor temperature is about 110 0C and reactor pressure is about 200 atm. What ought to be the control objectives ? Which process variables needs to be measured using sensors and suitable instrumentation ? Which of the process variables are input variables and which are manipulatble ? What are suitable process constraints ? Which is the best control strategy from among feedback, feedforword and the number of control methods discussed in Chapter 7.0. 8.0. Butadiene Monomer Reclaimation from Waste Tires Over 2 billion waste tires are estimated to be stockpiled in the landfills of this country, United States. The number of tires used in cars on the roads in the U.S. is more than one billion. China has 9.83 million passenger vehicles registered in the first eight months of 2011. 292 million automobile tires are discarded every year. In Texas state 32 million tires were thrown away this past year. 5 million tires form a mountain in Odessa and another 809,000 are piled on the side of Genoa Red Bluff Road in Southeast Houston, Texas. Tires thrown in landfills are against the law for the past two decades. Piled up tires collect water and leads to pandemic caused by spread of West Nile virus. A tire mountain in Ciudad Juarez, Mexico, across the Rio Grande from El Paso, TX has gone down some from 7 million tires to 2.5 million tires. Several old tire fires have been reported in Canada and U.S. The energy needed to make 1,3 butadiene monomer is 60,000 BTU per pound. The fuel energy value of tire is ~ 15,000 BTU per pound. Monomer reclamation from waste tires may be more profitable compared to its use as fuel.. Given the cost of generation of butadiene monomer and polybutadiene and tire manufacture good uses for waste tires are needed. High performance rubber asphalt concrete based on stone mosaic asphalt, SMA can be used to resist differential frost along highways. Tireadded latex concrete TALC was developed to include recycle tire rubber as part of concrete. The P-T diagrams for PBd can be constructed from the Tait equation of state. The ceiling temperature of PBd is 585 C at atmopsheric pressure. Butadiene monomer can be recovered by a two step process of swelling and de-vulcanization of cross-linked PBd rubber chains followed by unzipping of the linear PBd chains. The reactor temperature is about 635 0C and reactor pressure is 0.1 atm. NMP, N-methyl2-Pyrrolidone can be used to swell the rubber and acid may be used to unhook the sulfur used for cross-links amongst the PBd chains. What ought to be the control objectives ? Which process variables needs to be measured using sensors and suitable instrumentation ? Which of the process variables are input variables and which are manipulatble ? What are suitable process constraints ? Which is the best control strategy from among feedback, feedforword and the number of control methods discussed in Chapter 7.0. 9.0. Enzymatic Process for Production of Acrylamide Acrylamide can be made by hydrolysis of acrylonitrile with the aid of the enzyme nitrile hydratase. Metabolic pathways of sequential metabolism of nitriles in bacteria are due to nitrole hydratase and amidase enzymes. The genome of M. Brevicollis has

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a sequence that has been found to encode for nitrile hydratase. The gene transcription and translation to rto protein and the signals to effect the enzymatic function is mapped. The process comprises of a reactor and a separator. Which is the best control strategy from among feedback, feedforword and the number of control methods discussed in Chapter 7.0. 10.0. Hydrogen Production by Copper-Chlorine Thermochemical Cycle Thermal energy from nuclear or solar irradiance can be used for production of hydrogen. This is achieved using a hybrid copper-chlorine, CuCl cycle. 125 MT H2/day production rate requires 210 MW of thermal energy and 87.8 MW of electrical energy. There are three major reaction in the process; (i) Electrolysis Reaction; (ii) Hydrolysis of Cuproc Chloride leads to formation of Copper oxychloride and; (iii) Molten Cuprous Chloride is produced by decomposition. (i) 2CuCl + 2HCl + 4H2O  2CuCl2.2H2O (anode) + H2 (cathode) (ii) 2CuCl2 + H2O  CuOCl2 + 2HCl (iii) CuOCl2  0.5O2 + 2CuCl What ought to be the control objectives ? Which process variables needs to be measured using sensors and suitable instrumentation ? Which of the process variables are input variables and which are manipulafigure ? What are suitable process constraints ? Which is the best control strategy from among feedback, feedforword and the number of control methods discussed in Chapter 7.0. 11.0. Ethanol Production by Fermentation In order to deliver bioethanol to a newly proposed city distilleries were built. Starch is hydrolyzed and enzymatically saccharified in a series of agitated vessels. Glucose is sugar and Sacchromyces Cerevisiae is the yeast organism. pH of the vessel can be maintained in order to optimize the metabolism in the yeast organism. Oxygen is added to the vessel. The yeast cells are separated from the substrate and ethanol using a centrifuge. The ethanol is further refined using distillation. The reactor temperature is ~ 68 – 104 0F and pH range is ~ 2.5 – 5.5. Enzyme can catalyze the formation of ethanol. What ought to be the control objectives ? Which process variables needs to be measured using sensors and suitable instrumentation ? Which of the process variables are input variables and which are manipulatable ? What are suitable process constraints ? Which is the best control strategy from among feedback, feedforword and the number of control methods discussed in Chapter 7.0. 12.0. Hydrogenation, Deodourization and Decoloration Vanaspati is a solid and is used in households in India as oil needed in vegetarian cooking. Other similar products are vegetable ghee, hardened industrial oils and partially hydrogenated liquid oil. Well stirred reactors are used for hydrogenation of oils in order to remove the unstaturation in oils and to raise the melting point of the fats and improve the resistance of oils to rancid oxidation. Vacuum tower is used to deodourized the hydrogenated oil. Fuller‟s earth is used as in adsorption towers in order to decolorize the hydrogenated oil. The adsorption isotherm obeyed by the adsorbent is Langmuir isotherm. The color concentration is reduced from 20 to 5 color units in the adsorption tower.

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Kal Renganathan Sharma What ought to be the control objectives ? Which process variables needs to be measured using sensors and suitable instrumentation ? Which of the process variables are input variables and which are manipulatable ? What are suitable process constraints ? Which is the best control strategy from among feedback, feedforword and the number of control methods discussed in Chapter 7.0. 13.0. Adsorption Tower for Magnesium Chloride Sea water contains a number of salts. One example is Magnesium Chloride at 1.3 gm/liter concentration. The NaCl, common salt concentration is 3.6 wt% in sea water. Montmorillonite clay can be used as adsorbent. Sea water feed solution can be made to come in contact in a counter-current manner with monotmorllinite clay. Crystalline Magnesium Chloride can be made from adsorption process. The Langmuir type adsorption has been found to take place for this system. Upon the adsorbent reaching a saturated state the adsorbed Magnesium salts can be desorbed using pure water. A supersaturated solution of Magnesium chloride can be pumped across a crystallizer and Magnesium chloride crystals formed. What ought to be the control objectives for the adsorption tower and crystallizer ? Which is the best control strategy for the adsorption and crystallization unit operations ? What are the control variables and input variables ? Enumerate the process contraints. 14.0. Sugar Crystals Sugarcane from the farms are crushed using a jaw crusher and the sugar cane water is made. The bagasse is separated and sent to a facility as feedstock for enzymatic creation of bioethanol. The sugar water is pumped across a triple effect evaporator. The number of stages used in this step is optimized for minimal total cost. As the number of stages is increased the amount of steam used will increase running the utility bill up. Fewer stages would mean higher capital equipment cost and less utility cost. There exist an optimal number of stages for minimal total cost. The sugarcane water is decolorized in a adsorption tower. Linear isotherm is obeyed by the adsorbent. Upon evaporation sugar is harvested in crystalline form. What ought to be control objevtives for; (i) Crusher; (ii) Decolirizind Adorption tower; (iii) Triple Effect Evaporator and; (iv) Crystallizer. Which is the best control strategy that can be deployed for each of the unit operation. Discuss the process contraints for each unit operation. 15.0. Separation of p-Xylene from its Isomers The method of choice for separation of isomers is adsorption. Zeolite can be used as adsorbent in counter-current adsorption continuous operations. A mixture of para xylene, ortho xylene and metal xylene can be brought in contact in a counter-current manner with zeolite adsorbent. On account of the differences in molecular sizes of the three isomers, the bigger molecule, para xylene is retained by the adsorbent and o-xylene and m-xylene elutes through the tower. The inlet feed solution to the adsorption tower is ~ 9% and the eluted stream has 2000 ppm of p-xylene. The spent adsorbent concentration can be 2.5 weight %. The Langmuir isotherm is applicable for this system. What ought to be the control objectives for this process ? Which of the methods discussed in this chapter and in earlier chapters are best suited for this process ? What are the process constraints ? Which are the input variables and which are the manipulatable variables ?

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16.0. Polymer Composition Replaces Sutures American dental association patented a process (US Patent # 5320886, 1994) to make a polymeric composition that can be used in order to replace sutures. Organofunctional monomers with dimethacrylates or diacrylates are used. Statistics and probability are used to optimize the polymer reaction products. Materials having similar physiochemical characteristics have higher affinity and adhesive bonding between them. A batch polymerization reactor can be used for this purpose. The unreacted monomers and polymer product are separated using a centrifuge. What are the control objectives ? Which is the best control strategy for the polymerization and separation steps ? List the input and manipulatable variables. 17.0. SAN and PMMA Polymer Blend SAN, styrene and acrylonitrile copolymer with high molecular weight and PMMA, polymethacrylate are blended in a co-rotating, non-intermeshing twin screw extruder made by Berstoff, Germany. The system exhibits thermodynamic miscibility for certain compositions of SAN and temperature and pressure of the extruder. Design a feedforward control strategy in order to form a SAN/PMMA blend. How is this strategy different from a IMC, Internal model control strategy. The model for the SAN/PMMA blend can be found in Sharma [2012]. 18.0. Feedback Control A ternary alloy made of silicone rubber, polycarbonate and polybutylene terephthalate is decomposed at high temperature and pressure. Simultaneous measurements using MS,mass spectrometry and FTIR, Fourier Transfrom Infra Red spectrpscopy are used in order to characterize the decomposition products. Develop a feedback control strategy for the decomposition of PC/PBT/silicone rubber ternary alloy. 19.0. Durability of High Peformance Adhesive Joints The interface is an important consideration in adhesive systems. The Tg, glass transition temperature of polyamide used in adhesive systems is 260 0C. Notched coating adhesion tests can be used in order to study the debonding of the high performance adhesive joints. The strain at debonding can be indentied. X-ray photoelectron spectroscopy can be used in order to measure the ageing time and bonding energy. Design a hybrid feedback/feedforward control strategy for the process of making the adhesive joints. 20.0. IMC of High Temperature, Water-Gas Shift Inert Membrane Reactor An accurate model of a commercial scale membrane reactos is developed. A PFR, plug flow reactor with no back-mixing operated at constant temperature is used with no catalyst at high temperature. The feed mixture is pumped across a single tube membrane reactor. The permeate is hydrogen and the retentate is in the shell side of the tube. Ideal gas law is assumed to be applicable to the gas phase system. A first order kinetic model is applicable. The reaction rate constant varies with temperature and pressure and an expression is given to predict the reaction rate constant k as a function of T and P. When the temperature exceeds 600 0C unfavourable conditions can be expected for the formation of hydrogen. Ceramic-ion-transport membranes can be used in order to have hydrogen only as the permeate. An expression for the mass transfer rate can be written in terms of the concentration difference and area

486

Kal Renganathan Sharma across which the mass transfer occurs. The equilibrium conversion is ~ 30-40%. The membrane can be characterized using porosity, pore size distribution etc. A bundle of 7600 tubes can be used. Conversions greater than 90% can be attempted. Discuss the Internal Model Control strategy for this process ? Is a filter needed in order to make the control function “more proper” ?

Chapter 8

PHARMACOKINETIC ANALYSIS        

Single and Two Compartment Models Ethanol in Brain Tissue – Fourth Order Absorption First Order Absorption with Elimination Special Case when klumped = kinfusion Second Order Absorption with Elimination Michaelis and Menten Absorption with Elimination Krebs Cycle and Subcritical Damped Oscillations Periodic Absorption with Elimination

8.1. OVERVIEW Pharmacokinetics comes from the Greek words, pharmacon that means drugs and kinetikos that means setting in motion. The mathematical models, experimental trials and computer simulations that are undertaken to study the change of concentration of drug or other compounds of interest with time in the human physiology is called Pharmacokinetics. Application of pharmacokinetics allows for the processes of liberation, absorption, distribution, metabolism and excretion to be characterized mathematically. The absorption of drug can be affected by 13 different methods. These methods include; (i) sublingual entry (beneath the tongue) (ii) buccal cavity (via the mouth) (iii) gastric entry (through stomach) (iv) IV therapy (through veins) (v) intramuscular therapy (within the muscles) (vi) subcutaneous therapy (beneath the skin) (vii)intradermal therapy (within the dermis) (viii)percutaneous therapy (topical treatment of skin)

488

Kal Renganathan Sharma (ix) inhalation through mouth, nose, pharynx, trachea, bronchi, bronchioles, alveolar sacs, alveoli (x) intrarterial route (introduced to artery) (xi) intrathecal route (introduced to cerebrospinal fluid) (xii)vaginal route (through the vagina) (xiii)intraocular route (through eye) (xiv)splanchnic circulation (absorbed from stomach, intestines, colon and upper rectum)

The change with time of the concentration of the drug can be by three different types i) Slow absorption Type A; ii) maxima and rapid bolus, Type B; iii) constant rate delivery, Type C. Pharmacokinetics studies can be performed by 5 different methods including compartment methods [Cooney, 1976 ] the five different methods are; (i) Non-compartment method (ii) Compartment method (iii) Bio analytical method (iv) Mass spectrometry and (v) Population pharmacokinetic methods. According to Notari [1987] the drug concentration can be denoted by a simple weighted summation of exponential decays. The factors that affect how a particular drug is distributed throughout the anatomy are: i) rate of blood perfusion; ii) permeability of capillary; iii) biological affinity of drug; iv) rate of metabolism of drug and; v) rate of renal extraction. Drugs may bind to proteins sometimes. The distribution volume is restricted. An apparent distribution volume of the drug is defined in such cases [Cooney, 1976]. The kinetics of enzymatic reactions that depleted the drug in the tissues can be expected to obey the Michaelis-Menten kinetics. The elimination of drug is affected by kidneys in a big manner by enzymatic degradation and formation of water soluble drug products. Blood urea nitrogen, BUN, a waste product produced in the liver as the end product of protein metabolism is removed from the blood by the kidneys in the Bowman‟s capsule. BUN is removed along with creatinine, a waste product of creatinine phosphate as energy storing molecule produced largely from muscular breakdown. The kidney comprises of more than a million nephrons. The nephron is composed of a glomerulus, entering and exiting arteriole and a renal tubule. The glomerulus consists of a tuft of 20-40 capillary loops protruding into Bowman’s capsule. Bowman‟s capsule [Cooney, 1976] is a cup-like shaped extension of the renal tubule which is the beginning of the renal tubule. The epithelial layer of Bowman‟s capsule is about 40 nm in thickness and facilitates passage of water into inorganic and organic compounds. The renal tubule has several distinct regions which have different functions such as the proximal convoluted tubule, the loop of Henle, the distal convoluted tubule and the collecting duct that carries the final urine to the renal pelvis and the ureter. Glomerular filtration is the amount of fluid movement from the capillaries into the Bowman‟s capsule. Glomerular filtration rate, GFR, is about 125 ml/min or about 180 liters per day.

Pharmacokinetic Analysis

489

8.2. RENAL CLEARENCE The concept of renal clearance is introduced by performing a mass balance on the drug in the human anatomy‟s apparent distribution volume at transient state. The elimination of drug in urine is seen to be a first-order process

plasma Cdrug

plasma Cdrug 0 e

Frenal t Va

(8.1a)

The term plasma clearance represents all the drug elimination processes of the body. The primary elimination processes are from metabolism and GFR. The secondary processes can be from sweat, bile, respiration and feces. The rate constant for each secondary process is Fj denoted by k j Va A overall rate constant is defined that can be used to account for all the primary and secondary processes of elimination of drug in human anatomy; klumped

1 Va

kj j

Fj

(8.1b)

j

The change in concentration of drug with time can be written as; F plasma t plasma Cdrug

Where, F plasma

plasma Cdrug 0 e

Va

(8.1c)

Fj j

Eq. (8,1c) is an example of a pharmacokinetic model derived from first principles. The Type B response can be explained using the model. The area under the concentration of drug vs. time can be denoted by Area (Eq. (8.1d); Area 0..

0

Cdrug dt

(8.1d)

8.3. SINGLE COMPARTMENT MODELS The drug or metabolite that enters the human anatomy from external sources finds its way to the blood plasma by a process of diffusion from the tissue regions. The concentration of the drug in the plasma starts at zero, some times experiences a lag phase, and then undergoes a phase of increase in concentration of drug with time. A maximum is reached. Then the concentration of the drug begins to decrease with time. The decrease in concentration in the plasma can be attributed to the primary and secondary elimination reactions that are in place

490

Kal Renganathan Sharma

to deplete the drug concentration. Compartment models can be developed to describe the different phases of the dynamics of the concentration of the drug in the plasma. A schematic of a single compartment model with first order absorption with elimination is shown in Figure 8.1. The dose infused in given by Dose, D. A factor,  is introduced. This represents the fraction of dose that is absorbable. The mass balance equation on the concentration of the drug within a volume VA of the human anatomy can be written for Figure 8.1 as follows;

plasma Frenal Cdrug

kinf usion nanatomy

Va

plasma dCdrug

dt

(8.1)

Where kinfusion is the first-order rate constant of the absorption process and anatomy is the amount of drug that is available for absorption.

8.3.1. First Order Absorption with Elimination The first-order absorption of the drug process can be written as;

dnanatomy dt

kinf usion nanatomy

(8.2)

Where kinfusion is the absorption rate constant. Eq. (8.2) can be integrated to obtain; nanatomy D

e

kinf usion t

Obtaining the Laplace transforms of Eq. (8.1) and Eq. (8.2);

Figure 8.1. Single Compartment Model with First Order Absorption with Elimination.

(8.3)

491

Pharmacokinetic Analysis kinf usion nanatomy ( s )

plasma Frenal Cdrug (s )

nanatomy ( s )

kinf usion nanatomy ( s )

D

plasma sVa Cdrug

0

(8.4)

Eliminating nanatomy(s); plasma Cdrug (s )

kinf usion nanatomy ( s ) sVa

kinf usion D

Frenal

Va s

klumped

s

(8.5) kinf usion

Where, Frenal is given in terms of a klumped that serves as an overall rate constant that can be used to account for all of the primary and secondary processes of the elimination of the drug in the human anatomy. Thus Frenal = (klumpedVa). Eq. (8.5) may be inverted in the time domain [Sharma, 20.10] as follows;.

kinf usion

D Va

plasma Cdrug

kinf usion

e

klumped

klumped t

e

kinf usion t

(8.6)

Eq. (8.6) may be written in dimensionless form as follows;

u

Where,

plasma Cdrug

1

D Va

1

e

klumped t

e

kinf usion t

(8.6a)

klumped kinf usion

β can be seen to be the ratio of the overall elimination rate constant with the absorption rate constant. This is valid for, kinf usion

klumped . From Eq. (8.6) it can be seen that the

concentration drug as a function of time varies inversely with the apparent volume of distribution of the drug within the human anatomy and directly proportional to the amount of drug that is absorbable and depends on the first order rate constants of absorption and elimination. It can be seen that Eq. (8.6) exhibits a maxima. This occurs at; 1 m

kinf usion

klumped

ln

kinf usion klumped

(8.7)

The corresponding maximum concentration can be given by; klumped plasma ( m) Cdrug

D Va

kinf usion klumped

klumped kinf usion

(8.8)

492

Kal Renganathan Sharma The maxima vanish for large values of rate constant, kinfusion. The concentration will reach

the asymptotic maximum in these cases. Eqs. (8.7-8.8) are valid only when klumped

kinf usion .

Eq. (8.6a) is plotted in Figure 8.2 for different ratios of the rate constants, . It can be seen from Figure 8.2 that for very small values of  the dimensionless concentration does not exhibit maxima. It reaches the asymptic maximum limit of 1 at large time. For small values of  the maxima is sharper. A right skew can be seen in these decay curves. For the special case when the overall rate constant of the primary and secondary elimination processes is equal to the rate constant of absorption the following analysis would be applicable. Eq. (8.1) can be written when k

kinf usion

klumped as;

(k D )e

plasma (Va k )Cdrug

kt

plasma dCdrug

dt

plasma Cdrug

kt ; u

Let,

Va

(8.9)

(8.10)

D Va

Eq. (8.9) becomes; e

u

du d

(8.11)

Obtaining the Laplace transforms of Eq. (8.11) recognizing the initial concentration is u0 and the transformed variable can be solved for as; _

u

u0

1

(s 1)

( s 1)2

(8.12)

Obtaining the inverse of the Laplace transformed dimensionless concentration;

u u0 e

e

(8.13)

The solution for the dimensionless concentration given by Eq. (8.13) is shown in Figure 8.2 for the case of 0. initial concentrations. A maxima can be seen in the concentration vs. time graphs. Eq. (8.13) in the dimensional form can be written as; plasma Cdrug

D kte Va

kt

(8.14)

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Pharmacokinetic Analysis

Figure 8.2. Dimensionless Concentration u as a Function of Time for Systems that Obey first Order Kinetics with Elimination (kinfusion = 1 min-1;

klumped kinf usion

).

Eq. (8.14) can be integrated and given in terms of area under the concentration of the drug vs. time graph as; Area 0..

D Va klumped

D F plasma

(8.15)

Example 8.1 Ethanol in Brain Tissue Pharmacokinetic modeling of ethanol plays a salient role on brain‟s response to ethanol. Ethanol is a naturally produced drug used by humans for thousands of years because of its psychoactive properties. It is beneficial when used in moderation. Excessive use of ethanol can be devastating. Brain is a high blood flow, small water volume organ and ethanol readily crosses the blood-brain barrier. The ethanol gets eliminated by an oxidation reaction as follows:

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Kal Renganathan Sharma

Figure 8.3. Drug Concentration as a Function of Time for the Special Case when k = klumped = kinfusion.

C2 H 5OH

3O2

2CO2 3H 2 0

(8.16)

The stoichiometry of the reaction suggests that the rates of reaction are related by;

rA

rO2

1

3

(8.17)

The kinetics of the reaction can be expected to be of the form;

rA

kinf usion nanatomy nO3 2

(8.18)

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Pharmacokinetic Analysis

Lumping all the rate effects into the alcohol concentration assume that the rate th expression can be given by a 4 order rate expression such as;

dnanatomy

4 kinf usion nanatomy

rA

(8.19)

dt

Let the amount of alchohol ingested by given by D moles. Develop a single-compartment pharmacokinetic model for this 4th order absorption with elimination of ethanol. Plot the alchohol concentration as a function of time. Does it undergo maxima? Calculate the amount of ethanol in the urine. It appears that ethanol infusion in the brain tissue is a fourth order process of absorption. Eq. (8.19) is applicable. The elimination steps are also included. Integration of Eq. (8.19) realizing the initial condition at t = 0, nanatomy = (D) would result in; 1 3 D

3

kinf usion t

1

(8.20)

3 3nanatomy

Eq. (8.20) can also be written as; 3

D 1 3 D kinf udion t

nanatomy

1 3

(8.21)

Eq. (8.21) can be expressed as a Newton‟s generalized binomial series as follows;

nanatomy

14 33

D

D kinf usion

2

3

D 1

D kinf udion t

6

kinf usion t

1 3

2

2!

.......

(8.22)

....

(8.23)

Obtaning the Laplace transform of Eq. (8.22)

nanatomy ( s )

D

1 s

3

s2

D

6

2 kinf usion

9s 3

The mass balance of ethanol in the plasma distributed over a volume per the single compartment model shown in Figure 8.1 as given by Eq. (8.1) is applicable and can be written as in the Laplace domain as; kinf usion nanatomy (s )

plasma (s ) Frenal Cdrug

plasma sVa Cdrug

0

(8.24)

The concentration of drug in plasma can be solved for in the Laplace domain by elimination of nanatomy(s) between Eq. (8.23) and Eq. (8.24) as;

496

plasma Cdrug (s )

Kal Renganathan Sharma kinf usion nanatomy (s ) Va s

klumped

kinf usion Va s

D

klumped

1 s

D

3

kinf usion

2

D

s2

6

2 kinf usion

9s 3

(8.25)

Term by term in Eq. (8.25) can be inverted from the Laplace domain into the time domain as follows;

plasma Cdrug (t )

kinf usion

D

klumped Va

3

1 e

klumped t

D kinf usion klumped

1 e

klumped t

t klumped

.......

(8.26)

Eq. (8.26) is plotted in Figure 8.3 for a certain set of rate constants and dosage and factor of absorption. The set of parameters used to generate the concentration of drug in the plasma is given below in Table 8.1.

Figure 8.4. Concentration of Drug in Plasma with Fourth Order Absorption with Elimination.

497

Pharmacokinetic Analysis Table 8.1. Parameters used in Figure 8.3 Ethanol in Brain kinfusion 400 klumped 2 Va 260 D 0.2 0.8 

-1

hr -1 hr liter moles

8.3.2. Second-Order Absorption with Elimination A mass balance on the concentration of drug within the human anatomy for the case of second order absorption with elimination can be written for Figure 8.1;

2 k " nanatomy

plasma Frenal Cdrug

Va

plasma dCdrug

dt

(8.27)

Where k” is the second order rate constant of the absorption process and nanatomy is the amount of drug that is available for absorption. The second order absorption of the drug process can be described by;

dnanatomy

2 k " nanatomy

dt

(8.28)

The solution to Eq. (8.28) can be written as;

1 fDose

k "t

1

(8.29)

nanatomy

Eq. (8.29) and Eq. (8.27) can be combined and the concentration of drug as a function of time can be solved for from the first order ODE, ordinary differential equation and by use of particular integral as; plasma Cdrug

e

1

klumped t

1

k "t fDose

2

(8.30)

The initial condition is; plasma Cdrug 0

0, t 0

(8.31)

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Kal Renganathan Sharma

Figure 8.5. Second Order Absorption and Elimination -

Frenal Va

0.33;

k" fDose

5 .

Eq. (8.30.) can be seen to exhibit maxima. The solution for the maxima needs numerical methods as the resulting equation is transcendental. Eq. (8.30) can be seen to be interplay of the rate of absorption and rate of excretion. When the second order absorption processes are rapid and excretion is slow, the drug tends to accumulate in the blood plasma. When the rate of excretion is rapid the drug concentration tends to drop off rapidly.

8.3.3. Michaelis-Menten Absorption with Elimination A mass balance on the concentration of drug within the human anatomy for the case where the kinetics of absorption is in obeyance of Michaelis-Menten Kinetics with elimination can be written for Figure 8.1 as; kC E 0 nanatomy Va C M nanatomy

plasma Va klumped Cdrug

Va

plasma dCdrug

dt

(8.32)

Let nanatomy be the amount of drug that is available for absorption. The absorption of the drug process can be described by Michaelis-Menten kinetics;

499

Pharmacokinetic Analysis dnanatomy

kVa C E 0 nanatomy

dt

Va CM

(8.33)

nanatomy

Where k is the infusion Michaelis-Menten rate constant, CE0 is the total enzyme concentration and CM is the rate constant. It can be seen that (Levenspiel, [1999]) the Michaelis-Menten kinetics becomes independent of concentration at high drug concentration and becomes zeroth order and at the low concentration limit reverts to a simple first order rate expression. Integration of Eq. (6.54) can be seen to be [Michaelis and Menten 1913];

CM ln

nanatomy

fDose Va

fDose

nanatomy Va

kC E 0 t

(8.34)

It can be seen that Eq. (8.34) is in a form that is not readily usable in terms of a one-toone mapping between the independent variable t and dependent variable nanatomy. In order to combine Eq. (8.34) with Eq. (8.32) and then solved for, for the concentration of drug in the plasma, a more usable form of Eq. (8.34) is sought. This can be done by realizing that any arbitrary function can be represented using the Taylor series. Taylor series representation of any arbitrary function is an infinite series containing derivatives of the arbitrary function about a particular point. Prior to obtaining the Taylor series Eq. (8.33) is made dimensionless as follows [10-11]; nanatomy

u

fDose

;

k3u E 0t ; u E 0

CE 0Va fDose

; uM

CM Va fDose

(8.35)

Eq. (8.33) becomes;

du d

u u uM

(8.36)

Taylor series in terms of derivatives of u evaluated at the point =0 can be written as follows; 2

u

From the initial condition, From Eq. (8.36),

u (0)

u '(0)

2!

3

u "(0)

u(0) = 1

u '(0)

1 1 uM

3!

u "'(0) .........

(8.37) (8.38) (8.39)

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Kal Renganathan Sharma

The initial value of the second derivative of the dimensionless concentration, u” can be seen to be; d 2u d

u u M u ' uu '

2

uM

u

uM

2

1

uM

0

(8.40)

3

The initial value of the third derivative of the dimensionless concentration, u”‟ can be seen to be; d 3u d

u

uM

3

2

u M u " 2u M u '(u uM

u

2 uM

uM )

4

2u M

uM

0

1

5

(8.41)

Plugging Eqs. (8.39-8.41) in Eq. (8.37);

u 1

uM uM

1 2! u M

3

2

1

3

2 uM

3! u M

2u M 1

5

......

(8.42)

Eq. (8.42) and Eq. (8.34) are sketched for a particular value of uM =11, in Figure 8.6. It 25 fDose , the Taylor series expression evaluated near the origin can be seen that for times t < kC E 0Va up to the third derivative is a reasonable representation of the integrated solution given in Eq. (8.34). More terms in the Taylor series expression can be added to suit the application and the apparent volume, dosage, enzyme concentration, Michaelis constant and the desired accuracy level needed as shown above.

Figure 8.6. Michaelis-Menten Kinetics from the Integrated and Taylor Series Expressions.

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Pharmacokinetic Analysis

Eq. (8.37) and Eq. (8.34) can be combined and the concentration of drug as a function of time can be solved for from the resulting equation by the method of Laplace transforms as follows. The combined equation is also made dimensionless; u

klumped

uM u

ku E 0

plasma dudrug

plasma udrug

(8.43)

d

Combining Eq. (8.42) and Eq. (8.43); 1 uM

2

uM

1

uM

1

3

2 uM

2 uM

2u M 1

klumped

......

5

ku E 0

plasma udrug

plasma dudrug

(8.44)

d

Obtaining the Laplace transform of the governing equation for dimensionless drug concentration in the compartment (plasma), Eq. (8.44); 1 s (u M

2 uM

uM s 2 uM

1)

1

3

2u M

2s 3 u M

1

5

klumped

......

ku E 0

_

u

_

(8.45)

su 0

The transformed expression for dimensionless drug concentration in the compartment can be seen to be; 1

klumped   s (u M  1)  s   ku E 0  



uM

klumped  3 s 2  u M  1  s   ku E 0  



u

2 M

 2u M



klumped  5 2s 3  u M  1  s   ku E 0  

 ......  u (8.46) _

It can be seen that the inversion for each term in the infinite series is readily available from the tables [Mickley, Sherwood and Reid, 1957]. Thus a non-linear differential equation was transformed using Taylor series and some manipulations into an equation with closed form analytical solution. The term by term inversion of Eq. (8.46) can be looked from the Tables as; 2 klumped

klumped ku E 0

1 e (u M

ku E 0

1)

uM

k

k

klumped

klumped uM

klumped

1 e 1

3

k

...... u

(8.47)

The dimensionless drug concentration in the compartment or plasma is shown in Figure 8.7. It can be seen from Figure 8.7 that the dimensionless drug concentration in the compartment goes through a maxima. The curve is convex throughout the absorption and elimination processes. The drug gets completely depleted after a said time. The curve is

502

Kal Renganathan Sharma

asymmetrical with a right skew. The constants used to construct Figure 8.7 using Microsoft MS Excel Spreadsheet were; uM k

11; klumped 1sec 1; u E 0

2sec

1

4

8.4. ANALYSIS OF SIMPLE REACTIONS IN CIRCLE The mathematical model predictions for drug concentration as discussed above depend on the nature of kinetics of absorption. It can be simple zeroth order, first order, second order, fractional order any order n. It can also be reversible in nature. It can obey Michaelis-Menten kinetics. Sometimes in the absorption process the Krebs cycle [1953] may be encountered. Reactions such as these can be represented by a scheme of reactions in circle (Sharma, [20..10]). The essential steps in the Krebs cycle are formation of: i) A. Oxalic Acid; ii) B. Citric Acid; iii) C. Isocitric Acid; iv) D. -Ketoglutaric Acid; v) E. Succyl Coenzyme A; vi) F. Succinic Acid; vii) G. Fumaric Acid; viii) H. Maleic Acid. There are other sets of reactions in metabolic pathways that can be represented by a scheme of reactions in circle. Systems of reactions in series and reactions in parallel have been introduced [Levenspiel, 1999]. Consider a system of reactions in circle: i) System of 3 Reactants in Circle; ii) System of 4 Reactants in Circle; iii) System of 8 Reactants in Circle such as in Krebs Cycle and; iv) General Case. A scheme of reactants in circle is shown in Figure 8.8.

Figure 8.7. Michaelis-Menten Absorption and Elimination.

Pharmacokinetic Analysis

503

(i) 3 Reactions in Circle The simple first order irreversible rate expressions for (i) 3 Reactants in Circle can be written as; dC A dt dC B dt

dC C dt

k1C A

k3CC

(8.48)

k2C B

k1C A

(8.49)

k3CC

k3C B

(8.50.)

Where, CA, CB and CC are the concentrations of the reactants A, B and C at any instant in time, t. Let the initial concentrations of reactants A, B and C be given by CA0, CB0 =0 and CC0=0. The Laplace transforms of Eqs. (6.68-6.71) are obtained as; _

s

_

k1 C A

C A0 _

s

k2 C B

s

k3 CC

_

k3 CC

(8.51)

_

k2 C A

(8.52)

_

k2 CB

Figure 8.8. Simple Reactions in Circle Representation of Krebs Cycle.

(8.53)

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Kal Renganathan Sharma

Eliminating CB and CC between Eqs. (6.72-6.74) the transformed expression for the instantaneous concentration of reactant A can be written as; _ _

s

k1 C A

_

CA

Or,

k3 k2 k1 C A

C A0

s

C A0 s s s

2

s k1

k2

k3 s

k3 s k3

(8.54)

k2

k2

k1k2

(8.55) k1k3

k2 k3

The inversion of Eq. (6.76) can be obtained by use of the residue theorem. The 3 simple poles can be recognized in Eq. (6.76). Further it can be realized that when the poles are complex, subcritical damped oscillations can be expected in the concentration of the reactant. This is when the quadratic, b 2

0 . This can happen when;

4ac

k3

k2

k3

Or,

Or,

k1

k2

k3

2

k1 k2

4k2 k1 0

(8.56)

2 k1k2

(8.57)

k1

2

(8.58)

This expression is symmetrical with respect to reactants A, B and C. When the relation holds, i.e., when one reaction rate constant is less than the square of the sum of the square root of the rate constants of the other two reactions, the subcritical damped oscillations can be expected in the reactant concentration.

(ii) 4 Reactions in Circle The equivalent Laplace transformed expression for concentration of reactant A for a system of 4 reactions in circle assuming that all the reactions in the cycle obey simple, first order kinetics can be derived as; _

CA

C A0 s s s3

s 2 k1

k2

k3

k4

s (k1k2

k1k3

k3 s

k2 k3

k1k4

k2 s

(8.59)

k1

k2 k4

k3k4 ) k1k2 k3

k1k2 k4

k1k3k4

k2 k3k4

The conditions where the concentration can be expected to exhibit subcritical damped oscillations when the roots of the following equation becomes complex; s3

Where,

s2

k1

s

k2

k3

0

(8.60)

k4

(8.61)

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Pharmacokinetic Analysis k2 k3

k1k3

k1k2 k3

k1k2 k1k2 k4

k1k4 k1k3k4

k4 k3

k1k4

k2k3k4

(8.62) (8.63)

It can be seen that  is the sum of all the 4 reaction rate constants,  is the sum of product of all possible pairs of the reaction rate constants and  is the sum of product of All possible triple products of rate constants in the system of reactions in circle. Eq. (8.60) can be converted to the depressed cubic equation by use of the substitution, x s

(8.64)

3

This method was developed in the Renaissance period (Ans Magna, [1501]). The depressed cubic without the quadratic term will then be; 2

x3

3

2 3 27

x

2

Let,

3

2 3 27

B;

3

3

0

(8.65)

C

(8.66)

Then, Eq. (6.86) becomes; x3

Bx C

0

(8.67)

The complex roots to Eq. (6.88) shall occur when D > 0, where,

D

B3 27

C2 4

(8.68)

Thus the conditions when subcritical damped oscillations can be expected for a system of 4 reactions in circle are derived.

(iii) General Case of n Reactions in Circle For the general case, of which the Krebs cycle with 8 reactions in circle is a particular case can be obtained by extension of the expressions derived for 3 reactions in circle and 4 reactions in circle. Aliter to this would be the method of Eigen values and Eigen vectors. The cases when  is imaginary is when the concentration of the species will exhibit subcritical damped oscillations are given by the characteristic equation [Varma and Morbidelli, 1997];

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Kal Renganathan Sharma

Det K- I

0

(8.69)

The size of the K matrix depends on the number of reactions in circle. For n reactions in circle K would be a n x n matrix. For the case of Krebs cycle it would be a 8 x 8 matrix. Upon expansion an 8th order polynomial equation in  arises. Eight roots of the polynomial exist. Even if all the values in the characteristic matrix are real some roots may be complex. When complex roots occur they appear in pairs. The roots of the polynomial are called eigenvalues of the characteristic matrix. The polynomial equation is called the Eigen value equation.

8.5. SUBCRITICAL DAMPED OSCILLATIONS As was discussed in the previous section, the concentration of the drug during absorption on account of kinetics such as the reactions in circle can undergo subcritical damped oscillations. In such cases, how can the absorption with elimination process be modeled? The solution for the dosage drug concentration upon absorption that obeys a certain type of kinetics that results in a subcritical damped oscillations can be given by;

fDose e Va

nanatomy

kinf usion t

(2 Cos

kt

)

(8.70)

A mass balance on the concentration of drug within the human anatomy for the case of kinetics of absorption resulting in subcritical damped oscillation can be written for Figure 6.4 as; kinf usion ( fDose)e

kinf usion t

(2 Cos(

Va

k t ))

plasma Va klumped Cdrug

Va

plasma dCdrug

dt

(8.71)

Eq. (6.92) is the governing equation for concentration of drug in the single compartment. Eq. (6.92) is made dimensionless by the following substitutions;

u

plasma Cdrug

;

klumped t ; *

fDose Va

k

klumped

(8.72)

Plugging Eq. (8.72) in Eq. (8.71), Eq. (8.71) becomes; du d

kinf usion klumped

e

(2 Cos ( * )) u

(8.73)

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Pharmacokinetic Analysis

Figure 8.9. Characteristic Matrix for System of 8 Reactions in Circle.

The concentration of drug as a function of time can be solved for by the method of Laplace transforms as follows. Obtaining the Laplace transforms of Eq. (6.94); _

2

kinf usion

u

klumped

s 1

( s 1) 2

s 1

2

*2

(8.74)

Obtaining the inverse of the transformed expression by using the convolution property of Eq. (6.95);

u

e

kinf usion klumped

Sin ( * ) *

(8.75)

The solution for dimensionless concentration of the drug in the single compartment for different values of rate constants and dimensionless frequency are shown in Figure 8.10 – Figure 8.14. The drug profile reaches a maximum and drops to zero concentration after a said time. The fluctuations in concentration depend on the dimensionless frequency resulting from the subcritical damped oscillations during absorption. At low frequencies (* < 1.8) the fluctuations are absent. As the frequency is increased the fluctuations in concentration are pronounced. The frequency of fluctuations was found to increase with increase in frequency of oscillations during absorption. At high dimensionless frequencies a “saw-tooth” pattern could be seen in the dimensionless concentration. At even higher dimensionless frequencies the fluctuations in concentration again vanish.

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Kal Renganathan Sharma

Figure 8.10. Dimensionless Concentration of Drug in Compartment *=/klumped = 6;

Figure 8.11. Dimensionless Concentration of Drug in Compartment *=/klumped=2;

kinf usion klumped

kinf usion klumped

0.5 .

0.5 .

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Pharmacokinetic Analysis

Figure 8.12 Dimensionless Concentration of Drug in Compartment *=/klumped = 1.8;

kinf usion klumped

Figure 8.13. Dimensionless Concentration of Drug in Compartment, *= 0.5, kinfusion/klumped =0.4.

0.5 .

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Kal Renganathan Sharma

Figure 8.14. Dimensionless Concentration of Drug in Compartment *=/klumped=25;

kinf usion klumped

0.5 .

8.6. TWO COMPARTMENT MODELS Two and three compartment models are used when need to describe complex drug profiles. Such need arises especially when equilibrium between central compartment to describe the concentration of drug in blood and a peripheral tissue compartment is not rapid. A two-compartment model to model the absorption process with elimination is shown in Figure 8.15. The concentration that has diffused to the tissue region in the human anatomy is also accounted for in addition to the concentration of drug in the blood plasma. A bolus is administered intravenously. A mass balance on the concentration of drug within the human anatomy in the blood plasma and tissue compartments can be written for Figure 8.15 as follows;

plasma Vp Cdrug k pt

Frenal

plasma Vp Cdrug k pt

tissue ktpV T Cdrug

tissue ktpV T Cdrug

VT

Vp

plasma dCdrug

tissue dCdrug

dt

dt

(8.76)

(8.77)

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Pharmacokinetic Analysis

Figure 8.15. Two Compartment Models.

Where Vp and VT are the apparent distribution volumes of the blood plasma and tissue compartments. The initial concentrations of the drug in the compartments are; Dose VP

plasma Cdrug tissue Cdrug

(8.78)

0

Differentiating Eq. (6.98) with respect to time and eliminating the concentration of the drug in the tissue from Eq. (6.10..0) the governing equation for the concentration of drug in the blood plasma compartment can be written as; plasma d 2Cdrug

dt

2



 k pt  Frenal



plasma dCdrug

dt





plasma  ktp  k pt Cdrug 

ktp Dose Vp

0

(8.79)

Eq. (6.10.1) is an ordinary differential equation of the second order with constant coefficients. This can be solved for by obtaining the roots of the complementary function and adding a particular solution. The solution to Eq. (6.10.1) can be written after use of the initial condition can be written as; plasma Cdrug

Dose e Va

t

c2 e

t

e

t

The concentration of the drug in the tissue can be written as;

(8.80)

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Kal Renganathan Sharma

Dose (1 e t ) c2 e Va

tissue Cdrug

Where,

k pt

Frenal

k pt

Frenal

2

t

e

t

(8.81)

4 k pt

k tp

4 k pt

k tp

2 k pt

Frenal

k pt

Frenal

2

2

(8.82)

(8.83)

The solution of the integration constant needs another time condition in addition to the initial conditions given by Eq. (6.10.0) and by mass balance at any given instant in time the initial concentration of the drug in the plasma is the total of the concentration of the drug in the blood compartment plus the concentration of the drug in the tissue compartment. The 4th constraint can be that the initial rate of reaction in the tissue compartment is zero. i.e.

Dose Va

0

c2

Then,

Eq. (6.10.7) is valid only when,

c2

Dose VP

(8.84)

(8.85)

.

8.7. COMPUTER SIMULATION With the advent of personal computers, the pharmacokinetic models are implemented on computers. Both linear and nonlinear pharmacokinetic models can be simulated in the computer. This is especially the case when drug concentration throughout the body or a particular location is high. The possible reason for this situation is the capacity of a biochemical process to reduce the concentration of the drug becomes saturated. MichaelisMenten kinetics is used to capture the non-linear nature of the system. This involves mixtures of zero-order and first order kinetics. Experimental methods are deployed to collect data on the change in the concentration of drug with time from a patient who has been injected a particular dose of a drug. This is followed by interpreting the data by analysis. Analysis of data involves plotting the points of concentration of drug in a logarithmic graph. The slope and intercept of the best-fit, linear, regression line to the data can be used to obtain the rate constant and the initial concentration of the drug. These constants are used in the compartment models to describe the drug‟s time course for additional patients and dosing regimes. Experimental methods to study drug profiles affected by Michaelis-Menten kinetics are similar to those used in standard compartment models. The drug profiles are usually non-

Pharmacokinetic Analysis

513

linear. At high concentrations the drug concentration is linear. This is so, because the drug is eliminated at a maximal constant rate by a zero-order process. The data line then begins to curve in an asymptotic fashion with time until the drug concentration drops to a point where the rate process becomes proportional to the drug concentration via a first order process. Nonlinear pharmacokinetics can be used to describe solvation of the therapeutic ingredient from a drug formulation as well as metabolism and elimination processes. Toxicological events related to threshold dosing can be described using nonlinear pharmacokinetics. Single, two and three compartment pharmacokinetic models require in vivo blood data to obtain rate constants and other relevant parameters that are used to describe drug profiles. Further what may work for one drug may not be suitable for another drug. Blood profile data need be generated for each drug under scrutiny. In vivo state of a spectrum of drugs without experimental blood samples from animal testing cannot be predicted accurately using such models. Physiological pharmacokinetic models have been developed. These integrate the basic physiology and anatomy with drug distribution and disposition. The compartments used correspond to anatomic entities such as GI tract, liver, lungs, ocular, buttocks, etc, that are connected by passage of blood. However, a large body of physiological and physiochemical data is employed that is drug aspecific. The rate processes are lumped together in the physiological models. Computer systems have been used in pharmacokinetics to provide easy solutions to complex pharmacokinetic equations and modeling of pharmacokinetic processes. Other uses of computer in pharmacokinetics include: i) statistical design of experiments; ii) data manipulation; iii) graphical representation of data; iv) projection of drug action and; v) preparation of written reports or documents. Pharmacokinetic models are described by systems of differential equations. Computer systems and programming languages have been developed that are more amenable for the solution of differential equations. Graphics-oriented model development computer programs are designed for development of multi-compartment linear and nonlinear pharmacokinetic models. The user is allowed to interactively draw compartments and then link them with other iconic elements to develop integrated flow pathways using pre-defined symbols. The user assigns certain parameters and equations relating the parameters to the compartments and flow pathways and then the model development program generates the differential equations and interpretable code to reflect the integrated system in a computer-readable format. The resulting model can be used to simulate the system under scrutiny, when input values for parameters corresponding to the underlying equations of the model such as drug dose, etc are used. Tools are developed to implement pharmacokinetic models. However, the current state of the art does not permit predictability of the pharmacokinetic state of extravascularly administered drugs in a mammal from in vitro cell, tissue or compound structure-activity relationship SAR/QSAR data. The predictability is poor when attempting to predict absorption of drug in one mammal from data derived from a second mammal. Different approaches to predict oral administration and fraction dose absorbed are presented in the literature [Grass, 2003, Lvet-Trafit, 1996]. There are lacunae in these models as they make assumptions that limit the scope of prediction to a few specific compounds. These collection of compounds possess variable ranges of dosing requirements and of permeability, solubility, dissolution rates and transport mechanism properties. Other deficiencies include use of drugspecific parameters and values in pharmacokinetic models that limit the predictive capability

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Kal Renganathan Sharma

of the models. Generation of rules that may be universally applicable to drug disposition in a complex physiological system such as GI, gastrointestinal tract. Bioavailability of the drug includes the product price, patient compliance, ease of administration. Failure to identify promising, problematic drug candidates during the discovery and pre-clinical stages of drug development is a significant consequence of problems with drug bioavailability. There is a need to develop a comprehensive, physiologically-based pharmacokinetic model and computer system capable of predicting drug bioavailability and variability in humans that utilizes relatively straightforward input parameters. Computer based biopharmaceutical tools are needed for the medical community that encounters new therapeutic alternates and use of high throughput drug screening for selecting drug candidates. Lion Bioscience [Grass, 2003] has patented a pharmacokinetic-based design and selection PK tool. The tool can be used to predict absorption of a compound in a mammalian system of interest. The PK tool consists of: i) input/output system; ii) physiologic-based simulation model of one or more segments of a mammalian system of interest with physiological barriers to absorption based on route of administration; iii) simulation engine having a differential equation solver and a control statement module. The structure of the PK tool is shown in Figure 6.17. The PK tool is a multi-compartment mathematical model. Linked components include differential equations for one or more fluid transport, fluid absorption, mass transit, mass solvation, mass solubility and mass absorption for one or more segments of the human anatomy, input parameter values for the differential equations corresponding to physiological parameters and selectively optimized adjustable parameters for one or more segments of human anatomy and control statement rules for one or more transit, absorption, permeability, solubility, solvation, concentration and mathematical error correction for one or more segments of the human anatomy. The dose, permeability and solubility data of a drug or compound is received by the input/output system. Absorption profile for the compound is generated by application of physiological based simulation models. The PK tool also has a database that includes physiological based models, simulation model parameters, differential equations for one or more of fluid transport, fluid absorption, mass transport, mass solvation, mass absorption for different parts of the human anatomy, initial parameter values for the differential equations, optimized adjustable parameters, regional correlation parameters, control statement rules for transport, absorption, permeation, solvation, mathematical error corrections for different parts of human anatomy. The database also has a compartment-flow data structure that is portable into and readable by a simulation engine for calculation of rate of absorption, extent of absorption, concentration of a compound at a sampling site across physiological barriers in different parts of the human anatomy as a function of time. The PK tool can be used to predict accurately one or more in vivo pharmacokinetic parameters of a compound in human anatomy. The method uses a curve-fitting algorithm to obtain the fit of the model with one or more input variables. Then adjustable parameters are generated. These steps are repeated until the adjustable parameters are optimized. An example of simulation engine is STELLA® program from high performance systems Inc. It is an interpretive program that can use three different numerical schemes to evaluate differential equations: i) Euler‟s Method; ii) Runge-Kutta 2 or; iii) Runge-Kutta 4. The program KINETICATM is another differential equation solving program that can evaluate the equations of the model. By translation of the model from a STELLA® readable format to

Pharmacokinetic Analysis

515

KINETICATM readable format, physiological simulations can be constructed using KINETICATM which has various fitting algorithms. The PK tools.

8.8. SUMMARY The mathematical models, experimental trials and computer simulations that are undertaken to study the change of concentration of drug or other compounds of interest with time in the human physiology is called Pharmacokinetics. Application of pharmacokinetics allows for the processes of liberation, absorption, distribution, metabolism and excretion to be characterized mathematically. The absorption of drug can be affected by 14 different methods. The change with time of the concentration of the drug can be by three different types as shown in Figure 6.1: i) Slow absorption (A); ii) maxima and rapid bolus (B); iii) constant rate delivery (C). Pharmacokinetics studies can be performed by 5 different methods including compartment methods. The five different methods are non-compartment method, compartment method, bio analytical method, and mass spectrometry and population pharmacokinetic methods. The factors that affect how a particular drug is distributed throughout the anatomy are: i) rate of blood perfusion; ii) permeability of capillary; iii) biological affinity of drug; iv) rate of metabolism of drug and; v) rate of renal extraction. Drugs may bind to proteins sometimes. The distribution volume is restricted. The kinetics of enzymatic reactions that depleted the drug in the tissues can be expected to obey the Michaelis-Menten kinetics. The elimination of drug is affected by kidneys in a big manner by enzymatic degradation and formation of water soluble drug products. Blood urea nitrogen, BUN, a waste product produced in the liver as the end product of protein metabolism is removed from the blood by the kidneys in the Bowman‟s capsule along with creatinine, a waste product of creatinine phosphate as energy storing molecule produced largely from muscle breakdown. The kidney comprises of more than a million nephrons. The nephron is composed of a glomerulus, entering and exiting arteriole and a renal tubule. The glomerulus consists of a tuft of 20-40 capillary loops protruding into Bowman‟s capsule. The renal tubule has several distinct regions which have different functions such as the proximal convoluted tubule, the loop of Henle, the distal convoluted tubule and the collecting duct that carries the final urine to the renal pelvis and the ureter. Glomerular filtration is the amount of fluid movement from the capillaries into the Bowman‟s capsule. Glomerular filtration rate, GFR, is about 125 ml/min or about 180 liters per day. The concept of renal clearance is introduced by performing a mass balance on the drug in the human anatomy‟s apparent distribution volume at transient state. The elimination of drug in urine is seen to be a first-order process (Eq. 7.1a). The term plasma clearance represents all the drug elimination processes of the body. The primary elimination processes are from metabolism and GFR. The secondary processes can be from sweat, bile, respiration and feces. The rate constant for each secondary process is denoted by An overall rate constant is defined that can be used to account for all the primary and secondary processes of elimination of drug in human anatomy (Eq. 7.1b). Eq. (7.1) is an example of a pharmacokinetic model derived from first principles. Single compartment models were developed for:

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Kal Renganathan Sharma i)

firstrepresents the fraction of dose that is absorbable. The model equation is solved for by the method of Laplace transforms. (Eq. 7.5). The concentration of the drug as a function of time varies inversely proportional to the apparent distribution volume of the drug within the human anatomy and directly proportional to the amount of drug that is absorbable and depends on the first order rate constants of absorption and elimination. The time where the concentration reaches a maximum is identified (Eq. 7.7) and the corresponding maximum concentration given by Eq. (7.6). The dimensionless concentration of the drug in the plasma as a function of time for various values of  is given in Figure 7.2. The maxima vanish at small values of . A right tail is seen in the curve.  is the ratio of the lumped rate constant of excretionary processes to the rate constant of absorption. A separate model solution for the special case when the overall rate constant of the primary and secondary elimination processes is equal to the rate constant of absorption (Eq. 7.13) was developed. ii) fourth order absorption with elimination. The absorption of ethanol in brain tissue with elimination was discussed in Example 7.1. Newton‟s generalized binomial series expansion of the negative 0.33 power ponentiated term was used in order to obtain a closed form analytical solution. The model solution is given by Eq. (7.26). The concentration of ethanol in plasma was plotted in Figure 7.4 for a certain set of rate constants. (iii) second order absorption with elimination. The model solution was obtained by the method of particular integral for first order ODE and the model solution was given by Eq. (7.30). The concentration profile exhibited a maximum. The solution for the time at which maxima occurs needs numerical solution as the resulting equation in transcendental. Eq. (6.39) can be seen to be interplay of the rate of absorption and rate of excretion. When the second order absorption processes are rapid and excretion is slow, the drug tends to accumulate in the blood plasma. When the rate of excretion is rapid the drug concentration tends to drop off rapidly; iv) Michaelis-Menten absorption with elimination. The MM kinetics when integrated (Eq. 7.33) results in a transcendental equation. It is not in a form that is readily usable. A more usable form of Eq. (7.33) is developed. This was accomplished by using Taylor series expansion of dimensionless concentration u in terms of its derivatives. The infinite series expression for dimensionless concentration is given by (Eq. 7.57). It can be seen that for times t