Heat Exchangers: Types, Design, and Applications : Types, Design, and Applications [1 ed.] 9781617614255, 9781617613081

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Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved. Heat Exchangers: Types, Design, and Applications : Types, Design, and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved. Heat Exchangers: Types, Design, and Applications : Types, Design, and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest

ENERGY SCIENCE, ENGINEERING AND TECHNOLOGY

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

HEAT EXCHANGERS: TYPES, DESIGN, AND APPLICATIONS

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ENERGY SCIENCE, ENGINEERING AND TECHNOLOGY

HEAT EXCHANGERS: TYPES, DESIGN, AND APPLICATIONS

SPENCER T. BRANSON Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

EDITOR

Nova Science Publishers, Inc. New York

Heat Exchangers: Types, Design, and Applications : Types, Design, and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest

Copyright © 2011 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: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com 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.

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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 Branson, Spencer T. Heat exchangers : types, design, and applications / Spencer T. Branson. p. cm. Includes bibliographical references and index.

ISBN:  (eBook)

1. Heat exchangers. I. Title. TJ263.B72 2010 621.402'5--dc22

2010026937

Published by Nova Science Publishers, Inc. † New York Heat Exchangers: Types, Design, and Applications : Types, Design, and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest

CONTENTS Preface

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Chapter 1

vii  Lightweight Compact Heat Exchangers with Open-Cell Metal Foams S.S. Feng, J.J. Kuang, T. Kim and T.J. Lu 

Chapter 2

The NTU-Effectiveness Method Sylvain Lalot 

Chapter 3

Mathematical Model for Plate Heat Exchangers for Steam Generation in Absorption Systems Sotsil Silva - Sotelo, Rosenberg J. Romero and Roberto Best y Brown 

Chapter 4

Thermal Design of Compact Heat Exchangers Martín Picón-Núñez 

Chapter 5

Optimal Detailed Design of Shell-and-Tube Cooler Units Using Genetic Algorithms José María Ponce-Ortega, Medardo Serna-González and Arturo Jiménez-Gutiérrez 

Chapter 6

Chapter 7

1  43 

115 

147 

175 

Advances in Design Optimization of Shell and Tube Heat Exchangers Mohammad Fesanghary and Majid Rasouli 

199 

Thermal Design Methodology of Industrial Compact Heat Recovery with Helically Segmented Finned Tubes E. Martínez, W. Vicente, G. Soto and M. Salinas 

215 

Index

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PREFACE A heat exchanger is a device built for efficient heat transfer from one medium to another. The media may be separated by a solid wall, so that they never mix, or they may be in direct contact. They are widely used in space heating, refrigeration, air conditioning, power plants, chemical plants, petrochemical plants, petroleum refineries, natural gas processing, and sewage treatment. One common example of a heat exchanger is the radiator in a car, in which the heat source, being a hot engine-cooling fluid, water, transfers heat to air flowing through the radiator. This book presents current research data in the study of heat exchangers, including lightweight compact heat exchangers with open-cell metal; the NTU-effectiveness method to design and assess heat exchangers; a mathematical model for plate heat exchangers; and advances in design optimization of shell and tube heat exchangers. With rapid progress in materials processing, a number of low cost porous materials have been produced and employed as the core parts of lightweight structures due to their distinctive mechanical, thermal, electrical and acoustical properties. Highly porous cellular metallic foams with open cells provide all of these attributes and are under serious consideration for a variety of engineering applications. With open cells, the pores of the metal foam are interconnected, providing three-dimensional irregular and tortuous flow paths and substantially large extended heat transfer surfaces. These topological advantages of open-cell metal foams over conventional heat dissipation media can be tailored to make lightweight compact heat exchangers. Thus far, the fundamental flow and thermal transport behaviors of open-cell metal foam heat exchangers have been characterized based on both micro- and macro-structural properties under various heat transfer configurations. Chapter 1 aims to firstly review the recent progress and then present some new results on the fundamental research and applications regarding the thermo-fluidic properties of open-cell metal foam heat exchangers. In this work, the thermo-physical aspects of open-cell metal foams as heat exchangers are grouped into three different categorizes depending on the cooling strategy used: (i) forced liquid/air convection in open-cell metal foam channels/tubes, (ii) impinging single circular and annular jets on open-cell metal foams, and iii) impinging axial fan flow that swirls periodically on open-cell metal foams. The NTU-Effectiveness method is one of the most powerful methods to design heat exchangers or to assess their performance, when they are used in steady states. In Chapter 2, the analyses of the most common heat exchanger types are presented (parallel flow, counter flow, 1-2, cross flow with only one fluid mixed, cross flow with both fluids mixed). All these configurations are studied in detail, from a schematic drawing to the final equations. For more

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viii

Spencer T. Branson

complex configurations (1-2n, cross flow with both fluids unmixed), analytical equations are given along with approximations. Some examples are also given for heat exchanger networks. In all cases a Scilab script is given to compute the effectiveness from the number of transfer units and vice versa, and charts are plotted. For the analyses, the Number of Transfer Units, the effectiveness, and the heat capacity rate ratio are defined for the hot fluid as well as for the cold fluid, so that trial and error procedures are not necessary. The detailed solutions of some representative problems are given, and only the numerical results are given for the other exercises. Since its introduction in 1930, plate heat exchangers have become important components in the food, dairy and pharmaceutical industries and also in power generation. This type of heat exchangers offers significant advantages over traditional equipment (such as shell-andtube heat exchangers), including: great facility for maintenance, compactness, high heat transfer rates, the possibility of modifying the transfer area by removing some of the plates and minimum fouling. Recently, it have been used these heat exchangers in absorption systems. These systems can be used either for air conditioning, refrigeration or to provide thermal energy at high temperatures. It has been shown the advantage of using plate heat exchangers in these systems, because it has reduced the size of the facilities, without affecting the performance of the system. The steam generator is a key component of the absorption equipments. This component carries out the partial evaporation of the more volatile substance from an aqueous mixture. The efficiency in heat transfer in the generator has an impact on the efficiency of the entire system. It is therefore very important to know the heat transfer mechanisms dominant in this component and, moreover, in such type of heat exchangers. Chapter 3 of the book presents a mathematical model for a plate heat exchanger used as a steam generator in an absorption system. The model describes the two-phase flow characteristics in the plate heat exchanger (PHE). Heat transfer coefficients for the aqueous mixture, void fraction, flow quality and heat transferred were calculated. From experimental data has been possible to determine the temperature profiles in the plate heat exchanger. This chapter deals with the thermal sizing of compact heat exchangers. The fundamentals of a general approach for the development of new design methodologies for compact heat exchangers are outlined. The approach is based on the concept of full utilisation of available pressure drop and can be extended to most compact exchanger geometries. In some applications, depending on the exchanger geometry and flow arrangement, both streams can fully absorb the allowable pressure drop; however, in other cases it is only one stream that can fully utilise it. In Chapter 4, specific applications are given for the cases of: plate and fin, plate and frame and spiral heat exchangers. The design approach covers aspects such as the determination of the temperature correction factor derived from the thermal effectivenessnumber of heat transfer units method (ε-Ntu). The concept of the pictorial representation of the design space for these types of exchangers is presented. This design tool takes into consideration the non-continuous nature of the design variables associated with the exchanger geometry. Some insights towards the design of multi-stream units are also presented for the case of plate and frame and plate and fin geometries. The methodologies are demonstrated using case studies and the results compared with those reported in the literature. Chapter 5 presents an approach based on genetic algorithms for the optimal design of shell-and-tube coolers. The approach involves both discrete and continuous variables, and uses compact formulations based on the Bell-Delaware method. The objective is to minimize the total annual cost of the exchanger, which includes capital and operating costs. The capital

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Preface

ix

cost of the cooler is given by a detailed function that involves the cost of the exchanger components and manufacturing costs. The proposed methodology calculates the optimal mechanical and thermal-hydraulic variables, considering geometric constraints such as maximum shell diameter and tube length, and maximum and minimum ratios of baffle spacing to shell diameter and of cross-flow area to area in a window. In addition, maximum pressure drops and velocities for the fluids in both sides of the cooler, and maximum outlet temperature for the cooling water to avoid fouling conditions are considered. The procedure follows the standards of the Tubular Exchanger Manufacturers Association to select the external tube diameter, the tube length, the tube arrangement pattern, the tube pitch and the number of tube passes. Because the constraints of the model are highly nonlinear and nonconvex, genetic algorithms are used as a suitable optimization technique to provide a global or near-global optimum design. An example is used to show the applicability of this methodology. Shell and tube heat exchangers (STHEs) are the most widely used heat exchangers in process industries because of their relatively simple manufacturing and their adaptability to different operating conditions. The optimum thermal design of STHEs is a complex task and it involves the consideration of many interacting geometrical and operational design parameters and constraints. In the last few decades several methods for design optimization of STHEs have been developed. These methods can be classified into three main groups, namely thermodynamic approaches, mathematical programming methods and stochastic optimization methods. Chapter 6 covers recent developments and applications of these methods in design of STHEs. After a brief description of the basics of each method, the available literature in the field is reviewed. Also, an approach based on global sensitivity analysis (GSA) is presented to identify the most influential design parameters. The GSA results can provide designers with a broad view that is useful in the design process as well as the reduction of the optimization problem size. In Chapter 7, a thermal design methodology of compact heat recovery with helically segmented finned tubes in staggered layout on an industrial scale is proposed. The methodology is based in thermodynamic analyses and semiempirical models for heat transfer and pressure drop, coupled to the Logarithmic Mean Temperature Difference (LMTD) method. Some of the best models available in the open literature for heat transfer and pressure drop in helically segmented finned tubes are used. The methodology is validated with experimental data of industrial equipment under different operating conditions. Comparisons between predictions and experimental data show a precision greater than 95% in the heat transfer for Reynolds number, based on outside diameter of bare tube, of about 10000 (Reynolds number based on volume-equivalent diameter of 13751). In the case of pressure drop of gas phase, there is a precision of approximately 90% for a Reynolds number based on outside diameter of bare tube, of about 10000 (Reynolds number based on volume-equivalent diameter of 13751). Therefore, the results show that the best flow regime in which heat transfer and pressure drop are optimal is for Reynolds number based on outside diameter bare tube, of about 10000 (Reynolds number based on volume-equivalent diameter of 13751) with a length-width ratio (cross section area) around 2.5.

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In: Heat Exchangers Editor: Spencer T. Branson

ISBN: 978-1-61761-308-1 © 2011 Nova Science Publishers, Inc.

Chapter 1

LIGHTWEIGHT COMPACT HEAT EXCHANGERS WITH OPEN-CELL METAL FOAMS S.S. Feng1,2, J.J. Kuang1,2, T. Kim1,3 and T.J. Lu1 1

MOE Key Laboratory for Strength and Vibration, Xi’an Jiaotong University, Xi’an, People’s Republic of China 2 School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an, People’s Republic of China 3 School of Mechanical, Industrial and Aeronautical Engineering, University of the Witwatersrand, Johannesburg, Republic of South Africa

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ABSTRACT With rapid progress in materials processing, a number of low cost porous materials have been produced and employed as the core parts of lightweight structures due to their distinctive mechanical, thermal, electrical and acoustical properties. Highly porous cellular metallic foams with open cells provide all of these attributes and are under serious consideration for a variety of engineering applications. With open cells, the pores of the metal foam are interconnected, providing three-dimensional irregular and tortuous flow paths and substantially large extended heat transfer surfaces. These topological advantages of open-cell metal foams over conventional heat dissipation media can be tailored to make lightweight compact heat exchangers. Thus far, the fundamental flow and thermal transport behaviors of open-cell metal foam heat exchangers have been characterized based on both micro- and macro-structural properties under various heat transfer configurations. This chapter aims to firstly review the recent progress and then present some new results on the fundamental research and applications regarding the thermo-fluidic properties of open-cell metal foam heat exchangers. In this work, the thermo-physical aspects of open-cell metal foams as heat exchangers are grouped into three different categorizes depending on the cooling strategy used: (i) forced liquid/air convection in open-cell metal foam channels/tubes, (ii) impinging single circular and annular jets on open-cell metal foams, and iii) impinging axial fan flow that swirls periodically on open-cell metal foams.

Keywords: Electronics cooling; Forced convection; Heat exchanger; Impinging jet; Opencell metal foam

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1. INTRODUCTION For air-cooled heat exchangers, the airside performance is important since the dominant thermal resistance is usually on the airside. Therefore, for heat dissipation enhancement, extended surfaces are commonly used in the airside of the heat exchanger. The extended surfaces enhance the rate of heat transfer by increasing the surface area per unit volume. In the past century, a variety of techniques to increase the heat transfer area of heat exchangers without increasing their overall dimensions have been developed. In general, the extended surfaces properly selected have the advantage of reducing both the size and weight of a heat exchanger. Typical configurations of extended surfaces range from circular pin-fins, louvered and wavy fins to, more recently, cellular materials. Since the publication of the comprehensive review of heat transfer and pressure drop data across various heat exchanger geometries by Kays and London [1], new manufacturing techniques have been developed, introducing new heat exchange media to meet the requirement of emerging applications. Amongst these new heat exchange media, high porosity metal foams with closed or open cells have attracted much attention due to their novel thermal, mechanical, electrical and acoustic properties [2]. The metal foams have been actively studied by both the materials and structures community owing to the flexibility in design and mechanical strength/stiffness provided by these lightweight cellular structures. As a consequence, well-established data on the mechanical properties of metal foams have been reported [3,4]. Furthermore, these studies demonstrate that the relatively high stiffness and yield strength achievable at low density create an opportunity for lightweight structures. In addition, the metal foams with open interconnected pores have thermal attributes that may enable multifunctional applications which require a medium for heat dissipation as well as mechanical strength. The metal foams have a high surface area density and may be constructed out of high conductivity materials such as aluminum (Al) and copper (Cu). These combinations make the metal foams capable of forming heat dissipation (exchanger) media that can be used effectively for coupled thermal and structural applications. From the heat transfer point of view, even though metal foams can be broadly classified as a porous medium, they have very distinctive features such as high porosities and unique open-celled morphologies. Consequently, most of previous studies on packed beds and granular porous media with a medium porosity range of 0.3–0.6 are not directly applicable to metal foams. It is only during the past 30 years that the transport phenomena in highly porous open-celled metal foams with solid-cell ligaments have started to receive attention [5-15]. Under the assumption of local thermal equilibrium, Hunt and Tien [5] studied the effects of thermal dispersion on forced convection in metal foams with water as the fluid phase, and concluded that the conduction of metal foams may not be significant due to their thin cell ligaments and that dispersion may dominate the heat transport. Sathe et al. [6] studied the combustion in metal foams as applied to porous radiant burners. Younis and Viskanta [7] measured the volumetric heat transfer of ceramic foam materials and developed a Nusselt number correlation based on experimental data; the volumetric heat transfer rates measured were higher than those for packed beds. Lee et al. [8] investigated the application of metal foams as high-performance air-cooled heat sinks for electronics packaging, and found experimentally that Al foams could dissipate heat fluxes up to 100 W/cm2.

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With the metal foams idealized as interconnected cylinders, Lu et al. [9] developed an analytical model to predict their heat dissipation capability under forced convection, whilst Bastarows et al. [10] studied single-sided heating of a foam-filled plate channel for electronics cooling applications. Both conductive thermal epoxy bonding and brazing have been used to bond metal foam to a heated plate. The results revealed that brazed-foam samples are much more effective at heat removal than epoxy-bonded ones, with the former capable of removing three times more heat than a conventional fin–pin array. Calmidi and Mahajan [11,12] proposed an effective thermal conductivity model for metal foams having idealized cellular structures, and studied experimentally and numerically forced air convection in Al foams. In their numerical study, the two-equation heat transfer model was employed, and it is found that thermal dispersion effect is extremely small if the fluid phase is air, in contrast with the conclusion derived by Hunt and Tien [5] with water as the coolant. Kim et al. [13] experimentally studied laminar heat transport in Al foams and found that the foam offers a better heat transfer performance compared to that of a louvered array, albeit at the expense of a greater pressure drop. The effective conductivity and permeability of Al foams were studied by Paek et al. [14] and Boomsma and Poulikakos [15]. This chapter aims to review the recent progress and then to present some new results on the fundamental research and applications regarding the thermo-fluidic properties of open-cell metal foam heat exchangers. Depending on the cooling strategy used, the thermo-physical aspects of open-cell metal foams as heat exchangers are grouped into three different categorizes, as:

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(i) forced liquid/air convection in open-cell metal foam channels/tubes, (ii) impinging circular and annular jets on open-cell metal foams, and (iii) impinging axial fan flow that swirls periodically on open-cell metal foams.

2. TOPOLOGIES OF OPEN-CELL METAL FOAMS 2.1. Macro/Micro Topologies Metal foam cells are usually polyhedrons of 12-14 faces in which each face has a pentagonal or hexagonal shape (by five or six filaments). As argued by Mahjoob and Vafai [16], due to the geometric complexity and random orientation of the solid phase, the solution of the transport equations inside the foam pores is difficult to obtain. Typical macro/micro topologies of open-cell metal foams are shown in Figures 1 and 2(b), which are consisted of ligaments forming a network of inter-connected dodecahedral-like cells. As the cells are randomly oriented, metal foams are typically classified as a stochastic cellular material.

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(a)

(b)

Figure 1. Open-cell metal foams (provided by Prof. F. S. Han of Chinese Academy of Science, Hefei, P.R. China): (a) aluminum (Al) foam circular block; (b) aluminum (Al) foam square block brazed on to substrate.

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(a)

(b) Figure 2. Investment casting technique for processing metal foams with solid ligaments: (a) schematic representation by Banhart [20]; (b) scanning electron microscope (SEM) image of Al foam.

The topological parameters characterizing a metal foam are typically: (1) Pore size (Dp), which is specified by the diameter of the open space in each of the cell faces and, typically, the averaged value is used. (2) Pore density, defined as the number of pores per unit length of the material speficied as pores per inch (PPI). (3) Relative density (ρrel), being the volume fraction of solid foam material relative to the total volume of the foam. (4) Porosity (ε), defined as the void volume fraction, with ε = 1 − ρrel .

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(5) Surface area density (ρsurface), which is the ratio of the total surface area of the foam to its volume. For metal foams contemporarily used in engineering applications, the pore size may be varied from approximately 0.4 mm to 3 mm, and the relative density from 3% to 15 % (or the porosity from 85% to 97%) of the base material [2]. Whilst the strength of the foam is dependent mainly upon its base material and relative density, other properties including the above-listed parameters play important roles in the pressure drop and heat transfer performance of the foam [17,18].

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2.2. Fabrication Methods of Metal Foams A variety of techniques have been developed to create open-cell metallic foams, including metal sintering, metal deposition through evaporation, electrodeposition or chemical vapor deposition (CVD), and investment casting. Each technique leads to distinctive cellular morphology, cell size and relative density range of the produced metal foam. As these technique have been comprehensively addressed by several researchers [2,19-21], they are not repeated in this chapter. However, to illustrate how open-cell metal foams are made, the investment casting technique for processing the widely applied and studied Al foam with solid cell ligaments, brand named as “DUOCEL”, and the sintering technique for fabricating metal foams having hollow cell ligaments, are detailed below. As schematically illustrated in Figure 2(a), the first step of the investment casting technique is to form a mould by putting the precursor foam with open cells in a stainless steel casing, into which plaster slurry with appropriate composition and viscosity is poured to fill the volume fraction of the polymer foam. After the plaster slurry is naturally dried and hardened, the casting assembly is heated to approximately 250∼300°C to decompose the polymer foam. Molten metal is then infiltrated into the mould under pressure, filling the voids of the original matrix. After solidification of the molten metal, the resulted composite is leached in water to remove the plaster, retaining the metal skeleton that inherits the structure of the polymer foam. With this method, open-cell metallic foams can be produced with a wide range of pore sizes (5-40 PPI) and porosities from 80% to 97%. The other advantage of the method is that the morphology of the metal foam can be precisely controlled, matching well with that of the origin polymer foam. However, the price of the method is relatively high due to low production rate. Figure 2(b) shows the scanning electron microscope (SEM) image of an open-cell Al foam produced through this route. In the metal sintering process, metallic particles are suspended in a slurry. This slurry is then used to coat an open-cell polymer foam substrate (Figure 3(a)). Once coated, the precursor is put through a mechanical press in order to eliminate excess slurry. This procedure is repeated several times with the orientation of the material remaining constant. Once the polymer is coated with sufficient slurry to achieve the desired relative density, the material is placed on a conveyor and passed through a series of furnaces. The firing process serves to sinter the metal particles together while burning out the polymer precursor. The end result is an open-cell metal foam having hollow struts (cell ligaments) as shown in Figure 3(b). This method offers a cost-effective way for mass production compared to the more costly techniques of investment casting or chemical vapor deposition.

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(a)

(b)

Figure 3. Sintering technique for processing metal foams with hollow ligaments: (a) polymer precursor; (b) hollow ligament of a sintered steel alloy (FeCrAlY) foam due to vaporization of polymer precursor.

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3. SINGLE PHASE FORCED CONVECTION IN OPEN-CELL METAL FOAMS A simple yet fundamental flow condition in an open-cell metal foam (as heat exchanging medium) is the so-called “parallel flow forced convection” or simply “forced convection”, namely, a coolant convects parallel to a heat transfer surface (or heat source) where heat is imposed through (i.e., normal to the direction of heat flux). The majority of existing experimental and theoretical studies deal with this flow condition. The design parameters characterizing most of conventional heat exchangers include pressure drop (loss) and heat transfer. The former is related to the pumping power required for the forced flow motion and, therefore, plays an essential role in selecting a pump that drives the coolant across the heat exchange medium. As a consequence of the coolant flow, the latter refers to the heat removed by the medium. With air or water as the coolant, forced convection heat transfer in both rectangular and circular channels with/without extended surfaces, e.g., louvered fins, banks of cylinders and open-cell foams, have been extensively studied. This section reviews the recent progress on the fundamental research and applications regarding the thermo-fluidic properties of open-cell metal foam heat exchangers. In particular, the effects of topological parameters including porosity and pore size upon pressure drop and heat transfer characteristics are addressed.

3.1. Pressure Drop (Loss) 3.1.1. Overall Pressure drop Behavior The analysis of pressure drop in porous media including open-cell foams has been a popular research subject for its importance in numerous engineering applications [22-27]. Typically, for simplicity, open-cell foams are assumed to be isotropic. At the microscopic level, the size of individual pores often differs spatially. However, at the macroscopic level, the pores are taken as statistically and volumetrically isotropic. This assumption may be

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reasonable if the pores are randomly distributed within the bulk of the foam, as can be seen from Figure 1. Numerous efforts made hitherto have led to a multitude of comprehensive studies on pressure drop across various porous media (see, e.g., Trussell and Chang [22]). Following the classical Darcy experiment in 1856 for pressure drop through a unidirectional (isotropic) porous medium, the so-called “Darcy flow” has been extended to high flow velocity regimes such as Forchheimer, transitional, and turbulent regimes. This is due to the fact that Darcy’s empirical correlation is only valid in the creeping flow regime, i.e., ReDp < 1 where Dp is the representative pore diameter. It has been established that pressure drop through an open-cell metal foam is dependent mainly upon its porosity and representative pore size. Moreover, the pressure drop is proportional to the flow velocity in the Darcy flow regime and its square in the turbulent flow regime. For isotropic open-cell foams, it is nonetheless independent upon flow orientation. Reynolds numbers for flow through porous media such as soil and sand are often inhibited by low porosities, resulting in therefore Darcy flow. On the other hand, flow through open-cell metal foams with porosities as high as 0.99 falls typically in the turbulent regime. For turbulent flow, with viscous contribution neglected, pressure drop ΔP through length ΔL of the foam can be expressed as:

ΔP = βV 2 ΔL

(1)

where V is the discharge velocity through the foam (= U/ε) and β is the empirical constant, ε being the foam porosity.

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3.1.2. Empirical Correlations for Homogeneous Isotropic Foams with Open Cells For practical purposes, pressure drop across an open-cell metal foam block is often empirically correlated as a function of coolant flow velocity (or mass flow rate). For nondimensionalization, the concept of “friction factor” as a function of Reynolds number is conventionally used, similar to that used for traditional heat transfer surfaces, e.g., tube arrays, corrugated ducts, pin-fins, plate-fins (see Kays and London [1]). For open-cell metallic foams, the friction factor f is typically expressed as:

f =

Pinlet − Poutlet 1 Dp L ρU 2 / 2

(2)

where Dp is the representative pore diameter and U is the average coolant velocity before entering the foam block. In some cases, the square root of the permeability K, representing the imaginary pore diameter, is used as a characteristic length instead of the pore diameter. Alternatively, the pressure drop may then be expressed following the Forchheimer equation as:

−

ρF 2 dP μ = U+ U dx K K

(3)

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where F is the inertial coefficient. Once the empirical coefficients such as K and F are experimentally measured, the pressure drop can be calculated using Eq. (3). Mahioob and Vafai [16] presented a comprehensive review of existing pressure drop correlations for open-cell metal foams. Here, two empirical correlations are listed but more details can be found in reference [16]. (1) Adopting the definition of friction factor defined in Eq. (2), Liu et al. [28] obtained:

f = 22

1− ε + 0.22 Re D p

for 30 < ReDp< 300

f = 0.22 for ReDp > 300

where Re D p =

ρuD p μ

(4a) (4b)

3 3D p (1 − ε ) (1 − ε ) , f = ΔP D p ε , and D = . p 2 2ε L ρU 1 − ε

(2) With the Forchheimer extension to Darcy’s law, permeability and inertia coefficient are empirically correlated as functions of porosity (ε), ligament diameter (df), and pore diameter (Dp) as reported by Calmidi [11,12], as:

⎛ df K − 0.224 ⎜ = − 0 . 00073 ( 1 ε ) ⎜ Dp D p2 ⎝

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⎛ df F = 0.00212(1 − ε ) −0.132 ⎜ ⎜ Dp ⎝

⎞ ⎟ ⎟ ⎠

⎞ ⎟ ⎟ ⎠

−1.11

−1.63

(5)

Substitution of Eq. (5) into Eq. (3) yields the pressure drop across a given foam block with specific topological parameters.

3.1.3. Contribution of Overall Foam Shape and Size to Pressure Drop Pressure drop in a circular or rectangular flow channel fully filled with open-cell metal foam is determined by the cellular morphology of the inserted foam, whereas the contribution from the channel wall is typically considered negligible. For a metal foam with fixed porosity, pore size (diameter) and ligament diameter, the pressure drop is expected to be identical, regardless of the cross-sectional shape and overall size of the flow channel. This argument has been supported by numerous studies including Lu et al. [29] who found that the pressure drop across a foam-filled circular pipe remains unchanged although the diameter of the pipe varies. Salas and Waas [30] also performed experiments to examine the effect of rectangular channel height on pressure drop and reached the same conclusion as Lu et al. [29]. Thus, to calculate the pressure drop in a straight channel (with rectangular or circular cross-section), it is reasonable to conclude that the empirical correlations derived from homogeneous, isotropic open-cell foams could be used, although these correlations exclude the effects of flow channel (walls, cross-sectional shape, and overall dimension).

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3.1.4. Effect of Porosity on Pressure Drop With both the pore density (or pore size) and coolant mass flow rate (or flow velocity) fixed, the pressure drop across a foam block, in general, decreases with increasing porosity. However, only a few studies have attempted to explore systematically the dependence of pressure drop upon porosity. Assuming the foam is homogeneous and periodic, Irmay [31] presented mathematical derivations of pressure drop as a function of foam topological parameters for unidirectional flow:

⎛ 1 ⎞ βμ (1 − ε ) 2 α (1 − ε ) dP ⎟ D p ⎜⎜ = − 2 −2 2 3 ⎟ dx ρUε ε3 ⎝ ρU / 2 ⎠

(6)

where Dp is the mean particle diameter, ρ is the fluid density, and α and β are empirical constants. If high Reynolds number flow is considered, a further modification of Eq. (6) can be made to define pressure drop across a representative unit pore of the foam. With viscous effect ignored, Eq. (6) can be rewritten as:

⎞ 1 dP ⎛⎜ ⎟ = −2 α (1 − ε ) dp 2 ⎜ ρ fU / 2⎟ dx ε3 ⎝ ⎠

(7)

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Experimental validation of the above mathematical expressions has not been reported since most investigations of flow in porous media focused on applying Darcy’s equation and its extension using permeability rather than actual pore diameter (see Eq. (3)). Equation (7) reveals how the porosity plays a role in determining the pressure drop across a foam block:

dP (1 − ε ) ~ dx ε3

(8)

3.1.5. Effect of Pore Density (Pore Size) on Pressure Drop It has been widely accepted that, at fixed coolant flow velocity (or mass flow rate) and porosity, pressure drop increases as pore density is increased or, equivalently, as pore size is decreased. Again, adopting Irmay’s derivation of Eq. (7) for such conditions, one can see that the pressure drop is inversely proportional to pore size (Dp):

dP 1 ~ dx D p

(9)

Equation (9) states that as the pore size is decreased whilst the porosity is fixed, topologically the cell ligament diameter becomes thinner. Therefore, from microscopic point of view, a smaller form drag is expected to occur in a representative unit pore. However, for a given flow length (or foam thickness along flow direction, L), shear layers separated from the thinner but more foam ligaments interact with each other and thus increase the mixing loss. As a result, a higher overall pressure drop is induced as the pore size is decreased.

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3.2. Heat Transfer

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3.2.1. Forced Convection Heat Transfer in Open-Cell Metal Foams Consider single phase forced convection in a rectangular or circular flow channel fully filled with open-cell metal foam (Figure 4(a)). Constant and uniform heat flux is often imposed through the channel wall(s). Heat is then conducted via the channel wall(s), followed by forced convection in the foam-filled channel which takes two distinct routes: (1) direct convection from the channel wall(s), and (2) conduction via foam ligaments followed by forced convection from ligament surfaces. The three-dimensional (3D) topology of open-cell metal foam provides a high surface area density for heat transfer when high thermal conducting metals, e.g., aluminum (Al) and copper (Cu), are used. This leads to a high overall heat transfer coefficient due to low convective thermal resistance between foam ligaments and coolant flow. Therefore, the majority of heat is removed through foam ligaments, not from channel walls (Calmidi and Mahajan [12]). Similar to an empty channel, fluid flow and heat transfer in a foam-filled channel exhibit a developing region near the inlet. Dependent upon the channel length available, the hydraulic and thermal boundary layers then become fully developed as convecting downstream. The velocity profile normal to the channel wall is determined based on channel cross-sectional shape, stream-wise location in the channel, and flow conditions (e.g., developing or fully developed). For an empty channel, these features can be visualized using relatively simple techniques such as hydrogen bubble generation, and quantified by traversing a Pitot tube normal to the channel wall at a selected stream-wise position. On the other hand, thermal boundary layer governed by the momentum boundary layer mentioned above is not easy to visualize. Traversing a probe with a built-in thermocouple normal to the (heated) channel wall enables one to obtain the temperature profile (thermal boundary layer), but this is only feasible in the empty channel case. Figure 4(b) presents the thermal field captured by infrared camera in a rectangular flow channel filled with 30 pores per inch (PPI) FeCrAlY foam; the experimental setup is displayed in Figure 4(a). The measurement was carried out under steady state condition, and the foam sample was sprayed in black to increase the emissivity of the infrared ray. Thermal images of local solid (cell ligament) temperature distribution were captured by an infrared camera viewing through a quartz window. Heat conducted from the channel wall is spread through foam ligaments and then removed by coolant flow. It is assumed that the foam ligaments and fluid adjacent to the ligaments are in thermal equilibrium, so that the temperature of foam ligaments is the same as that of fluid flow near the ligaments. For ReH = 2500, it is seen from Figure 4(b) that a thermal boundary layer (thickness δ t ) is clearly developed with δ t ~ x where x is the axis along the direction of fluid flow, its origin coinciding with the inlet of the test section. Whilst pressure drop in open-cell metal foams is dependent mainly on foam topology (channel wall effects negligible), heat transfer depends both on channel dimensions and foam topology. Therefore, it is difficult to find a characteristic length to correlate the overall heat transfer performance of open-cell metal foams. Salas and Waas [30] attempted to use the height of a flat channel filled with metal foam as the characteristic length in defining Reynolds and Nusselt numbers. Their experimental results demonstrate that the Nusselt

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number increases with channel height even for a fixed Reynolds number. As is well known for empty channel flows, the Nusselt number is independent of channel height when the height is used as a characteristic length.

(a)

(b)

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Figure 4. Forced convection in a rectangular channel fully filled with open-cell metal foam: (a) schematic illustration; (b) foam ligament temperature field obtained using an infrared camera showing thermal boundary layer-like profile (darker color indicates lower temperature and brighter color represents higher temperature) [32]. Asymmetric heating from lower endwall is applied.

Figure 5. Forced convection in foam-filled channel showing notations and cubic unit cell [9].

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3.2.2. Analytical Modeling of Heat Transfer Efforts have been devoted to analytically estimating pressure drop and heat transfer in open-cell metal foams under single phase forced convection. The commonly used two approaches based separately on the classical fin analogy and the porous medium approach, are discussed below. Fin analogy approach Making use of the classical fin analogy, Lu et al. [9] developed an analytical model for forced convection heat transfer in a foam-filled rectangular channel (Figure 5). For simplicity, the metal foam is idealized as a periodical array of mutually perpendicular cylinders as shown in Figure 5. As the foam considered is highly porous so that the foam ligaments are thin and slender, heat transfer in the foam is analyzed using existing data on bank of cylinders in cross flow. The explicit dependence of the overall heat transfer coefficient on channel height and foam properties such as relative density ρ and ligament diameter d is obtained as [9]: h=

2hρ

⎛ b tanh⎜ 2 ⎝ d Bi

⎞ Bi ⎟ ⎠

(10)

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where h is the heat transfer coefficient representing local heat transfer between foam ligaments and fluid, and Bi is the Biot number defined as Bi = hd/ks, ks being the solid conductivity. Porous medium approach: Due to high porosity, open-cell metal foams differ from traditional porous media such as packed beds and granular materials, exhibiting high permeability and form drag dominant pressure drop. For heat transfer, as the thermal conductivity of the ligament material is typically several orders of magnitude higher than that of air or water, the assumption of thermal equilibrium between the solid and fluid phases for traditional porous media no longer holds. Consequently, thermal non-equilibrium in metal foams has been taken into consideration with the so-called “two equation model”. According to this model, the energy equations for solid and fluid phases may be expressed as: 0 = ∇ ⋅ {k se ⋅ ∇〈 T s 〉} − h sf ρ surface (〈T s 〉 − 〈T f 〉 )

{

}

〈 ρ 〉 f C f 〈V 〉 ⋅ ∇〈 T f 〉 = ∇ ⋅ k fe ⋅ ∇〈 T f 〉 + h sf ρ surface (〈T s 〉 − 〈T f 〉 )

(11) (12)

where 〈〉 denotes volume averaging; ρf, μf and Cf are the density, viscosity and specific heat of the fluid phase; kse and kfe are the effective thermal conductivity of the solid and fluid phases; hsf is the interfacial heat transfer coefficient between solid and fluid phases; ρ surface is the surface area per unit volume (surface area density); and V is the flow velocity vector. As the volume-averaged energy equations (11) and (12) have general forms, for a specific porous medium such as metal foam, empirical coefficients representing the effects of foam topology on fluid flow and heat transfer are required. These include the effective thermal conductivities, kse and kfe, the interstitial heat transfer coefficient hsf, and the surface area density ρsurface. The prediction accuracy of the porous medium model is often dictated by these empirical coefficients, determined either experimentally or theoretically.

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3.2.3. Effect of Porosity (ε) Pore size, surface area density and porosity are key topological parameters that influence pressure drop and heat transfer in metal foams. For fixed pore density (PPI) and coolant flow rate, it has been established that the overall heat transfer performance of a metal foam heat exchanger decreases as its porosity is increased. Under such conditions, a higher porosity implies thinner ligament thickness and hence reduced effective heat transfer area (or surface area density), leading to the decrease in overall heat transfer. However, pressure drop is also decreased as the porosity is increased. An optimal porosity of the foam may therefore exist, balancing heat transfer and pressure drop. Existing data that can be used to construct a relationship between porosity and overall heat transfer in open-cell metal foams are scarce, which may be attributed to the fact that the overall heat transfer depends not only on topological parameters such as porosity, pore size and surface area density but also on material properties (e.g., thermal conductivity) and flow conditions. Since many parameters are involved, the analysis of heat transfer in metal foams strongly depends on empirical correlations. For example, heat transfer is often correlated as a function of Reynolds number as follows:

Nu D p = b1 Re D p

b2

(13)

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where Dp is the mean pore diameter, and b1 and b2 are empirical constants. Sometimes, the Prandtl number is also included as a variable for the correlation. Hwang et al. [33] presented experimental results of forced convective heat transfer in Al foams. To define the Reynolds number, the flow velocity U was replaced by U/ε to take account of velocity change due to flow area change as the porosity is varied. However, this may be misleading since the porosity represents volume ratio, not (flow) area ratio.

Figure 6. Effect of porosity on heat transfer in open-cell metal foam [34].

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The relationship between porosity and volumetric heat transfer coefficient was reported by Fuller et al. [34] for open-cell FeCrAl foams, as shown in Figure 6. Within the porosity range of 0.82 < ε < 0.92, the experimental data of Figure 6 demonstrate that heat transfer increases with decreasing porosity, which is consistent with the general trend reported by others.

3.2.4. Effect of Pore Size (Dp) and Pore Density (PPI) It is not hard to understand that, at fixed flow rate and porosity, overall heat transfer increases as pore density is increased (or pore size decreased) due to increased surface area density. Correspondingly, pressure drop also increases. Younis and Viskanta [7] defined the volumetric heat transfer coefficient as:

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h' =

Q 1 [W/m3K] V ΔT

(14)

where Q is the input heat flow, V is the volume of the foam matrix and ΔT is the mean temperature difference between fluid and solid phases. They then experimentally observed that, for a fixed velocity, the volumetric heat transfer coefficient decreases with increasing pore diameter as shown in Figure 7. However, this result may be misleading since mean pore diameter is not the only variable that affects h ' . This is because a given pore size can be associated with both low and high porosities, depending on the size of foam ligament. The effects of mean pore diameter and pore ligament diameter should be considered simultaneously. A more systematic study on the effect of pore size upon pressure drop and heat transfer was performed by Kim et al. [35]. With the porosity of Al foam samples fixed at 0.92 and within the Reynolds number range of 200 < ReH < 2000, they found that foam sample with a larger pore size (i.e. 10 PPI) had a higher pressure drop than that with 20 or 40 PPI. This observation disagrees with the experimental results of Boomsma and Poulikakos [15] that, for a given porosity, decreasing the pore diameter by the use of larger PPI foams increases the flow resistance. This increase was attributed to the higher surface area density as a result of the smaller pore size. For heat transfer, Boomsma and Poulikakos [15] found that foams having larger pore diameters led to higher rate of heat transfer, although no clear physical explanation was given. As mentioned previously, surface area density (ρSA) refers to the compactness of a particular heat exchanger. When heat is transferred to the solid phase, the temperature difference within the solid phase drives conduction, with convection followed from the surface of the solid. This explains the importance of surface area available for convective heat transfer. Also, for example, when a single cylinder is immersed in fluid flow and heat transfer from the cylinder occurs, the downstream side of the cylinder (so-called “dead flow region”) does not provide a significant contribution to the transfer of heat. This fact complicates the examination of the contribution of surface area to heat transfer. To explore the relationship between surface area and heat transfer performance for a given heat exchanger medium, local flow field within the medium must be understood before investigating local heat transfer patterns.

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Figure 7. Effect of pore size Dp on volumeric heat transfer coefficient h’ [7].

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3.2.5. Effect of Thermal Conductivity Lu et al. [30] found that the overall heat transfer of a circular pipe filled with metal foam decreases as the fluid to solid thermal conductivity ration (kf/ks) is increased. It was further shown that when the ratio is sufficiently high, e.g., kf/ks>0.001, the influence of pore density on heat transfer is marginal. This is practically significant. For example, for air-ceramic foam combinations, a low pore density foam (e.g., 10PPI) is an ideal choice as it has low pressure drop but similar heat transfer performance relative to a high pore density foam (e.g., 40PPI). According to Fuller et al. [34], the experimental results obtained from metal foams with very different solid thermal conductivities such as copper (ks ~ 300 W/mK) and FeCrAlY (ks ~ 16W/mK) showed that, for a given Reynolds number, copper foams had a volumetric heat transfer coefficient about 2 or 3 times as high as that of FeCrAlY. They also found that the effect of solid conductivity on volumetric heat transfer coefficient is non-linear. A further systematic study needs to be conducted to examine the effects of solid conductivity on both the local and overall heat transfer behaviors of open-cell metal foams. Using water as coolant, Hunt and Tien [5] measured the heat transfer performance of open-cell foams made of carbon, Ni and Al that have very different conductivities. It was found that the overall heat transfer characteristics of these foams are similar if the porosity and pore density are the same. In comparison, when air is used as coolant, the conductivity effect is apparent and cannot be ignored (Zhao et al. [36]). For high porosity foam materials, the cell ligaments are thin and hence conduction may not be significant if water is used as coolant, as thermal dispersion dominates molecular conduction in this case. Thermal dispersion is referred to heat transfer induced by hydraulic mixing of interstitial fluid at the pore scale, which is known to be a function of Prantl number of fluid, Reynolds number and pore structure of foam material [5]. In contrast, for air cooling, molecular conduction dominates thermal dispersion. The above observation also holds for textile materials [37]. For practical application, when water is selected as coolant, due to the negligible role of molecular conduction, cheaper metal foams made of low conductivity materials such as FeCrAlY may be used to substitute the more expensive Al and Cu foams.

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3.3. Comparison of Overall Performance for Different Types of Heat Dissipation Medium A wide variety of heat dissipation media have been used to construct heat exchangers, as discussed in the previous sections. In this section, to compare the performance of different types of heat exchanger in single phase forced convection, the aspects considered are: (1) pressure loss through the heat exchanger; (2) overall heat transfer rate of the heat exchanger.

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Furthermore, the height of flow channel is selected as a characteristic length in defining the Reynolds number, friction factor and Nusselt number. When a wide channel having a relatively short height is of concern, the influence of channel wall on overall pressure drop and heat transfer is pronounced. In such cases, the use of channel height may be better than the hydraulic diameter, the latter being a function of both the channel height and width. For heat exchangers of sandwich type, by converting the pressure and heat transfer results into relationships based on the core thickness H, plots of Nusselt number NuH (Re H ) and friction factor f H (Re H ) encompassing all of the competing core topologies can be constructed. The results are summarized on Figures 8(a) and 8(b), and details of the test data used for the comparison can be found in Appendix A. Each topology category is encompassed by an ellipse, because both Nu H (Re H ) and f (ReH ) are affected by the cell size, the material, the relative density and the thermal boundary conditions. While the Reynolds number can in principle vary from arbitrarily small to arbitrarily large for each topology, the limits on Re shown in Figure 8 are set by experimental conditions: the lower limit by the accuracy of the pressure tappings and the upper by the pump capacity. These vary as the core topology is varied. Upon comparison of the two plots, note the countervailing influences of topology on the Nusselt number and friction factor. The implication is that the topology preference depends on the application. In most cases, the choice is dictated by a maximum allowable temperature in the metal and/or the fluid, subject to either a specified pressure drop or pumping power. A natural goal would be to maximize the total heat transferred to the fluid.

4. IMPINGING JETS ON OPEN-CELL METAL FOAMS 4.1. Impinging Round Jet The extensive use of an impinging jet for heating or cooling is attributed to its superior heat transfer performance over other schemes of single phase heat transfer. Numerous experimental and theoretical studies have been carried out to characterize the thermo-fluidic mechanisms of the impinging jet [39-46]. Its engineering implementation includes electronics cooling, turbine blade cooling as well as heating, cooling and drying of pulps, textiles and foods.

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Typically, an impinging round jet is configured by high momentum flow discharged through a round nozzle/orifice and its subsequent impingement on impingement (target) surface. In this case, the peak momentum region is coinciding with the jet axis.

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(a)

(b) Figure 8. Comparison amongst various lightweight heat dissipation media: (a) pressure drop (friction factor); (b) heat transfer (Nusselt number). Detailed of experimental data can be found in Appendix A [38]

For a single round cooling jet impinging normally on a flat plate experiencing constant heat flux (or temperature) boundary conditions, it has been shown that the radial heat transfer distribution on the plate exhibits either a single peak at the stagnation point or two peaks, with the secondary maximum located at approximately two nozzle diameters away from the

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stagnation point [39,42-46]. The appearance of the secondary maximum is dependent upon the spacing between the nozzle exit and target plate, also called the “separation distance.” When such a round jet impinges on an open-cell metal foam (Figure 9), the mechanics of fluid flow in the foam differs from that in “parallel flow forced convection.” Therefore, pressure drop and heat transfer in the foam is expected to behave in a different manner. However, the role of topological parameters in determining the pressure drop and heat transfer may be similar to that discussed in previous sections for parallel flow forced convection. This section discusses recent progress regarding the hydraulic and thermal behaviors of open-cell metal foams subjected to round impinging jet cooling. Particular focus is placed upon the effects of topological parameters such as porosity and pore density (size). Furthermore, impinging jets with either single round or annular jet nozzles are considered, and the separation distance H (see Figure 9) is systematically varied.

Figure 9. Schematic of open-cell metal foam subjected to circular single impinging jet cooling.

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4.2. Single Jet Flow Structures For reference, we discuss birefly the flow characteristics of a single round jet impinging on a flat plate (without open-cell foam attached). Figure 10 schematically illustrates the flow structures of the impinging jet containing a high momentum flow issued through a circular nozzle. According to Martin [47], three flow regions may be classified, as: (1) free jet region, (2) stagnation region, and (3) wall jet region. A potential core where the jet flow is undisturbed by flow interaction with the surrounding fluid appears in the inner region of the jet flow, which affects the stagnation heat transfer characteristics pronouncedly. Such flow characteristics are expected to vary when an open-cell foam block is mounted on the target plate, as shown in Figure 9.

4.3. Heat Transfer Enhancement

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Unlike forced convection in open-cell metal foams, only a few studies have been devoted to studying round impinging jets on metal foams. Amongst these, Hsieh et al. [39-41] carried out a series of experimental studies on the heat transfer performance of open-cell Al foams subjected to circular impinging jet cooling, Jeng and Tzeng [42-46] numerically and experimentally studied the thermal performance of Al foam heat sinks under slot and round impinging jets, whilst Kim et al. [46] investigated experimentally multi- and single-round jet impingement on Al foam heat sinks.

(a)

(b)

Figure 11. Open-cell Al foams under impinging slot jet [43,44]: (a) schematic; (b) Nusselt number ratio Nus/Nus0 plotted as a function of Reynolds number Re.

In the previously mentioned studies, high porosity open-cell metal foams were introduced as a heat spreading medium and found to enhance significantly the overall cooling performance of the heat sink compared to that in parallel flow forced convection. Figure 11 shows the heat transfer enhancement obtained from Al foam heat sinks under laminar slot jet (Figure 9) relative to that without using the foam as additional heat dissipation medium (Figure 10).

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The Nusselt number ratio Nus/Nuso indicates the heat transfer enhancement by the foam, where Re is the Reynolds number (= ρVjW/μ), W is the width of the slot jet, Nu is the Nusselt number (= hW/k), Nus is the Nusselt number at the stagnation point, Nuso is the Nusselt number at the stagnation point without foam attachment (reference). It is seen from Figure 11(b) that, upon attaching the Al foam to the target plate (Figure 9), the Nusselt number at the stagnation point is significantly enhanced, about 20 times higher at Re = 100 and 10 times higher at Re = 500.

4.4. Effects of Topological Parameters For metal foam heat sinks, the main parameter for heat transfer characterization, Nusselt number, is dependent on many topological parameters:

L W C ⎛ ⎞ Nu = f ⎜ ε , PPI , k *s , , , , Re ⎟ D D D ⎝ ⎠

(15)

where ε is foam porosity, PPI is pore density, L/D is dimensionless foam length, W/D is dimensionless foam width, C/D is dimensionless distance of jet nozzle to foam tip (see Figure 11(a)), and D is round jet diameter or slot jet width. The effects of these parameters on the overall heat transfer performance of the heat sink subjected to the round/slot impinging jet cooling are discussed below.

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4.4.1. Effect of Porosity For forced convective heat transfer in highly porous open-cell metal foams, porosity (or relative density) plays an important role. In general, reducing the porosity increases the interfacial heat transfer area, leading to a better cooling performance, albeit at the expense of increased flow resistance. For round impinging jet cooling with metal foams, Shih et al. [39,40] experimentally studied the porosity effect on overall heat transfer behavior. It was found that the Nusselt number increases with increasing porosity (Figure 12), consistent with the case in parallel flow forced convection (see Figure 6). In Figure 12, the Reynolds number is defined using representative pore diameter D and Darcy velocity U (equal to average inlet velocity) as ReD = ρUD/μ, and the Nusselt number is defined as Nu = hD/k, h being the convective heat transfer coefficient and k the effective thermal conductivity of the foam. For experimental measurements, the round jet diameter was D = 60 mm, the diameter and height of the circular foam block tested were Df = 65 mm and Hf = 60mm. The separation distance H between jet exit and target plate was fixed at H = Hf. All the Al foam samples tested had a pore density of 20 PPI. For a fixed Reynolds number (e.g., Re = 10,000), whilst the porosity is systematically varied in the range of ε = 0.86 to ε = 0.96, the results of Figure 12 reveal that the cooling performance of the Al foam sample with a lower porosity was better than that having a higher porosity. Shih et al. [40] argued that Al foams with a low porosity typically has a low Darcy number Da, a large fluid-to-solid Nusselt number Nufs, and a large effective conductivity k*s. A low Da prevents the jet flow from penetrating the foam to near the heated surface, resulting

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in poor cooling performance as observed. However, a large k*s promotes the transfer of heat to the tip region of the foam heat sink where the convective heat transfer greatly exceeds that in the bottom region of the sink. Accordingly, a large k*s enhances the thermal performance. A large Nufs, representing heat transfer between fluid and solid ligaments, also improves the thermal performance. Moreover, it was argued that k*s dominates among the three factors (Da, Nufs, and k*s), so that the foam samples having various pore densities had similar heat transfer characteristics because they had similar k*s.

4.4.2. Effect of Pore Density (Pore Size)

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Topologically, with fixed porosity, the heat transfer area of a foam heat sink increases as the pore density PPI is increased or, equivalently, as the pore size Dp is decreased. Therefore, the overall heat transfer performance of the sink is expected to increase with increasing pore density.

Figure 12. Effect of porosity on averaged Nusselt number of Al foam heat sink under round impinging jet cooling [39,40].

Figure 13. Effect of pore density on Nu for open-cell Al foam heat sink under circular impinging jet cooling [40]. Heat Exchangers: Types, Design, and Applications : Types, Design, and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest

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Under single round jet impingement cooling, Figure 13 plots the averaged Nusselt number Nu as a function of pore density for a given Reynolds number of 10,000 and selected separation distances [39,40]. Note the surface area of 10 PPI foam is approximately 46% of 20 PPI foam and 29 % of 40 PPI foam. When the separation distance is relatively short, e.g., H/D = 0.31, whilst the Nusselt number slightly increases as the pore density is increased from 10 PPI to 20 PPI, a further increase to 40 PPI causes an unexpected drop in Nu (Figure 13). Such decrease may be attributable to the fact that, for a given porosity, reducing the pore size further from a certain value leads to enlarged dead flow area that does not contribute to overall heat transfer. Under the conditions set in [40], one may conclude from Figure 13 that there exists an optimal pore size around 20 PPI when H/D = 0.31. In contrast, as the separation distance is increased from H/D = 0.31 to H/D = 0.92, the averaged Nu drops continuously as the pore density is increased (Figure 13).

4.4.3. Effect of Foam Height

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Configurationally, the height (Hf) of an open-cell foam block subjected to impingement cooling coincides with the nominal direction of flow discharged from the jet nozzle. It has been reported that the foam height has two conflicting effects on the cooling efficiency: on one hand an increase in Hf acts to enlarge the gas-solid interfacial heat-transfer area, resulting in enhanced cooling performance; on the other hand, it increases flow resistance, preventing the cooling flow from reaching the base plate which worsens the cooling performance. Consequently, an optimum foam height exists which maximizes the cooling performance. Slot jet: As reported by Jeng and Tzeng [43] for Al foam heat sinks under laminar slot impinging jet cooling, the Nusselt number increases as the foam height Hf/W is reduced, W being the width of the slot jet nozzle. Obtained results including contours of the stream lines and velocity vectors showed how the variation of foam height affects the overall thermal characteristics. They also demonstrated that the coolant flow rapidly becomes quasi-uniform due to the flow resistance of the foam in the transverse flow region (normal to nominal flow direction), with the coolant velocity decreasing as Hf/W is increased. As the velocity near the heated (base) surface is reduced, the overall heat transfer performance decreases as shown in Figure 14 where the Nusselt number is plotted as a function of Hf/W for Al foams having W = 5 mm, L/W = 10, C/W = 0 to 4, and Re = 100 to 500 [43]. It was suggested that the minimum Hf/W must exceed the pore diameter to ensure the completeness of the pore structure. Based on measured results like those of Figure 14, correlations for the averaged Nusselt number as a function of foam height and Reynolds number were obtained as: ⎛Hf Nu = 34.35⎜⎜ ⎝ W

⎞ ⎟ ⎟ ⎠

−0.847

Re 0.164

for ε = 0.97, 5 and 40 PPI Al foams

⎛Hf Nu = 133.39⎜⎜ ⎝ W

⎞ ⎟ ⎟ ⎠

(16)

−0.969

Re 0.0828

for ε = 0.91, 5 and 40 P PI Al foams

(17)

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23

Round jet: For Al foam heat sinks subjected to impinging round jet cooling, Shih et al. [40] experimentally studied the foam height effect on the cooling performance. For two selected Reynolds numbers, Figure 15 plots the averaged Nusselt number Nu as a function of Hf/D for 20 PPI Al foams of porosity 0.87. It is seen from Figure 15 that Nu monotonically increases till Hf/D = 0.23, followed by a steep decrease. It is then apparent that, for maximized cooling performance, Hf/D = 0.23 is preferred. The results of Figure 15 also show that Nu substantially increases as the Reynolds number is doubled.

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Figure 14. Averaged Nusselt number plotted as a function of foam height (Hf/W) for 5 PPI Al foams of porosity 0.91 under laminar impinging slot jet at Re = 500 [43].

Figure 15. Averaged Nusselt number plotted as a function of foam height (Hf/D) for 20 PPI Al foams of porosity 0.87 under round impinging jet cooling [40].

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4.4.4. Effect of Foam Width

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Under impingement jet cooling, the ratio of foam width-to-jet nozzle size (Lf/D or Lf/W) plays an important role in the heat transfer performance of metal foam heat sinks as it refers to the coverage of the coolant flow. Slot jet: For laminar slot jets, Jeng and Tzeng [43] numerically studied the effect of foam width on Al foam heat sink performance. As plotted in Figure 16(a), for Al foams with ε = 0.97 and 5 PPI, W = 10 mm, C/W = 2, Hf/W = 2, and Re = 100 to 500, both the local Nusselt number along the X-axis (see Figure 11(a)) and the stagnation Nusselt number increases as Lf/W is reduced. When Lf/W is relatively small (e.g., Lf/W = 2 or 3), the local Nusselt number was not maximal at the stagnation point but at the exit of the foam block. Figure 16 also shows that foams with larger Lf/W dissipate more heat than that with smaller Lf/W. To explain this, the stream lines and velocity vectors of the coolant flow were numerically calculated [43]. It is demonstrated that the jet flow initially penetrates the porous heat sink and then turns in the jet-transverse direction. A vortex is generated in the pure fluid region, next to the exit of the jet, due mainly to fluid entrainment. The length of the vortex typically increases as Lf/W is decreased. Increasing Lf/W causes the flow resistance within the foam to increase. When Lf/W is relatively small (e.g., Lf/W = 2), the flow goes downward at the exit of the foam in the jet-transverse direction because the flow resistance is low. Correspondingly, the local Nusselt number is maximized at the exit. When Lf/W becomes larger, such as Lf/W = 5 or 10, the flow goes upward at the exit of the foam, reducing the local Nusselt number in the jettransverse direction. Figure 16(b) shows a relationship between Lf/W and Nu. As discussed, the initial decrease in Lf/W from e.g., Lf/W = 2 causes steep decrease in Nu followed its gradual decreasing trend as Lf/W is decreased further. Round jet: For round impinging jets, Jeng and Tzeng [45] experimentally studied the effects of foam length Lf/D of a square Al foam and nozzle-to-foam tip distance C/D on the average Nusselt number. The Al foam had a porosity of 0.93 and a pore density of 10 PPI, the foam height was fixed at Hf = 25.4 mm, and the jet circular diameter D was fixed at 13 mm. The results showed that a larger Reynolds number corresponds to a larger average Nusselt number, the effect of C/D on the average Nusselt number was negligible when Lf/D = 3.0 or 4.4615, whilst the average Nusselt number slightly decreased with increasing C/D when Lf/D = 2.222. The following relationship between the average Nusselt number and jet Reynolds number was obtained:

(

Nu = 1.121 L f / D

)

1.076

× Re

(

0.777 L f / D

)−0.227

for Lf/D = 2.222–4.615 and C/D = 0–5.

(18)

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(a)

(b) Figure 16. Effect of Al foam block width (Lf/W) under laminar slot jet [43]: (a) local Nusselt number along X-axis (see Figure 11(a)); (b) averaged Nusselt number

4.4.5. Effect of Nozzle-to-Foam Tip Distance Unlike parallel flow forced convection in open-cell metallic foams, the relative position of slot/round impinging jet nozzle to both the base plate and the open-cell foam plays an important role, due mainly to the unique jet flow characteristics including the presence of a potential core. In this section, the influence of nozzle-to-foam tip position on the overall heat transfer of the foam heat sink is discussed separately for slot and round impinging jets.

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S.S. Feng, J.J. Kuang, T. Kim et al.

Figure 17. Effect of slot jet-to-sink tip distance C/W on average Nusselt number under laminar slot jet cooling [43].

Slot jet: Jeng and Tzeng [43] studied numerically the effect of jet-to-foam tip distance C/W on the averaged Nusselt number. As illustrated in Figure 17(a), a 5 PPI Al foam having ε = 0.91 was subjected to slot impinging jet cooling, with W = 5 mm, Lf/W = 10 and Hf/W = 2, and the Reynolds number was defined as a function of pore diameter, Dp. Heat was imposed from a multi-chip module having the size of 5 cm × 5 cm, attached to the base plate of the Al foam. Obtained results for laminar slot jet flow plotted in Figure 17(b) show that the jet-tofoam tip distance has negligible effect on the heat transfer characteristics. Jeng and Tzeng [43] also calculated the contours of streamlines and velocity vectors and found that, at a given Reynolds number, the flow patterns for large C/W values are similar to those for small C/W values. Round jet: Kim et al. [46] examined the effect of jet nozzle-to-heated surface distance, H/D, on the averaged Nusselt number of a 10 PPI Al foam heat sink subjected to round impinging jet cooling; see Figure 18(a). The foam block had a porosity of 0.92 and a size of

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100 mm × 90 mm×10 mm. As the foam height was fixed, the variation of H can be treated as varying the nozzle plate-to-foam tip distance, i.e., C/D. Figure 18(b) presents the experimentally measured results, where Hf/D = 0.83 and the round jet diameter D = 12 mm.

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(a)

(b) Figure 18. Effect of nozzle-to-foam tip separation distance C/D on average Nusselt number for Al foam with 10 PPI and ε = 0.92 at Re = 15,402 [46].

The results of Figure 18(b) show that the average Nusselt number initially decreases with increasing C/D in the range of 0 < C/D < 3. When C/D is increased beyond this range (up to C/D ~ 9.1), a substantial enhancement of the heat transfer is observed. With a further increase in C/D, the average Nusselt number gradually decreases due to the reduced jet momentum. Kim et al. [46] argued that the observed decrease in the heat transfer as the distance C/D is increased till 3 may be attributed to the weak recirculation of the jet flow within the foam and the reduced flow penetration into the foam. As the distance C/D exceeds 3, heat transfer increases owing to the increased momentum of the entrained air. Increasing C/D further (C/D > 9.1) causes the flow to behave as unconfined jet impingement. Consequently, the air jet

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penetrating into the foam becomes weak, resulting in weakened heat transfer. In summary, the nozzle-to-foam tip separation distance affects the overall heat transfer performance significantly when the Reynolds number is large and Lf/D (or Lf/W) is small.

4.5. Impinging Annular Jet 4.5.1. Exit Flow Structures

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Annular jet refers to a high momentum fluid stream forced out from a coaxial tube. One of the physical features differentiating the annular jet form the single round jet is the exit flow, i.e., a high momentum flow radially off from the jet axis. We have investigated the flow field of an annular jet impinging on a flat target plate with the technique of particle image velocimetry (PIV). Figure 19 displays the measured velocity vector fields in the r-z plane for Red = 7000. With the separation distance fixed at H/D2 = 1.0, the exit flow is deflected radially upon exiting the jet nozzle whilst forming a reverse stagnation region by the flow separated from the inner circular rod. The two exit flow streams, i.e., that exiting radially outward and that reversing inward, are separated by the stagnation line.

Figure 19. Velocity vector field of annular impinging jet obtained using particle image velocimetry for H/D2 = 1.0 and Red = 7000, where d = D2 - D1 [48].

4.5.2. Heat Transfer Enhancement Consider a Cu foam heat sink subjected to annular jet flow impingement. The open-cell foam has a porosity of 0.97 and a pore density of 8 PPI. Focus is placed upon the effect of foam height on the overall heat transfer performance of the heat sink. To generate the annular jet, a coaxial-inner circular road (with diameter D1) is inserted into a circular-outer tube (with inner diameter D2), as shown schematically in Figure 20.

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29

(a)

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(b) Figure 20. Schematic of open-cell metal foam heat sink subjected to annular impinging jet cooling: (a) annular jet setup; (b) heat transfer setup.

To characterize the cooling performance of the heat sink, its thermal resistance (Rθ) is defined as:

Rθ =

Tc − Tin Qnet

(19)

where Qnet is the net heat removed by the sink measured by a film heat flux gauge, Tc is the junction temperature measured at the interface between the modular heater and the base plate (at the center of the heater), and Tin is the inlet flow temperature (measured by a bead type thermocouple); see Figure 20. To exclude the heat lost out of the total heat generated by the modular heater, the heat flux is measured directly using the heat flux gauge.Figure 21 plots the measured thermal resistance as a function of foam height at a fixed volume flow rate. It is seen from Figure 21 that decreasing the foam height from Hf/D = 0.4 causes a monotonic decrease in the thermal resistance till Hf/D = 0.22. As the foam height is decreased further, the thermal resistance starts to increase slightly.

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Figure 21. Thermal resistance of Cu foam heat sink (ε = 0.97) under annular jet cooling plotted as a function of foam height at fixed volume flow rate.

5. OPEN-CELL METAL FOAMS FOR ELECTRONICS COOLING

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5.1. Background As modern electronic components such as central processing unit (CPU) tend to be operated at ever increasing high clock speed as well as miniaturized, thermal management becomes a major issue. Amongst various cooling techniques for such electronic components, single phase air supplied by axial flow fans impinging on extended surfaces (e.g., plate-fins and pin-fins) has been a popular choice due to its wide availability and low cost. In recent studies [39-41,49-52], open-celled high porosity aluminum and copper foams have been used to construct compact heat sinks for power electronics cooling. In the majority of the above-mentioned studies, heat conducted to the metal foam heat sink was cooled by circular/non-circular impinging jets or parallel forced convection flows. The effects of various topological parameters such as porosity, pore size and impinging jet velocity on the overall heat transfer characteristics of the heat sink were examined. Under impinging jet conditions, it has been established that foam height (parallel to the jet axis) is another key parameter affecting heat sink performance [41,52]. It has been argued that, due to increased interfacial heat transfer area, increasing the foam height would lead to enhanced cooling performance. However, the increase in foam height also increases flow resistance (pressure drop) and, therefore, less coolant flow reaches the base plate of the heat sink where heat is conducted from the electronic component. As a result, the overall cooling performance of the heat sink may deteriorate. Shih et al. [41] studied experimentally the two conflicting effects of foam height on the overall performance of Al foam heat sinks cooled with circular impinging jet. It is shown that an optimum foam height Hf,opt that provides the maximum cooling performance exists. For Al foams with a porosity of 0.87 and a pore density of 20

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PPI, by varying systematically the foam height from 0.15D to 0.92D, it was found that Hf,opt = 0.23D where D is the hydraulic diameter of the impinging jet [41].

5.2. Conventional Heat Sink (Exchanger) for Electronics Cooling

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In practice, an axial flow fan has been widely used to provide the coolant flow that impinges on heat sink media (e.g., plate/pin-fins) mounted on electronic devices. Sui et al. [53-55] studied the flow fields of axial fan flows as well as the pressure and heat transfer characteristics on a flat plate (or base plate of heat sink). It was shown that the swirling impinging jet supplied by an axial fan has distinct characteristics compared to conventional impinging jets: (i) a large flow area exists in the central portion of the impinging flow, which is displaced over roughly 25% of the total flow area; as a result, the impingement (direct impact of high momentum flow) occurs at the target surface area off from the driving fan axis, due to the presence of a hub within which the DC motor is housed; (ii) the impinging flow is highly unsteady (albeit periodic) caused by the rotating fan blades, in contrast to the mostly steady flow in conventional impinging jets; (iii) the impinging distance between the fan exit and base plate is relatively short, typically less than half the fan diameter due to heat sink size restrictions; (iv) no potential core exists, while its existence in conventional impinging jets is known to be influential on the heat transfer performance, especially for short impinging distances.

Figure 22. Conventional heat sink composed of plate-fin heat spreader and axial flow fan, contemporarily used for cooling a central processing unit (CPU).

5.3. Open-Cell Metal Foam Heat Sink: Flow Field and Overall Thermal Performance We experimentally examined the heat transfer characteristics of a high porosity open-cell foam heat sink made of pure copper (Cu) under axial fan flow impingement [58]. The effects of foam height and impinging distance on the overall heat transfer performance of the heat sink and the pressure distribution on its base plate were investigated. As reference, the overall heat dissipation performance of copper plate and pin-fins heat sinks was also measured under identical cooling flow conditions.

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In general, the fan exit flow decelerates axially and is deflected radially outward as it approaches the flat plate, as shown in Figure 24 by the ensemble average velocity vectors in the r-z plane. Note the presence of flow recirculation towards the central part of the plate (i.e., r/D = 0). The recirculated flows meet and interact at the central part of the plate. Conventionally, this region is called the “secondary stagnation region,” compared to the “primary stagnation region” where the exit flow impinges directly on the plate; see Can [56]. In the secondary stagnation region, the flow motion causes “suction” from the plate due to interaction of the recirculated flows.

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Figure 23. Metal foam heat sink, composed of metal foam and axial flow fan, subject to discrete heating from electronic component, e.g., central processing unit (CPU).

Figure 24. Velocity vector field of axial fan impinging jet obtained using particle image velocimetry for H/D2 = 1.0 and Red = 7000, where d = D2-D1 [58].

Consider next the radial distribution of the pressure coefficient (Cp) on the base plate of the Cu foam heat sink. As reference, the static pressure distribution on a bare (base) plate without foam attachment was also measured where the wall pressure coefficient (Cp) is defined as:

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Cp =

p s − p amb ρ V tip2 / 2

33

(20)

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where ps is the static pressure measured on the plate, pamb is the atmospheric pressure, and Vtip is the speed of fan blade tip calculated from the measured fan rotation speed. As shown in Figure 25, when the foam is relatively thick (e.g., Hf /D = 0.4), the pressure coefficient decreases slightly as moving away radially from the fan axis (r/D = 0) up to r/D = 0.45. It should be mentioned that the base plate has the same size as the fan diameter. Therefore, the outmost measurable location of the static pressure is r/D = 0.5. Reducing the foam height to Hf /D = 0.18 causes the static pressure in the central region to decrease whilst that near the edge of the base plate increases (Figure 25). A further decrease in the foam height such that Hf /D Æ 0 (i.e., bare plate) has similar effects. It is known that the impinging distance plays a crucial role in dictating the pressure and heat transfer characteristics: a small impinging distance provides a higher momentum (due to less entrainment, i.e., less flow interaction with the surrounding fluid at rest), leading to a higher pressure and better heat transfer on the target plate. The influence of impinging distance upon base plate pressure distribution is presented in Figure 26. As the impinging distance is increased, the static pressure in the central region coinciding with the axial fan axis (r/D = 0) monotonically increases whereas that near the edge of the base plate remains nearly unchanged. For large impinging distances (e.g., H/D = 0.5), the pressure near the edge of the base plate is higher than that in the central region.

Figure 25. Radial distribution of pressure coefficient Cp on base plate of Cu foam heat sinks for selected foam heights at fixed fan input voltage of 12 V and fixed impinging distance of H/D = 0.5.

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Figure 26. Distribution of pressure coefficient Cp on base plate of Cu foam heat sink for selected impinging distances with foam height fixed at Hf /D = 0.22 (Hfoam = 15 mm).

To evaluate the heat dissipation capability of Cu foam heat sinks having different foam heights, the heat sink thermal resistance is defined as:

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Rθ =

Tc − Tin Qnet

(21)

where Qnet is the net heat removed by the heat sink measured by film heat flux gauge, Tc is the junction temperature measured at the interface between the modular heater and the base plate (at the center of the heater), and Tin is the inlet flow temperature (measured by bead type thermocouple). Figure 27 plots the measured thermal resistance of Cu foam heat sinks as a function of foam height. It is seen that there exists an optimal foam thickness of approximately Hf,op/D = 0.22. This finding is consistent with that reported by Shih et al. [41] who examined the influence of foam height on the overall thermal performance of Al foam heat sinks under conventional impinging jet flow conditions. Whilst a thick foam with high heat dissipation capability causes high flow resistance (due to high momentum dissipation), a thin foam offers low flow resistance but has inadequate surface area for heat transfer, and hence an optimal foam thickness exists that minimizes the overall thermal resistance. As aforementioned, reducing the impinging distance down to the foam height may improve the overall heat dissipation. To confirm this, the impinging distance was systematically reduced from H/D = 0.5 (upper bound set by the distance typically used in plate fin and axial fan coolers for CPUs) down to H/D = 0.22 (lower bound set by foam height). It is seen from Figure 28 that the thermal resistance decreases monotonically with decreasing impinging distance till H/D = 0.25. This may be attributed to the fact that decreasing the impinging distance leads to less momentum loss as a result of fluid entrainment. In addition, as the impinging distance is reduced further (from H/D = 0.25), the thermal resistance starts to increase (Figure 28). This may be attributed to the fact that, upon exiting the axial fan, the flow needs a certain distance to develop as in conventional fin heat sink units.

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Figure 27. Thermal resistance of porous medium heat sink plotted as a function of medium height for fixed impinging distance.

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As porosity is an important topological parameter of open-cell metal foams, its influence on the cooling performance of Cu foam heat sinks was also investigated. Figure 29 shows that the foam heat sink with a porosity of 0.90 has the best cooling performance amongst the samples tested. This is may be attributed to the fact that Cu foams with a lower porosity (e.g. ε = 0.90) have a higher effective thermal conductivity which is beneficial for heat transfer, and those with a higher porosity (e.g., ε = 0.97) have lower flow resistance. Again, as foam height, the conflicting effects of porosity on flow resistance and heat transfer need to be balanced when designing an open-cell metal foam heat sink.

Figure 28. Thermal resistance of Cu foam heat sink plotted as a function of impinging distance for fixed foam height.

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Figure 29. Thermal resistance of Cu foam heat sink plotted as a function of porosity with foam height fixed at Hf /D = 0.4 and impinging distance at H/D = 0.5.

5.5. Lightweight Compact Heat Sinks

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The ever increasing heat dissipation emitted by power electronics has led to increased total size, volume and weight of the heat sinks. As a result, the mechanical loads exerted on a motherboard are increased.

(a)

(b) Figure 30. Schematic illustration of force (F) and moment (M) generated on a motherboard due to the weight of a heat sink unit (W) where g is the gravitational axis: (a) horizontal configuration; (b) vertical configuration.

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Figure 30 illustrates the two possible mechanical loading conditions depending on the relative placement of the motherboard to the direction of gravity. For the horizontal configuration (Figure 30(a)), the weight of the heat sink unit (W) placed at the center of the motherboard causes bending deformation of the latter. For vertical placement (Figure 30(b)), a moment (M) is generated by the heat sink on the motherboard. Together with the thermally induced mechanical loads, both may lead to serious reliability issues. With similar overall heat dissipation performance, Cu foam heat sinks require only 10% of the weight and 50% of the volume compared to plate/pin-fin heat sinks. The use of such lightweight compact heat sinks for electronics cooling may help to resolve the reliability concerns caused by mechanical loadings of the type envisioned in Figure 30.

Acknowledgments This work is supported by the National Basic Research Program of China (2006CB601203), the National 111 Project of China (B06024), the National Natural Science Foundation of China (10632060 and 10825210) and the National High Technology Research and Development Program (2006AA03Z519).

APPENDIX A A1. Test data for comparison (rectangular channel height H selected as characteristic length)

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Sample

Brazed Cu wire screens [37] Sintered FeCrAlY foams [34,36] Sintered Cu foams [36] LFM (LM25) [38] Kagomé (BeCu), [57] Al foams [12] Louvered fins [1] Corrugated ducts [58] Empty channel [59]

H [mm]

Porosity

Pore size [mm]

ReH

Friction factor

NuH

10

0.78

2.2

6355400

-5

64.6267

12

0.8220.917

1.0-3.0

200011000

4-80

57-217

12

0.8810.94

0.55-3.0

13-88

35-332

12

0.938

12.7-14.7

~0.61

57-220

12

0.9260.971

12.2-14.7

0.3-0.64

50-257

-45

0.92

1.6-3.9

6.1-13.5

39-140

0.06-0.4

8-53.5

0.22-0.4

9-30

0.020.64

8.235

0.020.04

12-781

-4.45 Laminar flow Turbulent flow

f = 64/Re, Nu ~ constant Nu=0.023(Re)0. 8 Pr0.4 by Dittus & Boelter

5508200 150018000 120027000 5702800 60010000 2502000 1002500 250030000

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S.S. Feng, J.J. Kuang, T. Kim et al. A2. Test data for sintered FeCrAlY and Cu foams (H = 12mm for all samples) [34,36] Sample

FeCrAlY foam

Cu foam

Sample number

U [m/s]

h [W/m2K]

ReH

Friction factor

NuH

S-1 S-2 S-3 S-4 S-5 S-6 S-7 S-9 S-10 S-11 S-12 S-13 S-14

2.6-14.2 4.4-10.9 3.64-10.7 2.8-8.71 2.66-6.89 2.2-5.3 2.25-8.3 1.96-10 1.72-8.9 1.64-7.1 0.97-4.52 0.67-4.86 0.81-4

130-393 293-477 147-302 174.7-349 124.8-281 173-341 138-312 217-728 176-686 132-552 102-533 98-653 77-578

2000-11000 3600-9000 3000-9000 2300-7200 2200-5500 2000-4400 2000-7000 1600-8200 1400-7400 1350-6000 800-4400 550-4000 600-3400

4 9.5 8.5 12.5 22 80 11.5 13 18 40 52 65 88

59-173 133-217 67-137 79-159 57-128 79-156 63-142 99-332 80-312 60-254 46-248 44-298 35-263

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NOMENCLATURE Ai Ao C Cp df D D1 D2 Df Dp Dh Dhub H Hf ks Lf Nu Nus Nus0 pamb ps Qnet

average displaced area within hollow foam ligaments, m2 average total area of foam ligament, m2 jet-to-foam tip distance, (= H - Hf), m wall static pressure coefficient (=(ps-pamb)/(ρVtip2/2)) ligament diameter (or thickness), m axial fan diameter (or circular jet diameter), m inner circular rod diameter of annular jet, m outer flow tube diameter of annular jet, m circular foam block diameter, m representative (average) pore diameter of open-cell foam, m hydraulic diameter of rectangular flow channel, m axial fan hub diameter, m distance between fan (jet nozzle) exit and base plate (and rectangular channel height), m metal foam height, m solid thermal conductivity, W/(mK) length of metal foam block, m Nusselt number Nusselt number at stagnation point under impinging jet Nusselt number at stagnation point under impinging jet for bare plate ambient pressure as reference, Pa static pressure on base plate, Pa net heat imposed on heat sink measured using film heat flux gauge, W

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Lightweight Compact Heat Exchangers with Open-Cell Metal Foams Re ReDp Rθ S Tin Tc VFR Vtip W Wf

Reynolds number Reynolds number based on pore diameter, Dp thermal resistance (=(Tc-Tin)/Qnet), K/W axial fan blade span, m inlet (ambient) temperature, K junction temperature between heat sink source at the center of modular heater, K volume flow rate, m3/s axial fan blade tip speed (=πDΩ / 60), m/s width of slot jet nozzle, m width of foam sample, m

base

plate

39

and

heat

Greek Symbols ρ ρrel ρsurface μ Ω

density of air, kg/m3 relative density surface area density, m2/m3 viscosity of air, kg/(ms) fan rotation speed, rpm

Abbreviations

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PPI rpm

pores per inch revolution per minute

REFERENCES [1] [2] [3] [4]

[5] [6] [7]

Kays, WM; London, AL. “Compact Heat Exchangers,” 3rd ed., McGraw-Hill, London, 1984. Lu, TJ. “Ultralight Porous Metals: from Fundamentals to Applications,” Acta Mechanica Sinica, 2002, 18, 457-479. Evans, AG; Hutchinson, JW; Ashby, MF. “Multifunctionality of Cellular Metal Systems,” Progress in Materials Science, 1999, 43, 171-221. Chiras, S; Mumm, DR; Evans, AG; Wicks, N; Hutchinson, JW; Dharmasena, K; Wadley, HNG; Fichter, S. “The Structural Performance of Near-Optimized Truss Core Panels,” Int. J. Solid Structures, 2000, 39 4093-4115. Hunt, ML; Tien, CL. “Effects of Thermal Dispersion on Forced Convection in Fibrous Media,” Int. J. Heat Mass Transfer, 1988, 31, 301-309. Sathe, SB; Peck, RE; Tong, TW. “A Numerical Analysis of Heat Transfer and Combustion in Porous Radiant Burners,” Int. J. Heat Mass Transf, 1990, 33 1331-1338. Younis, LB; Viskanta, R. “Experimental Determination of the Volumetric Heat

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S.S. Feng, J.J. Kuang, T. Kim et al. Transfer Coefficient Between Stream of Air and Ceramic Foam,” Int. J. Heat Mass Transfer, 1993, 36, 1425-1434. Lee, DY; Vafai, K. “Analytical Characterization and Conceptual Assessment of Solid and Fluid Temperature Differentials in Porous Media,” Int. J. Heat Mass Transfer, 1999, 42, 423-435. Lu, TJ; Stone, HA; Ashby, MF. “Heat Transfer in Open-Cell Metal Foams,” Acta Mater., 1998, 46, 3619-3635. Bastawros, AF; Evans, AG. “Proceedings of the Symposium on the Application of Heat Transfer in Microelectronics Packaging,” IMECE, Dallas, TX, 1997. Calmidi, VV; Mahajan, RL. “The Effective Thermal Conductivity of High Porosity Metal Foams,” J. Heat Transfer, 1999, 121, 466-471. Calmidi, VV; Mahajan, R.L. “Forced Convection in High Porosity Metal Foams,” J. Heat Transfer, 2000, 122, 557-565. Kim, SY; Kang, BH; Kim, JH. “Forced Convection from Aluminum Foam Materials in an Asymmetrically Heated Channel,” Int. J. Heat Mass Transfer, 2001, 44, 1451-1454. Paek, JW; Kang, BH; Kim, SY; Hyun, JM. “Effective Thermal Conductivity and Permeability of Aluminum Foam Materials,” Int. J. Thermophys, 2000, 21, 453-464. Boomsma, K; Poulikakos, D; Zwick, F. “Metal Foams as Compact High Performance Heat Exchangers,” Mech. Mater, 2003, 35, 1161-1176. Mahjoob, S; Vafai, K. “A Synthesis of Fluid and Thermal Transport for Metal Foam Heat Exchangers,” Int. J. Heat Mass Transfer, 2008, 51, 3701-3711. Alazmi, B; Vafai, K. “Analysis of Variants within the Porous Media Transport Models, J. Heat Transfer, 2000, 122, 303-326. Yu, Q; Straatman, AG. Thompson, BE. “Carbon-foam Finned Tubes in Air-water Heat Exchangers,” J. Appl. Thermal. Eng., 2006, 26, 131-143. Davies, GJ; Zhen, S. “Review: Metallic Foams: Their Productions, Properties, and Applications,” J. Material Sci., 1983, 18, 1899-1911. Banhart, J. “Manufacture, Characterization and Application of Cellular Metals and Metal Foams,” Prog. Mater. Sci., 2001, 46, 559-632. Ashby, MF; Evans, AG; Fleck, NA. “Metal Foam: a Design Guide”, ButterworthHeinemann, Boston, 2000. Trussell, RR; Chang, M. “Review of Flow through Porous Media as Applied to Head Loss in Water Filters,” J. Environ. Eng., 1999, 998-1006. Burcharth, HF; Andersen, OH. “On the One-dimensional Steady and Unsteady Porous Flow Equations,” Costal Eng., 1995, 24, 233-257. Lage, JL; Antohe, BV. “Darcy’s Experiments and the Deviation to Non-linear Flow Regime,” J. Fluids Eng., 2000, 122, 619-625. Sabiri, NE; Comiti, J. “Pressure Drop in Non-Newtonian Purely Viscous Fluid Flow Through Porous Media,” Chem. Eng. Sci., 1995, 50, 1193-1201. Venkataraman, P; Rao, PRM. “Darcian, Transitional, and Turbulent Flow Through Porous Media,” J. Hydra. Eng., 1998, 840-846. Legrand, J. “Revisited Analysis of Pressure Drop in Flow Through Crushed Rocks,” J. Hydra. Eng., 2002, 1027-1031. Liu, JF; Wu, WT; Chiu, WC; Hsieh, WH. “Measurement and Correlation of Friction Characteristic of Flow Through Foam Matrixes,” Exp. Thermal Fluid Sci., 2006, 30, 329-336.

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Lightweight Compact Heat Exchangers with Open-Cell Metal Foams

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[29] Lu, W; Zhao, CY; Tassou, SA. “Thermal Analysis on Metal-foam Filled Heat Exchangers. Part I: Metal-foam Filled Pipes,” Int. J. Heat Mass Transfer, 2006, 49, 2751-2761. [30] Salas, KI; Waas, AM. “Convective Heat Transfer in Open-cell Metal Foams,” J. Heat Transfer, 2007, 129, 1217-1229. [31] Irmay, S. “On the Theoretical Derivation of Darcy and Forchheimer Formulas,” Trans. Amer. Geophys. Union, 1958, 39, 702-707. [32] Kim, T; Song, SJ; Lu, TJ. “Fluid Flow and Heat Transfer Measurement Techniques,” Xi’an Jiaotong University Press, Xi’an, 2009. [33] Hwang, JJ; Hwang, GJ; Yeh, RH; Chao, CH. “Measurement of Interstitial Convective Heat Transfer and Frictional Drag for Flow across Metal Foams,” J. Heat Transfer, 2002, 124, 120-129. [34] Fuller, AJ; Kim, T; Hodson, HP; Lu, TJ. “Measurements and Interpretation of the Heat Transfer Coefficients of Metal Foams,” J. Mecha. Eng. Sci. Part C, 2005, 219, 183191. [35] Kim, SY; Paek, JW; Kang, BH. “Flow and Heat Transfer Correlations for Porous Fin in a Plate-fin Heat Exchanger,” J. Heat Transfer, 2000, 122, 572-578. [36] Zhao, CY; Kim, T; Lu, TJ; Hodson, HP. “Thermal Transport in High Porosity Cellular Metal Foams,” AIAA J. Thermofluid Phys., 2004, 18, 309-317. [37] Tian, J; Lu, TJ; Hodson, HP; Queheillalt, DT; Wadley, HNG. “Cross Flow Heat Exchange of Textile Cellular Metal Core Sandwich Panels,” Int. J. Heat Mass Transfer, 2007, 50, 2521-2536. [38] Kim, T. “Fluid-flow and Heat-transfer in a Lattice-Frame Material,” Ph.D. Thesis, University of Cambridge, UK 2004. [39] Hsieh, WH; Wu, JY; Shih, WH; Chiu, WC. “Experimental Investigation of Heattransfer Characteristics of Aluminum-foam Heat Sinks,” Int. J. Heat Mass Transfer, 2004, 47, 5149-5157. [40] Shih, WH; Chiu, WC; Hsieh, WH. “Height Effect on Heat-transfer Characteristics of Aluminum-foam Heat Sinks," J. Heat Transfer, 2006, 128, 530-537. [41] Shih, WH; Chou, FC; Hsieh, WH. “Experimental Investigation of the Heat transfer Characteristics of Aluminum-foam Heat Sinks with Restricted Flow Outlet,” J. Heat Transfer, 2007, 129, 1554-1563. [42] Jeng, TM; Tzeng, SC. “Numerical Study of Confined Slot Jet Impinging on Porous Metallic Foam Heat Sink,” Int. J. Heat Mass Transfer, 2005, 48, 4685-4694. [43] Jeng, TM; Tzeng, SC. “Forced Convection of Metallic Foam Heat Sink under Laminar Slot Jet Confined by Parallel Wall,” Heat Transfer Eng., 2007, 28, 484-495. [44] Jeng, TM; Tzeng, SC. “Experimental Study of Forced Convection in Metallic Porous Block Subject to a Confined Slot Jet,” Int. J. Thermal Sci., 2007, 46, 1242-1250. [45] Jeng, TM; Tzeng, SC; Liu, TC. “Heat Transfer Behavior in a Rotating Aluminum Foam Heat Sink with a Circular Impinging Jet,” Int. J. Heat Mass Transfer, 2008, 51, 12051215. [46] Kim, SY; Lee, MH; Lee, KS. "Heat removal by aluminum-foam heat sinks in a multiair jet impingement," Compo. Pack. Technol., 2005, 28, 142-148. [47] Martin, H. “Heat and Mass Transfer between Impinging Gas Jet and Solid Surface,” Adv Heat Transfer, 1977, 13, 1-60. [48] Yang, HQ; Kim, T; Lu, TJ; Ichimiya, K. “Flow Structure, Wall pressure, and Heat

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Transfer Characteristics of Impinging Annular Jet with/without Steady Swirling,” Int. J. Heat Mass Transfer, 2010, in press. Jeng, TM; Tzeng, SC. “Numerical Study of Confined Slot Jet Impinging on Porous Metallic Foam Heat Sink,” Int. J. Heat Mass Transfer, 48, 2005, 4685-4694. Kim, SY; Lee, MH; Lee, KS. “Heat Removal by Aluminum-foam Heat Sinks in a Multi-air Jet Impingement,” Compo. Pack. Technol., 2005. Mahdi, H; Lopez, P; Jr., Fuentes, A; Jones, R. “Thermal Performance of Aluminumfoam CPU Heat Exchangers,” Int. J. Energy Research, 2006, 30, 851-860. Jeng, TM; Tzeng, SC; Liu, TC. “Heat Transfer Behavior in a Rotating Aluminum Foam Heat Sink with a Circular Impinging Jet,” Int. J. Heat M ass Transfer, 2008, 51, 12051215. Sui, D; Kim, T;. Xu, ML; Lu, TJ. “Flow and Heat Transfer Characteristics of Impinging Axial Fan Flows on a Uniformly Heat Flat Plate,” Int. J. Transport Phenomena, 2008, 10, 353-363. Sui, D; Kim, T; Wang, SS; Mao, JR; Lu, TJ. “Exit Flow Behaviour of Axial Fan Flows with/without Impingement,” J. Fluids Eng., 2009, 131, 061103. Sui, D; Kim, T; Xu, ML; Lu, TJ. “Novel Hub Fins for Axial Flow Fans for the Enhancement of Impingement Heat Transfer on a Flat Plate,” J. Heat Transfer, 2009, 131, 074502. Can, M. “Experimental Optimization of Air Jets Impinging on a Continuously Moving Flat Plate,” Heat Mass Transfer, 2003, 39, 509-517. Hoffmann, F. “Heat Transfer Performance and Pressure Drop of Kagomé Core Metal Truss Panels,” M. Phil. Thesis, University of Cambridge, UK, 2002. Blomerius, H; Holsken, C; Mitra, NK. “Numerical Investigation of Flow Field and Heat Transfer in Cross-corrugated Ducts,” J. Heat Transfer, 1999, 121, 314-321. Bejan, A. “Convection Heat Transfer,” 2nd Ed. Wiley International Science, 1995.

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S.S. Feng, J.J. Kuang, T. Kim et al.

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In: Heat Exchangers Editor: Spencer T. Branson

ISBN: 978-1-61761-308-1 © 2011 Nova Science Publishers, Inc.

Chapter 2

THE NTU-EFFECTIVENESS METHOD Sylvain Lalot* Univ Lille Nord de France F-59000 Lille France

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ABSTRACT The NTU-Effectiveness method is one of the most powerful methods to design heat exchangers or to assess their performance, when they are used in steady states. In this chapter, the analyses of the most common heat exchanger types are presented (parallel flow, counter flow, 1-2, cross flow with only one fluid mixed, cross flow with both fluids mixed). All these configurations are studied in detail, from a schematic drawing to the final equations. For more complex configurations (1-2n, cross flow with both fluids unmixed), analytical equations are given along with approximations. Some examples are also given for heat exchanger networks. In all cases a Scilab script is given to compute the effectiveness from the number of transfer units and vice versa, and charts are plotted. For the analyses, the Number of Transfer Units, the effectiveness, and the heat capacity rate ratio are defined for the hot fluid as well as for the cold fluid, so that trial and error procedures are not necessary. The detailed solutions of some representative problems are given, and only the numerical results are given for the other exercises.

1. INTRODUCTION The NTU-Effectiveness method is efficient in both the design phase of heat exchangers and when it is necessary to check that the performance of a heat exchanger (when installed) corresponds to what was estimated. It is necessary to make the following assumptions to be able to develop the calculations: 1. the heat exchanger is in a steady state 2. the heat loss to the surroundings is negligible 3. there is no heat source or heat sink in the heat exchanger

*

E-mail address: [email protected] phone: +33 327 511 973 fax: +33 327 511 961. (Corresponding author)

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44

Sylvain Lalot 4. for flows inside a tube or a channel, the bulk temperature is representative, and only depends on the position along the fluid path 5. for one path, the fluid can be mixed (the temperature varies in one direction only), or it can be unmixed (the temperature depends on two directions) 6. the thermo-physical characteristics of the fluids are constant (do not vary with temperature) 7. the fluids are well distributed in the headers 8. the thermo-physical characteristics of the materials used for the walls between the fluids and for the extended surfaces are constant (do not vary with temperature) 9. the geometry of the tubes or channels does not vary along the fluid paths 10. the heat exchange surface area is evenly distributed along the fluid paths and between passes 11. there is no conduction in the fluids and in the separating walls in the flow direction 12. there are only two fluids in the heat exchanger

There are two important consequences that can be drawn immediately from these assumptions. Firstly, from the first, second, third, and twelfth assumptions, the heat that is lost by the hot fluid is gained by the cold fluid. This can be written as follows:

(

)

(

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& =m  Q & h c h Th ,in − Th ,out = m & c c c Tc,out − Tc,in

)

(1)

Secondly, from the first, sixth, seventh, eighth, ninth, and tenth assumptions, the Prandtl numbers and the Reynolds (or Grashof) numbers are constant. Hence, the Nusselt (or Stanton) numbers are also constant, leading to a constant overall heat transfer coefficient. It has to be noted that to calculate this overall heat transfer coefficient, the properties of the fluids are to be taken at the average temperature between the inlet and the outlet. Hence, it is necessary to know the four temperatures to accurately know the effectiveness of the heat exchanger. Considering one fluid, the comparison between the heat that is actually exchanged and what could be exchanged if the heat exchanger was perfect, leads to the definition of the effectiveness for this fluid. It is the ratio of these two quantities. So, for the hot fluid, the effectiveness is defined as follows: Eh =

& h c h (Th ,in − Th ,out ) m & h c h (Th ,in − Tc ,in ) m

⇒

Eh =

Th ,in − Th ,out , if the heat exchanger is perfect the outlet Th ,in − Tc ,in

temperature of the hot fluid is the inlet temperature of the cold fluid. The effectiveness for the cold fluid is defined in the same way: Ec =

& c c c (Tc ,out − Tc ,in ) m & c c c (Th ,in − Tc ,in ) m

⇒

Ec =

Tc ,out − Tc ,in , if the heat exchanger is perfect the outlet Th ,in − Tc ,in

temperature of the cold fluid is the inlet temperature of the hot fluid. & hch Introducing the heat capacity rate ratios defined as R h = m & ccc m

& ccc 1 , and R c = m = & hch m

Rh

and taking account of the energy balance (equation 1), it is possible to find a relation between the effectiveness for the hot fluid and the effectiveness for the cold fluid:

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The NTU-Effectiveness Method Eh =

45

& h c h (Th ,in − Th ,out ) m & cccm & h c h (Th ,in − Th ,out ) m & cccm & c c c (Tc ,out − Tc ,in ) m = = & h c h (Th ,in − Tc ,in ) & hch m & c c c (Th ,in − Tc ,in ) m & hch m & c c c (Th ,in − Tc ,in ) m m

⇒

Eh = R cEc

This relation can be reversed as follows: Ec =

1 Eh Rc

⇒

Ec = R hEh

It is important to note that if the fluids have very dissimilar heat capacity rates, the effectivenesses are also dissimilar. For example, if the cold fluid has a heat capacity rate twice as large as the heat capacity rate of the hot fluid, i.e. R h = 0.5 , and if the effectiveness for the hot fluid is E h = 0.8 , then the effectiveness for the cold fluid is E c = R h E h = 0.4 . To get a high effectiveness for both fluids, the heat capacity rate ratio should be close to unity. It is also possible to make the comparison between the heat that could be exchanged if the heat exchanger is perfect, and the heat that could be exchanged between the two fluids if they are at their inlet temperature all along the heat exchanger. This defines the Number of Transfer Units:

NTU h =

U A A (Th ,in − Tc ,in ) ⇒ NTU h = U A A for the hot fluid, and & hch & h c h (Th ,in − Tc ,in ) m m

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NTU c =

U A U A A (Th ,in − Tc ,in ) ⇒ NTU c = A for the cold fluid. & ccc m & c c c (Th ,in − Tc ,in ) m

These two Numbers of Transfer Units are linked by the following equation:

NTU h =

& ccc U A A m & ccc UAA UAA m = = & hch m & hch m & ccc m & ccc m & hch m

⇒

NTU h = R c NTU c

This equation can also be reversed as: NTU c =

1 NTU h Rc

⇒

NTU c = R h NTU h

It is then not necessary to find equations for both fluids. In this chapter, only the equations for one fluid (generally the hot fluid) will be given. The NTU-Effectiveness method can be used to determine the size of the heat exchanger (through the heat exchange surface area) knowing the inlet and outlet temperatures (sizing procedure). In this case, the equation needed is the one linking the Number of Transfer Units to the effectiveness. It can also be used to check the performance of a heat exchanger, once installed. In that case, the equation needed is the one linking the effectiveness to the Number

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466

Syylvain Lalot

off Transfer Units. U So, whhen possible, both equatioons will be given g in the following deevelopments. The first coonfiguration studied s in thiis chapter is the parallel flow heat exxchanger.

2. THE PARALLEL FLOW HEA AT EXCHA ANGER A simple example e of a parallel p flow heat exchangeer is a tube inn tube heat exxchanger as shhown in Figurre 1. It can be scchematized as shown in Figuure 2. The heat th hat is lost by the hot fluid in the elemenntary section of the heat exxchanger is gaained by the cold c fluid. Thiis heat is linkeed to the tempperature differrence. So, twoo equalities caan be written:

& c c c dTc = − m & h c h dT m Th = U A dA (Th − Tc ) ntary temperatture increases are expressedd as follows: The elemen

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dTc =

U UA Th = − A dA(Th − Tc ) . dA (Th − Tc ) , dT & hch & ccc m m

Fiigure 1. Cross section s of a tubee in tube heat exxchanger (the innner tube havinng extended surffaces).

Fiigure 2. schemaatic of a parallell flow heat exchhanger (the inleet is on the left hand h side).

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The NTU-Effectiveness Method

47

Subtracting the second expression to the first one: ⎛ UA U − A dTh − dTc = ⎜⎜ − & & m c m h h ccc ⎝

⎞ ⎟⎟dA(Th − Tc ) ⎠

leads to the differential equation for the temperature difference: ⎛ U d(Th − Tc ) U = −⎜⎜ A + A (Th − Tc ) ⎝ m& h c h m& c c c

⎞ ⎟⎟dA ⎠

It is then possible to integrate this equation between the inlet of the heat exchanger, where A = 0 , and the outlet of the heat exchanger, where the heat exchange surface area is the total area A .

⎛ Th ,out − Tc,out Ln⎜⎜ ⎝ Th ,in − Tc,in

⎞ ⎛ U A U A⎞ ⎟ = −⎜ A + A ⎟ ⎜m ⎟ & c c c ⎟⎠ ⎝ & hch m ⎠

This can be written as:

⎛ − Th ,in + Th ,out + Th ,in − Tc,in + Tc,in − Tc,out Ln⎜⎜ Th ,in − Tc,in ⎝

⎞ ⎟ = −(NTU h + NTU c ) ⎟ ⎠

or:

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Ln (− E h + 1 − E c ) = Ln (− E h + 1 − R h E h ) = − (NTU h + R h NTU h )

and finally: ⎛ ⎞ 1 ⎟ Ln⎜⎜ 1 − E h (1 + R h ) ⎟⎠ ⎛ ⎞ ⇒ 1 1 ⎝ ⎟ NTU h = NTU h = Ln⎜ 1+ Rh 1 + R h ⎜⎝ 1 − E h − R h E h ⎟⎠

Starting from Ln(− E h + 1 − R h E h ) = −(NTU h + R h NTU h ) , it is possible to find the reverse equation:

1 − E h (1 + R h ) = exp[− (NTU h + R h NTU h )] = exp[− NTU h (1 + R h )] that can be written as: Eh =

1 − exp[− NTU h (1 + R h )] 1+ R h

Figure 3 shows a representation of these two equations.

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488

Syylvain Lalot

Fiigure 3. E h verssus NTU h (as a function of R h ) for a paralleel flow heat excchanger.

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

The implan ntation in Scilaab1 is straightfforward:

3. THE COUNTER FLOW HEA AT EXCHAN NGER The constrruction of a counter c flow heat h exchangeer is similar to t the construuction of a paarallel flow heeat exchangerr; the only diff fference is thatt the fluids floow in oppositee direction, innstead of flowiing in the sam me direction. It can be scchematized as shown in Figuure 4. 1

F (open sourcee) scientific softw Free ware available at http://www.scilab h b.org/

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The NTU-E Effectiveness Method M

49

The heat th hat is lost by the hot fluid in the elemenntary section of the heat exxchanger is gaained by the cold c fluid. Thiis heat is linkeed to the tempperature differrence. So, twoo equalities caan be written: & c c c dT & h c h dTh = U A dA (Th − Tc ) m Tc = m

Fiigure 4. schemaatic of a counterr flow heat exchhanger (the inleet, where

A = 0 , is on the leftt hand side).

The elemen ntary temperatture increases are expressedd as follows: dTc =

UA UA d (Th − Tc ) , dT dA d h = dA A (Th − Tc ) . & hch & ccc m m

Subtracting g the second expression to thhe first one: ⎛ U U dTh − dT Tc = ⎜⎜ A − A & & ⎝ m h ch mccc

⎞ ⎟⎟dA(Th − Tc ) ⎠

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leeads to the diffferential equattion for the tem mperature diffference: d(Th − Tc ) ⎛ U A U = ⎜⎜ − A (Th − Tc ) ⎝ m& h c h m& c c c

⎞ ⎟⎟dA A ⎠

It is then possible p to inntegrate this equation e betw ween the inlett of the heat exchanger, w where A = 0 , Tc = Tc ,in , Th = Th ,out , and the outlet off the heat excchanger, wherre the heat exxchange surface area is the total t area A , Tc = Tc,out , annd Th = Th ,in . ⎛ Th ,in − Tc ,out Ln⎜⎜ ⎝ Th ,out − Tc ,in

⎞ ⎛ UAA UAA ⎞ ⎟=⎜ ⎟ − ⎟ ⎜m & c c c ⎟⎠ ⎠ ⎝ & hch m

This can bee written as: ⎛ Th ,in − Tc ,in + Tc ,in − Tc ,out Ln⎜⎜ ⎝ Th ,out − Th ,in + Th ,in − Tc ,in

⎞ ⎟ = (NTU h − NTU c ) ⎟ ⎠

orr:

⎛ (Th ,in − Tc,in ) − E c (Th ,in − Tc,ini ) ⎞ ⎛ 1− Ec ⎟ = Ln⎜ Ln⎜⎜ ⎜1− E ⎟ h ⎝ ⎝ − E h (Th ,in − Tc,ini ) + (Th ,in − Tc,in ) ⎠

⎞ ⎟⎟ = (NTU U h − R h NTU h ) ⎠

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50

Sylvain Lalot

and finally: ⎛ 1− Eh Ln⎜⎜ 1− EhR h ⎛1− R hEh ⎞ ⇒ 1 ⎟⎟ NTU h = ⎝ NTU h = Ln⎜⎜ R h −1 1− R h ⎝ 1− Eh ⎠

Starting from Ln⎛⎜ 1 − E c ⎜1− E h ⎝ equation:

⎞ ⎟⎟ ⎠

⎞ ⎟⎟ = (NTU h − R h NTU h ) , it is possible to find the reverse ⎠

1− EhR h = exp[(NTU h − R h NTU h )] = exp[NTU h (1 − R h )] ⇒ 1− Eh

1 − E h R h = (1 − E h ) exp[NTU h (1 − R h )] ⇒ 1 − exp[NTU h (1 − R h )] = E h R h − E h exp[NTU h (1 − R h )] that can be written as: Eh =

1 − exp[NTU h (1 − R h )] R h − exp[NTU h (1 − R h )]

It is important to note that the expressions obtained for the counter flow heat exchanger are not valid when the heat capacity rates are equal. In this special case, the starting equation & c c c dTc = m & h c h dTh = U A dA(Th − Tc ) shows that dTc = dTh , which can be written as m Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

dTh − dTc = 0 , or

d(Th − Tc ) = 0 . This means that the temperature difference all along the

heat exchanger is constant. It is then easy to integrate the second part of the starting equation.

& h c h dTh = U A dA(Th − Tc ) m ⇒ dTh =

U A dA (T − Tc ) = (Th − Tc )dNTU h & hch h m

(

)

⇒ Th ,in − Th ,out = Th ,out − Tc,in NTUh

The image part with relationship ID rId168 was not found in the file.

E h (Th ,in − Tc,in ) = (Th ,out − Th ,in + Th ,in − Tc,in )NTU h = (− E h + 1)(Th ,in − Tc,in )NTU h

and finally: NTU h (R h =1) =

Eh or E h (R =1) = NTU h h (1 − E h ) 1 + NTU h

Figure 5 shows a representation of these two equations.

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The NTU-E Effectiveness Method M

Fiigure 5. E h verssus NTU h (as a function of R h ) for a counteer flow heat excchanger.

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

The implan ntation in Scilaab is as follow ws:

Heat Exchangers: Types, Design, and Applications : Types, Design, and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest

51

522

Syylvain Lalot

4.. THE 1-2 HEAT EXC CHANGER 4.1. First Configuration n (Inlets on the t Same Siide) There are different d confiigurations for a 1-2 heat excchanger. Figurre 6 shows a 1 shell pass 2 tube passes co onfiguration for f which the inlets i are on thhe same side. This config guration can be schematizedd as shown in figure 7. In the elem mentary sectionn of the heat exchanger, threee basic equattions can be written w

& 2 c 2 dT2 = U A m

1 dA (T1 − T2 ) 2

(2)

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Fiigure 6. schemaatic representatiion of a 1 shell pass p 2 tube passes heat exchannger.

Fiigure 7. First 1--2 configurationn studied.

& 2 c 2 dT2' = − U A m

(

(

1 dA T1 − T2' 2

)

)

& 2 c 2 dT2 − dT2' = −m & 1c1dT m T1

(3) (4)

Between th his elementaryy section and the right hannd side of thee heat exchangger, all the d by fluid 1 is transferred to fluid 2. So, a fourth basic equation e can be b written: heeat exchanged

(

)

& 2 c 2 T2' − T2 = m & 1c1 (T1 − T1,out ) m

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(5)

The NTU-Effectiveness Method

53

Equation 2 can be written as:

dT2 =

& 1 c1 UA 1 U 1m R dA(T1 − T2 ) = A dA(T1 − T2 ) = 1 dNTU1 (T1 − T2 ) & 2c2 2 & 1 c1 2 m & 2c2 m m 2

⇒ with R 1 =

dT2 R = 1 (T1 − T2 ) dNTU 1 2

(2')

& 1 c1 m & 2c2 m

Equation 3 can be written as: dT2' = −

& 1 c1 UA 1 U 1 m R dA (T1 − T2' ) = A dA(T1 − T2' ) = − 1 dNTU 2 (T1 − T2' ) & 2c2 2 & 1 c1 2 m & 2c2 m m 2 ' ⇒ dT2

dNTU 1

=−

(

R1 T1 − T2' 2

)

(3')

Equation 4 can be written as: dT2 − dT2' = − R 1dT1

⇒

dT2 dT1 dT2' + R1 = dNTU 1 dNTU 1 dNTU 1

(4')

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Equation 5 can be written as:

T2' − T2 = R 1 (T1 − T1,out ) ⇒ T2' = T2 + R 1 (T1 − T1,out )

(5')

The introduction of equation 2' and 3' in equation 4' leads to:

(

)

R1 (T1 − T2 ) + R 1 dT1 = − R 1 T1 − T2' and using equation 5': 2 dNTU 1 2 R1 (T1 − T2 ) + R 1 dT1 = − R 1 (T1 − (T2 + R 1 (T1 − T1,out ))) or 2 dNTU 1 2

(T1 − T2 ) +

dT1 R = 1 (T1 − T1,out ) dNTU 1 2

that can be written as T2 = T1 +

dT1 R − 1 (T1 − T1,out ) and finally dNTU 1 2

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(6)

54

Sylvain Lalot T2 − T1,out = T1 − T1,out +

( ) d(T1 − T1,out ) R 1 (T1 − T1,out ) = d T1 − T1,out + ⎛⎜1 − R 1 ⎞⎟(T1 − T1,out ) − dNTU1 2 dNTU1 2 ⎠ ⎝

(7)

The derivation of this equation with respect to NTU1 leads to : d(T2 − T1,out ) dNTU1

=

d 2 (T1 − T1,out ) ⎛ R 1 ⎞ d(T1 − T1,out ) dT2 = + ⎜1 − ⎟ 2 dNTU1 2 ⎠ dNTU1 dNTU1 ⎝

The introduction of equation 2' in this equation leads to: d 2 (T1 − T1,out ) ⎛ R 1 ⎞ d(T1 − T1,out ) R1 and using equation 6: (T1 − T2 ) = + ⎜1 − ⎟ 2 2 ⎠ dNTU1 2 dNTU1 ⎝ 2 R1 ⎛ R1 dT1 ⎞ d (T1 − T1,out ) ⎛ R 1 ⎞ d(T1 − T1,out ) , or ⎜⎜ (T1 − T1,out ) − ⎟⎟ = + ⎜1 − ⎟ 2 2 ⎝ 2 dNTU1 ⎠ 2 ⎠ dNTU1 dNTU1 ⎝

d 2 (T1 − T1,out ) d(T1 − T1,out ) R 1 2 (T1 − T1,out ) = 0 + − 2 dNTU1 4 dNTU1

(8)

The general solution of this differential equation is:

(T

1

− T1,out ) = A exp(λ 1 NTU 1 ) + B exp(λ 2 NTU 1 )

with λ1 and λ 2 that are the roots of the following equation: 2

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x2 + x −

R1 = 0 (the coefficients are those of equation 7). Hence, these values are 4

known as: λ1 = −

1 + 1 + R1 2

2

and λ 2 =

−1 + 1 + R1

2

2

At the left hand side of the heat exchanger (inlet of the fluids), the temperature is T1,in , and the Number of Transfer Units is NTU 1 = 0 . At the right hand side of the heat exchanger, the temperature is T1,out , and the Number of Transfer Units is NTU 1 = NTU 1,1− 2 . Thus, the integration constants are the solutions of the following system: ⎧(T1,in − T1,out ) = A + B ⎨ ⎩0 = A exp(λ 1 NTU 1,1− 2 ) + B exp(λ 2 NTU 1,1− 2 )

(

)

(

)

Writing A = α T1,in − T1,out and B = β T1,in − T1,out , this system becomes: ⎧1 = α + β ⎨ ⎩0 = α exp(λ1 NTU1,1− 2 ) + β exp(λ 2 NTU1,1− 2 )

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The NTU-Effectiveness Method

55

Taking α as a function β from the first equation of this system, the second equation of the system is: 0 = (1 − β) exp(λ1 NTU1,1−2 ) + β exp(λ 2 NTU1,1−2 ) or β=

exp(λ1 NTU1,1−2 ) 1 1 = = exp(λ1 NTU1,1−2 ) − exp(λ 2 NTU1,1−2 ) 1 − exp((λ 2 − λ1 )NTU1,1−2 ) 1 − exp⎛⎜ 1 + R 2 NTU ⎞⎟ 1 1,1− 2 ⎝ ⎠

This leads to: 2 2 exp⎛⎜ 1 + R 1 NTU1,1−2 ⎞⎟ − exp⎛⎜ 1 + R 1 NTU1,1− 2 ⎞⎟ ⎠ ⎝ ⎠ ⎝ = = α = 1− 2 2 2 1 − exp⎛⎜ 1 + R 1 NTU1,1−2 ⎞⎟ 1 − exp⎛⎜ 1 + R 1 NTU1,1−2 ⎞⎟ exp⎛⎜ 1 + R 1 NTU1,1−2 ⎞⎟ − 1 ⎠ ⎝ ⎠ ⎝ ⎠ ⎝

1

α=

or

1 2 1 − exp⎛⎜ − 1 + R 1 NTU 1,1−2 ⎞⎟ ⎝ ⎠

With these values, equation 6, written at the inlet of the heat exchanger, becomes:

(T

1,in

⎛ dT1 ⎞ R ⎟⎟ = 1 (T1,in − T1,out ) = (T1,in − T2,in ) + Aλ 1 + Bλ 2 − T2,in ) + ⎜⎜ dNTU 2 1 ⎠ in ⎝

or

(T Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

1,in

− T2,in ) =

R1 (T1,in − T1,out ) − α(T1,in − T1,out )λ1 − β(T1,in − T1,out )λ 2 2

The effectiveness of the heat exchanger for fluid 1 is then E1 =

1 R1 − αλ 1 − βλ 2 2

=

1 R1 − (1 − β )λ 1 − βλ 2 2

=

1 R1 − λ 1 + β(λ 1 − λ 2 ) 2

and finally: E1 =

1 2

2

1 + R1 R1 1 + 1 + R1 + + 2 2 2 exp⎛⎜ 1 + R 1 NTU 1,1− 2 ⎞⎟ − 1 ⎠ ⎝

This can also be written as : E1 = R1 + 1 + 1 + R1

2

2 2 = 2 2 exp⎛⎜ 1 + R 1 NTU 1,1− 2 ⎟⎞ + 1 1 + exp⎛⎜ − 1 + R 1 NTU 1,1− 2 ⎞⎟ 2 ⎝ ⎠ ⎝ ⎠ R1 + 1 + 1 + R1 2 2 exp⎛⎜ 1 + R 1 NTU 1,1− 2 ⎞⎟ − 1 1 − exp⎛⎜ − 1 + R 1 NTU 1,1− 2 ⎞⎟ ⎝ ⎠ ⎝ ⎠

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566

Syylvain Lalot The reversee equation is easily e obtainedd from the firsst expression of o the effectiveeness: 1 + R1 R1 1 + 1 + R1 1 ⇒ + + = 2 2 2 exp⎛⎜ 1 + R 1 NTU 1,1− 2 ⎞⎟ − 1 E 1 ⎝ ⎠ 2

1 + R1

2

2

2 exp e ⎛⎜ 1 + R 1 NTU N 1,1− 2 ⎞⎟ − 1 ⎝ ⎠

2 exp⎛⎜ 1 + R 1 NTU 1,1− 2 ⎞⎟ = 1 + ⎝ ⎠

NTU1,1− 2 =

1 2

1 + R1

=

1 R1 1 + 1 + R1 − 2 E1 2 1 + R1

2

1 R1 1 + 1 + R1 − 2 E1 2

2

⇒

and finaally: 2

⎛ ⎜ 2 ⎜ 1 + R1 Ln⎜1 + 2 ⎜ 1 R1 1 + 1 + R 1 − ⎜ E 2 2 1 ⎝

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

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Figure 8 sh hows a represeentation of theese relations.

Fiigure 8. Effectiv veness versus thhe Number of Transfer T Units (as a function off the heat capaccity rate raatio) for the onee pass fluid of a 1-2 heat exchannger.

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The NTU-E Effectiveness Method M

57

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

The Scilab script for thiss configurationn is :

Fiigure 9. Second d 1-2 configurattion studied.

4.2. Second Configurati C on (Inlets on Opposite Sides) If the 1 passs fluid flows in the other direction, the schematizationn is represented on figure 9.. 2 2', 3, 3' are still valid. Equuation 4 becom mes: Equations 2,

(

)

& 2 c 2 dT2 − dT2' = m & 1c1dT1 , or m

dT2 dT2' dT1 − = R1 d dNTU dNTU U dNTU U1 1 1

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(4'')

58

Sylvain Lalot Equation 5 becomes:

(

)

& 2 c 2 T2 − T2' = m & 1c1 (T1 − T1,in ) , or T2' = T2 − R 1 (T1 − T1,in ) m

(5'')

The introduction of equation 2' and 3' in equation 4'' leads to: R1 (T1 − T2 ) + R 1 (T1 − T2' ) = R 1 dT1 and using equation 5'' 2 2 dNTU 1 R1 (T1 − T2 ) + R 1 (T1 − (T2 − R 1 (T1 − T1,in ))) = R 1 dT1 or 2 2 dNTU 1

(T1 − T2 ) + R 1 (T1 − T1,in ) =

d (T1 − T1,in )

2

(T

1

− T1,in ) +

dNTU 1

and finally

R1 (T1 − T1,in ) − d(T1 − T1,in ) = T2 − T1,in 2 dNTU 1

(6'')

The derivation of this equation with respect to NTU1 leads to: −

d 2 (T1 − T1,in ) dNTU1

2

+

d(T1 − T1,in ) dNTU1

+

R 1 d(T1 − T1,in ) d(T2 − T1,in ) = 2 dNTU1 dNTU1

The introduction of equation 2' in this equation leads to

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−

d 2 (T1 − T1,in ) d(T1 − T1,in ) R 1 d(T1 − T1,in ) R 1 (T1 − T2 ) , and using equation 6'' + + = 2 dNTU1 2 dNTU1 2 dNTU1 −

( ) d 2 (T1 − T1,in ) d(T1 − T1,in ) R 1 d(T1 − T1,in ) R 1 ⎛ R 1 ⎜− (T1 − T1,in ) + d T1 − T1,in ⎞⎟⎟ + + = 2 dNTU1 ⎠ dNTU1 2 dNTU1 2 ⎜⎝ 2 dNTU1

or d 2 (T1 − T1,in ) d(T1 − T1,in ) R 1 2 (T1 − T1,in ) = 0 − − 2 dNTU1 4 dNTU1

(8'')

The general solution of this equation is:

(T

1

− T1,in ) = A exp(λ1 NTU1 ) + B exp(λ 2 NTU1 )

with λ1 and λ 2 that are the roots of the following equation: 2

x2 − x −

R1 = 0 (the coefficients are those of equation 7''). Hence, these values are known 4

as:

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The NTU-Effectiveness Method

λ1 =

1 − 1 + R1 2

2

and λ 2 =

1 + 1 + R1

59

2

2

At the left hand side of the heat exchanger, the temperature is T1,out , and the Number of Transfer Units is NTU 1 = 0 . At the right hand side of the heat exchanger, the temperature is

T1,in , and the Number of Transfer Units is NTU 1 = NTU 1,1− 2 . Thus the integration constants are the solutions of the following system: ⎧T1,out − T1,in = A + B ⎨ ⎩0 = A exp(λ 1 NTU 1,1− 2 ) + B exp(λ 2 NTU 1,1− 2 )

Writing A = α(T1,out − T1,in ) and B = β(T1,out − T1,in ) , the system becomes: ⎧1 = α + β ⎨ ⎩0 = α exp(λ1 NTU1,1− 2 ) + β exp(λ 2 NTU1,1− 2 )

Taking α from the first equation of the system, the second equation of the system is: 0 = (1 − β) exp(λ1 NTU1,1−2 ) + β exp(λ 2 NTU1,1−2 ) or 0 = exp(λ1 NTU1,1−2 ) + β(exp(λ 2 NTU1,1−2 ) − exp(λ1 NTU1,1−2 )) ⇒ exp (λ 1 NTU 1,1− 2 )

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β=

exp (λ 1 NTU 1,1− 2 ) − exp (λ 2 NTU 1,1− 2 )

=

1 1 = 1 − exp ((λ 2 − λ 1 )NTU 1,1− 2 ) 1 − exp⎛⎜ 1 + R 2 NTU ⎞ 1 1,1− 2 ⎟ ⎝ ⎠

Then, equation 6'', written at the left hand side of the heat exchanger, becomes:

(T

1, out

− T1,in ) +

R1 (T1,out − T1,in ) − (T1,out − T1,in )(αλ 1 + βλ 2 ) = T2,in − T1,in 2

or

(T

1,in

⎛ R ⎞ − T1,out )⎜1 + 1 − (αλ 1 + βλ 2 )⎟ = T1,in − T2,in 2 ⎝ ⎠

The effectiveness for the 1 pass fluid is then E1 =

1 1 1 = = R1 R1 R1 − ((1 − β )λ 1 + βλ 2 ) 1 + − ((1 − β )λ 1 + βλ 2 ) 1 + − λ 1 + β(λ 1 − λ 2 ) 1+ 2 2 2

or

1

E1 =

2

1+

1 + R1 R1 1 − − 2 2 1 − exp⎛⎜ 1 + R 2 NTU ⎞ 1 1,1− 2 ⎟ ⎝ ⎠

Heat Exchangers: Types, Design, and Applications : Types, Design, and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest

60

Sylvain Lalot

⇒

E1 =

1 2

1 + R1 R1 1 + + 2 2 exp⎛⎜ 1 + R 2 NTU ⎞ 1 1,1− 2 ⎟ − 1 ⎝ ⎠

It can be concluded that the effectiveness of a 1-2 heat exchanger is the same for both fluid directions.

5. THE 1-2N HEAT EXCHANGER

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There are different configurations for a 1-2n heat exchanger. Figure 10 shows a 1 shell pass 4 tube passes configuration where the inlets are on the same side. This configuration can be schematized as shown in figure 11.

Figure 10. schematic representation of a 1 shell pass 4 tube passes heat exchanger.

Figure 11. 1-2n configuration studied.

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The NTU-Effectiveness Method

61

In the elementary section of the heat exchanger, the balances can be written as follows:

& 2 c 2 dT2,k = ε k U A m

1 dA (T1 − T2,k ) 2n

(9)

With ε k = 1 if k is odd, and ε k = 0 if k is even, which is equivalent to ε k = (− 1) . & 1c1 , and the If the heat capacity rate ratio, for the shell side fluid, is defined as R ss = m & 2c2 m k +1

Number of Transfer Units, still for the shell side fluid, as NTU ss = U A A , equation 9 can be & 1c1 m written as dT2,k = ε k R ss

⇒

1 dNTU ss (T1 − T2 ,k ) 2n

dT2 , k 2n + T2 , k = T1 ε k R ss dNTU ss

(10)

From the elementary section to the right hand side of the heat exchanger, the energy balance gives: n

& 2 c 2 ∑ (T2, 2 k − T2, 2 k −1 ) = m & 1c1 (T1 − T1,out ) m k =1

n

⇒ ∑ (− ε 2 k T2, 2 k − ε 2 k −1T2, 2 k −1 ) = R ss (T1 − T1,out ) ⇒ Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

k =1

2n

∑ (− ε k =1

k

T2,k ) = R ss (T1 − T1,out )

The derivative of this equation is: dT1 1 =− dNTU ss R ss

2n

⎛

∑ ⎜⎜ ε k =1

⎝

k

dT2,k ⎞ ⎟ dNTU ss ⎟⎠

The introduction of equation 10 leads to: dT1 1 =− dNTU ss R ss

2n

⎛

∑ ⎜⎝ ε k =1

k

εk

R ss (T1 − T2,k )⎞⎟ ⇒ 2n ⎠

2n dT1 1 R ss ⎛ 2 n ⎞ =− ⎜ ∑ T1 − ∑ T2,k ⎟ dNTU ss R ss 2n ⎝ k =1 k =1 ⎠

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(11)

62

Sylvain Lalot

⇒

dT1 1 2n + T1 = ∑ T2,k dNTU ss 2n k =1

The derivative of this equation is:

d 2 T1 dNTU ss

2

+

dT1 1 2 n dT2,k = ∑ dNTU ss 2n k =1 dNTU ss

The introduction of equation 10 leads to: d 2 T1 dNTU ss

2

+

2n 2n R R ⎛ 2n dT1 1 2n ⎞ R ⎛ ⎞ = ε k ss (T1 − T2 ,k ) = ss2 ⎜ ∑ ε k T1 − ∑ ε k T2 ,k ⎟ = ss2 ⎜ 0 − ∑ ε k T2 ,k ⎟ ∑ dNTU ss 2n k =1 2n 4n ⎝ k =1 k =1 k =1 ⎠ 4n ⎝ ⎠

The introduction of equation 11 leads to: d 2 T1 dNTU ss

2

+

d 2 (T1 − T1,out ) dNTU ss

2

R dT1 = ss2 (R ss (T1 − T1,out )) ⇒ dNTU ss 4n

d(T1 − T1,out ) 1 ⎛ R ss ⎞ − ⎜ ⎟ (T1 − T1,out ) = 0 dNTU ss 4⎝ n ⎠ 2

+

This is equivalent to equation 8. So, it is possible to write:

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T1 − T1,out = (T1,in − T1,out )(α exp(λ1 NTU1 ) + β exp(λ 2 NTU1 )) with ⎛R ⎞ 1 + 1 + ⎜ ss ⎟ ⎝ n ⎠ λ1 = − 2

β=

2

,

⎛R ⎞ − 1 + 1 + ⎜ ss ⎟ ⎝ n ⎠ λ2 = 2

1

2

,

, α = 1− β

2 ⎞ ⎛ ⎛R ⎞ 1 − exp⎜ 1 + ⎜ ss ⎟ NTU 1,1− 2 n ⎟ ⎟ ⎜ ⎝ n ⎠ ⎠ ⎝

Equation 10 can then be written as follows: 2n d (T2,k − T1,out ) + T2,k − T1,out = (T1 − T1,out ) = (T1,in − T1,out )(α exp(λ1 NTU 1 ) + β exp(λ 2 NTU 1 )) ε k R ss dNTU ss

Or ⎛ T2, k − T1,out d⎜⎜ 2n ⎝ T1,in − T1,out dNTU ss ε k R ss

⎞ ⎟ ⎟ T −T 1,out ⎠ + 2,k = α exp(λ 1 NTU 1 ) + β exp(λ 2 NTU 1 ) T1,in − T1,out

Heat Exchangers: Types, Design, and Applications : Types, Design, and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest

The NTU-Effectiveness Method Introducing θ 2 ,k =

T2,k − T1,out T1,in − T1,out

63

, this equation becomes:

dθ 2,k 2n + θ 2,k = α exp(λ 1 NTU 1 ) + β exp(λ 2 NTU 1 ) ε k R ss dNTU ss

The solution of this differential equation is: ⎛ ε R ⎞ θ 2 , k = A k exp ⎜ − k ss NTU 1 ⎟ + δ k exp (λ 1 NTU 1 ) + γ k exp (λ 2 NTU 1 ) 2n ⎝ ⎠ 2 n α β and γ k λ 2 2n + γ k = β ⇒ γ = With δ k λ 1 + δk = α ⇒ δk = k ε k R ss ε k R ss 2nλ1 2nλ 2 1+ 1+ ε k R ss ε k R ss

This leads to:

(T

1,in

T2,k = T1,out + (T1,in

(T

1,in

⎛ ε R ⎞ − T1,out )A k exp⎜ − k ss NTU 1 ⎟ 2n ⎝ ⎠ α − T1,out ) exp(λ 1 NTU 1 ) 2nλ 1 1+ ε k R ss β − T1,out ) exp(λ 2 NTU 1 ) 2nλ 2 1+ ε k R ss

Then, for k = 1 , A 1 is found by expressing the temperature on the left hand side of the Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

heat exchanger: A1 =

T2,in − T1,out

(T

1,in

− T1,out )

−

α β − 2 nλ 1 2 nλ 2 1+ 1+ ε1 R ss ε1 R ss

Hence, the temperature of the first pass is known all along the pass, and in particular at the right hand side of the heat exchanger. It is then possible to compute A 2 replacing NTU 1 by NTU ss,1−2 n . The temperature of the second pass is known, and especially at the left hand side of the heat exchanger. The process is repeated until the last pass. The generic expressions are as follows: k = 2p + 1 ⇒ A = T2,k −1 (0 ) − T1,out − k (T1,in − T1,out )

α β − 2 nλ 1 2 nλ 2 1+ 1+ R ss R ss

Heat Exchangers: Types, Design, and Applications : Types, Design, and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest

64

Sylvain Lalot T2,k −1 (NTU ss ,1− 2 n ) − T1,out

(T

1,in

k = 2p ⇒

− T1,out )

1 ⎛ R ss ⎞ exp⎜ NTU ss ,1− 2 n ⎟ ⎝ 2n ⎠ exp(λ 1 NTU ss ,1− 2 n )

α 2 nλ 1 ⎛R ⎞ 1− exp⎜ ss NTU ss ,1− 2 n ⎟ R ss 2 n ⎝ ⎠ exp(λ 2 NTU ss,1− 2 n ) β − 2 nλ 2 ⎛R ⎞ 1− exp⎜ ss NTU ss ,1− 2 n ⎟ R ss ⎝ 2n ⎠ −

Ak =

The temperature of the tube side fluid at the outlet is then obtained by:

T2,out = T2, 2 n (0 ) Then a numerical procedure is used to find the correct effectiveness: a first guess is made, the corresponding outlet temperature of the tube side shell is estimated, and then all temperatures are evaluated, until the outlet temperature of the tube side shell. If the difference between the two values is lower than a specified threshold, the effectiveness is correct. On the contrary, the effectiveness is adapted until convergence. Note that the procedure is not valid for a heat capacity rate ratio R ss = 0 . In that case, the temperature of the tube side fluid is constant all along the heat exchanger. It is then possible to write:

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

& 1c1dT1 = U A dA (T1 − T2 ) m

⎛ (T − T2 )out ln⎜⎜ 1 ⎝ (T1 − T2 )in

⇒

dT1 = dNTU 1 , and as (T1 − T2 )

⎞ ⎛ T − T2,in ⎟ = NTU ss ,1− 2 n ⇒ ln⎜ 1,out ⎟ ⎜ T −T 2 ,in ⎠ ⎝ 1,in

T1,out − T1,in + T1,in − T2 ,in T1,in − T2,in

T2 is constant:

⎞ ⎟ = NTU ss ,1− 2 n ⎟ ⎠

⇒

= exp (NTU ss ,1− 2 n )

⇒ 1 - E ss,1-2n = exp(NTU ss,1− 2 n ) ⇒ E ss,1-2n = 1 - exp(NTU ss,1− 2 n ) −6

For a threshold of 10 , Figure 12 shows the effectiveness of a 1-4 heat exchanger. It is important to note that the effectiveness has a maximum value, for a given heat capacity rate ratio. In a first approach, it can be considered that the effectiveness of a 1-2n heat exchanger, when the number of tube passes is small, is close to the effectiveness of a 1-2 heat exchanger. Figure 13 shows the relative error, in percentage, between the two expressions. Note that the error gets higher when the number of passes increases.

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Fiigure. 12. Effecctiveness versuss the Number off Transfer Unitss (as a function of the heat capaacity rate raatio) for the onee pass fluid of a 1-4 heat exchannger.

Fiigure 13. Relatiive error betweeen the effectivenness of a 1-4 heeat exchanger annd the effectiveeness of a 12 heat exchangerr.

Fiigure. 14. Effecctiveness versuss the Number off Transfer Unitss (as a function of the heat capaacity rate raatio) for the onee pass fluid of a 1-20 heat exchhanger. Heat Exchangers: Types, Design, and Applications : Types, Design, and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest

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Fiigure 15. relativ ve difference beetween the effecctiveness of a 1-20 heat exchannger and the efffectiveness off a cross flow heat exchanger with w both fluidss unmixed.

Fiigure 16. Temperature distribuution in a shell and a tube heat exxchanger havingg the inlets on thhe same side.

It can also o be considereed that when the number of o tube passess is large, thee 1-2n heat exxchanger is clo ose to a cross flow heat excchanger with both b fluids mixxed. Figure 144 shows the efffectiveness of a 1-20 heat exchanger; annd Figure 15 shows s the relaative differencce between thhe effectiveneess of a 1-200 heat exchannger and the effectivenesss of a cross flow heat exxchanger with h both fluids unnmixed. The tempeerature of the tube side fluuid and of thee shell fluid can c be compuuted at any loocation of thee heat exchannger. Figure 16 shows theese evolutionss for two vallues of the

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The NTU-E Effectiveness Method M

67

Number of Traansfer Units annd two valuess of the numbeer of tubes paasses (the hot fluid is the N shhell side fluid)). It can be seen s that in onne part of thee heat exchangger the "cold"" fluid is hotter than the "hhot" fluid. Thiis, of course, should s be avoiided. The Scilab b script necesssary for the coomputation off the temperatture distributioon is given heereafter. It begins with an initial part whhere the mainn characteristiccs of the heat exchanger arre defined:

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thhen the constan nt parameters are computedd at the beginnning of the iterrative loop:

thhen the temperrature of the shhell side fluidd is computed along with thee temperature of the tube siide fluid for th he first pass:

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Syylvain Lalot

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thhen the temperrature of the tuube side fluid is computed for f all passes:

Finally, thee effectivenesss is adapted iff necessary:

mpute the effeectiveness of a 1-2n heat Following the same proccedure, it is possible to com exxchanger haviing the inlets on o opposite siides. It is thenn found that thhe effectivenesss does not deepend on the configurationn. But it must be noted thatt the temperatture distributiion is quite diifferent. Figurre 17 shows the t temperaturre distributionn for a 1-2 heat exchanger having the innlets on oppossite sides. It iss comparable of the resultss of the first heat h exchanger in Figure 166.

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Fiigure. 17. Temp perature distribuution in a shell and a tube heat exxchanger havinng the inlets on opposite o sides.

6. THE CROSS FLOW HEAT EXCHANG GERS

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6.1. Unmixed d-Mixed Coonfiguration n There are different conffigurations forr a cross flow w heat exchannger. Figure 18 1 shows a heeat exchangerr with one fluid mixed, onee fluid unmixeed. One fluid flows inside a tube; the otther fluid flow ws between finns. This is the first configuraation studied. This configuration can be schematizzed as shownn in Figure 19. The subsccript "u" is reelative to the unmixed u fluid.. The subscrippt "m" is relativve to the mixeed fluid.

Fiigure 18. Schem matic representaation of a cross flow heat exchaanger with one fluid mixed andd one fluid unnmixed.

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Syylvain Lalot

Fiigure 19. First cross c flow confiiguration studieed.

On the seccond order eleementary heat exchange surrface

, the heat balance b can

bee written as fo ollows: ⎛ dx d ⎞ dTu U L &u ⎜⎜ m ⎟⎟c u dTu = − U A dxdy(Tu − Tm ) or = − A x dy & ucu (Tu − Tm ) m Lx ⎠ ⎝

The integration of this equation betw ween the inlett and the outllet of the unm mixed fluid leeads to: ⎛ Tu , out (x ) − Tm (y ) ⎞ U L L ⎟ = − A x y = − NT TU u ⎟ & u cu m ⎝ Tu ,in − Tm (y ) ⎠

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[Ln(Tu − Tm )]inout = Ln L ⎜ ⎜

( ) ( ) or Tu ,outt x − Tm y = exp (− NTU u ) ⇒ Tu ,in − Tm (y ) Tu ,out (x ) − Tm (y ) = (Tu ,in − Tm (y ))exp (− NTU u ) or N Tu ,ouut (x ) − Tu ,in = (Tu ,in − Tm (y ))(exp (− NTU u ) − 1) ⇒

Tu ,in − Tu ,out (x ) = (Tu ,in − Tm (y ))(1 − expp (− NTU u )) On the elem mentary surfacce

& m c m dT & ucu m Tm = m

(12)

, the heat balannce can be wrritten as follow ws:

ddx (Tu ,in − Tu ,out (x )) Lx

The introdu uction of equaation 12 in thiss equation leadds to: Heat Exchangers: Types, Design, and Applications : Types, Design, and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest

The NTU-Effectiveness Method

& m c m dTm = m & ucu m

71

dx (Tu ,in − Tm (y ))(1 − exp(− NTU u )) or Lx

dTm dx = − R u (1 − exp(− NTU u )) Lx Tm (y ) − Tu ,in

The integration of this equation gives:

[Ln(T (y) − T )]

out

m

u , in

in

Tm ,out − Tu ,in Tm ,in − Tu ,in

Lx

⎡x ⎤ = −R u (1 − exp(− NTU u ))⎢ ⎥ or ⎣ Lx ⎦0 = exp(− R u (1 − exp(− NTU u )))

But the effectiveness for the mixed fluid is linked to the effectiveness for the unmixed fluid: Tm ,out − Tu ,in Tm ,in − Tu ,in

=

Tm ,out − Tm ,in Tm ,in − Tu ,in

+

Tm ,in − Tu ,in Tm ,in − Tu ,in

= −E m + 1 = −R u E u + 1

This leads to : − R u E u + 1 = exp(− R u (1 − exp(− NTU u )))

⇒ E u = 1 − exp(− R u (1 − exp(− NTU u ))) Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

Ru

The reverse equation is: Ln(− R u E u + 1) = −R u (1 − exp(− NTU u ))

⇒ exp(− NTU u ) = 1 + 1 Ln (− R u E u + 1) ⇒ Ru

⎞ ⎛ 1 Ln(1 − R u E u )⎟⎟ NTUu = −Ln⎜⎜1 + ⎠ ⎝ Ru

Note that these two expressions are not valid for R u = 0 . In this special case, it is possible to write: Eu ≅

1 − (1 − (− R u (1 − exp (− NTU u )))) R u (1 − exp (− NTU u )) = = 1 − exp (− NTU u ) Ru Ru

and NTU u = −Ln(1 − E u )

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722

Syylvain Lalot Figure 20 shows s a repressentation of thhese relations.

Fiigure 20. Effecttiveness versus the Number off Transfer Units (as a function of o the heat capaacity rate raatio) for the unm mixed fluid of an a unmixed-mixxed cross flow heat h exchanger.

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The corresp ponding Scilabb script is as follows: f

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The NTU-E Effectiveness Method M

6.2. Mixed-M Mixed Confiiguration

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If both fluids are mixed, the schematizzation is as folllows:

Fiigure 21. Secon nd cross flow coonfiguration stuudied

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73

744

Syylvain Lalot On the elem mentary surfacce

, thee heat balance can be writtenn as follows: Lx

& 1c1dT m T1 = U A L x dy ∫ (T2 (x ) − T1 (y ))) 0

dx Lx

⇒ dT1 = dNTU 1 ⎛⎜ T2 (x ) dx − T1 (y )⎞⎟ ⇒ ⎜∫ ⎟ Lx

⎝

Lx

0

dT1 Lx

T1 (y ) − ∫ T2 (x ) 0

dx Lx

= −dNTU 1

Lx ⎛ d ⎜ T1, out − T2 (x ) dx ∫ ⎜ Lx 0 Ln⎜ Lx ⎜ T − T (x ) dxx ⎜ 1,in ∫ 2 L x 0 ⎝ Lx

⇒

⎞ ⎟ ⎟ ⇒ ⎟ = − NTU1 ⎟ ⎟ ⎠

T1,ouut = T1,in exp(− NTU1 ) + ∫ T2 (x ) 0

⎠

dx (1 − expp(− NTU1 )) ⇒ Lx

Lx ⎞ ⎛ ddx (1 − exp(− NTU1 )) = ⎜⎜ T1,in − ∫ T2 (x ) ddx ⎟⎟(1 − exp(− NTU1 )) Lx Lx ⎠ 0 ⎝

Lx

T1,in − T1,out = T1,in (1 − exp(− NTU1 )) − ∫ T2 (x ) 0

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This can allso be written as follows: Lx ⎛ dx T1,in − T1,out = ⎜ T1,in − T2,in − ∫ (T2 (x ) − T2 ,in ) ⎜ Lx 0 ⎝

On the elem mentary surfacce

⎞ ⎟(1 − exp(− NTU U 1 )) ⎟ ⎠

(13)

, the heat balannce can be wrritten as follow ws: Ly

& 2 c 2 dT m T2 = U A L y dx ∫ (T1 (y ) − T2 (x )) ) 0

⎛ Ly

dy Ly

⎞

⇒ dT2 = R 1dNTU 1 ⎜ T1 (y ) dy − T2 (x )⎟ ⇒ ∫ ⎜ ⎝

0

dT2 Ly

T2 (x ) − ∫ T1 (y ) 0

dyy Ly

Ly

⎟ ⎠

= − R 1dNTU U1

The integraation of this eqquation between the inlet annd the outlet leeads to: Heat Exchangers: Types, Design, and Applications : Types, Design, and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest

(14)

The NTU-Effectiveness Method

75

Ly ⎛ ⎞ ⎜ T − T (y ) dy ⎟ 2 , out 1 ∫ ⎜ Ly ⎟ ⇒ 0 Ln⎜ ⎟ = −R1NTU1 Ly dy ⎟ ⎜ ⎜ T2,in − ∫ T1 (y ) L ⎟ y ⎠ 0 ⎝ Ly

T2,out = T2,in exp(− R 1 NTU 1 ) + ∫ T1 (y ) 0

dy (1 − exp(− R 1 NTU 1 )) Ly Ly

T2,out − T2,in = −T2,in (1 − exp(− R 1 NTU 1 )) + ∫ T1 (y ) 0

⇒

dy (1 − exp(− R 1 NTU 1 )) Ly

⎛ Ly ⎞ ⎜ T (y ) dy − T ⎟(1 − exp(− R NTU )) ⇒ or T − = T 2 , out 2 ,in 1 2 ,in 1 1 ∫ ⎜0 ⎟ Ly ⎝ ⎠ Ly

dy

∫ T (y ) L 1

0

− T2,in =

y

T2,out − T2,in

(15)

1 − exp(− R 1 NTU 1 )

The integration of equation 14 between the inlet and any location leads to:

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Ly ⎛ ⎞ ⎜ T (x ) − T (y ) dy ⎟ 2 1 ∫ ⎜ Ly ⎟ x , as dNTU = NTU dx 0 1 1 Ln⎜ ⎟ = −R1NTU1 Ly Lx Lx dy ⎟ ⎜ ( ) − T T y 2 , in 1 ∫0 ⎜ L y ⎟⎠ ⎝

⇒ T2 (x ) = T2,in exp⎛⎜ − R 1 NTU 1 ⎜ ⎝

⇒ T (x ) − T 2 2 ,in

x Lx

L

⎞ y ⎛ x ⎞ ⎞⎟ dy ⎛ ⎟⎟ + ∫ T1 (y ) ⎜1 − exp⎜⎜ − R 1 NTU 1 ⎟⎟ ⎟ ⎜ L L y⎝ x ⎠⎠ ⎠ 0 ⎝ L

y ⎛ ⎛ ⎛ x ⎞ ⎞⎟ dy ⎛ x ⎞ ⎞⎟ ⎟⎟ + ∫ T1 (y ) ⎜1 − exp⎜⎜ − R 1 NTU 1 ⎟ = −T2,in ⎜⎜1 − exp⎜⎜ − R 1 NTU 1 ⎜ ⎟ Lx ⎠⎠ 0 Ly ⎝ L x ⎟⎠ ⎟⎠ ⎝ ⎝ ⎝

⎛ ⎞ ⇒ T2 (x ) − T2,in = ⎜ T1 (y ) dy − T2,in ⎟⎛⎜1 − exp⎛⎜ − R 1 NTU 1 x ⎞⎟ ⎞⎟ ∫ ⎜ ⎜ ⎟⎜ Ly L x ⎟⎠ ⎟ ⎝ Ly

⎝

0

⎠⎝

⎠

The introduction of equation 15 in this equation leads to: T2 (x ) − T2,in =

⎛ T2,out − T2,in ⎛ ⎞⎞ ⎜1 − exp⎜ − R 1 NTU 1 x ⎟ ⎟ ⎜ ⎟⎟ 1 − exp(− R 1 NTU 1 ) ⎜⎝ L x ⎠⎠ ⎝

In equation 13, the integral of this equation is needed.

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Sylvain Lalot Lx

dx ∫ (T (x ) − T ) 2

2 ,in

0

Lx

=

T2,out − T2,in

1 − exp(− R 1 NTU 1

Lx

⎛

⎛

⎜1 − exp⎜ − R NTU ⎜ ) ∫⎜ ⎝ 0

⎝

1

1

x Lx

⎞ ⎞ dx ⎟⎟ ⎟ ⎟ ⎠⎠ Lx Lx

⎡⎛ x ⎛ 1 x ⎞ ⎞⎟⎤ ⎟ ⎥ exp⎜⎜ − R 1 NTU 1 = + ⎢⎜⎜ 1 − exp(− R 1 NTU 1 ) ⎣⎢⎝ L x R 1 NTU 1 L x ⎟⎠ ⎟⎠⎥⎦ ⎝ 0 T2,out − T2,in

=

⎞ ⎛⎛ ⎞ 1 1 ⎟ ⎜ ⎜1 + exp(− R 1 NTU 1 )⎟⎟ − ⎜ ⎟ ⎜ 1 − exp(− R 1 NTU 1 ) ⎝ ⎝ R 1 NTU 1 ⎠ R 1 NTU 1 ⎠ T2,out − T2 ,in

=

⎛ 1 − exp(− R 1 NTU1 ) ⎞ ⎜1 − ⎟⎟ 1 − exp(− R 1 NTU1 ) ⎜⎝ R 1 NTU1 ⎠

=

R 1 (T1,in − T1,out ) ⎛ 1 − exp(− R 1 NTU1 ) ⎞ ⎟⎟ ⎜1 − 1 − exp(− R 1 NTU1 ) ⎜⎝ R 1 NTU1 ⎠

T2,out − T2,in

The introduction of this equation in equation 13 leads to: ⎛ R 1 (T1,in − T1,out ) ⎛ 1 − exp(− R 1 NTU 1 ) ⎞ ⎞ ⎟⎟ ⎟(1 − exp(− NTU 1 )) ⎜1 − T1,in − T1,out = ⎜⎜ T1,in − T2,in − ⎟ R 1 NTU 1 1 exp(− R 1 NTU 1 ) ⎜⎝ − ⎠⎠ ⎝

⇒

⎛ T1,in − T1,out T1,in − T1,out R 1 (T1,in − T1,out ) ⎛ 1 − exp(− R 1 NTU 1 ) ⎞ ⎞ ⎟⎟ ⎟ ⎜1 − =⎜ − ⎟ 1 − exp(− NTU 1 ) ⎜⎝ E1 1 − exp(− R 1 NTU 1 ) ⎜⎝ R 1 NTU 1 ⎠⎠

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⇒

⎛ 1 R1 1 1 ⎞ ⎟ = ⎜⎜ − + 1 − exp(− NTU 1 ) ⎝ E 1 1 − exp(− R 1 NTU 1 ) NTU 1 ⎟⎠

⇒ E = 1

1 R1 1 1 + − 1 − exp(− NTU 1 ) 1 − exp(− R 1 NTU 1 ) NTU 1

Note that this expression is not valid for R 1 = 0 . In this special case, it is possible to write: E1 ≅

1 1 1 = = R1 R1 1 1 1 1 1 + − + − 1 − exp(− NTU1 ) 1 − (1 − R 1 NTU 1 ) NTU 1 1 − exp(− NTU1 ) R 1 NTU1 NTU1 1 − exp(− NTU1 )

⇒ E1 = 1 − exp(− NTU1 ) and NTU1 = −Ln(1 − E1 ) Note also that the derivative of the effectiveness with respect to the Number of Transfer Units is not always positive. Its sign is the sign of 2 exp(− NTU 1 ) R exp(− R 1 NTU 1 ) 1 . This means that increasing the heat + 1 − 2 2 (1 − exp(− NTU 1 )) (1 − exp(− R 1 NTU 1 )) NTU 1 2 transfer surface area does not always increase the effectiveness.

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The NTU-E Effectiveness Method M

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Finally notte that the revverse equation does not exisst, so that onlyy a numericall procedure haas to be used.

Fiigure 22. Effecttiveness versus the Number off Transfer Units (as a function of o the heat capaacity rate raatio) for a cross flow heat exchanger with bothh fluids mixed.

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Figure 22 shows a repreesentation of the effectivenness. The greyy zone shows the region where the effecctiveness is decreasing with increasing Nuumber of Trannsfer Units. w The corressponding Scilaab script can be divided inn parts. The first part dealls with the diirect equation:

In a second d part, the maxximum effectivveness is com mputed for the heat h capacity rate ratios:

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788

Syylvain Lalot

In the last part, p the Numbber of Transfeer Units is deteermined usingg a simple routtine:

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6.1. Unmixed d-Unmixed Configurattion If both fluids are unmixeed, the heat exxchanger can be b schematizedd as shown in Figure 23. pe of heat exchhangers, the teemperature off each fluid deepends on bothh directions For this typ ass shown in Fig gure 24.

Fiigure 23. schem matic representaation of a cross flow f heat exchaanger with both fluids unmixedd.

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Syylvain Lalot

Fiigure 24. Third cross flow conffiguration studiied.

On the second order elem mentary heat exchange surface

, the heat baalances can

bee written as fo ollows: ⎛ ∂T NTU 1 dx d ⎞ &1 ⎜⎜ m ⎟⎟c1 dT1 = − U A dxdy(T1 − T2 ) or 1 = − (T1 − T2 ) Lx ⎠ ∂y Ly ⎝ ⎛ NTU 1 ∂T dy d ⎞⎟ ⎜m &2 (T1 − T2 ) c dT = U A dxdy (T1 − T2 ) or 2 = R 1 ⎟ 2 2 ⎜ x Lx ∂ L y ⎠ ⎝

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To solve th his set of equuations it is coonvenient to inntroduce the following dim mensionless vaariables: x + = R1

T1 − T2,in T − T2,in NTU 1 , + NT TU 1 , + N , and T2+ = 2 . y T1 = x y = Lx Ly T1,in − T2,in T1,inn − T2,in

The dimensionless tempeerature differeence is then wrritten as T1+ − T2+ =

T1 − T2 ,in T1,in − T2,in

−

T2 − T2,in T1,in − T2,in

=

T1 − T2 T1,in − T2,in

f The partiall derivatives arre written as follows: −1

∂T1+ ∂T1+ ∂T1 ∂y 1 = = ∂T1 ∂y ∂y + T1,in − T2,in ∂y +

⎞⎛ NT ⎛ NTU U1 TU 1 ⎞⎟ ⎜− (T1 − T2 )⎟⎟⎜⎜ ⎟ ⎜ L L y y ⎠⎝ ⎠ ⎝

∂T2+ ∂T2+ ∂T2 ∂x 1 = = ∂x + ∂T2 ∂x ∂x + T1,in − T2,in

⎛ NTU1 ⎞⎛ NTU1 ⎞ ⎟ ⎜⎜ R 1 (T1 − T2 )⎟⎟⎜⎜ R 1 Lx L x ⎟⎠ ⎝ ⎠⎝

=−

T1 − T2 = − T1+ − T2+ T1,in − T2,in

(

−1

=

)

T1 − T2 = T1+ − T2+ T1,in − T2,in

One way to o solve this seet of equation is to assume that T2+ = exp(− (x + + y + ))f (x + , y + ) . In thhat case, the seecond equationn gives:

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The NTU-Effectiveness Method

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∂T2+ ∂f ∂f = − exp − x + + y + f x + , y + + exp − x + + y + = −T2+ + exp − x + + y + ∂x + ∂x + ∂x +

( (

)) (

)

( (

))

( (

))

The comparison of this equation with the original equation gives: T1+ = exp(− (x + + y + ))

∂f ∂x +

It is then possible to evaluate the first derivative: ∂T1+ ∂f ∂ 2f ∂ 2f = − exp(− (x + + y + )) + + exp(− (x + + y + )) + + = −T1+ + exp(− (x + + y + )) + + + ∂y ∂x ∂x ∂y ∂x ∂y

The comparison of this equation with the original equation gives: T2+ = exp(− (x + + y + ))

∂ 2f ∂x + ∂y +

But this dimensionless temperature is defined as T2+ = exp(− (x + + y + ))f (x + , y + ) . Hence, the partial differential equation for the unknown function is: ∂ 2f =f ∂x + ∂y +

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It is then possible to use a Laplace technique where x+ plays the role of time. The inlet temperature of the second fluid is supposed to be uniform. So, the dimensionless temperature at X+ = 0 is: T2+ (x = 0 ) =

T2,in − T2,in T1,in − T2,in

= 0 = exp(− (x + + y + ))f (x + , y + )

(

)

⇒ f x + = 0, y + = 0 Noting F the Laplace transform of f , the transformed equation is: s + The solution is F = A exp⎛⎜ y ⎜ s ⎝

dF =F dy +

(

)

⎞ . The boundary condition y + = 0 is + T1,in − T2,in ⎟⎟ T1 = = 1 , or T − T 1, in 2 , in ⎠

⎛ ∂f ⎞ T1+ = exp(− (x + + 0 ))⎜ + ⎟ =1 ⎝ ∂x ⎠ y + = 0

⇒ ⎛⎜ ∂f ⎞⎟ +

⎝ ∂x ⎠ y + = 0

( )

= exp x +

⇒ sFy

+

=0

=

1 ⇒ 1 A= s −1 s(s − 1)

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82

Sylvain Lalot + ⇒ F = 1 1 exp⎛⎜ y ⎞⎟ ⎜ ⎟

s −1 s

⎝ s ⎠

( ) and of I

Hence, f is the convolution product of exp y

(

+

f x ,y

+

x+

) = ∫ exp(x

)

+

+

0

(2

)

y + x + , or:

)

(

− u I 0 2 y + u du

0

Finally, the dimensionless temperature of the second fluid is written as:

( (

T2+ = exp − x + + y +

x+

)) ∫ exp(x

+

)

)

(

(

− u I 0 2 y + u du = exp − y +

0

x+

) ∫ exp(− u )I

The effectiveness for the second fluid is defined as E 2 = Ly

∫ T (x = L + 2

E2 =

x

=

0

Ly NTU 1

E1 =

NTU 1

)

, y + dy

(

0

(2

)

y + u du

0

∫ T (x + 2

+

)

= R 1 NTU 1 , y + dy +

0

T2 ,out , avg − T2 ,in T1,in − T2 ,in

= T2+,out , avg or

, so the effectiveness for the first fluid is

NTU 1 NTU 1

)

+ + + + ∫ T2 x = R 1 NTU 1 , y dy 0

R 1 NTU 1

∫ 0

=

⎛ ⎜ exp − y + ⎜ ⎝

(

x+

)∫ exp(− u )I

0

(2

0

)

⎞ y + u du ⎟dy + ⎟ ⎠

R 1 NTU 1

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It is again convenient to use the Laplace transform to evaluate this expression. To do so, it is necessary to compute the expression considering that x this variable to the final value: x + = R 1 NTU 1 . NTU 1

L(E 1 ) =

∫ 0

+ ⎛ ⎜ exp − y + 1 ⎛⎜ 1 exp⎛⎜ y ⎜ ⎜ ⎜ s⎝s ⎝ s ⎝ R 1 NTU 1

(

)

⎞ ⎞ ⎞⎟ + ⎟⎟ ⎟ dy ⎟ ⎠ ⎠ s +1 ⎟⎠

=

1 1 s s +1

NTU 1

∫ 0

+

is a variable, and then to set

⎛ ⎛ 1 ⎞⎞⎞ + ⎜ exp⎜⎜ − y + ⎛⎜1 − ⎟ ⎟ ⎟dy ⎜ + 1 ⎠ ⎟⎠ ⎟⎠ s ⎝ ⎝ ⎝ R 1 NTU 1

The integration leads to: NTU1

⎡ ⎛ 1 ⎞ ⎞⎤ 1 1 1 +⎛ ⎟ ⎟⎥ ⎢exp⎜⎜ − y ⎜1 − s s s +1 1 + 1 ⎠ ⎟⎠⎦ 0 ⎝ −1 ⎣ ⎝ s +1 L(E 1 ) = R 1 NTU 1

=

⎛ 1 1 s +1⎛ 1 ⎞⎞ ⎞ ⎜ exp⎜⎜ − NTU 1 ⎛⎜1 − ⎟ ⎟ − 1⎟ s s + 1 − s ⎜⎝ s + 1 ⎠ ⎟⎠ ⎟⎠ ⎝ ⎝

or

R 1 NTU 1

1 ⎛1 1 ⎞ 1 ⎛ NTU 1 ⎞ − ⎜ + ⎟ exp(− NTU 1 ) exp⎜ ⎟ s +1 s2 ⎝ s s2 ⎠ ⎝ s +1 ⎠ L(E 1 ) = R 1 NTU 1

So, the effectiveness can be written using a convolution product:

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The NTU-E Effectiveness Method M

83

⎛ + x ⎞ ⎜ x − 1 + x + − τ exp(− NTU 1 ) exp(− τ )I 0 2 NTU N 1 τ dτ ⎟ ∫ ⎜ ⎟ 0 E1 = ⎜ ⎟ R NTU U 1 1 ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ x + = R 1 NTU 1 +

(

(

)

)

The final expression is thhen: R 1 NTU1

exp(− NTU U1 ) E1 = 1 −

∫ (1 + R

1

(

)

NT TU 1 − τ ) exp(− τ)I 0 2 NTU 1 τ dτ

0

R 1 NTU 1

This expresssion is not vaalid for R 1 = 0 . In that case, the integral can c be evaluatted: R1 NTU1

∫ (1 + R NTTU 1

1

(

)

− τ) exp(− τ)I 0 2 NTU1 τ dτ ≅

0

R 1NTU1

R 1NTU U1

∫ (1 + R NTU 1

0

1

⎡ τ2 ⎤ − τ)dτ = ⎢(1 + R 1 NTU1 )τ − ⎥ 2 ⎦0 ⎣

This leads to:

E1 ≅ 1 −

1 ⎛ 2⎞ exp(− NTU 1 )⎜ R 1 NTU N 1 + (R 1 NTU 1 ) ⎟ 1 2 ⎛ ⎞ ⎠ ⎝ = 1 − exp(− NT TU 1 )⎜1 + (R 1 NTU U 1 )⎟ R 1 NTU 1 2 ⎝ ⎠

And the lim mit is:

E1 = 1 − expp(− NTU1 ) annd NTU 1 = − Ln L (1 − E1 )

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he reverse equuation does noot exist, so thatt only a numerical procedurre has to be Note that th ussed. s a repressentation of thhe effectivenesss. Figure 25 shows

Fiigure 25. Effecttiveness versus the Number off Transfer Units (as a function of o the heat capaacity rate raatio) for a cross flow heat exchanger with bothh fluids unmixed.

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Syylvain Lalot

The corressponding Scilaab script can be divided inn parts. The first part dealls with the diirect equation:

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d part is a sim mple routine too determine thhe Number of Transfer Unitts when the The second efffectiveness iss known.

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The NTU-Effectiveness Method

85

For this type of heat exchangers both fluids play the same role. So, it is always possible to consider the fluid which has a heat capacity rate ratio lower than unity. In this case, an approximate expression for the effectiveness is: ⎛ 1 E 1 = 1 − exp⎜⎜ ⎝ R1

( (

) )

⎛ ⎛ 0.5 ⎞ ⎞⎟ ⎤ ⎞ ⎡ ⎟ 0.2 − 0.04 R 1 ⎜ 1− exp ⎜⎜ − 0.81 ⎟⎟ ⎜ − 1 ⎟⎟ ⎝ R 1 NTU 1 ⎠ ⎠ ⎥ exp − R 1 NTU 1 ⎝ ⎢ NTU 1 ⎣ ⎦ ⎠

The relative error between the exact effectiveness and the effectiveness obtained using this approximation is given in Figure 26. It can be seen that the maximum absolute relative error is less than two percent.

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Figure. 26. relative error between the approximated effectiveness and the exact value for a cross flow heat exchanger with both fluids unmixed.

7. HEAT EXCHANGER NETWORKS Only a few configurations are presented in this chapter.

7.1. Heat Exchangers Globally in a Counter Flow Configuration The first configuration is shown on Figure 27. The heat exchangers are globally in a counter flow configuration

Figure 27. First heat exchanger network.

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Sylvain Lalot

For the ith heat exchanger, the effectiveness and the heat capacity rate ratio are respectively: E 1,i =

(T (T

1, i −1

1, i −1

− T1,i )

& 1c1 , R 1,i = R 1 = m & 2c2 m

− T2 , i )

It can be noticed that: 1 − R 1 E 1,i = 1 −

(T (T

1 − E 1,i = 1 −

2 , i −1

− T2 , i )

1, i −1

− T2 , i

(T (T

(T ) (T

1, i −1

=

− T2 ,i −1 ) − T2 , i )

1, i −1

1, i −1

− T1,i )

(T ) (T

1, i −1

− T2 , i

− T2 , i )

1, i

=

and

− T2 , i )

1,i −1

Hence, the ratio of these two quantities is: 1 − R 1 E 1,i 1 − E 1,i

(T (T

− T2,i −1 )

1, i −1

=

− T2 , i )

1, i

So the product, for all heat exchangers, of these ratios is: n

1 − R 1 E 1,i

i =1

1 − E 1,i

∏

(T (T

1, 0

=

1, n

− T2 , 0 )

(16)

− T2 , n )

The effectiveness of the heat exchanger that is equivalent to the whole system is

(T (T

1, 0

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E 1⇔ =

1, 0

− T1, n )

− T2 , n )

Noticing that 1 − R 1 E 1⇔ = 1 −

(T (T

1 − E 1⇔ = 1 −

2, 0

− T2 , n )

1, 0

− T2 , n

(T (T

(T ) (T =

1, 0

1, 0

1, 0

− T1, n )

1, 0

− T2 , n

− T2 , 0 )

− T2 , n )

(T ) (T =

1, n 1, 0

and

− T2 , n )

− T2 , n )

Equation 16 can be written as: n

1 − R 1 E 1,i

i =1

1 − E 1,i

∏ or

n

1 − R 1 E 1,i

i =1

1 − E 1,i

∏

=

1 − R 1 E 1⇔ 1 − E 1⇔

n

1 − R 1 E 1,i

i =1

1 − E 1,i

− E 1⇔ ∏

= 1 − R 1 E 1⇔ , or

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(17)

The NTU-Effectiveness Method n

E 1⇔ =

1− ∏

87

1 − R 1 E 1,i

i =1

n

R1 − ∏

1 − E 1,i 1 − R 1 E 1,i 1 − E 1,i

i =1

It can be noticed that if all heat exchangers are counter flow heat exchangers, equation 16 can also be written as ⎛ 1 − R 1E1,i ⎞ ⎛ n 1 − R 1E1,i ⎞ ⎞ n ⎛ ⎟ ⎟ = Ln⎜ 1 − R 1E1⇔ ⎟ = ∑ Ln⎜ Ln⎜ ∏ ⎟ ⎜ ⎟ ⎜ ⎜ i =1 1 − E ⎟ 1, i ⎠ ⎝ 1 − E1⇔ ⎠ i =1 ⎝ 1 − E1,i ⎠ ⎝

The introduction of the heat capacity rate ratio leads to: ⎛ 1 − R1E1⇔ ⎞ ⎟⎟ Ln⎜⎜ ⎝ 1 − E1⇔ ⎠ = 1 − R1

⎛ 1 − R1E1,i ⎞ ⎟ ⎟ n ⎝ 1 − E1,i ⎠ = NTU1,i and finally ∑ 1 − R1 i =1

n

∑ Ln⎜⎜ i =1

n

NTU1⇔ = ∑ NTU1,i i =1

When the heat capacity rates are equal the expression of the equivalent effectiveness is not valid. In this special case, it can be noted that T1, i −1 − T1, i = T2 , i −1 − T2 , i can be written as

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T1, i −1 − T2 ,i −1 = T1,i − T2 ,i . The temperature difference at the left hand side of any heat exchanger is the temperature difference at the right hand side of this exchanger; and so is equal to the temperature at the right hand side of the nth heat exchanger. It can also be noted that E1,i 1 − E1,i

=

(T 1 − (T

1, i −1

− T1,i ) / (T1,i −1 − T2,i )

1, i −1

− T1,i ) / (T1,i −1 − T2,i )

=

T1,i −1 − T1,i T1,i − T2,i

=

T1,i −1 − T1,i T1, n − T2, n

The sum of these expressions is n

n

E1,i

∑ 1− E i =1

=

1, i

∑T i =1

1, i −1

− T1,i

T1, n − T2, n

=

T1, 0 − T1, n T1, n − T2, n

=

E1⇔ 1 − E1⇔

(18)

So it is possible to write: E1,i

n

E 1⇔ =

∑1− E i =1

1,i

n

E 1,i

i =1

1 − E 1,i

1+ ∑

The Number of Transfer Units of the network is deduced from equation 18. Heat Exchangers: Types, Design, and Applications : Types, Design, and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest

88

Sylvain Lalot n

NTU1⇔ = ∑ NTU1,i i =1

It is interesting to look at the evolution of the global effectiveness with the number of heat exchangers. Equation 17 can be written as n

1 − R 1E1,i

i =1

1 − E1,i

∏

=

1 − R 1E1, n 1 − E1, n

n −1

1 − R 1E1,i

i =1

1 − E1,i

∏

=

1 − R 1E1, n 1 − R 1E1⇔ n −1 1 − R 1E1⇔ n = 1 − E1, n 1 − E1⇔ n −1 1 − E1 ⇔ n

If the heat capacity rate ratio is lower than unity 1 − R 1E1, n is higher than unity and 1 − E1, n 1 − R 1E1⇔ n −1 1 − R 1E1⇔ n or < 1 − E1⇔ n −1 1 − E1⇔ n 1 − E1⇔ n − R1E1⇔ n −1 + R1E1⇔ n −1E1⇔ n < 1 − E1⇔ n −1 − R1E1⇔ n + R1E1⇔ n −1E1⇔ n or E1⇔ n −1 − R1E1⇔ n −1 < E1⇔ n − R1E1⇔ n

or

(1 − R1 )E1⇔ n −1 < (1 − R1 )E1⇔ n

and finally, as the heat capacity rate ratio is lower than unity: E1⇔ n −1 < E1⇔ n

If the heat capacity rate ratio is higher than unity 1 − R 1E1, n is lower than unity and 1 − E1, n

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1 − R 1E1⇔ n −1 1 − R 1E1⇔ n > 1 − E1⇔ n −1 1 − E 1⇔ n

or

1 − E1⇔ n − R1E1⇔ n −1 + R1E1⇔ n −1E1⇔ n > 1 − E1⇔ n −1 − R1E1⇔ n + R1E1⇔ n −1E1⇔ n or

R1E1⇔ n − E1⇔ n > R1E1⇔ n −1 − E1⇔ n −1 or (R1 − 1)E1⇔ n > (R1 − 1)E1⇔ n −1 and finally, as the heat

capacity rate ratio is higher than unity: E1⇔ n > E1⇔ n −1

When the heat capacity rates are equal, equation 18 shows that adding one heat exchanger increases E1⇔ . As the derivative of the function x is 1(1 − x ) − x (− 1) 1 , which is = 2 1− x 1 − E1⇔ (1 − x ) (1 − x )2 always positive, it can be concluded that E1⇔ n > E1⇔ n −1 .

7.2. Heat Exchangers Globally in a Parallel Flow Configuration The second configuration is shown on Figure 28. The heat exchangers are globally in a parallel flow configuration. For the ith heat exchanger, the effectiveness and the heat capacity rate ratio are respectively:

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The NTU-Effectiveness Method E1,i =

89

(T

− T1, i ) , T − T2,i −1 T2, n − T2, 0 & c m = R1,i = R1 = 1 1 = 2,i & 2c 2 T1,i −1 − T1,i (T1,i −1 − T2,i −1 ) m T1, 0 − T1, n 1, i −1

It can be noticed that: 1 = 1 − (R1 + 1)E1,i

T1,i −1 − T2,i −1 T −T 1 = = 1,i −1 2,i −1 ⎛ T2,i − T2,i −1 + T1,i −1 − T1,i ⎞ T1,i −1 − T1,i T1,i −1 − T2,i −1 − T2,i + T2,i −1 − T1,i −1 + T1,i T1,i − T2,i ⎟ 1− ⎜ ⎜ ⎟ T −T T1,i −1 − T1,i ⎝ ⎠ 1,i −1 2,i −1

Figure 28. Second heat exchanger network.

So the product, for all heat exchangers, of these quantities is: n

∏ 1 − (R i =1

(T − T2,0 ) 1 = 1, 0 (T1, n − T2, n ) 1 + 1)E1, i

The effectiveness of the heat exchanger that is equivalent to the whole system is E 1⇔ =

(T (T

1, 0

1, 0

− T1, n )

− T2, 0 )

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Taking the second expression of the heat capacity rate ratio, it is possible to write 1 = 1 − (R1 + 1)E1⇔

T1,0 − T2,0 T − T2,0 1 = = 1,0 T1, 0 − T2,0 − T2, n + T2,0 − T1, 0 + T1, n T1, n − T2, n ⎛ T2, n − T2,0 + T1,0 − T1, n ⎞ T1,0 − T1, n ⎟ 1− ⎜ ⎜ ⎟T −T T1,0 − T1, n 2,0 ⎝ ⎠ 1,0

and n

∏ 1 − (R i =1

1 1 = 1 − (R1 + 1)E1⇔ 1 + 1)E1, i

(19)

or n

E1⇔ =

1 − ∏ 1 − (R1 + 1)E1,i i =1

R1 + 1

It can be noticed that if all heat exchangers are parallel flow heat exchangers, Equation 19 can also be written as

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90

Sylvain Lalot n ⎛ n ⎞ ⎛ ⎞ ⎛ ⎞ n 1 1 1 ⎟ = Ln⎜ ⎜ ⎟ Ln⎜ ∏ ⎜ 1 − (R + 1)E ⎟⎟ = ∑ Ln⎜ 1 − (R + 1)E ⎟ = (R1 + 1)∑ NTUi ⎜ i =1 1 − (R + 1)E ⎟ i =1 i =1 1 1, i ⎠ 1 1⇔ ⎠ 1 1, i ⎠ ⎝ ⎝ ⎝

and finally n

NTU 1⇔ = ∑ NTU i i =1

7.3. Heat Exchangers in Paralle/Serial Flow Configuration The third configuration is shown on Figure 29. One of the fluids is in a parallel flow configuration; the second fluid is in a serial flow configuration. For the ith heat exchanger, the effectiveness is: E s ,i = Ts ,i −1 − Ts ,i . This can be written as Ts ,i −1 − T//, 0

(T

s ,i −1

− T//, 0 )E s ,i = Ts ,i −1 − Ts ,i , so the outlet temperature of this exchanger is

Ts,i = Ts,i−1 (1 − E s,i ) + T//,0 E1,i = Ts,i−1 (1 − E s,i ) + T//,0 − T//,0 + T//,0 E s,i and finally:

Ts,i = Ts,i −1 (1 − E s,i ) + T//,0 E s,i = Ts,i −1 (1 − E s,i ) + T//,0 (1 − (1 − E s,i )) .

Using the similar expression for Ts ,i −1 , i.e. Ts,i−1 = Ts,i −2 (1 − E s,i −1 ) + T//,0 (1 − (1 − E s,i−1 )) , Ts ,i becomes

Ts,i = (Ts,i−2 (1 − E s,i−1 ) + T//,0 (1 − (1 − E s,i −1 )))(1 − E s,i ) + T//,0 (1 − (1 − E s,i )) , or

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Ts,i = Ts,i−2 (1 − E s,i −1 )(1 − E s,i ) + T//,0 ((1 − E s,i ) − (1 − E s,i−1 )(1 − E s,i ) + 1 − (1 − E s,i )) , or Ts,i = Ts,i −2 (1 − E s,i −1 )(1 − E s,i ) + T//,0 (1 − (1 − E s,i−1 )(1 − E s,i ))

Repeating this process from the last heat exchanger to the first one leads to:

Figure 29. Third heat exchanger network. n n ⎛ ⎞ Ts , n = Ts , 0 ∏ (1 − E s ,i ) + T//, 0 ⎜⎜1 − ∏ (1 − E s ,i )⎟⎟ i =1 i =1 ⎝ ⎠

Hence the effectiveness of the whole system is

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The NTU-Effectiveness Method

E s⇔ =

E s⇔

Ts ,0 − Ts ,n Ts ,0 − T//,0

91

n n ⎞ ⎛ Ts ,0 − Ts ,0 ∏ (1 − E s ,i ) − T//, 0 ⎜⎜1 − ∏ (1 − E s ,i )⎟⎟ i =1 i =1 ⎠ or, ⎝ = Ts ,0 − T//, 0

n n ⎞ ⎛ ⎞ ⎛ Ts , 0 ⎜⎜1 − ∏ (1 − E s ,i )⎟⎟ − T//, 0 ⎜⎜1 − ∏ (1 − E s ,i )⎟⎟ i =1 i =1 ⎠ ⎝ ⎠ , and finally: ⎝ = Ts ,0 − T//, 0 n

E s ⇔ = 1 − ∏ (1 − E s ,i ) i =1

8. EXAMPLES 8.1. Comparison of Counter Flow and Parallel Flow Heat Exchangers A counter flow heat exchanger is used to preheat water from 15°C to 45°C. Oil is & w = 0.08 kg / s leading to a convection available at 90°C. The mass flow rate of water is m

& o = 0.17 kg / s leading to coefficient of h w = 2000 W / m 2 K . The mass flow rate of oil is m a convection coefficient of h o = 500 W / m 2 K . The heat capacities are considered constant at c w = 4200 J/kgK and c o = 2000 J/kgK for water and oil respectively. The separating

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material is supposed to get a very high conductivity. Extended surfaces are used on the oil side. The convection surface area is then five times higher on the oil side than on the water side. • • •

Determine the convection surface area on the water side Find the outlet temperature of oil Determine the total heat rate

After a cleaning operation, the connections are mismounted so that the heat exchanger is in a parallel flow the configuration. • • •

Find the outlet temperature of oil Determine the total heat rate Find the outlet temperature of water.

The solution procedure is shown on Figure 30: The Scilab scrip starts with the known variables:

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Fiigure 30. Solutiion procedure foor the first exam mple.

h capacity rate r ratio and the effectivenness are compputed for the water, and Then the heat deeduced for thee oil:

i possible to compute the Number N of Trransfer Units; and then the product of Then is it is thhe overall heatt transfer coeff fficient and thee reference areea:

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The NTU-E Effectiveness Method M

93

h transfer coefficients c The overalll heat transferr coefficient iss linked to thee convection heat 1 1 1 annd to the conv vection surfacee areas: . It is possible p to dettermine the = + UA h w A w h o Ar A oAw coonvection surfface area needded on the water side:

The oil outtlet temperatuure is computeed from the innlet temperaturres and the efffectiveness onn the oil side:

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ossible to checck that there iss no heat loss in the heat exxchanger: It is then po

o on the waterr side as done here: The total heat rate can bee computed onn the oil side or

d on a parallell flow configuuration, the coonvection heatt transfer coeffficients are When used h transfer coefficient reemains the saame. As the convection noot modified. The overall heat suurface areas are a constant, thhe Number of Transfer Unnits is the sam me. It is then possible to coompute the efffectiveness in the new confiiguration:

utlet temperatuures and the tootal heat rate, as a done is the first configuraation: annd then the ou

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Syylvain Lalot

The numerrical results aree: •

& = 10080 W the totaal heat rate is Q

•

the con nvection surfaace area on thee water side is A w = 0.2 m 2

•

the outtlet temperaturre of oil is To ,out = 60.4 °C

In a paralleel flow the connfiguration: •

the outtlet temperaturre of oil is To ,out // = 62.7 °C

•

& = 9290 W the totaal heat rate is Q //

•

the outtlet temperaturre of water is Tw ,out // = 42.6 °C

8.2. Counter Flow Heat Exchanger (Variable Properties) P

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A counter flow heat exxchanger is used u to preheat water from m 15°C to 455°C. Oil is & w = 0.08 kg / s leading to a convection avvailable at 90°C. The masss flow rate off water is m cooefficient of h w = 2000 W / m 2 K . The heat capacityy of water is considered constant c at & o = 0.17 kg / s . The prroperties of oil vary with c w = 4200 J/kgK K The mass flow rate of oill is m

teemperature, to a coonvection leading coefficiennt 2 2 heat cap oil . The acity of h o = 0.0312 To ,ref - 14.6 To,ref + 1725 W/m K

of is

m is suppposed to get a very high coonductivity. c o = 1190 + 4.25 5 T J/kgK . Thee separating material Extended surfaaces are used on the oil sidde. The conveection surfacee area is then five times hiigher on the oil side than onn the water sidde. • • •

Determ mine the conveection surface area on the water w side Find th he outlet temperature of oil Determ mine the total heat h rate

on procedure is i shown on Figure 31. It is very similar to t the solutionn procedure The solutio foor the first ex xample, but for fo the iterativve way of com mputing the convection c heeat transfer cooefficient on the oil side. In fact, it is necessary to make a first guess for thee reference teemperature (th he temperaturee taken into account for the determination d n of the properrties of oil).

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The NTU-E Effectiveness Method M

95

Fiigure 31. Solutiion procedure foor the second exxample.

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The Scilab scrip starts with w the knownn variables:

onstants are defined d that maake possible the t estimation of the heat caapacity and Then the co thhe convection heat transfer coefficient. c

Then there is the initializzation of the iterative i proceess. In this casse, the first guuess for the reeference tempeerature is the inlet i temperatuure of oil.

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And finally y the iterative process startss with the estim mation of the heat capacityy and of the coonvection heatt transfer coeffficient for oil::

h capacity rate r ratio and the effectivenness are compputed for the water, and Then the heat deeduced for thee oil:

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i possible to compute c the Number N of Traansfer Units, thhe product of the overall Then is it is heeat transfer co oefficient and the t reference area, a and finallly the convecction surface area: a

The oil outtlet temperatuure is computeed from the innlet temperaturres and the efffectiveness onn the oil side:

ossible to com mpute the absoolute difference between thee average tempperature of It is then po he reference tem mperature andd either to stopp the iterationss or to modifyy the thhe fluid and th reeference tempeerature.

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o on the waterr side as done here: The total heat rate can bee computed onn the oil side or

The numerrical results aree: •

& = 10080 W the totaal heat rate is Q

•

the con nvection surfaace area on thee water side is A w = 0.186 m 2

•

the outtlet temperaturre of oil is To ,out = 50.2 °C

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8.3. Cross Fllow Heat Exxchanger A 16 squarre meter thin walled w cross flow fl heat exchhanger is used to recover heat from the aiir extracted from fr a house. The volume of the housee is 250 cubiic meters andd the air is reenewed at a raate of one voluume per hour. The temperaature inside thhe house is maaintained at 21°C. Air is co onsidered as an a ideal gas having h a density of 1.2041 kg/m3 at 0°C C. The heat caapacity is conssidered constaant at 1005 J/kkg.K. The inleet temperature of the outdooor air varies frrom -10°C to +15°C. For both flows the convection heat h transfer cooefficient is suupposed to bee 20 W/m2K.

Fiigure 32. Solutiion procedure foor the third exam mple. Heat Exchangers: Types, Design, and Applications : Types, Design, and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest

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Syylvain Lalot • •

Plot th he heat rate reccovered versuss the inlet tem mperature of coold air Plot th he evolution of the outleet temperaturre of the cold air versuss the inlet temperrature of cold air

w the charactteristics of thee hot air (and of o the heat excchanger): The Scilab scrip starts with

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thhen the characteristics of thee cold fluid aree estimated:

p to com mpute the heaat capacity rate ratio, the Nuumber of Trannsfer Units It is then possible annd the effectiv veness:

utlet temperatture of the hott air is determiined, and finallly the total heeat rate and Then the ou thhe outlet tempeerature of the cold air:

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The NTU-E Effectiveness Method M

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It is then po ossible to plott the evolutionn of outlet tem mperature of the cold air

annd of the heat recovered:

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1000

Syylvain Lalot

8.4. Coupled d Heat Exchangers (Ideentical Effecctiveness)

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Water has to be heated from f 15°C to 45°C. Oil is available a at 900°C. Due to thhe distance beetween the tw wo tubing systtems, an interrmediate fluidd (water) has to be used ass shown on Fiigure 33. & w = 0.08 kg / s leading to a convection w is m c coeefficient of The mass flow rate of water

& o = 0.17 kg / s leading to a convection m flow ratee of oil is m h w = 2000 W / m 2 K . The mass cooefficient of h o = 500 W / m 2 K . Thee mass flow rate of thee intermediatte fluid is

& i = 0.12 kg / s leading to a convectionn coefficient of h i = 25000 W / m 2 K in both heat m exxchangers.

The T

heat

c capacities

arre

consideredd

constant

at

c w = 42200 J/kgK ,

c i = 4200 J/kg gK and c o = 2000 J/kgK for water, inttermediate fluuid and oil reespectively. The separating material is suupposed to get a very high conductivity in i both heat exchangers. e Extended surfaaces are used on the oil sidde for the firsst exchanger, and on the inntermediate r is five foor both heat fluid side in thee second heat exchanger. Thhe convection surface area ratio t effectiveneess of the heatt exchangers are a to be equall, exxchangers. If the • • •

Find th he effectivenesss of each heaat exchanger Determ mine the conveection surface area on the coold side for eaach heat exchaanger Find th he temperaturees of the interm mediate fluid

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Fiigure 33. Config guration for thee fourth example.

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Fiigure 34. Solutiion procedure foor the fourth exxample.

Before starrting the calcuulations, it is necessary to write the rellation linking the global efffectiveness (tthe effectiveneess of the equivalent heat exxchanger) andd the effectiveeness of the heeat exchangers. The definitiion of the globbal effectiveneess (for the colld fluid) is:

Eg =

Tw ,ouut − Tw ,in To ,inn − Tw ,in

t inlet tempeeratures can be b written The differeence between the

To ,in − Tw ,in = To ,in − Ti , 2 + Ti , 2 − Ti ,1 + Ti ,1 − Tw ,in The first teemperature difference can be b deduced frrom the effecttiveness of the first heat exxchanger (for the cold side):

E c ,1 =

Ti ,1 − Ti , 2 To ,in − Ti , 2

⇒ To ,in − Ti , 2 =

Ti ,1 − Ti , 2 E c,1

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Syylvain Lalot

The second d temperature difference is deduced d from the energy baalance of the second s heat exxchanger:

Ti ,1 − Ti , 2 = R c 2 (Tw ,out − Tw ,inn ) The last teemperature diifference is deeduced from the effectivenness of the seecond heat exxchanger (for the cold side):

E c,2 =

Tw ,out − Tw ,in Ti ,1 − Tw ,in

⇒ Ti ,1 − Tw ,in =

Tw ,out − Tw ,in E c,2

The introdu uction of thesee three equatioons in the globbal effectiveneess leads to:

Eg =

Tw ,ouut − Tw ,in R c,2 (Tw ,out − Tw ,in ) − R c,2 (Tw ,out − Tw ,in ) + 1 (Tw ,out − Tw ,in ) E c, 2 E c,1

⇒ E = g

1 ⎞ 1 ⎛ 1 − 1⎟⎟ + R c, 2 ⎜⎜ ⎝ E c,1 ⎠ E c, 2

If the effecctiveness of thhe first heat exxchanger are to t be equal too the effectiveness of the p to wriite: seecond heat excchanger, it is possible

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Eg =

1 ⎛ 1 ⎞ 1 R c, 2 ⎜⎜ − 1⎟⎟ + ⎝ Ec ⎠ Ec

⇒ 1 (R c , 2 + 1) − R c, 2 = 1 ⇒ E = R c, 2 + 1 c E E c

g

1 + R c, 2 Eg

The Scilab script starts with w the constaant values:

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The NTU-E Effectiveness Method M

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g effectivveness is com mputed, and thhe effectiveneess of each exxchanger is Then the global deeduced:

onvection surfface area is coomputed for thhe first heat exxchanger Then the co

annd for the seco ond heat exchaanger

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Syylvain Lalot

Finally the temperatures of the intermeediate fluid arre determined:

h exchangerr: It is possible to check thee effectiveness of the first heat

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The numerrical results aree: • • • •

the efffectiveness of each e heat exchhanger is 0.5226 the con nvection surfa face area on the t cold side for the first heat h exchangeer is 0.642 square meters the con nvection surfaace area on thhe cold side foor the second heat exchangger is 0.159 square meters the tem mperatures of the t intermediaate fluid are 722°C and 52°C

8.5. Coupled d Heat Exchangers (Ideentical Heat Exchangerrs) Water has to be heated from f 15°C to 45°C. Oil is available a at 900°C. Due to thhe distance beetween the tw wo tubing systtems, an interrmediate fluidd (water) has to be used ass shown on Fiigure 33. & w = 0.08 kg / s leading to a convection w is m c coeefficient of The mass flow rate of water

& o = 0.17 kg / s leading to a convection h w = 2000 W / m 2 K . The maass flow rate of oil is m cooefficient of h o = 500 W / m 2 K . Thee mass flow rate of thee intermediatte fluid is

& i = 0.12 kg / s leading to a convectionn coefficient of h i = 25000 W / m 2 K in both heat m exxchangers. Th he heat capacitties are considdered constantt at c w = 4200 J/ J kgK , c i = 42000 J/kgK and uid and oil resspectively. Thhe separating material is c o = 2000 J/kgK for water, inttermediate flu

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suupposed to gett a very high conductivity c in both heat exxchangers. Exxtended surfaces are used onn the oil side for the first exchanger, e annd on the interrmediate fluidd side in the second s heat exxchanger. Thee convection surface area ratio is five for f both heat exchangers. If the heat exxchangers are to be identicaal, • • • •

Find th he effectivenesss of each heaat exchanger Determ mine the conveection surface area on the coold side for eaach heat exchaanger Compaare the total convection suurface area too the one com mputed in thhe previous situatio on Find th he temperaturees of the interm mediate fluid

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The solution proceduree (Figure 35)) is very sim milar to the previous p one. The only diifference is du ue to an iteratiive process. A first guess is necessary forr the effectivenness of one heeat exchanger (here the firstt heat exchangger). But it is important i to note n that the whole w range [00, 1] is not posssible for this first guess as shown hereaftter.

Fiigure 35. Solutiion procedure foor the fifth exam mple.

1

The globall effectivenesss is E = g

. So, in general, knowingg the global

⎛ 1 ⎞ 1 R c, 2 ⎜⎜ − 1⎟⎟ + ⎝ E c,1 ⎠ E c, 2 efffectiveness th hat has to be achieved, a the effectiveness e o the first heaat exchanger is of i linked to thhe effectivenesss of the seconnd heat exchannger: ⎛ 1 ⎞ 1 1 = R c , 2 ⎜⎜ − 1⎟⎟ + Eg ⎝ E c,1 ⎠ E c , 2

⇒E = c ,1

⇒ 1 ⎛⎜ 1 − 1 ⎞⎟ = ⎛⎜ 1 − 1⎞⎟ R c, 2 ⎜⎝ E g E c, 2 ⎟⎠ ⎜⎝ E c,1 ⎟⎠ 1

1 ⎛⎜ 1 1 ⎞⎟ +1 − ⎜ R c , 2 ⎝ E g E c, 2 ⎟⎠

Hence, the minimum efffectiveness forr the first heatt exchanger iss linked to thee maximum h exchangerr: efffectiveness off the second heat Heat Exchangers: Types, Design, and Applications : Types, Design, and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest

1006

Syylvain Lalot E c ,1,min =

1 ⎛ 1 1 ⎜ 1 − R c , 2 ⎜⎝ E g E c , 2 ,max

⎞ ⎟ +1 ⎟ ⎠

It is also necessary to note that foor a counter flow heat exxchanger the maximum d on the t heat capacity rate raatio. The genneral expressiion of the efffectiveness depends U h (1 − R h )] . The introduction of the efffectiveness iss for the hot fluid is: E h = 1 − exp[NTU R h − exp[NT TU h (1 − R h )] heeat capacity raate ratio, the Number N of Traansfer Units, and a the effectiiveness for thee cold fluid ⎡ ⎛ 1 ⎞⎤ 1 − expp[NTU c (R c − 1)] ⎟⎟⎥ ⇒ ⇒ 1 − exp ⎢R c NTU c ⎜⎜1 − leeads to: Ec = R c ⎠ ⎝ ⎣ ⎦ ( ) 1 1 − R ex xp [ NTU R − ] R cEc = c c c ⎡ ⎛ 1 1 ⎞⎤ − expp ⎢R c NTU c ⎜⎜1 − ⎟⎟⎥ Rc R c ⎠⎦ ⎝ ⎣

Ec =

exp[NTU U c (1 − R c )] − 1 exp[NTU U c (1 − R c )] − R c

⇒ E c = 1 − exp[NTU c (1 − R c )]

R c − exp[NTU c (1 − R c )]

h capacity raate ratio is low wer than unityy, the effectiveeness tends to unity u when So, if the heat thhe Number off Transfer Units tends to innfinity (also true t when the heat capacityy rate ratio eqquals unity). If I the heat cappacity rate ratiio is higher thhan unity, thenn the maximuum value of thhe effectivenesss is E c , max = 1 . Rc

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The Scilab script begins as the previouus script:

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The NTU-E Effectiveness Method M

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Then the minimum m effecctiveness is determined for the t first heat exchanger e

The maxim mum effectivenness is computted for the firsst heat exchannger

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Then the iterative processs begins

mined from thee effectiveness tested for thhe first heat The convecction surface area is determ exxchanger

annd for the seco ond heat exchaanger

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Syylvain Lalot

d to decidee to stop the itterative processs The converrgence test is done

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mediate fluid arre determinedd Finally, thee temperaturess of the interm

h exchangerr It is possible to check thee effectiveness of the first heat

The numerrical results aree: • • • • •

the efffectiveness of the t first heat exchanger e is 0.381 0 the efffectiveness of the t second heaat exchanger is i 0.705 the con nvection surfaace area on thee cold side for both heat excchangers is 0.2295 m² the tottal convectionn surface areaa is 0.590 m² compared to 0.8 m² in thhe previous situatio on the tem mperatures of the t intermediaate fluid are 577.6°C and 37.66°C

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8.6. Determination of the Heat Exchanger Type and of the Overall Heat Transfer Coefficient Oil enters a heat exchanger at 90°C. Water enters the heat exchanger at 15°C. The heat capacity rate ratio is 0.5 (for the hot side). The effectiveness is 0.775. • • •

is the heat exchanger a parallel flow or a counter flow heat exchanger? what is the oil outlet temperature? what is the water outlet temperature?

If the total heat rate is 10 kW and the convection surface area is 0.895 m², what is the overall heat transfer coefficient? The answers are: • • • •

it is a counter flow heat exchanger (the Number of Transfer Units is 2) the oil outlet temperature is 31.9°C the water outlet temperature is 44°C the overall heat transfer coefficient is 385 W/m².K

8.7. Counter Flow Heat Exchanger

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Milk has to be cooled from 25°C to 5°C. Its specific heat varies with temperature: 2.814 T+3824 (in J/kg.K, with the temperature in °C). A cooling fluid at -5°C is available. Its specific heat is 3800 J/kg.K. The mass flow rate of milk is 0.1 kg/s while the mass flow rate of the cooling fluid is 0.2 kg/s. The overall heat transfer coefficient is known to be 1496 W/m².K. • • • • • •

what is the total heat rate of this heat exchanger? what is the cooling fluid outlet temperature? could it be possible to use a parallel flow heat exchanger? what is the effectiveness of the heat exchanger for the hot side? what is the effectiveness of the heat exchanger for the cold side? using a counter flow heat exchanger, what is its convection surface area? The answers are: • • • • • •

the total heat rate is 7732 W the cooling fluid outlet temperature is 5.2°C no. The cooling fluid outlet temperature is higher than the milk outlet temperature the effectiveness for the hot side is 0.67 the effectiveness for the cold side is 0.34 the convection surface area is 0.36 m²

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8.8. Cross Flow Heat Exchanger (Effect of Measurement Errors) A cross flow heat exchanger (thin walled tubes with no extended surfaces) having both fluids unmixed is used to preheat water from 20°C to 80°C, while cooling hot air from 500°C to 200°C. The specific heat is 1010 J/kg.K for air and 4200 J/kg.K for water. The design total heat rate is 100 kW and the mass flow rates are set to correspond to this value. The temperature of air is measured with an error of +/- 1°C. The temperature of water is measured with an error of +/- 0.5°C. The convection heat transfer coefficients used for the design of the heat exchanger are 200 W/m².K and 2000 W/m².K for air and water respectively. • • • • • •

•

find the mass flow rate of air find the mass flow rate of water determine the design effectiveness compute the design convection surface area find the maximum effectiveness and the minimum effectiveness taking account of the measurement errors considering that the convection heat transfer coefficient is the design one for water, find the convection coefficient for air corresponding to the maximum effectiveness and to the minimum effectiveness considering that the convection heat transfer coefficient is the design one for air, find the convection coefficient for water corresponding to the maximum effectiveness and to the minimum effectiveness

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The answers are: • • • • • • •

the mass flow rate of air is 0.33 kg/s the mass flow rate of water is 0.397 kg/s the design effectiveness is 0.625 the design convection surface area is 2 m² the effectiveness range is [0.6215 0.6285] the corresponding convection coefficient range is [197.7 202.3] W/m².K for air the corresponding convection coefficient range is [1790 2262] W/m².K for water

8.9. Counter Flow Heat Exchanger (Effect of Fouling) A counter flow heat exchanger is used to transfer heat (10 kW) from the district heating network to a detached house. The temperature of the district heating fluid is 90°C. In the house, the fluid has to be reheated from 50°C to 60°C. Its specific heat is 4200 J/kg.K. When the heat exchanger is clean, the overall heat transfer coefficient is 2000 W/m².K. After a long period, due to fouling, the overall heat transfer coefficient drops to 1900 W/m².K. The heat capacity rate ratio is unity. • • •

find the effectiveness of the clean heat exchanger compute the convection surface area find the outlet temperature of water after fouling

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The NTU-Effectiveness Method •

111

determine the heat rate in this condition

The answers are: • • • •

the effectiveness of the clean heat exchanger is 0.25 the convection surface area is 0.1667 m² the water outlet temperature after fouling is 59.6 °C the heat rate after fouling is 9620 W

9. CONCLUSION The most common configurations of heat exchangers have been studied in detail. It can be noted that it is possible to consider other configurations or to take account of the variation of the fluid properties, of maldistributions, of longitudinal conduction, as shown by the bibliography given hereafter. It has been shown that it is not necessary to consider the minimum and maximum heat capacity rates to use the NTU-Effectiveness method. An iterative process is only necessary when it is mandatory to take account of the variations of the properties of the fluids with temperature.

REFERENCES [1]

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

[2]

[3] [4]

[5] [6]

[7]

[8] [9]

Adamski, M. Heat transfer correlations and NTU number for the longitudinal flow spiral recuperators, Applied Thermal Engineering, 2009, 29, 591-596. Aparecido Navarro, H; Cabezas-Gomez, L. A new approach for thermal performance calculation of cross flow heat exchangers, International Journal of Heat and Mass Transfer, 2005, 48, 3880-3888. Baclic, BS. 1-2N shell and tube exchanger effectiveness: a simplified Fraus-Kern equation, Journal of heat Transfer, 1989, Vol. 111, 181-182. Bobbili, PR; Sunden, B; Das, SK. Thermal analysis of plate condensers in presence of flow maldistribution, International Journal of Heat and Mass Transfer, 2006, 49, 49664977. Browne, MW; Bansal, PK. An elemental NTU-e model for vapour-compression liquid chillers, International Journal of Refrigeration, 2001, 24, 612-627. Cabezas-Gomez, L; Aparecido Navarro, H; Machado de Godoy, S; Saiz-Jabardo, AJM. Thermal characterization of a cross-flow heat exchanger with a new flow arrangement, International Journal of Thermal Sciences, 2009, 48, 2165-2170. Chen, JD; Hsieh, SS. General procedure for effectiveness of complex assemblies of heat exchangers, International Journal of Heat and Mass Transfer, 1990, Vol. 33, No. 8, 1667-1674. Chowdhury, K; Sarangi, S. The effect of variable specific heat of the working fluids on the performance of counterflow heat exchangers, Cryogenics, 1984, 679-680. de Oliveira Filho, LO; Queiroz, EM; Costa, ALH. A matrix approach for steady-state simulation of heat exchanger networks, Applied Thermal Engineering, 2007, 27, 2385-

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[14]

[15] [16] [17]

[18] [19]

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[20]

[21]

[22]

[23]

[24] [25]

[26]

Sylvain Lalot 2393. Domingos, JD. Analysis of complex assemblies of heat exchangers, International Journal of Heat and Mass Transfer, 1969, Vol. 12, 537-548. Jung, J; Jeong, S. effect of flow mal-distribution on effective NTU in multi-channel counter flow heat exchanger of single body, Cryogenics, 2007, 47, 232-242. Lalot, S; Florent, P; lang, SK; Bergles, AE. Flow maldistribution in heat exchangers, Applied Thermal Engineering, 1999, 19, 847-863. London, AL; Seban, RA. Pioneer paper, A generalization of the methods of heat exchangers analysis, International Journal of Heat and Mass Transfer, 1980, Vol. 23, 5-16. Mathew, B; Hegab, H. Application of Effectiveness-NTU relationship to parallel flow microchannel heat exchangers subjected to external heat transfer, International Journal of Thermal Sciences, 2010, 49, 76-85. Noie, SH. Investigation of thermal performance of an air-to-air thermosyphon heat exchanger using ε-NTU method, Applied Thermal Engineering, 2006, 26, 559-567. On the search of new solutions of the single-pass crossflow heat exchanger problem, International Journal of Heat and Mass Transfer, 1985, Vol. 28, No. 10, 1965-1976. Pignotti, A; Shah, RK. Effectiveness- Number of Transfer Units relationships for complex flow arrangements, International Journal of Heat and Mass Transfer, 1992, Vol. 35 No. 5, 1275-1291. Pignotti, A; Tamborenea, PI. Thermal effectiveness of multipass plate exchangers, International Journal of heat and Mass Transfer, 1988, Vol. 31 No. 10, 1983-1991. Pradeep Narayanan, S; Venkatarathnam, G. Performance degradation due to longitudinal heat conduction in very high NTU counterflow heat exchangers, Cryogenics, 1998, Vol. 38, No. 8, 927-930. Raisul Islam, Md; Wijeysundera, NE; Ho, JC. Heat and mass transfer effectiveness and correlations for counter flow absorbers, International Journal of Heat and Mass Transfer, 2006, 49, 4171-4182. Ranganayakulu, C; Seetharamu, KN; Sreevatsan, KV. The effects of longitudinal heat conduction in compact plate-fin and tube-fin heat exchangers using a finite element method, International Journal of Heat and Mass Transfer, 1997, Vol. 40 No. 6, 12611277. Ranganayakulu, C; Seetharamu, KN; Sreevatsan, KV. The effects of inlet fluid flow nonuniformity on thermal performance and pressure drops in crossflow plate-fin compact heat exchangers, International Journal of Heat and Mass Transfer, 1997, Vol. 40 No. 1, 27-38. San, JY; Lin, GS; Pai, KL. Performance of a serpentine heat exchanger: Part I Effectiveness and heat transfer characteristics, Applied Thermal Engineering, 2009, 29, 3081-3087. Smith, EM. Effectiveness-NTU relationships for tubular exchangers, International Journal of Heat & Fluid Flow, 1979, Vol. 1 No 1, 43-46. Sphaier, LA; Worek, WM. Parametric analysis of heat and mass transfer regenerators using a generalized effectiveness-NTU method, International Journal of Heat and Mass Transfer, 2009, 52, 2265-2272. Tan, JO; Liu, CY. Predicting the performance of a heat pipe heat exchanger using the effectiveness-NTU method, International Journal of heat and Fluid Flow, 1990, Vol.

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11, No. 4, 376-379. Triboix, A. Exact and approximate formulas for cross flow heat exchangers with unmixed fluids, International Communications in Heat and Mass Transfer, 2009, 36, 121-124. van den Bulck, E; Mitchell, JM; Klein, SA. Design theory for rotary heat and mass exchangers- II Effectiveness-number of transfer units method for rotary heat and mass exchangers, International Journal of Heat and Mass Transfer, 1985, Vol. 28, No. 8, 1587-1595. Venkatarathnam, G; Pradeep Narayanan, S. Performance of a counter flow heat exchanger with longitudinal heat conduction through the wall separating the fluid streams form the environment, Cryogenics, 1999, 39, 811-819. Zhang, LZ. Flow maldistribution and thermal performance deterioration in a cross-flow air to air heat exchanger with plate fin cores, International Journal of Heat and Mass Transfer, 2009, 52, 4500-4509.

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In: Heat Exchangers Editor: Spencer T. Branson

ISBN: 978-1-61761-308-1 © 2011 Nova Science Publishers, Inc.

Chapter 3

MATHEMATICAL MODEL FOR PLATE HEAT EXCHANGERS FOR STEAM GENERATION IN ABSORPTION SYSTEMS Sotsil Silva - Sotelo1, Rosenberg J. Romero1,* and Roberto Best y Brown2 1

Engineering and Applied Sciences Research Centre (CIICAP) - UAEM, Morelos, México 2 Energy Research Centre (CIE) - UNAM, Morelos, México

ABSTRACT Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

Since its introduction in 1930, plate heat exchangers have become important components in the food, dairy and pharmaceutical industries and also in power generation. This type of heat exchangers offers significant advantages over traditional equipment (such as shell-and-tube heat exchangers), including: great facility for maintenance, compactness, high heat transfer rates, the possibility of modifying the transfer area by removing some of the plates and minimum fouling. Recently, it have been used these heat exchangers in absorption systems. These systems can be used either for air conditioning, refrigeration or to provide thermal energy at high temperatures. It has been shown the advantage of using plate heat exchangers in these systems, because it has reduced the size of the facilities, without affecting the performance of the system. The steam generator is a key component of the absorption equipments. This component carries out the partial evaporation of the more volatile substance from an aqueous mixture. The efficiency in heat transfer in the generator has an impact on the efficiency of the entire system. It is therefore very important to know the heat transfer mechanisms dominant in this component and, moreover, in such type of heat exchangers. This chapter of the book presents a mathematical model for a plate heat exchanger used as a steam generator in an absorption system. The model describes the two-phase flow characteristics in the plate heat exchanger (PHE). Heat transfer coefficients for the aqueous mixture, void fraction, flow quality and heat transferred were calculated. From experimental data has been possible to determine the temperature profiles in the plate heat exchanger.

*

E-mail address: [email protected] (Corresponding author)

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S. Silva - Sotelo, R.J. Romero and R. Best y Brown

1. INTRODUCTION One major challenge facing the world today is to reduce the negative effects to the environment as a result of generation and power consumption. There are four main sources of anthropogenic CO2 generation and other GHG, energy production to be the largest source of emission of these pollutants (Figure 1), as reported by the International Energy Agency [1]. Waste 3%

Agriculture Industrial processes 7% 7%

Energy 83%

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Figure 1. Major emitters of greenhouse gases.

Emissions of greenhouse gas’ 83% emissions are derived from energy production, 94 % corresponds to CO2, 5% to CH 4 and 1% N2O. As regards consumption, it is very important to make greater use of alternative energies such as solar, wind and geothermal energy. On the other hand it is important to make devices that transform energy with high efficient, which could be achieved take advantage of most of the supplied energy. This objective has focused studies for looking optimal designs with high rates of transfer heat that allow the efficient use of energy sources and also that must be compact equipment and easy device’s maintenance.

1.1. Industrial Process Heat Exchange Heat transfer processes are an essential part of many industrial processes. So there is heat transfer while two currents are in contact: a hot stream, which will give heat to another stream, called cold flow. This heat exchange is carried out in the heat exchangers and there are different types. Main selection will depend on several factors such as the process, the cost and maintenance needs. There are different types of heat exchangers; its classification depends on the process of transfer, number of fluids, construction, heat transfer mechanism the characteristics of the surface flows array and number of fluids in the process [2]. Classification of heat exchangers defined by trajectories shall be parallel and counter-current flow. In first type, both currents have their entries at the same device’s end. Flows are circulating in the same direction over the exchanger in a parallel way. In the counter-current way, flows have their entries on opposite sides of the exchanger, so moving in opposite directions. There are other classifications, such as crossed flow in a single step and crossed flow of several steps. Flows mode will depend of process as well as the characteristics of the temperature requirements.

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There are different types of heat exchangers, the best known are spiral, concentric double tube, pipe and armor, and tubes with fins. The plate heat exchangers (commonly abbreviated as PHE) have largely covered requirements for heat transfer, flexibility and compactness. This characteristic make them in a great position in the industry, mainly in the food industry, dairy equipment and generation of energy; also have also been used recently, with great success in absorption cooling and waste heat recovery systems called heat pumps by absorption. The PHE arose in 1923, by Dr. Richard Seligman, he was founder of VPA International [3]. For some time these devices were used only in industries with high hygiene requirements, but in the recent decades, the PHE have attracted the attention of power generation industries. It is possible to find PHE in secondary circuit of nuclear reactors [4]. There is a wide variety of PHE, different types and its main features will be discussed in the next section.

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2. THE PHE FEATURES The main elements of a PHE include heat transfer plates and separation of flows, boards and frames support boards; these last contain ports of entry and exit for the fluids (Figure 2). The PHE main body is made of thin rectangular metal sheets. They are interspersed between the perimeters of the joints and assembled together with screws in frames support. Boards have stamped channels that allow the passage of fluids and installation of joints. Depending on the fluid materials boards will be different, being the most common AISI 316, titanium, and SMO 254 [5] Joints on the shores of each plates pair confine each stream of fluid within the channel (i.e. should be as the space between two plates which circulates the fluid channel), preventing mixed flows, as well as leakage of fluid across the plates and the surrounding. These joints lead fluids through alternate channels from the PHE. Joints can be attached to the plate, or fixed with some clip or embedded on the board. There are various types of board materials (nitril, viton, ethylene propylene rubber etc), and the selection will depend on the process requirements. Some PHE do not require the bar carrying therefore only boards all covered with large screws as shown in Figure 3. Another type of plate heat exchangers does not include joints. The plates are welded each other, forming channels like the exchangers with joints. This type of exchangers is called brazed plate heat exchangers (BPHE) and his selection in general is for applications with high pressures and high temperatures. Plates of exchanger with joints can be used interchangeably as plates oriented right / left, turning 180 °. Same plate can be used as left or right: on right-oriented plate fluid goes from holes numbers 2 to 3 on front face and other fluid goes from holes number 3 to 2 on rear leftoriented plate. On the other way, fluid goes from hole number 1 to hole number 4 on front face and other fluid goes from hole number 4 to hole number 1 on the rear one (Figure 4).

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Figure 2. A PHE components.

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Figure 3. Plate heat exchanger. 1

2

1

2

4

3

4

3

Left

Right

Figure 4. Flow on the plates in the heat exchangers.

By their very nature, plate heat exchangers have some advantages over the conventional shell / tubes heat exchangers, such as:

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1. The space between plates creates channels where circulating fluids in films with very low thickness and great turbulence. These low thickness films also allows control of the process. The small volume of the fluid leads very quickly to changes in the process. Obtained heat transfer coefficients are significantly higher than those obtained in armor/tubes exchangers in comparable fluid conditions. Compared heat exchangers of spiral type the PHE have heat transfer coefficients values from two still three times higher [5]. 2. The PHE usually have a lower physical size due to its high heat transfer coefficients. For constant effective transfer area the weight and volume of a PHE are approximately only 20 or 30% compared with shell/tube exchangers; a PHE can perform the function of several heat exchangers in spiral type. In this way, not only saves space, also it saves costs. 3. PHE can operate at temperatures very close among its flows (~ 1 ° C). This is because the high transfer coefficients and their arrangement in cross flow. The result of this advantage is that PHE can be achieved more than 90% of the recovered heat, which is a thermal performance significantly high, compared with 50% that is recovered in the shell / tube heat exchangers. 4. The PHE are therefore highly suitable for use in thermal sources with very close temperatures or low temperature heat recovery. 5. In the PHE each fluid is confined into channels between boards with joints, and it is closed to the atmosphere, thus it is eliminating the possibility of contamination for the fluids. The plates in the PHE can be dissemble and assemble easily, for inspection, cleaning and replacement of the joints, conveniently. In particular hygiene can be maintained. Additionally, the easy dissemble and assemble process provides flexibility to change the thermal load of PHE, simply by adding or removing some plates [6]. 6. The construction of the plates with different corrugations causes turbulence in the fluid, which reduces the amount of inlays in the exchanger. 7. For the PHE with joints, it is easy the maintenance process. It is a great advantage. Remove the plates for proper cleaning is not greater problem. In spite of the above, the PHE have some disadvantages, such as: • • • •

Large pressure falls due to the corrugation plates, PHE costs might be higher than traditional shell / tube exchanger, PHE design does not have a known standard procedure; and Data designs are protected by the companies responsible of marketing.

2.1. The PHE Geometries There are a wide variety of commercial PHE with different patterns of corrugation. Chevron type geometry and its variation Herringbone type or zig - zag are the commonly used. Figure 5 shows the two types of corrugation. To describe the different PHE geometries and corrugation surfaces different nomenclatures in literature and the distributors’ catalogues are used. Common definitions are,

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Chevron angle θ, internal thickness between the limits of the joints (W), length (L = L p-p) and the effective projected area (W_x_L). For the calculation of the flow between channels, the convective heat transfer and equivalent diameter are used. The equivalent diameter is defined as [6]:

a)

b)

Figure 5. Plates with geometry type a) Chevron and b) Herringbone. 4WH i ≈ 2H i 2(W + H i )

(1)

4 × channel ⋅ free ⋅ flow ⋅ area ⋅ for ⋅ fluid 2 H i = wetted ⋅ perimeter ⋅ for ⋅ the ⋅ fluid φ

(2)

de =

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And hydraulic diameter d h is: dh =

Where φ is the heat transfer’s effective area factor defined as: φ=

developed ⋅ length projected ⋅ length

(3)

Should be noted that d h =(de/_φ), and both are used in the literature as a length scale to characterize the Reynolds number, Nusselt number and friction factor. It is preferable to use equivalent diameter de. The different geometric parameters of the PHE are shown in Figure 6. The flow velocity into the channel can be calculated with:

u=

m Afreeflow× n

Where: m is the total flow rate of the fluid under consideration A freeflow is the free flow area of the channel

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n is the number of channels to which the fluid is distributed

The free flow area can be estimated with:

A freeflow = b × w

(5)

b = P − Δz (6) where P is the PHE pitch, and Δz is the thickness of the plate.

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Figure 6. Geometric parameters of a plate heat exchanger.

In all PHE three configuration for hot and cold fluids are the following: (1) arrangement in parallel way with both currents flowing in the same direction, (2) cross flow with both currents flowing in opposite directions, and (3) arrays in multi-step where trajectory of one flow, at least, goes in a direction and on the end this flow is reversed over the same length; one or more times.

2.2. The PHE Applications At the beginning of this chapter, we mentioned that PHE have a large number of applications, mainly those processes with cleaning, flexibility and compactness for installations requirements. Recently these PHE has been evaluated for absorption cycles, which operate as condensers, evaporators, absorbers and steam generators. The following sections describe the absorption cycles characteristics.

2.2.1. Absorption Systems Absorption systems (also absorption called heat pumps) have been considered as an option instead of traditional compression systems. Considering a traditional compression cycle as illustrated in Figure 7.

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QCO

PCO

Condenser

Expansion valve W Compressor

PEV

QEV Evaporator

TEV

TCO

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Figure 7. Traditional compression cycle.

Into a compression cycle, a “working fluid” evaporates in the evaporator while it is extracting an amount of heat QEV from the environment, then the “working fluid” is compressed while it transfer latent heat (QCO) at higher temperature. The condensed “working fluid” expands its volume through the expansion valve and returns to the evaporator, starting the cycle again. In the case of absorption systems, the compressor is replaced by two components: an absorber and a steam generator. For an understanding of this substitution: there is a desorption process (or steam generation) and absorption heat rejection instead the compression process. Absorption systems use a mixture of a “working fluid” with an “absorbent” substance. There are two types of cycle: Type I, in which the evaporator temperature is lower than the temperature of the condenser (including this classification the compression and absorption systems). These absorption systems are called “absorption heat pumps”. Type II is a cycle where the temperature of the evaporator is greater than the temperature of the condenser [7], this last type of absorption system is called “absorption heat transformer”. Figures 8a and 8b show a diagram of both cycles for pressure against temperature. The type I cycle can be used for the conditioning of spaces, either providing heat at a temperature of the atmosphere (warming mode) or extracting heat at lower temperature than the environment (cooling mode). In the case of the absorption heat transformer, the thermodynamic cycle is as follow: in the generator is carried out the partial separation of “working fluid”. It is due to the supply of a heating source (costless energy from alternative energy or waste heat from an industrial process) to an intermediate temperature TGE. The evaporated “working fluid” of work is a correspondent to the condenser where it changes phase when rejecting a quantity of heat, the “working fluid” in liquid form is pumped to the evaporator. The evaporator is at higher pressure than condenser. Into the evaporator, the “working fluid” is changed to vapor phase supplying similar quantity of heat that the generator.

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Mathematical Model for Plate Heat Exchangers…

QCO

PCO

Expansion valve

PEV

QEV

Condenser

Generator

123

QGE

Pump

QAB

Evaporator

Absorber

TEV

TGE

TCO,TAB

Figure 8. a) Type I absorption cycle.

QAB

QEV

PEV

Evaporator

Absorber

Economizer

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Pump

PCO

Expansion valve

QGE

QCO Condenser

TCO

Generator

TEV,TGE

TAB

Figure 8. b). Type II absorption cycle (absorption heat transformer).

The vapor goes to the absorber, where that vapor enters in contact with the solution with high concentration of absorbent. This process produces diluted solution at high temperature (TAB) and then this goes to generator. Also this dilution phenomenon delivers heat. The mixture in the generator begins the cycle again. The economizer is a heat exchanger; it preheats the solution that goes from the generator to the absorber using the heat of diluted solution with temperature (TAB) at absorber conditions. That exchange increases the efficiency of cycle.

2.2.2. Mixtures and Pairs As mentioned earlier in this chapter, the absorption heat pumps use a mix of fluids (called also for engineers as pair) formed by “working fluid” and absorbent. The absorbent circulates

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in the circuit of the generator - economizer - absorber, and the “working fluid” goes into the condenser and the evaporator. Absorption cycle performance depends heavily of pair properties. For a mixture be able to use into an absorption heat pump it must have specific properties. The “working fluid” and absorbent properties are crucial in the design and cost of the absorption cycle. The properties that “working fluid” include are: • • • •

High latent heat of vaporization, Vapor pressure into the limits fixed by condensation conditions, Fusion temperatures lower than operating conditions and environment temperature, Thermodynamic critic point higher than any operating conditions.

Those characteristics prevent damage to components by dilation of the fluid at freezing temperature. And those avoid any undesirable phase change. On the other hand the absorbent must have:

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• • • •

High boiling point. Negligible vapor generation. Solubility into the “working fluid” in a large concentrations range. Vapor pressure lower than “working fluid” at absorption temperature.

Those characteristics prevent absorbent contamination into the condenser and also that prevent crystallization risk. There is not currently a mixture that has all the above requirements. Followed mixtures have some required properties for be able to use into absorption cycles. According the prices and features of heat and mass transfer the “water – lithium bromide” mixture is most popular. In this pair water is the “working fluid”. However this pair displays some drawbacks, such as solubility in water, which is limited to 70% by weight. Higher concentrations of aqueous lithium bromide have crystallization problem. Another undesirable feature is the corrosion at high temperatures, which has limited use into “heat transformers”. Another common mixtures is the pair of ammonia - water (NH3 - H2O), where ammonia is the “working fluid” and water is the absorbent. Ammonia – Water pair is used for cooling at temperatures below 0 ° C, the main advantages of this mixture are: wide range of solubility and many operating conditions. Ammonia has a high latent heat. Its disadvantages are its toxicity and its operation requires high pressure equipment.

2.2.3. Water - Carrol ™ A mixture that can solve the problem of crystallization water – lithium bromide is the water – Carrol™. This mixture was developed by Carrier Co., during his work for develop of solar cooling system. Carrier patent mixing water - Carrol ™ has expired, so it is possible to use unlicensed [8]. This mixture was previously tested for Carrier Co. in air conditioning by absorption systems and the pair did not report great benefits, it remains without commercial application. This mixture is an aqueous solution consisting of: lithium bromide and ethylene glycol as an additive, in relationship 4.5/1 by weight. The Carroll ™ advantage over the

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mixture of lithium - water bromide is because ethylene-glycol inhibits the crystallization; therefore it can be used at higher temperatures and higher concentrations. Carrol ™ has demonstrated better performance compared to systems that operate with the mixture of lithium - bromide water [9]. Vapor to the condenser

Weak solution from the absorber Generator

Strong solution to the absorber

Heater system Figure 9. Diagram of process of steam generation in the absorption heat pump.

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3. STEAM GENERATION For “absorption heat pumps” using mixtures water - Carrol ™ or water – lithium bromide, the term desorption means the “steam generation” from the pair “working fluid” and absorbent solution. By adding heat to the aqueous concentrated solution (called strong solution), generates a flow of steam, while a concentrated solution remains in generator (Figure 9). Many studies on “absorption heat pumps” have been detailed for the operation and its absorbers; however, today few researches have been conducted on generators in such systems. Most studies consider it a small portion of the entire system of “absorption heat pumps” [10]. There are two types of generators or desorbers: heated directly or indirectly. Directly heated generators use heat transferred by the combustion to heat the solution from the absorber. To do this, a gas burner is used as a direct heat source. Indirectly heated generators using a hot fluid heated previously using waste heat, solar collectors, [11] etc. Generator is one of the main components of “heat transformers”, its efficiency has a significant influence on the performance of the system. One factor that complicates the comprehensive study of this component is a problem of heat and mass transfer in two-phase flow. At present, workflow transfer mechanism is not fully realized in this type. The generator is also the component with more exergy losses in the system, as demonstrated by the literature review at beginning of this work [12]. Is important to say, in literature there are various studies about stagnant boiling of lithium bromide – water, however, there are very few work for this pair. Nevertheless there are no previous on flow forced boiling to Carrol ™ - water, therefore, this chapter represents the first approach to this pair. Now, we show most

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outstanding reports on generators’ absorption heat pump for ammonia – water, and lithium bromide – water with plate heat exchangers.

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4. STUDIES IN PLATES IN ABSORPTION SYSTEMS HEAT EXCHANGERS The PHE have recently been used in systems of absorption, mainly on absorption systems throughout the pilot plant in R&D centers at different countries. Ribeiro Jr. and Caño [13] Andrade presented a model for the simulation in a plate heat exchanger (PHE), which flows can go parallel or upstream; and different arrangements of flows distribution, all this for steady state in 2002. The authors estimate temperatures distribution in the PHE for several channels as a linear combination of exponential functions. The model is valid and accurate for analytical solutions but available for simple cases and some experimental data. Gut and Pinto [14] presented in 2003 a generalized model to analyze the influence performance by exchanger settings. Exchanger setting is described by number of channels, the number of steps on each side of the PHE. Authors considered in the model: a) steady state, b) no losses of heat with the surroundings, c) there is no exchange of flow in the direction of flow, d) type plug flow into channels, e) uniform distribution of flow through each channel, f) perfect mixing of the fluid at the end of plate, and g) there is no phase change. Vallès et al analyze in 2003 [15] absorption in jets for 2 organic fluids in a PHE. Experiments conducted with different types of jets (spray nozzles) and mixtures: methanoltetraethyleneglycol dimethylether (TEGDME) and trifluoroethanol (TFE) - TEGDME. The selected PHE consisted of 60 plates, particularly selected for lower pressure drop and with a transfer area of 5.5 m2. Selected sprayer considered the PHE geometry so they chose sprayer with a small spray angle. The authors found that absorption systems main limitation is the pressure drop in the heat exchanger. The absorber consisted of two parts, the absorption process takes place first in an adiabatic mixing chamber, where diluted solution is sprayed in the refrigerant vapor, then two – phases flow goes inlet PHE, where the solution is cooled and then this absorption process is complete. Vega M. [16] study an absorption system cooling, it was conducted with generator, evaporator and condenser were PHE. The authors propose using an equation previously proposed for a coolant for determining the heat transfer coefficient in the generator. This equation was validated experimentally whereas water as evaporated fluid. The authors show different values of COP (which is the effectiveness coefficient of this heat pump operation) for several condensing temperatures. García Cascales et al presented in 2007 [17] studies of plates condensers and evaporators in cooling systems. The authors made a comparison between several previously published for pairs in one phase and two-phase. The refrigerants R-22 and R-290 were used into PHE. In 2009 Marcos et al [11] identified the heat transfer coefficient boiling for a generator in a “double stage heat pump”. The evaluated generator experimentally was a vertical PHE. The solution was lithium bromide - water and heating fluid was oil. The generator was evaluated with a constant temperature of 55 ° C in the condenser. It was determined that there are two heat transfer mechanism in the high temperature, at the beginning, generator transfer heat in a phase liquid - liquid, (called zone I), then when steam generation begins there is heat transfer

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between liquid and liquid - vapor (zone II). Thus the generator was divided into two zones corresponding to these different mechanisms of heat transfer. The authors showed zone II heat transfer rates are higher than those obtained in zone I. The coolant quality is very small with values between 4 and 6%. The pressure drop in the area I is negligible compared to the zone II. One of main reasons for using PHE in absorption systems is the decreasing in the size (steam generator and absorber). Cerezo et al., presented in 2009 [18] a study of an absorber in a cooling system using the pair ammonia - water. The absorber is a three-channel, in the center channel PHE circulates diluted ammonia solution, and ammonia vapor is injected on a tube of 1.7 mm, this process release heat reaction. In the two remaining channels of the PHE circulating cooling water. Before to PHE evaluation with the pair, the authors characterized the PHE only with cooling water, and thus they were proposing two equations for the zones of transition and turbulent flow: Transition zone:

Nu c = 0.990 Re 0.530 Pr 0.330

(7)

Nu c = 0.339 Re 0.703 Pr 0.330

(8)

Turbulence zone:

The authors found that increase in the mass flow of cooling water increases mass flow absorption. They also found that increased pressure in the absorber improved heat transfer and absorption mass flow.

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5. TWO-PHASE FLOWS The steam generating process is carried out in the generator due to application of heat. If an amount of heat is applied to a surface with a fluid, to reach up a higher temperature than saturation temperature, the boiling phenomenon happen on surface. The boiling may occur under conditions of stationary fluid, which referred such as “stagnant boiling” (or pool boiling) or under conditions of forced flow, called “boiling forced convection” or boiling in forced flow (flow boiling). There are various parameters to characterize a biphasic flow: the flow pattern, flow quality, the vapor fraction and pressure drop.

5.1. Flow Pattern Two-phase flows are classified by the bubbles distribution. There are three basic types of bubbles distribution: bubbles suspended in the liquid, liquid droplets suspended in the flow of vapor – liquid and intermittently vapor stream. Commonly observed flow structures are known as two-phase flow patterns which have particular characteristics that identify them. Flow patterns in horizontal two-phase flow are influenced by gravity, which are in stratification way, with the liquid at the bottom and the gas at the top of the channel. Identifiable flow patterns are (Figure 10):

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Bubble Flow: bubbles are dispersed in the fluid with higher concentrations in the half top pipe. However high flow rates tend bubbles to disperse uniformly in the tube. Stratified flow. At liquid and gas low speeds, there is a complete separation of the two phases, with the gas at the top and the liquid in the bottom, separated by a horizontal interface without alterations.

(a)

(b)

(c)

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(d)

(e) Figure 10. Horizontal tubes flow patterns. a) bubble flow, b) stratified flow, c) stratified-wave flow, d) annular flow, e) mist flow.

•

•

•

Stratified – wave flow: When the gas speeds are increasing there is a complete phase separation. It forms waves in the gas - liquid interface and those travels in the direction of flow. The amplitude of wave depends on the relative speed of the two phases, but the crest does not reach the top of the tube. Waves tend to wrap up the sides of the tube leaving a thin liquid film on the wall after the waves pass. Intermittent flow: increasing the speed of the gas, the waves grow in magnitude until they reach the top of the tube. In this way, large scale waves dipped the top tube intermittently, while other waves with narrow and movement are evident. Waves with wide-ranging contain a large amount of fluid and commonly dragged bubbles. The top of the tube is constantly wet with a liquid film that left behind each wave of large amplitude. Annular flow: large amount of gas flow the liquid forms a continuous ring film around the film on the tube, which tends to be significantly thicker at the bottom of the tube. The film interface is altered by waves with small amplitude and drops of

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liquid can disperse in the gas. With high fraction of gas, the top tube eventually becomes dry, with flow returning to the flow stratified wave. Mist flow: in this type of flow liquid can be dragged as small droplets in a gas phase with continuous high-speed [2].

6. MATHEMATICAL MODEL FOR GENERATION OF STEAM IN A PHE To determine the operating condition in the PHE, a mathematical model based on the following control volume is defined (Figure 11):

weight weight

Figure 11. Control volume for the mathematical model.

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Table 1. Characteristics of the PHE Fluid flow length [mm] Distance between plates [mm] Plate tickness [mm] Type corrugation Angle of corrugation Number of plates Material of construction of the plate

380 6 0.5 Chevron 60° 7 AISI 316

The one-dimensional model considers that: 1. The first channel generator corresponds to the solution and the lower channel circulates heating water. 2. The heat is transferred by hot water to Carrol ™ - water which flows into the channel. 3. The heat losses to surroundings are negligible. 4. The generator is in steady state.

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S. Silva - Sotelo, R.J. Romero and R. Best y Brown 5. It is considered input flows (both working solution and water) are linearly distributed between the six heat exchanger channels (three channels of water and three of working solution). 6. It is considered steam constant temperature during the generation process. The study PHE features are shown in table 1.

6.1. The Energy Balance Equation The study of the channel the generator will be divided into two sections. Since steam generation is not happen in all length of channel. To define the time when generating begins it is consider initial concentration, equilibrium temperature and steam temperature. Because it has temperatures channel profile, it will be possible to locate length where equilibrium condition happens and the generation begins. For this reason we obtain two energy balances corresponding to the supplied heat to bring the solution to balance point (Qliq,eq) and later heat supplied to continue steam generation (Qliq,gen), these are obtained based on equations 9 and 10 respectively.

Qliq , eq = mliq ( H liq , eq − H liq ,in )

(9)

Qliq , gen = mliq ( H liq , out − H liq , eq ) + mvap (H vap − H liq , eq )

(10)

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6.2. Overall Heat Transfer Coefficient Two equations, the proposal by Bogaert & Bölcs were used for heating water transfer coefficient [19]: 1 ⎛ 6.4 ⎞

hheat = 0.2634

e⎜ ⎟ k heat Re 0.7152 Pr 3 ⎝ Pr + 30 ⎠ de

(11)

And the equation of Chisholm and Wanniarachchi [20]: hheat = 0.724

kheat ⎛ 6 β ⎞ ⎜ ⎟ de ⎝ π ⎠

0.646

1

Re 0.583 Pr 3

(12)

Overall heat transfer coefficient will be calculated with two equations, since a part of the solution is single phase flow, and it begins generation as two-phase flow. For the part in a single phase, use the equation reported by Hewitt [21] for a PHE angle of 60 ° from the vertical.

hliq ,eq = 0.4

kliq de

Re0.64 Pr 0.4

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(13)

Mathematical Model for Plate Heat Exchangers…

131

Pr < 200 Results for two-phase flow are calculated by two equations: equation used by M. de Vega [16] for the water - lithium bromide pair. ⎛ kliq hliq , gen = 0.2⎜⎜ ⎝ de

⎞ 0.7 13 ⎟⎟ Re Pr 88 Bo 0.5 ⎠

(

)

(14)

With boiling number defined as:

Bo =

qheat i fg ρvapVvap

(15)

And second equation by Han et al., [22] based on the refrigerant evaporation in PHE, for several PHE geometries:

hliq , gen = Ge1

kliq de

0.3 0.4 2 ReGe eq Boeq Pr

(16)

Where Ge1 and Ge 2 parameters are PHE geometry function. ⎛ p⎞ Ge1 = 2.81⎜⎜ ⎟⎟ ⎝ de ⎠

−0.041

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⎛ p⎞ Ge2 = 0.746⎜⎜ ⎟⎟ ⎝ de ⎠

⎛π ⎞ ⎜ −β⎟ ⎝2 ⎠

−0.082

−2.83

⎛π ⎞ ⎜ −β⎟ ⎝2 ⎠

0.61

(17)

(18)

Where “P” is the heat exchanger pitch and “β” is the chevron angle.

6.3. Vapor Fraction The “α“ represents the vapor fraction in the tube, and it is depending on the density and quality is determined by [23]: ⎛ ⎜ ⎛ 1 − Xf α = ⎜1 + ⎜⎜ ⎜ ⎝ Xf ⎝

⎞⎛⎜ ρvap ⎞⎟ ⎟⎟ ⎠⎜⎝ ρliq ⎟⎠

2

3

⎞ ⎟ ⎟ ⎟ ⎠

−1

6.4. Steam Quality The steam quality is estimated with:

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S. Silva - Sotelo, R.J. Romero and R. Best y Brown Xf =

⎤ 1 ⎡ Qheat − Cp liq (Tliq ,out − Tliq ,in ) ⎥ ⎢ i fg ⎢⎣ mliq ⎥⎦

(20)

6.5. Pressure Drop The pressure drop in a PHE is given by the following equation:

ΔPt = ΔPf + ΔPg + ΔPa + ΔPNi

(21)

Where ΔPf is the pressure drop due to friction

ΔPg is the pressure drop by static head

ΔPa is the pressure drop due to the fluid acceleration ΔPNi is the pressure drop in the ports of entry and exit of fluid. In this study case, it does not take into account the term ΔPg since it has a horizontal PHE and also are not considered losses at ports of entry and exit of flows ΔPNi , since this analysis only the first channel of the PHE. Again, since the PHE channel has two areas, one with single phase flow and another with a two-phase flow, equation 21 are used for the two cases. In the case of single phase flow the pressure drop by the fluid acceleration is negligible, so the equation 21 reduces a:

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ΔPt = ΔPf

(22)

The pressure drop due to the friction for single phase flow is given by the equation:

l G2 ΔPf = 4 f de ρ

(23)

Where “G” is the mass flow expressed in kg/m2s and “f” is a factor which depends on the PHE geometry, which is obtained by Muley and Manglik equation [24]: f=

(24)

[(40.32 / Re)0.5 + (8.21Re −0.5 )5 ]0.2 for 2 ≤ Re ≤ 200

1.274 Re −0.15 for Re ≥ 1000 The pressure drop for two-phase flow given by:

ΔPt = ΔPf + ΔPa

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Where frictional pressure drop is the two-phase multiplier function:

⎛

C

1 ⎞

⎟⎟ + φlo2 = ⎜⎜1 + X X tt tt ⎠ ⎝

(26)

Where “C” value depends of liquid and vapor flow regimes (see table 2). X tt is the Lockhart Martinelli parameter described by: 0. 9

⎡ Xf ⎤ ⎡ ρliq ⎤ 1 =⎢ ⎥ ⎢ X tt ⎣1 − Xf ⎥⎦ ⎢⎣ ρ vap ⎥⎦

0. 5

⎡ μvap ⎤ ⎥ ⎢ ⎢⎣ μliq ⎥⎦

(27)

Table 2. Values for “C” for two-phase multiplier Liquid regime Turbulent Laminar Turbulent Laminar

Steam regime Turbulent Turbulent Laminar Laminar

Channels for working solution

(C) 20 12 10 5

Channels for the heater system

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Figure 12. Plate edges form a regular honeycomb pattern.

6.6. Temperature Profiles The PHE longitudinal temperature profiles were determined experimentally. For the experimental setup, thermocouples type T were placed on the plates that form the first channel of the heater system and working solution. To fix the thermocouples to the wall of the plates was designed a mould with the figure of the plates edges, which corresponds to a honeycomb pattern, (Figure 12). For the mould plates was used a high viscosity silicone. This material is used for highprecision printings and is characterized by a high initial fluidity in the phase of work and a high final hardness. For final hardness, of the mould was added to the initial mixture eight drops of catalyst and thus, it was possible to obtain the printing of the plates. Once obtained the mould in the silicone, it was obtained the opposite image in a container with plaster; the image in plaster was covered with a sealer to eliminate the permeability of the material; this final mould was filled with polyester resin. This corresponds to the image of the plates; finally, it was drilled and in these holes the thermocouples were placed with cyanoacrylate glue.

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Moulds of resin with thermocouples were placed in the PHE edges and the thermocouples were connected to a data logger. In this way, the temperature profiles for the heater system and working solution were calculated. The equations of temperature profiles are presented in the experimental evaluation of the PHE.

7. EXPERIMENTAL EVALUATION OF THE PHE Several experiments were conducted in a heat transformer; the five main components were PHE, three different concentrations (54, 56 and 58 %) and several flows of the heater system (water) and the working solution were used. The heat transformer was instrumented with thermocouples type T. The working solution was water - Carrol ™.

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7.1. Instantaneous Determination of Heat Transfer Coefficients One objective to propose the mathematical model was to bring a tool to allows the modification of some parameters during the experimental test, for this, it was decided to make a program for determining instantaneously the heat transfer coefficient for the working solution and for the heater system. The software used for this program was Agilent Vee Pro © 7.5 by Agilent Technologies, which measured the temperatures in the generator with the thermocouples fixed in the PHE, figure 13 shows a first screen of the program. Once the program measured the temperatures, the calculation of the heat transfer coefficients was made. Currently, it is not possible to control mass flows through the PC, these data must be entered during the test. The algorithm for this program is shown in Figure 14. The second screen displays the panel, which only shows the plots of the heat transfer coefficients versus the plate length (Figure 15).

Figure 13. Program for the instantaneous determination of heat transfer coefficients.

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Figure 14. Algorithm for instantaneous determination of heat transfer coefficients in the PHE.

Figure 15. Panel whit the plots of the heat transfer coefficients in the PHE.

With this program was possible during the experimental tests, to use different mass flows, and thus determine the best operation conditions for the PHE [25]. 7.2. Experimental Results for the PHE

To determine the temperature profiles in both, heater system and working solution, experimental data were obtained in stable state. Figure 16 and 17 show temperature along the Heat Exchangers: Types, Design, and Applications : Types, Design, and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest

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plate and statistical adjustment, which shows that the temperature profiles for the PHE is described by a second order equation. The second degree equations presented in the plots will be used to calculate the conditions of equilibrium and concentrations profiles along the PHE. An experimental stable generation graphic is shown in figure 18. These temperatures are of thermocouples located in the ports of inlet and outlet of the PHE; do not correspond to the thermocouples placed in the walls of the plates for the determination of the temperature profiles. It has been experimentally evaluated three concentrations in the heat transformer, 54, 56 and 58 %, it was determined that, increasing the concentration, it was necessary to increase the heat input, the pressure in the system was fixed between 80 and 131 mm Hg (Figure 19).

Figure 16. Temperature profile of the heater system.

Figure 17. Temperature profile of the working solution. Heat Exchangers: Types, Design, and Applications : Types, Design, and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest

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90 80

Temperature [°C]

70 60 50 40 In heater system

30

Out heater system Out working mixture water - Carrol

20

TM

TM

In working mixture water - Carrol Out steam from the generator

10 0 0

1

2

3

4

5

Time [hours]

Figure 18. Experimental plot of temperature versus time for stable steam generation.

It is possible to see that in some points input heat is lower to a higher concentration, this is because different pressures were fixed in each test for each concentration. It should be noted that during experimental tests, pressure is one most critics parameters. Figure 20 shows the equilibrium diagram for the mixture water - Carrol ™. 2.00

1.60 1.40 Heat input [kW]

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1.80

1.20 1.00 0.80 0.60 0.40 0.20

54%

56%

58%

0.00 53

54

55

56

57

58

Concentration [% weigth]

Figure 19. Heat input to the generator versus concentration of the solution.

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Figure 20. Equilibrium diagram for Water - Carrol ™ solution [26].

8. THE PHE SIMULATION

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The mathematical model will be used to determine the conditions in the first channel of the PHE, i.e., it will only simulate the first channel of working solution, which receives heat from the heater system. Figures 21 and 22 show the temperature profiles obtained from the simulator using the experimental equation.

Figure 21. Temperature profile for the working solution in the PHE.

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Figure 22. Temperature profile for the heater system in the PHE.

Figure 23 shows the concentration profile for the working solution in the PHE. Because there are experimental equations to determine the temperature profile, it is possible to know the point of the plate length where the steam generation begins. To calculate this point, is obtained the equilibrium temperature, according to correlations reported by Reinmann and Bierman [26]. It is possible to observe an increase in the concentration of the solution, showing the started of water evaporation from the solution. Figure 24 presents the heat transfer coefficients for the working solution. As previously mentioned in the chapter, there are no previous reports about the water - Carrol ™ solution heat transfer coefficients; however, there are few works for the water – LiBr mixture, the work reported by Marcos et al., [11], also made in a PHE as steam generator in an air cooled absorption system, shows that the heat transfer coefficients for the solution water - LiBr are higher in the steam generation zone. The authors obtained experimentally the heat transfer coefficients, finding values of 0.75 to 1.3 kW/m 2 K for the zone previous to the steam generation (one phase flow) and values of 2.5 to 4.4 kW/m 2 K for the generation zone (twophase flow). It should be noted, that the system analyzed by Marcos et al., is a PHE with 30 plates and the heater system used oil. Other work reported for the water - LiBr solution is the work reported by Rivera and Xicale [27]. The authors evaluated the mixture with constant heat fluxes of 2 to 10 kW, and concentrations of 48 to 56%. The average transfer coefficients were 1 to 4 kW/m2 K.

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Figure 23. Concentration profile of the working solution along the PHE.

Figure 24. Heat transfer coefficients for Waster - Carrol ™ solution.

The heat transfer coefficients for the heater system were also calculated with two different equations, as shown in Figure 25. It is possible to observe that, there is not a significant decrease in the heat transfer coefficient along the PHE. The equation of Bogaert and Bölcs (Ec. 11) shows higher values, this could be explained by the exponential term in this equation

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Figure 25. Heat transfer coefficients for the heater system.

Table 3 shows a summary of the parameters calculated with the simulation program. It is possible to observe that the largest pressure drop is for steam generation area in the PHE, the same effect is for the input heat Qliq,gen, these behaviors are agree to the reported by Marcos et al., [11]. Figure 26 shows the program of the model in the Software Vee Pro ©, unlike the instantaneous determination of heat transfer coefficients program, the simulation program does not has a panel; all the data are sent to a file in Excel ™ (Figure 27). Table 3. Parameters calculated with the simulation program X %, in X %, out Qliq,eq Qliq,gen Pressure drop single phase flow Pressure drop two phase flow Quality

58% 59.7% 0.03 kW 0.20 kW 0.52 mbar 2.8 to 2.7 mbar 7%

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Figure 26. Simulation program for the PHE.

Figure 27. Results of the simulation in an Excel ™ file.

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CONCLUSION The plate heat exchangers (PHE) are frequently used in different processes industries such as food, dairy and power generation. Advantages such as flexibility, ease of maintenance and cleaning, compactness and high heat transfer rates, make these equipments a better option compared to traditional equipments like shell/tubes, spiral, etc. The PHE have been used recently in absorption heat pumps systems, with satisfactory results. One main component of these systems is the steam generator, where partial evaporation of more volatile component of the working mixture is carried out. This chapter had shown a mathematical model to determine the transfer heat coefficients of the mixture water - Carrol™ (lithium bromide + ethylene glycol) in a PHE functioning as steam generator in an absorption system. The PHE was instrumented with thermocouples to determine experimentally the temperature profiles, finding that they are described by a second order equation. The study of the generator was divided into two sections, before and during the steam generation. It was found that the heat transfer coefficients and the pressure drop in the first channel of the PHE are higher for steam generation zone. So far, there are no data reported on the heat transfer of the water - Carrol ™ mixture, so the results were compared with previously reported data for the water - lithium bromide mixture, finding that the magnitude of the heat transfer coefficients and trends for these parameters are similar.

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NOMENCLATURE Bo Cp de h G ifg k m Nu Pr Q q Re T Xf X% Xtt

Boiling number specific heat [kJ/kg °C] equivalent diameter [m] heat transfer coefficient [kW/m2 °C] mass flux [kg/m2s] enthalpy of vaporization [J/kg] thermal conductivity [W/m K] mass flow [kg/s] Nusselt number Prandtl number heat flow [kW] heat flux [kW/m2] Reynolds number temperature [°C] vapor quality concentration of the working solution Lockhart – Martinelli parameter

Greek Symbols α β

void fraction chevron angle

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S. Silva - Sotelo, R.J. Romero and R. Best y Brown ρ μ φ Δz ΔP

density [kg/m3] viscosity [Pa s] two-phase multiplier plate thickness pressure drop

Subscripts gen heat in liq lo out vap eq

condition in steam generation heater system inlet condition liquid working solution the total mass flux flowing with the liquid properties outlet condition vapor equilibrium condition

REFERENCES

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[1]

International Energy Agency, CO2 emissions from fuel combustion, highlights, 2009 edition, available on the web. http://www.iea.org/ publications/ free_new_Desc.asp? PUBS_ID=2143 [2] Bejan, A. & Kraus, A. (2003). Heat transfer handbook, John Wiley & sons. [3] Kuppan, T. (2000). Heat exchanger design handbook, Mercel Dekker Inc. [4] Sarit, K. (2005). Das, Process Heat Transfer, Alpha Science. [5] Plate heat exchangers manual, by Sondex, http://www.sondex.dk/archive/ downloads/manual/Sondex%20manual%20spansk.pdf, [6] Wang, L. & Sundén, B. (2007). Manglik, Plate Heat Exchangers: Design, applications and performance, WIT Press. [7] Herold, K. E., Radermacher, R. & Klein, S. A. (1996). Absorption chillers and Heat pumps, CRC press, USA. [8] Robert, A. & Zogg, Michael, Y. (2005). Feng, Detlef Westphalen, Guide to Developing Air-Cooled LiBr Absorption for Combined Heat and Power Applications, US department of energy, energy efficiency and renewable energy. [9] Rivera, W., Cardoso, M. J. & Romero, R. J. (1998). Theoretical comparison of single stage and advanced heat transformers operating with water/lithium bromide and water/Carrol mixtures, International Journal of Energy Research, 22, 5, 427-442. [10] Jurng Jongsoo, (1998). Park Chan Woo, On The Performance Of A Desorber For Absorption Heat Pumps With A Thermosyphon And A Surface-Flame Burner, Applied Thermal Engineering, 18, Nos 3-4, 73-83. [11] Marcos, D., Izquierdo, M., Lizarte, R., Palacios, E. & Infante Ferreira, C. A. (2009). Experimental boiling heat transfer coefficients in the high temperature generator of a double effect absorption machine for the lithium bromide/water mixture, International Journal of refrigeration, 32, 627-637.

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[12] Martínez, H. & Rivera, W. (2009). Energy and exergy analysis of a double absorption heat transformer operating with water/lithium bromide, Int. J. Energy Res., 33, 662674. [13] Ribeiro, Jr., C. P. & Caño Andrade, M. H. (2002). An algorithm for steady-state simulation of plate heat exchangers, Journal of Food Engineering, 53, 59-66. [14] Jorge, A. W. & Gut, José, M. (2003). Pinto, Modeling of plate heat exchangers with generalized configurations, International Journal of Heat and Mass Transfer, 46, 25712585. [15] Manel Vallès, (2003). Mahmoud Bourouis, Dieter Boer, Alberto Coronas, Absorption of organic fluid mixtures in plate heat exchangers, International Journal of Thermal Sciences, 42, 85-94. [16] M. De Vega, J. A. Almendros-Ibañez & G. Ruiz, (2006). Performance of a LiBr–water absorption chiller operating with plate heat exchangers, Energy Conversion and Management, 47, 3393-3407. [17] García-Cascales, J. R., Vera-García, F., Corberán-Salvador, J. M. & Gonzálvez-Maciá, J. (2007). Assessment of boiling and condensation heat transfer correlations in the modelling of plate heat exchangers, International Journal of Refrigeration, 30, 10291041. [18] Jesús Cerezo, (2009). Mahmoud Bourouis, Manel Vallès, Alberto Coronas, Roberto Best, Experimental study of an ammonia–water bubble absorber using a plate heat exchanger for absorption refrigeration machines, Applied Thermal Engineering, 29, 1005-1011. [19] Bogaert, R. & Bölcs, A. (1995). Global performance of a prototype brazed plate heat exchanger in a large Reynolds number range, Experimental Heat Transfer, 8, 293-311. [20] Chisholm, D. & Wanniarachchi, A. S. (1991). Layout of plate heat exchangers, in: ASME/JSME Thermal Engineering Proceedings, 4, ASME, New York, 433-438. [21] Hewitt, G., Shires, G. L. & Bott, T. R. (1994). Process Heat Transfer, Begell House. [22] Han Dong-Hyouck, (2003). Lee Kyu-Jung, Kim Yoon-Ho, Experiments on the characteristics of evaporation of R410A in brazed plate heat exchangers with different geometric configurations, Applied Thermal Engineering, 23, 1209-1225. [23] Yan Yi Yie, (1998). Lin Tsing Fa, Evaporation heat transfer and pressure drop of refrigerant, R123a in a small pipe, International Journal of Heat and Mass Transfer, 41, 1998, 4183-4194. [24] Muley, A. & Manglik, R. M. (1999). Experimental study of turbulent flow heat transfer and pressure drop in a plate heat exchanger with chevron plates, Journal of heat transfer, 121, No. 1, 110-117. [25] Sotsil Silva Sotelo, Rosenberg, J. (2009). Romero, Roberto Best, Instantaneous Determination of Heat Transfer Coefficients in a Steam Generator for an Alternative Energy Upgrade System, Open Renewable Energy Journal, 2, 116-123. [26] Reimann, R. C. & Biermann, W. J. (1984). Development of a single family absorption chiller for use in solar heating and cooling systems, Phase III Final Report. Prepared for the USA Department of Energy under contract EG-77-C-03-1587, Carrier Corporation. [27] Rivera, W. & Xicale, A. (2001). Heat transfer coefficients in two phase flow for the water/lithium bromide mixture used in solar absorption refrigeration systems, Solar Energy Materials and Solar Cells, 70, 309-320.

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In: Heat Exchangers Editor: Spencer T. Branson

ISBN: 978-1-61761-308-1 © 2011 Nova Science Publishers, Inc.

Chapter 4

THERMAL DESIGN OF COMPACT HEAT EXCHANGERS Martín Picón-Núñez Department of Chemical Engineering, University of Guanajuato, Guanajuato, México

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ABSTRACT This chapter deals with the thermal sizing of compact heat exchangers. The fundamentals of a general approach for the development of new design methodologies for compact heat exchangers are outlined. The approach is based on the concept of full utilisation of available pressure drop and can be extended to most compact exchanger geometries. In some applications, depending on the exchanger geometry and flow arrangement, both streams can fully absorb the allowable pressure drop; however, in other cases it is only one stream that can fully utilise it. In this chapter, specific applications are given for the cases of: plate and fin, plate and frame and spiral heat exchangers. The design approach covers aspects such as the determination of the temperature correction factor derived from the thermal effectivenessnumber of heat transfer units method (ε-Ntu). The concept of the pictorial representation of the design space for these types of exchangers is presented. This design tool takes into consideration the non-continuous nature of the design variables associated with the exchanger geometry. Some insights towards the design of multi-stream units are also presented for the case of plate and frame and plate and fin geometries. The methodologies are demonstrated using case studies and the results compared with those reported in the literature.

INTRODUCTION The key to the derivation of a design procedure for compact heat exchangers is based upon the concept of full use of available pressure drop (Picón-Núñez, et al, 1999). This concept gives rise to the derivation of a thermo-hydraulic model that relates the pressure drop of a process stream to the heat transfer coefficient of that side of the exchanger. This relation involves the stream physical properties and the exchanger geometry. The total pressure drop any stream experiences as it flows through a heat exchanger is made up of four main terms: (a) the pressure drop due to the entry effects (entry header), (b) the pressure drop due to friction in the core of the unit, (c) the pressure drop due to

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acceleration losses (resulting from changes in density), and (d) the pressure drop due to exit effects (exit header). The acceleration losses due to changes in density are small for the case of liquids and short flow length gas applications, and in the case of entry and exit losses they may account for as much as 10% of the whole pressure drop. Out of the various components of the total pressure drop, the one due to friction is effectively related to the fluid capacity for heat transfer. When this relation is expressed through a thermo-hydraulic model, it can be used to derive a design methodology for sizing the core of a heat exchanger. We start from a heat transfer correlation, for instance an expression of the Nusselt number as a function of the Reynolds and Prandtl numbers as given by equation (1):

Nu = a Re b Pr c

(1)

The heat transfer coefficient h, can be related to the fluid bulk velocity (v) through an expression of the form:

h = K1 v b

(2)

Where, the parameter K1 is a function of the physical properties and the exchanger geometry. The pressure loss through the exchanger core can be expressed as:

ΔP =

2 ρv 2 Lf dh

(3)

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If the functionality of the friction factor (f) with respect to the Reynolds number is known, then the pressure drop as a function of the fluid bulk velocity can be expressed as:

ΔP = K 2 v 2− y

(4)

Where, y is the exponent of the Reynolds number in the friction factor equation and K2 is a function of the physical properties and exchanger geometry. Combining equations (2) and (4) yields:

ΔP = K 3 h

2− y b

(5)

Equation (5) together with the heat exchanger design equation, constitute the thermohydraulic model. The application of the model and the manner to solve it depends on the geometrical characteristics and the way the heat transfer coefficients and the friction factor are correlated to the Reynolds number. The application to three exchanger geometries is described below.

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PLATE AND FIN HEAT EXCHANGERS A plate and fin heat exchanger is a compact heat exchanger that is mainly used in gas to gas applications. It consists of a stack of alternate plates called parting sheets and corrugated fins brazed together as a block as shown in Figure 1. Streams exchange heat as they flow through the passages created between the fins and the parting sheets. The fins have two functions: they serve as a secondary heat transfer surface and they provide mechanical support between layers. There are a number of different types of fins available for design (Kays and London, 1984). The main geometrical parameters of a plate and fin exchanger are: ratio of total surface area of one side of the exchanger to volume between plates (βs), plate spacing (δ), ratio of secondary surface area to total surface area (fs), hydraulic diameter (dh), fin thickness (τ) and fin thermal conductivity (k). The design of a heat exchanger requires the specification of the heat duty, stream allowable pressure drops and certain aspects of exchanger geometry. With a plate and fin exchanger the geometrical aspects are the type of heat transfer surfaces to be employed on each stream. The heat transfer performance of most compact surfaces can be correlated in terms of the Reynolds number. The correlating expressions take the form:

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j = a Re − b

(6)

(a)

(b) Figure 1. Plate and fin heat exchanger: (a) counter flow arrangement, (b) cross flow arrangement. Heat Exchangers: Types, Design, and Applications : Types, Design, and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest

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Martín Picón-Núñez The term j is the Colburn factor and is defined according to: 2

j = St Pr 3

(7)

Where Pr is the Prandtl number (Cpμ/k) and St is the Stanton number which is given by:

h Ac m Cp

St =

(8)

In the case of plate and fin exchangers, the Reynolds number can be expressed as a function of the hydraulic diameter according to:

Re =

m dh μ Ac

(9)

Combining Equations (6), (7), (8) and (9) and rearranging for the heat transfer coefficient we have:

⎛ 1 h = K h ⎜⎜ ⎝ Ac

⎞ ⎟⎟ ⎠

1−b

(10)

Where Ac is the stream free flow area and Kh is given by:

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Kh =

am1−b μ b C p b h

d Pr

2 3

(11)

The expression that represents the pressure drop across the core of a heat exchanger is:

ΔP =

f A m2 2 ρ Ac Ac2

(12)

For most secondary surfaces, the friction factor f can be correlated as a function of the Reynolds number through expressions of the form:

f = x Re − y

(13)

Where x and y are correlating constants. These expressions are valid for Reynolds numbers in the range of 500 to 10,000. The heat transfer surface area of one side of the exchanger can be expressed as a function of total exchanger volume as:

A1 = α1VT

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(14)

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Where, α1 is the ratio of the heat transfer surface area of one side of the exchanger to the total exchanger volume (VT). Combining equations (9), (12), (13) and (14) gives:

⎛ 1 ΔP = K pVT ⎜⎜ ⎝ Ac

⎞ ⎟⎟ ⎠

3− y

(15)

Where:

Kp =

xm 2− y μ yα 2 ρ d hy

(16)

Combining equations (10) and (16) yields:

⎛ Kp ΔP = ⎜⎜ ⎝ Kh

3− y ⎞ ⎟⎟VT h 1−b ⎠

(17)

The basic heat exchanger design equation (18) can be combined with the expression for the determination of the overall heat transfer coefficient to give equation (19):

Q = UAFΔTLM

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A1 =

Q FΔTLM

⎡1 ⎛1 ⎞ 1 ⎢ ⎜⎜ + R1 ⎟⎟ + ⎠ η2 ⎣η1 ⎝ h1

(18) ⎞⎤ ⎛ A1 1 ⎜⎜ + R2 ⎟⎟⎥ ⎠⎦ ⎝ A2 h2

(19)

The terms A1 and A2 represent the heat transfer surface area on sides 1 and 2 respectively. The total exchanger volume and heat transfer surface area of one side are related according to equation (14). After substitution of A1 and A2 into equation (19) yields: VT =

⎞ Q ⎡ 1 ⎛1 1 ⎜⎜ + R1 ⎟⎟ + ⎢ FΔTLM ⎣η1α 1 ⎝ h1 ⎠ η 2α 2

⎞⎤ ⎛ 1 ⎜⎜ + R2 ⎟⎟⎥ ⎠⎦ ⎝ h2

(20)

The log mean temperature difference (∆TLM) can be calculated from the end temperatures as:

ΔTLM =

(Tin − t out ) − (Tout ⎛ T − t out ln ⎜⎜ in ⎝ Tout − t in

− t in ) ⎞ ⎟⎟ ⎠

(21)

Equation (20) represents the total exchanger volume as a function of the heat duty, the surface geometry and the heat transfer coefficients. The terms η1 and η2 are the temperature effectiveness of each surface. Its value can be calculated from:

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152

Martín Picón-Núñez 1 ⎧ ⎫ 2 ⎪ Tanh ⎡ 2h ⎤ ⎛⎜ δ ⎞⎟ ⎪ ⎢ ⎥ ⎪ ⎣ kτ ⎦ ⎝ 2 ⎠ − 1⎪ η = 1 + fs ⎨ ⎬ 1 ⎪ ⎡ 2h ⎤ 2 ⎛ δ ⎞ ⎪ ⎪ ⎢ ⎥ ⎜ ⎟ ⎪ ⎩ ⎣ kτ ⎦ ⎝ 2 ⎠ ⎭

(22)

For a given heat load, pressure drop and a compact surface for each surface, equations (17) and (20) can be solved to yield the total exchanger volume. For instance, for a cross flow arrangement, as shown in Figure 1(b), the flow paths of the hot and cold fluids are independent; this particular feature renders each stream in a condition where full pressure utilisation can be achieved. The solution of the pressure drop equation for ach stream yields the exchanger length (L) and width (W). The number of passages (P) can be engineered by manipulating the width and height in order to achieve the number of passages that fit the desirable core dimensions. An approach for the determination of the exchanger dimensions is as follows: Applying the thermo-hydraulic model (equation 17) to both streams, the heat transfer coefficients (h1 and h2) are found as a function of the total exchanger volume. The surface temperature effectiveness (equation 22) can also be expressed as a function of the total exchanger volume. Substitution of all these parameters into the design equation (20) reduces to a single equation with the total exchanger volume (VT) as the only unknown. The correction factor for the logarithmic mean temperature difference (∆TLM) can be calculated from the following expression:

F=

NTU counterflow NTU cross flow

(23)

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Table 1. Process data and physical properties for case study Mass flow rate (kg/s) Core pressure drop (Pa) Inlet temperature (°C) Outlet temperature (°C) Density (kg/m3) Heat capacity (J/kg°C) Thermal conductivity (W/m °C) Viscosity (kg/m s) Surface description Coefficient in heat transfer correlation Exponent in heat transfer correlation Coefficient in friction factor correlation Exponent in friction factor correlation Fin thermal conductivity (W/m°C)

Cold stream 24.3 3,718 175 366 5.7 1,050 0.044 2.9x 10-6 Louvered 3/8-6.06 0.0879 0.30 0.1467 0.1627 20.8

Hot stream 24.66 2,688 430 247 0.58 1,080 3.084 3.0x10-6 Plain-fin 11.1 0.0396 0.281 0.4247 0.4403 20.8

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Table 2. Design results for a cross flow arrangement

Core volume (m3) Length (m) Width (m) Height (m) Reynolds number cold side Reynolds number hot side Heat transfer coefficient cold side (W/m2°C) Heat transfer coefficient hot side (W/m2°C) Surface thermal efficiency cold side Surface thermal efficiency hot side Overall heat transfer coefficient (W/m2°C)

New algorithm 3.84 1.83 0.85 2.46 3,981 1,275 255 91 0.787 0.88 73.1

Existing design 3.82 1.83 0.91 2.29 4,090 1,370 262 85 0.786 0.887 70.9

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Where NTUcounter flow and NTUcross flow are the number of heat transfer units for the counter flow and cross flow arrangements calculated for the same thermal effectiveness. The application of the design approach is demonstrated using a case study. The operating conditions and physical properties of a design problem taken form the literature (Kays and London, 1984) are given in Table 1. Table 2 shows a comparison between the block dimensions as reported by Kays and London (1984) and the block dimensions obtained through the application of the design approach based on a thermo-hydraulic model. Now, in the case of counter flow arrangements the flow path is the same for both streams. The free flow areas of each stream are related to one another according to the expression:

Ac 2 =

δ 2β s2dh2 A δ1β s1d h1 c1

(24)

Since pressure drop is directly related to free flow area, the smaller the free flow area the larger the pressure drop. From equation (24) it can be shown that both pressure drops are related; the consequence is that only one of the two pressure drops can be fully utilized. The other has to conform to the resulting dimensions.

PLATE AND FRAME HEAT EXCHANGERS Plate and frame heat exchangers (PFHE) are finding wider application in process industries for they are the best alternative to shell and tube exchangers for heating, cooling and heat recovery applications. This is mainly due to their construction characteristics that facilitate the increase or reduction of surface area and also due to their versatility in terms of the materials of construction. Some of their limitations are that they can not be used in applications with high pressure and temperature and in situations where a large difference in operating pressure between streams exit since this can cause plate deformations (Marriot, 1971; Hesselgreaves, 2001). Another limitation is that due to the small free flow area these

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exchangers are not suitable for application with dirty fluids since passages can easily be clogged. Plate and frame heat exchangers consist of a series of corrugated plates stacked in a bolted frame; one end of the frame is fixed and the other end is mobile allowing addition or removal plates. The plates are supported by shaped slots in the top and bottom of the plates that engage in an upper and lower guiding bars mounted on the frame. The plates are sealed by means of gaskets made of polymeric material. With PFHE, the need for distribution headers is eliminated since ports are an essential element of the plate design and are incorporated in it. The geometry of this type of heat exchangers is characterized by the following elements: number of plates (N), number of passages (P), plate length (L), plate width (W), chevron angle (β), plate spacing (b), port diameter (Dp), plate thickness (τ) and enlargement factor (φ). This last term refers to the ratio of the actual surface area of a plate to the area projected on the plane. The geometrical features of the plates and the whole exchanger assembly are shown in Figure 2. The thermal performance of PFHE is a strong function of the geometry of the corrugation of the plates. The most common corrugation is the chevron type that is characterised by an angle β with respect to the horizontal in the direction of the flow. Designs with a low β angle present high level of turbulence, high heat transfer coefficients and pressure drop, whereas designs with high values of β exhibit low turbulence and therefore lower heat transfer coefficients and pressure drops. The chevron angle is depicted in Figure 2(a).

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Table 3. Typical plate dimensions Dimension Length, L (m) Width W, (m) Port diameter, Dp (m) Chevron angle, β Plate spacing, b (mm) Plate thickness, τ (mm)

Range 0.5-3.0 0.2-1.5 0.254-0.4 25-65 1.5-5.0 0.5-1.2

Table 3 provides the typical range over which the geometrical dimensions of commercial plates are available.

 

(a)

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(b)

(c)

(d)

Figure 3. Flow passage arrangements in Plate and Frame Heat Exchangers: (a) Series, (b) U circuit , (c) Z circuit and , (d) single pass two pass.

When necessary, particularly during retrofit campaigns for increased throughput or maximisation of energy recovery, the flexible structure of a PFHE allows it to be adjusted in order to achieve the required heat load within the restrictions of the specified pressure drop. This can be done by: (a) increasing or reducing the number of thermal plates, (b) changing the type of plate according to its thermo-hydraulic performance, (c) modifying the plate dimensions, and (d) modifying the number of passes.

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There are a large number of flow pass arrangements combinations that can be achieved with this type of exchangers; however, all of them are the result of combining any of the basic three types that are shown in Figure 3: series, circuit and complex. The series arrangement contains 2P-1 thermal plates (N), where P is the total number of channels. This arrangement is used in applications with low flow rates and when close temperature approaches are required. The circuit arrangement is the most commonly used since it gives a pure counter flow arrangement; it is normally used in applications with large flow rates and with close temperature approaches. The complex arrangement is a combination of the circuit and series arrangement; it is characterized by more than one pass in at least one stream. The two main aspects to be considered in the design of PFHE are: (a) the geometry and the thermo-hydraulic performance of the plates, and (b) the sizing of the unit that will satisfy the heat duty within the specified pressure drops. There is a considerable amount of literature on the thermo-hydraulic performance of plate and frame exchangers, both for single and two phase flow (Kumar, 1993; Ayub, 2003). Most design methods for PFHE are based on the rating of a selected geometry. The most widely used methods are the thermal effectiveness –number of heat transfer units (ε-NTU) and the differential methods where each one of the channels is analyzed independently to give the channel outlet temperature. The differential methods are applied to account for the influence of end channels and channels between passes on the thermal performance (Gut and Pinto, 2003). Although flow passage arrangement in this type of exchangers can be countercurrent, the presence of flow mal-distribution and temperature distortions due to end effects make this type exchangers depart from ideal so that a correction factor for the log mean temperature difference has to be employed (Prabhakara et al., 2002). During design especial care must be taken when applying conventional fouling factors that are used for shell and tube exchangers. Fouling factors in PFHE are much smaller and the use of excessive values may lead to unnecessary large flow areas which in turn result in lower velocities and consequently higher tendency to foul. Panchal and Rabas (1999) reported a comparison of fouling factors for compact surfaces and shell and tube exchangers. These alternative values are more convenient to use in the design of this type of units. The free flow area in a plate heat exchanger (Ac) can be defined as:

Ac = bW

(25)

The wavy shape of the plates increases its surface area; the level of enlargement depends on the depth of its channels. The enlargement factor (φ) is defined as the ratio between the actual surface area (As) and the projected area (Ap):

ϕ=

As Ap

(26)

Where the projected area (Ap) is defined as:

Ap = W (L − Dp )

(27)

The sectional area for the flow of a fluid in a PFHE is irregular, so the hydraulic diameter (dh) can be expressed as: Heat Exchangers: Types, Design, and Applications : Types, Design, and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest

Thermal Design of Compact Heat Exchangers

dh =

4(δW ) 2(δ + ϕW )

157 (28)

Since δ ≤ W, then:

dh =

2δ

(29)

ϕ

The Reynolds number is defined as a function of the mass velocity rate (G) and the hydraulic diameter (dh) and is expressed as:

Re = G

dh

(30)

μ

The mass velocity rate (G) as a function of the mass flow rate (m) and the number of channels per stream (Nc) can be defined as:

G=

m N cδW

(31)

Equation (18) represents the design equation for heat exchangers. For single phase processes, the heat load (Q) can be determined from an energy balance as:

Q = (mC p )H (Tin − Tout ) = (mC p )C (t out − t in )

(32)

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The overall heat transfer coefficient under fouled conditions is expressed as:

U=

1 1 1 τ + + + R1 + R2 h1 h2 k w

(33)

In equation (32), the individual heat transfer coefficients (h1 and h2) are calculated using the appropriate correlations. For instance, equations (34) and (35) have been reported by Shah and Focke (1988) for the Chevron plate type P31 of Alfa Laval.:

⎧⎪0.729 Re1/ 3 Pr1/3 Nu = ⎨ ⎪⎩0.380 Re 2/3 Pr1/3

for Re ≤ 7 for Re〉 7

and

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⎧17.0 Re −1 ⎪ - 0.57 ⎪6.29Re f =⎨ − 0.20 ⎪1.141 Re ⎪0.581Re -0.10 ⎩

for Re 〈 10 for 10 〈 Re〈101 for 101 〈 Re 〈 855 for Re 〉 855

(35)

The geometrical characteristics of this plate are as follows: heat transfer surface area per plate (As), 0.30 m2; effective length (L), 0.904 m; Plate spacing (δ), 2.9 mm; hydraulic diameter (dh), 5.8 mm; Chevron angle (β), 60° and Port diameter (Dp), 0.125 m. The total surface area (A) can be calculated by multiplying the surface area (As) by the number of thermal plates (N):

A = As N

(36)

Where the total number of plates, which includes the end plates that have no thermal function but only of support, is obtained from:

NT = N + 2

(37)

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In a plate heat exchanger, the total pressure drop has the following components: (a) pressure drop in ports and fluid collectors, (b) pressure drop due to friction in the channels of the exchanger and (c) pressure drop due to changes in height Empirically, the pressure drop in ports and collectors has been calculated as a function of the velocity (v) as:

⎛ ρ v2 ⎞ ΔPports = 1.5 ⎜ ⎟ ⎝ 2 ⎠

(38)

Where the velocity (v) can be expressed as:

v=

4m m = ρAport ρπD p2

(39)

The pressure drop in channels is the result of friction and the contraction-expansion of the fluid due to temperature changes. Therefore, this component is expressed as:

ΔPchannels

2 N p fLG 2 ⎛ 1 ⎞ ⎛ 1 1 ⎞ ⎜⎜ ⎟⎟ + ⎜⎜ = − ⎟⎟G 2 dh ⎝ ρ ⎠ ⎝ ρo ρi ⎠

(40)

The pressure drop due to changes in height is given by:

ΔPelevation = ± ρgL

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In equation (41), the positive sign refers to ascending flow (pressure drop due to increase in height) and the negative sign is for descending flow; g is the gravitational constant and L is the plate length. Considering that the height change in a plate exchanger is relatively small and the momentum change due to changes in density is usually negligible, the total pressure drop in the core of the exchanger due to friction (for Np number of passes), can be simplified as:

ΔP =

2 N p fLG 2

(42)

ρd h

For a given configuration and for Reynolds numbers above 100, the viscous effects upon the pressure drop ΔP is no longer important and the effect of the mass flow rate becomes the most significant one. From equation (35), the friction factor (f) is seen to be related to the Reynolds number through an expression of the form of equation (13). So, ccombining equations (9), (13) and (42), we have:

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⎛ ⎞ • 2− y ⎜ 2x m μ y L⎟ ⎟ Ac = ⎜ ⎜ d h 1+ y ρ ΔP ⎟ ⎜ ⎟ ⎝ ⎠

1 / (2 − y )

(43)

Equation (43) shows that once the plate type and its geometrical characteristics are chosen, it is possible to determine the free flow area of a stream as a function of its allowable pressure drop. With the free flow area, the number of channels per stream can be computed. The number of channels thus calculated is termed: Hydraulic channels (NHydraulic). This value must be matched against the number of channels required to meet the heat duty which are termed as thermal channels (NThermal). When the design is complete, the following condition must be met:

N Hydraulc = N Thermal

(45)

Thermal plates are commercially available in fixed dimensions. So, for the equality given by equation (45) to be fulfilled, plate dimensions should be considered to be available at any dimensions required. In reality this is not the case for any plate geometry. So, as we seek to meet the required heat duty, the resulting number of channels not necessarily will satisfy the permitted pressure drop. The corresponding pressure drop for the given number of channels and flow area will be either lower or higher than the specified value. The design that must be accepted is that in which the calculated pressure drop does not surpass the value specified at the start of the design process. The number of hydraulic channels per pass can be calculated by combining equations (4), (11) and (12), from where we have: 1

N thermal , pass

• 2− y ⎡ ⎤ ( 2− y ) y 1 ⎢ 2 x m μ Np L ⎥ = (Wδ ) ⎢⎢ ΔP ρ d h1+ y ⎥⎥ ⎣ ⎦

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The total exchanger surface area (A) can be calculated from the general design equation (18). In the case of the series and circuit arrangements, the flow arrangement is purely counter-current. For complex arrangements with more than one stream passes, the correction factor F can be calculated form the expression:

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F=

NTU counter −current NTU other

(47)

The term NTUcounter-current is the number of heat transfer units that a counter-current unit has for the specified heat duty (or thermal effectiveness), whereas NTUother is the number of heat transfer units of the actual arrangement to be designed. In the case of multi-pass designs, the term “other” means a combination of countercurrent and parallel flow. Applications for two or more passes per streams are further described in (Picón-Núñez et al., 2006). The process data and physical properties of the streams involved in a case study form the literature (Marriot, 1971) are shown in Table 4. In the original problem, the type of plate used is not specified, so no heat transfer and friction data are known. In the present approach the design is carried out considering the chevron P31 Alfa Laval plate (Shah and Focke, 1988). The results of the design are compared to the base case in Table 5. The heat transfer surface area required is 18.7 m2 with 30 channels per stream and an overall heat transfer coefficient of 6,090.0 W/m2°C. In this case, the pressure drop on the hot stream was maximized. The base case design has a surface area of 25 m2; 40 channels per stream and an overall heat transfer coefficient of 4,610.7 W/m2°C. Comparing the results, despite the fact that the plate geometry and surface area is exactly the same in both cases (0.32 m2), the base case exchanger is slightly bigger than the new design. This may indicate that the base case design could have used a chevron plate with a poorer thermo-hydraulic performance. In the single pass design, the cold fluid pressure drop is 40,633 Pa. In the event of this pressure drop being higher than the permitted value which is not kwon for the base case, the pressure drop on the cold stream could have been maximized instead. An alternative available in design is the use of two or more flow passes (Picón-Núñez, 2006). Table 4. Operating conditions and physical properties for the case study Mass flow rate, (kg/s) Pressure drop, (Pa) Inlet temperature, (°C) Outlet temperature, (°C) ρ (kg/m3) Cp (J/kg°C) K (W/m°C) μ (kg/m s) Fouling, (m2°C/W )

Hot fluid 13.6 39,310 80 40 983.2 4,185.2 0.6536 4.67 E-4 1.03 E-5

Cold fluid 13.6 --20 60 992.2 4,178.6 0.6316 6.5 E-4 5.2 E-6

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Table 5. Design results compared with base case dimensions 2

Surface area (m ) Overall heat transfer coefficient (W/m2 °C) Number of passes (Hot side / cold side) Channels per stream Plate size (m2) Hot side film heat transfer coefficient (W/m2 °C) Cold side film heat transfer coefficient (W/m2 °C) Pressure drop on hot side (Pa) Pressure drop on cold side (Pa) Reynolds number on hot side Reynolds number on cold side Heat duty (MW)

Base case 25 4,610.76 1-1 40 0.32

39,310.0

2.274

New method 18.70 6,090.0 1-1 29.5 0.32 26,025 22,701 39,310.0 40,633.0 1,104.0 785.8 2.274

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SPIRAL HEAT EXCHANGERS Due to the geometrical features of a spiral heat exchanger (SHE), its application involves situations where highly viscous liquids and dirty fluids are involved. In a spiral heat exchanger, a fluid changes its direction continually as it flows in the outer or in the inner direction, as shown in Figure 4, creating a high shear stress that eliminates stagnant zones and increases heat transfer coefficients. The continual change of direction also maintains suspended solids in motion preventing their deposition (Wilhelmsson, 2005). Despite the fact that the two fluids move along the length of the exchanger is countercurrent fashion, the temperature driving forces at each location do not behave in the same way thus making this geometry depart from an ideal counter-current behavior. There are two thermal phenomena that take place during the operation. One phenomenon is related to what is called the end effect. This is, all along the length of the unit, each stream exchanges heat with two adjacent streams but in the innermost and outermost part of the unit, heat is transferred only to one side of the channel (see Figure 4). The second thermal effect is related to the exchange of heat with two adjacent streams. For instance, in internal channels, the hot fluid exchanges heat across two adjacent cold channels which are at different temperature giving rise to a disturbed temperature driving force. The determination of the correction factor for this geometry is quite elaborate; however, simplified equations (Martin, 1992) have been derived to approximate the correction factor as a function of the number of heat transfer units, the number of turns and the heat capacity rate ratio. Bes y Roetzel (1993) presented a different simplified expression for the correction factor as a function of the number of turns and a dimensionless parameter called the criterion number. This parameter is shown to be a function of the number of heat transfer units of the hot and cold side and the ratio of free flow area to surface area.

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Figure 4. Relative flow of fluids within a spiral heat exchanger.

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Empirical correlations for the determination of heat transfer coefficients based on the average curvature for this type of geometry have been presented by Minton (1970) and Martin (1992). The exchanger curvature represented by the Dean number (K) has an effect upon the heat transfer coefficient, this is, as curvature increases the heat transfer coefficient increases. It has been shown hat the effect is more significant in laminar flow than in turbulent flow (Egner and Burmeister, 2005).

Figure 5. Geometrical features of a spiral heat exchanger.

The design of a heat exchanger refers to the definition of the geometry that will transfer the required heat duty within the limitations of the allowable pressure drop. In the case of spiral exchangers, this geometry includes: plate spacing of the two streams (δ1 and δ2), plate width (W), inner diameter (Di), outer diameter (Ds) and plate length (L). The basic geometrical features of a spiral heat exchanger are depicted in Figure 5. A design methodology can be implemented by seeking the exchanger dimensions that lead to full pressure drop utilization. In a first approximation, plate width and plate spacing are considered to be continuous variables; however, these values are later changed according to the standard dimensions for the final design to be achieved. There are some assumptions that must be taken into consideration for the development of the approach, these are: the effects of the entrance regions are not considered, the heat

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transfer coefficient is constant along the length of the exchanger, losses to ambient are negligible; the bolts that separate the plates do not affect movement of the fluid. At the outset of the design approach, the following three dimensions are specified. These are: plate spacing for the cold and hot stream (δ1 and δ2) and the plate width (W). With these parameters a first approximation to the free flow area (Ac) and the Reynolds number can be calculated using equations (48) and (49).

Ac = δW

Re =

(48)

dhm μAc

(49)

The hydraulic diameter, dh is given by:

dh =

2δH δ +H

(50)

An expression for estimating the heat transfer coefficient valid for a Reynolds number range between 400 and 30,000 was presented by Martin (1992):

Nu = 0.04 Re 0.74 Pr 0.4

(51)

Where Pr is the Prandtl number is given by:

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Pr =

cpμ

(52)

k

The pressure drop across the core of the exchanger can be calculated from:

ΔP =

2 fLm 2 ρ d h Ac2

(53)

Where f is the friction factor, L is the exchanger length, m is the mass flow rate and ρ is the density. The friction factor for laminar, transitional and turbulent flow can respectively be estimated using the following expressions (Hesselgreaves, 2001): 2 3 4 5 ⎛ δ ⎛δ ⎞ ⎛δ ⎞ ⎛δ ⎞ ⎛δ ⎞ ⎞ f Re = 24⎜1 − 1.3553 + 1.9467⎜ ⎟ − 1.7012⎜ ⎟ + 0.9564⎜ ⎟ − 0.2537⎜ ⎟ ⎟ ⎜ H ⎝H ⎠ ⎝H ⎠ ⎝H ⎠ ⎝ H ⎠ ⎟⎠ ⎝

f = 0.0054 +

2.3 x10 −8 Re

−

(54)

(55)

3 2

1 = 1.56 ln(Re) − 3.00 f

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A simplified expression for the determination of the correction factor of the logarithmic mean temperature difference was presented by Martin (1992). The model can be applied to cases where the heat capacity rate ratio is different to unity. The expression is: F=

1

(1 + C ) NTU

⎛ 1+ C ⎞ ⎟ ln⎜1 + n ⎜⎝ (1 ε i ) − 1 ⎟⎠

(57)

Where C is the heat capacity rate ratio, NTU is the number of heat transfer units, n is the number of semi-turns and εi is the thermal effectiveness per turn. These terms are respectively given by:

C=

(mC ) (mC )

p min

(58)

p max

NTU =

UA (mC p )min

(59)

1 − e − (1+C ) NTU / n εi = 1+ C

(60)

The number of semi turns (n) and the exchanger external diameter (Ds) were presented by Dongwu (2003):

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− ⎛⎜ Di − τ / 2 + n= ⎝

(Di − τ / 2 )2 + 4τL / π ⎞⎟ τ

⎠

Ds = 1.28 τ L + D i2

(61)

(62)

Using the initial exchanger geometry, two values for the flow length can be found. One is the flow length that meets the required heat duty and is termed the thermal length, and the second is the length that meets the specified pressure drop and it is called the hydraulic length. The latter term is calculated based on the allowable pressure drop of only one of the streams; this is the stream where pressure drop is to be maximized. For the thermal and hydraulic lengths to equate, an iterative process is implemented where the value of the plate width is changed until convergence is achieved. Up to this point, only one stream makes full use of pressure drop, the degree of freedom that is left for the other stream to fully utilize its allowable pressure is plate spacing. So, an iterative process is also implemented whereby the plate spacing of the other stream is changed until the thermal length and the hydraulic length for each stream equate. The resulting dimensions are used to select, from the standard available dimensions, those that will best perform in terms of heat duty and pressure drop. Table 6 shows a set of standard dimensions reported originally in the Imperial system of units by Minton (1970). The approach is next demonstrated on a case study.

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Table 6. Typical standard dimensions for spiral plate exchangers Plate width [m]

0.102 0.152 0.305 0.305 0.457 0.457 0.61 0.61 0.762 0.914 1.219 1.524 1.778

Maximum external diameter [m] 0.813 0.813 0.813 1.473 0.813 1.473 0.813 1.473 1.473 1.473 1.473 1.473 1.473

Internal diameter [m] 0.203 0.203 0.203 0.305 0.203 0.305 0.203 0.305 0.305 0.305 0.305 0.305 0.305

Plate spacing [m]:4.762E-3 (for a maximum plate width of 0.305 m), 6.35E-3 (for a maximum plate width of 1.219 m), 7.938E-3, 9.525E-3, 0.013, 0.016, 0.019 and 0.025. Plate thickness [m]: Carbon steel, 3.175E-3, 4.762E-3, 6.35E-3 and 7.938E-3.

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Table 7. Stream data for case study Flow rate (kg/s) Inlet temperature (°C) Outlet temperature (°C) Heat capacity (J/kg °C) Thermal conductivity (W/m °C) Density (kg/m3) Pressure drop (Pa) Viscosity (kg/m s) Plate thickness (m) Internal diameter (m) Thermal conductivity of material of construction (W/m °C)

Hot Stream Cold Stream 0.7833 0.7444 200 60 120 150.4 2,973 2,763 0.348 0.322 843 843 6.89 x10-2 6.89 x10-2 3.35x10-3 8.0x10-3 3.175x10-3 0.203

17.3

The operating data and physical properties for the case study are shown in Table 7. The results in Table 8 show that both pressured drops can be fully utilized by relaxing plate spacing and plate width. Only a few iterations are needed for the model to converge. Table 8 also shows the geometrical features of the final design. They are obtained by choosing the nearest standard values from Table 6. The rating analysis of the new unit indicates that the exchanger meets the heat load and both pressure drops are within acceptable bounds.

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Martín Picón-Núñez Table 8. Design results for maximum use of allowable pressure drop and for standard dimensions.

2

Heat transfer area (m ) Re (Hot side) Re (Cold side) Heat load (W) Heat transfer coefficient. Hot side (W/m2 °C) Heat transfer coefficient. Cold side (W/m2 °C) Overall heat transfer coefficient (W/m2 °C) Plate length (m) Plate width (m) Spiral outer diameter (m) Pressure drop. Hot side (Pa) Pressure drop. Cold side (Pa) Plate spacing, Hot side (m) Plate spacing, Cold side (m)

Design with full pressure drop use on both streams 14.48 773 307 186,300

Final design with standard dimensions 18.50 759 301 186,174

771.9

573.1

390.1

304.6

247.4

192

12.06 0.6 0.56 6,890 6,890 4.77x10-3 6.27x10-3

15.16 0,61 0,66 3,651 4,249 6.35x10-3 7.938x10-3

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DESIGN SPACE The graphical representation of a design exercise can provide much and more valuable information at one sight than just numbers. A tool for the pictorial representation of the design space for the case of shell and tube exchangers was introduced by Poddar and Polley (2000) and is called: Parameter plot. With shell and tube exchangers once a set of preliminary decisions has been made (baffle type, ratio of shell diameter to baffle spacing, tube size, number of tube passes, tube pitch and bundle layout) the design can be characterised using two principal dimensions: the tube count and the tube length. The design objectives can then be displayed as three separate curves, one giving the tube length required for a given duty, and two giving tube lengths associated with the full absorption of allowable pressure drop. A typical parameter plot is displayed in Figure 6. In the case of plate and frame heat exchangers, the standard geometry for its construction is characterized by discrete values. Therefore, a graphical design tool for this type of exchangers must take into consideration such features and a bar chart has the construction features needed to display the available design options at a glance (Picón-Núñez et al., 2010). Some assumptions must be taken into consideration for this pictorial tool to be derived. These are: (a) the effects of the entrance regions are not considered, (b) the heat transfer coefficient is constant along the length of the exchanger, and (c) heat losses to ambient are negligible. A parameter plot is a pictorial representation of the design space where a set of exchanger geometries are rated to see whether the main process specifications, such as heat duty, cold

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side allowable pressure drop and hot side allowable pressure drop are met and what the area requirements are. The development of parameter plots for plate and frame heat exchangers is demonstrated with a case study. This same approach can readily be extended to the spiral exchanger technology. The operating conditions and physical properties for the case study are given in Table 9 whereas Table 10 shows the plate sizes to be used. The design approach of section 3 is applied. Figure 7 shows the design space for the operating conditions of the case study.

Figure 6. Parameter plot for the design of shell an tube heat exchangers.

Table 9. Physical properties for case study Mass flow rate (kg/s) Pressure drop (kPa) Inlet temperature (°C) Outlet temperature (°C) Density (kg/m3) Viscosity (kg/m s) Thermal conductivity (W/m ºC) Heat capacity (kJ/kg ºC)

Hot fluid 12 50 70 45 800 0.5x10-3 0.1 2.0

Cold fluid 80 20 35 1021 1x10-3 0.6 4.01

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Table 10. Plate dimensions used in the construction of the parameter plot Plate length (m) 0.3 0.5 0.8 1.0 1.2 1.5 1.8 2.0 2.4 3.0

Width (m) 0.1 0.2 0.3 0.4 0.4 0.5 0.6 0.7 0.8 1.0

Enlargement factor Pitch (m) Plate spacing (m) Plate thickness (m) Equivalent diameter (m) Thermal conductivity (W/m ºC)

1.17 0.0035 0.0029 0.0006 0.00495 17

Plate No.

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1 2 3 4 5 6 7 8 9 10

Plate surface Area (m2) 0.03 0.1 0.24 0.4 0.48 0.75 1.08 1.4 1.92 3.0

Port diameter (m) 0.025 0.03 0.05 0.08 0.08 0.12 0.15 0.20 0.25 0.30

The plot of Figure 7 shows the design results for all available plate sizes (specified length and width). Three columns are used for each plate size. The first column indicates the number of plates required to satisfy the pressure drop on the hot side, the second column indicates the number of plates required to meet the pressure drop on the cold side and the third one indicates the number of plates required to meet the heat duty. From Figure 7 we see that, with plate size 3, the heat duty is met with 40 plates; the pressure drop on the hot side is fully absorbed with this number of plates but the pressure drop on the cold side, which absorbs the full allowable pressure drop with only 25 plates, would exceed the permitted value if 40 plates were chosen. Any design from plate size 4 to 10 meet the thermal duty with lees pressure drop than specified. Plate size 4 is the better choice: with the 18 plates required to meet the heat duty, pressure drops on both streams are absorbed to an acceptable extent.

MULTI-STREAM HEAT EXCHANGER APPLICATIONS Multi-stream heat exchangers have been long used in the cryogenic industry. The types of exchangers employed for this purpose were shell and helical tubes and plate and fin. Shell and helical exchangers are able to handle one cold and two or more hot streams or vice versa, whereas the geometrical features of plate and fin exchangers and plate and frame make them suitable for handling more than two hot and more than two cold stream in the same unit. An attractive area of application of multi-stream heat exchangers is in the design of heat recovery networks in above ambient processes. There is a potential saving in terms of space,

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weight and supporting structure that could be achieved if all these heat duties were to be processed in a single unit. The main concerns regarding the widespread use of heat exchangers of the plate and fin type for multi-fluid applications are the limited range of temperature and pressure at which they can operate and the restrictions regarding their application to relatively clean fluids. On the other hand, a heat exchanger construction of the plate and frame type has a flexible geometry that allows for the incorporation of a number of streams into a single frame. Various approaches to the design of multi-stream heat exchangers have been proposed. Picón-Núñez et al.(2002) developed a design approach for plate and fin heat exchangers and that same method was extended to cover the design of multi-stream PFHE units (PicónNúñez, 2006). The method starts with the construction of the Temperature-Enthalpy diagrams or composite curves (Smith, 2005). Figure 8 shows the hot and cold composite curves, the minimum temperature approach (pinch point) and the enthalpy intervals. A grid diagram (Figure 9) shows the stream population per enthalpy interval.

Figure 7. Parameter plot for the design of Plate and Frame Heat Exchangers.

Figure 8. The composite curves show the inlet and outlet position of streams in a multi-stream exchanger application.

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Figure 9. A grid diagram shows the stream population per enthalpy interval.

Figure 10.

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The use of plate and fin technology for the design of multi-stream exchangers results in rather complex structures. One approach (Picón-Núñez, 2002) treats each enthalpy interval at a time and designs the corresponding multi-stream unit. For instance, from Figure 9, five multi-stream blocks are generated each with different dimensions. For all of them to be assembled into a single structure, a reconciliation technique is applied. This involves the relaxation of pressure drop so that the design of each block is revisited; in the final design all blocks must exhibit the same width and height. A design methodology like this would require the implementation of intermediate headers for the entry and exit of streams. The use of plate and frame exchangers in multi-stream applications can be viewed in a slightly different way. Instead of seeking to design a multi-stream unit according to the stream population in enthalpy intervals they could be used to merge, in a single unit, two or more independent duties (Polley and Haslego, 2002). Figure 10 shows a heat recovery network and the possible way in which a network structure composed of six exchangers can be simplified to three multi-stream units. In each of these units, two heat exchangers are merged into a single structure; this would involve the design of a three stream exchanger of the plate and frame type. Figure 11 shows a schematic of the way a three stream heat transfer process can be accommodated into a single frame.

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CONCLUSION This chapter has shown the use of the concept of full pressure drop utilization in the derivation of design approaches for compact heat exchangers. Such an approach gives rise to the thermo-hydraulic model which relates the pressure drop due to friction of a process stream to the heat transfer coefficient, the physical properties and the exchanger geometry. The application of the methodology to case studies indicates that this is a quick and convenient approach for estimating the size of the compact heat exchangers. Three types of compact exchangers have been analysed in detail in this chapter, namely: plate and fin, plate and frame and spiral heat exchangers. The methodology for the derivation of a sizing approach presented here can be extended to any other compact exchanger geometry. It has also been shown that a traditional design approach can be complemented through the use of a graphical representation of a design space. This type of representation, named parameter plot, gives an overall picture of all the design options available to the designer in a practical way. This is so since the standard dimensions available for the design are incorporated in the analysis through a rating exercise that is conducted to show the way each set of standard dimensions behave in terms of heat duty and pressure drop. The chapter also presents a section on multi-stream heat exchangers. This is an area that might find wider application particularly in the above ambient processing industry. Design approaches based on a combination of the thermo-hydraulic model and principles of process integration have been discussed.

NOMENCLATURE A As Ac

Heat transfer surface area (m2) Plate surface area (m2) Free flow area (m2)

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Martín Picón-Núñez Ap a b C Cp c Di Ds Dp dh F f G g H h K1 K2 K3 Kh Kp k kw L m N Nc NHydraulic NThermal Np NT NTU Nu n P Pr ∆P Q Re R T t ∆TLM U VT v

Projected surface area (m2) Constant in heat transfer correlation Exponent in heat transfer correlation Heat capacity rate (W/°C) Heat capacity (J/kg °C) Exponent in Nusselt number correlation Inner spiral diameter Outer spiral diameter Port diameter (m) Hydraulic diameter (m) Log mean temperature correction factor Friction factor Mass velocity rate (kg/m2 s) Gravitational constant Plate height (m) Film heat transfer coefficient (W/m2 °C) Constant Constant Constant Constant Constant Thermal conductivity of fluid (W/m °C) Thermal conductivity of material (W/m °C) Plate length (m) Mass flow rate (kg/s) Total number of thermal plates Number of channels per stream Hydaulic plates Thermal plates Number of passes Total number of plates Number of heat transfer units Nusselt number Number of semi-turns Number of channels Prandtl number Pressure drop (Pa) Heat duty (W) Reynolds number Fouling factor (m2 °C/W) Temperature of hot fluid (°C) Temperature of cold fluid (°C) Log mean temperature difference (°C) Overall heat transfer coefficient (W/m2 °C) Total exchanger volume (m3) Velocity (m/s)

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173

Plate width (m) Constant in friction factor correlation Exponent in friction factor correlation Exponent in thermo-hydraulic model

Greek Symbols α β βs δ ε εi η φ μ π ρ τ

Ratio of surface area of one side of the exchanger to the total exchanger volume Chevron angle Ratio of surface area to volume in one side of the exchanger (m2/m3) Plate spacing (m) Thermal effectiveness Thermal effectiveness per turn Fin temperature effectiveness Enlargement factor Viscosity (kg/m s) Constant Density (kg/m3) Plate thickness (m)

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Subscripts 1 2 o i max min

Side 1 Side 2 Initial conditions Final conditions Maximum Minimum

REFERENCES [1] [2] [3] [4] [5] [6]

Ayub, ZH. Heat Transfer Engineering, 2003, vol. 24, No. 5, 3-16. Bes,TH; Roetzel, W. International Journal of Heat and Mass Transfer, 1993, vol. 36, No. 3, 765-773. Dongwu, W. Chemical Engineering Technology, 2003, vol. 26, 592-598. Egner, MW; Burmeister, LC. Journal of Heat Transfer, March 2005, vol. 127, 352-356. Gut, JAW; Pinto, JM. International Journal of Heat and Mass Transfer, 2003, vol. 46, 2571-2585. Hesselgreaves, JE. Compact Heat Exchanger. Selection, Design and Operation, First Edition, Pergamon: Oxford, UK, 2001.

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174 [7] [8] [9] [10] [11] [12] [13]

[14] [15] [16] [17] [18] [19] [20] [21]

Kays, WM; London, AL. Compact Heat Exchangers, Third Edition; Mc Graw Hill: EUA, 1984. Kumar, H. Institute of Chemical Engineering Symposium Series, 1984, No. 86, 2751288, Marriott, J. Chemical Engineering, 1971, vol. 78,No. 8, 127-134. Martin, H. Heat Exchangers; Hemisphere Publishing Corporation, 1992, 73-82. Minton, PE. Chemical Engineering, 1970, May 4 issue, 103-112. (article) Palm, B; Claesson, J. Heat Transfer Engineering, 2006, vol. 27, No.4, 88-98. Panchal, CH; Rabas, TJ. Proc. International Conference on Compact Heat Exchaners and Enhanced Technology for the Process Industries, Banff, Canada. Begell House, New York. 1999. Picón-Núñez, M; López-Robles, JL; Martínez-Rodríguez, G. Heat Transfer Engineering, 2006, vol. 27, No. 6, 12-21. Picón-Núñez, M; Polley, GT; Jantes-Jaramillo, D. Heat Transfer Engineering, 2010, vol. 31, No. 9, 742-749. Picón Núñez, M; Polley, GT; Medina Flores, J. M. Applied Thermal Engineering, 2002, vol. 22, 1643-1660. Picón-Núnez, M; Polley, GT; Torres-Reyes, E; Gallegos-Muñoz, A. Applied Thermal Engineering, 1999, vol. 19, 917-931. Polley, GT; Haslego, C. Chemical Engineering Progress, 2002, vol. 98, No. 10, 48-51. Poddar, TK; Polley, GT. Chemical Engineering Progress. September 2000. Prabhakara, BR; Krishna, PK; Sarit, KD. Chemical Engineering and Processing, 2002, vol. 41, 49-58. Shah, RK; Focke, WW. Heat Transfer Equipment Design, RK; Shah, EC; Subbarao, RA. Mashelkar, Hemisphere Publishing, Washington D. C., 1998. Wilhelmsson, B. Hydrocarbon Processing, July 2005, 81-83.

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[22]

Martín Picón-Núñez

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In: Heat Exchangers Editor: Spencer T. Branson

ISBN: 978-1-61761-308-1 © 2011 Nova Science Publishers, Inc.

Chapter 5

OPTIMAL DETAILED DESIGN OF SHELL-AND-TUBE COOLER UNITS USING GENETIC ALGORITHMS José María Ponce-Ortega1, Medardo Serna-González1 and Arturo Jiménez-Gutiérrez2 1

Chemical Engineering Department, Universidad Michoacana de San Nicolás de Hidalgo, Morelia, México 2 Chemical Engineering Department, Instituto Tecnológico de Celaya, Celaya, México

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ABSTRACT This chapter presents an approach based on genetic algorithms for the optimal design of shell-and-tube coolers. The approach involves both discrete and continuous variables, and uses compact formulations based on the Bell-Delaware method. The objective is to minimize the total annual cost of the exchanger, which includes capital and operating costs. The capital cost of the cooler is given by a detailed function that involves the cost of the exchanger components and manufacturing costs. The proposed methodology calculates the optimal mechanical and thermal-hydraulic variables, considering geometric constraints such as maximum shell diameter and tube length, and maximum and minimum ratios of baffle spacing to shell diameter and of cross-flow area to area in a window. In addition, maximum pressure drops and velocities for the fluids in both sides of the cooler, and maximum outlet temperature for the cooling water to avoid fouling conditions are considered. The procedure follows the standards of the Tubular Exchanger Manufacturers Association to select the external tube diameter, the tube length, the tube arrangement pattern, the tube pitch and the number of tube passes. Because the constraints of the model are highly nonlinear and nonconvex, genetic algorithms are used as a suitable optimization technique to provide a global or near-global optimum design. An example is used to show the applicability of this methodology.

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1. INTRODUCTION Shell and tube is one of the most widely-used types of heat exchangers in the process industries to accomplish the task of transferring heat from a hot stream to a cold stream. They are resistant and flexible units that work in a wide range of pressures, flows and temperatures (Sinnot, 1996). Figure 1 shows a pictorial representation of a shell and tube heat exchanger. The fluid that goes through the tubes could go countercurrent, cocurrent, or part countercurrent and part cocurrent when space limitations exist in the plant layout. To increase the contact between both fluids through the transversal area of the tubes, several baffles are installed in the shell. This complicated geometry in the shell side yields several leakage and bypass streams (see Figure 2). Therefore, the problem of designing these types of units is a complicated task because of the complex nature of the flow pattern for a specific geometry (which includes number of tubes, diameter of tubes, diameter of shell, number of baffles, space between baffles, baffle cut, etc.) and a set of operational variables (which includes pressure drops, velocities for the tube and shell side streams, etc.). One approach for the design of shell and tube heat exchangers, known as rating, assumes a large set of geometrical configurations and operational variables to find the one that satisfies the heat duty requirement and a given set of constraints. Two major disadvantages are identified in the rating method, the first one is that it consumes a large CPU time, and the second one is that the resulting heat exchanger usually does not correspond to the optimal one (i. e., the one with the lowest total cost). Other design methodologies have been reported to determine the heat transfer area, the individual and overall heat transfer coefficients, and the pressure drops for the tube and the shell side (Taborek, 1983). To predict the behavior for the shell side fluid, Kern (1950) presented one of the most utilized methodologies; however, Kern’s model simplified significantly the flow pattern inside the shell, ignoring for instance leakage and bypass streams. In addition, Kern’s method is restricted to a fixed baffle cut of 25% and is not applicable in the laminar flow region. The Bell-Delaware method provides a more general design method (Taborek, 1983) since it accounts for leakage and bypass flow paths, can be applied in the laminar region and is valid for a wide range of baffle cuts, thus providing better estimates for the shell side pressure drop and film heat transfer coefficient than the Kern’s method.

Figure 1. Schematic representation of a shell and tube heat exchanger. Heat Exchangers: Types, Design, and Applications : Types, Design, and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest

Optimal Detailed Design of Shell-and-Tube Cooler Units…

Flowstreams A=Tube Baffle Leakage, B = Crossflow, E = Shell Baffle Leakage, F = Tube Pass Partition Bypass

C

=

Crossflow

177

Bypass,

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Figure 2. Flow model for the shell side flow.

For the case of shell and tube heat exchangers, Polley et al. (1991) and Jegede and Polley (1992) proposed different design algorithms based on the total use of allowable pressure drops for the hot and cold streams, in order to provide the highest film heat transfer coefficients for a given geometrical configuration of the unit. Jegede and Polley (1992) formulated simple equations for the tube and shell sides that relate the film heat transfer coefficients, the pressure drops and the heat transfer area. The combination of these compact relationships with the heat exchanger design equation yields a simple design algorithm that avoids the need for an iterative solution procedure. One limitation of this work is that the equations for the shell side were based on the Kern method. Polley et al. (1991) used the BellDelawere method, but the relationships are significantly complicated, which requires an iterative procedure involving detailed estimations for the heat exchanger geometry. In addition, the algorithm by Polley et al. (1991) is restricted to cross flow areas equal to the window areas and the spacing for the end baffles must be equal to the spacing of the central baffles. Other limitations for this method are that it usually yields fractional number of baffles and does not consider the pressure losses at the end of the tubes. Serna and Jimenez (2005) proposed an algorithm for the design of shell and tube heat exchangers that can use the allowable pressure drops without geometrical limitations; the algorithm uses two compact formulations to predict the pressure drops. The formulation for the shell side was based on the Bell-Delawre method, and the formulation for the tube side accounted for end effects. Muralikrishna and Shenoy (2000) generated graphically feasible exchanger design regions to identify the limits for the pressure drops that satisfy all the geometric and operational constraints for a given design problem. Their approach was based on the Kern method to predict the behavior for the shell side fluid. To overcome this limitation, SernaGonzalez et al. (2007) proposed a similar approach based on the Bell-Delaware method. None of the above methodologies considers the simultaneous optimization of the geometrical configuration of the exchanger and the pressure drops of the streams. Several techniques have been reported for the optimal design of shell and tube heat exchanger based on stochastic optimization methods. Chaudhuri and Diwaker (1997) developed an approach for optimal designs using the simulated annealing algorithm that is linked to the HTRI software. Selbas et al. (2006) used genetic algorithms to determine the smallest heat transfer area for a given exchanger configuration. Babu and Munawar (2007) applied an improved version of genetic algorithms called inferential evolution to determine the minimum heat

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transfer area. Recently, Ravagnani et al. (2009) proposed a methodology for the optimal design of shell and tube heat exchangers using the particle swarm optimization approach considering the TEMA standards, while Ponce-Ortega et al. (2009) proposed an approach based on genetic algorithms with detailed cost functions for the capital cost of the units. Several methods based on mathematical programming techniques have recently been reported. Mizutani et al. (2003) presented a disjunctive programming model that was reformulated as a Mixed Integer Non Linear Programming problem using the Bell-Delaware method. The objective function in this formulation was the minimization of the total annual costs, including the capital cost for the heat exchanger and the capital cost for the two pumps, and the operational cost for the electricity to operate the pumps. Ravagnani and Caballero (2007) improved the model by Mizutani et al. (2003) by including the TEMA standards for the shell diameter, tube bundle diameter, external tube diameter, tube pitch, tube arrangement pattern, number of tube passes and number of tubes. Reported methodologies have used as design parameters the inlet and outlet temperatures for the cold and hot streams as well as their flowrates. None of them considers the case of the optimal design of a cooler, in which the inlet and outlet temperature and the flowrate for the hot stream are known, while only the inlet temperature for the cold stream is specified, such that the outlet temperature and cooling stream flowrate must be optimized. The optimal design of coolers finds a special application in heat exchanger networks (see Ponce-Ortega et al., 2007). This work proposes a new strategy based on genetic algorithms for the optimal design of shell and tube cooler units considering detailed capital cost functions and standard dimensions for the units. This chapter is organized as follows. Section 2 presents a description of the problem. Section 3 shows the variables used as search parameters in the optimization approach. Sections 4 and 5 present the model formulation and solution approach. Finally, sections 6 and 7 present a numerical application and the conclusions of the chapter.

2. PROBLEM DESCRIPTION The problem addressed in this work is stated as follows.

Given A hot stream that requires cooling from its supply temperature to its target temperature, its flowrate and physical properties such as heat capacity, density, viscosity and thermal conductivity. A cold stream that is used as a cooling utility. The inlet temperature, heat capacity, density, viscosity and thermal conductivity, as well as its unit cost are known (notice that the flowrate and outlet temperature are not known). A set of geometric and operational constraints for the cooler unit.

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Determine The flowrate and outlet temperature for the cooling stream. The heat exchanger geometry (shell diameter, number of tubes, tubes length, tubes diameter, tubes pattern, pitch, baffle cut, baffle spacings, number of tubes passes). The operational variables for the units (streams pressure drops, velocities, film heat transfer coefficients).

Objective Function Minimization of the total annual cost, which is a function of the heat duty and the operational and geometric constraints.

3. SEARCH VARIABLES A vector X of search variables to complete the degrees of freedom is used as part of the optimization algorithm. The vector contains 11 components, as shown in Table 1. The algorithm described in section 4.1 is used to determine the cooler design model once the vector of search variables is defined. It is worth to notice that the selection of the search variables that satisfy the geometric and operational constraints and yield the cooler with the minimum total annual cost represents a major difficulty for the design problem.

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Table 1. Search optimization variables Variable X1 X2 X3 X4 X5

X6 X7 X8 X9 X10 X11

Definition

Tube-side pressure drop Shell-side pressure drop Baffle cut (between 15% and 45%) Number of tube passes (1, 2, 4, 6 or 8) Standard inside and outside tube diameters and pitch (80 standard possible combinations given by the TEMA) Tube pattern arrangement (triangular, square or rotated square) Hot fluid allocation (tubes or shell) Number of sealing strips (1, 2, 3 or 4) Tube bundle type (fixed-tube plate, packed-tube plate, floating head, pull-through floating head or U-tube bundle) Ratio inlet and outlet baffle spacing Outlet temperature for the cooling stream

4. MODEL FORMULATION The proposed methodology for the optimization task is based on genetic algorithms, which is a convenient approach because of the significant non convexities of the model; the use of genetic algorithms typically prevents the search from getting trapped in a local optimal

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solution that is away from the global optimal one. Genetic algorithms are stochastic search techniques based on the mechanism of natural selection and genetics, and use a black box model to determine the optimal solution after several iterations (evolutions). Genetic algorithms combine fittest string structures with a structured yet randomized information exchange to form a search algorithm based on the idea of the survival of the fittest. The solution methodology for this problem needs to determine the configuration of the cooler for a given set of design parameters, which may require high CPU times for each iteration; therefore, an efficient solution strategy is needed. The following section presents the strategy used for the solution of the heat exchanger design equations when the search variables are specified.

4.1. Heat Exchanger Model Given the input data described in section 3 that satisfy the degrees of freedom of the problem, the solution of the heat exchanger model determines the remaining variables. The main steps of the solution procedure are shown in Figure 3. Basically, the solution consists of the following steps.

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1. Specify the design data. The data include the mass flowrate, physical properties and inlet and outlet temperatures for the hot stream. The outlet temperature for the cooling water is an optimization variable provided by the genetic algorithms (x11), while the flowrate for the cooling water is calculated from the energy balance. The heat duty is calculated from the stream data. 2. Determine the initial values for KT, KS, n and m using the correlations of the Kern method: ⎛ 4.06 × 10 −9 C1 ⎞ ⎛ Ltp − DT KS = ⎜ ⎟⎜ gc ⎝ ⎠ ⎝ DT

 

1.109 1.297 ⎞ ⎞ ⎛ Ltp De μ S ⎜ ⎟⎜ 1.999 3.406 1.703 ⎟ ⎟ ⎠ ⎝ QS ρ S k S CpS ⎠

(1)

where C1 takes values of 0.866 and 1 for triangular and square tube layout, respectively.

⎛ 1.246 ×10−3 Dti2 μT6 KT = ⎜ ⎜ g Q ρ 2 k 73 Cp 72 c T T T T ⎝ 1

11

⎞⎛ D ⎞ ⎟⎟ ⎜ ti ⎟ ⎠ ⎝ DT ⎠

(2)

The initial values for m and n are 5.109 and 3.5, respectively. 3. Solve the following equation to find hT,  

1

⎛ ⎞n ⎛ ΔPT FT ΔTLM ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ KT Q ⎝ ⎠ f ( hT ) = hT − ⎜ ⎟ =0 1 ⎜ ⎛ K ΔP ⎞ m ⎟ ⎛ ⎞ D D D D ⎜ ⎜ S T n ⎟ + Rds + T ln ⎜ T ⎟ + T + T Rdt ⎟ ⎜ K ΔP h ⎟ 2 K w ⎝ Dti ⎠ Dti hT Dti ⎝⎝ T S T ⎠ ⎠

4. Determine hs from the following equation: Heat Exchangers: Types, Design, and Applications : Types, Design, and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest

(3)

Optimal Detailed Design of Shell-and-Tube Cooler Units…

181

1

⎛ K ΔP h n ⎞ m hS = ⎜ T S T ⎟ ⎝ K S ΔPT ⎠

(4)

The heat transfer area is calculated as follows,

A0 =

Q FT ΔTLM

⎛ 1 ⎛D Dt ln ⎜ T ⎜⎜ + Rds + 2k w ⎝ Dti ⎝ hS

⎞ ⎞ DT D + T Rdt ⎟⎟ ⎟+ ⎠ Dti hT Dti ⎠

(5)

5. Determine the geometric parameters as follows. a) Shell side fluid velocity. For the first iteration the following relationship based on the Kern’s method is used, 1

⎛ μ 6 De0.45 hS ⎞ 0.55 vS = ⎜ S 2 1 0.55 ⎟ 3 3 ⎝ 36kS CpS ρ S ⎠ 1.3

(6)

whereas for other iterations the following relationship is applied,

⎛ h ⎞ vS = ⎜ S ⎟ ⎝ KS 4 ⎠

1 1+ a2

(7)

where

⎛ k s ρ s ( PrS ) 3 jsi ⎞ ⎛ μ ⎞ 0.14 S ⎟ =⎜ Jb Jc Jl J r J s ⎜ 10 −3 vSa2 μ S ⎟ ⎜⎝ μ Sw ⎟⎠ ⎝ ⎠

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

1

KS 4

(8)

b) Tube side fluid velocity:

⎛ Dti μT hT ⎞ vT = ⎜ 2 4 1 ⎟ 3 5 3 ⎝ 2.3kT ρT CpT ⎠ 1

5

7

15

1 0.8

(9)

c) Total number of tubes:

NTT =

106 N tp QT ⎛ π Dti2 ⎞ ⎜ ⎟ vT ⎝ 4 ⎠

(10)

d) Tubes length:

Lta =

106 A0 π DT NTT

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e) Shell diameter:

DS = Dctl + Lbb + DT

(12)

where 1

⎛ 4 NTT C1 Dctl = Ltp ⎜⎜ ⎝ (1 −ψ n ) π

f)

⎞2 ⎟⎟ ⎠

(13)

and ψ n is the correction factor given by Taboreck (1983) that depends on the number of tubes passes. For single tube passes the correction factor is equal to 1, whereas for multiple passes the correction factor takes values between 0.013 and 0.243. C1 takes values of 0.866 and 1.0 for triangular and square tube layout, respectively. Central baffle spacing:

Lbc =

106 QS ⎛ ⎞ D vS ⎜ Lbb + ctl ( Ltp − DT ) ⎟ ⎜ ⎟ Ltpe ⎝ ⎠

(14)

Lta − ( Li + L0 ) Lbc +1 Lbc

(15)

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g) Number of baffles:

Nb =

h) Calculate the parameters a2, b2, jsi, fsi, as well as Jc, Jl, Jb, Jr, Js, Rl, Rb and Rs from the Bell Delaware method (Taborek, 1983). calc

6) Calculate K Tcalc , K Scalc , m First for K Tcalc and n

KTcalc =

calc

and n

calc

.

:

Dti ρT ( KT 1vTr ' ) + KT 2vTr '−0.2 KT−3n 8 gcQT DT

(16)

where

n=

3− r ' 0.8

KT 1 = K end

Dti Lta

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(17)

(18)

Optimal Detailed Design of Shell-and-Tube Cooler Units… 0.2

KT 2

⎛ 2L ⎞ ⎛ μ ⎞ ⎛ μ ⎞ = 0.184 ⎜ 1 + ts ⎟ ⎜ T ⎟ ⎜ tw ⎟ Lta ⎠ ⎝ Dti ρT ⎠ ⎝ μT ⎠ ⎝

KT 3

⎛ 103 kT ⎞ ⎛ Dti ρT ⎞ ⎛ μT ⎞ = 0.023 ⎜ ⎟⎜ ⎟ ⎟⎜ ⎝ Dti ⎠ ⎝ μT ⎠ ⎝ μtw ⎠ r'=

calc

For K Scalc and m

183

0.14

(19)

0.14 1

PrT 3

(20)

0.2 KT 1 +1 K T 2 vT−0.2

(21)

: r ' +b2

r'

K Scalc = K S1vS p + K S 2vS p

+ K S 3 K S−4m

(22)

3 − r 'p

m=

1 + a2

(23)

where

⎛ S ⎞ ⎡ (1 + 0.3 N tcw ) Rl N b ρ S ⎤ K S 1 = 10 −3 ⎜ m ⎟ ⎢ ⎥ ( N b + 1) DS ⎝ Sw ⎠ ⎣ ⎦ ⎡ ( N − 1) ( N + Ntcw ) R ⎤ ⎡1 − 2 ⎛ Bc ⎞ ⎤ ⎛ μsw ⎞ KS 2 = ⎢ b Rl + tcc ⎟ S⎥ ⎜ ⎟⎥ ⎜ Ntcc ( Nb + 1) ⎦ ⎢⎣ ⎝ 100 ⎠ ⎦ ⎝ μ s ⎠ ⎣ ( N b + 1)

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0.14

⎛ 210−3 Rb f si ρ S ⎞ ⎜⎜ ⎟⎟ L pp vSb2 ⎝ ⎠

⎛ ⎞⎛ ⎛ Ltp − DT DS ( N b + 1) Lbc KS 3 = ⎜ ⎟ ⎜ Lbb + Dctl ⎜⎜ ⎜ ⎟⎜ ⎝ Ltpe ⎝ π QS DT NTT ⎣⎡( N b − 1) Lbc + Lbi + Lbo ⎦⎤ ⎠ ⎝

 

KS 4

⎛ k ρ Pr 13 j = ⎜ S −S3 aS2 si ⎜ 10 vS μ S ⎝ r 'p =

(24)

⎞⎞ ⎟⎟ ⎟ ⎟ ⎠⎠

⎞ ⎛ μ ⎞0.14 ⎟ ⎜ S ⎟ Jb J c Jl J r J s ⎟ ⎝ μSw ⎠ ⎠

−b2 K S1 +1 K S 2 vSb2

(25)

(26)

(27)

(28)

7) Check for convergence. The algorithm finishes when the convergence criteria are satisfied:

KT − KTcalc ≤ ε Heat Exchangers: Types, Design, and Applications : Types, Design, and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest

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J. María Ponce-Ortega, M. Serna-González and A. Jiménez-Gutiérrez

where

K S − K Scalc ≤ ε

(30)

m − mcalc ≤ ε

(31)

n − ncalc ≤ ε

(32)

ε is a specified tolerance.

If convergence is not achieved, update the assumed values for K T , K S , m and n

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with the calculated values and repeat the algorithm from step 3. It is worth noting that a different numerical method (for example Newton) could be used instead; however, direct substitution has given good convergence results in the authors’ experience.

Figure 3. Solution approach.

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185

4.2. Feasible Constraints for the Cooler Design To yield a feasible heat exchanger design model, a set of geometric and operational constraints must be satisfied. The algorithm described in section 4.1 yields a configuration for the heat exchanger for the given design parameters; however, it does not ensure the feasibility constraints given by standard design codes. The geometric constraints are those related to the heat exchanger configuration, whereas the operational constraints are related to limits for several variables to allow a proper operation.

4.2.1. Operational Constraints The operational constraints considered in this work are the following. a) Maximum Pressure Drops For the design of shell and tube heat exchangers, limits for the pressure drops for the shell and the tube sides are required. These limits are imposed for the capacity of the external pumps and they are formulated as follows,

ΔPT ≤ ΔPTmax

(33)

ΔPS ≤ ΔPSmax

(34)

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b) Limits for the Velocities Upper limits for the velocities for the shell and tube fluids are required to prevent erosion and flow-induced tube vibration, whereas lower limits are used to prevent fouling. The limits for the velocities are states as follows:

vTmin ≤ vT ≤ vTmax

(35)

vSmin ≤ vS ≤ vSmax

(36)

Limits recommended by Sinnott (1996) for liquid stream velocities are from 1 to 2.5 m s-1 on the tube side and 0.3 to 1 m s-1 on the shell side. c) Limits for the Outlet Temperature for the Cooling Water An upper limit for the outlet temperature for the cooling stream is required to avoid fouling conditions because of salt deposits. This upper limit is also required for a proper operation of the cooling tower when water is used as a cooling utility. This constraint is modeled as follows, max tout ≤ tout

A common upper limit for cooling water is 55°C.

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4.2.2. Geometric Constraints a) Shell Diameter A set of standard dimensions for the shell diameter that are given by the TEMA standards is included,

DS = DSs tan dard

(38)

b) Tube Length The tube length is restricted to avoid transportation problems and to have exclusively standard dimensions, s tan dard LTT ≤ LTT

(39)

c) Ratio Baffle Spacing to Shell Diameter Close spacing for baffles leads to higher heat transfer coefficients and a smaller area at the expense of higher pressure drops. On the other hand, a wide baffle spacing yields bypassing and reduces the cross-flow, thus decreasing the heat transfer coefficient and increasing the heat exchanger area. Therefore, the ratio baffle spacing to shell diameter must be restricted as follows to ensure a proper behavior of the heat exchanger:

Rbsmin ≤ Rbs ≤ Rbsmax

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Typical values for

(40)

Rbsmin and Rbsmax are 0.2 and 1.0, respectively.

d) Ratio for the Cross-Flow Area to the Area in a Window To ensure a proper heat transfer process in the shell side, the following constraints must be used:

⎛ Sm ⎞ ⎛ Sm ⎞ ⎛ Sm ⎞ ⎜ ⎟ ≤⎜ ⎟≤⎜ ⎟ ⎝ S w ⎠ min ⎝ S w ⎠ ⎝ S w ⎠ max

(41)

⎛S ⎞ Common lower and upper values for ⎜ m ⎟ are 0.8 and 1.4, respectively. ⎝ Sw ⎠

4.3. Objective Function The objective function is the minimization of the total annual cost. The annual cost includes the capital cost for the exchanger and the two pumps, in addition to the cost for the electricity required to operate the pumps and the cost for the cooling water. The relationship for the total annual cost is stated as,

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Optimal Detailed Design of Shell-and-Tube Cooler Units…  

TAC = K F ( Cexc + CTpump + CSpump ) + C pow

H Y ⎛ mT ΔPT mS ΔPS ⎞ cooling + ⎜ ⎟+C η ⎝ ρT ρS ⎠

187

(42)

is the capital cost for the cooler, CTpump and C Spump are the capital costs for the where Cexc pumps, C pow is the electricity cost, and C

cooling

is the cost for the cooling water.

In this work, the capital cost for the cooler is split into the cost of component parts and manufacturing costs (Purohit, 1983),

Cexc = Cts + Csh + Cb + Ctd + Ctb + Cba

(43)

where, Cts is the tube-sheet cost based on weight (including cutting but not drilling), Csh is the shell cost (including fabrication) based on weight, Cb is the baffle cost based on weight (assuming ½-in thickness), Ctd is the tube-sheet, baffle drilling and bundle tubing cost (based on the number of tubes), Ctb is the cost of tubes (based on the outside heat transfer surface), and Cba is the cost for the overhead and labor costs (independent of the type of material). The following equations are used to determine the cost of component parts,  

Cts =

π ρ mat c1 ( Ds + 2t s ) 2 t s 3456

Csh =

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' b

π ρmat c2 DS Ltots 144

π ρmat c1DS2 Nb 13824

(44) (45)

(46)

Ctd = c4 N TT

(47)

Ctb = c3 A

(48)

Cba = c5

(49)

The cost data used in this work are: c1 = 0.5$/lb, c2 = 1.0 $/lb, c3 = 75 (A)-0.4 $/ft2, c4 = 2.0 $/tube, c5 = $30800. Also, ρmat= 486.954 lb/ft3, while ts and tt are computed from the TEMA (1988) standard. The cost for the cooling stream depends on its outlet temperature (which is an optimization variable in the model formulation), and is calculated as follows,

⎛ ⎞ 3600Q C cooling = Ccw H Y ⎜⎜ ⎟⎟ ⎝ Cpcw ( tout − tin ) ⎠

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5. SOLUTION METHODOLOGY USING GENETIC ALGORITHMS Figure 4 shows the major steps of the optimization procedure. These steps are described as follows. Step 0. The proposed methodology begins by creating randomly an initial population. The population is a set of designs (which use potential values for the search variables). Each design of the population is considered as an individual for the genetic algorithm. Even when the initial population is generated randomly, the values for each search variable are restricted to be between specified upper and lower limits. In addition, the population size (i.e., the number of coolers designs in each generation) is set as 100. Step 1. Each of the individuals in the current population (set of search variables) is fed to the design algorithm for heat exchangers described above to obtain a set of heat exchanger designs. For each one of these designs the total annual cost is evaluated. Step 2. The simulated designs may or may not satisfy the geometric and operational constraints given in section 4.2. For the cases in which such constraints are not satisfied, a penalty term is used to increase the value of the objective function using the following equation,

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0, when all constraints are satisfied ⎧ ⎪ penalty = ⎨ ⎛ ⎞ 2 2 ⎪r ⎜ ∑ g1i ( geometric ) + ∑ g 2i ( operational ) ⎟ , for other cases i ⎠ ⎩ ⎝ i

(51)

where r is a penalty coefficient (which was set as 1x10-3 in the numerical application of this work). The constraints g1i and g2i define the feasible region for the geometric and operational variables, respectively. Once all designs are tested, go to the next step. Step 3. Evaluate the fitness function of the genetic algorithm for each heat exchanger design based on the simulation results (for all members of the population) as follows,

fitness= TAC + penalty

(52)

Notice that the optimization function for the genetic algorithms is the fitness instead of the TAC to ensure that the feasibility constraints are satisfied. Step 4. Evaluate the stopping criteria. In this case, the algorithm stops and the best design is taken as the optimal solution if one of the following stopping criteria is satisfied: a) the maximum number of generations is reached (which was set as 1000), b) the computation time limit is reached (set as infinite), or c) there is no improved in the objective function in ten successive generations. If none of these criteria are met, the algorithm continues to select the next generation (i.e., a new set of heat exchangers).

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189

Step 5. Selection of the parents for the new generation. From the current generation, the selection function for the parents for the next generation was based on the scaled values from the fitness function using the roulette wheel proportional scheme (for details see Goolberg, 1989). The fitness scaling function converts the fitness scores given by the fitness function to ranked values, such that individuals with lower scaled values have higher probability to be selected as parents for the new generation Step 6. Obtain the new generation. Based on the parents selected in the previous step, three operations are used to yield the new generation: a) Elite count, b) Crossover and c) Mutation.

Elite count The elite count refers to the number of individuals with the best fitness values in the current generation that are guaranteed to survive into the next generation. In the present work, the elite count was set as two, which means that the two coolers (i.e., the set of design variables) with the smaller total annual costs will remain identical in the next generation (or iteration).

Crossover The crossover operation is defined as follows: if   parent1 = ( x p1 , x p1 , x p1 , x p1 , x p1 , x p1 , x p1 , x p1 , x p1 , x p1 , x p1 ) X 1 2 3 4 5 6 7 8 9 10 11

(

  parent1 = x p 2 , x p 2 , x p 2 , x p 2 , x p 2 , x p 2 , x p 2 , x p 2 , x p 2 , x p 2 , x p 2 X 1 2 3 4 5 6 7 8 9 10 11

)

and

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represent two parents (selected using the selection process described in step 5), and  offspring 1 = x o1 , x o1 , x o1 , x o1 , x o1 , x o1 , x o1 , x o1 , x o1 , x o1 , x o1 X 1 2 3 4 5 6 7 8 9 10 11

)

 offspring

o2 11

2 X

( = (x

o2 1

o2 2

o2 3

o2 4

o1 5

o2 6

o2 7

o2 8

o2 9

o2 10

,x ,x ,x ,x ,x ,x ,x ,x ,x ,x

)

and

represent two offspring, then the genes of the offspring are produced through linear  x o1 = α x p1 + (1 − α ) x p 2  x o 2 = α x p 2 + (1 − α ) x p1 i i, i i i i, i i combinations such that i and i , where α i is a uniform random number between [-0.5, 1.5] (see Michalewicz, 1996). The crossover fraction (i.e., the fraction of new designs generated by crossover) used in this work was 0.78 (Goldberg, 1989).

Mutation A non-uniform scheme is used for the mutation operation. Let   parent = ( x p , x p , x p , x p , x p , x p , x p , x p , x p , x p , x p ) X 1 2 3 4 5 6 7 8 9 10 11 be a chromosome and xip the element to p max min be mutated (where xi ∈ ⎡⎣ xi , xi ⎤⎦ ) to yield the chromosome offspring X = ( x1p ,L , xi0 ,L , x11p ) .

Let τ be a random number that takes values of zero or one. Then, the mutated offspring was obtained by

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190

J. María Ponce-Ortega, M. Serna-González and A. Jiménez-Gutiérrez ⎧ xip + Δ ( NumberGeneration, ximax − ximin ) , if τ = 0 ⎪ xio = ⎨ p p ⎪⎩ xi − Δ ( NumberGeneration, xi − 0 ) , if τ = 1

(53)

where NumberGeneration 1− ⎛ ⎞ 100 Δ ( NumberGeneration, y ) = y ⎜ 1 − rr ⎟ ⎝ ⎠

 

(54)

Here, rr is a random number in the interval [0,1]. The mutation fraction used in this work for the solution of the cases of study was 0.2 (Goldberg, 1989).

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Step 7. Check convergence. The following convergence criteria were used. a) The maximum number of 1000 generations, or b) the lack of improvement in the fitness function in 20 successive generations. If one of the convergence criteria is reached, the algorithm stops and reports the solution that correspond to the cooler with the smallest value for the fitness function; otherwise, a new generation is produced and we return to step 1.

Figure 4. Solution strategy.

It is worth nothing that one important advantage of the genetic algorithms is that one obtains a set of final solutions (the population for the last generation) usually with similar values for the fitness function, and one can select the one that represents the best design configuration using other practical design criteria.

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191

Figure 5. Data for the example.

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6. A NUMERICAL APPLICATION In this example, a shell-and-tube heat exchanger must be designed to cool down oil using cooling water. Figure 5 shows the design data. The tube wall thermal conductivity is neglected. Notice in Figure 5 that the outlet temperature and mass flowrate for the cooling water are unknown. The problem then consists in determining the heat exchanger configuration and operating conditions that meet the heat duty and satisfy the operational and geometric constraints at the lowest total annual cost. To begin the application of the genetic algorithm, a first generation with 100 sets of values for the cooler design variables are generated randomly, and only a set of upper and lower values for the design variables are specified. Table 2 shows the upper and lower limits for the design continuous variables. This yields 100 cooler designs that need to be simulated to determine the cooler configurations and the associated costs. Then, the feasibility constraints are evaluated; however, many of these coolers do not satisfy such constraints. Therefore, the penalty term is evaluated for the cases in which the feasible constraints are not satisfied, whereas the penalty term is equal to zero for the cases that meet the feasibility constraints. The fitness function is evaluated for each design adding the penalty term to the total annual cost. Table 3 shows the first population of this example for the twenty fittest coolers. Notice in Table 3 that most of the coolers of the first population do not satisfy the feasibility constraints because the penalty term is higher than zero; this also yields high values for the fitness function. One can also notice the high dispersion between the values of the fitness function in the first generation. To yield the next generation, the two best coolers (the ones with the lower fitness values) remain constant, 78 coolers are formed from crossover operations, and 20 coolers are formed by mutation as described in section 5. Table 4 shows the 20 best coolers for the 10th generation. Notice in this table that the dispersion of the fitness values for the coolers has been reduced significantly (the configuration and design parameters for these coolers are quite homogeneous), and that the fitness function for the best cooler has

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decreased drastically. However, the penalty term and the fitness values are still high. Therefore, the algorithm continues. The problem comes to an end in the 95th generation, consuming a total of 315 s of CPU time. In the last generation, most of the coolers of the population contain a similar configuration and therefore similar fitness values. Table 5 shows a summary of the results obtained for the geometric configuration, and Table 6 gives the results for the operational variables. Notice in Table 5 that the cooler design satisfies all the geometric and operational constraints given in section 4.2 of this chapter. In addition the shell diameter and the tube lengths are selected according to TEMA standard dimensions. The outlet temperature for the cooling stream was obtained as 40°C; this temperature provides a proper tradeoff between the heat transfer area and the required cooling water flowrate. It is also worth noting in Table 6 that the stream velocities and pressure drops for the tube and shell side streams satisfy the operational constraints. Table 2. Limits used for the design variables Design variable

Shell side pressure drop (Pa) Tube side pressure drop (Pa) Baffle cut (%) Number of tubes pases

Lower limit

Upper limit

1,000 1,000 20 1

50,000 50,000 30 8

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Table 3. First Population for the Example Design 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Total annual cost 2,294,993 2,294,993 2,294,993 2,294,993 4,559,053 4,679,445 2,294,993 2,294,993 2,294,993 2,294,993 2,294,993 2,294,993 2,294,993 2,294,993 16,523,500 2,294,993 4,559,053 2,294,993 4,559,053 2,294,993

Penalty

0 438,964 1,414,321 1,872,461 0 0 2,954,659 3,033,085 4,178,276 4,341,175 5,328,001 5,941,365 9,718,689 11,647,761 0 15,745,678 13,991,111 29,307,452 211,617,587 218,808,108

Fitness 2,294,993 2,733,958 3,709,315 4,167,455 4,559,053 4,679,445 5,249,652 5,328,079 6,473,270 6,636,169 7,622,995 8,236,359 12,013,683 13,942,755 16,523,500 18,040,672 18,550,164 31,602,446 216,176,641 221,103,102

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Optimal Detailed Design of Shell-and-Tube Cooler Units… Table 4. Population for the generation 10 for the Example Design 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Total annual cost 215,497 215,782 215,801 215,953 215,975 215,986 216,070 216,403 218,057 219,705 223,405 226,070 205,789 204,805 201,054 195,785 198,785 193,047 198,425 204,807

Penalty 1,371 1,086 1,067 914 893 882 798 465 7,082 33,576 34,348 361,218 2,046,410 2,918,663 6,498,835 6,646,427 9,997,388 11,958,313 12,090,099 14,816,519

Fitness 216,868 216,868 216,868 216,868 216,868 216,868 216,868 216,868 225,139 253,282 257,754 587,288 2,252,199 3,123,468 6,699,889 6,842,212 10,196,174 12,151,361 12,288,525 15,021,327

Table 5. Results for the geometric configuration for the cooler design

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Concept

Q (kW) area (m2) U (W/m2 K) Number of tubes Tubes arrangement Number of tube-passes Number of shells Dti (mm) Dt (mm) Tube pitch length (mm) Number of baffles Heat type Hot fluid allocation ΔTLM (°C) FT Ds (mm) Total tube length (mm) Baffle spacing (mm) Baffle cut (%) Sm (mm2) Sw (mm2)

tout

(°C)

Result

4,541.38 191.20 447.00 4,138 square, 90° 2 1 4.928 6.350 7.938 3 Fixed tubes shell 55.27 0.961 609.57 2,438.00 593.34 35 108,053.22 51,094.79 40.00

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193

194

J. María Ponce-Ortega, M. Serna-González and A. Jiménez-Gutiérrez Table 6. Results for operational variables for the cooler design Concept

ΔP (Pa) v (m/s) h (W/m2 K) Re Pr

Result Shell side Tube side 7,769.94 62,372.52 0.492 1.985 821.16 10,220.29 1,045.82 14,242.63 41.535 4.514

Table 7. Component parts for the cooler capital costs Cost Tube sheet cost, Cts ($) Shell cost, Csh ($) Baffle cost, Cb ($) Tube sheet, baffle drilling and tubing cost, Ctd ($) Tube cost, Ctb ($) Overhead and labor cost, Cba ($) Total capital cost for the cooler, Cexc ($)

Value 325.29 9,152.79 43,247.74 8,276.53 18,876.77 30,000.00 109,879.11

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Table 7 shows the component parts for the cooler cost. One important advantage of using genetic algorithms as optimization technique is that it is not restricted for convex models. The model presented here is quite complex, which may lead standard MINLP methods to get trapped into a local optimal solution that may be far away from the global optimal solution, a situation that is overcome with the application of genetic algorithms. Also, in the methodology presented in this chapter a detailed capital cost function for the coolers was easily incorporated to provide a more accurate optimization task. Table 8. Costs results for the cooler design. Cost Capital cost for the cooler ($/year) Capital cost for the pumps ($/year) Operational cost for the pumps ($year) Cost for the cooling water ($/year) Total annual costs ($/year)

Value 35,381.07 1,903.96 2,725.57 3,573.13 43,583.74

Finally, Table 8 shows the results for the annualized costs for the cooler design. In this example, the major contribution for the total cost is the capital cost of the cooler.

7. CONCLUSIONS This chapter has presented an optimization formulation for the design of shell-and-tube cooler units. The objective is to design the coolers with the minimum total annual cost, including the capital costs for the cooler and pumps as well as the operational costs for the electricity to operate the pumps and the cost of the cooling stream. The methodology has incorporated a detailed capital cost function for the cooler, which considers the cost by

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component parts that yields a more accurate prediction than the simplified functions based on the heat transfer area commonly used in the design stage. The proposed methodology has also incorporated the constraints given by the TEMA standards for the geometry and the exchanger operation; in addition, standard dimensions for the shell diameter and tube length have been included. The optimization of the flowrate and outlet temperature of the cooling water has been considered to account for the trade offs between capital and operating costs. It should be pointed out that no numerical complications were observed in the application of the proposed methodology, and the CPU time for the solution of the case study was not high.

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NOMENCLATURE a, a1, a2, a3 = correlational coefficients for the estimation of the heat transfer factor, jsi A = heat transfer surface area Af = annualization factor for capital cost b, b1, b2, b3 = correlational coefficients for the estimation of the friction factor, fsi Bc = baffle cut as percent of inside shell diameter C1 = tube count constant for multiple tube pass layouts Ca, Cb = cost coefficients in capital cost law for heat exchangers Cb = baffle cost assuming ½ in thickness, based on weight Cba = base cost to cover those overheat and labor cost Ccooling = cooling cost Ccw = unitary cost for the cooling stream Ce, Cf = cost coefficient in capital cost law for pumps Cexc = capital exchanger cost ch = coefficient in heat transfer factor relationship Cp = fluid specific heat at constant pressure and average temperature Cpow = cost per unit of power Cpump = capital pump cost Csh = shell cost including fabrication, based on weight Ctb = cost of tubes, based on outside heat transfer surface Ctd = tubesheet and baffle drilling, and bundle tubing cost, based on number of tubes Cts = tubesheet cost based on weight, including cutting but not drilling Dctl = diameter of the circle through the centers of the outermost tubes of a bundle De = equivalent diameter DS = inside shell diameter Dti = tube inside diameter Dt = tube outside diameter FT = correction factor to logarithmic mean temperature difference for non-countercurrent flow fsi = friction factor for an ideal tube bank g = cost coefficient (exponent) in capital cost law for pump gc = gravitational relationship h = clean heat transfer coefficient hsi = clean shell-side heat transfer coefficient for an ideal tube bundle

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HY = annual plant operation time jsi = heat transfer factor Jb = bundle bypass correction factor for heat transfer Jc = segmental baffle window correction factor for heat transfer Jl = baffle leakage correction factor for heat transfer Jr = laminar flow heat transfer correction factor Js = heat transfer correction factor for unequal baffle spacing ks, kt, kw = thermal conductivity of shell-side fluid, tube-side fluid, and tube wall KS = shell-side constant for pressure drop relationship KT = tube-side constant for pressure drop relationship Kend = pressure losses tube ends L = characteristic length Lbb = inside shell-to-bundle bypass clearance (diametric) Lbc = central baffle spacing Li = ratio inlet baffle spacing Lo = ratio outlet baffle spacing Lpp = tube layout pitch parallel to fluid flow Lpte = tube layout parameter Lta = effective tube length for heat transfer area calculations Lto = overall tube length Ltp = tube layout pitch Lts = tubesheet thickness Ltt = total tube length m = exponent for shell-side heat transfer coefficient in pressure drop relationship m& = fluid mass velocity .

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m' = flow rate n = exponent for tube-side heat transfer coefficient in pressure drop relationship Ns = number of shells Nb = number of baffles Ntp = number of tube passes Ntcc = number of tube rows crossed between baffle tips of one baffle compartment Ntcw = number of tube rows crossed in one baffle window NTT = total number of tubes Ntw = number of tubes in baffle window Pr = Prandtl number for fluid stream Q = heat duty Q’ = volumetric throughput for fluid stream r = penalty term

rp′ = exponent for velocity in pressure drop relationship Rb = bundle bypass correction factor for pressure drop Rbs = ratio baffle spacing to inside shell diameter Rds = shell-side fouling factor Rdw = combined resistance of tube wall and fouling factors Rdt = tube-side fouling factor

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Rl = baffle leakage correction factor for pressure drop

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Rs = baffle end zones correction factor for pressure drop Re = Reynolds number for fluid stream Sm = cross-flow area near shell centerline Sw = net cross-flow area through one baffle window TAC = total annual cost ts = shell thickness tt = tubesheet thickness v = velocity for fluid stream X = parameter in equation (23) x = vector of optimization variables μ = viscosity of fluid stream η = efficiency of pump ρ = density of fluid stream ψ = correction factor for the number of tube passes ΔP = pressure drop for fluid stream ΔTLM = log-mean temperature difference Subscripts max = maximum value mat = tubes material min = minimum value S = shell-side T = tube-side w = at wall temperature

REFERENCES [1]

[2] [3] [4] [5] [6] [7]

[8]

Babu, BV; Munawar, SA. Differential evolution strategies for the optimal design of shell and tube heat exchangers. Chemical Engineering Science, 2007, 62(14), 37203739. Chaudhuri, PD; Diwekar, UM. An automated approach for the optimal design of heat exchangers. Industrial and Engineering Chemistry Research, 1997, 36, 3685-3693. Gen, M; Cheng, R. Genetic algorithms and engineering design, John Wiley & Sons, New York, 1997. Goldberg, DE. Genetic algorithms in search, optimization and machine learning; Addison-Wesley: Reading, MA, 1989. Jegede, FO; Polley, GT. Optimum heat exchanger design. Chemical Engineering Research and Design, 1992, 70(A2), 133-141. Kern, DQ. Process heat transfer, McGraw-Hill: New York, 1950. Mizutani, FT; Pessoa, FLP; Queiroz, EM; Hauan, S; Grossmann, IE. Mathematical Programming model for heat exchanger network synthesis including detailed heatexchanger design. 1. Shell-and-tube heat exchanger design. Industrial and Engineering Chemistry Research, 2003, 42, 4009-4018. Muralikrishna, K; Shenoy, UV. Heat exchanger design targets for minimum area and

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[9]

[10]

[11]

[12] [13]

[14]

[15]

[16]

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[17]

[18] [19] [20]

J. María Ponce-Ortega, M. Serna-González and A. Jiménez-Gutiérrez cost. Chemical Engineering Research and Design, 2000, 78, 161-167. Polley, GT; Panjeh Shahi, MHP; Jegede, FO. Pressure drop considerations in the retrofit of heat exchanger networks. Chemical Engineering Research and Design, 1990, 68, 211-220. Ponce-Ortega, JM; Serna-González, M; Jiménez-Gutiérrez, A. Heat exchanger network synthesis including detailed heat exchanger design using genetic algorithms. Industrial and Engineering Chemistry Research, 2007, 46, 25, 8767-8780. Ponce-Ortega, JM; Serna-Gonzalez, M; Jimenez-Gutierrez, A. Use of genetic algorithms for the optimal design of shell and tube heat exchangers. Applied Thermal Engineering, 2009, 29, 203-209. Purohit, GP. Estimating costs of shell and tube heat exchangers. Chemical Engineering, 1983, 90(17), 57-67. Ravagnani, MASS; Caballero, JA. A MINLP model for the rigorous design of Shell and tuve heat exchangers using the TEMA standards. Chemical Engineering Research and Design, 2007, 85(A10), 1-13. Ravagnani, MASS; Silva, AP; Biscaia, EC; Caballero, JA. optimal design of shell and tube heat exchanger using particle swarm optimization. Industrial and Engineering Chemistry Research, 2009, 48, 6, 2927-2935. Selbas, R; Kizilkan, O; Reppich, M. A new design approach for shell and tube heat exchangers using genetic algorithms from economic point of view. Chemical Engineering and Processing, 2006, 45(4), 268-275. Serna-Gonzalez, M; Ponce-Ortega, JM; Castro-Montoya, AJ; Jimenez-Gutierrez, A. Feasible design space for shell and tube heat exchangers using the Bell-Delaware method. Industrial and Engineering Chemistry Research, 2007, 1, 143-155. Serna, M; Jimenez, A. A compact formulation of the Bell-Delaware method for heat exchanger design and optimization. Chemical Engineering Research and Design, 2005, 83(A5), 539-550. Sinnot, RK. Coulson & Richarson´s Chemical Engineering-Chemical Engineering Design, 2nd ed; Butter-worth-Heinemann: Oxford, U.K., 1996, Vol. 6. Taborek, J. Shell and tube heat exchangers: Single phase flow. Heat exchanger design handbook, Hemisphere publishing corp.: Bristol, PA, 1983; section 3.3. TEMA. Standards of the tubular heat exchanger manufactures association. 7th ed; Tubular heat exchanger manufactures association: New York, 1988.

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In: Heat Exchangers Editor: Spencer T. Branson

ISBN: 978-1-61761-308-1 © 2011 Nova Science Publishers, Inc.

Chapter 6

ADVANCES IN DESIGN OPTIMIZATION OF SHELL AND TUBE HEAT EXCHANGERS Mohammad Fesanghary and Majid Rasouli 1

Department of Mechanical Engineering, Louisiana State University, Baton Rouge, LA, USA 2 Department of Chemical Engineering, École Polytechnique de Montreal, Station Centre-Ville, Montreal, Quebec, Canada

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ABSTRACT Shell and tube heat exchangers (STHEs) are the most widely used heat exchangers in process industries because of their relatively simple manufacturing and their adaptability to different operating conditions. The optimum thermal design of STHEs is a complex task and it involves the consideration of many interacting geometrical and operational design parameters and constraints. In the last few decades several methods for design optimization of STHEs have been developed. These methods can be classified into three main groups, namely thermodynamic approaches, mathematical programming methods and stochastic optimization methods. This chapter covers recent developments and applications of these methods in design of STHEs. After a brief description of the basics of each method, the available literature in the field is reviewed. Also, an approach based on global sensitivity analysis (GSA) is presented to identify the most influential design parameters. The GSA results can provide designers with a broad view that is useful in the design process as well as the reduction of the optimization problem size.

1. INTRODUCTION Shell and tube heat exchangers are the most flexible type of heat exchangers. They are used in many process industries such as oil, conventional and nuclear power stations, steam generators and etc. These heat exchangers provide relatively great ratios of heat transfer area to volume in a form that are easy to construct in variety of sizes as well as that are mechanically strong enough to survive form stresses caused by shop fabrication, shipping, field erection and operating conditions. They can be easily cleaned and repaired, which are

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key requirement for some applications in the case of dirty fluids and tough operating conditions such as oil industries. They are used in almost any condition, including particularly high and low pressure and temperature, great temperature differentials, condensing, vaporizing and etc. Existing of good design methods, the expertise and shop facilities can provide us with successful design and construction of shell and tube heat exchangers[1]. A typical shell and tube heat exchanger is shown in Figure 1.

 

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Figure 1. Schematic of a shell and tube heat exchanger.

In the early 1900s, the condensers and feedwater heaters of power plants were designed by appearing the basic design of shell and tube heat exchangers. Both of these original applications continue to be used, but the designs have become more complicated and specialized, cause to develop different design methods. The tube side design is relatively straightforward, while different design methods differ in shell side design based on assumption and simplification made for flow pattern. In the early 1930s, the early design method was proposed based on flow over ideal tube banks without considering baffle, leakage and bypass effects. This assumption lead to a simple correlation (Colburn’s equation) for calculation of heat transfer coefficient. The simplicity of this method makes it suitable for very quick estimations, while it suffers from lack of accuracy[1]. Later, investigators took into account baffled cross flow but still without modifying leakage and bypass effects. The used correlations in this method are based on the overall data from actual baffled exchangers, because of that it is so-called integral approached. Donohue and Kern method [2]are two well-known approaches in this category. The Donohue method uses a Nusselt equation similar to Colburn’s correlation associated with a weak function of the baffle-to-shell leakage area, which obtains the coefficients of the Nusselt equation. However, the result is a single curve and cannot take into account the effect of tube layout configuration. For pressure drop, the baffle window effect is considered by assuming the window as an orifice with a discharge coefficient of 0.7. This method is easy to handle, involving minimum data input and is suited for quick estimation, but the results are not reliable by standards of today. The Kern [2] method was appeared in 1950 and employed as a virtual industrial standard for many years. In this method with an intelligent choice, the heat transfer and pressure drop are treated for 25% baffle cut only. In order to consider tube layout variations, an equivalent diameter based on longitudinal flow projection is applied as the

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length term in Nusselt and Reynolds numbers. These assumptions add to simplicity of calculation, whereas the results for pressure drop are almost on the safe side and for the heat transfer vary from unsafe to safe side. The practical comments of this method still remain qualitatively valid, while the method cannot be recommended any more[1]. Later, flow visualization studies and a systematic research at University of Delaware provided much more insight in the shell side flow. a new method, Bell-Delaware[3], appeared that was based on ideal tube bank flow and at the same time considered the leakage and bypass effects, but not the interaction between them. This method was able to account for baffle cut and adverse temperature gradient in laminar flow. The Bell-Delaware[4] method is recommended as the most suitable for general engineering application in handbooks and it will be explain more in detail in the next section. The next development emerged by using computational fluid dynamics (CFD) approaches to reach a velocity profile in the shell side. The heat transfer could then be obtained on a local basis. This method is very promising, but it is very difficult to apply for practical cases[1, 5].

2. DELAWARE METHOD 2.1. Shell Side Heat Transfer

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The basic principle of this method is based on the calculation of Colburn ( ) and friction ( ) factors for flow over ideal tube banks and then modifies directly the obtaining heat ) and pressure drop (∆ ) for deviation caused by the streams. In transfer coefficient ( general, Colburn ( ) and friction ( ) factors can be obtained from simply reading the value from the graphs or using a set of curve-fit constants for programming applications in the handbooks. Generally these two factors are function of Reynolds number and tub layout configuration [1]: ,

(1)

,

(2)

Note Reynolds number in the above equations is based on the outside tube diameter and the minimum cross-sectional flow area at the shell diameter. The ideal heat transfer coefficient can be calculated as follows[1]:  .

⁄

.

where: is heat transfer coefficient for ideal tube bank. is Colburn factor. is fluid specific heat at average temperature. . is mass velocity. is Prandtl number. is the viscosity correction factor.

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(3)

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2.2. Tube Side Heat Transfer The tube side heat transfer coefficient ( ht ) for the fluid inside the tube in the turbulent zone and the transitional zone, are given by the following correlations respectively [6]: ht = h′ +

0.14 ⎤ ⎛ μ ⎞ Re− 2100 ⎡ λ ⎟⎟ − h′⎥ ⎢0.23 Re 0.8 Pr1 3 ⎜⎜ Dt 10000 − Re ⎢ ⎝ μw ⎠ ⎣ ⎦⎥

⎡ 0.085Gz ⎛ μ ⎞ ⎜ ⎟ h′ = ⎢3.66 + 1 + 0.047Gz 2 3 ⎜⎝ μ w ⎟⎠ ⎢⎣ ht = 0 .23

⎛ μ ⎞ ⎟⎟ Re t0.8 Prt1 3 ⎜⎜ Dt ⎝ μw ⎠

λ

0.14

2100 ≤ Re t ≤ 10000

(4a)

⎤ λ ⎥ ⎥⎦ Dt

0 .14

Re t ≥ 10000

(4b)

2.3. Shell Side Pressure Drop The shell side pressure drop ( ΔPs ) includes the pressure loss of the pure transverse flow in the zone between the tops of the baffles ( ΔPc ), the pressure loss in the baffle windows (

ΔPw ) and the pressure loss in the end zones of the heat exchanger ( ΔPe ) [1]: ⎛ ⎡ ⎤ m N ⎞ + N b ⎢(2 + 0.6 N tcc ) s × 10 −3 ⎥ Rl + ΔPbi ⎜⎜1 + tcw ⎟⎟ Rb R s ρ N ⎠ 2 s ⎣ 4244 444⎦4 3 14⎝442tcc 4443 144424443 144444

ΔPs = ΔPbi ( N b − 1)Rb Rl

ΔPw

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ΔPc

ΔPe

(5)

is the revised factor, is the leakage factor and is the end zone correction where factor. Details of the individual pressure drop calculations and the related correction factors given in Eq. (5) can be found in [1].

2.4. Tube Side Pressure Drop The tube side pressure drop ( ΔPt ) is given by the following expression [7]: ΔPt =

2 f t G t2 L N tp

(6)

Dt ρ t (μ μ w )0.14

where G t is the tube side mass flow rate per unit cross-sectional area, N tp is the number of tube passes and ft is the tube side friction factor. The tube side friction factor for commercial pipe or slightly corroded tubes is given by Saunders [7]as:

f t = 0.0035 +

0.264 Re t0.42

Re t ≥ 2100

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3. OBJECTIVE FUNCTION A variety of objective functions for heat exchanger optimization design were proposed. In most cases the total annual cost is considered as the objective function. The overall cost associated with a heat exchanger may be categorized as the capital and operating costs. The capital cost includes the cost associated with design, materials, manufacturing, testing, shipping and installation. There are several different methods for cost estimations of heat exchangers. A good summary of the most common methods used for cost estimation of heat exchange equipment in the process industry is given in [8]. Table 1 shows heat exchanger cost based on the surface area and material type. The operating cost consists of the cost associated with electrical power consumed by fans which can be determined from the equation below [7]: EΔP =

m& t ΔPt

ηρ t

+

m& s ΔPs

(8)

ηρ s

where η is the pump efficiency. The annual operating, capital and total cost can be expressed by Eqs. (9) and (10), respectively. Table 1. Cost of Shtes for Different Materials [8]

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Material (shell – tube) Carbon Steel (CS) - CS CS – Stainless Steel (SS) SS -SS CS – Titanium (Ti) Ti - Ti

Cost ($) 7000 + 360 A0.8 8500 + 409 A0.85 10000 + 324 A0.91 14000 + 614 A0.92 17500 + 699 A0.93

Coperating = TP. ec . EΔP

(9)

i (1 + iR ) = Cc . R (1 + iR )TL − 1

(10)

CTotal = Coperating + CCapital

(11)

TL

CCapital

TP , iR ,

and ec are the period of the time of operation per year, interest rate,

technical life and the unit cost of the energy, respectively. The objective functions based on the second law of thermodynamics were also applied to the heat exchanger design optimization problem. These objective functions commonly consider the annual total cost as the sum of the costs associated with the irreversibities and annualized capital costs for equipment.

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4. DESIGN PARAMETERS 4.1. Tube Layout Patterns (

)

There are four tube layout patterns, triangular (30°), rotated triangular (60°), square (90°), and rotated square (45°). Smaller angels result is better heat transfer, but, more pressure drop.

4.2. Tube Pitch (

)

The tube layout pitch control the cross-flow area, the smaller the value the more tubes can be accommodated. Based on the HEDH [1] recommendations the ratio of tube pitch to tube diameter should be kept between 1.25 and 1.5.

4.3. Baffle Spacing (

)

For good thermohydraulic performance, the baffle spacing should be kept within reasonable limits. The practical values are from 0.2 to1 .

4.4. Baffle Cut (

)

The baffle cut is a crucial parameter in having efficient heat transfer. For single-phase applications, the recommended values are in the range of 15% to 40%.

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4.5. Tube Diameter (

)

Tubes are commercially available in specified diameters. Small tube diameters are more preferable due to thermohydraulic considerations. The available sizes range from 6mm to 51mm.

4.6. Tube Passes (

)

The number of tube passes is an important design parameter. It controls the tube-side pressure drop and heat transfer coefficient. The number of tube passes is typically between 2 and 10.

4.7. Tube Length ( ) Generally, STHE with long tubes have lower cost compared to exchangers with short tube length. This is because of that it results in smaller shell diameters, tubesheets and flanges. The performance is usually better, when the length to shell diameter ratio is between 5 and 10.

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205

)

The shell diameter is one the most essential parameters in design of STHEs since it affect the selection of the other design parameters like baffle spacing, baffle cut, tube length and etc. there is no accepted standard for shell diameters, but, the available commercial shells lay between 150 mm to 1200 mm.

4.9. Number of Sealing Strips (

)

Sealing strips are used when the bundle-to-shell bypass clearance becomes large. It is recommended to use one sealing strip for each approximately four to six tube rows crossed.

5. EFFECT OF DESIGN PARAMETERS ON HEAT EXCHANGER COST

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There are some parametric studies on shell and tube heat exchangers [6, 9-12] trying to investigate the influence of design parameters on their performance. These works just study the effect of change in a single design parameter while the other parameters are evaluated at a fixed value. For a complex design problem like heat exchanger design problem this type of parameter study would not be much helpful in understanding the sensitivity behavior of the heat exchanger over the entire domain of design parameters. In order to determine the influence of design parameters when all parameters can change in a predefined range GSA should be performed. The GSA results can serve as a useful guide in the identification of influential as well as non-influential parameters. The GSA results can also provide designers with a broad view to help them in design stage.

5.1. GSA Analysis Generally, to perform a GSA analysis model is considered in the form of Y = f (x1 , x 2 K , x k ) , where x1 , x2 K , xk are input factors and Y is the model output. In heat exchanger design problem Y represents the total cost of the STHE and related input factors are the geometrical design parameters discussed in section 3. Then, using Sobol method [13] which is a variance-based technique the total variance of Y , V (Y ) , is partitioned as follows [14]: k

V (Y ) = ∑ Vi + i =1

∑V

1≤i p j ≤ k

ij

+ K + V1, 2,K,k

(12)

where V (Y ) is the total variance of the output variable Y. This equation is used to derive two types of sensitivity indices defined by

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Si =

Vi V (Y )

ST i = 1 −

(13)

V−i V (Y )

(14)

where V −i is the sum of all the variance terms that not including the index . S i is the firstorder sensitivity index for the th parameter. This index represents the main effect of parameter xi on the output variable Y and measures the variance reduction that would be achieved by fixing (or reducing the range of) that parameter. ST i is the total sensitivity index for the th parameter and is the sum of all effects involving the parameter xi . ST i takes into account the interactions between the th parameter and the other parameters. It can be used for model reduction purposes; when a factor does not have any effect both on its own and in cooperation with others, it can be considered as non-influential and can be fixed to any value within its range of uncertainty. The sensitivity indices can be computed using the Monte Carlo method. The principle is to generate randomly samples of parameters within their permissible ranges and to estimate V (Y ) , Vi and V −i as follows: 1. Choose a base sample dimension .

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2. Generate two random input sample matrices M 1 and M 2 of dimension

′ x12 ′ K x1′i K x1′k ⎤ ⎡ x11 x12 K x1i K x1k ⎤ ⎡ x11 ⎢x x Kx Kx ⎥ ⎢ x′ x′ K x′ K x′ ⎥ 2i 2k ⎥ 2i 2k ⎥ M 1 = ⎢ 21 22 M 2 = ⎢ 21 22 ⎢ M M M M M M ⎥ ⎢ M M M M M M ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ x N 1 x N 2 K x Ni K x Nk ⎦ ⎣ x′N 1 x′N 2 K x′Ni K x′Nk ⎦

:

(15)

3. Define a matrix N i formed by all columns of M 2 , except the th column which is taken from M 1 , and a matrix NT i complementary to N i , formed with the th column of M 2 and with all the remaining columns of M 1 :

′ x12 ′ K x1i K x1′k ⎤ ⎡ x11 ⎡ x11 x12 K x1′i K x1k ⎤ ⎢ x′ x′ K x K x′ ⎥ ⎥ ⎢x x 2i 2k ⎥ 21 22 K x 2′ i K x 2 k ⎥ N i = ⎢ 21 22 ⎢ ⎢ M M M M M M ⎥ NT i = ⎢ M M M M M M ⎥ ⎥ ⎢ ⎥ ⎢ ⎣ x′N 1 x′N 2 K x Ni K x′Nk ⎦ ⎣ x N 1 x N 2 K x ′Ni K x Nk ⎦

(16)

4. Compute the model output for all the input values in the sample matrices M 1 , N i and

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5. The sensitivity indices are computed based on scalar products of the above defined vectors of model outputs.

1 δ0 = N V (Y ) =

Vi =

1 N

V−i =

1 N

1 N

N

∑Y

j

(18)

j =1

∑ (Y )

N

N

j 2

j =1

− δ 02

∑ Y ( )Y ′( ) − δ j

j

j =1 N

∑ Y ( )Y ′ j =1

j

T

( j)

2 0

− δ 02

(19)

(20)

(21)

6. Calculate the first-order and total sensitivity indices from Eqs. (13) and (14).

5.2. Case Study

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Fesanghary et al. [15] has done a GSA analysis for STHXs. The result of their study on a typical STHE design problem has shown in Figure 2. It can be seen that the most sensitive parameter is Ds and the parameters like N tp , Lbc D s , and Dt have significant effect on exchanger cost. The total cost is less sensitive to parameters like Bc , Ltp Dt and N ss .

Figure 2. First-order sensitivity indices of total cost [15].

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6. OPTIMIZATION METHODS In the last few decades several methods for optimization of heat exchanger have been developed. These methods can be classified into three main groups, namely thermodynamic approaches, deterministic methods and stochastic methods.

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6.1. Thermodynamic Approaches Thermodynamic approaches based on Second Law Analysis have been introduced by Bejan [16].The aim of these approaches is to minimize irreversibility. The irreversibility of a thermal system can be measured by the rate of exergy destruction. The exergy destruction shows the location, magnitude and source of the irreversibility in a system. In heat exchangers the exergy destruction comes from two primary sources: (i) thermal losses (ΔT-losses) which are associated with the temperature differences and (ii) pressure drop (ΔP-losses) which are because of frictional losses. The application of thermodynamic approaches for heat exchanger design has been addressed by Bagalagel and Sahin [17], Mansour et al.[18], Jassim et al.[19], Hesselgreaves [20], and others. Eryener [12] used thermoeconomic analysis to find optimum baffle spacing. The objective function was total cost and the parameters were baffle spacing, baffle spacing to shell diameter ratio, tube length, tube outer diameter, exchanger area and number of tube. The method applied on two different design cases: a) fixed heat exchanger area with specific flow rate and b) fixed thermal duty. More recently, Guo et al. [21]proposed the field synergy principle, which is an indication of entropy generation in the system to optimize STHEs. This principle indicates that the heat transfer rate depends not only on the magnitudes of the flow velocity and temperature gradient but also on their synergy. They considered tube outer diameter, number of tubes, baffle spacing, central angle of baffle cut, and output temperature as design variables and used a genetic algorithm to minimize the field synergy number. Guo et al. [22]also used dimensionless entropy generation rate to optimize STHEs. They showed that for the case that the heat duty is given, their method increase the heat exchanger effectiveness significantly, and also decrease the pumping power dramatically.

6.2. Stochastic Optimization Methods Generally, Stochastic optimization refers to the minimization (or maximization) of a given function in the presence of randomness in the optimization process. Stochastic optimization methods use randomness in setting parameter values and choice of solutions. In general, these methods are good at global searching. They can easily find the near-globaloptimum regions. Stochastic optimization methods are able to handle complex problems without being limited by the nonlinearities, the non-convexities and the discontinuities of the objective function. These capabilities make them best suited for in heat exchanger design optimization problems, where design variables and objective functions are usually discontinuous. However, the drawback of these methods is their slowness. Since these methods do not use gradient information to guide the search direction, they are relatively slow at finding the global optimum. Among this class of the optimization methods simulated

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annealing[23], genetic algorithms (GAs) [24-27], differential evolution [28] and harmony search algorithm [15, 29] are the most frequently used methods. Ozcelik [10] developed a genetic based algorithm to optimize the variables of the MINLP test problems. The algorithm was applied to find optimum configuration of shell and tube heat exchanger respect to the sum of the annual capital cost and exergetic cost. The chosen variables are tube length, tube outer diameter, pitch ratio and type, tube layout angle, number of tube passes, baffle spacing ratio and mass flow rate of the utility. Selbas et al. [30] considered tube outer diameter, tube layout, number of tube pass, shell diameter, baffle spacing and baffle cut percentage as optimization variables. The minimum heat exchanger area was obtained respect to the variables by using genetic algorithm. Caputo et al. [31] proposed a method based on genetic algorithm to minimize total cost, which is the sum of operating cost related to pumping and capital cost. The shell inside diameter, tube outer diameter and baffle spacing were taken into account as the optimization variables. Wang et al. [32] carried out a systematic experimental work to obtain correlations for relating Nusselt number and friction factor to Reynolds number in three different types of shell and tube heat exchangers. A genetic algorithm based procedure is used to obtain the correlations from experimental data. Fesanghary et al. [15] used a harmony search algorithm (HS) for design optimization of STHEs from economic point of view, First they performed a GSA analysis to recognize the most important parameters on the total cost of heat exchanger. In a case study they showed that the HS algorithm can perform better than GA in design of STHEs. A multi-objective genetic algorithm has been employed by Agarwal and Gupta[33]. They considered two multiobjective design problems. In the first problem, the cooling water is returned to its source after use, without cooling. They minimized the total annualized cost and the amount of required cooling water simultaneously. In their second problem, they assumed that the cooling water is recycled to the heat exchanger after it is cooled in a cooling tower. Then they minimized the total cost including the cooling tower expenses. Minimizing the cost of STHE with genetic algorithm considering the maintenance has been studied by Wildi-Tremblay and Gosselin [27]. They considered reliability and maintenance due to fouling by restraining the “coefficient of increase of surface” which account for fouling effects. Allen and Gosselin [26] have used a genetic algorithm op optimize shell and tube condensers. Eleven design parameters including tube pitch, tube layout patterns, baffle spacing at the center, baffle spacing at the inlet and outlet, baffle cut, tube-to- baffle diametrical clearance, shell-to-baffle diametrical clearance, tube bundle outer diameter, shell diameter, and tube outer diameter were considered and optimized for minimum total annual cost.

6.3. Deterministic Methods Mathematical programming methods are another main class of optimization methods. These methods are deterministic which it means they do not use randomness in their algorithms. Mathematical programming methods, especially gradient-based algorithms, are very effective in performing local search. In general, gradient-based algorithms converge faster and they can obtain solutions with higher accuracy compared to stochastic approaches. However, these approaches rely strictly on the initial starting point, the topology of the feasible region and the surface associated with the objective function. A good starting point is

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vital for these methods to be executed effectively. Also, in some problems gradient can be quite expensive to calculate numerically. These drawbacks decrease the effectiveness of these powerful methods in thermal system design problems with non-convex objective function and discrete design variables. Applications of mathematical programming methods for design optimization of heat exchangers has been reported by several researchers including Reppich and Kohoutek [34], Zalewski et al. [35], Gonzalez et al. [36], Unuvar and Kargici [37], and Soltan et al. [38]. Reppich et al.[39] selected the tube and shell side heat transfer coefficient as optimization variables to minimize the total annual cost, which is the sum of capital, operating and maintenance cost. Serna et al. [40] presented a compact formulation to connect the shell side pressure drop with the heat transfer coefficient and the exchanger area. The formulation leads to more convenience in optimization process. The obtained compact correlation was applied to problems with total annual cost as objective function and the shell and tube side heat transfer coefficients as the variables. Mizutani et al. [41] have used generalized disjunctive programming to minimize the total cost of STHE accounting for area and pumping expenses. In their design optimization problem they considered the number of tubes, number of passes, internal and external tube diameters, tube arrangement pattern, number of baffles and head type as design variables.

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7. CONCLUSIONS Advances in design optimization of shell and tube heat exchangers were studied in this chapter. The different methods used for design of STHEs reviewed followed by a brief description of well-known Bell-Delaware method. The design parameters were introduced as well as different objective functions. Also, an approach based on global sensitivity analysis (GSA) was presented to identify the most influential design parameters. Study of the current literature shows that stochastic optimization methods especially genetic algorithms are becoming more popular compared to deterministic methods during the recent years.

NOMENCLATURE A Bc Cc

heat transfer area (m2) baffle cut capital cost ($)

CCapital

annual capital cost ($/yr)

Coperating

annual operating cost ($/yr)

CTotal Cp Ds Dt Ec E ΔP

annual total cost ($/yr) Heat capacity (J/kg.K) inside shell diameter (m) tube outside diameter energy cost ($/kWh) pumping power (kW)

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Advances in Design Optimization… F G Gz H iR L Lbc Ltp L/ Ds Lbc/ Ds Ltp/ Dt m& Mat Nss Ntp Pr Re Si

friction factor mass flow rate per unit area (kg/m2.s) Graetz number heat transfer coefficient (W/m2K) interest rate length (m) central baffle spacing (m) tube pitch (m) length ratio baffle spacing ratio pitch ratio mass flow rate (kg/s) material type number of sealing strips (pairs) number of tube passes Prandtl number Reynolds number first-order sensitivity index

ST i

total sensitivity index

T TL TP

V()

temperature, (°C ,K) technical life (year) operating time (hr/yr) total variance

ΔPs

shell side pressure drop (Pa)

ΔPc

pressure loss of the pure transverse flow (Pa)

ΔPw

pressure loss in the baffle windows (Pa)

ΔPe

pressure loss in the end zones (Pa)

ΔPbi

ideal tube bank pressure loss (Pa)

Greek letters λ μ ρ

thermal conductivity (W/m.K) viscosity (kg/m.s)

θ tp

tube layout characteristic angle (deg)

η

pump efficiency

fluid density (kg/m3)

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Subscripts S T W

shell tube wall

Acknowledgments The author would like to thank Rina Mascarenhas for her constructive comments and suggestions.

REFERENCES [1] [2] [3] [4] [5] [6]

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[7] [8] [9] [10] [11]

[12] [13]

[14]

Schlèunder, EU. Heat exchanger design handbook. 1983, Washington: Hemisphere Pub. Corp. 5 v. in 7 (loose-leaf). Kern, DQ. Process heat transfer. first ed. 1950, McGraw-Hill Companies. Bell, KJ. Final Report of the Cooperative Research Program on Shell-and-Tube-Heat Exchangers. 1963, University of Delaware Eng. Bell, KJ. Thermal and hydraulic design of heat exchangers. Heat exchanger design handbook 3. 1983, Washington: Hemisphere Pub. Corp. 1 v.(loose-leaf). Mukherjee, R. Effectively design shell-and-tube heat exchangers. Chemical Engineering Progress, 1998. 94(2), 21-37. Lerou, PPPM., et al., Optimization of counterflow heat exchanger geometry through minimization of entropy generation. Cryogenics, 2005, 45(10-11), 659-669. Saunders, EAD. Heat exchangers : selection, design & construction. Designing for heat transfer. 1988, New York, NY: Longman Scientific & Technical. Taal, M. et al., Cost estimation and energy price forecasts for economic evaluation of retrofit projects. Applied Thermal Engineering, 2003. 23(14), 1819-1835. Qi, ZG; Chen, JP; Chen, J. Parametric study on the performance of a heat exchanger with corrugated louvered fins. Applied Thermal Engineering, 2007. 27(2-3), 539-544. Ozcelik, Y. Exergetic optimization of shell and tube heat exchangers using a genetic based algorithm. Applied Thermal Engineering, 2007, 27(11-12), 1849-1856. Saffaravval, M; Damangir, E. A General Correlation for Determining Optimum Baffle Spacing for All Types of Shell-and-Tube Exchangers. International Journal of Heat and Mass Transfer, 1995, 38(13), 2501-2506. Eryener, D. Thermoeconomic optimization of baffle spacing for shell and tube heat exchangers. Energy Conversion and Management, 2006, 47(11-12), 1478-1489. Sobol, IM. Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Mathematics and Computers in Simulation, 2001, 55(1-3), 271280. Saltelli, A; Tarantola, S; Campolongo, F. Sensitivity analysis as an ingredient of modeling. Statistical Science, 2000, 15(4), 377-395.

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[15] Fesanghary, M; Damangir, E; Soleimani, I. Design optimization of shell and tube heat exchangers using global sensitivity analysis and harmony search algorithm. Applied Thermal Engineering, 2009, 29(5-6), 1026-1031. [16] Bejan, A. Concept Of Irreversibility In Heat Exchanger Design: Counterflow Heat Exchangers For Gas-To-Gas Applications. Journal of Heat Transfer, 1977, 99 Ser C(3), 374-380. [17] Bagalagel, SM; Sahin, AZ. Design optimization of heat exchangers with high-viscosity fluids. International Journal of Energy Research, 2002, 26(10), 867-880. [18] Mansour, MK; Musa, MN; Hassan, MNW. Thermoeconomic optimization for a finnedtube evaporator configuration of a roof-top bus air-conditioning system. International Journal of Energy Research, 2008, 32(4), 290-305. [19] Jassim, RK; Khir, T; Ghaffour, N. Thermoeconomic optimization of the geometry of an air conditioning precooling air reheater dehumidifier. International Journal of Energy Research, 2006, 30(4), 237-258. [20] Hesselgreaves, JE. Rationalisation of second law analysis of heat exchangers. International Journal of Heat and Mass Transfer, 2000, 43(22), 4189-4204. [21] Guo, JF; Xu, MT; Cheng, L. The application of field synergy number in shell-and-tube heat exchanger optimization design. Applied Energy, 2009, 86(10), 2079-2087. [22] Guo, JF; Cheng, L; Xu, MT. Optimization design of shell-and-tube heat exchanger by entropy generation minimization and genetic algorithm. Applied Thermal Engineering, 2009, 29(14-15), 2954-2960. [23] Chaudhuri, PD; Diwekar, UM; Logsdon, JS. An automated approach for the optimal design of meat exchangers. Industrial & Engineering Chemistry Research, 1997, 36(9), 3685-3693. [24] Ozkol, I; Komurgoz, G. Determination of the optimum geometry of the heat exchanger body via a genetic algorithm. Numerical Heat Transfer Part a-Applications, 2005, 48(3), 283-296. [25] Gholap, AK; Khan, JA. Design and multi-objective optimization of heat exchangers for refrigerators. Applied Energy, 2007, 84(12), 1226-1239. [26] Allen, B; Gosselin, L. Optimal geometry and flow arrangement for minimizing the cost of shell-and-tube condensers. International Journal of Energy Research, 2008, 32(10), 958-969. [27] Wildi-Tremblay, P; Gosselin, L. Minimizing shell-and-tube heat exchanger cost with genetic algorithms and considering maintenance. International Journal of Energy Research, 2007, 31(9), 867-885. [28] Babu, BV; Munawar, SA. Differential evolution strategies for optimal design of shelland-tube heat exchangers. Chemical Engineering Science, 2007, 62(14), 3720-3739. [29] Doodman, AR; Fesanghary, M; Hosseini, R. A robust stochastic approach for design optimization of air cooled heat exchangers. Applied Energy, 2009, 86(7-8), 1240-1245. [30] Selbas, R; Kizilkan, O; Reppich, M. A new design approach for shell-and-tube heat exchangers using genetic algorithms from economic point of view. Chemical Engineering and Processing, 2006, 45(4), 268-275. [31] Caputo, AC; Pelagagge, PM. Salini, P. Heat exchanger design based on economic optimisation. Applied Thermal Engineering, 2008, 28(10), 1151-1159.

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[32] Wang, QW. et al., Experimental study and genetic-algorithm-based correlation on shellside heat transfer and flow performance of three different types of shell-and-tube heat exchangers. Journal of Heat Transfer, 2007, 129(9), 1277-1285. [33] Agarwal, A; Gupta, SK. Jumping gene adaptations of NSGA-II and their use in the multi-objective optimal design of shell and tube heat exchangers. Chemical Engineering Research & Design, 2008, 86(A2), 123-139. [34] Reppich, M; Kohoutek, J. Optimal-Design of Shell-and-Tube Heat-Exchangers. Computers & Chemical Engineering, 1994,18, S295-S299. [35] Zalewski, W; Niezgoda-Zelasko, B; Litwin,v Optimization of evaporative fluid coolers. International Journal of Refrigeration-Revue Internationale Du Froid, 2000, 23(7), 553-565. [36] Gonzalez, MT; Petracci, NC; Urbicain, MJ. Air-cooled heat exchanger design using successive quadratic programming [SQP]. Heat Transfer Engineering, 2001, 22(3), 1116. [37] Unuvar, A; Kargici, S. An approach for the optimum design of heat exchangers. International Journal of Energy Research, 2004, 28(15), 1379-1392. [38] Soltan, BK; Saffar-Avvwal, A; Damangir, E. Minimizing capital and operating costs of shell and tube condensers using optimum baffle spacing. Applied Thermal Engineering, 2004, 24(17-18), 2801-2810. [39] Reppich, M; Zagermann, S. New design method for segmentally baffled heat exchangers. Computers and Chemical Engineering, 1995, 19(Suppl), S137-S142. [40] Serna, M; Jimenez, A. A compact formulation of the Bell-Delaware method for heat exchanger design and optimization. Chemical Engineering Research and Design, 2005, 83(5 A), 539-550. [41] Mizutani, FT. et al., Mathematical programming model for heat-exchanger network synthesis including detailed heat-exchanger designs. 1. Shell-and-tube heat-exchanger design. Industrial & Engineering Chemistry Research, 2003, 42(17), 4009-4018.

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In: Heat Exchangers Editor: Spencer T. Branson

ISBN 978-1-61761-308-1 c 2011 Nova Science Publishers, Inc.

Chapter 7

T HERMAL D ESIGN M ETHODOLOGY OF I NDUSTRIAL C OMPACT H EAT R ECOVERY WITH H ELICALLY S EGMENTED F INNED T UBES E. Martínez1,∗, W. Vicente2,†, G. Soto1 and M. Salinas2 1 Universidad Autónoma Metropolitana Azcapotzalco. Av. San Pablo 180, Azcapotzalco 02200, Mexico City, México 2 Instituto de Ingeniería, Universidad Nacional Autónoma de México. Ciudad Universitaria, 04510 Mexico City, México

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Abstract A thermal design methodology of compact heat recovery with helically segmented finned tubes in staggered layout on an industrial scale is proposed. The methodology is based in thermodynamic analyses and semi-empirical models for heat transfer and pressure drop, coupled to the Logarithmic Mean Temperature Difference (LMTD) method. Some of the best models available in the open literature for heat transfer and pressure drop in helically segmented finned tubes are used. The methodology is validated with experimental data of industrial equipment under different operating conditions. Comparisons between predictions and experimental data show a precision greater than 95% in the heat transfer for Reynolds number, based on outside diameter of bare tube, of about 10000 (Reynolds number based on volume-equivalent diameter of 13751). In the case of pressure drop of gas phase, there is a precision of approximately 90% for a Reynolds number based on outside diameter of bare tube, of about 10000 (Reynolds number based on volume-equivalent diameter of 13751). Therefore, the results show that the best flow regime in which heat transfer and pressure drop are optimal is for Reynolds number based on outside diameter bare tube, of about 10000 (Reynolds number based on volume-equivalent diameter of 13751) with a length-width ratio (cross section area) around 2.5.

PACS 44.05.+e, 44.10.+i, 44.27.+g, Keywords: Compact heat exchangers, Segmented fins, Helical fins, Pressure drop, Heat transfer coefficient. ∗ E-mail address: † E-mail address:

[email protected] [email protected]

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1.

E. Martínez, W. Vicente, G. Soto et al.

Introduction

The design of compact heat recovery systems requires appropriate knowledge of heat transfer and fluid dynamics phenomena. Nowadays, there are two main design methods; the first uses Computational Fluid Dynamics (CFD) techniques, and the second uses semiempirical models. The CFD method provides complete and detailed information on thermophysical phenomena. However, it requires good computational support and long calculation times, which are not always available in industrial applications. Semi-empirical models allow a quick evaluation of thermo-physical phenomena with minimum computational infrastructure. Therefore, the technique chosen for industrial applications in the design of heat recovery is the semi-empirical method, which is analyzed in the present chapter. Helically segmented finned tubes (figure 1) are used in industrial applications for obtaining compact heat recoveries because gas phase turbulence and heat transfer surface are increased. However, pressure drop in the gas phase increases and, consequently, operational problems can emerge. Therefore, the use of appropriate predictive models for heat transfer and pressure drop is necessary. In the open literature, there are many studies on compact heat exchangers and some of them are focused on helically finned tubes. The majority of the papers have studied solid fins, like Genic et al. [1]. Only a few papers focus on segmented fins, and so, there are few correlations for the heat transfer and pressure drop. One of the most commonly used models was developed by Weierman [2,3], who developed heat transfer and friction factor correlations for different tube bundles (inline and staggered) with solid and serrated fins. These correlations were modified by ESCOA (Extended Surface Corporation of America) in order to obtain better predictive models (Ganapathy [4]). Later, Nir [5] analyzed heat transfer and pressure drop in helically finned tube bundles and the results were validated with experimental data; he found a maximum deviation of 10% in the predictions. Finally, Kawaguchi et al. [6] analyzed heat transfer and pressure drop in helically segmented finned tubes for Reynolds numbers of 7000 to 50000 and 2000 to 30000. On the other hand, there are some authors, like Martin [7, 8], who tried to use Lévêque’s [9] generalized equation in complex finned tubes with the model of Gaddis and Gnielinski [10]. However, the model of Gaddis and Gnielinski [10] has to be modified for helically segmented finned tubes and, therefore, the results are not conclusive yet. Some models from previous studies for helically segmented finned tubes have been analyzed and the correlations of Weierman [2,3], Nir [5], ESCOA [4], Kawaguchi et al. [6], and Lévêque [9] have been validated with experimental data. For example, Hofmann et al. [11] carried out a comparative analysis with academic equipment for heat transfer and pressure drop correlations of Weierman [2,3] and Lévêque [9]. Other authors, such as Naess [12, 13], analyzed the models of Weierman [2, 3] and Nir [5] with academic equipment. Finally, Martínez et al. [14] analyzed the models of Weierman [2, 3], ESCOA [4], Nir [5], and Kawaguchi et al. [6] and the predictions were validated with experimental data in heat recovery on an industrial scale. So, the objective in the present chapter is the proposal of a thermal design methodology of Helically Segmented Finned Tube Heat Exchanger (HSFHE) on an industrial scale. The models of Weierman [2, 3] and Kawaguchi et al. [6] are proposed for heat transfer analysis and the Weierman’s model [2, 3] is proposed for pressure drop study. The methodology was applied to industrial equipment and the results were compared with experimental data taken from Martínez et al. [14, 15].

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Figure 1. Helically segmented finned tube.

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2.

Methodology

A compact heat recovery (CHR) uses residual energy (flue gases) from any heat source (boilers, steam generators, furnaces, etc) and heats working fluids such as treated water, feedwater, air, oils, etc. The CHR can handle single-phase flows (heat recovery or economizer) or two-phase flows (heat recovery steam generator), which depend on flue gas temperature. Generally compact heat recovery systems are designed to handle single-phase flows, such as HSFHE, which are coupled to boilers or steam generators as heat sources (HS). The HSFHE uses residual energy from clean flue gases because fly ash or unburned particles could accumulate in the fin segment. These clean gases can come from gaseous fuel combustion or flue gas cleaning processes. The flue gas cleaning process is costly and so it is not considered for heat recovery projects. Thus, gaseous fuel combustion is considered as an energy source. In the present chapter, a methodology for the thermal design of HSFHE with staggered layout coupled to boilers or steam generators (SG), with combustion gas temperatures below 400 ◦C, is analyzed. The thermal design methodology for HSFHE is proposed in four steps: 1) Design data, 2) Thermodynamic analysis, 3) Thermal design of equipment, and 4) Optimum design and performance of compact heat recovery. The design data should be established according to maximum, minimum, and nominal operating conditions of the heat source. The design premises should be determined with industrial measurements of the heat source under stable operating conditions. Data could be taken from the control room panel or by measurements taken directly from the equipment. The main information required is mass flow, temperature, and flue gas pressure. The pressure and temperature of the residual gases are easily measurable by means of a thermometer and a differential manometer in one of the chimney’s ports. However, mass flow gases are not easy to obtain because the chimney’s port is not designed for the insertion of flow meters. Thus, there are two indirect ways to obtain flue gas mass flow: 1) non-reactive mass balance in the combustion chamber, and 2) global energy balance in the equipment. Fuel consumption, stoichiometric air-fuel ratio, and air excess (value reported

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Fluids Pressure (kPa) Temperature (◦ C) Mass flow (kg/h) Type LCV (kJ/Nm3) Others Efficiency (%) Air excess (%) Pressure (kPa) Gas leaks (%)

Feedwater 3000 105 23600 Undercooled

Steam 2800 311 23600 Superheated

SG 80 20 2800 0

SG with HSFHE 82.96 20 2800 3

Fuel 1542 Natural gas 34352

Flue gases 78 180 32400

in isokinetic analysis of flue gases) are required for a non-reactive mass balance. In the global energy balance, steam production conditions (mass flow, pressure, and temperature), feedwater conditions (mass flow and temperature), HS efficiency, and fuel characteristics (composition and low calorific value, LCV), are required. The flue gas mass flow obtained by means of a mass or energy balance needs to be corrected, because there are going to be leaks across the damper, and fuel consumption will be reduced due to an increase in HS efficiency. Flue gas mass flow leaks in the damper are easily corrected, since a leak of about 3-5% can be considered. In the case of a correction due to increased efficiency, the procedure is more complicated since the combustion gas mass flow needs to be corrected by means of an iterative procedure. Therefore, a new global energy balance in the HS is proposed for expected conditions with HSFHE coupled to the heat source. Once the new flue gas mass flow is obtained, an efficiency increase in HS by heating the working fluid (feedwater) is calculated. Later, a new mass flow is calculated by means of an energy balance, using the new HS efficiency. The procedure continues until the flue gas mass flow is the same in the last two iterations. An example of the basic information that is required for the design of a helically segmented finned tube heat exchanger is shown in Table 1. The table is divided into two sections: the first section includes all data regarding fluids (feedwater, steam, fuel, and flue gases), whereas the second section shows basic information on heat source performance, both before (HS without HSFHE) and after the heat recovery project (HS with HSFHE installed). The information in Table 1 is focused on flue gases from natural gas combustion and HSFHE coupled to steam generator (SG). Once the design parameters are available, a thermodynamic analysis on the combustion process is required, because some gases such as SO3 could react with H2 O to form H2 SO4 . This sulfuric acid could condense if flue gas temperature is cooled down to values below dew point. And so, calculation of dew point of the main condensable gases is very important. The thermodynamic analysis considers a stoichiometric combustion study and the calculation of dew point. The stoichiometric combustion analysis considers the evaluation of flue gas composition in order to obtain the saturation temperatures of sulfuric acid (H2 SO4 ) and steam water (H2 O). In the case of natural gas combustion, the flue gas

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Table 2. Example of flue gases composition and dew point of the main condensable gases

Natural gas composition Mol fraction CH4 (%) 91.7170 Mol fraction O2 (%) 0.2014 Mol fraction N2 (%) 5.0342 Mol fraction CO2 (%) 3.0205 Mol fraction H2 O (%) 0.0151 Mol fraction H2 S (%) 0.0004 Mol fraction S (%) 0.0114 Combustion parameters air-fuel ratio 17.2

Flue gas composition Mol fraction CO2 (%) 8.2622 Mol fraction H2 O (%) 15.9978 Mol fraction SO2 (%) 0.0010 Mol fraction N2 (%) 72.5429 Mol fraction O2 (%) 3.1961 % SO2 converted SO3 2.27 SO3 converted H2 SO4 (%) 100 Saturation temperatures (◦C) H2 SO4 111.3 H2 O 50.2

composition could be obtained with the following chemical reaction: YCH4 CH4 +YO2 O2 +YN2 N2 +YCO2 CO2 +YH2 O H2 O +YH2 S H2 S +YS S + (1 + ea )aO2 + (1 + ea )3.76aN2 → bCO2

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+ cH2 O + dSO2 + eN2 + f O2

(1)

where yi are mol the fractions for species and ea is the air excess. The terms a, b, c, d, e, and f are the stoichiometric coefficients. The dew point of main condensable gases, such as sulfuric acid and steam water, should be calculated by the method of Verhoff and Banchero [16] and with steam water charts [17], respectively. The Verhoff and Banchero [16] method is based on an empirical model as a function of partial pressures of sulfuric acid and water. The partial pressure of water ( PH2 O ) is obtained directly from the flue gas composition. The partial pressure of sulfuric acid (PH2 SO4 ) is obtained from the conversion of SO3 to H2 SO4 . A conversion factor of 100% is recommended because that is the critical situation. And so, the sulfuric acid dew point (TDP ) is obtained with the following equation: 1 = 0.002276 − 0.00002943LnPH2O − 0.0000858LnPH2SO4 + TDP + 0.0000062(LnPH2O )(LnPH2 SO4 )

(2)

where TDP in K and P in mm Hg. In the case of steam water, the dew point is obtained from steam water charts [17]. An example of sulfuric acid dew point calculated for flue gases from natural gas combustion is shown in Table 2. Fuel composition (natural gas), flue gas composition, and main condensable gas dew point are shown in Table 2. The results show that the minimum cooling gas temperature should be above 111 ◦C in order to avoid sulfuric acid condensation.

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The previous steps are the basis of thermal design method, which is based on the Logarithmic Mean Temperature Difference (LMTD) method coupled with heat transfer coefficients. The LMTD method considers the evaluation of overall heat transfer coefficient ( Uo ) which is a function of inside and outside convective coefficients of finned tubes, inside and outside fouling factors, geometry of finned tubes, thermal resistance, and fluid properties. The fouling factors could be evaluated [18] but its calculus is complicated because is a transient phenomenon. These factors could be obtained from tables (according to the fluid and its operating conditions) but its value is considered constant. Usually, the heat exchangers are designed with fouling factors from tables but its value must be chosen carefully. So, the inside fouling factors for treated feedwater could be obtained from ASME guidelines (Ganapathy [19]. The outside fouling factors could be taken from Shah and Sekuli´c [18] tables. The value of common fluids properties could be taken from any thermodynamic or fluid dynamic books. Finally, the flue gases properties may be obtained from Ganapathy [4] tables. Therefore, the Uo based on the outside surface is shown in the following equation:

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Uo =

1 Ao +Ao R f o (ho +hr (ho +hr )(η f cA f At

+

ew Ao kw Ai

+ ( h1i + R f i ) AAoi

(3)

where ho , hi , and hr are outside convective coefficient, inside convective coefficient, and radiation heat transfer coefficient, respectively. In the case of flue gas temperatures lower than 400 ◦C, the value of hr could be negligible [4], and so this value is considered zero. R f o and R f i , are the outside and inside fouling factors, respectively. η f c , A f , At , Ao , and Ai are fin corrected efficiency, fin surface area, bare tube surface area, total surface area, and inside surface area, respectively. Finally, ew and kw are tube wall thickness and tube material thermal conductivity, respectively. Surface areas, fin efficiency, heat transfer coefficients, and pressure drop are analyzed separately. The finned tube surface areas are usually expressed per unit length in order to simplify calculations. Thus, the inside surface and the bare tube outside areas are calculated with the following equations: Ai = πdi

(4)

At = πdo

(5)

where di and do are the inside and outside diameter tube. The calculation of fin surface area is complex because the fin is integrated by helically solid and helically segmented parts. Some authors consider a completely segmented fin for the calculation of this area because the base of the fin surface is solid (less than 25%) while the remainder part is segmented (greater than 75%). However, this consideration could cause significant error because the fin surface area is underestimated. On the other hand, considering the solid-segmented arrangement could lead to complicated calculations since fin efficiency should be calculated as a last arrangement. One way to solve this problem is by considering the fin surface area as a corrected helically segmented fin. So, the fin surface area is calculated with the ESCOA proposal [20] for 38.1 mm (1.5 in) to 60.96 mm (2.4 in)

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tube diameter: A f = .01016πn f (do + .00508) +

πn f (do + .00508) wf

[(2l f − .01016)(w f + t f ) + w f t f ]

(6)

where n f , w f , l f , and t f are fin density (fins/unit length), fin wide, fin height, and fin thickness, respectively. Finally, total outside surface is obtained with the following expression: Ao = A f + πdo (1 − n f t f )

(7)

In the case of fin efficiency, it could be obtained from the rectangular-fin theory, because helically segmented finned tubes are composed of rectangular fins placed helically around the tube. So, efficiency could be obtained with the one-dimensional rectangular fin theory for adiabatic boundary condition coupled to Harper and Brown’ s [21] proposal: ηf =

tanh mLc mLc

(8)

where m and Lc are parameter of fin equation and critical length, respectively. The values of m and Lc are defined in the following equations:   2ho (t f + w f ) 1/2 (9) m= kftfwf

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Lc = l f + t f /2

(10)

where k f is the fin thermal conductivity. Equation 9 is based on a constant convective heat transfer coefficient, but this heat transfer coefficient does not have a uniform profile, according to Gardner [22, 23] and Ghai [24]. There are some models of convective heat transfer profile, such as those developed by Gardner [23], Han et al. [25], and Chen et al. [26], but they require empirical values. One way to solve this problem is by using a mean convective heat transfer model, as proposed by Weierman [2]: η f c = η f (0.9 + 0.1η f )

(11)

The last part of the method is the calculation of heat transfer coefficients and pressure drop of flue gases. The convective heat transfer coefficients are proposed for the inside and outside of finned tubes. The inside convective coefficient ( hi ) proposed in the methodology is the Gnielinski’s [27] model, which, according to Bejan [28], is the best available in the open literature. This model has been validated with satisfactory results by Rane and Tandale [29] and others authors. Thus, the inside heat transfer coefficient is evaluated as a function of the Nusselt number, according to the following equation: Nu =

( f1 /8)(Rei − 1000)Pr hi di = k 1 + 12.7( f /8)1/2(Pr 2/3 − 1)

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(12)

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where k is thermal conductivity of fluid. Rei and Pr are the inside Reynolds Number and Prandtl Number. Finally, f i is the friction factor, which is defined in the following equation: fi =

1 (1.82log 10Rei − 1.64)2

(13)

In the case of outside convective coefficients (ho ) for staggered tube bundles and segmented fins, the models of Weierman [2, 3] and Kawaguchi et al. [6] are recommended, according to Martínez et al, [14] since both models are complemented, offering reliable results. On the other hand, these models should be used simultaneously in case one of them is outside the range of application. Therefore, the Weierman [2, 3] and Kawaguchi et al. [6] models are proposed in the thermal design methodology in order to compare results. The Weierman’s model [2, 3] was calculated in terms of the Colburn heat transfer factor, as shown in the following equation:  −0.35l    f ho Pr 2/3 −0.35 j= = 0.25Re 0.55 + 0.45e s f c p Go i  d 1/2  T 1/4 h  2 f bo (14) 0.7 + 0.7 − 0.8e(−0.15Nr ) e(Sl /St ) do Ts

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where Go and c p are gas mass flux and specific heat capacity at constant pressure, respectively. Tb and Ts are average outside fluid temperature and average fin temperature, respectively. d f is outside diameter of finned tube. The terms s f , Sl , St , and Nr are clear space between fins, longitudinal pitch, transversal pitch, and number of tube-rows, respectively. The average outside fluid temperature (Tbo ) is obtained from the arithmetic mean flue gases temperature. The average fin temperature (Ts ) could be calculated with the following expression: Ts = Tbi + 0.3(Tbo − Tbi )

(15)

where Tbi is the average inside fluid temperature which is calculated with arithmetic mean working fluid temperature. On the other hand, Kawaguchi’s model [6] for staggered tube banks and segmented fins is presented in terms of the Nusselt number, according to the following equation: Pr 1/3(s f /dv )−0.062 Nu = A2 Re0.784 v

(16)

where Rev is the Reynolds number based on volume-equivalent diameter. The terms A2 , s f , and dv are the experimental coefficient for tube rows, fin gap, and volume-equivalent diameter, respectively. The volume-equivalent diameter is defined by following equation:  1/2 dv = t f n f {(t f + 2l f )2 − do2 } + do2

(17)

Finally, the gas phase pressure drop was calculated with the model of Weierman [2, 3], according to the comparative analyses of Martínez et al. [14]. In the design of HSFHE, a maximum pressure drop of 248.9 Pa [30] (1 inch (in) of water column (wc)) should be Heat Exchangers: Types, Design, and Applications : Types, Design, and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest

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considered in order to avoid technical problems such as backpressure. The model of Weierman [2,3] calculates the friction factor of finned tube and the pressure drop is obtained with the following empirical equation: h i (1+B2 )ρ fo + A 4Nr gp G2o Nr ∆Pg = (18) 1.083x109ρgp

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where f o and ρgp are the friction factor and density of gas phase at average outside temperature, respectively. B is defined as the square relation between free gas area and total area. The gas phase friction factor ( fo ) is evaluated with Weierman’s model [2, 3], according to the following equation: "  −0.7(l f /s f )0.23 #  1/2   df 0.05S t 0.11 f o = 0.07 + 8Re− o 0.45 do do h      2 2 1.1 + 1.8 − 2.1e−0.15Nr e−2(Sl /St ) − 0.7 − 0.8e0.15Nr  i e−0.6(Sl /St ) (19) Heretofore, the LMTD method has been described but the procedure of the thermal design of HSFHE is an iterative process because equipment configuration and finned tube size must be proposed. Then, with a finned tube and an equipment proposal, the LMDT method is applied according to the operating conditions. Finned tube size and equipment configuration must be chosen carefully in order to prevent possible faults. For example, the length of the finned tube (Lt f ) should be as long as possible in order to have the least amount of joints. Thus, the construction and duration of the equipment will be better because fewer joints represent less manufacturing time and less failure. On the other hand, the number of tubes per line (Nt ) needs to be higher than the number of tube-rows ( Nr ) in order to have minimal gas-phase pressure drop. A configuration of Nr in even numbers is recommended because the finned-tube row is connected to another tube row by bends. Thus, location of inlet and exit headers will be on the same side. Once the equipment configuration has been proposed, an iterative process begins, with Nr as its main variable, for comparison between calculated and proposed Nr . The tube rows are calculated using the following equation: Nr =

Q˙ Uo Nt Ao Lt f ∆TML

(20)

where Q˙ and ∆TML are the heat transfer and mean logarithmic temperature difference, respectively. Once the calculated value of Nr is obtained, its value is compared with initial Nr . If tuberow values are different, then the thermodynamic condition of fluids should be changed. On the contrary, if Nr values are the same, the thermal design is completed. Finally, the last step of the methodology ends with an optimum design of the equipment, which is obtained by means of an iterative process, according to the maximum heat transfer that is possible with the maximum allowable pressure drop. Once the optimal design is obtained, HSFHE

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Figure 2. Finned tube geometry. performance must be obtained for different operating conditions. The last step is very important because this sensitivity analysis could avoid operational problems such as back pressure or sulfuric acid condensation. An example of the application of this methodology in the design of HSFHE is based on size and geometry of the finned-tube shown in figure 2. Moreover, design data and predictions for two different operating conditions are shown in Table 3 according to the following nomenclature: Twi,o (water temperature at inlet or outlet), Tgi,o (flue gas temperature at inlet or outlet), ∆Pg (gas side pressure drop), m ˙ a (water steam mass flow), m ˙ g (flue gas mass flow), Eao (feedwater final precision temperature), Eg f (flue gas final precision temperature), E pg (flue gas precision pressure drop), EU (overall heat transfer coefficient precision).

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Table 3. Results and design data Model

Tia (◦C)

Kawaguchi Weierman Experimental

105 105 105

Kawaguchi Weierman Experimental

108 108 108 (Ea (%)

Kawaguchi Weierman

99.45 98.89

Kawaguchi Weierman

99.30 99.69

Tf a (◦C)

Tig Tf g (◦C) (◦C) Condition 1 127.3 184 119.7 126.6 184 121.5 128.0 184 120.0 Condition 2 128.9 180 120.9 128.4 180 122.5 128.0 180 122.0 Precision analyses (Eg (%) (E pg (%) (EU (%) Condition 1 99.75 99.59 98.77 94.12 91.84 Condition 2 99.09 94.39 99.59 89.74 97.66

∆Pg Pa

m˙ a kg/h

m˙ g kg/h

211.58 199.14

24400 24400 24400

34003 34003 34003

194.16 174.14

22700 22700 22700

32330 32330 32330

Rei,o 10951 10438 10951 10438

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3.

225

Results

The thermal design methodology was applied in the design of industrial HSFHE, according to the geometry, configuration, and operating conditions of the industrial equipment tested by Martínez et al. [14, 15]. The equipment was designed with a maximum pressure drop of 248.9 Pa and a 5% safety factor. These predictions with the models of Weierman [2, 3] and Kawaguchi et al. [6] for heat transfer and Weierman [2, 3] for pressure drop of gas-phase were compared with experimental data. So, the results of pressure drop and Reynolds number for gas phase are shown in Table 3. The results show all values are always higher than the experimental data and the best precision (over 89%) for Reynolds numbers (based on the outside diameter of bare tube, Reo ) over 10000. These characteristics are essential because technical problems, like backpressure, will be avoided, guaranteeing equipment performance. The results show that predictions could be lower than experimental data for Reynolds numbers (Reo ) higher than 12000. The relationship between theoretical and experimental overall heat transfer coefficients are shown in table 3. The results show that the best predictions are obtained with the Kawaguchi-Gnielinski model, because their precision is 94.39% to 99.59% for Reynolds numbers (Reo ) between 10439 and 10951. The Weierman-Gnielinski model has a precision of 91.84% to 97.66% for Reynolds numbers ( Reo ) higher than 10439. The values are less precise than those obtained with the Kawaguchi-Gnielinski models, but are adequate in spite of some cooling-gas temperature predictions are lower than the experimental data. These models are a good option because their predictions of working fluid temperatures (feedwater and flue gases) are close to experimental values. Flue gas cooling temperatures and their Reynolds number based on the outside diameter of bare tube are shown in Table 3. The results show that the best combination of models is the Kawaguchi-Gnielinski, because these models have the best precision (higher than 99%). Moreover, the predictions of flue gas temperatures are lower than the experimental data which is the best situation because acid gases condensation could be avoided. The Weierman-Gnielinski model has a precision higher than 94.39% for Reynolds numbers (Reo ) over 10000. The comparative analysis of results and the experimental data show that the best predictions are obtained at a Reynolds number ( Reo ) of about 10000, because at this flow regime pressure drop is close to 248.9 Pa. On the other hand, the final water temperature is shown in table 3. The results show that the best models are the KawaguchiGnielinski, because they have a precision higher than of 99.3%. Thus, the maximum deviation of 0.7% could be compensated by a minimum-safety factor. The Weierman-Gnielinski model has a precision of 98.89 to 99.69% which are acceptable values.

4.

Conclusion

The HSFHE thermal methodology was applied in the design of industrial equipment and the results were validated with experimental data. The results of flue gas pressure drop show that the model used in the methodology is adequate, because all predictions are higher than the experimental values. Moreover, precision is over 89% for Reynolds numbers ( Reo ) over 10000. Therefore, the best flow regime occurs at a Reynolds number ( Reo ) of about 10000 with a length-width ratio (cross section area) of around 2.5, because at this zone, pressure

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drop is lower than 249 Pa (1 in water column) [30] and, consequently, the heat recoveries do not affect the process. Thus, operational problems such as backpressure could be avoided. However, a safety factor must be used in order to guarantee equipment performance. Heat transfer was analyzed for the overall heat transfer coefficient and final temperatures of fluids (feedwater and flue gases). The results of the overall heat transfer coefficient are precise since maximum deviation is 8.16% for models considered in the methodology (Kawaguchi-Gnielinski and Weierman-Gnielinski). However, the Kawaguchi-Gnielinski model has better predictions (94.39% to 99.59% precision). On the other hand, temperature evaluation has a minimum precision of 98.77% with the methodology, but the best model combination is the Kawaguchi-Gnielinski, although the Weierman-Gnielinski model should be considered if finned tube geometry and fluid conditions are outside the range of application of the Kawaguchi et al. model [6]. Finally, the results show that the proposed thermal methodology is adequate because pressure drop is overestimated and fluid temperatures are close to the experimental data. Therefore, the methodology can be applied for the effective design of helically segmented finned tube heat exchangers. Nevertheless, a 5-10% safety factor is recommended in order to guarantee equipment performance.

Acknowledgments We appreciate the support given to the research presented here by Consejo Nacional de Ciencia y Tecnología (CONACYT), Universidad Autónoma Metropolitana Azcapotzalco, and Universidad Nacional Autónoma de México (Direccion General de Asuntos del Personal Académico, PAPIIT-IN111709-3).

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References [1] Srbislav B. Genic, Branislav M. Jacimovic, and Boris R. Latinovic. Research on air pressure drop in helically-finned tube heat exchangers. Applied Thermal Engineering , 26:478–485, 2006. [2] C. Weierman. Correlations ease the selection of finned tubes. Oil and Gas Journal, 74:94–100, 1976. [3] C. Weierman. Correlations ease the selection of finned tubes. Oil and Gas Journal, 74:94–100, 1976. [4] V. Ganapathy. Industrial boilers and heat recovery steam generators: design, applications and calculations . Marcel Dekker, 2003. [5] A. Nir. Heat transfer and friction factor correlations for crossflow over staggered finned tube banks. Heat Transfer Engineering , 12(1):43–58, 1991. [6] Kiyoshi Kawaguchi and Kenichi Okui. Heat transfer and pressure drop characteristics of finned tube banks in forced convection. Journal of Enhanced Heat Transfer , 12(1):1–20, 2005. Heat Exchangers: Types, Design, and Applications : Types, Design, and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest

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[7] Holger Martin. How to predict heat and mass transfer from fluid friction. In 4th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics HEFAT, 2005. [8] Holger Martin. The generalized lévêque equation and its practical use for the prediction of heat and mass transfer rates from pressure drop. Chemical Engineering Science, 57:3217–3223, 2002. [9] A Lévêque. Les lois de la transmission de chaleur par convection. Annales des Mines, 13:201–299, 305–362, 381–415, 1928. [10] E. S. Gaddis and V. Gnielinski. Pressure drop in cross row across tube bundles. International Journal of Chemical Engineering , 25:1–15, 1985. [11] R. Hofmann, F. Frasz, and K. Ponweiser. Performance evaluation of solid and serrated finned-tube bundles with different fin geometries in forced convection. In Fifth European Thermal-Sciences Conference , 2008. [12] Erling Naess. Heat transfer in serrated-fin tube bundles with staggered tube layouts. In 9th U.K. National Heat Transfer Conference , 2005. [13] Erling Naess. Heat transfer and pressure drop in serrated-fin tube bundles for waste heat recovery applications. In 6th World Conference on Experimental Heat Transfer, Fluid Mechanics, and Thermodynamics , 2005.

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[14] E. Martínez, W. Vicente, G. Soto, and M. Salinas. Comparative analysis of heat transfer and pressure drop in helically segmented finned tube heat exchangers. 2010. [15] E. Martínez, W. Vicente, M. Salinas, and G. Soto. Single-phase experimental analysis of heat transfer in helically finned heat exchangers. Applied Thermal Engineering , 29:2205–2210, 2009. [16] Verhoff and Banchero. Predicting dew points of flue gases. Chemical Engineering Progress Journal , 70(8):70–71, 1974. [17] Yunus Cengel and Michael Boles. Thermodynamics: An Engineering Approach . McGraw Hill, 6th edition, 2007. [18] Shah R. K. and Sekuli´c D. P. Fundamentals of Heat Exchanger Design . John Wiley, 2003. [19] V Ganapathy. Fouling-the silent heat transfer thief. Hydrocarbon Processing , pages 49–52, 1992. [20] ESCOA. Escoa turb-x hf rating instructions. Technical report, 1979. [21] D. R. Harper and W. B. Brown. Mathematical equations for heat conduction in the fins of air cooled engines. Technical Report 158, NACA, 1923. [22] K. A. Gardner. Efficiency of extended surface. Trans. ASME, 67(8):621–631, 1945. Heat Exchangers: Types, Design, and Applications : Types, Design, and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest

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[23] K. A. Gardner. Discussion on paper by m. l. ghai, proc. Technical report, Institute of Mechanical Engineers, London, 1951. [24] M. L. Ghai. Heat transfer in straight finsheat transfer. proc. general discussion of heat transfer. Technical report, Institute of Mechanical Engineers, London, 1951. [25] L. S. Han and S. G. Lefkowitz. Constant cross section fin efficiencies for non-uniform surface heat transfer coefficients. Technical Report 60-WA-41, ASME, 1960. [26] S. Y. Chen and G. L. Zyskowski. Steady state heat conduction in a straight fin with variable heat transfer coefficient. In 6th National Heat Transfer Conference . ASMEAIChE, 1963. [27] V Gnielinski. New equations for heat and mass transfer in turbulent pipe and channel flow. Int. Chem. Eng., 16:359–366, 1976. [28] A Bejan. Convection Heat Transfer. John Wiley, 2nd edition, 1995. [29] M. V. Rane and S. Tandale. Water-to-water heat transfer in tube-tube heat exchanger: Experimental and analytical study. Applied Thermal Engineering , 25:2715–2729, 2005.

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[30] Jonh Weale, Peter H. Rumsey, Dale Sartor, and Lee Eng Lock. Laboratory lowpressure drop design. ASHRAE Journal, pages 38–42, August 2002.

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INDEX A  acid, 218, 225 adaptability, xi, 199 adaptations, 214 algorithm, 134, 145, 153, 177, 179, 180, 183, 184, 185, 188, 190, 191, 208, 209, 212, 213, 214 alternative energy, 122 ammonia, 124, 126, 127, 145 amplitude, 128 annealing, 177, 209 atmosphere, 119, 122 atmospheric pressure, 33

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B  banks, 6, 200, 201, 222, 226 base, 5, 22, 25, 26, 29, 30, 31, 32, 33, 34, 38, 39, 160, 161, 195, 206, 220 behaviors, ix, 1, 15, 18, 141 bending, 37 benefits, 124 boilers, 217, 226 bonding, 3 bounds, 165

C  calculus, 220 campaigns, 155 carbon, 15 case studies, x, 147, 171 case study, 152, 153, 160, 164, 165, 167, 195, 209 casting, 4, 5 catalyst, 133 cell size, 5, 16 ceramic, 2, 15 chemical, ix, 5, 219

chemical vapor deposition, 5 China, 1, 4, 37 chromosome, 189 circular flow, 10 classification, 116, 122 cleaning, 91, 119, 121, 143, 217 CO2, 116, 144 color, 11 column vectors, 206 combustion, 2, 125, 144, 217, 218, 219 commercial, 119, 124, 155, 202, 205 comparative analysis, 216, 225 complexity, 3 complications, 195 composition, 5, 218, 219 compression, 111, 121, 122 computation, 67, 188 computational fluid dynamics, 201 computing, 94 condensation, 124, 145, 219, 224, 225 conditioning, ix, x, 115, 122, 124, 213 conduction, 2, 10, 14, 15, 44, 111, 112, 113, 227, 228 conductivity, 2, 3, 12, 13, 15, 20, 35, 38, 91, 94, 100, 105, 143, 149, 152, 165, 167, 168, 172, 178, 191, 196, 211, 220, 221, 222 configuration, 36, 37, 46, 52, 57, 60, 68, 69, 70, 73, 80, 85, 88, 90, 91, 93, 94, 121, 159, 177, 180, 185, 190, 191, 193, 200, 201, 209, 213, 223, 225 construction, 48, 116, 119, 129, 153, 165, 166, 168, 169, 200, 212, 223 consumption, 116, 217 contamination, 119, 124 convergence, 64, 108, 164, 183, 184, 190 convergence criteria, 183, 190 cooling, ix, xi, 1, 3, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 29, 30, 31, 35, 37, 109, 110, 117, 122, 124, 126, 127, 145, 153, 175, 178, 179, 180, 185, 186, 187, 191, 192, 194, 195, 209, 219, 225 copper, 2, 10, 15, 30, 31

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230

Index

correction factors, 202 correlation, 2, 7, 13, 148, 152, 172, 173, 200, 210, 214 correlations, 8, 13, 22, 111, 112, 139, 145, 157, 162, 180, 200, 202, 209, 216, 226 corrosion, 124 cost, ix, x, 1, 5, 30, 116, 124, 175, 176, 178, 179, 186, 187, 188, 191, 192, 193, 194, 195, 197, 198, 203, 204, 205, 207, 208, 209, 210, 213 CPU, 30, 31, 32, 42, 176, 180, 192, 195 crystallization, 124 CVD, 5 cycles, 121, 122, 124

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D  deformation, 37 degradation, 112 Department of Energy, 145 deposition, 5, 161 deposits, 185 derivatives, 80 designers, xi, 199, 205 desorption, 122, 125 destruction, 208 deviation, 201, 216, 226 dew, 218, 219, 227 dilation, 124 dispersion, 2, 3, 15, 191 distortions, 156 distribution, 10, 17, 31, 32, 33, 66, 67, 68, 69, 112, 126, 127, 154, 156 district heating, 110 drying, 16

E  economic evaluation, 212 electricity, 178, 186, 187, 194 electrodeposition, 5 electron, 4, 5 emission, 116 emitters, 116 energy, 12, 44, 61, 102, 116, 117, 122, 130, 144, 155, 157, 180, 203, 210, 212, 217, 218 energy efficiency, 144 energy recovery, 155 engineering, ix, 1, 5, 6, 16, 197, 201 enlargement, 154, 156 entropy, 208, 212, 213 environment, 113, 116, 122, 124 equality, 159

equilibrium, 2, 10, 12, 130, 136, 137, 139, 144 equipment, x, xi, 115, 116, 117, 124, 203, 215, 216, 217, 223, 225, 226 erosion, 185 ethylene, 117, 124, 143 ethylene glycol, 124, 143 evaporation, x, 5, 115, 131, 143, 145 evolution, 88, 98, 99, 177, 197, 209, 213 exercise, 166, 171 experimental condition, 16 exponential functions, 126

F  fabrication, 5, 187, 195, 199 finite element method, 112 first generation, 191 fitness, 188, 189, 190, 191 flexibility, 2, 117, 119, 121, 143 flow field, 14, 28, 31 flue gas, 217, 218, 219, 220, 221, 224, 225, 226, 227 foams, ix, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 13, 14, 15, 18, 19, 20, 22, 23, 24, 25, 30, 35, 37, 38 food, x, 115, 117, 143 food industry, 117 force, 36, 161 fouling, x, xi, 110, 111, 115, 156, 175, 185, 196, 209, 220 France, 43 freedom, 164, 179, 180 freezing, 124 friction, 7, 8, 16, 17, 120, 132, 147, 148, 150, 152, 158, 159, 160, 163, 171, 173, 195, 201, 202, 209, 211, 216, 222, 223, 226, 227 fuel consumption, 218

G  genes, 189 genetics, 180 geometrical parameters, 149 geometry, x, 44, 119, 120, 126, 131, 132, 147, 148, 149, 151, 154, 156, 159, 160, 161, 162, 164, 166, 169, 171, 176, 177, 179, 195, 212, 213, 220, 224, 225, 226 glue, 133 glycol, 125 gravitational constant, 159 gravity, 37, 127 greenhouse gases, 116 GSA, xi, 199, 205, 207, 209, 210

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Index

H  hardness, 133 heat capacity, x, 43, 44, 45, 50, 56, 61, 64, 65, 72, 77, 83, 85, 86, 87, 88, 89, 92, 94, 95, 96, 97, 98, 106, 109, 110, 111, 161, 164, 178 heat loss, 43, 93, 129, 166 heat removal, 3 height, 8, 10, 12, 16, 20, 22, 23, 24, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 152, 158, 159, 171, 172, 221 hub, 31, 38 hydrogen, 10

I 

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identification, 205 image, 4, 5, 10, 28, 32, 133 individuals, 188, 189 industries, x, xi, 115, 117, 143, 153, 176, 199 industry, 117, 168, 171, 203 inertia, 8 information exchange, 180 insertion, 217 integration, 54, 59, 70, 71, 74, 75, 82, 171 interface, 29, 34, 128 investment, 5 iteration, 180, 181, 189 iterative solution, 177

J  joints, 117, 119, 120, 223

231

manufacturing, xi, 2, 175, 187, 199, 203, 223 marketing, 119 mass, 5, 7, 9, 91, 94, 100, 104, 109, 110, 112, 113, 124, 125, 127, 132, 134, 135, 143, 144, 157, 159, 163, 180, 191, 196, 201, 202, 209, 211, 217, 218, 222, 224, 227, 228 materials, ix, 1, 2, 12, 15, 44, 117, 153, 203 mathematical programming, xi, 178, 199, 210 matrix, 5, 14, 111, 206 matter, v measurement, 10, 110 measurements, 20, 217 meat, 213 mechanical loadings, 37 mechanical properties, 2 media, ix, 1, 2, 12, 16, 17, 31 metals, 10 meter, 97 methanol, 126 methodology, xi, 148, 162, 171, 175, 178, 179, 188, 194, 195, 215, 216, 217, 221, 222, 223, 224, 225, 226 Mexico, 215 microscope, 4, 5 mixing, 9, 15, 124, 126 model reduction, 206 modelling, 145 models, xi, 194, 212, 215, 216, 221, 222, 225, 226 momentum, 10, 17, 19, 27, 28, 31, 33, 34, 159 Monte Carlo method, 206 morphology, 5, 8 multiplier, 133, 144 mutation, 189, 190, 191

N  L 

laminar, 3, 19, 22, 23, 24, 25, 26, 162, 163, 176, 196, 201 lead, 30, 37, 117, 156, 162, 194, 200, 220 leakage, 117, 176, 196, 197, 200, 201, 202, 218 ligament, 6, 8, 9, 10, 11, 12, 13, 14, 38 liquids, 148, 161 lithium, 124, 125, 126, 131, 143, 144, 145

M  machine learning, 197 magnitude, 12, 128, 143, 208 majority, 6, 10, 30 management, 30

natural gas, ix, 218, 219 natural selection, 180 negative effects, 116 next generation, 188, 189, 191

O  oil, 91, 92, 93, 94, 95, 96, 97, 100, 104, 109, 126, 139, 191, 199 operating costs, x, 175, 195, 203, 214 operating data, 165 operations, 189, 191 optimization, ix, xi, 175, 177, 178, 179, 180, 187, 188, 194, 197, 198, 199, 203, 208, 209, 210, 212, 213, 214 optimization method, xi, 177, 199, 208, 209, 210

Heat Exchangers: Types, Design, and Applications : Types, Design, and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest

232

Index

P 

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

parallel, ix, 6, 18, 19, 20, 25, 30, 43, 46, 48, 88, 89, 90, 91, 93, 94, 109, 112, 116, 121, 126, 160, 196 permeability, 3, 7, 8, 9, 12, 133 pharmaceutical, x, 115 physical characteristics, 44 physical features, 28 physical phenomena, 216 physical properties, 147, 148, 152, 153, 160, 165, 167, 171, 178, 180 pitch, xi, 121, 131, 166, 175, 178, 179, 193, 196, 204, 209, 211, 222 plants, ix pollutants, 116 polymer, 5, 6 population, 169, 170, 171, 188, 190, 191 population size, 188 porosity, 2, 5, 6, 7, 8, 9, 12, 13, 14, 15, 18, 19, 20, 21, 22, 23, 24, 26, 28, 30, 31, 35, 36 porous materials, ix, 1 porous media, 2, 6, 7, 9, 12 power generation, x, 115, 117, 143 power plants, ix, 200 programming, 178, 201, 209, 210, 214 project, 218 propylene, 117 prototype, 145 publishing, 198 pumps, 117, 121, 122, 123, 125, 143, 144, 178, 185, 186, 187, 194, 195

Q  quadratic programming, 214 quartz, 10

R  radial distribution, 32 radiation, 220 recommendations, v, 204 reconciliation, 171 recovery, xi, 117, 119, 153, 168, 171, 215, 216, 217, 218, 226, 227 rejection, 122 relaxation, 171 reliability, 37, 209 renewable energy, 144 requirements, 116, 117, 121, 124, 167 researchers, 5, 210

resistance, 14, 20, 22, 24, 29, 30, 34, 35, 36, 196 restrictions, 31, 155, 169 risk, 124 root, 7 roots, 54, 58 rubber, 117

S  safety, 225, 226 saturation, 127, 218 scaling, 189 sensitivity, xi, 199, 205, 206, 207, 210, 211, 212, 213, 224 shape, 3, 8, 10, 156 shear, 9, 161 shores, 117 showing, 11, 139 simulation, 111, 126, 141, 142, 145, 188 sintering, 5 solar collectors, 125 solid phase, 3, 14 solidification, 5 solubility, 124 solution, 3, 54, 58, 63, 81, 91, 94, 105, 123, 124, 125, 126, 127, 129, 130, 133, 134, 135, 136, 137, 138, 139, 140, 143, 144, 152, 178, 180, 188, 190, 194, 195 South Africa, 1 species, 219 specific heat, 12, 109, 110, 111, 143, 195, 201, 222 specifications, 166 state, 10, 43, 111, 126, 129, 135, 145, 228 states, ix, 9, 43, 185 steel, 5, 6, 165 strategy use, ix, 1, 3, 180 stratification, 127 stress, 161 structure, 5, 15, 22, 155, 169, 171 substitution, 122, 151, 184 substrate, 4, 5 sulfuric acid, 218, 219, 224 surface area, 2, 5, 10, 12, 13, 14, 22, 31, 34, 39, 44, 45, 47, 49, 76, 91, 93, 94, 96, 97, 100, 103, 104, 105, 107, 108, 109, 110, 111, 149, 150, 151, 153, 154, 156, 158, 160, 161, 171, 172, 173, 195, 203, 220 survival, 180 synthesis, 197, 198, 214

Heat Exchangers: Types, Design, and Applications : Types, Design, and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest

Index

T  target, 17, 18, 19, 20, 28, 31, 33, 178 techniques, 2, 5, 10, 30, 177, 178, 180, 216 technology, 167, 171 test data, 16 testing, 203 textiles, 16 TFE, 126 thermal energy, x, 115 thermal resistance, 2, 10, 29, 34, 39, 220 thermodynamic cycle, 122 thermodynamics, 203 tics, 226 titanium, 117 topology, 10, 12, 16, 209 toxicity, 124 trajectory, 121 transfer performance, 3, 10, 14, 15, 21, 28, 31 transmission, 227 transport, ix, 1, 2, 3 transportation, 186 turbulence, 119, 154, 216

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

U  UK, 41, 42, 173 uniform, 10, 22, 81, 126, 189, 221, 228 unit cost, 178, 203 USA, 144, 145, 199 UV, 197

V 

233

vapor, 122, 123, 124, 126, 127, 131, 133, 143, 144 variables, x, 80, 91, 95, 147, 162, 175, 176, 178, 179, 180, 185, 188, 189, 191, 192, 194, 197, 208, 209, 210 variations, 111, 200 vector, 12, 28, 32, 179, 197 velocity, 7, 9, 10, 12, 13, 14, 20, 22, 24, 26, 28, 30, 32, 120, 148, 157, 158, 172, 181, 196, 197, 201, 208 versatility, 153 vibration, 185 viscosity, 5, 12, 39, 133, 144, 178, 197, 201, 211, 213 visualization, 201

W  wall temperature, 197 Washington, 174, 212 waste, 117, 122, 125, 227 waste heat, 117, 122, 125 water, ix, xi, 2, 3, 5, 6, 12, 15, 40, 91, 92, 93, 94, 96, 97, 100, 104, 109, 110, 111, 124, 125, 126, 127, 129, 130, 131, 134, 137, 139, 143, 144, 145, 175, 180, 185, 186, 187, 191, 192, 194, 195, 209, 217, 218, 219, 222, 224, 225, 226, 228 water absorption, 145 water evaporation, 139 windows, 202, 211

Y  yield, 2, 152, 179, 185, 189, 191

valve, 122

Heat Exchangers: Types, Design, and Applications : Types, Design, and Applications, Nova Science Publishers, Incorporated, 2010. ProQuest