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FLUID TRANSPORT: THEORY, DYNAMICS AND APPLICATIONS
EMMA T. BERG Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.
EDITOR
Nova Science Publishers, Inc. New York
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LIBRARY OF CONGRESS CATALOGINGINPUBLICATION DATA Fluid transport : theory, dynamics and applications / editors, Emma T. Berg. p. cm. Includes index. ISBN: H%RRN 1. Fluid dynamics. 2. Transport theory. I. Berg, Emma T. QC151.F635 2010 532dc22 2010041307
Published by Nova Science Publishers, Inc. + New York Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
CONTENTS Preface Chapter 1
Fluidodynamics Characteristics of a Vertical GasSolid and LiquidSolid Flow Radmila GarićGrulović and Željko Grbavčić
Chapter 2
Numerical Simulation on Flows Past Porous Bluff Bodies Peng Yu, Yan Zeng, Thong See Lee and Hong Tong Low
Chapter 3
Fluid Flow and Heat Transport: Theory, Numerical Modeling and Applications for the Formation of Mineral Deposits Jianwen Yang
Chapter 4
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vii
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Simulation of Velocity and Mass Transport Profiles in a Laboratory Electrolyser using Computational Fluid Dynamics Leticia Vázquez Gutiérrez, Alberto Alvarez Gallegos and Fernando Z. Sierra Computational Simulation of Instability Phenomena Associated with Mass and Energy Transport through Fluid Flow in Porous Media Chongbin Zhao The Dynamics of NMR – Diffusion Differential Equation for Qualitative Analysis of Hemodynamic and Metabolic Changes in Biological Tissue O. B. Awojoyogbe and M. Dada
1 45
75
121
157
183
Compressibility Effects on Peristaltic Flow of a NonNewtonian Maxwell Fluid through an Annulus Kh. S. Mekheimer and A. N. AbdelWahab
219
Microscale and Nanoscale Thermal and Fluid Transport Phenomena: Rapidly Developing Research Fields Lixin Cheng
237
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vi Chapter 9
Contents Transport Control of Fluid and Solutes in Microchannels Using AC Field and Semiconductor Diodes Dimiter N. Petsev and Orlin D. Velev
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Index
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PREFACE Applications of microscale and nanoscale thermal and fluid transport phenomena involved in traditional industries and highly specialized fields such as bioengineering, microfabricated fluidic systems, microelectronics, aerospace technology, micro heat pipes, and chips cooling, have become especially important since the late 20th century. This book presents topical research data in the study of fluid transport, including vertical twophase gassolid and liquidsolid flow in chemical, biochemical and mechanical processes; computational fluid dynamics; measuring diffusion of water protons and specific chemicals encountered in MRI methods; and microscale and nanoscale thermal and fluid transport phenomena. Chapter 1  The research of vertical twophase gassolid and liquidsolid flow is generally important in chemical, biochemical and mechanical processes. The modeling of a vertical gassolid and liquidsolid flow, where the solid is coarse spherical particles, is very important for practical applications such as spouted beds and modified spoutfluidized beds with draft tube. For modeling of these systems twophase flow equations must be used. For vertical nonaccelerating fluidsolids flow of coarse spherical particles the onedimensional steadystate model has been presented. The theoretical bases of the model are the continuity and momentum equations for the fluid and particle of Nakamura and Capes, and the variational model for calculating the fluidparticle interphase drag coefficient. The main model objective is establishing relations for: the fluidparticle interphase drag coefficient, the fluidwall friction coefficient and the particlewall friction coefficient. A new method for the indirect determination of the particlewall friction coefficient in the vertical pneumatic and hydraulic transport of coarse particles is presented. The proposed procedure simplifies experimental work since it does not require experimental determination of the voidage in the transport system. In vertical gassolid flow two major flow regimes are exist. There is dilute flow with an apparently uniform distribution of solid in the flowing mixture and the dense phase flow. In vertical liquidsolid flow two different flow regimes are identified: "turbulent" and "parallel". The method for predicting the regime transition for the gassolid flow is the choking criterion proposed by Day et al. Besides that, choking criterion for vertical gassolid flow could be used for predict of the regime transition in liquidsolid flow. Chapter 2  Flow past bluff porous bodies has attracted relatively less attention so far although such flow occurs widely in industries and everyday life. In the present study, computational fluid dynamics is applied to investigate this type of problems. The coupling flow in both porous medium and homogenous fluid regions is solved by a finite volume
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viii
Emma T. Berg
method based on the bodyfitted and multiblock grids. At the porousfluid interface, a shear stress jump condition that includes both the viscous and inertial effects is imposed, together with a continuity of normal stress. The steady flows are simulated in a wide range of the Reynolds and Darcy numbers. Three most commonly used shapes of the bluff body in the literature, that is, circular cylinder, square cylinder and sphere, are examined. Several interesting flow phenomena, which are different from those of flow past a solid bluff body, are revealed by the present simulations. It is found that the recirculating wake existing downstream of a porous bluff body may be completely detached from the body in a certain range of parameters. In a certain range of Darcy number, the wake may initially increase in size with an increase in Reynolds number but then decrease in size and eventually disappear when the Reynolds number is sufficiently large. There is only one ―separation‖ point along the horizontal axis of the porous bluff body, but not a pair of separation points on the surface of the solid one. The present findings may provide a starting point for reevaluating the flow around and through a porous bluff body and stimulate new studies to understand the underlying mechanics of the new flow phenomena which have not been answered here. Chapter 3  Subsurface fluid flow and heat transport play an important role in many geological processes, such as hydrocarbon migration, structural geology, plate tectonics, diagenesis, and metamorphism, and in particular they have major implications for both the formation and preservation of almost every class of economic ore deposit, especially those in sedimentary basins. This chapter is dedicated to addressing fundamental theory and numerical modeling technique of fluid flow and heat transport in subsurface porous media as well as presenting two application examples associated with ore genesis under different geological conditions. The chapter is organized into four sections. The first section is aimed at theoretical aspect of fluid flow and heat transport. It first introduces major physical processes involved in a hydrothermal flow system and outlines a variety of driving mechanisms deemed responsible for largescale groundwater flow (i.e., topography, buoyancy, tectonic deformation, and sediment compaction), and then describes the mathematical equations that govern and control the behavior of subsurface fluid migration and thermal regime. It ends up with deriving an analytical solution using the Laplace transformation to address the heat transport process subject to a constant fluid flow in a single fracture embedded in an impermeable host porous medium. The second section concentrates on the numerical modeling aspect of fluid flow and heat transport by detailing the Galerkin finite element technique that is capable of numerically simulating complex hydrothermal flow systems. Noticing that topography and buoyancy are the two top driving forces commonly encountered in reality and that the interactions between them are still poorly understood, the third section is therefore attempts to quantify the relative importance of these two mechanisms in driving fluid transport via a series of numerical experiments, and also to determine under what conditions both topography and buoyancydriven flows coexist and under what conditions one flow system dominates the other. The fourth section of this chapter focuses on the application aspect of fluid flow and heat transport associated with the formation of mineral deposits, including the Sedextype leadzinc deposits in northern Australia (with buoyancy as the primary driving force), and the Dachang polymetallic deposits in southern China (with tectonic deformation as the primary driving force). Chapter 4  Parallel plate reactors are prevalent both at laboratory and industrial scales. They are easy to construct and are versatile when choosing separators and electrodes as well as in the mode of electrical operation and flow manifolds. A successful scaleup procedure
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Preface
ix
includes several steps: the analysis of a variety of dimensionless groups which describe the geometric, kinematic, thermal, chemical and electrical characteristics. Accordingly, mass transport and fluid dynamics are among the main parameters in a reliable scaleup procedure. The hydrodynamic behaviour of a parallel plate reactor can be described the flow through a rectangular channel where the flow pattern is strongly linked to the rate of mass transport (km), according to the following equation. Chapter 5  Most mass and energy transport systems through fluid flow in porous media can be mathematically treated as coupled nonlinear multipleprocess problems, for which instabilities are common emerging phenomena that control the fundamental behaviours of such coupled nonlinear systems. This chapter deals with the computational simulation aspects of two common instability phenomena, namely thermodynamic instability and chemicaldissolution front instability, which are closely associated with energy (through heat transfer) and reactive mass (through chemical reaction) transport in fluidsaturated porous media. The physical appearance of thermodynamic instability displays actually a kind of convective porefluid flow in a fluidsaturated porous medium so that thermodynamic instability is also called the convective instability of porefluid flow. This kind of convective porefluid flow can play an important role in transporting heat energy and aqueous minerals in the Earth‘s crust that is often comprised of porous rocks. However, the physical appearance of chemicaldissolution front instability shows a fundamental change in the morphology of an initial chemicaldissolution front. This kind of instability is the direct consequence of an interaction between nonlinear porefluid flow and reactive mass transport through nonlinear porositypermeability feedback effects in the fluidsaturated porous medium. After the mathematical formulations for simulating both thermodynamic instability and chemicaldissolution front instability are described, computational simulation methods, such as the finite element method and the finite difference method, are used to solve the coupled nonlinear problems associated with energy and mass transport processes in fluidsaturated porous media. Chapter 6  During the past decade, major breakthroughs in magnetic resonance imaging (MRI) quality were made by means of great improvement in scanner hardware and pulse sequences. Some advanced MRI techniques have truly revolutionized the detection of disease states and MRI can nowwithin a few minutesacquire important quantitative information noninvasively from an individual in any plane or volume at comparatively high resolution. However, the very basic physics of this promising technological breakthrough is not well understood. Parameters that are measured from time to time in advanced MRI seem to be logically and functionally related but the theoretical facility to optimally explore them is still missing. In a single experimental investigation, for example, few of huge amount of information available are effectively used. This study intends to provide a very straightforward theoretical background for measuring diffusion of water protons and specific chemicals encountered in most common advanced MRI methods including diffusion MRI, perfusion MRI, functional MRI. Chapter 7 – It is accepted now that majority of the biological and industrial fluids are nonNewtonian. Unlike the Newtonian fluids, the nonNewtonian fluids cannot be described by a single constitutive relationship between stress and strain rate. Such constitutive equations give rise to complicated mathematical problems and thus, the mathematicians, modelers, physicists and computer scientists encounter wide variaty of challenges in constructing analytical and numerical solutions. Generally, the classification of nonNewtonian fluids is based into three categories, namely, the differential type, the rate type and an integral type.
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Maxwell model is the simplest subclass of rate type fluids. The objective of the present work is to analyze the effect of viscoelasticity on the dynamics of a nonNewtonian fluid by studying the peristaltic flow of a compressible Maxwell fluid through the gab between two coaxial tubes (annulus). This peristaltic flow is actually a result of the influence of ultrasonic radiation on the flow of a liquid through an annulus which deforms the walls of the outer tube in the shape of traveling transversal waves exactly like peristaltic pumping. Those traveling transversal waves induce a net flow of the liquid inside the annulus. This problem has numerous applications in various branches of science, including stimulation of fluid flow in annulus under the effect of elastic waves , the production process of oil, and studies of blood flow dynamics in living creatures(catheter in an artery). NavierStokes equations for an annulus are solved by means of a perturbation analysis, in which the ratio of the wave amplitude to the outer tube radius is a small parameter. In the second order approximation, a net flow induced by the travelling wave is calculated for various values of the Reynolds number, Maxwell relaxation time, the compressibility parameter and the annulus radius ratio. The calculation disclose that the compressibility of the liquid and the nonNewtonian effects in presence of peristaltic transport have a strong influence on the net flow induced. Variation of emerging parameters embedded in the flow system are discussed numerically and graphicly. Chapter 8  With the emergence of microscale and nanoscale thermal, fluidic and chemical systems, MEMS (MicroElectroMechanicalSystems), NEMS (NanoElectroMechanicalSystems), thermal management technologies for microelectronics, power electronics, aerospace and defence technologies etc., the development of ultracompact heat exchangers, miniature and micro pumps, miniature compressors, microturbines, micro thermal systems for distributed power production, microfluidic and nanofludic device, labonachip, microscale and nanoscale energy systems, nanofluid heat transfer technologies and others has become an important agenda of many researchers, research institutions and funding agencies. Applications of microscale and nanoscale thermal and fluid transport phenomena involved in traditional industries and highly specialized fields such as bioengineering, microfabricated fluidic systems, microelectronics, aerospace technology, micro heat pipes, chips cooling etc. have been becoming especially important since the late 20th century. However, microscale and nanoscale thermal and fluid transport phenomena are quite different from those at conventional scale or macroscale. For example, gas liquid twophase flow and flow boiling heat transfer characteristics in microcale channels are quite different from those in macroscale channels. Channel confinement also has a great effect on twophase flow and flow boiling heat transfer characteristics. Furthermore, the available experimental results of single phase heat transfer and fluid flow in microscale channels are quite contradictory from one study to another. Studies of supercritical fluid flow and heat transfer in microscale channels have also exhibited contradictory results by various researchers. Just to show several examples here. Therefore, there are many aspects to be clarified from both theoretical and applied aspects in microscale and nanoscale thermal and fluid transport phenomena. Furthermore, new research areas and technologies in the relevant fields have been emerging from time to time. For example, as a new research frontier of nanotechnology, the research of nanofluid twophase flow and thermal physics is growing rapidly, however it has also posed new challenges as there are quite contradictory results in the available research. For another example, advances in microelectronics technology continue to develop with surprisingly rapidity and the thermal energy density of electronic devices to be dissipated is becoming
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much higher and higher. Therefore, it is essential to develop new high heat flux cooling technology to meet the challenging heat dissipation requirements. All the newly emerging research areas require the understanding of fundamentals and applications of the microscale and nanoscale fluid and thermal transport phenomena. Chapter 9  This chapter presents an overview of recent results on the electric field control and manipulation of fluids in microfluidic devices. The newer approaches are based on using alternating or a combination of alternating and direct current fields. The alternating field can be locally converted to direct by semiconductor diodes that may be placed at key locations where an electroosmotic force has to be applied to the fluid. Such techniques allow to design and fabricate small micrometer sized pumps and mixers. The latter are important because of the inherent low Reynolds characteristics of the flow in microchannels. The diode mixers are simple to fabricate and can be turned on and off depending on the operational requirements. Combining alternate and direct current fields and diode pumps makes possible the decoupling of the electroosmotic fluid flow from the electrophoretic particle or macromolecular mass flux. This can be exploited for precise analyte focusing, preconcentration and separation.
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In: Fluid Transport: Theory, Dynamics and Applications ISBN: 9781611223170 Editor: Emma T. Berg, pp. 143 © 2011 Nova Science Publishers, Inc.
Chapter 1
FLUIDODYNAMICS CHARACTERISTICS OF A VERTICAL GASSOLID AND LIQUIDSOLID FLOW Radmila GarićGrulović1 and Željko Grbavčić2 1
Institute for Chemistry, Technology and Metallurgy, University of Belgrade, Njegoševa 12, 11000 Belgrade, Serbia 2 Faculty of Technology and Metallurgy, Department of Chemical Engineering, University of Belgrade, Karnegijeva 4, 11000 Belgrade, Serbia
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ABSTRACT The research of vertical twophase gassolid and liquidsolid flow is generally important in chemical, biochemical and mechanical processes. The modeling of a vertical gassolid and liquidsolid flow, where the solid is coarse spherical particles, is very important for practical applications such as spouted beds and modified spoutfluidized beds with draft tube. For modeling of these systems twophase flow equations must be used. For vertical nonaccelerating fluidsolids flow of coarse spherical particles the onedimensional steadystate model has been presented. The theoretical bases of the model are the continuity and momentum equations for the fluid and particle of Nakamura and Capes [1], and the variational model for calculating the fluidparticle interphase drag coefficient (Grbavčić et al. [2]). The main model objective is establishing relations for: the fluidparticle interphase drag coefficient, the fluidwall friction coefficient and the particlewall friction coefficient. A new method for the indirect determination of the particlewall friction coefficient in the vertical pneumatic and hydraulic transport of coarse particles is presented. The proposed procedure simplifies experimental work since it does not require experimental determination of the voidage in the transport system. In vertical gassolid flow two major flow regimes are exist. There is dilute flow with an apparently uniform distribution of solid in the flowing mixture and the dense phase flow. In vertical liquidsolid flow two different flow regimes are identified: "turbulent" and "parallel". The method for predicting the regime transition for the gassolid flow is the choking criterion proposed by Day et al. [3]. Besides that, choking criterion for vertical gassolid
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2
Radmila GarićGrulović and Željko Grbavčić flow could be used for predict of the regime transition in liquidsolid flow (Grbavčić et al. [4], GarićGrulović et al. [57]).
Keywords: vertical twophase flow, modeling, flow regimes
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1. INTRODUCTION Twophase flow is important in number of industries application. For example, accurate twophase pressure drop models are necessary to design the piping systems in the oil and gas industry. Also, twophase flow occurs in heat exchangers, steam generators, chemical reactors, oil transportation and many other processes. Accurate prediction of particle velocity, voidage and pressure gradient is critical for design of any transport line. The design of any other fluidsolids contacting systems which contains twophase vertical fluidsolids flow also require such knowledge (for example spouted and spoutfluid beds with draft tubes, circulating fluidized beds, etc.). In literature could be found four main fundamental equations for description of two phase vertical fluidsolids flow (Arastoopour and Gidaspow [8], Leung [9], Kmiec and Leshonski [10], Grbavčić et al. [2], Grbavčić et al. [4, 11], Littman et al. [12], Garić et al. [13, 14], GarićGrulović et al. [15], Grbavčić et al. [16] and other). All of mentioned investigators discuse the solids momentum equation and problems in obtaining good correlations for the drag coefficient and the particlewall friction factor. From different investigations of fluidodynamics parameters of twophase fluidsolid flows could be concluded that all these systems shows similar behavior which is characteristic for vertical two phase flow. In the design of equipment where vertical transport of solids occurs, it is necessary to be able to predict the relationship between the fluid and particle phase velocities, voidage and pressure drop. The modeling of nonaccelerating fluidsolids flow is crucial for practical applications of draft tube spouted and spoutfluid beds. The draft tube in these systems is essentially a pneumatic or hydraulic riser which is relatively short even in the industrial scale units, so that nonaccelerating twophase flow equations must be used to model the hydrodynamics of such systems. Note that, knowledge of the flow regime in both the pneumatic and hydraulic transport is necessity for development of the model for vertical twophase flow.
2. THEORETICAL BACKGROUND 2.1. Background of Hydrodynamic Models A hydrodynamic approach to twophase flow (fluidsolid), such as fluidization, was started by Davidson in 1961 [17]. He analyzed single bubble motion in an infinite fluid bed using two continuity equations and an expression for relative velocities in terms of Darcy's law for flow in porous media. Davidson [17] assumed that the solids flow around a bubble was irrotational. This assumption can be justified by the mixture momentum equation
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Fluidodynamics Characteristics of a Vertical GasSolid …
3
(Gidaspow and Solbrig [18]). It can also be shown that the use of Darcy's law and the mixture momentum equation is in the limit equivalent to the use of two separate phase momentum balances (Gidaspow and Solbrig [18]). A year after Davidson‘s solution, Jackson [19] formulated the more general equations of motion. Later Jackson [20] reviewed his work. In the last years interest in hydrodynamic modeling was renewed due to the energy crisis (Gidaspow and Solbrig [18]). When an attempt was made to solve the hydrodynamic models similar to those presented by Jackson numerically on highspeed computers, it was found that the differential equations are illposed as an initial value problem (Gidaspow [21], Lyczkowski et al. [22]). According to a theorem in the literature (Lax, [23]), it is impossible to find a numerically stable finite difference technique to solve such a set of equations. Several solutions to this dilemma are possible. One approach is to use a set of equations which neglects interaction of inertia between the gas and the particles. One simply writes Newton‘s second law of motion for particles. Such a set of equations had been used by aerodynamicists. It is not illposed. The second approach is to derive an equation of relative motion using the principles of nonequilibrium thermodynamics. This approach was described by Gidaspow [24]. The third approach is to modify Jackson‘s equations by dropping and adding proper terms so as to remedy the illposedness problem. This was done by a Systems, Science and Software group in a paper presented in December 1976 (Pritchett et al. [25]). For steadystate problems mathematical difficulties do not arise. In literature which is focused on the twephase flow a number of problems were solved using two continuity and two momentum equations. One of the earliest and best described problems is the flow through a nozzle (Soo [26]). For gassolids flow (vertical pneumatic transport), Nakamura and Capes [1] considered a similar approach. Vertical pneumatic conveying of solids of a reasonably uniform size can be described by means of onedimensional, steadystate momentum balances. Also, the assumption of a steadystate discards any fluctuations, such as those that occur in slug flow. The four differential equations can be written: two continuity equations with no phase changes, the mixture momentum equation, and the fourth equation which differs from model to model. The common equations are as follows:
Fluid continuity: d d G u f f dz dz At
(1)
d d G v 1 p p dz dz At
(2)
Solid continuity:
Mixture momentum: The onedimensional mixture momentum equation, with constant fluid properties including frictional effects between the bed and the wall, is (Lefroy and Davidson [27], Nakamura and Capes [1], Grbavčić et al. [4]):
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4
f
d d dp ( u 2 )+ p (1 )v 2 =  pg(1 ) f g  F dz dz dz
(3)
In equation (3) F is the frictional force between the mixture fluidsolids and the wall.
F
4 Dt
(4)
The onedimensional momentum equations for fluid and particles in vertical fluidparticle flow are (Lefroy and Davidson [27], Nakamura and Capes [1], Grbavčić et al. [4]):
for fluid:
f
p
d 2 dp ( u 2 ) =   u v f g  F f dz dz
(5)
for solid (particle):
d 2 dp (1 )v 2 = (1 )  u v  pg(1 )  F p dz dz
(6)
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where Ff, Fp and β are pressure losses due to fluidwall, particlewall friction and the fluidparticle interphase drag coefficient, respectively. In equations (3), (5) and (6), p is average local fluid pressure. By introduction dynamic pressure as:
P = p  f gz
(7)
where z the distance from any reference plane, derive is:

dp dP fg dz dz
(8)
We want to determine fluid and solid velocity, voidage and pressure drop. Therefore, besides eqns. (1), (2) and (3) we need one more equation. The four different momentum balances are used in the literature:
case A, pressure drop in both solid and fluid phases (annular flow model, Nakamura and Capes [1]), from equations (6) and (8),
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Fluidodynamics Characteristics of a Vertical GasSolid …
dv u v dP F pv p f g p dz 1 dz 1
5
2
case B, pressure drop in fluid phase only (Soo [26]),
pv
dv = F d  pg dz
(10)
case C, relative velocity (Gidaspow [24]),
1 d 2 p u v Fd p g 2 dz
(9)
(11)
case D, partial pressure drop in both phases (Deich et al. [28]),
pv
dv P d dP Fd p g dz dz 1 dz
(12)
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In the above equations (10), (11) and (12) Fd is hydrodynamic drag force per unit volume of particles, it can be expressed as showed by both Arastoopour and Gidaspow [8] and Kmiec and Leshonski [10]. In their calculations Fd is determinated as:
f 3 2 2.65 F d = C Ds (u  v ) 4 dp
(13)
Leung, 1980th [9] pointed out those only two continuity equations (i.e. eqns.1 and 2) and mixture momentum equation (eqn. 3), which are generally accepted by most investigators, while forth equation is choice of researchers. Four forms of momentum equations were proposed (Nakamura and Capes [1], Soo [29], Gidaspow and Solbrig [18] and Deich et al. [28]), as reviewed by Arastoopour and Gidaspow [8], Kmiec and Leshonski [10], and recently by Louge et al. [30] and Littman et al. [12]. All of these forms consider the particle phase to be a dilute system of colliding particles and proposing essentially a twodimensional model describing particle phase movement. In this study are taken into account only one dimensional models, the equation of Louge et al. [30] is excluded from the analysis. The model of Deich et al. [28] is also excluded since Arastoopour and Gidaspow [8] found that it incorrectly predicts increasing solids concentration with distance from the tube inlet. The analysis of the equations for momentum balance which are proposed by Soo [26] equation (10), Gidaspow [24] equation (11) and Deich et al. [28] equation (12), points that in these equations the friction force between particle and wall of transport tube is neglected. Since equation (9) of Nakamura and Capes [1] and Grbavčić et al. [4] does not neglect this friction, therefore this equation will be used as the most appropriate to define the model.
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Radmila GarićGrulović and Željko Grbavčić
The general behavior of two phase vertical fluidsolids flow depends primarily on fluid phase. Note that, when the fluid phase is known the knowledge of the flow regime in the relevant twophase flow is needed also. In vertical gassolids flow a wide variety of flow regimes are possible, depending on particle and fluid characteristic and tube diameter (Chong and Leung [31]). Two major flow regimes can be identified, i.e. the dilute phase flow and the dense phase flow regime. At high gas velocity the particles are carried up trough the transport tube as an apparently diluted uniform suspension with low volumetric concentration. If the gas velocity is gradually reduced at the same particle mass flowrate, the solids concentration in the tube increases. A border point will be reached when the gas velocity is not sufficient to support the stable dilute flow and a transition to the dense flow appears. Corresponding gas velocity is choking velocity. With further decrease in the gas velocity a wide variety of flow regimes are possible depending on particle and fluid characteristics and tube diameter. Below a certain gas velocity the entire suspension collapses and the solids in the tube form fluidized and finally packed bed (Garić et al. [13, 14], Grbavčić et al. [16]). In vertical fluidsolids flow, if a liquid is used as a fluid then there is continuous transition between very dense flow to the very dilute flow, since voidage in the transport tube can be from values close to minimum fluidizing voidage up to the 1 (Grbavčić et al. [4], Garić et al. [14]).
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2.2. A Study of the Models Parameters The general problem in applying the onedimensional, steady state continuity and momentum equations to any fluidparticle system is that the four equations contain four dependent variables (fluid and solid velocity, voidage and pressure drop), and a two parameter, β and F (fluidparticle interphase drag coefficient and pressure gradient due to the mixturewall friction).
2.2.1. Fluidparticle interphase drag coefficient For a proper use of the onedimensional momentum equations for fluid and particle in vertical fluidparticle flow (Lefroy and Davidson [27], Nakamura and Capes [1], Grbavčić et al. [4]) defined by equations (5) and (6), the fluidparticle interphase drag coefficient, should be known. For the spout of the spouted bed Lefroy and Davidson [27] proposed two relationships. The first relationship is based on the Richardson and Zaki [32] correlation:
d p / f 0.33 1 / 1.78
(14)
while the second uses only the second term of the Ergun [33] equation:
d p / f 1.75 1
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Fluidodynamics Characteristics of a Vertical GasSolid …
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Morgan et al. [34] and Day et al. [35] determined the drag coefficient for 2.76 and 5.0 mm glass particles flowing in the spout of an airspouted bed by a new technique. They formulated the axial voidage distribution at minimum spouting as an isoperimetric problem in calculus of variations and used that distribution to solve onedimensional, steadystate continuity and momentum equations of Lefroy and Davidson [27], to obtain the drag coefficient as a function of voidage. The results were shown to be in good agreement with experiments after corrections were made for the effect of the spout inlet region (Day et al. [35]). This could be well described by following equation:
d p / f 0.33 1
(16)
For calculations of the fluidparticle interphase drag coefficient Grbavčić et al. [2] variational model is proposed. This model is described and verified for particulate fluidization. The drag relationship is obtained by solving the appropriate EulerLagrange differential equation, satisfying three boundary conditions at minimum fluidization and thermal velocity. The final relationship for β applying this model is:
1  mF = 1 C 2 + 1 +C1 1 mF mF
2 1/ 2
(17)
where the constants C1, C2 and are:
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2 2 C 1 = 1+ U mF
C2=
1/ 2
1 1 C 12
= 1 C 12  C 1
(18)
(19)
(20)
In equation (17), βmF is fluidparticle interphase drag coefficient in a particulately fluidized bed at minimum fluidization, i.e.:
mF =
3mF(1 mF )g( p  f ) 2
(21)
U mF
The slip velocity between the fluid and particles in a particulately fluidized bed is
usF
UF
U mF
3 (1 ) mF 3 mF (1 mF )
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(22)
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Radmila GarićGrulović and Željko Grbavčić
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Figure 1. Comparison between experimental and variational model fluidparticle interphase drag coefficient in particulate fluidization and hydraulic transport (Grbavčić et al. [4]).
Figure 2. Comparison between experimental and variational fluidparticle interphase drag coefficients in pneumatic transport (Garić et al. [13]). Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
Fluidodynamics Characteristics of a Vertical GasSolid …
9
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Both the drag coefficient and velocityvoidage relationship were predicted very well in the particulately fluidized bed for voidages varying from minimum fluidization up to ε=0.96 (Grbavčić et al. [4]). Experimental verification of the variational formulation of the drag coefficient for hydraulic transport was done by Grbavčić et al. [4]. Remarkably, as seen in Figure 1, the drag coefficient for hydraulic transport and particulate fluidization follow the same relationship (eqn. 17). The mean deviation between all of the data and equation (17) is 18.2%. Also, the variational model (eqn. 17), is a tested by Garić et al. [13] in vertical gassolids flow of coarse spherical particles. Initial assumption was that the variational model would work in the dilute regime only. However, the data in Figure 2 show that the model gives good predictions for both dilute and stable slug flow regimes. The mean deviation between calculated and measured values of β for dilute flow is 11.6%. The limit between flow regimes is Gp/Gf< 15, which also corresponds to ε>0.95. Comparison between experimental values of fluidparticle interphase drag coefficient and different prediction methods is shown in Figure 3 (Garić et al. [13]). As can be seen, as it was excepted from equation (15), the drag coefficients predicted by all other equations are close to each other in the high voidage region. This value is most important for pneumatic transport.
Figure 3. Comparison between experimental values of fluidparticle interphase drag coefficient and different prediction methods in pneumatic transport (Garić et al. [13]). Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
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Radmila GarićGrulović and Željko Grbavčić
2.2.2. Fluidwall and particlewall friction The onedimensional momentum equations for fluid and particle in vertical fluidparticle flow (Lefroy and Davidson [27], Nakamura and Capes [1], Grbavčić et al. [4]) are defined by equations (5) and (6). In this equations Ff, and Fp are pressure gradient due to fluidwall and particlewall friction, respectively. The friction term Ff, due to conveying fluid alone is usually defined following the Fanning equation as:
F f =2 f f f
U2 Dt
(23)
The fluid friction coefficient for smooth transport tubes can be accurately predicted using the standard friction factor correlation (Bird et al. [36]),
ff
0.0791 Re 0.25
(24)
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However, the friction due to solid particles Fp is defined differently by many authors. It is important distinguish these differences whenever the solid frictional loss discussed. Stemerding [37], Konno and Saito [38], Capes and Nakamura [39] and Yousfi and Gau [40], defined the solid friction factor following the Fanning equation based on solid particle velocity and the dispersed solid density (Yang [41]):
F p= 2 f p p
1 v 2 Dt
(25)
Table 1 presents a review of the most commonly used correlations in vertical two phase vertical fluidsolids flow calculations. Note that in the literature there are numerous other correlations, based somewhat on different definitions of Fp. Using equation (26), Stemerding [37] and Yousfi and Gau [40] found for glass particles that solidwall friction coefficient, fp, is constant value (Yang, [41]). Several other correlations for fp for airparticle systems shown in Table 1 propose that fp is an inverse function of the solid velocity, but also propose different exponents on the velocity term. Yang [42] assumed that pneumatic conveying might behave similarly to moving beds if slip velocities were used in place of fluid velocities. A unique correlation relating a modified solidswall friction coefficient and a modified Reynolds number was generated as it shown in equation (31). Separation of Capes and Nakamura [39] data into two groups at U/Ut = 1.5, Yang [41] is recommended if the equation (33) will be used to calculate the solidswall friction coefficient. In the dilute phase vertical pneumatic transport (U/Ut > 1.5) could be excepted in cases where transporting velocities are close to the terminal velocities of the solid particles. Equation (34) should be used in those conditions.
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Fluidodynamics Characteristics of a Vertical GasSolid …
11
Table 1. Correlations for solidwall friction Reference Stemerding [37], Yousfi and Gau [40]: Reddy and Pei, [43]:
f p = 0.003
(26)
f p = 0.046 v 1
(27)
Konno and Saito [38]:
f p = 0.0285 g Dt v 1
(28)
van Swaaij et al. [44]:
f p = 0.080 v 1
(29)
Capes and Nakamura [39]:
f p = 0.048 v1.22
(30)
Yang [42]:
(Re )t (1 ) (1 ) f p = 0.0206 3 4 (Re ) p
de Jong [45]:
f p = 0.0012
Yang, [41] U/Ut>1.5
(Re )t (1 ) (1 ) f p = 0.0126 4 3 (Re ) p
U/Ut1 starts to decline slightly, which is probably a consequence of particle dosage (Figure 2 in Grbavčić et al. [11]). Dosage of solid particles – spouted bed with a draft tube, is characterized by a complex relationship between fluid flow, particle flow and geometry.
4.4.2. Model and model parameters The onedimensional suspension momentum balance outside acceleration zone in vertical liquid solids flow, through the transport tube is given by equation (44) (Nakamura and Capes [1], Grbavčić et al. [4]). In this equation, overall pressure gradient (dP/dz), can be obtained only experimentally, is sum of effective weight of the suspension (ρpρf)g(1ε)=Fe, pressure
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Radmila GarićGrulović and Željko Grbavčić
gradient due to the particlewall, Fp and fluidwall friction, Ff (Figure 18). Those friction terms represent overall suspensionwall friction, F (eqn. 45). Using experimental data for – dP/dz, U and , collected by Garić [69] vertical liquid solids flow, the experimental value of F was determined by equations (44) and (45).
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Figure 17. The loading ratio of liquid flow, depending on the fluid velocity in the vertical twophase liquidsolid flow (data of Garić [69]).
Figure 18. Variation of pressure drop ratios with superficial fluid velocity, (dp=2.98 mm, Dt=30mm, data of Garić [69] and GarićGrulović [74]). Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
Fluidodynamics Characteristics of a Vertical GasSolid …
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Figure 18 gives the relative magnitude of four momentum terms in equation (56) normalized with the overall pressure gradient, for glass sphere dp=2.98 mm (Garić [69], GarićGrulović [74]). Since in this system main flow parameters change with increase in U/Ut, a few characteristic sets of parameter values are indicated on the graph. It is seen that the liquidwall friction term slightly increases with U/Ut, consuming up to about 10% of the total pressure drop. At low liquid velocity, the major portion of the pressure drop is due to the static head of particles, Fe. With an increase in liquid velocity, solids concentration decreases, resulting in a decrease in the contribution of the static head of particles to the overall pressure drop. At the same time, with the increase in liquid velocity, the particlewall friction term rapidly increases and can equal up to 50% of the total pressure drop at a U/Ut value of about four.
4.4.3. Applying the model to predict the basic fluidodynamics parameters of the vertical liquid solids flow For practical application of the model in vertical twophase liquidsolid flow algorithms I and II are the most important, and they will be further discussed and compared with available experimental data.
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4.4.3.1. Prediction solids flowrate in the transport tube Algorithm I, could be used to calculate the particles velocity, voidage and particle mass flowrate (v, ε and Gp=f(v, )) in the vertical twophase liquidparticle mixtures, when are known fluid mass flowrate and overall pressure gradient (Gf, dP/dz) as it is shown in Table 3.
Figure 19. Comparison between experimental and calculated values of the model, for the particles velocity in the hydraulic transport (data of Garić [69]). Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
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Radmila GarićGrulović and Željko Grbavčić
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Figure 20. Comparison between experimental and calculated values of the model, for the particles mass flowrate in the hydraulic transport (data of Garić [69]).
On Figure 19 are compared the available experimental values of particle velocity and values calculated from the model. It could be observed that the agreement between the experimental and calculated values is very well (mean absolute deviation for parallel and turbulent flow is = 12.9%). Based on the calculated values of particle velocity and voidage is possible to calculate the particles mass flow rate. On Figure 20 are compared the available experimental values of particle mass flow rate and calculated values from the model. Agreement between calculated and experimental values is very good, in term of the mean absolute deviation between the experimental and calculated values = 5.9%.
4.4.3.2. Prediction pressure gradient in the transport tube Algorithm II, is used to calculate the voidage, pressure gradient, as well as particles velocity (,dP/dz , v=f(Gp, )) in the vertical twophase liquidparticle mixtures, when are known fluid and particle mass flowrate (Gf, i.e. U, Gp) as it is shown in Table 3. Figure 21 shows the excellcnt agreemcnt of calculated and experimental voidage (Garić [69], Kopko et al. [75]). The mean deviation for data of Garić [69] is 0.70%, and for data of Kopko et al. [75] is 1.6%. On Figure 22 comparasion of the available experimental values of pressure gradient and calculated values is shown. It can be observed that the agreement between the experimental and calculated values of Garić [69] is very good, with mean absolute deviation of = 3.69%. The agreement between the experimental and calculated values of Kopko et al. [75] with mean absolute deviation of = 19.5%.
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Fluidodynamics Characteristics of a Vertical GasSolid …
35
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Figure 21. Comparison of measured and calculated values of voidage in the transport tube (data of Garić [69] and Kopko et al. [75]).
Figure 22. Comparison between experimental and calculated values of the model, for the pressure gradient in the hydraulic transport (data of Garić [69] and Kopko et al. [75]). Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
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Radmila GarićGrulović and Željko Grbavčić
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4.4.3.3. Indirect determination of solidswall friction coefficient In a liquidsolids transport the friction between particles and wall has been often neglected. However, Grbavčić et al. [4] show that contribution of solids wall friction in hydraulic transport is not negligible. The proposed procedure is restricted to coarse particle systems where the onedimensional model has been verified (Grbavčić et al. [4], Garić et al. [13]). The mean deviations between the experimental and calculated values were 16.3% in the hydraulic experiments (Figure 23).
Figure 23. Comparison of the experimental and calculated values of the particlewall friction coefficient (hydraulic conveying system, Garić et al. [14]).
4.5. Comparison of a Vertical GasSolid and LiquidSolid Flow Accurate prediction of the pressure losses due to the particlewall friction is a very important problem in the modeling and design of vertical transport lines since the contribution of this friction to the overall pressure drop can be very high, especially in pneumatic transport lines. Figure 24 gives the contribution of the particle wall friction to the overall pressure gradient in the pneumatic and hydraulic conveying of spherical glass particles (Garić et al. [14]). As can be seen in hydraulic transport runs particlewall friction consumes about 10% of the total pressure drop over a wide range of fluid velocities. In pneumatic transport runs, however, the particlewall friction term rapidly increases with gas velocity and can be up to 60% of the total pressure drop at a value of U/Ut of about 2. Loading ratio of gas and liquid (Gp/Gf)is parameter studied by number of researchers in their flow regimes studies of gasparticles systems (Konno and Saito [38], Yang, [41], Capes
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Fluidodynamics Characteristics of a Vertical GasSolid …
37
and Nakamura [39]). It is found that this values of loading ratio have wide range (Garić [69]), while for the liquidparticles systems these values are even 40 times lower for the same velocity ratio U/Ut (U/Ut 1.12, Figure 25).
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Figure 24. The contribution of particlewall friction to the overall pressure gradient in the pneumatic and hydraulic conveying of spherical particles (Garić et al. [14]).
Figure 25. The loading ratio of gas and liquid flow, depending on the fluid velocity in the vertical twophase fluidsolid flow (data of Garić [69]). Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
Radmila GarićGrulović and Željko Grbavčić
38
CONCLUSION
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A model describing vertical pneumatic and hydraulic transport of coarse particles is verified experimentally. The model is based on momentum equations for fluid and solids phase of Nakamura and Capes [1] and Grbavčić et al. [2] variational model for predicting the fluidparticle interphase drag coefficient and a correlation for the solids wall friction coefficient (Grbavčić et al. [4], Garić et al. [13]). In pneumatic transport the proposed model agrees quite well with the experimental data for dilute flow and solids slug flow if ε>0.95. Mean deviation in between the calculations and experimental data for solids flow rates and pressure gradients are 9.0 and 6.7 %, respectively (data of Garić [69]). In hydraulic transport the proposed model agrees quite well with the experimental data for paralelel and turbulent flow (data of Garić [69] and Kopko et al. [75]). A comparison with several literature correlations for calculating the particlewall friction coefficient shows that each correlation is system specific, supporting Leung‘s [49] view that the use of empirical correlations outside the original range of conditions is quite uncertain. Much more research should be conducted in order to distinguish purely hydrodynamic interactions from other effects usually lumped together in particlewall friction terms (electrostatic effects, interparticle collisions, etc.). The proposed procedure for the indirect determination of the particlewall friction coefficients simplifies experimental work since it does not require the experimental determination of the voidage in the transport system. Indirectly determined values of the particlewall friction coefficient agree quite well with the measured values both for the hydraulic and pneumatic transport of coarse particles.
NOMENCLATURE At C1 C2 dp Dt F Fe ff Ff fp Fp g Gf Gp p P Re
crosssectional area of the transport tube (m2) variational constant in Eq. (18) variational constant in Eq. (19) particle diameter (m) diameter of the transport tube and fluidized bed (m) pressure gradient due to the suspensionwall friction (Pa/m) pressure gradient due to the effective weight of particles (Pa/m) fluidwall friction coefficient pressure gradient due to the fluidwall friction (Pa/m) particlewall friction coefficient pressure gradient due to the particlewall friction (Pa/m) gravitational acceleration, m/s2 fluid mass flowrate in the transport tube (kg/s) particle mass flowrate in the transport tube (kg/s) static fluid pressure in transport tube, Pa dynamic pressure in transport tube, Pa Reynolds number, =DtfU/μ
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Fluidodynamics Characteristics of a Vertical GasSolid … (Re)p (Re)t u U UF UmF Ut us usF v
39
slip Reynolds number (= dp (uv)f/) Reynolds number at terminal velocity (= dpUtf/) mean interstitial fluid velocity in the transport tube, =U/ε (m/s) superficial fluid velocity in the transport tube (m/s) superficial fluid velocity in a particulately fluidzed bed at a voidage , m/s superficial fluid velocity at minimum fluidization, m/s particle terminal velocity, m/s slip velocity between fluid and particles, (uv), m/s slip velocity between fluid and particles in a particulately fluidized bed at the same voidage as in the transport tube, UF/, m/s particle velocity in the transport tube, m/s
Greek Letters
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mF * mF f p
fluidparticle interphase drag coefficient, kg/m4 fluidparticle interphase drag coefficient in a particulately fluidized bed at minimum fluidization, kg/m4 criterion for regime transition from "turbulent" to the "parallel" percent relative deviation voidage in the transport tube bed voidage at minimum fluidization variational constant in Eqn. (20), Lagrange multiplier viscosity of the fluid (Ns/m2) fluid density (kg/m3) particle density (kg/m3) mathematical constant (=3.14)
ACKNOWLEDGMENTS Acknowledgment Financial support of the Research Council of Serbia is gratefully acknowledged.
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Nakamura, K. & Capes, C. E. (1973). Vertical Pneumatic Conveying: A Theoretical Study of Uniform and Annular Flow Models, Can. J. Chem. Engng., 51, 3946. Grbavčić, B. Ž., Garić, V. R., Hadžismajlović, E. Dž., Jovanović, Dj. S., Vuković, V. D., Littman, H. & Morgan III, H. M. (1991). Variational Model for Prediction of the FluidParticle Interphase Drag Coefficient and Particulate Expansion of Fluidized and Sedimenting Beds, Powder Technol., 68, 199211.
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[12]
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Radmila GarićGrulović and Željko Grbavčić Day, J.Y., Littman, H. & Morgan III, M. H. (1990). A New Choking Velocity Correlation for Vertical Pneumatic Conveying, Chem. Engng. Sci., 45 (1), 355360. Grbavčić, Ž. B. Vuković, D. V., Garić, R. V., Hadžismajlović, Dž. E., Littman, H., Jovanović, Dj. S., Littman, H. & Morgan III, M. H. (1992). Hydrodynamic Modeling of Vertical LiquidSolids Flow, Powder Technol., 72(2), 183191. GarićGrulović, R.V., Grbavčić, Ž. B. & Arsenijević, Z. Lj. (2004). Heat Transfer and Flow Pattern in Vertical LiquidSolids Flow, Powder Technol., 145(3), 163171. GarićGrulović, R., BoškovićVragolović, N., Grbavčić, Ž. & Arsenijević, Z. (2008). Walltobed heat transfer in vertical hydraulic transport and in particulate fluidized beds, International Journal of Heat and Mass Transfer, 51, 59425948. GarićGrulović, R. V., Grbavčić, Ž. B., BoškovićVragolović, N. & Arsenijević, Z. Lj. (2009). Mass transfer in vertical liquidsolids flow of coarse particles, Powder Technol., 189(1), 130136. Arastoopour, H. & Gidaspow, D. (1979). Vertical Pneumatic Conveying Using Four Hydrodynamic Models, Ind. Eng. Chem. Fundam., 18, 123130. Leung, L. S. (1980). The ups and down of gassolids flow  a review, in: J.R. Grace, J.M. Matsen (Eds.), Fluidization Technology, Plenum Press, New York, 2568. Kmiec, A. & Leschonski, K. (1987). Acceleration of the Solid Phase During Pneumatic Conveying in Vertical Pipes, Chem.Engng J., 36, 5970. Grbavčić, Ž. B., Garić, R. V., Jovanović, S. Dj. & Rožić, Lj. S. (1997). Hydrodynamic modeling of vertical accelerating gassolids flow, Powder Tehnol,. Elsevier Science S.A., 92(2), 155161, and 94(2), 9197. Littman, H., Morgan III, M. H., Pacione, J. D., Jovanovic, S. Dj. & Grbavcic, Ž. B. (1993). Modelling and Measurement of the Effective Drag Coefficient in Decelerating and NonAccelerating Turbulent GasSolids Dilute Phase Flow of Large Particles in a Vertical Transport Pipe, Powder Technol., 77, 267283. Garić, R. V., Grbavčić, Ž. B. & Jovanović, S .Dj. (1995). Hydrodynamic modeling of vertical nonaccelerating gassolids flow, Powder Tehnol., 84, 6574. Garić, R., Grbavčić, Ž. & Rožić, Lj. (1996). Indirect determination of the particlewall friction coefficient in the vertical pneumatic and hydraulic conveying of coarse particles, J.Serb.Chem.Soc., 61, 391400. GarićGrulović, R. V., Rožić, Lj. S. & Grbavčić, Ž. B. (1997). Onedimensional modeling of vertical accelerating gassolids flow, 1st SouthEast European Symposium on Fluidized Beds in Energy production, Chemical and Process Engineering and Ecology, Proceeding of Lectures, Ohrid, Volume 1, 6376. Grbavčić, Ž. B., GarićGrulović, R. V. & Arsenijević, Z. Lj. (2006). Prediction of choking velocity and voidage in vertical pneumatic conveying of coarse particles, Powder Technol., 161(1), 19. Davidson, J. F. (1961). Symposium on FluidizationDiscussion, Trans. Inst. Chem. Engrs., 39, 230232. Gidaspow, D. & Solbrig, C. W. (1976). Transient TwoPhase Flow Models in Energy Production, State of the Art Paper presented at the AIChE 81st National Meetings, Apr. 1114; in revised form preprinted for NATO Advanced Study Institute on TwoPhase Flows and Heat Transfer, Aug 1627, 1976, AS1 Proceedings, Istanbul, Turkey. Jackson, R. (1963). The mechanics of fluidized beds. I: the stability of the state of uniform fluidization, Transactions of the Institution of Chemical Engineers, 41, 1321.
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[20] Jackson, R. (1971). Chapter 3. Fluid Mechanical Theory. 651 19 in Fluidization, J. F. Davidson and D. Harrison, Ed., Academic Press, New York, N.Y. [21] Gidaspow, D. (1974). Round Table Discussion (RT12). Modeling of TwoPhase Flow, 5th International Heat Transfer Conference, Sept 3, Tokyo, Japan, in Heat Transfer, Vol. VII, 163168. [22] Lyczkowski, R. W., Gidaspow, D., Solbrig, C. W. & Hughes, E. C. (1978). Characteristics and Stability Analysis of Transient Onedimensional TwoPhase Flow Equations and their Finite Difference Approximations, Nuclear Science and Engineering, 66, 378396. [23] Lax, P. D. (1958). Differential equations, difference equations and matrix theory, Communications on Pure and Applied Mathematics, 11(2), 175194, Copyright © Wiley Periodicals, Inc., A Wiley Company. [24] Gidaspow, D. (1978). Hyperbolic Compressible TwoPhase Flow Equations Based on Stabbnary Principles and the Fick's Law, in Two phase Transport and Reactor Safety, S. Kakac and T. N. Veziroglu, Ed., Vol. I, pp 283298, Hemisphere Publishing Corp. [25] Pritchett, J. W., Levine, H. B., Blake, T. R. & Gary, S. K. (1976). Numerical Model of Gas Fluidized Beds", 69th AIChE Annual Meeting, Chicago. [26] Soo, S. L. (1967). Fluid Dynamics of Multiphase Systems, p 279, Blaisdell Publishing Co., Waltham, Massachusetts. [27] Lefroy, G. A. & Davidson, J. F. (1969). The Mechanics of Spouted Beds, Trans. Inst. Chem. Engrs., 47, T120. [28] Deich, M. E., Danilin, V. S., Sleznev, L. I., Solomko, V. I., Taiklouri, G. V. & Shannon, V. K. (1974). High Temp. 12(2), 299307 (translation of Teplofiz. Vys. Temp., 12(2), 34453 (1974) by Consultants Bureau, New York). [29] Soo S. L. (1980). Fluid Dynamics of Multiphase Systems, Blaisdell. Plenum, New York, 2668. [30] Louge, M. Y., Mastorakos, E., Jenkins, J. T. (1991). The role of particle collisions in pneumatic transport, Journal of Fluid Mechanics, 231, 345359. [31] Chong, Y. O. & Leung, L. S. (1986). Comparison of choking velocity correlations in vertical pneumatic conveying, Powder Technol., 47, 4350. [32] Richardson, J. F. & Zaki, W. N. (1954). Sedimentation and fluidisation: Part 1, Trans. Instn. Chem. Engrs., 32, 3553. [33] Ergun, S. (1952). Fluid Flow Through Packed Columns, Chem. Engng. Progr., 48(2), 89. [34] Morgan III, M. H., Day, J. Y. & Littman, H. (1985). Spout voidage distribution, stability and particle circulation rates in spouted bads of coarse particles  I. Theory, Chem. Engng. Sci., 40(8), 13671377. [35] Day, J. Y., Morgan III, M. H. & Littman, H. (1987). Measurements of spout voidage distributions, particle velocities and particle circulation rates in spouted beds of coarse particles  II. Experimental verification, Chem. Eng. Sci., 42(6), 14611470. [36] Bird, R. B., Stewart, W. E. & Lightfoot, E. N. (1960). Transport Phenomena, J. Wiley, New York, 26773, 27483. [37] Stemerding, S. (1962). The pneumatic transport of cracking catalyst in vertical risers, Chem. Engng. Sci., 17(8), 599608. [38] Konno, H., Saito, S. J. (1969). Pneumatic Conveying of Solids Through Straight Pipes, J. Chem. Engng. Japan., 2, 211217.
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[39] Capes, C. E. & Nakamura, K. (1973). Vertical Pneumatic Conveying: An Experimental Study with Particles in the Intermediate and Turbulent Flow Regimes, Can. J. Chem. Engng., 51(2), 3138. [40] Yousfi, Y. & Gau, G. (1974). Aerodinamique de l‘ecoulement vertical de suspensions concentrees gazsolides. I  Regimes d‘ecouelement et stabilite aerodinamique, Chem. Eng. Sci., 29, 19391946. [41] Yang, W. C. (1978). A Correlation for Solids Friction Factor in Vertical Pneumatic Conveying Lines, AIChE J., 24(3), 548552. [42] Yang, W. C. (1974). Correlations for solid friction factors in vertical and horizontal pneumatic conveyings, AIChE J., 20, 605607. [43] Reddy, K. V. S. & Pei, D. C. T. (1969). Particle dynamics in solidsgas flow in a vertical pipe, Industrial and Engineering Chemistry Fundamentals, 8(3), 490497. [44] Van Swaaij, W. P. M., Buurman, C. & van Breugel, J. W. (1970). Shear stresses on the wall of a dense gassolids riser, Short Commun., Chem. Engng. Sci., 18181820. [45] De Jong, J. A. H. (1975). Aerated solids flow through a vertical standpipe below a pneumatically discharged bunker, Powder Technol., 12(3), 197200. [46] Kato, K., Ozawa, Y., Endo, H., Hiroyasu, M. & Hanzawa, T. (1986). Particles Holdup and Axial Pressure Drop in Vertical Pneumatic Transport Reactor (Riser), Fith Engineering Foundation Conference on Fluidization, Elsinore, Denmark., 265272. [47] Matsumoto, S., Harakawa, H., Suzuki, M. & Ohtani, S. (1986). Solid particle velocity in vertical gaseous suspension flows, Int. J. Multiphase flow, 12, 445. [48] Matsumoto, S., Ohnishi, S. & Maeda, S. (1978). Heat Transfer to Vertical GasSolids Flows, J. Chem. Engng. Japan, 11, 8995. [49] Leung, L. S. (1980). Vertical pneumatic conveying: a flow regime diagram and a review of choking versus nonchoking systems, Powder Technol., 25, 185190. [50] Costa, I. A., Ferreira, M. C. & Freire, J. T. (2004). Analysis of regime transitions and flow instabilities in vertical conveying of coarse particles using different solids feeding systems, Can. J. Chem. Eng., 82, 4859. [51] Smith, T. N. (1978). Limiting volume fractions in vertical pneumatic transport, Chem. Eng. Sci., 33, 745749. [52] Yang, W. C. (1983). Criteria for choking in vertical pneumatic conveying lines, Powder Technol., 35, 143150. [53] Bi, H. T., Grace, J. R. & Zhu, J. X. (1993). On types of choking in pneumatic systems, Int. J. Multiph. Flow, 19, 10771092. [54] Leung, L. S., Wiles, R. J. & Nicklin, D. J. (1971). Correlation for predicting choking flowrates in vertical pneumatic conveying, Ind. Eng. Chem. Process Des. Dev., 10, 183189. [55] Yang, W. C. (1975). A mathematical definition of choking phenomena and a mathematical model for predicting choking velocity and choking voidage, AIChE J., 21, 10131015. [56] Knowlton, T. M. & Bachovchin, D. M. (1976). The determination of gas– solids pressure drop and choking velocity as a function of gas density in a vertical pneumatic conveying line, in: D.L. Keairns (Ed.), Fluidization Technology, Hemisphere Publishing Corp, Washington D.C., 253282. [57] Mink, W. H. (1983). Critical velocity in solids transport, in: J.K. Beddow (Ed.), Particulate Systems: Technology and Fundamentals, Hemisphere Publishing Corp,
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Washington D.C., 15771. Briens, C. L., Bergougnou, M. A. (1986). New model to calculate the choking velocity of monosize and multisize solids in vertical pneumatic transport lines, Can. J. Chem. Eng., 64, 196204. Bi, H. T. & Fan, L. S. (1991). Regime transitions in gas–solid circulating fluidized beds, AIChE Annual Meeting, Los Angeles, 1722. Yang, W. C. (2004). "Choking" revisited, Industrial and Engineering Chemistry Research, 43(18), 54965506. Geldart, D. (1973). Types of gas fluidization, Powder Technol., 7(5), 285292. Xu, G., Nomura, K., Gao, S. & Kato, K. (2001). More fundamentals of dilute suspension collapse and choking for vertical conveying systems, AIChE J., 47, 21772196. Zenz, F. A. & Othmer, D. F. (1960). Fluidization and FluidParticle Systems, Reinold Publ., Co., New York. Satija, S., Young, J. B. & Fan, L. S. (1985). Pressure fluctuations and choking criteria for vertical pneumatic conveying of fine particles, Powder Technol., 43, 257271. Mok, S. L. K., Molodtsof, Y., Large, J. F. & Bergougnou, M. A. (1989). Characterization of dilute and dense phase vertical upflow gassolid transport based on average concentration and velocity data, Can. J. Chem. Eng., 67, 1016. Bai, D., Issangya, A. S. & Grace, J. R. (1998). A novel method for determination of choking velocities, Powder Technol., 97, 5962. Bi, H. T. & Grace, J. R. (1995). Flow regime diagrams for gas–solids fluidization and upward transport, Int. J. Multiph. Flow, 21, 12291236. Ferreira, M. C. & Freire, J. T. (1992). Fluid Dynamics Characterization of a Pneumatic Bed Using a Spouted Bed Type Solid Feeding System, Can. J. Chem. Eng., 70, 905909. Garić, R. V. (1990). Istraživanje fluidomehaničkih karakteristika vertikalnog dvofaznog toka fluidčestice, Magistarski rad, Tehnološkometalurški fakultet, Beograd (in Serbian). Garside, J. & AlDibouni, M. R. (1977). VelocityVoidage Relationships for Fluidization and Sedimentation in SolidLiquid Systems, Ind. Eng. Chem. Process Des. Dev., 16(2), 206214. Riba, J. P. & Couderc, J. P. (1977). Expansion de couches fluidisées par des liquids, Can. J. Chem. Eng., 55, 118. Wilhelm R. H. & Kwauk, M. (1948). Fluidization of solid particles, Chem. Eng. Prog., 44(3), 201218. Klinzing, G. E. & i Mathur, M. P. (1981). The dense and extrusion flow regime in gassolid transport, Can. J. Chem. Engng, 59, 590594. GarićGrulović, R. V., Grbavčić, Ž. B. & Arsenijević, Z. Lj. (2005). A pseudofluid representation of vertical liquidcoarse solids flow, J.Serb.Chem.Soc., 70(5), 775784. Kopko, R. J., Barton, P. & McCormick, R. H. (1975). Hydrodynamics of vertical liquidsolids transport, Ind. Eng. Chem. Proc. Des. Dev., 14, 264269.
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In: Fluid Transport: Theory, Dynamics and Applications ISBN: 9781611223170 Editor: Emma T. Berg, pp. 4573 © 2011 Nova Science Publishers, Inc.
Chapter 2
NUMERICAL SIMULATION ON FLOWS PAST POROUS BLUFF BODIES Peng Yu1*, Yan Zeng1, Thong See Lee1 and Hong Tong Low2 1
Department of Mechanical Engineering, Universoty of Sydney, Australia Division of Bioengineering, National University of Singapore, Sinagpore
2
ABSTRACT Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.
Flow past bluff porous bodies has attracted relatively less attention so far although such flow occurs widely in industries and everyday life. In the present study, computational fluid dynamics is applied to investigate this type of problems. The coupling flow in both porous medium and homogenous fluid regions is solved by a finite volume method based on the bodyfitted and multiblock grids. At the porousfluid interface, a shear stress jump condition that includes both the viscous and inertial effects is imposed, together with a continuity of normal stress. The steady flows are simulated in a wide range of the Reynolds and Darcy numbers. Three most commonly used shapes of the bluff body in the literature, that is, circular cylinder, square cylinder and sphere, are examined. Several interesting flow phenomena, which are different from those of flow past a solid bluff body, are revealed by the present simulations. It is found that the recirculating wake existing downstream of a porous bluff body may be completely detached from the body in a certain range of parameters. In a certain range of Darcy number, the wake may initially increase in size with an increase in Reynolds number but then decrease in size and eventually disappear when the Reynolds number is sufficiently large. There is only one ―separation‖ point along the horizontal axis of the porous bluff body, but not a pair of separation points on the surface of the solid one. The present findings may provide a starting point for reevaluating the flow around and through a porous bluff body and stimulate new studies to understand the underlying mechanics of the new flow phenomena which have not been answered here.
*
Corresponding Author: Dr. P. Yu, Current Working University of Sydney, Email: [email protected]
Address:
School
of
Civil
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Engineering,
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Peng Yu, Yan Zeng, Thong See Lee et al.
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1. INTRODUCTION The problems on flow past bluff bodies are attractive not only due to its academic value but also the related engineering significance. Thus, this topic has been extensively studied for a long time, with a wide range of analytical, experimental and numerical methods. The most commonly used geometries of the bluff body in the literature are circular or square cylinders and sphere, which are confined in a channel or subjected to a free stream fluid flow. Generally, the Reynolds number is the most important parameter for the flows around the bluff body placed in a uniform stream. For a small Reynolds number (Re < ~1), the flow is fully attached to the surface of the bluff body. With an increase in Re, the flow may separate from the surface and form an attached recirculating wake behind the body. The wake becomes unsteady and/or three dimensional, and then transits to turbulence with a further increase in Re. The reader may refer to the books of Zdravkovich (1997; 2003) and Clift et al. (1978), and the paper of Williamson (1996) for a comprehensive review on this topic. Although the flow past bluff bodies has attracted extensive attention, a little effort has been taken to understand the effect when the solid bodies become permeable. However, such conditions universally occur in many practical applications. A typical application in bioengineering is the flows in bioreactor involving porous microcarriers (Braeckmans et al., 2002) or porous scaffold (Yu et al., 2009a), in which the surrounding flow should effectively transport nutrients and metabolites to and from the porous bodies where the cells are attached. Another interesting example can be found in chemical process industries, which involves the settling of ‗flocs‘ of material in liquidsolid rectors (Masliyah and Polikar, 1980; Noymer et al., 1998). Other examples include the nuclear biological chemical filters, which have been widely implemented in medical, chemical and pharmaceutical industries (Bhattacharyya et al., 2006). There are limited numerical works on flow around and through porous bluff bodies so far. The main obstacle is the coupling issue at the porousfluid interface. Noymer et al. (1998) has applied commercial software PHOENICS to simulate the flow around a permeable circular cylinder. The Darcy equation was used to govern the porous flow. The pressure and mass flow were matched at the interface of the two regions. Their computational results, substantiated by wind tunnel tests, indicated that the permeability has a significant effect on the drag coefficient. Bhattacharyya et al. (2006) investigated the steady flow motion around and through a porous circular cylinder by using finite volume method. The effect of the Darcy number on the drag coefficient, separation angle, and wake length, streamline and vorticity patterns was studied. The reductions in the drag coefficient, the wake length and separation angle are observed with an increase in the Darcy number. Jue (2004) applied finite element method to simulate vortex shedding behind a porous square cylinder, employing a general nonDarcy porous media model to describe the flows both inside and outside the cylinder. A harmonic mean was used to treat the sudden change between the fluid and porous medium. It was found that, besides the Reynolds number, the Darcy number also has a significant effect on the flow. However, the porosity shows negligible effect on the flow. Chen et al. (2008) numerically studied the flow past a porous square cylinder by using a finite volume method based on multiblock, bodyfitted grid. The stress jump condition including both viscous and inertial effects is imposed at the porousfluid interface (OchoaTapia and Whitaker, 1998). Chen et al. (2008) examined the flow
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Numerical Simulation on Flows Past Porous Bluff Bodies
47
behaviors for the Reynolds number within 200. It was found that, with a larger Darcy number, the Reynolds number has to be higher before the vortex shedding phenomena occurs. The results also showed that the interface stress jump parameters play an important role in the stability of the flow around a porous square cylinder, with noticeable effect of the first coefficient and small effect of the second coefficient. The objective of the present study is to simulate the steady flow around and through porous bodies. Three basic geometries of the porous bluff bodies are considered, which are circular cylinder, square cylinder, and sphere. It is worth mentioning that previous numerical investigations of this topic (Noymer et al., 1998; Jue, 2004; Bhattacharyya et al., 2006) mainly focused on the flow around the cylinder. The flow behavior within the cylinder was only touched for the case with high permeability. There is scarce information concerning the flow feature within the cylinder with low and moderate permeability. Thus, one of the motivations behind the present study is to provide detailed information in this aspect. However, what surprises us is that the wake existing downstream of the porous body is found to either penetrate into or be completely detached from, but not attached to it. Other interesting findings revealed herein include that the recirculating wake may develop downstream of or within the porous bodies, but not separated from the surface of it as what happens for the solid pair. The present study may thus provide a starting point for reevaluating the flow around and through a porous bluff body and stimulate new studies to understand the underlying mechanics of some new findings which have not been answered here. The remainder of this chapter is organized as follows: Section 2 provides a detailed description of the governing equations and boundary conditions, numerical techniques, as well as gridindependent study and validation. This is followed by a presentation and discussion of the numerical results in Section 3, which include the variations of flow patterns and closedwake geometrical parameters with the Reynolds and Darcy numbers. The new findings are also highlighted in this section. And finally, Section 4 provides the concluding remarks.
2. NUMERICAL METHOD 2.1. Governing Equations Generally, two approaches has been employed to model the coupled flow in both homogeneous fluid and porous regions: onedomain approach and twodomain approach. The details on these two approaches can be referred to the previous studies (Goyeau et al., 2003; Yu et al. 2007), which are not be illustrated for the sake of brevity. In the present study, we use a twodomain approach to solve such a coupled flow. In the twodomain approach, two sets of governing equations are applied to describe the flow in the two regions and additional boundary conditions are used at the interface to close the two sets of equations. Considering steady, laminar flow of an incompressible, viscous fluid, the governing equations for a homogenous fluid region, using vector form, can be written as:
u 0 Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
(1)
48
Peng Yu, Yan Zeng, Thong See Lee et al.
uu p 2u where p is the pressure;
(2)
is the mass density of the fluid; and is the fluid dynamic
viscosity. The porous medium is considered to be rigid, homogeneous and isotropic; and saturated with the same singlephase fluid as that in the homogenous fluid region. Considering viscous and inertial effects, the governing equations for porous region based on DarcyBrinkmanForchheimer extended model can be expressed as (Nithiarasu et al., 2002; Yu et al., 2007):
u 0
(3)
CF u uu 2 u u p u K K Brinkman Term
(4)
Convective Term
Pressure Term
Darcy Term
Forchheimer Term
where u is the local average velocity vector (Darcy velocity); p* is the intrinsic average pressure; is the fluid dynamic viscosity; is the porosity; K is the permeability; and CF is
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Forchheimer coefficient. Note that throughout the paper, viscosity means the dynamic viscosity of the fluid but not the effective (Brinkman) viscosity. The ―*‖ denotes the intrinsic average. The local average and intrinsic average can be linked by the DupuitForchheimer relationship, for example, p = p*. At the interface between the homogeneous fluid and porous medium regions, additional boundary conditions must be applied to couple the flows in the two regions. In the present study, the stress jump condition (OchoaTapia and Whitaker, 1998) is applied:
ut n
porous
ut n
fluid
K
1 ut2
ut
(5)
interface
where in the porous medium region, ut is the Darcy velocity component parallel to the interface aligned with the direction t and normal to the direction n while in the homogenous fluid region ut is the fluid velocity component parallel to the interface; β and β1 are adjustable parameters which account for the stress jump at the interface. It is noted that different boundary conditions at the interface have been proposed, which include semiempirical slip boundary condition (Beavers and Joseph, 1967), continuous boundary conditions in both stress and velocity (Neale and Nader, 1974), and stress jump conditions (OchoaTapia and Whitaker, 1995a; 1995b; 1998). A stress jump condition with one jump parameter was deduced by OchoaTapia and Whitaker (1995a; 1995b), using the nonlocal form of the volume averaged method. Based on the Forchheimer equation with the Brinkman correction and the NavierStokes equation, OchoaTapia and Whitaker (1998) developed another stress jump condition which includes the inertial effects. Two coefficients appear in this jump condition: one is associated with an excess viscous stress and the other is related to an excess inertial stress.
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The implementation of the numerical methodology on the stress jump condition with one jump parameter (OchoaTapia and Whitaker 1995a; 1995b) can be found in the work of Silva and de Lemos (2003). Yu et al. (2007) developed a numerical methodology using the finite volume method, based on a collocated variable arrangement, to treat the stress jump condition which includes two jump parameters (OchoaTapia and Whitaker, 1998). Another example on the implementation of the stress jump condition with two jump parameters is the controlvolume finiteelement method proposed by Costa et al. (2008). The applications involving stress jump condition can be also widely found in literatures. Kuznetsov (1998) has applied it to analytical investigation of flow in channel partially filled with porous medium. Bhattacharyya and Raja Sekhar (2004) have applied it to viscous flow past a porous sphere with an impermeable core. Partha et al. (2005) have employed it to simulate viscous flow past a porous spherical shell. Chen et al. (2008) and Yu et al. (2010) have used it to flow past a porous square cylinder. Yu et al. (2009a) have applied it to flow in a bioreactor with a porous scaffold. However, there are no explicit expressions for the parameters β and β1. OchoaTapia & Whitaker (1995a; 1998) derived analytical expressions for parameters β and β1, which indicate their dependence on permeability and porosity. The authors concluded that these two parameters are both of order one. OchoaTapia & Whitaker (1995b) experimentally determined that β varies from 1.0 to +0.7 for different materials with permeability varying from 15 × 106 to 127 × 106 in2 and average pore size from 0.016 to 0.045 in. In the present simulations, the jump parameters β and β1 are both set to zero, within the range of jump parameters of order 1. In addition to Equation (5), the continuity of velocity and normal stress prevailing at the interface is given by:
u fluid u porous vinterface
un n
porous
un n
0
(6)
(7)
fluid
where in the porous medium region, un is the Darcy velocity component normal to the interface; and in the homogenous fluid region, un is the fluid velocity component normal to the interface. By combining with the appropriate boundary conditions of the composite region, Equations (1)  (7) can be used to simulate the flow in a system composed of a porous medium and a homogenous fluid. For the present type of flow problems, the physical domain is infinite while the simulation must be performed on a confined computational domain (see Figure 1 for the case of flow past a circular cylinder). Thus the computational domain should be truncated from the real domain by using artificial open boundary conditions (Sohankar et al., 1998). To balance the computational performance and the effect of outer boundaries, the lengths of the computational domain are defined as 60 times the diameter of the cylinder. For the outer boundaries, the freestream condition on the velocity is imposed on the upstream boundary, a Neumann condition for the velocity is specified at the downstream boundary, and the slip boundary condition is applied on the lateral boundaries.
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Peng Yu, Yan Zeng, Thong See Lee et al.
For flow past a porous bluff body, the overall flow behaviour is determined by the interaction between the external flow around and the internal flow through it. Generally, the flow around a bluff body is mainly affected by the Reynolds number Re. The ability of a porous medium to conduct fluid flow can be quantitatively represented by the permeability, or the dimensionless Darcy number Da. Thus, the present flow is chiefly determined by Re and Da while other parameters have relatively small influences. The definitions of Re and Da can be written as:
Re U D
(8)
Da K D 2
(9)
where U∞ is the free stream velocity and D is the reference length, which represents the diameter of the circular cylinder, the side length of the square cylinder, and the diameter of the sphere, respectively.
30D
Computational Domain D
U∞
60D 30D
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Porous Circular Cylinder
60D Figure 1. Schematic of flow past a porous circular cylinder. A
B
• n
N L
e
vinterface
t
L•
el e
•
NR
S
ye yne yse le xe2 ye2
R
n
• R
s •
xe xne xse
y
e
• SR
ey ex
x
Interface
Figure 2. Interface between two blocks with matching grids. Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
Numerical Simulation on Flows Past Porous Bluff Bodies
51
It is well known that for a laminar, steady flow past a bluff body, a recirculating wake consisting of a counterrotating vortex pair may be observed behind the body in a certain range of Re. Usually, the recirculating wake is fully attached to the body. However, in the present study, it is found that a recirculating wake behind a bluff body is not attached to the body. The recirculating wake is either detached from or penetrates into it. This means that there is no reattachment of the recirculating wake on the surface of porous body. Thus, the geometrical parameters considered here are the downstream distances to the leading and trailing edges of the recirculating wake (LL and LR). The length of the recirculating wake then can be calculated as Lw = LR – LL. The negative value of LL means that the recirculating wake already penetrates into the bluff body.
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2.2. Numerical Techniques for FluidPorous Interface In the present study, the multiblock grid method is used to provide a compromise between the simplicity and wide variety of solvers available for structured grids and ability to handle complex geometries that unstructured grids allow. The FVM code based on nonorthogonal grid is used for simulation. The second order central difference scheme is applied to discretize the governing equations. The solution algorithm is based on SIMPLEC method. The nonstaggered grid arrangement is applied and the interpolation of Rhie and Chow (1983) is used to get a good coupling between pressure and velocity. The detailed numerical procedures on discretization can be found in the works of Ferziger and Perić (1999) and Yu et al. (2007). Figure 2 shows details of the interface between two different blocks. The neighboring control volumes lying in gridblocks A and B share the interface and the grids in these two neighboring blocks match at that interface. Generally, there are three types of interfaces when the blockstructured grids method is employed to calculate the flow in the composite region: (i) fluid  fluid interface, (ii) porous  porous interface, and (iii) fluid  porous interface. When both blocks A and B (Figure 2) represent the same media, either fluid or porous medium, the method proposed by Lilek et al. (1997) is applied to treat the block interface. A special data structure is designed to save the information at the interface, which consists of: the indices of the left (L) and right (R) neighboring cells, the surface vector (pointing from L to R) and the coordinates of cellsurface center. The interface cell surface shared by two control volumes is treated as a cell surface in the interior of the block. Thus, the convective and diffusive terms at the block interface can be calculated in the same way as that for the cell faces in the interior of the block. Now we focus on the numerical details when blocks A and B represent different media. The velocity vector at the interface vinterface can be written in either the xy or nt coordinate systems as:
vinterface uex vey un n ut t
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(10)
52
Peng Yu, Yan Zeng, Thong See Lee et al.
where u and v are the components of vinterface in the x and y directions while un and ut are the vinterface components along n and t directions respectively. And the component ut then can be written as:
ut uex t ve y t
(11)
By combining Equations (5), (7) and (10):
vinterface n
porous
vinterface n
fluid
K
ut t 1 ut2t
(12)
The unit vector ( t ) parallel to the interface (Figure 2) is calculated from:
t
xne xse ex yne yse ey 2 2 xne xse yne yse
xe ex ye ey
(13)
le
By substituting the components of vinterface in the x and y directions, the Equation (12)
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becomes: x uxe vye u u uxe xe vye xe 1 e 2 n porous n fluid l le3 K e
2
y uxe vye v v uxe ye vye ye 1 e 2 n porous n fluid le le3 K
2
(14)
(15)
The derivatives at the interface are calculated from the values at auxiliary nodes L and R ; these nodes lie at the intersection of the cell face normal n and straight lines connecting nodes L and N or R and NR, respectively, as shown in Figure 2. The normal gradients at the interface can be calculated by using the first order difference approximation:
u u e v v v e u R , R n porous LeR n porous LeR
(16)
v v L v v v L v e , e n fluid LLe n fluid LLe
(17)
The Cartesian velocity components at L and R can be calculated by using bilinear interpolation or by using the gradient at the control volume center: Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
Numerical Simulation on Flows Past Porous Bluff Bodies
u L u L gradu L LL
53 (18)
To obtain higher order approximation of the derivatives, the velocity components at more auxiliary nodes may be needed. Alternatively, the shape functions may be used, which produces a kind of combined Finite Element/Finite Volume method for calculating the higher order approximations. By using Equations (14) to (18) and explicitly calculating the terms at the right hand sides of Equations (14) and (15), the Cartesian velocity components u and v at the interface are obtained. Then the convective fluxes at the interface can be calculated. The diffusive fluxes are calculated from Equations (16)  (18). Then the coefficients AL and
AR can be obtained. The above treatment is suitable for the twodimensional flow past circular or square cylinders. However, when the cylinders are replaced by spheres, the flow is axisymmetric if the Reynolds number is less than a certain threshold. Axisymmetric flows are threedimensional with respect to Cartesian coordinates but they are only twodimensional in a cylindrical coordinate system since all derivatives with respect to the azimuthal direction are zero and all three velocity components are functions of only the axial and radial coordinates. Specially, vθ equation can be treated as a conservation equation for a general dependent variable, as there is no pressure term. It worth mentioning that, for the flow past sphere, vθ is always zero in the present range of Re investigated. Thus, the procedure described above is enough to simulate flow past sphere. However, we still provide the numerical treatment for the case of nonzero vθ. Now the velocity vector at the interface can be written as:
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vinterface vr er vz ez v e vn n vt t
(19)
The component vt then can be written as:
vt vr er vz ez v e t
(20)
By combining Equations (5), (6) and (19):
vinterface n
porous
vinterface n
fluid
K
vt t 1 vt2t
(21)
As the flow is axisymmetric, the unit vector ( t ) parallel to the interface is calculated from:
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vr r vz z rer zez v e l l t 2 vr r vz z 2 v l
(22)
By substituting the components of vinterface in z, r, and θ directions, Equation (21) can be rewritten as:
vz n
porous
vz n
fluid
vz zz vr r z l2
K
v z z vr r z vr r vz z 2 1 z v l2 l 2
vr n
porous
vr n
fluid
(23)
vz zr vr r r l2
K
v z r vr r r vr r vz z 2 1 z v 2 l l 2
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v n
v n porous
vr r vz z 2 v l 2
K
fluid
(24)
v 1 v
(25)
The discretization procedures of the velocity components vz and vr at the interface are the same as those in Cartesian coordinate. The derivatives of v at the interface are also calculated from the values at auxiliary nodes L and R ; these nodes lie at the intersection of the cell face normal n and straight lines connecting nodes L and N or R and NR, respectively. The normal gradients of v at the interface can be calculated by using the first order difference approximation:
v n
v
R
v
LeR
porous
e
and
v n
v e v
fluid
L
LLe
(26)
The values of v at L and R can be calculated by using bilinear interpolation or by using the gradient at the control volume center:
v
P
v p gradv P LL
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(27)
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55
By making use of Equations (25) to (27), the values of v at the interface are obtained. Then the convective fluxes at the interface can be calculated. The diffusive fluxes are calculated from Equations (26) and (27). Once the overall fluxes at interface are obtained, the equations for v can be solved.
Outer flow region
Porous Square Cylinder
a)
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Outer flow region
2 Porous sphere
1 b) Figure 3. Grid topology near the porousfluid interface, a) square cylinder, and b) sphere.
2.3. Grid Independent Study and Validations Typical examples of the mesh within and around the porousfluid interface for the porous square cylinder and sphere are shown in Figure 3. To generate a bodyfitted, structured grid with good quality in Cartesian coordinates, the whole computational domain was divided into a few subdomains, with one or two domains for the porous body and one domain for the outer flow region. The outer flow domain was meshed using Otype grid and the grids in the
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radial direction were stretched through an exponential progression to ensure a fine grid near the porousfluid interface. Note that only a small part of outer flow domain is shown in Figure 3 in order to clearly illustrate the grid topology. To ensure a gridindependent solution and accurate resolution in space, a series of studies has been performed with three sets of mesh for porous circular and square cylinders. The Reynolds number tested was 20 and the Darcy number for the porous cylinder was 1×105. Table 1 shows the comparisons of LR at various grid sizes. For the circular cylinder, the difference in LR between cases 2 and 3 is within 0.4%. For the square cylinder, the difference in LR between cases 5 and 6 is around 0.6%. To ensure gridindependent solution, the grids used in cases 3 and 6 were chosen for the final simulations. For the porous sphere, the grid numbers used for the final simulations are 121 × 61, 241 × 55, and 241 × 301 in each subdomain. The preliminary tests have confirmed that the solutions based on the present mesh are gridindependent. Table 1. Effect of grid size on the overall wake length LR. Case circular cylinder
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square cylinder
1 2 3 4 5 6
Grid size Porous Body Outer Flow Region 40 × 40 160 × 25 160 × 140 60 × 60 240 × 30 240 × 170 80 × 80 320 × 40 320 × 200 40 × 40 160 × 160 60 × 60 240 × 200 80 × 80 320 × 240
LR 0.885 0.899 0.902 1.260 1.283 1.291
To examine the accuracy of the present method, a study was performed to investigate the relationship between LR and Da for the porous circular and square cylinders at Re = 20 and ε = 0.7. The wake length becomes longer with a decrease in Da. It approaches a constant value at a low Da as the porous cylinder tends to a solid one. For the circular cylinder, the wake length at Da = 5 × 106 is about 0.906 as shown in Figure 4, which is rather close to the value of around 0.916 for the solid one. For the square cylinder, the overall wake length LR at Da = 1 × 106 is about 1.297 (Figure 4), which is rather close to the value of around 1.310 for the solid one.
3. RESULTS AND DISCUSSION 3.1. Flow Pattern The effects of Re and Da on the flow pattern are illustrated by presenting streamline plots at various values of Re and Da. As illustrated below, for different Da, the interactions between the external flow around and the internal flow within the porous bluff body are different, resulting in different flow behaviours. In the present study, Re is kept within certain ranges to ensure steady flows around the bluff bodies. Da is varied from 101 to 106. The porosity is fixed at 0.7 and the jump parameters are both set to zero.
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1.4
1.2
Solid Square Cylinder
1
0.8
LR
Solid Circular Cylinder 0.6
Circular Square
0.4
0.2
0 7 10
10
6
5
10
10
4
3
10
10
2
Da
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Figure 4. Variation of the overall wake length LR with Darcy number at Re = 20.
At a high Da (≥ 1 × 102), the flow experiences little resistance when it passes through the porous bluff body. As a significant portion of fluid penetrates through the porous bluff body, the streamlines around it exhibit a smaller deviation compared with those around the solid body. Figures 5a to 5d show the streamlines for flow past a porous square cylinder at Da = 1 × 102 for different Reynolds numbers. The streamlines become flatter with an increase in Re. This is expected when we examine the resistance force that the internal porous flow experiences. The Darcy drag force
u K
is the main source of the resistance force, which
decreases with an increase in Da or Re. As a result, the streamlines are less deviated with an increase in Da or Re. As shown in Figure 5, there is no recirculating wake behind the square cylinder in the range of Re investigated for a high Darcy number. Figure 6 presents the streamlines for flow past sphere and circular cylinder at a high Da = 5 × 102. Also, no recirculaing wake was observed behind the porous bluff body. The results (Figures 5 and 6) indicate that, for a high Da (≥ 1 × 102), the shape of the object has a small effect on streamline contours. In this sense, the overall flow behavior for a high Da is mainly determined by the internal porous flow. For an intermediate Da, (~ 1 × 103), the flow patterns are more complicated. The most striking flow feature at this range of Darcy number is the detached recirculating wake as reported in Yu et al. (2010). Figure 7 shows a series of streamline plots at various values of Re at Da = 7 × 103 for a porous square cylinder. The wake first appears at Re ~ 21, then grows to a maximum length at roughly Re = 30, after which it gradually decreases, and eventually completely disappears when the Reynolds number reaches 42. For the flow past a porous circular cylinder (Figure 8) or sphere (Figure 9), the similar phenomena can be observed. For the porous circular cylinder at Da = 7 × 103, the recirculating wake first
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appears at Re ~ 37, grows to a maximum length at roughly Re = 45, and then gradually decreases until finally completely disappears when the Reynolds number reaches 57. Similarly, for the porous sphere at Da = 5 × 103, the wake first appears at a Reynolds number of approximately 38, grows to a maximum length at roughly Re = 55, and then gradually shrinks and completely disappears when the Reynolds number reaches 71.
a) Re = 10
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b) Re = 20
c) Re = 30
d) Re = 40 Figure 5. Contours of streamline of flow past a porous square cylinder at a fixed Darcy number of Da = 1 × 102 for different Reynolds numbers as indicated.
There is a remarkable contrast between this detached wake for the porous bluff body and the wellknown attached wakes which exist on solid bodies (Zdravkovich, 1997; 2003; Clift Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
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59
et al. 1978). However, it is not new since the detached wake has already been observed in other flow conditions (Leal and Acrivos, 1969; Leal, 1989; Rivkind and Ryskin, 1976; Dandy and Leal, 1989). The first example is the flow past bluff bodies with base bleed, which has been studied by Leal and Acrivos (1969). Their experiments clearly demonstrated that at a Reynolds number of 260, there is a recirculating zone that is completely detached from the body when a modest flow of fluid is injected from the downstream surface of the body. Another example is the translational motion of a viscous drop (Rivkind and Ryskin, 1976; Dandy and Leal, 1989). The numerical solutions of Dandy and Leal (1989) showed that the recirculating eddy which appears downstream is completely detached from the drop for a wide range of parameter values. When the detached wake occurs, there is not even a separation or a detachment point on the bodies, but only one ―separation‖ point along the horizontal axis. It is also observed that the magnitude of the surface vorticity must exceed a minimum threshold level (dependent on Re) before a recirculating wake appears, which is approximately independent of the boundary condition at the surface (Leal, 1989). Based on the numerical and experimental observations of the detached recirculating wake (Leal and Acrivos, 1969; Rivkind and Ryskin, 1976; Dandy and Leal, 1989), Leal concluded in his pioneer work that 'Recirculating wakes form at finite Reynolds number due to vorticity accumulation, and this has nothing to do with the mechanics of boundary layer separation in the limit Re → ∞' (Leal, 1989).
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a)
b)
Figure 6. Contours of streamline for flow past porous bluff bodies at Da= 5 × 102; a) sphere, Re = 200; b) circular cylinder, Re = 20.
For flow past a smooth bluff body, vorticity is generated by two mechanisms, i.e. the noslip boundary condition and the surface curvature. It is shown that the dimensionless vorticities generated by surface curvature and by the noslip condition are O(1) and O(Re1/2) for large Re, respectively (Leal, 1989). For the porous circular cylinder, the rate of vorticity production results from the corporation/competition of the two mechanisms. For the porous square cylinder, there is no smooth surface curvature but sharp edge. The recirculating wake first occurs around Re = 1 for a solid square cylinder (Sharma and Eswaran, 2004; Zaki, 1994) whilst it first occurs around Re = 6 for a solid circular one (Underwood, 1969), which suggest that the sharp edge is a more effective source of vorticity generation compared with
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the noslip condition. Note that the flow passes through the porous bodies and the exiting flow at the leeward surface resembles ‗base bleed‘. This ‗base bleed‘ also has a significant effect on the flow behaviour.
a) Re = 10
b) Re = 15
c) Re = 20
d) Re = 25
e) Re = 30
f) Re = 35
g) Re = 40
h) Re = 45
Figure 7. Contours of streamline of flow past a porous square cylinder at a fixed Darcy number of Da = 7 × 103 for different Reynolds numbers as indicated.
Now we consider the flow field in the range of low Darcy number. The Darcy number of 105 was chosen to show the flow behavior at this low extreme. At the low Da, the flow behaviour is almost the same as that around a solid bluff body. As shown in Figure 10, the streamline contours of a porous square cylinder at different Re indicate that the flow field resembles that around a solid square cylinder. The recirculating wake occurs when Re > 1 and increases in size with an increase in Re. At higher Re, the wake is fixed by two of the shape edges. The lengths of the wake are 1.291 for Re = 20 and 2.772 for Re = 40, which are pretty close to those of a solid square cylinder (1.310 for Re = 20 and 2.802 for Re = 40). It is worth noting that the wake is detached from the porous square cylinder for the Reynolds number just above that for the onset of the wake. However, at high Re, the wake appears to be attached to the leeward surface so that no fluid passes through the leeward surface.
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a) Re = 30
b) Re = 40
c) Re = 50
d) Re = 60
61
Figure 8. Contours of streamline of flow past a porous circular cylinder at a fixed Darcy number of Da = 7 × 103 for different Reynolds numbers as indicated.
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a) Re = 30
b) Re = 50
c) Re = 60
d) Re = 80 Figure 9. Contours of streamlines of flow past a sphere at a fixed Darcy number of Da = 5 × 103 for different Reynolds numbers as indicated.
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a) Re = 1
b) Re = 5
c) Re = 10
d) Re = 20
e) Re = 30
f) Re = 40
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Figure 10. Contours of streamline of flow past a porous square cylinder at a fixed Darcy number of Da = 1 × 105 for different Reynolds numbers as indicated.
a) Re = 1
b) Re = 5
c) Re = 20
d) Re = 40
Figure 11. Contours of streamline of flow past a porous circular cylinder at a fixed Darcy number of Da = 1 × 105 for different Reynolds numbers as indicated.
Figures 11 and 12 show the contours of streamlines of flows past a porous circular cylinder and a porous sphere, respectively. The same as those of flow past a porous square cylinder, the contours of streamlines indicate that the flow patterns resemble those of their solid pairs. However, it is seen that the streamlines can still penetrate the porous body even Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
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the Darcy number is very small. In summary, the porous bluff body approaches a solid one in the low Darcy number range. The flow resistance is very large and little fluid can pass through the body. Thus, the overall flow behavior in this range of Da is chiefly determined by the Reynolds number.
a) Re = 5
b) Re = 10
c) Re = 30
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d) Re = 50
e) Re = 150 Figure 12. Contours of streamlines of flow past a porous sphere at a fixed Darcy number of Da = 1 × 105 for different Reynolds numbers as indicated.
3.2. Occurrence of Recirculating Wake Now the attention is focused on the effect of Da on the critical Reynolds number for the occurrence of a recirculating wake for the cases of porous cylinders. As shown in Figure 13, the critical Reynolds number decreases with decreasing Da for both porous square and circular cylinders. For the porous square cylinder, there is no recirculating wake in the range of Re investigated when Da > 7.5 × 103. The recirculating wake first appears at around Recr = 24.5 for Da = 7.4 × 103. The critical Reynolds number rapidly decreases to 5.5 for Da = 103 and slowly drops to 1.5 for Da = 106. Figure 13 suggests that when Da goes to an infinitely small value, the critical Reynolds number approaches an asymptote which equals to that of a solid cylinder. The present simulation indicates that the critical Reynolds number for a solid
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cylinder is around 1.5, which is consistent with the results of Zaki et al. (1994) and of Sharma and Eswaran (2004). For the flow around a solid circular cylinder, the calculation of Underwood (1969) showed that the critical Reynolds number at which separation first appears behind the cylinder is 5.75. The numerical stability analysis of Noack and Eckelmann (1994) pointed out that the separation first occurs at Recr = 5 ± 1. This critical Reynolds number is found to be around 7 by Bhattacharyya et al. (2006). The present numerical simulations indicate that the critical Reynolds number is around 6 as shown in Figure 13, which is consistent with the above values. For the porous circular cylinder, the recirculating wake first appears at Recr = 36.5 for Da = 7 × 103. However, the critical Reynolds number rapidly decreases to 8.5 for Da = 1 × 103 and slowly drops to 6.5 for Da = 1 × 104. Figure 13 also suggests that at an extreme high Da, the recirculating wake may not appear in the range of Re investigated and at an extreme low Da, the critical Reynolds number approaches to that of the solid cylinder. 50
40 Onset of wake, porous square cylinder Disappearance of wake, porous square cylinder Onset of wake, porous circular cylinder
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Re
30
20
Onset of wake, solid circular cylinder 10
0 7 10
Onset of wake, solid square cylinder
10
6
10
5
10
4
10
3
10
2
Da Figure 13. Variations of Critical Reynolds numbers for onset and disappearance of recirculating wake with Darcy number.
Figure 13 also presents the variations of the critical Reynolds number for the disappearance of a recirculating wake for the case of the porous square cylinder. For Da = 7.4 × 103, the recirculating wake disappears at Redis ≈ 33.5. With decreasing Da, the recirculating wake disappears at a higher Re, for example at Redis ≈ 48.5 for Da = 6.6 × 103. The disappearance phenomenon of a recirculating wake only occurs at a narrow range of Da. For a lower Da = 104, the recirculating wake does not disappear. For the case of the porous circular
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cylinder, the recirculating wake behind may also disappear in a certain range of parameters as shown in Figure 8. However, we did not summarize the data in Figure 13. Figure 14 presents the streamlines for the flow around and through the porous square cylinder at the Reynolds number slightly larger than the critical Re for the onset of the recirculating wake. The streamline contours clearly show that, for all the three Darcy numbers Da = 7 × 103, 1 × 103 and 1 × 105, the recirculating wake develops downstream of the cylinder but is not separated from the surface of it. For a lower Da, the wake initially occurs closer to the rear of square cylinder as shown in Figure 14b. When the Darcy number tends to an infinitely small value, that is the porous cylinder approaches a solid one, the recirculating wake eventually attaches to the rear of square cylinder (Figure 14b). Figure 15 presents the streamlines for the flow around and through the porous circular cylinder at the Reynolds number slightly larger than the critical Re for the onset of the recirculating wake. For the case of Re = 18 and Da = 5 × 10 3, the streamline contours clearly show that the recirculating wake develops downstream of the cylinder but is not separated from the surface of it (Figure 15a). However, at the case of Re = 6 and Da = 5 × 106, it is found that the recirculating wake is developing inside the cylinder (Figure 15b). In summary, our simulations show that when Da > 7 × 103, the recirculating wake does not appear in the present range of Re investigated. In the range of 2.5 × 103 ≤ Da ≤ 7 × 103, the recirculating wake first occurs downstream of the cylinder. For Da < 2.5 × 103, the recirculating wake initially develops inside the cylinder.
(a)
(b)
(c) Figure 14. Streamlines for the flow around and through the porous square cylinder just after the formation of recirculating wake; a) Re = 21, Da = 7 × 103; b) Re = 6, Da = 103; c) Re = 2, Da = 105.
The present simulations reveal that the recirculating wake existing behind the porous cylinders or sphere either penetrates into or is detached from the porous body, but is not attached to the porous body as which occurs behind a solid body. The recirculating wake is Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
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found to develop downstream of or within the porous body, but not from its surface as that of its solid pair. Also, there is only one ―separation‖ point along the horizontal axis, either inside or outside the porous body, but not a pair of separation points on the surface of the solid body. All these features support Leal‘s conclusion (Dandy and Leal, 1989; Leal, 1989) that the existence of recirculating wake behind any bluff body at large, but finite Reynolds number should be regarded as being a consequence of the accumulation of vorticity generated upstream on the body surface.
a)
b)
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Figure 15. Streamlines for the flow around and through the porous cylinder just after the formation of recirculating wake; a) Re = 18, Da = 5 × 103; b) Re = 6, Da = 5 × 106; the dash line represents the surface of porous cylinder.
From the point of view of vorticity generation, the present detached wake is similar to that behind a viscous drop. However, the difference is that the flow can pass through the porous bluff body while it recirculates inside the viscous drop. The flow exiting at the rear of the bluff body resembles that of ―base bleed‖, which also contributes to the occurrence of the detached wake as investigated by Leal and Arcrivos (1969). More interestingly, the wake penetrates into the porous circular cylinder when the Darcy number is lower than a certain value. In such an instance, the flow entering the rear of the cylinder resembles ―base suction‖. For flow past a bluff body, the flow pattern is chiefly controlled by the Reynolds number and the body shape (curvature). For the flow within a porous medium, it is mainly determined by Re and Da. However, for the flow past a porous bluff body, the situation is more complex due to the interaction of the external flow and the internal porous flow. The fluid velocity along the surface, which is mainly determined by the Darcy number and Reynolds number, is nonzero. Compared with that of the solid body, this velocity changes the rate of vorticity production on the surface. Also, it influences the transport of vorticity along the shear layer that exists behind the body. Thus, different from the attached wake existing on solid bodies, the recirculating wake is now detached from the porous surface. According to the entrainmentdetrainment mechanism (Leal and Acrivos, 1969), the recirculating wake behind bluff body is primarily due to the flow field that results from fluid being entrained into the inner side of this shear layer. The entrained fluid detrains and reverses itself in the direction to supply the entrainment needs of the shear layer. This description has been used to discuss the effects of base bleed on the wake structure by Leal and Acrivos (1969). It is also suitable to explain the effect of the exit velocity component
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67
along the rear surface (‗base bleed‘ velocity) in the present study. If the Darcy number is relatively small so that the exit velocity is small, and the amount of fluid being supplied into the nearwake is not sufficient to satisfy the entrainment needs of the shear layer along the detached streamline, the recirculating wake should form downstream. However, if the exit velocity is large enough due to a relatively large Da, the recirculating wake should disappear. On the other hand, if a significant portion of flow, characterized by a dimensionless value of q, passes through the porous body, the effective Reynolds number for flow around the cylinder is decreased to Reeff = (1q)Re. In this sense, a relatively larger Reynolds number is needed to induce the recirculating wake. Thus, at a high Da, both Da and Re itself primarily determine the critical Reynolds number for onset of a recirculating wake. However, at a very low Da, the ‗base bleed‘ rate is negligible and the detached wake is mainly influenced by the external flow. The limit Da → 0 can be regarded as a solid body and the critical Reynolds number for onset of a recirculating wake is equal to that of the solid body. It is worth noting that the noticeable differences exist between the flow behaviours behind the porous square and circular cylinders, due to the shape effect. Clearly, at the same Da, the critical Reynolds number for onset of a recirculating wake of the porous square cylinder is smaller than that of the porous circular cylinder as shown in Figure 13. As explained above, this is because the sharp edge of the square cylinder is more effective on vorticity generation compared with the smooth curvature of the circular one. Furthermore, the recirculating wake always initially develops outside of the porous square cylinder while may initially occur inside of the porous circular cylinder for a certain range of Da. Other minor differences include that the largest Darcy number of the onset of a recirculating wake for a porous square cylinder is slightly larger than that for a porous circular cylinder. As explained above, the sharp edge of the porous square cylinder might cause these differences. The detached wake has also been observed behind a viscous drop (Dandy and Leal, 1989). This detached wake may also disappear with a continuous increase in Re. Based on the detailed numerical simulations, Dandy and Leal (1989) concluded that these flow behaviours are controlled by the weak motion of the fluid inside the viscous drop. Similarly, in the present study, a small amount of fluid passes through the porous body, which causes the same behaviour of the recirculating wake. Both of the studies indicate that the necessary condition of the occurrence of detached wake is the weak motion of the fluid existing inside the bluff body, which appears to control the behaviour of the external flow near the body. The difference of the two flows is that the fluid only recirculates inside the viscous drop while it penetrates through the porous body. In this sense, this penetrating flow is more similar to the base bleed effect (Leal and Acrivos, 1969). The critical Reynolds number for onset of the detached wake behind the viscous drop occurs at Re ~ O(10), which is the same order as that of the solid sphere. The critical Reynolds numbers for the onset of a recirculating wake behind a porous circular/square cylinder and a porous sphere are Re ~ O(1) and Re ~ O(10) respectively, with the same order as those of their solid pairs. Specially, when Da → 0, the critical Reynolds numbers approach exactly those of the solid pairs. This suggests that, although the exact value of the critical Reynolds number for a porous bluff body is different from that of the solid pair, the underlying physics of the detached or penetrating wake behind the porous body should share common mechanism with that of the solid pair.
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3.3. Geometrical Parameters of Recirculating Wake
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The geometrical parameters considered here are the downstream distances to the leading and trailing edges of the recirculating wake (LL and LR). Note that the separation angle, which is often used to characterize the recirculating wake for flow past solid circular cylinder or sphere, is not considered in the present study because the recirculating wake does not develop from the surface of the porous circular cylinder or sphere. For the porous square cylinder, the downstream distances to the leading and trailing edges of the recirculating wake (LL and LR) are plotted against Re with Da as a parameter in Figure 16. The curves shown in Figure 16 indicate that for Da = 1 × 106 and 5 × 104, LL decreases slightly with increasing Re, while for Da = 7.4 × 103, LL initially decreases slightly and then increases with increasing Re. For a relative small Da ( = 5 × 104), LL becomes negative but rather close to zero for a certain range of Re, which means the recirculating wake slightly penetrates into the porous square cylinder. However, it is worth mentioning that the recirculating wake always initially develops downstream of the cylinder. Figure 16 shows that, for fixed Da, with increasing Re, LR increases linearly if Da = 1 × 106 and 5 × 104, while LR initially increases and then decreases if Da = 7.4 × 103. Figure 16 also indicates that for fixed Re, LR always decreases with a decrease in Da. For fixed Re, with a decrease in Da, LR increases when Re is less than ~30 while it may decreases if Re > ~30.
Figure 16. Downstream distances to the leading and trailing edges of the recirculating wake (L L and LR) against Re with Da as a parameter, porous square cylinder.
The lengths of the recirculating wake Lw for a porous square cylinder are plotted against Re with Da as a parameter in Figure 17. Also, Lw for the solid square cylinder obtained by the present simulations and by the simulations and the formulation of Sharma and Eswaran (2004) at different Re are presented in Figure 17. Apparently, for the solid cylinder, the present results agree well with those of Sharma and Eswaran (2004) and only a small discrepancy exists when Re is large (Re > 40). Also, as in the case of LR, with increasing Re,
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Lw linearly increases when Da is moderate (= 5 × 104) or small (= 1 × 106) while it initially increases and then decreases when Da is large (= 7.4 × 103). Generally, for fixed Re, the wake length increases with decreasing Da. However, when Re is large enough (~45), the wake length reaches its maximum (~3.27) at Da ~ 5 × 104 as shown in Figure 17.
2.4 LL, Da = 5e3 LL, Da = 1e4 LL, Da = 5e6 LR, Da = 5e3 LR, Da = 1e4 LR, Da = 5e6
2
1.6
LL, LR
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Figure 17. Length of the recirculating wake Lw against Re with Da as a parameter, porous square cylinder.
1.2
0.8
0.4
0
0.4
0
10
20
30
40
Re Figure 18. Downstream distances to the leading and trailing edges of the recirculating wake (L L and LR) against Re with Da as a parameter, porous circular cylinder. Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
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Peng Yu, Yan Zeng, Thong See Lee et al. 2.5 Da = 5e3 Da = 1e4 Da = 5e6 Solid, Present Simulation Solid, Coutanceau and Bouard
2
LW
1.5
1
0.5
0
0
10
20
30
40
Re Figure 19. Length of the recirculating wake Lw against Re with Da as a parameter, porous circular cylinder. 0.5 LR, solid sphere
0.3
LR
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0.4
0.2
0.1
0 7 10
10
6
5
10
10
4
3
10
10
2
Da Figure 20. Variation of LR as functions of Da, Re = 50, porous sphere.
For the porous circular cylinder, the downstream distances to the leading and trailing edges of the recirculating wake (LL and LR) are plotted against Re with Da as a parameter in Figure 18. Only the data for Re ≥ 10 are presented. It is worth mentioning that both L L and LR can be negative. The negative value of LL means that the rear of the recirculating wake is penetrated into the cylinder and the negative value of LR means that the whole recirculating wake is within the cylinder. As shown in Figure 18, LL initially decreases and then increases
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with Re for a fixed Da, which means that the leading edge of the recirculating wake moves first upstream and then downstream with an increase in Re. At the same Re, the leading edge moves downstream with an increase in Da. The value of LL increases from negative to positive when Da increases, which implies that the recirculating wake initially penetrates into and then is detached from the cylinder. Figure 18 also shows that within the range of Re investigated, LR almost increases linearly with Re. For a constant Re, LR increases with a decrease in Da. The lengths of the recirculating wake Lw for a porous circular cylinder are plotted against Re with Da as a parameter in Figure 19. Also, Lw for the solid circular cylinder obtained by the present simulations and by the experimental measurements of Coutanceau and Bouard (1977) at different Re are presented in Figure 19. It is shown that for the solid cylinder, the present results agree well with those of Coutanceau and Bouard (1977). For all the three Darcy numbers shown in Figure 19, Lw linearly increases with increasing Re. Generally, for fixed Re, the wake length increases with decreasing Da. However, it is seen that L w at a small Da (5 × 106 and 1 × 104) is larger than that of the solid cylinder when Re > ~15. As shown in Figure 18, a small part of the wake penetrates into the porous cylinder, thus resulting in a longer wake length. However, it is worth mentioning that at the same Re, the wake length of the solid cylinder is very close to LR when Da approaches 0. For the porous sphere, we only summarized the variation of the downstream distances to the trailing edge of the recirculating wake (LR) with Da at Re = 50. Figure 20 shows that LR quickly increases with decreasing Da and reaches the maximum value at Da ~ 2 × 10 3. LR then decreases with decreasing Da and reaches the minimum value at Da ~ 1 × 104. After that, it slowly increases with decreasing Da and asymptotically approaches a constant value as that of a solid sphere.
CONCLUSION The present study introduces a numerical method to simulate flows involving an interface between a homogenous fluid and a porous medium. The main novelty of the numerical method is to apply the theory developed by OchoaTapia and Whitaker (1998) to model the momentum jump condition at the boundary between a porous and fluid media, which includes both viscous and inertial jump parameters. A distinctive feature of the present method is the use of multiblock grids which, together with bodyfitted grids, makes it more suitable for handling complex geometries. The shear stress jump condition affects both the convective and diffusive fluxes. The normal stress condition, assumed continuous at the interface, is also needed in order to close the two sets of equations. The numerical method is applied to study the steady flow around and through permeable square and circular cylinders, as well as a permeable sphere over wide ranges of Darcy and Reynolds numbers. The effects of Darcy number on the flow behavior, particularly streamlines, wake geometry, and the critical Reynolds number for the onset of recirculating wake are investigated. For the porous square cylinder, the present simulations reveal that the recirculating wake existing behind is detached from the cylinder, but not attached to the cylinder as which occurs behind a solid cylinder. The recirculating wake is found to develop downstream of the porous
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square cylinder, but not from its surface as that of the solid cylinder. Also, there is only one ―separation‖ point along the horizontal axis, but not a pair of separation points as those on the surface of the solid square cylinder. For the porous circular cylinder and sphere, the flow phenomena revealed herein is rather similar to that of the porous square cylinder. The only difference is that the recirculating wake may develop inside of the porous circular cylinder or sphere under a certain range of Da. All these features support the conclusion of Leal (1989) that the existence of recirculating wake behind any body at large, but finite Reynolds number should be regarded as being a consequence of the accumulation of vorticity generated upstream on the body surface. When Da → 0, the wake behaviour of the porous body resembles that of the solid one, which suggests the underlying physics of the detached or penetrating wake behind the porous body may share common features with that of the solid one. The present study also suggests that the surface curvature has an important effect on the initial position of the onset of the recirculating wake as the wake only develops downstream of a porous square cylinder while it develops either downstream of or within a porous sphere or circular cylinder. The present wake behaviour can be explained from interaction between the external flow around and the internal flow within the porous bluff body. Obviously, the Reynolds number has an important effect on the wake structure. Also, the ‗base bleed‘ along the rear of the porous body is affected by both Re and Da, thus influencing the recirculating wake structure. The wake structure in turn affects the ‗base bleed‘, due to the ‗block‘ effect. Specially, the wake structure may also be influenced by the change of the effective Reynolds number due to the ‗base bleed‘ effect. The final wake structure is determined by the balance of all the effects.
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REFERENCES Beavers, GS; Joseph, DD. J. Fluid Mech., 1967, 30, 197207. Bhattacharyya, A; Raja Sekhar, GP. Chem. Eng. Sci., 2004, 59, 44814492. Bhattacharyya, S; Dhinakaran, S; Khalili, A. Chem. Eng. Sci., 2006, 61, 44514461. Braeckmans, K; De Smedt, SC; Leblans, M; Pauwels, R; Demeester, J. Nat. Rev. Drug Discovery, 2002, 1, 447456. Chen, XB; Yu, P; Winoto, SH; Low, HT. Int. J. Numer. Meth. Heat Fluid Flow, 2008, 18, 635655. Clift, R; Grace, JR; Weber, ME. Bubbles, drops, and particles; Academic Press: New York, USA, 1978. Costa, VAF; Oliveira, LA; Baliga, BR. Numer. Heat Transfer B: Fund. 2008, 53, 383411. Coutanceau, M; Bouard, RJ. Fluid Mech., 1977, 79, 231256. Dandy, D; Leal, LG. J. Fluid Mech., 1989, 208, 161192. Ferziger, JH; Perić, M. Computational Methods for Fluid Dynamics, 2nd ed; Springer: Berlin, German 1999. Goyeau, B; Lhuillier, D; Gobin, D; Velarde, MG. Int. J. Heat Mass Transfer, 2003, 46, 40714081. Jue, TC. Int. J. Numer. Meth. Heat Fluid Flow, 2004, 14, 649663. Kuznetsov, AV. Int. J. Heat Mass Transfer, 1998, 41, 25562560.
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Leal, LG. Phys. Fluids A, 1989, 1, 124131. Leal, LG; Acrivos, A. J. Fluid Mech., 1969, 38, 735752. Lilek, Ž; Muzaferija, S; Perić, M; Seidl, V. Numer. Heat Transfer B: Fund., 1997, 32, 385401. Masliyah, JH; Polikar, M; Can. J. Chem. Eng., 1980, 58, 299302. Neale, G; Nader, W. Can. J. Chem. Eng., 1974, 52, 475478. Nithiarasu, P; Seetharamu, KN; Sundararajan, T. Arch. Comput. Meth. Engng., 2002, 9, 342. Noack, B; Eckelmann, H. Phys. Fluids, 1994, 6, 124143. Noymer, PD; Glicksman, LR; Devendran, A. Chem. Eng. Sci., 1998, 53, 28592869. OchoaTapia, JA; Whitaker, S. Int. J. Heat Mass Transfer, 1995a, 38, 26352646. OchoaTapia, JA; Whitaker, S. Int. J. Heat Mass Transfer, 1995b, 38, 26472655. OchoaTapia, JA; Whitaker, S. J. Porous Media, 1998, 1, 201217. Partha, MK; Murthy, PVSN; Sekhar, GPR. J. Eng. Mech. ASCE, 2005, 131, 12911301. Rhie, CM; Chow, WL. AAIA J., 1983, 21, 15251532. Rivkind, VY; Ryskin, GM. Fluid Dyn., 1976, 11, 512. Sharma, A; Eswaran, V. Numer. Heat Tr. AAppl., 2004, 45, 247269. Silva, RA; de Lemos, MJS. Numer. Heat Transfer A: Appl., 2003, 43, 603617. Sohankar, A; Norberg, C; Davidson, L. Int. J. Numer. Meth. Fluids, 1998, 26, 3956. Underwood, RL. J. Fluid Mech., 1969, 37, 95114. Williamson, CHK. Annu. Rev. Fluid Mech., 1996, 28, 477539. Yu, P; Lee, TS; Zeng, Y; Low, HT. Int. J. Numer. Methods Fluids, 2007, 53, 17551775. Yu, P; Zeng, Y; Lee, TS; Low, HT. Int. J. Heat Mass Transfer, 2009a, 52, 316327. Yu, P; Lee, TS; Zeng, Y; Meguid, SA; Low, HT. Int. J. Numer. Methods Fluids, 2009b, 60, 337353. Yu, P; Zeng, Y; Lee, TS; Bai, HX; Low, HT. Int. J. Heat Fluid Flow, 2010, 31, 141153. Zaki, TG; Sen, M; GadelHak, M. J. Fluid Struct., 1994, 8, 555582. Zdravkovich, MM. Flow around circular cylinders: volume 1: fundamentals; Oxford University Press: New York, USA, 1997. Zdravkovich, MM. Flow around circular cylinders: volume 2: applications; Oxford University Press: New York, USA, 2003.
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In: Fluid Transport: Theory, Dynamics and Applications ISBN: 9781611223170 Editor: Emma T. Berg, pp. 75120 © 2011 Nova Science Publishers, Inc.
Chapter 3
FLUID FLOW AND HEAT TRANSPORT: THEORY, NUMERICAL MODELING AND APPLICATIONS FOR THE FORMATION OF MINERAL DEPOSITS Jianwen Yang* Department of Earth and Environmental Sciences, University of Windsor, Windsor, Ontario, Canada
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SUMMARY Subsurface fluid flow and heat transport play an important role in many geological processes, such as hydrocarbon migration, structural geology, plate tectonics, diagenesis, and metamorphism (Ingebritsen and Sanford, 1999), and in particular they have major implications for both the formation and preservation of almost every class of economic ore deposit, especially those in sedimentary basins. This chapter is dedicated to addressing fundamental theory and numerical modeling technique of fluid flow and heat transport in subsurface porous media as well as presenting two application examples associated with ore genesis under different geological conditions. The chapter is organized into four sections. The first section is aimed at theoretical aspect of fluid flow and heat transport. It first introduces major physical processes involved in a hydrothermal flow system and outlines a variety of driving mechanisms deemed responsible for largescale groundwater flow (i.e., topography, buoyancy, tectonic deformation, and sediment compaction), and then describes the mathematical equations that govern and control the behavior of subsurface fluid migration and thermal regime. It ends up with deriving an analytical solution using the Laplace transformation to address the heat transport process subject to a constant fluid flow in a single fracture embedded in an impermeable host porous medium. The second section concentrates on the numerical modeling aspect of fluid flow and heat transport by detailing the Galerkin finite element technique that is capable of numerically simulating complex hydrothermal flow systems. Noticing that topography and buoyancy are the two top driving forces commonly *
Corresponding author: Tel: (519)2533000/ext 2181; Fax: (519)9737081; Email: [email protected]
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encountered in reality and that the interactions between them are still poorly understood, the third section is therefore attempts to quantify the relative importance of these two mechanisms in driving fluid transport via a series of numerical experiments, and also to determine under what conditions both topography and buoyancydriven flows coexist and under what conditions one flow system dominates the other. The fourth section of this chapter focuses on the application aspect of fluid flow and heat transport associated with the formation of mineral deposits, including the Sedextype leadzinc deposits in northern Australia (with buoyancy as the primary driving force), and the Dachang polymetallic deposits in southern China (with tectonic deformation as the primary driving force).
1. THEORY This section aims at the theoretical aspect of fluid flow and heat transport in porous media. It starts with introducing major physical processes involved in a hydrothermal flow system and discussing different driving mechanisms for groundwater flow, then addresses governing and supplemental equations that fully describe the behavior of a hydrothermal fluid flow system, and finally presents an analytical solution of heat transport in a single fracture situated in an impermeable host porous medium.
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1.1. Physical Processes A hydrothermal fluid flow system involves complicated chemical and physical processes as well as various waterrock interactions. The major physical processes involved include heat transport and fluid flow.
1.1.1. Heat transport There are three types of heat transport as follows. 1. Heat conduction Conductive heat transport takes place via molecular collisions. It is a diffusive process wherein molecules transmit their kinetic energy to other molecules by colliding with them. Heat travels through a medium in which there is a spatial variation in temperature, diffusing from hightemperature area to lowtemperature area. 2. Heat convection Convective heat transport is associated with the motion of a fluid. It is an advective process wherein a fluid flow ‗carries‘ heat to travel with itself. Depending on the amplitude of fluid flow rate, heat convection is usually more efficient than heat conduction in transporting heat.
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3. Thermal radiation It is essentially a thermal electromagnetic radiation. All surfaces of finite temperature emit energy in the form of electromagnetic waves. For instance, heat from the Sun reaches the Earth by thermal radiation. Among these three types of heat transport, heat conduction and heat convection need to be considered for a hydrothermal fluid flow system. Thermal radiation is important only for atmospheric physics. 1.1.2. Fluid flow and driving forces A variety of mechanisms are deemed responsible for largescale fluid flow (e.g., Garven, 1995), including 1. Topography Topographydriven fluid flow is associated with the differences in elevation of water table that create differences in potential energy and hence drive fluid flow from high elevation to low elevation.
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2. Buoyancy Buoyancydriven fluid flow results from variations in fluid density due to spatial and temporal changes in temperature and/or salinity, and is usually named ‗free convection‘ because of the lack of external inputs and outputs. 3. Tectonic deformation Tectonic deformationdriven fluid flow is associated mainly with variations in pore fluid pressure due to the deformation. Surface topography resulting from tectonic deformation facilitates fluid flow as well. 4. Sediment compaction Sediment compactiondriven fluid flow results from variations in pore fluid pressure due to sediment loading with subsidence and burial. A rapid sedimentation can produce an overpressure zone at depth, which drives basinal fluids to move upwards. Among these driving mechanisms, topographydriven force is normally considered to overwhelm other forces since topography typically leads to a maximum flow rate of 1–10 m/year while buoyancy and tectonic deformation usually approach a maximum of 1 m/year and sediment compaction commonly reaches 1cm/year (Garven, 1995; Evans and Nunn, 1989). The relative importance of topography and buoyancy in driving groundwater flow will be addressed in section 3 by a series of numerical case studies.
1.2. Governing Equations Mathematical equations that govern the physical behaviour of fluid flow and heat transport in porous media can be derived by considering the continuity of fluid mass and heat energy, including the equation of fluid motion, the equation of fluid mass conservation, and the equation of energy conservation. Derivation of these equations is trivial if assuming that a
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porous medium's physical properties (e.g., density, permeability and porosity) are functions of space only, the medium is saturated by a singlecomponent fluid (water), and thermal equilibrium exists between water and solid matrix.
1.2.1. The equation of fluid motion In 1985, Henry Darcy found that the flow through porous media is linearly proportional to the applied pressure gradient and inversely proportional to the viscosity of the fluid. For a 1D geometry in which fluid is driven by the applied pressure gradient in xdirection, Darcy's law takes the form
k ( p1 p0 ) k dp w L w dx ,
(1)
where q is the Darcy flux in the xdirection
(ms 1 ) , k is the permeability
q
2
( m ),
w is the dynamic viscosity of fluid ( kg m1s1 ), p 0 is the pressure at the entrance to
the section ( kg m1s2 ), p1 is the pressure at the section exit, and L is the length of the section (m). The experimentally derived form of Darcy's law was limited to 1D flow. When the flow is threedimensional, it can be generalized as (Bear, 1972)
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q
k p p p nx ny ( w g )nz , w x y z
(2)
where q is the Darcy flux vector with three components ( qx , q y , qz ) in the x, y and zdirections, respectively, n x , n y and n z are three unit vectors in these directions, w is the 3
fluid density ( kg m ), and g is the gravitational acceleration ( ms
2
).
The pressure p in eq. (2) can be removed through the use of a ‗equivalent freshwater head‘ h defined as (Frind, 1982) h
where
p
0 g
z
,
(3)
0 is a reference density of fluid, and z is the elevation above a datum (m). Under
these rules, the Darcy equation (2) is of the form
h h h q K nx ny ( r )nz , y z x where K is the hydraulic conductivity (ms1) defined by
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(4)
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K
and
79
k 0 g
w
(5)
r is the relative density of fluid defined by r
w 1 0 .
(6)
1.2.2. The Equation of fluid mass If the porous medium is nondeformable, the fluid mass conservation equation can be written as (Bear, 1972) h h h h K r Ss 0, K K K x x y y z z t
(7)
where S s is the specific storage (dependent on the fluid density and compressibility, rock matrix compressibility, and porosity), and t is the time (s).
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1.2.3. The equation of thermal energy conservation If there exists a thermal equilibrium between the fluid and the solid matrix, the thermal energy conservation equation can be expressed as (Bear, 1972)
T T T m m m cw w qxT x x y y z z x T cw w q yT cw w qzT cw w cs s (1 ) 0, y z t where is the porosity,
(8)
m w s 1 ( w and s denote the thermal conductivity of the
fluid and solid phase) is the thermal conductivity of the porous medium, c w and c s are the specific heat capacity of the fluid and solid phase,
s is the density of the solid phase, and T
is the temperature. On the left hand side of (8), the first three terms represent the conductive heat transport, the second three terms express the convective heat transport related to the actual movement of fluid, and the last term is the heat stored in unit volume during the unit time due to the variation in temperature.
1.2.4. Supplemental equations Darcy's law, and the fluid mass and thermal energy conservation equations are not sufficient to describe a hydrothermal fluid flow system, and they have to be supplemented by the definition of the temperature dependence of fluid density and viscosity. In previous
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studies on dimensional analysis of hydrothermal systems, the fluid viscosity is usually assumed a constant, whereas the fluid density is assumed to be a linear function of temperature. In fact, the temperature dependence of fluid density and viscosity is very complex, generally nonlinear; thus more realistic supplementary equations must be defined. This can be achieved either using interpolation functions to fit the published data base (e.g., Weast, 1980) or carrying out explicit calculation using commercial properties software packages, e.g., NIST/ASME Steam Properties Code (Klein and Harvey, 1996). Either ways can be easily implemented in fluid flow modeling code. Over the temperature range 0100C, for instance, the function defining the temperature dependence of water density is given by a thirddegree polynomial of the form (Molson et al., 1992)
w (T ) 1000.0 1.0 0.435 105 0.838 108 (T 4.0) (T 4.0)2 ,
(9)
and the function defining the temperaturedependent fluid viscosity can be expressed as an exponential function of the form
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w (T ) 1.787 103 exp (0.0254 0.443 104 T )T
.
(10)
Functions (9) and (10) were obtained by a linear leastsquares regression fit to observed water density and viscosity data (Weast, 1980). For a wider range of temperature variations, similar methodology can be used to get the temperature dependence of fluid density and viscosity. Equations (1)(10) fully describe heat transport and fluid flow in porous media driven by topography and buoyancy forces. The governing equations for tectonicdriven fluid flow will be discussed in section 4.2.
1.3. Heat Transport in Fractured Porous Media: Analytical Solution for a Single Fracture Fractures play an important role in controlling fluid flow, heat transfer and contaminant transport, due to their commonly higher permeability compared with that of the host media. If fluid flow rate is assumed to be constant in a single fracture or a set of parallel fractures embedded in an impervious host rock, then analytical solutions are derivable to define nonreactive solute transport (e.g., Tang et al., 1981; Sudicky and Friend, 1982). It is well known that fluids flowing through fractures exchange heat with the surrounding medium. In this section, an analytical expression is developed by using the Laplace transform for the heat transport in a single fracture which is situated in an impervious host rock. More realistic boundary conditions and heat transport mechanism are taken into account. The analytical solution can be used in particular to evaluate the accuracy of numerical modeling results, and it is also useful in obtaining firstorder estimates of heat transport problems in which only limited data are available to constrain and justify more complex numerical models.
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Consider the case of a thin rigid fracture situated in a saturated porous rock, as illustrated in Figure 1. The water velocity in the fracture v z is assumed constant, and a heat source of
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the constant temperature T0 exists at the origin of the fracture. It is further assumed that the width of the fracture 2b is much smaller than its length so that the variation in temperature across the fracture aperture is negligible, the fluid is always instantaneously and thoroughly mixed across the fracture and thus thermal equilibrium exists between fluid and solid matrix, and the temperature in the rock mass, immediately outside the fracture, is approximately equal to the temperature of the fracture. The porous matrix is impervious and hence heat transport therein is mainly by conduction, whereas heat transport along the fracture is by both conduction and convection, much faster than that within the matrix. The temperature variation is assumed so small that w , w , m and cw are constant throughout the system.
Figure 1. A simplified 2D fracturematrix system..
These assumptions provide the basis for a 1D representation of heat transport along the fracture itself and for taking the direction of heat flux in the porous matrix to be perpendicular to the fracture. This results in the simplification of the basically 2D system to two orthogonal, coupled 1D systems.
1.3.1. Governing equations The heat transport processes in the system of Figure 1 can be described by two coupled, 1D equations, one for the fracture and one for the porous matrix. The coupling is provided by the continuity of the temperature along the interface. The differential equation of the heat transport for the fracture can be obtained by averaging the general energy conservation equation (8) at any point within the fracture across the fracture aperture. Referring to the geometry of Figure 1 and noticing that 1 and the
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Darcy flux is equal to the velocity of fluid particle for the open watersaturated fracture, one has
T T T b x w x y w y z w z x cw wvxT T cw wv yT cw wvzT (cw w ) dx 0, y z t b
(11)
where vx , v y and vz are the fluid velocity in the x, y and zdirections. Equation (11) can be rewritten as
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T T ( wcwv yT ) ( wcwvzT ) T 2b w wcw w y z t y y z z T T w x wcwvxT x b w x wcwvxT x b 0,
(12)
where the last two terms on the lefthand side of (12) represent influx heatflow and efflux heatflow through the two fracture walls due to fluid leakage and molecular diffusion. Also, these two terms together with an assumed continuity in temperature along the interface provide the link between energy conservation equations in porous medium and in the fracture. Equation (12) is a general expression of the energy conservation principle in the fracture. Since the facture is imbedded in an impervious rock, the normal component of fluid flow in the fracture vx should be zero. Notice that when the fracture aperture is very small, temperature distribution is symmetric about the facture, one has
T w x
x b
T w x
x b
. (13)
These two conditions together reduce (12) to T T ( wcwv yT ) ( wcwvzT ) T w T wcw w w y y z z y z t b x
x b
0.
(14)
Recognizing
T w x
x b
T ' m x
x b
,
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Fluid Flow and Heat Transport where T ' and
83
m are respectively the temperature and thermal conductivity of the host matrix
rock, thus (14) can be expressed m T ' T (vyT ) (vzT ) w T w T t y z y wcw y z wcw z wcwb x
x b
0.
For a 2D problem on the xz plane, T / y 0 and v y 0 . Also noticing that vz ,
(16)
w ,
w , m and cw are assumed constant, thus (16) becomes
w 2T T ' T T vz m 2 t z wcw z wcwb x
x b
0
z 0 . (17)
The differential equation of the heat transport for the host matrix rock can be obtained by substituting a zero Darcy flux into (8) and taking the direction of heat flux to be perpendicular to the fracture under the assumptions stated above. For a 2D problem on the xz plane, temperature in the host matrix T ' satisfies
T ' m 2T ' 0 t cm m x 2
(b x ), (18)
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where m and cm are respectively the density and specific heat of the host porous matrix, and
cm m cw w cs s (1 ) . The boundary and initial conditions for (17) are T (0, t ) T0 ,
(19)
T (, t ) 0,
(20)
T ( z , 0) 0.
(21)
The boundary and initial conditions for (18) are
T '(b, z, t ) T ( z , t ),
(22)
T '(, z, t ) 0,
(23)
T '( x, z , 0) 0.
(24)
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Clearly, the boundary condition (22) represents the coupling of the porous matrix to the fracture.
1.3.2. General transient solution Applying the Laplace transformation to (18) yields
m d 2T ' sT ' , cm m dx 2
(25)
where T ' is the Laplace transformation of T ' , defined as
T '( x, z, s) exp(st )T '( x, z, t )dt.
(26)
0
The only possible solution for (25) is of the form
c T '( x, z, s) c1 exp m m m
1/2 s ( x b) ,
(27)
where the constant c1 can be determined to equal to T ( z, s) , the Laplace transformation of
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T ( z , t ) , using the boundary condition (22). Thus (27) becomes c T '( x, z, s) T ( z, s) exp m m m
1/2 s ( x b) .
(28)
The partial gradient of T ' with respect to x at the interface x=b is
T '( x, z, s) x
x b
c T ( z , s ) m m m
1/2
s .
(29)
Applying the Laplace transformation to (17) yields
sT vz
d 2T T ' dT w m 2 dz cw w dx cw wb x
x b
0.
Substituting (29) into (30), one has
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(30)
Fluid Flow and Heat Transport
85
c dT cw w d 2T vz w w ( s s1/2 / A)T 0, 2 dz w dz w where
A bcw w / (m cm m )1/2
(31)
.
X w / cw w , then (31) becomes Let
d 2T vz dT 1 ( s s1/2 / A)T 0, 2 dz X dz X
(32)
which is a secondorder ordinary differential equation, and its solution has the form
T ( z, s) c2 exp( zr ) c3 exp( zr ),
(33)
where c2 and c3 are undetermined constants and r takes on the two forms
1/2
r 1 1 2 ( s s1/2 / A)
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where
,
(34)
vz / 2 X and 2 4 X / vz2 .
The boundary condition (20) requires c2 0 . Thus (33) becomes
1/2
T ( z, s) c3 exp( z ) exp z 1 2 ( s s1/2 / A)
.
(35)
Applying the Laplace transformation to the boundary condition (19) yields
T (0, s)
T0 , s
(36)
which is obviously the value of the constant c3 . Equation (35) therefore becomes
T ( z, s)
1/2 T0 exp( z ) exp z 1 2 ( s s1/2 / A) . s
Based on the identity
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(37)
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exp(2 )
2
1/2
0
2 exp 2 2 d ,
(38)
the exponential term in (37) can be converted into an integral form
1/2
exp z 1 2 (s s1/2 / A)
2
1/2
0
1/2 2 z2 exp 2 2 1 2 ( s s1/2 / A) d . 4
(39)
Substituting (39) into (37) yields 2 z2 2 1 T 2 2 z2 1/2 exp( z ) exp 2 2 exp ( As s1/2 ) d . 2 0 T0 4 4 A s
(40)
1
Thus the original temperature T will be given in terms of the inverse transform L as 2 z2 2 1 T 2 2 z 2 1 1/2 exp( z ) exp 2 L exp ( As s1/2 ) d . 2 2 0 T0 4 4 A s
(41)
To evaluate the inverse transformation, the following two identities are needed
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exp 2a s a L1 erfc , s t
(42)
L1 exp sE f ( s) f (t E )U (t E ),
(43)
and
where erfc is the complementary error function, defined as
erfc( x) 1
2
1/2
x
0
exp( 2 )d , (44)
and
U 0
(t E); U 1
(t E).
On the basis of (44) and (45), one has
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(45)
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2 z2 2 2 z 2 2 1/2 1 L1 exp s erfc 2 , 2 4 A s 8 A t
(46)
and 2 z2 2 1 2 z2 2 2 z2 2 L1 exp As s1/2 U t erfc 2 2 2 2 2 2 2 4 4 A s 8 A t z / 4
.
(47)
Substituting (46) and (47) into (41) yields the final normalized temperature solution in the fracture as 2 z T ( z, t ) 2 2 z2 2 1/2 exp( z ) z exp erfc 2 2 2 2 T0 4 2 t 8 2 A t z 4 2 2 2
d .
(48)
Let us now find the temperature distribution in the host porous matrix. Substituting (40) into (28) gives rise to
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T' 2 2 z2 1/2 exp( z ) exp 2 0 T0 4 2
c 1/2 2 z2 2 1/2 1 m m d . exp ( As s ) exp s ( x b ) 2 m 4 A s
(49)
Thus T' 2 2 z2 1/2 exp( z ) exp 2 2 0 T0 4
2 z 2 2 c 1/2 1 2 z 2 2 L exp m m ( x b) s1/2 exp s d . 2 2 s 4 A m 4 1
(50)
Once again, conducting the inverse Laplace transformation leads to the final normalized temperature solution in the host porous matrix as 2 z T '( x, z , t ) 2 Y 1/2 exp( z ) z exp erfc 2 2 2 2 T0 4 2 t 2 t z 4 2 2 2
d ,
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(51)
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2 z 2 2 cm m Y ( x b). 4 2 A m
(52)
1.3.3. Steady state solution Equations (48) and (51) provide the general transient solutions for the temperature distribution in the fracture and host porous matrix, respectively. Now let us investigate the steady state solution when the time reaches infinite. In (48), when t , the lower limit of integration is zero, and the complementary function is unity. Based on the identity (38), on has T ( z, t ) 2 2z2 1/2 exp( z ) exp 2 2 d 0 T0 4
2
exp( z ) 1/2
1/2 2
exp( z )
1.
(53)
Similarly,
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T '( x, z , t ) 1. T0
(54)
These results indicate that at the steady state ( t ), the whole system will reach the constant temperature T0 of the heat source.
1.3.4. Illustrative examples The fracturematrix is assigned the following thermal transport properties: cw 4147 J kg 1 o C1 cm 800 J kg 1 o C1 m 2650 kg m3 , w 1000 kg m3 , , , m 2.0 J m 1 s 1 oC 1 , and w 0.5 J m 1 s 1 o C1 . When the fluid velocity along the fracture
v 5 103 m s 1
is equal to z , the normalized temperature distribution in the fracture and within the host matrix is illustrated in Figure 2 at a time of 1.0 and 3.0 days. When the fluid v 102 m s 1 velocity is equal to z , the normalized temperature distribution are illustrated in Figure 3 at a time of 0.5 and 1.5 days. It can be seen from these results that fluid flowing a fracture in impermeable rock exchanges a great amount of heat with the surrounding medium. As fluid transports along the fracture, it warms the fracture itself, as well as the host matrix due to the heat released from the fracture by thermal diffusion (conduction). As time progresses, both the fracture and host medium becomes warmer. Comparing Figures 2 and 3 indicates that a greater fluid velocity leads to a quicker increase in temperature of the solution domain. The convective heat
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transport due to the fluid flow along the fracture is faster than the conductive heat transport perpendicular to the fracture in the host rock. Thus, the temperature contours are characterized by a trianglelike shape with the fracture as its symmetrical axis. The greater the fluid velocity, the sharper the ‗triangle‘ becomes.
Figure 2. Normalized temperature contours corresponding to a fluid velocity of vz 5.0 103 m s 1 at a time of: (a) 1 day, and (b) 3 days. The contour intervals are from 0.1 to 1.0.
2. NUMERICAL SIMULATION METHOD In above section, an analytical solution has been developed to describe heat transport in a highly conceptualized 2D fracturematrix system. In general, it is very difficult, if not
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impossible, to get analytical solutions due to the fact that fluid mass and thermal energy conservation equations (7) and (8) form a transient and nonlinear system, coupled through the Darcy's equation (4) and nonlinearities also exist in the temperaturedependent fluid density and viscosity (9) and (10). Therefore, numerical modeling becomes a must in order to fully understand realistic hydrothermal system. In this section, finite element modeling technique will be discussed in detail.
Figure 3. Normalized temperature contours corresponding to a fluid velocity of vz 1.0 102 m s1 at a time of: (a) 0.5 day, and (b) 1.5 days. The contour intervals are from 0.1 to 1.0. Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
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2.1. Finite Element Method In the finite element method with linear elements, a complex function is always approximated by means of a simple interpolation function, defined in terms of nodal values. The key to the method is a minimization principle which allows one to solve for the nodal values in such a way that the numerical error is minimized on average over the domain. To generate the algebraic equations of the unknown nodal values, the Weighted Residual Method is applied. Assume a partial differential equation of the form
(u ) 0,
(55)
where is a Cartesian differential operator. Let a trial solution be expressed in the form n
u( x, y, z ) uˆ( x, y, z ) u j N j ( x, y, z ),
(56)
j 1
where u j are nodal values, N j ( x, y , z ) are the basis functions, and n is the total number of nodes in the domain. Substituting the trial solution into the original differential equation (55) yields
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uˆ( x, y, z) R( x, y, z ) 0.
(57)
The nonzero residual R ( x, y, z ) on the righthand side expresses the error due to the approximate representation. According to the theory of weighted residuals, R ( x, y, z ) can be minimized on the average over the domain by satisfying a set of weighted residual equations, which are:
V
R( x, y, z)Wi ( x, y, z)dV 0, i 1, 2,..., n,
(58)
where V designates the solution domain, and Wi ( x, y, z ) are a set of n weighting functions corresponding to the n nodes. In the general weighted residual method, the weighting functions are independently chosen, and many forms of weighting functions are possible. A special variant of weighted residuals is the Galerkin Method, in which the weighting functions Wi are chosen to be identical to the basis N i . This choice has some advantages, for example, the coefficient matrix for the flow equation is symmetrical. The Galerkin Method has been well proven in groundwater flow and heat transport (e.g., Huyakorn and Pinder, 1983). By setting the integral to zero in (58), the weighted residual equation effectively forces the residual to zero since the integral of weighting function is always nonzero. But because
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the integral equation will also be satisfied if R were nonzero locally in such a way that the error would average out to zero over the domain, the error is forced to zero globally, that is, over the domain as a whole. By writing one weighted residual equation for each node, one requires that the error everywhere is forced to zero on the average. Substituting (56) and (57) into the Galerkin equation (58) yields
n u N ( x , y , z ) N i ( x, y, z )dV 0, i 1, 2,..., n, j j V j 1
(59)
representing n equations for n unknowns, which can be solved for the unknown nodal values uj .
2.2. The Galerkin Element Solutions of the Fluid and Energy Conservation Equations The solution domain is discretized into a network of finite elements. Again, the trial solutions are expressed as n
hˆ( x, y, z, t ) N j ( x, y, z )h j (t ), j 1
(60)
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n
Tˆ ( x, y, z, t ) N j ( x, y, z )T j (t ), j 1
(61)
where hˆ is the approximate hydraulic head, Tˆ is the approximate temperature, and h j and
T j are the hydraulic head and temperature at node j. The Galerkin method of weighted residuals is then used to transform these governing equations into integral equations of the form
Ni 1 (hˆ)dV 0,
Ni 2 (Tˆ )dV 0,
V
V
(62)
(63)
where 1 and 2 represent the differential operators of the lefthand side of equations (7) and (8).
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Since hˆ and Tˆ are expressed by linear basis functions (60) and (61), their second derivatives will be zero. In order to obtain a nontrivial solution, the second derivatives must therefore be eliminated through integration by parts using Green's theorem
V
adb a db B
V
dbda,
(64)
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where B denotes the boundary of the solution domain V. The integrals of the terms involving Nj and their firstorder derivatives over each element can be performed using the analytical influence coefficient method for rectangular elements (e.g., Huyakorn et al., 1986) or using the gauss quadrature numerical integration for irregular elements (e.g., Yang, 2006). Summing the elemental contributions yields
M ijh dh j / dt Sijh h j Fi h 0,
(65)
M ijT dT j / dt SijT T j FiT 0,
(66)
where i, j=1, 2, …, n (n is the total number of nodes in the system). In (65) and (66), the coefficient matrices [M], [S], and [F] can be obtained from the textbooks (e.g., Huyakorn and Pinder, 1983) for rectangular elements or using the gauss quadrature numerical integration for irregular elements. The above equations are coupled through the convective terms and the temperaturedependent variations of fluid density and viscosity. Approximating the temporal derivatives in (65) and (66) by a weighted firstorder finite difference approach, these equations can be rewritten as
h M ijh Sij t
t t
T M ijT Sij t
t
h
t t j
Fi
ht t
t t
M ijh (1 ) Fi (1 ) Sijh htj , t ht
(67)
t
T
t t j
Fi
T t t
M ijT (1 ) Fi (1 ) SijT T jt , t Tt
(68)
is the timeweighting factor, ranging from 0.5 to 1.0, and t is the size of time step. In particular, 1 and 0.5 correspond to the fully implicit and CrankNicholson time where
scheme, respectively. Equations (67) and (68) form a system of 2n, nonlinear simultaneous equations, which can be solved by updating the fluid properties at each timestepping cycle and by using the Picard iterative technique (Huyakorn and Pinder, 1983). Listed here are the major modeling i
steps: 1) update the fluid density and viscosity using the latest temperature T as defined in (9) and (10) from the previous time step; 2) determine the ‗equivalent freshwater‘ head Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
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distribution h based on the updated fluid properties by solving the fluid continuity matrix equation (67); 3) determine the Darcy flux based on Darcy's law (4) using the updated fluid i 1
i 1
properties and hydraulic head h ; 4) calculate the temperature distribution T by solving the thermal energy conservation matrix equation (68); and 5) check whether the preset convergence criteria
Max hij1 hij h and Max T ji 1 H ij T
(69)
are satisfied. If not, steps 1) through 4) are repeated; if yes, move on to the next time step. In designing the finite element grid, the primary considerations are the grid Peclet and Courant criteria, defined as (Daus et al., 1985)
Px
v y y vx x v z 2, Py 2, Pz z 2, km km km
(70)
v t vx t v t 1, C y y 1, Cz z 1, x y z
(71)
Cx
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where k m is thermal diffusivity of the porous medium, and x, y and z are grid spacings in the x, y and zdirections. These criteria should be used to determine the grid spacing and time step prior to numerical calculations for any given models. The finite element modeling method discussed in this section is capable of simulating heat transport and fluid flow driven by buoyancy and topography. For fluid flow driven by tectonic deformation, the governing equations and numerical modeling technique will be addressed in section 4.2.
3. RELATIVE IMPORTANCE OF TOPOGRAPHY AND BUOYANCY IN DRIVING GROUNDWATER FLOW As discussed in section 1, topography and buoyancy are the two driving forces commonly encountered in reality, particularly in largescale groundwater systems. Topographydriven force is normally considered to overwhelm buoyancy force since the former typically leads to a maximum flow rate of 110 m/year (Garven, 1995), while the latter usually approaches only 1 m/year (Evans and Nunn, 1989). But in certain circumstances, buoyancydriven flow may modify and/or limit topographydriven flow, as evidenced by the longterm presence of evaporates and brines in many sedimentary basins (Thornton and Wilson, 2007; Ranganathan, 1993). The interactions between topography and buoyancydriven flows are still not fully understood and hence need to be investigated further. Toth (1962, 1963] developed analytical solutions of topographydriven groundwater flow in small drainage basins of Central Alberta, Canada, but without considering the effect of
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buoyancy force. In their interpretation of seafloor heat flow variations, several researchers have investigated the influence of basement topography on hydrothermal circulation in oceanic crust. For instance, Hartline and Lister (1981] conducted laboratory HeleShaw cell experiments to investigate topographic focus of hydrothermal convection in porous media simulating oceanic crust, and found that topography controlled convection only for Rayleigh numbers less than a critical value. Fisher and Becker (1995] argued that the convection pattern in sedimentburied oceanic crust might be controlled only by basement topography, and indicated that the wavelength of observed seafloor heat flow anomalies was consistent with the wavelength of the basement relief. Wang et al. (1997] conducted a numerical modeling to address how basement topography controls steady state convection for different Rayleigh region. However, all these models are for submarine conditions, and hence water table is assumed to be flat. Recent numerical modeling studies have indicated that topographydriven flow and free convection can coexist under certain conditions. For instance, Lopez and Smith (1995) examined the interaction of thermally driven convection in a steeply dipping fault zone and groundwater flow in the surrounding host rock driven by a regional topographic gradient. Stanislavsky and Gvirtzman (1999) conducted hydrological modeling of the Dead Sea basin and found that buoyancydriven flow might have coexisted with topographydriven flow when evaporated seawater began to sink. Thornton and Wilson (2007) addressed the importance of buoyancydriven flow in an uplifted sedimentary basin where topographydriven has long been thought to be dominant. However, all these studies assumed steadystate convection conditions, which is not always the case. In particular, steadystate solutions usually do not exist since topographydriven flow can cause convective cells to migrate continuously with time in the downhill direction, as will be discussed in detail below. In addition, the previous modeling studies are casespecific, and their conclusions cannot be easily generalized. The purpose of this section is to quantify the relative importance of topography and buoyancy force in driving groundwater flow. A series of 2D numerical experiments are conducted to determine under what conditions both topography and buoyancydriven flows coexist and under what conditions one flow system dominates the other.
3.1. A 2D Conceptual Model To simplify the physics involved, let us consider a cross section between a topographic high (water divide) and an adjacent topographic low (valley bottom). As illustrated in Figure 4, the 2D model is bounded by two vertical planes beneath the valley bottom (righthand side boundary) and the water divide (lefthand side boundary) and by a horizontal impermeable boundary overlying the bedrock at bottom and a linearly sloping water table at top. The water table is assumed a straight line, which is reasonable when the angle of its slope dH/L is less than 1o (Toth, 1962). The temperature dependence of fluid density and viscosity is generally nonlinear, as shown in (9) and (10). In this section, however, it is assumed that the fluid viscosity is a constant and the fluid density is a linear function of temperature in order to introduce a dimensionless parameter, the Rayleigh number. The fluid density is assumed to have the form
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w 0 (1 T ),
(72)
where is the fluid volume thermal expansion coefficient. These assumptions are reasonable when the temperature range is small. Refer to Figure 4, hydraulic head on the top boundary decreases linearly from H+dH at the water divide to H at the valley bottom, and its normal gradient is equal to zero over the other boundaries. As for thermal boundary conditions, the side boundaries are assumed adiabatic, the top boundary is fixed at a constant temperature of 20oC, and the lower boundary is assigned a higher temperature, depending upon the depth H and geothermal gradients. Let us assign H=200m and L=800m, while change dH to reflect different slopes of the water table. The temperature on the lower boundary is fixed at 26oC corresponding to a typical geothermal gradient of 30oC/km. The solution domain is discretized by a nonorthogonal quadrilateral mesh consisting of 40 element columns and 10 rows in the horizontal and vertical directions. In addition, the 2D porous medium is homogeneous and assigned the following physical parameters, characterized by a typical unconfined sand aquifer:
cs 800 J kg 1 o C1 , cw 4147 J kg 1 o C1 , s 2630 kg m 3 , 0 1000 kg m3 , s 2.0 J m1 s1 o C1 , w 0.5 J m1 s1 o C1 , 5.0 104 o C1 , w 1.787 kg m1 s1 ,
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k 1011 m2 , and
0.1 .
Figure 4. 2D conceptual model simulating a cross section between a water divide and an adjacent valley bottom.
3.2. Numerical Modeling Results A variety of numerical case studies are conducted with different slopes of the water table and other conditions. Case 1: the water table is perfectly flat (i.e., dH=0 m). This implies that there is no topographic effect and buoyancy is the only driving force for fluid flow. With the given physical parameters and the model dimension as stated above, the dimensionless Rayleigh number is equal to 35, greater than the critical value 27.1 for the onset of free convection in a porous medium that is heated from below and bounded by a free surface with isothermal condition (Combarnous and Bories, 1975). Therefore, free convection can be initiated and maintained in the solution domain. Figure 5 shows the temperature distribution and fluid velocity vectors at steady state, from which it can be seen that four convection cells have developed with an aspect ratio of 1. The maximum fluid velocity is equal to 5.45×107m/s.
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Figure 5. Numerical results corresponding to the condition that the water table is perfectly flat: (a) temperature contours at steady state; (b) fluid velocity vectors at steady state (the maximum value is 5.45×107m/s). Notice that four convection cells have developed.
Case 2: the water table has a slope of 0.001m/m (i.e., dH=0.8m). Let us first consider topography only by turning off the buoyancy term through enforcing the fluid density to be temperatureindependent with a constant value of 1000kg/m3. Temperature distribution and fluid velocity vectors at steady state are shown in Figure 6. Similar to the previous results of Toth (1962), the topographydrive fluid flow (Figure 6b) is symmetrical with respect to the midline between the water divide and the valley bottom. The recharge is distributed over the area between the water divide and the midline, and the discharge is over the area between the midline and the valley bottom. The maximum fluid velocity is 1.11×106m/s. Thermal gradients (Figure 6a) increase with depth around the water divide (recharge zone) but decrease with depth around the valley bottom (discharge zone) due to the effect of fluid flow shown in Figure 6b. Let us then consider free convection only by allowing the fluid density to vary with temperature following eq. (72) but eliminate the topographic effect by assigning a constant hydraulic head of 200m over the upper boundary. Temperature and fluid velocity field under this condition are nearly identical to what shown in Figure 6, and the maximum fluid velocity of the buoyancydriven flow is equal to 5.45×107m/s. Clearly, the groundwater velocity due to the topography alone is greater than that due to free convection alone. Bearing this in mind, let us now take into account both the topography and buoyancy effects in our modeling experiment. Numerical results at steady state are almost the same as those shown in Figure 6, except for a greater maximum fluid velocity of 1.6×106m/s due to the mixed contribution from both topography and buoyancy in this case. No convection cells now develop and the topographydriven flow seems to have eliminated the buoyancydriven free
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convection, obviously due to the fact that the topographydriven flow rate exceeds the buoyancydriven flow rate. Similar results have been obtained by Raffensperger and Vlassopoulos (1999) who superimposed a horizontal groundwater flow on thermally driven free convection, and found that the horizontal flow would totally mask the free convection if the former has a greater rate than the latter does. When the water table slope is over 0.001m/m, the numerical results (now shown here) indicate that topography force always overwhelms buoyancy force.
Figure 6. Numerical results corresponding to the condition that the water table has a slope of 0.001m/m and the fluid density is constant: (a) temperature contours at steady state; (b) fluid velocity vectors at steady state (the maximum value is 1.11×106m/s, greater than that due to buoyancy in Figure 5b).
Case 3: the water table has a slope of 0.00025m/m (i.e., dH=0.2m). When only the topography is considered, numerical modeling results at steady state are shown in Figure 7. The maximum fluid velocity is 2.76×107m/s, less than the value in Case 2 where the water table slope is greater (0.001m/m); consequently the convective effect of groundwater flow on temperature distribution becomes weaker (refer to Figures 6a and 7a). Once again, when only the buoyancy is considered, temperature and fluid velocity field are nearly the same as what are shown in Figure 5 with a maximum fluid velocity of 5.45×107m/s. In this case study, therefore, the groundwater velocity due to the topography alone is less than that due to the free convection alone. Under this condition, it is discovered that hydrothermal fluid flow keeps changing periodically with time and no steady state solutions exist. Figures 8 and 9 illustrate, respectively, the timedependent temperature contours and groundwater velocity fields at every 50,000 days over a total simulation time of 500,000 days. It can be seen from these two figures that the transient hydrothermal system exhibits three evolution periodicities over the simulation time: the first periodicity ranging from Stages (b), (c) to (d), the second
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periodicity from Stages (e), (f), and (g), and the third periodicity from Stages (h), (i), and (j). It is also clear that groundwater receives initial recharge near the water divide, and this gravitydriven flow pushes the free convective cells to roll towards the valley bottom on the righthand side. The convection cells migrate laterally from the left to the right at a rate of 3.56×108 m/s, which is less than the fluid velocity due to pure topography. Similar periodical phenomenon has been also recognized when a horizontal flow is superimposed on a thermally induced convection (e.g., Combarbous and Bories, 1975; Prats, 1966). When dH=0.4m so that the water table has a slope of 0.0005m/m (results not shown here), the groundwater velocity due to free convection alone is slightly higher than that due to pure topography. Therefore, mixed free and forced convection occurs, and again the hydrothermal flow system evolves periodically with time in a manner similar to that discussed above.
Figure 7. Numerical results corresponding to the condition that the water table has a slope of 0.00025m/m and the fluid density is constant: (a) temperature contours at steady state; (b) fluid velocity vectors at steady state (the maximum value is 2.76×107m/s, less than that due to buoyancy in Figure 5b).
Case 4: the water table has a slope of 0.001m/m (i.e., dH=0.8m) like in Case 2, but there is a lowpermeability layer sitting at 20m below the water table and across the model width. The nearsurface layer is 20m thick and its permeability is three orders of magnitude less than that of the surrounding area. All other conditions remain the same as before. Temperature distribution and fluid velocity vectors at steady state are shown in Figure 10. It can be seen that the less permeable layer serves as a barrier separating two flow systems. Above the layer, fluid flow is driven entirely by the water table gradient with the maximum velocity of 5.47×107m/s; whereas below the layer the fluid flow is fully controlled by buoyancy with the maximum fluid velocity of 1.68×107m/s. Similar results are also obtained for even higher
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water table gradients (not shown here). However, in comparison with the hydrothermal free convection shown in Case 1, free convection in this case is relatively weak, as evidenced by the smaller curvature of the isotherms and the less flow rate. This is because the vertical dimension of the convection cells in this case is less so that the Rayleigh number becomes smaller.
Figure 8. Numerical results corresponding to the condition that the water table has a slope of 0.00025m/m and the fluid density is a function of temperature defined in eq. (3). Evolution of timedependent temperature distribution: (a) at a time of 5×104days; (b) at 105days; (c) at 1.5×105days; (d) at 2×105days; (e) at 2.5×105days; (f) at 3×105days; (g) at 3.5×105days; (h) at 4×105days; (i) at 4.5×105days; (j) at 5×105days. Notice that three evolution periodicities can be recognized over the simulation time (refer to the text for details).
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Figure 9. Numerical results corresponding to the condition that the water table has a slope of 0.00025m/m and the fluid density is a function of temperature defined in eq. (3). Evolution of timedependent fluid velocity field (the maximum value is 8.38×107m/s): (a) at a time of 5×104days; (b) at 105days; (c) at 1.5×105days; (d) at 2×105days; (e) at 2.5×105days; (f) at 3×105days; (g) at 3.5×105days; (h) at 4×105days; (i) at 4.5×105days; (j) at 5×105days. Notice that three evolution periodicities can be recognized over the simulation time (refer to the text for details).
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Figure 10. Numerical results corresponding to the condition that the water table has a slope of 0.001m/m and a lowpermeability layer exists near the surface: (a) temperature contours at steady state; (b) fluid velocity vectors at steady state. Notice that the less permeable layer serves as a barrier, isolating the effect of topographydriven flow (the maximum value is 5.47×107m/s) on the underlying buoyancydriven free convection (the maximum value is 1.68×107m/s).
3.3. Discussions and Conclusions In this section, a series of 2D numerical experiments has been conducted to quantify the relative importance of buoyancy and surface topography in driving groundwater flow. The numerical results have indicated that at low water table slopes (less than 0.0005m/m) so that fluid velocity due to buoyancy alone is higher than that due to topography alone, mixed free and forced convection occurs. Free convection is dominated, but the effect of topography cannot be ignored since the sloping water table ‗forces‘ the convection cells to roll laterally towards the valley bottom. As a result, hydrothermal fluid flow is characterized by a periodical variation with time and no steady state solutions exist. At large water table slopes (over 0.001m/m) so that fluid velocity due to pure topography is higher than that due to pure buoyancy, forced convection overwhelms free convection as if hydrothermal system were controlled entirely by topography. These results are consistent with the previous theoretical calculations of Prats (1966) and Combarbous and Bories (1975) who revealed that the presence of a horizontal current can cause temperature and fluid velocity fluctuations with time, and the pattern of hydrothermal fluid flow is dependent on the relative magnitude of the horizontal current rate and the fluid velocity due to buoyancy. The numerical results
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presented in this section therefore have confirmed that topography seems more important than buoyancy in controlling a hydrothermal fluid flow system. However, the effect of surface topography can be completely sealed off by the presence of a nearsurface lowpermeability layer, beneath which fluid flow is totally controlled by thermally driven free convection.
4. APPLICATION EXAMPLES OF HYDROTHERMAL FLUID FLOW MODELING FOR ORE GENESIS Subsurface fluid flow and heat transport have major implications for both the formation and preservation of almost every class of economic ore deposit, especially those in sedimentary basins. Computational models simulating heat and fluid flow in complex hydrothermal systems can provide considerable insight into how these systems operate to produce economic concentration of metals and hence can be used to test, compare and contrast various geological hypotheses. In many cases, numerical modeling may be the only available tool since the oreforming process occurs too slowly in time and too widely in space to be directly observed in the field or laboratory. This section presents two application examples of hydrothermal fluid flow modeling related to the formation of mineral deposits: buoyancydriven fluid flow for sedimentary exhalative (Sedex) leadzinc deposits in northern Australia, and tectonic deformationdriven fluid flow for the Dachang polymetallic deposits in southern China.
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4.1. Buoyancydriven Fluid Flow Associated with the Formation of Sedextype Deposits Sedimentaryexhalative (Sedex) deposits are a major source of lead and zinc, and an important source of silver. On the global scale, they account for about 40% of zinc production and about 60% of lead production (Goodfellow et al., 1993). Geological and sulphide paragenetic studies suggest that this type of deposits may form by hydrothermal exhalative processes of metalbearing brines discharging onto the basin floor. Despite this general understanding, geologists continue to debate the mechanisms of fluid migration, heat flow and mass transport for ore deposition. A buoyancy force is resulted from variations in fluid density due to spatial and temporal changes in temperature and/or salinity. Buoyancy driven flow is potentially of great importance because it can lead to fluid flow and mass transport over large distances and significantly shorter time scales, compared with diffusion alone. This is because diffusion results from molecular collision, representing a microscale process. In certain circumstances, buoyancy may become even more important than other forces in enhancing hydrodynamic mixing of the dense fluid with the less dense ambient groundwater (Raffensperger and Vlassopoulos, 1999). The importance of this type of fluid flow has long been recognized in midocean ridge hydrothermal systems (Fisher and Becker, 1995; Yang et al., 1998), seawater intrusion (Frind, 1982; Voss and Souza, 1987), and solute transport (Therrien and Sudicky, 1996; Shikaze et al., 1998). Recently, buoyancydriven free convection models become more popular as one of the most likely hydrological scenarios for Sedex formation (Solomon and
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Heinrich, 1992). For instance, Garven et al. (2001) investigated the likely role of free convective fluid migration in the formation of the HYC deposit. Their numerical results exhibit a dominant rechargedischarge pattern controlled by synsedimentary faults and a subsurface sandstone aquifer. Yang et al. (2004a) simulated timedependent fluid flow and developed criteria for efficient identification of potentially mineralizing discharge faults in sedimentary basis, and more recently Yang et al. (2004b) and Yang et al. (2010) investigated the effect of salinity on oreforming processes and fluid flow patterns. The major aim of this fluid flow modeling is to evaluate whether buoyancy driven fluid flow and heat transport was a viable mechanism for the generation of Sedextype deposits. This will be done in the subsequent sections by numerical simulations corresponding to a highly conceptualized paleohydrological system pertinent to the genesis of shalehosted leadzinc ores in the Mount Isa basin, northern Australia.
4.1.1. A 2D conceptualized hydrological model Figure 11 illustrates the 2D paleohydrological model, which is constrained by some of the common features of a sedimentary basin‘s riftandsag phase, and in particular by the reconstructions of the Mount Isa basin, northern Australia (O’dea et al., 1997; Betts et al., 2003). It represents the highly conceptualized and simplified subsurface stratigraphy and structure, which controlled the hydrological system when the Mount Isa Sedex deposits were formed early in the history of the Mount Isa basin. As shown in Figure 11a, the conceptual model involves a volcanic basement sequence of low permeability (Unit 4), a sandstone aquifer of high permeability (Unit 3), a rift cover sequence of intermediate permeability (Unit 2), and an upper cover sequence of shales and siltstones (Unit 1) that hosts mineral deposits formed during or soon after sediment deposition. Two more permeable faults (Fault 1 and Fault 2) are also included to penetrate from the upper sequence into the basement. The faults are 1 km wide and steeply westdipping, which is constrained by surface exposure and seismic profiling (Bierlein and Betts, 2004). Grid discretization of the faults can be found in Figure 11b. Similar general geological structure of the Mount IsaMcArthur basin region has been also used in previous fluid flow models (Garven et al., 2001; Yang et al., 2004a; Yang et al., 2004b; Yang, 2006; Oliver et al., 2006; McLellan et al., 2006). Like in the previous studies (Oliver et al., 2006; McLellan et al., 2006), Fault 1 on the left is equivalent to the Mount Isa fault system, and Fault 2 on the right simulates a fault zone to farther north, such as the Termite Range fault at the Century deposit. The conceptual model has a dimension of 60 km × 20 km and is discretised by a 2D nonorthogonal quadrilateral mesh consisting of 52 element columns and 19 rows (Figure 11b). The two steeplydipping faults cut the sandstone aquifer (Unit 3), which forms a favorable hydrological framework for regionalscale fluid flow. The permeabilities and thermal conductivities assigned to the stratigraphic elements and faults are given in Table 1, following the previous numerical investigations in the Mount IsaMcArthur basin region (Garven et al., 2001; Yang et al., 2004a; Yang et al., 2004b; Yang, 2006; Oliver et al., 2006; McLellan et al., 2006). The vertical permeability of the host rocks is assumed to be two orders of magnitude less than the horizontal permeability due to the bedded and stratified nature of these sedimentary rocks. In addition, it is assumed that cs 800 J kg 1 o C1 , cw 4147 J kg1 o C1 ,
s 2630 kg m3 ,
0 1000 kg m3 , w 0.5 J m 1 s 1 o C1 ,
0.1 . Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
and
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Figure 11. (a) Highly simplified model of titled and submerged rift blocks associated with the formation of Sedex deposits of the Mount Isa basin, northern Australia and (b) finite element mesh used in the 2D fluid flow and heat transport modeling. See Table 1 for the hydrological and thermal properties assigned to the faults and rock units.
The upper boundary beside the faults, maintained at 20°C following the previous numerical studies (Garven et al., 2001; Yang et al., 2004a; Yang et al., 2004b; Yang, 2006), is at the seafloor and permeable to fluid flow. Over the top of the faults, the vertical temperature gradient is fixed at zero. That is, the fluid is assumed to be isothermal near the top surface owing to fluid flow via the faults. The lower boundary is maintained at 450°C, justified by presentday heat flow measurements in northern Australia (S. McLaren and M. Sandiford, pers. comm.). The bottom of the model is assumed impermeable since it lies within the deep volcanic basement. The side boundaries are assumed to be adiabatic to heat transfer and impermeable to fluid flow. The initial fluid velocity is set to zero over the whole solution domain, and the initial temperature is assumed to vary linearly with depth. In the following numerical case studies, the faults‘ penetration depth and boundary conditions will be adjusted.
4.1.2. Numerical modeling results The first case study corresponds to the reference conditions specified above and given in Figure 11 and Table 1. Figure 12 shows the temperature and fluid velocity distribution at different time levels. The modeling reveals that the cold seawater penetrates downwards along Fault 2 and then flows laterally, mainly through the permeable aquifer but with a little
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via the less permeable basement sequence at depth. In the meantime, the fluid is heated up from below. Basinal fluid with elevated temperature ascends along Fault 1 driven by buoyancy force and ultimately discharges onto the seafloor where it could form a Sedextype deposit like the Mount Isa deposit adjacent to the Mount Isa fault. Two convection cells also develop in the aquifer unit near the side boundaries, which is likely due to the impermeable fluid conditions assigned for the sides. Comparison of Figures 12a with 12b indicates that as time progresses, more cold seawater recharges into the basin and hence cools down the system. As a result, the venting fluid temperature drops from 240oC at a time of 55,000 years to 160oC at 1,000,000 years, after which the hydrothermal fluid flow system reaches a steady state (i.e., temperature and fluid velocity no longer vary with time). The venting fluid velocity at steady state is 4.1 m/year. Table 1. Major physical parameters of the faults and host rocks Hydrological unit and formation
Horizontal Permeability (m2)
Vertical Permeability (m2)
Thermal Conductivity
(J m1 s1 oC1 )
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Fault Unit 1 Unit 2 Unit 3 Unit 4
4×1014 1×1015 2×1015 4×1014 2×1016
4×1014 1×1017 2×1017 4×1016 2×1018
2.0 2.0 2.5 3.0 3.0
The second case study considers the same conditions as case 1, except that the faults have been sealed by the uppermost cover sequence to test the concept that stratiform leadzinc deposits may form by replacement of chemically favorable black shale units below lowpermeability cap rocks adjacent to major discharge faults (Broadbent et al., 1998). As shown in Figure 13, no significant marine fluids recharge down into the basin and no venting of hydrothermal fluids at the seafloor. Fluid flow occurs within the aquifer and along the faults to form three 20kmscale hydrothermal circulation cells: one between the faults and two close to the side boundaries. It is also apparent from Figure 13b that significant hydrothermal fluid moves laterally below the uppermost caprock layer. Thus, the potential exists for shalehosted, replacementstyle, stratiform Zn deposits to form by this mechanism if a chemically suitable rock type (e.g., organicrich siltstone or reactive carbonate) is present. However, the fluid flow rates and the fluid temperatures are significantly less than case 1, suggesting that only small and low grade deposits are likely to form by this mechanism. The third case study involves a scenario that Fault 1 only penetrates 10 km deep (reaching the aquifer only) and Fault 2 remains the same as before (penetrating all the way to the base of the model). All the other conditions are kept the same as those in case 1. As illustrated in Figure 14, numerical results for this model are very similar to those for case 1 (refer to Figures 12b and 12c). Again, Fault 1 behaves as the discharge pathway and Fault 2 as the recharge pathway. However, the venting fluid temperature at the steady state in this case is 115oC, 45oC less than that in case 1. This is obviously due to the fact that the discharge fault only penetrates 10 km deep and does not reach the volcanic basement at bottom where temperature is very high. The venting fluid velocity at the steady state is 2.6 m/year.
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Figure 12. Case 1 corresponding to the reference conditions given in Figure 11 and Table 1: (a) temperature contours at a time of 55,000 years; (b) temperature contours at a time of 1,000,000 years, after which the hydrothermal system reaches a steady state; (c) fluid velocity vectors at steady state.
Figure 13. Case 2 corresponding to the condition that the faults have been sealed by the lowpermeability uppermost cover sequence: (a) temperature contours at steady state; (b) fluid velocity vectors at steady state. Note the lack of fluid discharge onto the basin floor. Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
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Figure 14. Case 3 corresponding to the condition that Fault 1 only penetrates 10 km deep and Fault 2 remains the same as shown in Figure 11: (a) temperature contours at steady state; (b) fluid velocity vectors at steady state. Note the venting fluid temperature in this case is lower than that in case 1.
Figure 15. Case 4 corresponding to the condition that both the faults only penetrate 10 km deep: (a) temperature contours at a time of 55,000 years; (b) fluid velocity vectors at steady state. Note Fault 1 now acts as the recharge fault and Fault 2 as the discharge fault. Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
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The final case study assumes both the faults to penetrate only 10 km deep, and all the other conditions remain the same as in case 1. Numerical results are shown in Figure 15. It can be seen that unlike the previous scenarios, this hydrological model facilitates and accommodates a ‗reversed‘ rechargedischarge pattern of hydrothermal fluid flow. Fault 1 now acts as the recharge fault and Fault 2 as the discharge fault. Cold seawater moves downwards from the seafloor along Fault 1, and it becomes divergent when reaching the aquifer. Little of the fluid migrates westbound and forms a clockwise convection cell to the left of the recharge fault; however, most of the fluid travels laterally along the permeable aquifer horizon towards the east side of the model. In the meantime, the thermal energy gained from the lower boundary heats up the fluid and eventually drives it to ascend along Fault 2 and discharge onto the basin floor where it could form a Sedextype deposit. The venting fluid temperature and velocity via the discharge fault at the steady state are 142oC and 3.5 m/year, respectively.
4.1.3. Discussions and conclusions Numerical experiments have been conducted in this section in order to investigate the role of buoyancy force in the formation of Sedextype deposits. The Mount Isa basin in northern Australia serves as the field setting. The numerical results corresponding to a highly conceptualized hydrological system have revealed that the interplay between active synsedimentary faults and clastic aquifer units controls the basinal fluid flow and hydrothermal discharge. Marine waters recharge the basin via one fault, move laterally through the clastic aquifer unit where they are heated by the geothermal gradient and leach metals, and then discharge to the surface via the other fault. Convective cells are established, based on the fault spacing and penetration depth. In all of the first three case studies, Fault 1 acted as the discharge conduit and Fault 2 as the recharge conduit. However, when both faults only penetrate down to 10 km deep as discussed in the forth case study, a ‗reversed‘ rechargedischarge pattern of hydrothermal fluid flow developed. The factors causing this change are not fully understood; nevertheless they are more likely controlled by the basin architecture and its spatial relation with the faults (e.g., Yang et al., 2004a). Although noticing the fact that one fault behaves as the recharge conduit and the other as the discharge conduit, or vice versa, this section does not intend to investigate the exact reasons leading to a certain rechargedischarge pattern of fluid flow. Rather, this study aims at evaluating the potential of buoyancy force in driving hydrothermal fluid flow associated with the formation of Sedextype deposits. All the numerical case studies described above have confirmed that buoyancy force is indeed an effective and efficient driving mechanism for basinscale hydrothermal fluid flow. The critical question is whether the fluid discharge velocities and temperatures predicted by the model have the potential to form worldclass supergiant deposits, like the Mount Isa deposit. The numerical results for different scenarios have revealed that the venting fluid velocities range from 2.6 m/year to 4.1 m/year over a period of 1,000,000 years. This gives rise to an average discharge fluid flux of about 300 m3 year per meter of fault strike (noticing the average porosity is 10 %). If the Mount Isa deposit formed from fluid discharge from the Mount Isa fault along a fault length of 2000 m, and the discharging fluids were carrying about 100 ppm Zn (Cooke et al., 2000), then the simulated venting fluid has the capacity to form a deposit containing 60 million tonnes of Zn metal over the discharge period of 1,000,000 years, provided that all the zinc carried in the fluid is precipitated upon exhalation to form the
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deposit. The defined zinc metal resource of 15.7 million tonnes (Goodfellow et al., 1993) at the Mount Isa deposit could be accounted for with a metal deposition efficiency of about 26 percent. Also given the temperature constraint of the Mount Isa mineralization and the simulated thermal patterns stated above, cases 1, 3 and 4 seem to be favorable for the formation of a Mount Isatype ore body adjacent to a major fault system. The numerical modeling studies presented here therefore support the following final conclusion: buoyancy force is an efficient and effective driving mechanism for basinscale fluid flow and heat transport, and it is viable enough to give rise to the generation of supergiant Sedex deposits like the Mount Isa deposit, northern Australia.
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4.2. Tectonic Deformationdriven Fluid Flow Related to Ore Genesis in the Dachang District, Southern China The polymetallic deposits in the Dachang ore district, Guangxi province, southern China are one of the most famous supergiant ore districts in the world due to abundant dispersed elements (e.g. In, Cd, Ca and Ge) and huge reserves. Polymetallic deposits in this area are hosted in the Yanshanian (Jurassic to Cretaceous) platform sedimentary successions that were intruded and thermally metamorphosed by Yanshanian granitoids. Tectonic activity was dominated by faulting in the DevonianCarboniferous times, compression during the Triassic and Jurassic and extension in the Cretaceous. The main structures include the parallel NWtrending Dachang anticline and Dachang fault. Structural studies indicate that compressional force was a major tectonic force when mineral deposits were formed during the Triassic (Chen, 1993; Cai et al., 2004). Structural control and fluid flow are interpreted to be two key factors contributing to the formation of ore deposits in this district (Cai et al., 2007). Despite this general understanding, the relationship between tectonic control and oreforming fluid flow are still poorly understood. Extensive numerical studies have been conducted to simulate tectonic deformationdriven fluid flow for a variety of oreforming environments, including the gold mineralization at Bendigo and western Victoria (Schaubs and Zhao, 2002), the Xiangshan uranium deposit of southern China (Lin et al., 2006), and the Mount Isa leadzinc deposits (Zhang et al., 2006; Feltrin et al., 2009). However, no attempt has been made to numerically investigate oreforming fluid flow driven by tectonic deformation in the Dachang district. This section aims at filling up this knowledge gap and achieving a better understanding of mineralization processes in this district. A series of numerical experiments of fully coupling tectonic deformation and fluid flow are conducted to quantify the role of structural control in governing fluid flow and the resultant oreforming processes in the targeted area.
4.2.1. A 2D conceptualized hydrological model A 2D conceptualized hydrological model is developed, on the basis of the existing geological, geophysical, and structural data available for the study area, to traverse the TongkengChangpo deposits. As illustrated in Figure 16, the model is 4000 m wide and 2600 m deep, involving five geological units: Upper Devonian argillaceous limestone (D33), Upper Devonian lenticular limestone (D32), Upper Devonian silicalite limestone (D31), Middle Devonian black Shale (D22), and the Dachang fault. In our numerical experiments, the
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Table 2. Initial material properties assigned to the different rock units and faults. Stratigraphic units
Shear modulus (Pa) 1.2E10
Friction angle (°) 30
Cohesion (Pa)
porosity
2694
Bulk modulus (Pa) 8.60E9
Dilation angle (°) 4
Permeability (m2)
0.18
Tension strength (Pa) 5.85E5
1.17E6
2740
2.04E10
3.24E10
35
2.97E6
0.22
1.44E6
3
4.00E16
2663
9.64E9
1.53E10
38
2.98E6
0.3
1.49E6
4
1.00E15
2844
1.15E10
1.83E10
42
2.50E6
0.12
1.25E6
3
3.00E17
2450
2.13E8
1.00E10
25
1.00E6
0.2
5.00E5
4
2.00E15
2.00E16
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Upper Devonian argillaceous limestone (D33) Upper Devonian lenticular limestone (D32) Upper Devonian silicalite (D31) Middle Devonian black shale (D22) fault
Density (kg m3)
cID=3018126.
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Dachang fault is simulated as weak narrow zones that channel fluid flow. The mechanical properties assigned to these units are tabulated in Table 2, based on the field data and literature (Luo and Ye, 1998; Hu et al., 2008).
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Figure 16. 2D conceptualized model (cross section) traversing the TongkengChangpo deposits in the Dachang district, southern China.
Fluid boundary conditions are expressed in terms of pore pressure. The sides and bottom of the model are assumed to be impermeable. The upper boundary is permeable to fluid flow. Initial pore pressure is equal to the hydrostatic value. Structural studies in this area indicate that compressional stress was predominant when ore deposits were formed during Triassic. Thus, a horizontal contractional velocity boundary condition reflecting the compression is applied to the side boundaries of the model.
.2.2. Governing equations and numerical scheme To simulate fluid flow driven by tectonic deformation, rock deformation is usually approximated by elasticplastic deformation, following MohrCoulomb constitutive laws. The rock deforms elastically up to a yield point, after which it deforms in a plastic manner. In the elastic deformation range, pore pressure is elevated due to the volume decrease by an imposed stress. Yield occurs when the following relationship is satisfied,
C n ,
(73)
is the maximum shear stress, C is the cohesive strength of porous rocks, is the coefficient of friction and n is the normal stress. The plastic deformation after yield point where
shows an enhancement to localized porosity due to the sliding of frictional surfaces past each other (Ord and Oliver, 1997). Such deformationinduced dilatancy is determined by a Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
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dilatancy angle in the MohrCoulomb model. For most common rocks, the dilation angle is between +10° and +20°. However, for some high porosity rocks or rocks confined by extremely high pressure, the plastic deformation will reduce localized porosity, for which a negative dilatancy angle must be employed (Ord, 1991). Numerical experiments are undertaken using a finite difference code FLAC (Fast Lagrangian Analysis of Continual, Itasca, 2002) that is capable of fully coupling fluid flow and tectonic deformation. The major governing equations include Darcy‘s law
qi kij
( P w g k xk ), x j
where q i is the specific discharge vector, k ij
(74) is the mobility coefficient (FLAC‘s
permeability tensor, i.e., the ratio of intrinsic permeability and fluid dynamic viscosity), P is the fluid pressure, and w is the fluid density. The fluid mass conservation equation
q i q , t xi where
(75)
is the variation of fluid content, t is time, and q is the volumetric fluid source
intensity. Momentum balance satisfies
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ij
dui' gi , x j dt
(76)
(1 n) s nw ,
(77)
and
where ij is the component of stress tensor,
is the solid bulk density, u i' is velocity, s
is dry rock density, and n is porosity. If rock grains are not compressible, the response equation of fluid due to deformation has the form
P K w t n t t where K w is the fluid bulk modulus and
,
(78)
is the volumetric strain.
In previous numerical studies, porosity and permeability assigned to each rock unit usually remain unchanging with tectonic deformation. In this study, a simple linkage is Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
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developed to reflect the variations of hydraulic properties with time due to deformation. Porosity n can be expressed as
n 1
V0 (1 n0 ), V
(79)
where V 0 is the initial element volume, V is total volume of element, and n 0 is the initial porosity. For large strain mode, the volume strain ev can be approximated by
ev
2(V V0 ) . V V0
(80)
Reorganizing (78) and (79) yields
2 ev n 1 (1 n0 ). 2 ev
(81)
According to Bear (1972), hydraulic conductivity K can be written as
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K (
w g n3 dm 2 ) , 1 n 2 180
(82)
where dm is grain size. This equation was derived from the KozenyCarman equation which expresses the relation between permeability and its relationship to porous medium properties. This study sets dm=0.002 mm for argillaceous limestone, 0.01 mm for lenticular limestone, 0.3 mm for silicalite, and 0.003 mm for black shale, based on the literature and field data. It can be seen from these equations that the change in volumetric strain due to tectonic deformation leads to the variation in porosity, which in turn results in the variation in hydraulic conductivity and permeability.
4.2.3. Numerical modeling results This study is concentrated on the effect of tectonic shortening on oreforming fluid flow since the compressional force is thought to be the principal tectonic force related to the ore genesis in the study area. Figure 17 shows the initial permeability distribution before deformation (a) and after a 3% shortening (b). Comparing Figs. 17(a) and (b) indicates that the shortening has significantly altered the original permeability prior to tectonic deformation. In particular, it has reduced the fault permeability from 2×1015 to 1.081×1015 m2 due to the reduction of volumetric strain in this area, but enhanced the permeability of the D31 unit (predominant in the hangingwall) from 1×1015 to 3.23×1015 m2.
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Figure 17. Permeability distribution: (a) prior to deformation; (b) after a 3% shortening. Note the difference in colour scheme for the permeability changes between (a) and (b).
Figure 18. Fluid pore pressure distribution and fluid flow velocity field corresponding to the 3% shortening stage under the condition of (a) unchangeable permeability and (b) changeable permeability. Arrows show fluid velocity vectors (the maximum fluid velocity is 0.326 m/year in model (a) and 0.233 m/year in model (b)). Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
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Figure 19. Volumetric strain distribution in the model (min=1.0×103, max=9.0×103).
Figure 18 illustrates pore pressure distribution and fluid flow velocity vectors corresponding to the 3% shortening stage. Pore pressure decreases rapidly as a result of the tectonic deformation, which drives fluid flow into the Dachang fault from deep levels and surrounding rocks. Comparing Figs. 18(a) with 18(b) reveals that the fluid flow in Upper Devonian silicate limestone (D31) in the case of changeable permeability due to deformation (Figure 18(b)) greatly exceeds that in the case of unchangeable permeability (Figure 18(a)), indicating that deformationrelated variations in permeability do influence the fluid flow pattern significantly. The fluid flow is mainly accommodated in the hanging wall Upper Devonian silicalite limestone (D31) unit where the permeability is enhanced most significantly by the shortening effect. This suggests that the D31 unit, in particular the hangingwall side of the Dachang Fault, is the favorable location for the formation of ore deposits. Figure 19 illustrates the distribution of volumetric strain. There is a volume increase or dilation surrounding the D31 unit and along the upper left side of the fault, indicating intensive shearing effect during deformation, which can lead to permeability variation. The volumetric increase or dilation around the D31 and D32 units enables the surrounding fluid flow to channel through therein. These modeling results are consistent with the real mineralization situation for the Dachang ore district. In fact, two mineral deposits (i.e., No.91 and No.92) were discovered in this hanging wall unit and are currently mined. Also note a downward fluid flow in the upper half of the fault, which is likely due to the surface topography and broad pore pressure reduction resulting from the tectonic deformation.
4.2.4. Conclusions Numerical modeling has been conducted to simulate fluid flow driven by tectonic deformation (in particular, the effect of tectonic shortening on oreforming fluid flow) in the Dachang district, southern China. Unlike many of previous modeling studies, this study has considered the variations of permeability with time due to tectonic deformation. Numerical modeling results based on a conceptualized 2D model indicate that the shortening
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deformation expels basal fluid to ascend from the basement along the Dachang fault; while the surface topography and pore pressure reduction resulting from the tectonic deformation allows the surficial fluid to descend along the upper part of the fault. These two fluids mix at depth and then travel laterally into the Upper Devonian silicalite limestone (D31) due to the elevated permeability caused by the shortening rock deformation. A large amount of fluids circulating in the hanging wall D31 unit implies that it is a favorable site to host ore deposits. Although No. 91 and No. 92 deposits are currently mined from this unit, our modeling results indicate that this unit may host other deposits, especially in its lower part.
ACKNOWLEDGMENTS This research was supported by the Natural Science and Engineering Research Council of Canada (NSERC) through a Discovery Grant (RGPIN 261283) to Jianwen Yang.
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REFERENCES Bear, J. (1972). Dynamics of Fluids in Porous Media, American Elsevier Publishing Company, New York, Betts, P. G., Giles, D. & Lister, G. S. (2003). Tectonic environment of shalehosted massive sulfide PbZnAg deposits of Proterozoic northeastern Australia. Econ Geol, 98, 557576. Bierlein, F. P. & Betts, P. G. (2004. The Proterozoic Mount Isa Fault Zone, northeastern Australia: is it really a ca. 1.9 Ga terranebounding suture?, Earth and Planetary Science Letters, 225, 279294. Broadbent, G. C., Myers, R. E. & Wright, J. V. (1998). Geology and origin of shalehosted Zn PbAg mineralization at the Century deposit, northwest Queensland, Australia, Economic Geology, 93, 12641294. Cai, M. H., Mao, J. W., Liang, T., Franco, P. & Huang, H. L. (2007). The origin of the TongkengChangpo tin deposit, Dachang metal district, Guangxi, China: Clues from fluid flow inclusions and He isotope systematics. Mineralium Deposita, 42, 613626. Cai, M. H., Liang, T., Wu, D. C. & Huang, H. L. (2004). Structural feature and its control of mineralization of the NandanHechi metallogenic belt in Guangxi Province. Geology and Exploration, 6, 510. Chen, Y. C. (1993). Tin Geology in Dachang. Geology Publishing Company, Beijing, 2647. Combarnous, M. A. & Bories, S. A. (1975). Hydrothermal convection in saturated porous media. Advances in Hydroscience, 10, 231307. Cooke, D. R., Bull, S. W. & Large, R. R. et al., (2000). The importance of oxidized brines for the formation of Australian Proterozoic stratiform sedimenthosted PbZn (Sedex) deposits, Economic Geology, 95, 118. Daus, A. D., Frind, E. O. & Sudicky, E. A. (1985). Comparative error analysis in finite element formulations of the advectiondispersion equation, Adv. Water Resour., 8, 8695. Evan, D. G., & Nunn, J. A. (1989). Free thermohaline convection in sediments surrounding a
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28572867. O‘dea, M. G., Lister, G. S. & MacCready, T. et al., (1997). Geodynamic evolution of the Proterozoic Mount Isa terrain. Geol Soc Spec Pub, 121, 99122. Oliver, N. H. S., McLellan, J. G. & Hobbs, B. E. et al., (2006). Numerical models of extensional deformation, heat transfer, and fluid flow across basementcover interfaces during basinrelated mineralization, Economic Geology, 101, 131. Ord, A. & Oliver, N. H .S. (1997). Mechanical controls on fluid flow during regional metamorphism: some numerical models, Journal of Metamorphic Geology, 15, 345359. Ord, A. (1991). Deformation of rock: a pressuresensitive, dilatant material, Pure and Applied Geophysics, 137, 337366. Prats, M. (1966). The effect of horizontal fluid flow on thermally induced convection currents in porous mediums, J Geophys Res, 71, 48354838. Raffensperger, J. P. & Vlassopoulos, D. (1999). The potential for free and mixed convection in sedimentary basins, Hydrol J, 7, 505520. Ranganathan, V. (1993). The maintenance of high salt concentrations in interstitial waters above the New Albany Shale of the Illinois Basin, Water Resour Res, 29, 36593670. Schaubs, P. M. & Zhao, C. (2002). Numerical models of golddeposit formation in the BendigoBallarat zone, Victoria, Australian Journal of Earth Sciences, 49, 10771096. Shikaze, S. G., Sudicky, E. A. & Schwartz, F. W. (1998). Densitydependent solute transport in discretelyfractured geological media: is prediction possible? J Contam Hydrol, 34, 273291. Solomon, M. & Heinrich, C. A. (1992). Are highheatproducing granites essential to the origin of giant leadzinc deposits at Mount Isa and McArthur River, Australia? Explor Min Geol, 1, 8591. Stanislavsky, E. & Gvirtzman, H. (1999). Basinscale migration of continentalrift brines: paleohydrologic modeling of the Dead Sea basin, Geology, 27, 791794. Sudicky, E. A. & Frind, E. O. (1982). Contaminant transport in fractured porous media: analytical solutions for a system of parallel fractures, Water Resources Research, 18, 16341642. Tang, D. H., Frind, E. O. & Sudicky, E. A. (1981). Contaminant transport in fractured porous media: analytical solution for a single fracture, Water Resour. Res., 17, 555564. Therrien, R. & Sudicky, E. A. (1996). Threedimensional analysis of variablysaturated flow and solute transport in discretelyfractured porous media, J Contam Hydrol, 23, 144. Thornton, M. M. & Wilson, A. M. (2007). Topographydriven flow versus buoyancydriven flow in the U.S. midcontinent: implications for the residence time of brines, Geofluids, 7, 6978. Toth, J. (1962). A theory of groundwater motion in small drainage basins in Central Alberta, Canada, J Geophys Res, 67, 43754387. Toth, J. (1963). A theoretical analysis of groundwater flow in small drainage basins, J Geophys Res, 68, 47954812. Voss, C. I. & Souza, W. R. (1987). Variable density flow and solute transport simulation of regional aquifers containing a narrow freshwatersaltwater transition zone, Water Resour Res, 23, 18511866. Wang, K., He, J. & Davis, E. E. (1997). Influence of basement topography on hydrothermal circulation in sedimentburies igneous oceanic crust, Earth Planet Sci Lett, 146, 151
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164. Weast, R. C. (Ed), (1980). CRC Handbook of Chemistry and Physics, CRC Press Inc., Boca Raton, Fla. Yang, J., Feng, Z., Luo, X. & Chen, Y. (2010). Threedimensional numerical modeling of salinity variations in driving basinscale oreforming fluid flow: Example from Mount Isa Basin, northern Australia, 106, 236243. Yang, J. (2006). Finite element modeling of transient saline hydrothermal fluids in multifaulted sedimentary basins: implications for oreforming processes, Can J Earth Sci, 43, 13311340. Yang, J., Large, R. R. & Bull, S. W. (2004a). Factors controlling free thermal convection in faults in sedimentary basins: implications for the formation of zinclead mineral deposits, Geofluids, 4, 237247. Yang, J., Bull, S. W. & Large, R. R. (2004b). Numerical investigation of salinity in controlling oreforming fluid transport in sedimentary basins: Example of the HYC deposit, Northern Australia, Mineralium Deposita, 39, 622631. Yang, J., Latychev, K. & Edwards, R. N. (1998). Numerical computation of hydrothermal fluid circulation in fractured earth structures, Geophys J Int, 135, 627649. Zhang, Y., SorjonenWard, P., Ord, A. & Southgate, P. N. (2006). Fluid flow during deformation associated with structural closure of the Isa superbasin at 1575 Ma in the central and northern Lawn Hill Platform, northern Australia, Economic Geology, 101, 12931312.
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Chapter 4
SIMULATION OF VELOCITY AND MASS TRANSPORT PROFILES IN A LABORATORY ELECTROLYSER USING COMPUTATIONAL FLUID DYNAMICS Leticia Vázquez Gutiérrez, Alberto Alvarez Gallegos and Fernando Z. Sierra Universidad Autonoma del Estado de Morelos, México
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INTRODUCTION Parallel plate reactors are prevalent both at laboratory [1, 2] and industrial scales [35]. They are easy to construct and are versatile when choosing separators and electrodes as well as in the mode of electrical operation and flow manifolds [1, 6, 7]. A successful scaleup procedure includes several steps: the analysis of a variety of dimensionless groups which describe the geometric, kinematic, thermal, chemical and electrical characteristics [8]. Accordingly, mass transport and fluid dynamics are among the main parameters in a reliable scaleup procedure [511]. The hydrodynamic behaviour of a parallel plate reactor can be described the flow through a rectangular channel where the flow pattern is strongly linked to the rate of mass transport (km), according to the following equation [10]:
k m a( v )b
(1)
Where v is the characteristic linear flow velocity, the ratio of mean volumetric flow rate, Q, and the cross sectional area, Ax, of the electrolyte channel: _
v
Q Ax
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(2)
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Leticia Vázquez Gutiérrez, Alberto Alvarez Gallegos and Fernando Z. Sierra
The empirical constants a and b can be evaluated under carefully controlled laboratory reactor conditions [10, 12]. Laboratory Parallel Plate Reactors. For academic studies, electrochemical cell design includes a calming zone between the electrolyte entrance and the electrodes to ensure a fully developed flow [10, 13]. Under these conditions km can be evaluated in a rectangular channel from correlations involving dimensionless parameters [5, 10, 1315]. Therefore, the mass transport coefficient is a function of fluid flow conditions and the transport properties of the electrolyte. Mass transport correlation may be written in the form [10]:
Sh a Re b Scc Lee
(3)
Where Sh is the Sherwood number and takes into account the forced convective mass transport, Re is the Reynolds number, which represents the fluid flow conditions, the transport properties of the electrolyte are represented in the Schmidt number (Sc), and the dimensionless length group, Le, is the ratio of hydraulic diameter (de) to the length of the electrode in the direction of the flow. Figure 1 shows a rectangular flow channel, representative of the most common electrochemical reactor design, including the calming zone between the electrolyte entrance/exit and the electrodes. Expansion of the dimensionless groups in correlation 3 gives:
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_ km de vd a e D
b c de D L
e (4)
Where D is the diffusion coefficient of an electroactive species, is the kinematic viscosity of solution, a, b, c, and e are empirical constants and the rest of the terms were already defined.
Figure 1. Idealized electrochemical reactor channel, including the calming zone between the electrolyte entrance/exit and the electrodes.
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In the case of a well developed laminar flow in a rectangular section channel, fluid flow behaviour is completely described by the NavierStokes and momentum conservation equations [13]. For simple cases, and taking into account appropriate assumptions, the set of such equations can be analytically solved. For electrodes of finite width, correlations are obtained as a function of (), the aspect ratio S/B, of the reactor channel [13]:
Sh 1.47[Re Sc Le ] 1.3 [ 2 /( 1 )] 1 / 3
(5)
In general, equation 5 agrees well with experimental data, although Pickett [15] suggested a correlation (see equation 6) that agrees better with experimental data:
Sh 2.54[Re Sc Le ]1 / 3
(6)
In the case of a well developed turbulent flow, the mass transfer is independent of the electrode length and can be described by the following correlation [13]:
Sh 0.023 Re 0.8 Sc1 / 3
(7)
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However, for much shorter electrodes the parameter Le should be included in the mass transfer correlations, several of them have been proposed [10, 13]:
Sh 0.145 Re 2 / 3 Sc1 / 3 Le1 / 4
(8)
Sh 0.276 Re 0.58 Sc1 / 3 Le1 / 3
(9)
As we can see, a multitude of empirical correlations can be found in the literature [5, 10, 11, 13, 15] however all of them are limited to well developed flows. Industrial Parallel Plate Reactors. In industrial electrochemical reactors a deliberate calming section is considered a luxury, therefore fully developed flow is rare because manifolds usually inject or extract the electrolyte very close to the electrodes [16, 17]. It is _
v does not represent the mean linear electrolyte velocity ( ) in the reactor channel. Because, it is not possible to have a detailed understanding of fluid dynamics ( obvious that
and Re) by means of equation 2, nor it is possible to attain a theoretical evaluation of km by combining equations 2 and 6, 7, 8, or 9. A successful scaleup procedure includes studies of mass transport that are normally carried out experimentally using a shorter version of equation 4, represented by equation 1. The volumetric flow rate is measured experimentally with a flow meter and a loglog plot of km vs. the characteristic linear flow velocity should yield a straight line with a slope b.
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Although this scaleup procedure can provide a highly accurate approximation of the reactor performance, it is expensive (several reactors should be constructed and tested before the better one is selected), time consuming and depends greatly on experimental skills. The objective of this work is to discuss the main tools available for evaluating flow pattern and its effect on mass transport in fluids flowing along electrochemical reactors. This evaluation will be validated against an industrial electrochemical reactor of well known performance: the FM01LC laboratoryscale cell. This reactor was introduced into the market in 1988 [1]. A number of papers on the characterization of the reaction environment of this cell have been published previously [1, 9, 1622]. Figure 2 shows (a) the main geometrical details (projected area 0.0064 m2) of such reactor and (b) the inlet manifolds.
Figure 2. FM01LC laboratoryscale cell.
FLOW REGIME ASSESSMENT Fluid flow similarities are among the key parameters for a reliable scaleup electrochemical reactor procedure. Mechanical similarities of two flows exist if their Reynolds numbers are equal for both flows [13]. In this way, the Reynolds number (the ratio of inertial forces to viscous forces) defines the flow conditions in a particular electrochemical reactor geometry. All flows encountered in engineer practice become unstable above a certain Reynolds number, known as critical Reynolds (2000 < ReCrit < 3000). At low Reynolds numbers (Re < 2000) flows are laminar and at higher Reynolds numbers (Re > 3000) flows are turbulent. At
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low Reynolds numbers, below ReCrit, the flow is smooth and adjacent layers of fluid slide past each other in an orderly fashion. If the applied boundary conditions do not change with time, the flow is steady and the regime is called laminar flow. However, above ReCrit flow behaviour is radically different and motion becomes intrinsically unsteady even with constant imposed boundary conditions. This regime is called turbulent flows and can be characterized in terms of the mean values of flow properties and some statistical properties of its fluctuations. Direct visualization of turbulent flow reveal rotational flow structures (turbulent eddies) with a wide range of length scales. In general, in both industrial and laboratory electrochemical reactors the laminar regime is preferred to the turbulent regime because, in the first case, the hydrodynamic behaviour is easier to understand. Additionally, the laminar regime requires less energy for pumping the fluid through an electrochemical reactor. Experimental works have demonstrated that the ReCrit is larger if the disturbances in the flow through the pipe are smaller. It is possible to reach ReCrit = 40 000 for a fluid flowing through a pipe without disturbances [23]. In general it is accepted that the numerical value of ReCrit lies at about 2000, however, turbulent flows with a Reynolds number well below of 2000 have been encountered. A huge amount of work has been done to understand the initial cause of the transition to turbulence; resulting in the development of the stability theory (Hydrodynamic Stability) [24]. The basic premise of this theory is that laminar flow is affected by turbulence such as those which could arise in the channel (pipe) entrance, or, in the boundary layer of a body, from the wall roughness or from irregularities in the outer flow. The decisive question is whether the disturbances subside or grow with time and/or distance. Should the first condition prevails the basic flow is considered to be stable; if the last conditions prevail, flow is considered unstable, i.e. it is possible that the laminarturbulent transition will occur. The numerical value of ReCrit depends quite strongly on the geometry of manifolds to introduce the fluid and to remove it from a duct (pipe, channel, etc) therefore the precise ReCrit value at which the flow conditions become turbulent is very important. The following criteria have been employed either to assess the minimum downstream length required for a fully developed flow, or to find a suitable model to explain fluid flow behavior which would lead to a reliable scaleup procedure in filterpress electrochemical reactors. Entrance Length. For a circular pipe of diameter d or a channel of hydraulic diameter de, the minimum entrance length (L*) was proposed [2628]: * Laminar flow L 0.06 Re d
(10)
* 0.167 d Turbulent flow L 4.4 Re
(11)
* Turbulent flow L (20 to 40) d
(12)
Turbulent flow (Channel) L* (20 to 100) d e
(13)
Turbulent flow (Channel) L* 6 d e
(14)
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The evaluation of L* is only a function of two parameters: the dimensionless Re number and the hydraulic diameter de. The first is evaluated by taking into account the characteristic
linear flow velocity ( v ) and the second one is related to the channel cross sectional area. Under these conditions, the flow behavior differences between electrode surface and bulk solution, as well as near manifolds are ignored. Therefore, the main drawbacks of those criteria are their inconsistencies as shown by [8, 28]. Flow Dispersion Models. In general, parallel plate reactors can be considered to be plug flow reactors due their geometrical characteristics. The hydrodynamic behaviour of these reactors assumes no longitudinal but complete radial mixing, implying an identical velocity profile and residence time for all elements of the flow [29]. This flow pattern is characterized by a Dirac delta function: an instantaneous impulse at the cell inlet should pass along the reactor and be detected at the cell outlet without alteration. However, flow behaviour deviates from the model due to several factors, among them are reactor geometry and velocity fluctuations. Therefore, the ideal residence time is altered to produce a actual residence time distribution (called curve C). Several approaches are available for the assessment of such fluid flow pattern [30]. The hydrodynamic behaviour of an FM01LC reactor has been studied using the stimulus–response technique where a small amount of a high conductivity solution (KCl) was injected in the reactor inlet [31, 32]. The conductivity of the solution at the reactor exit was monitored as a function of time to determine the flow dispersion pattern within the reactor. Due to the inherent difficulties of this flow model for the correct interpretations of the experimental data and the appearance of tailing in the tracer curves leading to error [33; 34]. The main conclusions that can be drawn are solely qualitative: the Peclet number increases as Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.
_
a function of the flow velocity (in this case velocity is v , from equation 2) [31, 32]. In order to minimize the tailing phenomenon colorants may be employed to assess the flow pattern [35, 36]. The main drawbacks of this approach are: 1) the necessity to have first an electrochemical reactor to test it, and then, assess its flow pattern, and 2) this information is not sufficient to better understand how the flow pattern affects the rate of mass transfer. Dimensionless Parameters. The sudden expansion of fluid flow is commonly used to introduce the electrolyte to an electrochemical reactor. As a result, at the cell entrance the flow pattern is characterized by the presence of recirculating fluid. Hence, the Reynolds number is a function of the expansion geometry and several attempts were made to evaluate this phenomenon. In general, the method consists of correlating the expansion factor (manifold geometries) with a measurable quantity such as heat [3740] or mass transfer coefficients [4144]. From some of these works [39, 42], the following general mass transport correlation was obtained as a function of different entry arrangements:
Sh Sc 1/ 3 0.11 Re 0.73 (d e / De ) 1
(15)
Where (de/De) is the ratio of entrance equivalent diameter to the electrochemical cell equivalent diameter. In general, equation 15 agrees well with experimental data, as long as Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
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the cell entry is limited to one circular bore whole. Unfortunately, equation 15 cannot be used for different manifold designs, i.e. slittype cell entrance. For this reason, equation 15 was improved by introducing a new expansion factor: s/S [45], where s and S are the crosssection areas of the entrance and the cell, respectively. The following empirical correlation was found to be valid for both tubetype entrance and slittype entrance and 400 < Re < 3500:
Sh Sc 1/ 3 0.068 Re 0.72 (s / S ) 1/ 2
(16)
Although equation 16 seems to work well for the manifolds previously described, it fails when applied to FM01LC (see Figure 2) because it does not take into account the distribution, number and type of ports. In an effort to circumvent the problem, a geometrical manifold parameter was proposed [46]. The dimensionless group is more developed than those described previously. It includes geometrical details of the manifold design such as thickness, width, distribution of the open spaces and the free area for the electrolyte entrance within electrochemical reactor. Mathematically, the manifold parameter is expressed as:
de L
(17)
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Where is the aspect ratio, de is the equivalent diameter, L is the channel length, λ considers the geometrical arrangement of the holes in the flow distributor and is the free area for liquid entrance within the reactor (expressed as the ratio of whole area of the electrode channel to the manifold cross sectional area). These parameters can be mathematically defined by the following equations:
nh nr
(18)
nh nr Ahole BS
(19)
S B
(20)
Where nh is the number of holes in one row and nr is the number of rows in the distribution manifold and Ahole is the crosssectional area ( d2/4, for a circular hole) of an individual hole. According to the authors equations 17 to 20 are interpreted as follow: large values of are associated with high entrance/exit effects and high mass transport rate; in a similar way, high values of Sh are associated with low values of . For a well designed
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reactor the ideal performance (low entrance/exit effects) value of
is 1. However, for < 1
a jet stream is expected at the cell inlet. Applying this method to the FM01LC the following numerical results are found: = 0.0027, = 0.105. Using this information the following diagnosis for the FM01LC can be made: i) the electrochemical cell design is far from to be ideal ( = 0.105) and, ii) at the cell entrance, jet stream, vortex and a high mass transport rate are expected. Although this information may be useful for an experienced designer, for others it is very subjective. In fact, for a theoretical design of an electrochemical reactor, the following key questions remain open: What is the minimum required length of the calming zone to subdue disturbances? How does the flow pattern affect the rate of mass transfer inside and out of the calming zone? In general, these questions can be answered by constructing and then testing an electrochemical reactor. During this task, the parameters ( and ) have no
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interest for the designer. Computational Fluid Dynamics. For almost 20 years, computational fluid dynamics (CFD) has been regarded as a feasible tool for reactor designing purposes. In this field, Patankar has developed one of the best numerical approaches: the Finite Volume Formulation [47]. This numerical method, with some modifications, has been applied to the treatment of electrode boundary conditions considering a 2dimensional, unsteady tertiary current distribution in laminar natural convective flow [48]. A hydrodynamic model for a tricklebed reactor was developed and simulated using a similar approach [49]. The limitation of this mathematical model is the difficulty to correctly visualize the whole numerical method and to solve it taking into account appropriate assumptions and restrictions that describe the hydrodynamic behaviour of the flow. This approach requires extra mathematical skills for electrochemists and, unfortunately, following this mathematical route, flow fields with turbulent effects are not included because the mathematical problem becomes too difficult. The problem can be overcome using commercial computational fluid dynamics (CFD) packages. Over the past 20 years they have been employed to investigate flow patterns in empty [50] and spacerfilled rectangular channels [51]. At present, CFD offers the possibility for evaluating, under realistic conditions, the flow pattern and its effect on mass transport in developing fluids along a reactor [16, 17, 5254]. Although several CFD packages are available, in this study Fluent (v6) was used to provide a numerical description of the flow patterns in the FM01LC reactor under the same experimental conditions as those documented elsewhere [19]. The following section provides a detailed description of CFD capabilities for assisting in electrochemical reactor design.
MODELLING THE FLOW PATTERN ALONG THE FM01LC The electrolyte is distributed to the reactor channel by internal manifolds incorporated within the cell spacers as shown in the Figure 2. When the electrolyte is pumped through the channel from left to right a flow pattern develops along the channel length. The fluid flow can be modelled from the basic principles of conservation of mass, momentum and energy [55]. The CFD package includes two main elements (preprocessor and postprocessor) structured around a numerical algorithm (solver) for fluid flow problems [25].
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Modelling Conditions. The main modelling conditions are fed via preprocessor (Gambit). At this stage the user normally perform the following activities:
Definition of the geometry of the FM01LC reactor (≈ 35.2 cm3). Subdivision of the geometry into nonoverlapping 206 272 tetrahedric cells via grid generation. Definition of fluid properties (see Table 1) and selection of appropriated boundary conditions such as the electrolyte mass flux (kg s1) at the main entrance of the reactor channel (before the manifold), see Table 2.
Governing Equations of Fluid Flow. In three dimensions, fluid flow is described by means of a system of five partial differential equations: the mass conservation, xy and z momentum and the energy balance equation. Given the experimental conditions described elsewhere [19] the following restrictions are allowed: i) steady state is assumed in the fluid flow equations, ii) the temperature and viscosity of the fluid are constant, iii) incompressibility is assumed without energy variations, and iv) natural convection is neglected due to the predominance of forced convection. Under these conditions, the system is reduced to four partial differential equations. The NavierStokes equations require mass (equation 21) and momentum (equation 22) conservation to be satisfied to simultaneously solve a coupled set of second order, nonlinear equations. Table 1. Physical properties of the electrolyte
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Physical property
Value
Fluid density ( )
1096 kg m3
Dynamic viscosity ( ) 0.00119 kg m1 s1
(
Schmidt number
/ DCu 2
)
Diffusion coefficient of Cu2+
(v
de / )
( DCu 2 ) 5.0 x 1010 m2 s1
2172
Reynolds number
197  918
Table 2. Characteristic linear flow velocities (defined by equation 2) of the electrolyte and Reynolds number in the FM01LC Characteristic linear flow velocity in the channel / m s1 Nominal Reynolds number Re
0.024
0.032
0.048
0.065
0.081
0.096
0.112
0.192
197
262
393
533
664
787
918
1574
( v ) 0
(21)
For simple cases (i.e. laminar regime) the resulting flow pattern is completely described by the set of previous equations and they can be either analytically or numerically solved
[55]. However, as was previously described, the FM01LC cell design is not ideal ( = 0.105) and irregular velocity gradients are expected at Reynolds number well below of 2000 Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
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along the channel length and width, and particularly near the channel entrance and exit due to the four jet streams from the flow distributors.
v v v 2 v x 2 v x 2 v x 1 p vx x v y x vz x 2 2 2 x x y z y z x
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2 2 2 v y v y v y v y v y v y 1 p vx vy vz x 2 x y z y 2 z 2 y v v v 2 v z 2 v z 2 v z 1 p vx z v y z vz z x 2 y 2 x y z z 2 z
(22)
As a result, the solely NavierStokes equations are not enough to describe a realistic flow pattern. Fortunately CFD packages offer the possibility of using turbulence models that can be solved alongside the set of equations (21) and (22). Several models are available, including the SpalartAllmarras model, k models (including the standard k model, renormalizationgroup k model, realizable k model), k models (including standard k model, shearstress transport k model), Reynolds stress model and large eddy simulation model. All of these turbulence models are fully described elsewhere [25, 55]. The choice of an appropriate turbulence model is not a simple task, it depends on several factors, i) the nature of the hydrodynamic problem to be solved, ii) the level of accuracy required, iii) the available computer resources, iv) the amount of time available for the simulation and v) the expertise of the user. The renormalizationgroup (RNG) turbulent model was chosen for modelling the expected flow pattern in the FM01LC reactor. Such a model coupled with NavierStokes equations is able to solve turbulent structures that appear in reactor regions where strong velocity gradients, like vortex flows or boundary layer separation flows are dominants. An advantage of this approach is that RNG model also uses the standard k model to resolve transport equations that represent the kinetic energy, k and its dissipation, . A detailed description of how the NavierStokes equations interact with the RNG model can be found elsewhere [25, 55]. Numerical Algorithm. The following steps are performed by the CFD package: i) integration of governing equations of fluid flow over all the (finite) control volumes of the domain, ii) discretization (conversion of the resulting integral equations into a system of algebraic equations) and iii) solution of algebraic equations by iteration. Once the velocity, pressure and other dependent variables are known for a particular cell centre, the solution is applied to the cell boundaries by interpolation. The information is then used to advance step by step to the neighbouring cell until the whole FM01LC reactor geometry is covered. Applying this method to the FM01LC the following results are expected: i) a well defined entry region where the flow develops (effect of the inlet manifolds to introduce electrolyte); ii) the region where the fluid flow is developed; iii) the region where manifolds remove electrolyte from the reactor and affect the fluid flow pattern. As a result, a detailed 3Dflow pattern is obtained as a function of the FM01LC length (0.160 m), width (0.040 m)
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and depth (0.0055 m). The length of the channel is divided in 7 parts corresponding to 0, 0.01, 0.04, 0.08, 0.12, 0.14 and 0.15 m (see Figure 3). The depth of the channel is divided in three planes (from the top: z1= 0.00125 m, z2 = 0.00275 m and z3 = 0.00425 m), parallel to the channel bottom. The postprocessor can display the flow pattern and graphically present it as contours of xvelocity, velocity vectors and velocity profiles for a given channel depth across the channel width. A detailed description of the effect of the manifold configuration on flow pattern is obtained along the channel length and it is discussed in the following section.
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Velocity Vectors. For a characteristic flow velocity of 0.024 m s1, Figure 4 shows how fluid flow develops along the first four centimeters of the channel length. As it was expected, jet streams and vortexes are formed at the cell entrance, as a result of the flow distributors. Vortexes are characterized by some negative velocities in the x direction (i.e. 0.0182 m s1) while jets are characterized by high velocities. The longest jet is formed at the flow distributor located at the channel bottom and they decrease progressively to the channel up side. In this order, each one of them reaches a maximum flow velocity ( 0.069, 0.052, 0.041, 0.022 m s1) at the centre of channel depth (z = 0.00275 m). For the same characteristic flow velocity, Figure 5 shows the velocity vectors indicating both magnitude and direction of the fluid flow at the centre of channel depth (0.00275 m) for 0, 0.01, 0.04 and 0.08 m from the channel entrance.
Figure 3. The length of the FM01LC channel is divided in 7 parts, and a detailed description of the flow pattern is obtained for each of them.
The major flow disturbances are very clear for the first fourth of the channel length but most of them gradually diminish as a function of the distance. Figure 6 shows the flow pattern in the second half of the channel. In this section, the fluid flow can be considered developed, except for the last centimetre of the channel length (0.15 and 0.16 m).In the region near the exit manifolds, the reactor fluid flow pattern is strongly affected by the formation of four jets, but in inverse order with respect to channel entrance; the strongest jet at the top and the weakest jet at the bottom.
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Figure 4. Fluid flow development along the first four centimeters of the channel length. Characteristic flow velocity of 0.024 m s1
Figure 5. Velocity vectors indicating both magnitude and direction of the fluid flow at the centre of channel depth for the first half of the reactor channel. Characteristic flow velocity of 0.024 m s1.
Similar velocity vectors (and flow contours) were obtained for five more flow rates (0.032 ms1, 0.065 ms1, 0.097 m s1, 0.112 m s1 and 0.192 m s1) along the channel length. The flow pattern gradually changes as a function of the flow velocity. In all cases, four jets alternate with three vortex zones formed at the channel entrance. The longest jet is always formed at the flow distributor located at the bottom of the channel and the jets decrease progressively towards the top of the channel. For a characteristic flow velocity of 0.192 m s1, Figure 7 shows how fluid flow develops along the first four centimeters of the channel length. As it was expected, at this flow velocity, jet streams and vortexes are the strongest at the cell entrance, as a result of the flow distributors.
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For the same characteristic flow velocity, Figure 8 shows the velocity vectors of the fluid flow at the centre of channel depth (0.00275 m) for 0, 0.01, 0.04 and 0.08 m from the channel entrance.
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Figure 6. Velocity vectors indicating both magnitude and direction of the fluid flow at the centre of channel depth for the second half of the reactor channel. Characteristic flow velocity of 0.024 m s1.
Figure 7. Fluid flow development along the first four centimeters of the channel length. Characteristic flow velocity of 0.192 m s1.
The major flow disturbances are very clear for the first half of the channel length. Figure 9 shows velocity vectors for a characteristic flow velocity of 0.192 m s1 at the centre of channel depth (0.00275 m) for the second half of the channel. Although the fluid flow is more uniform in the second half of the channel length, disturbances due to manifolds located at the channel exit can reach 0.15 m from the channel entrance. At high flow rates the zone free of disturbances is located between 0.12 and 0.14 m
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and this correspond to the ≈13% of the channel length. In contrast, at low flow rates the zone free of disturbances correspond to the ≈70% of the channel length.
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Figure 8. Velocity vectors indicating both magnitude and direction of the fluid flow at the centre of channel depth for the first half of the reactor channel. Characteristic flow velocity of 0.192 m s1.
Figure 9. Velocity vectors indicating both magnitude and direction of the fluid flow at the centre of channel depth for the second half of the reactor channel. Characteristic flow velocity of 0.192 m s1.
Velocity Profiles in the yDirection. For a characteristic flow velocity of 0.024 m s1, Figure 10 shows, at a distance of 0.01 m from the channel entrance, three simulated velocity profiles across the channel width as a function of its depth (z). The velocity profiles are uneven and have several points of inflexion across the channel width (in the ydirection). It can be seen that, across the channel depth, in the three planes considered (z1 = 0.00125 m, z2 = 0.00275 m and z3 = 0.00425 m), local flow velocities are also uneven. Figure 10 shows as
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well that moving away from the channel entrance, the flow disturbances decay downstream, although some of them are still seen at 0.04 m of the channel length. It can be seen from Figure 10 that, at 0.04 m from the channel entrance and in the central part of the channel width (0.01 m < y < 0.03 m) the three velocity profiles are more uniform than near the channel walls. In the first part of the channel depth (0 m < z < 0.00275 m) the local flow velocities are slower than in the second part of the channel depth (0.00275 m < z < 0.0055 m). In the FM01LC cell entrance, flow distributors are not located at the centre of its depth (z2 = 0.00275 m). This could explain why velocities are different at the top (z1 = 0.00125 m) compared with those at the bottom (z3 = 0.00425 m) of the channel. This suggests that fluid layers are mixing and the fluid flow is not fully developed for the first few centimetres of the channel length. For the same characteristic flow velocity (0.024 m s1), Figure 11 shows how the fluid flow develops in the first 0.08 m of the channel length by means of three simulated velocity profiles (averaged in the zdirection) across the channel width. The flow predictions show how the strong vortex structures at the channel entrance decrease downstream. At distances of 0.04 m and 0.08 m from channel entrance, local flow velocities are more uniform in the centre of the channel although some variations are still found near the channel walls.
Figure 10. Three simulated velocity profiles across the channel width at a distance of 0.01 and 0.04 m from the channel entrance, as a function of its depth (z): () z1= 0.00125 m, (●) z2 = 0.00275 m and (▲) z3 = 0.00425 m. Characteristic linear velocity = 0.024 m s1.
Similar velocity profiles across the channel width were obtained for five more flow rates (0.032 ms1, 0.065 ms1, 0.097 m s1, 0.112 m s1 and 0.192 m s1) along the channel length. It was verified that the flow pattern gradually changes as a function of the flow velocity, as it was previously discussed. At high flow rates the disturbances reach the 0.08 m of the channel length. Figure 12 shows how four simulated velocity profiles (averaged in the zdirection) across the channel width change as a function of the following characteristic flow rates: 0.024, 0.065, 0.097 and 0.192 m s1. It is clear that the flow pattern is affected by the magnitude of the local flow velocity.
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Figure 11. Vortex structures generated at the channel entrance decrease downstream. Velocity profiles develop at () 0.01 m, () 0.04 m and (▲) 0.08 m from the channel entrance. An averaged depth (z) was considered. Characteristic linear velocity = 0.024 m s1
Figure 12. Mean linear flow velocity profiles across the channel width for various characteristic linear flow velocities: () 0.024, () 0.065, (▲) 0.097 and (●) 0.192 m s1 at a distance of 0.08 m from the channel inlet. An averaged depth (z) was considered. Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
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Figure 13. Eight equidistant velocity profiles (VP) across the channel width (y dimension) at: () = 0.0006 m, () = 0.0066 m, () = 0.0126 m, () = 0.0186 m, () = 0.0246 m, () = 0.0306 m, () = 0.0366m and () = 0.0396m. The first ( ) and the eighth () velocity profiles are located only 0.0006 m and 0.0004 m from the bottom and top of the channel respectively. VP are located at three different distances from the channel entrance. Characteristic flow rate of 0.024 m s1.
Velocity Profiles in the zDirection. The width of the channel is divided in 8 points corresponding to 0.0006 m (), 0.0066 m (), 0.0126 m (), 0.0186 m (), 0.0246 m (), 0.0306 m (), 0.0366 m () and 0.0396 m (). For all widths, velocity profiles are evaluated at three characteristic flow velocities. Figure 13 shows, for a characteristic flow rate of 0.024 m s1, the corresponding velocity profiles across the channel depth at 0.01, 0.04 and 0.08 m away from the channel entrance. At 0.01 m from the channel entrance the first velocity profile () is slower than the eighth () due to the manifold configuration effects (see as well the description of Figures 4 and 5). In contrast, across the channel depth it can be seen that most of the velocity profiles have parabolic behaviour. However, for a given zposition, the local linear flow velocities are different across the channel width, indicating inflexion points across the channel width. This situation is better visualized in Figure 5.
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At 0.04 m from the channel entrance the first () and the eighth () velocity profiles have similar local velocities along the channel depth but its hydrodynamic behaviour is not parabolic due to a combination of the entrance and wall effects. The rest of the velocity profiles have parabolic behaviour along the channel depth. For a given zposition, local flow velocities are similar, implying that disturbances (inflection points across the channel width) decay downstream. This situation is better visualized in Figure 5.
Figure 14. Eight equidistant velocity profiles (VP) across the channel width (y dimension) at: () = 0.0006 m, () = 0.0066 m, () = 0.0126 m, () = 0.0186 m, () = 0.0246 m, () = 0.0306 m, () = 0.0366m and () = 0.0396m. The first ( ) and the eighth () velocity profiles are located only 0.0006 m and 0.0004 m from the bottom and top of the channel respectively. VP are located at three different distances from the channel entrance. Characteristic flow rate of 0.065 m s1.
At 0.08 m from the channel entrance, all the velocity profiles showed parabolic behaviour. The lowest velocity profiles are those located near the channel walls. At this point, the fluid flow can be considered well developed. Figure 14 shows, for a characteristic flow rate of 0.065 m s1, the corresponding velocity profiles across the channel depth at 0.01, 0.04 and 0.08 m away from the channel entrance. As we can expect, at 0.01 m the hydrodynamic behaviour confirms that the fluid flow is not developed at this point. However, the downstream parabolic profiles tend to develop progressively.
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Figure 15. Eight equidistant velocity profiles (VP) across the channel width (y dimension) at: () = 0.0006 m, () = 0.0066 m, () = 0.0126 m, () = 0.0186 m, () = 0.0246 m, () = 0.0306 m, () = 0.0366m and () = 0.0396m. The first ( ) and the eighth () velocity profiles are located only 0.0006 m and 0.0004 m from the bottom and top of the channel respectively. VP are located at three different distances from the channel entrance. Characteristic flow rate of 0.112 m s1, m s1.
At 0.04 m from the entrance, the first () and the eighth () velocity profiles have similar local velocities along the channel depth but the hydrodynamic behaviour is not parabolic due to a combination of the entrance and wall effects. Although the other velocity profiles show a parabolic behaviour along the channel depth, for a given zposition, local linear flow velocities are slight different and there are inflection points across the channel width. At 0.08 m from the entrance the fluid flow is considered fully developed across the channel width and depth. Figure 15 shows, for a characteristic flow rate of 0.112 m s1, the corresponding velocity profiles across the channel depth at 0.01, 0.04 and 0.08 m away from the channel entrance. As
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we can expect, at high flow rates all of them are uneven but downstream parabolic profiles tend to develop progressively. At high flow rates it is confirmed that the disturbances reach the 0.08 m of the channel length. According to the hydrodynamic analysis made it can be concluded that, for practical purposes, fluid flow can be considered to be developed at low flow rates (150 < Re < 550) from 0.04 m to 0.15 m. This distance corresponds to the ≈70 % of the channel length. Moreover it is clear that this conclusion cannot be drawn from the other criteria discussed in the previous sections.
MASS TRANSPORT AND FLUID DYNAMICS
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As it was discussed in the introduction, a correct reactor design requires fully developed flow conditions before the electrolyte contacts the electrode surface. The main parameters such as the Nernst diffusion layer thickness (δN), the local mass transport coefficient (km) and the limiting current (IL) should be uniform before contacting the electrode surface.) However, turbulent flows with a nominal Reynolds number well below of 2000 have been encountered. Consequently, fully developed flows are rare because manifolds usually inject or extract the electrolyte very close to the electrodes. As there is currently no mathematical equation that links the mass transport coefficient and the hydrodynamic behaviour for developing flows in a channel, the use of alternative routes is justified. The following section discusses how the flow pattern is substituted into a general mass transport correlation to calculate the mass transport coefficients. Modelling Mass Transport. The hydrodynamic behaviour within the FM01LC reactor influences the mass transport coefficient and the electrode current when the process is mass transport controlled. It was assumed that a Nernst type stagnant diffusion layer of thickness N was developed between the bulk solution and the electrode surface as shown in the following Figure 16:
Figure 16. Idealized development of the Nernst type stagnant diffusion layer,
N .
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Where Ax is the cross sectional area of the FM01LC channel, c is the concentration of the electroactive species i, A is the electrode area, IL is the transport controlled current, n is electron stoichiometry, F is Faraday constant, km is local mass transport coefficient and N is the flux of electroactive species and is defined as [56]:
N i Di ci
zi F Di ci ci v RT
(23)
Therefore, the flux of electroactive species i depends on their diffusion (first term on the righthand side), migration (second term on the righthand side) and convection (last term). Equation 23 can be simplified taking into account the following suppositions:
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1. Steady state is assumed during the FM01LC operation. 2. The migration term can be neglected because the solution contains an excess of supporting electrolyte. 3. In the bulk solution the diffusion term can be neglected and a realistic flow pattern can be obtained by applying a CFD technique (the solution of the NavierStokes equations coupled with the RNG model). This implies a uniform concentration in the bulk solution. 4. In the Nernst type stagnant diffusion layer, convection can be neglected and its thickness is determined by the flow velocity in the outer edge of the boundary layer. In the diffusion layer Ni can be expressed in terms of onedimensional linear molecular diffusion [13]. Therefore, the flux of electroactive species to the electrode surface is expressed by:
Ni
Di (c0 c) N
(24)
Where c0 and c are the bulk concentration and concentration respectively at the electrode surface of the species i, the Mass transfer coefficient, km, can be expressed as [13]:
km
Di N
(25)
Conversely, mass transport in electrochemical systems is often cast in terms of dimensionless numbers. Thus, dimensionless flux (Sh) is a function of the dimensionless fluid velocity (Re) multiplied by the dimensionless fluid properties (Sc). In forced convection reactors this correlation is often of the form of equation 3. Consequently, if the first supposition is true, mass transfer coefficient in the FM01LC must satisfy simultaneously both equations: 3 and 24.
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Therefore, if a realistic flow pattern is obtained from numerical calculation and an appropriated mass transport correlation is chosen for a given electrochemical reactor, then, the mass transport coefficient (km) can be numerically evaluated via the Sh number definition (see equation 3). For parallel plate reactors (including FM01LC) working at 60 < Re < 2000, a selected collection of empirical constants for equation 3 are summarized in Table 3. All these equations 6, 2629 were coupled with the theoretical flow pattern at different volumetric flow rates. Then, it was possible to estimate the main transport properties of the electrolyte, such as Re, Sh and km at different positions in the FM01LC reactor channel by taking into account the above assumptions, as well as those that follow: Table 3. Selected mass transport correlations for fluid flow through parallel plate reactors working at 60 < Re < 2000. The number of these correlations follows sequentially from equations (125) in the text.
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Equation
Sh a Re b Sc c Lee
6
a 2.54
b 0.33
c 0.33
e 0.33
26
1.85
0.33
0.33
0.33
27
0.38
0.68
0.33
28 29
0.39 0.22
0.58 0.71
0.33 0.33
Conditions
Ref
Fully developed laminar flow. 58 < Re < 2000, 2850 < Sc < 5140, 0.17 < Le < 12.5, S/B = 0.167 or 0.175 Theoretical derived for fully developed laminar flow assuming infinitely wide electrodes. ReSc(de/L) > 104 and Le 35 Baffled parallel plate reactor. Electrode area 225 cm2, Le = 15 cm/2.3 cm = 6.5 Flat plate, 70 < Re < 800 FM01LC reactor, 200 < Re < 1000, Le = 16 cm / 0.967 cm = 16.6, electrode area 64 cm2.
10, 15 57
58
59 14
Although the fluid flow is not fully developed, it is considered to be sufficiently uniform to apply to equations (6, 2629 at all volumetric flow rates described in Table 2. In order to evaluate the local mass transport coefficients, imaginary segmented electrodes were considered into an array of 100 x 32, across the channel width and length, respectively. Some of them are shown at 0.01 m, 0.04 m and 0.08 m from the channel entrance, see Figure 17. A surface of 0.003 m length (xdimension) x 0.00035 m width (ydimension) was considered for each imaginary segmented electrode and the separation between them was 0.00005 m along the channel width. Each imaginary segmented electrode surface is related to a channel volume (0.003 m x 0.00035 m x 0.0055 m = 5.8 x 109 m3). For each volume, a theoretical velocity profile was obtained with 10 local flow velocities along the channel depth (zdimension). For each velocity profile an arithmetical average fluid velocity was obtained and it was considered to be the local mean flow velocity (v) representative of its volume.
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A local mean Re was computed, for each corresponding local mean flow velocity (v), according to: Re
v de
(30)
For a given mass transport correlation 6, 2629, a local mean Sh was computed, from equations 6, 2629, for each corresponding local mean flow velocity (v) and local mean Re. For each corresponding local mean Sh obtained, an average km value was computed from equation 31 and it was considered as the averaged km to be observed on the corresponding electrode surfaces. Sh k m D de
143
(31)
The differences between theoretical and experimental mean mass transport coefficient were calculated according to:
k RE 1 theory x 100 kexp t
(32)
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Where RE is the relative error [60].
Figure 17. In order to evaluate the local km, imaginary segmented electrodes were considered into an array of 100 x 32, across the channel width and length, respectively.
From the detailed description of the numerical flow pattern in the FM01LC reactor, it can be seen that fluid flow cannot develop because major fluid streams gradually shifts from the bottom to the top of the channel, creating a low velocity region at the centre of the cell. This hydrodynamic behaviour is observed across the whole range of nominal Re considered Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
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in Table 2. As previously discussed, the formation of vortexes and jets are observed at the channel entrance due to the presence of four distinct jets that result from the flow distributors. Accordingly, mass transport coefficient will vary along the channel width and the highest values will coincide with the jet positions. Inversely, the lowest values will coincide with the vortex positions. As a result, it is expected to have a more uniform km distribution, along the channel width and length, as the fluid is moving away from the channel entrance. In this section a summary of the numerical evaluation of km, corresponding to three columns of 100 imaginary electrodes each and located at 0.01, 0.04 and 0.08 m from the channel entrance, will be analyzed and discussed. Main Results. For a characteristic flow velocity of 0.024 m s1, the effect of the manifold configuration on the mass transport coefficient is presented in Figure 18. At 0.01 m away from the channel entrance, local km values vary between 6 x 106 m s1 to 105 m s1 according to the location of the imaginary segmented electrode surface across the channel width. The average km value obtained at this distance is 8 x 106 m s1. Following the same procedure, numerical km values across the channel width were evaluated at a distance of 0.04 m and 0.08 m away from the channel entrance. From Figure 18, it is observed that, for these distances, local km values are more uniform across the channel width. These results are in agreement with the hydrodynamic behavior of the electrolyte. Over the remainder of the channel length, local km values remain constant. Under the same operational conditions, it has been documented [18] that the experimental local averaged km value, along the channel length was 7.7 x 106 m s1. The relative error (RE), evaluated by means of the equation 32, for the numerical prediction is 4.5 % higher. For a flow rate of 0.065 m s1, Figure 19 shows km values across the channel width at 0.01 m away from the channel entrance. As expected, local km values vary with local flow velocities, from 9 x 106 m s1 to 1.38 x 105 m s1. The average km value obtained at 0.01 m, 0.04 m and 0.08 m is the same: 1.1 x 105 m s1. For the rest of the channel length local km values are essentially the same. For the same operational conditions, Brown et al [18], have shown under the same operational conditions, that the experimental local averaged km value, along a channel length of 1.54 x 105 m s1. The relative error for the numerical prediction is 28.6 % lower. For a flow rate of 0.112 m s1, Figure 20 shows km values across the channel width at 0.01, 0.04 and 0.08 m away from the channel entrance. As expected, local km values vary with local flow velocities, from 1.1 x 105 m s1 to 1.6 x 105 m s1. The average km value along the channel length is the same: 1.3 x 105 m s1. No experimental value of km was reported in the work carried out by Brown et al, [18] at 0.110 m s1 but the extrapolation of the experimental curve shows that the value is 2.2 x 105 m s1, 41 % higher when compared with the numerical prediction in this paper. Applying the procedure described above, mass transport coefficients were numerically evaluated in the whole channel for a wide range of nominal Reynolds numbers (in the range of 197 and 918, see Table 2), using the mass transport correlations shown in Table 3. Figure 20 shows five numerical sets of –log(km) values (curves ◊, , , ▲, ●) and one experimental
set (curve ○) of –log(km) values vs. the –log( v ), characteristic mean linear electrolyte velocity. Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
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Figure 18. Local mass transport coefficients, for a characteristic flow velocity of 0.024 m s1 at a) 0.01 m, b) 0.04 m and c) 0.08 m from the channel entrance.
The best mass transport correlation is, obviously, the empirical correlation experimentally found for the FM01LC reactor (equation 29, curve ▲). When this correlation is coupled with a theoretical flow pattern, the mass transport coefficient (km) can be predicted within an experimental error: RE 6%. The second best equation is the flat plate mass transport
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correlation (equation 28, curve ). Using this equation, mass transport coefficient (km) can be predicted within 18% < RE < 28.7%.
Figure 19. Local mass transport coefficients, for a characteristic linear flow velocity of 0.065 m s1 at a) 0.01m, b) 0.04 m and c) 0.08 m from the channel entrance.
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●) 22.8% < RE < 36.6%; ideal fully developed laminar flow (equation 6, curve ) 5.4% < RE < 40.9% and finally, the highest rate of was found in the fully developed laminar flow,
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assuming infinitely wide electrodes (equation 26, curve ◊) 23.3% < RE < 56.9%.
Figure 20. Local mass transport coefficients, for a characteristic linear flow velocity of 0.112 m s1, at a) 0.01m, b) 0.04 m and c) 0.08 m from the channel entrance.
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log km
3 2.8 2.6 2.4 2.2 2 0.6
0.8
1
1.2
1.4
1.6
1.8
log linear velocity Figure 21. Five empirical sets of km values vs. the characteristic mean linear electrolyte velocity (equation 26, ◊, 23.3% < RE < 56.9%; equation 6, , 5.4% < RE < 40.9%; equation 28, , 18% < RE < 28.7%; equation 29, ▲, RE 6%; equation 27, ●, 22.8% < RE < 36.6%) and one experimental set (curve ○) of km values vs. the characteristic mean linear electrolyte velocity.
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PARTICLE IMAGING VELOCIMETRY As it was discussed above, the rate of mass transport is a function of the flow pattern, which is strongly affected by the manifold geometry of the electrochemical reactor. CFD offers the possibility to evaluate the flow pattern and its effect on mass transport in developing fluids along a reactor. However, CFD should be further validated, as a manifold design tool, against a flow visualisation technique. Particle Image Velocimetry (PIV) has been used in other hydrodynamic studies to qualitatively compare with the numerical results obtained by commercial CFD packages [61 63]. In this last section, CFD is used to assess a different manifold geometry and it effect on the flow pattern development along the electrochemical reactor. The fluid flow was modelled using the same numerical approach described previously. In order to validate the numerical results a laboratory electrochemical reactor was built and its flow pattern was evaluated by means of the particle image velocimetry (PIV) technique. Figure 22 shows the main geometrical details (projected electrode area 0.005 m2) of such electrochemical reactor and the inlet and outlet manifolds. The electrolyte is pumped to the cell entry by means of a circular pipe. At the cell entry, the geometry of the circular pipe is converted to a rectangular duct (0.02 m long) followed by a smooth expansion before the 0.10 m of the electrochemical channel is reached. After the channel, a smooth narrowing section is located before the exit. The electrolyte leaves the electrochemical reactor through a circular pipe normal to the electrochemical channel.
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Figure 22. Main geometrical details (projected electrode area 0.005 m2) of the electrochemical reactor, including the inlet and outlet manifolds.
Figure 23. Experimental setup of the PIV system and the working region (0.003 m2) located in the first 0.06 m of the channel at 0.005 m in the zdirection.
It is expected that the flow pattern is influenced by the change in shape of the flow crosssection (from a circular to a rectangular duct). In order to minimize this effect a calming sections is located after the change in shape to force the fluid flow to develop. After the channel reactor a second calming section is included to minimize the exit effects. Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
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The PIV technique was used to evaluate a realistic and instantaneous flow pattern along the electrochemical reactor depicted in Figure 22. The main electronic components of the PIV system are synchronized and include: the electrochemical reactor with a clear window at the channel side to ensure the optical access, a laser source, a Charge Coupled Device (CCD) camera, a PC to acquire and evaluate data and the hydraulic circuit with a pump to control fluid flow rate. Figure 23 shows the experimental setup of the PIV system and the working region (0.003 m2) located in the first 0.06 m of the channel at 0.005 m in the zdirection. This system works under the basic principles described briefly here: PIV technique is based on the well known equation:
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Speed
Dis tan ce Time
(33)
Therefore, the property actually measured is the distance travelled by particles in the flow within a known time interval. These particles are added to the flow and known as seeding. The particles follow the flow and the PIV system detects their movement because an area of the flow field is illuminated by a lightsheet. The lightsheet is not continuous, but pulsed to produce a stroboscopic effect, freezing the movement of the seeding particles. The time between the light pulses is the denominator in equation (33). To detect the position of the illuminated seeding particles, a CCDcamera is positioned at right angles (90°) to the lightsheet, and particle positions will appear as light specks on a dark background on each camera frame. The pulsing lightsheet and the camera are synchronized so that particle positions at the instant of light pulse number 1 are registered on frame 1 of the camera, and particle positions from pulse number 2 are on frame 2. The camera images are divided into rectangular regions and for each of these regions the image from the first and the second pulse of the lightsheet are correlated to produce an average particle displacement vector. Doing this for all regions a vector map of average particle displacements is obtained. Dividing the known time between two captured images, the displacement vectors are converted into a map of raw velocity vectors. Main Results. The images were taken from the whole working rectangular region located at the middle (0.005 cm) of the channel depth (zdirection). Path lines and flow contours were obtained for the flow rates shown in table 2. For all flow rates a big vortex was observed at the middle of the channel, implying that the change in shape of the flow crosssection prevents the fluid flow to be developed. Modelling instable flows at turbulent regimes are known to be especially challenging [61, 62] and Figure 24 shows a qualitative comparison between a calculated (CFD) and measured (PIV) path lines in the xdirection for a characteristic linear flow velocity of 0.06 m s1. As it can be seen from Figure 24, an instable flow patter along the channel length is observed. For the same characteristic flow velocity, Figure 25 shows a qualitative comparison between a calculated (CFD) and measured (PIV) flow contours. Some differences in velocities are evident; however, in general the both flow contours are in agreement: the highest velocity region is located at the right part of the channel and the lowest velocity region is at the left part. Moreover, the stagnation area is located in the same position in both approaches.
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Figure 24. Qualitative comparison between CFD and measured PIV path lines in the xdirection for a characteristic linear flow velocity of 0.06 ms1.
Figure 25. Qualitative comparison between a calculated (CFD) and measured (PIV) flow contours, for a characteristic linear flow velocity of 0.06 m s1.
For a characteristic linear flow velocity of 0.18 m s1, Figure 26a shows, at 0.005 m of the channel depth and at 0.06 m from the beginning of the channel, a quantitative comparison between a numerical (, CFD) and experimental (▲, PIV) velocity profiles across the channel width. The corresponding velocity profiles for characteristic linear flow velocity of 0.06 m s1 is shown in Figure 26b. In general, local flow velocities evaluated by CFD are higher than those evaluated by PIV. For a characteristic linear flow velocity of 0.06 m s1 the biggest differences (≈ 50%) in velocities are located between 0.0185 m and 0.025 m of the channel width. However, this
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difference gradually diminishes across the channel width. A similar situation was found for the corresponding characteristic linear flow velocity of 0.18 m s1. In this case, the biggest differences (≈ 50%) in velocities are shift to the left of the channel and they are located between 0.005 m and 0.018 m of the channel width.
Figure 26. Quantitative comparison between a numerical (, CFD) and experimental (▲, PIV) velocity profiles across the channel width at two different characteristic linear flow velocities.
Following the same method described above, the effect of the manifold configuration on the mass transport coefficient was estimated from local flow velocities. For a characteristic linear flow velocity of 0.18 m s1, Figure 27a shows the km estimated from CFD () and from PIV (▲). In general, km estimated from CFD are higher than those measured from PIV. The errors trend to increase toward the channel walls but they decrease in the stagnation area, where the flow velocity is almost zero. The corresponding km for a characteristic linear flow velocity of 0.06 m s1 is shown in Figure 27b. It can be seen that, in general, km estimated from CFD are higher (28%) than those measured from PIV. However, a different situation was found in the stagnation area: the errors trend to increase in this zone.
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Figure 27. Quantitative comparison between a numerical (, CFD) and experimental (▲, PIV) km across the channel width at two different characteristic linear flow velocities.
CONCLUSIONS Among the main available tools for evaluating the flow pattern in electrochemical reactors, CFD is one of the best tools. It was found that CFD provides a convenient and versatile tool to analyse the hydrodynamic behaviour along an electrochemical filterpress reactor. A major feature of this approach lies in the minimal number of physical parameters needed for a rapid reliability test to model the hydrodynamic behaviour in the electrolyte channel. If the fluid flow pattern is near to the steady state conditions, it is possible to estimate the main transport properties of the electrolyte, such as Re and Sh and combine them with equation (28) to obtain realistic km values along a reactor channel. If the fluid flow is instable, an acceptable qualitative description of the flow pattern along a reactor channel can still be obtained by CFD. This approach could be attractive to reactor designers since it can reduce the experimental efforts essential to develop more efficient and economic electrochemical reactors.
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[25] Fluent. User‘s Guide. Fluent Inc.: Lebanon, USA, 2003. [26] Janna, WS. In The Handbook of Fluid Dynamics. Johnson R. W; Ed; SpringerVerlag: Heidelberg. 1998. [27] Bogard , DG; Thole, KA. In Wallbounded turbulent flows. In The Handbook of Fluid Dynamics. Johnson RW. Ed; SpringerVerlag: Heidelberg. 1998. [28] Pickett, DJ; Wilson, CJ. Electrochim Acta., 1982, 27, 591594. [29] Levenspiel, O. Chemical Reaction Engineering, Wiley: New York, 1972. [30] Fahim, MA; Wakao, N. Chem Eng J., 1982, 25, 18. [31] Trinidad, P; Walsh, FC. Electrochim Acta., 1996, 4 493502. [32] Trinidad, P; Ponce de León, C; Walsh, FC. Electrochim Acta., 2006, 52, 604613. [33] Shah, YT; Stiegel, GJ; Sharma, U. J Am Inst Chem Engr., 1978, 24, 369401. [34] Riemer, M; Kristensen, G; Harremoes, HP. Wat Res., 1980, 14, 949958. [35] Jimenez, B; Noyola, A; Capdeville, B; Roustan, M; Faup, G. Wat Res. 1988, 22 , 12531257. [36] Jimenez, B; Noyola, A; Capdeville, B. Biotech Tech., 1988, 2, 7782. [37] Krall, KM; Sparrow, EM. J Heat Transfer., 1966, 88, 131138. [38] Filetti, EG; Kays, WM. J Heat Transfer., 1967, 89, 163. [39] Zemanick, PP; Dougall, RS. J Heat Transfer., 1970, 92, 5359. [40] Baughn, JW; Hoffman, NA; Takahashi, RK; Launder, BE. J Heat Transfer., 1984, 106, 789797 [41] Runchal, AK. Int J Heat Mass Transf, 1971, 14, 781788. [42] Tagg, DJ; Patrick, MA; Wragg, AA. Trans I Chem E., 1979, 57, 176. [43] Wragg, AA; Tagg, DJ; Patrick, MA. J Appl Electrochem., 1980, 10, 4349. [44] Rizk, TY; Thompson, GE; Dawson, JL. Corros Sci., 1996, 38, 18011808. [45] Djati, A; Brahimi, M; Legrand, J; Saidani, B. J Appl Electrochem, 2001, 31, 833837. [46] FríasFerrer, A; GonzálezGarcía, J; Sáez; Ponce de León C; Walsh FC. AIChE J., 2008, 54, 811823 [47] Patankar, SV. Numerical Heat Transfer and Fluid Flow. McGrawHill: New York, 1980. [48] Chung MH. Electrochim Acta., 2000, 45, 39593972. [49] Souadnia, A; Soltana, F; Lesage, F; Latifi, MA. Chem Eng Proc., 2005, 44, 847854. [50] Pellerin, E; Michelistsch, E; Darcovich, K; Lin, S; Tam. CM. J Membr Sci, 1995, 100, 139146. [51] Karniadakis, GE; Mikic, BB; Patera, AT. J Fluid Mech., 1988, 192, 365371. [52] Cao, Z; Wiley, DE; Fane, AG. J Membr Sci, 2001. 185, 157176. [53] Ahmad, AL; Lau, KK; Abu Bakar, MZ; Abd Shukor, SR. Comp Chem Eng., 2005, 29, 20872095. [54] Vivek. V; Ranade, W; Kumar, A. J. Membr Sci., 2006, 271, 115. [55] Versteeg, HK; W. Malalasekera, W. An Introduction to Computational Fluid Dynamics. Pearson Prentice Hall: Essex, 2007, 40114. [56] Bard, AJ; Faulkner, LR. Electrochemical Methods. Fundamentals and Applications. John Willey & Sons, 1980, 281290. [57] Ralph, TR; Hitchman, ML; Millington, JP; Walsh, FC. Electrochim Acta., 1996, 41, 591603 [58] Goodridge, F; Mamoor, GM; Plimley, RE. I Chem E Symp Ser., 1986, 98, 6168. [59] Ponce de León, C; Reade, GW; Whyte, I; Male, SE; Walsh, FC. Electrochim Acta.,
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2007, 52, 58155823. Rabinovich, S. Measurement Errors. Theory and Practice. AIP Press: New York, 1995, 120. Angioletti, M; Di Tommaso, RM; Nino, E; Ruocco, G. Int J Heat Mass Trans., 2003, 46, 1703–1713. Klemm, T; Gabi, M. Proceedings of PSFVIP4 (F4013), 2003, June 35, Chamonix, France. Angioletti, M; Nino, E; Ruocco, G. Int J Thermal Sci, 2005, 44,349356.
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Leticia Vázquez Gutiérrez, Alberto Alvarez Gallegos and Fernando Z. Sierra
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Chapter 5
COMPUTATIONAL SIMULATION OF INSTABILITY PHENOMENA ASSOCIATED WITH MASS AND ENERGY TRANSPORT THROUGH FLUID FLOW IN POROUS MEDIA Chongbin Zhao Computational Geosciences Research Centre, Central South University, Changsha, China
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ABSTRACT Most mass and energy transport systems through fluid flow in porous media can be mathematically treated as coupled nonlinear multipleprocess problems, for which instabilities are common emerging phenomena that control the fundamental behaviours of such coupled nonlinear systems. This chapter deals with the computational simulation aspects of two common instability phenomena, namely thermodynamic instability and chemicaldissolution front instability, which are closely associated with energy (through heat transfer) and reactive mass (through chemical reaction) transport in fluidsaturated porous media. The physical appearance of thermodynamic instability displays actually a kind of convective porefluid flow in a fluidsaturated porous medium so that thermodynamic instability is also called the convective instability of porefluid flow. This kind of convective porefluid flow can play an important role in transporting heat energy and aqueous minerals in the Earth‘s crust that is often comprised of porous rocks. However, the physical appearance of chemicaldissolution front instability shows a fundamental change in the morphology of an initial chemicaldissolution front. This kind of instability is the direct consequence of an interaction between nonlinear porefluid flow and reactive mass transport through nonlinear porositypermeability feedback effects in the fluidsaturated porous medium. After the mathematical formulations for simulating both thermodynamic instability and chemicaldissolution front instability are described, computational simulation methods, such as the finite element method and the finite difference method, are used to solve the coupled nonlinear problems associated with energy and mass transport processes in fluidsaturated porous media.
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Keywords: Computational simulation, energy and mass transport, fluid flow, heat transfer, instability phenomena, porous media.
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1. INTRODUCTION There are two common instability phenomena, namely thermodynamic instability and chemicaldissolution front instability, which are closely associated with energy (through heat transfer) and reactive mass (through chemical reaction) transport in fluidsaturated porous media. The physical appearance of thermodynamic instability displays actually a kind of convective porefluid flow in a fluidsaturated porous medium so that thermodynamic instability is also called the convective instability of porefluid flow. This kind of convective porefluid flow can play an important role in transporting heat energy and aqueous minerals in the Earth‘s crust that is often comprised of porous rocks. However, the physical appearance of chemicaldissolution front instability shows a fundamental change in the morphology of an initial chemicaldissolution front. This kind of instability is the direct consequence of an interaction between nonlinear porefluid flow and reactive mass transport through nonlinear porositypermeability feedback effects in the fluidsaturated porous medium. In terms of investigating thermodynamic instability phenomena in porous rocks, much has been done over the past years (Horton and Rogers 1945, Lapwood, 1948, Gasser and Kazimi 1976, Murphy 1979, Horne and Caltagirone 1980, Bau and Torrance 1982, Kaviany 1984, Caltagirone and Bories 1985, Jones and Persichetti 1986, Lebon and Cloot 1986, Pillatsis et al. 1987, Bjorlykke et al. 1988, Chen and Chen 1989, Phillips 1991, Nield and Bejan 1992, Alavyoon 1993, Chevalier et al. 1999, Tournier et al. 2000, Gow et al. 2002, Ord et al. 2002, Schaubs and Zhao 2002, SorjonenWard et al. 2002, Lin et al. 2003, Zhao et al. 1997, 1998ab, 1999ac, 2000ab, 2001ab, 2002ab, 2003ab, 2004, 2005ab, 2006ac, 2008ab). For example, a large amount of research has been carried out, both theoretically and numerically, to investigate the detailed physical mechanisms related to the condition to trigger convective porefluid flow, and the effects of geological heterogeneities on the convective heat transfer in the crust of the Earth. As a result, the following main conclusions have been made: (1) if the Rayleigh number of a system is either critical or supercritical, convective porefluid flow can take place in the porous rocks where the porefluid pressure gradient is close to the hydrostatic pressure gradient. (2) If the porefluid pressure gradient is close to the lithostatic pressure gradient, convective porefluid flow cannot take place in the porous medium that has constant temperature and impermeable boundary conditions at both the top and the bottom of the system, but it can take place if the system has a permeable top with a constant pressure and temperature, and a bottom with a constant upward porefluid velocity and conductive heat flux. (3) The heterogeneity of either medium permeability or thermal conductivity has a significant effect on the convective heat transfer within the system. The material thermoelasticity may affect the convective heat transfer, depending on the relative hardness of the rock masses. (4) Convective porefluid flow can also take place in a geological fault zone, depending on the Rayleigh number of the system, which is directly proportional to the ratio of the fault height to the fault thickness. For the theoretical analysis of instability problems, it is desirable to establish a criterion that can be used to assess whether or not the considered system is stable. Toward this end, a
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comprehensive dimensionless number is often derived to represent the driving mechanisms and controlling characteristics of a nonlinear dynamic system. In the case of chemicaldissolution front instability problems (Chadam et al. 1986, 1988, Ortoleva et al. 1987, Renard et al. 1998, Chen and Liu 2002, Zhao et al. 2008ce, 2009ab, 2010ac), the Zhao number is recently used to represent the porefluid flow that is the driving force, and other controlling characteristics such as solute diffusion/dispersion, chemical kinetics and particle shapes of minerals within the mineral dissolution system. Extensive research (Zhao et al. 2009a) has demonstrated that: (1) if the Zhao number of a chemicaldissolution system is smaller than the corresponding critical Zhao number, then the chemicaldissolution system is subcritical; (2) if the Zhao number of a chemicaldissolution system is greater than the corresponding critical Zhao number, then the chemicaldissolution system is supercritical, and (3) if the Zhao number of a chemicaldissolution system is equal to the corresponding critical Zhao number, then the chemicaldissolution system is in a critical state. As scientific conclusions are commonly derived from an understanding of basic laws that are supported by laboratory experiments, observations of nature, and mathematical and computational simulations, there are three major scientific methods, namely the experimental method, the theoretical analysis method and the computational simulation method, which are currently available for solving contemporary scientific and engineering problems. The experimental method is used to produce data for establishing fundamental physical and chemical laws, while theoretical analysis is used to establish the corresponding fundamental physical and chemical laws as well as governing equations for describing inherent characteristics of a problem. In addition, theoretical analysis can produce exact theoretical solutions for some kinds of scientific and engineering problems. With the advent of more powerful computers, applications of the computational simulation method have gained popularity in many scientific and engineering fields. Due to the complex and complicated nature of geoscientific problems, the computational simulation method has found more and more applications in this particular field. This gradually changes the traditional geoscience from a descriptiondominated empirical discipline into a computersimulationdominated scientific discipline. As a result, an emerging discipline, known as the computational geoscience, has been established in the recent years (Zhao et al. 2009a). The contextual arrangements of this chapter are as follows. In section 2, the mathematical formulation of thermodynamic instability in twodimensional fluidsaturated porous media is presented in detail. In section 3, the finite element method combined with the progressive asymptotic approach procedure is used to simulate mass and energy transport as a result of twodimensional convective porefluid flow in the North West Shelf basin. In section 4, the detailed mathematical formulation of chemicaldissolution front instability in twodimensional fluidsaturated porous media is derived. As an application example, the computational simulation method is employed, in section 5, to simulate the morphological evolution of a chemical dissolution front in the twodimensional fluidsaturated porous medium of a supercritical Zhao number. Finally, some conclusions are given in section 6.
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2. MATHEMATICAL MODEL OF THERMODYNAMIC INTABILITY PROBLEMS IN TWO DIMENSIONAL FLUIDSATURATED POROUS MEDIA The mathematical model of a thermodynamic instability problem is comprised of a set of partial differential equations, which are used to describe the physical processes in the fluidsaturated porous medium. In essence, the mathematical treatment of a thermodynamic instability problem involves a fully coupled problem between porefluid flow and heat transfer in a porous medium. In the process of deriving the mathematical formulation of a thermodynamic instability problem, the porous medium is often assumed to be undeformable (i.e. rigid), while the porefluid is assumed to be incompressible. Under these assumptions, the corresponding steadystate governing equations of a thermodynamic instability problem can be expressed as follows (Zhao et al. 2009a):
u v 0, x y
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v
f 0 c p (u
u
u
Kx
Ky
(
p ), x
(2)
p f g) , y
(3)
(
T T 2T 2T v ) ex 2 ey 2 , x y x y
C C 2C 2C , v Dex D ey x y x 2 y 2
f f 0 [1 (T T0 )] , ex fx (1 ) sx , Dex D fx ,
(1)
(4)
(5)
(6)
ey fy (1 )sy ,
(7)
Dey D fy ,
(8)
where u and v are the horizontal and vertical velocity components of the porefluid in the x and y directions respectively; p is the porefluid pressure; T is the temperature of the porous medium; C is the normalized concentration (in a mass fraction form relative to the porefluid density) of the nonreactive mass; K x and K y are the permeabilities of the porous
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medium in the x and y directions respectively; is the dynamic viscosity of the porefluid;
f is the density of the porefluid and g is the acceleration due to gravity; f 0 and T0 are the reference density and reference temperature used in the analysis; fx and
sx are the
thermal conductivities of the porefluid and solid matrix in the x direction; fy and sy are the thermal conductivities of the porefluid and solid matrix in the y direction; c p is the specific heat of the porefluid; D fx and D fx are the molecular diffusivities of the chemical species in the x and y directions respectively; is the porosity of the porous medium;
is
the thermal volumetric expansion coefficient of the porefluid. Note that since a nonreactive mass (i.e. a tracer) is considered in the fluidsaturated porous medium, there is no source/sink term in the mass transport equation (i.e. Equation (5)). On the other hand, Equation (6) indicates that only the temperaturedependent density of the porefluid, which can be determined from the OberbeckBoussinesq approximation, is considered in the thermodynamic system.
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3. APPLICATION EXAMPLES OF THERMODYNAMIC INSTABILITY PROBLEMS IN TWODIMENSIONAL FLUIDSATURATED POROUS MEDIA In recent years, the petroleum industry worldwide has rapidly adopted computational basin simulation as a standard exploration tool. Fluid flow simulation within the hydrocarbon industry is often considered purely by the various physical conditions (porosity, permeability, pressure) within a sedimentary basin. Thus, by building models of basin properties over time, explorationists can simulate hydrocarbon migration and predict the location, composition, and volume of a hydrocarbon entrapment. Recent advances in advanced computational simulation techniques have adequately demonstrated that fluid flow is not a simple process but is a result of the intimate coupling between fluid flow, mechanical, chemical and thermal processes. In order to build a complete and more accurate picture of fluid flow within a petroleum system, these processes must be thoroughly understood. Although current industry simulation techniques may offer fairly direct results for basinscale reservoir models that are simple to construct, the fundamentals which underlie these models are incomplete in that they lack true physical properties, in particular the coupled nature of fluid flow. Therefore, the considerable time and expense often put into the development of reservoir models may not reflect the true fluid flow scenario, because the potential of thermodynamic instability is often neglected. The main purpose of this section is to assess the role of temperature driven convective porefluid flow within a petroleum basin environment. No attempt has been made to simulate the complexities (e.g. reservoir permeability heterogeneities, varying fluid phases, changing heat flow etc.) of a detailed petroleum system at this stage, but to establish whether this type of convective porefluid flow may represent a significant mechanism for the transport of fluids (including hydrocarbons).
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To investigate convective porefluid flow and hydrocarbon transport in the North West Shelf basin (Zhao et al. 1999c), a simple basin geometry is set up (Figure 1) with a number of layers representing pre, syn, and postrift units. It should be noted that for the purpose of showing the detailed geological structure, Figure 1 was not drawn on scale. The structure of the basin is controlled by opposing polarity faults which dip towards each other generating a simple graben. A single layer within the graben is included to replicate a mature hydrocarbon source rock by generating fluid which can be traced within the model by its concentration. The general feature of the basin is that it is stratified by several layers in the vertical direction and separated by two major faults in the horizontal direction. In Figure 1, A, B, C, D and E are used to represent different layers, whereas F is used to denote the faults. The relative permeability between the different stratigraphic units and the faults is then varied to replicate a number of different permeability scenarios. The sensitivity of convective porefluid flow to these scenarios can then be assessed by the changes in the porefluid velocity, streamline distribution, temperature distribution and concentration distribution of the source fluids.
Figure 1. Geological model of the North West Shelf basin. Layer A represents postrift, layers B and C represent prerift, layer D represents synrift basin fill and layer E represents a source rock interval. Faults are denoted by F. The basin geometry represents a typical graben with footwall uplift in the hangingwall of the major faults. Permeabilities within the model can be varied allowing a number of different scenarios to be simulated.
The finite element method combined with the progressive asymptotic approach procedure (Zhao et al. 1997) is used to simulate convective porefluid flow and hydrocarbon transport in this basin. The computational domain of the basin is 13 km in depth and 110 km in length. Figure 2 shows the computational model of the basin, in which the computational domain of the basin is subdivided into 3672 fournode quadrilateral elements. The temperature at the top o
o
and the bottom of the basin is 20 C and 150 C respectively. The porous constituents of the basin are considered to be isotropic. For the purpose of investigating how hydrocarbon is transported in the basin, there is a unit (normalized) concentration of hydrocarbon located near the centre of the basin (Layer E in Figure 1). In the numerical computation, both the left and right lateral boundaries are considered to be isolated and impermeable in the horizontal direction.
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Figure 2. Computational model of the North West Shelf basin.
To facilitate the corresponding computations, the material within each fault and layer is assumed to be isotropic. Nevertheless, the material heterogeneity of the system can be represented by faults and different layers in the computational models. The following hydrothermal parameters are used for the porous rock mass in the computation. For the porefluid, dynamic viscosity 10 3 ( N s / m 2 ) , density 0 1000(kg / m 3 ) , thermal volumetric expansion coefficient 2.07 104 (1/ oC) , specific heat c p 4185( J / kgo C) , thermal
conductivity
fx fy f 0.6(W / m o C)
and
molecular
diffusivity
D fx D fy D f 0.736 10 6 (m 2 / s) . For the rock matrix, permeability of the reference rock mass (Layer C) is K r 10 14 (m 2 ) , thermal conductivity s 3.35(W / m o C ) and porosity
0.1 . In addition, different permeability ratios between layers, faults and rock
masses are considered to examine their effects on convective porefluid flow and hydrocarbon transport in the basin. The related results and their implications are reported below.
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3.1. Layered Model without Faults In order to investigate how two major faults affect the porefluid flow, heat transfer and hydrocarbon transport in the North West Shelf basin, a layered model without faults is considered by assuming that the two faults have the same material properties as the reference rock mass (Layer C). The numerical results from this model will provide a good basis, against which the numerical results from a layered model with permeable faults can be compared. Moreover, three submodels are considered for the layered model either with permeable faults or without permeable faults, to investigate the effect of basin stratification on the porefluid flow, heat transfer and hydrocarbon transport in the basin. Figures 3 and 4 show the porefluid velocity and streamline distributions, which exhibit the characteristics of porefluid flow in the basin without faults. In these figures, K A , K B ,
K D and K r are the permeabilities of Layers A, B, D and C respectively, whereas K f is the permeability of the faults. K f K r 1 means that the permeability of the fault is equal to that of the reference rock mass (Layer C). It is observed that the porefluid flow dominates within the more permeable layer (Layer B in this situation). o
Although the vertical temperature gradient considered is only 10 C / km , the convective porefluid flow occurs in the basin. There are three convective cells within the region bounded by the two faults and other six convective cells outside the region. As the top layers (Layer A and Layer D) become less permeable, the permeability of which is one fifth that of the reference rock mass (Layer C), the porefluid flow becomes more focused in Layer B,
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which is the most permeable layer in the basin. This phenomenon can be clearly seen from the results shown in both Figure 3 and Figure 4. Furthermore, it is also noted that the size of the convective cells within the region bounded by two faults is significantly different from that of the convective cells outside the region. This indicates that large dislocation of strata has a significant effect on the convective porefluid flow in the basin.
( K f / K r 1, K B / K r 5, K A / K r 1, K D / K r 1)
( K f / K r 1, K B / K r 5, K A / K r 0.2, K D / K r 1)
( K f / K r 1, K B / K r 5, K A / K r 0.2, K D / K r 0.2)
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Figure 3. Porefluid velocity distributions for the layered model without faults.
( K f / K r 1, K B / K r 5, K A / K r 1, K D / K r 1)
( K f / K r 1, K B / K r 5, K A / K r 0.2, K D / K r 1)
( K f / K r 1, K B / K r 5, K A / K r 0.2, K D / K r 0.2)
Figure 4. Streamline distributions for the layered model without faults. Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
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Figures 5 and 6 show the temperature and hydrocarbon concentration distributions, which characterize the heat transfer and hydrocarbon transport in the basin when the effect of the faults is neglected. As expected, owing to the formation of the convective porefluid flow, both the temperature and hydrocarbon concentration distributions are highly localized.
( K f / K r 1, K B / K r 5, K A / K r 1, K D / K r 1)
( K f / K r 1, K B / K r 5, K A / K r 0.2, K D / K r 1)
( K f / K r 1, K B / K r 5, K A / K r 0.2, K D / K r 0.2)
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Figure 5. Temperature distributions for the layered model without faults.
( K f / K r 1, K B / K r 5, K A / K r 1, K D / K r 1)
( K f / K r 1, K B / K r 5, K A / K r 0.2, K D / K r 1)
( K f / K r 1, K B / K r 5, K A / K r 0.2, K D / K r 0.2)
Figure 6. Concentration distributions for the layered model without faults.
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Since the localization of either temperature or hydrocarbon concentration plays an important role in the formation of economic minerals resources, the occurrence and the pattern of convective porefluid flow should be carefully investigated for hydrothermal basins. Clearly, the hydrocarbon concentration is mainly distributed in the region bounded by the two faults. This is the direct consequence of the formation of three convective cells within the region. Besides, the hydrocarbon concentration becomes more focused when the top layers (Layer A and Layer D) become less permeable.
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3.2. Layered Model with More Permeable Faults In this case, the faults are considered more permeable than the reference rock mass (Layer C). The permeability of the faults is 10 times that of the reference rock mass. To enable the numerical results to be directly comparable with those obtained from the layered model without permeable faults, other parameters are kept to be exactly the same as those used in the previous computations. Figures 7 and 8 show the distributions of the porefluid velocity and streamline for the layered model with more permeable faults. The results shown in these two figures are comparable with those shown in Figures 3 and 4. It has been recognized that the general pattern of the convective porefluid flow is very similar for the layered models with or without permeable faults. This indicates that the general pattern of the convective porefluid flow in the basin is controlled by the dislocated strata, rather than by the vertical permeable faults. Nevertheless, since the faults provide a good connection channel for the porefluid flow between the dislocated layers, the maximum velocity of the convective porefluid flow corresponding to the layered model with more permeable faults is increased by more than 50%, compared with that corresponding to the layered model without permeable faults in the previous computations. For example, in the case of K f K r 1 and KB Kr 5 (see Figure 3), the maximum velocity of the convective porefluid flow is about 1.92 109 m / s , whereas it is about 3.27 109 m / s in the case of K f K r 10 and K B K r 5 (see Figure 7). This implies that although the permeable faults have little influence on the general pattern of the convective porefluid flow, they do affect the formation of economic minerals resources because the rate of mineralization is strongly dependent on the velocity of the convective porefluid flow in hydrothermal systems. Similarly, Figures 9 and 10 show the temperature and hydrocarbon concentration distributions in the basin with more permeable faults. Although the temperature distribution for the layered model with more permeable faults (Figure 9) is very similar to that for the layered model without permeable faults (Figure 5), the hydrocarbon concentration distribution (Figure 10) is somewhat different from the previous computations (Figure 6). This indicates that the permeable faults have some considerable effects on the hydrocarbon transport in the North West Shelf basin.
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( K f / K r 10, K B / K r 5, K A / K r 1, K D / K r 1)
( K f / K r 10, K B / K r 5, K A / K r 0.2, K D / K r 1)
( K f / K r 10, K B / K r 5, K A / K r 0.2, K D / K r 0.2)
Figure 7. Porefluid velocity distributions for the layered model with high permeable faults.
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( K f / K r 10, K B / K r 5, K A / K r 1, K D / K r 1)
( K f / K r 10, K B / K r 5, K A / K r 0.2, K D / K r 1)
( K f / K r 10, K B / K r 5, K A / K r 0.2, K D / K r 0.2)
Figure 8. Streamline distributions for the layered model with high permeable faults.
3.3. Layered Model with Less Permeable Faults In this situation, the faults are considered less permeable than the reference rock mass (Layer C), so that they act as fluid barriers, rather than fluid channels that are considered in the previous subsection. The permeability of the faults is 0.1 times that of the reference rock mass. To enable the numerical results to be directly comparable with those obtained from the
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layered model without more permeable faults, other parameters are kept to be exactly the same as those used in the previous computations.
( K f / K r 10, K B / K r 5, K A / K r 1, K D / K r 1)
( K f / K r 10, K B / K r 5, K A / K r 0.2, K D / K r 1)
( K f / K r 10, K B / K r 5, K A / K r 0.2, K D / K r 0.2)
Figure 9. Temperature distributions for the layered model with high permeable faults.
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( K f / K r 10, K B / K r 5, K A / K r 1, K D / K r 1)
( K f / K r 10, K B / K r 5, K A / K r 0.2, K D / K r 1)
( K f / K r 10, K B / K r 5, K A / K r 0.2, K D / K r 0.2)
Figure 10. Concentration distributions for the layered model with high permeable faults.
Figures 11 and 12 show the distributions of the porefluid velocity and streamline for the layered model with less permeable faults. The results shown in these two figures are similar to those shown in Figures 7 and 8 outside the graben, but somewhat different from those shown in Figures 7 and 8 within the graben. This indicates that the general pattern of the convective porefluid flow outside the graben is controlled by the layered strata, rather than by the vertical permeable faults. Nevertheless, since the faults provide fluid barriers for the
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porefluid flow between the dislocated layers, the maximum velocity of the convective porefluid flow corresponding to the layered model with less permeable faults is significantly decreased, compared with that corresponding to the layered model with more permeable faults in the previous computations. For instance, in the case of K f K r 10 and K B K r 5 (see Figure 7), the maximum velocity of the convective porefluid flow is about 3.27 10 9 m / s , whereas it is about 1.5 109 m / s in the case of K f K r 0.1 and K B K r 5 (see Figure 11).
( K f / K r 0.1, K B / K r 5, K A / K r 1, K D / K r 1)
( K f / K r 0.1, K B / K r 5, K A / K r 0.2, K D / K r 1)
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( K f / K r 0.1, K B / K r 5, K A / K r 0.2, K D / K r 0.2)
Figure 11. Porefluid velocity distributions for the layered model with less permeable faults.
( K f / K r 0.1, K B / K r 5, K A / K r 1, K D / K r 1)
( K f / K r 0.1, K B / K r 5, K A / K r 0.2, K D / K r 1)
( K f / K r 0.1, K B / K r 5, K A / K r 0.2, K D / K r 0.2)
Figure 12. Streamline distributions for the layered model with less permeable faults. Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
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( K f / K r 0.1, K B / K r 5, K A / K r 1, K D / K r 1)
( K f / K r 0.1, K B / K r 5, K A / K r 0.2, K D / K r 1)
( K f / K r 0.1, K B / K r 5, K A / K r 0.2, K D / K r 0.2)
Figure 13. Temperature distributions for the layered model with less permeable faults.
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( K f / K r 0.1, K B / K r 5, K A / K r 1, K D / K r 1)
( K f / K r 0.1, K B / K r 5, K A / K r 0.2, K D / K r 1)
( K f / K r 0.1, K B / K r 5, K A / K r 0.2, K D / K r 0.2)
Figure 14. Concentration distributions for the layered model with less permeable faults.
Again, the related results indicate that although the less permeable faults have limited influence on the general pattern of the convective porefluid flow, they do affect the formation of economic minerals resources because the rate of mineralization is strongly dependent on the velocity of the convective porefluid flow in the hydrothermal systems. For this reason, caution should be taken in the case of determining the permeability of a fault for the quantitative analysis of an ore deposit in the upper crust of the Earth. Similarly, Figures 13 and 14 show the temperature and hydrocarbon concentration distributions in the basin with less permeable faults. Although the temperature distribution for
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the layered model with less permeable faults (Figure 13) is very similar to that for the layered model with more permeable faults (Figure 9), the hydrocarbon concentration distribution (Figure 14) is somewhat different from the previous computations (Figure 10). This indicates that the faults have some considerable effects on mass (i.e. hydrocarbon) and energy (i.e. heat) transport in the North West Shelf basin.
4. MATHEMATICAL MODEL OF CHEMICALDISSOLUTION FRONT INTABILITY PROBLEMS IN TWODIMENSIONAL FLUIDSATURATED POROUS MEDIA The mathematical model of a chemicaldissolution front instability problem in a fluidsaturated porous medium can be established by considering both the related chemical reaction kinetics and the mass conservation of the porefluid and dissolved minerals (i.e. solutes hereafter) in the fluidsaturated porous medium. For a fluidsaturated porous medium, Darcy‘s law is often used to describe porefluid flow, while Fick‘s law and the related chemical reaction kinetics (Zhao et al. 2009a) are commonly used to describe reactive mass transport phenomena. If the porefluid (i.e. water in this investigation) and solid matrix are incompressible, the governing equations of the coupled nonlinear problem between porosity evolution, porefluid flow and reactive singlesolute transport in the fluidsaturated (rigid) porous medium including solute dispersion effects can be expressed as follows:
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[ ( )p ] 0 , t A (C ) [D( ) C C ( )p] s kchemical p ( f )(C Ceq ) 0 , t Vp
Ap k chemical ( f )(C Ceq ) 0 , t Vp
( )
k ( )
,
(9)
(10)
(11)
(12)
where p and C are the porefluid pressure and the molar concentration of the solute; C eq is the equilibrium concentration of the solute;
is the dynamic viscosity of the porefluid;
is the porosity of the porous medium; f is the final (i.e. maximum) porosity of the porous medium after the completion of dissolvable mineral dissolution; D( ) is the general dispersion tensor of the solute; k ( ) is the permeability of the porous medium; V p is the average volume of the dissolvable grain; Ap is the average surface area of the dissolvable
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grain; k chemical is the rate constant of the chemical reaction; and
s is the molar density (i.e.
moles per volume) of the dissolvable grains. As the general dispersion tensor is of the secondorder for a twodimensional problem, it is usually defined by considering solute diffusion, longitudinal and transversal dispersion in a fluidsaturated porous medium.
ui u j D( ) Dij D( ) T u A ij ( L T ) uA where D ( ) is the diffusivity of the solute;
T
and
L
,
(13)
are the transversal and longitudinal
dispersivities of the solute; u A is the absolute magnitude of the mean average linear velocity;
u i is the average linear velocity of the porefluid in the i direction; and ij (i=1, 2 and j=1, 2 for a twodimensional problem) is the Kronecker delta. Based on Darcy‘s law, the average linear velocity vector of the porefluid can be expressed as follows:
u1 u k ( ) ulinear p , u 2
(14)
is the dynamic viscosity of the porefluid; is the porosity of the porous medium; and k ( ) is the permeability of the Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.
where u is the Darcy velocity vector of the porefluid;
porous medium. If the porous medium is comprised of spherical particles, then the shape factor which is defined as the ratio of the average volume to the average surface area (i.e. V p Ap ) of the soluble grain can be expressed as (Zhao et al. 2008c):
Vp Ap where
(3
( f ) ( f ) 1 , )3 sphere 3 36 Dp Dp
(15)
sphere 3 1 (36 ) is the shape coefficient of the spherical grains; D p is the density
of the soluble grains, which is defined as the number of the soluble gains per unit medium volume. Since the solute diffusivity is considered as a function of porosity in the chemicaldissolution front instability problem, a common phenomenological relation can be used for describing this function (Chadam et al. 1986).
D ( ) D0 q
(
3 5 q ), 2 2
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(16)
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where D 0 is the chemical species diffusivity in pure water. To consider the permeability change caused by a change in porosity, the CarmanKozeny law can be used to express the relationship between permeability and porosity (Nield and Bejan 1992) as follows:
k ( )
where
k 0 (1 0 ) 2 3 , 03 (1 ) 2
(17)
0 and k 0 are the initial reference porosity and permeability of the porous medium,
respectively. Substituting Equation (15) into Equations (10) and (11) yields the following equations: 2 3 D p (C ) [D( ) C C ( )p] s kchemical ( f ) 3 (C Ceq ) 0 , t sphere
2 3 D p k chemical ( f ) 3 (C Ceq ) 0 . t sphere
(18)
(19)
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If the x and y directions are coincident with i=1 and i=2 directions, respectively, then the general dispersion tensor expressed in Equation (13) can be simplified for some special cases (Zhao et al. 2009a, 2010c) as follows:
u D( ) D( )I D dispersion( ) D( )I L x 0
0 , T ux
(20)
where I is a unit secondorder tensor (i.e. a 2 by 2 unit matrix); u x is the average linear velocity of the porefluid in the x direction; D dispersion( ) is the secondorder tensor due to pure dispersion only. Considering Equations (12), (18) and (20) simultaneously yields the following expression:
L ( ) p x D dispersion( ) 0
. T ( ) p x 0
This enables Equation (18) to be rewritten in the following form:
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(21)
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(C ) D( )I D dispersion( ) C C ( )p t 2 3 D p s k chemical ( f ) 3 (C C eq ) 0.
(22)
sphere
To translate the dimensional governing equations of the problem (i.e. Equations (9), (19) and (22)) into dimensionless ones, the following dimensionless quantities are usually used in the theoretical analysis:
x , L*
x
L
L *
L
y
, T
y C , C , * L C eq
T *
L
,
p
p , p*
C t , eq 1 , * t s
(23)
where is a slow dimensionless time to describe the slowness of the chemical dissolution that takes place in the system; is the mineral dissolution ratio of the chemical dissolution system. Other characteristic parameters used in Equation (23) can be expressed as follows:
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t*
sphere
,
k chemicalC eq 3 D p
D * ( )
L* f D( f )t * ,
D( ) , f D( f )
* ( )
p*
f D( f ) , ( f )
( ) . ( f )
(24)
Inserting Equations (23) and (24) into Equations (9), (19) and (20) yields the following dimensionless equations:
[ * ( )p ] 0 ,
(25)
(C ) D * ( )I D *dispersion( ) C C * ( )p 0,
(26)
( f ) 3 (C 1) 0 ,
(27)
2
where Ddispersion( ) is the dimensionless secondorder tensor due to pure dispersion. *
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. p * T ( ) x 0
175
(28)
5. AN APPLICATION EXAMPLE OF CHEMICALDISSOLUTION FRONT INTABILITY PROBLEMS IN TWODIMENSIONAL FLUIDSATURATED POROUS MEDIA
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As an application example, the numerical procedure, which was presented in the previous study (Zhao et al. 2008d), is used to simulate the morphological evolution of a chemical dissolution front in a supercritical system. Figure 15 shows the geometry and boundary conditions of the chemicaldissolution front instability problem in a fluidsaturated porous medium. For this problem, the dimensionlesspressure gradient (i.e. p fx 10 ) is applied to the left boundary, implying that there is a horizontal throughflow from the left to the right of the computational model. The dimensionless height and width of the computational model are 5 and 10 respectively. Except for the left boundary, the initial porosity of the porous medium is 0.1, while the initial dimensionlessconcentration is one within the computational domain. The final porosity after the depletion of the soluble mineral is 0.2. This final porosity is applied to the left boundary as a boundary condition of the computational domain. The permeability of the porous medium is computed using the CarmanKozeny formula, which has the power of 3 in the power law. The diffusivity of chemical species is computed using the power law, which has the power of 2. Both the top and the bottom boundaries are assumed to be impermeable for the porefluid and chemical species. The ratio of the equilibrium concentration to the solid molar density of the chemical species is assumed to be 0.001, while the dimensionless timestep is set to be 0.001 in the computation. Using the abovementioned parameters, the critical Zhao number of the system is approximately equal to 1.77. Since the Zhao number of the system is equal to 10, which is greater than the critical value, the coupled system considered in this subsection is supercritical so that a planar dissolution front evolves into a complicated morphology during its propagation within the system. In order to simulate the instability of the chemical dissolution front, a small perturbation of 1% initial porosity is randomly added to the initial porosity field in the computational domain. To appropriately simulate the propagation of the dissolution front, the whole computational domain is simulated by 19701 fournode rectangular elements of 20000 nodal points in total. Figure 16 shows the porosity distributions due to the morphological evolution of the chemical dissolution front in the fluidsaturated porous medium. It is observed that after the dimensionless time is greater than 0.03, the initial planar dissolution front is gradually changed into an irregular one. With the increase of the dimensionless time, the amplitude of the resulting irregular dissolution front increases significantly, indicating that the chemical dissolution front is morphologically unstable during its propagation within the computational model.
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Figure 15. Geometry and boundary conditions of the chemicaldissolution front instability problem.
(Numerical, 0.03 )
(Numerical, 0.04 )
(Numerical, 0.05 )
(Numerical, 0.06 )
(Numerical, 0.07 )
(Numerical, 0.08 )
Figure 16. Porosity distributions due to morphological evolution of the chemical dissolution front in the fluidsaturated porous medium.
It is interesting to investigate how the porefluid flow evolves with time during the propagation of the unstable dissolution front in the computational model. Figure 17 shows the streamline evolution during the morphological evolution of the unstable chemical dissolution front within the coupled system. Due to the growth of the amplitude of the irregular dissolution front, the porefluid flow focusing takes place in the peak region of the porosity, which can be identified by the streamline density (in Figure 17).
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( 0.06 )
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( 0.07 )
( 0.08 ) Figure 17. Streamline distributions due to morphological evolution of the chemical dissolution front in the fluidsaturated porous medium..
Generally, the greater the streamline density, the stronger the porefluid flow focusing. It is obvious that the width of the flow focusing zone is clearly dependent on the peak and trough values of the irregular dissolution front in the computational model. Since the numerical procedure can be used to simulate both the porosity generation and the porefluid flow focusing, it provides a useful tool for the better understanding of the related physical and chemical mechanisms associated with chemicaldissolution front instability within fluidsaturated porous rocks.
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CONCLUSIONS Thermodynamic instability and chemicaldissolution front instability are important mechanisms to cause localizations of mass and energy (i.e. heat) distributions in a fluidsaturated porous medium. In terms of a thermodynamic instability problem, mass and energy are localized as a result of convective porefluid flow in the fluidsaturated porous medium, while in terms of a chemicaldissolutionfront instability problem, mass and energy are localized as a result of flow focusing in the fluidsaturated porous medium. These two mechanisms are very important to the understanding of distributions of mineral and geothermal resources within the upper crust of the Earth. Through such an understanding, it is possible to develop innovative exploration methods and techniques for locating new ore deposits in the deep Earth. From the mathematical point of view, a thermodynamic instability problem can be treated as a coupled problem between porefluid flow and heat transfer in the fluidsaturated porous medium. For this problem, both the deformability of the solid matrix and the compressibility of the porefluid are usually neglected. However, a chemicaldissolution front instability problem can be treated as a coupled problem between porosity, porefluid flow, mass transport and chemical reactions. For this problem, heat transfer is commonly neglected, but the effect of porosity variations on the permeability of the porous medium is considered in detail. To assess the occurrence likelihood of these two kinds of instabilities, the Rayleigh number is used to represent the geometrical and thermodynamic characteristics of a system, while the Zhao number is used to represent the geometrical, hydrodynamic and chemical kinetics of a system. If the Rayleigh number of a system is equal to or greater than its corresponding critical value, then convective porefluid flow can take place in the system. Similarly, if the Zhao number of a system is equal to or greater than its corresponding critical value, then chemicaldissolution fronts can be involved from simple shapes into complicated shapes in the system. Computational simulation methods are used to solve both thermodynamic instability and chemicaldissolution front instability problems in twodimensional fluidsaturated porous media. The related simulation results have demonstrated that: (1) faults and layered strata have significant effects on the mass and energy (i.e. heat) transport in the fluidsaturated porous medium. (2) The distribution of mass and energy transport can be remarkably localized by either fluid convection or fluid focusing in the fluidsaturated porous medium.
ACKNOWLEDGMENTS This work is financially supported by the Natural Science Foundation of China (Grant No: 10872219). The author is also very grateful to the Central South University for financial support during writing this chapter of the book.
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REFERENCES Alavyoon, F. (1993). On natural convection in vertical porous enclosures due to prescribed fluxes of heat and mass at the vertical boundaries, International Journal of Heat and Mass Transfer, 36, 24792498. Bau, H. H. & Torrance, K. E. (1982). Low Rayleigh number thermal convection in a vertical cylinder filled with porous materials and heated from below, ASME Journal of Heat Transfer, 104, 166172. Bjorlykke, K., Mo, A. & Palm, E. (1988). Modelling of thermal convection in sedimentary basins and its relevance to diagenetic reactions, Marine and Petroleum Geology, 5, 338351. Caltagirone, J. P. & Bories, S. (1985). Solutions and stability criteria of natural convective flow in an inclined porous layer, Journal of Fluid Mechanics, 155, 267287. Chadam, J., Hoff, D., Merino, E., Ortoleva, P. & Sen, A. (1986). Reactive infiltration instabilities, IMA Journal of Applied Mathematics, 36, 207221. Chadam, J., Ortoleva, P. & Sen, A. (1988). A weekly nonlinear stability analysis of the reactive infiltration interface, IMA Journal of Applied Mathematics, 48, 13621378. Chen, F. & Chen, C. F. (1989). Experimental investigation of convective stability in a superposed fluid and porous layer when heated from below, Journal of Fluid Mechanics, 207, 311321. Chen, J. S. & Liu, C. W. (2002). Numerical simulation of the evolution of aquifer porosity and species concentrations during reactive transport, Computers and Geosciences, 28, 485499. Chevalier, S., Bernard, D. & Joly, N. (1999). Natural convection in a porous layer bounded by impervious domains: from numerical approaches to experimental realization, International Journal of Heat and Mass Transfer, 42, 581597. Gasser, R. D. & Kazimi, M. S. (1976). Onset of convection in a porous medium with internal heat generation, ASME Journal of Heat Transfer, 98, 4954. Gow, P., Upton, P., Zhao, C. & Hill, K. (2002). CopperGold mineralization in the New Guinea: Numerical modeling of collision, fluid flow and intrusionrelated hydrothermal systems, Australian Journal of Earth Sciences, 49, 753771. Horne, R. N. & Caltagirone, J. P. (1980). On the evaluation of thermal disturbances during natural convection in a porous medium, Journal of Fluid Mechanics, 100, 385395. Horton, C. W. & Rogers, F. T. (1945). Convection currents in a porous medium, Journal of Applied Physics, 16, 367370. Jones, M. C. & Persichetti, J. M. (1986). Convective instability in packed beds with throughflow, AIChE Journal, 32, 15551557. Kaviany, M. (1984). Thermal convective instabilities in a porous medium, ASME Journal of Heat Transfer, 106, 137142. Lapwood, E. R. (1948). Convection of a fluid in a porous medium, Proc. Camb. Phil. Soc., 44, 508521. Lebon, G. & Cloot, A. (1986). A thermodynamical modeling of fluid flows through porous media: Application to natural convection, International Journal of Heat and Mass Transfer, 29, 381390. Lin, G., Zhao, C., Hobbs, B. E., Ord, A. & Mühlhaus, H. B. (2003). Theoretical and
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numerical analyses of convective instability in porous media with temperaturedependent viscosity, Communications in Numerical Methods in Engineering, 19, 787799. Murphy, H. D. (1979). Convective instabilities in vertical fractures and faults, Journal of Geoghysical Research, 84, 61216130. Nield, D. A. & Bejan, A. (1992). Convection in Porous Media, SpringerVerlag, New York. Ord, A., Hobbs, B. E., Zhang, Y., Broadbent, G. C., Brown, M., Willetts, G., SorjonenWard, P., Walshe, J. & Zhao, C. (2002). Geodynamic modelling of the Century deposit, Mt Isa Province, Queensland, Australian Journal of Earth Sciences, 49, 10111039. Ortoleva, P., Chadam, J., Merino, E. & Sen, A. (1987). Geochemical selforganization II: The reactiveinfiltration instability, American Journal of Science, 287, 10081040. Phillips, O. M. (1991). Flow and Reactions in Permeable Rocks, Cambridge University Press, Cambridge. Pillatsis, G., Taslim, M. E. & Narusawa, U. (1987). Thermal instability of a fluidsaturated porous medium bounded by thin fluid layers, ASME Journal of Heat Transfer, 109, 677682. Renard, F., Gratier, J. P., Ortoleva, P., Brosse, E. & Bazin, B. (1998). Selforganization during reactive fluid flow in a porous medium, Geophysical Research Letters, 25, 385388. Schaubs, P. & Zhao, C. (2002). Numerical modelling of golddeposit formation in the BendigoBallarat zone, Victoria, Australian Journal of Earth Sciences, 49, 10771096. SorjonenWard, P., Zhang, Y. & Zhao, C. (2002). Numerical modelling of orogenic processes and mineralization in the south eastern part of the Yilgarn Craton, Western Australia, Australian Journal of Earth Sciences, 49, 935964. Tournier, C., Genthon, P. & Rabinowicz, M. (2000). The onset of natural convection in vertical fault planes: Consequences for the thermal regime in crystalline basements and for heat recovery experiments, Geophysical Journal International, 140, 500508. Zhao C., Mühlhaus, H. B. & Hobbs, B. E. (1997). Finite element analysis of steadystate natural convection problems in fluidsaturated porous media heated from below, International Journal for Numerical and Analytical Methods in Geomechanics, 21, 863881. Zhao C., Hobbs, B. E. & Mühlhaus, H. B. (1998a). Finite element modelling of temperature gradient driven rock alteration and mineralization in porous rock masses, Computer Methods in Applied Mechanics and Engineering, 165, 175187. Zhao C., Mühlhaus, H. B. & Hobbs, B. E. (1998b). Effects of geological inhomogeneity on high Rayleigh number steadystate heat and mass transfer in fluidsaturated porous media heated from below, Internal Journal of Computation and Methodology: Numerical Heat Transfer, 33, 415431. Zhao C., Hobbs, B. E. & Mühlhaus, H. B. (1999a). Theoretical and numerical analyses of convective instability in porous media with upward throughflow, International Journal for Numerical and Analytical Methods in Geomechanics, 23, 629646. Zhao, C., Hobbs, B. E. & Mühlhaus, H. B. (1999b). Effects of medium thermoelasticity on high Rayleigh number steadystate heat transfer and mineralization in deformable fluidsaturated porous media heated from below, Computer Methods in Applied Mechanics and Engineering, 173, 4154. Zhao C., Hobbs, B. E., Baxter, K., Mühlhaus, H. B. & Ord, A. (1999c). A numerical study of
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porefluid, thermal and mass flow in fluidsaturated porous rock basins, Engineering Computations, 16, 202214. Zhao C., Hobbs, B. E. & Mühlhaus, H. B. (2000a). Finite element analysis of heat transfer and mineralization in layered hydrothermal systems with upward throughflow, Computer Methods in Applied Mechanics and Engineering, 186, 4964. Zhao, C., Hobbs, B. E., Mühlhaus, H. B., Ord, A. & Lin, G. (2000b). Numerical modelling of double diffusion driven reactive flow transport in deformable fluidsaturated porous media with particular consideration of temperaturedependent chemical reaction rates, Engineering Computations, 17, 367385. Zhao C., Hobbs, B. E., Mühlhaus, H. B., Ord, A. & Lin, G. (2001a). Finite element modelling of threedimensional convection problems in porefluid saturated porous media heated from below, Communications in Numerical Methods in engineering, 17, 101114. Zhao C., Lin, G., Hobbs, B. E., Mühlhaus, H. B., Ord, A. & Wang, Y. (2001b). Finite element modelling of heat transfer through permeable cracks in hydrothermal systems with upward throughflow, Engineering Computations, 18, 9961011. Zhao C., Hobbs, B. E., Mühlhaus, H. B., Ord, A. & Lin, G. (2002a). Analysis of steadystate heat transfer through midcrustal vertical cracks with upward throughflow in hydrothermal systems, International Journal for Numerical and Analytical Methods in Geomechanics, 26, 14771491. Zhao, C., Lin, G., Hobbs, B. E., Wang, Y., Mühlhaus, H. B. & Ord, A. (2002b). Finite element modelling of reactive fluids mixing and mineralization in porefluid saturated hydrothermal/sedimentary basins, Engineering Computations, 19, 364387. Zhao, C., Hobbs, B. E., Mühlhaus, H. B., Ord, A. & Lin, G. (2003a). Convective instability of threedimensional fluidsaturated geological fault zones heated from below, Geoghysical Journal International, 155, 213220. Zhao, C., Hobbs, B. E., Ord, A., Mühlhaus, H. B. & Lin, G. (2003b). Effect of material anisotropy on the onset of convective flow in threedimensional fluidsaturated faults, Mathematical Geology, 35, 141154. Zhao, C., Hobbs, B. E., Ord, A., Peng, S., Mühlhaus, H. B. & Liu, L. (2004). Theoretical investigation of convective instability in inclined and fluidsaturated threedimensional fault zones, Tectonophysics, 387, 4764. Zhao, C., Hobbs, B. E., Ord, A., Lin, G. & Mühlhaus, H. B. (2005a). Theoretical and numerical analysis of largescale heat transfer problems with temperaturedependent porefluid densities, Engineering Computations, 22, 232252. Zhao, C., Hobbs, B. E., Ord, A., Peng, S., Mühlhaus, H. B. & Liu, L. (2005b). Double diffusiondriven convective instability of threedimensional fluidsaturated geological fault zones heated from below, Mathematical Geology, 37, 373391. Zhao, C., Hobbs, B. E., Ord, A., Hornby, P., Peng, S. & Liu, L. (2006a). Theoretical and numerical analyses of porefluid flow patterns around and within inclined large cracks and faults, Geophysical Journal International, 166, 970988. Zhao, C., Hobbs, B. E., Ord, A., Peng, S., Liu, L. & Mühlhaus, H. B. (2006b). Analytical solutions for porefluid flow focusing within inclined elliptical inclusions in porefluidsaturated porous rocks: Solutions derived in an elliptical coordinate system, Mathematical Geology, 38, 9871010. Zhao, C., Hobbs, B. E., Ord, A., Kühn, M., Mühlhaus, H. B. & Peng, S. (2006c). Numerical simulation of doublediffusion driven convective flow and rock alteration in three
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dimensional fluidsaturated geological fault zones, Computer Methods in Applied Mechanics and Engineering, 195, 28162840. Zhao, C., Hobbs, B. E. & Ord, A. (2008a). Investigating dynamic mechanisms of geological phenomena using methodology of computational geosciences: an example of equaldistant mineralization in a fault, Science in China Series D: Earth Sciences, 51, 947954. Zhao, C., Hobbs, B. E. & Ord, A. (2008b). Convective and Advective Heat Transfer in Geological Systems, Springer, Berlin. Zhao, C., Hobbs, B. E., Ord, A., Hornby, P. & Peng, S. (2008c). Effect of reactive surface areas associated with different particle shapes on chemicaldissolution front instability in fluidsaturated porous rocks, Transport in Porous Media, 73, 7594. Zhao, C., Hobbs, B. E., Hornby, P., Ord, A., Peng, S. & Liu, L. (2008d). Theoretical and numerical analyses of chemicaldissolution front instability in fluidsaturated porous rocks, International Journal for Numerical and Analytical Methods in Geomechanics, 32, 11071130. Zhao, C., Hobbs, B. E., Ord, A., Hornby, P. & Peng, S. (2008e). Morphological evolution of threedimensional chemical dissolution front in fluidsaturated porous media: A numerical simulation approach, Geofluids, 8, 113127. Zhao, C., Hobbs, B. E. & Ord, A. (2009a). Fundamentals of Computational Geoscience: Numerical Methods and Algorithms, Springer, Berlin. Zhao, C., Hobbs, B. E. & Ord, A. (2009b). Theoretical analyses of nonaqueousphaseliquid dissolution induced instability in twodimensional fluidsaturated porous media, International Journal for Numerical and Analytical Methods in Geomechanics, doi: 10.1002/nag.880. Zhao, C., Hobbs, B. E., Ord, A. & Peng, S. (2010a). Effects of mineral dissolution ratios on chemicaldissolution front instability in fluidsaturated porous rocks, Transport in Porous Media, 82, 317335. Zhao, C., Hobbs, B. E. & Ord, A. (2010b). Theoretical and numerical investigation into rules of geofluid flow in ore forming systems: Integrated mass conservation and generic model approach, Journal of Geochemical Exploration, 106, 251260. Zhao, C., Hobbs, B. E. & Ord, A. (2010c). Theoretical analyses of the effects of solute dispersion on chemicaldissolution front instability in fluidsaturated porous rocks, Transport in Porous Media, 84, 629653.
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In: Fluid Transport: Theory, Dynamics and Applications ISBN: 9781611223170 Editor: Emma T. Berg, pp. 183217 © 2011 Nova Science Publishers, Inc.
Chapter 6
THE DYNAMICS OF NMR – DIFFUSION DIFFERENTIAL EQUATION FOR QUALITATIVE ANALYSIS OF HEMODYNAMIC AND METABOLIC CHANGES IN BIOLOGICAL TISSUE O. B. Awojoyogbe and M. Dada* Department of Physics, Federal University of Technology, Minna, NigerState, Nigeria
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ABSTRACT During the past decade, major breakthroughs in magnetic resonance imaging (MRI) quality were made by means of great improvement in scanner hardware and pulse sequences. Some advanced MRI techniques have truly revolutionized the detection of disease states and MRI can nowwithin a few minutesacquire important quantitative information noninvasively from an individual in any plane or volume at comparatively high resolution. However, the very basic physics of this promising technological breakthrough is not well understood. Parameters that are measured from time to time in advanced MRI seem to be logically and functionally related but the theoretical facility to optimally explore them is still missing. In a single experimental investigation, for example, few of huge amount of information available are effectively used. This study intends to provide a very straightforward theoretical background for measuring diffusion of water protons and specific chemicals encountered in most common advanced MRI methods including diffusion MRI, perfusion MRI, functional MRI.
Keywords: Bloch NMR flow equations, Diffusion, Brownian motion, Perfusion, fMRI, Biological flow.
*
Corresponding author: Email: [email protected]
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O. B. Awojoyogbe and M. Dada
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INTRODUCTION Magnetic resonance imaging (MRI) has been established for over a decade as a superior research and clinical modality for anatomical imaging. Noteworthy for exceptionally good submillimeter spatial and subsecond temporal resolution, MRI is now demonstrating the potential of tracing the links between tissue function, metabolism, blood flow, and hemodynamics in both normal and disease states. Furthermore, Magnetic resonance imaging could – within minutes – noninvasively acquire functional images in any plane or volume at comparatively high resolution. Functional MRI (fMRI) can image the hemodynamic and metabolic changes that are associated with human brain functions, such as vision, motor skills, language, memory, and mental processes. These techniques have also revolutionized detection of a wide variety of disease states, such as stroke, multiple sclerosis, and tumors. Much of the recent progress in MRI resulted because of highspeed imaging capabilities. This typically means that enhanced gradient strengths are used to produce images in one or a few shots or echotrains. Echoplanar and spiral imaging are two variations of such highspeed gradientecho MRI techniques, whereas spinecho variations of highspeed imaging fall into the class of fast spinecho MRI. Highspeed MRI suggests that images are acquired in several seconds or fewer and provide the means of producing motionfree images or a series of rapidly acquired images. In this study, the theoretical foundations of these advanced MRI techniques would be reviewed and we would attempt to address their potential strengths. Furthermore, we will provide an overview of the potential applications of advanced MRI in neurologic diagnosis, tissue engineering and therapeutic monitoring. Diffusion sets the basis for one of the most robust contrast mechanisms in MRI [1]. Diffusionweighted MRI (DWI) and more advanced methods, such as diffusion tensor imaging (DTI), [2, 3] have advanced from an experimental tool to a frequently used method for clinical evaluation and for a better understanding of the pathophysiology of certain diseases. Specifically, DWI now belongs to the standard arsenal of MRI pulses sequences, and the impact it has made for the early diagnosis of acute stroke [4, 5] seems to be unmatched by other MRI methods. The possibility of diagnosing stroke within the window of opportunity to treat patients while brain tissue is still salvageable and to triage patients with different etiology of a stroke is of great importance for treatment decisions in critically ill patients [6]. Although diffusion tensor imaging has not yet been able to match the impact of DWI in stroke, it has proven to be a sensitive tool for detecting subtle abnormalities in the white matter of patients with diseases like multiple sclerosis [7, 8] or Amyotrophic Lateral Sclerosis (ALS). Despite DTI's sensitivity to white matter abnormalities shown in group comparisons, further work in this area is warranted to demonstrate its value for diagnosing individual patients, which is a major aim of this study [9]. MRI allows one to measure the selfdiffusion of water—that is, the Brownian or random motion of the water molecules driven by their internal thermal energy [1]. By means of DWI, it is possible to take snapshots of the motion of water or any other chemical substance on a time scale of a few tens of milliseconds. Within pure water or CSF, diffusion is usually unrestricted and the measured diffusion constant is independent of any of the spatial directions. The diffusion coefficient that is obtained from a tissue voxel via MRI reflects the
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The Dynamics of NMR – Diffusion Differential Equation …
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contribution of different compartments. If diffusion is isotropic, there is no preferred direction of water motion for these tissues. However, for white matter, consisting of dense fiber bundles, water moves more easily parallel to the fibers than across them. Depending on the fiber packing density the displacement front of the water protons will be more or less cigar shaped (that is, a stretched ellipsoid) in contrast to a spherical shape for isotropic diffusion. Here, the orientation of its long axis will be dependent on the local fiber orientation. The diffusion ellipsoid [that is, a threedimensional ellipse] is parameterized with six elements [1]. The first three parameters are the radii along the principal axes of the ellipsoid. If they are all equal, this would represent a sphere and, hence, isotropic diffusion. The other three parameters are the rotation angles of the ellipsoid defining its orientation in space, relative to the laboratory frame of reference. Hence, in isotropic diffusion in which the fluid contained in the compartment under investigation is homogeneous, the diffusion coefficient is said to be constant (independent of the space coordinates). For this situation, the Bloch equation which properly describes the motion of the magnetic molecule is given as follows:
Mathematical Formulation For this investigation, we derive analytical expressions in spherical polar coordinate for the NMR transverse magnetization [913] which can be detected by the recovery unit in the MRI scanner based on the Bloch NMR flow equations with the assumption that resonance condition exists at Larmor frequency f o B 0
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The x, y, z components (in the rotating frame) of the magnetization of a particle may be given by the Bloch NMR flow equations which may be written as follows [12]:
V
2
2M y x 2
2 M y 2 M y 1 1 M y 1 1 M y 2 2 1 M B ( x, t ) M y o 1 2V 2 V B1 (t ) xt t T1 T2 x T1 T2 t T1T2 T1
(1a)
where is the gyromagnetic ratio of the material Mo is the equilibrium magnetization T1 and T2 are the spinlattice and spinspin relaxation parameters respectively The solution presented here is subject to the following two reasonable initial boundary conditions which may conform to the realtime experimental arrangements; i) Mo ≠ Mz, a situation which hold good in general and in particular when the rF B1(t) field is strong. ii) Before entering signal detector coil, the soft particle has Mx= 0, My = 0 and. 2B12 = 2π
Z
2π/α
g(r, z, t)dt.
(3.35)
0
Consequently, the net axial velocity < vz > reads < vz >= ε2 V20(r), under neglect of O(ε3 )terms, while the net flow rate < Q > is given by Z 1 V20(r)rdr, < Q >= 2πε2
(3.36)
(3.37)
δ
under neglect of O(ε3 ) terms. Thus, the traveling wave induces a net flow of the liquid, of which the (dimensionless) rate is expressed by (3.37). Hence, the net flow is an effect of order ε2 . Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
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4.
Kh. S. Mekheimer and A. N. AbdelWahab
Numerical Results and Discussion
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It is clear that we have to choose (ε < 1) because we have used the perturbation method with the amplitude ratio (ε) as a parameter. We consider the net flow rate < Q > given by (3.37).After one integration by parts, < Q > can be expressed as 2 Z 1 δ −1 2 1− V20(1) − Re (1 − iαtm ) r2 ξ(r)dr < Q > = πε 2 ln δ δ 2 Z 1 δ −1 +Re (1 − iαtm ) ξ(r)dr (4.1) 2 ln δ δ where the solution (3.33) for V20(r) is used. For the flow in an annulus we compare our results to that for Aarts and Ooms [12] which present the flow rate of comprisable fluid in tube and discuss the results numerically and graphically. It is known that the viscoelastic fluids, described by the Maxwell model, have different flow regimes depending on the value of the parameter De = tν /tm , which is called the Deborah number [37]. In fact, the Deborah number is a ratio of the characteristic time of viscous effects tν = ρR2/µ to the relaxation time tm , the value of the parameter De (which the authors of Ref.[37] actually call α) determines in which regime the system resides. Beyond a certain critical value (Dec = 11.64), the system is dissipative, and conventional viscous effects dominate. On the other hand, for small De(De < Dec ) the system exhibits viscoelastic behavior. Note that tm is dimensionless and scaled by R/c. After the above discussion, it is relevant to define quantitatively the transition point where the flow starts to exhibit (nonNewtonian) viscoelastic effects. Now, using the definition of Re = ρcR/µ, we can define the critical value of tm as Re R (4.2) tm.c = Dec c In all our figures we have used Re = 10000. If we put the latter value of Re and the critical value of the Deborah number 11.64 into Equ. 4.2, we obtain tm.c = 859.11 (measured in units of R/c). Therefore, the values of tm > tm.c (for a given Re) correspond to subcritical De < Dec Deborah numbers for which viscoelastic effects are pronounced. Now we start to explain this dissection numerically and graphically the main point is the effects of radius ratio δ on the flow rate. Firstly, Fig.2 investigate the dependence of the dimensionless flow rate < Q > on the compressible parameter χ for various values of relaxation time tm , we compar the flow pattern for δ = 0.0001 and δ = 0.2 at ε = 0.001, Re = 10000 and α = 0.001. For the two curves tm = 0 (Newtonian limiting), tm = 100 there is no noticeable change in the plot, where both curves coincides within a plotting accuracy. While, for tm = 1000 ( δ = 0.0001) there is a slight deviation from the Newtonian limiting (tm = 0) but for δ = 0.2 this deviation in the flow rate goes to a negative values, which means a back flow. The dependence of this flow rate deviation on δ, indicates that the critical value of the relaxation time tm (tm.c. ) is decreases as the radius ratio δ increases. So, tm.c. for an annulus is less than that for a tube, and large values of tm (tm = 10000) for an annulus will occurs more back flow (reflux), i.e, no back flow for a tube.
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3 5
10
3
(a)G=0.0001 tm =0 =100 =1000 =10000
(b)G=0.2 tm =0 =100 =1000 =10000
5
10
2
1
2
0
1
1 2
3
0
4 0.0
0.2
0.4
F
0.6
0.8
1.0
0.0
0.2
0.4
F
0.6
0.8
1.0
Figure 2. Dimensionless flow rate ¡Q¿ as a function of compressibility paremeter ? for e =0.001, Re =10000 and a =0.001 Fig.3 represents the variation of the flow rate < Q > with wave number α for different values of tm = 0, 100, 1000 at t = 0.001, Re = 10000, δ = 0.00001 and χ = 0.6. We observe that the flow rate < Q > attains a maximum value for a certain value of α, and this maximum decreases with increasing tm . The phenomena brought by introducing the nonNewtonian effect is investigated, where we compar the solid curve in Fig. 3 with the dashed curve in Fig.3 in ref[[12]] (Newtonian). We will note that no difference occur and the flow rate somewhat change attaining lower values as α increases. 1.6 4
tm =0 =100 =1000
10
1.2 1.0
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1.4
0.8 0.6 0.4 0.2 0.0 0.000
0.002
0.004
0.006
0.008
0.010
Figure 3. Dimensionless flow rate ¡Q¿ as a function of wave number α for =0.001, Re =10000, δ =0.00001 and χ =0.6 In Fig.4 the net flow rate < Q > is plotted versus the wave number α for the following set of parameters: = 0.001, Re = 10000, χ = 0.6 and tm = 1000 (nonNewtonian regime) and various values of δ, the curves are plotted for the interval of variation of α up
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to 0.05. 5
10
G=0.001
6
=0.2 4
2
0
2
4
0.00
0.01
0.02
0.03
0.04
0.05
We observe that the flow exhibit viscoelastic behaviour for small values of δ and as δ increases highly oscillatory pattern is appeared at small values of the wave number α. The negative flow rate means that flow occurs in the direction opposite to the direction of propagation of traveling wave on the tube wall (back flow) and as the annulus radius ratio increases, more back flow will occurred at certain values of α. Finally, we confirm our results by study the effect of δ on the flow rate < Q > for tm = 100 and tm = 1000. At tm = 100, see Fig.5. The flow rate increase by increasing δ , and the flow rate will attains a maximum value at a certain value of χ and then it will decreases as χ increases and negative values of < Q > are appeared. For a highly nonNewtonian regime tm = 10000, the maximum value of < Q > is increases, back flow appear and viscoelastic behaviour is expected at great values of δ. 5
5
10
10
4.0
120
(a)tm =100 G=0.001 =0.01 =0.1
3.5 3.0
(b)tm =10000 G=0.001 =0.01 =0.1
100
80 2.5 60
2.0
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Figure 4. Dimensionless flow rate ¡Q¿ as a function of wave number α on a larger (than in fig 3) interval of vaiation of α for =0.001, Re =10000, χ =0.6 and tm =1000.
1.5 1.0
40
20
0.5 0 0.0 20 0.5 0.0
0.2
0.4
0.6
F
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
F
Figure 5. Dimensionless flow rate ¡Q¿ as a function of compressibility paremeter χ for =0.001, Re =10000, α = 0.001
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[8] G.B. Bohme & R. Friedrich ,(1983) Peristaltic flow of viscoelastic liquids, Rheol. Acta, 128 ,109122,. [9] L. M. Srivastava & V. P. Srivastava ,(1984) Peristaltic transport of blood: Casson model II, J. Biomechanics 17 ,821829. [10] D. Tsiklawri, & I. Bresenev, (2001) NonNewtonian effects in the peristaltic flow of Maxwell fluid., Physical Review.,64 ,ID 036303. [11] L.K. Antanovskii & H. Ramkissoon , (1997) Long wave peristaltic transport of a compressible viscous fluid in a finit pipe subjct to a time dependent pressure drop, Fluid Dynamics Research, 19 , 115123. [12] A. C. T. Aarts & G. Ooms , (1998) Net flow of compressible viscous liquids induced by travelling waves in porous media, Journal of Engineering Mathematics, 34 , 435450. [13] R. S. Hankin, (2001) the Euler equations for multiphase compressible flow in conservation from simulation of shock bubble interaction, Journal Computational physics, 172 , 808826. [14] M. Kawashital, (2002) On globle solution of Cauchy problems for compressible Navierstocks equations, Journal of Nonlinear Analysis 48 , 894. [15] E. F. Elshehawy ,N.T. ElDabe & I. M. ElDesoky ,(2006) Slip effects on the Peristaltic flow of a nonNewtonian Maxweellian fluid, Acta Mechanica,186 , 141159. Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
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[16] N.T. ElDabe , A. Y. Ghaly & H. M. Sayed ,(2007) MHD Peristaltic flow of nonNewtonian fluid through a porous medium in circular cylindrical tube, Cal. Math. Soc., 169 , 123136. [17] Kh. S. Mekheimer & A. N. Abd ElWahab, (2009) Effect of wall compliance on compressible fluid transport induced by a surface acoustic wave in a micro channel, Numerical Methods of Partial Differential Equations Journal (NMPDE), to be appear . [18] R.B. Bird , R.C. Armstrong & O. Hassager ,(1987) Dynamics of Polymeric liquids, Vol.1, Fluid Mechanics,John WileySons,. [19] S. Deutsch , W.M. Phillips & J. Heist, (1976) An interpretation of low strain rate blood viscosity measurements: a continuum approach, Biorheology, 13 , 297307. [20] Y.C. Fung, (1984) Biodynamics: Circulation , SpringerVerlag, New York. [21] W.M. Phillips & S. Deutsch , (1975) Towarda constitutive equation for blood, Biorheology, 12 , 383389. [22] H. Chmiel , I. Anadere & E. Walitza , (1990) The determination of blood viscoelasticity in clinical hemorheology, Biorheology, 27 , 883894. [23] D. E. Mann & J.M. Tarbell , (1990) Flow of nonnewtonian blood analog fluids in rigid curved and straight artery models, Biorheology, 27 , 711733.
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[24] Kh. S. Mekheimer,(2005) Peristaltic transport of a Newtonian fluid through a uniform and nonuniform annulus.,the Arabian jor. for science and Engin, 30 , 6983 [25] P. Nagarani , G. Sarojamma & G. Jayaraman, (2008) Effect of boundary absorption on dispersion in Casson fluid flow in an annulus application to catheterized artery, Acta Mechanica, Published online, 4th Iune , (10.1007/s007070080013y). [26] P. Nagarani, G. Sarojamma & G. Jayaraman,(2006) Exact analysis of unsteady convective diffusion in Casson fluid flow in an annulus Application to catheterized artery, Acta Mechanica, 187 , 189202 [27] D. Srinivasacharya & D. Srikanth, (2008) Effect of couple stresses on the flow in a constricted annulus, Arch Appl Mech 78, 251257 [28] Kh. S. Mekheimer & Y. Abd elmaboud,(2008) Peristaltic flow of a couple stress fluid in an annulus: Application of an endoscope, Physica A., 387 , 24032415. [29] Kh. S. Mekheimer & Y. Abd elmaboud, (2008) The influences of a micropolar fluid on peristaltic transport in an annulus: Application of a clot model, J. Applied Bionic and Biomech., 5, No 1, , 1323. [30] Kh. S.Mekheimer & Y. Abd elmaboud, Peristaltic transport of a particlefluid suspension through a uniform and nonuniform annulus, J. Applied Bionic and Biomech, 5, No. 2 (2008), 4757 Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
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[31] Kh. S. Mekheimer & Y. Abd elmaboud, (2008) The influence of heat transfer and magnetic field on peristaltic transport of a Newtonian fluid in a vertical annulus: Application of an endoscope, Physics Letters A , 372, 16571665 [32] Kh. S. Mekheimer & M. A. ELkot, (2010) Suspension model for blood flow through arterial catheterization, Chemical Eng. Communication J.,GCEC #457984, 197, ISS 9. [33] R. Roose & S. Lykodis ,(1971) The fluid mechanics of the ureter with an inserted catheter. J. Fluid Mech., 46 , 625630. [34] D. MacDonald , (1986) Pulsatile flow in a catheterized artery.J.Biomech, 19 , 239249. [35] G. Karahalios,(1990) Some possible effects of a catheter on the arterial wall. Med. Phys., 17 , 922925. [36] R. Dash , G. Jayaraman & K.N. Mehta , (1996)Estimation of increased flow resistance in a narrow catheterized artery  a theoretical model. J. Biomech, 29 , 917930.
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[37] J. A. Rio , M. Haro Lopez & S. Whitaker, (1998)Enhancement in the dynamic response of a viscoelastic fluid flowing in a tube ,Phys. Rev. E, 58 , 63236327.
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In: Fluid Transport: Theory, Dynamics and Applications ISBN: 9781611223170 Editor: Emma T. Berg, pp. 237240 © 2011 Nova Science Publishers, Inc.
Chapter 8
MICROSCALE AND NANOSCALE THERMAL AND FLUID TRANSPORT PHENOMENA: RAPIDLY DEVELOPING RESEARCH FIELDS† Lixin Cheng*
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School of Engineering, University of Aberdeen, King‘s College, Aberdeen, Scotland, United Kingdom With the emergence of microscale and nanoscale thermal, fluidic and chemical systems, MEMS (MicroElectoMechanicalSystems), NEMS (NanoElectroMechanicalSystems), thermal management technologies for microelectronics, power electronics, aerospace and defence technologies etc., the development of ultracompact heat exchangers, miniature and micro pumps, miniature compressors, microturbines, micro thermal systems for distributed power production, microfluidic and nanofludic device, labonachip, microscale and nanoscale energy systems, nanofluid heat transfer technologies and others has become an important agenda of many researchers, research institutions and funding agencies. Applications of microscale and nanoscale thermal and fluid transport phenomena involved in traditional industries and highly specialized fields such as bioengineering, microfabricated fluidic systems, microelectronics, aerospace technology, micro heat pipes, chips cooling etc. have been becoming especially important since the late 20th century [111]. However, microscale and nanoscale thermal and fluid transport phenomena are quite different from those at conventional scale or macroscale [1214]. For example, gas liquid twophase flow and flow boiling heat transfer characteristics in microcale channels are quite different from those in macroscale channels [13, 68, 12, 13]. Channel confinement also has a great effect on twophase flow and flow boiling heat transfer characteristics [16, 17]. Furthermore, the available experimental results of single phase heat transfer and fluid flow in microscale channels are quite contradictory from one study to another [4, 5]. Studies of supercritical fluid †
A version of this chapter also appears in International Journal of Microscale and Nanoscale Thermal and Fluid Transport Phenomena, edited by Lixin Cheng, published by Nova Science Publishers, Inc. It was submitted for appropriate modifications in an effort to encourage wider dissemination of research. * Founder and EditorinChief of the IJMNTFTP, email: [email protected]. Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
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flow and heat transfer in microscale channels have also exhibited contradictory results by various researchers [14]. Just to show several examples here. Therefore, there are many aspects to be clarified from both theoretical and applied aspects in microscale and nanoscale thermal and fluid transport phenomena. Furthermore, new research areas and technologies in the relevant fields have been emerging from time to time. For example, as a new research frontier of nanotechnology, the research of nanofluid twophase flow and thermal physics is growing rapidly, however it has also posed new challenges as there are quite contradictory results in the available research [10]. For another example, advances in microelectronics technology continue to develop with surprisingly rapidity and the thermal energy density of electronic devices to be dissipated is becoming much higher and higher. Therefore, it is essential to develop new high heat flux cooling technology to meet the challenging heat dissipation requirements [15]. All the newly emerging research areas require the understanding of fundamentals and applications of the microscale and nanoscale fluid and thermal transport phenomena. Over the past years, a great amount of research work in the fields of microscale and nanoscale thermal and fluid transport phenomena has intensively been conducted. In response to the numerous evolving research topics, technologies and applications based on microscale and nanoscale thermal and fluid transport phenomena, it is highly the time to form a new journal to provide a highquality forum specially for a wide range of papers dealing with original research results, technical notes and stateoftheart reviews pertaining to thermal and fluid transport phenomena at microscale and nanoscale. Thus, this newly founded journal aims to meet such urgent needs and to bring worldwide these important frontier research works together. It covers a wide range of topics on fundamentals and applications of microscale and nanoscale transfer processes of mass, momentum and energy such as microscale and nanoscale heat transfer and fluid flow, nanofluid heat transfer and flow, microfluidcs, nanofludics and technologies based on these transport processes such as various microscale and nanoscale thermal and fluid devices, micro and nano energy systems, microcooling technology in the computer and electronics industries and information technologies etc., MEMS, NEMS and the interdisciplinary research related to microscale and nanoscale thermal and fluid transport phenomena in bioengineering, medical engineering and life engineering etc. Furthermore, proposals for special issues on selected topics reflecting recent microscale and nanoscale thermal and fluid flow advances or on selected articles of conferences are also encouraged and may be submitted directly to the EditorinChief. Reviews on important issues are invited by the EditorinChief from tme to time and may also be submitted to the EditorinChief in a normal way. One new feature of this journal is to present one leading research laboratory or group introduction in relevant fields in each issue. For example, a leading laboratory with interest in microscale twophase flow and heat transfer research, led by Prof. John R. Thome, the Heat and Mass Transfer Laboratory (LTCM) at the Swiss Federal Institute of Technology Lausanne (EPFL), Switzerland is presented in this issue. If anyone is interested in presenting his/her lab or group in the journal, please contact the EditorinChief directly for this. As the founder and EditorinChief, I would like to express my sincere thanks to all the editorial board members who have been giving their great support to this new journal. I welcome any suggestions and comments on improving this journal from our community. It is my greatest wish to make this journal a leading one in the microscale and nanoscale thermal
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and fluid transport phenomenon research fields in the near future. I am looking forward to receiving more support from our community, which I believe are invaluable.
REFERENCES [1] [2] [3] [4] [5] [6] [7] [8]
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[9]
[10]
[11] [12]
[13]
[14] [15]
[16]
Cheng, L. & Mewes, D. (2006). Review of twophase flow and flow boiling of mixtures in small and mini channels, Int. J. Multiphase Flow, 32, 183207. Kanlikar, S. G. (2002). Fundamental issues related to flow boiling in minichannels and microchannels, Exp. Therm. Fluid Sci., 26, 389407. Thome, J. R. (2006). The new frontier in heat transfer: microscale and nanoscale technologies, Heat Transfer Eng., 27(9), 13. Morini, G. L. (2004). Singlephase convective heat transfer in microchannels: a review of experimental results, Int. J. Therm. Sci., 43, 631651. Hetsroni, G., Mosyak, A., Pogrebnyak, E., Yarin, L. P. (2005). Fluid flow in microchannels, Int. J. Heat Mass Transfer, 48, 19821998. Thome, J. R. (2004). Boiling in microchannels: a review of experiment and theory, Int. J. Heat Fluid flow, 25, 128139. Thome, J. R. (2006). Stateofthe art overview of boiling and twophase flows in microschannels, Heat Transfer Eng., 27(9), 419. Cheng, L., Ribatski, G., Thome, J. R. (2008). Gasliquid twophase flow patterns and flow pattern maps: fundamentals and applications, ASME Appl. Mech. Rev., 61, 050802105080228. Cheng, L., Mewes, D., Luke, A. (2007). Boiling phenomena with surfactants and polymeric additives: a stateoftheart review, Int. J. Heat Mass Transfer, 50, 27442771. Cheng, L., Bandarra Filho, E. P. & Thome, J. R. (2008). Nanofluid twophase flow and thermal physics: a new research frontier of nanotechnology and its challenges, J. Nanosci. Nanotech., 8, 33153332. Cheng, L. (2009). Nanofluid heat transfer technologies, Recent Patents on Engineering, 3(1), 17. Ribatski, G., Wojtan, L. & Thome, J. R. (2006). An analysis of experimental data and prediction methods for twophase frictional pressure drop and flow boiling heat transfer in microscale channels, exp. Therm. Fluid Sci., 31, 119. Thome, J. R. & Ribatski, G. (2005). Stateoftheart of twophase flow and flow boiling heat transfer and pressure drop of CO2 in macro and micro channels, Int. J. Refrigeration, 28, 11491168. Cheng, L., Ribatski, G., Thome, J. R. (2008). Analysis of supercritical CO2 cooling in macro and micro channels, In. J. Refrigeration., 31, 13011316. Cheng, L. & Thome, J. R. (2009). Cooling of microprocessors using flow boiling of CO2 in a microevaporator: preliminary analysis and performance comparison, Appl. Therm. Eng., 29, 24262432. Moreno Quibén, J., Cheng, L., da Silva Lima, R. J. & Thome, J. R. (2009). Flow boiling in horizontal flattened tubes: part I twophase frictional pressure drop results and model, Int. J. Heat Mass Transfer, 52, 36343644.
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[17] Moreno Quibén, J., Cheng, L., da Silva Lima, R. J. & Thome, J. R. (2009). Flow boiling in horizontal flattened tubes: part II flow boiling heat transfer results and model, Int. J. Heat Mass Transfer, 52, 36453653.
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In: Fluid Transport: Theory, Dynamics and Applications ISBN: 9781611223170 Editor: Emma T. Berg, pp. 241258 © 2011 Nova Science Publishers, Inc.
Chapter 9
TRANSPORT CONTROL OF FLUID AND SOLUTES IN MICROCHANNELS USING AC FIELD * AND SEMICONDUCTOR DIODES Dimiter N. Petsev1 and Orlin D. Velev2 1
Center for Biomedical Engineering, Department for Chemical and Nuclear Engineering, University of New Mexico, Albuquerque, NM 871310001, USA 2 Department of Chemical and Biomolecular Engineering, North Carolina State University, Raleigh, NC 276957905, USA
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ABSTRACT This chapter presents an overview of recent results on the electric field control and manipulation of fluids in microfluidic devices. The newer approaches are based on using alternating or a combination of alternating and direct current fields. The alternating field can be locally converted to direct by semiconductor diodes that may be placed at key locations where an electroosmotic force has to be applied to the fluid. Such techniques allow to design and fabricate small micrometer sized pumps and mixers. The latter are important because of the inherent low Reynolds characteristics of the flow in microchannels. The diode mixers are simple to fabricate and can be turned on and off depending on the operational requirements. Combining alternate and direct current fields and diode pumps makes possible the decoupling of the electroosmotic fluid flow from the electrophoretic particle or macromolecular mass flux. This can be exploited for precise analyte focusing, preconcentration and separation.
*
A version of this chapter also appears in Continuum Mechanics, edited by Andrus Koppel and Jaak Oja, published by Nova Science Publishers, Inc. It was submitted for appropriate modifications in an effort to encourage wider dissemination of research. Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
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INTRODUCTION
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Micro and nanofluidics have great potential as core technologies in a variety of devices and applications. Examples include chemical[114] and biomolecular[1544] sensing, separation,[4,7,15,17,23,24,26,4570] DNA molecule manipulation,[54,7178] sample preconcentration and focusing,[21,49,7986] conducting small scale liquid reactions and manipulations,[8789] and formation and handling of monodisperse droplets and particles.[90108] The fabrication of microfluidic structures and patterns for these applications is based on the advancement of micro and nanofabrication techniques such as soft lithography,[109114] bulk[115120] and surface[1,74,121127] machining, and chemical and thermal oxidation.[128] Other methods employed in the making of microfluidic devices include electron beam technology,[129] interferometric lithography[130132] and patterning by selfassembly.[133135] As the channel dimensions become smaller, the Reynolds numbers of the flows becomes lower and viscous effects dominate.[136139] This fact leads to difficulties in the rapid processing of fluids and solutes. A typical example is mixing, which for micro and nanofluidics requires completely new engineering approaches.[47,140162] Manipulating the fluid pressure and flow and analytes distribution in fluidic systems also requires the development of pumps and valves to direct and transport the liquid and dissolved species.[113,114,163204]
Figure 1. Schematics of the geometry and electric field actuation modes of the three types of microfluidic pumps discussed here. (a) Electroosmotic DC pumping – the ions in the double layers move towards the electrode of opposite charge dragging the liquid. (b) Induced charge electroosmosis by AC field – the induced charges and the liquid near the surfaces move in the direction of the field Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
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gradient. The asymmetric configuration of the energized electrodes leads to unbalanced liquid flow. (c) Pumping by diodes embedded in the channel – the energy is provided by an external AC field, while the liquid moves because of DC electroosmosis localized between the diode electrodes.
The most facile way to manipulate the liquid flow and the distribution of charged molecular species is to use electric fields. The first widely used type of actuation relies on the use of direct current (DC) fields in effects such as electrophoresis and electroosmosis (Figure 1a). More recently, various phenomena based on alternating current (AC) fields have been investigated as means for pumping and manipulating particles in microfluidic devices. Here we focus on new techniques that combine both DC and AC fields in more sophisticated ways with the goal of achieving new functionality and levels of performance. Recently we developed a new method for fluid transport control and manipulation, which is based on AC field actuation in combination with semiconductor diodes placed at specific locations along microfluidic channels.[142,205] The diodes locally rectify the AC field into DC field which drives the fluid near the diodes by electroosmosis. In this Chapter we present an overview of the fluid transport driven by AC powered semiconductor diodes in the context of the earlier developments in DC and AC electrokinetics. In the next section we discuss the governing equations describing the fluid transport in microchannels. Section 3 presents some examples that illustrate prospective useful applications of the proposed approach. The potential for the development of these techniques is discussed briefly in the concluding remarks.
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GOVERNING EQUATIONS FOR FLUID AND ANALYTE FLOW IN MICROCHANNELS We consider here the fundamentals relations between DC and/or AC external fields and fluid flow. The fluid flux in the microchannels is given by the momentum balance equation at low Reynolds[206] flow regime[207,208]
2 v p e E p 02 E
(1)
where v is the velocity field, is the fluid viscosity, p is the pressure, e is the local charge density (due to the dissolved ionic species), E is the externally applied DC electric field, 0 = 8.8541012 F m1 is the dielectric constant for vacuum, is the relative dielectric permittivity for the solvent (~ 80 for water at room temperature) and is the potential of the electric double layer that is formed at wallsolution interface. For most cases E and can be assumed to be uncoupled.[207,209] The double layer potential is related to the local charge density via the PoissonBoltzmann equation[207,210214]
2
e e 0 0
z n
0 i i
i
z e exp i kT
(2)
where e = 1.6021019 C is the elementary charge, zi and ni0 are the charge number and bulk number concentration of ionic species i. The thermal energy is included in these equations by Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
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the kT product where k = 1.3811023 J K1 is the Boltzmann constant and T is the temperature. The electric double layer potential decays with the distance from the wall and its typical range is usually expressed by the socalled double layer thickness or Debye screening length 1/ which is defined by the equation
2
e2 0 kT
2
zi2 ni0 . i 1
(3)
The quantity has the dimension of inverse length. Therefore its product with the typical characteristic lengthscale is a dimensionless number that gives a qualitative measure of the relative importance of the double layer region. The double layer thickness 1 depends on the background electrolyte concentration. For example, aqueous solution of monovalent symmetric electrolyte at room temperature with concentrations ranging from 106 to 103 M results in 1 lengths varying between 300 and 10 nm respectively. These dimensions are often much smaller than the width of microchannels and wider capillaries and it is reasonable to assume that the electric double layer thickness is effectively zero (extremely thin double layer approximation).[207,208,213] Then the term that is proportional to the electric field can be omitted from Eq. (1) and the obtained result is simply
2 v p .
(4)
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However, the external field still applies a force on the liquid through the infinitely thin double layer at the channel wall. This effect can be accounted for by defining a boundary condition at the wall which reads
v wall
0
E.
(5)
The quantity is historically known as the electrokinetic zeta potential. It is defined at the "shear" surface near the wall where the fluid starts slipping.[207,208,213] In fluidic devices with narrow channels (in the nanometers range) the electrical double layer thickness might be comparable to the channel width (or radius). The infinitely thin double layer assumption may not be applicable for such small channels and Eqs. (4) and (5) do not adequately describe the fluid flow. Eq. (1) has to be used for such systems where the opposing ionic electric double layers overlap. The body force term can be calculated by using Eq. (2) and the problem becomes substantially more complex even for conventional DC field electroosmosis. This is overcome by introducing certain approximations like low potential,[215] assuming simple channel geometry (plane parallel slit) or weak double layer overlap.[216220] Recently the potential distribution and electroosmotic flow in a cylindrical capillary with arbitrarily high potential were also calculated using the methods of matched asymptotic expansions.[221] When the electric double layer is of the order of the channel width only numerical analysis is possible.[64]
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The problem becomes even more difficult to solve analytically if nanoparticles or macromolecules with sizes that are comparable to that of the channel are also transported. The overall electrokinetic problem presents considerable mathematical difficulties. However, there is an analytical analysis of this case offered by Brenner.[222224] It also employs the matched asymptotic expansion and the results are valid for relatively thin electric double layers around the particles and at the channel walls. Uniform AC fields applied normally or tangentially to a charged wall do not engender unidirectional fluid flows as do the DC fields. Fluid flows, however, are generated in areas where a strong gradient of a nonuniform electric field exists across a solidliquid interface. These flows, which have been referred to as AC electrohydrodynamics[225,226] or Induced Charge Electroosmosis (ICEO),[176,227] originate as a result of the interaction of the field with the fieldinduced component of the electric double layer near interfaces. The external applied voltage at these interfaces modifies the native charge on the surface thereby leading to an "induced" zeta potential added to the intrinsic zeta potential. The induced component is strongly dependent on the field frequency and electrolyte concentration. If the AC frequency is low enough, the induced double layer charge changes sign synchronously with the electric field. The counterions in the double layer move in and out of the layer during the halfcycles of the field. The induced component of the zeta potential is of the same sign as that of the applied field.[228] When the field has a component tangential to the surface a net flow towards the higher intensity area is induced. The ions in the double layer then react to tangential electric fields leading to bulk unidirectional liquid flow along the interface. Even though an AC field is applied, the bulk flow in different half cycles is always in the same direction along the field gradient. The AC electrohydrodynamics flow velocity is given by
vAC
0 ind
Et
(6)
where, ind is the induced zeta potential due to the applied external field and Et is the tangential component of the electric field.[229,230] The induced potential is usually proportional the field that generates it in a linear manner ind Et. Thus, the ICEO velocity is usually proportional to the squared potential, vAC Et2, a "signature" relation for AC electrohydrodynamic flows. [176,231] The AC electrokinetic flows can be used in microfluidic pumping in devices with asymmetric electrodes[178,185188,232,233] (Figure 1b) and in techniques for onchip manipulation and collection of particles.[142,234] The AC fields do not lead to electrophoretic motion of homogeneous particles or biological macromolecules. Thus, the use of alternating fields in microfluidic pumping and mixing avoids some of the problems arising from the joint action of electroosmosis and electrophoresis in microfluidic devices manipulating biomolecules. The AC field, however, can lead to Induced Charge Electrophoresis (IECP) of Janus and other anisotropic particles[235,236] and can lead to particle attraction or repulsion to electrodes by dielectrophoresis (DEP).[237] The use of AC fields in devices operating by ICEO, ICEP and DEP is problematic for a number of reasons. The design of devices operating on field gradients is complex and in many cases requires highend microfabrication. One factor that increases the complexity is the
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frequency dependence of the AC electrohydrodynamic flows occurring because the polarization effects in the electric double layer are timedependent.[209] The upper frequency limit of ICEO pumping is 510 kHz. This prevents the use of fields of higher frequencies, which can penetrate more materials and lead to low power dissipation. One additional problem with ICEO is that the presence of even millimolar electrolyte concentrations in combination with the low efficiency of most AC electrode pump configurations leads to liquid heating and potential chemical decomposition of the solution. Thus, a method combining some of the advantages of the DC and AC liquid handling with a simple straightforward design and fabrication can be of high value in many microfluidic devices. We present in the next section a new method based on semiconductor diode actuation bridging the DC and AC field electrokinetics techniques.
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SEMICONDUCTOR DIODE PUMPS AND MIXERS POWERED BY ALTERNATE CURRENT ELECTRIC FIELD The major advantage of the AC fields is the ability to transmit a relatively large power flux through the chip without engendering electrokinetic flows in the channels where the field is uniform. One way to use this power in a simple way is to rectify the AC field into DC voltage and use it in local pumping without applying DC field across the whole chip. We reported how such a process can be realized by using semiconductor diodes.[142,205] Consider a microchannel where semiconductor diodes are embedded in the walls while their electrodes are in contact with the fluid (Figure 1c). If alternate current (AC) electric field is applied it is locally converted into direct current (DC) field. Hence, half of the field is available to apply electroosmotic unidirectional force on the fluid at the diode surface.[142,205] This force is periodic in time with frequency that corresponds to that of the AC field. However, for observation time scales that are longer than the inverse AC frequency we can consider the force and the velocity to be uniform. In addition, the rectified voltage is likely to be further smoothened and averaged in time by the capacitance of the double layers, especially at high frequencies. The most important effect from practical point of view is that only half of the total AC field is utilized to directionally drive the fluid into the channel. The other half is shortened by the diode and does not drive the fluid backwards. The velocity of the liquid flows generated by the diodes can be estimated by relations similar to the ones derived for DC liquid actuation. The DC voltage, Vd, harvested by the diode is[142]
Vd
Ld Eext 2
Ed0
(7)
where Ld is the diode length, Eext is the magnitude of the external AC field and Ed0 is the offset field that characterizes the particular pnjunction.[238] Then the electroosmotic velocity generated by the external AC field between the diode electrodes is
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v
0
2
Eext
Ed0 .
247
(8)
The factor ―2‖ in the denominator is due to fact that only half of the total field is used; other half is ―shortened‖ by the diode and does not lead to fluid flow. The flow pattern in the vicinity of a diode ―pump‖ that has been powered by an AC field in the channel is illustrated by a computer simulation in Figure 2a and an experimental visualization in Figure 2b. The pump consists of two diodes oriented in parallel at the microchannel walls. The fluid near the diode surface moves due to electroosmosis generated by the local DC field. The rest of the fluid in the channel (away from the electrodes) exerts hydrodynamic resistance, which leads to a backflow in the middle of the channel. Still, there is a net flow in one direction (right to left in the figure), which quickly obtains parabolic shape that is typical for pressure driven fluid motion. The diode pumps and a combination of AC and DC fields allows for focusing and separation of charged species. Using a loopshaped channel (Figure 3) allows decoupling of the fluid flow from the particle motion. The applied AC field is rectified by the collinear diodes, which drive the fluid into a clockwise circulation. The DC field does not affect the fluid flow because of the loop shape of the channel is symmetric and global DC forces counterbalance each other. It, however, gives rise to a lefttoright electrophoretic motion of the particles. In the upper part of the loop both the electrophoresis and the convective particle transport are in the same direction. In the lower part of the loop the fluid flow and the particle electrophoresis are in opposite directions and particle with lower mobility might be swept righttoleft by the fluid flow. This concept has been proven by efficiently separating two types of particles of slightly different charge.[205] If the diodes have an antiparallel orientation they will generate a vortex instead of a directional flow and thus will act as an efficient microfluidic mixer. This is illustrated in Figure 4. The first one presents a computation and the second is again an experimental flow visualization of the pattern that develops at this configuration. We investigated the mixing efficiency of a diode device operating on a two concurrent laminar fluxes passing through a small chamber with oppositely facing diodes.[205] The mixing in the chamber occurs because of the vortex that is created between the oppositely aligned diodes. The efficiency of the diode mixer was tested in Yshaped microchannel configuration.[205] We showed that two completely separated streams (pure water and dye solution) become almost uniformly distributed 2.5 mm after the diode mixer. The fluorescence intensity averaged across the channel width obtained from the confocal micrographs confirmed that the vortex flow induced by the diodes dramatically enhanced the mixing of the two adjacent laminar streams. The degree of mixing was evaluated with a mixing index calculated as the standard deviation of the fluorescence intensity in the experimental confocal images[151]
Mixing Index
1 N
Ik k
I0
2
I0
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where N is the total number of pixels of the scanned image, Ik is the intensity of pixel ―k‖ and I0 is the average intensity over all pixels. Our experiments[205] showed that the resulting mixing index was 80% for Texas Red solution (see Figure 5). The same index would require channel length ymix 68 cm if the only driving force for mixing is molecular
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diffusion.[205] The diode mixer is very efficient for laminar flows with low Peclet numbers (Pe 2.2103). In contrast most passive microfluidic mixers work well only for high Peclet numbers (Pe > 104).[143,145,156] Besides the high mixing efficiency that this device could achieve it could be turned on and off remotely via the AC field and multiple mixers situated in different channels can be actuated simultaneously.
Figure 2. Schematics and flow visualization on a diode micropump. (a) A Computational Fluid Dynamics simulation of the flow field that develops in the vicinity of the AC powered diode. (b) An experimental visualization of the flow between the diodes (situated on the top and bottom but not visible in this micrograph, see Ref. [49,92,114].
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Figure 3. A sketch of the a device for particle focusing and preconcentration (see Ref. 142). This device allows decoupling of the fluid flow and electrophoretic particle motion as discussed the text.
Figure 4. AC Diode mixer. (a) and (b) A Computational Fluid Dynamics simulation of the flow fields in a diode mixer. (c) Experimental flow visualization (see Ref. 142). Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest
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CONCLUSIONS
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Mixing Index
We discuss here the origins of the DC and AC electroosmotic effects that can be used for pumping and mixing in microfluidic devices. The major effect routinely used in the present devices in DC electroosmosis, often combined with DC electrophoresis for processes of biomolecular separation. The use of AC fields in analogical applications is presently investigated intensively, but while the ICEO and ICEP effects are rather interesting from fundamental perspective, their application in engineered devices is not straightforward. We describe here a new technique combining the DC and AC effects in microfluidic devices based on semiconductor diode actuators. That technique benefits from the advantages of both
Axial distance from mixer, ymix (mm)
Figure 5. Mixing index at different AC fields calculated as a standard deviation of fluorescence intensity in crosssectional confocal imaging as a function of the distance from the Yshaped channel junction (see Ref. 205). The frequency of the AC field is 1 kHz
other classes of electric field effects, as the power is transported through the chip by AC fields, while the actuation is carried out by DC electroosmosis between the electrodes of the diodes, which generates direct current after rectifying the AC field propagating through the chip. In the absence of the nonlinear electric element of the diodes the AC field can not lead to directional fluid motion, so it can be applied over the whole microfluidic network and be used to provide power to multiple diode pumps and mixers. This allows convenient pumping and mixing by locating diodes in all positions of the channels where such functions are needed. One of the potentially interesting future developments in this area might be the creation of a new generation of "microfluidicelectronic" chips. The microfluidic channels in such devices might directly interface the surfaces of silicon chips that would not only host the diodes and other nonlinear elements used to drive and mix the liquid, but might also include sensors and be connected to underlying electronic circuits that might provide feedback, control and logic functions. The multiditude of complex engineered structures that might provide new microfluidic functionality has only begun to be investigated.
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INDEX # 20th century, vii, x, 237
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A access, 150 actuation, 242, 243, 246, 250 actuators, 250 additives, 239 advancement, 242 aerospace, vii, x, 237 aerospace technology, vii, x, 237 agencies, x, 237 algorithm, 23, 51, 128 alternating field, xi, 241, 245 amplitude, x, 76, 175, 176, 219, 222, 223, 230 amyotrophic lateral sclerosis (ALS), 184, 215 analyte focusing, xi, 241 analytical solution, viii, 75, 76, 80, 89, 94, 119 anisotropy, 181 annuals, 222 aqueous minerals, ix, 157, 158 aquifers, 119 artery, x, 219, 221, 234, 235 assessment, 126 Australia, viii, 76, 104, 105, 109
B barriers, 167, 168 base, 59, 60, 66, 67, 72, 80, 106 behaviors, 47 Beijing, 117 benefits, 250 biochemical, vii, 1 bioengineering, vii, x, 46, 237
biomolecules, 245 blood, x, 184, 186, 219, 220, 221, 233, 234, 235 blood flow, x, 184, 186, 219, 221, 235 bluff porous bodies, vii, 45 board members, 238 body shape, 66 Boltzmann constant, 244 brain, 184, 216 Brownian motion, 183 buoyancy, viii, 75, 77, 80, 94, 95, 96, 97, 98, 99, 102, 103, 104, 106, 109, 110, 119
C Cairo, 219 calculus, 7 capillary, 244 case studies, 77, 96, 105, 109 case study, 98, 105, 106, 109 catalyst, 41 catheter, x, 219, 221, 235 cation, 219 cauchy problem, 220, 233 cell surface, 51 challenges, ix, x, 219, 238, 239 channel confinement, x, 237 charge density, 243 chemical, vii, ix, x, 1, 2, 46, 76, 121, 157, 158, 159, 161, 171, 172, 173, 174, 175, 176, 177, 178, 181, 182, 184, 186, 237, 242, 246 chemical kinetics, 159, 178 chemical reactions, 178 chemicaldissolution front instability, ix, 157, 158, 159, 171, 172, 175, 176, 177, 178, 182 chemicals, vii, ix, 183 Chicago, 41 China, viii, 76, 103, 110, 112, 116, 117, 118, 157, 178, 182
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260
Index
chips cooling, vii, x, 237 choking criterion, vii, 1, 29 chyme, 220 circular cylinder, viii, 45, 46, 47, 49, 50, 56, 57, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 221 circulation, 41, 95, 106, 118, 119, 120, 247 city, 219 classes, 250 classification, ix, 19 closure, 120 coarse spherical particles, vii, 1, 9, 19 collisions, 29, 38, 41, 76 commercial, 46, 80, 128, 148 community, 238 compaction, viii, 75, 77 competition, 59 complex hydrothermal flow systems, viii, 75 complexity, 245 compliance, 234 composition, 161 compressibility, x, 79, 178, 219, 220, 221, 223, 231, 232 compression, 110, 112, 222 computation, 120, 162, 163, 175, 247 computational fluid dynamics, vii, 45, 121, 128 computational performance, 49 computer, ix, 130, 159, 219, 238, 247 conceptual model, 96, 104 conduction, 76, 77, 81, 88 conductivity, 78, 79, 83, 114, 126, 158, 163 conductor, 243 configuration, 131, 137, 144, 152, 243, 247 confinement, x, 202, 207, 237 conservation, 53, 77, 79, 81, 82, 90, 94, 113, 123, 128, 129, 171, 182 constituents, 162 contaminant, 80 continental, 118 contour, 89, 90 convection model, 103 convective porefluid flow, ix, 157, 158, 159, 161, 162, 163, 165, 166, 168, 170, 178 convergence, 94 convergence criteria, 94 cooling, vii, x, 237, 238, 239 copyright, 41 correlation, 6, 10, 11, 14, 19, 23, 24, 38, 122, 123, 126, 127, 140, 141, 142, 143, 145 correlations, 2, 10, 11, 12, 17, 19, 23, 30, 38, 41, 122, 123, 142, 144, 146 counterbalance, 247 cracks, 181
critical state, 159 critical value, 20, 95, 96, 175, 178, 230 crust, ix, 95, 118, 119, 157, 158, 170, 178 crystalline, 180 CSF, 184 cycles, 245
D Dachang polymetallic deposits, viii, 76, 103 data structure, 51 Dead Sea, 95, 119 decay, 135, 138 decomposition, 246 decoupling, xi, 241, 247, 249 defence, x, 237 deformability, 178 deformation, viii, 75, 77, 94, 103, 110, 112, 113, 114, 115, 116, 118, 119, 120 Denmark, 42 dense phase flow, vii, 1, 6, 18, 19 dependent variable, 6, 16, 53, 130 deposition, 103, 104, 110 deposits, viii, 76, 103, 104, 105, 106, 109, 110, 112, 116, 117, 118, 119, 120, 178 depth, 77, 96, 97, 105, 106, 109, 117, 131, 132, 133, 134, 135, 136, 137, 138, 139, 142, 150, 151, 162 derivatives, 52, 53, 54, 93, 227 designers, 153 detachment, 59 detection, ix, 183, 184 deviation, 9, 19, 21, 22, 26, 28, 34, 38, 39, 57, 230, 247, 250 diagenesis, viii, 75 dielectric constant, 243 dielectric permittivity, 243 differential equations, 3, 12, 129, 160, 190 diffusion, vii, ix, 82, 88, 103, 122, 140, 141, 159, 172, 181, 183, 184, 185, 186, 191, 192, 195, 199, 202, 207, 208, 209, 211, 212, 214, 215, 216, 234, 248 diffusion MRI, ix, 183 diffusion process, 192, 207, 211 diffusion time, 186 diffusionweighted imaging (DWI), 184, 215 diffusivities, 161 diffusivity, 94, 163, 172, 173, 175 dilation, 113, 116 dilute flow, vii, 1, 6, 9, 18, 19, 20, 22, 23, 24, 38 dimensionless groups, ix, 121, 122 diode mixers, xi, 241 diode pumps, xi, 241, 247, 250 diodes, xi, 241, 243, 246, 247, 248, 250
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Index direct current fields, xi, 241 direct measure, 17, 28 discharges, 106 discretization, 51, 54, 104, 130 diseases, 184 dislocation, 164 dispersion, 117, 126, 159, 171, 172, 173, 174, 182, 234 displacement, 150, 185, 214 distribution, vii, 1, 7, 20, 41, 82, 87, 88, 94, 96, 97, 98, 99, 100, 105, 114, 115, 116, 126, 127, 128, 144, 162, 166, 170, 178, 242, 243, 244 DNA, 242 dosage, 22, 31 draft, vii, 1, 2, 31 drainage, 94, 119 driving mechanisms, viii, 75, 76, 77, 159 dynamic viscosity, 48, 78, 113, 161, 163, 171, 172
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E edema, 186 elastic deformation, 112 electric field, xi, 241, 242, 243, 244, 245, 246, 250 electrical operation, viii, 121 electrode surface, 126, 140, 141, 142, 143, 144 electrodes, viii, 121, 122, 123, 140, 142, 143, 144, 147, 243, 245, 246, 247, 250 electrolyte, 121, 122, 123, 126, 127, 128, 129, 130, 140, 141, 142, 144, 148, 153, 244, 245, 246 electromagnetic, 77 electromagnetic waves, 77 electron, 141, 242 electronic circuits, 250 electroosmotic force, xi, 241 electrophoresis, 243, 245, 247, 250 electrophoretic particle, xi, 241, 249 endoscope, 221, 234, 235 energy, ix, x, 3, 76, 77, 79, 81, 82, 90, 94, 109, 118, 125, 128, 129, 130, 157, 158, 159, 171, 178, 207, 237, 238, 243 energy density, x, 238 engineering, 46, 159, 181, 184, 238, 242 entrapment, 161 environment, 117, 124, 161, 214 equilibrium, 78, 79, 81, 171, 175, 185 equipment, 2 etiology, 184 everyday life, vii, 45 evolution, 98, 100, 101, 119, 159, 171, 175, 176, 177, 179, 182 excitation, 186 experimental condition, 128, 129, 203
261
expertise, 130 exposure, 29, 104 extrusion, 43
F fabrication, 242, 246 fiber, 185 fiber bundles, 185 filters, 46 financial, 178 financial support, 178 finite element method, ix, 46, 91, 157, 159, 162 finite volume method, viii, 45, 46, 49 flow field, 60, 66, 128, 150, 221, 248, 249 flow manifolds, viii, 121 flow phenomena, viii, 45, 72 flow regimes, vii, 1, 2, 6, 9, 18, 20, 29, 36, 230 fluctuations, 3, 19, 21, 22, 43, 102, 125, 126 fluid dynamics, vii, ix, 45, 121, 123, 128 fluidized bed, vii, 1, 2, 7, 9, 15, 16, 19, 29, 38, 39, 40, 43 fluidparticle interphase drag coefficient, vii, 1, 4, 6, 7, 8, 9, 13, 14, 15, 16, 38, 39 fluidwall friction coefficient, vii, 1, 38 fluorescence, 247, 250 food, 220 force, viii, xi, 4, 5, 14, 57, 76, 77, 94, 95, 96, 98, 103, 106, 109, 110, 114, 149, 159, 186, 207, 208, 211, 212, 241, 244, 246, 248 formation, viii, 65, 66, 75, 103, 105, 106, 109, 110, 116, 117, 119, 120, 131, 144, 165, 166, 170, 180, 242 formula, 175, 210 foundations, 184 fractures, 80, 119, 180 France, 156 freezing, 150 freshwater, 78, 93, 119 friction, vii, 1, 2, 4, 5, 6, 10, 11, 12, 14, 15, 16, 17, 18, 22, 23, 24, 25, 28, 32, 33, 36, 37, 38, 40, 42, 112 functional MRI, ix, 183 fundamental theory, viii, 75 funding, x, 237
G Galerkin finite element technique, viii, 75 gassolid flow, vii, 1 geological processes, viii, 75 geology, viii, 75
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262
Index
geometric, ix, 47, 51, 68, 121, 124, 126, 127, 148, 149, 178, 186, 207, 211, 212, 221 geometrical parameters, 47, 51, 68 geometry, 22, 31, 71, 78, 81, 124, 125, 126, 129, 130, 148, 162, 175, 186, 187, 191, 192, 207, 222, 242, 244 global scale, 103 grain size, 114 graph, 22, 33 gravity, 99, 161 grid generation, 129 grids, viii, 45, 50, 51, 55, 56, 71 groundwater, viii, 75, 76, 77, 91, 94, 95, 97, 98, 102, 103, 118, 119 growth, 176 Guinea, 179
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H hardness, 158 heat capacity, 79 heat release, 88 heat transfer, ix, x, 40, 80, 105, 119, 157, 158, 160, 163, 165, 178, 180, 181, 221, 235, 237, 238, 239, 240 heat transport, viii, 75, 76, 77, 79, 80, 81, 83, 89, 91, 94, 103, 104, 105, 110 height, 158, 175 heterogeneity, 158, 163 history, 104 homogenous fluid, vii, 45, 47, 48, 49, 71 host, viii, 75, 76, 80, 83, 87, 88, 95, 104, 106, 117, 250 human, 184, 215, 216 human brain, 184, 215, 216 Hunter, 258 hydraulic transport, vii, 1, 2, 8, 9, 11, 15, 29, 30, 33, 34, 35, 36, 38, 40 hydrocarbon migration, viii, 75, 161 hydrocarbons, 161 hydrodynamic behaviour, ix, 121, 125, 126, 128, 138, 139, 140, 143, 153 hydrothermal flow system, viii, 75, 76, 99 hydrothermal system, 80, 90, 98, 102, 103, 107, 166, 170, 179, 181
I ideal, 126, 128, 129, 147 identification, 104 identity, 85, 88 IMA, 179
image, 148, 150, 184, 248 images, 150, 184, 215, 247 impermeable host, viii, 75, 76 in vivo, 215 induction, 118 industries, vii, x, 2, 45, 46, 237, 238 inertia, 3 inertial effects, viii, 45, 46, 48 infarction, 215 inhomogeneity, 180 insertion, 221 institutions, x, 237 integration, 88, 93, 130, 193, 200, 210, 229, 230 interface, viii, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 71, 81, 82, 84, 179, 243, 245, 250 interphase, vii, 1, 4, 6, 7, 8, 9, 13, 14, 15, 16, 38, 39 intestinal tract, 220 inversion, 216 ions, 242, 245 ischemia, 215 isotherms, 100 isotope, 117 issues, 238, 239 iteration, 130
J Japan, 41, 42
K kinematic, ix, 121, 122 kinetics, 159, 171, 178
L laminar, 47, 51, 123, 124, 125, 128, 129, 142, 147, 247, 248 Laplace transformation, viii, 75, 84, 85, 87 laws, 112, 159 lead, viii, 24, 76, 103, 104, 106, 110, 116, 118, 119, 120, 125, 209, 229, 245, 246, 247, 250 leakage, 82 Lebanon, 155 liberty, 199, 208, 212 light, 29, 150 limestone, 110, 111, 114, 116, 117 linear function, 80, 95 liquids, 43, 220, 227, 229, 233, 234 liquidsolid flow, vii, 1, 2, 31, 32, 33 lithography, 242 localization, 166
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Index lumen, 221 Luo, 112, 118, 120 lying, 51
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M macromolecular mass flux, xi, 241 macromolecules, 245 magnetic field, 207, 220, 221, 235 magnetic resonance, ix, 183, 215 magnetic resonance imaging, ix, 183, 215 magnetization, 185, 196, 207, 208, 211, 212, 214 magnitude, 17, 22, 33, 59, 99, 102, 104, 131, 132, 133, 134, 135, 172, 224, 246 majority, ix, 219 man, 220 management, x, 237 manifolds, viii, 121, 123, 124, 125, 126, 127, 128, 130, 131, 133, 140, 148, 149 manipulation, xi, 241, 242, 243, 245 Maryland, 118 mass, ix, xi, 6, 18, 19, 25, 26, 33, 34, 38, 46, 48, 77, 79, 81, 90, 103, 113, 121, 122, 123, 124, 126, 127, 128, 129, 140, 141, 142, 143, 144, 145, 146, 147, 148, 152, 157, 158, 159, 160, 161, 163, 166, 167, 171, 178, 179, 180, 181, 182, 222, 238, 241 mass transport, ix, 103, 121, 122, 123, 124, 126, 127, 128, 140, 141, 142, 143, 144, 145, 146, 147, 148, 152, 157, 158, 161, 171, 178 materials, 49, 179, 220, 246 mathematical equations, viii, 75 mathematics, 215, 221 matrix, 41, 78, 79, 81, 83, 84, 87, 88, 89, 91, 94, 161, 163, 171, 173, 178 matter, 184 measurement, 216 measurements, 11, 17, 19, 28, 71, 105, 234 mechanical processes, vii, 1 mechanical properties, 112, 118 media, viii, ix, 2, 46, 51, 71, 75, 76, 77, 78, 80, 95, 117, 119, 157, 158, 159, 178, 179, 180, 181, 182, 207 medical, 46, 238 memory, 184 MEMS (MicroElectoMechanicalSystems), 237 mental processes, 184 metabolic, v, 183 metabolic changes, 184 metabolism, 184 metabolites, 46 metals, 103, 109 metamorphism, viii, 75, 119 meter, 109, 123
263
methodology, 49, 80, 182 methylene blue, 29 Mexico, 241 micro heat pipes, vii, x, 237 microchannels, vi, 241, 243 microelectronics, vii, x, 237 microfabricated fluidic systems, vii, x, 237 microfabrication, 245 micrometer, xi, 241 microscale, vii, x, 237, 238, 239 migration, viii, 75, 103, 104, 119, 141, 161 mineral deposits, viii, 76, 103, 104, 110, 116, 120 mineralization, 110, 116, 117, 118, 119, 166, 170, 179, 180, 181, 182 miniature, x, 237 mixing, 103, 126, 135, 181, 242, 245, 247, 248, 250 modeling, vii, viii, 1, 2, 3, 17, 36, 40, 75, 80, 90, 93, 94, 95, 97, 98, 103, 104, 105, 110, 114, 116, 119, 120, 129, 130, 154, 179, 180, 181, 221 models, 2, 3, 5, 19, 80, 94, 95, 103, 104, 118, 119, 130, 161, 163, 166, 220, 234 modifications, 128, 237, 241 modified spoutfluidized bed, vii, 1 modulus, 111, 113 molecules, 76, 184, 214 momentum, vii, 1, 2, 3, 4, 5, 6, 7, 10, 12, 13, 14, 15, 16, 22, 23, 24, 31, 33, 38, 71, 123, 128, 129, 222, 223, 238, 243 Moon, 254 morphology, ix, 157, 158, 175 motor skills, 184 MRI, vii, ix, 183, 184, 185, 215 multiblock grids, viii, 45, 71 multiple sclerosis, 184, 215 multiples, 209 multiplier, 39
N Nakamura and Capes, vii, 1, 3, 4, 5, 6, 10, 13, 31, 38 nanofabrication, 242 nanometers, 244 nanoparticles, 245, 258 nanoscale, vii, x, 237, 238, 239 nanotechnology, x, 238, 239 NATO, 40 neglect, 5, 229 NEMS (NanoElectroMechanicalSystems), x, 237 Nigeria, 183 NMR, v, 183, 185, 186, 189, 191, 192, 195, 199, 207, 208, 211, 212, 214, 215, 216 nodes, 52, 53, 54, 91, 93 nonequilibrium, 3
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Index
nonlinear equations, 129 nonlinear systems, ix, 157 normal stress, viii, 45, 49, 71, 112 numerical analysis, 118, 181, 244 numerical modeling technique, viii, 75, 94 nutrients, 46
O oil, x, 2, 219 one dimension, 5, 13, 15 open spaces, 127 ordinary differential equations, 199, 208, 213 ore deposit, viii, 75, 103, 110, 112, 116, 117, 170, 178 ores, 104 overlap, 244 oxidation, 242
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P parallel, vii, ix, 1, 29, 30, 34, 39, 48, 52, 53, 80, 110, 119, 121, 126, 131, 142, 146, 185, 244, 247 parallel flow, 30 parallel plate reactors, viii, 121 parenchyma, 216 partial differential equations, 129, 160 particle collisions, 29, 41 particle mass, 6, 18, 25, 26, 33, 34, 38 particlewall friction coefficient, vii, 1, 23, 28, 36, 38, 40 pathophysiology, 184 pathways, 118 perfusion, ix, 183 periodicity, 98 peristalsis, 220 permeability, ix, 46, 47, 48, 49, 50, 78, 80, 99, 102, 103, 104, 106, 107, 113, 114, 115, 116, 118, 157, 158, 161, 162, 163, 166, 167, 170, 171, 172, 173, 175, 178 permittivity, 243 petroleum, 161, 179 pharmaceutical, 46 physical mechanisms, 158 physical properties, 78, 161 physics, ix, x, 67, 72, 77, 95, 183, 233, 238, 239 plastic deformation, 112 plate tectonics, viii, 75 platform, 110 polar, 185, 202, 203 polarity, 162 polarization, 246
polymers, 220 porosity, ix, 16, 46, 48, 49, 56, 78, 79, 109, 111, 112, 113, 114, 157, 158, 161, 163, 171, 172, 173, 175, 176, 177, 178, 179 porous materials, 179 porous media, viii, ix, 2, 46, 75, 76, 77, 78, 80, 95, 117, 119, 157, 158, 159, 178, 179, 180, 181, 182, 219, 220, 233 porous medium, vii, viii, ix, 45, 46, 48, 49, 50, 51, 66, 71, 75, 76, 78, 79, 82, 94, 96, 114, 118, 119, 157, 158, 159, 160, 161, 171, 172, 173, 175, 176, 177, 178, 179, 180, 220, 234 preconcentration, xi, 241, 249 preservation, viii, 75, 103 pressure gradient, 2, 6, 10, 13, 14, 16, 17, 19, 22, 25, 26, 28, 30, 31, 33, 34, 35, 36, 37, 38, 78, 158, 175, 221 principles, 3, 128, 150, 215 propagation, 175, 176, 222, 232 protons, vii, ix, 183, 185 pulse sequences, ix, 183 pumps, x, xi, 220, 237, 241, 242, 247, 250 pure water, 173, 184, 247 pyrolysis, 216
Q Queensland, 117, 118, 180
R radiation, x, 77, 219, 220 radius, x, 196, 203, 219, 220, 223, 230, 232, 244 reaction rate, 181 reactions, 178, 179, 242 reading, 224 reality, viii, 76, 94 reconstruction, 216 recovery, 180, 185 regression, 80 relaxation, x, 185, 219, 220, 222, 230 relevance, 179 reliability, 153 relief, 95, 118 renormalization, 130 repulsion, 245 requirements, xi, 238, 241 research institutions, x, 237 researchers, x, 5, 36, 95, 220, 221, 238 reserves, 110 residuals, 91, 92 resistance, 57, 63, 207, 221, 235, 247
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Index resolution, ix, 56, 183, 184 resources, 130, 166, 170, 178 response, 113, 216, 222, 238 restrictions, 128, 129 Reynolds and Darcy, viii, 45, 47 Reynolds characteristics, xi, 241 room temperature, 243, 244 roots, 196, 203 roughness, 125 routes, 140 rules, 78, 182
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S salinity, 77, 103, 120 salt concentration, 119 saltwater, 119 scaleup procedure, viii, 121, 123, 124, 125 scaling, 223, 224 scanner hardware, ix, 183 scattering, 17 science, x, 219, 234 scientific method, 159 Sedextype leadzinc deposits, viii, 76 sediment, viii, 75, 77, 95, 104, 117, 118, 119 sedimentary basins, viii, 75, 94, 103, 119, 120, 179, 181 sedimentation, 77 sediments, 117 seeding, 150 selfassembly, 242 selforganization, 180 semiconductor, xi, 241, 243, 246, 250 sensing, 242 sensitivity, 162, 184, 215 sensors, 250 separation, viii, xi, 45, 46, 59, 64, 66, 68, 72, 130, 142, 187, 192, 199, 203, 208, 212, 241, 242, 247, 250 Serbia, 1, 39 shape, x, 53, 57, 60, 66, 67, 89, 149, 150, 172, 185, 219, 247 shear, viii, 45, 66, 71, 112, 130, 221, 244 shock, 233 showing, 162 signals, 212 silicon, 250 silver, 103, 118 simulation, ix, 49, 51, 63, 98, 100, 101, 119, 130, 157, 158, 159, 161, 178, 179, 181, 182, 233, 247, 248, 249 simulations, viii, 45, 49, 56, 64, 65, 67, 68, 71, 104, 159
265
Singapore, 45, 217 soft lithography, 242 software, 46, 80 solid matrix, 78, 79, 81, 161, 171, 178 solid phase, 12, 79 solution, viii, 3, 51, 56, 75, 76, 80, 84, 85, 87, 88, 89, 91, 92, 93, 96, 105, 119, 122, 126, 130, 140, 141, 185, 187, 190, 191, 192, 193, 194, 195, 199, 200, 201, 202, 203, 209, 211, 213, 214, 223, 226, 227, 228, 229, 230, 233, 243, 244, 246, 247, 248 species, 122, 141, 161, 173, 175, 179, 242, 243, 247 specific heat, 79, 83, 161, 163 sphere, viii, 22, 33, 45, 46, 47, 49, 50, 53, 55, 56, 57, 59, 61, 62, 63, 65, 67, 68, 70, 71, 72, 185 spin, 184, 185, 186, 215 spouted beds, vii, 1, 41 square cylinder, viii, 45, 46, 47, 49, 50, 53, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 67, 68, 69, 71, 72 stability, 40, 41, 47, 64, 125, 179 standard deviation, 19, 21, 22, 247, 250 stars, 224 state, vii, 1, 3, 6, 7, 13, 15, 16, 19, 40, 88, 95, 96, 97, 98, 99, 102, 106, 107, 108, 109, 129, 141, 153, 159, 160, 180, 181, 186, 215, 238, 239 states, ix, 183, 184 steadystate model, vii, 1 steel, 19 stenosis, 221 stimulus, 126 stoichiometry, 141 storage, 79, 118 stratification, 163 stress, viii, ix, 45, 46, 48, 49, 71, 112, 113, 130, 219, 221, 222, 234 stroke, 184, 215, 216 structural geology, viii, 75 structure, 20, 51, 66, 72, 104, 162, 207, 211, 212, 221 style, 106 substitution, 208 subsurface fluid flow, viii, 75, 103 Sun, 77, 253, 256 supercritical fluid flow, x, 238 surface area, 171, 172, 182 surface layer, 99 surfactants, 239 susceptibility, 215 suspensions, 42, 220 suture, 117 Switzerland, 238
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Index
Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.
T techniques, ix, xi, 47, 161, 178, 183, 184, 241, 242, 243, 245, 246 technologies, x, 237, 238, 242 technology, vii, x, 220, 238, 242 tectonic deformation, viii, 75, 77, 94, 103, 110, 112, 113, 114, 116 temperature, 76, 77, 79, 80, 81, 82, 83, 86, 87, 88, 89, 90, 92, 93, 95, 96, 97, 98, 99, 100, 101, 102, 103, 105, 106, 107, 108, 109, 110, 129, 158, 160, 161, 162, 163, 165, 166, 170, 180, 181, 216, 243, 244 temperature dependence, 79, 80, 95 testing, 128 textbooks, 93 thermal, vii, viii, ix, x, 7, 75, 77, 78, 79, 81, 83, 88, 90, 94, 96, 104, 105, 109, 110, 118, 120, 121, 158, 161, 163, 179, 180, 181, 184, 216, 237, 238, 239, 242, 243 thermal and fluid transport phenomena, vii, x, 237, 238 thermal energy, x, 79, 90, 94, 109, 184, 238, 243 thermal expansion, 96 thermal management technologies, x, 237 thermal oxidation, 242 thermal properties, 105 thermal regime, viii, 75, 180 thermodynamic instability, ix, 157, 158, 159, 160, 161, 178 thermodynamics, 3 threshold level, 59 tin, 117 tissue, 184, 186, 187, 214, 215 topography, viii, 75, 77, 80, 94, 95, 97, 98, 102, 116, 117, 119 topology, 55, 56 tracks, 29 transformation, viii, 75, 84, 85, 86, 87, 186, 200, 201 translation, 41 transport processes, ix, 81, 157, 238 transport system, vii, ix, 1, 15, 22, 38, 157 transportation, 2 traveling waves, 220 treatment, 53, 128, 160, 184 trial, 91, 92 Trinidad, 154, 155 tumors, 184 turbulence, 46, 125, 130 turbulent flow, 22, 30, 34, 38, 123, 125, 140, 155 Turkey, 40 twophase flow equations, vii, 1, 2
U uniform, vii, 1, 3, 6, 18, 20, 40, 46, 133, 135, 140, 141, 142, 144, 216, 221, 234, 245, 246 united, 237 United Kingdom (UK), 118, 237 updating, 93 uranium, 110, 118 ureter, 220, 235 urine, 220 USA, 72, 73, 118, 155, 241, 253, 255, 256 USSR, 258
V vacuum, 243 validation, 47, 118 variables, 6, 16, 18, 130, 187, 192, 199, 208, 212, 223, 224 variations, 7, 19, 47, 64, 77, 80, 93, 95, 103, 114, 116, 120, 129, 135, 178, 184 vector, 47, 48, 51, 52, 53, 78, 113, 150, 172 vertical gassolid, vii, 1, 6, 9, 11, 18, 25 vertical nonaccelerating fluid, vii, 1 vertical pneumatic, vii, 1, 3, 10, 20, 29, 38, 40, 41, 42, 43 vertical twophase gassolid, vii, 1, 25 viscoelastic liquids, 233 viscoelastic properties, 222 viscosity, 39, 48, 78, 79, 80, 90, 93, 95, 113, 122, 129, 161, 163, 171, 172, 180, 221, 222, 234, 243 viscous, viii, 45, 46, 47, 48, 49, 59, 66, 67, 71, 124, 220, 221, 230, 233, 242 vision, 184 visualization, 125, 247, 248, 249 voidage, vii, 1, 2, 4, 6, 7, 9, 15, 16, 19, 20, 23, 25, 26, 27, 29, 33, 34, 35, 38, 39, 40, 41, 42
W Washington, 42, 43 water, vii, ix, 29, 76, 77, 78, 80, 81, 82, 95, 96, 97, 98, 99, 100, 101, 102, 171, 173, 183, 184, 221, 243, 247 wave number, 223, 231, 232 wear, 22 Western Australia, 180 white matter, 184, 185 worldwide, 161, 238
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Index
Y
Z zinc, viii, 76, 103, 104, 106, 109, 110, 118, 119, 120
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yield, 112, 123
267
Fluid Transport: Theory, Dynamics and Applications : Theory, Dynamics and Applications, Nova Science Publishers, Incorporated, 2011. ProQuest