Basic Equations of the Mass Transport Through a Membrane Layer

Basic Equations of the Mass Transport through a Membrane Layer

Basic Equations of the Mass Transport through a Membrane Layer

Endre Nagy Research Institute of Chemical and Process Engineering University of Pannonia, Veszpre´m, Hungary

AMSTERDAM BOSTON HEIDELBERG LONDON NEW YORK OXFORD PARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYO '

'

'

'

'

'

'

'

'

'

Elsevier 32 Jamestown Road, London NW1 7BY 225 Wyman Street, Waltham, MA 02451, USA First edition 2012 Copyright r 2012 Elsevier Inc. All rights reserved No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangement with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-12-416025-5 For information on all Elsevier publications visit our website at elsevierdirect.com This book has been manufactured using Print On Demand technology. Each copy is produced to order and is limited to black ink. The online version of this book will show color figures where appropriate.

Dedication

To my beloved wife, E´va, and my children, Anita and Andra´s.

Preface

My introduction to the membrane process, namely pervaporation, was as a newly graduated chemical engineer in 1969 in the Research Institute of Chemical and Process Engineering of the Hungarian Academy of Sciences. At that time, membrane separation technology made its first steps toward becoming a real industrial technology. My research group was the first in my home country that started with investigation of the pervaporation process for separation of alcohol (with low carbon number)/water binary mixture applying cellulose-hydrate, cellulose-acetate foils. Since then, pervaporation has become one of the most important separation processes for both aqueous/organic and organic/organic mixtures. During these early experiments, we learned that the mass transfer process cannot be described correctly by Fick’s diffusion, because the pretreatment of the membrane and/or the feed concentration of the alcohol strongly affected the separation efficiency. This experience encouraged me to study the mass transport phenomena during different membrane processes. The mass transport through membranes is affected by several factors originating mainly in the membrane/component interaction, membrane structure, and the operating conditions. All differential balance equations used for description of the transport (mass, momentum, energy) can be obtained by simplification of the Navier Stokes flow models. These equations are mostly adapted for membrane transport as well as flow models for the fluid phases flowing on the feed and/or permeate sides of an active or supporting membrane layer. The existing texts primarily focus on the theoretical description of the mass transport processes through a separating and/or reacting membrane layer. It defines the membrane inlet/outlet mass transfer rates depending on the constant or variable mass transfer parameters (e.g., diffusivity, convective flow, external mass transfer resistances), membrane curvature (plane, capillary), and on chemical/biochemical reactions (analytical solution for first- and zero-order reactions, analytical approach for the second-order reaction, or Michaelis Menten kinetics). Many equations have been developed by the author and published here for the first time. Several of them should then be adapted to real situations before their application. The effect of constant and variable transport parameters is illustrated in several figures in this book. The first two chapters give a brief survey of the transport equations through membranes considering the diffusive and the convective transport applying the solution-diffusion model, Maxwell Stefan, Flory Huggins approaches (Chapter 1), and equations to predict the diffusion coefficient in the fluid and membrane phases (Chapter 2). In Chapters 3 6, we give the most important mass

xii

Preface

transfer rate and concentration distribution expressions, mostly developed by the author. Diffusion with constant and variable diffusion or solubility coefficients is modeled for plane interface (rectangular coordinate, Chapter 3), and for capillary/ tube membrane (Chapter 4), as well as mass transfer accompanied by chemical reaction in plane sheet membranes (Chapter 5) and cylindrical membranes (Chapter 6). These chapters also discuss the mass transport through asymmetric membranes due to its importance in the membrane separation processes. The flow models of the fluid phases in capillary or plane membranes are summarized in Chapter 7. Starting from the general Navier Stokes equations, we give simplified variations of these equations and some special cases considering the convective flows discussed in this section. The application of the mass transfer expression previously developed and balance equation listed is discussed in Chapters 8 12 in the cases of membrane technologies as membrane reactor (Chapter 8), membrane bioreactor (Chapter 9), nanofiltration (Chapter 10), pervaporation (Chapter 11), and membrane contactors (Chapter 12). The radial flow rate is expressed under different operation conditions in enzyme membrane reactors (Chapter 9). Unified models have been developed and discussed which take into account the simultaneous effect of the membrane and the polarization layers in Chapters 10 and 11, respectively. Chapter 12 gives a brief overview of mass transfer equations for the membrane contactors, applying them to absorption, extraction, or distillation. Mass transfer equations defining in explicit, closed forms are important to predict easily the effectiveness of a separation and/or chemical reaction process or for planning devices or technologies. This book intends to make easier the work of the engineers and technologists who are working, researching, and learning the membrane separation processes.

1 On Mass Transport Through a Membrane Layer

1.1

General Remarks

1.1.1

Transport of Dilute Solution

A membrane may be defined as a permselective barrier between two homogeneous phases. Two main potential differences are important in membrane processes, namely the chemical potential difference (Δμ) and the electrical potential difference (ΔF) (the electrochemical potential is the sum of the chemical potential and the electrical potential). Other possible forces such as magnetic fields, centrifugal fields, and gravity will not be considered here. In passive transport, components or particles are transferred from high potential to low potential. In the case of multicomponent mixtures, fluxes often cannot be described by simple phenomenological equations because the driving forces and fluxes are coupled. In practice, that means that the individual components do not permeate independently from each other. For example, a pressure difference across the membrane not only results in a solvent flux, but also leads to mass flux and the development of a solute concentration gradient. On the other hand, a concentration gradient not only results in diffusive mass transfer, but also leads to a building up of hydrostatic pressure. Potential difference arises as a result of differences in pressure, concentration, temperature, or electrical potential. Only charged molecules or ions are affected by the electrical field. The chemical potential in the presence of electrical field can be given as μi 5 μ0i 1RT ln ai 1Vi p1Fzi E

ð1:1Þ

where R is the gas constant (0.082 atm m /kmol K); T is the temperature (K); μ is the chemical potential (atm m3/kmol); ai is the activity of component i; Vi is the molar volume of component i (m3/kmol); p is the pressure (atm); F is the Faraday constant (96.5 kC/mol); and E is the electrical potential (V). The electrical field forces on the charged species only. Most of the transport processes take place because of a difference in chemical potential Δμ. Under isothermal conditions (constant T), pressure and concentration contribute to the chemical potential of a component according to 3

μi 5 μ0i 1 RT ln ai 1Vi p Basic Equations of the Mass Transport through a Membrane Layer. DOI: 10.1016/B978-0-12-416025-5.00001-6 © 2012 Elsevier Inc. All rights reserved.

ð1:2Þ

2

Basic Equations of the Mass Transport through a Membrane Layer

The concentration or vapor composition is given in terms of activities ai in order to express nonideality: ai 5 γ i x i

ð1:3Þ

where xi is the mole fraction of component i and γ i is the activity coefficient of component i (kmol/kmol). The chemical potential change dμi 5 RT d lnðγ i xi Þ 1 Vi dp

ð1:4Þ

Taking into account that the driving force is the potential gradient, and there is no coupling of flows, the mass transfer rate, Ji (Flux, kg/m2s, kmol/m2s, or in cgs unit gmol/cm2 s) perpendicular to the membrane interface (y is the distance of diffusion in the membrane in meters) can be given as (Wijmans and Baker, 1995; Baker, 2004): Ji 52 Li

dμi dy

ð1:5Þ

where dμi /dy is the chemical potential gradient of component i and Li is a coefficient of proportionality linking this chemical potential driving force to flux. Driving forces such as gradients in concentration, pressure, temperature, and electrical potential can be expressed as chemical potential gradients, and their effect on flux is expressed by Eq. (1.5). If the pressure gradient inside the membrane is negligible (e.g., mass transport through dense membrane), one can get Ji 5 2RTLi

d ln ai RTLi dai 2 dy ai dy

ð1:6Þ

For ideal solutions, the activity coefficient γ i-1, and the activity ai becomes equal to the mole fraction xi. Introducing the more practical concentration, φi (kg/m3) is defined as φi 5 Mi ρxi

ð1:7Þ

where Mi denotes the molecular weight of i (kg/kmol) and ρ is the molar density of the membrane (kmol/m3). From the value of φi defined by Eq. (1.7), the concentration in weight fraction ( 5 φi/ρ) or in volume fraction [5 φiVi/Mi, where Vi is molar volume (m3/kmol) and Mi is molar weight (kg/kmol)] can easily be gotten. Thus, from Eq. (1.6), the Fick’s law in a stationary fluid (i.e., there has been no net movement of the bulk fluid or convection flow of the entire phase) can be given as Ji 5 2

RTLi dφi dφ  2Di i φi dy dy

ð1:8Þ

On Mass Transport Through a Membrane Layer

3

with Di 5

RTLi φi

ð1:9aÞ

where Di is called the diffusion coefficient (m2/s) and is a measure of the mobility of the component i. For single-component permeation where the activity coefficient is not constant, the mass transfer rate can be derived from Eqs (1.5) and (1.6) as Ji 5 2 Li RT

@ ln ai @φi dφ  2Di ðφi Þ i @φi @y dy

ð1:9bÞ

That means that the diffusion coefficient can be expressed as Di ðφi Þ 5 2Li RT

@ ln ai @φi

ð1:10Þ

The function of ai 5 f(φi) can be given by applying the FloryHuggins thermodynamics, as will be shown later. It should also be expressed as the link between the thermodynamic diffusion coefficient DTi and Di. According to Lonsdale (1982), the thermodynamic diffusion coefficient can be expressed as Ji 5 2

DTi dμi φ RT i dy

ð1:11Þ

Now, taking into account Eqs (1.5), (1.10), and (1.11), one can get (Meuleman et al., 1999): Di ðφi Þ 5 DTi φi

@ ln ai @ ln ai  DTi @φi @ ln φi

. DTi 5 Di

@ ln φi @ ln ai

ð1:12Þ

During separation of binary mixtures, often coupled transport may occur, which means that the flux of a component of a mixture may change not only by the presence of the other component, but also by its movement. Both the solubility and the diffusivity of component i can depend on the concentration of both components. According to Eq. (1.5), the following equation system can be given for a binary system with components i and j: Ji 5 2Lii

dμj dμi 2 Lij dy dy

ð1:13Þ

Jj 5 2Lji

dμj dμi 2 Ljj dy dy

ð1:14Þ

4

Basic Equations of the Mass Transport through a Membrane Layer

The first term on the right-hand side of Eq. (1.13) describes the flux of component i due to its own gradient, and the second term of this equation describes the flux of component i due to the gradient of component j. This second term represents the coupling effect. Taking into account Eqs (1.9a), (1.9b), and (1.10), that the permeation rate of component i can be given as (Mulder, 1981; Meuleman et al., 1999): ! ! @ ln ai @φi @ ln ai @φj @ ln aj @φi @ ln aj @φj 1 1 2 Lij RT Ji 5 2Lii RT @φi @y @φj @y @φi @y @φj @y ð1:15Þ Rewriting Eq. (1.13), one can get ! )  @ ln ai @ ln aj @φi @ ln ai @ ln aj @φj 1 Lii Lii 1Lij 1Lij @φi @φi @y @φj @φj @y

( Ji 5 2RT

ð1:16Þ thus,   dφj dφi 1 Aij Ji 5 2 Aii dy dy

ð1:17Þ

  dφj dφ Jj 5 2 Aji i 1 Ajj dy dy

ð1:18Þ

  @ ln ai @ ln aj Aii 5 RT Lii 1 Lij @φi @φi

ð1:19Þ

and

with

@ ln ai @ ln aj Aij 5 RT Lii 1 Lij @φj @φj

1.1.2

! ð1:20Þ

Transfer Rate of Concentrated Feed Solution

For more concentrated feed solutions, bulk convection terms must be included. When describing the transport process, it is necessary to specify the frame of reference of the transport process (Bird et al., 1960). In the case of fixed frame of reference transport, the membrane can be used as the frame of reference because

On Mass Transport Through a Membrane Layer

5

the membrane is stationary at steady state. For this static reference frame, the transport process is a sum of diffusional and convective flux. Generally, the following simplified equations can be used for binary (components i and j) mixture, assuming no coupling of the transport process: Ji 5 2Di

dφi φ 1 i ðJi 1Jj Þ dy ρ

ð1:21aÞ

Jj 5 2Dj

dφj φj 1 ðJi 1Jj Þ dy ρ

ð1:21bÞ

and

as well as from Eq. (1.21a), Ji can be expressed as   1 dφi φi Jj Ji 5 2Di 1 dy ρ Ji 1 2 φj =ρ

ð1:21cÞ

where φi(Ji 1 Jj)/ρ is the bulk flow term (convective mass transfer); Di and Dj are the effective binary diffusion coefficients of components; and J is the total mass flux (diffusive 1 convective flux) (kmol/m2 s) of a component with respect to a fixed frame of reference. (Note that here the diffusive flux and the total mass flux is not distinguished in this chapter as it is generally made in the literature where the total flux is mostly denoted by N). In the case of binary mixture the mass flux of the membrane material also has to be given. The polymer mass flux will be zero, while the diffusional mass flux of membrane is equal in value and opposite in the direction to the flow with respect to its bulk mass flux (not given here) (Kamaruddin and Koros, 1997). Note that the fluxes in Eq. (1.21a) or (1.21b) are the sum of the diffusive mass transfer rate and the convective one, accordingly the values of J in Eqs (1.8), (1.21a), and (1.21b) should be different in presence of convective velocity. The convective velocity can be expressed from Eq. (1.21a) or (1.21b) as (Bird et al., 1960; Geankoplis, 2003): υ5

Ji 1 Jj ρ

ð1:22Þ

The υ convective velocity in Eq. (1.22) is a product of diffusion of components in the membrane matrix. Accordingly, the total flux can be given as Ji 5 2Di

dφi 1 φi υ dy

ð1:23Þ

The convective velocity can exist also due to the pressure difference on the two sides of a porous membrane which has to be involved in the υ value. Diffusive mass

6

Basic Equations of the Mass Transport through a Membrane Layer

transport takes place in a dense polymer layer with no visible pores. The transport can be described well by the solution-diffusion model that will be discussed briefly in Section 1.2. This is the case for reverse osmosis, pervaporation, and polymeric gas separation (Baker, 2004). The spaces between the polymer chains in these membranes are less about 0.5 nm in diameter, and these spaces are caused by the thermal motion of the polymer chains. Ultrafiltration, microfiltration, and microporous Knudsen-flow gas separation membranes are all clearly microporous, and transport occurs by convective flow through the pores (see Section 1.3). The pore size can be stated to be larger than 1 nm for these separation processes. The mass transport equations of the diffusive plus convective mass transport will be separately discussed in Chapter 3 and 5, respectively. The extended NernstPlank equation, for the membrane in electrical field, is as (Geraldes and Brites, 2008): Ji 2 Di;p

dφ zi φ Di;p dψ F 1 Ki;c Jv 2 i RT dy dy

ð1:24Þ

where Ji is solute flux of component i (kg/m2 s); Jv is permeate volume flux (m3/ m2 s); Di,p is hindered diffusion coefficient (m2/s); Ki,c is hindrance coefficient for convection; zi is charge number; F is Faraday constant (9.64867 3 104 C/eq); R is ideal gas constant (8.314 J/kmol); ψ is electric potential (V); and Di,N is the diffusivity of the i species in water at infinite dilution (m2/s).

1.2

Transport Through Dense Membrane: Solution-Diffusion Theory

We can broadly classify transport in solids into two types of diffusion: diffusion that can be considered to follow Fick’s law and does not depend primarily on the actual structure of the solid membrane (e.g., dense polymer membrane); and diffusion in porous solids (e.g., zeolite membrane), where the actual structure and channels are important. Both types will be briefly considered. The separation process is illustrated in Figure 1.1. The solution-diffusion is the generally accepted mechanism of mass transport through nonporous membranes (Lonsdale, 1982). It is held that the component permeation through a homogeneous membrane consists of five fundamental processes: (1) the solute molecules must first be transported or diffused through the liquid (or gas; in this latter case the mass transfer resistance of the boundary layer is negligible, consequently this step can also be omitted) film (Figure 1.1B) of the feed phase on the feed side of the solid membrane; (2) the solution of the solute molecule in the upstream surface of the membrane matrix; (3) diffusion of the dissolved species across the membrane matrix; (4) desorption of the solute molecules in the downstream side (permeate side) of the membrane; and (5) diffusion through the boundary layer of the permeate phase. The first and the fifth steps can be omitted when there are no mass transfer resistances between the

On Mass Transport Through a Membrane Layer

(A)

7

(B)

δ

φ*

Co C* Ji

Ji φ*δ

C*δ

o

Cδ

Figure 1.1 (A) Transport through dense membrane. (B) Concentration profile and the important notations.

continuous and membrane phases. This can often be the case, especially for gas permeation or liquid permeation with high flow rates in the fluid phases on the two sides of membrane. The thermodynamic equilibrium between the continuous phase and the membrane interface assumes that the chemical potentials are equal to each other, therefore: μfi 5 μmi

ð1:25Þ

where subscripts f and m denote fluid and membrane, respectively. According to Eq. (1.25) for liquid phase in contact with membrane phase (Wijmans and Baker, 1995; Baker, 2004; Wijmans, 2004): sat RTd lnðγ if xif Þ 1 Vif ðp 2 psat i Þ 5 RTd lnðγ im xim Þ 1 Vim ðp 2 pi Þ

ð1:26Þ

Assuming that the molar volume difference is negligible, it can be understood from Eq. (1.26) as γ if xif 5 γ im xim

ð1:27Þ

Taking into account Eq. (1.7) and applying it for the membrane phase, we can obtain φi 5

γ if ρm ci  H i ci γ im ρf

ð1:28Þ

8

Basic Equations of the Mass Transport through a Membrane Layer

with Hi 5

γ if ρm γ im ρf

ð1:29Þ

where ρ is the density (kg/m3). The value of H solubility coefficient depends on the separation process to be applied according the chemical potential. Baker (2004) and (in part) Wijmans (2004) define this solubility coefficient for different membrane processes as dialysis, reverse osmosis, pervaporation, gas separation, and hyperfiltration. Accordingly, let us define the partition coefficient as φi =ci 5 Hi 

ð1:30Þ 

where c and φ represent the concentration of species in the membrane surface and the feed, respectively, and H is the equilibrium distribution coefficient (similar to Henry’s law coefficient for gas and liquid; note that the inverse of H is often defined in the literature as equilibrium coefficient), which is a characteristic parameter dependent upon interaction of the species with the membrane. Taking into account the H partition coefficient, and integrating Fick’s first law, given by Eq. (1.8) with boundary conditions illustrated in Figure 1.1B, one can obtain Ji 5 Di

φi 2 φδi Di Hi o Pi  ðci 2 coδi Þ  ðcoi 2 coδi Þ δ δ δ

ð1:31Þ

where Pi is the permeability coefficient (m2/s) and ðcoi 2 coδi Þ is the driving force. The Di and Hi and thus Pi are concentration dependent for many systems. Thus, these values should be considered as averaged over the membrane thickness in Eq. (1.31).

1.3

Convective Transport Through a Porous Membrane Layer

Pressure-driven convective flow, the basis of the pore flow model, is most commonly used to describe flow in a capillary or a porous medium. Look first at the flow through cylindrical capillary tubes. Let us assume that there are cylindrical capillaries in the membrane matrix perpendicular to the interface. The laminar fluid flow through these capillaries can be described by means of the differential momentum balance equation, Newton’s law of viscosity as (Bird et al., 1960; Geankoplis, 2003): τ ry 5 2μ

dυy dr

ð1:32Þ

On Mass Transport Through a Membrane Layer

9

where τ ry is shear stress or force per unit area [((kg m/s)/m2 s)  N/m2]; μ is the viscosity of the flowing fluid (Pa s or kg/m s), r denotes cylindrical coordinate in the capillary tube (Figure 1.2; note, assuming that the capillary pores is perpendicular to the membrane interface, the r space coordinate is parallel to the interface, while y is perpendicular to it). On the other hand, the τ ry shear stress can be expressed as τ ry 5

p 2 pδ r 2δ

ð1:33Þ

where p 2 pδ is the pressure drop for a capillary length δ (N/m2 or Pa); δ is the thickness of the membrane layer; p is the feed side, while pδ is the permeate side pressure (Pa). Substituting Eq. (1.32) into Eq. (1.33), we obtain the following differential equation for the convective velocity: dυy p 2 pδ 52 r dr 2μδ

ð1:34Þ

After integrating Eq. (1.34) using suitable boundary conditions (Bird et al., 1960), we get the well-known HagenPoiseuille equation for the average convective velocity through the porous membrane: υ5

Δprp2 8μδ

ð1:35Þ

where Δp is the pressure difference between the two sides of the membrane (Pa) and rp is the pore radius (m). Figure 1.2 Transport through a porous membrane.

10

Basic Equations of the Mass Transport through a Membrane Layer

The specific volume flow rate can also be calculated by means of Eq. (1.36), namely (Baker, 2004): υo 5

Δpεrp2 8μδ

ð1:36Þ

where υo is the flow rate related to the membrane total area (m3/m2 s) and ε is the porosity of the membrane (ε 5 Nπrp2 ; where N is the number of capillaries related to the membrane surface). For porous medium, Darcy’s law is a basic equation to describe the rate of fluid flow [Eq. (1.37)] and the mass transport through it [Eq. (1.38)]. It can be written as υ5

K 0 dp μ dy

ð1:37Þ

where υ is the convective velocity (m/s); μ is the fluid dynamic viscosity (Pa s); K0 is an empirical constant (m2); p is the pressure (Pa); and y is the space coordinate perpendicular to the porous layer’s interface. Ji 5

K 0 dp φ μ i dy

ð1:38Þ

where dp/dy is the pressure gradient existing in the porous membrane; φi is the concentration of component i in the membrane; K0 is a coefficient reflecting the nature of the membrane; and Ji is the mass transfer rate of component i (kg/m2 s). This equation defines the mass transfer rate through porous membrane as it is given in Eq. (1.8) for diffusive mass transport through dense membrane. The transition between the two models depends on the size of free-volume elements (pores). According to Baker (2004), this transition regime can exist in the pore size range of 0.51 nm (1 nm 5 1 3 1029 m). For the sake of completeness, let us give the following often-used form of the convective velocity, in its one-dimensional form, namely: υ52

Bo dp μ dy

ð1:39Þ

where the permeability Bo is characteristic of the membrane structure and has to be determined experimentally, along with the porositytortuosity factor (ε/τ); μ is the viscosity of the fluid mixture (Pa s). The permeability coefficient Bo can be calculated for some typical structures. For a cylindrical pore, the permeability is calculated from the Poiseuille flow relationship [see Eq. (1.35)]: Bo 5

rp2 8

ð1:40Þ

On Mass Transport Through a Membrane Layer

11

For a suspension of spheres of diameter do, the RichardsonZaki correlation gives Bo 5

d2o 2:7 ε 18

ð1:41Þ

where ε is the porosity of the suspension. The CarmanKozeny equation also should be mentioned for cases in which if the membrane consists of a packed bed of particles (Silva et al., 2008): Bo 5

d2o ε2 180 ð1 2 εÞ2

ð1:42Þ

Thus, υ52

d2p

ε3 dp 180ð1 2 εÞ η τ dy 2

ð1:43Þ

where dp is the particle size (m); ε is the porosity (); τ is the tortuosity factor (); and υ is total volumetric flux (m/s). For a corresponding bed of fibers (Krishna and Wesselingh, 1997): Bo 5

do2 ε2 80 ð1 2 εÞ2

ð1:44Þ

The viscous flow contribution is important for transport in membrane with open structures and it is relatively unimportant for transport in membranes with dense structures, such as those present in gas permeation and pervaporation membranes.

1.4

Component Transport Through a Porous membrane

The mass transport through a porous layer is extensively studied. Its type depends on the pore size comparing it with the mean path of the transported molecules. In the case of gas transport, three regions are distinguished regarding the transport (Seidel-Morgenstern, 2010): 1. Molecular and/or viscous flow, when э , dp 2. Knudsen diffusion, when ds , dp , э 3. Molecular sieving transport (configurational diffusion), dsDdp

12

Basic Equations of the Mass Transport through a Membrane Layer

where dp is the pore size; ds is the size of transported species; and э is the free path of molecule. The value of the mean free path of the gas molecule i, эi, can be calculated as (Lawson and Lloyd, 1997): kB T 3i 5 pffiffiffi 2 2πpds;i

ð1:45Þ

where kB is the Boltzman constant (kB 5 1.34 3 10223 J/K); p is the mean pressure within the pore (Pa); and T is the absolute temperature (K). Molecular diffusion and/or viscous flow are dominated by the interaction between the transported molecules when the pore diameter is considerably larger than the mean free path. The real diffusivity is, however, reduced by the ratio between porosity, ε, and tortuosity, τ. Thus, we can get the effective diffusion coefficients De, as ε De;i 5 Di ð1:46Þ τ By application of kinetic gas theory to a single, straight, and cylindrical pore, the coefficient of Knudsen diffusion can be derived as (Mason and Malinauskas, 1983): rffiffiffiffiffiffiffiffiffi dp ε 8RT ð1:47Þ DK;i 5 3 τ πMi where R is the gas constant (R 5 8.314 J/mol K); Mi is the molecular weight of species i (kg/kmol); and dp is the pore size (m). By analog to the molecular diffusion, the Knudsen diffusion flux can be calculated to Ji 5 2DK;i

dci DK;i dpi  2 dy RT dy

ð1:48Þ

where Ji the mass transfer rate (kmol/m2 s). The gradient of the partial pressure, dpi/dy, is the driving force of the transport. Note the diffusion is regarded as one dimensional. The viscous flow can exist when the pore size is larger than the mean free path. For a single capillary, the viscous flow can be calculated according to HagenPoiseuille [see Eq. (1.35)], thus: Ji 5 υ

pi pi ε dp2 dp 52 RT RT τ 32 dy

ð1:49Þ

where p is the total pressure and pi is the partial pressure. Note that the viscous flow does not contribute to the separation of different species. [But as discussed by Eqs (10.3)(10.5), in the case of liquid phase, the fluid viscosity will be increased in the nanopores, and the convective velocity of the molecules will also be hindered depending on the ratio of ds (solute molecule size) to dp.]

On Mass Transport Through a Membrane Layer

13

When both the molecular and the viscous flows exist, the total mass transfer rate is expressed by the sum of these two rates, as ! 1 ε dp2 dp dpi pi 1 DK;i ð1:50Þ Ji 5 2 dy RT τ 32 dy The total (p) and the partial pressure of species i (pi) can vary partly independently. This change can be given as   dp 1 dpi dxi 2p 5 dy dy xi dy

ð1:51Þ

where xi is the mol fraction of species i. Assuming that the mol fraction of the diffusing component does not change during the transport (there is no source or ^ sink term), from Eq. (1.51) it can be obtained that ðci 5 ct xi ; ci 5 pi =RTÞ: dp 1 dpi ct RT dpi  5 dy xi dy pi dy

ð1:52Þ

Replacing Eq. (1.52) into Eq. (1.50), one can obtain 1 Ji 5 2 RT

! ε dp2 dpi ct RT 1 DK;i dy τ 32

ð1:53Þ

After integration of Eq. (1.53) with y 5 0 pi 5 pe,i and with y 5 δ pi 5 pt,i (the thickness of the catalyst sponge layer is neglected here; the thickness of the active layer is generally a few micrometers, while that of the distributor a few hundred micrometers), it can be obtained for the mass transfer rate: ( ) 1 ε dp2 DK;i ct 1 ðpe;i 2 pt;i Þ Ji 5 RT δ τ 32

ð1:54Þ

where subscript e and t denote the shell side and the tube side and ct is the total concentration (kmol/m3). The thickness of a porous gas distributor in a membrane reactor [e.g., in case of partial oxidation in an oxygen permeable membrane reactor (Hoang et al., 2005)] can be relatively large, a few hundred micrometers. In the case of low feed side radius, the cylindrical effect should also be taken into account. This effect can be predicted applying the following equation:   1d dJi 50 r dr r dr

ð1:55Þ

14

Basic Equations of the Mass Transport through a Membrane Layer

After integration of Eq. (1.55), applying the boundary conditions at r 5 ro then pi 5 pt,I and r 5 ro 1 δ then pi 5 pe,I, one sees ( ) 1 ε dp2 DK;i ct 1 ðpe;i 2 pt;i Þ Ji 5 RT ro lnð1 1 δ=ro Þ τ 32

ð1:56Þ

where pt and pe are total pressure on the tube and shell side (Pa). Equations (1.54) and (1.56) do not involve the molecular diffusion in the membrane pores. The molecular diffusion coefficient often is combined with the Knudsen diffusion coefficient. Thus, the total diffusion flux will be the sum of molecular and Knudsen diffusion. The overall diffusion coefficient, Di,eff, is given by Bosanquet equation (Seidel-Morgenstern, 2010, p. 41): Di;eff 5

1 ð1=Dei;j Þ 1 ð1=DK;i Þ

ð1:57Þ

where Dei;j is the effective molecular diffusion coefficient of component i (its value involves the ε/τ factor) (m2/s) and DK,i is the effective Knudsen diffusion coefficient [Eq. (1.47), m2/s]. The value of Dei;j should then be substituted into Eq. (1.54) or (1.56), replacing DK,i with it. If the pore diameters approach the range of the sizes of the molecules, the configurational diffusion regime is reached. The corresponding flux density can be described by (Thomas et al., 2001): Ji 5 2

1 ε c dpi D RT τ i dz

ð1:58Þ

with Dci

rffiffiffiffiffiffiffiffiffi 8RT 2ðEic =RTÞ 5 ρ g dp e πMi

ð1:59Þ

where Dci is the diffusion factor (m2/s) and Eic is activation energy of diffusion (kJ/ kmol). There are no well-accepted formulas to predict the Dci diffusivity (Thomas et al., 2001). In the region of very small pores, the effect of the surface diffusion might be significant. The surface diffusion flux can also be important in the mass transport through pores in the Knudsen regime due to the adsorption, consequently this flux will be added to the above-discussed molecular and Knudsen diffusion and the viscous flow (Tuchlenski et al., 1998; Thomas et al., 2001). Physically adsorbed molecules are relatively mobile and, although the surface mobility is in general several orders of magnitude lower compared to the mobility in fluid phase, surface fluxes might be relatively high due to larger molecular densities

On Mass Transport Through a Membrane Layer

15

on the pore surface. The most general approach for multicomponent surface diffusion was proposed by Krishna and Wesselingh (1997). Based on the physical picture of molecules moving on the surface, the generalized MaxwellStefan equations were applied to describe interactions mechanistically (Tuchlenski et al., 2001). This model is applied for bulk diffusion through a membrane layer in the next section. For adaptation of these equations, see Krishna and Wesselingh (1997) or Krishna (1990).

1.5

Application of the MaxwellStefan Equations

The generalized MaxwellStefan equations are based on the assumption that movement of species is caused by a driving force which is balanced by the friction that the moving species experience from each other and their surroundings. The generalized form of this equation applying it for multicomponent fluid mixtures is given as (Krishna and Wesselingh, 1997; Wesselingh and Krishna, 2000; Amundsen et al., 2003): 2

j5n X rμi xj ðui 2 uj Þ 5 ; RT Dij j51

i 5 1; 2; . . . ; n

ð1:60Þ

j 6¼ i

where μi is the chemical potential of species i (kJ/kmol); R is the gas constant (8.314 J/mol K); T is the absolute temperature (K); xj is the mol fraction of species j; ui, uj are molar-averaged mixture velocity of species i or j (m/s); Dij is the MaxwellStefan diffusivity (m2/s) and represents inverse friction factor between molecules i and j. Accordingly the force balance on the species 1. e.g. in a ternary fluid mixture takes the form for one-dimensional diffusion as 2

dμ1 x2 ðu1 2 u2 Þ x3 ðu1 2 u3 Þ 5 1 RTdy D12 D13

ð1:61Þ

where y is the space variable (m); ternary system is a membrane with two-component separation, thus this system can be originated by the above equation. The molar flux Ji, with respect to a laboratory-fixed coordinate reference frame, can be given by the following equation (Krishna and Wesselingh, 1997): J i 5 c i ui  c t x i ui

ð1:62Þ

where ct is the total molar concentration of the fluid mixture (kmol/m3); ci is the molar concentration of species i (kmol/m3), n is the number of diffusing component. Because motion of molecules must be balanced, i.e., movement of one

16

Basic Equations of the Mass Transport through a Membrane Layer

molecule must be balanced by the motion of another molecule, thus it can be given as i5n X

i5n X

Ji 5 0;

i51

xi 5 1

ð1:63Þ

i51

Thus, any one of the differential equations can be replaced by one of the two conditions above. Taking into account Eq. (1.62), we can obtain for a multicomponent system as 2

j5n X rμi xj Ji 2 xi Jj 5 RT ct Dij j51

ð1:64Þ

j 6¼ i

Accordingly, for component 1 in a ternary system can be given as (Amundsen et al., 2003):   dx1 x2 x3 x1 x1 52 J1 1 1 J2 1 J3 ct dy D12 D13 D12 D13

ð1:65Þ

The thermodynamic potential gradient rμi can be expressed in terms of the mole fraction gradients (Krishna and Wesselingh, 1997) or in terms of the gradients of the surface occupancies (Kapteijn et al., 1995; Krishna and Baur, 2003) by the matrix of thermodynamic factors. For a fluid mixture, it will be j5 n 21 X xi rμi 5 Γij rxi ; RT j51

Γij 5 δij 1 xi

d ln γ i dxj

ð1:66Þ

6 j) and γ is the activity where δij is the Kronecker delta (δij 5 1 for i 5 j, δij 5 0 for i ¼ coefficient of species i. Combining Eqs (1.64) and (1.66), it can be obtained (Krishna and Wesselingh, 1997): ðJÞ 5 2ct ½B 21 ½ΓðrxÞ

ð1:67Þ

with Bii 5

k5 n 21 X xi xk 1 ; Din D ik k51

 Bijði 6¼ jÞ 2 xi

1 1 2 Dij Din



k 6¼ i

where (J) represents the column vector (n 21) diffusion fluxes (kmol/m2 s); the element of the matrix [B] can be derived by the above equation. The application

On Mass Transport Through a Membrane Layer

17

of the above equation for zeolitic membrane is discussed in Section 5.2. It is common to define a matrix of Fick diffusivities [D] by using (n 21) 3 (n 21) matrix notation: ½D 5 ½B 21 ½Γ

ð1:68Þ

Applying the above basic equation given for a fluid system, the generalized MaxwellStefan equation can be reformulated for membrane separation systems. Application of the above equation will be discussed briefly in the next sections.

1.5.1

The MaxwellStefan Approach to Mass Transfer in a Polymeric, Dense Membrane

The various species transport through a polymer matrix or inorganic ceramic layer involves different transport mechanisms, different diffusion mechanism as bulk gas diffusion, bulk liquid diffusion, Knudsen diffusion inside pores, interface diffusion, and so forth. The Fick’s law of diffusion is a basic design procedure and postulates a linear dependence of flux on its composition gradient as it is given by Eq. (1.8). The deviation from the linearity is taken into account by variable diffusion coefficient. After the literature data, the MaxwellStefan formulation provides the most general approach for describing mass transport for nonideal systems, thus, for the mass transport during membrane separation (Krishna and Wesselingh, 1997; Wesselingh and Krishna, 2000). This approach can be applied for mass transport through both the dense membrane [Eqs (1.17)(1.20)] and inorganic membrane [Eqs (1.13)(1.16); Krishna and Baur, 2003]. The following equation can be obtained for multicomponent mass transfer through solid membrane layer, applying the MaxwellStefan approach regarding the membrane as a motionless (n 11)th component and (Krishna and Wesselingh, 1997): 2

n X φj Ji 2 φi Jj φi dμi φ Ji 5 1 m φt RT dy φ D φ ij t t Dim j51

ð1:69Þ

j 6¼ i

with j 5X n11

φj 5 φt ð ρÞ

j51

where φi is the concentration of ith species in the membrane (kg/m3 or kmol/m3); Ji is the mass transfer rate of species i (kg/m2 s or kmol/m2 s); Dij is the MaxwellStefan counterexchange diffusivity of components i, j, and it represents the inverse friction factor between molecules (m2/s); Dim is the diffusivity of i in the membrane (m2/s).

18

Basic Equations of the Mass Transport through a Membrane Layer

For diffusion of binary mixture in a membrane (μi 5 μoi 1 RT ln xi ; φ1 1 φ2 1 φm 5 ρ): 2ρ

dμi φ J1 2 φ1 J2 φm J1 5 2 2 dy D12 D1m

ð1:70Þ

The driving force for diffusion is the chemical potential gradient (dμi/dy) on the left-hand side of Eq. (1.70). Friction is the result of interaction between transported (adsorbed) molecules (first term on the right-hand side) and interaction between a solute molecule and the membrane matrix (second term on the right-hand side). For details, see Krishna and Wesselingh (1997) or Wesselingh and Krishna (2000). For binary mixture (i 5 1, j 5 2): 2ρ

dμ1 φ J1 2 φ 1 J2 φ J1 5 2 1 m dy D12 D1m

ð1:71Þ

2ρ

dφ2 φ J2 2 φ2 J1 φ J2 5 1 1 m dy D21 D2m

ð1:72Þ

and

The mass transfer rate of the individual components can be expressed by means of Eqs (1.71) and (1.72). Assuming that D12 5 D21 and dμi 5 d ln φi, one can obtain (Heintz and Stephan, 1994a,b; Iza´k et al., 2003; Nagy, 2004):   dφ dφ J1 5 2 A11 1 1 A12 2 dy dy

ð1:73Þ

with 

D12 1 φ1 D2m A11 5 D1m ρ D12m 1 φ1 D2m 1 φ2 D1m  A12 5 D1m ρ

φ1 D2m D12 1 φ1 D2m 1 φ2 D1m

 ð1:74Þ

 ð1:75Þ

as well as

  dφ1 dφ2 1 A22 J2 5 2 A21 dy dy

ð1:76Þ

with  A21 5 D2m ρ

φ2 D1m D12 1 φ1 D2m 1 φ2 D1m

 ð1:77Þ

On Mass Transport Through a Membrane Layer

and



D12 1 φ2 D1m A22 5 D2m ρ D12 1 φ1 D2m 1 φ2 D1m

19

 ð1:78Þ

The separation of ethanolwater binary mixture by pervaporation, using poly (vinyl)alcohol membrane is illustrated briefly in Section 3.2.5.2. How can we express the link between the MaxwellStefan diffusion coefficient and the Fickian diffusion coefficient? This can be made by the so-called Darken correction factor, namely: D 5 DΓ

ð1:79Þ

The Γ thermodynamic factor can be given of the difference of chemical potential change in real and ideal system. In a real system, the mass transfer rate is (D 5 RTLi/φ): Ji 5 2

RTLi d ln ai dφi dφ  Di i φi d ln φi dy dy

ð1:80Þ

Accordingly, Γ5

d ln ai φi dai  d ln φi ai dy

ð1:81Þ

Note, if we assume that ai 5 γ ixi [Eq. (1.3)], it easily can be understood that (Wesselingh and Krishna, 2000): Γ 5 1 1 xi

dðln xi Þ dxi

ð1:82Þ

Note that Eq. (1.80) defines the mass transfer rate for constant H partition (solubility) coefficient. The solubility coefficient is determined by the fact that the chemical potential of the continuous fluid and the membrane phases have to be equal in equilibrium [Eq. (1.30)]. If we want to estimate the solubility coefficient, the value of chemical potential as a function of the concentration both in the fluid and membrane phases have to be predicted. For the latter, several methods can be used. The most-often recommended approach for it is the FloryHuggins theory briefly shown in Section 1.6.

1.5.2

The MaxwellStefan Approach to Mass Transfer in a Ceramic (Zeolite) Membrane

Aluminosilicate (e.g., zeolites, silicalite) and other inorganic membrane layers have high chemical and thermal stability. They operate as molecular sieves and

20

Basic Equations of the Mass Transport through a Membrane Layer

have very promising industrial potential as material for separation (Gardner et al., 2002). They separate molecules based on differences in adsorptive and diffusive properties of the membrane. Mostly the Langmuir adsorption isotherms and the MaxwellStefan diffusion model are applied to describe the transport through an aluminosilicate membrane (Kapteijn et al., 1995; van de Graaf et al., 1999; Krishna and Baur, 2003). The diffusion in ceramic (mostly zeolite) membrane takes place in narrow channels both in the bulk channel phase and on the channel surface. The general form of the generalized MaxwellStefan equation applied to surface diffusion is given as (Wesselingh and Krishna, 1990; Krishna et al., 1999; van de Graaf et al., 1999): 2

n X θi dμi θ j Ji 2 θ i Jj Ji 5 1 ; RT dy ρθ D ρθ max ij max Di j51

i 5 1; 2; . . . ; n

ð1:83Þ

j 6¼ i

where θi is the molecular loading within the zeolite, expressed in molecules per unit cell; θmax is the maximum molecular loading; θmax 5 θA 1 θB for twocomponent diffusion; ρ represents here the number of unit cells per m3 of membrane; D1 is the diffusivity of species i in the membrane (m2/s) (note that D1  D1m Þ: The often-used form of the MaxwellStefan equation is given as (Kapteijn et al., 2000): 2ρ

n θi dμi X qj J i 2 qi J j Ji 5 1 ; RT dy q q D q i;sat Di j 5 1 i;sat j;sat ij

i 5 1; 2; . . . ; n

ð1:84Þ

j 6¼ i

where θi is the fractional occupancy defined as θi 5 qi/qi,sat; qi is adsorbed concentration in zeolite (cell or kmol/kg), qi,sat is saturation loading (cell or kmol/kg), ρ is the density of membrane, number of unit cell per m3 (kg/m3). The gradient of the thermodynamic potential can be expressed in terms of thermodynamic factors as it was shown at beginning of this section (Krishna, 1993; Kapteijn et al., 1995; Martinek et al., 2006) as n X θi rμi 5 Γij rθj ; RT j51

Γij 5

θi @pi ; pi @θj

i; j 5 1; 2; . . . ; n

ð1:85Þ

where pi is the partial pressure of species i (Pa). The combination of Eqs (1.84) and (1.85) gives the surface flux of a component through the membrane in multicomponent system. Thus, the mass transfer rates can be given in matrix form as ðJÞ 5 2ρ½qsat ½B 21 ½ΓðrθÞ

ð1:86Þ

On Mass Transport Through a Membrane Layer

21

The value of [B]21 matrix can be obtained by inversion of the [B] matrix as j5n X 1 θj 1 ; D Di ij j51

Bii 5

Bij 5 2

θi ; i 6¼ j Dij

j 6¼ i

The surface occupancy, θi, is related to the partial pressure by the adsorption isotherm. The choice of the adsorption model determines the mathematical form of the thermodynamic factor obtaining according to sense from Eq. (1.79). Applying the extended single-site Langmuir equation, the fractional occupation is as θi 5

11

bp Pin i

j51

bj pj

;

i; j 5 1; 2; . . . ; n

ð1:87Þ

Note that the dual-site (Krishna et al., 1999) or triple-site (Krishna et al., 2007) Langmuir models take into account the preferential adsorption or location of molecules at certain sites in the membrane structure. The sorption characteristics of the different sites can differ from each other (Krishna et al., 1999). The Langmuir isotherm can be given for single-component adsorption as θ1 5

b1 p 1 1 b1 p

ð1:88Þ

Applying Eq. (1.85), one can get Γ5

1 1 2 θ1

ð1:89Þ

Thus, the flux through the membrane for single-component adsorption is given by (Kapteijn et al., 1995; van de Graaf et al., 1999): Ji 5 qi;sat ρ

Di dθi 1 2 θi dy

ð1:90Þ

For a single-site, extended Langmuir isotherm, the thermodynamic correction factors for a binary mixture are (i 5 1; j 5 2): Γ11 5

1 2 θ2 ; 1 2 θ1 2 θ2

Γ11 5

θ1 1 2 θ1 2 θ2

Γ21 5

θ2 ; 1 2 θ1 2 θ2

Γ22 5

1 2 θ1 1 2 θ1 2 θ2

ð1:91Þ

22

Basic Equations of the Mass Transport through a Membrane Layer

The Γ factor gives much more complex equation, e.g., for dual-site adsorption isotherm (Krishna et al., 1999). Combining Eqs (1.84)(1.90) and assuming D12 5 D21 and solving the fluxes yield (Kapteijn et al., 2000; Martinek et al., 2006): J1 5 q1;sat ρD1

    Γ11 1 θ1 ðD2 =D12 ÞðΓ11 1 Γ21 Þ ðdθ1 =dyÞ 1 Γ12 1 θ1 ðD2 =D12 ÞðΓ12 1 Γ22 Þ ðdθ2 =dyÞ Ξ

ð1:92Þ

J2 5 q2;sat ρD2

    Γ22 1 θ2 ðD1 =D12 ÞðΓ22 1 Γ12 Þ ðdθ2 =dyÞ 1 Γ21 1 θ2 ðD1 =D12 ÞðΓ21 1 Γ11 Þ ðdθ1 =dyÞ Ξ

ð1:93Þ

where Ξ 5 1 1 θ2

D1 D2 1 θ1 D12 D12

Substituting Γ values from Eq. (1.47) into Eq. (1.48), relatively simple expression of the fluxes, J1 and J2, will be obtained (Kapteijn et al., 1995; van de Graaf et al., 1999; Nagy, 2004), taking also into account the role of the porosity of the support layer, ε, as:   dθ1 dθ2 1 A12 ð1:94Þ J1 5 2 A11 dy dy with   D2 Q; A11 5 q1;sat ερD1 1 2 θ2 1 θ1 D12 and

  D2 A12 5 q1;sat ερD1 θ1 1 θ1 Q D12

  dθ1 dθ2 1 A21 J2 5 2 A22 dy dy

ð1:95Þ

with   D1 Q; A21 5 q2;sat ερD2 θ2 1 θ2 D12

  D1 A22 5 q2;sat ερD2 1 2 θ1 1 θ2 Q D12

where Q5

1   ð1 2 θ1 2 θ2 Þ 1 1 θ1 ðD2 =D12 Þ 1 θ2 ðD1 =D12 Þ

ð1:96Þ

On Mass Transport Through a Membrane Layer

23

An important simplification is that there is no interaction between the components, that is, the terms with the D12 vanish (numerically D12 -NÞ: The flux expression for component 1 becomes  J1 2 ρεD1 ½Γ11 rθ1 1 Γ12 θ1  5 2 ρD1

  

θ1 @p1 θ2 @p1 rθ1 1 rθ2 p1 @θ1 p1 @θ2

ð1:97Þ

For application of the above equation in order to predict the binary gas separation by zeolite membrane, see Section 3.2.5.1. A single-component adsorption also can be described by dual-site Langmuir model (Zhu et al., 1998; Krishna et al., 1999; Kapteijn et al., 2000) as qi 5

θiA biA pi θiB biB pi biA pi biB pi 1  qiA;sat 1 qiB;sat 1 1 biA pi 1 1 biB pi 1 1 biA pi 1 1 biB;sat pi

qi;sat 5 qiA;sat 1 qiB;sat

ð1:98Þ ð1:99Þ

where the different sites A and B may be represented by channel interiors (e.g., site A) and the intersections (site B). For a binary mixture (components 1 and 2), the dual-site Langmuir isotherm can be given as (Krishna et al., 1999): θ1 5

ðθ1A b1A 1 θ1B b1B Þp1 1ðθ1A 1 θ1B Þb1A b1B p21 1 1ðb1A 1 b1B Þp1 1 b2A b2B p21 1ðb2A 1 b2B Þp2 1 b2A b2B p22

ð1:100Þ

The Γ thermodynamic factor can be obtained by means of Eq. (1.44). For it, the pi 5 f(θi) function should be expressed, then it is relatively easy to get @p/@θ differential quotient. For details regarding the Γ values, see Krishna et al. (1999). According to paper of Krishna and Baur (2003), the thermodynamic correction factor for the dual-site Langmuir model is Γ5

Θ1A Θ1



1  1 2ðΘ1A =Θ1A;sat Þ 1

Θ1B Θ1



 1 2ðΘ1B =Θ1B;sat Þ

ð1:101Þ

with θ1 

Θ1 q1 5 Θ1;sat q1;sat

The interchange coefficient, D12 ; is not measurable. Its value can be predicted by means of measured data, as it was done for polymeric mass transport by Heintz and Stephan (1994a,b); Iza´k et al. (2003); and Nagy (2006); or it can be calculated as a logarithmic average of the single-component MaxwellStefan diffusivities, as

24

Basic Equations of the Mass Transport through a Membrane Layer

it is recommended by Krishna and Baur (2003) for mass transport through a zeolite membrane layer, as D12 5 D1 θ1 =ðθ1 1 θ2 Þ D2 θ1 =ðθ1 1 θ2 Þ

ð1:102Þ

When the saturation loading of the two components differs, and the singlecomponent MaxwellStefan diffusivity is occupancy dependent, Skoulidas et al. (2003) suggest a modified formula in which D12 6¼ D21 (this is not shown here). Equation (1.102) is based on the empirical approach of Vignes (1966) for describing the concentration dependence of the diffusivity in a binary liquid mixture, using as inputs the infinite dilution diffusivity values. Note that Eq. (1.102) is similar to the modified Vignes equation published by Bitter (1991), which defines the real diffusion coefficient of components in a (polymeric) membrane, but they must not be confused [see Eq. (3.111)]. Recent experiments show that the MaxwellStefan diffusion coefficient (Di ; note that Di  Dim Þ of gases as CO2, CO, O2, CH4, and so forth in zeolites that consist of cages separated by narrow windows, such as e.g. CHA chabasite), DDR (decadodecasil R), LTA (Linde Type A), ERI (erionite), MFI mordenite framework inverted), can be strongly loading dependent, especially at high pressure, i.e. at high loading (Krishna et al., 2007; Li et al., 2007a,b). To quantify the loading dependence, the model developed by Reed and Ehrlich (1981) is the mostoften applied one. This model gives the following expression: Di 5 Di ð0Þ

ð1 1 εi Þz 21 ð1 1 εi =φi Þ2

ð1:103Þ

where z is the coordination number, representing the maximum number of neighbors within a cage; φi is the ReedEhrlich parameter, dimensionless; Di ð0Þ is the zero-loading MaxwellStefan diffusivity of species i (m2/s); the other parameters are defined as [see Krishna et al. (2007) for more details]: εi 5

ðβ i 21 1 2θi Þφi ; 2ð1 2 θi Þ

βi 5

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 4θi ð1 2 θi Þð1 2 1=φi Þ;

θi 5

qi qi;sat

Results obtained by the above model can be found in the papers of Li et al. (2007a,b) and Krishna et al. (2007).

1.5.3

The MaxwellStefan Approach for Mass Transfer in Porous Media

There are several excellent reviews available analyzing the mass transfer in a porous layer (Mason and Malinauskas, 1983; Wesselingh and Krishna, 1997). It is widely accepted (Thomas et al., 2001) that the mass transfer in a large range of pore sizes can be quantified using the dusty gas model (Mason and Malinauskas, 1983). This model takes into account molecular and Knudsen diffusion, as well as the contributions of viscous flux. However, in the region

On Mass Transport Through a Membrane Layer

25

of micropores, other effects began to dominate and different models have been suggested. One of the most advanced concepts is based on the assumption of activated configuration diffusion in the small pores [see Eq. (1.58)]. Accordingly, the molar flux densities Ji for all components are as (Thomas et al., 2001; Pedernera et al. 2002): 2

  n X p xi Beo xj Ji 2 xi Jj Ji p rp 5 rxi 2 1 11 e ^ ^ DK;i μ Dij DK;i RT RT j51

ð1:104Þ

j 6¼ i

where p is the total pressure (Pa); Beo is effective permeability constant (m2); μ is the viscosity (Pa s); Ji is molar flux density (kmol/m2 s); DK,i and Deij are as sffiffiffiffiffiffiffiffiffi ^ 8RT ; DK;i 5 Ko πMi

Deij 5

ε Dij τ

ð1:105aÞ

with [for the values of Ko and Beo , see Eqs (1.47) and (1.40)]: Ko 5

2 ε rp ; 3 τ

Beo 5

1 2ε r 8 pτ

ð1:105bÞ

Generally, three gas phase transport mechanisms should be considered by the above model (see 1.4 for details): 1. Molecular diffusion 2. Knudsen diffusion 3. Viscous flux

For a single gas permeation through a membrane, the following flux can be obtained from Eq. (1.104) (Thomas et al., 2001): 0

1 sffiffiffiffiffiffiffiffiffi e ^ 1 @ 8RT Bo A p rp 1 Ji 5 2 Ko ^ μ πMi RT

ð1:106Þ

Eq. (1.106) is in harmony with Eq. (1.50).

1.6

FloryHuggins Theory for Prediction of the Activity

The FloryHuggins equation deals with molecules that are similar chemically, but differ greatly in length. The model is based on the idea that the chain elements arrange themselves randomly on a three-dimensional structure. The resulting equation for the activity of the solvent is a simple proportional function of the volume fraction of the solvent. Note that the volume fraction is

26

Basic Equations of the Mass Transport through a Membrane Layer

denoted here by ε to distinguish it from the φ concentration [εi 5 φiVi/Mi, where Mi is molar weight (kg/kmol); Vi is molar volume of i (kmol/m3); and εi is the volume fraction (m3/m3)]. The activity of a component in the membrane can be described according to FloryHuggins thermodynamics (Flory, 1963; Lue et al., 2004) by   Vi 1 χim ð1 2 εi Þ2 ln ai 5 ln εi 1ð1 2 εi Þ 1 2 Vm

ð1:107Þ

where χim is an interaction parameter between the component i and the membrane called the FloryHuggins interaction parameter; Vi and Vm are molar volumes of solvent and membrane, respectively (kmol/m3). The χ interaction parameter is a dimensionless quantity characterizing the difference in interaction energy of a solvent molecule immersed in pure polymer compared with one in pure solvent. Its value can be positive or negative. If χ . 0, then the solvent and polymer “dislike” each other; if χ 5 0, then the solvent and polymer are similar; and if χ , 0, the solvent and polymer attract each other (Wesselingh and Krishna, 2000). After differentiating Eq. (1.107), it can be shown as   d ln ai 1 Vi 22χim ð1 2 εi Þ 5 1 12 dεi Vm εi

ð1:108Þ

The mass transfer rate can be given according to Eqs (1.9a) and (1.9b) as [ε Pi 5 φiρ/ρi, that is, dφi 5 ρidεi/ρ, where ρ is the membrane’s density, ρ 5 nj5 1 ρi φi (kg/m3)]: Ji 5 DTi φi

     d ln ai dφi ρi 1 Vi ρ dφi T 2 2χim 1 2 φi 5 Di φi 1 12 dφi dy ρ φi Vm dy ρi ð1:109Þ

where Ji is here the mass transfer rate (kg/m2 s).  Integration of Eq. (1.109) over the membrane layer (with φ 5 φ and φ 5 φδ at y=0 and y=δ, respectively), the mass transfer rate can be: DT Ji 5 i δ

(

! !)   2 3 3 φi 2 φ2 ρi  Vi ρ φi 2 φδ;i δ;i  ðφ 2φδ;i Þ 1 1 2 22χim 1 2χim ρ i Vm 2 3 ρi ð1:110Þ

In the case of polymeric membrane and a binary liquid mixture, a ternary system (solvent components and membrane matrix), the activity ai and aj of liquid

On Mass Transport Through a Membrane Layer

27

components i and j in the polymeric membrane are given by (Mulder and Smolders, 1984): Vi Vi 2 εm 1ðχij εj 1 χim εm Þðεj 1 εm Þ Vj Vm @χij Vi 2χjm εj ε 1 u1 u2 ε1 Vj @uj

ln ai 5 ln εi 1ð1 2 εi Þ 2 εj

ð1:111Þ

  Vj Vj Vj ln aj 5 ln εj 1ð1 2 εj Þ 2 εi 2 εm 1 χij εi 1 χjm εm ðεi 1 εm Þ Vi Vm Vi Vj Vj 2 @χij 2χim εi εm 1 u1 εj Vi Vi @u2 ð1:112Þ with ui 5

εi ; εi 1 εj

uj 5

εj εi 1 εj

where subscripts i, j, and m denote solvent components and membrane, respectively; χim and χjm are interaction parameters between components and membrane. Substituting Eq. (1.111) into Eqs (1.9a) and (1.9b), one can get the mass transfer rate for component i as

Ji 52 φi DTi ðφi ;φj Þ

9 > > > =

8 Vi Vi > ln εi 1ð12 εi Þ 2 εj 2 εm > > < V V

j m d dφi @χ V > > i ij dy dφi > 1ðχ ε 1 χ ε Þðε 1 ε Þ2 χ > εj εm 2 ui uj εj > > j m ij j im m jm : V @u ; j

j

ð1:113Þ

For the sake of completeness, the differential quotient of Eqs (1.111) and (1.112) will here be given for separation of binary mixture. Note that ε1 1 ε2 1 εm 5 1 for this system (Mulder, 1984; Meuleman et al., 1999): @ ln a1 1 V1 V1 5 21 1 2 χ12 ε2 2 χ1m εm 1χ2m ε2 @ε1 Vm V2 ε1   @χ12 ðu2 Þ @χ @2 χ12 1 u22 ðu1 2 u2 Þ 12 1 u1 u32 1ðε2 1 εm Þ 2χ1m 1 ε2 @ε1 @u2 @u22 ð1:114Þ

28

Basic Equations of the Mass Transport through a Membrane Layer

@ ln a1 V1 V1 @χ 52 1 1 χ12 ð12 ε1 Þ 1χ1m ðε1 21Þ1ð12 ε1 Þε2 12 @ε2 V2 Vm @ε2 V1 @χ @2 χ12 1 χ2m ðε1 1 2ε2 21Þ2ðu21 u2 2u1 u22 1 u1 u2 Þ 12 2 u21 u22 V2 @u2 @u22 ð1:115Þ

@ ln a2 V2 V2 V2 V2 @χ12 52 1 1 χij ð1 2ε2 Þ 2χ12 ð1 2ε2 Þ 1ð12 ε2 Þε1 @ε1 V1 Vm V1 V1 @ε1   2 V2 V2 2 @χ12 2 2 @ χ12 2 χ1m ð12 2ε1 2 ε2 Þ 1 22u1 u2 2 u1 u 2 V1 V1 @u2 @u22 ð1:116Þ

@ ln a2 1 V2 V2 V2 @χ12 5 21 1 2 χ12 ε1 2χ2m ð12 ε1 2 ε2 Þ 1ð1 2 ε2 Þε1 @ε2 Vm V1 V1 @ε2 ε2 V2 2 V2 @χ12 2ð1 2ε2 Þχ2m 1ε1 χ1m 22u2 u1 V1 V1 @u2 2 @ χ12 V2 2 @χ12 1 u1 u32 1 u1 V1 @u2 @u22 ð1:117Þ

1.6.1

MaxwellStefan Equation with the FloryHuggins Theory

Let us start with the general expression of the MaxwellStefan equation that can be applied for a multicomponent mixture, e.g. in the dusty gas model (Krishna and Wesselingh, 1997; Schaetzel et al., 2001), namely:

2



n X 1 dμi cj Ji Jj 5 2 ; c D ci cj RT dy j 5 1 t ij j 6¼ i

where

ct 5

j5n X

cj

ð1:118Þ

j51

where n is the number of species in the mixture; Ji is the average diffusive molar flux of species i (kmol/m2 s); ci is the molarity of species i (kmol/m3); Dij are generalizations of MaxwellStefan diffusivities (m2/s); μi is the chemical potential of species i (J/mol or kJ/kmol). Adapting Eq. (1.118) to a polymer (m) and single solvent (i) system, the MaxwellStefan equation (1.118)

On Mass Transport Through a Membrane Layer

29

becomes (Schaetzel et al., 2001) as (Jm 5 0; ct 5 cm 1 ci; Mmcm 5 φm; Mici 5 φi; ct M 5 ρÞ: d ln ai cm Ji φm M Ji 5  dy ct Dim ci Mm ρDim φi

ð1:119Þ

where Mm is the molar weight of membrane (kg/kmol); M is average molar weight of the system (e.g., M 5 ðcm Mm 1 ci Mi Þ=ρÞ; ρ is the density of the membrane/component system (it is assumed that its value is constant, kg/m3); φi is the concentration of diffusing species (kg/m3); Ji is the mass transfer rate (kg/m2 s). The relationship between φi and φm is Vi Vm φi 1 φ 51 Mi Mm m

ð1:120Þ

Inserting Eq. (1.108) and Eq. (1.120) into Eq. (1.109) and taking into account that d ln ai /dy 5 (d ln ai /dφi)dφi /dy, the following differential expression can be obtained: Ji 5 Dim

     ρ φi ρi 1 Vi ρ dφi 22χim 1 2 φi 1 12 Vm dy ρi M ð1 2 Vi φi =Mi Þ ρ φi ð1:121Þ

Let us integrate Eq. (1.121) term by term in order to check the solution. Thus, one can get 0 1 0 1 0 2 31 9φδ;i 8 > > > > ρ M V V M M V i i i i i i > i > > ln@12 φi A 1 @1 2 22χim A @ 2φi 1 ln41 2 φi 5A > > > > > = < ρ V M V V V M i i m i i i Dim ρ 0 1 Ji 52 * + > δ M> 2 > > ρ Mi Mi @ Vi φ > > > > > > 2φi 1 ln 1 2 φi A2 i 12χim > > ; : Vi Mi 2 ρi Vi φi

ð1:122Þ

Rearranging Eq. (1.122), it can be obtained as 9 8   = 1 2ðVi =Mi Þφi Dim ρ < ρ M i 2  1 Bðφi 2 φδ;i Þ 2 χim Ji 5 ðφ2 2 φ Þ A ln δ;i ; ρ i Vi i δM : 1 2ðVi =Mi Þφ δ;i

ð1:123Þ where 



 2  ρi Mi Vi ρ Mi A5 1 12 22χim 1 2 ρ Vi Vm ρi Vi2

ð1:124Þ

30

Basic Equations of the Mass Transport through a Membrane Layer



 Vi Mi ρ B52 12 22χim 12 Vm Vi ρi

ð1:125Þ

It was assumed during the solution of Eq. (1.119) or (1.121) that the value of ct ðct 5 MρÞ should be constant and consequently, the values of M; ρ should also be constant. In reality, according to Eq. (1.120):   Vi Mm φm 5 1 2 φi Mi Vm

ð1:126Þ

Accordingly, Eq. (1.121) can be rewritten as D  E  1 2 VMi φi i Mm =Vm 1 φi φi d ln a i

 Ji 5 2 Dim  V i φi  dy 1 2 Mi Mm =Vm

ð1:127Þ

Inserting Eq. (1.108) into Eq. (1.127), the mass transfer rate can be given by the following differential equation:  Ji 52Dim

ρi 1 φi φ2i φ3i dφi 1A 1B 2C ρ 12ðVi φi =Mi Þ 12ðVi φi =Mi Þ 12ðVi φi =Mi Þ 12ðVi φi =Mi Þ dy ð1:128Þ

where A512

  Vi Vi Vm ρ i 22χim 2 2 Mi Mi Mm ρ

   ρ Vi Vm Vi 12 B 5 2χim 2 2 22χim Mi Mm Mi ρi C 5 2χim

ρ ρi

Solving Eq. (1.128), one can get a rather complex expression, namely: Dim Ji 5 δ

( *

)  2 +   Mi ρi Mi Mi 12ðVi =Mi Þφi   2 A1B2C 1 A 1 B 2C ln 1T ðφi 2 φδ;i Þ 1 Vi ρ Vi Vi 12ðVi =Mi Þφδ;i

ð1:129Þ

On Mass Transport Through a Membrane Layer

31

where ! !    2 3 2 2 φi 2 φ3 Mi Mi φi 2 φδ;i Mi δ;i T 52 B2C 1C Vi Vi 2 Vi 3

1.7

UNIQUAC Model

According to the well-known, solution-diffusion model (Paul, 1976), sorption is usually considered one of the key steps for mass transfer through dense polymeric membranes. Often recommended model is the UNIQUAC (short of UNIversal QUAsiChemical) model, originally proposed by Abrams and Prausnitz (1975), for the prediction of the nonideal solubilities of mixtures in a dense active layer (Heintz and Stephan, 1994a,b; Janquie´res et al., 2000). UNIQUAC is the initial approach for predicting the activity of the solvent components in the membrane from which several other models have been developed as UNIQUAC-HB (Janquie´res et al., 2000). The detailed discussion of these models is not a topic of this material, only the UNIQUAC approach, originally proposed by Abrams and Prausnitz (1975), will be shown briefly. This model accounts for the different sizes and shapes of the molecules, as well as for the different intermolecular interactions between the mixture components, including polymeric components. The UNIQUAC model requires only binary interaction parameters for the description of multicomponent mixtures. The thermodynamic activity ai of a component i dissolved in a polymer membrane material, in presence of n solvent components, is given by (Heintz and Stephan, 1994a,b; Janquie´res et al., 2000):    n X Z θi ri Z qm 12 εj lj 2 ri εm 21 ln ai 5 ln εi 1 qi ln 1 li 2 εi rj rm 2 2 j51 ð1:130Þ

j 6¼ m

1 qi 2 qi ln

n X j51

!

θj χij

2 qi

n X j51

θj χij k 5 1 θk χkj

Pn

with li 5

Z ðri 2 qi Þ 2ðri 21Þ 2

ð1:131Þ

where εi is volume fraction of i, θi is the surface fraction of component i, χij, τ ji are binary interaction parameters; Z is the coordination number; ri and qi are dimensionless parameters for the relative molecular size and surface of component i related to the size and surface of a CH2 segment in polymer membrane,

32

Basic Equations of the Mass Transport through a Membrane Layer

respectively; n is number of components including the membrane as well. Altogether, n(n 21) χ parameters are needed for describing an n-component mixture with the UNIQUAC model. The volume fraction can be calculated by φ =ρ εi 5 Pn i i j 5 1 φj =ρj

ð1:132Þ

The UNIQUAC model gives excellent results in the case of polar liquid mixtures and hydrophilic membranes (Heintz and Stephan, 1994a,b).

References Abrams, D.S., and Prausnitz, J.M. (1975) Statistical thermodynamics of liquid mixtures: a new conception for the excess gibbs energy of partly or completely miscible mixtures. Am. Inst. Chem. 21, 116128. Amundsen, N.R., Pan, T.-W., and Paulsen, V.I. (2003) Diffusing with Stefan and Maxwell. AIChE J. 49, 813830. Baker, R.W. (2004) Membrane Technology and Application, 2nd ed. John Wiley and Sons, Chichester. Bird, R.B., Stewart, W.E., and Lightfoot, E.N. (1960) Transport Phenomena. John Wiley and Sons, New York. Bitter, J.G.A. (1991) Transport Mechanisms in Membrane Separation Processes. Shell-Laboratorium, Amsterdam. Flory, P. (1963) Principles of Polymer Chemistry. Cornell University Press, New York. Gardner, T.Q., Falconer, J.L., and Noble, R.D. (2002) Adsorption and diffusion properties of zeolite membranes by transient permeation. Desalination 149, 435440. Geankoplis, C.J. (2003) Transport Processes and Separation Process Principles, 4th ed. Prentice Hall, New Jersey. Geraldes, V., and Brites, A.M. (2008) Computer program for simulation of mass transport in nanofiltration membranes. J. Membr. Sci. 321, 172182. Heintz, A., and Stephan, W. (1994a) A generalized solution-diffusion model of the pervaporation process through composite membrane, Part I. Prediction of mixture solubilities in the dense active layer using the UNIQUAC model. J. Membr. Sci. 89, 143151. Heintz, A., and Stephan, W. (1994b) A generalized solution-diffusion model of the pervaporation process through composite membrane, Part II. Concentration polarization, coupled diffusion and the influence of the porous layer. J. Membr. Sci. 89, 153169. Hoang, D.L., Chan, S.H., and Ding, O.L. (2005) Kinetic modeling of partial oxidation of methane in an oxygen permeable membrane reactor. Chem. Eng. Res. Design 82, 177186. Iza´k, P., Bartovska´, L., Friess, K., Sipek, M., and Uchytil, P. (2003) Description of binary liquid mixtures transport through non-porous membrane by modified MaxwellStefan equations. J. Membr. Sci. 214, 293309. Janquie´res, A., Perrin, L., Arnold, S., Cle´ment, R., and Lochon, P. (2000) From binary to ternary systems: general behavior and modeling of membrane sorption in purely organic systems strongly deviating from ideality by UNIQUAC and related models. J. Membr. Sci. 174, 255275.

On Mass Transport Through a Membrane Layer

33

Kamaruddin, H.D., and Koros, W.J. (1997) Some observation about the application of Fick’s first law for membrane separation of multicomponent mixtures. J. Membr. Sci. 135, 147159. Kapteijn, F., Bakker, W.J.W., Zheng, G., and Poppe, J. (1995) Permeation and separation of light hydrocarbons through a silicalite-1 membrane. Application of the generalized MaxwellStefan equations. Chem. Eng. J. 57, 145153. Kapteijn, F., Moulijn, J.A., and Krishna, R. (2000) The generalized MaxwellStefan model for diffusion in zeolites: sorbate molecules with different saturation loadings. Chem. Eng. Sci. 55, 29232930. Krishna, R. (1990) Multicomponent surface diffusion of adsorbed species: a description based on the generalized MaxwellStefan equations. Chem. Eng. Sci. 45, 17791791. Krishna, R. (1993) A unified approach to the modeling of intraparticle diffusion in adsorption processes. Gas. Sep. Purif. 7, 91. Krishna, R., and Baur, R. (2003) Modeling issues in zeolite based separation processes. Sep. Purif. Technol. 33, 213254. Krishna, R., and Wesselingh, J.A. (1997) The Maxwell-Stefan approach to mass transfer. Chem. Eng. Sci. 52, 861911. Krishna, R., Vlugt, T.J., and Smit, B. (1999) Influence of isotherm inflection on diffusion in silicalite. Chem. Eng. Sci. 54, 17511757. Krishna, R., van Baten, J.M., Garcia-Perez, E., and Calero, S. (2007) Incorporating the loading dependence of the MaxwellStefan diffusivity in the modeling of CH4 and CO2 permeation across zeolite membranes. Ind. Eng. Chem. Res. 46, 29742986. Lawson, K.W., and Lloyd, D.R. (1997) Membrane distillation: review. J. Membr. Sci. 124, 125. Lonsdale, H.K. (1982) The growth of membrane and technology. J. Membr. Sci. 10, 81181. Li, S., Falconer, J.I., Noble, R.D., and Krishna, R. (2007a) Modeling permeation of CO2/CH4, CO2/N2 and N2/CH4 mixtures across SAPO-34 membrane with the MaxwellStefan equations. Ind. Eng. Chem. Res. 46, 39043911. Li, S., Falconer, J.I., Noble, R.D., and Krishna, R. (2007b) Interpreting unary, binary and ternary mixture permeation cross a SAPO-34 membrane with loading-dependent MaxwellStefan diffusivities. J. Phys. Chem. C 111, 50755082. Lue, S.J., Wang, F.J., and Hsiaw, S.-Y. (2004) Pervaporation of benzene/cyclohexane mixtures using ion-exchange membrane containing copper ions. J. Membr. Sci. 240, 149158. Martinek, J.G., Gardener, T.Q., Noble, R.D., and Falconer, J.L. (2006) Modelling transient permeation of binary mixtures through zeolite membrane. Ind. Eng. Chem. Res. 45, 60326043. Mason, E.A., and Malinauskas, A.P. (1983) Gas Transport in Porous Media: The Dusty Gas Model. Elsevier, Amsterdam. Meuleman, E.E.B., Bosch, B., Mulder, M.H.V., and Stratman, H. (1999) Modeling of liquid/ liquid separation by pervaporation: toluene from water. AIChE J. 45, 21532160. Mulder M. (1981) Pervaporation Separation of EthanolWater and Isomeric Xylenes. PhD Thesis, University of Twente, The Netherlands. Mulder, M.H.V., and Smolders, C.A. (1984) On the mechanism of separation of ethanol/ water mixtures by pervaporation I. Calculations of concentration profiles. J. Membr. Sci. 17, 289307. Nagy, E. (2004) Nonlinear, coupled mass transfer through a dense membrane. Desalination 163, 345354. Nagy, E. (2006) Binary, coupled mass transfer with variable diffusivity through cylindrical dense membrane. J. Membr. Sci. 274, 159168. Paul, D.R. (1976) The solution-diffusion model for swollen membranes. Sep. Purif. Methods 5, 35.

34

Basic Equations of the Mass Transport through a Membrane Layer

Pedernera, M., Alfonso, M.J., Mene´ndez, M., and Santamaria, J. (2002) Simulation of a catalytic membrane reactor for the oxidative dehydrogenation of butane. Chem. Eng. Sci. 57, 25312544. Reed, D., and Ehrlich, G. (1981) Surface diffusion, atomic jump rates and thermodynamics. Surf. Sci. 102, 588609. Schaetzel, P., Bendjama, Z., Vauclair, C., and Nguyen, Q.T. (2001) Ideal and non-ideal diffusion through polymers: application to pervaporation. J. Membr. Sci. 191, 95102. Seidel-Morgenstern, A. (2010) Membrane Reactors. Wiley-VCH, Weinheim. Silva, P., Peeva, L.G., and Livingston, A.G. (2008) Nanofiltration in organic solvents, in Advanced Membrane Technology and Applications, Ed. by, N.N., Li, A.G., Fane, W.S.W., Ho, and T., Matsuura. John Wiley and Sons, New Jersey, pp. 451468. Skoulidas, A.I., Sholl, D.S., and Krishna, R. (2003) Correlation effects in diffusion of CH4/CF4 mixtures in MFI zeolite. A study linking MD simulation with the MaxwellStefan formulation. Langmuir 19, 79777988. Thomas, S., Schafer, R., Caro, J., and Seidel-Morgenstern, A. (2001) Investigation of mass transfer through inorganic membrane with several layers. Catalysis Today 67, 205216. Tuchlenski, A., Uchytil, P., and Seidel-Morgenstern, A. (2001) An experimental study of combined gas phase and surface diffusion in porous glass. J. Membr. Sci. 140, 165184. Van de Graaf, J.M., Kapteijn, F., and Moulijn, J.A. (1999) Modeling permeation of binary mixtures through zeolite membranes. AIChE J. 45, 497511. Vignes, A. (1966) Diffusion in binary solutions. Ind. Eng. Chem. Fund. 5, 189199. Wesselingh, J.A., and Krishna, R. (1990) Mass Transfer. Ellis Horwood, Chichester, U.K. Wesselingh, J.A., and Krishna, R. (2000) Mass Transfer in Multicomponent Mixture. Delft University Press, Delft. Wijmans, J.G. (2004) The role of permeant molar volume in the solution-diffusion model transport equations. J. Membr. Sci. 237, 3950. Wijmans, J.G., and Baker, R.W. (1995) The solution-diffusion model: a review. J. Membr. Sci. 107, 121. Zhu, W., Graaf, J.M., Broeke, L.J.P., Kapteijn, F., and Moulijn, J.A. (1998) TEOM: a unique technique for measuring adsorption properties. Light alkanes in silicalite-1. Ind. Eng. Chem. Res. 37, 19341942.

2 Molecular Diffusion 2.1

Introduction

Molecular diffusion of molecular transport can be defined as the transfer or movement of individual molecules through a fluid by means of the random, Brownian, or thermal motion of the molecules. Molecules move at high speeds but travel extremely short distances before colliding with other molecules. The migration of individual molecules, therefore, is slow except at quite low molecular densities. Because the molecules travel in random paths, mainly due to the frequent collision with other moving molecules, molecular diffusion is often called a random-walk process (Geankoplis, 2003). The diffusion coefficient is often defined as the ratio of flux density to the negative of the concentration gradient in direction of diffusion, then according to Fick’s law: Ji 5 2Di

dφ dy

ð2:1Þ

where J is the transfer rate (kg/m2 s); φ is the concentration in the membrane (kg/m3); and y is the space coordinate in the direction of diffusion (m). The quantity, Di, is the diffusion coefficient, dφ/dy is the concentration gradient, and it can be regarded as the driving force for diffusion. But the real force can be obtained by means of thermodynamic consideration. The mean energy per mole in a uniform system is the partial molar Gibbs function, Gi ; or chemical potential, μi. The mean energy per molecule is μ=N; where N is Avogadro’s number. If energy were dissipated in moving a molecule down the chemical potential gradient, the driving force, Fi, per molecule of species, i, would be (Bungay et al., 1983): Fi 5 2

1 dμ N dy

ð2:2Þ

Assuming a velocity-dependent frictional resistance, fi, then the mean velocity, υi, from one plane to the next will be
 1 dμ ð2:3Þ υi 5 2 Nfi dy If the molar concentration of species i is φi, then the number of moles passing through unit area in unit time, Ji, will be Ji 5 υ i φ i Basic Equations of the Mass Transport through a Membrane Layer. DOI: 10.1016/B978-0-12-416025-5.00002-8 © 2012 Elsevier Inc. All rights reserved.

ð2:4Þ

36

Basic Equations of the Mass Transport through a Membrane Layer

For a nonideal medium, the chemical potential is as μi 5 μoi 1 RT ln ai

ð2:5Þ

Equation (2.5) involves the concentration effect; it does not contain the effect of temperature, pressure, or electrical field. Thus, Ji 5 2

RT d ln ai φi dy Nfi

ð2:6Þ

Ji 5 2

RT @ ln ai @φi RT @ ln ai @φi 2 φi @φ @y Nfi Nfi @ ln φi @y i

ð2:7Þ

or

Considering Fick’s law [Eq. (2.1)], one can get Di 5

RT @ ln ai RT φi @ai φi  @φi Nfi Nfi ai @φi

ð2:8aÞ

Di 5

RT @ ln ai Nfi @ ln φi

ð2:8bÞ

or

This latter equation can be rewritten as (ai 5 γ iφi):
 RT φi @γ i 11 Di 5 γ i @φi Nfi

ð2:8cÞ

For ideal solution, Ji 5 2 Di 5

RT d ln φi RT dφi 2 φi dy Nfi Nfi dy

RT Nfi

ð2:9Þ ð2:10aÞ

In the ideal solution, the quantity RT=Nfi is sometimes called the thermodynamic diffusion coefficient, DT. Taking into account Eq. (2.8b), the link between the Di and DT is as, see Eq. (1.12): DT 5 Di

@ ln ai @ ln φi

ð2:10bÞ

Molecular Diffusion

2.2

37

Gas Diffusivities

Gas diffusivities can be estimated quite well from the kinetic theory of gases. In the simplest form of this theory, molecules are hard spheres. In a simplified treatment, it is assumed that there are no attractive or repulsive forces between molecules. The derivation uses the mean free path λ, which is the average distance that a molecule has traveled between collisions. The final equation is DAB 5

1 uλ 3

ð2:11Þ

where u is the average velocity of the molecules. A more accurate and rigorous treatment must consider the intermolecular forces, attraction and repulsion between molecules as well as different sizes of molecules A and B. Chapman and Enskog (Hirschfelder et al., 1954) solved the Boltzmann equation, which uses a distribution function instead of the mean free path, λ. For a pair of nonpolar molecules, a reasonable approximation to the forces is the LennardJones function. The final relation for predicting the diffusivity of a binary pair of A and B molecules is
 1:8583 3 1027 T 3=2 1 1 1=2 1 ð2:12Þ DAB 5 MA MB Pσ2AB ΩD;AB where DAB is the diffusivity (m2/s); T is temperature (K); MA and MB are the molecular weights of components A and B (kg/kg mol); and P is absolute pressure (atm). The term σAB is an “average collision diameter”; ΩD,AB is a collision integral based on the LennardJones potential. The effect of concentration of components is not included in Eq. (2.12), but this effect can be neglected in most cases. Equation (2.12) is relatively complicated to use, and often some of the constants such as σAB are not available or are difficult to estimate. Hence, the semiempirical method of Fuller et al. (1966), which is much more convenient to use, often is preferred. The equation is
 1:0 3 1027 T 1:75 1 1 1=2 ð2:13Þ DAB 5 P  1=3 2 MA 1 MB 1=3 P P υA 1 υB P where υA 5 sum of structural volume increments, DAB is in m2/s. This method can be used for mixtures of nonpolar gases or for a polarnonpolar mixture.

2.3

Prediction of Diffusivities in Liquids

The equations for predicting diffusivities of dilute solutes in liquids are by necessity semiempirical, as the theory for diffusion in liquids is not well established as yet.

38

Basic Equations of the Mass Transport through a Membrane Layer

The StokesEinstein equation, one of the first theories, was derived for a very large spherical molecule (A) diffusing in a liquid solvent (B) of small molecules. Stokes’ law was used to describe the drag on the moving solute molecule. Accordingly, the diffusion coefficient (DAB, m2/s) can be estimated by DAB 5

kT 6πrs η

ð2:14Þ

where k is Boltzmann’s constant; rs is the radius of solute (m); and η is the solute dynamic viscosity (cP). The equation is a good approximation for large solutes with radii greater than ˚ (Baker, 2004). But, as the solute becomes smaller, the approximation of 510 A the solvent as a continuous fluid becomes less valid. Thus, the above equation was modified by assuming that all molecules are alike and arranged in a cubic lattice, and by expressing the molecular radius in therms of molar volume (Geankoplis, 2003): DAB 5

9:96 3 10216 T

ð2:15Þ

1=3

ηVA

where T is temperature (K); η is viscosity of solution (Pa s); and VA is the solute molar volume at its normal boiling point (m3 kg mol). The modified WilkeChang equation is as DAB 5 1:173 3 10216 ðψMB Þ1=2

T ηVA0:6

ð2:16Þ

where MB is the molecular weight of solvent B; η is viscosity (Pa s); ψ is an “association parameter” of the solvent, where ψ is 2.6 for water, 1.9 for methanol, 1.5 for ethanol, etc. (Geankoplis, 2003, p. 432).

2.4

Diffusion of an Electrolyte Solution

Electrolytes in aqueous solution dissociate into anions and cations. Each ion diffuses at a different rate. If the solution is to remain electrically neutral at each point, the cations and anions diffuse effectively as one component, and the ions have the same net flux. The well-known NernstHaskell equation for a dilute, single-salt solution can be used to predict the overall diffusivity DAB of the salt A in the solvent B (Geankoplis, 2003, p. 434): DAB 5 8:928 3 1010 T

ð1=n1 Þ 1ð1=n2 Þ ð1=λ1 Þ 1ð1=λ2 Þ

ð2:17Þ

Molecular Diffusion

39

where DAB is in cm2/s; n1 is the valence of the cation; n2 is the valence of the anion; λ1 and λ2 are the limiting ionic conductances in very dilute solutions ((A/cm2)(V/cm) and (g eq/cm2), respectively). The diffusion coefficient of an individual ion i at 25 C can be calculated by Di 5 2:662 3 1027

λi ni

ð2:18Þ

Substituting Eq. (2.9) into Eqs (2.8a)(2.8c), one can obtain DAB 5

2.5

n1 1 n2 ðn2 =D1 Þ 1ðn1 =D2 Þ

ð2:19Þ

Diffusion in a Membrane

2.5.1

Diffusion in a Dense Membrane

One can broadly classify transport in solids into two types of diffusion: diffusion that can be considered to follow Fick’s law and does not depend primarily on the actual structure of the solid, and diffusion in porous solids where the actual structure and void channels are important. This latter process is discussed in the next section. If the diffusion coefficient is independent of penetrant concentration, it can be determined by time lag or by equilibrium sorption measurements (Vieth et al., 1976). In the case of time lag measurements, flux through a membrane is measured as a function of time when pressure is applied to one side of the membrane and vacuum is pulled at the other. Extrapolation of the linear region of a plot of the steady-state flux versus time provides an intercept with the time axis called the time lag, θ, from which D is obtained as follows (Barrer, 1939): D5

δ 6θ

ð2:20Þ

where δ is the membrane thickness (m) and D is diffusion coefficient (m2/s). In the case of gas or vapor sorption by a membrane, D can be obtained from the ratio of the mass sorbed gas at time t(Mt) to the equilibrium sorption mass (MN) by the following relationship (Crank and Parks, 1968): Mt 4 5 1=2 MN π

rffiffiffiffiffi Dt δ2

ð2:21Þ

The diffusion coefficient can be obtained from the initial gradient of a plot of qffiffiffiffiffiffiffiffiffi Mt/MN versus t=δ2 :

40

Basic Equations of the Mass Transport through a Membrane Layer

As concentration dependency of the diffusion coefficient, the most often suggested function is the exponential and linear concentration dependency of the diffusion coefficient (Mulder, 1981; Qin and Cabral, 1998). Equations for these cases are discussed in detail in Chapters 3 and 11. In the two- or multicomponent separation, the coupling of the diffusion of the components occurs in most cases. The MaxwellStefan approach for coupling is discussed in Chapter 3 in cases of pervaporation and gas separation by application of zeolite membrane (Nagy, 2004). Another often recommended mass transport theory is the so-called FloryHuggins approach (Meuleman et al., 1999). This theory is especially applicable for organophilic pervaporation of organic compounds in water. The FloryHuggins theory that gives the chemical potential as a function of the components’ concentration, and consequently to give the mass flux through the membrane, is also recommended to describe the binary separation by pervaporation (Smart et al., 1998; Meuleman et al., 1999; Nagy, 2006). For polymers with high molecular weight, the activity in a membrane, according to the FloryHuggins approach, in a case of single-component transport, can be given as [see Eq. (1.107)]:
 Vi 1 χim ð12εi Þ2 ln ai 5 ln εi 1ð12εi Þ 12 Vm

ð2:22Þ

Expressing the value of d ln a/dε, the concentration dependence of the diffusion coefficient can be expressed [Eq. (1.108)]. For detailed analysis of the mass transport for unary and binary systems, see Chapter 1.

2.5.2

Diffusion in a Porous Membrane

Mass transfer mechanisms depend strongly on the pore size during membrane distillation; accordingly, different flux equations can be obtained. Basically, the following transfer rate can be defined depending on the pore size (Soni et al., 2009).

2.5.2.1 Knudsen-Limited Diffusion In membranes with small pores (r{λ; λ is the mean free path of the diffusing molecules), the molecule-pore wall collisions are dominant, therefore the Knudsen equation can used to describe the transport; the mass transfer rate can be given by Eq. (2.23). The flux is directly proportional to the difference of partial pressures. Ji 5 2DK;i

dci DK;i dpi 2 dy RT dy

ð2:23Þ

Molecular Diffusion

41

2.5.2.2 Knudsen-Viscous Transition Diffusion The mean free path of diffusing chemical component, λ, is similar to the membrane pore diameter, consequently the convective flow also can exist during the transport. The Knudsen-viscous transition equation is (see Chapter 1):
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi  8RTavg pi;avg 1 Ko ΔP Δpi 1 Bo Ji 5 πMi μ RTavg δ

ð2:24Þ

with Bo 5

r2 ε ; 8τ

Ko 5

1 dp 3

ð2:25Þ

where Δpi is the partial pressure difference of transported component on the two sides of the membrane (Pa); ΔP is the pressure difference (Pa); μ is the viscosity (kg/ms); and δ is the membrane thickness (m).

2.5.2.3 Knudsen-Molecular Diffusion For membranes with smaller air-filled pores (about , 0.5 μm; Soni et al., 2009), even the molecular diffusion also can affect the transport rate, thus both mechanisms should be taken into account. Accordingly: Ji 5 2

2.6



 1 1 1 21 dpi 1 1 21 dci 2 1 1 dy dy RT Do;i DK;i Do;i DK;i

ð2:26Þ

Transport with Convective Velocity Due to the Component Diffusion

Due to the diffusion, the movement of the whole phase also can be moved especially in the case of a high concentration of the diffusing component. The rate at which moles of A passed a fixed point (JAo kmol/m2 s) can be converted to a velocity of diffusion of A by o 5 υAd cA JAc

ð2:27Þ

where υAd is the diffusion velocity of A (m/s) and cA is the concentration of A (kmol/m3). Look at briefly the situation when the whole fluid is moving in bulk or convective velocity. The molar average velocity of the whole fluid relative to a stationary point is υm (m/s). The velocity of A relative to the stationary point

42

Basic Equations of the Mass Transport through a Membrane Layer

is the sum of the diffusion velocity and the average or convective velocity (Bird et al., 1960; Geankoplis, 2003): υA 5 υAd 1 υM

ð2:28Þ

Multiplying Eq. (2.28) by cA, it can be obtained as υA cA 5 υAd cA 1 υM cA

ð2:29Þ

The term υAcA can be represented by the flux JAo as follows: o o o JAo 5 JAd 1 υM cA  JAd 1 JAc

ð2:30Þ

o o where JAd is the diffusion flux relative to the moving fluid (kmol/m2 s) and JAc is 2 the convective flow of component A (kmol/m s). Let Jo be the total convective flux of the whole binary stream relative to the stationary point, then:

J o 5 υM c 5 JAo 1 JBo

ð2:31Þ

JAo 1 JBo c

ð2:32Þ

or υM 5

where c is the total concentration of the fluid (kmol/m3). Substituting Eq. (2.32) into Eq. (2.28), one can obtain o 1 JAo 5 JAd

cA o ðJ 1 JBo Þ c A

ð2:33Þ

Applying the known diffusive flow, the general equation for diffusion plus convection can be given as JAo 5 2DAB

dcA cA 1 ðJAo 1 JBo Þ dy c

ð2:34Þ

Equation (2.34) is valid for mass transport in solid phase as well assuming the stationary coordinate is fixed to the laboratory device or to the membrane module. Thus, for the mass transport through a membrane layer [see also Eqs (1.21a) and (1.21b)]: JAo 5 2DAB

dφA cA 1 ðJAo 1 JBo Þ dy c

ð2:35Þ

Molecular Diffusion

2.7

43

Ion Transport and Hindrance Factors

The extended NerstPlank equation for the membrane is as (Geraldes and Brites, 2008): Ji 2 Di;p

zi φ Di;p dψ dφ F 1 Ki;c Jv 2 i RT dy dy

ð2:36Þ

where Ji is the solute flux of component i (kg/m2 s); Jv is the permeate volume flux (m3/m2 s); Di,p is the hindered diffusion coefficient (m2/s); Ki,c is the hindrance coefficient for convection; zi is the charge number; F is the Faraday constant (9.64867 3 104 C/eq); R is the ideal gas constant (8.314 J/kmol); ψ is the electric potential (V); and Di,N is the diffusivity of the i species in water at infinite dilution (m2/s): Di;p 5 Ki;d Di;N

ð2:37Þ

Hindrance factors Ki,d, Ki,c are functions of λi and are related to hydrodynamic coefficients such as the enhanced drag and lag coefficients. According to Dechadilok and Deen (2006), for λi # 0.95: A22:81903λ4i 1 0:270788λ5i 21:10115λ6i 20:435933λ7i ð12λi Þ2

ð2:38Þ

A 5 1:01ð9=8Þλi ln λi 21:56034λi 1 0:528155λ2i 11:91521λ3i

ð2:39Þ

Ki;d 5 with

For λi . 0.95, the results of Mavrovouniotis and Brenner (1988) can be applied:
 12λi 5=2 ð2:40Þ Ki;d 5 0:984 λi For convection, the hindrance factor can be as: Ki;c 5

11 3:867λi 21:907λ2i 20:834λ3i 111:867λi 20:741λ2i

ð2:41Þ

or as given by Deen (1987): ^ i gð11 0:054λi 20:988λi 2 1 0:44λi 3 Þ Ki;c 5 f2 2 Φ where λi 5

rs;i rp

and

^ i 5 ½12λi 2 Φ

ð2:42Þ

44

Basic Equations of the Mass Transport through a Membrane Layer

References Baker, R.W. (2004) Membrane Technology and Application, 2nd ed. John Wiley and Sons, Chichester. Bird, B., Stewart, W.E., and Lightfoot, E.N. (1960) Transport Phenomena. John Wiley and Sons, New York. Bungay, P.M., Lonsdale, H.K., and de Pinho, M.N. (1983) Synthetic Membranes: Science, Engineering and Application, NATO ASI Series, D. Reidel Publishing Company, Vol 181. Barrer, R.M. (1939) Permeation, diffusion and solution of gases in organic polymer. Trans. Faraday Soc. 35, 629643. Crank, J., and Parks, G.S. (1968) Diffusion in Polymers. Academic Press, London. Dechadilok, P., and Deen, W. (2006) Hindrance factors for diffusion and convection in pores. Ind. Eng. Chem. Res. 45, 6953. Deen, W.M. (1987) Hindered transport of large molecules in liquid-filled pores. AIChE J. 33, 14091425. Fuller, E.N., Schettler, P.D., and Giddings, J.C. (1966) Ind. Eng. Chem. 58, 19. Geankoplis, Ch.J. (2003) Transport Process and Separation Process Principles, 4th ed. Prentice Hall, Upper Saddle River. Geraldes, V., and Brites, A.M. (2008) Computer program for simulation of mass transport in nanofiltration membranes. J. Membr. Sci. 321, 172182. Hirschfelder, J.Q., Curtiss, C.F., and Bird, R.B. (1954) Molecular Theory of Gases and Liquids. John Wiley and Sons, New York. Mavrovouniotis, G.M., and Brenner, H. (1988) Hindered sedimentation diffusion and dispersion coefficient for Brownian spheres in circular cylindrical pores. J. Colloid Interface Sci. 124, 269. Meuleman, E.E.B., Bosch, B., Mulder, M.H.V., and Strathmann, H. (1999) Modeling of liquid/liquid separation by pervaporation: toulene from water. AIChE J. 45, 21532160. Mulder, M. (1981). Pervaporation Separation of EthanolWater and Isomeric Xylenes. PhD Thesis, University of Twente, The Netherlands. Nagy, E. (2004) Nonlinear, coupled mass transfer through a dense membrane. Desalination 163, 345354. Nagy, E. (2006) Binary, coupled mass transfer with variable diffusivity through cylindrical membrane. J. Membr. Sci. 274, 159168. Qin, Y., and Cabral, J.M.S. (1998) Lumen mass transfer in hollow fiber membrane processes with nonlinear boundary conditions. AIChE J. 41, 836848. Smart, J., Starov, V.M., Schucker, R.C., and Lloyd, D.R. (1998) Pervaporative extraction of volatile organic compounds from aqueous systems with use of a tubular transverse flow module. Part II. Experimental results. J. Membr. Sci. 143, 159179. Soni, V., Abildskov, J., Jonsson, G., and Gani, R. (2009) A general model for membrane-based separation process. Computers Chem. Eng. 33, 644659. Vieth, W.R., Howel, J.M., and Hsieh, J.H. (1976) Dual sorption theory. J. Membr. Sci. 1, 177220.

3 Diffusion Through a Plane Membrane Layer

3.1

Introduction

In this chapter, we consider various cases of one-dimensional diffusion, perpendicular to the membrane surface, into a solid layer with thickness of δ bounded by two parallel plane surfaces. For a description of this process, the solution-diffusion model is applied. That means that the transported component is adsorbing on the solid interface and then it is transported by molecular diffusion through the membrane layer to its external surface, and from here (through an external mass transfer resistance) to the bulk permeate phase. Note that the concentrations are in equilibrium at the interface, between the continuous phase and membrane layer, and that this sorption process is an instantaneous process. Most of the membrane modules used for separation are capillary membrane (hollow-fiber) modules. The question arises, under which conditions the mass transport through a cylindrical membrane layer can be considered as a plane layer. The diffusional mass transfer equation through a cylindrical membrane is defined in Chapter 6. For limiting cases, this condition can be obtained as if δ=ro -0

then

Di Di Δφi - Δφi ro lnð11δ=ro Þ δ

ð3:1Þ

where ro is the internal radius of fibers, δ is the membrane thickness, Di is the diffusion coefficient of component i (m2/s), and φi is the concentration in the membrane (kg/m3). The selective (or skin) layer of an asymmetric membrane is very thin; its thickness falls generally in the range of 0.12 μm, while the radius of a capillary is about 100150 μm. Thus, the value of δ/ro equals about 1022, and consequently, the cylindrical effect can generally be neglected in this case. The support or spongy layer is much thicker; it is about 100 μm; thus in this case, the cylindrical effect must not be neglected. Besides these special cases, the mass transport through a membrane layer through plate-and-frame, tubular, and spiral-wound modules should be regarded as that through a plane interface.

Basic Equations of the Mass Transport through a Membrane Layer. DOI: 10.1016/B978-0-12-416025-5.00003-X © 2012 Elsevier Inc. All rights reserved.

46

Basic Equations of the Mass Transport through a Membrane Layer

3.2

Steady-State Diffusion

Let us consider the case of diffusion through a plane membrane of thickness δ and diffusion coefficient D, whose surfaces, y 5 0 and y 5 δ, are maintained at constant concentrations, Hco 5 φ and Hcoδ 5 φδ ; respectively (Figure 3.1). When the value of c . Cδ ; then the diffusion process occurs from one side of the membrane to the other side due to the driving force existing in the membrane layer. For a longer time, a steady state is reached in which the concentration in the membrane remains constant. The one-dimensional, steady-state diffusion equation in one dimension then can be given, in dimensionless variables [Φ 5 φ/φ  φ/(Hco), where c denotes the membrane concentration, co is the fluid phase’s concentration] with c H 5 φ , Y 5 y/δ as D

d2 φ 50 dY 2

ð3:2Þ

The membrane concentration, c, is given here in a unit of measure of kgmol/m3. This can be easily obtained by means of the usually applied g/g unit of measure with the equation of φ 5 wρ/M, where w is concentration (kg/kg); ρ is membrane density (kg/m3); and M is molar weight (kg/mol). Its dimensionless form can be given by Φ 5 φ/(coH), where H is the partition coefficient of the transported component in the membrane phase. After integration of Eq. (3.2) (its general integration is well known: Φ 5 TY 1 S) and using the boundary conditions without external mass transfer resistance (continuous line in the fluid phase boundary layer) in Figure 3.1, one can get the concentration as Y5

φ 2 φ φ 2 Hco Φ21 5  o   o o φδ 2 φ Hcδ 2 Hc Cδ 2 1

ð3:3Þ

Figure 3.1 Illustration of the concentration distribution and the important variables through mass transport through a membrane layer.

Membrane βo βofδ

βof Co

C* Φ* Φ*δ ,

C*δ Cδo

δf

δfδ δ

Diffusion Through a Plane Membrane Layer

47

According to Eq. (3.3), the concentration inside the membrane layer changes linearly from φ to φδ through the layer that is from Hco to Hcoδ or from H to HCδo : The mass transfer rate can be given as (note that the value of J here, in this chapter, denotes the specific mass transfer rate without chemical reaction, kg/m2s or kmol/m2s) J5 2

DHco dΦ D  D 5 ðφ 2 φδ Þ  Hðco 2 coδ Þ  β o Hco ð12Cδo Þ δ dY δ δ

ð3:4aÞ

Introducing the permeability constant P with P 5 DH, one can get from the mass transfer rate: J5

P o c ð12Cδo Þ δ

ð3:4bÞ

According to Eq. (3.4b), the permeability coefficient can be expressed here in units such as m2/s, because H is dimensionless due to its definition. Because different investigators can use different units, the permeability constant can have different units and even different definitions. Look at the situation when feed phase and also sweep fluid (permeate side; this is the case during dialysis or often during pervaporation) are flowing on the two sides of the membrane layer. Then a polarization layer can be formed in the fluid phases, which can essentially affect the mass transfer rate into and/or out of the membrane. This situation is illustrated by Figure 3.1 plotted by dotted line. The general solution of Eq. (3.1) is well known, φ 5 TY 1 S, where the constants T and S should be determined by means of the suitable boundary conditions. Taking into account the mass transfer resistance on both sides of the membrane, the boundary conditions can be given as at Y 5 0

then

β of ðco 2c Þ 5 2β o T

ð3:5Þ

at Y 5 1

then

β ofδ ðcδ 2coδ Þ 5 2β o T

ð3:6Þ

where β o 5 D/δ and β of 5 Df =δf as well as β ofδ 5 Dfδ =δfδ : After solution, the values of T and S can be expressed, for φ 5 TY1S, as T5

β oov o o ðc 2c Þ βo δ

ð3:7Þ

and S 5 Hcoδ 1

 β oov
o o o cδ 2 c βf

ð3:8Þ

where 1 1 1 1 o 5 o1 o1 o β ov β f Hβ β fδ

ð3:9Þ

48

Basic Equations of the Mass Transport through a Membrane Layer

The mass transfer rate can be given as: J 5 β oov ðco 2coδ Þ

ð3:10Þ

3.2.1 Concentration-Dependent Diffusion Coefficient It is clear that the classic solution-diffusion theory is only valid for governing permeation through essentially nonswollen membranes, for example, in the case of removal of dilute organics from water. When membranes are used for pervaporation dehydration, or organicorganic separation, appreciable membrane swelling usually occurs, and both the partition and diffusion coefficient become concentration dependent. Therefore, the classic solution-diffusion theory should be modified to adapt to the generally swollen pervaporation membranes. When a membrane is swollen or plasticized by transporting species, the interaction between polymer chains tends to be diminished, and the membrane matrix will therefore experience an increase in free volume. It is generally true that in a given membrane, increased free volumes correspond to increased diffusion coefficients of the penetrants. When a membrane is plasticized by more than one species, the diffusion coefficient of a species is facilitated by all the plasticizants. Many membranologists have found that the diffusion coefficient of species i in a ternary system of membrane/species i/species j could generally be expressed as (Schaetzel et al., 2004): Di 5 Dio expðα~ i φi 1 β~ j φj Þ

ð3:11aÞ

or in dimensionless form as: Di 5 Dio expðαi Φi 1β j Φj Þ

ð3:11bÞ

where Dio represents the diffusion coefficients of species i at infinite solution; φi and φj represent the local concentrations of the species of i and j in the membrane, respectively (kg/m3); and α~ and β~ are usually interpreted as the plasticization coefficients of the two species for the membrane (m3/kg). The plasticization coefficient of the less-permeable species can be neglected during dehydration processes, because dehydration membranes generally show overwhelming affinity for water, and the concentration of the less-permeable species in the membrane is negligibly small. The diffusion coefficients of both the species in the membrane are thus dependent on the concentration of water in the membrane phase alone. Thus, the diffusion coefficient of species i can be written as (Mulder, 1984): Di 5 Dio expðα~ i φi Þ

ð3:12Þ

Diffusion Through a Plane Membrane Layer

49

Some other relationships were also found to be adequate for depicting the concentration-dependent diffusion coefficient as Di 5 Dio ð11 α~ i φi 1 β~ j φj Þ Both concentration dependencies are discussed briefly in this section. If the D is concentration dependent, Eq. (3.1) is to be replaced by   d dφ D 50 dy dy

ð3:13Þ

Integrating between φ and φδ ; the two surface concentrations (Figure 3.1), one can get the mass transfer rate as 1 J 52 δ

φðδ

D dφ 5 φ



DH o ðc 2 coδ Þ δ

ð3:14Þ

where D5

1 ðφ 2 φδ Þ

φðδ

D dφ φ

ð3:15Þ



where D is the average value of the diffusion coefficient. It is important to note that the concentration depends on the concentration dependency of D, thus its value is not known, and thus, the real concentration distribution can be calculated by iteration, supposing a concentration distribution at beginning. Starting from this value, we should then calculate the value of Dm and then the concentration distribution. About three to four calculation steps are enough to get the correct concentration distribution by means of the real diffusion coefficients.

3.2.1.1 Exponential Concentration Dependency, ~  Do expðαΦÞ D 5 Do expðαφÞ Integrating Eq. (3.13) and applying Eq. (3.12), the mass transfer rate can be given ~  Þ: as ðα 5 αφ J 52

Do co H αΦ dΦ e δ dY

ð3:16Þ

After integration of Eq. (3.16), one can get as Φ5

  1 αδ ln ðS2JYÞ α Do

ð3:17Þ

50

Basic Equations of the Mass Transport through a Membrane Layer

where S and J parameters should be determined by means of the boundary conditions as at Y50, Φ51 and y51, Φ 5 Φδ : Accordingly, one can get for the concentration distribution as Φ5

 1  α ln e ð12 YÞ 1 YeαΦδ α

ð3:18Þ

The concentration distribution is plotted in Figure 3.2 as a function of the α exponent. The mass transfer rate can be given, applying the measurable concentra~ o Þ: ~   αHc tions, as ðα 5 αφ J5

Do Hco α o ðe 2 eαCδ Þ αδ

ð3:19Þ

If one takes into account the mass transfer resistances on both sides of the membrane, ~ o : it can be calculated applying the following equations ½Φ 5 φ=ðHco Þ; α 5 αHc J 5 β of ðco 2 c Þ

ð3:20Þ

J 5 β ofδ ðcδ 2 coδ Þ

ð3:21Þ

and J5

Do Hco αΦ αΦδ ðe 2e Þ αδ

ð3:22Þ

Taking into account the equality of the above equations, one can get as: J5

o o o Do Hco n αð12J=½βof co Þ o e 2 eαðJ=½βfδ c 1Cδ Þ αδ

ð3:23Þ

Figure 3.2 Concentration distribution with variable diffusion coefficient, D 5 Do eαΦ, without external mass transfer resistance.

Concentration distribution

1.0 α=10 0.8

5 3

0.6 1 0

0.4 0.2 0.0 0.0

0.2

0.4

0.6

Membrane layer Y

0.8

1.0

Diffusion Through a Plane Membrane Layer

51

By Eq. (3.23), the J value can be obtained by simple iterative manner. Knowing the value of J, the S value can be calculated (in order to predict the concentration distribution) by one of the boundary conditions, for example replacing the limiting case of Eq. (3.17), namely if Y 5 0, into Eq. (3.20), one can get as: J

5 β of co

   1 αδ 12 ln S α Do

ð3:24Þ

and from that becomes: S5

Do αð12J=½β of co Þ e δα

ð3:25Þ

Thus, the concentration distribution can be calculated in presence of mass transfer resistance in the feed and permeate phases by means of Eqs (3.17), (3.23), and (3.25). The average value of the diffusion coefficient can be calculated by Eq. (3.15) and thus, the mass transfer rate can be calculated by Eq. (3.14). Using this equation, the external mass transfer resistance can easily be taken into account by the following equation: J5

co ð1 2 Cδo Þ 1 δ 1 1 1 β of HD β ofδ

ð3:26Þ

The effect of the external mass coefficient is illustrated in Figure 3.3 on the mass transfer enhancement. It can be seen that the relatively large value of the external mass transfer resistance can strongly diminish the effect of the exponentially increasing diffusion coefficient.

Figure 3.3 The effect of the external mass transfer resistances on the mass transfer rate (β o 5 Do/δ; Jo denotes here the overall mass transfer rate with α 5 0 and J with α . 0, and β with α . 0; β of 5 β ofδ Þ:

3.0 Enhancement (J/J*)

o o βf /β →∞

2.5

10

5

2.0 1

1.5 1.0 0.1

0.1 1.0 Exponent (α)

10.0

52

Basic Equations of the Mass Transport through a Membrane Layer

~  Do ð1 1 αΦÞ 3.2.1.2 Linear Concentration Dependency, D 5 Do ð11 αφÞ 

The mass transfer rate can be given, in this case, as follows (Φ 5 φ/φ ): J 52

Do  dΦ φ ð11αΦÞ dY δ

ð3:27Þ

After solution of Eq. (3.13) with suitable boundary conditions, as were used for the case of exponential concentration dependence, the concentration distribution for the membrane layer can be obtained as (φ 5 Hco): φ 1 Φ   52 1 φ α

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2δ S2JY 1 2 α Do φ α

ð3:28Þ

with S5

Do o α Hc 11 δ 2

ð3:29Þ

J5

  o  αCδ Do o α c H 11 11 2Cδo δ 2 2

ð3:30Þ

and

Note that S and J values in Eqs (3.28)(3.30) have a dimension of kg/(m2 s), and α is dimensionless. Knowing the values of S and J, the concentration distribution can be calculated by means of Eq. (3.28). In order to take into account the external mass transfer resistances, the following algebraic equations system should be solved [Φ 5 φ /(Hco) where due to the external mass transfer resistance φ , Hco]:        Do o  αΦ  αΦδ c H Φ 11 2 Φδ 11 J5 δ 2 2

ð3:31Þ

with C 5 1 2

J β of co

Cδ 5 Cδo 1

J β ofδ co

ð3:32Þ

ð3:33Þ

Diffusion Through a Plane Membrane Layer

53

Replacing the C and Cδ (C 5 Φ ; Cδ 5 Φδ Þ from Eqs (3.32) and (3.33), respectively into Eq. (3.31), one gets a second-order algebraic equation to be solved. This becomes: J5

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  2B6 B2 24AE 2A

ð3:34Þ

with ! 1 1 α A5 2 o2 o2 βf β fδ 2co2      1 1 1 Cδo 1 1 1 α 1 1 1 B5 β ofδ β of β ofδ β of β o H co

ð3:35Þ ð3:36Þ

as well as E5

α ð12Cδo2 Þ 2 ð12Cδo Þ 2

ð3:37Þ

The effect of the external mass transfer resistances can be predicted by means of Eq. (3.34).

3.2.1.3 Optional Concentration Dependency of the Diffusion Coefficient The concentration dependency of the diffusion coefficient can be essentially different from the above-discussed function. Thus, a general solution of the problem will be shown in this section. The solution methodology will be discussed in this chapter. The membrane is divided into N sublayers with ΔY thickness, and the diffusion coefficient Di was assumed to be constant for every sublayer. See e.g. section A2 for the solution’s methodology. The concentration distribution for ith sublayer can be given as Φ 5 Ti Y 1 Si

at

Yi21 # Y # Yi

ð3:38Þ

where Ti 5 2

ð12 Cδo Þ H PN Di j 5 1 ðΔY=Dj Þ

ð3:39Þ

and Si 5 ΔY

i 21 X j51

Tj 2ði21ÞΔYTi 11

ð3:40Þ

54

Basic Equations of the Mass Transport through a Membrane Layer

The mass transfer rate can be given, for example, as co ð12Cδo Þ J 5 H PN δ j 5 1 ðΔX=Dj Þ

ð3:41Þ

As mentioned, the Di value depends on the concentration, thus the real concentration distribution and the mass transfer rate can be obtained by iterative method. The external mass transfer resistance can easily be taken into account in the mass transfer rate as J 5H

co ð12Cδo Þ PN o ðH=β f Þ1δ j51 ðΔY=Dj Þ1ðH=β of Þ

ð3:42Þ

3.2.2 Concentration-Dependent Solubility Coefficient, H 5 H(c) The second key factor determining permeability in a membrane is the sorption coefficient. The sorption coefficient can also vary as a function of the concentration, but this dependency is much lower than that of the diffusion coefficient (Chandak et al., 1998). The main types of this dependency are discussed briefly here, namely linear and the Langmuir-type absorption isotherms. Accordingly, assuming linear concentration dependency, one can get as ~ φ 5 Ho ð11αcÞc

ð3:43Þ

Thus in dimensionless form, it will be as (C 5 c/co): φ  Φ 5 ð11αCÞC Ho co

ð3:44Þ

or in case of Langmuir-type sorption property (φsat is saturation sorption concentration at which all excess free volume sites are filled, kg/m3): φ 5 φsat

~ αc ~ 11 αc

ð3:45aÞ

~ o ; Φ 5 φ=φsat Þ : or ðα 5 αc Φ5

αC 11αC

ð3:45bÞ

According to the dual-sorption model, gas sorption in polymer occurs in two types of sites (Baker, 2004; Follain et al., 2010; Koros, 1980; Vieth, 1988). The total sorption can be written as: φ 5 Ho c 1

~ φsat αc ~ 1 1 αc

ð3:46Þ

Diffusion Through a Plane Membrane Layer

55

According to the ENSIC (engaged species induced clustering) model (Favre et al. 1996; Shah et al., 2007), the activity of the solvent in the polymer is given by   1 A2B a 5 γc 5 log 11 φρ ð3:47Þ A2B B From Eq. (3.47), the φ(c) function can be expressed as φ5

B ðexp½γcðA2BÞ21Þ A2B

ð3:48Þ

where γ is the activity coefficient; ρ is the density of the swollen polymer membrane (kg/m3); and A and B are the parameters of the model. The ENSIC model is essentially an empirical model, but it can fit the sorption data for most polymer/solvent systems very well (Favre et al., 1996). Alternatively, a more thermodynamically rigorous model like the extended FloryHuggins model can also be used. According to the extended FloryHuggins model for a single component, the activity can be given as (Mulder, 1984) log a 5 logðφρÞ1ð12φρÞ1ðχ0 1χ1 φρÞφρ

ð3:49Þ

It can be obtained for φ(c) as φ 5 γc expð11½χ0 21φ1χ1 φ2 Þ

ð3:50Þ

where χ0 and χ1 are interaction parameters of the model. The role of the concentration dependency is especially important when the external mass transfer resistance cannot be neglected. In this case, the membrane’s interface concentration depends on the external mass transfer coefficients; thus, the H value changes according to the interface concentrations, namely to φ and φδ : Look briefly at how the concentration distribution can be obtained in the presence of limiting values of β of and β ofδ (see Figure 3.1, dotted line).

3.2.2.1 Linear Solubility Dependency The mass transfer rate, when the solubility coefficient changes according to Eq. (3.43), can be expressed as follows, without external mass transfer resistance, β of 5 β ofδ -N: J 5 β o co Ho ð11α 2 ½11αCδo Cδo Þ where βo 5

D δ

ð3:51Þ

56

Basic Equations of the Mass Transport through a Membrane Layer

A more important case is when the mass transfer resistances are not negligible. For that case, the mass transfer rate can be given by the following expression: 
 J 5 β o Hco ð1 1 αC  ÞC 2 1 1 αCδ Cδ ð3:52aÞ Replacing e.g. Eq. (3.32) and (3.33) into Eq. (3.52a) one can obtain a second order algebraic equation for J to be solved, namely: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2B6 B2 24AE ð3:52bÞ J5 2A with 1 1 A5 2 o2 o2 βf β fδ

!

α co2

  1 112α 1 1 2αCδo 1 1 1 β ofδ co β of βo H
 E 5 1 1 α 2 1 1 αCδo Cδo B5 2

ð3:53Þ ð3:54Þ ð3:55Þ

3.2.2.2 Langmuir-Type Dependency ~ o; The mass transfer rate, without external mass transfer resistance, is as (α 5 αc 3 o o o kg/m ; Cδ 5 cδ =c Þ :     ~ oδ ~ o αc α αCδo αc o sat  β ð3:56Þ 2 2 φ J 5 β o φsat ~ o 1 1 αc ~ oδ 11 αc 11α 11αCδo The situation is rather complicated in this case; thus, the mass transfer rate is only defined here taking into account the external mass transfer resistance of the fluid phases as Cδo 1J=½β ofδ co  J 12J=½β of co  2 5 β o φsat α 11αð12J=½β of co Þ 11αðCδo 1J=½β ofδ co Þ

ð3:57Þ

By means of Eq. (3.57), the J value can be easily obtained, for example, by iterative method. The other possibility is that by rearranging this equation, one can get a third-order algebraic equation that has an analytical solution.

3.2.2.3 Dual-Sorption Model The linear concentration distribution is not changed due to the sorption isotherm, thus the mass transfer rate can be given as    ~ oδ ~ o αc αc 2 J 5 β o ðφ 2 φδ Þ 5 β o ðco 2 coδ ÞHo 1 φsat ~ o 1 1 αc ~ oδ 1 1 αc

ð3:58Þ

Diffusion Through a Plane Membrane Layer

57

The mass transfer rate in presence of external mass transfer resistances can be given similarly as it was made previously: J 5 J1 1 J2

ð3:59Þ

J1 5 β oov ðco 2 coδ Þ

ð3:60Þ

with

 J2 5 β o φsat α

 12J=½β of co  Cδo 1J=½β ofδ co  2 11αð12J=½β of co Þ 1 1 αðCδo 1J=½β ofδ co Þ

ð3:61Þ

with 1 1 1 1 o 5 o 1 o 1 o β ov β f Ho β β fδ where J1 and J2 denote the mass transfer rate according to the Henry’s law and to the Langmuir’s equation, respectively [kg/(m2 s)]. It has been demonstrated that the sorption isotherms of small molecule penetrant gases such as carbon dioxide, methane, argon, etc. in glassy polymers are generally concave to the pressure axis (Koros, 1980; Vieth, 1988). The dual mode sorption model is widely used to describe such behavior. The adsorbed amount will be the sum of the Henry’s and the Langmuir’s isotherm, namely: 0

φHbp φ 5 φD 1φH 5 kD p1 11bp

ð3:62Þ

where φ is the solubility (cm3 (STP)/cm3 polymer); kD is the Henry’s law dissolution constant (cm3 (STP)/cm3 polymer); b is the hole, microvoid, affinity constant 0 (1/Pa); φH is the hole saturation constant (cm3 (STP)/cm3 polymer); and p is the pressure (Pa). The following equation can be obtained from Eq. (3.61) (Koros et al., 1981; Vieth, 1988): φ 5 φD 1

KφD 11αφD

ð3:63Þ

with 0

φ b K5 H ; kD

p5

φD ; kD

α5

b kD

58

Basic Equations of the Mass Transport through a Membrane Layer

Dual mobility transport of a gas in a glassy polymer is described by Fick’s law, assuming that the two modes of sorption occur simultaneously and the diffusion can occur in both modes, though the diffusion coefficients are different in the two modes, as follows (Paul and Koros, 1976): J 5 JD 1JH 5 2DD

dφD dφ 2 DH H dy dy

ð3:64Þ

where φD and φH are given in Eq. (3.62); DD and DH are the respective diffusion coefficients of the two sorbed populations; J is the total diffusional flux; and JD and JH are the respective fluxes of the two populations. Another approach for penetrant transport is to define a phenomenological diffusion coefficient as the “effective diffusivity,” which expresses the flux in terms of the total concentration, φ: J 5 2Deff

dφ dy

ð3:65Þ

The gradient of φH can be expressed as a function of φD, according to Eq. (3.62): dφH K dφD 5 2 dy ð11αφD Þ dy

ð3:66Þ

Accordingly, the total mass transfer rate can be expressed as   DH K=DD dφD J 5 2DD 11 ð11αφD Þ2 dy

ð3:67Þ

The question arises how the J mass transfer rate can be expressed as a function of the φ total concentration. Differentiating Eq. (3.63), one can get as:   dφ K dφD 5 11 2 dy dy ð11αφD Þ

ð3:68Þ

Replacing Eq. (3.68) into Eq. (3.67), one can get the value of the Deff given in Eq. (3.69) as (Vieth, 1988; Follain et al., 2010):  Deff 5 DD

 DH K=DD 1 11 K 2 11 ð11αφD Þ ð1 1 αφ

ð3:69Þ DÞ

2

Diffusion Through a Plane Membrane Layer

59

Deff diffusion constant contains the value of φD. For its elimination, its value should be expressed as a function of φ concentration. Applying Eq. (3.63), one can get as:

φD 5

2ð11K2 αφÞ 1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð11K2αφÞ2 14αφ 2α

ð3:70Þ

Replacing Eq. (3.70) into Eq. (3.68), now the mass transfer rate can be estimated as a function of the total concentration. To develop Deff as a function of position in the membrane layer, after integration of Eq. (3.65), one can obtain 2Jy 1 S 5 DD φD 1DH φH

ð3:71Þ

The values of J and T can be determined by the boundary conditions as (Vieth, 1988): y50

then

S 5 DD φD 1DH φH

ð3:72aÞ

y5δ

then

2Jδ1T 5 0

ð3:72bÞ

Thus, J5

DD φD 1DH φH δ

ð3:73Þ

as well as  y DD φD 1DH φH 5 ðDD φD 1DH φH Þ 12 δ

ð3:74Þ

where φD and φH are the values at the boundary. Expressing φH in terms of φD [Eqs. (3.60) and (3.61)], φD can be expressed as a function of y, and the effective diffusivity can subsequently be expressed as a function of y, namely: φD 5

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T1 T 2 1 4αξð12y=δÞ 2α

where T 5 11

 DH y K2αξ 1 2 DD δ

ð3:75Þ

60

Basic Equations of the Mass Transport through a Membrane Layer

and ξ5

DD φD 1DH φH DD

Substituting Eq. (3.75) into Eq. (3.69), φD can be expressed as a function of y, and the effective diffusivity can then be expressed also as a function of position in the membrane layer.

3.2.2.4 ENSIC Model The membrane surface concentration can be obtained by this model as φ5

B ðeγcðA2BÞ 21Þ A2B

ð3:76Þ

Thus, the mass transfer rate without external mass transfer resistance can be expressed as J 5 βo

B o o ðeγðA2BÞc 2eγðA2BÞcδ Þ A2B

ð3:77Þ

The mass transfer rate with external mass transfer resistances will be as J 5 βo

B   ðeγðA2BÞc 2eγðA2BÞcδ Þ A2B

ð3:78Þ

with c 5 co 2

J β of

ð3:79Þ

cδ 5 coδ 1

J β ofδ

ð3:80Þ

and

Applying Eqs (3.78)(3.80), the J mass transfer rate can be determined by trial-and-error method.

3.2.2.5 FloryHuggins Model According to this model, the φ(c) function is 

    1 2 φ exp 2 1 1 χ0 21 φ 1 χ1 φ 5c γ

ð3:50aÞ

Diffusion Through a Plane Membrane Layer

61

In the presence of the external mass transfer resistance, the values of the membrane interface concentrations have to be determined, because J 5 β o ðφ 2φδ Þ: Taking into account Eqs (3.50a), (3.79), and (3.80), the surface membrane concentrations can be expressed in implicit equations. Thus: φδ 5 φ 2

β of φ exp ψ βo 1 fo co o β γ β

ð3:81aÞ

φ 5 φδ 1

β ofδ φδ exp ψδ β ofδ o 2 o cδ βo β γ

ð3:81bÞ

ψ 5 11ðχ0 2 1Þφ 1 χ1 φ2 ψδ 5 11ðχ0 2 1Þφδ 1 χ1 φ2 δ The values of φ and φδ can be determined by iteration. e.g. replacing φδ in Eq. (3.81a) into Eq. (3.81b), the value of φ (then that of φδ ) can be obtained. By means of Eq. (3.81), the value of φδ can be calculated by trial and error.

3.2.3

Mass Transfer Through a Composite Membrane

Two essential conditions should be taken into account: the thickness of the sublayers can be different and the solubility coefficient can also be different. The diffusion coefficient is also different, but its value is constant inside of every sublayer. It is assumed that the solubility coefficient for the sublayers can be defined as follows: Hi ci 5 Hi11 ci11

for i51 to N

ð3:82Þ

For the external phases: Hoco5H1c1 and HN cN 5 Hδ coδ (or in the case of external mass transfer resistances, according to Figure 3.1, Hoc 5H1φ and HN φδ 5 Hδ cδ Þ: The values of the parameters in Eq. (3.38) can be given as (C 5 c/co): Ti 5 2

1 ðHo 2 Hδ Cδo Þ N Di P ðHj ΔYj Þ j51

ð3:83Þ

Dj

and Si 5

i21 X Ho Hj 1 Tj ΔYj 2 Ti Yi21 Hi Hi j51

ð3:84Þ

Accordingly, the mass transfer rate is J 52

D1 T1 5 β o co ðHo 2 Hδ Cδo Þ δΔY1

ð3:85Þ

62

Basic Equations of the Mass Transport through a Membrane Layer

Taking into account the external mass transfer resistances on the both sides of the membrane, the mass transfer rate can be given as follows: J 5 co

3.2.4

Ho =H1 C o 2 Hδ =HN Cδo Ho =H1 ko 1 1=β o 1 Hδ =HN kδo

ð3:87Þ

Binary, Coupled Component Diffusion Transport

In addition, coupling phenomena could occur in most cases during the separation of liquid mixtures. A strong coupling of diffusion of components to be separated takes place in the MaxwellStefan approach (Wesselingh and Krishna, 2000). This approach could be applied to describe mass transport, with strongly concentrationdependent diffusion, during pervaporation of binary wateralcohol mixture, with low carbon number (Heintz and Stephan, 1994), as well as during membrane extraction of organic components (Kubaczka et al., 1998). Separation by means of zeolite, or, generally, of inorganic membranes is another important group of membrane separation processes where the MaxwellStefan approach to mass transfer is recommended (Krishna and Wesselingh, 1997; van den Graaf et al., 1999; van den Broeke et al., 1999). Another often recommended mass transport theory is the so-called FloryHuggins approach (Meuleman et al., 1999; Ghoreyshi et al., 2008). This theory is especially applicable for organophilic pervaporation of organic compounds in water. For the estimation of the countersorption diffusivity, Krishna and Wesselingh (1997) and Bitter (1991) proposed to use a generalized form of Vignes equation, which also means strong concentrationdependent diffusion coefficients. The differential mass balance equations, for binary, diffusional mass transport processes, can generally be given by Eqs (3.1) and (3.2) for all theories discussed earlier. The solution of these equations can only be done by numerical methods, which makes their applications difficult. Recently, Nagy (2006) gave an analytical approach for the solution, but the equations developed are very complex. The aim of this work was to develop a much simpler mathematical model that defines the concentration distribution of components in the cases of single component as well as that of binary, coupled diffusion processes in the membrane layer and the mass transfer rates of the diffusing components at the membrane interface, by means of explicit mathematical equations. This approach should generally be used independently of what function the diffusion coefficient of components could depend on the concentrations. As a practical example, the binary mass transport through zeolite membrane will be shown by applying the well-known MaxwellStefan approach. Obviously, the model can be used for multilayered membranes having space-dependent mass transport parameters.

Diffusion Through a Plane Membrane Layer

63

3.2.4.1 Modeling of the Coupled Diffusion Some important assumptions of the mass transport for modeling (see also Chapter 1): (1) the binary mass transport takes place by diffusion only; (2) concentrations at the interfaces are in equilibrium with the gas (or liquid) phase; (3) the membrane can be plane or can have a cylindrical interface and diffusion in direction of the radius is taken into account; (4) mass transport is a steady-state process; and (5) diffusion coefficients are concentration/space dependent and the diffusion of the components is coupled to each other. The differential mass balance equations for binary, coupled diffusion through a plane membrane layer can generally be given as follows:     d dφ d dφ LA A 1 LB B 5 0 dy dy dy dy

ð3:88Þ

    d d dφB  dφA 1 50 L LB dy dy A dy dy

ð3:89Þ

The membrane concentration is given here in a unit of measure of kgmol/m3. This can easily be obtained by means of the usually applied kg/kg unit of measure with the equation of φ 5 wρ/M, where w is the concentration (kg/kg); ρ denotes the membrane density (kg/m3); M is the molar weight (kg/mol). For a binary mixture, we have four effective diffusion coefficients, LA, LB, LA ; LB (Wesselingh and Krishna, 2009) that could strongly depend on both concentrations and on the real Fickian diffusion coefficients. Their values can be given, as we can see it later, by the given function of the concentrations in the membrane matrix or in membrane pores, φA, φB, and the diffusivities, DA, DB, DAB, depending on the approaches used for describing the mass transport. Basically, there are two main theories for the coupling diffusion process, namely the Maxwell Stefan theory (Heintz and Stephan, 1994; van den Graaf et al., 1999), and the FloryHuggins theory (Meuleman et al., 1999). It also should be mentioned that the Vignes and the modified Vignes equations can be used (Bitter, 1991) for the concentration dependency of the real diffusion coefficients given by Eq. (3.32). All theories mentioned above involve the above effective diffusion coefficients, and the component transport can be described by the differential equations (3.88) and (3.89). The specific mass transfer rate in dimension of m/s, at the membrane interface, can be given as follows:   dφ dφ JA 5 2 LA A 1 LB B dy dy   dφ dφ JB 5 2 LA A 1 LB B dy dy

at y 5 0

ð3:90Þ

at y 5 0

ð3:91Þ

64

Basic Equations of the Mass Transport through a Membrane Layer

In order to receive the mass transfer rates of component A and B, the differential equation system, given by Eqs (3.88) and (3.89), is to be solved. The schematic diagram of the physical model applied in order to get an analytical approach of the solution is illustrated in Figure 3.4. The membrane is divided into N sublayers with thickness of ΔY perpendicular to the membrane interface (ΔY 5 δ/N), with constant diffusion coefficient in all these sublayers. Consequently, the differential mass balance equations of the diffusing components for the nth sublayer becomes (Yn 21 , Y , Yn): LAn

2 d2 φAn  d φBn 1L 5 0; Bn dY 2 dY 2

n51; . . . ; N

ð3:92Þ

LAn

d2 φAn d2 φBn 1LBn 5 0; 2 dY dY 2

n51; . . . ; N

ð3:93Þ

As general solutions of Eqs (3.92) and (3.93), the concentration distribution for components A and B can be given as follows: φAn 5 Tn Y 1Qn

ð3:94Þ

φBn 5 Sn Y 1En

ð3:95Þ

and

Figure 3.4 Schematic figure of concentration distribution of membrane sublayers and notations (for flat membrane, its value will be infinite).

δ

ο

CA

φ*

φ1

φn

φN ο

β fδ ο

βf

L1

Ln

L1

∗

Ln

Ln

ΔYn

ΔYn

ΔYN

∗

L N φδ* ∗

ο

Cδ A

Diffusion Through a Plane Membrane Layer

65

The boundary conditions for the interface of the sublayers, at Y 5 Yn, are as LAn

LAn

dφAn dφ dφ dφ 1LBn Bn 5 LAn11 An11 1LBn11 Bn11 dY dY dY dY at Y5Yn ; n51; . . . ; N21

ð3:96Þ

dφAn dφ dφ dφ X1LBn Bn 5 LAn11 An11 1LBn11 Bn11 dY dY dY dY at Y5Yn ; n51; . . . ; N21

ð3:97Þ

as well as φAn 5 φAn11

at Y5Yn ;

n51; . . . ; N21

ð3:98Þ

φBn 5 φBn11

at Y5Yn ;

n51; . . . ; N21

ð3:99Þ

and

The boundary conditions at the external interfaces of the membrane layer can be distinguished in two cases, namely the external mass transfer resistances could be neglected or not. Both cases involve large parts of membrane processes depending on the hydrodynamic conditions of the streaming phases, as well as the transfer rate inside of the membrane. During gas mixture separation, it could often be neglected, while during liquid separation process often not. In this text, the mass transfer rate without external mass transfer resistance will be shown. This case can be used for mass transport of gas components through a membrane. The external boundary conditions will be: cA HA 5 φA ;

cB HB 5 φB

at Y 5 0

ð3:100Þ

and cAδ HAδ 5 φAδ ;

cBδ HBδ 5 φBδ

at X 51

ð3:101Þ

Through some mathematical manipulation, it can be obtained that the two differential equations [Eqs (3.77) and (3.78)] can be solved separately. Their solution is well known, and they are given in Eqs (3.79) and (3.80). Assuming N sublayers in the membrane, one can get N algebraic equations for the concentration distribution of both components. Each component has 2N parameters to be determined (Nagy, 2006). These parameters can be calculated by means of the 4N boundary conditions defined by Eqs (3.96)(3.99). The algebraic equation system containing 2N equations, obtained using Eqs (3.96)(3.99), was solved by means of the well-known Cramer rules using properties of determinants. Only the end results of this procedure will be

66

Basic Equations of the Mass Transport through a Membrane Layer

given here. The mass transfer rate of the transporting components can be given as follows at Y 5 0: JA 5 2ðLA1 P1 1LB1 S1 Þ

ð3:102aÞ

JB 5 2ðLA1 P1 1LB1 S1 Þ

ð3:102bÞ

where P1 5

MN ðφAδ 2φA Þ2FN ðφBδ 2φB Þ LA1 γ 1 ðVN ZN 2WN YN Þ

ð3:103Þ

with  N N  X X 1 βi ; 1ΔY 2  MN 5 ΔY LBi γ i L i51 Bi i52

FN 5 ΔY

N X βi L γ i52 Ai i

as well as VN 5

 N  X ΔY 1 αi ; 1ΔY 1 LA1 LAi LAi γ i i52

WN 5 2

αi 5

N X ΔY 1 1ΔY ; LA1 γ 1 L γ i52 Ai i

LA1 LAi 1 ; LA1 LAi

βi 5 2

ZN 5

N X ΔY αi 2ΔY  γ L LB1 i52 Bi i

UN 5 2ΔY

LB1 LBi 1 ; LB1 LAi

γi 5

N X αi  L γ i52 Bi i

LBi LAi 2 ; LBi LAi

i 5 22N

The value of S1 for the component B is as follows: S1 5

VN ðφAδ 2φA Þ 2 YN ðφBδ 2φB Þ LB1 ðVN ZN 2WN YN Þγ 1

ð3:104Þ

By means of Eqs (3.102)(3.104), the mass transfer rate, at the membrane interface, can be predicted for both components. If one wants to calculate the concentration distribution through the membrane layer, it can be done by means of the following equations according to the internal boundary conditions for components A and B, respectively: Pi11 5

JAi LBi11 2 JBi LBi11 ; LAi11 LBi11 2 LAi11 LBi11

2 # i # N21

ð3:105Þ

Diffusion Through a Plane Membrane Layer

Si11 5

JBi LAi11 2 JAi LAi11 ; LAi11 LBi11 2 LAi11 LBi11

67

2 # i # N21

ð3:106Þ

as well as (Yi 5 iΔY) Qi11 5 φAi 2 Pi11 Yi

ð3:107Þ

Ei11 5 φBi 2 Si11 Yi

ð3:108Þ

and

3.2.5

Case Studies

We will show two examples for mass transfer through membrane using the MaxwellStefan approach: (1) separation of binary liquid mixture by pervaporation (Heintz and Stephan, 1994; Iza´k et al., 2003; Nagy, 2004, 2008) and (2) binary gas separation by zeolite membrane (van den Graaf et al., 1999; Wesselingh and Krishna, 2000).

3.2.5.1 Binary Gas Separation by Zeolite Membrane The mass transfer through solid membrane and, thus through zeolite layer, was extensively investigated in the literature (van den Broeke et al., 1999; van den Graaf et al., 1999). The diffusion coefficients for coupled diffusion, taking also into account the friction between the diffusing molecules (this effect is represented by the value of DAB), on the pore interface can be given as follows (Nagy, 2004):   DB Qn ; LAn 5 qsat ερDA 12θBn 1θAn DABn   DA Qn ; LAn 5 qsat ερDB θBn 1θBn DABn

  DB Qn LBn 5 qsat ερDA θAn 1θAn DABn ð3:109aÞ

  DA Qn LBn 5 qsat ερDA 12θAn 1θBn DABn ð3:109bÞ

and Qn 5

1    B A ð12θAn 2θBn Þ 11θAn DDABn 1θBn DDABn

ð3:110Þ

where θ is fractional loading (5 q/qsat); DA and DB are the diffusion coefficients of components in zeolite layer (m2/s); DABn is the MaxwellStefan interaction

68

Basic Equations of the Mass Transport through a Membrane Layer

parameter (m2/s); ε is porosity; qsat is the saturated concentration (kmol/kg); and ρ is the density (kg/m3). The values of DA and DB are assumed to be constant while the value of DAB is concentration dependent. The value of DABn can be given by the modified Vignes equation (Bitter, 1991): ðθ =ðθAn 1θBn ÞÞ

DABn 5 DA An

ðθ =ðθAn 1θBn ÞÞ

ð3:111Þ

DB Bn

The concentration distribution and the mass transfer rate of methane (component A) and ethane (component B) mixture were calculated and illustrated in two figures (Nagy, 2004). Both the concentrations and mass transfer rates were calculated by means of the predicted value of DAB, using the modified Vignes equation given by Eq. (3.96) (Figure 3.5; dotted lines), and for the case when DAB -N (Figure 3.5; continuous lines; this is the so-called single-file diffusion). The data used for calculation are given in Table 3.1 (van den Graaf et al., 1999). Values of KA and KB are parameters of the Langmuir isotherm, θA 5 qA/qsat. The overall pressure was kept to be 1 kPa in this side. Both the concentration and mass transfer rates (Figure 3.5) were calculated by the predicted value of DABn using the above-mentioned Vignes equation (dotted lines) and those without interaction, namely in the case when DABn -N (continuous lines). As can be seen, the mass transfer rate of methane is strongly affected by the friction between the molecules (Figure 3.3); consequently, the separation factor is also strongly altered as a function of the membrane concentration. 40.0 - - - - - - Coupled _____ Noncoupled

Flux (mmol m–2 s–1)

30.0

Methane

20.0

Ethane

10.0

0.0

1

25

P ethane (kPa)

75

99

Figure 3.5 The mass transfer rates of methane and ethane as a function of ethane partial pressure with adsorbateadsorbate interactions (dotted lines) and without that interaction (continuous lines; single-file diffusion, DAB-N).

Diffusion Through a Plane Membrane Layer

69

Table 3.1 Parameters for the Zeolite Layer Applied for Calculation DA 5 21.4 3 10210 m2/s KA 5 3.1 3 1026 Pa21 θsat 5 1.85 mmol/g Psum 5 100 kPa Pδ 5 1 kPa

DB 5 5 3 10210 m2/s KB 5 57 3 1026 Pa21 ρ 5 1.8 3 106 g/m3 ε 5 0.2 δ 5 10 μm

It also is interesting to compare the change of the diffusion coefficients of the adsorbed components, in zeolite layer (not shown here). All L values, LA, LB, LA ; and LB ; change strongly in the membrane, and consequently, with the concentration. The concentrations of the components alter drastically the diffusion rate of the other components. The effect of the adsorbateadsorbate interaction is also essential.

3.2.5.2 Binary Transport for Pervaporation The separation of ethanol/water by poly(vinyl alcohol) membrane was discussed in detail by Heintz and Stephan (1994). The effective diffusion coefficients for the nth section of the membrane, applying the MaxwellStefan approach, can be given, for components A and B, as follows (Heintz and Stephan, 1994):  LAn 5 DAn ρ  LBn 5 DAn ρ LAn

 5 DBn ρ

D12 1φAn DBn D12 1φAn DBn 1φBn DAn φAn DBn D12 1φAn DBn 1φBn DAn φBn DAn D12 1φAn DBn 1φBn DAn

 ð3:112Þ  ð3:113Þ  ð3:114Þ

and  LBn 5 DBn ρ

D12 1φBn DAn D12 1φ1 DBn 1φBn DAn

 ð3:115Þ

where DAn and DBn are the diffusion coefficients of components in the membrane given for the concentration in the nth sublayer (m2/s); DAB is the MaxwellStefan interaction parameter (m2/s); and ρ is density of swollen membrane (kg/m3). Note that here DAn and DBn diffusion coefficients are not average values, their values should be fitted to the real concentration in the membrane. Thus, the calculation of the concentration distribution or the mass transfer rate needs a few iteration steps in order to obtain the real diffusion coefficients to the real concentration values. Equations (3.111)(3.113) can easily be obtained by the MaxwellStefan theory.

70

Basic Equations of the Mass Transport through a Membrane Layer

Diffusion coefficient (10–10 m2/s)

20.0

15.0 B: Ethanol

L

A: Water

A

10.0

LB

L* B

5.0

0.0 0.0

L* A

0.2

0.3

0.4

0.6

0.8

0.9

1.0

Membrane thickness

Figure 3.6 Change of effective diffusion coefficients obtained by Eqs (3.113)(3.115) in the membrane top layer. Coupled diffusion coefficient: D12 5 25 3 10215 m2 =s; noncoupled diffusion: D12 -N; DB 5 30 3 10215 m2 =s; DA φwater-0 5 10 3 10215 m2 =s (the water diffusivity is increased with its concentration; see Heintz and Stephan, 1994).

As illustration, we show typical concentration distributions of water and ethanol for two cases, namely with coupled (D12 5 25 3 10 211 cm2/s) and uncoupled ðif D12 -NÞ diffusion. The diffusion coefficient of water and ethanol in the membrane used for our calculations, as a function of concentrations, were measured by Hauser et al. (1989). The value of D12 was also predicted by Heintz and Stephan (1994). The interface concentrations of the upstream side (at y 5 0) were also used measured values. Concentrations were chosen to be zero for both components at the permeate side (at y5δ). The external mass transfer resistance and that for the porous support layer were neglected. As seen in Figure 3.2, the coupling of diffusion decreases the concentration change of components and, thus changes the mass transfer rates. The effective different diffusion coefficients, according to Eqs (3.112)(3.115), are plotted in Figure 3.6 as a function of the space coordinate in the membrane. [Measured diffusion coefficients of water and ethanol in the membrane, DA and DB and the calculated MaxwellStefan interaction parameter, DAB, used by us, are given by Heintz and Stephan (1994) and their related articles.] Their strong changes can be observed as a function of x, thus, as a function of the concentrations.

3.3

Nonsteady-State Diffusion

Let us consider a membrane layer with constant, uniform concentration. Its value can be equal to that existing on the membrane side with lower concentration; thus,

Diffusion Through a Plane Membrane Layer

71

at t 5 0 the membrane concentration is kept at a concentration of φoδ 5 Hcoδ (Figure 3.1). Keeping the concentration on the other side at higher value, φ 5 Hc , the diffusion process will be started through the membrane toward the lower concentration side. To calculate the amount of species transferred into the membrane, it is first necessary to determine the concentration distribution of the transferred species within the membrane layer as a function of position and time. The differential mass balance equation can be the well-known Fick II equation, for the membrane layer under unsteady conditions as @φ @2 φ 5D 2 @t @y

ð3:116Þ

in which the diffusion coefficient is considered constant. The initial and the boundary conditions are at at at

t50 y50 y5δ

φ 5 Hcoδ φ 5 Hco φδ 5 Hcoδ

for all y t.0 t.0

ð3:117Þ

The concentration difference on the two sides of the membrane is kept constant; thus, the mass transport through the membrane is permanent, and after a certain time it will be constant, that is, steady state. This transitional state is described by Eq. (3.116), while the steady-state process is described by Eq. (3.1). Let us introduce the following dimensionless quantities: C o 2C Co 2Cδo

Φ5

c ; Hco

Y5

y δ

ð3:119Þ

τ5

Dt δ2

ð3:120Þ

Φ5

ð3:118Þ

Thus, one can obtain a dimensionless form of Eq. (3.121) as @Φ @2 Φ 5 2 @τ @Y at at at

τ 50 Y 50 Y 51

ð3:121Þ Φ 5 Cδo Φ51 Φ 5 Cδo

ð3:122Þ

72

Basic Equations of the Mass Transport through a Membrane Layer

Note that the boundary condition at Y 5 1 is not homogeneous and, as a result, the method of separation of variables cannot be applied. To overcome this problem, a solution is sought in the following form: Φðτ; YÞ 5 ΦN ðYÞ 2 Φt ðτ; YÞ

ð3:123Þ

in which ΦN(Y) is the steady-state solution, i.e.: d2 Φ N 50 dY 2

ð3:124Þ

with the following conditions: at Y 5 0 ΦN 5 1 at Y 5 1 ΦN 5 Cδo

ð3:125Þ

The solution will be as ΦN 5 11ðCδo 21ÞY

ð3:126Þ

However, the transient contribution of Φt(τ,Y) satisfies Eq. (3.121): @Φt @ 2 Φt 5 @τ @Y 2

ð3:127Þ

From Eqs (3.122) and (3.126), Φt 5 Y2Φ, thus the initial and boundary conditions become: at at at

τ 50 Y 50 Y 51

Φt 5 Cδo 2ΦN Φt 5 0 Φt 5 0

ð3:128Þ

Now the differential equation, Eq. (3.127), can be solved by the method of separation of variables that assumes that the solution can be represented as a product of two functions of the form Φt ðτ; YÞ 5 f ðτÞgðYÞ

ð3:129Þ

Substitution of Eq. (3.129) into Eq. (3.127) and rearranging it, one can get 1 df 1 d2 g 5 f dτ g dY 2

ð3:130Þ

Diffusion Through a Plane Membrane Layer

73

The left side of Eq. (3.130) is a function of τ only, the right side depends on Y only. This is possible only if both sides of Eq. (3.130) are equal to a constant, 2λ2: 1 df 1 d2 g 5 2λ2 5 f dτ g dY 2

ð3:131Þ

The choice of a negative constant is due to the fact that the solution will decay to zero as time increases. Equation (3.131) results in two ordinary differential equations. The equation for f is given by df 1λ2 f 5 0 dτ

ð3:132Þ

The solution of Eq. (3.132) is f ðτÞ 5 e2λ

2

τ

ð3:133Þ

However, the equation g is d2 g 1λ2 g 5 0 dY 2

ð3:134Þ

with the boundary conditions at at

Y 50 Y 51

g50 g50

ð3:135Þ

The solution of Eq. (3.134) is gðYÞ 5 T sinðλYÞ 1 S cosðλYÞ

ð3:136Þ

where T and S parameters should be determined by means of boundary conditions given by Eq. (3.136). The boundary conditions given at Y51 in Eq. (3.135) give that S 5 0. The use of the boundary condition defined at Y51 in Eq. (3.135) results in T sin λ 5 0

ð3:137Þ

For nontrivial solution, the eigenvalues are given by λn 5 nπ;

n 5 1; 2; 3; . . .

ð3:138Þ

74

Basic Equations of the Mass Transport through a Membrane Layer

The most general solution of the Φt value is obtained by adding the solutions for all integrals from n 51 to n 5N; thus, the transient solution is Φt 5

N X

An e2n

π τ

2 2

ð3:139Þ

sinðnπYÞ

n51

The unknown coefficient An can be determined by using the initial condition in Eq. (3.128) at τ 5 0. The result is Cδo 2ΦN 5

N X

ð3:140Þ

An sinðnπYÞ

n51

Because the eigenfunctions are simply orthogonal, multiplication of Eq. (3.140) by sin(mπY) and integration from Y50 to Y51 gives ð1 ðCδo 2ΦNÞsinðmπYÞdY

5

N X

ð1 An sinðnπYÞsinðmπYÞdY

n51

0

ð3:141Þ

0

Note that the integral on the right side of Eq. (3.141) is zero when n6¼ m and nonzero when n 5 m. Therefore, when n 5 m, the summation drops out and Eq. (3.141) reduces to the form ð1

ð1 ðCδo 2ΦN ÞsinðnπYÞdY

5 An sin2 ðnπYÞdY

0

ð3:142Þ

0

Eigenvalues of the integral give An 5

2 ð12Cδo ÞcosðnπÞ; nπ

n 5 61; 62; 63; . . . ; 6N

ð3:143Þ

The transient solution takes the form Φt 5 2

N 2X 12Cδo 2 2 cosðnπÞe2n π τ sinðnπYÞ π n51 n

ð3:144aÞ

In limiting case, if Cδo -0, Eq. (3.144a) will be as Φt 5 2

N 2X 1 2n2 π2 τ e sinðnπYÞ π n51 n

ð3:144bÞ

Diffusion Through a Plane Membrane Layer

75

Substitution of the steady-state and the transient solutions, Eqs (3.126) and (3.144) into Eq. (3.104) and taking into account Eqs (3.118) and (3.119) gives the solution as (Carslaw and Jaeger, 1959; Crank, 1975; Slattery, 1999): N
 2 Cδo 21 2X 2 2 cosðnπÞe2n π Dm t=δm sinðnπYÞ Φ 5 1 2 1 2 Cδo Y 1 π n51 n

ð3:145Þ

The specific mass transfer rate at the membrane interface, at Y 5 0, as a function of time, can be given as: J 52

3.3.1

N X 2 DHco 2 2 ð12Cδo Þ e2ðn π D=δ Þt δ n51

ð3:146Þ

Mass Transport with External Mass Transfer Resistance on the Feed Side

In this case, the permeate side mass transfer resistance is neglected, β oδ -N: Thus, the initial and boundary conditions to be applied for the solution of Eq. (3.116) are as follows: at

t50

at

y50

at

y5δ

φ 5 φ @φ 2D 5 β of ðco 2c Þ @y φ50

for all y

ð3:147Þ

Applying dimensionless variables, given in Eqs (3.118) and (3.119) with φ 5 0 at y 5 δ, the above conditions can be expressed as follows: τ 5 0;

Φ 5 1 for all Y

ð3:148aÞ

Y 5 0;

Φ50

ð3:148bÞ

Y 5 0;

@Φ β of 5 o Φ 5 BiΦ β @Y

ð3:148cÞ

with Bi 5

β of βo

With solution by the separation of variables, the general solution can be given, similar to that discussed in Eqs (3.107)(3.134), as Θ 5 e2λ τ fT sinðλYÞ1S cosðλYÞg 2

ð3:149Þ

76

Basic Equations of the Mass Transport through a Membrane Layer

According to boundary conditions given by Eqs (3.148b) and (3.148c), one can get as T sin λ1S cos λ 5 0

ð3:150Þ

2Tλ cos 01Sλ cos 0 5 BiðT sin 01S cos 0Þ

ð3:151Þ

From Eqs (3.150) and (3.151), the following expression between the parameters T and S can be obtained: S 5 2T tan λ

ð3:152Þ

The eigenvalues of λ can be obtained by the following expression: Bi 5

λn tan λn

ð3:153Þ

Therefore, the transient solution after a few manipulations is Θ5

N X

An e2λn τ

n51

2

sin½λn ðY 21Þ cos λn

ð3:154Þ

Applying the initial boundary condition, Eq. (3.148a), the unknown An coefficient can be obtained by the following equation: 15

N X n51

An

sin½λn ðY 21Þ cos λn

ð3:155Þ

Applying the methodology used by Eqs (3.141) and (3.142), the An can be given as   λn 1 sinð2λn Þ 2 An 5 cos λn ðcos λn 21Þ 2 4λn

ð3:156Þ

Thus, the transient solution takes the form   N X co 2c λn 1 sinð2λn Þ 2λ2n τ sin½λn ðY 21Þ 5 e ϕ o c 20 n 5 1 cos λn ðcos λn 21Þ 2 4λn cos λn with λn given by Eq. (3.153).

ð3:157Þ

Diffusion Through a Plane Membrane Layer

3.3.2

77

Solution of Fickian Diffusion by Boltzmann’s Transformation

Solution of Eq. (3.116) for short time can be made by transformation. At small values of time, the component does not penetrate very far into the membrane layer. Under these circumstances, it is possible to consider the slab as a semiinfinite medium in y-direction. The initial and boundary conditions become: at

t50

φ 5 φo

ð3:158aÞ

at

z50

φ 5 φ

ð3:158bÞ

at

z 5 N φ 5 φo

ð3:158cÞ

Introduction of the dimensionless concentration: φ2φo φ 2φo

Φ5

ð3:159Þ

Equation (3.116) reduces to @Φ @2 Φ 5D 2 @t @y

ð3:160Þ

at

t50

ð3:161aÞ

at

Y 50

at

Y 5N Φ50

Φ50 Φ51

ð3:161bÞ ð3:161cÞ

Introducing the variable η as pffiffiffiffiffi η  y=ð2 DtÞ

ð3:162Þ

The chain rule of differentiation gives @Φ @f dη 1 η df 5 52 @t @η dt 2 t dη

ð3:163Þ

as well as   @2 Φ d2 f @η 2 df @2 η 1 d2 f 5 1 5 2 2 2 @y dη @y dη @y 4Dt dη2

ð3:164Þ

78

Basic Equations of the Mass Transport through a Membrane Layer

Substitution of Eqs (3.163) and (3.164) into Eq. (3.160) gives d2 f df 12η 50 2 dη dη

ð3:165Þ

The boundary conditions, from Eqs (3.161b) and (3.161c), will be as at

η50

f 51

at

η5N

ð3:166aÞ

f 50

ð3:166bÞ 2

The integrating factor for Eq. (3.165) is exp(η ). Multiplication of Eq. (3.163) by the integrating factor yields   d η2 df e 50 dη dη

ð3:167Þ

which gives that df 2 5 C1 e2η dη

ð3:168Þ

After integration of Eq. (3.168), one can get as f 5 C1

ðη

e2u du1C2 2

ð3:169Þ

0

where u is a dummy variable of integration. Application of the boundary condition at η50 gives C251. However, application of the boundary condition at η 51 gives pffiffiffi C1 5 22= π: Therefore, the solution becomes ðη 2 2 f 5 12 pffiffiffi e2u du 5 12erfðηÞ π

ð3:170Þ

  Φ2Φo y p ffiffiffiffiffiffiffi ffi 5 12erf Φ 2Φo 4Dt

ð3:171Þ

0

or

The specific mass transfer rate will be as (Tosun, 2002):   Φ 2Φo @φ  5 pffiffiffiffiffiffiffiffi J 5 2D @y y 5 0  Dπt

ð3:172Þ

Diffusion Through a Plane Membrane Layer

3.3.3

79

Solution with a Variable Diffusion Coefficient

There is a variable diffusion coefficient, and its value can be changed as a function of concentration and/or space coordinate. The Fick’s law to be solved is as   @φ @ @φ 5 D @t @y @y

ð3:173Þ

where D is a function of φ, e.g., D 5 D0 eαφ. In certain cases, the Boltzmann transformation may be employed to convert this to an ordinary differential equation as it is done in Eqs (3.162) and (3.164). The solution is discussed by Follain et al. (2010). For a numerical solution, see Section A.1.

References Baker, R.W. (2006) Membrane Technology and Applications, 2nd ed. Wiley, Chichester. Bitter, J.G.A. (1991) Transport Mechanisms in Membrane Separation Processes. ShellLaboratorium, Amsterdam. Chandak, M.V., Lin, Y.S., Ji, W., and Higgins, R.J. (1998) Sorption and diffusion of volatile organic compounds in polydimethylsiloxane membranes. J. Appl. Polym. Sci. 67, 165175. Carslaw, H.S., and Jaeger, J.C. (1959) Conduction of Heat in Solids. Clarendon Press, Oxford. Crank, J. (1975) Mathematics of Diffusion. Clarendon Press, Oxford. Favre, F., Nguyen, Q.T., Cle´ment, R., and Ne´el, J. (1996) The engaged species induced clustering (ENSIC) model: a unified mechanistic approach of sorption phenomena in polymers. J. Membr. Sci. 117, 227236. Follain, N., Valleton, J.-M., Lebrun, L., Alexandre, B., Schaetzel, P., and Metayer, M., et al., (2010) Simulation of kinetic curves in mass transport phenomena for a concentrationdependent diffusion coefficient in polymer membranes. J. Membr. Sci. 349, 195207. Ghoreyshi, A.A., Jahanshahi, M., and Peyvandi, K. (2008) Modeling of volatile organic compounds removal from water by pervaporation process. Desalination 222, 410418. Hauser, J., Heintz, A., Schmitttecker, B., and Lichtenthaler, R.N. (1989) Sorption equilibria and diffusion in polymeric membranes. Fluid Phase Equilib. 51, 369381. Heintz, A., and Stephan, W. (1994) A generalized solution-diffusion model of the pervaporation process through composite membrane. J. Membr. Sci. 89, 153169. Iza´k, P., Bartovska´, L., Friess, K., Sipek, M., and Uchytil, P. (2003) Description of binary liquid mixtures transport through nonporous membrane by modified MaxwellStefan equation. J. Membr. Sci. 214, 293309. Koros, W.J. (1980) Model for sorption of mixed gases in glassy polymers. J. Polym. Sci. Phys. Ed. 18, 981992. Koros, W.J., Smith, G.N., and Stanett, V.T. (1981) High-pressure sorption of carbon dioxide in solvent-cast poly(methyl methacrylate) and poly(ethyl methacrylate) films. J. Appl. Polym. Sci. 26, 159170.

80

Basic Equations of the Mass Transport through a Membrane Layer

Krishna, R., and Wesselingh, J.A. (1997) The MaxwellStefan approach to mass transfer. Chem. Eng. Sci. 52, 862906. Kubaczka, A., Burghardt, A., and Mokrosz, T. (1998) Membrane-based solvent extraction in multicomponent systems. Chem. Eng. Sci. 53, 899917. Meuleman, E., Bosch, B.B., Mulder, M.H.V., and Strathmann, H. (1999) Modeling of liquid/ liquid separation by pervaporation: toulene from water. AIChE J. 45, 21532160. Mulder, M.H.V. (1984) Pervaporation Separation of EthanolWater and of Isomeric Xylenes. PhD Thesis, University of Twente, The Netherlands. Nagy, E. (2004) Nonlinear, coupled mass transfer through a dense membrane. Desalination 163, 345354. Nagy, E. (2006) Binary, coupled mass transfer with variable diffusivity through cylindrical membrane. J. Membr. Sci. 274, 159168. Nagy, E. (2008) Mass transport with varying diffusion and solubility coefficient through a catalytic membrane layer. Chem. Eng. Res. Des. 86, 723730. Paul, D.R., and Koros, W.J. (1976) Effect of partially immobilizing sorption on permeability and the diffusion time lag. J. Polym. Sci. Phys. Ed. 14, 675685. Schaetzel, P., Vauclair, C., Nguyen, Q.T., and Bouzerar, R. (2004) A simplified solutiondiffusion theory in pervaporation: the total solvent volume fraction model. J. Membr. Sci. 244, 117127. Shah, M.R., Noble, R.D., and Clough, D.E. (2007) Measurement of sorption and diffusion in nonporous membranes by transient permeation experiments. J. Membr. Sci. 287, 111118. Slattery, J.C. (1999) Advanced Transport Phenomena. Cambridge University Press, Cambridge. Tosun, ´I. (2002) Modelling in Transport Phenomena. Elsevier, New York. van den Broeke, L.J.P., Bakker, W.J.W., Kapteijn, F., and Moulijn, J.A. (1999) Binary permeation through a silicalite-1 membrane. AIChE J. 45, 977985. van den Graaf, J.M., Kapteijn, F., and Moulijn, J.A. (1999) Modeling permeation of binary mixtures through zeolite membranes. AIChE J. 45, 497. Vieth, W.R. (1988) Membrane Systems: Analysis and Design. Hanser Publisher, Munich. Wesselingh, J.A., and Krishna, R. (2000) Mass Transport in Multicomponent Mixtures. Delft University Press, The Netherlands.

4 Diffusion Accompanied by Chemical Reaction Through a Plane Sheet

4.1

Introduction

The catalytic membrane reactor as a promising novel technology is widely recommended for carrying out heterogeneous reactions. A number of reactions have been investigated by means of this process, such as dehydrogenation of alkanes to alkenes, partial oxidation reactions using inorganic or organic peroxides, as well as partial hydrogenations and hydration. As catalytic membrane reactors for these reactions, intrinsically catalytic membranes can be used (e.g., zeolite or metallic membranes), or membranes that have been made catalytic by dispersion or impregnation of catalytically active particles such as metallic complexes, metallic clusters or activated carbon, and zeolite particles throughout dense polymeric or inorganic membrane layers (Markano and Tsotsis, 2002). In the majority of the above experiments, the reactants are separated from each other by the catalytic membrane layer. In this case, the reactants are absorbed into the catalytic membrane matrix and then transported by diffusion (and in special cases by convection) from the membrane interface into catalyst particles where they react. Mass transport limitation can be experienced with this method, which can also reduce selectivity. The application of a sweep gas on the permeate side dilutes the permeating component, thus increasing the chemical reaction gradient and the driving force for permeation (Westermann and Melin, 2009). For their description, two types of membrane reactors should generally be distinguished, namely intrinsically catalytic membranes and membrane layers with dispersed catalyst particle, either nanometer-sized or micrometer-sized catalyst particles. Basically, in order to describe the mass transfer rate, a heterogeneous model can be used for larger particles and/or a pseudohomogeneous one for very fine catalyst particles (Nagy, 2007). Both approaches, namely the heterogeneous model for larger catalyst particles and the homogeneous one for submicron particles, will be applied for mass transfer through a catalytic membrane layer.

Basic Equations of the Mass Transport through a Membrane Layer. DOI: 10.1016/B978-0-12-416025-5.00005-3 © 2012 Elsevier Inc. All rights reserved.

82

Basic Equations of the Mass Transport through a Membrane Layer

4.2

Steady-State Condition

The differential mass balance equation can generally be given by the following equation for the catalytic membrane layer with various geometries, perpendicular to the membrane interface (Ferreira et al., 2001; Julbe et al., 2001):
 d dφ D 2Q50 dy dy

ð4:1Þ

The membrane concentration, φ, is given here in a unit of measure of kmol/m3 or kg/m3. This can easily be obtained by means of the usually applied g/g unit of measure with the equation of φ 5 wρ/M, where w is the concentration (kg/kg); ρ is the membrane density (kg/m3); and M is the molar weight (kg/mol). The boundary conditions can depend on the external mass transfer resistance as discussed here. As mentioned, the catalytic membrane can be intrinsically catalytic or the membrane matrix can be made catalytic by dispersed catalytic particles. For a membrane with dispersed catalyst particles, the Q source term should involve the mass transport in the membrane matrix to the catalyst particle and the simultaneous internal transport, as well as the internal chemical reaction. Accordingly, the source term can be strongly different for a membrane reactor with dispersed catalyst particles or that for an intrinsically catalytic membrane layer. The mathematical description of the mass transport through these membrane layers can be different depending on the size of the catalytic particle. Thus, presentation of the mass transport equations is divided into two parts, namely: 1. Mass transport through intrinsically catalytic or nanometer-sized catalytic particles are dispersed in the membrane layer; in this case, it can be assumed that the mass transport inside the catalytic particles or the mass transport to the catalytic interface is instantaneous and catalytic particles can be located in every differential volume element of the membrane; accordingly, the membrane can be regarded as a continuous catalytic layer. 2. The dispersed catalytic particles fall into the micrometer-sized regime, the internal mass transport mechanism, inside of catalyst particles, must be taken into account. In this case, the so-called heterogeneous model should be used, which takes into account the internal mass transport as well.

4.2.1

Mass Transport with an Intrinsically Catalytic Layer or a Membrane with Fine (Nanometer-Sized) Catalyst Particles (Pseudohomogeneous Model)

In both cases, the membrane matrix is regarded as a continuous phase for the mass transport. Assumptions made for expression of the differential mass balance equation to the catalytic membrane layer are: G

G

Reaction occurs at every position within the catalyst layer. Mass transport through the catalyst layer occurs by diffusion.

Diffusion Accompanied by Chemical Reaction Through a Plane Sheet

Figure 4.1 Illustration of a catalytic membrane layer with concentration distribution and with important notations [layer with dispersed fine catalysts (points in the membrane layer) or intrinsically catalytic membrane].

Membrane

βfo

βo

83

βofδ φδ∗

J Cο

φ∗

C*δ o

Cδ

C∗ Jδ

x

G

G

G

G

y

The partitioning of the components (substrate, product) is taken into account (thus, c H 5 φ , where φ denotes membrane concentration on the feed interface; see Figure 4.1). The mass transport parameters (diffusion coefficient, partitioning coefficient) are constant. The effect of the external mass transfer resistance should also be taken into account. The mass transport is steady state and one dimensional.

In the case of dispersed catalyst particles, they are uniformly distributed and they are very fine particles with a size of less than 1 μm, i.e., they are nanometer-sized particles. It is assumed that catalyst particles are placed in every differential volume element of the membrane reactor. The reactant first enters in the membrane layer and from that, it enters into the catalyst particles where the reaction of particles is porous as active carbon, zeolite (Vital et al., 2001), occurs or enters onto the particle interface and reacts [the particle is nonporous as a metal cluster (Vancelecom and Jacobs, 2000)]. Consequently, the mass transfer rate into the catalyst particles has to be defined first. In this case, the whole amount of the reactant transported in or on the catalyst particle will be reacted. Then this term should be placed into the mass balance equation of the catalytic membrane layer as a source term. Thus, the differential mass balance equation for intrinsically catalytic membranes and membranes with dispersed nanometer-sized particles differ only by their source term. The cylindrical effect can be significant only when the thickness of a capillary membrane can be compared to the internal radius of the capillary tube as shown by Nagy (2006). On the other hand, the application of a cylindrical coordinate hinders the analytical solution for first- or zero-order reactions as well. Thus, the basic equations will be shown here for plane interface and in the section 6.3.2, an analytical approach will be presented for a cylindrical tube as well.

4.2.1.1 Reaction Terms for a First-Order Reaction Intrinsically catalytic membrane: The reaction term can be expressed by the following equation, where k1 is the reaction rate constant: Q 5 k1 φ

ð4:2Þ

84

Basic Equations of the Mass Transport through a Membrane Layer

where φ is the reactant concentration in membrane (kg/m3) and k1 is the first-order reaction rate constant (1/s). Catalyst with dispersed particles; reaction takes place inside of the porous particles: The differential mass balance equation for the catalytic membrane can be given as Dp

! dφ2p 2 dφp 2 k1 φp 5 0 1 dr 2 r dr

ð4:3Þ

with boundary conditions at r 5 Rp

φp 5 φp

ð4:4Þ

at r 5 0

dφp 50 dr

ð4:5Þ

where φp is the particle concentration (kg/m3); subscript “p” denotes the catalyst particles; r is the radial coordinate (m); Rp 5 r/rp; rp is the particle radius (m); and Dp is the diffusion coefficient in the catalytic particles (m2/s). It is assumed in Eq. (4.3) that the reaction takes place inside of the catalyst particles. The reactant diffuses in the catalyst particle and it reacts. This case, when the reaction occurs at the catalyst surface, is also briefly discussed later. The membrane layer with nanometer-sized catalyst particles is illustrated in Figure 4.1 where the very fine particles, illustrated by points, are uniformly distributed in the membrane matrix. For a catalytic membrane with dispersed nanometer-sized particles, the mass transfer rate into the spherical catalyst particle must be defined. The internal specific mass transfer rate in spherical catalyst particles, for steady-state conditions and when the mass transport is accompanied by a first-order chemical reaction, can be given as follows (Nagy and Moser, 1995): j 5 β p φp

ð4:6Þ

where βp 5 and ϑp 5


 Dp ϑp 21 rp tanh ðϑp Þ

ð4:7Þ

sffiffiffiffiffiffiffiffi k1 rp2 Dp

where j is the mass transfer rate into catalytic particles (kg/m2 s) and φp is the concentration on the catalyst surface (kg/m3).

Diffusion Accompanied by Chemical Reaction Through a Plane Sheet

φ*

φ

j

βpo

δp

j

j

Figure 4.2 Mass transfer into the nanometer-sized catalyst particles in the membrane layer and the important notations.

dp

rp

βp

85

j h

The external mass transfer resistance through the catalyst particle depends on the diffusion boundary layer thickness, δp. According to Figure 4.2, the value of δp can be estimated from the distance of particles from each other, h [e.g., h 5 0.806 dp /ε1/3, where ε denotes the holdup of spherical catalyst in the membrane layer; the h value is discussed by Nagy (2007), in the case of cubic particles]. Namely, its value is limited by the neighboring particles; thus, the value of β op will be slightly higher than that which follows from the well-known equation of 2 5 β op dp =D; where the value of δp is supposed to be infinite. Thus, one can obtain (Nagy et al., 1989): β op 5

2D D 1 dp δ p

ð4:8Þ

h 2dp 2

ð4:9Þ

where δp 5

where D is the diffusion coefficient in the membrane around particles (m2/s); δp is the thickness of the boundary layer around particle (m); and h is the distance of particles modeled as cubic ones (m). From Eqs (4.6) and (4.8), one can obtain for the mass transfer rate with the overall mass transfer resistance j 5 β p tot φ 5

φ ð1=β op Þ1ð1=Hp β p Þ

ð4:10Þ

Accordingly, the k1 value in Eq. (4.2) can be expressed as follows (Nagy et al., 1989; Nagy, 2007): k1 5

ω β 1 2 ε p tot

ð4:11Þ

86

Basic Equations of the Mass Transport through a Membrane Layer

The ω value denotes the specific particle interface (ω 5 6ε/dp) in the membrane, (1 2 ε) means the portion of the membrane layer for mass transport, and ε denotes the catalyst particle holdup. Reaction occurs on the interface of the catalytic particles (Nagy, 2007): It often might occur that the chemical reaction takes place on the interface of the particles, for example, in cases of metallic clusters; the diffusion inside the dense particles is negligible. Assuming the Henry’s sorption isotherm of the reacting component onto the spherical catalytic surface, Hs (CHs 5 qs), applying D dφ/dr 5 ks1Hsφp boundary condition at the catalyst’s interface, at r 5 rp, the k1 reaction rate constant can be given according to Eq. (4.11) with the following β tot value: β p tot 5

1 ð1=β op Þ1ð1=ks1 Hs Þ

ð4:12Þ

where ks1 is the interface reaction rate constant (m3/m2 s). The above model is obviously a simplified one. Figure 4.3A illustrates the effect of catalyst phase holdup on the inlet mass transfer rate into the catalytic membrane layer, applying the pseudohomogeneous model (fine particles are dispersed in the membrane structure). The k1 value was calculated by Eq. (4.11). Both the reaction modulus and the catalyst holdup can strongly affect the inlet mass transfer rate of the membrane layer.

4.2.1.2 Mass Transfer Accompanied by First-Order Reaction Herewith, first the reaction source term will be defined in the case of intrinsically catalytic membrane and the solution of the differential mass balance equation under different boundary conditions. The differential mass balance equation for the catalytic membrane layer becomes D

d2 φ 2Q50 dy2

ð4:13Þ

The differential mass balance equation for the reactant entering the catalytic membrane layer is in dimensionless form of the space coordinate (Y 5 y/δ): d2 φ 2 ϑ2 φ 5 0 dY 2

ð4:14Þ

with sffiffiffiffiffiffiffiffiffi k1 δ 2 ϑ5 D

ð4:15Þ

Diffusion Accompanied by Chemical Reaction Through a Plane Sheet

J

31

87

(A) ϑP = 5

J0

1

21

0.5

11

0.15 1 0.0

0.1

0.2

0.3

0.4

Catalyst phase holdup, ε (B) 9.0 Enrichment (J/J o)

ο

ο

β f = β fδ → ∞ 7.0 5.0

ο

ο

β f = β fδ = 10 1

3.0

0.1

1.0 0.1

1.0

10.0

Reaction modulus

Figure 4.3 (A) The mass transfer rate as a function of the catalyst phase holdup obtained by the pseudohomogeneous model (D 5 1 3 10210 m2/s; Cδo 5 0; β of 5 β of;δ -N; dp 5 2 μm; δ 5 30 μm; δ is the thickness of the membrane layer). (B) The change of the relative value of the mass transfer rate as a function of the reaction modulus at different values of the external mass transfer coefficient coδ 5 0; β of 5 β ofδ :

Solution of Eq. (4.14) is well known: φ 5 TeϑY 1Se2ϑY

ð4:16Þ

For the sake of generalization, in the boundary conditions we should take into account the external mass transfer resistance on both sides of the membrane, though it should be noted that the role of the β oδ will be gradually diminished with the increase of the reaction rate. At the end of this subsection, the limiting cases will also be given briefly. Thus (both sides of boundary conditions are divided by co) it becomes

88

Basic Equations of the Mass Transport through a Membrane Layer

Y 5 0;

Y 5 1;

β of ðco

  2 c Þ 5 2β dY  

o dφ 

β ofδ ðcδ 2 coδ Þ 5 2β o

ð4:17aÞ Y50

 dφ  dY Y51

ð4:17bÞ

Parameters of Eq. (4.16), applying Eqs. (4.17a) and (4.17b), will be as (they can easily be obtained by means of the known Cramer rules):

  1 β ofδ coδ β of β of co β ofδ 1 o 11 o eϑ 12 o S5 βo β Hϑ β β Hϑ N

ð4:18Þ

with N 52


 β of 1 β ofδ β of β ofδ 2sinh ϑ cosh ϑ 1 ϑ 1 βoH ðβ o H Þ2 ϑ

 

o  1 β ofδ coδ β of β of co β fδ 2ϑ T5 2 o 21 e 11 o βo β Hϑ β β o Hϑ N

ð4:19Þ

The mass transfer rate on the upstream side of the membrane can be given as follows (Nagy, 2007): Jov 5 βco ð1 2 FCδo Þ

ð4:20Þ

with


β 5 βo

 β o ϑH 1 1 o tanh ϑ β ! fδ
 o2 1 β ϑH 1 1 1 o o tanh ϑ 1 β o o 1 o Hϑ β f β fδ βf β fδ

ð4:21Þ

and F5

1
 β o ϑH tanh ϑ cosh ϑ 1 1 β ofδ

ð4:22Þ

with β of 5

Df ; δf

βo 5

D δ

The strong influence of the external mass transfer resistances is illustrated on the inlet enhancement in Figure 4.3B.

Diffusion Accompanied by Chemical Reaction Through a Plane Sheet

89

Similarly, the mass transfer rate for the downstream side of the membrane at Y51:
  β o ϑH o Cδo ð4:23Þ Jδ 5 β δ Hc 1 2 cosh ϑ tanh ϑ 1 β of with βδ 5

βo ϑ cosh ϑ

1

 ðβ o ϑÞ2 2 1 1 Hβ o ϑ tanh ϑ 1 1 o o H 1 1 β f β fδ β of β ofδ

ð4:24Þ

Limiting cases: The transfer rate without external mass transfer resistances, namely when β of -N and β ofδ -N; can easily be obtained from Eq. (4.22) as a limiting case:
 β o ϑHco Cδo 12 ð4:25aÞ J5 tanh ϑ cosh ϑ Similarly, the outlet mass transfer rate without external mass transfer resistance will be as: Jδ 5

 β o ϑHco

1 2 Cδo cosh ϑ sinh ϑ

ð4:25bÞ

Equation (4.25) is a well known mass transfer equation for liquid mass transfer accompanied by first-order reaction. Note that the overall mass transfer rate can also be obtained by means of resistance-in-series model. For that the J inlet mass transfer rates for the boundary layer of the feed side [J 5 β of ðco 2 c Þ] and that for the membrane layer Eq. (4.25a) as well as for the Jδ values for the outlet membrane layer

 Eq. (4.25b) and for the boundary layer on the downstream side [Jδ 5 β ofδ cδ 2 coδ ] should be applied. The mass transfer rate can similarly be obtained for the case when the outlet concentration.

 ð4:26aÞ J 5 β tot co 2 coδ where β tot 5

1 tanh ϑ 1 o 1 o Hϑβ βf

ð4:26bÞ

Avoiding the outlet flow of reactant is an important requirement for the membrane reactors. For this, the operating conditions should be chosen correctly. The concentration distribution is illustrated in Figure 4.4 without external mass transfer resistances, that is, β ofδ -N and β of -N:

4.2.1.3 Mass Transfer in an Ultrafiltration Operating Mode, i.e., dφ/dy 5 0 at y 5 δ For the sake of completeness, this mode will be shown here briefly. The solution of the mass balance equation is as given in Eq. (4.16). Boundary conditions neglecting the external mass transfer resistances are

90

Basic Equations of the Mass Transport through a Membrane Layer

Concentration distribution

1.0 0.8 ϑ=0

0.6

1

0.4

3

0.2 0.0 0.0

5

0.2

0.4

0.6

0.8

1.0

Catalytic membrane layer

Figure 4.4 Concentration distribution p at ffiffiffiffiffiffiffiffiffiffiffiffi different ffi values of reaction modulus, without external mass transfer resistances ϑ 5 k1 δ=D; Cδo 5 0:5:

φ 5 φ

at y 5 0

ð4:27aÞ

dφ 50 dy

at y 5 δ

ð4:27bÞ

The concentration distribution will be as φ 5 φ

cosh½ϑð1 2 YÞ cosh ϑ

ð4:28Þ

where sffiffiffiffiffiffiffiffiffi k1 δ 2 ; ϑ5 D

Y 5 y=δ

The inlet mass transfer rate is as J 5 β o φ ϑ tanh ϑ

ð4:29Þ

Figure 4.5 illustrates the effect of the reaction rate on the concentration at zero outlet mass transfer rate. Figures 4.4 and 4.5 clearly demonstrate the significant effect of the operation modes on the concentration distribution and, consequently on the inlet mass transfer rate.

4.2.2

Mass Transfer Accompanied by Zero-Order Reaction

4.2.2.1 Source Term for Zero-Order Reaction The zero-order reaction is important primarily for biocatalytic reactions. That is why it will be briefly discussed here for the sake of completeness.

Diffusion Accompanied by Chemical Reaction Through a Plane Sheet

91

Concentration distribution

1.0 0.8 ϑ=0 0.6 0.4 3 0.2 5 0.0

0.0

0.2

0.4

0.6

0.8

1.0

Catalytic membrane layer

Figure 4.5 Typical concentration distribution for ultrafiltration mode at different values of the reaction modulus dφ/dy50 at y5δ.

For intrinsically catalytic membranes, the source term is equal to k0 (Q 5 k0), where it is the reaction rate constant with a dimension of mol/m3/s. In the case of dispersed fine catalyst particles, the specific mass transfer rate should be given in order to use it as a reaction term in the mass balance equation given for the membrane phase. The differential mass balance equation for a spherical catalyst particle is as
2  dφ 2 dφ 1 2 k0 5 0 D dr 2 r dr

ð4:30Þ

with boundary conditions at r 5 Rp

φ 5 φ

ð4:30aÞ

at r 5 0

dφ 50 dr

ð4:30bÞ

Solving Eq. (4.28) with the boundary conditions (4.29) and (4.30), the concentration distribution can be given as φp 5 φp 2

k0 2 ðr 2 r 2 Þ Dp p

ð4:31Þ

The specific mass transfer rate, j, related to the particle interface can be expressed as j 5 2k0 rp

ð4:32Þ

92

Basic Equations of the Mass Transport through a Membrane Layer

Taking into account that the catalyst-specific interface in the membrane layer is equal to 3ε/rp, the reaction term for the mass balance equation of the membrane, Eq. (4.1), can be given as Q5j

ω 6k0 ε 5 12ε 12ε

ð4:33Þ

When the zero-order reaction takes place on the spherical catalyst surface, the reacted amount related to the surface can be given as j 5 ks0, thus the mass transfer rate due to the reaction can be given as (the dimension of ks0 is mol/m2/s): Q5

ω ks0 12ε

ð4:34Þ

The external mass transfer resistance can also affect the reacted amount. As applied, the mass transfer rate through the boundary layer around the catalyst particles is j 5 β op ðφ 2 φp Þ: Accordingly, the maximum value of the reaction term is determined by the external transfer rate, jmax 5 β op φ; thus Q # jmax.

4.2.2.2 Mass Transfer for Zero-Order Reaction In this case, the reaction rate is independent of the concentration of reactant in the membrane layer. The differential mass balance equation can be given as d2 Φ 5 ϑ2 dY 2

ð4:35Þ

The value of ϑ can be given for an intrinsically catalytic membrane as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6k0 δ2 ε ϑ5 DHco ð12 εÞ

ð4:36Þ

The general solution of Eq. (4.35) gives Φ 5 TY 1 S 1

ϑ2 Y 2

ð4:37Þ

Applying the known boundary conditions without external mass transfer resistance  (at Y50, Φ5Φ 51 and at Y51, Φ 5 Φδ  Cδo Þ; one can obtain for the concentration distribution in the membrane as φ 5 ðφδ 2 φ ÞY 1 φ 1 φ ðY 2 1ÞY

ϑ2 2

ð4:38aÞ

Diffusion Accompanied by Chemical Reaction Through a Plane Sheet

93

or in dimensionless concentrations: Φ 5 ðΦδ 2 1ÞY 1 1 1

ϑ2 YðY 2 1Þ 2

ð4:38bÞ

The mass transfer rate entering the catalytic membrane (at Y 5 0) is as J5

  D  ϑ2 φ ð1 2 Φδ Þ 1 2 δ

ð4:39aÞ

The outlet mass transfer rate can be given as   D  ϑ2  Jδ 5 φ ð1 2 Φδ Þ 2 2 δ

ð4:39bÞ

Let us look at the mass transfer with external mass transfer resistances. For the solution of Eq. (4.37), let us use the following boundary conditions:
 D ϑ2    Φ 2 Φδ 1 2 δ

at Y 5 0

β of ð1 2 C Þ 5 J

at Y 5 1

β ofδ ðCδ 2 Cδo Þ 5 Jδ

ð4:40aÞ ð4:40bÞ

as well as [from Eqs (4.39a) and (4.39b)]: Jδ 5 J 2

D 2 ϑ δ

ð4:40cÞ

From the above equations, it can be shown:  J

5 β oov Hco

1 2 Cδo

β o ϑ2 1 o β fδ

 ð4:41Þ

where 1 1 1 1 5 o1 1 o β oov βf Hβ o β fδ In order to determine the T and S parameters, Eqs (4.40a)(4.40c) can be rewritten, taking into account Eq. (4.37), as follows: 12

S βo 5 2T o βf H

ð4:42aÞ

94

Basic Equations of the Mass Transport through a Membrane Layer


 1 ϑ2 βo 2 Cδo 5 2 o ðT 1 ϑ2 Þ T 1S2 2 β fδ H

ð4:42bÞ

Then, from Eqs (4.42a) and (4.42b), one can obtain for T and S:
 β oov βo 2 o T 5 o Cδ 2 12 o ϑ β β fδ

ð4:43aÞ

and S5

4.2.3

βoH 1H β of

ð4:43bÞ

Mass Transfer Accompanied by Second-Order Reaction

It is assumed that the reagents (components A and B) are fed separately on the feed (component A) and on the shell sides of the membrane reactor (component B), and that they are diffusing through the membrane layer concurrently (Figure 4.6) or in the countercurrent. The reaction term can be given for an intrinsically catalytic membrane as follows: Q 5 k2 φ A φ B

ð4:44Þ

Substituting the reaction term into Eq. (4.1) for both reactants and the plane interface, as well as steady-state condition (DA, DB are the diffusion coefficients for components A and B, respectively; their values can be constant or variable), one can get
 d dφA 2 k2 φA φB 5 0 DA dy dy

ð4:45aÞ


 d dφB 2 k2 φA φB 5 0 DB dy dy

ð4:45bÞ

Figure 4.6 Illustration of the concentrations for a second-order reaction.

δ

o

CA

o

φBi

CB φAi

o CBδ

i

o

CAδ

Diffusion Accompanied by Chemical Reaction Through a Plane Sheet

Figure 4.7 Division of the catalytic membrane layer assuming constant parameters in these sublayers including the external mass transfer coefficients as well (Nagy, 2006, 2010).

δ

φ*

ο

CA

φ1 φi

D1

Di

95

φN

DN φδ* ο

CδA

ΔYl

ΔYi

ΔYN

This equation can be solved either by numerical method or by developing an analytical approach. The physical model to get an analytical approach is illustrated in Figure 4.7. Essential to this method is that the membrane layer is divided into N very thin sublayers (N $ 100) and parameters ϑAi ϑBi are assumed to be constant in every sublayer. Thus, one can get a second-order differential equation with a linear source term that can be solved analytically. In dimensionless space coordinate, the mass balance equation, for the ith sublayer, will be as d2 φ A 2 ϑ2Ai φA 5 0 for Yi 2 1 # Y # Yi dY 2

ð4:46aÞ

where ϑAi

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 δ2 φBi 5 DA;i

d2 φ B 2 ϑ2Bi φB 5 0 dY 2

for Yi 2 1 # Y # Yi

ð4:46bÞ

where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 δ2 φAi ϑBi 5 DB;i where φBi and φAi denote the average concentration of components in the ith sublayer (Figure 4.6). The general solution of Eq. (4.46) is well known [Eq. (4.16)]. The solution for every sublayer has two parameters that can be determined by

96

Basic Equations of the Mass Transport through a Membrane Layer

the suitable boundary conditions. The boundary conditions will be for model A as follows: at Y 5 0

φA 5 φA ;

at Yi 2 1 # Y # Yi at Yi 2 1 # Y # Yi at Y 5 1

φ 5 φBδ

ð4:47Þ

  Di dφA  Di 1 1 dφA  5 with i 5 1; 2; . . . ; N δ dY Yi2 δ dY Yi1     φA Yi2 5 φA Yi1 with i 5 1; 2; . . . ; N

φ 5 φAδ ;

φB 5 φB

ð4:48Þ ð4:49Þ ð4:50Þ

The internal boundary conditions for component B will be the same as for component A, adapting them for component B [Eqs (4.48) and (4.49)]. The N algebraic equations obtained can be solved using the well-known Cramer rules. After solution of the N differential equation with 2N parameters to be determined, the T1 and S1 parameters for the first sublayer can be obtained as (ΔY is the thickness of the sublayers): ! 1 φAδ T  ξN φA 2 N T1 5 2 O 2ξN coshðϑA1 ΔYÞ Li 5 2 coshðϑAi ΔYÞ

ð4:51Þ

! 1 φAδ S  ξN φA 2 N S1 5 O 2ξN coshðϑA1 ΔYÞ Li 5 2 coshðϑAi ΔYÞ

ð4:52Þ

and

For details of the general solution, see Section A.2. Note that in order to determine the T1 and S1 parameters, the average concentration of the B component should be known. For this, its correct value should be used. At the starting calculation, we should assume values for φBi ; it can also be zero. After differentiating Eq. (4.16) and applying it for the first sublayer, the mass transfer rate of component A can be expressed as 0 J5

1

B DA1 ϑA1 ξSN 2 ξTN B  BφA 2 @ δ 2ξO coshðϑ ΔYÞ A1 N

C C C N A S T ðξN 2 ξ N Þ L coshðϑAj ΔYÞ φAδ

ð4:53Þ

j52

where ξ ij 5 ξ ij2 1 1 κij2 1

tanhðϑAi ΔYÞ zi 2 1

for i 5 2; 3; . . . ; N

and j 5 S; T; O

ð4:54Þ

Diffusion Accompanied by Chemical Reaction Through a Plane Sheet

97

and κij 5 ξ ij2 1 tanhðϑAi ΔYÞ 1

κij2 1 zi 2 1

for i 5 2; 3; . . . ; N 2 1 and j 5 S; T; O ð4:55Þ

The starting values of ξ1j and κ1j are as follows: ξT1 5 e 2 ϑA1 ΔY ;

ξS1 5 eϑA1 ΔY ;

ξO 1 5 tanhðϑA1 ΔYÞ

and κT1 5 2 e 2 ϑA1 ΔY ;

κS1 5 eϑA1 ΔY ;

κO 1 51

as well as zi 2 1 5

DAi ϑAi DAi 2 1 ϑAi 2 1

ð4:56Þ

Obviously, in order to get the inlet mass transfer rate of component A, the exact concentration distribution of component B is needed. Thus, for prediction of the J value, the concentration of component B must be known. It is easy to learn that a trial-and-error method should be used to get the component concentrations alternately. Three to four calculation steps are enough to get the correct value of one of the components. The concentration of a component gradually and automatically approaches its correct value belonging to a given concentration of the other component. Steps of calculation of concentration of both components can be as follows: 1. A starting concentration distribution for component B should be given and one calculates the concentration distribution of component A applying Eqs. (4.16) and (4.51)(4.57d) in three to four calculation steps, using the previously obtained concentration for every new step. 2. The indices of sublayers for component A have to be changed and adjusted to that of B starting from the permeate side of membrane, i.e., at Y 5 Yi, thus, subscript i of A, namely Ai, should be replaced by Y 5 YN 2 i. 3. Now applying the previously calculated averaged value of concentration of A ðΦAi Þ; one can predict the concentration distribution of component B, using Eqs. (4.51)(4.57d), adapting them to component B; ϑB,i parameters can significantly differ from the value of ϑA,i depending on the DB and DA values. 4. These three steps should be repeated three to four times until concentrations of both components do not change anymore.

Knowing the T1 and S1, the other parameters, namely Ti and Si (i 5 2, 3, . . . ,N), can be easily calculated by means of the internal boundary conditions given by

98

Basic Equations of the Mass Transport through a Membrane Layer

Eqs (4.48) and (4.49), starting from T2 and S2 up to TN and SN. Thus, one can get the following equations for prediction of the Ti and Si from Ti21 and Si21, for the component A: TAi eϑAi YAi 1 SAi e 2 ϑAi Yi 5 ΓAi 2 1

ð4:57aÞ

DAi ϑAi ðTAi eϑAi Yi 2 SAi e 2 ϑAi Yi Þ 5 ΞAi 2 1

ð4:57bÞ

ΓAi 2 1 5 TAi 2 1 eϑAi 2 1 Yi 1 SAi 2 1 e 2 ϑAi 2 1 Yi

ð4:57cÞ

ΞAi 2 1 5 DAi 2 1 ϑAi 2 1 ðTAi 2 1 eϑAi 2 1 Yi 2 SAi 2 1 e 2 ϑAi 2 1 Yi Þ

ð4:57dÞ

with

Now knowing the Ti and Si (with i 5 1, 2, . . . , N) parameters, the concentration distribution can be calculated easily through the membrane, its value at Y5Yi with i51 N: φA 5 TAi eϑAi Yi 1 SAi e 2 ϑAi Yi ;

Yi  iΔY

ð4:58Þ

Now, knowing the value of TN and SN, the outlet mass transfer rate can be predicted by the following equation: JAδ 5 2

DAN ϑAN ðTN eϑAN 2 SN e 2 ϑAN Þ δ

ð4:59Þ

During the calculation, the concentration distribution of the component B will also be known. From that, its mass transfer rate can easily be calculated by a similar manner as for the component A. The concentration distribution is illustrated in Figure 4.8 with the sweep phase on the permeate side (model A, i.e., dφ/dy . 0). The reaction modulus was chosen five times higher for component B, thus its value of component A was relatively low; it varied between 0.2 and 1.2. As can be seen, the concentration of component A varies only in lesser extent.

4.2.3.1 The Concentration Gradient Is Zero on the Outlet Surface, dφB/dY 5 0 at Y 5 0 It can often occur during membrane processes that one of the reactants cannot enter the flowing (sweep) phase on the other side of the catalytic membrane, when it does not solve in that phase. Accordingly, its concentration gradient will be zero at Y51 (model B). Therefore, it seems important to analyze this mass transport process. The starting equations are the same as in the previous case. Only one of

Diffusion Accompanied by Chemical Reaction Through a Plane Sheet

99

1.0

Concentration distribution

0.8

ϑBi = 1 2

4

0.6 6 0.4 0.2 ϑAi = ϑBi / 5 0.0 0.0

0.2

0.4

0.6

0.8

1.0

Catalytic membrane layer

Figure 4.8 Concentration distribution for a second-order chemical reaction ðφAδ =φA 5 φBδ =φB 5 0:2Þ (continuous lines: component B; dotted lines: component A).

the external boundary conditions, namely that at Y 5 1, is different from the previous case, namely: at Y 5 1

then

dφB 50 dY

ð4:60Þ

The solution methodology remains the same as used for the previous case or for the second-order reaction. The values of T1 and S1 obtained will be as T1 5 2

κTN φB 2κO N coshðϑB1 ΔYÞ

ð4:61Þ

and S1 5

κSN φB 2κO N coshðϑB1 ΔYÞ

ð4:62Þ

j The value of κTN ; κSN ; and κO N should be calculated by Eq. (4.55), as well as ξ i j and κi (j5T, S, O; j512N21) and transforming it for component B [Eqs (4.61) and (4.62) are valid for component B, only while for the component A Eqs (4.51)(4.57) remain valid due to its unchanged boundary conditions]. The concentration distribution is plotted in Figure 4.9 at different values of reaction modulus for the case when the B reactant cannot leave the catalytic membrane

100

Basic Equations of the Mass Transport through a Membrane Layer

Concentration distribution

1.0

ϑBi = 1 2

0.8

4

0.6

6

0.4 0.2 0.0

ϑAi = ϑBi / 5 0.0

0.2

0.4

0.6

0.8

1.0

Catalytic membrane layer

Figure 4.9 Concentration distribution of a second-order chemical reaction with no outlet flow of component B, dφB/dy50 at y50 (continuous lines: component B; dotted lines: component A).

layer. Strong effects of the reaction rate can be observed for component B. Accordingly, the inlet mass transfer rate of component B significantly increases with increasing values of the reaction modulus. On the other hand, applying the above expressions, the concentration level of the B reactant can be adjusted to the technological requirements. The concentration change of the component A is rather low.

4.2.4

Mass Transfer Accompanied by MichaelisMenten (or Monod) Kinetics

This example of reaction kinetics is especially important in the case of biochemical reactions, which is why this case will be discussed briefly.

4.2.4.1 With Sweep Phase on the Permeate Side, dφ/dY . 0 at Y51 This assumption is important regarding the boundary condition on the permeate side. Namely, it often occurs during catalytic or biocatalytic membrane reactor that there are feeding phases on both sides of the membrane reactor. Considering the membrane bioreactors, in order to increase the membrane transport rate, phases can be circulated on both sides of membrane. Thus, the external boundary conditions given by Eqs (4.47) and (4.50) should be used, assuming negligible external mass transfer resistances. The role of the general MichaelisMenten kinetics (or the Monod kinetics, in the case of living cells) is crucially important (Moser,

Diffusion Accompanied by Chemical Reaction Through a Plane Sheet

101

1988; Nagy, 2009) for biochemical reactions. Therefore, the analytical approach of this reaction will also be briefly discussed here, applying the approach used in the case of second-order reaction with constant concentration B. The reaction rate of the substrate in the catalytic membrane layer can be given as Q5

rmax φ KM 1 φ

ð4:63Þ

The source term can be rewritten for the ith sublayer as follows: Qi 5

rmax φ D ki φ KM 1 φi

ð4:64Þ

ki 5

rmax KM 1 φi

ð4:65Þ

with

Note that the value of φi denotes the average concentration of the substrate in the ith sublayer. It can be applied for its algebraic average value. In dimensionless form, one can get the following equation (Y5y/δ): d2 φ 2 ϑ2i φ 5 0 dY 2

for

Yi # φ # Yi 1 1

ð4:66Þ

with sffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi δ 2 ki δ2 vmax 5 ϑi 5 Di Di KM 1 φi The solution of Eq. (4.66) is well known [Eq. (4.16)]. For calculation of the φi value, we should use its previously predicted value. This calculation should be repeated until the concentration will be constant in every sublayer. The prediction will automatically tend to its correct value, as this problem is a self-adjusted one. The concentration distributions are illustrated in Figure 4.10 at the case of firstorder, zero-order reactions, and an intermediate reactor rate. The starting value of ϑ was chosen to be constant for every case. Both parameters, namely vmax and KM, were chosen in such a way that the starting value of ϑ remains the same. During the process, its value changed according to the actual substrate concentration. Figure 4.10 clearly shows that there is an essential difference between results obtained by the limiting cases. On the other hand, the limiting cases, namely the first-order or zero-order reaction, might be used in special cases, only if someone wants to obtain real results.

102

Basic Equations of the Mass Transport through a Membrane Layer

Concentration distribution

1.0

0.8

0.6

0.4 First order 0.2 KM / φ*= 0.01

Zeroorder 0.0 0.0

0.2

0.4

0.6

0.8

1.0

Catalytic membrane layer

Figure 4.10 Concentration change applying the MichaelisMenten kinetics (the starting pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi value of reaction modulus: ϑ 5 ðvmax =φ Þ=ðKM =φ 1 1Þ 5 2Þ:

4.2.4.2 The Concentration Gradient Is Zero on the Outlet Surface, dφ/dY 5 0 If the liquid phases, flowing on the two sides of the membrane, are immiscible and the reactant does not dissolve in the liquid on the permeate side, then its concentration gradient will be zero at Y 5 1. The same is true when there is no sweep phase. This is often the case for biocatalytic processes, e.g., hydrolysis of organic esters. Therefore, it seems important to analyze this mass transport process. The starting equations are the same as in the previous case. Only the external boundary condition is different from the previous case, namely: at Y 5 1 then

dφ 50 dY

ð4:67Þ

The solution methodology remains the same as used for the previous case or for the second-order reaction. The values of T1 and S1 obtained will be as T1 5 2

2κO N

κTN φ coshðϑ1 ΔYÞ

ð4:68Þ

and S1 5

2κO N

κSN φ coshðϑ1 ΔYÞ

ð4:69Þ

Diffusion Accompanied by Chemical Reaction Through a Plane Sheet

103

The values of κTN ; κSN ; κO N ; and those values necessary for the prediction of these three values are given by Eqs (4.54)(4.56). The inlet mass transfer rate then will be as J5 2

D1 κS 2 κTN φ ϑ1 O N δ 2κN coshðϑ1 ΔYÞ

ð4:70Þ

The concentration distribution can be calculated applying the internal boundary conditions as done in the previous case.

4.2.5

Mass Transfer Through an Asymmetric Catalytic Membrane

The mass transport through an asymmetric membrane is an important process in membrane separation. Most often the skin layer is noncatalytic; it serves as a layer retaining the catalytic particles in the sponge side of the membrane, while the sponge layer contains the catalytic particles, e.g., enzymes and microorganisms. Thus, chemical or biochemical reactions occur in the sponge layer only. In principle, feeding of the reactant substrate can be carried out on both sides of the membrane (Figure 4.11). Because the diffusion coefficient can be essentially different between the two membrane layers (mostly the diffusivity of the skin layer is one to two orders of magnitude less than that in the sponge layer), the inlet mass transfer rates can be different on the two sides of membrane.

4.2.5.1 Reactant Is Fed on the Sponge Side (Figure 4.11A) The differential mass balance equations for the sponge and the skin layers, respectively for first-order chemical reaction in the sponge layer, will be as D1

d2 φ 2 k1 φ 5 0; dy2

D2

d2 φ 5 0; dy2

Catalytic layer (1) C

J φ*1

C C C

(A)

ð4:71Þ

δ1 # y # δ

ð4:72Þ

Figure 4.11 Illustration of the mass transport through an asymmetric, catalytic membrane layer (for the internal interface: e.g., H1 φ1;δ 5 H2 φ2 Þ:

Noncatalytic layer (skin)

C

J φ*2

C C

C C C

0

0 # y # δ1

δ1 δ

C C C C

Jδ ,φ*2,δ (B)

C C C

C C C

0 δ2

δ

Jδ ,φ*1,δ

104

Basic Equations of the Mass Transport through a Membrane Layer

The general solution of the above equation system, with dimensionless space coordinate (Y 5 y/δ with δ 5 δ1 1δ2, where the subscripts 1, 2 denote the sponge and skin layers, respectively): φ 5 Aeðϑ



YÞ

1 Be 2 ðϑ



ð4:73Þ

YÞ

and φ 5 EY 1 F

ð4:74Þ

Boundary conditions for defining the parameters are as at Y 5 0

then φ1 5 A 1 B

ð4:75Þ

at Y 5

δ1 δ


 δ1 then H1 ðAeϑ 1 Be 2 ϑ Þ 5 H2 E 1 F δ

ð4:76Þ

at Y 5

δ1 δ

then

D1  ϑ D2 ϑ ðAe 2 Be 2 ϑ Þ 5 E δ δ

ð4:77Þ

at Y 5 1

then φ2;δ 5 E 1 F

ð4:78Þ

with sffiffiffiffiffiffiffiffiffi k1 δ 2 δ ϑ ; ϑ 5 D1 δ1

sffiffiffiffiffiffiffiffiffi k1 δ21 ϑ5 D1

where H1 and H2 are the solubility constants, D1 and D2 are the diffusivities for the sponge and the skin layers, respectively, δ1 is the thickness of the sponge layer, δ denotes the overall thickness of the layers, and φ and φδ are the membrane surface concentrations as they are given in Figure 4.11. After solution, one can get the following parameter values:
 H2 δ2 2 ϑ  D2  D2 2 ϑ  ϑ e φ1 2 φ2;δ 2 e φ1 DA 5 H1 δ1 D1 D1

DB 5


 H2 δ2 D2  D2 ϑ  ϑ eϑ φ1 2 φ2;δ 1 e φ1 H1 δ1 D1 D1

δ DE 5 ϑ δ1




2 φ1

H2  1 φ cosh ϑ H1 2;δ

ð4:79Þ

ð4:80Þ

 ð4:81Þ

Diffusion Accompanied by Chemical Reaction Through a Plane Sheet

DF 5 D5

D2  H2  δ φ2;δ sinh ϑ 2 ϑφ2;δ cosh ϑ 1 ϑ φ1 δ1 D1 H1

D2 H2 δ 2 sinh ϑ 1 ϑ cosh ϑ D1 H1 δ 1

105

ð4:82Þ ð4:83Þ

with I5

Di ; D

where I 5 A; B; E; F

ð4:84Þ

Let us give the inlet, J, and the outlet mass transfer rate, Jδ, which can be used in the differential mass balance equation given for the lumen and the shell sides:  β o2 H1 1  β o2 1 1 φ1 2 o φ2;δ tanh ϑ 1 o β H β ϑ ϑ cosh ϑ 1 2 1 J 5 β o1 ϑ ð4:85Þ β o2 H1 1 11 o tanh ϑ β 1 H2 ϑ The outlet mass transfer rate, that at y5δ, is as H1 φ1 2 φ2;δ o H2 cosh ϑ Jδ 5 β 2 β o H1 1 tanh ϑ 1 1 2o β 1 H2 ϑ

ð4:86Þ

with sffiffiffiffiffiffiffiffiffi k1 δ21 ϕ5 ; D1

β o1 5

D1 ; δ1

β o2 5

D2 ; δ2

δ2 5 δ 2 δ1

where subscripts 1 and 2 represent the sponge (catalytic layer) and the skin (noncatalytic) layers, respectively. The overall mass transfer rate J given by Eq. (4.85) can also be obtained by the mass transfer rates given for the resistance-in-series model, as well. For it, the inlet and outlet mass transfer rates of the sponge layer, J1 and J1,δ, as well as the inlet mass transfer rate of the skin layer, J2, should be given, namely [these equations are partly known at gasliquid systems (Danckwerts, 1970; Nagy et al., 1982)]: J  J1 5 β o1 J1;δ 5 β o1


 φ1;δ ϑ φ1 2 cosh ϑ tanh ϑ

ϑ ðφ 2 cosh ϑφ1;δ Þ sinh ϑ 1

J2 5 β o2 ðφ2 2 φ2;δ Þ

ð4:87aÞ ð4:87bÞ ð4:87cÞ

106

Basic Equations of the Mass Transport through a Membrane Layer

Taking into account that J2 5 J1,δ, and H1 φ1;δ 5 H2 φ2 ; the value of φ1;δ can be expressed and it can be replaced into Eq. (4.87a). Thus, one can get Eq. (4.85).

4.2.5.2 Reactant Is Fed on the Skin Side Figure 4.11B illustrates this situation. Equations obtained for this case serve as a good opportunity to see the difference between the two feeding modes regarding the mass transfer rates. The differential balance equations and boundary conditions remain the same. The absolute values of the boundary conditions should be changed to get the desired transport direction. The concentration distribution in the skin layer was obtained to be as φ1;δ H2  φ2 2 δ H1 cosh ϑ y 1 φ2 φ2 5 2 ϑ D2 δ1 H2 δ2 δ ϑ 1 tanh ϑ H1 δ1 D1

with

0 # y # δ2

ð4:88Þ

That for the sponge layer is as 

φ1 5 Aeϑ Y 1 Be 2 ϑ with



Y


  1 D2 H2 2 ϑδ=δ1  D2 H2 δ2 2 ϑδ2 =δ1  e 2 e φ2 1 1 ϑ φ1;δ D1 H1 D1 H1 δ 1 G 
  1 D2 H2 ϑδ=δ1  D2 H2 δ2 ϑδ2 =δ1  B5 e e φ2 2 2 ϑ φ1;δ D1 H1 δ1 G D1 H1 A5

ð4:89Þ

ð4:90Þ ð4:91Þ

and 

 D2 H2 G52 sinh ϑ 1 ϑδ2 =δ1 cosh ϑ D1 H1

ð4:92Þ

The mass transfer rates on the feed side is as   H1 φ1;δ H1 φ1;δ φ2 2 H2 cosh ϑ H2 cosh ϑ J 5 β o2  1 1 H1 1 β o2 H1 1 tanh ϑ 1 o tanh ϑ 11 o βo β 1 H2 ϑ β 1 H2 ϑ

φ2 2

ð4:93aÞ

The second form of Eq. (4.93a) is a well-known expression in the gasliquid systems where the first-order chemical reaction takes place in the liquid phase (Westerterp et al., 1984). The ratio of the mass transfer rates given by Eqs (4.85) and (4.93a) is plotted in Figure 9.9. This figure illustrates very well the difference between the two operation modes.

Diffusion Accompanied by Chemical Reaction Through a Plane Sheet

107

The outlet mass transfer rate is as
o  β o2 1 1 β 2 H1 1  φ2 2 1 tanh ϑ φ1;δ o β o1 H2 ϑ o β 1 ϑ cosh ϑ Jδ 5 β 1 ϑ β o H1 1 1 1 2o tanh ϑ β 1 H2 ϑ

ð4:93bÞ

The values of ϑ, β o1 ; β o2 are the same as in Eq. (4.86). The subscripts denote the same layer as the previous one, namely 1 denotes the spongy layer, while 2 denotes the skin layer. As expected, the mass transfer rates are independent of the direction of the transport process. Equations (4.85) and (4.86) are, by suitable application, the “same” as it was obtained by means of the countercurrent diffusion expressed by Eqs (4.93a) and (4.93b).

4.2.6

Mass Transfer with Micrometer-Sized, Dispersed, Catalyst Particles: Applying the Heterogeneous Model

Mostly, depending also on the membrane thickness, when particles fall into the micrometer-sized regime, the internal mass transport mechanism, inside of catalyst particles, must be taken into account. A simple physical model could be applied for the description of the process in this case, as schematically illustrated in Figure 4.12. The gas (or liquid) reactant enters first the catalytic membrane layer and then diffuses to the first catalytic particle, perpendicular to the membrane interface. The chemical reaction, namely a first-order, irreversible chemical reaction, takes place in the catalyst particles only. It is assumed that the concentration of the organic reactant should be much higher in a hydrophobic polymer membrane than that of the reactant investigated, such as peroxides, oxygen, hydrogen, and so forth. Then the nonreacted reactant diffuses through the first catalytic particle to its other side and enters again the βof

βofδ

O2

Jo

CH H2

J

Co

or

J

Coδ

Product

Figure 4.12 Membrane reactor with dispersed catalyst particles (for heterogeneous model, the spherical particles are modeled as cubic ones).

108

Basic Equations of the Mass Transport through a Membrane Layer

polymer membrane matrix, and so on (the route of this mass transfer process is illustrated by an arrow denoted by J in Figure 4.12). This diffusion path exists only for the heterogeneous part of the membrane interface (which is the projection of the cubic catalyst particle onto the membrane interface). There can be a portion of membrane interface, that is the so-called homogeneous part of the interface, where the diffusing reactant does not cross any catalyst particle (this mass stream is denoted by Jo in Figure 4.12). This also affects the resultant mass transfer rate. The assumed cubic (Mehra, 1999; Yawalkar et al., 2001; Nagy, 2002) catalyst particles are supposed to be uniformly distributed in the polymer membrane matrix. For the description of this transport process, the catalyst membrane layer should be divided into 2N 11 sublayers, perpendicular to the membrane interface. Namely, N sublayers for catalyst particles located perpendicular to the membrane interface, N 11 sublayers for the polymer membrane matrix between particles (ΔY) and between the first particle (Y1) and the last particle ð1 2 YN Þ and the membrane interfaces (Figure 4.13B). In order to get a mathematical expression for the mass transfer rates, a differential mass balance equation should be given for each sublayer. Thus, one can obtain a differential equation system containing 2N 11 second-order differential equations. This equation system with suitable boundary conditions can be solved analytically, which is also demonstrated in this chapter. The number of particles, N, and the distance between them, ΔY, can be calculated from the particle size, dp and the catalyst phase holdup, ε (Nagy, 2007). The distance of the first particle from the membrane interface, Y1 (Y1 5 y1/δ), can be regulated by the preparation method of the catalytic membrane layer. The differential mass balance equations for the sections of the polymer membrane phase and for that of the catalyst particles can be given, in dimensionless form, as follows, respectively: D d2 Φ 5 0; δ2 dY 2

0 # Y # Y1 ;

d2 Φ p k1 δ2 2 Φp 5 0; dY 2 Dp

(A)

Co C*

Yi 1

d # Y # Yi 1 1 ; δ

Yi # Y # Yi 1

Yi # Y # 1

ð4:94Þ

d δ

ð4:95Þ

(B)

φ∗

1

Cδo

Y1 0Y

1

i

N

ΔY

d Y •1

Yi

Y •i

YN

Y •N1

Figure 4.13 Concentration distribution in the membrane reactor (A) and a particle line with notations (B).

Diffusion Accompanied by Chemical Reaction Through a Plane Sheet

109

The solutions of the above differential equations, for the ith sections, are well known, namely: Φ 5 Ti Y 1 S i ;

1#i#N 11

ð4:96Þ

Φp 5 Ei expðϑYÞ 1 Fi expð2 ϑYÞ;

1#i#N

ð4:97Þ

with sffiffiffiffiffiffiffiffiffi k1 δ 2 ϑ5 Dp Thus, one can obtain 2N11 algebraic equations with twice as many parameters, Ti, Si (i 5 1, 2, 3, . . ., N11), as well as Ei and Fi (i 5 1, 2, 3, . . ., N), which are to be determined. Their values can be determined by means of suitable boundary conditions at the external interfaces of the membrane, at Y 5 0 and Y 5 1, as well as at the internal interfaces of every segment in the membrane matrix, at Yi and Yi with i 5 1, 2, 3, . . ., N. The effect of the external mass transfer resistances should be taken into account: β of ðco






2c Þ 

β of

at Y 5 0

then

at Y 5 1

then β ofδ ðcδ 2 coδ Þ  β ofδ

 T1 DH dΦ 52  2β o HT1 c 2 H δ dY o




 TN 1 1 1 SN 1 1 DH dΦ 2 coδ 5 2 H δ dY

ð4:98Þ

ð4:99Þ

 2β HTN 1 1 o

The boundary conditions for the internal interfaces of the sublayers are also well known (Figure 4.11B): at Y 5 Yi

ðTi Yi 1 Si ÞHp 5 Ei eϑYi 1 Fi e2ϑYi

ð4:100Þ

at Y 5 Yi

DTi 5 Dp ϑðEi eΦYi 2 Fi e2ΦYi Þ

ð4:101Þ

as well as for the other side of the catalyst particles, namely at Y 5 Yi : at Y 5 Yi 1

d 5 Yi δ

ðTi 1 1 Yi 1 Si 1 1 Þ Hp 5 Ei eϑYi 1 Fi e2ϑYi

ð4:102Þ

at Y 5 Yi 1

d 5 Yi δ

  DTi 1 1 5 Dp ϑ Ei eϑYi 2 Fi e2ϑYi

ð4:103Þ





110

Basic Equations of the Mass Transport through a Membrane Layer

Equations (4.100) and (4.102) express that there is equilibrium on the sublayer interfaces, while Eqs (4.101) and (4.103) show that there is no accumulation or source at the internal interfaces. Thus, an algebraic equation system with 2(N 11) equations can be obtained that can be solved analytically with a traditional method using the Cramer rules. The solution is briefly discussed in Nagy’s papers (2007, 2008) and in Section A.2. As a result of this solution, the mass transfer rate on the upstream side of the membrane interface, related to its heterogeneous part (which is the projection of the cubic catalyst particle onto the membrane interface), can be given as follows:

 J 5 βHco 1 2 ΨCδo

ð4:104Þ

where β 5 β o Hp

UNT 1 1 1 Hp βo

N

L

UNO 1 1 βofδH 1 Hp i 5 1




αTi αO i

 ð4:105Þ

as well as Ψ5

β ofδ

T  N  UN 1 1 1 Hp L αTi cosh ϑp

ð4:106Þ

i51

with " UNj 1 1

5

Hp ð1 2 YN Þ 1

ξ jN

#

αjN

YN 5 Y1 1 ðN 2 1ÞΔY 1 N

with J 5 T; O

d δ

ð4:107Þ

ð4:108Þ

as well as Hp φ 5 φp ;

Hco 5 φ ;

Hcδ 5 φδ

where φ , φδ denote the membrane concentration at membrane interfaces, subscript p denotes the catalyst particles in the membrane layer; The values of Uij , αji , and ξ ji with j 5 T, O should be calculated from sub-layer to sub-layer, that is from 1 to N (αji , ξji ) or N+1 (Uij ) from equations given in Table 4.1. Also it may be important to know the portion of the reactant that reacts in the catalytic membrane layer during its diffusion, or that there is an unreacted portion of the diffused reactant that passes on the downstream side of the membrane into

Diffusion Accompanied by Chemical Reaction Through a Plane Sheet

111

Table 4.1 The List of Variables That Should Be Applied for the Calculation of the Mass Transfer Rates For i 5 2 to N and j 5 T, O

 αji 5 1 1 κUij tanh ϑp ; ξ ji 5

tanhðϑp Þ 1Uij ; κ

the starting values of parameters:

 U1T 5 1; αT1 5 κ tanh ϑp ; ξ T1 5 1;

 O αO 1 5 1 1 κU1 tanh ϑp ; as well as qffiffiffiffiffiffi2ffi ϑp 5 kD1 dp ;

κ5

Dp D

qffiffiffiffi k1 Dp

ξO 1 5

tanhðϑp Þ κ

βo

i21



o U1O 5 Hp Y1 1 Hp H ββ o ; f

1 U1O ;

ΔY 5 Yi 2 Yi2 1 ;

 β po ϑp ;

ξj

Uij 5 Hp ΔY 1 αi21 ; j

β op 5

Dp d

;

βo 5

D δ

the continuous phase. This outlet mass transfer rate, for the heterogeneous part of the membrane interface, at X 5 1, can be given as follows: 0

1

J δ 5 β δ @1 2

Hp Cδo D E A ξO β o BN 2 AN αNO βfo N

ð4:109Þ

with β δ 5 β Hc o

o

ξO

BN 2 AN αNO N

β of βo

cosh N ϑp

UNO 1 1 1 Hp H ββo

o

ð4:110Þ

fδ

with Ai 5 2



 1 βO tanh ϑp βO i 21 i21 ; B i5 2; . .. ; N B 1 A 5 2 B 1 A i 21 i21 O i i21 i 21 O αi 2 1 κUiO αi 2 1 UiO

The starting values of A and B will be as: Hp Y1 A1 5 1 2 O ; U1


 tanh ϑp Hp Y1 B1 5 21 κ U1O

O O The values of αO i , β i and Ui (i 5 1, . . . ,N) are given in Table 4.1. The physical mass transfer rate for the heterogeneous part of the interface is as follows: o 5 β otot Hco ð1 2 Cδo Þ Jtot

ð4:111Þ

112

Basic Equations of the Mass Transport through a Membrane Layer

The physical mass transfer coefficient, with external mass transfer resistances, for the portion of the membrane interface where there are particles in the diffusion path, taking into account the effect of the catalyst particles as well, can be given by the following equation: β otot 5

1 H β of 1 β ofδ

1

1 βo

1

Nd Dp



1 Hp

2

Dp D

ð4:112Þ

Depending on the value of the diffusion coefficient Dp, solubility coefficient H, (φp5Hpφ), as well as the number of particles perpendicular to the interface, N, the value of the physical mass transfer coefficient of the membrane with catalytic particles, β otot , might be completely different from that of the membrane layer without catalyst particles, β o (β o5D/δ). For example, in the case of a polymer membrane filled with zeolite particles as catalyst, the value of Dp can be lowered by about four orders of magnitude than that in the polymer matrix (Yawalkar et al., 2001). The specific mass transfer rate related to the total catalytic membrane interface (Nagy, 2007) can be given as Jave 5 KJε2=3 1 J o ð1 2 Kε2=3 Þ

ð4:113Þ

or that for the outlet mass transfer rate: Jave δ 5 KJδ ε2=3 1 J o ð1 2 Kε2=3 Þ

ð4:114Þ

The value of the mass transfer rate can be easily obtained for the homogeneous part of the interface, Jo, namely: J o 5 β o Hco ð1 2 Cδo Þ

ð4:115Þ

In order to calculate the enhancement during the mass transfer accompanied by chemical reaction, the physical mass transfer rate related to the total membrane interface also should be defined: o o 2=3 5 KJtot ε 1 J o ð1 2 Kε2=3 Þ Jave

ð4:116Þ

The value of the factor K can be obtained from the distribution of catalyst particles in the polymer membrane matrix. Its value, depending on the particles distribution in the membrane matrix, should be K51 or K51.8715 (for details, see Nagy, 2007). The effect of the catalyst particle size is illustrated in Figure 4.14 at different reaction rate, applying the heterogeneous model. The mass transfer rate is very sensitive to the particle size, namely its value strongly decreases with the increase of the size. With increasing size decreases the number of particles, because the distance between them increases. Accordingly, the particle size should be decreasing as

Diffusion Accompanied by Chemical Reaction Through a Plane Sheet

13

113

Limiting value

11

J 0ave

Jave

Had=10

3

9

1

7

0.5

5

0.2

3 1 0.01

0.1 0.10

1.00

10.00

Catalyst particle size, dp, (μm)

Figure 4.14 Effect of the catalyst particle size on the mass transfer rate related to the total membrane interface, as a function of the reaction modulus (H 5 Hp 5 1; D 5 1 3 10210 m2/s; o Cfδ 5 0; β of 5 β ofδ -N; y1 5 1 μm; δ 5 30 μm; ε 5 0.1).

low as possible. Detailed analysis of this model and its comparison to the homogeneous one is given in Nagy’s paper (2007). The effect of the catalyst phase holdup is illustrated in Figure 4.15 at different reaction rates by heterogeneous (continuous lines) and the homogeneous (dotted lines) models. The mass transfer rates obtained by the homogeneous and the heterogeneous models are plotted in Figure 4.15 as a function of the catalyst holdup, at different reaction modulus. The results are in good agreement up to about ϑp 5 Had 5 1, proving that the two models give practically the same results at low particles size (dp , about 1 μm) and reaction rate. But they can differ from each other in the fast reaction rate regime.

4.2.7

Approaching an Analytical Solution of the Mass Transport with Variable Parameters

It is assumed that both the diffusion coefficient and reaction rate constant can vary as a function of space coordinate or concentration. The differential mass balance equation for first-order chemical reaction can be given as
 d dφ D½φ; y 2 kðyÞφ 5 0 dy dy

ð4:117Þ

Let us consider negligible external mass transfer resistance for the fluid phases,  thus the usual boundary conditions will be used: at y 5 0, φ 5 φ and at y5δ, φ 5 φδ : For the solution of Eq. (4.117), the catalytic membrane should be divided into N sublayer (Figure 4.7), in the direction of the mass transport, that is

114

Basic Equations of the Mass Transport through a Membrane Layer

31 ϑp = 5 1

J 0ave

Jave

21

0.5

11

0.15 1 0.0

0.1

0.2

0.3

0.4

Catalyst phase holdup, ε

Figure 4.15 Enhancement as a function of the catalyst phase holdup at different reaction rates for heterogeneous model (continuous lines) and the pseudohomogeneous one (dotted lines) (Dp 5 D 5 1 3 10210 m2/s, β of 5 β of;δ -N; coδ 5 0; ϑp 5 Had, x1 5 0.01 μm) (x1 denotes the distance of the first particle to the fluid/membrane interface), δ 5 30 μm (δ is the membrane thickness), dp 5 2 μm (dp is the catalytic particle size).

perpendicular to the membrane interface, with thickness of Δδ (Δδ5δ/N) and with constant transport parameters in every sublayer for details of a general solution (see Nagy, 2008, 2009). Thus, for the nth sublayer of the membrane layer, using dimensionless quantities, it can be obtained: Dn

d2 φn 2 kn φn 5 0; dy2

yn 2 1 , y , yn

ð4:118Þ

In dimensionless form, one can get the following equation: d2 φ n 2 ϑ2n φn 5 0 dY 2

ð4:119Þ

where ϑn 5

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi δ2 kn =Dn

The solution of Eq. (4.118) is well known [see Eq. (4.16)]: φ 5 Tn eðϑn YÞ 1 Sn eð 2 ϑn YÞ ;

Y n 2 1 , Y , Yn

ð4:120Þ

Parameters Tn and Pn of Eq. (4.120) can be determined by means of the boundary conditions for the nth sublayer (with 1 # n # N). The boundary conditions at the

Diffusion Accompanied by Chemical Reaction Through a Plane Sheet

115

internal interfaces of the sub-layers (1 # n # N 2 1; Yn 5 nΔY; ΔY 5 1/N) can be obtained from the following two equations [Eqs (4.121) and (4.122)]: dφn Dn 1 1 dφn 1 1 5 dY Dn dY

at Y 5 Yn

  φn Y 5 Y 1 5 φn 1 1 Y 5 Y 2 n

n

ð4:121Þ

at Y 5 Yn

ð4:122Þ

After solution of the N differential equation with 2N parameters to be determined, the T1 and S1 parameters for the first sublayer can be obtained as (ΔY is the thickness of the sublayers): !  1 φ δ φ ζ TN 2 N ð4:123Þ T1 5 2 O 2ζ N coshðϑ1 ΔYÞ Li 5 2 coshðϑi ΔYÞ and 1 φδ S1 5 O φ ζ SN 2 N 2ζ N coshðϑ1 ΔYÞ Li 5 2 coshðϑi ΔYÞ

! ð4:124Þ

For details of the general solution, see Section A.2. Knowing the T1 and S1, the other parameters, namely Tn and Sn (n 5 2, 3, . . ., N), can be easily calculated by means of the internal boundary conditions given by Eqs (4.121) and (4.122) starting from T2 and S2 up to TN and SN, as it is given by Eqs (4.57a)(4.57d) in subscript i parameter. After differentiating Eq. (4.120) and applying it for the first sublayer, the mass transfer rate of the reactant component can be expressed as: ! D1 ϑ1 ζ SN 2 ζ TN φδ  φ 2 S ð4:125Þ J5 N δ 2ζ O ðζ N 2 ζ TN ÞLj 5 2 coshðϑj ΔYÞ N coshðϑ1 ΔYÞ where ζ ji 5 ζ ji 2 1 1 κji 2 1

tanhðϑi ΔYÞ zi 2 1

for i 5 2; 3; . . . ; N

and j 5 S; T; O

ð4:126Þ

κji 2 1 zi 2 1

for i 5 2; 3; . . . ; N

and j 5 S; T; O

ð4:127Þ

and κji 5 ζ ji 2 1 tanhðϑi ΔYÞ 1

The starting values of ξj1 and κj1 are as follows:

116

Basic Equations of the Mass Transport through a Membrane Layer

Figure 4.16 Typical curves for illustrating the effect of variable diffuion coefficient (δ 5 100 μm; k1 5 0.1 s21; D 5 5 3 1029 25 3 10210 m2/s) [e.g., when D 5 (5 24.5i/N) 3 1029, N 5 100 then ϑ varies between 0.6 and 1.9].

Concentration distribution

1.0 0.8 0.6 0.4 ϑ = 0.6 − 1.91

0.2 0.0

ϑ = 0.6 − 1.91 0.0

0.2

0.4

0.6

0.8

1.0

Catalytic membrane layer

ζ T1 5 e 2 ϑ1 ΔY ;

ζ S1 5 eϑ1 ΔY ;

ζO 1 5 tanhðϑ1 ΔYÞ

and κT1 5 2 e 2 ϑ1 ΔY ;

κS1 5 eϑ1 ΔY ;

κO 1 51

as well as zi 2 1 5

Di ϑi Di 2 1 ϑi 2 1

ð4:128Þ

For illustration, the concentration distribution is plotted with variable diffusion coefficients and consequently with variable reaction rate modulus (Figure 4.16). The diffusion coefficient was varied about one order of magnitude in the two cases, namely with increasing or decreasing values. The curves do not show significant difference with varying diffusion coefficients.

4.2.7.1 Mass Transfer in Ultrafiltration Mode, dφ/dy 5 0 at y 5 δ The general solution is given by Eq. (4.97) for that case as well. After solution of the N differential equation with 2N parameters to be determined, the T1 and S1 parameters for the first sublayer can be obtained as (ΔY is the thickness of sublayers): T1 5 φ and

κTN O 2κN coshðϑi ΔYÞ

ð4:129Þ

Diffusion Accompanied by Chemical Reaction Through a Plane Sheet

Figure 4.17 Concentration variation obtained by increasing reaction modulus [D 5 (2.5 20.25 3 9 3 i/ N) 3 1029], at the average ϑ value (D 5 1.25 3 1029), as well as by decreasing reaction modulus D5 (0.25 10.25 3 9 3 i/N) 3 1029, δ 5 100 μm; k1 5 0.1 s21.

Concentration distribution

1.0 0.9 0.8

ϑ = 1.9 → 0.63

0.7

ϑ = 0.89

0.6 ϑ = 0.63 → 1.9 0.5

0.0

0.2

0.4

0.6

0.8

117

1.0

Catalytic membrane layer

S1 5 φ

κSN 2κO N coshðϑi ΔYÞ

ð4:130Þ

where κjN 5 tanhðϑi ΔYÞ 1

κji 2 1 zi 2 1

with j 5 T; S; O

ð4:131Þ

The other values of κji (i 5 2 2 N 21) are the same as it given by Eq. (4.55). Accordingly, other parameters are also the same as in Eqs. (4.126)(4.128). Figure 4.17 illustrates the concentration distribution with increasing and decreasing reaction modulus due to the change of the diffusion coefficient in the membrane, as well as at the average value of ϑ. In this case, the difference between the curves is relatively large. This indicates that the transport essentially can be affected by the anisotropy of a membrane.

4.3

Unsteady-State Diffusion and Reaction

Here we consider a case where dilute A is absorbed at the surface of a solid, catalytic membrane layer and then unsteady-state diffusion and reaction occur in the membrane. The absorbed A reacts by a first-order reaction: A 1 B-C

ð4:132Þ

and the rate of generation is k1φ. The mass balance equation will be then as

118

Basic Equations of the Mass Transport through a Membrane Layer

@φ d2 φ 5 D 2 2 k1 φ 5 0 @t dy

ð4:133Þ

The initial and boundary conditions are t 5 0;

φ50

for y . 0

ð4:134aÞ

y 5 0;

φ 5 φ

for t . 0

ð4:134bÞ

y 5 δ;

φ 5 φδ

for t . 0

ð4:134cÞ

Let us solve Eq. (4.133) with the conditions (4.134a)(4.134c) by Laplace transformation. The Laplace-transformed form of Eq. (4.133) will be as ~ d2 Φ sδ2 ~ 2~ Φ 2 ϑ Φ 5 D dY 2

ð4:135Þ

 ~ is the Laplace transform where Φ is the dimensionless concentration, Φ 5 φ/φ , Φ of Φ(s) and Y 5 y/δ. After integration of Eq. (4.135), one can get

~ 5 TeΘY 1 Se 2 ΘY Φ

ð4:136Þ

where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sδ2 Θ 5 ϑ2 1 D The T and S parameters can be determined by the following boundary conditions: Y 5 0;

~ 5 1=s Φ

ð4:137aÞ

Y 5 1;

  ~ 5 φ δ  Φδ Φ φ s s

ð4:137bÞ


~ Φ5 12


  Φδ Φδ 2 eΘ ΘY e 1 e 2 ΘY 2s sinh Θ 2s sinh Θ

ð4:138Þ

The dimensionless concentration distribution can be obtained by the inverse transformation of Eq. (4.138) as

Diffusion Accompanied by Chemical Reaction Through a Plane Sheet

119

~ ΦðtÞ 5 L 2 1 ½ΦðsÞ The solution for fast reaction where the catalytic membrane can be considered as a semiinfinite medium (Danckwerts, 1950):
 pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi φ 1 y p ffiffiffiffiffi expð 2 y k 5 =D Þerfc t 2 k 1 1 φ 2 2 tD
pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 1 y 1 expðy k1 =DÞerfc pffiffiffiffiffi 1 k1 t 2 2 tD

ð4:139Þ

References Danckwerts, P.V. (1950) Absorption by simultaneous diffusion and chemical reaction. Trans. Faraday Soc. 46, 300305. Danckwerts, P.V. (1970) Gas-Liquid Reactions, McGraw-Hill, New York. Ferreira, B.S., Fernandes, P., and Cabral, J.M.S. (2001) Design and modeling of immobilized biocatalytic reactors, in Multiphase bioreactor design. Ed. by J.M.S. Cabral, M. Mota, J. Tramper), Taylor & Francis, London, pp. 85180. Julbe, A., Farusseng, D., and Guizard, C. (2001) Porous ceramic membranes for catalytic reactors—overview and new ideas. J. Membr. Sci. 181, 320. Marcano, J.G.S., and Tsotsis, T.T. (2002) Catalytic Membranes and Membrane Reactions. Wiley-VCH, Weinheim. Mehra, A. (1999) Heterogeneous modeling of gas absorption in emulsion. Ind. Eng. Chem. Res. 38, 24602468. Moser, A. (1988) Bioprocess Technology. Springer, Wien. Nagy, E. (2002) Three-phase oxygen absorption and its effect on fermentation. Adv. Biochem. Eng. Biotechnol. 75, 5181. Nagy, E. (2006) Binary, coupled mass transfer with variable diffusivity through cylindrical membrane. J. Membr. Sci. 274, 159168. Nagy, E. (2007) Mass transfer through a dense, polymeric, catalytic membrane layer with dispersed catalyst. Ind. Eng. Chem. Res. 46, 22952306. Nagy, E. (2008) Mass transport with varying diffusion- and solubility coefficient through a catalytic membrane layer. Chem. Eng. Res. Design 86, 723730. Nagy, E. (2009) Mathematical Modeling of Biochemical Membrane Reactors, in Membrane Operations, Innovative Separations and Transformations, Ed. by, E., Drioli, and L., Giorno. Wiley-VCH, Weinheim, pp. 309334. Nagy, E. (2010) Convective and diffusive mass transport through anisotropic, capillary membrane. Chem. Eng. Process. Process Intens. 49, 716721. Nagy, E., and Moser, A. (1995) Three-phase mass transfer: Improved pseudo-homogeneous model. AIChE J. 41, 2334. Nagy, E., Blickle, T., and Ujhidy, A. (1989) Spherical effect on mass transfer between fine solid particles and liquid accompanied by chemical reaction. Chem. Eng. Sci. 44, 198201.

120

Basic Equations of the Mass Transport through a Membrane Layer

Nagy, E., Blickle, T., Ujhidy, A., and Horva´th, K. (1982) Mass transfer accompanied by first order intermediate reaction rate in two phase cocurrent flow with axial dispersion. Chem. Eng. Sci. 37, 18171819. Vancelecom, I.F.J., and Jacobs, P.A. (2000) Dense organic catalytic membrane for fine chemical synthesis. Catalysis Today 56, 147157. Vital, J., Ramos, A.M., Silva, I.F., Valenete, H., and Castanheiro, J.E. (2001) Hydration of α-pinene over zeolites and activated carbons dispersed in polymeric membranes. Catalysis Today 67, 217223. Yawalkar, A.A., Pangarkar, V.G., and Baron, G.V. (2001) Alkene epoxidation with peroxide in a catalytic membrane reactor: a theoretical study. J. Membr. Sci. 182, 129213. Westermann, T., and Melin, T. (2009) Flow-through catalytic membrane reactors—principles and applications. Chem. Eng. Process. 48, 1728. Westerterp, K.R., van Swaaij, W.P.M, and Beenackers, A.A.C.M. (1984) Chemical Reactor Design and Operation. John Wiley and Sons, New York.

5 Diffusive Plus Convective Mass Transport Through a Plane Membrane Layer

5.1

Introduction

Convective mass transport can take place if the transmembrane pressure difference exists between the two membrane sides. There are several membrane processes when the diffusive driving force, for the mass transport, is enlarged by the pressure difference between the two sides of membrane, causing convection flow as well. The investigation of the simultaneous effect of the diffusive and convective flows is especially important when the measures of the two flows are comparable with each other. This can be the case by nanofiltration membrane bioreactor and so forth.

5.2

Mass Transport Without Chemical Reaction

The mass balance equation for a plane membrane layer with constant transport parameters to be solved as (D denotes here the effective diffusion coefficient in the membrane layer): D

d2 φ dφ 50 2υ 2 dy dy

ð5:1Þ

where φ is the concentration in the membrane layer (kg/m3); υ is the convective velocity (m/s); y denotes space coordinate, perpendicular to the membrane surface (m); D is the diffusion coefficient in the membrane (m2/s) (note that the membrane diffusion coefficient means the effective one in the membrane which should involve the effect of the membrane structure, the interaction between the transported molecules and membrane material, etc.; see Chapters 3 and 4). In dimensionless space, coordinate (Y 5 y/δ): d2 φ dφ 50 2 Pe 2 dY dY Basic Equations of the Mass Transport through a Membrane Layer. DOI: 10.1016/B978-0-12-416025-5.00006-5 © 2012 Elsevier Inc. All rights reserved.

ð5:2Þ

122

Basic Equations of the Mass Transport through a Membrane Layer

where (Pe denotes the Peclet number) Pe 5

υδ D

The concentration distribution, without chemical reaction, is very simple to obtain. The solution of the differential equation, Eq. (5.2), without source term is as follows: φ 5 TePeY 1 S

ð5:3Þ

where Y denotes the dimensionless space coordinate (Y 5 y/δ) and δ is the thickness of the membrane layer (m). Values of parameters T and S can be determined, without external diffusive mass transfer resistances, by the boundary conditions according to the value of concentration gradient on the downstream (permeate) side of membrane. There can be diffusive flux into the fluid phase on the permeate side, i.e. dφ/dy . 0 at y 5 0. The boundary conditions are as (see Fig. 11.1 for denotes and the possible concentration distribution): at Y 5 0

φ 5 φ

ð5:4aÞ

at Y 5 1

φ 5 φδ

ð5:4bÞ

The concentration distribution, applying Eqs (5.4a) and (5.4b) as boundary conditions, is as follows: 

  ePeY=2 Pe PeY   2Pe=2 ½1 2 Y  φ 1 e sinh sinh φδ φ5 sinhðPe=2Þ 2 2

ð5:5Þ

The concentration distribution is illustrated in the membrane layer as a function of the Peclet number in Figure 5.1. The outlet membrane concentration was chosen to be 0.1. The overall mass transfer rate, namely the sum of the convective and the diffusive flows, can be expressed as J 5 2D

dφ 1 υφ dy

ð5:6Þ

Thus, the physical mass transfer rate, through the membrane, can be obtained by means of Eqs (5.6) and (5.7) as follows: J o 5 β}o ðφ 2 e2Pe φδ Þ

ð5:7Þ

where β}o 5

D ePe Pe ePe=2  βo Pe Pe e 21 δ 2 sinhðPe=2Þ

ð5:8Þ

Diffusive Plus Convective Mass Transport Through a Plane Membrane Layer

Figure 5.1 Concentration distribution in a layer as a function of the Peclet number, without external mass transfer coefficient.

1.0 Concentration distribution

10 0.8

3 1

Pe = 0.01

0.6

123

5

2

0.1 0.4 0.2 0.0 0.0

φδ*/ φ* = 0.1 0.2

0.4

0.6

0.8

1.0

Membrane thickness

Note that the physical mass transfer coefficient in the case of the simultaneous effect of the diffusion and convection, β}o (it denotes the overall mass transfer coefficient in the presence of convective and diffusive flows), can be much higher than that of the diffusion mass transfer coefficient, β o 5 D/δ. The β}o value increases linearly if Pe . about 3. It should be noted also that the driving force depends on the Peclet number as it is given by Eq. (5.7). With the increase of the Pe number, the driving force also increases. In a limiting case, namely if Pe-~, the value of e2Pe φδ 5 0: The physical mass transfer rate can also be given by diffusive mass transfer resistances in the boundary layers of fluid phases (see Figure 5.3A). The mass transfer rate for the boundary layers can be expressed as follows: if Y 5 0

then J o 5 β of} ðco 2 e2Pef c Þ

ð5:9Þ

where, according to Eq. (5.8) (Pef 5 υδf/Df; f subscript denotes the external feed phase, while δ subscript denotes the permeate or shell phase): β of} 5 β of

Pef ePef =2 2 sinhðPef =2Þ

ð5:10Þ

as well as for the permeate side (with subscript fδ) boundary layer ðPefδ 5 υδfδ =Dfδ  υ=β ofδ Þ: if

Y 5 1 then

J o 5 β ofδ} ðcδ 2 e2Pefδ coδ Þ

ð5:11Þ

The value of β ofδ} can be obtained with the same equation than β of} [Eq. (5.10)] with β ofδ 5 Df δ =δf δ :

124

Basic Equations of the Mass Transport through a Membrane Layer

Applying Eqs (5.7), (5.8), and (5.11), the mass transfer rate for the overall mass transfer resistances will be as   J o 5 β oov co 2 e2ðPef 1 Pe 1 Pefδ Þ coδ

ð5:12Þ

1 1 e2Pef e2ðPef 1 PeÞ 1 o 5 o 1 o β ov} β f} β} H β ofδ}

ð5:13Þ

with

Concentration distribution with concentration boundary layer on the feed side with dφ/dy . 0 at Y 5 1: Equation (5.12) gives the mass transfer rate with mass transfer resistances on both sides of the membrane. Regarding its importance, the concentration distribution will be given here also in the presence of a concentration polarization layer on the feed phase only. The mass transfer resistance is neglected on the permeate phase. For it, the mass balance equation for both layers should be solved as a differential equation system. Equation (5.2) should also be adapted to the feed boundary layer as well. Its solution is the same as in Eq. (5.3). The external boundary conditions for the solution of the parameters are as usual: y 5 0, c 5 co and at y 5 δf 1 δ then c 5 coδ (β ofδ -N; φδ 5 Hcoδ ). The internal boundary conditions express that the total transfer rate and the concentration are equal to each other on the inner edge of the boundary layer and on the feed side of the membrane, that is, at y 5 δ. The parameters obtained will be as (Hc 5 φ). For the liquid (fluid) phase (Nagy and Borbe´ly, 2007): c 5 Tf ePef 1 Sf

ð5:14Þ

where Tf 5

co ð1 2 e2Pe 2 HÞ 1 Hcoδ e2Pe HePef 2 e2Pe 1 1 2 H

ð5:15aÞ

and Sf 5 H

co ePef 2 coδ e2Pe HePef 2 e2Pe 1 1 2 H

ð5:15bÞ

For the membrane phase with parameters T, S in Eq. (5.3): T 5H S 5 Sf

2co ePef 1 coδ ðHePef 1 1 2 HÞ Pe ePeð11δf =δÞ½He f 2ePe112H

ð5:16aÞ ð5:16bÞ

Diffusive Plus Convective Mass Transport Through a Plane Membrane Layer

125

1.0 Concentration distribution

Pe = 10 0.8 Pef = 1

0.6

1

o β f = 1×10 –4 m/s

0.4

0.5

δf / δ = 1

0.2

2

0.1

0.0 0.0

0.4

0.8

1.2

1.6

2.0

Boundary layer and membrane thickness

Figure 5.2 Concentration distribution in the boundary layer and membrane layer at different Pe numbers.

The concentration distribution in both layers is illustrated in Figure 5.2. It is clearly shown that the value of the concentration on the membrane interface, namely at y 5 δ (dotted vertical line), strongly depends on the Pe value. The overall mass transfer rate in the presence of the feed side mass transfer resistance also will be increased. For two-layer diffusive plus convective mass transfer, the overall transfer rate will be as [see Eq. (5.12)] (Nagy, 2007a; Nagy and Kulcsa´r, 2009): J o 5 β oov ðco 2 e2ðPef1PeÞ coδ Þ

ð5:17Þ

with β oov 5

1 ð1=β of} Þ 1ðe2Pef =β}o Þ

where Pe and Pef are the Peclet numbers of the membrane and the polarization layers, respectively. Mass transfer without diffusive flux into the permeate side, i.e., dφ/dy 5 0 at Y 5 1. This situation can exist when there is no sweep phase on the downstream side or the transported component is immiscible in the sweep phase. This letter case can often occur, for example, at enzymatic bireactions, when the two reactants are in immiscible phases. Let us look first the case when the membrane is considered as a “black box,” and the boundary conditions of the boundary layer at memo o 5 JY51 Þ: brane interfaces are as ðJY50 2D

dc 9 1 υc9Y50 5 υcp dy Y50

ð5:18aÞ

126

Basic Equations of the Mass Transport through a Membrane Layer

and c 5 co

Y 50

ð5:18bÞ

Applying the general solution of the concentration, given by Eq. (5.3) in the membrane, for the boundary layer, one can get the concentration distribution of the boundary layer as: c 2 cp 5 ePef Y co 2 cp

ð5:18cÞ

From the above expression, the following well-known equation can be obtained at Y 5 1 (where c 5 c ): c 2 cp 5 ePef Y co 2 cp

ð5:18dÞ

Details about the origin of the above expression are given in Section 11.4. Equations (5.18c) and (5.18d) do not contain any information on the concentration distribution or its gradient inside the membrane (it can be larger than zero as it is used for description of pervaporation process; see Chapter 11). Assuming that dφ/dy 5 0 at Y 5 1 (here at y 5 δf 1 δ; see Figure 11.1), then the concentration gradient and, due to it, the diffusive flow will be zero along the whole membrane, accordingly the outlet mass transfer rate ðJδo Þ as well. Thus, Jδo 5 υφ

ð5:18eÞ

The mass transfer rate, given by Eq. (5.9) for the boundary (polarization) layer, and that for the membrane layer [Eq. (5.18e)] are equal to each other. Applying the resistance-in-series model, one can get for the overall mass transfer rate as: o 5 Jov }

υePef β of Pef ePef o c co  o HυðePef 2 1Þ 1 1 Hβ f Pef ðePef 2 1Þ 1 1

ð5:18fÞ

It is easy to see when the solubility coefficient is equal to unit, i.e., H 5 1 (the transported component is not dissolved in the membrane matrix), then o 5 β of Pef co  υco Jov }

ð5:18gÞ

Thus, there is diffusive flux neither in the membrane layer and nor in the boundary layer. Note that it was assumed that the convective velocity has the same value in both layers. But it can also occur that the υ value differs in the membrane and the boundary layers. Especially, when the diffusive and the convective flows have similar values. This can be the case of the nanofiltration which is discussed in detail in Chapter 10. This case is not discussed here.

Diffusive Plus Convective Mass Transport Through a Plane Membrane Layer

5.3

127

Diffusive Plus Convective Mass Transport with an Intrinsic Catalytic Layer or with Fine Catalytic Particles

Recently, it was proved in the literature (Ilinitch et al., 2000; Nagy, 2010) that the presence of convective flow can improve the efficiency of the membrane reactor. Thus, the study of the mass transport in the presence of convective mass flow can be important in order to predict the reaction process. On the other hand, the use of convective flow is rather rare, because the aim is mostly to minimize the outlet rate of the reactant on the permeate side.

5.3.1

Mass Transport Accompanied by First-Order Reaction

The differential mass balance equation for the polymeric or macroporous ceramic catalytic membrane layer, for steady state, taking both diffusive and convective flow into account, can be given as D

d2 φ dφ 2υ 2 k1 φ 5 0 2 dy dy

ð5:19Þ

or in a dimensionless form of space coordinate d2 φ dφ 2 Pe 2 ϑ2 φ 5 0 dY 2 dY

ð5:20Þ

where υδ Pe 5 ; D

sffiffiffiffiffiffiffiffiffi k1 δ 2 ϑ5 D

where υ denotes the convective velocity; D is the diffusion coefficient of the membrane; and δ is the membrane thickness. In the case of dispersed catalyst particle in the membrane matrix, the reaction takes place in the particles or at their interface. In this case, the k1 “reaction rate” can be defined by the following equation, as discussed in detail in Chapter 4 (Nagy, 2007b): k1 5

ω β 1 2 ε ptot

ð5:21Þ

where β ptot is determined by the mass transfer rate into the catalyst particles [see Eq. (4.12) or Eq. (28) in Nagy, 2007b]; ω is the specific interface of catalyst particles in the membrane ( 5 6ε/dp, where dp is the particle size, m) (m2/m3); and ε is the catalyst phase holdup.

128

Basic Equations of the Mass Transport through a Membrane Layer

φ~ 5 φe2PeY=2

ð5:22Þ

Introducing a new variable, φ~ [Eq. (5.17)], the following differential equation is obtained from Eq. (5.20): d2 φ~ 2 Θ2 φ~ 5 0 dY 2

ð5:23Þ

where rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pe2 1 ϑ2 Θ5 4

ð5:24Þ

The general solution of Eq. (5.23) is well known, so the concentration distribution in the catalytic membrane layer can be given as follows: ~

φ 5 TeλY 1 SeλY

ð5:25Þ

with Pe λ~ 5 2 Θ; 2

λ5

Pe 1Θ 2

The inlet and the outlet mass transfer rate can easily be expressed by means of Eq. (5.25). The overall inlet mass transfer rate, namely the sum of the diffusive and convective mass transfer rates, is given by J 5 υφ9Y50 2

D dφ ~ 1 λSÞ 9 5 β o ðλT δ dY Y50

ð5:26Þ

with βo 5

D δ

The outlet mass transfer rate is obtained in a similar way as Eq. (5.26) for Y 5 1: ~ λ 1 λSeλÞ ~ Jδ 5 β o ðλTe

ð5:27Þ

The value of parameters T and S can be determined from the boundary conditions. For the sake of generality, two models, namely models A and B, will be distinguished according to Figure 5.3 (for details, see Nagy, 2010). The essential difference between the models is that, in the case of model A, there is a sweep

Diffusive Plus Convective Mass Transport Through a Plane Membrane Layer

(A)

Figure 5.3 Illustration of the concentration distribution for mass transport with diffusion 1 convection for models A (there is a diffusive outlet flux, dφ/dy . 0 at y 5 δ) and B (there is no diffusive outlet flux dφ/dy 5 0 at y 5 δ).

Catalyst particles (B)

φ∗δ

Cο

Cο

ο

Cδ

129

Cδ

φ∗

βof βo βfoδ

phase that can remove the transported component from the downstream side, providing the low concentration of the reacted component in the outlet phase and due to it, a high diffusive mass transfer rate. There is no sweep phase on the permeate side, in the case of model B, thus the outlet phase is moving from the membrane due to the lower pressure on the permeate side, that is, due to convective flow, only because there is no diffusive outlet flow. Mass transport without external resistances, for dφ/dy . 0 at y 5 δ. An important limiting case should also be mentioned, namely the case when the external diffusive mass transfer resistances on both sides of membrane can be neglected, i.e., when β of -N and β ofδ -N: Accordingly, the boundary conditions will be: at   Y 5 0, φ 5 φ (φ 5 Hco) and at Y 5 1 then φδ 5 Hcoδ : For that case, the concentration distribution and the inlet mass transfer rate can be expressed by Eqs (5.28) and (5.29), respectively. o ePeðY21Þ=2 n Pe=2 e sinh½Θð1 2 YÞφ 1 φδ sinhðΘYÞ sinh Θ
 Θ J 5 β φ 2 Pe=2 φδ e ½ðPe=2Þsinh Θ 1 Θ cosh Θ

φ5

ð5:28Þ ð5:29Þ

with β5

β o ð½Pe=2tanh Θ 1 ΘÞ tanh Θ

ð5:30Þ

In several cases, the value of the outlet mass transfer rate, Jδ, can be important during catalytic or biocatalytic reactions in a membrane reactor. Its value will be as
 Θ 2ððPe=2Þtanh ΘÞ   cosh Θφδ Jδ 5 β δ φ 2 ΘePe=2

ð5:31Þ

130

Basic Equations of the Mass Transport through a Membrane Layer

1.0 Concentration distribution

Pe = 30 0.8 10

0.6 5 0.4 1 0

0.2 0.0

3

0.0

0.2

0.4

0.6

0.8

1.0

Catalytic membrane layer

Figure 5.4 Concentration distribution in a catalytic membrane by means of Eq. (5.28); ϑ 5 3.

with βδ 5

D ΘePe=2 δ sinh Θ

ð5:32Þ

The effect of the convective flow, namely the Peclet number, is illustrated in Figure 5.4. The outlet concentration was chosen to be zero, while the reaction modulus was 3 for this calculation (ϑ 5 3). This is a typical figure regarding the concentration distribution. By increasing the Peclet number, the trend of curves will gradually change from convex to concave ones. Looking at the concentration gradients close to the inlet interface, it can be seen clearly that the inlet diffusive flow strongly decreases with the increase of the Peclet number. It is also obvious from this figure that the outlet diffusive flow also strongly increases with the increase of the Peclet number, owing to the increasing portion of the unreacted amount of reactant in the catalytic membrane. This value can be predicted exactly by means of Eq. (5.32) for the plotted case, namely at φδ 5 0: The membrane reactor makes it possible to supply the reactant on both sides of the membrane layer. This can be useful when the reactant concentration should be maintained above a critical level, such as in case of biocatalytic processes, especially in the case of sparingly soluble oxygen supply for cell culture. This situation is illustrated in Figure 5.5 where the effect of the reaction modulus is plotted at given interface concentrations. Obviously, the mass transport takes place by diffusion, only (Pe 5 0) in countercurrent mode. The application of this mode can be useful, only if the two reactants are fed, separatedly, on the two sides of the catalytic membrane layer. How the Peclet number affects the enhancement as a function of the reaction rate is illustrated in

Diffusive Plus Convective Mass Transport Through a Plane Membrane Layer

131

Concentration distribution

1.0 ϑ=0

0.8

1

0.6

3

0.4

5

0.2 0.0

10

Pe = 0 0.0

0.2

0.4

0.6

0.8

1.0

Catalytic membrane layer

Figure 5.5 The reactant is fed on both sides of the catalytic membrane layer.

Pe = 0.1

Enhancement (β/βo)

17.0

1 13.0

1 2 4

9.0 10

5.0 1.0 1.0

25 10.0

100.0

Reaction modulus

Figure 5.6 Effect of the reaction modulus and Pe number on the enhancement of the mass transfer rate.

Figure 5.6. As expected, the increase of the convective velocity can significantly decrease the effect of the chemical and biochemical reactions. The reacted amount of the reactant can also be given by the following expression: ð1 o ePeðY 21Þ=2 n  Pe=2 Jreact 5 δ k1 φ e sinh½Θð1 2 YÞ 1 φδ sinhðΘYÞ dY ð5:33Þ sinh Θ 0 Thus, by integration of the above equation, one can obtain Jreact 5 β r ðφ 2 F2 φδ Þ

ð5:34Þ

132

Basic Equations of the Mass Transport through a Membrane Layer

with βr 5

D ΘððPe=2Þ 1 1Þ 2ðΘePe=2 =cosh ΘÞ δ tanh Θ

ð5:35Þ

F2 5

Θ½ðe2Pe1 =2 =cosh ΘÞ 2 1 1ðPe1 =2Þtanh Θ Θ½1 2ðePe1 =2 =cosh ΘÞ 1ðPe1 =2Þtanh Θ

ð5:36Þ

The same result can be obtained by the difference of J 2 Jδ, applying Eqs (5.29) and (5.31). Mass transfer resistance is at both sides of membrane: In this case, due to the effect of the sweeping phase, the external mass transfer resistance on both sides of the membrane should be taken into account in the boundary conditions, though the role of β oδ is gradually diminished as the catalytic reaction rate increases (model A). The concentration distribution in the catalytic membrane when applying a sweep phase on the two sides of the membrane is illustrated as well in Figure 5.3A. On the upper part of the catalytic membrane layer, in Figure 5.3A, the fine catalyst particles are illustrated with black dots. It is assumed that these particles are homogeneously distributed in the membrane matrix. Due to sweeping phase, the concentration of the bulk phase on the permeate side may be lower than that on the membrane interface. The boundary conditions can be given for that case as υc 1 β of ðco 2 c Þ 5 J

at Y 5 0

υcδ 1 β ofδ ðcδ 2 coδ Þ 5 Jδ

at Y 5 1

ð5:37Þ ð5:38Þ 

where the values of J and Jδ are expressed in Eqs (5.26) and (5.27), respectively; c and cδ denote the fluid phase concentration on the feed side and permeate side of the membrane, respectively (kg/m3). Boundary conditions, given by Eqs (5.37) and (5.38), are only valid in the case of two external flowing phase flows. Where coδ denotes the concentration on the downstream side of the continuous phase, β of and β ofδ are the diffusive mass transfer coefficients in the continuous phases. The solution of the algebraic equations, applying Eq (5.25) as well as Eqs (5.37) and (5.38), can be obtained by means of known mathematical manipulations (not shown here). Thus, the values of T and S obtained for Eq. (5.25) in order to describe the concentration distribution are as fol  lows (Hc 5 φ ; here β of 5 Df =δf ; β ofδ 5 Dfδ =δfδ Þ; β o 5 D/δ: T 52

β of ξ2 co 2 β ofδ ξ 4 coδ 1 ξ2 ξ3 2 ξ1 ξ4 β o

ð5:39Þ

and S5

β of ξ 1 co 1 β ofδ ξ3 coδ 1 ξ2 ξ3 2 ξ1 ξ4 β o

ð5:40Þ

Diffusive Plus Convective Mass Transport Through a Plane Membrane Layer

133

1.5 Pe = 0.01

J/J o

1.4

Pe f =1

0.1

1.3 1

1.2 1.1

10

1.0 0.10

1.00

10.00

Reaction modulus

Figure 5.7 The ratio of J/Jo as a function of first-order reaction modulus, ϑ, at different values of Pe number and in the presence of external mass transfer resistances ðβ of 5 β o 5 1 3 1024 m=sÞ:

where
 Pe β oδ ~ 1 o 2 λ eλ ; ξ1 5 β H H ξ3 5

Pe βo ~ 2 o f 2 λ; β H H

ξ4 5


 Pe β oδ ~ 1 o 2 λ eλ; ξ2 5 β H H Pe βo 2 o f 2 λ; β H H

Knowing the value of T and S from Eqs (5.39) and (5.40), the inlet and outlet mass transfer rates can be calculated by Eq. (5.26) and Eq. (5.27), respectively (not written here). The effect of the enhancement is illustrated at different Peclet numbers, in Figure 5.7 in presence of the external mass transfer resistance, according to Eqs (5.26), (5.39), and (5.40). Jo denotes the physical (mass transfer without chemical reaction) mass transfer rate. The curves tend to a limiting value as it is expected. This value is limited by the external mass transfer resistances. No sweep phase on the one side of membrane, thus dφ/dy 5 0 at Y 5 δ. This case of catalytic membrane reactor is operating in dead-end mode as in Figure 5.3B. In this case, the concentration of the permeate phase does not change during the mass transport from the membrane interface (there is only convective outlet flow). If there is no sweep phase on the downstream side, then the correct boundary conditions will be as υc 1 β of ðco 2 c Þ 5 J υφδ 5 Jδ

at Y 5 1

at Y 5 0

ð5:41Þ ð5:42Þ

134

Basic Equations of the Mass Transport through a Membrane Layer

After solution, one obtains T 52 S5

β of ξ2 co 1 ξ2 ξ3 2 ξ1 ξ4 β o

ð5:43Þ

β of ξ1 co 1 ξ2 ξ3 2 ξ1 ξ4 β o

where ξ1 5


 Pe ~ λ 2λ e ; H

ð5:44Þ

ξ2 5


 Pe ~ 2 λ eλ H

The values of ξ 3 and ξ 4 are the same as they are given after Eq. (5.40). Mass transport without mass transfer resistances. During the catalytic membrane processes, it can often occur that the transported component cannot enter the permeate side, accordingly dφ/dy 5 0 at y 5 δ. Assuming that there is no mass  transfer resistance in the feed phase, thus, at y 5 0 then φ 5 φ , the solution of the differential equation given by Eq. (5.19) gives the following concentration distribution in the membrane and mass transfer rate: φ 5 φ ePeY=2 with

½  sinh½Θð1 2 YÞ Pe 2 2 Θ cosh Θð1 2 YÞ Pe 2 sinh Θ 2 Θ cosh Θ

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pe2 Θ5 1 ϑ2 ; 4

ϑi 5

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi δ2 k1 =D;

Pe 5

ð5:45Þ

υδ D

Note that Eq. (5.45) does not involve the external mass transfer resistances. The inlet mass transfer rate, namely the sum of the diffusive and the convective flows, can be given as J 5 βφ with

ð5:46Þ



2 2 Θ tanh Θ D Pe  β5 δ 2 tanh Θ 2 Θ Pe2 4

The outlet mass transfer rate, Jδ, is the convective flow; thus, Jδ 5 υφδ, because the diffusive flow is zero in this case. The concentration distribution is plotted at different values of the Peclet number with zero outlet diffusive flows for mass transfer accompanied by a first-order reaction (Figure 5.8). The value of reaction modulus chosen was relatively high for better illustration the effect of the Peclet number. The concentration level of the

Diffusive Plus Convective Mass Transport Through a Plane Membrane Layer

135

1.0 Concentration distribution

Pe = 30 0.8 10

0.6 5 0.4

1

3

0

0.2

ϑ=3 0.0

0.0

0.2

0.4

0.6

0.8

1.0

Catalytic membrane layer

Figure 5.8 Concentration distribution when there is no diffusive outlet flow on the permeate side (dφ/dY 5 0 at Y 5 1); ϑ 5 3.

reactant essentially can be affected by the Peclet number. Obviously, this level can strongly depend on the ϑ value. Let us compare the concentration change close to the inlet region of the catalytic membrane in Figures 5.4 and 5.8. At a given value of Pe, the concentration has practically the same values between Y 5 0 and 0.2 for the two operating modes, namely when the outlet diffusion flow can be larger than zero (Figure 5.4) or it is equal to zero (Figure 5.8). This fact should mean that the values of the inlet diffusive flows are practically the same in the two operating modes. Accordingly, the same mass transfer rate can be achieved with a higher concentration level by the operating mode when dφ/dY 5 0 at Y 5 1. The operating mode, when a reactant cannot diffuse into the permeate phase (e.g., because it is not dissolvable in it) might often exist. On the other hand, this is not discussed in detail in the literature. Figure 5.9 illustrates how the reaction modulus can affect the concentration distribution at a given Peclet number. In the slow reaction rate regime, ϑ , 0.30.5, the concentration remains above about 0.9. In the intermediate reaction regime, ϑ 5 0.33, the value of φδ will generally be higher than zero, and its value can strongly depend on the Peclet number as well. If ϑ . 3, then the concentration level can be maintained above zero by higher Peclet value only.

5.3.1.1 Mass transport with polarization layer Mass transport in the case of dφ/dy . 0 at y 5 δ. It is assumed the concentration polarization affects the mass transfer rate of the inlet reactant. In the boundary layer, there is no chemical reaction, thus the mass transfer rate through it can be given according to Eq. (5.7). Thus, the mass transfer rate in the liquid phase, at the membrane interface, will be as J o 5 β of} ðco 2 e2Pef c Þ

ð5:47Þ

136

Basic Equations of the Mass Transport through a Membrane Layer

1.0

0.3

Concentration distribution

0.5 0.8

1

0.6 ϑ=2

0.4

ϑ=3

0.2 0.0

0.0

0.2

0.4 0.6 Catalytic membrane layer

0.8

1.0

Figure 5.9 Effect of the reaction modulus on the concentration distribution (dφ/dY 5 0 at Y 5 1), Pe 5 1.

where β of} 5

Df ePef Pef ePef =2  β of Pef Pe f δf e 21 2 sinhðPef =2Þ

ð5:48Þ

The co denotes the bulk concentration in the fluid, c represents the liquid concen  tration at the membrane interface (Hc 5 φ ). The J value defined by Eq. (5.29) and the Jo in Eq. (5.47) are equal to each other, thus the overall mass transfer rate can be given by means of the resistance-in-series principle. The overall inlet mass transfer rate is, for a first-order chemical reaction, as follows (note dφ/dY . 0 at Y 5 1 by this model): J 5 β ov} ðco 2 Ke2Pef coδ Þ

ð5:49Þ

where β}tot 5

1 1 e 2 Pef o 1 Hβ β f}

ð5:50Þ

and K5

Θ Θ 1 Θ cosh Θ

ePe=2 ½ðPe=2Þsinh

ð5:51Þ

The value of β is defined by Eq. (5.30). The parameter values in order to predict the concentration distribution are as follows:   ePef e 2 Θ Peco 2 ePef 2 1 λH 1 Pe e2Pe=2 coδ 1 T 5 λδ =δ ð1 2 e2Pef ÞA 2 Pe 2e f cosh Θ H tanh Θ

ð5:52aÞ

Diffusive Plus Convective Mass Transport Through a Plane Membrane Layer

137

with A5Θ1

Pe tanhΘ 2

  ~ 1 Pe e 2 Pe=2 co ePef eΘ Peco 2 ePef 2 1 λH 1 δ S5 ~ ð1 2 e 2 Pef ÞA 2 Pe 2eλδf =δ coshΘ H tanhΘ   o 2Pe=2 o A 2 Pe cδ =coshΘ H tanhΘ c 2 Θe Tf 5 ð1 2 ePef ÞA 2 Pe tanhΘ H

ð5:52bÞ ð5:53aÞ

and S f 5 c o 2 Tf

ð5:53bÞ

with Pe λ~ 5 2 Θ; 2

λ5

Pe 1 Θ; 2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pe ϑ2 Θ5 1 1 4 2; Pe 2

sffiffiffiffiffiffiffiffiffi k1 δ2 ϑ5 D

δf denotes the thickness of the polarization layer, m; δ is the membrane thickness, m. Formulas given by Eqs (5.52a)(5.53b) are rather complex ones and it is rather difficult to handle them. That is why it is important to note that if one knows the mass transfer rate defined by Eq. (5.46), there is a much easier way to get the values of T, S, Tf and Sf parameters. Using, e.g, Eq. (5.46) and the boundary condition at Y 5 0, namely C 5 Tf 1 Sf, the values of Tf and Sf can simply be predicted. The same method can be used to get the values of T and S, namely applying Eq. (5.46) again and the boundary condition at Y 5 1 1 δf/δ [Eq. (5.4b) and Eq. ~ (5.25), that is HCoδ 5 Teλ 1 Seλ ], the T and S values can be obtained. Typical concentration distributions are shown in Figure 5.10A at different values of the reaction modulus. The ϑ can significantly lower the concentration level not only in the membrane layer, but also in the concentration boundary layer as well. At larger values of ϑ, the concentration curves have inflexion points, at Y 5 1, at the feed membrane interface, where the curves change their concave trends to convex. The relative value of the outlet concentration was chosen to be 0.1. As can be seen, the concentration can decrease below its outlet value, at a high value of the reaction modulus. That should mean that to maintain the given outlet concentration, additional feeding of the reactant is needed on the outlet side of the catalytic needed on the outlet side of the catalytic membrane. Mass transport in the case of dφ/dy 5 0 at y 5 δ. The mass transfer rate is given here. Applying Eqs. (5.46) and (5.47), one can easily get the overall mass transfer rate as: Jov} 5

1 β of}

co 2 Pe 1 e Hβ f

ð5:54Þ

138

Basic Equations of the Mass Transport through a Membrane Layer

1.0

Concentration distribution

ϑ = 0.01 0.8

1

Pef = Pe = 5

0.6

2

δf / δ = 1

5

0.4 βfo = βo = 1×10–4 m/s 0.2 Boundary layer 0.4

10 Membrane layer

0.0 0.0

0.1

0.8

1.2

1.6

2.0

Boundary layer + membrane

Figure 5.10A The effect of a first-order chemical reaction on the concentration distribution in the two layers at different values of reaction modulus ðcoδ =co 5 0:1; β of 5 β o 5 1 3 1024 m=sÞ:

with β [according to Eq. (5.46)] as

2 2 Θ tanh Θ D Pe  β5 δ 2 tanh Θ 2 Θ Pe2 4

ð5:46Þ

where the value of β of} is given by Eq. (5.48).

5.3.1.2 Remarks for Application of the Fick’s Diffusive Transfer Plus Convective Flow The mass transfer resistance given by Fick’s law is often applied for diffusive mass transport through a boundary layer ( 5 β oΔc). The question arises whether this can also be used when there is convective flow in the boundary layer as well as it was done the mass transport is accompanied by chemical reaction in the membrane phase [see boundary conditions given by Eqs. (5.37) and (5.38)]. The Fick’s law is true in case of linear concentration distribution. This is not fulfilled when convective velocity also exists in the boundary layer (see for example Figure 5.2). The diffusive flow, JD, can be expressed, in the presence of convective flow as JD} 5 2β of Pef

Δc ePef 2 1

ð5:55aÞ

Diffusive Plus Convective Mass Transport Through a Plane Membrane Layer

Ratio of mass tranfer rates,–

1.0

139

C* = 0.90 0.75

0.8

0.50

0.6

0.25

0.4

0.10

0.2 0.0 0.0

1.0

10.0

Peclet number, Pef – o o Figure 5.10B The value of JFi /J EX as a function of the Pef at different value of C interface concentration.

According to Eq. (5.55a), the diffusive mass transfer coefficient ðβ oD} Þ for diffusive plus convective flow can be expressed as (here Y 5 y/δf). At Y 5 Y, then β oD} 5 2β of

Pef ePef Y ePef 2 1

ð5:55bÞ

At Y 5 0, then β oD} 5 2β of

Pef 21

ð5:55cÞ

Pef ePef ePef 2 1

ð5:55dÞ

ePef

At Y 5 1, then β oD} 5 2β of

Accordingly, the diffusive mass transfer coefficient, β oD} ; continuously increases as a function of space coordinate in the case of so-called exact solution. Thus, the overall mass transfer rate will be unchanged with the decrease of the concentration. The value of β oD} # β of at Y 5 0 and β oD} $ β of at Y 5 1. Let us compare the mass transfer rates of the boundary layer, entering the membrane phase, namely at y 5 δf, obtained the exact solution and by usage of Fickian mass transfer coefficient

140

Basic Equations of the Mass Transport through a Membrane Layer

for expressing the mass transfer resistance. Look at the ratio of these two mass transfer rates, namely: o JFi β of ðco 2 c Þ 1 υc f1 1 ðPef 2 1ÞC gðePef 2 1Þ 5  o JEx β of} ðco 2 e2Pef c Þ Pef ePef ð1 2 e2Pef C  Þ

ð5:55eÞ

where subscripts Fi and Ex denote the Fick’s and the exact solutions, C  5 c  /co. It is obvious that in limiting case, namely if Pef tends to zero, the above ratio tends to unit. On the other hand, when the Pef value increases the difference between the o o =JEx ratio is decreasing, in the range of C  two models also increases, i.e., the JFi between 0 and 1. Figure 5.10B illustrates the change of the ratio of mass transfer rates as a function of Pef at different values of C  . It can be stated that the approach by the Fick’s law can only be used at very low values of Pef and relatively large values of C  .

5.3.2

Mass Transport Accompanied by Zero-Order Reaction

The effect of the zero-order reaction will be discussed here for an intrinsically catalytic membrane layer only. This reaction has no important role in the case of a membrane reactor. The differential mass balance equation to be solved is as D

d2 φ dφ 2 k0 5 0 2υ 2 dy dy

ð5:56Þ

Similar to Eq. (5.2), the differential mass balance equation for the catalytic membrane can be given as d2 φ dφ 5 φ ϑ2 2 Pe dY 2 dY

ð5:57Þ

where sffiffiffiffiffiffiffiffiffi k0 δ 2 ϑ5 Dφ

ð5:58Þ

Mass transport with dφ/dy . 0 at Y 5 δ. First look at the solution with the following boundary conditions: Y 50

then φ 5 φ

ð5:59aÞ

Y 51

then φ 5 φδ

ð5:59bÞ

Diffusive Plus Convective Mass Transport Through a Plane Membrane Layer

141

The general solution of Eq. (5.56) is as φ 5 TePeY 2φ

ϑ2 Y 1S Pe

ð5:60Þ

Applying the boundary conditions [Eqs (5.59a) and (5.59b)], one can get 


 
  Pe PeY  ð1 2 YÞ 1 S φ 1 e2Pe=2 sinh φδ 2 2

φ5

ePeY=2 sinhðPe=2Þ

S5



 ϑ2 2Pe=2 PeY Pe e 2Y Sinh sinh Pe 2 2

with

sinh

ð5:61Þ

ð5:62Þ

The mass transfer rate can be given as
 e2Pe  φδ J 5 β φ 2 11T

ð5:63Þ

where β 5 β}o ð1 1 T Þ

ð5:64Þ

and T 52

 ϑ2 e2Pe  Pe 1 1 2 ePe 2 Pe

ð5:65Þ

and according to Eq. (5.48) β}o 5

D ePe Pe Pe e 21 δ

The outlet mass transfer rate should also be given as
 e2Pe   φ Jδ 5 β} ζ φ 2 ζ δ o

ð5:66Þ

where ζ 512



 ϑ2 2Pe=2 ϑ2 Pe Pe 2Pe=2 ½ Þ e ð1 2 e 1 2 Pe  1 2 Pe 22e sinh Pe2 Pe2 2 ð5:67Þ

142

Basic Equations of the Mass Transport through a Membrane Layer

Mass transfer rate with polarization layer. Now look at the solution with mass transfer resistance in the feed phase, applying the resistance-in-series model: υC 2 β of

dc 5J dY

ð5:68Þ

The physical mass transfer rate through the boundary layer can be expressed, in the case of convective and diffusive flows, as follows (Nagy and Kulcsa´r, 2009): J o 5 β of} ðco 2 e2Pef c Þ

ð5:69Þ

with β of} 5

Df ePef Pef Pe δf e f 21

ð5:70Þ

where Pef 5

υδf Df

The overall mass transfer rate can be given, applying Eqs. (5.63) and (5.69), which two mass transfer rate are equal to each other, as follows: J 5 β}ov ðco 2 Te2Pef coδ Þ

ð5:71Þ

with β}ov 5

1 ð1=β of} Þ 1ðe2Pef =HβÞ

ð5:72Þ

The values of β ofδ and β are defined in Eqs (5.70) and (5.64). Because the zero-order reaction rate is independent of the concentration, the effect of the chemical reaction, on the concentration level, can be much higher than in the case of the first-order reaction (Figure 5.11). The concentration lowers quickly down to zero when the ϑ value is larger than two to three. Comparing these results to that of the first-order reaction (Figure 5.10A),there can be a huge difference between them (Nagy and Kulcsa´r, 2009). There is no diffusive mass transport into the permeate side, dφ/dY 5 0 at Y 5 1. For example, the dimensionless concentration in the membrane layer is as Φ5

 2  1 ϑ PeY ϑ2 Pe 2 Y ðe 2 1Þ 1 Pee Pe Pe Pe Pe e

ð5:73Þ

Diffusive Plus Convective Mass Transport Through a Plane Membrane Layer

143

1.0

Concentration distribution

ϑ = 0.01 0.8

1

Pef = Pe = 5 0.6

δf / δ = 1

2

0.4 βfo = βo = 1×10–4 m/s

3

0.2 5 Boundary layer

10

0.0 0.0

0.4

0.8

Membrane layer 1.2

1.6

2.0

Boundary layer + membrane

Figure 5.11 The effect of the zero-order reaction on the two-layer concentration distribution ðcoδ =co 5 0:1; β of 5 β o 5 1 3 1024 m=sÞ:

with sffiffiffiffiffiffiffiffiffi k0 δ 2 ϑ5 ; Dφ

Pe 5

υδ D

The inlet mass transfer rate is as 
 1 J 5 β o φ Pe 2 ϑ2 Pe 2 1 e

ð5:74Þ

The above equation gives the mass transfer rate without external mass transfer resistances. The overall mass transfer rate can similarly be obtained as it was done to get Eq. (5.71).

5.3.3

Mass Transport with Variable Parameters

It is assumed that both the diffusion coefficient and reaction rate constant can vary as a function of space coordinate or concentration. The differential mass balance equation for a first-order chemical reaction can be given as
 d dφ dφ D½φ; y 2υ 2 kðyÞφ 5 0 ð5:75Þ dy dy dy

144

Basic Equations of the Mass Transport through a Membrane Layer

For analytical solution of Eq. (5.75), it should be linearized. The solution methodology of this type of differential equation was given by Nagy (2009a) for diffusional mass transport through a membrane reactor and by Nagy (2007b) for diffusive plus convective mass transport with variable parameters. Essentially, this solution methodology serves the mass transfer rate and the concentration distribution in a closed, explicit mathematical expression. For the solution of Eq. (5.75), the catalytic membrane should be divided by the N sublayer, in the direction of the mass transport, that is perpendicular to the membrane interface, with thickness of Δδ (Δδ 5 δ/N) and with constant transport parameters in every sublayer. Thus, for the ith sublayer of the membrane layer, using dimensionless quantities, it can be obtained as Di

d2 φ dφ 2 ki φ 5 0; 2υ 2 dy dy

yi21 , y , yi

ð5:76Þ

In dimensionless space coordinate d2 φ dφ 2 Pei 2 ϑ2 i φ 5 0 dY 2 dr

ð5:77Þ

where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϑi 5 δ2 ki =Di ;

Pei 5

υδ2 Di

~ Let us introduce the following variable φ: φ~ 5 φe2Pei Y=2

ð5:78Þ

After a few manipulations, one can get the following differential equation to be solved: d2 φ~ 2 Θ2i φ~ 5 0 dR2

ð5:79Þ

where rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pei 1 ϑ2i Θi 5 4 The solution of Eq. (5.79) can be easily obtained by well-known mathematical methods as follows: ~

φ 5 Ti eλi Y 1 Si eλi Y

ð5:80Þ

Diffusive Plus Convective Mass Transport Through a Plane Membrane Layer

145

with Pei λ~ i 5 1 Θi ; 2

λi 5

Pei 2 Θi 2

The solution of the parameters Ti and Si strongly depends on the operating mode and thus, on the external boundary conditions. Accordingly, see models A and B (Figure 5.3). This will be shown in the next solution and several figures illustrate the effect of variable parameters. Model Adφ/dY . 0 at Y 5 1. The Ti and Pi parameters of Eq. (5.80) can be determined by means of the external [Eqs (5.81a) and (5.81d)] and the internal boundary conditions [Eqs (5.81b) and (5.81c)] for the ith sublayer (with 1 # i # N 2 1; Yi 5 iΔY; ΔY 5 1/N). These can be given by the following equations for model A as φ 5 φ ;

Y 50

ð5:81aÞ

dφi dφ 1 Pei φi 5 2 i 1 1 1 Pei11 φi11 dY dY     φi  1 5 φi11  2 at Y 5 Yi Y5Y Y5Y 2

i

at Y 5 Yi

ð5:81bÞ ð5:81cÞ

i

φ 5 φδ ;

Y 51

ð5:81dÞ

Taking into account the concentration distribution [Eq. (5.80)], the algebraic equation system to be solved will be as φ 5 T1 1 S1

ð5:82Þ

~

~

Ti eλi Yi 1 Si eλi Yi 5 Ti11 eλi11 Yi 1 Si11 eλi11 Yi

ð5:83Þ

~ ~ Di ðTi λi eλi Yi 1 Si λ~ i eλi Yi Þ 5 Di11 ðTi11 λi11 eλi11 Yi 1 Si11 λ~ i11 eλi11 Yi Þ

ð5:84Þ

~

φδ 5 TN eλN 1 SN eλN

ð5:85Þ

After solution of the algebraic equation system containing 2N equations, applying the well-known Cramer rules, the value of the integration parameters for the first sublayer can be expressed as ζT T1 5 NO ζN



φ 2

φδ

!

1 o N ζ TN eðPeN =2Þ Li52 coshðΘi ΔYÞ 2coshðΘ1 ΔYÞ

ð5:86Þ

146

Basic Equations of the Mass Transport through a Membrane Layer

The value of ξ i 21 (with i 5 1 2 N 2 1), ζ ji and ψji (with j 5 T, O and i 5 1 2 N) can be calculated by the following expressions: ξi21 5 ehPei2Pei21 iYi =2

ð5:87Þ

*

ζ ji

tanhðϑi ΔY Þ ψji21 Di 2 1 Pei 5 12 2 j 2 ϑi ζ i21 Di

!+

ζ ji21 ξi21

with j 5 T; O;

i 5 2; . . . ; N ð5:88Þ

* ψji

5 A i 2 Bi

ψji21 Di21 Pei 2 j D 2 ζ i21 i

!+

ζ ji21 ξi21

with j 5 T; O;

i 5 2; . . . ; N 2 1 ð5:89Þ

Ai 5

Pei tanhðϑi ΔYÞ 2 Θi ; 2

i 5 1; . . . ; N 2 1

ð5:90Þ

Bi 5

Pei 2 Θi tanhðϑi ΔYÞ; 2

i 5 2; . . . ; N 2 1

ð5:91Þ

The initial values of ζ ji and ψji ; namely ζ j1 and ψj1 (j 5 T, O), are as ζ T1 5 e2Θ1 ΔY ; ψT1 5 λ~ 1 e2Θ1 ΔY ;

ζO 1 5 2tanhðΘ1 ΔYÞ ψO 1 5 2A1

The value of S1 can be predicted by means of Eq. (5.82). For details of this solution, see the Appendix. The mass transfer rate at place of Y 5 0, as a sum of the diffusive and the convective flows, can be given as J5

D1 ðλ1 T1 1 λ~ 1 S1 Þ δ

ð5:92Þ

Knowing T1 and S1, the other parameters, namely Ti and Si (i 5 2,3,. . .,N), can be easily calculated by means of the internal boundary conditions given by Eqs (5.83) and (5.84), starting from T2 and S2 up to TN and SN. Thus, one can get the following equations for prediction of the Ti and Si from Ti 21 and Si 21, for the component A: ~

Ti eλi Yi 1 Si eλi Yi 5 Γi21 ~

Di ðTi λi eλi Yi 1 Si λ~ i eλi Yi Þ 5 Ξi21

ð5:93Þ ð5:94Þ

Diffusive Plus Convective Mass Transport Through a Plane Membrane Layer

147

with ~

Γi21 5 Ti21 eλi21 Yi 1 Si21 eλi21 Yi

ð5:95Þ

~ ~ λi21 Yi Þ Ξi21 5 Di21 ðTi21 λi21 eλi21 Yi 1 Si21 λe

ð5:96Þ

Now knowing the Ti and Si (with i 5 1,2,. . .,N) parameters, the concentration distribution can be calculated easily through the membrane, i.e., its value for every sublayer. The outlet mass transfer rate can also be expressed as Jδ 5

DN ðλN TN eλ~ 1 λ~ N SN eλ Þ δ

ð5:97Þ

Some typical figures will be shown in order to illustrate the effect of the variable Peclet number. Figure 5.12 shows how the concentration distribution changes at different types of an anisotropic membrane where the Peclet number changes, due to the variable diffusion coefficient. The dotted lines give the concentration distribution at a low reaction rate, namely at ϑ 5 0.2, while the continuous lines show the fast reaction rate, namely at ϑ 5 3. The change of Pe as a function of the space coordinate can significantly affect the concentration distribution, as consequently the value of the diffusive inlet mass transfer rate. Decreasing the Pe number with high initial value provides a high inlet mass transfer rate, accordingly,

1.0

Concentration distribution

1 0.8 2

1 0.6

2 3

0.4 0.2 ............. 0.0

0.0

0.2

3

ϑ=3 ϑ = 0.2 0.4

0.6

0.8

1.0

Catalytic membrane layer

Figure 5.12 Concentration distribution with variable Peclet number and at constant reaction modulus φδ 5 0:1: Curves: (1) Pe decreases from Pe 5 10 (at Y 5 0) down to 1 (at Y 5 1); (2) Pe 5 5.5; and (3) Pe increases from Pe 5 1 (at Y 5 0) to Pe 5 10 (at Y 5 1).

148

Basic Equations of the Mass Transport through a Membrane Layer

higher concentration level in the catalytic membrane. Increasing the Pe number with low initial value serves a much lower concentration level in the catalytic membrane. Line 2 gives the concentration distribution by constant, by the average Peclet number, namely Pe 5 5.5. In the case of a variable Peclet number, the Pe was varied between 1 and 10, with expression of Pe 5 1 19i/N or with Pe 5 10 29i/N, for its increasing and decreasing values, respectively. Mass transport with boundary condition of dφ/dY 5 0 at Y 5 1. The starting expressions are the same as that in φ 5 φ ;

Y 50

ð5:98aÞ



 dφ dφ 1 Pei φ 5 Di11 2 1 Pei11 φ Di 2 dY dY     φ 1 5 φ 2 Y5Y Y5Y

at Y 5 Yi

ð5:98bÞ ð5:98cÞ

at Y 5 Yi

i11

i

dφ 5 0; dY

Y 51

ð5:98dÞ

The solution of this problem differs from the model A, due to the difference in the last, namely Nth, sublayer regarding the two models. Accordingly, one can get the following solution for T1 and S1: *

A^ B^ 2 ΘN

ψTN21 DN21 PeN 2 T 2 ζ N21 DN

A^ B^ 2 ΘN

ψSN21 DN21 PeN 2 S 2 ζ N21 DN



T1 5 φ

* 

S1 5 φ

!+

!+

ζ TN 2 1 1 1 ξN 2 1 Ω 2coshðΘ1 ΔY Þ

ð5:99Þ

ζ SN 2 1 1 1 ξN 2 1 Ω 2coshðΘ1 ΔY Þ

ð5:100Þ

with A^ Ω 5 B^ 2 ΘN

ψO PeN N21 DN 2 1 2 O D 2 ζ N21 N

!

ζO N21 ξN21

ð5:101Þ

where PeN tanhðΘN ΔYÞ 1 ΘN A^ 5 2

ð5:102Þ

PeN B^ 5 1 ΘN tanhðΘN ΔYÞ 2

ð5:103Þ

Diffusive Plus Convective Mass Transport Through a Plane Membrane Layer

149

and ξi21 5 ehPei 2 Pei21 iYi =2

ð5:104Þ

* ζ ji 5

tanhðϑi ΔY Þ ψji21 Di21 Pei 12 2 2 ϑi ζ ji21 Di

!+

ζ ji21 ξ i21

with j 5 T; S; O; i 5 2 2 N 2 1 ð5:105Þ

* ψji 5

A i 2 Bi

ψji 2 1 Di21 Pei 2 2 ζ ji 2 1 Di

!+

ζ ji21 ξ i21

with j 5 T; S; O; i 5 2 2 N 2 1 ð5:106Þ

Ai 5

Pei tanhðϑi ΔYÞ 2 Θi ; i 5 2 2 N 21 2

ð5:107Þ

Bi 5

Pei 2 Θi tanhðϑi ΔYÞ; i 5 2 2 N 21 2

ð5:108Þ

The initial values of ζ ji and ψji ; namely ζ j1 and ψj1 (j 5 T, S, O), are as ζ T1 5 e2Θ1 ΔY ; ψT1 5 λ~ 1 e2Θ1 ΔY ;

ζ S1 5 eΘ1 ΔY ;

ζO 1 5 2tanhðΘ1 ΔYÞ

ψS1 5 λ1 eΘ1 ΔY ;

ψO 1 5 2A1

The concentration distribution is plotted in Figure 5.13 at different membrane properties, at ϑ 5 3. As can be seen, there is a significant difference in the concentration depending on the variable Peclet number. With a decreasing Peclet number, the concentration can be maintained at very high level even in the fast reaction rate regime as well. For comparison, the concentration distribution is plotted at a low value of the reaction rate, ϑ 5 0.2 (Figure 5.14). Surprisingly, the concentration of the reactant strongly increases far above unit, in the catalytic membrane layer. This should mean that the diffusive mass transfer flows in countercurrent direction to the convective one, decreasing the overall transport rate. At constant value of Pe, Pe 5 5.5, the concentration falls close to unit, its value is more than 0.99. In the case of an increasing Peclet number, the concentration decreases down to 0.2, even at low reaction rate. In reality the convective velocity, related to the total membrane interface, is generally not changed during a process. The υ value in Eq. (5.75), as can be seen in Chapter 10, should mean that the convective velocity is related to the total membrane interface. The change of the Peclet number can occur due to the change of the diffusion coefficient as a result of the membrane anisotropy. The D value can change depending on the membrane structure’s properties. Accordingly, the value of reaction

150

Basic Equations of the Mass Transport through a Membrane Layer

1.0

Concentration distribution

Pe = 10 → 1 0.8 0.6 Pe = 5.5 0.4 0.2 Pe = 1 → 10

ϑ=3 0.0 0.0

0.2

0.4

0.6

0.8

1.0

Catalytic membrane layer

Figure 5.13 Concentration distribution with variable Peclet number and with boundary condition of dφ/dY 5 0 at Y 5 1 (ϑ 5 3). Pe 5 10 29i/N for Pe 5 10-1 and Pe 5 1 19i/N for Pe 5 1-10.

10.0

Concentration distribution

ϑ = 0.2 8.0 Pe = 10 → 1 6.0

4.0

2.0

Pe = 5.5 Pe = 1 → 10

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Catalytic membrane layer

Figure 5.14 Concentration distribution with variable Peclet number and with boundary condition of dφ/dY 5 0 at Y 5 1 at lower value of ϑ; (ϑ 5 0.2). Pe 5 10 29i/N for Pe 5 10-1 and Pe 5 1 19i/N for i 5 1-10.

Diffusive Plus Convective Mass Transport Through a Plane Membrane Layer

151

3.2 ϑ = 0.087 − 0.27

Concentration distribution

2.8

Pe = 10 → 1

2.4 2.0 1.6

Pe = 2.5

1.2 0.8

ϑ = 0.13

Pe = 1 → 10

0.4 0.0 0.0

0.2

0.4

0.6

0.8

1.0

Catalytic membrane layer

Figure 5.15 Concentration distribution with variable Peclet number at low reaction rate (dφ/dY 5 0 at Y 5 1; Pe 5 10 29i/N for Pe 5 10-1 and Pe 5 1 19i/N for Pe 5 1-10); (D 5 1 3 10291 3 10210 m2/s; υ 5 1 3 1024 m/s; δ 5 10 3 1026 m; k1 5 0.075 s21).

modulus should also change at a constant reaction rate. Now, let us look at the concentration distribution when the diffusion coefficient changes both the values of Peclet number and ϑ reaction modulus. Figures 5.15 and 5.16 illustrate this situation at two different values of reaction modulus, namely with its average values of 0.13 (Figure 5.15) and 1.25 (Figure 5.16). The Peclet number was changed between 1 and 10 in an increasing or decreasing manner. The effect of the decreasing diffusion coefficient, that is, decreasing Pe and ϑ, is basically different from that of increasing the diffusion coefficient. These figures prove that the transport process essentially can be affected by the anisotropy structure of the membrane.

5.3.4

Mass Transport Through an Asymmetric Catalytic Membrane

A two-layer membrane, where the catalytic particles are distributed in one of these layers while the other layer is noncatalytic is assumed (Figure 5.17). This can serve as a support layer or a separating layer. In the membrane reactors, the catalytic layer is often a very thin layer, and the noncatalytic layer is a distributor and/or support layer. Or in the case of biocatalyzed membrane reactor, the sponge layer holds the biocatalytic particles, enzymes, or living cells, while the skin layer separates the catalyst particles from the permeate side. That is why the mass transport through an asymmetric membrane reactor will be analyzed. We consider here a general case, namely when diffusion plus convection can take place in both layers. From that, the limiting cases, in which there is only diffusion mass transport in the skin layer, can easily be obtained.

152

Basic Equations of the Mass Transport through a Membrane Layer

1.4 Pe = 10 → 1

Concentration distribution

1.2 1.0 0.8

Pe = 2.5

0.6

ϑ = 1.85

0.4 Pe = 1 → 10

0.2 ϑ = 1.2 − 3.8

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Catalytic membrane layer

Figure 5.16 Concentration change with varying Peclet number and accordingly with varying reaction modulus; Pe 5 110; ϑ 5 1.23.8 (D 5 1 3 10291 3 10210 m2/s; υ 5 1 3 1024 m/s; δ 5 10 3 1026 m; k1 5 15 s21).

Noncatalytic Figure 5.17 Schematic illustration of an asymmetric catalytic layer (skin) membrane layer.

Catalytic layer (1) C

J φ∗1

C C C

C C C

C C C

0

Jδ , φ∗2,δ

δ1 δ

5.3.4.1 Reactant Is Fed on the Catalyst Layer Look at a first-order reaction. If you do not want to know the concentration distribution of reactants in the membrane, the overall mass transfer rates can be expressed by means of those that are given for both the single layers. The starting expressions are the mass transfer rate of a symmetric membrane layer, as discussed previously. The inlet and the outlet rates of a single catalytic membrane layer are as follows (Nagy, 2009a, b): Inlet stream [see Eqs (5.28)(5.32)]: J 5 βðφ1 2 E1 φ1;δ Þ

ð5:109Þ

Diffusive Plus Convective Mass Transport Through a Plane Membrane Layer

β5

D1 β o ð½Pe1 =2tanh Θ 1 ΘÞ δ1 tanh Θ

E1 5

ePe1 =2 ½ðPe

153

ð5:110Þ

Θ 1 =2Þsinh Θ 1 Θ cosh Θ

ð5:111Þ

Outlet stream [see Eqs (5.30) and (5.31)]: J1;δ 5 β δ ðφ1 2 E2 φ1;δ Þ

ð5:112Þ

βδ 5

D1 ΘePe1 =2 δ1 sinh Θ

ð5:113Þ

E2 5

Θ 2ðPe1 =2Þtanh Θ cosh Θ ΘePe1 =2

ð5:114Þ

and rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pe21 Θ5 1 ϑ2 ; 4

sffiffiffiffiffiffiffiffiffi k1 δ21 ϑ5 ; D1

Pe1 5

υδ1 ; D1

βo 5

D δ

The mass transfer rate for the skin layer, assuming that there is no chemical reaction in this layer, is as [see Eqs (5.69) and (5.70)]: J2o 5 β o2 ðφ2 2 e2Pe2 φ2;δ Þ

ð5:115Þ

where β o2} 5

D2 ePe2 Pe2 Pe δ2 e 2 21

ð5:116Þ

where the subscripts 1 and 2 denote the sponge and the skin layers, respectively; D1 and D2 represent the effective diffusivities of the layers (m2/s); and Pei 5 υiδi =Di with i 5 1, 2. The overall inlet mass transfer rate can be expressed by combinations of Eqs (5.110), (5.112), and (5.115), for example, by the following way: the φ1;δ value is expressed by Eqs (5.112) and (5.115) ðJδ 5 J2o ; H1 φ1;δ 5 H2 φ2 Þ and replaced into Eq. (5.110). Thus, one can obtain J 5 β ov ðφ1 2 Gφ2;δ Þ

ð5:117Þ

154

Basic Equations of the Mass Transport through a Membrane Layer

with β ov 5 β

β o2} ðH1 =H2 Þ 1 β δ ðE2 2 E1 Þ β o2} ðH1 =H2 Þ 1 β δ E2

ð5:118Þ

and G5

ββ o2} e2Pe2 o β 2} ðH1 =H2 Þ 1 β δ ðE2

2 E1 Þ

ð5:119Þ

The overall outlet mass transfer rate at δ 5 δ1 1 δ2 can be obtained by Eqs (5.112) and (5.115) with expression H1 φ1;δ 5 H2 φ2 (note J1;δ 5 J2o  Jδ Þ as
 H2 2Pe2   Jδ 5 β δ;ov φ1 2 E2 e φ2;δ H1

ð5:120Þ

with β δ;ov 5

1 ð1=β δ Þ 1ðE2 H2 =H1 Þð1=β o2} Þ

ð5:121Þ

and Pe2 5

υδ2 D2

5.3.4.2 Reactant Is Fed on the Noncatalytic Layer For the sake of completeness, we give here the overall mass transfer rate when the feed of the reactant is carried out through the nonreactive skin layer. This can be done more easily than in the previous case, though the reacted amount is also needed to give the outlet rate in the spongy layer. This mass transfer is illustrated in Figure 5.18. Note that the subscripts 1 and 2 denote here also the skin and the catalytic sponge layers as it was made in section 5.3.4.1. Equations (5.115 )(5.117) and (5.120) were applied in order to express the mass transfer rates at y 5 0 and y 5 δ1 1 δ2 5 δ. The inlet mass transfer rate will be as
 H2  2Pe2  φ1;δ J 5 β ov φ2 2 E1 e H1

ð5:122Þ

with β ov 5

1 ð1=β o2} Þ 1ð1=βÞðH2 =H1 Þe2Pe2

ð5:123Þ

Diffusive Plus Convective Mass Transport Through a Plane Membrane Layer

C

J φ∗2

C C C

Figure 5.18 Illustration of the mass transport in the case when the substrate is fed on the skin side of the asymmetric catalytic membrane.

C C C C

C C 0 δ2

155

Jδ , φ∗2,δ

δ

The value of β and β o2} are given in Eqs (5.110) and (5.116), respectively. The outlet mass transfer rate, namely that on the permeate side, at y 5 δ [taking into account that the value of overall mass transfer rate, J, in Eq. (5.122) is equal to J2o given by Eq. (5.115), the value of φ1 can be expressed; then it should be replaced into Eq. (5.112)] can be given as Jδ 5 β δ;ov ðφ2 2 E3 φ1;δ Þ

ð5:124Þ

where
 β H2 Pe2 e β δ;ov 5 β δ 1 2 ov o β 2} H1

ð5:125Þ

and E3 5

E2 2 E1 ððβ o2} =β ov Þ 21ÞðH2 =H1 ÞePe2

ð5:126Þ

The values of β ov, β δ, E1, and E2 are given by Eqs (5.123), (5.113), (5.111), and (5.114), respectively. The mass transfer rate through asymmetric membrane, when the transport is accompanied by zero-order reaction is given in Chapter 9 because this reaction often can be important in the case of biocatalytic reactions applying asymmetric membranes.

References Ilinitch, O.M., Cuperus, F.P., Nosova, L.V., and Gribov, E.N. (2000) Catalytic membrane in reduction of aqueous nitrates: operational principles and catalytic performance. Catalysis Today 56, 137145. Nagy, E. (2007a) The effect of the concentration polarization and a membrane layer mass transport on the membrane separation. J. Appl. Membr. Sci. 6, 18. Nagy, E. (2007b) Mass transport through a dense, polymeric, catalytic membrane layer with dispersed catalyst. Ind. Eng. Chem. Res. 46, 22952306.

156

Basic Equations of the Mass Transport through a Membrane Layer

Nagy, E. (2009a) Mathematical Modeling of Biochemical Membrane Reactors, in Membrane Operations, Innovative Separations and Transformations, Ed. by, E. Drioli, and L. Giorno. Wiley-VCH, Weinheim, pp. 309334. Nagy, E. (2009b) Basic equations of mass transfer through biocatalytic membrane layer. Asia-Pacific J. Chem. Eng. 4, 270278. Nagy, E. (2010) Mass transport through a convection flow catalytic membrane layer with dispersed nanometer-sized catalyst. Ind. Eng. Chem. Res. 49, 10571062. Nagy, E., and Borbe´ly, G. (2007) The effect of the concentration polarization and the membrane layer mass transport on the membrane separation. J. Appl. Membr. Sci. Technol. 6, 916. Nagy, E., and Kulcsa´r, E. (2009) Mass transport through biocatalytic membrane reactors. Desalination 245, 422436.

6 Diffusion in a Cylindrical Membrane Layer

6.1

Introduction

The capillary, a hollow fiber membrane, is the most often-applied membrane module in the separation industry. This proves the importance of the mass transport through a cylindrical membrane layer. The schematic illustration of a capillary is given in Figure 6.1.

6.2

Steady-State Diffusion

We consider a circular cylinder, a capillary membrane in which diffusion everywhere is radial. Concentration is then a function of radius r, only under steady-state conditions. Thus, the mass balance equation will be as
 d dφ rD 50 dr dr

ð6:1Þ

Rearranging Eq. (6.1) in dimensionless form with R 5 r/ro and Φ 5 φ/φ 5 φ/(Hco), C=c/co (where co is the bulk fluid concentration, kg/m3 or kmol/m3; H is the solubility coefficient, -; φ is concentration in membrane, kg/m3 or kmol/m3, see Figure 3.1) and assuming that D is constant, it can be get as:
 d dΦ R 50 dR dR

ð6:2Þ

Assuming a hollow cylinder whose membrane thickness is δ and the inner radius is ro, as well as considering the D value as a constant, the general solution of Eq. (6.1) is Φ 5 T ln R 1 S

ð6:3Þ

where T and S are constants to be determined from the boundary conditions at R 5 1 and R 5 1 1 δ/ro. If the surface of the membrane at R 5 1 is kept at a Basic Equations of the Mass Transport through a Membrane Layer. DOI: 10.1016/B978-0-12-416025-5.00004-1 © 2012 Elsevier Inc. All rights reserved.

158

Basic Equations of the Mass Transport through a Membrane Layer

δ

J

r0

J

z

r

r0 A

δ

βof

βo βofδ

Figure 6.1 Schematic representation of the transfer conditions and important notations for a capillary membrane.

constant concentration φ , Φ 5 φ /(Hco) 5 1 and R 5 1 1 δ/ro at Φoδ ; then (note that there is no mass transfer resistance in the fluid boundary layer): φ5

φ ln½ðro 1 δÞ=r 1 φδ lnðr=ro Þ ln½ðro 1 δÞ=ro 

ð6:4Þ

In dimensionless form (without external resistance Φ 5 1): Φ5

Φ ln½ð1 1 δ=ro Þ=R 1 Φδ ln R ln½ð1 1 δ=ro Þ=R 1 Cδo ln R  ln½ð1 1 δ=ro Þ ln½ð1 1 δ=ro Þ

The mass transfer rate can be defined as  dφ  D Dco H o J 52D  ðφ 2 φδ Þ  ð1 2 Cδo Þ 5 dr r5ro ro lnð1 1 δ=ro Þ ro lnð1 1 δ=ro Þ

ð6:5Þ

ð6:6Þ

It is easy to see that ro ln(1 1 δ/ro)-δ if ro-N, i.e., the mass transfer rate tends to that of plane interface in limiting cases. Defining the β o mass transfer coefficient for cylindrical interface, the mass transfer rate can be written in a form obtained for plane interface, without external mass transfer resistance, as J o 5 β o ðφ 2 φδ Þ  β o Hðco 2 coδ Þ

ð6:7Þ

with βo 5

D ro lnð1 1 δ=ro Þ

ð6:8Þ

Diffusion in a Cylindrical Membrane Layer

159

Note that, according to Eq. (6.2), the specific mass transfer rate depends on the value of the r space coordinate. Thus, the connection between the inlet, at R 5 1, and the outlet mass transfer rate, at R 5 1 1 δ/ro is as   J o R51 5 ð1 1 δ=ro ÞJ o R511δ=ro ð6:9Þ Look at the mass transfer rate when there is external mass transfer resistance on both sides of a cylindrical (capillary) membrane. The mass transfer rate can be given for the fluid phase boundary layer as at r 5 ro

β of ðco 2 c Þ  β of co ð1 2 C  Þ 5 2D

at r 5 ro 1 δ

 dφ  Dφ dΦ  5 Jo ro dR dr r5ro

β ofδ ðcδ 2 coδ Þ  β ofδ co ðCδ 2 Cδo Þ 5 J o

1 ð1 1δ=ro Þ

ð6:10Þ ð6:11Þ

The overall mass transfer rate and mass transfer coefficient, β oov ; taking into account Eqs (6.7) and (6.8), are easy to express, similarly to that for plane interface: J o 5 β oov ðco 2 coδ Þ  β oov co ð1 2 Cδo Þ

ð6:12Þ

1 1 1 1 5 o1 1 o β oov βf Hβ o β fδ ð1 1 δ=ro Þ

ð6:13aÞ

1 1 1 1 5 o1 1 o β oov βf Hβ o ðβ fδ Þr5ro

ð6:13bÞ

with

or

where β of 5

Df ro lnðro =½ro 2 δf Þ

β ofδ 5

Dfδ ro ð1 1 δ=ro Þlnð½1 1 δ 1 δfδ =½ro 1 δÞ

ð6:14Þ

and ð6:15aÞ

where δf and δfδ are the thickness of the boundary layer for feed and permeate side, respectively (m), and Df and Dfδ are the diffusivity of the continuous phase (m2/s). Note that the β ofδ gives the mass transfer coefficient at r 5 ro 1 δ. If this coefficient is related to the place of r 5 ro, its value will be as ðβ ofδ Þr 5 ro 5

Dfδ ro lnð½1 1 δ 1 δfδ =½ro 1 δÞ

ð6:15bÞ

160

Basic Equations of the Mass Transport through a Membrane Layer

Note that the value of βof in Eq. (6.14) is also related here to the place of r 5 ro. The T and S parameters, for calculation of the concentration distribution through the membrane, can be expressed, in dimensionless form, for Φ 5 T ln R 1 S, with Φ 5 φ/(Hco), as follows: ro ð6:16Þ T 5 β oov ðCδo 21Þ D and


S 5 Hβ oov

  1 1 Cδo 1 o 1 βf ð1 1 δ=ro Þβ ofδ Hβ o

ð6:17Þ

6.2.1 Concentration-Dependent Diffusion Coefficient Two important cases will be discussed in this section, namely the exponential and linear concentration dependency.

~ 6.2.1.1 Exponential Concentration Dependency, D 5 Doexp(αφ) Here also two cases will be discussed briefly, namely exponential and linear concentration dependency of the diffusion coefficient. Look at first an exponential ~ ~  : Replacing it or D 5 Doexp(αΦ) with α 5 αφ function, namely D 5 Doexp(αφ) into Eq. (6.1), one can get
 d ~ dφ αφ e 50 ð6:18Þ Do dr dr After integration of Eq. (6.18), we get ~ dφ T~ 5 reαφ dr

ð6:19aÞ

Separating the variables, we get T~ ~ dr 5 eαφ dφ r

ð6:19bÞ

Integrating Eq. (6.19a), one can get a general solution of this equation as ~ eαφ 5 T~ ln r 1 S~ α~

ð6:20Þ

In Eq. (6.20), parameters T~ and S~ have a dimension of kg/m3. Rewriting the membrane concentration and the space coordinate of Eq. (6.19b) into dimensionless form, it becomes eαΦ 5 T ln R 1 S α

ð6:21Þ

Diffusion in a Cylindrical Membrane Layer

161

Regarding negligible mass transfer resistance in the fluid phase boundary layer, assuming that fluid phase concentration is kept constant, namely at r 5 roφ 5 φ and r 5 ro 1 δφ 5 φδ ; the concentration distribution can be expressed as (Φ 5 φ/φ ; here Φδ 5 Cδo Þ:    1 ln R ð6:22Þ 1 eα Φ 5 ln ðeαΦδ 2 eα Þ α lnð1 1 δ=ro Þ The mass transfer rate expressing with measurable concentrations, at R 5 1, can be expressed as Jo 5

β o Hco α o ðe 2 eαCδ Þ α

βo 5

Do ro lnð1 1 δ=ro Þ

ð6:23Þ

The mass transfer rate can be expressed, in the presence of external mass transfer resistances, considering Eqs (6.10) and (6.11), as Jo 5

β o Hco αA ðe 2 eαB Þ α

ð6:24Þ

with A512

Jo β of co

B 5 Cδo 1

Jo ½1 1 δ=ro β of δ co

Accordingly, the Jo value can easily be determined by trial-error method by means of Eq. (6.24).

6.2.1.2 Linear Concentration Dependency, D 5 Do(1 1 αΦ) The differential mass transfer equation to be solved becomes
 d dΦ Rð1 1 αΦÞ 50 dR dR

ð6:25Þ

From that, after integration Rð1 1 αΦÞ

dΦ 5T dR

ð6:26Þ

162

Basic Equations of the Mass Transport through a Membrane Layer

The general solution of Eq. (6.26) is rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 T ln R 1 S 12 Φ52 1 α α2 α

ð6:27Þ

with (note Φ 5 1, Φδ 5 Cδo Þ:  α S5 11 2 and T5

ð6:28Þ

n o 1 α ðCδo 21Þ 1 ðCδo2 21Þ lnð1 1 δ=ro Þ 2

ð6:29Þ

The mass transfer rate can be given, taking into account Eqs (6.26) and (6.29), as J o 5 2Do ð1 1 αΦÞ

n o Hco dΦ α 5 β o Hco ð1 2 Cδo Þ 1 ð1 2 Cδo2 Þ ro dR 2

ð6:30Þ

Look at the mass transfer rate with external mass transfer resistance in the fluid phases flowing in the cylindrical membrane. The mass transfer rates in the external boundary layer can be given as it is done in Eqs (6.10) and (6.11). The mass transfer rate through the membrane, taking into account that the interface concentration can change according to the membrane resistance, can be given as n o α ð6:31Þ J o 5 β o φ ðΦ 2 Φδ Þ 1 ðΦ2 2 Φ2 δ Þ 2 According to Eqs (6.10), (6.11), and (6.31), the mass transfer rate with external mass transfer resistance can be expressed as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  2B 6 B2 2 4AE ð6:32Þ Jo 5 2A with ( ) α 1 1 2 A5 2 ðβ of co Þ2 ð½β ofδ r5ro co Þ2 (

1 1 αCδo 11α 1 B5 2 1 o o 1 o ½β f δ r5ro βf β H E 5 1 2 Cδo 1

 α

1 2 Cδo2 2

ð6:33Þ ) 1 co

ð6:34Þ ð6:35Þ

The values of β o, β of and (β ofd)r5ro are defined by Eqs. (6.8), (6.14) and (6.15b), respectively.

Diffusion in a Cylindrical Membrane Layer

163

6.2.1.3 Optional Concentration and/or Local-Coordinate Dependency of the Diffusion Coefficient The concentration dependency of the diffusion coefficient can be essentially different from the above discussed function. On the other hand, preparation and application of anisotropic membrane become more important. The diffusion coefficient can vary from place to place. Note that in the case of D 5 D(R), the concentration distribution can be directly calculated, while in the case of D 5 D(Φ), a few iteration steps are needed to get the correct D and Φ values in the membrane. Thus, a general solution of the problem will be shown in this section. The membrane is divided into N sublayers with ΔR thickness and the diffusion coefficient Di is assumed to be constant for every sub-layer (see Fig.4.7). The concentration distribution for ith sublayer, without external mass transfer resistance on the boundary layer, can be given as Ri 5 1 1 iΔR and ΔR 5 δ/(roN) (Nagy, 2006): Φ 5 Ti ln R 1 Si

at

Ri21 # R # Ri

ð6:36Þ

where (Ri 5 ri/ro) Ti 5 2

1 ðΦ 2 Φδ Þ PN Dj j51 ð1=Dj ÞlnðRj =Rj21 Þ

1 ð1 2 Cδo Þ  2 PN Dj j51 ð1=Dj ÞlnðRj =Rj21 Þ

ð6:37Þ with i 5 1; . . . ; N

and (with S1 5 Φ 5 1) Si 5

i21 X

Tj ln Rj 2 Ti ln Ri 1 Φ  1 1

j51

i21 X

Tj ln Rj 2 Ti ln Ri with i 5 2; . . . ; N

j51

ð6:38Þ The mass transfer rate can be given as:   D1 dΦ o 5 β o Hco ð1 2 Cδo Þ J  5 2 Hco ro dR R51 βo 5

1 1 P ro Nj51 ð1=Dj ÞlnðRj =Rj21 Þ

ð6:39Þ ð6:40Þ

As mentioned, when the Di values depend on the concentration, the real concentration distribution and the mass transfer rate can be obtained by iterative method. The external mass transfer resistance can easily be taken into account applying Eqs. (6.13b) and (6.40). Thus, the mass transfer rate can be obtained as:  ð6:41Þ J o R51 5 β oov co ð1 2 Cδo Þ

164

Basic Equations of the Mass Transport through a Membrane Layer

with N 1 1 1 ro X 1 Rj 1 ln o 5 o 1 o H j51 Dj Rj21 β ov βf ðβ f δ Þr5ro

ð6:42Þ

For the value of ðβ ofδ Þr5ro ; see Eq. (6.15b).

6.2.2 Concentration-Dependent Solubility Coefficient, H Two cases are discussed, namely linear and Langmuir-type concentration dependency, as given in Sections 3.2.2.1 and 3.2.2.2.

6.2.2.1 Linear Concentration Dependency of the Sorption Coefficient, ~ with Ho ð1 1 αc ~  Þc 5 φ H 5 Ho ð1 1 αcÞ The sorption isotherm can be given, in dimensionless form {C 5 φ =ðHo co Þ  Φ ; ~ o g; as α 5 αc Φ 5 ð1 1 αCÞC

ð6:43Þ

According to Eqs (6.5) and (6.43), the concentration distribution in the membrane, without external mass transfer resistance, can be given as Φ5

ð1 1 αÞln½ð1 1 δ=ro Þ=R 1 ð1 1 αCδo ÞCδo ln R ln½ð1 1 δ=ro Þ

ð6:44Þ

The mass transfer rate will be as J o 5 β o Ho co ð1 1 α 2 ð1 1 αCδo ÞCδo Þ

ð6:45Þ

The overall mass transfer rate with mass transfer resistance, applying Eqs (6.10) and (6.11) as well as (6.45) can be given, after some manipulation of the equation system, as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  ð6:46Þ 2B6 B2 2 4AE J5 2A with (

)

1 1 A5α 2 o o 2 o ðβ f c Þ ð½β f δ r5ro co Þ2 1 B5 2 o c

(

) 1 1 1 2αCδo 1 1 2α 1 1 ðβ ofδ Þr5ro β o Ho β of

ð6:47Þ

ð6:48Þ

Diffusion in a Cylindrical Membrane Layer

165

E 5 1 1 α 2 Cδo ð1 1 αCδo Þ

ð6:49Þ

6.2.2.2 Langmuir-Type Sorption Isotherm φ 5 Ho

Φ5

~ αc ~ 1 1 αc

ð6:50Þ

φ αC  Ho 1 1 αC

ð6:51Þ

Note that the Ho value has a dimension of kg/m3 here. The concentration distribution is not affected by the isotherm used, the interface concentration depends on the fluid concentration only. Thus, the concentration distribution can be given when there is no external mass transfer resistance, as Φ 5 T ln R 1 S

ð6:52Þ

where
 Cδo α 1 2 T5 lnð1 1 δ=ro Þ 1 1 αCδo 1 1 α

ð6:53Þ

and S5

α 11α

ð6:54Þ

The mass transfer rate can be expressed as


Cδo 1 2 J 5 β Ho c α 1 1 α 1 1 αCδo o

o

 ð6:55Þ

o

Look at the concentration distribution regarding the external mass transfer resistance as well. The mass transfer rate through the fluid boundary layer can be given as it is done in Eqs (6.10) and (6.11). The mass transfer rate through the membrane will be according to Eq. (6.55):


C Cδ 2 J 5 β Ho c α 1 1 αC 1 1 αCδ o

o

o

 ð6:56Þ

166

Basic Equations of the Mass Transport through a Membrane Layer

Expressing the value of C and Cδ from Eqs (6.10) and (6.11), respectively, and replacing them into Eq. (6.56), the following equation can be obtained: 1 2 J o =ðβ of co Þ Cδo 1 J o =hðβ ofδ Þr 5 ro co i Jo 5 2 β o Ho co α 1 1 αð1 2 J o =½β of co Þ 1 1 α½Cδo 1 J o =ð½β ofδ r 5 ro co Þ

ð6:57Þ

The Jo value can easily be obtained by an iteration method. It is also possible to rearrange Eq. (6.57) to get the Jo value in explicit form. Thus, one can get a 3rd order algebraic equation from that the Jo can be calculated by the well-known traditional method.

6.2.2.3 Dual-Sorption Isotherm According to the dual-sorption isotherm: φ 5 Ho c 1

~ φsat αc ~ 1 1 αc

ð6:58Þ

or in dimensionless concentration, it will be (Φsat 5 φsat/co) Φ5

φ Φsat αC 5 C 1 Hco 1 1 αC

ð6:59Þ

The mass transfer rate without external mass transfer resistance will be as apply~ o ; C 5 c/co), assuming that the diffusion coefficients are the ing Eq. (6.7), (α 5 αc same in the two models (see also section 3.2.2.3):   φsat αCδo φsat α ð6:60Þ 2 Cδo 2 J o 5 β o H o co 1 1 1 1 αCδo 11α Taking into account the external mass transfer resistance, we have three equations in order to determine the mass transfer rate, namely [see Eqs (6.10) and (6.11); Φ 5 φ /(Hoco)]: Jo 11 co β of   Φsat αC Φsat αCδo o o o  o 2 Ho Cδ 2 J 5 β c Ho C 1 1 1 αC  1 1 αCδo C 5 2

Cδ 5 J o

1 1 Cδo ð1 1 δ=ro Þβ ofδ

ð6:61aÞ ð6:61bÞ ð6:61cÞ

Substituting Eqs (6.61a) and (6.61c) into Eq. (6.61b), the following implicit equation can be obtained for calculating the mass transfer rate at R 5 1: Jo Ho J o Φsat αð1 2 J o =β of Þ 2A 5 2 1 1 1 β o co β of 1 1 αð1 2 J o =β of Þ

ð6:62Þ

Diffusion in a Cylindrical Membrane Layer

A5 2

Φsat αð1 2 J o =½β ofδ r5ro 1 Cδo Þ Ho J o 1 Cδo 1 o ½β f δ r5ro 1 1 αð1 2 J o =½β ofδ r5ro 1 Cδo Þ

167

ð6:63Þ

The Jo mass transfer rate can be calculated by trial-and-error method by means of Eqs (6.62) and (6.63).

6.2.2.4 Freundlich Sorption Isotherm The sorption isotherm of the species can be written in this case as φ 5 Hcn

or

Φ5

φ 5 Cn Hcon

ð6:64Þ

The dimension of H is (m3/kg)n 21 in the Freundlich equation. Consider the fluid phase concentration without external mass transfer resistance (see Figure 6.1) and Eq. (6.5), the concentration distribution inside the membrane is as Φ5

Φ ln½ð1 1 δ=ro Þ=R 1 Φδ ln R ln½ð1 1 δ=ro Þ=R 1 Cδon ln R  ln½ð1 1 δ=ro Þ ln½ð1 1 δ=ro Þ

ð6:65Þ

Thus, the mass transfer rate is as J 5 β o Hcon ðΦ 2 Φδ Þ  β o Hcon ð1 2 Cδon Þ

ð6:66Þ

The expression of β o is given in Eq. (6.8). The mass transfer rate can be similarly given in the presence of mass transfer resistance as it was made previously. Applying Eqs (6.10), (6.11), and (6.66), the mass transfer rate is as J o 5 β o Hcon fðC o 2 J o=½β of Hcon Þn 2 ðCδo 1 J o=½β ofδ h1 1 δ=ro iHcon Þn g

ð6:67Þ

The Jo value can be determined by e.g. iteration.

6.2.3

Mass Transfer Through a Composite Membrane

Two essential conditions should be taken into account: the thickness of the sublayers can be different and the solubility coefficient can also be different. The diffusion coefficient is also different but its value is constant inside of every single sublayer. It is assumed that the solubility coefficient for the sublayers can be defined as follows: Hmi φi 5 Hmi11 φi11

for i 5 1 to N 2 1

ð6:68Þ

168

Basic Equations of the Mass Transport through a Membrane Layer

For the external phases: Hoco 5 H1φ and HN φN 5 Hδ cofδ (or in the case of external mass transfer resistances, according to Figure 6.1, Hoc 5 H1φ and HN φN 5 Hδ cfδ Þ: After solution of the algebraic equation system obtained by the boundary conditions, the parameters have been determined. The values of the parameters in Eq. (6.36) can be given as: Ti 5 2

ðHo Co 2 Hδ Cδo Þ 1 PN Di j51 ðHj =Dj ÞlnðRj =Rj 21 Þ

with i 5 1; . . . ; N

ð6:69Þ

and (note that S1 5 Ho/H1) i21 X Ho Hj Rj 1 Tj ln 2 Ti ln Ri21 Hi H R i j 21 j51

Si 5

with i 5 2; . . . ; N

Accordingly, the mass transfer rate is   D1 H1 co o J  52 T1 5 β oov H1 co ðHo 2 Hδ Cδo Þ ro R51

ð6:70Þ

ð6:71Þ

where βo 5

1 1 P ro Nj51 ðHj =Dj ÞlnðRj =Rj21 Þ

ð6:72Þ

Taking into account the external mass transfer resistances on the both sides of the membrane, the mass transfer rate can be given as follows: J5

Ho 2 Hδ Cδo D 1 co 1 Hδ ro Ho o 1 o 1 o βf ½β fδ r5r0 β

ð6:73Þ

The β o value is expressed in Eq. (6.72), while the β of and ½β ofδ r5r0 are given by Eqs. (6.14) and (6.15b), respectively.

6.3

Diffusion Accompanied by Chemical Reaction

6.3.1

Solution as a Bessel Function

Let us look at the solution of mass transfer accompanied by first-order chemical reaction in a cylindrical hollow fiber membrane. The mass balance equation for the catalytic membrane will be as d2 φ 1 dφ 1 2 ϑ2 φ 5 0 dR2 R dR

ð6:74Þ

Diffusion in a Cylindrical Membrane Layer

169

with rffiffiffiffiffiffiffiffi k1 ro2 ϑ5 D where k1 is the reaction rate constant (1/s); ro is the capillary radius (m); and D is the diffusion coefficient in the capillary membrane (m2/s). The general solution of Eq. (6.74) will be as (O’Neil, 1987): φ 5 TIo ðϑRÞ 1 SKo ðϑRÞ

ð6:75Þ

where Io is a Bessel function of the first kind of order zero and Ko is a Bessel function of the second kind of order zero. The parameters T and S should be determined by the suitable boundary conditions as R 5 1;

φ 5 φ

R 5 1 1 δ=ro ;

ð6:76aÞ φ 5 φδ

ð6:76bÞ

After solution, we get T5

φ φ Io ðϑ½1 1 δ=ro Þ 2 φδ Io ðϑÞ Ko ðϑÞ 2 Io ðϑÞ Ko ðϑÞIo ðϑ½1 1 δ=ro Þ 2 Ko ðϑ½1 1 δ=ro ÞIo ðϑÞ Io ðϑÞ

ð6:77Þ

S5

φ Io ðϑ½1 1 δ=ro Þ 2 φδ Io ðϑÞ Ko ðϑÞIo ðϑ½1 1 δ=ro Þ 2 Ko ðϑ½1 1 δ=ro ÞIo ðϑÞ

ð6:78Þ

and

The Io and Ko are well-known functions and their values can be found in O’Neil (1987, p. 311).

6.3.2

Analytical Approach for Solution

The analytical solution of the Bessel function is a rather complex task. On the other hand, the parameters are often concentration- or local-coordinate dependent. Thus, an analytical approach for the solution can be very useful. Look at a general case here, namely when there is diffusion plus convection, as well as a chemical reaction. Let us look at a first-order reaction. The methodology of the solution is discussed in details in Sections 4.2.3 and A2. How this methodology can be extended to the solution of second-order or

170

Basic Equations of the Mass Transport through a Membrane Layer

Michaelis-Menten reactions is also discussed in Chapter 4. Accordingly, the ith sublayer of the membrane can be given in the following balance equation: Di


 d2 φ 1 dφ 1 2 υ 2 ki φ 5 0; dr 2 ri dr

ri21 , r , ri ;

ð6:79Þ

where the value of ki, first-order reaction constant, can be regarded as constant or variable. Equation (6.79) is in dimensionless form as [R 5 r/ro, Φ 5 φ/φ , Φi 5 ðΦi21 1 Φi Þ=2:
 d2 Φ 1 dΦ 2 Pei 2 2 ϑ2 i Φ 5 0 dR2 Ri dR

ð6:80Þ

where Ri denotes the average value of Ri in the ith sublayer of the catalytic membrane layer [Ri 5 1 1ði 20:5ÞΔR; ΔR 5 δ/(Nro)], ro is inner radius of the cylindrical membrane (m). ϑi 5

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ro2 ki =Di ;

Pei 5

υ i ro Di

~ Let us introduce the following new variable Φ: ~ 5 Φe2Peoi R=2 Φ

ð6:81Þ

with Peoi 5 ðPei 2 1=Ri Þ After a few manipulations, one can get the following differential equation to be solved: ~ d2 Φ ~ 50 2 Θ2i Φ dR2

ð6:82Þ

where rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Peo2 i 1 ϑ2i Θi 5 4 The solution of Eq. (6.82) can be easily obtained by well-known mathematical methods as it follows (Nagy, 2008, 2009; Nagy and Borbe´ly, 2009): ~

Φ 5 Ti eλi R 1 Si eλi R

ð6:83Þ

Diffusion in a Cylindrical Membrane Layer

171

with Peo λ~ i 5 i 1 Θi ; 2

λi 5

Peoi 2 Θi 2

Parameters Ti and Pi of Eq. (6.83) can be determined by means of the boundary conditions for the ith sublayer (with 1 # i # N). The boundary conditions at the internal interfaces of the sublayers (1 # i # N 21; Ri 5 1 1 iΔR; ΔR 5 δ/[Nro]) can be obtained from the following two equations [Eqs (6.84a) and (6.84b)]:

 dΦi dΦi11   1 Pei Φi  1 Pei 1 1 Φi11 R5R1i at R 5 Ri 5 Di11 2 Di 2 R5R2i dR dR ð6:84aÞ   Φi 

R5R2 i

  5 Φi11 

R5R1 i

at R 5 Ri with i 5 1; . . . ; N 2 1

ð6:84bÞ

where Φi is the substrate concentration of the ith sublayer in the biocatalytic membrane at R 5 Ri. The external boundary conditions, namely that at R 5 1 and R 5 1 1 δ/ro can depend on the operating conditions. Two cases will be discussed here: (1) the concentrations on the both sides of the membrane are determined by the feed and sweep phases (there is diffusion transport into the sweep phase, thus, generally saying, dφ/dR . 0 at R 5 1 1 δ/ro); and (2) the permeate compound is immiscible in the sweep phase or there is no sweep phase, accordingly dφ/dR 5 0 at R 5 1 1 δ/ro. Mass transfer with sweep phase on the permeate side, dΦ/dR . 0 at R 5 1 1 δ/ro (Case A). Thus, the external boundary conditions will be as ~

at R 5 1 then Φ 5 1  T1 eλ1 1 S1 eλ1

ð6:84cÞ

~

ð6:84dÞ

at R 5 1 1 δ=ro then Φ 5 Φδ  TN eλN ð11δ=ro Þ 1 SN eλN ð11δ=ro Þ

The solution methodology is discussed in Nagy’s papers (Nagy, 2008; Nagy and Borbe´ly, 2009). After solution of the algebraic equation system containing 2N equations, the value of the integration parameters for the first sublayer can be expressed as 0 1 T1 5

ΩTN B B  Bφ 2 @ ΩO N

e2PeN ðδ=ro Þ=2 o

ΩTN

N

L coshðΘi ΔRÞ

~ C e2λ1 C φδ C A 2 coshðΘ1 ΔRÞ

ð6:85Þ

i52

0 S1 5

ΩSN B B  Bφ 2 @ ΩO N

1 e2PeN ðδ=ro Þ=2 o

ΩSN

N

L coshðΘi ΔRÞ i52

C e2λ1 C φδ C A 2 coshðΘ1 ΔRÞ

ð6:86Þ

172

Basic Equations of the Mass Transport through a Membrane Layer

with *

ΩNj

j tanhðΘN ΔRÞ ψN21 DN21 PeoN 5 12 2 j 2 ΘN ΩN21 DN

!+

j ΩN21 ξ N21

with j 5 T; S; O ð6:87Þ

The value of ξ i21, Ai, and Bi (with i 5 1 2 N 21) as well as Ωij (with j 5 T, S, O and i 5 1 2 N 21), and ψij (with j 5 T, S, O and i 5 1 2 N) can be calculated by the expressions given below: ξ i21 5 ehPei2Pei21 iRi21 =2

with i 5 1 2 N

* Ωij 5

j tanhðΘi ΔRÞ ψi21 Di21 Peoi 12 2 j 2 Θi Ωi21 Di

ð6:88Þ

!+

j Ωi21 with j 5 T; S; O; i 5 2 2 N 21 ξi21

ð6:89Þ * j 5 ψi21

Bi21 2 Ai21

j ψi21 Di21 Peoi 2 j Di 2 Ωi21

!+

j Ωi21 with j 5 T; S; O; i 5 2 2 N 21 ξ i21

ð6:90Þ Ai 5

Peoi Peo tanhðΘi ΔRÞ 2 Θi ; Bi 5 i 2 Θi tanhðΘi ΔRÞ with i 5 1 2 N 21 2 2 ð6:91Þ

The initial values of Ωij and ψij ; namely Ω1j and ψ1j (j 5 T, S, O), are as ΩT1 5 e2Θ1 ΔR ; ψT1 5 λ~ 1 e2Θ1 ΔR ;

ΩS1 5 eΘ1 ΔR ;

ΩO 1 5 2tanhðΘ1 ΔRÞ

ψS1 5 λ1 eΘ1 ΔR ;

ψO 1 5 2A1

ð6:92Þ ð6:93Þ

It is important to note that the calculation of theΩij and ψij values for i 5 1 to N 21 (or to N) requires a very accurate process. All calculated variables should be given or calculated, even the value of ΔR ( 5 δ/(roN) with accuracy. Each step of the calculation should be carried out by a quick basic computer program with an accuracy of 14 decimals. This is the maximal accuracy of this program. To get the concentration distribution, the values of Ti and/or Si, with i 5 2 2 N, should be determined. [The N value should be chosen to be not less than 100 during calculation; note that in reality it is enough to predict either T1 or S1 because if you know one of these two parameters, the other one can be obtained from the boundary condition given by Eq. (6.84c).]

Diffusion in a Cylindrical Membrane Layer

173

From the internal boundary conditions given by Eqs (6.84a) and (6.84b), the parameters Ti and Si (i 5 2 2 N) can be calculated by the following expressions: Ti 5

ðΓi21 λ~ i 2 Ξi21 Þ λi ½11ði 21ÞΔR e 2Θi ePei ½1 1ði 21ÞΔR

with i 5 2 2 N

ð6:94Þ

Si 5

Ξi21 2 λi Γi21 λ~ i ½11ði 21ÞΔR e 2Θi ePei ½1 1ði 21ÞΔR

with i 5 2 2 N

ð6:95Þ

where ~

Γi 5 Ti eλi ð11 iΔRÞ 1 Si eλi ð11iΔRÞ Ξi 5

ð6:96Þ

o Di n ~ Ti λi eλi ð11iΔRÞ 1 Si λ~ i eλi ð11iΔRÞ Di11

ð6:97Þ

The mass transfer rate at place of Y 5 0, as a sum of the diffusive and the convective flows, can be given as J5

D1 ðλ1 T1 eλ~ 1 λ~ 1 S1 eλ Þ ro

ð6:98Þ

Mass transfer without sweep phase on the permeate side, dΦ/dR 5 0 at R 5 1 1 δ/ro (Case B). ~

at R 5 1 then Φ 5 1  T1 eλ1 1 S1 eλ1

ð6:99aÞ

~ at R 5 1 1 δ=ro then Φ 5 Φδ  TN λN eλN ð11δ=ro Þ 1 SN λ~ N eλN ð11δ=ro Þ

ð6:99bÞ

The internal ones with i 5 2, 3, . . ., N 21 are given by Eqs (6.84a) and (6.84b). After the solution, one can obtain for the values of T1 and S1 as T1 5 φ e2λ1

ΨTN 1 2 coshðΘ ΨO 1 ΔRÞ N

ð6:100Þ

S1 5 φ e2λ1

ΨSN 1 2 coshðΘ ΨO 1 ΔRÞ N

ð6:101Þ

~

where * ψjN

5 B N 2 AN

ψjN21 DN21 Peoi 2 2 ΩjN21 DN

!+

ΩjN21 ξN21

with j 5 T; S; O

ð6:102Þ

174

Basic Equations of the Mass Transport through a Membrane Layer

and AN 5

PeoN tanhðΘN ΔRÞ 1 ΘN ; 2

BN 5

PeoN 1 ΘN tanhðΘN ΔRÞ 2

ð6:103Þ

Similar to the previous case, the value of ξi 21, Ai, and Bi [with j 5 T, S, O; i 5 1 2 (N 21)] as well as Ωji and ψji [with j 5 T, S, O and i 5 1 2 (N 2 1)] can be calculated by expressions given by Eqs (6.87)(6.91). The difference between the two models is caused by the outlet boundary condition, thus, the expressions differ at expression related to place of r 5 ro 1 δ. Knowing T1 and S1, the other parameters, namely Ti and Si (i 5 2, 3, . . ., N), can easily be calculated by means of the internal boundary conditions given by Eqs (6.84a) and (6.84b), starting from T2 and S2 up to TN and SN. Thus, the values of Ti and Si can be predicted from Ti21 and Si21 using Eqs. (6.94)(6.97).

6.3.3

Diffusion Accompanied by Zero-Order Reaction

Mass transfer without sweep phase on the permeate side, dΦ/dR . 0 at R 5 1 1 δ/ro. The mass balance equation to be solved will be, for steady-state, as:
2  d φ 1 dφ 1 2 k0 5 0 D dr 2 r dr

ð6:104Þ

In dimensionless form d2 Φ 1 dΦ 2 ϑ2 5 0 1 dR2 R dR

ð6:105Þ

where

sffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi k0 ro2 k0 ro2 ϑ5   Dφ DHco

The solution of Eq. (6.105), with boundary conditions of Φ 5 1 at R 5 1 and Φ 5 Φδ at R 5 1 1 δ/ro, will be as Φ 5 T ln R 1 S 1 where

T5

Φδ

ϑ2 2 R 4


212

ϑ2 4

2δ ro

 ln 1 1

1

δ ro

ð6:106Þ h i2  δ ro

ð6:107Þ

Diffusion in a Cylindrical Membrane Layer

175

and S512

ϑ2 4

ð6:108Þ

The inlet mass transfer rate is as:   D o dΦ  J 5 2 Hc ro dR 

 8 9  ϑ2 δ δ < 1 2 Φ 1 2 1 δ ro 4 ro D o ϑ2 =  5 Hc 2 : 2; ro ln 1 1 rδo R51

ð6:109Þ

The outlet mass transfer rate, Jδ, i.e., at R 5 1 1 δ/ro is as:  8 9 
 ϑ2 δ δ < 1 2 Φ 1 2 1 δ ro 4 ro D ϑ2 δ =   2 11 Jδ 5 2 ro : 1 1 δ ln 1 1 δ ro ; ro ro

ð6:110Þ

Mass transfer without sweep phase on the permeate side, dΦ/dR 5 0 at R 5 1 1 δ/ro. After solution of Eq. (6.105) with this new boundary condition on the downstream side of the capillary membrane, the concentration distribution can be expressed as follows: Φ5


 ϑ2 δ 2 ϑ2 11 ð1 2 ln RÞ 1 1 1 ðR2 2 1Þ 2 4 ro

ð6:111Þ

The inlet mass transfer rate will be as:
 D o 2δ δ 11 J 5 Hc ϑ ro ro 2ro

ð6:112Þ

References Nagy, E. (2006) Binary, coupled mass transfer with variable diffusivity through cylindrical membrane. J. Membr. Sci. 274, 159168. Nagy, E. (2008) Mass transport with varying diffusion- and solubility coefficient through a catalytic membrane layer. Chem. Eng. Res. Design 86, 723730. Nagy, E. (2009) Basic equations of mass transfer through biocatalytic membrane layer. Asia-Pacific J. Chem. Eng. 4, 270278. Nagy, E., and Borbe´ly, G. (2009) Mass transport through anisotropic membrane layer. Desalination 240, 5463. O’Neil, P.V. (1987) Advanced Engineering Mathematics. Wadsworth Inc., Belmont.

7 Transport of Fluid Phase in a Capillary Membrane

7.1

Introduction

A complete description of a membrane module requires that local transport equations that describe the flow and transport conditions should be simultaneously solved. There has been a substantial amount of published research on developing equations to characterize the transport properties (Kelsey et al., 1990; Mondor and Moresoli, 1999; Geraldes et al., 2001; Calabro´ et al., 2002; Long et al., 2003; Marriott and Sorensen, 2003; Richardson and Nassehi, 2003; Damak et al., 2004; Godongwana et al., 2007). There are different regions (lumen side, membrane matrix, and shell side), all of which should be modeled. A structure of the model system is illustrated in Figure 7.1 (Marriott and Sorensen, 2003). As it will be shown, the starting Navier
Stokes flow models with component and energy equations are very complex equation systems. Thus, significant simplification is needed to obtain more easily applicable model equations. Careful analysis of the separation equipment should be carried out regarding the flow and operating conditions. The pressure distribution in the membrane can be determined by application of the overall balance of linear momentum. The utilization of this conservation principle is complicated here by the fact that an external force must be applied to keep the polymer membrane stationary. Consequently, the balance of linear momentum must include the effects of this externally applied force in addition to inertia, viscous forces, the gravitational force, and pressure forces. Inertia effects are negligible in the membrane because velocities are small, and the effects of viscous forces are also negligible because deformation gradients are small. Consequently, the externally applied force is effectively used to balance pressure and gravitational forces. For example, in a capillary membrane module with permeable membrane wall (Figure 7.2), three regions of flow should be considered: flow in the lumen (0 # r # ro), flow within the membrane matrix (ro # r # ro 1δ), and flow in the extracapillary space (ro 1δ # r). The differential balance equation should be given for every region, thus, the continuity, momentum equations, as well as the component mass balance and/or energy balance equations. Thus, in order to describe behavior of a membrane module, three submodels are required: two that describe the flow or transport on either side of the membrane and a third model that characterizes the separation properties of the membrane and any porous support material (for details of the membrane transport, see Chapters 3
6). Basic Equations of the Mass Transport through a Membrane Layer. DOI: 10.1016/B978-0-12-416025-5.00007-7 © 2012 Elsevier Inc. All rights reserved.

178

Basic Equations of the Mass Transport through a Membrane Layer

Operating conditions

Transport parameters (Pe, Re, etc.)

Model equation

Physical properties

Simplified equations

Feed side Flow model

Qi

Transport models for membrane

Qi

Permeate side Flow model

Principal model

Figure 7.1 A model structure for modeling a membrane module.

7.2

Flow Models for Fluid Phases on Both Sides of Capillary Membrane Modules

The general model equations listed by Eqs (7.1)
(7.8) should be adapted for fluids in the lumen and shell-side regions. Note that the balance equations given below do not contain any source terms. In given cases, the production or consumption of momentum, component, or heat should additionally be taken into account in their balance equations, as will be made in a few chapters. The equation of continuity (conservation of mass) in a cylindrical coordinate system (Bird et al., 1960) (note there is no source term in the balance equations for the fluid phase) is: @ρ 1 @ 1 @ @ 5 ðρrυr Þ1 ðρυθ Þ1 ðρυz Þ50 @t r @r r @θ @z

ð7:1Þ

where ρ is the mass density (kg/m3); υ is the convective velocity (m/s); t is the time (s); r is the radial coordinate (m); z is the axial coordinate (m); and subscripts r, θ, and z denote the different coordinate directions.

Transport of Fluid Phase in a Capillary Membrane

Membrane matrix

179

Figure 7.2 Schematic diagram of the flow in a capillary tube with a permeable wall.

Membrane lumen Shell

ro

δ

The equation of motion in cylindrical coordinates (in terms of Newtonian fluid with constant ρ and μ): r-component: ρ


 @υr @υr υθ @υr υ2θ @υr 1υr 1 2 1υz @t @r r @θ r @r
   @p 1@ @υr υr 1 @2 υr 2 @υθ @2 υr 2 21 2 2 2 2 1 2 1ρgr 5 2 1μ r @r r @z @r r @r r @θ r @θ

ð7:2Þ

θ-component: ρ


 @υθ @υθ υθ @υθ υr υθ @υθ 1υr 1 2 1υz @t @r r @θ r @r 
  1 @p @ 1@ 1 @2 υθ 2 @υr @2 υθ 1 2 1ρgθ 52 1μ ðrυθ Þ 1 2 2 2 2 @z r @θ @r r @r r @θ r @θ

ð7:3Þ

180

Basic Equations of the Mass Transport through a Membrane Layer

z-component:  

 @υz @υz υθ @υz @υz @p 1@ @υz 1 @2 υz @2 υz ρ 1υr 1 1υz 52 1μ 1 2 2 1 2 1ρgz r @t @r r @θ @z @r @z @z r @r r @θ ð7:4Þ where p is the pressure (Pa); μ is the dynamic viscosity (Pa s 5 kg/ms); and g is the gravitational acceleration (9.80665 m/s2). Equation (7.1) defines the mass balance in the cylindrical system (tube, membrane). According to Eq. (7.1), the component balance equation can be defined as well. The convective transported components, inside the flowing phase, also can be transported by molecular diffusion. Let Ji define the overall transport rate of the component i, Eq. (7.1) can be rewritten for component i, defining the three-dimensional distribution of i, as (Labecki et al., 2004): @ci 1 @ðrJr;i Þ 1 @Jθ;i @Jz;i 5 1 1 @t @z r @r r @θ

ð7:5Þ

where ci is the local concentration of component i (kg/m3); Jr,i, Jθ,i, and Jz,i are component flux in r, θ, and z direction, respectively (kg/m2 s). Upon expansion of the component flux into diffusive and convective terms, the following equation governing the i solute transport is obtained:


 @ci 1@ @ci 1 @ Dθ;i @ci @ @ci 5 2rυr;i ci 1 2υθ;i ci 1 2υz;i ci rDr;i Dz;i @t @r @z r @r r @θ r @θ @z ð7:6Þ Equation (7.6) considers variable diffusion coefficient and convective flow rate. In the case of constant diffusivity and convective velocities, one can get the following, known equation (Bird et al., 1960, p. 559): 
  @ci @ci @ci υθ @ci 1@ @ci 1 @2 c i @2 c i 1u 1υr 1 5 Di 1 2 21 r @t @z @r r @θ @r @z r @r r @θ

ð7:7Þ

The equations of energy in terms of the transport properties (with Newtonian fluids of constant ρ and k):
 
  @T @T @T υθ @T @ 1 @T 1 @2 T @2 T ^ 1u 1υr 1 5k 1 2 21 1ℑ ρC p r @θ @t @z @r @r r @r r @θ @z ð7:8aÞ where C^ p denotes the heat capacity at constant pressure, per unit mass (kJ/kg K  m2/s2 K); k is the thermal conductivity (kW/m K  kg m/s3 K); and

Transport of Fluid Phase in a Capillary Membrane

181

u is the axial convective velocity (m/s). Note that k 5 λρC^ p ; where λ is the thermal diffusivity (m2/s), ρ is the fluid density (kg/m3). In Eq. (7.8a), the terms associated with viscous dissipation also are given according to Bird et al. (1960, p. 319). These terms [Eq. (7.8b)] should be involved for systems with large velocity gradients only (for details, see Bird et al., 1960), thus this term can be neglected, but in special cases should be taken into account. In a capillary and even in a tubular membrane, these terms are generally not important. (

2
2 ) @υr 2 1 @υθ @u 1υr 1 1 ℑ 5 2μ r @θ @z @r (

2
 ) @υθ 1 @u @u @υr 2 1 @υr @ υθ 2 1 1 1 1 1 1μ @z @z r @θ @r r @θ @r r

ð7:8bÞ

Let us consider a steady-state, laminar, incompressible, viscous, and isothermal flow with a permeable wall. The flow is axisymmetric, so only half of the tube (capillary) should be considered. Moreover, with a sufficiently long entrance region in the tube, the flow is fully developed at the permeable tube entrance. A schematic diagram of the physical model and coordinate system is given in Figure 7.2 for a vertically oriented tube. Thus, the above Navier
Stokes equations [Eqs (7.1)
(7.4)] can be simplified for the tube side of a capillary as follows (Wiley and Flechter, 2003; Damak et al., 2004): 1@ @u ðrυÞ1 50 r @r @z

ð7:9Þ

  
  @u @u @p @ 1 @ðruÞ @2 u ρ υ 1u 5 2 1μ 1 2 @r @z @z @r r @r @z

ð7:10Þ

  
  @υ @υ @p @ 1 @ðrυÞ @2 υ ρ υ 1u 5 2 1μ 1 2 @r @z @z @r r @r @z

ð7:11Þ

and for solute concentration in the case of constant density u



 @c @c @ @c 1 @ðDcÞ @ @c 1υ 5 D 1 1 D @z @r @r @r r @r @z @z

ð7:12Þ

as well as for the temperature distribution
  2  @T @T @ T 1 @T @2 T ^ ρC p u 1 1υ 5k 1 @z @r @r 2 r @r @z

ð7:13Þ

182

Basic Equations of the Mass Transport through a Membrane Layer

where u and υ are the axial and radial flow rates, respectively; the diffusion coefficient can be varied as a function of the concentration. Equations (7.9)
(7.13) can further be reduced depending on the transmembrane convective velocity, υ. Its value depends strongly on the transmembrane pressure [see Eqs 1.35
1.37]. In the case of ultrafiltration or nanofiltration, the permeation velocity falls between about 1024 and 1026 m/s, while the axial convective velocity may be between 1 3 1023 and 3 3 1021 m/s, depending strongly on the necessary operating conditions, that is, depending on the membrane process itself. As the reader can see, the axial velocity often can be much larger than that of the permeation rate. Look at the value of the axial (Peax 5 uL/Df, where L is the capillary length, and Df is diffusivity in fluid) and radial Peclet numbers (Perad 5 υro/Df, where ro is the internal radius of capillary); their values can help for the further simplification of Eqs (7.8)
(7.10). The Peclet number is the ratio of the convective velocity to the diffusive mass transfer coefficient. Let the radius of the capillary be 150 μm, while the diffusion coefficient in the flowing liquid changes between 1028 and 1029 m2/s. Thus, we can see the axial Peclet number is between about 10 and 4.5 3 104, thus the axial diffusion term and the axial thermal conductivity term can be neglected. On the other side, the radial Peclet number can change between about 1025 and 1 thus, the radial concentration gradient is rather low, accordingly, momentum equation of the transverse velocity can be neglected. The question is that the momentum radial convective velocity term in Eqs (7.9) and (7.10) as well as Eqs (7.12) and (7.13) can be neglected when the radial velocity is relatively low. This can only be done when the axial volumetric flow rate does not change significantly during the flow trough of a capillary tube with a length of L. The ratio of the volumetric permeation rate to the volumetric axial rate, that is υro/uL, can orient us. If this ratio is much less then 1, the transverse term can be neglected. Accordingly, Eqs (7.8)
(7.11) can be simplified as @u 1 @ðrυÞ 1 50 @z r @r

ð7:14Þ

In the most viscous flows, normal stress effects, @2u/@z2, are not as important as shear stresses (Godongwana et al., 2007) and thus, Eq. (7.10) becomes 052


 @p 1@ @u 1μ r @z r @r @r

ð7:15Þ

The component and energy balance will be as

u


2  @ci @ ci 1 @ci 5D 1 @z @r 2 r @r

ð7:16Þ

Transport of Fluid Phase in a Capillary Membrane

183

and
2  @T @ T 1 @T ^ ρC p u 1 5k @z @r 2 r @r

ð7:17Þ

Similarly, the balance equations can be simplified for the membrane and for the shell side, as well. In the following sections, some special cases will be shown briefly.

7.3

Special Cases

7.3.1

For the Axial Flow of an Incompressible Fluid in a Circular Tube with an Impermeable Wall (Figure 7.3)

We consider a long tube and set υθ and υr equal to zero. The remaining velocity component υz (let it be denoted by u) will not be a function of θ because of cylindrical symmetry. The z-component of the equation of motion for constant ρ and μ may then be written as  
 @u @p 1@ @u @2 u r 1 2 ρu 5 2 1μ @z @z r @r @r @z

ð7:18Þ

This equation may be further simplified by taking advantage of the equation of continuity, which reduces here to @u 50 @z

ð7:19Þ

In the most viscous flows, normal stress effects, @2u/@z2, are not as important as shear stresses (Godongwana et al., 2007), and thus Eq. (7.12) becomes
 @p 1@ @u r 0 5 2 1μ @z r @r @r

ð7:20Þ

Figure 7.3 Flow in a capillary tube with an impermeable wall. r z

z=0

z=L

184

Basic Equations of the Mass Transport through a Membrane Layer

Integration twice with respect to r and use of the boundary conditions u50 at r5R, and u is finite at r50 gives ðp0 2pδ Þro2 u5 4μδ

7.3.2

(


2 ) r 12 ro

ð7:21Þ

Flow Equations for Ultrafiltration in a Capillary Tube with a Permeable Wall (Song, 1998)

During ultrafiltration, the axial volumetric velocity will change in an axial direction, depending on the transverse volumetric flow rate. Consequently, both the radial and axial convective velocities have to be taken into account (Figure 7.4). The equations to be solved for isothermal ultrafiltration are as follows: Continuity equation: 1@ @u ðrυÞ1 50 r @r @z

ð7:22Þ

Momentum equation:
    @u @u @P @ 1 @ðruÞ ρ υ 1u 5 2 1μ @r @z @z @r r @r

ð7:23Þ

Mass balance equation: υ

  @c @c @ 1 @ðrcÞ 1u 5 D @r @z @r r @r

ð7:24Þ

It is worth noting here that u and υ are the bulk flow velocities, that is, the convective transport of the fluid and particles is the same. These equations are subject to the following boundary conditions: At the inlet of the capillary lumen side, at z50: c5co ;

z50;

υ50;

u5uo

Figure 7.4 Transport in a capillary tube with a permeable wall.

ν u r z

z=0

for all z

z=L

Transport of Fluid Phase in a Capillary Membrane

185

At the membrane wall, r5ro: at r5ro

u50;

at r50

@υf 5 0; @r

υw 5 Δp=μRm ; @c 50 @r

uc2Df

@c @φ 5 υw φ2Dm @r @r

for all z

for all z

where subscripts w, f and m denote wall, fluid and membrane, respectively, R is membrane resistance, μ is viscosity. Parameters without subscripts are related to the flowing fluid phase.

7.3.3

Capillary Transport with Low Transverse Convective Velocity

Several membrane separation processes often involve low convective velocity in a radial direction such as nanofiltration, membrane reactors, gas separation, pervaporation in porous ceramic membrane where the momentum and continuity equations, and consequently, the component and energy equations can significantly be reduced or neglected, as briefly discussed by Eqs (7.14) and (7.15). In this case, the membrane is relatively impermeable, that is, the membrane is the main resistance to flow. Regarding its importance, a solution of this case will be shown here. The continuity and momentum equations to be solved are (Kelsey et al., 1990; Piret and Cooney, 1990; Mondor and Moresoli, 1999): @u 1 @ðrυÞ 1 50 @z r @r 052

ð7:14Þ


 @p 1@ @u 1μ r @z r @r @r

ð7:15Þ

When the momentum equation [Eq. (7.15)] is integrated twice with respect to r, one can get u5

dp r 2 1A ln r1B dz 4

ð7:25Þ

The integration constants A and B can be determined by the following boundary conditions (Bird et al., 1960; Mondor and Moresoli, 1999): at r 5 ro

u50

at r 5 0

@u 5 0 for all r @z

for all r

ð7:26Þ ð7:27Þ

186

Basic Equations of the Mass Transport through a Membrane Layer

Consequently, it can be obtained as  2 ! ro2 r dpf 12 u52 4μ dz ro

ð7:28Þ

Replacing Eq. (7.28) into Eq. (7.15), one can get for the radial convective velocity in the lumen side (for distinction for the membrane’s velocity, subscript f denotes the fluid phase here in the lumen side) (
 ) ro3 r 1 r 3 d2 pf 2 ð7:29Þ υf 5 8μ ro 2 ro dz2 From Eq. (7.29), the wall velocity, υw, can be given as (Brotheton and Chau, 1990): υw 5

ro3 @2 pf 16μ @z2

ð7:30Þ

Look at the radial convective velocity in the membrane itself. The flow through this membrane is proportional to the transmembrane pressure drop and inversely proportional to the fluid viscosity. The continuity equation for the membrane (ro #r#ro 1δ) is as follows:
 1@ @υ r 50 ð7:31Þ r @r @r Let us apply Darcy’s law to obtain the radial convective in the membrane: υ52

K dp μ dr

ð7:32Þ

where K is the Darcy permeability (m2); μ denotes the viscosity (kg/ms); ε is the porosity of the membrane; υ is the radial convective velocity in the membrane (m/s); and p is the pressure (Pa  kg/m s2). Replacing Eq. (7.32) into Eq. (7.31), one can obtain the following equation (the constant factor can be omitted):
 1@ @p r 50 ð7:33Þ r @r @r When Eq. (7.33) is integrated twice with respect to r using the boundary conditions, and when r 5 ro then p 5 po and when r 5 ro 1 δ then p 5 pδ, the following membrane pressure is obtained: p5

ΔpðzÞ r ln 1po ðzÞ lnð11δ=ro Þ ro

ð7:34Þ

Transport of Fluid Phase in a Capillary Membrane

187

where Δp 5 po 2 pδ. Replacing Eq. (7.34), after derivation, into Eq. (7.32), the radial velocity can be obtained as υ5

K ro po ðzÞ2pδ ðzÞ εμ r lnð11δ=ro Þ

ð7:35Þ

Note that pressures are a function of the axial coordinate according to the change of the boundary conditions in axial direction, thus po(z), Δp(z), and pδ(z) should be considered. Equation (7.35) expresses the radial convective velocity distribution in the membrane at a given axial point. The wall velocity will be from this equation as υw 5

K po ðzÞ2pδ ðzÞ εμ lnð11δ=ro Þ

ð7:36Þ

In the following case, we assume that the pressure in the shell side does not change as an axial direction, that is, pδ 5 constant. From equality of Eqs (7.30) and (7.36), one can obtain a second-order differential equation as (the radial pressure gradient in the lumen is negligible) d2 p 2A2 p 5 2A2 pδ dz2

ð7:37Þ

where

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16K A5 3 ε ro lnð11δ=ro Þ

The following boundary conditions can be used (Brotheton and Chau, 1990): a. In an open system:

at z50 at z5L

p5pin p5pout

ð7:38Þ

b. In a closed system:

at z50 at z5L

p5pin dp 50 dz

ð7:39Þ

The main resistance is created by the capillary membrane in the closed system. In an open system, the outlet pressure is mostly zero. The solution of Eq. (7.37) is well known p 5 T eAz 1Se2Az 1pδ

ð7:40Þ

188

Basic Equations of the Mass Transport through a Membrane Layer

The pressure axial distribution will be as: a. For an open system:

p5

 pin 2pout cosh ALð12z=LÞ 1pδ coshðALÞ

ð7:41Þ

b. For a closed system:

p5

 pin 2pout sinh ALð12z=LÞ 1pδ sinhðALÞ

ð7:42Þ

With the knowledge of p(z), now one can determine the wall velocity in an axial direction applying Eqs (7.36) and (7.41) or (7.42), for the open and closed systems, respectively, as υw 5

 K pin 2pout cosh ALð12z=LÞ εμ lnð11δ=ro Þ coshðALÞ

ð7:43Þ

υw 5

 K pin 2pout sinh ALð12z=LÞ εμ lnð11δ=ro Þ sinhðALÞ

ð7:44Þ

or

The mass balance over a slice dz of the fiber can give du 2 5 2υw dz ro

ð7:45Þ

where u denotes the average axial velocity (for laminar flow u 5 umax =2; where umax is the maximal velocity of the parabolic flow profile). Note that the value of u is radial average one; it depends on the axial coordinate. By means of Eq. (7.45), the distribution of the average axial lumen velocity can be determined, namely ðu uo

du 5 2

2 ro

ðL

υw dz

ð7:46Þ

0

Accordingly, we can get for the open and closed systems, respectively. u 5 uo 2Ξ tanh½AL with

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ro KΔp Ξ5 2 4μ εL2 lnð11δ=ro Þ

ð7:47Þ

ð7:48Þ

Transport of Fluid Phase in a Capillary Membrane

189

and u 5 uo 2Ξ

½12coshðALÞ coshðALÞ

ð7:49Þ

The above methodology was used and discussed with varying axial pressure on the shell side of the capillary membrane (Brotheton and Chau, 1990; Kelsey et al., 1990; Mondor and Moresoli, 1999).

7.3.4

Mass Transport Through a Capillary Membrane: Component Model for the Feed Side

Most of the membrane separation processes involve the component transport as well. One hundred percent retain of a component, particle, colloids by a membrane layer, is very rare. However, the membrane enables the selective retain and permeation of solute components (e.g., gas separation, pervaporation, nanofiltration, dialysis, and membrane reactor) depending on the operating conditions, membrane structure, and so on. The description of such processes needs the solution of the differential balance equations, discussed above, for both the feed and the permeate phases. The connection between the two departments is determined by the transport through the membrane. Thus, the balance equation also should be solved for the membrane phase. These results are discussed in Chapters 3
6. General balance equations [e.g., Eqs (7.1)
(7.8)] can be essentially simplified in most of the membrane processes. Let us assume isothermal process with zero radial convective velocity and steady-state conditions. This situation is illustrated schematically in Figure 7.5. In the case of a dilute solution, the differential component balance equation can be given as for the feed side (fiber lumen) of a capillary membrane, assuming a fully developed parabolic velocity profile (cf,i denotes the fluid concentration while the concentration in membrane will be given without subscript) as
2 
 r 2 @cf;i @ cf;i 1 @cf;i 2 5 Di ð7:50Þ 1 2uo ro 12 2 ro @z @r 2 r @r Let it be given in dimensionless form with R5r/ro; Z5z/L; Ci 5ci =coi
2  2uo ro2 @Cf;i @ Cf;i 1 @Cf;i ð12R2 Þ 5 1 Df;i L @Z @R2 R @R ν=0

Ji uo

uo

cio

r,y z

Ci,out Ji

Figure 7.5 Illustration of a component transport through a membrane.

ð7:51Þ

190

Basic Equations of the Mass Transport through a Membrane Layer

One should apply the common boundary conditions (  ) as follows: at Z 50 Ci 51 for all R at R 5 0

@Ci 5 0 for all Z @R

at R 5 1

Df;i

ð7:52aÞ ð7:52bÞ

@Cf;i @Φi 5 Di  2Ji @R @R

ð7:52cÞ 





where Φ is the dimensionless concentration in the membrane Φ 5 φ/φ (Hc 5 φ where concentration with stars are interface concentrations); Df and D are diffusivity in the fluid phase and in the membrane layer (m2/s); and J denotes the specific mass transfer rate (kg/m2 s). Note the third boundary condition contains the connection between the fluid phase and the membrane layer. Equation (7.52c) expresses that there is no accumulation or consumption of component on the membrane interface. In Chapters 3 and 6, Ji values have been defined under different conditions. One of these equations should be substituted in Eq. (7.52c). Average concentration distribution was calculated by numerical solution of Eq. (7.51) (Nagy and Hadik, 2002). The Ji mass transfer rate is given by Eq (3.23) or (3.26). Figure 7.6 clearly demonstrates how strongly the membrane mass 1.0

1.0

0.8

0.8 α=0

0.6

0.6

Ci

0.3 0.5

0.4

0.4

1.0

0.2

0.0

0.2

0

0.2

0.4

0.6

0.8

1

0.0

Z

Figure 7.6 Average concentration as a function of axial direction when the diffusion coefficient exponentially varies with the membrane concentration ðDi 5Do;i eαCi ; H55, ro5100 μm, Do,i51 3 1029 m2/s, uo55 3 1022 m, Sh5βroH/Di50.5, β5Do,i/δ55 3 1026 m/s).

Transport of Fluid Phase in a Capillary Membrane

191

transport parameters can influence the axial average concentration distribution, and further, the Ji mass transfer rate as well.

7.3.5

Flow Models for Plane Membrane Modules (Rectangular Coordinates, x, y, z)

As mentioned previously, there are a lot of membrane modules where rectangular coordinates should be used. In this section, the basic balance equations for these cases are given. These general forms of equations should then be simplified according to the operating conditions. According to Eqs (7.1)
(7.8), the balance equations for a rectangular coordinate system are as follows: For z-component (note the axial velocity is denoted by u, while that for perpendicular direction, y, is by υy, similar to the cylindrical coordinate; in x direction, the convective velocity is denoted by υx):
2 
 @u @u @u @u @p @ u @2 u @2 u 1ρgz 1υy 1υx 1u 5 2 1μ 1 1 ρ @t @y @x @z @z @x2 @y2 @z2

ð7:53Þ

For y-component:

2  @υy @υy @υy @υy @p @ υ y @2 υ y @2 υ y 1υy 1υx 1u 5 2 1μ 1 2 1 2 1ρgy ρ @t @y @x @z @x2 @y @z @y ð7:54Þ For x-component:

2  @υx @υx @υx @υx @p @ υ x @2 υ x @2 υ x 1υy 1υx 1u 5 2 1μ ρ 1 2 1 2 1ρgx @t @y @x @z @x2 @y @z @x ð7:55Þ For component balance:
2  @ci @ci @ci @ci @ ci @2 ci @2 ci 1υx 1υy 1u 5 Di 1 21 2 @t @x @y @z @x2 @y @z

ð7:56Þ

For energy balance:  2 
 @T @T @T @T @ T @ 2 T @2 T ^ ρC p 1 1 2 1ℑ 1υx 1υy 1u 5k @t @x @y @z @x2 @y2 @z

ð7:57Þ

192

Basic Equations of the Mass Transport through a Membrane Layer

(


 ) @υx 2 @υy 2 @u 2 ℑ 5 2μ 1 1 @x @y @z (


 ) @υx @υy 2 @υx @u 2 @υy @u 2 1μ 1 1 1 1 1 @y @x @z @z @x @y

ð7:58Þ

The value of ℑ that is associated with viscous dissipation can be neglected, as it can be done usually (Bird et al., 1960, p. 319). These terms should be involved for systems with large velocity gradients (for details, see Bird et al., 1960).

References Bird, R.B., Stewart, E., and Lightfoot, E.N. (1960) Transport Phenomena. John Wiley and Sons, New York. Brotheton, J.D., and Chau, P.C. (1990) Modeling analysis of an intercalated-spiral alternate-dead-ended hollow-fiber bioreactor for mammalian cell culture. Biotechnol. Bioeng. 35, 375
394. Calabro´, V., Curcio, S., and Iorio, G. (2002) A theoretical analysis of transport phenomena in a hollow fiber membrane bioreactor with immobilized biocatalyst. J. Membr. Sci. 206, 217
241. Damak, K., Ayadi, A., Zeghmati, B., and Schmitz, P. (2004) A new Navier
Stokes and Darcy’s law combined model for fluid flow in crossflow filtration tubular membrane. Desalination 161, 67
77. Geraldes, V., Semiato, V., and de Pinho, M.N. (2001) Flow and mass transfer modeling of nanofiltration. J. Membr. Sci. 191, 109
128. Godongwana, B., Sheldon, M.S., and Solomons, D.M. (2007) Momentum transfer inside a vertically oriented capillary membrane bioreactor. J. Membr. Sci. 303, 86
99. Kelsey, L.J., Pillarella, M.R., and Zydney, A.L. (1990) Theoretical analysis of convective flow profiles in a hollow-fiber membrane bioreactor. Chem. Eng. Sci. 45, 3211
3220. Labecki, M., Piret, J.M., and Bowen, B.D. (2004) Effects of free convection on three-dimensional protein transport in hollow-fiber bioreactor. AIChE J. 50, 1974
1990. Long, W.S., Bhatia, S., and Kamaruddin, A. (2003) Modeling and simulation of enzymatic membrane reactor for kinetic resolution of ibuprofen ester. J. Membr. Sci. 209, 69
88. Marriott, J., and Sorensen, E. (2003) A general approach to modeling membrane modules. Chem. Eng. Sci. 58, 4975
4990. Mondor, M., and Moresoli, C. (1999) Theoretical analysis of the influence of the axial variation of the transmembrane pressure in cross-flow filtration of rigid spheres. J. Membr. Sci. 152, 71
87. Nagy E., and Hadik P. (2002) Analysis of mass transfer in hollow-fiber membranes, Desalination. 145, 147
152. Piret, J.M., and Cooney, C.L. (1990) Model of oxygen transport limitations in hollow fiber bioreactors. Biotechnol. Bioeng. 37, 80
92. Richardson, C.J., and Nassehi, V. (2003) Finite element modeling of concentration profiles in flow domains with curved porous boundaries. Chem. Eng. Sci. 58, 2491
2503. Song, L. (1998) A new model for the calculation of the limiting flux in ultrafiltration. J. Membr. Sci. 144, 173
185. Wiley, D.E., and Flechter, D.F. (2003) Techniques for computational fluid dynamics modeling of flow in membrane channels. J. Membr. Sci. 211, 127
137.

8 Membrane Reactor 8.1

Introduction

The laboratory application and investigation of different types of membrane reactors as a promising unit operation started in the 1980s (Marcano and Tsotsis, 2002; Seidel-Morgenstern, 2010). It is well known that the selectivity in reaction networks toward a target compound can be increased by properly adjusting the local concentration of the reactants involved. A membrane separation unit can be applied especially for adjustment of the reactant’s and/or product’s concentration simply by coupling the membrane separation unit with a chemical/biochemical reactor. This coupling or integration often is made in the same unit. During the last three decades, this technical concept has attracted substantial worldwide research and process development efforts (Marcano and Tsotsis, 2002). There are several books and reviews published in the fields of both membrane reactors (Reij et al., 1998; Coronas and Santamarı´a, 1999; Saracco et al., 1999; Julbe et al., 2001; Marcano and Tsotsis, 2002; Paturzo et al., 2002; Dittmeyer et al., 2004; Charcosset, 2006; Judd, 2006; McLearly et al., 2006; Ozdemir et al., 2006; Seidel-Morgenstern, 2010) and membrane bioreactors (Marcano and Tsotsis, 2002; Rios et al., 2004; Fenu et al., 2010; Santos et al., 2010). It is not the aim of this work to discuss in detail the properties, applications, or devices of this process. We look at this unit operation as deeply as needed for its mathematical modeling or description. On the other hand, considering the essential differences in the operation and behaviors between the membrane reactors and the membrane bioreactors, mainly due to the other kinetic models, especially for the living organisms, these two reactor types will be discussed separately. Basically, two configurations of the membrane reactor system can be applied: in the first case, the reactor and the membrane separation equipment are simply connecting in series, while in the second case, the real membrane reactor concept combines these two different processing units (namely, reactor and a membrane separator) into a single unit (Figure 8.1). The membrane can serve as a distributor of one of the reactants or as an active catalyst and permselective layer. The subject of this chapter is to analyze briefly the mass transport in this latter membrane reactor configuration.

8.2

Membrane Reactor Configurations

Membrane-based reactive separation processes are mostly applying thin permselective, porous, or dense layers prepared by means of materials that are organic, inorganic, metal, and so on. The choice of a porous or a dense film and the type of material Basic Equations of the Mass Transport through a Membrane Layer. DOI: 10.1016/B978-0-12-416025-5.00008-9 © 2012 Elsevier Inc. All rights reserved.

194

Basic Equations of the Mass Transport through a Membrane Layer

used for manufacturing depends on the desired separation process, operating temperature, and driving force used for separation; the choice of material depends on the desired permeance and selectivity, and on thermal and mechanical stability requirements (Marcano and Tsotsis, 2002). There are various membrane reactor configurations at laboratory scale that focus on the reactant/product distribution in order to improve selectivity-conversion performances. According to Saracco et al. (1999) and Seidel-Morgenstern (2010), the membrane reactor concept can be divided into six groups which are schematically illustrated in Figure 8.2. Products in sweep fluid

Sweep fluid Reactants

Reactants

Catalytic and permselective membrane

Figure 8.1 Integrated membrane reactor system. (I)

(II)

Reactants

Homogeneous catalyst Ji

A

C

B

C

(III) A

(IV)

A

B⫹C

Sweep

B

C

A

B

D

(V) A Sweep

Catalytic membrane layer (A ⫹ B C)

A D⫹B

B⫹C

E

C E

B

(VI) A⫹B D⫹B JD

D E

E

A

D

B

A⫹B D⫹B

D E

JB

Figure 8.2 Illustration of most often-applied membrane reactor concepts (I
VI).

D

Membrane Reactor

195

According to Figure 8.2, the essentials of this membrane concept are: 1. The reaction product is separated continuously from the homogeneous catalyst, thus, this process can be operating in continuous mode. 2. The membrane serves as a catalyst layer; reactants can supply in regular manner which enables the avoidance of side reactions. 3. The selective removal of the product B enables the enhancement of the productivity to shift the reaction to production of compound B. 4. This is a realization of the selective transport of product B which participates in a second reaction. 5. This concept enables the removal of the product in order to avoid the undesirable consecutive reaction. 6. This is a controlled addition of a reactant through the membrane in order to achieve higher selectivities and yields.

All these reactor configurations serve for better selectivity and yield of the chemical reaction. Thus, it is particularly important to understand the relation between local concentrations, temperatures, and the selectivity-conversion behavior. In the following section, a few basic expressions of chemical reaction engineering that are important for understanding how membrane reactor should act to achieve higher reaction efficiency are discussed.

8.3

Reaction Rate

The reaction rates are the key information required to quantify chemical reactions and to describe the performance of chemical reactors. The specific rate of a single reaction in which N components are involved is defined; for details, see Levenspiel (1999) and Westerterp et al. (1984): Qi 5

1 dci ; ξ i dt

i 5 1; . . . ; N

ð8:1Þ

where Qi is the reaction rate of component i (kg/m3s, kmol/m3s); c is the concentration (kg/m3, kmol/m3); and ξ i is the stoichiometric coefficient (equal to zero for inerts or diluents). Equation (8.1) is applied for a homogeneous catalyst reaction. In heterogeneous catalysis, often the mass or surface area of the catalyst is used for relation, thus the reaction rate measures as kmol/kgs, kmol/m2s, and so forth. Obviously, the chosen scaling quantity should be used consistently for calculation of the Qi reaction rate. The reaction rates can depend on temperature and the molar concentration change of reactants. Conversion for constant volume can be defined as Xi 5

coi 2ci coi

ð8:2Þ

196

Basic Equations of the Mass Transport through a Membrane Layer

where coi is the initial or inlet concentration and Xi is the conversion. The reaction rate can be given for reaction of A 1 B2E 1 F reversible reaction as
 cA cB 2cE cF ð8:3Þ Q 5 k2 Keq where k2 is the reaction rate constant (m3/kmols); Keq is the equilibrium constant; and Q is the reaction rate (kmol/m3s). Selectivity of component E, σE, is the ratio between the amount of desired product E obtained and the amount of a key reactant, A, converted: σE 5

cP 2coPξ A coA 2cAξ P

ð8:4Þ

where coi is the initial concentration of components (kmol/m3) and ξi is the stoichiometric coefficient. Thiele modulus (ϑ), which is the ratio of characteristic time for radial diffusion to the characteristic time for reaction in the membrane, is described by the following equation: sffiffiffiffiffiffiffiffiffi δ2 Qi ð8:5Þ ϑ5 D i ci where δ is the membrane thickness; c-i is the average concentration of i (kmol/3); Qi is the reaction rate (kmol/m3s); and Di is the effective diffusion coefficient in the membrane layer (m2/s). For first-order chemical reaction, Q 5 k1c (k1 is the reaction rate constant (1/s)), thus the value of ϑ is as sffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi k1 D k1 δ 2 5 ϑ5 D βo

ð8:6Þ

Equation (8.6) is the well-known Ha-number for fluid phase, Ha (Ha  ϑ, with β o5D/δ). Let us express ϑ for cylindrical space. Taking into account Eq. (3.8), the Thiele modulus for cylindrical space can be expressed as follows: rffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi k1 D k1 ro2 ϑ5 5 o D β lnð11δ=ro Þ

ð8:7Þ

with βo 5

D ro lnð11δ=ro Þ

ð8:8Þ

Membrane Reactor

197

The cylindrical mass transfer coefficient, β o, tends to the plane mass transfer coefficient, β o 5 D/δ, in limiting cases, namely when ro-N.

8.4

Modeling of Membrane Reactors

As seen in Figure 8.1, different membrane’s configurations can be obtained depending on the placement of a catalyst (tube and/or shell side or in/on membrane), and motion or not of catalyst particles (packed beds, fluidized beds). The mass balance equations should be given according to real operating conditions. The basis of the continuum models are the differential balance equation given by Eqs (7.1)
(7.8) for the fluid zones, completing them by source or sink terms. Because fluid phases and catalyst and/or permselective membrane layers must be treated simultaneously, a multiphase approach is necessary (Seidel-Morgenstern, 2010). Accordingly, the balance equations should be given every three sections of a capillary membrane reactor, namely for the lumen and shell fluid phases as well as for the catalytic (or noncatalytic) membrane. Here we give the general equations that should be applied for every section, taking into account operating conditions of every section. The complete equations for conservation of momentum, total mass, component mass, and energy must be considered in every real fluid and membrane matrix. Thus, one needs for description of such a system continuity equation, momentum transport energy balance, and (n 21) species balance equations (n is number of components of the system). Accordingly, the starting equation system to be used is as it is given in Chapter 7. The equation of continuity (conservation of mass) in cylindrical coordinate system and with variable μ, ε, and ρ (Bird et al., 1960): @ðρεÞ 1 @ðrερυr Þ 1 @ðερυθ Þ @ðερυz Þ 1 1 1 50 @t r @r r @θ @z

ð8:9Þ

The equation of motion in cylindrical coordinates (in terms of Newtonian fluid with constant ρ and μ and porosity ε; these equations are assumed to be valid for membrane or packed-bed reactors as well; the fluid phase in packed membrane reactors are considered as quasihomogeneous flowing phase): r-component: @ðρευr Þ υr @ðrρευr Þ υθ @ðρευr Þ ρευ2θ @ðρευr Þ @ðεpÞ 1υz 1 1 2 1 r r @θ r @t @r @r @r 2 0 1 0 1 0 13 1 @ @ @ðευr ÞA 1 @ @ @ðευr ÞA 2 @ðευθ Þ @ @ @ðευr ÞA5 μr 1 2 μ 2 2μ 1 μ 24 r @r @r r @θ @θ r @θ @z @z 2ερgr 5 2εfr ð8:10Þ

198

Basic Equations of the Mass Transport through a Membrane Layer

θ-component: 0

1 @@ðρευθ Þ 1υr @ðρευθ Þ 1 υθ @ðρευθ Þ 2ρε υr υθ 1υz @ðρευθ ÞA1 1 @ðεpÞ r r @t @r @θ @r r @θ 2 0 0 1 0 13 1 @ 1 @ 1 @ @ðευθ ÞA 2 @ðευr Þ @ @ @ðευθ ÞA5 2 2μ 1 μ 24 @ μr ðrυθ ÞA1 2 @μ @r r @r r @θ @θ r @θ @z @z 2ερgθ 5 2εfθ ð8:11Þ z-component: 0

1 @ðρευ Þ υ @ðrρευ Þ υ @ðρευ Þ @ðρευ Þ z r z θ z z @ A1 @ðεpÞ 1 1 1 υz r r @θ @t @r @z @z 2 0 1 0 1 0 13 1 @ @ðευ Þ 1 @ @ðευ Þ @ @ðευ Þ z z z @μr A1 @μ A1 @μ A52 ερgz 52εfz 24 r @r @r r 2 @θ @θ @z @z ð8:12Þ where the source terms in the momentum conservation equations (SeidelMorgenstern, 2010): fj 5 f1 υj 1 f2 υj jυj with j 5 r; q; z

ð8:13Þ

The fr ; fz source terms in the momentum conservation equations can be neglected for lumen and shell without solid particles. This is not the case in porous membranes or in packed-bed lumens or shells. Equation (8.13) expresses that friction and inertial forces caused by flow through pores lead to an additional loss of momentum, accounted for by the source term f. The parameters f1 and f2, taking into account the pressure drop during transport through membrane layer, can be calculated using the coefficient determined by application of the dusty gas model (Wesselingh and Krishna, 2000) in the case of gas
fluid phase: f1 5

μf ; 2 =τ32Þ1ðDμ =pÞ ðεdpore f

f2 5 0

ð8:14Þ

where μf is the dynamic viscosity of the gas phase (Pa s); dpore is the pore size; p is pressure (Pa), ε is porosity; τ denotes the tortuosity; and D is diffusivity of the key component (m2/s). The factor f1 takes into account the viscous slip at pore walls, the parameter f2 can be set to zero due to the laminar character of the flow (SeidelMorgenstern, 2010).

Membrane Reactor

199

The component balance equation will be as: 0

1

@@ðερxi Þ 1υz @ðερxi Þ 1υr @ðερxi Þ 1 υθ @ðερxi ÞA r @θ @t @z @r 8 0 1 0 1 0 19 = <1 @ @rDi @ðερxi ÞA1 1 @ @Di @ðερxi ÞA1 @ @Di @ðερxi ÞA 5 Mi Q^ i 2 ; : r @r @r r 2 @θ @θ @z @z ð8:15Þ where Q^ i denotes the reaction rate of component i related to the total volume of membrane (kmol/m3s); xi is the molfraction of component i; Mi is the mole weight of i (kg/kmol); and ρ is the fluid density (kg/m3). The equations of energy in terms of the transport properties (with Newtonian fluids of variable parameters) are 0 1 ^ ^ ^ ^ @@ðρC p εTÞ 1υz @ðρCp εTÞ 1 υr @ðrρC p εTÞ 1 υθ @ðρC p εTÞA @t @z @r @θ r r 8 0 1 0 1 0 19 ð8:16Þ n = X < @ kε @ðrTÞ 1 @ @T @ @T @ A1 @kε A1 @kε A 5 2 hi Q^ i :@r r @r r 2 @θ @θ @z @z ; i51 where C^ p denotes the heat capacity at constant pressure, per unit mass (kJ/kgK); k is the thermal conductivity (kJ/ms); hi is the molar enthalpy of formation of species i (kJ/ kmol); and M is the molecular weight (kg/kmol). Note that k 5 λρC^ p ; where λ is the thermal diffusivity (m2/s). Applying Eq. (8.16) for the membrane, it should be noted that Eq. (8.16) does not involve the heat capacity of the membrane matrix; its value assumed to be negligible. If it is not the case, this should also be taken into account. The relation between the superficial and interstitial (real) velocity in a porous medium is υz;o 5 ευz

ð8:17Þ

The same is true for the transverse velocity as well (υo 5 ευ). If we assume that the porosity on the membrane interface is equal to that in the membrane matrix, then υz,o or υr,o gives the velocity related to the total interface. It should be noted that the axial velocity is generally negligible in the membrane matrix (υz 5 0); only the transverse velocity should be taken into account. The conservation equations for mass and energy should be used in the reaction zone only. Chemical reactions can take place outside the membrane (lumen and/or shell sides—cases III
VI in Figure 8.2, these are so called packed-bed membrane reactors) or in the catalytic membrane layer (case II in Figure 8.2, called catalytic membrane reactors). Simplification of the equation system, from Eqs (8.9)
(8.17), is necessary for all practical cases in order to get the model as simple as possible. Equations (8.9)
(8.17) can easily be given by variable parameters as well (Geraldes et al., 2001; Marriott and Sorensen, 2003; Seidel-Morgenstern, 2010, pp. 33
34).

200

Basic Equations of the Mass Transport through a Membrane Layer

8.4.1

Modeling of a Membrane Reactor with a Catalytic Membrane Layer

Because of its importance, let us look at the model equations of a catalytic membrane layer (case II, in Figure 8.2), assuming steady-state, highly diluted gas reaction system, and no gravity influence. A chemical reaction takes place in the membrane layer only. Model equations for cylindrical catalytic membrane layer for Newtonian fluid (steady state, with constant ρ, μ, C^ p ; k values) are listed in Table 8.1a with boundary conditions in Table 8.1b. Note that υr5ro 50 and υr5rδ 50 when the transmembrane pressure is equal to zero in every axial position. If there is a large volume change during the reaction, then this can create pressure difference between the membrane layer and the lumen and/or shell side, which can cause transverse and axial convective velocities in the membrane. Accordingly, the values of υr5ro and υr5rδ will differ from zero. On the other hand, looking at case II, in Figure 8.2, there is mass transfer of reactants on both sides of membrane; this fact is not involved in the boundary conditions for ro and rδ. Thus, the boundary can be expressed for mass and energy conservation of component B as follows: for r5rδ

DB;f

@cB @φ 5DB B ; @r @r

kB;f

@Tf @T 5kB @r @r

ð8:18Þ

Table 8.1a Model Equations for Cylindrical Catalytic Membrane Layer for Newtonian Fluid (Steady State, with Constant ρ, μ, C^ p , k values) Conversion of total mass 1@ @ ðρrυr Þ1 ðρυz Þ 5 0 r @r @z Momentum conservation equations r-coordinate:

   @υr @υr 1@ @υr υr @2 υr 1υz 2μ 2 2 1 2 5 2fr r ρ υr @r @r @r r @z r @r z-coordinate:  

 @υz @υz @p 1@ @υz @2 υz ρ υr 1υz 1 2μ 1 2 52fz r @r @z @r @z @z r @r Transport equation for species i: 
  @φi @φi 1@ @φi @ 2 φi υz 1υr 2Di 1 5 Q^ i r @z @r @r @z r @r Energy conservation equation
 
  X n @T @T @ 1 @T @2 T hi Q^ i 1υr 1 2k 1 5 ρC^ p υz @z @r @r r @r @z 1

Membrane Reactor

201

Table 8.1b Boundary Conditions for Balance Equations in Table 8.1a Boundary Conditions Geometric Position

Momentum

Mass

Energy

Shell side, z50, υz5υδ,o, υr50 ro#r#rδ

φi 5φδ;i

T5Tδ

Tube side, z50, υz5υz,o, υr50 ro#r#rδ

φi 5φoi

T 5T o

Tube side, z50, ro#r#rδ

p5po

Shell side, z50, ro#r#rδ

p5poδ @υz 50 @z

@φi 50 @z

z5L, ro#r#rδ

υr50,

r5ro, all z

ρυr5roro 5 ρδυδ(ro1δ) Equation (8.18) or (8.19)

@φi 50 @z rδ 5ro1δ; re is radius of shell side (see Figure 8.3)

r5re, all z

υr5υz50

@T 50 @z Equation (8.18) or (8.20) @T 50 @z

where DB,f and DB are the diffusion coefficients in the fluid and membrane phases, respectively (m2/s); cB,f and φB are the concentration in the fluid phase and membrane layers, respectively (kmol/m3); and k is the thermal conductivity [kmol m/s3K or J/(Kms)]. If the radial convective velocity is higher than zero, then this fact should also be taken into account in Eq. (8.18) as for r 5 ro for the component balance: υw cB 2DB;f

@cB @φ 5 υw φ2DB @r @r

ð8:19Þ

or for heat transport at r5ro: ρf C^ p;f υw T 2kf

@T @T 5 ρC^ p υw T 2k @r @r

ð8:20Þ

where υw is the radial convective wall velocity at r 5 ro (m/s) and subscript f denotes the fluid phase. For the complex description of the system, the transport equations also must be given for the lumen and the shell side of the membrane. The exact absolute values of the boundary conditions, such as φoi or Tio or the differential quotient at the wall, can only be obtained by the simultaneous solution of the equation system given in all three regions. In the case of constant transport parameters, the inlet and outlet

202

Basic Equations of the Mass Transport through a Membrane Layer

mass transfer rates of the membrane, J and Jδ, respectively (for details of these transfer rates see Chapters 3
6) given in Eqs (8.18)
(8.20) can be expressed separately, then the lumen and shell sides of the balance equations can be solved separately. This solution should incorporate the mass transfer rates in the boundary conditions of these equations given for the membrane interfaces.

8.4.2

Some Typical Reactor Configurations: Packed-Bed Membrane Reactor

Let us look at an often-applied membrane reactor configuration where the membrane is not catalytic, and it serves as a separation layer. Its task is to remove one of the product components in order to improve the separation efficiency by shifting the reaction to higher conversion (e.g., case III in Figure 8.2). Such a process is the hydrogen enrichment during the water
gas-shift reaction, namely, H2O 1 CO2CO2 1 H2 (Brunetti et al., 2007, 2009; Gosiewski and Tanczyk, 2010; Gosiewski et al., 2010). The steam methane (Pieterse et al., 2010) or ethane reforming (Mendes et al., 2010) is also used for hydrogen production, CH4 1 H2O2CO 13H2 and CH4 1 2H2O2CO2 14H2. The membrane reactor configuration is illustrated in Figure 8.3. This is the so-called packed-bed reactor, where the reforming reaction takes place on the catalyst particles filled in either the tube side, as the case in Figure 8.3, or on the shell side (Brunetti et al., 2007, 2009). Typically, the feed pressure of the membrane reactor equals the pressure at which natural gas is available from high-pressure pipelines, i.e., 40
45 bars (Pieterse et al., 2010). Permeate pressure typically is in the range of 5
10 bars. To the permeate side, an inert sweep gas can be introduced either in cocurrent or in countercurrent configuration with respect to the flow direction in the reaction side. The removal of one of the reaction products, namely H2, shifts the reaction in production’s direction. Inorganic membranes that Pd–Ag membrane

Sweep ⫹ H2

Sweep H2

δ Feed

re

r0 H2O ⫹ CO

CO2 ⫹ H2

H2

Figure 8.3 Schematic illustration of the packed-bed membrane reactor with hydrogen removal by a selective membrane.

Membrane Reactor

203

can be applied for hydrogen removal are reviewed by Lu et al. (2007). For the mathematical description of the process, every three sections, namely the tube and shell sides (reaction and permeation sides) and the membrane layer should be modeled. The mathematical analysis can be based on the differential conservation equations of total mass, momentum chemical species, and enthalpy in steady, twodimensional, cylindrical flow. Thus, the balance equations listed in Table 8.1a can be regarded as starting equations adapting for flowing fluid phase in the packed-bed column. The reaction rate can be based on the Langmuir
Hinshelwood kinetic model (Brunetti et al., 2007; Mendes et al., 2010; Pieterse et al., 2010): 2QCO 5

kKCO2 KH2 O ðpCO pH2 O 2½ pCO2 pH2 =Ke Þ ð11KCO pCO 1KH2 O pH2 O 1KCO2 pCO2 Þ2

ð8:21Þ

where
QCO is the rate of carbon monoxide consumption (kmol/skgcat); k is the rate constant; Ke is the equilibrium constant; Ki is the adsorption constant for species i (i 5 CO, H2O, H2, and CO2); and pi is the partial pressure of the component i. The values of Ki, k, and Ke can be given in papers of Brunetti et al. (2007) and Mendes et al. (2010). The k reaction rate is a strong function of temperature. This function can be expressed as k 5 ko e 2Ecat =RT

ð8:22Þ

where k is preexponential factor (kmol/skg); Ecat is activation energy of catalyst reaction (kJ/kmol); and R is the gas constant (kJ/kmol K). The rate of hydrogen permeation generally is calculated by the Sieverts’ equation (Brunetti et al., 2007; Lu et al., 2007; Ziaka and Vasieiadis, 2011): JH2 5

Do H n 2E=RT n n e ðp 2pδ Þ δ

ð8:23Þ

where JH2 is the permeation rate of H2 (kmol/m2s); P^H2 is the membrane permeability of H2 (kmol/[msPan]); E is the activation energy for permeation (equal to the sum of the diffusion energy and the heat dissolution) (kJ/kmol); p and pδ are the partial pressures of H2 on the reaction and permeate side, respectively, n 5 0.5
1; R is the gas constant; Do is diffusivity of hydrogen (m2/s); Hn is the hydrogen solubility in metal film (kmol/m3Pan); and δ is the membrane thickness (m). According to Eq. (8.8), the mass transfer rate, JH2 ; can be rewritten for cylindrical space as JH2 5

Do H n e 2E=RT ðpn 2pnδ Þ ro lnð11δ=ro Þ

ð8:24Þ

204

Basic Equations of the Mass Transport through a Membrane Layer

The f1 and f2 friction coefficient for momentum loss in the catalytic bed can be calculated according to Ergun equations (Koukou et al., 2001; Brunetti et al., 2007; Seidel-Morgenstern, 2010): f1 5 150

ð12εÞ2 μ ; ε3 dp2

f2 5 1:75

ð12εÞ2 ρ ε3 dp

ð8:25Þ

where ε is the bed porosity (2); μ is the viscosity (Pas); dp is the particle size (m); and ρ is the fluid density (kg/m3). The question arises, how can we simplify the relatively complex equation system given for the reaction side. This simplified model should be valid for practical applications. Let us take the following main assumption for it: plug flow in the catalyst bed, negligible radial temperature and concentration gradients (one-dimensional model with negligible concentration polarization), negligible pressure drop in the reactor in the both sides, thus the momentum balance equation can also be neglected. The model equations are listed in Tables 8.2a and 8.2b. Note that the axial dispersion term is also neglected because of the relatively large axial convective velocity. Some remarks are needed for the balance equations in Table 8.2a. The Qi reaction rate does not involve the mass transport inside of the catalyst particle. The mass balance equation of the reaction side considers a particle as a point source or sink term. If the Thiele modulus is low, then the internal mass transport should also be taken into account (see Chapter 4). The axial diffusion or dispersion term (this latter can be formed during turbulent flow) often can be neglected due to the relatively high axial convective velocity. Estimation of the axial Peclet number can help to this decision. Similar consideration should be done in the radial direction. Regarding the relatively high permeation rate of hydrogen, it is also recommended to take into account the change of the volumetric flow rate. This is done by the total mass balance equation. Obviously, when the volumetric flow rate of the sweep gas is large enough, then this effect could also be neglected. The permeation rate is given here by the Sieverts’ equation [Eq. (8.23) or (8.24)] (see also Chapters 1.5 and 12.2). The superficial axial convective velocity is denoted here by u. The parameter ρcat denotes the density of catalyst particles (kg/m3), thus (12ε)ρcat defines the amount of catalyst related to the reactor volume.

Membrane Reactor

205

Table 8.2a Basic Balance Equations of Modeling the Packed-Bed Membrane Reactor Investigated Reaction (lumen) side—: mass balances of reactive components (ξi is the stoichiometric factor):
 @ðuci Þ 1@ @ci 5 6ð12εÞρcat ξ i Qi 1ε rDf;i ε @r @z r @r Total mass balance: @ðρuÞ 2 5 2 Ji ; @z ro

i 5 H2

Boundary conditions: z50 ρu5ρouo

for all r

@ci 50 for all z for all reactants @r @ci 5Ji for all z with i5H2 2 Df;i @r @ci 50 for all z; i5CO, CO2, H2O Df;i @r

r50 Df;i r5ro r5ro

Permeate (shell) side:
 @ðue ce;i Þ 1 @ @ce;i 5 0; 1 rDf;i @r @z r @r

i5H2

Total mass balance: @ðρe ue Þ 2 5 2 Ji @z ro Boundary conditions: z50 ρeue 5ρe,oue,o for all r @φi @ce;i 5 Def;i for all z for i5H2 @r @r @ce;i Def;i 5 0 for all z for i5H2 @r

r5ro1δ r5re

Di

206

Basic Equations of the Mass Transport through a Membrane Layer

Table 8.2b Heat Balance of the Packed-Bed Reactor Reaction (lumen) side: 
  @T @ 1 @T @2 T 1 5 2ΔHð12εÞρcat QCO ρC^ p υz ε 2kf @z @r r @r @z Boundary conditions: z50 T5To

for all r

@T 5 0 for all z @r @T @T kf 5k for all z @r @r

r50 kf r5ro

For the membrane layer: 
 2  @ 1 @T @ T 1 50 k @r r @r @z Boundary conditions: z50 T5To

for all r

@T 5 0 for all r @z @T @T r5ro1δ k 5 ke for all z @r @r

z5L

k

Shell (permeate) side:
 @T @ 1 @T 2ke 50 ðρC^ p uεÞe @z @r r @r Boundary condition: z50 T5To z5L r5re

for all r

@T 5 0 for all r @z @T ke 5 αe ðTe 2Tsurr Þ @r

ke

for all z

ΔH denotes the heat of reaction (kJ/kmol); Q is the reaction rate (kmol/skgcat); ρcat 5 mcat/Vcat (kg/m3); Te and Tsurr are the wall temperature at r5re and the outside temperature, respectively; αe is the heat transport coefficient through the wall at r5ro (kJ/m2Ks); and L is the length of the membrane tube (m). The k thermal conductivity for the catalytic membrane can be estimated as k 5 εkf 1 ð12εÞkp

ð8:26Þ

Membrane Reactor

207

r0 δ1

CH4,

Butane

Ji

δ

Catalytic layer Distributor

re O2

Je , O2

Figure 8.4 Membrane reactor for oxidative dehydrogenation.

where kf is the fluid phase conductivity (kJ/ms) and kp is the conductivity of the catalyst particles (kJ/ms). This means further simplifications when the radial component diffusivity and the thermal conductivity as well as the axial thermal conductivity are neglected, as done by Brunetti et al. (2007) and Gosiewski et al. (2010).

8.4.3

Catalytic Membrane Reactor

As an example, the oxidative dehydrogenation of paraffin hydrocarbons, from methane to butane, was chosen (Te´llez et al., 1999; Bottino et al., 2002; Pedernera et al., 2002; Rodriguez et al., 2010). This reaction can also be carried out in a packed-bed reactor (van Dyk et al., 2003; Rodriguez et al., 2010). The mass transfer equations given in Section 8.4.2 can also be used here. The oxygen transport through the membrane can be estimated by Eqs (1.48)
(1.56). In this section, the reactor configuration is discussed where the chemical reaction takes place in the catalytic membrane layer as illustrated in Figure 8.4. The membrane layer is asymmetric: it contains a diffuser layer for distribution of the oxygen (diffusion zone) and a thin active layer where the dehydrogenation reaction takes place (catalytic zone; Julbe et al., 2001; Bottino et al., 2002; Pedernera et al., 2002). For the mathematical model, let us make the following simplification (Table 8.3): steady-state process; one-dimensional diffusion in the membrane; pressure gradient, along the tube and the shell side, is neglected; the radial velocity (plug flow), mass, and temperature gradient are neglected on the tube and shell sides; there is no reactant transport from the tube side to the shell side, pe2pt . 0 (pe and pt are the shell and the tube side pressures, respectively); the kinetics of dehydrogenation reactions are not discussed here; see Te´llez et al. (1999) and Heracleous and Lemonidou (2006).

208

Basic Equations of the Mass Transport through a Membrane Layer

Table 8.3 Model Equation of Oxidative Dehydrogenation Applying Catalytic Membrane Layer Lumen side mass balance Total mass balance :

n dðut ρÞ 2X Jj ; 5 dz ro j51

Component balance :

dðut ci Þ 2 5 2 Ji dz ro

j 6¼ O2 with i 5 1; . . . ; n; i 6¼ O2

dT 2 ðρC^ p uÞt 5 αðTt 2Tw;ro Þ dz ro

Energy balance :

Shell side mass balance due 1 2 5 Je;i ; dz ρ ro 1δ

Total mass balance :

dðue ce;i Þ 2 Je;i ; 52 dz ro 1δ

Species mass transport : ðρC^ p uÞe

Energy balance :

i 5 O2 i 5 O2

dT 2 5 αðTw;r1δ 2Te Þ dz ro

(Note: t denotes tube (lumen), e denotes shell) Membrane catalyst layer Component balance :

Di

ρC^ p u

Energy balance :

n X dφi ξ i Qi 5 dy i51

nr X @T d2 T Qj ð2ΔHj Þ 1 k 2 5 ρcat @z dy j51

Membrane porous layer Component balance ði 5 O2 Þ : ^ 5 ci Þ (Note: pi =RT Energy balance :


kpor

with i 5 1; . . . ; n

Ji 5


 ε dp2 DK;i pe;i ln ðpe;i 2pt;i Þ1 ^ pt;i δτ 32 δRT


 1 @ @ðrT Þ 50 r @r @r

Boundary conditions, e.g., concurrent mode ´ ;o ; z50 then ut 5 ut r5ro

then c5ct;

ue 5ue;o ;

φ5Hct,;

T 5To ;

for all r

Ji 5 2Di

dφi dy

T5Tt

r5r1δ1 then H1φ 5 Hporcpor; T 5 Tpor; r5r 1 δ1 then k

ci 5coi ;

dT dT 5 kpor dy dr

(Note: porous layer is cylindrical, catalyst layer is plane)

i 5 O2

Membrane Reactor

209

Where n is the number of reactant (e.g., in case of ethane dehydrogenation the components are: C2H6, C2H4, O2, H2O, CO2; Rodriguez et al., 2010); Di is the effective diffusion coefficient for i (m2/s); δ is the thickness of the catalyst layer (m); α is the heat transfer coefficient (kJ/m2sK); subscript “e” denotes the shell side, subscript “por” is the porous distributor layer; Jj is the component transport rate (kmol/m2 s) (the oxygen transfer rate depends on the membrane properties, thus the value of Ji, i 5 O2, in the porous layer is valid for the Knudsen flow regime; see Chapter 12), that can enter or leave the membrane or the lumen fluid; δ is the thickness of the porous layer of the membrane (m); δ1 is the thickness of the catalyst layer (m); kpor is the thermal conductivity in the porous membrane layer for oxygen distribution (kJ/K m s); and H is the solubility coefficient (2). Practically, the overall membrane thickness is the same as the thickness of the membrane because the catalyst layer is generally very thin. Assuming that the catalyst membrane layer is very thin (δ/ro{1), the cylindrical effect is negligible. On the other hand, if the thickness of the membrane distributor is relatively large, the cylindrical effect can have significant effect of the transport process. In the energy balance of the porous layer, the effect of the countercurrent viscous flow of oxygen is neglected. The component balance equation of the porous membrane layer does not involve the cylindrical effect. Its thickness can be several hundred μm. In the case of a tubular inorganic membrane, where the inner diameter of the membrane is about 0.008
0.01 m, this effect is negligible. When one applies a capillary membrane with an inner radius of 100
300 μm (Bottino et al., 2002; van Dyk et al., 2003), this effect can be significant, thus Eq. (1.56) can be applied for description of the membrane mass transport.

References Bird, R.B., Stewart, W.E., and Lightfoot, E.N. (1960). Transport Phenomena, John Wiley and Sons, New York. Bottino, A., Capannelli, G., and Comite, A. (2002) Catalytic membrane reactors for the oxidehydrogenation of propane: experimental and modeling study. J. Membr. Sci. 197, 75
88. Brunetti, A., Caravella, A., Barbieri, G., and Drioli, E. (2007) Simulation study of water gas shift reaction in a membrane reactor. J. Membr. Sci. 306, 329
340. Brunetti, A., Barbieri, G., and Drioli, E. (2009) Upgrading of syngas mixture for pure hydrogen production in a Pd
Ag membrane reactor. Chem. Eng. Sci. 64, 3448
3454. Charcosset, C. (2006) Membrane processes in biotechnology. Biotechnol. Adv. 24, 482
492. Coronas, J., and Santamarı´a, J. (1999) Catalytic reactors based on porous ceramic membranes. Catalysis Today 51, 377
389. Dittmeyer, R., Avajda, K., and Reif, M. (2004) A review of catalytic membrane layers for gas/liquid reactions. Topics Catalysis 29, 3
27. Fenu, A., Guglielmi, G., Jimenez, J., Spe´randio, M., Saroj, D., Lesjean, B., et al. (2010) Activated sludge model based on modelling of membrane bioreactor (MBR) processes: a critical review with special regards to MBR specificities. Water Research 44, 4272
4294.

210

Basic Equations of the Mass Transport through a Membrane Layer

Geraldes, V., Semiao, V., and de Pinho, N.M. (2001) Flow and mass transfer modeling of nanofiltration. J. Membr. Sci. 191, 109
128. Gosiewski, K., and Tanczyk, M. (2010) Applicability of membrane reactor for WGS coal derived gas processing: simulation-based analysis. Catalysis Today (doi:10.016/j. cattod.2010.11.042). Gosiewski, K., Warmuzinski, K., and Tanczyk, M. (2010) Mathematical simulation of WGS membrane reactor for gas from coal gasification. Catalysis Today 156, 229
236. Heracleous, E., and Lemonidou, A.A. (2006) Ni
Nb
O mixed oxides as highly active and selective catalysts for ethene production via ethane oxidative dehydrogenation. Part II: Mechanistic aspects and kinetic modeling. J. Catalysis 237, 175
189. Judd, S. (2006) The MBR Book: Principles and Application of Membrane Bioreactors in Water and Wastewater Treatment. Elsevier, New York. Julbe, A., Farrusseng, D., and Guizard, Ch. (2001) Porous ceramic membranes for catalytic reactors—overview and new ideas. J. Membr. Sci. 181, 3
20. Koukou, M.K., Papayannakos, N., and Markatos, N.C. (2001) On the importance of non-ideal flow effects in the operation of industrial scale adiabatic membrane reactors. Chem. Eng. Sci. 83, 95
105. Levenspiel, O. (1999) Chemical Reaction Engineering, 3rd ed. John Wiley and Sons, New York. Lu, G.Q., Diniz da Costa, J.C., Duke, M., Giessler, S., Socolow, R., Williams, R.H., et al. (2007) Inorganic membranes for hydrogen production and purification: a critical review and perspective. J. Colloid Interface Sci. 314, 589
603. Marcano, J.G.S., and Tsotsis, T.T. (2002) Catalytic Membranes and Membrane Reactors. Wiley-VCH, Weinheim. Marriott, J., and Sorensen, E. (2003) A general approach to modeling membrane modules. Chem. Eng. Sci. 58, 4975
4990. McLearly, E.E., Jansen, J.C., and Kapteijn, F. (2006) Zeolite based films, membranes and membrane reactors: progress and prospects. Microporous Macroporous Mater. 90, 198
220. Mendes, D., Tobsti, S., Borgognomi, F., Mendes, A., and Madeira, L.M. (2010) Integrated analysis of a membrane-based process for hydrogen production from ethanol steam reforming. Catalysis Today 156, 102
117. Ozdemir, S.S., Buonomenna, M.G., and Drioli, E. (2006) Catalytic polymer membranes: preparation and application. Applied Catalysis A General 307, 167
183. Paturzo, L., Basile, A., and Drioli, E. (2002) High temperature membrane reactors and integrated membrane operations. Rev. Chem. Eng. 18, 511
551. Pedernera, M., Alfonso, M.J., Mene´ndez, M., and Santamarı´a, J. (2002) Simulation of a catalytic membrane reactor for the oxidative dehydrogenation of butane. Chem. Eng. Sci. 57, 2531
2544. Pieterse, J.A.Z., Boon, J., van Delft, Y.C., Dijkstra, J.W., and van den Brink, R.W. (2010) On the potential of nickel catalysts for steam reforming in membrane reactors. Catalysis Today 156, 153
164. Reij, M.W., Keurentjes, J.T.F., and Hartmans, S. (1998) Membrane bioreactors for waste gas treatment. J. Biotechnol. 59, 155
167. Rios, G.M., Belleville, M.P., Paolucci, D., and Sachez, J. (2004) Progress in enzymatic membrane reactors—a review. J. Membr. Sci. 242, 189
196. Rodriguez, M.L., Ardissone, D.E., Heracleous, E., Lemonidou, A.A., Lo´pez, E., Pedernera, N., et al. (2010) Oxidative dehydrogenation of ethane to ethylene in a membrane reactor: a theoretical study. Catalysis Today 157, 303
309.

Membrane Reactor

211

Santos, A., Ma, W., and Judd, S.J. (2010) Membrane bioreactors: two decades of research and implementation. Desalination (10.1016/j.desal.2010.07.063). Saracco, G., Neomagnus, H.W.J.P., Versteeg, G.F., and van Swaaij, W.P.M. (1999) Hightemperature membrane reactors: potential and problems. Chem Eng. Sci. 54, 1997
2017. Seidel-Morgenstern, A. (2010) Membrane Reactors. Wiley-WCH, Weinheim. Te´llez, C., Mene´dez, M., and Santamarı´a, J. (1999) Kinetic study of oxidative dehydrogenation of butane on V/MgO catalysts. J. Catalysis 183, 210
221. van Dyk, L., Miachon, S., Lorenzen, L., Torres, M., Fiaty, K., and Dalmon, J.-A. (2003) Catalysis Today 82, 167
177. Wesselingh, J.A., and Krishna, R. (2000) Mass Transfer in Multicomponent Mixture. Delft University Press, Delft. Westerterp, K.R., van Swaaij, W.P.M., and Beenackers, A.A.C.M. (1984) Chemical Reaction Design and Operation. John Wiley and Sons, New York. Ziaka, Z., and Vasieiadis, S. (2011) New integrated catalytic membrane process for enhanced propylene and polypropylene production. Sep. Sci. Technol. 46, 224
233.

9 Membrane Bioreactor 9.1

Introduction

Membrane bioreactor technology is advancing rapidly around the world in both research and commercial applications (Giorno and Drioli, 2000; Marcano and Tsotsis, 2002; Rios et al., 2004; Charcosset, 2006; Strathmann et al., 2006; Yang et al., 2006; Santos et al., 2010). Integrating the properties of membranes with biological catalysts such as cells or enzymes forms the basis of an important new technology, called a membrane bioreactor. The membrane layer is useful especially for immobilizing whole cells [bacteria, yeast, mammalian, and plant cells (Brotherton and Chau, 1990; Sheldon and Small, 2005)], bioactive molecules such as enzymes (Charcosset, 2006; Rios et al., 2007) to produce a wide variety of chemicals and substances. Membrane bioreactors were introduced over 30 years ago and until now they were recommended or applied to the production of foods, biofuels, plant metabolites, amino acids, antibiotics, anti-inflammatories, anticancer drugs, vitamins, proteins, optically pure enantimers, isomers, fine chemicals, as well as for treatment of wastewater [e.g., industrial, domestic, and municipal (Belfort, 1989; Yang et al., 2006)]. Membrane bioreactors for immobilized whole cells can provide a suitable environment for high-cell densities (Schonberg and Belfort, 1987; Kelsey et al., 1990). Cells are grown either in the extracapillary space with medium flow through the fibers and supplied with oxygen and nutrients, or within the fibers with medium flow outside or across the fibers while wastes and desired products are removed. The main advantages of the hollow-fiber bioreactor are the large specific surface area (internal and external surface of the membrane) for cell adhesion or enzyme immobilization; the ability to grow cells to high density; the possibility for simultaneous reaction and separation; the relatively short diffusion path in the membrane layer; the presence of convective velocity through the membrane if it is necessary to avoid the nutrient limitation (Piret and Cooney, 1991; Sardonini and DiBiasio, 1992). This work is focused primarily on the hollow-fiber bioreactor with biocatalyst, either live cells or enzymes, inoculated into the shell and immobilized within the membrane matrix or in a thin layer at the membrane matrix
shell interface. The performance of a hollow-fiber or sheet bioreactor is primarily determined by the momentum and mass transport rate (Calabro et al., 2002; Godongwana et al., 2007) of the key nutrients through the biocatalytic membrane layer. Thus, the operating conditions (transmembrane pressure, feed velocity), the physical properties of membrane (porosity, wall thickness, lumen radius, matrix structure, etc.) can influence considerably the performance of a bioreactor and the effectiveness of the reaction. Basic Equations of the Mass Transport through a Membrane Layer. DOI: 10.1016/B978-0-12-416025-5.00009-0 © 2012 Elsevier Inc. All rights reserved.

214

Basic Equations of the Mass Transport through a Membrane Layer

The main technological difficulties in using membrane bioreactors on an industrial level are related to rate-limiting aspects and scale-up difficulties of this technology. The limited transport of nutrients can cause serious damage in production (Schonberg and Belfort, 1987; Sardonini and DiBiasio, 1992). The introduction of convective transport is crucial in overcoming diffusive mass transport limitation of nutrients (Nakajima and Cardoso, 1989), especially of the sparingly soluble oxygen. The mathematical description of the transport processes enables us to predict the concentration distributions of nutrients in the catalyst membrane layer, and thus makes it possible to choose correctly the operating conditions that provide a sufficient level of nutrient concentration in the membrane layer. The main aim of this study is to give closed, as simple as possible, mathematical equations in order to predict the concentration distribution and the mass transfer rate through biocatalyst membrane layer as well as the concentration variation in the lumen (or shell) side of a capillary membrane, with special regard to the variable transport parameters (diffusion coefficient, convective velocity, reaction rate constant) due to the anisotropy of the membrane and/or cell colony in or around the membrane (Piret and Cooney, 1991; Sardonini and DiBiasio, 1992; Giorno and Drioli, 2000; Giorno et al., 2007).

9.2

Configurations of Membrane Bioreactors

Membrane bioreactors have been reviewed previously in every detail (Giorno and Drioli, 2000; Marcano and Tsotsis, 2002; Rios et al., 2004; Charcosset, 2006). There are two main types of membrane bioreactors: (1) the system consists of a traditional stirred tank reactor combined with membrane separation unit (Figure 9.1); and (2) the membrane contains the immobilized biocatalysts such as enzymes, microorganisms, and antibodies and thus acts as a support and a separation unit (Figure 9.2). The biocatalyst can be immobilized in or on the membrane by entrapment, gelification, physical adsorption, ionic binding, covalent binding, or cross-linking (Giorno and Figure 9.1 Schematic illustration of the external membrane bioreactor Permeate configuration.

Reactants

Reactor

Membrane Bioreactor

215

Product

C E E E

E E E

E E

E E

Substrate

C C C C C C C

C C

C C

C C C C

E

Figure 9.2 Membrane bioreactor with immobilized biocatalysts (enzyme or microorganism).

Drioli, 2000; Giorno et al., 2003; Rios et al., 2007). Our attention will be focused primarily on the second case where the membrane acts as a support for a biocatalyst and as a separation unit, in this study. The momentum and mass transport process, in principle, are the same in both cases, namely when there is not a biochemical reaction in the membrane (Figure 9.1) or when a biochemical reaction occurs in the membrane due to its catalytic activity (Figure 9.2). Applications of whole cell biocatalytic membrane reactors, in the agro-food industry and in pharmaceutical and biomedical treatments are listed by Giorno and Drioli (2000). Frazeres and Cabral (2001) have reviewed the most important applications of enzyme membrane reactors such as hydrolysis of macromolecules, biotransformation of lipids, reactions with cofactors, synthesis of peptides, and optical resolution of amino acids. Another widespread application of the membrane bioreactor is wastewater treatment. This will be discussed only briefly as a membrane-aerated biofilm reactor.

9.3

Enzyme Membrane Reactor

The enzymatic bioconversion processes are of increasing use in the production, transformation, and valorization of raw materials (Calabro et al., 2002; Lozano et al., 2002; Long et al., 2003; Rios et al., 2004, 2007; Be´lafi-Bako´ et al., 2006; Giorno et al., 2007; Andric et al., 2010). Important applications have been developed in the field of food industries, fine chemicals, or even for environmental purposes. Several important applications of the enzyme membrane reactor are given in Charcosset’s (2006) and Rios’s (2004, 2007) papers, as well as an excellent summary given by

216

Basic Equations of the Mass Transport through a Membrane Layer

Giorno and Drioli (2000) and Frazeres and Cabral (2001). A new application of the enzyme membrane reactor by immobilizing lipase in presence of organic emulsion was suggested by Giorno et al. (2007). They discussed the different enzyme membrane configurations (recycle, dialysis, diffusion, multiphase membrane reactors) and the different types of enzyme retention (trapping, chemical coupling, adsorption electrostatic interactions, etc.). The enzyme membrane reactor has several advantages (continuous mode, retention and reuse of catalyst, reduction in substrate/product inhibition, enzyme-free product, integrated process, etc.) and also some disadvantages (decreasing enzyme activity as a function of time, membrane fouling, low substrate concentration, etc.). The enzyme can be immobilized into the membrane matrix or on the membrane interface. In this latter case, the membrane surface is covered by a gel layer and the enzyme is binding to this layer (Habulin and Knez, 1991; Lozano et al., 2002). In this case, the substrate solution has direct contact with the catalyst, while in the other case, namely when the enzyme is placed in the membrane matrix, the substrate has to flow through the membrane layer to contact the biocatalyst. The operating mode of this latter bioreactor could be different. The most important possible flow configurations of its realization are illustrated in Figure 9.3. Each mode of operation can be characterized by the quantity f ðf 5 ut;L =uo Þ; where ut,L denotes the outlet flow rate of the lumen (m/s), while uo is the inlet flow of lumen (m/s). Applying the f quantity, the conservation balance equations, as will be seen later, can be handled by the same methodology. Depending on the transmembrane pressure and operating mode, the substrate flows with different convective velocity through the biocatalytic membrane affecting the reaction efficiency. In the case of c, convective flow through the membrane can also be provided according to the transmembrane pressure (Bruining, 1989; Kelsey et al., 1990). Concerning the mathematical description of these enzymatic processes, basically two different cases can be distinguished, namely the enzyme or live cells are immobilized onto or in the membrane layer or the biocatalyst is dissolved homogeneously (A)

(B)

Ut,L < U0

U0

Ue Ut,L

(C)

(D)

(E)

Figure 9.3 Some modes of operation on an enzyme membrane reactor.

Membrane Bioreactor

217

or mixed heterogeneously (it is immobilized in porous particles) in the feed phase. The description of these two biochemical processes is different. This text focuses on the biocatalytic or catalytic membrane layer and does not deal with biocatalytic processes where a membrane layer is a selective layer retaining the catalyst particles. The structure of membrane matrix does not practically change during the biocatalytic process, thus, the transport parameters remain constant. Obviously, due to the fouling or cake forming, if it is the case, the increase of the external mass transfer resistance can alter parameters such as convective velocity and external mass transfer coefficient. This change has to be taken into account to describe the transport process.

9.3.1

Modeling of a Capillary Enzyme Membrane Reactor

Independent of the configuration mode, for the bioreactor applications, mass transfer and flow phenomena, as well as pressure drop data are of major importance. Taking into account the operating conditions, the mass transfer does not affect significantly the flow conditions through the membrane. Consequently, the mass and momentum transfer can be discussed separately. The laminar flow condition, the axial and the radial velocity profiles, and the main notations are illustrated in an open shell mode in Figure 9.4. Typical arrangement is where enzyme or cell is immobilized in the spongy layer of the membrane layer. In open lumen and shell modes, the substrate flow occurs into shell and partly back to lumen, depending on the value of f. The value of f can be controlled by the transmembrane pressure (Bruining, 1989). In the following section, we give the velocity and pressure profiles for any values of f in range between 0 and 1. The flow streamlines are shown by Kelsey et al. (1990) or Labecki et al. (1995) at different f values in the hollow-fiber bioreactor.

Open shell Product δ Feeding substrate

ut

νt

re

r0 Closed lumen

E E E E E E

Figure 9.4 Velocity profiles and important notations in the lumen for open shell mode of membrane bioreactor with immobilized enzyme in the membrane matrix.

218

Basic Equations of the Mass Transport through a Membrane Layer

The system of interest consists of a bundle of fiber spotted at both ends and encased by a concentric cylindrical cartridge (Figure 9.5A) or a single hollow-fiber membrane (Figure 9.5B). In the former case, the fibers are assumed to be arranged in a hexagonal array (Figure 9.5A) with the triangular voids between the shell regions neglected (Waterland et al., 1974; Kelsey et al., 1990). Even though the concept of a capillary hollow-fiber enzyme bioreactor is relatively simple, several assumptions must be made to simplify the problem to a set of model equations appropriate for engineering analysis, especially if we are to establish useful design equations (Brotherton and Chau, 1990).

9.3.1.1 Momentum Transfer Inside a Membrane Bioreactor The model solutions were based on the following imposed operating conditions: (1) the system is isotherm; (2) the flow regime within the fiber lumen and shell side (the extracapillary space) is fully developed; (3) physical transport parameters (density, viscosity, and diffusivity) are constant; (4) the substrate flows in axial and radial directions in the lumen and shell, whereas in the dense and in the spongy matrix of the membrane the flux is only in the radial direction; (5) the momentum transfer can be solved independently of the mass transfer (Damak et al., 2004; Godongwana et al., 2007); (6) wall flux is pressure driven, the pressure drop in the

Lumen Matrix Shell

re

rk

r0 (B)

r0⫹δ

r0 (A)

r0 ⫹ δ

Figure 9.5 Schematic representations of a hollow-fiber cartridge and of a hexagonal array of hollow fibers (A) and a module with a single capillary membrane (B).

Membrane Bioreactor

219

spongy matrix is negligible, the velocity profiles adjust instantaneously to the changing conditions. The continuity equation is 1@ @u ðrυr Þ 1 50 r @r @z

ð9:1Þ

where u is the axial convective velocity (m/s); υr is the transverse convective velocity (m/s); and z is the axial space coordinate (m). Momentum conservation equations are r-coordinate:



   @υr @υr @p 1@ @υr @2 υr 1u 5 2 1μ 1 2 r ρ υr @r @r @r @z @r r @r

ð9:2Þ

z-coordinate:

 

 @u @u @p 1@ @u @2 u ρ υr 1 u 5 2 1μ r 1 2 @r @z @z r @r @r @z

ð9:3Þ

where p is the pressure (Pa); μ is the viscosity (Pa s); and ρ is the density (kg/m3). Equations (9.2) and (9.3) may be reduced by considering the fact that for a small wall Reynolds number (Rew 5 ρυwro/μ), the inertial terms will be negligible (Bruining, 1989; Brotherton and Chau, 1990). In most viscous flows, normal stress effects, @2 u=@z2 ; are not as important as shear stresses and are negligible when the ratio, ro/L (L is length of capillary), is less than 1022, a condition that is satisfied in almost all hollow-fiber membrane devices. Accordingly, the momentum equation for steady-state flow in the fiber lumen and shell are given as 
 1@ @u @p r 5 μ r @r @r @z

ð9:4Þ

Note if you use a vertically oriented bioreactor, the effect of the hydrostatic pressure change (in this case ptot 5 p1ρgz, with gravitational acceleration g (m/s2); subscript “tot” denotes the total pressure) should also be taken into account (Godongwana et al., 2007). The membrane is relatively impermeable, i.e., the membrane is the main resistance to flow. For such a situation, the axial pressure gradient within the membrane is negligible compared to the radial pressure gradient. The axial velocity within the membrane is negligible compared to the radial velocity (uM 5 0, here subscript M denotes the membrane layer) because the pore walls in the macroporous matrix region provide a very large resistance to axial flow in this region (Kelsey et al., 1990). Only the radial component of the continuity equation should be considered

220

Basic Equations of the Mass Transport through a Membrane Layer

(Mondor and Moresoli, 1999). In addition, because the pores in the ultrathin skin are several orders of magnitude smaller than those in the spongy matrix, the radial pressure drop in the matrix is very small compared to that in the skin, thus pM(z) pe(z) (subscript “e” denotes the shell or extracapillary space). The flow through this membrane is proportional to the transmembrane pressure drop and inversely proportional to the fluid viscosity. Applying the Darcy and the continuity equations, one gets the following equation system to the membrane layer: υM 5 2

κ dpM μ dr

ð9:5Þ

where κ is the permeability constant of the fiber’s membrane layer (m2); μ is the fluid viscosity (Pa s); and subscript “M” denotes the membrane matrix.

 1@ @ðrυM Þ κ1 @ @pM 50 r  2 r @r r @r @r μ r @r

ð9:6Þ

When Eq. (9.6) is integrated twice with respect to r with boundary conditions: at r 5 ro then pM 5 pt and at r 5 ro 1 δ then pM 5 pe, one gets the radial pressure and velocity distributions [see Eqs (7.34) and (7.35)] within the membrane as pM ðr; zÞ 5

pt ðzÞ 2 pe ðzÞ r ln 1 pt ðzÞ lnð1 1 δ=ro Þ ro

ð9:7Þ

υM ðr; zÞ 5

K ro pt ðzÞ 2 pe ðzÞ εμ r lnð11 δ=ro Þ

ð9:8Þ

For the hydrodynamic analysis in the lumen and in the shell, the continuity equation (9.1) and the momentum equation (9.4) have to be solved with suitable boundary conditions. In the lumen (t), these are @u 5 0; @z

at r 5 0

then

at r 5 ro

then υ 5 0

υ50

ð9:9Þ ð9:10Þ

Regarding the extracapillary space, two cases will be distinguished, namely the hollow fibers in a cartridge are arranged in a hexagonal array (case A, Figure 9.5A). The actual arrangement of fibers in a typical cartridge is somewhat random. Thus, this model provides a description of flow through some average fiber which is representative of the overall system geometry. The model neglects the small “triangular” voids between the adjacent fibers (Kelsey et al., 1990; Mondor and Moresoli, 1999). The second case (case B) when a single hollow-fiber membrane is investigated, where permeate is flowing in a cylindrical annulus (Figure 9.5B).

Membrane Bioreactor

221

Boundary conditions for case A: at r 5 ro 1 δ at r 5 rk

then ue 5 0

then

@u 5 0; @r

ð9:11Þ υ50

ð9:12Þ

where rk is the Krogh radius (m), according to Figure 9.5A. Boundary conditions for case B: at r 5 ro 1 δ at r 5 re

then ue 5 0

then ue 5 0;

υ50

ð9:13Þ ð9:14Þ

where re is the radius of the extracapillary space (Figure 9.5B). The general solution of Eq. (9.4) is as u5

r 2 @p 1 K1 ln r 1 K2 4μ @z

ð9:15Þ

The integration constants K1 and K2 can be determined by means of the boundary conditions given by Eqs (9.9)
(9.14). Thus, the axial flow rate for the tube side is as
 ro2 r 2 dpt 12 2 ut 5 2 4μ ro dz

ð9:16Þ

The axial velocities for the shell side are as: For case A (Kelsey et al., 1990; Mondor and Moresoli, 1999): (

2
 ) ro2 rk 2 r r δ 2 dpe 2 2 ue 5 2 ln 1 11 4μ ro dz ro 1 δ ro ro

ð9:17Þ

For case B (see Bird et al., 1960, p. 53): r2 ue 5 2 o 4μ

(


) re r 2 ðro 1 δÞ2 2 re2 r dpe ln 2 1 2 ro ro lnðre =½ro 1 δÞ dz ro re

ð9:18Þ

Substitution of these expressions in the continuity equation [Eq. (9.1)] and after integration with the above boundary condition given for the radial velocities, the

222

Basic Equations of the Mass Transport through a Membrane Layer

radial velocity profiles can be obtained both in the lumen and shell. Substituting Eq. (9.16) after differentiation of the velocity, one can obtain "
2 # 2 ro2 r d pt 1@ 12 ðrυt Þ 5 0 1 2 4μ dz2 ro r @r

ð9:19Þ

Applying boundary condition given by Eq. (9.9) to determine the integral constant, the radial lumen velocity will be as   ! ro2 r 1 1 r 2 d2 pt υt ðr; zÞ 5 2 dz2 4μ 2 4 ro

ð9:20Þ

Radial velocity of the extracapillary space for case A: υe ðr; zÞ 5

ro2 r d2 pe Ψ 4μ dz2

ð9:21Þ

with 
2

 rk r 3 r 2 1 ðro 1 δÞ2 2 rk2 rk4 2 ln 1 1 χ Ψ5 ro ro2 r 2 ro2 ro 1 δ 4 ro 2

ð9:22Þ

where
χ 5 ln

 rk 1 1 ðro 1 δÞ2 2 1 ro 1 δ 4 2 rk2

The radial velocity of the extracapillary space for case B: ( )
hr i2  1
r 2 1 r 4 ro2 1 re d2 pe e e υe 5 2 12 1 2 2 1ς 4μ 2 ro r dz2 4 ro 4 r ro

ð9:23Þ

where 
  ðro 1 δÞ2 2 re2 r re2 211 2 ln ς5 2 ro lnðre =½ro 1 δ Þ r re [note that Eq. (9.23) differs from that of Mondor and Moresoli (1999, Eq. (21) in their paper)], because the starting equations are different (they applied Eq. (9.17), for that case). The other significant difference between our work and the works of Mondor and Moresoli (1999) or Kelsey et al. (1990) is that we took into account the cylindrical effect in the Darcy equations [see Eqs (9.5) and (9.6)].

Membrane Bioreactor

223

The effect of the transport in cylindrical space can be significant here because the measure of thickness of the spongy layer is similar to the lumen radius. Knowing the radial velocities in all three sections [Eqs (9.8), (9.20), and (9.21)], the axial pressure gradient can be expressed in both the lumen and shell sides. To it, let us give the boundary conditions at ro and ro 1 δ [Eqs (9.24) and (9.25)] regarding the radial velocities as at r 5 r;

υM 5 υt

ð9:24Þ

and at r 5 ro 1 δ;

υe 5 υM

ro ro 1 δ

ð9:25Þ

The limiting cases of the radial velocities (namely at r 5 ro for υM and υt and r 5 ro 1 δ for υe) are as follows: υt 5

ro3 d2 pt 16μ dz2

υM 5

ð9:26Þ

K pt 2 pe εμ lnð11 δ=ro Þ

ð9:27Þ

as well as (for case A): r 2 ðro 1 δÞ υe 5 o 16μ

(


δ 11 ro

2




rk 2 ro 1 δ

2 )

d2 pe dz2

ð9:28Þ

Substituting Eqs (9.26)
(9.28) in Eqs (9.24) and (9.25), one can get the following differential equations for the axial pressure gradients: For the lumen: d2 pt 16κ ðpt 2 pe Þ 5 3 dz2 ro lnð11 δ=ro Þ

ð9:29Þ

For the shell: d2 pe 16κ ðpt 2 pe Þ 52 3 2 dz ro lnð11 δ=rÞς with 2

δ 2 rk 2 ς 5 11 ro 1 δ ro

ð9:30Þ

224

Basic Equations of the Mass Transport through a Membrane Layer

From Eqs (9.29) and (9.30), the following differential equation can be expressed, for the lumen (Kelsey et al., 1990): 2 d4 pt 2 d pt 2 λ 50 dz4 dz2

ð9:31Þ

where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 16κ 1 λ5 11 ro3 lnð11 δ=ro Þ ς

ð9:32Þ

The general solution of Eq. (9.31) is well known; it is pt ðzÞ 5 T1 eλz 1 T2 e 2λz 1 T3 z 1 T4

ð9:33Þ

The pressure gradient in the shell side can also be given, applying Eq. (9.29) as pe ðzÞ 5 2

T1 λz T2 2λz e 2 e 1 T 3 z 1 T4 ς ς

ð9:34Þ

Several possibilities exist for boundary conditions depending upon the mode of operation for the hollow-fiber bioreactor. In closed shell mode, there is no inlet and outlet flow, the dpe/dz 5 0 at both ends of the shell side. In open shell (filtration mode), the boundary condition at z 5 L could be represented by the fraction (f) of the inlet feed directed out through the shell, thus f5

ut;L uo

ð9:35Þ

where both ut,Land uo are average values, thus uo 5

ð ro 0

umax ro2

 2 ! r r 12 dr ro ro

ð9:36Þ

The following four boundary conditions were used to determine the T1 2 T4 integration constants: at z 5 0

then pt 5 po ;

at z 5 L

then ut 5 fuo

ut 5 uo

and

dpe 50 dz

ð9:37Þ ð9:38Þ

Membrane Bioreactor

225

According to Eq. (9.16), the tube side pressure gradient at the inlet for parabolic flow at z 5 0 can be expressed as dpt 4μ 8μ 5 2umax 2  uo 2 dz z50 ro ro

ð9:39Þ

The algebraic equation system obtained can be solved easily by means of the well-known Cramer rules, thus B1 5

8uo μ ςð2 ½1 1 f ð1 1 ςÞ 1 e2λL Þ ro2 2λðς 1 1ÞsinhðλLÞ

ð9:40Þ

B2 5

8uo μ ςð2 ½1 1 f ð1 1 ςÞ 1 eλL Þ ro2 2λðς 1 1ÞsinhðλLÞ

ð9:41Þ

B3 5 2 B4 5

8uo μ ς ro2 ðς 1 1Þ

8uo μ ςð11 f ð1 1 ςÞ 2 cosh½λLÞ 1 po ro2 λðς 1 1ÞsinhðλLÞ

ð9:42Þ ð9:43Þ

Substituting Eqs (9.40)
(9.43) in Eq. (9.33), the pressure distribution of the lumen will be as pt 5

 8uo μς cosh½λðL 2 zÞ 2 h11 f ð11 ςÞicoshðλzÞ 1 B3 z 1 B4 2 ro λð11 ςÞsinhðλLÞ ð9:44Þ

Knowing the pressure distribution, we can now calculate the velocity distribution by means of Eqs (9.16)
(9.22). The pressure gradient similarly can be calculated using the case B (not shown here). Let us look at, as an example, the radial convective velocity given by Eq. (9.20). After derivation of Eq. (9.44), one can get as  d2 pt 8uo μς 1

5 2 cosh½λðL 2 zÞ 2 h11 f ð11 ςÞicoshðλzÞ 2 2 dz ro λð11 ςÞsinhðλLÞ λ ð9:45Þ Accordingly,   ! ro2 r 1 1 r 2 2 A υt ðr; zÞ 5 4μ 2 4 ro

ð9:46Þ

226

Basic Equations of the Mass Transport through a Membrane Layer

with A5

 8uo μς 1

cosh½λðL 2 zÞ 2 h1 1 f ð1 1 ςÞicoshðλzÞ ro2 λð1 1 ςÞsinhðλLÞ λ2

ð9:47Þ

The weakness of the above models is that they cannot account for the macroscopic radial pressure gradient and the radial interfiber flows that can exist on the shell side (Labecki et al., 1995, 1996). Labecki et al. (1995, 1996) developed a so-called porous medium model which takes into account the radial pressure gradient in the modules, as well as the longitudinal expansion of a fiber due to the swelling in the aqueous solution. They proved that their model provides more accurate results in the case of highly permeable membranes as the hydraulic resistances of the lumen and the extracapillary space become relatively more significant. In limiting cases, this model and the models given above give the same results. Note the expressions given in this chapter can easily be rewritten in dimensionless forms as generally made in the literature. The author feels that the forms used here give more information to the reader.

9.3.1.2 Component Mass Balances Look at the fully open system as it is illustrated in Figure 9.6. The reaction takes place inside the catalytic membrane layer. It is assumed that the reaction itself does not affect the velocity profiles in the system. The fluid phases can be recirculated, as it is often the case, during the process (the model does not involve this condition). Starting from momentum equations, Eqs (9.2) and (9.3), it is relatively easy to get the mass balance equation for the substrate components which have to be given for all three sections of a capillary membrane module, namely lumen, shell, and membrane layer. The biochemical reaction takes place in sections, only where the enzyme (or cell) is immobilized. Note that the principle of the mass transport of substrates/nutrients into the immobilized enzyme/cells, through a solid, porous layer (membrane, biofilm) or through a gel layer of enzyme/cells is the same. The biochemical processes are regarded as isotherm processes, thus heat balance is not needed to describe these processes. Considering that we know the radial and axial Figure 9.6 Schematic illustration of an open membrane reactor.

Membrane Bioreactor

227

convective velocities in the lumen and in the extracapillary space, the mass balance equations will be as υðr; zÞ



 @cj @cj 1@ @cj @2 c j 1 uðr; zÞ 5 Dj 1 2 r @r @z @r @z r @r

with j 5 t; e

ð9:48Þ

Equation (9.48) can further be simplified. For it, the radial and axial Peclet numbers (they are Pe 5 υro/D and Pe 5 uL/D, respectively) should be analyzed. The radial convective term and the axial diffusion terms often can be neglected (Piret and Cooney, 1991). For the fiber (substrate is feeding the lumen, on the skin side): z 5 0; z 5 L; r 5 0;

ct 5 cot @ct 50 @z @ct 50 @r

for all r for all z

ð9:49Þ

for all z

Let us accept that the cylindrical effect in the membrane layer can be neglected, so the mass transfer rate defined can be applied for boundary condition in the lumen or extracapillary space (Qin and Cabral, 1998). If the substrate is feeding on the skin side then [J is defined e.g. by Eq. (5.122), for first-order reaction]: r 5 ro ;

υct 2 Dt

@ct 5J @r

ð9:50Þ

If the substrate is feeding on the sponge side, then [Jδ is defined by Eq. (5.120), it enters the permeate (tube) phase from the membrane]: r 5 ro ;

υct 2 Dt

@ct 5 Jδ @r

ð9:51Þ

For the extracapillary space [substrate is fed in tube (lumen) side]: @ce 50 @z @ce 50 z 5 L; @z @ce 50 r 5 rk ; @r

z 5 0;

for all r for all z for all z

ð9:52Þ

228

Basic Equations of the Mass Transport through a Membrane Layer

r 5 ro 1 δ;

υce 2 De

@ce 5 Jδ;ov @r

ð9:53Þ

where rk is the Krogh radius (Figure 9.5A). The value of Jδ,ov is defined by Eq. (5.124), for this case. Obviously, the boundary conditions should always be suited the special, if it is the case, operating conditions. If the substrate is feeding on the sponge (extracapillary space) side, then [J is defined in Eq. (9.67)]: r 5 ro ;

υce 2 De

@ce 5J @r

ð9:54Þ

Note that according to Eq. (9.8), the radial convective velocity in the membrane depends on the radial space coordinate. Assuming plane membrane interface, this value will be constant in the membrane matrix as it is used for the mass transfer equations given in Eqs (9.50), (9.51), (9.53), and (9.54). But the υM value can change as a function of the axial coordinate due to the variable transmembrane pressure. This fact should be taken into account during integration of the mass balance equations [Eq. (9.48)] given for the lumen and the extracapillary space.

9.4

Mass Transfer Through a Biocatalytic Membrane Layer

Let us assume that enzyme is immobilized inside of membrane matrix or onto the membrane interface. If we define the mass transfer rate into the membrane layer, then we can apply it for describing the component balance of the fluid phases in the lumen and shell side. The structure, the thickness of this mass transport layer, can be very different; thus, the mass transport parameters, namely diffusion coefficient, convective velocity, the bioreaction rate constant, their dependency on the concentration and/or space coordinate, are characteristics of the porous layer and the nature of the biocatalysts. Several investigators modeled the mass transport through this biocatalyst layer, through enzyme membrane layer (Schonberg and Belfort, 1987; Hossain and Do, 1989; Salzman et al., 1999; Ferreira et al., 2001; Calabro et al., 2002; Long et al., 2003; Nagy 2009a,b; Nagy and Kulcsa´r, 2009). Some assumptions were made for expression of the differential mass balance equation to the biocatalytic membrane layer: G

G

G

G

G

G

Reaction occurs at every position within the biocatalyst layer. Reaction has one rate-limiting substrate/nutrient. Mass transport through the biocatalyst layer occurs by diffusion and convection. The partitioning of the components (substrate, product) is negligible. The mass transport parameters (diffusion coefficient, convective velocity, bioreaction rate constant) can vary as a function of the space coordinate. The external mass transfer resistance is to be taken into account.

Membrane Bioreactor G

G

G

229

There is no axial flow and component transport inside the membrane layer. Asymmetric membrane is applied. Process is unsteady state.

The differential component balance equation, for radial coordinate, perpendicular to the membrane interface, can generally be given, according to chapter of membrane reactor (see Chapter 8), in Table 8.1a, by the following equation, for substrate component i and for unsteady-state condition, as (Brotherton and Chau, 1990; Calabro et al., 2002):
 @φi @φi 1@ @φi 1 υr 5 2 Q^ i Di r @t @r @r r @r

ð9:55Þ

where φi is the substrate and/or product concentration in the biocatalytic membrane layer (kg/m3, kmol/m3); D is the diffusion coefficient (m2/s); r is the radial space coordinate (m); t is time (s); and Q^ i is the reaction rate (kmol/m3s, kg/m3s). The source term can be different in biocatalytic reactions, the most often applied equations are listed in Table 9.1. The inhibited reaction can take place in both the enzymatic and microbial reactions. In Table 9.1, φp is the product concentration, (kmol/m3); φ is the substrate concentration; Km is the Michaelis
Menten coefficient (kmol/m3); Kp, Ks (product, substrate, respectively) are the inhibition coefficients (kmol/m3); vmax is the maximum reaction rate (kmol/m3s); and Q^ is the reaction rate (kmol/m3s). The boundary conditions of Eq. (9.55) depend on whether the external mass transfer resistances are negligible or not. The initial condition is t50

then φi 5 φi;o

ð9:56Þ

Table 9.1 Expressions of the Important Biocatalytic Reactions vmax φ Michaelis
Menten kinetics Q^ 5 Km;i 1 φ Substrate inhibition

Q^ 5

vmax φ Km;i 1 φ 1 φ2 =Ks

Substrate inhibition and competitive product inhibition

Q^ 5

vmax φ Km;i ð11 φp =Kp Þ 1 φ 1 φ2 =Ks

Competitive product inhibition

Q^ 5

vmax φ Km;i ð11 φp =Kp Þ 1 φ

Noncompetitive product inhibition

Q^ 5

vmax φ ðKm;i 1 φÞð11 φp =Kp Þ

230

Basic Equations of the Mass Transport through a Membrane Layer

The boundary conditions are as t . 0 r 5 ro and r 5 ro 1 δ then @ci;j @φ 5 υM φi 2 Di;M i with j 5 t; e υj ci;j 2 Di;j @r @r

ð9:57Þ

or υj ci;j 2 β j ðci;j 2 ci;j Þ 5 υM φi 2 Di;M

@φi @r

ð9:58Þ

where subscripts t and e represent the lumen and shell, respectively, while subscript M denotes the membrane layer; c is the radial average concentration; and c is the interface concentration (kmol/m3). If a product component cannot enter the fluid phase from the catalytic membrane layer, then its concentration gradient will be zero at this interface. Note that the radial velocities, υj and υM, are equal to each other at r 5 ro or r 5 ro 1 δ. The υM values are defined by Eqs (9.20) and (9.21). The values can be determined by application of Eqs (9.20) and (9.21) as well as Eqs (9.44) and (9.34) for the lumen and shell, respectively.

9.4.1

Mass Transfer Through an Asymmetric Enzyme Membrane Layer

9.4.1.1 Feeding Is on the Sponge Side The enzyme is immobilized in the spongy layer of the asymmetric membrane (Figure 9.7). The substrate enters the spongy layer of the biocatalytic membrane and diffuses across the asymmetric membrane. The unreacted substrate can permeate into the other fluid phase flowing in the permeate side. All assumptions given in subsection 9.4 is true with exceptions as: the external mass transfer resistances are negligible; the transport parameters (υ, D1, D2) are constant; the curvature, cylindrical effect also is neglected.

G

G

G

Sponge skin φ*1

E E E E

φ*1,δ , φ*2

E E E

φ*2,δ

E

E E D1 ν

δ1 δ D2 Η1 φ*1,δ = Η2 φ *2 ν

δ − δ 1 = δ2

Figure 9.7 Schematic illustration of an asymmetric enzyme membrane layer.

Membrane Bioreactor

231

Look at a first-order reaction as a limiting case of Michaelis
Menten kinetics. Let us express the mass transfer rates into the membrane layer in case of first- and zero-order reactions as a limiting case of Michaelis
Menten kinetics. First-order reaction: The inlet and the outlet mass transfer rates were discussed in detail in Chapter 4 (mass transport without convective velocity) and in Chapter 5 (transport by diffusion plus convection). The overall inlet mass transfer rate was defined as [see Eq. (5.117)]: J12 5 β ov ðφ1 2 Gφ2;δ Þ

ð9:59Þ

with β ov 5 β

β o2} ðH1 =H2 Þ 1 β δ ðE2 2 E1 Þ β o2} ðH1 =H2 Þ 1 β δ E2

ð9:60Þ

and G5

ββ o2} e 2 Pe2 β o2} ðH1 =H2 Þ 1 β δ ðE2 2 E1 Þ

ð9:61Þ

The values of β, β δ, and β o2} as well as E1 and E2 are given by Eqs (5.110), (5.113), and (5.116) as well as Eqs (5.111) and (5.114). Zero-order reaction: The starting mass transfer rate equation was discussed in Chapter 6, namely: For the catalytic sponge layer, see Eqs (5.63)
(5.67): J1 5 βðφ1 2 Tφ1;δ Þ J1;δ 5 β o1} ζ


 e 2 Pe1   φ1;δ φ1 2 ζ

ð9:62Þ ð9:63Þ

For the noncatalytic skin layer, see Eqs (5.47) and (5.48): J2o 5 β o2} ðφ2 2 e 2 Pe2 φ2;δ Þ

ð9:64Þ

where J1 and J1,δ are the inlet and outlet mass transfer rates of the catalytic sponge layer, respectively (kg/m2s); and J2o is the inlet mass transfer rate of the catalytically inactive skin layer (kg/m2s). The expression of β, T, ζ, and β o2} are given by Eqs (5.64), (5.65), (5.67), and (5.48), respectively. For their usage, the subscripts of the layers should be taken into account. According to Eqs (9.62)
(9.64), the overall inlet mass transfer rate of the sponge layer, for zero-order reaction, can be given as J12 5 β ov ðφ1 2 Fφ2;δ Þ

ð9:65Þ

232

Basic Equations of the Mass Transport through a Membrane Layer

with
 ϑ2 Tβ o1 ς β ov 5 β o1 1 2 2 2 o β 2} ðH1 =H2 Þ 1 β o1 e 2 Pe1 Pe1

ð9:66Þ

and F5

Tβ o2 ePe2 ð1 2ðς 2 =Pe21 ÞÞ ð1 2ðς 2 =Pe21 ÞÞðβ o2} ðH1 =H2 Þ 1 β o1 e 2 Pe1 Þ 2 Tς

ð9:67Þ

9.4.1.2 Feeding Is Done on the Skin Side of the Catalytic Membrane For the sake of completeness, we give here the overall mass transfer rate when the feeding of the substrate is carried out through the nonreactive skin layer. This can be done more easily than that in the previous case, though the reacted amount is also needed to give the outlet rate in the spongy layer. The situation is illustrated in Figure 9.8. Note that the subscripts 1 and 2 also denote the skin and the catalytic sponge layers, as it was made in subsection 9.4.1.1. First-order reaction: The inlet mass transfer rate will be as (for details, see Chapter 5, subsection 5.3.4.2):
 H1 E1 e 2Pe2 φ1;δ J21 5 β ov φ2 2 H2

ð9:68Þ

with β ov 5

1 ð1=β o2} Þ 1 ð1=βÞðH1 =H2 Þe 2Pe2

ð9:69Þ

As an illustration, the ratio of J1 (the reactant enter the biocatalytic sponge layer) and J2 (the reactant can enter the inactive skin layer of the asymmetric membrane) is plotted in Figure 9.9 for the case when Pe2 5 Pe1 5 0. This limiting Skin sponge E φ 2*

E E E E

E

φ 2*,δ , φ *1

E E E E

φ 1*,δ

Figure 9.8 Illustration of the mass transport when the substrate is fed on the skin side of the asymmetric catalytic membrane.

Membrane Bioreactor

233

100.0

Ratio of fluxes, J1/J2

H1/H2 = 10 10.0 1

1.0 Pe1 = Pe2 = 0 0.1

1.0

0.1

10.0

Reaction modulus, ϑ

Figure 9.9 The ratio of mass transfer rates as a function of the reaction modulus at different values of H1/H2 [δ1 5 80 μm (the catalytic sponge layer); δ2 5 0.5 μm (skin layer); D2 =D1 5 10:

case can also be obtained by Eqs (9.59) and (9.68). The mass transfer rate equations are given by Eq (4.85) and (4.93a) for that case. The ratio of the mass transfer rates can be given as β o1 ϑfðtanh ϑ 1ðβ o2 =β o1 ÞðH1 =H2 Þð1=ϑÞÞφ1 2 ðβ o2 =β o1 Þð1=ðϑ cosh ϑÞÞφ2;δ g J1 5 β o2 ðφ2;δ 2 ðH1 =H2 Þð1=cosh ϑÞφ1 Þ J2 ð9:70Þ Let us regard the outlet concentrations to be zero [according to Eq. (9.70) φ2;δ 5 0 in the numerator as well as φ1 5 0 in the denominator] and take the same values for the entering concentrations, thus Eq. (9.70) will be as J1 β o ϑðtanh ϑ 1ðβ o2 =β o1 ÞðH1 =H2 Þð1=ϑÞÞ 5 1 J2 β o2

ð9:71Þ

The calculated data are plotted in Figure 9.9. Calculation was made at three different solbility coefficients. Figure 9.9 illustrates well that in the case of an asymmetric, catalytic, or biocatalytic membrane reactor, the upstream phase should be chosen on the catalytic, sponge side of the membrane. Zero-order reaction: The overall mass transfer rate, J21, can be obtained by application of Eqs (9.61) and (9.64). Accordingly,
J21 5 β

φ2

2Pe2

2 Te

H1  φ H2 1;δ

 ð9:72Þ

234

Basic Equations of the Mass Transport through a Membrane Layer

φ1*

E E

J1

E

φ2*,δ

E

E E E E

Figure 9.10 Substrate transport by feeding it on both sides of the membrane reactor.

J2

E E

0

δ D1

D2

with β ov 5

1 ð1=β o2} Þ 1ðH1 =H2 Þe2Pe2 ð1=βÞ

ð9:73Þ

9.4.1.3 Substrate Is Feeding on Both Sides of the Catalytic, Asymmetric Membrane The sparingly soluble oxygen is often supplied on both sides of the membrane in the case of cell culture in the membrane matrix (Gonzalez-Brambila et al., 2006). This transport process is illustrated in Figure 9.10. For determination of the mass transfer rates, it was assumed the substrate transport takes place by diffusion, only there is no convective substrate transport through the membrane bioreactor. The mass transfer rates will be given here for both limiting cases of the Michaelis
Menten kinetics. The sum of J12 and J21 gives the total mass transfer rate when the substrate can enter on both sides of the two-layer mass transfer. This can be the situation in the case of a biofilm formed on an oxygen permeable membrane (see the next section).

9.4.2

Modeling of Whole Cell Membrane/Biofilm Reactor

Membrane bioreactors for immobilized whole cells (Schonberg and Belfort, 1987; Chung, 2005; Sheldon and Small, 2005) provide an advantageous environment for increased cell densities. The cells are perfused via a membrane with a steady continuous flow of medium containing the oxygen and other nutrients. The cells are grown either in the extracapillary space (to form biofilm), or within the fibers (Figure 9.2). It was shown that a mass transfer limitation for oxygen or other nutrient could be occurred, especially at higher cell density (Brotherton and Chau, 1990; Sardonini and DiBiasio, 1992; Chung et al., 2005). Due to the change of the nutrient concentration in axial direction, the density of cell culture, the thickness of the biofilm on the membrane interface, can also change. This fact can alter the value of transport parameters (diffusion coefficient, convective velocity); it can

Membrane Bioreactor

235

alter even the biochemical reaction rate (the consumption rate of nutrient). The density of the cell can change not only the values of transport parameters but the value of the reaction rate constant (Schonberg and Belfort, 1987). The variation of cell density is also true in the biocatalyst membrane layer perpendicular to the inlet surface. Increasing distance from the surface can mean decreasing nutrient concentration. That is why the variability of the transport parameters should also be taken into account. The conventional biofilm or the membrane-aerated biofilm processes are increasingly used for wastewater purification as they are environmentally friendly and less energy intensive (Nicolella et al., 2000; Gonzalez-Brambila et al., 2006; Matsumoto et al., 2007; Juang and Kao, 2009; Merkey et al., 2009; Lackner et al., 2010; Wang and Zhang, 2010). Biofilms are agglomerations of microorganisms or bacteria, which is due to their metabolic activity converting the contaminant components of the wastewaters into harmless products (Rao et al., 2010). Contrary to the suspended cell systems, biofilm reactors provide high mass concentration and higher substrate conversion without the need for solid separation devices (Kermanshaipour et al., 2006). Important reactions investigated are biodegradation of phenol (Juang and Kao, 2009), p-xylene (Kermanshahi et al., 2006), other aromatics (Farhadian et al., 2008), or nitrification and denitrification (Satoh et al., 2004; Matsumoto et al., 2007; Lackner et al., 2010). Biofilms consist of cells immobilized in an organic polymer matrix. The structure of a biofilm can be variable. When a biofilm is strongly diffusion limited, it will tend to become a heterogeneous and porous structure (Loosdrecht et al., 2002). When the conversion is the rate-limiting step, the biofilm will tend to become homogeneous and compact. Biofilm structure results from the interplay between these interactions (mass transfer, conversion rates, detachment forces). Biofilm formation is a complicated dynamic process governed by various physical and chemical principles and biological protocols (Wang and Zhang, 2010). Biofilm development depends heavily on the environmental conditions and types and properties of the bacteria inside the biofilm. Several investigators modeled the mass transport through cell culture membrane layer (Schonberg and Belfort, 1987; Brotherton and Chau, 1990, 1996; Piret and Cooney, 1991; Sardonini and DiBiasio, 1992; Cabral and Tramper, 1994; Melo and Oliveira, 2001; Lu et al., 2011). Mathematical modeling of biofilms is crucial to attain a broader and deeper understanding of this complex microorganism. The mathematical model can be used to make qualitative and quantitative predictions that might serve well as guidelines for experimental design. Theoretical studies also confirm that nutrient limitation can often occur in hollow-fiber biocatalytic membranes (Belfort, 1989; Piret and Cooney, 1991; Calabro et al., 2002). To avoid this limitation, the mass transfer rate through the cell culture should be increased. This can be achieved by construction change, by change of the membrane structure, thickness, or by the increase of the transmembrane pressure. Due to it, the radial convective velocity also increases. Development of the models, which mostly have been successfully applied, is reviewed by Wang and Zhang (2010).

236

Basic Equations of the Mass Transport through a Membrane Layer

Membrane

O2

Figure 9.11 Illustration of biofilm attached to a permeable, silicon rubber, membrane layer with the concentration profiles; homogeneous model is applied for the biofilm.

Biofilm

C oOG

C oSL JS

JOG

C oOL JOL

Air/O2 Aqueous liquid y 0

Interphase gas membrane

δ

Interphase membrane biofilm

δf

Interphase biofilm liquid

Two important, often applied cases will be briefly analyzed here. These are illustrated in Figures 9.11 and 9.12. In both cases, two substrates, S (e.g., carbon source) and O (oxygen), are fed into the biofilm in the same fluid as the other substrate (they are diffusing cocurrently into the biofilm (Figure 9.11), or the substrates are diffusing countercurrently in the biofilm (they are fed in separate phases; Figure 9.12).

9.4.2.1 Countercurrent Operating Mode As can be seen in Figure 9.11, oxygen is supplied through a silicone membrane using a given pressure. The oxygen enters the liquid in the biofilm layer in dissolved form, where it is consumed by the cells. The other substrate enters the biofilm on the other side. The mass transport occurs by diffusion for both components. The limiting cases of such a system are analyzed briefly, and the mass transfer rates are defined by Eqs (9.77)
(9.80). The solution of the general case, namely the Monod kinetics, is a complex task and needs an approaching treatment.

9.4.2.2 Mass Transfer Rate for First- and Zero-Order Reactions According to the mass transfer rate equations given previously in this or other chapters as well as in Figures 9.11 and 9.12, the mass transfer rate into the biofilm under different conditions can be given easily: here we shoe the mass transfer rate for 4 typical cases (case 1
4).

Membrane Bioreactor

Support

237

Figure 9.12 Biofilm attached to a support, impermeable layer, and the substrate concentration profiles.

Biofilm

CoSL CoOL

0

δf

Case 1: The substrate enters the biofilm from the bulk phase and cannot enetr the impermeable support layer, in the case of first-order [Eq. (5.46)] and zero-order [Eq. (5.74)] reactions, respectively:



Pe2f 2 Θ 2 f 4 tanh Θf Df   J5 φ Pef δf 2 tanh Θf 2 Θf rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pe2f 1 ϑ2f ; Θf 5 4

sffiffiffiffiffiffiffiffiffiffiffi δ2f k1 ; ϑf 5 Df

ð9:74Þ

Pe 5

υδf Df

where subscript f represents the biofilm layer; J is the mass transfer rate at y 5 δf (accord ing to Figure 9.12); and φ is the substrate concentration in the biofilm at y 5 δf (kg/m3, 3 or kmol/m ):

J5


 Df ϑ2f 1 1 2 Pe φ δf Pef e f

ð9:75Þ

with

sffiffiffiffiffiffiffiffiffiffi k0 δ2f ϑf 5 Df φ Case 2: The substrate can enter the permeable support layer. In this case, the effect of both layer, namely the biofilm and the permeable support layer, should be taken into account. This J12 mass transfer rate is defined by Eqs (9.59) and (9.68) for firstorder reaction, and by Eqs (9.65) and (9.72) for zero-order reactions. Note that the substrate enters the biocatalytic biofilm layer.

238

Basic Equations of the Mass Transport through a Membrane Layer

Case 3: In the case of sparingly soluble oxygen, it is often supplies on both sides of the membrane/biofilm system. An important question to be answered is the total amount of the oxygen transfer. If the substrate concentration is lower than the interface concentration, due to the biochemical reaction in the biofilm, then the inlet mass transfer can occur on both sides of this system (see Figure 5.5 in where the effect of a first-order chemical reaction on the concentration distribution is illustrated). Then the total mass transfer rate can be obtained by sum of J12 and J21. This will be true in the case of first-order reaction as [JΣ 5 J12 1 J21]:

  H1 2Pe2  φ1 2 hðβ ov Þ12 G 2 ðβ ov Þ21 iφ2;δ JΣ 5 ðβ ov Þ12 2 ðβ ov Þ21 E1 e H2

ð9:76Þ

with

E1 5

ePef =2 ½ðPe

Θ f =2Þsinh Θ 1 Θcosh Θ

where subscripts 12 and 21 relate to the value of β ov in Eqs (9.60) and (9.73), respectively; the parameter G is given in Eq. (9.61). The total mass transfer rate can likewise be given similar in the case of zero-order reaction. Case 4: The microorganism is immobilized in a porous membrane and it forms a colony on the membrane interface. This situation is illustrated in Figure 9.13. The substrate can enter either the membrane or the biofilm. It can, at least partly, be transported through both layers. The substrate partly reacts in the feed side biocatalytic layer, only the remaining part gets in the other biocatalytic layer. The mass transfer rate, taking into account the effect of both layers, can be expressed similarly to the methodology used for the asymmetric membranes. Look at it here for the first-order

δ1 ⫹ δ2

Biofilm

δ1

Membrane

Figure 9.13 Schematic diagram of mass transfer for two biocatalytic layers (biofilm and membrane).

0

Co J

Co Cδ Jδ

Cδ

0

δ1

δ2 ⫹ δ1

Membrane Bioreactor

239

reaction. Let us look at a situation when the substrate enters the catalytic biofilm layer (continuous line in Figure 9.13). The starting equations are as follows [the necessary mass transfer rates are defined by Eqs (5.29) and (5.31)]: The inlet mass transfer rate:

J1 5 β 1 ðHco 2 E1 φ1 Þ

ð9:77Þ

with β1 5

β o1 ð½Pe1 =2tanh Θ1 1 Θ1 Þ ; tanh Θ1

E1 5

Θ1 ePe1 =2 ½ðPe1 =2Þsinh Θ1 1 Θ1 cosh Θ

with rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pe21 Θ1 5 1 ϑ21 ; 4

sffiffiffiffiffiffiffiffiffiffiffi k1 δ21 υδ1 ϑ1 5 ; Pe 5 D1 D1

The effect of the reaction rate is illustrated in Figure 14 at different values of the Peclet number. As there is no external mass transfer resistance, the mass transfer rate monotonously increases as a function of the reaction modulus. The outlet mass transfer rate entering the membrane layer:

Jδ 5 β δ ðHco 2 Aφ1 Þ

ð9:78Þ

with

βδ 5

D1 Θ1 ePe1 =2 ; δ1 sinh Θ1

A5

Θ1 2ðPe1 =2Þtanh Θ1 cosh Θ1 Θ1 ePe1 =2

where subscript 1 denotes the biofilm layer, and φ1 represents the concentration at the internal biofilm surface, at y 5 δ1 (kg/m3). It is assumed that the substrate concentrations are equal to each other in the two layers, thus φ1 5 φ2 : It is easy to consider when the partition coefficients of these layers are different. According to Eq. (9.77), the inlet mass transfer rate of the biofilm can be given as

J2 5 β 2 ðφ2 2 E2 cδ Þ

ð9:79Þ

with β2 5

β o2 ð½Pe2 =2tanh Θ2 1 Θ2 Þ ; tanh Θ2

E2 5

ePe2 =2 ½ðPe

Θ2 2 =2Þsinh Θ2 1 Θ2 cosh Θ2 

240

Basic Equations of the Mass Transport through a Membrane Layer

Applying Eqs (9.77)
(9.79) and taking into account that J1 5 Jδ, the overall mass transfer rate can be given as

J12 5 β ov ðco 2 Fcδ Þ

ð9:80Þ

with

β ov 5 β 1 H

1 2 E1 β δ ; β2 1 βδ A

F5

E1 β 2 E2 12 E1 β δ

The mass transfer rate, involving the effect of both layers, can similarly be obtained for zero-order reactions. For it, Eqs (5.63) and (5.66) should be adapted to this mass transport process. The change of the mass transfer coefficient is plotted for single catalytic layer in Figure 9.14 and for two catalytic layers in Figure 9.15 as a function of the reaction modulus. The properties of the two catalytic layers were chosen to be the same. The value of β oov was obtained as a limiting case of β ov ; namely if ϑ1 -0, ϑ2 -0: Tendency of curves obtained by two layers are similar to that obtained for single layer mass transport. Difference between the two figures, namely Figures 9.14 and 9.15, is that the value of β ov =β oov is much higher in the case of two-layer mass transport at low values of Peclet number. This is due to the different values of the physical mass transfer coefficient of the two cases.

9.4.2.3 Mass Transport Accompanied by Bioreaction of Monod Kinetics Mass transport with outlet mass transfer, dφ=dy . 0: We offer a relatively simple mathematical approach for the solution of Monod kinetics applying for two

11

Enhancement, β/β°

9 D/δ = 1 x 10–5 m/s 7

Pe = 0.1 0.3 1

5 3

3

10 1 1

10 Reaction modulus, ϑ

Figure 9.14 Enhancement as a function of reaction modulus in the case of a single catalytic layer.

Membrane Bioreactor

241

Enhancement, βOV/βtOV°

11 Pe1 = Pe2

9

Pe1 = 0.1

D1/δ1 = D2/δ2 7

0.3

D1/δ1 = 1 ⫻ 10–5 m/s 1

5

3

3 1

10 0

1

10

Reaction modulus, ϑ

Figure 9.15 Enhancement as a function of the reaction modulus, taking into account the simultaneous effect of the two catalytic layers (β 1 5 β 2).

substrates. It is assumed that the reagents (components O and S, namely oxygen and carbon source, respectively) are fed on the both sides of the membrane reactor and they are diffusing through the membrane layer countercurrently (Figure 9.11). The biofilm is considered as a homogeneous layer with constant mass transfer parameters. In the membrane matrix, cell culture is growing. The reaction term can be given for this intrinsically catalytic membrane layer applying the Monod kinetics as follows: Q5

1 μmaxB φB φA YXB KMB 1 φB KMA 1 φA

ð9:81Þ

We assume that the biomass concentration does not change during the process, thus, the value of μmaxB =YXB can be regarded to be constant during the process. This assumption can be made because the growth of the microorganism is much less than the mass transport in the biofilm. Applying the methodology showed in Chapter 4, the source term of Eq. (9.81) can be approached by a first-order reaction term, for the ith sublayer: Qi 5 k~Ai φSi φO

ð9:82Þ

with k~Oi 5

1 ; KMO 1 φOi

φSi 5

1 μmaxS φSi YXS KMS 1 φSi

242

Basic Equations of the Mass Transport through a Membrane Layer

Accordingly, one can get a second-order differential equation with linear source term that can be solved analytically for plane membrane layer. The balance equation in dimensionless form for the ith sublayer is as d2 φ O 2 ϑ2Oi φO 5 0 dY 2

for Yi21 #Y #Yi

ð9:83Þ

where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k~Oi δ2 φSi ϑOi 5 DO where φSi denotes the average concentration of component S in the ith sublayer. Solution of Eq. (9.83) is well known, it will be as Φ 5 Ti eϑOi Yi 1 Pi e 2ϑOi Yi

Yi21 #Y #Yi

ð9:84Þ

The general solution for every sublayer has two parameters that should be determined by the suitable boundary conditions. The external and the internal boundary conditions, applicable for that case, are defined by Eqs (4.47)
(4.50). After solution of the 2N algebraic equations with 2N parameters (the value of N is generally above 100), obtained by means of the boundary conditions, the value of the parameters T1 and P1 for the first sublayer can be obtained as (ΔY is the thickness of the sublayers): 1 φOδ ξTN φO 2 N T1 5 2 O 2ξN coshðϑO1 ΔYÞ Li52 coshðϑOi ΔYÞ

! ð9:85Þ

and 1 φOδ ξPN φO 2 N P1 5 O 2ξ N coshðϑO1 ΔYÞ Li52 coshðϑOi ΔYÞ

! ð9:86Þ

Knowing T1 and S1, the other parameters, namely Ti and Pi (i 5 2, 3, . . ., N), can be calculated easily by means of the internal boundary conditions given by Eqs (4.48) and (4.49), starting from T2 and S2 up to TN and SN. The values of ξ TN ; ξ PN ; and ξO N are as follows: j j ξ ij 5 ξ i21 1 κi21

tanhðϑOi ΔYÞ zi 2 1

for i 5 2; 3; . . . N;

j 5 P; T; O

ð9:87Þ

Membrane Bioreactor

243

and j tanhðϑOi ΔYÞ 1 κij 5 ξ i21

j κi21 zi21

for i 5 2; 3; . . . N 2 1;

j 5 P; T; O

ð9:88Þ

The starting values of ξj1 and κj1 are as follows: ξT1 5 e2ϑO1 ΔY ;

ξP1 5 eϑO1 ΔY ;

ξO 1 5 tanhðϑO1 ΔYÞ

and κT1 5 2e2ϑO1 ΔY ;

κP1 5 eϑO1 ΔY ;

κO 1 51

as well as zi21 5

DOi ϑOi DOi 2 1 ϑOi 2 1

ð9:89Þ

The mass transfer rate of component O, from the gas and liquid phases, can be expressed as JOG 5 2

DO1 ϑO1 ðT1 2 P1 Þ δ

ð9:90Þ

and JOL 5

DOL ϑON ðTN 2 PN Þ δ

ð9:91Þ

Obviously, in order to get the concentration of component O, the concentration distribution of component S is also needed. It is easy to learn that a trial-and-error method should be used to get the component concentrations alternately. Steps of calculation of concentration of both components can be as follows: 1. Starting concentration distribution, such as for component S, it should be given and one calculates the concentration distribution of component O applying Eqs (9.84)
(9.89), as well as boundary conditions by Eqs (9.92)
(9.95). 2. The indices of sublayer of component O have to be changed, adjusted them to that of S started from the permeate side of membrane, i.e., at Y 5 1, thus, i subscript of O, namely Oi should be replaced by N 2 i. 3. Now applying the previously calculated averaged value of concentration of OðφOi Þ; one can predict the concentration distribution of component S, using Eqs (9.84)
(9.89) as well as boundary conditions by Eqs (9.92)
(9.95), adapted them to component S. 4. These three steps should be repeated until concentrations of components O and S do not change anymore.

244

Basic Equations of the Mass Transport through a Membrane Layer

Note that the above problem is a so-called self-adjusted one, thus, the true concentration can be obtained after three to four iteration steps. Knowing T1 and S1, the other parameters, namely Ti and Si (i 5 2, 3, . . ., N), can be easily be calculated by means of the internal boundary conditions given by Eqs (4.48) and (4.49), starting from T2 and S2 up to TN and SN. Thus, one can get the following equations for prediction of the Ti and Si from Ti 21 and Si 21, for the component A: TOi eϑOi YOi 1 POi e2ϑOi Yi 5 ΓOi 2 1

ð9:92Þ

DOi ϑOi ðTOi eϑOi Yi 2 SOi e2ϑOi Yi Þ 5 ΞOi21

ð9:93Þ

ΓOi21 5 TOi21 eϑOi21 Yi 1 SOi21 e2ϑOi21 Yi

ð9:94Þ

ΞOi21 5 DOi21 ϑOi21 ðTOi21 eϑOi 2 1 Yi 2 SOi21 e2ϑOi 2 1 Yi Þ

ð9:95Þ

with

Now knowing the Ti and Si (with i 5 1, 2, . . ., N) parameters, the concentration distribution can be calculated easily through the membrane, i.e., its value for every sublayer. The same methodology can be used when the two substrates, oxygen and glucose, are fed on the same side. This method can also easily be applied for product inhibition when the second component will be the product. Biofilm attached to an impermeable membrane layer, dφ=dy 5 0: Both substrates, namely oxygen and the carbon source, are fed according to Figure 9.11. Neither of the reactants can permeate through the impermeable membrane layer. Accordingly, the concentration gradient of the substrates will be zero at y 5 0 as it is given in Figure 9.12. The solution of Eq. (9.83) with source term of Eq. (9.82) should be carried out similarly as in the previous case. The values of T1 and S1 and the necessary parameters are given by Eqs (4.68)
(4.70). The solution methodology is the same as it is written in Section 9.4.2.3. For this case, when not only diffusive flow but also convective flow is occurring during the transport process, the solution can be found in Eqs (5.98a)
(5.108).

References Andric, P., Meyer, A.S., Jensen, P.A., and Dam-Johanson, K. (2010) Reactor design for minimizing product inhibition during enzymatic lignocelluloses hydrolysis. II. Quantification of inhibition and suitability of membrane reactors. Biotechnol. Adv. 28, 407
425. Be´lafi-Bako´, K., Koutinas, A., Nemesto´thy, N., Gubicza, L., and Webb, C. (2006) Continuous enzymatic cellulose hydrolysis in a tubular membrane bioreactor. Enzyme Microb. Technol. 38, 155
161. Belfort, G. (1989) Membranes and bioreactors: a technical challenge in biotechnology. Biotechnol. Bioeng. 33, 1047
1066.

Membrane Bioreactor

245

Bird, R.B., Stewart, W.E., and Lightfoot, E.N. (1960). Transport Phenomena, John Wiley and Sons, New York. Brotherton, J.D., and Chau, P.C. (1990) Modeling analysis of an intercalated-spiral alternatedead-ended hollow fiber bioreactor for mammalian cell cultures. Biotechnol. Bioeng. 35, 375
394. Brotherton, J.D., and Chau, P.C. (1996) Biotechnol. Prog. 12 (5), 575
590. Bruining, W.J. (1989) A general description of flows and pressure in hollow fiber membrane modules. Chem. Eng. Sci. 44, 1441
1447. Cabral, J.M.S., and Tramper, J. (1994) Bioreactor design, in Applied Biocatalysis, Ed. by J.M.S., Cabral, D., Best, L., Boross, and J., Tramper. Harwood Academic Publishers, Switzerland, pp. 330
370. Calabro, V., Curcio, S., and Iorio, G. (2002) A theoretical analysis of transport phenomena in a hollow fiber membrane bioreactor with immobilized biocatalyst. J. Membr. Sci. 206, 217
241. Charcosset, C. (2006) Membrane processes in biotechnology: an overview. Biotechnol. Adv. 24, 482
492. Chung, T.P., Wu, P.C., and Jung, R.S. (2005) J. Membr. Sci. 258, 55
63. Damak, K., Ayadi, A., Zeghmati, B., and Schmitz, P. (2004) A new Navier
Stokes and Darcy’s law combined model for fluid flow in cross-flow filtration tubular membrane. Desalination 161, 67
77. Farhadian, M., Duchez, D., Vachelard, C., and Larroche, C. (2008) Monoaromatics removal from polluted water through bioreactors—a review. Water Res. 42, 1325
1341. Ferreira, B.S., Fernandes, P., and Cabral, J.M.S. (2001) Design and modeling of immobilized biocatalytic reactors, in Multiphase Bioreactor Design, Ed. by J.M.S., Cabral, M., Mota, and J., Tramper. Taylor & Francis, London, pp. 85
180. Frazeres, D.M.F., and Cabral, J.M.S. (2001) Enzymatic membrane reactors. in Ed. by J.M.S., Cabral, M., Mota, and J., Tramper, 2001. Taylor & Francis, London, pp. 135
184. Giorno, L., and Drioli, E. (2000) Biocatalytic membrane reactors: applications and perspectives. Trends Biotechnol. 18, 339
349. Giorno, L., De Bartolo, L., and Drioli, E. (2003) Membrane bioreactors for biotechnology and medical applications, in New Insight into Membrane Science and Technology: Polymeric and Bifunctional Membranes, Ed. by D., Bhattacharyya, and D.A., Butterfiled. Elsevier, Oxford, (Chapter 9). Giorno, L., Amore, E.D., Mazzei, R., Piacentini, E., Zhang, J., and Drioli, E., et al. (2007) An innovative approach to improve the performance of a two separate phase enzyme membrane reactor by immobilizing lipase in presence of emulsion. J. Membr. Sci. 295, 95
101. Godongwana, B., Sheldon, M.S., and Solomons, D.M. (2007) Momentum transfer inside a vertically oriented capillary membrane bioreactor. J. Membr. Sci. 303, 86
99. Gonza´lez-Brambila, M., Monroy, O., and Lo´pez-Isunza (2006) Experimental and theoretical study of membrane-aerated biofilm reactor behavior under different modes of oxygen supply for the treatment of synthetic wastewater. Chem. Eng. Sci., 61, 5268
5281. Habulin, M., and Knez, K. (1991) Enzymatic synthesis of n-butil oleate in a hollow fiber membrane reactor. J. Membr. Sci. 61, 315
324. Hossain, M.M., and Do, D.D. (1989) Biotechnol. Bioeng. 1989 (33), 963
975. Juang, R.-S., and Kao, H.-C. (2009) Estimation of the contribution of immobilized biofilm and suspended biomass to the biodegradation of phenol in membrane contactors. Biochem. Eng. J. 43, 122
128. Kelsey, L.J., Pillarella, M.R., and Zydney, A.L. (1990) Theoretical analysis of convective flow profiles in a hollow fiber membrane bioreactors. Chem. Eng. Sci. 45, 3211
3220.

246

Basic Equations of the Mass Transport through a Membrane Layer

Kermanshaipour, A., Karamanev, D., and Margaritis, A. (2006) Kinetic modeling of the biodegradation of the aqueous p-xylene in the immobilized soil bioreactor. Biochem. Eng. J. 27, 204
211. Labecki, M., Bowen, D., and Piret, J.M. (1996) Two-dimensional analysis of protein transport in the extracapillary space of hollow-fibre bioreactors. Chem. Eng. Sci. 51, 4197
4213. Labecki, M., Piret, J.M., and Bowen, D. (1995) Two-dimensional analysis of fluid flow in hollow-fibre modules. Chem. Eng. Sci. 50, 3369
3384. Lackner, S., Terada, A., Horn, H., Henze, M., and Smets, B. (2010) Nitrification performance in membrane-aerated biofilm reactors differs from conventional biofilm systems. Water Res. 44, 6073
6084. Long, W.S., Bhatia, S., and Kamaruddin, A. (2003) Modeling and simulation of enzymatic membrane reactor for kinetic resolution of ibuprofen ester. J. Membr. Sci. 219, 69
88. Loosdrecht, M.C.M., Heijnen, J.J., Eberl, H., Kreft, J., and Picioreanu, C. (2002) Mathematical modeling of biofilm structures. Antonie van Leeuwenhoek 81, 245
256. Lozano, P., Perez-Marin, A.B., De Diego, T., Gomez, D., Paolucci-Jeajean, D., and Belleville, M.P., et al. (2002) Active membranes coated with immobilized Candida Antarctica lipase B: preparation and application for continuous butyl butyrate synthesis in organic media. J. Membr. Sci. 201, 55
64. Lu, S.G., Imai, T., Ukita, M., Sekine, M., Higouchi, T., and Fukagawa, M. (2011) Water Res. 35 (8), 2038
2043. Marcano, J.G.S., and Tsotsis, T.T. (2002) Catalytic Membranes and Membrane Reactores. Wiley-VCH, Weinheim. Matsumoto, S., Terada, A., and Tsuneda, S. (2007) Modeling of membrane-aerated biofilm: effect of C/N ratio, biofilm thickness and surface loading of oxygen on feasibility of simultaneous nitrification and denitrification. Biochem. Eng. J. 37, 98
107. Melo, L.F., and Oliveira, R. (2001) Biofilm reactors, in Multiphase Bioreactor Design, Ed. by J.M.S., Cabral, and M., Mota, J. Tramper. Taylor & Francis, London, pp. 271
308. Merkey, B.V., Rittmann, B.E., and Chopp, D.L. (2009) Modeling how soluble microbial products (SMP) support heterotropic bacteria in autotroph-based biofilms. J. Theor. Biol. 259, 670
683. Mondor, M., and Moresoli, C. (1999) Theoretical analysis of the influence of the axial variation of the transmembrane pressure in cross-flow filtration of rigid spheres. J. Membr. Sci. 152, 71
87. Nagy, E. (2009a) Mathematical modeling of biochemical membrane reactors, in Membrane Operations, Innovative Separations and Transformations, Ed. by E., Drioli, and L., Giorno. Wiley-VCH, Weinheim, pp. 309
334. Nagy, E. (2009b) Basic equations of mass transfer through biocatalytic membrane layer. Asia-Pacific J. Chem. Eng. 4, 270
278. Nagy, E., and Kulcsa´r, E. (2009) Mass transport through biocatalytic membrane reactors. Desalination 245, 422
436. Nakajima, M., and Cardoso, J.P. (1989) Forced-flow bioreactor for sucrose inversion using ceramic membrane activated by silanization. Biotechnol. Bioeng. 33, 856. Nicolella, C., Pavasant, P., and Livingston, A.G. (2000) Substrate counter diffusion and reaction in membrane attached biofilms: mathematical analysis of rate limiting mechanisms. Chem. Eng. Sci. 55, 1385
1398. Piret, J.M., and Cooney, C.L. (1991) Model of oxygen transport limitations in hollow fiber bioreactors. Biotechnol. Bioeng. 37, 80
92. Qin, V., and Cabral, M.S. (1998) Lumen mass transfer in hollow fiber membrane processes with nonlinear boundary conditions. AIChE J. 44, 836
848.

Membrane Bioreactor

247

Rao, K.R., Srinivasan, T., and Venkateswarlu, Ch. (2010) Mathematical and kinetic modeling of biofilm reactor based on ant colony optimization. Process Biochem. 45, 961
972. Rios, G.M., Belleville, M.P., Paolucci, D., and Sanchez, J. (2004) Progress in enzymatic reactors—a review. J. Membr. Sci. 242, 189
196. Rios, G.M., Belleville, M-P., and Paolucci, D. (2007) Membrane engineering in biotechnology: quo vamus? Trends Biotechnol. 25, 242
246. Salzman, G., Tadmor, R., Guzy, S., Sideman, S., and Lotan, N. (1999) Hollow fiber enzymic reactors for a two substrate process: analytical modeling and numerical simulations. Chem. Eng. Progress 38, 289
299. Santos, A., Ma, I.W., and Judd, S.J. (2010) Membrane bioreactors: two decades of research and implementation. Desalination 273, 148
154. Sardonini, C.A., and DiBiasio, D. (1992) An investigation of the diffusion-limited growth of animal cells around single hollow fibers. Biotechnol. Bioeng. 40, 1233
1242. Satoh, H., Ono, H., Rulin, B., Kamo, J., Okabe, S., and Fukushi, K.-I. (2004) Macroscale and microscale analyses of nitrification and denitrification in biofilms attached on membrane aerated biofilm reactors. Water Res. 38, 1633
1641. Schonberg, J.A., and Belfort, G. (1987) Enhanced nutrient transport in hollow fiber perfusion bioreactors. Biotechnol. Prog. 3 (2), 81
89. Sheldon, M.S., and Small, H.J. (2005) Immobilisation and biofilm development of Phanerochaete chrysosporium on polysulphone and ceramic membrane. J. Membr. Sci. 263, 30
37. Strathmann, H., Giorno, L., and Drioli, E. (2006) An Introduction to Membrane Science and Technology. Institute on Membrane Technology, Italy. Yang, W., Cicek, N., and Ilg, J. (2006) State-of-the-art of membrane bioreactors: Worldwide research and commercial applications in North America. J. Membr. Sci. 2006 (270), 201
211. Wang, Q., and Zhang, T. (2010) Review of mathematical models for biofilm. Solid State Commun. 150, 1009
1022. Waterland, L.R., Michaels, A.S., and Robertson, C.R. (1974) A theoretical model for enzymatic catalysis using asymmetric hollow-fiber bioreactors. AIChE J. 20, 50
59.

10 Nanofiltration 10.1

Introduction

Nanofiltration is a pressure-driven membrane process that lies between ultrafiltration and reverse osmosis in terms of its ability to reject molecular or ionic species. Nanofiltration membranes, organic membranes, or ceramic membranes can be either dense or porous. Nanofiltration membranes may have a larger free space, small pores, or nanovoids (Baker, 2004; Van der Bruggen, 2009; Crespo, 2010) available for transport. The size of these nanovoids forms a transition between microporous and dense membranes that can be in the range of 0.5
1 nm (Baker, 2004). Usually it is considered that nanofiltration membranes may exhibit nominal cutoff between 1,000 and 200 Da. In comparison with ultrafiltration and reverse osmosis, nanofiltration has always been a difficult process to define and describe (Van der Bruggen et al., 2008; Geens, et al., 2006). The specific features of nanofiltration membranes are mainly the combination of very high rejections for multivalent ions (.99%) with low to moderate rejections for monovalent ions (, about 5 70%), and high rejection (.90%) of organic compounds with molecular weight above the molecular weight of membrane. The mechanism of the mass transport depends strongly on the membrane structure, on the interactions between the membrane and transported molecules. The separation efficiency can be governed by the sieving effect (when the size of the nanopores and that of the solute molecules have the main effect) or by the solution and diffusion properties of the solute molecules. In the case of charged molecules, the electrical field has a determined role in the transport. Three parameters are crucial for operation of a nanofiltration unit: solvent permeability or flux through the membrane, rejection of solutes, and yield or recovery. Let us look at the most important expressions in order to be able to predict the separation efficiency during nanofiltration.

10.2

Transport of Uncharged Solutes in Aqueous Solution

Most nanofiltration membranes are hydrophilic, thus they are used for charged molecules or uncharged (mostly organic) compounds from aqueous solution. The separation mostly occurs by a sieving effect where the solute’s molecule size should be larger than that of the pore size in the nanofiltration membrane. It can be assumed that there is no essential interaction between the solute molecules and membrane molecules. As a pressure-driven process, the known Hagen
Poiseuille equation Basic Equations of the Mass Transport through a Membrane Layer. DOI: 10.1016/B978-0-12-416025-5.00010-7 © 2012 Elsevier Inc. All rights reserved.

250

Basic Equations of the Mass Transport through a Membrane Layer

[Eq. (1.35)] is recommended for the solvent transport, though strictly considered, this equation is valid for porous membranes (Brian, 1966; Bowen and Welfoot, 2002; Baker, 2004; Nagy et al., 2011): υo 5

rp2 ðΔp 2 ΔπÞε rp2 ðΔp 2 ΔπÞ 1  N 8μδτ 8μo

N5

ψδτ ε

ð10:1Þ

with

where solvent viscosity within pores, μ (Pa s), denotes its real value that can be much higher, as will be seen later, in the narrow pores of the membrane than its bulk value, the rp is the pore radius; μo is the bulk solvent viscosity; υo is the volume flow rate related to the total membrane interface (m/s); Δπ is the osmotic pressure (Pa); δ is the membrane thickness (m); and N involves the membrane structures properties as the product of ψ is the ratio of pore to bulk viscosity (this value can be predicted by the value of λ), membrane thickness, δ, porosity, ε, and tortuosity, τ, i.e., N 5 δτψ/ε. During filtration processes, the uncharged solute particles (macromolecules, submicron- or nanometer-sized particles, colloids, etc.), their size, and the steric (sieving) effects may have essential influence on the separation efficiency (Deen, 1987; Bowen and Mohammad, 1998; Bowen and Welfoot, 2002; Kova´cs and Samhaber, 2008). Due to the steric effects, a hindered diffusion and convection can take place in the membrane matrix. Accordingly, the specific overall mass transfer rate, through a membrane matrix, can be defined in relation to the total membrane interface, as follows (Bowen and Welfoot, 2002): dci dy

ð10:2Þ

Di Vi 8μ RT rp2

ð10:3Þ

Ji 5 αi υo ci 2 Di where αi 5 ξc;i 2 and Di 5 Do;i ξd;i where ψ5

μ μo

ε ψτ

ð10:4Þ

Nanofiltration

251

where ξd,i and ξc,i are called hindrance factors for diffusion and convection, respectively; Vi is the partial molar volume of solute (m3/kmol); Do,i and Di are solute diffusivity in bulk solvent and in the pores of the membrane, respectively (m2/s); rp is the pore radius (m); and μ and μo are solvent viscosity within pores and the bulk solvent, respectively (Pa s). The Di diffusion coefficient contains both the membrane’s porosity, ε (the free membrane area is assumed to be equal to the holdup of free membrane’s volume), and tortuosity, τ. The parameters ξ d,i and ξc,i are related to the hydrodynamic coefficient and the lag coefficient of the spherical solute moving inside a cylindrical pore of infinite length (Deen, 1987; Bowen and Welfoot, 2002; Geraldes and Alves, 2008). Deen (1987) reviewed various equations that were used to predict the value of the hindrance factors. The hindered nature of diffusion and convection within the membrane is accounted for using the hindrance factors, ξd,i and ξ c,i: ^ i gð11 0:054λi 2 0:988λ2 1 0:44λ3 Þ ξc;i 5 f2 2 Φ i i

ð10:5Þ

ξd;i 5 12 2:3λi 1 1:154λ2i 1 0:224λ3i

ð10:6Þ

ψ 5 11 18d=rp 2 9ðd=rp Þ2

ð10:7Þ

with λi 5

rs;i rp

and

^ i 5 ½1 2 λi 2 Φ

ð10:8Þ

where rs is radius of solute molecule and rp is effective pore radius. For larger particles such as proteins, λi $ 0.95, diffusion hindrance factor ξd,i is as (Mavrovouniotis and Brenner,1988): ξd;i 5 0:984

  1 2 λi 5=2 λi

ð10:9Þ

For convection, the hindrance factor for larger molecules, λi $ 0.95 is (Ennis et al., 1996): ξ c;i 5

1 1 3:867λi 2 1:907λ2i 2 0:834λ3i 11 1:867λi 2 0:741λ2i

ð10:10Þ

The second additional term of Eq. (10.3) involves the effect of pressure on the chemical potential (Bowen and Welfoot, 2002). At relatively low pressures, this term can be neglected; as it is mostly the case for nanofiltration, and accordingly α 5 ξc. The υo convective velocity is also related to the whole membrane interface (υo 5 υε, where ε denotes the porosity; its value is assumed to be equal to the

252

Basic Equations of the Mass Transport through a Membrane Layer

interface porosity of the membrane, while υ gives the real solvent velocity through the pores). The rs solute radius can be predicted by the Stokes equation as 2rs 5

kT 6πμDs

where k is the Boltzmann constant (J/mol K); T is the temperature (K); μ is the dynamic (absolute) viscosity (Pa s or kg/m s); and Ds is the solute diffusion coefficient (m2/s). The Stokes is an indirect parameter to describe the molecular size. Boundary conditions for differential equation (10.2) are as (for harmony with the literature, the subscript of the outlet concentration is p in this chapter): cy 5 0 5 c0i

and

cy 5 δ 5 cp;i

ð10:11Þ

Integration of Eq. (10.2) with boundary conditions (10.11) can obtain (Bowen and Welfoot, 2002): ^ i αi ePei cp;i Φ 5 ^ i αi 2 1 1 ePei coi Φ

ð10:12Þ

with Pei 5

υo δαi υo  αi Di βi

ð10:13Þ

The rejection coefficient can be given as Ri 5 12

^ i αi ePei cp;i Φ 5 12 ^ i αi 2 1 1 ePei coi Φ

ð10:14Þ

According to Eqs (10.1), (10.4), and (10.13), further expressions can be obtained for Pei and β i (Nagy et al., 2011). Substituting the value of υo from Eq. (10.1) and the Di value from Eq. (10.4) into the expression (10.13), the Peclet number of the membrane can also be expressed as follows: Pei 5

rp2 ðΔP 2 ΔπÞ αi 8Do;i μo ξ d;i

ð10:15Þ

Note that regarding the membrane structure, Eq. (10.15) does not involve the parameter δτψ/ε but the pore radius only. Accordingly, if the pore radius is known,

Nanofiltration

253

the diffusion membrane mass transfer coefficient can be predicted by means of Pei, namely, according to Eq. (10.13), one can get βi 5

υo αi Pei

ð10:16Þ

or applying Eq. (10.4), one can obtain β i 5 Do;i ξ d;i

ε 1  Do;i ξd;i τδψ N

ð10:17Þ

Equations (10.16) and (10.17) should give the same results, thus parameter N (N 5 δτψ/ε) can be calculated, for example, by the following equation, as well [similarly, this parameter can easily be predicted by Eq. (10.1) as well] (Nagy et al., 2011): N5

Do;i ξd;i Pei υo αi

ð10:18Þ

Note that the permeate concentration, cp,i, and the rejection coefficient defined by Eqs (10.12) and (10.14), respectively, are given without taking into account the possible effect of the polarization layer. Most of the models applied neglect the effect of the concentration polarization layer, assuming that the boundary layer’s Peclet number is close to zero, and, consequently, it is negligible (Bowen and Mohammad, 1998; Bowen and Welfoot, 2002; Kova´cs and Samhaber, 2008). The question arises under which operating conditions the role of the polarization layer should be taken into account. To discover this, the simultaneous effect of the two layers should be analyzed. Gradually improving the diffusive mass transfer coefficient of the membrane layer by the industry, preparing a thinner separation layer with larger porosity, the effect of the boundary layer on the separation efficiency can also be gradually increased. Nagy et al. (2011) and Nagy (2011) analyzed the role of the polarization layer under different operating conditions. They proved that the effect of the polarization layer on the separation efficiency can be neglected under certain conditions only. For it, the mass transfer resistance should be much less than that of the membrane layer.

10.3

Two-Layer Mass Transport: Coupled Effect of the Polarization and Membrane Layers (Nagy et al., 2011)

The physical model of the mass transport through the boundary layer and through the asymmetric membrane layer is illustrated in Figure 10.1. The mass transfer resistance of the support layer is neglected, as seen in this figure. In the presence of the mass transfer resistance in the boundary layer, one can get a convex

254

Basic Equations of the Mass Transport through a Membrane Layer

Polarization layer

Selective layer

Bulk feed

Figure 10.1 Concentration distribution of the concentration boundary layer and of the asymmetric membrane layer Porous and important notations support for nanofiltration. layer

Cm φ* Cb

*

Cp

φp

J

J

δ

δf

concentration curve in it (if cp , co), while a concave concentration curve can exist in the membrane layer if cp . co as it plotted in Fig. 11.1 (Baker et al., 1997; Bhattacharya and Hwang, 1997; De and Bhattacharya, 1997; Kim and Hoek, 2005). The sum of the diffusive and the convective mass transport, denoted by J, related to the total membrane interface, can be given by the well-known expression, e.g., for the concentration polarization layer, as follows (Brian, 1966; Bowen and Welfoot, 2002): Ji 5 υo ci 2 Do;i

dci dy

ð10:19Þ

During filtration processes, the uncharged solute particles (macromolecules, submicron- or nanometer-sized particles, colloids, etc.), their size, the steric (sieving) effects may have essential influence on the separation efficiency. Due to the steric effects, a hindered diffusion and convection can take place in the membrane matrix. Accordingly, the specific, overall mass transfer rate, through a membrane matrix, can be defined, in relation to the total membrane interface, as follows: Ji 5 αi υo ci 2 Di

dci dy

ð10:2Þ

For the sake of a simpler and general mathematical treatment, let us differentiate Eqs (10.19) and (10.2), thus one can obtain the differential mass balance equation

Nanofiltration

255

for both the concentration boundary layer and in the pores of the membrane layer, respectively, as follows: υo

dci d2 c i 2 Do;i 2 5 0 dy dy

ð10:20Þ

dci d 2 ci 2 Di 2 5 0 dy dy

ð10:21Þ

and αi υo

It is important to note that the solution of Eqs (10.20) and (10.21) should give the same results as those of Eqs (10.19) and (10.2). The integration of Eqs (10.20) and (10.21), the concentration distribution of the boundary layer and the membrane layer can be given (applying dimensionless space coordinate, Y 5 y/δf) as follows (Baker, 2004; Nagy, 2009, 2010; Nagy and Kulcsa´r, 2009): For the boundary layer: ci 5 Ωf ePef;i Y 1 Ψf

ð10:22Þ

For the membrane layer: φi 5 ΩePei δf Y=δ 1 Ψ

ð10:23Þ

where Pef;i 5

υo δ f υo  ; Do;i β f;i

Pei 5

υo δαi υo  αi Di βi

The boundary conditions to determine the values of parameters Ωf, Ψf, Ω, and Ψ in Eqs (10.22) and (10.23) are as follows (Nagy, 2009, 2010; Nagy and Kulcsa´r, 2009): Ωf 1 Ψf 5 coi ; Ψf 5 αi Ω;

Y 50

ð10:24Þ

Y 51


^ Ωf ePef;i 1 Ψf 5 ΩePei δf =δ 1 Ψ; Φ αi υo Ψ 5 υo cp;i ;

Y 5 11 δ=δf

ð10:25Þ Y 51

ð10:26Þ ð10:27Þ

Equation (10.24) defines the equality of concentration on the external interface, namely between the bulk liquid phase and the concentration polarization layer.

256

Basic Equations of the Mass Transport through a Membrane Layer

Equation (10.25) defines equality of mass transfer rates on the two sides of the ^ Y 5 21 5 cY 5 11. membrane interface, while Eq. (10.26) gives the expression of Φc The equality of the mass transfer rates, given in Eq. (10.25), according to Eqs (10.2) and (10.19), can easily be obtained from Eqs (10.22) and (10.23), respectively. The boundary condition by Eq. (10.27) expresses that the sum of the diffusive and the convective flows in the outlet membrane phase, αi υo Ψ; is equal to the well-known permeation rate, namely toυocp. This permeation rate, however, will only be valid if there is no sweep phase on the permeate side of the membrane. In this case, the liquid mass transfer coefficient on the permeate side should also be taken into account. This latter operating mode is not used during nanofiltration. An important question is how the main parameters of a nanofiltration membrane can be evaluated in order to predict its performance for uncharged solute. The following data of the transported molecule should be taken from the literature: diffusion coefficient in the solvent, Do,i, and molecule size, rS,i. Regarding the membrane, one should measure the fluxes versus the pressure drop of the pure solvent, in most cases water. Additionally, the mass transfer coefficient of the polarization layer also should be predicted. Now, if the pore size, rp, is known, the rejection coefficient, R, can be calculated by Eqs (10.15) and (10.35). For this, the β i value can be predicted by Eq. (10.16). If the rp value is not known, then it can be fitted to the measured rejection coefficient. In this case, only one parameter, namely the rp value, should be fitted. The value of N 5 ψδτ=ε can be calculated by Eq. (10.17) or (10.18). Obviously, the separate determination of δ, ε, and τ values needs additional measurements.

10.3.1 Determination of the Concentration Distribution Values of parameters Ωf, Ψf, Ω, and Ψ can be gotten relatively easily from the algebraic equation system of Eqs (10.24)
(10.27). Thus, the Ω value can be given as follows: Ω5

^ i cp 2 cp =αi ^ i ðco 2 cp Þ ePef;i 1 Φ Φ ePei δf =δ

ð10:28Þ

Let us express the Ψ value by means of Eq. (10.27), in order to get the overall mass transfer rate J (Ji 5 αiυoΨ) Ω5

cp αi

ð10:29Þ

From Eqs (10.24) and (10.25), the values of Ψf and Ωf can be obtained as Ψf 5 c p

ð10:30Þ

Nanofiltration

257

and Ωf 5 co 2 cp

ð10:31Þ

10.3.2 Determination of the Permeate Concentration, cp With the knowledge of the values of Ω and Ψ, the membrane concentration on the permeate side (at Y 5 11 δ/δf), φδ can be given from Eq. (10.23) as φδ;i

 
ð1 2 ePei Þ o Pef;i 1 Pei Pef;i Pei ^ ^ e 1 5 Φi ci e 1 cp;i Φi 12 e αi

ð10:32Þ

The boundary condition of Eq. (10.27) does not involve the φp;i value; it expresses the mass transfer rates on the two sides of the outlet membrane interface. Equation (10.32) gives a link between the permeate concentration and the membrane concentration on the downstream side. It is important to determine the permeate concentration, cp, directly from the mass transport data of the boundary layer and of the membrane layer. For this, the link between the two concentrations must be given. We assume ^ i cp ; at Y 5 1 1 δ/δf, is correct. This means that there is no concentration that φp;i 5 Φ jump on the outlet membrane interface. Accepting this equation, now the permeate concentration can be given as cp;i 5

^ i αi co ePef;i 1 Pei Φ i ^ i αi 2 1Þ 1 Φ ^ i αi ePef;i 1 Pei ð12 ePei ÞðΦ

ð10:33Þ

According to Eq. (10.33), the value of cp will tend to unit, if Φ-1 and αi-1. This means that there is no any separation in this case, cp 5 coi ; independent of the diffusion coefficient in the membrane matrix. It is easy to see from Eq. (10.33) that the cp;i =coi value will tend to the literature expression, if Pef,i-0, i.e., the effect of the polarization layer is negligible (Bowen and Welfoot, 2002). If Pef;i then

^ i αi ePei cp;i Φ 5 ^ i αi 211 ePei coi Φ

ð10:12Þ

Taking into account that the convective velocity and the value of υo are equal to each other in both layers, thus Pef,i 5 Peiβ i/{β f,iαi}, Eq. (10.33) can be rewritten as cp;i 5

^ i αi co ePeo ð11 βi =½β f;i αi Þ Φ i Pe ^ ^ i αi ePei ð11 βi =½βf;i αi Þ i ð12 e ÞðΦi αi 21Þ 1 Φ

ð10:34Þ

258

Basic Equations of the Mass Transport through a Membrane Layer

The rejection coefficient is easy to get from Eq. (10.34) as follows: Ri  12

^ i αi 2 1Þð12 ePei Þ cp;i ðΦ 5 o ^ i αi 21Þð12 ePei Þ 1 Φ ^ i αi ePei ð11 β i =½βf;i αi Þ ci ðΦ

ð10:35Þ

The polarization modulus, ci =coi ; is an important factor; it can strongly influence the permeate concentration. Thus, its value can be defined, by means of Eqs (10.28)
(10.31), as well (note that cm is generally applied instead of c , in the literature): ^ i αi ePei ð11 βi =½βf;i αi Þ ci Φ 5 ePef;i 1 ð12 ePef;i Þ o ^ i αi 21Þ 1 Φ ^ i αi ePei ð11 β i =½β f;i αi Þ ci ð12 ePei ÞðΦ

ð10:36Þ

10.3.3 Solution by Means of Mass Transfer Rates of Layers It is important to note that the cp,i value also can be obtained by separate calculation of parameters of Eqs (10.22) and (10.23) by common boundary conditions, namely at y 5 0 then c 5 coi and y 5 δf then c 5 ci for the polarization layer as well ^ i c 5 φ Þ: as y 5 0 then φi 5 φi and y 5 δ then φi 5 φδ;i for the membrane layer ðΦ i i By these boundary conditions, every parameter can be expressed (Nagy, 2011). Now one can determine the mass transfer rates entering the layers, namely: For the polarization layer: J i 5 Ψf υ o 5 υ o

ci 2 coi ePef ;i 12 ePef ;i

ð10:37Þ

For the membrane layer: Ji 5 αi υo Ψ 5 αi υo

φδ;i 2 φi ePei 12 ePei

ð10:38Þ

Taking also into account Ji 5 υo cp;i

ð10:39Þ

the value of cp,i and the rejection coefficient, Ri, can be expressed using the above three equations. Obviously, one gets the same equations for the value of cp,i and Ri as the Eqs (10.33) and (10.35), respectively. Nagy et al. (2011) have discussed in detail the effect of the polarization layer under different conditions, as well as compared the measured and experimental ^ 5 0:14Þ: data evaluating them with and without polarization layer ðΦ A typical figure illustrates the value of the rejection coefficient as a function ^ i can be of the membrane’s Peclet number in Figure 10.2. The value of αi and Φ

Nanofiltration

259

Rejection coefficient, R

1.0 0.001

0.8

0.1 0.6

0.25

0.4 0.2

β/β f = 0.5 α i = 1.39 Φ = 0.14

0.0 1.0E–01

1.0E+00

1.0E+01

Membrane’s peclet number, Pe

^ 5 0:14Þ: Figure 10.2 Typical curves of rejection as a function of Pe (α 5 1.39; Φ

regarded as typical values, considering for example the separation of glycerol, glucose, lactose, or ribose (Nagy et al., 2011). With the increase of the permeation rate, namely of the Pe, the R value tends to a maximum value depending strongly on the β/β f ratio. If the external mass transfer resistance is negligible, namely β/β f-0, the R rejection coefficient has limiting value. It clearly shows the strong effect of the polarization layer with increasing value of β/β f. The membrane’s mass transfer coefficient, β, should be as low as possible from industrial points of view. For this purpose, the industry produces better and better membranes regarding their mass transfer resistance. Accordingly, a relatively high value of β/β f can occur easily during nanofiltration processes. Nagy et al. (2011) discussed the effect of the polarization layer, with its relatively low mass transfer resistance, for separation of dilute aqueous solutions of glycerol, ribose, lactose, and glucose, applying experimental data. For illustration, the effect of the liquid side mass transfer coefficient is shown in Figure 10.3, on the rejection coefficient. The liquid film thickness was varied between 2.5 μm (β f 5 2.08 3 1024 m/s) and 30 μm (β f 5 1.73 3 1025 m/s). The rejection coefficient increases in the whole range of β f value investigated, though in decreasing measure. It is true that the effect of the concentration polarization layer often can be neglected (e.g., in cases of MAT 30 and NTR 7410 between β f 5 9 3 1025 and 20.8 3 1025 m/s), while the change of R value is relatively high for membrane MPT 36 in the above range. The effect of the permeation rate strongly affects the rejection. A larger transmembrane pressure difference causes larger convective velocity and, thus decreasing rejection depending on the mass transfer resistance of the boundary layer (Figure 10.4). The data used for the calculation were taken from the paper of Bowen and Mohammad (1998) who investigated, among other components, the separation of lactose (for details, see Nagy, 2011).

260

Basic Equations of the Mass Transport through a Membrane Layer

1.0

MAT 30

Rejection, R

0.8

MPT 36

0.6 NTR 7410

0.4 0.2 0.0

1.0

10.0

Figure 10.3 Applying the model presented at different values of liquid side mass transfer coefficient for calculation of R value of sucrose using membranes as MPT 30, MPT 36 (Koch membrane), NTR 7450, NTR 7410 (Nitto
Denko membrane) (Do 5 0.52 3 1029 m2/s; rS 5 0.471 nm) (for details about membranes, see Bowen 100.0 and Mohammad, 1998).

Mass transfer coefficient, βf (10–5 m/s)

1.0

Rejection, R,-

0.8 0.6 0.4 0.2 0.0

Figure 10.4 Nanofiltration of sucrose through MPT 36 membrane βf = 1.1⫻ 10–4 m/s (Bowen and Mohammad, 0.5 ⫻ 10–4 1998) in presence of increasing mass transfer resistance 0.25 ⫻ 10–4 of the boundary layer [υo 5 55.6 3 1026 m/s at ΔP 5 3.03 MPa; Sucrose rp 5 0.69 nm (rs is the D = 0.52 ⫻ 10–9 m2/s pore radius); Rmeasured 5 0.7; 20.0 40.0 60.0 rs 5 0.49 nm; rs is the molecule size of sucrose]. –6 βf

0.0

∞

Permeate flux of water, νo (10 m/s)

10.4

Solvent-Resistant Nanofiltration

Nanofiltration recently emerged from its traditional application area with the development of solvent-resistant nanofiltration or, most often, organophilic nanofiltration (Peeva et al., 2004; Braeken et al., 2006; Dijkstra et al., 2006; Darvishmanesh et al., 2009). The essential difference, in this case from the simple filtration, is that there is strong interaction between solvents and membranes that should be taken into account for description of the mass transport. Swelling of the polymer network due to the sorption of the solvent, the surface energy varying in different solvents, the diffusivity varying the swelling’s degree should be taken into account (Braeken et al., 2006; Darvishmanesh et al., 2009). Retention of organic components is determined by different membrane properties, such as molecular weight cutoff, membrane charge, and so forth, and component properties, such as molecular weight, hydrophobicity, and so on (Van der Bruggen et al., 1999).

Nanofiltration

261

For relatively porous nanofiltration membranes, simple pore-flow models on convective flow can be adapted to incorporate the influence of the solvent or solute properties. The Hagen
Poiseuille model, which is commonly used for aqueous systems permeating through porous media, takes no interaction parameters into account, and the viscosity as the only solvent parameter. This equation will be insufficient to describe the performance of solvent-resistant nanofiltration membranes (Van der Bruggen, 2009). Machado et al. (2000) developed a resistance-in-series model based on convective transport of the solvent for the permeation of pure solvent and solvent mixtures Δp
 J5  φ γ c 2 γ s 1 f1 η 1 f2 η

ð10:40Þ

where f1 and f2 are solvent independent parameters characterizing the nanofiltration (selective skin layer, m/s) and ultrafiltration (porous support layer, 1/m) sublayers; φ is a solvent parameter (s/m2); γ c is the critical surface tension of the membrane material; γ s is the surface tension of the solvent (mN/m); η is the dynamic viscosity of the solvent (Pa s); Δp is transmembrane pressure (Pa); and J is the solvent convective velocity (m/s). This model is also based on the dependence of the flux on two parameters, namely the solvent viscosity and the difference in surface tension between the solid membrane material and the liquid solvent. This model seems to be inadequate for the description of fluxes through hydrophilic membranes (Yang et al., 2001). Polymeric membranes with less porous structure can be described by a solutiondiffusion mechanism, taking into account also the convective transport according to the transmembrane pressure difference as (Bhanushali et al., 2001):    Vm 1 J~ η φn γ sv

ð10:41Þ

where Vm is the molar volume of membrane (m3/kmol); η is the dynamic viscosity of solvent; γ sv is solid
vapor surface tension; φ is a measure for membrane
solvent interaction; n is an exponent used as an empirical constant to obtain the best possible fit for the above equation. An alternative equation is given as (Geens et al., 2006): J~

Vm ηΔγ

ð10:42Þ

where Δγ is the difference in surface tension (mN/m) and η and Vm are as in Eqs (10.41). The solution-diffusion model is recommended often for description of the mass transport (Peeva et al., 2004; Braeken et al., 2006; Darvishmanesh et al., 2009). This states that solute/solvent diffuse across the membrane under chemical potential gradient. The flux can be given in this case by Eq. (1.5). Assuming incompressibility,

262

Basic Equations of the Mass Transport through a Membrane Layer

it follows from the equality of the interface’s chemical potentials that (see Chapter 1), for the feed side o sat μoi 1 RT d lnðγ f;i xf;i Þ 1 υi ðpi 2 psat i Þ 5 μi 1 RT d lnðγ m;i xm;i Þ 1 υi ðpi 2 pi Þ

ð10:43Þ where pi and pm,i are the component’s pressure (Pa). From Eq. (10.4), it can be obtained (Baker, 2004; Peeva et al., 2004) as xm;i 5

γ f;i xf;i  Ki xf;i γ m;i

ð10:44Þ

A similar equation can be given for the permeate side: γ fp;i xmp;i 5 xfp;i  Kp;i xfp;i γ mp;i

10.5

ð10:45Þ

Spiegler
Kedem Transport Model

Transport of solutes through nanofiltration membranes can be described by the equations of Spiegler and Kedem, which combine both diffusive and convective effects (Li et al., 2008; Van der Bruggen et al., 2008; Peng et al., 2010): Jv 5 LðΔP 2 σΔπÞ e:g:; Jv 5 Js 5 P s δ

dc 1 ð1 2 σÞJv c dy

εr 2 ΔP 8ητ Δy

ð10:46aÞ ð10:46bÞ

where Js is the solute flux (kg/m2 s); Jv is the water flux (m3/m2 s); L is the solvent permeability (m/s Pa); δ is the membrane thickness (m); σ denotes the reflexion coefficient (%); c is the concentration (kg/m3); η is viscosity (Pa s); r is the pore radius (m); ε is the membrane porosity; and τ is tortuosity. The transport of diffusion is represented by the first term of Eq. (10.46b). The second term stands for the contribution of convection to the transport of uncharged molecules. The osmotic pressure can be calculated by using the Van’t Hoff equation (Mulder, 1996). The rejection of component i in percent is defined as  cp  Ri 5 1 2 o 100 ð10:47Þ c where cp is the permeate concentration (kmol/m3) and co is the feed concentration (kg/m3).

Nanofiltration

The above equations lead to an expression for the rejection R:   σð12 FÞ 12 σ with F 5 exp 2 Jv R5 12 σF P

263

ð10:48Þ

The Spiegler
Kedem model can be combined by the film theory. Based on film theory and the relevant boundary conditions (c 5 co at y 5 0, c 5 c at y 5 δf, where δf is the liquid boundary layer thickness), a solute mass balance across the membrane gives (Peng et al., 2010):   c 2 cδ Jw ð10:49Þ 5 exp  o c 2 cδ β of where cδ is permeate concentration (kg/m3); c is liquid concentration on the membrane interface (usually denoted by cm; for harmony with the previous notations c is used here, kg/m3); Jw is water convective velocity (m3/m2 s); β of is liquid film mass transfer coefficient β of 5 Df =δf ; (m/s); and δf is film thickness (m). The true rejection coefficient can be given as   c Ro R Jw exp 2 o with Ro 5 12 δo 5 ð10:50Þ 12 Ro βf c 12 R The intrinsic transport properties of a membrane are determined by rearranging Eq. (10.48) to Ro σ 5 ð12 FÞ 12 Ro 12 σ

ð10:51Þ

and then, substituting Eq. (10.51) and F from Eq. (10.48) into Eq. (10.50) to yield      Ro σ Jw ð12 σÞ Jw 12 exp 2 exp 2 o 5 12 σ Ps 12 Ro βf

ð10:52Þ

Equation (10.52) is the working equation for the combined Spiegler
Kedem film theory model. Darvishmanesh et al. (2009) developed a new flux model based upon the solutiondiffusion model and pore-flow model. The membrane is considered as a parallel connection of a matrix having the solution-diffusion mechanism of solvent transport and of pores, the solvent is convectively transported without change of concentration. Viscous transport is taken into account through imperfections in the membrane beside diffusive transport. The resultant transport equations are written as Jv 5 Ld ðΔP 2 σΔπÞ 1 Lv ΔP

ð10:53Þ

Js 5 Pðco 2 cp Þ 1 Li co ΔP

ð10:54Þ

264

Basic Equations of the Mass Transport through a Membrane Layer

where Jv and Js are the volume and solute fluxes, Ld and Lv are the diffusional (the molecules that diffuse through the matrix) and viscous (for those that pass through the pores) permeabilities (m/s Pa); P is the partial diffusional permeability of the matrix (m/s); and co and cp are the solute concentrations in the feed and permeate (kg/m3). Taking into account the surface tension, dielectric constant of a solvent, the final equation has the following format for the water flux (Darvishmanesh et al., 2009): Jv 5

ao α bo ΔP ðΔP 2 ΔπÞ 1 ηexpð12 βÞ ηexpð12 βÞ

ð10:55Þ

where ao and bo are specific diffusivity and permeability values, which are determined during the modeling; β is the ratio of the surface tension of the membranes; α is the nondimensional polarity coefficient; and η is viscosity (Pa s).

10.6

Nanofiltration of Ionic Components

Prediction of solute rejections in nanofiltration of electrolyte solutions requires the modeling of species transport through both the concentration polarization boundary layer and the active layer of the membrane. Considering the unidimensional continuity equation, for the ith ion, in the steady state, the molar flux of the ion i, Ji, through the polarization layer can be given as (Geraldes and Alves, 2008): Ji 5 β of;i ðcoi 2 ci Þ 1 Jv ci 2 zi ci Df;i

F dψ y 5 δf RT dy

ð10:56Þ

where Ji is solute flux of component i (kmol/m2 s); β of is the mass transfer coefficient of the polarization boundary layer (m/s); Jv is permeate volume flux (m3/m2 s); zi is charge number; F is Faraday constant (9.64867 3 104 C/eq); R is ideal gas constant (8.314 J/kmol); ψ is electric potential (V); Df,i is the diffusivity of the species i in water phase (m2/s); ci is solvent concentration at the feed membrane interface (kmol/m3); δf is thickness of the boundary layer (m); and y 5 δf means the membrane surface. The symbol β of;i is the mass transfer coefficient of the ion i for high mass transfer rates at permeable walls (Bird et al., 1960). The ion mass transfer coefficient can differ from the conventional one. This difference can be taken into account by a correction factor (Geraldes and Alves, 2008). The extended Nerst
Plank equation for the membrane is as (Geraldes and Brites, 2008): Ji 2 Di;p

zi φi Di;p dψ dφ F 1 Ki;c Jv 2 RT dy dy

ð10:57Þ

Nanofiltration

265

where Di,p is the hindered diffusion coefficient (m2/s) and Ki,c is the hindrance coefficient for convection: Di;p 5 Ki;d Di;N

ð10:58Þ

Hindrance factors, Ki,d and Ki,c, are functions of λi and are related to hydrodynamic coefficient such as the enhanced drag and lag coefficient (detailed analysis of hindrance factors are given in Chapter 2).

References Baker, R.W. (2004) Membrane Technology and Applications, 2nd ed. John Wiley and Sons, Chichester. Baker, R.W., Wijmans, J.G., Athayde, A.L., Daniels, R., Ly, J.H., and Le, M. (1997) The effect of concentration polarization on the separation of volatile organic compounds from water by pervaporation. J. Membr. Sci. 137, 159
172. Bhanushali, D., Kloos, S., Kurth, C., and Bhattacharyya, D. (2001) Performance of solventresistant membranes for non-aqueous systems: solvent permeation results and modeling. J. Membr. Sci. 189, 1
21. Bhattacharya, S., and Hwang, S.-T. (1997) Concentration polarization, separation factor, and Peclet number in membrane processes. J. Membr. Sci. 173, 73
90. Bird, R.B., Stewart, W.E., and Lightfoot, E.N. (1960). Transport Phenomena, John Wiley and Sons, New York. Bowen, W.R., and Mohammad, A.W. (1998) Characterization and prediction of nanofiltration membrane performance—a general assessment. Trans IChemE 76 (Part A), 885
893. Bowen, W.R., and Welfoot, J.S. (2002) Modelling the performance of membrane nanofiltration —critical assessment and model development. Chem. Eng. Sci. 57, 1121
1137. Braeken, L., Bettens, B., Boussu, K., Van der Meeren, P., Cocquyt, J., and Vermant, J., et al., (2006) Transport mechanism of dissolved organic compounds in aqueous solution during nanofiltration. J. Membr. Sci. 279, 311
319. Brian, P.L.T. (1966) Mass transport in reverse osmosis, in Desalination by Reverse Osmosis, U., Merten Ed. MIT Press, Cambridge, p. 181. Crespo, J. (2010) Development in Membrane Science for Downstream Processing, in Membrane Operations, Ed. by, E., Drioli, and L., Giorno. Wiley-VCH, Weinheim, pp. 245
263. Darvishmanesh, S., Buekenhoudt, A., Degre´ve, J., and Van der Bruggen, B. (2009) General model for prediction of solvent permeation through organic and inorganic solvent resistant nanofiltration membranes. J. Membr. Sci. 334, 43
49. De, S., and Bhattacharya, P.K. (1997) Modeling of ultrafiltration process for a two-component aqueous solution of low and high (gel-forming) molecular weight solutes. J. Membr. Sci. 136, 57
69. Deen, W.M. (1987) Hindered transport of large molecules in liquid-filled pores. AIChE J. 33, 1409
1425. Dijkstra, M.F.J., Bach, S., and Ebert, K. (2006) A transport model for organophilic nanofiltration. J. Membr. Sci. 286, 60
68.

266

Basic Equations of the Mass Transport through a Membrane Layer

Ennis, J., Zhang, H., Stevens, G., Perera, J., Scales, P., and Carnie, S. (1996) Mobility of protein through a porous membrane. J. Membr. Sci. 119, 47
58. Geens, J., Boussu, K., Vandecasteele, C., and Van der Bruggen, D. (2006) Modeling of solute transport in non-aqueous nanofiltration. J. Membr. Sci. 281, 139
148. Geraldes, V., and Alves, A.M.B. (2008) Computer program for simulation of mass transport in nanofiltration membranes. J. Membr. Sci. 321, 172
182. Kim, S., and Hoek, E.M.V. (2005) Modeling concentration polarization in reverse osmosis processes. Desalination 186, 11
128. Li, N., Fane, A.G., Ho, W.S.W., and Matsuura, T. (2008) Advanced Membrane Technology and Application. John Wiley and Sons, New York. Kova´cs, Z., and Samhaber, W. (2008) Characterization of nanofiltration membranes with uncharged solutes. Membra´ntechnika (in English) 12 (2), 22
36. Machado, D.R., Hasson, D., and Semiat, R. (2000) Effect of solvent properties on permeate flow through nanofiltration membranes. Part II. Transport model. J. Membr. Sci. 166, 63
69. Mavrovouniotis, G.M., and Brenner, H. (1988) Hindered sedimentation diffusion and dispersion coefficients for Brownian spheres in circular cylindrical pores. J. Colloid. Interface Sci. 124, 269. Mulder, M. (1996) Basic Principles of Membrane Technology, 2nd ed. Kluwer Academic, Dordrecht. Nagy, E. (2009) Basic equations of mass transfer through biocatalytic membrane layer. Asia-Pacific J. Chem. Eng. 4, 270
278. Nagy, E. (2010) Coupled effect of the membrane properties and concentration polarization in pervaporation: unified mass transport model. Sep. Purif. Technol. 73, 194
201. Nagy, E. (2011) Nanofiltration of uncharged solutes: simultaneous effect of the polarization and membrane layers on separation. Desalin. Water Treat. 34, 70
74. Nagy, E., and Kulcsa´r, E. (2009) Mass transport through biocatalytic membrane reactors. Desalination 245, 422
436. Nagy, E., Kulcsa´r, E., and Nagy, A. (2011) Membrane mass transport by nanofiltration: coupled effect of the polarization and membrane layer. J. Membr. Sci. 368, 215
222. Peeva, L.G., Gibbins, E., Luthra, S., Lloyd, S.W., Stateva, R.P., and Livingston, G. (2004) Effect of concentration polarization and osmotic pressure on flux in organic solvent nanofiltration. J. Membr. Sci. 236, 121
136. Peng, F., Huang, X., Jawor, A., and Hoek, E.M.V. (2010) Transport, structural, and interfacial properties of poly (vinyl alcohol)-polysulfone composite nanofiltration membranes. J. Membr. Sci. 353, 169
176. Van der Bruggen, B. (2009) Fundamentals of membrane solvent separation and pervaporation, in Membrane Operations, Ed. by, E., Drioli, and L., Giorno. Wiley-VCH, Weinheim, pp. 45
62. Van der Bruggen, B., Schaep, J., Wilms, D., and Vandecasteele, C. (1999) Influence of molecular size, polarity and charge on the retention of organic molecules by nanofiltration. J. Membr. Sci. 156, 29
41. Van der Bruggen, B., Manttari, M., and Nystro¨m, M. (2008) Drawbacks of applying nanofiltration and how to avoid them: a review. Sep. Purif. Technol. 63, 251
263. Yang, X.J., Livingston, A.G., and dos Santos, L.F. (2001) Experimental observation of nanofiltration with organic solvents. J. Membr. Sci. 190, 45
55.

11 Pervaporation 11.1

Introduction

During the past few decades, pervaporation has become one of the emerging technologies that has undergone rapid growth. There are several excellent review works discussing the pervaporation process for both its fundamental aspects and practical applications (Karlesson and Tragardh, 1993; Feng and Huang, 1997; Semenova, 1997; Lipnizki et al., 1999; Villaluenga and Tabe-Mohammadi, 2000; Bowen et al., 2004; Smitha et al., 2004; Vane, 2005; Shao and Huang, 2007; Wee et al., 2008; Ravanchi et al., 2009). The main point of this section is to discuss pervaporation from the point of view of mass transport expressions, consequently the process is only briefly discussed. Pervaporation allows separation of mixtures that are difficult to separate by distillation, extraction, adsorption, and absorption. Pervaporation has advantages in the separation of azeotrope mixtures, close-boiling mixtures, thermally sensitive compounds, molecules that are similar in shape or weight, and removing species present in low concentrations (Bowen et al., 2004). During pervaporation, a phase change from liquid to vapor takes place. Processes involving phase changes are generally energy intensive, but pervaporation cleverly survives the challenge of phase change by two features (Shao and Huang, 2007): (1) pervaporation deals only with minor components, and (2) pervaporation uses the most selective membranes. The first feature effectively reduces the energy consumption of the process. The second feature generally allows pervaporation, the most efficient liquid-separating technology. More recently, the hybrid processes (Lipnizki, 1999) integrating pervaporation with other viable liquid-separating technologies and processes seem to be very promising for practical purposes. Pervaporation has three main areas of separation (Shao and Huang, 2007): (1) dehydration of organic solvents (e.g., alcohols, acids, esters, and so on); (2) removal of dilute organic compounds from aqueous mixtures; and (3) organic
organic mixture separation. There are two main types of membrane material, namely polymeric or inorganic membrane. The mass transport equations will be discussed briefly in the next sections.

11.2

Fundamentals of Pervaporation

Solution-diffusion is the generally accepted mechanism of mass transport through nonporous membranes. The basic equations for the solution-diffusion model are Basic Equations of the Mass Transport through a Membrane Layer. DOI: 10.1016/B978-0-12-416025-5.00011-9 © 2012 Elsevier Inc. All rights reserved.

268

Basic Equations of the Mass Transport through a Membrane Layer

given in Section 11.3. There are two phases that affect essentially the mass transfer and the separation efficiency, namely the concentration boundary layer in the feed liquid phase on the feed side of the membrane and the membrane layer itself. Thus, a full description of the mass transport mechanism of the pervaporation systems and thus, to be able to predict the separation efficiency, the discussion should involve mass transfer in both the liquid feed boundary layer and the membrane layer (Baker, 2004). It is interesting to note that almost all of these studies consider the mass transport equations in the boundary layer only, defining its Peclet number (Pef 5 υδf/Df) and its effect on the outlet concentration and on the interface concentration on the feed side of membrane, but they do not discuss the effect of the mass transport through the membrane layer and its back effect on the concentration polarization layer, and consequently, its effect on the membrane separation. The membrane is regarded as a black box; there is inlet and outlet mass transfer in that box, but how its mass transport properties affect the inlet and the outlet mass flows is not discussed in most of these papers. This situation is illustrated by Figure 11.1A, where concentration profiles are not given in the membrane layer. Thus, the transport equations given for the feed side boundary layer do not involve mass transport parameters of the membrane, as diffusion coefficient in the membrane phase, solubility coefficient, membrane thickness, and so forth. The question arises, how can the outlet concentration or the concentration distribution be calculated directly, taking into account the membrane mass transport parameters as well. Van der Bruggen et al. (2004) studied the mass transfer mechanism intensely through dense and nanoporous membranes during pervaporation. They concluded that the diffusion dominates the mass transport through even (A)

Boundary layer

Cp

C*

Bulk feed

(B)

Membrane layer

Boundary layer C*

Co

Membrane Cp φ*

Co Cp

Co

Cp C*

y=0

Bulk feed δf

φδ*

δf + δ

Figure 11.1 Concentration profiles of the concentration boundary layer and of the membrane layer and important notations. (A) The mass transport parameters, thus concentration profile, are unknown in the membrane. (B) Concentrations are illustrated in both layers.

Pervaporation

269

porous membrane due to its swelling, which could eliminate the nanopores. The convection also can affect the mass transfer rate, in special cases, namely when one of the transferred components has a much higher permeation rate, about one order of magnitude or more higher, than the other one, in the case of binary mixture separation (Kamaruddin and Koros, 1997; Cle´men et al., 2004). This latter case will not be discussed here. The sorption-diffusion transport mechanism generally is accepted and mostly used for the description of transport through dense pervaporation membranes. All of these models are based on the Fick’s law for diffusion (Olsson and Tra¨gardh, 2001; Schafer and Grespo, 2007; Delgado et al., 2008; Fouad and Feng, 2008). The essential of this formalism is that the simple diffusive flux (J 5 β oΔC, where ΔC denotes the driving force in the membrane layer, and β o denotes the membrane diffusive mass transfer coefficient, β o 5 D/δ) is given for the membrane layer (Trifunovic and Tra¨gardh, 2006; Schafer and Grespo, 2007), or this diffusion flux is combined with the concentration polarization equation in order to incorporate the contribution of the convective transport in the mass transfer rate (Michels, 1995; Wijmans et al., 1996; Bhattacharya and Hwang, 1997; Baker, 2004; She and Hwang, 2004; Jiraratananon et al., 2008). Another often-applied approach is the resistance-in-series model in that the mass transport is regarded as diffusive one in both the boundary layer and membrane layer (Schafer and Grespo, 2007; Fouad and Feng, 2008). Strictly taken, the resistance-in-series model can be used in case of conductive flows in the layers only. This condition is often not fulfilled in the presence of convective flow in the boundary layer. The concentration-dependent diffusion coefficient, a strong coupling of diffusion of permeating components, can make the process description much more complex (Heintz and Stephan, 1994; Smart et al., 1998; Iza´k et al., 2003; Nagy, 2006). In the case of dilute solutions, the coupling effect and/or concentration dependency of the diffusion can be neglected (Shaetzel et al., 2001), which also was done in this paper. To our knowledge, there is no paper that presents a coupled model of concentration polarization and membrane transport in pervaporation processes. Recently, Nagy and Kulcsar (2009) and Nagy and Borbely (2007) have studied the simultaneous effect of the convective and diffusive mass transport in a single mass transport layer. They showed that the mass transfer rate can be given by J 5 β}o ðco 2 exp½2 Pef c Þ expression (β}o denotes here the so-called convective mass transfer coefficient defined in presence of convective velocity) [see Eq. (11.26) or Eq. (5.7) or (5.10) as well as section 5.3.1.2]. According to this equation, the driving force is not a simple concentration difference in the presence of convective flow in the layer investigated, but it depends on the Peclet number as well. Obviously, the expressions of the two driving forces by convection plus diffusion flows or only by diffusion flow, tends to each other lowering the value of the Pef number. An important question is the real value of the ratio of the diffusive flow and the convective one, exactly stating the measure of the Peclet number. During pervaporation, the value of Pef can be said to be relatively low; it is mostly lower than 0.1. According to Baker (2004) and Baker et al. (1997), Pef is about 1
3 3 1023 during pervaporation of volatile organic components with silicone rubber. Wijmans et al. (1996) predict this value between about 1023 and 1024 for pervaporation of volatile organic components. A coupled

270

Basic Equations of the Mass Transport through a Membrane Layer

model might offer a single means to predict the exact effect of the two layers, namely the concentration polarization layer and the membrane layer on the pervaporation performance. The main aim of this work is to develop general equations of mass transfer rates, of the concentration profiles of the permeated concentration, and so on in closed, explicit forms, which contain also the simultaneous effect of the membrane’s mass transport parameters (diffusion coefficient, solubility, and membrane thickness) and of the boundary layer’s transport parameters as convective and diffusive flows. This general equation should also fulfill the J 5 υCp boundary condition applied for pervaporation. This can enable us to predict directly the Cp outlet concentration of the liquid permeate by means of transport parameters of both layers.

11.3

Solution-Diffusion Model for Pervaporation

Applying the chemical potential gradient as a driving force, the mass transfer rate can be expressed by Fick’s law. The general approach is that the chemical potential of the feed and permeate fluids are in equilibrium with the adjacent membrane surfaces. The chemical potential of an incompressible liquid and membrane can be given as (Wijmans and Baker, 1995; Baker, 2004; Wijmans, 2004): μi 5 μoi 1 RT d lnðγ i xi Þ 1 Vi ðp 2 psat i Þ

ð11:1Þ

For the compressible gases as μi 5 μoi 1 RT d lnðγ i xi Þ 1 RT ln

p psat i

ð11:2Þ

o where psat i is defined as the saturation pressure of component i (Pa), μi is the chemical potential of pure i at a reference pressure; R is the gas constant (0.082 atm m3/kmol K); T is the temperature (K); μ is the chemical potential (atm m3/kmol); p is the pressure (atm); xi is the mole fraction of component i; γ i is the activity coefficient of component i (kmol/kmol); and Vi is the molar volume of component i (m3/kmol). At the liquid solution
membrane feed interface, assuming that the molar volume of i is the same in the liquid and in the membrane interface, Eq. (11.1) gives as

xm;i 5

γ f;i xfi 5 Hf;i xf;i γ m;i

ð11:3Þ

or with xf,i 5 ci/ρf and xm,i 5 φi/ρm, then φi 5

γ f;i ρm ci  Hf;i ci γ m;i ρf

ð11:4Þ

Pervaporation

271

where subscripts m,i and f,i denote the component i in the membrane and liquid phases; Hf,i is the sorption coefficient for sorption between the liquid phase and membrane phase; and ρf and ρm are the molar densities of the membrane with the dissolved component i and that of the liquid (kmol/m3). At the permeate
membrane interface, the pressure drops from po in the membrane permeate interface (note that there is no pressure drop inside the membrane, po is the feed side pressure) to pp in the permeate vapor. The expression of the chemical potential in each phase is then μoi 1 RT d lnðγ G;i xG;i Þ 1 RT ln

pp 5 μoi 1 RT d lnðγ m;i xm;i Þ 1 Vi ðpo 2 psat i Þ psat i ð11:5Þ

Rearranging Eq. (11.5) gives
 γ G;i pp Vi ðpo 2 psat i Þ xG;i exp 2 xm;i 5 γ m;i psat RT i

ð11:6Þ

The exponential term is close to unity (Baker, 2004); thus, the concentration in kg/m3 is as φp;i 5 ρm

γ G;i pi  HG;i pi γ m;i psat i

ð11:7Þ

where HG,i is the gas-phase sorption coefficient. Substituting the concentration terms in Eqs (11.4) and (11.6) into Eq. (11.26), one can get the membrane flux as Ji 5 2Di

Δφi Di 5 ðHf;i coi 2 HG;i pi Þ δ δ

ð11:8Þ

where coi is the feed concentration of i and δ is the membrane thickness (m). It is easy to get the link between coi and pi, namely (Baker, 2004): coi 5

HG;i pi Hf;i

ð11:9Þ

Thus, Ji 5 2Di

Δφi Di Hf;i o Di HG;i o 5 ðci 2 cp;i Þ  ðpi 2 pp Þ δ δ δ

ð11:10Þ

where poi and pp,i are the partial pressures of component i on the either side of membrane (Pa), while coi and cp,i are the concentrations of i on either side of membrane (kg/m3).

272

Basic Equations of the Mass Transport through a Membrane Layer

From Eq. (10) it follows that the Hi Henry coefficient is equal to HG,i/Hf,i. Equation (11.10) gives the mass transfer rate for the ideal case when there is no strong interaction between solvent molecules and the membrane chain molecules. The other important effect on the mass transfer rate can be the boundary layer or the concentration polarization layer of the fluid phase. Depending on the sorption coefficient and the diffusivity, the mass transfer resistance of the fluid phase can strongly affect the separation efficiency of the pervaporation process. In the next section, the basic equation of the boundary layer will be shown.

11.4

Basic Equations of the Polarization Model

Due to the selective mass transport through the membrane material, a concentration gradient can form in the fluid phase boundary layer. If the permeate concentration is less than that of the feed phase (e.g., salt concentration during desalination by reverse osmosis), a convex concentration curve will be formed (Figure 11.1A, continuous line) and when the permeate concentration is higher than that in the feed phase (this is the case for pervaporation) than the concentration of the transported component will be lower at membrane surface (Figure 11.1A, dotted line). At steady state, the sum of the convective and diffusive transport in the boundary layer equals the amount permeated through the membrane (Wijmans et al., 1996; Bhattacharya and Hwang, 1997): J  υc 2 D

dc 5 υcp dy

ð11:11Þ

where υ is convective transverse velocity (m/s) and cp is permeate concentration (kmol/m3). Equation (11.11) is valid for both the increasing (dc/dy . 0) and decreasing concentration (dc/dy , 0) in the concentration boundary layer as they are illustrated in Figure 11.1A and B. After integration of Eq. (11.11) with the boundary conditions (at y 5 0, c 5 co), and replacing the boundary condition at y 5 δf, c 5 c into the solution, one can get (Baker et al., 1997):
 c 2 cp 1=Eo 21 υδf  ePef  5 exp o c 2 cp 1=E 21 Df

ð11:12Þ

where the concentration terms are replaced by an enrichment factor, namely by E defined as cp/co and intrinsic enrichment factor, Eo, obtained in the absence of a boundary layer defined as cp/c . The ratio c /co is a useful measure of the extent of concentration polarization. From Eq. (11.12), this quantity can be written as (Baker, 2004): c E ePef  I  5 co 1 1 Eo ½ePef 21 Eo

ð11:13Þ

Pervaporation

273

Equation (11.13) allows the concentration polarization modulus to be calculated as a function of Pe using given values of the intrinsic enrichment factor. The concentration polarization modulus should be larger than the unit if the permeating compound is depleted in the membrane (Figure 11.1A, dotted line) and less than the unit if the permeating compound is enriched in the permeate (Figure 11.1A, continuous lines) (Olsson and Tra¨gardh, 2001). Typical equation for the boundary layer’s mass transfer coefficient and the rate during pervaporation process can be given by Eq. (11.14), where xo and xp are weight fractions of the inlet and outlet phases (Jiraratananon et al., 2002): J 5 kðco 2 c Þ 

υ ðco 2 c Þ ð1 2 ðxo =xp ÞÞ½ePef 21

ð11:14Þ

The question arises looking at this equation whether the driving force given as concentration difference is correct also in those cases when there is convective mass transport, as well.

11.5

Simultaneous Effect of the Polarization and Membrane Layers

For the sake of a general solution, let us introduce the second-order differential mass balance equations to be solved for both layers as follows (for details of solution and its discussion, see Nagy, 2010): υ

dC d2 C 2 Df 2 5 0 dy dy

ð11:15aÞ

d2 φ 50 dy2

ð11:15bÞ

D

where subscript f denotes the polarization layer (membrane is not subscripted here). Figure 11.1B illustrates the concentration distribution in the membrane matrix due to the diffusive flow in it. The concentration distribution can be convex or concave curves in the boundary layer due to the convective velocity. The case when there is a convective flow in a porous membrane layer (e.g., membrane reactors) was discussed by Nagy and Kulcsar (2009) and Nagy (2009). After integration of Eqs (11.15a) and (11.15b), the concentration distribution of the boundary layer and the membrane layer can be given, respectively, as follows (Nagy and Borbely, 2007; Nagy and Kulcsar, 2009): c 5 Ωf eυy=Df 1 Ψf φ 5 Ωy 1 Ψ;

0 # y # δf

ð11:16Þ

δ # y # δf 1 δ

ð11:17Þ

274

Basic Equations of the Mass Transport through a Membrane Layer

The solution of second order differential equations, Eqs. (11.16) and (11.17), has two parameters, namely Ωf and Ψf and Ω and Ψ for the polarization layer and the membrane layer, respectively. These parameters can be determined by the well-known boundary conditions listed as follows: Ωf 1 Ψf 5 co ;

y50

ð11:18Þ

υΨf 5 2DΩ;

y 5 δf

ð11:19Þ

HðΩf ePef 1 Ψf Þ 5 Ωδf 1 Ψ; Ωðδf 1 δÞ 1 Ψ 5 cp ;

y 5 δf

Y 5 δf 1 δ

ð11:20Þ ð11:21Þ

where Pef 5 υδf/Df and Hc 5 φ (for notations, see Figure 11.1). Equations (11.18) and (11.21) define the equality of concentration on the external interfaces, between bulk liquid phase and boundary layer as well as between the membrane layer and liquid phase on the permeate side, respectively. Eq. (11.20) gives expression of Hc 5 φ while Eq. (11.19) defines equality of the mass transfer rates on two sides of the internal interface. Solving the algebraic system of equations (11.18)
(11.21), by means of well-known mathematical methods, one can get the following values of parameters ΩL, ΨL, Ωm, and Ψm (Nagy, 2010): ΩL 5

co ð12 NÞ 2 φδ =H 12 ePef 2 N

ð11:22aÞ

ΨL 5

co ePef 2 φδ =H ePef 211 N

ð11:22bÞ

Ωm 5

β f Pef co ePef 2 φδ =H D 12 ePef 2 N

ð11:22cÞ

Ψm 5 φδ


 β f Pe δf δ ePef co 2 φδ =H 11 2 β δ δf 12 ePef 1 N

ð11:22dÞ

with N5

β f Pef ; βH

βf 5

Df D ; β5 δf δ

Applying the well-known boundary condition for pervaporation processes, namely that the overall mass transfer rate through the boundary layer is equal to the permeate rate, one gets the following equalities: υΨf  υcp 5 2DΩ

ð11:23Þ

Pervaporation

275

Replacing Eq. (11.22b) into Eq. (11.23), one can express the liquid permeate concentration as follows: cp 5

co ePef 2 φδ =H ePef 211 N

ð11:24Þ

This is a new equation that offers the possibility to predict the permeate concentration as a function of the two-layer mass transport parameters. Thus, this equation involves the characteristics of the membrane layer and its effect on the pervaporation’s efficiency as it was mentioned as the main aim of this chapter. By means of Eqs (11.11), (11.19), and (11.24), the mass transfer rate through the membrane can be given as follows: J 5 2DΩm 5 υcp 5 β f Pef

co ePef 2 φδ =H ePef 211 N

ð11:25Þ

The mass transfer rate can also be given using the resistance-in-series concept. The J value can be obtained, for the boundary layer, by the solution of Eq. (11.15a) obtaining Eq. (11.16) with boundary conditions at y 5 0 (c 5 co) and y 5 δf (c 5 c ), as J 5 β f Pef

co ePef 2 c ePef  β ðco 2 e2Pef c Þ Pe f f ePef 21 ePef 21

ð11:26Þ

Taking into account the mass transfer rate for the membrane layer ½J 5 βðφ 2 φp Þ; as well, the mass transfer rate, applying the overall mass transfer coefficient can be expressed as
 φ J 5 β ov co ePef 2 δ H

ð11:27Þ

1 1 1 5 1 β ov β f} Hβ

ð11:28Þ

with

with β f} 5

β f Pef β f Pef  e2Pef =2 Pe f e 21 2 sinhðPef =2Þ

ð11:29Þ

From Eq. (11.29), it is clear that the β f} =β f value is practically equal to unit if Pe , 0.01 which in equality is almost always fulfilled during pervaporation (Baker, 2004).

276

Basic Equations of the Mass Transport through a Membrane Layer

The enrichment factor of a pervaporation process, E, can easily be obtained as E

cp ePef 2 φδ =ðHco Þ 5 co ePef 21 1 N

ð11:30Þ

The polarization modulus can also be expressed by means of values of ΩL and ΨL [see Eqs (11.22a) and (11.22b)], applying Eq. (11.13) in limiting case, namely at y 5 δ, as follows: I

c NePef 1ðePef 21Þφδ =ðco HÞ 5 co ePef 211 N

ð11:31Þ

The internal enrichment factor, Eo (Eo 5 cp/c ) can be obtained by means of the ratio of Eqs (11.30) and (11.31). A significant difference between Eqs (11.13) and (11.31) is that the polarization modulus can directly be calculated in the latter case, while in the former one should know the intrinsic enrichment factor, Eo, as well, which is not known generally. The value of the outlet membrane concentration, φδ is often also not known. But decreasing the total pressure of the permeate side vapor phase, its value can tend to decrease close to zero. This is often the case during pervaporation. Thus, assuming that φδ 5 0, the E and I or Eo values can be calculated easily. The same is true regarding the permeate rates comparing Eqs (11.14) and (11.25).

11.5.1 Looking for the Connection Between φδ and cp It is obvious that there is strict function between the outlet membrane concentration, φδ ; and the condensed, liquid permeate concentration, cp. In order to obtain it, it will be assumed that: (1) the permeate vapor concentration is in equilibrium to the concentration in membrane interface on the permeate side; and (2) the vapor phase mol fraction of the permeated component, Yi (the subscript i is applied in this chapter, only as it is made often in the literature, in order to indicate that one of the permeated components is chosen for the investigation), is equal to that of the condensed permeate concentration, namely to Xi. The activity of the liquid phase’s component and the fugacity of its vapor phase are equal to each other; thus, the following equality can be defined between the liquid and gas phases (Bowen et al., 2003): pi φi  po Yi f^i 5 psat i γ i Xi

ð11:32Þ

The mol fractions can easily be expressed in the common concentrations in both the vapor and liquid phases, namely poYi 5 cG,iRT and Xi 5 ci,pMi/ρp, respectively (po denotes the total vapor pressure; Mi is mol weight of the permeated component; ρp is density of the condensed permeate; and ci is the concentration in mol/dm3; R is the gas constant; T is the temperature; f^i is the fugacity coefficient; and Yi

Pervaporation

277

and Xi denote the mol fraction of component i in the vapor and liquid phases, respectively). Accordingly, Eq. (11.32) can be rewritten to the following equation: φδ 5 HcG;i 5

psat i γ i Mi H cp;i f^i ρp RT

ð11:33Þ

The value of φδ can be now replaced into Eq. (11.24) and thus, one can obtain the following expression: cp 5 co

ePef ePef 211 N

1ϑ

ð11:34Þ

where ϑ5

γ i Mi psat i f^i ρ RT

ð11:35Þ

p

Note that the ϑ value does not depend on the total or partial pressure of the permeated component investigated. Accepting it, it can be stated that the φδ value can never be zero if cp,i . 0. Thus, the value of ϑ should be taken into account. Its value depends on the properties of the component investigated. For example, the 24 at 333 K. The fugacity value of Mi psat i =ðρp RTÞ for water is equal to 1.295 3 10 coefficient, f^i ; can be regarded to unit at low pressure of the permeate phase. Strongly hydrophobic components can be solved in a limiting amount in water only. Accordingly, the condensed liquid permeate often will be separated into two nonmiscible phases. Equation (11.34) cannot be applied for this case. The following simple equation can be recommended for that case. Accepting that the molar fraction of component in the vapor phase and the liquid permeate should be equal to each other (Yi 5 Xi), one can obtain the connection between φδ and cp,i as follows: φδ 5 HcGi 5

pp M p H cpi ρp RT

ð11:36Þ

where subscript p denotes the liquid permeate. In this case, the cp,i concentration might be a virtual one due to the possibility of the liquid permeate phase separation. According to Eq. (11.36), the ϑ value will be as ϑ5

M p pp ρp RT

ð11:37Þ

The membrane properties are key factors in the efficiency of the separation by pervaporation process. A typical figure (Figure 11.2) illustrates the effect of the

Concentration distribution

278

Basic Equations of the Mass Transport through a Membrane Layer

1.0

δ = 100 μm (N = 0.01, E = 91)

0.8 0.6

10 (0.001; 500.4)

0.4 0.2 0.0

Pef = 0.001 Boundary layer 0.0

Figure 11.2 Concentration distribution as a function of the thickness of the layers, at different values of membrane thickness (φδ 5 0; β f 5 1 3 1025 m2/s; Df 5 1 3 1029 m2/s; D 5 1 3 10212 m2/s; H 5 100).

1 (0.0001; 909.6) Membrane layer

0.4 0.8 1.2 1.6 Boundary layer + membrane

2.0

value of the membrane thickness as one of the most important property (or indirectly that of the parameter value of N) on the concentration distribution in both layers. Besides the N value that involves all important parameters that influence the separation efficiency (N 5 β fPef/βH), the υ/D factor separately affects the concentration distribution. The values of parameters used for this calculation are listed in the caption to Figure 11.2. The δ value was changed between 1 and 100 μm; accordingly the N value was changed between 1024 and 1022, while the Pef number was kept to be 0.001. Note that the normalized value of the thickness of the membrane layer is plotted, that is, the space coordinate in the membrane was multiplied with the value of δf/δ, obtaining the same thickness for the both layers in the figure. Similarly, the concentration within the membrane layer was divided by H for its illustration; thus, there is no concentration jump on the membrane interface. The outlet membrane concentration, φδ ; was chosen to be zero for this calculation. The values of enhancement obtained are also given in Figure 11.2. As can be seen, the concentration distribution is practically linear in both layers. This is due to the low value of the Pef number regarding the boundary layer. By increasing the value of membrane thickness, the values of β decrease, causing a strong change in the value of N. The separation of a membrane layer with given diffusion coefficient, solubility, can be improved strongly by decreasing the value of N. Enrichment is an important parameter for pervaporation; thus, its value change as a function of different parameters, namely Pe number, and intrinsic enrichment has been intensively analyzed in the literature (Baker et al., 1997; Baker, 2004). Enrichment is plotted in Figure 11.3 as a function of parameter N at different values of Pef. The parameter N involves four important variables of the membrane layer and the concentration boundary layer (mass transfer coefficient of layers, Pef, H; interestingly the diffusion coefficient of the boundary layer is missing from this dimensionless parameter because β fPef  υ), which can alter the separation efficiency. The parameter N has a significant effect on the enrichment. The enrichment expression [Eq. (11.30)] can be simplified essentially at low value of Pef number and φδ 5 0; namely

Pervaporation

279

Figure 11.3 The change of the enrichment factor as a function of the N value at different Peclet numbers, φδ 5 0:

1.E+03

Enrichment, E

0.001

0.0001

1.E+02 0.01 1.E+01 Pef = 0.1 1.E+00 1.E–04

E5

1.E–03

1.E–02

11Pef 1 β D  ov Pef 1 N Pef 1 N υ

1.E–01 N

1.E+00

if Pef # 0:01

ð11:38Þ

with 1 1 1 5 1 β ov βf βH where β f and β are defined after Eq. (11.22d). At the end, the ratio of the overall diffusion mass transfer coefficient to the convective velocity could determine the value of enrichment according to Eq. (11.38). Assuming that the condition defined in Eq. (11.38) can fulfill in almost all pervaporation processes, this equation offers a simple means to predict the enrichment. The important limiting value of this equation is: when Pef-0 then E-N. Figure 11.3 shows the effect of both the parameter N and the Peclet number. The film thickness may be between 10 and 50 μm, the membrane thickness about 1 and 100 μm, the Pef number, as it was mentioned, ranges between 0.01 to 0.0001 and the solubility; H may have a relatively large value, about 10
100 or more; the Df value may be about 1 3 1028
1029 m2/s; and while that for the membrane is about 1 3 10211
10212 m2/s. Let us calculate an average value of N (with Df 5 5 3 1029 m2/s, D 5 5 3 10212 m2/s, Pef 5 0.001, δ 5 10 μm, δf 5 10 μm, and H 5 100); one could get that N 5 0.005. The application of Eq. (11.24) makes it possible to choose easily the right membrane in order to reach the desired enrichment. It is important to be able to predict the role of the boundary layer on the separation efficiency. Nagy investigated the concentration distribution in the mass transfer layer in cases of ethanol
water as well as toluene
water binary mixture separations (Nagy, 2010). It was shown that in the case of toluene
water mixtures, the concentration polarization layer can more drastically affect the separation efficiency. Baker et al. (1997) investigated the pervaporation of toluene through silicon rubber with different thickness. These measured data were also used to show the agreement between the measured and the calculated results (Table 11.1). Water and toluene fluxes, the liquid side mass transfer coefficient (β f) were given by the

280

Basic Equations of the Mass Transport through a Membrane Layer

Table 11.1 Parameter Values Used for the Calculation DA 5 1 3 1028 cm2/s χ12 5 12.2 V1 5 107 cm3/mol ρ1 5 0.87 g/cm3

χ13 5 0.3 V2 5 18 cm3/mol ρ2 5 1 g/cm3

DB 5 0.5 3 1027 cm2/s χ23 5 6.3 V3 5 0.116 cm3/mol ρ3 5 0.86 g/cm3

For details, see Meuleman et al. (1999).

Table 11.2 Pervaporation Performance of Silicon Rubber with Toluene
Water Mixtures δ μm

3.5 5.5 10 20

Flux (1027 kg/m2s) Water

Toluene

3.4 2.2 1.1 0.39

0.119 0.114 0.11 0.086

Pef 3 103

β (1025 m/s)

N 3 104

E Measured

E Calculated

2.59 1.71 0.89 0.35

4.2 2.7 1.5 0.75

1.74 1.79 1.68 1.32

338 493 909 1812

360 529 945 2074

Feed concentration: 100 ppm toluene; β f 5 1.36 3 1024 m/s; H 5 48; ϑ 5 (1.14
1.36) 3 1025; D 5 0.15 3 1029 m2/s. Data taken from Baker et al. (1997).

authors, toluene diffusion coefficient in the membrane and the solubility were taken from Nagy’s paper (2006) that analyzed the coupling effect during pervaporation of toluene
water binary mixture also through silicon rubber membrane. The ϑ value was obtained for 40 C applying Eq. (11.27) for toluene with ρp 5 1,333 Pa (e.g., ρp 5 950 kg/m3; M 5 25:4 for 10 wt% toluene in the vapor phase). Accordingly, its value was obtained to be between 1.14 3 1025 and 1.36 3 1025. Comparing its value to the Pe number, its effect can be neglected on the enrichment. Values of parameter N are listed in Table 11.2; its value falls between 1.32 3 1024 and 1.74 3 1024; consequently, it can influence the E value in increasing extent as a function of the membrane thickness. Comparing the measured and calculated data, it can be stated that there is excellent agreement between them (the difference ranges between 3% and 12%) proving that the model developed is suitable to describe the performance of the pervaporation process. [The measured E values were obtained by means of the separation factor given by Baker et al., (1997; Table 2). Separation factors given by authors are somewhat higher than the measured E values.] Obviously, real values of the mass transport parameters are needed for correct estimation. The concentration distribution is plotted in Figure 11.4, at δ 5 3.5 μm and δ 5 20 μm, in order to illustrate the strong effect of the polarization layer. The value of φδ was regarded to be close to zero for this calculation (the partial pressure of toluene was less than 133 Pa in the vapor phase). As can be seen, the mass transfer resistance of the boundary layer

Toluene concentration (ppm)

Pervaporation

100.0

281

Figure 11.4 Calculated concentration distribution of toluene during toluene
water binary mixture’s pervaporation applying silicon rubber membrane using data of Baker et al. (1997) listed in Table 11.1.

Toluene–water, silicon rubber

80.0 60.0 40.0

δ = 20 μm; E = 2074

20.0

δ = 3.5 μm; E = 360

0.0 0.0

0.4 0.8 1.2 1.6 Boundary layer + membrane

2.0

significantly decreases the interface concentration on the feed side (c ) and thus, the permeation rate as well. However, the selectivity increases with the increase of the membrane thickness.

11.6

Concentration-Dependent Diffusivity

The diffusional mass transport through a homogeneous or asymmetric, dense membrane layer could often be nonlinear due to the concentration and/or space dependence of the diffusion coefficient and/or partition coefficient (solubility). In an earlier study, Mulder (1981) showed that the diffusion of ethanol and water through cellulose acetate membrane is strongly concentration dependent. He obtained that this dependence could be given by simple, exponential functions with two parameters for both components, independently from each other. Qin and Cabral (1998) modeled the mass transport through a hollow-fiber membrane using the widely applied solution-diffusion model. They simulated the concentration distribution in the lumen side of the membrane. They supposed exponential or linear dependency of the diffusion coefficients inside the membrane as most often occurring cases during different membrane processes as membrane-based extraction, the supported liquid membrane, gas membrane, membrane-based absorption
desorption, membrane distillation, pervaporation, etc. In addition, coupling phenomena could occur, in the most cases, during the separation of liquid mixtures. A strong coupling of diffusion of components to be separated takes place in the Maxwell
Stefan approach (Wesselingh and Krishna, 2000). This approach could be applied to describe mass transport, with strong concentration-dependent diffusion, during pervaporation of binary water
alcohol mixture, with low carbon number (Heinz and Stephan, 1994; Schaetzel et al., 2001; Iza´k et al., 2003). Separation by means of zeolite, or, generally, of inorganic membranes is another important group of membrane separation processes where the Maxwell
Stefan approach to mass transfer is recommended (Krishna and Wesselingh, 1997; van de Graaf et al., 1999; van den Broeke, 1999).

282

Basic Equations of the Mass Transport through a Membrane Layer

Depending on the interaction between the solvent molecules and the membrane molecules, the diffusion coefficient can strongly depend on the solvent concentration. In general, two concentration dependency functions are considered, namely exponential [Eq. (11.39)] and linear [Eq. (11.40)] ones (Mulder, 1981; Meuleman et al., 2001; Schaetzel, 2001; Jiraratananon et al., 2002; Schaetzel et al., 2010). Di 5 Di;0 expðαi1 φ1 1 αi2 φ2 1 ? 1 αiN φN Þ

ð11:39Þ

Di 5 Di;0 ð1 1 αi;1 φ1 1 αi;2 φ2 1 ? 1 αi;N φN Þ

ð11:40Þ

where Di,0 is the limiting diffusivity of the pure component at zero concentration; φi is the membrane concentration of component i; and αi is the plasticization coefficient. Assuming that one of the components as a key component has a much higher concentration than all other components together, we can express this limiting condition by φi;i 6¼ key comp: 5 0

ð11:41Þ

Accordingly, Eqs (11.39) and (11.40) will lead to Di 5 Di;0 eαi φi

ð11:42Þ

Di 5 Di;0 ð11 αi φi Þ

ð11:43Þ

and

The mass transfer rates for these cases are discussed in Chapters 3 and 6. The question to be answered is how the mass transfer resistance affects the mass transport and the separation in the case of the concentration-dependent diffusivity.

11.6.1 Exponential Concentration Dependency, D 5 D0eαφ The mass transfer without external mass transfer resistance rate can be given as (see Chapter 3): J 5 2D0 eαφ

dφ dy

ð11:44Þ

The concentration distribution can be given, after generally integrating Eq. (11.44), as: φ5

  1 α ðΨ 2 JyÞ ln α D0

ð11:45Þ

Pervaporation

283

Note that the J as a parameter means the real mass transfer rate inside the membrane (kg/m2s). The mass transport through both the polarization and membrane layers, taking into account the simultaneous effect of both layers, also can be expressed. The concentration distribution of the boundary layer is extensively discussed in the literature (Baker, 2004; Nagy, 2010). It can be given, with parameters Ωf and Ψf, as follows: c 5 Ωf eυy=Df 1 Ψf

0 # y # δf

ð11:46Þ

For the membrane layer, at δf # y # δf 1 δ, the concentration distribution is given by Eq. (11.17). Parameters of Eqs (11.45) and (11.46) should be determined by suitable boundary conditions as Ωf 1 Ψf 5 co ; υΨf 5 2J; Pef

HðΩf e

φδ 5

y50

ð11:47Þ

y 5 δf

ð11:48Þ

  1 α 1 Ψf Þ 5 ln ðΨ 2 JδÞ ; α D0

  1 α ðΨ 2 J ½δ f 1 δ Þ ; ln α D0

y 5 δf

y 5 δf 1 δ

ð11:49Þ

ð11:50Þ

with Pef 5 υδf/Df. After solution of the algebraic equation system of Eqs (11.47)
(11.50), one can get the following implicit equation for calculation of the value of the mass transfer rate, J, as    J 1 Jα ln 1 eαφδ ð12 ePef Þ 1 co ePef 5 β f Pef αH β

ð11:51Þ

with β f 5 Df/δf and β 5 D0/δ. Taking into account the well-known condition that should be fulfilled during the pervaporation, namely that J 5 cpυ, the cp is the liquid permeate concentration, can be obtained as cp ð12 e

Pef

Þ1c e

o Pef

  1 βf αφδ Pef αcp 1 e 5 ln β αH

ð11:52Þ

Both the mass transfer rate, J, and the permeate concentration, cp can be calculated by trial-and-error method by means of Eqs (11.51) and (11.52), respectively. The link between cp and φδ is given in Eq. (11.36).

284

Basic Equations of the Mass Transport through a Membrane Layer

As an illustration, Figure 11.5 shows the concentration distribution in the polarization and the membrane layers. As can be seen the role of the concentration polarization layer on the concentration distribution, and consequently ~ on the separation efficiency strongly depends on the value of the exponent, α: With increasing α~ values decreases the concentration on the inlet membrane interface, decreasing both the enrichment and the permeation rate. How the α~ value affects the enrichment is illustrated in Figure 11.6. During the calculation, the Peclet number of the polarization layer was kept to be constant, Pef 5 0.001. Accordingly, with the increase of the β f diffusive mass transfer coefficient, the convective flow should also proportionally be increased. As a consequence, the enrichment decreases with the increase of the β f value. The increase of the membrane diffusivity, due to the increasing α~ value, the enrichment also increases.

Concentration distribution

1.0 0.8 ∼=0 α

0.6

0.3

0.4 0.2 0.0

1

Pef = 0.001 βf = 1 ×

10–4

m/s

Boundary layer 0.0

1

10

5 Membrane layer

0.4 0.8 1.2 1.6 Boundary layer + membrane

1000

Figure 11.5 Concentration distribution with exponentially concentration-dependent diffusion coefficient. [Di,0 5 3 3 1029 m2/s; β 5 2.5 3 1026 m2/s (β is the membrane mass transfer coefficient); β f 5 1 3 1024 m2/s (β f 5 Df/δf, where subscript f denotes the polarization layer); α~ is the dimensionless  exponent, α~ 5 αφ ; where φ 2.0 is the membrane interface concentration on the feed side].

∼ = 10 α

Enrichment, E

1 100 Pef = 0.001

10

β = 2.5 × 10–6 m/s

1 1.00E–05

0

1.00E–04 1.00E–03 βf (m/s)

1.00E–02

Figure 11.6 Enrichment as a function of the diffusive mass transfer coefficient of the polarization layer (Di,0 5 3 3 1029 m2/s; H 5 10).

Pervaporation

285

11.6.2 Linear Concentration Dependency, D 5 D0 (1 1 αφ) The mass transfer rate without external mass transfer resistance can be given, in this case, as follows: J 5 2 Do ð11 αφÞ

dφ dy

ð11:53Þ

After integration of Eq. (11.53), the concentration distribution for the membrane layer can be obtained as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1 4 8 Ψ 2 Jy 1 1 φ5 2 6 2 α α α D0 2

ð11:54Þ

The parameters Ψ (denoted by S in Chapter 3) and J are given by Eqs. (3.29) and (3.30) for the membrane layer. Look now at how the mass transfer rate and the concentration distribution can be given in the presence of external mass transfer resistance in the feed phase. The concentration distribution for the membrane and the concentration boundary layer can be defined by Eqs (11.54) and (11.46), respectively. The parameters Ψ, J, Ωf, and Ψf can be determined by means of the following boundary conditions: Ωf 1 Ψf 5 co ; υΨj 5 2J;

HðΩf e

φδ

Pef

y50

ð11:55Þ

y 5 δf

ð11:56Þ

1 1 1 Ψf Þ 5 2 1 α 2

1 1 52 1 α 2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 8 Ψ 2 Jδ 1 ; α2 α Do

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 8 Ψ 2 Jðδf 1 δÞ 1 ; α2 α D0

y 5 δf

y 5 δf 1 δ

ð11:57Þ

ð11:58Þ

Solving the above algebraic equation system, the mass transfer rate can be obtained by a second-order algebraic equation as follows: J 2 ξ 1 Jϑ 1 ζ 5 0

ð11:59Þ

with  ξ5

H ð1 2 ePef Þ β f Pef

2

286

Basic Equations of the Mass Transport through a Membrane Layer


 2H 1 21 o ϑ5 Hc 1 ð12 ePef Þ 2 β f Pef α αβ

 1 2 1 2  o 2 φδ 1 ζ 5 Hc 1 α α From Eq. (11.59), the value of J can be obtained by the well-known solution of a second-order algebraic equation. The permeate concentration, cp, can here also be given by replacing J with υcp (J 5 υcp) into Eq. (11.59) and replacing φδ with cp using Eq. (11.36), thus it can be obtained c2p ξ^ 1 cp ψ 1 ζ^ 5 0

ð11:60Þ

where ξ^ 5 υ2 ξ 2 A2 ψ 5 υϑ 2

A α


2 ^ζ 5 Hco 1 1 2 1 α α2 and according to Eq. (11.36) with φδ 5 Acp A5

pp M p H ρp RT

ð11:61Þ

As an illustration, the concentration distribution is plotted in Figure 11.7 at different α~ values for the case of linearly dependent diffusion coefficient.

Concentration distribution

1.0

Figure 11.7 Concentration distribution in case of linearly dependent diffusion coefficient [Di,0 5 3 3 1029 m2/s; β 5 2.5 3 1026 m2/s; β f 5 1 3 1024 m2/s (β f 5 Df/δf, where subscript f denotes the polarization layer), α~ is the dimensionless exponent, α~ 5 αφ ; where φ is the membrane interface concentration on the feed side].

∼=0 α

0.8 0.6 0.4 0.2

10 Pef = 0.001 βf = 1×10–4 m/s

1

100

Boundary layer Membrane layer 0.0 0.4 0.8 1.2 1.6 0.0 Boundary layer + membrane

2.0

Pervaporation

11.7

287

Coupled Diffusion

A strong coupling of diffusion of components to be separated takes place in the Maxwell
Stefan approach. This approach could be applied to describe mass transport, with strong concentration-dependent diffusion, during pervaporation of binary wateralcohol mixture, with low carbon number (Heinz and Stephan, 1994; Schaetzel et al., 2001; Iza´k et al., 2003; Nagy, 2006). Another often recommended mass transport theory is the so-called Flory
Huggins approach (Meuleman et al., 1999). This theory is especially applicable for organophilic pervaporation of organic compounds in water. The Flory
Huggins theory that gives the chemical potential as function of the components’ concentration and, consequently to give the mass flux through the membrane, is also recommended to describe the binary separation by pervaporation (Smart et al., 1998, Meuleman et al., 1999). Meuleman et al. (1999) investigated separation of toluene and ethanol by means of ethylene
propylene
diene terpolymers (EPDM). This theory that basically differs from the Maxwell
Stefan approach, also gives strong coupling of diffusion of components. We calculated the concentration profiles and the mass transfer rate of components to be separated under different inlet conditions on the tube side of a capillary membrane. As a model system, the separation of toluene
water binary mixture by means of EPDM, (Meuleman et al., 1999), using pervaporation, was chosen. The values of LA, LB, LA ; and LB (A corresponds to toluene, while B corresponds to water) were predicted by means of Eqs (11.62)
(11.64) using the data of Table 11.1. Three to five calculation steps were needed to obtain the correct concentration distribution and the mass transfer rate using Eqs (3.79)
(3.94). During these steps, the values LA, LB, LA ; and LB were suited to the concentration obtained during the calculation steps. Equilibrium volume fraction of components was calculated using Eqs (11.62) and (11.63) by iteration method. The measured (Meuleman et al., 1999) and the calculated results are shown in Figure 11.8 (Nagy, 2004, 2006). For the prediction of the values of LA, LB, LA ; and LB ; we applied the Flory
Huggins theory. According to it, the chemical potential of two components, 1 and 2, to be separated is as follows: V1 V1 2 ε3 1ðχ12 ε2 1 χ13 ε3 Þðε2 1 ε3 Þ V2 V3 

V1 V1 ρ3 2Mc 1 1=3 2χ23 ε2 ε3 1 12 ε3 2 ε3 V2 Mc M 2

ln a1 5 ln ε1 1ð12 ε1 Þ 2 ε2

ð11:62Þ


 V2 V2 V2 2 ε3 1 χ12 ε1 1 χ23 ε3 ðε1 1 ε3 Þ V1 V3 V1 

V2 V2 ρ3 2Mc 1 1=3 2χ13 ε1 ε3 1 ε3 2 ε3 12 V1 Mc M 2 ð11:63Þ

ln a2 5 ln ε2 1ð1 2 ε2 Þ 2 ε1

288

Basic Equations of the Mass Transport through a Membrane Layer

3.5

Flux (107 cm/s)

3.0 2.5

Toluene

2.0 1.5 1.0 0.5 0.0

Water

0

50

100 150 Membrane thickness (10–4 cm)

200

250

Figure 11.8 Experimental flux (points) and the calculated one (continuous lines) as a function of the membrane thickness (feed toluene concentration: 250 ppm; β f 5 1.3 3 1023 cm/s; β δ 5 0; DA 5 1 3 1028 cm2/s; DB 5 0.5 3 1027 cm2/s; r0-N; Cδ 5 0).

with ξ i 5 ððVi ρ3 Þ=Mc Þð12ð2Mc =MÞÞ (Meuleman et al. 1999) or ξ i 5 υe λi (Smart 1997) and 1 5 ε1 1 ε2 1 ε3 where ε1, ε2, and ε3 are volume fraction of components (note that εi 5 ρφi with ρ as a density of the swollen membrane, kg/m3); V1, V2, and V3 are molar volume of components and the membrane, respectively (cm3/mol); χ12, χ13, and χ23 are interaction parameters; Mc and M are molecular weight between two cross-links and polymer, respectively; and ρ3 is density of the membrane. The value of the LA, LB, LA ; and LB can be calculated by the following equations [A and B in index correspond to 1 and 2 in Eqs (11.62) and (11.63)]: @ ln aA ; @φA @ ln aB LA 5 φB DB ; @φA

LA 5 φA DA

@ ln aA @φB @ ln aB LB 5 φB DB @φB LB 5 φA DA

ð11:64Þ

The modified Vignes equation can be given as (Bitter, 1991) e.g., for component A: DA 5 DA eðσAA φA 1 σAB φB Þ

ð11:65Þ

Pervaporation

289

with σAA 5 ln DAA 2 ln DA ;

σAA 5 ln DAB 2 ln DA

where DAA is the coefficient of self-diffusion of component A in pure A; DA is the binary diffusion coefficient of A in membrane in infinite dilution of A; DAB is the coefficient of the diffusion of component A in pure B; and DA is the coefficient of self-diffusion of component A in the mixture. Separation of toluene
water binary mixture, by pervaporation, was analyzed by Nagy (2006) using transport data obtained by EPDM membrane (Meuleman et al., 1999), as well as polymethylsiloxan membrane (Smart, 1997; Smart et al., 1998). For calculation, the analytical approach was used which is given in detail by Eqs (3.90)
(3.106). It shows the concentration distribution, mass transfer rate, and how the coupling effect affects the separation efficiency.

References Baker, R.W. (2004) Membrane Technology and Applications, 2nd ed. John Wiley and Sons, Chichester. Baker, R.W., Wijmans, J.G., Athayde, A.L., Daniels, R., Ly, J.H., and Le, M. (1997) The effect of concentration polarization on the separation of volatile organic compounds from water by pervaporation. J. Membr. Sci. 137, 159
172. Bhattacharya, S., and Hwang, S.-T. (1997) Concentration polarization, separation factor, and Peclet number in membrane processes. J. Membr. Sci. 132, 73
90. Bitter, J.G.A. (1991) Transport Mechanisms in Membrane Separation processes. ShellLaboratorium, Amsterdam. Bowen, T.C., Li, S., Noble, R.D., and Falconer, J.L. (2003) Driving force for pervaporation through zeolite membranes. J. Membr. Sci. 225, 165
176. Bowen, T.C., Noble, R.D., and Falconer, J.L. (2004) Fundamental and application of pervaporation through zeolite membranes. J. Membr. Sci. 245, 1
33. Cle´men, R., Jonquie´res, A., Sarti, H., Sosata, M.F., Teixidor, M.A.C., and Lochon, P. (2004) Original structure-property relationships derived from a new modeling of diffusion of pure solvents through polymer membranes. J. Membr. Sci. 232, 141
152. Delgado, P., Sanz, M.T., and Beltra´n, S. (2008) Pervaporation study for different binary mixtures in the esterification system of lactic acid with ethanol. Sep. Purif. Technol. 64, 78
87. Feng, X., and Huang, R.Y.M. (1997) Liquid separation by membrane pervaporation: a review. Ind. Eng. Chem. Res. 36, 1048
1066. Fouad, E.A., and Feng, X. (2008) Use of pervaporation to separate butanol from dilute aqueous solutions: Effect of operating conditions and concentration polarization. J. Membr. Sci. 323, 428
435. Heintz, A., and Stephan, W. (1994) A generalized solution-diffusion model of the pervaporation process through composite membrane. J. Membr. Sci. 89, 153
169. Iza´k, P., Bartovska´, L., Friess, K., Sipek, M., and Uchytil, P. (2003) Description of binary liquid mixtures transport non-porous membrane by modified Maxwell
Stefan equation. J. Membr. Sci. 214, 293
309.

290

Basic Equations of the Mass Transport through a Membrane Layer

Jiraratananon, R., Chanachi, A., and Huang, R.Y.M. (2002) Pervaporation dehydration of ethanol
water mixtures with chitosan/hydroxyethylcellulose (CS/HEC) composite membranes. II. Analysis of mass transport. J. Membr. Sci. 199, 211
222. Jiraratananon, R., Chanachai, A., and Huang, R.Y.M. (2008) Pervaporation dehydration of ethanol
water mixtures with chitosan/hydroxyethylcellulose (CS/HEC) composite membranes: II. Analysis of mass transport. J. Membr. Sci. 199, 211
222. Karlesson, H.O.E., and Tragardh, G. (1993) Pervaporation of dilute organic
water mixtures. A literature review on modeling studies and applications to aroma compound recovery. J. Membr. Sci. 76, 121
146. Kamaruddin, H.D., and Koros, W.J. (1997) Some observation about the application of Fick’s first law for membrane separation of multicomponent mixtures. J. Membr. Sci. 135, 147
159. Krishna, R., and Wesselingh, J.A. (1997) The Maxwell
Stefan approach to mass transfer. Chem. Eng. Sci. 52, 862
906. Lipnizki, F., Field, R.W., and Ten, P.K. (1999) Pervaporation-based hybrid process: review of process design, application and economics. J. Membr. Sci. 153, 183
210. Meuleman, E.E.B., Bosch, B., Mulder, M.H.V., and Strathmann, H. (1999) Modeling of liquid/ liquid separation by pervaporation: toulene from water. AIChE J. 45, 2153
2160. Meuleman, E.E.B., Willemsen, J.H.A., Mulder, M.H.V., and Strathmann, H. (2001) EPDM as a selective membrane material in pervaporation. J. Membr. Sci. 188, 235
249. Michels, A.S. (1995) Effects of feed-side polarization on pervaporative stripping of volatile organic solutes from dilute solutions: a generalized analytical treatment. J. Membr. Sci. 101, 117
126. Mulder, M. (1981) Pervaporation Separation of Ethanol
Water and Isomeric Xylenes. PhD Thesis, University of Twente, The Netherlands. Nagy, E. (2004) Nonlinear, coupled mass transfer through a dense membrane. Desalination 163, 345
354. Nagy, E. (2006) Binary, coupled mass transfer with variable diffusivity through cylindrical membrane. J. Membr. Sci. 274, 159
168. Nagy, E. (2009) Basic equations of mass transport through biocatalytic membrane layer. Asia-Pacific J. Chem. Eng. 4, 270
278. Nagy, E. (2010) Coupled effect of the membrane properties and concentration polarization in pervaporation: unified mass transport model. Sep. Purif. Technol. 73, 194
201. Nagy, E., and Borbely, G. (2007) Effect of the concentration polarization and the membrane layer mass transport on the separation. J. Appl. Membr. Sci. 6, 1
8. Nagy, E., and Kulcsar, E. (2009) Mass transport through biocatalytic membrane reactor. Desalination 245, 422
436. Olsson, J., and Tra¨gardh, G. (2001) Pervaporation of volatile organic compounds from water, I Influence of permeate pressure on selectivity. J. Membr. Sci. 187, 25
37. Qin, Y., and Cabral, J.M.S. (1988) Lumen mass transfer in hollow fiber membrane processes with nonlinear boundary conditions. AIChE J. 41, 836
848. Ravanchi, M.T., Kaghazchi, T., and Kargari, A. (2009) Application of membrane separation processes in petrochemical industry: a review. Desalination 235, 199
244. Schaetzel, P., Vauclair, C., Luo, G., and Nguyen, Q.T. (2001) The solution-diffusion model, order of magnitude calculation of coupling between the fluxes in pervaporation. J. Membr. Sci. 191, 103
108. Schaetzel, P., Bouallouche, R., Amar, H.A., Nguyen, T., Riffault, B., and Marais, S. (2010) Mass transfer in pervaporation: the key component approximation for the solution-diffusion model. Desalination 251, 161
166.

Pervaporation

291

Schafer, T., and Crespo, J. (2007) Study and optimization of the hydrodynamic upstream conditions during recovery of a complex aroma profile by pervaporation. J. Membr. Sci. 301, 46
56. Semenova, S.I., Ohya, H., and Soontarapa, K. (1997) Hydrophilic membranes for pervaporation: an analytical review. Desalination 110, 251
286. Shao, P., and Huang, R.Y.M. (2007) Polymeric membrane pervaporation. J. Membr. Sci. 287, 162
179. She, M., and Hwang, S.-T. (2004) Concentration od dilute flavor compounds by pervaporation: permeate pressure effect and boundary layer resistance modeling. J. Membr. Sci. 236, 193
202. Smart, J.L. (1997) Pervaporative Extraction of Volatile Organic Compounds from Aqueous Systems with Use of a Tubular Transverse Flow Module. PhD Thesis, University of Texas. Smart, J., Starov, V.M., Schucker, R.C., and Lloyd, D.R. (1998) Pervaporative extraction of volatile organic compounds from aqueous systems with use of a tubular transverse flow module. Part II. Experimental results. J. Membr. Sci. 143, 159
179. Smitha, B., Suhanya, D., Sridhar, S., and Ramakrishna, M. (2004) Separation of organic
organic mixtures by pervaporation: a review. J. Membr. Sci. 241, 1
21. Trifunovic, O., and Tra¨gardh, G. (2006) Mass transport of aliphatic alcohols and esters through hydrophobic pervaporation membranes. Sep. Purif. Technol. 50, 51
61. Vane, A. (2005) A review of pervaporation for product recovery from biomass fermentation processes. J. Chem. Technol. Biotechnol. 80, 603
629. Van de Graaf, J.M., Kapteijn, F., and Moulijn, J.A. (1999) Modeling permeation of binary mixtures through zeolite membranes. AIChE J. 45, 497. Van den Broeke, L.J.P., Bakker, W.J.W., Kapteijn, F., and Moulijn, A. (1999) Binary permeation through silicalte-1 membrane. AIChE J. 45, 977
985. Van der Bruggen, B., Jansen, J.C., Figoli, A., Geens, J., Van Baelen, D., Drioli, E., and Vandecasteele, C. (2004) Determination of parameters affecting transport in polymeric membranes: parallels between pervaporation and nanofiltration. J. Phys. Chem. B 108, 13273
13279. Villaluenga, J.P.G., and Tabe-Mohammadi, A. (2000) A review on the separation of benzene/ cyclohexane mixtures by pervaporation process. J. Membr. Sci. 169, 159
174. Wee, Sh.-L., Tye, Ch.-Th., and Bhatia, S. (2008) Membrane separation process. Pervaporation through zeolite membrane. Sep. Sci. Technol. 63, 500
516. Wesselingh, J.A., and Krishna, R. (2000) Mass Transport in Multicomponent Mixtures. Delft University Press, The Netherlands. Wijmans, J.G. (2004) The role of permeant molar volume in the solution-diffusion model transport equations. J. Membr. Sci. 237, 39
50. Wijmans, J.G., and Baker, R.W. (1995) The solution-diffusion model: a review. J. Membr. Sci. 107, 1
21. Wijmans, G., Athayde, A.L., Daniels, R., Ly, J.H., Kamanaddin, H.D., and Pinnau, I. (1996) The role of boundary layers in the removal of volatile organic compounds from water by pervaporation. J. Membr. Sci. 109 (1996), 135.

12 Membrane Contactors 12.1

Introduction

Membrane contactors are membrane systems mainly used to contact two phases in order to promote the mass transfer between them. The membrane can be considered a nonselective barrier, allowing relatively free passage of one component and the transport of species occurs through the membrane pores mainly by diffusion. The membrane functions as an interface between two phases but does not control the rate of passage of permeants across the membrane. The membrane should be hydrophobic in the case of the aqueous phases to be separated, or it can be hydrophilic and hydrophobic in cases of two immiscible phases (Drioli et al., 2006; Strahmann et al., 2006). Membrane contactors can be used in several industrial areas: liquid/liquid extraction, gas absorption and stripping, wastewater treatment, emulsification, membrane distillation, membrane crystallization, and membrane chemical and biochemical reactors (Criscuoli, 2009; Gaeta, 2009). An important advantage of the membrane contactor is its high contact area; it is about 10-fold higher than that of an equivalent-sized tower. Another important advantage of it is that it physically separates the counter-flowing phases. The membrane area between them is independent of their relative flow rates, so large flow ratio differences can be used without producing operating difficulties. The membrane contactors are typically shell-and-tube devices containing microporous capillary hollow-fiber membranes. The membrane pores are sufficiently small that capillary forces prevent direct mixing of the phases on either side of the membrane. The breakthrough transmembrane pressure difference can be given by the Laplace equation, as a function of the pore size, as Δp 5

4σ cos θ dp

with σ, surface tension; θ, contact angle; and dp, pore diameter. According to the above equation, size of the pores is an important property, with its increase decreases the mass transfer resistance of the membrane but decreases also the capillary pressure. The mass transfer resistance is the disadvantage of the membrane contactor. In order to reduce it, the membrane thickness should be chosen as thin, while the pore size should be as large as possible. Their values are limited by the need of sufficient mechanical stability and the hindrance of wetting of the membrane pores. The detailed discussion of the properties, of the application of the membrane contactors is not the aim of this material. The literature cited above gives sufficient Basic Equations of the Mass Transport through a Membrane Layer. DOI: 10.1016/B978-0-12-416025-5.00012-0 © 2012 Elsevier Inc. All rights reserved.

294

Basic Equations of the Mass Transport through a Membrane Layer

Nonpolar

polar

(A)

Nonpolar

polar

(B)

Figure 12.1 The wetting of hydrophobic (A) or hydrophilic (B) membranes by nonpolar or polar phases, respectively.

Vapor/gas-filled pores Figure 12.2 Scheme of the membrane

distillation (P1.P2) and (T1.T2) and osmotic distillation (P1.P2) and (T1 5 T2).

P1

P2

T1

T2

Feed

Strip

information about it. Here we focus on the mass transport’s description through the porous membrane during these processes using membrane contactors. The main types of membrane contactors, regarding the phase distribution, are illustrated in Figures 12.1 and 12.2. The pores of membrane contactors are filled with liquid that has the same polarity as that of membrane or with gas phase. Accordingly, the main groups of membrane contactors considering the mass transport inside the pores: 1. Liquid phase a. Pores are filled with liquid, liquid/liquid system, the mass transport (extraction) takes place between immiscible liquid phase, there is a direct contact between the immiscible phases and the pressure difference between the two sides of membrane is less than the capillary pressure, the mass transport takes place by diffusion through the pores due to the concentration difference as driving force.

Membrane Contactors

295

b. A special case of this system is the liquid membrane process, where the pores are filled by liquid which separates two miscible liquid phases, the mass transport is diffusive and often is accompanied by chemical reaction with the so-called carrier component. 2. Gas phase a. The pores are filled by gas phase, gas/liquid systems (stripper, scrubber), the driving force is the concentration difference; thus, the diffusion is the rate determining transport. b. The pores are filled again with vapor phase separating the two liquid phases (membrane distillation); there is a pressure and temperature difference between the two sides of the membrane; thus, the transport of the evaporated component can be partly diffusion, partly convection.

Referring to the description of the membrane mass transport, it can strongly depend on the membrane structure properties and the mass and heat transport resistances of the external phases. The right choice of operating conditions is also the basis of a good performance of the membrane contactor. Higher flow rates can significantly reduce the mass and heat transport resistances in the external phases. However, they have to be carefully defined, in order to avoid the pressure reaching the breakthrough values. For this reason, the pressure of the flowing phases to be processed must be controlled, and necessarily varied in a given case, during the whole process.

12.2

Mass Transport

In order to describe the mass transport, an easy way is to apply the resistance-in-series model. The external mass transfer resistance, depending on the flow conditions, is well known. These are briefly summarized at the end of this chapter. As in the other chapters, we try to define the mass transfer rates (inlet and outlet rates) in the membrane contactors, which then can be applied to the description of the concentration distribution in the external phases by means of suitable conversation balance equation. Here, first the mass transport inside the membrane will be mathematically modeled and briefly discussed.

12.2.1 Diffusive Mass Transport Through the Membrane Pores The pores are filled either by gas (e.g., absorption, desorption) or liquid (e.g., extraction). Because there can exist relatively low transmembrane pressure, thus the transport can take place mainly by diffusion. If the volume of the vapor phase can change inside the pores, during its condensation in the case of membrane distillation, convective mass transport can exist as well. In some cases, the mass transport can be accompanied by chemical reaction as is the case often during absorption or extraction.

296

Basic Equations of the Mass Transport through a Membrane Layer

12.2.1.1 Physical Mass Transport The diffusion rate through a porous membrane layer is influenced by several factors, and the effective diffusion coefficient can be written as Di 5 Do;i

εp ξ τ d;i

ð12:1Þ

where εp is the porosity or the area of support/area of pores (only part of the support is void and available for diffusive transport, while a large part is occupied by solid matter); τ is the tortuosity (the pore network is complex and not straight, diffusion occurs changing direction continuously). The tortuosity factor is assumed to be in the range of 1.47. Parameters of ξ d,i are called hindrance factor for diffusion. Do,i and D are solute diffusivity in bulk solvent and in the pores of the membrane, respectively (m2/s). Detailed analysis of the role of hindrance factor is discussed in Chapter 10. The membrane diffusivity coefficient is a combination of bulk and Knudsen diffusion coefficients in case of gas phase (Li and Cheng, 2005; Zhang et al., 2008; Seidel-Morgenstern, 2010): 1 1 1 5 1 Di Db;i DK;i

ð12:2Þ

where subscripts b and K denote bulk and Knudsen, respectively. For gas-filled pores, the diffusion through membrane can be either bulk diffusion or Knudsen diffusion, depending on the pore diameter, dp. Obviously, to predict the effective diffusion coefficient given by Eq. (12.1) should also be taken into account. The limiting values of the pore size regarding the diffusion types can be estimated as (Luis et al., 2009) G

G

G

If dp , 1 3 1027 m, Knudsen diffusion dominates. If dp . 1 3 1025 m, bulk diffusion dominates. If 13 1027 m ,dp, 13 1025 m, both types of diffusion can exist.

The Knudsen diffusion coefficient can be determined as (Drioli et al., 2006): 1 DK;i 5 dp 3

rffiffiffiffiffiffiffiffiffi 8RT πM

ð12:3Þ

where dp is the pore diameter (m); R is the gas constant (J/kmol K); T is the temperature (K); M is the molecular weight (kg/kmol); and diffusivity is obtained in m2/s. In gas/liquid membrane contactors, typical absorption processes are the removal of natural gases, such as CO2 using water (Zhang et al., 2006; Faiz and Al-Marzouqi, 2010, 2011) or chemical absorbent (Wang et al., 2004). A review of CO2 absorption by chemical solvents, mostly different amines, is published by Li and Cheng (2005). The process can be carried out at variable pressures (Marzouk et al., 2010). Many

Membrane Contactors

297

Gas (or liquid)

Liquid

c1o

Figure 12.3 Concentration profile for a species that moves from phase 1 toward phase 2.

c2*

c1*

c2o

0

c2*

δ

parameters can change with pressure, such as gas diffusion coefficients, concentration, and flow rates. The membrane introduces a mass transfer resistance which should be reduced as low as possible. The best operation mode is the nonwetted one, where the membrane pores are filled by gas. The gas phase has several orders of magnitude higher diffusion coefficient than that of the liquid phase. Obviously, in liquid/ liquid systems the pores are filled by one of the liquid phase. The membrane material should be chosen according to the solvent property: a hydrophilic membrane is used for hydrophobic liquids and a hydrophobic membrane for hydrophilic liquids (Drioli et al., 2006). However, in the technical operation of membrane gas absorption, solvent could wet the membrane penetrating partially into the membrane pores. Partial wetting can significantly increase the resistance to mass transfer of the membrane (Malek et al., 1997; Mavroudi et al., 2006; Zhang et al., 2008). That is why this situation must be avoided. Mass transfer in nonwetted membrane: The concentration distribution is illustrated in Figure 12.3. The pores are filled by gas and the diffusive mass transport takes place from the gas phase into the liquid one. Assuming constant mass transport parameters as diffusivity, average value of pore size, porosity, partial pressure, membrane thickness, and no source term, the mass transfer rate can be expressed, for nonwetted contactor, as (for details, see Chapter 1): Ji 5 2DK;i

dci DK;i dpi 2 dy RT dy

ð12:4Þ

The mass transfer rate can be effectively modeled as proposed by Bandini and Sarti (1999): Ji 5

Ko RT

rffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffi 8RT Δpi 8RT Δci  Ko πMi δ πMi δ

ð12:5Þ

298

Basic Equations of the Mass Transport through a Membrane Layer

where the value of Ko, according to Eqs (12.3) and (12.4), is as Ko 5

1 dp 3

where δ is the membrane thickness (m); Δpi is the partial pressure difference of component i; and Δci is the concentration difference between the two sides of the membrane. The membrane contactors have relatively large thickness compared to the inner radius of a capillary membrane. Thus, the cylindrical space can significantly affect the mass transfer rate. According to Eq. (6.6), the mass transfer rate can be given as Ko Ji 5 RT

rffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffi 8RT Δpi 8RT Δci  Ko πMi δ πMi ro lnð1 1 δ=ro Þ

ð12:6Þ

where ro denotes the lumen radius of capillary membrane (m). Using the resistance-in-series conception, the liquid side mass transfer resistance, o , can be taken into account as ðφ 5 Hc Þ: β fδ Ji 5

co1 2 Hco2 ðH=β ofδ Þ1ð1=β o Þ

ð12:7Þ

where rffiffiffiffiffiffiffiffiffi 8RT 1 β  Ko πMi ro lnð1 1 δ=ro Þ o

ð12:8Þ

and β ofδ 5

Df;i ro ðro 1 δÞlnð1 1 δf =½ro 1 δÞ

ð12:9Þ

The value of Ji continuously decreases in direction of the shell side of membrane due to the cylindrical effect. Accordingly, the outlet mass transfer rate can be expressed as Ji;δ 5 Ji

1 1 1 δ=ro

ð12:10Þ

Equation (12.6) involves the radial diffusive flow inside the membrane contactor, assuming that the diffusion occurs in the pores only and the orientation of pores is perpendicular to the membrane surface. In principle, diffusion can also

Membrane Contactors

299

exist in an axial direction as well. The differential mass balance equation when the axial diffusivity also is taken into account will be as (Faiz and Al-Marzouqi, 2010).  2  @ φi 1 @φi @2 φi 1 2 50 1 Di @r 2 @z r @r

ð12:11Þ

The J and Jδ mass transfer rates can be given by means of numerical solution of Eq. (12.11). Mass transfer in partially wetted membrane (Figure 12.4): The partial wetting of the membrane pores can significantly increase the membrane’s mass transfer resistance (Randwala, 1996). In this case, the transported component should diffuse through the liquid layer existing in the membrane pores as well. The mass transfer rate inside the pores can be given simply by the resistance-in-series model as it was done for the expression of Eq. (12.7). Only the physical mass transfer coefficients can be given correctly. These can be given easily according to Eqs (12.8) and (12.9). Thus, the mass transfer coefficient in the wetted pore will be as (δ2 5 δ 2 δ1). β ofδ 5

Df;i ðro 1 δ1 Þlnð11 δ2 =½ro 1 δ1 Þ

ð12:12Þ

where δ1 and δ2 are the thickness of the gas and liquid layers in the membrane pores, respectively. In the most cases, chemical absorption takes place during the gas component’s removal. Therefore, the effect of the irreversible and reversible first-order reactions will be briefly discussed. The concentration distribution and the mass transfer rates were developed for gas/liquid or liquid/liquid systems long ago (Danckwerts, 1970; Westerterp et al., 1993).

Gas (or liquid)

Liquid

c1o

c2*

c2o c1*

βf0 0

δ1

δβ fδ

Figure 12.4 Illustration of the concentration distribution in partially wetted pores.

300

Basic Equations of the Mass Transport through a Membrane Layer

12.2.1.2 Mass Transfer Rate Accompanied by Chemical Reaction The main analytical equations will be shown here briefly, namely mass transfer accompanied by first-order reaction. The reaction takes place in the liquid phase, thus the expressions are similar to that obtained for two-layer mass transport. Basically, the necessary mass transfer rate’s equations are known in the cases of gas/liquid systems, and only a few modifications are needed. First-order irreversible reaction: The effect of the cylindrical space is neglected here. Its role strongly decreases with the increasing reaction rate. The mass transfer rates, on the two sides of the wetted pores, are as follows: J2 5 β of

  ϑ co2 c2 2 cosh ϑ tanh ϑ

J2;δ 5 β of

ϑ ðc 2 cosh ϑco2 Þ sinh ϑ 2

ð12:13Þ

ð12:14Þ

with sffiffiffiffiffiffiffiffiffi k1 δ22 ; ϑ5 Df

δ2 5 δ 2 δ1

The important notations are given in Figure 12.3 or 12.4. The mass transfer rates taking into account both layers in the pores, and taking into account that Jδ1  Ji =ð11 δ1 =ro Þ 5 J2 [see Eq. (12.10)], are as follows:  o  Hc1 2 co2 ð12:15Þ J 5 β ov cosh ϑ with β ov 5

1 11ðδ1 =ro Þ=β o1 1 H=β 2

ð12:16Þ

where β2 5

Df ϑ δ2 tanh ϑ

ð12:17Þ

The value of β o1 is given in Eq. (12.8), replacing δ with δ1. The expression of the outlet mass transfer rate, taking into account the mass transfer resistance of both layers in the pores, is much more complex. For it, the reacted amount of the reactant in the wetted layer has to be defined [the difference of Eqs (12.13)

Membrane Contactors

301

and (12.14) gives that in the form of mass transfer rate, Jr 5 J2 2 J2,δ]. According to Eq. (4.93b), the outlet mass transfer rate can be given as (Hc1 5 c2): Jδ 5 β o1 ϑ

ðβ o2 =β o1 Þð1=ϑÞð1=cosh ϑÞco1 2 ððβ o2 =β o1 Þð1=HϑÞ 1 tanh ϑÞco2 11 ðβ o2 =β o1 Þð1=HϑÞtanh ϑ

ð12:18Þ

where sffiffiffiffiffiffiffiffiffi k1 δ22 ϑ5 ; D2

β o1 5

D1 ; δ1

β o2 5

D2 ; δ2

δ2 5 δ 2 δ1

where δ1 and δ2 represent the thickness of the gas and liquid layers in the pores, respectively; co1 and co2 denote the gas and liquid concentration at the outer membrane interface, respectively (kmol/m3); and D1 and D2 are the diffusion coefficients in the gas and liquid phase, in the pores. Then using the expression of Ji 2 Jr,i 5 J2,δ,i the outlet mass transfer rate can be given by the overall mass transfer resistances [the subscript i means the reactant here; for simplification, this subscript is left out from Eqs. (12.13) to (12.17)] (Nagy et al., 1983; Nagy and Ujhidy, 1989). First-order reversible reaction A3B : Several chemical absorption processes are accompanied by reversible reaction. The mass transfer rates will be briefly shown here. It is assumed that the reaction will get in equilibrium at the outlet membrane interface (this condition is involved in Eq. (12.23)).  cE  Q 5 k1 cA 2 K

with K 5

k1 k21

ð12:19Þ

where k1 and k21 are the reaction rate constants for forward and backward direction (1/s), respectively. DA

 d2 c A cE  2 k c 2 50 1 A dy2 K

ð12:20Þ

DE

 d2 c E cE  1 k c 2 50 1 E dy2 K

ð12:21Þ

y50

then cA 5 cA ;

dcE 50 dy

ð12:22Þ

y5δ

then cA 5 coA ;

cE 5 KcoA

ð12:23Þ

302

Basic Equations of the Mass Transport through a Membrane Layer

According to the boundary condition at y 5 δ, the components are in chemical equilibrium in the bulk liquid phase. The inlet mass transfer rate can be given for DA 5 DB, as (Danckwerts, 1970): JA 5

β o ð11 KÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 11fK tanhð ðDA k1 ð11 KÞÞ=ðβ o2 KÞÞ=ð ðDA k1 ð11 KÞÞ=ðβ o2 KÞÞg ð12:24Þ

The mass transfer rates for different diffusion coefficients and also for the case when the components are not in equilibrium at the inner edge of the boundary layer, can also be similarly expressed and easy to calculate (Nagy et al., 1983; Nagy and Ujhidy, 1989). The mass transfer rate entering the bulk phase can be expressed as JAδ 5 J2 JEδ

ð12:25Þ

where JEδ 5 KJAδ : Mass transfer rates with external mass transfer resistances: Note that Eq. (12.22) or (12.23) does not involve the external mass transfer resistances. If the flow is laminar, the thickness of the laminar boundary layer involves the whole lumen. Accordingly, the mass balance equation for lumen involves the effect of the external mass transfer resistance. When the flow is turbulent or intermediate from laminar to turbulent, the mass transfer resistance of the external phases can be characterized and described by the external mass transfer coefficient. The overall mass transfer rate can be expressed by the resistance-in-series model for the inlet phase. For the liquid phase, the reacted amount in the external boundary layer should be given and subtracting it from the Jδ value given by Eq. (12.18), the mass transfer rate entering the bulk liquid phase will be obtained. Note that the reaction modulus, ϑ; can differ essentially in the wetted pore and the laminar boundary layer, from each other, due to the different thicknesses and diffusivities (Figure 12.5).

12.2.1.3 Mass Balance Equations for the Lumen and Shell When the volume rate and the axial pressure drop is significant, the continuity, the motion, the mass, and the heat balance equations should be taken into account (see Chapter 7) for description of the mass transfer process. The simplified mass balance equations of the feed (lumen) and the shell sides (here gas phase), considering cylindrical coordinates, can be given as follows (Faiz and Al-Marzouqi, 2010; Sohrabi et al., 2011): For the lumen side, in steady-state condition and for component i:  2  @ ci 1 @ci @2 c i @ci 1 2 2u 2 Qi 5 0 1 Df;i @r 2 @z @z r @r

ð12:26Þ

Membrane Contactors

303

Gas (or liquid)

Liquid

c1o

c2*z

Figure 12.5 Mass transfer accompanied by chemical reaction with external mass transfer resistances.

c2o c1*

0

βf

0

δ1

δ β fδ

with, for fully developed laminar velocity profile, u 5 2uavg

 2 ! r 12 ro

ð12:27Þ

where u is the axial convective velocity (m/s); Qi is the reaction rate (kmol/m3 s); z is the axial space coordinate (m); uavg is the average axial velocity (m/s); ro is the inner radius of the hollow fiber (m); Df,i is the diffusion coefficient of i in liquid (m2/s); and L is the length of the hollow fiber (m). Boundary conditions for the lumen side are ci 5 coi @ci 50 @r Df;i

at z 5 0 or z 5 L at r 5 ro

@ci 5 Ji;δ @r

at r 5 ro 1 δ

ð12:28Þ ð12:29Þ

ð12:30Þ

The mass transfer rate of reactant entering the lumen phase, Ji,δ, depends on whether the membrane is nonwetted or partially wetted. For the nonwetted case, there is no reaction in the pores, thus, Ji 5 Ji,δ. Obviously, the cylindrical effect should also be taken into account [Eq. (12.10)] for capillary membrane. For a partially wetted case, the reactant will be reacted, partially in the pores, thus, Ji,δ , Ji. (For the case of fast, irreversible reaction Ji,δ 5 0 can often occur.)

304

Basic Equations of the Mass Transport through a Membrane Layer

For the shell side:  Dg;i

 @2 c i 1 @ci @ 2 ci @ci 1 2 2 us 50 1 2 @r @z @z r @r

ð12:31Þ

with (Faiz and Al-Marzouqi, 2010; Sohrabi et al., 2011): us 5 2us;avg

 2 ! r ðr=r3 Þ2 2 ðr2 =r3 Þ2 1 2 lnðr2 =rÞ 12 ro 3 1 ðr2 =r3 Þ4 1 4 lnðr2 =r3 Þ

ð12:32Þ

where Dg,i denotes the gas phase diffusion coefficient (m2/s); us is the shell side linear velocity (m/s); r2 represents the shell side’s radius (r2 5 ro 1 δ, where δ is the membrane thickness); and r3 is the Krogh radius (see Figure 9.5A; r3  rk). Boundary conditions for the shell side can be given as ci 5 cog;i

at z 5 0 or z 5 L

ð12:33Þ

@ci 50 @r

at r 5 r3

ð12:34Þ

2Dg;i

@ci 5 Ji @r

at r 5 r2

ð12:35Þ

Membrane Distillation 12.3

Introduction

Membrane distillation is a relatively new process that is attracting increasing interest because of its relatively low cost (Calabro et al., 1994; Jorgensen et al., 2004). In a membrane distillation process, a heated feed solution is brought into contact on the feed side of a hydrophobic, microporous membrane (Lawson and Lloyd, 1997). Due to the capillary force, water only enters the membrane pores if the pressure exceeds the so-called capillary force, which is determined by the membrane material, the pore size, and the surface tension of the water. The hydrophobic nature of the membrane prevents penetration of the aqueous solution into the pores, resulting in a vaporliquid interface at each pore entrance (Otaishat et al., 2008). In a membrane distillation process, the difference in temperatures at the membrane interfaces causes the differences in the vapor pressures of the components to be separated and thereby generate driving force for separation (Calabro et al., 1994; Soni et al., 2009). The liquid on the hot side (the feed side) vaporizes from the feed

Membrane Contactors

305

membrane surface, transports through the membrane pores to the membrane surface at the cold side, and condenses on the cold side. The separation can take place partly during the diffusion and partly due to the vaporliquid equilibrium conditions at the membranesolution interface. Any model for membrane distillation needs to be able to describe the effects of the boundary layers, transmembrane flux, vaporliquid equilibrium at the membrane interface and temperature effect. Vacuum membrane distillation process is a pressure-driven process, where the partial pressure difference across the membrane is maintained by creating a vacuum on the permeate side of the membrane. Depending on the pore size, the transport of components is due to the Knudsen diffusion or Knudsen-viscous diffusion (Bandini and Sarti, 2002; Soni et al., 2009). The boundary layer resistance can be modeled by temperature polarization and concentration polarization (Izquierdo-Gil and Jonsson, 2003). In a sweeping gas membrane distillation process, gas is passed on the permeate side to reduce the partial pressure of the permeant molecules on the permeate side. In osmotic membrane distillation, a hydrophobic membrane separates two aqueous solutions having different osmotic pressure. Water evaporates in the solution of higher chemical potential and the vapor crosses the membrane before being condensed in the solution of lower water potential (Nagaraj et al., 2006).

12.4

Mass Transport Through the Membrane

Mass transfer mechanism depends strongly on the pore size during membrane distillation, accordingly, different flux equations can be obtained. Basically, the following transfer rate can be defined, depending on the pore size (Soni et al., 2009).

12.4.1 Knudsen Limited Diffusion In a membrane with small pores (r{λ; λ is the mean free path of the diffusing molecules) the moleculepore wall collisions are dominant; therefore, the Knudsen equation can be used to describe the transport; the mass transfer rate can be given by Eq. (12.5). The flux is directly proportional to the difference of partial pressures.

12.4.2 Knudsen-Viscous Transition Diffusion The mean free path of diffusing chemical component, λ, is similar to the membrane pore diameter, consequently the convective flow can also exist during the transport. The Knudsen-viscous transition equation is as  rffiffiffiffiffiffiffiffiffiffiffiffiffiffi  8RTavg pi;avg 1 Ko ΔP Δpi 1 Bo Ji 5 πMi μ RTavg δ

ð12:36Þ

306

Basic Equations of the Mass Transport through a Membrane Layer

with Bo 5

r2 ε ; 8τ

Ko 5

1 dp 3

where Δpi is the partial pressure difference of transported component on the two sides of membrane (Pa); ΔP is the pressure difference (Pa); μ is the viscosity (kg/ms); and δ is the membrane thickness (m).

12.4.3 Knudsen-Molecular Diffusion For membranes with smaller air-filled pores (about ,0.5 μm, Soni et al., 2009), even the molecular diffusion can also effect the transport rate, thus both mechanisms should be taken into account. Accordingly [see Eq. (12.2)], Ji 5 2

    1 1 1 21 dpi 1 1 21 dci  2 1 1 dy dy RT Db;i DK;i Db;i DK;i

ð12:37Þ

12.4.4 Mass Transfer Correlation for Boundary Layer The external mass and heat transfer resistance can be especially important in turbulent flow regime. This is often the case during membrane distillation (Figure 12.6). Concentration and temperature polarization in the boundary layers are often the rate-limiting steps in liquid separations using membranes for mass and heat transfer, respectively. The β of mass transfer coefficient can be calculated from the Sherwood number as β of 5

DAB Sh dh

ð12:38Þ

where dh is hydraulic diameter. The Sherwood correlations can take three different forms depending on whether the flow is laminar, transitional, or turbulent, Sh 5 f ðRe; Sc; dh ; L; μÞ

ð12:39Þ

Re , 2,100 laminar regime:     dh 0:33 μbf 0:14 Sh 5 1:86 Re Sc L μwf

ð12:40Þ

Membrane Contactors

307

Cool water

Hot water Τ1o q

Τ2*

(heat flux) Τ1*

Τ2o

C1o J

C2* (Pm2)

(mass flux)

C2o

C1* (Pm1)

Hydrophobic membrane

Figure 12.6 Concentration and temperature distribution, with external mass transfer resistances, in membrane distillation process.

or the known Le´ve´que correlation (Skelland, 1974; Sisak et al., 2000):   dh 0:33 Sh 5 1:62 Re Sc L

ð12:41Þ

2,100 , Re , 10,000 transitional regime: Sh 5 0:116ðRe

2=3

2125ÞSc

0:33

 2=3 ! 0:14 dh μbf 11 L μwf

ð12:42Þ

Re.10,000 turbulent regime:  Sh 5 0:023Re2=3 Sc0:33 Re 5

uρdh ; μ

Sc 5

μbf μwf

ν ; DAB

0:14 ð12:43Þ dh 5

2wL w1L

where u is the convective velocity (m/s); μ is the dynamic viscosity (kg/ms); ν is viscosity (m2/s); w is the width of the membrane module (m); L is the length of the

308

Basic Equations of the Mass Transport through a Membrane Layer

Table 12.1 Values for Constant in Eq. (12.44) a 0.023 0.34 0.0096 0.0096

b

c

Re number

Author

0.8 0.75 0.931 0.33

0.33 0.33 0.346 0.33

Re . 105 104 , Re , 105 104 , Re , 105 Re . 104

Gekas and Hallstro¨m (1987) Gekas and Hallstro¨m (1987) Gekas and Hallstro¨m (1987) Berger and Hau (1977)

Table 12.2 Empirical Correlations for Shell Side Mass Transfer (Zeng et al., 2003) Author

Packing Density (%)

Re Number Range

Correlations

Wu and Chen (2000)

870

321,287

Sh 5 (0.3φ22 0.34φ 1 0.15) Re0.9 Sc0.33 Sh 5 [5.8(12 φ)dh/L] Re0.6 Sc0.33 Sh 5 (0.53 2 0.58φ) Re0.53 Sc0.33 Sh 5 0.09(12 φ) Re(0.48 1 0.16φ) Sc0.33 Sh 5 (0.163 1 0.27φ)Gz0.60 Sh 5 8(Re dh/L)Sc0.33

Prasad and Sirkar (1988) 4, 8, 20, 40

0500

Costello et al. (1993)

3276

25300

Gawronski and Wrzesinska (2000) Zeng et al. (2003) Dahuron and Cussler (1998)

21, 28, 52, 55, 65

0 10

20, 30, 40, 50 15

1781,194

φ is the packing density in percent; Gz is the Graetz number, Gz 5 (dh/L)Re Sc for mass transfer, or Gz 5 (dh/L)Re Pr for heat transfer.

membrane module (m); ρ is the density (kg/m3); and subscripts b, f, and w denote bulk, feed, and wall, respectively. For turbulent flow, several other correlations exist in the following form (Brookes and Livingston, 1995): Sh 5 a Reb Shc

ð12:44Þ

Values for a, b, and c are listed in Table 12.1. The correlations for the shell side mass transfer coefficient are listed in Table 12.2.

12.4.5 Heat Transfer Correlations The Nusselt number correlations again depend on the flow regime divided into three regions (Soni et al., 2009): Nu 5 f ðRe; Pr; dh ; L; μÞ

ð12:45Þ

Membrane Contactors

309

with hf 5

λNu ; dh

μC^ p λ

Pr 5

where hf is the heat transfer coefficient (J/Ksm2); λ is the thermal conductivity (J/mKs); C^ p is specific heat (J/kg K); and subscript f denotes the fluid phase. Re , 2,100 laminar regime:     dh 0:33 μbf 0:14 Nu 5 1:86 Re Pr L μwf

ð12:46Þ

2,100 , Re , 10,000 transitional regime:  2=3 ! 0:14   dh μbf 2=3 0:33 11 Nu 5 0:116 Re 2125 Pr L μwf

ð12:47Þ

Re.10,000 turbulent regime:  Nu 5 0:023Re

2=3

Pr

0:33

μbf μwf

0:14 ð12:48Þ

Geankoplis (1993) recommends the following correlation for turbulent flow regime:  Nu 5 0:027Re Pr 0:8

c

μbf μwf

0:14 ð12:49Þ

where c is 0.4 in case of heating and 0.3 for cooling. The heat transfer coefficient for the membrane can be calculated as h5

λ δ

ð12:50Þ

where λ and δ are thermal conductivity and thickness of the membrane, respectively. The heat transfer rate can be given through the membrane as (Martı´nez-Diez and Va´zquez-Gonza´lez, 2000; Otaishat et al., 2008): q 5 hðT1 2 T2 Þ 1 JðCp Tavg 1 ΔHe Þ

ð12:51Þ

where J is the total mass transfer rate (kg/m2 s); Cp is specific heat of vapor (J/kg K); Tavg is the average temperature in the pores (K); h is the membrane heat transfer coefficient; and ΔHe denotes the latent heat of vaporization (J/kg).

310

Basic Equations of the Mass Transport through a Membrane Layer

The heat transfer coefficient of the hydrophobic membrane can be calculated from the thermal conductivities of the hydrophobic membrane polymer (λ) and air trapped inside the membrane (λg). Thus (Otaishat et al., 2008), h5

λg ε 1 λð12 εÞ δ

ð12:52Þ

where δ and ε are the thickness and porosity of the hydrophobic membrane, respectively. The heat transfer rates for the boundary layers will be as follows: Through the feed solution thermal boundary layer (Figure 12.6):

q 5 hf T1o 2 T1 1 J C^ p Tavg;1 Through the permeate solution thermal boundary layer:

q 5 hf T2o 2 T2o 1 JC€p Tavg;2

ð12:53Þ

ð12:54Þ

Average temperature will be as TðT1o 1 T1 Þ=2:

12.5

Mass and Heat Balance Equations for the Lumen and Shell

The mass balance equations can be the same as given by Eqs (12.26) and (12.31) taken into account that there is no chemical reaction here (Qi 5 0). The inlet and the outlet mass transfer rates, in boundary conditions given by Eqs (12.30) and (12.35), are the same. This is determined by the mass transfer mechanisms through the membrane transport. The heat balance equation, for the both sides of membrane, can be given, for constant parameters, as  2 @T @ T 1 @T @2 T ^ 5λ 1 1 ρC p u @z @r 2 r @r @z

ð12:55Þ

Boundary conditions for the lumen (feed) side: T 5 T1o

at z 5 0

for all r

ð12:56aÞ

@T 50 @r

at r 5 0

for all z

ð12:56bÞ

2λf;1

@T 5 q at r 5 ro @r

for all z

ð12:56cÞ

Membrane Contactors

311

For the shell side: T 5 T2o

at z 5 0

@T 50 @r

at r 5 rk

λf;2

@T 5q @r

for all r for all z

at r 5 ro

for all z

ð12:57aÞ ð12:57bÞ ð12:57cÞ

where rk is the Krogh radius (m); λf,i is the thermal conductivity in the lumen (i 5 1) and the shell sides (i 5 2). When the radial convective velocity can also be taken into account, then the heat balance equation applied is as    2 @T @T @T 1 @T @2 T ^ 1 1υ 5λ 1 ρC p u @z @r @r 2 r @r @z

ð12:58Þ

Accordingly, the mass balance equation should also be modified. For boundary conditions, see Chapter 7.

References Bandini, S., and Sarti, G.C., (1999) Heat and mass transport in vacuum membrane distillation. AIChE J., 45, 14221433. Bandini, S., and Sarti, G.C. (2002) Concentration of must through vacuum membrane distillation. J. Membr. Sci. 149, 253259. Berger, F.P., and Hau, K.F. (1977) Mass transfer in turbulent flow measured by the electrochemical method. Int. J. Heat Mass Transfer 20, 11851194. Brookes, P.R., and Livingston, A.G. (1995) Aqueousaqueous extraction of organic pollutants through tubular silicone rubber membranes. J. Membr. Sci. 104, 119137. Calabro, V., Jiao, B.L., and Drioli, E. (1994) Theoretical and experimental study on membrane distillation in the concentration of orange juice. Ind. Eng. Chem. Res. 33, 18031808. Costello, M.J., Fane, A.G., Hogan, P.A., and Schofield, R.W. (1993) The effect of shell side hydrodynamics on the performance of axial flow hollow fiber modules. J. Membr. Sci. 80, 111. Criscuoli, A. (2009) Membrane contactors, in Membrane Operations, Ed. by, E., Drioli, and L., Giorno. Wiley-VCH, Weinheim, pp. 449460. Dahuron, L., and Cussler, E.L. (1998) Protein extraction with hollow fibers. AIChE J. 34, 130136. Danckwerts, P.V. (1970) Gas-Liquid Reactions, McGraw-Hill, Book Company, New York. Drioli, E., Criscuoli, A., and Curcio, E. (2006) Membrane Contactors: Fundamentals, Applications and Potentialities. Elsevier, Amsterdam.

312

Basic Equations of the Mass Transport through a Membrane Layer

Faiz, R., and Al-Marzouqi, M. (2010) CO2 removal from natural gas at high pressure using membrane contactors: model validation and membrane parametric studies. J. Membr. Sci. 365, 232241. Faiz, R., and Al-Marzouqi, M. (2011) Insights on natural gas purification: simultaneous absorption of CO2 and H2S using membrane contactors. Sep. Purif. Technol. 76, 351361. Gaeta, S. (2009) Membrane contactors in industrial applications, in Membrane Operations, Ed. by, E., Drioli, and L., Giorno. Wiley-VCH, Weinheim, pp. 499512. Gawronski, R., and Wrzesinska, B. (2000) Kinetics of solvent extraction in hollow-fiber contactors. J. Membr. Sci. 168, 213222. Geankoplis, C.J. (1993) Transport Processes and Unit Operations, 3rd ed. Prentice Hall, New York. Gekas, V., and Hallstro¨m, B. (1987) Mass transfer in the membrane concentration polarization layer under turbulent cross flow. I. Critical literature review and adaptation of existing Sherwood correlations to membrane operation. J. Membr. Sci. 30, 153157. Izquierdo-Gil, M.A., and Jonsson, G. (2003) Factors affecting flux and ethanol separation performance in vacuum membrane distillation. J. Membr. Sci. 214, 113130. Jorgensen, R.B., Meyer, A.S., Varming, C., and Johsson, G. (2004) Recovery of volatile aroma compounds from black currant juice by vacuum membrane distillation. J. Food Eng. 64, 2331. Lawson, K.W., and Lloyd, D.R. (1997) Review: membrane distillation. J. Membr. Sci. 124, 125. Li, J.L., and Cheng, B.H. (2005) Review of CO2 absorption using chemical solvents in hollow fiber membrane contactors. Sep. Purif. Technol. 41, 109122. Luis, P., Garea, A., and Irabien, A. (2009) Zero solvent emission process for sulfur dioxide recovery using a membrane contactor and ionic liquids. J. Membr. Sci. 330, 8089. Malek, A., Li, K., and Teo, W.K. (1997) Modeling of microporous hollow fiber membrane modules operated under partially wetted conditions. Ind. Eng. Chem. Res. 36, 784793. Martı´nez-Diez, L., and Va´zquez-Gonza´lez, M.I. (2000) A method to evaluate coefficients affecting flux in membrane distillation. J. Membr. Sci. 173, 225234. Marzouk, S.A.M., Al-Marzouqi, M.H., El-Naas, M.H., Abdullahtif, N., and Ismail, Z.M. (2010) Removal of carbon dioxide from pressurized CO2CH4 gas mixture using hollow fiber membrane contactors. J. Membr. Sci. 351, 2127. Mavroudi, M., Kaldis, S.P., and Sakellaropoulos, G.P. (2006) A study of mass transfer resistance in membrane gasliquid contacting processes. J. Membr. Sci. 272, 103115. Nagaraj, N., Patil, G., Babu, P.R., Hebbar, U.H., Raghavarao, K.S.M.S., and Neneb, S. (2006) Mass transfer in osmotic membrane distillation. J. Membr. Sci. 268, 4856. Nagy, E., and Ujhidy, A. (1989) Model of the effect of chemical reaction on bulk-phase concentrations. AIChE J. 35, 15641568. Nagy, E., Blickle, T., and Ujhidy, A. (1983) Untersuchung der Stoffu¨bertragung mit einer chemischen Reaktion erster Ordnung von endlicher Geswindigkeit nach einem einheitlichen Modell, I. Definition und Untersuchung der auf dem Haupstrom der Phasen bezogenen Stoffstro¨me bei irreversiblen und reversiblen Reaktionen mit Hilfe der Zweifilmtheorie. Chem. Techn. 35, 307310. Otaishat, M., Matsuura, T., Kruczek, B., and Khayet, M. (2008) Heat and mass transfer analysis in dierect contact membrane distillation. Desalination 219, 272292. Prasad, R., and Sirkar, K.K. (1988) Dispersion-free solvent extraction with microporous hollow-fiber modules. AIChE J. 34, 177188. Randwala, H.A. (1996) Absorption of carbon dioxide into aqueous solutions using hollow fiber membrane contactors. J. Membr. Sci. 112, 229240.

Membrane Contactors

313

Sisak, C., Nagy, E., Burfeind, J., and Schu¨gerl, K. (2000) Technical aspects of separation and simultaneous enzymatic reaction in multiphase enzyme membrane reactors. Bioprocess Eng. 23, 503512. Skelland, A.H.P. (1974) Diffusional Mass Transfer. John Wiley and Sons, New York. Sohrabi, M.R., Marjai, A., Moradi, S., Davallo, M., and Shirazian, S. (2011) Mathematical modeling and numerical simulation of CO2 transport through hollow-fiber membranes. Appl. Math. Model. 35, 174188. Soni, V., Abildskov, J., Jonsson, G., and Gani, R. (2009) A general model for membrane-based separation process. Comput. Chem. Eng. 33, 644659. Strahmann, H., Giorno, L., and Drioli, E. (2006) An Introduction to Membrane Science and Technology. Institute on Membrane Technology, CNR-ITM, Rende, Italy. Seidel-Morgenstern, A. (2010) Membrane Reactors. Wiley-VCH, Weinheim. Wang, R., Li, D., and Liang, D. (2004) Modeling of CO2 capture by three typical amine solutions in hollow fiber membrane contactors. Chem. Eng. Process 43, 849856. Westerterp, K.R., Swaaij, W.P.M., and Beenackers, A.A.C.M. (1993) Chemical Reactor Design and Operation. John Wiley and Sons, New York. Wu, J., and Chen, V. (2000) Shell-side mass transfer performance of randomly packed hollow fiber modules. J. Membr. Sci. 172, 5974. Zeng, J.M., Xu, Y.Y., and Xu, Z.-K. (2003) Shell side mass transfer characteristics in a parallel flow hollow fiber membrane module. Sep. Sci. Technol. 38, 12471267. Zhang, H.-Y., Wang, R., Liang, D.T., and Tay, J.H. (2006) Modeling and experimental study of CO2 absorption in a hollow fiber membrane contactor. J. Membr. Sci. 279, 301310. Zhang, H.-Y., Wang, R., Liang, D.T., and Tay, J.H. (2008) Theoretical and experimental studies of membrane wetting in the membrane gasliquid contacting process for CO2 absorption. J. Membr. Sci. 308, 162170.

Appendix

A.1

Numerical Solution of Parabolic Equations

Look at the so-called convection
diffusion mass balance equation as (Cebeci et al., 2005): D

@2 c @c @c 2u 5 @y2 @y @t

ðA:1Þ

If the convective velocity is zero, then Eq. (A.1) will be the Fick’s II equation. Applying the finite difference method (central difference, forward difference, and backward difference, Cebeci et al., 2005, p. 99), the difference approximation of @c=@y follows as in the case of backward difference:  @c 1
n 5 ci 2 cn21 i @t Δt

ðA:2Þ

Similarly, the difference approximations of @2 c=@y2 and @c=@y follow as  @2 c 1
n 5 ci11 2 2cni 1 cni21 2 2 @y Δy

ðA:3Þ

 @c 1
n 5 ci11 2 cni @y Δy

ðA:4Þ

and

where subscripts n denote the time steps, while i denotes the steps for the space coordinate. Numerical solution of a second-order, unsteady-state differential equation with constant parameters can be given by the following algebraic equation system given in vector notation: Ac 5 s Basic Equations of the Mass Transport through a Membrane Layer. DOI: 10.1016/B978-0-12-416025-5.00019-3 © 2012 Elsevier Inc. All rights reserved.

ðA:5Þ

316

Appendix

where the (N 11)-dimensional vectors are      c0   s0       c1   s1       c   ^ s 5  ^    cN21   sN21       cN   sN 

ðA:6Þ

Note that the concentration on the membrane interface differs from that of bulk concentration because of the external mass transfer resistances on both sides of the membrane layer; consequently, the algebraic equations at place of y 5 0 and y 5 δ are also necessary for the correct solution. The Thomas algorithm (Cebeci et al., 2005, p. 108) contains N 21 algebraic equation because it is not calculated with the concentration change in the fluid boundary layer. The (N 11)-order matrix for A, with nonzero elements (called the tridiagonal matrix) is   0   b0 e 0    a1 b1 e 1    ?   ðA:7Þ A5  ?     aN21 bN21 eN21     0 aN bN The solution of Eq. (A.5) can be obtained by the Thomas algorithm, which has two sweeps, namely a so-called forward and backward sweep (for details, see Cebeci et al., 2005, p. 108). We recommend a more simple solution for the value of concentration at y 5 0, namely c0, which contains forward sweep only (N is the number of the sublayer, δ/Δy): c0 5

ξN ð21ÞN ξN

ðA:8Þ

where ξi 5 ri 2

ai  bi  ξi22 2 ξ ei22 ei21 i21

with i 5 2; . . . ; N

ðA:9Þ

with ξ0 5 s0 ;

ξ1 5 s1 2

b1  ξ e0 0

and ξi 5 2

ai bi ξ 1 ξ ei 22 i 22 ei21 i21

with i 5 2; . . . ; N

ðA:10Þ

Appendix

317

with ξ 0 5 b0 ;

ξ1 5

b1 ξ 2 a1 e0 0

Note that knowing the value of c0, the value of c1 to cN can now be calculated by the algebraic equations given by Eq. (A.5), from step by step. It is important to note that the calculation should be made with very high accuracy (it is recommended accuracy of 10-14 decimals) in order to get back the correct value of the bulk concentration defined by the boundary condition at y 5 δ. Example A.1. Partial differential equation to be solved is as D

@2 φ @φ 5 @y2 @t

ðA:11Þ

The initial and boundary conditions are (the concentration distribution and the important notations are illustrated in Figure A.1; Hc 5 φ as well as Hcδ 5 φδ Þ : t 5 0;

c 5 cst

t . 0;

β o1 ðco 2 c Þ 5 2D

t . 0;


 @φ β of δ cδ 2 coδ 5 2D @y

ðA:12aÞ

for all y @φ @y

for y 5 0

ðA:12bÞ

for y 5 δ

ðA:12cÞ

Figure A.1 Illustration of mass transport with external resistances through a solid membrane layer.

δ

CAο

φ* φ1 φi

D1

Di

K1

Ki

ΔY1

ΔYi

φN

* DN φδ

Kn

ΔYN

ο CδA

318

Appendix

The numerical form of Eq. (A.11) can be given as j j 2ð2 1 Ψi Þφi j 1 φi21 5 2 Ψi φi j21 φi11

for i 5 1 to N 21

ðA:13Þ

with ψi 5

δ2 ΔY 2 D Δt

where i and j subscripts represent the steps in the space and time, respectively; Y 5 y/δ; δ is the membrane thickness (m). It is easy to get the necessary equation for a, b, e, and r, namely: b0 5

D 1 ; o 1 δΔYβ f H

ai 5 1;

aN 5 2

e0 5 2

bi 5 2ð2 1 ψi Þ; D ; δΔYβ o2

bN 5

D ; δΔYβ of

ei 5 1;

s0 5 c o

si 5 2 ψi φij21

1 D ; 1 H δΔYβ o2

with i 5 1; . . . ; N 21 ðA:14Þ

sN 5 coδ

Note that that concentrations obtained by Eq. (A.14) are not dimensionless concentrations. It is easy to get its dimensionless concentration by dividing by co. Example A.2. Let us look at the diffusion with cylindrical coordinates, namely:  2  @ φ 1 @φ @φ D 1 5 @r 2 r @r @t

ðA:15Þ

The initial and the boundary conditions can be given as t 5 0;

c 5 cst

t . 0;

β o1 ðco 2 c Þ 5 2D

t . 0;


 @φ β ofδ cδ 2 coδ 5 2D @r

ðA:16aÞ

for all r @φ @r

for r 5 r0 for r 5 r0 1 δ

ðA:16bÞ ðA:16cÞ

Applying the same difference quotient as in the case of the plane membrane layer, one can get the following parameter values for cylindrical coordinates as b0 5

βo 1 ; o 1 ΔRβ f H

e0 5 2

βo ; ΔRβ of

s0 5 c o

Appendix

319

ai 5 1 2 ei 5 1; aN 5 2

  ΔR bi 5 2 2 2 1 ψi ; Ri

ΔR ; Ri

si 5 2 ψi φij21 βo ; ΔRβ o2

bN 5

ðA:17Þ

with i 5 1; . . . ; I 21 1 βo 1 ; ΔRβ o2 H

sN 5 coδ

with [see Eqs. (6.14)
(6.15b)] β o1 5

Df ; r1 lnðro =r1 Þ

β ofδ 5

Df ; ro ðro 1 δÞlnð1 1 δ2 =r2 Þ

βo 5

D ; ro lnð1 1 δ=ro Þ ΔR 5

δ 1 ro N

Example A.3. The transport model of a fluid phase in a capillary membrane can be expressed, under steady-state conditions, as follows (see Eq. (7.50)):  Df

   @2 c 1 @c r 2 @c 2 5 2uo ro 1 2 2 1 ro @z @r 2 r @r

ðA:18Þ

where z is an axial coordinate (m); uo the inlet axial convective velocity related to the total membrane area (m/s); c is the concentration in the fluid phase (kg/m3 or kmol/m3); and Df is the diffusivity of component in the fluid phase (m2/s). The numerical form of Eq. (A.18) can be treated similarly to that of the unsteady ones as Eq. (A.11) or (A.15). The initial and the boundary conditions can be given as z 5 0;

c 5 co

r 5 0;

@c 50 @r

r 5 r0 ;

β of ðco 2 c Þ 5 2 D

ðA:19aÞ

for all r

ðA:19bÞ

for all z @φ @r

for all z

ðA:19cÞ

Equation (A.18) in dimensionless form is given as Δc2 1 Δc 2υo ro2 Δc ð1 2 R2 Þ 1 2 50 ΔR2 DL R ΔR ΔZ where Z 5 z/L; L is the length of the capillary membrane (m); and R 5 r/ro.

ðA:20Þ

320

Appendix

For the ith sublayer, the following difference equation can be obtained by applying Eqs (A.2)
(A.4):     ΔR j ΔR j ci21 2 2 2 1 ψi cij 1 ci11 5 ψi cij21 12 Ri Ri

ðA:21Þ

Applying Eq. (A.21) and taking into account the external boundary conditions, parameters of the solution will be b0 5 1;

e0 5 21;

s0 5 0

ai 5 1 2

ΔR ; Ri

ei 5 1;

si 5 ψi cij21 ; i 5 1; . . . ; N 21

aN 5 1;

bN 5 21

  ΔR bi 5 2 2 2 1 Ψi ; Ri

where ψi 5

2  2ro2 υo  2 ðΔRÞ 1 2 Ri ; DL ΔZ

Ri 5 Ri21 1

ΔR  ΔRði 2 0:5Þ 2

This method gives the same results as the Thomas algorithm (Cebeci et al., 2005) but seems to the author that the calculation is more simple.

A.2

Analytical Approach to a Solution with Variable Parameters: Diffusion Plus Convection Accompanied by First-Order Chemical Reaction

Let us look at the solution of the diffusion plus convection accompanied by firstorder chemical reaction under steady-state conditions. Parameters of this differential equation can be variable, thus it will be as   d dφ dφ D 2υ 2 k1 φ 5 0 dy dy dy

ðA:22Þ

For the analytical solution of Eq. (A.22), it should be linearized (Figure A.2). The solution methodology of this type of differential equation was given by Nagy (2009a) for diffusive mass transport through membrane reactor and by Nagy (2008, 2009b) for diffusive plus convective mass transport with variable and constant parameters, respectively. In essentials, this solution methodology serves the mass transfer rate and the concentration distribution in closed, explicit mathematical

Appendix

321

Figure A.2 Illustration of the linearization’s method for analytical approach of the solution.

Membrane βo βof C*

Co

βfδo Φ*

Φ*δ , C*δ Cδo δf

δ fδ

δ

expression. For the solution of Eq. (A.22), the catalytic membrane should be divided by N sublayer, in the direction of the mass transport, that is perpendicular to the membrane interface, with thickness of Δδ (Δδ 5 δ/N) and with constant transport parameters in every sublayer. Thus, for the ith sublayer of the membrane layer, using dimensionless quantities, it can be obtained: Di

d2 φ dφ 2 ki φ 5 0; 1υ dy2 dy

yi21 , y , yi

ðA:23Þ

In dimensionless space coordinate: d2 φ dφ 2 Pei 2 ϑ2i φ 5 0 2 dY dr

ðA:24Þ

where ϑi 5

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi δ2 ki =Di ;

Pei 5

υδ2 Di

Let us introduce the following φ~ variable: φ~ 5 φe 2 Pei Y=2

ðA:25Þ

After a few manipulations, one can get the following differential equation to be solved: d2 φ~ 2 Θ2i φ~ 5 0 dR2

ðA:26Þ

322

Appendix

where rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pei 1 ϑ2i Θi 5 4 The solution of Eq. (A.26) can be easily obtained by well-known mathematical methods as follows: ~

φ 5 Ti eλi Y 1 Si eλi Y

ðA:27Þ

with Pei λ~ i 5 1 Θi ; 2

λi 5

Pei 2 Θi 2

The inlet and the outlet mass transfer rate can be easily expressed by means of Eq. (A.27). The overall inlet mass transfer rate, namely the sum of the diffusive and convective mass transfer rates, is given by   J 5 υφ

Y 50

2

 D1 dφ  D1 ðλ1 T1 1 λ~ 1 S1 Þ 5 δ dY Y 5 0 δ

ðA:28Þ

The outlet mass transfer rate is obtained in a similar way as Eq. (A.28) for Y 5 1: Jδ 5

 DN  ~ λN TN eλN 1 λ~ N SN eλN δ

ðA:29Þ

Parameters Ti and Si (i 5 1 to N) should be determined using the external and internal boundary conditions. Let us neglect the external mass transfer resistances, thus the algebraic equations to be obtained will be as [see also Eqs. (5.82)
(5.84)]: at y 5 0

φ  5 T1 1 S 1

at y 5 yi

Ti eλi Yi 1 Si eλi Yi 5 Ti11 eλi11 Yi 1 Si11 eλi11 Yi

at y 5 yi

at y 5 d

~



ðA:30Þ ~

ðA:31Þ

 D   ~ ~ i11 Ti λi eλi Yi 1 Si λ~ i eλi Yi 5 Ti11 λi11 eλi11 Yi 1 Si11 λ~ i11 eλi11 Yi Di ðA:32Þ ~

φδ 5 TN eλN 1 SN eλN

ðA:33Þ

Appendix

323

The value of T1 can be obtained by solving Eq. (A.34) according to the wellknown Cramer rule, namely: T1 5

Ω1 Ω

ðA:34Þ

where, applying the boundary conditions listed by Eqs (A.30)
(A.33), the values of Ω1 and Ω are as follows: Parameters of Charts A.1 and A.2 are as follows: ξi21 5 ehPei 2 Pei21 iYi =2 ;

εi21 5

Di ξ Di21 i21

ðA:35Þ

Values of λi, λ~ i ; and Θi are expressed by Eqs (A.26) and (A.27). The solution of Eq. (A.34) with Charts A.1 and A.2 needs several steps. For them, the well-known

1

1

eΘ1Y1

e− Θ1Y1

λ1eΘ1Y1

~ −Θ1Y1 λ 1e

− ξ1e Θ2 Y1 − ε1λ 2 e Θ2 Y1 eΘ2 Y 2

λ 1eΘ1Y1

− ξ1e− Θ2 Y1 ~ − ε1λ 2 e Θ2 Y1 e− Θ2 Y 2

~ λ 2 e−Θ1Y1

Ω=

− ξ 2 e Θ3 Y 2 − ε2 λ 2 e Θ3 Y 2

− ξ 2 e −Θ3 Y 2 ~ − ε2 λ 2 e Θ3 Y 2

eΘ3 Y 3

e−Θ3 Y 3

λ 3 eΘ3 Y 3

~ −Θ3 Y 3 λ 3e

− ξN −1e Θ N Y N −1 − ε n −1 λ N e PeN

Θ Y e N N −1

/ 2e Θ N

− ξN −1e −Θ N Yn −11 ~ − εN −1 λ N e Θ N Y N −1 e PeN / 2e − Θ N

Chart A.1 Determinant of the denomination of Eq. (A.34).

co

1

0

e−Θ1Y1

0

~ −Θ Y λ1e 1 1

− ξ1eΘ2 Y1 − ε1λ 2 eΘ2 Y1 eΘ2 Y2

λ1eΘ1Y1

− ξ1e−Θ2 Y1 ~ − ε1λ 2 eΘ2 Y1 e−Θ2 Y2

~ −Θ Y λ2 e 1 1

− ξ 2 e Θ3 Y 2 − ε2 λ 2 e Θ3 Y 2 eΘ3 Y 3

Ω1 =

λ3 eΘ3 Y 3

− ξ 2 e−Θ3 Y2 ~ − ε2 λ 2 e Θ3 Y2 e− Θ3 Y3

~ −Θ Y λ3e 3 3

− ξ N − 1e Θ N Y N − 1 coδ

Chart A.2 The numerator of Eq. (A.34).

− εn −1λ N e Θ N Y N − 1 e Pe N / 2e Θ N

− ξ N − 1e−Θ N Yn −11 ~ − ε N −1λ N e Θ N Y N −1 e

Pe N / 2e− Θ N

324

Appendix

Cramer rules should be applied. For these steps, let us reform Chart A.1, obtaining the following expression: Ω 5 2N ePeN =2 Ω

ðA:36Þ

where the determinant Ω is given by Chart A.3.

Chart A.3 Simplification of determinant given in Chart A.1.

where notations ch(ΘiΔY) and sh(ΘiΔY) denote cosh(ΘiΔY) and sinh(ΘiΔY), respectively, and with: Ai 5

Pei sinhðΘi ΔYÞ 2 Θi coshðΘi ΔYÞ; 2

i 5 1; . . . ; N 2 1

ðA:37Þ

Bi 5

Pei coshðΘi ΔYÞ 2 Θi sinhðΘi ΔYÞ; 2

i 5 1; . . . ; N 2 1

ðA:38Þ

Finally, decreasing from step to step the number of rows and columns, essentially using the Gauss elimination, the value of Ω will be as O N PeN =2 Ω 5 ζO N ζ N21 2 e

  Dj11 2 L ξj Θj11 Dj j51

N21

ðA:39Þ

Appendix

325

This should be done with Chart A.2 and finally, one can get the value of T1 as ζT 1 φδ φ 2 T T1 5 NO N ζ N 2 coshðΘ1 ΔYÞ ζ N ePeN =2 Lj 5 2 coshðΘj ΔYÞ

! ðA:40Þ

where *

j tanhðΘi ΔY Þ ψi21 Di21 Pei ζ ij 5 1 2 2 j 2 Θi ζ i21 Di

!+

ζ ji21 with j 5 T; S; O; i 5 2; . . . ; N ξi21 ðA:41Þ

* ψij 5

j Ai ψi21 Di21 Pei Bi 2 2 j Θi ζ i21 Di 2

!+

ζ ji21 with j 5 T; S; O; i 5 2; . . . ; N 2 1 ξ i21 ðA:42Þ

Ai 5

Pei tanhðϑi ΔYÞ 2 Θi ; i 5 1; . . . ; N 2 1 2

ðA:43Þ

Bi 5

Pei 2 Θi tanhðϑi ΔYÞ; i 5 2; . . . ; N 2 1 2

ðA:44Þ

as well as ξi21 5 ehPei 2 Pei21 iYi =2

ðA:45Þ

The initial values of ζ ji and ψji ; namely ζ 1j and ψ1j ( j 5 T, O) are as ξT1 5 e2Θ1 ΔY ; ψT1 5 λ~ 1 e2Θ1 ΔY ;

ζO 1 5 2tanhðΘ1 ΔYÞ

ðA:46Þ

ψO 1 5 2A1

ðA:47Þ

Now, knowing the value of parameter T1, the S1 value can be calculated by Eq. (A.30) and from that all Ti and Si parameters (i 5 2, . . ., N) can be predicted by means of the internal boundary conditions, namely by Eqs (A.31) and (A.32). The values of Ti and Si should be calculated with great accuracy (it is recommended to calculate all parameters used to about 10
14 decimals in our experience). Additionally, we give the value of S1 predicted similarly to that of T1 as ζS 1 φδ φ 2 S S1 5 NO N ζ N 2 coshðΘ1 ΔYÞ ζ N ePeN =2 Lj 5 2 coshðΘj ΔYÞ

! ðA:48Þ

326

Appendix

The value of ζ SN can be calculated by Eq. (A.41) with the following initial values: ζ S1 5 eΘ1 ΔY ;

ψS1 5 λ1 eΘ1 ΔY

ðA:49Þ

Other parameters are the same as in the case of T1.

A.2.1

The General Solution of This Type of Differential Equation with Variable Parameters

Applying a general second-order differential equation with variable parameters and with boundary conditions expressed by Eqs. (A.50)
(A.53), the general form of algebraic equation: φ 5 φ ;

Y 50

dφi dφ 1 Pei φi 5 2 i11 1 Pei11 φi11 dY dY    φi Y 5 Yi1 5 φi11 Y 5 Yi2 at Y 5 Yi 2

φ 5 φδ ;

ðA:50Þ at Y 5 Yi

Y 51

ðA:51Þ ðA:52Þ ðA:53Þ

algebraic equation system can be given as follows: P0 5 a1 T1 1 a~1 S1

ðA:54aÞ

P1 5 b1 T1 1 b~1 S1 1 a2 T2 1 a~2 S2

ðA:54bÞ

P~1 5 c1 T1 1 c~1 S1 1 b2 T2 1 b~2 S2

ðA:54cÞ

P2 5 c2 T2 1 c~2 S2 1 a3 T3 1 a~3 S3

ðA:54dÞ

P~2 5 d2 T2 1 d~2 S2 1 b3 T3 1 b~3 S3 ^

ðA:54eÞ

Pi 5 ci Ti 1 c~i Si 1 ai11 Ti11 1 a~i11 Si11

ðA:54fÞ

P~i 5 di Ti 1 d~i Si 1 bi11 Ti11 1 b~i11 Si11 ^

ðA:54gÞ

PN21 5 cN21 TN21 1 c~N21 SN21 1 aN TN 1 a~N SN

ðA:54hÞ

P~N21 5 dN21 TN21 1 d~N21 SN21 1 bN TN 1 b~N SN

ðA:54iÞ

Appendix

327

PN 5 cN TN 1 c~N SN

ðA:54jÞ

The value of T1 can be obtained by solving Eq. (A.55) using the solution of the determinant of the denominator of Eq. (A.55) and of the determinant of the nominator of Eq. (A.55), according to the well-known Cramer rule, namely: T1 5

Ψ1 Ψ

ðA:55Þ

where ψ and ψ1 are given in Charts A.4 and A.5.

a1 b1 c1

a~1 ~ b1 a2 ~ c1 b2 c2 d2

a~2 ~ b2 ~ c2 a3 ~ d2 b3

a~3 b3

Ψ= cN−2 dN−2

c~N−2 ~ dN−2

aN−1 bN−1 cN cN

a~N−1 ~ bN−1 c~N ~ dN

a~N ~ bN c~

aN bN cN

N

Chart A.4 Determinant of the denomination of Eq. (A55)

P0 P1 ~ P1 P2 ~ P2

a~1 ~ b1 a2 c~1 b2 c2 d2

a~2 ~ b2 c~2 a3 ~ d2 b3

a~3 b3

Ψ1 = PN−2 ~ PN−2 PN−1 ~ PN−1 PN

Chart A.5 The numerator of the Eq. (A.55)

cN−2 dN−2

c~N−2 ~ dN−2

aN−1 bN−1 cN−1 cN−1

a~N−1 ~ bN−1 ~ cN−1 ~ dN−1

aN bN cN

a~N ~ bN ~ c N

328

Appendix

Applying the well-known rules of a determinant, the value of T1 can be obtained as follows: T1 5

ζ TN a1 ζ O N

ζ Ti 5

    β i bi T ci T ζ i21 2 ψTi21 2 ζ i 22 1 Si ; α i ai ai

with

ðA:56Þ

i 5 2; 3; . . . ; N

ðA:57Þ

i 5 2; 3; . . . ; N 2 1

ðA:58Þ

and ψTi

    γ i bi T di T T ~ 5 ζ 2 ψi21 2 ζ 1 Si ; αi ai i21 ai i 22

where the starting values of ζ Ti and ψTi are as b~1 S0 2 S1 ; a~1

ζ T1 5

ψT1 5

c~1 S0 2 S~1 a~1

ðA:59Þ

The value of ζ O i has a similar form getting from solution of ψ determinant, namely: ζO i

  β i bi O ci O 5 ζ i21 2 ψi21 2 ζ O ; α i ai ai i21

i 5 2; 3; . . . ; N

ðA:60Þ

i 5 2; 3; . . . ; N 2 1

ðA:61Þ

and ψO i 5

  γ i bi O di O ζ i21 2 ψO ζ ; i21 2 α i ai ai i21

O with starting values of ζ O i and ψi are as

ζO i 5 α1 ;

ψO i 5 β1

ðA:62Þ

Values of αi, β i, and γ i are as αi 5

b~i bi 2 ; a~i ai

βi 5

γi 5

d~i di 2 ; a~i ai

i 5 2; 3; . . . ; N 2 1

c~i ci 2 ; a~i ai

i 5 1; 2; . . . ; N

ðA:63Þ

ðA:64Þ

Appendix

329

Another solution methodology is presented in Nagy’s paper (2008) where the determinants were solved starting from the Nth sublayer. Then, applying the known value of T1, the value of S1 can be obtained by means of the first boundary condition at X 5 0, namely: S1 5

P0 2 aT1 a~1

ðA:65Þ

The values of Ti and Si can then be obtained by means of the internal boundary conditions (i 5 2, . . ., N), the forward sweep. Thus, the values of Ti and Si from the equation system of (A.66) and (A.67) should be determined applying Eqs (A.68)
(A.70), as well. The following equations can be obtained easily with i 5 1, 2, 3,. . ., N 21 (see Sections 4.2.3 and 5.3.3): Ti11 5

ηi b~i11 2 η~ i a~i11 Yi11

ðA:66Þ

Si11 5

ηi ai11 2 η~ i bi11 Yi 1 1

ðA:67Þ

with ηi 5 Pi 2 ci Ti 2 c~i Si

ðA:68Þ

η~ i 5 P~i 2 di Ti 2 d~i Si

ðA:69Þ

as well as Yi11 5 ai11 b~i11 2 bi11 a~i11

ðA:70Þ

References Nagy, E. (2008) Mass transport with varying diffusion- and solubility coefficient through a catalytic membrane layer. Chem. Eng. Res. Design 86, 723
730. Nagy, E. (2009a) Mathematical Modeling of Biochemical Membrane Reactors, in Membrane Operations, Innovative Separations and Transformations, Ed. by, E., Drioli, and L., Giorno. Wiley-VCH, Weinheim, pp. 309
334. Nagy, E. (2009b) Basic equations of mass transfer through biocatalytic membrane layer. Asia-Pacific J. Chem. Eng. 4, 270
278. Cebeci, C., Shao, J.P., Kafyeke, F., and Laurendeau, E. (2005) Computational Fluid Dynamics for Engineers. Horizons-Springer, CA.