Basic equations of the mass transport through a membrane layer

001science......Page 1
Basic Equations of the Mass Transport Through a Membrane Layer......Page 2
Copyright......Page 3
Dedication......Page 4
Preface......Page 5
1.1.1 Transport of Dilute Solution......Page 7
1.1.2 Transfer Rate of Concentrated Feed Solution......Page 10
1.2 Transport Through Dense Membrane: Solution-Diffusion Theory......Page 12
1.3 Convective Transport Through a Porous Membrane Layer......Page 14
1.4 Component Transport Through a Porous membrane......Page 17
1.5 Application of the Maxwell–Stefan Equations......Page 21
1.5.1 The Maxwell–Stefan Approach to Mass Transfer in a Polymeric, Dense Membrane......Page 23
1.5.2 The Maxwell–Stefan Approach to Mass Transfer in a Ceramic (Zeolite) Membrane......Page 25
1.5.3 The Maxwell–Stefan Approach for Mass Transfer in Porous Media......Page 30
1.6 Flory–Huggins Theory for Prediction of the Activity......Page 31
1.6.1 Maxwell–Stefan Equation with the Flory–Huggins Theory......Page 34
1.7 UNIQUAC Model......Page 37
References......Page 38
2.1 Introduction......Page 41
2.3 Prediction of Diffusivities in Liquids......Page 43
2.4 Diffusion of an Electrolyte Solution......Page 44
2.5.1 Diffusion in a Dense Membrane......Page 45
2.5.2.1 Knudsen-Limited Diffusion......Page 46
2.6 Transport with Convective Velocity Due to the Component Diffusion......Page 47
2.7 Ion Transport and Hindrance Factors......Page 49
References......Page 50
3.1 Introduction......Page 51
3.2 Steady-State Diffusion......Page 52
3.2.1 Concentration-Dependent Diffusion Coefficient......Page 54
3.2.1.1 Exponential Concentration Dependency, D=Doexp(α˜ϕ)≡Doexp(αΦ)......Page 55
3.2.1.2 Linear Concentration Dependency, D=Do(1+α˜ϕ)≡Do(1+αΦ)......Page 58
3.2.1.3 Optional Concentration Dependency of the Diffusion Coefficient......Page 59
3.2.2 Concentration-Dependent Solubility Coefficient, H=H(c)......Page 60
3.2.2.1 Linear Solubility Dependency......Page 61
3.2.2.3 Dual-Sorption Model......Page 62
3.2.2.5 Flory–Huggins Model......Page 66
3.2.3 Mass Transfer Through a Composite Membrane......Page 67
3.2.4 Binary, Coupled Component Diffusion Transport......Page 68
3.2.4.1 Modeling of the Coupled Diffusion......Page 69
3.2.5.1 Binary Gas Separation by Zeolite Membrane......Page 73
3.2.5.2 Binary Transport for Pervaporation......Page 75
3.3 Nonsteady-State Diffusion......Page 76
3.3.1 Mass Transport with External Mass Transfer Resistance on the Feed Side......Page 81
3.3.2 Solution of Fickian Diffusion by Boltzmann’s Transformation......Page 83
References......Page 85
4.1 Introduction......Page 87
4.2.1 Mass Transport with an Intrinsically Catalytic Layer or a Membrane with Fine (Nanometer-Sized) Catalyst Particles (Ps.........Page 88
4.2.1.1 Reaction Terms for a First-Order Reaction......Page 89
4.2.1.2 Mass Transfer Accompanied by First-Order Reaction......Page 92
4.2.1.3 Mass Transfer in an Ultrafiltration Operating Mode, i.e., dϕ/dy=0 at y=δ......Page 95
4.2.2.1 Source Term for Zero-Order Reaction......Page 96
4.2.2.2 Mass Transfer for Zero-Order Reaction......Page 98
4.2.3 Mass Transfer Accompanied by Second-Order Reaction......Page 100
4.2.3.1 The Concentration Gradient Is Zero on the Outlet Surface, dϕB/dY=0 at Y=0......Page 104
4.2.4.1 With Sweep Phase on the Permeate Side, dϕ/dY0 at Y=1......Page 106
4.2.4.2 The Concentration Gradient Is Zero on the Outlet Surface, dϕ/dY=0......Page 108
4.2.5.1 Reactant Is Fed on the Sponge Side (Figure 4.11A)......Page 109
4.2.5.2 Reactant Is Fed on the Skin Side......Page 112
4.2.6 Mass Transfer with Micrometer-Sized, Dispersed, Catalyst Particles: Applying the Heterogeneous Model......Page 113
4.2.7 Approaching an Analytical Solution of the Mass Transport with Variable Parameters......Page 119
4.2.7.1 Mass Transfer in Ultrafiltration Mode, dϕ/dy=0 at y=δ......Page 122
4.3 Unsteady-State Diffusion and Reaction......Page 123
References......Page 125
5.2 Mass Transport Without Chemical Reaction......Page 127
5.3.1 Mass Transport Accompanied by First-Order Reaction......Page 133
5.3.1.1 Mass transport with polarization layer......Page 141
5.3.1.2 Remarks for Application of the Fick’s Diffusive Transfer Plus Convective Flow......Page 144
5.3.2 Mass Transport Accompanied by Zero-Order Reaction......Page 146
5.3.3 Mass Transport with Variable Parameters......Page 149
5.3.4 Mass Transport Through an Asymmetric Catalytic Membrane......Page 157
5.3.4.1 Reactant Is Fed on the Catalyst Layer......Page 158
5.3.4.2 Reactant Is Fed on the Noncatalytic Layer......Page 160
References......Page 161
6.2 Steady-State Diffusion......Page 163
6.2.1.1 Exponential Concentration Dependency, D=Doexp(α˜ϕ)......Page 166
6.2.1.2 Linear Concentration Dependency, D=Do(1+αΦ)......Page 167
6.2.1.3 Optional Concentration and/or Local-Coordinate Dependency of the Diffusion Coefficient......Page 169
6.2.2.1 Linear Concentration Dependency of the Sorption Coefficient, H=Ho(1+α˜c) with Ho(1+α˜c*)c*=ϕ*......Page 170
6.2.2.2 Langmuir-Type Sorption Isotherm......Page 171
6.2.2.3 Dual-Sorption Isotherm......Page 172
6.2.3 Mass Transfer Through a Composite Membrane......Page 173
6.3.1 Solution as a Bessel Function......Page 174
6.3.2 Analytical Approach for Solution......Page 175
6.3.3 Diffusion Accompanied by Zero-Order Reaction......Page 180
References......Page 181
7.1 Introduction......Page 182
7.2 Flow Models for Fluid Phases on Both Sides of Capillary Membrane Modules......Page 183
7.3.1 For the Axial Flow of an Incompressible Fluid in a Circular Tube with an Impermeable Wall (Figure 7.3)......Page 188
7.3.2 Flow Equations for Ultrafiltration in a Capillary Tube with a Permeable Wall (Song, 1998)......Page 189
7.3.3 Capillary Transport with Low Transverse Convective Velocity......Page 190
7.3.4 Mass Transport Through a Capillary Membrane: Component Model for the Feed Side......Page 194
7.3.5 Flow Models for Plane Membrane Modules (Rectangular Coordinates, x, y, z)......Page 196
References......Page 197
8.2 Membrane Reactor Configurations......Page 198
8.3 Reaction Rate......Page 200
8.4 Modeling of Membrane Reactors......Page 202
8.4.1 Modeling of a Membrane Reactor with a Catalytic Membrane Layer......Page 205
8.4.2 Some Typical Reactor Configurations: Packed-Bed Membrane Reactor......Page 207
8.4.3 Catalytic Membrane Reactor......Page 212
References......Page 214
9.1 Introduction......Page 217
9.2 Configurations of Membrane Bioreactors......Page 218
9.3 Enzyme Membrane Reactor......Page 219
9.3.1 Modeling of a Capillary Enzyme Membrane Reactor......Page 221
9.3.1.1 Momentum Transfer Inside a Membrane Bioreactor......Page 222
9.3.1.2 Component Mass Balances......Page 230
9.4 Mass Transfer Through a Biocatalytic Membrane Layer......Page 232
9.4.1.1 Feeding Is on the Sponge Side......Page 234
9.4.1.2 Feeding Is Done on the Skin Side of the Catalytic Membrane......Page 236
9.4.2 Modeling of Whole Cell Membrane/Biofilm Reactor......Page 238
9.4.2.2 Mass Transfer Rate for First- and Zero-Order Reactions......Page 240
9.4.2.3 Mass Transport Accompanied by Bioreaction of Monod Kinetics......Page 244
References......Page 248
10.2 Transport of Uncharged Solutes in Aqueous Solution......Page 252
10.3 Two-Layer Mass Transport: Coupled Effect of the Polarization and Membrane Layers (Nagy et al., 2011)......Page 256
10.3.1 Determination of the Concentration Distribution......Page 259
10.3.2 Determination of the Permeate Concentration, cp......Page 260
10.3.3 Solution by Means of Mass Transfer Rates of Layers......Page 261
10.4 Solvent-Resistant Nanofiltration......Page 263
10.5 Spiegler–Kedem Transport Model......Page 265
10.6 Nanofiltration of Ionic Components......Page 267
References......Page 268
11.2 Fundamentals of Pervaporation......Page 270
11.3 Solution-Diffusion Model for Pervaporation......Page 273
11.4 Basic Equations of the Polarization Model......Page 275
11.5 Simultaneous Effect of the Polarization and Membrane Layers......Page 276
11.5.1 Looking for the Connection Between ϕδ* and cp......Page 279
11.6 Concentration-Dependent Diffusivity......Page 284
11.6.1 Exponential Concentration Dependency, D=D0eαϕ......Page 285
11.6.2 Linear Concentration Dependency, D=D0 (1+αϕ)......Page 288
11.7 Coupled Diffusion......Page 290
References......Page 292
12.1 Introduction......Page 295
12.2.1 Diffusive Mass Transport Through the Membrane Pores......Page 297
12.2.1.1 Physical Mass Transport......Page 298
12.2.1.2 Mass Transfer Rate Accompanied by Chemical Reaction......Page 302
12.2.1.3 Mass Balance Equations for the Lumen and Shell......Page 304
12.3 Introduction......Page 306
12.4.2 Knudsen-Viscous Transition Diffusion......Page 307
12.4.4 Mass Transfer Correlation for Boundary Layer......Page 308
12.4.5 Heat Transfer Correlations......Page 310
12.5 Mass and Heat Balance Equations for the Lumen and Shell......Page 312
References......Page 313
A.1 Numerical Solution of Parabolic Equations......Page 316
A.2 Analytical Approach to a Solution with Variable Parameters: Diffusion Plus Convection Accompanied by First-Order Chemic.........Page 321
A.2.1 The General Solution of This Type of Differential Equation with Variable Parameters......Page 327
References......Page 330