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Table of Contents 1. General Introduction ..............................................................2 1.1 Industrial interest of Soret effect ........................................................... 4 1.2 Theoretical Direct numerical solution (DNS) ....................................... 6 1.3 Theoretical upscaling methods .............................................................. 6 1.3.1 Multi-scale, hierarchical system .............................................................. 6 1.3.2 Upscaling tools for porous media ............................................................ 9 1.4 Experimental methods ......................................................................... 10 1.4.1 Two-bulb method................................................................................... 10 1.4.2 The Thermogravitational Column.......................................................... 12 1.4.3 Thermal Field-Flow Fractionation (ThFFF) .......................................... 13 1.4.4 Forced Rayleigh-Scattering Technique.................................................. 13 1.4.5 The single-beam Z-scan or thermal lens technique ............................... 14 1.5 Concentration measurement ................................................................ 14 1.5.1 From the variation of thermal conductivity ........................................... 15 1.5.2 From the variation of viscosity .............................................................. 16 1.5.3 Gas Chromatography (GC) .................................................................... 16 1.5.4 Analysis by mass spectrometer.............................................................. 18 1.6 Conclusion ........................................................................................... 19 2. Theoretical predictions of the effective diffusion and thermal diffusion coefficients in porous media ..........................21 2.1 Introduction.......................................................................................... 25 2.2 Governing microscopic equation......................................................... 27 2.3 Volume averaging method................................................................... 29 2.4 Darcy’s law .......................................................................................... 31 2.4.1 Brinkman term ....................................................................................... 31 2.4.2 No-linear case ........................................................................................ 32 2.4.3 Low permeability correction.................................................................. 33 XI
2.5 Transient conduction and convection heat transport ........................... 34 2.5.1 One equation local thermal equilibrium ................................................ 36 2.5.2 Two equation model............................................................................... 48 2.5.3 Non-equilibrium one-equation model.................................................... 49 2.6 Transient diffusion and convection mass transport ............................. 51 2.6.1 Local closure problem............................................................................ 53 2.6.2 Closed form............................................................................................ 56 2.6.3 Non thermal equilibrium model............................................................. 57 2.7 Results.................................................................................................. 59 2.7.1 Non-conductive solid-phase ( k ≈ 0 ) .................................................... 60 σ 2.7.2 Conductive solid-phase ( k ≠ 0)............................................................ 67 σ 2.7.3 Solid-solid contact effect ....................................................................... 71 2.8 Conclusion ........................................................................................... 76 3. Microscopic simulation and validation................................78 3.1 Microscopic geometry and boundary conditions ................................ 79 3.2 Non-conductive solid-phase ( k ≈ 0)................................................... 80 3.2.1 Pure diffusion ( 0, k ≈ 0) ≈ σ σ Pe ................................................................ 80 3.2.2 Diffusion and convection ( 0, k ≈ 0) Pe ............................................. 83 ≠ σ 3.3 Conductive solid-phase ( k ≠ 0) .......................................................... 85 3.3.1 Pure diffusion ( 0, k ≠ 0) σ Pe .............................................................. 85 ≈ σ 3.3.2 Diffusion and convection ( 0, k ≠ 0) Pe ............................................. 92 ≠ σ 3.4 Conclusion ........................................................................................... 97 4. A new experimental setup to determine the effective coefficients .....................................................................................99 4.1 Introduction........................................................................................ 102 4.2 Experimental setup ............................................................................ 103 4.2.1 Diffusion in a two-bulb cell ................................................................. 106 4.2.2 Two-bulb apparatus end correction ..................................................... 109 XII
Page 1: ��
Page 4 and 5: while the effective thermal conduct
Page 6 and 7: propriétés thermiques, sont ainsi
Page 8 and 9: Remerciements Ma thèse, comme bien
Page 10: Toute mon amitié à Yohan Davit (l
Page 15 and 16: List of tables Table 1-1. Flux-forc
Page 17 and 18: Fig. 2-11. Comparison of closure va
Page 19 and 20: Fig. 4-15. Composition-time history
Page 21 and 22: Chapter 1 General Introduction
Page 23 and 24: interested in the upscaling of mass
Page 25 and 26: possible to ignore the important ve
Page 27 and 28: • mesoscopic or macroscopic scale
Page 29 and 30: macroscopic scale. In these macrosc
Page 31 and 32: different temperatures. A temperatu
Page 33 and 34: 1.4.3 Thermal Field-Flow Fractionat
Page 35 and 36: 1.5.1 From the variation of thermal
Page 37 and 38: separated, the compounds are fed in
Page 39 and 40: 1.6 Conclusion From the discussion
Page 41 and 42: 2. Theoretical predictions of the e
Page 43 and 44: g Gravitational acceleration, m 2 /
Page 45 and 46: 2.1 Introduction It is well establi
Page 47 and 48: However, for all these models many
Page 49 and 50: At the fluid-solid interfaces there
Page 51 and 52: 2.4 Darcy’s law If we assume that
Page 53 and 54: where F is the Forchheimer correcti
Page 55 and 56: ε β ∂ β ( ρc ) T + ( ρc ) p
Page 57 and 58: Finally, we postulate the functiona
Page 59 and 60: ⎛ ⎜ k ~ . βσ ⎜ p ⎝ β Aβ
Page 61 and 62:
− ε V −1 σ ∫ Aβσ n βσ 2
Page 63 and 64:
( ε + ε k ) ( k − k ) ( ρcp )
Page 65 and 66:
We see on these figures a very good
Page 67 and 68:
If the local equilibrium assumption
Page 69 and 70:
epresentation of these coefficients
Page 71 and 72:
2.6 Transient diffusion and convect
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ε ε −1 β −1 β ⎛ ⎜ Dβ
Page 75 and 76:
Problem I for Chang’s unit cell 2
Page 77 and 78:
1 ∫ Vβ A βσ n βσ b Sβ dA =
Page 79 and 80:
2.7 Results In order to illustrate
Page 81 and 82:
Problem IIa ( k ≈ 0 ): closure pr
Page 83 and 84:
We have therefore found that the ef
Page 85 and 86:
a b c d β D * β D β xx * β ε
Page 87 and 88:
2.7.2 Conductive solid-phase ( k
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diffusion. Fig. 2-13 shows the effe
Page 91 and 92:
β ϕ ϕ * 1.6 1.4 1.2 1.0 0.8 0.6
Page 93 and 94:
Unfortunately we cannot use this ty
Page 95 and 96:
a) κ = 0. 001 , b Tβ b Sβ b) κ
Page 97 and 98:
Chapter 3 Microscopic simulation an
Page 99 and 100:
3.1 Microscopic geometry and bounda
Page 101 and 102:
in the dimensionless system, for th
Page 103 and 104:
Fig. 3-2b and Fig. 3-2c show the di
Page 105 and 106:
3.3 Conductive solid-phase ( k ≠
Page 107 and 108:
Temperature Concentration 1 0.9 0.8
Page 109 and 110:
a b k * = 1. 72ε k . Fig. 3-7a and
Page 111 and 112:
c a Volume averaged temperature. Vo
Page 113 and 114:
c a Volume averaged concentration.
Page 115 and 116:
If we define a new parameter, A S ,
Page 117 and 118:
3.4 Conclusion In order to validate
Page 119 and 120:
4. A new experimental setup to dete
Page 121 and 122:
N N i n The number of chemical spec
Page 123 and 124:
The main purpose of this part is to
Page 125 and 126:
Table 4-1. Thermal conductivity and
Page 127 and 128:
Two vessels containing gases with d
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4.2.2 Two-bulb apparatus end correc
Page 131 and 132:
then the thermal diffusion factor
Page 133 and 134:
4.3 Experimental setup for porous m
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Table 4-2. The properties of CO2, N
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Table 4-3. Molecular weight and Len
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Glass spheres d= 700-1000 μm ε=42
Page 141 and 142:
Concentration of CO2 in bottom bulb
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increases with increasing concentra
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Concentration of CO 2 in bottom bul
Page 147 and 148:
Concentration of N2 in bottom bulb
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convection between the two bulbs th
Page 151 and 152:
Table 4-11. The solid (spheres) and
Page 153 and 154:
Table 4-12. Porous medium tortuosit
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in Fig. 2-6. In this system, the ef
Page 157 and 158:
4.6 Conclusion In this chapter, we
Page 159 and 160:
5. General conclusions and perspect
Page 161 and 162:
this model. Therefore, the next ste
Page 163 and 164:
Temperature Concentration 1 0.8 0.6
Page 165 and 166:
moyennes sont comparées avec les p
Page 167 and 168:
dispositif expérimental avec une c
Page 169 and 170:
Appendix B. Estimation of the therm
Page 171 and 172:
References [1] A. Ahmadi, A. Aiguep
Page 173 and 174:
porous media. Lecture Notes of the
Page 175 and 176:
[51] B. Lacabanne, S. Blancher, R.
Page 177 and 178:
[80] M. Quintard, L. Bletzaker, D.
Page 179:
[106] H. Watts. Diffusion of krypto
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