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Fig. 3-8. Effect of thermal conductivity ratio at diffusive regime on (a and b) instantaneous temperature and concentration field at t=10 and (b) time evolution of the concentration at x = 15 ....................................................................................................... 91 Fig. 3-9. Comparison between theoretical and numerical results, κ=10 and Pe=1, (a) time evolution of the concentration at x = 0.5, 7.5 and 13.5 (b and c) instantaneous temperature and concentration field ........................................................................................................ 93 Fig. 3-10. Influence of Péclet number on steady-state (a) temperature and (b) concentration profiles (κ=10) .............................................................................................. 94 Fig. 3-11. Influence of Péclet number on steady-state concentration at the exit (κ=10).... 95 Fig. 3-12. Influence of (a) separation factor and (b) conductivity ratio on pick point of the concentration profile............................................................................................................ 96 Fig. 4-1. Sketch of the two-bulb experimental set-up used for the diffusion and thermal diffusion tests..................................................................................................................... 104 Fig. 4-2. Dimensions of the designed two-bulb apparatus used in this study .................. 104 Fig. 4-3. Katharometer used in this study (CATARC MP – R) ....................................... 105 Fig. 4-4. A schematic of katharometer connection to the bulb......................................... 106 Fig. 4-5. Two-bulb apparatus ........................................................................................... 106 Fig. 4-6. Katharometer calibration curve with related estimation of thermal conductivity values for the system He-CO2 ........................................................................................... 114 Fig. 4-7. Solute transport process in porous media .......................................................... 115 Fig. 4-8. Cylindrical samples filled with glass sphere..................................................... 118 Fig. 4-9. X-ray tomography device (Skyscan 1174 type) used in this study.................... 119 Fig. 4-10. Section images of the tube (inner diameter d = 0. 795cm) filled by different materials obtained by an X-ray tomography device (Skyscan 1174 type)........................ 119 Fig. 4-11. Composition-time history in two-bulb diffusion cell for He-N2 system for different medium. ( 300K T C 0 = and 100% c )..................................................................... 120 1 = b Fig. 4-12. Composition-time history in two-bulb diffusion cell for He-CO2 system for 0 different medium ( T = 300K and 100% c )....................................................................... 121 1 b = Fig. 4-13. Schematic diagram of two bulb a) diffusion and b) thermal diffusion processes ........................................................................................................................................... 122 Fig. 4-14. Composition-time history in two-bulb thermal diffusion cell for He-N2 binary mixture for different media ( T = 50K 0 Δ , T = 323. 7K and 50% XVII c ) .................................... 124 1 b =
Fig. 4-15. Composition-time history in two-bulb thermal diffusion cell for He-CO2 binary mixture for different media .( T = 50K 0 Δ , T = 323. 7K and 50% XVIII c ) ................................... 125 Fig. 4-16. New experimental thermal diffusion setup without the valve between the two bulbs .................................................................................................................................. 126 Fig. 4-17. Composition-time history in two-bulb thermal diffusion cell for He-N2 binary mixture for different media ( T = 50K 1 b = 0 Δ , T = 323. 7K and 61. 25% c ) ................................ 127 Fig. 4-18. Cylindrical samples filled with different materials (H: stainless steal, G: glass spheres and ε=42.5) ........................................................................................................... 128 Fig. 4-19. Katharometer reading time history in two-bulb thermal diffusion cell for He- CO2 binary mixture for porous media having different thermal conductivity (3 samples of stainless steal and 3 samples of glass spheres) ( T = 50K 1 b = 0 Δ , T = 323. 7K and 50% c )....... 129 Fig. 4-20. Cylindrical samples filled with different materials (A: glass spheres, B: aluminium spheres and ε=0.56)......................................................................................... 130 Fig. 4-21. Katharometer time history in two-bulb thermal diffusion cell for He-CO2 binary mixture for porous media made of different thermal conductivity (aluminum and glass spheres) ( Δ T = 50K , T 323. 7K 0 = and 50% 1 b = 1 = b c ) .................................................................. 131 Fig. 4-22. Definition of tortuosity coefficient in porous media, L= straight line and L’= real path length .................................................................................................................. 132 Fig. 4-23. Cylindrical samples filled with different materials producing different tortuosity but the same porosity ε=66% (E: cylindrical material and F: glass wool)........ 133 Fig. 4-24. Composition time history in two-bulb thermal diffusion cell for He-CO2 binary mixture in porous media made of the same porosity (ε=66% ) but different tortuosity (cylindrical materials and glass wool) ( T = 50K 0 Δ , T = 323. 7K and 50% c ) ................... 134 Fig. 4-25. Comparison of experimental effective diffusion coefficient data with the theoretical one obtained from volume averaging technique for different porosity and a specific unit cell................................................................................................................. 135 Fig. 4-26. Comparison of experimental effective thermal diffusion coefficient data with theoretical one obtained from volume averaging technique for different porosity and a specific unit cell................................................................................................................. 136 Fig. 4-27. Comparison of the experimental thermal diffusion ratio data with theoretical one obtained from volume averaging technique for different porosity and a specific unit cell ..................................................................................................................................... 136 1 b =
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 13 and 14: 2.5 Transient conduction and convec
Page 15 and 16: List of tables Table 1-1. Flux-forc
Page 17: Fig. 2-11. Comparison of closure va
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
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epresentation of these coefficients
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2.6 Transient diffusion and convect
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ε ε −1 β −1 β ⎛ ⎜ Dβ
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Problem I for Chang’s unit cell 2
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1 ∫ Vβ A βσ n βσ b Sβ dA =
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2.7 Results In order to illustrate
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Problem IIa ( k ≈ 0 ): closure pr
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We have therefore found that the ef
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a b c d β D * β D β xx * β ε
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2.7.2 Conductive solid-phase ( k
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diffusion. Fig. 2-13 shows the effe
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β ϕ ϕ * 1.6 1.4 1.2 1.0 0.8 0.6
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Unfortunately we cannot use this ty
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a) κ = 0. 001 , b Tβ b Sβ b) κ
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Chapter 3 Microscopic simulation an
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3.1 Microscopic geometry and bounda
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in the dimensionless system, for th
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Fig. 3-2b and Fig. 3-2c show the di
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3.3 Conductive solid-phase ( k ≠
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Temperature Concentration 1 0.9 0.8
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a b k * = 1. 72ε k . Fig. 3-7a and
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c a Volume averaged temperature. Vo
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c a Volume averaged concentration.
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If we define a new parameter, A S ,
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3.4 Conclusion In order to validate
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4. A new experimental setup to dete
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N N i n The number of chemical spec
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The main purpose of this part is to
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Table 4-1. Thermal conductivity and
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Two vessels containing gases with d
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4.2.2 Two-bulb apparatus end correc
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then the thermal diffusion factor
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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
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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
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Concentration of N2 in bottom bulb
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convection between the two bulbs th
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Table 4-11. The solid (spheres) and
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Table 4-12. Porous medium tortuosit
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in Fig. 2-6. In this system, the ef
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4.6 Conclusion In this chapter, we
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5. General conclusions and perspect
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this model. Therefore, the next ste
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Temperature Concentration 1 0.8 0.6
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moyennes sont comparées avec les p
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dispositif expérimental avec une c
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Appendix B. Estimation of the therm
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References [1] A. Ahmadi, A. Aiguep
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porous media. Lecture Notes of the
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[51] B. Lacabanne, S. Blancher, R.
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[80] M. Quintard, L. Bletzaker, D.
Page 179:
[106] H. Watts. Diffusion of krypto
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