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List of figures Fig. 1-1 Example of a multi-scale system ........................................................................... 7 Fig. 1-2. A schematic diagram of the two-bulb apparatus used to determine the thermal diffusion factors for binary gas mixtures ............................................................................ 11 Fig. 1-3. Principle of Thermogravitational Cell with a horizontal temperature gradient... 12 Fig. 1-4. Principle of Thermal Field-Flow Fractionation (ThFFF) .................................... 13 Fig. 1-5. Principle of forced Rayleigh scattering ............................................................... 14 Fig. 1-6. Diagram showing vertical section of the katharometer ...................................... 15 Fig. 1-7. Schematics of a Gas Chromatograph Flame Ionization Detector (GC-FID)...... 17 Fig. 1-8. Schematics of a Gas Chromatograph Electron Capture Detector (GC-ECD) ... 17 Fig. 1-9. Schematics of a simple mass spectrometer.......................................................... 18 Fig. 2-1. Problem configuration ......................................................................................... 28 Fig. 2-2. Normalized temperature versus position, for three different times (triangle, Direct β Numerical Simulation= ( T −T ) ( T −T ) σ = ( T −T ) ( T −T ) C H C β C H C ; circles, Direct Numerical Simulation ; solid line, Local-equilibrium model= ( T T ) ( T −T ) σ C H C XV − ...................... 44 Fig. 2-3. Normalized temperature versus position, for three different times (triangle, Direct β Numerical Simulation= ( T −T ) ( T −T ) σ = ( T −T ) ( T −T ) β C H C ; circles, Direct Numerical Simulation ; solid line, Local-equilibrium model= ( ) ( ) σ C H C T TC TH −TC − ...................... 46 Fig. 2-4. Chang’s unit cell .................................................................................................. 55 Fig. 2-5. Spatially periodic arrangement of the phases ...................................................... 59 Fig. 2-6. Representative unit cell (εβ=0.8).......................................................................... 60 Fig. 2-7. Effective diffusion, thermal diffusion and thermal conductivity coefficients at Pe=0..................................................................................................................................... 62 Fig. 2-8. Effective, longitudinal coefficients as a function of Péclet number ( k ≈ 0 and ε 0. 8 ): (a) mass dispersion , (b) thermal dispersion , (c) thermal diffusion and (d) Soret = β number................................................................................................................................. 65 Fig. 2-9. Comparison of closure variables b and Sβ x b for εβ=0.8 ............................... 66 Fig. 2-10. The influence of conductivity ratio (κ ) on (a) effective, longitudinal thermal conductivity and (b) effective thermal diffusion coefficients (εβ=0.8) ............................... 68 Tβ x σ
Fig. 2-11. Comparison of closure variables fields b Tβ and b Sβ for different thermal conductivity ratio ( ) κ at pure diffusion ( 0 & ε = 0. 8) Pe ................................................... 69 = β Fig. 2-12. Comparison of closure variables fields conductivity ratio ( ) XVI b and Tβ x κ at convective regime ( 14 & ε = 0. 8) = β b for different thermal Sβ x Pe ........................................... 70 Fig. 2-13. The influence of conductivity ratio (κ ) on the effective coefficients by resolution of the closure problem in a Chang’s unit cell (εβ=0.8 , Pe=0)............................ 71 Fig. 2-14. Spatially periodic model for solid-solid contact ................................................ 72 Fig. 2-15. Effective thermal conductivity for (a) non-touching particles, a/d=0 (b) touching particles, a/d=0.002, (εβ=0.36, Pe=0) .................................................................................. 72 Fig. 2-16. Spatially periodic unit cell to solve the thermal diffusion closure problem with solid-solid connections a/d=0.002, (εβ=0.36, Pe=0)............................................................ 73 Fig. 2-17. Effective thermal conductivity and thermal diffusion coefficient for touching particles, a/d=0.002, εβ=0.36, Pe=0..................................................................................... 74 Fig. 2-18. Comparison of closure variables fields b Tβ and b Sβ when the solid phase is continue, for different thermal conductivity ratio ( κ ) at pure diffusion................................ 75 Fig. 2-19. Effective thermal conductivity and thermal diffusion coefficient for touching particles, a/d=0.002, εβ=0.36 ............................................................................................... 76 Fig. 3-1. Schematic of a spatially periodic porous medium ( T H : Hot Temperature and T C : Cold Temperature)............................................................................................................... 79 Fig. 3-2. Comparison between theoretical and numerical results at diffusive regime and κ=0, (a) time evolution of the concentration at x = 15 and (b and c) instantaneous temperature and concentration field .................................................................................... 82 Fig. 3-3. Comparison between theoretical and numerical results, κ=0 and Pe=1, (a and b) instantaneous temperature and concentration field, (c) time evolution of the concentration at x = 0.5, 7.5 and 13.5 ....................................................................................................... 84 Fig. 3-4. Influence of the thermal conductivity ratio on the temperature and concentration fields .................................................................................................................................... 86 Fig. 3-5. (a) Temperature and (b) concentration profiles for different conductivity ratio 87 Fig. 3-6. Temporal evolution of the separation profiles for different thermal conductivity ratio...................................................................................................................................... 88 Fig. 3-7. Comparison between theoretical and numerical results at diffusive regime and κ=10, temporal evolution of (a) temperature and (b) concentration profiles ...................... 89
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: List of tables Table 1-1. Flux-forc
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
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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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