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a c Mass fraction at the exit Volume averaged temperature. Volume averaged concentration. 0.5 0.4 0.3 0.2 0.1 0.0 -0.1 0.9 0.7 0.5 0.3 0.1 -0.1 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 ψ=1 ψ=0 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 t=1 t=0 Time t=10 t=30 82 Prediction macro Averaged micro 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Lines: prediction macro Points: averaged micro t=1 t=0 x t=10 t=30 Lines: prediction macro Points: averaged micro t=300 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 x t=300 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
Fig. 3-2b and Fig. 3-2c show the distribution of temperature and concentration in the medium at given times. Here also, one can observe the change of the concentration profile generated by the Soret effect compared with the isothermal case (c=0), and the microscopic model also perfectly fits the macroscopic results. These modifications are well matched with temperature profiles for each given time. 3.2.2 Diffusion and convection ( Pe 0, k ≈ 0) ≠ σ Next, we have imposed different pressure gradients on the system shown in Fig. 3-3. The temperature and concentration profiles at Pe = 1 for different times are shown in Fig. 3-3a and Fig. 3-3b, respectively. The results show a significant change in the concentration profile because of species separation when imposing a thermal gradient. Here, also, the theoretical predictions are in very good agreement with the direct simulation of the micro-scale problem. Comparison of the concentration elution curves at x=0.5, 7.5 and 13.5 in Fig. 3-3c between the two regimes (with and without thermal diffusion) also shows that the elution curve for no-thermal diffusion is different from the one with thermal diffusion. The shape of these curves is very different from the pure diffusion case (Fig. 3-2a) because, in this case, the thermal diffusion process is changed by forced convection. One also observes a very good agreement between the micro-scale simulations and the macro-scale predictions. 83
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while the effective thermal conduct
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propriétés thermiques, sont ainsi
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Remerciements Ma thèse, comme bien
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Toute mon amitié à Yohan Davit (l
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2.5 Transient conduction and convec
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List of tables Table 1-1. Flux-forc
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Fig. 2-11. Comparison of closure va
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Fig. 4-15. Composition-time history
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Chapter 1 General Introduction
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interested in the upscaling of mass
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possible to ignore the important ve
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• mesoscopic or macroscopic scale
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macroscopic scale. In these macrosc
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different temperatures. A temperatu
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1.4.3 Thermal Field-Flow Fractionat
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1.5.1 From the variation of thermal
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separated, the compounds are fed in
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1.6 Conclusion From the discussion
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2. Theoretical predictions of the e
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g Gravitational acceleration, m 2 /
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2.1 Introduction It is well establi
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However, for all these models many
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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
Page 73 and 74: 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
Page 89 and 90: 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: in the dimensionless system, for th
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
Page 129 and 130: 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
Page 135 and 136: Table 4-2. The properties of CO2, N
Page 137 and 138: Table 4-3. Molecular weight and Len
Page 139 and 140: Glass spheres d= 700-1000 μm ε=42
Page 141 and 142: Concentration of CO2 in bottom bulb
Page 143 and 144: increases with increasing concentra
Page 145 and 146: Concentration of CO 2 in bottom bul
Page 147 and 148: Concentration of N2 in bottom bulb
Page 149 and 150: convection between the two bulbs th
Page 151 and 152: 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.
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[106] H. Watts. Diffusion of krypto
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