Diphasic equilibrium and chemical engineering

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Niveau: Supérieur, Doctorat, Bac+8
Diphasic equilibrium and chemical engineering Franc¸ois JAMES ? Mathematical topics in fluid mechanics (Lisbon, 1991) J.-F. Rodrigues and A. Sequeira Eds., Pitman Res. Notes Math. Ser., 274 Longman Sci. Tech., Harlow, 1992, 246-250 1 Introduction Many processes in Chemical Engineering involve matter exchange between two phases in view of separate or analyze multicomponent mixtures. One can mention chromatography [5], [3], distillation [1], or electrophoresis. It is possible, under several hypothesis, to model these processes by a system of first order conservation laws. Consider a 1-dimensional diphasic medium in which phase 1 is moving with a velocity u, and phase 2 with velocity v. Assume u and v to be constant, u > 0 and v ≤ 0: we deal with a countercurrent process. We shall assume also that the whole process is isothermal. Thus the equations of momentum and energy are useless, and we are left with the conservation of matter. So, let c1 and c2 be vector-valued functions of x and t, related to the concentrations in phase 1 and 2 respectively. We have ∂t(c 1 + c2) + ∂x(uc 1 + vc2) = 0 (1) The system is presently underdetermined: we have n equations for 2n unknowns.

• modelisation mathematique des equilibres diphasiques et des colonnes de chromatographie


• diphasic equilibrium


• since g?

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• École polytechnique

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• Publié par: profil-zyak-2012
• Nombre de lectures: 7
• Langue: English

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Diphasic equilibrium and chemical engineering

∗ Fran¸coisJAMES

Mathematical topics in fluid mechanics (Lisbon, 1991) J.-F. Rodrigues and A. Sequeira Eds., Pitman Res.Notes Math.Ser.,274 Longman Sci.Tech., Harlow, 1992, 246-250

1 Introduction Many processes in Chemical Engineering involve matter exchange between two phases in view of separate or analyze multicomponent mixtures.One can mention chromatography [5], [3], distillation [1], or electrophoresis.It is possible, under several hypothesis, to model these processes by a system of first order conservation laws.Consider a 1-dimensional diphasic medium in which phase 1 is moving with a velocityu, and phase 2 with velocity v. Assumeuandvto be constant,u >0 andv≤deal with a countercurrent0: we process. Weshall assume also that the whole process is isothermal.Thus the equations of momentum and energy are useless, and we are left with the conservation of matter.So, 1 2 letcandcbe vector-valued functions ofxandt, related to the concentrations in phase 1 and 2 respectively.We have 1 21 2 ∂ t (c+c) +∂ x (uc+vc) = 0(1) The system is presently underdetermined:we havenequations for 2nunknowns. The closure is obtained by a fundamental assumption:we suppose the process to bequasi- static. Thismeans that, at each time, the two phases are at stable thermodynamical equilibrium. 1 2 This hypothesis introduces a non linear relation betweencandc, which we investigate in the next section.As we shall see, the system (1) will become a nonlinear system of conservation laws, which is proved to be hyperbolic.

2 Diphasicequilibrium We give here a few basic thermodynamical tools we shall use widely in the following. Consider two phases, denoted byi= 1,2, and M chemical species, or components, 1≤m≤ M. Weadopt the following convention throughout this paper:asuperscriptwill denote a i phase, and asubscripta chemical species.Namely,cis the amount of componentmin m phasei, fori= 1,2 and 1≤m≤M. Assume that both phases are at thermodynamical equilibrium.According to [2], this means 1. eachphase, considered as a simple thermodynamical system, is in equilibrium; ∗ CentredeMathe´matiquesApplique´es,EcolePolytechnique,F-91128PalaiseauCedex,FRANCE

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