Diphasic equilibrium and chemical engineering
François 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 (c1 + c2 ) + ∂x (uc1 + vc2 ) = 0
(1)
The system is presently underdetermined: we have n equations for 2n unknowns. The
closure is obtained by a fundamental assumption: we suppose the process to be quasistatic. This means that, at each time, the two phases are at stable thermodynamical
equilibrium.
This hypothesis introduces a non linear relation between c1 and c2 , 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
Diphasic equilibrium
We give here a few basic thermodynamical tools we shall use widely in the following.
Consider two phases, denoted by i = 1, 2, and M chemical species, or components, 1 ≤ m ≤
M . We adopt the following convention throughout this paper: a superscript will denote a
phase, and a subscript a chemical species. Namely, cim is the amount of component m in
phase i, for i = 1, 2 and 1 ≤ m ≤ M .
Assume that both phases are at thermodynamical equilibrium. According to [2], this
means
1. each phase, considered as a simple thermodynamical system, is in equilibrium;
∗
Centre de Mathématiques Appliquées, Ecole Polytechnique, F-91128 Palaiseau Cedex, FRANCE
1
2. the internal energy of the system constituted by the two phases is minimum, with
respects to several constraints.
It can be given a precise mathematical meaning to these two assumptions, following
the classical formalism of Gibbs. We cannot go into the details of modelling here, and
we refer for instance to [3] for such a work in the case of chromatography. Therefore our
starting point will be the following
Basic assumption. There exist two functions ηi : RM → RM , strictly convex, of class
C 2 , such that the equilibrium state is the unique solution of the constrained minimization
problem
min
η1 (c1 ) + η2 (c2 ).
1
2
c +c =const.
Let us just say that the existence of the ηi -s corresponds to assumption 1, and that the
constrained minimum property to assumption 2.
We shall denote in the following by µi the gradient of ηi , and by Dµi the matrix of its
second derivative:
µi (ci ) = ηi0 (ci ),
Dµi (ci ) = ηi00 (ci ).
By introducing the Lagrange multipliers corresponding to the constraint c1 + c2 =
const., one easily check that the equilibrium state is characterized by
µ1 (c1 ) = µ2 (c2 ).
(2)
This equality is nothing but the well-known equality of chemical potentials at equilibrium.
We intend to study the mathematical properties of this relation, and its consequences on
the system (1).
Notice first that, since η2 is strictly convex, µ2 is monotone on its domain of definition,
so that (2) can be solved as
c2 = h(c1 ).
(3)
Since the functions ηi are twice continuously differentiable, the function h is of class C 1 ,
and we denote by J(c1 ) = h0 (c1 ) its jacobian matrix, which will be called equilibrium
matrix of the system. But one has a little more.
Theorem 2.1 The equilibrium matrix is diagonable, and its eigenvalues αi , 1 ≤ i ≤ n,
are positive.
Proof. The matrix J can be written as the product of two symmetric positive definite
−1
Dµ1 (c1 ). Since η2 is strictly convex, the relation <
matrices: J(c1 ) = Dµ2 h(c1 )
u, v >2 = Dµ2 h(c1 ) u · v for u and v in Rn defines a scalar product on Rn . It is now easy
to prove that J is self-adjoint with respect to this scalar product, and therefore diagonable.
Let ri be an eigenvector of J, αi the corresponding eigenvalue: we have Dµ2 Jri = αi Dµ1 ri .
Taking the scalar product of this relation with ri leads to
αi =
Dµ1 ri · ri
,
Dµ2 ri · ri
and the positivity immediately follows from the strict convexity of η1 .
We have the following corollary, which allows us to deal with the total amount of
matter in the system, namely w = c1 + c2 .
Corollary 2.1 The mapping w → c1 + h(c1 ) is a C 1 -diffeomorphism from the equilibrium
manifold on itself.
2
Proof. The result comes from the properties of J. Indeed, one has dw = I + J(c1 ) dc1 ,
I being the identity matrix of Rn . Since J(c1 ) is diagonable, with positive eigenvalues,
the matrix I + J(c1 ) is also diagonable, with eigenvalues greater than 1. It is therefore
invertible, and there exists a function g of class C 1 such that c1 = g(w). The proof is
complete.
3
Conservation equations
Let us go back to system (1), to introduce the function h of (3):
∂t c1 + h(c1 ) + ∂x uc1 + vh(c1 ) = 0.
(4)
By Corollary 2.1, we can perform the variable change w = c1 +h(c1 ), which consists simply
in writing the conservation of total amount. The system (4) then becomes
∂t w + ∂x ug(w) + vh g(w) = 0.
(5)
Since g 0 (w) = (I + J)−1 , the jacobian matrix A(w) of (5) is given by A(w) = (uI +
vJ)(I + J)−1 . One deduces easily that A is diagonable, which insures the hyperbolicity of
(5). Moreover, the eigenvalues λi of A are given by λi = φ(αi ), with
φ(α) =
u + vα
.
1+α
The function φ is strictly monotone as soon as u 6= v, and, since α takes its values between
0 and +∞, the λi -s are uniformly bounded with respect to w. More precisely, we have the
Theorem 3.1 The system (5) is hyperbolic, its eigenvalues are uniformly bounded with
respect to w by
min(u, v) < λi (w) < max(u, v)
(6)
for w such that g(w) is in the domain of definition of h.
This result is natural: it means that non linear phenomena of interaction between
phases do slow the components with respect to the purely hydrodynamical propagation
which is linear (with velocity u or v). In other respects, we do not have any result about
strict hyperbolicity (distinct eigenvalues). The system (5) is well posed in the following
sense:
Theorem 3.2 The function η(w) = η1 g(w) + η2 h g(w) is a strictly convex mathematical entropy for system (5).
Proof. Let us first compute the first derivative of η. We have, after @Equil,
η 0 (w) = µ1 g(w) I + h0 g(w) g 0 (w)
= µ1 g(w)
by definition of g 0 .
We now want to verify that there exists an entropy flux q(w), i.e. a real function on
Rn such that
η 0 (w)A(w) = q 0 (w).
Replace A(w) by its value and write h0 (g(w)) = J. One obtains
η 0 (w)A(w) = µ1 g(w) (uI + vJ)(I + J)−1 .
3
Again apply the definition of g 0 (w), and use (2):
η 0 (w)A(w) = uµ1 g(w) g 0 (w) + vµ2 h g(w) h0 g(w) g 0 (w).
The form of the function q is now obvious:
q(w) = uη1 g(w) + vη2 h g(w) ,
and η is indeed an entropy of (1).
For the sake of brevity, we skip the proof of the convexity of η, which is obtained by
straightforward computation.
We give now, without any proof, two remarks concerning discontinuous solutions of (5).
First, consider the Riemann problem associated to (1). The behaviour of the characteristic
fields of A is therefore of some interest. Recall that the i-th field is genuinely non linear
(GNL) if λ0i (w) · ri (w) 6= 0, and linerarly degenerate if λ0i (w) · ri (w) ≡ 0. More generally,
we are interested in the behaviour of the eigenvalue
λi along the integral curve of the
0
0
0
eigenvector ri . We have λi (w) · ri (w) = φ α(w) α (w) · r(w). Since φ is strictly monotone,
we can state
Lemma 3.1 The i-th characteristic field of A has the same behaviour as the i-th characteristic field of the equilibrium matrix J.
Next, consider a piecewise C 1 weak solution propagating with velocity σ. One can
prove in a similar way as Theorem 3.1 the following
Theorem 3.3 The propagation velocity of discontinuities σ satisfies
min(u, v) < σ < max(u, v).
Again, the nonlinear propagation cannot be faster than the hydrodynamical linear propagation: we model retention phenomena.
References
[1] Canon E., Étude de deux modèles de colonne à distiller, Thèse de l’Université de
Saint-Etienne, 1990
[2] Gibbs J.W., On the Equilibrium of Heterogeneous Substances, Trans. Connecticut
Academy III, (1876), 108-248; (1878), 343-524 & Coll. Works, 55-353
[3] James F., Sur la modélisation mathématique des équilibres diphasiques et des colonnes
de chromatographie, Thèse de l’Ecole Polytechnique, 1990
[4] Kvaalen E., Neel L., Tondeur D., Directions of Quasi-static Mass and Energy Transfer
Between Phases in Multicomponent Open Systems, Chem. Eng. Sc., 40 (1985), no 7,
1191-1204
[5] Valentin P., Guiochon G., Propagation of Finite Concentration in Gas Chromatography, Separation Science, 10 (1975), 245-305
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