Directions of quasi-static mass and energy transfer between phases in multicomponent open systems

Chemical EngimeringScience Printed in Great Britain.

Vol. 40. No. 7. pp. I191-1204.

1985

OOOS-2509/85 $3.00 + .OO Per&mm Press Ltd.

DIRECTIONS OF QUASI-STATIC MASS AND ENERGY TRANSFER BETWEEN PHASES IN MULTICOMPONENT OPEN SYSTEMS IMPLICATIONS ERIC KVAALEN,

IN

LAURENT

Laboratoire des Sciences du Genie Chimique,

SEPARATION NEEL

SCIENCE

and DANIEL

CNRS-ENSIC, France

TONDEUR*

1 rue Grandville,

(Received 30 September 1983; accepted 15 May

54042 Nancy,

Cedex,

1984)

Abstract-The present work deals with the dynamic and thermodynamic properties of multicomponent open systems involving equilibrium between two phases. We consider “quasi-static” composition changes that maintain equihbrium between these phases. Among all such possible changes at constant P and T, we define the so-calIed “characteristic directions” in the composition space, by the fact that along them, the ratio of the changes in mole number in the two phases is the same for all components. These directions are shown to arise naturally and to play an important role in the dynamics of multicomponent mass and energy transfer operations. The essential purpose of this paper is to demonstrate that these directions are always real, distinct, in number equal to the variance of Gibbs phase rule, and given by the eigenvectors of the matrix of second derivatives (the Hessian) of free enthalpy. The result extends to variable pressure and temperature, with entropy replacing free enthalpy. This property is established under the assumption of uniqueness and stability of the thermodynamic equilibrium in the domain considered. The relation of this problem with non-equilibrium multicomponent situations, such as diffusion, is discussed. INTRODUCTION: STATEMENTS OF THE FUNDAMENTAL RESULT

The purpose of the present paper is to demonstrate and illustrate a property of multicomponent twophases systems which has fundamental implications in multicomponent separation processes such as distillation, chromatography, extraction, etc. _. _ Let us first present a non-mathematical formulation of this property. Consider two phases a and b at thermodynamic equilibrium, and p different chemical species distributed between these phases. By adding differentially matter, energy or volume to the system, it is possible to change the state of the two phases while maintaining them at equilibrium with each other. Let us specify in addition that the ratio of the changes in state variables in the two phases is the same for all components. Then the possible changes are restricted to a finite number of directions in the compositionenergy-volume space (the state space) called churacteristic directions. These characteristic directions arise naturally in the solution of multicomponent mass and energy transfer problems. The fundamental result which we propose to discuss may then be expressed as: The number of distinct characteristic directions is equal IO she variance of the sysrem, as given b_v Gibb’s phase rule

*Author to whom correspondenceshould be addressed. 1191

A more mathematical formulation will result from the fact that the characteristic directions are given by the eigenvectors of a matrix derived from a thermodynamic function, such as the free enthalpy (at constant P and T), or more generally the entropy. Consider the special case where the two phases are composed of two totally immiscible “solvents” and p distributed solutes, and the equilibrium is considered at constant pressure and temperature. Let ci and qi be the molar concentrations of species i in phases a and b respectively defined with respect to the solvent. The p x p matrix (J) having as element

is called the Jacobian matrix of the equilibrium. Its fundamental property may be stated as follows: The Jacobian matrix of the equilibrium is diagonaeigenvalues, and p correbie; it has p real positive sponding independent eigenvectors. characteristic directions

which

define

the

In the following, we shall first discuss the significance of this property in relation to various physical situations, and specially to mass transfer and separation processes. Next we shall present a full demonstration of the property. Finally, we discuss further implications of this result, and illustrate it on an example of adsorption.

1192

E. SIGNIFICANCE

OF THE

KVAALEN

PROPERTY

To illustrate how this kind of problems arises, let us consider the example of multicomponent fixed-bed isothermal and isobaric adsorption, for which the so-called “equilibrium theory” has been extensively worked out, specially for Langmuir isotherms[l-51. The material balance equations are written as (l--n)$$+c~+u~=O

(i=1,2

)_..)

p).

(2)

Where t is the packing porosity, and u the apparent velocity, both assumed constant. These equations also imply radial homogeneity, unidirectional flow, no axial diffusion, and conservation of species. The adsorbed phase concentrations q, may be eliminated, assuming a known and differentiable equilibrium relationship: 4i = 4‘(C,, cz, . . . 3CJ-

(3)

The system of material balance becomes, in matrix-vector notations:

equations then

.$+ with C = (c,, c2, . . . , c,)‘,

(lM)+z

(M)$=O

(4)

ui = U/E and

(J)

+ (1)

(J) being the Jacobian matrix of equilibrium, of element qii = aq,/&, and (I) the unit matrix. The eigenvalues I of (TM) are simply related to the eigenvalues cr of (J) by 1 = ~7+ [e/(1 - =z)]. This system of equation is first-order, homog+ neous and quasi-linear. An important mathematical issue for constructing solutions is tien whether the system is totalIy hyperbolic, in other words, whether the eigenvectors are real and distinct. If this is the case, one is assured of the existence of non-trivial piecewise continuous solutions for important classes of initial and boundary conditions. Powerful mathematical tools may then be used to construct these solutions, specially the method of characteristics[6,7]. The existence and nature of singularities, for example loci or isolated points where two eigenvectors become

parallel (and the system ceases to be totally hyperbolic) strongly determines the overall dynamics of adsorption, as one of the present authors has shown [8-lo]. Some of these singularities may even furnish the basis of efficient separation processes[ 111. A similar problem arises in countercurrent operations, such as distillation: the number and nature of steady-states and of pinch points that may occur in a section of distillation column are dependent on the topology of the characteristic directions[l2], and the stability analysis of the steady-states relies on the positive definiteness of the Jacobian matrix[l3]. In most studies involving multicomponent equilibria, the hyperbolic character of the conservation equations is implicitely or explicitely assumed[l&ld].

et al.

In the special cases of Langmuir adsorption equilibrium, and of constant relative volatility in vapour-liquid equilibrium, the eigenvectors can be shown explicitly to be real and distinct [S, 171 except at isolated points on the borders of the composition space. In a quite general fashion, the dynamics of multicomponent separation processes involving phase equilibria are strongly determined by the topology of the characteristic directions, and it is therefore essential to establish the properties of these directions on thorough fundamental bases. DEMONSTRATION

The model

OF THE

PROPERTY

system

We propose to investigate quasi-static changes in an open two-phase multicomponent system, in the absence of chemical reaction. The first task is to define such a process and to make sure it is thermodynamically meaningful and consistent. Consider the system of Fig. 1, consisting of a vessel, in which two phases a and b are assumed to be in thermodynamic equilibrium with each other. A Aow of both phases in and out of the vessel is possible, as well as heat exchange with the surroundings. Phases a and b may undergo changes in composition, energy, volume, but we impose that they stay in equilibrium with each other; in other words, the state of the system goes through a continuous series of internal equilibrium states. Such a process is called a “quasi-static process”[l S]. Note that a reversible process is quasistatic, since it goes by definition through a series of equilibrium states. The reciprocal is not true: in the quasi-static process considered here, the phases may well not be in equilibrium with the surroundings, and some irreversible processes may occur. For example heat transfer from the surroundings could occur irreversibly (by radiation for example) while equilibrium between the phases is maintained. A second assumption is that each phase be uniform at all times (that is, have the same composition, temperature and pressure at all of its points). External fields (gravity, magnetic, etc. . .) are neglected. For these two assumptions to hold, the incoming flows may not be arbitrary. They must be slow enough and/or their composition must vary continuously in order to avoid non-uniformity of the phases or departure from equilibrium. Such ideal situations are more or less approached for example in chromatographic systems, or systems involving a large number of theoretical stages or of elementary contacts.

t Fig. 1. The open two-phase system.

Directions Internal

equilibrium

criteria for

of quasi-static

dS(t) = 0

(6)

< 0

(7)

being understood that the differential is taken at time t with respect to the variables, U, V and n, in other words:

+[ . . . . . . . . Iphase*=

1193

Owing to eqn (9), we have

open systems

Defining thermodynamic equilibrium in an open system is not a trivial matter and needs a careful discussion. In the present development, we make use of the notion of internal equilibrium, meaning that the phases present at any time in the vessel are in equilibrium with each other (according to criteria to be defined), but not necessarily with the surroundings. We could take a priori as criteria for this internal equilibrium the equality of chemical potentials, temperatures and pressures of the phases. However, it is not obvious that these criteria are meaningful under all conditions, and we shall gain insight into that matter by starting from a more general condition, stemming more directly from the second law of thermodynamics, and expressed for example as an extremal property of entropy or internal energy. Let us thus choose a set of independent extensive state variables (such as internal energy, volume and number of moles of each component), and define the state of internal equilibrium at any given time as the distribution between the phases of the matter, energy and volume present in the system at that time which maximises the entropy of the system. In other words, S is maximum with respect to all other possible states of the system involving the same overall composition, the same total internal energy, and the same total volume. But at any other time, the overall composition, total energy and volume may be different, since the system is open, and therefore the value of S _x will be different, and change with time. This can be expressed as:

d’s(t)

transfer

(8)

dU”+dUb=O;

dV”+dV’=O;

dn,+dm,=O

(11)

and there results that eqn (10) is identically verified only if T”=Tb;

J’“=Pb;

p;=p;

(12)

all these quantities being functions of time. We therefore conclude that it is legitimate to consider equalities 12 as definitions of internal phase equilibrium at any time for an open system, just as for a closed system, keeping in mind that in both cases, uniformity of each phase is assumed. The difference between closed and open systems thus lies in the fact that in the latter, the state of the system may vary in time as fluxes enter and exit the system, and S is therefore not maximal with respect to the variable t. Stability

criterion

of internal equilibrium

Inequality (7) expresses the local stability of the phase equilibrium with respect to perturbations of the independent variables U, V, n constrained by eqn (9) In other words, we express that any deviation from the equilibrium distribution of matter, energy and volume between phases (at constant total content) will decrease entropy. Let us emphasize two assumptions inherent to this treatment. First, we consider here local stability, that is stability with respect to small perturbations. Second, we assume stability with respect to the disappearance of a phase or the appearance of a new one: we consider there is always the same number of phases, specifically two. The important matter of phase stability has been discussed in detail for closed systems by Othmer[l9] and specially, Doherty[20, 211 from whom we shall borrow certain results. To express explicitly eqn (7), we differentiate eqn (8) with respect to the variables U,, U,, V,, V,, n, mi, constrained by eqn (9). The result may be expressed in compact form by writing: d2S = d’s” + d2Sb =

c c g, ij r,

d
1 Phase0

together with the constraints: U” + 157 = constant (t); ni + mi = constant

1Phase b

V” + Vb = constant (t);

(t)

(9)

Developing eqn (8) by using the relations between thermodynamic functions, we obtain:

dS(t)=~dUa+~dUb+~dV“+~dYb L? b -+dn,-x$dm,_O

(10)

40

(13)

where & may designate any of the variables U,, V,,ni and &any of the variables U,, Vb, mi. The summations thus extend over p + 2 variables, where p is the number of distributed components. In order to simplify the notations, we shall in the following restrict ourselves without loss of generality, to systems at constant temperature and pressure. Entropy may then be replaced by Gibbs free enthalpy in the formulation of the equilibrium criteria, since then dG = - T dS. Equations (8) and (13) thus become:

E. K~AALJZN et al.

1194

(14)

and d2G = 11

G$ dni dnj

t i

where

dG ani

(-1

P* TV=)

=

Pi

(16)

k Fig. 2. The composition space for a 3-component system, illustrating the relation between n and x vectors, and

showing the mole-fraction simplex.

is the chemical potential, and (17) Equation (15) may be expressed more concisely if we observe that the r.h.s. consists of two quadratic forms, which may be written in vector-matrix notation: d2G=dnT.(Gu)-dn+dm’.(G’).dm>O

(18)

(G) is the matrix of element G, given by eqn (17), in other words, the so-called Hessian of function G. This matrix is symmetric, owing to the commutativity of the differentiation in eqn (17). Geometric representation of phase equilibrium relatiqnships In the following, it will be useful to be able to visualize and give a geometric interpretation of some of the relations. We shall discuss this in a general manner, but illustrate it on the case of a threecomponent system. A mixture of p components at given P and T is described by a vector n = (n,, n2, . . . , nJ in the pdimensional space of mole numbers, or more accurately, in the part of that space restricted to the positive nj’s. Alternately, a mixture may be characterized by the end point of the vector. Letting n =Cn,

(19)

the set of all possible n’s defines a p-l dimensional hypersurface, which we call the a-manifold. Similarly, the possible m’s define the b-manifold (Fig. 3). The intersections of these manifolds with the mole fraction simplex define respectively the E-manifold, and the P-manifold, of dimension p - 2 (that is the so-called binodal curves for p = 3, as shown on Fig. 3). The equilibrium condition of eqn (12), +.J: - r” establishes a one-to-one correspondence (a bijection) between the a-manifold, and the &manifold. Geometrically, this bijection is represented by a set of tie lines. Composition vectors located in the space between the two manifolds represent unstable mixtures which split into equilibrated phases, and composition vectors outside this space and not on a manifold represent stable single phases (that is, not in equilibrium with another phase). In the space of the chemical potentials (Fig. 4), the end point of the p-dimensional vector pa - (cc,“, _ . . , p,,“) describes a p - 1 dimensional manifold. which we call the w”-manifold. Similarly, we define the hb-manifold for phase b. The intersection of cc“and pb is a p - 2 dimensional manifold, on which pia - Crib[22].

/I

A

%.m, t

5

we define the mole fractions xi and the corresponding vector x by x1=:;

x=(x

,,...,

x/J; Xx,=1.

(20)

Figure 2 shows, for p = 3, the relationship between the n and x vectors. The latter has its end point in the hyperplane X xi = 1, which we call, for brevity, the mole fraction- or x-simplex. Consider now two phases a and b in equilibrium with each other, and let n and x be respectively the mole numbers and mole fractions for phase a, and m and y that for phase b. Since the phases are constrained to be equilibrated,

n,.n, Fig. 3. The manifolds corresponding to two phases a and b in equilibrium (3 component system).

1195

Directions of quasi-static transfer

differential changes in composition in one phase and changes in composition in the other phase when equilibrium between these phases is maintained. Two questions then arise. The first is whether there is a one-to-one mapping (a bijcction) between the dm* and the dn*. This requires that the linear mappings associated with (GO) and (Gb) be invertible, and therefore be everywhere non-singular. The second question is whether dn* and dm* may be parallel. As we shall see, this condition defines the so-called characteristic directions. Fig. 4. The pa and p b manifoldsand theirintersectionin the space of chemical potentials. Relations

for

quasi-static

displacements

of an open

sys tern

Let n* and m* be two composition vectors reprtsenting phases a and b at equilibrium and lying in the a and b manifolds respectively, so that pP(n*) = p,*(m*)

(all i)

(21)

We define an infinitesimal quasi-static displacement as the set of vectors dn*, dm*, such that the compositions m* + dm* and n* •t dn* are also at equilibrium: pi”@* + dn*) = p”(m* + dm*)

(all i)

*

dn f = dpP

(23)

Substituting this expression, written for phase a and b, into eqn (22), and using (21), we obtain:

directions

Let us assume, for the time being, that the mappings defined by (Gil) and (G*) are invertible, implying the positive definiteness of matrices (Gd) and (G”) (we shall return to this delicate question later on). With this assumption, (G”) and (G”) have an inverse, and we may rewrite eqn (25) as: dm* = (G”-’

. (G”) . dn* = (M)

. dn*

(26)

and therefore there is a linear mapping between dm* and dn*. The characteristic directions are defined by the fact that dm* and dn* are parallel, in other words that the

ratio of the change in mole number in the two phases is the same for all components

(22)

All the possible vectors dn* are tangent to the a-manifold, and form a p - 1 dimensional subspace containing n*. (Similarly does the set of all possible dm*). Such quasi-static displacements are obtained as by adding or subtracting already mentioned, differentially matter to each phase. Let us now establish a general condition for quasistatic displacements. For this purpose, we develop the expressions in eqn (22) to the first order, in the form: p,(n* + dn*) - p,(n*) = 5 $f

Characteristic

dm*=,Idn*.

(27)

Combining eqns (26) and (27), we obtain (M) . dn* = 1 dn*

(29

an eigenvalue problem. The characteristic directions

are thus defined by the eigenvectors of matrix (M) = (G*)-’ - (G”). The issue is then whether these characteristic directions are always real and distinct, and form a basis of the corresponding vector space, implying that the matrix (M) is diagonable. Another issue is the positiveness of the eigenvalues, which implies (see eqn 27) that the mole numbers in the two phases vary in the same direction. Let us now examine these questions. of (M jpositiveness of the eigenvalues We know that real symmetric matrices have real eigenvalues and are diagonable. Unfortunately, (M) is not in general symmetric (unless (63-l and (G”) commute, which would be an extremely special case). In order to assess the diagonability, we need a deeper investigation.? Diagonabiiity

The vectorial form of this equality is dF” = dp* = dp * = (G”) . dn* = (G*) _dm*

(25)

The vector d/r* represents the chemical potential displacement corresponding to dn* and dm*, that is the image in the p-space of the quasi-static displacement. This vector is tangent to both the ,u” and p* manifolds, thus to their intersection which is a p - 2 dimensional manifold (Fig. 4) The vectors dn* and dm* are tangent to the a and b manifolds respectively. Equation (25) establishes a relationship between

TDr. Doherty (University of Massachusetts, Amherst) recently drew our attention on an unpublished demonstration by Prof. Davis (Univ. of Minnesota) that the product of two symmetric matrices is diagonable. Somewhat surprisingly, we have not found this property in usual advanced books on matrices. Hence the present demonstration, which also shows the special orthogonality property of the eigenvectors and the positiveness of the eigenvalues.

E. KVAALEN

1196

The following demonstration was inspired by Ramkrishna and Amundson[23] and is based on the so-called “spectral theorem”[24, 25] which states that self-adjoint compact operators are diagonable. This implies that the eigenvalues are all real. The corresponding eigenvectors form an orthogonal basis with respect to the inner product defining self-adjointness. Let us recall that an operator Z is said to be self-adjoint with respect to an inner product in an Euclidian space, if for any two vectors u and v of this space, one has (Z - u,v> = {u,Z

orthogonal with respect to the scalar product, that is: (zi, z,) = zir. (G”) . zj = 0 unless i =i.

where (w, w’> designates the inner product of w and w’. In particular, it is easily seen that real symmetric matrices represent self-adjoint operators with respect to the ordinary scalar product (u,v)=ur.v=~u,~i-

(30)

Other inner products may be defined, for example, “weighted” scalar products (u. v> = c riupi

(31)

such that some non-symmetric matrices define selfadjoint operators with respect to this product. For our present purpose, we shall use the scalar product defined by the bilinear form: (u, v) = UT - (G”) - v

(32)

(in the rest of this section, we omit the subscripts ij on (C)l. It is easily

verified that eqn (32) satisfies the requirements for an inner product in an Euclidian space, owing to the symmetric positive definite character of (GO), namely: (u,v)=u~-(G”)~v=v~.(G”).u=(v,u)

(33)

(u,u>=uT-(G=). u > 0 unless u = 0. ((M)

(M) . zi = (G*)-’ . (G=) . zi = ljzp

as

(37)

Multiplying this equality on the left successively by (Gb) and by ziT, we obtain z.r. , (G*) . zi = &zj’. (G*) . z,

(3’3)

from which we obtain: #JiZ

Z-~. (G") . zi > o ’

z,=. (G*)

. zi

.

The numerator and denominator are positive definite quadratic forms, by our initial assumption. Therefore the &‘s are positive definite; this establishes the second part of the sought result. The special case of immiscible matrix

solvents-the

Jacobian

of equilibrium

Consider the case where the two phases a and b are composed of two totally immiscible solvents, with p solutes distributed between the phases. Then we may define the concentration of each solute with respect to the volume or mass of the solvent in the respective phase (moles of solute per liter of solvent, for example). Relations analogous to eqns (25) and (26) may then be expressed in terms of the Jacobian matrix of equilibrium. Let us differentiate the equilibrium condition (21) with respect to n, for example along a quasi-static displacement (we leave out the *, keeping in mind that all the development below implies compositions m and n at equilibrium):

=c--. aw

LJ’G”

-=

am,am,

an,anj . u, v},

am k an,

(40)

There are p* such relations, one for each couple (i,j), which may be rewritten in matrix notation: (G$ = (GFk).

((M)~u,v)=(M-u)~.(G~).v =u T - @QT. (GO) . v

(36)

Let us show that the eigenvalues li of (M) are positive. By definition of the eigenvector zi, and using eqn (26), we have:

(29)

* v)

Now, let us consider the product defined by eqn (32):

ez al.

(41)

(34) or, in view of eqn (26):

Using eqn (26) to develop (M), we obtain ((M)

. u, v> = uF. (G”) . (G*)-’ . (G“). v = IJ=. (C”) . (M) . v = (u, (M)

(M) = . v). (35)

Therefore, by the definition of eqn (29), matrix (M) defines a self-adjoint operator with respect to the scalar product eqn (32). The spectral theorem thus guarantees that the eigenvalues are real and that the matrix is diagonable. This establishes the Iirst part of the sought result. In addition, the eigenvectors zi are

(42)

(mkjl

the matrices being of dimension p x p. With our assumption of immiscibility, the amounts A and B of solvent in each phase a and b are constant, and the quantities mki are simply related to the concentrations at equilibrium by m

Cif,.

(43)

1197

Directionsof quasi-statictransfer If the equilibrium distribution of the solutes is described by a vectorial relation between the concentrations in each phase

then the partial derivatives on the r.h.s. of eqn (43) are the elements of the Jacobian’ matrix (J) of this equilibrium relation and thus = (C*)-’

- (C=) =

E (J).

(4%

The properties established for (M), such as diagonability and positiveness of eigenvalues, also hold for the Jacobian (J). The characteristic directions are thus given by the p eigenvectors of (J); they are distinct and form a basis in the vectorial space of concentrations. The multicomponent

Gibbs-Duhem

relations

and their

implications

In the section on characteristic directions, we asthat (G”) and (G’) were positive definite, in other words that the quadratic form v.(G) . v was strictly positive for any non-zero vector v. We shall now see that this is not in general the case. The classical form for the Gibbs-Duhem relation, in the absence of external fields, is

sumed

-_ u

*dT+

(G”).a=n(G”).x=O (Gb) . m = m(G*) * y = 0.

(9

Q = Q(C)

(M)

The vectorial form of this set of equations, valid for both phases a and b is

VdP-_Cnidpc,=O

(46)

which for an isothermal, isobaric system, reduces to: xn,dpi=O.

(47)

This is not a thermodynamic property, but merely a property of the differential of a homogeneous function, the free enthalpy G, a linear form of ni and piA more explicit form is obtained by developing the differentials dpi as:

(Sl)

The first equality results from n = nx (eqn 20). These relations will play an essential role in the next sections. One of their consequences is that any bilinear form built from the above by multiplying on the left by some vector, is zero, and so are specially the quadratic forms n’.(GO)-n

= 0; mT-(G*)-m

xT-(G“).x

= 0; y’.(G&).y

= 0 = 0.

(52) (53)

Equations (52) imply that the matrices (G”) and (G*) are not positive definite and therefore not invertible on the whole space of mole numbers. The treatment leading from eqn (25) to eqns (26)-(28) is therefore not possible, and characteristic directions may not be defined in the vector space of mole numbers. We shall see below why this problem does not appear in the case of immiscible solvents and how to resolve it in the general case. Relation tions

between

Gibbs-Duhem

and stability

condi-

The stability of a single phase a may be expressed by the positiveness of the quadratic form built on (GO): d2G” = dnT - (Ga) a dn z 0

for dn # 0.

(54)

The meaning of this inequality appears more clearly when one decomposes dn into a change of total mole number n at constant composition x, and a change of composition at constant mole number: dn=ndx+xdn.

(55)

Substituting into eqn (54), we obtain: d2G”= ndxT - G” - ndx

and substituting in eqn (47), we obtain

CniCG,-dn, i j

=~dnj~n,G,=O

i

1

(49)

Since in eqns (48) and (49), the bfi’s are independent and arbitrary (there is no constraint of constant n), eqn (47) can only be identically verified if all the coefficients of the dnj’s are zero; in other words: c niGii = 0

for all j.

+ dnxT.

G” . xdn

+ndxT-GLI.xdn+dnx7.G“.ndx.

(56)

The second and third term on the r.h.s. are obviously zero by virtue of the Gibbs-Duhem equation (51). The last term is equal to its transpose, owing to the symmetry of (CT, and is thus also zero by virtue of Gibbs-Duhem. Inequality (54) then splits into dZG”=nZdx7~G”~dx>0 d’G’=O

fordx=O,

fordx#O dn #O.

(57) (58)

1199

Directionsof quasi-statictransfer independent extensive variables, in addition to the number of moles nj. Alternately, U could be used as the thermodynamic function, and S, I/ and q as the variables [ 181. The vector space to consider is then the p + 2 dimensional space of U, V, n, and the matrices (G) are replaced by matrices (S) which contain the derivatives of generalized chemical potentials:

a*s ap a(n, u, v) = a(n,u, v)*. The Gibbs-Duhem equation must be taken in its general form, eqn (46). The demonstration is otherwise unchanged, the only difference being that the number of characteristic directions is increased by two, with respect to the case at constant P and T. A well known special case of non-isothermal nonisobaric quasi-static displacement is that corresponding to the vapour-liquid equilibrium of a single component, governed by the Clapeyron equation: dP

-=-

dZf-

AH TdV

where AN is the enthalpy of vaporization, and AV the volume change corresponding to the vaporization of one mole at T. The Clapeyron equation defines the unique characteristic direction in the state space (P. T)Relation

with Gibbs’ phase

ruie

We have seen that, in the case of immiscible solvents, at constant P and T, there are p characteristic directions, that is a number equal to the dimension of the matrix (M), and equal to the number of independent components distributed between the two phases. The two immiscible solvents are thus not counted as components. In the case where all components are distributed, we have been led to define the characteristic directions in a subspace of dimensions p - 2. Considering P and T as variables each increases by one the dimension of the space to be considered, and therefore increases likewise the number of characteristic directions. It appears clearly that the number of characteristic directions is equal to the “variance”, as given by Gibbs’ phase rule for a two-phase system; that is, equal to the total number of independent components when P and T are variable (recall that the number of independent components is the number of components minus the number of chemical reactions relating them). The non-distributed components (immiscible solvents, insoluble adsorbent, etc. . .) are also counted as components in this general statement. The idea of a connection between Gibbs’ phase rule and the number of fronts generated by a step input in a chromatographic column, the notion of variance of a chromatographic system are due to Klein[28,29]. Let us further discuss the relation between the phase rule and the present properties. Recall that in its classical formulation, Gibbs’ phase rule, by count-

ing variables and equations, establishes the number of variables that may be, and must be chosen by the experimentalist in order for a closed system to be in a determinate equilibrium state[22, 301. This number is the variance of the system. This result extends to open systems ([18], Chap. 2). The implications of the present theorem that are not explicit in Gibbs’ phase rule are the following: (1) By addition or subtraction of matter, energy or volume, an open system may undergo quasi-static changes, through a continuous succession of equilibrium states; in these changes, one is allowed to change freely a number of variables equal to the variance, the change of all other variables being then determined. (2) The changes in variables in a quasi-static change are related by eqns (25), (41), or (66) (or by Clapeyron’s equation in the special case of a single component vapour-liquid equilibrium). (3) Special directions of quasi-static change appear to play an important role in transfer processes implying phase equilibria. These are the characteristic directions, defined by the property that the ratios of the changes of a variable in the two phases is the same for all components. These characteristic directions are always real and distinct, and in number equal to the variance of the system. The present theorem may thus in a sense be considered as generalizing Gibbs’ phase rule from the statics of closed systems to the dynamics of open systems. I[LLIJSTRATIONs: NON-ISOTHERMAL

ISOTHERMAL AND ADSORPTION

Consider the case of isothermal, isobaric twocomponent adsorption governed by Langmuir type equilibrium NK,C, 41 -

1 + K,C,

+ K,C,

i = 1,2.

(67)

The treatment outlined earlier in this text leads us to calculate the eigenvalues CTof matrix (J) using the characteristic equation:

0*-

WI,, + 922) + 411922- 912921= 0.

(68)

It is easy to verify that the coefficients of this equation are always positive (assuming K, positive) and so is its discriminant, for any non zero C, and C,. Therefore, the eigenvalues are real, distinct and positive for C, # 0, C, # 0. Complete analytical solutions for the eigenvalues, eigenvectors, and characteristic directions have been developed for this case[2-51. Figure 5 shows the network of characteristic lines in the (C,, C,) plane (envelopes of characteristic directions at any point) as presented by Glueckauf[2]. The characteristic lines are straight, the r + corresponding to the largest root 0 + of eqn (68) and r - to the smallest root cr-. Along these lines, the following

E. KVAALEN

1200

Fig. 5. Characteristic directions r for two-component adsorption following composite Langmuir isotherms, showing envelope P and watershed point W (after Glueckauf[2]).

relationships

hold

The family of characteristic lines has an envelope, the parabola P, on which the two roots Q + and cr- are equal, in other words, the discriminant of eqn (68) is zero. The parabola has a tangency point W with the side C, = 0, C, > 0 of the state plane. This point, called “watershed point”[5] is thus a singular point of the state plane.

et al.

Figure 6 shows a concentration profile resulting from the elution by an inert eluent of a bed initially saturated by two components. On this profile, the fast moving front (on the right) corresponds to the smallest root CT-, and the slow moving front to u +. The representation of this profile on Fig. 5 is a traject composed of a segment of r+ (between 0 and I on the C, axis, corresponding to the slow front), followed by a segment of r -, between I and II corresponding to the fast front. Note that an entirely similar treatment can be applied to stoichiometric three-component ionexchange[3, 51. The second illustration concerns temperatureswing adsorption. Figure 7 represents the diagram of characteristics for the non-isothermal adsorption of n-pentane on 5 A molecular sieves, in the presence of non-adsorbed iso-pentane. The coordinates are here n-pentane concentration and temperature. It can be seen that the characteristic lines are no longer is somewhat straight, and that their topology different from that of Fig. 5. A watershed point W exists on the temperature axis; at this point, the two characteristic lines coincide. This is the only singularity in the range of conditions considered: at all other points of the diagram, there are two distinct characteristic lines. Figure 8 shows the concentration and temperature breakthrough curves corresponding to the traject IP,PJZ on Fig. 7; this behaviour is generated by a rapid cooling of the gaseous feed to the column from 300 to 60°C. Note that the sorption front actually follows the characteristic lines only approximately, because this front has a shock behaviour, and obeys integral conservation laws, which are discussed elsewhere[ 111.

I

CONCENTRATION

0’

! 1

I

SLOW DESORPTION

I I

I I

FAST

1 I

ABSCISSA

IN

BE

FRONT

Fig. 6. Schematic composition profile in adsorption

column, during the elution df two adsorbed by an inert eluent.

components

Directions of quasi-static transfer

1201

i

-200 SJ

Fig. 7. Characteristic directions for non-isothermal adsorption of n-pentane on 5A molecular sieves (from [ 1 I]). FURTHER MATHEMATICAL ASPECTS AND RELATED PROBLEMS The eigenvector problem in terms of mole fractions We mentioned earlier that the mole fraction sim-

plex was not, in general, a suitable vectorial subspace of the space of mole numbers, for the purpose of defining characteristic directions. The reason for this can be visualized on Fig. 3; the components dx* and dy* in the mole fraction simplex of quasi-static changes dn* and dm* are not in genera1 parallel; dx and dy are tangent respectively to the binodal manifolds CYand jl; only if these manifolds were parallel would dx and dy be parallel and define characteristic directions. A well known physical situation where this is true is binary liquid-vapour equilibria? with composition independent molar heat of vaporization (the saturated liquid line and the saturated vapour line are parallel on the enthalpy-composition diagram). This situation leads to distillation with constant molar overflow. The equivalent in liquid-liquid extraction is the situation when the mutual solubility of the two solvents is independent of the solute concentration; the saturated extract line and the saturated raffinate line are then parallel on a Janecke diagram. This also leads to constant molar flow rates is a contacting apparatus. A third such situation is for example stoichiometric ion-exchange or stoichiometric exchange adsorption, in which the sorption of ions or molecules is compensated by the desorption of an equal number of ions or molecules. In the genera1 case, where the conditions above are not satisfied, it can be shown (31) that the characteristic directions are given by the vectors z solution of the following equation (at constant P and T):

Time(h)

Fig. 8. Concentration and temperature breakthrough curves corresponding to traject IP,PJI of Fig. 7 (obtained by rapid cooling from 300 to 60°C of a gaseous feed containing 5% n-pentane in 95% non-adsorbed iso-pentane) ----, theoretical. (from [l 11). 0, V, experimental; -,

Y of matrix (M’) is parallel to the mole fraction simplex, z becomes an eigenvector of (yV). and dx* and dy* are parallel. In the situations considered above of equimolar counter-transfer or stoichiometric exchange, all R’s are zero. When R # 0, a geometric interpretation of eqn (70) is the following: a change in composition dx in phase a in the direction z generates a change dy in phase b such that dy and dx are “coplanar”, that is their extensions intersect. eigenvector

Multiple eigenvalues We have seen on the example of adsorption

where (y,.,) is the Jacobian matrix of element (dy,/ ax,),,,,; pk is a scalar related to the eigenvalues Xk of matrix (M’) by p* = &n/m, and Rk = Xv, is the sum of the components of the eigenvector vk corresponding to

that eigenvalues of the Jacobian matrix may be multiple at singular points on the borders of the composition space, implying local coincidence of eigenvectors. These so-called “Watershed points” play an important role with respect to the topology of characteristic directions, and therefore with respect to the dynamics of the sorption process. From the mathematical point of view, no difficulty is introduced by these singularities, inasmuch as they are located on a border or sub-border of the composition space: the number of independent eigenvectors is reduced to the dimension of the border. The existence of multiple roots of order n cp inside the composition space is not totally excluded by the present theorem, inasmuch as multiple eigenvalues may still in some cases define independent eigenvectors, and allow diagonalization of the matrix. However, this is possible only in extremely special cases, or under conditions of phase instability, or instabilities related to multiple equilibria. Similar considerations have been discussed in the theory of diffusion (see below).

(70) does not define an eigenvalue problem unless R = 0. When R = 0, the corresponding

The notion

tNote added in prooJ Clearly, in this case, the vectors dn and dm include enthalpy as a component

The notion of “asymptotic coherence” has been introduced by Helfferich and Klein[S] as a limiting regime reached by a long chromatographic column,

(Yi,)

. 2 = p,z

+

R,(x

-

y)

(70)

Xk. Equation

of coherence

1202

E.

KVAALEN

after long times, as a result of a change in inlet conditions between two constant states. If the change is instantaneous, the coherent regime is reached immediately. More precisely, consider the concentration profile of Fig. 6 which represents the state of a chromatographic column at a given instant. At a given abscissa, the local state is defined by the local concentrations cl’, cZo. At a later instant, it may be that these values are no longer found together, at any downstream abscissa. The state (c,O, c2”> is then said to be non-coherent. On the contrary, if this state propagates as such, and the same values of the variables are found further downstream at a later time, the state is coherent : the values of the variables “stick together”. A finite part, or even the totality of the chromatogram, may be composed of a succession of coherent states. Its representation in the statespace of concentrations (and temperature) is then a constant, time-independent path. It turns out that the possible coherent paths coincide with the characteristic curves. This connection between coherence and characteristic directions is related to the fact that the latter are defined as directions of parallel quasi-static displacements. Coherence, as introduced by Helfferich and Klein, can be considered as a generalization of the notion of steady-state, in the sense that a coherent path is time-independent in state-space and also in the sense that it is the regime toward which the system tends after a perturbation, in the absence of further perturbations. The shape of the profile in “real space” (Fig. 6) need not be time independent: although coherence is preserved, the profile may spread out as it propagates along the column, because all the states do not have the same velocity. This “spreading” or “dispersive” behaviour does represent a more general situation than the common steady-state. The relation

with counter-current

et al.

law may be written as N=

-C(D)-l’X

(71)

where N is the diffusion flux vector, pX the composition gradient, and (D) a matrix of diffusion coefficients. Cullinan[33] has shown that (D) is always diagonable, which implies that the eigenvalues are all real and that the eigenvectors form a base (even though there may be multiple eigenvalues at singular points)_ In addition, the eigenvalues are all positive. The set of eqns (71) may then be uncoupled into pseudo-binary equations. A recent graphical representation of the solutions to multicomponent diffusion problems 1381 is completely analogous (at least formally) to the characteristic diagram of multicomponent adsorption (Fig. 5 for two components). Some of the arguments developed in the present text appear to have much resemblance with the ones used by Cullinan, and illustrate “the interrelationship between the diffusional behaviour and the thermodynamic properties” [371. Clearly, the same type of problems will be met in all multicomponent coupled transfer processes, since the dynamics of such processes will depend on the structure and properties of the coupling matrix of the system of conservation equation. But in purely kinetic processes, such as electrophoresis[39], sedimentation or traffic flow, where no phase equilibria are involved, one may ask whether constraints are implied by the second law of thermodynamics. In this respect, let us mention an important general result established recently by Sursock[40], concerning the quasi-linear first order differential equations describing two-phase flow:

(A) * ;

+(Eq.z

=(C).W+E

(72)

operations

We have chosen as illustrations cases of fixed-bed adsorption, for which the use of the concepts of quasi-static displacements and characteristic direction is classical, in the framework of the so-called “equilibrium theory”. However, this approach can equally well be used, with some modifications, to describe continuous counter-current operations, as shown elsewhere[4143]. The transient response of a counter-current system to a step change in the feed conditions shows the same pattern of fronts and plateaus as a chromatographic column. The profile that stabilizes in the counter-current column at steady-state comprises a part of the chromatographic pattern; a pinch point corresponds to a plateau of this pattern. These features have been analyzed in detail by some of us[ 12,44,45]. As a result, the dynamics of counter-current operations are determined as well by the eigenvectors of the equilibrium matrix. The relation wirh dtgiiuion and other kinetic phenomena

Very similar issues arise in the theory of multicomponent diffusion[32-381. The generalized Fick’s

where W is an n-dimensional vector, (A), (B), (C) are n x n matrices and E is a n-dimensional source vector. The result is that the principle of causality is violated, unless the characteristics are all real (more explicitely, the solution of the equation at a time t would depend on the solutions at all times larger than t, if the characteristics were complex). This result is obviously related to the type of situation considered here, and can be considered as a reinforcement of the present theorem. In addition, it may possibly suggest new connections between causality and classical thermodynamics. However, it gives no insight into the properties of thermodynamic equilibrium, which are independent of the particular form of eqn (72). CONCLUSIONS

As a conclusion, let us now restate in a synthetic way, but in broad perspective, the concepts and properties discussed in this text. (1) We define characteristic directions as paths of parallel quasi-static changes of the state of a multicomponent, two-phase open system. Parallel means

Directionsof quasi-statictransfer here that the ratios of the changes in the two phases is the same for all components when expressed in consistent units. Energy and volume may be treated as components in these considerations. (2) Under the assumption of local uniqueness and stability of the thermodynamic equilibrium, the number of characteristic directions is given by the variance of the system, as defined by Gibbs’ phase rule. The characteristic directions are then real and distinct for any composition, and allow local decoupling of the conservation equations. (3) The characteristic directions are given by the eigenvectors of the Jacobian matrix of equilibrium, in the case each phase contains at least one component that is completely absent from the other phase (immiscible “solvents”), or in the case where such components are present in constant proportions in the other phase (constant mutual solubility, constant enthalpy of phase change). These latter cases lead to situations of constant flow (molar or mass) of each phase in a separator. (4) When the above conditions are not satisfied, the characteristic directions are still defined by the eigenvectors of a more general matrix, constructed from the second derivatives of the entropy. (5) Characteristic directions play an essential role in the dynamics of many separation processes such as adsorption, extraction, distillation. In particular, they determine the coherent composition profiles which appear in fixed-bed sorption operations as a response to a step change in inlet conditions, and also the multiplicity and nature of the steady-states that may be established in a counter-current operation. (6) Similar issues arise in other areas of multicomponent processes, not involving phase equilibria, such as diffusion [34], electrophoresis [39], two-phase flow [40], leading to similar results: the eigenvectors of some characteristic matrix are all real and possibly distinct. This “universal” property appears to result from basic thermodynamics, through the stability properties of thermodynamic equilibrium, or through causality, as different expressions of the second principle.

Acknowledgement-The authors thank Prof. H. Renon, Ecole des Mines de Paris, and Dr. M. Doherty, Universityof

Massachusetts, Amherst, for pertinent and constructive cism.

criti-

NOTATION Vectors (first-order tensors) are designated by bold-face letters, for example, II, de, dm, Y. Matrices (second-order

tensors) are designated by bold-face letters in parenthesis, for example (G”), (Gb), (D), (J). The tensor products used in this text are all “dot” products, and identified by a single dot. The order of the product tensor is X - Zd, where Z is the sum of the orders of the factors. and d the number of dots. For example, n.(G) is of o;der 1 (a vector), whereas the bilinear f&& II. (G) . m is of order 0 (a scalar). The absence of dot implies the‘c&ventional multi&ation by a scalar, which doe; not change the tensorial order.

1203

superscriptidentifying the two phases amounts of solvent in phase a and spectively molar concentration

of component

b

re-

i in phase

a

concentration vector, of components c, matrix of diffusion coefficients in eqn (71) restriction of II or m to a submace Gibbs free enthalpy second order derivative of G (eqn ._ 17) matrices of elements G, restrictions of the above to a subspace enthalpy indices identifying the components identity matrix Jacobian matrix of equilibrium (see eqn 5) total number of moles in phase b and a respectively number of moles of component i in phase b and a respectively vectors of the number of moles (m,,+,..., mp> and (n,, n2. . . . , n,) derivative of m, with respect to 5 (eqn 43) matrix in eqns (4), (26) and (45) restriction of (M) to a subspace monolayer capacity in Langmuir isotherm (eqn 67) diffusion flux vector (eqn 71) total number of chemical components pressure molar concentration of component i in phase b

partial derivative of 4i with respect to c, concentration vector of components 4i = Z vi, sum of components of eigenvector, in eqn (48) entropy time temperature apparent and interstitial flow velocity respectively components of arbitrary vectors u and v in cqns (29), (30) and (31) internal energy volume mole fraction of component i in phase a and 6, respectively distance variable in adsorbent bed ith eigenvcctor of matrix (.I) Greek

symbols

intersections of the a and b manifolds with the mole fraction simplex characteristics of the adsorption problem, in composition space (Fig. 5) void fraction (porosity) of the adsorbent bed ith eimnvalue of matrix 0 , or &I’ ., ) chemi:al potential of sp&es i = &(n/m

X,) scalar in eqn (70)

eigenvalues of the Jacobian matrix roots of eqn (68) eigenvector of m’atrix (M’) in eqn (66) general state variables for phase a and b respectively superscript indicating equilibrium values superscript indicating the transpose of a vector

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WI