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Ind. Eng. Chem. Res. 2005, 44, 1112-1119
Theoretical Aspects of Ternary Diffusion James S. Vrentas* and Christine M. Vrentas Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802
Four theoretical aspects of ternary diffusion are investigated. Restrictions imposed on the ternary diffusion process by the entropy inequality are formulated, and their effect on the eigenvalues of the diffusion coefficient matrix is determined. Additional restrictions are formulated by accepting the Onsager reciprocal relations as valid postulates, yielding a relationship that can be regarded as a sufficient condition for the existence of real eigenvalues for the diffusion coefficient matrix. The stability of the ternary system with respect to the equilibrium state is investigated both with and without the Onsager reciprocal relations. The existence of negative concentrations is analyzed, and interpretations of the reasons for the appearance of such nonphysical results are proposed. 1. Introduction
2. Definitions of Diffusion Coefficients
The majority of diffusion problems analyzed in transport processes utilize the assumption that diffusion takes place in a binary system. Typically, either there actually are only two components in the system, or, for dilute multicomponent systems, the diffusion of each of the solutes can be analyzed by considering binary solute-solvent pairs. However, for concentrated multicomponent systems, often the concentration profiles cannot be accurately computed by means of a binary analysis, and so multicomponent diffusion equations must be utilized. For an N-component system, the diffusion process can be described by an overall continuity equation and N-1 independent species continuity equations which contain (N-1)2 diffusion coefficients. Although it can be expected that the four diffusion coefficients needed to analyze diffusion in a threecomponent system can be either estimated or measured, it would be difficult to obtain the nine diffusion coefficients needed for the analysis of a four-component system. Consequently, reasonable progress can be made in analyzing multicomponent diffusion problems if true ternary systems or systems with more than three components that can be adequately modeled using a ternary representation are considered. It is therefore reasonable to place special emphasis on analyzing the diffusion process in ternary systems. The objective of this paper is the examination of various theoretical aspects of ternary diffusion. After definitions of multicomponent diffusion coefficients are discussed in the second section of this paper, the following theoretical aspects of ternary diffusion are considered in the remaining sections: (1) restrictions based on the entropy inequality, (2) application of the Onsager reciprocal relations, (3) stability in ternary systems, and (4) negative concentrations arising in the solution of ternary diffusion problems. The first two topics deal with restrictions placed on diffusion coefficients, while the third and fourth topics are concerned with the properties of solutions to diffusion problems.
In analyzing binary and multicomponent diffusion problems, it is necessary to define various reference velocities which are weighted averages of the velocities of the components in the mixture. Diffusion flows can then be defined relative to each of these reference velocities. Four frequently used reference velocities are the mass average velocity v, the molar average velocity v*, the volume average velocity v*, and the velocity of component N, vN. For binary systems, the four diffusion fluxes relative to these four reference velocities can all be expressed in terms of the same mutual diffusion coefficient, but, for multicomponent systems, different sets of diffusion coefficients are used to represent the various diffusion fluxes. Consequently, these different sets of diffusion coefficients must be carefully defined, and the relationships between the different diffusion coefficients need to be established. In the analysis of three-dimensional transport processes for an isothermal ternary system, it is necessary to solve seven coupled equations (two species continuity equations, three equations of motion, an overall continuity equation, and a thermal equation of state) for seven unknowns (two mass fractions, three components of a reference velocity, the total density, and pressure). Since the equations of motion will, in general, be coupled to the species continuity equations, the solution process is facilitated if the mass average velocity is utilized. Similarly, the mass average velocity is typically used for two-dimensional systems. However, for a onedimensional diffusion process, some flexibility in the choice of a reference velocity is possible because, in certain cases, the chosen reference velocity will be zero everywhere in the diffusion field. For example, for massdiluted, isomer, or various isotopic mixtures, the total mass density, F, is approximately constant so that the overall continuity equation reduces to
* To whom correspondence should be addressed. Tel.: (814) 863-4808. Fax: (814) 865-7846. E-mail:
[email protected].
∂vx )0 ∂x
(1)
If vx is zero somewhere in the flow field for all time (as at an impermeable solid wall), then vx ) 0 everywhere for all time. Similarly, the total molar density is constant for perfect gas mixtures at constant temper-
10.1021/ie040076u CCC: $30.25 © 2005 American Chemical Society Published on Web 07/03/2004
Ind. Eng. Chem. Res., Vol. 44, No. 5, 2005 1113
ature and pressure and, approximately, for molar diluted systems. For this case, if there are no chemical reactions
∂v/x ∂x
)0
(2)
N-1
D h IS ) DIS -
(
V ˆS -
∑
FI(V ˆJ J)1 V ˆN N-1
V ˆN
)
-V ˆ N)DJS +
N-1
v/x
Again, it is possible that ) 0 everywhere in the flow field. In addition, for perfect gas mixtures at constant temperature and pressure, the partial specific volumes of the components in the system are constant. The partial specific volumes of the components of many liquid systems and of mixtures of isomers and isotopes also do not vary appreciably with concentration and pressure. It can be shown that, for constant partial specific volumes and no reactions,
∂v* x )0 ∂x
(3)
so that again there exists the possibility that v* x ) 0. For all of the above one-dimensional diffusion processes, the two mass fractions can be obtained directly from the two species continuity equations if pressure effects on the diffusion process can be considered to be negligible. This is because there are no convective terms in the species continuity equations since the x component of the appropriate reference velocity is zero. The single remaining equation of motion and the thermal equation of state can then be used to determine the pressure field and the distribution of the total mass density of the mixture, F, in the system. Although at least four different sets of ternary diffusion coefficients (for the four reference velocities mentioned above) can be defined, it appears reasonable to focus on just the two diffusion coefficient sets associated with the mass average and volume average velocities. The mass average velocity is useful when analyzing twoand three- dimensional diffusion problems and in massdiluted one-dimensional diffusion problems. The volume average velocity is a better choice for one-dimensional diffusion problems involving mixtures of perfect gases, liquids, isotopes, or isomers. For a ternary mixture, the linear constitutive equation for the mass diffusion flux jI of component I (I ) 1 or 2) relative to the mass average velocity can be expressed as
DIK ) D h IK + ωI
∑ DIK∇ωK + φpI∇p K)1
(4)
Similarly, the corresponding linear constitutive equation for the mass diffusion flux j* I relative to the volume average velocity can be written as
j* I
)-
∑ Dh IK∇FK + φh pI∇p
The two sets of diffusion coefficients DIK and D h IK can be related using the following equations which are applicable for an N-component system:1
V ˆN
)
N-1
ωSD h IS +
∑
+
(
ωSωID h JS
J)1
)]
V ˆJ - V ˆN V ˆN
(7)
3. Restrictions of Entropy Inequality Restrictions can be imposed on the ternary diffusion process by using the entropy inequality which reflects the observation that transport processes are dissipative and irreversible.2 Such restrictions can be formulated for a ternary system by extending an earlier formulation carried out on a binary system.2 A derivation similar to the analysis for the binary system produces the following generalization (for a ternary system with components 1, 2, and 3) of eq 69 of the earlier investigation:
( )
j1‚∇
∂G ˆ ∂ω1
+ j2‚∇
p,ω2
( ) ∂G ˆ ∂ω2
- tr(S‚∇v) e 0
(8)
p,ω1
In addition, introduction of the following definitions
( ) ( ) ∂G ˆ ∂ω1
p,ω2
∂G ˆ ∂ω2
p,ω1
) µ1 - µ3
(9)
) µ2 - µ3
(10)
and utilization of a standard thermodynamic analysis produces the following results which are extensions of the binary result (eq 77 of the previous investigation2):
ˆ1 - V ˆ 3)∇p ∇(µ1 - µ3) ) A11∇ω1 + A12∇ω2 + (V
(11)
ˆ2 - V ˆ 3)∇p ∇(µ2 - µ3) ) A22∇ω2 + A21∇ω1 + (V
(12)
A11 ) A12 )
(5)
K)1
[
(
V ˆJ - V ˆN
(6)
In this paper, we concentrate on results for the DIK since the above equations can be used to derive the corresponding results for the D h IK.
A22 )
2
∑
S)1
2
jI ) -F
∑ Dh JK J)1
N-1
ˆ K) F(V ˆN - V
N-1
∑ [ωJDIJ - K)1 ∑ FIωK(Vˆ J - Vˆ N)DJK] J)1
A21 )
( ( ( (
∂µ1 ∂ω1 ∂µ1 ∂ω2 ∂µ2 ∂ω2 ∂µ2 ∂ω1
) ) ) )
[ [ [ [
1+
p,ω2
1+
p,ω1
1+
p,ω1
p,ω2
1+
] ] ] ]
( ( ( (
ω1 ω2 ∂µ2 + ω3 ω3 ∂ω1 ω1 ω2 ∂µ2 + ω3 ω3 ∂ω2 ω2 ω1 ∂µ1 + ω3 ω3 ∂ω2 ω2 ω1 ∂µ1 + ω3 ω3 ∂ω1
It also can easily be shown that
) ) ) )
(13)
p,ω2
(14)
p,ω1
(15)
p,ω1
p,ω2
(16)
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Ind. Eng. Chem. Res., Vol. 44, No. 5, 2005
( ) [ ∂µ2 ∂ω1
p,ω2
1+
] ( ) ( ) [
ω2 ω1 ∂µ1 + ω3 ω3 ∂ω1 ∂µ1 ∂ω2
p,ω1
M11 ) A11D11 + A21D21
) p,ω2
] ( )
ω1 ω2 ∂µ2 1+ + ω3 ω3 ∂ω2
(17)
M12 ) M21 )
p,ω1
A11D12 + A12D11 + A22D21 + A21D22 2 (28) M22 ) A12D12 + A22D22
so that
A21 ) A12
(18)
and the matrix A with elements defined by eqs 13-16 is symmetric. For a linear or first-order theory of diffusion in a purely viscous ternary fluid mixture, the diffusion fluxes are given by eq 4, and the extra stress S can be expressed as follows:
S ) λ(∇‚v)I + 2µD
(19)
Therefore, utilization of eqs 4, 9-12, and 19 in eq 8 produces the following form of the entropy inequality:
ˆ1 - V ˆ 3)∇p]‚[FD11∇ω1 + [A11∇ω1 + A12∇ω2 + (V FD12∇ω2 - φp1∇p] + [A22∇ω2 + A21∇ω1 + ˆ 3)∇p]‚[FD21∇ω1 + FD22∇ω2 - φp2∇p] + (V ˆ2 - V λ(trD)2 + 2µtr(D‚D) g 0 (20) For ∇p ) 0, D ) 0, ∇ω2 ) 0, and arbitrary ∇ω1, eq 20 requires that
A11D11 + A21D21 g 0
(21)
(22)
|A| * 0, |D| * 0, |M| * 0
An equation such as eq 23 can be considered to be a positive semidefinite quadratic form if and only if all of the eigenvalues of the coefficient matrix are nonnegative. This requirement is satisfied if
|A| > 0, tr A > 0
(31)
|M| > 0, tr M > 0
(32)
The inequality of eq 25 can be satisfied only if
|D||A| g 0
The above expression can also be written as
[(A11D12 + A21D22) - (A12D11 + A22D21)]2 4|D||A| e 0 (25) Here, |A| is the determinant of the matrix A, and |D| is the determinant of the diffusion coefficient matrix D:
[
D D D ) D11 D12 21 22
]
(26)
Associated with the positive semidefinite quadratic form of eq 23 is the positive semidefinite symmetric matrix M with elements defined as follows:
(33)
so that eqs 31 and 33 require that
|D| ) D11D22 - D12D21 > 0
(34)
Also, it can easily be shown that
M)
AD + DTA 2
(35)
so that
2
[A11D12 + A12D11 + A22D21 + A21D22] 4[A11D11 + A21D21][A12D12 + A22D22] e 0 (24)
(30)
The thermodynamic stability condition with respect to ternary diffusion4 requires that A be a positive semidefinite matrix. The two symmetric matrices A and M are thus both positive semidefinite and are assumed to be nonsingular. Hence, they are also positive definite.5 Consequently, the following mathematical restrictions6 are applicable to the determinants and the traces tr of these two matrices:
For the special case of ∇p ) 0, D ) 0, and arbitrary ∇ω1 and ∇ω2, eq 20 requires that
[A11D11 + A21D21][∇ω1‚∇ω1] + [A11D12 + A12D11 + A22D21 + A21D22][∇ω1‚∇ω2] + [A12D12 + A22D22][∇ω2‚∇ω2] g 0 (23)
(29)
Equations 21, 22, and 25 are restrictions placed on the combined thermodynamic-diffusion behavior of the ternary system. It is also possible to use the above results to derive additional restrictions on the system involving only the four diffusion coefficients of the mixture. Restrictions of this type have been previously derived3 using the assumption that the Onsager reciprocal relations are valid. It is of interest here to see if similar restrictions involving the diffusion coefficients can be derived without assuming the validity of the Onsager reciprocal relations. The derivation is carried out by assuming that the matrices A, D, and M are not singular:
and, for ∇p ) 0, D ) 0, ∇ω1 ) 0, and arbitrary ∇ω2, eq 20 yields the following restriction:
A12D12 + A22D22 g 0
(27)
A-1M )
D A-1DTA + 2 2
(36)
Now, since A is positive definite, it follows6 that A-1 is also positive definite, and A-1M is thus a product of two positive definite matrices. It has been proved7 that the product of two positive definite matrices must have real and positive eigenvalues. One of the necessary conditions6 for positive eigenvalues is that
tr (A-1 M) > 0
(37)
If we take the trace of eq 36, we obtain
tr (A-1 M) ) tr D ) D11 + D22
(38)
and the combination of eqs 37 and 38 produces the
Ind. Eng. Chem. Res., Vol. 44, No. 5, 2005 1115
following inequality:
D11 + D22 > 0
then it would constitute an alternative sufficient condition for the existence of real eigenvalues for D.
(39) 4. Onsager Reciprocal Relations
The two eigenvalues λ1,2 of the diffusion coefficient matrix D can be expressed as
λ1,2 )
tr D ( x(tr D)2 - 4|D| 2
(40)
and the eigenvalues will be real if
(tr D)2 - 4|D| ) (D22 - D11)2 + 4D12D21 g 0
(41)
If the diffusion coefficient matrix is such that eq 41 is satisfied, then it follows from eqs 34, 39, and 40 that the eigenvalues λ1 and λ2 are real and positive. If, on the other hand,
(tr D)2 - 4|D| < 0
(42)
then the eigenvalues λ1 and λ2 are complex with positive real parts. It has been shown previously3 that restrictions on diffusion coefficients (which are based on the volume average velocity) of the form of eqs 34, 39, and 41 can be derived by assuming that the Onsager reciprocal relations are valid. We have shown here, by simply using the entropy inequality, that two of these three restrictions (eqs 34 and 39) must always be applicable even in the absence of the Onsager relations resulting in eigenvalues that will either be complex with positive real parts or real and positive. Since the matrix A-1 M has real and positive eigenvalues, the following inequality must be satisfied:
[tr (A-1M)]2 - 4|A-1M| g 0
(43)
Also, the determinant of the matrix M can be expressed as follows:
|M| ) |D||A| [(A11D12 + A21D22) - (A12D11 + A22D21)]2 (44) 4 Combination of eqs 38, 43, and 44 produces the following additional restriction based on the entropy inequality:
(tr D)2 - 4|D| g [(A11D12 + A21D22) - (A12D11 + A22D21)]2 (45) |A| This restriction does not guarantee that the eigenvalues are real nor does it appear that the entropy inequality alone can produce eq 41. It will be shown in the next section that, by also assuming that the Onsager relations are valid, it is possible to guarantee that the eigenvalues of the diffusion coefficient matrix are real and positive. The Onsager reciprocal relations can thus be viewed as a sufficient condition for the existence of real eigenvalues for D. Note, however, from eq 41, that if the following inequality were valid
D12D21 g 0
(46)
Most proofs of the Onsager reciprocal relations are based on a microscopic analysis, such as the statistical mechanics of fluctuations or kinetic theory. Consequently, there are some questions as to whether the Onsager relations are correct at a macroscopic level.8 It appears that no general macroscopic proof of the Onsager reciprocal relations exists,9,10 and it has been suggested11 that the Onsager relations should be considered as postulates at the macroscopic level until a general macroscopic proof is proposed. Here, we suppose that the Onsager relations are valid postulates and see what effect they have on the analysis of a ternary diffusion process. A specific form of the Onsager reciprocal relations for a ternary diffusion process can be derived either by using a standard nonequilibrium thermodynamic analysis4 or by combining a friction coefficient analysis with statistical mechanics.12 Both approaches lead to the following result for the ternary system considered in this study:
A11D12 + A21D22 ) A22D21 + A12D11
(47)
The first contribution of the Onsager relation responsible for eq 47 is the reduction of the number of independent diffusion coefficients from four to three. In addition, substitution of eq 47 into eq 45 produces eq 41, which is the condition required for the existence of real eigenvalues for the diffusion coefficient matrix. Hence, the second contribution of the Onsager relation is that it provides a sufficient condition for the existence of real, positive eigenvalues of the matrix D. This result can also be seen in the following manner. When eq 47 is applicable, it can easily be shown that eq 36 reduces to the following result:
D ) A-1 M
(48)
Consequently, the matrix D is now the product of two positive definite matrices and hence must have real, positive eigenvalues. 5. Stability in Ternary Systems One important aspect of unsteady-state diffusion processes is the stability of a ternary system with respect to an equilibrium state. This can be investigated by perturbing the component concentrations in the system by a suitably small amount (so that all system properties are independent of concentration) and then determining if and how the system returns to its initial state. Here, we examine stability by considering the case of isothermal, one-dimensional diffusion in a ternary system with constant mass density F and constant diffusion coefficients D11, D12, D21, and D22. Pressure effects on diffusion fluxes can be considered to be negligible. The diffusion process takes place in the region x ) 0 to x ) L which is bounded by impermeable solid walls. The following initial conditions can be used to examine the stability of the system to a mass fraction perturbation:
ω1(x,0) ) ω10 + cos
πx L
(49)
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Ind. Eng. Chem. Res., Vol. 44, No. 5, 2005
ω2(x,0) ) ω20
(50)
Here, ω10 and ω20 represent equilibrium mass fractions for the system, and is a suitably small parameter which describes a small perturbation in the mass fraction of component one. In addition to the above initial conditions, the diffusion process for the ternary system is described by the following set of equations:
∂2ω1 ∂2ω2 ∂ω1 ) D11 2 + D12 2 ∂t ∂x ∂x
(51)
∂ω2 ∂2ω1 ∂2ω2 ) D21 2 + D22 2 ∂t ∂x ∂x
(52)
∂ω1 ∂ω2 ) ) 0,x ) 0 ∂x ∂x
(54)
The solution of the above two partial differential equations is carried out using the fact that the entropy inequality requires that the constant diffusion coefficients satisfy both eqs 34 and 39. However, the Onsager reciprocal relations are not utilized so that either eq 41 (λ1 and λ2 are real and positive) or eq 42 (λ1 and λ2 are complex with positive real parts) may apply. The solutions of the partial differential equations take the following form:
ω1(x,t) ) ω10 + T1(t)cos
πx L
T1(0) ) 1
(55) (56)
ω2(x,t) ) ω20 + T2(t)cos
πx L
T2(0) ) 0
(57) (58)
Introduction of eqs 55 and 57 into eqs 51 and 52 and straightforward application of the Laplace transform method to the resulting ordinary differential equations yield the required expressions for T1(t) and T2(t). If the parameter Q is defined as
Q ) (D11 + D22)2 - 4(D11D22 - D12D21)
(59)
then, when Q > 0 (real and positive eigenvalues), T1 and T2 are given by the following expressions
T1 )
λh1exp(-λh1t) - λh2exp(-λh2t)
π2 1/2 Q L2 D22[exp(-λh1t) - exp(-λh2t)] Q1/2
T2 )
D21[exp(-λh1t) - exp(-λh2t)] Q1/2 λh1,2 )
π2 λ1,2 L2
[ ( )
T1 ) e-γt cos
(53)
∂ω1 ∂ω2 ) ) 0,x ) L ∂x ∂x
decay nonperiodically to ω1 ) ω10 and ω2 ) ω20 as t f ∞. This is, of course, the expected result which has been stated to be true when the eigenvalues of D are real and positive.3,7 It has further been stated3 that real, positive eigenvalues are the conditions that solutions of the diffusion equation be real. The validity of this statement can be ascertained by considering the case Q < 0 (complex eigenvalues) for which T1 and T2 can be expressed as follows:
(60) (61) (62)
From eqs 55, 57, 60, and 61, it is clear that ω1 and ω2
( )
( )]
D11 D22 π2ξt π2ξt π2ξt sin sin + ξ ξ 2L2 2L2 2L2 (63)
T2 ) γ)
2D21 π2ξt -γt e sin ξ 2L2
( )
(64)
π2 (D11 + D22) 2L2
(65)
ξ ) (-Q)1/2
(66)
As is evident from eqs 55, 57, 63, and 64, the solutions for ω1 and ω2 are real and decay periodically to ω1 ) ω10 and ω2 ) ω20 as t f ∞ since, for long times, the damping term exp (-γt) becomes dominant. Clearly, the existence of real eigenvalues for D is a sufficient but not a necessary condition for real solutions as would be expected from the basic theory of solutions to differential equations. Consequently, application of the entropy inequality is sufficient by itself to guarantee that solutions to unsteady diffusion problems are real and to guarantee that the system is stable with respect to concentration perturbations. The introduction of the Onsager reciprocal relations guarantees that the eigenvalues are real and positive and that there is a nonperiodic decay of any perturbation. If the Onsager reciprocal relations are not introduced, the eigenvalues could be complex and there could be a periodic decay of any perturbation since the real parts of the eigenvalues are positive. The applicability of the Onsager relations is a sufficient condition for nonperiodic decay. 6. Negative Concentrations in Ternary Diffusion Another important aspect of unsteady-state diffusion in ternary mixtures is the appearance of negative concentrations in the solution of some diffusion problems.13,14 Nauman and Savoca13 discussed ternary diffusion analyses which yielded negative concentrations, and they argued that such nonphysical results are caused by the assumption of constant diffusion coefficients. However, they also noted that this assumption produced negative concentrations only for certain initial conditions for unsteady-state diffusion problems. It thus appears that the presence of negative concentrations involves more than just the assumption of constant diffusion coefficients. Price and Romdhane14 also reported examples of negative concentrations for unsteady ternary diffusion, and they attributed these results to the utilization of improper ratios of self-diffusion coefficients in the application of theories for determining the diffusion coefficient matrix. It certainly is possible that the choice of diffusion coefficients for a ternary system can affect the existence of completely positive
Ind. Eng. Chem. Res., Vol. 44, No. 5, 2005 1117
solutions to an unsteady diffusion problem. For example, difficulties could arise if the restrictions imposed by the entropy inequality (eqs 34 and 39) are not satisfied by the diffusion coefficient matrix. Also, it is possible that restrictions imposed by the Onsager reciprocal relations (eqs 41 and 47) need to be satisfied by the elements of D if negative concentrations are to be avoided. It seems somewhat less likely that restrictions introduced by empirical theories formulated to relate the elements of D to friction coefficients will lead to nonphysical results. Additionally, it appears that the existence of negative concentrations involves the form of the initial conditions as well as the characteristics of the diffusion coefficient matrix. The purpose of this section is to conduct a somewhat more comprehensive analysis of the appearance of negative concentrations and to propose somewhat different interpretations of the reasons for the existence of such nonphysical results. To investigate the existence of negative concentrations, we consider isothermal, onedimensional diffusion in a ternary system with constant mass density F and constant diffusion coefficients D11, D12, D21, and D22 and negligible pressure effects on the diffusion flux. The diffusion process is free diffusion in the region - ∞ < x < ∞ with an impermeable solid wall at x ) - ∞ and with step change initial mass fraction profiles for components 1 and 2. We further assume that D11 > 0, D22 > 0, and D12 ) D21 ) 0. Clearly, the elements of the matrix D satisfy eqs 34, 39, and 41. The Onsager relation, eq 47, is satisfied only for the special case D11 ) D22. The unsteady diffusion process for components 1 and 2 is described by the following set of equations:
component 3 (except for the special case D11 ) D22) even though there are no cross diffusion effects for components 1 and 2. The solutions to the above equation sets are as follows:
[ [
W x 1 + erf 2 2(D11t)1/2
(78)
ω2 )
W x 1 - erf 2 2(D22t)1/2
(79)
[ {
}]
ω3 ) 1 - W -
W x x erf - erf 1/2 2 2(D11t) 2(D22t)1/2 (80)
Clearly, ω1 g 0 and ω2 g 0. When D11 ) D22, ω3 > 0 since there is no cross diffusion in this case for component 3. However, when D11 * D22, there is cross diffusion for component 3, and there exists the possibility that ω3 < 0 somewhere in the diffusion field. We consider the case when D11 > D22 and rewrite eq 80 in the following form:
ω3 ) 1 - W -
(67)
∂2ω2 ∂ω2 ) D22 2 ∂t ∂x
(68)
ω1 (∞,t) ) ω2 (-∞, t) ) W
(69)
ω1 (-∞,t) ) ω2 (∞, t) ) 0
(70)
ω1 (x,0) ) W, x > 0
(71)
ω1 (x,0) ) 0, x < 0
(72)
ω2 (x,0) ) 0, x > 0
(73)
ω2 (x,0) ) W, x < 0
(74)
Here, the parameter W (0 < W < 1) is a constant mass fraction. The diffusion process for component 3 is described by the following set of equations: 2
W [erf η - erf (Rη)] 2
(81)
( )
(82)
R)
η)
2
∂ω1 ∂ ω1 ) D11 2 ∂t ∂x
ω3 (∞,t) ) ω3 (-∞,t) ) 1 - W
(76)
ω3 (x,0) ) 1 - W, -∞ < x < ∞
(77)
η)(
(83)
lnR R -1
(84)
2
where the plus sign in eq 84 is associated with a maximum value of ω3 and the minus sign with a minimum value of ω3. The minimum value of ω3 can be evaluated using eqs 81 and 84 to arrive at the following result:
ω3(min ) ) 1 - W +
[ (
)
(
)]
ln R RlnR W erf 2 - erf 2 2 R -1 R -1 (85)
It follows that ω3 (min) > 0 only if the mass fraction W satisfies the following inequality for a given value of R:
β ) erf
For component 3, the initial mass fraction is uniform. Also, it is evident that there is cross diffusion for
1/2
x 2(D11t)1/2
We (75)
D11 D22
From eqs 76, 77, 81, and 83, it is clear that ω3 (η) must achieve maximum and minimum values in the interval η ) - ∞ to η ) ∞ as it deviates from ω3 ) 1 - W. The extremum points for ω3 ) ω3 (η) occur at
2
∂ ω1 ∂ ω3 ∂ω3 ) D22 2 + (D22 - D11) 2 ∂t ∂x ∂x
}] }] } {
{ {
ω1 )
(
2 β+2
)
(86)
(
)
Rln R ln R - erf 2 2 R -1 R -1
(87)
Negative values for the concentration will be obtained when
W>
2 β+2
(88)
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Ind. Eng. Chem. Res., Vol. 44, No. 5, 2005
valid, in agreement with the numerical results of a previous investigation.14 However, the results of this study are for a reduced form of the Onsager reciprocal relations, and it would be of interest to see if negative concentrations are excluded when a more general form of the Onsager reciprocal relations is studied. In addition, the mass transfer problem in this study utilized Dirichlet boundary conditions, and it would be useful to investigate a mass transfer problem with a so-called radiation boundary condition which is used in drying calculations.14 In this case, the external mass transfer coefficient might influence the appearance of negative concentrations. 7. Summary of Results
Figure 1. Dependence of mass fraction W (which separates completely positive solutions for ω3 from solutions with negative ω3) on R ) (D11/D22)1/2.
The above results indicate that the appearance of negative concentrations in the analytical solution depends on the characteristics of the diffusion coefficient matrix D (through R) and on the initial conditions for the free diffusion process (through W). The value of the mass fraction W which separates completely positive solutions for ω3 from solutions with negative ω3 is illustrated in Figure 1 as a function of R, the square root of the ratio of the main-term diffusion coefficients. Negative concentrations exist for all finite R > 1 but only for relatively large W (small initial mass fractions of component 3). There are no negative concentrations for any value of W for R ) 1 and for R ) ∞. The smallest value of W for negative concentrations is approximately W ) 0.87 near R ) 4. From a mathematical point of view, solutions to the ternary diffusion problem exist for all values of W and R. Most of these solutions lead to positive values of ω3 everywhere in the diffusion field for all time, but there are also solutions (for sufficiently small 1-W) which yield negative values of ω3 at certain places in the diffusion field for certain values of time. The above results indicate that the existence of negative mass fractions is governed not just by the properties of the diffusion coefficient matrix since, depending on the value of W, there are both completely positive solutions for ω3 and solutions with some negative values of ω3 at the same value of R. Previous interpretations13,14 have suggested that it is the utilization of an inappropriate diffusion coefficient matrix that has produced negative concentrations. The above results suggest an alternative point of view, namely that completely positive solutions do not appear to be possible for certain diffusion coefficient matrices and certain initial conditions. Presumably, this would mean that it would be impossible to conduct a meaningful experiment under such conditions. This conclusion is based on the fact that there is a unique solution to the above linear set of equations. There, of course, exists the possibility that different, completely positive solutions can be derived if the constraints ω1 g 0, ω2 g 0, and ω3 g 0 are somehow introduced into the analysis. The technique used in this section could be applied to carry out a more comprehensive study of the appearance of negative concentrations. For example, it was noted above that there were no negative concentrations for R ) 1 when the Onsager reciprocal relations are
The results of this study can be summarized as follows: 1. Imposition of the entropy inequality to the linear, purely viscous theory of ternary diffusion yields both combined thermodynamic-diffusion restrictions (eqs 21, 22, 25, and 45) and diffusion coefficient restrictions (eqs 34 and 39). The two eigenvalues of the diffusion coefficient matrix are either complex with positive real parts or real and positive. 2. Imposition of the Onsager reciprocal relations yields another diffusion coefficient restriction (eq 41) and also an equation (eq 47) which reduces the number of independent diffusion coefficients from four to three. The two eigenvalues of the diffusion coefficient matrix are real and positive. 3. Ternary diffusion systems are always stable to small perturbations in the equilibrium component concentrations. If the parameter Q (defined by eq 59) is greater than zero, the component concentrations are real and decay nonperiodically as t f ∞. If Q < 0, the component concentrations are real and decay periodically as t f ∞. The applicability of the Onsager relations is a sufficient condition for nonperiodic decay. 4. The appearance of negative concentrations in unsteady ternary diffusion problems depends on the characteristics of the diffusion coefficient matrix as well as on the form of the initial conditions for the diffusion process. The results of this study suggest that either it is not possible to conduct experiments under certain conditions or that different solutions must be derived subject to the constraint that all concentrations are positive. Acknowledgment We are pleased to contribute this paper in honor of the 65th birthday of Professor Darsh Wasan. This study was supported by funds provided by the Dow Chemical Company. Nomenclature A ) matrix with elements defined by eqs 13-16 (m2/s2) AIJ ) elements of matrix A (m2/s2) D ) diffusion coefficient matrix defined by eq 26 (m2/s) DIJ ) diffusion coefficients used for fluxes relative to mass average velocity (m2/s) D h IJ ) diffusion coefficients used for fluxes relative to volume average velocity (m2/s) D ) rate of strain tensor (s-1) G ˆ ) specific free energy (m2/s2) I ) identity or unit tensor
Ind. Eng. Chem. Res., Vol. 44, No. 5, 2005 1119 jI ) mass diffusion flux of component I relative to mass average velocity (kg m-2 s-1) jqI ) mass diffusion flux of component I relative to volume average velocity (kg m-2 s-1) L ) thickness of diffusion field (m) M ) matrix with elements defined by eqs 27-29 (m4/s3) p ) pressure (kg m-1 s-2) Q ) parameter defined by eq 59 (m4/s2) S ) extra stress tensor (kg m-1 s-2) t ) time (s) T1, T2 ) functions defined by eqs 55 and 57 v ) mass average velocity (m/s) vx ) x component of mass average velocity (m/s) v* ) molar average velocity (m/s) vx* ) x component of molar average velocity (m/s) v* ) volume average velocity (m/s) vx* ) x component of volume average velocity (m/s) vN ) velocity of component N (m/s) V ˆ I ) partial specific volume of component I (m3/kg) W ) mass fraction appearing in eqs 69, 71, 74, 76, and 77 x ) spatial variable (m) Greek Letters R ) parameter defined by eq 82 β ) parameter defined by eq 87 γ ) parameter defined by eq 65 (s-1) ) small parameter in eq 49 η ) dimensionless independent variable defined by eq 83 λ ) second viscosity coefficient (kg m-1 s-1) λ1, λ2 ) eigenvalues of D defined by eq 40 (m2/s) λh1, λh2 ) parameters defined by eq 62 (s-1) µ ) shear viscosity (kg m-1 s-1) µI ) chemical potential of component I (m2/s2) ξ ) parameter defined by eq 66 (m2/s) F ) mass density of mixture (kg/m3) FI ) mass density of component I (kg/m3) φpI ) pressure coefficient in diffusion flux for component I relative to mass average velocity (s) φ h pI ) pressure coefficient in diffusion flux for component I relative to volume average velocity (s)
ωI ) mass fraction of component I ωI0 ) equilibrium mass fraction of component I
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Received for review March 5, 2004 Revised manuscript received April 13, 2004 Accepted May 21, 2004 IE040076U
