407
Ind. Eng. Chem. Fundam. 1982, 21, 407-413
Solution of the Linearized Equations of Multicomponent Mass Transfer Ross Taylor Department of Chemical Engineering, Clarkson College of Technology, Potsdam, New York 13676
We consider the solution of the coupled linearized equations of multicomponent mass transfer. Most previous attempts at solving these equations first uncouple them and then present results in terms of the eigenvalues of the matrix of multicomponent diffusion coefficients, [ D ] . In this paper it is noted that if the linearized differential mass balance can be reduced (by whatever means) to the form d(y)lda = [ A ( q ) ] ( y ) (a(v)),then the solution can be obtained without uncoupling in terms of the matrizant of [ A ] . The method is illustrated with the solution
+
of the film and penetration models of multicomponent mass transfer, yielding solution in complete matrix form. Many other, related, problems can be solved in a similar way. The advantages of this approach include independence of the computation of the eigenvalues of [ D ] . A new algorithm for calculating the rates of mass transfer from the multicomponent penetration model is given.
Introduction and Review The analysis of mass transfer in nonreacting multicomponent mixtures starts from the equations of continuity of species (Bird et al., 1960) aci - + V.Ni = 0 (i = 1, 2, ..., n ) (1) at The Ni are the constituent molar fluxes and are made up of a diffusive flux and a convective flux as Ni = Ji+ xiNt (i = 1, 2, ..., n) (2) which may be inserted into eq 1 to give aci - + V.(Ntxi)= -0.J; (i = 1, 2, ..., n ) (3) at The description of multicomponent diffusion is completed with an appropriate constitutive equation for the fluxes, Ji. The n - 1 independent diffusion fluxes can be related to the independent concentration gradients by a generalization of Fick’s law as n-1
4 = -CxDikVXk k=l
(i = 1, 2, ..., ?2 - 1)
(4)
which is usually used for diffusion in gas mixtures, or n-1
Ji = -xDikVCk k=l
(i = 1, 2, ..., n - 1)
(5)
which is used to describe diffusion in liquid systems. Here, the Dikare the concentration-dependent multicomponent diffusion coefficients. For gas mixtures these coefficients can be accurately predicted from binary diffusion coefficients through the Maxwell-Stefan equations (Burchard and Toor, 1962; Stewart and Prober, 1964; Krishna and Standart, 1979). Prediction methods for the Dik in liquid systems are much less well developed, and experimental measurements of these coefficients are often required (Cussler, 1976). Solutions of the linearized equations obtained in this paper use eq 4 for the Ji. A similar development using eq 5 is also possible. The calculation of the rates of diffusion in multicomponent mixtures is complicated by the coupling which exists between the individual diffusion fluxes (D;k # 0, i # k). A consequence of this coupling is that a particular species may diffuse in opposition to its own concentration gradient or may not diffuse at all even though a gradient for that species exists (Toor, 1957). 0196-4313/82/1021~0407$01.25/0
Writing the first n - 1 of eq 3 and 4 in column matrix form and combining the two results gives (4 = -c[DIV(x) (6) (7) which, together with the momentum and energy equations, are the equations to be solved for any given situation. The nonlinearity of eq 7 is such that exact analytical solutions for the general n component case are known only for a film model of steady-state unidirectional mass transfer in ideal gas mixtures where [D]is given by the Maxwell-Stefan equations (Krishna and Standart, 1979; Taylor, 1981a). In 1964 both Toor and Stewart and Prober independently put forward a general method of solution to eq 7. The essence of what has since become known as the linearized theory of multicomponent mass transfer are the assumptions that c and the matrix c[D] can be considered constant over the diffusion path. With these assumptions eq 7 simplify to
w + V-(N,(x)) = c [ D ] V 2 ( x ) at
C-
The utility of the linearized eq 8 would seem, on the face of things, to be limited to situations in which the diffusion coefficients do not change significantly with concentration. This is not entirely true, however. Accordingly, the goodness of the assumption of constant [ D ] ,central to the theory, has been fairly thoroughly tested. Obviously, [ D ] , if evaluated at the average concentration as usually recommended, will not change significantly if the concentration gradients are small (this being the basis on which Toor (1964) and Stewart and Prober (1964) first presented their theory). Cussler (1976) reviews the experimental and theoretical evidence for the concentration dependence of the Dikin liquid systems and points out that this dependence is often small. More significant variations of the Dik with concentration are found in gas mixtures. Here, it is possible to compare the predictions of the linearized equations against exact solutions of the Maxwell-Stefan equations. The calculations of Stewart and Prober (1964)-mass transfer from a surface into a boundary layer flow; Arnold and Toor (1967)-unsteady-~tate diffusion; Cotrone and de Giorgi (1971), Taylor and Webb (1980, 1981), Taylor (1982)-steady-state diffusion in mixtures containing from three to ten components; Krishna et al. 0 1982 American Chemical Society
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Ind. Eng. Chem. Fundam., Vol. 21, No. 4, 1982
(1976), Bandrowski and Kubaczka (1981), Webb and Sardesai (1981)-mass transfer during condensation in the presence of an inert gas, all show the linearized equations to be of excellent accuracy. Examples showing larger discrepancies between predicted fluxes have been published (e.g., Krishna and Standart, 19761, but they do not appear to be truly representative of the behavior of the linearized equations. Carty and Schrodt (1975) show that the linearized equations are much less satisfactory for calculating composition profiles than they are for predicting rates of mass transfer. To try and determine the range of validity of the linearized equations for steady diffusion in gas mixtures, Smith and Taylor (1982) calculated rates of mass tranfer for some 10500 problems with each of twenty-three different real ternary gas mixtures displaying a wide range in their binary diffusion coefficients. The average discrepancy in the constituent N , was only 1.4%. Problems involving mixtures of high concentration and the highest possible rates of mass transfer were included just as frequently as problems with low mass transfer rates. Predictions of high mass transfer rates were usually in excellent agreement with more exact calculations even in cases where [ D ] changes significantly over the diffusion path. It can be concluded from these studies that, for the purposes of calculating rates of mass transfer rather than composition profiles, the linearized equations are almost always adequate for design purposes. This is a comforting conclusion and certainly justifies continued interest in, and applications of, the linearized theory. The general method of solution that was developed by Toor (1964) and by Stewart and Prober (1964) exploits the properties of the matrix [PI whose columns are the eigenvectors of [ D ] . The matrix product IP1-l [DI[Pl =
@I
(9)
is a diagonal matrix with elements that are the eigenvalues of [D]. Cullinan (1965) has shownAthatthis transformation is always possible and that the Di are real, positive, and invariant under changes of reference velocity. Equations 8 may therefore be uncoupled by premultiplying by [PI-' and inserting the identity matrix, [PI [PI-', between [Dl and V 2 ( x ) as (Stewart and Prober, 1964; Krishna and Standart. 1979)
aa
+
V-N,X^,= cBiV'2, (i = 1, 2, ..., n - 1) (10) at where the f i , which have been called "pseudo compositions," are defined by C-
( 2 ) = [P]-'(x)
(11)
Now eq 10 have exactly the same form as eq 8 when written purely for a binary system. Thus any solution of eq 8 for diffusion in binary mixtures, of which there are many (Bird et al., 1960; Crank, 1975), may immediately be generalized to the multicomponent case by replacing the real mole fraction by the pseudo composition, f, and the binary diffusion coefficient by the eigenvalue D ,
where fi0and fi, are suitably transformed bpundary conditions. The precise form of the functions f i depends on the assumed model of mass transfer. Equation 4 may be uncoupled in exactly the same way as eq 8 and yields (Stewart and Prober, 1964; Toor, 1964; Krishna and Standart, 1979) (i = 1,2, ..., rz - 1) j , = -cBpa, (13)
where the 4 are the "diffusion fluxes" of the pseudo species i and are given by
(3) = [PI-' (4
(14) To calculate these "pseudo-fluxes'' the gradients of "pseudo-composition" are obtained by differentiation of eq 12 and combined with eq 13 in the form
ki (r,t,Bi,NJ(fi0- f i n ) (i = 1, 2, ..., n - 1) (15) where the k*i are "pseudo mass transfer coefficients."
ji=
To recover the real fluxes eq 15 are arranged in matrix form and multiplied by [PI (see eq 11 and 14)
(4 = [k'l(Ax)
(16)
where [k'] is a square matrix of finite flux mass transfer coefficents defined by (Stewart and Prober, 1964; Krishna and Standart, 1979) (17) [k'] = [PI [k'] [PI-' which is the inverse of the transformation performed in eq 9. The purpose of this somewhat lengthy introduction is to place in better perspective the remarks which follow. Transformation methods are widely used in science and engineering but, unless there are insurmountable mathematical difficulties, solutions are rarely presented in terms of the transformed variables. For example, we might consider solving a problem using the Laplace transform technique but we would not normally leave the final result in terms of the complex inversion integral, particularly when the solution can be found in a table of transforms. The inverse transformation of eq 17 to the original variables ( x i , [ D ] ,J,, [ k ] )can almost always be carried out completely but, for some reason, this is hardly ever done and most authors stop at this point. This includes Toor (1964), Stewart and Prober (1964) and, among many others, Cussler (1976) and Krishna and Standart (1979) who, in recent surveys have reviewed the methods and some of the more important applications of the linearized theory. Diffusion is a real process engendered by real concentration gradients; the_re are -no pseudo species and no pseudo fluxes. The Di, fi, Ji,and ki are simply mathematical entities defined as an aide to developing eq 17 and, as such, it is quite unnecessary to ascribe any physical interpretation to them or, for that matter, to leave the results in the form of eq 17. The solution to the first-order matrix differential equation d b ) / d v = [A(v)I(u) + (4s)) (18) with variable coefficient matrices [A(v)] and (a(q)) can be obtained using a generalization of the method of successive substitutions and this method does not require the uncoupling of eq 18 (and, therefore, the introduction of any "pseudo quantities"). The solution is (see, e.g., Amundson, 1966)
b)= [ Q o ' W I C ~ O )+ [Qo?(A)I
Lv
[ Q o U ) I - ' ( ~ ( T ) ) d7
(19) where (yo) is the "initial condition", (y(v = O)), and where [Q,.(A)] is the matrizant given by the iterated integral
Ind. Eng. Chem. Fundam., Vol. 21, No. 4, 1982 409
If [A] is independent of q then [Qov(A)]is the exponential matrix exp([Alq). It follows that if the linearized differential mass balance (8) can be simplified or transformed to the form of eq 18, then the solution is given immediately by eq 19. The rather large number of integrations required by eq 20 cannot always be carried out to give a useful result. However, the assumption of constant [D] often ensures that [QO'J]has a particularly simple form, an exponential matrix. An exponential matrix is relatively straightforward to compute, even in mixtures containing a large number of components. A good number of important models of multicomponent mass transfer can, in fact, be solved by making use of eq 18-20. The purpose of this paper is to provide some illustrations of this method and to point out other cases that can be handled in the same way. This introductory review would not be complete if we failed to point out that Toor (1964) noted that uncoupling was not strictly necessary. This paper employs a different method of expanding a matrix function. In the following we will focus attention on the film and penetration theories of multicomponent mass transfer. The physical basis of these models is discussed elsewhere (e.g., Bird et al., 1960) and is not considered here. The Film Model Steady-state one-dimensional mass transfer processes are examples for which the linearized equations can be reduced to the form of eq 18 by direct integration. With these assumptions the linearized eq 8 becomes
(x
- xo) = (exp[*Iq - [IlIlexp[*l - [Ill-' (xa - xo)
(28)
The diffusion fluxes at z = 0 can now be calculated directly from eq 4 with the mole fraction gradients obtained by differentiation of eq 28 as
(Jo)=
Pl[*Ilex~[*l -
[W( x o - xa)
(29)
If the total flux, N,, vanishes, the development above does not hold. For this special case eq 21 can be integrated to give eq 22 and then integrated again to give, on eliminating the matrix of constants using the boundary condition at z = 6 (q = 1) (x - XI)) = dxa - xo)
(Nt = 0)
(30)
The Ji follow by differentiation of eq 30 and combination with eq 4 C
(4 = j [Dl(.o - xa)
(Nt = 0)
(31)
The matrix function [\k](exp[*] - [4]-lis a generalization of eq 21.5-48 of Bird et al. (1960) and accounts for the influence of a nonzero total flux on the diffusion fluxes predicted by eq 31. The matrix c [ D ] / 6is a matrix of "low flux" mass transfer coefficients, the matrix (c/6) [D][*I(exp[*] is the matrix of "high flux" mass transfer coefficients or [k'] in eq 16. Equations 28 and 29 were given in this form by Taylor and Webb (1980). The present derivation is slightly simpler, however, and has been given in more detail for comparison with the solution of the penetration model which we consider now.
Since Nt, c, and [D] are all constant, the first integral of eq 21 may be written immediately as
T h e Penetration Model For unsteady-state diffusion in a semiinfinite field the linearized eq 8 simplifies to
It is not strictly necessary to calculate the column matrix of constants. However, from the form of the solution it is possible to deduce that they are in fact the constituent molar fluxes, Ni. Although eq 22 is already in the form of eq 18, it is convenient to rearrange it to the even simpler form
which is to be solved subject to the boundary conditions t I O , 2 L 0, (x) = (xm)
where q is a dimensionless distance defined by q = z / 6 in which 6 is the film thickness for mass transer. [*I is a matrix of rate factors defined by
[*I
Nd
= - [Dl-' C
and the qi are the flux ratios Ni/N,. The *ij and $i are of course constant (because [D] and the Ni are). Comparing eq 23 to eq 18 we see that [A] = [Q] and ( a ) = (0). Thus [Q0'J(9)] = exp{[*]q) and the solution is (x - 9) = expN*1d(xo - 9) (25) where (xo) me the mole fractions at z = 0. Subtracting (x,) from eq 25 leaves (x - xo) = lexp[*Iv - [AI b o - $) (26) The column matrix ( x o - $) can be eliminated from eq 25 and 26 by introducing the boundary condition at z = 6 (x, - xo) = (exp[*I - [IIl(x0 - $) (27) and combining this result with eq 26 gives the real composition profiles
t > 0, z = 0, ( x ) = (x,) t > 0, z =
m,
(x) = (x,)
(33)
u in eq 32 is the molar average velocity (= Nt/c). With the assumption of a constant molar density, c, the velocity u is a function only of time. The solution to the analogous binary problem is obtained most easily by defining a new independent variable ~ / ( 4 D ~ ~ t(Bird ) ' / ~et al., 1960). In the uncoupled multicomponent case the eigenvalue generalization z / (4Dit)'i2 may be used. However, this variable is different for each species and in matrix terms is without any formal meaning. To solve eq 32 directly, we use the combined variable q = z/(4D,t)'l2 which is the same for all species (real or otherwise) in the mixture. In terms of q eq 32 becomes (34) where
[a1= [Dl/Dr
(35)
Here, D, is an arbitrary reference diffusivity and 4 = u,(t/Dr)1/2. C$ is not a function of q (see Bird et al., 1960). The boundary conditions that apply to eq 34 are 7 = 0, (x) = b o ) ; q = m, (x) = (xm) (36)
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Ind. Eng. Chem. Fundam.. Vol. 21,
No. 4, 1982
If we denote d(x)/dq by ( i )then eq 34 becomes (cf. eq 23) (37) with [B(s)l = 4x11 - 4 ) [ W
(38)
The solution to eq 38 is given by eq 19 with (a) = (0) d(x) - = ( i )= [Q,"B)](i,) (39) dll where (io) is the column matrix of composition gradients at 9 = 0. The solution to eq 39 can be obtained by direct integration as
--
,'/2
2
ex~1[81~Ha)l'~~ lerflh - 6)x
[B]-1/2} + erf[B]) (48)
Thus the composition profiles are ( x - xo) =
{erfl(a- 4)[91-'/21 + erf[ell{[rl + erf[6Il-'(xm - xo) (49) The column matrix ( i ocan ) be eliminated using the boundary condition at q = m (x,
- xo) = (Lm[Qon(B)l dsJ(i0)
(41)
( x - x,) =
which can be combined with eq 40 to give
{[rl- erfl(v - 4)[Wi2JJ([rl+ erf[el)-'(xo-
d?J(Jm[QoW1 dq1-l ( x , - xo)
( x - X O ) = {J'[Qo'(B)l
(42) So far we have not said anything about the form of the matrizant [Q] or of its integral in eq 40-42. The integrations required by eq 21 can be performed quite straightforwardly since the dependence of [B] on q is so simple. The result is
c
-
(43)
The last part of eq 43 follows because functions of one matrix ([D] here) all commute. Consider now the integral exp(-t2[AI2)dt
where [A] is any arbitrary, but constant, matrix. Expanding the exponential function in a power series and integrating term by term yields 2 -J' e ~ p ( - t ~ [ Adt ] ~=) [A]-' erqq[A]) (45) 0
erfla[A])is the matrix error function defined by erfldA11 = 713
q[A]
- -[AI3 3.1!
1 d(x)/ (4Drt)1'2 d11 '=o
C
(Jo)= -[D]'/2
(7rt)'iZ
(44)
,112
z=o
which may be combined with eq 4 to give the following fluxes at t = 0
where we have defined
0
dz
24T)k[W
= exp{-(q2 - ~@v)[BI-~I = exp(-((q - 4)2- @2)[a)l-11 = exp([eI2lexp(-(v - 4 ) 2 [ W 1
*1/2
(50) which is the complete matrix generalization of eq 19.1-16 given by Bird et al. (1960). The diffusion fluxes are calculated as before. The composition derivatives are obtained by differentiation of eq 50 as
(-l)k
[QOV(B)1= [I1+ k = l -+I2k.
"1'
where we have made use of the first part of eq 47 and the fact that [D]1/2,e~p[O]~, and erq[e])all commute. The more usual way of expressing the composition profiles is in terms of the difference ( x - xJ. Thus on subtracting (x,) from eq 49 we find, after some manipulation
v5
+ -[A5] 5.2!
-
11'
-[A]' 7.3!
+ ...
which, if [A] is positive definite and diagonalizable, has the properties erfla[Al) = [I];erfl-v[A]) = -erf[q[A]] (47) The integral of the matrizant in eq 40-42 can now be derived from eq 45-47 as
e~p{-[8]~){[1] + erf[e])-'(xo - x , )
(52) The preceding development holds for the special case of N, = 0, simply by setting @ = 0 in eq 34 et seq. The composition profiles become
{ [ I ]-erflq[B]-1/2))(xo- x,) (Nt = 0) and the diffusion fluxes are given simply by ( x - x,) =
(53)
The matrix ~ [ D ] ' / ~ / ( a t ) a' / ~ generalization , of eq 21.6-7 of Bird et al. (1960), is the matrix of "low flux" mass transfer coefficients. The matrix function e~p(-[O]~)([Il + erf[O]I-', a generalization of eq 21.6-13 of Bird et al. (1960), accounts for the effect of a nonzero total flux. The product c [ D ] ' / ~ / ( T ~ )with ' / ~ the function of [e] is the matrix of "finite flux" mass transfer coefficients, [k']. The penetration model predicts a weaker coupling between the fluxes than does the film model. Computational Methods The computation of the diffusion fluxes, Ji,requires an iterative calculation unless the total flux, N,, is specified. The most time consuming part of the calculation is the repeated calculation of the matrix [k'] (here defined by eq 16, 29, and 52). Algorithms for these calculations have been given in the recent review paper by Krishna and
Ind. Eng. Chem. Fundam., Vol. 21, No. 4, 1982
Standart (1979), who suggeht the use of Sylvester's theorem for calculating [k']. This method requires the evaluation of the eigenvalues of [D] but, unfortunately, the repeated use of this formula for mixtures with more than three or four components demands excessive amounts of computation time. A stable and much more efficient algorithm for calculating the rates of transfer from the film theory result, eq 29, has been described by Taylor (1982). Many of the computational techniques that were developed for the film model may be used with the penetration model for which we provide a new algorithm below. The required calculations fall conveniently into three parts; (i) [D]1/2= Dr1/2[a)]1/2; (ii) exp(-[eI2};(iii) erf[e]. It is possible to compute all three matrix functions from convergent power series without the evaluation of the eigenvalues of [D] at any stage. (i) The function [D]1/2may be computed from a matrix generalization of the binomial series
411
definite [A] which, in the present case, requires Nt to be positive. If Nt is negative in sign [A] is replaced by -Nt[e] and the computed series multiplied by -1 (see eq 47). Indeed, if Nt is negative it becomes unnecessary even to compute exp(-[0l2)since the function of [e] in eq 52 is given approximately by exP(-[e121([rl+ erf[OIl-l
+ a2[T12+ dT131-l
(59) As a result of eq 59 the algorithm which follows is particularly efficient if Nt < 0. A knowledge of the multicomponent diffusion coefficients is assumed. Step 1. Calculate from the series (55) with [A] given by ([a)] - [A).Invert this matrix. D,is calculated from fl-1
Dr = m a x { C IDikIl I
k=l
Step 2. In the first iteration (the only one if Nt = 0) calculate the Ji from eq 54. In subsequent iterations calculate the Jifrom right t o left from: (i) eq 52 if Nt > 0 or (ii)
(55) The series (55) converges only if the absolute value of the dominant eigenvalue of the arbitrary matrix [A] is less than one. Convergence slows as this limit is reached. Thus the computation of [D]1/2will be expedited if, in place of [A], we use ([a)] - [Il}with D, best chosen as an over estimate of the dominant eigenvalue of [D]. Gershgorin's theorem (Wilkinson, 1965) may be used to estimate D,(see below). (ii) The matrix exponential function can be calculated from the uery rapidly convergent power series (Buffham and Kropholler, 1971; Taylor and Webb, 1981)
[m
if Nt < 0 where is defined in step 5. Do not complete the calculation of the products of the square matrices in the expressions above until convergence, and then only if required for some other purpose. Step 3. Calculate Nt from an appropriate determinacy condition (see Krishna and Standart, 1979). Check the current and previous estimates of Nt for convergence. If convergence has not been obtained continue with step 4. Step 4. Calculate [elz and [e] by scalar multiplication of the appropriate matrices. Step 5. Calculate [Z'I = {[I] + a4[6']]-'. Complete the say: (a) calculation of ( u l [ q + a z [ q 2+ a3[TI3)= if Nt > 0 compute exp(-[eI2}from the power series (56). Complete the calculation of ([q+ erf[B])-' from eq 57; (b) if Nt < 0 invert Return to step 2. Numerical testing has shown the power series methods described above to be distinctly superior to Sylvester's theorem for systems with large numbers of components. At all rates of mass transfer convergence of the exponential function is uery rapid. At very high rates of transfer the algorithm given above will not converge unless the current and previous estimates of Nt are averaged. This failing is common to other published algorithms for this model. This minor problem with the linearized theory has been noted previously only in connection with the film model (Taylor, 1982). Further Applications of the Method The two examples considered above illustrate direct integration of the simplified linearized equations and combination of variables to obtain a first-order matrix differential equation. Other applications of direct integration are the following. (i) Solution of the time-averaged equations for steadystate one-dimensional diffusion in turbulent flow. In this situation the diffusivity matrix is augmented by a position-dependent diffusion coefficient that arise when eq 8 are time averaged (Stewart, 1973). A solution to Stewart's equations obtained by uncoupling is given by Standart and Krishna (1979). The complete solution based on the calculation of the matrizant is given by Taylor (1981b). (ii) Steady one-dimensional coupled heat and mass transfer (i.e., including the Soret and Dufour effects) (Taylor, 1981~).
[m-l
where q is an appropriately chosen integer (see references cited above eq 56). Typically, fewer than six terms of this series are required. (iii) The matrix error function is given by the series (46). If Iddl (where dd is the dominant eigenvalue of [e]) is of the order one (or less) then only a few terms_ofthis series are required. At high rates of mass transfer lodl may be greater than one. In these cases the direct computation of erf[e] from the series (46) is not recommended and the following matrix generalization of a formula given by Abramowitz and Stegun (1964) is suggested erf[AI = [I1- h[Tl + a z [ q 2+ a3[5"l3Iexp{-[Al2l (57) where
[q= (14 + adA1l-l
(58)
with al = 0.3480242, a2 = -0.0958798, a3 = 0.7478556, and a4 = 0.47047. The approximation (57) requires only four matrix multiplications and additions plus one inversion regardless of the number of components or of the total rate of mass transfer (notice that with [A] = [e] the exponential matrix in eq 57 has already been calculated). On the other hand, the time for the computation of matrix functions using Sylvester's theorem increases rapidly as the number of constituents rises. The scalar equivalent of (57) has an which should be sufficiently accurate accuracy of = for most cases. The series (57) is valid only for positive
412
Ind. Eng. Chem. Fundam., Vol. 21, No. 4, 1982
(iii) Unsteady-state diffusion in the diaphragm cell. The solution to the uncoupled equations can be found in the review by Cussler (1976). A solution obtained without uncoupling is given by Delancey (1969). Some other problems that can be handled using combination of variables are: (i) steady evaporation of a multicomponent liquid from a plane liquid surface into a gas stream moving tangentially to the liquid with a velocity that varies linearly with distance from the interface. (The analogous binary problem is exercise 17.5 in Bird et al., 1960.); (ii) absorption of a multicomponent gas into a falling liquid film; (iii) unsteady coupled heat and mass transfer in a semiinfinite medium. A solution to this last problem obtained by uncoupling is given by Delancey and Chiang (1970a) and Delancey (1972). There are, of course, a very large number of situations where these two methods cannot be usefully employed. One example of such a problem is the combined filmpenetration model of Toor and Marchello (1958). This model is a generalization of both film and penetration models discussed above (which, therefore, are the infinite time and short time limiting cases, respectively). The mathematical description of this problem is provided by eq 32 and 33 with the omission of the convective term (u = 0). (The more general film-penetration model with a nonzero flux has not yet found a complete solution except for the case Nt = constant (Thambynayagam et al., 1980). Nt is, in fact, a function of time). The boundary conditions at z is applied at a finite distance ( z = 6) from the origin (z = 0). The multicomponent generalization of the film-penetration model can be solved by uncoupling eq 32 and using the Laplace transform (the results are given by Krishna, 1978). The Laplace transform removes the time dependence from eq 32 but leaves a second-order matrix differential equation that is not amenable to solution using either of the foregoing techniques. A Fourier transform, on the other hand, will remove the spatial dependence from the partial differential equation to leave a first-order matrix differential equation with time as the independent variable. Equations 18-20 and an inverse Fourier transform then provide the complete solution without having to uncouple the governing equations. Integral transforms can be used to solve many related problems, two particularly important ones being multicomponent diffusion with a time-dependent diffusivity matrix and multicomponent diffusion accompanied by homogeneous first-order chemical reactions. This last example cannot generally be solved using the uncoupling methods since the diffusivity matrix and the rate constant matrix cannot be diagonalized by the same modal matrix except in special circumstances (Toor, 1965). Solutions to this third class of problems (including the examples above) are developed in detail by Delancey and Chiang (1970b) and by Delancey (1974). Our final example of a problem that would seem to admit a solution by integral transform methods concerns diffusion in the Loschmidt tube. Solutions to the uncoupled equations are given by Arnold and Toor (1967) and by Krishna and Standart (1979). An important restriction on the integral transform method is that the boundary conditions must be of the same form for all species.
in the form of a matrix similarity transformation. Frequently, however, it is possible to solve the linearized equations without first uncoupling them. Solutions are obtained as full matrix analogues of classical results obtained for binary mixtures. The method of solution used here can be employed only if the linearized eq 8 can be reduced (by some means) to the first-order matrix differential equation with a variable coefficient matrix
Final Remarks The linearized theory is undeniably one of the most important methods of calculating rates of mass transfer in multicomponent mixtures. With the assumptions leading to eq 8 it is possible to formulate the mass transfer coefficients for a wide variety of important hydrocynamic models using the "pseudo variable" approach of the uncoupled equations. Final results have frequently been left
Matrix Notation
-
0)
(60) In this paper we have shown that the film and penetration models of multicomponent mass transfer can be reduced to this form and solved by using the properties of the matrizant. We have also identified many other models that can be solved in this way. If the linearized eq 8 cannot be written in the form (60), the diagonalization procedures of Toor (1964) and of Stewart and Prober (1964) may be necessary to obtain a solution. However, even if we must uncouple the equations there is no real need to leave the composition profiles in the form of a similarity transformation; the solution can always be written out explicitly in matrix terms. Finally, it is never necessary to uncouple eq 4 or 5 for the diffusion fluxes; composition gradients can be obtained by differentiation of the composition profiles written in terms of the original variables. Nomenclature [A(n)]= matrix in eq 18 [ A ]= arbitrary square matrix ( a ) = arbitrary column matrix [ B ]= matrix defined by eq 38 c = molar concentration [D] = matrix of multicomponent diffusion coefficients
[a1= [DI/Q D , = reference diffusivity [ I ] = identity matrix J , = diffusion flux of species i [ k ] = matrix of multicomponent mass transfer coefficients N, = molar flux of species i N , = total molar flux [PI = modal matrix of [D] r = position vector u = molar average velocity = Nt/c x , = mole fraction of species i z = distance Greek Letters 6 = film thickness 9 =
independent variable
u(t/D,)'iZ [ q ]= matrix defined by eq 24
f$ =
& = flux ratio N i / N t
[e] = matrix defined by eq 44
[Q]= matrizant defined by eq 20 Subscripts i, j , k , n = index denoting component number 0, 6, m = pertaining to position Superscripts = quantity modified by finite rates of transfer ^= eigenvalue of corresponding square matrix; pseudo variable
1, { ] = square matrix of dimension n-1 [_I-', { j-l = inverse of a square matrix [
X
n-1
[ ] = diagonal matrix of eigenvalues matrix with n - 1elements ( ) = column matrix with n - 1 elements
Literature Cited Abramowitz, M.; Stegun, L. A. "Handbook of Mathematical Functions"; National Bureau of Standards, 1964.
Ind. Eng. Chem. Fundam. 1982, 2 1 , 413-416 Amundson, N. R. “Mathematical Methods in Chemical Engineering, Matrices and their Applications”; Prentlce Hall: Englewood Cliffs, NJ, 1966. Arnold, K. R.; Toor, H. L. AIChE J. 1987, 13, 909. Bandrowski, J.; Kubaczka. A. Int. J. Heat Mass Transfer 1981, 2 4 , 147. Bird, R. B.; Stewart, W. E.;Lightfoot, E. N. ”Transport Phenomena”; Wiley: New York. 1960. Buffham, 8. A.; Kropholler, H. W. Conference On Line Computer Methods Relevant to Chemical Englneering, UnkrersRy of Nottingham, 1971. Burchard, J. K.; Toor, H. L. J. W y s . Chem. 1982,66, 2015. Catty, R.; Schrodt, T. I n d . Eng. Chem. Fundam. 1975, 14, 276. Cotrone, A.; deGlorgi, C. Ing. Chim. Ita/. 1971, 7 , 84. Cullinan. H. T. Ind. Eng. Chem. Fundam. 1885,4 , 133. Cussler, E. L. “Multicomponent Diffusion”; Elsevier: Amsterdam, 1976. Crank, J. “The Mathematics of Diffusion”, 2nd ed.; Ciarendon: Oxford, 1975. Dekncey, G. B. J. Phys. Chem. 1989, 73, 1591. Deiancey, G. B. Chem. Eng. Sci. 1972,2 7 , 555. Dekncey, 0. B. Chem. Eng. Sci. 1974,2 9 , 2307. Dekncey, G. B.; Chiang, S. H. Ind. Eng. Chem. Fundam. 197Oa, 9 , 138. Dekncey, G. B.; Chkng, S. H. Ind. Eng. Chem. Fundam. 197Ob, 9 , 344. Krishna. R. Chem. Eng. Sci. 1978,33, 765. Krishna, R.; Panchai. C. B.; Webb, D. R.; Coward, I . C. Lett. Heat Mass Transfer 1978,3 , 163. Krishna, R.; Standart, G. L. AIChE J. 1978,22, 383. Krishna, R.; Standart, G. L. Chem. Eng. Commun. 1979,3 , 201.
413
Smith, L. W.; Taylor, R. Ind. Eng. Chem. Fundam. 1982,in press. Standart, G. L.; Krishna, R. Lett. Heat Mass Transfer 1979,6 , 35. Stewart, W. E. AIChE J. 1973, 19, 398. Stewart, W. E.; Prober, R. Ind. Eng. Chem. Fundam. 1984,3 , 224. Taylor, R. Chem. Eng. Commun. 1981a, 10, 61. Taylor, R. Lett. Heat Mass Transfer 1981b,8 , 397. Taylor, R. Lett. Heat Mass Transfer 1981c,8 , 405. Taylor, R. Comput. Chem. Eng. 1982,6 , 69. Taylor, R.; Webb, D. R. Chem. Eng. Commun. 1980, 7 , 287. Taylor, R.; Webb, D. R. Comput. Chem. Eng. 1981, 5 , 61. Thambvnavaaam. R. K.: Winter, P.; Branch, S. W. Trans. I . Chem. E . 1980, 58,’27?. Toor, H. L. AIChE J. 1957,3 , 197. Toor, H. L. AIChE J. 1984, 10, 448, 460. Toor, H. L. Chem. Eng. Sci. 1985,20. 945. Toor, H. L.; Marchello, J. M. AIChEJ. 1958, 4 , 97. Webb, D. R.; Sardesai, R. G. Int. J. Multiphase Flow 1981, 7 , 507. Wilkinson, J. H. “The Algebraic Eigenvalue Problem”; Clarendon: Oxford, 1965.
Received for review July 13, Revised manuscript received May 13, Accepted July 28,
1981 1982 1982
Estimating Gaseous Diffusion Coefficients from Passive Dosimeter Sampling Rates, with Application to HF Michael S. Young and Jamie P. Monat* WaMen Division of Abcor, Inc., Wilmington, Massachusetts 0 1887
Knowledge of gaseous diffusion coefficients is important for many gas-phase mass-transfer operations. I n many instances, these diffusion coefficients are difficult to measure with simple techniques. I n this paper, it is shown that passive dosimeters may be used experimentally to determine diffusion coefficients for species that are efficiently sampled by this technique. Examples are presented, and the method is applied to the determination of the HF-inair diffusivity, a quantity for which no previous experimental data have been published.
Introduction Gaseous diffusion coefficients are important for the quantitative description of many gas-phase mass-transfer operations, among them gas chromatography, diffusion separations, and gas-phase catalysis. In one recent application, predictable molecular diffusion has been used for the quantitative sampling of work-place atmospheric pollutants. In this correspondence it is shown that good estimates of gaseous diffusivities can be obtained from experimentally determined sampling rates for such diffusion-controlled sampling devices. Because the calibration procedure employs a dynamic test atmosphere, refinements of the method described herein may be of use in determination of gaseous diffusion coefficients for materials that are too reactive for normal static or chromatographic measurements, or for such materials that show large variations of diffusivity with relative concentration of the diffusing species. Good diffusivity estimates may be obtainable for binary mixtures where one of the diffusing species exists for as short as a few seconds. One must exercise caution in the application of this technique, however; it is valid only for those species that are efficiently sampled by passive dosimetry, and blind application of it to species for which collection efficiency has not been validated may lead to erroneous conclusions. Theory A number of devices have been marketed in recent years in which the limiting resistance to mass transfer is con0196-4313/82/ 1021-0413$0 1.25/0
tained in a current-free air layer of fixed geometry located between the diffusion opening of the device and an efficient sorbent located within the device. Palmes and Gunnison (1973) and Tompkins and Goldsmith (1977) have described in detail this mass-transport to two similar types of diffusion samplers. They have shown that the sampling rate for a diffusion sampler is given by eq 1for 100% collection efficiency and ideal response. DAC, N = - L (1) where N = diffusive sampling rate (mol/s), A = crosssectional area of diffusion path (cm2),L = diffusion path length (cm), C , = ambient concentration of pollutant (mol/cm3),and D = diffusivity of pollutant in air (cm2/s). N is measured by chemical analysis subsequent to controlled exposure of the device to known test atmospheres. For the Gasbadge dosimeter, an experimental calibration factor (R)is generally used to account for such parameters as less than 100% collection efficiency and adsorptive losses internal to the device. Equation 1is thus modified DRAC, N=Ir
where R is the experimental calibration factor. It is clear from eq 2 that the product D X R can be determined in a controlled test atmosphere experiment in which A and L are known, C , is measured by independent sampling 0 1982 American Chemical Society
