Ind. Eng. Chem. Fundam'. 1983, 22, 97-104 Rastorguev, Yu. L.: Pugach, V. V. 1970, 13. 69. Rastorguev, Yu. L.; Pugach, V. V.
1971,1 4 . a i .
Izv. Vyssh. Ucheb. Zaved., NeftiGaz Izv. Vyssh. Ucheb. Zaved., N e f t i G a z
Roder, H.; de Castro, C. A. N. J. Chem. Eng. Data 1982,2 7 , 12. Roder, H. M. J. Res. Nstl. Bur. Stand. ( U . S . ) 1982,8 7 , 279. Rowlinson, J. S. "LiquMs and LiquM Mixtures", 2nd ed.; Plenum Press: New York, 1969. Ryabtsev, N. I.; Kazaryan, V. A. Gazov. Prom. 1969, 1 4 , 46. Ryabtsev, N. I.; Kararyan. V. A. Gazov. &lo. 1970,2 , 36. Sakiadis, 8. C.: Coates, J. AIChE J. 1955, 1 , 275. Sakiadis. B. C.; Coates, J. AIChE J. 1957,3 , 121. Sengers, J. V. "Critical Phenomena": Proc. Int. School of Physics "Enrico Fermi", Course LI, Green, M. S.,Ed.; Academic Press: New York, 1971. Sengers, J. V. Ber. Bunsenges. Phys. Chem. 1972, 76, 234.
97
Swlnney, H. L.; Henry, D. L. Phys. Rev. 1973,AB, 2566. Teja, A. S.; Rice, P. Chem. Eng. Sci. 1981,3 6 , 417. Tsederberg. N. V. "Thermal Conductivlty of Gases and Liquids"; MIT Press: CambrMge, MA, 1965; Chapter 7. Vargaftik, N. B. "Tables on the Thermophysical Properties of Liquids and Gases", 2nd ed.: Wlley: New York, 1975. Vernart, J. E. S. "The Thermal Conductlvity of Binary Organic Liquid Mlxtures"; Proc. 4th Symposium on Thermophysical Properties, 1968. Villm, 0. Collect. Czech. Chem. Commun. 1960,25, 993. Ziebland, H.: Burton. J. T. A. J. Chem. Eng. Data 1961,6 , 579.
Received for reuiew F e b r u a r y 16, 1982 Accepted O c t o b e r 1, 1982
Film Models for Multicomponent Mass Transfer: A Statistical Comparison Lawrence W. Smith' and Ross Taylor' Department of Chemical Engineering, Ciarkson College of Technology, Potsdam, New York 13676
The rates of multicomponent mass transfer predicted from several approximate solutions of the Maxwell-Stefan equations for a film model of steady-state diffusion are compared with the fluxes calculated from an exact solution for many hundreds of thousands of example problems. It is found that the assumption that the matrix of multicomponent diffusion coefficients remains constant is an excellent one and that the solution of the linearized equations due to Toor and to Stewart and Prober always provides adequate estimates of the mass-transfer rates. The method of Taylor and Smith (a generalization of a method by Burghardt and Krupiczka) is the better of two explicit methods, bettering even the solution of the linearized equations in a number of cases. The explicit method of Krishna performs well if the rates of mass transfer are low. The simple effective diffusivity methods are woefully
inadequate.
Introduction Multicomponent mass transfer is encountered in many important processes such BS condensation, distillation, and gas absorption. The rigorous calculation of the rates of mass transfer in mixtures with three or more components is complicated by the coupling, or interaction, between individual concentration gradients. Possible consequences of this coupling are that a species may transfer in the direction opposite to that expected (reverse diffusion), may not diffuse at all even though a concentration gradient for that species exists (a diffusion barrier), or may diffuse in the absence of any driving force (osmotic diffusion) [see, e.g., Toor (1957), Krishna and Standart (1979),Reinhardt and Dialer (198l)l. It is now well established that diffusion in multicomponent ideal gas mixtures is accurately described by the Maxwell-Stefan equations. Exact analytical solutions of these equations for the general n-component case are known only for a film model of steady-state one-dimensional mass transfer [Krishna and Standart (1976, 1979), Taylor (1981a, 1982a)l. In addition to these exact solutions, a number of approximate solutions have also been published. The simplest and most widely known are those based on the concept of an "effective diffusivity" [Wilke (1950), Bird et al. (1960)l. The drawback of the effective diffusivity formulas as they are frequently used is that they do not accurately reflect the character of multicomponent diffusion. More rigorous methods that are able to predict the various interaction phenomena, based on the assumption that the matrix of multicompo'Indiana Institute
of
Technology,
Fort
Wayne,
IN 46803.
nent diffusion coefficients remains constant over the diffusion path, are due to Toor (1964) and to Stewart and Prober (1964). Notwithstanding the present availability of a generally applicable exact solution, approximate methods continue to be published [Burghardt and Krupiczka (1975), Krishna (l979,1981a), Taylor and Smith (1982)l and to find applications in design calculations [Bandrowski and Kubaczka (1981), Hegner and Molzahn (1979), Webb et al. (1981)l. Even as recently as 1975, Sherwood et al., in discussing three of the effective diffusivity methods, were able to write that "the procedures have not been tested by comparison with the rigorous equations over a wide enough range to provide an adequate judgement of their relative merits". Their remarks remain true today. Indeed, they could be applied with considerable justification to most of the other approximate methods cited above. The objective of the present paper is to provide a comprehensive statistical comparison of many of the approximate solutions of the Maxwell-Stefan equations for a film model of steady-state diffusion. All of the methods included in this comparison have the important property of being applicable to mixtures with any number of constituents. The Equations of Multicomponent Mass Transfer For steady-state unidirectional diffusion under isobaric, isothermal conditions the Maxwell-Stefan equations can be written as dy. ' Y i N k - Y k N i ' Y i J k - ykJi (1) do k = l C a i k / b k=l ca,k/h
-L=c k#i
0196-4313/83/1022-0097$01.50/00 1983 American Chemical Society
=c k#i
98
Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983
Table I. Determinacy Coefficients "i
1. equimolar counter transfer
Table 11. Definition General of Matrices with Elements [ M i j ] qmi n mk M..--+ c - - i = l , 2 ,..., n - 1 11 -
Vn
1 1
1
ai,,
2. diffusion through a stagnant 0 n t h species 3. simultaneous heat and mass g? - HF Eny H," transfer 4. diffusion in a carillan d z - tube JfcT; 5. zero mass average velocity hi ' M n ''
Mi=-m.(' I
where 7 is a dimensionless distance defined by q = z / 6 , in which 6 is the film thickness. The a)ik are the binary diffusion coefficients of the i-k pairs. The molar fluxes, Ni, which, for steady diffusion through a plane film are position invariant, are made up of diffusive and convective terms as
matrix [B]
[B] [A] [Q]
n
Ni = Ji+ yiNt;Nt = E N i i=l
(2)
i = 1, 2, ..., n where Nt is the total molar flux. Only n - 1 of eq 1 are independent since the mole fraction gradients and the molar diffusion fluxes, Ji, sum over the n species to zero
(3) From a practical engineering viewpoint it is the fluxes
Ni and Ji which are of interest. Equations 1 must therefore be 3nverted" in order to relate these fluxes to the mole fraction gradients. This is most easily done in n - 1 dimensional matrix form (although we should note that other, less useful, methods have also been employed (see below)). The n - 1independent Ji can be related to the mole fraction gradients as
[U ]
k=l k #i
i
' j )
.nu a,
#
j = 1, 2,
meaning inverted matrix of diffusive mass transfer coefficients matrix of inverse diffusion coefficients matrix of inverse total mass transfer coefficients matrix of rate factors in the exact solution matrix of rate factors in the linearized theory
..., n - 1 mi
Qi
y i / ( c / 8)
1
yi
1
y i / ( c / 8)
Vi/vn
Ni/(c/&)
1
Ntyiav/(c/8)
1
placing Nt in eq 2 by eq 7 and employing the second summation in eq 3 to give (8) ui = 1 - ( y i / Z i ) i = 1, 2, ..., n which should be evaluated using the compositions at one of the boundaries Cyo or ya). Since this type of problem involves the iterative calculation of some of the boundary compositions the vi may be subject to change. This does not pose any particular computational problem. With this additional relationship we can relate the N , to the independent Ji in matrix form as [Krishna and Standart (1979)]
where the "bootstrap" matrix [ p ] has elements where [ D ]is a matrix of multicomponent diffusion coefficients. The multicomponent Dik can be related to the binary a ) ; k by [D] = [B]-'
(5)
the elements of which matrix are defined in Table I. The matrix [B] in eq 4 is a matrix of reciprocal binary mass transfer coefficients and is also defined in Table I. The linear dependence among the mole fraction gradients means that an additional relationship, called the determinacy condition, is required before the n Ni can be calculated. Two types of determinacy conditions are encountered in practice. In some cases a linear relationship between the fluxes exists as [Krishna and Standart (197911 CuiNi = 0 i=l
Values of the vi for several special cases are given in Table 11. In other situations such as vapor condensation to an unmixed liquid or in diffusion-controlled heterogeneous chemical reactions, the determinacy condition can be written as N J N , = zi (7) i = 1, 2, ..., n In this case, however, only one of the fluxes can be considered independent and some of the boundary conditions (see below) must be separately determined to satisfy eq 7. This second class of problems can be fitted into the general (and, as we shall see, useful) form of eq 6 by re-
pik = aik - (vk - un)-Yi or 6ik Y*
1- Yi/Zi
(10)
y* in eq 10 is a weighted mole fraction defined by n
y* = cuyi i=l
A null value of y* also serves to identify combinations of boundary and determinacy conditions for which no solution can be found. As an alternative to eq 9 we may choose to relate the Nidirectly to the mole fraction gradients by [Taylor and Smith (1982)]
with the elements of the matrix [A] given in Table I. We note here that although [A]-' is equivalent to [p][BI-', the two matrices are not, in general, equal to each other. Equations 1-12 are the basis for many of the published solutions of the Maxwell-Stefan equations for a fiim model of steady-state mass transfer. The boundary conditions of this model are (13) t) = 0, Yi = Yio; t) = 1, yi = Yia i = 1, 2, ..., n The solutions are reviewed below. An Exact Solution Exact solutions of eq 1 for certain special cases have been known for a long time [Gilliland (1937),Pratt (1950),
Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983 9S
Table 111. Physical Properties of Ternary Mixtures binary diffusion coefficients, determinacy condition mmz s-' from system
I I1 I11
IV V VI VI1 VI11 IX X XI XI1 XI11 XI v
xv XVI
ammonia (l), water ( 2), air ( 3 ) acetone ( l ) ,benzene ( 2 ) , helium ( 3 ) ammonia ( l ) ,water (2), hydrogen ( 3 ) butane (l), octane ( 2), hydrogen ( 3 ) propane (l), undecane ( 2), hydrogen ( 3 ) acetone ( l )methanol , (2), air ( 3 ) carbon dioxide ( l ) water , (2), hydrogen (3) pentane (l),ethanol ( 2 ) , water ( 3 ) ethanol (l), 1-butanol ( 2 ) , water ( 3 ) helium (l), nitrogen (2), oxygen ( 3 ) helium (l), sulfur hexafluoride (2), oxygen ( 3 ) nitrogen (l), sulfur hexafluoride (2), oxygen propane (l), butane (2), pentane ( 3 ) helium (l), neon ( 2 ) , argon ( 3 ) hydrogen (l), styrene ( 2 ) , ethylbenzene ( 3 ) methanol (l), ethanol (2), water ( 3 )
aza
2) 11
a,13
147.0
107.5
124.5
2
4.0
41.0
39.0
2
29.4
113.0
130.0
2
42.0
589.9
402.0
2
Sherwood (1937); Krishna (1979) Krishna and Standart (1976); Krishna (1979) Sherwood e t al. (1975); Krishna (1979) Taylor and Webb (1980a,b)
Table I
sources
8.17
110.83
52.15
2
Taylor and Webb (1981)
8.48
13.72
19.91
2
Carty and Schrodt (1975)
27 0.6
345.8
1, 2
Wilke (1950); Toor (1957)
7.27
14.4
20.9
1,3
Krishna (1981a)
7.99
21.4
16.5
1
Krishna et al. (1981)
73.6
79.2
21.9
194, 5
Tai and Chang (1979)
43.5
79.2
9.99
1,4, 5
Tai and Chang (1979)
10.5
21.9
9.99
194, 5
Tai and Chang (1979)
15.81
13.72
11.47
1
Taylor and Webb (1981)
112.6
72.9
32.2
4
12.8
12.6
4
Remick and Geankoplis (1970) Krishna (1977)
13.4
26.8
3
Rohm and Vogelpohl(l980)
92.2
Cichelli et al. (1951), Toor (1957), Keyes and Pigford (1957), Hsu and Bird (1961),Johns and DeGance (1975)l. A general solution applicable to mixtures with any number of constituents seems first to have been developed by Turevskii et al. (1973) and independently by Krishna and Standart (1976). The solution is
b - YO) = (exp[@Io- [41J((Yo)+ [@I-'($))
(14)
The elements of the square matrix [@I are defined in Table I; the elements of the column matrix (6) are
1.17 20.6
caijremain constant over the diffusion path.
This is true for ideal gases at constant temperature and pressure. For nonisothermal conditions suitably averaged properties are required. The computation of the fluxes from eq 9 and 17 requires an iterative approach. Taylor and Webb (1980a) have shown that eq 17 may not converge if the eigenvalues of [ a ] are large and negative. To overcome this problem the diffusion fluxes must be recalculated at 17 = 1 from (Ja) = [~~l-l[@llex~[@lJlex~[@l -
[~ll-'(YO- Y&
(17)
(18) which may not converge if the eigenvalues of [@I are positive. Stable and efficient algorithms for computing the fluxes from an exact solution of the Maxwell-Stefan equations have been developed by Taylor and Webb (1981) and by Krishnamurthy and Taylor (1982). The latter algorithm was used in the present study. The Effective Diffusivity Methods Approximate solutions of the Maxwell-Stefan equations are of considerable historical importance in chemical engineering (and also because generally applicable exact solutions have not yet been derived for other models of mass transfer, e.g., penetration and boundary layer). The oldest and simplest method, pioneered by Hougen and Watson (1947) and by Wilke (1950) employs the concept of an effective diffusion coefficient. The diffusion flux of species i is written as the product of an effective diffusion coefficient and the negative of the composition gradient as
The subscript 0 indicates the position at which (4and [B] are calculated. The N i are then obtained by combining eq 9 and 17. The only assumptions required for the development of eq 14-17 (other than those of a film model) are that the
i = 1, 2, ..., n - 1 The effective diffusivity, B a , is then defined by comparing
$i
= -Ni/(Cain/6)
i = 1, 2,
(15)
..., n - 1
The first of two important contributions made by Krishna and Standart (1976) was to substitute the boundary condition for q = 1 into eq 14 and combine the result with eq 14 to obtain the composition profiles through the film as (Y - YO) = lexp[@Io- [41Jlex~[@l - VII-YY~- Y O )
(16)
Alternative forms and derivations of eq 14 and 16 valid when the matrix [a]is singular (when Nt = 0) and for when [a] has complex eigenvalues (see below) have been given by Taylor (1981a, 1982a). The second contribution of Krishna and Standart (1976) was to derive an expression that permits the diffusion fluxes to be calculated ( J o )= [Bol-l[@llex~[@l - [Who - Y ~ )
100
Ind.
Eng. Chem. Fundam., Vol. 22, No.
1, 1983
eq 19 to eq 1 or 4. Two general expressions for this coefficient are [Bird et al. (1960)l (20)
..., n - 1
with the Ni then given by eq 9. An iterative approach to the determination of the fluxes is required unless Nt = 0 in which case
j#i
j#i
i = 1, 2,
i = 1, 2, ..., n - 1 and [Stewart (1954)J i = 1, 2, i = 1, 2, ..., n - 1 Actually, Stewart presents only the expanded form of eq 21 for a ternary mixture. Stewart suggests that for practical calculations the relative mole fraction gradients dyk/dyi be replaced by the ratio of differences (YkO Yka)/(Yio - yia). The Dik are calculated from eq 5 with [B] evaluated a t the arithmetic average composition. In general, aidis not a physical property of the mixture and is dependent on the molar transfer rates which, in turn, cannot be calculated without knowing 21iefp If the flux ratios are known or can be approximated, then the Bieff may be calculated. Some special cases of eq 20 are of particular importance. (i) All binary diffusion coefficients equal
Bieff= Dii =
=
B
(22)
(ii) n - 1species present in such low concentrations that the approximations yi = 0, i = 1,2, ..., n - 1,yn = 1 can be made. In this case aieff
=
(23)
Bin
i = 1, 2, ..., n - 1 (iii) When species i diffuses through n - 1stagnant gases, N j = 0, j # i [Wilke (1950)]
An alternative formula for Qeff has been used by Burghardt and Krupiczka (1975). Their approach is to set
k#i
which is equivalent to setting BieB= l/Bii with Bii defined in Table I. This really amounts to neglecting the off-diagonal terms in the matrix [B]. If Bieffcan be assumed constant a t some suitably averaged composition, then the composition profiles are easily obtained as
i = 1, 2, ..., n
-1
where @ieff
= Nd/CBieff
i = 1, 2, ..., n - 1 The diffusion fluxes at 7 = 0 are calculated from
(27)
..., n - I
The greatest drawback to eq 23-29 is that they completely fail to account for the various interaction phenomena (reverse or osmotic diffusion) that can arise in systems with three or more components. Stewart's definition of Bieff is the best in this connection. However, in view of the current availability of the more rigorous (and, indeed, sometimes even simpler) methods described elsewhere in this paper and the relative ease with which elementary matrix operations can be performed on a computer, there seems to be little justification in the continued use of the effective-diffusivity formulation. Yet these methods continue to find employment in engineering design calculations. Hegner and Molzahn (1979), for example, considered the use of one of the matrix methods but then opted for an effective diffusivity method in their design of a packed tower. Webb et al. (1981) have recently advocated eq 28 with aieff given by eq 23 for the process of condensation in the presence of an inert gas. Webb et al. impose a constraint on this method that I@ieffl should not be greater than 0.4. This is, however, a fairly conservative limit. We have included the simpler effective diffusivity methods (23,24,25) and Stewart's (1954) method in our has been evaluated at the comparison. In all cases aieff arithmetic mean composition. More complicated averaging procedures, sometimes requiring iteration, or methods postulating a variation of aieff on y, have been discussed by Wilke (19501, Toor (1957), Shain (1961), and by Hsu and Bird (1961). In view of the many better methods that have been developed since then they have not been included in this exercise. It shouId be recognized that we have used each of these effective diffusivity methods for all the problems considered (see below) even if conditions are such that the assumptions leading to the simplifications of (20) are not even approximately satisfied. In this way we have deliberately overextended the generally acceptable range of their usefulness. This is, however, surely not the first work to do this. The Linearized Theory The f i t successful attempt to develop a method capable of reliably predicting coupled diffusion effects and which was also applicable to mixtures with any number of components led to the linearized theory of multicomponent mass transfer, proposed independently by Toor (1964) and by Stewart and Prober (1964). Their method is based essentially on the assumption that the matrix of multicomponent diffusion coefficients, [D], remains constant over the diffusion path. Solutions of the coupled diffusion equations were then obtained by uncoupling the equations and expressing the results in terms of the eigenvalues of [ D ] . For several important models of multicomponent mass transfer the solution of the linearized equations can in fact be obtained without uncoupling the equations [Toor (1964), Taylor and Webb (1980b), Taylor (1981b, 1982c)J. Thus, for a f i i model of mass transfer the diffusion fluxes can be calculated from [Taylor and Webb (1980b)l
Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983
101
f(Y*) is given by
N& [@I = -[D]-l C
= NJB,,]
The subscript av indicates that [B] and [@I are calculated at the arithmetic average composition. As with the effective diffusivity methods and the exact solutions, an iterative approach to the computation of the fluxes is required. The formal similarity between eq 17 and 30 is noteworthy and has been exploited in the computational algorithm of Taylor (1982b). If Nt is known, then the calculation of the Jifrom eq 30 is explicit. A particularly simple case is that of equimolar counter transfer, Nt = 0, for which eq 30 simplifies to (N) = (JO) = [BavI-'(YO - Y6)
INt = 01 (32) The assumption of constant [D] is plainly incorrect, as inspection of Table I shows. Nevertheless, the results of a considerable number of authors [Toor (1964), Stewart and Prober (1964), Arnold and Toor (1967), Cotrone and de Giorgi (1970, Carty and Schrodt (1975), Krishna et al. (1976), Taylor and Webb (1980b), Bandrowski and Kubaczka (1981), Krishna (1981b),Webb and Sardesai (1981), Taylor and Krishnamurthy (1982)] suggest that this assumption is certainly a good approximation at low rates of mass transfer and may also be a good approximation in mixtures of high concentration and at high transfer rates. However, examples suggesting the contrary can also be found in the literature [e.g., Krishna and Standart (197611. One of the objectives of the present work is to try and reconcile these two somewhat divergent opinions. Explicit Approximate Solutions The last two methods in our survey have been developed specifically to eliminate the iteration that is required in the other methods described above. Explicit solutions are possible if the composition profiles of all species are similar (Le., the composition profiles are independent of species index) or by making some assumptions which lead to similar composition profiles. The two methods that fall into this category, developed by Krishna (1979,1981a)and by Taylor and Smith (1982), can be written in a common format as (N) = f(Y*)[AavI-l(Yo- ~ a E) f(u*)[PavI[BavI-'(Y~ - YJ
(33) where f(Y*) is a single valued function of y* defined in eq 11. The direct method of Krishna (1979,1981a)is based on the assumption that the matrix [A]-' (or, rather, [P][B]-') can be considered constant over the film. Thus the composition profiles are linear and f(Y*) is simply unity. In deriving eq 33, Krishna (1981a) has argued that there is no theoretical evidence for considering the matrix [D] any less composition dependent than the matrix [AI-'. One of our objectives is to test the validity of this argument. The explicit method of Taylor and Smith (1982), a generalization of a method developed by Burghardt and Krupiczka (1975) for diffusion through a number of stagnant gases, is based on the assumption that [Ally* is constant over the film. With this assumption the composition profiles are found to be ~t io - Y* -YO* = -e*" - 1 -(34) Yi6 - Yio Y6* - YO* e* - 1 where
For the special case of equimolar counter transfer (a = 0), both explicit methods become equal to eq 32 obtained from the linearized theory. A Basis for Comparison All of the methods described above with the exception of the direct method of Krishna (1979, 1981a) are exact for binary mixtures and for multicomponent mixtures where all the Bij are equal. It is an interesting exercise to prove the exactness of eq 33-36 for this case and to compare these equations with eq 8-9 of Krishna (1981~). Equation 35 is noteworthy since it permits the immediate calculation of the mass transfer rate factor for all determinacy conditions encountered in practice. Krishna's explicit method is exact only if all the 2Jijare equal and all the ui are equal (Nt= 0). Any comparison of the methods must therefore focus on systems in which the binary Bij and/or the vi coefficients differ. A great many of the papers cited above include examples of real systems illustrating the characteristics of one or more of the approximate methods. Many of these systems have been included in what we believe to be the most comprehensive evaluation of these methods ever undertaken. These systems exhibit a very wide range of binary diffusion coefficients and determinacy coefficients which are summarized in Table 111. The reader will observe that only ternary systems are used in the comparison. Although some examples with more than three components have been published [Cotrone and de Giorgi (1971); Taylor and Webb (1981)] they have not been included in our calculations. We do not believe, however, that our conclusions would in any way be changed by their inclusion. The approximate methods have been evaluated by computing the discrepancies between the molar fluxes predicted from the exact solution and an approximate method [Taylor and Webb (1980b)l n ,
i = 1, 2, ..., n We have also computed an average discrepancy from
and the discrepancy between the predicted total fluxes. This last, denoted by et, is just the sum of the n ti. We have solved many thousands of example problems for each of the systems described in Table 111. Thus, something over two million separate calculations of mass transfer rates have been made. Any two points on the ternary triangular composition diagram constitute appropriate boundary conditions. The boundary conditions of the examples used in this study were generalized automatically by varying ya2,yal, yo2,and yol in turn over the range 0 5 y 5 1 (illegal combinations being identifiable by having yaaor ym negative or by having yo* or ya*equal to zero). A fixed incremental change, Ay, in ya and yo was used to generate each new set of boundary conditions. For example, a Ay of 0.25 yields about 150 possible problems. A Ay of 0.1 gives 3660 problems and a Ay of 0.075 gives more than loo00 problems. The results reported here were obtained using this last value of Ay.
Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983
102
Table IV. Statistical Comparison of Film- Models. Mean Discrepancies in Individual Fluxes ( c X 100) method“
ED1
system
__
-.
I I1 I11 IV V VI
~~
__-
ED2
ED3
TS
ED4 KR79 _.
Transfer through a Stagnant Gas 14.1 10.0 12.8 1.4 12.2 98.7 74.7 101.2 24.1 22.7 62.2 52.2 64.9 15.6 17.6 >,250 77.3 105.9 25.2 23.6 >400 71.8 95.6 23.4 22.5 54.4 26.0 31.9 5.9 14.0
LIN
---.--
1.2 16.6 10.2 17.6 16.5 4.5
0.1 3.5 2.0 5.2 6.5 1.1
Equimolar Counter Transfer 73.8 38.2 40.1 0.6 56.8 27.0 29.2 0.2 0.2 51.0 26.7 28.8 33.8 20.5 25.5 0.6 69.9 27.8 39.8 2.3 0.1 40.9 10.2 15.3 10.9 6.5 7.6 0.0 Nonequimolar Transfer 57.6 27.5 29.4 3.2 0.7 0.6 0.5 32.9 16.9 18.8 2.0 0.1 0.0 0.0
VI1 VI11 IX
x
XI XI1 XI11 VI11 XVI
Graham’s Law 32.3 24.9 26.8 3.9 0.9 0.0 0.5 3.2 1.6 2.0 0.2 2.3 0.0 0.0 35.3 21.6 27.0 3.0 0.8 0.1 0.7 67.2 31.4 36.1 5.7 2.1 0.4 2.3 40.7 9.9 1‘1.5 1.5 0.8 0.0 0.2 Zero Mass Average Velocity X 34.1 21.0 26.4 3.5 2.4 0.3 1.4 XI 38.0 9.0 12.8 1.7 2.9 0.4 0.4 XI1 57.1 33.8 35.7 5.8 5.7 1.5 4.2 ( I Key t o methods: ED1,effective diffusivity given by eq 23;ED2,effective diffusivity given by eq 24;ED3, effective diffusivity given by e q 25;ED4,effective diffusivity given by eq 20;KR79,explicit method of Krishna (1979);TS, explicit method of Taylor and Smith (1982) and for condition 2 from Table I only, of Burghardt and Krupiczka (1975);LIN, the linearized theory of Toor (1964)and Stewart and Prober (1964).
XIV x1‘ X XI XI1
To keep the quantity of data down to manageable proportions we report the values o f t and tt averaged arithmetically over all the problems considered for each system (Tables IV and V, respectively). The “maximum” indi-
vidual flux error that we found is given in Table VI. These results are meant to give an indication of the magnitude of the errors that may accrue with the use of one of the approximate methods. Note that the actual values of the binary allor of the vI play no role in the computation of the e,. Rather, it is the ratios of these coefficients that are important. Thus the results are readily extrapolated to other systems not given here. In addition to the calculation of these overall errors, each problem solved has been classified by the nature of any coupling effects which may take place. Problem number averaged z and tt have been calculated for examples where osmotic mass transfer occurs (N, # 0, Asl = 0), for examples where reverse mass transfer takes place ( N J A y , < 0) and for examples of unidirectional mass transfer ( N , / N , 1 0, all i tt j ) . This latter case is of importance in multicomponent condensation. Any particular problem may fall into more than one group or even into all three. The four major groups of problems described above have been further divided according to the magnitude of the dominant eigenvalue of [a]. This dimensionless quantity is a goodjndicator of the rates of mass transfer. Large values of adare characteristic of problems which may have convergence difficulties [Taylor and Webb (1980a, 198l)l. Reasons of space have kept us from tabulating here this further body of data and we can comment only briefly on the results. Complete results are available from the authors. While on the subject of the eigenvalues of [a],we should report the “experimental” discovery of problems in which these eigenvalues are complex. The conditions necessary for the existence of such entities were first discussed by Johns and de Gance (1975). They did not, however, find any examples of real systems which exhibited complex roots. We have found that system VI11 for determinacy condition 3 and system XI for determinacy condition 5 display complex roots for certain boundary conditions only. Other, related, systems probably do so too, but we have not searched for them. This finding is of mathematical interest only; no difficulties of physical interpretation or of computation results. The most important results of our extensive computations, t, et, and emax, are summarized in Tables IV, V, and VI for our readers to draw their own conclusions. We
Table V. Statistical Comparision of Film Models. Discrepancies in Total Fluxes ( e t method“ ~
~
.___
system I I1 I11 1V V VI
~
ED1
ED 2
VI11 XVI
21.5 5.2
XI v XV X XI XI1
3.9 1.0 8.9 16.3
10.6 2.8
0.9
x
15.7 0.4 37 9 For LPL to methods, sce footnote
ED 3
ED4
Transfer through a Stagnant Gas 29.5 36.8 4.1 >, 200 300 68.9 i150 i 150 45.4 :> 200 > 300 71.9 > 2000 .> 200 66.4 55.8 :. 200 27.1
39.0 83.5 > 100 2 700 2 1000 350
XI XI1
a
X
100)
~~~~
Nonequimolar Transfer 12.1 1.3 3.1 0.3
Graham’s Law 16.6 11.6 2.9 0.4 0.6 0.1 2.9 5.2 0.6 41.1 36.7 5.9 1.3 0.2 1.6 Zero Mass Average Velocity 5.7 9. Ti 1.4 2.7 2.3 0.3 71.2 72.3 14.2 in Table IV
~
~~
__
KR79
TS
LIN
36.1 37.1 37.2 39.1 11.9 37.2
0.3 1.2 2.1 10.0 17.8 2.5
0.4 1.3 2.4 10.2 16.8 3.0
0.4 0.0
0.2 0.0
0.2 0.0
1.2
4.2 1.1 3.7
0.0 0.0 0.1 0.6
0.1 0.0 0.1 2.5
0.8
0.0
0.0
4.8 5.5 14.1
0.4 0.0 2.4
0.6 0.0
8.1
Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983
Table VI. Statistical Comparison of Film Models: “Maximum” Errors, All Problems methoda system
ED 1
ED 2
I I1 I11 IV V VI
7.06 39.9 34.0 > 200 > 350 41.5
7.32 28.1 22.8 32.6 34.6 12.6
___
ED4
Transfer through an Inert Gas 14.3 0.922 > 200 3.67 97.4 2.90 > 200 3.62 >150 3.68 44.2 1.50
___
KR79
TS
LIN
1.41 1.74 1.65 1.91 2.06 1.54
0.202 1.23 1.00 1.51 1.73 0.71
0.053 0.369 0.22 0.64 1.00 0.31
VI1 VI11 IX X XI XI1 XI11
4.03 2.74 2.45 1.14 2.32 1.66 0.398
1.74 1.32 1.25 1.33 1.73 0.87 0.377
Equimolar Counter Transfer 3.49 2.44 2.20 1.02 1.37 1.44 0.387
VI11 XVI
2.68 1.61
1.46 0.824
Nonequimolar Distillation 3.12 1.01 1.21 0.745
0.096 0.007
0.064 0.006
0.048 0.007
XIV
X XI XI1
1.37 0.182 1.67 3.86 2.26
2.00 0.128 1.50 2.06 1.00
Graham’s Law 1.96 1.99 0.204 0.091 1.55 1.50 1.92 1.67 1.00 1.00
0.196 0.556 0.167 0.492 0.108
0.004 0.005 0.010 0.047 0.001
0.054 0.001 0.066 0.318 0.018
X XI XI1
2.14 4.37 2.68
1.56 2.61 0.817
Zero Mass Average Velocity 1.95 1.46 1.75 2.17 0.647 0.638
0.595 1.28 0.432
0.033 0.195 0.040
0.126 0.736 0.045
xv
a
ED3
103
0.056 0.023 0.020 0.052 0.226 0.006 0.001
For key to methods, see footnote a in Table IV.
apologize to them for the rather dry reading of the tables. We cannot, however, refrain entirely from making a few observations of our own. The solution of the linearized equations always provides good estimated of the individual fluxes. The largest discrepancies occur when inert species are present (Z < 0.07, tmaxI: 1.00) and, for this particular special case, this solution is, by a significant margin, the better of the approximate methods. We regard this finding as among our most important. The implication is that if the assumption of constant [D] is a good one for a film model of steadystate diffusion, then it should also be good for other models of mass transfer for which exact analytical solutions of the Maxwell-Stefan equations have not yet been obtained. It should be noted that examples involving mixtures with the largest possible concentration changes and high rates of mass transfer occur just as frequency in our survey as problems with low rates of mass transfer. We note with pleasure (and not a little surprise) that for determinacy conditions 3 to 5 from Table I (Graham’s law, zero mass average velocity, and nonequimolar distillation) our own explicit method is actually the better of the seven approximate methods although the advantage over the linearized theory is usually small. This explicit formulation rates second best if stagnant components are in the mixture. The average discrepancy in the predicted total flux is comparable to the discrepancies in this flux obtained from the linearized theory. The advantage of an explicit method should not be overlooked. In fact, we have used eq 33 and 36 to provide initial estimates of the fluxes to accelerate convergence of all the iterative methods. The reduction in computation time made possible in this way has been so marked [see Krishnamurthy and Taylor (1982)] that very many more problems could be included in our calculations. Further, no convergence difficulties were then encountered with any solution.
The direct method of Krishna (1979,1981a)is successful only if the vi are close together and, therefore (or for other reasons) the total transfer rate is low. At high rates of mass transfer the assumption of constant [A]-’ (or of [p][Bl-’) is a poor one, particularly in cases involving an inert species. System I displays a relatively narrow range of binary Dij,and the direct method of Krishna (1979) suffers in comparlson with the results of other methods as a result. The reason for this failure lies in the fact this explicit formulation is the only one of the methods tested here that is not exact for binary mixtures or for multicomponent mixtures with all Bijequal (except when Nt = 0) for this latter case. The classification of problems according to the magnitude of adclearly shows that the direct method of Krishna (1979) is much less good at the higher rates of mass transfer. All of the approximate methods are somewhat lesa successful in predicting, accurately, the rates of osmotic or reverse mass transfer. See Toor (1957) and Taylor and Webb (1980b) for further discussion of this point. The simple effective diffusivity methods (23-25) can be used with confidence for systems where the binary Bij display little or no variation. They must also provide good results for all systems if conditions are such that the appropriate limiting forms of eq 20 and 21 apply. However, the identification of these conditions requires some further computational tests which none of the more rigorous methods requires and which we have not, for this reason, performed. To use these formulas, knowingly or unknowingly, in situations such that these limiting cases do not apply (as we have, of course, done) would stand about as much chance of successfully predicting the fluxes as the throw of a dice to predict the magnitude and the toss of a coin to determine the sign. The effective diffusivity formula of Stewart (1954) is by far the best of these methods. This should not come as
104
Ind. Eng.
Chem. Fundam., Vol. 22, No. 1, 1983
a surprise since this method is capable of correctly identifying the various interaction phenomena possible in multicomponent systems. Indeed, for equimolar counter transfer this effective diffusivity method becomes equivalent to the linearized theory and to both explicit methods discussed above. In fact, for some systems Stewart's effective diffusivity method is superior to Krishna's explicit method. However, since the explicit methods are basically simpler to use (requiring the same basic data) than Stewart's effective diffusivity method and, in general, provide such superior results, we see no reason for any further use of the effective diffusivity. Conclusions and Recommendations We have compared the rates of mass transfer predicted from several approximate solutions of the Maxwell-Stefan equations for a film model of steady-state diffusion with the fluxes predicted by an exact solution. Our ranking of the methods in order of goodness would read: (i) the solution of the linearized equations by Toor (1964) and by Stewart and Prober (1964); (ii) an explicit method due to Taylor and Smith (1982) and for the special case of diffusion through an inert, to Burghardt and Krupiczka (1975); (iii) an explicit method due to Krishna (1979, 1981a); (iv) an effective diffusivity method due to Stewart (1954);(v) three other simple effective diffusivity methods. We find the assumption that the matrix of multicomponent diffusion coefficients [Toor (1964); Stewart and Prober (1964)] remains constant over the diffusion path is generally an excellent one, even in mixtures of high concentration and at the highest possible rates of mass transfer. If a film model of multicomponent mass transfer is to be used in process design calculations we would suggest that: (i) an effective diffusivity method is not used; (ii) the exact solution is used with initial estimates of the fluxes provided by the explicit method of Taylor and Smith (1982); (iii) if a "simple" approach is desired then one of the explicit methods be employed. Acknowledgment The support of the National Science Foundation through grants No. SPI-8026155 to L.W.S. and CPE8105516 to R.T. is sincerely appreciated. Nomenclature [A] = matrix of inverse mass transfer coefficients defined in Table I1 [B] = matrix of inverse diffusion coefficients defined in Table I1 [B]= matrix of inverse mass transfer coefficients defined in Table I1 c = molar concentrations a,)= diffusion coefficient for the binary i-j pair areff = effective diffusion coefficient = matrix of multicomponent diffusion coefficients H, = partial molar enthalpy J, = diffusion flux of species i M , = molecular weight of species i N , = molar flux of species i Nt = total molar flux .y* = weighted mole fraction defined by eq 11 z, = ratio of fluxes, N , / N ,
[e]
Greek Letters [ p ] =. "bootstrap" matrix defined by eq 10
6 = film thickness 6rk = Kronecker data ti = discrepancy in flux of species i
e = average discrepancy et = discrepancy in total flux 7 = dimensionless distance 4,
= rate factor in the explicit method defined by eq 34
La] = matrix of rate factors defined in Table I1 = dominant eigenvalue of [a]
= rate factor in the effective diffusivity methods (4) = matrix with elements defined by eq 15 [*] = matrix of rate factors in the linearized equations v i = determinacy coefficients
Matrix Notation. [ ] = square matrix of dimension [
I-'
R
- 1X n - 1
= inverse of a square matrix
( I ] = diagonal identity matrix
( ) = column matrix with n - 1 elements
Subscripts O,6 = pertaining to the position t) = 0, t) = 1, respectively av = quantity to be evaluated at the arithmetic average mole
fraction Literature Cited Arnold, K. R.; Toor, H. L. AIChE J. 1967, 73,909. Bandrowski, J.; Kubaczka, A. Int. J. Heat Mass Transfer 1981, 2 4 , 147. Bird, R. B.; Stewart, W. E.; Lightfood, E. N. "TransDort Phenomena"; Wiley: New York, 1960; p 571. Burghardt, A.; Krupiczka, R. In.?. Chem. 1975, 5 , 487, 717. Carty, R.; Schrodt, T. Ind. Eng. Chem. Fundam. 1975, 74, 276. Cichelli, M. T.; Weatherford, W. D.; Bowman, J. R. Chem. Eng. Prog. 1951, 4. .7 . 63. 123. Cotrone, A.; DeGiorgi, C. Ing. Chim. Ita/. 1971, 7 , 84. Gilliland, E. R. I n Sherwood, T. K. "Absorption and Extraction", 1st ed.; McGraw-Hill: New York, 1937. Hegner, B.; Molzahn, M. Third International Symposium on Distillation, London, April 1979; Institution of Chemical Engineers. Hougen, 0. A.; Watson, K. M. "Chemlcal Process Principles III", 1st ed.; Wlley: New York, 1947. Hsu, H-W.; Bird, R. B. AIChE J. 1961, 6 , 516. Johns, L.; DeGance, A. E. Ind. Eng. Chem. Fundam. 1975, 7 4 , 215. Keyes, J. J.; Pigford, R. L. Chem. Eng. Sci. 1957, 6, 215. Krishna, R. Ind. Eng. Chem. Fundam. 1977, 76, 228. Krishna, R. Lett. Heat Mass Transfer 1979, 6 , 439. Krishna, R. Chem. Eng. Sci. 198la, 36, 219. Krishna, R. Trans. I . Chem. E . 1981b, 5 9 , 35. Krishna, R. Chem. Eng. J. 198lc, 22, 251. Krishna, R.; Panchal, C. E.;Webb, D. R.; Coward, I . Lett. Heat Mass Transfer 1976, 3 , 163. Krishna, R.;Rahman, M. A.; Salomo, R. S. Trans. I . Chem. E. 1981, 59, 44. Krishna. R.; Standart, G. L. AIChE J. 1976, 22, 383. Krishna, R.; Standart, G. L. Chem. Eng. Commun. 1979, 3 . 201. Krishnamurthy, R.; Taylor, R. Chem. Eng. J. 1982, 25, 47. Pratt, H. R . C. Ind. Chem. 1950, p 470. Reinhardt, D.; Dialer, K. Chem. Eng. Sci. 1981, 3 6 , 1557. Remick, R. R.; Geankoplls, C. J. Ind. Eng. Chem. Fundam. 1970, 9, 206. Rohm, H. J.; Vogelpohl, A. Warme Stoffubertrag. 1980, 231. S h a h S. A. AIChE J. 1961, 7 , 17. Sherwood, T. K. "Absorption and Extraction", 1st ed.; McGraw-Hill; New York, 1937. Sherwood, T. K.; Pigford, R . L.; Wilke, C. R. "Mass Transfer", McGraw-Hill: New York, 1975. Stewart, W. E. NACA Technical Note 3208, 1954. Stewart, W. E.; Prober, R. Ind. Eng. Chem. Fundam. 1964. 3 , 224. Tai, R. C.; Chang, H-K. Bull. Math. Biol. 1979, 4 7 , 591. Taylor, R. Chem. Eng. Commun. 1961a, 70, 61. Taylor, R. Lett. Heat Mass Transfer 198lb, 8 , 397. Taylor, R . Chem. Eng. Commun. 1962a, 74, 121. Taylor, R. Comput. Chem. Eng. 1982b, 6 , 69. Taylor, R. Ind. Eng. Chem. Fundam. 1982c, 27, 407. Taylor, R.; Krishnamurthy, R. Bull. Math. Biol. 1982, 4 4 , 381. Taylor, R.; Smith, L. W. Chem. Eng. Commun. 1982, 1 4 , 361. Taylor, R.; Webb, D. R. Chem. Eng. Commun. 1980a, 6 , 175. Taylor, R.; Webb, D. R. Chem. Eng. Commun. 198Ob, 7 , 287. Taylor, R.; Webb, D. R. Comput. Chem. Eng. 1961. 5 , 61. Toor, H. L. AIChE J. 1957, 3. 197. Toor, H. L. AIChE J. 1964, 10, 448, 460. Turevskii, E. N.; Aleksandrov, I. A.; Gorechenkov, V. G. Khim. Tekhnol. Topliv; Masel 1973, 5 , 34 (see, also, Int. Chem. Eng. 1974, 74. 112). Webb, D. R.; Panchal, C. B.; Coward, I. Chem. Eng. Scl. 1981, 36, 87. Webb, D. R.; Sardesai, R. G. Int. J . Multiphase Flow 1981, 7 , 507. Wilke, C. R. Chem. Eng. Prog. 1950, 46, 95.
. --.
Received for review March 1, 1982 Accepted August 26, 1982
