smoothly to the boiling point, and, in addition, leads to a realistic estimate of the occupied pore volume. It does, however, lead to separate curves for oxygen and nitrogen. Of further interest in this regard are the data of van Dingenen and van Itterbeek (8) on the adsorption of hydrogen on Carbotox activated carbon over the temperature range 17' to 90' K. They correlated their low-temperature data by the Polanyi theory available a t that time but reported they could not correlate their high-temperature data, whereas Maslan et ai.,as we have seen. correlated their high-temperature data but could not correlate their low-temperature data. Figure 9 shows the successful correlation of all the isotherms of van Dingenen and van Itterbeek, using the modified LGCC method. Although there is no overlap of data above and below critical temperature, the correlation curve shows a smooth transition between the two regions and again a realistic estimate of the occupied pore volume. This hydrogen correlation resembles the corresponding hydrogen curve on BPL carbon (Figure 5)? suggesting a similarity between the two carbons. The extrapolation above critical temperature, therefore, appears to work for hydrogen as well as for oxygen, nitrogen, methane, and argon, and apparently has general applicability. There appear to be good reasons for supposing that a liquid-like state may persist on a n adsorbent a t temperatures far above the critical temperature of the free liquid-i.e., for supposing that the critical temperature of an adsorbed liquid is to be sharply distinguished from that of a free liquid. T h e attractive force of the adsorbent for the molecules may be considered as reinforcing the attraction of these molecules for each other, so that the over-all effect is in the direction of increasing the magnitude of the van der Waals constant. a, a t constant b . Lvhich immediately raises the critical temperature.
An increased value of a also can account for liquid-like properties for the adsorbate. If we assume a liquid that retains its liquid-like properties-i.e., enthalpy, as well as density-as we raise the temperature beyond the (normal) critical temperature. we may (by the Clausius-Clapeyron equation) similarly justify the extrapolation of its vapor pressure. Under these considerations. any expectation of a discontinuity in adsorption behavior as one passes through the (normal) critical temperature is based on neglect of the influence of the force field on the properties of the liquid phase. 'These effects were implicitly recognized in the LGCC assumption that one of the properties of the liquid phase-Le., its densityshould be evaluated a t some temperature below the adsorption temperature. Acknowledgment
The experimental assistance of J. R. Krause is gratefully acknowledged. literature Cited (1) Altman, M., D.Sc. thesis, New York University, New York;
1952. ( 2 ) Brunauer, S., "Adsorption of Gases and Vapors," Princeton University Press, Princeton, N. J., 1943.
(3) Grant, R. J.. Manes, M., Smith, S. B., A.I.Ch.E.J. 8, 403 (1962). (4) Kaganer, M. G., Dokl. Akad. .VQU~ SSSR 138, 405 (1961). (5) Lewis. \V. K., Gilliland. E. R., Chertow, B., Cadogan, LV. P., Ind. Eng. Chem. 42, 1326 (1950). (6) Lopez-Gonzalez, .J. de D., Carpenter, F. G., Deitz, V. R., J . Res. .Vat/. Bur. Std. 5 5 , 11 (1955). (7) Maslan, F. D.; rlltman, M., Aberth, E. R., J . Phys. Chem. 57, 106 (1953). (8) van Dingenen, \V., van Itterbeek, A , , Physic0 6, 49 (1939). RECEIVED for review October 7, 1963 , ~ C C E P TMarch ~ 27, 1964
MATRIX CALCULATION OF MULTICOMPONENT MASS TRANSFER IN ISOTHERMAL SYSTEMS WARREN E. STEWART A N D R I C H A R D P R O B E R 1 Department of Chemical Engineering, 1 %.iversity of Wisconsin, Madison, Wis.
Matrix methods are developed for solving multicomponent diffusion problems in terms of related binary problems. The diffusion coefficients of the binary problems are determined b y the eigenvalues of a multicomponent diffusivity matrix, and are shown to be real and positive. When the diffusivity matrix i s known, multicomponent concentration profiles and mass transfer rates can be calculated for systems of any geometry for which the corresponding binary functions are available. The treatment holds for nonideal mixtures in laminar or turbulent isothermal flow with moderate concentration differences and no homogeneous chemical reactions. The theory i s tested b y comparison with exact boundary-layer calculations for mass transfer between a flat plate and a three-component gas stream.
M
diffusion problems are important in many chemical and physical processes. Exact treatment of such problems is seldom feasible? since the governing differential equations are complicated and nonlinear. For small concentration differences. however. a linearized treatment of the differential equations leads to simple and accurate ULTICOMPONENT
1
Present address. Shell Dewlopment Co., Emeryville. Calif.
224
I&EC FUNDAMENTALS
results. The present paper develops and tests this treatment for isothermal systems. Tlvo concepts of linearization are applied jointly in this work. First. we use Onsager's postulate (73! 7 4 that in ordinary diffusion the chemical potential gradients and diffu. sion fluxes are linearly related. Second, we write the differential equations in their limiting forms for nearly constant physical properties. The use of these concepts to analyze multi-
component diffusion in nonflow systems was suggested by Onsager (73) and has recently been carried out for certain cases (2. 3 ) . Related linearization techniques \vere used by Hirschfelder. Curtiss. and Bird (70) to zolve multicomponent one-dimensional flame problems, by Prober (76) to solve multicomponent diffusion and flokv problems in laminar boundary layers? and by Toor (22) to analyze the multicomponent diffusion equations in flobv systems. T h e present work is a continuation of the studies of Prober (76). Here the linearized theory is extended to turbulent flo~v.nonideal mixtures. and high mass transfer rates. T h e properties of the multicomponent diffusion coefficients are established by nonequilibrium thermodynamics, and are used to simplify the matrix calculations. T h e linearized theory is also compared Lvith finite-difference calculations for threecomponent laminar boundary layers. 'The selection of a systsem of units for this work was not easy. T h e equation of motion takes its simplest form in terms of the mass average velocity (7: 7Oj. T h e laws of diffusion. for gases a t least. are best written in terms of molar units and the molar average velocity. T h e equations of continuity for isothermal. isobaric problems are conveniently Lvritten in terms of volumetric units and a volumetric average velocity (2. 78). T h e system preferred here is a hybrid: Mass units are used for the equation of mot'ion? and molar units for the diffusion equatioris. This system gives linearized solutions of good accuracy and involves the simplest matrix calculations.
a t moderate pressures the ID,, can be accurately predicted by replacing them with the binary diffusion coefficients Dt3= DJt defined by Bird and coworkers ( 7 , 70). \Vhen the ID,, are thus predicted, Equation 6 corresponds to the StefanMaxwell equation ( 7 ) of low density gaseous diffusion. For dense gases and liquids the ID,, are difficult to predict. and are best determined from multicomponent diffusion measurements. linearized Equations
T h e presence of variable physical properties in the above equations makes them difficult to apply, even in simple flow situations. Ho\vever, useful approximate solutions can be obtained by reIvriting the equations as if the changes in pressure and composition were small. This gives the equations
2 3 1
Conservation and Diffu!iion Equations
Consider the f'loiv and diffusion of a n n-component isothermal gas or liquid phase. in vr.hich no homogeneous chemical reactions occur and in which gravity is the only external field of force. Thermal diffusion and forced diffusion are absent, and pressure diffusion is neglected. T h e equations of motion and continuity are ( 7 ) :
3 7 4- (e at ax< G
---
at
+ (cv*
cy,)
dc at
==
-(T
+ (v
pv) = 0
Ji*) z
CV*)
=
0
=
1.
n - 1
(3)
(4)
Here v is the mass-average velocity of the mixture and v* is the molar average velocity. The two velocities are related by p(v - v*) =
which are linear. except for Equation 7. Here the properties with subscript a are evaluated a t a reference state p a . T. x a l . x ~ , 1,~ based on average conditions in the region of interest. T h e stress function, e,, is expressed in terms of the rheological properties of this state; thus for a Sewtonian fluid [y e,] = - I J ~ vP. T h e coefficients A , , are given by
2
A{, Jt*
(5)
i=1
and are thus nearly eqtial except in regions of high diffusion fluxes. A multicomponent diffusion laiv is needed to relate the fluxes and compositions in Cquation 3. T h e follow4ng diffusion la\v, derived in the .\ppendix,
and are properties of the reference state. T;Z'hen n = 2, Equation 12 reduce:: properly to Fick's first law of binary diffusion, with A l l = (l/ID12)a. Equations 7 through 12 give a realistic description of multicomponent forced convection, free convection, and nonflow diffusion in fluids with nearly constant properties. T h e extent to which these equations can describe systems with variable properties is best determined by numerical examples ; this is done later in the paper. Solutions of Equations 7 through 12 are frequently obtainable by analogy \vith known results on binary systems; hoivever? this fact is obscured by the coupled form of the multicomponent diffusion law. Equation 12. A simpler diffusion law can be obtained by matrix analysis, as sho\vn below. Matrix Analysis of Diffusion Equations
Equation 12 can be Fvritten in matrix form as is consistent ivith tkc results of Onsager (73. 74). T h e diffusion coefficients: IDt,. are functions of the local pressure? temperature. and composition; for ideal solutions and binary solutions ID,, = I D j T .This diffusion lakv is convenient because for gases
~ a [ r h l=
- [.ll[J*l
(1 3)
in \\hich [T'x] and [J*] are column vectors with elements T\1. . T t n - 1 and J 1 * . . J n - I * . respectively. and [ A ] VOL. 3
NO. 3 A U G U S T 1 9 6 4
225
is the (n - 1) X ( n - 1) matrix with A , , as the element in the zth row and I t h column. Taking the divergence of this equation and inserting Equation 9 gives a matrix differential equation for the compositions X I , ,x n - ] : [Px] =
[A]
:[
+ ("*
4
k
=
1, . . . n - 1
(1 5)
with the convention that a q-fold root is listed q times, then the properties of [ A ] ensure the existence of modal matrices [PI and [PI-' with the property [Pl-'[-4IIPl = [ X I
(16)
Furthermore, under the same requirements, the eigenvalues XI: are real and greater than zero. These properties of [ A ] are proved in the Appendix ; they hold, whatever the multiplicity of the eigenvalues. T h e columns of [PI are the right-hand eigenvectors (4,5, 8, 9. 7 7 . 75, 23) of [A]. Equation 16 determines only the ratios of the elements in each column; to determine the elements uniquely one may use the convention that the largest element in each column shall be unity. N o ~ v ,if one defines new compositions a n d fluxes by the transformations
and introduces these new variables into Equations 13 and 14, then with the aid of Equation 16 one obtains the uncoupled equations
1 - v2ff At
=
br 1 - + (v* bt
~ 2 , )i
=
1, . . . n - 1
[PI-' [ N ] - [2]
=
-
(19)
Thus, each pair of transformed variables, N i , Ji* satisfies the differential equations of a binary diffusion problem, with diffusion coefficient 1 ' X i and \vith the same v* function as the multicomponent system. T h e boundary conditions on .Ti and J,*are obtainable by applying Equations 1 7 , a and b , to the multicomponent boundary conditions. We refer to the diffusion problem or problems thus defined as the transformed problem(s) and to .Ti and Jz*as transformed variables, of the multicomponent system. I t is oftcn convenient to Lvork bvith transformed fluxes relative to stationary coordinates. Starting tvith the identity
Ni; k=t
T h e transformed fluxes gl, . .N,-l relative to stationary coordinates may accordingly be defined by ,
[PI-'[N]
[w] =
Since none of the elements of [A] need be zero, the fluxes a n d mole fractions of unlike species may be coupled strongly in mixtures with n > 2. However, the properties of [ A ] a r e such that Equations 13 and 14 are usually reducible to uncoupled sets. T o permit this reduction it is sufficient to require that the reference mole fractions x l a , . . x n a all be greater than zero, and that the reference state not be a critical solution state nor one of maximum attainable supersaturation with respect to any species. These requirements are easily met in most practical problems. When the reference state meets the requirements just given, the matrix [ A ] is diagonable-that is, if one defines the matrix [ X I as a diagonal array of the roots of the characteristic equation
'A - I h k 8 = 0
n
[j*]
(17c)
Equation 21 then becomes:
n*]
=
[R] - [e]
Ni; k=l
which is analogous to Equation 20.
T h e quantity
2
Nk
k=l
remains unchanged in the transformation, as does the equivalent quantity c,v* in the linearized equations. T h e matrices [XI, [PI, and [PI-' provide a general means, though not always the simplest one, for organizing the solution of multicomponent diffusion problems. T h e procedure involves the following steps : 1. Computation of [XI, [PI, a n d [PI-'. 2. Transformation of the multicomponent problem into new variables via Equations 17. If the boundary conditions are simple enough and if free convection is not involved. the transformation yields a set of n - 1 independent binary diffusion problems. 3 . Solution of the transformed problem or problems by any available means-e.g., empirical correlations or numerical solutions of the differential equations. 4. Calculation of the multicomponent solution by the inverse transformations
Step 1 can be accomplished by standard numerical techniques (4, 5, 9 ) 7 7 ) . Steps 2, 3, and 4 lead to a variety of possibilities depending on the form of the boundary conditions and the importance of the free convection term (pa - p)Y& in Equation 7. T h e examples that follow are restricted, for simplicity, to forced-convection problems solvable by simple generalizations of binary functions. Matrix Generalization of Binary Functions. A multicomponent generalization of any binary mass transfer function consistent with Equation 14 can be derived by a modification of the procedure just given. I n brief, one writes a set of transformed functions consistent with the binary one and then transforms them into the multicomponent notation according to Equations 23. If the boundary conditions for the binary function are known, one should transform them in like manner to determine the conditions for which the multicomponent function is valid. T h e matrix transformations are illustrated here; allowance for the flow conditions is discussed in the next section. It is convenient to begin by analyzing the solution of a standard binary problem. Consider the binary diffusion of species A and B! without homogeneous chemical reaction, in a one-phase region V bounded by surfaces S o and SI.T h e initial and boundary conditions are taken to be At t
5
t', xA
At t At t
=
x A l throughout V
> t', xA > t', x A
=
(244
SI
(24b)
xAo on SO
(24c)
= x A l on
For this system Equation 14 gives:
a n d applying thr tranqformations (17a, b). one obtains 226
I&EC
FUNDAMENTALS
bx.4 bt
+ ("*
YXA)
=
IDABY%,
(25)
X great variety of transient and steady-state diffusion processes are described by these equations. Here ID,, is evaluated a t state a. T h e surfaces Soand S1 may or may not completely enclose V . depending on the completeness with lvhich G ~ x , is represented ; thus in flow problems, if one neglects axial diffusion it is usually necessary to leave x g unspecified on the do\vnstream boundaries of I;. So\t? t': IDAB)
(29)
when the v * function and the contours So, SIare specified. It should be noted in particular that k,' does not depend on composition! total presscre, or -Vao .YBOexcept as they influence the variables cited above; in these respects k,' is superior to other mass transfer coefficients, as stated elsewhere ( 7 , 79). To extend these results to multicomponent mixtures we define an analogous set of n - 1 transformed problems by changing x A to T ? : to l i A f , etc. This gives the initial and boundary conditions At t
5 t ' ? .Fi
= Til throughout
> t', t > t',
At t
j., =
At
f f
i
V
=
1, , . ,, n - 1
f f l on .SO
i
=
1, . , ,, n
Si
i
=
1, . .
= ?io
on
.:
-
(30a)
1
(30b)
n - 1
(30c)
and gives Equations 19 as the differential equations. T h e solutions of these n -. 1 diffusion problems are given by Equations 26 and 28 with the same notational changes :
=
i
=
j.01
=
-
[RI][j.1
I, . , n - I
201
LVriting this in matrix form and applying Equations 23, one obtains [xi] =
[to]
- [P][l/k,'][P]-'
I n the same way, for unknown compositions x I O . one obtains
Here [l
i,'is]a
diagonal element
diagonal matrix. with 1 T h e matrix
(33) *A,
[ E , ]= [ I ] in Xihich [fi] and [F,'] are diagonal matrices. with the ith diagonal element given by IIi and krl', respectively. Appiication of Equations 23 then gives
-
,YO]
=
[PJ[E][P]-' [xl -
the tth
fined analogously. These results are, of course: strictly valid only if the position-dependent fluxes .YIOsatisfy Equation 36 with constant [ , k o J and [ X I ] . For internal flows the binary transfer coefficient can be defined in terms of ( y A o - x A b ) if preferred. Fvhere x.qo is the bulk composition in the given flow cross section. This change can be effected by changing subscript 1 to b in Equation 28. and in all multicomponent formulas that contain 6='. An alternative procedure for the above transformations is provided by the expansion (8,75, 23)
a
[y
iZt'as
This holds for any diagonal matrix [ f ( A ) ] in which the elements J ( A r ) for equal eigenvalues are equal. Here s is the number of different values of Ak that satisfy Equation 15. and is less than n - 1 if repeated eigenvalues occur. T h e matrices [E,] are the principal idempotent elements of [.1] and are given by
These results can he reu.ritten as matrix equations [Z -
(37)
xO]
( 3 5)
if s
=
1
(41b)
in \vhich [ I ]is the unit matrix of the same order as [ A ] . Equations 40 and 41 are related to the Sylvester formula (5.9) but are simpler to use because they hold regardless of the multiplicity of the eigenvalues VOL. 3
NO. 3
AUGUST 1964
227
Equations 35, 36, 38, and 39 all meet the conditions of Equation 40, and can accordingly be rewritten as follows:
(42) r
n
Here ATI is the change in the TI function of Equation 24 when the element of area AS, located around the point i ’ . 0 : ( 2 , - f j l ) = JJJL’=I
s
F(Y,Y:
2.
t ; t.’, 0.
At t
=
x A l throughout L’ and S
y A = .yA0
(t:
s
(47b)
ivhere S is a portion of the boundary of V . sufficient to define the problem Lvithout overdetermining it. The solution of Equation 14 under these conditions isj fort > 0,
t’=O-
F ( x , y , z , t?t.’?j - ’ , t’, A,) [dxi,(t’)]dS (52) for the multicomponent composition profiles. Thase results hold for laminar or turbulent flow; they are expected to be useful in studies of heterogeneous reactions. T h e effect of nonuniform initial conditions can be analogously calculated. Thus, under the conditions At t = O? sA - x A l At t
= 6(x
> 0, x A
- x ’ ; y - y’?
t
--
2’)
- xAl = 0 on S
(53a) (53b)
lvhere S(x - , t ’ ; y - y ’ , z - z’) is a Dirac &function. the solution of Equation 2 5 for a given geometry and v* function is: fort > 0:
-
xAl
=
Q ( x . y : 2, t . x ’ , y ’ ,
2’:
DAB)
(54)
By superposition, the solution for an arbitrary initial composition distribution, .x4(x,y,z,0), is
[xa(x,y,
z , 0) - xa!]dV (55)
Converting this to multicomponent form in the usual \yay. one obtains the solution
(474
j - ) on S
(51)
t’=O-
and, via Equation 40, the idempotent solution
xA
x, T A T
=JJJ”=‘ [ P I [ F I [ P ] - [d.yiO(t’)ldS ~
[xi
- xi11
=
JJJ
[ p l [ Q l [ p l - ’ [(~x ,i y , z , 0) - x1,1dC7 (56)
I’
in which [ Q ] is the diagonal matrix of functions Q ( t , j , z , t , t ’ , v’,a’,X The corresponding idempotent solution is
JJJ t’=O-
s
(48) T h e notation ~8\.~,,(/‘) signifies the differential of xA0 ( t ’ , i f ,C‘) \ 0 under the general conditions .At t =
0. [ k l ]
= [ t i ( \ , y.
z . O ) ] in I’
(58a)
At t
2 0. [ y l I
=
[.Y~($.
{, t ) ] on
S
(58b)
in terms of the multicomponent diffusion coefficients and the binary functions F and Q a t the same flow conditions. I n this section we have treated the mole fractions xi. or fluxes as known quantities in the initial and boundary conditions. Actually, the boundary conditions are not usually specified so directly. Often they take the form of simultaneous equations bet\veen the fluxes andpor compositions of the various species a t the interfaces of the system. T h e analysis of such coupled boundary conditions is feasible Lvith the methods given here; however, it appears likely that more direct methods can be developed. 51any other types of formulas based on Equation 25 can be generalized in the manner indicated here ; however? certain notarional requirements should be observed. First, the compositions and fluxes in i.he binary formulas must all refer to species A : except that total fluxes such as (N, N,) and (N,.ZI, N,.Z.i,) are allowed. Second, numerical values of x A . .Yao/(.Vao .Vao). cetc.! should be replaced by symbols as far as possible in the formulas and in the initial and boundary conditions. These requirements are easily met by writing the binary expressions properly: and lead to the most convenient generalization of a given binary result.
+
+
which can be converted into ordinary mole fractions by either the modal or idempotent method. This example sho\vs the importance of replacing numbers by symbols Lvherever possible before transforming a binary problem. One-Dimensional Convection. I n a number of idealized flows the composition .yA and velocity v* depend on only one space coordinate, in addition to a possible time dependence. ,4 related class of flo\vs is that for \vhich v* is directed along a given coordinate, and depends just on that coordinate and time. In either situation?Equation 10 and a one-point boundary condition define v* sufficiently that Equation 2 5 can be integrated. T h e quantities used to evaluate v* appear automatically in the binary result, and the extension to multicomponent flow proceeds as before. Thus, the function ( 7 )
+
for diffusion in a fluid film in which the composition and velocity depend only on y , implies the multicomponent expression :
Allowance for Flow Coinditions
If a binary diffusion function includes all the system variables that affect the v* profile-e.g.: if it is a complete dimensionless expression--the methods of the preceding section will yield the corresponding multicomponent function together with its conditions of validity. I n cases of large diffusion fluxes? a complete binary function is not usually available, but one can estimate it from related information on systems \vith small fluxes. Here the governing equations and assumptions are reviewed for several flo\v situations. v* = 0, the methods of the precedSystems at Rest. \%lien ing section apply direct1.i. T h e substitutions x A -+ f t ,JA* -+ j l * . N, + Ni>ID,, l,.'Xi> for i = 1, , . . n - 1: convert any such solution of Equatio.7 25 into a set of corresponding multicomponent functions. Thus, the binary function
-
X A = 1,
*
k
2
n=l
(-1)n-I
~
sx
Yn
T h e dimensionless interfacial velocity
may be correspondingly generalized to give :
T h e quantities k , and E,< are the mass transfer coefficients in the limit as Z;O* -+ 0: and are given here by (c IDAa)a,'6and c, .'Xi6, respectively, where 6 is the film thickness. T h e penetration theory for rapid mass transfer ( 7 ) deals with unsteady-state binary diffusion in a one-dimensional system. 'The results can be analogously extended to give the multicomponent function
2 =
( 7 ! p , 625) yields the transformed functions
,
~
.
.n - 1
(66)
where in this case 1 and for the compositions in a pair of stirred reservoirs connected by a tube of length 215 in which v* = 0, after removal of a transverse diaphragm from i:he midplane of the tube a t t = 0 . Here y k is the kl_h root of y tan y = .V> and 2.V is the ratio of the tube volume to the volume of either reservoir. Equation 60 is. unfortunately, of limited utility for n > 2 bel , ' ? in the t\vo cause it requires the initial conditions T i = l '! halves of the system for each of the transformed functions. Reivriting Equation i9 with the initial compositions t,+ and i A leads 10 the more general multicomponent solution
*
(6')
is calculated from Equation 65 bvith this expression for
Ezi. Forced Convection with Small Fluxes, ATra. This is the caw of small mass transfer rates treated by Bird. Stev-art. and Lightfoot ( 7 , section 21.2). It is closely approximated in many engineering situations, and most of the published studiey of binary forced convection are for this case. The governing differential equations are 7 ,8. 25. and lick's first laiv for binary systems. or 7 . 8: 19, and 18 for multicomponent systrms. \vilh v* replaced by v except in the definition of J 1 * . 'Thc approximation v* = v follo\vs from Equation 11 and thc smallness of the fluxes J r * in f l o ~ v sof this type. From the ymallness of the fluxeP it also follo\\.j that the ma+ flus acres, the interface can be neglected in Lvriting boundary conditions VOL.
3
NO. 3
AUGUST
1964
229
for v . The extension of binary functions to multicomponent diffusion can be made in the usual way, unless mctions induced by composition-dependent density or interfacial tension are important; the latter effects are not considered here. As a simple example, the formula of Gilliland and Sherlvood ( 7 . 7)
for a binary gas stream in turbulent flow through a wettedwall column of inside diameter D . implies the multicomponent function
iziDXi ~-
(x) )(-; DG
=
0.023
La
0.83
i
=
0.44
1, . . . n - 1
(69)
provided that the interfacial liquid composition is nearly constant, as it was in the binar)- experiments, and that the motion of the liquid surface is comparable. In writing Equation 68 we have substituted k , for the quantity k,P,, used by Gilliland and Shenvood (7). T h e same substitution can be made in other mass transfer correlations, and \vi11 simplify their use. Forced Convection with Large Fluxes hiin. I n one-dimensional flow systems, large fluxes .Yio are easily dealt Lvith. in Thus, Equations 62 through 67 are valid for large fluxes the systems for which they Lvere derived. In hvo-dimensional or three-dimensional flow systems, the treatment of large fluxes is more difficult. Here the full set of linearized equations, 7 through 12, comes into the diffusion problem and approximations are necessary to obtain practical results. O n e approximate treatment, the film theory, assumes that a correlation of k , for any flow system can be extrapolated to large fluxes by means of Equation 62. This procedure is a n oversimplification but! being easy to apply, it is widely used. For multicomponent flow the analogous procedure is to use Equation 63 to extrapolate the corresponding correlation of iZi to large fluxes .YIO. A second approximate treatment: the penetration theory, describes fluid-fluid mass transfer in terms of the exposure time of each element of area in the interface. A corresponding multicomponent treatment, for nonreacting fluids with large fluxes .Tic. is provided by Equations 66 arid 67 Lvith t replaced by the local effective age of the interface. This treatment should be useful for multicomponent liquid diffusion in packed toLvers and around freely circulating bubbles. T h e influence of mass transfer on the v profile is not considered in the diffusional calcularions of the film and penetration theories. A more general method which includes this influence is as follo\vs: O n e first evaluates the desired binary or transformed function assuming v* = v? and then makes a correction for (v* - v ) , T h e first step is done by the methods for small fluxes, except that the boundary condition
where A
in place of the initial assumption for v*; a method for doing this is presented belo\v. Here n is the local unit vector normal to the interface or reference surface, and v* and v are the velocities a t any neighboring point along n . Equation 71 is a compromise solution for v*. It satisfies Equation 11 in the limits of vanishing fluxes J i * or equal molecular \\,eights M I . and preserves the most important features of the v* profile in other cases. It satisfies Equation 10 \tithin a distance from the boundary comparable \vith the local radii of boundary curvature, as it should to be compatible lvith Equation 9. It satisfies the normal component of Equation l l a t the interface. as required by the boundary conditions on v and v*. It satisfies the components of Equation 11 normal to n if the diffusion fluxes Ji* are directed along n ; this is very nearly true in many applications. Equations 70 and 72 equate the interfacial fluxes: pacO and taco*: of the linearized equations to the corresponding quantities of the actual multicomponent system. From a materialbalance vieivpoint this is preferable to equating the interfacial velocities of the linearized and actual systems; it is also necessary for consistency with Equations 11. 20: and 22 a t the interface. .4s an example of the general method, consider the steadystate transfer of species A and B , in any fixed ratio .\-AO .\-BO. between a flat plate and a tangential binary gas stream \vith a uniform upstream velocity c' and upstream composition Assuming first that v* = v , the position-dependent mass transfer coefficient in the laminar region is evaluated from Stewart and Prober ( 2 0 ) :
ivhere c'is the velocity a t the outer edge of the boundary layer, rn and /3 are shape factors ivhich vanish for the present geometry, A is the distance downstream from the leading edge of the plate, and K is a dimensionless interfacial velocity:
Table I . Interfacial Velocity Correction for Three-Dimensional, Steady-State Convection along Stationary Surfaces
X +o 00
0 10 0 20 0 30
v* 230
l&EC
=
v
+n
FIJNDAMENTALS
-
GO)
(71)
0
1059 2246 3580
40
n
0 0 0 0 0 1
50 60 70 80
0 6764 0 8665
90 00 1 10
1 20 1 30 1 40 1 50 1 60 1 70 1 80 2 2 3 4
00 SO 00 00 m
(to*
B 0 0 0 0
n
1 90
is used in place of c0 = 0 a t each interface. T h e second step is to compute the affect of substituting the improved approximation
n
5078
1 0808 1 3230 1 59'0 1 90'2 2 2590 2 6584 3 1124 3 6291 4 2!81 4 8001 5 65-9 6 5363 ' 5424 8 6963 1' 654 36 144 162 03 m
Y
X
1 0000 -0 00 0 9443 -0 10 0 8903 -0 20 0 8381 -0 30 o -8-7 -0 40 0 '392 0 50 0 6925 -0 60 0 6476 -0 0 ' 0 604' -0 80 0 5636 1 -0 90 0 5243 -1 00 0 48'0 -1 10 0 4514 -1 20 0 4177 -1 30 0 3858 -1 40 0 3556 -1 50 0 3272 -1 60 0 3005 -1 0 ' 0 2'54 -1 80 0 2519 -1 90 0 2300 -2 00 0 1416 -2 50 0 0830 -3 00 0 0247 -4 00
B 00 -0 0946 -0 1791 -0 2348 -0 3226 -0 3835 -0 4382 -0 4873 -0 5316 -0 5'16 -0 60'6 -0 6402 -0 6696 -0 6964 - 0 '206 -0 '425 -0 7625 -0 '80' -0 -'92 ' -0 8123 -0 8261 - 0 8'94 -0 9142 -0 9535
0 0
-I
- m
0
Y 1 0000 1 0575
1 1166 1 17'4 1 2398 1 3038 1 3694 1 4364 1 5048 1 5-46 1 64.58 1.7183 1 ,7920 1 ,8669 1 ,9429 2 0201 2 0983 2 1'76 2 2578 2 3389 2 4209 2.8430 3,2817 4.1951 m
Table II.
Properties of Gaseous Hz, Np,and COZ at 300'
HP
K. and Moderate Pressuresa
co 2
N q
Equimolar Mixture 24,68 1.620 X ... ...
Molecular weight 2.016 28.02 44.01 p , g. cm. -l sec. 9.03 X 1.778 x 10-4 1 . 4 9 4 x 10-4 Index j 1 2 3 cIDI,, g.-mole cm.-'sec. 3.133 X 10-5 2 . 6 0 0 X 10-6 clDsj, g.-mole c m - I set:-' 3.133' X 10-6 , . . 6 . 2 4 9 X 10-6 cIDD,,g.-mole cm.? sec. 2.600 X 6 . 2 4 9 X 10-6 ... a Values of p i and cIDi, calculated from Equations 8.2-78 and 8.2-44 of (lo),assuming ID,j = Dij.
...
...
= k, and Y(uo) = 1. The members of Equation 77 then become equal to $ A B , and Y(uo*) beromes equal to the binary correction factor, k,'/k,, as given for high Schmidt numbers in Bird. Stewart, and Lightfoot ( 7 ) . The same simplifications apply generally in steady flows with high Schmidt numbers, stationary interfaces, and finite B, since 00 0 as Sc 03 a t any finite value of B. The boundary-layer diffusion calculations for rapid mass transfer in ( 7 , 72. 77, 20) are based on the assumption that uo = uo*, and are therefore applicable as written only when the concentration differences are small. With the correction in Equation 74 these earlier results can be extended to high concentration differences. A corresponding correction is evidently necessary for heat transfer; however, this has not been tested as yet and will be discussed a t another time. T o extend the above procedure to multicomponent systems one simply makes the usual notational changes. For the. flat-plate problem one then obtains
k,,,,
Since K and U are independent of x under these conditions,
00
varies as l / d L Here ug is given by Equation 70 with n = 2. T h e coefficient n'(0, 8, K: Sc) is tabulated as a function of p , K , and Sc by Stewart and Prober (20); related tables for flat-plate flow are given by Bird, ;Stewart, and Lightfoot ( 7 ) and Mickley et al. ( 7 2 ) . The coefficient k,,,, is equal to k,' only if v* = v . If one knew the v profile, the calculation of k,, vo could be bypassed ; one could instead integrate Equation 25 with the v* profile computed from v and :Equation 71. This is, in fact, possible for the present geometry with the data given by Stewart and Prober (20). A simpler and more widely useful method, however, is to compute a correction for (uo* - u g ) from Table I. Table I gives the effect of ug* on the mass transfer coefficient? k,': for laminar flow along a stationary three-dimensional surface, with constant xAo and :VAo/iVBo, downstream of a step change in the interfacial composition. It is taken with notational changes from Stewart (78), and provides solutions of Equation 2 5 with the v* function of Equation 71. These solutions are for linear variation of the tangential velocity with distance along n. but in the present context the shape of the velocity profile is rather unimportant, so long as the correct k,,,, and interfacial velocities are used. T h e correction for-(uo* - 00) in a binary system is as follows:
-+
-+
i
=
1, . . . n - 1
(78)
in which hi = g a h i / p o is a generalized Schmidt number, and the mass transfer coefficients for the transformed problems become
(74)
T h e quantities Yi(v0)are found from Table I at the Bt values Here Y(uo*) and Y(u0) are dimensionless concentration gradients obtained from Table I. T o find Y(u0)one computes the quantity (75)
and uses this as B in Table I. The value of Y a t this value of B is Y(v0). T h e effective boundary layer thickness, 6, assigned
and the quantities Yi(vo*) are found a t the
Xi values
n
by this calculation is
and represents either a local or average value depending on the definition of k,,vo. The factor 1.1198 = l / r ( 4 / 3 ) arises from the treatment by Stewart (78), where aAB is evaluated for a linear velocity profile. O n e then computes the quantity (77) and finds I7 in 'I'able I a t this value of A '. The resulting Y value is inserted as Y([,0lk) in Equation 74. In problems with u 0 = O? such as occur in heterogeneous catalysis, the procrdure becomes simpler. In such cases
T h e numerical application of these results is considered below. An alternate to the use of Equations 71 et seq. is to formulate the diffusion problem in mass units, and thus eliminate v* from the calculations entirely. Such an approach would simplify the binary functions, but would require added calculations to transform the [ A ] matrix into mass units. A detailed comparison of these two approaches will be made a t another time. Numerical Comparisons
It is instructive to compare the matrix method with a more exact analysis given by Prober (76), who integrated Equations 1 through 6 numerically for the ternary system H2--N2--C02 VOL. 3
NO. 3
AUGUST 1964
231
Table 111.
Comparison of linearized Solutions with Exact Solutions
Exact Vaiues of
Ki,o 0,001
K2.0
-0.002 0.0045 -0,009
-0.097
0.196
0.194
-0,392
0.1955
-0,100
-0,391
0.192
0,028
K3.0
- XL,I -0 0028 -0 1906 0 0039 0 3071 0 0035 0 2538 -0 0030 -0 2854 0 20’1 0 178’ -0 1148 -0 0508 -0 0556 -0 0407 0 0452 0 0224 0 1362 -0 2411 -0 0907 0 1315 -0 0566 0 09’8 0 0448 -0 065’
[x,,
0,200 -0.020
-0.028
-0.192
0.020
-0.002
0.038
0.164
0.002
-0.038
-0,164
0.020
-0.022
0.202
-0.020
0.022
-0.202
-0,0012
0 120
0.0812
0.0012
-0,120
-0.0812
in laminar isothermal flo1.r. aloilg a flat plate. T h e composition dependence of p and p \vas computed from the ideal gas formula and the M’ilke viscosity formula (2.1). using the data of Table 11. A constant interfacial composition. ~ 1 . 0 = x2.0 = y3.0 = 1, 3 \vas specified in each problem. and the interfacial fluxes of the species Lvere given in the form
which is analogous to the binary boundary conditions associated Lvith Equation 73.a and b. T h e boundary layer forms of Equations 1 through 6 \vrre integrated numerically to obtain the profiles of velocity and composition for various sets of values of Kl.0. KY>o. and K3.0. T h e calculated composition differare listed in Table I I I ? ences, i i o - . t t q , for several sets of and are used here to test the linearized theory. Linearized solutions have been computed for comparison using Equation 44, Lvith the mass transfer coefficient given by Equations 78 and 79. T h c results can be ivritten in the dimensionlrss form
where
of [xLo - x , ] ~ Lineorired _ _ I’aiilts _ 1st opprox. 2nd approx.
-0,0028 -0.1893 0.0042 0,3095 0 0021 0,2578 -0 0007 -0.2868 0.2094 0.2326 -0.1149 -0,0440 -0.0552 -0,0377 0,0452 0,0238 0 1364 -0 2574 -0 0904 0 12’4 -0.0562 0,0950 0.0449 -0 0668
-0.0027 -0.1917 0.0040 0.3056 0.0018 0.2530 -0.0007 -0.2887 0,2029 0.1446 -0,1158 -0.0483 -0.0556 -0 0405 0 0452 0.0226 0 1347 -0 2338 -0 0908 0 1305 -0.0564 0.0977 0.0449 -0 0659
Error in
X ~ O-
1st approx.
CI ,o
x,,,
2nd approx.
-0.7
0.6
...
, . .
-0.5
0.8 ...
...
-0.3
1.6
...
...
12 -2.0 -19.1 0.9 -4.9 0.0 -0.5 0.0 1 .0 -1.1 -3 . o 0.1 -0.8 -0.4 -0.1 0.2 0.3
0.5 1.1 30.2 0.1 -13.4 -0.7 -7.4 0.0 6.3 0.1 6.8 -0.3 -3.1 -0.7 -2.9 0.2 1,-
as the values of B , for evaluation of Y,(uo) from Table I. and Equations 78. 81, and 82 give -
7
=1: . .
(86)
.J
as the values c f X,for evaluation of I’,(va*). Equation 82 requires an iterative solution for best accuracy. since the reference composition is best taken as a mean of the terminal compositions. Choosing y10 = i Y a = l a a = 113 for the first iteration, and using the data of Table 11: one finds =
IO
0 02181 1.1732
3628
, O 4270
x
105 G0
For a ternary mixture Equation 1 5 gives the quadratic equation X2 - (.411
+ A??)X f
(L4ii24ri- Aig.421)
=
0
(87)
r h e roots of this equation in the present case are X1
=
1.18444 X 1O5c,
Xg
=
0.3514’ X l o 5 C,
Equation 41a then gives the idempotent matrices
[Ell
=
[ A - XJ]
(XI Here -2, I< the rth distinct value of from Equation\ ’0. ’?b. and 82
p o x , pa,
and
-
-
[0.0136
0,02621
0.5126 0 . 9 8 6 4
A?)
R is obtained
-
ivhich are independent of Equations -3b. 78. and 80 qive
232
l&EC
FUNDAMENTALS
ca.
[E,]
The identity T=
=
[I]
I
provides a convenient check 011 rhe Lvork: it is sarisfied to four decimals here. These matrices ivere used to obtain first approximations for all the problems in Table 111.
T h e calculation of the functionsJ(.i,) for the first problem of Table 111 is as follows: 7 = 2
7 = l
A7 .1, = pJ1.4Iaca II’10, 0. K . .I,)
0 0 0 0
B,
100 7775 3717 2092
0 100 0.2307 0.2552 0.0904
0 03672
0 036’2
n
31,
I.tf,
0.0689 0.9616 48.18
0.0316 0.9824 21.63
Insertion of these results in Equation 83a gives
to
-
.tal
=
+
(48 18)
(21 63)
IO
0136 0 5126
i -0
r-0 0028
0 9864
0 026Z1
0 9864
-0
5126
0262,[ 0 0136
0.3333 [0,3333]
003958
0 OOOOOO] -0 003958
I
T h e predicted mole fractions of H ? and stream then become
‘‘‘cl =
-0
-0
- [-O
iY2
0028 18,3]
Only isothermal systems with no homogeneous chemical reactions have been considered here. T h e isothermal limitation is not a stringent one; the results may be applied to nonisothermal forced convection in the absence of significant thermal diffusion., T h e theory is currently being extended in these respects. Acknowledgment
T h e financial support of this \cork under National Science Foundation Grant 86-3354 is gratefully acknowledged. T h e authors also thank E. N . Lightfoot and E. L. Cussler, Jr., for helpful information, and Hans Schneider for supplying earlier references to Equation 41a, which was rediscovered in this work. Appendix.
Diffusion Laws
Equations 13 and 16 are derivable from nonequilibrium thermodynamics. T h e local rate of entropy production by ordinary diffusion in an n-component medium is (70, 7 1 ) :
in the approaching
=
188)
0 3361 [ O 52261
and the mole fraction of C 0 2 may be found by difference. Equivalent results would be obtained if H2 or N?were chosen as the nth species. Table 111 compares the linearized calculations kvith exact values for various combinations of mass transfer rates Kto. T h e linearized calculations \ v e x done twice. first using [ I ia I = [ x i o ] and then using [ x i , ] = 1# 2 [ x i o x I m ] computed from the first approximation. The agreement with exact calculations is remarkably good ; the linearized values are accurate Lyithin 3yc after two iterations. except in problems 5 and 6, ivhere the convergence is less rapid. These results give confidence in the soundness of the linearized method. T h e comparisons could equally well have been made by starting u i t h the exact values of [ . y j 0 - x , = ] and calculating the dimensionless mass transfer rates Kio via Equation 43. Since only n - 1 independent fluxes can be calculated from Equation 43. it would then have been necessary to specify one additional equation involving the fluxes AFt0in order to obtain a solutione.g.. one could specify h 1 . d or K I , o . ’ K or?K , oI , ~ K2.o K3,o. T h e need for an nth relation in addition to the diffusional equations is a general feature of problems with unknoivn fluxes -Vl0; examples of such relations are given by Bird, Stetvart. and Lightfoot ( 7 : section 21.1).
+
+
+
Conclusion
T h e matrix treatment given here allows a straightfonvard generalization of many mass transfer correlations from binary to multicomponent systems. T h e functions so obtained should be useful in studies of heterogeneous reactions and multicomponent separation processes. T h e linearization method provides a simple allo\cance for variable physical properties in binary and multicomponent systems. T h e method has been given here in terms of a single reference state. but can be written Lcith a separate reference for the properties p and p if desired. This modification should be useful a t high Schmidt numbers, \There the flo\v may be determined largely by conditions outside the diffusional boundary la)-er.
Taking the fluxes can write
Jn-l*
J1*.
as independent variables. one
as the diffusion la\\ for an isotropic fluid with small fluxes and small departures from a given equilibrium state p a . T.xja. x , - ~ . ~ . Inserting this in 88 and using the second law of thermod)namics. one finds 0-1
n-1
for any set of fluxes Jl*: . .J,-l*. T h e equality- in equation 90 applies only when all the fluxes vanish; thus. the quadratic form in equation 90. and the matrix in equation 89. are positive definite: provided that the fluxes .J1* . . Jn..l* are linearly independent. In addition, [HI is symmetric ( H z J= H I * )in the absence of external magnetic fields or Coriolis forces. according to the Onsager relations ( 7 1 ) for physical laics of this type. T h e chemical potentials G,are defined by ,
n-1
and satisfy the relations
withp. T.‘1% . . Y ~ as - the ~ independent variables. Therefore, in the neighborhood of state a. the chemical potential gradients are given by [(VE, -
VCrr)D,T1 =
(93)
[G,,I[~u,l
in which [C,,] = [$G a \ , a t , ] is the symmetric matrix formed by the derivatives in Equation 92. evaluated a t state a . O n physical grounds it is clear that the full set of composition gradients T i l 3 . . t u , , - 1 can be independently varied in diffusion experiments. Consequently. from Equation 93. the VOL. 3
NO. 3
AUGUST
1964
233
number of linearly independent potential gradients is equal to the rank, r , of the matrix [GI. Equation 88, with u 2 0, then requires that there be just r independent diffusion fluxes. The assumption of independent fluxes J I * . . .Jn- I* in Equations 89 and 90 corresponds to r = n - 1, and is thus equivalent to the assumption that [GIis nonsingular. Thermodynamic stability criteria (6, 27) indicate that [GI is either positive definite or singular for all observable equilibrium states. T h e only singular cases known are critical solution states, states of maximum attainable supersaturation, and states where one or more of the x 1 vanish. Hereafter we assume that a reference state with nonsingular [GI is chosen. Equations 89 and 93 can then be combined to give c,[VxI
=
- [GI-'[HI[J*l
permutation, they do not depend on the assignment of the subscript n in a given system, and their composition dependence tends to be simpler. Nomenclature
Symbols used repeatedly are listed here; others are defined a t the point of use. Dimensions are given in terms of mass M , length L , and time t . = coefficients in Equation 12,
(94)
which is equivalent to Equation 13, with
[AI
=
[GI-'[HI
(95)
Equation 6 follows'as a consequence of Equations 12, 12a, and
12b and the identity
J3*
=
0.
]=I
The symmetry and positive definiteness of [GI and [HI in Equation 94 have very important consequences. These properties assure the existence of a matrix [Q] with the properties ( 9 , 75) [Ql'[Gl[QI = VI (96) [Ql'[HI[Ql
=
[Dl
(97)
in which [D] is a diagonal matrix of real numbers greater than zero. Application of these relations to Equation 95 gives [ Q l ' [ ~ l [ A I [ Q I = [QI'[Hl[QI
=
[Dl
(98)
Premultiplying this equation by a diagonal matrix of constants [L], and postmultiplying by the inverse matrix [LIP' = [1/L], gives [~I[Ql'[~I[~I[Q1~~1-' = [LlIDI[Ll-' = [Dl (99)
(101a)
-
i = 1, . . . n - 1 j # i,n
(101b) (101c)
which are consequences of Equations 6 and 12 and the identity
.k
Jr*
'=
0.
The IDtj are convenient for presentation of
j=l
experimental results since, unlike the Afj, they are applicable for an arbitrary reference velocity. Furthermore, aside from 234
I&EC FUNDAMENTALS
-
-
-
-
and the diffusion coefficients IDi, are obtained, if desired, from the equations
=
-
+
This is equivalent to Equation 16 with [PI-' = [L][QITIG], [PI = [Q][L]-', and [ A ] = [ D ] . T h e choice of unity as the largest element in the zth column of [PI corresponds to the insertion of the largest element of the ith column of [Q]as the zth diagonal element of [ L ] . Equation 16 thus holds whenever [GI is nonsingular; the conditions for this are given above Equation 94. T o determine coefficients Ai, or IDtj experimentally, one first determines matrices [PI,[PI-',and [ A ] . The experimental methods used by Dunlop and Costing ( 3 ) .for example, yield the necessary information. The matrix [A] is then obtained from [AI = [ P I b I [ P I - ' (100)
(z),
L-Q
TA,;l. matrix in Eauation 13. LP2t kj?,' flux ratio in' Table I, 'see Equation 75 or 80, dimensionless diffusion coefficients in Equation 6, L2t-l diffusion coefficient for binarv mixture of species i and j, L2t-' idemootent matrices in Eauation 40. dimensionless molar Gibbs free energy, M L P mole-' molar chemical potential of component i, M L 2 t P mole -l matrix in Equation 93, ML2tw2mole-' matrix in Equation 89, Mt-l mole-' unit matrix ci(vi - v*), molar diffusion flux of species i relative to v*> moles L-2t-1 column vector with elements J1* . , . Jn- I* transformed column vector in Equation 17b with elements J1*: , . , Jn- I* interfacial velocity parameter in Equation 73b, dimensionless interfacial flux parameter in Equation 82, dimensionless number-mean molecular weight of a mixture molecular weight of species i civi, flux of species i relative to stationary coordinates, moles L-2t-I column vector with elements N1, , , . N,- 1 transformed column vector in Equation 17c with elements N I , . . .N,- 1 normal component of N , into given one-phase region at a boundary surface, moles L-2t-1 matrix in Equation 16, dimensionless Sa S1,boundary surface in Equations 24b through 58b molar entropy of mixture temperature flow velocity a t outer limit of boundary layer region bounded by S dimensionless interfacial velocity in Table I dimensionless composition gradient in Table I molar concentration of species i, moles Lw3 total molar concentration of fluid, moles L P 3 binary transfer coefficient, moles L - V 1 limit of k; as .VAo 0 and NBO 0 binary transfer coefficient evaluated at 00 assuming
x , y. X
2
-
v* = v multicomponent transfer coefficient in Equation 32 wedge-shape factor, zero for flat-plate flow number of chemical components in mixture unit vector normal to system boundary and inwardly directed static pressure, ML-1t-2 number of distinct values of Ak that satisfy Equation 15 time mass average velocity of mixture molar average'velocity of mixture mass average velocity normal to a mass transfer surface, Equation 70 molar average velocity normal to a mass transfer surface, Equation 72 mole fraction of component i column vector with elements X I , , . .x,- 1 transformed column vector in Equation 17a with elements XI, . . . X,- I = space coordinates - distance downstream from leading edge of flat plate or vertex of wedge
GREEKSYMBOLS Ait
II
&
multicomponent Schmidt number, dimensionless = composition function in Equation 26. dimensionless = gravitational potential energy function, L2t-2 = 2m,’(m 1) = binary boundary-layer thickness in Equation 76, L = multicomponent boundary-layer thickness in Equation 81, L = position coordinates on S = eigenvalues of [ A ] , n - 1 in number? L-2t = distinct eigenvalues of [AI, J in number, L-2t = viscosity, h4L-lt-l = density, M L - 3 = local rate of entropy production, e.u. Lp3t-I = viscous stress tensor, = dimensionless interfacial velocity in Equation 64 = dimensionless interfacial velocity in Equation 65 = p.+,/pa:
+
p
6i ( >4
Ai.
A, P U
..
y.4B Q~
-
OVERLIYES
=
per mole
A
= per unit mass
-
=
-
=
transformed according to Equations 17a, b, c partial molar in Equation 88
SUBSCRIPTS A. . . B = species in binary system z, 1. k = species, transformed functions or transformed variables in multicomponent system a = reference state 0. 1 = surface conditions MATHEMATICAL NOTATIONS
[ ] 0m
= column vector or matrix, depending on context = negative side of zero = upstream conditions
literature Cited
(2) Cussler, E. L., Jr., Lightfoot, E. K., A.I.Ch.E. J . 9, 702, 783 11963). (3) Dunlop, P. J.: Gosting, L. J., J . Phys. Chem. 63, 86 (1359). (4) Fadeev, D. K.: Fadeeva, V. N., “Computational Methods of Linear Alcebra.” translated bv R. C. LVilliams. Freeman. San Franciscoo, 1963. (5) Frazer, R. A,: Duncan, LV. J., Collar. A. R.. “Elementary Matrices,” L-niversity Press, Cambridge, England, 1938. (6) Gibbs. J. LY.. ”Collected LVorks,” Vol. 1. Yale Universitv Press, k e w Haven. Conn., 1948. (7) Gilliland, E. R., Sherwood: T. K.; Ind. En?. Chem. 26, 516 (1934). (8) Halmos. P. R.: ”Finite Dimensional Vector Spaces:” p. 156, Princeton Cniversity Press, Princeton, N. J.. 1942. (9) Hildebrand. F. B.: “Methods of Applied ,Mathematics,“ Prentice-Hall, Englewood Cliffs, N. J., 1952. (10) Hirschfelder, J. 0.:Curtiss, C. F.: Bird, R. B., “Molecular Theory of Gases and Liquids,” Chap. 11, LViley: N e b 7 York, 1954. (11) Marcus, M.: ‘.Basic Theorems in Matrix Theory,” National Bureau of Standards Applied Mathematics Series, No. 57, 1960. (12) Mickley, H . S., Ross? R. C.: Squyers, A. L., Stewart, 11.. E., Natl. Advisory Comm. Aeronaut. Tech. Note 3208 (1954). (13) Onsager, L., Ann. >V.Y.Acad. Sci.46, 241 (1945). (14) Onsager. L., Phys. ReLm. 37, 405 (1931); 38, 2265 (1931). (15) Perlis, S.; “Theory of Matrices,” 3rd printing, Chap. 9, Addison-\Yesky, Reading Mass.. 1958. (16) Prober, R.: Ph.D. thesis, University of LVisconsin, 1961. (17) Prober, R.: Stewart, I V . E., Intern. J . Heat ‘Mass Transfer 6, 221 (1963); Corrigenda, Ibzd., 6, 872 (1963). (18) Stewart, LV. E.: A.I.Ch.E. J . 9, 528 (1963). (19) Stewart, LV. E., Sc.D. thesis, Massachusetts Institute of Technology, 1951. (20) Stewart, LV. E.. Prober, R.. Intern. J . Heat ~ti‘ass Transfer 5 , 1149 (1962); Corrigenda, Ibid., 6, 872 (1963). (21) Tisza, L.. “General Theory of Phase Transitions,” Chap. 1: “Phase Transformations in Solids,” R. Smoluchowski, J. E. Mayer, and LV, A. LVeyl, eds., LViley, New York, 1951. (22) Toor, H. L., A.I.Ch.E. J. (to be published). (23) IVedderburn, J. H., “Lectures on Matrices,’’ Vol. 17, p. 28, American Mathematical Society Colloquium Publications, 1934. (24) W‘ilke, C. R., J . Chem. Phys. 18, 517 (1950). >
,
(1) Bird, R. B., Stewart, LV. E., Lightfoot, E. N., “Transport Phenomena,” 4th printing with corrections, LViley, New York, 1964.
RECEIVED for review March 9, 1964 ACCEPTED April 14, 1964
FRACTIONAL SOLIDIFICATION OF EUTECTIC-FORMING MIXTURES W
.
R
.
W I L C 0 X, Aerospace Corp., El Segundo, Calif.
The solute redistribution resulting from progressive freezing or zone melting of a simple eutectic-forming mixture has been solved analytically for pure diffusional mass transfer. Comparison with experimental results shows ,that constitutional subcooling usually occurs and produces considerable trapping of impurity.
zone melting work was done on high-temperature, For these materials the equilibrium ratio of solid to liquid solubility of impurity (the distribution coefficient) was often a constant. For this reason and for simplicity, nearly all theoretical treatments of solute redistribution in fractional solidification processes have assumed a constant distribution coefficient (3, 9 ) . Recently. however, a considerable amount of zone melting work has been done on organic systems (2, 9). Most organic systems d o not have a constant distribution coefficient but have phase diagrams of the eutectic type, as shown in Figure 1 ( 7 , 9). Concentrated mixtures of metals or inorganic compounds also usually have eutectic-type phase diagrams : either of the simple eutectic type or the limited solid-solubility type. I n this paper simple eutectic behavior (no solid solubility) is considered in detail. ARLY
E high-purity materials--principally semiconductors.
Assuming ideal conditions the concentration profile resulting from the zone melting of a mixture of composition ze, in Figure 1 can easily be predicted (9). Initially the solid freezing out would contain no impurity and so the solute \\.odd accumulate in the melt. M’hen the melt reaches the eutectic composition w e , the solid composition would j u m p in a step function to the original concentration, E,. Each additional zone pass of a semi-infinite solid would produce an added amount of pure material equal to that of the first pass. Experimentally such concentration profiles have not been observed (5. 6, 9). T h e composition deviates much earlier from pure material. the approach to E, is not a step function, but is gradual. and the zone composition is rarely E,. TIVO mechanisms could account for this. O n e is the occurrence of constitutional subcooling, which produces a rough freezing interface, which in turn tends to trap the melt (9). T h e exact effect of coiistitutional subVOL. 3
NO. 3
AUGUST
1964
235
