is high enough so that negligible temperature differences occur within it. The local heat transfer coefficient is expected to vary from point to point on the sphere surface, and this will influence the local heat flux, but will be of no consequence to the measurement of the average heat transfer coefficient so long as the shell conductivity is high enough to provide an essentially uniform shell temperature. Assuming the device has an average density of 2.9 grams per cc., the temperature difference between the sphere surface and a fluidizing current of atmospheric air a t 75' F. is estimated to be 7.3' F., for a 0.5-watt heat source. The longitudinal temperature distribution, t (above ambient), in a thin spherical shell which is maintained a t temperature T o n a latitude and steadily cooled by a uniform heat transfer coefficient, h, over its surface is given in terms of the hypergeometric series, F,as (2) : t = T(F(cr,P,l,x)/F(ff,P,l,x,)j
where m-1
F(cu,&l,x) = 1
A h r k 6
= = = = =
+
II n=o m=l
(A
+ n + nz) xm
(mi)'
hr2/k6 heat transfer coefficient shell radius thermal conductivity of shell thickness of shell x = (!J 1)/2
= COS
e
0 = angle from pole above isothermal latitude to a latitude a t which temperature is t. T corresponds to xo.
Using this expression for a sphere of unit average density, for which the average temperature difference with a 0.5-watt source would be 10' F. from sphere to fluidizing air, the maximum temperature difference in the shell shown in Figures 1 and 2 is 0.37' F. This indicates a 3.7% error in the heat transfer coefficient from this source. However, the error can be made smaller by correcting +hereading of the sensor for the anticipated error at its location. A similarly small error is anticipated for the actual device. Acknowledgment
The author is grateful to the following people for help with various portions of the development of this device: University of Michigan, Major Ash and Ian Scott; Brookhaven National Laboratory, Warren Winsche, George Lindauer, Bob Chase, and Dmitri Stephani; Nuclear Materials and Equipment Corp., Karl Puechl; National Beryllia Corp., Eugene Ryskewitch; Monsanto Research (Mound Laboratory), Dale Coffey and Murph Jones. literature Cited
(1) Barker, J. J., Znd. Eng. Chem. 57, No. 4, 43-51; No. 5, 33-9 (1965). (2) Barker, J. J., IND.ENG.CHEM.FUNDAMENTALS 6, 154 (1967).
RECEIVED for review October 31, 1966 ACCEPTED December 16, 1966
+
CO M M UN ICATIONS
CONVERGENCE OF APPROXIMATION METHODS USED IN MULTICOMPONENT MASS TRANSPORT A method is given for approximating the flux expressions that can b e derived from the Stefan-Maxwell equations or other similar models for multicomponent mass transfer in liquids and gases. convergence o f the method are given.
RECESTLY, there has been considerable
interest in simplified approaches to multicomponent mass transport (7, 7, 8). In general, the multicomponent flux equations have been linearized, and then reduced to diagonal form. The treatment here also is concerned with simplifications in multicomponent mass transport. However, a different method of approximation is used, and the uses to which the results are to be put are different. Here the emphasis is upon simplified multicomponent flux expressions which should prove useful where one wishes either to assess or to take into account the concentration dependence of the coefficients multiplying the forces in the usual irreversible thermodynamic flux expressions. A specialization of the general method given here was used previously to correlate certain transport phenomena in membranes ( 9 ) . In this paper, the previous treatment i s generalized and extended to higher order approximations, and the conditions for convergence are given. 142
I&EC FUNDAMENTALS
The conditions for
Theory
The generalized Stefan-Maxwell equations are the generally accepted model for multicomponent mass transport in gases. I n liquids and other condensed phases the models for multicomponent mass transport are less clearly defined. However, while a number of models have been proposed (2-6), these are substantially the same as the generalized Stefan-Maxwell expressions except for the nomenclature and the concentration scales employed. Though, essentially, the same set of flux expressions results from different viewpoints, the derivations are different. The point to be made is that the generalized Stefan-Maxwell equations, in one form or another, have been suggested for use in condensed phases as well as in the gas phase. This general form is the model used here, written as (1) Ft = Nt,C5(ut u5)
c
-
where Fi = generalized force on species i ?’..13 - a coefficient characteristic of the interaction between species i and species j = concentration of species j, molesjunit volume
c.
n
C = total molar concentration u 1. ) u3. =
a single type of experimental data over a small range of concentrations. For reasons that will become apparent later, Equation 1 is rewritten so that each species velocity appears but once.
cj
Fi
=
velocity of species i and j, respectively, with respect to a frame of reference, usually the molar, mass, or volume average velocity of the system but often with respect to space fixed coordinates, especially in membranes number of species present in system flux with respect to the frame of reference
The Aril’s used her? are those of Kressman (3) and are related to Spiegler’s Xi,( 6 ) , Lightfoot’s D i j ( 5 ) , Laity’s Rij (4),and Klemm’s r z , (2) as follows:
C,NiI
ui
j
-
I
i
CjNijuj j # i
(3)
The first sum in Equation 3 is defined to be a generalized and an ideal flux, Nr*, is introduced and defined mobility, Ui, by Equation 5.
(4)
i
(5)
iV 0.5. In the general multicomponent case, there need be no cy greater than unity. Frequently, however, the solvent species mole fraction will be nearly unity, in which case the cy for the solvent species will be large. Substitution of Equation 16 into Equation 12 gives: 144
I&EC FUNDAMENTALS
Again, clearly, the method will converge if the c y i s and aJ)s are small, and the cyk is large. However, the simple interpretation that the aj are equal to CiUiiV, is no longer valid. literature Cited
(1) Cullinan, H. T., IND. ENG. CHEM.FUNDAMENTALS 4, 139 (1965). (2) Klemm, A,, Z. Naturforxh. 8a, 397 (1953). ( 3 ) Kressman, T. R. E., Stanbridge, P. A., Tye, F. L., Trans. Faraday Sac. 59, 2133 (1963). (4) Laity, R. W., J . Phys. Chem. 63, 80 (1959). (5) Lightfoot, E. N., Cussler, E. L., Rettig, R. L., A.Z.Ch.E. J., 8, 708 (1962). (6) Spiegler, K. S., Trans. Faraday Sac. 54, 1408 (1958). (7) Stewart, W. E., Prober, R., IND.END.CHEM.FUNDAMENTALS 3. 224 (1964). (8)’Toor,‘H. L:, A.I.Ch.E. J . 10,448 (1964). (9) Wills, G. B., Lightfoot, E. N., IND.ENG.CHEM.FUNDAMENTALS 5 , 114 (1966). GEORGE B. WILLS Virginia Polytechnic Institute Blacksburg, Va.
RECEIVED for review February 28, 1966 ACCEPTEDSeptember 6, 1966 Research project supported by the Office of Water Resources Research, Department of the Interior, pursuant to Water Resources Research Act of 1964 (Public Law 88-379).
