Rosen’s results indicated 114 functional evaluations were needed for convergence to eight significant figures when numerical derivatives were used. The use of analytic derivatives required 31 functional evaluations plus 68 derivative evaluations to achieve the same convergence. If one derivative evaluation can be equated to one functional evaluation, the comparison would be 99 to 114 in favor of analytic derivatives. As could be expected, the analytic derivatives were of particular advantage close to the final solution. I n this region
the difference approximations were based on step sizes larger than the computed corrections. William E. Ball Washington University St. Louis, M o . Literature Cited
(l) W‘ E’3Groenweghe’ L‘ * ’ D*’IND‘ FUNDAMENTALS 5 , 181 (1966). ( 2 ) Marqiiardt, D. W., J . SOC. Ind. Appl. Math. 11, 431 (1963).
VELOCITY PROFILE I N THE STEFAN DIFFUSION TUBE SIR: Recent papers (2, 3) have raised the question of nonuniform velocity profiles in the Stefan diffusion tube experiment. Mathematical analysis has been suggested ( 2 ) and performed ( 2 , 3) making use of the “no-slip” condition a t the tube wall. This letter points out a contradiction in these solutions, and strengthens the argument for a uniform velocity profile across the tube. While one hesitates to rederive the diffusion equation, the derivation would seem to clarify the matter under consideration. Conservation of the ith species in a multicomponent system may be represented as
Here we depart somewhat from traditional continuum concepts in that we do not require the velocity vector ui to be zero a t fixed, impermeable surfaces. Designating n and 3, as the unit normal and tangent vectors respectively, we require only that
u i . n = 0, a t fixed, impermeable surfaces while the tangential component is given by
I n the Stefan diffusion tube the diffusion velocity is directly related to the mass average velocity by
ut = where p i represents the mass density of the ith species, D[,/Dt is the ith species material derivative, and V , ( t )is the material volume of the ith species. Application of the species transport theorem ( 5 ) yields
(9)
(k -
1) v
where the ith species is now considered to be the vaporizing liquid. Here we see that the previous suggestion that v = 0, a t the tube wall
(12)
quickly leads us (via Equations 8 and 11) to the conclusion that where v i is the velocity of the ith species. Since V i ( t ) is arbitrary, the species continuity equation results. bpi
dt
+v .
Vi) = 0
(Pi
(3)
Defining the diffusion velocity as
where I
v = -
pivi
P i=l P = C P t i=l
leads us to a modified form of Equation 3.
V
(9 -
= 0, a t the tube wall
This is certainly not the case, for Equation 13 would require the concentration to be constant along the wall. The explanation of this difficulty is that if Equation 8 is used, one is committed to allowing nonzero values of 31 ui a t solid surfaces. This is not to say that the equations of motion do not apply; it is only that the traditional no-slip condition is not valid for that motion occurring because of concentration gradients. Use of Equation 8 as a constitutive equation may be open to question; however, there appears to be ample evidence that it is valid, provided the mean free path is small compared to the tube diameter. With no restrictions on the tangential velocity a t the tube wall, there is no reason to believe that the velocity profile is anything but uniform, provided one is willing to neglect entrance and end effects. a
literature Cited
To this point the analysis follows a strict continuum point of view-Le., a continuous and invertible relationship must exist between the spatial coordinates and the ith species material coordinates (4) in order to obtain Equation 2 from Equation 1, and the integrand in Equation 2 must be continuous if Equation 3 is to be extracted from Equation 2. If we now restrict the analysis to Fickian diffusion, the diffusive mass flux vector is expressed as (7) :
476
l&EC FUNDAMENTALS
(1) Bird, R. B., Stewart, W. E., Lightfoot, E. N., “Transport Phenomena,” p. 502, Wiley, New York, 1960. ( 2 ) Heinzelmann, F. J., Wasan, D. T., Wilke, C. R.,IND.END. CHEM.FUNDAMENTALS 4, 55 (1965). (3) Rao, S. S., Bennett, C. O., Ibid.,5 , 573 (1966). (4) Truesdell, C., “Principles of Continuum Mechanics,” p. 10, Colloquium Lectures in Pure and Applied Science, No. 5, Socony Mobil Oil Co., New York, 1961. ( 5 ) Truesdell, C., Toupin, R.,“The Classical Field Theories,” pp. 347, 470, “Handbuch der Physik,” Vol. 3, Part 1, Springer Verlag, Berlin, 1960. Stephen Whitaker Univer.vity of California Davis, Calif.
