Noticing that
d
iix?
Conclusion
a
1 =-
dTP
1
(14)
and writing
The problem of the optimal design of adiabatic reactor sequence with cold shot cooling seems of interest to many “computer chemists,” probably because of its numerous industrial applications. The sophisticated optimization methods available sometimes cause authors to ignore the essential rule that the first derivatives vanish a t the optimum. Using this rule correctly, one can indeed solve this complicated problem in an elegant and efficient manner (Malenge, 1968). Literature Cited
one obtains r l ( F l )= r 2 ( x 2 )since , the other terms in Equation 15 cancel two by two. Next, Bhandarkar and Narsimhan derive the relation
(Equation 1 2 in their correspondence.) This relation does not seem correct, taking into account Equations 11 and 12 above, and its use therefore would seem risky t o us. Finally, in the definition of P r , Bhandarkar and Narsimhan omitted to multiply the last term by Xr-only the A? fraction of the feed goes through the preheater. T o obtain the final correct expressions for P L , dP2/ax2 and dPr/aT2, the coefficient ,U in Equations 2, 10, and 12 above should be replaced by A?,u.
Bhandarkar, P. G., Narsimhan, G., IND. ENG. CHEM. PROCESS DESIGNDEVELOP. 8, 142 (1969). Buzzi Ferraris, G., Ing. Chim. Ital. 4, 5 (1968a). Buzzi Ferraris, G., Ing. Chirn. Ital. 4, 31 (1968b). Hellinckx, L. G., Van Rompay, P. V., IND.ENG.CHEM. PROCESS DESIGNDEVELOP.7, 595 (1968). DESIGN Lee, K. Y., Ark, R., IND. ENG. CHEM.PROCESS DEVELOP. 2, 300 (1963). Malenge, J. P., C . R . Acad. Sci. Paris 267, 1651 (1968). Malenge, J. P., Villermaux, J., IND.ENG.CHEM.PROCESS DESIGNDEVELOP. 6,535 (1967).
J . P . Malengk Centre de Cinktique Physique ut Chimique C.N.R.S. 54-Nancy, France
SIMULATION OF THE HEAT TRANSFER PHENOMENA IN A ROTARY KILN SIR: A recent paper (Sass, 1967) reported a method of solving the equations of heat transfer which takes place in rotary kilns. Having made use of the methods described in this paper ourselves, we would like to report that we have found the method described to be unsatisfactory. Sass advises that the mathematical model should use the Runge-Kutta method of solving differential equations with the aid of a digital computer. We followed these recommendations, but found that the rate of rise of gas temperature along the kiln was dependent on the magnitude if the integration step size chosen, which is, of course, incorrect. Analysis has shown that the Runge-Kutta method is the cause of the instability. Collatz (1951) gives a rough rule of thumb that a certain ratio of integration increments should not be allowed to increase in magnitude to more than a few hundredths. If it does, the integration step size should be reduced. We found that with step sizes of 1 to 10 feet for a 200-foot long kiln of about 10-foot diameter the ratio referred to varied between 0.5 and 15. I t was only when we had reduced the step size to about 1 x lo-’” foot that this ratio fell to a magnitude of a few hundredths. We caution readers who have used this method or are considering using it to examine the ratio referred to above for their particular conditions and to be wary of the accuracy of‘ their solutions if the ratio is more than a few per cent. We found the discussion on the stability of the Runge-Kutta method given by the National Physical Laboratory (1961) and McCracken and Dorn (1964) very helpful.
Literature Cited
Collatz, L., “Xumerical Treatment of Differential Equations,” Springer Verlag, Berlin, 1951 (English translation available). McCracken, D., Dorn, W., “Numerical Methods and Fortran Programming,” Wiley, New York, 1964. National Physical Laboratory, “Modern Computing Methods,” Notes on Applied Science, No. 16, H. M. Stationery Office, London, 1961. Sass, A., IND. ENG. CHEM. PROCESS DESIGNDEVELOP. 6, 532 (1967); 7, 318 (1968).
J . L . Cribb P . A . Langley
C . S . R . Research Laboratories Roseville, N . S. W . ,Australia SIR: The comments by Cribb and Langley are helpful and I agree that the stability of the Runge-Kutta solution should be checked. I n my work, I did not adopt the procedure they recommend, but I did vary the integration step size to determine if this would alter the solution. For a change in step size from 0.1 t o 1.0 foot for the two kilns discussed in the paper, I did not detect any significant change in the solution. Allan Sass Garrett Research & Development Co., Inc. 1855 Carrion Road La Verne, Calif. 91750 VOL. 8 N O . 4 OCTOBER 1969
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