CORRESPONDENCE NUMERICAL SOLUTION OF COUPLED, ORDINARY DIFFERENTIAL EQUATIONS SIR: The following observations from the field of numerical analysis have some bearing on Newman’s communication (1968) :
First- and second-order finite difference approximations of ordinary differential equations of the form of Newman’s Equation 1 lead to tridiagonal algebraic equations (Lapidus, 1962, pp. 108-13). Efficient methods of solution of such tridiagonal systems have been extensively developed and are readily available (Funderlic, 1968). In particular, Thomas’ explicit method is well known (Lapidus, 1962, pp. 254-5). Boundary conditions involving linear combinations of the problem system-dependent variable (s) and its derivative (s) such as Newman’s Equation 11 can be handled by a straightforward application of finite difference techniques (Lapidus, 1962, pp. 112-13). Nonlinear boundary-value problems can be solved by iterative solution of the corresponding linear problem obtained by the generalized Newton-Raphson or quasilinearization method. Newman’s Equations 23 and 24 are a straightforward application of this technique, which was previously treated theoretically (Bellman and Kalaba, 1965; Kenneth
and hlcGil1, 1966) and applied to problems in chemical engineering (Lee, 1966, 1968). In particular, the rapid (quadratic) convergence is well known. I t appears to me that all of the essential points of Newman’s development have been reported previously and are fully documented in the literature. literature Cited
Bellman, R., Kalaba, R., “Quasilinearization and Nonlinear Boundary-Value Problems,” American Elsevier, New York, 1965.
Funderlic, R. E., “The Programmer’s Handbook,” pp. 153-5, U.S. Atomic Energy Commission, Rept. K1729 (Feb. 8, 1968). Kenneth, P., McGill, R., “Two-Point Boundary-Value-Problem Techniques,” Vol. 3, Chap. 2, “Advances in Control Systems,” Academic Press, New York, 1966. Lapidus, L., “Digital Computation for Chemical Engineers,” McGraw-Hill, New York, 1962. Lee, E. S., Chem. Eng. Sci. 21, 183 (1966). Lee, E. S.A.I.Ch.E. J . 14, 490 (1968). Newman, John, IND.ENG.CHEY.FUNDAMENTALS 7, 514 (1968).
W . E. Schksser Lehigh University Bethlehem, Pa. 18015
NUMERICAL SOLUTION OF COUPLED ORDINARY DIFFERENTIAL EQUATIONS SIR:Schiesser has compiled an interesting collection of useful and pertinent references. I have evidently been remiss in supplying references for standard finite-difference methods. I originally formulated in 1963 the techniques used in my paper and have developed them over the years, only recently having published a complete description of the method. I have treated coupled equations, whereas tridiagonal matrices and Thomas’s method apply to single equations. I should be gratified that Schiesser has found no serious error in my work and that several elements of my method have substantial backing in the literature. These have been synthesized into a useful method for the problems outlined. Lee (1968), we may note, shied away from solving coupled tridiagonal matrices.
I am indebted to E. A. Grens for other relevant references (Fox, 1957; Holt, 1964; Sylvester and Meyer, 1965). literature Cited
Fox, L., “Numerical Solution of Two-Point Boundary Value Problems in Ordinary Differential Equations,” Clarendon, Oxford, 1957. Holt, J. F., Commun. ACM 7, 366-73 (1964). Lee, E. S., A.I.Ch.E.J. 14, 490 (1968). Sylvester, R. J., Meyer, F., J. SOC.Ind. Appl. Math. 13, 586-602 (1965).
John Newman University of California Berkeley, Calif. 947.20
ESTIMATION OF IDEAL GAS ENTROPY OF ORGANIC COMPOUNDS SIR: Rihani and Doraiswamy [IND.ENG.CHEM.FUNDA7, 375 (1968)l propose that the standard molar
MENTALS
entropy of gaseous compounds should be estimated by the equation :
+ +
So = contributions of composing groups corrections where necessary R T In u
- RT In 7
whereo is the total symmetry number and 7 is the number of optical isomers. This equation is obviously incorrect dimensionally, the
last two terms on the right having the dimensions of energy. But even when the intrusive factor, TI is removed, the signs of these two terms are incorrect. The effect of molecular symmetry on entropy was stated by Mayer et al. (1933) as “The molar entropy of a molecule having a symmetry number u is R In u less than the entropy of the same molecule were it entirely nonsymmetrical.” Thus, to correlate entropy with structure, aside from symmetry, the effects of symmetry should first be removed by adding R In u to the actual molar entropy. VOL.
a
NO.
3
AUGUST
1969
599
