135
Ind. Eng. Chem. Fundam. 1984. 23, 135
F(a) =
Fp =
m
m
1 - 192b/a/a5C tanh [(2n - l)a/2b/a)]/(2n - l I 5
a/12[(1 - 1 9 2 / a s 5 C (tanh (2n - l)aa/2)/(2n - 1)5] n=l
n=l
(19a)
(2)
The corresponding equations from Sen are Q = [a4/k1(-@/dx)F(a)
(74
m
[(coth (2n - l)aa -
F(a) = ‘/12a - 16/a5 n=l
cosech a(2n - l)a]/(2n - U5 (19) A comparison of Squires’ and Sen’s equations can be made by solving each rate equation for a normalized flow rate 4 = pQ/-dp/dx from eq 1 4 = b3aFp/12 from eq 7a
4 = a4F(a)
The results for the first 5 terms of the series are shown to be equivalent in Table I. It is seen that convergence is a function of a and may not necessarily occur to the fifth decimal place in five terms as stated by Sen. This is not a problem with the use of a high-speed computer. Sen’s eq 19 can be simplified by the substitution of the identity coth (2n - 1)aa - cosech (2n - 1)aa = tanh (2n - 1 ) ~ ( ~ / 2
Equation 19a is easier to compute than eq 19. It would appear that Squires’ solution is not only identical with Sen’s but also is simpler to use because only one hyperbolic function (tanh)is required in place of Sen’s coth and cosech functions. Also it shows a direct relation to the asymptotic one-dimensional solution of the NavierStokes equations for the aspect ratio b/a = a = 0 for which Fp = 1.0.
Literature Cited Bernhardt, E. C., Ed. “Processing of Thermoplastic Materials”; Reinhold: New York, 1959. Edwards, R. B.; Fogg, J. E.; Kraybill, R. R.; Regan, J. T. SPE J . Mar 1984, 2 0 , 234-243. Johnston, A. K. I n d . Eng. Chem. Fundem. 1973, 12, 482. Sen, A. K. I n d . Eng. Chem. Fundem. 1982, 2 1 , 488-408. Squires, P. H. SPEJ. May 1058, 14, 24-30. Srlnivasan, S.;Bobba, K. M.; Stenger, L. A. Ind. Eng. Chem. Fundam. 1979, 18, 130.
Manufacturing Technology Division Kodak Park Eastman Kodak Company Rochester, New York 14650
Richard R. Kraybill*
Address correspondence to 1704 Laurie Lane, Beileair, Clearwater, FL 33516.
Response to Comments on “Isothermal Creeping Flow In Rectangular Channels” Sir: As pointed out by Dr. Kraybill in his correspondence, Poisson’s equation with appropriate boundary conditions has been solved many years ago and has been used extensively in connection with screw extrusion of plastics. In fact, Poisson’s equation appears quite frequently in many other areas of applications such as solid mechanics (e.g., torsion of shafts), heat transfer, and electrostatics. A solution of the problem in the form of a single Fourier series has been repeatedly used by several researchers in these various areas. The purpose of the communication “Isothermal Creeping Flow in Rectangular Channels” was merely to show
that such a single Fourier series solution is much more rapidly convergent than the double Fourier series presented by Srinivasan et al., for the problem of vapor condensation inside a rotating tube. The reader should not be misled by the fact that the above solution is entirely original; however, it demonstrates considerable improvement over the double Fourier series solution of Srinivasan et al. in terms of convergence and accuracy. Department of Mathematical Sciences Purdue School of Science at Indianapolis Indianapolis, Indiana 46205
CORRECTION The Relative Thermal Value of Tomorrow’s Fuels, Hoyt C. Hottel, Ind. Eng. Chem. Fundam. 1983,22, 271. Page 273. In Table 11, all numbers in the last column contained 1before the decimal. All but the three following 1’s should be deleted (1.039); 1.000 [opposite l/zCO, lj2H2];and 1.000 [opposite “no. 6 resid. fuel oil.”]. Page 275. Clarity can be added to the first sentence two paragraphs above eq 4; in “the enthalpy of 60 OF-halfsaturated air”, insert “at 1800 K” after “enthalpy”.
Asok K. Sen
