Ind. Eng. Chem. Fundam. 1984, 23, 134-135
134
-
the pore mouth becomes infinitely large a t pore closure (D(u) un,n > 0) implying that, within the limits of the quasi-steady-state approximation, reaction practically ceases within the pore. The conversion q(t) for a finite-length pore can be appropriately defined as
where X denotes the normalized amount of reactant originally available (A = 1 for catalytic reactions, X = [(lto)(a- l)]/2kto for noncatalytic gas-solid reactions with structural changes). Substitution of (22) into (24) eventually leads to the conversion profile
and the final conversion a t pore closure time
of validity of the asymptotic results is expected to be independent of k. Nomenclature D = diffusion coefficient function, dimensionless De = effective diffusivity through solid layer, cmz/s D, = molecular diffusivity within pore, cmz/s G = reaction rate function, dimensionless k = K r o / D , = Biot modulus, dimensionles K = reaction rate constant, cm/s L = length of pore, cm ro = initial pore radius, cm t = time, dimensionless u = reactant concentration, dimensionless u = solid product extent, dimensionless x = axial distance, dimensionless Greek Letters a = chemical parameter to account for structural changes, dimensionless to = initial porosity, dimensionless 7 = conversion, dimensionless 4 = (KL2/r,Jlp)1/z = Thiele modulus, dimensionless
Literature Cited The above asymptotic results simulate accurately the conversion performance of a reaction scheme leading to pore closure in a finite pore provided that condition (412) is not violated. Due to the continuous shrinking of the pore radius and the resultant steepening of the concentration profiles, the accuracy of the asymptotic analysis improves with time. Thus, it normally suffices to test the zero-flux condition during the initial stages of the process. A sensitivity analysis carried out by Shankar and Yortsos (1983) for gas-solid reactions has shown that good agreement is observed for values of 4 as low as 3 and k values not exceeding -O(lO). It should be noted that G(u) is also a function of k for gas-solid reactions; thus large values of k lead to significant consumption of the solid reactant beyond x = 1initially, implying a violation of the zero-flux condition. On the other hand, G(u) does not depend on k for most catalyst-deactivation reactions; thus the domain
Androutsopouios, G. P. ; Mann, R. Chem. Eng. Sci. 1878, 33, 673. Christman, P. G.; Edgar, T. F. 73rd AIChE Annual Meeting, Chicago, IL, 1980; Paper No. 10f. Chrostowski, J. W.; Georgakis, C. Proceedings of the Fifth International Symposium on Chemical Reaction Engineering, Houston, TX, 1978. DudukoviE, M. P. AIChE J. 1878, 22(5).945. Hartman, M.; Coughlin, R. W. Ind. Eng. Chem. Process Des. Dev. 1874, 13, 24%. Shankar, K. ; Yortsos, Y. C. Chem. Eng. S d . 1883, 38(8).1159. Szekely, J.; Evans, J. W.; Sohn, H. Y. ”Gas-Solid Reactions”; Academic Press: New York, 1976. Uirichson, D. L.; Mahoney, D. J. Chem. Eng. Sci. 1980, 35, 567.
Departments of Chemical and Yanis C. Yortsos* Petroleum Engineering Krishnamurthy Shankar University of Southern California Los Angeles, California 90089 Received for review November 2, 1982 Revised manuscript received September 12, 1983 Accepted October 12, 1983
CORRESPONDENCE Comments on “ Isothermal Creeping Flow In Rectangular Channels” Sir: Sen (1982) claims to have found a solution in terms of a single Fourier series for steady isothermal creeping
flow in rectangular channels. His solution is said to converge much more rapidly than a double Fourier series obtained earlier by Srinivasan et al. (1979). The analytical solution by Srinivasan is said to be more convenient than a numerical solution of Johnston (1973). This problem has been one of particular interest for screw extrusion of plastics, and a rapid-converging solution equivalent to Sen’s has been used for many years as discussed in Bernhardt (1959) and applied in Edwards et al. (1964). A shape factor F, as given by Squires (1958) clearly demonstrates the effect of the drag of the sidewalls upon the one-dimensional pressure-induced flow between two
Table I. Typical Convergence Patterns for @ e q 1(Squires), e q 7a (Sen), b 3a F~ i 12 dF(a) n a = 3 a = ‘13 a=3 CY = ’ I 3 1
2 3 4 5
0.21519 0.19920 0.19786 0.19761 0.19754
0.19772 0.19751 0.19749 0.19749 0.19749
0.19772 0.19751 0.19749 0.19749 0.19749
0.21519 0.19920 0.19786 0.19761 0.19754
horizontal parallel plates of infinite width or zero aspect ratio, a = b / a . Equations 1 and 2 from Squires are rewritten as Q = (b3a/12p)(-~/dx)~p (1) 0 1984 American Chemical Society
135
Ind. Eng. Chem. Fundam. 1984. 23, 135
F(a) =
Fp =
m
m
1 - 192b/a/.n5C 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 ) ~ -a
F(a) = ‘/12a - 16/a5 n=l
cosech ~ ( 2 -nl)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 - l)m - 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
