There appears to be available now a n increasing amount of evidence that inlet velocity profiles of the type inferred above do occur in the moderate to high Reynolds number flow of Newtonian fluids through sharp-edged entrances. Burke and Berman (1969) detected such velocity distributions experimentally. I n addition, recent finite-difference solutions of the present authors and of Christiansen, et al. (1971), indicate the presence of symmetrical maxima which become more pronounced with increasing Reynolds number and increasing p. Finally, for somewhat different reasons, profiles of this type are observed in the entrance flow solutions of Vrentas, et al. (1966), and Friedmann, et al. (1968), both of which are based on simplified models of a true entry region. Nomenclature
C,
D, J,(y)
R1 R2 r T,,, I; V z
= = = = = = = = = =
eigenfunction coefficient eigenfunction coefficient Bessel function of the first kind of order n radius of smaller tube radius of larger tube dimensionless radial distance quantity defined by eq 22 dimensionless axial velocity dimensionless radial velocity dimensionless axial distance
/3 = radius ratio defined by eq 14
4= w =
dimensionless stream function dimensionless vorticity
literature Cited
Burke, J. P., Berman, N. S., Paper 69-WA/FE-13, ASME Meeting, Los Angeles, Calif., Nov 1969. Christiansen, E. B., Kelsey, S. J., Carter, T. R., University of Utah, Salt Lake City, Utah, private communication, 1971. Friedmann, M., Gillis, J., Liron, N., A p p l . Sci. Res. A 19, 426 (1968). Goldstein, S., “Lectures on Fluid Mechanics,” Chapter 8, Interscience, New York, N. Y., 1960. Kantorovich, L. V Krylov, V. I., “Approximate Methods of Higher Analysis,” 4th ed, Chapter 1, Interscience, New York, N. Y., 1964. Van Dyke, M., J. F h i d Mech. 44, 813 (1970). Vrentas, J. S., Duda, J. L., Bargeron, K. G., A.Z.Ch.E. J . 12, 837 (1966). Wilson, S. D. R., J. Fluid Mech. 46, 787 (1971). Yih, C.-S., J. Fluid Mech. 5 , 36 (1959).
J. LARRY DLTDrl* Department of Chemical Engineering The Pennsylvania State University Cniversity Park, Pa. 16802 JAMES S. VREKTAS Department of Chemical Engineering Illinois Institute of Technology Chicago, Ill. 60616
GREEKLETTERS a, = zeros of eq 17
RECEIVED for review October 7 , 1971 ACCEPTED August 2, 1972
Simpler Solutions for Some Well-Known Boundary Value Problems Expressions for the effectiveness factor of a finite cylinder and rectangular parallelopiped presented previously in the literature as multiply summed infinite series are simplified b y summation over one index. In both cases the result suggests the separation of variables technique for the solution of the differential equations.
T h e important effect of pellet shape on the effectiveness factor 11 has been the motivation for solutions of the differential equation for diffusion with a n isothermal first-order reaction in several finite geometries. Expressions for the effectiveness factor of a finite cylinder (Aris, 1957) of radius ro and height a , and of a rectangular parallelopiped (Luss and Amundson, 1967) of sides a, b, and c have already been presented in the literature as a n infinite series summed over two and three indices, respectively. It is the purpose of this note to demonstrate that in each case one of the indices can be eliminated. For a cylinder with porous ends the effectiveness factor as given by Aris (1957) is qc(cylinder)
=
1-
1
32
-
n2 m = l
(2n
+ l ) 2 j m 2X X2r02
x2r02
+ j m 2 + (an + 1 ) 2 P 2 ~ 2( 1 )
where h 2 = k u / D is the product of u, the active catalyst surface per unit catalyst volume, and k , the reaction rate conqtant per unit active catalyst area, divided by the effective
diffusion coefficient D of the reactant within the particle, lm is the mth zero of the Bessel function J o ( a ) , and P = ro/a is the radius t o length ratio for the cylinder. On the other hand, for a rectangular parallelopiped Luss and Aniundson (1967) give the expression ar(rect.)
=
1
-
512
=
n6 n,m,p=O
x
where the aspect ratios p = a / b and 6 = a / c . Yumerical values for q from ( 1 ) and ( 2 ) are given by Luss and Xmundson. We introduce the expansion of the hyperbolic Bessel function in terms of the rootsj, m
11(a)’2alo(a) =
[a’ m=l
+
(3)
.7m21-’
Ind. Eng. Chem. Fundam., Vol. 11, No.
4, 1972
593
Note that the expansion holds in the limit gest that qe is more readily calculable as ac(cylinder)
=
16Xzr02 1 - __ a*
(Y
1
n = (an ~
+
1)2cun%
-+
0, and sug-
X
of variables technique; the separation of variables when applied to steady state problems always gives one summation index less than the number of dimensions. Indeed if C(r) the dimensionless concentration variable is taken to be unity on the boundaries the substitutions (Bird, et al., 1960) C(r)
an2 = ( 2 n
+ 1)2+P2 +
{
x2r02
On the other hand, the hyperbolic tangent can be expanded in terms of its argument as
=
+
v(r,x) w(z,y,z)
42)
+ s(x,y) +
(cylinder) u(z) (rectangular parallelopiped)
(7)
where
(8) and
b2s ax2
b2s + by2 - = xzs
(s(z,O) = s(x,b) = 1
- u(z);
and we offer in place of eq 2 the simpler form render the differential equation separable. literature Cited
CY,,,* = h2c2
+ ( 2 n + 1)2a26-* + (am+ 1)2~2p26-2
The functions introduced in (4)and (6) are easily obtained, and these summations can be evaluated more readily than the original expressions (1) and (2) for 9 . A summation over one index in (4) and two indices in (6) for the two-dimensional cylindrical and three-dimensional rectangular parallelopiped strongly suggests the separation
Ark. R.. Chem. Ena. Sci. 6. 262 (1957). Bird, R. B., SteGart, W.’ E., ‘Lightfoot, E. N., “Transport Phenomena,’’ pp 126-128, Wiley, New York, N. Y., 1960. Luss, D., Amundson, N. R., A.Z.Ch.E. J . 13, 759 (1967).
JAMES P. KO”* WILLIhhLI STRIEDER Department of Chemical Engineering University of 9otre Dame Xotre Dame, I n d . 46556 RECEIVED for review March 10, 1972 ACCEPTED August 7, 1972
Oxidation of Hydrogen Chloride with Molecular Oxygen in a Silent Electrical Discharge
+
+
The reaction 4HCI 0 2 --t 2H20 2C12 in a silent discharge was investigated. At atmospheric pressure yields up to 18% per cycle are obtainable. Energy yields are in the order of magnitude of 1.5 moles of Ch/ kWh. Results indicate that at lower pressures higher conversions and probably higher energy yields can b e obtained.
T h e promising aspects of low-temperature plasmas in both inorganic and organic preparative chemistry have been pointed out by several authors (Bradley, 1971; Cooper, 1968; Drake, 1972; Suhr, 1970). Such plasmas are obtained in a high-frequency microwave discharge a t reduced pressure, but also in a low-frequency silent discharge or in a glow discharge. The plasmas obtained in these ways have one characteristic feature in common: a very high ratio of electron temperature to gas temperature. This provides a means of performing high-temperature reactions uithout the need for a subsequent quench step. Cooper, et al. (1968), investigated the oxidation of hydrogen chloride in a microwave discharge at 13-27 mbars. They found the temperature of the gas mixture leaving the reactor to lie in the range 500-700°C. Though these temperatures 594
Ind. Eng. Chern. Fundarn., Vol. 1 1 , No. 4, 1972
are relatively low compared to the electron temperatures (over 10,OOO°C) they still have a considerable effect on the kinetics of the reaction. We investigated the same reaction in the plasma of a silent electrical discharge of atmospheric pressure in which the gas temperature never exceeded 4OOC. Experimental Section
The plasma was generated by a 50-H~voltage which was variable up to 20 kV. The discharge reactor was of a modified Siemens type and is described in detail elseiyhere (van Drumpt, 1972). The gap between the electrodes amounted to either 1.0 i 0.1 or 1.5 + 0.1 mm, the corresponding effective reactor volumes being 15.0 and 27.5 ml, respectively. Gases were commercial grade and were used without further purification. Flow rates were measured with calibrated flow
