Ind. Eng. Chem. Fundam.
C(B) is the value of S w less one for a Newtonian fluid at the same value of B; C(B)is obtained from Table I. The computed swell ratios for an Oldroyd fluid B are compared to eq 8.2 in Figure 13 for B = 0.25. Agreement is good up to SR of about 2, after which the analytical solution underestimates the actual swell. The deviation is quite large at S R = 4. The comparison is similar to that between the Tanner (1970) theory and the free extrudate swell calculations of Crochet and Keunings (1982b). Results for S R = 2 are shown in Figure 14 up to B = 1.25. There is a small difference, and the computed swell drops off more rapidly with force than eq 8.2, but the general trend is consistent. In view of the difference between the position of maximum swell and the position of velocity uniformity, and the inability to make an a priori estimate of the location of maximum swell, the use of eq 8.2 seems to offer no advantage over spinneret conditions as initial conditions for the thin filament equations. 9. Conclusion The elasticity levels that have been reached in these calculations are in a range that is relevant to polymer processing practice. The velocity and stress arrangement from spinneret to extensional flow always takes place in a small region, and the uncertainty about the location of the origin of the thin filament equations will not be im-
1983, 22,
355-357
355
portant on long spinlines. The maximum swell is adequately predicted by the White-Roman equation up to a recoverable shear of 2, but it is not evident that this equation provides any advantage over use of the spinneret area and velocity as initial conditions for the thin filament equations. The calculations generally support use of a zero initial condition for the ratio of transverse to axial extrastresses in the thin filament equations. Literature Cited Crochet, M. J. "The Flow of a Maxwell Fluid around a Sphere"; I n Gallagher, R. H., ed.; "Finlte Elements in Fluids IV"; Wlley: New York, 1982. Crochet, M. J.; Davles, A. R.; Walters, K. "Numerical Simulation of Non-Newtonlan Flow"; Elsevier: London, 1983. Crochet, M. J.; Keunings, R. J. Non-Newtonian F/u/d Mech. 1980, 7 , 199. Crochet, M. J.; Keunlngs, R. J . Non-Newtonkn Nu/d Mech. 1982a, 10, 85. Crochet, M. J.; Keunings, R. J . Non-NewtonianFluidMech. 1982b, 10, 339. Denn, M. M. Ann. Rev. F/u/dMech. 1980, 12, 365. Denn, M. M.; Petrie, C. J. S.; Avenas, P. AIChE J . 1975, 21, 791. Fisher, R. J.; Denn, M. M.; Tanner, R. I. Ind. Eng. Chem. Fundam. 1980, 19, 195. Gagon, D. K.; Denn, M. M. Polym. Eng. Sci. 1981, 27, 844. Nickeil. R. E.; Tanner, R. I.; Caswell, B. J. F/u/d Mech. 1974, 65, 189. Petrie, C. J. S. "Elongational Flows"; Pitman: London, 1979. Phan-Thien, N. J . Rheol. 1978, 22, 259. Phan-Thlen, N.; Tanner, R. I. J . Non-Newtonlan F/u/dMech. 1977, 2,353. Tanner, R. I. J . Polym. Sci. 1970, 8 , 2067. Whlte, J. L. Po/ym. Eng. Rev. 1962, 1, 297. Whlte, J. L.; Roman, J. F. J. Appl. Polym. Sci. 1976, 20, 1005. White, J. L.; Roman, J. F. J . Appl. Po/ym. Sci. 1977, 21, 869. Ziabicki, A. "Fundamentals of Fibre Formation": Wiley: New York, 1976.
Received for review August 18, 1982 Revised manuscript received March 16, 1983 Accepted March 28, 1983
COMMUNICATIONS Effect of Vapor Efflux from a Spherical Particle on Heat Transfer from a Hot Gas
The efflux of vapor from a particle reduces the heat transfer to the particle. This effect was evaluated for spherical and elongated (cylindrical) particles. One example of this phenomenon is the devolatilizationof small coal particles, for which previous investigators have used the Ackermann correction which was derived for flat-plate geometry. Computation shows that for the case of 74-pm coal particles in an entrained-flow gasifier, the flat-plate solution yields significantly greater values for this effect, and hence significantly lower rates of heat transfer, than the spherical-geometry solution.
Introduction The heat transfer rate to a surface is reduced by the presence of a mass flux from that surface. When the mass flux is independent of the heat transfer, this effect can be incorporated into the boundary conditions as the Ackermann correction to the heat transfer coefficient, that is
h = Ah-ho
(1)
where
0196-4313/83/1022-0355$01.50/0
For derivation of this solution, the reader is referred to Sherwood et al. (1975). One practical problem which exhibits this physics is the thermal decomposition of solids with the generation of gaseous products. Coal devolatilization is an example of such a decomposition. A number of investigators have employed the factor C,, or a factor derived from it, in their studies of the devolatilization of powdered coal in pyrolysis, gasification, or combustion processes. For example, it was used by James and Mills (1976) and Sprouse (1979) in its original form, by Ubhayakar et al. (1977) after assumption of the small-particle limit (Nu = 21, and by Coates and Glassett (1974) after further assumption of uniform first-order reaction within the particle. However, all the above authors employed eq 2 and 3, which were derived for a flat-plate geometry, for devolatilizing particles which were in all other respects considered spherical. 0 1983 American Chemical Society
356
Ind. Eng. Chem. Fundam., Vol. 22, No.
3, 1983
\,
\',
These solutions may find use, for example, in wood pyrolysis problems.
, /
Figure 1. Coordinate system for spherical devolatilizing particle.
The Effect of Volatiles Efflux in Spherical Geometry The correction factor was therefore re-derived for spherical geometry (See Figure 1). A heat balance yields
-"[ dr
r2(-Ag!$)]
~d( r * c p g N=T )0
-
Solution with the appropriate boundary conditions (T(r = R,) = TR, T(r = R,) = T J yields
T - TR -T, - TH
-
exp(-a/r) -exp(-a/R,) exp(-a/RJ -- exp(-a/RJ
Substitution of ho = A,/6. 2h&,/Ag yields
R, = R,
(5)
+ 6 , and Nuo =
Example Consider a 74-pm coal particle devolatilizing at 600 "C in an entrained-flow gasifier. The terminal velocity is approximately 11.8cm/s and Re = 0.092. Based upon eq 12 (Ranz and Marshall, 1952)
Nu, = 2.0
+ 0.6Re1f2Pr1f3
(12)
Nuo = 2.16, and Bio = Nuo(A,/2A,,d) = 0.27. From eq 7, the spherical Co (Cos) is 1/(1 + 2/2.16) or 52% of that predicted for the flat-plate solution. The Ackermann correction, hence the heat flux, can be calculated if the magnitude of the efflux can be estimated. Theoretical considerations suggest that a Peclet number of the order of unity is an upper limit for the volatile matter efflux (Kalson, 1981). Taking Peh = 1,the flat-plate solution (eq 2 and 3) yields Co = Peh/Bio = 1/0.27 = 3.70, and Ah = 0.094. The spherical-geometry solution (eq 7 and 8) yields COa= 0.52 (3.70) = 1.92 and AhS = 0.33. The calculated heat transfer rates differ by a factor of 0.33/ 0.094 = 3.5. Taking a vapor efflux one-tenth as great (Peh = O.l), the Ackermann corrections for the flat-plate and spherical geometries are 0.83 and 0.93, respectively. For powdered coal particles undergoing devolatilization, the reduction in heat transfer due to volatiles efflux thus may be significant. Moreover, it is overestimated (and the heat transfer rate underestimated) by use of the flat-plate solution. Acknowledgment
As the mass flux approaches zero, q (using ex approaches NRcPg( T , - TR)/ Cos, whence
-
1+ x )
This work was begun while the author was a Michigan Gas Association Fellow a t the University of Michigan. Completion of this work was made possible by award of the Lady Davis Post-Doctoral Fellowship a t the Technion-Israel Institute of Technology.
Nomenclature = Ackermann correction factor (defined by eq 1) Co = dimensionless parameter (heat transferred by bulk
Ah
This solution can be checked for its limiting values. At large R,, Nuo approaches w ; hence
which is the planar solution. For small R,, Nu, approaches 2; hence
The Effect of Volatiles Efflux in Cylindrical Geometry A similar analysis was also performed for cylindrical geometry; again a multiplicative correction factor arises which differs from the planar case T - TR
--
T- Ahr
TR
=
(r/RJCGC -1
-
(1 -I-2 / N ~ , ) ~ o - '1
CocIn (1
+-___ 2/Nu0)
+ 2 / N ~ o ) ~ c '11
[(l
-
(9)
flow/conductive heat flux) cp = heat capacity at constant pressure, J/(kg K) h = heat transfer coefficient, J/(m2 s K) N = mass flux, kg/(m2 s) g = heat flux, J/(m2s) r , R = radial position, m T = temperature, K u = velocity, mls 6 = boundary layer thickness, m h = thermal conductivity, J/(m s K) p = density, kg/m3
Subscripts g = gas R = solid surface 6 = edge of the boundary film 0 = condition of no mass efflux
ambient Superscripts c = cylindrical s = spherical m
=
Literature Cited (10)
Coates. R . M.; Giassett, J. M. "High-Rate High-Temperature Pyrolysis of Coal"; Final Technical Report to Bituminous Coal Research Inc., OCR Contract No. 14-32-001-1207 (1974). James, R. K.; Mills, A. F. Left. Heat M8SS Transfer 1976, 3 , 1.
357
Ind. Eng. Chem. Fundam. 1983,22, 357-358 Kalm, P. A. Ph.D. Dissertation, University of Michigan, Ann Arbor, MI, 1981. Ranr, W. E.; Marshall, R. E. Jr. Chem. fng. frog. 1952, 48, 173. sherwood. T. K.; Pigford, R. L.; Wllke, c. R. “Mass Transfer”; McGraw-Hill: New York, 1975; Chapter 7, pp 257-8. Sprouse, K. M. AIChE 87th Natlonai Meeting, Boston, MA, Aug 1979. Ubhayakar, S. K.; Stickler, D. B.; Gannon, R. E. Fuel 1977, 56(3), 281.
’ Faculty of Chemical Engineering, Technion-Israel Institute of Technolo-
gy, Haifa 32000, Israel.
Department of Chemical Engineering The University of Michigan Ann Arbor, Michigan 48109
Philip A. Kalson*
Received for review March 5 , 1982 Accepted February 28, 1983
Influence of Thermal Shocks on the Catalytic Activity of Palladium Pretreatment steps involving abrupt cooling increased for several hours the catalytic activity of palladium wires toward the oxidation of hydrogen. This “Superactivity” decayed slowly under reaction conditions and may disguise the intrinsic catalytic activity.
Introduction Measurements of the electrical power required to maintain a catalytic wire at a preset resistance (temperature) is a convenientmethod of determining the isothermal catalytic activity (Zuniga and Luss, 1978; Rajagopalan et al., 1980). When the wire is heated electrically abrupt changes in the temperature may occur. The purpose of this work is to study the influence of thermal shocks on the oxidation activity and resistance of palladium wires. Experimental System and Methods A. Reaction Rate Measurements. A 5 cm long and 0.005 cm diameter palladium wire (Goodfellow Metals, Cambridge, England) of 99.9% purity was placed in a flow reactor and maintained at a constant preset resistance and hence temperature by a constant-temperature anemometer (Thermo-Systems, Inc., Minneapolis, MN). The resistance of the catalytic wire can be set by the anemometer to within 0.01 R. Extra-dry grade oxygen and hydrogen and high-purity grade nitrogen from cylinders (Linde Inc.) were passed through activated charcoal beds, mixed, and dried by passing through two beds packed with glass beads and Drierite pellets, respectively. The dry, mixed gases were then fed to the reactor. In all the experiments nitrogen was used as the inert gas. The isothermal reaction rate was determined by measuring the electric power required to maintain the wire at a preset resistance with and without reaction. All reaction rate measurements reported here are normalized per unit geometric area of the catalyst. Details of the experimental system and instrumentation are described by Zuniga and Luss (1978). B. Resistance Measurements. The electrical resistance of the catalytic wire was measured by passing a small current ( w 100 mA) through it from a 6-V battery with a calibrated decade resistance box in series. The voltage drop across the catalyst and the known resistance were compared to compute the catalyst resistance. All the resistance measurements were made with the catalyst kept in a stream of nitrogen flowing at a velocity of 3 cm/s at room temperature. Hence, the reference temperature of the catalyst for all resistance measurements is the same. Estimated values of the heat transfer coefficient and the current through the wire were used to compute the reference temperature as 304 K. The exact temperature is not of concern since we only compare resistance values at the same standard temperature. The estimated time constant for cooling of the wire is 4 s. Resistance measurements were made 30 s after the electirc heating was stopped. This time was considered 0196-4313/83/1022-0357$01.50/0
sufficient for the catalyst to thermally equilibrate with the nitrogen stream. The voltage drop values were measured by a digital voltmeter (Non-Linear Systems Inc., Del Mar, CA). The estimated maximum error in the resistance measurement is 0.01 52. C. Activation of the Catalyst. The commercial cold drawn palladium wire exhibited immeasurably small activity under the conditions employed in this work (T,= 85 to 200 “C, H2= 0 to 2% v and excess 02). It was activated using the procedure described by Zuniga and Luss (1978). The activated catalytic wire was always kept in an atmosphere of flowing N2between experimental runs to prevent poisoning.
Results Heating the activated catalyst to 900 “C for 10 min in N2 and quenching it to room temperature increased the resistance at the reference temperature of 304 K by 0.1 9. Scanning electron micrographs showed that the heat treatment increased the average size of the crystalline grains on the Pd surface. In all the quenching experiments, cooling was by radiation and it lasted about 0.5 s. Thus, the quench rate was less than 1.8 X lo3 K/s. After quenching from 900 “C, the catalyst was left at room temperature in an atmosphere of N2. The resistance at the reference temperature dropped exponentially until all the residual resistance disappeared. The response of the resistance to quenching from different temperatures is shown in Figure 1. In all these cases the resistance of the catalyst before the heat treatment was 2.46 9 and the residual resistance due to quenching disappeared completely at room temperature. The durations of heating for the different temperatures are also specified in Figure 1. The processes that increase the resistance are slower at temperatures lower than 900 “C. Hence the catalyst was kept for a longer time at the lower temperatures before quenching. The transient activity of the catalyst toward H2 oxidation could be influenced by its thermal history. Quenching from a high temperature to the reaction temperature caused a temporary activity enhancement, hereafter referred to as transient superactivity. The higher activity disappeared in about 2 h under reaction conditions. The influence of the thermal history on the catalytic activity is shown in Figure 2. The superactivity due to the quenching from 900 “C and the subsequent deactivation under reaction conditions occurred for all reactant compositions in the range studied, i.e., O2= 10 to 40% v and H2 = 0 to 1% V. 0 1983 American Chemical Society
