Ind. Eng. Chem. Res. 1987,26, 1048-1050
1048
Anwar, M. M.; Hanson, C.; Pratt, M. W. T. Trans. Inst. Chem. Eng. 1971a, 49, 95. Anwar, M. M.; Hanson, C.; Pratt, M. W. T. Proc. Znt. Soluent Extr. Conf., 1971 1971b, 2, 911. Gaikar, V. G.; Sharma, M. M. J. Sep. Process Technol. 1984a, 5,45. Gaikar, V. G.; Sharma, M. M. J.Sep. Process Technol. 1984b,5,53. Gaikar, V. G.; Sharma, M. M. Solvent Extr. Ion Exchange 1985,3(5), 679. Jagirdar, G. C.; Sharma, M. M. J . Sep. Process Technol. 1980,1(2), 40. Jagirdar, G. C.; Sharma, M. M. J. Sep. Process Technol. 1981a, 2(3), 37. Jagirdar, G. C.; Sharma, M. M. J. Sep. Process Technol. 1981b, 2(4), 7. Vasermans, H.; Hillers, S.;Avots, A. Latu. PSR Zinat., Akad. Vestis, Kim. Ser. 1958, 5, 79; Chem. Abstr. 1959, 53, 13349.
Wadekar, V. V.; Sharma, M. M. J.Sep. Process Technol. 1981a, 2(1), 1. Wadekar, V. V.; Sharma, M. M. J.Sep. Process Technol. 1981b, 2(2), 28. Wadekar, V. V.; Sharma, M. M. J . Chem. Technol. Biotechnol. 1981c, 31, 279. Vilae G. Gaikar, Man Mohan Sharma* Department of Chemical Technology University of Bombay Matunga, Bombay 400 019, India Received for review December 23, 1985 Revised manuscript received September 5, 1986 Accepted December 15, 1986
An Expanding Core Model for a Heterogeneous, Noncatalytic, Gas-Solid Reaction An analysis is presented for a gas-solid heterogeneous, noncatalytic reaction which propagates from the center. The results are compared with those for a shrinking-core model. The analysis reveals that the reaction rate.can exhibit a maximum or a discontinuity. For a first-order reaction, the moving reaction interface is always stable toward a one-dimensional perturbation. The shrinking-core model has been frequently used in modeling gas-solid noncatalytic reactions. While most of the past work has dealt with a simple first-order reaction, it has been shown that a number of noncatalytic systems may follow the Langmuir-Hinshelwood form of kinetics (Kurosawa et al., 1970; Sohn and Szekely, 1973). Recently, Erk and Dudukovich (1983, 1984) applied this form of reaction kinetics to both shrinking-core and volume-reaction models. They found that multiple reaction pathways are possible. To our knowledge, no attempts have been made to study the behavior of noncatalytic systems with the reaction starting from the center. The expanding reaction interface is important for the underground coal gasification problem, combustion of solid propellants, and synthesis of ceramic materials by the self-propagating high-temperature method.
Model Equations A noncatalytic heterogeneous reaction of the general form aA(g) + W s ) sS(s) + gG(g) (1) is considered to occur in a porous solid cylinder. The assumptions of Erk and Dudukovich (1984) hold good for the present case analyzed in this work. The balance equations for the expanding interface model can be written in a dimensionless form as
-
x = xo
dY
-=
dx
--E&
-y)
(3)
x = x,2 - xo2
(7) and the corresponding rate of change of conversion is given by dXC -dX- - 22: d8 de When eq 2 is integrated with boundary conditions (eq 3 and 4), the gas concentration at the reaction interface, yo can be related to the position of the reaction interface, xc,
(1 - Y c F1 =
where y is a function of x , and is given by
As discussed by Erk and Dudukovich (1984), multiple solutions for eq 9 are possible whenever Ybl
_ dXC -d8
-sc
(1
+ KyC)'
~ b 1 , 2=
xc=xo The particle conversion can be expressed as 8=0
(6)
0888-5885/87/2626-1048$01.50/0
1 -[P + 20K - 8 f ( K ( K - 8)3)'/2]
(11) 8 It is also evident from eq 11that multiple solutions are not possible for K I8 (Perlmutter, 1972). The present work aims at investigating the behavior of the system with the reaction initiated at the center and analyzing the differences with the shrinking-core model. Equation 9 can be differentiated, and after combination with eq 10 and 5, we have
r (5)
I?' IYb2
where
Y
x = x,
Yc
(1 + KYJ'
Y
d6'
X.
1
2Ks: - Ky,
11
+1
with 6' = 0
yc = O(t) Yc = 1
0 1987 American Chemical Society
Y.
\3
Ind. Eng. Chem. Res., Vol. 26, No. 5, 1987 1049 Y
'b2
'bl X
X
X
U X
Figure 1. Variation of
X
y
X
e
during reaction for expansion interface.
Simultaneous integration of eq 12 and 5 will determine the position of the reaction interface, x,, and the corresponding gas concentration, y,, at any instant. Equation 12 is similar in form to the equation obtained by Erk and Dudukovich (1984). A discontinuity occurs when
Figure 2. Variation of reaction interface and gas concentration during reaction for shrinking-core model ( K = 20, Bi, = 10,Da = 400). 0
0
.? CI
yc 0
?
'0
(13) 0
,?
When the jump occurs, the new value of y, has to be provided by solving eq 9 to facilitate continuation of integration of eq 12.
0
0
Results and Discussion For a particle of a cylindrical or spherical geometry, the earlier analysis of Erk and Dudukovich (1984) showed for Bi, > 1 that the parameter, y, can increase and then decrease during the course of the reaction. For the expanding reaction interface, eq 10 relates y with the reaction interface position, x,. It can be easily shown that, for all values of Bi,, y always increases monotonically as the reaction proceeds. Consequently, the possible variations of y during the reaction which are different for the expanding reaction interface are shown in Figure 1. Unique solutions are possible only for the cases represented by parts a and b of Figure 1. The other situations, shown in parts c-f of Figure 1, lead to multiple pathways. Consider the situation given in Figure Id. It would be instructive to track the events occurring during the reaction. With the progress of reaction, y increases with x , and yb.1 is reached when x , = xC1. Multiple solutions are now possible, but y, is still higher than Y,,~, and yc will continue to decrease smoothly. When x , , ~i s reached, y b attains yb,Z exactly and y, attains the value of Y,,~. A sharp jump now occurs for smaller values of yc. The reaction interface is now in the region of an unique reaction pathway, and y c continues to decrease smoothly to the end of the reaction, without any further jump. When the reaction is initiated with y, = O(4, y, values will always decrease smoothly and will not have any discontinuity a t all if the reaction is unperturbed. Thus, it is clear that for the expanding interface with the self-inhibited rate form, y can enter the multiple solution region only once. As a direct consequence, the gas
f
'0
0
o! 0
0 0 '
0
9 I
I
I
I
120
240
380
480
0
00 E
Figure 3. Variation of reaction interface and gas concentration during reaction for expansion interface (K = 20, Bi, = 10,Da = 400).
concentration can experience only one jump, whereas for the shrinking-core model two jumps can occur. Figure 2 shows gas concentration and reaction interface profiles for a particle of cylindrical geometry and for the case considered by Erk and Dudukovich (1984). Figure 3 displays the corresponding profiles for the same set of parameters for the expanding interface. When the reaction is started with yc = 1, then the only jump occurs at yc = 0.444, as in Figure 3. When the reaction is started with y, = 0, there is no jump in the gas concentration. Figures 4 and 5 show the dependence of the conversion on the dimensionless time.
Geometric Instability It has been shown earlier that geometrical instability can occur for first-order reactions with particles of cylindrical and spherical geometry (Beveridge and Goldie, 1968; Ishida and Wen, 1968). This can occur if, with progress of the
1050 Ind. Eng. Chem. Res., Vol. 26, No. 5, 1987
IO2
The quantity dR,/dx, will always be negative. This brings us to an important conclusion that, for cylindrical geometry and a first-order reaction, geometrical instability can never occur if the reaction interface is propagating from inside.
e
Figure 4. Variation of reaction rate and solid conversion with time for-shrinking-core model ( K = 20, Bi, = 10,Da = 400). 0
3.0 103
2 a4
1.8
1.2
0 .e
: y , , 0
1
~,
0
120
240
360
480
e
0 600
Figure 5. Variation reaction rate and solid conversion with time for expansion interface ( K = 20, Bi, = 10, Da = 400).
reaction, the reaction rate increases. The criterion for instability is
dRa>o dXC For the expanding interface we have
Equation 14 indicates that geometrical instability can occur. The instability of the reaction interface can be avoided only for the cases of parts a and b of Figure 1. Interestingly, for a first-order reaction, the quantity dR,/dx, is given by
Conclusions With self-inhibited rate forms, solid particles may exhibit multiple reaction pathways for both shrinking and expanding reaction-interface models. The former can exhibit two discontinuities, whereas only one discontinuity may occur for expanding interphase situation. This is due to the fact that the parameter, y, can increase and then decrease for the shrinking-core model, while for the expanding interface this parameter increases monotonically as the reaction proceeds. For a cylindrical geometry with self-inhibited rate forms, both shrinking and expanding models predict geometrical instability at certain levels of conversion. For a first-order reaction, geometrical instability does not exist for the expanding interface. Nomenclature a , b, g , s = stoichiometric coefficients Bi, = Biot number for mass transfer CA = gas reactant concentration CAb= gas reactant concentration in the bulk CBo= initial solid reactant concentration De = effective diffusivity in solid product layer Da = k,K/De = Damkohler number K = k,CAo = dimensionless inhibition constant k, = mass-transfer coefficient k , = surface reaction rate constant L = half thickness of the solid particle R = position of the solid particle measured from center line R, = reaction rate x = R / L = dimensionless position in the solid particle X =: solid reactant conversion y = C A / C A b = dimensionless gas concentration Greek Symbols y = parameter defined by eq 10 6' = aCB&/bk,cAo = dimensionless time Subscripts b = bifurcation values c = at the reaction interface 0 = initial position
Literature Cited Beveridge, G. S. G.; Goldie, P. J. Chem. Eng. Sci. 1968, 23, 913. Erk, H. F.; Dudukovich, M. P. Ind. Eng. Chem. Fundam. 1983,22, 55. Erk, H. F.; Dudukovich, M. P. Ind. Eng. Chem. Fundam. 1984,23, 49. Ishida, M.; Wen, C. Y. Chem. Eng. Sci. 1968, 23, 125. Kurosawa, T.; Hasegawa, R.; Yagihashi, T. Nippon Kinzolu G a k kaishi 1970,34, 481. Perlmutter, D. D. Stability of Chemical Reactors; Prentice Hall: Englewood Cliffs, NJ, 1972; p 19. Sohn, H. Y.; Szekely, J. Chem. Eng. Sci. 1973, 28, 1169.
V. K. J a y a r a m a n , V. Hl@vacek,*J. P u s z y n s k i Department of Chemical Engineering State University of New York at Buffalo Amherst Campus Buffalo, New York 14260 Received for review February 3, 1986 Accepted January 21, 1987
