Ind. Eng. Chem. Res. 1987, 26, 1262-1264
1262
Nomenclature B = column bottoms flow rate D = column distillate flow rate F = reactor effluent flow rate F,, = fresh feed flow rate k = dimensionless reaction constant k l = reaction constant L = column reflux flow rate MIMO = multiinput, multioutput RGA = Relative Gain Array SISO = single-input, single-output V = column vapor flow rate V , = reactor volume xD = distillate mole fraction X F = column feed mole fraction xw = bottoms mole fraction x o = fresh feed mole fraction Greek Symbols a = relative volatility X = relative gain p = reaction mixture molar density
Bristol, E. H. IEEE Trans. Autom. Control 1966, AC-11, 133. Cohen, G. H.; Coon, G. A. Trans. ASME 1953, 75, 827. Georgakis, C.; Papadourakis, A. Chem. Eng. Sci. 1982, 37, 1585. Grosdidier, P.; Morari, M.; Holt, B. R. Ind. Eng. Chem. Fundam. 1985, 24, 221. Jafarey, A.; McAvoy, T. J.; Douglas, J. M. Ind. Eng. Chem. Fundam. 1979, 18, 181. Lau, H.; Alvarez, J.; Jensen, K. F. AICHE J . 1985, 31, 427. McAvoy, T. Interaction Analysis: Principles and Applications; Instrument Society of America: Research Triangle Park, NC, 1983. Papadourakis, A. Ph.D. Dissertation, University of Massachusetts, Amherst, 1985. Shinskey, F. G. Distillation Control, 2nd ed.; McGraw-Hill: New York, 1979. Shinskey, F. G. Hydrocarbon Process. 1981, 60, 196. Stanley, G.; Marino-Galarraga, M.; McAvoy, T. J. Ind. Eng. Chem. Process Des. Deu. 1985, 24, 1181.
Antonis Papadourakis, Michael F, Doherty James M. Douglas* Department of Chemical Engineering Goessmann Laboratory University o f Massachusetts Amherst, Massachusetts 01003
Literature Cited Boyle, T. J. Ph.D. Dissertation, Massachusetts Institute of Tech. nology, Cambridge, 1963.
Received for review August 16, 1985 Accepted March 16, 1987
Analysis of Gas-Solid Reactions with Initially Nonporous Solid Using Dusty Gas Model T h e dusty gas model is applied t o the theoretical analysis of gas-solid reactions with initially nonporous solid. The models are formulated in general terms, and the solutions are given partially in analytical form. T h e results are compared with those obtained by Sohn and Sohn. Sohn and Sohn (1980) studied the effect of bulk flow due to volume change in the gas phase on the rate of gas-solid reaction. The so-called shrinking unreacted core model (solid initially nonporous) and standard single reaction scheme was used there, and the solutions were given partially in analytical form. They assumed that the product layer was sufficiently permeable and the mass transport process occurred in the regime of bulk diffusion control. However, for the product layer with moderate pores, their solutions are no longer valid, and therefore, a more general model which describes multicomponent diffusion in the intermediate regime between Knudsen and molecular diffusion should be applied. This paper is concerned with such a case. The dusty gas model is used here to describe the mass transport phenomena in the “ash” layer. The analysis is performed in general case. We will deal with a gas-solid reaction of the form uAA(g) + VBWS) vcC(g) + U D D ~ ) taking place in a solid pellet of symmetrical shape (flat plate, infinite cylinder, sphere). The simplifying assumptions are summarized as follows: ( 1 ) The solid is initially nonporous. (2) The pseudosteady-state approximation is valid. ( 3 ) The system is isothermal. ( 4 )The solid retains its original shape and size during the course of chemical reaction. (5) The physical properties of “ash” remain constant during the process. (6) The pressure changes inside the product layer can be neglected. On the basis of Sohn and S o h (1980), Kaza and Jackson (1980), Burghardt and Patzek (1983), and Skrzypek et al. (1984, 1985),the following model may be derived to describe mass transport phenomena in the product layer of
-
0888-5885/87/2626-1262$01.50/0
solid pellet (in dimensionless form): -dA= o dz
where f ( x ) is functional expression characterizing the mass transport phenomena as shown in Table I. It should be noted that because of further algebraical transformation, the function f ( x ) was presented as a five-coefficient function (general case). However, in fact, it only depends on four unknown coefficients, vc, K,, 8, and
P.
The boundary conditions are derived for the four following cases, depending on the relative values of the resistance of diffusion and chemical reaction. (a) The influence of external diffusion is significant. 1dx = Bi(1 - x) at z = 1 (3) f ( x ) dz at z = z,
(4)
(b) The external diffusion resistance is negligible. x = l at z = 1 (5)
-
(c) 9’ negligible.
~0
and the external diffusion resistance is x = l
at z = 1
(7)
x=O
at z = z c
(8)
0 1987 American Chemical Society
Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987 1263 Table I. Coefficients a-e of the Function f ( x )
f ( x )=
a
+ bx + ex2 c
+ dx
a=ZzZ
f(x) = a
b=-
+ bx
/3
-
m
general case
c = 1+ 8
a = -vAZZZ
and after elimination of Cz
+ /3K2
1 (Zd - Z1(Kl - K,)) 1+8
(c,
( z , ' - ~ - 1) (18) l-m Hence, to find X vs. t*, we must solve the algebraic equation 18 with respect to C1 for each integration step of differential equation (11). When conditions 3 and 8 are applicable, eq 18 reduces to
d as above)
(case considered by Sohn and Sohn, 1980)
For the more general case as r = r ( x ) , it is better first to eliminate C1and then to solve the resulting equation with respect to x,. In this case, the following equations may be used: q(x,) - q
-
(d) (P2 negligible.
0 and the external diffusion resistance is
x = 1 at z = 1 (9) All cases (a-d) can be analytically treated by using the method similar to that of Sohn and Sohn (1980). Solving eq 1 and inserting the result in eq 2, we obtain dx 1 = -CcJ(x) dz ~m
( Zi) 1- -
C1 = 2(m
=
---(Z;-m C1
- 1)
1-m
+ 1)z,m@2r(xc)
(20) (21)
Case b. Making use of eq 5 , 6, and 15, the following
expression is obtained:
where C1 is expressed by eq 21. In the case where r ( x ) = x n , eq 22 takes the form
The relation between overall conversion of solid reactant
B, X, and dimensionless time, t*,has the form (Sohn and Sohn, 1980)
This equation can easily be derived by using (1) the equation expressing the rate of consumption of the solid. dz, dt* (2) the well-known relation between X and z,, x = 1 - z,m+l
(23) So to determine the relation between X and t*, one must solve numerically the differential equation (ll),where C1 is calculated from eq 21 and 22 or 23, respectively. Case c. Here the solution of eq 10 with conditions 7 and 8 has the form
C1
- = -r(x,)
q(0) - q(1) = -l( Z-c m l-m
- 1)
Using this equation and eq 11,we can obtain (13)
and (3) condition 4 where eq 10 was applied, and after integration ( z , = 1 (m Iq(1) - q(O)lt+ = 1 -
Solving eq 10, we can get
,-.
where q ( x ) = J dx/f(x). Case a. For z = 1 (x = x,, x, is still unknown), eq 15 can be combined with condition 3 to give
For z = z,, combining eq 4 and 15 and assuming that r ( x ) = x", we obtain
+ 1)(1 - X)z/(m+l)- 2(1 - X) m-1
(26) Case d. When the Thiele modulus approaches zero, it is evident that x , approaches 1. We then find that t* = 1 - (1 - X)l/(m+l) (27) From eq 28 it is seen that the dimensionless time, t*, does not depend on the function f ( x ) . Figures 1 and 2 show overall conversion of component B, X, vs. dimensionless time, t*, for various vc and K2. The dashed lines in Figures 1 and 2 were obtained by using eq 11 and 23, where f ( x ) = a + bx ( p m), n = 1 (first-order reaction), m = 2 (sphere), and = 1. The dashed line in Figure 1was obtained for uc = 2 (see Figure 2 in the paper by Sohn and Sohn (1980)) and in Figure 2 for vc = 3.
-
1264 Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987 kA
_----Kf0.5 K2= 2
5' 4
*
I
0.2
0.4
0,6
0.8
10
1.2
1.4
1.6
1,6
2.0
t*
Figure 1. Overall conversion of solid component B, X,vs. dimensionless time, t*, for vc = 2.
corporating chemical reaction, mass transport phenomena in the "ash" layer and external mass transfer were treated here by a more general and accurate method.
Acknowledgment The financial support of a scholarship for M. Grzesik from the Ministry of Education, Science and Culture of Japan is gratefully acknowledged. Nomenclature a = characteristic size of solid pellet, m a-e = coefficients in the function f ( x ) Bo = permeability factor, m2 Bi = Biot number C , = total concentration, mol/m3 C,, C, = integration constants D = DiDiz/(Di.+ Did D,, D , = effective Knudsen diffusion coefficients, mz/s D12= effective binary pair bulk diffusion coefficient, mz/s f ( n ) = auxiliary function characterizing the mass transport phenomena K,-= D,/D, = 1 K 2 = D1/Dz = (M2/M1)1/2 1 = space coordinate, m m = 0, 1, or 2 for slab, long cylinder, or sphere, respectively M = molecular weight, kg/mol N = molar flux, mol/(m2 s) P = pressure, N/m2 q ( x ) = J dx/f(x) = auxiliary function r ( x ) = R ( x ) / R ( l )= dimensionless reaction rate R ( x ) = reaction rate, mol/(m2 s) t* = ~ g R ( l ) t / ( ~ p g = ~ Adimensionless ) time t = time, s x = mole fraction of component A X = overall conversion of component B z = l / a = dimensionless coordinate Greek Symbols
P = B80I(rD)
a, = D,,)Dl2 = 1 0 = Dl/D12 X = lmN,/(C$VAam-') = viscosity, N s/m2 v = v'/IV/AI U' = stoichiometric coefficient p B = number of moles of B per unit volume of the solid, p
mol/m3 (aR(1)/(2(m + 1)C@))1/2 = modified Thiele modulus
@ =
Subscripts
,
.
.
.
.
0,2
0,4
0.6
0.8
1.0
. 1,2
. 1.4
. 1.6
. 1.8
. 20
t*
Figure 2. Overall conversion of solid component B, X,vs. dimensionless time, t*, for uc = 3.
The full lines were obtained with the aid of the same equations as dashed lines and general function f ( x ) = ( a + bx + ex2)/(c d x ) for the following values of coefficients: Figures 1 and 2, n = 1, m = 2, @ = 1, B = 0.01, @ = 1, K 2 = 0.5, 2, and 4; Figure 1, uc = 2; and Figure 2, uc = 3. From Figures 1 and 2, it is seen that the values of overall conversion, X,represented by full lines differ up to 25% from respective values of conversion represented by dashed lines. Furthermore, these deviations only slightly depend on the value of uc. In summation, it should be pointed out that eq 23 and 26 have, respectively, similar forms to eq 18 and 26 in the paper by Sohn and Sohn (1980). However, they were derived there under assumption that the process takes place in the regime of bulk diffusion control ( f ( x ) = a + b x ) , so they can only be regarded as the particular cases of general solutions (23) and (26). Besides, the cases in-
+
A-D = refer to the chemical compounds c = refers to the reaction interface 0 = refers to the bulk conditions s = refers to the surface of the solid pellet 1, 2 = refer to the gaseous components
Literature Cited Burghardt, A.; Patzek, T. W. Znt. Chem. Eng. 1983, 23, 739. Kaza, K. R.; Jackson, R. Chem. Eng. Sci. 1980,35,1179. Skrzypek, J.; Grzesik, M.; Szopa, R. Chem. Eng. Sci. 1984,39, 515. Skrzypek, J.; Grzesik, M.; Szopa, R. Chem. Eng. Sci. 1985, 40, 671. Sohn, H. Y.; Sohn, H. J. Ind. Eng. Chem. Process Des. Deu. 1980, 19, 237.
Miroslaw Grzesik* Faculty of Mechanical Engineering Cracow Technical University Cracow, Poland Eiichi Kunugita Department of Chemical Engineering Osaka University Toyonaka, Japan Received for review February 13, 1985 Accepted February 9, 1987
