Ind. Eng. Chem. Res. 1991,30,430-434
430
Bi
= Biot number C = NO gas-phase concentration, mol/cm3 D = effective diffusion coefficient, cm2/s d = pore diameter, cm F = volumetric gas flow rate, cm3/s k = external mass transfer coefficient, cm/s L = length of the channel, cm S = catalyst internal surface area, cm2/cm3 x = axial coordinate, cm W = catalyst wall thickness, cm
catalyst
0
x
x+dx
L
__. Figure 3. Diffusion cell schematic.
Greek Symbols
specific molar flux, mol/(sec-cm2) = porosity, cm3/cm3
@ = t 41
Subscripts m = micropore M = macropore 1 = inside of the channel 2 = outside of the channel
01 50
100
150
i
200
Superscripts b = bulk gas phase s = interface e = exit o = inlet
Registry No. NH3, 7664-41-7; NO, 10102-43-9; T i 0 2 , 13463Flow rate (cm3/rec at STP)
Figure 4. Specific molar flux dependence on gas flow rate.
The experimental technique presented in this Research Note can be extended for high-temperature measurements using an adequate sealant material between the catalyst and the stainless steel tubing and inserting the entire diffusion cell in a temperature-controlled furnace. Effective diffusion coefficients for other support materials besides titania can also be determined with this technique as long as they can be extruded as uniform straight hollow bodies. Acknowledgment
67-7.
Literature Cited Hegedus, L. L. Catalyst Pore Structures by Constrained Nonlinear Optimization. Znd. Eng. Chem. Prod. Res. Deo. 1980,19,533-537. Satterfield, C. N. Mass Transfer in Heterogeneous Catalysis; Krieger: Malabar, FL, 1970; Chapter 1. Wakao. N.: Smith, J. M. Diffusion in Catalvst Pellets. Chem. Eng. Sci. '1962, 17, 825. Wicke, E.; Kallenbach, R. Die Oberflachendiffusion von Kohlendioxyd in Aktiven Kohlen. Kolloid 2. 1941, 97, 135-151.
Jean W.Beeckman
I thank Ronald J. Majeran for setting up the diffusion cell and performing the experiments.
Research Division W . R. Grace & Company-Conn. 7379 Route 32 Columbia, Maryland 21044
Nomenclature A = specific geometric area of the catalyst channel, cmz/cm
Received for review February 21, 1990 Accepted October 8, 1990
Steady-State Cubic Autocatalysis in an Isothermal Stirred Tank In this paper, we demonstrate how the recently proposed procedure for global steady-state multiplicity analysis by Balakotaiah and Luss may be used to map regions in parameter space having a specific number of feasible solutions. The system chosen is a generalized scheme of autocatalytic reaction taking place in an isothermal CSTR. Both reversible and irreversible cases were considered. For each case, two possible forms of kinetic rate expressions leading to cubic steady-state manifold equations were examined. The influence of the Damkohler number (residence time and feed concentration) and stoichiometry of the autocatalyst have been studied. While stoichiometry may influence the steady-state solution pattern of the irreversible autocatalytic reaction, it was found that the behavior of the reversible reaction is independent of the stoichiometric coefficient of the autocatalyst. Introduction The genesis of the investigation of chemically reacting systems characterized by steady-state multiplicity is probably found in Aris and Amundson (1958). Further research has since shown that what would have easily passed for mere experimental artifacts are indeed real 0888-5885/91/2630-0430$02.50/0
phenomena not uncommon in the analysis of nonlinear systems. It is now becoming increasingly acceptable that not only does preliminary analysis of steady-state multiplicity serve as a useful guide in delineating regions of interest for experimental investigation, but also the associated question of the stability of such solutions is of equal
0 1991 American Chemical Society
Ind. Eng. Chem. Res., Vol. 30, No. 2, 1991 431 importance in the optimal reactor design (Morbidelli et al., 1987). Luss and co-workers (Balakotaiah and Luss, 1988, 1984, 1983, 1982a,b, 1981; Leib and Luss, 1981) among others have made signal contributions in this respect. Balakotaiah and Luss (1988, 1984) developed a procedure for steady-state multiplicity analysis based on singularity theory. With this method, it is possible to (1) determine the regions in parameter space having a specific number of feasible solutions and (2) map regions with different types of bifurcation diagrams without the tedium of an extensive parametric study. These features should naturally arouse the interest of experimentalisits since they may be used in selecting the range of operating variables and probably in locating regions of hysteresis for some systems. In order to demonstrate the utility of this procedure, it is applied to the analysis of autocatalytic reactions occurring in an isothermal CSTR. Autocatalysis is a common phenomenon in many chemical reaction systems, especially in organic syntheses and biochemical processes. Experimental observations of hysteresis have also been reported in some cases, hence the rationale for the choice of our reaction scheme. Additionally, Professor Gray's team at Leeds (Gray and Scott, l984,1983a,b) and Scott et al. (Kaas-Peterson et al., 1989; Kay et al., 1989; 1988) have also studied the nature of the oscillations and the stability to small perturbations for reactions of this type (item 2 above). In a sense, these contributions provided the impetus for the present work. Here, however, we seek to map, in parameter space, regions having a specific number of feasible solutions. The reactions of interest and the relevant rate expressions are irreversible reaction A + nB ( n 1)B n = 1,2 (1)
-
+
with two possible forms of rate equations and r = k1CA2CB
(3)
reversible reaction A+nB+(n+l)B n=l,2 with two possible forms of rate expressions
(4)
and
r = k1CA2CB - k-,CB3
(6)
Equations 2 and 5 have been frequently used to describe cubic kinetics for reactions 1 and 4, respectively. While this may suggest to the casual reader that orders of reaction and stoichiometric coefficients have some correlation, in general this is not so unless the reaction step is taken to be an elementary one. Since the reactions under investigation are not necessarily elementary, we have introduced the companion rate expressions, eqs 3 and 6 {to eqs 2 and 5, respectively],which are important and equally admissible cubic rate laws from a kinetic viewpoint, to provide generality of discussion in the forthcoming analysis.
Theory Consider the system represented by the single-variable nonlinear equation F(x,p) = 0 (7) where n is the unknown (state variable) and p is the n-
length vector of parameters such that XI I x I xu
(84
and (8b) Pi1 5 Pi 5 ~ i u are constraints often encountered in practical situations. In what follows, eq 7 is referred to as the steady-state manifold. According to Thom's theorem, singular points of codimension k exist at the point (xo,po)when
is satisfied. Suppose g(x) is an Mth-order polynomial with N distinct roots, xl, x2, ..., XN having multiplicity ml, m2, ..., mN, respectively, it is easily shown that at the roots gb1) =
= o; dx -- ... -- dm1-'gW dxml-l M X 1 )
d"g(x1) dY'"
# O
(loa) g(xJ =
&(xi) -- ... -- d"-'g(x2) dx dxmz-1
dg(XN) g(xN) = -dx
- d""g(xN) "*
dn"l
=
o;
= 0;
dm2g(x,) # O dxm2 (lob) d"g(xN) dXN"
# O
(10c) Catastrophe theory allows the conditions for singular points given by eqs 10 to be applied to any arbitrary nonlinear F(x,p),provided (1)it is a single-valued function of x , (2) it is smooth with respect to x and p, (3) it is nonvanishing at x1and xu, and (4) both F ( x , p ) and dF/dx do not simultaneously vanish on a boundary of the feasible parameter space for any x within the feasible region. Pritzker and Fahidy (1989) have discussed the implications of these conditions, especially 3 and 4 which are often violated in bounded systems. In the notation of Balakotaiah and Luss (1988), SkM(ml,m2,...,mN)denotes the set of parameter points corresponding to a particular type of singularity where M is the total number of solutions and k is the codimension of the singular point. In anticipation of the analysis for a cubic autocatalytic system, let us consider the case where M is 3 and p is a vector of two elements p1and p 2 . Clearly, the steady-state manifold may be written in expanded form as
F ( x ) = x 3 + Bx2 + Cx + D = 0 (11) where B, C, and D bear simple relations to the system parameters p . Analytical advantages accrue if eq 11 is transformed to the well-known standard form
Fb,P) = Y 3 + p l y + p2 = 0
(12)
where y =x
+ B/3
(134
p1 = (1/3)(3C - B 2 )
p 2 = (1/27)(27D - 9BC
+ 2B3)
(1%)
(13~)
Indeed, the transformation to eq 12 ensures that the number of parameters in the vector p is the same as the codimension of highest singularity, i.e., 2. With the use of eq 11 in eqs 10, and after considerable algebraic manipulations, the relations between the coefficients B, C, and D (and invariably the parameters p1and p 2 in eqs 13)
432 Ind. Eng. Chem. Res., Vol. 30, No. 2, 1991 Table I. Relations between B , C, and D (cf. Eq 11) for Singular Points of Codimension 2 singular points relations S3(0,0,3) C = B2/3, D = B3/27, triple root a t x = -B/3 C = B2/4, D = 0, double root at x = -B/2 S3(1,0,2, C = -B2/4 - B/2 - 314, D = -B2/4 - B/2 SH(0,1,2) 1/4, double root a t x = -1/[2(B + I)] S3(2,0,1), S$2,0,0) C = 0, D = 0, single root a t x = -B S!(0,2,1), S2(0,2,0) C = -2B - 3, D = B + 2, single root a t x = -B - 2 Si(l,l,l), Si(1,1,0) C = -B - 1, D = 0, single root a t x = -B - 1
b
0 2.51,;
1 SOLUTKWI
0
2
Table 11. Relations between B , C, and D (cf. Eq 11) for Singular Points of Codimension 1 singular mints relationso Si(0,0,2,1),S7(0,0,2,0) (2B3- 9BC + 27D)2 = 4(B2 - 3C)3, double root a t x = (B2/9- C/3)1/2- B/3 Si(l,O,I,l),S~(l,O,l,O),(2B2/9) - C = (W + B/3)2 + B/3(w + R / 3 ) , 0 5 w 5 1, D = 0, single roots a t st (1,O,O,O) x=wandx=-B-w Si(O,1,1,1),S~(o,l,l,O),R + C + D + 1 = 0, single roots at x = w and x = 1 - B - w s:(o,l,O,o) is the root of the equation w 2 + (B + l ) w 0 2 ( I ' 2 1.
+ (B + C + 1) = 0,
for singular points of codimension 1and 2 may be derived and summarized as shown in Tables I and 11. Results and Discussion Irreversible Reaction. Case i: Rate Expression r = klCACB2. In an isothermal CSTR, the steady-state mass balance yields vk1cAcB2-
q(CAo - c,) = 0
(14)
where V and q are reactor volume and fluid volumetric flow rate, respectively. For a constant-density system, the concentrations of species A and B (CA and CB) are related to the conversion x by
CA
c&(1- x)
CB = CA,,(~B + VBX)
(154 (15b)
with the variables as defined in the Nomenclature section. Substituting eqs 15 into eq 14 and regrouping gives
F(x,P) = Da(1 - x ) ( a - x ) =~ 0
(16)
as the steady-state manifold, whence DU = vk,cA,2VB2/q a =
e,/
(-UBI
(17a) (1%)
Observe that Da (modified Damkohler number) and a are nonnegative elements. Also the conversion, x , must be a positive number less than unity. Obviously, eq 16 is a cubic expression of the form given in eq 11, where B = -(2a + 1) (18a)
C = a2 + 2a
+ Da-'
(18b)
(184 D = -a2 After recasting into the standard form, parameters p1and p 2 are Da-' and a , respectively. Employing the relations provided in Tables I and 11, the variation of 1/Da with a for each singular point was established. The results are as shown in Figure la. The pair of parameters that defines any point in region H yields only one feasible solution for the steady-state manifold. The other two values of x are a pair of conjugates. However, in traversing AB to region J, the conjugate pair becomes two distinct real positive
8
10-
a
d
1
C
' :;::::
O w
6
4
a
0.12
0
Lo,, 3U)L0.04
1 SOL"
0.00 0.00 0.01 0.02 0.03 0.04 0.05
A
--
O.8.0
0.2
0.4
0.6
el3
88
Figure 1. Regions of steady-state feasible solutions.
values less than unity. Although it is possible to find values of x in the interval 0 Ix I1that satisfy the steady-state manifold in the second, third, and fourth quadrants of the l/Da-a plane, the first quadrant is considered the only feasible parameter space because of the physical oddity associated with negative Da and a. Notice that since a naturally emerges as a parameter in the analysis, Figure l a is applicable to two common re2B and A + 2B 3B because the actions A + B stoichiometric coeficient of B, vB, has been incorporated into both parameters (a and Da-'). It is apparent that stoichiometric coefficient of the autocatalyst as well as feed concentrations and fluid flow rate determine the steadystate multiplicity for a given reactor size. The importance of this type of analysis in preliminary experimental work is self-evident. For example, to avoid steady-state hysteresis, operating variables that define a point in J may not be used. However, when the reaction rate is second order in the concentration of A and first order in B, steady-state multiplicity persists but for different choices of parameters. Case ii: Rate Expression r = klCA2CB.Without going through the formal motions, it is easily shown that the steady-state manifold for the isothermal CSTR is F ( x ) = x 3 - (a + 2)x2 + (2a + 1 + D u - ' ) ~- a = 0 (19) where Da = vklc$2(-VB)/q and a = BB/(-vB). Using the provision set forth in Tables I and 11, Figure l b was produced. In this situation for as long as a is positive and less than 1, there will always be only one feasible conversion a t steady state (region H)for any positive value of the modified Damkohler number. Nevertheless, if B is supplied in excess of stoichiometric requirements (i.e., a > 1) it may be possible to have multiple solutions (region J). Incidentally, this may not be desirable in practice since B being a catalytic species would always be fed in relatively smaller concentrations than A (i.e., OB or a < 1). Reversible Reaction. Case i: Rate Expression r = klCACB2 - k_,CB3.For this kinetics, steady-state mass balance yields klCA:(1 - x)(eB + x ) 2 - k_lCA;(8B x)3 - q x / v = 0 (20)
-
-
+
Ind. Eng. Chem. Res., Vol. 30, No. 2, 1991 433 Note that because of the reversibility and stoichiometry of the reaction (eq 4), CB is now (0, + x). At equilibrium, r = 0 and x = x,; thus
Combining eqs 20 and 21, we secure the steady-state manifold Da[(l - x)(OB
+ x)'
- (OB
+ x ) ~ / K ,=] 0
(22)
where Da = Vk,C,'/q. Reorganizing eq 22 into the familiar form of eq 11, we get
B=
30B/K, 1
30B2/K,
C=
+ 2 0 ~- 1
+ l/Ke
+ OB2 - 20B + Da-' 1 + l/Ke
Acknowledgment (23b)
The singular points for the system represented by eq 22 were evaluated using K, = 1. Observe that in this case, alongside 1/Da, OB (ratio of the feed concentration of B to A) rather than N emerged as a parameter in the analysis, suggesting that the solution behavior is independent of the reaction stoichiometry (unlike the previous cases for irreversible reaction). As may be seen from Figure k , there are two distinct regimes (H and J) in the first quadrant of the parameter space where only one feasible solution exists and region I, which contains three admissible conversion values. To strengthen confidence in the validity of the analysis, a check was carried out which confirmed that the estimates of the conversion are not only less than unity but are also smaller than the calculated equilibrium conversion. An autocatalytic reaction with this form of rate law requires that the feed concentration of B be about 2 orders of magnitude lower than that of the coreactant
A.
Case ii: Rate Expression r = k1CA'CB - k-,CB3.The steady-state lumped parameter model with this kinetics may be routinely obtained in the form of eq 11where B, C, and D are now redefined as
B= C=
38,'
D=
20B
+2
1/K, - 1
+ 2 8 +~ DU-' 1/K, - 1
~B(&'/K, - 1) 1/K, - 1
Concluding Remarks We have explored the steady-state behavior of an isothermal CSTR inside which either a reversible or irreversible autocatalysis takes place. Two distinct but important types of cubic rate laws were examined. The effects of feed concentration and reaction stoichiometry (hitherto overlooked in the literature) have been considered in detail. For an irreversible autocatalytic reaction, it was found that the stoichiometric coefficient of the autocatalyst affects steady-state multiplicity. Also, it is manifest that, having mapped the regions for feasible solution(s) in parameter space, it is now possible to devise good experimental strategies to study the reaction. The steady-state analysis of the same cubic autocatalytic scheme under nonisothermal conditions is the subject of future research endeavors.
(24b) (244
The difficulty that arises when K , = 1 in eqs 24 is easily avoided by substituting this value into the steps involved in the derivation of the model equation. The exercise yields the situation for quadratic autocatalysis. For the purpose of our discussion, though, let K, = 0.5 and, with the relations given in Tables I and 11, the results plotted in Figure Id were generated. It seems that the effect of switching the second-order dependency from species B to A in the forward rate step leads to an "inversion" of the solution behavior (cf. Figure IC).Moreover, in comparison with case i, OB is about an order of magnitude greater while the modified Damkohler number experiences a decrease of the same magnitude.
Financial support for A.A.A. during the course of this work by the Natural Sciences and Engineering Research Council of Canada is greatly appreciated. Thanks are also due to Professor M. Pritzker for useful and encouraging discussions. Nomenclature A, B = reacting species CA, CB = concentration of species A and B, respectively CAo= feed (inlet) concentration of species A Da = modified Damkohler number k , , k-l = forward and backward rate constants, respectively K , = thermodynamic equilibrium constant pi,,pi,, = lower and upper limits for parameter pi q = fluid volumetric flow rate V = reactor volume x , , xu = lower and upper limits of the variable x Greek Symbols a = OB/ (-UBI uB = ratio of the stoichiometric coefficient of B to A, taken
as negative by convention if B is a reactant OB = ratio of the feed concentration of B to A, i.e., CB,/C, Literature Cited Aris, R.; and Amundson, N. R. An analysis of chemical reactor stability and control-I: The possibility of local control with perfect and imperfect control mechanisms. Chem. Eng. Sci. 1958, 7, 121-131. Balakotaiah, V.; Luss, D. Analysis of multiplicity patterns of a CSTR. Chem. Eng. Commun. 1981,13, 111-132. Balakotaiah, V.; Luss, D. Exact steady-state multiplicity criteria for two consecutive or parallel reactions in lumped parameter systems. Chem. Eng. Sci. 1982a, 37, 433-445. Balakotaiah, V.; Luss, D. Structure of the steady-state solutions of lumped-parameter chemically reacting systems. Chem. Eng. Sci. 1982b, 37,1611-1623. Balakotaiah, V.; Luss, D. Multiplicity features of reacting systems-Dependence of steady-states of a CSTR on the residence time. Chem. Eng. Sci. 1983,38, 1709-1721. Balakotaiah, V.; Luss, D. Global analysis of multiplicity features on multi-reaction lumped parameter systems. Chem. Eng. Sci. 1984, 39, 865-881. Balakotaiah, V.; Luss, D., Global mapping of parameter regions with a specific number of solutions. Chem. Eng. Sci. 1988,43,957-964. Gray, P.; Scott, S. K. The approach to stationary states of autocatalytic systems in the continuous stirred tank reactor. Ber. Bunsen-Ges. Phys. Chem. 1983a, 87, 379-382. Gray, P.; Scott, S. K. Autocatalytic reaction in the isothermal continuous stirred tank-isolas and other forms of multistability. Chem. Eng. Sci. 1983b, 38, 29-43. Gray, P.; Scott, S. K. Autocatalytic reactions in the isothermal continuous stirred tank reactor-oscillations and instabilities in the 3B; B C. Chem. Eng. Sci. 1984, 39, system A + 2B 1087-1097.
-
-
434 Ind. Eng. Chem. Res., Vol. 30, No.2, 1991 Kaas-Petersen, C.; McGarry, J. K.; Scott, S. K. A two-step model for cubic autocatalysis in an open system. J . Chem. SOC.Faraday Trans. 2 1989,85(111,1831-1835. Kay, S. R.; Scott, S. K.; Tomlin, A. S. Quadratic autocatalysis in a non-isothermal CSTR. Chem. Eng. Sci. 1989,44(5),1129-1137. Leib, T.;Luss, D. Exact uniqueness and multiplicity criteria for an nth order reaction in a CSTR. Chem. Eng. Sci. 1981,36,21@-211. Morbidelli, M.; Varma, A,; Aris, R. Reactor steady-state multiplicity and stability. In Chemical Reaction and Reactor Engineering; Carberry, J. J., Varma, A., Eds.; Marcel1 Dekker: New York, 1987. Pritzker, M.; Fahidy, T. Z. A steady-state multiplicity analysis of a kinetic model for zinc electrodeposition. J . Electrochem. SOC. 1989,136 (8), 2238-2250. Scott, S. K. Reversible autocatalytic reactions in an isothermal CSTR-Multiplicity, stability and relaxation times. Chem. Eng. Sci. 1983,38,1701-1708.
* To whom correspondence should be addressed a t the Department of Chemical & Petroleum Engineering, University of Calgary, Calgary, Alberta, Canada T2N 1N4. Adesoji A. Adesina* Department of Chemical Engineering University of Waterloo Waterloo, Ontario, Canada N2L 3Gl Kolapo E. P. Adewale Department of Polymer Engineering University of Akron, Akron, Ohio 44325
Received for review August 9,1990 Accepted December 3, 1990
