I n d . Eng. Chem. Res. 1987,26, 794-804
794
Hu, R.; Balakotaiah, V.; Luss, D. Chem. Eng. Sci. 1985a,40,589; 1985b,40, 599. Ibbotson, D.E.; Wittrig, T. S.; Weinberg, W. H. Surf. Sci. 1981,111, 149. Jiracek, F.; Havlicek, M.; Horak, J. Collect. Czech. Chem. Commun. 1971,36,64. Jiracek, F.; Horak, J.; Hajkova, M. Collect. Czech. Chem. Commun. 1973,38, 185. Lindstrom, T . H.; Tsotsis, T. T. Surf. Sci. 1984,146, L569. Rader, C. G.; Weller, S. W. AZChE J . 1974,20,515. Rajagopalan, K.; Luss, D. J . Catal. 1980,61, 289. Sheintuch, M.; Schmidt, J. Chem. Eng. Commun. 1986,46, 289.
Schmitz, R. A. Adv. Chem. Ser. 1975,148,156. Schwartz, A.; Holbrook, L.; Wise, H. J . Catal. 1971,21,199. Takoudis, C.G.; Schmidt, L. D. J. Phys. Chem. 1983,87,958. Taylor, J. L.; Ibbotson, D. E.; Weinberg, W. H. J . Catal. 1980,62, 1.
Tsotsis, T. T.; Haderi, A. E.; Schmitz, R. A. Chem. Eng. Sci. 1982, 37, 1235. Xiou, R. R.; Sheintuch, M.; Luss, D. J. Catal. 1986,100,552. Zuniga, J. E. Znt. Chem. Eng. 1977,17, 650. Received for review June 21, 1985 Accepted November 5, 1986
Analysis and Modeling of Multiplicity Features. 2. Isothermal Experiments Michael P. Harold,+Moshe Sheintuch,*and Dan Luss* Department of Chemical Engineering, University of Houston, Houston, Texas 77004
A methodology is presented for elucidating the functional features of the intrinsic kinetic expression from an analysis of the multiplicity features of isothermal experiments. The analysis makes use of the sensitive dependence of the multiplicity features on the form of the rate expression and the presence of inter- and/or intraparticle transport limitations. Simple criteria and graphical tests are developed for the analysis of experimental diagrams of rate vs. limiting reactant concentrations. Three classes of kinetic models are shown to be capable of accounting for the observation of multivalued dependence of the rate on the limiting reactant concentration. The methodology is illustrated by analyzing several published isothermal multiplicity data and developing kinetic models which describe the observed features. Observed steady-state multiplicity features of a chemically reacting system provide information about the functional form of the kinetic rate expression. In the first part of this study (Harold et al., 1987) a method was presented for analyzing and modeling multiplicity features of nonisothermal, single-reaction data. In this part, we discuss the analysis and modeling of isothermal multiplicity data. Isothermal rate data are usually presented as a function of the limiting reactant feed concentration ( c b ) or mole fraction (xb) at a constant catalyst temperature (T,). This set of measurements is denoted as (r, C b (or x b ) ; T,). The assumed or known intrinsic rate model depends on a vector of surface concentrations or mole fractions x , a vector of surface coverages 8, and a vector of parameters p . The overall rate is determined by a solution of species balances which account for all the elementary steps. In many limiting cases, as when a single rate-limiting step exists, the overall rate can be expressed as an explicit function of x b and p . Consider such a case in which the overall single reaction CglujAj= 0 occurs in a symmetric catalytic pellet (vi is positive for a product and negative for a reactant). An intraparticle concentration balance, which ignores the influence of volume change and total pressure gradient, is
tS(2 )+
DeCT-- :s
sg
ujr(x) = 0
j = 1, 2, ..., N (1)
subject to the surface boundary condition
* Author
t o whom correspondence should be addressed. Present address: Department of Chemical Engineering, University of Massachusetts, Amherst, MA 01003. Present address: Department of Chemical Engineering, Technion Haifa, Israel.
kcj(Xbj
- x,j) - D p V x j = 0
(2)
Equations 1 and 2 can be combined to get a relation between x s j 0‘ = 2, 3, ..., N) and the limiting reactant xS1
Similarly, eq 1 and 3 can be solved to give the relation
j = 2, 3, ...,N
This relation may be substituted into r ( x ) in eq 1, which can then be solved to determine the dependence of the observed rate on the feed concentration of the limiting species 1 (r, xbl; T,, xb,, j = 2, ...,N). When intraparticle gradients are negligible or the pellet is nonporous, eq 2 is replaced by kCjC,(xbj
- xSj)
+ vjr(x,) = 0
(2’)
By use of relations 3 and 3’, 2’ can be solved for ( r ,xbl; T,). Our goal is to develop a scheme for solving the inverse problem, i.e., deducing information about the functional form of the intrinsic rate expression (r, x,; T,) from the observed bifurcation diagram of (r,xb; T,). We drop the subscript 1from now on for the sake of brevity. We limit the analysis to single-reaction cases in which at most two stable states exist for any operating conditions and in which at most four limit points exist in the bifurcation diagram of (r, A; T,), where X is either x b or x,. Figure 1 contains 13 schematic ( r , ?+,; T,)diagrams. Diagrams similar to case (a-d, f, h) have been observed experimentally, while those corresponding to case (e, g, i-m) have not yet been reported but are considered in the 0 1987 American Chemical Society
Ind. Eng. Chem. Res., Vol. 26, No. 4, 1987 795
Xb
Figure 1. Thirteen types of (r, X b ; T,)bifurcation diagrams. Dashed branches describe unstable states. Cases a-d, f, and h have been observed experimentally.
analysis. Other diagrams having at most three solutions (two stable states) and four limit points may be constructed but are not shown in Figure 1. The diagrams in Figure 1 have three distinguishing features-multiplicity, inverse multiplicity (the existence of the same isothermal rate for two or more values of xb), and isolas (isolated or disconnected branches of states). These features may be caused by either the intrinsic kinetics or the interaction between the kinetics and transport rate processes. Some of the observed features, such as the existence of isolated branches, cannot be explained by an intrinsic rate expression which is an explicit function of the temperature and concentration of one reactant. We present in this work a methodology for elucidating the essential features of the intrinsic kinetics by an analysis of either ( r , A; T,) bifurcation diagrams or a (A, T,) bifurcation map where A is either x b or x,. To simplify the application of the results by experimentalists, we state briefly each analysis and its main conclusions, which are very helpful in the discrimination among rival kinetic models. Experimental observations are used to illustrate several of the predictions. In the last section, we use the proposed methodology to find the type of rate expression which can describe published multiplicity data.
Analysis of (I,xb; T,)Bifurcation Diagrams In a large number of reactions, the rate depends primarily on the surface concentration of a limiting reactant as the other reactants are in excess. Consider a case in which the surface rate expression depends only on the surface concentration of a single species, x,. Rates which depend on the surface concentrations of two species are considered later. For a specified bulk concentration, x, satisfies the relation k c c T ( X b - X,)
- r(Xs,P) = 0
where r(xs,p ) need not be an explicit function of p is a vector of kinetic parameters.
(4) x, and
The observed diagram of ( r , xb; T,) is essentially a solution of eq 4. To infer the functional form of r(x,, p ) , we need to solve the equation kcCT(Xb - xs) - r(xb, P) = 0
(5)
Thus, we need to construct a positively sloped supply line (slope of kcCT)and look for the number of the intersections it has with ( r , xb; T,)for any positive intercept x,. When multiple intersections occur for some x, (Figure 2a-c), then the (r, x,; T,) diagram is multivalued. This situation always occurs for a single-valued ( r , x b ; T,) diagram with an apparent order exceeding unity (see Harold et al. (1987) for details) and for multivalued ( r , xb; T,) diagrams having either an isolated branch of states (Figures 2c, If-m) or a counterclockwise (S type) hysteresis (parts c and e of Figure 1).
Figure 2. Transformation of three ( r , n b ; T,) diagrams into ( r , x,; TJ diagrams.
Figure 2a shows that a monotonic single-valued ( r , 3tb;
T,)diagram which has an inflection point is mapped into a multivalued ( r , x,; T,) diagram (Figure 2al) whenever k,CT
max
(E) 9
where the right-hand side of (6) is the maximum slope of the rate curve, i.e., the slope at the positively sloped inflection point on the stable branch. Figure 2b shows that a clockwise hysteresis loop is transformed into a multivalued ( r , x,; T,) if
kcCT > min
(E)
(7)
U
where the right-hand side is the slope at the positively sloped inflection point of the unstable branch of solutions. That slope is not observable. A more conservation criterion is
k,CT >
rE - rI ~
XbE
- XbI
(8)
where the right-hand side of (8) is the slope of the chord connecting the observed extinction (E) and ignition (I) points, which is obviously larger than the right-hand side of (7). Parts b and bl of Figures 2 illustrate this situation. When kcCT is not known, a conservative lower bound on its value is the maximum slope of any chord connecting the origin to a point on the observable (R, x b , T,) diagram as shown in Figure 2b. Thus, condition 8 is certainly satisfied if
(:)
rE - r1
> XbE_ICbI
Note that, if condition 9 is not satisfied, no definite conclusion can be drawn, even though this usually implies that the rate is a single-valued function of x,. The transformation of (r, xb; T,) diagrams into (r, x,; T,) is much more intricate when two positively sloped inflection points exist (parts b and c of Figure 3). The transformation depends on the relative magnitudes of kcCT,the slope at the stable inflection point (dr/dxb),, and the slope of the ignition-extinction chord. Various cases are shown in Figure 3. For example, when the ( r , x b ; T,) diagram satisfies conditions 6 and 9, Figure 3c is transformed into a mushroom (r,x,; T,) diagram (Figure 39. A novel feature is obtained when the (r, x,; T,) diagram has five solutions for a narrow x, range (Figure 3j). A detailed analysis of these cases is given elsewhere by Harold (1985).
796 Ind. Eng. Chem. Res., Vol. 26,No. 4,1987 Table I. Pairs of Diagrams from Figure 1 which Infer Crossing One of the Varieties and the Corresponding Local Singular Diagram singular Figure 1 combinations variety diagram a-c, W, be,d-m, e m , f-i, g-h, j-k, j-1 c k , d-I, f-g, h-i, j-m
hysteresis
f
isola (type a)
a-j, c k , d-1
iaola (type b)
G
Figure 3. Transformation of inverse-S (cluckwire) ( r , zb; TJ diaT3 diagrams (ahown in circles).
grams (shown in squares) into ( r , x.;
We conclude that when the rate is a function of the surface concentration of a single species, the transformation of (r,xb; T J into (r,x.; 7'") has the following features: 1. A monotonic (r, xb: TJ diagram with a positively sloped inflection point may be mapped into a multivalued S-type ( r , xI; T J diagram for a sufficiently small k,CT ualue. 2. An ( r , xb; TJ diagram with an apparent order exceeding unity is always mapped into an S-type (r.x.: T J diagram. 3. A (r, xb: TJ diagram with an isolated branch or a counterclockwise hysteresis is always transformed into an (r, xg; T,) diagram having the same multiplicity features. 4. The transformation may preserve or eliminate a clockwise hysteresis observed in the (r, xb; TJ diagram. Analysis of (T,, x,) Bifurcation Maps A (Ts, xb) bifurcation map, which describes the loci of the observed limit (ignition or extinction) points, can be constructed from a series of bifurcation diagrams using either T. or xb as the bifurcation variable. While the construction of the map may be tedious, it contains very useful information about the features of the intrinsic rate. An important advantage of constructing a map is that it enables a rapid detection of a nonmonotonic dependence of an ignition or extinction branch on parameter pi. This feature may be easily missed from the analysis of (r, X; p ) diaprams, where X is the bifurcation variable and .. pi is an element Of p. The loci of the limit (ienition and extinction). .Doints satisfy the defining condkons
where y is the state variable. By use of eq 4 and y = xB, eq 10 predicts that at the limit point the supply line is tangential to (r, 2.; T J . T w o branches of limit nointa with sloDes of eaual sign coalesce typically at a cusp point in the bifurcatfon map. At this point, the following conditions are satisfied ~~
~
Using eq 4,we find that a t a cusp point the supply line is tangent to (r, x.; TJ at its inflection point. The set of all the cusp points forms the hysteresis uariety. When the parameter vector crosses this variety, two limit points appear or disappear so that a hysteresis loop emerges or collapses in the bifurcation diagram.
Xb
Figure 4. Seven typical (T8,xd bifurcation maps. Cases a-c and e have been observed experimentally.
An isola point appears typically in a bifurcation map as a local extremum point with respect to one of the two parameters. At a local extremum with respect to A,
The locus of all isola points is called the isola uariety. When the parameter vector crosses this variety, two limit points typically appear or disappear and the bifurcation diagram separates into two isolated curves (isola type a) if
or an isolated branch of solutions appears (isola type b) if
A 0, if for some parameters
where
1
K = KO exp(-AH,)/RT, (19) where E and -AHH,are positive. It can be shown that for a mildly activated reaction, dD/dT, > 0. Also, dL/dx, > 0, so that the transformation to observable variables amounts to a stretching of the coordinates, and the cusp in the (Ta,xJ plane has the same orientation as that in the (0, L ) plane. In this case the (r,xb; TJ has an inverses hysteresis loop, while the (r, T.; xb) diagram is of the S type. If, on the other hand, the activation energy is very small so that dD/dT. < 0, both ignition and extinction branches become negatively sloped (as in case c in Figure 4) but with TsE> T:. In this case, both (r,x,; TJ and (r, T; xb) are of the inverse4 type. No experimental observation of such a (Ts, xb) map has been reported so far. Analysis of the case in which only intraparticle concentration gradients exist (Harold, 1985) shows that for
Condition 22 can be satisfied when x . ~is either a reactant that inhibits the rate or a Droduct which accelerates the rate, i.e.,
_ _ar
