Ind. Eng. Chem. Res. 1987, 26, 786-794
786
= 0, the initial decomposition rate can be given by dXT/dt = 0.7808(dXi/dt) 032)
where t, is the experimentally determined time to complete the decomposition.
On the other hand, dXl/dtl,,o is represented from eq 1 and 9 as dXl/dtlt=o = 3/P,t0 (B3)
Literature Cited
Then eq B2 is rewritten as dXT/dtl,=, = 2.342/Pito
(B4)
If eq B4 is substituted into eq 6, the following equation can be obtained I( = [2.56P,t,(dX,/dtl,,o) - (2P1 + 1)1T*,2/b - 20, 035)
Hill, J. M.; Kucera, A. Znt. J. Heat Mass Transfer 1983, 26, 1631-1637. Itoh. N.: Obata. K.: Hakuta. T.: Yoshitome. H. KuPaku Koeaku Ronbunshu 1983; 9, 434-440. ’ Itoh, N.; Obata, K.; Yoshitome, H. Kagaku Kogaku Ronbunshu 1981, 7, 377-383. Tao, L. C. AIChEJ. 1967, 13, 165-169. Wen, C. Y. Ind. Eng. Chem. 1968, 60, 34-54.
.,
-
Received for review November 5, 1984 Revised manuscript received July 8, 1986 Accepted November 20, 1986
Analysis and Modeling of Multiplicity Features. 1. Nonisothermal Experiments Michael P. Harold,+Moshe Sheintuch,t and Dan Luss* Department of Chemical Engineering, University of Houston, Houston, Texas 77004
A methodology is presented for elucidating the functional features of the isothermal rate expression from an analysis of the multiplicity features of nonisothermal experiments. The necessary kinetic features are predicted from an analysis of either a bifurcation diagram (dependence of surface temperature on bulk temperature or concentration) or a bifurcation map (dependence of ignition and extinction points on two operating conditions). The methodology is illustrated by the analysis of several published nonisothermal bifurcation diagrams to gain information about the functional form of an isothermal rate expression, which can describe the observed data. The methodology of kinetic model discrimination and parameter estimation is by now well developed (see, for example: Froment and Bischoff, 1979; Froment and Hosten, 1984). Chemically reacting systems may attain multiple steady states under certain conditions. The ability of a steady-state model to predict observed multiplicity features may be used for qualitative kinetic model discrimination. Moreover, the transition (bifurcation) to steady-state multiplicity may be used as a sensitive tool for fitting kinetic parameters (e.g.: Beskov et al., 1976; Herskowitz and Kenney, 1983). A steady-state model which includes the kinetic rate expression must be able to predict any observed bifurcat i o n diagram, i.e., the sequence of states observed when an operating or bifurcation variable is changed. A more discriminating test is its ability to fit a bifurcation map which describes the transitions (boundaries) between parameter regions with a different number of solutions. These boundaries are the loci of the ignition and extinction points. This work presents a systematic procedure for finding the simplest rate expression which can describe observed bifurcation diagrams and maps. This procedure enables a rapid elimination of classes of candidate rate expressions. Thus, the finding of a proper rate expression is significantly simplified. We consider in this work only systems in which a single overall chemical reaction occurs and in which no more than
* Author to whom
correspondence should be addressed. Department of Chemical Engineering, University of Massachusetts, Amherst, MA 01003. Present address: Department of Chemical Engineering, Technion Haifa, Israel. t Present address:
*
0888-5885/87/2626-0786$01.50/0
three steady-state solutions exist, at least one of which is unstable. Theoretical analyses (Tsotsis et al., 1982; Hu et al., 1985a,b)and experimental studies (Harold and Luss, 1985) indicate that more than three steady-state solutions may exist for certain single reaction systems. Our analysis does not cover these rather uncommon cases. In some specifically mentioned cases, however, the results apply also when an arbitrary number of stable states exist. Most single chemical reactions consist of several elementary steps involving either adsorbed intermediates in a heterogeneous catalytic reaction or free radicals in a homogeneous gas-phase reaction. It is usually possible to describe the rate by an overall expression which does not account for the details of the intermediate steps but relates it to measurable quantities. This expression may be obtained by using physiochemical arguments such as the existence of a single rate-determining step. The functional form of an overall rate expression is often rather different from that describing a single kinetic step. This phenomological approach is usually adequate for design purposes even though the simplified kinetic model may disguise some multiplicity or dynamic features. The qualitative features that differentiate among various isothermal rate expressions (Figure 1)are (i) multiplicity or uniqueness of the rate; (ii) local extremum points or inverse multiplicity of the rate, i.e., the existence of more than one concentration giving the same isothermal rate; and (iii) continuity or discontinuity of the rate expression, i.e., the existence of isolated branch(es) of states (isolas). Using these yardsticks, we classify in Table I the eight functional forms shown in Figure 1. These eight bifurcation diagrams of isothermal rate ( r ) vs. limiting reactant feed mole fraction (xb) (denoted as (r, xb; T,))describe the simplest form of isothermal rate dependence, and all have 0 1987 American Chemical Society
Ind. Eng. Chem. Res., Vol. 26, No. 4, 1987 787 Table 1. Qualitative Classification of the Eight Rate Forms in Figure 1" Figure 1 discriminating feature (i) multivalued (ii) inversely multivalued (iii) existence of isola
a
b
C
N N N
N Y N
Y Y N
d Y N N
e
f
g
h
N Y
Y Y Y
Y Y Y
Y Y Y
N
" N = no; Y = yes.
occurs, where vi is the stoichiometric coefficient of Ai (negative for reactants and positive for products). The mass balance for species i gives
xb
Figure 1. Eight types of experimentally observed (P, XI,; T,)bifurcation diagrams. Dashed lines represent unstable states.
been observed experimentally. Of course, more complex (r, xb; T,) diagrams with up to three stable states may be devised, but these have not yet been observed. Parts a and b of Figure 1 describe the common single-valued dependence which is encountered in many applications. For some conditions, a multivalued (r, xb; Tb diagram (Figure lc-h) may imply intrinsic rate multiplicity of (r, x,; T,). This occurs usually when several reaction steps are needed to account for the overall reaction. In solving design problems, we are interested in finding the dependence of measurable state variables, such as catalyst temperature, conversion, or observed reaction rate, on a measured operating condition such as the bulk (feed) temperature ( T >,feed concentration ( c b ) , or feed mole fraction (xb). &is work outlines a systematic approach for solving the inverse problem, i.e., finding the simplest intrinsic rate expression which can describe the steadystate multiplicity features observed in bifurcation diagrams and maps. Experimental multiplicity features of nonisothermal catalytic reacting systems are usually reported as bifurcation diagrams of either catalyst temperature vs. feed temperature for a fixed limiting reactant mole fraction (denoted as (T,, T,; xb)) or catalyst temperature vs. the limiting reactant mole fraction for a fixed feed temperature (denoted by (T,, xb; T,)). Isothermal experimental data are usually reported as (r, xb; T,) or (r, T,; i b ) bifurcation diagrams. This work is the first of a two-part study (Harold et al., 1987). In this part we present a systematic approach to predicting the qualitative features of the rate expression from nonisothermal experiments in which steady-state multiplicity is observed. The method is based on a series of analyses of the (T,, Tg; xb) and (T,, xb; T,) bifurcation xb) bifurcation map. Whenever diagrams and the (Tg, possible, we cite experimental examples. The main assumption implicit in the analysis is that the transport properties are constant and uncoupled from the reaction rate and temperature and concentration gradients. Thus, the convective heat and mass fluxes are assumed to depend linearly on the temperature and concentration differences between the bulk and catalyst surface, respectively. We assume that a single overall reaction N
CviAi = 0
i=l
(1)
where kCi is the mass-transfer coefficient for species i. Thus, the rate expression r ( x 8 ,T,, p ) can be expressed as r(xsl,x b , k,,u, T,, p ) . This relation need not be explicit and may include 8 , the vector of surface coverages of each of the N species. Hence, the determination of the rate may involve solving a set of N surface coverage balances, the surface balance of the limiting species 1, and eq 2. The temperature and limiting reactant concentration difference between the surface and bulk satisfy the relation
(3)
where ATadis the adiabatic temperature rise which is the maximal temperature rise for a given xb and CT is the total concentration. The determination of the functional form of the rate expression is constructed by a series of data analyses which examine the qualitative features of different bifurcation diagrams or maps. To simplify the application of the results by experimentalists, we print in italics the main conclusions of each analysis and cite some experimental illustrations. In the final section of this paper, we use this method to predict the functional form of the rate expression for various reactions based on inspection of published nonisothermal multiplicity features. Use of Bifurcation Diagrams To Predict t h e Reaction Rate Expression Steady-state multiplicity data are usually reported in the form of bifurcation diagrams. We present two analyses which enable one to conclude from the features of either (T,, T,; xb) or (T,, x b ; T g )bifurcation diagrams about the functional dependence of the rate on the feed mole fraction of the limiting reactant and the catalyst temperature. I. Analysis of (T,, T,;x b ) Bifurcation Diagram. The analysis of the (T8,T g ;xb) diagram is carried out by transforming it into an (r, T,; xb) diagram. This transformation involves taking isothermal (fixed T,) cross sections of the (Ta,T,; xb) diagram. When such a horizontal line intersects the (T,, Tg;xb) graph more than once, Le., inverse multiplicity exists, several feed temperatures can sustain the same surface temperature at a fixed feed composition. The rate is proportional to T , - T g since
Thus, inverse multiplicity implies that the isothermal rate is a multivalued function of xb for some surface temperatures. A monotonic rate expression leads to (Ta,T,; xb) diagrams of the type shown in parts a and b of Figure 2 and cannot account for isothermal multiplicity.
788 Ind. Eng. Chem. Res., Vol. 26, No. 4, 1987
Figure 3. Two possible types of (QR, T.; xb) diagrams that predict a decrease in feed extinction temperature with increasing x b ( x b l
1%vol (Figure 5b). Reconstruction of the CO oxidation data in the form of (T,, xb; T ) diagrams (the study employed T, as the bifurcation variatle) reveals two types of diagrams (parts a and d of Figure 6). The lower (upper) branch is a stable continuous branch for gas temperatures below approximately 305 "C (above 320 "C) as seen in Figure 6a (6d). The transition between these two types of bifurcation diagrams occurs at a singular diagram shown in Figure 6c at which an upper isola and a continuous lower branch with an inverse S pattern (Figure 6b) coalesce. This singular point at Tg= T,*is called an isola, and it satisfies the conditions aF aF F(y, X , p ) = - = - = 0 (12) ay aX where X is the bifurcation variable (in this example X = xb). The singular diagram (Figure 6c) is structurally
790 Ind. Eng. Chem. Res., Vol. 26, No. 4, 1987 Rate At Fixed x b Rate A t Fixed T,
Ti Ts
'bl
'b2
Xb
Figure 8. (a) Two ( r , T,; Zb) curves for which the feed ignition temperature is an increasing function of Xb. (b) The corresponding ( r , x b ; Tsi)diagram.
b'
Figure 7. Bifurcation maps in (Tg, Zb) plane and corresponding ( r , Zb; T,) diagrams.
unstable and it splits for Tg # Tg*into either diagram 6b or 6d. In order to account for the transition from part b to part a of Figure 6, another transition through a cusp point is necessary. The isola and cusp are generally distinct points in the bifurcation map. The type of isola point depicted in Figure 6c appears as a local maximum (minimum) along the extinction (ignition) branch (Figure 5b). Figures similar to parts b and d of Figure 6 were observed by Harold and Luss (1985) in another study of CO oxidation on Pt/Al2O3 The cusp and isola points were clearly identified. The coalescence of the cusp and isola points creates a singular point of higher codimension. At this pitch-fork point, conditions 11and 12 are satisfied and the bifurcation diagram has a pitch-fork shape. Another type of isola point, also defined by eq 12, exists a t the point for which the isola (Figure 4eJ shrinks and disappears. 111. Analysis of ( T g ,x b ) Bifurcation Maps. The most revealing feature in the bifurcation map is an isola point with respect to feed concentration or gas temperature. At an kola point with xb being the bifurcation variable, eq 12 is satisfied withy = x , and X = xb. Applying these conditions to the steady-state eq 5 and noting that T, - Tg = ATad(xb- x , ) gives
and
Since dT,/dx, = -dT,/dxb, we conclude that d r l a x , = 0 at an isola point. Moreover, since dx,/dxb # 0, we conclude that dr/dxb = 0; Le., ( r ,xb; T,)has a local extremum so that the reaction rate is locally of zero order with respect t o both x , and xb at t h e k o l a point. Therefore, a transition from positive- to negative-order kinetics occurs on a point along the ignition or extinction branch. Several schematic maps are shown in Figure 7 with the corresponding inferred ( r , xb; T,) diagrams. These maps differ in the orientation of the ignition and/or the extinction lines. The analysis of the (Tg,xb) maps is best explained by considering the influence of xb on the heat generation (QG, T,; xb) curves. All (QG; T,; xb) curves have usually a sigmoidal shape with asymptotes Q G ( T, 0 ) = 0 and Q G ( T , m) = (-AI-I)k,CTxb. The ignition and ex-
-
-
tinction feed temperatures ( T l and TgE)are those for which the heat removal line of slope h is tangential to QG at T,' and TSE.If the rate (or, equivalently, QG) is an increasing or decreasing monotonic function of x b for a certain range of xb and T,, then the family of heat generation curves for different do not intersect in that range of conditions. Clearly, in the former (latter) case, T i and TgEdecrease (increase) as xb is increased as shown in Figure 7a (Figure 7b). In order to satisfy QG(T, a) = ( - m k c C T X b in the latter case, a transition from a monotonic decreasing to a monotonic increasing function of x b must occur for a sufficiently large T,. Consider two feed compositions xb2 > xbl with TgI(xb2) > TgI(xbl)(see Figure 8a). For sufficiently large T,, the rate becomes transport limited and is a h e a r function of xb (apparent positiveorder reaction). Therefore, the two heat generation curves must intersect at some T, = TSi.At this intersection, two feed compositions sustain the same rate (Figure 8b) and the isothermal rate must have a local maximum. We conclude that if the ignition temperatures increase with increasing reactant concentration, t h e n the isothermal rate c w u e has a local maximum. This argument applies to heat generation curves with several inflection points which for a certain range of conditions may exhibit more than three solutions. If the locus of ignition feed temperatures from any branch of states is an increasing function of Zb, then the family of heat generation curves must intersect and the isothermal rate must exhibit a local maximum for some xb. With these features in mind, the maps in Figure 7 can be analyzed to get the following conclusions: (a) The monotonic (Figure 7aJ and S-type (Figure 7aJ rate curves generate families of noninteresecting singlevalued (Figure 3a) and multivalued (Figure 3b) ( Q G , T,; xb) curves if (dr/dT,), > 0 (i-e.,the apparent activation energy is positive). In both cases the maps are cusps with negatively sloped ignition and extinction branches. (b) A cusp with positively sloped ignition and extinction branches exists if the rate is of negative order for some xb. In order to satisfy r ( x b = 0 ) = 0, the rate expression must have a local maximum for some xb (Figure 7bl-bz). (c and d) A local m a x i m u m (minimum) in the extinction (ignition) branch implies a transition from negative(positive-) to positive- (negative-) order kinetics. In either case t h e isothermal rate m u s t exhibit a local m a x i m u m for some xb. An extinction branch with a local maximum probably implies a transition from a kinetic-limited negative-order rate to a transport-limited positive-order rate. (e) A local maximum along the ignition branch implies a transition from negative- to positive-order kinetics. Near this kola point, the (r, xb; T,) diagram must therefore exhibit a local minimum and a local maximum (to satisfy r(xb = 0) = 0). The shape of a (Tg, xb) bifurcation map gives useful information about the inverse multiplicity features of the corresponding (r,xb; T,) diagram. Knowledge of this shape
-
Ind. Eng. Chem. Res., Vol. 26, No. 4,1987 791
'b
XS
Figure 9. Transformation of a ( r , xb; T,) diagram into a ( r , x,; T,) diagram.
is not sufficient to discriminate between single-valued (a1, bl, and el of Figure 7) or multivalued (a2, b2 and e2 of Figure 7) ( r , xb; T,) diagrams. To assess the multiplicity features of a ( r , xb; T,) diagram, one must carry out a cross-sectional analysis of (T,, Tg; xb) diagrams to check if all infer a single-valued ( r , xb; T,) diagram for all T,. The last case to be considered is an isola point with respect to Tg. Applying condition 1 2 withy = x , and X = Tg to eq 5 gives
Simultaneous solution of eq 15 and 16 implies that
i.e., at a n isola point with respect t o Tg,the rate is temeprature independent and of negative order. No multiplicity studies have revealed such an extremum with respect to Tgalong an ignition or extinction branch. Mass-Transfer Effects. Isothermal rate multiplicity in the form of ( r ,xb; T,) is detected by inverse multiplicity of one or more (T,, Tg;xb) diagrams. The functional dependence of the isothermal rate on xb may be affected by external mass transfer or intraparticle diffusion limitations. Therefore, a single-valued ( r ,xb; T,) curve, which may be either monotonic or have some local extremum points, may correspond to multivalued intrinsic kinetics. That is, intraparticle diffusion limitations and external mass-transfer resistance may mask intrinsic rate multiplicity. We consider here only continuous, single-valued (r, xb; T,) curves. Other cases which imply ( r , x,; T,) multiplicity are considered in the second part of this study. With this in mind, we propose the following analysis. IV. Analysis of ( r ,x b ; T,)Diagram. We are interested in transforming a measured single-valued (r,xb; T,) diagram into ( r , x,; T,) to find if this transformation may introduce multiple solutions. At steady state the rate of mass transfer ( r M )is equal to the reaction rate at the catalyst surface (r):
rM = kccT(xb - X s ) = r(Xb, TJ
(18)
rM is a straight line of slope k,CT and intercept x,. The reaction rate may be an implicit function of x, which passes through the origin. Now, if three xb values satisfy eq 18 for a given k , c ~ and x,O, then the three rates are the intersections of the (r, xb; T,) diagram with a line of slope k c c T and intercept xSoas shown in Figure 9. A ( r , xb; T,) diagram can be transformed into a ( r , x,; T,) diagram by finding the intersections of a line with a slope of kcCT and intercept xSowith the ( r , xb; T,) curve. Multiple intersections are possible only if the positive slope of a ( r , xb; T,) diagram at its inflection point exceeds k,CT (see Figure 9). In such a case,
'b
xs
Figure 10. (r, xb; T,) diagrams and corresponding type of (r, x,; T,) diagrams.
three different rates occur for x,O; i.e., ( r , x,; T,) is multivalued. Thus, a (r,Xb; T,) diagram with a positively sloped inflection point is mapped into a multiualued ( r , x,; T,) diagram if kccT C m a x ( d r / d x b ) (Figure 10c-e). When a ( r , xb; T,) diagram does not have a positively sloped inflection point, the transformation of (r, xb; T,) to (r,x,; T,) does not alter the features of the rate curve (parts and and b of Figure 10). When a ( r , xb; T,) diagram exhibits an apparent reaction order exceeding unity (Figure 10d), an inflection point exists at some reactant concentrations since the rate is bounded by kcC+b. In such cases the reactant supply line intersects the high rate portion of r which asymptotically approaches kcCTXb so that the ( r ,x,; T,) diagram is multivalued. Multivalued ( r , xb; T,) diagrams which exhibit an S-type hysteresis loop or isolas are always mapped into multivalued ( r , x,; T,) diagrams. These cases are considered in the second part of this study. Systematic Analysis of Some Nonisothermal Data. We analyze here reported nonisothermal multiplicity data using the rules developed in this work. The method leads to useful insight about the type of rate expression which governs certain classes of reactions and points out useful experiments which should be carried out to gain more information about the kinetics. Many studies of steady-state multiplicity have been reported in the literature. Table I1 lists nonisothermal steady-state multiplicity studies in which Tgand/or xb was the bifurcation variable. We do not list studies that used the flow rate (or equivalently the residence time) as the bifurcation variable, although the suggested methodology applies to such systems as well. Schmitz (1975) tabulated reports of multiplicity of homogeneous reactions in a CSTR, many of which employed the residence time as the bifurcation variable. All the studies reported in Table I1 were carried out at close to atmospheric pressure. With the exception of ethylene hydrogenation (Furusawa and Kunii, 1971), all the studies consisted of oxidation on various noble metal catlaysts. Table I1 summarizes the important multiplicity features for the particular reaction, catalyst, and reactor type and refers to a schematic figure(s) which describes the observations. Oxidation of Paraffins. All reported nonisothermal catalytic oxidations of paraffins on Pt revealed similar multiplicity features, i.e., inversely unique, S-type (T,, Tg; xb) diagrams with ignition and extinction temperatures which descend with increasing xb. Thus, the ( r , xb; T,) relationship should be single-valued and monotonic. These trends were reported among others by Cardoso and Luss (1969) for butane oxidation and by Harold and Luss (1985) for ethane oxidation. The surface ignition temperature was observed to be a decreasing function of xb for ethane, propane, isobutane (Hiam et al., 1968), and butane oxi-
792 Ind. Eng. Chem. Res., Vol. 26, No. 4, 1987 Table 11. Published Nonisothermal Steady-State Multiplicity Data multiplicity features diagram type no.
reaction, catalyst, reactola
1 ethane, Pt/A1203, S P
2 propane, Pt, SW 3 butane, Pt, SW 4 butane, Pt, SW
(Ts, Tg; xb)
(T8,xb; T,) dT,'/dxb dT,'/dxb Hvdrocarbon Oxidations
S
S
dT:/dxb
