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Effectiveness factors for the general non-isothermal case of bimolecular Langmuir-Hinshelwood kinetics
EFFECTIVENESS FACTORS FOR THE GENERAL NON-ISOTHERMAL CASE OF BIMOLECULAR LANGMUIR-HINSHELWOOD KINETICS SAMUEL
H. WONGt
and STEPHEN
SZl?PE*
University of Illinois at Chicago Circle, Chicago, IL 60680. U.S.A. (Received 21 August 1981; accepted 2 April 1982) Abstract-Non-isothermal catalyst effectiveness factors are determined for both exothermic and endothermic reactions with rate equations of the form r, = /&,,/(I + K,,CJ’. It is shown that with such rate equations, multiplicities may occur even for endothermic reactions. In addition, multiple states may arise in regions where the effectiveness factor is less than one. Regions of unique and multiple steady states of bimolecular Langmuir-Hinshelwood kinetics have also been defined. Contrary to previous findings for exothermic and/or isothermal reactions, multiplicities in endothermic reactions are likely to occur in systems showing higher reaction temperatures and stronger surface chemisorption characteristics. Criteria (developed by previous investigators) sufficient for uniqueness and multiplicity of catalyst particle steady state are compared and discussed. INTRODUCTION
reaction takes place within a porous catalyst, multiplicity of the steady state solutions may occur at some operating conditions. This phenomenon was first demonstrated for a firs? order exothermic reaction by Weisz and Hicks[l], then for a negative order isothermal reaction[2], and was also shown to occur for more complex rate equations of the Langmuir-Hinshelwood type [3-71. These results can be expressed in terms of the catalyst effectiveness factor, 7). which is in general a function of the following four dimensionless parameters: the Thiele modulus, &; the Arrhenius number, 4; the Prater number, p; and, the adsorption parameter, &CA,. The major conclusions of the above authors can be summarized as follows:
When
a chemical
(l)Thereisonlyonesteadystatesolution(a)aleithervery
high or very low values of the Thiele modulus, &, or (b) when the values of all three parameters, c_ 8, and K,C,,, are small. (2) In certain ranges of 9.. there may exist three or more different solutions for a given & (a) for positive nth order exothermic reactions (@ > 0), when the values of Q and B are quite large; and (b) for isothermal (p = 0) bimolecular Langmuir-Hinshelwood rate equations [3,4] when the values of the adsorption parameter are sufficiently large. Although these findings set an upper limit of three steady-state solutions for infinite flat plates and for the infinite cylindrical geometries, this is not the case for a sphere[8]. The investigation of Hlavacek and Marek[9] revealed the occurrence of five steady state profiles in the case of spherical geometry for a zeroth order reaction. Others[ lo] reported that an infinite number of steady states may exist for highly exothermic first order reactions when values of /3 approach m. The existence of five steady states has also been shown for the modestly *Author to whom correspondence should be addressed. tPresent address: Northern Petrochemical Company, Technical Center, Morris, IL 60450, U.S.A.
non-isothermal bimolecular Langmuir-Hinshelwood reaction[5,6, Ill. It should be emphasized that all steady state solutions, except for the lower “quenched” and the upper “burnt out” profiles resulting from these analyses, can be shown to be unstable[l2]. The objectives of this investigation are: (I) to study the effectiveness factor for all three cases of thermal conditions (exothermic, isothermal, and endothermic) in reactions that have the following Langmuir-Hinshelwood type of rate equation:
(2) to examine the stability for each of the calculated steady state profiles; and (3) to determine the boundary, in parametric space (c, 0, and &CA*), below which uniqueness of the steady state for the system is generated for all operating conditions. The results obtained from the investigation of the general case of bimolecular Langmuir-Hinshelwood kinetics include as special cases all of the following previously reported studiesrl-71: (1) isothermal and exothermic fist order reactions (p 2 0 and K,C, = 0); (2) Isothermal negative first order reactions (/3 = 0 and &CA B 1); and (3) isothermal Langmuir-Hinshelwood kinetics (fl = 0, any &CA). Problems relating to the direct application of this type of rate equation can be found in many chemical engineering reaction systems, such as the hydrogenation of codimer[13]; the formation of Phosgene[l4]; the dehydration of ethanol[l5]; the decomposition of N,O[l6]; the oxidation of CO and Hl[17a-i]; and the hydrogenation of ole!ins[ll, 181. PROBLEMSTATEMENT
Consider a porous, fully impregnated, spherical catalyst. Reactant A is in the fluid surrounding the pellet at concentration C ,+ and temperature T,. Since essentially all of the active surface is internal, molecules of A must diffuse into the catalyst in order to react. At steady state, 1629
1630
SAMUELH. WONGand STEPHENSZBPE
the mass and heat transfer processes that occur simultaneously within the pellet are described by the following two equations which are presented in dimensionless form:
with the boundary
The above criterion, though easy to use, provides only the sufficient conditions for uniqueness and tends to be conservative in its prediction. Both the necessary and sufficient conditions are provided by the criterion of Michelsen and Villadscn[lZ]. Although this method has been demonstrated only for a first order exothermic reaction, the procedure can be applied quite generally for any form of the rate expression, as shown belnw. By differentiating eqn (2) with respect to the center concentration, yO, the following is obtained:
conditions: dy dz -_=-_=o dx dx
atx=O
atx=l
y=z=
It was shown by Michelsen and Villadsen[lZ] given steady state is asymptotically stable if 1.
(5) o(x) > 0
Combining eqns (2) and (3), and integrating the resulting equation twice, one obtains, with the boundary conditions given, (z-
l)=P(l
-y).
(6)
This result, first derived by Damkoehlerll91, is valid for any form of the rate equation and for any particle geometry[20]. The use of this relationship permits the rate equation to be written in terms of one variable only, either y or .z. Thus, the rate equation as expressed in eqn (1) can be reduced to a function of concentration only. In dimensionless form, one obtains:
(l+d
This criterion
also implies that any steady
dx
SUP
(y-l)6!!G!U [
(11)
(12) state with
dyo,o d&
assures local asymptotic stability, a condition earlier by Luss and Lee1221 and by Aris[23].
shown
RESULTS AND DISCUSSION It
is well recognized that certain reactions following the bimolecular Langmuir-Hinshelwood type of kinetics can exhibit unique or multiple steady states, depending on the system parameters and the inherent bounds on system variables. Starting with eqn (9), it is possible to determine the condition under which uniqueness of the steady state can exist. For a first order kinetics, this is [21]: (13)
x=,’
As illustrated by the earlier workers, the steady state, computed for a catalyst under some conditions, may have more than one solution. The transitional point, which represents the separation between the two regimes, one regime with unique solution and the other regime with multiple states, is called the bifurcation point. To determine the conditions under which no bifurcation point can occur, the following criterion proposed by Luss[21] can be supplied: 05y51
or
Q? 5 4(1+ p).
(-dy >
1.
O,=, >O
dR
This approach reduces the system of equations to a single differential equation describing the change in y with respect to radial position. Since the reaction rate constant is usually a much stronger function of the temperature than of the physical parameters involved (KA, fl, and A,), the temperature dependence of the latter group of variables is neglected in this study. To provide a measure of the actual catalyst performance in a specific condition, an effectiveness factor, 7, is defined:
05x<
This is true regardless of the number of steady states that may arise when solving the system equations. In regions where a maximum of only three multiple steady states can occur, a minor simplification of Criterion (11) becomes possible. Instead of checking the sign for every value in o along the x-direction, stability is assured when:
!a<0
q=m
that a
1
dy
Since the upper limit for the absolute value of p has been estimated to be approximately unity, eqn (13) assures uniqueness for any first order isothermal (p = 0) or endothermic (p < 0) reaction. In the case of isothermal bimolecular Langmuir kinetics, substituting fl = 0 into the criterion would yield: %C,.v
=g
(14)
under which no bifurcation can occur. For more general nonisothermal cases, no analytical solution can be found, and the problem has to be solved numerically[24]. The results of the calculation are illus-
The general non-isothermal case of bimolecular Langmuir-Hinshelwood trated in Fig. 1. The scale of the vertical axis has been chosen as In(1 + K,C,,) to accommodate the case where KAC,, = 0. The region below each curve, drawn for constant /S-value, provides the condition for uniqueness. For the isothermal conditions, the inequality in eqn (14) holds. This is the point a in Fig. I and corresponds to &CA, = 8. For fi = 0, P, has no effect on this criterion; therefore, a horizontal line can be drawn through point a. Thus all curves converge to the point (or line) of K,,C,,, = 8 as the values of either 6, or 6 approach zero. Multiple solutions exist for the exothermic reactions when any one of the three parameters E,, 8, and K,C,, attains a high value. Since Criterion (9) o&y provides the sufficient condition which assures uniqueness of the steady state solutions, one would expect that the values calculated by this criterion would be conservative in comparison to the numerical solution obtained by solving eqn (2), using the applicable boundary conditions shown in eqns (4) and (5). Nevertheless, as shown by Drott and A&[251 in the case of first order exothermic reactions, the agreement between Luss’s prediction and the exact values over a flat plate is excellent. With @= 0.1, the critical value of s, predicted by Luss is 44 [inequality (14)], while the exact value calculated by Drott and Aris is 45.39. However, this difference becomes larger when a different particle geometry, such as a spherical pellet, is considered. For a spherical geometry and a p value of 0.1, one obtains 48.86 as the critical value for l, as shown on Fig. 2. The difference is even bigger for reactions which have the bimolecular Langmuir-Hinshelwood form (&CAS > 0). For instance, in isothermal conditions, Luss’s prediction yields 8 as the critical value of KACA,,while uniqueness is in fact obtained for K,,C,, < 13.65. Figure 2 is a comparison of the criterion and the “exact” solution. Two lines are drawn for each value of 0. The dotted lines connect the results from the “exact” solution (open circle symbol in this figure). Due to the excessive amount of computational time required to locate these “exact” solution, only a few of these points
kinetics
1631
Fig. 2. Boundaries of regions of unique solution for adiabatic reactions obtained by solving egn (16) and compared with Luss’s criterion. Point (I and a’: isothermal complex reactions (fi =O). Line b: first order exothermic reactions (I&C,,, = 0). are calculated. Figure 2 indicates that the prediction from Criterion (9) becomes much more conservative for bimolecular Langmuir-Hinshelwood kinetics as KACA, increases, particularly for endothermic reactions where B is negative. Figure 3 is a plot of 7 versus &.S for endothermic reactions (@ = - 0.1) illustrating the influence of c, under conditions of strong surface chemisorption characteristics (K,C,, = 100). The shape of the curves indicate that when c, 5 20, up to three values of q exist for one value of .$.. The values representing the three possible steady state solutions correspond to conditions under which the rate of heat required for the reaction equals the rate
3.c
‘I
,-
I., ,P
0.1
0.
Fig. I. Boundaries of regions of unique solution for adiabatic complex reactions. as predicted from Luss’s criterion. Point a: isothermal comolex reactions IL7= 0) I.ine h: firstorder exo&rmic reactions(KAkAs = 0).
Fig. 3. Effectiveness factor as function of Thiele modulus for endothermic reactions.
1632
SAMUELH. WONGand STEPHENSZ&E
Fig. 4. Values of OJ~_, as function of the Thiele modulus in nonisothermal complex reactions. of heat transferred by conduction through the pellet. Multiple solutions with 1) > 1 have previously been calculated for many different cases[l-4] but not for endothermic reactions. Moreover, as this curve (c, = 20) in Fig. 3 shows, multiplicity may occur even for the case where the effectiveness factor is less than unity-a result not discovered in any previous investigations for catalyst under moderate pore diffusion limitation. When the operating parameters (em p, and K,C,,) exceed the critical point shown in Fig. 2, multiple steady state solutions may exist at certain ranges of & values. To determine the stability or instability of any steady state solution, the criteria in eqns (I 1) and (12) can be used. In Fig. 4, the values of w,_, are plotted as a function of 4. at selected values of p. For E, = 20 and &CA, = 20, numerical integration of eqn (16) yields three steady state solutions at various /3 values ranging from -0.1 to 0.01. Tbe plot of w,_, vs 4. would, therefore, reveal the stability status for each of the points in this o,=, vs $. curve. Below the dashed line are negative CO,_, values. They represent those profiles which are unstable with respect to perturbations. The portion of the curves which lie between the two bifurcations just above the dashed line represent the minimal solution in which 11is near unity. The maximal solutions are represented by that portion of the curves where o,=, P 1. For @ = - 0.034, the plot of w,_, with 4” reveals one point in this figure where o,-, = 0. It represents the critical value of p above which all computed solutions are stable. In the case of p = - 0.1, all the points along the TJvs $. curve are unique. Figure 4 shows a monotonic increase of w,=, values as 4, value increases. SUMMARY AND CONCLUSIONS
The results of this study show the existence of multiplicity in the non-isothermal bimolecular LangmuirHinshelwood kinetic model. It is demonstrated that for a sufficiently lugh value of KACAS, multiplicity of steady state solutions can occur in all three thermal cases: exothermic, isothermal, and endothermic. In determining whether a given system may or may not have multiplicities, two methods are used based respectively on (a) Luss’s criterion for uniqueness, and on (b) Michelsen and Villadsen’s criterion for local stability. Luss’s criterion provides sufficient conditions but not
the necessary conditions for uniqueness, thus giving a somewhat conservative evaluation. This method, however, has the great advantage that it does not require knowledge of the concentration profiles. Therefore, it can be used for a priori estimates without solving the differential equations. On the other hand, Michelsen and Villadsen’s criterion provides both necessary and sufficient conditions, although at the cost of considerable computational time. The calculations obtained for predicting multiplicities can be exhibited in compact form in a parameter space showing the boundary between regions of uniqueness and regions of multiplicity. Having three parameters, this can be reduced to a single chart, showing this boundary on the (I + KACAs) vs E, plane for various values of & This chart includes, as special cases, previously obtained results for exothermic first order reactions, and for isothermal Langmuir-Hinshelwood reactions. The chart can be directly used in estimating whether multiplicities will or will not occur in various systems. An interesting finding of this study is the possibility that even endothermic reactions have multiple solutions that can occur even in regions where the effectiveness factor is less than one. Acknowledgements-Computer services used in this research were provided by the Computer Center of the University of Illinois at Chicago Circle. The assistance of the Center is gratefully acknowledged. NOTATION
C.4 concentration for reactant De effective mass diffusivity
species
A
E activation energy F function defined in eqn (7) heat of reaction rate constant pre-exponential factor in the Arrhenius equation for rate constant K* adsorption equilibrium constant for species A rA rate of chemical reaction with respect to reactant species A radial distance from the center of pellet R’ radius of pellet R, gas constant T temperature x dimensionless radial distance = r/R concentration = CA/CA, Y dimensionless center concentration YU dimensionless 2 dimensionless temperature = T/T,
-AH
k k:
Greek symbols p Prater number = (- AW)D,C,,lh.T, -yS reciprocal of adsorption parameter = l/&C,+, c, Arrhenius number = E,/R,T, 7 catalyst effectiveness factor, equation (8) A. effective thermal conductivity & Thiele modulus = R/3~/(kJD,) OJ derivative of concentration with respect to center concentration = dy/dy, Subscripts r quantity related to reaction S at surface, or based on surface
The general non-isothermal case of bimolecular Langmuir-Hinshelwood REF8RENCES
[I] Weiss P. B. and Hicks J. S., Chem. Engng Sci 1962 17 265. 121 Hegedus L. L. and Petersen E. E., Cat. Rev. 1974 9 245. I31Hutchings J. and Carberry J. J., A.LClrE. I. 1966 12 20. 141Roberts G. W. and Satterfield C. N., bd. Engng Chem. 1966 5 317. IS] Wei J. and Becker E. R., Adunn. Chem. Ser. 1975 143 116. I61 Smith T. G., Zahradnik J. and Carberry J. I., Chem. Engng Sci. 1975 30 763. [7] Perelra C. J. and Varma A., Chem. Bngng Sci 1978 33 1645. [B] Schmitz R. A., Advan. Chem. Ser. 1975 148 156. 191 Hlavacek V. and Marek M., C/rem. Engug Sci 1968 23 865. [IO] Villadsen 1. and Ivanov E., Chem. Engng Sci. 1978 33 41. [I 11 Elnashaie S. S. and Mahofouz A. T., Chem. Engng Sci 1978 33 386. 1121 Michelsen M. L. and Villadsen J., Chem. Engng Sci. 1972 27 751. 1131 Tschernitz J., Bornstein 8, Beckman R. and Hougen D. A., Trans. Am. Inst. Chem. Engrs 1946 42 883. 1141 Potter C. and Baron S., Chem. Engng Prog. 1951 47 473. I151 Kabel R. L. and Johanson L. N., A.LChE. I. 1962 8 621. 1161 Hugo P., 4th European Symposium on Chemical Reaction Engineering, Brussd 1968, 459. 1971; Ber. Bunsenges. Phys. Chem. 1970 74 121.
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[I71 (a) Beusch IL, Fieguth P. and Wicke E., Ado. Chem. Ser. 1972 109 615 and Chem. -1ng. -Tech 197244445; (b] Eckert E., Hlavacek V. and Mare): hf., Chem. Engng Commun. 1973 189; (c) Belyaev V. D.. Slinko hf. hf., Timoshenko V. 1. and Slink0 hf. G.. Kinet. K&al. 1973 14 810: (d) Belyaev V. D., Sliiko M. hf., Slink0 hf. Cl. and Timoshenko V. I., Do&l. A&ad. Nauk SSSR 1974 214 1298; (e) Belyaev V. D., Slinko M. G. and Timosheuko V. I., Kinet. K&al. 1975 16 555; (f) Volta S. E., Morgan C. R., Liederman D. and Jacob S. M., Ind. Eng. Chem. Proc. Des. Dev. 1973 12 294; (g) McCarthy E., Zahradnik J., Kuczynski G. C. and Carberry J. J., J. Cnrol. 1975 39 29; (h) Huang T. and Carberry I. J., 69th Annual A.I.Ch.E. Meeting Chicago, Paper No. 109a, 1976; (i) Hegedus L. L., Oh S. H. and Baron K., AILhE. J. 1977 23 632. [IS] Uppal A. and Roy W. IL, Chem. Engng Sci. 1977 32 649. [19] Damkoehler G., Z. Phys. Chem. 1943 Al93 16. [20] Prater C. D., Chem. Engng Sci 1958 8 294. [21] Luss D., Chem. Engng Sci. 1971 8 294. [22] Luss D. and Lee J: C., Chem. Engng Sci 1%8 23 1237. [23] Aris R., Chem. Engng Sci I%9 24 149. [24] Wong S. H., Ph.D. Dissertation, Univ. of Illinois, Chicago 1975. [25] Drott D. W. and Aris R.. Chem. Engng. Sci. 1969 24 541.
H. WONGt
and STEPHEN
SZl?PE*
University of Illinois at Chicago Circle, Chicago, IL 60680. U.S.A. (Received 21 August 1981; accepted 2 April 1982) Abstract-Non-isothermal catalyst effectiveness factors are determined for both exothermic and endothermic reactions with rate equations of the form r, = /&,,/(I + K,,CJ’. It is shown that with such rate equations, multiplicities may occur even for endothermic reactions. In addition, multiple states may arise in regions where the effectiveness factor is less than one. Regions of unique and multiple steady states of bimolecular Langmuir-Hinshelwood kinetics have also been defined. Contrary to previous findings for exothermic and/or isothermal reactions, multiplicities in endothermic reactions are likely to occur in systems showing higher reaction temperatures and stronger surface chemisorption characteristics. Criteria (developed by previous investigators) sufficient for uniqueness and multiplicity of catalyst particle steady state are compared and discussed. INTRODUCTION
reaction takes place within a porous catalyst, multiplicity of the steady state solutions may occur at some operating conditions. This phenomenon was first demonstrated for a firs? order exothermic reaction by Weisz and Hicks[l], then for a negative order isothermal reaction[2], and was also shown to occur for more complex rate equations of the Langmuir-Hinshelwood type [3-71. These results can be expressed in terms of the catalyst effectiveness factor, 7). which is in general a function of the following four dimensionless parameters: the Thiele modulus, &; the Arrhenius number, 4; the Prater number, p; and, the adsorption parameter, &CA,. The major conclusions of the above authors can be summarized as follows:
When
a chemical
(l)Thereisonlyonesteadystatesolution(a)aleithervery
high or very low values of the Thiele modulus, &, or (b) when the values of all three parameters, c_ 8, and K,C,,, are small. (2) In certain ranges of 9.. there may exist three or more different solutions for a given & (a) for positive nth order exothermic reactions (@ > 0), when the values of Q and B are quite large; and (b) for isothermal (p = 0) bimolecular Langmuir-Hinshelwood rate equations [3,4] when the values of the adsorption parameter are sufficiently large. Although these findings set an upper limit of three steady-state solutions for infinite flat plates and for the infinite cylindrical geometries, this is not the case for a sphere[8]. The investigation of Hlavacek and Marek[9] revealed the occurrence of five steady state profiles in the case of spherical geometry for a zeroth order reaction. Others[ lo] reported that an infinite number of steady states may exist for highly exothermic first order reactions when values of /3 approach m. The existence of five steady states has also been shown for the modestly *Author to whom correspondence should be addressed. tPresent address: Northern Petrochemical Company, Technical Center, Morris, IL 60450, U.S.A.
non-isothermal bimolecular Langmuir-Hinshelwood reaction[5,6, Ill. It should be emphasized that all steady state solutions, except for the lower “quenched” and the upper “burnt out” profiles resulting from these analyses, can be shown to be unstable[l2]. The objectives of this investigation are: (I) to study the effectiveness factor for all three cases of thermal conditions (exothermic, isothermal, and endothermic) in reactions that have the following Langmuir-Hinshelwood type of rate equation:
(2) to examine the stability for each of the calculated steady state profiles; and (3) to determine the boundary, in parametric space (c, 0, and &CA*), below which uniqueness of the steady state for the system is generated for all operating conditions. The results obtained from the investigation of the general case of bimolecular Langmuir-Hinshelwood kinetics include as special cases all of the following previously reported studiesrl-71: (1) isothermal and exothermic fist order reactions (p 2 0 and K,C, = 0); (2) Isothermal negative first order reactions (/3 = 0 and &CA B 1); and (3) isothermal Langmuir-Hinshelwood kinetics (fl = 0, any &CA). Problems relating to the direct application of this type of rate equation can be found in many chemical engineering reaction systems, such as the hydrogenation of codimer[13]; the formation of Phosgene[l4]; the dehydration of ethanol[l5]; the decomposition of N,O[l6]; the oxidation of CO and Hl[17a-i]; and the hydrogenation of ole!ins[ll, 181. PROBLEMSTATEMENT
Consider a porous, fully impregnated, spherical catalyst. Reactant A is in the fluid surrounding the pellet at concentration C ,+ and temperature T,. Since essentially all of the active surface is internal, molecules of A must diffuse into the catalyst in order to react. At steady state, 1629
1630
SAMUELH. WONGand STEPHENSZBPE
the mass and heat transfer processes that occur simultaneously within the pellet are described by the following two equations which are presented in dimensionless form:
with the boundary
The above criterion, though easy to use, provides only the sufficient conditions for uniqueness and tends to be conservative in its prediction. Both the necessary and sufficient conditions are provided by the criterion of Michelsen and Villadscn[lZ]. Although this method has been demonstrated only for a first order exothermic reaction, the procedure can be applied quite generally for any form of the rate expression, as shown belnw. By differentiating eqn (2) with respect to the center concentration, yO, the following is obtained:
conditions: dy dz -_=-_=o dx dx
atx=O
atx=l
y=z=
It was shown by Michelsen and Villadsen[lZ] given steady state is asymptotically stable if 1.
(5) o(x) > 0
Combining eqns (2) and (3), and integrating the resulting equation twice, one obtains, with the boundary conditions given, (z-
l)=P(l
-y).
(6)
This result, first derived by Damkoehlerll91, is valid for any form of the rate equation and for any particle geometry[20]. The use of this relationship permits the rate equation to be written in terms of one variable only, either y or .z. Thus, the rate equation as expressed in eqn (1) can be reduced to a function of concentration only. In dimensionless form, one obtains:
(l+d
This criterion
also implies that any steady
dx
SUP
(y-l)6!!G!U [
(11)
(12) state with
dyo,o d&
assures local asymptotic stability, a condition earlier by Luss and Lee1221 and by Aris[23].
shown
RESULTS AND DISCUSSION It
is well recognized that certain reactions following the bimolecular Langmuir-Hinshelwood type of kinetics can exhibit unique or multiple steady states, depending on the system parameters and the inherent bounds on system variables. Starting with eqn (9), it is possible to determine the condition under which uniqueness of the steady state can exist. For a first order kinetics, this is [21]: (13)
x=,’
As illustrated by the earlier workers, the steady state, computed for a catalyst under some conditions, may have more than one solution. The transitional point, which represents the separation between the two regimes, one regime with unique solution and the other regime with multiple states, is called the bifurcation point. To determine the conditions under which no bifurcation point can occur, the following criterion proposed by Luss[21] can be supplied: 05y51
or
Q? 5 4(1+ p).
(-dy >
1.
O,=, >O
dR
This approach reduces the system of equations to a single differential equation describing the change in y with respect to radial position. Since the reaction rate constant is usually a much stronger function of the temperature than of the physical parameters involved (KA, fl, and A,), the temperature dependence of the latter group of variables is neglected in this study. To provide a measure of the actual catalyst performance in a specific condition, an effectiveness factor, 7, is defined:
05x<
This is true regardless of the number of steady states that may arise when solving the system equations. In regions where a maximum of only three multiple steady states can occur, a minor simplification of Criterion (11) becomes possible. Instead of checking the sign for every value in o along the x-direction, stability is assured when:
!a<0
q=m
that a
1
dy
Since the upper limit for the absolute value of p has been estimated to be approximately unity, eqn (13) assures uniqueness for any first order isothermal (p = 0) or endothermic (p < 0) reaction. In the case of isothermal bimolecular Langmuir kinetics, substituting fl = 0 into the criterion would yield: %C,.v
=g
(14)
under which no bifurcation can occur. For more general nonisothermal cases, no analytical solution can be found, and the problem has to be solved numerically[24]. The results of the calculation are illus-
The general non-isothermal case of bimolecular Langmuir-Hinshelwood trated in Fig. 1. The scale of the vertical axis has been chosen as In(1 + K,C,,) to accommodate the case where KAC,, = 0. The region below each curve, drawn for constant /S-value, provides the condition for uniqueness. For the isothermal conditions, the inequality in eqn (14) holds. This is the point a in Fig. I and corresponds to &CA, = 8. For fi = 0, P, has no effect on this criterion; therefore, a horizontal line can be drawn through point a. Thus all curves converge to the point (or line) of K,,C,,, = 8 as the values of either 6, or 6 approach zero. Multiple solutions exist for the exothermic reactions when any one of the three parameters E,, 8, and K,C,, attains a high value. Since Criterion (9) o&y provides the sufficient condition which assures uniqueness of the steady state solutions, one would expect that the values calculated by this criterion would be conservative in comparison to the numerical solution obtained by solving eqn (2), using the applicable boundary conditions shown in eqns (4) and (5). Nevertheless, as shown by Drott and A&[251 in the case of first order exothermic reactions, the agreement between Luss’s prediction and the exact values over a flat plate is excellent. With @= 0.1, the critical value of s, predicted by Luss is 44 [inequality (14)], while the exact value calculated by Drott and Aris is 45.39. However, this difference becomes larger when a different particle geometry, such as a spherical pellet, is considered. For a spherical geometry and a p value of 0.1, one obtains 48.86 as the critical value for l, as shown on Fig. 2. The difference is even bigger for reactions which have the bimolecular Langmuir-Hinshelwood form (&CAS > 0). For instance, in isothermal conditions, Luss’s prediction yields 8 as the critical value of KACA,,while uniqueness is in fact obtained for K,,C,, < 13.65. Figure 2 is a comparison of the criterion and the “exact” solution. Two lines are drawn for each value of 0. The dotted lines connect the results from the “exact” solution (open circle symbol in this figure). Due to the excessive amount of computational time required to locate these “exact” solution, only a few of these points
kinetics
1631
Fig. 2. Boundaries of regions of unique solution for adiabatic reactions obtained by solving egn (16) and compared with Luss’s criterion. Point (I and a’: isothermal complex reactions (fi =O). Line b: first order exothermic reactions (I&C,,, = 0). are calculated. Figure 2 indicates that the prediction from Criterion (9) becomes much more conservative for bimolecular Langmuir-Hinshelwood kinetics as KACA, increases, particularly for endothermic reactions where B is negative. Figure 3 is a plot of 7 versus &.S for endothermic reactions (@ = - 0.1) illustrating the influence of c, under conditions of strong surface chemisorption characteristics (K,C,, = 100). The shape of the curves indicate that when c, 5 20, up to three values of q exist for one value of .$.. The values representing the three possible steady state solutions correspond to conditions under which the rate of heat required for the reaction equals the rate
3.c
‘I
,-
I., ,P
0.1
0.
Fig. I. Boundaries of regions of unique solution for adiabatic complex reactions. as predicted from Luss’s criterion. Point a: isothermal comolex reactions IL7= 0) I.ine h: firstorder exo&rmic reactions(KAkAs = 0).
Fig. 3. Effectiveness factor as function of Thiele modulus for endothermic reactions.
1632
SAMUELH. WONGand STEPHENSZ&E
Fig. 4. Values of OJ~_, as function of the Thiele modulus in nonisothermal complex reactions. of heat transferred by conduction through the pellet. Multiple solutions with 1) > 1 have previously been calculated for many different cases[l-4] but not for endothermic reactions. Moreover, as this curve (c, = 20) in Fig. 3 shows, multiplicity may occur even for the case where the effectiveness factor is less than unity-a result not discovered in any previous investigations for catalyst under moderate pore diffusion limitation. When the operating parameters (em p, and K,C,,) exceed the critical point shown in Fig. 2, multiple steady state solutions may exist at certain ranges of & values. To determine the stability or instability of any steady state solution, the criteria in eqns (I 1) and (12) can be used. In Fig. 4, the values of w,_, are plotted as a function of 4. at selected values of p. For E, = 20 and &CA, = 20, numerical integration of eqn (16) yields three steady state solutions at various /3 values ranging from -0.1 to 0.01. Tbe plot of w,_, vs 4. would, therefore, reveal the stability status for each of the points in this o,=, vs $. curve. Below the dashed line are negative CO,_, values. They represent those profiles which are unstable with respect to perturbations. The portion of the curves which lie between the two bifurcations just above the dashed line represent the minimal solution in which 11is near unity. The maximal solutions are represented by that portion of the curves where o,=, P 1. For @ = - 0.034, the plot of w,_, with 4” reveals one point in this figure where o,-, = 0. It represents the critical value of p above which all computed solutions are stable. In the case of p = - 0.1, all the points along the TJvs $. curve are unique. Figure 4 shows a monotonic increase of w,=, values as 4, value increases. SUMMARY AND CONCLUSIONS
The results of this study show the existence of multiplicity in the non-isothermal bimolecular LangmuirHinshelwood kinetic model. It is demonstrated that for a sufficiently lugh value of KACAS, multiplicity of steady state solutions can occur in all three thermal cases: exothermic, isothermal, and endothermic. In determining whether a given system may or may not have multiplicities, two methods are used based respectively on (a) Luss’s criterion for uniqueness, and on (b) Michelsen and Villadsen’s criterion for local stability. Luss’s criterion provides sufficient conditions but not
the necessary conditions for uniqueness, thus giving a somewhat conservative evaluation. This method, however, has the great advantage that it does not require knowledge of the concentration profiles. Therefore, it can be used for a priori estimates without solving the differential equations. On the other hand, Michelsen and Villadsen’s criterion provides both necessary and sufficient conditions, although at the cost of considerable computational time. The calculations obtained for predicting multiplicities can be exhibited in compact form in a parameter space showing the boundary between regions of uniqueness and regions of multiplicity. Having three parameters, this can be reduced to a single chart, showing this boundary on the (I + KACAs) vs E, plane for various values of & This chart includes, as special cases, previously obtained results for exothermic first order reactions, and for isothermal Langmuir-Hinshelwood reactions. The chart can be directly used in estimating whether multiplicities will or will not occur in various systems. An interesting finding of this study is the possibility that even endothermic reactions have multiple solutions that can occur even in regions where the effectiveness factor is less than one. Acknowledgements-Computer services used in this research were provided by the Computer Center of the University of Illinois at Chicago Circle. The assistance of the Center is gratefully acknowledged. NOTATION
C.4 concentration for reactant De effective mass diffusivity
species
A
E activation energy F function defined in eqn (7) heat of reaction rate constant pre-exponential factor in the Arrhenius equation for rate constant K* adsorption equilibrium constant for species A rA rate of chemical reaction with respect to reactant species A radial distance from the center of pellet R’ radius of pellet R, gas constant T temperature x dimensionless radial distance = r/R concentration = CA/CA, Y dimensionless center concentration YU dimensionless 2 dimensionless temperature = T/T,
-AH
k k:
Greek symbols p Prater number = (- AW)D,C,,lh.T, -yS reciprocal of adsorption parameter = l/&C,+, c, Arrhenius number = E,/R,T, 7 catalyst effectiveness factor, equation (8) A. effective thermal conductivity & Thiele modulus = R/3~/(kJD,) OJ derivative of concentration with respect to center concentration = dy/dy, Subscripts r quantity related to reaction S at surface, or based on surface
The general non-isothermal case of bimolecular Langmuir-Hinshelwood REF8RENCES
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