Ind. Eng. Chem. Rod. Res. Dev. 1082, 21, 29-37
29
CATALYST SECTION
Transients of Monolithic Catalytic Converters: Response to Step Changes in Feedstream Temperature as Related to Controlling Automobile Emissions Se H. Oh* and Jamee C. Cavendbh General Motors Research laboratorks, Wanen, Mlchbn 48090
The transient behavior of automobile monoiithic converters has been studied using a mathematical model which accounts for the simultaneous processes of heat transfer, mass transfer, and chemical reaction. The monolith response to a step increase in the feedstream temperature was examined as a function of catalyst design parameters and operating conditions in order to analyze their effects on the lightoff behavior of monoiithic catalysts. Also, our simulation results showed that a step decrease in the feedstream temperature can lead to a somewhat unexpected transient temperature rise in the solid phase (that is, the catalytic wail) above Rs initial temperature, provided that the reactant concentrations in the feedstream are sufficiently high. This phenomenon is similar to the “wrong-way’’behavior in packed-bed reactors previously reported in the literature.
Introduction Catalytic monoliths used for controlling automobile emissions operate under highly transient conditions. One important mode of transients is a rapidly varying inlet exhaust temperature. At the time of a cold start, for example, a cold monolith is suddenly exposed to exhaust gas a t an elevated temperature. The temperature transients of catalytic monoliths have direct implications in automobile emission control. For instance, converter performance during ita warm-up period is an important factor in the design of catalytic converters, because the Federal Test Procedure (FTP) driving cycle requires a cold start of a vehicle at the beginning of the test. Furthermore, automobile monolithic catalysts often encounter unexplained temperature excursions which can result in thermal degradation or even melting of the monolithic converters (e.g., Morgan et al., 1973). Therefore, it is of practical importance to understand how heat is generated and distributed in catalytic monoliths during exothermic chemical reactions under transient feedstream conditions. It is hoped that such an understanding will aid in the development of monolithic converters which give improved warm-up performance and are less prone to overtemperature problems. In this paper a transient mathematical model of catalytic monoliths is developed which accounts for the simultaneous processes of heat transfer, mass transfer, and chemical reaction. The model is then used to simulate the dynamic behavior of a catalytic monolith following a step change in the feedstream temperature. The Pt-catalyzed oxidation reactions of CO, hydrocarbons,and H2 are considered. The monolith response to a step increase in the feedstream temperature was examined as a function of catalyst design parameters and operating conditions in order to analyze their effects on the lightoff behavior of monolithic catalysts. We also simulated the transient
response of a catalytic monolith after a hot monolith has been subjected to a cool feedstream containing high concentrations of combustible species. Of particular interest is the prediction that a step decrease in the feedstream temperature can lead to a temporary temperature increase in the solid phase (Le., catalytic wall) of the monolith well above ita initial temperature, even though the feedstream temperature assumed after the step change is low enough to quench the reaction completely at steady state. The prediction of such temperature excursions is interesting in that the system’s variable (wall temperature) moves temporarily in the opposite direction to the perturbation in the input variable (feedstream temperature). The occurrence of this “wrong-way” phenomenon in packed-bed reactors has been observed experimentally by many investigators (e.g., Hoiberg et al., 1971; Hansen and Jorgensen, 1974; Van Doesburg and De Jong, 1976a, 1976b; Sharma and Hughes, 1979) and has recently been analyzed in detail by Mehta et al. (1981). Our simulation results suggest that similar phenomena can also occur in monolithic reactors when the reactant concentrations in the feed are sufficiently high and the feedstream temperature is rapidly decreased. It has been observed (Mondt, 1976, 1981) that similar operating conditions (high unburned hydrocarbon concentration and decreasing exhaust temperature) are encountered during sudden vehicle deceleration or prolonged high-speed, closed-throttle coasts. Our model predicts, for a step decrease in the exhaust temperature, a substantial transient temperature rise in the solid phase of the monolithic converter under certain conditions; however, the predicted temperature rise was found to be too small to cause monolith melting, in accordance with a recent experimentalobservation (Mondt, 1981). In succeeding sections a transient, one-dimensional monolith model is developed and used for parametric
Ol96-4321/82/I22I-OO29$Ol.25/0 0 1982 American Chemical Society
30
Ind. Eng. Chem. Prod. Res. Dev., Vol. 21, No. 1, 1982
calculations. Emphasis will be given to the effects of converter design parameters and operating conditions on both the lightoff behavior and temperature excursions of monolith converters. Development of the Mathematical Model In our model we consider the following oxidation reactions over Pt
-
co + ‘/202 cop C3H6 + ’/& 3cop + 3Hp0 CH4 + 2 0 2 Cop + 2Hp0
-
Basic Equations and Assumptions. In this paper we adopt a transient, one-dimensional model to study the dynamic response of a catalytic monolith following a step change in the feed temperature. The model neglects the radial variations of the gas-phase temperature, concentration, and velocity within the individual channels so that these variables are to be interpreted as cross-sectional averages. The material and energy balances for the gas phase are
-+
Here propylene is assumed to be representative of “fastoxidizing hydrocarbons” in automobile exhaust and methane of “slow-oxidizing hydrocarbons” (Kuo et al., 1971). Also, hydrogen is included as a separate species because it significantly influences the behavior of a monolith during the warm-up process, as will be shown later. The specific reaction rate expressions (i.e., rates per unit Pt surface area) for the oxidation of CO, C3H6,and CH4 were obtained by calibrating the rate equations of Voltz et al. (1973) against the recycle reactor data obtained in our laboratory (Schlatter and Chou, 1978). In this calibration, only the preexponential factors of the rate constants were adjusted, without changing the activation energies and adsorption equilibrium constants given by Voltz et al. The rate expressions used in the computations are given below.
A, (1- t
d2T, ) y
ax
4
+ hS(T, - T,) + ~ ( xi=l) C ( - A H ) ~ B ~ ( C ~ , T , ) (12)
The axial dependence of a(x) is explicitly shown in eq 11 and 12 to stress the fact that the noble metal surface area is permitted to vary along the reactor length. The subscript i in eq 9 and 11refers to the species of interest: i = 1, CO; i = 2, C3H6; i = 3, CH4; i = 4, Hp; and i = 5, O D Rc0 = klccoco,/G mol CO/(cm2 Pt-s) (1) Other limiting assumptions invoked in formulating the above equations include (1)negligible temperature gra& s ~ B = kpcc3~co2/G mol C,H6/(Cm2 Pt-s) (2) dients in the solid phase in the transverse direction, (2) negligible axial diffusion of maw and heat in the gas phase, &I, = ~~CCH,CO~/Gmol CH4/(cm2 PbS) (3) and (3) the occurrence of chemical reactions only on the where external surface of the catalytic wall. We remark that axial heat conduction in the solid phase G= is in our model because, as we shall see, it has , )K3cco2cc,~2)(1 ~(~ + K~CNO~’’) anincluded T(1 + Kicco K ~ C C ~ H+ important effect on the wall temperature profile. (4) For our calculations,the accumulation of mass and heat in the gas phase was neglected (that is, d c g j / d t = dT,/dt kl = 6.699 X lo9 exp(-l2556/T) = 0 in eq 9 and lo), since their time constants are typically k, = 1.392 X 10” exp(-14556/T) much smaller than that of the solid thermal response (Young and Finlayson, 1976a). With this quasi-static apk3 = 7.326 X lo6 exp(-l9OOO/T) (5) proximation, then, the appropriate boundary and initial conditions are and K1 = 65.5 exp(961/T)
K 2 = 2.08 X
lo3 exp(361/T)
cg,i(O,t) = cg,iin
(13)
T,(O,t) = Tgh
(14)
K3 = 3.98 exp(ll611/T) K4 = 4.79
X
lo5 exp(-3733/T)
(6)
The rate of hydrogen oxidation in the exhaust gas is assumed to be the same as that for carbon monoxide (Kuo et ai., 1971; Young and Finlayson, 1976b). That is to say aH2 = ~~CH,CO,/G mol H,/(cm2 Ptss) (7) This assumption is compatible with earlier experimental observations (Dabill et al., 1978; Stetter and Blurton, 1980) that in the presence of CO, H2 oxidation rate is inhibited by CO to approximately the same extent as the CO oxidation rate. From stoichiometry, the reaction rate for oxygen is given by BO, = O.5Bco + 4.5aCaH6+ 2 a C H , + 0.5aH, mol 02/(cm2 Pt-s) (8)
TS(x,O) T,o(x) (16) The boundary conditions given by eq 15 indicate that no heat exchange is assumed between the solid and the surroundings both at the inlet and the outlet of the monolith. The values of gas/solid heat and mass transfer coefficients were estimated based on the Nusselt and Sherwood numbers obtained from analytical solutions for fully developed laminar flow with constant wall heat flux (Shah and London, 1971). These Nuaselt and Sherwood numbers are the asymptotic values reached downstream of the developing laminar flow regime located near the monolith inlet; however, the length of this hydrodynamic entrance region is only a small fraction of the monolith length under typical converter operating conditions (Sherony and Sol-
Ind. Eng. Chem. Prod. Res. Dev., Vol. 21, No. 1, 1982
brig, 1970), and thus ita contribution was neglected in the calculations. The heat and mass transfer coefficienta were calculated from
h = NumXg/(mh)
(17)
kqi = ShmDi/ (2RJ
(18)
where the molecular diffusivity of species i, Di, was estimated using the Slattery-Bird formula (Bird et al., 1960), and the thermal conductivity of the reaction mixture, was approximated by that of Nz (=2.269 X 10*T,0.832 cm.s.K). Numerical Solution of the Equations. The purpose of this section is to provide a brief description of the numerical methods used tQ approximate the variables T,,Tg, E, = ( c ~ ,cg,2, ~ , ...,c , , ~ )and ~ E, whose transient behavior is govemed by eq 12,10,9, and 11, respectively. To that end we express these equations in the following compact form
fsi
31
Table I. Standard Set of Parameter Values Pwt= 101.3 kPa (1atm) p s = 2.5 g/cm3 W, = 40 g/s a = 268.95 cm2Pt/cm3reactor A = 60 cmz
L=10cm Rh = 0.06062 cm E
(square channel) = 0.6836 case 1. step increase in feed temperature
Tso= 300 K
1.675 X J/cm.s.K C,, = 1.089 J / g K C,,= 1.071 + 1.56 X T,3.435 x 1 0 4 / ~ , 2J / ~ K h, =
case 2. step decrease in feed temperature Ts!, = 750 K
Note that eq 24 is an initial-value problem which requires evaluation of T,and t?,. Approximations to these values are provided by replacing eq 20, 21, and 22 by spatially discrete analogues. (Tg(xi+l,t)- Tg(xi,t))/h = J'z((T,' + Tsi+l)/29 (Tg(xj+l,t) + Tg(xi,t))/2) (25)
0 = F4(Tg,T,,Eg,G)
(22)
where \k(T,) and Fi (1 I i I 4) are defined in the Nomenclature section. Notice that eq 19-21 represent a set of coupled nonlinear differential equations while eq 22 is an algebraic equation relating T,,T,,E,, and E,. Also, eq 19 is a transient partial differential equation while eq 20 and 21 are ordinary differential equations in the independent variable x with the time variable, t, regarded as a parameter. If the solid temperature T,(r,t)is known at a given value of time, then eq 20 (with appropriate inlet boundary conditions) can be solved for T,and the coupled system of eq 21/22 can then be solved for concentrations E, and e,. These simple observations provide the basis for the numerical methods used to approximate eq 19-22. Our approach to solving eq 19-22 was to first partition the spatial domain (OJ) with a uniform spatial grid n, where II,: 0 = x1 C x z C ... C X N = L. We remark that for the calculations reported here, N = 81. Next, we approximate T,in eq 19 by piecewise linear trial functions $j(x) associated with II, (see Cavendish and Oh, 1979, for details) N
T,(x,t) = c T j ( t ) + j ( x )
(23)
1-1
Galerkin's method is then used to replace eq 19 by a semi-discrete space, continuous-time matrix differential equation of the form d Ts (24) B ( T , ) x = ET', + F1(TB,Tg,t?,) where
T,= (TSvt),T,2(t),..., TsN(t))T T,= (T,(x,,t),T,(%,t), .*., T , ( X N , W
e, = (E,(Xl,t),
E S ( X 2 , t ) , ***,
G,(xN,t))T
The initial condition [T8(0)lj= [T,(xj)] is imposed on T',(t) to complete the system.
(Eg(xi+l,t)- cg(xi,t))/h = Fa((Tg(xi+l,t)+ Tg(xi,t))/2, (Eg(xi+l,t)+ cg(xi,t))/2, (cs(xi+l&)+ E,(xi,t))/2) (26) F4((Tg(xi+l,t) + Tg(xi,t))/Z, (T,'+ T,'+')/2, (Eg(xi+l,t)+ Eg(xi$))/2, (E,(xi+l,t)+ E,(xi,t))/2) = 0 (27) Equation 24 is an ordinary matrix differential equation while eq 25-27 represent a nonlinear set of algebraic Eg, equations in which t is a parameter. If we regard Tg, and E, in eq 25-27 as implicit functions of TB, then eq 24 is of the form - dTs B(T,)-&- = F(T,,t) t >0 (28)
In order to evaluate the right-hand side of eq 28 for a given vector Tsand a given value of t, the nonlinear system of eq 25 must be solved for T,(xi,t), 1 C i IN followed by the solution of the coupled nonlinear system of eq 26/27 for and E,(xi,t). Hence, eq 28 represents a nonlinear initial-value problem in which evaluation of the farcing term requires the solution of a nonlinear set of equations. To integrate eq 28 we used a banded version of the GEAR codes (Hindmarsh, 1976), a carefully engineered Fortran subroutine for the integration of stiff ordinary matrix differential equations. High, controllable numerical accuracy (in the time variable, t ) is achieved at minimum computer cost by the dynamic variation of both time steps and multistep integration methods used in the code. To solve the nonlinear systems of eq 25 and 26/27 for T , ( X ~ + ~and , ~ ) E, , ( X ~ + ~ , ~ E,s(~~+~,t), ), respectively, Newton's method was used. I t should be noted that strictly speaking, the Jacobian of F in eq 28 is not a banded matrix. However, because of the high, local spatial dependence of T,(x,t),~ , ( x , t ) ,and C,(x,t) on T,(x,t),the Jacobian of F is well approximated by a banded tridiagonal matrix. Results and Discussion Table I shows a standard set of parameter values used in the computations for both a step increase (case 1) and a step decrease (case2) in the feedstream temperature. For each of these two cases, to be reported here separately, we will first examine the standard case in detail to gain insight into the system's behavior. Then, the parametric sensi-
32 Ind. Eng. Chem. Prod. Res. Dev., Vol. 21, No. 1, 1982 1000 r
(8) A s (Standard)
p
700
47.2 E
E 0
0.2 0.4 0.6 0.8 Normalized Axial Distance
1.0
Figure 1. Time variation of the wall temperature profile following a step increase in the feedstream temperature, at the standard conditions listed in Table I (case 1).
tivity of the system will be investigated by systematically perturbing the key design and operating parameters around the standard values. The monolith properties listed in Table I correspond to a square-channel ceramic monolith (cordierite) having a cell density of 46.5 openings/cm2 (300 openings/in.2) and a wall thickness of 0.0254 cm. The exhaust gas composition and flow rate for case 1are similar to the test conditions of an engine dynamometer system developed for the evaluation of converter lightoff (Herod et al., 1973). The exhaust compositions used for case 2 were obtained by measuring the engine-out concentrations of a 1979 California Buick 3.8-L V6 engine when the vehicle suddenly decelerated after stable engine operation at 96 km/h (Mondt, 1981). The initial converter temperature for m e 2 is assumed to be uniform, because the exothermic temperature rise in the converter after the stable engine operation was observed to be minimal as a result of the attendant low CO and hydrocarbon concentrations in the exhaust (Mondt, 1981). As will become apparent from the discussion of the results for case 2, the combination of the high reactant (primarily hydrocarbon) concentrations and the high initial monolith temperature listed in Table I results in a temporary temperature rise in the monolith during the transients. During vehicle deceleration, exhaust flow rate also decreases with time; its effect on the temperature excursions will be examined later. Case 1. A Step Increase in the Feed Temperature. In this case we simulate the transient response of a catalytic monolith, initially at room temperature, following a step flow of stabilized exhaust gas at an elevated temperature. Figure 1 shows the time variation of the wall temperature profile in the monolith for the standard conditions listed in Table I. At early times, the hot exhaust heats up the upstream portion of the monolith primarily by convective heat transfer (see t = 15.7 s). As time elapses, however, the downstream section of the monolith becomes hotter than the upstream section (see t = 47.2 s) because the reaction exotherm generated in the upstream section is constantly carried downstream by the exhaust flow. This leads to the development of a temperature peak in the downstream section as a result of a vigorous reaction in that region, and then this temperature peak moves rather slowly toward the inlet of the monolith (see t = 71.8 s, 124.1 s, and 300 8 ) . Such movement of the temperature peak in the solid phase during the converter warmup process is clearly illustrated by Curve B in Figure 2. Also shown in Figure 2 are the results obtained when the thermal conductivity of the solid is perturbed about its standard value (A, = 1.675 X J/cm*s.K) by a factor of 10 (see cymes A and C). The thermal conductivity value assumed in curve C is representative of a typical metal-substrate monolith. Of particular interest is the observation that athough the
zoo[ / a
,
,\
\,
I
50
0
100 Time ( 5 )
150
200
Figure 2. Normalized axial position of wall temperature peak as a function of time for three different values of solid thermal conductivity. Other parameter values are listed in Table I (case 1).
2ot / t // O
I
0
K
I
40
CH4 I
,
/ I
SO
I
120
,
I
160
,
I
200
Time 1s)
Figure 3. Conversions of the individual species as a function of time at the standard conditions listed in Table I (case 1).
location of the temperature peak at early times (for t < 60 s) is independent of the thermal conductivity of the solid phase, it significantly influences the wall temperature profile in the later stages of the warm-up process. This demonstrates the importance of including the axial heat conduction in the solid phase in the modeling of monolith lightoff. It is worth mentioning that a monolithic converter exhibits a different mode of lightoff behavior depending on the temperature level of the exhaust gas. At sufficiently high exhaust temperature (e.g., 700 K), the lightoff occurs at the monolith entrance after the converter is heated up by convective heat transfer, so that the solid temperature remains highest at the inlet of the monolith throughout the transient period. This is in contrast to the case of Tgh = 600K, where the location of the solid temperature peak varies with time along the reactor length, as was illustrated in Figure 2. Figure 3 shows the conversions of the individual species as a function of time for the standard conditions listed in = 600K). Although the conversions at a given Table I (Tgh time are somewhat different among the species of interest here, their lightoff times (say, the time required for 50% conversion) are similar except for CHI, because of the kinetic coupling between the species. For clarity, then, we will use a plot of CO conversion v8. time only in subsequent figures in order to illustrate the warm-up performance of the monolithic catalytic converter. Figure 4 shows the effects of the CO concentration in the exhaust on the lightoff behavior of a monolith. The catalyst lightoff is delayed substantially upon increasing the CO concentration, as can be anticipated from the negative-order kinetics of the CO oxidation reaction over Pt. The variation of the C3Hs concentration similarly affects the catalyst lightoff, because the oxidation rates are also inhibited by C3H, as a result of its relatively strong chemisorption on Pt. Increasing the O2concentration in the exhaust, on the other hand, improves both the lightoff and steady state oxidation performance.
Ind. Eng. Chem. Prod. Res. Dev., Vol.
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33
Table 11. Channel Size and Void Fraction as a Function of Cell Density
0
0
20
40 SO Time Is)
SO
cell density, openings/cmZ
Rh,cm
E
31.0 46.5 (standard) 93.0
0.07710 0.06062 0.03915
0.7371 0.6836 0.5696
100
Figure 4. Effecta of CO concentration in the exhaust on catalyst lightoff. Other parameter values are listed in Table I (case 1). loo
c
!40K 00
20
60
40
Time
Without H2
80
100
(SI
Figure 7. Effects of cell density on catalyst lightoff at Tg*= 600 K. Other parameter values are listed in Table I (case 1).
20
0
20
40
SO
SO
100
Time 1s)
Figure 5. Catalyst lightoff behavior with and without H2in the exhaust. Other parameter values are listed in Table I (case 1).
0 0
0 0
401
2
4
6
8
10 1 2 Time Is)
14
16
16
20
Figure 8. Effects of cell density on catalyst lightoff at T * = 700 K. Other parameter values are listed in Table I (case 1). (kote the change in time scale.)
20
0 0
20
40 Time
SO
80
100
Is)
Figure 6. Effects of exhaust gas flow rate on catalyst lightoff. Other parameter values are listed in Table I (case 1).
The concentration of H2 in the exhaust gas is usually assumed to be determined by the equilibrium of the water-gas shift reaction, which gives a mole ratio of approximately 1:3 between H2and CO (Wei, 1975). As shown in Figure 5, the model predicts considerably faster catalyst lightoff when the presence of H2 in the exhaust is accounted for. It is also interesting to note that H2 in the exhaust leads to a solid temperature exceeding the adiabatic reaction temperature even at steady-state conditions (Heck et al., 1976; Young and Finlayson, 1976b), as illustrated in Figure 1 (compare the wall temperature at x = 0 and x = 1when t = 300 s). Such an overshoot in the solid temperature profile was observed experimentally by Hegedus (1975) during steady-state oxidation of H2. This is so because, as indicated by its high Lewis number ( E Dip,C /%= 4), H2can diffuse to the catalytic wall faster than %e eat generated as a result of reaction can be conducted away. The effect of exhaust gas flow rate on the lightoff behavior of a monolith is depicted in Figure 6. The monolith lightoff is delayed upon increasing the flow rate, primarily due to the attendant decrease in the contact time between the catalyst and exhaust gas. It is interesting to note that the opposite effect is predicted for a packed-bed converter as the flow rate is varied; that is, increasing the flow rate
shortens its lightoff time (Oh et al., 1980). This difference in lightoff behavior can be explained on the basis of a difference in the heat transfer characteristics between the two converter configurations; in contrast to monoliths (where the heat transfer coefficient is virtually independent of gas flow rate as can be seen from eq 17), the gas/solid heat transfer coefficient for packed-bed converters increases with increasing gas flow rate (De Acetis and Thodos, 1960). As the exhaust flow rate through a packed-bed converter increases, this improvement in the heat transfer characteristics dominates the competitive effect of the decreased residence time of gas within the reactor, resulting in the enhancement of the converter lightoff performance. The cell density (that is, the number of openings per unit cross-sectional area) is an important design parameter in developing a monolithic catalyst with improved performance. In actual catalyst design, the wall thickness is usually held constant when perturbing the cell density. This results in the simultaneous variation of both the channel size and the void fraction of the monolith, as the cell density is varied (see Table 11;square channels with a fixed wall thickness of 0.0254 cm). As shown in Figure 7 (Tgh= 600 K), the lightoff characteristics of the catalyst are improved as the cell density is decreased. The improved lightoff performance predicted for monoliths having lower cell densities can be attributed to their lower solid fraction (1 - E), and thus the reduced thermal capacity of the monolith. It is important, however, to emphasize that decreasing the cell density does not always improve catalyst lightoff
Prod. Res. Dev., Vol. 21, No. 1, 1982 100
-
Ts,max,
r--
- E 80
,E 60 f
-
400
-
0
0
20
0
40 Time
60
(SI
80
100
Figure 9. Effects of monolith length on catalyst lightoff. Other parameter values are listed in Table I (case 1).
'"F -
- 80 -
z
I
,i 6 0 E 3
6 400
(AI a = 537.9 ( l - x / L l ( 5 ) a = 268.95 (Standard) ( C ) a = 537 9 x/L
-
20 -
0
0
20
40 Time
(SI
60
80
100
Figure 10. Effects of noble metal activity profile on catalyst lightoff. Other parameter values are listed in Table I (case 1).
performance. When the inlet gas temperature is increased from 600 K to 700 K, for example, the order of catalyst lightoff is reversed; that is, faster lightoff occurs with higher cell density, as illustrated in Figure 8. This can be attributed to the fact that with increasing exhaust temperature, the catalyst's lightoff behavior is increasingly dominated by the convective gas/solid heat transfer (hS(T T,)in eq 12) rather than by the heat capacity of the sofid ((1- e)p,C, in eq 12). In this case, then, rapid catalyst lightoff is favored by high cell density as a result of the attendant decrease in the channel size. (Note that both h and S are inversely proportional to the channel size.) Figure 9 compares the transient responses of catalytic monoliths having different lengths, following a step increase in the feedstream temperature (300 K to 600 K). Here the frontal area of the monolith is kept constant (A = 60 cm2),and a fixed amount of Pt (i.e., constant a0A.L) is distributed in the monoliths of three different lengths. It can be seen that the catalyst lightoff is enhanced upon decreasing the reactor length. Notice, however, that if the reactor length is too short, the catalyst's activity after lightoff (i.e., the steady-state performance) is considerably degraded, as illustrated by the curve for L = 2.5 cm. The computational results also show that the variation of the frontal area of the monolith (while keeping its length and Pt content constant) similarly affects the monolith performance; that is, as the frontal area decreases, a tradeoff again exists between improving the lightoff behavior and impairing the steady-state performance. It should be noted, however, that the simultaneous variation of the frontal area and length of the monolith while keeping its volume constant does not significantly change the lightoff characteristics of the monolith. This is not surprising, because the basic model equations, when appropriately nondimensionalized, indicate that the transient behavior of the monolith depends primarily on the product of A and L,and not their individual values. Figure 10 shows how the lightoff behavior of a monolithic catalyst is influenced by the Pt activity profile along the reactor length. Both the linearly decreasing (curve A)
02
04 0.6 08 Normalized Axial Distance
10
Figure 11. Time variation of wall temperature profile following a step decrease in the feedstream temperature, at the standard conditions listed in Table I (case 2).
and the linearly increasing (curve C) activity profiles are considered, and the computational result for the standard case (uniform activity profile, curve B) is also reproduced in the same figure for a direct comparison. Note that the Pt content in the monolith is kept constant in all three cases that are considered here. It can be seen that the catalyst's lightoff performance as well as the overall performance is improved substantially when the noble metal is concentrated in the upstream section of the monolith (compare curves A and B). In the case of the linearly increasing Pt profile, on the other hand, the lightoff time (say, the time required for 50% CO conversion) is predicted to be slightly shorter than the standard case, but the catalyst's activity before and after the lightoff is lower, resulting in the deterioration of the overall performance (compare curves B and C). Case 2. A Step Decrease in the Feed Temperature. In this case we examine temperature excursions encountered in a catalytic monolith after a hot monolith has been exposed to a step flow of cool exhaust gas containing high concentrations of reactants. The possibility that such exhaust temperature perturbations can lead to a transient temperature rise in the solid phase above its initial temperature (that is, an overtemperature) has been pointed out by Young and Finlayson (1976b). The objective here is to identify conditions under which the overtemperature phenomenon occurs and to determine the magnitude and location of the temperature peak as a function of converter design and operating parameters. As mentioned earlier, case 2 approximates a situation which may occur in a fully warmed-up monolith when a vehicle suddenly decelerates. During deceleration of a vehicle, both exhaust flow rate and temperature decrease. However, since the overtemperature phenomenon is caused primarily by a reduction in the exhaust temperature, we will focus on its effect first. As will be shown later, a simultaneous reduction of the exhaust flow rate and temperature does not change the essential features of the overtemperature phenomenon; that is, a reduction in the exhaust flow rate simply shifts the location of the wall temperature peak toward the monolith inlet without significantly changing its magnitude. Figure 11shows how the wall temperature profile varies with time, following a step decrease in the feedstream temperature. The parameter values were taken to be the same as those listed in Table I. As we expect, the upstream section of the monolith is constantly cooled by the cold feed exhaust. In the downstream portion of the monolith, however, a temperature peak develops above the initial wall temperature ( t = 2.1 s), grows in magnitude, and moves toward the inlet of the monolith ( t = 4.7 s). A t t = 10.1 s, the highest wall temperature (T,,, = 991 K) is attained at a normalized axial location of 0.675. This "hot spot", of course, cannot sustain itself indefinitely; hence
Ind. Eng. Chem. Prod. Res. Dev., Vol.
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0.2
0.4
0.6
0.8
21,
No. 1,
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35
1.0
Normalized Axial Distance
Figure 12. Time variation of gas-phase CO concentration profile following a step decrease in the feedstream temperature, at the standard conditions listed in Table I (case 2).
the temperature peak moves downstream with decreasing magnitude (t = 25.6 s and t = 64.7 s) and eventually leaves the reactor (t = 200 8 ) . The quantity T,,,, shown in Figure 11, is the highest wall temperature reached during the transient process. This will be referred to as the maximum wall temperature, and its magnitude and axial location will be a focal point of our discussion in the sequel. Notice in Figure 11that the exhaust temperature after the step change was chosen to be sufficiently low (Tgh= 300 K), so that the reaction is quenched over the entire length of the monolith at steady state (see t = 200 s in Figure 11). In this paper we will limit our attention to such cases. The simulation results for Tgh = 400 K showed similar temperature excursions in the monolith following the exhaust temperature perturbation. It is important to emphasize that the presence of H2in the feed is not a necessary condition for the occurrence of the temperature rise reported here; the model predicts similar overtemperature phenomena even when the presence of H2in the exhaust is not accounted for. As will be explained in the following paragraph, the transient temperature rise results from the combination of high exhaust concentrations of combustible species and high initial monolith temperature. In order to understand this transient temperature rise, it is instructive to examine the gas-phase concentration profile of the reactant (e.g., CO) as a function of time, for the standard case considered in Figure 11. As Figure 1 2 indicates, the gas-phase CO concentration in the upstream section of the monolith initially increases with time (compare t = 0 and t = 2.1 s), because the reaction rate (and thus the conversion of the reactant) in that region decreases as a result of the cooling of the catalytic wall by the cold exhaust. Because of the relatively large heat capacity of the solid, however, the wall temperature in the downstream section of the reactor continues to remain high. As a result, the gas stream containing high concentration of the unconverted reactants eventually contacts the hot catalytic wall in the downstream section of the monolith. This leads to vigorous reactions in the downstream section of the reactor, as evidenced by a rapid decrease in the gas-phase CO concentration profile at t = 10.1 s. The attendant reaction exotherm, then, causes a transient temperature rise above the initial solid temperature. This condition persists until the downstream portion of the monolith is cooled by the exhaust far enough so that a decrease in the rate of heat generation due to the temperature decrease overshadows the competitive effect of the increased reactant concentration. The appearance of the transient temperature rise is limited to only some parts of the monolith as a result of the spatial variation of the competing effects between the heat generation and
750
0
200 400 600 a (cm2 Pt/cm3 Reactor)
Figure 13. Maximum wall temperature as a function of the local Pt surface area. Other parameter values are listed in Table I (case 2). IA) a = 268.95 cm2 Pt/cmJ Reactor IS1 a = 9 0 . l 0 c m 2 Pt/cm3 Reactor
Y
700 700
750 800 850 Initial Wall Temperature (K)
Figure 14. Effects of initial wall temperature on maximum wall temperature for two different local Pt surface areas. Other parameter values are listed in Table I (case 2).
the gas/solid heat transfer process. It can be deduced from the above discussion that in order for the overtemperature phenomenon to occur, the following two conditions must be satisfied simultaneously: (1) sufficient cooling of the upstream section of the reactor (so that very little reactant is consumed by reactions taking place there), and (2) sufficient reactivity in the downstream section (sothat,the unconverted reactants carried from the upstream section can be readily reacted to generate heat). The delicate nature of the process is illustrated in Figure 13, where the maximum wall temperature encountered during the transient is plotted as a function of the local Pt surface area. Interestingly enough, the largest temperature rise is attained at an intermediate value of Pt surface area. If the Pt surface area is too low, the rate of heat generation in the downstream portion of the reactor is not high enough to cause a significant temperature rise. If the Pt surface area is too high, on the other hand, considerable reaction occurs in the upstream section despite the cooling in that region, so that only a relatively small amount of unconverted reactants contacts the hot catalytic wall in the downstream section of the monolith. For this reason, the resulting temperature rise is smaller than that corresponding to the intermediate Pt surface area, as illustrated in Figure 13. The sharp change in the maximum wall temperature that occurs just on the lefthand side of the peak in Figure 13 can be attributed to the fact that for this range of Pt surface areas, the maximum solid temperature is encountered at the reactor exit and thus its magnitude is limited by the monolith length. Also, we remark that besides its magnitude variation, the location of the maximum wall temperature shifts toward the monolith inlet as the local Pt surface area is increased. Figure 14 shows how the initial wall temperature influences the transient temperature rise following a step flow of cool exhaust gas at 300 K, for two different local Pt
38
Ind. Eng. Chem. Prod. Res. Dev., Vol. 21, No. 1, 1982
- 1150 YI
-
t c
1050
-
E -
'
5 400
950
300 0 850
0
40 Monolith Length (cm)
10
20
30
50
Figure 15. Effects of monolith length on maximum wall temperature. Other parameter values are listed in Table I (case 2).
0.2 0.4 0.6 0.8 Normalized Axial Distance
10
Figure 17. Time variation of wall temperature profile for a linearly decreasing Pt activity profile. Other parameter values are listed in Table I (case 2). 1050
aE
950
wg'log/s 2 0 /
3:
40
y
o
0
I 64
7
Axial Position of Maximum Wall Temperature
o-oo* Cell Density IOpenmgs/cm2)
Figure 16. Effects of cell density on maximum wall temperature. Other parameter values are listed in Table I (case 2).
surface areas. For each of the two Pt surface areas considered, the highest wall temperature (and thus the largest temperature rise, as determined by the vertical distance between the solid and the dotted curves in Figure 14) is reached at an intermediate initial wall temperature; both higher and lower initial wall temperatures lead to lower wall temperatures during the transient. The similarity between Figures 13 and 14 is not surprising, because the Pt surface area and the specific reaction rate (which depends on the wall temperature) appear in the model equations as their product only. In addition, the regime of overtemperature tends to shift toward higher initial wall temperatures as the Pt surface area is decreased (e.g., by sintering). Notice also that the peak temperature of curve B (lower Pt surface area) is higher than that of curve A. Figure 15 shows the effects of the monolith length (L) on the maximum wall temperature. Here, the length was perturbed while keeping both the frontal area of the monolith ( A ) and the Pt content in the reactor (a.A.L) constant at their standard values. As indicated in Figure 15, a higher wall temperature is encountered following a rapid decrease in the feed temperature, as the monolith length is increased. It is interesting to note that the variation of the monolith frontal area (with reactor length and Pt content fixed) similarly affects the temperature excursions of the monolith; that is to say, increasing the frontal area tends to increase the transient temperature rise. However, the simultaneous variation of frontal area and length while keeping their product (Le., monolith volume) constant does not significantly change the magnitude and normalized axial location of the maximum wall temperature. The magnitude of the temperature rise is also affected by the cell density of the monolith. As shown in Figure 16, the maximdm wall temperature increases with increasing cell density. In perturbing the cell density, square channels with a fixed wall thickness of 0.0254 cm were assumed. This resulta in the simultaneous variation of the channel size and void fraction of the monolith, as illustrated in Table TI for three selected cell densities. Also, our calculations show that for a given channel size and void
Figure 18. Effects of exhaust flow rate on the magnitude and location of maximum wall temperature. Other parameter values are listed in Table I (case 2).
fraction, the monolith with circular channels leads to a higher solid overtemperature than the square-channel monolith. This can be attributed to the fact that circular channels provide more effective cooling of the upstream section of the monolith as a result of the higher gas/solid heat transfer coefficient. The noble metal activity profile along the reactor length has an important effect on the solid temperature profile developed within the monolith (again,after a step decrease in the feed temperature). Figure 17 shows how the solid temperature profile varies with time when the Pt activity profile is a linearly decreasing function of axial distance. In this case virtually no overtemperature is predicted, in contrast to the considerable temperature rise (241 "C) observed in the case of the uniform activity profile (see Figure 11). We remark that the total Pt surface area in the monolith is the same for both cases. The resulta displayed in Figure 17 suggest the possibility of minimizing the temperature excursions in catalytic monoliths by a proper choice of the noble metal activity profile. During a sudden vehicle deceleration, the exhaust flow rate is also rapidly decreased, in addition to a reduction in the exhaust temperature. Therefore, it is of practical interest to investigate how the magnitude and location of the maximum wall temperature change with the exhaust flow rate. Figure 18 summarizes the results of a set of calculations for a range of exhaust flow rates. The same step decrease in the exhaust temperature as given in Table I was considered in the calculations. For flow rates lower than 60 g/s, the magnitude of the maximum wall temperature remains more or less constant, independent of the exhaust flow rate, whereas its location tends to shift closer to the monolith inlet with decreasing flow rate. For flow rates exceeding 60 g/s, the overtemperature phenomenon becomes limited by the monolith length, so that the maximum solid temperature is encountered at the reactor exit and the magnitude of the temperature rise decreases rapidly with increasing exhaust flow rate. The effect of the solid thermal conductivity was also investigated by perturbing its value about the standard value (see Table 111). As might be expected, increasing the thermal conductivity decreases the magnitude of the
Ind. Eng. Chem. Prod. Res. Dev., Vol. 21,
Table 111. The Magnitude and Location of Maximum Wall Temperature as a Function of Solid Thermal
Conductivity
As,
J/cm-s.K
1.675 X 1.675 X loF2(standard) 1.675 X lo-‘ 1000
900
Tsmax,K 996 991 915
0.6625 0.6750 0.7625
1
400
300
~XnaxlL
200 s
o
02
04 os o Normalized Axial Distance
I
a
l
l
i o
Figure 19. Temperature excursions in a monolith having a nonuniform initial wall temperature profiie. Other parameter values are listed in Table I (case 2). temperature rise during the transient. Interestingly, however, the magnitude and location of the temperature peak were found to be rather insensitive to the variation of the solid thermal conductivity, and this is indicated in Table 111. We conclude this paper by considering the effects on transient converter response caused by nonuniformities in the initial wall temperature profile. Recall that our discussion so far has been limited to the case where the initial wall temperature profile is uniform over the entire length of the monolith. Under certain operating conditions, however, the reaction lights off a t some point within the monolith, resulting in a nonuniform axial wall temperature profile (e.g., Heck et al., 1976). Figure 19 shows how the wall temperature profile varies with time after a monolith with a nonuniform initial wall temperature profile (Tao= 550 K,x / L I 0.2; Tso= 750 K,x / L > 0.2) has been exposed to a step flow of cold exhaust gas at 300 K. As can be seen from the comparison of Figure 11and Figure 19, the nonuniform initial wall temperature profile does not alter the essential features of the transient response, except that the location of the maximum wall temperature is shifted somewhat downstream. Nomenclature a(%)= catalytic surface area per unit reactor volume, cm2/cm3 A 5 frontal area of the monolith, cm2 B(TJ = mass matrix in Galerkin method, eq 28 c, = concentration of species i, mole fraction c,,, = concentration of species i in bulk gas stream, mole fraction inlet concentration of species i, mole fraction ==specific heat of gas, J/gK = specific heat of solid, J / g K c,”= concentration of species i in the solid phase, mole fraction ea = vector with entries ca,,, mole fraction D, = diffusivity of species in the reactive mixture, cm2/s E = stiffness matrix in Galerkin method, eq 24 Fi = hS(T + a(x)C4,=1(-AH)W(Es,Ts) FZ = hs(
