Znd. Eng. Chem. Res. 1993,32, 1081-1086
1081
Methods To Increase the Efficiency of a Metallic Monolithic Catalyst S. Lennart Andereson' and Nils-Herman Schson Department of Chemical Reaction Engineering, Chalmers University of Technology, S-412 96 Gbteborg, Sweden
The increasing interest in using metallic monolithic catalysts for automotive emission control raises certain optimization possibilities. Creating obstacles in the channels by disfiguring the channel wall in the manufacturing process can give rise to an increased transport rate to the catalyst wall and an increased reaction rate in the wash-coat. In the present mathematical simulation study, the decrease in the residual carbon monoxide a t the outlet and the accompaning pressure drop increase were calculated for different designs of the metallic monolithic catalyst. A standard unsegmented monolith without obstacles was compared with monoliths with two obstacles, monoliths with changed dimensions of the channels, and monoliths segmented into three separate pieces in series. Introduction Monolithic catalysts have their most important use in automotiveemission control, but they are also widely used in catalytic combustion with applications in stationary gas turbines, aircraft gas turbines, highway vehicle turbines, and boilers. For references to nonautomobile combustion applications of monolithic catalysts see, for example, Sinha et al. (1985). Monolithic catalysts have also been used for pure chemical process applications for both gae phase reactions and gas-liquid phase processes (Irandoust and Andersson, 1988). In gas-liquid phase processes, the advantages of the monolithic catalyst over the trickle-bed have resulted in some full-scale process planta (Irandoust et al. 1989). In automotiveemission control applications,the ceramic monolithic carriershave been predominant for a long time, but recently the metallic monoliths have attracted increasing interest. This interest is connected with the use of electrically heated precatalysts for decreasing pollution when starting cold motors. It is also argued that the risk of frost injuries from frozen condensate in severe winter cold is much less with a metallic monolithic catalyst than a ceramic one (Backmansson, 1989). Despite the fact that the gas flow is turbulent in the exhaust pipe upstream from the catalyst, with a Reynolds number of 5 ~ 8 o o o (Wei, O 1975),the flow in the narrow channels of the converter corresponds to a rather low Reynolds number value of only 75-600 (Young and Finlayson, 1976)for typical converters at highway crusing speeds. Since the transition to turbulent flow occurs at approximately a Reynolds number of 2000 for many duct shapee,it istherefore reasonable to assumefully developed laminar flow conditions in the catalyst channels with a considerable resistance against mass and heat transfer to and from the catalyst channel wall. Very few studies have been published about how to decrease the resistance against the transport steps in the monolithiccatalyst. Since themass and heat transfer rates are known to be higher in the entrance region of the channel before the laminar flow boundary layers have fully developed,Wendland (1980)proposed that the monolithic catalyst should be segmented normal to the flow direction and separated into pieces in order to utilize this increased rate of mass and heat transfer. A monolithic catalyst segmented into four separate pieces proved experimentally to give a reduction of the outlet content of hydrocarbons and carbon monoxide by 33% and 4775, respectively,
* To whom correspondence should be addressed.
compared to an unsegmented monolithic catalyst. This effect was also found in a simulation calculation by Wendland (1980). By using a metallic monolithic catalyst the laminar flow can easily be disturbed, thereby increasingthe rate of mass and heat transfer to the channel wall, by introducing obstaclesin the channels. The obstacles are readily created by mechanically disfiguring the channel wall during the manufacturing process. In the present work the effect of these obstacles on the transport processes is studied on the basisof mathematicalsimulationsof the concentration, temperature, and pressure fields of the flow by numerically solving the Navier-Stokes equations for these phenomena. The simulation will also include the oxidation of carbon monoxide proceeding in the wash-coat,whose rate is closely dependent on the external mass and heat transfer conditions. The effect of the obstacles on the degree of conversion will be compared with the corresponding effect found by changing the dimensions of the channels in the absence of obstaclesand also after segmenting the channels in separated pieces connected in series. Mathematical Model The monolithic catalytic converter is assumed to be wellinsulated and, moreover, the inflow gas is assumed to be equally distributed over the entire cross section, which means that the study of the reactor can be reduced to a study of a single channel. Channel Flow Model. In the laminar flow case the equations for transport of mass, heat, and momentum are as follows: Mass flow dPco dt = -V(pcov) .f DV2pco + r Heat flow
+ V2((UT) + A-
= -V(Tv)
dt Momentum flow
p
dVY = -pvTVvy + pV2v, -
cPp
+ pgy -
*
dt dY dv p 2 = -pvTvv, + pV%, + pg, - 3 2 dt dz Moreover, the continuity equation gives
0888-5885/93/2632-1081$04.00/0 0 1993 American Chemical Society
(3a) (3b) (3c)
1082 Ind. Eng. Chem. Res., Vol. 32, No. 6,1993
dp/dt = -V(PV) (4) The symbols are given under the heading Nomenclature. Rate of the Oxidation of Carbon Monoxide in the Wash-Coat. The oxidation of carbon monoxide is assumed to proceed entirely in the wash-coat, since unlike other applications of catalytic combustionthe temperature is too low for the homogeneous oxidation to proceed in the gas phase. Like most simulation calculations of catalytic oxidation of carbon monoxide in monolithic catalysts, the kinetic properties are based on the rate equation given by Voltz et al. (1973)
r = kccoco,/(l
+~
c
~
~ (5) )
with the modifications ink andK introduced and discussed by Oh and Cavendish (1982). The values of the parameters in the rate equation are given in Table I. Internal Transport in the Wash-Coat Layer. In order not to overload the mathematical model and to limit the extent of the numerical calculation, it is important to estimate whether the gradients of concentration and temperature in the wash-coat can be omitted or not. We have to consider gradients in the axial direction (z), perpendicular to the wash-coat surface direction ( x ) , and in the peripheral direction (y) in the wash-coat layer. The gradients perpendicular to the external catalyst surface are the most often discussed gradients. Due to the negative value of the effectivereaction order in carbon monoxide above 1%of this concentration, the catalytic reaction system was shown by Smith et al. (1975) to be self-amplifying with respect to the reaction rate, rather than the more common self-limiting behavior, resulting in a maximum value of the effectiveness factor q much greater than unity. This high value was dependent in a complicated way on the Thiele modulus (d8),the Arrhenius number (y),the dimensionless adiabatic temperature rise (B), and the Biot numbers for heat and mass transfer (Bib and B i m , respectively). Using values of the transport and flow properties given in Table I, it was possible to estimate the following values of these characteristics at 850 K and for 1.2% carbon monoxide, giving: d8= 4.6,/3 = 1.5 X 10-6, y = 14.8, B i m = 0.42,and B i h = 0.003. By comparing these values with those given by Smith et al. (1975), it is found that the effectiveness factor is probably a monotonic decreasing function with respect to C$8 and does not attain values greater than unity. Since the values of the rate constant and the activation energy given by Smith et al. do not agree with the values given in literature, the value of has to be recalculated in order to demonstrate the extent of the concentration gradients. (Only the value of d8above was calculated with the rate constant given by Smith et al.) For 1.2% CO and 0.65% 02 corresponding to stoichiometric conditions, of more interest for an automotive emission control study than the lean burning conditions studied by Smith et al., we obtained for the mass transport modulus according to Weisz and Prater (1954): @M = 3540 at 850 K and CPM = 0.35 at 550 K. This demonstrates that there are important concentration gradients at 850 K, whereas at 550 K the effectiveness factor q is close to unity. A similar test for the temperature gradients, using the Anderson (1963) criterion CPH < 0.75 for isothermal conditions, gave the values CPH = 9.6 at 850 K and @H = 103 at 550 K. This indicates a deviation from isothermal conditions at 850 K, giving a reaction rate that according to Andersson (1963) differs more than 55% from the rate prevailing a t isothermal conditions. The maximal temperature rise in the wash-
Table I. Properties of Gar, Warh-Coat, and Chemical Reaction Gar kinematic viscosity, Y = (-37.45 + 0.15T) X 10-8 m2 8-l thermal diffusivity, a = ~10.7m2 8-1 molecular diffusivity, D = ~10.72m2 8-1 thermal conductivity, X = 2 X 1V kJ m-18-1 K-1 temperature at inlet, T = 550 K War h-Coat' thickness, L = 25 x 10-8 m effective molecular diffusivity at 838 K, 101.3 Ha,b D, = 4.87 X 10-8m28-l effective thermal conductivity: A, = 1.8 X lo-' kJ m-1 8-l K-l Chemical Reaction ~rate constant, k = ko exp(-E/RT) ko = 1.7 X 10'8 kg*kmol-1 m-9 8-1 EIR = 12600K adsorption equilibrium constant, K =KO exp(-AH/RT) KO = 1.87 X 108kg kmol-l -AH/R 96OK enthalpy of reaction (-AHr) = 280 0oO kJ kmol-1
0 T h e name for the t hinporous inert carrier of the Catalytic active layer fiied to the channel wall. According to Oh et al. (1980).
*
coat at 850 K was estimated from AT- = D,(-PH)cco JL to be 1.3 K, which corresponds to a 2.3% increase of the reaction rate. The investigation above thus shows important concentration gradients in the x-direction but rather small temperature gradients in thie direction. Consequentlythe mass transport in peripheral and axial directions may be neglected, while the corresponding heat transport in these directions must be considered. The influence of the concentration gradient in the x-direction on the rate of reaction is described by the effectiveness factor q. The boundary condition at the external surface of the wash-coat gives
D . 3 = qrL 8
The element of the wash-coat is considered to be a slab with closed ends. Since the mass transport in the x- and y-directions is assumed to proceed a t isothermal conditions with a great influence of the pore transport, the effectiveness factor may be written 1 = l/d (7) where the normalized Thiele modulus was calculated according to Bischoffs equation (1965)
and where r is a function of cco and c% according to the rate equation (5). At conditions in which the pore transport has less influence, the effectiveness factor can be approximated by the equation 9 (tanh 9)/d (9) within an error of 5 % . In order to calculate the temperature at the external surface of the wash-coat, the heat transport in the y- and z-directions must be considered, giving
The numerical calculation was performed using a com-
Ind. Eng. Chem. Res., Vol. 32, No. 6,1993 1083
-
E 0.02-0.04 mm
CIW,.,cC"O"
L, I mm Figure 1.Obatacledongthechannel-&dire&ion and inthemiddle
of the channel.
n
-
: 25 mln
Flow velocity field dong the channelaria direction and in the section given by the tine in the crw section of the channel.
Figure 3.
-. . 7 mb Figure 2. Flow velocity field perpendicukr to the channel aris at 2.6 mm behind the fmt obstacle.
mercial 'finite volume" program, PHOENICS (CHAM Ltd) on a SUN SPARC station 1. The program used the solution method described by Patankar (1988). The boundary condition (eq 10) was, moreover, solved by a tridiagonal matrix method based on an algorithm by Thomas (1949).
Figure 4. Carbon monoxide concentration levels perpendicular to thechannel-arisat2.5mmbehindthefmtobstade.Themaximum concentrationis 0.28 mol of COIkg and the concentrationdifference between the level a w e s ia 0.019 mol of COIkg. Table 11. Influence of Obstacles flow rats. 30 z 8-I
flowrate. 61.33 z r'e
OhaaClS
dimensions.m m height distance
Results and Discussion
Dimensionsofthe Channels and the Obstacles. The simulation calculations were performed for channels with sinusoidal-shaped periphery with respect to the channel cross section, different base widths (1.6-4.0 mm) and heights (0.875-1.975 mm), and different lengths (41.9100 mm). The obstacles consist of two parallelepipeds (Figure 1). The fust one was placed at different distances from the channel inlet (5.5-21.5 mm) and the other one 75 mm from the inlet. Some monolithic catalysts segmented into three separate pieces in series contained only one obstacle per segment. The obstacles filled up the whole width of the channel. Figure 1 shows the form of the obstacle along the axis of the channel. In order to simplify the calculations, the top of the obstacle was smoothed with a low hill. The height of this hill corresponded to 1 4 % of the total height. Influence of the Obstacle on Velocity and Concentration. The result of simulations given in Figure 2 shows that the first obstacle causes twooppositelyrotating flow loops. The velocity of the axial flow at different distances from the obstacle is given in Figure 3,showing a characteristic increase of the flow rate near the center of the channel. This type of flow circulation is in aggrement with the experimental results obtained by Mason and Morton (1987).who studied the laminar flow past a sinusoidal obstacle on a plane. They found that the obstacle gave rise to loops whose direction of rotation was dependent on the heightoftheobstacle. Schlichting(1979)foundalready in 1937 a similar flow phenomenon in studying the flow past a row of balls. Figure 4 shows the simulated concentration levels of carbon monoxide behind the obstacle. Owing to the
1
0
2
0.67 0.67 0.67 0.81 0.96
3 4 5 6
0
0.28
2.19
5.6 11.5 21.5 21.6 21.6
a The distance between the fmt obstacle and the inlet. *Inflow rate 12.6 -01 CO e-]. Inflow rate 26.8 mmol CO s '
reaction of carbon monoxide in the wash-coat, a great concentrationdifferenceis created between the center and thewallofthechannel,andthisd~erenceisnotelimiiated by the induced circulation of the gas. In undisturbed laminar flow in front of the obstacle, the difference of concentration is much higher partly due to the fact that thedeereeofconversionislower. resultineinamuchhieher rate oT reaction. Influence of T w o Obstacles on Degree of Conversion and Pressure Drou. In all these simulations the dimensions of the channels were 1.975 mm height, 4 mm width, and 100 mm length, corresponding to a hydraulic diameter of 1.789mm. (Thetheoreticalhydraulicdiameter was 1.61 mm for a perfect sinusoidal cross section of these dimensions, but due to a small truncation of the ends of the sinusoidal shape (cf. Figure 2) the hydraulic diameter is somewhat greater.) The total cross section area was A = 8.08 X 1W m2 and the number of channels was n = 2128. The total channel wall area was S = 1.88 m2. A comparison is made at the constant mass flow rates 30 and 61.33 g s-l, corresponding to vehicle speeds of about 90 and 130 km h-l, respectively, based on a gasoline consumption of 0.1 dm3 km-'(0.043gal mi-% With the hydraulic diameter as the linear quantity, the mass flow rates correspond to Reynolds number values 236 and 482 I
I
1084 Ind. Eng. Chem. Res., Vol. 32, No. 6, 1993 Table 111. Influence of the Dimensions of the Channels
runno. 1 2 3 4 5
6 7 8 9 10 0
channel dimensions,mm height width length 4 100 1.975 3.8 100 1.825 100 1.625 3.4 100 1.575 3.2 3.0 100 1.525 1.475 1.275 1.075 0.975 0.875
3.0 2.6 2.2 1.8 1.6
72 64.3 54.1 47.1 41.9
dh,mm 109A,m2 1.789 8.08 1.669 7.53 1.486 6.71 1.424 6.44 1.388 6.16 1.333 1.150 1.027 0.957 0.883
8.08 8.08 8.08 8.08 8.08
flow rata., 30 e 8-l - _ - I, CO out, pressure drop, n mmols-1 Pamm-l 2.19 2128 0.28 2.80 2267 0.23 2546 0.17 3.93 2682 0.15 4.34 2832 0.13 4.93 3850 5715 7340 10010 12689
0.15 0.07 0.06 0.05 0.03
3.83 5.29 7.06 9.32 11.81
-
flow rate. - . -- - r 61.33 P -8-1 -.
CO out, ReC mmols-l 236 3.17 220 2.87 196 2.36 188 2.12 183 1.97 176 152 136 126 117
2.12 1.53 1.10 0.83 0.65
*
pressure drop, Pamm-1 5.83 8.79 15.44 19.13 22.63
Re 482 450 401 384 374
11.08 11.01 15.30 19.94 25.20
360 311 278 258 239
Inflow rate 12.6 mmol of CO 8-1. b Inflow rate 25.8 mmol of CO s-l. Re = Gddp at 550 K and 100 e a .
at the inflow temperature 550 K and the pressure 105 Pa. The inflow gas contained 1.2% CO and 0.6% 0 2 . The content of CO in the outflow and the pressure drop are given in Table I1 for different heights of the obstacles and for different placings of the first obstacle with respect to the inlet. It is seen that the influence of the obstacles is moderate at the low flow rate, giving at best a reduction of the carbon monoxide unconverted fraction by 32 % . This is, however, obtained at the expense of a 126 7% higher pressure drop. At the high flow rate, the reduction of the residual carbon monoxide is at best 50% with a pressure drop increase of 432% compared to a channel without obstacles (run no. 1). The relatively simple modification of the channel geometry, with two obstacles each corresponding to 7 % of the total channel length and 48% of ita height, thus halves the unconverted fraction of carbon monoxide at the high flow rate, compared to a channel without these obstacles. The pollution consequences of this reduction and the corresponding influence on the engine power caused by the increased pressure drop are, however, outside the subject of this paper. Influence of Channel Dimensions on Degree of Conversionand PressureDrop. These simulations were performed at reaction conditions similar to those given above. The total channel wall area S = 1.88 m2 was kept constant, so the change of the channel dimensions was possible only if the number (n)of channels was varied. A change of the channel dimensionswill therefore also result in a change of the cross-section area (A), which will consequently give rise to an extra pressure drop. The results are given in Table I11 and the simulations concerning 100-mm channels are directly comparable with those given in Table 11. At the low flow rates it is seen that the reduction of carbon monoxide is, at most, 54% (comparerun no. 5 and no. 1 in Table 111)for a channel without obstacles but with smaller height and width. For a channel with obstacles, we have a corresponding reduction of 32% (run no. 6, Table 11). The pressure drop increase is 125% for the channel without obstacles, compared with 126% for the channel with two obstacles. This comparison makes clear that introducing two obstacles is not more effective than changing the height and width of the channel. A t the high flow rate the reduction of the carbon monoxide outflow was only 38% (run no. 5, Table 111)in channels with changed dimensions but without obstacles, compared with 50% reduction in the case of two obstacles (run no. 6, Table 11). The pressure drop increase for the obstacle-free channel was 288 % compared with 432 % for the channel with two obstacles. Also at high flow rates,
the obstacle variant is obviouslynot preferable to a variant where the height and width of the channels have been reduced. The comparison of the results between Tables I1 and I11has hitherto only included channels of the same length. In the second part of Table 111,the length has also been changed. In the first run (no. 6, Table 111)with the length of 72 mm, the residual carbon monoxide is reduced by 46% at the low flow rate and by 33 % at the high flow rate, when compared with the standard channel of 100mm and the same cross-sectional area (run no. 1, Table 111). The pressure drop was increased by 75 % and 90 ?6 at low and high flow rates, respectively. A further change of all dimensions resultad in a very rapid increase of the pressure drop. For example, the channels with the height of 0.975 mm (run no. 9) will give a pressure drop of 326% at the low flow rate and a reduction of the carbon monoxide residual as high as 82 % when compared to reference run no. 1 in Table 111. Since the channel wall area is constant (S = 1.88 m2) in all these simulations, the number of channels will increase very much when changing the dimensions of the channels, and for channels with the height of 0.975 mm, this number has increased from 2128 to 10 010. This shows that it is hardly possible to decrease the dimensions of the channels beyond a certain limit, since the pressure drop will increase greatly without any correspondingreduction of the carbon monoxide residual. Very small channels will also lead to increased practical problems in the manufacture of the metallic monolith. Influence of Segmenting the Monolithic Catalyst into Three Separate Pieces in Series. The influence of segmenting the monolith was simulated for channels with the standard dimensions 1.975 mm height, 4 mm width, and 100 mm total length, and with a cross-section area of A = 8.08 X 1 0 3 m2 and a total channel wall area of S = 1.88 m2. The simulation also included three segmented channels in series with one obstacle in each. The effect of segmenting the monolithic catalyst is given in Table IV. Segmenting without obstacles will increase the reduction of the residual carbon monoxide by 43% at the two flow rates with 16% increased pressure drop at the low flow rate and only 7 % increased pressure drop at the high flow rate. Obstacles in the segmented catalyst will increase the reduction of carbon monoxide only moderatelycompared with the segmented catalyst without obstacles. A more precise comparison between the alternative ways to design the metallic monolithic catalysts is obtained by calculating the pressure drop at the same increased reduction of carbon monoxide. This calculation is based on an interpolation of data and the increased reduction of carbon monoxide is compared with the standard
Ind. Eng. Chem. Res., Vol. 32, No. 6,1993 1085 Table IV. Influence of Segmenting the Converter obstacle dimensions, m m flow rate, 30 g 8-1 flow rate, 61.33 g s - l c run no. height distance CO out, mmol s-1 pressure drop, Pa mm-1 CO out, mmol s-1 pressure drop, Pa mm-1 1,nonsegmentad 0.28 2.19 3.17 5.83 2,segmentad 0.16 2.53 1.77 6.21 3,nonaegmentad 0.67 11.6 0.23 2.69 2.15 12.98 4,segmented 0.67 11.5 0.13 3.38 1.26 8.37 a Distance from the inlet. Inflow rate 12.6 mmol of CO s-l. Inflow rate 26.8 mmol of CO 8-1.
*
Table V. A Comparison of Pressure Drop at Different Increased Reductions of CO Relative to the Standard Monolith for Alternative Catalysts increased type of catalyst with two obstacles changed dimensions (length 100 mm)
with two obstacles changed dimensions (length 100 mm) changed dimensions (length < 100 mm) segmented, without obstacles
reduction of CO, % 33 33
pressure drop, Pa mm-1 flow rate, flow rate, 30 g 8-1 61.33 g s-l 4.9 14 3.4 19
44 44
4.1
20 28
44
3.7
11
44
2.5
6.4
unsegmented catalyst without obstacles (cf. Table 111,run no. 1). The results are given in Table V. At the increased reduction of 33% ! ,only data from two alternative designs are available since the other alternatives are more efficient, giving a higher reduction of carbon monoxide at the flow rates of interest. From Table V it is seen that the two obstacle alternatives give a lower pressure drop than the alternative with changed channel dimensions at the high flow rate. At a higher degree of carbon monoxide reduction (44% 1, all the design alternatives can be compared. The differences between these are particularly marked a t the high flow rate. The results show that the monolithic catalyst divided into three segments will give only a small pressure drop compared to the other alternatives, and this advantage remains at the low flow rate.
Conclusions Compared to the standard monolithic catalyst, the unsegmented monolithic catalyst with two obstacles is still an effective way to attain a reduction of the residual carbon monoxide, although a t the cost of increase pressure drop. This may be a simpler method of achievingsuch a reduction of the pollution, in terms of the manufacturing of the monolith, than the alternatives with a segmented catalyst or a catalyst with more narrower channels. The design of obstacles, and their combination with segments, should evidently be investigated further as means of improving the effectiveness of catalysts. It is, however, important to add that these different alternatives of the common monolithic catalyst do not decrease the tailpipe emissions during the cold start-up due to the fact that the bulk gas mixing and the transport processes do not have the same decisive effect at low temperatures. This means that the total emissions collected during the EPA-prescribed driving cycle will not be noticeably decreased. It should also be mentioned that the heat capacity of the corrugated monolithic catalyst with obstacles probably is 1 to 2% higher than without these obstacles. This higher value depends on the fact that the underside of the obstacles are filled with wash-coat substance. This will contribute to a somewhat slower temperature rise during the cold start-up.
Acknowledgment The financial support of Volvo Research Foundation and Volvo Educational Foundation, Gijteborg, Sweden, is gratefully acknowledged. Nomenclature A = cross-section area, m Bih = Biot number for heat transfer (hL/&) Bi, = Biot number for mass transfer (k&/D,) c = concentration, kmol kg1 c, = specific heat, kJ kgl K-1 cpw = specific heat of wash-coat, kJ kgl K-1 D = molecular diffusivity, m2 5-1 dh = hydraulic diameter, m E = Arrhenius activation energy, kJ kmol-1 g = acceleration of gravity, m 5-2 AH = enthalpy of adsorption, kJ kmol-1 AHr = enthalpy of reaction, kJ kmol-1 h = heat transfer coefficient, kJ m-2 5-1 K-1 h = height of the obstacle, mm k = rate constant, kg2 kmol-1 m-S 8-1 K = absorption equilibrium constant, kg kmol-1 k, = mass transfer coefficient, m 5-1 L = thickness of wash-coat layer, m n = number of channels p = pressure, Pa q = heat of reaction, k J ma 5-1 r = rate of reaction, kmol ma 5-1 R = gas constant, kJ kmol-1 K-1 S = channel wall area, m2 T = temperature, K t = time, s u = velocity, m 5-1 Subscripts CO = carbon monoxide e = effective value h, H = heat m, M = mass s = external surface conditions w = average wash-coat property x = direction perpendicular to the wash-coat surface y = peripheral direction z = axial direction Superscripts O = pre-exponential coefficient T = transpose of a velocity vector Greek Symbols a! = thermal diffusivity, m2 5-1 j3 = the dimensionless adiabatic temperature rise (cco.(-~r)DdTJ.3) y = the Arrhenius number (EIRT,) c = height (Figure l),m t) = effectiveness factor X = thermal conductivity, kJ m-1 s-1 K-' p = dynamic viscosity, kg m-1 8-1 Y = kinematic viscosity, m2 5-1 p = density, kg ma 4 = Thiele modulus (eq 8)
1086 Ind. Eng. Chem. Res., Vol. 32, No. 6,1993 @H = heat transport modulus according to Anderson @M = mass transport modulus according to Weisz and Prater
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Sparrow, E. M., Schneider, G. E., Pletcher, R.H., Eds.; John Wiley & Sone, Inc.: New York, 1988; Chapter 6. SchlichtinE.H. Boundary-Laver Theory. - .7th ed.:McGraw-Hill Book Co.: New York, 197%-p 656. Sinha, N.; Bruno, C.; Bracco, F. V. Two Dimensional, Transient Catalytic Combustion of CO-Air on Platinum. PCHPhys. Chem. Hydrodyn. 1985,6,373-391. Thomas, L. H. Elliptic Problems in Linear Difference Equations over a Network. Wateon Science Computer Laboratory Report; Columbia University: New York, 1949. Voltz, S. E.;Morgan, C. R.; Liederman, D.; Jacob, S. M. Kinetic Study of Carbon Monoxide and Propylene Oxidation on Platinum Catalysts. Znd. Eng. Chem. Rod. Res. Deu. 1973,12,294-301. Wei, J. The Catalytic Muffler. In Chemical Reaction Engineering; Hulburt, H. M., Ed.; Adv. Chem. Series; American Chemical Society: Washington, DC, 1975;Vol 148,pp 1-25. Weisz, P. B.; Prater, C. D. Interpretation of Measurements in Experimental Catalysis. Adv. Catal. 1954,6,144-196. Wendland, D. W. T h e Segmented Oxidizing Monolith Catalytic Converter, Theory and Performance. Trans. ASME. 1980,102, 194-198. Young,L. C.; Finlayaon, B. A. Mathematical Models of the Monolith Catalytic Converter: Part I. Development of Model and Application of Orthogonal Collocation. MChE J. 1976,22,331-343. Received for review September 24, 1992 Revised manuscript received February 5, 1993 Accepted March 6,1993
