An alternative explanation of the results obtained must take into consideration the state of flocculation in the ball mill. This is very likely because of the influence of acidity or alkalinity on the state of flocculation. I t is only a special case of the effect of electrolytes and is a n important one. T h e ions of water, H’ and OH-, are more effective control ions than other monovalent ions. Hence, the increased production of fines during grinding in a neutral medium compared with a n acidic or alkaline medium is probably due to the fact that the greater flocculation in a neutral medium makes the flocculated material more amenable to receiving the impacts. From the present work, it is not possible to say which explanation holds good. Probably both are true to a varying degree.
I
Conclusions
T h e initial rate of formation of fine particles smaller than a given size is a function of the given size T h e kinetics of comminution of quartz is a zero-order reaction. T h e same is true for wet grinding (pH 5 to 9). T h e initial rate of formation of fines is faster when the grinding is performed in a medium which is almost neutral ( p H 6.8). In weakly acidic and basic medium, the rate is lower compared with that a t p H 6.8.
OH
OH
literature Cited
Arbiter, N.,Bhrany, U. N., Trans. A I M E 217, 245 (1960). Berry, C. E., Kamack, H. J., Proc. Second Intern. Congr. Surface Actioity IV,196 (1957). Engelhardt, W. V., h’aturwissen. 33, 195 (1946). Frangiskos, A. Z., Smith, H. G., Progr. Mineral Dressing, Trans. Intern. Mineral Dressing Congr. Stockholm 1957, p. 67. Gaudin, A. M., Trans. A I M E 202, 561 (1955). Gilbert, I,. A., Hughes, T. H., Vortr. Diskussionen Europaeischen Symp. Zerkleinern, I , Frankfurt am M a i n 1962, p. 170. Gomer, R., Smith, C. S., Ed., “Structure and Properties of Solid Surfaces,” p. 164, University of Chicago Press, Chicago, Ill., 1953.
0PRODUCTS
RECEIVED for review J u n e 6, 1966 RESUBMITTED March 9, 1967 ACCEPTEDSeptember 25, 1967
Figure 5. Reactant, activated complex, and products in the wet grinding of quartz
THERMAL BEHAVIOR OF AN EXHAUST GAS CATALYTIC CONVERTOR JOSEPH V A R D I AND W I L L I A M F. B I L L E R
Products Research Division, ESSOResearch and Engineering Co., Linden, .V. J .
aim of this work was to examine the transient thermal behavior of a fixed-bed catalyst reactor for treating the exhaust gases of an internal combustion engine. T h e analysis and the original data presented will predict the behavior of the reactor and, therefore, provide a n approach for an effective reactor design. Although in most chemical processes the temperature and the flow rate of the feed are adjusted to produce optimal process conditions, the exhaust gas conversion process provides a unique situation in which the feed cannot be modified easily and thus feed conditions predetermine process operation. T h e operation of the exhaust gas reactor HE
07036
includes continuous variation of the temperature and the flow rate of the feed in the range of 0’ to 1600’ F. and 0 to 200 cu. feet per minute, respectively. This results in variable physical properties of gas and catalyst and a changing heat transfer coefficient. The changing feed conditions impose a specialized empirical boundary condition in the mathematical description of the behavior of the reactor. Therefore, the problem a t hand is far more complicated than the simplified situations of constant properties, a constant heat transfer coefficient, and a constant temperature feed: the unsteady-state operation of a packed bed with the assumption of a step change in the VOL. 7
NO. 1
JANUARY 1 9 6 8
83
~~~
~
A combined theoretical-experimental study of the thermal behavior of a catalytic convertor for controlling air-polluting emissions in the exhaust gases of motor vehicles consisted of devising a numerical solution to a mathematical model describing the transient temperature profiles in the convertor and experimentally measuring the flow rates and temperatures of the exhaust gases to b e treated. Data were obtained for two vehicles operating on a chassis dynamometer over the driving regime prescribed by the California Motor Vehicle Pollution Control Board for exhaust device testing. The measured exhaust gas data and the mathematical model were combined to predict the dynamic behavior of a catalytic convertor for various sets of conditions. Excellent agreement was obtained when predicted temperature profiles were compared with measured profiles in an experimental convertor. The study enabled a quantitative evaluation of the restrictions imposed on the usefulness of the catalytic convertor approach to reduce air pollution due to warm-up problems.
temperature of the incoming gas. This was considered by Furnas (1930). The packed-bed heat regenerator problem was investigated by Anzelius (1926) and Schumann (1929) and some numerical solutions were described by Mickley et ai. (1957) and Willmott (1964). Analysis
Concentrations of pollutants in the exhaust gas of an internal combustion engine of a motor vehicle can be reduced by passing the gases through a fixed-bed catalyst reactor, which is referred to generally as a catalytic convertor or simply a convertor. The convertor is imagined to be placed a t or after the exhaust manifold of the engine. Since a catalyst bed can be designed to perform the functions of the muffler of the motor vehicle too, the catalytic convertor can be thought of as a special muffler-namely, a catalytic muffler. A practical volume for a catalytic convertor is in the range of 5 to 1 0 liters. Only a small pressure drop in the convertor is tolerated without affecting the normal operation of the vehicle; consequently, the depth of the catalyst bed is about 4 inches. I t is believed that the primary mode of heat transfer between the gas stream and the catalyst is forced convection and the heat loss from the outer surfaces of the convertor to the surroundings is negligible because of insulation. T o provide a general method of convertor design and not restrict the utility of the analysis to a specific catalyst system, the heat generation due to the chemical reactions is considered negligible. This situation corresponds to the adaptation of the catalytic convertor to the new motor vehicle which is either equipped with an intermediate mechanical control device or has new engine design and operation aimed at reducing the original exhaust emissions. The analysis is, of course, precise for the type of catalytic reactions producing little or no heat. Moreover, though this approach will predict low steady-state convertor temperatures for exothermic reactions coupled with high concentration of pollutants, the analysis will hold even for this case during warm-up, because no appreciable chemical reaction on the surface of the catalyst can take place before warm-
mathematical analysis is confined to the axial direction alone and the volume element between x and x dx may be analyzed. By writing the expressions for heat conduction into the element, the heat conduction out, the gas stream in, the gas stream out, the heat transferred to the solid, and the heat accumulated into the volume element, the following partial differential equation is obtained for the gas phase.
+
where h is the local heat transfer coefficient obtained from available literature (Bird et al., 1960).
h = 0.91 C,GRe-oJ1Pr--2/a
h = 0.61 CQGRe-o.41Pr-2/a
< 50) (Re > 50) (Re
(2) (3)
Equations 2 and 3 consider the heat transfer process as taking place on the catalyst outer surface. The equation for the solid phase is:
ke, is the point-contact conductivity between the catalyst pellets and may also be obtained from available literature (Smith, 1956). In Equation 1, the conduction term, b/bx [keg (bT,/bx)], and the accumulation term, pepCg(bT,/br), are very small in comparison to the other two terms and thus both may be neglected-for example, at 68' F. and 30 cu. feet per minute the values of the coefficients of the terms dropped are k , , = 0.006, pLgCg= 0.007, and k,, = 0.022. This is in contrast with the following coefficients of the terms left: pesCs = 12, GC, = 38, and ash = 3330. Similarly in Equation 4, the point-contact conduction term, d/bx [k,, (bT,/bx)], is also very small. After rearrangement, Equations 1 and 4 may be written as follows:
TS bt
- = a,h(T,
bT, bz ~
- T,)
(5)
+ a,hL - ( T , - T,) = 0 GC,
The simultaneous solution of Equations 5 and 6 may be obtained numerically by replacing the partial derivative terms in Equations 5 and 6 by finite difference approximations and the following equations are derived. z = O
i-1 i iil
z = 1
According to the afore-mentioned assumptions, there are no radial temperature gradients in the convertor. Thus, the 84
I L E C PROCESS DESIGN A N D DEVELOPMENT
(7)
The primes are used to indicate quantities a t time t 4- A t . I n order to apply Equations 7 and 8, the convertor is divided into a large number of sections. T h e temperature of the At is computed from the temsolid in section i and time t perature of gas a t section i and the solid temperature a t time t by Equation 7. Similarly, the temperature of the gas in contact with the solid is obtained by Equation 8. In order that Equation 7 may approximate partial differential Equation 5 , there is a restriction on the time step-namely,
+
(9) Equation 8 is obtained from Equation 6 by employing backward differences. T h e choice of backward differences is prescribed by the nature of partial differential Equation 6 a n d involves mathematical stability considerations for the numerical solution (Hellums et al., 1961). T h e use of other types of finite difference approximations for the partial derivativesfor example, central differences as implied by the derivation of Mickley et al. (1957)-results in two restrictions imposed on the numerical solution, both on the time step, At, and on the axial interval, A z . For a particular simultaneous solution of Equations 5 and 6, initial and boundary conditions have to be stated: the initial temperature of the bed and the temperature and flow rates of the incoming gas.
into the exhaust manifold and along the exhaust pipe. Consequently, it was possible to measure the temperature close to the exhaust valve. Figures 1, 2, and 3 show measured temperatures and engine air-intake rates. I n the California test (State of California Motor Vehicle Pollution Control Board, 1964), the vehicle is driven in a prescribed cycle which lasts for 137 seconds (Figures 1, 2, and 3). The California test starts with the engine a t a nominal ambient temperature of 60’ F. and terminates after the vehicle is driven seven of the 137-second cycles. (The first four cycles in the test are referred to as the “cold cycles” and the sixth and seventh cycles are called the “hot cycles.”) Only the first and seventh cycles of the California test are presented in Figures 1 to 3. However, a computer program was written in order to process all the recorded data; thus temperature and flow rates for all seven cycles were actually tabulated. Figure 1 is for the maximum temperature available at the exhaust manifold of vehicle A, and Figure 2 is for the temperature a t the muffler inlet of the same vehicle. Figure 3 is for vehicle B, which was equipped with a n experimental catalytic convertor instead of the standard muffler. Computation of Temperature Profiles
I n the numerical computation transient temperature profiles of the catalyst bed were evaluated. I n addition, an average catalyst bed temperature was obtained by integration of the local profiles as follows: Pl
T,,may be assumed constant and equal to the ambient temperature. Similarly, To,has a constant value a t steady speed of vehicle operation. However, in urban driving, the temperatures, the flow rates, and the chemical composition of the exhaust gases do not remain constant, because of the continuously changing conditions of the vehicle operation. Similarly, the operation of a vehicle on a chassis dynamometer over the driving regime prescribed by the California Motor Vehicle Pollution Control Board for exhaust device testing produces similar changing conditions. This procedure of testing is, in effect, a simulation of an average trip in the Los Angeles area and is referred to as the “California test.” Accordingly, T,, = T,,(s) and G = the feed is varying in time-namely, G( 7)-and this variation needs to be determined experimentally.
While considering a chemical reaction progress on the catalyst bed, the average temperature concept is of very little utility. However, the average temperatures are helpful for the design and comparison of various systems. For the numerical computation, a digital computer program was written in Fortran for the IBM-7094 computer. I n the program, the exhaust gas was assumed to have the physical properties of nitrogen-namely,
Experimental Setup and Measurements
Equation 13 is an empirical expression of viscosity of nitrogen (Hodgman et al., 1962). For a simulation of a convertor in the California cycle, the following data had to be furnished:
T h e dynamic determination of T g o ( r and ) G ( r ) in a California test requires a measuring and recording system with a fast response. Such a system was assembled. T h e test vehicle was operated on the chassis dynamometer for the prescribed California test. The components of the experimental system were a Meriam laminar flow element, a Decker 306 pressure transducer, and a Sanborn direct-writing optical oscillograph Model 650 with a medium gain amplifier Model 658-3400. T h e Meriam element was connected to the carburetor of the vehicle to measure the air intake. The exhaust gas temperature was measured a t different locations of the exhaust system with very sensitive micro-miniature thermocouples manufactured by Baldwin-Lima-Hamilton. Experimental data were obtained on two vehicles which had their exhaust pipes insulated from the exhaust manifold to the muffler. Vehicle A had a 283-cu. inch engine and vehicle B had a n engine displacement of 318 cu. inches. Each of vehicles A and B was equipped with a V8 engine, a n automatic transmission, and a two-barrel carburetor. T h e muffler in vehicle A was closer to the engine than in vehicle B. Thus, greater heat losses would be anticipated in vehicle B than in vehicle A. I n vehicle A, thermocouple inlets were inserted
Jo
p = 7.185 X
C, = 0.2330
To.655
(lb.msBJft. hr.)
+ 0.0248 X 10-8 T (B.t.u./lb. ” R.)
(1 3) (14)
1.
Superficial surface of catalyst, as, sq. foot per cu. foot Dimensions of convertor Cross-section area, S,sq. feet Depth of convertor, L , feet 3. Size of “time” step, At = Ar/pulCa,hr. ” F. cu. ft./B.t.u. 4. Initial catalyst temperature, T,,(” R.) 5. Specification of axial interval, A z 6. Ambient conditions for feed measurements-namely, outside and inside temperatures (” F.), and the barometric pressure (inches of Hg) 7. Shape factor for catalyst particles ($ = 1.0 for spheres, 0.91 for cylinders) 8. Prandtl number for the gas, P r (Pr = 0.73)
2.
The accuracy of the numerical solution was investigated and it was found that a time step of 1.0 X 10-6 or 2.0 X 10-5 hr. ” F. cu. ft./B.t.u. produced similar results. O n the other hand, the size of the axial interval, A z , influenced the value of the computed temperatures somewhat. Computations with Az = VOL 7
NO. 1
JANUARY 1960
05
I
1600
I
I
I
I
I
I'
>I
I
I
1st C Y C L E
I
I
I
I
7th C Y C L E
Figure 1. Measured exhaust gas in California test Vehicle A, 283-cu. inch engine, temperatures at manifold location
20
0
60
40
80
100
120
14b TIME, SECONDS
I
1600
1400
I
I
1
I
I
I
1st C Y C L E
7111 C Y C L E
1200
Figure 2. Measured exhaust gas in California test Vehicle A, 283-cu. inch engine, temperatures at muffler location
a
I w W
T I M E , SECONDS
I 1
Figure 3. Measured exhaust gas in California test
k W K
+
3
Vehicle 8, 31 8-cu. inch engine, temperatures at muffler location
a K W a
I !W
4
0 1st C Y C L E
0
I
I
I
I
I
h 7th C Y C L E
800
6ool P-J1 2
N
TEMPERATE
x
200
I
a
400
150 W
VI a
200
100
6 E '2 0
50
w* Y
2 O 0 T I M E , SECONDS
86
I L E C P R O C E S S D E S I G N A N D DEVELOPMENT
S :
0.025 produced reliable quantitative results. Moreover, computations with Az = 0.05, or even with Az = 0.10, produced accurate qualitative results. Computed vs, Measured Temperature Profiles in an Experimental Convertor
T o test the agreement between the computed temperatures and the actual developed profiles, vehicle B was equipped with an experimental convertor, in which the shell of the vehicle standard muffler housed the catalyst pellets. T h e convertor was well insulated. Its cross section was 0.256 sq. foot and its depth 13 inches. Eight liters of chemically inert cylindrical pellets were used. T h e measurements were taken while running a standard California test. At the time of the test, the dynamic air intake to the carburetor and the temperatures of the exhaust gases before entering the convertor were also recorded: Table I represents the measured and the computed temperatures inside the catalyst bed a t various fractional depths of the convertor. The data listed are for the end of each of the seven cycles in the California test. Inspection of the listed temperatures in Table I indicates a n excellent agreement between the measured and computed temperatures. The deviation of the computed profiles is about 10%. In the low temperature range during the first cycle of the California test the measured are higher than the computed results. This is expected, since the exhaust gas was assumed 'to be nitrogen ; thus condensation and evaporation of water vapors in the actual exhaust gas stream are neglected. T h e range of 100" to 300" F. is not important for any catalyst operation. By the end of the test the computed temperatures are slightly higher. This is attributed to the thermal inertia of the shell of the convertor and some heat losses through the insulation. However, if the container had a substantial thermal inertia, a larger deviation between measured and computed results would be expected. In such a case, a correction for the thermal inertia of the vessel can be included.
Results of Investigation
Temperature profiles and average catalyst bed temperatures were investigated for several cases of convertors in a California test operation. (The simpler case of steady speed operation, step change in the temperature, is mentioned in the appendix.) Since in practice the volume of the convertor is determined by back-pressure and available space considerations, a convertor with a volume of 0.25 cu. foot (7 liters) was selected. (An anticipated configuration of 1-sq. foot cross section and 3-inch depth is used.) This volume is reasonable for a practical catalyst convertor. The variables investigated were: Dynamic exhaust gas data from vehicles Vehicle A for exhaust manifold location Vehicle A for muffler location Vehicle B for muffler location 2. Catalyst systems Catalyst A, cylindrical pellets, Q, = 300 s q . ft./cu. ft., p e a = 0.83 g./cc. Catalyst B, cylindrical pellets, a, = 270 sq. ft./cu. ft., p e a = 0.57 g./cc. 1.
I
,
Figures 4, 5, and 6 are computed transient temperature profiles for catalyst A and Figure 7 is for catalyst B. Figure 4 i
7nn
~ 6 . 8 5M i t i (3rd C y c l e )
a
,>I
L
=
0.25 ft
Pr
=0.,73
1
0.2
0
End of Cycle in California Test
a
1.0
Figure 4.
Computed profiles
Manifold location, vehicle A, catalyst A I
Table 1.
0.8
A X I A L DISTANCE, z
l
l
1
I
1
I
,6.58 M i n (3rrl C y c l e )
1400
~~~~~
0.6
0.4
Transient Temperature Profiles in Experimental Convertor
Measured Temp, in Experimental Time, Min. Convertor, F. At Fractional Axial Distance of 0.19 2.3 510 1 4.6 660 2 6.9 730 3 9.2 785 4 11.4 808 5 13.7 825 6 16.0 870 7 At Fractional Axial Distance of 0.34 2.3 460 1 4.6 635 2 6.9 735 3 9.2 795 4 11.4 835 5 13.7 855 6 16.0 870 7 At Fractional Axial Distance of 0.65 1 2.3 270-245= 4.6 570-560 2 3 6.9 690-685 4 9.2 775-765 11.4 805-805 5 13.7 835-835 6 16.0 865-850 7 Repeated experimental run.
Predicted TomP., F.
577 735 846 864 869 903 924 431 642 754 807 835 852 870 168 549 725 805 844 867 884
L
=
0.25 ft
0.4
0.2
0
0.6
0.8
1.0
I
1
A X I A L DISTANCE, L
Figure 5.
Computed profiles
Muffler location, vehicle A, catalyst A
1000,
I
a. 01
0
'
=
300
I
I
ft2/ft3
I
I 0.2
0.4
1
0.6
I
0.8
I 1.0
A X I A L DISTANCE, L
Figure 6.
Computed profiles
Muffler location, vehicle 8, catalyst A
VOL. 7
NO. 1
JANUARY 1968
07
3
s
.
.u. 1000
Pr
= 1 ft2
6.9 M i n (3rd Cyclel
l? 800
5
0.91 0.73
9 . 2 M i n (4th C y c l e l
K
d
= =
same conditions. The difference between curves 2 and 4 is attributed to a longer exhaust pipe in vehicle B and possibly a lower temperature output than vehicle A. Curves 3 and 4 show the difference in warm-up time due to a different catalyst material. The important factor here is primarily the packed-bed density. Although Figure 8 was computed for a specific geometry of bed, it can still provide an estimate for other geometrical arrangements, if the volume of the bed remains constant. Table I1 shows average catalyst bed temperatures for two convertors operating under the same conditions but with different configurations. The data are for a convertor operated on the vehicle B-catalyst system B a t the muffler location. The reason for the close agreement of the average temperatures is primarily the small size of the catalyst pellets considered here. Accordingly, the large superficial surface of the catalyst allowed an effective heat transfer between the gas and the solid in both configurations.
600 400
200 01
0
I
I
I
0.2
I 0.4
I
I 0.6
1.0
0.8
AXIAL DISTANCE, z
Figure 7.
Computed profiles
Muffler location, vehicle B, catalyst B
represents the highest achievable temperature profiles if the convertor is placed a t the exhaust manifold location. However, Figures 5, 6, and 7 are for the muffler location. The trend of the profiles is similar except for the different temperature level of the profiles. T h e temperatures of the gas and the solid in contact were close to each other. This is expected because of the small dimensions of the catalyst pellets used in the examples treated. These pellets provided a very large specific surface area of about 300 sq. feet per cu. foot. T h e resistance for the heat transfer between the gas and the solid was small. Therefore, it may seem tempting a t first glance to set this resistance to zero and obtain a simpler theoretical description of the convertor. Such a description will require only one differential equation. However, this approach is risky and its results have to be compared with the mathematical model employed in this work. As far as the exhaust gas convertor is considered, the simplified approach is not even practical, because a larger catalyst particle size can be employed for the conversion process. Moreover, since the time-dependent feed entering the convertor is empirically described, the solution of any mathematical model will require a numerical solution of a kind, and in turn, as much effort as the solution described. Figure 8 is a summary of integrated average catalyst bed temperatures for different systems. The solid curves are drawn from computed temperature values obtained a t the end of each cycle of the California test. The actual curves are not smooth and are similar to the broken line drawn for curve 1. Curve 1 represents the maximum temperature available if the catalyst is placed in the manifold. Curve 2 shows the average catalyst bed temperature for vehicle A a t the muffler location. Curve 2 should be compared with curve 4 for vehicle B a t the
Discussion and Application
T h e theoretical analysis and the experimental data in Figures
1 to 3 provide a general method for predicting the temperature profiles developed in the exhaust gas catalytic reactor. These temperature profiles are essential data for the design of the catalytic convertor. A warm-up design according to the method proposed in this paper will ensure a safe warm-up design without recourse to experimentation on a “prototype.” When the conversion reactions produce no heat, the computed temperature profiles may be integrated with kineticsconversion data specific to a certain catalyst. Thus, concentration profiles may be evaluated easily. If the amount of heat of reaction is substantial, only a minimum conversion can be computed. Equation 5 may be expanded to include a heat generation term as follows:
T h e use of this equation is limited, since rare kinetics data are needed to calculate the rate of heat generation, Q7. I n the California test, the “cold cycles” last 9.2 minutes. (The contribution of the cold cycles to the average pollutant concentration is 35% and that of the hot cycles is 65%.) The amount of pollution emissions due to inactive cold catalyst may be estimated from the data that can be extracted from Figures 4 to 7. Table I11 is a n example. 7
1
I
I
I
I
I
I
1000
1
i
LL
a W E 3
d
Figure 8. Computed average g catalyst bed temperature $
1. 2. 3. 4.
600
A-I 4
M a n i f o l d Location, Vehicle A - Catalyst A M u f f l e r Location, Vehicle A C a t a l y s t A , M u f f l e r Location, Vehicle B - C a t a l y s t B M u f f l e r Location, Vehicle B - C a t a l y s t A
-
1
i
Convertor Volume 0 . 2 5 it3 1-
I
I
1
12
14
16
201
0
ot 0
0 I 2
I 4
I 6
I
a T I M E , MINUTES
88
I&EC PROCESS DESIGN A N D DEVELOPMENT
I 10
Table II. Influence of Convertor Configuration on Average Catalyst Bed Temperatures Average Bed Temp. ( ' F . ) in End of 0.25-Cu. Ft. Convertor Cycle in California Time, I sa. f t . X 0.25 sq. f t . X ?est Mini j iiches 1 2 E'nihes 1 2.3 340 345 624 2 4.6 605 761 3 6.9 75 1 820 4 9.2 855 5 11.4 6 13.7 876 7 16.0 895 896
Warm-up of Catalyst Bed to 700' F. Time, Minutes for Temperature to Reach Case z = 0.20 z = 0.50 z = 0.80 1. Manifold location, vehicle A, catalyst A 1.2 1.8 3.7 2 . Muffler location, vehicle A, 1.6 3.6 4.6 catalyst A 3. Muffler location, vehicle B, 3.8 6.1 7.2 catalyst B 4. Muffler location, vehicle B, 4.6 6.9 8.8 catalyst A Table 111.
Table I11 lists the times needed for 20, 50, and 80% of the bed to assume a temperature of 700' F. If the catalyst has only a little activity below 700' F., no appreciable conversion will take place until about 5070 of the bed is warmed up. Therefore, during the times listed in the middle column of Table 111, uncontrolled exhaust will be ejected to the atmosphere. T h e weight of this warm-up contribution to the averaged emission of the vehicle depends on the concentration of the pollutants before entering the reactor. Today's uncontrolled vehicle emits 860 p.p.m. of unburned hydrocarbons (Maga, 1966). T h e warm-up contributions will be about 60 p.p.m. (1.8/9.2 X 0.35 X 860) for Case 1 (Table 111), in contrast with 120 p.p.m. for Case 2 and higher contributions for Cases 3 and 4. Because of this, it is evident that if the catalytic convertor is to meet future strict emission standards, it either has to be placed near the exhaust manifold or combined in series with other pollution-control devices. I n this investigation attention was given to the development of temperature profiles in the exhaust gas catalytic convertor. Thus the behavior of the catalyst bed during warm-up was predicted. However, although the warm-up behavior imposes a real restriction on the operation and design of the catalytic convertor, we do not pretend that it is in the only consideration necessary for a complete design of such a reactor. Localized and post-warm-up overheating can develop in the catalyst bed when high concentration of pollutants are allowed to undergo oxidative reactions. This is more of a problem of the past, which can be solved by an overheat bypass. I n the future, vehicles will have lower concentration of pollutants and oxidative reactions are not necessarily the only reactions that can be adapted to the exhaust gas conversion process.
A convenient boundary condition is to assume that the exhaust gas maintains a constant temperature when the operation of the vehicle is started. Mathematically, this is expressed as a step change in the inlet gas temperature to the convertor-namely, T,(r,O) = To,a t r 2 0. Since no experimental data are needed for the simulation of this case, the computer program was developed first for this situation and extended later for the dynamic California test. Figure 9 presents transient temperature profiles for a constant inlet temperature of 1160' F. a t a space velocity of 10,000 v./v./hr. and Figure 10 presents corresponding average catalyst bed temperatures. As expected, the curves here are very smooth. Included in Figures 9 and 10 are the gas temperatures in contact with the solid. As can be seen, no great temperature difference exists between solid and gas. This is due to the small size of the catalyst pellets.
1200
I
I
01 0
1
1
0.2
0.4
I t can be shown that for this case Equation 6 can be written in terms of the space velocity, U , as follows:
17.6 M i n .
I 0.6
1
I
0.8
1.0
AXIAL D I S T A N C E , L
Figure 9. Computed temperature feed
profiles
for
constant
1 160' F., space velocity 10,000 v./v./hour 300 rq. ft/cu. ft. 1000 Ib./sq. ft. hr. I. 1.38 ft. r8,,. 8 0 0 F. $, 0.91 Pr. 0.73
oa.
G.
1200 1000
-
U
.
-S o l i d Temperature --- Gas
-
Temperature -
800
W K 3
I-
2 % w
600
W
400
+
200
Appendix. Transient Temperature Behavior of Convertor for Steady-Speed Operation of a Vehicle
I
-
+
= 0.91
Pr
= 0.73
-
or
I
I
I
I
I
0
1
2
3
4
5
6
TIME, MINUTES
Figure 10. Computed average bed temperature for constant feed 1 160' F., space velocity 10,000 v./v./hour
VOL 7
NO. 1
JANUARY 1968
89
Nomenclature
superficial surface of catalyst, surface per unit volume of bed, sq. ft./cu. ft. = heat capacity of gas a t constant pressure, B.t.u./lb. O R . = heat capacity of solid, B.t.u./lb. O R. = rate of mass flow of gas, lb./sq. ft. hr. = heat transfer coefficient between gas and solid, B.t.u./ sq. ft. hr. O R. = equivalent thermal conductivity of gas based on total cross section of convertor (kea = 6 k g ) , B.t.u./ft. hr. a R. = thermal conductivity of gas, B.t.u./ft. hr. O R. = equivalent point-contact thermal conductivity of solid, B.t.u./ft. hr. R . = depth of catalyst bed, ft. = Prandtl number for gas, &,/kg = heat generation due to a chemical reaction, B.t.u./ cu. ft. hr. = Reynolds number, G/Q,$~). = cross-sectional area of convertor, sq. ft. - “time” ( T = ~ j p & J , hr. O F. cu. ft./B.t.u. = average catalyst bed temperature [ j ;T(T,z)dz], a R . = gas temperature, a R . = gas temperature a t z = 0, O R. = solid temperature, a R . = initial solid temperature, O R . = axial distance, ft. = dimensionless axial distance, z = x / L
= c 9
C, G h kea
X
Z
GREEKLETTERS
6
= void fraction of bed
At
= =
A2
?
=
P *a
= = =
PP
=
Peg
size of “time” step, hr. O F. cu. ft./B.t.u. size of axial interval viscosity of nitrogen, lb.mJft. hr. shape factor for catalyst particles equivalent density of gas, (pea = 6p,), lb./cu. ft. density of catalyst bed [ p e s = (1 - 6 ) p , ] , lb./cu. ft. density of gas, lb./cu. ft.
pa
=
7
= =
density of catalyst particles, Ib./cu. ft.
p s r p = density of gas a t 1 atm., 32’ F., lb./cu. ft. u
time, hr. space velocity a t 1 atm., 32’ F., cu. ft./cu. ft./hr.
SUBSCRIPTS = at axial point i inside packed bed
i
SUPERSCRIPTS I = quantity a t time t
+ At
literature Cited
Anzelius, A., 2.Angew. M a t h . Mech. 6 , 291 (1926). Bird, K. B., Stewart, W. E., Lightfoot, E. N., “Transport Phenomena,” p. 41 1, W’iley, New York, 1960. Furnas, C. C., Trans. A m . Inst. Chem. Engrs. 24, 142 (1930). Hellums, J. D., Churchill, S. W., “International Development in Heat Transfer,” p. 985, Am. SOC.Mech. Engrs., New York, 1961. Hodgman, C. D., et al., “Handbook of Chemistry and Physics,” p. 2267, Chemical Rubber Publishing Co., Cleveland, 1962. Maga, J . A,, Kinosian, J. R., “Motor Vehicle Standards, Present and Future,” SAE Automotive Engineering Congress, Detroit, 1966. Mickley, H. S.,Sherwood, T. K., Reed, C. E., “Applied Mathematics in Chemical Engineering,” p. 376, McGraw-Hill, New York, 1957. Schumann, T . E. W., J . Franklin Inst. 208, 405 (1929). Smith, J. D., “Chemical Engineering Kinetics,” p. 391, McGrawHill, New York, 1956. State of California Motor Vehicle Pollution Control Board, Los Angeles, Calif., “California Procedure for Testing Motor Vehicle Exhaust Emissions,” 1964. Willmott, A. J., Intern. J . Heat Mass Transfer 7, 1291 (1964). RECEIVED for review July 11, 1966 RESUBMITTED July 12, 1967 ACCEPTED September 21, 1967 Division of Petroleum Chemistry, 152nd Meeting, ACS, New York, N. Y., September 1966.
A MODEL OF CATALYTIC CRACKING
CONVERSION IN FIXED, MOVING, AND FLUID-BED REACTORS VERN W. WEEKMAN, JR.
Applied Research & Development Division, Mobil Research and Development Corp., Paulsboro, N . J .
deal of exploratory work in catalytic cracking takes place in fixed-bed reactors, which are cheap and easy to operate on a laboratory scale. Most potential industrial applications, however, lie in moving or fluid-bed reactors. T h e present work provides a method for relating catalytic cracking conversion from one reactor configuration to another, based on the principles of reaction kinetics. Within the last 20 years various mathematical models have been proposed for catalytic cracking. Voorhies (1945) presented a model of coking and conversion behavior. Blanding (1953) gave a model of cracking behavior based on simple reaction kinetics. Using the Voorhies expressions, Andrews
A
90
GREAT
l&EC PROCESS DESIGN A N D DEVELOPMENT
(1959) compared the various reactor types. Froment and Bischoff (1961) treated catalyst decay in fixed-bed systems which result from series or parallel fouling reactions. Masamune and Smith (1966) described theoretically the interaction between catalyst fouling and diffusion. This paper describes the development of mathematical models that account for the temporary decay of the catalyst during use. The models are presented in terms of a dimensionless decay group and a dimensionless reaction group. Comparisons among isothermal, fixed, fluid, or moving-bed reactors reduce simply to comparisons among the appropriate dimensionless groups. Finally the proposed model is shown
