Two-Stage Catalytic Converter Steady-State Operation as a Limiting Condition George 1. Bauerle and Ken Nobe* School of Engineering and Applied Science, University of California, Los Angeles, 9002.4
Steady-state modeling of a two-stage catalytic exhaust converter may be used in preliminary design to establish limiting conversion for a given set of operating conditions. The adiabatic assumption i s justified for converters of practical size. A graphical method for solution of the material and energy balances has been developed; conversion and temperature determined b y the graphical method are in close agreement with values calculated b y numerical methods.
C a t a l y s i s is a promising approach to the control of auto exhaust emissions. The concept of the two-stage catalytic converter suggested by Sourirajan and Blumenthal (1961)is currently receiving considerable attention by automobile manufacturers as a means to control the primary auto pollutants, NO, CO, and hydrocarbons. Ideally, nitric oxide is completely reduced by the CO, H I , and hydrocarbons in the first stage and the remaining CO, H I , and hydrocarbons are oxidized to completion with addition of secondary air in the second stage. The purpose of this study is to develop some of the mathematical relationships required for the preliminary design of two-stage catalytic converters. Although the exhaust conditions of an automobile are highly transient in nature, the development of the steady-state model is primarily to predict limiting attainable conversion. The present mathematical model is applied to a cylindrical packed bed catalytic converter considering only axial flow of gases. Processes occurring in the reactor can be described by differential energy and material balances. The differential material balance for each reacting species is :
In this study, the catalyst in the first stage will be assumed to be highly selective for the reaction: NO
+ CO
+
’/zNz
+ CO2
(3)
It is also assumed that the oxidation reactions in the second stage can be represented by the reaction:
co +
‘/no2
-
con
(4)
Inhibition effects (e.g., HzO and COz) can be taken into account in the rate expressions selected. Gas-particle and intraparticle diffusion are not considered in this study, but can be taken into account), if necessary, in the analysis. Longitudinal diffusion is not significant a t the high volumetric flow rates in converters of practical size. Modified forms of the rate expressions obtained by Baker and Doerr (1964)for the NO-CO reaction on copper chromite and by Blumenthal and Nobe (1966)for the CO-0, reaction on copper oxide have been selected to illustrate the method of analysis. For the first stage,
R
=
3.9(10)12e-3wO’R’TC~~C~o, mol/(hr g cat)
(5)
and for the second stage,
R
When we assume that no temperature gradients exist between the solid and gas (a reasonable assumption for steady-state analysis) and that the packed bed can be considered a hypothetical solid continuum, the energy balance for steady state is :
=
1.4(10)17e-24,WO’R’TC~oCo,, mol/(hr g cat)
(6)
A computer program was written for the numerical solution of the material and energy balances (Equations 1 and 2, respectively) for a variety of conditions including isothermal operation , adiabatic operation, and operation with heat loss to the surroundings. Subroutines containing appropriate boundary conditions were included to handle each case. For illustrative calculations, typical flow and composition parameters determined experimentally for actual automobile exhaust under several driving modes have been selected and are given in Table I. The concentrations given are representative of fuel-rich operation necessary for proper first-stage operation. Nonadiabatic Operation
The material and energy balances with the appropriate boundary and initial conditions must be solved simultaneously to determine concentration and temperature profiles as functions of radius and length.
A lower conversion will be obtained with a reactor operating with heat loss than with a reactor operating adiabatically. Calculations for the CO-NO reaction during the acceleration mode were performed. Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 2, 1973
137
Table 1. Exhaust Conditions for Design Calculations Mode
Variable inlet temp., O K Flow, scfm M,, (mol wt) G," g/hr sq cm YNO(mol frac) YCO(mol frac) Yo2(mol frac) Based on a l j c m diameter reactor.
Acceleration
Cruise
Deceleration
Cold idle
Hot idle
1033 325 30 4060 0.002 0.01 0.002
900 100 30 1250 0.001 0.01 0.002
900 20 30 250 0.0002 0.08 0.008
300 40 30 500 0.0002 0.04 0.008
900 20 30 250 0.0002 0.04 0.008
Table 11. Conversion and Temperature for Nonadiabatic Operation of NO Reduction Converter in Acceleration Mode
Conversion, yo length, crn.
Adia. b atic operation
0
0.2
0.4
0.6
0.8
Wall
0 2 10 20 40 60 80 100
0 3 13 24 41 55 64 72
0 3 13 24 41 55 64 72
0 3 13 24 41 55 64 72
0 3 13 24 41 55 64 72
0 3 13 24 42 54 64 71
0 3 13 24 42 54 63 70
0 3 14 25 40 51 58 65
Radius, r / R
Integrated average final conversion = 70 Temperature, O K 1033 1033 1033 1033 1033 1033 1033 1034 1034 1034 1034 1034 1034 97 1 1036 1036 1036 1036 1036 1036 771 1039 1039 1039 1039 1039 1039 62 1 1043 1043 1043 1043 1043 1043 425 1046 1046 1046 1046 1046 1046 352 1048 1048 1048 1048 1048 1048 32 1 1050 1050 1050 1050 1050 1050 308 I
0 2 10 20 40 60 80 100
adiabatic case was quite close to that determined for adiabatic operation. Little radial variation in temperature occurred except in the immediate vicinity of the wall. Thus, lateral heat transfer is shown to be negligible when compared with axial convection. For all practical purposes, the adiabatic assumption gives an adequate estimate of the conversions and temperatures for preliminary design purposes. An even closer approximation of adiabatic operation would be achieved by decreasing the overall heat transfer coefficient to a more realistic value. Adiabatic Operation
Each operating condition in Table I was applied to a full two-stage converter (Le., an adiabatic NO reduction reactor in series with an adiabatic CO oxidation reactor). Twenty percent secondary air was assumed to be added to the latter. Tables I11 and IV show the results of these calculations. The CO content in the reactant stream was adequate to convert essentially all the NO within 100 cm of reactor length for all operational modes with the exception of acceleration. The high mass velocity and pi0 concentration both contributed to incomplete conversion in the latter case. For CO oxidation with added 20Y0 secondary air, Table IV shows that complete conversion was achieved for all operational modes except cold idle. The low temperature a t cold idle completely precluded significant reaction. Adiabatic Operation-Graphical Solution
Required input data to the computer program for the determination of conversion and temperature a t these conditions were: heat transfer coefficient, k; particle diameter of the catalyst; ambient temperature; and radial Peclet numbers for heat and mass transfer, Peh and Pem, respectively, where
Pen
dpu/De,
(7)
dpGCp/ker
(8)
=
Peh =
B particle diameter of 0.318 cm was selected. Smith (1956)
e)*]
showed that for Reynolds number above 40,
P,,/[
1
+ 19.4
=9
Ra
=
kCa"Cbm, g mol/(hr g catalyst)
(10)
is applicable, Equations 1 and 2 may be solved analytically for the adiabatic case. Concentrations can be expressed more conveniently as, C,
Yo(l - z)P/RT
(11)
Yo(S - Bx)P/RT
(12)
=
and Cb
(9)
Since D was taken as 15 cm, P,, 9. It was assumed that P e h was numerically equivalent to P,,, which implies that turbulent diffusion is the dominant mechanism for both heat and mass transfer in the radial direction. A heat transfer coefficient of 30 cal/hr sq cm O C was selected. Table I1 presents conversion and temperature calculations for the NO reactor as functions of radius and length during the acceleration mode. Also shown are results of calculations for adiabatic operation. The average final conversion for the non138 Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 2, 1973
A graphical method to determine conversion in an adiabatic reactor was devised. If a rate expression of the form
=
By defining a reduced temperature X = E/R'T = E/R'(To
+
LYX)
(13)
where 01
=
- A H Yo/CpM
substituting Equations 10-13 in Equation 1gives
(14)
'I= 0.70
0-
,'
Note
and v Interchanged
(I
-120
I
I
I
I
I I IIII
0-5
0-4
I
I I 1 1111
P
1
1
,
io-'
10-2
Figure 1 . Example of graphical construction: $t as a function of p for various values of v with constant 7 of 0.70 (7 and v interchanged if 7 > v) Table 111. Conversion and Temperature at Various Reactor Lengthsa
-
(Adiabatic operation) Reaction 1: CO Mode Length, cm
1 2 10 20 100
Acceleration
Cruise
3 (4) 13 (13) 24 (25) 71 (71)
7 (5) 9 (11) 38 (42) 61 (62) 99 (99)
+ NO
COZ
+ '/zNZ
Deceleration Conversion, %
100 (84) 100 (96) 100 (100) 100 (100) 100 (100) Temperature,
and v = R'(aS/B
+ TJ/E
(17)
51 (56) 98 (83) 100 (100) 100 (100) 100 (100)
300 (301) 301 (301) 302 (302) 302 (302) 302 (302)
901 (901) 902 (902) 902 (902) 902 (902) 902 (902)
Ei(x)
-
--MpAR'a GY,E
14 (30) 26 (56) 79 (86) 96 (98) 100 (100)
to determine XZ a t a specified value of the right-hand side of Equation 18. For integral values of vz and n, analytical solutions of Equation 18 may be obtained in terms of various forms of the exponential intcgral,
and substituting into Equation 15, we arrive at eXdX
Hot idle
O K
1 1033 (1033) 901 (901) 902 (902) 2 1034 (1034) 901 (901) 902 (902) 10 1036 (1036) 905 (905) 902 (902) 20 1039 (1039) 907 (907) 902 (902) 100 1050 (1050) 912 (912) 902 (902) * Numbers in parentheses indicate values calculated by graphical method.
By defining two additional variables, q and v ,
Cold idle
=
Jm ;
As an example for m and n equal to one, the integral on the
(7;)"t"L -
left side of Equation 18 becomes e'ddh
( V B ) ~ (18) ~ ~ Z S*:'h'(h
- l/T)(X - l/v)
-
If it is desired to calculate the length of reactor required to obtain a specific final conversion, the integral in Equation 18 can be evaluated graphically or by quadrature methods, enabling direct evaluation of z. If the conversion a t a specified z is desired, a plot of the integral as a function of Xz can be used Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 2, 1973
139
Table IV. Conversion and Temperatures in Oxidation Stage with Addition of 20% Secondary AiP Mode
Accelera tion
Cruise
Deceleration Conversion, %
. . . (1) 8 (7) 38 (33) 66 (61) 100 (100)
. . . (. . . ) 100 (100) 100 (100) 100 (100) 100 (100)
length, cm
. . . (4)
1 2 10 20 100
10 (12) 45 (42) 74 (69) 100 (100)
Temperature,
Cold idle
Hot idle
. . . (0) 0 (0) 0 (0) 0 (0) 0 (0)
. . . (54) 100 (92) 100 (100) 100 (100) 100 (loo)
. . . (300) 300 (300) 300 (300) 300 (300) 300 (300)
. . . (1062) 1200 (1176) 1200 (1200) 1200 (1200) 1200 (1200)
O K
1 . . . (1036) . . . (901) . , . ( . . .) 2 1041 (1042) 906 (905) 1502 (1502) 928 (925) 1502 (1502) 10 1067 (1064) 20 1088 (1085) 949 (946) 1502 (1502) 975 (975) 1502 (1502) 100 1108 (1109) a Numbers in parentheses indicate values calculated by graphical method.
Table V. Values of Design Parameters for Graphical Method of Solution Mode
Acceleration
x
Cruise
5 1 -89,250 0.25 23.8 0.70 0.76 1.58 0.225
b AH
CP CY
9 Y Pl
Ao/z
s AH
CP ff
9 Y
PI Adz
It is convenient to define
where the subscripts 1 and 2 refer to point values of $ a t the lower and upper limits of integration, respectively. General solutions for the adiabatic reactor can be presented in graphical form. It is convenient to define a new variable, p : For
Y
> 7,
For
Y
< 7,
p p
=
=
R’a(1
- x)/E
R’a(S/B
- x)/E
(22)
(23) For various combinations of 7, Y, XI, and hz (preselected as typical values for automotive converters), A$ values were determined using six-point, double-precision quadrature integration of Equation 21. The reference point for each graph was established by determining $ using Equation 20 and a series representation for exponential integrals (Abramowita and Stegun, 1964),
where r
=
Deceleration
+
4.22 0.5 - 67,640 0.25 74.85 0.092 0.138 6 . 2 (10)-3 11.2
b
+
+
Cold idle
Reaction 1: CO NO -,COS l/&z 10 400 200 1 1 1 - 89,250 -89,250 -89,250 0.25 0.25 0.25 11.9 2.39 2.38 0.60 0.60 0.20 0.68 1.23 0.51 7 . 9 (lo)+ 1 . 5 8 (10) -a 1.58 (10) -3 0.477 4.32 0.30 Reaction 2: CO 1 / 2 0 2 = C02 1.20 4.22 0.72 0.5 0.5 0.5 - 67,640 - 67,640 - 67,640 0.25 0.25 0.25 300.3 74,85 601.6 0.081 0.124 0.050 0,084 0.127 0.146 2.5 6.2 5 . 0 (lo)+ 30.06 33.4 259.2
0,5772156649
140 Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 2, 1973
Hot idle
200 1 - 89,250 0.25 2.38 0.60 0.91 1.58 3.2 1.20 0.5 - 67,640 0.25 300.3 0.099 0.134 2 . 5 (10) -2 190,7
The A$ values determined by integration were then plotted relative to the reference value, $1, to complete the graph of $ vs. p . A typical graph is shown in Figure 1. A sample set of curves for m and n equal to unity has been given elsewhere (Bauerle and Nobe, 1971). The procedure for handling a given set of conditions is summarized below : 1. Determine whether 9 or Y is smaller. Consider the graphs to be a t constant values of the smaller variable. If: la. 9 is smaller, select the lb. Y is smaller, select the graph corresponding graph corresponding to to the calculated q value the calculated Y value 2b. Calculate p1 (Equation 2a. Calculate p l (Equation 22) 23) 3b. Calculate A+ for a given 3a. Calculate A$ for a z (Equation 18) given z (Equation 18) 4b. Lay off A$ on line of 4a. Lay off A+ on line of constant Y constant q 5b. Determine p2 5a. Determine p 2 6b. Calculate z (Equation 6a. Calculate z (Equation 22) 23) The value of the integral in Equation 21 is unchanged if 9 and Y are interchanged-that is, after construction of graphs
similar t o Figure 1, it is immaterial whether the complete graph is considered a t constant q or at constant Y for a given set of reaction conditions; the other quantity then becomes any one of the family of parametric curves on that graph. For the typical conditions given in Table I, the graphical method was used to determine conversion. The results are compared with the computer solutions in Tables I11 and IV. Table V shows values of the design parameters and the determined values of A+/z. Table I11 and IV show that agreement was good after a few length increments are traversed. Disagreement at initial length increments can be attributed to uncertainty in reading the +-q charts at low A@ increments. The negligible conversion for the cold idle reaction of CO and 02, as calculated by the numerical method, was verified by the graphical method. T o calculate the length of reactor required to get a higher conversion of NO in the acceleration mode, for example, one must merely determine the A+ necessary to affect this conversion. Using a plot of Q vs. p at q = 0.7 and Y = 0.76, a A+ of about 84 or larger gives a n 2: value of at least 99%. Since A+/z is 0.225 (Table V), a length of 373 cm is required. Except for the cruise and hot idle modes, converter conditions will not be a t steady state. However, the graphical method of solution can be used to predict rapidly the maximum attainable conversion for a given set of inlet conditions. Nomenclature
A A, B
pre-exponential factor in rate equation external area of reactor, sq cm/cm = moles of second component reacting per mole of key component C = concentration, mol/cm3 C, = specific heat of reacting gas mixture, cal/g “C D = reactor diameter, em De, = effective diffusivity in radial direction, cm2/hr De, = effective diffusivity in axial direction, cm2/hr d, = particle diameter, cm E = activation energy, cal/mol G = mass velocity, g/hr em2 AH = heat of reaction, cal/mol h = film coefficient, cal/cm2 “Chr k,, = effective thermal conductivity in radial direction, cal/”C cm hr k,, = effective thermal conductivity in axial direction, cal/”C cm hr m = order for key component in power-law rate equation = number of reactions occurring JI = av mol wt, g/mol n = order for second component in power-law rate equation = number of species present P = pressure, atm =
=
Peh = Peclet no. for heat transfer P,, = Peclet no. for mass transfer p = R’a(1 - z ) / E and R’a(S/B - $ ) / E r = radius, cm RZK = average rate of reaction, mol/g cat hr R’ = gas constant, cal/mol OK R = gas constant, em3 atm/mol O K S = moles second component per mole key component entering reactor T = temperature, O K u = velocity, cm/hr z = conversion of key component Y = mole fraction z = axial length, em
GREEKLETTERS a = - AH Y0/C& 7 = R’(To a)/E Y
p
= = = =
+
dimensionless temperature, EIR’T R’(T,, a S / B ) / E density of catalyst, g/cm3 value of design equation integral
+
SUBSCRIPTS o = inlet conditions for key component a = key component am = ambient 6 = second component CO = carbon monoxide i = index of reactant k = index of reaction S O = nitric oxide 0 2 = oxygen Literature Cited
Abramowitz, &I.,Stegun, I. .4.,Eds., “Handbook of Mathematical Functions,” U. S. Department of Commerce, Nat. Bur Stds., Appl. Math. Ser. 53, p 229, 1964. Baker, R. A,, Doerr, R. C., “Catalyzed Nitric Oxide Reduction with Carbon Alonoxide,” Rept from Franklin Institute (1964), reproduced in part in Ind. Eng. Chem. Process Des. Develop., 4, 189 (1965). Bauerle, G. L., Sobe, K., “Catalytic Afterburner Design Studies, Part I-Steady State,” Rept UCLA-EiVG-7103, University of California, Los Angeles, January 1971. Blumenthal, J. L., Nobe, K., Ind. Eng. Chem. Process Des. Develop. 5 , 177 (1966). Smith, J. AI., “Chemical Engineering Kinetics,” p 393, McGrawHill, New York, N. Y., 1956. Sourirajan, S., Blumenthal, J. L., “i2ctes du Deuxieme Congres International de Catalyse,” Paris, 1960; Tome 11, Ed. Technip, p 2521, Paris, 1961. RECEIVED for review February 28, 1972 ACCEPTED January 5, 1973 Work supported by grant no. AP 00913, Air Pollution Control Office, Environmental Protection Agency.
Ind. Eng. Chern. Process Des. Develop., Vol. 12, No. 2, 1973
141
