47 Mathematical Modeling of the Monolith Converter LARRY
C. YOUNG
and BRUCE
A.
FINLAYSON
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Department of C h e m i c a l Engineering, University of Washington, Seattle, W a s h . 98195
Mathematical models for the monolith converter are developed and applied to the oxidation of carbon monoxide in automobile exhausts. The converter consists of an array of ducts with catalyst coating the walls. The exhaust gas flows axially through the ducts in laminar flow. Two models for this type of converter are proposed and solved numerically. The simpler model uses heat and mass transfer coefficients for fully developed flow in ducts to account for the resistance to transfer between the fluid and the catalytic wall. The more complicated model accounts for the distribution of mass and energy in the duct cross-section, which is assumed to be circular. Both transient and steady-state calculations are performed for conditions which would be expected during vehicle operation. Mass and energy distribution in the duct cross-section is important. Transient calculations predict that the wall temperature can overshoot the steady-state adiabatic temperature and thus contribute to converter failure.
O
n e of t h e c o n t e m p o r a r y c h a l l e n g e s t o t e c h n o l o g y is t h e c o n t r o l of a u t o m o b i l e emissions. A p r o m i s i n g m e t h o d of m e e t i n g emission standards i n t h e s h o r t t e r m is t o u s e c a t a l y t i c c o n v e r t e r s . I n t h i s p r e l i m i n a r y s t u d y t h e a u t h o r s p r o p o s e a n d s o l v e t w o m a t h e m a t i c a l m o d e l s f o r a p a r t i c u l a r t y p e of c o n v e r t e r , t h e m o n o l i t h c o n v e r t e r . P a r t i c u l a r a t t e n t i o n is g i v e n t o t h e u s e o f efficient c o m p u t i n g t e c h n i q u e s . B o t h t r a n s i e n t a n d s t e a d y - s t a t e c a l c u l a t i o n s are p e r f o r m e d for conditions w h i c h c o u l d be expected i n automobile operation, a n d t h e e s s e n t i a l f e a t u r e s of t h e p r o b l e m a r e o u t l i n e d . T w o types of catalytic converters, p a c k e d beds a n d monoliths, h a v e been p r o p o s e d f o r t h e o x i d a t i o n of c a r b o n m o n o x i d e a n d h y d r o c a r b o n s i n a u t o m o b i l e exhausts. M a t h e m a t i c a l models h a v e b e e n p r o p o s e d a n d solved for p a c k e d b e d c o n v e r t e r s (1,2, 3, 4) w h i l e K u o ( 5 ) h a s d e v e l o p e d a m o d e l f o r t h e m o n o l i t h c o n v e r t e r w h i c h is s i m i l a r t o t h e s i m p l e r o n e s o l v e d h e r e . T h e m o n o l i t h c o n v e r t e r c o n s i s t s o f a n a r r a y o f d u c t s or c e l l s t h r o u g h w h i c h t h e e x h a u s t gas flows a x i a l l y . T h e i n t e r n a l s u r f a c e a r e a o f t h e s u b s t r a t e m a t e r i a l is r e l a t i v e l y l o w ; t h e r e f o r e , a w a s h c o a t of c a t a l y s t d i s p e r s e d o n a c t i v a t e d a l u m i n a is d e p o s i t e d o n t h e s u b s t r a t e . B e c a u s e of a m u c h s m a l l e r volumetric heat capacity, m o n o l i t h converters w a r m u p more q u i c k l y than 629
Hulburt; Chemical Reaction Engineering—II Advances in Chemistry; American Chemical Society: Washington, DC, 1974.
630
C H E M I C A L REACTION ENGINEERING
II
p a c k e d b e d devices, g i v i n g the monolith an important advantage d u r i n g cold starts o f t h e e n g i n e . T h e m o n o l i t h s h a v e d i f f i c u l t i e s , h o w e v e r , c a u s e d b y t h e r m a l expansion, a n d i n certain d r i v i n g modes the m o n o l i t h substrate can m e l t , r e s u l t i n g i n c o n v e r t e r f a i l u r e (6). T h e mathematical models developed b e l o w i l l u s t r a t e i m p o r t a n t f e a t u r e s of t h e t h e r m a l b e h a v i o r of t h e m o n o l i t h converter. Model
Development
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A v a i l a b l e i n f o r m a t i o n o n t h e r e a c t i o n k i n e t i c s is r e v i e w e d i n o r d e r t o m a k e realistic a p p r o x i m a t i o n s to the k i n e t i c parameters w h i c h m i g h t be ex p e c t e d for the m o n o l i t h converter u s i n g either base m e t a l oxide or p l a t i n u m catalysts. T h e b a s e m e t a l o x i d a t i o n o f c a r b o n m o n o x i d e is a f i r s t - o r d e r r e a c t i o n i n c a r b o n m o n i x i d e a n d i n d e p e n d e n t of o x y g e n f o r o x y g e n c o n c e n t r a t i o n s g r e a t e r t h a n 2% ( 3 ) . T h e base m e t a l oxide rate expressions u s e d i n the calculations w e r e e s t i m a t e d f r o m t h o s e g i v e n b y K u o et al. ( 3 ) , w h o u s e d c a t a l y s t s w i t h b a s e m e t a l a p p l i e d o n t h e o u t e r p o r t i o n of a l u m i n a s p h e r e s . A r a t e e x p r e s s i o n w a s d e t e r m i n e d b y assuming the reaction rate for the m o n o l i t h converter w o u l d b e t h e s a m e b a s e d o n t h e v o l u m e of a c t i v e c a t a l y s t a n d b y e s t i m a t i n g t h e thickness of the catalytic layer o n the m o n o l i t h w a l l . Since the pre-exponential factor i n the rate constant c a n v a r y a c c o r d i n g to catalyst d e a c t i v a t i o n , a m o u n t o f a c t i v e c a t a l y s t , etc., w e u s e t h e t w o r a t e e x p r e s s i o n s , B l a n d B 2 , l i s t e d i n Table
I. Table I.
PI: -
Bl: -
r
c o
B2: -
r
c o
r
= Y
c o
Reaction Rate Expressions
= 7.8 X 10 ^ 5
T
= 3.9 X 10* ^
Q2
T
F / ( 0 . 5 F + c o
2
exp ( ^
16000 °R/T)
exp ( -
16000 °R/T)
1.33 F
c o
2
l
b
'
l b
m
o
l
e
mm ft
" !° mm ft m o 1
exp (11000 ° R / T ) )
°
R
2
R
2
j^ffi
T h e o x i d a t i o n o f c a r b o n m o n o x i d e o n p l a t i n u m is g e n e r a l l y a g r e e d to b e a first-order r e a c t i o n i n o x y g e n a n d i n v e r s e l y p r o p o r t i o n a l to c a r b o n m o n o x i d e (2, 7, 8). T h e r a t e e x p r e s s i o n is n e g a t i v e o r d e r i n c a r b o n m o n o x i d e b e c a u s e o f t h e s t r o n g a d s o r p t i o n of c a r b o n m o n o x i d e o n p l a t i n u m . A s p o i n t e d o u t b y S c h l a t t e r et al. (8) t h e r a t e e x p r e s s i o n w o u l d n o t a p p l y a t l o w c a r b o n m o n o x i d e concentrations. F o r the o x i d a t i o n of c a r b o n m o n o x i d e o n p l a t i n u m w i r e , S k l y a r o v et al. ( 9 ) r e p o r t a r a t e e x p r e s s i o n s i m i l a r t o t h e L a n g m u i r H i n s h e l w o o d t y p e . F o r the conditions of the m o n o l i t h converter, t h e i r rate expression can be accurately approximated b y : r
= F
c o
0
2
Yoo/ik, Y
Q2
+ k Fco ) 2
2
(1)
T h e first t e r m i n t h e d e n o m i n a t o r is i m p o r t a n t o n l y at h i g h t e m p e r a t u r e s a n d l o w carbon monoxide concentrations. U n d e r other conditions the reaction w o u l d a p p e a r to be inversely p r o p o r t i o n a l to c a r b o n m o n o x i d e . A rate ex p r e s s i o n o f t h e f o r m o f E q u a t i o n 1 is u s e d i n t h e c a l c u l a t i o n s . T h e a c t i v a t i o n energy i n the constant k w a s t a k e n f r o m H a r n e d (2), although H a r n e d a s s u m e d k w a s z e r o . S k l y a r o v et al. ( 9 ) r e p o r t a z e r o a c t i v a t i o n e n e r g y f o r 2
x
Hulburt; Chemical Reaction Engineering—II Advances in Chemistry; American Chemical Society: Washington, DC, 1974.
47. k
l9
Monolith
YOUNG AND FINLAYSON a n d this v a l u e is u s e d here.
Converter
T h e ratio of k
to k
x
s a m e as f o u n d b y S k l y a r o v et al. (9)
631
2
at 6 0 0 ° F .
w a s assumed to be the
T h e m a g n i t u d e s of t h e rate
c o n s t a n t s w e r e e s t i m a t e d so t h a t t h e p l a t i n u m k i n e t i c s w o u l d g i v e q u a l i t a t i v e l y t h e s a m e c o m p a r i s o n t o t h e b a s e m e t a l k i n e t i c s as w a s o b s e r v e d b y S c h l a t t e r et al. (8). T h e p l a t i n u m r a t e e x p r e s s i o n , so d e t e r m i n e d , i s d e s i g n a t e d P I a n d listed i n Table I. Space l i m i t a t i o n s p r e c l u d e a c o m p l e t e discussion of t h e justification of t h e assumptions. references.
W e therefore
list the important assumptions
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1. T h e flow is l a m i n a r s i n c e R e < 2 0 0 0
a n d the available
(10).
2. E x c e p t u n d e r c o n d i t i o n s o f v e r y h i g h flow r a t e , t h e v e l o c i t y e n t r y l e n g t h is a s m a l l f r a c t i o n o f t h e c o n v e r t e r l e n g t h ( I I ) , a n d i t s effect o n h e a t a n d mass transfer is s m a l l (12). 3. A x i a l d i f f u s i o n i n t h e fluid p h a s e is n e g l i g i b l e s i n c e P e > 5 0
(13).
4. T h e t r a n s i e n t r e s p o n s e o f t h e c o n v e r t e r is c o n t r o l l e d b y t h e t h e r m a l r e s p o n s e o f t h e s o l i d . A l l o t h e r t i m e d e r i v a t i v e s a r e set t o z e r o ( I ) . 5. D i f f u s i o n c o u p l i n g c a n b e n e g l e c t e d
(14,
15).
6. T h e c o n v e r t e r is a d i a b a t i c (6), a n d o n l y a s i n g l e c e l l is m o d e l e d . I f t h e flow r a t e is d i f f e r e n t i n a d j a c e n t c e l l s , t h e n a c o m p l e t e m o d e l m u s t a c c o u n t f o r t h i s flow d i s t r i b u t i o n a n d r e p e a t t h e s i n g l e - c e l l c a l c u l a t i o n s f o r e a c h c e l l . T h i s is b e y o n d t h e s c o p e o f t h e p r e s e n t w o r k . fluid
7. T h e s o l i d t e m p e r a t u r e a t a n y a x i a l p o s i t i o n is u n i f o r m a n d e q u a l t o t h e t e m p e r a t u r e a t t h e fluid-solid i n t e r f a c e . 8. A x i a l c o n d u c t i o n i n t h e s o l i d is u n i m p o r t a n t p r o v i d e d : A* . d T* 2
«
AH r
c o
9. B e c a u s e t h e c a t a l y t i c l a y e r is v e r y t h i n (16), i n t e r n a l d i f f u s i o n effects i n t h e catalytic layer are small a n d are i n c l u d e d i n t h e reaction rate expressions. 1 0 . T h e c o n c e n t r a t i o n r a t i o o f h y d r o g e n t o c a r b o n m o n o x i d e is 1 : 3 e v e r y w h e r e i n t h e r e a c t o r (3, 7). 1 1 . T h e p r e s e n c e o f h y d r o c a r b o n s d o e s n o t a p p r e c i a b l y affect t h e c o n v e r sion of carbon monoxide or t h e heat generated i n the converter. 1 2 . T h e m o l e f r a c t i o n o f o x y g e n is d e t e r m i n e d b y s t o i c h i o m e t r y . A s s u m p t i o n s 9 - 1 2 are m a d e b e c a u s e of a l a c k of sufficient i n f o r m a t i o n o n the reaction kinetics. T h e calculations for w h i c h the inequality i n assumption 8 d o e s n o t h o l d a r e i d e n t i f i e d b e l o w . A l t h o u g h w o r k is p r e s e n t l y i n p r o g r e s s t o solve t h e p r o b l e m r e l a x i n g assumptions 7 a n d 8 for t y p i c a l irregular d u c t g e o m e t r i e s , w e s o l v e t h e p r o b l e m h e r e f o r t h e case o f a c i r c u l a r as a n i n i t i a l s t u d y . T h e g o v e r n i n g e q u a t i o n s a r e t h e n : Model II: 2r. P . , (1 -
y
-
J|
cross-section
(r ^)
(2.) < 2 b
A r h
8
P
8
C
A< C
P
p
8
dT*
f
dt
"
C
(z, t)) P e / ( 2 0
A
)
N+ltN+l
m
1
+ Downloaded by CORNELL UNIV on September 6, 2016 | http://pubs.acs.org Publication Date: June 1, 1975 | doi: 10.1021/ba-1974-0133.ch047
k
+ £
0 - T.W
(s^fe)/'
[7*
