27 Yield Optimizationina Tube-Wall Reactor
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DANIEL TA-JEN HUANG and ARVIN VARMA Department of Chemical Engineering, University of Notre Dame, Notre Dame, IN 46556
A tube-wall reactor, in which the catalyst is coated on the tube wall, is conceptually
ideally
suited
for highly
and equilibrium-limited
reactions
the wall can be rapidly
taken away by the coolant.
(1) has numerically selectivity
demonstrated that for highly
reactions,
from both steady state the fixed-bed reactor. vanced as a possible gasification
plants
view, it therefore the analytically
exothermic
because the heat generated at Previous work
exothermic
the optimized tube-wall reactor is superior production and dynamic points
of view to
Also, the tube-wall reactor is being ad-
reactor
for carrying
(2). From a reaction seems
appropriate
resolvable
out methanation in coal engineering point of
to analyze the reactor for
case of complex first-order
isothermal
reactions. In the case of methanation, the tube-wall reactor the advantage of being able to accommodate much as 25% CO without recycle thermal conditions matically material
eases reactor
control.
iso-
Mathe-
speaking, an isothermal model is one which uses only the
(5),
exothermic oxidation
using the tube-wall reactor
of naphthalene on V 0^ 2
dropping the energy balance equation still
accurate picture Also
has
balance equations and drops the energy balance equation.
the highly
yst,
and yet operate under nearly
which greatly
As shown by Smith and Carberry for
(3,4)
feeds containing as
of the reactor
catal-
provides a rather
conversion and yield
behavior.
Senkan et al. (6) and Schehl et al. (_3) have shown that
for
methanation, the material balance equation can be solved independently of the energy balance equation cases, because the effects ties
essentially
consider
in
diffusion-limited
of temperature variation
cancel each other.
on gas proper-
It is therefore
justified
the isothermal model for the purpose of yield
to
optimiza-
tion. It is our purpose to optimize the performance of a tube-wall reactor
with the reaction
rate
constants as parameters, and also
with respect to some other key design parameters. Tube-Wall Reactor Model The following
assumptions have been made:
0-8412-0549-3/80/47-124-469$05.00/0 © 1980 American Chemical Society
Squires and Reklaitis; Computer Applications to Chemical Engineering ACS Symposium Series; American Chemical Society: Washington, DC, 1980.
470
COMPUTER
APPLICATIONS TO
CHEMICAL
ENGINEERING
(1) Steady s t a t e operation is achieved. (2) To enhance mass t r a n s f e r to and from the w a l l s u r f a c e , the tube is packed with i n e r t p e l l e t s . Thus, plug-flow in the a x i a l d i r e c t i o n is assumed and only r a d i a l d i s p e r s i o n is considered, which are v a l i d not only f o r h i g h l y turbulent flow but even f o r r e l a t i v e l y low Reynolds number flow compared to the empty tube. (3) The c a t a l y s t a c t i v i t y is assumed to be constant. Because of e x c e l l e n t temperature c o n t r o l achievable in tube-wall r e a c t o r s , thermal s i n t e r i n g of the c a t a l y s t is l e s s l i k e l y than in a fixed-bed r e a c t o r . Some methods of avoiding c a t a l y s t de a c t i v a t i o n or reducing the d e a c t i v a t i o n r a t e f o r methanation are described by M i l l s and Steffgen (7). (4) The p h y s i c a l p r o p e r t i e s of the gas and the mass t r a n s f e r c o e f f i c i e n t are assumed to be independent of p o s i t i o n in the r e a c t o r . (5) Since the c a t a l y s t f i l m coated is u s u a l l y very t h i n , the c a t a l y s t may be considered to be only an a c t i v e s u p e r f i c i a l surface and thus the g a s / c a t a l y s t i n t e r f a c e is assumed to c o i n c i d e with the w a l l . The f o l l o w i n g f i r s t - o r d e r c o n s e c u t i v e - p a r a l l e l r e a c t i o n scheme is used, which is in accord with the o x i d a t i o n of naphtha lene (1): l 2 3 A —>B —> C ; A —> D I f C and D are the same, it is a t r i a n g u l a r r e a c t i o n , e.g.,inthe s i l v e r - c a t a l y z e d p a r t i a l o x i d a t i o n of ethylene. By s e t t i n g k^ equal to zero, we get a consecutive r e a c t i o n : k
k
k
k
k
l 2 A —> Β —> C And by s e t t i n g k equal to zero instead, we get a p a r a l l e l reaction: . , 1 3 A —> Β ; A —> D In a l l cases, Β is the d e s i r e d product. In the f o l l o w i n g d e r i v a t i o n s , only the c o n s e c u t i v e - p a r a l l e l r e a c t i o n scheme is considered, s i n c e the other two a r e j u s t s p e c i a l cases of this more general scheme. With the l i s t of symbols given at the end, At the C a t a l y s t Surface
C
=
wA
(
1 + (k-îkj/k 1 3 gw C_ + (k-/k )C _ Β 1 gw wA wB " 1 + Wk 2 gw M a t e r i a l Balance
1
)
τ
A
Γ
2
3 C. V
at
7 ^
=
e r i
dz z* = 0,
r 3 r
;
Ι " Α. Β
n
Squires and Reklaitis; Computer Applications to Chemical Engineering ACS Symposium Series; American Chemical Society: Washington, DC, 1980.
(3) (4a)
27.
HUANG AND VARMA
Tube-Wall
471
Reactor
3 C. at
(4b)
= 0
0,
at the g a s / c a t a l y s t i n t e r f a c e , i . e . , at r 3C - D . —I eri ^ *
=k
= R,
(C.-C .) ι wi
gw
(4c)
With the dimensionless v a r i a b l e s defined in the l i s t of symbols at the end, and s u b s t i t u t i n g the expressions (1) and (2) f o r and C i n t o the boundary c o n d i t i o n (4c), we o b t a i n the f o l l o w i n g dimensionless equations: For species A
3z
~Pe
%
A
2
r
(5)
r 3r
= 1
(5a)
-^=0
(5b)
at
ζ = 0,
u
at
r - 0 ,
3
A
at r = a ( g a s / c a t a l y s t
interface),
3u.
Tf
(5c) +
K
U
0
A =
where k
d i (6)
ι + (k k )/k _ 1+
erA
3
gw
For species Β 2
* d u.
(7) 9ζ
Pe. Β
at
ζ = 0,
at
r-0,
at
r = a,
9r B
=
B
^ d —r 9 u
_ 1 ~
k
r 3r ' (7a)
U
3 u
where
2 U
B0
=
(7b)
0
B
T7
d gw p ( D _ erB
(7c) +
K
U
K
U
1 B - 2 A Ί
_
1 1
(8)
+ k2/k 1 0
iV
gw
k d 1
D
( 1
+
k
Ρ ) [ 1 +
(9) ( k
+ k
) / k
]
erB V gw l 3 gw The c o r r e l a t i o n f o r the pressure drop through the packed bed
Squires and Reklaitis; Computer Applications to Chemical Engineering ACS Symposium Series; American Chemical Society: Washington, DC, 1980.
472
COMPUTER
APPLICATIONS TO
CHEMICAL
ENGINEERING
is the modified Ergun equation by Tallmadge, as described by Smith and Carberry (1): 2
pv (l-e)bM*
T
~
ΔΡ = S
ε
3
150(1-ε)Μ* Re
.1/61 4 # Z
(
f o r 10
Re - I
;
10
f = '
Re
1 0 0
s=
1 1 p The f o l l o w i n g observations can be made from the c a l c u l a t i o n s : 1. From Figures l a and 2a, as k^/k_ decreases Y increases at 3 1 max the expense o f smaller conversion, and a l s o at tne expense of a longer r e a c t o r and higher pressure drop. From Eq. (10), pressure drop is a l i n e a r f u n c t i o n of the r e a c t o r length b, and a non l i n e a r f u n c t i o n of Re . Thus i f Re is f i x e d , ΔΡ is a l i n e a r f u n c t i o n of b , and longer reactors mean higher pressure drop, where ΔΡ denotes ΔΡ at b . m m 2. From Figures l b and 2b, reducing ^ / k , i H increase Y l a r g e l y with an i n c r e a s e o f X u n t i l a point is reached, from which f u r t h e r reducing k?/k^ ^ increase Y^ much, but makes X so c l o s e to 1 tnat the r e q u i r e d reactor îength b (and hence Δ? ) are very l a r g e , m 3. From Figures l c and 2c, we note that as R/d decreases, Y increases s l i g h t l y with very small i n c r e a s e of X but at the b e n e f i t of l a r g e l y reducing b . Results of the ?ype shown in F i g u r e 2c are v a l u a b l e in making p r a c t i c a l judgements concerning r e a c t o r s , f o r they show the i n t e r p l a y between r e a c t o r diameter, length, pressure drop and achievable product y i e l d . 4. From F i g u r e s Id and 2d, as we increase Re , Y increases s l i g h t l y and r a p i d l y reaches a p o i n t , from which îurther increase of Re will only add to the r e a c t o r length needed and l a r g e l y i n c r e a s e ΔΡ . T h i s is because as Re increases the gas residence time decreases r e l a t i v e l y , and thus a longer r e a c t o r is r e q u i r e d to achieve equivalent conversion. S
w
m a x
o
e
s
n
o
t
χ
m
Squires and Reklaitis; Computer Applications to Chemical Engineering ACS Symposium Series; American Chemical Society: Washington, DC, 1980.
27.
HUANG AND VARMA
Figure 1.
Tube-Wall
Reactor
Product (B) yield versus reactant (A) conversion, varying (a) (b) k /k (c) R/d , and (d) Re 2
u
p
8
Squires and Reklaitis; Computer Applications to Chemical Engineering ACS Symposium Series; American Chemical Society: Washington, DC, 1980.
ki/k
u
476
COMPUTER APPLICATIONS TO CHEMICAL ENGINEERING
Squires and Reklaitis; Computer Applications to Chemical Engineering ACS Symposium Series; American Chemical Society: Washington, DC, 1980.
27. H U A N G AND V A R M A
Tube-Wall
Reactor
477
Figure 2. Optimal reactor length and corresponding pressure drop versus optimal yield, (a) k /k (b) k /k (c) R/d , and (d) Re as implicit parameter s
1}
2
ly
p
s
Squires and Reklaitis; Computer Applications to Chemical Engineering ACS Symposium Series; American Chemical Society: Washington, DC, 1980.
COMPUTER APPLICATIONS TO CHEMICAL ENGINEERING
Squires and Reklaitis; Computer Applications to Chemical Engineering ACS Symposium Series; American Chemical Society: Washington, DC, 1980.
27.
H U A N G AND V A R M A
Tube-Wall
479
Reactor
I t is thus c l e a r that l a r g e r maximum d e s i r e d product y i e l d , max' ^ obtained by reducing k^/k^, k^/k , or R/d , and increasing Re . But it is b e t t e r to keep Re w i t h i n a r i a s o n a b l e range from the point of r e a c t o r length and pressure drop. I t should be noted that although higher Re will provide b e t t e r c o o l i n g e f f e c t , it is at the expense o f a longer r e a c t o r . Also note that plug-flow has been assumed, and in a packed r e a c t o r turbulent flow occurs f o r Re greater than 100, and f o r the t r a n s i t i o n zone, 10 < Re < Ï00. I t is thus found that the most b e n e f i t of a higher Y is obtained by reducing k2/k^, then by reducing R/d and kg/ïc and relatively little from i n c r e a s i n g Re . Since ^ F k ] , ^3 1 u s u a l l y f u n c t i o n s o f temperature, t h l s e l e c t i o n of an optimal operating temperature is very important f o r y i e l d o p t i m i z a t i o n . k k The Consecutive Reaction: A > Β > C . T h i s is a s p e c i a l case of the c o n s e c u t i v e - p a r a l l e l r e a c t i o n , and i t s behavior is s i m i l a r to it — as shown in Figures 2b,c,d. However, the values o f R/d and Re have little e f f e c t on X in ., . p s m this case.
Y
m
a
y
e
S
S
aX
a
n
d
a
r
e
v
k
\ 3 The P a r a l l e l Reaction: A —> Β ; A —> D For this case, since the d e s i r e d product Β does not r e a c t f u r t h e r , i t s y i e l d versus conversion depends only on the r a t i o k /k :
IS&.h X(b)
k
(21) U
i
;
3
We t h e r e f o r e have to solve only f o r X(b), which is e x a c t l y the same as Eq. (16). We see that y i e l d increases as k^/k^ decreases, and have found that the values of R/d and Re have no e f f e c t on the y i e l d versus conversion functionaïity. i f is apparent that no l o c a l maximum occurs on the Y(b) versus X(b) p l o t . Concluding Remarks A n a l y t i c s o l u t i o n s f o r reactant (A) conversion and d e s i r e d product (B) y i e l d f o r the f i r s t - o r d e r r e a c t i o n schemes: A •> B, A -> C Parallel A •> Β C Consecutive A -> Β + C., A + D Consecutive-parallel o c c u r r i n g i s o t h e r m a l l y in a packed tube-wall r e a c t o r a r e reported. These s o l u t i o n s take the form o f r a p i d l y convergent i n f i n i t e s e r i e s o f Bessel f u n c t i o n s . The y i e l d versus conversion f u n c t i o n a l i t y is thus deduced, from which the reactant conversion (and hence the r e a c t o r length) leading to an optimum y i e l d of Β is identified. I t is found that s e l e c t i o n of the operating tempera ture is very important f o r the purpose of y i e l d o p t i m i z a t i o n . The design of the optimal r a t i o of the r e a c t o r diameter (2R) to the i n e r t p a r t i c l e diameter is a l s o important f o r both the con s e c u t i v e - p a r a l l e l and consecutive r e a c t i o n s . This r a t i o can be v a r i e d by changing e i t h e r p a r t i c l e o r r e a c t o r diameter. Ob v i o u s l y , l a r g e r packing p a r t i c l e s will lead to lower pressure
Squires and Reklaitis; Computer Applications to Chemical Engineering ACS Symposium Series; American Chemical Society: Washington, DC, 1980.
COMPUTER APPLICATIONS T O CHEMICAL ENGINEERINC
480
drop but poorer e f f e c t i v e r a d i a l d i f f u s i v i t y . L i s t o f Symbols
LUI
b G d D
C
u X
C
/ A0 a
| [l-u (z,r)]rdr/J A
e
p e l l e t diameter e f f e c t i v e r a d i a l mass diffusivity
r
D
molecular d i f f u s i v i t y s p e c i f i c r a t e constant gas-wall mass t r a n s f e r coefficient r e a c t o r length P e c l e t number, d v/D r a d i a l coordinate* r*/d
k g
w
L Pe r* r
e
r
Re
[\ ( β
γ
/f^dr
ζ j r ) r d r
J
z
z
Y max *
%> 0 maximum Y a t r e a c t o r e x i t a
z
x
±
a
l
c
o
o
r
d
l
n
a
t
e
* ^
ρ Greek Symbols ε void f r a c t i o n μ viscosity ρ gas d e n s i t y
insiBe tube r a d i u s
R
rdr
o
0
concentration p
a
Subscript^ ^
r
e
a
z
=
Q
Reynolds number, d v ^ / u , at r e a c t o r i n l e t " 3 s i r e d product Β Se Schmidt number, μ/pD o n d i n g to Y ν i n t e r s t i t i a l gas v e l o c i t y wall v s u p e r f i c i a l v e l o c i t y , εν Acknowledgment The Union O i l Fellowship in Reaction Engineering and a R e i l l y t u i t i o n s c h o l a r s h i p f o r D. T . - J . Huang a r e g r a t e f u l l y acknowledged. Literature Cited 1. Smith, T. G.; Carberry, J. J. Chem. Eng. Sci., 1975, 30, 221. 2. Haynes, W. P.; Schehl, R. R.; Weber, J. K.; Forney, A. J. Ind. Eng. Chem. Proc. Des. Dev., 1977, 16, 113. 3. Schehl, R. R.; Weber, J. K.; Kuchta, M. J.; Haynes, W. P. Ind. Eng. Chem. Proc. Des. Dev., 1977, 16, 227. 4. Pennline, W. H.; Schehl, R. R.; Haynes, W. P. Ind. Eng. Chem. Proc. Des. Dev., 1979, 18, 156. 5. Smith, T. G.; Carberry, J. J. Chem. Eng. Sci., 1976, 31, 1071. 6. Senkan, S. M.; Evans, L. B.; Howard, J. B. Ind. Eng. Chem. Proc. Des. Dev., 1976, 15, 184. 7. Mills, G. Α.; Steffgen, F. W. Catalysis Reviews, 1973, 8, 159. 8. Yagi, S.; Wakao, N. AIChE J., 1959, 5, 79. 9. Amundson, N. R. Ind. Eng. Chem., 1956, 48, 26. s
A
r
e
a
c
t
a
n
t
A
d e
m
c o r r e s p
m
g
RECEIVED
November
5,
1979.
Squires and Reklaitis; Computer Applications to Chemical Engineering ACS Symposium Series; American Chemical Society: Washington, DC, 1980.
a
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