(acs symposium volume 168) h. scott fogler (eds.)-chemical reactors-american chemical society (1981)...

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In Chemical Reactors; Fogler, H.; ACS Symposium Series; American Chemical Society: Washington, DC, 1981.
8/18/2019 (ACS Symposium Volume 168) H. Scott Fogler (Eds.)-Chemical Reactors-American Chemical Society (1981) (1)
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The University of Michigan
Based on a symposium
Industrial and Engineering Chemistry
(180th ACS National Meeting),
6 8
AMERICAN CHEMICAL SOCIETY
W A S H I N G T O N , D. C
1981
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Chemical
reactors.
Industrial and Engineering Chemistry at the Second
Chemical
August
American
III. Chemical Congress of the North
American
Con
TP157.C423
A ll
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PREFACE
T
he symposium upon which this volume is based focused on three areas
in
and
packed bed reactors. Each area comprises a section of this book.
Professor J. R.
chaired and coordinated the fluidized bed sessions;
Professors Y. T. Shah and A. Bishop, the bubble column reactor session;
and
Fluidized
bed reactors have received increased interest in recent years
owing to their application in coal gasification. The section on fluidized
beds
areas in fluid bed reactor modeling. Computer simu
lation of both solid-catalyzed gas phase reactions as well as gas-solid
reactions are included.
ters
needed for scale-up are presented along with models for reaction and
heat transfer. The mixing characteristics of columns are described as are
the directions for future research work on bubble column reactors.
The
presents
niques for solving the differential
mass
reactors. The advantages and limitations of various models
(e.g.,
pseudo-
homogeneous
vs.
heterogeneous)
are
also
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l i c a t i o n D a t e : S e p t e m b e r 2 1 , 1 9 8 1 | d o i : 1 0 . 1 0 2 1 / b k - 1 9 8 1 - 0 1 6 8 . p r 0
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Department
Vancouver, Canada V6T
1W5
C r i t i c a l a r e a s in f l u i d bed r e a c t o r m o d e l i n g
are
dis
cussed
in
in this
symposium.
T h e r e c o n t i n u e s to be a wide d i v e r s i t y of
a s s u m p t i o n s
u n d e r l y i n g
m o d e l s .
However,
it is now
c l e a r t h a t p r e
d i c t i o n s are
g e n e r a l ly
much more
s e n s i t i v e to
some
a s s u m p t i o n s t h a n
to
For
mod
e l i n g of i n t e r p h a s e exchange is g e n e r a l l y
more
cal
the
a s s u m p t i o n s a d o p t e d
to
a x i a l
g a s d i s p e r s i o n
in
the
e m u l s i o n
p h a s e . For
t he 1980's a d v a n c e s
are
looked
for
e s p e c i a l l y
in
more
s o p h i s t i c a t e d
computer
mod
e l s ,
u n s t e a d y s t a t e r e p r e s e n t a t i o n s s u i t a b l e
for
con
w h i c h
d e s c r i b e h i g h
v e l o c i t y
r e g i m e s
of f l u i d i z a t i o n ,
i n c l u s i o n
of g r i d and
f r e e
e f f e c t s , and
s t u d y
of radial
T h i s
volume
b r i n g s t o g et h e r
a
number
of
p a p e r s u n d e r
the
theme of f l u i d i z e d bed r e a c t o r m o d e l i n g . T h i s f i e l d is of r e l a
t i v e l y r e c e n t
o r i g i n .
T a b l e I g i v e s
the
emphasis in r e s e a r c h in
s u c c e s s i v e
decade s
b e gi n n i n g w i t h the 1940's.
It is s e e n
t h a t
r e s e a r c h
was
p r i m a r i l y
to
p r o b l e m s
a s s o c i
a t e d w i t h
the
o pe r a t i o n of f l u i d i z e d bed r e a c t o r s and to v e r y
s i m p l e m o d e l s . W i t h
the
p a s s a g e of t i m e models have been d e v i s e d
w h i c h
are i n c r e a s i n g l y s o p h i s t i c a t e d . Reviews
of
de velopme n t
f l u i d i z e d beds
as
r e a c t o r s have been p r e p a r e d
by
the
1970
1
were
a
number
(3-7)
w h i c h c o n s i d e r e d f l u i d i z e d bed r e a c t o r m o d e l i n g .
I n o r d e r
to be
to
r e p r e s e n t the b e h a v i o u r
of f l u i d i z e d
bed r e a c t o r s w i t h c o n f i d e n c e ,
one
a
t h o r o u g h u n d e r s t a n d
i n g of
the
bed h y d r o d y n a m i c s and of
the
r e a c t i o n k i n e t i c s . A l m o s t
a l l of
the r e a c t i o n s
c ar r i e d
out
in f l u i d i z e d beds
are e i t h e r
s o l i d - c a t a l y s e d
gas
phase
or
g a s - s o l i d r e a c t i o n s .
(We
not
c o n s i d e r h e r e
homogeneous
gas
phase r e a c t i o n s , r e a c t i o n s
in l i q u i d f l u i d i z e d beds or r e a c t i o n s in t h r e e phase f l u i d i z e d
beds.) W h i l e
the c h e m i c a l
k i n e t i c s
can o f t e n
be
complex,
0097-6156/81/0168-0003$05.00/0
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i n t h e
g a s i f i c a t i o n
o r c om b u s t i o n o f c o a l , t h e h y d r o -
dynamic a s p e c t s have g i v e n t h e g r e a t e s t
d i f f i c u l t y
and have been
s u b j e c t t o t h e g r e a t e s t d e b at e . W h i l e c o n s i d e r a b l e p r o g r e s s h as
been made i n a c h i e v i n g an u n d e r s t a n d i n g o f many a sp e c t s o f b ed
b e h a v i o u r , t h e r e a r e
many
w h i c h
r e m a i n p o o r l y u n d e r
s t o o d . Some
of t h e s e ( e. g. r eg im e s o f bed beha v i o u r , ga s m i x i n g
p a t t e r n s , and exchange of gas between p h a s e s ) can a f f e c t p r o
f o u n d l y t h e n a t u r e o f t h e model adopted.
T a b l e I : Focu s of Res ear ch on
F l u i d i z e d
Bed R ea c t o r s
Decade Emphasis
1940's P r a c t i c a l d e s i g n an d o p e r a t i o n p r o b le ms . S i n g l e
phase models o n l y .
1950
1
g a s - p h a s e
s o l i d - c a t a l y s e d
r e a c t i o n s .
I 9 6 0
1
s I n c o r p o r a t i o n o f p r o p e r t i e s o f
s i n g l e
b u b b l e s . E a r l y
models
f o r g a s - s o l i d
r e a c t i o n s .
1970
T
s A d d i t i o n of end ( g r i d a n d f r e e b o a r d )
e f f e c t s . More
s o p h i s t i c a t e d models f o r s p e c i f i c g a s - s o l i d r e a c t i o n s
i n c l u d i n g e n er gy b a l an c e s . C o n s i d e r a t i o n o f complex
k i n e t i c s .
1980
1
emphasis
o n n o n - b u b b l i n g
( t u r b u l e n t
and
f a s t
fluidization) r e g i m e s . P r o b a b l e c o n s i d e r a t i o n
o f e f f e c t s o f a i d s t o fluidization ( e. g. c e n t r i f u g a l ,
magnetic and e l e c t r i c a l f i e l d s ,
b a f f l e s ) .
I n c r e a s i n g
emphasis
on more complex h y d r o d y n a m i c s and
k i n e t i c s ,
w i t h models r e q u i r i n g
compu t e r s
f o r
s o l u t i o n .
The p ap e r s p r e s e n t ed a t t h e L a s
Vegas
symposium, most o f
w h i c h a r e r e p r o d u c e d i n
this
volume, b o t h i l l u s t r a t e t h e d i v e r
s i t y of modeli ng approac h and show some new
d i r e c t i o n s
f o r r e a c t o r
model i n g i n t h e 1980's. B e f o r e t u r n i n g t o t h e s e m a t t e r s i n d e
t a i l , i t is n e c e s s a r y t o d i s c u s s b r i e f l y t h r e e of t h e p ap e r s
w h i c h a r e f u n d a m e n t a l l y d i f f e r e n t i n f oc u s f r om t h e o t h e r e i g h t .
The paper by
Rami r e z
jet aJL (8) co n s i de r s th e i mport ant qu es
t i o n
of p a r t i c l e- t o - g a s heat t r a n s f e r in f l u i d i z e d beds. I n ad
d i t i o n t o t h e i mp or t a n c e of this q u e s t i o n i n i t s own r i g h t , p a r
t i c l e - t o - g a s hea t t r a n s f e r c a n b e i m po r t a n t f o r f l u i d b ed r e a c
t o r s ,
f o r example i n de t e r m i n i n g t h e r m a l g r a d i e n t s in t h e e n t r y
( g r i d ) r e g i o n , in
e s t a b l i s h i n g
t h e s u r f a c e t e m pe r a t u r e o f
p a r t i
c l e s
u n d e r g o i n g
r e a c t i o n s ,
a n d v i a t h e a n a l o go u s c a s e o f g a s - t o -
p a r t i c l e
mass
t r a n s f e r .
Ther e has been c o n s i d e r a b l e c o n t r o v e r s y
o v e r t h e f a c t t h a t N u s s e l t an d Sherwood numbers
have
been fou nd
t o f a l l w e l l below 2, the lower l i m i t f o r a
s i n g l e
sphere in a
medium.
f u r t h e r
e v i d e n c e of S h
« 2 and Nu « 2 and co n s i de r th es e
r e s u l t s
i n t h e
l i g h t
o f
models i n t h e
l i t e r a t u r e .
The paper by Blake and
Chen
( 9) r e p r e s e n t s a n e x t e n s i o n o f
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Reactors
5
S y s t e m s ,
S c i e n c e a n d S o f t w a r e
group. In what
must
be the most a mb i t i o u s a nd c o m p r e h e n s i v e
fluidization m o d e l i n g e f f o r t t o d a t e , this g r o u p has used modern
c o m pu t a t i o n a l t e c h n i q u e s t o s o l v e a s e t o f e q u a t i o n s r e p r e s e n t i n g
t h e p h y s i c s an d c h e m i s t r y o f
f l u i d i z e d
bed
Hy-
d r o d y n a m i c f i x t u r e s a r e r e p r e s e n t e d by a s e t o f c o n t i n u u m equa
t i o n s and
c o n s t i t u t i v e r e l a t i o n s h i p s ,
w h i l e c h e m i c a l
k i n e t i c s
e q u a t i o n s a r e w r i t t e n f o r k ey h e t er o g en e o u s an d
homogeneous
r e
a c t i o n s b as ed o n s t u d i e s r e p o r t e d i n t h e l i t e r a t u r e . I n p r e v i o u s
p a p e r s , t h e a u t h o r s have
shown
t h a t t h e model g i v e s a r e a l i s t i c
s i m u l a t i o n
o f a j e t o f gas
i s s u i n g i n t o
a bed of s o l i d s . I n t h e
p r e s e n t pape r t h e y s e e k t o d u p l i c a t e
r e s u l t s
o b t a i n e d i n t h e I G T
and W e s t i n g h o u s e p i l o t s c a l e r e a c t o r s . The r e s u l t s a r e o f co n
s i d e r a b l e
i n t e r e s t , g i v i n g a good match w i t h most o f t h e
e x p e r i
m e n t a l r e s u l t s .
A
f u r t h e r
pape r b y G i b b s ( 10) d e a l s w i t h d e s i g n a nd m o d e l i n g
o f
c e n t r i f u g a l f l u i d i z e d b e d s . I n
this
cas e gas i s fed r a d i a l l y
i nward s
i n t o
a s p i n n i n g b e d. On acco u n t o f t h e g r e a t l y augmented
e f f e c t i v e g r a v i t y
f o r c e ,
g r e a t e r t h r o u g h - p u t s o f g as c a n be a c
commodated
a n d e n t r a i n m e n t i s
g r e a t l y
l ow e r e d . T h i s new t e c h
n i q u e h a s r e c e i v e d a t t e n t i o n i n t h e l a t e 1970
1
s e s p e c i a l l y i n
c o n n e c t i o n w i t h c o a l c o m b u s t i o n . Some u n i q u e p r o b l e m s a r e e n
c o u n t e r e d , e .g. t h e
minimum
fluidization
becomes a
of bed depth, while
e j e c t e d
i n t o
1 1
by bubbles b u r s t i n g a t t h e b ed s u r f a c e t r a v e l
i n i t i a l l y
n e a r l y
a t
r i g h t
a n gl e s t o t h e ga s
e x i t d i r e c t i o n .
T h i s pape r
g i v e s a p r e l i m i n a r y scheme f o r d e a l i n g w i t h some o f t h e s e
problem s.
C l a s s i f i c a t i o n o f R e ac t o r Models
T h e r e a r e many c h o i c e s t o b e
made
in
bed r e a c t o r mod
e l i n g
and
l i t t l e
u n a n i m i t y among t h o s e who d e v i s e s u ch models on
t h e b e s t c h o i c e s . T a b l e I I
l i s t s
some of the p r i n c i p a l a r e a s
f o r d e c i s i o n
an d t h e c o r r e s p o n d i n g c h o i c e s o f t h e o t h e r e i g h t
pape r s a t this
symposium
(11-18).
P h a s e s . B o t h two-phase an d t h r e e- p h as e r e p r e s e n t a t i o n s a r e
w i d e l y u s e d a s
shown
s c h e m a t i c a l l y i n F i g u r e 1. I n two-phase
r e p r e s e n t a t i o n s t h e d i l u t e phase may r e p r e s e n t bu b b l e s a l o n e , j e t s
( i n
r e g i o n ) ,
o r b u bb l es p l u s c l o u d s . T h r e e - p h a s e r e p r e
s e n t a t i o n s g e n e r a l l y u s e t h e scheme f o l l ow e d b y
K u n i i
and Leve n -
whereby
b u b b l es , c l o u d s , an d " e m u l s i o n " ( i . e . t h a t
p a r t o f t h e n o n - b u b b l e b ed n o t i n cl u d e d i n t h e cl o u d s ) a r e e ach
t r e a t e d
as s e p a r a t e r e g i o n s . As
shown
i n T a b le I I , a l l o f t h e s e
p o s s i b i l i t i e s a r e r e p r e s e n t e d in t h e
models
a u t h o r s i n
this
symposium.
however,
t o be an in
c r e a s i n g t e n d e n c y t o ado p t t h r e e phase m o d e l s , p r o b a b l y as a r e
s u l t o f e x p e r i m e n t a l r e s u l t s (20) w h i c h
showed
t h a t t h e K u n i i and
L e v e n s p i e l model gave a b e t t e r r e p r e s e n t a t i o n o f
measured
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D,
dense
phase;
E,
b;
three-phase models, c
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B ed
Reactors
7
t r a t i o n p r o f i l e s f o r a p a r t i c u l a r p a r t i c l e s i z e t h a n o t h e r
models
t e s t e d . The b u bbl e s t h em s el v e s a r e u s u a l l y t r e a t e d a s b e i n g
com
p l e t e l y d e v o i d o f
p a r t i c l e s ,
bu t it is important (21) t h a t s o l i d s
d i s p e r s e d i n t h e b u bbl e s b e i n c l u d e d w i t h t h e b u b b l e phase f o r
f a s t
r e a c t i o n s ,
e v e n t ho u g h t h e i r c o n c e n t r a t i o n i s s m a ll ( t y p i
c a l l y
< 1% by v olu me).
G a s M i x i n g i n t h e Dense o r E m u l s i o n P h a s e , No o t h e r f e a t u r e
of f l u i d i z e d bed r e a c t o r modeli ng has been s u b j e c t e d t o s o
many
a l t e r n a t i v e
a s s u m p t i o n s a s a x i a l mi x i n g i n t h e d e n s e p h a s e . At
l e a s t e i g h t p o s s i b i l i t i e s
have
shown
i n F i g u r e 2 .
T h e s e r a n g e f r om upward p l u g f l o w , t h r o u g h p e r f e c t m i x i n g and
s t a g n a n t g a s , t o downflow. I n t e r m e d i a t e d eg r e e s o f m i x i n g have
been r e p r e s e n t e d by a x i a l
d i s p e r s i o n models
a nd well - m i x ed
s t a g e s
in s e r i e s . As shown in
Tabe
p o s s i b i l i
t i e s
have been
c o v e r e d in t h e p r e s e n t
symposium.
I n v i ew of t h e l a r g e number o f d i s p a r a t e r e p r e s e n t a t i o n s o f
d e n s e
phase
a x i a l
m i x i n g , o n e m i g h t e a s i l y c o n c l u d e t h a t
this
is
more
i m p o r t a n t m o d e l i n g f e a t u r e s . I n p r a c t i c e this is
n o t t h e c as e , u n l e s s h i g h c o n v e r s i o n s ( e . g . 90 % o r g r e a t e r i n a
s i n g l e s t a g e ) a r e s o u g h t . F o r l ow er c o n v e r s i o n s , o v e r a l l r e a c t o r
p e r fo rm a n c e t e n d s t o b e i n s e n s i t i v e t o t h e p a t t e r n o f
a x i a l
mix
s e v e r a l
i l l u s t r a t i o n s o f this p o i n t
in
this
al
( 1 6 ), r e
placement of th e downflow c o n d i t i o n a d op t e d b y F r y e r an d P o t t e r
(22) by
p e r f e c t
m i x i n g i n t h e em u ls i o n l e d t o c o n v e r s i o n s
w h i c h
were b a r e l y
d i s t i n g u i s h a b l e
f r o m t h o s e g i v e n b y t h e e a r l i e r mod
e l . ( A t t h e same t i m e s o l u t i o n became
much
s i m p l e r . ) J a f f r e s £t
a l (15) show t h a t the two extreme c a s e s o f p e r f e c t m i x i n g a nd
p l u g f l o w i n t h e O r c u t t models (23) l e a d t o s i m i l a r r e s u l t s . ( I n
t h e i r c a s e , however, b u b b l e p r o p e r t i e s were
v a r i e d
t o g e t h e r w i t h
k i n e t i c
c o n s t a n t s i n
t h e i r
o p t i m i z a t i o n s o i t i s h a rd e r t o
d i s
t i n g u i s h t h e i n f l u e n c e o f t h e m i x i n g as s u m p t i o n s a l o n e . ) E l n a -
s h a i e and E l s h i s h i n i (12) f u r t h e r show t h a t t he e f f e c t of a x i a l
d i s p e r s i o n
i s n o t o n l y r e l a t i v e l y s l i g h t in
t e r m s
o f o v e r a l l c o n
v e r s i o n ,
bu t
t h a t d e n s e phase
m i x i n g a l s o p l a y s a r e l a t i v e l y
m i n o r
r o l e
i n d e t e r m i n i n g
s e l e c t i v i t y
f o r c o n s e c u t i v e
r e a c t i o n s
and m u l t i p l i c i t y of s t ea d y s t a t e s .
I n a lm os t a l l p r e v i o u s mo de li n g work, o n e - d i m e n s i o n a l f l o w
has been assumed in each p h a s e , radial g r a d i e n t s b e i n g t a k e n as
n e g l i g i b l e . There is some e x p e r i m e n t a l e v i d e n c e (24) t h a t s u b
s t a n t i a l r a d ia l g r a d i e n t s may e x i s t , however. R a d i a l g r a d i e n t s
a r e
e s p e c i a l l y i m p o r t a n t f o r
f l u i d
bed c o m b u s t o r s wi t h in-bed
f e e d i n g o f f r e s h c o a l v i a a s e r i e s o f n o z z l e s . I n
this
c a s e t h e
r a p i d d e v o la t i l i z a t i o n r e a c t i o n s
w i l l
o c c u r
c l o s e
t o t h e d i s t r i
b u t ed feed p o i n t s , and radial
d i s p e r s i o n
o f v o l a t i l e s away from
t h e s e p o i n t s an d
oxygen
them
w i l l
b e e x t r e m e l y i m p o r t a n t
i f t h e v o l a t i l e s a r e t o b u r n o u t w i t h i n t he bed. Fan and Chang
(13) have c o n s i d e r e d
this
p r o b le m , c o u p l i n g a n a s s u m p t i o n o f p e r
f e c t a x i a l m i x i n g w i t h a
d i f f u s i o n - t y p e
m i x i n g model i n t h e r a -
P u b
| d o i : 1 0 . 1 0 2 1 / b k - 1 9 8 1 - 0 1 6 8 . c h 0
0 1

P u b l i c a t i o n D a t e : S e p t e m b e r 2 1 , 1 9 8 1
| d o i : 1 0 . 1 0 2 1 / b k - 1 9 8 1 - 0 1 6 8 . c h 0 0 1
In Chemical Reactors; Fogler, H.;
ACS Symposium Series; American Chemical Society: Washington, DC, 1981.

P u b l i c a t i o n D a t e : S e p t e m b e r 2 1 , 1 9 8 1
| d o i : 1 0 . 1 0 2 1 / b k - 1 9 8 1 - 0 1 6 8 . c h 0 0 1
In Chemical Reactors; Fogler, H.;
ACS Symposium Series; American Chemical Society: Washington, DC, 1981.
P u b
| d o i : 1 0 . 1 0 2 1 / b k - 1 9 8 1 - 0 1 6 8 . c h 0
0 1
11
d i a l d i r e c t i o n . A l t h o u g h
the u n d e r l y i n g
of d i f f u s i o n -
t y p e models o f t e n seem i n a p p r o p r i a t e
to a x i a l
m i x i n g
in
f l u i d i z e d
b e d s (25) , t h e r e is some e v i d e n c e ( e . g . 26) t h a t l a t e r a l m i x i n g
ca n
be d es c r i b e d in this
manner. Hence
a
a p p r o a c h
to the
d e s c r i p t i o n
of
a n i m p o r t a n t p r o b l e m .
I n t e r p h a s e
Gas
the
in this
symposium ( s e e T a b l e I I )
on
t h e mass t r a n s f e r e q u a t i o n s p r o p o s e d
b y D a v i d s o n and H ar r i s o n (27) and by
K u n i i
and L e v e n s p i e l ( 1 9 ) ,
o ne m i gh t r e a s o n a b l y c o n c l u d e
t h a t
t h e s e a p p r o a c h e s have b e e n s u p
p o r t e d
by at
l e a s t t h e m aj o r i t y
of
e x p e r i m e n t a l e v i d e n c e . N o t h i n g
c o u l d
be
f u r t h e r f r o m
the
t r u t h .
T h e D a v i d s o n
and
H a r r i s o n a p p r o a c h c o n c e n t r a t e s s o l e l y
on the
r e s i s t a n c e
at the
b u b b l e / c l o u d b o u n da r y ( o r bu b b l e / de n s e p h as e
b o u n d a r y f o r
α < 1 ) . The t r a n s f e r c o e f f i c i e n t , r e f e r r e d to bubble
s u r f a c e a r e a , is
\ r ° -
( 1 )
A l t e r n a t i v e l y , on a bubble volume
b a s i s ,
this becomes
w v o * (2)
Th e f i r s t t e r m in e a c h c a s e a r i s e s f r o m b u l k f l o w of gas i n t o the
f l o o r of an i s o l a t e d
bubble
by the
h y d r o d y n a m i c
model of
and
. The
of
e x p e r i m e n t a l e v i d e n c e , f r om
s t u d i e s
of
c l o u d
s i z e ( 2 8 , 2 9 ) , f r o m
c h e m i c a l r e a c t i o n
s t u d i e s
( e . g . 3 0 ) , and f r o m i n t e r p h a s e t r a n s f e r
s t u d i e s
(e .g. 31 ,32) ,
is
be t t e r d e s c r i b e d
by the
p r o p o s e d
by
. The
l a t t e r l e a d s
to a
r e d u c t i o n
in t h e f i r s t t e r m by a f a c t o r of 3. Some enhancement of the b u l k
f l o w
component
o c c u r s f o r i n t e r a c t i n g b u b b l e s ( 3 4 , 3 5 ) , but this
enhancement
a
f r e e l y b u b b l i n g
bed is
of 20-
30% (35),
would
be
r e q u i r e d f o r
the
b u l k f l o w
t e r m E q ua t i o n s (1) and (2) to be v a l i d .
Th e s e c o n d t e r m
in
(1) and (2)
a c c o u n t s f o r
d i f f u -
s i o n a l t r a n s f e r a c r o s s the b u b bl e b o u n d a r y . (A f a c t o r ε
f / i l + ^ f )
in the
of Eq. 2 Ψο
f o r the
d e n s e p h a s e
d i f f u s i o n a l
r e s i s t a n c e . ) T h e r e
is some
q u e s
t i o n (30) of the e x t e n t to wh i c h t h e r e is i n t e r f e r e n c e between
t h e b u l k f l o w and
d i f f u s i o n
t e r ms . N e v e r t h e l e s s ,
most
e x p e r i m e n t
a l
e v i d e n c e s u gg e s t s t h a t
the two
are
and
the
d i f f u s i o n a l
t e r m
by the
p e n e t r a t i o n t h e o r y . W i t h
t h e s e c h a n g e s , and i n c l ud i n g a s m a l l
enhancement
f a c t o r f o r b u bb l e
i n t e r a c t i o n , S i t and G r a c e ( 35 ) have
recommended
the f o l l o w i n g
e q u a t i o n s
as
b e i n g
in b e s t a c c o r d
=
(3)
P u b
| d o i : 1 0 . 1 0 2 1 / b k - 1 9 8 1 - 0 1 6 8 . c h 0
0 1
3
K u n i i
a n d
L e v e l s p i e l
(19) a g a i n u s e E q u a t i o n (2) t o d e s c r i b e
b u b b l e / c l o u d
t r a n s f e r . B a s e d
o n t h e p e n e t r a t i o n t h e o r y , t h e y
p r o p o s e t h e
f o l lo w i n g
e x p r e s s i o n f o r c l o ud / e m uI s i o n
t r a n s f e r :
k
1
c e mf b b
E q u a t i o n (5) c o n s i d e r s ga s d i f f u s i o n t o be t h e o n l y mechanism o f
t r a n s f e r a c r o s s t h e o u t e r c l o u d b o u n d a r y . I n p r a c t i c e t h e r e a r e
a t
l e a s t t h r e e o t h e r i m po r t a n t
mechanisms
n o t a c c o u n t e d f o r :
( a) Th e c l o u d b o u n d a r y i s a s t r e a m l i n e f o r ga s e l e me n t s b ut n o t
f o r
s o l i d p a r t i c l e s .
P a r t i c l e s e n t e r i n g an d
le a v i n g
t h e c l o u d
b o u n d a r y w i l l c a r r y a d s o r be d s p e c i e s w i t h them. (b) Th e re i s
s t r o n g e v i d e n c e o f s h e d d i n g o f e le m e n t s f r o m t h e
wakes.
P h o t o
grap h s (28) i n d i c a t e t h a t t h e s e sh e d e l e me n t s r e s u l t s i n t r a n s f e r
o f
c l o u d ga s t o t h e e m u l s i o n . ( c ) Th e c o n c e p t o f t h e c l o u d is
b a s e d o n s t e a d y
s t a t e
a n a l y s e s
(27,33)
.
W h i l e a m a n t e l o f g a s
a p pe a r s t o r e m ai n a s s o c i a t e d w i t h b ub bl e s a s t h e y c o a l e s c e , t h e s e
" c l o u d s '
1
l i k e
t h e b ub bl e s t h e m s e l v e s ,
d i s t o r t
a n d
c h a n g e s
d u r i n g b ub bl e
i n t e r a c t i o n
a nd c o a l e s c e n c e ( 28 , 36 ). T h i s
no
d o u b t f u r t h e r
enhances
c l o u d / e m u ls i o n
t r a n s f e r .
F o r most
p r a c t i c a l c o n d i t i o n s ,
a c o m p a r i s o n o f k ^ a n d k ^
f r o m E q u a t i o n s (4) a n d (5)
w oul d
t h a t
t h e p r i n c i p a l r e
s i s t a n c e
t o
r e s i d e s
a t t h e o u t e r c l o u d
bo u n d a r y .
How
e v e r , when ( a ) , ( b) a n d ( c ) a r e t a k e n
i n t o
this
t h e c a s e . I n
f a c t ,
e x p e r i m e n t a l e v i d e n c e ( e .g .
30,31,32)
i n d i c a t e s s t r o n g ly t h a t
t h e p r i n c i p a l
r e s i s t a n c e
i s a t t h e b u b bl e /
c l o u d
i n t e r f a c e .
W i t h
mind,
i t i s p r o b a b l y
more s e n s i b le
t o i n c l u d e t h e c l o u d w i t h t h e
d e n s e phase
( a s i n t h e O r c u t t ( 23 ,
27)
models)
r a t h e r t h a n w i t h t h e bu bb le s ( a s i n t h e P a r t r i d g e a nd
Rowe (37) model) i f a two-phase r e p r e s e n t a t i o n i s t o be a d o pt e d
( s e e
F i g u r e 1 ) . I f t h r e e - p h a s e
models
a r e u s e d , t h e n E q u a t i o n s
(2) an d (5) a p p e a r t o be a p o o r b a s i s f o r p r e d i c t i o n . F o r t u n a t e
l y
e r r o r s
go i n o p p o s i t e
d i r e c t i o n s ,
E q u a t i o n (2) o v e r p r e -
d i c t i n g
t h e b u bb l e / c l o u d
t r a n s f e r
c o e f f i c i e n t ,
w h i l e E q u a t i o n ( 5)
u n d e r e s t i m a t e s t h e c l o u d / e m u l s i o n
t r a n s f e r c o e f f i c i e n t .
T h i s
p r o b a b l y a c c o u n t s f o r t h e f a c t t h a t t h e
K u n i i
a nd L e v e n s p i e l model
(19) c a n g i v e r e a s o n a b l e
p r e d i c t i o n s
i n
s p e c i f
i c i n s t a n c e s (e , g.20).
F l o w D i s t r i b u t i o n between P h a s e s . On e o f t h e p r i n c i p a l
a s s ump ti o n s un d e r l y i n g
many
models
o f f l u i d i z e d be d
r e a c t
o r s i s t h e "two-phase t h e o r y o f f l u i d i z a t i o n . T h i s t h e o r y ,
r e a l l y no
more
t h a n a p o s t u l a t e , h o l d s
t h a t
beyond t h a t
r e q u i r e d
f o r
minimum fluidization
p a s s e s t h r o u gh t h e be d a s
t r a n s
l a t i n g
v o i d u n i t s .
A l t h o u gh n o t i n c l u d e d i n what t h e o r i g i n a t o r s
o f
this
p o s t u l a t e (38) a p p e a r e d t o
have
phase
t h e o r y i s o f t e n h e l d t o i m pl y , i n
a d d i t i o n , t h a t
t h e
d e n s e phase
v o i d a g e re m a i n s c o n s t a n t a nd e q u a l t o ε f o r a l l U > U .
Much
s a i d
about th e two
P u b
| d o i : 1 0 . 1 0 2 1 / b k - 1 9 8 1 - 0 1 6 8 . c h 0
0 1
Fluidized Bed
see 39). For our purpos e s here i t
s u f f i c e s
t o n o t e t h a t
t h e r e i s v e r y l i t t l e e v i d e n ce i n d e e d t h a t t h e f l o w d i s t r i b u t i o n
r e a l l y f o l l o w s t h e t h e o r y . I n f a c t , t he
w e i g h t
o f e v i d e n ce ( s e e
39) s u gge s t s t h a t t h e t h e o r y
s e r i o u s l y
o v e r e s t i m a t e s t h e f l o w a c
coun t ed f or b y t r a n s l a t i o n of bub bles , e x cept i n the l i m i t as s lug
f l ow c o n d i t i o n s a r e
a p p r o a c h e d .
Y et , d e s p i t e a l l
this
e v i d e n c e ,
the two phase t h e or y c o n t i n u e s as an u n d e r p i n n i n g f o r
much
model i ng
work,
a s is a g a i n e v i d e n t f r om T a b l e I I .
T h e r e
a r e s e v e r a l p r o b a b le r e a s o n s f o r t he c o n t i n u i n g p o p u l a r i t y o f t h e
two phase t h e o r y i n t h e f ace o f
c o n t r a d i c t i n g
e v i d e n c e :
( i ) T h e r e
i s a l a c k o f
al t e r n a t i v e
approaches.
( i i ) T h e r e
i s c o n f u s i o n
between
v i s i b l e and i n v i s i b l e
( i . e . b u l k f l o w o r " t h r o u gh f l ow" )
t e r m s .
Toomey and John s t o ne
(38) appear ed t o
have
in
mind
o n l y t h e v i s i b l e ( i . e . f l o w du e t o
v o i d u n i t t r a n s l a t i o n )
t e r m . As no t ed ab o v e , t h e t h e o r y t h e n
o v e r e s t i m a t e s t h e b u b bl e f l o w .
However,
i f the bubble flow i s
t a k e n t o i n c l u d e t h e i n v i s i b l e th r ough flow, t he the or y may do
b e t t e r and may
e v e n
u n d e r e s t i m a t e t h e b u b b l e f l ow. Many
w o r k e r s
f a i l
t o
d i s t i n g u i s h
c l e a r l y
whether
of v i s i b l e or
t o t a l
bub ble f low.
al
i n th at i t at t empt s
a new
to the two
f l o w
d i s t r i b u t i o n p r o b l e m .
F u r
t h e r
d e t a i l s a r e g i v e n i n a n o t h e r
pape r
by the
a u t h o r s ( 4 0) .
However, t h e a u t h o r s f a i l t o
d i s t i n g u i s h
c l e a r l y between v i s i b l e
and i n v i s i b l e f l o w
components
At
this
a p p r o a c h
must
be r e g a r d e d a s a p u r e l y e m p i r i c a l
method
w h i c h ap pe a r s t o g i v e a r e a s o n a b l e match w i t h
s e l e c t e d
ex
p e r i m e n t a l d a t a .
B u b b l e S i z e .
A
number
o f e m p i r i c a l an d s e m i - e m p i r i c a l a p
p r o ach e s a r e
a v a i l a b l e
f o r p r e d i c t i n g
mean
b u b b l e s i z e a s a f u n c
t i o n o f h e i g h t an d o t h e r c o n d i t i o n s i n g as
f l u i d i z e d
b e d s .
J u d g
a p p r o a c h
f ol lo we d by Mo r i and Wen (41) a p
p e a r s t o have
become
t h e f a v o r e d method of p r e d i c t i n g d^. Th i s
e q u a t i o n is s e m i - e m p i r i c a l ; p r e d i c t i o n s a r e
bounded
between an
i n i t i a l s i z e
produced at a
di s t r i b u t o r
and a
a c h i e v
ed o nl y
u n d e r s l u g
f lo w c o n d i t i o n s .
A n o t h e r
r e c e n t m e c h a n i s t i
c a l l y
based e q ua t io n due to Dar t o n e t a l (42) i s
a l s o r e c e i v i n g
c o n s i d e r a b l e
a t t e n t i o n ,
but has no t been t e s t e d by a ny o f t h e a u
t h o r s in this
symposium. B o t h a p p r o a c h e s seem
t o r e p r e s e n t
marked
improvements
o v e r p r e v i o u s e q u a t i o n s o f a
s o l e l y
e m p i r i c a l n a t u r e
i n t h e l i t e r a t u r e . A method is s t i l l r e q u i r e d f o r p r e d i c t i n g
bubble
in
beds
c o n t a i n i n g t u b es a s i n t h e
Type Β
c o m b u s t o r
c o n s i d e r e d by Fa n a n d
Chang
(13).
F i v e o f t h e p a p e r s s u r v e y e d i n Tabe I I t r e a t t h e bubbl e s i z e
a s i f i t were i n d e p e n d e n t o f h e i g h t . I n two c a s e s ( 14 , 17) d ^ i s
a l l o w e d t o v a r y w i t h h e i g h t . W h i l e t h e
l a t t e r
a s s um pt i o n i s c e r
t a i n l y
more
r e a l i s t i c ,
a s s u mp t i o n o f a c o n s t a n t bu bb l e
s i z e
i s d e
f e n s i b l e o n t h e g r o u n d s o f s i m p l i c i t y a nd l i m i t e d
s e n s i t i v i t y
r e l
t o
o f t h e o t h e r
d i s c u s s e d i n
this
H e a t
B a l a n c e . F o r
many
y e a r s i t h a s
been c u s t o ma r y
t o
P u b
| d o i : 1 0 . 1 0 2 1 / b k - 1 9 8 1 - 0 1 6 8 . c h 0
0 1
f l u i d
be d r e a c t o r s a s i s o t h e r m a l a nd t o i g n o r e e n e r g y b a la n c e s in
t h e m o d e l i n g p r o c e s s . R e c e n t emphasis on c o a l c o m b u s t i o n and
o t h e r g a s - s o l i d r e a c t i o n s w i t h h i g h h e a t s o f r e a c t i o n h as l e d t o
t h e i n c l u s i o n o f h e a t
b a la n c e s
w i t h more and more
m o d e l s . H e a t
b a l a n c e s
a r e i g n o r e d i n o n l y t h r e e o f t h e e i g h t
p a p e r s
s u r v e y e d
i n T a b le I I c o n f i r m i n g this t r e n d .
S t e a d y
v e r s u s
U n s t e a d y
S t a t e
M o d e l s .
U n t i l v e r y r e c e n t l y ,
f l u i d i z e d
bed r e a c t o r models have d e a l t a l m o s t e x c l u s i v e l y w i t h
s t e a d y s t a t e c o n d i t i o n s . S t e a d y s t a t e models a r e u n s u i t a b l e f o r
c o n t r o l p u r po s e s , f o r l o a d f o l l o w i n g i n
f l u i d
bed c o m b u s t o r s , and
f o r s t a r t - u p a nd shutdown p ur p o s e s . I t i s a welcome s i g n t h a t
two o f t h e p a p e r s i n this
symposium
(13,15) d e r i v e models w h i c h
a r e p o t e n t i a l l y s u i t a b l e f o r t h e s e
p u r p o s e s .
Type o f R e a c t i o n and A p p l i c a t i o n . An i n c r e a s e d
emphasis
o n
g a s - s o l i d r e a c t i o n s h as been e v i d e n t f o r a b o u t a
d e c a d e .
T h r e e
o f t h e p a p e r s i n this
symposium
t r e a t g a s - s o l i d r e a c t i o n s , two
(13,18) d e a l i n g w i t h c o a l c o m b u s t i o n a nd t h e o t h e r (11) w i t h c a t
a l y s t r e g e n e r a t i o n . Of t h e f o u r
p a p e r s wh i c h
c o n s i d e r s o l i d - c a t
a l y s e d
g a s - p h a s e
r e a c t i o n s , o n e (15) d e a l s w i t h a
s p e c i f i c
a p p l i
c a t i o n ( p r o d u c t i o n o f m a l e i c a n h y d r i d e ) , a nd o ne (12) t r e a t s a n
u n s p e c i f i e d
c o n s e c u t i v e r e a c t i o n o f t h e
t y p e
A •> Β C; th e o t h e r
two (14,16) a r e c o n c e r n e d wi t h u n s p e c i f i e d f i r s t o r d e r
i r r e v e r s i
b l e r e a c t i o n s . Th e f i n a l p a p e r (17) c o n s i d e r s a r e l a t i v e l y r e
c e n t a p p l i c a t i o n ,
f l u i d i z e d
bed a e r o s o l
f i l t r a t i o n .
P r i n c i p l e s
o f
be d r e a c t o r
m o d e l i n g
a r e d i r e c t l y a p p l i c a b l e t o
s u c h
a
c a s e : A e r o s o l
p a r t i c l e s
d i s a p pe a r by a d s o r p t i o n on t h e
c o l l e c t o r
( f lu i d i z e d ) p a r t i c l e s much a s a g a s e o u s
component
d i s a p p e a r s by
r e a c t i o n in t h e c a s e o f a s o l i d - c a t a l y s e d r e a c t i o n .
E x p e r i m e n t a l D a t a . W h i l e t h e emphasis in this s e s s i o n was
o n r e a c t o r m o d e l i n g , models c a n o n ly u l t i m a t e l y p r o v e s u c c e s s f u l
i f
t h e y
a r e compared t o e x p e r i m e n t a l d a t a . T h i s p o i n t may seem
o b v i o u s ,
but it is
making s i n c e
m o d e l i n g
e f f o r t s to o o f t e n
seem to be i n t e l l e c t u a l e x e r c i s e s r a t h e r t h a n e f f o r t s t o r e p r e s e n t
r e a l i t y . W h i l e t h e r e i s a n e e d t o v e r i f y some o f t h e models p r e
s e n t e d a t this symposium, it is g r a t i f y i n g t h a t t h r e e o f t h e p a
p e r s (11,15,17) have a l r e a d y been
e x p o s e d
t o t h e t e s t o f e x p e r i
m e n t a l d a t a .
O t h e r Model
F e a t u r e s
Some o f t h e p r i n c i p a l f e a t u r e s common t o t h e d i f f e r e n t models
a r e d i s c u s s e d a b o v e . I n this s e c t i o n some f u r t h e r f e a t u r e s o f r e
a c t o r models a r e c o n s i d e r e d b r i e f l y w i t h r e f e r e n c e t o i n d i v i d u a l
p a p e r s
i n this symposium.
O nl y t h e p a p e r by de L a s a ^ t a l (11) e x p l i c i t l y t r e a t s t h e
e n t r y o r g r i d r e g i o n a s a n o n - b u bb li n g r e g i o n . T h i s r e g i o n is
modeled
t e r m s
o f d i s c r e t e ga s
j e t s ,
a n i d e a o r i g i n a t e d by Be -
h i e a n d Kehoe ( 43 ), bu t c o n t e s t e d a c t i v e l y by Rowe e t al (44). As
i n d i c a t e d in t h e p a p e r s by J a f f r e s e t a l (15) a nd Rehmat et. a l
( 18) , t h e g r i d r e g i o n is c l e a r l y a
zone
o f e f f e c t i v e ga s - s o l i d
P u b
| d o i : 1 0 . 1 0 2 1 / b k - 1 9 8 1 - 0 1 6 8 . c h 0
0 1
Fluidized
c o n t a c t i n g ,
bu t c o n s i d e r a b l e
work
is r e q u i r e d t o a c h i e v e a n u n
d e r s t a n d i n g o f t h e h y d r o d y n a m i c s and gas exchange p r o c e s s e s
t h e r e i n .
None
this
e x p l i c i t l y
c o n s i d e r s t h e
f r e e b o a r d r e g i o n al t h o u g h b o t h de L a s a £ t a l ( 11 ) an d
J a f f r e s
e t
a l
t o p r e v i o u s
work w h i c h
has
shown
t h a t t h e f r e e b o a r d
ca n pl a y a n i m po r t a n t
r o l e
i n d e t e r m i n i n g t h e
o v e r a l l
r e a c t o r pe r
f o r m a n c e . None of the paper s
t r e a t s
d i r e c t l y f l o w
r e g i m e s
o t h e r
t h a n t h e b u b b l i n g
r e g i m e ,
a l t h o u g h
Rehmat
et a l (18) m e n t i o n the
t u r b u l e n t
f l o w
r e g i m e
( t o g e t h e r w i t h
r a p i d
i n t e r p h a s e
exchange
in
t he g r i d r e g i o n ) a s j u s t i f i c a t i o n f o r u s i n g a
model w h i c h
t r e a t s
the gas as a
s i n g l e
phase i n plu g flow. As al r ea d y s ugges ted i n
T ab l e I , e f f o r t s t o model t u r b u l e n t an d f a s t f l u i d i z e d
beds
a r e
l i k e l y t o be i m p o r t a n t f e a t u r e s o f t h e
1980
f
s.
I n model i ng g a s - s o l i d r e a c t i o n s in f l u i d b e d s ,
p r o v i s i o n
must
be
made
f o r d e a l i n g w i t h
p a r t i c l e s i z e d i s t r i b u t i o n s
and w i t h
s o l i d s
m i x i n g .
S o l i d s
mi x i n g i s
u s u a l l y
a d e q u a t e l y d e s c r i b e d i n
t e r m s o f
p e r f e c t
m i x i n g . To a c co u n t f o r s i z e d i s t r i b u t i o n e f
f e c t s , p o p u l a t i o n b a la n c e s a r e g e n e r a l l y r e q u i r e d .
T h e s e
must
s i z e
d i s t r i b u t i o n o f t h e f e e d ,
e l u t r i a t i o n
an d l o s s e s o f
f i n e s , a t t r i t i o n
( i f a p p r e c i a b l e ) , an d an y
c h a n g e s
in p a r t i c l e s i z e due t o che m ic al r e a c t i o n . The
pape r
by
Rehmat
et
a l (18) i l l u s t r a t e s how t he s e f a c t o r s can be take n
i n t o
O v e r a l l
s o l i d
r e a c t i o n
r a t e s must be dete rmi ned by
sum
ming
ov er a l l p a r t i c l e
s i z e s ,
an d c o n v e r s i o n must be
r e l a t e d
t o
gas c o n v e r s i o n v i a t h e s t o i c h i o m e t r y o f t h e r e a c t i o n s .
C o n c l u d i n g Remarks
I t is
c l e a r
t h a t t h e r e a r e many u n r e s o l v e d q u e s t i o n s i n t h e
f i e l d
o f
f l u i d i z e d
bed r ea ct o r model i n g. Onl y the bubble and
s l u g
f l o w
have
r e c e i v e d s i g n i f i c a n t
a t t e n t i o n .
Whi le end
e f f e c t s ( g r i d
zone an d f r e e b o a r d r e g i o n e f f e c t s ) a r e b e g i n n i n g t o
b e t r e a t e d , a lm os t n o e f f o r t s
have been made
t o model h i g h v e l o c
i t y f l u i d i z e d beds
o p e r a t i n g in t h e t u r b u l e n t an d f a s t
f l u i d i z a -
t i o n r e g i m e s . T h e s e
r e g i m e s
a r e o f g r e a t i m por t a n ce i n d u s t r i a l l y
an d f o r f u t u r e a pp l i c a t i o n s .
E v e n
i n the bubble flow r e g i m e ,
where t h e r e i s a w e al t h o f
h y d r o d y n a m i c
a n d o t h e r d a t a, some o f
t h e k e y a s pe c t s o f b e h a v i o r
r e m a i n
p o o r l y u n d e r s t o o d .
I t
c l e a r
f r o m p r e v i o u s
work
symposium t h a t
models
more s e n s i t i v e
t o
in
some a r e a s t h a n i n o t h e r s . F o r v e r y
s l o w
r e a c t i o n s , r a t e s
become
c o n t r o l l e d
b y ch e m i cal
k i n e t i c s
and
i n s e n s i t i v e
t o
h y d r o -
dynamic a s s u m p t i o n s
a r e ad op t e d ( 1 4 , 48). F o r i n t e r m e d i a t e r e a c
t i o n s ,
i n t e r p h a s e t r a n s f e r g e n e r a l l y becomes the key
f a c t o r
c o n
t r o l l i n g t h e r e a c t o r
p e r f o r m a n c e ,
wi t h t h e d i s t r i b u t i o n of gas
between
a l s o
p l a y i n g a
s i g n i f i c a n t r o l e .
As
o u t l i n e d
a b o v e , a d v a n c e s have been made
i n u n d e r s t a n d i n g b o t h a r e a s , b u t
models have
g e n e r a l l y been
s l o w
to adopt c h a n g e s i n t h e b a s i c
used in
e a r l y
b u b b l e m o d e l s . For f a s t r e a c t i o n s , t h e
P u b
| d o i : 1 0 . 1 0 2 1 / b k - 1 9 8 1 - 0 1 6 8 . c h 0
0 1
CHEMICAL REACTORS
e x t e n t of a x i a l m i x i n g of gas in the d e n s e or e m ul s i o n phase and
t h e
of
a s s i g n e d to the
d i l u t e
phase a l s o become
i m p o r t a n t . However, a x i a l gas m i x i n g is
l e s s
i m p o r t a n t in g e n e r
a l t h a n m i gh t be
i n d i c a t e d
by the d e g r e e of a t t e n t i o n d e v o t e d to
this f e a t u r e . On the o t h e r hand,
radial
m i x i n g has r e c e i v e d too
l i t t l e a t t e n t i o n .
T h e p a p e r s p r e s e n t e d in this
symposium
a d v a n c e s
t h a t
w i l l be i m p o r t a n t in the
1980
f
s.
T h e s e i n c l u d e :
( a ) f u n d a m e n t a l l y new t y p e s of models u s i n g the power of modern
c o m p u t e r s to s o l v e c o m pr e h e n s i v e g o v e r n i n g e q u a t i o n s (9);
(b) c o n t i n u i n g s t r o n g
a t t e n t i o n
on g a s - s o l i d r e a c t i o n s as w e l l as
g a s - p h a s e s o l i d - c a t a l y s e d r e a c t i o n s ; (c) un s t e a d y s t a t e models
s u i t a b l e
for
c o n t r o l
pu r p o s e s (13 , 15); (d) a t t e n t i o n to
r a t e -
s t e p s
and to s e n s i t i v i t y
a n a l y s e s ;
(e)
of g r i d
a nd f r e e b o a r d e f f e c t s ; (f) i n c l u s i o n of e n e r gy b a l a n c e s ; and
(g) stud y
and
radial
d i s p e r s i o n
( 1 3 ) .
r e a c t o r models have
c h i e f l y been u s e f u l
in the p a s t as an
e d u c a t i o n a l t o o l in
a i d i n g
u n d e r s t a n d i n g of
f l u i d
bed p r o c e s s e s
a n d , to a
l i m i t e d
e x t e n t , for
s i m ul a t i o n
of e x i s t i n g
r e a c t o r s
and
c h e m i c a l p r o c e s s e s . If t h e s e
models
are to become u s e f u l a l s o for
d e s i g n , s c a l e - up and c o n t r o l of new
equipment
and p r o c e s s e s , ad
v a n c e

(acs symposium volume 168) h. scott fogler (eds.)-chemical reactors-american chemical society (1981)...

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