4 Physical Processes in Chemical Reactor Engineering
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E.
WICKE
Institut für Physikalische Chemie, Westfälische Wilhelms-Universität, 4400 Münster, Germany
The
scope
of this review
examination
of the
pseudo-homogeneous systems. SLP
catalysts,
are discussed this principle processes
this principle in detail.
Taylor
are indicated
with
the mass transfer beds.
of mass in packed This difference
boundary
and Wilhelm,
heat transfer
its
flow The
conditions
at the bed
ad-
systems, elementary
are
illustrated; in f l u i d i z e d
in packed
problems
Differentiation
and
applicability
The limits of pseudo-homogeneous
and of shallow
emphasized. known
dispersion
the
gas-solid
diffusivity;
For heterogeneous
the
critical
multiphase
and the limits of its
beds as well as to interphase
dispersion
i.e. porous
in
to gas phase and to solids dispersion
are presented. screens
processes
is effective
and
that underlie
is radial and axial dispersion.
including
applications
for
systems,
its shortcomings,
presentation
principles
models
For diffusion
vantages,
is the
physical
beds
modeling of
catalyst
between
axial
beds and real backmixing requires
a change
of Dankwerts,
of the
and of
is well
Wehner
entrance.
A l t h o u g h the papers i n the session o n " P h y s i c a l Processes" (see A D V A N C E S I N C H E M I S T R Y S E R I E S N O . 1 3 3 ) are different i n scope a n d n a t u r e , t h e y c o n t a i n a n u m b e r of c o m m o n v i e w p o i n t s that m a y serve as a basis f o r this r e v i e w .
O n e of t h e most i m p o r t a n t v i e w p o i n t s seems to b e t h e
o b v i o u s t e n d e n c y to d e s c r i b e processes t h a t r u n i n heterogeneous m u l t i p h a s e systems b y p s e u d o - h o m o g e n e o u s
m o d e l s a n d to c o m p r e h e n d t h e
b e h a v i o r of s u c h processes, s t a t i o n a r y o r d y n a m i c , i n terms of s i n g l e p h a s e concepts. principles
H e n c e t h e t h e m e of this r e v i e w w i l l b e to i n d i c a t e t h e
a n d the methods
of d e v e l o p i n g h o m o g e n e o u s
models for
heterogeneous processes a n d to present t h e usefulness of those concepts as w e l l as t h e l i m i t s of t h e i r a p p l i c a b i l i t y . 75 In Chemical Reaction Engineering Reviews; Hulburt, H.; Advances in Chemistry; American Chemical Society: Washington, DC, 1975.
76
CHEMICAL
REACTION ENGINEERING REVIEWS
T h e r e are t w o p r i n c i p a l types of heterogeneous systems i n c h e m i c a l reactors ( T a b l e I ) : the f l o w systems w h e r e h y d r o d y n a m i c flow is t h e p r e d o m i n a n t process of mass transport, a n d the d i f f u s i o n systems w h e r e
Table I. Type of System
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Flow
Two
Heterogeneous Reaction Systems Three
Phases
Phases
t r i c k l e beds bubble columns ( w i t h suspended c a t a l y s t s )
p a c k e d beds fluidized beds
macroheterogeneity:
e
m
microheterogeneity :
u
i j s
o
n
\ p h a s e - b u b b l e phase
grains-fluid
grids, screens, gauzes Diffusion
porous solids
| S L P catalysts
m a c r o p o r e a n d m i c r o p o r e systems
m o l e c u l a r d i f f u s i o n p r e v a i l s . T y p i c a l flow systems are r e p r e s e n t e d b y p a c k e d beds,
fluidized
b e d s , g r i d s , a n d screens, a n d those w i t h three
phases b y t r i c k l e beds a n d b u b b l e c o l u m n s w i t h s u s p e n d e d catalysts. D i f f u s i o n systems are the porous gas—solid catalysts a n d t h e s u p p o r t e d l i q u i d p h a s e ( S L P ) catalysts. I n some cases, different levels of h e t e r o geneity can be discerned. I n
fluidized
beds a n d i n three-phase b u b b l e
c o l u m n s there are the m a c r o l e v e l of h e t e r o g e n e i t y ( d e n s e or e m u l s i o n phase a n d b u b b l e p h a s e )
a n d the m i c r o l e v e l ( g r a i n s a n d
fluid);
in
d i f f u s i o n systems, these levels of h e t e r o g e n e i t y are t h e m a c r o p o r e a n d the m i c r o p o r e arrays. I n flow systems, the h e t e r o g e n e i t y leads to the w e l l k n o w n d i s p e r s i o n effects.
P a r a l l e l to the m a i n flow d i r e c t i o n — a x i a l or l o n g i t u d i n a l — t h e
d i s p e r s i o n is a p p r e c i a b l y b r o a d e r t h a n i t is n o r m a l to i t — r a d i a l or transversal. I n o r d e r to d e s c r i b e these effects i n a n a l o g y w i t h d i f f u s i o n p r o c esses, m e a n i n g f u l coefficients
for r a d i a l a n d a x i a l d i s p e r s i o n m a y
be
m e a s u r e d a n d defined o n l y i f t h e system is l a r g e e n o u g h to c o v e r the l e n g t h of at least 10 c h a r a c t e r i s t i c d i s p e r s i o n p a t h w a y s i n either d i r e c t i o n . I f this c o n d i t i o n is n o t satisfied, t h e m o d e l of t h e i d e a l single m i x i n g stage or the cascade of m i x i n g cells m a y b e a p p l i c a b l e to t h e system. I f this fails also, t h e n a p s e u d o - h o m o g e n e o u s
m o d e l does not seem to b e
a d e q u a t e for the flow system. T h i s is t r u e , for instance, for screen c a t a lysts a n d for s h a l l o w p a c k e d beds as w e l l as for c e r t a i n (see
fluidized
below).
In Chemical Reaction Engineering Reviews; Hulburt, H.; Advances in Chemistry; American Chemical Society: Washington, DC, 1975.
beds
4.
Physical
wiCKE
Diffusion
Processes
Systems
T h e pseudo-homogeneous m o d e l can be a p p l i e d
i n heterogeneous
diffusion systems i f the units of h e t e r o g e n e i t y — p o r e s ,
holes, p a r t i c l e s ,
e t c . — a r e s m a l l c o m p a r e d w i t h the l e n g t h of the diffusion p a t h s , i.e. t h e extension of the c o n c e n t r a t i o n profiles. T h e most s i m p l e a p p l i c a t i o n of this m o d e l is to m a k e use of a n effective diffusion coefficient:
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D
e f
, =
ψ · An
(la)
w h e r e φ is the p e r m e a b i l i t y of the porous b o d y a n d D coefficient i n the free gas phase.
is t h e d i f f u s i o n
m
I n o r d e r to measure D
accordingly,
ett
steady state d i f f u s i o n m u s t be a p p l i e d ( t h e t e r m p e r m e a b i l i t y i n d i c a t e s steady state c o n d i t i o n s ) , a n d the gas pressure m u s t b e sufficiently h i g h i n o r d e r to justify the neglect of K n u d s e n diffusion.
T h e permeability
c a n t h e n b e u s e d to define a l a b y r i n t h f a c t o r χ (or t o r t u o s i t y factor l / χ ) by: φ =
c · χ w h e r e c is t h e p o r o s i t y . T h e system of transport pores p
p
for p e r m e a t i o n t h r o u g h the porous b o d y is n o r m a l l y the same as t h e macropore
system a n d has a f a i r l y w e l l defined m e a n p o r e r a d i u s , r;
the m i c r o p o r e s are t h e n s i m i l a r to pockets i n t h e w a l l s of the m a c r o p o r e s . I n s u c h a case, the p e r m e a b i l i t y i n t h e K n u d s e n r e g i o n , a c c o r d i n g to ϋ\
=
ΐ ί
φ
κ
(lb)
· D (r) K
w i t h D ( r ) = K n u d s e n diffusion coefficient, c a n b e c o n s i d e r e d e q u a l to K
t h e p e r m e a b i l i t y for b u l k d i f f u s i o n : ψ
κ
«
φ.
I f o n the other h a n d , t h e
m i c r o p o r e s c o n t r i b u t e m a r k e d l y to the d i f f u s i o n resistance for p e r m e a t i o n , ψ =τ^ψ h o l d s , w h i c h means t h a t the p e r m e a b i l i t y i n the t r a n s i t i o n r e g i o n κ
b e c o m e s d e p e n d e n t o n t o t a l gas pressure a n d o n the t y p e of m o l e c u l a r species t h a t is diffusing. F o r n o n s t e a d y state diffusion i n porous systems, the mass
balance
e q u a t i o n is
—
(c c + n ) &
p
=
d i v (xf/D · g r a d c)
(2)
m
w h e r e n is the a m o u n t of gas a d s o r b e d at the p o r e w a l l s p e r u n i t v o l u m e a
of the porous b o d y . P s e u d o - h o m o g e n e o u s
treatment is possible i f n
a
can
b e t a k e n as the v a l u e i n a d s o r p t i o n e q u i l i b r i u m w i t h c. I n the r e g i o n of linear adsorption isotherm and w i t h D
m
i n d e p e n d e n t of
concentration,
E q u a t i o n 2 t h e n simplifies to
IT dt
=
, ^ / τ div grad c = c + dn /dc m
p
D'
e f f
· div grad c
a
In Chemical Reaction Engineering Reviews; Hulburt, H.; Advances in Chemistry; American Chemical Society: Washington, DC, 1975.
(3)
78
CHEMICAL
w i t h a n effective coefficient D '
e f f
REACTION ENGINEERING REVIEWS
for n o n s t e a d y state d i f f u s i o n .
When
a d s o r p t i o n is n e g l i g i b l e
D ' e f f - ^ D m - — C
P
C
=χΑη
P
(4)
r a t h e r t h a n E q u a t i o n l a for steady state c o n d i t i o n s . If, o n the other h a n d , a d s o r p t i o n w i t h c u r v e d isotherms has to be c o n s i d e r e d , t h e mass b a l a n c e i n t h e g e n e r a l f o r m of E q u a t i o n 2 m u s t b e u s e d , a n d the s o l u t i o n of
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diffusion problems becomes very complicated although still w i t h i n the c o n c e p t of
pseudo-homogeneity.
Wakao's paper ( I )
starts w i t h the q u e s t i o n i f the b i n a r y c o u n t e r -
diffusion of inert, n o n a d s o r b a b l e gases t h r o u g h p o r o u s m e d i a r u n s for steady state a n d for n o n s t e a d y state c o n d i t i o n s w i t h effective d i f f u s i o n coefficients t h a t differ o n l y b y the f a c t o r e , i.e. D f = p
to E q u a t i o n 4.
e
e · D' f
f
p
e
£
according
I n fact, the v a l i d i t y of this r e l a t i o n c o u l d b e c o n f i r m e d
b y measurements w i t h N - H 2
2
m i x t u r e s i n p o r o u s r e f r a c t o r y m a t e r i a l as
w e l l as b y c a l c u l a t i o n s of t h e d i f f u s i o n t h r o u g h a t w o - d i m e n s i o n a l n e t w o r k of i n t e r c o n n e c t e d m a c r o - a n d m i c r o p o r e s . T h i s n e t w o r k m o d e l w a s also u s e d b y W a k a o to discuss the b a s i c q u e s t i o n i f t h e effective d i f f u s i o n coefficient, as m e a s u r e d b y m e t h o d s of steady state p e r m e a t i o n t h r o u g h the porous m e d i u m , D
D
e f £
— b i n a r y b u l k d i f f u s i o n or K n u d s e n d i f f u s i o n —
w i l l b e t h e same as the coefficient, D f , that c a n be o b t a i n e d f r o m t h e R
effectiveness
f a c t o r of a
first-order
e
f
r e a c t i o n r u n n i n g at the p o r e w a l l s .
I n fact, t h e c a l c u l a t i o n y i e l d s differences for the same m o l e c u l a r species w i t h and without reaction: D
R
e f f
