15 Problems in the Design of the Countercurrent Moving-Bed Catalytic Reactor Downloaded by UNIV OF MICHIGAN ANN ARBOR on November 9, 2014 | http://pubs.acs.org Publication Date: December 9, 1984 | doi: 10.1021/bk-1984-0237.ch015
A Geometric Approach DIMITRY ALTSHULLER
1
School of Chemical Engineering, Purdue University, West Lafayette,IN47907
We consider a reactor with a bed of solid catalyst moving i n the direction opposite to the reacting fluid. The assumptions are that the reaction is irreversible and that adsorption equilibrium is maintained everywhere i n the reactor. It i s shown that discontinuous behavior may occur. The condi tions necessary and sufficient for the development of the internal discontinuities are derived. We also develop a geometric construction useful in c l a s s i f i c a t i o n , analysis and prediction of discon tinuous behavior. This construction is based on the study of the topological structure of the phase plane of the reactor and i t s modification, the input-output space. Consider a p h y s i c a l system shown s c h e m a t i c a l l y i n Figure 1. A f l u i d stream c o n t a i n i n g reactant A i s moving upwards i n plug flow with a constant v e l o c i t y U. The reactant i s adsorbed by a stream of s o l i d c a t a l y t i c p a r t i c l e s f a l l i n g downwards with a constant v e l o c i t y V and occupying the v o i d f r a c t i o n of 1 - ε. On the surface of c a t a l y s t an i r r e v e r s i b l e chemical r e a c t i o n A ·> Β i s o c c u r r i n g and the product Β i s then r a p i d l y desorbed back i n t o the f l u i d phase. Instantaneous adsorption e q u i l i b r i u m f o r the species A i s assumed. The advantage of the moving-bed r e a c t o r over the convention a l fixed-bed r e a c t o r i s the p o s s i b i l i t y o f continuous recharging of f r e s h c a t a l y s t , thus e l i m i n a t i n g the need f o r shutdown to replace i t . I t has a l s o been shown (1, 2) that the residence time required f o r a given conversion i s shorter i n a moving-bed r e a c t o r . This system has received a c o n s i d e r a b l e a t t e n t i o n a t the U n i v e r s i t y of Minnesota. Viswanathan and A r i s (1) studied the behavior of the r e a c t o r assuming that the adsorption isotherm was convex and using the Langmuir equation as an example. The 'Current address: Dept. of Mathematics, University of Missouri, Columbia, MO 65211. 0097-6156/84/0237-0271$09.00/0 © 1984 American Chemical Society In Chemical and Catalytic Reactor Modeling; Dudukovi, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
Downloaded by UNIV OF MICHIGAN ANN ARBOR on November 9, 2014 | http://pubs.acs.org Publication Date: December 9, 1984 | doi: 10.1021/bk-1984-0237.ch015
CHEMICAL AND CATALYTIC REACTOR MODELING
Z= L
Z= 0 solids out Figure 1. Moving bed r e a c t o r schematic. (Reproduced with permission from Ref. 8. Copyright 1983, Chem. Eng. Commun.)
In Chemical and Catalytic Reactor Modeling; Dudukovi, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
Downloaded by UNIV OF MICHIGAN ANN ARBOR on November 9, 2014 | http://pubs.acs.org Publication Date: December 9, 1984 | doi: 10.1021/bk-1984-0237.ch015
15.
ALTSHULLER
Countercurrent Moving-Bed
Catalytic Reactor
273
r e s u l t s f o r continuous steady states f o r a r b i t r a r y form of ad s o r p t i o n e q u i l i b r i u m were derived i n the e a r l i e r paper ( 2 ) . In that paper we have a l s o discussed some features of t r a n s i e n t and discontinuous behavior. This work may be viewed as a c o n t i n u a t i o n of the previous work ( 2 ) . We want to d e s c r i b e the steady-state discontinuous behavior of the r e a c t o r , thus g e n e r a l i z i n g some of the r e s u l t s obtained by Viswanathan and A r i s (J.) . We want to d e r i v e the necessary and s u f f i c i e n t c o n d i t i o n s f o r the development of the i n t e r n a l d i s c o n t i n u i t i e s . Then, we s h a l l see how the geometric methods can be used i n a n a l y s i s and p r e d i c t i o n of the d i s c o n t i n u ous behavior. The presence of the i n t e r n a l d i s c o n t i n u i t i e s makes the use of the standard design equations impossible. Therefore i t i s neces sary to know at what degree of conversion w i l l the d i s c o n t i n u i t y occur. Then the design equation w i l l t e l l us at what point i n the r e a c t o r t h i s happens. A f t e r these p r e l i m i n a r y remarks we now proceed to construct the mathematical model of the system. The Equations of the System We begin our a n a l y s i s by making a mass balance f o r the r e a c t a n t . It leads to the f o l l o w i n g equation f o r the steady s t a t e ~
[ϋεο
Α
- V ( l - ε)η ] Α
+ R(n ) A
= 0
(1)
The concentration of the product Β can be e s t a b l i s h e d , i f d e s i r e d , by o v e r a l l mass balance. The above equation can be made dimen s i o n l e s s by i n t r o d u c i n g the new v a r i a b l e s u = c / c £ , w = n /n°
(2)
x = kz/ευ, X = kL/εϋ
(3)
A
A
A
where k i s the c o e f f i c i e n t which has a dimension of the f i r s t order r a t e constant and r e l a t e s the f u n c t i o n R(n^) to the dimensionless f u n c t i o n F ( v ) : R(n ) A
= kn° F(v)
Therefore we have two
(4) important dimensionless parameters
μ = η χ and,
In Chemical and Catalytic Reactor Modeling; Dudukovi, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
(5)
274
CHEMICAL AND CATALYTIC REACTOR MODELING γ = p ( l - ε)ν/εϋ
(6)
The parameter γ i s a r a t i o of the t r a n s p o r t c a p a c i t i e s of the two phases, while μ i s a r a t i o of the reference c o n c e n t r a t i o n of the reactant i n these phases. The Equations 1 and 2 then become
Downloaded by UNIV OF MICHIGAN ANN ARBOR on November 9, 2014 | http://pubs.acs.org Publication Date: December 9, 1984 | doi: 10.1021/bk-1984-0237.ch015
|^{u The
- γν] + yF(v) = 1
(7)
f o l l o w i n g boundary c o n d i t i o n s can be imposed on u and
ν
u(o_)
= u
b
(8)
v(x )
= v
t
(9)
+
i . e . , we s p e c i f y the c o n d i t i o n s j u s t outside of the r e a c t o r which do not have to c o i n c i d e with those j u s t i n s i d e of the r e a c t o r s i n c e the adsorption e q u i l i b r i u m i s a t t a i n e d instantaneously. Adsorption e q u i l i b r i u m which we have assumed can be described i n two ways: u = f ( v ) and ν = g(u). The advantages and l i m i t a t i o n s of these two expressions were discussed e a r l i e r ( 2 ) . In t h i s work we s h a l l assume that the f u n c t i o n f ( v ) i s monotonie and use p r i m a r i l y the f u n c t i o n ν = g(u). The i n t r o d u c t i o n of the adsorption e q u i l i b r i u m leads to the equation: [1 - Yg'(u)] g
+ yGiu) = 0
where G(u) = F(g(u)) In order to solve the Equation du dx
=
(10)
10, we
rearrange
i t to o b t a i n
yG(u) Yg'(u) - 1
U
S o l u t i o n of Equation 11 seems to be q u i t e easy. t i o n of v a r i a b l e s and i n t e g r a t i o n leads to :
The
i
;
separa
ο The d e s i r e d r e a c t o r length t h e r e f o r e i s μΧ = |z(u _) t
- Z(u )|
(13)
b +
where Z(u) i s an a u x i l i a r y f u n c t i o n defined as an
integral
In Chemical and Catalytic Reactor Modeling; Dudukovi, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
Countercurrent Moving-Bed
15. ALTSHULLER
Z(u) - / lower-limit
v
1
275
Catalytic Reactor
du
(14)
J
The lower l i m i t of i n t e g r a t i o n may be omitted s i n c e the f u n c t i o n Z(u) can appear only i n a d i f f e r e n c e r e l a t i o n s h i p . However, i f we look at Equation 11 more c a r e f u l l y , we n o t i c e that i t s right-hand s i d e becomes discontinuous at the points where yg^(u) = 1 . I f such a p o i n t i s between u ^ and u , continuous s o l u t i o n s (and only f o r them Equation 13 i s v a l i d ) f a i l to e x i s t . Therefore, we need to analyze the d i s continuous behavior of the r e a c t o r . In order to do t h a t , we must f i r s t e s t a b l i s h some general p r o p e r t i e s of the d i s c o n t i n u i t i e s o f t e n r e f e r r e d to as shocks. +
Downloaded by UNIV OF MICHIGAN ANN ARBOR on November 9, 2014 | http://pubs.acs.org Publication Date: December 9, 1984 | doi: 10.1021/bk-1984-0237.ch015
t
The General P r o p e r t i e s of Shocks Let us make a mass balance over the plane of the d i s c o n t i n u i t y as shown i n F i g u r e 2. I n s i s t i n g that there i s no accumulation of matter i n the shock plane, we o b t a i n u
l
-
Ύ ν
u
1
= 2
( 1 5 )
- ^2
Therefore, every d i s c o n t i n u i t y can be represented on a plane ν v s . u by a segment of a s t r a i g h t l i n e having a slope of 1/y o f t e n c a l l e d a shock path. We can d i s t i n g u i s h between two kinds of shocks depending on where they occur: boundary and i n t e r n a l . The former ones are caused by the d i f f e r e n c e between the i n l e t and the e q u i l i b r i u m c o n c e n t r a t i o n s . The nature of the l a t t e r ones i s more s u b t l e . For the boundary d i s c o n t i n u i t i e s , we have the f o l l o w i n g two equations : U
b
~
u _ t
Y V
= U
b
- yv _
b+-
= u
t
t
Y V
( 1 6 )
b+
- yv
(17)
t
From the boundary c o n d i t i o n s , we know u and v . Since e q u i l i b rium i s maintained j u s t i n s i d e the r e a c t o r ^ we have b
v v
b +
t
t
- g(u )
(18)
= g(u _)
(19)
b +
t
Therefore>one end of the path of the boundary d i s c o n t i n u i t y must l i e on the isotherm. The o u t l e t c o n c e n t r a t i o n s u and v^ can be determined by combining Equations 16-19. t
In Chemical and Catalytic Reactor Modeling; Dudukovi, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
Downloaded by UNIV OF MICHIGAN ANN ARBOR on November 9, 2014 | http://pubs.acs.org Publication Date: December 9, 1984 | doi: 10.1021/bk-1984-0237.ch015
CHEMICAL AND CATALYTIC REACTOR MODELING
>. X Figure 2. The shock plane. (Reproduced with permission from Ref. 8. Copyright 1983, Chem. Eng. Commun.)
In Chemical and Catalytic Reactor Modeling; Dudukovi, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
15.
Counter current Moving-Bed
ALTSHULLER
v
b
u
=
[
7
=
u
t
t
-
u
b -
u
+
b
Y
g
(
u
+
) b
" ^g(u J t
Catalytic Reactor
]
l / g ' ( 0 ) ; c, l/g'(u) < γ < U*/g(u*); and d, u*/g( < γ < 1/g'(0).
In Chemical and Catalytic Reactor Modeling; Dudukovi, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
CHEMICAL AND CATALYTIC REACTOR MODELING
Downloaded by UNIV OF MICHIGAN ANN ARBOR on November 9, 2014 | http://pubs.acs.org Publication Date: December 9, 1984 | doi: 10.1021/bk-1984-0237.ch015
280
d i r e c t e d upwards above the isotherm and downwards below i t . Therefore, we can d i s t i n g u i s h between s t a b l e and unstable parts of the isotherm depending upon whether the shock paths point towards o r away from i t . In Figures 4a - 4c, the s t a b l e parts of the isotherm are shown with the s o l i d l i n e and the unstable ones with the broken l i n e . I f there are no tangency p o i n t s , the e n t i r e isotherm i s unstable (Figure 4a). I f there i s only one tangency p o i n t , the part below i t i s s t a b l e (Figure 4b). I f there are two tangency p o i n t s , the s t a b l e part l i e s between them (Figure 4 c ) . From Equation 11, we know that du/dx < 0 i f g^(u) < l/γ and du/dx > 0 otherwise. Therefore, the t r a j e c t o r i e s are d i r e c t e d towards the upper tangency p o i n t and away from the lower tangency point (Figures 4b and 4 c ) . A s t a b l e part of the isotherm has an a t t r a c t i o n area a s s o c i ated with i t . By a t t r a c t i o n area we mean the region i n which a l l the d i s c o n t i n u i t y paths and on the isotherm. The a t t r a c t i o n areas are bounded by the tangents as w e l l as some p o r t i o n s of the unstable p a r t s . I f a tangent i n t e r s e c t s the isotherm at some p o i n t other than the tangency p o i n t , such an i n t e r s e c t i o n w i l l be c a l l e d a l i m i t a t i o n point. I f t h i s does not happen, the l i m i t a t i o n p o i n t w i l l be defined as an end o f the chord beginning a t the o r i g i n . The former s i t u a t i o n occurs i f u*/g(u*) < γ < l/g^(0) and the l a t t e r one i f l/g^(u) < γ < u*/g(u*). In any case, we have two l i m i t a t i o n points u = u ^ and u = U£. Figures 5a and 5b show the complete t o p o l o g i c a l s t r u c t u r e of the phase plane when i/g'(S) < γ < i/g'(o). Every chord drawn t o the isotherm a s s o c i a t e s a p o i n t on the unstable part with a point on the s t a b l e p a r t . Mathematically, t h i s a s s o c i a t i o n can be expressed by a f u n c t i o n J(u) which we s h a l l c a l l a jump f u n c t i o n . T h i s f u n c t i o n can be defined i m p l i c i t l y using Equation 22, but then we would have two values U£ corresponding to some values of u]_. This d i f f i c u l t y i s eliminated i f an a d d i t i o n a l c o n d i t i o n that u* < J(u) < u* f o r every u i s imposed. The same d i f f i c u l t y a r i s e s i f we attempt to d e f i n e an i n verse of the f u n c t i o n J(u) a l s o needed i n our a n a l y s i s . Two inverses must be defined which we s h a l l c a l l upper and lower inverses denoted by. J ( u ) and J ^ ( u ) , r e s p e c t i v e l y . The d i s t i n c t i o n i s that J-*-(u) > u and J-^(u) < u* f o r every u . The f u n c t i o n JMu) always maps the i n t e r v a l ( u ^ u*) onto the i n t e r v a l (u*, u^) and J ^ u * ) = u^. For the f u n c t i o n Jj_(u), the domain and the range depend on whether γ i s greater or l e s s than u*/g(u*). In the former case, t h i s f u n c t i o n maps the i n t e r v a l (u*, u*) onto the i n t e r v a l u^, u*) and J^(u*) = U£. In the l a t t e r case, the domain i s (u*, u^) and the range i s (0, u*) with u ^ = J ( 0 ) . Having e s t a b l i s h e d the geometric p r o p e r t i e s of shocks and the s t r u c t u r e of the phase plane, i n c l u d i n g appropriate f u n c t i o n s needed i n the a n a l y s i s , we may look at the discontinuous s o l u t i o n s as a whole. x
5
In Chemical and Catalytic Reactor Modeling; Dudukovi, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
Countercurrenî
Moving-Bed
Catalytic
Reactor
Downloaded by UNIV OF MICHIGAN ANN ARBOR on November 9, 2014 | http://pubs.acs.org Publication Date: December 9, 1984 | doi: 10.1021/bk-1984-0237.ch015
AITSHULLER
In Chemical and Catalytic Reactor Modeling; Dudukovi, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
CHEMICAL AND CATALYTIC REACTOR MODELING
Downloaded by UNIV OF MICHIGAN ANN ARBOR on November 9, 2014 | http://pubs.acs.org Publication Date: December 9, 1984 | doi: 10.1021/bk-1984-0237.ch015
282
In Chemical and Catalytic Reactor Modeling; Dudukovi, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
Downloaded by UNIV OF MICHIGAN ANN ARBOR on November 9, 2014 | http://pubs.acs.org Publication Date: December 9, 1984 | doi: 10.1021/bk-1984-0237.ch015
Counter current Moving-Bed
15.
ALTSHULLER
The
Structure of the Discontinuous
Catalytic
Reactor
283
Solutions
From the previous two s e c t i o n s , i t i s c l e a r that every phase t r a j e c t o r y of the moving-bed r e a c t o r c o n s i s t s of the segments of the isotherm and the segments of the s t r a i g h t l i n e having a slope of l/γ. Let us c l a s s i f y the s o l u t i o n s which may be encountered. F i r s t , we have two t r i v i a l types: the t o t a l l y discontinuous s o l u t i o n denoted D and a s o l u t i o n continuous everywhere, denoted C. These two s o l u t i o n s are shown on the Figures 6a and 6b, r e s p e c t i v e l y . The t r a j e c t o r y f o r the s o l u t i o n D i s a s t r a i g h t segment which e i t h e r s t a r t s or ends on the isotherm. The length of the r e a c t o r i s i n t h i s case zero. The t r a j e c t o r y of the s o l u t i o n C l i e s e n t i r e l y on the isotherm and may c o n t a i n a tangency p o i n t only at the end. The r e a c t o r length corresponding to t h i s s o l u t i o n can be c a l c u l a t e d i n a s t r a i g h t f o r w a r d way from Equation .13 % UX
= |Z(f(v )) - Z(u )|
C
t
b
(25)
Among the s o l u t i o n s with one d i s c o n t i n u i t y , we may d i s t i n guish between three types corresponding to t h e i r l o c a t i o n which may be bottom, top or middle. We s h a l l denote them Β, Τ and M r e s p e c t i v e l y . Figures 7a-c show some examples of these s o l u t i o n s . The s o l u t i o n of the type Β s t a r t s i n the a t t r a c t i o n area of the s t a b l e part of the isotherm. I t c o n s i s t s of a s t r a i g h t seg ment followed by segment of the isotherm l y i n g on i t s s t a b l e p a r t . The end p o i n t of the shock path w i l l be c a l l e d a drop p o i n t and denoted (u^, g ( u ) ) . The length of the r e a c t o r i s c a l c u l a t e d by applying the design equation to the continuous part: d
μΧυ-
|z(f(v )) - Z(u )| t
d
(26)
w i t h u^ = u ^ i n t h i s case. The s o l u t i o n of type Τ (Figure 7b) i s to some extent a r e verse of the s o l u t i o n of type B. I t c o n s i s t s of an arc of the unstable part of the isotherm followed by a shock path. The s t a r t i n g point of the shock path w i l l be c a l l e d a j u n c t i o n p o i n t , (uj,g(u..)), the equation f o r the r e a c t o r l e n g t h i s μΧ = τ
|Z(uj) - Z(u )| b
(27)
where Uj = u t - . Type M (Figure 7c) i s i n some sense a combination of the types Β and T. I t s s t r u c t u r e contains an unstable p a r t , j u n c t i o n p o i n t , drop p o i n t , and s t a b l e p a r t . Of course, the j u n c t i o n p o i n t must belong to the a t t r a c t i o n area. We may compute the r e a c t o r l e n g t h as f o l l o w s :
In Chemical and Catalytic Reactor Modeling; Dudukovi, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
Downloaded by UNIV OF MICHIGAN ANN ARBOR on November 9, 2014 | http://pubs.acs.org Publication Date: December 9, 1984 | doi: 10.1021/bk-1984-0237.ch015
CHEMICAL AND CATALYTIC REACTOR MODELING
Figure 6. S o l u t i o n types. b, s o l u t i o n of type C.
Key:
a, s o l u t i o n o f type D;
In Chemical and Catalytic Reactor Modeling; Dudukovi, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
Catalytic
Reactor
Figure 7. S o l u t i o n s with one d i s c o n t i n u i t y . B; b, type Τ ; and d, type M.
Key:
Countercurrent
Moving-Bed
285
Downloaded by UNIV OF MICHIGAN ANN ARBOR on November 9, 2014 | http://pubs.acs.org Publication Date: December 9, 1984 | doi: 10.1021/bk-1984-0237.ch015
15. ALTSHULLER
C. a, type
In Chemical and Catalytic Reactor Modeling; Dudukovi, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
286
CHEMICAL AND CATALYTIC REACTOR MODELING
h
*M "
+
X
(
T
8
)
but the e q u a l i t i e s u^ = u, and u^ = u _ are no longer v a l i d . Let us now look at the s o l u t i o n s w i t h the two d i s c o n t i n u i t i e s . Here we may d i s t i n g u i s h four types: BT, MT, BM and MM. Some examples are shown on the Figures 8a-d. A l l of these t r a j e c t o r i e s i n v o l v e the t r a n s i t i o n from s t a b l e to the unstable part of the isotherm. The BT type has the f o l l o w i n g components. Shock path - drop point - s t a b l e part - unstable part - j u n c t i o n point - shook path. Therefore, the only point which can be a j u n c t i o n point i s the upper tangency, i . e . u _ = u* (Figure 8a). The length of the reactor i s +
Downloaded by UNIV OF MICHIGAN ANN ARBOR on November 9, 2014 | http://pubs.acs.org Publication Date: December 9, 1984 | doi: 10.1021/bk-1984-0237.ch015
2
t
t
μΧ
= |z(u*) - Z ( u ) |
β Τ
(29)
b +
The s t r u c t u r e of the MT s o l u t i o n combines the s t r u c t u r e s of the types M and T. The sequence of the components i s : unstable part - j u n c t i o n point - drop point - s t a b l e part - unstable part j u n c t i o n point - shock path. This s t r u c t u r e contains two j u n c t i o n points U j i and u _ . Some r e s t r i c t i o n s apply to the l o c a t i o n s of these p o i n t s . The f i r s t one must l i e i n the a t t r a c t i o n area of the s t a b l e part and the second must be the upper tangency point so that u _ = u*. The equation f o r the r e a c t o r length i s t
t
μΧ^
= |Z(
U j l
) - Z ( u ) | + | Z ( u ) - Z(u*)| b
(30)
dl
The s o l u t i o n s of the types BM (Figure 8c) and MM (Figure 8d) do not occur i f the isotherm i s convex. The reason f o r that i s that convex isotherms do not have lower tangency p o i n t which serves as a drop point f o r both of these types. The r e a c t o r length may be c a l c u l a t e d f o r BM type from the f o l l o w i n g equation μ Χ
ΒΜ
=
| Z ( U
J1
}
"
Z ( u
)
b+ l
+
Z
f
v
l < ( t>
)
-
Z
(
u
dl
)
|
(
3
1
)
The s o l u t i o n of the type MM includes three continuous p a r t s . The r e a c t o r length i s computed by applying the design equation to each of them and adding the r e s u l t s : μ Χ
ΜΜ
=
l ^ j P - ^ V '
+
I ("j2 z
)
Z ( u
dl)l
+
|Z(f(v )-Z(u )| t
d 2
(32)
We must a l s o note that U j ^ = u ^ f o r MM type of s o l u t i o n s . L i k e BM and MM types,the s o l u t i o n s , with three d i s c o n t i n u i t i e s never occur i f the isotherm i s convex. Only two kinds belonging to t h i s group are p o s s i b l e . They are BMT (Figure 9a) and MMT (Figure 9b). They have very r e s t r i c t i v e conditions f o r t h e i r j u n c t i o n and drop p o i n t s . For BMT s o l u t i o n we must have = u*
In Chemical and Catalytic Reactor Modeling; Dudukovi, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
ALTSHULLER
Countercurrent
Moving-Bed
Catalytic
Reactor
Downloaded by UNIV OF MICHIGAN ANN ARBOR on November 9, 2014 | http://pubs.acs.org Publication Date: December 9, 1984 | doi: 10.1021/bk-1984-0237.ch015
15.
In Chemical and Catalytic Reactor Modeling; Dudukovi, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
287
Downloaded by UNIV OF MICHIGAN ANN ARBOR on November 9, 2014 | http://pubs.acs.org Publication Date: December 9, 1984 | doi: 10.1021/bk-1984-0237.ch015
288
CHEMICAL AND CATALYTIC REACTOR MODELING
V
and b, type
MMT.
In Chemical and Catalytic Reactor Modeling; Dudukovi, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
15.
Countercurrent Moving-Bed
ALTSHULLER
289
Catalytic Reactor
and u _ = u*. Likewise, f o r MMT s o l u t i o n both tangency p o i n t s are involved as drop (lower) and j u n c t i o n (upper) p o i n t s . The equations f o r the length of the reactor are analogous to the previous case. They are: t
μ χ
Downloaded by UNIV OF MICHIGAN ANN ARBOR on November 9, 2014 | http://pubs.acs.org Publication Date: December 9, 1984 | doi: 10.1021/bk-1984-0237.ch015
μ Χ
ΜΜΤ
=
ΒΜτ z
=
Ι
ζ
(
ι
ν
-
z ( u
*>l
+
lz(u*> "
Ζ (
^1}1
( 3 3 )
u
l ( ) - Z ( u ) | + |Z3 3 3 3 3 3
•i->3 3
3,ο^
3 °^
-K
3
-K *
3
3
*
3
CO β
Ο •Η +J
Ο
CO
3"^
r-H •I)
ι—I
-
3
1^-
3
3
3
3
eu
^ 2 2 1 ι1
I
2
+J
ce 4J CO
1 T3 cd
ω
+
4-J
ί
^
ί
ί
^
^3 3
CO
3
3
3
3
3
3
3
3
Xi Xl Xi Xl Xi Xi XiXi 2 2 2 2 2 2 2 2
3
Η
CU iH ,Q cd Η
QJ
Η
Q H H H
Q H U H H H
Q
H H
S
S
es Ο
•Η W)
eu
In Chemical and Catalytic Reactor Modeling; Dudukovi, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
15.
291
Counter current Moving-Bed Catalytic Reactor
ALTSHULLER
eu 00 ce
— 1
/«-\
^ X 1
2
3 θ * * 3
Downloaded by UNIV OF MICHIGAN ANN ARBOR on November 9, 2014 | http://pubs.acs.org Publication Date: December 9, 1984 | doi: 10.1021/bk-1984-0237.ch015
ci
^ *
^-N
>
> w
3
rû 3
3
«
o
1
*
* * * * * * 3 3 3 3 3
ci 3
Ci 3 Λ
3
*
Xi
* > ^
O
1
O
^
3
4 J m
>
M-l
Ο
/-s ci
* * * * * * *
c i 3
2 2 3 2 3 2 2 ~ 1
I
1
I
ci ci ci 3 3 3
1
1 ^
1
1
1 ^
rû
ci 3
3
I I I
*
/"-\
rH r H
«m 1
1
1
•r->
1
1
1
1
1
1
1
1
1
1
*
* ι
ι
1
3
3 *
3
τ-)
1
1
1
•Κ
3
3
3
3
1
3
3
3
3
3
- I I I
3
/—s /-Ν ^
Xi Xi Xi
3
3
-K
- I I I
3
3
3 3
1
rû 3 1
/—s
1
1
^
-K * 3 3
* 3
* 3
* 3
rÛ , Ο
ί Ν—• ""3
3
r.
*
*
*
*
*
3
Q
1
H
H
H
S
|f
H
Q
Q
H
H
Ε-· 8 H H H g H H H H SpQPQPQPQpqpQPQ
W
Q
B
U
H
H
H H H H § H H H SpqpqpQFppqpqpq
Pu
In Chemical and Catalytic Reactor Modeling; Dudukovi, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
*
In Chemical and Catalytic Reactor Modeling; Dudukovi, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
D ΒΤ Τ ΒΤ ΒΤ ΒΜΤ ΒΤ ΒΤ ΒΤορ
I
J
BTQO
D Β Β ΒΜ ΒΤ ΒΤ
D Τ Τ M Β Β Β ΒΜ ΒΤ ΒΤ ΒΤοο
Type
D ΒΤ Β ΒΤ ΒΤ
H
G
Region
u
*
b
t
-
*
*
u* u*
u
b
b (u* , u )
u
(u ,u*)
b
u
-
(u*,f(v )) u* u* u* u* u*
-
u*
u
u
u
u
u
b +
b b b J( b> (u*,J(u )) u* u*
Table I. Continued
b
t
t
t
(
u
j l
_
_
J
-
-
)
(u*,f(v ))
_
A
(u ,f(v ))
J(UJI)
-
-
(u*(f ( v ) , U * )
-
(jiCfCVt)),!!*)
t
(Ji(f(v ),u )
ujχ
-
-
_
-
_
-
1
-
_
-
_
-
_
:
Downloaded by UNIV OF MICHIGAN ANN ARBOR on November 9, 2014 | http://pubs.acs.org Publication Date: December 9, 1984 | doi: 10.1021/bk-1984-0237.ch015
t
t
t
t
t
t
t
t
t
) ) ) )) ))
* *
0
u
z
u u* u* u
u
_
(f(v ),u*) f(v ) (0,u*) 0
_
f(v f(v f(v Ji(f(v 0,Ji(f(v 0
_
t
t
t
t
t
t
(0,Ji(f(v ))) 0
l
(u*,ji(f(v ))) J (f(v )) f(v ) f(v ) f(v ) f(v ) . f(v )
15.
4-» 43 +J +J - U > > > > > > >-/
I
Counter current Moving-Bed
ALTSHULLER
M_|
m m
> W
M-l tW '
3
3
3
o*
2
I
Catalytic
4 J 4-1 4-» 4-> 4-> > > > > > \»«/ v^x i «
X
^ ^ 2 _
O
* O
4-1 M-l
o
>
cT
o
Downloaded by UNIV OF MICHIGAN ANN ARBOR on November 9, 2014 | http://pubs.acs.org Publication Date: December 9, 1984 | doi: 10.1021/bk-1984-0237.ch015
CD
ι
ι
ι ι
ι ι
I
I
I
I
I
I
M-l
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
ι ι ι
3
3
X
2
3 4-1
X
3
I
I
>
I
I
I
M-l •H •-3
w M-4
rQ 3
rû 3 * 3 " 3
* 3
* 3
* 3
* 3
3
,ΰ X X
3
^
3
3
3
3 ^
pq pq pq pq
Q
HH H PQ PQ
pq pq S
H
3
χ
X Xi
Xi Xi Xi Xi Xi
Xi Xi
•K
χ χ
X
2 2 22
2 2
3
3
H g
O
H
ef
Ρ
H
H
Q
H
U
H
In Chemical and Catalytic Reactor Modeling; Dudukovi, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
H
294
CHEMICAL AND CATALYTIC REACTOR MODELING
Let us examine the r e g i o n G with some care. I t i s s u f f i c i e n t l y complex to c l a r i f y the method being used. F i g u r e 11 shows the types of the phase t r a j e c t o r i e s which may occur. First, we have a t r i v i a l discontinuous s o l u t i o n l a b e l l e d 1^. Then we may have a continuous path followed by a jump (Type T, t r a j e c t o r y à ) . The t r a j e c t o r y 3 i s a l i m i t i n g case of _2 corresponding to the longest r e a c t o r f o r which we may have a type Τ behavior. We must have u _ = (f(v )). Longer r e a c t o r s w i l l have a d i s c o n t i n u i t y i n the middle ( t r a j e c t o r y 4_) or i n the bottom ( t r a j e c t o r i e s J3, 7 ) . T r a j e c t o r y _5 represents the t r a n s i t i o n between the type M and type Β while 7 i s a l i m i t i n g case of 6^ with u = u*. The l a t t e r must hold f o r a l l the remaining t r a j e c t o r i e s (7_ - 10). The t r a j e c t o r y 1_ i s a BM type and 8^ i s i t s l i m i t i n g case with a drop point at u = J i ( f ( v ) ) belonging to the type BT. The t r a j e c t o r i e s 9^ and 10 are a l s o of type BT with JLO being the l i m i t i n g case corresponding to the i n f i n i t e l y long r e a c t o r with u^_ = 0. The input - output space when u*/g(u*) /-N r-s 1 >
3
M-l
4-1 4-J 4-1 4 J 4-1 4-1 4-1 + J > > . > > > > > v-^v-^ w w
4_t 4-1 4-1 4-» 4-1 4-1 4-J > 4 _ ! > > > > > > > w O J> V s^ (4_| w i W l W M - J t w M - I M - I M - l w M-l w w -H
Downloaded by UNIV OF MICHIGAN ANN ARBOR on November 9, 2014 | http://pubs.acs.org Publication Date: December 9, 1984 | doi: 10.1021/bk-1984-0237.ch015
•Η *~) ^ ν—'
CN
1 > > 1> >
-^-s
1
ι
I
4-1 4-1 4-1 4->
s-^ IW M-l t w M-l
*~3 « Ο ν - '
11 1 1 1 1 1 1 1 ι
/-"s 4->4-l >
I
I
I
I
I
O
O
I
I
I
1 1 1 11 11 1 1 1 1
I
I
I
1 1 1 11 11 1 1 1 1
3
3
1 1 1 ! 1 1 1 1 1 I
4-»
> 3 ,
ι
^
^ M-1
ι ι *ι ι ι /-Ν
I
I
I
I
I
I
/»-S 4-»
^-N 4-1
M-l
M-l
> > > 0 1 1 1 * 1 1 ** 1 1 1* ·->
*
3 3 W
4-»
•—Ν rH
I
rH ·Γ->
I
I
I
-
3
I
1
I
I
M-l
-
I
I
I
=5° /—s
/—s
3
4J
* ^* 1 1 1 1 1 1 1"Ο 1 >1 1 1 1 ο > > 3
3
1 1^> 1 * ο
/-s
ι ι ι I I». Iο
/—s
*
jû
s-\
3
4->
4J
I
I
'JJ
I
M-l
M-l
V»/ M-l v—' •H
•H
•H
V—' M-l v_x ·Η
/"-\
Χ X xQwb * * *
rQ 1
*
3 3
XlQa 3
3
* * * * * 3
3
0
3
3
3
I
•-ο ^
*
/—^ /~\ 4-» ρ> * w 3 M-l
/—s /-~\ 4-1
*
> *** ** *
3
3
* 3
* 3
M-l
3
PQ PQ
Q H H S P Q P Q P Q S H H H pq pq pq PQ
Ρ
PQ pq £ JE? pq PQ
P
pq pq
ω r-l
Cd H
C Ο
-H «)
PU
H
In Chemical and Catalytic Reactor Modeling; Dudukovi, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
X
pq
8 H H H pq pq pq
15.
Countercurrent
ALTSHULLER
Moving-Bed
Catalytic Reactor
299
/-s •K 3 1
4->
4-1
* 3
~ £ o ° 4-1 «4-1
1
3
3
3
3
3
3
4-1
4-J
4-1
4J
1
4-1
>,
1
«4-1
> > > > > ^ O w w w w w «4_( «4_l «4-4 «4-4 «4-4 «4-1 w ^ ·Η
w
4C * * * 1 3 3 3 3
Ο
Downloaded by UNIV OF MICHIGAN ANN ARBOR on November 9, 2014 | http://pubs.acs.org Publication Date: December 9, 1984 | doi: 10.1021/bk-1984-0237.ch015
«4-4
1
1
1
1
1
1
I
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
'
1
1
1
1
1
ι
ι ι ι ι ι ι ι
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
4-» rH
/—s rH
1
1
1
1
1
I
1
I
1
I
3
3
«4-4
1
I I
1
1
3
3
3
1
1
1
1
"
I
I
1
*
I
^3
•-3
3 XI 3
•-3 v—'
* 3^ 3
3 1
1
1
1
1
1
1
1
1
1
-
ο
ο
1
1
1
1
1
ο
* ο
1
1
1
1
ν—/
>
/~\ Xl 3 1
1
1
ι
ι
- ο
1
ο
«4-1
•H
/-S
* * 1 3
* # 3 3
* 3
3 1
Xi 3 - 3
8 H PQ H H PQ PQ PQ
I
W
£ '
ί
Q
* * * - 3 3 3
/—\
/"Ν
Xl Χ 3 * * * 3 * 3 3 Xi -Κ 3 3
3
* * 3
Ο
pq ç j pq pq
pQ pq pQ PQ
s
•K * 3
* 3
!3
S pq PQ
8 Q p Q O p q S H H H pq pq pq pq
^-s
? Ρ Ρ s° I
V—'
8 pq
/-s
/—s
Xi X 3 4C * - 3 * 3 3 Xi * 3 3 3
1
°
3 Xi»-) 3
3
3
s
ë
ë
«
Ο
In Chemical and Catalytic Reactor Modeling; Dudukovi, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
«
In Chemical and Catalytic Reactor Modeling; Dudukovi, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
D Τ C Τ Too
S
Τ
D Β Β Μ Τ Τ Τ
D Β Β M Moo
Type
D Τ Too
R
Q
gion
b
u
b+
b
t
b
u
u
u
u
u
u
u
u
u
u
b b b b
b b
b b b b
b
b b
t
(J(u ),f(v )) J(u )
u
u
(J(u ),f(v )) J(u )
Table I I . Continued
b
-
-
t
b
(J.(f(v )),u )
(0,u ) 0
ujl
u
b
(j(u),U£) b
b
dl
(j(u),U£>
U
-
J2
U
-
_
_
-
d2
U
Downloaded by UNIV OF MICHIGAN ANN ARBOR on November 9, 2014 | http://pubs.acs.org Publication Date: December 9, 1984 | doi: 10.1021/bk-1984-0237.ch015
±
f(v f(v f(v f(v
t-
t
t
b
b
(f(v ),u ) f(v ) (0,u ) 0
b
t
) ) ) )
(0,u ) 0
t
t
t
t
t
t
t
f(v ) f(v ) f(v ) J (f(vp (0,Ji(f(v ))) 0
u
m r
α
Ο
m
70
Η η
5
η
σ
>
> r
η
m
η χ
Countercurrent
ALTSHULLER
Moving-Bed
Catalytic
Reactor
V
/
/ Ν
Downloaded by UNIV OF MICHIGAN ANN ARBOR on November 9, 2014 | http://pubs.acs.org Publication Date: December 9, 1984 | doi: 10.1021/bk-1984-0237.ch015
Q
/
"
θ
R
s \
/
/
υ*
/
/ /
\
/ J1
\
1 1
j
H
C
\
\ ! Κ > l
!
Β
G
/ \
A
F
M
Ρ
\
1
j
L
υ,
υ*
l \ \ \ \
D
Ε
υ
υ
1
1
Figure 12. Input-output space f o r u*/g(ii*) < γ < 1/g (0)
2,6
u
*
u
i
u
*
u
b
U
Figure 13. T r a j e c t o r i e s f o r r e g i o n G (see Figure 12)
In Chemical and Catalytic Reactor Modeling; Dudukovi, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
302
C H E M I C A L A N D CATALYTIC REACTOR
MODELING
Downloaded by UNIV OF MICHIGAN ANN ARBOR on November 9, 2014 | http://pubs.acs.org Publication Date: December 9, 1984 | doi: 10.1021/bk-1984-0237.ch015
I f the r e a c t o r operates c l o s e to,but not q u i t e at adsorption e q u i l i b r i u m , or i f small d i f f u s i o n e f f e c t s are present, then i n stead of the d i s c o n t i n u i t i e s we may expect very r a p i d changes of c o n c e n t r a t i o n . This can s t i l l be a problem f o r numerical c a l c u l a t i o n s and the study of the d i s c o n t i n u i t i e s i n the i d e a l model can s t i l l be very u s e f u l . Moving-bed r e a c t o r s are probably not the only systems i n which the discontinuous behavior can occur. The techniques developed i n t h i s paper can a l s o be a p p l i e d to other such systems, AcknowTe dgment T h i s research was supported i n part by the N a t i o n a l Science Foundation i n the form of a grant CPE 7918902-02 to whom I am very g r a t e f u l . Thanks are a l s o due to P r o f e s s o r Henry C. Lim f o r his help and encouragement. Legend of Symbols c F(v) f(v) G(u) g(u) J(u) J (u) Ji(u) k L η R(n ) r ( u , v) U u u x
a
V ν X χ Z(u) ζ
Concentration i n the f l u i d phase Dimensionless r e a c t i o n r a t e expression Dimensionless expression f o r adsorption isotherm Dimensionless r e a c t i o n r a t e expression Dimensionless expression f o r a d s o r p t i o n isotherm Jump f u n c t i o n d e f i n e d i n the t e x t Upper i n v e r s e of the jump f u n c t i o n Lower i n v e r s e of the jump f u n c t i o n C o e f f i c i e n t i n Equation 5 Length of the r e a c t o r Concentration i n the s o l i d phase Reaction r a t e expression Dimensionless adsorption r a t e expression Interstitial fluid velocity Dimensionless c o n c e n t r a t i o n of A i n the f l u i d phase Dimensionless c o n c e n t r a t i o n of A a t the isotherm i n f l e c t i o n point S o l i d phase v e l o c i t y Dimensionless c o n c e n t r a t i o n of A i n the s o l i d phase Dimensionless r e a c t o r l e n g t h Dimensionless d i s t a n c e Function defined by the Equation 21 Distance
Greek L e t t e r s γ ε γ μ
μ(1 - ε)ν/ευ Void f r a c t i o n of the f l u i d Parameter i n Equation 7 n°/c°
In Chemical and Catalytic Reactor Modeling; Dudukovi, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
15.
ALTSHULLER
Countercurrent Moving-Bed Catalytic Reactor
303
Subscripts A, Β b
Downloaded by UNIV OF MICHIGAN ANN ARBOR on November 9, 2014 | http://pubs.acs.org Publication Date: December 9, 1984 | doi: 10.1021/bk-1984-0237.ch015
t 1, 2 ο
+» -
Chemical species F l u i d phase entrance Lower l i m i t a t i o n p o i n t Entrance of the s o l i d phase L e f t - and right-hand s i d e s of the shock I n i t i a l value or reference p o i n t Lower tangency p o i n t Sides
Superscripts
ο
Upper l i m i t a t i o n p o i n t s Reference c o n c e n t r a t i o n Derivative Upper tangency p o i n t
Literature Cited 1.
2.
3. 4.
5. 6. 7.
Viswanathan, S. and R. A r i s , "An Analysis of the Counter current Moving Bed Reactor," SIAM-AMS Proceedings, v. 8, pp. 99-124 (1974). Altshuller, D . , "Design Equations and Transient Behavior of the Countercurrent Moving-Bed Chromatographic Reactor," Chem. Eng. Commun., v. 19, pp. 3630375 (183). Smirnov, V . , A Course i n Higher Mathematics, v. 1, Pergamon Press, 1964. Cronin, J., Differential Equations. Introduction and Qualitative Theory, Marcel Dekker, Inc., New York, Ν. Y . , 1980. Sircar, S.; R. Gupta AIChE J, 27, No. 4, pp. 806-812 (1981). Brunauer, S. Adsorption of Gases and Vapors, v. 1, Princeton University Press, Princeton, N. J., 1945. Adamson, A. W., Physical Chemistry of Surfaces, 3rd ed., Wiley, 1976.
R E C E I V E D September 30, 1983
In Chemical and Catalytic Reactor Modeling; Dudukovi, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.