Chemical and Catalytic Reactor Modeling - American Chemical Society

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.


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,


(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


|^{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



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. +


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




>. X Figure 2. The shock plane. (Reproduced with permission from Ref. 8. Copyright 1983, Chem. Eng. Commun.)


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).




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


Countercurrenî

Moving-Bed

Catalytic

Reactor


AITSHULLER




282




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 :




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;


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


15. ALTSHULLER

C. a, type


286


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 +


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*


ALTSHULLER

Countercurrent

Moving-Bed

Catalytic

Reactor


15.


287


288


V

and b, type

MMT.


15.


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


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


15.

291

Counter current Moving-Bed Catalytic Reactor

ALTSHULLER

eu 00 ce

— 1

/«-\

^ X 1

2

3 θ * * 3


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


*


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

-

_

-

_

-

_

:


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


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


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


H

294


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


•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


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


«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


«


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


±

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


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)


302

C H E M I C A L A N D CATALYTIC REACTOR

MODELING


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°


15.

ALTSHULLER

Countercurrent Moving-Bed Catalytic Reactor

303

Subscripts A, Β b


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


Chemical and Catalytic Reactor Modeling - American Chemical Society

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