Modeling of Gas-Liquid Continuous-Stirred Tank Reactors (CSTRs

6 Modeling of Gas-Liquid Continuous-Stirred Tank Reactors (CSTRs) A. A. SHAIKH and A. VARMA

Downloaded by UNIV OF MINNESOTA on August 10, 2013 | http://pubs.acs.org Publication Date: December 9, 1984 | doi: 10.1021/bk-1984-0237.ch006

Department of Chemical Engineering, University of Notre Dame, Notre Dame, IN 46556

Some specific aspects in the modeling of gas-liquid continuous-stirred tank reactors are considered. The influence of volatility of the liquid reactant on the enhancement of gas absorption is analyzed for irreversible second-order reactions. The impact of liquid evaporation on the behavior of a nonadiabatic gas-liquid CSTR where steady-state multiplicity occurs is also examined. Previous analyses of the effect of chemical reaction on the rate of gas absorption have almost exclusively used film- or penetration-theory models (1-3) which assume the liquid reactant to be non-volatile. In some industrial applications (4), however, the liquid is volatile and evaporation of the liquid may therefore have important implications. In a previous work (5), the film theory was used to analyze special cases of gas absorption with an irreversible secondorder reaction for the case involving a volatile liquid reactant. Specifically, fast and instantaneous reactions were considered. Assessment of the relative importance of liquid reactant volatility from a local (i.e., enhancement) and a global (i.e., reactor behavior) viewpoint, however, necessitates consideration of this problem without limitation on the reaction regime. In the first part of this paper, we therefore generalize the analysis of the above-mentioned reaction system. An approximate reaction factor expression is derived without restriction on the reaction regime, and the accuracy of this factor is tested by comparison with numerical solutions of the film-theory model. The relative importance of volatility of the liquid reactant with respect to the enhancement of gas absorption is analyzed by using this reaction factor. The second part of this paper is devoted to assessing the influence of liquid evaporation on the steady-state behavior of gas-liquid CSTRs. The reaction factor expression developed in the first part is utilized for this purpose. 0097-6156/84/0237-0095$06.00/0 © 1984 American Chemical Society

In Chemical and Catalytic Reactor Modeling; Dudukovi, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

CHEMICAL AND CATALYTIC REACTOR MODELING

96

Enhancement of Gas Absorption For

the i r r e v e r s i b l e second-order r e a c t i o n : A

+

(g+£)

V B

P

W * W

the f i l m - t h e o r y model can be cast i n dimensionless form as f o l lows: 2


d a _ „2 M ab

(1)

2

d b _ ,„2 VM ab dÇ

(2)

2

;

5| = Y'b

ζ = 0:

a = 1

ζ = 1:

| | = [Μ (α'-1) + 3']a

2

and the r e a c t i o n

(3) ;

b = 1

(4)

f a c t o r i s d e f i n e d by:

E* = - ^2. |ς=ο A

ά ζ

(5)

Note that v o l a t i l i t y of the l i q u i d reactant i s accounted f o r by the parameter γ which r e f l e c t s the magnitude of the e q u i l i b r i u m d i s t r i b u t i o n c o e f f i c i e n t r e l a t i v e to that of the l i q u i d - s i d e mass t r a n s f e r c o e f f i c i e n t . 1

Approximate S o l u t i o n . Assuming a l i n e a r c o n c e n t r a t i o n p r o f i l e f o r reactant Β i n the l i q u i d - s i d e f i l m ( 5 ) : b

=

b. + ( l - b . K

an approximate s o l u t i o n =

,

ζε[0,ΐ]

(6)

of the preceding model i s ( 6 ) :

MvbT

(ψ + ψ χ

2

a )

(7)

£

* b

i

a. *

(

(1+γ·) ( E - l )

8

)

±

=

— § [M ( a ' - l ) + 3 ' ] »^b7 + Μ νΈ7 Ψ

(9) 4


Continuous-Stirred

6. SHAIKH AND VARMA where ψ_., j = 1

( I

*1 -

Tank Reactors

97

4, are parameters d e f i n e d below:

T

- l / 3 ^2/3 " l / 3 Π

M:

-

D )

1

1


tes»)"

*4 Γ) υ

=

- l / 3 ^ 2 / 3 ' 4/3

( I

I

2/3

) / D

(10)

= j l ^ _ j i ^ " -1/3 1/3 l / 3 -1/3 T Χ

T

I

1 = 1 (M!) Ξ I y μ i y

I

(Μ') Ξ I

3

(1-b.)

where μ = + 1/3, + 2/3. Note that expression 7 f o r the r e a c t i o n f a c t o r i s i m p l i c i t i n nature. However, f o r r e l a t i v e l y h i g h values of the Hatta number: a^, ψ and -»· 0, w h i l e and -> 1; thus the r e a c t i o n f a c t o r becomes e x p l i c i t i n t h i s l i m i t . 2

Approximate v s . Numerical S o l u t i o n . The accuracy of the a p p r o x i mate r e a c t i o n f a c t o r expression has been t e s t e d over wide ranges of parameter values by comparison with numerical s o l u t i o n s of the f i l m - t h e o r y model. The methods of orthogonal c o l l o c a t i o n and orthogonal c o l l o c a t i o n on f i n i t e elements (7,8) were used to o b t a i n the numerical s o l u t i o n s ( d e t a i l s are g i v e n by Shaikh and Varma ( 6 ) ) . Comparisons i n d i c a t e that d e v i a t i o n s i n the a p p r o x i mate f a c t o r are w i t h i n few percents (< 5%). I t should be men tioned that f o r r e l a t i v e l y h i g h values of Hatta number (Μ >20) , the asymptotic form of Equation 7 was used i n those comparisons. Volatile meter γ reaction when γ'= 1

vs. N o n - V o l a t i l e L i q u i d Case. The e f f e c t of the para on the r e a c t i o n f a c t o r i s shown i n F i g u r e 1, where the f a c t o r of the corresponding n o n - v o l a t i l e case ( i . e . , 0 ) , i s a l s o shown. I t i s c l e a r that the v o l a t i l e nature


98


of the l i q u i d reactant i s d e t r i m e n t a l to the enhancement of gas absorption. Figures 2a and 2b r e v e a l the i n f l u e n c e of the para meter r q on enhancement, as w e l l as the r e l a t i v e e f f e c t of v o l a t i l i t y of the l i q u i d r e a c t a n t . Note that while v o l a t i l i t y can be markedly d e t r i m e n t a l to enhancement, i t s e f f e c t i s not s i g n i f i c a n t i n the slow- and i n s t a n t a n e o u s - r e a c t i o n regimes. The regions of i n f l u e n c e of l i q u i d reactant v o l a t i l i t y a r e shown more c l e a r l y i n F i g u r e 3.


Steady-State

Reactor

Behavior

The occurrence of steady-state m u l t i p l i c i t y i n g a s - l i q u i d CSTRs has been demonstrated i n experimental (9) and t h e o r e t i c a l i n v e s t i g a t i o n s ( c f . , 10). The i r r e v e r s i b l e second-order r e a c t i o n system, i n p a r t i c u l a r , has been t r e a t e d e x t e n s i v e l y i n s e v e r a l t h e o r e t i c a l s t u d i e s (10-15). These s t u d i e s are however based on n e g l e c t i n g energy and m a t e r i a l l o s s e s which r e s u l t from evapora t i o n of the l i q u i d . In t h i s s e c t i o n we develop a model to simulate the behavior of a nonadiabatic g a s - l i q u i d CSTR taking i n t o account v o l a t i l i t y of the l i q u i d r e a c t a n t . The model i s then tested f o r i t s capa b i l i t y t o p r e d i c t some experimental r e s u l t s . Reactor Model. The f o l l o w i n g assumptions are made i n the de velopment of the model: 1. The p h y s i c a l and thermal p r o p e r t i e s o f the gas and l i q u i d , i n t e r f a c i a l area and l i q u i d holdup, p h y s i c a l mass t r a n s f e r c o e f f i c i e n t s , d i f f u s i o n c o e f f i c i e n t s , and volumetric flow r a t e of the l i q u i d are independent o f temperature and con version. 2. The l i q u i d feed i s f r e e of any d i s s o l v e d gaseous r e a c t a n t . The gas feed contains species A and other i n e r t s p e c i e s . 3. The gas and l i q u i d are a t the same temperature i n s i d e the r e a c t o r . The gas e f f l u e n t i s saturated with vapor of spe c i e s B, and passes through an i d e a l condenser. The conden sate i s returned c o n t i n o u s l y to the r e a c t o r . 4. The t o t a l pressure i s independent of p o s i t i o n and tempera ture. 5. Phase e q u i l i b r i a of the gaseous reactant f o l l o w s Henry's law, and that of the l i q u i d reactant follows Raoult's law. 6. The gas-side mass-transfer r e s i s t a n c e and v o l a t i l i t y of the l i q u i d product are both n e g l i g i b l e . 7. The i d e a l gas law and Arrhenius' law hold. The steady-state reactant conversions, x^ and χ , and dimen s i o n l e s s r e a c t o r temperature, Θ, a r e thus given by Isee(16) f o r details): Χ

Α

( 1

Υ

Χ

" Αί Α

}

D

E

" g A

( 1

y

" B

) ( 1

X

' A

)

β

Χ

ρ

Ρ

1 4

9

1

" )"" !

=

0


( 1 1

>

Continuous-Stirred

Tank Reactors


SHAIKH AND VARMA

Figure 2a. E f f e c t of the group (rq) on the r e a c t i o n f a c t o r ; Q!' = 50, p = 0.01, and T' = 1.5 ( 6 ) .




5

10

M

•

100

Figure 2b. E f f e c t of the group (rq) on the r e a c t i o n f a c t o r ; W = 50, β' = 0.01, and Τ = 1.5 ( 6 ) .

Figure 3. E f f e c t of Hatta number and the group (rq) on the r a t i o of r e a c t i o n f a c t o r s f o r v o l a t i l e and n o n - v o l a t i l e l i q u i d (6).


6.

SHAIKH AND

q (x +Va ) f

B

£ f

-

VQx

Q-Wy^ ( 1

y

X

)

' Af A "


=

A

0

(12)

g

[Γ(1+θ)"^[ B

101

Tank Reactors

θ + BDE2 d-y ) d - x )

3

1

+ R(6 -1) - v y r

Continuous-Stirred

VARMA

- pq a f

B

A

+ (R+vy ) B

y^x^

- 6(6-9^ = 0

£ f

(13)

where y_ i s mole f r a c t i o n of the l i q u i d vapor and y=y~

(1-y,,)

Note that now i s given by Equation 7. I t i s a l s o worth men t i o n i n g that Equations 11-13 reduce to the corresponding expres sions presented e a r l i e r (10) when y - « 1, y_ = 0, and θ =1. — Αχ Β r Equations 11 and 13 can be combined to y i e l d a s i n g l e non l i n e a r equation f o r the r e a c t o r temperature that can be solved n u m e r i c a l l y . I t can a l s o be shown (16) that the s t e a d y - s t a t e temperature i s bounded by: A

Βθ

+ Κ(θ -1) - v v , α + 3

1

Β + Ry

Modeling of Gas-Liquid Continuous-Stirred Tank Reactors (CSTRs

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