Stability of Adiabatic Reactors - American Chemical Society

18 Stability of Adiabatic Reactors C.H.BARKELEW

Downloaded by UNIV LAVAL on October 23, 2015 | http://pubs.acs.org Publication Date: December 9, 1984 | doi: 10.1021/bk-1984-0237.ch018

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An adiabatic exothermic reaction i n a continuous flow system can develop uncontrollable "runaways" or "hot-spots" i f the extent of reaction is allowed to increase beyond a c r i t i c a l l i m i t . This limit depends strongly on the thermal parameters of the reaction and on mixing of the reacting f l u i d in the direction of i t s flow. This paper provides broadly applicable rules for designing such reactors and operating them safely.

A runaway exotherm i n a flow reactor can occur l o c a l l y i f there are l a t e r a l i r r e g u l a r i t i e s and i n s u f f i c i e n t mixing of adjacent portions of the reactants. I t can also occur g l o b a l l y , i n which s i t u a t i o n a l l of the f l u i d passes through the runaway zone before i t leaves the r e a c t o r . This l a t t e r type of runaway r e a c t i o n i s the subject of t h i s paper. I f the k i n e t i c s and thermal parameters of a r e a c t i o n are known, and i f the c h a r a c t e r i s t i c s of the f l u i d flow through a reactor can be described, a set of d i f f e r e n t i a l equations can be formulated, whose s o l u t i o n w i l l give an estimate of the temperature p r o f i l e i n that r e a c t o r , f o r a v a r i e t y of operating c o n d i t i o n s . A c o l l e c t i o n of such temperature p r o f i l e s could i n p r i n c i p l e be used to design a safe reactor or to provide operating g u i d e l i n e s f o r an e x i s t i n g r e a c t o r . I t i s not the purpose of t h i s paper to e x h i b i t examples of t h i s sort of computational e x e r c i s e . Rather, i t i s to show that with understanding of the p h y s i c a l circumstances which can lead to runaway r e a c t i o n , a simple, accurate, and broadly a p p l i c a b l e set of r u l e s can be developed, which can be used without the need f o r extensive numerical a n a l y s i s . In a s u f f i c i e n t l y long a d i a b a t i c r e a c t o r , the temperature can u l t i m a t e l y increase so s t r o n g l y with length that i t w i l l appear to be discontinuous. The p o s i t i o n of t h i s d i s c o n t i n u i t y 0097-6156/84/0237-0337$06.75/0 © 1984 American Chemical Society

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


338

CHEMICAL AND CATALYTIC REACTOR MODELING

i s r a r e l y f i x e d ; instead i t wanders slowly back and f o r t h as operating conditions vary. A safe reactor i s one which i s short enough so that a wandering p o t e n t i a l d i s c o n t i n u i t y never becomes r e a l . "Short enough" thus means the p o s i t i o n of the d i s c o n t i n u i t y i n the steady s t a t e at the average operating c o n d i t i o n , m u l t i p l i e d by a f a c t o r (

P

, C" )

=

1

and SD \Γ~

l e

N-l -

e x

1 Λ 1

This l a t t e r form i s equivalent to


348

CHEMICAL AND CATALYTIC REACTOR MODELING SD* = Ν exp (~τ *)

(16)

Ν

This leads t o : Δ τ

Δ τ

Δ τ

1

κ 4-

= exp

Ν-1

=

Ν*

= exp

Ν-2

(17)

τ


*

Ν-2

) ]

* = exp [ - < v

Note that T

=

N " \

Δ τ

Ν

+

Δ τ

Ν-1

=

+

* * ·

A T

(

i+l

1

8

)

and, t h e r e f o r e , that Δ τ

Δ τ

Δ τ

Ν-3

=

e

x

Ν-2

p

[

~

=

Ν-1

e

( Δ τ

x

Ν

p

+

=

e

[

"

Δ τ

x

p

( Δ τ

(

A

~ V

+

Ν

Δ τ

+

Δ τ

Ν-1

Ν-1

Ν-2

) ]

) ]

(

'

e

t

C

1

9

)

#

Table V, constructed from Equation (19) and Equation (^6), gives values of the c r i t i c a l parameters. The sequence of τ 's i s the sequence of exponents i n Equation (19).

Table V.

C r i t i c a l Values f o r a Series of Equal

Ν 1 2 3 4 5 6 7 8 9 10

Δ τ

Ν

1.0000 .3679 .2547 .1974 .1620 .1378 .1201 .1065 .0957 .0872

τ

Ν

1.0000 1.3679 1.6225 1.8199 1.9820 2.1198 2.2398 2.3463 2.4402 2.5290

Stages

SD* .3679 .5093 .5922 .6482 .6890 .7204 .7453 .7658 .7827 .7974


18.

Stability of Adiabatic

BARKELEW

349

Reactors

Decreasing the backmixing by i n c r e a s i n g the staging s t a b i l i z e s the a d i a b a t i c r e a c t i o n i n tjje same^sense as f o r the c i r c u l a t i n g loop. Larger values of SD and τ can be t o l e r a t e d f o r l a r g e r values of N.


Interstage Mixing A bubble tower r e a c t o r e x h i b i t s some degree of s t a g i n g , but the dominant mixing p a t t e r n i s an i n t e r m i t t e n t l a r g e - s c a l e c i r c u l a t i o n caused by r i s i n g clumps of bubbles. An approximate mathematical d e s c r i p t i o n i s s i m i l a r to that of the staged r e a c t o r , but with added terms to account f o r mixing between adjacent stages. In t h i s d e s c r i p t i o n , a model f o r the nth of Ν stages can be w r i t t e n as (1 + 2 ε ) τ

η

- (1 + e >

V

l

- ετ

η + 1

= f - exp (τ >

(20)

η

where ε i s the volumetric r a t i o of c i r c u l a t i o n to feed. For the top stage, (1 + ε) ( τ Ν

V

l

)

= ψ- exp (τ„)

(21)

For the bottom stage, (1 + ε ) τ - ε τ = ψ- exp (τ^) 1

(22)

2

For a given N, the c r i t i c a l values of SD and τ can be c a l c u l a t e d by assuming a s e r i e s of values of τ - _ ^ = > then c a l c u l a t i n g the corresponding values of Δ τ ,, Ν=2* * * Δτχ by a cascading procedure s i m i l a r to that used f o r Equation (19). With the c a l c u l a t e d Ax's, SD i s given by Equation (21). The c r i t i c a l value i s the maximum p o s s i b l e value of SD, determined by i n t e r p o l a t i o n among the assumed Δτ 's. A p l o t of SD vs. τ resembles that of Figure 2. A few c r i t i c a l values are given i n Tables VI and V I I . Again, i n c r e a s i n g mixing i n the flow d i r e c t i o n d e s t a b i l i z e s the reaction. τ

Ν

Δ τ

Ν

Ν

Ν

Δτ

#

t

0

ΧΤ

Mixing by V e l o c i t y

Distribution

In a packed-bed r e a c t o r , the backward f l u x of energy i s l i m i t e d to the r e l a t i v e l y small e f f e c t of conduction through the packing. The p r i n c i p a l d i s p e r s i o n mechanism i s v a r i a t i o n i n flow rate from point to p o i n t , coupled with l a t e r a l mixing. The extent of a x i a l mixing i s t y p i c a l l y small, compared with other types of r e a c t o r s .


350



Table VI.

N_

_ε=

2 4 6 8 10

.1 .4923 .6260 .6984 .7457 .7787

Table VII.

N_ 2 4 6 8 10

JE1

C r i t i c a l Values of SD

.2 .4784 .6051 .6782 .7262 .7605

.5 .4498 .5544 .6244 .6744 .7111

f o r Interstage Mixing

1

2

5

.4246 .5025 .5621 .6092 .6470

.4030 .4534 .4958 .5530 .5658

.3842 .4084 .4300 .4504 .4698

C r i t i c a l Values of τ

10 .3766 .3984 .4012 .4125 .4235

f o r Interstage Mixing

.1

.2

.5

1

2

5

10

1.3430 1.7874 2.0826 2.3055 2.4853

1.3147 1.7414 2.0315 2.2507 2.4278

1.2397 1.5886 1.8537 2.0614 2.2307

1.1648 1.4124 1.6231 1.8042 1.9592

1.0998 1.2514 1.3890 1.5180 1.6380

1.0454 1.1144 1.1784 1.2409 1.3025

1.0238 1.0548 1.0932 1.1261 1.1586


18.


BARKELEW

351

Reactors

A model by which t h i s can be represented was proposed some years ago by Deans and Lapidus ( 3 ) . A simple r e a l i z a t i o n of t h i s model i s to consider the reactant as being d i v i d e d i n t o two p a r a l l e l streams with d i f f e r e n t v e l o c i t i e s , with the two streams being mixed and r e d i v i d e d at i n t e r v a l s . If this r e p r e s e n t a t i o n i s to resemble a r e a l packed r e a c t o r , the number of m i x i n g - d i v i d i n g points must be f a i r l y l a r g e , say greater than 8 or 10. With the same n o t a t i o n as before, the r e a c t i o n i n one of the p a r a l l e l streams can be described by


i l = i|° dz aN

(23)

where α i s the f r a c t i o n of the t o t a l flow i n the stream, 3 i s the f r a c t i o n of the c r o s s - s e c t i o n i t occupies, and Ν i s the number of mixing p o i n t s . There are 2N such equations i n t h i s r e p r e s e n t a t i o n . T h e i r s o l u t i o n can be summarized by

exp

(τ ) -^1 η

-

—

—

J

^1 -

( 1

_

α ) Ν

j

(24)

The c r i t i c a l value of SD i s that f o r which the second f a c t o r becomes e x a c t l y zero f o r η = N. I t can be r e a d i l y c a l c u l a t e d by a one-variable search, f o r any assumed values of the parameters. ^ A few values of SD , c a l c u l a t e d i n t h i s way, are given i n Table VIII f o r the s p e c i a l case of 3 = 1/2.

Table V I I I .

Ν 8 10 12

C r i t i c a l SD* Values f o r Mix-Separate

α=

.6

.7

.9641 .9706 .9751

.9026 .9188 .9302

Model

.8 .8003 .8296 .8509

Decreasing Ν and i n c r e a s i n g α correspond to i n c r e a s i n g degrees of mixing. In t h i s model there i s no backmixing whatsoever, i n the sense that no energy flows upstream. The temperature p r o f i l e s of t h i s r e p r e s e n t a t i o n resemble those of Figure 1, with a v e r t i c a l asymptote r a j h e r than a maximum i n ^ SD. Formally, the c r i t i c a l value of τ , corresponding to SD , would be that corresponding to maximum r a t e , j u s t as i t i s i n plug flow. The c r i t i c a l temperature r i s e i n a r e a l r e a c t o r


352


described by t h i s model would be determined by the disturbances i n the process. For t h i s reason a t a b l e of τ i s not appropriate f o r t h i s model.


Cross-Mixing with Channeling An a l t e r n a t i v e d e s c r i p t i o n of cross-mixing i s to assume that the reactant i s d i v i d e d i n t o two p a r a l l e l streams which do not remix u n t i l the reactor o u t l e t , but which exchange energy and matter l a t e r a l l y as they move downstream. Something l i k e t h i s could happen i n a packed bed i f there were gross m a l d i s t r i b u t i o n of the packing. The mathematical r e p r e s e n t a t i o n of p a r a l l e l flow with d i f f u s i o n crossmixing i s d T

A = 3SD exp (τ > + κ ( τ Α

(1 - a)

j^-

β

- τ )

(25)

Α

= (1 - 3) SD exp ( τ ) + κ ( τ - τ ) β

Α

β

where κ i s the appropriate Stanton number and the other v a r i a b l e s have been defined e a r i 1 e r . C a l c u l a t i o n of c r i t i c a l values of SD from Equation (25) r e q u i r e s numerical i n t e g r a t i o n . I used τ as the independent v a r i a b l e (the temperature of the slower-moving stream) and i n t e g r a t e d to the point where dXg/dz ^ 1 0 . For a s e r i e s of assumed values of SD, there i s a unique v^lue at t h i s point f o r which ζ = 1.000. This i s the c r i t i c a l SD . As wijh the p l u g flow r e a c t o r , there i s no p h y s i c a l l y s i g n i f i c a n t τ i n the absence of disturbances^ A few values of SD f o r t h i s model are given i n Table IX. β

6

Table IX.

C r i t i c a l SD*

Values f o r the P a r a l l e l - F l o w Model

* α .75 .75 .6 .6

κ .5 .5 .2 .2

2 4 4 8

SD .8359 .9012 .8570 .9219

With t h i s model, i f the value of κ i s small, l o c a l hot-spots w i l l develop w i t h i n the bed. This type of reactor i n s t a b i l i t y i s the subject of another paper. The values from which Table IX was constructed are s u f f i c i e n t l y l a r g e , as determined by a c r i t e r i o n to be described below.


18.


BARKELEW

Reactors

353


A Common C h a r a c t e r i z a t i o n of Mixing With each of the s i x r e a c t o r models j u s t described, the transport of energy from regions with high conversion to those with low conversion causes d e s t a b i l i z a t i o n . Each model has i t s own unique way of c h a r a c t e r i z i n g the transport process. For these r e s u l t s to be of p r a c t i c a l use, a common measure of energy transport i s r e q u i r e d . This common measure i s the variance of the residence-time distribution. In the absence of r e a c t i o n , a sudden change i n i n l e t conditions w i l l be followed by a spread-out change i n o u t l e t c o n d i t i o n s . The spreading can be described by common s t a t i s t i c a l parameters, the mean, variance, skewness, and so on. Ways of c a l c u l a t i n g these parameters are well-known. For example, they are simply r e l a t e d to the c o e f f i c i e n t s i n the T a y l o r ' s Series expansion of the Laplace Transform of the equation which describes the temperature t r a n s i e n t without r e a c t i o n . With each of the s i x r e a c t o r models, an expression for the r a t i o of the variance of the residence-time d i s t r i b u t i o n to the square of the mean can be derived a n a l y t i c a l l y by f i n d i n g the Laplace Transform. The r e s u l t s of such an a n a l y s i s are l i s t e d i n Table X. In the channel model, the residence-time d i s t r i b u t i o n f u n c t i o n i s bimodal i f the streams are not s u f f i c i e n t l y mixed, a s i t u a t i o n which occurs i f σ / μ > about 1/20. For l a r g e r values, l o c a l hot-spots are l i k e l y to develop, and the c o r r e l a t i o n to be described does not apply. This puts a lower l i m i t on the parameter κ of that model. Figure 3 i s a p l o t of the c a l c u l a t e d values of SD , f o r a l l s i x models, v s . σ / μ . With a d e v i a t i o n of only a few percent, a s i n g l e curve can be passed through a l l the p o i n t s . To t h i s degree of p r e c i s i o n , the c r i t i c a l value of SD f o r any r e a c t o r i s independent of i t s mechanism of l o n g i t u d i n a l mixing. Values of σ / μ can be found by using e x i s t i n g c o r r e l a t i o n s , by d i r e c t measurement, by modeling, by s i m u l a t i o n , or by using one of the expression^ of Table X. Figure 4 shows a p l o t of the c r i t i c a l τ vs. σ /μ . This graph can be used to p r e d i c t s t a b l g operating l i m i t s . The s c a t t e r i s a l i t t l e greater than f o r SD , p a r t i c u l a r l y f o r low values of σ / μ ,^where the r e a c t o r i s nearly i n plug flow and where a c r i t i a l τ may not e x i s t f o r m a l l y . A n a l y t i c a l forms f o r the c o r r e l a t i n g curves of Figures ^ and 4^can be s p e c i f i e d by using the exact expressions f o r SD and τ f o r the c i r c u l a t i n g loop, derived from Equation (14). 2

2

2

2

2

2

2

2

2

2


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

Interstage Mixing

N, the number of stages a, the f r a c t i o n of flow i n one stream 3, the f r a c t i o n of c r o s s s e c t i o n of that stream a, the f r a c t i o n of flow i n one stream 3, the f r a c t i o n of c r o s s s e c t i o n of that stream κ, the cross-conduction Stanton number

Mix-Separate

Channel, w/conduction

ε, mixing flow/feed flow

N,the number of stages

p, c i r c u l a t i o n flow/feed flow

P, the a x i a l P e c l e t number

Mixing Parameters

Variance of the Residence-Time

Stages i n S e r i e s

Loop

diffusion

Circulating

Axial

Model

TABLE X.

N

2(α-3)'

(3-a) Na( 1-a)

i=l

(N-i)

F

(-^-) ^Ι+ε'

;

1-exp (HP) χ p

1 2 N-1 ^r + ^r V Ν 2

1/N

P/(1+P)

Ρ

In-

2 2 Variance R a t i o , σ /μ

Distribution



BARKELEW

355

Reactors

1.0 MODEL Ο A X I A L DIFFUSION • CIRCULATION Ο STAGED Δ STAGES WITH MIXING


0.9

0

^"

8

^ MIX-SEPARATE ψ CHANNEL

J / K

0.7

8>o

0.6

0.5 D

Δ

0.4

[£Δ

0.3

Q Ο .

" 0

—

01

0.2

0.3

0.4

0.5 σ /μ 2

Figure 3.

0.6

0.7

0.8

2

C r i t i c a l Residenee-Time

Parameter.


0.9

1.0

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

356

MODELING

4.0


MODEL Ο A X I A L DIFFUSION • CIRCULATION Ο STAGED Δ STAGES WITH MIXING 3.0

Ο

ο Ο Θ

2.0

ο

4Δ

•

ε&Δ

1.0 0.1

0.2

0.3

0.4

2

σ /μ Figure 4.

Critical

0.6

0.5

0.7

0.8

2

Temperature-Rise

Parameter.


n i 0.9

1.0

18.

BARKELEW


357

Reactors

They are

2 (26)

SD

*

1

τ 2

log

(27)

(σ /μ )

2


ι //μ 1-σ Use of these expressions avoids the inconvenience of attempting to read values o f f the g r i d l e s s p l o t s of Figures 3 and 4. A few comments regarding Equation (26) and (27) are a p p r o p r i a t e . They should not be extrapolated beyond σ / μ > 1. Strongly-skewed d i s t r i b u t i o n s f o r which the i n e q u a l i t y holds are p h y s i c a l l y p o s s i b l e , but they are f o r systems i n which some of the reactants are trapped i n a "dead-zone." A runaway r e a c t i o n i n such a system would be l o c a l rather than g l o b a l , and a d i f f e r e n t procedure f o r assessing s t a b i l i t y would be r e q u i r e d . Otherwise, the expressions are conservative. They were derived from zero-order k i n e t i c s and the exponential approximation of the Arrhenius expression, both of which overstate the tendency of a r e a c t i o n to run away. The conservatism i s m i l d , however, amounting to a few percent at most. Nevertheless, i t i s p o s s i b l e to compensate f o r a l a r g e part of i t and s t i l l r e t a i n the s i m p l i c i t y of Equation (26) and Equation (27). This can be done by making a minor adjustment to the value of the parameter γ. In Equation (4), γ was determined by approximating the Arrhenius expression by an exponential, i n which the s p e c i f i c rate constant and i t s temperature d e r i v a t i v e were matched at the r e a c t o r i n l e t . A l e s s conservative, but s t i l l s a t i s f a c t o r y , approximation can be found by matching rates at two d i f f e r e n t temperatures. My preference i s to make t h i s match at Τ and T + l/γ (at τ = 0 and τ = 1), because the p o t e n t i a l f o r a runaway i s s t r o n g l y i n f l u e n c e d by the course of the r e a c t i o n i n t h i s temperature range, and rather weakly outside i t . In the same way, Equation (26) and (27) can be r e t a i n e d i f the order of the r e a c t i o n i s not zero by using the f a c t that (1 - τ/S) * exp (-τη/S) f o r small τ/S. Since the safe upper l i m i t on τ i s i n the range of 1 or 2, and t y p i c a l values of S are 20 or g r e a t e r , t h i s approximation i s not unreasonable. Taken together, these two adjustments define an e f f e c t i v e γ to be used i n (26) and (27), as γ(1 - n / S ) / ( l + RT /E) . The e f f e c t i v e S i s the defined S reduced by the same f a c t o r s . The u l t i m a t e r e s u l t i s that the c o r r e l a t i o n s f o r maximum safe SD and τ can be a p p l i e d to a much broader set of k i n e t i c expressions than the simple one from which they were d e r i v e d . 2

Q

11


2

358


Concluding Remarks


Although much has been published on the s t a b i l i t y of a d i a b a t i c r e a c t o r s , most of i t has dealt with conclusions to be drawn from the assumption that backmixing can be described i n a p a r t i c u l a r way. Perlmutter (4) has given a review of the s u b j e c t , i n which the strong e f f e c t of backmixing on s t a b i l i t y i s c l e a r l y recognized. This paper goes beyond the p r i o r a r t i n that i t provides a reasonably convincing demonstration of the i n t u i t i v e l y reasonable p r o p o s i t i o n that the mechanism of mixing i s r e l a t i v e l y unimportant i n determining the l i m i t s of stable operation of a d i a b a t i c r e a c t o r s . Table of Nomenclature α 3 Y C d D ε Ε F H h κ k

2

l> y n N P P R σ

2

S

*

SD τ

*

τ Τ T

F r a c t i o n of flow i n one of two p a r a l l e l streams F r a c t i o n of r e a c t o r c r o s s - s e c t i o n occupied by a stream E/ RT Heat capacity Axial diffusion coefficient kV/F Mixing flow/feed flow A c t i v a t i o n energy Mass flow r a t e Heat of r e a c t i o n Heat t r a n s f e r c o e f f i c i e n t between streams h/CF S p e c i f i c r e a c t i o n rate constant at i n l e t conditions Rate constant Reactor length V/F Reaction order Number of stages FL/d C i r c u l a t i o n flow/feed flow Gas constant Variance of residence-time d i s t r i b u t i o n

o V χ ζ

γΗ/c C r i t i c a l value of SD Ύ(Τ-Τ ) C r i t i c a l value of τ Temperature I n l e t temperature Reactor volume Conversion, 0 < ζ

Stability of Adiabatic Reactors - American Chemical Society

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