Stability of Trickle-Bed Reactors - ACS Symposium Series (ACS

4 Stability of Trickle-Bed Reactors C. H. BARKELEW and B. S. GAMBHIR Downloaded by UNIV OF CALIFORNIA SAN DIEGO on January 2, 2017 | http://pubs.acs.org Publication Date: December 9, 1984 | doi: 10.1021/bk-1984-0237.ch004

Shell Development Company, Houston, TX 77001 Trickle-bed reactors, in which gas and liquid flow co-currently downwards through a bed of catalyst particles, are commonly used for hydrogenation of various hydrocarbons. These reactors can develop local exotherms in which undesirably high temperatures occur. In this paper a rule for prevention of these hot spots is developed. Its form has been derived from theory and its two parameters have been estimated from real process data. Its use assures safe operation of adiabatic trickle-bed reactors. Trickle-bed reactors, which are widely used for hydrogénation of refinery hydrocarbon streams, are known to be prone to development of irregular flow patterns. In extreme cases, these irregularities can lead to large variability of the extent of reaction, with temperatures rising hundreds of degrees above normal, sintering the catalyst into massive clumps. This usually destroys the catalyst, but more seriously, it can lead to structural damage of the reactor vessel, with attendant personnel hazards. Safety requires that reactors be operated in such a way that they are free of hot spots at all times. Although many process operators have learned by experience how to run their units safely, there has as yet been no analytical treatment of the subject which gives them firm guidelines. A Model Of A Trickle Bed A suitable model of a trickle bed, with which its stability can be analyzed, can be devised by assuming that there are 0097-6156/84/0237-0061$06.00/0 © 1984 American Chemical Society

Dudukovi and Mills; Chemical and Catalytic Reactor Modeling ACS Symposium Series; American Chemical Society: Washington, DC, 1984.

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62

CHEMICAL AND CATALYTIC REACTOR MODELING

i r r e g u l a r i t i e s i n packing d e n s i t y and i n the flows of the phases, with c h a r a c t e r i s t i c dimensions on the order of a few tens of p a r t i c l e diameters. Observations on l a b o r a t o r y models of t r i c k l e - b e d reactors confirm t h i s , and furthermore i n d i c a t e that i r r e g u l a r i t i e s , once formed, tend to p e r s i s t f o r a s u b s t a n t i a l f r a c t i o n of the bed height. We s t a r t by c o n s i d e r i n g the steady state i n a c y l i n d r i c a l i r r e g u l a r i t y , extending downward i n t o the bed from near i t s upper s u r f a c e . Outside the d i s t u r b e d region, the r e a c t i o n can be described by the usual heat and m a t e r i a l balance equations f o r a r e a c t o r i n plug flow, CF^f = k Hf(x) exp (-E/RT)

(1)

F^j- = k f ( x ) exp (-E/RT)

(2)

g

The symbols are defined under "Nomenclature", below. The u n i t s of these q u a n t i t i e s are i r r e l e v a n t , so long as they are mutually c o n s i s t e n t . Equations (1) and (2) can be combined into a s i n g l e dimensionless equation,

j£

- SD exp (τ)

(3)

Equation (3) i n c l u d e s the approximation: 2

exp(-E/RT) « exp(-E/RT ) e x p [ E ( T - T ) / R T ] Q

o

Q

(4)

and i s s t r i c t l y v a l i d only f o r the known zero-order k i n e t i c s of hydrotreating. The s o l u t i o n of Equation (3) i s SDz

= 1 - exp (-τ)

(5)

Within the c y l i n d r i c a l d i s t u r b e d region, the temperature and composition vary with l a t e r a l p o s i t i o n . For our purposes, t h i s v a r i a t i o n can be adequately described by adding another term to Equation (3), p u t t i n g i t i n t o the form: 2 - g = SD*exp (ψ) + \

ν*ψ

(6)

S t a b i l i t y of the Model The behavior of systems described by Equation (6) i s w e l l known. I t was f i r s t analyzed by Frank-Kamenetskii (1) i n the 1940 s i n a study of the theory of thermal e x p l o s i o n s . He was i n t e r e s t e d i n the t r a n s i e n t behavior of a f i x e d volume of r e a c t a n t , which i s also d e s c r i b a b l e by Equation (6), with ζ 1


4.

Stability of Trickle-Bed

BARKELEW AND GAMBHIR

i n t e r p r e t e d as time. He showed that there i s or i s not steady s t a t e , according to the value of a dimensionless^ which includes the q u a n t i t i e s we have c a l l e d SD and Ρ which i s p r o p o r t i o n a l to the square of the diameter of c y l i n d r i c a l region. In our n o t a t i o n , h i s c r i t e r i o n i s 2

2


S D * P * R / L « ) 2 exp

63

Reactors a number , and the

(-τ)

(7)

The dimensionless r a t i o Fd/κ, where d i s the p a r t i c l e diameter, i s known to be c o r r e l a t e d with p a r t i c l e Reynolds number, s o l i d c o n d u c t i v i t y , void f r a c t i o n , and p a r t i c l e shape (2). I t i s a complicated but weak f u n c t i o n of these q u a n t i t i e s , which approaches a constant value of about 10 f o r large Reynolds numbers and low s o l i d c o n d u c t i v i t y . This l i m i t i n g value can be used f o r our purpose, leading to, Ρ =10 L/d. Write SD = Q* SD, d e f i n i n g G as the disturbance r a t i o . Then combination of Equations (5) and (7), using the l i m i t i n g value of Ρ , gives 2

GR [exp U )

" H/Ld

< 1/5

(8)

This can be i n t e r p r e t e d as the c o n d i t i o n that the temperature w i t h i n the d i s t u r b e d region w i l l not diverge e x p o n e n t i a l l y from that i n the normal region of the bed. G, R, L and d are f i x e d parameters f o r a given reactor c o n f i g u r a t i o n and flow r a t e , hence Equation (8) can be i n t e r p r e t e d as a s t a b i l i t y c r i t e r i o n which simply s t a t e s that the normal-bed temperature r i s e must not exceed some l i m i t i n g value, otherwise hot spots w i l l develop. Development Of A S t a b i l i t y

Criterion

We now turn to d e v i s i n g a procedure f o r estimating the maximum allowable temperature r i s e . Equation (8) i s not u s e f u l i n a q u a n t i t a t i v e sense because the parameters G and R cannot be measured or estimated. However, i t i s p o s s i b l e to deduce from elementary c o n s i d e r a t i o n s how they might vary with r e a c t i o n conditions. G i s the r a t i o of the dimensionless number SD i n the disturbed region to i t s value i n the normally-operating part of the bed. SD contains the a c t i v a t i o n energy, heat of r e a c t i o n , i n l e t temperature and bed height, a l l of which have f i x e d constant values i n a l l regions of the bed. I t also contains the p o s s i b l y v a r i a b l e q u a n t i t i e s C, k and F. C i s the average heat capacity of the f l u i d , and depends on the l o c a l phase r a t i o . k i s the s p e c i f i c rate constant, and depends on the l o c a l c a t a l y s t density and the phase holdup. F i s the l o c a l average l i n e a r v e l o c i t y , which can vary from point to point f o r a v a r i e t y of reasons. g



64


The phase flow v a r i e s from the top to the bottom of the bed, p r i m a r i l y because v o l a t i l e components vaporize w i t h r i s i n g temperature. We expect the l i n e a r v e l o c i t y of the gas phase to increase r e l a t i v e to that of the l i q u i d , and furthermore we expect i n c r e a s i n g v e l o c i t y to smooth out those f l u c t u a t i o n s i n G which are dependent on phase r a t i o . There i s a part of G which should not depend on gas v e l o c i t y , namely that which i s a t t r i b u t e d to f l u c t u a t i o n s i n packing d e n s i t y . This q u a l i t a t i v e reasoning leads us to suspect that the quantity G depends on the gas flow rate l i k e A + B / f ( F ) , where f i s some i n c r e a s i n g f u n c t i o n of gas flow. R, the radius of the d i s t u r b e d c y l i n d r i c a l region, i s l i k e l y to be p r o p o r t i o n a l to the p a r t i c l e diameter. We also suspect that i t may not be a constant over the bed, but may vary with height. In p a r t i c u l a r i t may vary i n such a way that the e f f e c t i v e value of R to use i n Equation (8) i s l i k e L , where L i s the t o t a l bed height and η i s a small p o s i t i v e number. This assumes that the d i s t u r b e d region grows as the f l u i d s move downward. P u t t i n g a l l t h i s i n t o Equation (8), we f i n d that something l i k e Κ 1 (9) exp (τ) - 1 < 2

n

ought to c o r r e l a t e with freedom from hot spots, i f we can f i n d appropriate values f o r the K s and can s p e c i f y a form f o r f . We have t r i e d a number of forms f o r f , i n the procedure to be described below, and concluded that p F at the reactor o u t l e t i s the form we p r e f e r to use. This i s the k i n e t i c energy density of the gas, commonly used to c o r r e l a t e drag, and i t seems reasonable that i t i s a reasonable way f o r v e l o c i t y to a f f e c t l o c a l f l u c t u a t i o n s of phase r a t i o . In the absence of more data, we cannot be more p r e c i s e about f , except to say that p F works. We do take note of the f a c t that the form does not e x t r a p o l a t e n e a t l y to zero gas v e l o c i t y . This i s not a significant limitation. Some i n t e r e s t i n g conclusions can be drawn from the form of Equation ( 9 ) . The K's can be determined only by comparison with data; they cannot, to our knowledge, be p r e d i c t e d from f i r s t p r i n c i p l e s or e x i s t i n g c o r r e l a t i o n s . The assumed value of 10 f o r the p a r t i c l e Peclet number i s completely absorbed i n t o K}, making that an u n c r i t i c a l assumption. If the development of hot spots were independent of gas flow, then Equation (9) could be transformed to f

2

2

log

[exp (τ) - 1] < l o g

K3 -

(n-1) l o g L


(10)

4.



65

Reactors

with K a d i f f e r e n t constant. A s t r a i g h t l i n e on a p l o t of log [exp (τ) - 1] vs. log L would separate reactors that develop hot spots from those which do not. I f the cross s e c t i o n of the d i s t u r b e d region were to vary l i n e a r l y with length, then a l l the tendency toward unstable behavior would be caused by flow v a r i a b i l i t y . E q u a t i o n (9) could then be w r i t t e n : 3

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exp

(τ) - 1

K

1

>

K

, 2 l

K

|

l

1 pF

(

u

)

2

2

and a s t r a i g h t l i n e on a p l o t of l/[exp (τ) - 1] vs l / ( p F ) would separate safe and unsafe operation. The l i m i t i n g cases, Equations (10) and (11), w i l l be considered l a t e r . Determination of Parameters From Process Data We have c o l l e c t e d operating data over a period of 15 years from four separate t r i c k l e - b e d hydrotreater r e a c t o r s . These four have operated with a number of d i f f e r e n t c a t a l y s t l o a d i n g s , have processed a v a r i e t y of feedstocks, and have operated over a wide range of average temperatures and pressures. P h y s i c a l d e s c r i p t i o n s of the four reactors are given i n Table I.

Table I.

P h y s i c a l D e s c r i p t i o n s of Reactors

Reactor

Diameter (ft)

1 2 3 4

11 11 13.5 13.5

Range of Bed (ft) 13-22 30-48 32.5 40.5

Height

C a t a l y s t Diameter (in.) 1/16 1/16 1/16 1/16

During the period of observation, s i x separate i n c i d e n t s occurred i n which hot spots s t a r t e d to develop. The conditions that led to these i n c i d e n t s are summarized i n Table I I . The o u t l e t vapors include components o r i g i n a l l y i n the l i q u i d feed, vaporized during passage through the bed. The amounts of these components were estimated using c o r r e l a t e d e q u i l i b r i u m r a t i o s . A set of operating c o n d i t i o n s known to be s a f e , from the same four r e a c t o r s , i s given i n Table I I I . The data of Tables II and I I I can be compared with the s t a b i l i t y c r i t e r i o n Equation (9) by making p l o t s of the sort suggested by Equations (10) and (11). The intermediate data



500 535 643 620 610 660 600

22 48 22 13 30 32.5 40.5

1 2 1 1 2 3 4

Reactor No.

Case No.

7 8 9 10 11 12 13

Inlet Temp. (°F)

Bed Height (ft)

2 1 2 1 3 4

1 2 3 4 5 6

7 9 57 50 30 50 15

Temp. Rise (°F) 1295 1565 1615 1550 1550 1200 900

Pressure (psia)

1.37 1.52 4.50 4.43 2.92 2.92 .41 8.8 10.2 16.0 17.2 17.2 31.6 32.8

4

Outlet Vapor Flow (lb/hrX10" )

Feed Liquid Flow (lb/hrX10" )

4

1.36 2.79 2.30 3.70 2.40 1.30

8.8 10.2 16.0 16.4 22.9 32.8

1295 1565 1615 1600 1200 900

4

35 60 40 68 40 40

4

Pressure (psia)

Temp. Rise (°F)

T a b l e I I I . O p e r a t i n g C o n d i t i o n s Known t o Be S a f e

480 575 620 627 680 643

48 22 48 15 32.5 40.5

Reactor No.

Case No.

Inlet Temp. (°F)

Bed Height (ft)

Outlet Vapor Flow (lb/hrX10" )

Feed Liquid Flow (lb/hrX10" )

T a b l e I I . C o n d i t i o n s L e a d i n g t o Hot S p o t s


151 115 94 126 97 140 58

Superficial Vapor Velocity (ft/hr)

144 133 61 86 79 61

Superficial Vapor Velocity (ft/hr)

5.3 5.3 5.3 5.3 5.3 7.4 5.8

Feed Gas Mol. Wt.

5.3 5.3 5.3 5.3 13.0 23.0

Feed Gas Mol. Wt.


4. BARKELEW AND GAMBHIR

Reactors

67

f o r these p l o t s are l i s t e d i n Table IV. An a c t i v a t i o n energy of 27 kcal/mole ( 3 ) was used f o r c a l c u l a t i n g the values i n the t h i r d column. Figure 1 i s the p l o t suggested by Equation ( 1 0 ) , which assumes that flow rate does not a f f e c t the disturbance r a t i o G. C l e a r l y , the map does not separate i n t o "safe" and "unsafe" regions, although i f points 9 and 12 could be lowered and points 5 and 6 r a i s e d , i t might do so. Inspection of Table IV shows that 9 and 12 have r e l a t i v e l y high values of p F , and that 5 and 6 have r e l a t i v e l y low values. This suggests that the form of Equation ( 1 1 ) , which allows f o r the e f f e c t of flow on G but also assumes that R i s p r o p o r t i o n a l to L , be t r i e d . Figure 2 i s the map suggested by Equation ( 1 1 ) . The separation i s almost clean, with point 13 being the sole " s a f e " c o n d i t i o n that f a l l s i n t o the "unsafe" zone. Case 13 has an unusually low vapor flow r a t e , and thus the discrepancy suggests that flow i s weighted too h e a v i l y by Equation ( 1 1 ) . This form does c o r r e c t the d e f i c i e n c i e s of Equation ( 1 0 ) , however. If the dashed l i n e on Figure 1, with slope = - 1 , were accepted as an approximate border, the exponent, n, would be 2. In Figure 2 , t h i s exponent i s 1 by hypothesis. Since these two f i g u r e s are f o r opposite kinds of l i m i t s , there i s a suggestion that an intermediate exponent, say 1 . 5 , should be considered. S i m i l a r l y , the d i s c r e p a n c i e s of Figure 1 i n d i c a t e that the dependence of G on flow ought to be greater than what was assumed i n drawing that f i g u r e . The s i n g l e discrepancy on Figure 2 suggests that i t s assumed flow dependence i s too l a r g e . Again, something i n between i s suggested, say


2

2

G =

[1

+

(WPF2)]1/2.

Together,

these transform Equation (9) t o :

exp

(τ) - 1
K +-Ar 2 ' 5 2 (τ) - 1 Γ pF

(13)

C

Ltexp

2

2

This says that a p l o t of l/[L(exp (τ) - l ) ] v s . 1/pF should be separable i n t o safe and unsafe regions by a s t r a i g h t l i n e . Figure 3 i s such a p l o t , with values of the ordinate taken from the l a s t column of Table I I I . The dashed l i n e i s represented by

io L[exp

3

(τ) - l ]

2

1

+

3JL10l pF

2


(

1

4

)


log L

3.87 3.09 3.87 2.71 3.48 3.70 3.09 3.87 3.09 2.56 3.40 3.48 3.70

Case No.

1 2 3 4 5 6 7 8 9 10 11 12 13

Q

2

.0277 .0228 .0210 .0207 .0188 .0200 .0265 .0247 .0201 .0210 .0214 .0195 .0218

E/RT 4.85 2.55 6.62 2.98 7.49 18.2 4.59 5.43 2.24 1.70 3.34 3.46 59.2 .611 .341 .761 .324 .890 .816 4.89 4.02 .466 .539 1.113 .605 2.59

2

.492 1.077 .273 1.127 .116 .203 -1.588 -1.392 .763 .617 -.107 .502 -.951

z

.969 1.370 .839 1.408 .753 .800 .186 .222 1.146 1.049 .641 .975 .327

5

(10 )/pF ft hr /lb

l/[exp(x)-l]

log[exp(x)-l]

I n t e r m e d i a t e Data o f P l o t s

τ

Table IV.

3

1

7.79 5.27 12.07 7.00 24.4 16.4 1094. 337. 9.87 22.4 41.3 11.3 165

(10 )/L[exp(x) ft"





,

m

m

\ \

\

•

HOT SPOT

Ο

SAFE

Reactors

©

\

\

m 0D\

03)

2.5

3

τ

3.5 Log L

Figure 1.

•*4

S t a b i l i t y Map with no Flow E f f e c t .



70


Figure 2.

S t a b i l i t y Map with Dominating Flow E f f e c t .





SAFE

/ /

Reactors

/

/ /

/

HOT SPOTS /

1

'

1

•

1 ι

ι ι

10 10 /pF 5

Figure 3;

15

2

Recommended S t a b i l i t y

Criterion.



72


Two of the p o i n t s , 5 and 12, are close to the l i n e , and point 13 i s s l i g h t l y on the wrong s i d e , but these discrepancies are minor and w i t h i n the u n c e r t a i n t y of the data. Further refinement i s p o s s i b l e but hardly warranted with the l i m i t e d data. One might be tempted to draw the separating l i n e of Figure 3 with a s l i g h t l y steeper slope, passing c l o s e r to points 2 and 4. However, t h i s must not be done i n such a way that the l i n e i n t e r s e c t s the h o r i z o n t a l axis at a p o s i t i v e value. Unphysical, imaginary values of temperature would be i n d i c a t e d i f i t were to do so. Equation (14) i s equivalent to

τ < τ

v

10

log

2

(15)

3 X 10 »P

2

which i s our suggested s t a b i l i t y c r i t e r i o n f o r t r i c k l e beds with c a t a l y s t p a r t i c l e s 1/16" i n diameter. Operation of a reactor with a temperature r i s e a few degrees less than that suggested by Equation (15) should be f r e e of hot spots. Table V compares observed temperature r i s e s f o r each of the cases with the maximum allowable value c a l c u l a t e d from Equation (15).

Table V. Comparison of Observed Temperature Rises f o r Each Case Case

Hot Spot' 1 2 3 4 5 6 7 8 9 10 11 12 13

Yes Yes Yes Yes Yes Yes No No No No No No No

Observed AT

Maximum Safe AT

35 60 40 68 40 40 7 9 57 50 30 50 15

The c r i t e r i o n i s i n s a t i s f a c t o r y correspondence data.

28 41 33 62 40 26 38 30 61 72 47 50 15

with the


4.

BARKEl EW AND GAMBHIR


C h a r a c t e r i s t i c Features Of the S t a b i l i t y

Reactors

73

Criterion

Dimensions The numerical constants i n Equation (15) are dimensional, with length i n f e e t , mass i n pounds, and time i n hours. This unfortunate b i t of u n t i d i n e s s does not detract from the u t i l i t y of Equation (15), however. A proper dimen s i o n a l a n a l y s i s must wait u n t i l we have a b e t t e r p h y s i c a l understanding of the mechanics of t r i c k l e flow. Presumably the c h a r a c t e r i s t i c length needed to make "10 " dimensionless would be the diameter of a disturbance that spontaneously forms i n the upper part of the bed. The c h a r a c t e r i s t i c drag force f o r "3 X 10 " would presumably i n v o l v e l i q u i d - s o l i d c a p i l l a r y effects.


3

5

E f f e c t s of P a r t i c l e Size The dependence of these two parameters on the dimension of the c a t a l y s t p a r t i c l e s can be i n f e r r e d from elementary reasoning. Equation (9) i n d i c a t e s d i r e c t l y that "10 " should be i n v e r s e l y p r o p o r t i o n a l to the square of d. This involved the assumption that both the s i z e of the disturbances and the e f f e c t i v e c o n d u c t i v i t y are p r o p o r t i o n a l to d. On the other hand, the drag constant ought to be r e l a t i v e l y i n s e n s i t i v e to d. This can be deduced by supposing that the force a c t i n g to break up accumulated pockets of excess r e a c t i v i t y i s p r o p o r t i o n a l to p F and acts over the circumference of the pocket, making the t o t a l d i s r u p t i v e f o r c e p r o p o r t i o n a l to p a r t i c l e diameter. The amount of m a t e r i a l i n the pocket i s p r o p o r t i o n a l to the cross s e c t i o n , and hence to d , but the r e s i s t a n c e to breakup i s p r o p o r t i o n a l to the s o l i d surface per unit volume, or to 1/d. The net i s again p r o p o r t i o n a l to d, making the balance between d i s r u p t i v e and cohesive f o r c e s , and hence the parameter "3 X 10 ," independent of diameter. Although t h i s a n a l y s i s i n d i c a t e s how the c r i t e r i o n could be used f o r d i f f e r e n t p a r t i c l e s i z e s , some degree of caution i s advisable u n t i l data with other p a r t i c l e s become a v a i l a b l e . 3

2

2

5

E f f e c t s of Reaction Parameters An unusual feature of Equation (15) i s that i t does not appear to i n v o l v e the heat of r e a c t i o n and heat c a p a c i t y . This i s because they were a l g e b r a i c a l l y e l i m i n a t e d by combining the Frank-Kamenetskii r e l a t i o n , Equation (7), with the a d i a b a t i c r e l a t i o n , Equation (3). Both are important i n determining s t a b i l i t y , of course, and t h i s importance can be demonstrated by e l i m i n a t i n g τ between Equation (15) and (3). This gives an e q u a l l y v a l i d but less useful s t a b i l i t y c r i t e r i o n : ^

(16)


74


Estimation of Unknown Parameters Approximate values of R and G can be estimated from the numerical constants of Equation(15) and (8)· We have \. 5\'2 3 X 10 G = 11 + 2 1 (17) m

and


R

2

5

(18)

7?

Excluding the two l o w - v e l o c i t y cases, G v a r i e s from 2.5 f o r case 10 to 4.8 f o r case 5. The l a r g e s t G i s 13, f o r case 13. The disturbance diameters, from Equation (18), range from 1" f o r case 10 to 2 1/2" f o r cases 1, 3 and 8. These estimates of R are rather s e n s i t i v e to the way the l i n e i s drawn i n Figure 3, since they are p r o p o r t i o n a l to the i n t e r c e p t on the v e r t i c a l a x i s . They should be considered as very crude estimates. E f f e c t of I n l e t Temperature

I f Equation (15) i s w r i t t e n as

/

RT ΔΤ

log

10"

1 + L

1 +

(19)

3 X 10" PF

2

7j

a curious feature appears. The s t a b i l i t y seems to be enhanced as the i n l e t temperature i n c r e a s e s , contrary to i n t u i t i o n . This i s a d i r e c t consequence of using the Arrhenius r a t e expression. The r e l a t i v e change i n rate per degree of temperature increase becomes smaller as the average temperature rises. The feature i s l e s s p u z z l i n g i f i t i s considered as an e f f e c t of d e c l i n i n g c a t a l y s t a c t i v i t y . As the c a t a l y s t g r a d u a l l y ages, the i n l e t temperature must be increased to maintain conversion. The l e s s - a c t i v e c a t a l y s t has a lower temperature dependence of r a t e , as w e l l as a lower r a t e . This r e s u l t s i n an i n c r e a s i n g tolerance to disturbances. Temperature F l u c t a t i o n s The above p h y s i c a l d e s c r i p t i o n of a t r i c k l e - b e d reactor does not include the assumption that the temperature i n the "normal" region i s uniform i n a cross s e c t i o n . In an a d i a b a t i c r e a c t o r , a l l f l u c t a t i o n s grow i n amplitude and s i z e , even the i n f i n i t e s i m a l ones. A mottled temperature s t r u c t u r e cannot be avoided, and a set of temperature sensors at a given height w i l l not n e c e s s a r i l y be c o n s i s t e n t . A band of temperature readings w i l l always occur. A rough estimate of the width of t h i s band can be found from the steady-state s o l u t i o n of Equation ( 6 ) . The


4. BARKELEW AND GAMBHIR

Stability of Trickle-Bed Reactors

75

temperature p r o f i l e w i t h i n a p o s i t i v e f l u c t a t i o n i s given by the s o l u t i o n of .2 GSDPR V ψ = (20) exp (ψ) 2

with ψ = τ at r = 1, which i s


ψ

- ψ = 2 log

1 + exp (ψ ) r

2 GSDPR

(21)

8L The boundary

condition i s

ψ - τ = 2 l o g (1 + exp ( ψ ) ^ § 2 ™ ° V ° 8ΙΓ η

(22)

or GSDPR

exp ( Ψ - τ) = I l + exp ( ψ - τ) ο

ο

2

(23)

8L exp (-τ)y a quadratic equation i n exp ( ψ - τ ) . ο

If a disturbance i s on the verge of developing i n t o a hot spot, the parameter group (GSDPR ) / [ L exp (-τ )] i s near i t s c r i t i c a l value of 2, at which point the quadratic equation i s 2

2

exp ( ψ - τΥ ο

exp (ψ

(24)

- τ)

whose s o l u t i o n i s exp (ψ - τ ) = 4. The magnitude of the temperature f l u c t u a t i o n i s thus ( R T )/E l o g 4, about 60°F f o r t y p i c a l conditions from Table I. Negative f l u c t u a t i o n s are never very l a r g e , so t h i s corresponds roughly to the maximum p o s s i b l e safe width of the temperature band. Operators w i l l often watch the readings from an array of thermocouples i n a r e a c t o r , as an i n d i c a t i o n of imminent hot spot formation. The h i s t o r i c a l danger point f o r the reactors of Table I has been a band of 35°F. I t i s i n t e r e s t i n g to note that the average p o s i t i v e temperature f l u c t u a t i o n w i t h i n a disturbance, from the i n t e g r a l of Equation (21), i s about 30°F. 2

Q

E f f e c t s of Assumptions The s t a b i l i t y c r i t e r i o n Equation (15) has two d i f f e r e n t meanings. F i r s t , i t i s an i n t e r p o l a t i o n procedure, guided by t h e o r e t i c a l c o n s i d e r a t i o n s , f o r p r e d i c t i n g the future behavior of a group of reactors from h i s t o r i c a l observations on those same r e a c t o r s . As such i t cannot be p a r t i c u l a r l y s e n s i t i v e to


76



assumptions made i n developing the theory. The second i n t e r p r e t a t i o n of Equation (15) i s that i t can be used f o r other r e a c t i o n s , with d i f f e r e n t k i n e t i c parameters, f o r example. In t h i s sense, i t i s an e x t r a p o l a t i o n procedure, and may be s e n s i t i v e to assumptions made about the model. We have made simple t e s t s of the e f f e c t s of three of the assumptions. They are a) that the r e a c t o r i s i n plug flow, b) that the a c t i v a t i o n energy to be used with the process data i s 27 kcal/mole, and c) that the r e a c t i o n i s of zero order. Each of these i s discussed b r i e f l y below. Plug Flow With a s i g n i f i c a n t amount of a x i a l d i s p e r s i o n , Equation (3), d e s c r i b i n g the normal bed temperature p r o f i l e , must be modified to account f o r t h i s d i s p e r s i o n . The e f f e c t of t h i s m o d i f i c a t i o n i s that the u l t i m a t e v e r t i c a l asymptote i n temperature i s moved forward i n the extended bed. D i s p e r s i o n enhances the tendency of a r e a c t o r to run away. However, with the type of d i s p e r s i o n that occurs i n a t r i c k l e bed, by v a r i a t i o n s i n v e l o c i t y from point to p o i n t , the p r o f i l e r e t a i n s i t s v e r t i c a l asymptote. The s o l u t i o n of Equation (3) plus d i s p e r s i o n i s almost i d e n t i c a l with Equation (5), but with a d i f f e r e n t value of SD. Since SD drops out i n the ultimate s t a b i l i t y c r i t e r i o n , a x i a l d i s p e r s i o n cannot be of any p a r t i c u l a r s i g n i f i c a n c e i n the development of l o c a l hot spots. I t a f f e c t s the g l o b a l s t a b i l i t y of the normal part of the r e a c t o r , but i t has l i t t l e i n f l u e n c e on the way disturbances grow, r e l a t i v e to the normal regions. A c t i v a t i o n Energy Although the assumed a c t i v a t i o n energy of 27 kcal/mole i s based on measurement, and i s probably about r i g h t , i t s use could conceivably be questioned because the c a t a l y s t s may have been d i f f e r e n t . I t has been r e a s s u r i n g to f i n d that our conclusions are not s e n s i t i v e to t h i s assumed value. To assess the e f f e c t of the assumption, we have repeated the e n t i r e d a t a - f i t t i n g procedure f o r two other values of E, 13.5 and 18 K cal/mole. For the lower value, the procedure does not work. The model i n e f f e c t says that i f the a c t i v a t i o n energy were that low, the observed hot spots could not have occurred. For Ε = 18, however, a c r i t e r i o n equivalent to Equation (15) can be derived i n the same way as before. We give the r e s u l t , without d e t a i l s . Γ

τ < τ

log

-

1^

3

1 + L

16 +

1(T 5.4 X PF

10 2

:

)J

(25)

The comparison equivalent to Table IV i s e q u a l l y good; however, the estimates of R and G are changed. R doubles, and G decreases s l i g h t l y . P h y s i c a l l y , t h i s can be i n t e r p r e t e d as


4.



Reactors


saying that i f s e n s i t i v i t y to temperature f l u c t u a t i o n s i s decreased, the disturbances must be bigger to account f o r the observed hot spots. The e f f e c t of assuming a p a r t i c u l a r a c t i v a t i o n energy i s thus i n s i g n i f i c a n t f o r the i n t e r p o l a t i v e aspect of Equation (15). For e x t r a p o l a t i o n to other r e a c t i o n s , however, there w i l l be an e r r o r i n the estimated c r i t i c a l temperature r i s e which i s p r o p o r t i o n a l to and opposite i n d i r e c t i o n to the e r r o r i n the assumed a c t i v a t i o n energy of h y d r o t r e a t i n g . Reaction Order The assumption that the r e a c t i o n i s of zero order i s reasonable f o r a s t a b i l i t y a n a l y s i s because of i t s conservatism. Extension of Equation (15) to other systems, however, may require c o n s i d e r a t i o n of other k i n e t i c s . We show how t h i s can be done f o r a f i r s t - o r d e r r e a c t i o n . For other orders, the procedure i s e q u i v a l e n t , and with about the same degree of complexity. The c o n d i t i o n f o r s t a b i l i t y of a d i s t u r b e d region i s that a s o l u t i o n e x i s t f o r the system of equations: Λ , ν~ψ =

2

GSDPR , ,v 5 — exp (ψ) χ L 2

ψ = τ at r = 1

2

2

(

V GDPR V~x = — r — IT

m

exp (ψ)

2

6

)

X

r

χ = x

Q

at r - 1

The n o t a t i o n i s the same as before, with ψ representing the v a r i a b l e dimensionless temperature w i t h i n the disturbance, χ r e p r e s e n t i n g composition v a r i a b l e . The r e a c t i o n i s f i r s t order and the Peclet numbers f o r heat and mass t r a n s f e r are assumed to be equal. x represents the composition i n the normal r e g i o n , equal to 1 - (τ/S) with τ and S defined as before. The d i f f e r e n t i a l equations of (26) can be combined to give Q

+ S^x r

r

= 0

(27)

r

from which i t follows that ψ + Sx = S

(28)

i f c y l i n d r i c a l symmetry holds w i t h i n the disturbance. Combination of Equation (28) with the f i r s t equation of (24)


78


gives

ο ^

GS2|R_

=

(

ψ

£

)

(

2

9

)

A c h a r a c t e r i s t i c of runaway r e a c t i o n s i s that they s t a r t with the dimensionless temperature, ψ, c l o s e to u n i t y . S i s t y p i c a l l y 15 to 20 or higher. Hence, (ψ/S) « 1, and 1 (ψ /S) » exp (-ψ/S), to a s u f f i c i e n t degree of approximation. Then, 2


ν

ψ=

GSDPR_

exp

[ ψ

( 1

_

1 / s ) ]

2 [1 - (1/S)] =

with ψ (1 - 1/S)

G

S

D

P

R

(

* " L

1 / S )

= τ (1 - 1/S)

· exp

[ψ (1 - 1/S)]

(30)

at r = 1.

Equation (30) i s i d e n t i c a l i n form with the steady state of Equation (6), hence i t has e x a c t l y the same c o n d i t i o n f o r stability, 2

GSDPR

(1 -

1/S) < 2

L

2

exp

[-τ

(1 -

(31)

1/S)]

For a f i r s t - o r d e r a d i a b a t i c r e a c t i o n i n the normal zone, with plug flow,

4

1

= SD exp

(τ) χ

(32)

and χ = 1 - τ/S Together

(33)

these give dx g = SDe

T

(1 - τ/S)

(34)

[ E i ( S ) - E i (S - τ)]

(35)

whose s o l u t i o n i s SD = Se"

S

where E i ( - ) i s the Exponential I n t e g r a l Function. and τ < S , Ei(S)

« exp

For large S,

(S)/S

E i ( S - r ) « exp ( S - T ) / ( S - T ) which together give


4.


SD « 1


exp

* 1 - exp

(-τ)/(1 [-τ

(1 -

79

Reactors

τ/S)

1/S)]

(36)

S u b s t i t u t i o n of Equation (36) i n t o (31) gives


(37)

This i s of e x a c t l y the same form as the s t a b i l i t y c r i t e r i o n f o r a zero-order r e a c t i o n , except that G and τ are both m u l t i p l i e d by the f a c t o r [1 - (1/S)], which i s not g r e a t l y d i f f e r e n t from unity. The ultimate form of the c r i t e r i o n by which the s t a b i l i t y of an a d i a b a t i c f i r s t - o r d e r r e a c t i o n can be judged i s i d e n t i c a l with Equation (19) except that R T /E i s replaced by RT /E[l-(1/S)]. In t h i s respect, a f i r s t - o r d e r r e a c t i o n behaves l i k e a zero-order r e a c t i o n with a s l i g h t l y - l o w e r a c t i v a t i o n energy. With these comments, we may c o n f i d e n t l y conclude that none of the assumptions used i n d e r i v i n g our model f o r hot spot development has a s i g n i f i c a n t e f f e c t on the s t a b i l i t y conditions deduced from that model. 2

Q

2

Some Ideas That Did Not Work During the course of the development described above, we t r i e d a number of a l t e r n a t i v e models, none of which l e d to a s a t i s f a c t o r y correspondence between theory and observations. These are l i s t e d below, with b r i e f comments. Liquid-phase flow does not c o r r e l a t e i n any way with hot spot development. A l l our attempts to f i n d such a c o r r e l a t i o n l e d only to random s c a t t e r . This i s c o n s i s t e n t with the hypothesis that there i s a minimum flow necessary to d i s t r i b u t e the l i q u i d over the cross s e c t i o n of the r e a c t o r , and that above the minimum the e f f e c t i s only one of holdup. A l l the observed data appear to be above t h i s minimum. Functions of gas flow other than p F do not c o r r e l a t e the data. In p a r t i c u l a r , we t r i e d to w r i t e G as a f u n c t i o n of p F and F, s e p a r a t e l y , and to use i n l e t flows. We found no way of d e v i s i n g a separable map l i k e Figure 3 with these r e l a t i o n s . Attempts to solve Equation (6) n u m e r i c a l l y , using the e x i s t i n g "software" of high-powered computers, are u s e l e s s . S o l u t i o n s can be r e a d i l y obtained, assuming a v a r i e t y of i n i t i a l conditions and disturbance p a t t e r n s , but they cannot be f i t t e d i n t o the patterns of Tables I I and I I I , without the i n t r o d u c t i o n of "fudge f a c t o r s . " The essence of our d e v e l opment i s that i t i s necessary to introduce such f a c t o r s , l i k e the parameters η, K , and Κ of Equation (9), to e x p l a i n the 2

1

2


80


observations, f o r instance; the apparent inverse c o r r e l a t i o n of allowable temperature r i s e and bed h e i g h t . The computers confirm t h i s , but give no deeper i n s i g h t i n t o the nature of the phenomena.


Concluding

Comments

We have used a rather crude d e s c r i p t i o n of two-phase flow through packing, invoking imprecise notions l i k e "regions of disturbance," s o l v i n g boundary-value problems with no d e f i n a b l e boundary, using extremely s i m p l i f i e d d e s c r i p t i o n s of the r e a c t i o n s that take place, and yet from t h i s we have devised what appears to be an accurate r u l e f o r p r e d i c t i n g the onset of runaway exotherms i n t r i c k l e beds, given by Equation (15). I t i s not that u n c e r t a i n t i e s and assumptions have disappeared or have been hidden, but rather that they have been d i r e c t e d i n t o estimates of q u a n t i t i e s whose values are unimportant i n defining s t a b i l i t y limits. The important q u a n t i t i e s are s t r o n g l y r e l a t e d to d i r e c t measurement and only weakly to u n c e r t a i n i n f e r e n c e s . The s t a b i l i t y r u l e i s intended to apply to a l l exothermic r e a c t i o n s i n t r i c k l e - b e d r e a c t o r s , i n p a r t i c u l a r to hydrogénation of petroleum f r a c t i o n s , the primary data source. Nomenclature A,Β C γ α D Ε f F G H k k g

κ 1 L η Ρ Ρ r R ρ S SD

Adjustable constants Heat c a p a c i t y E/RT P a r t i c l e diameter kV/F A c t i v a t i o n energy Symbol f o r f u n c t i o n gas v e l o c i t y at the r e a c t o r o u t l e t , f t / h r Disturbance r a t i o , SD /SD Heat of r e a c t i o n S p e c i f i c rate constant at i n l e t - k exp (S p e c i f i c rate constant i = 1,2, . . . Adjustable constants E f f e c t i v e thermal c o n d u c t i v i t y Length or height v a r i a b l e Bed height, f t Exponent i n r e l a t i o n of R to L Peclet number Fd/κ FL/κ Radial variable Radius of disturbance, gas constant Gas d e n s i t y , l b / f t EH/RT C SD i n disturbed region 2

E/RT)

2

3

2

Q


4.


Τ


Stability of Trickle-Bed Reactors

Temperature I n l e t temperature Temperature r i s e Maximum safe temperature r i s e γΔΤ C r i t i c a l value of τ Bed volume Conversion or composition v a r i a b l e F r a c t i o n of height, 1/L Symbol f o r τ i n a disturbance Peak value of ψ

Literature Cited 1. 2. 3.

Frank-Kamenetskii, D. Α., "Diffusion and Heat Transfer in Chemical Kinetics;" Plenum Press, 1969. Beek, J., "Design of Packed Catalytic Reactors;" Advances in Chemical Engineering, Vol. 3, p. 231, Academic Press, 1962. Beuther, Η., and Β. K. Schmid, "Reaction Mechanisms and Rates in Residue Hydrodesulphurization;" Proc. 6th World Petroleum Congress, Section III, 1964, p. 297.

RECEIVED July 19, 1983


81

Stability of Trickle-Bed Reactors - ACS Symposium Series (ACS

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