An economic model of a reactor containing decaying catalyst pellets

Engineering

Costs and Production Economics,

8 (1984)

33

3 3 -43

Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

AN ECONOMIC MODEL OF A REACTOR CONTAINING DECAYING CATALYST PELLETS Hongju Chang, Kenneth Smith

Kline and French Laboratories,

Kamholz

Swedeland,

PA 19479

(U.S.A.)

and Robert D. Tanner Vanderbilt

University,

Nashville,

TN 37235

(U.S.A.)

ABSTRACT

The best economic operation policy for a continuous stirred tank reactor with a batch of decaying solid catalyst pellets has been studied. The effect of operation--regeneration cycles, and the annual effect on a reactor’s profit was illustrated. The results were compared to the case of non-decaying catalyst with pellet size as a parameter, and showed

INTRODUCTION

In operating a catalytic reactor the optimal production rate is usually determined by maximizing the profitability. Under ideal situations (i.e. no catalyst deactivation), this optimal production rate can be achieved with a certain feed rate. However, catalyst deactivation is always a major concern, especially over a long operating period. In this case, the determination of an optimal operation policy becomes more complicated, and the maximal profitability is often achieved by varying either or both of the feed rate and the production time between catalyst regenerations, as well as the temperature. In heterogeneous catalysis, catalyst deactivation can be due to sintering, poisoning (reversible or irreversible) and/or fouling. 0167-188X/84/$03.00

0 1984 Elsevier Science Publishers B.V.

that the reactor’s profit decreased linearly with increasing pellet size in both cases. This study revealed that although a larger pellet catalyst has more stable activity as cited in the literature in the case of deactivation, it offers no economic advantage due to its lower effective reaction rate.

Sin tering is a phenomenon of structure change of the catalyst due to exposure to high temperatures. Poisoning and fouling result from the blocking of active catalytic sites by impurities from the feed or undesirable products of side reactions. The processes and rate equations for decaying reactions have been described by Chang et al. [ 1 I for hydrotreatment, and by Levenspiel 121 for general cases. Following their terminology, these decaying processes can be categorized as parallel, series, side-by-side and independent deactivation. If the poison is a result of a side reaction from a reactant, it is parallel deactivation; if the poison forms from further reaction of the product, it is series deactivation; if the poison is a result of impurities in the feedstock, it is side-byside deactivation: and if the deactivation is

34 non-concentration related, then it is independent fouling. Masamune and Smith [ 31 and Lee and Butt [4] have analyzed theoretically the deactivation of pellet catalysts in which diffusional resistance is significant. These authors have concluded that a catalyst with higher diffusional resistance has a slower decaying rate and thus is more stable than the one with lower resistance. Therefore, in addition to the optimal production rate, in the case of heterogeneous catalyst deactivation there arise important questions concerning the most protitable operation of a reactor containing such catalysts: (1) how to best operate the reactor during a run; (2) when to replace or regenerate the catalyst; and (3) how to select the best intraparticle diffusional resistance. Since adjusting temperature is one of the most common ways to compensate for catalyst activity decay, the best temperature policy has been investigated by several authors (Chou et al. [ 53, Ogunye and Ray [63, Crowe [ 71, Park and Levenspiel [8] ). The finding of the best operation-regeneration cycles have also been studied (Weekman [9] , Park and Levenspiel [ 81, Smith and Dresser [ lo], Walton [ 111, Kramers and Westerterp [ 121, Miertschin and Jackson [ 1311. In a pellet (sometimes called “heterogeneous”) catalyst, its deactivation is strongly dependent on the intraparticle diffusional resistance. As a result of the changing rate of reaction, the operation-regeneration cycle and the plant economics are also affected by that resistance. To date, however, this diffusional resistance has not received much attention. Polinski et al. [ 141 and Stiegel et al. [ 151 have recently demonstrated that, during catalytic coal liquefaction, the use of larger diameter catalysts tends to convert more coal to liquid products per unit mass of catalyst than does the use of smaller diameters. In the modeling aspect of this paper, a

shell (also called a shrinking core) reactiondiffusion model [3] is employed. That model includes both catalyst activity decay as a function of time and intraparticle diffusional resistance for a spherical catalyst. An economic criterion (maximal profit) is applied to select the best operation-regeneration cycles and optimal production rate. The effects of catalyst pellet size (related to diffusional resistance) on the operation policy and a reactor’s profit are highlighted, and these are compared to the cases where the catalyst does not decay and is not affected by the diffusional resistance. Prior to this study, the effect of pellet size has not been considered in this solid catalyst decay problem. ECONOMIC OBJECTIVE A major object of a firm is to maximize its profit. A firm’s profit on an annual basis can be expressed as II = (Tr) - (Tc)

(1)

In a perfect competition market, we may assume that a firm’s output constitutes such a small part of the total market supply that any single firm has no influence on the product’s market price [ 161. In this case, a firm’s total income, TI, can simply be calculated from the equation (Tr) = QP

(2)

where the price, P, is a constant in a perfect market, and Q is the total product production per year. The total annual production cost, TC, generally consists of two components: fixed, FC, and variable, VC, costs. Details of these costs have been discussed by Peters and Timmerhaus [ 171. An example of determining production cost for a continuous stirred tank reactor has been illustrated by Wei et al. [ 161, where it is assumed that the dominant cost of running the chemical plant is that associated with

35 the operation of the principal chemical reactor. For convenience and comparison purposes, the same cost function will be used in this report suitably modified for catalytic activity decay: (3)

(Tc) = (Fc) + (VO

where the fixed cost, FC, is related to the rate. It equipment size, not the production can be expressed as (4)

(Fc)=Pp

where p is a constant and V is the reactor size. factor ranging from 7 is the size component 0.5 to 1 .O [ 181. The variable cost, VC, includes the raw material, RC, product proCC, costs. The cessing, PC, and catalyst, catalyst cost is included in the variable cost since decaying catalysts need to be regenerated or replaced frequently in order to yield a maximal profit. The variable cost is, therefore,

and the total cost function

dQ

dQ

=

(10)

CckA 77cA

(11)

where n is an effectiveness factor accounting for diffusional resistance and activity decay which is a function of reaction time. For a continuous stirred tank reactor, as sketched below,

Therefore, since the marginal cost, MC, is similarly defined, at the optimum, the marginal cost equals the selling price:

WV = p dQ

the component mass balances can be expressed as follows for a fixed total volumetric flow rate, W: A q CA v= (dCA /dt) v( 12)

WCAF - WC, - C,k,

(8)

WC) dQ

rA

for reactant

o

From eqn. (2), the marginal income, MI, is defined and equated to the product market price in a perfectly competitive market:

(MC)=------=-

Catalyst A-D

(6)

To maximize a tirm’s profit with respect to the production rate, the following necessary condition developed from eqn. (1) must be met:

dQ

Following the example by Wei et al. [ 161, suppose we have a reaction converting A to D using a pellet catalyst; the bulk rate equation can be written as

is

(Tc) = (r;c) + CRC)+ (PC)+ (CO

d(TC)

FORMULATION

(5)

(vc)=(Rc)+(Pc)+(Cc)

dn _=__-=d(T0

The last equation means that, in a perfect competition market, a firm can receive its maximal profit if its output is such that the marginal cost is equal to the product market price.

(9)

and for the product -WC0

D

+ cck_,j 7) CA v = (dco /dt)V

(13)

In deriving eqns. (12) and (13), volume change due to the reaction was assumed to be negligible. The terms on the right side of both equations account for the concentration change of A and D with time due to catalyst deactivation. For simplicity, we further assume that rates of concentration change inside the reactor are slow compared to the rate of the reaction. The devia-

36

tion from steady-state is small in short time period. Solving for CA and CD, we get: CA =

c

=

D

wcAF

c&A

341

7) VCA

(15)

W

=

D

n=h2(1

3a2 + 1)

- 49)

(18)

-a)

1

CckA’?CAFV

+i =

w+-CckAqv

h If catalyst poison precursors come from the feedstock, and the diffusion rates of the reactants are slow with respect to the reaction, then a shell poisoning model is a good approximation. For parallel fouling, the case considered here, the effectiveness factor as a function of both time and the poisonedunpoisoned boundary position has been studied by Masamune and Smith [3]. From their work, for the reactions: D (desired product) (16) P (poison

inside a spherical below,

catalyst

1 +(hacoth(ha)-

,g =_k/wCA 40

Catalyst pellet equations

AG

(17)

where

or c

h cash (h) - sinh (h)

+h f (2LY3-

(14)

w+cck,jqv

ha! cash (/KY) - sinh (ha)

0 =ln

deposit) pellet,

the equations resulting from in terms of dimensionless

as sketched

a shell model, variables, are

=R&-i?%

t

l)(l

-cz)

(19)

(20) (21)

In the derivation, a spherical model shape was selected because this model is also a good approximation for other types of commonly used catalyst pellets, such as powders and cylindrical extrudates. In eqn. (19), OLis the dimensionless radius of the unpoisoned inner core of the sphere defining the interface between the poisoned outer shell and the clean inner core (the primary parameter of the shrinking core model); and Qi is the dimensionless concentration at the interface, (Y. The effectiveness factor, n, defined as the ratio of overall reaction rate under poisoned conditions to the reaction rate without diffusional resistance at non-poisoned conditions, is to account for catalyst activity decay and diffusional resistance. This effectiveness factor vs. dimensionless time is presented in Fig. 1. Note that the Thiele modulus, h, is a parameter to account for the relative diffusional resistance, and is expressed in terms of catalyst pellet size and effective diffusivity inside a pellet, D, (see eqn. 21). Since the effective diffusivity is related to the catalyst pore structure which is difficult to change after being manufactured, the easiest way to change a catalyst diffusional resistance, and thus Thiele modulus, is to vary its pellet

37 size. Figure 1 clearly shows that although a catalyst with larger pellet size (thus larger Thiele modulus and larger diffusional resistance) has a lower initial activity, it tends to have a slower decay rate, and therefore, a longer total life.

Production

function

The total production of D, in moles, from a CSTR reactor with a batch of catalyst in a period of time, tp, can now be calculated using the following equation:

&,=sp

(22)

WCDdt

0

from eqn. (15)

Qp = CckA CAF VW J*’ 0

(23)

Since n is not an explicit function of time, referring to eqns. (17)-(21), an iterative method must be used to obtain (Y at a corresponding time, t. The time required for catalyst regeneration, tR , may be considered as a fixed time, tRF, plus variable time, tRV, which iS a fUrKtion of the degree of poisoning, that is:

0.04 -

0.1

’ dt W+CckA vrl

02

0.4

0.6

0.6

1.0

2

4

6

8

DIMENSIONLESS TIME. 8

Fig. 1. Effectiveness factor versus time on stream for parallel deactivation mechanism using a Shell (Shrinking Core) model.

tR

(24)

+ tRV(a)

Since (Yis related to the time of a production cycle, tp , eqn. (24) can also be expressed as: tR

In addition to the Thiele modulus, the ratio of kAF/q,, is also an important factor affecting the rate of deactivation. This ratio is expressed in Fig. 1 through the dimensionless time variable, 8, as shown in eqn. (20). From eqn. (20), one can see that for a fixed real time, t, when kAF/qo decreases, dimensionless variable, 8, also decreases proportionally, and from Fig. 1, the effectiveness Physically decreasing factor, 77, increases. kAF/qo means that either the poisoning rate constant, kAF, decreases, or the amount of available sites on the catalyst surface, qo, increases (i.e., the poison tolerance capacity increases). Therefore, a low value of kAF/qo can give a low deactivation rate, and at extremely low values, (kAF/qo * 0) the catalyst deactivation is negligible.

= tRF

= tRF

(25)

+ tRV(tp)

Now, the total time required for an operation cycle can be calculated as the sum of the process time and the catalyst regeneration time: t,=tp+tR

=tp

+tRF+tRV(tp)

(26)

and the total number of operation cycles per year becomes the ratio of available time to one cycle time: N = t, /to The total production

(27) of D per year is:

Q=Q,N and the total feed of A to the reactor is:

f = WC,&N

(28) per year

(2%

38 Total variable cost can now be calculated from eqns. (28), (29), and (5) as the sum of the cost of A, the product processing costs, and the catalyst costs: (vc)=fuR

+Qu,

+(cc)

(30)

Here, UR and UP are fixed, but, in general, they are a function of quantity, Q [ 161. Catalyst regeneration or replacement cost can be considered as fixed cost proportional to the amount of catalyst used plus a variable cost proportional to the variable regeneration time. Thus, (Cc) = C,vu, Combining yields

+ c’t,, eqns.

(TC)=bVr+fUR

(l)-(6),

(31) (30)

and

(3 l),

+QUp+CcVUc+C’tRV (32)

and n=PQm-(TC)

= 45 P = 1.0 Y C’ = 0 and tRv = 0 mean that the catalyst is replaced or regenerated off line after each production period. Note that in the heterogeneous catalysis example here, the term CckA 77 is used in place of the first order reaction rate constant for the homogeneous reaction case. The new grouping is also in units of reciprocal minutes. In the case of negligible diffusional resistance (i.e. h < l), the initial effectiveness factor, no, which can be calculated using the limiting case of eqn. (18) as shown by Smith [ 191, is close to unity. Thus in the case of negligible diffusional resistance, CckAqo in this example is equal to 0.005 min-’ which is the value used by Wei et al. [ 161. The fixed costs, FC, are assumed to be 45 V (0~45 and r= 1 .O) for the illustration purposes [ 161. Other varied parameters, h and kAp/qO, will be discussed later.

(33) DISCUSSION

EXAMPLE Methods of Search The example previously developed by Wei et al. [ 161 is used, except that a batch of heterogeneous catalyst is used in the well stirred tank reactor and the catalyst is decaying according to the parallel mechanism shown earlier in eqn. ( 16). We use the same constants as those used by Wei et al. [ 161 whenever applicable. These are V = 30,000 1 = 0.2 mole/l CAF = 0.1 kg/mole MD = 504,000 min/year tY = 0.1 kg/l Cc = 0.05 l/min/kg-catalyst kA = $0.1 S/mole UR = $O.O5/mole u, = $20/kg U, P = $lO/kg C’ =o = 1440 min tRF

To maximize the reactor’s profit, two variables must be considered, namely the production time, tp , and the feed rate, W. The maximal profit can be determined by using either a numerical iterative method or a graphical technique. The graphical search method is used in this report to better picture how the profit is affected by the operation cycle and the feed rate. This one-dimensional search method can be summarized as follows: (1) At a fixed feed rate, W, vary production cycle time (the inverse regeneration frequency, N, given on the abscissa of Fig. 2), tp , to get the most profitable cycle time (local maximum), namely tp,m . This results in a constant feed rate line in Fig. 2; (2) change feed rate, W, to a new level and repeat the search of a new tp,m ; (3) plot the profit at tp,,,, for each feed rate to get a trajectory of local maxima, and a global maximal profit

39 should then appear on the graph. This is illustrated in Fig. 2: at W = 150 l/min and tp = 120,000 min (N = 4 cycles/yr), II = $2.1 X 106/yr.

k,,/qo

= 10m5L/mole/mm

(23)~-(28). Therefore, in plotting Fig. 3, the annual output, Q, which is often more useful, is used instead of the feed rate, W. However, several feed rates are indicated on the figure for illustration. Note that the catalyst pellet size is used in Fig. 3, in place of h, since the Thiele modulus is directly proportional to the pellet size and is related to diffusional resistance. Also note that the catalyst deactivation rate is determined by the parameter kAF/qo. In the example described earlier, a value of lo-’ l/mole/min for kAF/qo gives a moderate decay rate; in contrast, a value of 1O-* l/mole/min gives negligible deactivation within a finite time period.

w

I 0

I 10

5

REGENERATION

15 FREQUENCY.

20

25

CASE 1 R = 0.05 mm. k,,lq, = 10BL/moklmr NON-DIFFUSIONALLY LIMITED AND NON--DECAYING CASE 2 R = 0.06 mm, k.,/q, = 10~5L/moklmin NON-DIFFUSIONALLY LIMITED, BUT DECAYING CASE 3: R = 1 0 mm. k,,/qo = 105L/molelmm DIFFUSIONALLY LIMITED AND DECAYING

3 B ;

1

z t+ 9 d

B B ;

;

I I

30

CYCLESIYR

0

LOCAL MAXIMA

Fig. 2. Graphical search for the locally and globally maximal profits.

Figure 2 shows that for a constant feed rate there is an optimal regeneration frequency which gives the maximal profit. This is because more frequent regeneration can increase average catalyst activity and thus the productivity and profitability. However, too frequent regeneration can cause time loss in production and increase total catalyst cost, which in turn can result in profit loss. This figure also shows that the profit is not only a function of the regeneration frequency, but of the feed rate as well. Production

PRODUCTION(O).

lO%g/yr

Fig. 3. Profit, income and cost versus production rate for three cases: nondiffusionally limited and non-decaying; nondiffusionally limited but decaying; and diffusionally limited and decaying.

cycle

For each feed rate operated at an optimal regeneration/replacement frequency, a unique quantity of output is obtained from eqns.

An interesting result is revealed in Fig. 3. In the base case of using non-diffusionally non-decaying catalyst (Case 1: affected, R = 0.05 mm and kAF/qo = 10e8 l/mole/min),

40 the optimal feed rate is 180 l/min yielding 0.83 X lo6 kg of product annually; however, if the catalyst is decaying gradually with time without diffusional limitation (case 2: R = 0.05 mm and kAF/qo = lo-’ l/mole/min), the same feed rate can produce only 0.76 X lo6 kg per year. Furthermore, if with diffusionally affected and decaying catalyst (Case 3: R = 1 mm and kAF/q,, = 10m5 l/mole/min), the same feed rate (180 I/min) can produce even less product (0.66 X lo6 kg/yr). In order to meet the firm’s goal of maximal profit, the feed rate has to be increased to 200 l/min in Case 2 and decreased to 150 l/min in Case 3; these feed rates yield 0.80 X 10” and 0.62 X IO6 kg/yr of product, respectively, in Cases 2 and 3. The corresponding values for both total annual production costs are also shown in Fig. 3. Effect of catalyst diameter

The relative diffusional resistance can be expressed by the Thiele modulus as defined

in eqn. (21): the larger the Thield modulus, the lower the catalyst activity, and hence, the greater the diffusional resistance. For a given catalyst the larger pellet size gives a larger Thiele modulus, thus it should give a more stable activity (decrease of the effectiveness factor, n, is slower with respect to time), as shown in Fig. 1. Because of its more stable activity, this larger size catalyst can give more output through its whole life, as reported by Stiegel et al. [ 151 and discussed earlier. However, as shown in Fig. 4, due to catalyst deactivation, after a certain time on stream the production rate of a catalyst saturates at such a low level that the catalyst becomes practically unusable after that time. This is especially serious for the larger size catalyst, since its activity is lower initially and remains lower during most of its useful lifetime. Extending the same approach to the protitto-catalyst size relationship, one would expect that a smaller pellet catalyst can give a greater profit than does the larger pellet

1.4

1.2 -

P 1.0 -

k&q,,

= 10m5Llmolelmin

5 5 2

0.8 -

TIME

Fig. 4. One cycle production

ON STREAM,

DAYS

versus cycle length for three sizes of catalysts.

41 catalyst. This follows because the time required for a smaller size catalyst to produce a certain amount of product is shorter than for the larger one. This is illustrated in Fig. 5, which compares total annual costs for three different sized catalysts in the cases of both decaying and non-decaying situations. In both cases, smaller size catalysts always give lower production costs which, in turn, give higher profitability.

negligible deactivation (~AF/CJ~ = 1O-a l/mole/ min, dashed lines in Fig. 5), a=$3SX

and

106/yrandb=$1.17X

in the

(kAF/qo Fig. 5),

=

106/yr/mm

case of significant deactivation 10m5 l/mole/min, solid lines in

a = $3.1 X 106/yr and b = $1.03 X 106/yr/mm.

4

-

1

-

16 -

-

DECAYING CATALYST k,,/q,, = 10~5L/mole/min

-‘-

NON-DECAYING CATALYST k,,lq, = 10~8L/molelm~n

14 s. “0 z

5

lo-

i E

Fig. 6. Effects

0

0.2

0.4 ANNUAL

Fig. 5. Effects profit.

06 PRODUCTION,

of the catalyst

0.8

1.0

1.2

Q. lO’kg/yr

pellet size on the total cost and

In order to show the relationship between the reactor profit, II, and the catalyst pellet size, R, a regressional analysis was performed to fit a polynomial function. As Fig. 6 shows, the results can well be represented by the following linear relationship: n=a-bR

for 0.05 mm < R < 2 mm

in which a and b are constants.

(34)

In the case of

of the catalyst

pellet size on the reactor

profit.

The linear relationship in eqn. (34) means that the reactor would generate maximal profits when the catalyst pellet sizes approach a minimum. An increase of pellet size results in a decrease of reactor profits. If the pellet sizes are further increased to about 3 mm, the catalyst’s effective activity is so low that practically no profit can be expected from the operation of the reactor in this example. SUMMARY

By using an economic criterion (maximal profit), the operating policy of using a batch of decaying pellet catalyst of varying size has been studied. A shell (shrinking core) model has been used to describe the diffusionreaction-deactivation of the pellet catalyst.

42

A graphical method of searching for the maximal profit has been demonstrated. Prior to this study, the effect of pellet size has not been considered in this solid catalyst decay problem. Results of this study show that increasing pellet size decreases the reactor’s profit according to the following linear relationship: n=a-bR

(34)

Although recent literature reports that the larger pellet catalyst has a lower decay rate, eqn. (34), however, reveals that it offers no economic advantage over the smaller size whenever mechanical catalyst. Therefore, problems (e.g. excess pressure drop, agitation, or catalyst filtering) are not a major concern, catalyst size should be reduced to the point where diffusional resistance is not significant. Although only a stirred tank reactor is illustrated in this paper, the strategy of searching for the maximal profit and the conclusion about catalyst sizes would be expected to carry over to other types of reactors, such as batch and fixed bed reactors.

(MO MD N

P (PO fi 40

R rA ri

WC) UC) (Tr> tP

tp,m tR tRF

NOTATION tRV

A (Cc) CA cc CD CAF

C’

D D,

f V'c> h k/i

(MO

reactant catalyst cost $/yr cont. of reactant A, moles/l cont. of catalyst in CSTR, kg/l cont. of D, moles/l cont. of feed, moles/l constant in eqn. (3 1) product effective diffusivity in catalyst pores, mm2/min total feed rate, moles/yr fixed cost, $/yr Thiele modulus, defined in eqn. (2 1) main intrinsic rate constant, l/min/kgcatalyst intrinsic fouling rate constant, l/min/ kg-catalyst marginal cost, $/kg

marginal income, $/kg molecular weight of product D number of production-regeneration cycles per year market price of D, $/kg Product processing cost, $/yr total produc production, moles/yr total product production per operation cycle, moles maximum active sites available on the catalyst surface, moles/kg-catalyst radius of the spherical catalyst pellet, mm rate of reaction of A, moles/l-min radius of unpoisoned catalyst core within the spherical catalyst. It is the boundary between poisoned outer shell and unpoisoned core, mm total raw material cost per year, $/yr total annual cost, $/yr total annual income, $/yr time of a production cycle, min time of an optimal production cycle at a certain feed rate, min time of a regeneration cycle, min fixed time required for a regeneration, min variable time required for a regeneration, min total time required for one production-regeneration cycle, min total operation time available per year, min unit cost of catalyst, $/kg unit cost of product processing, $/mole of D unit cost of raw material, $/mole of A reactor volume, 1 variable cost, $/yr feed rate, l/min profit of firm or reaction process, $/yr dimensionless time defined in eqn. (20) dimensionless radius, ri/R constant, defined in eqn. (4)

43

Y @i

77 P

size exponent factor, defined in eqn. (4) dimensionless cont. at (Y,CAi/CA effectiveness factor, defined in eqn. (18) catalyst density, kg/l

9 10 11 12

13

REFERENCES Chang, H., Seapan, M. and Crynes, B.L., 1982. In: J. Wei and C. Georgakis (Ed%), Chemical Reaction Engineering-Boston, ACS, Washington, D.C., p. 309. Levenspiel, O., 1972. Chemical Reaction Engineering. Wiley, New York, p. 531. Masamune, S. and Smith, J.M., 1966. AIChE J., 12(2): 385. Lee, J.W. and Butt, J.B., 1973. Chem. Eng. J., 6: 111. Chou, A., Ray, W.H. and Ans, R., 1967. Trans. Inst. Chem. Eng., 45: T153. Ogunye, A.F. and Ray, W.H., 1968. Trans. Inst. Chem. Eng., 46: T225. Crowe, C.M., 1970. Can. J. Chem. Eng., 48: 576. Park, J.Y. and Levenspiel, O., 1976. Ind. Eng. Chem. Process. Des. Dev., 15(4): 534.

14 15 16

17

18 19

Weekman, V.W. Jr., 1968. Ind. Eng. Chem. Process Des. Dev., 7: 252. Smith, R.B. and Dresser, J., 1957. Petrol Refining, 36: 199. Walton, P.R., 1961. Chem. Eng. Prog., 57: 42. Kramers, H. and Westerterp, K.P., 1963. Elements of Chemical Reactor Design and Operation. Academic Press, New York. Miertschin, G.N. and Jackson, R., 1970. Can. J. Chem. Eng., 48: 702. PoIinski, L.M., Stiegel, G.J. and Saroff, L., 1981. Ind. Eng. Chem. Process Des. Dev., 20: 470. Stiegel, G.J., Polinski, L.M. and Tischer, R.E., 1982. Ind. Eng. Chem. Process Des. Dev., 21: 477. Wei, J., Russell, T.W.F. and Swartzlander, M.W., 1979. The Structure of the Chemical Processing Industries. McGraw HiIl, New York, p. 51. Peters, M.S. and Timmerhaus, K.D., 1968. Plant Design and Economics forThemical Engineering. 2nd Edition. McGraw HilI, Chap. 5. Guthrie, K.M., 1969. Chemical Engineering, March 24: 114. Smith, J.M., 1970. Chemical Reaction Engineering. McGraw Hill, New York, p. 428.

(Received

July 5, 1983; accepted

October

28,

1983)