written in four days-an experienced programmer should be able to improve on this. T h e program for three-stage optimization was about 900 words in length and could be accommodated within the high speed memory. Calculation times were approximately 30 minutes for one stage, 100 minutes for two stages, and 300 minutes for the complete three-stage problem. These times would be greatly reduced on an up-to-date machine Nomenclature
A
= cross-sectional area of bed
Ai
= ith chemical species
c,
=
f,, G g g,, gn’ g, gio
= maximum value of P, = total mass flow rate
Hi AH J
bH. zgi 2, specific heat of reaction mixture
bT
c,
= total number of beds = partial pressure of Ai
R
= reaction rate, rate of change of g per unit weight
p,,
= total net profit from n stages
T
=
To
= = =
t f.\!
= inlet stream temperature to bed n = outlet stream temperature from bed n
preheating temperature value of one mole of A i weight of catalyst per unit length of be3 weight of catalyst in bed n = distance from inlet = = = =
GREEKLETTERS = stoichiometric coefficient of Ai P = zaivi re = equilibrium curve = optimal inlet conditions for bed n I?, rn’ = optimal outlet conditions for bed n = amortized cost of a catalyst bed 8 = r/P 0% = holding time of bed n = fraction of total flow through bed n A, ai
Z
= wC,J/R = g*/t, =&l/t;+l = summation
w
= cost of unit amount of heat
Literature Cited
k , , k p = reaction rate constants
N
W, x
Pn
-AH
pi
ze,
u
= conversion or extent of reaction = conversion in inlet stream to bed n = conversion in outlet stream from bed n = number of moles of Ai per unit mass of mixture = basic feed composition = specific partial enthalpy of A , = ZcuiHi, heat of reaction
=--
t, t,‘ t , ~ Vi
of
catalyst temperature basic feed temperature new temperature variable or reduced temperature maximum permitted temperature
(1) Aris, R., “Optimal Design of Chemical Reactors,” Academic Press, New York, 1961. (2) Bellman, R., “Dynamic Programming,” Princeton University Press. Princeton, 1957. (3) Calderbank, P. H., Chem. Eng. Proqr. 49, 585 (1953). (4) Horn, F., Kuchler, L.. Chem. Z g . Tech. 31, 1 (1959). (5) Hougen, 0. -4., Watson, K. M., “Chemical Process Principles,” Part I, p. 255, Wiley, New York, 1954. (6) Kirk, R. E.. Othmer, D. F., “Encyclopedia of Chemical Technology,” Vol. 13, p. 488, Interscience, New York, 1954. (7) Krabetz, R., Bakemeier, H., Chem. Zngr. Tech. 34, 1 (1962).
RECEIVED for review November 26, 1962 ACCEPTEDMarch 13, 1963
CONTROL OF A COLD S H O T ADIABATIC BED REACTOR WITH A DECAYING C A T A L Y S T K U N G - Y O U LEE A N D
R U T H E R F O R D A R l S
Department of Chemical Engineering, liniumity of ‘tlinnesotn, Minneapolis 74,Minn.
As the activity of the catalyst in a reactor decays, the operating conditions should be changed to maintain the most productive operation. It i s shown how this can be done for a multibed adiabatic reactor and a specific example of a two-bed reactor for the oxidation of sulfur dioxide i s given. In general it i s necessary to judge the activity b y the behavior of the reactor; this may be done b y temperature measurements on the reactor
beds. HE deactivation of catalyst. a phenomenon of widespread Timportance in the chemical industry, falls under three heads. There is, first, the deactivation of catalyst by some fouling reaction which goeion simultaneously or consecutively with the main process reaction (2, 3 ) . Such fouling can be very rapid-for example, with catalytic cracking--and be accompanied by a change i n the pattern of the main kinetics. Secondly, there is deactivation of a pzrmanent sort due to the presence of trace poisons in the process stream which gradually block out the active centers of catalysis. This is a relatively slower process and may involve on!): a reduction of the reaction
306
l & E C PROCESS D E S I G N A N D DEVELOPMENT
rate without any change in the form of the kinetics (4). Thirdly, temporary deactivation may occur, requiring no great adjustment for its correction, as when, for instance, the de:eterious effect of a high water vapor concentration in ammonia synthesis gas is corrected by drying. We are concerned with the second form of dextivation, though the first form is still often represented by some over-all deactivation factor. Our hypothesis is that the rate of reaction can at all times be represented in the form EZ?, where R is the reaction rate as a function of composition, temperature, and pressure and E is an activity factor (0 5 E 5 1) representing
the current state of the catalyst. 4 is a function of time and if known exactly it allows us to prescribe a n optimal operating schedule over the economic life of the catalyst and to determine what this life is. If E is not known as a function of time, it must be determined from observation of the behavior of the reactor. W e consider these two problems here-namely, when 5 is a known function of time and when it is not. A full numerical study of the adiabatic bed reactor for the oxidation of sulfur dioxide has been made ( 5 ) . W e use the experience gained with this system (and the computer programs that were compiled for it) in the present study. Poisoning of vanadium catalyst by traces of arsenic occurs (6) and takes the form given here, T h e method developed is not in any way peculiar to this system, however, and may be used whenever the deactivation factor, E, is applicable. A somewhat different catalyst replacement problem has been studied by Roberts (7, 8 ) using the methods of dynamic programming.
T h e reactor system to be studied consists of two adiabatic beds in series (Figure 1). A fraction, 11,of the total flow rate, G, of the feed is preheated from a temperature TOto Tp and enters bed 2. I n the bed, the weight of which is W2, the conversion increases from g2 == 0 to g2’ and the temperature rises to T2’, T h e stream from bed 2 is then mixed with the remainder of the feed and passes to bed 1, of weight W1,where the conversion increases from g, to gl’. If the reaction rate, R, is a known function of conversion, g, and temperature, T , a complete, optimal design car: be found, giving the values of 11’1, Wp, XI? and Tz that maximize. = G.67;
- 6 ( WI + W v )
- X?p‘G( Ts - To)
This profit function repre:;ents the difference between the value of the product, Ggl‘, and the cost of the beds, S(W1 f W Z ) , and of preheating XzGp’(T2 - To). T h e cost of the beds is taken to be proportional to their weight, but the constant of proportionality can be adjusted to suit a more realistic estimate as closely as possible. I t is convenient to wo.rk with a reduced temperature t = C,(T T o ) / ( - A H ) , whose zero is the temperature of the cold feed, and unit corresponds to the adiabatic temperature rise accompanying unit conversion. T h e following equations are derived by Lee and Aris (7,5). For conversion g2’, attained in bed 2, we have
-
where
For the mixing with the bypass of cold feed we have
G
mtr
9.. 0
Convrnlon
T t m p r a t u r e T.
(J2= 0
9;
9,
9;
TZ
1; TI
TI
Figure 1 .
and ported tp
The reactor system
In the case for which numerical values are re-
X2.
el
= 0.563,
x2e2 = 0.093
Le., G = 7731 kg. per hour, W‘I = 4353 kg., and ?+‘2 = 719 kg. T h e cold feed temperature, To = 37’ C. and the reduced temperature, t, is related to T by T = 37 96.9.
(4)
R l k ) = R(g,&?
+ tl -
g1)
(5)
+ x 2 t u - x21*t2
(6)
T h e profit to be maximized is
- qel
P,/G = gli
Economic Life with Known Deactivation
W e divide the life of the catalyst into a succession of equal periods. Depending on the rate of deactivation these may be days, weeks, months, or even years. \%thin any one period, say the mth, we take the activity factor to be constant, say tm. Then in Equations 1 and 4, Rz and R1 should be replaced by Tp = 439‘ C. (point W ) . The optimal bypass ratio is now A ? = 0.654 (point X). This graphical construction
2.4
2.2
We turn now to the case in which E is not known to follow a prescribed course in time but must be determined by observation of the reactor. We still retain the assumption that the deactivation can be satisfactorily described by such a n over-all ratio as 4; temporary poisoning in which 4 may increase as well as decrease can be handled by this method. The principle is that the temperature rise in the catalyst bed of fixed size is itself a measure of 4. The curves in Figure 5 show the temperature profiles in bed 2 for various inlet temperatures, tz. They are not plots of temperature against bed length but rather of exit temperature that Xvould be attained in the bed if the activity factor were (. The upper curve shows the optimal exit temperature for various f and following back the steep curves to the line 5 = 0 gives the corresponding inlet temperature. Thus if 4 = 0.78 (point P ) the optimal exit temperature is t,’ = 5.819 (Tz’ = 598’ C.) at point Q . Then following back the curve through Q the optimal inlet temperature should be a t R?t z = 3.989 (Tp = 422’ C,). The optimal bypass ratio corresponds to point S on the Xs curve for this value of E and is A ? = 0.640. The operator may guess that the current state of the catalyst is E = 0.78 and accordingly set the preheating temperature at TZ = 422’ C. and bypass ratio X, = 0.640. However, he observes
r
1 1
2.01
I
Figure 3. Optimal outlet conditions in bed 1 for various activity factors
Ql
i
OZ7
3.9
4.1
43
,
4.5
4.7
4.9
’2
Figure 2. Optimal conditions in bed 2 for various activity factors 308
l&EC
PROCESS DESIGN A N D DEVELOPMENT
tl
Figure 4. Optimal inlet conditions in bed 1 for various activity factors
6.5
-
/
f
E,
It;
n
- 6 00-
(1099 0.91 0 78 10-
--
0815901
0 6 k
-
--_ _ _ _ _
0.4 t 5 BO-
0.2
c
/
I
tl
0 15.70
3.8
3.9
4.0
4.1
1 42
1 4.3
I 44
11
Figure 6. Optimal conditions with different activities in each bed
g,
= conversion in inlet stream to bed n , n = 1, 2. moles per
kg. gn‘ = conversion in outlet stream from bed n, moles per kg. A4 = total number of periods for expressing life of catalyst, dimensionless
P2
Pm p.tf
= total net profit from 2 beds, moles per hour = P?/G. moles per kg. = maximum value of p for mth period, moles per = average profit per period, moles per kg.
R,
=
T
= = =
p
Figure 5. Chart for determining the activity and optimal control from observations
To
t tn = t,&’ =
W,
=
kg.
rate of change of g per unit weight of catalyst in bed n, moles per hour per kg. of catalyst temperature, O C. basic feed temperature, ’ C. reduced temperature, moles per kg. inlet stream temperature to bed n, moles per kg. outlet stream temperature from bed n. moles per kg. weight of catalyst in bed n, kg.
might be tabulated in ]:he computer and all the necessary adjustments printed out for the operator or even ordered by the computer itself.
GREEKLErTERS 6
= ratio of cost of a catalyst bed to net increase in value per
Discussion
8,
= holding time
Since the deactivation envisioned may come about by the adsorption of trace quantities of poison, the deactivation in earlier beds may be more severe than in later ones. To see how important this effect might be? a number of calculations were done with different deactivation factors for each bed. Two general conclusions were reached. First, a decrease in catalyst activity in either bed called for compensating increase in preheating temperature. Secondly, the plots of the optimal were so relations of t2’ and t~ for all combinations of {I and close together that a single curve could be drawn, as in Figure 6. With known activities in each bed it is easv to determine the optimal operating conditions from Figures 4 and 6 . For example, with = o.c> = Ive have from Figure 6 (shown by the broken lines) t 2 = 4.09 ( T 2 = 432’ C.), tz’ = 5.88 ( p z = 6040 c.),arid l y i = t p i - t 2 = 5.88 - 4.09 = 1.79. Hence PI = &’/h’ = 0.304 and returning to Figure 4 with this value of PI and [I = 0.7 gives gl = 1.142, tl = 3.75 (TI = 399O ‘2.). The bypass ratio, h2, can then be calculatednamely, X z = pi/gl’ = 0.638.
h
=
P
=
P’
=
[
= = =
Nomenclature
G
= total mass flow rare, kg. per hour
g
=
conversion or extent of reaction, moles per kg. of mixture
trn pn
unit conversion, moles per kg. of catalyst per hour of bed n, (kg. of catalyst)(hour) per kg. of mixture bypass fracrion, dimensionless (cost of unit amount of heat) (heat of reaction) , dimennet increase in value per unit conversion sionless (cost of unit amount of heat) (specific heat of mixture) , net increase in value per unit conversion mole per kg. per C. catalyst activity factor, dimensionless catalyst activity factor in mth period g,,j’t, = g,_, ’ t,,_,, ’ dimensionless
’
literature Cited
(1) Aris, K., “The Optimal Design of Chemical Reactors,” Academic Press, New York, 1961. (2) Froment, G. F., Bischoff, K. B., Chem. Eng. Sci. 16, 189 (1961). (3) Ibid., 17, 105 (1962). (4) Griffith, R. H., “The Mechanism of Contact Catalysis,” Clarendon Press, Oxford, 1946. (5) Lee, K. Y . , Aris, R., IND. ENG. CHEM.,PROCESS DESIGN DEVELOP. 2, 300 (1963). (6) Olsen, J. C., Maimer, H., Znd. Eng. Chem. 29, 254 (1937). (7) Roberts, S. M., Chem. Eng. Progr. Symp. Ser. No. 31, 5 6 , 103 (1960). (8) Roberts, S. M., Symposium on Optimization, New York University, May 1960, p. 171. RECEIVED for review November 26, 1962 ACCEPTED March 13, 1963 VOL. 2
NO. 4
OCTOBER 1 9 6 3
309
