u,,,* = pure water permeation velocity, cm/s X A = mole fraction of total (dissociated and undissociated) solute
Greek Letters = degree of dissociation y+ = mean molal activity coefficient for the solute n ( X A ) = osmotic pressure of solution corresponding to (Y
X A
Subscripts 1 = bulk solution phase 2 = concentrated boundary solution on the high pressure side of the membrane 3 = membrane permeated product solution on the low pressure side of the membrane Literature Cited Dickson, J. M., Matsuura. T., Blais, P., Sourirajan, S., J. Appl. Polym. Sci., 19, 801 (1975). Fang, H. H. P., Chian, E. S. K., J. Appl. Polym. Sci., 19, 2889 (1975a). Fang, H. H.P., Chian, E. S. K., "Removal of Dissolved Solid by Reverse Osmosis," paper presented at the A.1.Ch.E. 68th Annual Meeting in Los Angeles, Calif.. Nov 16-20, 1975b. Heyde. M. E., Peters, C. R., Anderson, J. E., J. Colloidlnferface Sci., 50, 467 (1975).
Johnston, H. K., Desalination, 16, 205 (1975). Matsuura, T., Bednas, M. E., Sourirajan, S., J. Appl. Polym. Sci., 18, 567 (1974) Matsuura, T., Dickson, J. M., Sourirajan, S., hd. Eng. Chem., Process Des. Dev., 15, 149 (1976a). Matsuura, T., Dickson. J. M., Sourirajan, S., lnd Eng. Chem., Process Des. Dev., 15, 350 (1976b). Matsuura, T., Pageau, L., Sourirajan, S.. J. Appl. Polym. Sci., 19, 179 (1975). Matsuura, T., Sourirajan, S., J. Appl. Polym. Sci., 17, 1043 (1973). Pageau. L., Sourirajan, S., J. Appl. Polym. Sci. 16, 3185 (1972). Parsons, R., "Handbook of Electrochemical Constants," pp 20-27,54, Butterworths, London, 1959. Reid, R. C., Sherwood, T.K., "The Properties of Gases and Liquids," p 295, McGraw-Hill, New York. N.Y., 1958. Robinson, R. A., Stokes, R. H.,"Electrolyte Solutions," 2nd ed, (a) p 32: (b) p 39: (c) p 229; (d) p 392; (e) p 396; (f) p 513; Butterworths. London, 1959. Sillen, L. G . , Martell, A. E., "Stability Constants of Metal-Ion Complexes," (a) p x; (b) pp 166-178: Special Publication No. 17, The Chemical Society, London, 1964. Sourirajan, S., "Reverse Osmosis," Chapter 3, Academic Press, New York, N.Y., 1970. Yatsimirskii. K. B., Vasil'ev, "Instability Constants of Complex Compounds," p 120, Pergamon Press, London, 1960.
Received for review November 12, 1975 Accepted April 20, 1976
Issued as NRC No. 15428.
Optimum Operating Cycle for Systems with Deactivating Catalysts. 1. General Formulation and Method of Solution Jin Y. Park and Octave Levenspiel" Deparlment of Chemical Engineering, Oregon State University, Corvallis, Oregon, 9733 7
This paper shows how to find the best way of running a reactor whose catalyst decays with use and must consequently either be regenerated or replaced at regular intervals. Part 1 develops a one-variable search method which avoids the previously proposed multidimensional search procedures. This method is quite general and can be applied to any process or operation where one can run the unit hard or gently and where when run hard it gives a larger profit, but it also wears out more rapidly. Part 2 applies this technique specifically to catalytic reactors with both mixed flow and plug flow of fluid.
When reacting fluid flows through a batch of slowly deactivating catalyst, the activity of catalyst drops progressively to a point a t which, because of economic profitability considerations, the run is terminated, the catalyst is regenerated or replaced, and the cycle is repeated; see Figure 1. There are two important and real questions concerning the cyclic operation: (1)how to best operate the reactor during a run (the operational problem) and (2) when to stop a run and regenerate or replace the catalyst (the regeneration problem). These problems, although closely connected, can always be treated separately by means of some relay variable (or variables) which sufficiently characterizes both phases of the entire production cycle (the overall problem). In the past the operational problem has been studied by Szepe (1966) and others (Chou e t al., 1967; Ogunye and Ray, 1968; Crowe, 1970). Since temperature is the most important variable affecting reaction and deactivation of catalyst, these authors have primarily been concerned with finding the best temperature progression during the operational phase. The regeneration phase of the cycle is strongly dependent on the temperature progression during the operational phase; however, this has not yet received due attention. Nevertheless, 534
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 4, 1976
the essential dependency of regeneration on the operational problem has recently been demonstrated in a numerical example by Miertschin and Jackson (1970). In the overall optimization where both problems are to be solved, selection of the relay variable (or variables) as the means for separation, for recombination, and for computation of the two problems is of major importance. Szepe (1966) suggested a three-variable formulation consisting of initial .catalyst activity ai, final catalyst activity af,and operational time t o pas the relay variables. He further suggested that the overall problem be solved in two steps as follows: (1)solving for the operational problem with ai, af,and t o pas parameters, and then (2) solving the regeneration problem to find the optimum set of those parameters with the corresponding previously obtained optimum operation policy. Miertschin and Jackson (1970) showed that the number of the independent variables need only be 2, with ai and t o pbeing the convenient choice for their numerical computation. The present study shows that only one independent relay variable is in fact needed. In Part 1we develop the general one-variable formulation for catalyst deactivation and other types of cyclic operation.
Since only one relay variable is involved, the search for the optimum production cycle proceeds directly without the excessive computation inherent in the multivariable formulations. In Part 2 we apply this formulation to find the optimum policy for reactors having mixed or plug flow of gas and for reasonable reaction rate and deactivation rate forms.
A tI
a,
+
production cycle
I
I
I
I
,,’ ,,’
z
operational J phase I .C
.-+-> ._ 4-
Economic Objective The economic objective for cyclic operations, although differing in detail from process to process, may be expressed (see Miertschin and Jackson, 1970; a similar profit criterion was used earlier by Weekman, 1968) as maximizing the profit which, in the absence of intercyclic dependency, can be defined as
time for operational phase
(
+
m +-
1 a,+-
m 0
I I
I I
-I
I
I
operational time, top
I
1
I
I
+
!-
regeneration time, t R
Ti me
net income per production cycle profit = time for production cycle net income during) (operational phase
I- regeneration I--regeneratton phase II phase
Figure 1. Sketch of the activity of
a deactiving catalyst through a
production cycle.
- (regeneration) cost
where the varying regeneration time may be expressed as
time for regeneration phase
(
or equivalently in mathematical terms
Let us explain the various terms appearing in this expression. Net Income Rate Gop. At any instant during the operational phase =
total operational income rate
total operational costrate
1- (
(
)
c
E
( P r r @ y ) (value Of) (cost) (2) product ,“L rate in which the total operational cost rate in general consists of cost of feed, cost of operation, cost of maintenance, cost of separation, etc. Regeneration Cost C R ( U ~ a, f ) . In general this consists of two independent and additive costs: the fixed regeneration cost, present for each shutdown regardless of initial and final activities, and the varying regeneration cost, dependent on the extent of regeneration. Thus =
products a11
SoYTCes
C R ( a i , af) = C R v ( a i , af) -k
CRf
in which subscripts f and v stand for the fixed and the varying cost, respectively. Furthermore, since the extent of regeneration proceeds from activity to activity (see Levenspiel, 1972), it is assumed in this study that the varying regeneration cost is an exact function of catalyst activity. Thus if we let C R ~ ( Q , 0) be the cost for regeneration from activity zero to a , and if we let C’R,(U, 0) be its derivative with respect to a , we have, for the varying regeneration cost
C R V ( ~ Ia,d = C R ~ ( 0~) -~ CRv(af, , 0)=
&r
C ’ R , ( ~0, ) da
The total regeneration cost then becomes CR(alr
af) =
J:’C ’ R v ( a , 0 ) da + C R f
(3)
Regeneration Time ~ R ( u ~af).As , with regeneration cost we may write tR(ai, a f ) = tRv(ai, a f )
+ tRf
Concerning the regeneration cost and time, many variations arise in practice. For instance, only the fixed contributions are present for regeneration by straight replacement where the catalyst is discarded at the end of run. On the other hand, sometimes the fixed regeneration time is much smaller than the varying regeneration time, as in in situ regeneration where the reactant is switched off and replaced by the regeneration fluid. Since all of these extremes are mathematically degenerate cases of the general formulation, we here deal with both contributions. This format of regeneration also fits the common industrial problem where longer operation times lead to more carbon deposition which in turn requires more burning time.
Formulation of the Problem Let us confine our attention t o isothermal reactors operating under the following conditions: (1)The reaction proceeds on a single catalyst (as opposed t0.a multifunctional mixture of various catalysts). (2) The batch of catalyst solids is well mixed; hence the activity of catalyst is uniform everywhere in the reactor. (3) The rate of catalyst deactivation a t any time is dependent on the activity of catalyst at that instant and on temperature. (4) Reactant feed is continuous and remains constant during the operational phase. ( 5 ) Deactivation of catalyst is slow enough that the reactor can be considered to be at steady state at any instant of time. Furthermore, since temperature is the most important variable affecting the reaction and deactivation rates, we will consider it as the unique control variable. Since deactivation rate and the profit rate of eq 2 are explicitly independent of time (autonomous) and are entirely determined by catalyst activity a and temperature of reactor T , we may write da dt
- -= p
( ~T ,
)
Go, = Gop(a, T )
(5b)
where p(a, T )and Gop(a, T ) indicate known a priori functional dependencies of the rates on a and T . Ind. Eng. Chem., Process
Des. Dev., Vol. 15, No. 4, 1976 535
The profit of eq 1 may then be evaluated along the decaying activity of catalyst instead of time. Thus substituting eq 5 in eq 1 and changing the bounds of the integrals gives
Defining the operation index I ( a , T , P)as
and the optimum operation index i ( a , P ) as
which shows that the profit P is dependent on the interval of catalyst activity (ai, af) as well as the temperature progression T ( a )in that interval. The overall optimization then concerns finding both the best interval of catalyst activity (the regeneration problem) and the best temperature progression in that interval (the operational problem).
Optimal Temperature Policy T o find the optimal temperature progression during the operational phase, consider a production cycle on any arbitrary but known interval of catalyst activity (ai, Cf). If we let p(a)be the optimal temperature progression and P the largest profit that can be derived for that particular interval of catalyst activity (henceforth called the optimum profit), these are related by eq 6 as
Any other temperature progressions will then yield smaller profits. Consider in particular any temperature progression that is identical to p(a)everywhere in that particular interval of catalyst activity except for a sufficiently small but arbitrary subinterval (a0 6/2, a0 - 6/2). The profit for-this temperature progression, differing only slightly from P, may then be evaluated from eq 6 and 7 as P=
+
the best operating temperature a t any catalyst activity is that which satisfies the following condition
R a , P ) = max { ~ ( aT, , P)ls::a,ble
(10)
Thus the optimal temperature progression to use is that where the operation index stays as high as possible as the catalyst deactivates. This condition can also be derived by the maximum principle described elsewhere (Denn, 1969). In the format of this criterion, any value P then fixes a temperature progression which is optimal for azy interyal of catalyst activity where the optimum profit is P. Thus P becomes the single relay variable, and th_eregeneration problem simply reduces to finding the largest P and the corresponding interval of catalyst activity.
Optimal Regeneration Policy Turning our attention to the regeneration problem, consider the temperature progression p(a)as determined by eq 10 for any selected P. If P is realizable at all, or if there exists at least one interval of catalyst activity where the optimum profit is P, then the family of intervals of catalyst activity ( u , , a f ) and P are related as follows: by substituting eq 9 in eq 7 and rearranging f:’J
i ( a , P ) da
+ cR(al,a f ) + ~ t R ( a ,a,f ) = o
(11)
Substituting eq 3 and 4 in eq 11and defining the regeneration index J ( a ,P ) as ~ ( aP,) = i ( a , P ) - C ’ R V (0~),- P)t’RI.(a,0 )
(12)
allows us to rewrite the condition of eq 11 as
Jr’
where T and T ( a o )are respectively the nonoptimal and optimal temperatures when the activity of catalyst in the reactor is ao. Expanding and neglecting the higher order terms with respect to 6 then gives
6
p-p,
G o p ( a ~T, ) - P - G,,,(ao, F ( a o ) ) - P ~ ( a oTI , P(ao, n a o ) ) Since P 5 P by definition of the optimality, ?(ao) is distinguished from T as that satisfying
I
1
G,p(ao,
T ( a o ) )- P -, Gop(ao, TI
T’(u)) - P , G,,(a, T)- P -
p(a, n a ) ) 536
-
P ) d a = CRf + PtRf
(13)
Our aim now is to find the highest value for the profit, or P,,,, with its corresponding activity values, and af,optwhich will satisfy eq 13. This may, at least conceptually, be best performed by trial and error procedures. Thus the se_arch procedure basically consists of making a guess on P,,,, checking whether the guess is indeed the maximum profit attainable, and then iterating on different guesses. T o illustrate the search let (a,+, a f + )be the interval of catalyst activity where the left-hand side integral of eqJ3 becomes maximum for any selected_P. Then since J ( a , P) uniformly decreases with respect to P everwhere in 0 5 a 5 1,see eq 3, 4, 12, and 13, the left-hand side integral of eq 13 also decreases while the right-hand side increases with increasing P. Thus depending on the selected value P, one of the following three conditions holds:
-P
p(ao, n a o ) ) P(ao> T ) Furthermore, since a0 is arbitrary, then for i’(a) to be indeed the optimal, the inequality must be satisfied a t any point in the entire interval of catalyst activity. Thus G,,(u,
J(U,
p(a, T )
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 4, 1976
decreases with rising P
increases with rising P
This particular behavior of eq 14 then allows us to establish the following criterion for the search: for any guess Pguess, if 1.h.s. < r.h.s., then since there is no interval of catalyst activity Pguess > P,,,; and where eq 13 is satisfied for any P IPguess,
if 1.h.s. > r.h.s., then since there is a t least one interval of catalyst activity where eq 13 is satisfied for some P > Pguess, Pguess< p,,,. Thus for the maximum profit P,,, we have
1
a , "pt
J ( a , p m a x ) da = CRf
+ PmaxtRf
t
(15)
afupt
where a,,optand af,optare a,+ and a f + when p = P,,,. The search for the maximum profit P,,, is then summarized as follows: (1) guess p,,, = Pguess;-(2)findT ( a ) a t various a according to eq 10; (3) evaluate [ ( a , Pguess)according to eq 9; (4) evaluate J ( a ,pguess) according to eq 12; ( 5 ) repeat procedures ( I ) to (4) until the criterion for the optimal regeneration, or eq 15, is satisfied. Figure 2 graphically illustrates the search for a typical regeneration index. Notice here that the left-hand side integral of eq 14 simply represents the shaded area above the activity axis. In the majority of practical cases where the regeneration index J ( a ,P ) is a well behaved function of a , such as shown in Figure 2, finding (a,+, a f + ) is a very simple matter and usually can be accomplished by inspection. This remarkably reduces the computation. R e m a r k s a n d Discussions Optimum Scheduling. Once p,,, as well as ai,optand af,",, is known, optimum scheduling of the entire production cycle can be found without additional difficulty. Thus if we let T ( a ) be the optimal temperature progression determined by P,,,, and if we let n ( t ) be the optimal activity progression with time t , these are related by eq 5a as
The optimal temperature progression may also be expressed in terms off. Thus replacing a by 6 ( t )as determined by eq 16, we have
T ( a ) = T @ ( t ) =) Tit) Extensions a n d Limitations. The optimal criterion derived in terms of temperature of the reactor as the unique control variable can be extended to include other variables such as feed rate to the reactor, composition of the feed, pressure in the reactor, etc. The method of optimization, however, has certain limitations: first, although some modifications are still possible its use is primarily.limited to those cases where the catalyst activity is uniform in the reactor; secondly, it does not apply where uncontrollable time-dependent factors, such as varying but uncontrollable feed rate to the reactor, intrude. Finally, the one-parameter formulation developed here is rather general and can be used for all sorts of situations where the operating unit loses efficiency or effectiveness with use. The scheduling of a taxi fleet, the management of a brothel with its virgins or near virgins, the deployment in war of various kinds of military units are some possible areas of application. The following example illustrates this procedure for an artifically simple situation. For more examples, see Park (1976) or Part 2. Example. Our taxi fleet operates twenty-four hours a day seven days a week, and the taxis can work hard or not depending on their daily mileage. On the toughest schedule, say 500 miles a day, they bring in more money but they wear out quickly, and as they age their efficiency and income drops. Given the following information, how should we schedule a taxi? In other words, how should we operate our taxis and when should we replace them? (1)The taxi efficiency, a,is mainly dependent on the total mileage, M , as follows: a = e ~ p ( - l O - ~ M )(2) . On a daily basis the money brought in by a taxi is 0.15Ta dollars while the
3rd
~uess,$RsIS just
\ I
\
right
J
0
1 Activity, a
Figure-2. This graph illustrates the graphical search for the maximum profit P,,,.
+
maintenance cost is 10 O.lT(1 - a ) dollars where T is the daily mileage. (3) A new taxi costs $5000 while the trade-in price of a used vehicle is directly proportional to its condition, as measured by its efficiency. Other costs (license, tax, dealer preparation, lost time, etc.) amount to $409.70 plus 24 h down time for each vehicle. Solution. The information given above transforms itself into our mathematical format as follows da - 10-jTa (l/day) dt
(0 2
T
500 (miledday))
(i)
G,,,, = 0.15Ta - O.lT(1 - a ) - 10 (dollardday)
(ii)
C f i , ( a , 0) = 5000a (dollars); C R =~ 409.70 (dollars); f K v ( a , 0 ) = 0 (day); t R f = 1 (day) (iii) Operation Index. From eq 8, i, and ii
I ( a , T ,P ) = Go, - P
= lo4 2.5
[
--
b(
1
+ 100 + lop)] (iv) T
Optimal Operation Policy. Since I ( a , T,p ) increases as we increase T,the best policy is to run taxis on the toughest schedule, or
T ( a ) = 500 (miledday)
(VI
Optimum Operation Index. From eq 9, iv, and v
Regeneration Index. From eq 12, iii, and vi ~ ( aP ,) = ?(a,P ) -
C'I
