Ghosh, B . . lndian Chem. Eng. Trans., 2, (1963). Graham. W . . Lama, R., Can. J. Chem. Eng., 41, 31 (1963). inoge, K.. Uchibori, T., Katsuai. T., KolloidZ., 139, 167 (1954) Kinosita. K.. Mem. Fac. Eng. Nagoya, 1, 15 (1949a). Kinosita. K , J. Colloid Sci.. 4, 525 (1949b), Kynch, G. K.. Trans. Faraday Soc., 48, 166 (1952). Linzenmeier. G., Munchen. Med. Wochenschr,. 5 (1925). Lundgren, R.. Acta. Med. Scand.. 67, 63 (1927). Lundgren. R . . Acta Med. Scand.. 69, 405 (1928)
Nakamura H., Kuroda, K., Keiio J. Med.. 8, 256 (1937). Oliver, D. R., Jenson. V . G., Can. J. Chem. Eng., 42, 191 (1964) Pearce, K. W., Third Congress of European Federation of Chemical Engineering London, A30 (1962). Vohra. 0. K., Ghosh, B., lndianchem. Eng.. 13, 32 (1971).
Received for reuieu: .January 17, 1974 Accepted August 27, 1974
Design of a Catalytic Reactor-Separator System with Uncertainty in Catalyst Activity and Selectivity Gary J. Powers* and Jerald F. Mayer' Department of Chemical Engineering. Carnegie-Meilon University. Pittsburgh, Pennsylvania 75273
Uncertainty in the design of a piece of process equipment often resides in a number of design parameters such as physical property data and design correlations. The uncertainty in these parameters affects both the equipment's cost and performance and the cost of other processing units to which it is connected. A method based on the expected value criterion is used to obtain the optimum overdesign factor for a catalytic reactor-separator system. In t h e example the catalyst activity and selectivity are uncertain
Introduction The design of chemical processes requires knowledge of thermodynamic properties, rate constants, transport coefficients, etc. Quite often, however, exact values for these parameters are not available, and uncertainty must be taken into account in the design. Kittrell and Watson (1966) considered the design of a chemical reactor where the value of the reaction rate constant was uncertain. They used the expected value criterion and minimized expected cost. Wen and Chang (1968) applied the expected relative sensitivity criterion to the same problem. Watanabe, et al. (1973), analyzed the same problem using a more sophisticated technique based on statistical decision theory. All these studies indicated that the reactor should be overdesigned to compensate for the uncertainty in the rate constant. The optimum amount of overdesign varied with the relative uncertainty and also with the objective function used by the various authors. Villadsen (1967) has studied the overdesign of distillation columns. These studies considered uncertainty in only one parameter and determined the effect of this uncertainty on the design of a single item of processing equipment. In the design of a complete process it is important to consider how uncertainty in the design of one part of the process will affect other parts of the system. In certain instances propagation of uncertainty may make it necessary to add or remove complete parts of a process. These design changes can be very costly both in terms of equipment and installation costs and the losses due to business interruption. Hence, it is important to consider more than one part of a process when analyzing the effects of uncertainty on the design. In the design of a process uncertainty is very seldom confined to just one variable in one piece of equipment. The design of even a simple piece of equipment often in-
volves considering several uncertain parameters whose effects may be in opposition. For example, consider the design of a vacuum distillation column separating a temperature sensitive mixture. Uncertainty in tray efficiency leads to overdesign through the use of more than the optimum number of trays computed assuming complete knowledge. These trays are added to avoid penalties associated with using reflux ratios higher than the optimum. However, in the separation of temperature sensitive mixtures the temperature in the reboiler is often limiting. The uncertainty in the temperature a t which decomposition occurs often suggests using a temperature lower than the maximum allowed. The temperature in the reboiler of vacuum distillation columns is fixed via the phase rule by the pressure in the reboiler. The higher the pressure the higher the temperature. The pressure in the reboiler depends on the number of trays in the column. The more trays the higher the pressure in the reboiler. Hence, for this situation, uncertainty in tray efficiency indicates that more trays should be used while uncertainty in the decomposition temperature requires fewer trays. The relative costs of trays and utilities us. product decomposition will determine the overdesign required. The purpose of this paper is to illustrate the effect of uncertainty in two parameters on the optimum size for a single item of equipment which has an effect on the other pieces of equipment in the process. An example is presented for a catalytic reactor in which both the activity and selectivity of the catalyst are uncertain. The problem discussed is greatly simplified, but it demonstrates how uncertainty in a second variable can significantly affect design decisions. The interaction of the reactor with a separator in the process illustrates how uncertainty in one piece of equipment can propagate through a complete process. Problem Statement
' Present address, Chevron Research, Richmond, Calif
Consider a reaction in which a catalyst is used to conInd. Eng. Chem., P r o c e s s Des. Develop., Vol. 14, No. i , 1975
41
.-
I
the reactor and separator when the decision variable is the pounds of catalyst D used in the reactor. -
C
Figure 1. A catalytic reactor coupled to a separator. T h e reaction converts compound X into the desired product Y and by-product Z . Any Z in excess of t h e amount allowed in the product is removed in the separator.
,
XR,SEP
= inin
{ r e a c t o r cost 'R
+
'SEP
s e p a r a t o r cost
}
(2)
Reactor Costs The cost of the reactor Cli depends on the reactor fabrication costs, the catalyst cost per pound c,, the number of pounds of catalyst D, and the catalyst activity A ) . In this example the catalyst costs are assumed to dominate the costs so that the reactor fabrication costs can be neglected. Equation 3 gives the reactor cost for two situations. C,(D,Ay)
=
( c , D , i f D A ~z- qY s u c c e s s on first attempt
jc,, + c,
I ~~
vert compound X into a desired product Y and a byproduct Z. The amount of Z that is produced is very small so that the decrease in yield due to the formation of Z can be neglected. However, the byproduct Z is not allowed in the product Y in an amount greater than a lb of Z/lb of Y. It is necessary to produce Q> pounds per day of Y. A flowsheet for the process is given in Figure 1. The catalyst activity for the reactions producing Y and Z are uncertain. The activity is defined as the pounds of Y or Z produced per pound of catalyst in the reactor. The distribution of the activities are uniform as illustrated in Figure 2. The activities are assumed to be independent. If Z is present in the reactor effluent a t a concentration greater than a, it is necessary to install a separator to remove the excess Z . The problem is to determine the number of pounds of catalyst D to be used in the reactor. This problem has been solved by Rudd and Watson (1968) for the case of uncertainty in only the Y activity. In the following we have adopted their nomenclature.
Expected Value as the Objective Function for Design with Uncertainty The objective function for decision-making in uncertain situations depends on the degree of risk-aversion of the decision-maker. The degree of risk-aversion which is acceptable depends on the market position of the firm, its capital position, and the magnitude of the project being considered (Luce and Raiffa, 1957). In most cases a decision-maker will value an alternative a t less than its expected value. The expected value is defined as the return from each possible outcome multiplied by its probability of occurence. F o r a system in which the outcomes and probabilities are a function of a continuous variable, eq 1 is used to define expected value. -
D)
-+
c,,
(31 if DA < Q y (redesign r e q u i r e d )
In the first situation the reactor is large enough to produce Qb pounds per day. In the second situation the reactor is not large enough and must be rebuilt so that more catalyst can be added. In this situation a penalty C,, is incurred due to business interruption. The uniform distribution of the activity for Y is given by eq 4 where A Y I .and .4\H are the low and high activities, respectively.
1
for A y L
5
A, S A y H
(41
otherwise
Separator Cost The separator cost is given by eq 5 .
No separator is required if the activity of the catalyst for producing Z is such that the concentration of Z obtained for D pounds of catalyst is less than a If a concentration greater than a is obtained it is necessary to construct a separator. In eq 5 it is assumed that the major costs will be due to the operation of the separator. The costs are assumed to depend directly on the concentration of Z produced. The probability of the catalyst having an activity A, is given in eq 6
( 0 otherwise The expected value definition in eq 1 is used to give the expected costs ufthe reactor and separator.
C ( B ) P ( B )dB
The variable is the expected value of the cost C. The cost C depends on an uncertain variable B . The probabilit y of C ( B ) having a given value is P ( B ) . The objective function used in this problem will be based on the expected value of the costs. For this design problem it is desirable to minimize the expected cost for 42
-
-___---
Figure 2. The probability distribution for catalyst activity
C =
(2
Ind. Ena. C h e m . , Process D e s . Develop., Vol. 14, No. 1, 1975
The expected cost of the reactor is a function of the number of pounds of catalyst used in the reactor. Integration and simplification of eq 7 gives eq 8.
b
for -4- y= __ D
Q
5 -2-
(8a)
AYL
AZL
(4, n
