OPTIMAL DESIGN OF SYSTEMS INVOLVING PARAMETER UNCERTAINTY C. Y . W E N A N D T. M. C H A N G
Chemical Engineering Department, West Virginia University, Morgantown, W . Va. Two design criteria are proposed for optimal design of systems involving parameter uncertainty. One may b e used to obtain an appropriate decision which will keep the deviation of the objective from the optimal behavior within a certain tolerance. The other assures a minimum average normalized deviation of the objective from the optima over the range of uncertainty. These criteria are compared with the other existing criteria. Examples demonstrate the applicability of these criteria to the optimal reactor design with uncertainty in kinetic constants.
o ACHIEVE the optimal design of a process, one seeks the T b e s t design which will result in a maximum profit. Usually, this is done by applying an optimization technique to search for a set of optimal decisions-Le., design and/or operating variables-based on the specific values of system parameters. T h e optimal decisions so obtained will yield the best design if the values of parameters used are accurate. But often, the system parameters of chemical processes such as kinetic constants, heat and mass transfer coefficient, etc., are unknown or kno\rn only approximately, or vary within a certain range. Such uncertainty is mainly due to the errors associated with the experimental or empirical evaluation of parameters or. in some cases, to the changing operating conditions. ,4lthough an accurate evaluation of parameters is necessary so the result obtained by optimization based on the specific values of parameters can be useful, the time and money that must be spent may not be justified. Moreover, in process optimization, cost must be estimated before deciding whether to construct a certain plant. T o be meaningful, the estimation must be obtained under optimal conditions. In other words, to determine whether a certain process will be more profitable than other processes, an economic comparison should be made based on each process operating under optimal conditions. Hence, it often becomes necessary to optimize a process without accurate information of system parameters. Consequently, the problem of parameter uncertainty often exists in process optimization. The optimization based on specific values of parameters cannot take into account the uncertainties and: therefore, would not necessarily give the best profit obtainable. T o achieve the best design of processes involving parameter uncertainty, design criteria taking into account the uncertainties must be developed.
where & is a p-dimensional parameter vector and 0 is an r-dimensional decision vector. This criterion is not applicable to systems involving parameter uncertainty. Suppose that J ( w , e) is the cost of a process and the system parameter, w , has a range of uncertainty from w1 to w2. As shown in Figure 1, curve A is the locus of the minimum cost corresponding to each w. When the criterion as given in Equation 1 is used, an optimal decision, 81, based on a specific value of the parameter, say, the average value, ~ ' 3 is : obtained. Then, the process is designed based on 81, and the corresponding cost is J(w3, 81). If the true value of ze happens to be a t 2 ~ 3 , a n optimal design is obtained. If u! w3, the design is no longer optimal, as shown by comparing curve B with curve A in Figure 1. I n other words, a n overdesign or an underdesign, depending on the true value of the parameter, may be obtained. Since it is impossible to obtain an optimal decision which will result in a minimum cost for each =-namely, a curve coinciding with curve A-an overdesign or a n underdesign is unavoidable. Consequently: the problem of optimal design under uncertainty becomes that of minimizing overdesign or underdesign. Several decision criteria have been proposed to provide economic strategies for decision making under uncertainty (Luce and Raiffe, 1957). T h e most commonly used criterion for the design of processes involving parameter uncertainty is to maximize expected profits or minimize expected costs (Kittrell and \Vatson, 1966). According to this criterion, an optimal decision is the one which gives the largest (smallest)
+
Design Criteria
The objective in process design is usually the cost or profit of a process and is a function of a number of variables related by equations resulting from material and energy balances, etc. The number of variables less the number of equations is the degree of freedom (or the number of independent variables, termed decisions) of a process. The objective, J , is then a function of decisions, 0, and system parameters, w . T h e design criterion for a process when parameters are known accurately is to find a set of decisions, i, based on these parameters, zcj such that the objective is minimized or maximized-namely.
J(E,i) = min { J(W, e) } e
or max 8
{ J(E,e) }
(11
+ --_____
I Wl
wFigure
.
w2
w3
Minimization of deviation from optimal behavior VOL. 7
NO. 1
JANUARY
1960
49
among all possible expected profits (costs). cision, e*, is obtained from
J ( w , e) f ( w ) d w ]
min (expected J = e
An optimal de-
(2)
S-mm
where J ( w , B ) is the cost, f ( w ) is a density function of w , and f ( w ) dw is defined as the probability that w will fall between w and w dw (Figure 1). According to Equation 2, the optimal decision, e*, results in a curve, say curve C, which is so close to the line of zero cost (maxis) that the average dif-
+
ference between these two curves weighted by the probability function is minimized. Since curve A is the locus of minimum costs, no curve can lie below it. Therefore, the criterion as given in Equation 2 is equivalent to obtaining a n optimal decision, e*, by minimizing the expected difference between curves C and A-namely, min e
{
S_a
- 4%i)lf(w)d w }
[J(w,e)
(3)
e
(
s_9.
[ ~ ( we),
-~
e).
s(e*) =
This equivalence can also be verified by the following relationship, min
tion from the optimal behavior. However, the deviation in the criterion given in Equation 8 is weighted by J ( w , In other words, a larger deviation may be allowed when the value of J ( w , e) is larger. This means that the normalized deviation from the optimal behavior is emphasized instead of deviation itself in obtaining 8* by the criterion given in Equation 8. This criterion is useful for the design of a process with parameters that are inaccurate or varying because of changing operating conditions. I n some cases, an optimal decision is desired which will ensure that regardless of the true value of a parameter the deviation from the optimum will be within a certain tolerance. This is a problem of critical tolerance. The criteria presented above cannot handle this type of problem. However, a reasonable design criterion for the problem can be conjectured-that is, to minimize the maximum deviation from the optimum. I n terms of the relative sensitivity given in Equation 5 , the decision O* is obtained from
( wi ), ~ f ( w ) d w= )
Although this criterion will yield a curve, J ( w , e*), which has a minimum average deviation from the locus of minimum costs, it does not consider the magnitude of the lowest cost a t each w . However, this can be done by normalizing the deviation of J ( w , e) from J ( w , 8 ) as (5)
min {max ~ ( we), e
(9)
w
I n choosing e*, first, the largest normalized deviation is found over the range of the parameter for a given 0. This ensures that regardless of the true value of the parameter as long as the true value falls in this range, the normalized deviation will not be larger than this largest value. Consequently, the decision that minimizes the maximum relative sensitivity gives a reasonable solution to the problem of critical tolerance. Example 1. Optimum Overdesign of a Reactor. To illustrate the application of the expected relative sensitivity criterion of Equation 8 and to compare it with the expected cost criterion in Equation 2, the following numerical example originally considered by Levenspiel (1 962) and later used by Kittrell and Watson (1966) is presented to illustrate the expected cost criterion. k
where
e)
e)}
~ ( w , = min { ~ ( w ,
e
(6)
T h e normalized deviation S(w, e) as given in Equation 5 is called relative sensitivity (Rohrer and Sobral, 1965), which is a normalized measure of optimality. Based on the relative sensitivity function, Rohrer and Sobral (1965) proposed design criteria to help designers find a simple controller for many similar plants or a controller for a single changeable plant. These criteria may be adapted in optimal process design involving parameter uncertainty. One of the design criteria is to find a decision by minimizing the expected relative sensitivity, which may be defined according to the probability distribution of the system parameter,
SE@) =
m-J
(7)
S(w, e ) f ( W ) dw
T h e optimal decision according to this criterion is then obtained from S(e*) = min
e
($"(e)
S ( W ,O ) f ( w ) d w ]
=
(8)
S-mm
In view of the equivalent relationship in Equation 4 and the definition of S(w, e) in Equation 5, the difference between the criteria in Equations 2 and 8 lies in the fact that the latter for each value takes into account the minimum value, J ( w , of a parameter. I n both criteria, we are trying to make J ( w , e*) as close as possible to J ( w , e) by minimizing the devia-
e),
50
l&EC PROCESS D E S I G N AND DEVELOPMENT
Consider a reaction, A + R, with r = kCA,where k = 0.2 (hour)-'. The production rate of R is 100 gram moles per hour. The feed consists of a saturated solution of A(CAo = 0.1 gram mole per liter). T h e cost of the reactant in the saturated solution is c, = $0.50 per gram mole of A . The cost of the complete mixing reactor, includirlg installation, auxiliary equipment, instrumentation overhead, labor, depreciation, etc., is Gb = $0.01 per hour per liter. What Size of reactor, feed rate, F A O , and conversion, XA, should be used for optimum operations? T h e total hourly cost is cy = v c b
+
FA0
c, =
0 01
v + 0.5
(10)
FA0
For the complete mixing reactor,
Noting that FA0 = 10O/XA, the total cost can be expressed as a function of V and k,
cI(V, k ) = 0.01 V The optimum volume,
4-50 kV/(kV - 1000)
(12)
v, is obtained by letting dcdV - = 0I
namely, -
v=
1000 (1
+ 1/51c)/k
(13)
Based on the expected value of k, 0.2 (hour)-', the optima are: XA = 0.5, F A 0 = 200 moles o f A per hour, = 10,000 liters,
v
and
ct =
$200 per hour.
T h e optimum reactor size is 10,000 liters if one has confidence in the accuracy of the rate constant, k . However, k is determined experimentally and is known to fall i n a certain range with some probability density function. Kittrell and Watson (1966) have solved such a parameter uncertainty problem by applying the expected cost criterion. T h e optimum designed volume, V*, was obtained from min
{ J-f if i
CAV! k ) f ( k )
4
500 I
- V*= 12,425 1. b y eq. 14
;4001,
c \
+
{
0.2
2bk
0
- Ak_< k 5
0.2
+ Ak
V
(1 5)
otherwise
{
J---w, k)f(k)4
150
’
1.
eq. 16
by
I
(16)
where S ( V , k ) is the relative sensitivity of the total costnamely,
T h e reactor volume obtained from either Equation 14 or Equation 16 is different from that obtained by Equation 13 based on the expected value of k. T h e difference may be called the optimum overdesign (Kittrell and Watson, 1966). T h e optimum designed volume for various ranges of uncertainty was calculated by the expected relative sensitivity criterion of Equation 16 (Table I). Table I also shows the result obtained by Kittrell and Watson (1966). The present criterion gives lower optimum overdesign than the expected cost criterion. However, such a comparison based on the overdesign corresponding to the approximated value of a parameter cannot show the superiority of one criterion over another. Instead, overdesign or underdesign, defined as the difference between the optimum designed volume, V*, and the optimal volume, 8, corresponding to the true value of the parameter, should be used to compare the design criteria. We have a n overdesign if the difference is positive and a n underdesign if it is negative. For the present problem, if the interval of uncertainty in k is 0 . 2 , the total cost as a function of k is given in Figure 2. T h e cost resulting from the expected relative sensitivity criterion is closer to the locus of minimum costs over the range of k = 0.15 to k = 0.3 where the minimum cost is lower. From Equation 13, it can be calculated that the optimal volume is larger than the optimum designed volume when k is smaller than 0.15 and smaller
0.3
0.2
0.I
While applying the expected relative sensitivity criterion, the optimum designed volume, V*, is obtained from min
12,075
(14)
where f ( k ) is a density function for k . If k is distributed rectangularly in the interval, ( 0 . 2 - A k ) to (0.2 Ak), f ( k ) is defined as:
f(k) =
-- V.’
Rate constant,
Figure 2.
(hr. j’
Deviation from minimum total costs
when k is larger than 0.1 5. Therefore, there is a n underdesign if k falls in the higher cost region and an overdesign if k falls in the lower cost region. Both underdesign and overdesign result in a cost higher than the minimum cost. However, in choosing a n appropriate design criterion, the one which results in a larger underdesign in the higher cost region and a smaller overdesign in the lower cost region is usually preferred when comparing the results obtained by the two design criteria. As shown in Figure 2, the present criterion based on the relative sensitivity meets such a preference because it takes into account the magnitude of minimum costs. Example 2. Optimal Temperature for an Exothermic Reversible Reaction. Recently, Ray and Aris (1966) investigated the effect of the errors in the kinetic constants on the maximum reaction rate for a general homogeneous exothermic reaction. Their results allow one to decide, keeping the deviation from the maximum reaction rate within a certain tolerance, how tight the temperature control should be if kinetic constants are known exactly, how accurate the estimates of kinetic constants should be if the temperature is perfectly controlled a t its optimum corresponding to the approximated values of kinetic constants, and the combined effect of parameter and control errors. However, a question may be asked: “What will be the best choice of the reactor temperature so that the deviation from the maximum reaction rate is within a critical tolerance when the kinetic constants are not known exactly but are known to fall within a certain range?’’ A reasonable criterion is to choose a temperature which minimizes the maximum deviation from the maximum
Table 1. Optimum Overdesign Obtained from Expected Cost Criterion and Expected Relative Sensitivity Criterion Expected Relative Expected Cost Criterion Sensitivity Criterion Range of Optimum Amount of Optimum Amount of Uncerlainty, designed overdesign, designed overdesign, 2Ak, Hour-‘ 70 volume, liters volume, liters % 0.05 0.15 0.19 0.20
10,110 11,140 12,100 12,425
1.1 11.4 21 .o 24.3
10,090 10,950 11,780 12,075
0.9 9.5 17.8 20.8
4
I2
8 Activation Energy,
Figure 3.
E
x
16
(5t u./ib mole)
Relative sensitivity o f maximum reaction rate VOL. 7
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1968
51
reaction rate-Le., the criterion as given in Equation 9. An illustration is shown below. Consider a first-order reversible reaction, A B. Let x be the mole fraction of B, and k and k' be the forward and backward reaction rate constants, respectively. If the Arrhenius form is employed, they are expressed as
The necessary equations for finding an appropriate temperature for a general homogeneous reaction may be derived. Ray and Aris (1966) have shown that the optimal temperature for a general homogeneous, reversible, exothermic, and elementary reaction can be obtained from 7 ,
k = A exp (-E/RT),
k' = A'exp(-E'/RT)
X)
= k(1
X
+ 1/n ln(1 + l/p)
(21)
and the relative sensitivity of the maximum reaction rate is
Then the reaction rate expression is r(T,
=
S(r;
- X ) - k'x
= 1
(18)
For an exothermic reaction, E' > E, a t a given composition, there exists a n optimal temperature for the maximum reaction rate, obtained from br/bT = 0, and given by
1
- r(7; n , p ) / r ( r , ; .,PI
=
+ p expL-41
4-p) (7 - T,)] (1 + P ) exp [-nP (7 - 7 m ) I
(22)
where at
T , = (E'
-
A'E' E)/(Rln[ A E (1 ~
~
r =
x
- X ) 11
(yj - / 3 5 ) / ( ~ 5 and p = -EE/nAH
The maximum reaction rate is then expressed as r(T,, X ) by substituting Equation 19 into Equation 18. The system parameters of this reaction are E, E', A, and A', but they are related to the thermodynamics constants. The activation energies, E and E', are related by E - E' = AH, where AH is the heat of reaction, and A / A ' = K* can be obtained from the equilibrium constant of the reaction, K,, as
K,(T)
=
-
K* exp(-AH/RT)
T h e thermodynamic constants, AH and K,, are easier to obtain than the kinetic constants and are usually known. If a n experiment is performed and a rate constant, k , a t a certain temperature is obtained. then A can be related to E. Consequently, the four parameters are now reduced to one independent parameter, E. As a numerical example, let AH = -10,000 B.t.u./lb. mole, K* = A/A' = 0.0126, and from an experiment, A exp (-E/ 2000) = 1690. Consider E as the independent parameter which is estimated to fall in the range from 4000 to 16,000 B.t.u.,'lb. mole. For a specific value of E and for a specified yield of B, say x = 0.8,the optimal temperature, T,, can be calculated from Equation 19 and the reaction rate from Equation 18. Now the appropriate temperature for this reactor design with uncertainty in E may be found from Equation 9 as S(T*) = min T
(
max [I 4000 5 E 5 16,000
- r(T,E)/r(T,,
E)]}
(20)
T o obtain the minimax of the relative sensitivity, some optimization techniques may be used. For illustration, the relative sensitivity as a function of E is calculated for a given T and the maximum of the curve obtained is located. As shown in Figure 3, the minimax of the relative sensitivity, S(T*), is 10% occurring a t the intersection of E = 4000 and T * = 775' R. This simply means that if the reaction temperature is controlled a t 775' R., a deviation of at most 10% is obtained from the maximum reaction rate for the range of E considered. Any other choice of temperature will cause larger maximum deviation. T h e design criterion given above can provide not only an appropriate choice of the decision variable but also the maximum deviation from the optimal value. If this maximum deviation is more than can be tolerated, a more accurate estimate of the parameters is needed. Using the above example, if the maximum deviation of 10% is too large, the range of E must be further narrowed-for instance, to between 6000 and 14,000-then T * becomes780' R., and the maximum deviation is reduced to 5%. 52
-AH/RT, X = ln(IICj/K*), n =
l&EC PROCESS DESIGN A N D DEVELOPMENT
Substituting Equation 21 in Equation 22, one obtains
S(r; n,p) = 1
+ (1 + -
pY+P
exp[-np(r
-X)],
{exp[- n(r
-X)]-
PP
11
(23)
where n and p are independent parameters of the reaction which are related to the reaction orders and the activation energy, respectively. These two parameters must be obtained experimentally and are usually very difficult to determine accurately. Their values, however, may be estimated within a certain range, as n, 5 n 5 n* andp, 5 p < p * , Then the design criterion of Equation 9 becomes
4
S(T*) = min n*
max n 5 n*
5
P* IP SP*
[S(7;
PI
1
741
(24)
Numerical computation has shown that the maximum of S ( r ; p, n) for any r occurs a t one of the four points: (p*, n*), (p*, n*), (p*, n*), and (p*, n*), since it is a convex surface. This makes the task of finding a minimax much easier, since Equation 24 may be reduced to ~ ( r *= )
min (max[S(r; p*, n*), S ( r ; p*, n*), 7
S ( r ; p*, n*), S ( r ; P * , n * ) l ]
(25)
T o obtain r*, one has only to calculate those four values of S for each r and find r* which corresponds to the smallest value among the maximum S. Conclusions
The design criteria of Equations 8 and 9 are proposed for the optimal design of systems involving uncertainty in the values of parameters. Using the relative sensitivity function, the design criteria take into account the magnitude of the minimum cost for each value of a parameter. Unlike the expected cost criterion, the expected relative sensitivity criterion gives the minimum average normalized deviation from the optimal behavior over the range of uncertainty. This criterion should be particularly effective when one is seeking a decision for the system with inaccurate or changing values of the parameters in which no critical tolerance problem is present. When it is desirable to keep the deviation from the optimum within a certain tolerance, the minimax criterion of Equation 9 is especially useful. This criterion gives the minimax of the deviation from the optimal behavior, from which one can decide whether a more accurate estimate is needed by comparing the minimax with the tolerance limit.
Two examples illustrated here have demonstrated the usefulness of the proposed design criteria. T h e optimum designed volume for a reactor was obtained by the expected relative sensitivity criterion. T h e result so obtained gives a larger underdesign in the higher cost region and a smaller overdesign in the lower cost region in comparison with that obtained by expected cost criterion. By using this criterion, designers can make sure that the result obtained has a smallest average normalized deviation from the optimal behavior over the range of uncertainty. T h e optimal temperature for a reversible and exothermic reaction has been determined by the proposed design criterion. T h e optimal temperature to keep the deviation from the maximum reaction rate within a certain limit is found over the range of uncertainty in activation energy. I n addition, the necessary equations for finding a n appropriate temperature for a general homogeneous, reversible, and exothermic reaction by using the proposed design criterion have been obtained as indicated by Equations 23 and 25.
T h e authors express their gratitude to the Office of Coal Research, Department of the Interior, Washington, D. C., for financial support. Nomenclature
frequency factor for forward, backward reaction cost CA, C, concentration E, E’ activation energy for forward, backward reaction f(w) = density function for w FA0 = feed rate of component A J = objective function G
= = = =
= reaction rate constant for forward, backward reaction
= A/A’ = chemical equilibrium constant - reaction rate r R = gas constant s = relative sensitivity SE = expected value of S, Equation 7 T , T* = temperature T, = optimal temperature = reactor volume - parameter W - mole fraction x dimensionless extent of reaction = conversion of component A
v,v*
x x,
=
GREEKLETTERS pj
&
Ak e0 7
Acknowledgment
A,A’
k, k‘ K* K,
= = = = = = = =
stoichiometric coefficient of species j forward reaction order with respect to species j backward reaction order with respect to species j heat of reaction range o f k decision variable optimal decision dimensionless temperature
literature Cited
Kittrell, J. R., Watson, C. C., Chem. Eng. Progr. 62 (4), 79 (1966). Levenspiel, O., “Chemical Reaction Engineering,” p. 134, Wiley, New York. 1962. Luce, R. D.,’Raiffa, H., “Games and Decisions,” Chap. 13, Wiley, New York, 1957. Ray, W. H., Aris, R., IND.ENG. CHEM.FUNDAMENTALS 5 , 478 ( 1966). Rohrer,’ R. A., Sobral, M., Jr., ZEEE Trans. Automatic Control AC-10, No. I , 43 (1965). RECEIVED for review March 3, 1967 ACCEPTED July 31, 1967
OPTIMIZATION OF YIELD THROUGH
FEED COMPOSITION HCN Process B. Y . K . PAN AND R. G . ROTH Hydrocarbons and Polymers Division, Monsanto Co., Texas City,Tax.
Andrussow H C N process has been widely used to proThis is a n autothermal process involving the reactions of ammonia, methane, and air over a platinum and rhodium gauze catalyst. T h e over-all reaction can be expressed as: HE
T duce hydrogen cyanide. CHI
3 + NHI + 1
0 2 -+
HCN
+ 3Hz0
(1)
AH = -102,464 B.t.u./lb. mole of H C N a t 1150’ C. Several investigators, notably Chrttien and Thomas (1948), Maffezonni (1953), and Mihail (1957), have done a considerable amount of work to determine the effects of feed composition on H C N formation. They used various compositions by C H I , and air or adjusting the.relative concentrations of “3, oxygen in the feed. Lack of fundamental knowledge and
effective techniques, however, prevented optimization of yield. Usually a number of feed ratios were tested in the reactor, samples of both feed and off-gas were analyzed, and the reactor performance was evaluated from these data. A favorable feed composition was then chosen, based on these trial and error tests. Such techniques required much labor and time and gave no assurance of optimum conditions. This investigation was oriented to gaining insight into the reaction system and thereby establishing a method for the optimization of H C N yield. Reaction temperatures in conjunction with air/(CHI “3) mole ratio were the most reliable variables in controlling the reaction system, and a simple and effective method of optimization was developed. This was accomplished by theoretical considerations, followed by experimentation, mathematical correlation, statistical analysis, and chemical interpretation.
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