Efficiency Uncertainty and Distillation Column Overdesign Factors Peter K. Lashmet* and Slawomir 2. Szczepanski Systems Engineering Division, Rensseiaer Polytechnic institute, Troy, New York 12181
Probabilistics are applied to binary column design to estimate overdesign factors which account for uncertainty in stage efficiencies. Murphree vapor efficiencies derived from the American Institute of Chemical Engineers correlation are shown to overpredict the observed efficiencies by about 5% on the average. The ratio Eobd/Epred has an estimated standard deviation of about 0.11. Efficiencies are assumed to be normally distributed, and the approximate statistical distribution of the number of stages required for a desired separation is obtained by Monte Carlo simulation. Overdesign factors ranging from 1.07 to 1.28 for 90% confidence in system design are shown to vary with the deterministic number of stages required, and to be approximately independent of system characteristics.
Introduction The design of chemical processes requires much data, most of which are obtained from specific experiments or are estimated from correlations. The resulting design contains uncertainties, the magnitudes of which may be unknown. To reduce the possibility of inadequate system performance, overdesign factors are applied to account for these, uncertainties in the design parameters. Often, these safety factors are based on experience or tradition and are only partially related quantitatively to possible causes of system malfunctions. This investigation considers overdesign factors for the design of distillation columns to account for uncertainty in stage efficiencies. A prior investigation (Rudd and Watson, 1968) considered overdesign factors which minimize expected annual costs in such systems. In a related study Saletan (1969) demonstrated the use of probability distributions for design parameters in estimating the cumulative distribution function for distillation column throughput. The present work considers estimating the number of real stages required to effect a given separation under specified operating conditions with 90% confidence assuming uncertain stage efficiencies and conventional stage-to-stage calculations as the process model. Uncertainty in Stage Efficiencies In the 1950’s the American Institute of Chemical Engineers undertook a research program to develop methods for predicting Murphree stage efficiencies for bubble-cap trays using fundamental mass transfer concepts. The resulting correlations for efficiencies, defined for the vapor phase as
E k ( ~ n Yn+i)/(Yn* - ~ n + 1 ) (1) include effects of physical properties, tray hydraulics, liquid mixing, and entrainment (Gerster, et al., 1958). The procedures are well documented in a design manual (AIChE Distillation Subcomdttee, 1958) as well as in standard texts (Smith, 1963; Van Winkle, 1967). However, little comparison between the predicted and observed efficiencies exists. Table I summarizes experimental efficiency data presented in the final research report (Gerster, et al., 1958). Data points (125) from commercial as well as research columns and for chemical systems having relative volatilities ranging from 1.2 to 12 are available for analyzing uncertainty in the predictions. Vapor efficiencies predicted for the operating conditions of the systems of Table I are compared in Figure 1 with the experimentally determined efficiencies. Two qualita-
tive trends are evident in Figure 1: (1) the correlation slightly overpredicts the efficiency on the average; (2) data at high efficiency are more dispersed than those at low efficiency. Quantitatively, the 95% confidence intervals of the statistical parameters of the data using a zeroorder regression are 0.9304 I Eobsd/Epred I 0.9682 with mean 0.9493, and 0.009012 I u2 I0.01485 with data variance of 0.01139. These estimates of the statistical parameters assume normally distributed behavior at each efficiency level and increased standard deviation for the larger efficiencies. This behavior of the prediction uncertainty is shown more clearly in Figure 1 by the estimated 99% tolerance limits on 95% of the data (Owen, 1962). Statistically, the efficiency ratio is significantly less than unity, thereby supporting the observation of overprediction by the correlation. The above information is applicable to trays operating under conditions for which they were designed. In this situation, an estimate of the actual vapor efficiency may be derived from the reported correlations and
(2)
= Epred(a-k
where unbiased estimates of a and u2 are 0.9493 and 0.01139, respectively, and z is the random normal deviate with zero mean and unit standard deviation. As noted by Gerster, et al. (1958), efficiencies of trays of unusual design or operated under unusual conditions may deviate significantly from the predicted values. This study considers those systems for which the correlations are intended. System Model and Simulation The binary distillation column is assumed to follow the McCabe-Thiele difference equations (Smith, 1963) ~n
+i
= (L/V)xn
- -
Y n + 1 = (L/V)X,
+ (D/V)X, -w
n x ,
(3)
and all streams including the reflux and reboiled vapor are assumed saturated. Equilibrium is represented by the relative volatility equation y* = a x / [ l
+ (a - 1)x]
(4)
and performance of each tray is represented by the Murphree vapor efficiency (eq 1). The simulation evaluates the number of stages required for a given separation under specified conditions: (1) relative volatility, (2) Murphree vapor efficiency, (3) reflux ratio, (4) distillate composition, (5) feed composition, (6) recovery of light component, (7) quality of feed. Table I1 summarizes parameter Ind. Eng. Chem., Process Des. Develop., Vol. 13, No. 2, 1974
103
Table I. Sources of Efficiency Data No. of points,
Source
Trav
Column
Relative volatility
Chemical system
115, 3-in. round
Methylene dichloridebubble caps ethylene dichloride 88 (or 78), 4.375-in. Benzene-toluene round bubble caps 17,3-in. round bubble n-Pentane-p-xylene caps 17,3-in. round bubble Acetone-benzene caps 37,4-in. round bubble Cyclohexanecaps n-heptane
Eastman Kodak Co.
5.5-ft diameter with 60 trays Sun Oil Co. 8 f t diameter with 43 trays (split) University of Delaware 2-ft diameter with 2 trays University of Delaware 2-ft diameter with 5 trays Fractionation Research, Inc. 4-ft diameter with 10 trays
3 .6-4.3
9,O
1.2-2.5
6,o
12
12, A
1.4-3.6
34,V
1.5-1.7
64,O
I
I-
I
-
51.4/
I
I
I
I
I
I
I
I
I
I
I
I
I
1
1
1
I
1
1
symbol (Figure 1)
I
1
,
I
00
t
p 1.2 w k 1.0 W
;
0.8
U
> n 0.6
>
0.1
a
w
v)
0.4 0.4
0.6 0.8 PREDICTED
1.0 1.2 1.4 1.6 1.8 VAPOR EFFICIENCY (WET)
Figure 1. Comparison of observed and predicted efficiencies showing estimated 99% tolerance limits on 95% of the data. ranges considered, Feed quality is taken as unity and light component recovery is taken as 0.8 or 0.9. The nominal efficiencies of all trays, whether in the rectifying or stripping sections, are assumed the same and equal to values predicted from the reported correlations (Gerster, et al., 1958). The stage requirement based on this predicted or nominal efficiency is considered the base or deterministic value for determining overdesign factors: In the simulation considering efficiency uncertainty, the actual efficiency for each stage is derived from the predicted value using eq 2. Thus a single mean efficiency applies to all stages in the column, although individual tray efficiencies differ in a random but normally distributed manner. This procedure is supported by the following arguments. (1) While trays of a specific design are expected to behave similarly, identical performances are unlikely because of differences arising in manufacture. Efficiency data (Table I) were obtained on single trays or by averaging the performance of a few. Thus the comparison (Figure 1) and eq 2 are not expected to apply to large groups of trays but to reasofiably represent single trays. (2) Although the actual performance of each tray is unknown, the best estimate of its behavior for use in process design is obtained from the correlations as reported. To approximate actual behavior, the overprediction of the correlations, noted in this study and generally not included in the process design, is properly included in the simulation and not in the base case. Although the probability of eq 2 providing a negative efficiency is low, a lower bound on all efficiencies used in the simulation is taken as 0.1. No upper bound on efficiencies is required as efficiencies greater than unity are possible. However, the probability of obtaining large efficiencies from eq 2 for the nominal efficiencies selected (0.2 to 0.9) is small. For example, the probability of observing an efficiency greater than 1.25 for a nominal efficiency of 0.9 is less than 0.01. 104
Ind. Eng. Chem., Process Des. Develop., Vol. 13, No. 2, 1974
1
I
32
W
1
1
1
1
1
1
1
I
I
I
I IO 3 0 5 0 70 90 99 99.9 CUMULATIVE PER CENT OF OBSERVATIONS
Figure 2. Typical cumulative probability distribution for required number of stages: relative volatility, 1.5; Murphree vapor efficiency, 0.7; reflux ratio, 5.0; distillate composition, 0.95; feed composition, 0.5; fraction recovery of light component, 0.9; liquid fraction in feed, 1.0. Table 11. Ranges of Parameters Studied ~~
Relative volatility Reflux ratio = 10 Reflux ratio = 1.3 (minimum reflux ratio) Reflux ratio = 2.0 (minimum reflux ratio) Murphree vapor efficiency Reflux ratio Distillate composition Feed composii,ion
1 . 2 to 100 1 . 2 to 8 1 . 2 to 8 0.2 t o 0 . 9 5 to 100 0 . 6 to 0.99 0 . 4 to 0 . 7
It is evident from the following that the relation between stage efficiencies and number of stages is nonlinear. (1)For a system having linear equilibrium y* = mx
+b
an analytical solution to eq 1 and 3 is obtainable. Introduction of individual stage efficiencies into the stage-to-stage procedure of Souders and Brown (1932) yields for each section of the column an equation of the form Y.Y+l
- Y1
where s
Il = Il [ E ,
+ A(l
- E,)]
1-1
According to this model, Ej’s have the same mean and variance, although they are independent of each other. There appears to be no simple relation between these statistics and the mean and variance of N . (2) The equilibrium relation used in the simulation, eq 4, is nonlinear. The absence of a linear relation between the efficiencies and number of stages precludes analytical treatment of the statistics, so Monte Carlo simulation using an avail-
Table 111. Effect of Relative Volatility on Overdesign Factor at Constant R/R,i, System parameters Murphree vapor efficiency 0.90 1.3 (minimum reflux ratio) Reflux ratio Distillate composition 0.90 Feed composition 0.50 1.oo Fraction liquid in feed 0.90 Fraction recovery of light component Overdesign factors obtained from cumulative distribution Estimated required stages, Reflux Deterministic 90% probability Overdesign Relative of succew factor volatility ratio no. of stages 1.2 1.25 1.3 1.4 1.5 1.6 1.8 2 .o 2.5 2.8 3 .O 3.5 4 .O 5 .O 8.0
10.14 8.06 6.67 4.94 3.90 3.21 2.34 1.82 1.13 0.896 0.780 0.572 0.433 0.260 0.0371
51.7 42.3 36.3 28.6 24 .O 20.9 17 .O 14.7 11.4 10.4 9.78 8.76 8.20 7.14 6.46
able generator of random normal deviates (Ellsworth, 1972) is employed. Monte Carlo simulation of a typical system is shown in Figure 2. Column calculations were repeated 200 times, the resulting stage requirements were ranked, and the cumulative fraction of observations (abscissa) were estimated from
F = ( i - 0.5)/200 The adjustment constant 0.5 has been proposed (Gumbel, 1958; Hahn and Shapiro, 1967) to account for noncontinuous data which form a finite sample of the population. Figure 2 is an approximation to the cumulative distribution function for stage requirements, so the number of stages which provide a given confidence for success can be estimated directly. Thus for the system considered, an estimated 90% confidence in the system performing as well as expected requires the minimum specification of 35.6 stages. The nominal efficiency of 0.7 gives a deterministic design requirement of 33.0 stages, so an overdesign factor to account for overprediction and uncertainty in the efficiency correlations is about 1.08. Selection of the 90 percentile to estimate overdesign factors is arbitrary. However, confidence in the number of stages corresponding to the selected percentile depends largely upon the number of points produced in the simulation. Cumulative distribution functions obtained by Monte Carlo simulation are not well defined at low and high percentiles (cf. Figure 2). In this study use of 200 experimen'ts per simulation provides 20 points to describe the distribution function for percentiles greater than 90%, so the 90% point is fairly well defined. An analysis of variance technique designed to test normality (Shapiro and Wilk, 1965) shows high probability (about 0.95) that the data of Figure 2 are drawn from a normally distributed population. Hence statistical procedures based on the normal distribution are applicable to estimate the overdesign factor. However, the basic stageto-stage calculation procedure is nonlinear with respect to the random variable (stage efficiency), so the approximately normal distribution obtained in this case is perhaps fortuitous. Tests on other Monte Carlo simulations
55.6 45.6 39.2 30.9 26 .O 22.6 18.5 16.1 12.6 11.5 10.9 9.89 9.04 8.11 7.49
1.07 1.08 1.08 1.08 1.08 1.08 1.08 1.09 1.11 1.10 1.11 1.13 1.10 1.14 1.16
L
c L3 1.15 0 K
w
2 1.10 I.0 5
2
IO
100
NUMBER OF STAGES
Figure 3. Overdesign factors for estimated 90% probability of success: 0, relative volatility, R = 10; C ) , relative volatility, R / R,i, = 1.3; 8 , relative volatility, R/R,,, = 2.0; 0,Murphree vapor efficiency; 0 , reflux ratio; A, distillate composition; v, feed composition; approximate overdesign factor, eq 6, --. in this study show probabilities of samples being withdrawn from normal populations to be as low as 0.3. As a normal approximation is not assured, the cumulative distribution plots are used directly.
Results and Discussion Monte Carlo simulations using 200 experiments per study were conducted for parameter ranges given in Table 11. As an example, the effect of different relative volatilities for reflux ratios 30% greater than the minimum, a typical design value, is given in Table 111. These are consistent with similar studies conducted for twice the minimum reflux ratio and for a constant reflux ratio of 10, and show that overdesign factors for an estimated 90% confidence in the design increase as deterministic stage requirements decrease. The same trend is exhibited by the other design parameters as seen in Figure 3. It appears that the factor, which varies from about 1.07 to 1.28, is dependent primarily upon the conventional process design stage requirement and is reasonably independent of the separation desired or the operating conditions employed. Ind. Eng. C h e m . , Process Des. Develop., Vol. 13, No. 2, 1974
105
Examination of systems having a large number of stages yields an insight into the observed behavior of the overdesign factor. As the number of required stages becomes large, the equilibrium relation, eq 4, becomes approximately linear with respect to the operating lines, eq 3, and these two sets of equations become approximately parallel; Le., mV/L approaches unity. In this situation, eq 5 simplifies, and the column is approximated by an effective overall efficiency having approximately the same mean as the stage efficiencies but a variance of $ / N (Deley, 1973). Thus overdesign factor Y
(Eobd
/ Epred)-'
and for 90% confidence, the approximation becomes overdesign factor
ll(0.9493 - 1 . 2 8 2 w N )
(6)
This overdesign factor approaches an asymptote of 1.053 for infinite stages and about 1.065 for 100 stages, in agreement with the data of Figure 3. As the number of stages decreases, the overdesign factor increases because of the increased variance in eq 6. Furthermore, as the number of stages becomes small, the operating lines, eq 3, and the equilibrium relation, eq 4, diverge, the parameter m VIL deviates further from unity on the average, and eq 6 becomes a poorer representation of system behavior as seen in Figure 3. Thus while approximate analysis yields qualitative behavior, Monte Carlo simulation is required to prepare the correlation because of system complexity. Conclusions This study demonstrates the use of Monte Carlo simulation in estimating process overdesign factors. Approximate cumulative distribution functions resulting from uncertainty in design parameters are derived. These distribution functions permit estimation of overdesign factors at the desired level of confidence without the assumption of Gaussian or normal behavior. Thus the method is applicable to complex, nonlinear systems in which analytical evaluation of the statistical properties is difficult or impossible. Although a single uncertain parameter is considered, the method is applicable in multivariable systems to estimate overall behavior and interacting effects between the uncertain parameters. Nomenclature A = mV/L a = mean of E o b s d / E p r e d
106
Ind. Eng. Chern.,
Process Des. Develop.,
B = flow rate, bottoms product b = intercept of linear equilibrium relation D = flow rate, distillate product E = Murphree vapor efficiency of stage F = cumulative fraction of observations i = rank of calculated stage requirement in simulation 4 = liquid flow rate, rectifying section L = liquid flow rate, stripping section m = slope of linear equilibrium relation N = numberofstages R = refluxratio V = vapor flow rate, rectifying section V = vapor flow rate, stripping section x = liquid composition y = vapor composition y* = vapor composition in equilibrium with x z = standard random normal deviate Greek Letters a = relative volatility Il = product u = standard deviation Subscripts B = bottoms product D = distillate product min = minimum n = position or stage number in column obsd = observed pred = predicted Literature Cited AlChE Distillation Subcommittee, "Bubble-Tray Design Manual," American institute of Chemical Engineers, New York, N. Y., 1958. Deley, A. W., M.Eng. Project Report, Rensselaer Polytechnic Institute, Troy, N. Y., 1973. Ellsworth, J. H., private communication. Gerster, J. A,, Hill, A. B., Hochgraf, N. N., Robinson, D. G., "Tray Efficiencies in Distillation Columns, Final Report, University of Delaware," American institute of Chemical Engineers, New York, N. y . , 1958. Gumbel, E. J., "Statistics of Extremes," Columbia University Press, New York, N. Y . , 1958, pp 29-31. Hahn, G. J., Shapiro. S. S., "Statistical Models in Engineering," Wiley, New York, N. Y., 1967, pp 292-294. Owen, D. B., "Handbook of Statistical Tables," Addison-Wesley, Reading, Mass., 1962, p 127. Rudd, D. F., Watson, C. C., "Strategy of Process Engineering," Wiley, New York, N. Y., 1968, pp 348-352. Saletan, D. I., Chern. Eng. Progr., 65 (5), 80 (1969). Shapiro, S. S., Wilk, M . B., Biornetrika, 52, 591 (1965). Smith, B. D., Design of Equilibrium Stage Processes," McGraw-Hili, New York, N. Y., 1963, pp 124-127,577-611. Souders, Mott, Jr., Brown, G. G., Ind. Eng. Chern., 24, 519 (1932). Van Winkle, M., "Distillation." McGraw-Hili, New York. N. Y.. 1967, pp 553-555.
Receiued for reuiew January 17, 1973 Accepted November 7,1973
Vol. 13,
No. 2, 1 9 7 4
