Enthalpy Prediction of Mixtures Using BWR Equation of State with Additional Analytical Functions Jacob F. Ruf Information Systems Development, Inc., 3430 Broadway, Kansas City, Jfo. 64111
Fred Kurata Department of Chemical and Petroleum Engineering, University of Kansas, Lawrence, Kan. 66044
Thomas F. McCall C. TP. Sofsinger Co., 307 East 63rd St., Kansas City, Mo. 64113
A method i s proposed and checked against published data for calculating the effect of pressure on enthalpy using a slightly modified Benedict-Webb-Rubin equation of state. Polynomial functions are used for the coefficients Co and y and their corresponding derivatives. The results agree well with the previously published experimental data.
T h e Uenedict,-Webb-Rubin (1940) equation of st’ate is commonly used to derive the enthalpy departure from ideal gas to real vapor and liquids, both for mixtures and pure components. The originators of the BWR equation emphasized t’he importance of adjusting Cofor low reduced temperatures. Stotler and 13enedict (1953) reported COvalues a t several temperatures which they recommended for use in the prediction of methane/riitrogen vapor-liquid equilibrium. Earner and Schreiner (1966) and Barner arid Xdler (1968) presented Coas a function of temperature in graphical form for a number of light hydrocarbons. The same authors emphasized the importance of evaluating the derivatives of Co and y to evaluate enthalpies a t lo^ reduced t,emperatures. This method apparently relies on using tabular values of Co and y and finite difference techniques. Sehgal e t al. (1968), using data from the Thermal Properties of Fluids Laboratory a t the University of Michigan, have tested t’he values calculated from the BWR relationship, as well as from other methods, against experimental data for a number of systems. They showed that the enthalpy departure values predicted from BWR in its original form disagreed with the experimental values significantly a t low temperatures. Orye (1969) published a set of polynomial coefficients he recommended to represent Ca as a function of temperature for 16 low-molecular-weight hydrocarbons and four light gases. A part of Orye’s recommended procedure involves modifying the mixing rules for A,,. Mot’ard and Organick (1960) recommended making y a function of temperature for hydrogen to fit vapor-liquid equilibrium data for hydrogen with hydrocarbons. I n fact, it was this work which led the authors to consider a similar procedure to improve the fit of the helium data published in Ruf’s 11sthesis (1967). Starling and Powers (1970) published still another modification of the BWR equation. However, to use this method, all parameters must be redetermined simultaneously. It is imTo whom correspondence should be addressed.
portant to note that in this present work none of the original B K R mixing rules have been changed. Likewise, the methane and propane coefficients, excepting Co, are those published by the original authors. Polynomial coefficients of Co for methane and propane were determined from a curve fit of tabular data supplied by Xotard (1961). The helium and nitrogen coefficients were determined by a special technique which could well be applied to other systems. 1 detailed discussion of this technique is presented in Ruf’s M S thesis (1967). Very briefly, the technique involves a n iterative process whereby alternate sets of BWR coefficients derived from P V T data are tested against vapor-liquid equilibrium data to select the “best” set of basic coefficieiits, after which the polynomial coefficients, expressing y as a function of temperature, are determined with the use of a n iterative leastsquares approach. The objective of the work present’ed here was to incorporate the polynomial eq7ations for Co and y as functions of temperature into the enthalpy departure equation to evaluate the derivatives of Co and y with temperature and to compare the results with the conveiiient set of data presented by Sehgal et al. (1968). The data for helium and nitrogen were of particular interest to the authors because of their previous study of the phase behavior of systems containing these components in cooperation with The 11.IT. Kellogg Co. The results of their work were used in the design of the Satioiial Helium Corp. plant a t Liberal, Kan., in 1961. Derivation of Equations
The basic BM’Requation of state is:
P
+ (BoRT - - Co/Tz)d2+ (bRT - a ) d 3 + aQd6 + cd3/T2(1 + yd2) exp ( - y d 2 ) =
RTd
A 0
(1)
Earner and Adler (1968) presented the following equation for calculating the enthalpy departure with pressure a t constant temperature using the BWR equation: Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 1 , 1973
1
Table I. BWR Coefficients Used in This Work (Ruf, 1967)
(All in English units) Component
Helium
Ao”’ Bo
Nitrogen
7 6604720 0 20149362 0 40612568 1 4939885 0 97893347 29 151477 235 76287 0 0 0 6 5448492 0 80959674 (lo-’) 0 37252451 0 55990479 (lo+) Less than -100°F
bll3 a113 a1/3
c1i3
c1 c2 c3 c4
v1
68 71432 0 79401623 0 67627846 5 4847274 1 7801443 441 05677 0 81173498 0
0 0
v2 v3 v4 C or V coefficient temperature rangea Co1/2 = CI, a Above the indicated temperature:
3 8903218 0 27161818 (10-I) 0 13829655 0 24756510 Less than - 100°F
y1/2 =
Methane
Propane
83.6376 0.682401 0.953644 14.3970 0.799567 792.684 1.7122 11.134 0,81499 0.21044 1.24082 0 0 0 Less than - 100°F
160 983 1 55884 1 79396 38 5408 1 35644 2933 60 7 8708 (lo4) 7 0617 -0.73517 (lo-’) -0 15587 2 37597 0 0 0 Less than 50°F
VI.
Table II. Isobaric Enthalpy Differences
(5.2 Mol 7% propane in methane)
H12
ti,
OF
-250
f2, O F
P, psia
- 150
500 1000 1500 2000 500 1000 1500 2000 500 1000 1500 2000 500 1000 1500 2000 500 1000 1500 2000
- 150
- 50
- 50
+50
+50
+150
+150
+250
H T ~ H$’
=
(2 bRT
0.1850
{
- 3 a ) d 2 / 2 + 6 uad5/5 (-yd2)
rd2
TrZ
[
1
- (exp -
y d 2 ) ad2
+
+
- exp ( - y d 2 ) 2
+1
+
dCo ( z) + + y)]$} (2)
rd2exp i-rd2)] + C -
- Hii,
-31.04 -31.11 -31.19 -31.28 -3.40 -3.83 -3.87 -3.92 -0.40 -0.07 -0.12 -0.15 0 0 0 0 0 0 0 0
(BoRT - 2 Ao - 4 Co/T2)d
e[3 1 - exp T i
Colcd AH, H i 2 From dCo/dT
d
A similar equation was derived during this work from the fundamental equation presented by Beattie (1949) : 2 Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 1, 1973
BTU/lb Total
96 81 94 i 6 93 38 92 39 195 59 153 78 111 39 96 90 66 07 88 60 110 60 106 73 58 00 65 14 73 03 79 41 59 64 63 22 66 89 70 32
- Hut This work
Original BWR per Sehgal et al.
82.3 81.5 80.0 77.5 194.0 148.5 106.5 94.0 63.0 89.0 111.5 105.0 57.8 65.25 72.63 79.32 60.0 63.65 67.15 71.30
+17. 6 +16.3 +16.7 f19.2 +0.8 +3.6 $4.6 $3.1 +4.9 -0.4 -0.8 +l.6 +0.3 -0.2 +0.5 $0.1 -0.6 -0.7 -0.4 -1.4
+99.08 $98.77 100.98 +106.39 +4.38 $8.23 + l o . 71 t10.38 +8.05 -0.82 $0.41 +2.46 +0.21 +O.ll +0.74 +0,37 -0.48 -0.53 -0.19 -1.20
sH:ys; =
% Deviation
BTU/lb per Sehgal et al.
[P - T(dP/dT),]dV
+
+ PV
- RT
(3)
The result proved to be identical to the equation of Barner and Adler, Equation 2. The question then arises concerning how the terms dCo/dT and d+y/dT are to be evaluated. The method of Barner and Adler apparently uses the variation of Co and y for a small change in temperature to determine these values. I n the work presented here, the coefficients Co and y have been fitted to polynomials as a function of T and differentials of the resultant equations are used. A least-squares procedure was used to fit Co and y as functions of temperature for a number of components. They are
Table 111. Isobaric Enthalpy Differences
(11.7 Mol % propane in methane)
f l , OF
t2, OF
P , pria
-250
- 150
-150
- 50
500 1000 1500 2000 500 1000 1500 2000 500 1000 1500 2000 500 1000 1500 2000
- 50
+50
+50
+l50
Calcd AH, Ht2 From dCo/dT
- Htl, BTU/lb Total
-35.90 -35.93 -35.98 -36.04 -4.94 -6.27 -6.94 -6.97 -2.40 -1.20 -0.65
E x d AH. Ht2 - Hti, BTU/lb per Sehgal et al.
76 75 73 74 168 126 90 83 91 110 116 103 55 64 76 82
99.97 98.91 98,14 97,58 172,66 127.91 95.60 88.30 90,22 110.82 118,lO 104.96 57,43 66.73 77.07 83.72
-0.71 0 0 0 0
5 6 6 0 60
80 90 70 05 25 7 3 0 25 80 30
% Deviation Original BWR per Sehgal et al.
This work
+149.24 +151.03 +156.98 +154.99 $11.32 f4.92 +19.29 +21.45 +4.11 f2.24 +3.25 $3.16 f4.36 f4.03 +0.82 +2.29
+30.7 +30.8 $33.3 f31.9 +2.41 + O . 87 f5.17 f5.50 -0.91 +o. 52 +1.20 +1.61 +4.42 +3.86 +0.35 +1.73
Table IV. Isobaric Enthalpy Differences
(28 Mol
yo propane in methane) Exptl AH,
11,
t2,
OF
O F
-250
- 150
-150
- 50
+50
- 50
+50
+l50
+150
+250
Calcd AH, Hi2 From dCo/dT
P, pria
500 1000 1500 2000 500 1000 1500 2000 500 1000 1500 2000 500 1000 1500 2000 500 1000 1500 2000
-
H t i , BTU/Ib
Total
-46.14 -46.14 -46.16 -46.17 -10.87 -13.93 -13.94 -13.97 -6.13 -3.24 -3.34 -3.44 f0.03 +0.06 +0.04 +o. 05 0 0 0 0
109 108 108 108 127 80
14 85 65 56 58 81 78 25 76 59 122 53 123 61 102 67 86 97 66 79 91 53 93 99 93 50 57 68 64 46 71 65 76 72
Htz
% Deviation
- Hi,,
BTU/lb per Sehgal et al.
65 10 64 50 64 0 66 0 121 5 74 5 71 4 69 0 117 0 118 55 98 6 85 0 62 0 89 65 95 7 91 7 58 0 64 6 71 7 76 9
Original BWR per Sehgal et al.
This work
$67 +68 +69 +64 +5 +8 $9
6 8 8 5 0 5 6 +ll 0
+297 $300 +303 $291 +27 $48 +51 +54 +13
+4 7
+4 3 +4 1
$8 +7 +7
+2 3 +7 7 +2 1
+6
fl
-1 8
+o
0 5 2 1 2
$2 -0
+2 -0 -0 -0 -0
+o $0 +o
0 4 4 1 65 31 56 71 21 81 56 67 77 73 45 45 41 12 46 59
expressed by the following equations:
Co”’ 71’2
+ C2T + C3T2 + C4T3 = Vl + V2T + V2T2 + V4T3 =
C1
(4)
(5)
The coefficients obtained, along with the other B K R coefficients used for this study, are presented as Table I. Differentiating Equations 4 and 5 results in the following expressions : dCo/dT
=
2 (Cl)(C2)
6 [(CI)(C4)
+ [2 jC2)’ + 4 ICl)(C3)]T +
+ (C3)iC2)1T2 +
4 [(C3)*
+ 2(C2)(C4)1T3+
+ 6 (C4)T
10 (C3)(C4)T4
For pure components, the Equations 6 and 7 may be used directly. For mixtures, only the components for which Co and/or y vary with temperature are considered 111 order to calculate molal average coefficients. When J\ e use the coefficients V1 as anexample: n
(6)
V1 (liquid phase mixture)
= 1
X,V1,
Ind. Eng. Chem. Process Des. Develop., Vol. 1 2 , No. 1 , 1 9 7 3
3
Table V. Isobaric Enthalpy Differences (43.4 Mol % nitrogen in methane)
tl,
O F
f2,
P, psia
O F
-250
- 150
- 150
- 50
500 1000 1500 2000 500 1000 1500 2000 500 1000 1500 2000 500 1000 1500 2000
- 50
+50
+50
+EO
Exptl A H , Htz Hti, BTU/lb per Sehgal et 01.
-5.27 -5.20 -5.22 -5.30 -0.06 -0.22 -0.26 -0.2i 0 0 0 0 0 0 0 0
-4.55 -4.15 -3.91 -3,84 0 -0.55 -0.91 -1.08 0 0 0 0 0 0 0
0
% Deviation
-
Calcd A H , Ht2 - Hti, BTU/lb From dCa/dT From dy/dT Total
137,45 85.05 73.0 65,25 49.25 87.45
132.98 7 i . 13 69.07 66.20 49. i 2 89.67 82.53 74.13 41,04 47.09 52.80 55.43 39.87 42.60 45,23 47.44
84.50 72,25 40.8 47.25 53.50 57.50 39.40 42.0 455 47.5
This work
-3.25 -9.31 -5.38 +1.45 $0.95 +2.54 -2.33 +2.60 t0.59 -0.34 -1.31 -3.60 +1.19 +1.43 -0.59 -0.13
Original BWR per Sehgol et al.
+16,33 f 2 l . 24 +25.27 +34.18 +1.58 f4.40
+3.58
+1.84 + O . 03
-0.44 -0.019 -0.58 $0,652 $0,612 + O , 37 0.42
+
Table VI. Isobaric Enthalpy Differences (24.6 Mol % helium in nitrogen)
fl,
O F
-250
12,
- 150
- 50
-150
- 50
Calcd AH, H I Z From dy/dT
P, psia
OF
+50
40,18 65.56 65.19 61.07 32.27 35.34 38.16 40.64 30,7l 32.09 33,34 34.40
41 .O 64.0 60.8 58.0 32.6 34.9 37.7 40.0 30.4 32.0 33.2 34.4
-2.00 +2.44 +7.22 $5.29 -1.01 f1.26 +1.22 +1.60 $1.02 + O , 28 + O . 42 0
- Hti, BTU/lb
0 -1.26
500 1000 1500 2000 500 1000 1500 2000 500 1000 1500 2000
% Deviation
Total
Exptl A H , Htz - Hti, BTU/lb per Sehgal et a1,
-1.i7
-1.80 0 0 -0.05 -0.21 -0.01 0 0 0
for this work
Table VII. Isobaric Enthalpy Differences
(49.9 Mol % helium in nitrogen)
tl,
4
O F
t2,
O F
- 250
- 150
- 150
- 50
- 50
+50
P, pria
500 1000 1500 2000 500 1000 1500 2000 500 1000 1500 2000
Ind. Eng. Chem. Process Der. Develop., Vol. 12, No. 1, 1973
Calcd AH, Hi2 From dy/dT
0 -0.01 -0.08 -0,24 0 0 -0.02 -0.05 0 0 0 0
- Hti,
BTU/lb Total
42,61 47.65 51.84 55.08 39,08 40,76 42.20 43.42 38.15 39,02 39, i8 40,45
Exptl A H , Htz - H t i , BTU/lb per Sehgal et al.
Deviation for this work
42.0 47.3 56.5 60.0 39.5 41.6 42.4 43.4 38.6 40.0 40.2 40.5
$1.45 + O , 74 -8.25 -8.20 -1.06 -2.02 -0.47 $0.05 -1.16 -2.45 -1.04 -0,12
yo
Table VIII. Comparison of Co as f(t) from This Work with Values Calculated by Authors for Orye's (1969) Coefficients
co x
System and source o f data
10-9
Methane
t,
O F
50 25 0 - 25 - 50 - 75 - 100 - 125 - 150 - 200 -250 - 300
0.2758 0.2758 0.2758 0,2758 0.2758 0.2758 0.2760 0.2753 0.2747 0.2713 0.2606 0.2382
This work
0,2758 0.2758 0,2758 0.2758 0,2758 0.2758 0.2758 0,2758 0.2735 0.2726 0,2595 0,2388
From Orye's coefficients
6.2213 6.2155 6.1950 6.1600 6.1110 6.0484 5.9726 5.8840 5,7832 5.5472 5.2695 4.9553
6.2099 6.2060 6,1917 6.1637 6.1202 6.0604 5.9840 5,8921 5.7859 5.5387 5.2600 4.9682
n
VI (vapor phase miiture)
% dev
Av dev, BTU/lb
M a x dev, BTU/lb
5.8 a
12.2 14.9 82.5 2.6 24.5 115.5 8.1
corrections. These contributions are tabulated in Tables I1 t'lirough T'II. These corrections are opposit'e in sign to the total correction and thus reduce the predicted enthalpy departure and bring the results more in line with the real behavior. The corrections are most significant a t the lowest temperature studied. In the methane-propane system, the contribution due to dCo/dT is 30-4070 of the total correction a t the lowest temperature, whereas in the helium-nitrogen system a t the same temperature level, the dy/dT correction is of no consequence. This method appears to have about the same degree of accuracy as the method of Darner and Schreiner (1966) who state that for the 5.2 mol yo propane in methane system in the range of - 160" to +70"F and pressures from 260-1500 p i a , the absolute average deviation was 5 2 BTt-/lb. Starliiig and Powers (1970) obtained a somewhat better fit of the same methaiie-propane data over the entire temperature and composition range of the data by modifying the mixing rules. This method requires interaction parameters which are not generally available for all components. Table VI11 compares the Co values for methane and propane calculated from coefficients presented in this paper vs. Orye's (1969) coefficients. 111Table IX, the latent heats for methane and propane calculated from methods preseiited in this work are compared with experimental values from .1PI
I'%VI%
=
Av
5 . 2 mol yopropane in methane (-160 t o +70"F) This work 1.5 1.0 Barner and Schreiner (1966) a 2.0 Sehgal e t al. (1968)* 3.0 2.7 5 , 2 mol yopropane in methane (-250 to +250°F) This work 4.7 4.2 Sehgal et al. (1968)* 22.7 20.3 Starling and Powers (1970) 0.8 0.6 1 1 , 7mol yopropane in methane (-250 t o +250°F) This work 7.8 5.7 Sehgal et al. (1968)b 34.7 25.5 Starling and Po\yers (1970) 1.2 1.5 a Not stated. * Based on original BWR coefficients.
Propane
From Orye's coefficients
This work
Table X. Comparison of This Work with Results of Similar Correlations
1
Where n is the iiumber of components and components where y does not vary n i t h temperature are skipped. Comparison of Calculated Values with Experimental Results
I n this study the experimental data from the University of Michigall Thermodynamics Laboratory, as presented by Sehgal et al. (1968), were compared to the values as calculated by the method described. .In additional step, the evaluatioii of the ideal gas enthalpy for the mixtures, was also required. The calculations made for this study used a special curve fit of heat' capacity data from XPI Project 44 (1952) for the particular components. Others interested in using this procedure would probably prefer to use some standard procedure such as is published in the Technical Data Book (1966) of the .Imerican Petroleum Institute. The results of the recommended procedure, presented in Tables I1 through VII, indicate a significant improvement over the values obtained with the original BSTR. .Ilso, the agreement between experimental and calculat'ed values for the helium-nitrogen system were excellent. It is interesting to note the contribution of the terms resulting from the dCo/dT and/or dr/dT to the total enthalpg-
Table IX. Low-Temperature Latent Heats for Methane and Propane Latent heats of vaporization, BTU/lb-mol Compd
Xethane
Propane
f,
O F
-189 -207 -225 -243 -252 -261 -279 22 7 -13 -43 -69 -118
69 69 69 69 69 69 69 8 104 182 71 918 59
Exptl API, 1 9 6 6
2847.42 3071.70 3253,50 3409.20 3480.12 3546.54 3659,40 7257.6 7468,2 7720.2 8076.6 8357.4 8823,6
Orye, 1 9 6 9
2873 3052 3208 3371 3450 3528 3639 7201 7404 7667 8044 8337 8795
52 08 14 04 42 00 42 08 66 10 02 24 52
This work
2832 3079 3285 3444 3508 3561 3635 7264 7452 7671 7982 8240 8764
89 58 55 67 54 72 52 53 37 34 90 85 99
Error in latent heat, BTU/lb-mol Orye, 1 9 6 9 This work
+26 -19 -45 -38 -29 -18 -19 -56 -63 -53 -32 -20 -28
10 62 36 16
70 54 98 5 5 1 6 2 1
-14 $7 '32 +35 f28 +l5 -23 +6 -15 -48 -93 -116 -53
53 88 05 47 42 18 88 9 8 9 7 5 6
Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 1, 1973
5
data and also against tabular values presented by Orye. The two methods have about comparable accuracy. A more detailed comparison of the results of this work with previously published correlations is presented as Table X. Conclusions
The method presented here is convenient for machine computations. The results confirm previous work which indicated t h a t dCo/dT and d y / d T effects on enthalpy should not be neglected a t low temperatures. On the basis of this work a more promising procedure for calculating BWR coefficients would be t o subject all available data for mixtures to the regression analysis method as developed by Ruf (196i) and to express y as a function of temperature rather than Ca for all components. The escellent agreement between experimental and calculated values for the helium-nitrogen system where this method was employed demonstrates the promise and reliability of this approach. Nomenclature
A o , BO,CO,a, b, c, a, y = coefficients of BWR equation in English units C1, C2, C3, C4 = coefficients for the Co polynomial function d = density, lb mol/ft3 H t , = enthalpy a t temperature tl in BTU/lb a t indicated pressure H,, = enthalpy a t temperature t2 in BTU/lb a t indicated pressure = enthalpy a t a given pressure, P , and temperature, T in BTT:/lh-mol H T o = enthalpy of ideal gas (zero pressure) and a given temperature, T in BTU/lb-mol
HiP
- I
P
=
R
=
absolute pressure, psia universal gas constant, 10.7335 (fta) (psia)/(lb-mol)
(OF)
temperature, O F absolute temperature, OR V specific volume, ft3/lb-mol V1, V2, V3, V4 = coefficients for the y polynomial function t
T
=
= =
References
Barner, H. E., Adler, S. B., Hydrocarbon Process., 47 (lo), 150, (1968). Barner, H. E., Schreiner, W. C., ibzd., 45 (6), 161 (1966). Beattie, J. A., Chem. Rev., 44, 141 (1949). Benedict, ll., Webb, G. B., Rubin, L. C., J . Chem. Phys., 8, 344 (1940). Johnson, D. W., Colver, C. P., HydrocarbonProcess., 48 (l),127 (1969). lfotard, R. L., private communication, 1961. Motard, R. L., Organick, E. I., “Tffmodynamic Behavior of Hydrogen-Hydrocarbon AIixtures, A.I.Ch.E. J., 6 (l), 39 (1960). Orye, R. V., Ind. Eng. Chem. Process Des. Develop., 8 (4), 579 (1969). Ruf, J. F., “Low-Temperature Phase Equilibrium for Helium and Nitrogen in Mixtures of Natural Gas Components Using the Benedict-Webb-Rubin Equation of State,” 11s thesis, University of Kansas, 1967. Sehgal, I. J. S., Yesavage, V. F., Mather, A. E., Powers, J. E., Hydrocarbon Process., 47 (8), 137 (1968). “Selected Values of Physical and Thermodynamic Properties of Hydrocarbons and Related Compounds,” API Project 44, A m p r i r i n -PPtrolmim T n q._.t i t i i t~ e -119.52) -_ ,, ---,. ~ Starling, K. E., Powers, J. E., Ind. Eng. Chem. Fundam., 9 (4), 531 (1970). Stotler, H. H., Benedict, M.,Chem. Eng. Symp. Ser., 49 (6), 25 ~
---1
1195.11.
Technical Data Book-Petroleum Refining, American Petroleum Institute, 1966. RECEIVED for review February 22, 1971 ACCEPTED July 17, 1972
REFLEX Method for Empirical Optimization Ronald W. Glass1 and Duane F. Bruley’ Department of Cliewiical Engineering, Clemson University, Clemson, S.C.
While model development i s rarely expedient, practical techniques for computer-based control and optimization without such models are notably few, and the unique character of model-based process control and optimization has prompted considerable theoretical attention. Empirical optimization methods are typically less restricted in applicability than error-free methods, but it i s customarily assumed that the empirical methods are inherently inefficient. The REFLEX empirical optimization method presented i s shown to be both efficient and directly applicable to model-free process control optimization. Both on-line and off-line tests of the REFLEX operation are given as appropriate measures of the performance of the optimization method.
w i t h o u t notable exception the most generally applicable concept for empirical optimization is that of evolutionary operation. G. E. P.Box (1957) introduced the use of EVOP as a means of increasing industrial productivity, while Spendley et al. (1962) adapted the EVOP concept for computer utilizaPresent address. Oak Ridge National Laboratory,
Box X, Oak Ridge, Tenn. 37830. T o whom correspondence should be addressed. 6 Ind.
Eng. Chem. Process Des. Develop, Vol. 12, No. 1 , 1973
tion. -1more recent modificat!on of Spendley’s “sequential” operation is that of Xelder and Mead (1965) which they gener.cally term a “simples” method. Although the method as presented by Selder and Mead deals exclusively with errorfree function m nimization, the concept of the method is EVOI’ procedure. clearly in the spirit of Box’s original Evolutionary operations are basically gradient techniques, but they do not require precise functlorl evaluations as 0111p relative magnitudes are considered. Inasmuch as the gradient
