I
ARTHUR C. HOMMEI and DONALD
F. OTHMER
Polytechnic Institute of Brooklyn, Brooklyn, N. Y.
Sulfuric Acid
. . . Optimized
Conditions in Contact Manufacfure
The most profitable operation can be achieved through a simulation program designed to reflect shifting uncontrollable variables H O W TO MAKE THE MOST MONEY POSsible with existing equipment is the prime concern of every chemical producer, but there are factors over which he has little or no control even in the newest plants, from optimal computer design. Little can be done to govern raw material costs and quality, cost of services, equipment efficiencies, and even ambient temperature. The problem is resolved to finding combinations of factor settings, which can be controlled to result in the most profitable operation, and fixing these
settings a t the optimum under shifting uncontrollable variables. This problem may be of such magnitude that it can be solved only with an electronic computer. I n any case, simple mathematical correlations are needed to determine the best control settings. The production of HzS04 from pyrites through contact catalysis is one important case requiring study. Several steps in its analysis are discussed here : .Development of a method for correlating performance of one, catalyst with reference to others
0 Construction of a mathematical model of a contact HzS04 plant, in the form of computer programs which can be “run” or optimized on an IBM 650 computer 0 Generation of the response surface by the method of steepest ascent
The purpose of the study was to learn more about the derivation and interrelation of such control equations and the techniques of computer plant simulation. In this study, the best control settings are those producing maximum profit, defined as sales of acid produced less raw material and other costs.
Catalyst Rate Correlations In setting up a plant simulation computer program, it is advisable to build as much versatility into it as possible. An ideal programmed model of a plant would be able to simulate any number of different catalysts with only minor adjustments. T h e oxidation of SO2 to SO8 has received considerable attention from kineticists, and practically all of the experimental data have been represented by empirical equations. Many of the early correlations ( 7 , 2, 72, 78) cannot be applied with confidence to flow reactors, since they were derived from static system data. Equations of more recent vintage have appeared in the literature ( 7 7 , 74, 79) and can be used for specific platinum catalysts. Calderbank ( 5 , 6) has proposed a rate equation for a catalyst of VZO~-VZ(SO~) (VO), (SO4)3-KaZSO4 on silica gel support: r =
where r = rate of reaction, gram moles S O ~ / g r a mcatalyst-second k l = forward reaction velocity constant kz = reverse reaction velocity constant P = partial pressure, atm. This rate equation was programmed into the plant model. To permit simula1 Present address, Allied Chemical Corp., General Chemical Division, Morristown, N. J.
tion of operations on a catalyst other than Calderbank’s it is necessary either to have a rate equation specific for the other catalyst or find a correlation between Calderbank’s rates and the other catalyst’s rates. The latter approach was taken. Previous work (15, 76) has shown that plotting the property of one substance against the corresponding property of another, under similar conditions, may be used with considerable advantage, since it tends to compensate the mutual irregularities or deviations from ideality
of one “imperfect” material by those of another. Such plots of equilibrium constants and reaction rate constants us. the log of vapor pressure of a reference substance have given straight lines in ranges over 1000” C. when plotted a t the same temperature as the vapor pressure of the reference substance. Later work (77) has shown the velocity constants for numerous reactions to follow this same pattern. Because of the influence of gas flow rates, temperatures, and pressures on the reaction rate itself, it seemed desirable
The Authors C o m m e n t Lately, the EVOP and CHEOPS procedures have been much discussed as a means of optimizing a n operating process. Some comparison of these techniques with the technique described here would seem to be in order: EVOP, applied to manufacturing operations, requires experimentation in the actual plant to determine the best set of operating conditions. This seems to have many practical applications, but it also has its limitations. One would be failure to account for subtle uncontrolled economic factors affecting the plant and, therefore, failure to locate the true optimum point. Another disadvantage is the length of time required to perform plant experiments, not to mention the problem of plant upsets. The computer technique does not have these limitations. CHEOPS is concerned with “design” of a plant, punching out a set o f “best operating conditions” to go along with the optimum designed plant. Unfortunately, in actual operation there are always uncontrolled variables to shift the location of the best operating point. When this happens, the design conditions are obsolete and are no longer the best settings.
VOL. 53,
NO. 12
0
DECEMBER 1961
979
H / l /
1 I x 108
I
CALDERBANK
x
RATE
gm. mol S02/grn.-cat.-sec.
Figure 1. Log plot of various catalyst rates vs. Calderbank rate at same operating conditions provides basis for comparison to plot the reaction rate for one catalyst
tion was made that these expressions are valid for any gas composition. Lending support to the assumption is previous work ( 9 ) using the “catalystactivity factor” correlation of Hougen and Watson (70). Sets of experimental data were obtained for a series of temperatures and space velocities from a laboratory integral reactor operating isothermally. Gas composition ranged from 2.1 to 14.3% SO*. Activity factors were calculated by the Hougen and Watson method (70) and plotted against temperature. Figure 2, a similar plot, shows all points lying close to the curve.
Converter Simulation Program Rate of reaction is related to the weight of catalyst required by the fundamental expression :
us. the reaction rate for another catalyst,
always a t the same operating conditions of temperature, pressure, and gas flow rate. On this basis comparison was possible. A short computer program was written to facilitate comparison of derived reaction rates with rates predicted for the Calderbank catalyst under identical operating conditions. Figure 1 is a log plot of the various rates plotted against Calderbank rates.
Using a least square fit, Olson’s (74) and Eklund’s (8) rates were related to Calderbank’s rates by the following expressions:
+
log (Olson r ) = -2.02852 0.514385 log (Calderbank r )
(2)
and
+
log (Eklund r ) = 0.333571 1.01902 log (Calderbank r ) (3) Olson’s work was done a t constant gas strength, while Eklund varied his gas from 3.9 to 9.0% SOZ. The assumpN
1.3
e
..
1.2
’
L
-I
(4)
where
H
= enthalpy of reactants a t given
m = =
x
n =
cp
=
T = q =
inlet temperature referred to some base temperature T I gram moles SO2 fraction converted moles of various gas components specific heat equation for gases temperature, O K. radiation loss, cal.
Figure 3 represents the equilibrium curve for a gas containing 7% SOZ. Three computer derived temperatureconversion points are plotted for adiabatic conditions. Eklund’s adiabatic path is shown in good agreement. Through heat balance Equation 6 the temperature is calculated stepwise through the converter. Knowing the temperature corresponding to the conversion, the rate must then be calculated with the modified Calderbank equation. Forward and reverse reaction velocity constants k l and kz are calculated by ki
where W = grams catalyst F = feed, grams/second r = reaction rate, gram moles/gram catalyst-sec. x = conversion, moles/gram feed Equation 5 is Calderbank’s rate equation modified for Eklund’s catalyst:
Since the rate is affected by gas composition and temperature, how these variables change through the catalyst bed must be known. To find the temperature-conversion relationship, the following heat balance is set up in the computer:
+
Enthalpy of reactants Heat of reaction = Enthalpy of products Heat losses Hreaot. rnx(22,980 cal./gram mole) =
+
+
log (4.75
x
10-5)
=
9.63514
(
-,,’.I)
(7)
ki/K
k2
log K = -4.740
+ (4970.9/T)
(8 1
where
K = over-all equilibrium constant T = temperature, ’K. The first values of x and l/r calculated are stored magnetically on a “table” set aside on the IBM 650 computer’s rotating drum, and as each successive value is calculated it also is stored. The cycle of calculating x and l / r is repeated until equilibrium is reached, the net rate, r, being equal to zero. When simulating a given reactor’s performance, the amount of catalyst and feed rate are known, therefore W / F , the left hand side of Equation 5, is known. Taking successive values of x and l / r from the drum table, the computer integrates, between XI and some final xz, to satisfy Equation 5. 7.0% SO,
loot
,
10.9% O,,
\
82.1 %
N,
x = Eklund = computer
o
0 1.1
>
1.0
-
I-
>
0.9
L
I
0.8 700
I
I
800
900
TEMPERATURE,
1000
1100
“E
Figure 2. Activity factor v5. temperature plot was constructed using Hougen and Watson method
980
INDUSTRIAL AND ENGINEERING CHEMISTRY
800
1000 1100 TEMPERATURE,
900
1200
1300
OF:
Figure 3. Equilibrium curve for gas containing 7% SOzshows good agreement of computer derived points with Eklund’s
SULFURIC ACID o f f gas
Contact Plant Description T h e simple flow diagram of a contact converter plant (right), from gas blowers through absorbing towers, is a deliberate attempt to avoid any resemblance to known operating systems. The layout described herein has served merely as an aid in studying simulation techniques.
t air Typical contact plant flowsheet does not represent any actual plant operation 1. 2. 3.
Plant Operating Parameters The study outlined here has been made mainly from the manufacturer’s viewpoint. A given set of equipment must be operated to best advantage. The equipment (below) was sized more or less at random. Whether or not blower horsepower was adequate for the amount of gas the catalyst could handle was unknown. The test of simulation and optimization techniques would be more rigorous if there were no preconceived ideas as to how the equipment should be operated. The decision was made to study four uncontrollable variables and four controllable variables. These were: Uncontrollable Variables gas blower efficiency, To cost of electricity, $/kw.-hr. xs = sulfur content of pyrite ore, % x4 = cooling air temperature, O F. XI = xt =
Controllable Variables xlr = feed, tons sulfur per day x2’ = mole fraction SO2 XS’ = inlet temperature, converter A, O F. xq‘ = cooling air dilution rate, moles/min.
After selecting the variables to be sudied it was necessary to set certain limits on some of the variables. It may be assumed, from past experience that the optimum first reactor inlet temperature would never exceed 900” F. By defining this value as an upper limit, all values greater than 900’ F. are excluded from consideration. The ignition temperature of the platinum catalyst was assumed to be 670’ F.: so that temperature became the lower limit. Air dilution cooling stream was limited only by the 100 hp. available. Special mention should be made of the inlet temperature to vanadiumfilled converter B. This temperature is a result of: the inlet temperature to converter A, the amount of conversion in converter A or temperature rise, the initial gas composition, and the amount of air dilution cooling used. Being a result of variables over which there is control, it is, in effect, an indirect controllable variable. It has a t least a lower limit. This limit is the ignition temperature of vanadium catalyst, which is assumed in this case to be 800’ F.
4. 5. 6.
Gas blower Air blower Heat exchangers
Table I. Upper and Lower Limits for Controllable Variables Variable
’ 35 ’
XI X2
X4’
Lower Limit None 0.040
670 0
Upper Limit None 0.125 900 100 HP
The limits of the controlled and uncontrolled variables are given in Tables I and 11, respectively.
Method of Steepest Ascent The goal was maximum profit from the plant, expressed as net dollars per day. I n this study, the net is a dollar index of operation. It is not a true profit, since labor, maintenance, depreciation, selling expense, and the like are not considered. A second computer
Converter A Converter B Absorption tower
Table II. Upper and Lower Limits for Uncontrollable Variables Variable XI
Lower Limit 70
Upper Limit
0.008 46 80
0.012 51 120
52
Xa 24
90
program was used to evaluate this index for the various steps taken during the investigation. This program simulated the catalyst beds, heat exchangers, absorbers, and flues by means of a number of pressure drop correlations before arriving at the profit index. Mathematically, the problem consists of determining the particular values of the variables xl’, x2’, . . x,’ which will maximize the profit index, y.
.
Y
= f(x1’,
x2’, Xa’,
(9)
xPr)
-Equipment List for Contact Plant Converter B (vertical, cylindrical,
Blowers 300 hp.. 100 hp.
Gas Air
Heat Exchangers (two) Tube size 12 gage, 1 8 ft. long No. of tubes 400 Tube arrangement 60°, 3-in. triangular pitch No. of baffles 4 Free opening of 6.67 sq. ft. baffle segment 70 in. Shell diameter
Converter A (vertical, cylindrical, insulated) Diameter Catalyst type Density Charge Depth
12 ft. Olson Pt 60.5 Ib./cu. ft. 10 tons 35 in.
insulated) Diameter Catalyst type Density Charge Depth
14 ft. Eklund V 40.7 Ib./cu. ft. 10 tons 38.75 in.
Absorbing Tower (cross section assumed adequate to ensure operating below loading point at all times)
Packing
Dumped spiral rings
Miscellaneous Flues and Valves Equivalent to 4 6 0 0 ft. of 30-in. Pipe
VOL. 53, NO. 12
DECEMBER 1961
981
range within which they may change in relation to one another. This range was kept small so that they function of Equation 9 could be expressed linearly. If all combinations of the high and low values of the four controllable variables were to be run, 16 problems would be required. Davies (7) has provided a half replicate (eight problem) array suitable for four variable problems. This array is indicated in Table 111. Table I V contains values of the variables as they were run for the base case and the eight problefis of the array. Using the Box-Wilson method, the following ascent equation was derived:
Table 111. Half Replicate Array” Is Suitable for Four-Variable Problems VariProblems able 1 2 3 4 5 6 7 8
- + - + - + - +
- + + - - - -
- -+ ++ ++ + - + + - + - - +
3 7
xz xa 2 4
+ designates upper values, - designates
lower values of variables.
The method of steepest ascent, also known as the Box-Wilson method, is a technique recently developed to handle this type of problem (3, 4 ) . A complete discussion of the method is available (7). Neuwirth and Naphtali (73) have described a chemical engineering application. As a base starting point, random values within the limits of Table I were assigned to the controlled variables. These first values are referred to as the base case. For equation derivation purposes, figures slightly above and below the base case values were selected for these same variables to describe the
Table IV.
y =
-
1.144
+ 10.25 8x1’ - 22.655 x%’ 9.221
xS’
+ 22.49 xq’
The coefficient of a variable is an estimate of the partial derivative of the function with respect to that variable. To find the point where y is a maximum, it is necessary to move stepwise away from the arbitrary base case. The fastest increase in y is realized by changing each variable by an amount proportional to its derivative and in the direction indicated by its sign. Table V lists variable step changes for the steepest ascent.
Uncontrolled variable situation 1 First ascent
Problems 21
Base
’
+50.5 50 -4. 9.. .. 5 4-0.072 0.07 -0.068 833 830 - 827 f2.25 2.00 -1.75
XZ’
+
X3’
X4 I
1
2
3
4
5
6
7
8
49.5
50.5
49.5
50.5
49.5
50.5
49.5
50.5
0.068
0.068
0.072
0.072
0.068
0.068
0.072
0.072
827
827
827
827
833
833
833
833
1.75
2.25
2.25
1.75
2.25
1.75
1.75
2.25
Variable Step Changes for Steepest Ascent
For fastest increase, amount of variable change i s proportional to its derivative
Step-
Variable bf/bx XI’ + 1 0 . 2 5 8 ZZ‘ ZS’ 34‘
0.04 0.00016 0.24 0.02
+0.41 -0.0036 -2.2 f0.45
70% $0.008/kw.-hr. = 46% 8OOF.
XI =
=
~p $3
XP
Figure 4 shows how the net profit and the air and gas horsepower changed during the steepest ascent investigation. This particular ascent is only of academic interest, since it could not be made because of equipment limitations. Main blower horsepower was 288 a t the base case, very close to its limit. At the first step away from the base case the main blower was overloaded by 34 hp. This demonstrates a major weakness in the steepest ascent method, when applied to plant simulation problems. The method does not take into account equipment limitations. It developed that electricity was cheap and that the quickest way to improve the plant’s net involved, among other things, diluting the gas, regardless of how much more horsepower was needed. At step 8 a peak was found, but, of course, it was unobtainable. Conversions in the vessels were as follows :
verter
Conversion, % ’ Base case Step 8 Step 1 0
1 2
1000
1.0 0.004 6.0 0.5
I t was decided to hold all uncontrolled variables a t their lower values during the derivation of the steepest ascent equation-Le. :
Con-
500
Ranpe wise Range Fraction Change
-22.655 -9.221 f22.49
(10)
These Values Were Used for Base Case and Controlled Variables
Variable
Table V.
75.7 SO. 1
89.8 94.6
92.6 96.1
r 360
r
2 800
400
f
f
300
280
a a
K
a
;200
3
0
n
100
2
600
- limit
0 0
I 2 3 4
5
6
7
8
9
240
n
IO
TRIAL INTERVAL
Figure 4. Steepest ascent could not be made because of equipment limitations INDUSTRIAL AND ENGINEERING CHEMISTRY
200
sm v)’
z
200
-0
a w
400
0
982
600
v)
v)
t,
320
a
a
4 0
K
40
g J
m
- O
L
a
n
6
I2
18
24
30
TRIAL INTERVAL
Figure 5. First controlled ascent was made b y increasing variables in direction shown b y eighth array problem
SULFURIC A C I D
A Controlled Ascent When it was discovered that the steepest ascent method led out of the practical operating area, the answers to the eight problems of the derivation array were re-examined.
-log of First Ascent Trial
Description
Base
Variable steps:
16 Main Blower
Revised steps:
xl’
Net $/Day 0.00 - 1.073 6.181 - 1.469 -26.95 24.38 0.93 -67.44 0.67
The eighth array problem showed a slight increase in profit while horsepower decreased. An ascent was started by stepping all variables in the direction suggested by problem 8. The results of these changes are shown in Figure 5 . During this ascent, equipment limitations and operating parameters were watched. A log of the ascent is shown (right). Second and third “controlled” ascents were made in the same manner as the first. Answers to arrays of problems run about the peak of the first and second ascents were examined to determine the initial direction to be taken. Figures 6 and 7 depict the course of these runs. Perhaps the third run (below) should not be called an “ascent,” since it was devoted to reducing main blower horsepower to within limits, a t minimum loss of profit. It would appear that final conversion must be added to the list of constraints when making a controlled ascent. The final conversion, 92.3y0, would not
+0.5
=
$1 .O
= ’ +0.004 x;’ = +6.0 xq’ = $0.5 ~
HP. 288.9 299.0 316.7 267.5 275.0 303.0 313.9 264.1 279.7
Problem Base 1 2 3 4 5 6 7 8
=
XI’
xz’ = $0.002 X Z ’ = +3.0° F. xq’ = +0.25 2
19
Reached maximum gas strength available a t contact plant 1 1.8% SOz. Continued ascent at constant gas strength. Revised step: X I ’ = +2 .O
25
Rapidly approaching maximum air blower hp. Now at 93.39 hp. Continued ascent a t constant air dilution.
27
Reached peak of ascent at this point. Conversions: Converter A = 42.9% Converter B = 81 .3%
/ 1loo
I
I
2
950
90
i; 9 0 0
80
5
70
wa 5 0
I-
a
W
0 0
m 850 Q
a
V
p . 800
60
+
50
750
W
A
I
700
40 I
0
2
3
4
5
6
7
8
9 IO
TRIAL INTERVAL
Figure 6. Second ascent was based on arrays of problems arising from peak of first controlled ascent
.Final Conditions, Run 3, Trial 20, Variable
Final Value 72.9 0 . 1 142 7 5 3 “ F. 12.72
XI’
X2
’
’
XQ
X4’
Blower Power Air
B
% Conv. B
= 92.3y0
Profit Improvement Net $/day = $1 173.14
100
- 90 2
0 ‘
8>
I
125
(I)
a
(3
12c
- u
115
- t5
v)
0 7 . 4 hp.
-
..-
w
Gas 3 0 4 . 5 hp. Conversion Converter A = ~ 5 0 . 7 7 ~ Converter
I
a
Results
bp
5I-
a
d-820
a
B t; z
-
rn780L 0
I
4
I
’
8 12 16 TRIAL INTERVAL
P W
- 70 %8 bp
_-
% Cow A
110
-80
60
-.
- 50 I
I
I
20
Figure 7. Aim of third run was to reduce main blower horsepower while minimizing profit loss VOL. 53, NO. 12
DECEMBER 1961
983
be acceptable considering the effect on air pollution.
Correlation of Controllable and Uncontrollable Variables Direct computer controlled plants require faster operating programs than those used in this research. Simplified empirical correlations developed from slower, more rigorous programs would appear suitable for use in the computercontroller. Ideally, these control equations could be used to find the optimum setting for any control variable for any combination of uncontrolled variable values. The controllable variable would be shown as a dependent function of the uncontrollable variables. XIf
=
XI’
Q
= f(X1,
+ XI
22,
xs,
(11)
x4)
(12)
rt cxz f dxg rt ex4
An example of the derivations can be given using a closely related problem. Four control equations are required, one for each controllable variable. I n this example, each equation will tell how to adjust its controllable independent variable to make a steepest ascent, rather than directly describe the optimum value of the variable. All work reported u p to this point had been done with the uncontrolled variables at their lower limits. Assume that the conditions of Trial 20, run 3, represent the optimum characteristic of the first problem of the half replicate array of eight problems. Assume also that the method of steepest ascent can be used to locate better settings of the controls should any of the uncontrolled variables shift. For each combination of uncontrolled variables called for by the half replicate array of problems, an ascent equation, similar to that of Equation 10, is derived. A set of eight coefficients is provided for each controllable variable. Table VI is a summary of the least squares fit manipulation showing the Zy functions to derive the four separate control equations. The lower half of the
table contains the coefficients for the uncontrolled variables. Of course, the optimum settings could have been determined for each of the uncontrolled situations. With this knowledge, control equations explicit for optimum settings could have been derived rather than the equations below which only indicate how to approach the correct setting.
?a
=
2.18446
+ 0 . 0 1 9 4 6 ~-~
0.03165~2- 0.00647~3- 0.02008~4 (13)
rn
=
-3.58437
- 0.12375~1+
0.19781~2- 0.00094~a- 1.14438~4 (14)
a 7-
-5.73281
+ 0.0009375~1-
0.002500~2- 0.000625~a- 0.0003125~4 (15)
3 = 14.13687 + 0.0925~1aX41
0.14781~2f 0.00094~3- 0.00063~4 (16) I t is possible to relate all the variables affecting an H z S 0 4 plant by means of simple equations. Using these correlations, the exact and most profitable operation may be determined with any given catalyst and under any given values of the controllable variables. The profit improvement possible for a given plant, when following the methods presented herein, depends upon a number of things including the size of the plant, and the level of efficiency that the plant has been running at. I t is highly improbable that a plant operator regulating the plant of the flow sheet shown (p. 981) could make a profit improvement of $1170 per day. The conditions of the original base case were selected at random to provide a good test of the simulation and optimization techniques. Simulation study being the subject a t hand, no attempt has been made to document accuracy of the kinetic logplot correlations used. This may be the basis for a later report. The log plots in this work served to evaluate performance of catalysts described in the liter-
ature but not readily available. Hougen and Watson’s activity factors are useful when there is an actual catalyst to work with. To take full advantage of computer guidance systems, HzS04 plants of the future should be equipped with adequately sensitive temperature and flow measuring instruments. I n the plant shown, a 10” F. change in the inlet temperature of the first converter affects the daily profit by $57 per day. A cooling air rate change of 1 mole per minute represents $14 per day. The future looks bright for both open and closed loop control of chemical plants by computers. Faster operating machines with larger memories are becoming available. They make possible the development of integrated programs containing internally stored operating parameter limits and the development of more efficient routines for searching out the locus of optimum conditions.
Acknowledgment The assistance of the Allied Chemical Corp., General Chemical Division, given in supplying data and computer facilities for this work is greatly appreciated.
Literature Cited (1) Benton, A. F., IND. ENG. CHEM.19,
494 (1927).
(2) Bodenstein, M., Fink, C. G., Z. physik Chem. 60, 1 (1907). (3) Box, G. E. P., Biometrics 10, 16 (1954). (4) Box, G. E. P., Wilson, K. B., J . Royal &?tiStiCQ!! SOC. 13B, 1 (1951). (5) Calderbank, P. H., Chem. Eng. Progr.
49, 585 (1953). (6) Calderbank, P. H., J . Appl. Chem. (London) 2. Pt. 8 . 482 (1952). (7)’ Davies, 0. L., “The ’Design and Analysis of Industrial Experiments,” 2nd ed., London, Oliver & Boyd, 1956. (8) Eklund, R. B.? “The Rate of Oxidation of Sulfur Dioxide with a Commcrcial Vanadium Catalyst,” Almquist & Wiksell, Stockholm, 1956. (9) Homme, A. C., Allied Chemical Corp., General Chemical Division, Morristown, N. J., unpublished work, 1954. (10) Howen. 0. A.. Watson. K. M.. “Chem~cal’Process Principles,”’ Pt. 111; p. 935, Wiley, New York, 1947. (11) Hurt, D. M., IND. ENG.CHEM.35, 522 (1Qdq) ., .“,. (12) K.1iietsch, R., Ber. 34, 4069-4115 ,
,
I
I
(1901\ .
Table VI.
Summary of Least Squares Fit Manipulation Shows Zy Function for Deriving Control Equations Controllable Variables $2
21 I
Z
Functions ZY
ZZlY ZZZY ZX3Y
ZXW
4-17.475 +O. 1557 - 0.2532 -0.0518 -0.1607
Variables zo 21
XZ
x3 2 4
984
2.1844
+0.01946 - 0.03165
- 0.00647 - 0.02008
X3 I
Sigma Function Values - 28.675 -45.862 - 0.9900 +O. 0075
+ 1.5825 -0.0075 -9.1550
-0.0200 - 0.00500 -0.00250
Coefficients of Uncontrolled Variables - 3.5843 - 5.7328 -0.12375 +O. 19781 -0.00094 1.144375
-
INDUSTRIAL AND ENGINEERING CHEMISTRY
f 0.00093 -0.002500 - 0.000625 -0.0003125
21’
+113.09 +0.7400 - 1.1825 +O. 0075 -0.0050
14.136
+0.0925
-0.14781
+0.00094 -0.00063
(li) Nluwirth, S. I., Naphtali, L. M.
Chem. Eng. 64, 238 (June 1957). (14) Olson, R. W., Schuler, R. W., Smith, J. M., Chem. Eng. Progr. 46, 614 (1950). (15) Othmer, D. F., IND.ENC.CHEM.32, 841 (19401. (16) Othmgr, D. F., Luley, A. H., Ibid., 38, 408 (1946 (17) Othmer, . F., Zudkevitch, D., Barona, N., Polytechnic Institute of Brooklyn, Brooklyn, h-.Y . , unpublished ~
L
(18 Taylor, G. B., Leuher, G., Z. physik j:ez.) ::::stein Festband. 30-43 (1931). (19) Uyehara, 0. A., Watson, K. M., IND.ENG.CHEM.35, 541 (1943).
RECEIVED for review February 15, 1961. ACCEPTED June I, 1961 53rd Annual Meeting, A.I.Ch.E., Washington, D. C., December 1960.
