I
DONALD A. DECHMANI and MATTHEW VAN WINKLE
The University of Texas, Austin 12, Tex.
Perforated Plate Column Studies by
the Box Method of Experimentation An example of arriving at central composite design for four independent variables
Tm
efficiency of a perforated plate column and the pressure drop of the vapor through each plate can be expected to vary with reflux ratio, vapor mass velocity, weir height, hole diameter, and free area as well as with the system properties. Correlation of efficiency with operating and design variables and system properties would be invaluable in the design of perforated plate columns. Correlating experimental data by crossplotting these variables would be extremely difficult. The ideal solution would be a relatively simple equation which would relate these variables. A logical method involves selection of a type equation in which the variables correspond to the operating and design variables, solving the equation to evaluate constants, and repeating the procedure for several systems composed of different mixtures of compounds to be separated in the fractionating column. The constants of the equations for each system could then be related to the system properties. The response surface or Box method of experimental design can be used with relatively few experimental data points, and was chosen for closer examination. Braulick studied the effects of operating and design variables on efficiency and pressure drop of a perforated plate column with the toluene-n-octane system ( 5 ) , including effects of varying reflux ratio, vapor mass velocity, weir height, and hole diameter a t constant free area. A Box-type experiment was designed wherein about one tenth of the original data points were selected in accordance with the design technique and fitted to the equation by least squares methods.
Box Method of Experimentation The basic ideas of the Box method for exploring response surfaces are not new. Box described concisely a method for multifactor experimental design (7-4). H e proposed primary experimentation
of the experimenta1 error. Thus, additional runs should be made at:
to fit the first-order equation: Y* =
Pox0
+
131x1
+
132x2
+
. . . . -I-BkXk (1) X Ois always unity and k is the number of
Run
zl Level
9
-2
0
0
independent variables to be studied. For the case of three independent variables, four constants need to be determined. A 23-factorial design was proposed to give extra points needed to determine the least squares fitted equation accurately. In a 23-factorial experiment the data are determined at the eight corners of a cube inserted into the three-dimensional factor space. The axes of the design pass through the center of the cube and experimental points should be determined at equidistances from the axes. Thus for three independent variables, the 23-factorial design is described in t e r m of levels as follows:
10
+2
0 -2
0
Run
21 Level
x2 Level
1 2
-1
-1
3
4 5 6
7 8
$1 -1 +1 -1 $1
-1 +I
-1
za Level -1 -1
+1 +l -1 -1
-1 -1 +1 $1
tl
tl
+I
$1
Present address, Union Carbide Chemicals Co., Texas City, Tex.
Experimental error
SE = Z(yio
--z
0 0 0 0 0 0
+Z
0 0
0 0
The central composite design enables evaluation of the constants of the second-order equation : Y* =
80x0
+ 81x1 + 82x2 +
812Xlx2
f 8l*Xlx3
+
84x3
821X2X3
814
+
+ +
P 3 3 4
(2)
The lack of fit variance of the experimental data from the fitted second-order equation could be compared to the estimated experimental error variance. If the lack of fit variance was excessive, a different model should be proposed (7). In the present study, the secondorder equation was satisfactory. Experimental data deviate from a fitted equation for two reasons: experimental error and lack of fit of the equation to the data. The latter error can be minimized by an equation that suitably represents the data. Variance u;,,~ is a combined measure of these errors. Box proposed repeating the four runs in the center of the design to evaluate the experimental error variance. Then the sum of these four residuals squared divided by 3 would give the experimental error variance, u&. Lack of fit variance, uEOF,is then found by difference of the sum of the residuals squared and the degrees of freedom for the total and experimental error variance. Visual or statistical comparison of oEos and u& will allow the best evaluation of the applicability of the equation. Following is a general tabulation:
Sum of
SLoF =
0. 0
+2
0 0 0
18
m Level
Level
0
17
Degrees of
Freedom
Squares
Lack of fit
0 0 0 0
11 12 13 14 15 16
The constants in Equation 1 should be determined from these eight experimental points by the method of least squares. The variance of the points from the fitted equation could then be compared to the estimated experimental error variance visually or by the statistical F rest (6). If the first-order representacion of response illustrated by Equation 2 was not satisfactory, Box proposed a central composite design. The central composite design includes experimental points on the axes through the center of the cube at a & 2 level. The level &2 is not essential for a central composite design, but it is generally used. Repeat points should be determined a t the center of the cube and at other points, if possible, to enable estimation
Residualsfrornfittedequation SR = Z(ga
zz
- ji)2
SR - SE
- Ti,)'
n
(n -
P
Variance
-1
0
-
-1
VOL. 51, NO. 9
(P - 1)
SLOF (n
-
- 0 - (P 1) SE/(P - 1)
SEPTEMBER 1959
1015
where n = number of observations, I = number of constants in fitted model, p = number of repeated observations at center point, yio = observation at center point, and j , = average ofp observations at center point. Repeat runs could not be made, because the column was in other service. Thus, although a u& was estimated from knowledge of column operation, this method of evaluating results is not reported. The total standard deviation, which is the square root of the total variance? and the maximum deviate are reported as a vardstick. Davies (6) explains the method of analyzing results by comparison of uLop and u&. The Box method is not limited by numher of independent variables. Tables I and I1 represent the central composite design for four independent variables, the problem handled in this article.
Table I. Ideal Box Central Composite Design in Four Dimensions Run
The tabulated data of Braulick consisted of 246 experimental runs in a 5plate, 6-inch glass perforated plate column. Tray spacing was 12 inches and the perforated plate hole free area \vas 12.5Yo of the column cross-sectional area ( R - 6). Efficiencies below 20% were discarded, as a study of column operation beiow the weep point was not desired. Fall-off due to excessive entrainment was not encountered with the vapor m a s velocities studied. Efficiencyandpressure drop were the dependent variables. Levels of independent variables studied by Braulick were:
XP
z3
xb
$1
+1
+1
+l +I +I +I +1
+1 +1 +1 -1 -1 -1 -1
+l
+1 -1 +1 -1
-1
+1 +1
+1 +1
-1
$1
-1 -1
+1
-1 -1 ‘1 +I -1 -1 0 0
x1
1 2 3 4 5 6 7 8 9 10 11 12 13
+1
+1 -1
14 15
-1
-1 -1 -1
-1
-1
-1
16 17 18 19 20 21 22 23 24
Procedure
Coded Levels
_-
NO.
0 0
+2
-2 0 0 0 0 0
-1 -1 +1 +I
t2 -2 0 0
-1 0
0 0 0 0
$2
0
0
25
0
0
0
26 27 28
0
0 0 0
0
0 0
-1 +I
0 0
-2 0 0
0
+I -1 +I -1 +I -1 +I -1 +1
-1 -1
0
+2 -2
0 0
0
0
0
0
selection of the &1 levels at values for which there were experimental data. The following Box experiment design levels allowed 16 experimental points to be used without interpolation: _____ T*miable XI 22
Design Level
-
-2
0.79545
500
53
1/2
xb
a/;2
to use all 246 experimental points for a least squares analysis of efficiency. The 28 design points selected by the Box technique were also fitted by the least squares method for both efficiency and pressure drop. The two equations for efficiency were compared in the range of the Box design and the accuracy of extending the Box design equation beyond its indicated range to a reflux ratio of 1.00 (+3.4 level) was tested. Pressure drop was correlated using only the data points selected by the Box design technique. The 16 points read directly were given their measured values of vapor mass velocity, which were not exactly at the integral design levels. Integral levels are better for a more balanced design. Linear interpolation (and extrapolation for two points) over a relatively short range was used to calculate response surface design points that could not be taken directly from the data. Interpolation was in no instance greater than rhe estimated experimental error. Three sets of constants were evaluated by the method of least squares. One set was found from the least squares fit of the entire set of data relating efficiency. The other two were obtained from the 28 data points selected by the design technique relating efficiency for one set of constants and pressure drop for the other.
-1
0.83333 700 1 118
0 0.87121 900 1 I/? 5/32
41
0.90909
1100 2 3 ‘10
+2
0.94697 1300 2112 i f
i 32
Vari-
able XI
Units Symbol Levels L / V 0.83333, 0.90909,
none
Lb./sq. ft. hr. x3 Inches x) Inches
2%
G h, DO
1.0000 200 at odd levels to 1500 1, 2, 3 I/N,
l/8,
3/16
Reflux ratio was reported by Braulick as L I D values of 5 , 10, and total. Thus the significant places in the levels of xI are unwarranted and were used onlv because of no extra load on the computer. The Box-type experiment of a central composite design in four dimensions was used to determine the design levels. A simpler 24-factorial experiment of first-order analysis was eliminated, since preliminary contour plots of the original data showed curvilinear effects for both efficiency and pressure drop. .4 central composite hypercube design in four dimensions calls for 28 runs16 at the 16 coyners of the four-dimensional cube at levels with respect to the center of the cube of +l, eight on the axes of the cube at f 2 levels: and four repeated a t the center. It was desired to take the maximum number of points from the 246 experimental data points, to minimize interpolation of Braulick’s data. The best design resulted from
101 6
The reflux ratio of 1.000was disregarded as a possible +l level because a + 2 level would be impossible to obtain. Since about 40y0of Braulick’s data were at a reflux ratio of 1.00. i t was decided
Table 11. This i s the Box design in f o u r dimensions derived from the experimental runs
INDUSTRIAL AND ENGINEERING CHEMISTRY
The constants are listed in both coded and uncoded level form. The uncoded form is more adapted for calculations because the independent variables need not he coded. The coded form gives
P E R F O R A T E D PLATE COLUMNS Table 111. Const ant bo
0.1056 2.2075 1.8888 - 1.5425
-
biz bia
- 0.0041
- 0.5431 X
bir bn bzk bad
-
-
0.0006 0.0822 0.7809 0.5561 0.0418
bii
-
0.0681
b29 b33
- 0.6763 - 0.2161
bdi
-
bi
b?
ba br
a
Eff. Eq. Using Entire Data Coded Uncoded 36.0506 - 17.28
0.1570
Box Pressure Drop Equation Coded Uncoded
Box Efficiency Equation
Coded
96.81 16.33 X 10-3 0.2693 14.66
0.0339 - 69.45 7.809 X 88.97 X - 2.672
- 47.48
-
Constants for the Final Equations"
16.91 X 10-6 - 0.8642 -160.8
37.0000 0.1879 2.3874 1.9299 - 1.7819 - 0.1761 - 0.2379 0.1973 0.4361 - 0.1957 - 0.7704
-
0.0406 1.2969 - 0.2969 - 0.1469
-
Uncoded
3.5000
0.5438
- 40.54
- 0,0408
88.90 X 10-3 22.14 - 53.08
0.5884 0.3507 -0.2615
- 23.24 X lo-' - 12.56
- 0.0537 -0.0192 - 0.0448 0.1418 0.0381 0.0274
-
166.7 4.361 X 31.31 X lo-' 49.31
28.29 32.42 X - 1.188 -150.4
-22.17 32.89 15.44 X 0.9316 38.03 -
x
10-3
-37.85 1.418 X l o - & - 6.096 X 10-3 - 1.754
0.0166 0.1666 0.0291 0.0166
-
7.088
- 1.014
-11.57 - 4.165 X 10-6 - 0,1164 - 17.00
I n using coded constants, each value of independent variable must be converted to coded level form
H O L E D I A M E T E R :3 INCH
R E F L U X R A T I O : 10
I - 2
-
2
%
Figure 1 . Contour plot of pressure drop as a function of weir height, vapor mass velocity, and reflux ratio
b
BULK EQUATION
?
500
I
1
600
700
800
900
V A P O R MASS VELOCITY,
equal weight to each independent bariable and thus a basis for comparing independent variables by visual comparison of the constanrs of the equations. Contour plot points were evaluated for the three equations to facilitate contour plotting of the curves of the fitted equations The plots give graphical comparison of the results. The sum of the residuals squared for the residuals of efficiency from the equation using the entire set of data and the residuals of efficiency and pressure drop from the equations fitted using the Box design specified points werc calculated for 65 points within f - 2 design levels for each variable. The standard deviation was calculated and the maximum deviation reported for the three cases.
Discussion of Results The results from the computer consisted of three sets of constants for the second-order equation, contour plot points for the three equations, individual residuals of 65 experimental points, and the sum of the residuals squared. (The equation determined from utilizing the entire set of data is called the "bulk" equation.)
B O X EQUATION
Figure 2. Contour plot of efficiency illustrates deviation of experimental data from fitted equations
Equation Constants. Table 111 lists the coded and uncoded level constants for the three equations. Coded level constants have more utility with respect to comparison, as each independent variable has equal weight. The constants for the bulk and Box efficiency equations compare favorably; this is not evident when comparing the uncoded level constants. The indicated major effects on efficiency are mass velocity, weir height, and hole diameter, in that order, reflux ratio having much less effect. Increasing mass velocity and weir height increases efficiency; increasing hole diameter decreases efficiency. This follows intuitive reasoning of tray action effects on efficiency. The values of the constants of the squared and interaction terms compared to the constants of the linear t e r m of each of these three variables were small. However, it is difficult to predict the effect of reflux ratio on efficiency, for the constants of the squared term and interaction terms are appreciable. T h e same analysis can be applied to pressure drop by consulting Table 111. Again reflux ratio effect is indeterminable in direction and small in magnitude. Increased mass velocity or weir
1000
ti 1100
+i
1200
1300
1
LBI:T~-HR
* A S il V.%
t
c
-I
L3
FT1
Figure 3. Contour plot of efficiency as a function of weir height and vapor mass velocity Hole diameter,
000
eo0
100
'/*inch; reflux ratio, 5
100
800
SAPO" ULI3 " L L O C I - I
300 La,+
30
8200
330
*I)
Figure 4. Contour plot of efficiency illustrates adverse effects of extrapolation of Box-type equation VOL. 51, NO. 9
SEPTEMBER 1953
1017
height increases pressure drop. Increase in hole size decreases the pressure drop, as deduced from analysis of b 4 , 6 1 4 , 6 2 4 , b 3 4 : and 6 4 4 . This again agrees with intuitive reasoning concerning pressure drop variation in a perforated plate column. No attempt was made to eliminate constants as unnecessary. Statistical methods are available but were not utilized. Curves. The main purpose was to prove the accuracy of the Box design, to minimize the number of experimental points necessary to establish the relation among experimental variables within specified limits of accuracy. Therefore, the curves included d o not represent complete results, but compare the predicted response surface for eEciency in terms of the variables from each of the two equations. T h e curves representing the pressure drop equation illustrate the effects of the independent variables on pressure drop. Figure 1 represents graphically the equation for pressure drop obtained from fitting the Box design selected points by the method of least squares. ~ ~ Hole diameter is constant at 3 / inch. External reflux ratio is plotted a t 5, 10, and 17.9, corresponding to coded levels of internal reflux ratios of -1, +1, and Figure 2 shows graphically the deviation of experimental data from curves representing the two equations for efficiency. Figure 3 compares the two equations for efficiency; they agree. T h e general shapes of both sets of curves indicate the same trends. Figure 4 shows the fallacy of extending the levels of the designed experiment to increase its scope. T h e convergence of the response surface as developed in the Box-designed equation would be much like a football in three dimensions. Extrapolation of reflux ratio to the 4-3.4 level shows incorrectly a convergence of the response surface. Applicability of Equations. The values of the standard deviation of the residuals of the equations can be used to judge the applicability of the equations. I t was believed that the experimental error could be estimated to be h 2 . 0 for efficiency or 3 ~ 0 . 3for pressure drop.
Table IV. Analysis of Final Equations (Study of 65 experimental points) for Efficiency
Using Entire Data Z (R,)
246.0
Std. dev.
2.2P
Max. dev.
7.86“
Box Method
Equations_ Press. ciency drop 7.703 162.7 1.80” 0.39Zb 4.75a 1.0546 Effi-
Absolute units of efficiency.
units of pressure drop, in. HgO.
1018
Table V. Final Working Equations System, toluene-r,-octane General equation. 4 = 6 0 x ~ blxl b?x?
++ + + + biaxixef b+w x s x+r 4-+biix:+ + hizx2, + b 3 3 4 + bu.2 b 3 ~ 3
64x4
623~1.~3
* Absolute
614~1x4
bi9.Yixa
~ Y ~ X Z X ~
dummy variable
xo,
= 1.0
Units of Variables and Range of A4pplication Range of Application Box designed Eq. using Symbol Units eq. entire d a t a
Variable 21
LIV
None
22
G
z3
hw
Lb./hr. sq. ft. Inches Inches 70 In. H20
Do
X4
yl
E
Efficiency
u2 Press. drop
JP
0.8 to 0.95 500 to 1300 1/2 to 21/2 8 / 3 2 to ’/32
0.83 to 1.0 200 to 1500 1/2 t o 3
....
....
....
‘/le
to
....
’/11
Constants for General Eauation Constant
bo bi
Eff. eq. using entire
data -
17.28 96.81
b? X 16-33 103 - 0.2693 bs bd - 14.66 biz X IO3 b13
bir
x
b23
0.0339 - 69.45 7.809
103 X
b?4
88.97
bsr
- 2.672
bii
-
bz? X
Box-Designed Equation For efficiency -
47.48
drop
15.44 0.9316 38.03
88.90 22.14 - 53.08
-
12.56 166.7
7.088 1.014 -37.85 -
4.361
- 31.31 - 49.31 28.29
baa
- 16.91 - 32.42 - 0.8642 - 1.188
bri
-160.8
106
For pres=.
0.5438 -22.17 40.54 32.89
- 0.5431 - 23.24
IO3
+2.
a
~~
-150.4
1.418
- 6.096 - 1.754 -11.57 4.165 0.1164 -17.00 -
-
For example, a measurement of 40% efEciency and 3.0 inches of water pressure drop could be expected to be 40 Sc 2.0 and 3.0 & 0.3, because of possible errors involved in obtaining data on a fractionating column. T h e maximum deviation in each case is high (Table IV). A few erroneous results are to be expected and a rerun of the point might give a different value. The standard deviation is in each case satisfactorily low. The average deviation for 65 residuals for the equation utilizing the Box method was less than 65 residuals for the equation utilizing the entire data; the equation evaluated from Box design methods was necessarily limited in scope and thus, although it is superior to the bulk equation in its own range, the Box equation cannot be extrapolated into the wider range of the bulk equation. However, considering the standard deviations and the individual residuals (too cumbersome to report here) it is concluded that all three equations satisfactorily represent experimental findings. Final Working Equations. T h e final working equations in Table V are in terms of the iincoded levels of the independent variables. This form is cumber-
INDUSTRIAL AND ENGINEERING CHEMISTRY
some as a working equation. T h e equations listed may thus be employed over the indicated range of the independent variables using the correct units for each variable. Acknowledgment
T h e assistance of R. S. Schechter and E . S. Aldredge in programming the calculations, and the Cnion Carbide Chemicals Co., Texas City, Tex., in providing computer time and furnishing the fellowship under which this work is conducted, is gratefully acknowledged. Nomenclature
R
= constants to be estimated = estimated constants = residual error,y - j
x0
= dummy variable, always equal
Y,
= reflux ratio, moles liquid/mole
3
b
to unity
vapor vapor mass velocity, Ib./hr.-sq. ft. xI = weir height, inches xq = hole diameter, inches ,y = experimental value of dependent variable p = calculated value of dependent variable u&= total variance of data & = variance due to experimental error .EOF = variance due to lack of fit uTOT = standard deviation S =sum x2
=
literature Cited (1) Box, G. E. P., Biometrtcs 10, 16 (19541. (2) Box, G. E. P., Biometrika 39, 49 (1752). (3) Box, G. E. P., W7ilson. K. B., J . Roy. %at. Soc. B 13, 1 (1951). 14) Box, G. E. P., Youle, P. V.. Biometrics 11, 287 (1955’1. ( 5 ) Braulick, LV. J., dissertation in chem-
ical engineering, University of Texas,
Austin,-Januarf 1956. (6) Davies, 0. L., ed., “Design and Analy-
sis of Industrial Experiments, “Hafner Publishing Co., New York, 1956. (,7) Hunter, J . S., “Box Method of Es-
perimentation,” Harvard University Graduate School of Business .Idministration, May 20, 1956. RECEIVED for review May 19, 1958 ACCEPTED .4pril 15, 1953
