A Numerical Solution to
imensional
OCTAVE LEVENSPIEL’,NORMAN J. VEINSTEIN, AND JEROME C. R. LI Oregon State College, Corvallis, Ore. RITERIA in the choice of methods employed in arriving a t the solution to problems in the applied sciences arc simplicity, reliability, accuracy, economy, and applicability to available computing tools. As new calculating devices become readily available, new procedures for problem solutions should he developed, if they afford advantages not met by usual methods of solution.
applicable where one dependent vdriable is a linear function oi any number of independent variublcs. Therefore, this tcclinique may be used for finding the constants in any equation 1%hcir a transformation to linear form ie possible. Its application to tlir general solution of problems in dirnemionai analysis is strnightforward. METHOD
~~
The familiar form of an equation resulting from diniensioiiel analysis is:
~
D
Engineers who are curious about automatic calculators and statistical design have an opportunity to see how these tools sperate on Q series of experimental data, with real time- and labor-saving advantages
U ( D ~ ) ” ( D Z ).*.(Dii)bk ~.
(1)
.
in which D,D1,D 2 > . ., Dk are dimensionless groups and a, b l bz,. , ., b k are constants m-hich are to be estimated. Ry a logarithmic transformation, Equation 1 becomes
NOWlet y = log D, 51 = log Di,x2 = log Dz,. . bo = log a. Then Equation 2 beconics
Y
= bo
+ b i ~ i+ b a ~ z+ . . . +
.)
xk
=
log k , a11t1
(3)
bkxk
It remains to find the values of bo, bl, bs,. . b k . These X: 4 I constants resulting from the dimensionless groups are found by the method of least squares-that is, by solving the follon in,: kl 1 simultaneous equations:
Today electric desk calculators w e found in most engineering laboratories, justifying the utilization of methods peculiarly fitted to their use. One such method is presented here a? a solution to a familiar problem, that of fitting a dimensionless equation t o a set of experimental data. The authors do not claim originality in developing the numerical solution presented. In fact, this is a well known method of multiple regression and is described in many books on statistics ( g , 6, 7). The intention of the authors is not only to introduce a useful technique to engineers who may not already be familiar Kith it, but also to stimulate interest in the application of this and other statistical methods to the fields of engineering and chemistry.
.)
+
where the summation is over S, the number of runs. The following is an outline of the procedure, which is ne11 adapted t o desk calculator use. Procedure. From the data of a series of experiments, calculate the values of the dimensionlesq groups for each run and tabulate them. Find the logarithm of each dimensionless group. These logarithms become the variablep, u, 5 , ) 2 2 , . , ., xk, as shown in Table I.
THE PROBLEM
I n analyzing phenomena \\-here complexity or insufficient information does not allon- a rigid theoretical treatment) dimensional analysis is often employed. Hop-ever, in most cases the estimation of constants in the equations thus derived involves considerable time and effort, as the method used in engineering is essentially the determination of the interrelationships of the variables two a t a time, with the remaining variables held constant. The interrelationships are found either graphically or numerically by the methods of “least squares” or “group averages” ( 9 ) . This paper presents a numerical solutio11 to such problems, which is objective and rapid, and which, based on sound statistical theory, provides reliable estimates of the true values for the constants in the dimensionless equations. The method, which is known in statistics as multiple regresqion, is generally 1
=
Table I. Run
Illustration of Procedure ?I
51
X8
8 2x1
z;r*
S 5
10 6
Sum Notation
Present address, Buoknell University, Lewsburg, Pa
324
7 36
rz/
1I
INDUSTRIAL AND ENGINEERING CHEMISTRY
February 1956
325
Find the sum and the sum of the squares of each variable. Find the
‘*p
sums of products of all combinations of vari-
ables, taken two at a time. The summation is t o be taken over all the N runs, as shown in Table 11.
Example. As an example of the application of the method, the equation and data of Sieder and Tate (10) for heat transfer of liquids in tubes have been chosen.
NU Table 11. Details of Computation Run
x2u
m2/
21 2
22
=
XI22
=
Zy
=
ZXi ZX,
=
Z(YX1) Z(yX2) Z(yX3) Z(X1X2) Z(X1Xs) Z(XzX3) Z(X12) Z(~22) Z(X3z)
+
5 bo
+
8
bi
+ 11 b, = 36
bo bl be ba
bi
Solving the above equations simultaneously, one finds
= = = =
= = = =
=
= = =
-0.011848 0.288054 0.242888 0.142379
Since bl = log a, then a resulting equation is
I n Table I1 only the quantities in the last row are needed as coefficients in Equations 4. As these quantities can be obtained in one continuous operation, the individual squares and products need not be tabulated. They are listed in Table I1 only as an illustration of the computing procedure. Equation 5 is a modification of their recommended equation, with ( L / D ) - l J $included in the constant 0.402, as the original data do not include this group as a variable.
=
0.97308, Therefore the
as compared to the result of Sieder and Tate : Nu
=
0.402 (Re)1’3(Pr)l’3
(:)“*‘
Because Sieder and Tate did not actually determine the esponent of the Prandtl group, but assumed it t o be 1/s as found by previous investigators, the constants found in Equation 5 are somewhat different from those of Equation 6. Figure 1 shows that the results obtained by Equation 6 agree very closely with those calculated from Equation 5. However, this is the result of the use, by Sieder and Tate, of a trial and error method of least squares. They tried a number of coefficients until they found a set of constants which gave a minimum deviation. Thus, their equation is the result of a considerable amount of careful work. Of course, by the nature of the least squares method, the constants given in Equation 5 show a minimum root mean square deviation of log Nu, which for Equation 5 is st0.0469 versus 10.0535 for Equation 6. These two values are measures of the per cent deviation of the calculated from the observed Nu values. Of interest is the root mean square deviation for Nu, which for Equation 5 is 1 2 . 5 8 and for Equation 6 is 1 2 . 9 6 .
40
30
20
IO 9
-
= = =
84.5616 148.1025 201.3341 43.6002 190.717107 251.789129 44.518615 422.295834 - 100,993332 - 133.940573 362.017620 624.159862 116.512330
where y = log (Nu); x1 = log (Re); xz = log(Pr); x3 = log ( p a / p W ) . The substitution of these values in Equations 4 and the solution by the abbreviated Doolittle method (1,4 ) gave the following b values :
+ 14 + 18 b? = 58 I1 bo + 18 bl + 27 ba = 74 8 bo
b3
By means of a calculator the following summations were obtained :
ZX3
Substitute these values in the k 1 simultaneous Equations 4 and solve for the b values. A simple illustration is shown in Table I and the details of the computation are shown in Table 11. Substituting the values obtained in Table I1 in Equations 4, one has
(E)
U(Re)b,(Pr)bz
SUGGESTIONS AND CAUTIONS
8 9 IO
20
30
40
50
NU observed Figure 1. Application of method of calculation
In order t o avoid unwarranted extrapolation, the dimensionlesv groups used in calculating the constants should be allowed to take extreme values in the range for which the equation is t o be
326
I N D U S T R I A L A N D E N G I N E E R I N G C H E MI S T R Y
used. Using extreme values also assures gyeatest accuracy of the estimates of the b’s ( 4 ) . Better results may be obtained by the use of efficient experimental designs ( 2 , 3). The technique of multiple regression or any other statistical method does not overcome the disadvantages of haphazard selection of experimental data. If a number of calculations are to be done, it may be advantageous to learn and employ the abbreviated Doolittle method (1, 4 ) involving matrices for solving the k $. 1 simultaneoue Equat’ions 4. However, as the solution to the simultaneous equations involves differences of numbers which are small compared to the numbers themselves, the calculations should be carried to three more decimal places than required in the b values in problems involving up t o six dimensionless groups (6). This method in no way answers the question of whether the dimensionless equation is an adequate generalization of the facts or is of the right type. It determines the constants in Equation 1, once the type of equation has been chosen and decided upon. The use of the method minimizes t h e root mean square deviation of the logarithm of the dimensionless group rather than of the variable itself. ADVARTTAGES
Use of a graphical method for determining the constants in a dimensionless equation involves either a trial and error solution such as that outlined by Kern (8) or varying the dimensionless groups two a t a time while keeping the others constant. The trial and error solution entails a considerable amount of work even for three variables and becomes too complex t o handle for more than three. The practice of varying two groups a t a time requires consideriible experimental control and the accumulation of large quantities of data for reasonable accuracy. For comparable accuracy the method presented here requires less experimental data and control. Given a set of data, this method can be shown t o give a reliable estimate of the constants in the dimensionless equation, minimizing the per cent deviation of the dimensionless groups according to the root mean square criterion ( 7 ) .
Vol. 48,No. 2
The computations are completely objective, vhereaa the success of the graphical method involving curve fitting depends on experience and is subject to unavoidable bias. The computing method is straightforward and, after k i n g outlined, can be done by nontechnical personnel. It has real labor-saving advantages, although results almost as good can be obtained by very careful graphical analysis. The computations involved can be useful in appropriating statistical developments for more complete analysis of the experimental data if desired. Coii-cLusioN
The method is useful because it is rapid, straightforward, and objective; economical as it alloms for better experimental dwign, resulting in the elimination of a large fraction of experimental work; efficient because it obtains the most reliable information from the available data; and progressive because it is the starting point for statistical analyses that will be used more and more in engineering as increased accuracy of result is demanded of experiment a1 equipment. LITERATURE CI‘I‘ED
(1) Anderson, R. I,., and Bancroft, 11. A., “Statistical T ~ C O Iin ,J, Research,” pp. 168-206, RlcGraw-Hill, New York, 1952. (2) Bennett, C. A . , and Franklin, N. L., “Statistical Analysis in Chemistry and the Chemical Industry,“ Chap. 6 arid 8 ,
Wiley, New York, 1954. (3) Cochran, W. G., and Cox, G. M , “Experimental Designs,” Wiley. Sew York, 1950. (4) Dmyer, P. S., Psychometrika 6, 101-29 (1941). (5) Hald, d.,“Statistical Theory with Engineering Applicatioiis,” Chap. 20, TViley, Sew York, 1952. (6) Hotelling, H., Ann. Math. Stat. 14, 1-34 (1943). (7) Kendall, bI. G., “Advanced Theory of Statistics,” Vol. I . Chap. 15, Charles Griffin, London, 1946. (8) Kern, D. Q . , “Process Heat Transfer,” Chap. 111, b 1 ~ G i . a ~ Hill, h‘ew York, 1960. (9) Ifackey, C. O., “Graphical Solutions,” 2nd ed., Chap. 5, p. 105, Wiley, Sew k‘ork, 1944. (10) Sieder, E. X., and Tate, G. E., IND.ESG. CHEM.28, 1429-38 (1936). RECEIVED for review February 12, 1055.
. ~ C C C I ~ T E September D
18, 1955.
