Empiric 1 Equatio Variable HARRY STERN Department of Chemical Engineering, The Stute College of washing ton, Pullman, Wash.
I
T IS often desirable to express three-variable experimental data in the form of an empirical equation. This has the advantage of allowing accurate interpolation between a few scattered points and is a step preliminary to construction of a conventional nomograph relating the three variables. In addition an empirical equation often suggests theoretical explanations of the relationship.
Any function in three variables X , Y , and Z may he represented by the equation
where the right side is an infinite series in X,Y , and Z and the k's are constants such that the series converges. This series may be reduced to a finite polynominal of five terms by making the restrictions that X be linear with Y when 2 is held constant and 2 be linear with Y when X is held constant. Under such conditions no term can be higher than the first degree except the term &XZ. This term can exist because X is not required to be linear with Z. The general equation relating X , II', and %:isthen
0
=
ko
+ klX + kzY + ksZ + ksXZ
(3'
The five constants may be reduced to four by transposing ICo and dividing by -ko. The equation is thus reduced to the form of Equation 1.
I
I
I
I
0 20 40 60 80 100 C-GRAMS AMMONIA PER loo0 GRAMS SOLUTION Figure 1. Plot of p versus c
The method commonly used to obtain an analytical equation representing tables of data in three variables is first to seek functions of two of the variables such that these functions will plot aa a family of straight lines for various constant values of the third variable. Each straight line in the family has a slope, a, and an intercept, b, corresponding to a value of the third variable. Since a and b are secondary variables, a plot of a or b versus the third variable often results in irregular curves for which simple functions are very difficult if not impossible to find.
0' 0
In the method proposed here, it is not neceasaq- to attempt to correlate secondary variables. Instead, functions X, Y , and 2 are sought for the variables z, p, and z, respectively, such that a plot of X versus Y gives a family of straight lines and a plot of 2 versus Y gives another family of straight lines. The three functions X, Y , and Z are then automatically related by the equation
+ CZ + D X Z
T
I
20
30
40
50
T- TEMPERATURE 0 e. Figure 2. Plot of p v e r s u s t
MErHOD
A X -f- BI'
I IO
= 1
(1)
where A , B , G, and D are constants easily obtained from four representative points of experimental data. Proof of the validity of Equation 1 is as follows:
If these restrictions are complied with, Equation 1 is an exact representation of the function and not merely an approximation of Equation 2. Since in the general case three-variable data in x, y , and z will not obey the restrictions imposed, it is necessary t o find functions X = f(z) , Y = g(y), and 2 = h(z), such that the restrictions of linearity will be obeyed. The process of finding such functions is a matter of trial and is similar t o the procedure followed in the two-variable correlation (1, b), except that in this case families of curves rather than single curves must be rectified. This step in the method requires some experience in two-variable correlation. 2442
2.01
,
2443
INDUSTRIAL AND ENGINEERING CHEMISTRY
October 1952
I
I
I
L a ( y
Figure 3.
-
I
1.6)
1000 T m 0
Rectified Plot Figure 4.
The method is applicable to any three-variable data for which functions X , Y,and 2 can be found to meet the linearity restrictions. In general such functions can be found for any data which do not exhibit discontinuities either in the function or in the first partial derivatives of the function. If such discontinuities do occur it may be possible to divide the data into regions bounded by them and then find a different analytical equat,ion for each region. EXAMPLE
The method of correlation is illustrated by the following numerical example: The data (8)of Table I give the equilibrium partial pressure, p , of ammonia over a water solution as a function of the temperature, t , and the concentration, c. It is desired to represent the function by an empirical equation. In the table, the numbers in parentheses are values of p calculated by Equation 7, which is the empirical equation found by the proposed method. STEP1. Plot p versus c a t constant 1 (Figure 1) and p versus 1 a t constant c (Figure 2). STEP2. Using well-known methods for two-variable correlation (1, B ) , find functions of p , c, and t such that the curves of Figures 1 and 2 will be rectified. After some experimentation
indicated by large circles in Figures 1 through 4. All points must obey Equation 1, which in this case has the form
1000 c log +D P
1000 (im)
X log
1000 ( 7 - 1.6)
=
*
(4)
STEP4. Evaluate the constants A , B , G, and D by substituting the four arbitrarily selected points into Equation 4, thus obtaining four equations which follow.
OF AMMOXTIA OVER WATER TABLE I. PARTIALPRESSURE SOLUTION
G.
c, “I/
1000 G. Soln. 100 75
50
--
these functions were found to be log -, ‘Oo0 log l,,>, and P ‘Oo0 respectively. These rectified plots are shown in Figt 230’ ures 3 and 4. STEP 3. Select four arbitrary but representative points from the plots of Figures 3 and 4. These points should be widely separated throughout the data. (Note that it is not necessary to find the slopes or intercepts of any of these lines. The lines are drawn t o show that the relationship between the functions is linear and also to facilitate choosing the four representative points.) The points selected in this case together with the values of the corresponding functions are given in Table I1 and are
Rectified Plot
+
a
-
0
10
25.1 (25.1) 17.7 (17.8) 11.2 (11.0)
41.8 (42.8) 29.9 (30.5) l9,l (19.3) 16.1 (16.1) 11.3 (11.2)
p‘, hlm. He, a t t (” C.)----------
20
69.6 (69.9) 50.0 (50.0) 31.7 (31.8) 24.9 (24.9) 18.2 (18.6) 15.0 (15.2) 12.0 (12.0)
40
..
30
..
25
..
..
20
..
..
16
..
..
..
12
..
..
, .
10
..
..
..
30
110 (110.0) 79.7 (78.8) 51.0 (50.4) 40.1 (39.6) 29.6 (29.6) 24.4 (24.2) 19.3 (19.2) 15.3 (15.3) 11.5 , (11.4)
..
40
50
167 (167.2) 120 (120.2) 76.5 (77.1) 60.8 (60.8) 45.0 (45.5) 37.6 (37.3) 30.0 (29.7) 24.1 (23.6) 18.3 (17.7) 15.4 (14.7)
247 (247.7) 179 (177.9) 115 (114.5) 91.1 (90.3) 67.1 (67.9) 55.7 (58.7) 44.5 (44.4) 35.5 (35.5) 26.7 (26.6) 22.2 (22.2)
Values in parentheses were calculated by Equation 7.
2444
Vol. 44, No. 10
INDUSTRIAL AND ENGINEERING CHEMISTRY 4.348 A 3.571 A 3.704 A 4.000 A
+ 0.9243 B + 1.5999 C + 4.019 D 1 + 1.9930 B + 1.6532 C + 7.117 D = 1 ++ 1.6848 1.0694 B + 0,9191 C + 3.967 D = 1 B + 1.9206 C + 6.739 D = 1 =
(5)
Simultaneous solution of Equation 5 yields A = 0.2621, B = 0.1611, C = -0.2152, and D = 0.01388. When these values are substituted into Equation 4 the final equation becomes 0.2621
(tm) 1000 + 0.1611 log (y- 1.6) - 0.2152 log
(y) +
0.01388
(&o)
X log
(y-
1.6) = 1
+ 540.85 t + 230 0.7486 t + 236.68 ( t + 230 ) X log (F- 1.6)
t
Point 1 2 3
4
(6)
which may be rearranged t o give Iogp =
TABLE11. VALUESOF FOURSELECTED REPRESENTATIVE POINTS 1000 t 4-230 4.348 3.571 3.704 4.000
log
1000
(c0,9243 1.9930 1.0694 1,6848
log
1000
7 1,5999 1,6532 0.9191
1.9206
+ log 230
x
(y-
1.6) 4.019 7.117 3.957 6.739
=t1.1mm. of mercury and the average deviation is =k 0.3 mm. of mercury. LITERATURE CITED
7.6468 t
(7)
Values of p cakulated by Equation 7 are listed in parentheses in Table I. The maximum deviation from the original data is
(1) Davis, D. S., “Empirical Equations and Komography,” pp. 3-29, New York, McGraw-Hill Book Co., Inc., 1943. ( 2 ) Hoelscher, R. P., Arnold, J. S.,and Pierce, S. H., “Graphic Aids in Engineering Computation,” pp. 27-61, New York, McGrawHill Book Co., Inc., 1952. (3) Sherrood, T . K., IND. ENG.CHEM.,17, 745 (1925).
RECEIVED for review March 14, 1952.
ACCEPTEDM a y 12, 1952.
Variables Controlling the CrossLinking Reactions in Rubber U
BERNARD C. BARTON
AND EDWIN J. HART1
General Laboratories, United States Rubber Co., Passaic, N . J.
T
HERE is general acceptance of the theory that vulcanization
of rubber with sulfur results from achemical reaction in which intermolecular sulfur cross links are formed. Supplementary chemicals such as accelerators, metal oxides, and fatty acids are employed to make the reaction faster and t o increase the yield of cross links. That sulfur may react n-ith rubber and compounding ingredients in various ways t o give diversified reaction products is well known. Reactions of sulfur v-hich lead to products other than intermolecular cross links do not enhance vulcanization and may be considered side reactions of secondary importance. Insufficient knowledge of the manner in which temperature of cure and concentration of compounding ingredients affect the yield of cross links in the vulcanization reaction has led to the general belief that little or no dependence exists between physical properties and combined sulfur. In the present paper the relationship betveen the vulcanization variables and yield of cross links is described. I t is shown that when these variables are controlled in such a way as t o give the greatest yield of cross links, a definite reproducible relationship exists between physical cure (stress a t 2007, elongation) and combined sulfur. DETERMINATION OF YIELD OF CROSS LINKS IN VU LCANl ZATION REACTION
In general, the more highly vulcanized the rubber compound, the higher is its modulus. I t is, therefore, natural to suppose that modulus, or retractive force, is directly related to the number of cross links. Wall (6) and Flory ( 3 ) have reduced these generalities to precise mathematical terms. Based on the entropy changewhich occurs Then a vulcanized polymer is stretched, an equation has been derived which shows that the equilibrium 1
Present address, Argonne National Laboratories, Chicago, Ill.
retractive force-in other words, stress a t a given elongation-is a linear function of the number of cross links per unit volume. For pure gum compounds a constant relationship has been found to exist between equilibrium retractive force a t 200% elongation and the retractive force a t 200y0 elongation determined in the first elongation cycle. Since all the compounds employed in the investigations described a t this time are pure gum compounds, the nonequilibrium stress a t 20001, elongation obtained on the first elongation cycle is used as a measure of the relative number of cross links present. EFFECT OF VULCANIZATION V4RIABLES ON YIELD OF CROSS LINKS
In EFFECTOF TIMEAXD TEMPERATCRE OF VULCANIZATIOX. a study of the effect of vulcanization reactants on the yield of
sulfur cross-linked products it is desirable t o select time and temperature of reaction which are most favorable for a high yield of cross links. The effect of the temperature of cure on the relative yield of cross links was determined in a compound containing one part of sulfur and quantities of zinc oxide, lauric acid, and 2-mercaptobenzothiazole in excess of the amounts necessary for the greatest yield of cross links. The choice of composition is made clear in the folloming sections. Times of cure were varied a t each temperature, as shown in Table I, so as to make sure that the vulcanization reaction was carried t o completion. The maximum stress obtained a t each curing temperature is shown in Figure 1. This figure shonx that the yield of cross links increases with decreasing temperature of cure until 90” C. is reached. Curing temperatures lower than 90” C. result in lower yields of cross links. Searly maximum cross linking is obtained a t 100” C. and, since it is more convenient to cure a t this temperature than a t 90” C.,
