Upon application of elementary row transformations the reduced form of Equations 31 is
-ki
0
0
0
is used to indicate the implicit function of a . Clearly a can be evaluated from data. Knowledge of this ratio and x 3 = f (XI, a ) can be used to proceed. For example, from Equations 33d and 33a
‘
0
0
0
-k4
0
-k2
0
0
0
0
0
0
0
0
-kn
0
0
0
0
0
0
0
0
0 ,
Upon elimination of dt and insertion of dxg = f’ (XI: a) dxl there results the differential equation (oXl[l
xg}
dxi
+ PXldX6 = 0
(40)
Summary of Procedure The basic rationale of the method can be summarized as follows:
dxi dt = -k1xix2
Reduce the chemical equations to differential equation form. Express the differential equations in “matrix” form as explained in the examples. Develop a canonical form by elementary row transformations alone. This canonical form is generally not unique, because the redundancies may be used alternatively in the equations. Obtain the ratios of rate constants by utilizing the canonical form and experimental data. If n rate constants are involved, n - 1 ratios are required to reduce the number of computational parameters from n to 1.
The redundancies are recognizable as x4 =
c4
- x2
+ 2x3 + 3x3 + 2x6 f
xi = 3x1 - x2 %Z
-
which is again first-order. Upon solution p = k l / k 3 may be obtained. Proceeding thereby all the necessary ratios may be obtained.
which immediately gives as a canonical set
4x1 -
+ f’(x1,a)l
f
(34) X6 x6
f
67 66
(35)
literature Cited
(36)
(1) Albert, A. A , , “Introduction to Algebraic Theories:” Chap. 2, Univ. Chicago Press, Chicago, 1940. (2) Ames, 1%‘. F., IND.ENG.CHEM.52, 517 (1960). (3) Benson, S. W., “Foundations of Chemical Kinetics,” Chap. 2, McGraw-Hill, New York? 1960. (4) Hildebrand, F. B., “Methods of Applied Mathematics,” Chap. 1, Prentice-Hall, Englewood Cliffs, N. J.: 1952. (5) Johnson, P. R., Parsons, J. L., Roberts, J. B.: IND.ENG.CHEM. 51, 499 (1959). (6) MacDuffee, C. C., “Vectors and Matrices,” Chap. 2, Mathematics Association of .-\merica, 1943. (7) Tong, K. N., “Theory of Mechanical Vibration,” Chap. 3, Wiley, New York! 1960. RECEIVED for review February 12, 1962 A C C E P T E D March 15, 1962
Many alternatives are available in the system of Equations 33 through 36 for the development of implicit functions of rate constant ratios. For example, division of Equation 33c by 33a yields
which is a first-order linear differential equation. The solution for x 3 as a function of x1 involves the ratio a = k2/kl, so x3
= f(X1, a )
(38)
COM MU N IC A T ION
GENERAL RELATIONSHIP FOR EFFECT OF ENTRAINMENT ON DISTILLATION COLUMN PLATE EFFICIENCY Many users of Colburn’s equation for correcting distillation column plate efficiencies for the effect of entrainment are unaware that it i s strictly valid only when the enrichment per plate i s essentially constant. A general expression for E, is presented here, which removes this restriction.
THE importance
of correcting distillation column plate efficiencies for the effect of entrainment has been appreciated for many years. Sumerous investigators (7-5) considered
on distillation contains his approximate equation El’ E, = ___ eE, 1+x
this problem in the early thirties, but the work of Colburn ( 7 ) has emerged as the classical treatment and nearly every text
This relationship is strictly valid only when the enrichment per plate is essentially constant.
218
I&EC
FUNDAMENTALS
(1)
X
Figure 1. McCabe-Thiele diagram showing operating and pseudo-equilibrium lines
If the number of moles of liquid entrained per mole of vapor is denoted by e: the vapor and liquid streams are each increased by a n amount of eV and the corresponding material balance relationship considering entrainment becomes :
On writing an expressi.on for the apparent efficiency, E,, in terms of the apparent vapor composition, Yn = y n &,+I
,E
Figure 2. Ratio of exact to approximate expression for E, for e/R = 0.5
+
Substituting P = E,m (1 - E,)R together with Equation 8 into Equation 4 and collecting terms,
- xJ.
R
(9)
(3) Hence, Substitution of the defining relationships for the of x and I gives E4
-
y n - yJ,- 1 - e[(.+ yn* .- yn-
- x T L )- (z,~c ( x , - x,& 1 )
+
1
1
x,
Y's in terms
111
(4)
E,
=
-B + d B 2 2A
- 4dC
(10)
\\.here
Colburn neglected the bracketed term in the numerator to arrive at his well knoLvn approximate relationship for E,. From a material balance around plate n 1:
+
f = ? - _ l
R
At this point it is necessary to assume that the operating and equilibrium lines can be approximated by straight lines over the range of two plates (Figure 1). L-nder these conditions it follows that for constant E, the pseudo-equilibrium line (plot of Y, us. x,) will also be straight over this range and its slope, P. will be given by:
The ratio of E, as calculated from Equation 10 to that given by Colburn's approximate relationship is plotted in Figure 2. I t is seen that whenmjR is less than unity, the Colburn equation gives values of E, which are too low, while high values of E, result if m/R is greater than 1. In general, the deviation increases with elR and is most significant for values of E, in the vicinity of 0.5. literature Cited
Defining R = L I;, (7) Dividing Equation 5 through by (x,+l Equations 6 and 7 into the result,
(1) Colburn, A. P., IND.ENG.CHEM.28, 526-30 (1936). (2) Rhodes, F. H., Zbid., 26, 1333-5 (1934) ; 27, 272 (1935). (3) Sherwood, T. K., Jenny, F. J., Zbid.,27, 265-72 (1935). (4) Souders, M., Brown, G: G., Zbid.,26, 98-103 (1934). (5) Underwood, A. J. V., Trans. Znst.'Chein. Engrs. (London) 12, 169-78 (1934).
- x,) and substituting
D. E. DANLY
The Chemstrand Corp., Pensaco la, Fla . RECEIVED for review September 15. 1961 May 3, 1962 ACCEPTED VOL. 1
NO. 3
AUGUST 1962
219
