Nomenclature
Y
A,, Bi
$AO
A (COz), B (SH3) in liquid A I , B1 = initial concentrations of A, B in liquid = bulk-average concentration of A in jet emerging A2 from absorption chamber Dd,DS = diffusivities of dissolved gases A, B in liquid = diffusivity of B in gas DvB d = jet diameter h = jet length K,, KB, K1, K2 = mass action equilibrium constants k = second-order reaction rate constant for reaction between A and B in liquid phase = cumulative average mass transfer coefficient for A without chemical reaction = cumulative time-average mass transfer coefficient TLA for A = average mass transfer coefficient for B k*GB m = vA,/B, = average rate of absorption of A in j e t Z A = average rate of absorption of a nonreacting gas in JVAO jet P = total pressure = arithmetic average partial pressure of inert gas PAW = partial pressure of B p~~ PA;, p~~ = interfacial partial pressures of A and B = average absorption rate with reaction/rate without Q reaction = ratio of mass transfer coefficients for A QA qL = volumetric flow rate of liquid stream through nozzle = volumetric flow rate of gas mixture into apparatus
F r t
z
e
= interfacial concentrations of
= stoichiometric coefficient = absorption rate for whole jet
Literature Cifed
(1) Faurholt, C., J . Chem. Phys. 22, 1 (1925). 2) Fields, M. C., M. Ch. E. thesis, Univ. of Delaware, 1958. 3 Harned, H. S., Davis, R., J . Am. Chem. SOC.65,2030 (1940). 4 Harned, H. S., Robinson, R. A., Trans. Faraday SOC.36, 977 (1940). (5) Harned, H. S., Scholes, S. R., J . Am. Chem. SOC.63, 1706 (1941). (6) Hatch, T., M. Ch. E. thesis, Univ. of Delaware, 1958. (7) MacInnes, D. A., “The Principles of Electrochemistry,” p. 339, Reinhold, New York, 1939. (8) MacInnes, D. A., Shedlovsky, T., Longsworth, L. G., J . Am. Chem. SOL.54, 2758 (1932). (9) Manogue, W. H., Ph.D. thesis in Chem. Eng., Univ. of Delaware, 1953. (10) Nijsing, R. A. T. O., dissertation, Delft, 1957; Nijsing, R. A. T. O., Kramers, H., Chem. Eng. Sci.8, 81-9 (1958). Pearson, L., Roughton, F. J. \V., Trans. (11) Pinsent. B. R. W., ‘ Faraday SOC. 52, 1594 (1956). (12 Roper, G. H., Hatch, T. F., Jr., Pigford, R. L., IND.ENG. &EM., FUNDAhlENTALS 1, 144-52 (1962). (13) Scriven, L. E., Pigford, R. L., A . I. Ch. E. Journal 5, 397-402 (1959). (14) Zbid., 4, 439 (1958). (15) Shedlovsky, T., MacInnes, D. A., J . Am. Chem. SOG.57, 1705 (1935). (16) Sherwood, T. K., Pigford, R. L., “Absorption and Extraction,” 2nd ed., McGraw-Hill, New York, 1952. (17) Shokin, I. N., Solov’eva, A. S., Zhur. Priklad. Khim. 26, 584 (19 53).
Ii
RECEIVED for review January 26, 1960 RESUBMITTED December 1, 1961 ACCEPTEDFebruary 20, 1962
= DB/DA = constant in equation for interfacial concentration
26th Annual Chemical Engineering Symposium, Division of Industrial and Engineering Chemistry, ACS, Baltimore, Md., December 1959. Work supported by a National Science Foundation postgraduate fellowship to T. F. Hatch, Jr., 1956-58.
= elapsed time of exposure of liquid to gas = distance along surface of jet
= kBit
CANONICAL FORMS FOR NONLINEAR KINETIC DIFFERENTIAL EQUATIONS W.
F. AMES
Department of Mechanical Engineering, Uniiiersity of Delaware, Newark, Del.
The evaluation of relative rate constants is difficult for complex chemical reactions because of the nonlinearity of the describing system of differential equations. A useful “matrix” method is described for the development of a canonical form for the kinetic differential equations. Three examples are drawn from the literature. The advantages of the procedure are simplification of the system of equations; automatic determination and elimination of redundancies; applicability to relative rate constant determination; and reduction of computation complexity.
PREVIOUS
paper (2) considered the problem of evaluating
A ratios of rate constants for systems of differential equations arising as the mathematical model of chemical reactions. The technique described was, generally, a n implicit procedure whereby the ratios of the rate constants were obtained in the form of implicit algebraic functions. These functions usually require solution by numerical means. While the problems solved (2) were real problems and the methods used are capable of generalization, no unifying pro214
I&EC FUNDAMENTALS
cedure for attacking another problem of similar type was included. This paper presents a useful general procedure for developing a canonical form for the system of differential equations. In turn this canonical form is of great use in evaluation of rate constant ratios. The motivation has been the application of matrix theory in the development of “normal” coordinates for the vibrations of linear elastic systems with many degrees of freedom, The normal coordinate procedure is based on the reduction to diagonal form of the coefficient matrix of
the system of differeniial equations (7). This matrix contains the physical parameters of the system. I n the strictest sense this application of matrix algebra is useful only for linear systems. However, the essentials of the idea can be transferred to certain systems of nonlinear differential equations, providing precautions are observed. Matrix Transformations
Three types of elementary operations or transformations upon the rows of a mairix can be defined ( 7 , 4 , 6 ):
1. Interchange of t\vo rows 2. Multiplication (of the elements of any row by the same nonzero number. a 3. Addition (or subtraction) to any row of a multiple of any other row M‘hile it is possible to define equivalent column operations, these are not allowable here because of the nonlinearity of the equations. I n addition to operations 1, 2, and 3, several theorems, whose proofs may be found in the literature ( 7 , 4, 6), are important for the development of the canonical form.
Theorem I. If A is a nonsingular (determinant of A4# 0) square matrix, then A can be carried into the identity matrix by elementary row transformations alone. Theorem 2. Every nonzero m by n matrix A whose rank is r ( r # 0) can be transformed by elementary row transformations alone into a matrix of the form
There are redundancies in the system of Equations 3 ( 3 ) . No special treatment is needed to pick them out as seen in the development of the canonical form for Equation 3. However, useful information in the development of the canonical form is available in the selection of “dead end” variables: products of reaction which do not themselves react in so far as this model is concerned. The dead end variables in system 3 are clearly x4 and x i . The system has four reactions and therefore only four differential equations are required ( 3 ) .
To discover another redundancy (not unique), a most efficient method (which also develops the canonical form) is to arrange the complete system (Equations 3) in “matrix” form as follows: Along the top of the matrix place the individual terms in the right side of Equation 3. Along the side of the array place the derivative to be considered. At those positions where the differential equation has a term, insert the appropriate coefficient. O n proceeding thus, System 3 becomes %.
where Z, is the I by r identity matrix and 0 represents some zero matrix. (A matrix is said to have rank r if and only if it has a nonvanishing determinant of order r and no nonvanishing determinant of order greater than r.) Application of Theorem 2 is fundamental to the development of the canonical form, because the ”matrix form” of the nonlinear kinetic equations will invariably be initially a nonsquare matrix. The following examples illustrate the canonical system approach. Illustrative Examples
Example 1. tions
A chemical reaction is governed by the equa-
(2)
where A4 is the desired product and A6 and A7 are undesirable by-products. These reactions are assumed to proceed at constant volume and temperature. With x i = concentration of .4i (moles per unit volume), and with k i = rate constant (liters/gram-mole sec.) the differential equations for the chemical reactions are :
XZx5,
22x6
X I 0
0
0
-kz
-kr
-ki
-ka
0
0
x2,
-Li
ki
-
XZx3,
0
k
0
0
0
0
0
0
z
O
-k3
di dt
1
O
(4)
0
k3
-kr
0
Keeping in mind that elimination of the dead end variables may be expedient, the matrix Equation 4 is now reduced to diagonal form by elementary matrix operations on the rows alone. These elementary operations have already been described. T h e same operations are also carried out on the column vector in Equation 4. If any row is reducible to zero, a redundancy has been discovered and this row needs no further consideration. Further, by Theorem 2, with the exception of the redundant rows, the remainder of the matrix (4 X 4 in this case) always permits reduction to a diagonal matrix. When interpreted, this constitutes the canonical form. A variety of reduction approaches are possible, but some systematic scheme is preferred. To this end arrange the system so that the element in the upper left-hand corner is not zero. This can always be done by rearranging the rows. All elements in the first column are now eliminated by adding (or subtracting) a suitable multiple of the first ro\v to every other row which has a nonzero element in the first column. Thus for System 4 there results
(5:
VOL. 1
NO. 3
AUGUST 1962
215
S o w consider the second column, second row position, usually designated by (2, 2) position. This row has three elements. I n this problem (keeping in mind that x 4 and x, are dead end) interchange the third and second rows, thus utilizing x l ) contains only one the fact that the third row (for x 3 element. Further, exchange of the new third row with the sixth and the fourth with the fifth gives the new system:
termed the canonical form of the system. equation form this reads
In differential
+
i 13b)
(d dt
-ki
0
0
0
0
0
0
-kp
0
0
0
0
ks
k4
0
-ks
0
k
0
-k2
O
O
z
O
O
-kc
-k, O
k
4
Using the new element in the (2, 2) position of Equation 6, all other elements in the second column can be eliminated by elementary transformations. Continuing this procedure the final reduction becomes (d x2x3,
x2x6,
X u 6
-ki
0
0
0
0
-kz
0
0
0
0
-ka
0
0
0
0
--kr
0
0
0
0
0
0
0
0
0
0
0
0
XIXZ,
dt
Division of Equation 13d by 13c yields an equation of the same form as 14-namely (7
T h e 4 X 4 diagonal matrix in Equation 7 can be further reduced to the identity matrix Z4by dividing each row by -ki. This in practice may or may not be helpful. From the last three rows of Equation 7 the redundancies are easily seen to be d
$. x6 dt
(f7
where xz is expressible in terms of x , , x3, x g , and xf, by means of Equation 10. The canonical set (Equations 13) does not have all the advantages of a corresponding linear system. However. there are multiple advantages for proceeding with the analysis via the canonical form instead of the unreduced system. The first advantage is the ease with which the ADVANTAGES. rate constant ratios can be obtained from the data via the implicit function method. T o illustrate this approach divide Equation 13b by Equation 13a to obtain
+
Both Equations 14 and 15 are simple linear first-order ordinary differential equations involving the ratios CY = k z / k l and P = k J k 3 . The solutions of Equations 14 and 15 yield implicit equations in CY and p :
and
x5) =
or and similarly x4 = c4 x2
- XI
XI
=
XI
x3
c4
(9)
XI
+ + 2x5 + + + - + + - + - x4 + - + + +
= 2x1 =
-
(10:
CZ
x6
Yet a third ratio is required and it can be supplied by an examination of Equations 13a and 13c, which may be rewritten in the form
x4
x5
ci
xi
c2
x5
x.i
c4
ci
c2
(11)
which is equivalent to the new equation (y = kI/k3)
In Equations 8 to 11 the c's are combinations of the initial conditions-for example c2
- 2x1(0)
= xz(0)
- XP(0)
-240) -
XE!0)
with solution
where it is understood that some of these initial values may be zero. Equations 8, 9, and 10 are the required expressions for x,, x4, and x2 in terms of the remaining variables of the system. Similar relations can also be inferred from the stoichiometry of the system. Upon elimination of the redundancies from Equation 7 there results
-kl
0
0
0
0
-k2
0
0
0
0
-ka
0
0
0
0
-k4
+.;I i: XI
x5
216
l&EC FUNDAMENTALS
=
+
x6
(12)
Equations 16 and 17 are clearly implicit relations for CY and /3 which require numerical solutions from observed values of xl. XI: x j , and xg. Equation 20 is an explicit relation for y. A second advantage accrues from this technique from a computational point of view. If the canonical equations are used for computing after the rate constants ratios have been determined, the number of parameters is reduced from 4 to 1, for setting of one rate constant fixes the remaining three through the known ratios. In addition, the amount of computation is generally reduced.
Example 2. T h e gas phase pyrolysis of toluene is described by Benson ( 3 ) by the chemical equations
The redundancies are immediately obvious from Equation 24 and are in fact
x5
=
cg
-
k:,
2(k2
+
+ 2kr)
c'
(27)
where C- = x3 2x1 f x 2 - X g has been introduced for convenience. A canonical set of differential equations has the form Again setting ,t = concentration of A, and k , = rate constants shown in Equation 21, the kinetic differential equations become d- x_i -hixi dt dx _2 = kixi dt
+
- krxixa - k 3 ~ 1 ~-3 k4XiX6 k2x1xa
+ kaxixs - k5xZ2
(22a) (22b)
(224
(22f)
dx7 dt
=
(28a)
dt
wherecu = k2/k3. Example 3. A chlorination reaction has been characterized by Johnson, Parsons, and Roberts ( 5 ) by the chemical equations
(2%)
kaxixe
and in the matrix form, adopted here, these equations read
+ A4
A1
+ A2
A3
+ A2 +A s + A4
AI kf
(d dt
From the differential equations (Equations 2 2 ) or the chemical equations (Equations 21) it is clear that the dead end variables are x q , x j , and x,. With these in mind, application of elementary row transformations yields the following reduced form of Equations 23, after rearrangement of rows:
The matrix form of the differential Equations 30 is
d ( XIX2
xzxa
XBXB
VOL. 1
X?Yf,
dt
NO. 3 A U G U S T 1 9 6 2
217
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,
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)
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
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)
