ENGINEERING, DESIGN, AND EQUIPMENT Refractive Index Curve. Small weighing bottles of about 30ml. capacity, with female-joint tops, were found convenient for weighing out the liquids in the determination of t h e refractive index curve. T h e fact t h a t the grinding on the bottle proper was on t h e outside prevented accidental wetting of t h e ground portion as t h e liquids were poured in from small graduated cylinders. The smallest quantity of a n y one liquid weighed in these determinations was 1 gram, and because the balance was accurate to 0.0002 gram, t h e combined error for any sample could not be over 0.04%. Pressure-Regulating System. The barostat used was that designed by Willingham and Rossini ( 2 0 ) a t t h e National Bureau of Standards. It was thoroughly cleaned, then heated under high vacuum and filled with clean, redistilled mercury, a vacuum being obtained b y carefully boiling out dissolved air under vacuum IThile the filling was in progress. The barostat was installed in an air thermostat, the temperature of which was controlled t o u-ithin 3" C. b y a Fenwall thermoswitch, which was surrounded by the circulating air stream. T h e temperature of the mercury lagged behind t h a t of the air, and consequently did not change over as a i d e a range. 4 dibutyl phthalate manometer registered the difference between the system pressure and t h a t of t h e atmosphere. B y reading this manometer and the barometer, it was possible t o calculate the pressure within the apparatus. Kitrogen was supplied t o the barostat through a diaphragm valve, from a standard nitrogen cylinder. When the mercury column reached the upper contact of the barostat, a circuit was completed which activated a relay t h a t opened a solenoid valve and allowed escape of nitrogen through a needle valve. All runs were made a t 760 mm. of mercury T h e prevailing pressure in State College is about 730 mm. The pressure variation due t o the variation of the temperature of the mercury was about 0.1 mm. This far exceeds pressure variation due t o the opening and closing of the solenoid valve. It was easy to adjust t h e diaphragm valve on the nitrogen cylinder and the final outlet needle valve, so that the pressure variation could not be detected on the dibutyl phthalate manometer. Temperature Measurement. The temperature-measuring system consisted of a calibrated single-junction copper-constantan thermocouple, a Rubicon galvanometer with a sensitivity of 1.1 pv. per mm., and a Leeds & Northrup Type K-2 potentiometer, which could be read to 0.1 mv. The calibration was accomplished b y reading the electromotive force with carefully purified n-heptane, and with pure water boiling in the apparatus. The difference between the observed voltage and the voltage of a standard sample was taken a t each point and plotted against the observed voltage. A straight line was drawn from the zero point ( A E = 0, EOb4 = 0) which passed between the plotted points. The distance of t h e
calibration points from the line was within the limits of accuracy (0.05 O C. ) of t h e temperature-measuring equipment. Subsequent voltage readings were corrected b y adding t h e AE for each When the boiling point of the carefully puSified toluene was measured, it was 110.59",as compared t o a National Bureau of Standards value of 110.62'. This serves as a further indication of the accuracy of the temperature measurements. Literature cited
Bromiley, E. C., and Quiggle, D., IND.ENG.CHEX.,25, 1136 (1933).
Brown, J., and Ewald, A . H., Australian J . Sci. Research, 3A, 306-23 (1950); 4A, 198-212 (1951).
Cornell, L. W., and Montonna, R. E., IKD.ENQ.CHEM.,25, 1331-35 (1933).
Fowler, R. T., J. SOC.Chem. Ind. (London), 68, 131-2 (1949); Ind. Chemist, 24, 717, 824 (1948).
Garner, F. H., Ibid., 25, 238 (1948). Gillespie, D. T. C., IND.ENG.CHEM.,ANAL.ED.,18, 575 (1940). Jones, C. A., Schoenborn, E. >I., and Colburn, A. P., IND. ENG.CHEM.,35, 666-72 (1943). Kortiim, G., Moegling, D., and Woerner, F., Chem. Eng. Tech., 22, 453-7 (1950).
Natl. Bur. Standards, Circ. C461, 39, 43, 123, 126 (1947). Othmer, D. F., IND.ENQ.CHEM.,20,743-6 (1928); Anat. Chem., 20, 763 (1948).
Regnault, Mem. mad. inst. France, 26, 727 (1862). Rose, Arthur, Williams, E. T., Sanders, W. W., Heiny, R. L., and Ryan, J. F., IND.ENG.CHEW,45, 1568-72 (1963). Rose-Innes and Young, Phil. Mag., (V) 47, 353 (1899). Saddington, A. W., and Krase, N. W., J . Am. Chem. Soc., 56, 353-61 (1934).
Sage, B. H., and Lacey, W. N., Trans. Am. Inst. Mining Met. Engrs., 136, 13G-57 (1940).
Sieg, L., Chem. Eng. Tech., 22, 322-6 (1950). Steinhauser, H. H., and White, R. R., IND.ENG.CHEY.,41, 2912-20 (1949).
Swietoslawski. W., J . Chem. Educ., 5, 469 (1928). Williams, E. T., Ph.D. thesis, Pennsylvania State College, University Park, Pa., 1952. Willingham, C. B., and Rossini, F. D., API Research Project 6, December 1945. Zawideki, J., 2. physik. Chem., 35, 129 (1900). ACCEPTED February 2 , 1955. R E C E I V Efor D review April 21, 1953. Presented before the Division of Industrial end Engineering Chemistry a t CHEMICAL SOCIETY,Los .4ngeles, the 123rd Meeting of the AMERICAN Calif,
Applications of Matrix Mathematics to Chemical Engineering Problems ANDREAS ACRlVOSl
AND
NEAL R. AMUNDSON2
Departmenf of Chemical Engineering, University of Minnesofa, Minneapolis 14, Minn.
R
ECENTLY, chemic81 engineers have become increasingly aware of the usefulness of the calculus of finite differences as a mathematical tool for solving problems in stagewise operations. This particular section of mathematics has remained dormant for some years, because of the restricted use it has found in most branches of applied mathematics, where mostly continuous, rather than discrete, phenomena are studied. There are, however, many textbooks on the calculus of finite differences, including those of Boole ( 3 ) , Jordan ( I S ) , Milne-Thompson ( I $ ) , Wallenberg and Guldberg (271, and Norlund (21). Applications of the calculus of finite differences t o a variety of relatively simple chemical engineering problems have been made 1 Present address, Department of Chemistry and Chemical Engineering, Univeisity of California, Berkeley, Calif. * Present address (on leave), Cambridge University, Cambridge, England.
August 1955
by Tiller and Tour ( 2 5 ) , Tiller (ZC), Mason and Piret ( l 7 ) , Lapidus and Amundson (16), and Amundson ( I ). This article presents a method t h a t will enable the chemical engineer t o deal with more involved problems t h a t arise in connection with the unsteady-state behavior of stagewise operations; there has been, of late, considerable interest in these chemical engineering problems. This new approach will make extensive use of matrix algebra, the salient features of which are described. It is surprising t o learn t h a t most results and theorems of practical interest concerning matrices were discovered almost a century ago. However, their usefulness in many branches of applied mathematics was not appreciated until very recently. The present interest in matrices was first stimulated by the work of Born, Heisenberg and P. Jordan in quantum mechanics, a n d
INDUSTRIAL AND ENGINEERING CHEMISTRY
1533
ENGINEERING, DESIGN, AND EQUIPMENT it seems t h a t the aeronautical and electrical engineers were the first among t h e engineers t o have employed matrices extensively i n their researches. At present, of course, this interest is widespread among applied mathematicians, as revealed b y the appropriate literature. The present article will, i t is hoped, serve two purposes-to present t h e essential features of matrix algebra, and thus make them known to a larger circle of chemical engineers; and t o illustrate the usefulness of this new approach by means of some rather interesting examples. N o general theorems will be proved, except in some special cases, and constant reference t o the textbooks on the subject will be made. Some of the recommended textbooks in English are those of Perlis (22), Ferrar (6),and the first chapter of Courant and Hilbert ( 4 ) , which are primarily devoted t o the basic principles; those of Michal (18),Ferrar (?'), and especially Dwyer (6) and Frazer, Duncan, and Collar (9),which contain material of considerable practical interest ; and t h a t of MacDuffee (16), in which most of the finer points are examined.
where t h e Then
ai's
are scalars, is, of course, another matrix, P(A).
is said t o b e a polynomial of A of t h e n t h degree. This, of course, is a n extension of the definition of polynomials for scalars. Note that A", A"-',. . ., A2, A, Zare all of the same order. Power Series in Matrices. A natural extension of the above definition of a polynomial of a matrix t o include infinite series in matrices can now be put forth. Let there be a sequence, Ao, AI, A2,. . ., of matrices t h a t are all of t h e same order, and let P
m
Ai.
S, =
Then the infinite series,
0
vergent if every element in S, converges t o a bounded limit as p approaches infinity. Of particular interest in applications are the infinite power m
is, matrices of the form, Xa,Al,
series in a matrix, A-that
0
Properties of matrices
Some of the simple rules obeyed by matrices, namely addition and multiplicatiori, have already been stated by Amundson ( I ) . I n the present article, use is made only of square matrices, which are denoted by capital letters, and column matrices, called vectors, which are represented by lower ca8e letters. Boldface type is used for matrices t o dist,inguish them from scalars. Matrices with real elements will be studied. The following definitions apply t o a matrix:
A = [ u c ~=]
(211
(212
(221
(222
.
(Un1
. .
(213 (223
.
.
@;)
where the a 1 ' s are scalars. The discussion of the convergence of such power series and of their summation is presented with t h e formula of Sylvester. One of the most important matrix power series is the matrix exponential function, eA, which is defined, in complete analogy with the scalar exponential function, b y the power series m
A2
eA=Z+Affl-+...
an2
.
an3
1
ann
= [aii]. The adjoint of A, adj A, is equal t o [ A i < ]where , Aii is the cofactor of aii in A . The matrix, I = [&,I, where 869 is t h e Kronecker symbol ( S i j = 0 if i j and 6 ~ = i l),is called the unit matrix, since
A" m!
It can easily be seen from the above definition t h a t and eAeB -.
sinA=ACOS
A I = ZA = A for an arbitrary A of the same order as I . T h e matrix, 0,all the elements of which are equal to zero, is called the null matrix, since
A 0 = OA = 0 again for a n arbitrary A. The inverse matrix, A-1, of a matrix A has the property t h a t
AA-1 = A-'A
=
A = I -
A matrix, A, for which I A 1 = 0, is said to be singular. If A' = A, A is said t o be symmetric. If A' = A-I, A is said to be orthogonal. The matrix, B = C-'AC, where C is any nonsingular matrix, is said t o be obtained from A by a similarity transformation. Matrix Polynomials. The finite sum
+ oriA"-l + . . . 4- or,-~A + ornI
A3
3!
~
A2 2!
A5 +% ...
+ 4!
A4 -
(4)
If A is a square matrix of order n,the determinantal equation / X I - AI = 0
(5)
is called the characteristic (or secular) equation of A. X is a scalar, and b y expanding the above determinant it can be reduced t o a polynominal in X o€ degree n. Thus,
1XI- A l ~ P ( X ) ~ X " + p i X " - ' + p z X n - 2 +
++
,..
pn-1X
I
A necessary and sufficient condition for the existence of A-' is t h a t the determinant of A, denoted b y 1 A I or det A, be different from zero. I n t h a t case
eA+B
only if A and B are commutative matrices-that is, only if AB = BA. Other examples of matrix power series are the functions, sin A and cos A, defined by
+
1534
=c-
e*e-A =
A square matrix is of order n if it has n rows and n columns. The transposed A' of a matrix, A, is defined t o be the matrix A' which has its rows identical with t h e columns of A-Le.,
wA"
Ai, is said t o be con0
pa
0 (6)
where the p ' s are scalar coefficients. The algebraic equation, which are called t h e ( k = 1,2,. . Equation 6, has n roots, characteristic, or latent, roots of the square matrix, A. The Cagley-Hamilton theorem states t h a t P( A ) is the null matrix, or, in other words, t h a t P(A)
E
A"
+ PIA"..' + pzAn-2 + . . . + pn-jA + p n I =
0
(7)
where t h e p l ' s are identical with the corresponding coefficients of the characteristic equation. ilnother formulation of the above theorem is t h a t any square matrix, A, satisfies its own characteristic equation. Sylvester Formula. The Sylvester formula, a n exceedingly useful expression, can readily be derived from the Cayley-Hamilton theorem and the Lagrange interpolation formula. It states
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 47, No. 8
ENGINEERING, DESIGN, AND EQUIPMENT that, if the characteristic roots of a square matrix, A, of order
n are all distinct, and F( A ) is any matrix polynomial, or any con-
in the vapor leaving the nth plate, and z n ( i j , the mole fraction of the ith component in the liquid leaving the n t h plate, is given by
vergent power series, in A ,
Ydi)
=
Kn(i)zn(i)
(12)
where the function, K n ( i ) has , to be estimated a priori. The state of the rectification column can then be described b y the following equation, which can be obtained b y combining a material balance around a n y plate in t h e tower with Equation 12:
Thus
Ln-izn-~(i)- [Ln
+ V n K n ( i ) l ~ n (+i ) Vn+1K,+1(i)sn+.(i)= 0 (13)
(9)
from which it follows t h a t e A , is defined b y the infinite series, exists for all square matrices, A, since ex exists for all scalars, A. A similar argument can be used also for any power series in A, and it follows t h a t a power series in A , F ( A j , converges only as long as all the characteristic roots of A lie within the radius of convergence of F( A). The usefulness of the Sylvester theorem cannot be overestimated. Fractional powers of a matrix can now be defined without difficulty, and it can be stated that, under suitable conditions, a n y analytic function of a square matrix, A , of order n can be expressed as a polynominal in A of degree n - 1. -4s a matter of fact, many functions of a matrix A can be formulated satisfactorily only by means of the Sylvester formula. When multiplicities in the latent roots occur, the Sylvester formula has to be modified. Thus if, for example, a set of roots, X1, b, . . . , A,, are ~ ( iE) equal, Sylvester's formula can be stated in its confluent form ( 9 )
for 0 5 n iN , b u t n # f, where f denotes the feed tray. I n Equation 13, L, and V, denote the total moles of liquid and denotes the mole vapor, respectively, leaving plate n. If z~(i) fraction of the i t h component in the feed, when n = f,
- [ L , + V / K / ( i ) l ~ t (+i )
Lj-l~j-,(i)
+
Vj+iKj+i(ihj+i(i)
Z F ( ~ )=
0
(14)
Equations 13 and 14 can next be combined into a single matrix equation b y defining t h e vectors
0
0
LZ
where D is the nusnber of moles of overhead product. The matrix equation, which takes the place of Equations 13 and 14, is where &(A) = ( A -A, + 1) (X - A, + 5). . .(X - A,) and the sum is taken over all the distinct roots of A . It is worth remembering t h a t the following identity exists adj ( A i l
-
A ) = (-l)n-l
?r
i #j
(Ail
-
A)
(11)
Differentiation and Integration of Matrices. If u ( t ) is a square matrix or a vector, the elements of which are fractions of a parameter, t, the derivative of u ( t ) with respect to t is defined as the matrix obtained by differentiating each element of u(t). thus
A(i)x(ij
+ z(ij = 0
(15)
and its solution is
~ ( i= )-A-'(i)Z(ij
(16)
I n practice it is best t o obtain zo(i)from Equation 16 and then to calculate z,(i) for n > 0 b y iterating Equation 13. Example I11 will demonstrate the method of evaluating the determinant and the appropriate cofactor of a matrix of same form as A . Suppose that, as a result of the preceding calculations, the estimate in the function, K n ( i ) ,has t o be revised slightly so t h a t the new matrix thus obtained is
A i = A + B where A is identical with t h a t defined previously and B is another square matrix, the elements of which are much smaller in magnitude than t h e corresponding elements of A .
similarly
+
+
(A B ) - l = A-I(Z BA-l)-' = A-'[I - BA-' (BA-')Z - . . . j I n addition, there exists a definition for the derivative of a matrix with respect t o a matrix, but this concept will not be used in this discussion. Example 1.
Inversion of a matrix by iteration
It often happens t h a t the investigator is able to discover a n approximate solution t o a given problem, and then is faced with t h e task of devising a simple iterative procedure which will converge and which, moreover, will improve the answer rapidly. Such a n example is presented in the design of fractionating columns, if use is made of the Geddes and Thiele method (25). It is assumed t h a t the column has a total of N plates, including the reboiler, and a condenser, the plate 0, and that, moreover, t h e relation between yn(i), the mole fraction of the i t h component August 1955
+
or, approximately,
~ i ( i=) - A i - ' ~ ( i )
( I - A-'Bjx(i)
(17)
where ~ ( iis )given by Equation 16. This method appears attractive, in t h a t once a good approximation to the answer has been arrived at-that is, once A and A-I have been determinedan iteration can be set up b y varying Bin Equation 17. Example II. Analytical solution for equations of multicomponent rectifscation
Some years ago Underwood ($6) and Murdoch (60),among others, arrived a t an analytic solution for the equations of multicomponent rectification. Their solution can be derived b y the use of matrix algebra in a n elegant way. If constant molal overflow is assumed, constant relative
INDUSTRIAL AND ENGINEERING CHEMISTRY
1535
ENGINEERING, DESIGN, AND EQUIPMENT volatilities and a relation between x , ( i ) and y,(i) of the form
1 A =
+1
a1
1
1
+11 an
i=l
th*en a material balance around an arbitrary section of the enriching column results in (23)
= ?r i
ai j l
1 1 1 f a a
.
+ 7d;i
Therefore,
i= 1
where R is the reflux ratio, x o ( i ) denotes the composition of the overhead product, m the total number of components in the mixture and p ( i ) 2 [ a ( i ) ]-I, where a ( i ) is the relative volatility of t h e i t h component. Equation 18 can now be linearized b y a change in the dependent variable-that is, by defining a new function, X n ( i ) ,as
and the roots of A are identical with those of
It is, of course, very easy to establish t h a t the roots of Equation 26 are all positive and distinct (20). Moreover, it can be shown
that
j= 1
in which case Equation 18 can be transformed into where the representative term, bi, is equal t o together with the condit,ion t h a t and that, therefore, The substitution given b y Equation 20 follows as a logical consequence of the plate-by-plate method and will be described in detail in a subsequent publication. However, Equation 21 can be expressed as
where
and Equation 28 can be shown t o be identical to the formulas derived by Underwood (26) and Murdoch ( 2 0 ) . It is also apparent that, in general, if n is small, Equation 24 is t h e more convenient form of the solution, while if n is large, Equation 27 is preferable.
X , is a vector and A is a square matrix.
The solution of Equa-
tion 23 is
which expresses the result of t h e iterative method of solution, the so-called plate-by-plate method, in the convenient matrix notation. However, Equation 24 can be transformed by means of the Sylvester formula so t h a t
i f it is assumed, for the moment, t h a t the characteristic roots of A are all distinct. This equation can be simplified considerably. By subtracting the first column from each of the others and then expanding the resulting determinant, the determinant 1536
Example 111. Transient behavior of some stagewise systems
Mathematical Model and Its Solution. The following section considers, in some detail, a simple mathematical model t h a t appears wherever the unsteady-state behavior of absorption, extraction, and distillation columns, or other kinds of stagewise systems with reflux, is investigated. For the sake of illustration, a distillation column is considered, but, of course, this restriction should not be interpreted literally. If a material balance is made around a representative plate, n, in the column the following is obtained:
Again, as in Example I, i is the component-designating symbol; z n ( i , t ) and yn(i,t) are, respectively, the mole fraction of the ith
INDUSTRIAL AND ENGINEERING CHEMISTRY
voi. 47,
NO.
a
ENGINEERING, DESIGN. AND EQUIPMENT component in the liquid and the vapor stream leaving the n t h plate; L,(t) and V,(t) are, respectively, the moles of liquid and vapor leaving the nth plate; while h, is the liquid holdup on the nth plate. In constructing the above model, any time lags that are always present have been neglected because, of course, the various streams are moving across the column a t a finite speed. These time lags would have to be taken into account in any more exact calculations, because of the pronounced effects they have on the response of the column to an applied change in the operating conditions. Another simplification occurred when the vapor holdup was neglected. The vapor holdup may be included and more general models may be constructed but the solution, in any event, follows t h a t presented in this discussion. It is assumed that
where K,(i) is a function which can be determined a priori. I n particular, if the solution t o Equation 29 is t o be used to predict the transient characteristics of the column, following a small fluctuation of the various parameters of the system, such as the composition of the feed, the heat content of the feed, the reflux ratios, etc., then y,(i) and x,(i) can be made to denote the deviations from the corresponding steady-state values in the vapor and liquid, respectively, and Equation 30 may be written
By combining Equat,ions 29 and 30, and by restriction to a representative component, i, a single matrix equation can be obtained.
Similarly,
which is but a product of c square matrices.
[M;(A)]-I =
&1 [I-
A(t,-,)
:]
(38)
so that, in this way, Equation 33 can be evaluated numericaily with the aid of a sufficiently powerful digital computer, by performing 2c multiplications of a vector b y a square matrix. However, the problem under consideration has certain simplifying features, because the vector, z ( t ) , has all of its elements b u t one equal to zero, unless side streams are introduced into or removed from the column, and A ( t ) is o€ the form a12
0
as2
-aar a23
0 a 0a4
:)
(39)
where
It is important to note that the form of A ( t ) remains unchanged if side streams are introduced into or removed from the column. I n the following discussion the restriction will be to problems where the square matrix, A , is of the form given by Equation 39. Special Case. I n the special problem A ( t ) E A, if i t is assumed that the various flows in the column are independent of the time, so t h a t A ( t ) is a constant square matrix and if, in addition, it is supposed that z ( t ) = z(0) for t
0
+ J' e A ( l - a ) i ( s )ds
Equation 41 is equivalent, b y the Sylvester formula, to
-
x ( t ) = ~ ( 0 ) A-lz
M h ( A ) = M ; , ( A )34:1(A)
(35)
+ A-'eAtz
(44)
or, by the Sylvester formula,
Actually, the form of the matrizant given by Equation 34 is useless for computational purposes. However, i t is equivalent t o
Stability of the Solution. It is of some interest t o examine the solution as given by Equation 45. The following become obvious:
and, therefore, if c is fairly large,
(37) August 1955
1. T h e elements of x ( t ) will increase indefinitely if even one of the roots of A is positive.
INDUSTRIAL AND ENGINEERING CHEMISTRY
1537
ENGINEERING, DESIGN, AND EQUIPMENT 2. T h e elements of x ( t ) will exhibit a damped harmonic motion if all the roots of A have negative real parts and if a t least two of them are complex. 3. T h e elements of x ( t ) will be monotonic increasing functions of t, but bounded, for m(i)> 0, if all the roots of A are real and negative.
Table I.
1 2 3 4 5 6 7 8 9 10 11 12
AI = 0
are real and negative. T o prove this assertion, however, the following important theorems must be stated:
1. T h e matrix, S-lAS-ihat is, the matrix obtained from A b y a similarity transformation, where S is any lionsingular matrix-has t h e same characteristic roots as A. 2. All the roots of a symmetric matrix are real. 3. A sufficient condition for all the roots of a symmetric matrix, H, to be negative, is t h a t quadratic form, H ( z , z ) , be negative definite.
5 6 7 8 9 10 11 12
The first step in the proof would be to reduce A by means of a similarity transformation to a symmetric form. A is assumed t o be of the form given by Equation 39, where the elements, a S j are , positive real members. The matrices, S I , SB,. . .,S, + I , are defined as follows:
SGil . . . S;'S;'ASiSz
. . S Y + ~= H
5,j
6
1 0.500000 0.333333 - 0.250000 0.200000 -0.166667 0.142857 -0.125000 0.111111
-
5=4
0.500000
- 0.500000
0.468333 -0.416667 0.380566 0.380000 0.324100 -0.301984 0.282897 -0.271360 0.251656
-
- 0,100000 0.090909 - 0,083333 s = 5 ( X 10') 0,083333 - 0.20833 0.347222 - 0.486111 0,618634 - 0,742188 0.856013 - 0,960242
s = 6
( X 102)
0.166667 -0.250000 0.291667 - 0.312500 0.322222 -0.325694 0.325518 -0,323165 0.319504 -0,315068
0.118056 -0.145833 0.167882 -0.185417 0.199427
5=7 ( X 108)
5=8 ( X 109
0.041667
- 0.083333
-0.210676 0.219745
...
0 .' i i s s s g -0.416667 0.79861 1 - 1.250000 1.743634 2.269838 2.784862
...
o.'iQs413 - 0.694444 -2.604167 1.504630
-
0.248016 -0.992063 2.397487 - 4.546958 7.461833
3.952546
- 5.606366
s = 11
(x 9 10 11 12
bSi for
12
Values of 1.j s = 2 8 = 3
s = 1
39, the elements of which are given by Equation 40, all the roots of the characteristic equation
and so on. These matrices, S,, are all diagonal, and since, if Ki is the matrix derived from Z by replacing the 1 in the i t h diagonal position by IC, then K,A is the result of multiplying the ith row of A by k , and, similarly, AK, is the result of multiplying the i t h column of A b y k , then it can be shown without difficulty t h a t
k2
-kz
0
Because of t h e simplicity of A , a n expression for t h e characteristic roots, Ah, in a closed form can easily b e found. For illustrative purposes, however, the characteristic equation will be obtained by the method recommended above. A(A)
(
+ ;)
(kl
(k:
+ f>
If it is supposed, for simplicity, t h a t a t t = 0 the concentration of all the substances in all the tanks is equal to zero, and that, moreover, c(0) is independent of t , the solution t o Equation 55 is given b y
Ai
and by using Equation 47,
x
A(X)
A'A(X)
A2A(X)
A'A(h)
A4A(X)
A5A(X)
ASA(h)
0 0.5
0.29430 0,01984 -0.02044 0,01670 -0.04002 1.35608 20.45213
-0.27446 -0.04028 0.03714 -0,05672 1.39610
0.23418 0.07742 -0.09386 1.45282 17.69995
-0.15676 -0.17128 1.54668 16.24713
-0.01452 1.71796 14.70045
1.73248 12.98249
11.25001
1.0 1.0 2.0 2.5 3.0
19,09605
The more general problem, for which the concentrations in all t h e tanks are not all equal t o zero a t t = 0, can easily be reduced t o the one under consideration b y changing t h e dependent variables. For t h e second tank reactor
Because of Equations 48 and 49, the characteristic equation is A(A)
X6 - 7.0380A'
+ 18.9304A' - 24.2625Xa + 14.9928Xe - 3.9374A + 0.2943
=
1 ; c(1)
- Bc(2)
so t h a t =
0
the roots of which are A I = 0.1196, Xz = 0.4441,
-d dt
0.9129, Aa = 1.4333, h j = 1.9021, Xe = 2.2266
42) =
1
-
02
[ I - e-Bt - e - B f ( B t ) ] B - 2 ~ ( 0 )
Similarly, for any tank n,
Finally, substitution of the above into Equation 53 yields z g
=
+ +
0.0397(1 - e - a . 1 1 g 6 f ) 0.0347(1 0.0262(1 - e - 0 . 9 1 2 8 t ) 0.0167(1 0.0081(1 - e - 1 . 9 0 2 1 t )-I- 0.0021(1
- e-0.44416) - e--1.4333i) - e--2.22662)
+
which, because of t h e Sylvester theorem, can be expressed as
+ k= 1
which is the composition of t h e fat oil leaving the sixth plate.
August 1955
INDUSTRIAL AND ENGINEERING CHEMISTRY
0
afk
(60)
1539
ENGINEERING, DESIGN, AND EQUIPMENT
6 6
where Xk, 1 k 3, are the characteristic roots of B . Of course, Equation 59 holds no matter what t h e order of the matrix, B; hence any number of consecutive reactions may be considered As an example, consider the following:
e
= 1, kl = 0.20,
= 0.05, C2(0) =
k: C i ( 0 ) = 1,
k z = 0.10, k: = 0.05 ($0) = 0
wise process can be described by the matrix Equation 36, with 39. There is one difference, the matrix, A ( t ) , given by . Equation however, in t h a t the elements of A ( t ) are not numbers any longer but are themselves square matrices of order m. Thus, the elements, A l j , of A are
Then
-0.05 1.15 -0.10
1.20
B = (-0.20 0
-0.05 1.05 O )
T h e characteristic equation for B is
/XI - BI
Xa
-
+ 3.8325X - 1.4325 = 0
3.4000X'
\.
which has a s roots XI = 1.2866, XZ = 1.1133, and X 3 = 1.0000
and, therefore, from Equation 60, and
1
-
e-1.2868t
n - 1 ___ (1.2866t)i
c 0
+ (0.8982)"
{
1-
e-l.1133
