to the conventional mass transport phenomenon, except that the sonic pulsations may induce vortices to compress the boundary layer more than that predicted by the steady flow mechanics. Mass transfer is also affected by an interaction between the fluctuating components of velocity and concentration. The sound pressure has a directly proportional effect in this connection. A rapid decay of sound pressure limits the effective space in the neighborhood of a sound generator for improving the interfacial transport rates. A high power generator would be required to increase interfacial transport rates significantly.
R. L. Hummel, University of Toronto, offered many useful suggestions and comments during the preparation of this paper. Nomenclature
b
= defined in Equations 6A and 8A = velocity gradient a t interface = defined by Equation 5B
B . E = amplitudes of fluctuating components
C’ D J L N P Pe t
T
V 2,
X
Y a
v w
= concentration boundary layer thickness = gamma function = kinematic viscosity of air = angular velocity or 2n (pulsation frequency)
SUBSCRIPTS = a t interface x = a t position x y = at position y SUPERSCRIPTS m = defined in equation 5B o = components of B and E fluctuating component - = = average
o
literature Cited
Ac knowledgrnent
A a
A
r
concentration of naphthalene vapor diffusion coefficient of naphthalene in air molar flux of naphthalene relative to average diffusion velocity length of sample holder molar flux of naphthalene relative to fixed coordinat.es root-mean-square sound pressure Peclet number, VL/D time time period of one pulsation cycle superficial velocity of air in duct local, instantaneous fluid velocity longitudinal distance parallel to flat plate from leading edge vertical distance from plate surface to fluid stream phase angle between fluctuating components
Bird, R. B., Stewart, W. E., Lightfoot, E. N., “Transport Phenomena,” p. 552, Wiley, New York, 1960. Boucher, R. M. G., Chem. Eng. 68, No. 20, 83 (1961). Boucher, R. M.G., Brun, J., J. Acoust. SOC.Amer. 29,573 (1957). Colburn, A. P., Trans. Amer. Inst. Chem. Eng. 30, 187 (1933). Fand, R. M., Kaye, J., AWDC Tech. Note 59-18 (1959). Fussell, D. D., Tao, L. C., Chem. Eng. Progr. Symp. Ser. 59 (41), 180 (1963). General Radio Co., “Handbook of Noise Measurement,” p. 7, 1963. Hartman, J., Trolle, B., J . Sci. Instr. 4, 101 (1927). Honaker, D. E., M. S. thesis, Department of Chemical Engineering, University of Nebraska, pp. 42-7, 1965. Jackson, T. W., Purdy, K. R., J . Heat Transfer, A S M E Ser. C, 87,507 (1965). Johnston, H. L., McCloskey, K. E., J. Phys. Chem. 44, 1038 I1940). Kay, W’. M., Trans. A S M E 7 7 , 1265 (1955). Kubanskii, P. N., J. Tech. Phys. ( U S S R ) 22, No. 4, 593-601 (19.52). \-__-
Lernlich:’R., Hwu, C. K., A.1.Ch.E. J. 7 , 102 (1961). Lin, C.C Proceedings of 9th International Congress of Applied MechaGes (Brussels), Vol. 4, p. 155, 1957. Mack. E.. Jr.. J. Amer. Chem. SOC.49. 135 (1927). Schlichting, H., “Boundary Layer Theory, 4th ed. pp. 123, 537, McGraw-Hill, New York, 1960. Sokol, R.T., Department of Chemical Engineering, University of Nebraska, M.S. thesis, p. 39, 1963. RECEIVED for review August 2, 1968 ACCEPTEDJune 18, 1970
Determination of Relative Variance and Other Moments for Generalized Flow Networks or System Transfer Functions H. W. Kropholler Department of Chemical Engineering, Loughhough University of Technology, Leics., England A method described by Klinkenberg for determining the variance and other moments for a flow network consisting of a series of vessels with backflow is extended to general flow networks and also transfer functions for dynamic systems. The usefulness of the cumulants and a method for their indirect evaluation are described. The use of these methods is illustrated by their application to the derivation of a simplified model for the response to reflux changes in a packed distillation column section.
SEVERAL
AUTHORS have obtained expressions for residence time distribution and relative variance for stages in series with backflow. Klinkenberg (1966) has described a n elegant method for handling the recurrence relation that can be used to describe this flow network. However, for generalized flow networks or transfer functions the above short-cut method
does not apply. Methods for obtaining moments, hitherto unpublished, from a matrix formulation of the problem are given below. Gibilaro and Lees (1968) have shown how the moments of a complex model may be matched with those of a simpler model, when such a simple model can be used as a sufficiently accurate representation of the original model. Ind. Eng. Chem. Fundarn., Vol. 9, No. 3, 1970 329
A finite number of moments or cumulants of a distribution do not uniquely determine the distribution itself. I n general, one may use a n approximate model which will give conservative results for the particular application to be considered. Satisfactory results are usually obtained in practice, as discussed by Kendall (Kendall and Stuart, 1958). The results developed above are applied to a model of a batch distillation column section (Kropholler et al., 1968). The end to end change for the column section is easily approximated, but if input and output are adjacent, more complicated approximate models are required.
eters to handle. The relationships between cumulants and moments are summarized in Appendix 2. Only the first three central moments and cumulants are identical. The transfer function for any one of the state variables given by Equation 1 can be found explicitly in terms of the solution of Equation 2 using Cramer's rule-Le.,
(7) Successive differentiation of the transfer function, Equation 7 , will result in the usual formula for the moments; however, another method for arriving a t the cumulants directly is available. Both the numerator and the denominator of Equation 7 can be expanded to form polynominals in s, so that Equation 7 can be written as the ratio of the products of the first-order systems as given by Equation 8.
Derivation of Moments
The residence time distribution or impulse response of a linear system, u-hich may be represented by a lumped parameter system, is equivalent to the matrix representation given by Equation 1.
x
= Ax; x = x(O) a t t =
O
(1)
The Laplace transform of Equation 1
sX(s) = AX(s)
+~(0)
n
M
F,(s) =
IT (Tis + I)/
AT
k=l
i=l
(Tks
+ 1)
(8)
The time constants may be real or complex. Using the relationships given in Appendix 2, the cumulants may be obtained directly from Equation 8 and are given by
(2)
can be used to obtain the moments in the usual way, as given by Equations 3, by repeated differentiation with respect to s and solving for s = 0. k = 1, 2, 3 . . . .
Mk = -kA-'Mk-i; Mo
=
By making use of the properties of the trace of a matrix, the cumulants of Equation 9 may be derived directly. The sum of the eigenvalues of a matrix and the sums of the nth integer power of the eigenvalues may be obtained from the sum of the diagonal elements (or trace) of the matrix and the trace of the nth power of the matrix, respectively (Pipes, 1958). If A is nonsingular, this is also true for negative n.
(34
-A-'x(O)
(3b)
Equations 3 are similar to the equations given by Klinkenberg (1966), if the nonzero elements of the matrix A are taken to represent the coefficients of the recurrence relationsi.e.,
A
= -(1
+ a) o...
0
+ a
(1+a)
-(If
2 a)
...
0
.
0
CY..
(4)
0..
I n this case, the result is not very interesting, as the moments are more simply found by the methods proposed by Klinkenberg (1966, 1968). However, in using Equations 3 there is no restriction on the format of matrix A, and any system whose moments exist may be represented, albeit only numerical solutions may now be obtained. It is also possible to determine the moments simply by writing
Y(0) = -A-'x(0)
(5)
=
(-l)kk.W(k)
(6)
Derivation of Cumulants
For many uses, particularly if the transfer function consists of the product of a number of terms, all of which are a function of the complex variable s, the cumulants are simpler paramInd. Eng. Chem. Fundam., Vol. 9, No. 3, 1970
Xj
j=1 N
N
T,(A")
=
2 Xj" i=l
where X I is a n eigenvalue of the matrix A . If the inverse of A exists, then
T,[(A-')"] =
which is simple to compute.
330
=
N
and Y(l) = -A-l Y(O), etc. and therefore Equation 3a becomes MI,
T,(A)
2 Tj"
j=1
where the time constants are the reciprocals of the eigenvalues, T , = l / A j . B y applying the technique given by Equation 11 and relating the result to Equations 7, 8, and 9 it is seen that the contribution of the numerator of the transfer function has not been taken into account. TWO simple procedures are available for completing the solution; either the total cumulants can be obtained from Equations 3 using the relationships given in Appendix 1, or the contri-
Table 1.
-0.024 0 0.64 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0016 0 0.64 0 -0.024 0 0.64 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0.0146 0 -1.37 0 0,0146 0 -0.0896 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0146 0 -1.37 0 0.0146 0 -0.0896 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0.0146 0 -1.37 0 0.0146 0 -0.0896 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0146 0 -1.37 0 0.0146 0 -0.0896 0 0 0 0 0 0 0 0
-0.0016 0 0.64 0 -0.024 0 0.64 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0016 0 0.64 0 -0.024 0 0.64 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 -0.0016 0 0.64 0 -0.024 0 0.64 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0016 0 0.64 0 -0.024 0 0.64 0 0 0 0 0 0 0 0
bution of the numerator can be evaluated directly, using a procedure described by Davison (1968). Davison (1968) showed that the zeroes of the numerator of Equation 7 could be evaluated by solving a dummy eigenvalue problem, in which the j t h column of the determinant sej, and then the [which was x(O)] is replaced by K x ( 0 ) eigenvalues are determined by one of the standard methods. A number of extraneous eigenvalues are introduced by this method, but these can be shown to have values proportional to as K becomes very large. These ideas can be used as follows: A new matrix is introduced which is derived from the determinant of Equation 7, but this matrix has t h e j t h column replaced by K x ( 0 ) . This matrix is defined as B. As the spurious eigenvalues are large for large values of K , they will contribute very little to the trace of the inverse of B-i.e.,
+
n-mdg
N
M
Matrix A
0 0 0 0 0 0 0 0 0.0146 0 -1.37 0 0.0146 0 -0.0896 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0146 0 -1.37 0 0.0146 0 -0.0896 0 0 0 0
0 0 0 0 0 0 0 0 -0.0016 0 0.64 0 -0.024 0 0.64 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0016 0 0.64 0 -0.024 0 0.64 0 0 0 0
CI, = (-I)I,-'(k
0 0 0 0 0 0 0 0 0 0 0 0 0.0146 0 -1.37 0 0.0146 0 -0.0896 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0146 0 -1.37 0 0.0146 0 -0.0896
0 0 0 0 0 0 0 0 0 0 0 0 -0.0016 0 0.64 0 -0.024 0 0.64 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0016 0 0.64 0 -0,024 0 0.64
- I ) ! { T,(A-")
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0146 0 -1.37 0 0.0146 0 -0.0896 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0,013 0 -0.73
- T,(B-")}
(13)
or, more usefully, define one cumulant for the numerator and another for the denominator of the transfer function. This method is particularly useful in model simplification for the cases in which a lead-lag approximation is required. Application of Methods
To illustrate the use of the methods described above, they have been applied to the matrix model representing the response of a distillation column section. The column and dynamic response have been described by Kropholler el al. (1968). The behavior of the system to a step disturbance in reflux is approximated by the matrix equation
X=Ax+R
(14)
where A is a 20 X 20 matrix given in Table I and RT is given by the row vector. 0.01376 0 0.0104 0 0 00784 0 0.00592 0 0.00448 0
which is the desired result. Substituting the results of Equations 11 and 12 into Equation 9, a relationship for the direct evaluation of cumulants is obtained.
0.00336 0 0.00251 0 0,00190 0 0.00144 0 0.00108 0 I n this particular problem, simplified transfer functions, F t s ( s )and F l ( s ) ,are to be derived for the liquid near the top of Ind. Eng. Chem. Fundam., Vol. 9, No. 3, 1970
331
Table 11.
Cumulants Determined from Equations 3, 11, and 12
Cumulants from Moments TOP, xi9 Bottom,
CO C1 C2
CB
2.26 27.2 536.3 23,000
-
Top, XIS
XI
1.3 11.6 339.2 18 ,440 Table 111.
Cumulants Evaluated Directly Numerotor Bottom, XI
..
-
-
6.05 - 3.10 -61.64
21.52 198.4 5028
33.2 536.3 23,000
Parameters for Various Simple Models Parameters from Moments
No.
1
2
Model
Parameters
1
+1 3 (Ts + Ts
Top,
N
1.38 19.68
0.4 29.2
The cuniulants determined from the moments are given in the first two columns of Table 11. llodels 1 to 4 of Table I11 have been obtained for both the top and bottom of the distillation column. In this case, to enable objective assessment of the different simplified models, it should be noted that the dominant time constants of matrix -4 are 22.4, 4.94, 2.05, and 1.14. For the top of the column, model 1 is
70
332
90
Response of top of column section to reflux change Equation 14
Ind. Eng. Chem. Fundam., Vol. 9, No. 3, 1970
XI
11.6
Zl(S) = F l ( S ) R ( S )
Response predicted from - - - Model 3, Table II - - - - -Model - . 4, Table II
Bottom,
27
the column, z19, and the liquid leaving the column section or the bottom of the column section, xl,as given by Equation 15 x19(s) = FiQ(s)R(S) (15)
Figure 1.
XIS
T
T
1)N
Denominator
-
Parameters from Cumulantr Numerator TOP, XIS Bottom, x i
6.05
21.5 2.34 9.20
Denominator
33.2 2 16.6
inaccurate, but models 2, 3, and 4 all give a good fit, and the dominant time constant is close to that of the original system. The gamma distribution model 2 is a t times somewhat awkward to handle, and is therefore the least attractive. The other two models are compared with the actual response in Figure 1. For the bottom of the column, however, only the gamma distribution gives any sort of fit. Model 3 results in a negative time constant (which means that the system is unstable), and model 4 gives a negative time constant for the time delay and is physically impossible. The gamma distribution with a n exponent of less than unity would result in an infinite initial response to a delta function, and hence is not attractive. More complicated models are required and these are developed using the cumulants. The cumulants of the numerator and denominator are of comparable order of magnitude, in other words some sort of lead-lag model is required (Table 111, fourth and fifth columns). A further restriction is added that the denominator must represent a higher order transfer function than the numerator. For the denominator either model 5 or 6 would appear to be suitable. I n Figure 2 the actual system response is compared with the model.
Conclusions
By using the properties of the Laplace transform and the trace of a matrix, it is possible to obtain either the moments or cumulants of a transfer function directly. These operations can be carried out using standard computer routines for
(Wilkinson, 1965). If all the Jordan submatrices are of order unity, matrix c is the diagonal matrix of eigenvalues of A.
H-’AH
=
C
(A.1.2)
I n addition,
. H.-‘
H-‘AnH = H.-‘A.H.H.-‘A.H..
X
A.H.
C”
(A.1.3)
where n can be any positive or negative integer. Thus the sum of the nth power of the eigenvalues
c‘
N
0.
N
(A.1.4)
I
IO
Figure 2.
_______
=
30
5 0 TIME MINUTES
70
90
Response of bottom of column section to reflux change
Response predicted from Equation 14 Response predicted using Model 3 for numerator and Model 6 for denomnator from Table 11, Equation 1 6
where the Atn are the eigenvalues of the new matrix B. Appendix 2
The moments are defined by the equation (A.2.1)
matrix inversion and powering. The applications of these parameters for deriving simplified models have been illustrated. Although the moments of cumulants do not define a specific model, the suitability of a model may be further assessed in terms of its physical realizability.
s=o
and the cumulants by the equation
(A.2.2) From the above definition it can be shown that
No men c lature
N x N matrix N X N matrix derived from A kth cumulant, k = 0, 1, 2, 3 . . . , column vector
transfer function for j t h state variable unit matrix integers 0, 1, 2, 3 . . . large dimensionless ronstant kth moment of transfer function, column vector forcing function in Equation 14 column vector complex variable used in Laplace transform i or j t h time constant trace matrix-i.e., sum of diagonal elements column vector of state variables Laplace transform of x. column vector of transition variables constant j t h eigenvalue of matrix A determinant of ( I s - A ) with j t h column replaced by initial conditions
Co = logeMo
c1 = J!fl/llfO
cz = Mz/Mo - (Ml/O)Z c 3
=
kf3/~kfo
- 3;M2~u1/(;?fo)~ f 2(Iv1/11fo)a
(A.2.2)
c4 = M ; / M o - 4iM3M1/(M0)2
+ 6Mz(M1) ‘/(Mo)- 3( J I a
1)
‘/ ( M o )
- 3(MZ/lvo - ( M 1 ) z / ( M o ) z ) z etc. Additional relationFhips are given by Paynter (1957). I n general, the zeroth and first three moments or cumulants are adequate for model-fitting purposes (Gibilaro and Lees, 1968). literature Cited
Appendix
I
Some useful properties of the trace of a matrix are given below. The trace of matrix A-Le., the sum of the diagonal elements-is equal to the sum of the eigenvalues of the matrix (Pipes, 1958). N
c
i=l
IV
aft =
c
i=l
kt
(A.I.1)
I n addition, all matrices can be transformed by a similarity transform in such a way that matrix c of Equation A.1.2 consists only of Jordan submatrices isolated along the diagonal elements, with all the other elements equal t o zero
Davison, E. J., A.Z.Ch.E.J. 14, 46 (1968). Gibilaro, L. G., Lees, F. P., Chem. Eng. Sci. 24,85 (1968). Kendall, M. G. and Stuart, A., “The Advanced Theory of Statistics,” Vol. I, p. 87, Charles Griffin, London, 1958. Klinkenburg, A., Chem. Eng. Sci. 23, 175 (1968). Klinkenburg, A., IND. ENG.CHEM.FUKDAM. 5 , 283 (1966). Kropholler, H., Spikins, D. J., Whalley, F., Measurement Control 1, T55 (1968). Paynter, H. M., in “Regelungstechnik,” G. Muller, ed., p. 243, R. Older burg Verlag, Munich, 1957. Pipes, L. A., “Applied Mathematics for Engineers and Physicists,” p. 90, McGraw-Hill, New York, 1958. Wilkinson, J. H., “The Algebraic Eigenvalue Problem,” Clarendon Press, New York, 1965. RECEIVED for review September 19, 1968 ACCEPTEDFebruary 16, 1970
Ind. Eng. Chem. Fundam., Vol. 9,
No. 3, 1 9 7 0 333
