Munro, A. J. E., Madsin, E. G., Brit. Chem. Eng., 12 (3), 364 (1967). Parish, G. S. Patent 3,416,893 (1966). Pearson, T. G. et al., J . Chem. SOC.,1932,p 660. Perry, R. H., bhilton, C. H., Kirkpatrick, S. D., Eds., “Chemical Engineering Handbook,” RlcGraw-Hill, Kew York, N.Y.,
U.S. Department of Health Education and Welfare, “Air Quality Criteria for Sulfur Oxides,” Washington, D.C., 1967. lJ.S. Department of Health Education and Welfare, “Control Techniques for Sulfur Oxide Air Pollutants,” Washington, D.C. 1968. White, W.S., et al., J . Chem. Phys., 28 ( 5 ) ,751 (1958).
QuL;Y&, R., PhD Thesis, University of hIassachusetts, 1971. Ryason, P. It., Harkins, J., J . APCA, 17 (12), 796 (1967). Savin, V. P., et al., (U.S.S.R.), Khim. Prom., 44 (7), 550 (1968). Saxtaxtinskij, G., Bajand-Guliev, A. I., Azer. Khim. Zh. (6), 123 (1965). Slhck, A. V., Chem. Eng. (December 4, 1967).
RECEIVED for review July 29, 1971 ACCEPTED September 8, 1972
1a m
Work supported by Contract PHS-AP00791 with the Public Health Service, Department of Health, Education and Welfare.
Design of Closed-Circuit Grinding System with Tube Mill and Nonideal Classifier Masaaki Furuyafl Yoji Nakajimaf2and Tatsuo Tanaka Department of Chemical Process Engineering, Hokkaido Cniversity, Sapporo, Japan
Based on the comminution kinetics and material balances, the finished product size distribution of a typical closed-circuit grinding system with a tube mill and a nonideal classifier are calculated for a number of system parameters. The calculated results are summarized as simply and generally as possible with the aid of analytical consideration. Some simplified correlations are presented in a few figures which may be useful for designing new system or for improving old system, if data on mill and classifier performances are available. In connection to it, a simple method for determining the parameters of the commiwtion kinetics from open-circuit milling data i s suggested.
T h e authors (1971) reported an analytical discussion on closed-circuit grinding system with a n ideal classifier. As a matter of fact, however, a n ideal or clean-cut separation of particles cannot be expected to occur in any industrial classifier, and therefore the system analysis described in the previous paper is not always applicable to design purposes. I n this paper, a certain improvement is made in the comminution kinetics and the fractional recovery curves of industrial classifiers are represented in a simple function having several parameters. The finished product size distributions of a typical closed-circuit grinding system are numerically calculated for a number of combinations of the system parameters such as circulating load, sharpness of classification, and so on. However, the main purpose of this work does not consist in a n accurate prediction of the performance of a given closedcircuit grinding system, but consists in a general extension of the comminution kinetics for solving the system design problem; the accuracy of the prediction may be sacrificed to some extent for this purpose. I n this paper, the calculated results are summarized as generally and simply as possible with the aid of analytical consideration, and some simplified correlations represented in a few figures are presented which may be useful for designing a new system or improving the old system. 1
Present address, Shin-etsu Chemical Co., Gumma, Japan.
* T o whom correapondence shoi11d be addressed.
18 Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 1 , 1973
Mathematical Preparation
For the steady state of the operation of the grinding system shown in Figure 1, the following equations are derived from material balances :
+ + + CLfTII(1 + = (1 + C d ( 1 - avo
fE = (fF
fD = (fP fP
CLfT)/O
CL)
(1)
CL)
(2)
(3)
where f denotes the size frequency distribution by weight and the subscripts, D, E , F , P , and T , respectively, refer to the mill discharge, mill entry, make-up feed, finished product, and tailings; CL is the circulating load; and 7 is the fractional recovery of the classifier as a function of particle size. As suggested in the previous paper (Furuya et al., 1971), the flow mechanism in a n industrial tube mill may reasonably be assumed as a plug flow so far as the product size distribution is in question. Consequently, the batch milling equation (Austin and Klimpel, 1964), Equation 4, is directly usable to relate fE and jD without any consideration on the residence time distribution of particles in the mill:
where R(y,t) is the cumulative oversize by weight a t the size y and the milling time t , x m the maximum size in feed material,
-jF-z+yL*ss,F,ER I
MIXER
E=(1+
)F
20
0 04
1 5
003;
10
L” 002”
‘ . r
t
2
TAILINGS,
r,, Figure 1 system
.
FLF RT
5
-2
Typical connection of closed-circuit grinding
E
05
001
00
00 04
z the original size of particle selected for crushing, S(z) the
selection function, and B(y,z) the breakage function. For the simplest case where the product of S(z) and B(y,z) can be assumed to be Kyn (independent of original size), the solution of Equation 4 becomes
R(y,t) = R(y,O) exp (--Ky”t)
( 5)
The same assumption was employed by Herbst and Fuerstenau (1968), although they used discrete expressions with respect to particle size. An extensive study using a large experimental ball mill (93 cm i.d. X 110 cm) and a n industrial tube mill (Chiijo, 1949) suggests that Equation 5 becomes a fairly good approximation for ball milling process when the mill is operated properly. In contrast to it, several papers have been published (e.g., Harris, 1968 and 1969/70; Klimpel and Austin, 1970; Kapur, 1970; Grandy and Fuerstenau, 1970) showing t h a t Equation 5 is not always justified. However, the mills used in those experiments were rather too small and might be operated in quite different situations from industrial mills. Therefore, there remains still some possibility in applicability of Equation 5. Although the details are not presented here, some of the authors found the approximate solution for S(z) .B(y,z) = K z n . ( ~ / z (Kakajima, ) ~ Tanaka, 1973)-i.e.,
where r j z , p ) is the incomplete gamma function of second kind and r(z)is the gamma function, defined by the following equations :
Lm s,=
r (z,p) =
r(z) =
tz-1
E“
06
10 12 m /n
08
16
14
18
Figure 2. Approximation of Equation 6 by exponential function
for normal milling condition; if the mill is too small or incorrectly loaded, or if the size of grinding media is improper, the value of Y departs from unity. Harris (1968) also suggested that Y is in the vicinity of unity and the extreme value is probably 1.3, 1.05 being more typical. l-sing Equation 9, one obtains the relationship between fE andfD:
where n‘ =
nv,K’
=
(PLY)”,
7’
=
(W/E)” = [TV/(l
+ CL)F]”
(11)
in which E is the throughput, F is the make-up feed, and W is the holdup of the mill. If we combine Equations 1, 2, 3, and 10, the cumulative oversize distribution of the mill discharge may be written as follows, provided the size reduction ratio is so large that fF is negligible as compared with f p in the product size range
RD = exp
[
-
1
~ ( z d) z ]
(12)
where t h e function, G(z), is exp (- t)dt
tZ-’ exp ( - t ) d t
(7) G(z) (8)
exp (K’r’zn’) exp (K’7’zn‘) - ~ ( z )
n ’ ~ ’ 7 ’ ~ ‘ - 1
=
__
(13)
From Equations 3 and 12
Equation 6 can be closely approximated by Equation 9, which is called Andreev’s experimental equation (hIiwa, 1961) : Hence the cumulative undersize distribution of the finished product, P p , becomes where I* and Y are the constants determined by m/n. The values of p and v are shown in Figure 2 together with the niaximum discrepancy between Equations 6 and 9. As is readily seen from the figure, Equation 9 becomes identical to Equation 5 when m / n approaches unity. Illoreover, most esperimental data on laboratory scale mills, some of which conflict with Equation 5 as stated above, can be approximated by Equation 9 escept for very short milling time. Therefore Equation 9 may be taken as a general espression of batch milling processes. However, it should be noted that there are reliable data on a semi-industrial scale ball mill reported by Chujo (1949) who gives evidence that v = 1
exp
[ fb: -
G(s’) dz’ldz
(15)
h s seen from Equations 13 and 15, if the fractional recovery curve can be expressed in terms of (z/;tJn’, the integrating variable, z, may be replaced by z (= K ’ T ’ z ~ ;’ x) c is the cutoff size. Then the integrand of Equation 15 does not explicitly involve the parameter, n’, and the numerical iiitegration can simply be performed regardless of n’. I n this sense, the folInd. Eng. Chem. Process Des. Develop., Vol. 12, No. 1 , 1973
19
1 (x/x,)
Figure 3. Fractional recovery curves calculated from the mathematical model (Equations 16 and 17)
>
v
na
012
05
03
07
10
20
30
Y“‘ Figure 4. Standard size distributions of finished product 10.
0
A‘
2
4
4 0 1 6 3 2 CIRCULATING LOAD
Figure 6. Relationship between PP* and of S’ and A’ = 0.5
C L for
a few values
The parameter, S, is the slope of the fractional recovery at (z/x,) = 1 and the other parameter, A , affects the shape of the curve in the smaller and the larger size ranges as shown in Figure 3. The flexibility of this mathematical model is wide enough t o cover actual data on industrial separators. For instance, some nominal values of A and S are roughly estimated from available field data as follows: A = 1.5,S = 1 for centrifugal air separator, A = 0, S = 0.5 for cyclone, and A = 1.0, S = 0.5 for hydraulic cyclone. Molerus (1967) theoretically derived the same form as Equation 16 from a stochastic diffusional model of air and hydraulic classifiers. However, the effect of agglomeration of particles iri the classifier, which was not taken into account in the stochastic model, may play a n important role in industrial classifiers. Therefore the parameters, A and S, do not necessarily have physical meaning, and they should be determined from industrial field data for individual classifiers and operational conditions. After the analytical coilsiderations mentioned above, we have the final equation for Pp:
where the upper limit, 2, is $5 I
1
2
L 8 1 6 3 2 CIRCULATING LOAD
Figure 5. Relationship between Pp* and of S’ ( = S / n ’ ) and A’(= A/n’) = 0
C L for
a few values
lowing mathematical model for the fractional recovery is employed here :
The characteristic diameter, x,*, is introduced in n similar way to the previous paper; for open-circuit grinding, RD(xc*)/RE(x,*)= 0.362, as seen from Equation 9. Numerical Calculations and Results
and if we take the limit of the above equation as A approaches zero,
20 Ind. Eng. Chem.
Process Des. Develop., Vol. 12, No. 1 , 1973
Equation 19 shows clearly that if the calculated results the curves for P p become independent are plotted against P’, of n’. Further, since P p = 1 for sufficiently large y or 2, CL for a given fractional recovery curve is calculable from Equation 19, or in other words, one of the four parameters, CL, A’, S’, and Z,, is automatically determined from the rest of three parameters. I n the present discussion, CL, A ’ , aiid S’ are selected as the independent parameters for practical convenience of the system design. The numerical calculations were performed bj- a digital computer in the follo\\ing way: By putting so large a balue of 2 in Equation 19 that P p a t 2 could be assumed as unity, the parameter, Z,, for given values of C L , A’, and S’, was calculated from the combination of Equatioii 19 and Equation 16
I I
1
2
4 8 16 CIRCULATING LOAD
Figure 7. Relationship between Pp* and of S' and A' = 1 .O
32
C L for
a few values
or 17, using the trial and error method. Then the cumulative undersize of the finished product', Pp,was calculated from Equation 19 for various values of Y"', using the relationship between 2 and I.'"' in Equation 20. The ranges of parameters being esamined were from 1 to 48 for CL,from 0 to 3 for A', and l/8 to 3 for S'. These ranges are wide enough to cover most of practical cases. The numerical integrations were carried out by Gaussian method (e.g., Lapidus, 1962) and the masimum error wvas estimated to be less than 0.5%. From the calculated results of more than 600 combinations = 1 (this will be denoted by of the parameters, if P p a t P p * ) is the same, the finished product size distribution is virtually the same regardless of A', S', and CL except a few "inefficient" cases which will be pointed out later. The standard product size distributions are plotted against Y"' on Figure 4. Figures 5-9 show the relationship among Pp* (= P pat' I-"' = I), S', and C L for several values of A'. The broken portions of the curves in t,hese figures are the 'linefficient" regions because the effect of a n increase in CL is much lessened. Using Figures 4-9, one can estimat,e the finished product size distribution for given values of CL,S', and A'. For esample, when C L = 2, A ' = (A/n') = 1.5, and S' = ( S l n ' ) = 1, we find Pp* is about 0.8 from Figure 8. Then the size dist,ribution is the curve in Figure 4 which passes the point P p = 0.8 at I.'"' = 1. For converting the dimensionless particle size Y into the usual dimensional size y, Equations 21 and 22 are usable though a few data on comminution kinetics are necessary. Note that if the point corresponding to the given values of CL, S ' , and A ' falls on a broken portion of the curves in these figures, the condition may be rather unfavorable and the size distribution thus obtained becomes unreliable (the error in P p may exceed 0.025). The interrelationship between the dimensionless cutoff size, S, (= s,/L,*), and the other parameters, A', S', CL, Y , and n', is very complicat,ed, but X,"' mainly depends on S' and CL. I n Figure 10, a typical relationship between S,"' and C L is presented for several values of S ' ( A ' = 1, v = 1). This figure is usable for other values of A' and v for very rough estimation. Extension to System Design
I n this section, it is presumed that a good deal of accumulated data on mills, classifiers, arid other equipments are available; the parameters required for the system design have alreidy been determined. X method for est'imating the parameters of comminution kinetics from open-circuit milling data will be presented in the last, part of this section. The first step of the system design is to plot the desired size distributioii against y"' on a semilog sheet, which looks i n the form like Figure 4. Compare the plot with the curves
1
2
L 8 1 6 3 2 CIRCULATING LOAD
Figure 8. Relationship between of S' and A' = 1.5
1
2
L
Pp*
and
8
CL
for a few values
1 6 3 2
CIRCULATING LOAD
Figure 9. Relationship between PP* and of S' and A ' = 2.5
C L for
a few values
in Figure 4 and find a curve closely parallel to the plot. Read the value of P p at Yn' = 1 on the best fit curve, which was referred to as Pp* in the preceding section. The circulating load and the classification parameters, S (= 71's') and A ( = %'A'), are determinable from Pp*by using Figures 5-9. Of course many combinations are possible. For esample, if Pp* = 0.8, all the following combinations are usable: C L = 2 mit,h A" = 1, A ' = 0-2; CL = 4 with S' = 0.5, A' = 0.5-1.5, and so 011. Then the rates of part,icles flowing through the classifier and the transportation equipment' can be determined for every combination, and several available sets of a classifier and a conveyor may be picked up from conventional equipments, referring to field data. If necessary, the cutoff size can be roughly estimated from Figure 10. The characteristic diameter of the system, s,*,is also determinable from Pp* as below: Read the particle size from the original plot of desired size distribution a t which the fracThus obtained particle size, tional undersize is equal to Pp*. y*, is related to z,*by Equation 21-i.e., 2,* =
y*j1
+ CL)(l--V)ln'
(23)
When pK is known, the holdup of the mill, W, required for sat,isfying the given dut,y is calculable from Equation 22 and the mill scale may be determined. I n this way, several combinations of a mill, a classifier, and other auxiliary equipments are selected. The most efficient combination should be decided from economical consideration. Here some restriction on the choice of the circulating load is presented. As is seen from the calculated results, the use of poor performance classifier may be compensated to some estent by increasing the circulating load; however, for system stability it is recommended to operate the system with comparatively low circulating load, say u p to the order of 5 . Figure 10 shows that the circulating load becomes sensitive to the dimensionless cutoff size, X,( = E,/z,*), with increasing Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 1, 1973
21
relationship, n = n'/v. Figure 2 is usable to obtain (rn/n)and from Y. Finally, K can be evaluated from Equation 22. A t this time, the value of W may well be considered as the holdup with which the filling degree is equal to unity. I n this way, the parameters, K , rn, n, n', p , and Y are roughly determinable from open-circuit grinding data; however, these values are usable only for the mills whose diameter and grinding media are the same as the mill being examined. Therefore only the mill length is changeable for adjusting the holdup. T o estimate the mill scale from the holdup required for satisfying the given duty, the filling degree of the mill may be assumed to be unity regardless of the throughput. This is because the product, K W , of a given mill may be independent of the throughput unless the filling degree is too small and because K is estimated for the filling degree of unity. p"
Conclusion
Figure 10. Relationship between X,.' and v = 1
and
C L for A'
= 1
CL.If we consider t h a t z,* is affected by the make-up feed arid the grindability of the raw material, and if the Circulating load is set a t a very high level, t'he circulating load changes are susceptible to the possible disturbances-e.g., a n increase in the make-up feed. A t best, the finished product size distribution changes intolerably owing to the changes in classification characterist'ics caused by the pronounced increase in the throughput; a t worst, the system becomes unstable and overflowing. Obviously, the applicability of the present procedure to actual cases depends on whether the parameters of comminution kinetics are available. Cnfortunately, the existing comminution theory is incapable of estimating the values of the parameters for a given mill arid feed material so far. I n the present stage of the development, they can be determined only from experimental data. A n accurate method for determining the selection and breakage functions in numerical forms was discussed by Klimpel and Austin (1970) ; however, the procedure is not easy to carry out for a number of industrial mills and raw materials. As for syst'em design purposes, very accurate values of the parameters are not necessarily required, because in the actual closed-circuit grinding system there may be one or more adjustable variables, e.g., the cutoff size, with which we can compensate the possible error to some extent. Therefore a simple method for the estimation of the parameters from several data on open-circuit grinding is described below, though it may be rather crude. Since the plug flow assumption is reasonable for industrial tube mills, Equation 9 is usable for open-circuit grinding;
Aisis seen from the above equation, the Rosin-Ramrnler plot ) y may give a straight line and the of R D ( y ) j R B ( y against parameter, n', is the slope. Further, the characteristic diameter, z,*, for a given feed rate, E', is the size a t which the ratio of R,(y) to R,(y) is equal t'o 0.362. Hence, v can be estimated from the values o€ 5," for a few different F, using the proportionality between x , * ~ ' and F" implied by Equation 24 under the assumption that the product, KFt-, is indeperiderit of F . This assumption may be reasonable when the filling degree of particles iii t,he mill is nearly equal to or larger t'han unity (LIika et al., 1966); the filling degree is defined as the ratio of the bulk volume of partirles in the mill to the interstitial volume between grinding media. 'l'hen n is calculable from the 22
Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 1 , 1973
The equations derived from the comminution kinetics and material balances for a closed-circuit grinding system with a lionideal classifier are developed, and a method for designing practical systems is proposed. The method provides a rapid estimation of the system variables for satisfying the given duty and make it possible to select the most effective combination of a tube mill, a classifier, and other auxiliary equipments with the aid of several field data on individual equipmeiits arid economical considerations. The present procedure for system design purposes requires the parameters of the comminution kinetics. Although x simple method for estimating t,he parameters from some data o n open-circuit grinding is presented, the method is not usable for the case Tvhen the mill diameter or grinding media are to be changed. Therefore, the most important thing in the field of comminution theory may consist in the theoretical and experimental study of the grinding mechanism. Nomenclature = parameter appearing in fractional recovery, see Equation 16, dimensionless A' = A / n ' , dimensionless B(y,z) = breakage function C, = circulating load, dimensionless E = throughput of mill, tons/hr E' = make-up feed, tons/hr f = size frequency distribution, p-' G ( x ) = function of 5 defined by Equation 13 K = parameter of comminution kinetics, p-n miii-' K' = (NK)", p-n' min-p m = parameter of comminution kinetics, dimensionless n = parameter of comminution kinetics, dimensionless n' = vn, dimensionless P ( y ) = cumulative undersize by weight, dimensionless Pp* = P p at I .'.' = 1, dimensionless R(y), R(y,t) = cumulative oversize by weight, dimensionless S = parameter representing sharpness of classification, see Equations 16 and 17, dimensionless S' = S/n', dimensionless S(z) = selection €unction, mi1i-l t = milling time, mill; dummy variable for integration 11- = holdup of particles in mill, tons z, z' = dummy \-ariable for particle size or original size of particle selected for crushing, p 2, = cutoff size of classifier, p z,, = maximum size in make-up feed material, X, = (xcJxc*), dimensionless cutoff size I,* = characteristic diameter, see Equation 22, ,u y = particle size, p I* = dimensionless particle size defined by Equation 21
-4
particle size corresponding t o P , = P,*,p K‘T‘x~’ and K’dz’n’, respectively, variable for integration, dimensionless Z = K ’ T ’ Y ~ ‘ ,dimensionless particle size defined by Equation 20 Z, = K’T’x,~’, dimensionless cutoff size defined b y Equation 18 y* 2,
=
,z’
=
SUBSCRIPTS
D E F P T
mill discharge mill entry make-up feed finished product tailings
= = = = =
GREEKLETTERS
r (2)
gamma function incomplete gamma function of second kind ~ ( z ) = fractional recovery as a function of particle size, dimensionless 0 = dummy variable for time, min p ; v = constants determined from (m/n),dimensionless T = (W/E)”,min” =
r(z,p)
=
literature Cited
Austin, L. G., Klimpel, R. R., Ind. Eng. Chem., 56 ( I l ) , 18 (1964). ChGjo, K., Kagaku Kogaku (formerly Kagaku Kikai) (Ann. Rept. Japan SOC.Chem. Engrs.), 7, 1 (1949). Furuya, ?\.I.,Nakajima, Y., Tanaka, T., Ind. Eng. Chem. Process Des. Develop., 10 (4), 449 (1971). Grandy, G. A., Fuerstenau, D. W., Trans. AIME, 247, 348 (1970). Harris, C. C., ibid., 241, 449 (1968). Harris, C. C., Powder Tech., 3, 309 (1969/70). Herbst, J. A., Fuerstenau, D. W., Trans. A I M E , 241, 538 (1968). Kapur, P. C., ibid.,247,299 and 309 (1970). Klimpel, R. R., Austin, L. G., Ind. Eng. Chem. Fundam., 9 (2), 230 (1970). Lapidus, L., “Digital Computation for Chemical Engineers,” p 56, 1lcGraw-Hill, Kew York, N.Y., 1962. M k a , T. S., Berlioz, L. M., Fuerstenau, D. W., Preprints for the 2nd European Symposium on Comminution., Verlag Chemie, Weinheim, p 147 (1966). hIiwa, S., Funsai (Micromeritics), 7, 30 (1961) translated from Russian ,;ext) by Andreev, S. E., Towarov, W. W., Perov, W. A., Regularity of Size Reduction and Calculation of Characteristic Particle Size Distribution,” Moscow, 1959. SIolerus, O., Chem.-Ing.-Tech., 39, 792 (1967). Nakajima, Y., Tanaka, T., Ind. Eng. Chem. Process Des. Develop., 12, 23 (1973). RECEIVED for review September 30, 1971 ACCEPTED July 24, 1972
Solution of Batch Grinding Equation Yoji Nakajima’ and Tatsuo Tanaka Department of Chemical Process Engineering, Hokkaido Cniversity, Sapporo, Japan
An analytical solution of the batch grinding equation for S ( x ) B ( y , x ) = K ~ ~ ( y / i sx found ) ~ by using a few approximations; S ( x ) i s the selection function, B ( y , x ) the breakage function, x the original size, y the particle size, and K, m, n are the constants. Except for very short milling time, the solution i s virtually the same as Harris’ solution, which i s derived by assuming that the selection function is time-dependent and correlated to the breakage function.
As is well known, the fundamental equation of
batch
grinding may be written in a n integral form:
where R(y,t) is the cumulative oversize at the particle size, y, and the milling time, t , and Ro(y) is R(y,t)at t = 0. The funcrespectively, are the selection and tions, S(z) and B(y,x), the breakage functions usually assumed to be independent of the environmental size distribution; z the original size of particles selected for crushing, and xm the maximum size in a raw material initially charged in the mill. Usually, Equation 1 is solved numerically by giving S(T) and B(y,z) for individual mills and materials, however, if some analytical solution for certain special forms of the selection and the breakage functions is found reasonable, it may be much more convenient and useful than numerical solutions. Harris (1968) skillfully found a n analytical solution which fits esperimental data fairly well by assuming t h a t the selection function is To whom correspondence should be addressed.
correlated to the breakage function and, moreover, it depends also on the milling time. A possible esplanation of the time dependency of the selection function may be resolved into the statement that the selection function is affected by the environmental size distribution. This statement, however, conflicts with the works of many other investigators who have shown by simulation that the selection function is virtually independent of the environmental size distribution. Furthermore, the applicability of the comniinutiori kinetics to actual milling processes may mainly depend on the validity of the assumption t h a t S(z) and B(y,z) are independent of environmental size distribution. I n this paper, a n approsimate solution, close to the solution proposed by Harris, is derived without the assumption of the time dependency of the selection function. The product of the selection and the breakage functions is assumed in the following form: S(z)
B(y,z)= k‘znjy/zy
(2)
where K is the constant, and m and n are the constants in the vicinity of unity. From Equations 1 and 2 , one obtains the Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 1 , 1973
23
