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, J a p a n
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 functions, S(z) and B(y,x), respectively, are the selection and 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
20
The inverse Laplace transformation of the first and the second terms is easy, but the third term is rather difficult and requires a slight approximation. For the case where 0 < m / n < 1,the inverse transformation of the third term is performed as follows:
10
07 =05
03
.;02
* s
01
CF,
8 007
2
exp ( --Kynt) ; m/n
005 003
where
r(z)is the gamma function defined
>0
(7)
by
002
001
003 005
01
0 2 03
05
2
1
3
5
Kynt
On the other hand,
Figure 1. Rosin-Rammler plot of Equation 17 2 Or
,,
,
,
,
.
,
io04
where y ( z , p ) is the incomplete gamma function of first kind defined by 00'
'
1
04
06
./ 08
10
I
12
14
.
I
16
,
1
loo
18
Jo
m/n
Figure 2. p" and Y as a function of m/n; exp [-(pKy"t)"I becomes a linear line in Figure 1 which intersects the curve for a given m/n at the points corresponding to R*(y,t) = 0.2 and 0.8
Then when we use the Bore1 theorem,
c-1
partial differential equation:
K ymxmn-m
s(s
+ Ky")m'" (s + Kx,n)l-(m'")
-
(3) The Laplace transformation is made with respect to t and, after rearranging, i t yields
where s is the Laplace variable for t. Solving the above ordinary differential equation under the condition that R(y,s) = 0 a t y = xm, we obtain
Noting that 1 - ( m / n ) is a positive number close to zero arid that drn/")-1 exp ( -Kynr) is a rapidly decreasing function of 7, we may replace the incomplete gamma function in
{ -:-, Kxmnit -Kx,"t 4,
Equation 11, y 1
Cn
(S
+ K X " ) ( ~ ' " ) -R' ~ ( x ) ~(5)x
xn-m-l
Here a n approximation is made. When y is sufficiently small compared to zm, the term, xn-m-l (s + K X ~ ) ( " / ~in) -the ~, integral decays very rapidly with increasing x as compared to Ro(x); on the other hand, when y approaches x,, the integral term vanishes. Therefore, Ro(x) in the integrand may be replaced by Ro(y) without much error. Then Equation 5 is reduced to Equation 6:
24 Ind.
Eng. Chern. Process Des. Develop., Vol. 1 2 No. 1, 1973
by thegammafunc-
tion, r[l - ( m / n ) ]provided , that is sufficiently large; this approximation may be recognized by comparing Equation 8 with Equation 10. Then one can reduce Equation 11 to the following simple form:
The possible error never exceeds [l - (m/n)]/Kzmnt;the actual error is espected to be much smaller than this value. Then R(y,t) becomes
where r ( z , p ) is the incomplete gamma function of second kind defined b y
r(z,p)
=
lm
uz-lexp ( - u ) du =
r(z) - y ( z , p )
or
(14)
For the case where m = n, the following solution is readily obtained without any approximation:
R(y,t) = Ro(y)exp ( - K y n t ) ; m = n (15) For the last case where m / n > 1, Equation 16 is obtained in the same manner used for deriving Equation 13.
I n Figure 1 (log-log grid), log&*(y,t) is plotted against Kynt for several values of m/n. As is readily seen from t h e figure, Equation 17 is closely approximated by Equation 18:
R(y,t) = expi - (pKynt)"} Ro(Y)
where ,U and v are the constants determinable from m / n as shown in Figure 2, in which the maximum discrepancy between Equations 17 and 18 is also given. The solution proposed by Harris (1968) can be written in the form s.5
Equation 16 is rather inaccurate because the same approsimation applied to Equation 11 is used twice for this case. I n fact, Equation 13 reduces to the strict solution, Equation 15, when m / n approaches unity, but Equation 16 does not. Therefore, the second term in Equation 16 may originate mainly from the approximation used in the inverse transformation. I n any may, the second term dies out for xm >> y , or for large t . Then we may conclude t h a t the approximate solution of batch grinding equation for S(x)B(y,x) = Kxn( y / ~is)expressed ~ in the following equation:
where K ' , m', n', CY are the constants. Therefore the present solution being derived without the assumption of the time dependency of the selection function is virtually the same as Harris' solution escept for very short milling time. This suggests that Equation 17 is reasonable. literature Cited
Harris, C. C., Trans. AI:ME, 241 (12), 449 (1968) RECEIVED for review Xarch 31, 1972 ACCEPTED July 24, 1972
Generalized Correlation for Fugacity Coefficients in Mixtures at Moderate Pressures Application of Chemical Theory of Vapor Imperfections Karl-Heinz Nothnagel,' Denis S. Abrams, and John M. Prausnitz2 Department os Chemical Engineering, C'niversity of California, Berkeley, Calif. 94720
I n vapor-liquid equilibrium calculations, liquid-phase nonidealities are described by activity coefficients while deviations from ideal gas behavior in the vapor phase are described b y fugacity coefficients (Prausnitz, 1969). I n this work, we present a generalized method for estimating fugacity coefficients for a wide variety of mixtures including those containing polar and hydrogen-bonded components. This method is applicable a t moderate pressures as commonly encountered in separations of petrochemical mistures; the upper limit of a moderate pressure varies with the temperature and with the
2
Present address, Roehm GmbH, Darmstadt, Germany. To whom correspondence should be addressed.
chemical nature of the mixture's components but for many typical industrial applications the method discussed here is useful a t pressures up to perhaps 5 or 8 atm. I n special casesfor example, if the vapor contains an excess of a light component such as hydrogen, nitrogen, or methane-the upper pressure limit may be considerably higher. The method presented here IS based on a chemical theory of vapor imperfections in contrast to the physical theory (equation of state) which forms the basis of most earlier work on fugacity coefficients (Leland and Reid, 1965; Chueh and Prausnitz, 1967; Redlich et al. 1965). Several authors (Tsonopoulos and Prausnitz, 1970; Sebastiani and Lacquaniti, 1967, lIarek, 1955) have used the chemical theory to calculate Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 1,
1973 25
