RANR = UNIFORMLY D I S T R I B U T E D RANDOM NUMBER X O = NORMALLY D I S T R I B U T E D RAKDOM NUMBER X T = NORMALLY D I S T R I B U T E D RAXDOM NUMBER literature Cited
Adomian, G., Rev. Mod. Phys., 35, 185 (1963); Proc. Symp. A p p l . Math, 16, 1 (1964). Ariaratnam, S . T., Graefe, P. W. U., Int. J . Control, 1, 230 (1965a), 2, 161 (1965b); 2,205 (1965~). Caughey, T. K., J. Acoust. SOC.Amer., 35, 1706 (1963). Caughey, T. K., Iliens, J. R., J . Math. Phys., 41, 300 (1962). Gray, A. H., J . Acoust. SOC.Amer., 37, 235 (1965). Gray, A. H., Caughey, T. K., J.Math. Phys., 44, 288 (1965,). Himmelblau, D. M., Bischoff, K. B., “Process Analysis and Simulation,” Wiley, New York, N.Y., 1968.
Jordan, J. F., Ph.D. Dissertation, University of Texas, 1966. King, R. P., Chem. Eng. Sei., 23,1035 (1968). Mortensen, R. E., J . Statis. Phys., 1, 271 (1969). Pell, T. XI., Ark, R., Ind. Eng. Chem. Fundam., 8 , 339 (1969). Pike, &I. C., Hill, I. O., “Collected Algorithms from CAChl,” Algorithms No. 226 and 227, Ass. Comp. Mach., New York, 1968.
Rosenbloom, A., Heilfron, J , Trautman, D. I,.,I R E Convention Record, Part 4, 106 (1955). Samuels, J. C., Eringen, A. C., J . Muth. Phys., 38, 83 (1959). Wax, N., Ed., “Selected Papers on Noise and Stochastic Processes,” Dover Publications, New York, N. Y., 1954. RI.CEIVI:Dfor review October 15, 1970 ACCI:PTI:DMay 4, 1971
Presented at the Division of Industrial and Engineering Chemistry, 160th meeting, ACS, Chicago, Ill., September 1970.
Theoretical Analysis of Closed-Circuit Grinding System Based on Comminution Kinetics Masaaki Furuya,’ Yoji Nakajima,Z and Tatsuo Tanaka Department of Chemical Process Engineering, Hokkaido Unicersity, Sapporo, J a p a n
The basic integro-differential equation describing the comminution process i s applied to continuous milling by assuming breakage, selection, and retention time distribution functions to examine the effect of mixing on the product size distribution of open-circuit milling equipment. The theoretical result, as well as industrial data, suggests that the plug flow can be reasonably assumed in industrial tube mills. Then the theory i s extended to the analysis of a closed-circuit grinding system with an ideal classifier. A new parameter, x,*, i s introduced to characterize the circuit, and the finished product size distribution i s expressed in a general form. The calculated results show that the circulating load larger than 3 has no practical significance, if a clean-cut separation is attained in the classifier. An application of the theory to the design of a closed-circuit grinding system with a clean-cut separator i s also presented.
S i z e distribution has a great influence upon the physical and chemical properties of part,iculate materials. One of the possibilities to govern the size distribut’ion of finished product is to adopt the closed-circuit grinding system, on which several empirical reports and theoretical analyses have been published during the past two decades. Tanaka (1957, 1958) reported on the theoret’icalrelationship among various process variables specifying a closed-circuit grinding system and material constants to characterize the fineness of each stream in the system. On the basis of this relationship, lie proposed a nomograph for the practical design purpose of a closed-circuit system to produce a desired size distribution of finished product. However, here more at,tention was paid to the classifier, not t o the behavior of particles in the mill, so that the theory was not always clear in describing the interrelationship between classifier and mill performances in the circuit. Since Broadbent and Callcott (1956) applied a matrix algebra to the commiiiution process, comminution kinetics has Present address, Shin-etsu Chemical Co., Gumma, Japan. To whom correspondence should be addressed.
become a very exciting topic, being developed by many recent workers. However, the matrix representation is not cuiivenierit for the theoretical analysis of milling processes. 0 1 1 the other hand, Gaudin and Meloy (1962), Xustiii and Klimpel (1964), Reid (1965), and many other investigators expressed the comminution process quantitatively in an iiitegro-different,ial equation derived from a material balaiicc. The equation usually involves two fuiictions-i.e., the selection arid the breakage (or distribution) functions. As was pointed out. by Reid (1965), to esteiid the theory to a continuous milling process (and hence to a closed-circuit grinding system): it is necessary to deal with the retention time distribution of particles flowing through a mill. Several papers on this problem have been published, such as t.he works of Mori et nl. (1966), and Kelsall et al. (1968). However, their experimental apparatus were rather small, and no theoretical discussions on t’heeffect of mixing degree upoii t,he size distribution of product were presented. In this part’, certain forms of the selection a i d breakage functions are assumed and their validity is confirmed b y the experimental data reported by Chajo (1949), who carried out the experiments very carefully with a large ball mill and an inInd. Eng. Chem. Process Des. Develop., Vol. 10, No.
4, 1971 449
dustrial tube mill. B y using the assumed selection and breakage functions, the effect of mixing or retention time distribution is theoretically dealt with. The conclusion will be applied directly to the analysis of closed-circuit grinding systems in the later portion of this paper. Mathematical Description of Milling Process. Austin and Klimpel (1964) stated exactly t h e derivation of t h e basic equation describing a batch ball milling process; t h e resulting equation becomes
As t o ball milling and vibration milling, Tamura and Tanaka (1970) suggested the physical picture of S(z) as a function of various operating variables, though their treatment might be rather bold. Another constant in the breakage function, n, may be understood in association with physical breakage process as clearly stated in Gilvarry's deductive study (1961, 1962), but the reason the exponents in S(z) and B(y,r) can be assumed to have the same value has no theoretical basis. Under the assumption, we have Equation 4 as a solution of Equation 1.
R(y,t) = R(y,t).exP(--ty") where z m is the maximum particle size, P(y,t) is the cumulative undersize corresponding to the particle size y at the time t , S(z) is the selection function defined as the weight fraction of material of size z selected for crushing per unit time. B ( y , z ) is termed the breakage function defined as the weight fract,ion smaller than y produced by a breakage of material of original size z. N a n y investigators have solved Equation 1 numerically by giving S(z) and B(y,z) for their individual milling equipment and have shown good agreement between calculated and experimental results. However, if some analytical solution of Equation 1 for certain forms of the selection and breakage fuiictions is found reasonable to describe the comminution kinetics, it is much more favorable than numerical solutions for applying the commiiiution theory to a general treatment of milling systems; even though the accuracy of prediction is more or less sacrificed, the results may offer a means for rapid inspection of the system performalice. The simplest case where a n a,nalytical solution is readily fouiid is to assume the selection and breakage functions in the following forms (Harris, 1968) :
S(z) B(y,z)
=
K#(z)
=
$(Y)/$b)
where #(z) is an arbitrary function of 5 . The above assumption has no physical background, but several investigators employed it for mathematical simplicity and suggested its applicability as a first approximation. For instance,
$(z) =
$(z) = zn (Herbst and Fuerst'eiiau, 1968) $(z) = -b.ln{ 1 -
R(y,t) = exp(-KKty")
( z / ~ ~ (Harris, ) ~ ) 1968)
(5)
This is empirically evidenced in practice of batch ball milling. The validity of Equation 4 will be examined later, referring to the experimental data reported by Chajo (1949). For continuous milling a t steady state, let + ( t ) be the frequency distribution of retention time of particles in a continuous mill, then the size distribution of the product, R D ( y ) , may be written as follows (Reid, 1965), provided that @ ( t ) is independent of particle size. RD(y)
=
s,'
R(?/,t)@(t)dt
(2) (3)
(6)
For example, if a plug flow can be assumed, then 4(t) = 6 ( t
-
7)
= 6(t -
W/E)
where 6 ( t ) is the delta function (unit impulse function), and T ( = W / E ) is the retention time of particles in the mill, W and E being the hold-up and the throughput, respectively. Therefore, RD(y)
=
lm R(y,t)'6(t - W/E)dt
=
R(y,O).exp(-Ky"W/E)
R(y,W/E) = =
R d y ).exp(-WJv/E)
in which n and b are the constants, and x m is the maximum size of feed material. For such a case, t,he solution is generally writ'teii as ~ ( y , t )=
From Equation 4 it can be said that when the size reduction ratio is sufficiently large-Le., R(y,O) = 1 for y in the size range of product -then the final size distribution of product fits well with the Rosin-Rammler exponential form. T h a t is
(Reid, 1965)
z
(4)
(plug flow)
(7)
where the subscripts, E and D, refer to the mill inlet and the outlet. The Rosin-Rammler exponential law for product size distribution is also valid for this case, so long as R&) can be regarded as unity in the size range of product. On the other hand, if the completely mixing flow is assumed, then
R(Y,o). e w ( --K~$(II)}
where R(y,t) is the cumulative oversize fraction corresponding to the indicated size y a t the time t . 111 the present discussion, Equation 2 is assumed for $(z), although whichever Equation 2 or 3 may be employed, the resultant, solut,ions are virtually the same when the size reduction ratio is sufficiently large. Then,
S(z) B(y,z)
=
Kxn
= (V./.)n
At the present stage, K is merely a constant of the mathematical model, but it should be interpreted in the future in relation to the physical meaning of the comminution process. 450 Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 4, 1971
Hence RD(Y) =
RE(y) W 1+K-yy" E
(completely mixing flow)
(8)
Furthermore, it may be reasonable to assume that # ( t ) representing the actual mixing mechanism in a continuous mill can be approximated by the following equation: +(t) = 0 for 0 +(t) =
1
-
rc
t
e s p { - (t
