Ind. Eng. Chem. Process Des. Dev. 1982, 21, 29-35
area. The model fits the data in the complete range of operating variables used and gives activation energies which are conversion independent and similar for both AC and coal. Work is underway to extend this work to high pressures. Literature Cited Calvelo, A.; Cunnigham, R. E. J . Catal. 1970, 17, 1. Carberry, J. J. "Chemical and Catalytic Reaction Engineering";McGraw Hill: New York, 1976. Epperfy, W. R.; Slegei, H. M. 11th Intersociety Energy Engineering Conference, Lake Tahoe, NV, 1976. Wolf, E. E. Chem. Eng. Sci. 1978, 33, 1557. Ourman, 0.; Guzman, 0.;Wolf, E. E. Ind. Eng. Chem. Fundam. 1979, 18, 7. Haynes, W. P.; Gasior, S. J.; Fwney, A. J. Adv. Chem. Ser. 1974, No. 131, 179. Johnson, J. L. Catal. Rev. 1978, 14, 131.
29
Lefrancois, P. A.; Carclay, K. M.; Skoperdas, G. T. Adv. Chem. Ser. 1987, No. 69, 64. Lewis, W. K.; Gillitand, E. R.; Howard, H. Ind. Eng. Chem. 1953, 45, 1697. Long, F. J.; Sykes, K. W. Roc. R . SOC. London Ser. A 1052, 215, 100. McKee, D. W.; Chatterji, D. Carbon 1975, 13, 381-390. Otto, K.; Sheief, M. "Proceedings of the 6th International Congress on Catalysis", London, 1976, Paper 847. Szekely, J.; Evans, J. W. Chem. Eng. Sci. 1970, 25, 1091. Veraa, M. J.; Bell. A. T. Fuel, 1978, 57, 149. Watanabe, K.; Kondow, T.; Soma, M.; Onishi, T.; Tamaru, K. Nature(London) 1971, 233, 160. Wen, Wen-Young Catal. Rev.-Sci. Eng. 1980, 22(1), 1-28. Willson, W. G.; Seaiock, L. J.; Hoodmaker. F. C.; Hoffman, R. W.; Stinson, D. L.; Cox, J. L. Adv. Chem. Ser. 1979, No. 131, 203.
Received for review December 13, 1979 Revised manuscript received June 15, 1981 Accepted July 10,1981
Kinetic Analysis and Simulation of Batch Granulation Norlo Ouchlyama National Industrial Research Institute of Kyushu, Toso, Saga-ken, Japan
Tatsuo Tanaka' h p a r t m n t of Chemical Process Engineering, Hokkaido University, Sapporo, &pan
Based upon a kinetic theory of granulation, change of the size distribution of granules with the lapse of time was analyzed and simulated throughout a batch granulation. The theoretical treatment enabled one to give physical explanations for the rate of granule growth and for the gradual sharpening of cumulative granule size distribution regarding the dimensionless size. The theoretical results were in generally good agreement with past experimental data.
Introduction The primary purpose of granulation is to produce granules of the required size and strength from powder materials by use of some binding liquid. The strength of a produced granule was studied in our recent paper (Ouchiyama and Tanaka, 1980b) in connection with the compaction process of a granule. The present paper is concerned with the size distribution of granules in batch granulation. In our first paper on granulation (Ouchiyama and Tanaka, 1974), we developed a kinetic equation based upon the coalescence mechanism. In that treatment, however, a mathematical model was involved regarding the coalescence probability, which is more clearly presented as a physical formula in the present paper. The present treatment involves analysis and simulation of the size distribution of granules throughout a batch granulation and enables one to give physical explanations for the rate of granule growth and for the gradual sharpening of the cumulative size distribution of granules. The theoretical results will be examined by using past experimental data. Theory According to our first paper on granulation (Ouchiyama and Tanaka, 1974), the kinetic equation for batch granu0196-4305/82/1121-0029$01.25/0
lation in pan and drum type granulators can be described as
where N ( t ) is the total number of granules in the device at time t , f ( D , t ) is the frequency size distribution of granules, and q is the loading frequency. C(D,d), the function regarding the number of contacts in beds of granules, which was rather roughly taken as proportional to (D + d ) 2 in the paper, was exactly expressed in our recent paper (Ouchiyama and Tanaka, 1980a) as (2) 0 1981 American Chemical Society
30
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 1, 1982
where
and C’is a characteristic packing parameter, which is assumed to be constant in the present paper. According to our second paper on granulation (Ouchiyama and Tanaka, 1975), on the other hand, the coalescence probability, P(D,d),can here be given based upon a physical model as
Combining eq 6 with eq 10 leads to the partial differential equation regarding the size distribution of granules alone. That is
a a7
- f ( u , ~ ) = -SomC(u,u)P(u,u)f(u,r)f(u,7) do
1“c(U : d E ) P ( U : d n ) / ‘ d n ) ’ f ( (U
2
P(D,d) =
+ U,T).
0
P[ 1-
where 6 is a characteristic limiting size, and 7 are parameters regarding the mechanical properties of granules, and other parameters, A, y, and n, are the model parameters. In deriving eq 4,however, we made the mistake that the coalescence probability was affected by the number n of separating forces in spite of a deterministic, not statistical, treatment of the separating forces. This mistake is corrected here. The number of separating forces is fixed to be one instead of n, and the cumulative distribution of stochastic compressive force Q is interchangeablyassumed as 1 - KQM - Q ) / ( Q M - Qm)P instead of 1 - KQM - Q ) / ( Q M - &A)of the preceding paper. The corrections above mean a change of the physical meaning of n but need no other changes regarding the mathematical descriptions of the preceding paper. The dimensionless time, 7, and the dimensionless sizes, u and u, are defined as = C’qt; u = D / 6 ; u = d / 6
7
(5)
Newly denoting the frequency size distribution of granules regarding the dimensionless time and size by f(u,7) and the total number of granules regarding the dimensionless time by N(T)gives the following dimensionless expressions from eq 1 to 4
a dT -[N(7)f(U,7)1
= -N(T)~mC(u,u)P(u,u)f(u,r)f(u,~) du
Equation 11 enables one to simulate change of the size distribution of granules throughout a batch granulation. In the following section, we examine the two parameters characterizing a size distribution of granules, instead of direct examinatio! of the size distribution of granules, i.e., the average size, U , defined by eq 9 and the sharpness, 12. Here, the sharpness k means the slope of a straight line approximating the cumulative sizejistribution curve regarding the dimensionless size u / U , and is defined by a ratio of the smallest size to the largest size in the approximated size distribution of granules. The value of k can be determined by the following equation, as was already shown in our preceding paper (Ouchiyama and Tanaka 1974).
Simple expressions will be attempted for the rate of change of 0 and k. Two density functions, p(u,u) and +(u,u), characteristic of a granulation are defined below.
C ( u , ~ ) P ( u , u ) f ( u , ~ ) du f ( udu ,~)
#(u,u) du du =
~mSomC(u,~)P(u,~)fof(u,r) du du
(14)
-
Equation 13 is concerned with the loading probability that a pair between granules of the size fractions u u du and u u + du is subjected to a loading, whereas eq 14 with the loading-coalescence probability that a pair, hence two contacting granules belonging to the specified size fractions are loaded and coalesced together into a single unit. Using the above density functions, we can define the average coalescence probability, P(u,u),and the two average sizes characteristic of a binary coalescence, (u + u ) and (u3+ u ~ ) ’ /which ~ , are the average size of disappearance and the average size of new birth, respectively. That is
-
The mth-order average diameter, u”,of granules can generally be defined by
Multiplying both sides of eq 6 by du and integrating gives (see Appendix I)
+
P(u,u) = jmjmP(u,u)cp(u,u) 0 0 du du
(15)
(u + u ) = J m L m ( u+ u)#(u,u) du du
(16)
(u3 + u3)1/3
LmLm(u3 + u ~ ) ~ / ~ # ( udu, u )du (17)
I
Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 1, 1982 31
0
Figure 1. A typical simulation result regarding cumulative size distribution of granules.
50
Z = Cq'R
100
150
Figure 2. Simulation results regarding granule growth. Small circles show initial states.
Then, multiplying both sides of eq 11 by u du and integrating gives (see Appendix I)
(u3+ ~ ~ ) ~ / ~ ] C ( u , u ) P ( u , u ) f (du u , ~du) f ( u , ~ ) 1 - = - [ U - (u u ) 2
+ +
(u3+ u ~ ) ~JmJmC(u,u)p(u,u)f(u,i)f(u,~) /~] du du
lo$
'
'
'
- 1b
IO
u=ElS Figure 3. Simulation results regarding granule growth rate. Small circles show initial states.
= '2z U P ~ S - S0 - C0 ( u , u ) f ( u , ~ ) f (du u ,du ~)
r
,
I
where
z = 1 - ( u p + (u3+ u 3 ) 1 / 3 / O
(19)
On the other hand, taking the natural logarithm of both sides of eq 12 and differentiating it with respect to time yields (see Appendix I)
I
0
I
04 06 0.8 1.0 iY=B/S Figure 4. Simulation results regarding sharpness. Small circles show initial states.
A similar treatment also gives another expression for eq 10 as
Discussion By giving the initial size distribution, f(u,O),of granules as well as the values for parameters l, 77, A, y,n of coalescence probability, we can now numerically solve the basic partial differential eq 11 regarding the size distribution of granules (see Appendix 11). The characteristic parameters, 0and k, of the size distribution of granules and the granule growth rate dOf d r can also be calculated by using eq 9, 12, and 18, respectively. The initial size distribution of granules was given for every simulation as
02
the uniform size distribution, whose characteristic parameters fr(0)and k(0)were shown in the figures concerned together with the parameter values used. Figure 1shows a typical simulation result regarding the cumulative size distribution, F(u,T),of granules throughout a batch granulation, which is theoretically represented here for the first time. A series of the cumulative size distribution curves has an S-shaped configuration, which is likely to be in qualitatively good agreement with past experiments (Kapur and Fuerstenau, 1964; Sastry and Fuerstenau, 1971; Kapur, 1972; Linkson et al., 1973). The dimensionless granulating time T in the theory can easily be related to the total number, R, of revolutions of a granulator by assuming that the loading frequency q is proportional to the rotating speed, N , of the device. That is T = C'qt = C'q'N,t = C'qB (22) where q' is a proportionality constant. The dimensionless size can also be converted into the real size by multiplying the former by 6, the characteristic limiting size. In the
32 Ind. Eng. Chem. Process Des. Dev., Vol. 21, No. 1, 1982
,+-T-n-$ R
x10
Figure 5. Influence of moisture content on granule growth. Experiments of Linkson et al. (1973).
following, the two parameters of D and k characterizing the size distribution of granules and the granule growth rate dD/dr will be examined instead of direct examination of the size distribution of granules. Figures 2, 3, and 4 illustrate the simulation results regarding D,dD/dr, and k, respectively. First, general examination of the present theory will be made by using past experimental data regarding the granule growth. In reference to the mechanisms so far reported by Capes and Danckwerts (1965, 1967) and Kapur and Fuerstenau (1964,1969), Linkson et al. (1973) observed that for materials having a wide size distribution, coalescence tended to occur, whereas crushing and layering occurred for sands of similar sizes. In view of a possibility of the remarkable crushing effect, therefore, the experimental data for closely sized sand will be excepted in the present paper as well as in our preceding paper (Ouchiyama and Tanaka, 1974). Sastry and Fuerstenau (1971) investigated the influences of operational variables such as the rotating speed N , of a granulator and the holdup W in the device on the granule growth. According to their experimental results (Figure 6 and 7), the granule growth related to the total number of revolutions R was nearly independent of the rotating speed and of the holdup. According to the present theory, on the other hand, the granulating time r can be expressed in a proportional relationship to the total number of revolutions, as was already mentioned, and furthermore, the basic differential eq 11 regarding the size distribution of granules could be constructed without including the total number of granules, N ( 7 ) ,which depends on the holdup in the granulator. Both these theoretical findings and the above mentioned experimental data of Sastry and Fuerstenau suggest that all the parameters involved in the theory are independent of these operational variables. Figure 5 shows the effect of the moisture content, w,of the load on the granule growth as obtained in the experiments of Linkson et al. (1973). The larger growth rate of granule combined with the larger limiting size of growth can be seen as the moisture content increases. Comparison of Figure 5 with Figure 2 suggests that a characteristic limiting size 6 should increase with increasing moisture content for fixed values of the other parameters. Detailed analysis of the characteristic limiting size will be presented in the near future. According to the past experimental results summarized in our preceding paper (Ouchiyama and Tanaka, 19741, the granule growth rate and the sharpness varied depending upon the granulation stage as follows early later a / d Ra (D)? a = 1, simple decrease, E -4
-
k: = 0.2-0.3, simple increase, = 0.5 (23) The corresponding simulation results are illustrated in Figures 3 and 4. Regarding the granule growth rate, good agreement between the theory and the experimental relations above can be seen. As to the sharpness, on the
10''
1b-2
1.0
U=B/S
Figure 6. Granule growth rate for uniform sized granules.
10-21 10'
'
" '
1.0
1 04
a
Figure 7. Coalescence probability for uniform sized granules. 1.0 I
, Y /
A
Figure 8. A simulation result showing a trend of gradual sharpening.
other hand, some examples showed positive minimum values of k at the beginning stage of granulation, but others made it impossible to obtain positive values of k through 03, as shown in eq 12 because of too large values of g / ( Figure 4. Nevertheless, a qualitatively good agreement can be seen between the theory and the experiments except at the very early stages of granulation. The impossible negative values of k will be discussed below. The granule growth rate of eq 18 can be rewritten for uniform sized granules as
This relation is illustrated in Figure 6, which gives a trend similar to the simulation results of Figure 3. Equation 24 suggests that the granule growth rate should be proportional to the product of the average size of granules and the probability of coalescence for the average size. According to the present theory, therefore, the varying dependence of the granule growth rate on granulation stage can principally be ascribed to a marked decrease of the average coalescence probability, which is illustrated in Figure 7 for uniform sized granules. The average probability of coalescence can roughly be regarded as constant
Ind. Eng. Chem. Process Des. Dev., Vol. 21, 10
I
'
" 'A
Table I
I
restrictions in model
parameter
f
'13
u / U = D/D
Figure 9. Size dependence in different granule growth on function P(u,u).
for small sizes of granules, which leads to near a first-order rate of granule growth in the early stages of granulation. On the other hand, an increase of sharpness, k, means a gradual sharpening of the cumulative size distribution curve, as can be seen in a simulation result of Figure 8. According to eq 20, a positive sign-of the rate of change The of sharpness necessitates a value of 2 greater than value of Z is concerned with the average sizes of coalescence, which, in turn, are concerned with the loadingcoalescence probability of eq 14 and, hence, with the function C(u,u) P(u,u) for a specified size distribution of granules. As a simple approach, let us consider coalescences between granules of the same size, u = u. Then
2 = 1 - (2 - 3 ~ ) (>r1/3 0 (dk/dr > 0)
(25)
and
(26) of Equation 25 shows that a positive rate of change sharpness is brought about by a smaller average size (u/0 of coalescence, whereas eq 26 suggests, as is illustrated in Figure 9, that the granule growth tends to give a higher loading-coalescence probability for the granules of a smaller size u / U , which leads to the smaller average size (u/O)of coalescence. According to the present theory, therefore, a gradual increase of sharpness can also be ascribed to an increasing negative slope of the coalescence probability curve of Figure 7. Higher values of k observed in Figure 4 at the beginning stage of granulation merely originated from the assumed higher values, k(O), which were given as initial conditions of the theory and have no particular meaning. The negative values of k in the simulation examples are physically unreasonable, but apparently resulted from the corresponding higher values of ( Q3 as seen in Figure 4. In simulation of aerosol coagulation, Hulburt and Akiyama (1969) found nonphysical results for too wide initial size distributions. In order to determine whether this accounts for the present case or not, however, further investigations are needed. Furthermore, past investigations have sometimes referred to the effect of the transfer of binding liquid, accompanied by the compaction of granules, on the granule growth (Sastry and Fuerstenau, 1973). In our view, this effect seems to be interpreted by the dependence of the characteristic limiting size on the porosity, the strength and the deformability of granule, and so forth. Detailed discussions on the effect are beyond the scope of the present paper. However, when the strength of produced granules is of concern, we have to
s/
S f < 1 17 4 1 1 3
17 h
0
