Ind. Eng. Chern. Process Des. Dev. 1980, 19, 555-560
constants are about 3.5-4 times as large as those of Soni and Thomson. The reason for this difference is not certain. As mentioned earlier, a complete description of the combustion of oil shale char must incorporate the effects of mass and heat transfer effects as well as intrinsic reaction kinetics. The results of this investigation will be used in the subsequent work to analyze the combustion of large pieces of retorted shale. These additional effects are more important for larger particle sizes. The temperature within the solid will be uniform if the rate of heat generation is relatively slow, as observed by Dockter and Turner (1978). In such a case, diffusional and mass transfer effects can be incorporated by applying an approximate relationship obtained from the law of additive reactions times previously proposed (Sohn, 1978). This results in the following expression for the overall rate of reaction in a spheroidlal sample kPOP
d X -_ dt
g'(X)
+ [ k R 2 ' C ~ R , 2 / ( 2 F ~ , ) ] [ p '+( X2/Sh*] )
(19) where the prime designates derivative with respect to X. This expression will enable one to predict the temperature a t which char combustion is essentially extinguished or ignited depending on whether the temperature is decreasing or increasing:, respectively, with time. Under the conditions of significant temperature gradients, analysis of thle problem is rather complex and usually requires nume:rical solution of governing equations (Szekely e t al., 1976; Wen and Wang, 1970). Acknowledgments The authors express sincere thanks to Dr. J. J. Duvall of Laramie Energy Technology Center for providing oil shale samples, and to Dr. J. J. Duvall and Mr. L. Dockter of Laramie Energy Technology Center and Dr. R. L. Braun of Lawrence Livermore Laboratory for helpful discussions. Nomenclature a = heating rate 'C/min
555
A = preexponential factor, min-' atm-' C, = initial content of char in retorted shale, g-mol/cm3 De = effective diffusivity in porous solid, cm2/s E = activation energy, cal/g-mol F, = solid shape factor (= 1, 2, and 3 for finite slabs, long cylinders, and spheres, respectively) g(X) = conversion function defined by eq 3-6 k = reaction rate constant, m i d atm-' k , = external-mass-transfer coefficient, cm/s n = constant in eq 6 p = pressure, atm p(X) = conversion function: = X2 for Ff = 1;= X + (1- X) In (1 - X ) for F , = 2; = 1 - 3(1 - X) l 3 + 2(1 - X)for Fp =3 R = universal gas constant = 1.987 cal/ (g-mol K) or 82.06 (atm cm3)/(g-molK) R, = half-thickness of a slab or radius of a long cylinder or a sphere, cm Sh* = modified Sherwood number k,R, ID, T = temperature t = time X = fractional conversion of char
Literature Cited Burnham, A. K., Sutbblefield, C. T., Campbell, J. H., Lawrence Livermore Laboratory, Rept. UCRL-81951, 1978. Coates. A. W., Redfern, J. P., Nature(London),201, 68 (1964). Dockter, L., AIChE Symp. Ser., 72, No. 155, 24 (1976). Dockter. L., Turner, T. F., In Situ, 2, 197 (1978). Mallon, R. G., Braun. R. L.. Proc. 9th Oil Shale Symp., Colo. Sch. Mines Q . , 71(4), 309 (1976). Sohn. H. Y., Met. Trans. 6 ,9B, 89 (1978). Soni, Y., Thomson, W. J., Proc. 17th Oil Shale Symp., 364 (1978). Soni, Y., Thomson, W. J., paper presented at the 86th AIChE National Meeting, Houston, Texas, Apr 1-5, 1979. Szekely, J., Evans, J. W., Sohn, H. Y., "Gas-SolM Reactions", Academic Press, New York, 1976. Szekely, J., Lin, C. I., Sohn, H. Y., Chem. Eng. Sci., 28, 1975 (1973). Tyler, L.. paper presented at the 2nd Rocky Mountain Fuel Symposium, Salt Lake City, Utah, 1977. Wen, C. Y., Wang, S. C., Ind. Eng. Chem., 62(8), 30 (1970).
Received f o r review August 27, 1979 Accepted May 23, 1980 This work was supported by the U.S. Department of Energy under Contract No. DE-AS03-78ET 13095.
Stochastic Model for Compaction of Pellets in Granulation Norio Ouchiyama National Industrial Research Institute of Kyushu, Tosu, Saga-ken, Japan
Tatsuo Tanaka' DepaHment of Chemical Process Engineering, Hokkaido University, Sapporo, Japan
Based upon a stochastic model, the compaction process of a pellet in granulation is theoretically described for the first time in this paper. The present treatment reveals a theoretical relationship between the pellet porosity, the pellet strength, and the operating condition. Past experimental data on the influence of operational variables on the porosity and the strength of a pellet are in good agreement with the present theory. These results suggest possibilities of practical usage of the present theory for adjustments of the produced pellet porosity and strength.
Introduction Strength of the produced pellet is an important requirement for granulation other than the pellet size (Ouchiyama and Tanaka., 1974, 1975). For green balls in a 0196-4305/80/1119-0555$01.00/0
relatively higher degree of saturation, Tigershiold and Ilmoni (19501, Newitt and Conway-Jones (19581, and Rumpf (1958, 1962) showed that the tensile strength and the crushing strength of a pellet depended upon the ca0 1980 American
Chemical Society
556
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 4, 1980
-
c = =(7
85)
0
Experimental
fl
Calculated
--+-+-+---t4 0
02
04c0.6
08
10
Figure 2. Relationship between coordination number and porosity. Experimental data for random packing by Smith et al. (1929) and Arakawa and Nishino (1973). Calculated data for regular packings by Manegold et al. (1931).
Figure 1. Simplified states of contact in which a stochastic model for compaction of pellets is set forth.
porosity within the spherical volume of diameter 2d (Ouchiyama and Tanaka, 1980). That is ~ ( 2 d ) ~ -/ 6(7d3/6 + C ( 1 3 / 1 6 ) ( ~ d ~ / 1 2 ) ) € =
pillary negative pressure of binding liquid, which is often expressed as an inversely proportional relationship to the pore radius in the assemblage of particles. For agglomerates with localized bondings, Rumpf (1958, 1962) presented a theoretical formula for the tensile strength, which also recognized the porosity dependence of the pellet strength. On the other hand, Kono et al. (1959), Kono (1959, 1960), and Endell and Wagenknecht (1968) experimentally examined the influence of the operational variables on the produced pellet porosity in continuous granulation. Especially, Kono (1959, 1960) graphically represented the experimental relationship between the crushing force, the porosity and the retention time. Until now, however, no theoretical treatments on the compaction process of a pellet have been available to our knowledge. Based upon a stochastic model in the present paper, the compaction process of a pellet in granulation is theoretically described for the first time. The present treatment reveals a theoretical relationship between the porosity, the strength, and the operating condition. Several operational variables involved in the model are examined by using past experimental data.
Theoretical Treatment In the present paper, a pellet is assumed to be composed of uniform sized spheres of diameter d. For the packing of uniform sized spheres, the packing porosity has a fixed relationship to the average coordination number of the packing. Figure 1shows simplified states of contact around a particle, in which the central sphere particle is in direct contact with C number of neighboring spheres; C refers to the average coordination number. The compaction process that increases the coordination number can be described by a stochastic model for compaction; that is to say, another new particle comes into contact with the central particle in some cases and does not in others, depending upon whether or not the applied external force P overcomes the resisting force R; see Figure 1. Denoting the loading frequency (the number of the external forces applied on a particle in a unit time) by q , the increasing rate of the coordination number can be described as dC/dt = q
X
probability(P L R )
(1)
where the de-neighboring of particles from the circumference was discarded. The relationship between the coordination number and the porosity can also be obtained from Figure 1, by approximating the pellet overall porosity t by the volume
~(2d)~/6
or 32 C = -(7 - 8t) 13 where (13/16)(ad3/12) is the exactly calculated volume contribution by a neighboring sphere. The theoretical relationship of eq 2 is plotted in Figure 2, together with past experimental data for random packing and the calculated data for the regular packings. Good agreement between the theory and the known data can be observed. In the following treatment, the coordination number C is converted into the porosity E by eq 2. The angle 8 shown in Figure 1is the apex angle occupied by each neighboring sphere and can be related to the coordination number as follows 4ad2 C
- = 2 a S B d 2sin '6 d8 0
for a / 6 5 8 Ia / 2
or (3) From now on, simple expressions for P, R , q in eq 1will be attempted. (1) External Force Applied on a Particle: P . For a free fall of a pellet and a collision onto the rigid surface, followed by the ideal transformation of the kinetic energy into the deformation work MgH = M V / 2 =
s6=Q 0
d6
(4)
where M is the mass of a pellet, V is the attainable velocity at a fall of height H under gravitational acceleration g, and Q and 6 are the compressive force and the deformation at the surface of contact, respectively (see Figure 3). The expression for the deformation work in eq 4 neglects deformations other than the increase in the surface area of contact. For a pellet of completely plastic matter, denoting the yield stress by K gives Q = KS (5) N aD6K for small deformation where D is the diameter of a pellet. Combining eq 4 with eq 5 and substituting M by p , ( l - t)aD3/6 give the maximum compressive force, Q,,
Q,,
= a D 2 d p g ( l - 4HK/3
(6)
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 4, 1980
Figure 3. Force and deformation at the surface of contact.
557
Figure 5. Functional relationship for and 8 in the right-hand side of eq 8 where 0 is converted into t through eq 9.
relationship for p and 8 in the right hand side of eq 8 is shown in Figure 5, where 8 is converted into c through eq 2 and 3, as
1
99 - 1 2 8 ~ = cos-1( 16(7 - 8t)
Mv
Nw
Nw a
b
Figure 4. Simplified state of particles' binding: (a) general description; (b) free-body diagrams of particles A and B.
where p e is the density of particle. An external force applied, on an average, on a particle within a pellet is the concern of this theory. Dividing Q,, by the number of particles a t the cross-sectional area through the pellet center gives an expression for the external force P as
where the area porosity is assumed equal to the volume porosity. (2) Resisting Force: R. Particle assemblages in capillary attraction is concerned with in the present paper. Figure 4 shows a simplified state of particles' binding in a static equilibrium of forces, in which the resisting force R is formulated by equating the angle 8 to the apex angle 0 of Figure 1,for simplicity. In Figure 4, N, is the attractive force due to capillary negative pressure, F, is the external force applied on a particle, and F, and N , are the tangential force (frictionall force) and the normal force between a particle and a fixed surface, respectively. Furthermore, No is the force necessary for the initial equilibrium of forces at F, = 0, Fi and Ni are the frictional force and the normal force between particles, respectively. Increasing the external force F, brings about the critical state of incipient motion of particles. The external force at the critical state, F,, can be regarded as the maximum resisting force, above which the particles B and C cannot withstand separating apart from each other. Taking the critical external force F,, as R , the resisting force, gives (see Appendix) 2p(1 + cos 8) cos 8 = (8) "(sin 8 - p(1 + cos e)) sin 0 where p is the coefficient of sliding friction. The functional
Simplifying the relationship gives 2p(1 + COS e) COS 8 N (sin 0 - p(1 + cos 8)) sin 8 (1- t)"f(p)
where m e 2 (10)
where the function of p , f(p), increases with increasing p. Combining eq 8 with eq 10 and approximating N , as ird2P,/4 yields
R = F,, = (ad2/4)PC(l- c)"f(p)
where m
N
2
(11)
where P, is the capillary negative pressure of binding liquid in particle assemblages. (3) Loading Frequency of the External Force P : q . Granules within a granulator alternatively repeat a constant angular velocity motion and a tumbling-down motion. A collision onto the device surface or the bottom beds of granules after the tumbling-down motion appears most effective for compaction of pellets. Linear dependence of the loading frequency on the rotating speed N of a granulator is assumed for simplicity. Letting go be a proportionality constant (12)
9=4&
Prior to formulating the compaction process of a pellet, expressions for K and P, will be mentioned. The critical external force F,, derived above can be regarded as the critical bearing force of a particle (particle A in Figure 4) against the external force. Then, the compressive force Q in eq 5 becomes equal to the product of the critical bearing force F,, of a particle and the number of particles at the surface of contact.
Q = Fec(1 c ~ ) S / ( ~ d ~ / 4 1
(13)
Combining eq 5,11, and 13 together with expression for the yield stress K.
= t gives an
K = Q / S = P,(1 - t)'+"f(p)
(14)
According to the dimensional analysis by Kubota et al.
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Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 4, 1980
;:p
I
I
where AI is the proportionality constant. For agglomerates with localized bondings, on the other hand, Rumpf (1958, 1962) presented a theoretical formula for the tensile strength Ust
= 9(1 - t)CHb/(&d2)
or if eq 2 is used gst
I 2.0 x
104
1.6
1.2
0.8 04
651
0
0.2 04 0.6 0.8 1.0 x 256/(13~) N t = NWIF
Figure 6. Theoretical relationship between strength, porosity and dimensionless retention time. Two different initial porosities, 0.75 and 0.50, are assumed in simulations of eq 17.
(1968), the capillary negative pressure P, for sphere assemblages can be described as 1 - t u cos a p , = -___ ds) (15) t d where u is the surface tension of a binding liquid in ambient air, a is the angle of contact, and 4(s) is the function of saturation degree s. Now, we can fully describe the compaction process of a pellet by adopting the height H in eq 7 as the random variable. A simple density function for H (0 I H IH-) is assumed in the present paper. That is probability(H IHoJ= (1 - H0/Hm,)" for 0 IHo IH,,,
(16)
where n is a constant regarding the distribution of H . By combining eq 1, 2, 7, 11, 12, 14, 15, and 16, a final expression for the compaction process of a pellet can be obtained
for 1-t 0 I~ (- t)"/K, 1 I1
= (32/13)(7 - 8t)(l - t)*9Hb/(8ad2)
1- E P s s = -W P1
where ps and p1 are the densities of solid particles and the binding liquid, respectively. Since the porosity changes with the time, the saturation degree, hence @(s)in K , changes in a precise sense during the compaction process even for a fixed liquid content. For spray granulation, the liquid content of a pellet would also change during granulation. Taking these effects into consideration, however, makes the problems too complex for understanding compaction mechanisms. Therefore, an averaged time-independent saturation degree proportional to the added liquid content will be assumed in the following discussions. The numerical order of the compaction rate K , can roughly be estimated as follows; for example 16P&Hma d --Kf = Tf(p)f#J(s)u cos a
m
= 2; r
16PgHmax d = (13/256)q,JVt; K , = 3 f(p)@(s)u cos a
Equation 17 can numerically be solved by assuming the values of K,, n, and the initial porosity of feed pellets, tin. Here, a small agglomerate composed of original particles is supposed as the "feed pellet" irrespective of the feed conditions. Some simulation results are illustrated in the right hand half of Figure 6. The produced pellet porosity, tout, of a continuous granulation can be evaluated by replacing the compaction time t of the theory by the average retention time WIF. Equation 17 gives a theoretical relationship between the pellet porosity and the operating condition. The strength of the produced pellet can also be related through the porosity to the operating condition. For green balls in a relatively higher degree of saturation, Newitt and Conway-Jones (1958) showed that the crushing strength was proportional to sP,. By using eq 15
16 (2.5)(980)(50) (0.0040) -
t
where
(19)
where Hb is the bonding force at a point of contact and C is the coordination number around a particle. Both relations of eq 18 and 19 are illustrated in the left-hand half of Figure 6. The graphical expression of Figure 6 reveals a theoretical reiationship between the pellet porosity, the pellet strength, and the operating condition. Figure 6 shows that, for the constant parameters specified, increasing the dimensionless retention time NWIF decreases the pellet porosity tout and increases the pellet strength ust. Discussion The relationship of eq 20 should hold between the porosity t, the saturation degree s, and the added liquid content (liquid weightlsolid weight) w
3
(4) (6)
-
o(1.0)- 1.5
where following values are assumed: ps = 2.5 g/cm3; g = 980 cm/s2; H,, = 50 cm; p = 0.1 f(p) 4 from Figure 5; s = 0.9 @(s) 6 from experiments of Kubota et al. (1968); u = 72 dyn/cm for water at 20 "C;CY = 0 for a complete wetting; and d = 40 pm = 0.0040 cm. The other unknown parameters, n and qo, of eq 17 are assumed constant for simplicity. As can be seen in the simulation results of Figure 6, eq 17 shows that, for the constant parameters specified, the produced pellet porosity, tout, of continuous grandation should decrease with (1)increasing the compaction rate K,, (2) decreasing the porosity of feed pellets tin, and (3) increasing the dimensionless retention time NWIF. These dependences of the pellet porosity on K,, ejn, and NW/F can approximately be expressed as
-
where A2 = proportionality constant; a, b, c > 0; W = p,JaD;L/4 = holdup; F = mass feed rate; pB = bulk density of holdup; J = filling degree of holdup in a granualtor; D, = diameter of disk or drum granulator; and L = rim height of disk or length of drum.
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 4, 1980
2.0
1.5
xl&gf/d?
0 2 5L- .3
5 F
7 x(1/60)
1-1
\
1 1 02
5
h
16
N
0.5
0
0.2
04
NWlF
08
0.6
X(s;J)
Figure 9. Experimental relationship between crushing strength, porosity, and dimensionless retention time. Replotted from experiments of Kono (1959, 1960).
Figure 7. Effects of liquid content and feed rate on pellet porosity. Replotted from experiments of Kono (1960): dotted lines = raw material A‘; solid lines = raw material A.
6 0,351
1.0
6st
559
23 X(1160)
Figure 8. Effect of rotating speed on pellet porosity. Replotted from experiments of Endell and Wagenknecht (1968).
Kono et al. (1959) and Endell and Wagenknecht (1968) performed continuous granulation experiments in disks and obtained the experimental results of Figure 7 and Figure 8. They show that the produced pellet porosity iwreases with (1)increasing the feed rate of raw materials, (2) decreasing the liquid content, and (3) decreasing the rotating speed of the disk granulator. According to the present theory, these dependences should be interpreted by eq 21. For simplicity, secondary effects of F, N, w through Wand H,, on the porosity are discarded. Then, the observed positive or negative dependence on the feed rate or the rotating speed, respectively, can easily be explained as eout E (NIF)”. The liquid content dependence observed in Figure 7 can also be accounted for without resorting to the change of fin by the fact that @(s)decreases with increasing saturation degree as proved by Kubota et al. (1968), though some additional effect would be expected due to the decreased friction, hence to the decreased f(p.). Since a completely saturated pellet cannot really be further compacted, contrary to the liquid content dependence mentioned above, an existence of a higher liquid content would give the minimum porosity. The contrary liquid content dependence a t the near saturation would be attributed to the deneighboring of particles which was discarded in the derivation of eq 1. Kono (1959, 1960) also performed continuous granulations by varying disk dimensions as well as the feed rate, and graphically represented an experimental relationship between the crushing force, the porosity, and the retention time, through the particular parameters defined. The crushing force (Kg-force) was divided by the cross sectional area of a pellet xD2/4 (D = 10.5 mm) to give the crushing strength rst(Kg-force,/m2)as was done by Rumpf (1962), and the K value definled by Kono was converted into the retention time by a relation N W / F = pBJ(x/4)(N/K value)
(22)
where the experimented condition of N = 11rpm was used. Kono related the pellet porosity to the K value through another particular parameter (y value). Direct connection
gives Figure 9. A conversion factor pBJ in the figure could be regarded as constant in the reported experimental condition. Figure 9 shows that increasing the dimensionless retention time decreases the pellet porosity and increases the pellet strength. According to the present theory, on the other hand, the corresponding relationship was given by Figure 6. Good agreement can be observed between the theory and the experiments. The maximum falling height, H,,,, of the theory may roughly be proportionated to the device diameter for a fixed setting. Theoretically possible variations of eout through H- (ED,) for a constant N W I F did not strongly appear in Figure 9, presumably because of the small changes, from 0.48 m to 0.98 m, of disk diameters used by Kono (1959, 1960). For a more precise discussion, further experimental investigations are needed including effects of other variables of the theory. The present paper reveals a theoretical relationship between the pellet porosity, the strength, and the operating condition. Those results above suggest possibilities of practical usage of the present theory for adjusting the produced pellet porosity and strength. Acknowledgment Miss Chieko Wakamatsu is gratefully acknowledged for typing the manuscript. Appendix Equations for the static equilibrium of forces shown in Figure 4 are described as follows: particle A vertical direction: 2(Ni cos 8 + Fi sin 8) = F, No (A-1)
+
particle B or C horizontaldirection: Ni sin 8 - Fi cos 8 = N, vertical direction: Ni cos 8 Fi sin 8 = N, rotation:
+ Fw (A-2)
+
(A-3)
Fid/2 = F,d/2
(A-4)
From eq A-1 to A-4
F e + No sin 8 - N, cos 8 2 _ -(A-5) Ni F, + No (1 cos 8) + N, sin 8 2 Fe + No sin 8 - N, cos 8 U F W L 2 L _(-4-6) N W Fe + NO (1 + cos 8e)) 2 By comparing the right-hand sides of eq A-5 and A-6 Fi/Ni < F w / N w (A-7) On the other hand, from the law of sliding friction Fi/Ni Ip.; F,/Nw Ip. (A-8) where 1 is the coefficient of sliding friction. Taking the Fi
+
560
Ind. Eng. Chem. Process Des. Dev., Vol. 19,No. 4, 1980
inqualities of (A-7) and (A-8) into consideration, it can be shown that the critical state of incipient motion of particles should occur at the condition of
Fw/Nw = P
64-9)
From eq A-6 and A-9
+ No {sin 0 - p(1 + cos e)] = N, cos 8 (A-10) 2 where F, is the external force at the critical state. On the other hand, from the initial equilibrium condition of Fi = F, = 0 at F , = 0 N 0 / 2 = N, cos 8/sin 0 (A-11) Fec
Combining (A-10) with (A-11) gives (A-12)
Nomenclature a, b, c = positive constants, dimensionless C = coordination number, dimensionless d = diameter of particle, m D = diameter of pellet, m D, = diameter of disk or drum, m F = mass feed rate, kg F,, = critical external force, kg m s - ~ f(F) = function of p , dimensionless g = acceleration of gravity force, m H = falling height of pellet = random variable, m H,, = maximum falling height of pellet, m Hb = bonding force at a point of contact, kg m s - ~ J = filling degree of hold up in a granulator, dimensionless K = yield stress, kg sm2 K , = compaction rate, dimensionless L = depth of disk or length of drum, m N = rotating speed of disk or drum, s-' N , = 1/4ad2P,= capillary force, kg m s - ~ n = constant regarding distribution of H, dimensionless P = external force applied on a particle, k m s-' P, = capillary negative pressure, kg m- 1 s Q = compressive force at surface of contact, kg m s - ~
-f
q = loading frequency of external force P , s-] qo = proportionality constant, dimensionless
R = resisting force, kg m S = surface area of contact at collision, m2 s = saturation degree, dimensionless t = compaction or granulation time, s W = holdup, kg w = liquid content of pellet, dimensionless Greek Letters = angle of contact, dimensionless 6 = deformation of pellet at collision, m cy E
= porosity of pellet, dimensionless
= produced pellet porosity in continuous granulation, dimensionless cin = porosity of feed pellets, dimensionless 0 = apex angle occupied by a neighboring particle, dimensionless p = coefficient of friction, dimensionless pa = density of particle, kg m-3 pB = bulk density of hold up, kg m-3 a = surface tension, kg s-2 cat = tensile or crushing strength of a pellet, kg T = dimensionless compaction time 4(s) = function of s, dimensionless Literature Cited eout
Arakawa, M., Nishino, M., Zahyo, 22, 658 (1973). Endeii, J., Wagenknecht, P., Aufberefl. Tech., Q, 77 (1968). Kono, H., Iwamoto, Y., Nachi, T., Semento GJutsu Nenpo, 13, 90 (1959). Kono, H., Zement-Kalk-Glps, 12, 549 (1959). Kono, H., Semento GJutsu Nenpo, 14, 146 (1960). Kubota, N., Kawakami. T., Ohtani, S., Kagku Kogaku, 32, 822 (1968). Manegold, E., Hofman, R., Solf, K., KolloMZ., 56, 143 (1931). Newitt, M., ConwayJones, J. M., Trans. Inst. Chem. Eng., 36, 422 (1958). Ouchiyama, N., Tanaka, T., Ind. Eng. Chem. Process Des. Dev., 13, 383 (1974). Ouchiyama, N., Tanaka, T., Ind. Eng. Chem. Process Des. D e v . , 14, 286 (1975). Ouchlyama, N., Tanaka, T., Ind. Eng. Chem. Fundam., in press, 1980. Rumpf, H., Chem. Ing. Tech., 30, 144 (1958). Rumpf, H., "Agglomeration", Knepper, Ed., p 379, Interscience, New York, 1962. Smith, W. O., Foote, P. D.,Busang, P. F., Phys. Rev., 34, 1271 (1929). TigershW, M., Iimoni, P. A., "Proceedings, Blast Furnace, Coke Oven,and Raw Materials Conference", p 18, 1950.
Received f o r review September 5, 1979 Accepted April 24, 1980
