Mathematical Modeling of Steady-State Behavior in Industrial Granulators Chang Dae Han and Israel Wilenitz Department of Chemical Engineering, Polytechnic Institute of Brooklyn, 333 J a y St., Brooklyn, N . Y . 1 l d O l
A mathematical model is based on three assumptions: that the crusher, present in all granulation plants, i s the main source of seed particles, that the residence time of particles in the granulator i s a probabilistic function, and that particle growth rate i s constant. Using these assumptions, the model takes data for the particle size distribution of the granulator discharge stream and calculates a distribution for the recycle stream, in good agreement with plant data. From the two distributions, the model then calculates a value of recycle ratio, which again agrees closely with plant data. Having thus satisfactorily tested the model, the authors demonstrate the relationships among plant recycle ratio, product recycle fraction, and crusher speed.
Tm
term I‘granulation” is applied generally to many processes in the fields of fertilizers, pharmaceuticals, ceramics, and others. These processes differ markedly in the types and forms of feed streams, the kind of equipment in which particles are grown, and the manner in which individual particles increase their size. However, they all have in common the fact that the granulator product is a stream of particles having a comparatively wide size range, that this stream is screened to remove particles whose sizes fall in a range acceptable for packaging and sale, aiid those particles rejected are recycled to the granulator inlet after a size reduction stage if necessary. For this first approach, we confine our study to the fertilizer industry, aiid, further, we consider only rotating drum granulators. This still leaves a rather wide choice, and if we seek a fairly simple process, with, say, only one solid feed stream aiid one fluid stream, we find that the process for the manufacture of diammonium phosphate (Davis et al., 1965; Hail, 1970; Horii and Fouser, 1966; Young et al., 1962) fits the bill. I n diammonium phosphate (DAP) plants, it is customary to make phosphoric acid (in the region of 75y0 HJ‘O4 concentration) react wiih amnioiiia vapor to produce a fine suspension of mixed monoanimonium and diammonium phosphates in water (referred to as “slurry,” although the particle size is in the submicron range). The slurry is then piped into the granulator so as t o spray upon a tumbling bed of D A P granules. Ammonia vapor is also piped into the granulator, but the distributor is located in the bottom of the drum, so that it is always submerged in the granule bed. At first glance, it appears that we have two fluid streams, the slurry and the ammonia vapor. However, these two streams stand in a fixed relation to each other and cau be regarded its one. In general, the literature shows very little attention paid to mathematical analysis. A great deal of thought has been devoted to the chemistry of the syqtems, with pragmatic determinations of such parameters as the mole ratio of ammonia to phosphoric arid in the slurry (Davis el al., 1968; Horn and Fousar, 1966; Young et al., 1962). Further attention has been devoted to the manual operation of unite, and the fluctuations and shutdowns they are prone to. The first real attempt a t analysis is Han’s (1970). Jf the over-all process is considered, analogies can be drawn between granulation and crystallization. I n both processes, nucleation takes place and particles grow by deposition of
on nuclei either from solution (cryst’allization) or b y wetting with slurry followed by contact wit,li ammonia (granu1at)ion). If the crystallizer has facilities for screening of product and recycling of off-size, the analogy is closer still. In the light of t,his resemblance, the approach used hy Han (1967, 1965) in crystallizer studies is worth consideration. Consider a typical granulation process as sketched in Figure 1. Only granulators of the rotating cylinder type, rotat.ing about the axis, which is slightly inclined to the horizontal, are considered. The granulator exit stream goes to the screen, through which particles are classified into three streams : undersized particles which pass both t’he upper and lower sieve; oii-sized particles which pass the upper sieve but are retained on the lower; and oversized particles which are retained on the upper sieve. The undersized particles are recycled to the inlet of the granulator. Oversized particles are crushed in the mill and the crushed particles recycled. Depending on the particular process, a portion of the on-sized particles may also be recycled (sornet,irnes after crushing, although this is not shown in Figure 1). Two basic niechanisms can be postulated for part,icle growth in environments iri which particles tumble in a liquid medium. I n one case, wetted part’iclescollide and adhere t,o each other, possibly because of the surface tension of the interstitial liquid--Le., the “snowball” effect. In the other case, wetted particles absorb solute from the liquid, with excess liquid driven off by some means, thus depositing a layer. Successive wettings and dryings will then build the part,icle size up to the required value. I’articles grown under these conditions show typical pattern of concentric rings when sectioned, and the mechanism is t,herefore called the iionionskin’J mechanism. In the manufacture of mixed fertilizers, by rotatirig granulator where no chemical react>ion occurs b e h e e n the coniponents, the only mechanism for growth is t’he snowball one. However, in D.4P unit,s, where material is deposit,ed as the result of a chemical react,ion, particles can grow by both the snowball and the onionskin methods. The factor t,hat determines which takes place is the ratio of solid to liquid feed; in L‘\yetjter”systems, part’icles teiid to grow by agglonieration, in “dryer” ones by deposition (Capes, 1967a,b; Capes and Danckwert’s, 1965; Farnand et al., 1961 ; Newitt and Conway-Jones, 1958; Sutherland, 1962). The latter case is assumed here. Under st’eady-state conditions, the granules leaving the Ind. Eng. Cham. Fundam., Vol. 9, No. 3, 1970
401
granulator will have a definite size distribution and the rate of formation of granules will be equal to the rate a t which granules are withdrawn from the granulation plant as product. Also, the rate of growth will adjust itself in such a manner that granules formed will grow to the desired size in the time available for their growth. This paper sets up a mathematical model of the rotatingdrum diammoniuni phosphate granulation process in the steady state. From this model we compute values for certain performance criteria and compare these values with those measured in an actual pilot plant.
MAP SLURRY FEED
Theoretical Development
The granulation process shown schematically in Figure 1 is considered, and the following assumptions are made. The granule growth rate is uniform. Thus, if the granule size is represented by some dimension D, dD/dt is constant and independent of D-for example, for a sphere D is diameter; for a cube D is the length of a side. The crusher is the only source of new granules, or, in other words, any new particles formed by attrition are too small to serve as nuclei for new granu!es. Sieve efficiency is 100’%-i.e., size separation is ideal.
Particle Balance in System. We are assuming t h a t the number of particles entering the granulator per unit time is equal to the number of particles leaving in the same period-Le., by the nomenclature of Figure 1
N E = Na
(1)
Summing the particles in the recycle stream:
NR = Nc
+ + aNp
(2)
N U
A particle balance across the sieve gives
NG= N o
+
N U
4-Np
(3)
By virtue of Equation 1 we can equate the right-hand sides of Equations 2 and 3 :
Nc
+ + N u
aNp =
No
+ Nu +
N p
or
Nc
- No =
(1
- a)NP
=
Schematic of DAP granulation process
per cent basis-Le., for a given interval of particle size, the value plotted is the ratio to the total sample weight of the weight of granules whose sizes fall within that interval. This procedure thus defines a density function by weight and normalized. This is discussed below. Classification of Granulator Discharge. Material leaving the granulator is separated into three functions (see Figure 1): undersize, product, and oversize. The over-all range of granule size-Le., D1 to D4-is determined by plant operation. The product size range-Le., DZ to DI-is fixed by marketing requirements. Figure 2 shows ideally how the density function for the granulator discharge particle size distribution is divided into three sections by particle size. The three fractions have density functions fo(D),jp(D), and fu(D) for oversize, product size, and undersize, respectively. These functions will have the f o r m s f 0 D > DJ,fa(Dl DZ< D I D d , andfdDI D 5 D z ) , respectively. However, above we defined j ( D ) per unit mass of the particular stream. Consequently, each of the three functions must be multiplied by a different scale factor. Thus:
(5)
Seed particles are required as feed to the granulator. These are supplied as recycle, the total recycle stream comprising all
No. of particles in a stream whose size lies between D and D
+ dD
unit mass of stream
Figure 2 illustrates such a function for the granulator discharge stream. The size of the smallest particle present is D I ; that of the largest is Dd. Apertures in the sieve screens are DZand Da. The essence of the approach followed in this study is the particle balance in the system-Le., the number of granules circulating. Thus, the density functions referred to here are by number and unnormalized. I n plant practice, the histograms of particle size distribut,ion are determined on a weight 402
Figure 1 .
(4)
The quantity (1 - a ) N P is the number of particles withdrawn as product. The quantity ( N c - N o ) is the number of particles generated by the crusher-Le., the difference between the number leaving the crusher and the number entering it. Thus, we see that the specific function of the crusher is to provide the seed particles for the product. Density Function of Granule Size Distribution. This function designates the number of particles (in a given sample) having a given size. T o be precise, the density functionf(D) is defined by (D)dD
SIEVE -
Ind. Eng. Chcm. Fundam., Vol. 9, No. 3, 1970
the undersize from the screens, all the oversize after it has been reduced in size in the crusher, and a fraction cy of the product size stream. The density function for the recycle is then 1
. ~ R ( D=) - [MUfu(D) MR
+ MofC(D) + aMdp(D)l
which, when we apply Equation 5, becomes
UNDERSIZE -1.
PRODUCT
I r
1
The recycle stream mass flow rate, M E ,is given by
OVERSIZE I 1
M R
= hfQ
-
=
(1
- h)MQ
(15)
The recycle ratio R is given by Equations 13 and 15:
R
=
1--x
MR/R= -
x
(16)
Substituting into Equation 6 from Equations 11, 12, and 15,
Figure 2. Density function of granule size distribution at granulator outlet
1 ~ R ( D= ) -X MR
[llfafo(.l>ID
I D2) + lvofc(D) -t
ffJfofc(D IDz
< D _
m 2
g
0.8
0.7
-
-
Figure 5. Typical plot of recycle particle size distribution by weight
4.0 0
0
P L A N T DATA CALCULATED VALUES R U N No. I I € = 0 . 4 mm./HR.
€
E
a W
Nu)
Figure 6. Typical cumulative plot of calculated recycle size distribution by weight
W -I
9
I-
a: 9.0
K
n a
t 0.1
50
IO
~
CUMULATIVE
0 0
P L A N T DATA CALCULATED
PERCENT
90
LESS
99
9'
SIZE
THAN
VALUES
RUN N o . l l a = 4.0 HRS:'
i
E
a w N
UI
Figure 7. Typical cumulative plot of calculated recycle size distribution by weight
w
-t
0 Ia
2
- t 0.1
€=0.4 € = O S
I
1
I
I
CUMULATIVE
I
I
I
IO
I
I
I
I
,
50 PERCENT
LESS
, 90
THAN
1
1
,
99
9
SIZE
Ind. Eng. Chem. Fundom., Vol. 9, No. 3, 1970
407
Crusher Product Stream. T h e crushers most often found in granulation plants (Hignett, 1957; Horii and Fousrr, 1966) are of t h e dynamic stress type--e.g., hamnwr mills, cage mills, chain mills, etc. In such unitS, t h e individual particle is whjected t o a large number of impacts in passing through t h e mill (Ferry, 1963). Empirical d a t a show that for such mills, the density function of the particle size distribution of the outlet stream is independent of t h a t for t h e inlet stream. Furthermore, as would he expected, an increase in speed gives an increase in the number of product particles and a decrease in their average size. We are concerned particularly with the number of particles circulating in the system, so that niill speed is a variable of coiisidernhle interest. Furthermore, available data @eke, 1964) indicate that the percentage variation in number of particles is fairly large for a small change in speed. Consequently, we can assume a simple linear relation between mill speed and the parameters of the density function, thus, In I), = K l / r
(42)
Kz/r
(43)
In
ug =
With this in mind, the crusher output density function,
fC(D), can be rewritten as f c ( D Jr ) , where r is the rotational speed of the crusher. Similarly, $ p ( U ) can be rewritten as &,(I), r ) . The actual form of the size distribution density * function for this stream also turns out to be closely approximated by the lognormal. In fact, the foim has been derived theoretically for the product of dynamic-stress size-reducers (Beke, 1964). The final expression will have the form
and from plant data we obtain K1
=
Product Recycle Fraction a. A t this point, a h t value of a has been determilied. I t i i a i w m c d that the rnechanicnl operation of the granulator is iiever chaiiged, so t h a t a is now a fixed plant constant. However, E cnii change, and docs Nhhen the ratio of liquid feed to recycle flow changes. Coiisequently, we calculate a sequence of $R(D)by number for fixed a and a range of E . Of course, the “best” value of E , as previously obtained, lies iii that range. Results
Generation of Recycle Particle Size Distribution. Using an algorithm bawd on Equation 41 , the recycle particle size ciistiibiitioir 1 ) ~neiglit 7 \ 8 3 calculated for eight T alues each of residciice time paranietcr a a i l t i particle grok\th rate E 13y means of the procedure descri1)erl previously, the followiiig best values were selected : u = a =
4 hours-’ 4 hours-’
E
=
E =
0.4 nim per hour 0.2 inni per hour
Frorri thiq we see that the mean residence time of particles in the gi anulator m as 1.25 ta,
=
7n
=
0.625 hoiir
=
37.5 minutes
Sirice the interiial conqtruciion and angle of inclination of such equipment as granulators are not operating variables, and rotational speed is rarely iised as surh, two runs in the 408
Particle Growth Rate, Run No.
E
Mm/Hr
Product Size Specifications
- ~ _ _ _ _ _ (Tyler Mesh) - 6 f 10 -8
+ 10
-6
+8
-84- 12
11
0.2 0.3 0.4 0.5 0.6
0.634 0.491 0.384 0.305 0.250
0.634 0.489 0.381 0.303 0.247
0.635 0.493 0.386 0.307 0.251
0.636 0.491 0.383 0.305 0.249
12A
0.2 0.3 0.4 0.5 0.6
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
657 516 410 332 275
656 515 408 330 273
657 516 412 334 276
Table IV. Over-all Plant Recycle Ratio Particle Growth Rate, Run No.
6
Mm/Hr
11
0 0 0 0 0
2 3 4 5 6
1211
0.2 0.3 0.4 0 . .i 0.6
R
Product Size Specifications (Tyler Mesh)
__--___-_.__
-6 f 10 - 8 + l O
ci 4 3 2 2
00 03 15 68 41
5.60 3.68 2.84 2.39 2.12
9 6 5 4 4
49 51 20 51 10
9.65 6.56 5.19 4.47 4.04
-6+8
20 14 11 10 9
02 13 49 08 25
16.32 11.26 9.09 7.91 7.20
659 518 411 333 275
-8
5 3 2 2 2
+ 12 63 75 91 47 21
6.05 3.99 3.08 2.60 2.32
-4865
li? = 1444
Rim 11 Run 12>1
Table 111. Product Recycle Fraction a
Ind. Eng. Chem. Fundam., Vol. 9, No. 3, 1970
same plant would be expected to have the same residence time. Furthermore, the average residence time of approximately 40 minutes is consonant with the authors’ experience. ‘The choice of growth rates is also coiisist8eiitwith the plant data. First, the recycle ratio of 3.3 pounds per pound for run 12.4 is greater than that of 2.7 pounds per pound for run 11, and t,liis implies n smaller e for the same production rate. Secondly, inspection of the cumulative plots (Figures 3 and 4) shows that data points for granulator discharge and recycle stream lie closer t’ogether for run 12A than for run 11, indicating that the part’icle buildup per pass through the granulator was less. Reference to Table ZV gives a value of recycle ratio R of 3.15 for e = 0.4, product size specifications being -6 10 mesh. This is close to tlie plant data value, 2.7. However, a t E = 0.5 calculated R is 2.68. Certainly, a growth rate in the range 0.4 to 0.5 inm per hour appears to have been the case in ruii 11. Similarly, Table IV iiidioates that for run 12A, the growth rate fell in the range 0.3 to 0.35 tnm per hour. With the value of a fixed at 4 hours-1, and using the parameters of t8he size distribution by number of tlie granulator discharge stream, the recycle size distribution by number was generated for use in calculat’iiig C Y . Calculation of Product Recycle Fraction and Over-all Recycle Ratio. As tlic ticrioniiriator s h o w , Equation 26 holds for values of particle size D lying n-it’hin the range of the product size specifcatioiis. For all rims iii the T V A pilot plant tlicsc were - 6 $- 10 mesh. To see the effect of changiiig tlic product size specifications, product re-
+
cycle fraction a was calculated for four combinations in all: -6 10 mesh, -8 10 mesh, - 6 .f 8 mesh, and -8 12 mesh. The first three are commonly encountered. Hitherto, crushers have usually been run a t constant speed, in this case 1500 rpm. To study the effect of varying the crusher speed, the computation was performed for five values covering the range 1300 to 1700 rpm. The over-all plant recycle ratio, R, was calculated for each value of a. It was found that crusher speed had no influence on a , a t least, to three significant figures. Size specifications had a small effect, but change in B was very important. Equation 26, by which CY is calculated, is based on the consideration of the particle size distributions of the recycle stream and the streams which compose it. The values of CY tabulated in Table I11 are actually averages of individual values obtained for a number of particle sizes. These part’icle sizes are equally spaced a t intervals of 0.2 nim and t8henumber of individual values calculated for each set of paranieters is determined by the “width” of the product, size specifications, -6 10 mesh, etc. There is no space to t,abulate all the individual values here, but Table V shows two typical sets, both for the same set of parameters except for product size specifications. Individual values of CY are remarkably constant, so that “wider” or “narrower” product specifications have very little effect on the average value. The negligible ,influence of crusher speed is also explicable from inspection of Equation 26. The second term in the nuic.7 merator is - $ e @ ) , and for values of D in the product
+
+
+
Table V. Typical Sets of Recycle Fraction Values for Run 11
Residence time parameter, a. Particle growth rate, E . Mill speed, r .
__ __
Particle Size, M m
-6
4 hr --I 0 2 mni/hr 1700 rprn
Product Size SpeciRcatiotis_-
+ 10 mesh
1.70 1.90 2.10 2.30 2 .50 2.70 2.90 3.10 3.30
0.631 0,633 0.632 0 . 635 0.636 0 . 637
Av .
0.634
0.638 0.634 0.632
-8
+ 10 mesh
.-
0.638 0.634 0.632 0.631
0.634
+
From this can be seen that II depends on both a and product size specifications. Hence R, like CY,is insensitive to crusher speed r and sensitive to particle growth rate E, but it is also sensit’ive to changes in the product size, as is shown in Table IV. R is inversely proportional to the third moment of that portion of the granulator discharge size distribution falling inside the product size range (Figure 2). The trends are:
I.( C3
size range t,he values of cpc(D) are very small. This is the dist’ribution of the crusher output, having a very small mean particle size, meaning a distribution skewed very strongly in the direction of zero size, and having very small values a t large sizes. The factor $/plea is never large enough to make a n appreciable difference. The very strong influence of values of E on the values of ct comes from the term cpE(D),which means, after all, that for a given particle size distribution of graiiuliitor discharge stream, the recycle distribut’ion is a strong function of particle growth rate E. From Equations 14, 16, and 20, we have
The smaller the lower product size, the larger the value of the integral and the smaller the value of R . The narrower t’he product size range, the smaller the value of the integral and the larger the value of R. Both trends are as may be expect’ed from plant operatioil experience. Plots of CY os. R are shown in Figure 8 for the two pilot plant runs, for size specifications -8 f.10 mesh. Operating Line. Utilizing Equal ion 36, values of CY were calculated a t five crusher speeds for run 1 1 . The pilot plant size specifications w r e used--i.e., - 6 -t 10 mesh. Figure 9 shows product recycle fraction CY plvtted against crusher speed T for a value of 0.775-mni “minimum particle size” of crusher output,, latxled in t’he figure as D e . When computat,ion of a was first attempted, large negative values were obtained. Inspection of all variables showed that p’ca
(45)
__ PRODUCT - 6 * I O
MESH
0.7 CI
RUN No. 12A
z
2 0.6 .
+
-
RUN No.11
U
a
E 0.5 . W _J
0.4
’
u L1: W
I-
0.3.
U
3 0
g 0.2 . a
2.0
4.0
3.0 OVERALL
Figure 8.
PLANT
5.0 RECYCLE RATIO
6.0 R
Product recycle fraction a vs. over-all plant recycle ratio R Ind. Eng. Chem. Fundam., Vol. 9, No. 3, 1970
409
0.6
0.7
0.5
0
U
2-
z
0.E
2
9
2
0.4
V
\
(L
6-0.4
W
U e
W 1
U
y
U
U
W
0.5
(r W
0,3 I-
I-
u
V
3
3
0
0
0.4 0.2
\
D c = IOOmm-
0.3 0.1
I
I
I300 1400
I
I
1500
1600
CRUSHER
Figure 9.
I
1700
SPEED, r
I
I800
I
1300
I
+
+
Ind. Eng. Chem. Fundam., Vol. 9, No. 3, 1970
I
1400
(RPMI
Operating line for run 1 1
had extremely small values, so that the term in brackets had extremely large ones. Furthermore, it was determined that p ’ c 3 had such small values because +c(D) was very strongly skewed, with a mean of less than 0.05 mm. According to the plant model developed here, the function of the crusher is to provide particles to replace those withdrawn as productLe., to provide seed particles for granulation. It is clear that a particle stream having a mean particle size of less than 0.05 mm contains very many particles too small to act as seed particles. If the particle size distribution of such a stream is truncated a t some size-that is, if particles smaller than that size are disregarded-a new size distribution is defined and it has a larger third moment than the original. Table I11 provided a value of CY to aim for, in run 11. Having previously selected 0.4 as the best value of particle growth rate e, a value of 0.384 had been calculated for CY (product size range -6 10 mesh). Inserting this value in Equation 36 gave a value of p ’ c s , which required a minimum particle size of 0.775 mm. Similarly, an operating line was calculated for run 124, for D C = 1.00 mm, and is shown in Figure 10 with lines of constant t values taken from Table 111. This computation was restricted to the product size range of -6 10 mesh, because actual operating data were available only for that size. Operating Considerations. All equations t h a t have been developed here for calculating product recycle fraction CY and over-all recycle ratio R have been entirely independent of production rate. It would therefore appear that, in the steady state, cyI r , and E are unchanged when passing from one product rate to another, if the product size specifications remain unchanged. What then determines the choice of product recycle fraction a,crusher speed r, and thus particle growth rate t ? 410
I
1900
€.OS
1500
CRUSHER
Figure 10.
I
1600
I
1700
SPEED, r
I
1800
I
1900
(RPMI
Operating line for run 12A
Presumably, the recycle flow rate is maintained as low as possible, in order to utilize equipment capacity to the greatest extent. On the other hand, if recycle ratio R is too low, the solid-liquid ratio a t the granulator inlet is too low and the growth mechanism may change from onionskin to snowball. This, a t least, establishes a lower limit for CY. At the same time, for a given production rate, the number of particles withdrawn per unit time is determined, and an equal number of seed particles must be generated by the crusher. So, if a particular value of recycle ratio R is required (for whatever plant reasons), the corresponding value of product recycle fraction CY is thereby fixed, and this value of CY then determines the required value of crusher speed r . Example. Assuming the operating conditions of run 11, let us say t h a t certain considerations require a recycle ratio R of 4.0. Reference to Figure 8 shows t h a t this is equivalent to a product recycle fraction CY of 0.489. Entering this value of CY in Figure 9, we read off a crusher speed r of 1360 rpm. It is then seen that the resulting particle growth rate t is very close to 0.3 mm per hour. If any other operating factors necessitate a change in R, and hence in CY, r would have to change to maintain the required production rate of seed particles, and the result is a change in E . Conclusions
The mathematical model developed here is based on the important assumption that the crusher’s specific function in granulation plants is to generate seed particles for granulation a t a rate exactly equal to that a t which product-sized particles are withdrawn. Furthermore] particles reside in the granulator for times which are clearly not uniform but are probabilistically distributed. On these bases, the calculated
recycle distribution was found to be in good agreement with plant data for the recycle stream. Going a step further, and using plant data for the granulator discharge distribution and calculated values for the recycle distribution, a value of plant recycle ratio was calculated and found, again, to agree closely with the plant data. A very simple linear relationship was assumed between the speed of the crusher and the parameters of the lognormal function which represents the particle size distribution of its product. The literature strongly supports the assumption of lognormality, but experimental work would be required to substantiate or supplant the linear relationships. However, there is no question that the particle size distribution of crusher product does vary, at least qualitatively, Ivith crusher speed as described here. Thus, changes in recycle ratio must always be made in conjunction with changes in crusher speed if steady-state operation is to be maintained. This interdependence holds regardless of the actual plant production rate. The computations performed on the plant particle balance have disclosed that there is a lower limit to the size of particles entering the granulator, below which they are too small to serve as seed particles for granulation. The crusher is the source of these fines, and it becomes evident that their presence is due to the fact that the crusher operates a t too high a speed. To the authors’ knowledge, no one, hitherto, has operated a crusher a t varying speed. Such equipment is not very flexible in this respect, but it is suggested as a possibly useful point to investigate. Another area for experimental study is to correlate granulator speed and angle of inclination to an expression for the residence time distribution of particles in the granulator. However, on the basis of the results reported here, it is believed that this model provides many insights into the known behavior of granulation plants and will be of considerable help for plant operation. Furthermore, the model will eliminate some of the guesswork currently involved in the specification of equipment and thus make plant design more systematic.
SUBSCRIPTS
C G n 0
= = = =
i U
= = =
F
=
crusher discharge stream granulator discharge stream number index (distribution moments) oversize particle stream product size particle stream recycle particle stream undersize particle stream fluid feed streams
Literature Cited
Beke, Bela, “Principles of Comminution,” pp. 44, 63, 120, Akademiai Kiado, Budapest, 1964. Bouloucon, P., American Cyanamid Co., Wayne, N. J., private communication. 1968. Cadle, R. “ P a h l e Size. Theory and Industrial Applications,” pp. 10, 28, 33, 40, Reinhold, New York, 1965, Capes, C. E., Ind. Eng. Chem. Proc. Des. Develop. 6 , 390 (1967a). Capes, C. E., Trans. Znst. Chem. Eng. 45, CE78 (196713). Capes, C. E., Danckwerts, P. V., Trans. Znst. Chem. Eng. 43, T116 , I V ” V , .
Davis, C. H., Meline, R. S.,Graham, H. G., Chem. Eng. Progr. 64,
The authors express their thanks to T. P. Hignett, Tennessee Valley Authority, Muscle Shoals, Ala., for permission to use their diammonium phosphate pilot plant data in this study. Our thanks go, also, to Charles H. Davis, of the same organization, for providing the material. Nomenclature
parameter in residence time distribution function as given by Equation 50 D = particle size = minimum particle size of crusher output DC D o = geometric mean particle size, by weight = geometric mean particle size, by number = particle size dktribution function. Thus: KO. of particles of size between D and ( D d D ) f(D)dD = (unit mass of stream) particle size distribution function of recycle stream redefined per unit mass of granulator discharge granulator residence time distribution function crusher proportionality constants mass flow rate of a particle stream product mass flow rate particle flow rate of particle stream number of particles per unit mass of particle stream
Yo
+
(1QGK)
Acknowledgment
a
GREEKLETTERS fraction of product size material recycled zeroth moment of oversize portion of granulator discharge particle size distribution zeroth moment of product size portion of granulator discharge particle size distribution random variable of particle size particle growth rate third moment of product size portion of granulator discharge particle size distribution fraction of granulator discharge stream withdrawn as product moment of unnormalized particle size distribution function moment of normalized particle size distribution function particle volume shape factor density of particle material third moment of oversize portion of granulator discharge particle size distribution normalized particle size distribution function. Thus: + ( D ) d D = fraction of particles of size between D and ( D dD) arithmetic standard deviation geometric standard deviation residence time in granulator. Also random variable of residence time
=
+
75 11968). Farnand, J. R., Smith, H. M., Ruddington, 1:. E., Can. J .. Chem. Eng. 39,94 (1961). Han, C. D., Chem. Eng. Sci. 22, 611 (1967). Han, C. D., Chem. Eng. Sci. 23, 321 (1968). Han, C. D., Chem. Eng. Sci. 25, 875 (1970). Herdan, G,., “Small Particle Statistics,” 2nd ed., PP. 81 et seq., Academic Press. 1960. Hignett, T. P., A$. Chem. 12, 30 (1957). Horn, W. R., Fouser, N. D., Chem. Eng. Progr. 62, 109 (1966). Kottler, F., J . Franklin Znst. 250, 339 (1950). hIiskell, F., llarshall, W. It.,Chem. Eng. Progr. 52, 35-5(1956). Xewitt, D. AI., Conway-Jones, J. JI., Trans. Znst. Chem. Eng. 36,423 (1958). Perry, J. H., “Chemical Engineers’ Handbook,” section 8, “Size Reduction and Size Enlargement,” 4th ed., NcGraw Hill, Yew York, 1963. Saeman, W. C., Chem. Eng. Progr. 47, 508 (1951). Schofield, F. R., Glikin, P. G., Trans. Inst. Chem. Eng. 42, 183 f,----,. lQG3) Sutherland, J. P., Can. J . Chem. Eng. 40, 268 (1962). Turner, G. A., Can. J . Chem. Eng. 44, 13 (1966). Young, R. D., Hicks, G. C., Davis, C. H., J . Agr. Food Chem. 10, 442 (1962). RECEIVED for review July 28, 1969 ACCEPTEDJune 5, 1970 Taken in part from the dissertation of I. Wilenitz, submitted to the Faculty of the Polytechnic Institute of Brooklyn in partial fulfillment of the requirements for the degree of doctor of philosophy, 1970. Ind. Eng. Chem. Fundam., Vol. 9, No.
3, 1970 41 1
