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Steady-state behavior of continuous granulators — an elementary mathematical analysis
Pergamon Press.
Chemical Engineering Science, 1970, Vol. 25, pp. 875-883.
Printed in Great Britain.
Steady-state behavior of continuous granulators - an elementary mathematical analysis Department
CHANG DAE HAN of Chemical Engineering, Polytechnic Institute of Brooklyn, 333 Jay Street, Brooklyn, N.Y. II2OI,U.S.A. (First received
23 May
1969; accepted
20 October
1969)
Abstract-A
mathematical analysis of the steady-state behavior of a continuous granulator is presented. The analysis is based on the following main assumptions that granules grow at a uniform rate, independent of granule size, and that the crusher is the main source of new granule formation. Special attention is given to the balance of the number of granules in the process loop, and the role of the crusher which receives the oversized granules from the sieve is emphasized. Some numerical results from the model developed are given; these are the effect of the granule size of the crusher discharge stream on the recycle ratio of granules (the ratio of the recycle rate to the production rate), and the effect of the granule size of the crusher discharge stream on the fraction of the on-sized granules to be recycled. 1. INTRODUCTION GRANULATION
processes are becoming more and more important in the chemical and pharmaceutical industries, to name only two. Although granulation has been employed for the past few decades as an industrial manufacturing process, in particular in the fertilizer industry, it has received very little attention from the point of view of process analysis. At a first glance this kind of process may look rather unsusceptible to an analytical treatment. However, recent developments of the chemical engineering technology in this and related areas appear to have given us a better understanding of the main features of the granulation process. Several authors, (Brook [I], Hignett [5], Hignett [6]), have discussed general techniques of manufacturing fertilizer granules. A literature survey however indicates that continuous operation of industrial granulators suffers from, for instance, a sudden increase in recycle rate which leads very frequently to plant shut-down. Control of the recycle rate seems to be related to the control of the ratio of the rate of liquid to solid and the crushing operation; the former relates to the growth of granules and the latter to the formation of new granules. Of course, control of the recycle rate is one of many important
problems encountered in the operation of granulation processes. The purpose of this paper is to present an elementary mathematical analysis of a granulation process, in which the main assumptions made are that granules grow at a uniform rate, i.e., independent of granule size, and that the crusher is the main source of new granules. Special attention is given to the balance of the number of granules in the process loop, and the role of the crusher which receives the over-sized granules is emphasized. Also assumed is the plug flow of granules through the granulator, an assumption which very much simplifies the mathematical treatment. However, the removal of this assumption is not difficult and an analysis taking into account the residence time distribution of granules in the granulator is under way and this will be published in a forthcoming publication. 2. DESCRIPTION
OF PROCESS
Consider a typical granulation process as sketched in Fig. 1. The granulator exit stream goes to the screen through which granules are classified into three streams: under-size granules which pass both the upper and lower sieve, onsized granules which pass the upper sieve but
875
HAN
----_-
L,Q”lO
STREAM
Sam
STREAM
GM
STREAM
Fig. 1. Schematic diagram of a typical granulation process.
are retained on the lower sieve, and over-sized granules which are retained on the upper sieve. The over-sized granules are crushed in the crusher and the crusher discharge stream is recycled to the inlet of the granulator. Although it depends on the particular process, in some cases a part of the on-sized granules is also recycled. In other cases, a part of the on-sized granules is also crushed and sent to the granulator inlet as a part of the recycle stream, although this is not shown in Fig. 1. The continuous granulation process has a certain similarity to the continuous crystallization process in the sense that both new particle formation and the growth of existing particles should occur in the process loop, characterizing the granule size distribution of the granulator exit stream. Therefore the rate expression of both new particle formation and growth must be assigned to solve the material balance equation. Under steady-state conditions, the number of granules in the granulator should be constant at all times. These granules have a definite size distribution and a steady state is reached if the rate of new granule formation and growth does not change with time. The rate of new granule formation must be equal to the rate at which granules are withdrawn from the granulator as a final product. In the past, several authors (Newitt and Conway-Jones [7], Farnand, Smith and Puddington [4], Sutherland [8], Capes and Danckwerts[2], Capes[3] have examined the mechanism of granule growth, but little attention has been given to the mechanism of new granule formation. Two kinds of growth mechanism may be
considered in the granulation process. One is the “Onion-skin mechanism”, in which granules are assumed to grow by deposition of successive layers, each layer formed by picking up a coating of liquid feed, which subsequently dries (for example, the diammonium phosphate process). Another growth mechanism is the “Snow-ball mechanism”, in which recycling granules agglomerate as a result of cohesive forces in the presence of a liquid medium (for example, the triple superphosphate process). The use of a liquid medium provides a great increase in contact area by carrying dissolved or suspended material into the voids. The important contribution of the liquid phase is due to a combination of its surface tension and its high mobility. Two sources of new granule formation may be considered in general, regardless of the kind of process; one is to be found inside the granulator, and the other one is from outside the granulator, namely the crusher. The former source may be attributed to the attrition of tumbling granules within the granulator, the degree of attrition depending on the loading of the granulator, the rotational speed of thegranulator and the physical properties of the granule itself. On the other hand, the latter is due to the crushing operation of the crusher. From the limited experience of the author, the latter source of new granule formation appears to be a quite important one, which has received very little attention in the past in most industrial granulation processes, in particular, in the manufacturing of granular fertilizers. In the present study, we are concerned with the process which follows the “Onion-skin mechanism” of granule growth and we consider the crusher as the main source of new granule formation. 3. MATHEMATICAL
DEVELOPMENT
In developing a mathematical model the granulation process shown in Fig. 1 is considered and the following assumptions are made. (a) The granule growth rate is uniform, i.e., granules grow at the same rate, independent of their size.
876
f-i il;i
(b) Granules travel through the granulator at a constant forward velocity. (c) The crusher is the predominant source of new granule formation, i.e., the formation of new granules due to the attrition of granules within the granulator is assumed to be negligibly small compared with that due to crushing. (d) The granule size from the crusher discharge is uniform and it is smaller than the smallest granule size of the granulator discharge stream. (e) Sieve efficiency is 100 per cent.
f,?(D)
-1 - -- --
I ;
I ’ 7-
__I -
-
-
- _
-
I ’
.L’L
L
9
3.1 Density function of granule size distribution Throughout the system, there are a number of streams of granules, for example recycle, crusher output, etc. Each has its own particle size distribution, which is a function which determines the number of particles of every size to be found in a given sample of the stream. In order to relate all streams to a common basis, we define a particle size distribution as follows:
Fig. 2. The size distribution
L
iii
L
L
D
?J
of&(D)
and_&(D).
of the lower sieve, and D, is the opening of the upper sieve. The number of “stairs” between DI, and D, depends on the granule growth rate per pass. It is seen from Fig. I that the stream entering the granulator is the combination of three separate streams, i.e., MD)
=.MD; D = DL) +q&(D; + MD
f(D) = number of particles of size D in a stream (unit time) (unit volume of one particular stream)’
DL < D c D,)
- DA
(2)
in which, on the right hand side, the first term represents the under-sized granules, the second term represents the on-sized granules where CY is the fraction of the on-sized granules recycled, and the third term represents the crushed oversized granules in which S is the delta function having the following property:
For convenience, the particular stream we select here is the granulator discharge. It is to be understood that when we speak of the unit volume of a stream of solid particles we mean intrinsic not bulk volume. Based on assumptions (a) and (b), one can relate the size distribution of granules of the stream leaving the granulator to that entering the granulator by .&(D - L) =.&CD)
to(D)
I 1 I I I I
@D--D,)
1;D z= D, = O;D < D,
(3)
p in Eq. (2) represents the total number of crushed granules per unit time per unit volume of granulator discharge which can be obtained by
(1)
in which D is the granule size here taken as equivalent diameter and L is the increment in size of granule per pass. Equation (I) determines the shape of the granule size distribution as shown in Fig. 2, where D, is the granule size of the crusher discharge stream, DL is the opening
P=
&I,>, DYdD)dD ”
and it is the height of the distribution function f&D; D, =s D < D,) shown in Fig. 2. Since granules grow with a finite incremental
877
C. D. HAN
diameter L per pass, the following “integers” m and n may be defined in terms of D,, DL, and D,:
k--D,
m=
[ n=
[
L
b-41 1
(5)
{D,+mL}
s D,, < {D,+(m+
l)L}
{D,+nL}
c D, < {D,+(n+
1)L).
one can obtain by substituting (4): a”-‘”
~I~(~-DC-iL)+p
y
1)L 3=
1
(11)
Equation ( 1 1) is a very important relation from which L can be calculated for given values of D,, D,,, D, and (Y since integers m and n are defined by Eq. (5).
&-m--l
i=m+1
xS(D-DC-iL)
(6)
and use of Eqs. (I-6) gives m-1
fR(D) = /3 zO 6(D - D, - iL) +/3 i cG’+’ i= m
x6(0-DC-iL).
(7)
From Eq. (6) one obtains
3.2 Formation of new granules through crusher According to assumption tc) that crushing is the predominant source of new granule formation, the crusher should be operated in such a way that at steady-state the number of granules to be created in the crusher should be the same as the number of granules to be withdrawn as a final product; i.e., Number of granules = in product stream
D,)=pgS(D-DC-iL)
(8)
i=l
&(D;
D,+(n+
Eq. ( IO) into Eq.
DC
The size distribution of granules fG’(D) depicted in Fig. 2 may be represented by
fc(D;Ds
fc(D; D > 0,) = @a!“-“WD - D, - (n + l)L) (10)
L
in which the square bracket denotes the “integer”, and therefore one can write
fC(~)=p
according to Eq. (7). Therefore the size distribution function of granules, either Eq. (6) or Eq. (7) since one can be derived from another, are proved to describe what is shown in Fig. 2. Since it follows from Eq. (6)
D,* < D s D,) = p
2
Number of granules Number of granules from crusher to the crusher.
c?--‘~
i=m+l
XF(D-D-iL)
(9)
and thus making use of Eqs. (8) and (9) in the right hand side of Eq. (2) gives ptzs(D-D,--iL)+p
5
Knowing the size distribution of granules given by Eq. (6), the net number of new granules formed in the crusher can be calculated by either 1
D3f;(D)dD Dc3 JD>Du
&%(D-DC-iL)
i=m+l
=pc1 -(Y n--m)
+@CD - D,)
(12)
or
m-1
= p izO 6(D -DC - iL) + p i
cP%(D - D, - iL)
(1 -(Y) I,,,,,,,,“fG(D)dD
i=m
= fR(D)
where p is defined by Eq. (4). 878
= PC1-@+)
!13)
Steady state behavior of continuous granulators
3.3 Overall material balance At steady state the total feed rate in form of a liquid should be equal to the rate of granule product, and this may be expressed by F = ( 1 -a)
JD
Y
total recirculating rate of granules to the rate of production withdrawn as a final product, which is expressed by Recycle ratio = E
(14)
,k3
where F is the volumetric feed rate of solute per unit volume of granulator discharge, and v is the shape factor of a granule. Substituting Eq. (6) into the right-hand side of Eq. ( 14) and integrating, one obtains the result F =p(l
--a)~
=
PG3
=
--f,(D))dD.
-pR3
=
3L,.&‘.
(20)
D3f,(D)dD = /3 2 (D, + iL)3 i=l
i=
ai-~tr~l(Dc
I: 111+
+
iL)S
(21)
1
ll,-1 pR2
=
/
a,,D
+P
( 17)
(18)
Wi(DW
=
/3
E
(D,
+
iL12
i=O
Ii &“‘(DC
+ iL)2.
i=nf
4. RESULTS
(16)
3 5 Further useful relations One of the most important operating variables is the recycle ratio, defined as the ratio of the
Ii (Y~-“YD~+ iL)3
i=nr
a,,D
s
+P
in which f$D> is the first derivative of f&D) with respect to D. Equation (17) can then be expressed in terms of moments of the distribution functions of& andf, by k3
i=O
n+1
The left-hand side of Eq. (16) represents the total feed rate of solute and the right-hand side represents the net increase of mass within the granulator due, to granule growth by the postulated “Onion-skin mechanism”. Using Eq. (I), the right-hand side of Eq. (16) can be further rewritten as:
x P&(D)dD
D3fR(D)dD = p z (DC+ iL)”
a,,D
/
+P
3.4 Material balance around the granulator The material balance around the granulator may be obtained by
vD3(jdD) sm
(19)
pR3
m-1
pR3
Equation (15) is a useful relation for obtaining the quantitative nature of the size distribution, Eq. (6), since p can be calculated from it.
F=
-
where
c-u~-“‘-~(D,+~L)~. (15)
2
i=m+l
z sR2
PR3
=
AND
(22)
DISCUSSION
Equation ( 1 1) is a most fundamental relation, which allows us to calculate, for instance, the incremental size of granule per pass (L) as a function of the fraction of on-sized granules to be recycled (cu) for given sieve specifications (D, and &,) and crusher operation (D,). However, from the computational point of view it is rather difficult to find L from Eq. ( 11) because L appears in Eq. (5) which defines the two integers m and n. Therefore, in the present study, for convenience (Y was computed as a function of D, for fixed values of L, D,, and D,. Some of the computed results are shown in Figs. 3 and 4. The exact shape of the size distribution function f&D) can be obtained by determining p from Eq. (15) when the feed rate F and the shape factor I, are specified. In Fig. 5 is shown the computed recycle ratio as a function of D, which was obtained using Eqs. (19-22). In the numerical computation granules are assumed to be spherical, i.e., v = 7r/6. It is seen from Fig. 3 that for a given crusher operation (a fixed D,) the narrower the sieve
879
C. D. HAN k0 SINE SPECIFICATION: 7-8 MESH ID,. ~8mrn. DL. 2.37mm I FEED RATE: 30 T/DAY Ip =k6pr./ccl
L 406mn
FEEDRATE:
3OT/DAY(p~ffi.ykc)
0”. 2.28 mm
04-
06 -
iO.’ 8 = l
06-
I P
;;i 0.5 5 w 0.4 a sl Y + @3b [0.2-
04 -
0.1 -
04 -
0”“’
DO 02
04
8
0’
0 ” ’ ” IA 06 06 IO I~2 t4 M m 20 22 z4 26 PARTICLE SIZE FROM THE CRUSHER OUTLET D&n,
oc
Fig. 4. Steady-state
Fig. 3. Effect of the sieve specification on (Yand D,.
specification, the smaller the fraction of onsized granules to be recycled ((u), and for a given sieve specification the coarser the granules from the crusher (larger D,), the larger the fraction of on-sized granules to be recycled. In these figures D,* is the allowable upper limit of the granule size from the crusher outlet according to assumption (d). Of course D,* depends on the specification of the lower sieve DL. These results are as may be expected from actual plant experience. This is because when the crushed granules are coarser, the number of granules generated in the crusher is smaller and thus a larger fraction of on-sized granules must be recycled to provide the constant number of granules needed in the combined recycle stream at the granulator inlet. As a result, more oversized granules will be produced and thus a larger number of granules will be generated by the crusher. This is of course under the assump-
I
I
04 PARTlCL~SlZE
F&
I
TM
C%IWER
I
30 24 OUTLET D~~WIII
1
2.8
growth of granules upon the choice of (Yand D,.
tion of constant growth rate. In the actual operation of granulators one could adjust both LY and D, and then L is determined accordingly, as shown in Fig. 4. For instance, if (Yis fixed, then L depends on D,; that is, the coarser the crushed granules, the larger the growth rate (larger L) for a fixed feed rate because fewer granules are available at the inlet to the granulator. Another important operating variable is the recycle ratio. For a given sieve specification and fixed values of (Y and D,, the growth rate (or L) is uniquely determined according to Eq. ( 1 I), as shown in Fig. 4. Consequently, the recycle ratio is also uniquely determined according to Eq. (19). If the recycle ratio starts to increase, either of two corrective actions may be attempted; these are (i) to decrease the fraction of the onsized granules to be recycled (a), and (ii) to increase the feed rate. Here note that the growth
880
Steady state behavior of continuous granulators
Sieve rpcificoiion: 7-IOmwh (D, =2%1m, Food Role: SOT/day (P.l~Sg’lcC) L=008mm
DL=l65mm)
0”=I57mm 1
then the recycle ratio can be arbitrarily adjusted independently of either (Yor D,. This requires a more complicated analysis which is not considered in the present study. The second corrective action, in increasing the feed rate with fixed cz and D,, will initially increase the growth rate (or L). However, an excessive feed rate can introduce agglomeration of granules rather than the layer build-up of granules. Therefore there appears to be a maximum allowable L which still gives the onion skin growth mechanism. Increased growth rate by the additional feed will shift the size distribution of the granulator discharge stream, giving more over-sized granules to pass through the crusher. Therefore the speed of the crusher should be adjusted so that the number of particles is maintained (constant), which can be accomplished only by producing coarser granules. Therefore an increase in the feed rate should also require additional adjustment of either the operation of the crusher (D,) or the fraction of on-sized granules to be recycled (a).
1 L&++--J I6
I4
2-O
2.2
24
2.6
PARTICLE SIZE FROM THE CRUSHER OUTLET 0, fmm)
Fig. 5. Steady-state recycle ratio upon the choice of a and D,.
rate is assumed to be uniform, independent of granule size. The first corrective action should accompany a decrease of the particle size from the crusher outlet (D,) according to Fig. 4 as an example, otherwise the conservation of the number of the particle in the system will be destroyed. In other words, more particles will be withdrawn from the system as a final product than the new granules formed in the crusher. Therefore in this case an increase of the speed of the crusher will be necessary to generate the finer particles from the crusher, which naturally brings more new particles so that the number of particles can be maintained constant in each stream. Upon the choice of 1y and D, the increment in size of granule per pass (L) will be fixed, and then the recycle ratio is determined, as shown in Fig. 5. Therefore, in order to adjust the recycle ratio, control of both (Yand D, in a predetermined way according to Fig. 4 for instance is necessary, otherwise the plant will never reach the steady state. It may also be worth mentioning that if a part of the on-sized granules is crushed to adjust the steady-state number of granules, in the system,
5. CONCLUDING REMARKS An attempt at describing the rather complicated granulation process under certain assumptions has resulted in quite valuable information on the relationships between the operating variables. The results obtained in this study do not seem to be obvious from intuition or operating experiences. For instance, in common practice, an operator would attempt to adjust only the fraction of on-size granules to be recycled ((u) when the recycle ratio starts to increase, as opposed to the result of the present study suggesting that both cx and D, should be adjusted at the same time. This rather surprising result comes mainly from the consideration of a constant number of granules in the system. It appears that the concept of new granule formation in granulation plants has been overlooked, at least in the process where the onion skin growth mechanism is applicable, for instance in the production of diammonium phosphate (DAP). It is emphasized in this study that the role of the crusher is rather important from the point of view of 881
CESVOL.25.
NO. 5-I
C. D. HAN
generating new granules and consequently for controlling the size distribution of the various process streams. Continuation of the present study by removing some of the assumptions made in it, namely the plug flow of granular materials in the granulator and uniform granule size from the crusher, is currently under way by introducing the residence time distribution of granules in the granulator and the size distribution of granules from the crusher. Also a study on the dynamic and control aspects of the granulation process is under way. These will be subjects of future publications. Mathematical modelling, although simplified in this study, has much to offer as a scientific method of simulating the rather complicated behavior of industrial granulators. The results obtained here give some insights into this cornplicated process, which otherwise would be very difficult to understand, and future studies promise still further insights. Acknowledgment-The author wishes to express his acknowledgement to S. Katz and R. Shinnar for helpful discussions at the inception of the study at the American Cyanamid Company. NOTATION
D D,
the granule size taken as equivalent diameter the granular size of the crusher discharge stream
the opening of the lower sieve the opening of the upper sieve no. of particles of size D in a stream per unit time per unit volume of one particular stream fc(D) the size distribution of granules of the stream leaving the granulator ~RW the size distribution of granules of the stream entering the granulator F the volumetric feed rate of solute per unit volume of granulator discharge L the increment in size of granule per pass DL
fpD;
m
integer as defined by DL-D, L [
1 1
D -D n integer as defined by 7 [ Greek symbols __ the fraction of the on-sized granules reU cycled P the total number of crushed granules per unit time per unit volume of granulator discharge shape factor of a granule ; dirac delta function as defined by Eq. (3) the third moment off,(D) as defined by PC3 S,” D%(DW the third moment offR(D) as defined by PR3 S,” WR(DW the second moment off,(D) as defined by PR2 I,” D’f,(DW
REFERENCES
[II BROOK A. T., Proc. No. 47, The Fertil. Sot. 1957. PI CAPES C. E. and DANCKWERTS P. V., Trans. Instn them. Engrs 1965 43 Tl16. [31 CAPES C. E., Ind. Engng Chem. Dev. 1967 6 390. I. E., dan.J. them. Engng 196139 94. [41 FARNAND J. R., SMITH H. M. and PUDDINGTON PI HIGNETT T. P., General considerations on operating techniques, equipment and practices in manufacture of granular mixed fertilizers, in Chemistry and Technology ofFertilizers, (Edited by SAUCHELLI V.). Reinhold 1960. P., Fm Chem. 1963 126 34 (Jan.); 1963 126 14 (Feb.); 1963 126 30(March); 1963 126 32 (April). [61 HIGNETTT. [71 NEWITT D. M. and CONWAY-JONES J. M., Trans. Znstn them. Engng 1958 36 422. 181 SUTHERLAND J. P., Can. J. them. Engng 1962 40 268. R&me - Une analyse mathematique du comportement a l’etat stable dun granulateur en continu est presentee. L’analyse est basee sur les hypotheses principales par lesquelles les granules augmentent a un taux uniforme, independant de leur dimension, et que le broyeur est la principale source de formation de nouveaux granules. On p&e une attention particuliere a l’equilibre du nombre de granulCs dans la boucle du pro&de, et on insiste sur le role du broyeur quc recoit les granules hors dimension du tamis. Certains r&mats numiriques obtenus a partir du modtle dtveloppe sont foumis; ces resultats sont I’effet de,la taille des granules du courant de decharge du broyeur sur le taux de recyclage des gram&% (le rapport du taux de recyclage au taux de production), et aussi l’effet de la taille des granules du courant de d&charge du broyeur sur la fraction des granulks “on-sized” Arecycler.
882
Steady
state behavior
of continuous
granulators
ZusammenfassungEs wird eine mathematische Anaiyse des Verhaltens eines kontinuierlichen Granulators im stationaren Zustand dargelegt. Die Analyse basiert auf den haupts2ichlichen Annahmen, dass die Granula mit gleichmassiger Geschwindigkeit unabhingig von der Kijmchengrosse wachsen, und dass der Zerkleinerer die Hauptursache der Bildung neuer Kiimchen ist. Besondere Aufmerksamkeit wird dem Gleichgewicht der Anzahl von Granula in der Prozessschleife gewidmet, und die Rolle des Zerkleinerers, der die iibergrossen Granula vom Sieb erh5lt wird betont. Es werden einige numerische Ergebnisse, die mit dem Model1 erhahen wurden angegeben; diese beziehen sich auf den Effekt der K6mchengriisse des Abgabestroms des Zerkleinerers auf das Umlaufverhaltnis der Kiirnchen (das Verhahnis der Umlaufgeschwindigkeit zur Bildungsgeschwindigkeit), und den Effekt der Kiimchengrijsse des Abgabestroms des Zerkleinerers auf den Bruchteil der Kiimchen richtiger Griisse, die in den Umlauf gelangen.
883
Chemical Engineering Science, 1970, Vol. 25, pp. 875-883.
Printed in Great Britain.
Steady-state behavior of continuous granulators - an elementary mathematical analysis Department
CHANG DAE HAN of Chemical Engineering, Polytechnic Institute of Brooklyn, 333 Jay Street, Brooklyn, N.Y. II2OI,U.S.A. (First received
23 May
1969; accepted
20 October
1969)
Abstract-A
mathematical analysis of the steady-state behavior of a continuous granulator is presented. The analysis is based on the following main assumptions that granules grow at a uniform rate, independent of granule size, and that the crusher is the main source of new granule formation. Special attention is given to the balance of the number of granules in the process loop, and the role of the crusher which receives the oversized granules from the sieve is emphasized. Some numerical results from the model developed are given; these are the effect of the granule size of the crusher discharge stream on the recycle ratio of granules (the ratio of the recycle rate to the production rate), and the effect of the granule size of the crusher discharge stream on the fraction of the on-sized granules to be recycled. 1. INTRODUCTION GRANULATION
processes are becoming more and more important in the chemical and pharmaceutical industries, to name only two. Although granulation has been employed for the past few decades as an industrial manufacturing process, in particular in the fertilizer industry, it has received very little attention from the point of view of process analysis. At a first glance this kind of process may look rather unsusceptible to an analytical treatment. However, recent developments of the chemical engineering technology in this and related areas appear to have given us a better understanding of the main features of the granulation process. Several authors, (Brook [I], Hignett [5], Hignett [6]), have discussed general techniques of manufacturing fertilizer granules. A literature survey however indicates that continuous operation of industrial granulators suffers from, for instance, a sudden increase in recycle rate which leads very frequently to plant shut-down. Control of the recycle rate seems to be related to the control of the ratio of the rate of liquid to solid and the crushing operation; the former relates to the growth of granules and the latter to the formation of new granules. Of course, control of the recycle rate is one of many important
problems encountered in the operation of granulation processes. The purpose of this paper is to present an elementary mathematical analysis of a granulation process, in which the main assumptions made are that granules grow at a uniform rate, i.e., independent of granule size, and that the crusher is the main source of new granules. Special attention is given to the balance of the number of granules in the process loop, and the role of the crusher which receives the over-sized granules is emphasized. Also assumed is the plug flow of granules through the granulator, an assumption which very much simplifies the mathematical treatment. However, the removal of this assumption is not difficult and an analysis taking into account the residence time distribution of granules in the granulator is under way and this will be published in a forthcoming publication. 2. DESCRIPTION
OF PROCESS
Consider a typical granulation process as sketched in Fig. 1. The granulator exit stream goes to the screen through which granules are classified into three streams: under-size granules which pass both the upper and lower sieve, onsized granules which pass the upper sieve but
875
HAN
----_-
L,Q”lO
STREAM
Sam
STREAM
GM
STREAM
Fig. 1. Schematic diagram of a typical granulation process.
are retained on the lower sieve, and over-sized granules which are retained on the upper sieve. The over-sized granules are crushed in the crusher and the crusher discharge stream is recycled to the inlet of the granulator. Although it depends on the particular process, in some cases a part of the on-sized granules is also recycled. In other cases, a part of the on-sized granules is also crushed and sent to the granulator inlet as a part of the recycle stream, although this is not shown in Fig. 1. The continuous granulation process has a certain similarity to the continuous crystallization process in the sense that both new particle formation and the growth of existing particles should occur in the process loop, characterizing the granule size distribution of the granulator exit stream. Therefore the rate expression of both new particle formation and growth must be assigned to solve the material balance equation. Under steady-state conditions, the number of granules in the granulator should be constant at all times. These granules have a definite size distribution and a steady state is reached if the rate of new granule formation and growth does not change with time. The rate of new granule formation must be equal to the rate at which granules are withdrawn from the granulator as a final product. In the past, several authors (Newitt and Conway-Jones [7], Farnand, Smith and Puddington [4], Sutherland [8], Capes and Danckwerts[2], Capes[3] have examined the mechanism of granule growth, but little attention has been given to the mechanism of new granule formation. Two kinds of growth mechanism may be
considered in the granulation process. One is the “Onion-skin mechanism”, in which granules are assumed to grow by deposition of successive layers, each layer formed by picking up a coating of liquid feed, which subsequently dries (for example, the diammonium phosphate process). Another growth mechanism is the “Snow-ball mechanism”, in which recycling granules agglomerate as a result of cohesive forces in the presence of a liquid medium (for example, the triple superphosphate process). The use of a liquid medium provides a great increase in contact area by carrying dissolved or suspended material into the voids. The important contribution of the liquid phase is due to a combination of its surface tension and its high mobility. Two sources of new granule formation may be considered in general, regardless of the kind of process; one is to be found inside the granulator, and the other one is from outside the granulator, namely the crusher. The former source may be attributed to the attrition of tumbling granules within the granulator, the degree of attrition depending on the loading of the granulator, the rotational speed of thegranulator and the physical properties of the granule itself. On the other hand, the latter is due to the crushing operation of the crusher. From the limited experience of the author, the latter source of new granule formation appears to be a quite important one, which has received very little attention in the past in most industrial granulation processes, in particular, in the manufacturing of granular fertilizers. In the present study, we are concerned with the process which follows the “Onion-skin mechanism” of granule growth and we consider the crusher as the main source of new granule formation. 3. MATHEMATICAL
DEVELOPMENT
In developing a mathematical model the granulation process shown in Fig. 1 is considered and the following assumptions are made. (a) The granule growth rate is uniform, i.e., granules grow at the same rate, independent of their size.
876
f-i il;i
(b) Granules travel through the granulator at a constant forward velocity. (c) The crusher is the predominant source of new granule formation, i.e., the formation of new granules due to the attrition of granules within the granulator is assumed to be negligibly small compared with that due to crushing. (d) The granule size from the crusher discharge is uniform and it is smaller than the smallest granule size of the granulator discharge stream. (e) Sieve efficiency is 100 per cent.
f,?(D)
-1 - -- --
I ;
I ’ 7-
__I -
-
-
- _
-
I ’
.L’L
L
9
3.1 Density function of granule size distribution Throughout the system, there are a number of streams of granules, for example recycle, crusher output, etc. Each has its own particle size distribution, which is a function which determines the number of particles of every size to be found in a given sample of the stream. In order to relate all streams to a common basis, we define a particle size distribution as follows:
Fig. 2. The size distribution
L
iii
L
L
D
?J
of&(D)
and_&(D).
of the lower sieve, and D, is the opening of the upper sieve. The number of “stairs” between DI, and D, depends on the granule growth rate per pass. It is seen from Fig. I that the stream entering the granulator is the combination of three separate streams, i.e., MD)
=.MD; D = DL) +q&(D; + MD
f(D) = number of particles of size D in a stream (unit time) (unit volume of one particular stream)’
DL < D c D,)
- DA
(2)
in which, on the right hand side, the first term represents the under-sized granules, the second term represents the on-sized granules where CY is the fraction of the on-sized granules recycled, and the third term represents the crushed oversized granules in which S is the delta function having the following property:
For convenience, the particular stream we select here is the granulator discharge. It is to be understood that when we speak of the unit volume of a stream of solid particles we mean intrinsic not bulk volume. Based on assumptions (a) and (b), one can relate the size distribution of granules of the stream leaving the granulator to that entering the granulator by .&(D - L) =.&CD)
to(D)
I 1 I I I I
@D--D,)
1;D z= D, = O;D < D,
(3)
p in Eq. (2) represents the total number of crushed granules per unit time per unit volume of granulator discharge which can be obtained by
(1)
in which D is the granule size here taken as equivalent diameter and L is the increment in size of granule per pass. Equation (I) determines the shape of the granule size distribution as shown in Fig. 2, where D, is the granule size of the crusher discharge stream, DL is the opening
P=
&I,>, DYdD)dD ”
and it is the height of the distribution function f&D; D, =s D < D,) shown in Fig. 2. Since granules grow with a finite incremental
877
C. D. HAN
diameter L per pass, the following “integers” m and n may be defined in terms of D,, DL, and D,:
k--D,
m=
[ n=
[
L
b-41 1
(5)
{D,+mL}
s D,, < {D,+(m+
l)L}
{D,+nL}
c D, < {D,+(n+
1)L).
one can obtain by substituting (4): a”-‘”
~I~(~-DC-iL)+p
y
1)L 3=
1
(11)
Equation ( 1 1) is a very important relation from which L can be calculated for given values of D,, D,,, D, and (Y since integers m and n are defined by Eq. (5).
&-m--l
i=m+1
xS(D-DC-iL)
(6)
and use of Eqs. (I-6) gives m-1
fR(D) = /3 zO 6(D - D, - iL) +/3 i cG’+’ i= m
x6(0-DC-iL).
(7)
From Eq. (6) one obtains
3.2 Formation of new granules through crusher According to assumption tc) that crushing is the predominant source of new granule formation, the crusher should be operated in such a way that at steady-state the number of granules to be created in the crusher should be the same as the number of granules to be withdrawn as a final product; i.e., Number of granules = in product stream
D,)=pgS(D-DC-iL)
(8)
i=l
&(D;
D,+(n+
Eq. ( IO) into Eq.
DC
The size distribution of granules fG’(D) depicted in Fig. 2 may be represented by
fc(D;Ds
fc(D; D > 0,) = @a!“-“WD - D, - (n + l)L) (10)
L
in which the square bracket denotes the “integer”, and therefore one can write
fC(~)=p
according to Eq. (7). Therefore the size distribution function of granules, either Eq. (6) or Eq. (7) since one can be derived from another, are proved to describe what is shown in Fig. 2. Since it follows from Eq. (6)
D,* < D s D,) = p
2
Number of granules Number of granules from crusher to the crusher.
c?--‘~
i=m+l
XF(D-D-iL)
(9)
and thus making use of Eqs. (8) and (9) in the right hand side of Eq. (2) gives ptzs(D-D,--iL)+p
5
Knowing the size distribution of granules given by Eq. (6), the net number of new granules formed in the crusher can be calculated by either 1
D3f;(D)dD Dc3 JD>Du
&%(D-DC-iL)
i=m+l
=pc1 -(Y n--m)
+@CD - D,)
(12)
or
m-1
= p izO 6(D -DC - iL) + p i
cP%(D - D, - iL)
(1 -(Y) I,,,,,,,,“fG(D)dD
i=m
= fR(D)
where p is defined by Eq. (4). 878
= PC1-@+)
!13)
Steady state behavior of continuous granulators
3.3 Overall material balance At steady state the total feed rate in form of a liquid should be equal to the rate of granule product, and this may be expressed by F = ( 1 -a)
JD
Y
total recirculating rate of granules to the rate of production withdrawn as a final product, which is expressed by Recycle ratio = E
(14)
,k3
where F is the volumetric feed rate of solute per unit volume of granulator discharge, and v is the shape factor of a granule. Substituting Eq. (6) into the right-hand side of Eq. ( 14) and integrating, one obtains the result F =p(l
--a)~
=
PG3
=
--f,(D))dD.
-pR3
=
3L,.&‘.
(20)
D3f,(D)dD = /3 2 (D, + iL)3 i=l
i=
ai-~tr~l(Dc
I: 111+
+
iL)S
(21)
1
ll,-1 pR2
=
/
a,,D
+P
( 17)
(18)
Wi(DW
=
/3
E
(D,
+
iL12
i=O
Ii &“‘(DC
+ iL)2.
i=nf
4. RESULTS
(16)
3 5 Further useful relations One of the most important operating variables is the recycle ratio, defined as the ratio of the
Ii (Y~-“YD~+ iL)3
i=nr
a,,D
s
+P
in which f$D> is the first derivative of f&D) with respect to D. Equation (17) can then be expressed in terms of moments of the distribution functions of& andf, by k3
i=O
n+1
The left-hand side of Eq. (16) represents the total feed rate of solute and the right-hand side represents the net increase of mass within the granulator due, to granule growth by the postulated “Onion-skin mechanism”. Using Eq. (I), the right-hand side of Eq. (16) can be further rewritten as:
x P&(D)dD
D3fR(D)dD = p z (DC+ iL)”
a,,D
/
+P
3.4 Material balance around the granulator The material balance around the granulator may be obtained by
vD3(jdD) sm
(19)
pR3
m-1
pR3
Equation (15) is a useful relation for obtaining the quantitative nature of the size distribution, Eq. (6), since p can be calculated from it.
F=
-
where
c-u~-“‘-~(D,+~L)~. (15)
2
i=m+l
z sR2
PR3
=
AND
(22)
DISCUSSION
Equation ( 1 1) is a most fundamental relation, which allows us to calculate, for instance, the incremental size of granule per pass (L) as a function of the fraction of on-sized granules to be recycled (cu) for given sieve specifications (D, and &,) and crusher operation (D,). However, from the computational point of view it is rather difficult to find L from Eq. ( 11) because L appears in Eq. (5) which defines the two integers m and n. Therefore, in the present study, for convenience (Y was computed as a function of D, for fixed values of L, D,, and D,. Some of the computed results are shown in Figs. 3 and 4. The exact shape of the size distribution function f&D) can be obtained by determining p from Eq. (15) when the feed rate F and the shape factor I, are specified. In Fig. 5 is shown the computed recycle ratio as a function of D, which was obtained using Eqs. (19-22). In the numerical computation granules are assumed to be spherical, i.e., v = 7r/6. It is seen from Fig. 3 that for a given crusher operation (a fixed D,) the narrower the sieve
879
C. D. HAN k0 SINE SPECIFICATION: 7-8 MESH ID,. ~8mrn. DL. 2.37mm I FEED RATE: 30 T/DAY Ip =k6pr./ccl
L 406mn
FEEDRATE:
3OT/DAY(p~ffi.ykc)
0”. 2.28 mm
04-
06 -
iO.’ 8 = l
06-
I P
;;i 0.5 5 w 0.4 a sl Y + @3b [0.2-
04 -
0.1 -
04 -
0”“’
DO 02
04
8
0’
0 ” ’ ” IA 06 06 IO I~2 t4 M m 20 22 z4 26 PARTICLE SIZE FROM THE CRUSHER OUTLET D&n,
oc
Fig. 4. Steady-state
Fig. 3. Effect of the sieve specification on (Yand D,.
specification, the smaller the fraction of onsized granules to be recycled ((u), and for a given sieve specification the coarser the granules from the crusher (larger D,), the larger the fraction of on-sized granules to be recycled. In these figures D,* is the allowable upper limit of the granule size from the crusher outlet according to assumption (d). Of course D,* depends on the specification of the lower sieve DL. These results are as may be expected from actual plant experience. This is because when the crushed granules are coarser, the number of granules generated in the crusher is smaller and thus a larger fraction of on-sized granules must be recycled to provide the constant number of granules needed in the combined recycle stream at the granulator inlet. As a result, more oversized granules will be produced and thus a larger number of granules will be generated by the crusher. This is of course under the assump-
I
I
04 PARTlCL~SlZE
F&
I
TM
C%IWER
I
30 24 OUTLET D~~WIII
1
2.8
growth of granules upon the choice of (Yand D,.
tion of constant growth rate. In the actual operation of granulators one could adjust both LY and D, and then L is determined accordingly, as shown in Fig. 4. For instance, if (Yis fixed, then L depends on D,; that is, the coarser the crushed granules, the larger the growth rate (larger L) for a fixed feed rate because fewer granules are available at the inlet to the granulator. Another important operating variable is the recycle ratio. For a given sieve specification and fixed values of (Y and D,, the growth rate (or L) is uniquely determined according to Eq. ( 1 I), as shown in Fig. 4. Consequently, the recycle ratio is also uniquely determined according to Eq. (19). If the recycle ratio starts to increase, either of two corrective actions may be attempted; these are (i) to decrease the fraction of the onsized granules to be recycled (a), and (ii) to increase the feed rate. Here note that the growth
880
Steady state behavior of continuous granulators
Sieve rpcificoiion: 7-IOmwh (D, =2%1m, Food Role: SOT/day (P.l~Sg’lcC) L=008mm
DL=l65mm)
0”=I57mm 1
then the recycle ratio can be arbitrarily adjusted independently of either (Yor D,. This requires a more complicated analysis which is not considered in the present study. The second corrective action, in increasing the feed rate with fixed cz and D,, will initially increase the growth rate (or L). However, an excessive feed rate can introduce agglomeration of granules rather than the layer build-up of granules. Therefore there appears to be a maximum allowable L which still gives the onion skin growth mechanism. Increased growth rate by the additional feed will shift the size distribution of the granulator discharge stream, giving more over-sized granules to pass through the crusher. Therefore the speed of the crusher should be adjusted so that the number of particles is maintained (constant), which can be accomplished only by producing coarser granules. Therefore an increase in the feed rate should also require additional adjustment of either the operation of the crusher (D,) or the fraction of on-sized granules to be recycled (a).
1 L&++--J I6
I4
2-O
2.2
24
2.6
PARTICLE SIZE FROM THE CRUSHER OUTLET 0, fmm)
Fig. 5. Steady-state recycle ratio upon the choice of a and D,.
rate is assumed to be uniform, independent of granule size. The first corrective action should accompany a decrease of the particle size from the crusher outlet (D,) according to Fig. 4 as an example, otherwise the conservation of the number of the particle in the system will be destroyed. In other words, more particles will be withdrawn from the system as a final product than the new granules formed in the crusher. Therefore in this case an increase of the speed of the crusher will be necessary to generate the finer particles from the crusher, which naturally brings more new particles so that the number of particles can be maintained constant in each stream. Upon the choice of 1y and D, the increment in size of granule per pass (L) will be fixed, and then the recycle ratio is determined, as shown in Fig. 5. Therefore, in order to adjust the recycle ratio, control of both (Yand D, in a predetermined way according to Fig. 4 for instance is necessary, otherwise the plant will never reach the steady state. It may also be worth mentioning that if a part of the on-sized granules is crushed to adjust the steady-state number of granules, in the system,
5. CONCLUDING REMARKS An attempt at describing the rather complicated granulation process under certain assumptions has resulted in quite valuable information on the relationships between the operating variables. The results obtained in this study do not seem to be obvious from intuition or operating experiences. For instance, in common practice, an operator would attempt to adjust only the fraction of on-size granules to be recycled ((u) when the recycle ratio starts to increase, as opposed to the result of the present study suggesting that both cx and D, should be adjusted at the same time. This rather surprising result comes mainly from the consideration of a constant number of granules in the system. It appears that the concept of new granule formation in granulation plants has been overlooked, at least in the process where the onion skin growth mechanism is applicable, for instance in the production of diammonium phosphate (DAP). It is emphasized in this study that the role of the crusher is rather important from the point of view of 881
CESVOL.25.
NO. 5-I
C. D. HAN
generating new granules and consequently for controlling the size distribution of the various process streams. Continuation of the present study by removing some of the assumptions made in it, namely the plug flow of granular materials in the granulator and uniform granule size from the crusher, is currently under way by introducing the residence time distribution of granules in the granulator and the size distribution of granules from the crusher. Also a study on the dynamic and control aspects of the granulation process is under way. These will be subjects of future publications. Mathematical modelling, although simplified in this study, has much to offer as a scientific method of simulating the rather complicated behavior of industrial granulators. The results obtained here give some insights into this cornplicated process, which otherwise would be very difficult to understand, and future studies promise still further insights. Acknowledgment-The author wishes to express his acknowledgement to S. Katz and R. Shinnar for helpful discussions at the inception of the study at the American Cyanamid Company. NOTATION
D D,
the granule size taken as equivalent diameter the granular size of the crusher discharge stream
the opening of the lower sieve the opening of the upper sieve no. of particles of size D in a stream per unit time per unit volume of one particular stream fc(D) the size distribution of granules of the stream leaving the granulator ~RW the size distribution of granules of the stream entering the granulator F the volumetric feed rate of solute per unit volume of granulator discharge L the increment in size of granule per pass DL
fpD;
m
integer as defined by DL-D, L [
1 1
D -D n integer as defined by 7 [ Greek symbols __ the fraction of the on-sized granules reU cycled P the total number of crushed granules per unit time per unit volume of granulator discharge shape factor of a granule ; dirac delta function as defined by Eq. (3) the third moment off,(D) as defined by PC3 S,” D%(DW the third moment offR(D) as defined by PR3 S,” WR(DW the second moment off,(D) as defined by PR2 I,” D’f,(DW
REFERENCES
[II BROOK A. T., Proc. No. 47, The Fertil. Sot. 1957. PI CAPES C. E. and DANCKWERTS P. V., Trans. Instn them. Engrs 1965 43 Tl16. [31 CAPES C. E., Ind. Engng Chem. Dev. 1967 6 390. I. E., dan.J. them. Engng 196139 94. [41 FARNAND J. R., SMITH H. M. and PUDDINGTON PI HIGNETT T. P., General considerations on operating techniques, equipment and practices in manufacture of granular mixed fertilizers, in Chemistry and Technology ofFertilizers, (Edited by SAUCHELLI V.). Reinhold 1960. P., Fm Chem. 1963 126 34 (Jan.); 1963 126 14 (Feb.); 1963 126 30(March); 1963 126 32 (April). [61 HIGNETTT. [71 NEWITT D. M. and CONWAY-JONES J. M., Trans. Znstn them. Engng 1958 36 422. 181 SUTHERLAND J. P., Can. J. them. Engng 1962 40 268. R&me - Une analyse mathematique du comportement a l’etat stable dun granulateur en continu est presentee. L’analyse est basee sur les hypotheses principales par lesquelles les granules augmentent a un taux uniforme, independant de leur dimension, et que le broyeur est la principale source de formation de nouveaux granules. On p&e une attention particuliere a l’equilibre du nombre de granulCs dans la boucle du pro&de, et on insiste sur le role du broyeur quc recoit les granules hors dimension du tamis. Certains r&mats numiriques obtenus a partir du modtle dtveloppe sont foumis; ces resultats sont I’effet de,la taille des granules du courant de decharge du broyeur sur le taux de recyclage des gram&% (le rapport du taux de recyclage au taux de production), et aussi l’effet de la taille des granules du courant de d&charge du broyeur sur la fraction des granulks “on-sized” Arecycler.
882
Steady
state behavior
of continuous
granulators
ZusammenfassungEs wird eine mathematische Anaiyse des Verhaltens eines kontinuierlichen Granulators im stationaren Zustand dargelegt. Die Analyse basiert auf den haupts2ichlichen Annahmen, dass die Granula mit gleichmassiger Geschwindigkeit unabhingig von der Kijmchengrosse wachsen, und dass der Zerkleinerer die Hauptursache der Bildung neuer Kiimchen ist. Besondere Aufmerksamkeit wird dem Gleichgewicht der Anzahl von Granula in der Prozessschleife gewidmet, und die Rolle des Zerkleinerers, der die iibergrossen Granula vom Sieb erh5lt wird betont. Es werden einige numerische Ergebnisse, die mit dem Model1 erhahen wurden angegeben; diese beziehen sich auf den Effekt der K6mchengriisse des Abgabestroms des Zerkleinerers auf das Umlaufverhaltnis der Kiirnchen (das Verhahnis der Umlaufgeschwindigkeit zur Bildungsgeschwindigkeit), und den Effekt der Kiimchengrijsse des Abgabestroms des Zerkleinerers auf den Bruchteil der Kiimchen richtiger Griisse, die in den Umlauf gelangen.
883