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Mathematical model for the calculation of internal granule porosity
Powder
Technology.
33 (1982)
257
- 263
257
Mathematical Model for the Calculation of Internal Granule Porosity J. GVENIS, T. BLICKL?X and R. HAJDU Research Institute for Technical hem Zu.2. (HungaryJ (Received
March 1, 19s’;
Chemistry
of the Hungarian Academy
of Sciences.
Veszpr&n.
H-S201.
S&&z-
in revised form May 2.19S2)
SumMARY
A mathematical model was derived for the ca2cuZation of the average internal porosity value of granules produced in a fIkdised bed granulator. The mode2 allows the description of the variation of granrr2e porosity as a function of the granule size. Granules were formed in a batch-wise operated granulator from lactose, sodium nitrate, sodium chloride7 sand, polyethylene, two pharmaceutical products - Halidor@ and Chloricide@ powders - and glass beads, and their measured internal porosity values were compared with the calculated data. The measured and caiculated data agreed within t207o.
Ii\iTROfiUCTION
The quality of granules produced by the forced agglomeration of elementary particles can be described by their physical properties such as density, internal porosity, size distribution, mechanical strength, etc. [l] . One of the most important physical properties of a vule is its internal porosity, i.e. the total volume of the internal pores related to the volume of a granule. The internal granule porosity influences other properties, e.g. the density and mechanical strength_ Several methods are used for the determination of the internal granule porosity [I, 2] _ -4 simple and fast method was applied by Ormos [1] for granules of narrow size distribution_ This method is based on the difference existing between the total porosity of a heap of granules and a heap of non-porous re%rence material, the shape and size distribution of the examined granules and the reference material being the same. Ormos found that the internal porosity values of
various granules agreed within +-X0% with the results of control determinations. Moreover Ormos et al. 13 - 51 studied the variation of internal porosity of granules prepared from different powders. A fluidised bed granulator equipped with a blade stirrer was used in this experiment_ The authors determined the effects of the raw material quahty, of the state of the fluidised bed, etc., on the internal granule porosity and found that the porosity approaches a limit with increase of granule size. This limit is inversely affec&d by increasing the number of revolutions of the stirring blades. It was also found that the porosity of the granu!es depended on the type of stirrer used. The tendencies found in the analysis of the above-mentioned results led the authors to derive a mathematical mode1 for the prediction of the internal granule porosity.
PREMISE
AND
LIMITS
OF THE MODEL
(1) It is assumed that granule formation takes place via the adherence of single particles and that the inner structure of the granules remains unchanged during their growth. The primary particles form tetrahedrons in the granule; the first tetrahedron, formed by four elementary particles at the beginning of the granulation, grows by the successive adherence of primary particles and after a short time the granule becomes nearly spherical in shape. (2) By definition, the pore volume of a granule means the empty space enclosed by the adhering primary particles. ftegarding a tetrahedron formed by four spheres, the actual volume of the ‘empty space’ is hard to define due to the lack of a well-defined 0 Elsevier Sequoia/Printed
in The Netherlands
envelope surface_ This uncertainty is significant only in the case of very small granules. If the granules grow and their shape beeomes spherical, an envelope surface can be defined. (3) The primary particles can be charcterized by their mean particle size do_ (4) The granules sre assumed to be spherical with a shape factor as a measure of nonsphericity- It is assumed that the shape is independent of size_ (5) The granulation is an extremely complex process influenced by many factors. The mathematical model therefore consists of several param eters whose esact values have to be determined experimentally_
where uOis the volume In the case of s$erical
&T lJo=
-
OF THE MODEL
As was mentioned previously, four primary particles have to adhere to form a tetrahedron and to create a ‘pore’_ The pore formed by four tetrahedrahy arranged primary spherical pazticles (n = 4) is shown in Fig. 1_ The
d2b “P
+-c-y*
do
“0
Fig. 1. Pore formation in the case of tetrahedral arrangement of the particles, ~0, volume of the prima? particle; up, volume of the pore; do, diameter of the primary partic.e_
volume volume
of the pore (v,) is proportional of the primary particle (uO):
(1)
1: the next, i-e_ the fifth, priiary particle adheres to the previousIy mentioned tetrahedron, a new pore forms having the same volume; therefore the total pore volume (vi) in this case is zpu,
(la)
The ‘granule’ now contains five primary particles (n = 5)_ If further primary particles attach to the growing ‘granule’, the pore volume can be described as VP = (22 -3)(-Q
VP
(n--M n+(n-3)9
nuo
+
3lwo (R
-
=
3)u, (3)
where n is the number of primary particles in the given granule. Equation (3) is more or less valid for granules, even if the primary particles are not regular spheres and their sizes vary; moreover, if their contact with each other is not regular, the structure of the granule is somewhat looser_ The volume of a granule is the sum of the volumes of the primary particles and the pores: VP -I-
v,
(4)
If CT0is the mean particle size of the primary particles and d, the characteristic mean diameter of the granules with w. and w, as their respective shape factors, eqn. (4) may be written:
to the
L’p = gu,
“P* =
= +
of a given granule is (n -
v, v.0
by the primary
6
Thus, the porosity EP=
occupied
&r
VO=nvo=n-
v, =
\_-
(2)
6
and the total volume particles is
= DERIVATIOM
of the primary particle_ particles this volume is
(lb)
The actual values of d, and C& depend on both the selected characteristic mean diameter and the determination method used. For example, if an optical method is used for the determination of the particle diameter, the following possibilities arise_ The diameter can be determined using the equivalent circle having the same area as the projected particle, the algebraic or geometric mean of the smallest and largest diameters, or the diameter of the enclosing sphere. In the third case, the maximum value of the shape factor is 1. With sieve analysis, the granules are characterized by three diameters, e-g_ a, b and c, taken in the direction of the space coordinates Usually the values of these diam-
eters differ from each other, e-g_ a < b < c, where b is taken as the characteristic size of the granule. In this case the shape factor is greater than I_ The number of primary particles, n, forming a granule of diameter d, can be calculated from eqn. (6 jr
n=
&--[~($$+3y]
(6)
The ratio of the shape factors of primary particles and ihe granule gives a relative shape factor: &Jr= -0s (7) as If eqn_ (6) is substituted in eqn_ (3), the granule porosity takes the following form:
It is evid.ent from eqn. (8) that the granule porosity approaches a limit value with increase of granule diameter_ lh
L
e,=
1+y
dg--=
=e_
lun
(9)
where clim is the limit value of porosity_ Equation (8) can be written in the following form:
(10) Equation (10) can be used for the calculation of the average porosity of granules having a wide range of particle size. Taking the relative freq’crency, ni, of a_narrow fraction of average granule diameter dgsi where
Ni
ni = -
(11)
N
gives 112
5,
=
C Ep.iUini i= 1 (12)
m C i=l
uini
where Sp is the average porosity of the granules, Vi is the average volume of the granules characterized by the average diameter of d,, ir and m is the number of size groups considered_ Using the Mationship
(13) eqn. (10) can be substituted for ep,i in eqn. (12), so that it can be written (14)
where n, is the specific frequency of the granules, related to the total volume of the granules heap, expressed as n, =
N
-
(15)
V
Equation (10) can be used to determine the limit value of granule porosity, eLim,and the relative shape factor, w,, if the porosities of the various size fractions are known. On the other hand, eqn. (14) yields the average porosity of granules, if the limit value of porosity, the volume of the primary particles and the data of specific frequency n, are known.
COhlPARISON OF EzXPERIMENTAL LATED D_4T_4
AND
CALCU-
In the following, the experimental data of Orm6s et al. are compared with calculated ones. The results are as follows: On the diagrams presented in Fig. 2, the porosity values determined experimentally (epSm)and calculated by eqn. (10) marked as (Ep.c)are plotted against the increasing size of granules_ The agreement between the measured and calculated data is fairly good. The agreement is remarkable in the case of granules made from HalidorB, polyethylene, Chlorocide@ and lactose. Figure 3 shows the deviation existing between the measured and calculated data; generally the error is less than *20%.. The possible reason for the deviation lies in the experimental determination method, which gives smaller porosity values in the case of tiny granules due to their rough and mottled surface. This notion is also supported by Ormos ef al. [I], who compared the porosity data determined by their and other methods. Equation (10) was used for the calculation of the limit value of porosity, elimr using experimentally determined data. Plotting E,
260
as
02
0.4 06
a2
a_4 a.6 0.8 l-0 12 1.4 dgx103,m b.POLYETHYLENE
0.8 l.0 t2 14 dg=103,rn
@
.HALIDOR
6, 0.8 0.6 0.4
02 .
.
:
0.2 0.4 0.6 0 8
.
.
1.2 1.4
0.2 0.4 0.6 0.8 1.0 1.2 1.4 dg’
dgx103,m
@
c.CHLOROCIDE &P
.
10
103,
m
d _ LACTOSE
I
i
&P
a.8
0.8 t ! 0.6 t
I
I
-__.-----
I 4
0.6 0.4
X--a ,- &-Lx_ d , 0.2 L((.; a.-:
1
0.4 +i
0.2
i:
0.2 0.4 0.6 08 s.GLASS
1.0 1.2 1.4 dg':03,,
0.2 0.4 0.6 0.8 1.0 1.2 1.4 deX103,.n f-SODIUM NITRATE
BEADS
&, 0.6 $
r o;g=j
a.2 0.4 0 6 0.8 1.0 1.2 1.4 d+la3,m g.SODIUIH
CHLORIDE
a.2 a.4 0.6 0.8 1.0 12 1.4 dSXla3,m h. SAND
Fig. 2. Comparison of the porosities measured by OrmBs et al. [4 1 and the calculated values in function of the granule size; 0, measured; X, calculated_
values against. l/a&, and applying the method of least squares, the intercept gives the value of eIim and the slope can be expressed as a=-------
Q&m a,
(16)
from which w, can be calculated after determining the slope value a. The meas-ured and calculated data listed in ihe first four rows of Table 1 show a fairly good agreement, wk5le in the last four rows
261
0.1
0.2
0.3
OA
05
0.6
mcaramd
0.7
0.8
pomi,y
0.9
1.0
EP.”
Fig_ 3. Comparison of the measured 14 ] and cakuIated porosities; *, HaIidor@; 0, polyethylene; *, Chlorocide@ ; +, lactose; l, glass beads; a, sodium nitrate; 0, sodium chloride; X, sand.
TABLE
1
Characteristic
values of the granules produced
Model material
%m.m
%m.c
Halidor@ PoIyethylene Chlorocide@ Lactose Glass beads NaN03 NaCI Sand
0.70 0.66 0.62 0.55 0.52 0.49 O-47 0.45
0.72 0.63 0.5s 0.53 0.39 0.39 0.37 0.34
2.0 3.1 1.8 3.2 4.4 4.5 4.5 4.5
2.3 2.0 1.6 1.4 1-l 1.0 0.9 0.8
2.6 1.7 l-4 1.1 0.6 0.6 0.6 0.5
they differ from each other by as much as +20%. The discrepancy between the measured and calctdated volume fractions, ym and pC, is even larger.
EXPERIMENTAL
Experiments were carried out in a continuously operated fluidised bed granulator, i-d. 0.109 m and bed height 0.2 m. The granulator was equipped with a mechanical stirrer_ Hot air was blown through the granulator to fluidise the granules and evaporate the solvent of the binder liquid_ During the experiments the number of rotations of the stirrer (Q = 100 rniri-I), the minimum height of the fluidised bed (Y, = 8.5 X lo-’ m), the bed expansion (Y/Y, = 1.6) and the temperature of the exit air (Tk = 308 K) were kept constant.
Sand with mean particle diameter of & = 1.5 X 10e3 m was used as raw material and the binder was gelatine solution of different concentrations_ During the experiments, the following major param eters were altered: the feed rate of the gelatine solution ( W), the concentration of the gelatine in the solution (c), and the feed rate of the sand to be granulated (G). During the first run, the above parameters were kept constant and the relationship between the product quality and operation time was examined. In runs Nos_ 2,3 and 4, two of the parameters W, c and G, were held constant and the third varied. In the fifth run all the above parameters were altered but the ratio of the mass of the gelatine to the mass of the granules (c, = 0.45 X 10-z kg/kg) was held constant_ During the sixth run, the feed rates of both the binder liquid and the sand were altered; this caused a change in the relative binder concentration_ The data are summarized in Table 2_ Figure 4 shows how the determined average internal porosity values, E~,~, vary with the specific frequency, R,. As can be seen, a linear relationship exists between cp., and n,. The average porosity value seems to be independent of the variables and parameters tested, depending only on the specific frequency of the produced granules_ This proves the application of eqn. (14) for the description of the average porosity of granules_ The method of least squares was used for determining the slope and intercept of the EP versus n, relationship. The limit value of porosity, eLim, of granuIes produced from sand was determined from the intercept value E&
=
0.49
as well as the volume (?=
clim l-cElim
fraction
= 0.96
and the shape factor 3uccli, ax = -_ a
=
15-3
where the slope is a = 1.67 X lo-l2 m3 and the average volume of the primary particles is ua = 1.77 X lo-l2 m3. The limiting value of porosity (clim = 0.49) agrees well with that determined by Orm&
262 TABLE
2
Constant
and variable param eters of the experiments 1
2
3
4
5
6
5.83
from l-69 to 10.1
5-91
6.66
from 4.16 to 10.0
from 2.5 to 8.33
5.92
5-92
from l-99 to 7.86
5.92
from 3.97 to 7.37
5.92
G x lO"(kg/s)
8.33
4.5
4.5
from 2.83 to 15-O
from 3.7 to 13.6
from 3.0 to l-;-i3
t x 10-3
from 0.9 to 11-4
Time required to reach the steady state
Run IV x
10’ (kg/s)
c x 10’
(kg/kg)
(s)
The value of the relative shape factor, w,, indicates that the produced granuIes are almost spherical_ In Fig. 5 the measured and caIculated average porosity values using eqn. (13) are compared_ It can be seen that the majority of the catcillated and measured values agree within ?20%.
I
2
4
6
8
10 12
l4 16 18 20 22 24 26 rpuilic frepvencyn.-tP.rn3
Fig_ 4. The average porosity measured in a continuously operated fluidised bed granulator as a function of the specific frequency nv; 0, run 1; f, run2;x,~n3;0.run4;0,~n5;A,,n6.
CONCLUSIONS
Summarizing the results discussed above, it can be stated that the developed mathematical model is sufficiently adequate for the description of the internal granule porosity as a function of the characteristic granuIe diameter. The model allows the derivation of a relationship existing between the average internal porosity of granules and the specific frequency_
LIST OF SYMBOLS a
c 01
02 0.3 0.4 0.5 mea5.srcd anage pmaury Epm
Fig. 5. Comparison of the cakuiated and measured ~&~es of average porosity obtained in a continuously operated fluidised bed granulator; 0, run 1; +, run 2; x,run3;0,run4;2run5;A,run6_
5
do d
& G
et al [4] (eLim= 0.45). The latter authors carried otlt experiments with a batch-wise operated fluid&d bed granulator.
m
constant dermed by eqn_ (16), m3 concentration of the binder solution, kg/kg relative amount of binder = cm/G, kg/kg characteristic size of the primary particle, m characteristic size of the granule, m mean diameter of granules belonging to the ith size fraction, m feed rate of raw material to be granulatea, kg/s number of size fractions considered, dimensionless
263
n
ni
n, N
Ni t
Tk UO "P I.?:
V
vo K
v, W Y
Y, Y
number of primary particles in a granule, dimensionless relative frequency defined by eqn. (ll), dimensionless specific frequency of granules defined by eqn. (ES), mm3 number of granules in the granule heap considered, dimensionless number of granules in the ith narrow fraction in the granule heap considered, dimensionless time, s temperature of air stream leaving the granulator, K volume of the primary particle, m3 volume of the pore, m3 average volume of the granules in the ith narrow fraction, m3 total volume of the granules in the granule heap considered, m3 total volume of primary particles in a granule, m3 total vohune of pores in a granule considered, m3 volume of a granule, m3 feed rate of binder solution, kg/s height of the fluidised bed, m minimum height of the fluidised bed, m volume fraction defined by eqn. (I), dimensionless
"P W,
Q
porosity defined by eqn. (9), dimcnsionless porosity of the ith narrow size fraction, dimensionless average porosity, dimensionless limit value of porosity, dimensionless shape factor of primary particles, dimensionless shape factor of granules, dimensionless rleative shape factor defmed by eqn_ (7), dimensionless number of revolution of the stirrer, min-*
Subscripts
m C
measured calculated
REFERENCES
Hung. J. Ind. C&m_, I (1973) 207 - 2%. R. Polke, W. Kerrmann and K. Sommer, Chem_ Zng_ Tech.. 51 4 (1979) 283 - 288. 2. Orm&, R. Csukk and K. Pataki, Hung. J. fnd. Chem.. 3 (1975) 631- 646. 2. Orm6s and K. Pataki, Hung_ J. Znd. Chem.. 7 (1979) 89 - 101. 2. OrmBs, M. Mach&x and K. Pataki, Hung_ J. Znd. Chem.. 7 (1979) 341- 350. Z. Ormbs,
Technology.
33 (1982)
257
- 263
257
Mathematical Model for the Calculation of Internal Granule Porosity J. GVENIS, T. BLICKL?X and R. HAJDU Research Institute for Technical hem Zu.2. (HungaryJ (Received
March 1, 19s’;
Chemistry
of the Hungarian Academy
of Sciences.
Veszpr&n.
H-S201.
S&&z-
in revised form May 2.19S2)
SumMARY
A mathematical model was derived for the ca2cuZation of the average internal porosity value of granules produced in a fIkdised bed granulator. The mode2 allows the description of the variation of granrr2e porosity as a function of the granule size. Granules were formed in a batch-wise operated granulator from lactose, sodium nitrate, sodium chloride7 sand, polyethylene, two pharmaceutical products - Halidor@ and Chloricide@ powders - and glass beads, and their measured internal porosity values were compared with the calculated data. The measured and caiculated data agreed within t207o.
Ii\iTROfiUCTION
The quality of granules produced by the forced agglomeration of elementary particles can be described by their physical properties such as density, internal porosity, size distribution, mechanical strength, etc. [l] . One of the most important physical properties of a vule is its internal porosity, i.e. the total volume of the internal pores related to the volume of a granule. The internal granule porosity influences other properties, e.g. the density and mechanical strength_ Several methods are used for the determination of the internal granule porosity [I, 2] _ -4 simple and fast method was applied by Ormos [1] for granules of narrow size distribution_ This method is based on the difference existing between the total porosity of a heap of granules and a heap of non-porous re%rence material, the shape and size distribution of the examined granules and the reference material being the same. Ormos found that the internal porosity values of
various granules agreed within +-X0% with the results of control determinations. Moreover Ormos et al. 13 - 51 studied the variation of internal porosity of granules prepared from different powders. A fluidised bed granulator equipped with a blade stirrer was used in this experiment_ The authors determined the effects of the raw material quahty, of the state of the fluidised bed, etc., on the internal granule porosity and found that the porosity approaches a limit with increase of granule size. This limit is inversely affec&d by increasing the number of revolutions of the stirring blades. It was also found that the porosity of the granu!es depended on the type of stirrer used. The tendencies found in the analysis of the above-mentioned results led the authors to derive a mathematical mode1 for the prediction of the internal granule porosity.
PREMISE
AND
LIMITS
OF THE MODEL
(1) It is assumed that granule formation takes place via the adherence of single particles and that the inner structure of the granules remains unchanged during their growth. The primary particles form tetrahedrons in the granule; the first tetrahedron, formed by four elementary particles at the beginning of the granulation, grows by the successive adherence of primary particles and after a short time the granule becomes nearly spherical in shape. (2) By definition, the pore volume of a granule means the empty space enclosed by the adhering primary particles. ftegarding a tetrahedron formed by four spheres, the actual volume of the ‘empty space’ is hard to define due to the lack of a well-defined 0 Elsevier Sequoia/Printed
in The Netherlands
envelope surface_ This uncertainty is significant only in the case of very small granules. If the granules grow and their shape beeomes spherical, an envelope surface can be defined. (3) The primary particles can be charcterized by their mean particle size do_ (4) The granules sre assumed to be spherical with a shape factor as a measure of nonsphericity- It is assumed that the shape is independent of size_ (5) The granulation is an extremely complex process influenced by many factors. The mathematical model therefore consists of several param eters whose esact values have to be determined experimentally_
where uOis the volume In the case of s$erical
&T lJo=
-
OF THE MODEL
As was mentioned previously, four primary particles have to adhere to form a tetrahedron and to create a ‘pore’_ The pore formed by four tetrahedrahy arranged primary spherical pazticles (n = 4) is shown in Fig. 1_ The
d2b “P
+-c-y*
do
“0
Fig. 1. Pore formation in the case of tetrahedral arrangement of the particles, ~0, volume of the prima? particle; up, volume of the pore; do, diameter of the primary partic.e_
volume volume
of the pore (v,) is proportional of the primary particle (uO):
(1)
1: the next, i-e_ the fifth, priiary particle adheres to the previousIy mentioned tetrahedron, a new pore forms having the same volume; therefore the total pore volume (vi) in this case is zpu,
(la)
The ‘granule’ now contains five primary particles (n = 5)_ If further primary particles attach to the growing ‘granule’, the pore volume can be described as VP = (22 -3)(-Q
VP
(n--M n+(n-3)9
nuo
+
3lwo (R
-
=
3)u, (3)
where n is the number of primary particles in the given granule. Equation (3) is more or less valid for granules, even if the primary particles are not regular spheres and their sizes vary; moreover, if their contact with each other is not regular, the structure of the granule is somewhat looser_ The volume of a granule is the sum of the volumes of the primary particles and the pores: VP -I-
v,
(4)
If CT0is the mean particle size of the primary particles and d, the characteristic mean diameter of the granules with w. and w, as their respective shape factors, eqn. (4) may be written:
to the
L’p = gu,
“P* =
= +
of a given granule is (n -
v, v.0
by the primary
6
Thus, the porosity EP=
occupied
&r
VO=nvo=n-
v, =
\_-
(2)
6
and the total volume particles is
= DERIVATIOM
of the primary particle_ particles this volume is
(lb)
The actual values of d, and C& depend on both the selected characteristic mean diameter and the determination method used. For example, if an optical method is used for the determination of the particle diameter, the following possibilities arise_ The diameter can be determined using the equivalent circle having the same area as the projected particle, the algebraic or geometric mean of the smallest and largest diameters, or the diameter of the enclosing sphere. In the third case, the maximum value of the shape factor is 1. With sieve analysis, the granules are characterized by three diameters, e-g_ a, b and c, taken in the direction of the space coordinates Usually the values of these diam-
eters differ from each other, e-g_ a < b < c, where b is taken as the characteristic size of the granule. In this case the shape factor is greater than I_ The number of primary particles, n, forming a granule of diameter d, can be calculated from eqn. (6 jr
n=
&--[~($$+3y]
(6)
The ratio of the shape factors of primary particles and ihe granule gives a relative shape factor: &Jr= -0s (7) as If eqn_ (6) is substituted in eqn_ (3), the granule porosity takes the following form:
It is evid.ent from eqn. (8) that the granule porosity approaches a limit value with increase of granule diameter_ lh
L
e,=
1+y
dg--=
=e_
lun
(9)
where clim is the limit value of porosity_ Equation (8) can be written in the following form:
(10) Equation (10) can be used for the calculation of the average porosity of granules having a wide range of particle size. Taking the relative freq’crency, ni, of a_narrow fraction of average granule diameter dgsi where
Ni
ni = -
(11)
N
gives 112
5,
=
C Ep.iUini i= 1 (12)
m C i=l
uini
where Sp is the average porosity of the granules, Vi is the average volume of the granules characterized by the average diameter of d,, ir and m is the number of size groups considered_ Using the Mationship
(13) eqn. (10) can be substituted for ep,i in eqn. (12), so that it can be written (14)
where n, is the specific frequency of the granules, related to the total volume of the granules heap, expressed as n, =
N
-
(15)
V
Equation (10) can be used to determine the limit value of granule porosity, eLim,and the relative shape factor, w,, if the porosities of the various size fractions are known. On the other hand, eqn. (14) yields the average porosity of granules, if the limit value of porosity, the volume of the primary particles and the data of specific frequency n, are known.
COhlPARISON OF EzXPERIMENTAL LATED D_4T_4
AND
CALCU-
In the following, the experimental data of Orm6s et al. are compared with calculated ones. The results are as follows: On the diagrams presented in Fig. 2, the porosity values determined experimentally (epSm)and calculated by eqn. (10) marked as (Ep.c)are plotted against the increasing size of granules_ The agreement between the measured and calculated data is fairly good. The agreement is remarkable in the case of granules made from HalidorB, polyethylene, Chlorocide@ and lactose. Figure 3 shows the deviation existing between the measured and calculated data; generally the error is less than *20%.. The possible reason for the deviation lies in the experimental determination method, which gives smaller porosity values in the case of tiny granules due to their rough and mottled surface. This notion is also supported by Ormos ef al. [I], who compared the porosity data determined by their and other methods. Equation (10) was used for the calculation of the limit value of porosity, elimr using experimentally determined data. Plotting E,
260
as
02
0.4 06
a2
a_4 a.6 0.8 l-0 12 1.4 dgx103,m b.POLYETHYLENE
0.8 l.0 t2 14 dg=103,rn
@
.HALIDOR
6, 0.8 0.6 0.4
02 .
.
:
0.2 0.4 0.6 0 8
.
.
1.2 1.4
0.2 0.4 0.6 0.8 1.0 1.2 1.4 dg’
dgx103,m
@
c.CHLOROCIDE &P
.
10
103,
m
d _ LACTOSE
I
i
&P
a.8
0.8 t ! 0.6 t
I
I
-__.-----
I 4
0.6 0.4
X--a ,- &-Lx_ d , 0.2 L((.; a.-:
1
0.4 +i
0.2
i:
0.2 0.4 0.6 08 s.GLASS
1.0 1.2 1.4 dg':03,,
0.2 0.4 0.6 0.8 1.0 1.2 1.4 deX103,.n f-SODIUM NITRATE
BEADS
&, 0.6 $
r o;g=j
a.2 0.4 0 6 0.8 1.0 1.2 1.4 d+la3,m g.SODIUIH
CHLORIDE
a.2 a.4 0.6 0.8 1.0 12 1.4 dSXla3,m h. SAND
Fig. 2. Comparison of the porosities measured by OrmBs et al. [4 1 and the calculated values in function of the granule size; 0, measured; X, calculated_
values against. l/a&, and applying the method of least squares, the intercept gives the value of eIim and the slope can be expressed as a=-------
Q&m a,
(16)
from which w, can be calculated after determining the slope value a. The meas-ured and calculated data listed in ihe first four rows of Table 1 show a fairly good agreement, wk5le in the last four rows
261
0.1
0.2
0.3
OA
05
0.6
mcaramd
0.7
0.8
pomi,y
0.9
1.0
EP.”
Fig_ 3. Comparison of the measured 14 ] and cakuIated porosities; *, HaIidor@; 0, polyethylene; *, Chlorocide@ ; +, lactose; l, glass beads; a, sodium nitrate; 0, sodium chloride; X, sand.
TABLE
1
Characteristic
values of the granules produced
Model material
%m.m
%m.c
Halidor@ PoIyethylene Chlorocide@ Lactose Glass beads NaN03 NaCI Sand
0.70 0.66 0.62 0.55 0.52 0.49 O-47 0.45
0.72 0.63 0.5s 0.53 0.39 0.39 0.37 0.34
2.0 3.1 1.8 3.2 4.4 4.5 4.5 4.5
2.3 2.0 1.6 1.4 1-l 1.0 0.9 0.8
2.6 1.7 l-4 1.1 0.6 0.6 0.6 0.5
they differ from each other by as much as +20%. The discrepancy between the measured and calctdated volume fractions, ym and pC, is even larger.
EXPERIMENTAL
Experiments were carried out in a continuously operated fluidised bed granulator, i-d. 0.109 m and bed height 0.2 m. The granulator was equipped with a mechanical stirrer_ Hot air was blown through the granulator to fluidise the granules and evaporate the solvent of the binder liquid_ During the experiments the number of rotations of the stirrer (Q = 100 rniri-I), the minimum height of the fluidised bed (Y, = 8.5 X lo-’ m), the bed expansion (Y/Y, = 1.6) and the temperature of the exit air (Tk = 308 K) were kept constant.
Sand with mean particle diameter of & = 1.5 X 10e3 m was used as raw material and the binder was gelatine solution of different concentrations_ During the experiments, the following major param eters were altered: the feed rate of the gelatine solution ( W), the concentration of the gelatine in the solution (c), and the feed rate of the sand to be granulated (G). During the first run, the above parameters were kept constant and the relationship between the product quality and operation time was examined. In runs Nos_ 2,3 and 4, two of the parameters W, c and G, were held constant and the third varied. In the fifth run all the above parameters were altered but the ratio of the mass of the gelatine to the mass of the granules (c, = 0.45 X 10-z kg/kg) was held constant_ During the sixth run, the feed rates of both the binder liquid and the sand were altered; this caused a change in the relative binder concentration_ The data are summarized in Table 2_ Figure 4 shows how the determined average internal porosity values, E~,~, vary with the specific frequency, R,. As can be seen, a linear relationship exists between cp., and n,. The average porosity value seems to be independent of the variables and parameters tested, depending only on the specific frequency of the produced granules_ This proves the application of eqn. (14) for the description of the average porosity of granules_ The method of least squares was used for determining the slope and intercept of the EP versus n, relationship. The limit value of porosity, eLim, of granuIes produced from sand was determined from the intercept value E&
=
0.49
as well as the volume (?=
clim l-cElim
fraction
= 0.96
and the shape factor 3uccli, ax = -_ a
=
15-3
where the slope is a = 1.67 X lo-l2 m3 and the average volume of the primary particles is ua = 1.77 X lo-l2 m3. The limiting value of porosity (clim = 0.49) agrees well with that determined by Orm&
262 TABLE
2
Constant
and variable param eters of the experiments 1
2
3
4
5
6
5.83
from l-69 to 10.1
5-91
6.66
from 4.16 to 10.0
from 2.5 to 8.33
5.92
5-92
from l-99 to 7.86
5.92
from 3.97 to 7.37
5.92
G x lO"(kg/s)
8.33
4.5
4.5
from 2.83 to 15-O
from 3.7 to 13.6
from 3.0 to l-;-i3
t x 10-3
from 0.9 to 11-4
Time required to reach the steady state
Run IV x
10’ (kg/s)
c x 10’
(kg/kg)
(s)
The value of the relative shape factor, w,, indicates that the produced granuIes are almost spherical_ In Fig. 5 the measured and caIculated average porosity values using eqn. (13) are compared_ It can be seen that the majority of the catcillated and measured values agree within ?20%.
I
2
4
6
8
10 12
l4 16 18 20 22 24 26 rpuilic frepvencyn.-tP.rn3
Fig_ 4. The average porosity measured in a continuously operated fluidised bed granulator as a function of the specific frequency nv; 0, run 1; f, run2;x,~n3;0.run4;0,~n5;A,,n6.
CONCLUSIONS
Summarizing the results discussed above, it can be stated that the developed mathematical model is sufficiently adequate for the description of the internal granule porosity as a function of the characteristic granuIe diameter. The model allows the derivation of a relationship existing between the average internal porosity of granules and the specific frequency_
LIST OF SYMBOLS a
c 01
02 0.3 0.4 0.5 mea5.srcd anage pmaury Epm
Fig. 5. Comparison of the cakuiated and measured ~&~es of average porosity obtained in a continuously operated fluidised bed granulator; 0, run 1; +, run 2; x,run3;0,run4;2run5;A,run6_
5
do d
& G
et al [4] (eLim= 0.45). The latter authors carried otlt experiments with a batch-wise operated fluid&d bed granulator.
m
constant dermed by eqn_ (16), m3 concentration of the binder solution, kg/kg relative amount of binder = cm/G, kg/kg characteristic size of the primary particle, m characteristic size of the granule, m mean diameter of granules belonging to the ith size fraction, m feed rate of raw material to be granulatea, kg/s number of size fractions considered, dimensionless
263
n
ni
n, N
Ni t
Tk UO "P I.?:
V
vo K
v, W Y
Y, Y
number of primary particles in a granule, dimensionless relative frequency defined by eqn. (ll), dimensionless specific frequency of granules defined by eqn. (ES), mm3 number of granules in the granule heap considered, dimensionless number of granules in the ith narrow fraction in the granule heap considered, dimensionless time, s temperature of air stream leaving the granulator, K volume of the primary particle, m3 volume of the pore, m3 average volume of the granules in the ith narrow fraction, m3 total volume of the granules in the granule heap considered, m3 total volume of primary particles in a granule, m3 total vohune of pores in a granule considered, m3 volume of a granule, m3 feed rate of binder solution, kg/s height of the fluidised bed, m minimum height of the fluidised bed, m volume fraction defined by eqn. (I), dimensionless
"P W,
Q
porosity defined by eqn. (9), dimcnsionless porosity of the ith narrow size fraction, dimensionless average porosity, dimensionless limit value of porosity, dimensionless shape factor of primary particles, dimensionless shape factor of granules, dimensionless rleative shape factor defmed by eqn_ (7), dimensionless number of revolution of the stirrer, min-*
Subscripts
m C
measured calculated
REFERENCES
Hung. J. Ind. C&m_, I (1973) 207 - 2%. R. Polke, W. Kerrmann and K. Sommer, Chem_ Zng_ Tech.. 51 4 (1979) 283 - 288. 2. Orm&, R. Csukk and K. Pataki, Hung. J. fnd. Chem.. 3 (1975) 631- 646. 2. Orm6s and K. Pataki, Hung_ J. Znd. Chem.. 7 (1979) 89 - 101. 2. OrmBs, M. Mach&x and K. Pataki, Hung_ J. Znd. Chem.. 7 (1979) 341- 350. Z. Ormbs,