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Estimation of the co-ordination number in a two-component mixture of cohesive spheres
i’oocctdcr Technology. 36 (1983)
181
18 1 - 188
Estimation of the Co-ordination Cohesive Spheres
Number in a Two-component
Mixture of
--
SUhlMARY A model is proposed for estimating ihe co-ordination number of a mixture of untformsized cohesive particles, using the angle of friction between spheres. Combining this model wtth the model for a two-component mixture of spheres, which was proposed in a previous paper. the authors obtained new equations for a two-component mrxture of cohesive spheres. To verify the validity of the model, the co-ordination numbers calculated from the model are compared with those obtained from the computer simulation of a twocomponent mixture of cohesive spheres and fairly good agreement is found between the calculated results and the simulated data.
INTRODUCTION
A model for estimating the co-ordination number in a two-component mixture was proposed by Ouchiyama and Tanaka [l], Okazaki et al. [ 21 and the authors [ 31. These models, however, did not take account of the dynamic interaction between particles_ A packed bed is formed by the accumulation of many particles, so that the structure of particle arrangement in a packed bed is determined by the dynamic properties of each constituent particle, particularly the angle of friction or the cohesive force between particles. In this paper, the authors propose a simple model for estimating the co-ordination number in a non-compressed bed of uniformsized spherical particles, using the angle of friction between the particles. Furthermore, combining this model with the model for a two-
component misturc of splrtbrt*s.as proposed in the previous paper 13 1. WC* obtain model equations for estimating then co-ordination numhcr in a two-componr*nt misturc of cohesive spheres. ‘I’hc results calculattd from the equations l~asecl on the model arc* in agrccmcnt with the three-drnlcnslonal computer-simulate3 data.
DERIVATION OF MODEL EQUATIONS CO-ORDINATION NUXIBER
FOR TliE
Model for a bed of uniform-sized coi~cs~c particles Considering the packing process. we propose a simple model for estimating the co-ordination number from the angle of friction between particles. Generally, irregularities esist in the surface of the random packmg particle bed as shown in Fig. 1. First. we discuss a particle A m Fig. 1 which is situated on a peak in the surface of the particle bed. This particle A is
Fig. 1. Concept spheres.
of the surface of the random
0 Elsevier Sequoia/Printed
packing
in The Netherlands
182
cohesive region on the reference particle is the area of a circle of diameter D, sin @,_ We assume that a very large number of particles is dropped randomly, and that the number of particles which fall in one region is proportional to the area of this region. So, (A) the probability that an incoming particle will cohere parttcle
I
= (a/4)@,
sin Q,)‘/(n/4)(D,
(B) the probability will roll down = 1 -
Fig. 2. Concept of the model for estimating the co-ordination number N, from the angle of friction %-
able to contact an incoming particle in a wide range of values of 0, the zenithal angle of contact (0 < Q =G x/2)_ Secondly, we discuss a particle B in Fig_ 1 which is situated at a low point in the surface of the bed. This particle B contacts an incoming particle at only one point (Q = 0) given the geometric conditions. We consider all the particles on the random rough surface of the bed. and propose the simple model shown in Fig_ 2. _4n incoming particle can contact a reference particle of type A over a range of zenithal angles (9) lying between 0 and 7r/2. For a reference particle of type B, the zenithal angle will be zero. It is assumed that the mean range of values of 4 for all surface particles is 0 < Q =G a/4. Therefore, from a top view of the reference particle as shown in Fig_ 2, the point at which an incoming particle will make contact lies inside a circle of diametersDP sin(P/4)_ The assumption is also made that an incoming particle in the region of 0 < Q < 9, will cohere in this position but that it will roll down in the region of Q > Cp,; & is the angle of friction between the particles_ So, from the top view of a reference particle, the
(s/4)&
sin(sr/4))’
that an incoming
sin 0,)Z/(a/4)(D,
particle
sin(7r/4))’
In case (A). the number of possible contact points on a hemispherical surface is 1, but in case (B) this number is 3, because it is possible for an incoming particle to roll over the surface of the packed particles until it is stably supported at three points. The positions of the two contacting particles are symmetrical about the horizontal plane, so the number of contact points on the surface of a spherical particle in a packed bed is twice the number on the hemispherical surface of a reference sphere, and the following equation is obtained: NC=2
1I 3
~1
l-
(x/4)@, (a/4)(D,
(Z-/4)&
I (X/4)(0,
= 2(3 -
sin A)* sin(Z/4))*
1
sin 6,)’ sin(sr/4))’
4 sin2@,)
II (O,G
x14)
(1)
The co-ordination number NC for a bed of uniform-sized spherical particles can be calculated from the angle of friction between the packing particles by this simple model equation. Model for a two-component mixture of cohesive spheres The authors proposed a simple model for estimating the co-ordination number in a twocomponent mixture of non-cohesive spheres in the previous paper [ 3]_ According to this model, the co-ordination number NC, on a particle 1, of diameter DPI, in a twocomponent mixture comprising particles 1 and particles 2, can be calculated from the ~_ equation
lS3
NC,= SaN,. 2
+
(1 -
saw,.
(2)
1
where N,. Z is the co-ordination number on a particle 1 in direct contact with particles 2, calculated using eqn_ (9) which is given later; S, is the fractional area of the particles 2, which can be calculated from the equation (3) where S, is the fractional number of the particles 2. Similarly, we can determine the coordination number on a particle 2, NC=, from the equation NC, = S,N,.
1 +
(I-
S,)N,.
(4)
,
The average co-ordination number in a two-component misture, I\7,, can be espressed as a summation of the products of NC, or NC2 and the fractional number of each component, (1 -S,) orS,. N,=
(1 -
S,)N,,
(5)
+ SpNc=
Ni_ , to Nz_z in eqns_ (2) and (4) can be calculated by the following model. If we assume that the co-ordination number N,., is proportional to the ratio of that part of the surface area of the hypothetical sphere occupied by a contact particle 2 to the total surface area of the hypothetical sphere of radius OX, shown in Fig_ 3, then the following equation holds geometrically [ 31:
(6)
The proportional constant Q in eqn. (6) can be calculated from the co-ordination number N, for a uniform-sized spherical particle, as fOllOWS~ cr = =
Hy&lhel:caI
sphere
particle
1 in direct
contact
-
fl)
(7)
0_067N,
The proportional constant Q is calculated using the following equation, which is obtained by substituting eqn. (1) in eqn (7)4 sin’o,
3-
o=
(2 -
2 = 0.402
-
0.536
Substituting N1.Z =
4)
sin’d,
(6)
eqn_ (8) in eqn. (6). we obtain
(0604
-
1 +Dp,IDp2-
l-07
sin’o,)(DP,/DPZ
x/ (D&‘r+f(%,lD~:
i 1) +2)
(9) NV,. i, NZ_ , and ATZ_, are given by eqn. (9), with appropriate subscripts for D,. It should be noted that N,_z must not have a value less than 2, even as the value of DP,/DP2 approaches zero [ 31 I thus N,_,>
TION
(10)
2
COMPAR;SON
KITH
THE
COMPUTER
SIMULX-
RESULTS
Computer simulation programme for a tmoconzponent mixture of cohesive spheres It is a laborious process to measure the co-ordination number in a two-component mixture of cohesive spheres and it is very difficult to obtain accurate esperimental data. Tory et al. i-l] proposed the computer simulation programme for a random packing of uniform-sized spheres, and we developed this programme for a two-component misture
r31-
Fig. 3. Model of a reference with particles 3.
(&/4)(2
To espress the cohesion of particles, an improvement was made on the packing simulation in the previous paper [ 33. in which it was assumed that an incoming particle will roll down until it is supported at three points_ The model used in this paper is as followsConsidering an incoming particle at its moment of initial contact as shown in Fig_ 4, if the zenithal angle Q of the contact point is less than a critical value +,, the particle will
184
particles_ The total number of packing particles is 4000 to 6000. Two typical cross-sections of twocomponent mixtures of cohesive spheres as obtained by the computer simulation are shown in Fig_ 5(a) (9, = 0.2618 (Le. x/12), Dp,/Dp,= 2, S,= 0.5, E = 0.4356) and Fig. 5(b) (9, = 0.5236 (i.e. n/6), DpJDp3= 2, S, = 0.5, E = O-5462).
*
Fig.
Incomng
partu1ecm=
4. Concept particle bed_
of
~cIn;Dmng
first contact
px,lCle
rolls
cb.vn
in random packing
in this position; but if Q is greater than @,, the particle will roll down [ 5]_ The critical value 9, is the angle of friction between the
stay
Comparison wiih the computer-simulated results for a bed of uniform-stied cohesive spheres The co-ordination number for a bed of uniform-sized spheres was obtained by computer simulation with various angles of friction. The results are compared with those predicted by eqn. (l), as shown in Fig. 6. There is good agreement between the model and the Yt+_2618fIf/I2) DprIl&=2 sp co.5 & Al4356
lFig. 5(a). Caption on Facing page_
185
950_-5236m5~ D&m&=2 *=o.s E ~0.5462
Fig. 5_ Cross-section of a two-component 0, = n/12;(b), oC = ~16.
mixtures
produced
by computer
simulation;
(a),
186
1
0 0
01
02
03
10 20 Angle Of lrlcllon
04
05
OS 30
OBd, LO Cd@
FL
Fig. 6. Comparison between N,calculated and computersimulated results.
by eqn. (1)
simulated results with the angle of friction 9, ranging from 0 to 0.6_ Comparison with compker-simulated resulfs for a two-component mixfure of cohesive spheres The computer-simulated results are compared with those predicted by eqn. (9), which gives the relationship between N,., and D,,/D,_ for various values of the angle of interparticle friction_ In Fig_ 7, the results calculated from eqn. (9) with 9, = 0,0.2618 (i-e_ x/12) and 0.5236 (i.e. s/6) are compared with the computer-simulated results NC, when S, = 0_99(DP,/DP1> 1) and with NC2 when 3, = O_Ol(DP,/DP,< 1). There is good agreement between the model and the simulated resuits. _ _ By substrtutmg Nr. I to N2, 2, calculated from eqn. (9), into eqns. (2), (4) and (5), the co-ordination numbers NC,, NC, and flC are
s=O
(b) Fig. 8. Comparison between calculated co-ordination numbers N,,, NC=, f?c and computersimulated results; (a). 0, =x/l%(b), 0, = ~16.
obtained_ Figures 8 to 10 show the comparisons between the calculated results based on these model equations and the computersimulated results. The results calculated by the model equations agree sufficiently well with the computer-simulated results of DP,/DP, = l-5,2 and 3.
0 Ip,=o 2618 uvm .
Ipc=0 52360TE) corQ%terszzmlklhcn
5236VTf6)
CONCLUSION
Fig- 7. Comparison computer-simulated
between results.
N,,z
by eqn. (9) and
The co-ordination number for a bed of uniform-sized cohesive spheres can be estimated using the angle of friction between spheres from a simple model equation_ The
187
(b)
Fb-mtwxd
.xrraev
.2
t.%?reacIran5*-1
Fig. 10. Comparison between calcutated co-ordination numbers A-=,. A-,_+ _y=a,andcomputer-simulated results; (a), 0, = iill2; (b). 13, = iif6.
Fig. 9_ Comparison between calculated co-ordination numbers Nc,, N,,. Ec and computer-simulated results; (a), 0, = r/13; (b), sjc = 7716.
validity of the equation was verified by comparing the calculated results with those obtained by computer simulation. Corresponding equations for a two-component mixture of cohesive spheres, which is derived by substituting the model equation for a bed of uniform-sized cohesive spheres into the equations for a two-component misture of non-cohesive spheres, as proposed in a previous paper, are also obtained_ The equations are compared with the computersimulated results for co-ordination numbers in a two-component mixture of cohesive spheres and the calculated results obtained using the model equations found to be in good agreement with the simulated data.
LIST OF SYMBOLS
DP DD,, DP: N 1.2 N, Rz %, . NC,
%
diameter of particles in a bed of uniform-sized particles, m diameters of particle 1 or 2 in a two-component misture, m co-ordination number on particle 1 in direct contact with particles 2, co-ordination number in a bed of uniform-sized particles, average co-ordination number in a two-component misture, co-ordination number on particle 1 or 2 in a txvo-component mixture, fractional area of particles 2 in a two-component misture, -
188
S,
fractional number of particles 2 in a two-component mixture, proportionality constant in eqn. (61, void fraction, zenithal angle of contact point, as shown in Figs. 1 and 4, rad angle of friction between particles, as shown in Figs. 1 and 4, rad
REFERENCES 1 N. Guchiyama and T. Tanaka. Ind. Eng. Chem. Fundam.. 19 (1980) 338. 2 M. Gkazaki. T. Yamazaki, S. Gotoh and R. Toei, J. Chem. Eng Japan. 14 (1981) 183. 3 M. Suzuki and T. Gshima, Powder Technol.. 35 (1983) 159. .S E. M. Tory, B. H. Church. M. K_ Tam and K. Ratner, Can J_ Chem. Eng . 51 (1973) 384. 5 hl. Suzuki, K. Makino, Al. Yamada and K. Iinoya, Kagaku Kogaku Ronbunshu. 6 (1980) 59. (Int. Chem. Eng.. 21 (1981) 482 )
181
18 1 - 188
Estimation of the Co-ordination Cohesive Spheres
Number in a Two-component
Mixture of
--
SUhlMARY A model is proposed for estimating ihe co-ordination number of a mixture of untformsized cohesive particles, using the angle of friction between spheres. Combining this model wtth the model for a two-component mixture of spheres, which was proposed in a previous paper. the authors obtained new equations for a two-component mrxture of cohesive spheres. To verify the validity of the model, the co-ordination numbers calculated from the model are compared with those obtained from the computer simulation of a twocomponent mixture of cohesive spheres and fairly good agreement is found between the calculated results and the simulated data.
INTRODUCTION
A model for estimating the co-ordination number in a two-component mixture was proposed by Ouchiyama and Tanaka [l], Okazaki et al. [ 21 and the authors [ 31. These models, however, did not take account of the dynamic interaction between particles_ A packed bed is formed by the accumulation of many particles, so that the structure of particle arrangement in a packed bed is determined by the dynamic properties of each constituent particle, particularly the angle of friction or the cohesive force between particles. In this paper, the authors propose a simple model for estimating the co-ordination number in a non-compressed bed of uniformsized spherical particles, using the angle of friction between the particles. Furthermore, combining this model with the model for a two-
component misturc of splrtbrt*s.as proposed in the previous paper 13 1. WC* obtain model equations for estimating then co-ordination numhcr in a two-componr*nt misturc of cohesive spheres. ‘I’hc results calculattd from the equations l~asecl on the model arc* in agrccmcnt with the three-drnlcnslonal computer-simulate3 data.
DERIVATION OF MODEL EQUATIONS CO-ORDINATION NUXIBER
FOR TliE
Model for a bed of uniform-sized coi~cs~c particles Considering the packing process. we propose a simple model for estimating the co-ordination number from the angle of friction between particles. Generally, irregularities esist in the surface of the random packmg particle bed as shown in Fig. 1. First. we discuss a particle A m Fig. 1 which is situated on a peak in the surface of the particle bed. This particle A is
Fig. 1. Concept spheres.
of the surface of the random
0 Elsevier Sequoia/Printed
packing
in The Netherlands
182
cohesive region on the reference particle is the area of a circle of diameter D, sin @,_ We assume that a very large number of particles is dropped randomly, and that the number of particles which fall in one region is proportional to the area of this region. So, (A) the probability that an incoming particle will cohere parttcle
I
= (a/4)@,
sin Q,)‘/(n/4)(D,
(B) the probability will roll down = 1 -
Fig. 2. Concept of the model for estimating the co-ordination number N, from the angle of friction %-
able to contact an incoming particle in a wide range of values of 0, the zenithal angle of contact (0 < Q =G x/2)_ Secondly, we discuss a particle B in Fig_ 1 which is situated at a low point in the surface of the bed. This particle B contacts an incoming particle at only one point (Q = 0) given the geometric conditions. We consider all the particles on the random rough surface of the bed. and propose the simple model shown in Fig_ 2. _4n incoming particle can contact a reference particle of type A over a range of zenithal angles (9) lying between 0 and 7r/2. For a reference particle of type B, the zenithal angle will be zero. It is assumed that the mean range of values of 4 for all surface particles is 0 < Q =G a/4. Therefore, from a top view of the reference particle as shown in Fig_ 2, the point at which an incoming particle will make contact lies inside a circle of diametersDP sin(P/4)_ The assumption is also made that an incoming particle in the region of 0 < Q < 9, will cohere in this position but that it will roll down in the region of Q > Cp,; & is the angle of friction between the particles_ So, from the top view of a reference particle, the
(s/4)&
sin(sr/4))’
that an incoming
sin 0,)Z/(a/4)(D,
particle
sin(7r/4))’
In case (A). the number of possible contact points on a hemispherical surface is 1, but in case (B) this number is 3, because it is possible for an incoming particle to roll over the surface of the packed particles until it is stably supported at three points. The positions of the two contacting particles are symmetrical about the horizontal plane, so the number of contact points on the surface of a spherical particle in a packed bed is twice the number on the hemispherical surface of a reference sphere, and the following equation is obtained: NC=2
1I 3
~1
l-
(x/4)@, (a/4)(D,
(Z-/4)&
I (X/4)(0,
= 2(3 -
sin A)* sin(Z/4))*
1
sin 6,)’ sin(sr/4))’
4 sin2@,)
II (O,G
x14)
(1)
The co-ordination number NC for a bed of uniform-sized spherical particles can be calculated from the angle of friction between the packing particles by this simple model equation. Model for a two-component mixture of cohesive spheres The authors proposed a simple model for estimating the co-ordination number in a twocomponent mixture of non-cohesive spheres in the previous paper [ 3]_ According to this model, the co-ordination number NC, on a particle 1, of diameter DPI, in a twocomponent mixture comprising particles 1 and particles 2, can be calculated from the ~_ equation
lS3
NC,= SaN,. 2
+
(1 -
saw,.
(2)
1
where N,. Z is the co-ordination number on a particle 1 in direct contact with particles 2, calculated using eqn_ (9) which is given later; S, is the fractional area of the particles 2, which can be calculated from the equation (3) where S, is the fractional number of the particles 2. Similarly, we can determine the coordination number on a particle 2, NC=, from the equation NC, = S,N,.
1 +
(I-
S,)N,.
(4)
,
The average co-ordination number in a two-component misture, I\7,, can be espressed as a summation of the products of NC, or NC2 and the fractional number of each component, (1 -S,) orS,. N,=
(1 -
S,)N,,
(5)
+ SpNc=
Ni_ , to Nz_z in eqns_ (2) and (4) can be calculated by the following model. If we assume that the co-ordination number N,., is proportional to the ratio of that part of the surface area of the hypothetical sphere occupied by a contact particle 2 to the total surface area of the hypothetical sphere of radius OX, shown in Fig_ 3, then the following equation holds geometrically [ 31:
(6)
The proportional constant Q in eqn. (6) can be calculated from the co-ordination number N, for a uniform-sized spherical particle, as fOllOWS~ cr = =
Hy&lhel:caI
sphere
particle
1 in direct
contact
-
fl)
(7)
0_067N,
The proportional constant Q is calculated using the following equation, which is obtained by substituting eqn. (1) in eqn (7)4 sin’o,
3-
o=
(2 -
2 = 0.402
-
0.536
Substituting N1.Z =
4)
sin’d,
(6)
eqn_ (8) in eqn. (6). we obtain
(0604
-
1 +Dp,IDp2-
l-07
sin’o,)(DP,/DPZ
x/ (D&‘r+f(%,lD~:
i 1) +2)
(9) NV,. i, NZ_ , and ATZ_, are given by eqn. (9), with appropriate subscripts for D,. It should be noted that N,_z must not have a value less than 2, even as the value of DP,/DP2 approaches zero [ 31 I thus N,_,>
TION
(10)
2
COMPAR;SON
KITH
THE
COMPUTER
SIMULX-
RESULTS
Computer simulation programme for a tmoconzponent mixture of cohesive spheres It is a laborious process to measure the co-ordination number in a two-component mixture of cohesive spheres and it is very difficult to obtain accurate esperimental data. Tory et al. i-l] proposed the computer simulation programme for a random packing of uniform-sized spheres, and we developed this programme for a two-component misture
r31-
Fig. 3. Model of a reference with particles 3.
(&/4)(2
To espress the cohesion of particles, an improvement was made on the packing simulation in the previous paper [ 33. in which it was assumed that an incoming particle will roll down until it is supported at three points_ The model used in this paper is as followsConsidering an incoming particle at its moment of initial contact as shown in Fig_ 4, if the zenithal angle Q of the contact point is less than a critical value +,, the particle will
184
particles_ The total number of packing particles is 4000 to 6000. Two typical cross-sections of twocomponent mixtures of cohesive spheres as obtained by the computer simulation are shown in Fig_ 5(a) (9, = 0.2618 (Le. x/12), Dp,/Dp,= 2, S,= 0.5, E = 0.4356) and Fig. 5(b) (9, = 0.5236 (i.e. n/6), DpJDp3= 2, S, = 0.5, E = O-5462).
*
Fig.
Incomng
partu1ecm=
4. Concept particle bed_
of
~cIn;Dmng
first contact
px,lCle
rolls
cb.vn
in random packing
in this position; but if Q is greater than @,, the particle will roll down [ 5]_ The critical value 9, is the angle of friction between the
stay
Comparison wiih the computer-simulated results for a bed of uniform-stied cohesive spheres The co-ordination number for a bed of uniform-sized spheres was obtained by computer simulation with various angles of friction. The results are compared with those predicted by eqn. (l), as shown in Fig. 6. There is good agreement between the model and the Yt+_2618fIf/I2) DprIl&=2 sp co.5 & Al4356
lFig. 5(a). Caption on Facing page_
185
950_-5236m5~ D&m&=2 *=o.s E ~0.5462
Fig. 5_ Cross-section of a two-component 0, = n/12;(b), oC = ~16.
mixtures
produced
by computer
simulation;
(a),
186
1
0 0
01
02
03
10 20 Angle Of lrlcllon
04
05
OS 30
OBd, LO Cd@
FL
Fig. 6. Comparison between N,calculated and computersimulated results.
by eqn. (1)
simulated results with the angle of friction 9, ranging from 0 to 0.6_ Comparison with compker-simulated resulfs for a two-component mixfure of cohesive spheres The computer-simulated results are compared with those predicted by eqn. (9), which gives the relationship between N,., and D,,/D,_ for various values of the angle of interparticle friction_ In Fig_ 7, the results calculated from eqn. (9) with 9, = 0,0.2618 (i-e_ x/12) and 0.5236 (i.e. s/6) are compared with the computer-simulated results NC, when S, = 0_99(DP,/DP1> 1) and with NC2 when 3, = O_Ol(DP,/DP,< 1). There is good agreement between the model and the simulated resuits. _ _ By substrtutmg Nr. I to N2, 2, calculated from eqn. (9), into eqns. (2), (4) and (5), the co-ordination numbers NC,, NC, and flC are
s=O
(b) Fig. 8. Comparison between calculated co-ordination numbers N,,, NC=, f?c and computersimulated results; (a). 0, =x/l%(b), 0, = ~16.
obtained_ Figures 8 to 10 show the comparisons between the calculated results based on these model equations and the computersimulated results. The results calculated by the model equations agree sufficiently well with the computer-simulated results of DP,/DP, = l-5,2 and 3.
0 Ip,=o 2618 uvm .
Ipc=0 52360TE) corQ%terszzmlklhcn
5236VTf6)
CONCLUSION
Fig- 7. Comparison computer-simulated
between results.
N,,z
by eqn. (9) and
The co-ordination number for a bed of uniform-sized cohesive spheres can be estimated using the angle of friction between spheres from a simple model equation_ The
187
(b)
Fb-mtwxd
.xrraev
.2
t.%?reacIran5*-1
Fig. 10. Comparison between calcutated co-ordination numbers A-=,. A-,_+ _y=a,andcomputer-simulated results; (a), 0, = iill2; (b). 13, = iif6.
Fig. 9_ Comparison between calculated co-ordination numbers Nc,, N,,. Ec and computer-simulated results; (a), 0, = r/13; (b), sjc = 7716.
validity of the equation was verified by comparing the calculated results with those obtained by computer simulation. Corresponding equations for a two-component mixture of cohesive spheres, which is derived by substituting the model equation for a bed of uniform-sized cohesive spheres into the equations for a two-component misture of non-cohesive spheres, as proposed in a previous paper, are also obtained_ The equations are compared with the computersimulated results for co-ordination numbers in a two-component mixture of cohesive spheres and the calculated results obtained using the model equations found to be in good agreement with the simulated data.
LIST OF SYMBOLS
DP DD,, DP: N 1.2 N, Rz %, . NC,
%
diameter of particles in a bed of uniform-sized particles, m diameters of particle 1 or 2 in a two-component misture, m co-ordination number on particle 1 in direct contact with particles 2, co-ordination number in a bed of uniform-sized particles, average co-ordination number in a two-component misture, co-ordination number on particle 1 or 2 in a txvo-component mixture, fractional area of particles 2 in a two-component misture, -
188
S,
fractional number of particles 2 in a two-component mixture, proportionality constant in eqn. (61, void fraction, zenithal angle of contact point, as shown in Figs. 1 and 4, rad angle of friction between particles, as shown in Figs. 1 and 4, rad
REFERENCES 1 N. Guchiyama and T. Tanaka. Ind. Eng. Chem. Fundam.. 19 (1980) 338. 2 M. Gkazaki. T. Yamazaki, S. Gotoh and R. Toei, J. Chem. Eng Japan. 14 (1981) 183. 3 M. Suzuki and T. Gshima, Powder Technol.. 35 (1983) 159. .S E. M. Tory, B. H. Church. M. K_ Tam and K. Ratner, Can J_ Chem. Eng . 51 (1973) 384. 5 hl. Suzuki, K. Makino, Al. Yamada and K. Iinoya, Kagaku Kogaku Ronbunshu. 6 (1980) 59. (Int. Chem. Eng.. 21 (1981) 482 )