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A general particle stress equation for fixed and fluidised beds
Powder
Technology,
38 (1984)
3.3
33 - 38
A General Particle Stress Equation R. Y_ QASSIM EE/UFRJond (Received
for Fixed and Fluidised
Beds
and R. DE SOUZA COPPEIUFRJ.
February
15.1983;
CP 1191.
Rio de Janeiro (Brazd)
in revised form June Ii,
1983)
SUhIMARY
THEORY
A general constitutive equation is proposed for the solid particle stress in a fluidised bed_ This equation reduces in a straightforward manner to describe the particle stress in a fixed bed, and it implies a Mohr-Coulomb yieid criterion for the transition from the fued-bed to the fiuidised-bed state. Experimental evidence is given to show that at incipient fluidisation, the particle stress assumes a form agreeing with that predicted by theory_
Considering the bed of solid particles through which fluid is flowing as a binary misture of interacting interpenetrating continuous media, the mass and linear momentum balance equations may be written as
2 + -7 -(p,r,) ah -
at
+ T-(pfl'f)=o
= 0
(1) (2)
INTRODUCTION
When fluid flows vertically upwards through a bed of solid particles, the system can either be in the fixed-bed regime (at rather low fluid velocities) or in the fluidised-bed regime (at higher fluid velocities), transition occurring at the fluid velocity known as the velocity of incipient fluidisation The mechanical response of the solid particle phase of the bed is qualitatively different in the two regimes, being solid-like in the fixed-bed range [l, 2,3] and fluid-like when the bed is fluidised 14, 5]_ The transition from one state to another is considered as a type of Mohr-Coulomb yield phenomenon [I, 2, 33. Although theory is not lacking to describe each of these features separately, there is no general constitutive equation for the solid particle stress which describes in a unified way the f=ed-bed state, the fluidised-bed state, and the transition from one to the other_ In this paper, we propose such an equation and we supply esperimental support for the existence of a stress difference implied by this equation. 0032-5910/84/$3.00
This system of equations is closed with the aid of constitutive equations for T,_ Ti and m. We address ourselves specifically to the solid stress constitutive equation_ Suppose that when there is no fluid flow in the bed, the particle phase behaves as a rigid body_ Then, when fluid flows through the bed in the fised-bed range, we may write for T, the constitutive equation T, = F(Z)
(5)
where we have assumed constant bed voidage. thus eliminating the necessity for including E, VE, etc_, in the list of arguments of the tensor function F_ We note, however, that relaxing the assumption of constant bed voidage does not introduce any conceptual difficulties in the analysisWhen the flow of fluid is such that the bed is fluidised, the particle phase behaves mechanically as a fluid characterised by a pres0 Elsevier Sequoia/Printed in The Setherlands
34
sure and a coefficient of shear viscosity 14, 51. Then, in the fluidised-bed range, we expect the particle-stress constitutive equation to be of the form
and the force per unit area exerted on a surface placed perpendicular to the flow is given by
T, = G(Z, 0,)
CT,), =
(6)
where the inclusion of the deformation rate tensor in the list of arguments of the tensor function G is made to describe the fluid-like behaviour mentioned above. When D, vanishes, eqn. (6) must reduce to eqn. (5). This requirement ensures that eqn. (6) describes the particle-phase response in both the frsedbed and the fluidised-bed regimes. In addition to the preceeding particularisation, eqn. (6) implies when D, vanishes that the particle phase satisfies a generalised Mohr-Coulomb criterion_ This has been shown by Cowin [S] in a general forzn. Cowin demonstrates that at equilibrium, which corresponds in our case to the fised bed regime. the stress tensor obeys an equation, whi& in our case may be written as
T, = @(lzzl) -
r(lul))
I +
1 7
lzzl
r(lul)zz@zz
(7)
Further, following Cowin [S], eqn. (7) implies that under limiting equilibrium conditions, which corresponds to the fised bed-fluidised bed transition, the particle stress components obey the relation r.(T,),
--P(1zz1)1’
7 (T,)$ = Iz(lzzl)12
(8)
Equation (8) is a generalised Mohr-Coulomb criterion; i.e., at incipient fluidisation conditions, the bed yields according to the Mohr-Coulomb criterion_
EXPERIMENT
The theory described above requires that the dependence of the solid stress tensor on the fluid-solid relative velocity vector in the fixed bed range be given by eqn. (7). In order to verify this, we measured the forces exerted by the solid, both parallel and perpendicular to uniform flow in a bed in incipient fluidisation conditions. In such an experiment, the force per unit area exerted on a surface aligned parallel to the flow is given by VA,
=
(PWI))
-mm)
(9)
@(lzzl) -
z(lzzl)) + -
1
I4
r(lzzl)
jzzl’
(10)
The experimental set-up consists of a bed through which air or water is metered by rotameters. Operating conditions were given by ambient temperature and pressure. The bed material used is sand with properties summarized in Table 1. The experimental scheme is shown in Fig. 1. TABLE
1
Sand properties Density (g/cm3) Mean particle size (pm) Incipient fiuidisation velocity Incipient fluidisation velocity (cm/s)
in air (cm/s) in water
2.65 512 13.47 0.25
Two measurements were made: total stress and fluid pressure, the difference between them giving the solid-particle stress. In order to measure the total stress in the direction parallel to the relative velocity, the weight of a thin cylinder immersed in the bed was determined by suspending it from one arzn of a pan balance. The total stress in the direction perpendicular to the relative velocity was measured by a strain gauge with its sensing element aligned parallel to the flow. The fluid pressure was read through taps in the bed wall. Probe details are given in Figs_ 2 and 3_ The perturbing effect of the cylinder on flow in the bed was evaluated by inserting an identical hollow cylinder whose lower face consisted of a porous disk and whose hollow interior was connected to a micromanometer, to allow fluid pressure measurement at the lower face of the hollow cylinder_ Similarly, to asess the perturbation caused in the flow by the strain gauge element, a porous disk of the same size and geometry as the gauge element was used to measure the fluid pressure at the gauge element position. The values of fluid pressure measured at the lower face of the cylinder and at the strain gauge element showed insignificant differences from
r !iIr ‘T!ll I
6
9
N
II: E
MATERIAL
I
I
_=
+I :
-z :_ -
zL____:-_ _-2-L _g --_
Fig. 3_ Normal stress probe details.
4
=-Lx ---_--
----z- _-_. -----
the value measured at the wall tap, thus demonstrating that the cylinder and the strain gauge element did not significantly disturb the flow in the bed_ __ _ _. Readings mere taken at several bed anal positions, anticipating non-uniform conditions, in particular voidage- At each position, three measurements were made: fluid pressure, total pressure in the direction of flow_ and total pressure normal to fluid flax.
3
_----
---
POROUS
@
-
Fig- I Experimental scheme; 1, manometer; 2, rotameter; 3, fluid&d bed; 4, pressure tap; 5, cylnder probe; 6, analytical balance.
TO
TRANSDUCER
t
EITHER
STEEL
DIMENSIONS FOR
USE
FOR
TOTAL
MENT FOR MENT
Fig. 2. Parallel stress probe details.
LAMINA. 0 32
WITH
OR FLUID
X
0.1
STRAIN
STRESS WIRE
GRlD
PRESURE
m-n GAUGE
MEASURESCREEU MEASURE
36
Depth
Fig_ 4_ Solid stress difference golid stress normal to flow.
o
(
variation
cm) with
depth:
gas-fluidised
bed;
(Ts)p, solid
to flow;
VAN=
4.5
I” 5
4.0.
-z -z IL ’
3.5
3.0-
E 0 t
235.
2.0
1.5.
‘SO
0.5
5
6 Depth
Fig. 5. Solid stress difference solid stress normal to flow.
7
6
3
10
(cm)
variation with depth: liquid-fluidised
bed; (Ts)p solid stress parallel to flow;
(T.)N
3i
TABLE2 Totalstress,fluidpressure,andparticlestressrGas-fluidised bed Depth
m,
c%.J
Tf
(TSIP
Vsh
(cm)
(cmHz0)
(cmH20)
(cmH20)
(cmH2G)
(cmHzO)
1-o 1.5 2.0 2.5
0.76 1.51 2.27 3.02 3.78
0.88 1.77 2.65 3.53 4.41
0.76 1.51 2.26 3.01
0.00 0.00 0.01 0.01
0.12 O-20 0.39 0.52
-0.12 -0.20 -0.35 -0.51
3.0
4.53
5.30
3-77 4.49
O_Ol 0.04
O-64 0.89
-0.63 --OS5
5.29 6.04 6.80 7.55 8.31 9.06 9.82 10.57 11.33 12.08 12.84 13.59 14.35 15.10
6.18 7.06 7.94 8.83 9-71 10.59 11-47 12.36 13.24 14.12 15.00 15.89 16.77 17.65
5.25 5-99 6-74 7.49 S-24 8.99 9.74 10.49 11.34 11.99 12.73 13.48 14.23 14.96
0.04 0.05 0.06 0.06 o.oi 0.07 0.08 0.08 0.09 0.09 0.11 O-11 0.12 0.14
0.93 1.05 l-20 1.34 1.4'7 1.60 l-73 1-s 2.00 2.13 227 2-41 2.54 2.69
-0.95 -1.02 -1.14 -1.2'8 -1.40 -1.53 -1.65 --1_i9 -l-S9 -2.01 -2.16 -2.30 -2.42 -2.55
o-5
3.5 4.0 4.5 5.0 5.5 6.0 6.5 5.0 7.5 8.0 8.5 9.0 9.5 10.0
TABLE3 TotaIstrw,fluidpressure,andparticIestress:Liquid-fluidisedbed Depth
(TIP
(TIN
Tf
VSIP
V*)s
Vs)P
(cm)
(cmH20)
(cmH20)
(cmH2W
(~HzO)
(cmHzO)
(cmH=O)
O-30 O-i0 1.30 1.95 2.40
o-50 0.30 0.70 0.95 1.20
-0.20 0.40 0.60 1.00 1.20
1.60 ISO
1.10 O-70
2.15 2.35 2.65 2.s5 3.05 3-40 3.60 3.75 4.05 4.30 4_55 4-75
l-70 2.00 2.50 1.60 2.80 3.60 3.90
O-5 1.0 1.5 2.0 2.5 3.0
0.90 l-80 2.80 3-70 4.80 5.50
0.80 1.40 2.20 2.80 3.60 4.40
O-60 1.10 l-50 1.85 2.40 2.80
3.5 4.0
5.80 7.50
5.10 5.80
3.30 3.65
2.70 2.50 3.85
4.5 5.0 5.5 6.0 7-o 7.5 8.0 S-5 9.0 9.5 10.0
8.50 9.70 10.50 11.40 13.40 14.40 15.30 16.20 17-20 18.20 19.10
6-50 7.20 7.90 8.60 9.80 10.50 11so 11-80 12.50 13.20 13.90
4.15 4.55 5.05 3.33 _ -6-40 6.90 7.35 7-75 8.20 8.65 9-15
4.35 5.15 4.45 5.85 7.00 7.50 7.95 8.45 9.00 9.55 9.95
RESULTSANDDISCUSSION The experimental results obtained are summarized in Tables 2 and 3 for the gasand the liquid-fluidied beds respectively, and the solid stress difference variation with depth is shown in Fig. 4 and 5.
-
V,k
420
4.40 +.iO 5.00 5-20
In the gas-fluidised bed, the particle stress component parallel to flow is insignificant in comparison to the fluid pressure (or order l%), whilst the component perpendicular to flow is small but significant (of order 10% of the fluid pressure)_ Consequently, the difference between *he two stress components is
appreciable and readily detectable. In the liquid-fluidised bed, both stress components are appreciable (typically 30% of fluid pressure) and so is the difference between them. It is readily seen from Tables 2 and 3 that fluid pressure, both particle stress components and the particle stress difference all increase with increasing bed depth. The fact that the fluid pressure and the particle stresses increase with depth is expected, as each of these has a gravity (hydrostatic head) component. For fluid velocities in the fixed-bed range, as the cylinder weight technique fails when the bed is not fluidised, a different technique has to be developed for the particle stress component parallel to the direction of flow.
LIST OF SYMBOLS
D f g F, G
m
rate of deformation tensor, s-l subscript denoting fluid phase gravitational acceleration, cm s-’ tensor function specific fluid-solid particle interaction force, dyn g-i
subscript denoting direction normal to flow scalar functions subscript denoting direction parallel to flow subscript denoting solid particle phase time, s stress tensor, dyn cm-’ relative velocity vector, cm s-l velocity vector, cm 55-l density, g cme3 voidage
REFERENCES K. Rietema, Proc Intern. Symp. Nuidisation. Netherlands Univ_ Press, Amsterdam, 1967. In_ P. Gcpalo and G. P. Chevenkov, I Appl. Math. .%Tech PMM. 31 (1967) 603. G. P. Cherepanov, Ind. Eng. Chem. Fundam.. 1I (1972) 9. J_ F. Da&ison and D_ Harrison, Fluidised Particles. Cambridge Univ. Press. Cambridge, 1963. J. F. Davidson, D. Harrison and J. R. F. Guedes de Carvalho, Ann. Rev. Fluid Jfech.. 9 (1977) 55. S. C. Cowin. Acta
Mech..
20
(1974)
41.
Technology,
38 (1984)
3.3
33 - 38
A General Particle Stress Equation R. Y_ QASSIM EE/UFRJond (Received
for Fixed and Fluidised
Beds
and R. DE SOUZA COPPEIUFRJ.
February
15.1983;
CP 1191.
Rio de Janeiro (Brazd)
in revised form June Ii,
1983)
SUhIMARY
THEORY
A general constitutive equation is proposed for the solid particle stress in a fluidised bed_ This equation reduces in a straightforward manner to describe the particle stress in a fixed bed, and it implies a Mohr-Coulomb yieid criterion for the transition from the fued-bed to the fiuidised-bed state. Experimental evidence is given to show that at incipient fluidisation, the particle stress assumes a form agreeing with that predicted by theory_
Considering the bed of solid particles through which fluid is flowing as a binary misture of interacting interpenetrating continuous media, the mass and linear momentum balance equations may be written as
2 + -7 -(p,r,) ah -
at
+ T-(pfl'f)=o
= 0
(1) (2)
INTRODUCTION
When fluid flows vertically upwards through a bed of solid particles, the system can either be in the fixed-bed regime (at rather low fluid velocities) or in the fluidised-bed regime (at higher fluid velocities), transition occurring at the fluid velocity known as the velocity of incipient fluidisation The mechanical response of the solid particle phase of the bed is qualitatively different in the two regimes, being solid-like in the fixed-bed range [l, 2,3] and fluid-like when the bed is fluidised 14, 5]_ The transition from one state to another is considered as a type of Mohr-Coulomb yield phenomenon [I, 2, 33. Although theory is not lacking to describe each of these features separately, there is no general constitutive equation for the solid particle stress which describes in a unified way the f=ed-bed state, the fluidised-bed state, and the transition from one to the other_ In this paper, we propose such an equation and we supply esperimental support for the existence of a stress difference implied by this equation. 0032-5910/84/$3.00
This system of equations is closed with the aid of constitutive equations for T,_ Ti and m. We address ourselves specifically to the solid stress constitutive equation_ Suppose that when there is no fluid flow in the bed, the particle phase behaves as a rigid body_ Then, when fluid flows through the bed in the fised-bed range, we may write for T, the constitutive equation T, = F(Z)
(5)
where we have assumed constant bed voidage. thus eliminating the necessity for including E, VE, etc_, in the list of arguments of the tensor function F_ We note, however, that relaxing the assumption of constant bed voidage does not introduce any conceptual difficulties in the analysisWhen the flow of fluid is such that the bed is fluidised, the particle phase behaves mechanically as a fluid characterised by a pres0 Elsevier Sequoia/Printed in The Setherlands
34
sure and a coefficient of shear viscosity 14, 51. Then, in the fluidised-bed range, we expect the particle-stress constitutive equation to be of the form
and the force per unit area exerted on a surface placed perpendicular to the flow is given by
T, = G(Z, 0,)
CT,), =
(6)
where the inclusion of the deformation rate tensor in the list of arguments of the tensor function G is made to describe the fluid-like behaviour mentioned above. When D, vanishes, eqn. (6) must reduce to eqn. (5). This requirement ensures that eqn. (6) describes the particle-phase response in both the frsedbed and the fluidised-bed regimes. In addition to the preceeding particularisation, eqn. (6) implies when D, vanishes that the particle phase satisfies a generalised Mohr-Coulomb criterion_ This has been shown by Cowin [S] in a general forzn. Cowin demonstrates that at equilibrium, which corresponds in our case to the fised bed regime. the stress tensor obeys an equation, whi& in our case may be written as
T, = @(lzzl) -
r(lul))
I +
1 7
lzzl
r(lul)zz@zz
(7)
Further, following Cowin [S], eqn. (7) implies that under limiting equilibrium conditions, which corresponds to the fised bed-fluidised bed transition, the particle stress components obey the relation r.(T,),
--P(1zz1)1’
7 (T,)$ = Iz(lzzl)12
(8)
Equation (8) is a generalised Mohr-Coulomb criterion; i.e., at incipient fluidisation conditions, the bed yields according to the Mohr-Coulomb criterion_
EXPERIMENT
The theory described above requires that the dependence of the solid stress tensor on the fluid-solid relative velocity vector in the fixed bed range be given by eqn. (7). In order to verify this, we measured the forces exerted by the solid, both parallel and perpendicular to uniform flow in a bed in incipient fluidisation conditions. In such an experiment, the force per unit area exerted on a surface aligned parallel to the flow is given by VA,
=
(PWI))
-mm)
(9)
@(lzzl) -
z(lzzl)) + -
1
I4
r(lzzl)
jzzl’
(10)
The experimental set-up consists of a bed through which air or water is metered by rotameters. Operating conditions were given by ambient temperature and pressure. The bed material used is sand with properties summarized in Table 1. The experimental scheme is shown in Fig. 1. TABLE
1
Sand properties Density (g/cm3) Mean particle size (pm) Incipient fiuidisation velocity Incipient fluidisation velocity (cm/s)
in air (cm/s) in water
2.65 512 13.47 0.25
Two measurements were made: total stress and fluid pressure, the difference between them giving the solid-particle stress. In order to measure the total stress in the direction parallel to the relative velocity, the weight of a thin cylinder immersed in the bed was determined by suspending it from one arzn of a pan balance. The total stress in the direction perpendicular to the relative velocity was measured by a strain gauge with its sensing element aligned parallel to the flow. The fluid pressure was read through taps in the bed wall. Probe details are given in Figs_ 2 and 3_ The perturbing effect of the cylinder on flow in the bed was evaluated by inserting an identical hollow cylinder whose lower face consisted of a porous disk and whose hollow interior was connected to a micromanometer, to allow fluid pressure measurement at the lower face of the hollow cylinder_ Similarly, to asess the perturbation caused in the flow by the strain gauge element, a porous disk of the same size and geometry as the gauge element was used to measure the fluid pressure at the gauge element position. The values of fluid pressure measured at the lower face of the cylinder and at the strain gauge element showed insignificant differences from
r !iIr ‘T!ll I
6
9
N
II: E
MATERIAL
I
I
_=
+I :
-z :_ -
zL____:-_ _-2-L _g --_
Fig. 3_ Normal stress probe details.
4
=-Lx ---_--
----z- _-_. -----
the value measured at the wall tap, thus demonstrating that the cylinder and the strain gauge element did not significantly disturb the flow in the bed_ __ _ _. Readings mere taken at several bed anal positions, anticipating non-uniform conditions, in particular voidage- At each position, three measurements were made: fluid pressure, total pressure in the direction of flow_ and total pressure normal to fluid flax.
3
_----
---
POROUS
@
-
Fig- I Experimental scheme; 1, manometer; 2, rotameter; 3, fluid&d bed; 4, pressure tap; 5, cylnder probe; 6, analytical balance.
TO
TRANSDUCER
t
EITHER
STEEL
DIMENSIONS FOR
USE
FOR
TOTAL
MENT FOR MENT
Fig. 2. Parallel stress probe details.
LAMINA. 0 32
WITH
OR FLUID
X
0.1
STRAIN
STRESS WIRE
GRlD
PRESURE
m-n GAUGE
MEASURESCREEU MEASURE
36
Depth
Fig_ 4_ Solid stress difference golid stress normal to flow.
o
(
variation
cm) with
depth:
gas-fluidised
bed;
(Ts)p, solid
to flow;
VAN=
4.5
I” 5
4.0.
-z -z IL ’
3.5
3.0-
E 0 t
235.
2.0
1.5.
‘SO
0.5
5
6 Depth
Fig. 5. Solid stress difference solid stress normal to flow.
7
6
3
10
(cm)
variation with depth: liquid-fluidised
bed; (Ts)p solid stress parallel to flow;
(T.)N
3i
TABLE2 Totalstress,fluidpressure,andparticlestressrGas-fluidised bed Depth
m,
c%.J
Tf
(TSIP
Vsh
(cm)
(cmHz0)
(cmH20)
(cmH20)
(cmH2G)
(cmHzO)
1-o 1.5 2.0 2.5
0.76 1.51 2.27 3.02 3.78
0.88 1.77 2.65 3.53 4.41
0.76 1.51 2.26 3.01
0.00 0.00 0.01 0.01
0.12 O-20 0.39 0.52
-0.12 -0.20 -0.35 -0.51
3.0
4.53
5.30
3-77 4.49
O_Ol 0.04
O-64 0.89
-0.63 --OS5
5.29 6.04 6.80 7.55 8.31 9.06 9.82 10.57 11.33 12.08 12.84 13.59 14.35 15.10
6.18 7.06 7.94 8.83 9-71 10.59 11-47 12.36 13.24 14.12 15.00 15.89 16.77 17.65
5.25 5-99 6-74 7.49 S-24 8.99 9.74 10.49 11.34 11.99 12.73 13.48 14.23 14.96
0.04 0.05 0.06 0.06 o.oi 0.07 0.08 0.08 0.09 0.09 0.11 O-11 0.12 0.14
0.93 1.05 l-20 1.34 1.4'7 1.60 l-73 1-s 2.00 2.13 227 2-41 2.54 2.69
-0.95 -1.02 -1.14 -1.2'8 -1.40 -1.53 -1.65 --1_i9 -l-S9 -2.01 -2.16 -2.30 -2.42 -2.55
o-5
3.5 4.0 4.5 5.0 5.5 6.0 6.5 5.0 7.5 8.0 8.5 9.0 9.5 10.0
TABLE3 TotaIstrw,fluidpressure,andparticIestress:Liquid-fluidisedbed Depth
(TIP
(TIN
Tf
VSIP
V*)s
Vs)P
(cm)
(cmH20)
(cmH20)
(cmH2W
(~HzO)
(cmHzO)
(cmH=O)
O-30 O-i0 1.30 1.95 2.40
o-50 0.30 0.70 0.95 1.20
-0.20 0.40 0.60 1.00 1.20
1.60 ISO
1.10 O-70
2.15 2.35 2.65 2.s5 3.05 3-40 3.60 3.75 4.05 4.30 4_55 4-75
l-70 2.00 2.50 1.60 2.80 3.60 3.90
O-5 1.0 1.5 2.0 2.5 3.0
0.90 l-80 2.80 3-70 4.80 5.50
0.80 1.40 2.20 2.80 3.60 4.40
O-60 1.10 l-50 1.85 2.40 2.80
3.5 4.0
5.80 7.50
5.10 5.80
3.30 3.65
2.70 2.50 3.85
4.5 5.0 5.5 6.0 7-o 7.5 8.0 S-5 9.0 9.5 10.0
8.50 9.70 10.50 11.40 13.40 14.40 15.30 16.20 17-20 18.20 19.10
6-50 7.20 7.90 8.60 9.80 10.50 11so 11-80 12.50 13.20 13.90
4.15 4.55 5.05 3.33 _ -6-40 6.90 7.35 7-75 8.20 8.65 9-15
4.35 5.15 4.45 5.85 7.00 7.50 7.95 8.45 9.00 9.55 9.95
RESULTSANDDISCUSSION The experimental results obtained are summarized in Tables 2 and 3 for the gasand the liquid-fluidied beds respectively, and the solid stress difference variation with depth is shown in Fig. 4 and 5.
-
V,k
420
4.40 +.iO 5.00 5-20
In the gas-fluidised bed, the particle stress component parallel to flow is insignificant in comparison to the fluid pressure (or order l%), whilst the component perpendicular to flow is small but significant (of order 10% of the fluid pressure)_ Consequently, the difference between *he two stress components is
appreciable and readily detectable. In the liquid-fluidised bed, both stress components are appreciable (typically 30% of fluid pressure) and so is the difference between them. It is readily seen from Tables 2 and 3 that fluid pressure, both particle stress components and the particle stress difference all increase with increasing bed depth. The fact that the fluid pressure and the particle stresses increase with depth is expected, as each of these has a gravity (hydrostatic head) component. For fluid velocities in the fixed-bed range, as the cylinder weight technique fails when the bed is not fluidised, a different technique has to be developed for the particle stress component parallel to the direction of flow.
LIST OF SYMBOLS
D f g F, G
m
rate of deformation tensor, s-l subscript denoting fluid phase gravitational acceleration, cm s-’ tensor function specific fluid-solid particle interaction force, dyn g-i
subscript denoting direction normal to flow scalar functions subscript denoting direction parallel to flow subscript denoting solid particle phase time, s stress tensor, dyn cm-’ relative velocity vector, cm s-l velocity vector, cm 55-l density, g cme3 voidage
REFERENCES K. Rietema, Proc Intern. Symp. Nuidisation. Netherlands Univ_ Press, Amsterdam, 1967. In_ P. Gcpalo and G. P. Chevenkov, I Appl. Math. .%Tech PMM. 31 (1967) 603. G. P. Cherepanov, Ind. Eng. Chem. Fundam.. 1I (1972) 9. J_ F. Da&ison and D_ Harrison, Fluidised Particles. Cambridge Univ. Press. Cambridge, 1963. J. F. Davidson, D. Harrison and J. R. F. Guedes de Carvalho, Ann. Rev. Fluid Jfech.. 9 (1977) 55. S. C. Cowin. Acta
Mech..
20
(1974)
41.