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A note on the initial motion of fluidizatin bubble
Shorter Communications CO? total reactant concentration
in the bulk fluid phase
Greek 0 8 A A*
symbols fraction of bulk fluid reactant which is AI fraction of bulk fluid reactant which is As adjoint variable particular solution to adjoint equations l dimensionless distance (=2/L) I$ ditfusivity ratio (=DpJDIe) #J ditfusivity ratio (=DJD&
Dit? effective diffusion coefficient of species Al h Thiele modulus [=L@j 7; moditied Thiele modulus (= Lgg k L R’ XIX2 Y 2
]ll PI [31 t41
rate constant catalyst half-thickness local reaction rate overall reaction rate dimensionless concentrations extent of reaction distance from catalyst centerline
Subscripts 0 evaluated in bulk fluid m limitingcaseoff$+ m
REFERENCES MAYM6 J. 0. and CUNNINGHAM R. E.,J. Cafaf. 1967 6 186. CARBERRY J. J. and WHITE D., Ind. Engng Chem. 1969 6127. FROBERG C. E., Introduction to NumericalAnalysis. Addison-Wesley, LUSS D. and AMUNDSON N. R. Chem. Engng Sci. 1967 22 253.
Pergamon Press.
Chemical Engineering Science, 197 1, Vol. 26, pp. 995-997.
Reading, Mass. 1965.
Printed in Great Britain,
A note on the initial motion of a fiuidiition
bubble
(Received 16 October 1970) INTRODUCTION MURRAY[ I] has analysed the initial motion of a cylindrical or spherical bubble in a fluidized bed and has shown that a cylindrical bubble accelerates from rest with an acceleration g, while a spherical bubble has an initial acceleration 2 R. He also studied the distortion of these idealized shaped into the characteristic kidney shape possessed by fluidization bubbles and he was able to demonstrate that the typical shape was achieved after the bubble had moved through a distance of the order of one bubble radius. Partridge and Lyall’s experiments[2] were not inconsistent with Murray’s theoretical findings. In this note similar results are produced from a rather simpler analysis. THEORY 1. Initinl motion Imagine a bubble of circular cross section constrained to be stationary in a bed of identical particles fluidized incipiently, as shown in Fig. I. In the three-dimensional situation the bubble is spherical. Rowe [3] has previously considered this situation and he showed that if the constraint is removed from the bubble, then, assuming that their spacing remains unaltered, the particles will begin to move relative to an observer fixed in space along streamlines defined by the irrotational dipole. The flow pattern associated with the initial motion is thus the same as would be produced by a solid cylindrical or spherical body moving through a perfect fluid. The immediate implication of this result is that in the initial motion the bubble retains its form so that the initial
Fig. I. Forces on particles at the poles of a stationary bubble. acceleration of its centre can be identified with the initial acceleration of any particle which subsequently maintains its position relative to the bubble. There are two such particles located at the poles of the bubble, which are stagnation points in the resulting particle flow relative to the bubble. Consider the forces experienced by the particles at the stagnation points as shown in Fig. 1. Because there is an influx of fluid into the stationary bubble, the superficial gas
995
Shorter Communications velocity within the bubble exceeds Umf, the superficial gas velocity in the bed remote from the bubble. It can be shown that the velocity at the poles of a stationary bubble is numi vertically upwards, where n is the number of space dimensions involved in the problem. Rowe and Henwood [4] have shown that the drag force on a particle in an array of particles at a given spacing is a constant multiple of that which would act on the same particle at the same superficial velocity if all its neighbours were removed. At low particle Reynolds number where Stokes’ law applies, the drag will be directly proportional to the fluid velocity Ufi Since those particles remote from the bubble are incipiently fluidized, a force balance on such a particle of mass m shows
forces showed that the drag on a particle in the first layer of an array depends upon the direction of flow. When the array is disposed downstream of the particle, as occurs at the upper stagnation point of a bubble, the drag on a particle in the first layer is typical of that on a particle in the interior of the array and Rowe and Henwood’s results give kl = O-98k.
(5)
When the array is disposed upstream, however, as at the lower stagnation point, then the drag on the particle is higher than on a particle in the interior of the array and kp = 1.33 k.
mg = kU,,,,
(1)
where k is constant for a given particle spacing. The drag forces on the particles situated at the upper and lower stagnation points (denoted by suffixes I and 2 respectively) may be written as (2) where k, and kz have been distinguished from k in order to allow for possible changes in the constant of proportionality brought about by the fact that the particles under consideration are situated in the surface layers and not in the interior of an array. The resultant forces accelerating these particles upwards are then (3) so that on using Eq. (I) the initial accelerations particles are fi.e = (&,/k-
l)g.
of these
Thus while the upper margin of the bubble will accelerate with a value very close to (n- l)g, the lower margin will accelerate more rapidly. The acceleration of the lower margin relative to the upper may be seen to be f = ng(& - k,)/k = 0.35 ng.
2. Distortion Murray’s analysis described the distortion of the bubble during the initial motion whereas the considerations of the previous section specifically exclude this possibility. This exclusion, however, arises from the restriction k = k, = k. and if this is slightly relaxed an estimate of distortion is obtainable. Rowe and Henwood’s experimental measurements of drag l
996
(7)
A displacement of the lower margin relative to the upper of say ia, where a is the bubble radius, would produce a characteristic fluidization bubble shape, and would be achieved in a time t = (a/f) U* = (2*86a/ng) *12.
(8)
This shows, as Murray found, that three-dimensional bubbles distort more rapidly than two-dimensional bubbles, and that the time for distortion decreases with decreasing bubble size. If we retain the estimate of (n - I)g for the acceleration of the bubble as a whole, it is seen that during this distortion, the bubble moves a distance s = 1*43a(n- 1)/n.
(4)
In applying his theory to the initial motion problem, Murray assumed the particle flow to be incompressible and implicitly assumed that his equations, which include a term for the drag experienced by the particles, remained unchanged at the bubble surface where boundary conditions on gas pressure were applied. The same considerations applied here would imply k = k, = & and in fact the production of Rowe’s result concerning the dipole character of particle motion also requires this equivalence. With k = k, = kz then, Eq. (4) becomes an expression for the initial acceleration of the bubble. It follows that for a cylindrical bubble where n = 2, the initial acceleration is g, while for a spherical bubble where n = 3, the initial acceleration is 2 g. These results were also produced by Murray.
(6)
(9)
The distance travelled is thus of the order of one bubble radius in both the two-dimensional and three-dimensional situations. This again reproduces Murray’s result, although the mechanism for distortion is different in the two analyses. In Murray’s analysis the bubble distorts in order to preserve a uniform distribution of gas pressure on its boundary.
Department of Mechanical Engineering University College London London, England
0” f i m n s u:
R. COLLINS
NOTATION bubble radius drag force on particle acceleration acceleration due to gravity constant of proportionality in drag law mass of particle number of space dimensions distance time superticial velocity at minimum fluidization
Shorter Communications
[l] [2] [3] [4]
REFERENCES MURRAYJ. D.,J. FluidMech. 196728417. PARTRIDGE 9. A. and LYALL E.,J. Fluid Mech. 1967 28 429. ROWE P. N., Chem. Engng Sci. 1964 19 75. ROWE P. N. and HENWOOD G. A., Trans. Instn them. Engrs 196139 43.
Chemical Engineering Science, 197 1, Vol. 26, pp. 997-999.
Pergamon Press.
Printed in Great Britain.
Disappearance kinetics for coarse particles in comminution (Received 8 October 1970) INTRODUCTION IT HAS BEEN proposed[
AND THEORY Y’(% 1) =
I, 21 that comminution kinetics in a
batch process is best described differential equation: dM(x, t) ---=--s(x)M(x,r)+ dt
w’ (x. t) w,
(4)
by the following integroand (1)
(5)
where M (x. t) is the mass fraction in size range x to x + dx at grinding time t, S the selection function, B(v, x) the breakage function is the mass fraction of daughter particles reporting to size x when particles of size o are broken. For the feed size, the second term drops out and Eq. (1) reduces to a disappearance rate equation of tirst order for the feed size particles. This suggests that in lieu of the above integro-differential equation, a less complicated (though probably less accurate) mathematical model may be built in order to describe the disappearance of coarse particles as function of time. Such a model tested through some data on dry grinding of quartz in a rod mill is given below. The model rests on a simple rate equation:
where y’(x, r) corresponds to mass fraction of material coarser than size x at time t. If the feed consists of one uniform size between X, and R. x, where R (> 1) is the screen ratio, then for all lower sizes, (x < x,) we have w’ (x. 0) = w,, and Eq. (3) may be re-written:
s(u)B(u,x)M(v,
dw’(x 1) = dr
A
-
r) du
l-8
Also,
w’l+ (x, t) -w”-@ (x. 0)] = - KJ
1 -y’“(x,
t)] = Kxt.
1 -wo’-E[l-{l-y(~,f)}1-8] I-P
(3)
where w’ (x. 0) is the initial amount of material coarser than size x. Furthermore, if the total material being ground is w,,, then
(7)
Y'(X,f)= l-y(x,r)
where y(x, 1) is cumulative fractionfiner (6) then reduces to:
K, . do (x, z)
where w’(x, r) represents the amount of material coarser than size x at time r, K, is size-dependent but time-independent rate constant, and fi is an exponent depending upon material and mill conditions. If )3 = 1, we have a simple first order disappearance kinetics. However, fi need not be 1 and is just an empirical constant not to be confused with the order of a chemical reaction. Similar rate expressions have been used for flotation kinetics model[3] although physical significance of the exponent or “order” is not finally established. Integration of Eq. (2) leads to
-+
1 l_gwol+[
than size x. Equation
= Kxt.
(8)
Kapur[Z] has recently noted that the selection function S(X) in Eq. (1) is dependent on x. Following the analogy, one may propose that Kx in Eq. (8) depends on x and decreases with decrease in size: K, = K,.x”.
(9)
Thus, the final form of Eq. (8) is: 1 ----wo’-~[I 1-B
- { 1 -y(x,
From the derived Eq. x-y-r relationship, s ev e ral (1) For shorter period y(x, I) is a small fraction,
997
r)}l”]
= Koxm~t.
(10)
(10) which is a three parameter useful conclusions may be drawn: of grinding and low value of x, in which case Eq. (8) reduces to:
w,,'-@y (x.
1) = K,t
(11)
in the bulk fluid phase
Greek 0 8 A A*
symbols fraction of bulk fluid reactant which is AI fraction of bulk fluid reactant which is As adjoint variable particular solution to adjoint equations l dimensionless distance (=2/L) I$ ditfusivity ratio (=DpJDIe) #J ditfusivity ratio (=DJD&
Dit? effective diffusion coefficient of species Al h Thiele modulus [=L@j 7; moditied Thiele modulus (= Lgg k L R’ XIX2 Y 2
]ll PI [31 t41
rate constant catalyst half-thickness local reaction rate overall reaction rate dimensionless concentrations extent of reaction distance from catalyst centerline
Subscripts 0 evaluated in bulk fluid m limitingcaseoff$+ m
REFERENCES MAYM6 J. 0. and CUNNINGHAM R. E.,J. Cafaf. 1967 6 186. CARBERRY J. J. and WHITE D., Ind. Engng Chem. 1969 6127. FROBERG C. E., Introduction to NumericalAnalysis. Addison-Wesley, LUSS D. and AMUNDSON N. R. Chem. Engng Sci. 1967 22 253.
Pergamon Press.
Chemical Engineering Science, 197 1, Vol. 26, pp. 995-997.
Reading, Mass. 1965.
Printed in Great Britain,
A note on the initial motion of a fiuidiition
bubble
(Received 16 October 1970) INTRODUCTION MURRAY[ I] has analysed the initial motion of a cylindrical or spherical bubble in a fluidized bed and has shown that a cylindrical bubble accelerates from rest with an acceleration g, while a spherical bubble has an initial acceleration 2 R. He also studied the distortion of these idealized shaped into the characteristic kidney shape possessed by fluidization bubbles and he was able to demonstrate that the typical shape was achieved after the bubble had moved through a distance of the order of one bubble radius. Partridge and Lyall’s experiments[2] were not inconsistent with Murray’s theoretical findings. In this note similar results are produced from a rather simpler analysis. THEORY 1. Initinl motion Imagine a bubble of circular cross section constrained to be stationary in a bed of identical particles fluidized incipiently, as shown in Fig. I. In the three-dimensional situation the bubble is spherical. Rowe [3] has previously considered this situation and he showed that if the constraint is removed from the bubble, then, assuming that their spacing remains unaltered, the particles will begin to move relative to an observer fixed in space along streamlines defined by the irrotational dipole. The flow pattern associated with the initial motion is thus the same as would be produced by a solid cylindrical or spherical body moving through a perfect fluid. The immediate implication of this result is that in the initial motion the bubble retains its form so that the initial
Fig. I. Forces on particles at the poles of a stationary bubble. acceleration of its centre can be identified with the initial acceleration of any particle which subsequently maintains its position relative to the bubble. There are two such particles located at the poles of the bubble, which are stagnation points in the resulting particle flow relative to the bubble. Consider the forces experienced by the particles at the stagnation points as shown in Fig. 1. Because there is an influx of fluid into the stationary bubble, the superficial gas
995
Shorter Communications velocity within the bubble exceeds Umf, the superficial gas velocity in the bed remote from the bubble. It can be shown that the velocity at the poles of a stationary bubble is numi vertically upwards, where n is the number of space dimensions involved in the problem. Rowe and Henwood [4] have shown that the drag force on a particle in an array of particles at a given spacing is a constant multiple of that which would act on the same particle at the same superficial velocity if all its neighbours were removed. At low particle Reynolds number where Stokes’ law applies, the drag will be directly proportional to the fluid velocity Ufi Since those particles remote from the bubble are incipiently fluidized, a force balance on such a particle of mass m shows
forces showed that the drag on a particle in the first layer of an array depends upon the direction of flow. When the array is disposed downstream of the particle, as occurs at the upper stagnation point of a bubble, the drag on a particle in the first layer is typical of that on a particle in the interior of the array and Rowe and Henwood’s results give kl = O-98k.
(5)
When the array is disposed upstream, however, as at the lower stagnation point, then the drag on the particle is higher than on a particle in the interior of the array and kp = 1.33 k.
mg = kU,,,,
(1)
where k is constant for a given particle spacing. The drag forces on the particles situated at the upper and lower stagnation points (denoted by suffixes I and 2 respectively) may be written as (2) where k, and kz have been distinguished from k in order to allow for possible changes in the constant of proportionality brought about by the fact that the particles under consideration are situated in the surface layers and not in the interior of an array. The resultant forces accelerating these particles upwards are then (3) so that on using Eq. (I) the initial accelerations particles are fi.e = (&,/k-
l)g.
of these
Thus while the upper margin of the bubble will accelerate with a value very close to (n- l)g, the lower margin will accelerate more rapidly. The acceleration of the lower margin relative to the upper may be seen to be f = ng(& - k,)/k = 0.35 ng.
2. Distortion Murray’s analysis described the distortion of the bubble during the initial motion whereas the considerations of the previous section specifically exclude this possibility. This exclusion, however, arises from the restriction k = k, = k. and if this is slightly relaxed an estimate of distortion is obtainable. Rowe and Henwood’s experimental measurements of drag l
996
(7)
A displacement of the lower margin relative to the upper of say ia, where a is the bubble radius, would produce a characteristic fluidization bubble shape, and would be achieved in a time t = (a/f) U* = (2*86a/ng) *12.
(8)
This shows, as Murray found, that three-dimensional bubbles distort more rapidly than two-dimensional bubbles, and that the time for distortion decreases with decreasing bubble size. If we retain the estimate of (n - I)g for the acceleration of the bubble as a whole, it is seen that during this distortion, the bubble moves a distance s = 1*43a(n- 1)/n.
(4)
In applying his theory to the initial motion problem, Murray assumed the particle flow to be incompressible and implicitly assumed that his equations, which include a term for the drag experienced by the particles, remained unchanged at the bubble surface where boundary conditions on gas pressure were applied. The same considerations applied here would imply k = k, = & and in fact the production of Rowe’s result concerning the dipole character of particle motion also requires this equivalence. With k = k, = kz then, Eq. (4) becomes an expression for the initial acceleration of the bubble. It follows that for a cylindrical bubble where n = 2, the initial acceleration is g, while for a spherical bubble where n = 3, the initial acceleration is 2 g. These results were also produced by Murray.
(6)
(9)
The distance travelled is thus of the order of one bubble radius in both the two-dimensional and three-dimensional situations. This again reproduces Murray’s result, although the mechanism for distortion is different in the two analyses. In Murray’s analysis the bubble distorts in order to preserve a uniform distribution of gas pressure on its boundary.
Department of Mechanical Engineering University College London London, England
0” f i m n s u:
R. COLLINS
NOTATION bubble radius drag force on particle acceleration acceleration due to gravity constant of proportionality in drag law mass of particle number of space dimensions distance time superticial velocity at minimum fluidization
Shorter Communications
[l] [2] [3] [4]
REFERENCES MURRAYJ. D.,J. FluidMech. 196728417. PARTRIDGE 9. A. and LYALL E.,J. Fluid Mech. 1967 28 429. ROWE P. N., Chem. Engng Sci. 1964 19 75. ROWE P. N. and HENWOOD G. A., Trans. Instn them. Engrs 196139 43.
Chemical Engineering Science, 197 1, Vol. 26, pp. 997-999.
Pergamon Press.
Printed in Great Britain.
Disappearance kinetics for coarse particles in comminution (Received 8 October 1970) INTRODUCTION IT HAS BEEN proposed[
AND THEORY Y’(% 1) =
I, 21 that comminution kinetics in a
batch process is best described differential equation: dM(x, t) ---=--s(x)M(x,r)+ dt
w’ (x. t) w,
(4)
by the following integroand (1)
(5)
where M (x. t) is the mass fraction in size range x to x + dx at grinding time t, S the selection function, B(v, x) the breakage function is the mass fraction of daughter particles reporting to size x when particles of size o are broken. For the feed size, the second term drops out and Eq. (1) reduces to a disappearance rate equation of tirst order for the feed size particles. This suggests that in lieu of the above integro-differential equation, a less complicated (though probably less accurate) mathematical model may be built in order to describe the disappearance of coarse particles as function of time. Such a model tested through some data on dry grinding of quartz in a rod mill is given below. The model rests on a simple rate equation:
where y’(x, r) corresponds to mass fraction of material coarser than size x at time t. If the feed consists of one uniform size between X, and R. x, where R (> 1) is the screen ratio, then for all lower sizes, (x < x,) we have w’ (x. 0) = w,, and Eq. (3) may be re-written:
s(u)B(u,x)M(v,
dw’(x 1) = dr
A
-
r) du
l-8
Also,
w’l+ (x, t) -w”-@ (x. 0)] = - KJ
1 -y’“(x,
t)] = Kxt.
1 -wo’-E[l-{l-y(~,f)}1-8] I-P
(3)
where w’ (x. 0) is the initial amount of material coarser than size x. Furthermore, if the total material being ground is w,,, then
(7)
Y'(X,f)= l-y(x,r)
where y(x, 1) is cumulative fractionfiner (6) then reduces to:
K, . do (x, z)
where w’(x, r) represents the amount of material coarser than size x at time r, K, is size-dependent but time-independent rate constant, and fi is an exponent depending upon material and mill conditions. If )3 = 1, we have a simple first order disappearance kinetics. However, fi need not be 1 and is just an empirical constant not to be confused with the order of a chemical reaction. Similar rate expressions have been used for flotation kinetics model[3] although physical significance of the exponent or “order” is not finally established. Integration of Eq. (2) leads to
-+
1 l_gwol+[
than size x. Equation
= Kxt.
(8)
Kapur[Z] has recently noted that the selection function S(X) in Eq. (1) is dependent on x. Following the analogy, one may propose that Kx in Eq. (8) depends on x and decreases with decrease in size: K, = K,.x”.
(9)
Thus, the final form of Eq. (8) is: 1 ----wo’-~[I 1-B
- { 1 -y(x,
From the derived Eq. x-y-r relationship, s ev e ral (1) For shorter period y(x, I) is a small fraction,
997
r)}l”]
= Koxm~t.
(10)
(10) which is a three parameter useful conclusions may be drawn: of grinding and low value of x, in which case Eq. (8) reduces to:
w,,'-@y (x.
1) = K,t
(11)