- Email: [email protected]
The behaviour of a continuously bubbling fluidised bed
Chemical Engineering Science, 1966, Vol. 21, pp. 731-738. Pergamon Press Ltd., Oxford.
Printed in Great Britain.
The behaviour of a continuously bubbling fluidised bed J. F. DAVIDSON Department
and D. HARRISON
of Chemical Engineering, Pembroke St., Cambridge (Received 18 April 1966)
Abstract-This paper considers two aspects of the behaviour of a bubbling fluidised bed, namely (a) How is the rising velocity of a bubble in a cloud of bubbles related to the velocity of the same bubble in complete isolation? This is important in the consideration of the expansion of a bed of particles dud to bubbles rising through it. (b) How is the total flow of fluidising fluid to the bed divided between the bubble and particulate phases? New theory is given which partly justifies the assumption that the total flow is equal to the flow due to the rising bubbles plus the flow required for incipient fluidisation.
Uh
INTRODUCTION
II I
THIS PAPERdeals
with two aspects of the behaviour of a bubbling fluidised bed, namely (a) the expansion of the bed of particles due to the rising bubbles, and (b) the division of total flow of fluidising fluid between the bubble and particulate phases. New theory is given here which partly justifies the assumption that the total flow is equal to the flow due to the rising bubbles plus the flow required for incipient fluidisation. These two questions are discussed by TURNER [l] in a note criticising results given in [2]; he is mainly concerned with Question (a), and in particular with the rising velocity of bubbles in a continuously bubbling fluidised or gas-liquid system. THE RATE OF RISE OF BUBBLES IN A CLOUD
We shall begin with the gas-liquid system since it is easier to think about than the fluidised bed, and the simplest flow situation to visualise is when bubbles are held tied by a downward flow of liquid as shown in Fig. 1. The superficial liquid velocity is then U,; this is also the rising velocity of the same bubble cloud in a fixed vertical cyclinder closed at both ends as in [2] Fig. 10(b), since the cloud can be held fixed by moving the cylinder downwards at velocity U,. With the cloud fixed, and putting sb = (bubble volume)/(total volume),
FIG. 1. Bubbles fixed in position by downward flow of liquid.
the interstitial liquid velocity is UJ(1 - s&, and Turner’s discussion [l] can be summarised in the single question: what is the ratio of U, to the rising velocity U,,, of a single bubble in an infinite stagnant liquid? Some possible answers to this question are as follows. (1) TURNER[l] assumes that it would be consistent with the assumptions in [2] to put the interstitial velocity equal to U,, and therefore u, = (1 - Eb)Uboo.
(1)
(2) In [2] it was assumed, by analogy with slug Bow that u, = u,,.
(2)
(3) If the bubbles are assumed to behave like solid particles of zero density we could use the result given by RICHARDSON and ZAKI [3], which in the present notation is 731
J. F. DAVIDSONand D.
G
-TT
HARRISON
and from Eq. (1) U = &bUbm,
= (1 - &J,
“bco
where n is a function of the Reynolds number Re = Uba,Delv, D, being the diameter of the bubbles (assumed spherical), and v the kinematic viscosity of the liquid. Which of the assumptions (l), (2) or (3) is more likely to be valid will only be settled by experiment, and the experiments are difficult, because it is necessary to measure the rising velocity of a bubble cloud and the diameters of the individual bubbles. However, data for clouds of small continuously generated bubbles are given by BRIDGE et al. [4]. To compare these data for zero liquid flow with Eqs. (I), (2) and (3), we add an upward velocity of U,/(l - eb) to the flow situation of Fig. 1, giving a net interstitial liquid velocity of zero. This means that the bubbles are being continuously generated, and to obtain the resulting flow of rate of gas, its velocity &/(l - Ed) is multiplied by the area .sb available for gas flow in a unit horizontal crosssection, giving the superficial gas velocity U = Ubgb/(l
-
&b)r
from Eq. (2) u = &b&,/(1 - &b),
(5)
from Eq. (3) u = &b&,(1 - &$-I
(6)
BRIDGE et al. [4] give data for air bubbling through glycerine-water, water containing surfactant, and pure water; of the three systems, the last is most likely to be relevant to the fluidised bed for which viscous effects are small and surface active effects do not apply. Figure 2 shows Bridge et d’s data for air-water with no water flow compared with curves based on Eq. (4) (5) and (6), using ub, = 23.5 cm/set [4], and n = 2.39 [3] since Re -h 800. Equation (6) agrees remarkably well with the data in Fig. 2 (the equation also agrees well with Bridge et d’s data in their Fig. 9 for glycerine-water, in conflict with Bridge et al.3 finding that their data do not agree with the correlation of RICHARDSON and ZAKI). The agreement with RICHARDSON and ZAKI’S correlation in Fig. 2 shows that the small bubbles (== 3.5 mm dia. for air-water) do indeed behave like solid spheres of zero density. For these bubbles the
0 .----it7 /4
0.20
0.16
-i
(6)
Equation Ref.
[3]
’
0.04
.O
FIG. 2.
(4) [2]
1 : 3
Superficial
(4)
velocity
U,
Data for air bubbling through water [4]. 0 57.25 cm,
732
cm/set
l
103.15 cm, A 168.7
cm above sparger.
The behaviour of a continuously bubbling fluidised bed
Weber number = D,pUjf ,/T = 2.8 where p is the liquid density and Tthe surface tension; referring to HABERMANand MORTON’S[5] Fig. 17, it would appear that these bubbles are in the uncertain transition region between the spherical and spherical-cap shapes. It seems unlikely that such bubbles would have fully developed wakes; their rate of rise would be determined partly by viscous forces [6] and they would probably be of ellipsoidal shape. It is not at all clear that the behaviour of a cloud of such bubbles is relevant to the fluidised bed problem, where we are concerned with the behaviour of spherical-cap bubbles, and it is quite conceivable that spherical-cap bubbles would rise as fast in a cloud as in isolation; in the absence of experimental evidence, one can only ask whether a given bubble in a cloud is speeded by being in the wake of bubbles above, or slowed by the bubbles on either side. At present this is an open question, but it is noteworthy that another critic [7] has said that the formula in [2] gives a rising velocity too low, while in [I] the formula is said to give a velocity which is too high! We now turn to the question of the division of the total flow of fluidising fluid between the bubble and particulate phases. THE DIVISIONOF FLOW BETWEENBUBBLE ANDPARTICULATE PHASES The problem here is to decide for a continuously bubbling fluidised bed, how much of the total flowU per unit cross sectional area-shall be allotted to the bubble phase and how much to the particulate phase. The division of flow is an important factor in determining the behaviour of catalytic reactors, because the fluid in the bubble phase makes less direct contact with the particles than the fluid in the particulate phase. Some suggestions as to the division of flow are as follows. (1) It has been proposed that U = U, + bubble flow where U = U,, at incipient fluidisation, and “bubble flow” means the flow accounted for by empty bubbles moving up the bed. This proposal is supported by certain evidence [2] p. 42 and 61, though this is slender. (2) It has been suggested that the flow allotted to the particulate phase should be U,(l - sb) because
in a typical cross-section, a fraction .sbof the area is occupied by bubbles, and consequently the area of particles, seen in plan to be fluidised, is less than at incipient fluidisation when there are no bubbles. This argument overlooks the flow of fluidising fluid through the bubbles, which will be discussed below. [3]. Another equally plausible suggestion is that each rising bubble carries up not only the fluid within the space empty of particles, but also the fluid within the “cloud’ associated with the rising bubble. Ample evidence [8] shows that the cloud does indeed move with the bubble. But each cloud owes its existence partly to the upward movement of particles near a bubble, and since there is no net flow of particles there must be particles moving downwards in regions away from bubbles; these downward moving particles will tend to drag fluid with them, and offset the upward fluid movement due to bubbles. Analysis of flow
We now present an analysis of the fluid flow in a continuously bubbling fluidised bed, based on the following assumptions : (a) The voidage fraction in the particulate phase is assumed constant = sO, the voidage fraction at incipient fluidisation. (b) Within the particulate phase, the relative motion between the fluid and the particles is governed by Darcy’s law, so that neglecting the inertia of the fluid, its velocity is u”= v”- Kgradp,
(7)
where i; is the particle velocity, p the fluid pressure and K the constant of Darcy’s law. (c) The fluidising fluid is assumed to be incompressible. These assumptions have been used to analyse the motion of fluid near a rising bubble [2] p. 63-79, and to obtain the degree of conversion with a chemical reaction in the particulate phase [9]. Other examples include flow near a rising slug in a tube [lo] and in the neighbourhood of an orifice through which fluidised particles are flowing [ll]. In all these cases there was reasonable agreement between theory and experiment. But for the single rising bubble, cloud diameters measured by ROWE et al. [8] are in slightly better agreement with the more
733
J. F. DAVIDSONand D. HARRISON
good for unsteady motions, because with a fixed frame of reference and freely rising bubbles, we are dealing with a time-dependent motion at all points. With this result in mind, we compare the flow situation in Fig. 3(a) and (b).
complex theories Of JACKSON[ 121and MURRAY[ 131 than with the simple theory based on the above assumptions. Jackson’s theory allows for the variation in voidage near a bubble, and ,for the consequent variation in K, but the variations in voidage are not large, typically from E,, = 0.5 to about E = O-6 just above a rising bubble [lo] ; moreover the theory cannot allow for interparticle forces in regions of the bed where E tries to become smaller than so. MURRAY’S[13] theory in contrast assumes constant voidage and DAKCYconstant K, but because of a linearisation of the particle mass-acceleration equation, gives cloud diameters almost the same as Jackson’s theory.
Particles fixed
Figure 3(a) shows a normal freely bubbling bed of particles fluidised by a superficial fluid velocity U, the diameter of the bed being much larger than the diameter of any of the bubbles. Figure 3(b) shows the particles, and hence the bubbles, in the same position, but each particle is held fixed by an external force, and the fluid pressure immediately above the distributor is tied at p. as in Fig. 3(a); p0 is just enough to support the weight of the particles, as it must be for a freely bubbling fluidised bed at all values of U > U,, from a momentum balance on the particles. Then from the above theorem, the fluid pressure at every point in the particulate phase must be the same in Figs. 3(a) and (b). Thus we can get the total fluid flow in the freely bubbling bed, Fig. 3(a), by first of all finding the flow with the bubbles tied, and then adding on the effect of the bubble motion. To get the superficial velocity kU,, we have to solve Laplace’s Eq. (8) governing percolation between the fixed bubbles, within each of which there is a (different) constant pressure. The flow pattern in this case can be compared with the flow at incipient fluidisation represented in Fig. 3(c). In Figs. 3(b) and (c) we have the same volume of particulate phase and the same base pressure p,,, and it is not altogether clear whether the presence of the
Pressure distribution theorem
Thus there is much to be said for the simple theory based on assumptions (a) (b) and (c) above, and a variety of experimental evidence shows that the theory is not seriously in error. For our purpose the most important consequence of the simple theory is that the pressure distribution is independent of the particle motion for given pressure boundary conditions. This follows immediately by using Eq. (7) with the continuity equations for particles and fluid, div fi = 0 and div fi = 0, which hold good even for unsteady motion because both the particulate phase and the fluid are effectively incompressible; combining these continuity equations with Eq. [7] we have div grad p = 0,
(8)
the equation for percolation through either fixed or moving particles. It is important that Eq. (8) holds
l-----l OOO
0
000
x--
x--
&~_“* 00
0
0
00
Ora 0
--_--
Superficial velocity
FIG. 3.
U
(a)
--x
Ocl
0
,“,o
ttt
W
(b)
p0
-----
~ ttt
UO
po
(cl
(a) Freely bubbling bed. (b) Particles and bubbles held fixed in position (a). (c) Incipient fluidisation.
734
The behaviour of a continuously
fixed bubbles will increase or decrease the flow, i.e. whether k is greater or less than unity. It seems probable that k will in fact be greater than unity. Certainly with bubbles of finite size and close together, the particles between the bubbles would be by-passed and k > 1. Calculations are in progress to find k with evenly spaced bubbles of uniform size. Through each fixed bubble in Fig. 3(b) there will be a fluid flow, and we can get the contribution of bubble and particulate phases by integrating across a typical section XX to get the total flow through the bed, whose cross-sectional area is A, so that ALU,
= .cO u” .dx+B,=F,+B,,, s PJ
bubble surface, and i& is parallel to da”because the pressure is constant within the bubble. With the bubble moving, particles at the surface have velocity ii, and the fluid there has velocity ij -I- &, from the pressure distribution theorem. Hence for a single bubble the total fluid flow across XX between A and C in Fig. 4 must be ABM = Ed ii, + i-Q.da+ sA
(9)
cles,
‘a.dC, s
(12)
A
and the integral in Eq. (12) represents the flow between A and C due to movement of the bubble surface if it were continuous. This is true even when the bubble velocity is unsteady, as when it is about to coalesce with another bubble. Summing equations like (12) for all bubbles cut by XX in Fig. 3(a) gives the flow in bubbles
moving
Fig. 3(a), the section cutting the same bubbles, but each is moving upwards with its appropriate velocity, not necessarily the same for every bubble. In the particulate phase, the particles are moving with velocity v”, but from the above theorem, the pressure distribution is unaffected by the motion. Hence the fluid velocity at any point is ~7~+ ii and the flow through the particulate phase cut by XX s
(11)
s
ABM = AB, +
We now consider the same cross-section XX in
is E,, p(z2f + @.dA”.
(I - E,,) j.dE.
The first term is due to the flow of fluid at the bubble surface, and the second represents the displacement of fluid caused by the motion of the particles at the bubble surface. From Eq. (10) and (11) we have
where u”/ is the interstitial velocity (not necessarily vertical) and dA” is an upwards pointing vector representing an element of area; B, is the flow through bubbles intersected by XX, and the integral is over the area of particles cut by XX. Particles
bubbling fluidised bed
C
=B,+
c bubbla
sA
ii.da”.
(13)
It was shown above that the flow in the particulate phase is Fp, whether or not the bubbles are moving, and adding this on to Eq. (13) and then using Eq. (9) gives the total flow for a freely bubbling bed,
But for no net motion of partiAU
=
AkU,
+
C
:.dS = ALU,, + Qrr .
bubbles s A
i7d.A” = 0, since we are integrating across an
s area lirge compared with the size of the bubbles, and hence when the latter are moving, the flow through the particulate phase is still Fp. We now consider the bubbles cut by XX in Fig. 3(a), and Fig. 4 shows a typical bubble. At the surface in Fig. 4, the fluid interstitial velocity is z& when the bubble is jxed, and consequently the flow across XX between points A and C is for one bubble
(14) In Eq. (14), QB is the observed bubble flow even if the bubbles are not all the same size, nor need they
n
+__A___________
C
AB, = Ed i&d& (10) sA where da” is a vector representing an element of 735
FIG.
4.
c-_-x
Section through typical bubble.
J.
F. DAVIDSONand D. HARRISON
Since k is unlikely to be very different from unity, unless the bubbles are extremely close together, the rather widely used assumption that the total flow = (flow at incipient fluidisation) plus (bubble flow) would seem to have good theoretical justification. move with steady velocities.
Fluid velocities within the bubbles
The derivation of Eq. (14) depends on the theorem that the pressure distribution in the particulate phase is independent of particle motion, so that (fluid velocity) = (particle velocity) plus (fluid velocity with fixed particles). This does not apply inside the bubbles, where the motion of the particles forming the bubble surface may set up a vortex motion [2] p. 95, which will not, however, invalidate the above argument based as it was on a discussion of the particle and fluid motions within the particulate phase up to the surface of each bubble but not inside it. Bubble and bed diameters comparable
The above derivation requires modification when the ratio (bubble diameter D,)/(bed diameter 0) is not small. In this case, qP =
s
5. dx is not zero at
any instant, because althoughPthe mean particle Sow across XX in Fig. 3 is zero when averaged over a long period, the instantaneous particle flow is upwards at one instant and downwards at another. This is clear for the limiting case of slug flow as shown in Fig. 5; when XX cuts through a gas slug, position 1, particle flow may be entirely downwards, but when XX cuts the particulate phase between slugs, position 2, the particle flow must be upwards. Similar considerations will apply when 0,/D is such that the bubbles are large but not large enough to be termed slugs. However, when DJD is not small, the above theory can be appropriately modified. Equation (9) still holds good with k constant, because with the bubbles fixed, the total flow must be independent of the position of XX, but FP and B, will be functions of the position of XX, so that for slug flow, BO is finite in position 1 but zero in position 2. When the bubbles move, the fluid flow rate through the particulate phase cut by XX will be
FIG. 5.
Slug flow in a fluidised bed.
co $j s
+ iif). dA” = eoqp + FP.
WI
At any instant the flow rate through bubbles cut by XX is still given by Eq. (13), because the arguments leading to this equation do not depend upon 0,/D being small. Hence we can combine Eqs. (9), (13) and (15) to give the instantaneous fluid flow rate through section XX AU’ = eoqp + ARU, + QB.
(16)
s T
Now the mean superficial velocity U =
_n -
U’dtlT,
where T is a time long compared with the time for one large bubble to pass a fixed position. But T
q,dt = 0, since there is no net flow of particles,
s0 and since
‘Q8dt/T means the average flow due to
bubbles crossing a fixed position we deduce that U = kUo + bubble flow as before. For slug flow it seems certain that k will not be greatly different from unity; with small slugs far apart, k must be very nearly unity, and
736
The behaviour
of a continuously
with large slugs close together, k > 1 because the particles falling down the wall at the sides of the slug will be by-passed.
bubbling fluidised bed
D DC FP
k K n P PO
CONCLUSION The theory given partly justifies the assumption that the total flow of fluidising fluid to the bed is equal to the flow due to the rising bubbles plus the flow required for incipient fluidisation. The complete analysis waits upon the determination of the constant k. It is shown that the assumption also applies when the bubbles vary in size, or move with unsteady velocities, or when the ratio (bubble dia.)/(bed dia.) is not small.
4p
instantaneous
QB
observed bubble flow detined by Eq. (14).
Re
u u
NOTATION
ub
A Bo
cross-sectional area of the bed flow per unit cross-section through bubbles intersected by XX (Fig. 3(a)) ABo Bow in a fixed bubble across XX between A and C (Fig. 4) ABM flow in a moving bubble across XX between A and C (Fig. 4) vector area of element of bubble surface vector of an element of area
bed dia. equivalent bubble diameter = (6 bubble volume/#/s flow through particulate phase constant constant of Darcy’s law exponent in equation of RICHARDSON and ZAKI fluid pressure pressure just sufficient to support the weight of particles (Fig. 3)
ubcc
uo e & co Eb v
P
particle flow =
sP
a.dG.
UbmDc/V
time surface tension; long period of time interstitial fluid velocity interstitial fluid velocity with particles fixed interstitial fluid velocity at surface of fixed bubble (Fig. 4) superficial fluid velocity instantaneous superficial fluid velocity superficial liquid velocity to hold a bubble cloud fixed (Fig. 1) rising velocity of a single bubble in an intinite stagnant liquid superficial fluid velocity at incipient fluidisation particle velocity voidage fraction voidage fraction at incipient fluidisation bubble volume per unit volume kinematic viscosity of liquid liquid density
REFEREXES TURNER,J. C. R., Chem. Engng Sci., to be published. DAVIDSONJ. F. and HARRISOND., Fluidixed Particles, Cambridge University Press, 1963. RICHARDSONJ. F. and ZAKI W. N., Trans. In&n. Chem. Engrs, 1954 32 35. BRIDGEA. G., LAPIDUSL. and ELGIN J. C., A. I. Ch. E. J., 1964 10 819. HABERMANW. L. and MORTONR. K., U.S. Navy Dept (D. W. Taylor Model Basin) Report 802 1953. Mooas D. W., J. FIuid Mech., 1963 16 161. GILLILANDE. R., A. I. Ch. E. J., 1964 10 783. ROWE P. N., PARTRIDGEB. A. and LYALLE., Chem. Engng Sci., 1964 19 973. ROSE P. L., Ph.D dissertation, University of Cambridge 1965. STEWARTP. S. B., Ph.D. dissertation, University of Cambridge 1965. JONESD. R. M. and DAVIDSONJ. F., Rheol. Acta, 1965 4 180. JACKSONR., Trans. Znstn. Chem. Engrs., 1963 41 22. MURRAYJ. D., J. Fluid. Mech., 1965 21 465; 1965 22 57.
R&u&-Cet article ttudie deux aspects du comportement d’un lit fluidisd en bullage, a savoir: (a) Quelle est la relation entre la vitesse d’ascension dune bulle dans un nuage de bulles et la vitesse de la msme bulle isol& Cette question est importante lorsqu’on considere l’expansion d’un lit de particules due au passage de bulles. (b) Comment se partage le debit total de fluide, utilise pour la mise en fluidisation du lit, entre les bulles et la phase dense. Les auteurs proposent une nouvelle theorie qui justi6e partiellement l’hypothese que le debit total est 6gal a la somme des debits dus aux bulles et du debit necessaire au minimum de fluidisation.
737
J. F. DAVIDKINand D. HARRISON Zusammenfassung-Dieser Bericht beantwortet zwei Fragen zur Verhaltungsweise einer brodelnden Wirbelschicht : (a) Wei verh5lt sich die Steiggeschwindigkeit einer Blase innerhalb einer Blasenschau zur Steiggeschwindigkeit emer einzelnen, isolierten Blase? Betrachtungen dieser Art sind wichtig im Zusammenhang mit der Bettausdehnung durch aufsteigende Blasen. (b) Wie verteilt sich das aufwirbelnde Medium zwischen einzelnen Blasen und zusammenharrgender Phase? Neueren Theorien zufolge scheint der Gesamtdurchsatz aus der Summe von Blasendurchsatz und Mindestwirbeldurchsatz zu bestehen.
738
Printed in Great Britain.
The behaviour of a continuously bubbling fluidised bed J. F. DAVIDSON Department
and D. HARRISON
of Chemical Engineering, Pembroke St., Cambridge (Received 18 April 1966)
Abstract-This paper considers two aspects of the behaviour of a bubbling fluidised bed, namely (a) How is the rising velocity of a bubble in a cloud of bubbles related to the velocity of the same bubble in complete isolation? This is important in the consideration of the expansion of a bed of particles dud to bubbles rising through it. (b) How is the total flow of fluidising fluid to the bed divided between the bubble and particulate phases? New theory is given which partly justifies the assumption that the total flow is equal to the flow due to the rising bubbles plus the flow required for incipient fluidisation.
Uh
INTRODUCTION
II I
THIS PAPERdeals
with two aspects of the behaviour of a bubbling fluidised bed, namely (a) the expansion of the bed of particles due to the rising bubbles, and (b) the division of total flow of fluidising fluid between the bubble and particulate phases. New theory is given here which partly justifies the assumption that the total flow is equal to the flow due to the rising bubbles plus the flow required for incipient fluidisation. These two questions are discussed by TURNER [l] in a note criticising results given in [2]; he is mainly concerned with Question (a), and in particular with the rising velocity of bubbles in a continuously bubbling fluidised or gas-liquid system. THE RATE OF RISE OF BUBBLES IN A CLOUD
We shall begin with the gas-liquid system since it is easier to think about than the fluidised bed, and the simplest flow situation to visualise is when bubbles are held tied by a downward flow of liquid as shown in Fig. 1. The superficial liquid velocity is then U,; this is also the rising velocity of the same bubble cloud in a fixed vertical cyclinder closed at both ends as in [2] Fig. 10(b), since the cloud can be held fixed by moving the cylinder downwards at velocity U,. With the cloud fixed, and putting sb = (bubble volume)/(total volume),
FIG. 1. Bubbles fixed in position by downward flow of liquid.
the interstitial liquid velocity is UJ(1 - s&, and Turner’s discussion [l] can be summarised in the single question: what is the ratio of U, to the rising velocity U,,, of a single bubble in an infinite stagnant liquid? Some possible answers to this question are as follows. (1) TURNER[l] assumes that it would be consistent with the assumptions in [2] to put the interstitial velocity equal to U,, and therefore u, = (1 - Eb)Uboo.
(1)
(2) In [2] it was assumed, by analogy with slug Bow that u, = u,,.
(2)
(3) If the bubbles are assumed to behave like solid particles of zero density we could use the result given by RICHARDSON and ZAKI [3], which in the present notation is 731
J. F. DAVIDSONand D.
G
-TT
HARRISON
and from Eq. (1) U = &bUbm,
= (1 - &J,
“bco
where n is a function of the Reynolds number Re = Uba,Delv, D, being the diameter of the bubbles (assumed spherical), and v the kinematic viscosity of the liquid. Which of the assumptions (l), (2) or (3) is more likely to be valid will only be settled by experiment, and the experiments are difficult, because it is necessary to measure the rising velocity of a bubble cloud and the diameters of the individual bubbles. However, data for clouds of small continuously generated bubbles are given by BRIDGE et al. [4]. To compare these data for zero liquid flow with Eqs. (I), (2) and (3), we add an upward velocity of U,/(l - eb) to the flow situation of Fig. 1, giving a net interstitial liquid velocity of zero. This means that the bubbles are being continuously generated, and to obtain the resulting flow of rate of gas, its velocity &/(l - Ed) is multiplied by the area .sb available for gas flow in a unit horizontal crosssection, giving the superficial gas velocity U = Ubgb/(l
-
&b)r
from Eq. (2) u = &b&,/(1 - &b),
(5)
from Eq. (3) u = &b&,(1 - &$-I
(6)
BRIDGE et al. [4] give data for air bubbling through glycerine-water, water containing surfactant, and pure water; of the three systems, the last is most likely to be relevant to the fluidised bed for which viscous effects are small and surface active effects do not apply. Figure 2 shows Bridge et d’s data for air-water with no water flow compared with curves based on Eq. (4) (5) and (6), using ub, = 23.5 cm/set [4], and n = 2.39 [3] since Re -h 800. Equation (6) agrees remarkably well with the data in Fig. 2 (the equation also agrees well with Bridge et d’s data in their Fig. 9 for glycerine-water, in conflict with Bridge et al.3 finding that their data do not agree with the correlation of RICHARDSON and ZAKI). The agreement with RICHARDSON and ZAKI’S correlation in Fig. 2 shows that the small bubbles (== 3.5 mm dia. for air-water) do indeed behave like solid spheres of zero density. For these bubbles the
0 .----it7 /4
0.20
0.16
-i
(6)
Equation Ref.
[3]
’
0.04
.O
FIG. 2.
(4) [2]
1 : 3
Superficial
(4)
velocity
U,
Data for air bubbling through water [4]. 0 57.25 cm,
732
cm/set
l
103.15 cm, A 168.7
cm above sparger.
The behaviour of a continuously bubbling fluidised bed
Weber number = D,pUjf ,/T = 2.8 where p is the liquid density and Tthe surface tension; referring to HABERMANand MORTON’S[5] Fig. 17, it would appear that these bubbles are in the uncertain transition region between the spherical and spherical-cap shapes. It seems unlikely that such bubbles would have fully developed wakes; their rate of rise would be determined partly by viscous forces [6] and they would probably be of ellipsoidal shape. It is not at all clear that the behaviour of a cloud of such bubbles is relevant to the fluidised bed problem, where we are concerned with the behaviour of spherical-cap bubbles, and it is quite conceivable that spherical-cap bubbles would rise as fast in a cloud as in isolation; in the absence of experimental evidence, one can only ask whether a given bubble in a cloud is speeded by being in the wake of bubbles above, or slowed by the bubbles on either side. At present this is an open question, but it is noteworthy that another critic [7] has said that the formula in [2] gives a rising velocity too low, while in [I] the formula is said to give a velocity which is too high! We now turn to the question of the division of the total flow of fluidising fluid between the bubble and particulate phases. THE DIVISIONOF FLOW BETWEENBUBBLE ANDPARTICULATE PHASES The problem here is to decide for a continuously bubbling fluidised bed, how much of the total flowU per unit cross sectional area-shall be allotted to the bubble phase and how much to the particulate phase. The division of flow is an important factor in determining the behaviour of catalytic reactors, because the fluid in the bubble phase makes less direct contact with the particles than the fluid in the particulate phase. Some suggestions as to the division of flow are as follows. (1) It has been proposed that U = U, + bubble flow where U = U,, at incipient fluidisation, and “bubble flow” means the flow accounted for by empty bubbles moving up the bed. This proposal is supported by certain evidence [2] p. 42 and 61, though this is slender. (2) It has been suggested that the flow allotted to the particulate phase should be U,(l - sb) because
in a typical cross-section, a fraction .sbof the area is occupied by bubbles, and consequently the area of particles, seen in plan to be fluidised, is less than at incipient fluidisation when there are no bubbles. This argument overlooks the flow of fluidising fluid through the bubbles, which will be discussed below. [3]. Another equally plausible suggestion is that each rising bubble carries up not only the fluid within the space empty of particles, but also the fluid within the “cloud’ associated with the rising bubble. Ample evidence [8] shows that the cloud does indeed move with the bubble. But each cloud owes its existence partly to the upward movement of particles near a bubble, and since there is no net flow of particles there must be particles moving downwards in regions away from bubbles; these downward moving particles will tend to drag fluid with them, and offset the upward fluid movement due to bubbles. Analysis of flow
We now present an analysis of the fluid flow in a continuously bubbling fluidised bed, based on the following assumptions : (a) The voidage fraction in the particulate phase is assumed constant = sO, the voidage fraction at incipient fluidisation. (b) Within the particulate phase, the relative motion between the fluid and the particles is governed by Darcy’s law, so that neglecting the inertia of the fluid, its velocity is u”= v”- Kgradp,
(7)
where i; is the particle velocity, p the fluid pressure and K the constant of Darcy’s law. (c) The fluidising fluid is assumed to be incompressible. These assumptions have been used to analyse the motion of fluid near a rising bubble [2] p. 63-79, and to obtain the degree of conversion with a chemical reaction in the particulate phase [9]. Other examples include flow near a rising slug in a tube [lo] and in the neighbourhood of an orifice through which fluidised particles are flowing [ll]. In all these cases there was reasonable agreement between theory and experiment. But for the single rising bubble, cloud diameters measured by ROWE et al. [8] are in slightly better agreement with the more
733
J. F. DAVIDSONand D. HARRISON
good for unsteady motions, because with a fixed frame of reference and freely rising bubbles, we are dealing with a time-dependent motion at all points. With this result in mind, we compare the flow situation in Fig. 3(a) and (b).
complex theories Of JACKSON[ 121and MURRAY[ 131 than with the simple theory based on the above assumptions. Jackson’s theory allows for the variation in voidage near a bubble, and ,for the consequent variation in K, but the variations in voidage are not large, typically from E,, = 0.5 to about E = O-6 just above a rising bubble [lo] ; moreover the theory cannot allow for interparticle forces in regions of the bed where E tries to become smaller than so. MURRAY’S[13] theory in contrast assumes constant voidage and DAKCYconstant K, but because of a linearisation of the particle mass-acceleration equation, gives cloud diameters almost the same as Jackson’s theory.
Particles fixed
Figure 3(a) shows a normal freely bubbling bed of particles fluidised by a superficial fluid velocity U, the diameter of the bed being much larger than the diameter of any of the bubbles. Figure 3(b) shows the particles, and hence the bubbles, in the same position, but each particle is held fixed by an external force, and the fluid pressure immediately above the distributor is tied at p. as in Fig. 3(a); p0 is just enough to support the weight of the particles, as it must be for a freely bubbling fluidised bed at all values of U > U,, from a momentum balance on the particles. Then from the above theorem, the fluid pressure at every point in the particulate phase must be the same in Figs. 3(a) and (b). Thus we can get the total fluid flow in the freely bubbling bed, Fig. 3(a), by first of all finding the flow with the bubbles tied, and then adding on the effect of the bubble motion. To get the superficial velocity kU,, we have to solve Laplace’s Eq. (8) governing percolation between the fixed bubbles, within each of which there is a (different) constant pressure. The flow pattern in this case can be compared with the flow at incipient fluidisation represented in Fig. 3(c). In Figs. 3(b) and (c) we have the same volume of particulate phase and the same base pressure p,,, and it is not altogether clear whether the presence of the
Pressure distribution theorem
Thus there is much to be said for the simple theory based on assumptions (a) (b) and (c) above, and a variety of experimental evidence shows that the theory is not seriously in error. For our purpose the most important consequence of the simple theory is that the pressure distribution is independent of the particle motion for given pressure boundary conditions. This follows immediately by using Eq. (7) with the continuity equations for particles and fluid, div fi = 0 and div fi = 0, which hold good even for unsteady motion because both the particulate phase and the fluid are effectively incompressible; combining these continuity equations with Eq. [7] we have div grad p = 0,
(8)
the equation for percolation through either fixed or moving particles. It is important that Eq. (8) holds
l-----l OOO
0
000
x--
x--
&~_“* 00
0
0
00
Ora 0
--_--
Superficial velocity
FIG. 3.
U
(a)
--x
Ocl
0
,“,o
ttt
W
(b)
p0
-----
~ ttt
UO
po
(cl
(a) Freely bubbling bed. (b) Particles and bubbles held fixed in position (a). (c) Incipient fluidisation.
734
The behaviour of a continuously
fixed bubbles will increase or decrease the flow, i.e. whether k is greater or less than unity. It seems probable that k will in fact be greater than unity. Certainly with bubbles of finite size and close together, the particles between the bubbles would be by-passed and k > 1. Calculations are in progress to find k with evenly spaced bubbles of uniform size. Through each fixed bubble in Fig. 3(b) there will be a fluid flow, and we can get the contribution of bubble and particulate phases by integrating across a typical section XX to get the total flow through the bed, whose cross-sectional area is A, so that ALU,
= .cO u” .dx+B,=F,+B,,, s PJ
bubble surface, and i& is parallel to da”because the pressure is constant within the bubble. With the bubble moving, particles at the surface have velocity ii, and the fluid there has velocity ij -I- &, from the pressure distribution theorem. Hence for a single bubble the total fluid flow across XX between A and C in Fig. 4 must be ABM = Ed ii, + i-Q.da+ sA
(9)
cles,
‘a.dC, s
(12)
A
and the integral in Eq. (12) represents the flow between A and C due to movement of the bubble surface if it were continuous. This is true even when the bubble velocity is unsteady, as when it is about to coalesce with another bubble. Summing equations like (12) for all bubbles cut by XX in Fig. 3(a) gives the flow in bubbles
moving
Fig. 3(a), the section cutting the same bubbles, but each is moving upwards with its appropriate velocity, not necessarily the same for every bubble. In the particulate phase, the particles are moving with velocity v”, but from the above theorem, the pressure distribution is unaffected by the motion. Hence the fluid velocity at any point is ~7~+ ii and the flow through the particulate phase cut by XX s
(11)
s
ABM = AB, +
We now consider the same cross-section XX in
is E,, p(z2f + @.dA”.
(I - E,,) j.dE.
The first term is due to the flow of fluid at the bubble surface, and the second represents the displacement of fluid caused by the motion of the particles at the bubble surface. From Eq. (10) and (11) we have
where u”/ is the interstitial velocity (not necessarily vertical) and dA” is an upwards pointing vector representing an element of area; B, is the flow through bubbles intersected by XX, and the integral is over the area of particles cut by XX. Particles
bubbling fluidised bed
C
=B,+
c bubbla
sA
ii.da”.
(13)
It was shown above that the flow in the particulate phase is Fp, whether or not the bubbles are moving, and adding this on to Eq. (13) and then using Eq. (9) gives the total flow for a freely bubbling bed,
But for no net motion of partiAU
=
AkU,
+
C
:.dS = ALU,, + Qrr .
bubbles s A
i7d.A” = 0, since we are integrating across an
s area lirge compared with the size of the bubbles, and hence when the latter are moving, the flow through the particulate phase is still Fp. We now consider the bubbles cut by XX in Fig. 3(a), and Fig. 4 shows a typical bubble. At the surface in Fig. 4, the fluid interstitial velocity is z& when the bubble is jxed, and consequently the flow across XX between points A and C is for one bubble
(14) In Eq. (14), QB is the observed bubble flow even if the bubbles are not all the same size, nor need they
n
+__A___________
C
AB, = Ed i&d& (10) sA where da” is a vector representing an element of 735
FIG.
4.
c-_-x
Section through typical bubble.
J.
F. DAVIDSONand D. HARRISON
Since k is unlikely to be very different from unity, unless the bubbles are extremely close together, the rather widely used assumption that the total flow = (flow at incipient fluidisation) plus (bubble flow) would seem to have good theoretical justification. move with steady velocities.
Fluid velocities within the bubbles
The derivation of Eq. (14) depends on the theorem that the pressure distribution in the particulate phase is independent of particle motion, so that (fluid velocity) = (particle velocity) plus (fluid velocity with fixed particles). This does not apply inside the bubbles, where the motion of the particles forming the bubble surface may set up a vortex motion [2] p. 95, which will not, however, invalidate the above argument based as it was on a discussion of the particle and fluid motions within the particulate phase up to the surface of each bubble but not inside it. Bubble and bed diameters comparable
The above derivation requires modification when the ratio (bubble diameter D,)/(bed diameter 0) is not small. In this case, qP =
s
5. dx is not zero at
any instant, because althoughPthe mean particle Sow across XX in Fig. 3 is zero when averaged over a long period, the instantaneous particle flow is upwards at one instant and downwards at another. This is clear for the limiting case of slug flow as shown in Fig. 5; when XX cuts through a gas slug, position 1, particle flow may be entirely downwards, but when XX cuts the particulate phase between slugs, position 2, the particle flow must be upwards. Similar considerations will apply when 0,/D is such that the bubbles are large but not large enough to be termed slugs. However, when DJD is not small, the above theory can be appropriately modified. Equation (9) still holds good with k constant, because with the bubbles fixed, the total flow must be independent of the position of XX, but FP and B, will be functions of the position of XX, so that for slug flow, BO is finite in position 1 but zero in position 2. When the bubbles move, the fluid flow rate through the particulate phase cut by XX will be
FIG. 5.
Slug flow in a fluidised bed.
co $j s
+ iif). dA” = eoqp + FP.
WI
At any instant the flow rate through bubbles cut by XX is still given by Eq. (13), because the arguments leading to this equation do not depend upon 0,/D being small. Hence we can combine Eqs. (9), (13) and (15) to give the instantaneous fluid flow rate through section XX AU’ = eoqp + ARU, + QB.
(16)
s T
Now the mean superficial velocity U =
_n -
U’dtlT,
where T is a time long compared with the time for one large bubble to pass a fixed position. But T
q,dt = 0, since there is no net flow of particles,
s0 and since
‘Q8dt/T means the average flow due to
bubbles crossing a fixed position we deduce that U = kUo + bubble flow as before. For slug flow it seems certain that k will not be greatly different from unity; with small slugs far apart, k must be very nearly unity, and
736
The behaviour
of a continuously
with large slugs close together, k > 1 because the particles falling down the wall at the sides of the slug will be by-passed.
bubbling fluidised bed
D DC FP
k K n P PO
CONCLUSION The theory given partly justifies the assumption that the total flow of fluidising fluid to the bed is equal to the flow due to the rising bubbles plus the flow required for incipient fluidisation. The complete analysis waits upon the determination of the constant k. It is shown that the assumption also applies when the bubbles vary in size, or move with unsteady velocities, or when the ratio (bubble dia.)/(bed dia.) is not small.
4p
instantaneous
QB
observed bubble flow detined by Eq. (14).
Re
u u
NOTATION
ub
A Bo
cross-sectional area of the bed flow per unit cross-section through bubbles intersected by XX (Fig. 3(a)) ABo Bow in a fixed bubble across XX between A and C (Fig. 4) ABM flow in a moving bubble across XX between A and C (Fig. 4) vector area of element of bubble surface vector of an element of area
bed dia. equivalent bubble diameter = (6 bubble volume/#/s flow through particulate phase constant constant of Darcy’s law exponent in equation of RICHARDSON and ZAKI fluid pressure pressure just sufficient to support the weight of particles (Fig. 3)
ubcc
uo e & co Eb v
P
particle flow =
sP
a.dG.
UbmDc/V
time surface tension; long period of time interstitial fluid velocity interstitial fluid velocity with particles fixed interstitial fluid velocity at surface of fixed bubble (Fig. 4) superficial fluid velocity instantaneous superficial fluid velocity superficial liquid velocity to hold a bubble cloud fixed (Fig. 1) rising velocity of a single bubble in an intinite stagnant liquid superficial fluid velocity at incipient fluidisation particle velocity voidage fraction voidage fraction at incipient fluidisation bubble volume per unit volume kinematic viscosity of liquid liquid density
REFEREXES TURNER,J. C. R., Chem. Engng Sci., to be published. DAVIDSONJ. F. and HARRISOND., Fluidixed Particles, Cambridge University Press, 1963. RICHARDSONJ. F. and ZAKI W. N., Trans. In&n. Chem. Engrs, 1954 32 35. BRIDGEA. G., LAPIDUSL. and ELGIN J. C., A. I. Ch. E. J., 1964 10 819. HABERMANW. L. and MORTONR. K., U.S. Navy Dept (D. W. Taylor Model Basin) Report 802 1953. Mooas D. W., J. FIuid Mech., 1963 16 161. GILLILANDE. R., A. I. Ch. E. J., 1964 10 783. ROWE P. N., PARTRIDGEB. A. and LYALLE., Chem. Engng Sci., 1964 19 973. ROSE P. L., Ph.D dissertation, University of Cambridge 1965. STEWARTP. S. B., Ph.D. dissertation, University of Cambridge 1965. JONESD. R. M. and DAVIDSONJ. F., Rheol. Acta, 1965 4 180. JACKSONR., Trans. Znstn. Chem. Engrs., 1963 41 22. MURRAYJ. D., J. Fluid. Mech., 1965 21 465; 1965 22 57.
R&u&-Cet article ttudie deux aspects du comportement d’un lit fluidisd en bullage, a savoir: (a) Quelle est la relation entre la vitesse d’ascension dune bulle dans un nuage de bulles et la vitesse de la msme bulle isol& Cette question est importante lorsqu’on considere l’expansion d’un lit de particules due au passage de bulles. (b) Comment se partage le debit total de fluide, utilise pour la mise en fluidisation du lit, entre les bulles et la phase dense. Les auteurs proposent une nouvelle theorie qui justi6e partiellement l’hypothese que le debit total est 6gal a la somme des debits dus aux bulles et du debit necessaire au minimum de fluidisation.
737
J. F. DAVIDKINand D. HARRISON Zusammenfassung-Dieser Bericht beantwortet zwei Fragen zur Verhaltungsweise einer brodelnden Wirbelschicht : (a) Wei verh5lt sich die Steiggeschwindigkeit einer Blase innerhalb einer Blasenschau zur Steiggeschwindigkeit emer einzelnen, isolierten Blase? Betrachtungen dieser Art sind wichtig im Zusammenhang mit der Bettausdehnung durch aufsteigende Blasen. (b) Wie verteilt sich das aufwirbelnde Medium zwischen einzelnen Blasen und zusammenharrgender Phase? Neueren Theorien zufolge scheint der Gesamtdurchsatz aus der Summe von Blasendurchsatz und Mindestwirbeldurchsatz zu bestehen.
738