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Mechanisms of choking and entrainment
Powder
Technology.
Mechanisms JOHN
32 (1982)
21-
of Choking
33
21
and Entrainment
M. MATSEN
Exxo-z Research (Received
and Engineering
December
Co., P. 0
Box
IOI.
Florham
Park, NJ 07932
(V S-A.)
8, 1980; in revised form July 6, 1981)
SUMMARY
The related phenomena of choking in pneumatic transport systems and of entrainment from fluid beds have been the subject of considerable empirical correlation and theoretical speculation. For dilute suspensions, such as occur in transport and entrainment, literature data indicate that slip velocity increases as voidage decreases_ This paper demonstrates that such a slip-voidage relation leads to a limiting envelope of lines of constant voidage on coordinates of volumetric particle flux versus fluid velocity_ The limiting envelope defines the boundary between dilute phase and dense phase flow and determines the solids flux-gas velocity-voidage relationships at choking or entrainment conditions_
INTRODUCTION
A prominent characteristic of solid particles suspended in vertical gaseous flow is their abrhty to exist in both dense phase and dilute phase configurationsThe dense regxme is usually viewed as discrete bubbles or pockets of gas rising through a continuous emulsion phase of particles and associated interstitial gas. The dilute regime, on the other hand, consists of single particles or clusters of particles falling with respect tc a continuous gas phase. It is also geometrically possible for a continuous gas phase and a continuous emulsion phase to coexist, a condition which seems to occur in turbulent fluidization_ Pneumatic transport lines normally operate as dilute phase systems. If gas velocity is slowly decreased at constant solids feed rate or if solids rate is increased at constant gas velocity, however, the volume &action of solids in the suspension wiIl suddenly increase along with pressure drop. This sudden
increase in solids concentration and pressure drop is usually called ‘choking’. Solids transport continues in the dense phase mode below the choking point, but at lower gas velocities or higher solids flow rates. Either dilute phase or dense phase behavior is thus possible in a gas-solids transport system. Pneumatic transport lines are usually only a few centimeters in diameter and normally operate with relatively coarse material_ Under such circumstances the dense phase formed under choking conditrons LSin slug flow, and experimentalists have often characterized choking as the transition from dilute to slug flow_ Onset of slug flow, however, must be recognized only as one common manifestation of choking and not as the defmitive criterion_ Dense phase flow can also occur with small non-slugging bubbles or with turbulent fluidization, in which the bubble population has become too concentrated for clear individual sphere cap bubbles to be apparent, due to rapid coalescence and splitting_ Chokmg is the abrupt transition from dilute to dense flow regardless of the nature of the dense flow_ In a fluid bed operating at low gas velocity, only a dense phase is present. As velocity is increased above terminal velocity, however, particles begin to be entrained, and a dilute phase of low but fmite particle concentration is formed above the dense bed. Particle concentration decreases with height immediately above the bed, but eventually becomes constant_ At this point the dilute phase above the fluid bed is exactly the same as that beyond the particle acceleration zone in a dilute phase pneumatic transport Line operating at incipient choking velocity_ Zenz and Weil [l] pointed out the equivalence in 1959, but except for their own correlation, no use has been made of the identity of choking and entrainment_ For entrainment of 0 Ekevier Sequoia/Printed
in The Netherlands
22
a mixture of particle sizes, it is not the bed size distribution but the entrained size distribution which gives equivalence to dilute pneumatic transport at incipient chokingA phenomenon related to entrainment and pneumatic transport is that of the ‘fast fIuidized bed’_ Yerushalmi ef al [2] coined the term for their observations of gas-particle transport systems at velocities and densities intermediate between those of conventional dense phase fluid beds and those of dilute phase pneumatic transport_ The term generates considerable debate as to whether it is truly a separate regime apart from dilute and dense phase transport and has also caused speculation as to theoretical criteria which might differentiate it from other forms of fluidization and transport. Yerushalmi and Cankurt [ 31 make it clear that choking and their ‘fast fluidization’ occur m the same system at different gas and solids flow rates. SLIP VELOCITY-VOIDAGE
RELATIONSHIPS
The forces exerted on particles by a gas are a direct function of the slip velocity, i.e. the relative velocity between particles and gas. The concept of slip velocity is therefore useful in describing and analyzing gasparticle flow. Slip velocity is defmed mathematically as
(-p-u_
G PPV --E)
E
(1)
Slip velocity is independent of solids and gas flow rates for the limiting case where particle size is uniform and the effects of acceleration and walI diction are negligible_ In this limiting case and for a given ga* particle system, slip velocity is a function only of the voidage of the suspension. For analyzing solids transport rates, eqn. (1) may be rewritten as G -= PP
1-E -
All analysis in this paper assumes negligible wall friction, fully accelerated flow and uniform particie size. These limiting conditions are realistic for commercial entrainment, where gas velocity is low and vessel diameter is large. They are somewhat less reahstic but useful for pneumatic transport, which occurs at high velocity in relatively small pipes Particulate fluidtiation Particulate fluidization occurs when a bed of particles expands uniformly with increasing fluid velocity, without the formation of bubbles of fluid or clusters of particles. Particulate behavior is characteristic of fluidized beds of fine, low density particles especially at high pressures, of magnetically stabilized beds, and of liquid fluidized beds. It is also often though erroneously assumed to occur in dilute g-particle suspensions An example of a particulate fluidlzation relation between slip and voidage is the Richardson-Zaki [4] equation I for fluidization,
u = E-l&
us = E355uT when written in terms of slip velocity_ Substituting eqn. (3) into eqn. (2), and reducing to dimensionless form by dividing both sides by UT, the flow map shown in Fig. 1 results. As the figure shows, a particulate system will expand steadily with fluid velocity without any voidage discontinuity, as is expected for this type of fluidization Bubbling dense phase flow In gassolids systems, partrcles do not usually remain uniformly dispersed. In relatively dense suspensions, most of the gas flows as bubbles, while the sohd particles are present as an emulsion at minimum fluidization conditions- The equation governing density in such flow situations has been previously developed [5] I
(U--ELI,)
E
Since U, is a function only of E, a plot of G/pp wrsus U consists of a family of straight lines of constant E having slopes of (1 - E)/E and intercepts of EU, Such plots define the particlegas flow system and are developed below for some major voidage-slip relationships.
or
u - urn,- ~,&I~rnr u, + GJ~rnr
pmf=l+ P
This can be re arranged to give G -= PP
l--E E
U-U~f-UB(y---~f)]~5~ [
(4)
23
l-O-
0-B -
0
0.2
0.4
0.6
0.6
10
1.2
1.4
l-6
18
U/LIT
Fig. 1. Gas-solids
flow map. Richardson-Zaki
slip velocity_
The term Us is the velocity at which a single bubble will rise through non-flowing solids. For small diameter pipes or beds, the bubbles will be present as wall to wall slugs and Us = 0.3-, while for larger diameter vessels the relationship is Us = 0 71=, where D is the vessel diameter and Da is the diameter of a non-slug bubble. Davidson and Harrison [S] have proposed that the maximum stable diameter which a bubble can reach is approximately that required to make Ua equal to U, The fIow map for dense phase bubbhng or sIugging flow is obtained by dividing both sides of eqn. (5) by_ Va and plotting as in Fig. 2. Slip velocity in the bubbling dense phase flow regime is obtamed from eqns. (2) and (5) and may be written as
Equation (6) for the dense phase slipvoidage relation is used in this paper for the entire range of voidages, from E = E, f = 0.45 to a maxunum of E = 0.906. A~J earlier analysis [7] has shown data for some slugging systems to indicate that eqn (6) can be valid up to E = 0.9 while for non-slugging systems there is little direct evidence of its validity. Of particular interest is a region called ‘turbulent fluidrzation’ in which bubbles do not exist as discrete entities but rather coalesce and break at a ve_y rapid r+e_ Yerushahni and Cankurt [3] show that slip velocity follows the same qualitative trends in the bubbling and the ‘turbulent’ regions, so that use of a single slip equation seems to be a reasonable approximation. Dilute phase flow
Slip velocity in bubbling systems can readily exceed UT since slip is determined by the rise velocity of large bubbles rather than the falling velocity of single particles. Note that in this case slip velocity is the relative velocity averaged over time or space.
A third slip velocity relation applies in dilute suspensions. Many investigators have assumed that at high void fractions (e.g. E > 0.99) particles behave independently, and slip velocity should equal terminal velocity. Others have assumed that the RichardsonZaki relationship or a similar one applies in this region, predicting that slip decreases from
24
O-5
04
0.3 1” T 0 0.2
O_!
(
Fig_ 2_ Gas-solids
flow map_ Dense phase flow_
LlreS
OF Constant
Slow
= (I -E/E
1
Envelaw Defines Mannurn G/PD Possible Without Abrupt Change In E
Fig. 3. G as-solids concentration.
flow map_ SchematIc
behavior for dilute phase flow when slip velocity
increases with particle
25
V, as E decreases from 1.0. Neither of these slip velocity-voidage relationships leads to an adequate explanation of choking or entrainment, because flow maps constructed using these relationships do not predict the experimentalIy observed voidagediscontinuity and coexistence of dilute and dense phase for choking and entrainment. Suppose, however, that a third type of relationship exists in which as voidage decreases the slip velocity steadily increases above V, so as to give a gas flow-solids flow-voidage map as shown in Fig. 3. This map shows a very significant difference from Figs. 1 and 2. It exhibits an envelope which limits the possible solids rate at any given gas rate. In a pneumatic transport system below the envelope, either gas or solids rate may be changed slightly with a corresponding slight change in voidage. As the envelope is reached, by decreasing gas rate or increasing solids rate, the limiting or choking voidage is reached. In order to accommodate any further decrease in gas rate or increase in solids rate the system must change to a totally different voidageslip line_ The volume fraction of solids must make the sudden step increase, which is called ‘choking’. Because of the equivalence of the rate of dilute transport at choking and the rate of entrainment above a fluid bed, the chokmg envelope is also the entrainment curve for the system. The slope of tangc&s to the entrainment curve determines the dilute phase density above transport disengaging height from the relation: slope = (1 - E)/E_ The intercept of the tangents is eUs_ It is an important finding in this work that a limiting envelope formed by lines of constant voidage does exist for dilute suspensions. This envelope affords the physical mechanism which explains choking m pneumatic transport and permits coexistence of dilute and dense phases in entraining fluid beds. Envelope formation arises when slip velocity increases as voidage decreases
CLUSTER FORMATION The explanation for a sytem in which slip increases w-ith decreasing voidage probably lies in the phenomenon of cluster formation. It has long been recognized that
clusters will have slip velocity which increases with the number of particles in the cluster. If the size of the clusters increases with particle concentration, then the necessary slipvoidage relationship can exist. Clusters of particles will obviously fall somewhat faster than V,, the terminal velocity of a single particle. A review of a number of pertinent theoretical calculations appears in Happel and Brenner [S] _ For example, on page 277, they present equations derived by Burgers for the falling velocity of clusters of two, three, four, and eight particles, results of which are shown in Fig. 4. Jayaweera et al. [9] made experimental measurements of the falling behavior of clusters, verifying an increase m slip velocity and showing that such clusters would achieve stable, regular configurations. Particles introduced as a cluster would under most circumstances fall together as a cluster rather than becoming dispersed widely throughout the surrounding fluid. Kaye and Boardman [lo] reported quantitative data on falling behavior of marked particles in dilute suspensions of many particles. They observed definite cluster formation of a statistical nature and measured falling velocities well in excess of U,. Johne [ll] made similar but more accurate measurements_ Results of both investigations are shown in Fig_ 5. In both cases slip velocity is very nearly V, at E > 0.9997 and then rises with decreasing E to a maximum. With further decreases in E, V, decreases steadily, becoming less than U, in the range of 0.99 > E > 0.90. Both the above experiments measured settling velocity, with particles displacing liquid. Slip velocity in such a case is
In correlating data as in Fig. 5, the degree of clustering is implicitly a function of E. As voidage decreases, it is probable that the fraction of particles in clusters increases, that the number of particles in each cluster increases, and that the spacing between particles in a cluster decreases_ Throughout the rest of this paper, it will be assumed that V, is a function only of V, and E. It seems entirely reasonable, however, to suspect that other factors would have a secondary effect on V,. These factors might include: (1) particle Reynolds number in the region above
2 NUM8ER
Fig. 4. Cluster slip velocity
(I -
according
3 OF
PARTICLES
4 IN
5
6
I
I
7
89
I 3
CLUSTER
to Burgers.
0
Fig. 5. Slip velocity from sedimentation measurements_ Kaye and Board[lol.~JohneL11]_
creeping flow, and (2) particle impaction parameter which might be relevant to the rate at which clusters could form. Grace and Tuot [12] recently made a theoretical stability analysis of dilute suspensions and concluded that a uniform dispersion is unstable. Their calculations of disturbance growth rates showed that a suspension becomes more stable with increases in fluid density and viscosity and with a decrease in particle density_ Grace and Tuot treat only the tendency towanls cluster formation, but it is clear that beyond some degree of aggregation the tendency for
clusters to break up must exceed their tendency to form. Otherwise long vertical pneumatic transport lines would not work and entrainment from a fluid bed could be reduced to any desired level merely by increasing the disengaging height _ If an ent rainment cume is plotted on linear coordinates as G/pp uersus U, according to the present theory, a tangent to the curve at any point will have a slope (1 - E)/E, and the intercept with the X axis will be eUS_ Such a curve, taken by Wen and Hashinger [13] on uniformly sized glass beads, is shown in Fig. 6. The resulting slip uersus voidage relationship, obtained born tangents and intercepts, is shown in Fig_ 7. It is virtually identical to the Johne [ll] data of Fig. 5 up to solids concentrations of (1 - E) = 0 004, but continues to increase above that concentration while the Johne data level off. This may reflect the common observation that liquidsolids systems (Johne) have a greater tendency to form uniform dispersions while gas-solids systems (Wen [ 131) tend to cluster. A single data point from Koglin 1143 is also shown on Fig. 7. Koglin’s data are for sedimentation, and he claims that Johne’s data were influenced by wall effects to cause the deviations shown.
27
.a04
-003
5
G= ”
-Ooi
.OOl
uT
=0.326
rds
E =_99838 Inrerceot=EUS =0.5
r/s
15
Fig_ 6. Ent rainment
I
curve from Wen and Hashinger
I
I
[13]_
I
IllIll
I
I
I
Illll, LlmlL
1
I
I
IIll_
of Data
654-
Data
2
Of
Kcdm(l4)
7i
3 2-
1 .OOOl
-001
-01 (I-El
Fig_ 7. Slip velocity
US. voidage
for sedimentation
III hquids and entrahm
ent by gas_
.l
28 FLOW
MAP
FOR
VERTICAL
GAS-SOLIDS
FLOW
Data in Fig 7, for liquid-solid sedimentation and for gas-solid entrainment, suggest the following empirical expressions for slip versus voidage:
(1 -
US/U,
= 1
USI&
= 10.8(1-
E) < 0.0003 (8)
E)O=
(1 - E) > 0.0003 (9)
These expressions may be combined with eqn. (2) to give the dilute phase flow map. The dense phase map is defined by eqn (5) in terms of U, and Umr, but in order to be presented on the same dimensionless plot as dilute flow, U, must be used as the reference velocity. To provide a quantitative example it will be assumed throughout the rest of this paper that U, = l.OU, and Urn,= l/SO&_ With
a specific
bed
diameter
(if slug flow
exists) and a given gas-particle system, more accurate values for U, and U,, could be used, but the principle would remain the same_ The r
-45 -12 -
-10 -
_06-
_04-
.02-
o-
Fig
8. Flow
map for gas-solids
upflow.
graphic result is shown in Fig. 8, which gives in a generalized dimensionless form the interdependence of solids transport, gas velocity, and voidage for entrainment and choking, dilute pneumatic transport, and dense phase bubbhng flow The dilute pneumatic transport region is in the lower nght and the bubbling dense region is in the upper left of the figure_ The entrainment and choking curve is the line or envelope which separates the dense and dilute regions Below the entrainment curve in Fig_ 8, lines of constant voidage cross one another_ In thk region, the steady-state condition consism of only the lines of lesser slope as shown in Fig. 9. The lines of steeper slope represent a condition in which solids flux (at constant gas velocity) increases as voidage increases, a situation which is not stable to small perturbations in gas velocity, solids rate, or voidage. Figure 8 may be cross-plotted to demonstrate an analogy with a single-component liquid-vapor phase diagram, as shown in Fig_ 10. Here E is regarded as an independent variable, and one may imagine the physical
29
-06. d-
2
Fig 9
Stable solutions
4
3
5
for flow map.
I
a
1 i
6
3 3
\’
\ kLJ\
+ 4
2
C
!ny
B
boo4 .0002
1
\?A
I 0
6
0.01
I*
II,III
0.1
0.5 Cl-El
1
2 ,
1
5
VOLUME
, 10
.\ 20
PERCENT
Fig ‘_ 10. Phase diagram for vertical gas-solids
‘J I
30 SOLIDS
flow_
I
40
I
I
50 6(I 70
eo
30
system as being a fluidization column with a cyclone to recycle entrained solids back to the bottom of the column, and charged with enough particles to give any desired overall voidage E_ Inventory of particles in the cyclone and recycle line is negligible_ For most values of overall E at low velocities two phases will form, a dense bed of voidage defied by the right-hand porticn of the phase curve and a dilute entrained phase defined by the left side of the curve. As gas velocity is increased at constant overall E, however, the system will eventually become a single-phase system (i.e., there will be no horizontal interface having a dilute phase above and a dense phase below, although the column may have intermixed bubble and emulsion phases)_ For instance, the dashed line A - A in Fig. 10 represents a recycling fluid bed with an overall voidage of 98%. At a velocity U/U, = 2, the system will consist of a dense bed of voidage 82% (Point B) and a dilute phase of voidage 99.85% (Point C), and the entrainment rate G/ppUT will be about 0.0006. As velocity is increased the dilute voidage will decrease and the dense voidage and entrainment rate will increase, and at U/U, = 4.25 and G/ppUT = O-019 the dense phase will disappear If one operates at a velocity above the top of the phase curve, i_e at U/U, > 5.73 or at a solid rate G/pp > 0_0865Ur, it will be impossible to have two
Fig. ll_
Qualitative
slip us_ voidage relationships
phases at any value of E. The system may be thought of as ‘supercritical’. It is proposed that such supercritical behavior is responsible for the phenomenon of ‘fast fluidization’ Li and Kwauk [15] have recently published an experimental figure similar to Fig_ 10. Details of their figure are somewhat different because of acceleration and friction effects, and their interest was in the ‘fast fluidization’ area rather than in the choking and entrainment aspects of the diagram_ Yerushalmi and Cankurt [3] also show similar behavior on a somewhat different type of plot. The phenomenon of ‘fast fluidization’ has been studied extensively by Yerushalmi and co-workers [2, 31. Figure 11 shows their qualitative observations of slip velocity uers~s voidage relations for a wide range of gas particle behavior, including the fast fluidlzed bed and pneumatic transport. Figure 12 gives their quantitative data on slip uersus voidage for a specific system. Figure 12 may be compared with Fig. 13 drawn according to the theory developed in this paper. Both figures show a maxunum slip velocity at a voidage of about 0.9 and clearly represent the same phenomenon. Figure 12 includes non-ideal effects of: (1) mixed particle size, (2) wall friction, (3) particle acceleration, and (4) over-estimation of slip because voidage was inferred from pressure drop rather than being
from Yerushalmi
ef aL [3].
31
dilute suspensions, eqn. (8) applies and entrainment occurs at the limiting sohds concentration of (1 - l) = 0.0003. Substituting this value for (1 - E) and US = U, into eqn. (2) and dividing by U, gives G&UT
= 0.0003((
U/U,)
-
1)
(1 - E) = 0.0003
(10)
In the slightly higher solids concentration region, eqn. (9) applies- This may be substituted for U, in eqn. (2), which IS differentiated with respect to E and the derivative set equal to zero0 = g measured directly. For this reason Fig. 12 shows a family of slip-voidage curves with solids flux as a parameter, rather than the single curve of Fig_ 13_ NUMERICAL
(1 -
APPROXIMATIONS
Based on the present theory, simple equations for entramment and the associated dilute phase voidage may be derived For very
5
4 Dense Phase Riser Theory
> =
Phzse
3
Fig_ 13_ Slip velocity
us_ voidage relationships
E) >
0_0003
-
E)‘-~~& (11)
This gives the relationship between U and E which describes the entrainment envelope. In the region where E z 1 eqn (11) simplifies to u xa1 (le)=L26x 1O-4 vT
6
Dilute
+ (1.293)(10.8)(1
from present theory.
(1 - E) > 0.0003
(12)
32
Equation (12) is an approximate expression for dilute phase holdup when entrainment is occurring_ Substituting eqns_ (9) and (12) back into eqn. (2) and dividing by CT gives the generalized entrainment curve of G uersus u: G/p,&
= 2.84 (1 -
x 10-5(
tY/U,)3”1
E) > 0.0003
(13)
From eqn. (ll), the condition (1 -- E) > 0.0003 is equivalent to U[U, > 1.29_ Limitations of the numerical results expressed in eqns_ (S) - (13) and Figs. S and 10 must be emphasized_ These should not be used for quantitative analyses of systems of mixed particle sizes or where acceleration or wall friction effects are important Even for monosized, accelerationless, frictionless systems the numerical relations of eqns. (8) and (9) are only tentative_ Note for instance that for low values of Reynolds number based on U, this approach predicts no effect of gas density, while several empirical correlations indicate entrainment rate proportional to gas density. Nevertheless the numerical results do promise a useful form for correlation of choking, emrainment, and pneumatic transport data and a starting point for analysis of more complex systems.
DISCUSSION
It is seen that choking is a function of dilute phase slip-voidage behavior and does not depend on the dense phase. The bubble velocity in the dense phase affects the dense phase voidage and slip, however_ If the bubble velocity is reduced (e g. by using a small tube so as to cause slugs), the density difference between dense and dilute phases at choking will be reduced, as will the critical velocity above which choking cannot occur. Figure 9 demonstrates that a captive fluidized bed (ie_ with G = 0) cannot exist with finite (1 - E) at fluidlzing velocities greater than U,. Likewise, in a sedimentation experiment the top surface or sludge line cannot fall at a velocity greater than UT. The empirical equation of Richardson and Zaki [4] was based on sludge-line and bed expansion data and necessarily indicated
that Us/UT decreased monotonically from 1 as voidage decreased from 1. Figure 7 shows on the other hand that U’/U, increases above 1 as voidage decreases from 1. The reason is that in the sedunentation data in Fig. 7, the settling velocity of tracer particles within the suspension was measured rather than the settling velocity of the top surface_ The Richardson-Zaki equation is inappropriate for voidage close to 1 and should not be used in choking calculations This analysis would indicate that choking can occur in liquid-particle suspensions as well as gasparticle systems. In the liquid system, the dense phase expansion will be much greater, thus substantialiy lowering the value of U/UT above which two phases cannot coexist and also greatly decreasing the difference between dense phase and dilute phase voidage when choking does occur. Thus choking in a liquid system would be difficult to detect experimentally and would be of little practical consequence. For cases where particle to wall friction is significant, an approach analogous to that of Fig. 8 may be used. Lines of constant E will have a slope less than (1 - E)/E and may be curved or straight depending on the nature of the friction correlation_ These lines wrll have the same X intercept as in the frictionless case, since particle velocity and friction are zero at the intercept_ The result is to lower the choking curve so that higher gas velocities or lower solids flow rates are necessary to avoid choking_
CONCLUSIONS
For systems of uniform particle size and velocity profue, with negligible wall diction and inertial effects: (1) In dilute phase suspensions, actual particle slip velocity is greater than single particle terminal velocity even at very low sohds concentrations_ This occurs because the particles tend to cluster rather than behaving as individual particles_ (2) A provisional numerical expression for slip is Us = UT for (1 - E) < 0.0003 U, = lO.SU,(l
-- c)Oa3
for
(1 - E) > 0.0003
33
(3) Because slip increases with solids concentration, straight lines of constant voldage on a plot of G/p, UersU.s tY form an envelope. This envelope defines a maximum G/p, at a given U which IS equivalent to the choking curve for pneumatic transport or the entramment curve for fluid beds. (4) A tangent to the entrainment curve of G/p, uerSUs U has a slope of (1 -- E)/E and an intercept of lUs_ Thus, the entrainment curve can yield dilute phase holdup and slip velocities. (5) Using the newly derived relationships for dilute phase behavior combined with earlier expressions for dense phase behavior, a complete map of G/pP uersus U with lines of constant voidage can be drawn. This czn be cross-plotted in a phase diagram of U Uersus (1 - e) with G/D~ as a parameter. This phase diagram clearly shows the coexistence of dense and dilute phase fluld beds at low velocities and the existence of a ‘supercritical’ region at high velocities in the zone ‘fastfluidization’. (6) For entraining or chokmg conditions, the voidage may be represented as (1 -
E’ -= 0.0003
U/UT
< 1.29
(1 - E) = 1 26 X 1O-4 ( U/UT)3 UfU,
>
and the solids transport G/P,
= 0.0003(
GIP~
=
2.84 u/u,
LIST
D DB
g G U
X
41
1.29
U/U,)
rate as u/u*
< 1.29
10-5(U/UT)341U > 1.29
U nlf US
m
of isolated bubble, m/s mimmum fluldization velocity, m/s slip velocity between fluid and particle, m/s terminal velocity of isolated particle, m/s void fraction void fraction at minimum fluidization density of expanded bed, kg/m3 density of bed at minimum fluidization, kg/m3 particle density, kg/m3
E Emf
P Pmf PP
REFERENCES
1
F_ A_ Zenz and N_
A_
Weil. AZChE
J_, Q (1958)
472.
R. A Graff, S_ 2 J_ Yerushalmi, M. J_ Gluckman. Dobner and A_ M_ Squires, in D_ L_ Keairw (Ed ). Fluidization Technology. Vol 2, Hemisphere Pub1 Carp_, Washington, 1976. p_ 437. J. Yerushalmi and N. T Cankurt, Powder Technol. 24 (1979) 187. J F_ Richardson and W_ N Z&i, Trarrs. Znst Chem Eng , 35 (1954) 35. J M_ Matsen, Powder TechnoL. 7 (1973) 9% J. F. Davidson and D_ Harrison, Fluidrzed Particles, Cambridge Univ Press, Cambridge, 1963. 7 J_ M_ Matsen. S. Hovmand and J_ F. Davidson, Chem. Eng. Sci, 24 (1969) 1743. h’umber 8 J. Happel and H. Brenner, Low Reynolds Hydrodynamics, Prentice-Hall, Englewood Cliffs, NJ, 1965_ 9 K_ 0. L_ F_ Jayaweera, B_ J_ Mason and G. W_ Slack, J. Fluid Mech., 20 10 the
OF SYAIBOLS
bed diameter (for slug flow), bubble diameter, m acceleration of gravity, m/s’ solids mass flux, kg/m% superficial gas velocity, m/s
rise velocity
uB
Interaction
between
Fluids
and
Particles.
11 12
Inst. Chem. Eng., London, 1962, p_ 17. R. Johne, Chem. Zng. Tech, 38 (1966) 428_ J. R. Grace and J. Tuot, Trans_ Inst. Chem Eng.,
13
57 (1979) 4. C Y Wen and R. F_ Hashinger,
14 15
AIChE
J_
6
(1960) 220. B. Koglin, D. Ing. Diss_, Univ. of Karlsruhe. 1971 Youchou Li and M Kwauk. in J. R. Grace and J. M. Matsen (Eds.), Fluiduation. Plenum Pres, New York, 1980.
Technology.
Mechanisms JOHN
32 (1982)
21-
of Choking
33
21
and Entrainment
M. MATSEN
Exxo-z Research (Received
and Engineering
December
Co., P. 0
Box
IOI.
Florham
Park, NJ 07932
(V S-A.)
8, 1980; in revised form July 6, 1981)
SUMMARY
The related phenomena of choking in pneumatic transport systems and of entrainment from fluid beds have been the subject of considerable empirical correlation and theoretical speculation. For dilute suspensions, such as occur in transport and entrainment, literature data indicate that slip velocity increases as voidage decreases_ This paper demonstrates that such a slip-voidage relation leads to a limiting envelope of lines of constant voidage on coordinates of volumetric particle flux versus fluid velocity_ The limiting envelope defines the boundary between dilute phase and dense phase flow and determines the solids flux-gas velocity-voidage relationships at choking or entrainment conditions_
INTRODUCTION
A prominent characteristic of solid particles suspended in vertical gaseous flow is their abrhty to exist in both dense phase and dilute phase configurationsThe dense regxme is usually viewed as discrete bubbles or pockets of gas rising through a continuous emulsion phase of particles and associated interstitial gas. The dilute regime, on the other hand, consists of single particles or clusters of particles falling with respect tc a continuous gas phase. It is also geometrically possible for a continuous gas phase and a continuous emulsion phase to coexist, a condition which seems to occur in turbulent fluidization_ Pneumatic transport lines normally operate as dilute phase systems. If gas velocity is slowly decreased at constant solids feed rate or if solids rate is increased at constant gas velocity, however, the volume &action of solids in the suspension wiIl suddenly increase along with pressure drop. This sudden
increase in solids concentration and pressure drop is usually called ‘choking’. Solids transport continues in the dense phase mode below the choking point, but at lower gas velocities or higher solids flow rates. Either dilute phase or dense phase behavior is thus possible in a gas-solids transport system. Pneumatic transport lines are usually only a few centimeters in diameter and normally operate with relatively coarse material_ Under such circumstances the dense phase formed under choking conditrons LSin slug flow, and experimentalists have often characterized choking as the transition from dilute to slug flow_ Onset of slug flow, however, must be recognized only as one common manifestation of choking and not as the defmitive criterion_ Dense phase flow can also occur with small non-slugging bubbles or with turbulent fluidization, in which the bubble population has become too concentrated for clear individual sphere cap bubbles to be apparent, due to rapid coalescence and splitting_ Chokmg is the abrupt transition from dilute to dense flow regardless of the nature of the dense flow_ In a fluid bed operating at low gas velocity, only a dense phase is present. As velocity is increased above terminal velocity, however, particles begin to be entrained, and a dilute phase of low but fmite particle concentration is formed above the dense bed. Particle concentration decreases with height immediately above the bed, but eventually becomes constant_ At this point the dilute phase above the fluid bed is exactly the same as that beyond the particle acceleration zone in a dilute phase pneumatic transport Line operating at incipient choking velocity_ Zenz and Weil [l] pointed out the equivalence in 1959, but except for their own correlation, no use has been made of the identity of choking and entrainment_ For entrainment of 0 Ekevier Sequoia/Printed
in The Netherlands
22
a mixture of particle sizes, it is not the bed size distribution but the entrained size distribution which gives equivalence to dilute pneumatic transport at incipient chokingA phenomenon related to entrainment and pneumatic transport is that of the ‘fast fIuidized bed’_ Yerushalmi ef al [2] coined the term for their observations of gas-particle transport systems at velocities and densities intermediate between those of conventional dense phase fluid beds and those of dilute phase pneumatic transport_ The term generates considerable debate as to whether it is truly a separate regime apart from dilute and dense phase transport and has also caused speculation as to theoretical criteria which might differentiate it from other forms of fluidization and transport. Yerushalmi and Cankurt [ 31 make it clear that choking and their ‘fast fluidization’ occur m the same system at different gas and solids flow rates. SLIP VELOCITY-VOIDAGE
RELATIONSHIPS
The forces exerted on particles by a gas are a direct function of the slip velocity, i.e. the relative velocity between particles and gas. The concept of slip velocity is therefore useful in describing and analyzing gasparticle flow. Slip velocity is defmed mathematically as
(-p-u_
G PPV --E)
E
(1)
Slip velocity is independent of solids and gas flow rates for the limiting case where particle size is uniform and the effects of acceleration and walI diction are negligible_ In this limiting case and for a given ga* particle system, slip velocity is a function only of the voidage of the suspension. For analyzing solids transport rates, eqn. (1) may be rewritten as G -= PP
1-E -
All analysis in this paper assumes negligible wall friction, fully accelerated flow and uniform particie size. These limiting conditions are realistic for commercial entrainment, where gas velocity is low and vessel diameter is large. They are somewhat less reahstic but useful for pneumatic transport, which occurs at high velocity in relatively small pipes Particulate fluidtiation Particulate fluidization occurs when a bed of particles expands uniformly with increasing fluid velocity, without the formation of bubbles of fluid or clusters of particles. Particulate behavior is characteristic of fluidized beds of fine, low density particles especially at high pressures, of magnetically stabilized beds, and of liquid fluidized beds. It is also often though erroneously assumed to occur in dilute g-particle suspensions An example of a particulate fluidlzation relation between slip and voidage is the Richardson-Zaki [4] equation I for fluidization,
u = E-l&
us = E355uT when written in terms of slip velocity_ Substituting eqn. (3) into eqn. (2), and reducing to dimensionless form by dividing both sides by UT, the flow map shown in Fig. 1 results. As the figure shows, a particulate system will expand steadily with fluid velocity without any voidage discontinuity, as is expected for this type of fluidization Bubbling dense phase flow In gassolids systems, partrcles do not usually remain uniformly dispersed. In relatively dense suspensions, most of the gas flows as bubbles, while the sohd particles are present as an emulsion at minimum fluidization conditions- The equation governing density in such flow situations has been previously developed [5] I
(U--ELI,)
E
Since U, is a function only of E, a plot of G/pp wrsus U consists of a family of straight lines of constant E having slopes of (1 - E)/E and intercepts of EU, Such plots define the particlegas flow system and are developed below for some major voidage-slip relationships.
or
u - urn,- ~,&I~rnr u, + GJ~rnr
pmf=l+ P
This can be re arranged to give G -= PP
l--E E
U-U~f-UB(y---~f)]~5~ [
(4)
23
l-O-
0-B -
0
0.2
0.4
0.6
0.6
10
1.2
1.4
l-6
18
U/LIT
Fig. 1. Gas-solids
flow map. Richardson-Zaki
slip velocity_
The term Us is the velocity at which a single bubble will rise through non-flowing solids. For small diameter pipes or beds, the bubbles will be present as wall to wall slugs and Us = 0.3-, while for larger diameter vessels the relationship is Us = 0 71=, where D is the vessel diameter and Da is the diameter of a non-slug bubble. Davidson and Harrison [S] have proposed that the maximum stable diameter which a bubble can reach is approximately that required to make Ua equal to U, The fIow map for dense phase bubbhng or sIugging flow is obtained by dividing both sides of eqn. (5) by_ Va and plotting as in Fig. 2. Slip velocity in the bubbling dense phase flow regime is obtamed from eqns. (2) and (5) and may be written as
Equation (6) for the dense phase slipvoidage relation is used in this paper for the entire range of voidages, from E = E, f = 0.45 to a maxunum of E = 0.906. A~J earlier analysis [7] has shown data for some slugging systems to indicate that eqn (6) can be valid up to E = 0.9 while for non-slugging systems there is little direct evidence of its validity. Of particular interest is a region called ‘turbulent fluidrzation’ in which bubbles do not exist as discrete entities but rather coalesce and break at a ve_y rapid r+e_ Yerushahni and Cankurt [3] show that slip velocity follows the same qualitative trends in the bubbling and the ‘turbulent’ regions, so that use of a single slip equation seems to be a reasonable approximation. Dilute phase flow
Slip velocity in bubbling systems can readily exceed UT since slip is determined by the rise velocity of large bubbles rather than the falling velocity of single particles. Note that in this case slip velocity is the relative velocity averaged over time or space.
A third slip velocity relation applies in dilute suspensions. Many investigators have assumed that at high void fractions (e.g. E > 0.99) particles behave independently, and slip velocity should equal terminal velocity. Others have assumed that the RichardsonZaki relationship or a similar one applies in this region, predicting that slip decreases from
24
O-5
04
0.3 1” T 0 0.2
O_!
(
Fig_ 2_ Gas-solids
flow map_ Dense phase flow_
LlreS
OF Constant
Slow
= (I -E/E
1
Envelaw Defines Mannurn G/PD Possible Without Abrupt Change In E
Fig. 3. G as-solids concentration.
flow map_ SchematIc
behavior for dilute phase flow when slip velocity
increases with particle
25
V, as E decreases from 1.0. Neither of these slip velocity-voidage relationships leads to an adequate explanation of choking or entrainment, because flow maps constructed using these relationships do not predict the experimentalIy observed voidagediscontinuity and coexistence of dilute and dense phase for choking and entrainment. Suppose, however, that a third type of relationship exists in which as voidage decreases the slip velocity steadily increases above V, so as to give a gas flow-solids flow-voidage map as shown in Fig. 3. This map shows a very significant difference from Figs. 1 and 2. It exhibits an envelope which limits the possible solids rate at any given gas rate. In a pneumatic transport system below the envelope, either gas or solids rate may be changed slightly with a corresponding slight change in voidage. As the envelope is reached, by decreasing gas rate or increasing solids rate, the limiting or choking voidage is reached. In order to accommodate any further decrease in gas rate or increase in solids rate the system must change to a totally different voidageslip line_ The volume fraction of solids must make the sudden step increase, which is called ‘choking’. Because of the equivalence of the rate of dilute transport at choking and the rate of entrainment above a fluid bed, the chokmg envelope is also the entrainment curve for the system. The slope of tangc&s to the entrainment curve determines the dilute phase density above transport disengaging height from the relation: slope = (1 - E)/E_ The intercept of the tangents is eUs_ It is an important finding in this work that a limiting envelope formed by lines of constant voidage does exist for dilute suspensions. This envelope affords the physical mechanism which explains choking m pneumatic transport and permits coexistence of dilute and dense phases in entraining fluid beds. Envelope formation arises when slip velocity increases as voidage decreases
CLUSTER FORMATION The explanation for a sytem in which slip increases w-ith decreasing voidage probably lies in the phenomenon of cluster formation. It has long been recognized that
clusters will have slip velocity which increases with the number of particles in the cluster. If the size of the clusters increases with particle concentration, then the necessary slipvoidage relationship can exist. Clusters of particles will obviously fall somewhat faster than V,, the terminal velocity of a single particle. A review of a number of pertinent theoretical calculations appears in Happel and Brenner [S] _ For example, on page 277, they present equations derived by Burgers for the falling velocity of clusters of two, three, four, and eight particles, results of which are shown in Fig. 4. Jayaweera et al. [9] made experimental measurements of the falling behavior of clusters, verifying an increase m slip velocity and showing that such clusters would achieve stable, regular configurations. Particles introduced as a cluster would under most circumstances fall together as a cluster rather than becoming dispersed widely throughout the surrounding fluid. Kaye and Boardman [lo] reported quantitative data on falling behavior of marked particles in dilute suspensions of many particles. They observed definite cluster formation of a statistical nature and measured falling velocities well in excess of U,. Johne [ll] made similar but more accurate measurements_ Results of both investigations are shown in Fig_ 5. In both cases slip velocity is very nearly V, at E > 0.9997 and then rises with decreasing E to a maximum. With further decreases in E, V, decreases steadily, becoming less than U, in the range of 0.99 > E > 0.90. Both the above experiments measured settling velocity, with particles displacing liquid. Slip velocity in such a case is
In correlating data as in Fig. 5, the degree of clustering is implicitly a function of E. As voidage decreases, it is probable that the fraction of particles in clusters increases, that the number of particles in each cluster increases, and that the spacing between particles in a cluster decreases_ Throughout the rest of this paper, it will be assumed that V, is a function only of V, and E. It seems entirely reasonable, however, to suspect that other factors would have a secondary effect on V,. These factors might include: (1) particle Reynolds number in the region above
2 NUM8ER
Fig. 4. Cluster slip velocity
(I -
according
3 OF
PARTICLES
4 IN
5
6
I
I
7
89
I 3
CLUSTER
to Burgers.
0
Fig. 5. Slip velocity from sedimentation measurements_ Kaye and Board[lol.~JohneL11]_
creeping flow, and (2) particle impaction parameter which might be relevant to the rate at which clusters could form. Grace and Tuot [12] recently made a theoretical stability analysis of dilute suspensions and concluded that a uniform dispersion is unstable. Their calculations of disturbance growth rates showed that a suspension becomes more stable with increases in fluid density and viscosity and with a decrease in particle density_ Grace and Tuot treat only the tendency towanls cluster formation, but it is clear that beyond some degree of aggregation the tendency for
clusters to break up must exceed their tendency to form. Otherwise long vertical pneumatic transport lines would not work and entrainment from a fluid bed could be reduced to any desired level merely by increasing the disengaging height _ If an ent rainment cume is plotted on linear coordinates as G/pp uersus U, according to the present theory, a tangent to the curve at any point will have a slope (1 - E)/E, and the intercept with the X axis will be eUS_ Such a curve, taken by Wen and Hashinger [13] on uniformly sized glass beads, is shown in Fig. 6. The resulting slip uersus voidage relationship, obtained born tangents and intercepts, is shown in Fig_ 7. It is virtually identical to the Johne [ll] data of Fig. 5 up to solids concentrations of (1 - E) = 0 004, but continues to increase above that concentration while the Johne data level off. This may reflect the common observation that liquidsolids systems (Johne) have a greater tendency to form uniform dispersions while gas-solids systems (Wen [ 131) tend to cluster. A single data point from Koglin 1143 is also shown on Fig. 7. Koglin’s data are for sedimentation, and he claims that Johne’s data were influenced by wall effects to cause the deviations shown.
27
.a04
-003
5
G= ”
-Ooi
.OOl
uT
=0.326
rds
E =_99838 Inrerceot=EUS =0.5
r/s
15
Fig_ 6. Ent rainment
I
curve from Wen and Hashinger
I
I
[13]_
I
IllIll
I
I
I
Illll, LlmlL
1
I
I
IIll_
of Data
654-
Data
2
Of
Kcdm(l4)
7i
3 2-
1 .OOOl
-001
-01 (I-El
Fig_ 7. Slip velocity
US. voidage
for sedimentation
III hquids and entrahm
ent by gas_
.l
28 FLOW
MAP
FOR
VERTICAL
GAS-SOLIDS
FLOW
Data in Fig 7, for liquid-solid sedimentation and for gas-solid entrainment, suggest the following empirical expressions for slip versus voidage:
(1 -
US/U,
= 1
USI&
= 10.8(1-
E) < 0.0003 (8)
E)O=
(1 - E) > 0.0003 (9)
These expressions may be combined with eqn. (2) to give the dilute phase flow map. The dense phase map is defined by eqn (5) in terms of U, and Umr, but in order to be presented on the same dimensionless plot as dilute flow, U, must be used as the reference velocity. To provide a quantitative example it will be assumed throughout the rest of this paper that U, = l.OU, and Urn,= l/SO&_ With
a specific
bed
diameter
(if slug flow
exists) and a given gas-particle system, more accurate values for U, and U,, could be used, but the principle would remain the same_ The r
-45 -12 -
-10 -
_06-
_04-
.02-
o-
Fig
8. Flow
map for gas-solids
upflow.
graphic result is shown in Fig. 8, which gives in a generalized dimensionless form the interdependence of solids transport, gas velocity, and voidage for entrainment and choking, dilute pneumatic transport, and dense phase bubbhng flow The dilute pneumatic transport region is in the lower nght and the bubbling dense region is in the upper left of the figure_ The entrainment and choking curve is the line or envelope which separates the dense and dilute regions Below the entrainment curve in Fig_ 8, lines of constant voidage cross one another_ In thk region, the steady-state condition consism of only the lines of lesser slope as shown in Fig. 9. The lines of steeper slope represent a condition in which solids flux (at constant gas velocity) increases as voidage increases, a situation which is not stable to small perturbations in gas velocity, solids rate, or voidage. Figure 8 may be cross-plotted to demonstrate an analogy with a single-component liquid-vapor phase diagram, as shown in Fig_ 10. Here E is regarded as an independent variable, and one may imagine the physical
29
-06. d-
2
Fig 9
Stable solutions
4
3
5
for flow map.
I
a
1 i
6
3 3
\’
\ kLJ\
+ 4
2
C
!ny
B
boo4 .0002
1
\?A
I 0
6
0.01
I*
II,III
0.1
0.5 Cl-El
1
2 ,
1
5
VOLUME
, 10
.\ 20
PERCENT
Fig ‘_ 10. Phase diagram for vertical gas-solids
‘J I
30 SOLIDS
flow_
I
40
I
I
50 6(I 70
eo
30
system as being a fluidization column with a cyclone to recycle entrained solids back to the bottom of the column, and charged with enough particles to give any desired overall voidage E_ Inventory of particles in the cyclone and recycle line is negligible_ For most values of overall E at low velocities two phases will form, a dense bed of voidage defied by the right-hand porticn of the phase curve and a dilute entrained phase defined by the left side of the curve. As gas velocity is increased at constant overall E, however, the system will eventually become a single-phase system (i.e., there will be no horizontal interface having a dilute phase above and a dense phase below, although the column may have intermixed bubble and emulsion phases)_ For instance, the dashed line A - A in Fig. 10 represents a recycling fluid bed with an overall voidage of 98%. At a velocity U/U, = 2, the system will consist of a dense bed of voidage 82% (Point B) and a dilute phase of voidage 99.85% (Point C), and the entrainment rate G/ppUT will be about 0.0006. As velocity is increased the dilute voidage will decrease and the dense voidage and entrainment rate will increase, and at U/U, = 4.25 and G/ppUT = O-019 the dense phase will disappear If one operates at a velocity above the top of the phase curve, i_e at U/U, > 5.73 or at a solid rate G/pp > 0_0865Ur, it will be impossible to have two
Fig. ll_
Qualitative
slip us_ voidage relationships
phases at any value of E. The system may be thought of as ‘supercritical’. It is proposed that such supercritical behavior is responsible for the phenomenon of ‘fast fluidization’ Li and Kwauk [15] have recently published an experimental figure similar to Fig_ 10. Details of their figure are somewhat different because of acceleration and friction effects, and their interest was in the ‘fast fluidization’ area rather than in the choking and entrainment aspects of the diagram_ Yerushalmi and Cankurt [3] also show similar behavior on a somewhat different type of plot. The phenomenon of ‘fast fluidization’ has been studied extensively by Yerushalmi and co-workers [2, 31. Figure 11 shows their qualitative observations of slip velocity uers~s voidage relations for a wide range of gas particle behavior, including the fast fluidlzed bed and pneumatic transport. Figure 12 gives their quantitative data on slip uersus voidage for a specific system. Figure 12 may be compared with Fig. 13 drawn according to the theory developed in this paper. Both figures show a maxunum slip velocity at a voidage of about 0.9 and clearly represent the same phenomenon. Figure 12 includes non-ideal effects of: (1) mixed particle size, (2) wall friction, (3) particle acceleration, and (4) over-estimation of slip because voidage was inferred from pressure drop rather than being
from Yerushalmi
ef aL [3].
31
dilute suspensions, eqn. (8) applies and entrainment occurs at the limiting sohds concentration of (1 - l) = 0.0003. Substituting this value for (1 - E) and US = U, into eqn. (2) and dividing by U, gives G&UT
= 0.0003((
U/U,)
-
1)
(1 - E) = 0.0003
(10)
In the slightly higher solids concentration region, eqn. (9) applies- This may be substituted for U, in eqn. (2), which IS differentiated with respect to E and the derivative set equal to zero0 = g measured directly. For this reason Fig. 12 shows a family of slip-voidage curves with solids flux as a parameter, rather than the single curve of Fig_ 13_ NUMERICAL
(1 -
APPROXIMATIONS
Based on the present theory, simple equations for entramment and the associated dilute phase voidage may be derived For very
5
4 Dense Phase Riser Theory
> =
Phzse
3
Fig_ 13_ Slip velocity
us_ voidage relationships
E) >
0_0003
-
E)‘-~~& (11)
This gives the relationship between U and E which describes the entrainment envelope. In the region where E z 1 eqn (11) simplifies to u xa1 (le)=L26x 1O-4 vT
6
Dilute
+ (1.293)(10.8)(1
from present theory.
(1 - E) > 0.0003
(12)
32
Equation (12) is an approximate expression for dilute phase holdup when entrainment is occurring_ Substituting eqns_ (9) and (12) back into eqn. (2) and dividing by CT gives the generalized entrainment curve of G uersus u: G/p,&
= 2.84 (1 -
x 10-5(
tY/U,)3”1
E) > 0.0003
(13)
From eqn. (ll), the condition (1 -- E) > 0.0003 is equivalent to U[U, > 1.29_ Limitations of the numerical results expressed in eqns_ (S) - (13) and Figs. S and 10 must be emphasized_ These should not be used for quantitative analyses of systems of mixed particle sizes or where acceleration or wall friction effects are important Even for monosized, accelerationless, frictionless systems the numerical relations of eqns. (8) and (9) are only tentative_ Note for instance that for low values of Reynolds number based on U, this approach predicts no effect of gas density, while several empirical correlations indicate entrainment rate proportional to gas density. Nevertheless the numerical results do promise a useful form for correlation of choking, emrainment, and pneumatic transport data and a starting point for analysis of more complex systems.
DISCUSSION
It is seen that choking is a function of dilute phase slip-voidage behavior and does not depend on the dense phase. The bubble velocity in the dense phase affects the dense phase voidage and slip, however_ If the bubble velocity is reduced (e g. by using a small tube so as to cause slugs), the density difference between dense and dilute phases at choking will be reduced, as will the critical velocity above which choking cannot occur. Figure 9 demonstrates that a captive fluidized bed (ie_ with G = 0) cannot exist with finite (1 - E) at fluidlzing velocities greater than U,. Likewise, in a sedimentation experiment the top surface or sludge line cannot fall at a velocity greater than UT. The empirical equation of Richardson and Zaki [4] was based on sludge-line and bed expansion data and necessarily indicated
that Us/UT decreased monotonically from 1 as voidage decreased from 1. Figure 7 shows on the other hand that U’/U, increases above 1 as voidage decreases from 1. The reason is that in the sedunentation data in Fig. 7, the settling velocity of tracer particles within the suspension was measured rather than the settling velocity of the top surface_ The Richardson-Zaki equation is inappropriate for voidage close to 1 and should not be used in choking calculations This analysis would indicate that choking can occur in liquid-particle suspensions as well as gasparticle systems. In the liquid system, the dense phase expansion will be much greater, thus substantialiy lowering the value of U/UT above which two phases cannot coexist and also greatly decreasing the difference between dense phase and dilute phase voidage when choking does occur. Thus choking in a liquid system would be difficult to detect experimentally and would be of little practical consequence. For cases where particle to wall friction is significant, an approach analogous to that of Fig. 8 may be used. Lines of constant E will have a slope less than (1 - E)/E and may be curved or straight depending on the nature of the friction correlation_ These lines wrll have the same X intercept as in the frictionless case, since particle velocity and friction are zero at the intercept_ The result is to lower the choking curve so that higher gas velocities or lower solids flow rates are necessary to avoid choking_
CONCLUSIONS
For systems of uniform particle size and velocity profue, with negligible wall diction and inertial effects: (1) In dilute phase suspensions, actual particle slip velocity is greater than single particle terminal velocity even at very low sohds concentrations_ This occurs because the particles tend to cluster rather than behaving as individual particles_ (2) A provisional numerical expression for slip is Us = UT for (1 - E) < 0.0003 U, = lO.SU,(l
-- c)Oa3
for
(1 - E) > 0.0003
33
(3) Because slip increases with solids concentration, straight lines of constant voldage on a plot of G/p, UersU.s tY form an envelope. This envelope defines a maximum G/p, at a given U which IS equivalent to the choking curve for pneumatic transport or the entramment curve for fluid beds. (4) A tangent to the entrainment curve of G/p, uerSUs U has a slope of (1 -- E)/E and an intercept of lUs_ Thus, the entrainment curve can yield dilute phase holdup and slip velocities. (5) Using the newly derived relationships for dilute phase behavior combined with earlier expressions for dense phase behavior, a complete map of G/pP uersus U with lines of constant voidage can be drawn. This czn be cross-plotted in a phase diagram of U Uersus (1 - e) with G/D~ as a parameter. This phase diagram clearly shows the coexistence of dense and dilute phase fluld beds at low velocities and the existence of a ‘supercritical’ region at high velocities in the zone ‘fastfluidization’. (6) For entraining or chokmg conditions, the voidage may be represented as (1 -
E’ -= 0.0003
U/UT
< 1.29
(1 - E) = 1 26 X 1O-4 ( U/UT)3 UfU,
>
and the solids transport G/P,
= 0.0003(
GIP~
=
2.84 u/u,
LIST
D DB
g G U
X
41
1.29
U/U,)
rate as u/u*
< 1.29
10-5(U/UT)341U > 1.29
U nlf US
m
of isolated bubble, m/s mimmum fluldization velocity, m/s slip velocity between fluid and particle, m/s terminal velocity of isolated particle, m/s void fraction void fraction at minimum fluidization density of expanded bed, kg/m3 density of bed at minimum fluidization, kg/m3 particle density, kg/m3
E Emf
P Pmf PP
REFERENCES
1
F_ A_ Zenz and N_
A_
Weil. AZChE
J_, Q (1958)
472.
R. A Graff, S_ 2 J_ Yerushalmi, M. J_ Gluckman. Dobner and A_ M_ Squires, in D_ L_ Keairw (Ed ). Fluidization Technology. Vol 2, Hemisphere Pub1 Carp_, Washington, 1976. p_ 437. J. Yerushalmi and N. T Cankurt, Powder Technol. 24 (1979) 187. J F_ Richardson and W_ N Z&i, Trarrs. Znst Chem Eng , 35 (1954) 35. J M_ Matsen, Powder TechnoL. 7 (1973) 9% J. F. Davidson and D_ Harrison, Fluidrzed Particles, Cambridge Univ Press, Cambridge, 1963. 7 J_ M_ Matsen. S. Hovmand and J_ F. Davidson, Chem. Eng. Sci, 24 (1969) 1743. h’umber 8 J. Happel and H. Brenner, Low Reynolds Hydrodynamics, Prentice-Hall, Englewood Cliffs, NJ, 1965_ 9 K_ 0. L_ F_ Jayaweera, B_ J_ Mason and G. W_ Slack, J. Fluid Mech., 20 10 the
OF SYAIBOLS
bed diameter (for slug flow), bubble diameter, m acceleration of gravity, m/s’ solids mass flux, kg/m% superficial gas velocity, m/s
rise velocity
uB
Interaction
between
Fluids
and
Particles.
11 12
Inst. Chem. Eng., London, 1962, p_ 17. R. Johne, Chem. Zng. Tech, 38 (1966) 428_ J. R. Grace and J. Tuot, Trans_ Inst. Chem Eng.,
13
57 (1979) 4. C Y Wen and R. F_ Hashinger,
14 15
AIChE
J_
6
(1960) 220. B. Koglin, D. Ing. Diss_, Univ. of Karlsruhe. 1971 Youchou Li and M Kwauk. in J. R. Grace and J. M. Matsen (Eds.), Fluiduation. Plenum Pres, New York, 1980.