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Entrainment of solids from fluidized beds I. Hold-up of solids in the freeboard II. Operation of fast fluidized beds
Powder Technology,
193
61 (1990) 193 - 206
Entrainment of Solids from Fluidized Beds I. Hold-Up of Solids in the Freeboard II. Operation of Fast Fluidized Beds D. KUNII Combustion and Fluidization Engineers, Tokyo 154 (Japan)
401 Olympic Mansion, 1-33-l 7 Sangenjaya, Setagaya-ku,
and 0. LEVENSPIEL Chemical Engineering
Department,
Oregon State Uniuersity, Coruallis, OR 97331-2702
(U.S.A.)
(Received December 1, 1989)
SUMMARY
PART I - HOLD-UP OF SOLIDS IN THE FREEBOARD
Part I: Kunii and Levenspiel proposed a flow model to represent the complex phenomena occurring in the freeboard above a fluidized bed. A somewhat more generalized version of this model is developed in this paper, to explain and integrate the diverse experimental findings on freeboard behavior reported to date. The key parameter emerging from this model is the decay constant for the fall off of bulk density of solids with height in the freeboard. Values for this decay constant are extracted from the experimental data reported in the literature and a correlation for the decay constant, useful for predictive and design purposes, is found to reasonably represent the many findings reported to date. Part II: Viewing the fast fluidized column as having a lower region of approximately constant bulk density and an upper region wherein the bulk density decreases prog’ressively with height allows the model of Part I to be used directly for these con tat tors. As a consequence, the peculiar characteristics of fast fluidization systems are explained, and a correlation for the decay constant is obtained to represent the findings reported to date. An example of a design calculation shows how the fast fluidized system responds to changes in operating variables. More importantly, it makes clear what factors are needed to determine the flow characteristics of these systems.
Experimental findings There have been many studies, both steady and unsteady state, focusing on the lean zone above fluidized beds. These are summarized by Kunii and Levenspiel [ 11. Since the purpose of this paper is to develop a model to usefully represent these findings, we start by highlighting some of the pertinent studies in the literature. First, Zenz and Weil [2] proposed a convenient correlation for estimating the transport disengaging height (TDH) as well as the saturation carrying capacity for the pneumatic transport of FCC catalyst. Fournol et al. [ 31 later correlated the TDH for FCC catalyst with the Froude number of the solids. In one of the earliest studies on freeboard behavior, Lewis et al. [ 41 proposed that bubbles bursting at the surface of the bed was the means for getting particles into the freeboard. Let us designate the initial upward flux of these solids from the bed surface by G,, (kg/(m2*s)). Further in this line, Caram et al. [ 51 found that the initial velocity of particles ejected from fluidized beds depended only on bubble size, hence velocity of bubble, and was independent of size and density of particles. Horio et al. [6] also found that the intensity of turbulence at the bed surface was related directly to bubble size. Focus attention on the region below the TDH. According to Lewis et al. [4], the 0 Elsevier Sequoia/Printed in The Netherlands
194 Height
Hf.
A--’
zf
l-
.___. A
TDH --
density is also assumed to fall off exponentially, or
in
freeboard, A
G, = GsOeeaHf
Gas exit above the TDH
c--
At complete
\ \ \
reflux
FR = FRO exp I-azfI, Eq. (1)
___________________
Gs = Gsu
0. _. T---
l-l
F
D
B
Bed ._t_____ _--
Holdup or density of solids in the freeboard, p
Fig. 1. Density distribution of solids in the freeboard for three different freeboard heights, according to Lewis et al. [4].
density of solids present at various levels zf as the freeboard height Hf is changed can be represented by the sketch of Fig. 1. Here, curve AGB represents the solids hold-up when the freeboard height is at Hf, A > TDH. This may be called the condition of complete refhx. Curves CD and EF are for freeboard heights H, c and Hi, E, respectively, both below the TDH. From this sketch we see: (i) At complete reflux (curve AGB), the density of solids in the freeboard falls off exponentially from the value at the bed surface, or jja = PRO eearf
(1)
(ii) At smaller Hf, below TDH (curves CD and EF), the solid density is some constant value less than at complete reflux, or Pa - p = constant
throughout
(3)
Note that although the density of solids at any level in the freeboard goes down as Hi is lowered, the density at the exit does increase. Hence, entrainment increases as the freeboard height is reduced. Since the clumps of solids thrown into the freeboard by the bursting bubbles are representative of the bed material and since the larger solids in these clumps fall back to the bed, there is an upflow flux G, and downflow flux G so of solids everywhere in the freeboard below the TDH. The net flux or carryover out of the vessel G, is related to these up- and downflows at any level in the bed by
H/, c---
Hf, E ---
(kg/(m2-s))
the freeboard
(2) (iii) The entrainment G, from the bed is proportional to the density of solids at the gas exit (see dashed line AGCEH). This exit
-
Gsa
(4)
Kunii and Levenspiel [7] presented a simple flow model to represent the complex phenomena occurring in the freeboard. We now present a more generalized version of this model.
Freeboard-entrainment model Consider the freeboard above a bubbling or turbulent fluidized bed, and let x be the fraction of bed solids for which Ut < u,. This is the entrainable fraction, and here we call them the fines. Postulate 1. Three distinct phases are present in the freeboard. Phase 1: Gas stream with completely dispersed solids. The fines are carried upward and out of the bed at velocity ul. Phase 2: Agglomerates, coming from the bed, and moving upward at velocity u2. Phase 3: Agglomerates and thin wall layers of particles moving downward at velocity u3. Post&ate 2. At any level in the freeboard, the rate of removal of fines from the agglomerates to form dispersed solid of phase 1 is proportional to the volume. fraction (or solid density) of agglomerates at that level. Postulate 3. Upward-moving agglomerates will eventually reverse direction and move downward, the frequency of change from phase 2 to phase 3 being proportional to the volume fraction of phase 2 at that level.
195
TDH Density distribution for Hf >TDH, jjR Eq. (24) for
H
1._. ..._
Splash zone
____
r-------
G, = xG;
(b)
(a)
Fig. 2. Sketch and various terms and expressions of the freeboard model.
Figure 2(a) sketches the freeboard as viewed by this model. Now, at any level zf in the freeboard, let Gsl, Gs2, Gs3 (kg/(m’-s)) be the mass flux of each phase, and let pl, p2, p3 (kg/m3) be the mass of each phase per unit volume of freeboard. Then, at steady state conditions the net upward flux of solids at any level in the freeboard is given by G, = G,i +
Gs2
-
Gs3
=
PIUI +
~2~2
-
~3~3
independent of zf.
(5)
Also, the average (or bulk) density of solids at any level in the bed is P=pl+pZ+p3
Gsa
= Gs3
increase of solids in phase 3
i
= 1
transfer of solids i from phase 2 to 3
transfer of solids
-
i from phase 3 to 11
Introduce the rate coefficients 3tKi for the transfer of fines from phases 2 and 3 to 1, and K, for the transfer from phase 2 to phase 3. In these expressions, x is the fraction of fines in the dense fluidized bed. Then the above mass balances at level zf become
(6)
Relating these quantities to the terms previously used G,, = G,, +
for phase 3
UlZ
=3Cmp2+
P3)
dP2
Gs2
(7)
i
Now, a mass balance for phase 1, and this only concerns the fines, gives transfer of solids from phases 2 and 3 to 1 for phase 2 transfer of solids phase 2 to 1 and 3
-U2z-
= w,+K2)P2
dp3 -u3 =
Kg,
- xK,p,
(10)
dzf
where ui and u2 are the upward velocities of phases 1 and 2, and u3 is the downward velocity of phase 3. Since all solids which reach height Hi leave the vessel, there is no downflow there, so p3=0
atzf=Hf
(11)
196
Noting that at the bed surface, subscript 0, the only upflow is by clumps of solids projected into the freeboard, we have
(17)
u2
Thus, eqn. (13) reduces to
Pl=O G P2=
K2 -
a=
-fy-
atzf=O
(12)
G, -
xG,* E
e-WJUzWf
(18)
bSU0
I
Net upflow and carryover of solids Solving eqns. (8), (9), and (10) with the boundary conditions of eqns. (11) and (12) gives pl, p2, and p3 in terms of exponential functions of zf. With eqn. (5), the net flux of solids at any freeboard level and also at the vessel outlet is then found to be
G, - xG,* = -oHf e G suo - xG,*
(13)
where G,* is the flux of carryover from a very tall vessel fluidizing only entrainable solids - in practical terms, when Hf > TDH. The ratio of upward flux out of a tall vessel to the upward flux at its surface is also found to be
These expressions provide a physical interpretation of the equations reported by Lewis et al. [4], in eqn. (3). Upflow term alone In the present model, the upward flux is given by Gsu =
plu, +
~2~2
(19)
Substituting the expressions found for p1 and p2 into eqn. (19) gives
G, - xG,* G sue -
xG,*
=e
-aHf
+ e-bzf
_
e-[aHf-(a-
b)zf]
(20)
where b=
xK1 + K, u2
xG,*
xK1 -
a-b= =-
(14)
G sue
(a - b)zf < aHf (m-l)
(15)
In the special case of a vigorously bubbling bed or a turbulent bed with not too many fines, the amount of solids thrown into the freeboard is very much larger than those eventually removed from the bed. Thus, xG,* Q GsUo. In this situation, eqn. (14) indicates that
or
and from eqn. (15)
u3
Again, in the special case where xG,* < G,, we find xK1 Q K,, thus a - b Q a. Since we always have zf 5 Hf, all these inequalities lead us to conclude that
and
a=$.[1+~jl+~)]
(21)
(16)
ar
b
(22)
I Thus, in this special case eqn. (20) reduces to
(23) At zf = Hf there is no downward flux of solid. At this location, eqn. (23) reduces to eqn. (18), thus confirming that G, = G,.
Solids hold-up in the freeboard This model also gives the distribution of bulk density of solids in the freeboard. Substituting the expressions found for pl, p2, and p3 into eqn. (6) gives a bulky expression; however, in the special case where xG,* 4 Gsuo, this expression simplifies to two special cases. For very high freeboard, or in practical terms, for Hf > TDH, the density
197
ford, = 150 - 212 pm at u,, = 0.61 m/s and zf = 3 m. Geldart and Pope [9] measured j!~of fines elutriated from fluidized bed of coarse sands at very high air velocity, u, = 2 - 4 m/s, and found
of solids p, at any height zf is given by the expression
for Hf > TDH
(24)
_!!_ r 10-Z PO
where the lowest density of solids in the freeboard, that above TDH and at pneumatic transport conditions, is given by
at G, = 1.53 kg/(m2.s) and zf = 3 m. Comparing the above findings with eqns. (23) and (29), we estimate that
XG,* .Qj* = -
e_aHf= 10-Z
(25)
in case Hf is high enough, for example 3 m at the above conditions. Thus, at a level zf not far above the bed surface, eqn. (29) reduces to
Ul
For not very high freeboard, or Hf < TDH, the density of solids p is given by the following equation, again in the special case where xG,* Q G,s: e-azf -
P - xp* z Gsuo
p
1
1 _ e-‘JHf u3
(26) Combining eqn. (24) with eqn. (26) gives GSUO
fiR--P=
-e
--aHf
u3 =
constant throughout the freeboard
(27)
Equation (27) explains theoretically eqn. (2) given by Lewis et al. [ 41. At the surface of the fluidized bed, eqn. (26) gives PO
-xp-PGsuo[($
+
-3cp*
po- xp*
se
-0zf
(30)
Discussion of the freeboard-entrainment model The upward and downward mass fluxes of solids at the apparent surface of fluidized beds, G,, and GsdOrespectively, have been measured by several workers [8 - 131 i as shown in Fig. 3. We see appreciable variation in these values. This may be because GsuO is found by extrapolation back to zf = 0, and different assumptions were made regarding the location of zf = 0 in the vessel. Wen and Chen [13] took the zf = 0 plane at the mean value of the bed surface; others took it somewhere within the splash zone.
d)- d e_aHf] (28)
Thus, combination of eqns. (26) and (28) gives II e-a=f
p -3cp* po -
e_aHf
1 + u2/u3
2
xp*
-
l-
1
(2%
e_aHf
1 + u2t”3
Hoggen et al. [8] measured G, us. zf in a large industrial roaster, finding that
-GSUz 10-Z G SUO
+.+ +*--I I I Ill11 I 1 I IIIII 0.05 0.1 0.5 0.2 0.02 u. - Umf (m/s)
10-Z -
1
Fig. 3. Mass flux of particles just above the bed surface; from Walsh et al. [ 121. Sources of information: A, Gsuo from [la]; A Gsuofrom [14]; VI GsdO from [ll]; V GsdOfrom 1121; A, GsuOfrom [4]; A Gsuo from [15].
198
In this model, since Gsl= 0 at zf = 0, we have G, = GSz-- Gs3 atzf= 0 (31) = Gsuo- Gscio For a vigorously bubbling fluidized bed, we also have xG,* < Gsuo, in which case, just above the bed surface G SUO- Gsdo
atzf=
0
7 c
(33)
Walsh et al. [12] found this relationship to hold also for coarse particles. Now if we assume that u2, the upward velocity of agglomerates, is proportional to gives an immediate U 07 then eqn. (17) meaning to the empirical expression of eqn. (33). Equation (30) was used to obtain values of the decay constant a from experimental data of solids hold-up us. height at levels in the freeboard not close to Hf. Values of a were also obtained from experimental data of G, us. zf. Figure 4 correlates au0 thus obtained us. mean size of the entrained solids at velocities u, < 1.25 m/s. Since the decay constant for solids density is the key parameter needed to develop predictions of fast fluidized bed behavior, we urge further experimental studies to obtain more reliable values for a for a wide range of solids and a wider range of gas velocities.
PART II - OPERATIONS OF FAST FLUIDIZED BEDS
In the fast fluidization regime, carryover of solids is very large; hence, fresh solids have
2
3" c.
(32)
Therefore, Fig. 3 can be used to estimate both G,, and Gscio. In almost all cases of elutriation from a bubbling or turbulent fluidized bed, it is reasonable to assume that xG,* < GSUo.Thus, one is justified in using the special case simplifications, or eqn. (30) for the distribution of solid density in the freeboard, eqn. (23) for the flux of only upward moving solids in the freeboard, and eqn. (18) for the carryover as a function of various freeboard heights. Finally, for fine particles, Lewis et al. [ 41 found the following relationship to hold, au, Z constant
3
1 -I
‘c---q
o 0
I
I
400
200 dp
I
I 600
I
600
(pm)
Fig. 4. Decay constant for the freeboard agglomerates for Hf < TDH and u0 < 1.25 m/s. 0, From [14]; @, from [12]; 0, from @), from [16]; @, from [17]; 8, from [lo]; , from [8].
to be introduced continuously and at a significant rate to make up for the loss of bed solids and to achieve steady state operations. Figure 5 illustrates a typical fast fluidized bed with its various regions. (i) At the bottom one sees a relatively short entry zone having a solid fraction E, = p/p, = 1 - ef = 0.2 - 0.4. (ii) Then there is a portion of the vessel of almost constant solid fraction of about E, = 0.2. Regions (i) and (ii) may be called the dense region. (iii) Above this is an upper entrained region where the solid fraction decreases progressively to about e, = 0.02 - 0.05. These regions correspond somewhat to the dense turbulent bed and its freeboard. Also, the transition between regions is smooth. Because of the large entrainment of solids from fast fluidized beds, inner cyclones and diplegs are too small to handle the solid load. Thus, large cyclones located outside the column are used and these require careful design for proper operation. Overall, we may want to run a fast fluidized circulation system in one of four ways: Mode I. Keep a constant inventory of solids in the bed, even though u, may change. Mode II. Keep a constant throughflow of solids G,(kg/(m2.s)), even though u, may change. Mode III. Keep a constant gas flow rate u,, while changing the solid throughflow G,.
199
Esd Z 0.16 - 0.22
(4 Fig. 5. The fast fluidized bed and its regions of different fraction of solids.
Mode IV. G, and u, can be changed independently. Figure 6 illustrates the two basic types of solid circulation systems: those which do not include a reservoir of solids, and those which do. Without the reservoir, one can only operate according to Mode I. With the reservoir, the system becomes much more flexible in that it can operate in any of the four modes listed above. Experimental
findings
With the works of Yerushalmi and coworkers [ 181 as a catalyst, there has been a sharp increase in interest in investigating the characteristics of the fast fluidization flow regime, see [ 11. The following studies are of particular interest here. Vertical distribution
of solids
Li and Kwauk [ 191, Weinstein et al. [ 201, and Hartge et al. [ 211 all found an S-shaped
(b)
Fig. 6. Idealized solid circulation systems for fast fluidized operations. Scheme (b) contains a reservoir for solids and is much more flexible than scheme (a).
solid fraction curve, as shown by the solid line of Fig. 5. This S-shaped curve moved up or down the column depending on the solid and gas flow rates. Furthermore, it was found that the gas and solid favored flowing in the central core of the column and that at the wall there is a dense slow-moving layer of solid. In addition, Matsen [22] found an appreciable maldistribution of solids in large diameter beds - nonsymmetrical and nonreproducible. Consider a fast fluidized bed as having a lower region of constant solid fraction es,-, and an upper region wherein the solid density decreases progressively to its exit value E,,. This leaner region starts at the level given by the intersection of the two dashed lines, shown as point A in Fig. 5. Ignoring the short entry zone of Fig. 5, the sketches of Fig. 7, prepared from the published data, show that the flow rate of solids and of gas in the range U, = 1.5 - 5.0 m/s do not seem
200 0.4 u. = 3 m/s s w
0.3 -
~._..~~~__.~___~~~~~~..~_~_ ..._*..~_..‘.....l ..............L........
4) 0.2 -
4 m/s J pS = 1450 kg/m3
0.1 50
100
150
G, (kg/m2. s) 0.3
cl
A 2
0.2 -
lA
I
7 %
l+
.
w”
I
I
4, X lt
‘0
a I@ ;
$
-0
::*
l
.
Q 0.1
V
I
V
I
(a)
QVI
100
0 G,
0
I
200
(kg/m2* s)
0
/
I
2
h
(b)
I
I
4
6
(m/s)
Fig. 7. Volume fraction of solids in the lower dense region of the fast fluidized bed. (a), Various solid flows, see Table for references; (b), various gas flows, from Schnitzlein [23]. TABLE References and experimental conditions for the data of Figures 7(a), 8, and 9 Key
Reporters
dt (cm)
Type of solid
+ D
Hartge and Werther [ 211 Li and Kwauk [19]
5 40 9.0
@ @
Weinstein et al. [ 201 Rhodos and
15.2 15.2
Quartz Quartz, FCC FCC Alumina Pyrite cinder HFZ-20 cat. -
Geldart [ 241 Takeuchi et al. [ 251 Schnitzlein [ 231 Kato et al. [ 261 Horioetal. [27] Arena et al. [28] Li and Kwauk [19] Yang et al. [29] Arena et al. [ 28 ]
10.0 15.2 6.6, 9.7 5.0, 20 4.1,12 9.0 11.5 4.1
FCC HRZ-33 cat. FCC, cat. FCC FCC, glass Iron Silica gel Glass
61 59 61 60 70 105 220 88
1.7 1.5 2 1.1 4-5 5.3 5,7
20
Sand
170-650
-
0
Brereton and Stromberg [ 301 Furchi et al. [31]
$
Lu and Wang[32]
10
Glass Glass Sand, glass
196 269 230, 369
7.2 8.3 2.9-4.9
X
A
0l 0
O+ q
V
B Wq @
7.2
to appreciably affect the fraction of solids in the lower dense region of the vessel. It is interesting to compare eSdin the various fluidizing regimes. bubbling bed: E,d = 0.55 - 0.40 turbulent bed: f,d = 0.40 - 0.22 fast fluidization: E,d = 0.22 - 0.16
&, (pm) 56 56 54,58 54 56 49 64
UO (m/s)
Gs (kg/(m’*s))
3.4 72 1.2 - 4 7 0.8 - 2.1 14 2.2-4 73 1.5 - 2.5 129 2.9, 3.4 712.5 - 4.5 86 - 2.9 -5 4.4 - 1.6
Remarks
40 16 118 115
8.3 - 79 89 - 133 48 - 50 12 - 19 49,120 135 44 - 146 80 - 600
Fine particles, d,<:70pm
Larger particles, dp > 88 I_lrn
64 - 146 88 127 -
Keys to Fig. 7 as well as Figs. 8 and 9 are shown in the Table. Fraction
of solids at the column
exit, E,,
First of all, we note that E,, is somewhat larger than the saturation carrying capacity of the gas E,*. Next, the data of Fig. 8 show
201
3
4
5
G, (kg/m2.s)
50
50
50
100
100
up(mW
0.1
0.5
1.0
0.5
1.0
12
Line
200
0
400
G, (kg/m2
(a)
0
600 l
s)
2
4
h
(b)
6
(m/s)
Fig. 8. Volume fraction of solids at the exit of fast fluidized beds; experiment compared with curves of eqn. (36). (a), E,~ at high solid flows, G, = 100 - 600 kg/( rn*-s); from Arena et al. [ 281; (b); (b), ese at low solid flows, G, = 50 - 100 kg/(m*.s); from various sources, see Table for references.
that at given solid flux E, is higher at low U, than at high u,,. This seemingly puzzling finding can be explained in terms of the slip velocity between gas and solid, or up = ug - u,. Thus, letting u, be the mean velocity of solids at the exit level of the column, a mass balance gives G, = G,, (34) = PSES&S Introducing the slip velocity gives the solid velocity as UO
us= i-y--/P For fine particles at high u,, E,~ Q 1, and thus, the above expressions become G, = G,, = P&e(u,
-up)
or Es,
s
GS Ps(u0 - up)
(36)
If the fine particles are completely dispersed in the gas stream, then we can reasonably assume that up E ut and ut < uO. The two lines drawn in Fig. 8(a) are based on eqn. (36) with this assumption and are seen to be consistent with the data of this figure. In trying to find the saturation carrying capacity of gas es*, values of ese were taken from reported data at low G, or high enough columns such that e, seemed to level off in the upper portion of the column. Figure 8(b) summarizes these data from various sources and compares this with the curves calculated
from eqn. (36), using appropriate values of G, and up. Although there is considerable scatter in the experimental values, eqn. (36) seems to pass through the main body of this data. The freeboard-entrainment model applied to fast fluidization Consider a fast fluidization column as having a lower region of constant solid fraction &d and an upper leaner region wherein the solid density decreases progressively to its exit value E,,. This leaner region starts at the level given by the intersection of the two extrapolated dashed lines, shown as point A of Fig. 5. With this as the picture of a fast fluidization column, the decrease or decay of E, in the lean entrainment region can be treated with the model of Part I which was used to represent the freeboard above bubbling and turbulent beds, but with the simplification here that all the solids are entrainable, or x = 1. On the basis of this model, eqn. (30) predicts an exponential decay in solid density between eSd and a limiting value of E,*, with a decay constant a related to gas velocity u, by eqn. (33). Thus, a higher gas velocity u, means a slower decrease of solid concentration with height in the entrainment section of the column. This is to be expected. Figures 9(a) and 9(b) present values of the decay constant calculated from various literature sources for fine particle systems (d, < ‘70 pm) and for coarser particle systems (dp > 88 pm). Lines representing eqn. (33) for different values of the decay constant are also shown in these figures. Despite consid-
202 Smaller
OL 0
I
I
d, -Z70 pm
I
u. Larger
particles
I
4
2
(a)
(b)
particles
I
I
basically similar to the decay above bubbling and turbulent beds. The decay constant is important for design. Since reported values of this parameter are scattered and sketchy, more precise data on its proper value and how it changes with the imposed system conditions would be most welcome.
6
VW
d, > 86 pm
u. VW
Fig. 9. Decay coefficient a which represents the changing solid fraction in the entrainment region; eqn. (37) was used to calculate a from experimental data. See Table for references.
Design considerations In design, one needs to know the solid hold-up in the vessel as a function of the flow conditions, U, and G,. This, in turn, requires knowing the location of the top of the dense region and solid densities throughout. These values can be found from the information given so far in this paper, as we will now show. First consider the upper entrainment zone of the column. Referring to Fig. 5, eqn. (30) becomes f, - e,* Esd -
=e -&Zr
The fraction erable scatter in the reported data, this figure suggests the following: (i) The decay constant a seems to increase with decreasing c&. This may be explained by noting that in narrow columns the rising agglomerates are more likely to hit the wall surface and then be removed from the rising gas stream. (ii) The decay constant a seems to increase with increasing d,. This may be explained by noting that with coarser and denser particles the agglomerates are more likely to change direction and return to the dense region of the vessel. In the model this is represented by a larger K2-value. (iii) For the fine particle systems of Fig. 9(a), we see that higher gas velocities give lower values for the decay constant, as predicted by eqn. (33). On the contrary, the data on the coarse particle system reported by Brereton and Stromberg [30] do not seem to fit this relationship. (iv) The decay constant a for bubbling and turbulent beds and for fast fluidized beds (compare Fig. 4 with Fig. 9) all fall in the same range of values, suggesting that the mechanism of decay of solid fraction in the lean region of fast fluidization columns is
(37)
Es*
of solids at the vessel exit is
Ese = e, * + (e,d - e,*) eeaHf
(38)
and the mean value of E, in the upper entrainment region of height Hf is calculated by Hf
A-sf,dzf
T, =
Hf
(39)
0
Inserting eqn. (38) integrating gives cs
=
E,*
+
eqn.
(39)
and
‘sdaHf - %*(1 - eCaHf)
s* + sdE
=e
into
Ese
aHf
(40)
Finally, the total inventory of solids in the column of height Ht = Hf + Hd is then -
W
= I&-
Em)
AtPs =
Lnf(l-
Gnf)
=
H&d
+ HfZ,
=
Esd -
a
ese
+ H&d - Hf(e,d - ES*) (41)
To use this freeboard-entrainment model, one needs vahies of a, es*, and e,d. Let us
203
briefly point out where these values can be obtained. (i) First of all, a is estimated from Fig. 9. (ii) As for E,*, according to Monceaux et al. [33], E,* e 0.01 for the pneumatic transport of particles. Phenomenologically, this value may correspond to the transition between dilute and concentrated transportation of solids. Alternatively, from the definition of mass velocity and with U, 3 Ut we get
GS* es*= _ P&s
=-
Gs*
(42) PSUO where G,* is found by the method of Zenz and Weil [ 21. (iii) Finally, c,d is found from Fig. 7. The various quantities in this model are sketched in Fig. 10.
Height, z
Find a from Fig. 9; then get this curve
I
I
I
I
I
I
I I I I ._._ I__+_-__--_ . .. . . .._ 1
Includes the entry zone / Fraction in G/S
t
_ Lrom
Eq. (36)
K from
of sohds mixture
Fig.7
Fig. 10. Solids distribution in fast fluidization, from the freeboard entrainment model.
Let us now see how to calculate the performance of the fast fluidized bed in the four modes of operation outlined earlier in this chapter. Mode I. Constant inventory of solids (no reservoir of solids in the circulation system) (i) For given u,, estimate csd from Fig. 7(b), and estimate a from Fig. 9.
(ii) Calculate c,, as a function of Hf, using eqn. (38). (iii) For the desired inventory of solids Lmi (1 - E,~) and given height of vessel Ht = Hf + Hd, determine Hf by substituting all values known so far into eqn. (41). (iv) Calculate G, from eqn. (36). The mass flux of circulating particles from bubbling or turbulent fluidized beds through an inner cyclone collector is calculated in the same way. Modes II, III, and IV. Constant G, or uO, or both changing (i) Estimate csd and a, as above. (ii) Calculate E, for given G, using eqn. (36). (iii) Determine Hf and Hd = Ht -~-Hf from eqn. (30 (iv) Determine J& (1 -- e,t) with eqn. (41). The example below illustrates this calculation procedure and Fig. 11 displays the results of these calculations. Note that as u, is changed, mode I and mode II give opposite progressions of solid fraction vs. height in the bed. Example. Performance of a fast fluidized vessel Determine the performance characteristics of a fast fluidized column when operated in the following four modes Mode I: Constant solid inventory corresponding to j&f = 2.4 m, with variable gas flow of u, = 2, 4,6 m/s Mode II: Constant solid flow at G, = 100 kg/(m2*s), with variable gas flow of u, = 2, 4, 6 m/s Mode III: Constant gas velocity u, = 4 m/s, with changing solid flow G, = 42, 50, 100, 200, 400 kg/(m’. s) Mode IV: G, and u0 both vary as follows 2 4 6 u. (m/s) 100 120 I 70 Gs (kg/(m’- s)) For all modes determine the vertical distribution of solids, es. Data Column: dt = 0.4 m, Ht = 10 m Particles: catalyst, ps = 1000 kg/m3, d, = 55 pm, f,f = 0.5 Gas: ambient conditions Solution A straight application of the calculation procedure described above gives the performance characteristics shown in Fig. 11. For
Mode
Mode
I 10
II
G, - 100 kg/m2.s
Mode Ill
Mode IV
un = 4mls
u0 (m/s) 1G, (kg/m2.s) 10
u,(mls) 2 4.100
4 5
6,120
6
5 0 1 0
0.1
0 R
0.2
Es l-1
Es (-) (W
Cd)
Fig. 11. Solutions to the example; d, = 0.4 m, Ht = 10 m. (a), Mode I; (b), Mode II; (c), Mode III; (d), Mode IV.
detailed numerical calculations see [ 11. An examination of this figure shows that, despite the many simplifications made, the performance characteristics of this model do seem to be reasonable. LIST OF SYMBOLS
At a
Gs GS* Gsa,Gsu
Gsd0, Gsuo Hd,
Hf
Ht = Hd + Hf K,, K,, K3
L,
Lnf
TDH % UO
cross-sectional area of column, m* decay constant for fall off of solid density in freeboard, m-l net upward mass flux of solids in freeboard, kg/(m**s) saturation carrying capacity of upflowing gas for solids, kg/(m*+s) downflow and upflow flux of solids, respectively, kg/(m**s) Gsd, G, at surface of fluidized bed, respectively, kg/(m*- s) height of lower dense bed and freeboard, respectively, m height of column, m rate constant for interchange of solids in freeboard, see above eqn. (8), s-l height of fixed bed, bed at minimum fluidization, respectively, m transport disengaging height, m velocity of gas, m/s superficial gas velocity on an empty vessel basis, m/s
UP
US
Ut Ul,
u2,
u3
X Zf
Greek symbols Ef,
fS
ES*
Esd
e 88
Ern, fmf
slip velocity between gas and solid, m/s velocity of dispersed solids in a pneumatic line, m/s terminal velocity of a falling particle, m mean velocity of dispersed solids, of upward moving clusters, and of downward moving clusters, respectively, m/s weight fraction of fines distance above mean surface of fluidized bed or above lower dense region of a fast fluidized bed, m void fraction in a fluidized bed as a whole, in a fixed bed, and in a bed at minimum fluidization conditions, respectively volume fraction of solids, =1---r saturated carrying capacity of a gas, or maximum volume fraction of solids that can be pneumatically conveyed by a gas volume fraction of solids in lower dense region of a fast fluidized bed volume fraction of solids at column exit
205
Pl,PZ,
P3
P PO
PR
PRO
PS
bulk density of dispersed solids, of upward moving clusters, and of downward moving clusters, respectively, based on unit volume of freeboard, kg/m3 mean density of a gas-solid mixture, kg/m3 mean density of gas-solid mixture at surface of fluidized bed or of lower dense region of a fast fluidized bed, kg/m3 mean density of a gas-solid mixture at condition of complete reflux, kg/m3 ,8R at the surface of fluidized bed or of lower dense region of a fast fluidized bed, kg/m3 density of solids, kg/m3
REFERENCES
5
6
7
8
9 10 11
12
D. Kunii and 0. Levenspiel, Fluidization Engineering, Butterworths, 2nd edn., 1990. F. A. Zenz and N. A. Weil, AZChE J., 4 (1958) 472. A. B. Fournol, M. A. Bergounou and C. G. J. Baker, Can. J. Chem. Eng., 51 (1973) 401. W. K. Lewis, E. R. Gilliland and P. M. Lang, Chem. Eng. Prog. Symp. Ser., 58(38) (1962) 65. H. S. Caram, S. Edelstein and E. K. Levy, in D. Kunii and R. Toei (eds.), Fluidization ZV, Engineering Foundation, New York, 1983, p. 265; H. S. Caram, Z. Efes and E. K. Levy, AZChE Symp. Ser., 80(234) (1984) 106. M. Horio, A. Taki, Y. S. Hsieh and I. Muchi, in J. R. Grace and J. M. Matsen (eds.), Fluidization ZZZ, Plenum, 1980, p. 509; in D. Kunii and R. Toei (eds.), Fluidization IV, Engineering Foundation, New York, 1983, p. 307; in M Kwauk, D. Kunii, J. Zheng and M. Hasetani (eds.), Fluidization ‘85, Science and Technology, Science Press, Beijing, 1985, p. 124. D. Kunii and 0. Levenspiel, J. Chem. Eng. Japan, in M. Kwauk et al. (eds.), Fluidization ‘85, Science and Technology, Science Press, Beijing, 1985, p. 95. B. Hoggen, T. Lendstad and T. A. Engh, in K. Qstergaard and A. Sqhensen (eds.), Fluidization V, Engineering Foundation, New York, 1986, p. 297. D. Geldart and D. J. Pope, Powder Technol., 34 (1983) 95. D. V. Bachovchin, J. M. Beer and A. F. Sarofim, AZChE Symp. Ser., 77(205) (1981) 76. F. J. A. Martens, Proc. Colloquium on Pressurized Fluidized Bed Combustion, Delft Univ. of Tech., April 1983. P. M. Walsh, J. E. Mayo and J. M. Beer, AZChE Symp. Ser., 80(234) (1984) 119.
13 C. Y. Wen and L. H. Chen, AZChE J., 28 (1982) 117. 14 Zhang Qi et al., in Proc. CZESCIAZChE Joint Meeting, Chem. Ind. Press, Beijing, 1982, p. 374; in M. Kwauk et al. (eds.), Fluidization ‘85, Science and Technology, Science Press, Beijing, 1985, p. 95. 15 S. T. Pemberton and J. F. Davidson, in D Kunii and R. Toei (eds.), Fluidization ZV, Engineering Foundation, New York, 1983, p. 275. 16 A. Nazemi, M. A. Bergougnou and C. G. J. Baker, AIChE Symp. Ser., 70(141) (1974) 98. 17 G. Chen, G. Sun and G. T. Chen, in K. Ostergaard and A. Sdrensen (eds.), Fluidization V, Engineering Foundation, New York, 1986, p. 305. 18 J. Yerushalmi and A. Avidan, in J. F. Davidson, R. Clift and D. Harrison (eds.), Fluidization, Academic Press, New York, 2nd edn., 1985, p. 226; J. Yerushalmi, in D. Geldart (ed.), Gas Fluidization Technology, Wiley, New York, 1986, p. 155; Powder Technol., 24 (1986) 187. 19 Y. Li and M. Kwauk, in J. R. Grace and J. Matsen (eds.), Fluidization ZZZ, Plenum, New York, 1980, p. 537; M. Kwauk, N. Wang, Y. Li, B. Chen and Z. Shen, in P. Basu (ea.), Circulating Fluidized Bed Technology, Pergamon, 1986, p. 33; Y. Li, Y. Tung and M. Kwauk, in P. Basu (ed.), Circulating Fluidized Bed Technology ZZ, Compiegne, 1988, poster paper No. 5. 20 H. Weinstein, R. A. Graf, M. Meller and M. J. Shao, in D. Kunii and R. Toei (eds.), Fluidization ZV, Engineering Foundation, New York, 1983, p. 299; AZChE Symp. Ser., 80(234) (1984) 52; 80(241) (1984) 117; in K. ostergaard and A. Sdrensen (eds.), Fluidization V, Engineering Foundation, 1986, p. 329. 21 E. U. Hartge, Y. Li and J. Werther, in P. Basu (ed.), Circulating Fluidized Bed Technology, Pergamon, 1986, p. 153; in K. Qstergaard and A. S$rensen (eds.), Fluidization V; Engineering Foundation, New York, 1986, p. 345; E. U. Harge, D. Rensner and J. Werther, in P. Basu (ed.), Circulating Fluidized Bed Technology II, Compiegne, 1988, poster paper No. 7. 22 J. M. Matsen, in D. L. Keairns (ed.), Fluidization Technology, vol. 2, Hemisphere, 1976, p. 135. Ph.D. Dissertation, City Uni23 M. G. Schnitzlein, versity of New York (1987). 24 M. J. Rhodes and D. Geldart, in K. Qstergaard and A. S$rensen (eds.), Fluidization V, Engineering Foundation, New York, 1986, p. 193; in P. Basu (ed.), Circulating Fluidized Bed Technology ZZ, Compiigne, 1988; Powder Technol., 53 (1987) 155. T. Hirama, T. Chiba and L. S. 25 H. Takeuchi, Leung, in Proc. 3rd World Cong. Chem. Em?., Tokyo, 1986, p. 477. 26 K. Kato, H. Shibazaki, K. Tamura, C. Wang and T. Takarada, Paper, Sot. Chem. Eng. Japan, Annual Meeting, April 1987. 27 M. Horio, K. Morishita, 0. Tachibana and M. Murata, in P. Basu (ed.), Circulating Fluidized Bed Technology ZZ,1988, p. 13.
206 28 U. Arena, A. Cammarota and L. Pistone, in P. Basu (ed.), Circulating Fluidized Bed Technology, Pergamon,l986, p. 119; U. Arena, A. Cammarota, L. Massimilla and D. Pirozzi, in P. Basu (ed.), Circulating Fluidized Bed Technology II, Compiegne, 1988, p. 16. 29 G. Yang, Z. Huang and L. Zhao, in D. Kunii and R. Toei (eds.), Fluidization ZV, Engineering Foundation, New York, 1983, p. 145; R. Zhang, D. Chen and G. Yang, in M. Kwauk, D. Kunii, J. Zheng and H. Hasatani (eds.), Fluidization ‘85, Science and Technology, Science Press, Beijing, 1985, p. 148. 30 C. Brereton and L. Stromberg, in P. Basu (ed.),
Circulating Fluidized Bed Technology, Pergamon, New York, 1986, p. 133. 31 J. C. L. Furchi, L. Goldstein Jr., M. Mohseni and G. Lombardi, in P. Basu (ed.), Circulating Fluidized Bed Technology II, Compiegne, 1988, p. 69. 32 Q. Lu and Y. Wang, Private communication, 1985. 33 L. Monceaux, M. Azzi, Y. Molodtsof and J. F. Large, in P. Basu (ed.), Circulating Fluidized Bed Technology, Pergamon, 1986, p. 185; in K. Bstergaard and A. SQrensen (eds.), Fluidization V, Engineering Foundation, New York, 1986, p. 337.
193
61 (1990) 193 - 206
Entrainment of Solids from Fluidized Beds I. Hold-Up of Solids in the Freeboard II. Operation of Fast Fluidized Beds D. KUNII Combustion and Fluidization Engineers, Tokyo 154 (Japan)
401 Olympic Mansion, 1-33-l 7 Sangenjaya, Setagaya-ku,
and 0. LEVENSPIEL Chemical Engineering
Department,
Oregon State Uniuersity, Coruallis, OR 97331-2702
(U.S.A.)
(Received December 1, 1989)
SUMMARY
PART I - HOLD-UP OF SOLIDS IN THE FREEBOARD
Part I: Kunii and Levenspiel proposed a flow model to represent the complex phenomena occurring in the freeboard above a fluidized bed. A somewhat more generalized version of this model is developed in this paper, to explain and integrate the diverse experimental findings on freeboard behavior reported to date. The key parameter emerging from this model is the decay constant for the fall off of bulk density of solids with height in the freeboard. Values for this decay constant are extracted from the experimental data reported in the literature and a correlation for the decay constant, useful for predictive and design purposes, is found to reasonably represent the many findings reported to date. Part II: Viewing the fast fluidized column as having a lower region of approximately constant bulk density and an upper region wherein the bulk density decreases prog’ressively with height allows the model of Part I to be used directly for these con tat tors. As a consequence, the peculiar characteristics of fast fluidization systems are explained, and a correlation for the decay constant is obtained to represent the findings reported to date. An example of a design calculation shows how the fast fluidized system responds to changes in operating variables. More importantly, it makes clear what factors are needed to determine the flow characteristics of these systems.
Experimental findings There have been many studies, both steady and unsteady state, focusing on the lean zone above fluidized beds. These are summarized by Kunii and Levenspiel [ 11. Since the purpose of this paper is to develop a model to usefully represent these findings, we start by highlighting some of the pertinent studies in the literature. First, Zenz and Weil [2] proposed a convenient correlation for estimating the transport disengaging height (TDH) as well as the saturation carrying capacity for the pneumatic transport of FCC catalyst. Fournol et al. [ 31 later correlated the TDH for FCC catalyst with the Froude number of the solids. In one of the earliest studies on freeboard behavior, Lewis et al. [ 41 proposed that bubbles bursting at the surface of the bed was the means for getting particles into the freeboard. Let us designate the initial upward flux of these solids from the bed surface by G,, (kg/(m2*s)). Further in this line, Caram et al. [ 51 found that the initial velocity of particles ejected from fluidized beds depended only on bubble size, hence velocity of bubble, and was independent of size and density of particles. Horio et al. [6] also found that the intensity of turbulence at the bed surface was related directly to bubble size. Focus attention on the region below the TDH. According to Lewis et al. [4], the 0 Elsevier Sequoia/Printed in The Netherlands
194 Height
Hf.
A--’
zf
l-
.___. A
TDH --
density is also assumed to fall off exponentially, or
in
freeboard, A
G, = GsOeeaHf
Gas exit above the TDH
c--
At complete
\ \ \
reflux
FR = FRO exp I-azfI, Eq. (1)
___________________
Gs = Gsu
0. _. T---
l-l
F
D
B
Bed ._t_____ _--
Holdup or density of solids in the freeboard, p
Fig. 1. Density distribution of solids in the freeboard for three different freeboard heights, according to Lewis et al. [4].
density of solids present at various levels zf as the freeboard height Hf is changed can be represented by the sketch of Fig. 1. Here, curve AGB represents the solids hold-up when the freeboard height is at Hf, A > TDH. This may be called the condition of complete refhx. Curves CD and EF are for freeboard heights H, c and Hi, E, respectively, both below the TDH. From this sketch we see: (i) At complete reflux (curve AGB), the density of solids in the freeboard falls off exponentially from the value at the bed surface, or jja = PRO eearf
(1)
(ii) At smaller Hf, below TDH (curves CD and EF), the solid density is some constant value less than at complete reflux, or Pa - p = constant
throughout
(3)
Note that although the density of solids at any level in the freeboard goes down as Hi is lowered, the density at the exit does increase. Hence, entrainment increases as the freeboard height is reduced. Since the clumps of solids thrown into the freeboard by the bursting bubbles are representative of the bed material and since the larger solids in these clumps fall back to the bed, there is an upflow flux G, and downflow flux G so of solids everywhere in the freeboard below the TDH. The net flux or carryover out of the vessel G, is related to these up- and downflows at any level in the bed by
H/, c---
Hf, E ---
(kg/(m2-s))
the freeboard
(2) (iii) The entrainment G, from the bed is proportional to the density of solids at the gas exit (see dashed line AGCEH). This exit
-
Gsa
(4)
Kunii and Levenspiel [7] presented a simple flow model to represent the complex phenomena occurring in the freeboard. We now present a more generalized version of this model.
Freeboard-entrainment model Consider the freeboard above a bubbling or turbulent fluidized bed, and let x be the fraction of bed solids for which Ut < u,. This is the entrainable fraction, and here we call them the fines. Postulate 1. Three distinct phases are present in the freeboard. Phase 1: Gas stream with completely dispersed solids. The fines are carried upward and out of the bed at velocity ul. Phase 2: Agglomerates, coming from the bed, and moving upward at velocity u2. Phase 3: Agglomerates and thin wall layers of particles moving downward at velocity u3. Post&ate 2. At any level in the freeboard, the rate of removal of fines from the agglomerates to form dispersed solid of phase 1 is proportional to the volume. fraction (or solid density) of agglomerates at that level. Postulate 3. Upward-moving agglomerates will eventually reverse direction and move downward, the frequency of change from phase 2 to phase 3 being proportional to the volume fraction of phase 2 at that level.
195
TDH Density distribution for Hf >TDH, jjR Eq. (24) for
H
1._. ..._
Splash zone
____
r-------
G, = xG;
(b)
(a)
Fig. 2. Sketch and various terms and expressions of the freeboard model.
Figure 2(a) sketches the freeboard as viewed by this model. Now, at any level zf in the freeboard, let Gsl, Gs2, Gs3 (kg/(m’-s)) be the mass flux of each phase, and let pl, p2, p3 (kg/m3) be the mass of each phase per unit volume of freeboard. Then, at steady state conditions the net upward flux of solids at any level in the freeboard is given by G, = G,i +
Gs2
-
Gs3
=
PIUI +
~2~2
-
~3~3
independent of zf.
(5)
Also, the average (or bulk) density of solids at any level in the bed is P=pl+pZ+p3
Gsa
= Gs3
increase of solids in phase 3
i
= 1
transfer of solids i from phase 2 to 3
transfer of solids
-
i from phase 3 to 11
Introduce the rate coefficients 3tKi for the transfer of fines from phases 2 and 3 to 1, and K, for the transfer from phase 2 to phase 3. In these expressions, x is the fraction of fines in the dense fluidized bed. Then the above mass balances at level zf become
(6)
Relating these quantities to the terms previously used G,, = G,, +
for phase 3
UlZ
=3Cmp2+
P3)
dP2
Gs2
(7)
i
Now, a mass balance for phase 1, and this only concerns the fines, gives transfer of solids from phases 2 and 3 to 1 for phase 2 transfer of solids phase 2 to 1 and 3
-U2z-
= w,+K2)P2
dp3 -u3 =
Kg,
- xK,p,
(10)
dzf
where ui and u2 are the upward velocities of phases 1 and 2, and u3 is the downward velocity of phase 3. Since all solids which reach height Hi leave the vessel, there is no downflow there, so p3=0
atzf=Hf
(11)
196
Noting that at the bed surface, subscript 0, the only upflow is by clumps of solids projected into the freeboard, we have
(17)
u2
Thus, eqn. (13) reduces to
Pl=O G P2=
K2 -
a=
-fy-
atzf=O
(12)
G, -
xG,* E
e-WJUzWf
(18)
bSU0
I
Net upflow and carryover of solids Solving eqns. (8), (9), and (10) with the boundary conditions of eqns. (11) and (12) gives pl, p2, and p3 in terms of exponential functions of zf. With eqn. (5), the net flux of solids at any freeboard level and also at the vessel outlet is then found to be
G, - xG,* = -oHf e G suo - xG,*
(13)
where G,* is the flux of carryover from a very tall vessel fluidizing only entrainable solids - in practical terms, when Hf > TDH. The ratio of upward flux out of a tall vessel to the upward flux at its surface is also found to be
These expressions provide a physical interpretation of the equations reported by Lewis et al. [4], in eqn. (3). Upflow term alone In the present model, the upward flux is given by Gsu =
plu, +
~2~2
(19)
Substituting the expressions found for p1 and p2 into eqn. (19) gives
G, - xG,* G sue -
xG,*
=e
-aHf
+ e-bzf
_
e-[aHf-(a-
b)zf]
(20)
where b=
xK1 + K, u2
xG,*
xK1 -
a-b= =-
(14)
G sue
(a - b)zf < aHf (m-l)
(15)
In the special case of a vigorously bubbling bed or a turbulent bed with not too many fines, the amount of solids thrown into the freeboard is very much larger than those eventually removed from the bed. Thus, xG,* Q GsUo. In this situation, eqn. (14) indicates that
or
and from eqn. (15)
u3
Again, in the special case where xG,* < G,, we find xK1 Q K,, thus a - b Q a. Since we always have zf 5 Hf, all these inequalities lead us to conclude that
and
a=$.[1+~jl+~)]
(21)
(16)
ar
b
(22)
I Thus, in this special case eqn. (20) reduces to
(23) At zf = Hf there is no downward flux of solid. At this location, eqn. (23) reduces to eqn. (18), thus confirming that G, = G,.
Solids hold-up in the freeboard This model also gives the distribution of bulk density of solids in the freeboard. Substituting the expressions found for pl, p2, and p3 into eqn. (6) gives a bulky expression; however, in the special case where xG,* 4 Gsuo, this expression simplifies to two special cases. For very high freeboard, or in practical terms, for Hf > TDH, the density
197
ford, = 150 - 212 pm at u,, = 0.61 m/s and zf = 3 m. Geldart and Pope [9] measured j!~of fines elutriated from fluidized bed of coarse sands at very high air velocity, u, = 2 - 4 m/s, and found
of solids p, at any height zf is given by the expression
for Hf > TDH
(24)
_!!_ r 10-Z PO
where the lowest density of solids in the freeboard, that above TDH and at pneumatic transport conditions, is given by
at G, = 1.53 kg/(m2.s) and zf = 3 m. Comparing the above findings with eqns. (23) and (29), we estimate that
XG,* .Qj* = -
e_aHf= 10-Z
(25)
in case Hf is high enough, for example 3 m at the above conditions. Thus, at a level zf not far above the bed surface, eqn. (29) reduces to
Ul
For not very high freeboard, or Hf < TDH, the density of solids p is given by the following equation, again in the special case where xG,* Q G,s: e-azf -
P - xp* z Gsuo
p
1
1 _ e-‘JHf u3
(26) Combining eqn. (24) with eqn. (26) gives GSUO
fiR--P=
-e
--aHf
u3 =
constant throughout the freeboard
(27)
Equation (27) explains theoretically eqn. (2) given by Lewis et al. [ 41. At the surface of the fluidized bed, eqn. (26) gives PO
-xp-PGsuo[($
+
-3cp*
po- xp*
se
-0zf
(30)
Discussion of the freeboard-entrainment model The upward and downward mass fluxes of solids at the apparent surface of fluidized beds, G,, and GsdOrespectively, have been measured by several workers [8 - 131 i as shown in Fig. 3. We see appreciable variation in these values. This may be because GsuO is found by extrapolation back to zf = 0, and different assumptions were made regarding the location of zf = 0 in the vessel. Wen and Chen [13] took the zf = 0 plane at the mean value of the bed surface; others took it somewhere within the splash zone.
d)- d e_aHf] (28)
Thus, combination of eqns. (26) and (28) gives II e-a=f
p -3cp* po -
e_aHf
1 + u2/u3
2
xp*
-
l-
1
(2%
e_aHf
1 + u2t”3
Hoggen et al. [8] measured G, us. zf in a large industrial roaster, finding that
-GSUz 10-Z G SUO
+.+ +*--I I I Ill11 I 1 I IIIII 0.05 0.1 0.5 0.2 0.02 u. - Umf (m/s)
10-Z -
1
Fig. 3. Mass flux of particles just above the bed surface; from Walsh et al. [ 121. Sources of information: A, Gsuo from [la]; A Gsuofrom [14]; VI GsdO from [ll]; V GsdOfrom 1121; A, GsuOfrom [4]; A Gsuo from [15].
198
In this model, since Gsl= 0 at zf = 0, we have G, = GSz-- Gs3 atzf= 0 (31) = Gsuo- Gscio For a vigorously bubbling fluidized bed, we also have xG,* < Gsuo, in which case, just above the bed surface G SUO- Gsdo
atzf=
0
7 c
(33)
Walsh et al. [12] found this relationship to hold also for coarse particles. Now if we assume that u2, the upward velocity of agglomerates, is proportional to gives an immediate U 07 then eqn. (17) meaning to the empirical expression of eqn. (33). Equation (30) was used to obtain values of the decay constant a from experimental data of solids hold-up us. height at levels in the freeboard not close to Hf. Values of a were also obtained from experimental data of G, us. zf. Figure 4 correlates au0 thus obtained us. mean size of the entrained solids at velocities u, < 1.25 m/s. Since the decay constant for solids density is the key parameter needed to develop predictions of fast fluidized bed behavior, we urge further experimental studies to obtain more reliable values for a for a wide range of solids and a wider range of gas velocities.
PART II - OPERATIONS OF FAST FLUIDIZED BEDS
In the fast fluidization regime, carryover of solids is very large; hence, fresh solids have
2
3" c.
(32)
Therefore, Fig. 3 can be used to estimate both G,, and Gscio. In almost all cases of elutriation from a bubbling or turbulent fluidized bed, it is reasonable to assume that xG,* < GSUo.Thus, one is justified in using the special case simplifications, or eqn. (30) for the distribution of solid density in the freeboard, eqn. (23) for the flux of only upward moving solids in the freeboard, and eqn. (18) for the carryover as a function of various freeboard heights. Finally, for fine particles, Lewis et al. [ 41 found the following relationship to hold, au, Z constant
3
1 -I
‘c---q
o 0
I
I
400
200 dp
I
I 600
I
600
(pm)
Fig. 4. Decay constant for the freeboard agglomerates for Hf < TDH and u0 < 1.25 m/s. 0, From [14]; @, from [12]; 0, from @), from [16]; @, from [17]; 8, from [lo]; , from [8].
to be introduced continuously and at a significant rate to make up for the loss of bed solids and to achieve steady state operations. Figure 5 illustrates a typical fast fluidized bed with its various regions. (i) At the bottom one sees a relatively short entry zone having a solid fraction E, = p/p, = 1 - ef = 0.2 - 0.4. (ii) Then there is a portion of the vessel of almost constant solid fraction of about E, = 0.2. Regions (i) and (ii) may be called the dense region. (iii) Above this is an upper entrained region where the solid fraction decreases progressively to about e, = 0.02 - 0.05. These regions correspond somewhat to the dense turbulent bed and its freeboard. Also, the transition between regions is smooth. Because of the large entrainment of solids from fast fluidized beds, inner cyclones and diplegs are too small to handle the solid load. Thus, large cyclones located outside the column are used and these require careful design for proper operation. Overall, we may want to run a fast fluidized circulation system in one of four ways: Mode I. Keep a constant inventory of solids in the bed, even though u, may change. Mode II. Keep a constant throughflow of solids G,(kg/(m2.s)), even though u, may change. Mode III. Keep a constant gas flow rate u,, while changing the solid throughflow G,.
199
Esd Z 0.16 - 0.22
(4 Fig. 5. The fast fluidized bed and its regions of different fraction of solids.
Mode IV. G, and u, can be changed independently. Figure 6 illustrates the two basic types of solid circulation systems: those which do not include a reservoir of solids, and those which do. Without the reservoir, one can only operate according to Mode I. With the reservoir, the system becomes much more flexible in that it can operate in any of the four modes listed above. Experimental
findings
With the works of Yerushalmi and coworkers [ 181 as a catalyst, there has been a sharp increase in interest in investigating the characteristics of the fast fluidization flow regime, see [ 11. The following studies are of particular interest here. Vertical distribution
of solids
Li and Kwauk [ 191, Weinstein et al. [ 201, and Hartge et al. [ 211 all found an S-shaped
(b)
Fig. 6. Idealized solid circulation systems for fast fluidized operations. Scheme (b) contains a reservoir for solids and is much more flexible than scheme (a).
solid fraction curve, as shown by the solid line of Fig. 5. This S-shaped curve moved up or down the column depending on the solid and gas flow rates. Furthermore, it was found that the gas and solid favored flowing in the central core of the column and that at the wall there is a dense slow-moving layer of solid. In addition, Matsen [22] found an appreciable maldistribution of solids in large diameter beds - nonsymmetrical and nonreproducible. Consider a fast fluidized bed as having a lower region of constant solid fraction es,-, and an upper region wherein the solid density decreases progressively to its exit value E,,. This leaner region starts at the level given by the intersection of the two dashed lines, shown as point A in Fig. 5. Ignoring the short entry zone of Fig. 5, the sketches of Fig. 7, prepared from the published data, show that the flow rate of solids and of gas in the range U, = 1.5 - 5.0 m/s do not seem
200 0.4 u. = 3 m/s s w
0.3 -
~._..~~~__.~___~~~~~~..~_~_ ..._*..~_..‘.....l ..............L........
4) 0.2 -
4 m/s J pS = 1450 kg/m3
0.1 50
100
150
G, (kg/m2. s) 0.3
cl
A 2
0.2 -
lA
I
7 %
l+
.
w”
I
I
4, X lt
‘0
a I@ ;
$
-0
::*
l
.
Q 0.1
V
I
V
I
(a)
QVI
100
0 G,
0
I
200
(kg/m2* s)
0
/
I
2
h
(b)
I
I
4
6
(m/s)
Fig. 7. Volume fraction of solids in the lower dense region of the fast fluidized bed. (a), Various solid flows, see Table for references; (b), various gas flows, from Schnitzlein [23]. TABLE References and experimental conditions for the data of Figures 7(a), 8, and 9 Key
Reporters
dt (cm)
Type of solid
+ D
Hartge and Werther [ 211 Li and Kwauk [19]
5 40 9.0
@ @
Weinstein et al. [ 201 Rhodos and
15.2 15.2
Quartz Quartz, FCC FCC Alumina Pyrite cinder HFZ-20 cat. -
Geldart [ 241 Takeuchi et al. [ 251 Schnitzlein [ 231 Kato et al. [ 261 Horioetal. [27] Arena et al. [28] Li and Kwauk [19] Yang et al. [29] Arena et al. [ 28 ]
10.0 15.2 6.6, 9.7 5.0, 20 4.1,12 9.0 11.5 4.1
FCC HRZ-33 cat. FCC, cat. FCC FCC, glass Iron Silica gel Glass
61 59 61 60 70 105 220 88
1.7 1.5 2 1.1 4-5 5.3 5,7
20
Sand
170-650
-
0
Brereton and Stromberg [ 301 Furchi et al. [31]
$
Lu and Wang[32]
10
Glass Glass Sand, glass
196 269 230, 369
7.2 8.3 2.9-4.9
X
A
0l 0
O+ q
V
B Wq @
7.2
to appreciably affect the fraction of solids in the lower dense region of the vessel. It is interesting to compare eSdin the various fluidizing regimes. bubbling bed: E,d = 0.55 - 0.40 turbulent bed: f,d = 0.40 - 0.22 fast fluidization: E,d = 0.22 - 0.16
&, (pm) 56 56 54,58 54 56 49 64
UO (m/s)
Gs (kg/(m’*s))
3.4 72 1.2 - 4 7 0.8 - 2.1 14 2.2-4 73 1.5 - 2.5 129 2.9, 3.4 712.5 - 4.5 86 - 2.9 -5 4.4 - 1.6
Remarks
40 16 118 115
8.3 - 79 89 - 133 48 - 50 12 - 19 49,120 135 44 - 146 80 - 600
Fine particles, d,<:70pm
Larger particles, dp > 88 I_lrn
64 - 146 88 127 -
Keys to Fig. 7 as well as Figs. 8 and 9 are shown in the Table. Fraction
of solids at the column
exit, E,,
First of all, we note that E,, is somewhat larger than the saturation carrying capacity of the gas E,*. Next, the data of Fig. 8 show
201
3
4
5
G, (kg/m2.s)
50
50
50
100
100
up(mW
0.1
0.5
1.0
0.5
1.0
12
Line
200
0
400
G, (kg/m2
(a)
0
600 l
s)
2
4
h
(b)
6
(m/s)
Fig. 8. Volume fraction of solids at the exit of fast fluidized beds; experiment compared with curves of eqn. (36). (a), E,~ at high solid flows, G, = 100 - 600 kg/( rn*-s); from Arena et al. [ 281; (b); (b), ese at low solid flows, G, = 50 - 100 kg/(m*.s); from various sources, see Table for references.
that at given solid flux E, is higher at low U, than at high u,,. This seemingly puzzling finding can be explained in terms of the slip velocity between gas and solid, or up = ug - u,. Thus, letting u, be the mean velocity of solids at the exit level of the column, a mass balance gives G, = G,, (34) = PSES&S Introducing the slip velocity gives the solid velocity as UO
us= i-y--/P For fine particles at high u,, E,~ Q 1, and thus, the above expressions become G, = G,, = P&e(u,
-up)
or Es,
s
GS Ps(u0 - up)
(36)
If the fine particles are completely dispersed in the gas stream, then we can reasonably assume that up E ut and ut < uO. The two lines drawn in Fig. 8(a) are based on eqn. (36) with this assumption and are seen to be consistent with the data of this figure. In trying to find the saturation carrying capacity of gas es*, values of ese were taken from reported data at low G, or high enough columns such that e, seemed to level off in the upper portion of the column. Figure 8(b) summarizes these data from various sources and compares this with the curves calculated
from eqn. (36), using appropriate values of G, and up. Although there is considerable scatter in the experimental values, eqn. (36) seems to pass through the main body of this data. The freeboard-entrainment model applied to fast fluidization Consider a fast fluidization column as having a lower region of constant solid fraction &d and an upper leaner region wherein the solid density decreases progressively to its exit value E,,. This leaner region starts at the level given by the intersection of the two extrapolated dashed lines, shown as point A of Fig. 5. With this as the picture of a fast fluidization column, the decrease or decay of E, in the lean entrainment region can be treated with the model of Part I which was used to represent the freeboard above bubbling and turbulent beds, but with the simplification here that all the solids are entrainable, or x = 1. On the basis of this model, eqn. (30) predicts an exponential decay in solid density between eSd and a limiting value of E,*, with a decay constant a related to gas velocity u, by eqn. (33). Thus, a higher gas velocity u, means a slower decrease of solid concentration with height in the entrainment section of the column. This is to be expected. Figures 9(a) and 9(b) present values of the decay constant calculated from various literature sources for fine particle systems (d, < ‘70 pm) and for coarser particle systems (dp > 88 pm). Lines representing eqn. (33) for different values of the decay constant are also shown in these figures. Despite consid-
202 Smaller
OL 0
I
I
d, -Z70 pm
I
u. Larger
particles
I
4
2
(a)
(b)
particles
I
I
basically similar to the decay above bubbling and turbulent beds. The decay constant is important for design. Since reported values of this parameter are scattered and sketchy, more precise data on its proper value and how it changes with the imposed system conditions would be most welcome.
6
VW
d, > 86 pm
u. VW
Fig. 9. Decay coefficient a which represents the changing solid fraction in the entrainment region; eqn. (37) was used to calculate a from experimental data. See Table for references.
Design considerations In design, one needs to know the solid hold-up in the vessel as a function of the flow conditions, U, and G,. This, in turn, requires knowing the location of the top of the dense region and solid densities throughout. These values can be found from the information given so far in this paper, as we will now show. First consider the upper entrainment zone of the column. Referring to Fig. 5, eqn. (30) becomes f, - e,* Esd -
=e -&Zr
The fraction erable scatter in the reported data, this figure suggests the following: (i) The decay constant a seems to increase with decreasing c&. This may be explained by noting that in narrow columns the rising agglomerates are more likely to hit the wall surface and then be removed from the rising gas stream. (ii) The decay constant a seems to increase with increasing d,. This may be explained by noting that with coarser and denser particles the agglomerates are more likely to change direction and return to the dense region of the vessel. In the model this is represented by a larger K2-value. (iii) For the fine particle systems of Fig. 9(a), we see that higher gas velocities give lower values for the decay constant, as predicted by eqn. (33). On the contrary, the data on the coarse particle system reported by Brereton and Stromberg [30] do not seem to fit this relationship. (iv) The decay constant a for bubbling and turbulent beds and for fast fluidized beds (compare Fig. 4 with Fig. 9) all fall in the same range of values, suggesting that the mechanism of decay of solid fraction in the lean region of fast fluidization columns is
(37)
Es*
of solids at the vessel exit is
Ese = e, * + (e,d - e,*) eeaHf
(38)
and the mean value of E, in the upper entrainment region of height Hf is calculated by Hf
A-sf,dzf
T, =
Hf
(39)
0
Inserting eqn. (38) integrating gives cs
=
E,*
+
eqn.
(39)
and
‘sdaHf - %*(1 - eCaHf)
s* + sdE
=e
into
Ese
aHf
(40)
Finally, the total inventory of solids in the column of height Ht = Hf + Hd is then -
W
= I&-
Em)
AtPs =
Lnf(l-
Gnf)
=
H&d
+ HfZ,
=
Esd -
a
ese
+ H&d - Hf(e,d - ES*) (41)
To use this freeboard-entrainment model, one needs vahies of a, es*, and e,d. Let us
203
briefly point out where these values can be obtained. (i) First of all, a is estimated from Fig. 9. (ii) As for E,*, according to Monceaux et al. [33], E,* e 0.01 for the pneumatic transport of particles. Phenomenologically, this value may correspond to the transition between dilute and concentrated transportation of solids. Alternatively, from the definition of mass velocity and with U, 3 Ut we get
GS* es*= _ P&s
=-
Gs*
(42) PSUO where G,* is found by the method of Zenz and Weil [ 21. (iii) Finally, c,d is found from Fig. 7. The various quantities in this model are sketched in Fig. 10.
Height, z
Find a from Fig. 9; then get this curve
I
I
I
I
I
I
I I I I ._._ I__+_-__--_ . .. . . .._ 1
Includes the entry zone / Fraction in G/S
t
_ Lrom
Eq. (36)
K from
of sohds mixture
Fig.7
Fig. 10. Solids distribution in fast fluidization, from the freeboard entrainment model.
Let us now see how to calculate the performance of the fast fluidized bed in the four modes of operation outlined earlier in this chapter. Mode I. Constant inventory of solids (no reservoir of solids in the circulation system) (i) For given u,, estimate csd from Fig. 7(b), and estimate a from Fig. 9.
(ii) Calculate c,, as a function of Hf, using eqn. (38). (iii) For the desired inventory of solids Lmi (1 - E,~) and given height of vessel Ht = Hf + Hd, determine Hf by substituting all values known so far into eqn. (41). (iv) Calculate G, from eqn. (36). The mass flux of circulating particles from bubbling or turbulent fluidized beds through an inner cyclone collector is calculated in the same way. Modes II, III, and IV. Constant G, or uO, or both changing (i) Estimate csd and a, as above. (ii) Calculate E, for given G, using eqn. (36). (iii) Determine Hf and Hd = Ht -~-Hf from eqn. (30 (iv) Determine J& (1 -- e,t) with eqn. (41). The example below illustrates this calculation procedure and Fig. 11 displays the results of these calculations. Note that as u, is changed, mode I and mode II give opposite progressions of solid fraction vs. height in the bed. Example. Performance of a fast fluidized vessel Determine the performance characteristics of a fast fluidized column when operated in the following four modes Mode I: Constant solid inventory corresponding to j&f = 2.4 m, with variable gas flow of u, = 2, 4,6 m/s Mode II: Constant solid flow at G, = 100 kg/(m2*s), with variable gas flow of u, = 2, 4, 6 m/s Mode III: Constant gas velocity u, = 4 m/s, with changing solid flow G, = 42, 50, 100, 200, 400 kg/(m’. s) Mode IV: G, and u0 both vary as follows 2 4 6 u. (m/s) 100 120 I 70 Gs (kg/(m’- s)) For all modes determine the vertical distribution of solids, es. Data Column: dt = 0.4 m, Ht = 10 m Particles: catalyst, ps = 1000 kg/m3, d, = 55 pm, f,f = 0.5 Gas: ambient conditions Solution A straight application of the calculation procedure described above gives the performance characteristics shown in Fig. 11. For
Mode
Mode
I 10
II
G, - 100 kg/m2.s
Mode Ill
Mode IV
un = 4mls
u0 (m/s) 1G, (kg/m2.s) 10
u,(mls) 2 4.100
4 5
6,120
6
5 0 1 0
0.1
0 R
0.2
Es l-1
Es (-) (W
Cd)
Fig. 11. Solutions to the example; d, = 0.4 m, Ht = 10 m. (a), Mode I; (b), Mode II; (c), Mode III; (d), Mode IV.
detailed numerical calculations see [ 11. An examination of this figure shows that, despite the many simplifications made, the performance characteristics of this model do seem to be reasonable. LIST OF SYMBOLS
At a
Gs GS* Gsa,Gsu
Gsd0, Gsuo Hd,
Hf
Ht = Hd + Hf K,, K,, K3
L,
Lnf
TDH % UO
cross-sectional area of column, m* decay constant for fall off of solid density in freeboard, m-l net upward mass flux of solids in freeboard, kg/(m**s) saturation carrying capacity of upflowing gas for solids, kg/(m*+s) downflow and upflow flux of solids, respectively, kg/(m**s) Gsd, G, at surface of fluidized bed, respectively, kg/(m*- s) height of lower dense bed and freeboard, respectively, m height of column, m rate constant for interchange of solids in freeboard, see above eqn. (8), s-l height of fixed bed, bed at minimum fluidization, respectively, m transport disengaging height, m velocity of gas, m/s superficial gas velocity on an empty vessel basis, m/s
UP
US
Ut Ul,
u2,
u3
X Zf
Greek symbols Ef,
fS
ES*
Esd
e 88
Ern, fmf
slip velocity between gas and solid, m/s velocity of dispersed solids in a pneumatic line, m/s terminal velocity of a falling particle, m mean velocity of dispersed solids, of upward moving clusters, and of downward moving clusters, respectively, m/s weight fraction of fines distance above mean surface of fluidized bed or above lower dense region of a fast fluidized bed, m void fraction in a fluidized bed as a whole, in a fixed bed, and in a bed at minimum fluidization conditions, respectively volume fraction of solids, =1---r saturated carrying capacity of a gas, or maximum volume fraction of solids that can be pneumatically conveyed by a gas volume fraction of solids in lower dense region of a fast fluidized bed volume fraction of solids at column exit
205
Pl,PZ,
P3
P PO
PR
PRO
PS
bulk density of dispersed solids, of upward moving clusters, and of downward moving clusters, respectively, based on unit volume of freeboard, kg/m3 mean density of a gas-solid mixture, kg/m3 mean density of gas-solid mixture at surface of fluidized bed or of lower dense region of a fast fluidized bed, kg/m3 mean density of a gas-solid mixture at condition of complete reflux, kg/m3 ,8R at the surface of fluidized bed or of lower dense region of a fast fluidized bed, kg/m3 density of solids, kg/m3
REFERENCES
5
6
7
8
9 10 11
12
D. Kunii and 0. Levenspiel, Fluidization Engineering, Butterworths, 2nd edn., 1990. F. A. Zenz and N. A. Weil, AZChE J., 4 (1958) 472. A. B. Fournol, M. A. Bergounou and C. G. J. Baker, Can. J. Chem. Eng., 51 (1973) 401. W. K. Lewis, E. R. Gilliland and P. M. Lang, Chem. Eng. Prog. Symp. Ser., 58(38) (1962) 65. H. S. Caram, S. Edelstein and E. K. Levy, in D. Kunii and R. Toei (eds.), Fluidization ZV, Engineering Foundation, New York, 1983, p. 265; H. S. Caram, Z. Efes and E. K. Levy, AZChE Symp. Ser., 80(234) (1984) 106. M. Horio, A. Taki, Y. S. Hsieh and I. Muchi, in J. R. Grace and J. M. Matsen (eds.), Fluidization ZZZ, Plenum, 1980, p. 509; in D. Kunii and R. Toei (eds.), Fluidization IV, Engineering Foundation, New York, 1983, p. 307; in M Kwauk, D. Kunii, J. Zheng and M. Hasetani (eds.), Fluidization ‘85, Science and Technology, Science Press, Beijing, 1985, p. 124. D. Kunii and 0. Levenspiel, J. Chem. Eng. Japan, in M. Kwauk et al. (eds.), Fluidization ‘85, Science and Technology, Science Press, Beijing, 1985, p. 95. B. Hoggen, T. Lendstad and T. A. Engh, in K. Qstergaard and A. Sqhensen (eds.), Fluidization V, Engineering Foundation, New York, 1986, p. 297. D. Geldart and D. J. Pope, Powder Technol., 34 (1983) 95. D. V. Bachovchin, J. M. Beer and A. F. Sarofim, AZChE Symp. Ser., 77(205) (1981) 76. F. J. A. Martens, Proc. Colloquium on Pressurized Fluidized Bed Combustion, Delft Univ. of Tech., April 1983. P. M. Walsh, J. E. Mayo and J. M. Beer, AZChE Symp. Ser., 80(234) (1984) 119.
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Circulating Fluidized Bed Technology, Pergamon, New York, 1986, p. 133. 31 J. C. L. Furchi, L. Goldstein Jr., M. Mohseni and G. Lombardi, in P. Basu (ed.), Circulating Fluidized Bed Technology II, Compiegne, 1988, p. 69. 32 Q. Lu and Y. Wang, Private communication, 1985. 33 L. Monceaux, M. Azzi, Y. Molodtsof and J. F. Large, in P. Basu (ed.), Circulating Fluidized Bed Technology, Pergamon, 1986, p. 185; in K. Bstergaard and A. SQrensen (eds.), Fluidization V, Engineering Foundation, New York, 1986, p. 337.