Epstein N., Grace J.R. Spouted and Spout-Fluid Beds: Fundamentals and Applications
Подождите немного. Документ загружается.
Particle mixing and segregation 157
“spout mixing number,”
α = (NLH
0
U)/(F
s
/ρ
b
), (8.7)
which accounts for mixing, was proposed. An excessively high fountain height, as well
as fountain wandering, was found to increase the volume fraction of the mixed region
in their system.
A recent paper
31
examined the hydrodynamics of a multiple square-based reactor
for solid state polymerization. There was difficulty in controlling the levels in the
stages. This negative feature did not affect the chemical performance. The difficulty was
ascribable to a very high solids throughput, related to the mixing and solids circulation
of each cell. The findings suggest that it may be important to consider changes in the
angle of repose, at least in instances in which the solids properties vary during the
process.
8.8 Mixing and segregation in a modified spouted bed
Modification of the inlet geometry from a plain circular orifice to an annular one, in
which a dead inner zone communicates with a bottom dead collector,
32
indicates that
it may be possible to segregate and selectively discharge particles (or aggregates) that
are much bigger than the average bed par ticles. The overall mechanism, although not
fully investigated, is likely to involve (1) gas percolation in the annulus maintaining the
solids circulation, together with entrainment by the central spout, and (2) congregation
of the coarser component within inner circulation loops, triggering particle transfer
probability between annulus and spout (defined as “jump probability” by Larachi et al.
7
),
dropping the heavy body into the dead zone. It was demonstrated that a smaller dead
zone diameter (corresponding to a broader annular area and a reduced gas flow at the
inlet) assisted the separation/discharge effectiveness. This phenomenon, interesting for
some applications (such as ash agglomerate removal and nonhomogeneous granulation),
requires corroboration with materials of lesser density difference.
Waldie
33
simulated the discharge of large particle agglomerates from the central
orifice of a spout-fluid bed by controlling both the central and auxiliary gas flowrates.
A relatively small column (D = 0.1 m) was provided with an external porous conical
distributor for fluidization air. A phase diagram was traced to select possible oper-
ating conditions for spout-fluidization of glass ballotini (d
p
= 380 μm). The dilute
coarse fraction was characterized by a particle size ratio between 5 and 16. Tests con-
firmed that once the coarser material entered the upper inner zone of the bed, it tended
to cycle repeatedly there, with a small possibility of dropping out from the central
orifice.
Acknowledgment
The authors thank Ms. Marta Barberis Pinlung for her effective collaboration in editing.
158 Giorgio Rovero and Norberto Piccinini
Chapter-specific nomenclature
C
b
concentration of solids tracer in bed, mol/m
3
C
b0
C
b
at t = 0, mol/m
3
C
f
fractional concentration of solids tracer in feed after step change
C
p
fractional concentration of solids tracer in product after step change
C
r
radial coordinate, mm
C
z
axial coordinate, mm
C
θ
azimuthal coordinate, deg
d
R
particle diameter ratio, larger to smaller, in binary mixtures
D
disp
axial dispersion coefficient of solids, m
2
/s
D
p
pipe diameter, mm
f
n
(d
S
)
W
average value of Sauter mean diameter for the whole bed
F feed rate of solids, m
3
/s
F
s
solids flow rate, g/s
H
d
height of drying chamber, m
I(θ) internal age distribution function
l length, m
L cell size, m
M
b
mixing index for binary solids mixtures
M
t
mixing index for tertiary solids mixtures
N number of spout cells in the volume
N
s
spouting number = U/U
ms
Pe Peclet number, = v
z
H
d
/D
disp
Q
s
mass flow rate, kg/s
r
FR
flow reversal radius, mm
R
cfr
internal circulation to feed rate ratio
t
c
cycle time, s
V total bed volume, m
3
V
m
mixed flow volume, m
3
V
PF
plug flow volume, m
3
v
r
mean radial particle velocity, m/s
v
z
mean vertical particle velocity, m/s
W circulation rate of solids in annulus and spout, m
3
/s
X mass fraction of coarse component in bed
Y mass fraction of coarse component in discharge stream
Greek letters
ρ
b
bulk density of solids, kg/m
3
particle sphericity
Subscripts
i initial conditions
f final conditions
Particle mixing and segregation 159
References
1. K. B. Mathur and N. Epstein. Spouted Beds (New York: Academic Press, 1974).
2. D. Van Velzen, H. J. Flamm, H. Langenkamp, and A. Casile. Motion of solids in spouted
beds. Can. J. Chem. Eng., 52 (1974), 156–161.
3. U. Mann and E. J. Crosby. Cycle time distribution measurements in spouted beds. Can. J.
Chem. Eng., 53 (1975), 579–581.
4. H. Takeda and Y. Yamamoto. Mixing of particles in a spouted bed. J. Nucl. Sci. Technol., 13
(1976), 372–381.
5. G. S. McNab. Prediction of spout diameter. Brit. Chem. Eng. Proc. Tech., 17 (1972), 532.
6. D. Roy, F. Larachi, R. Legros, and J. Chaouki. A study of solid behavior in spouted beds using
3-D particle tracking. Can. J. Chem. Eng., 72 (1994), 945–952.
7. F. Larachi, B. P. A. Grandjean, and J. Chaouki. Mixing and circulation of solids in spouted
beds: particle tracking and Monte Carlo emulation of the gross flow pattern. Chem. Eng. Sci.,
58 (2003), 1497–1507.
8. M. J. San Jos
´
e, M. Olazar, M. A. Izquierdo, S. Alvarez, and J. Bilbao. Solid trajectories and
cycle times in spouted beds. Ind. Eng. Chem. Res., 43 (2004), 3433–3438.
9. J. Berghel, L. Nilsson, and R. Renstrom. Particle mixing and residence time when drying
sawdust in a continuous spouted bed. Chem. Eng. Proc., 47 (2008), 1246–1251.
10. T. Ishikura, I. Tanaka, and H. Shinohara. Residence time distribution of particles in the
continuous spouted bed of solid-gas system with elutriation of fines. J. Soc. Powder Tech.
Japan, 15 (1978), 453–457.
11. N. Piccinini. Coated nuclear fuel particles. In Advances in Nuclear Science and Tech-
nology, Volume 8, ed. E. J. Henley and J. Lewins (New York: Academic Press, 1975),
pp. 304–341.
12. S. Devahastin and A. S. Mujumdar. Some hydrodynamic and mixing characteristics of a
pulsed spouted bed dryer. Powder Technol., 117 (2001), 189–197.
13. N. Piccinini, A. Bernhard, P. Campagna, and F. Vallana. Segregation phenomenon in spouted
beds. Can. J. Chem. Eng., 55 (1977), 122–125.
14. N. Piccinini. Particle segregation in continuously operating spouted beds. In Fluidization III,
ed. J. R. Grace and J. M. Matsen (New York: Plenum Press, 1980), pp. 279–285.
15. G. S. McNab and J. Bridgwater. Solids mixing and segregation in spouted beds. In Third
European Conference on Mixing (York, UK: BHRA, 1979), pp. 125–140.
16. N. Piccinini and G. Rovero. Thick coatings by thermo-chemical deposition in spouted beds.
Can. J. Chem. Eng., 61 (1983), 448–453.
17. H. H. Cook and J. Bridgwater. Segregation in spouted beds. Can. J. Chem. Eng., 56 (1978),
636–638.
18. E. Kutluoglu, J. R. Grace, K. W. Murchie, and P. H. Cavanagh. Particle segregation in spouted
beds. Can. J. Chem. Eng., 61
(1983), 308–316.
19. O. Uemaki, R. Yamada, and M. Kugo. Particle segregation in a spouted bed of binary mixtures
of particles. Can. J. Chem. Eng., 61 (1983), 303–307.
20. G. Rovero. Baffle-induced segregation in spouted beds. Can. J. Chem. Eng., 61 (1983),
325–330.
21. G. Rovero and A. P. Watkinson. A two stage spouted bed process for autothermal pyrolysis
or retorting. Fuel Process. Technol., 26 (1990), 221–238.
22. R. Aguado, M. Olazar, M. J. San Jos
´
e, G. Aguirre, and J. Bilbao. Pyrolysis of sawdust in
a conical spouted bed reactor. Yields and product composition. Ind. Eng. Chem. Res., 39
(2000), 1925–1933.
160 Giorgio Rovero and Norberto Piccinini
23. M. Olazar, M. J. San Jos
´
e, F. J. Penas, A. T. Aguayo, and J. Bilbao. Stability and hydrodynamics
of conical spouted beds with binary mixtures. Ind. Eng. Chem. Res., 32 (1993), 2826–2834.
24. M. J. San Jos
´
e, M. Olazar, F. J. Penas, and J. Bilbao. Segregation in conical spouted beds
with binary and ternary mixtures of equidensity spherical particles. Ind. Eng. Chem. Res., 33
(1994), 1838–1844.
25. M. J. San Jos
´
e, S. Alvarez, A. Morales, M. Olazar, and J. Bilbao. Solid cross-flow into the
spout and particle trajectories in conical spouted beds consisting of solids of different density
and shape. Chem. Eng. Res. Des., 84 (2006), 487–494.
26. M. S. Bacelos and J. T. Freire. Stability of spouting regimes in conical spouted beds with inert
particle mixtures. Ind. Eng. Chem. Res., 45 (2006), 808–817.
27. H. Huang and G. Hu. Mixing characteristics of a novel annular spouted bed with several
angled air nozzles. Ind. Eng. Chem. Res., 46 (2007), 8248–8254.
28. H. Huang, G. Hu, and F. Wang. Experimental study on particles mixing in an annular spouted
bed. Energy Convers. Mgmt, 49 (2008), 257–266.
29. B. Taha and A. Koniuta. Hydrodynamics and segregation from the Cerchar FBC fluidization
grid. In Poster Session, Fluidization VI Conference (Banff, Canada, 1989).
30. M. B. Saidutta and D. V. R. Murthy. Mixing behavior of solids in multiple spouted beds. Can.
J. Chem. Eng., 78 (2000), 382–385.
31. C. Beltramo, G. Rovero, and G. Cavagli
`
a. Hydrodynamics and thermal experimentation on
square-based spouted bed for polymer upgrading and unit scale-up. Can. J. Chem. Eng., 87
(2009), 394–402.
32. T.-C. Chou and Y.-M. Uang. Separation of particles in a modified fluidized bed by a distributor
with dead-zone collector. Ind. Eng. Chem. Process Des. Dev., 24 (1985), 683–686.
33. B. Waldie. Separation and residence times of larger particles in a spout-fluid bed. Can. J.
Chem. Eng., 70 (1992), 873–879.
9 Heat and mass transfer
Andrzej Kmiec and Sebastian Englart
9.1 Introduction
A great number of processes carried out in spouted beds require the application of
different modes of heat and/or mass transfer. We may distinguish among the following
modes
1,2
:
r
Heat transfer, mass transfer, simultaneous heat and mass transfer – between fluid and
particles,
r
Heat transfer between wall and bed, and
r
Heat transfer between submerged object and bed.
For each mode, transfer mechanisms are examined; then experimental findings and,
in some cases, theoretical studies are discussed.
9.2 Between fluid and particles
9.2.1 Transfer mechanisms and models
Quite often, the basic assumption for analysis of heat or mass exchanged between
fluid and particles is that heat is transferred to the particles under conditions of external
control, neglecting heat transmission within the particles. For heat transfer in the absence
of mass transfer, this is justified when the particle heat transfer Biot number is sufficiently
small (e.g., <0.1) and the corresponding Fourier number exceeds 0.2.
2
For simultaneous
heat and mass transfer, such as when the particles are well wetted at the surface, the
average temperature at the particle surfaces is substantially uniform, and external control
again prevails.
Assuming plug flow conditions through the bed, the axial fluid temperature distribution
can be described by a dimensionless function.
2
Because t he spouted bed consists of two
distinct regions, with the average gas velocity in the spout being one or two orders of
Spouted and Spout-Fluid Beds: Fundamentals and Applications, eds. Norman Epstein and John R. Grace.
Published by Cambridge University Press.
C
Cambridge University Press 2011.
162 Andrzej Kmiec and Sebastian Englart
magnitude greater than that in the annulus, the decline of gas temperature in these two
zones is quite different; it is slight in the s pout and considerable in the annular zone.
In fact, the annulus gas attains thermal equilibrium with the bed solids within a few
centimeters of its entry into the bed.
2,3
Mass transfer of a vapor from particles to the surrounding gas is under external control
conditions if the drying rate is constant. It can usually be assumed that the particles
under these conditions remain at a constant uniform temperature – namely, the adiabatic
saturation temperature of the inlet gas. Although the vapor transfer mechanism from
an individual particle is analogous to that of heat transfer, an analogy is not fulfilled
for the bed as a whole, as pointed out by Mathur and Epstein,
2
Kmiec,
4–6
Oliveira
et al.,
7
and Kmiec et al.
8
In some processes – for example, solids drying during the falling rate period –
considerable temperature gradients in some parts of the bed can occur. For unsteady
state heating or cooling of a particle, the resulting intraparticle temperature differ-
ences relative to the differences between the particle surface and fluid are solely
related to the Biot number, Bi
H
= h
p
r
p
/λ
p
, provided that the Fourier number, Fo
H
=
αt/r
p
,
2
exceeds a minimum value of 0.2.
9
A number of mathematical models have
been developed for heat transfer in spouted bed drying,
10–14
but they require sophis-
ticated mathematical or numerical methods to solve a system of differential equa-
tions. Devahastin et al.
10
assumed no conduction of heat and moisture between
particles, no heat losses, and no effects of particle shrinkage. Markowski
11
devel-
oped a model in which he introduced the additional heat flux connected with the
wet film evaporation from the surface of an inert par ticle. Another heat- and mass-
transfer model was developed by Feng et al.
12
to simulate combined microwave
and spouted-bed drying of diced apples, a hygroscopic porous material. In the solu-
tions of both these models, the surface heat transfer coefficient, h
p
,usedforthe
boundary condition, was calculated from Kmiec’s second correlation.
6
On the other
hand, Jumah and Raghavan,
13
in their numerical solution of the differential equa-
tion for heat transfer in spouted bed microwave-assisted drying, used Kmiec’s first
correlation
4
to calculate the convective heat transfer coefficient in the boundary
condition.
Ando et al.
15
developed a so-called cell model, which was defined to include average
air volume associated with a single seed particle. On the other hand, Niamnuy et al.
16
analyzed coupled transport phenomena and mechanical deformation of shrimp during
drying in a jet spouted bed dryer. The model consisted of coupled heat conduction and
mass diffusion equations coupled with static solid mechanics equations. The simulated
results, in terms of the shrimp moisture content, mid-layer temperature, and shrink-
age, were compared with the experimental results, and good agreement was generally
observed.
Currently, CFD modeling is used more and more often to describe the heat and mass
transfer during the drying of grain in spouted bed dryers.
17,18
The Eulerian-Eulerian
multifluid modeling approach has been applied i n those studies to predict gas–solid flow
behavior.
Heat and mass transfer 163
Table 9.1. Correlations of fluid-to-particle heat transfer in spouted beds.
Author Correlation
Uemaki and
Kugo
19
Nu = 0.0005
d
p
U
ms
ρ
g
μ
1.46
U
s
U
ms
1.30
(9.1)
Reger et al.
22
Nu = 0.0597 Re
2.0
i
Ar
−0.438
Gu
0.61
(H
o
/d
p
)
−1.0
(9.2)
Kmiec
4,5
Nu = 0.897 Re
0.464
Pr
0.333
Ar
0.116
tan
θ
2
−0.813
H
o
d
p
−1.19
φ
2.261
(9.3)
Kmiec
6
Nu = 0.0451 Re
0.644
Pr
0.333
Ar
0.226
tan
θ
2
−0.852
H
o
d
p
−1.47
D
i
d
p
0.947
φ
2.304
(9.4)
Kmiec and
Kucharski
26
Nu = 9.4723 Re
0.6128
Pr
0.333
Ar
0.2302
H
o
d
p
−1.031
.
m
s
.
m
g
0.8135
(
1 − C
s
)
0.795
φ
0.8326
(9.5)
Kmiec and
Jabarin
27
Nu = 2.673 Re
0.516
Pr
0.333
Ar
0.033
tan
θ
2
−2.331
H
o
d
p
−1.334
D
i
d
p
0.602
φ
2.102
(9.6)
Englartetal.
29
Nu = 0.0030 Re
0.836
Pr
0.333
Ar
0.236
Gu
−2.527
D
i
d
p
3.356
D
H
d
p
−4.121
.
m
w
.
m
g
0.600
φ
−0.918
(9.7)
Kudra et al.
45
Nu = 1.975 Re
0.64
H
o
d
p
−1.20
H
o
w
0.45
s
d
p
0.26
(9.8)
Rocha et al.
47
Nu = 0.9892 Re
1.6421
Pr
1/3
H
o
φd
p
−1.3363
.
m
s
.
m
g
0.71
tan
θ
2
0.1806
(9.9)
9.2.2 Experimental findings
The mechanism of the fluid-to-particle heat transfer process outlined theoretically by
Mathur and Epstein
1,2
has been confirmed to some extent by a number of experimental
studies. Table 9.1 shows a series of empirical correlations. A pioneering investigation
was carried out by Uemaki and Kugo,
19
who developed an empirical cor relation for air
(Pr = 0.7) – Eq. (9.1) in Table 9.1. The values of h
p
obtained are rather low as a result
of overestimation of the temperature driving force, which was attributed to the spout gas
alone.
A number of investigations of particle-to-fluid mass transfer in spouted beds have
also been reported. Table 9.2 shows a series of equations. The first work was again
by Uemaki and Kugo,
20
whose correlation was Eq. (9.10). Numerical values of the
Sherwood number for their range of conditions were found to be at least an order of
magnitude smaller than calculated values for comparable fixed and fluidized beds.
2
El-Naas et al.
21
found that the Uemaki-Kugo correlation underestimated the convective
mass transfer coefficient by an average factor of 3.
Another early study, on heat transfer during drying of organic dyes on glass beads,
was carried out by Reger et al.
22
A correlation, Eq. (9.2) in Table 9.1, was developed on
the assumption that the overall process is controlled by the rate of drying of the surface
film rather than by the rate of its attrition. A recent analysis of heat and mass transfer
of a liquid film at the surface of a single inert particle was given by Leontieva et al.
23
Further research on the drying process for various kinds of particles was carried out
164 Andrzej Kmiec and Sebastian Englart
Table 9.2. Correlations of particle-to-fluid mass transfer in spouted beds.
Author Correlation
Uemaki and
Kugo
20
Sh = 0.00022 · Re
1.45
·
D
H
o
(9.10)
Kmiec
4,5
Sh = 0.829 · Re
0.687
· Sc
0.333
· Ar
0.031
· (tan
θ
2
)
−0.915
·
H
o
d
p
−1.227
· φ
1.754
(9.11)
Kmiec
6
Sh = 0.01173 · Re
0.8
· Sc
0.333
· Ar
0.229
· (tan
θ
2
)
−0.961
·
H
o
d
p
−1.446
·
D
i
d
p
1.036
· φ
1.922
(9.12)
Kmiec and
Kucharski
26
Sh = 5.314 · Re
1.072
· Sc
0.333
·
H
o
d
p
−0.901
·
˙
m
s
˙
m
g
0.687
· (1 − C
s
)
0.533
· φ
0.164
(9.13)
Kmiec and
Jabarin
27
Sh = 1.371Re
0.567
Sc
0.333
Ar
0.044
tan
θ
2
−2.442
H
o
d
p
−1.353
D
i
d
p
0.735
φ
2.252
(9.14)
Kmiec et al.
8
Sh = 0.0095Re
0.843
Sc
0.333
Ar
0.172
Gu
−2.586
D
i
d
p
3.124
D
H
d
p
−4.078
˙
m
w
˙
m
g
0.605
φ
−1.310
(9.15)
El-Naas
et al.
21
Sh = 0.000258Re
1.66
(9.16)
by Kmiec.
4–6
Measurements of the constant-rate drying rate of solids yielded heat and
mass transfer coefficients. The experimental data were correlated by Eq. (9.3). Then
correlations listed in Table 9.1 by Kmiec and colleagues were developed to describe heat
transfer in coating processes,
24–27
coal drying,
28
or air humidification.
8,29
Kmiec and
Jabarin
27
were able to describe the heat transfer coefficient h
p
in the coating of shaped
rings by Eq. (9.6). In all studies, an analogy between heat and mass transfer was not
fulfilled, as discussed by Kmiec et al.
8
in detail.
These correlations and the same body of experimental data were used for analysis in
the development of a new system, a spouted bed with a side swirling stream,
5,30
which
appeared to induce a considerable increase in the heat transfer coefficient, h
p
, especially
at higher airflow rates.
Extensive investigations of gas-to-particle heat transfer were also carried out by Kilkis
and Kakac,
31
Dolidovich and Efremtsev,
32,33
and Freitas and Freire.
34
No correlations
were developed, however. Heat transfer in various new types of spouted beds have been
investigated.
35–37
Khoe and Van Brakel
35
showed that there was a linear relationship
between the quantity h
p
(1 − ε) and the drying efficiency η of a spouted bed dryer
with a draft tube. Nemeth et al.
38
reported experimental studies on heat transfer in
their “mechanical” spouted bed, in which solids are conveyed mechanically in the spout
region.
Chatterjee and Diwekar
39
developed a model to estimate the gas-to-particle heat
transfer coefficient using the bubbling bed concept for both spout-fluid and spouted
beds. The heat transfer coefficients h
p
for spouted beds were found to be comparable to
the data calculated from Kmiec’s first correlation.
4
Dolidovich
40
investigated interphase
heat transfer in a “swirled” spouted bed system. It was shown that pressure drop was 20
to 30 percent lower and the incipient spouting velocity 40 to 50 percent lower, whereas
interphase heat transfer rate was 15 to 25 percent higher, than their respective values
Heat and mass transfer 165
in a classical spouted bed. Also, Akulich et al.
41
showed that peripheral gas jets could
intensify gas-to-particle heat transfer in spouted beds.
Many papers have been concerned with heat transfer in drying of liquid solutions on
inert particles.
8,29,42–44
Oliveira and Freire
43
developed a three-region model for analysis
of evaporation rate in a spouted bed sprayed with liquids. Littman et al.
44
modeled water
evaporation in a pneumatic transport system of glass beads, which is mechanistically
similar to the spout in a spouted bed. Kmiec et al.
8,29
carried out both experimental and
modeling studies of heat transfer in sprayed spouted beds. Their experimental data could
be correlated by Eq. (9.7) in Table 9.1.
Investigations of heat transfer in two-dimensional spouted beds were carried out by
Kudra et al.,
45
Swasdisevi et al.,
46
and Rocha et al.
47
Kudra et al.
45
studied convective
heat transfer to particulate material during drying. The experimental data were correlated
by Eq. (9.8). Swasdisevi et al.
46
developed a model for both gas-particle dynamics and
heat transfer. Rocha et al.
47
correlated heat transfer coefficients during coating of tablets
with solutes of solutions sprayed on them in two-dimensional spouted beds, by Eq. (9.9).
Their Reynolds number exponent was more than twice as high as that in Kudra’s equation.
9.2.3 Recommended design equations
As mentioned earlier and as shown by Mathur and Epstein,
2
gas in the annulus would
achieve thermal equilibrium with bed solids even in the shallowest of beds. For suffi-
ciently deep beds, as would be usual on an industrial scale, the spout gas would also
attain equilibrium with the surface of the particles, allowing the development of design
formulas for gas cooling and solids heating. In the case of batch heating of cold solids,
the heat balance takes the form:
W
b
c
ps
dT
b
= UAρ
g
c
pg
(T
gi
− T
b
)dt, (9.17)
which, for initial conditions t = 0, T
b
= T
b0
,isintegratedto
t =
W
b
c
ps
UAρ
g
c
pg
ln
T
gi
− T
b0
T
gi
− T
b
(
t
)
. (9.18)
For the case in which thermal equilibrium between exit gas and bed solids is not achieved,
it is recommended that a gas-to-particle heat transfer coefficient h
p
for the bed as a whole
be defined by the equation
2
q = ρ
g
UAc
pg
(T
gi
− T
ge
) = h
p
S
p
T
Lm
, (9.19)
which is equivalent to
ln
T
gi
− T
b
T
ge
− T
b
= St
S
p
A
, (9.20)
where the Stanton number, St,isgivenbyh
p
/(c
pg
Uρ
g
). The term x, which is the fractional
temperature approach to equilibrium, (T
gi
−T
ge
)/(T
gi
−T
b
), can be seen to depend only
on St (Re, Pr) and S
p
/A,asx =1 −(T
ge
−T
b
)/(T
gi
−T
b
). Thus, in the absence of thermal
166 Andrzej Kmiec and Sebastian Englart
equilibrium between exit gas and bed solids, Eqs. (9.17) and (9.18) can be generalized,
respectively, to
W
s
c
ps
(T
b
− T
si
) = UAρ
g
c
pg
(T
gi
− T
b
)x (9.21)
and
t =
W
b
c
ps
xU Aρ
g
c
pg
ln
T
gi
− T
b0
T
gi
− T
b
(
t
)
=
W
b
c
ps
x
.
m
g
c
pg
ln
T
gi
− T
b0
T
gi
− T
b
(
t
)
. (9.22)
Similar formulas were developed by Claflin and Fane,
36
but they also introduced other
parameters, such as drying efficiency and heat transfer losses. Their equation has the
form
t =
W
b
c
ps
.
m
g
c
pg
η + K
(9.23)
×ln
.
m
g
c
pg
η
T
gi
− T
b0
+ K
(
T
a
− T
b0
)
'
.
m
g
c
pg
η
T
gi
− T
b
+ K
(
T
a
− T
b
)
,
where T
a
is ambient air temperature, K denotes heat losses, and contacting efficiency η
is defined by
η =
K
.
m
g
c
pg
T
b
− T
a
T
gi
− T
b
. (9.24)
Assuming no heat losses, K = 0J/s·K, and with contacting efficiency η equal to unity,
Eqs. (9.23) and (9.18) are identical.
9.3 Between wall and bed
A number of processes carried out in spouted beds require either supplying supplemen-
tary heat to the bed or removing heat from the bed. We therefore address the problem of
heat transfer between column wall and bed.
9.3.1 Transfer mechanisms
Heat transfer from (or to) the wall takes place through transverse conduction in conjunc-
tion with convective transport by the particles moving downward in the annulus, whereas
inside the spout, the governing mechanism is more or less the same as in dilute fluidized
beds.
48
For spouting with gas (air), a characteristic thermal boundary layer about 1 cm
in thickness is developed along the wall vertically downward.
1,2
9.3.2 Experimental findings
Numerous studies on heat transfer between column wall and bed have been carried
out.
19,49–52
A number of different empirical equations for gas spouting, based on dimen-
sional analysis, have been developed. First, Malek and Lu,
49
working with their data
and those of Klassen and Gishler,
50
obtained Eq. (9.25), Table 9.3, whereas Uemaki