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A single particle model for surface-to-bed heat transfer in fluidized beds

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A single particle model for surface-to-bed heat transfer in fluidized beds
Francesco Di Natale
⁎
, Amedeo Lancia, Roberto Nigro
Dipartimento di Ingegneria Chimica, Università di Napoli
“
Federico II
”
P.le Tecchio, 80, 80125 Napoli, Italy
Received 12 March 2007; received in revised form 8 November 2007; accepted 17 January 2008Available online 2 February 2008
Abstract
This paper presents a semi empirical single particle model for the description of heat transfer coefficient between a submerged surface and afluidized bed. The model is applied to several experimental data and gives a satisfactory description of the effect of pressure, temperature and bedmaterial properties on the heat transfer coefficient either in bubbling or slugging fluidized beds. The model considers the averaged surface voidfraction as the only regression parameter for the description of experimental data. Surface void fraction results to be a function of Archimedesnumber and minimum fluidization bed voidage and its value is consistent with the numerical and experimental data reported in literature.© 2008 Elsevier B.V. All rights reserved.
Keywords:
Fluidized beds; Heat transfer model; Surface void fraction
1. Introduction
Fluidized granular materials are characterised by high valuesof heat and mass transfer rates, allowing the use of smaller exchange surfaces and, together with the high solid mixing, therealization of nearly isothermal conditions. These features are at the basis of many industrial applications such as fluidized bed boilers, quenched reactors, heat exchangers, reactors for surfacecoating and decoating, food drying and freezing, treatments onmetallic and polymeric surfaces etc [1,2].It is almost universally recognised that the surface-to-bedheat transfer rate is mainly connected to the unsteady contacts between the fluidized particles and the submerged surface,which are directly correlated to the overall fluidization regimeand to the local fluid dynamic field around the surface. Boththese phenomena can be related to particle Archimedes number and gas velocity, but while the former has been largely studied,the latter has been the subject of very few works [3
–
5].Since the early sixties, many experimental works on heat transfer coefficient in bubbling and circulating fluidized bedshave been carried out. The main objective of these studies wasthe description of heat transfer coefficient for immersed tube banks, due to their extensive applications to heat exchangersand fluidized bed boilers. For this reason, experimental studiesconcerned the effects of particle diameter [6
–
8], pressure [8,9]and temperature [10] on heat transfer coefficient and, moregenerally, on its dependence on the overall fluidization regime.However, Buyevich et al. [3] reported X-ray photos of bedstructure around different surfaces, highlighting the existence of a layer of higher gas velocity and bed porosity in the proximityof the immersed surface, whose characteristics are stronglydependent on the surface shape. This bed structure influencesthe heat exchange process as well as all the other transport phenomena between the surface and the fluidized bed.It is generally accepted that the overall surface-to-bed heat transfer coefficient can be considered to be made up of threeadditive components:
h
¼
h
gc
þ
h
pc
þ
h
r
;
ð
1
Þ
where
h
gc
,
h
pc
,
h
r
are the gas convective, the particle convectiveand the radiative heat transfer coefficients. In this notation,
h
gc
accounts for both the contributions of the bubbles and of the gasthat percolates in the emulsion phase.Several models have been developed for the evaluation of thegasconvectivecomponent [11
–
14]anditiswellacceptedthatfor
Available online at www.sciencedirect.com
Powder Technology 187 (2008) 68
–
78www.elsevier.com/locate/powtec
⁎
Corresponding author. Tel.: +39 081 7682246; fax: +39 081 5936936.
E-mail address:
fdinatal@unina.it (F. Di Natale).0032-5910/$ - see front matter © 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.powtec.2008.01.014
silicasandparticlesfinerthan1mm,itisoftheorderofmagnitudeof the heat transfer coefficient at minimum fluidization velocity,when particle motion does not exists yet [12]. However, it has to be observed that experimental evidences for low conductivity particles are not available at the moment.On the contrary, the properties of the particle convective heat transfer mechanism have not been still addressed. At themoment, in spite of the development of several models for thedescription of heat transfer coefficient, none of them is still ableto describe the experimental results obtained for different temperatures, pressures and physical properties of the granular material. Indeed, predictive models are usually based onempirical correlations or mechanistic approaches. Althoughthe former can be correctly used only within the experimentalconditions fit by the existing data, the available mechanisticmodels are not sufficiently complete and empirical models arestill the most used for reactor design and optimization.Albeit a definite model for particle convection is not stillavailable, experimental evaluations and theoretical analyseshave defined the main features of this mechanism. In particular,
Nomenclature
A
Area (m
2
)
α
Parameter of Gamma function (
–
)
c
Specific heat (J/g K)
d
Diameter (m)
D
Exchange surface diameter (m)
δ
Gas layer thickness (m)
E(t)
Residence time distribution (Hz)
ε
Bed voidage (
–
)
ε
w
Average surface void fraction (
–
)
f
Renewal frequency (Hz)
δ
b
Bubble fraction (
–
)
g
Acceleration of gravity (m/s
2
)
h
Heat transfer coefficient (W/m
2
K)
k
Coefficient of proportionality in Eq. (7) (m/s)
K
Thermal conductivity (W/m K)
L
Exchange surface length (m)
λ
Thermal gradient thickness (m)
µ
Gas viscosity (kg/m s)
n
Number of particles in contact with the surface (
–
)
q(t)
Particle heat flux (W)
R
w
Surface-to-particle thermal resistance (K m
2
/W)
ρ
Density (kg/m
3
)
T
Temperature (K)
t
Time (s)
b
t
N
=1/
f
Average surface renewal time (s)
τ
Characteristic time for particle heating/cooling (s)
U
Gas velocity (m/s)
V
Volume (m
3
)
x
Radial distance form the exchange surface (m)
y
Vertical position on the exchange surface (m)
Dimensionless numbers Ar
=
ρ
g
(
ρ
p
−
ρ
g
)
d
p3
g
/
µ
2
Archimedes number
Nu
=
hd
p
/
K
g
Nusselt number
Re
=
ρ
g
d
p
(
U
−
U
mf
)/
µ
Reynolds number
Pr
=
µc
g
/
K
g
Prandtl number
Subscript
∞
Bulk exp Experimental valueg Gasgc Gas convectivemax Averaged maximummf Minimum fluidization p Particle pc Particle convectiver Radiativew Wall
69
F. Di Natale et al. / Powder Technology 187 (2008) 68
–
78
it is generally accepted that particle convection can be describedstarting from the following considerations:1. The heat flow between the surface and a single contacting particle depends on the ratio between the contact time and acharacteristic particle heating time [15,16], which resumes themain features of the contact itself (number of contact points,extensionofthegaslayeraroundtheparticle,thermalpropertiesof the solid and the surrounding gas, particle diameter, etc.).2. Thethermalgradientbetweenthesurfaceandthefluidizedbedis restricted to a small region adjacent to the submergedsurface.Theamplitudeofthisregionistypicallyoftheorderof 0.5
–
1 mm [17
–
19,32]. For sandorglass particles coarser than300 µm it is of the order of one particle diameter [18].3. In proximity of an immersed surface, it is observed of a boundary layer region characterised by higher shear rates, gasvelocities and bed voidage [3,20] that induce a radialcomponent in the motion of the fluidized particles [21]. This phenomenon can be related to the surface shape dependencyon heat transfer coefficient [22
–
24].4. The overall heat transfer due to particle convection can bemechanistically considered as the sum of the contribution of each contacting particle, weighted according to their contact time with the exchange surface. For particle coarser than300
μ
m, the heat transfer coefficient appears to be limited tothose particles that directly contact the exchange surfacewhile, for the finer ones, the transmission of heat alsoinvolves the other surrounding particles, giving rise to amechanism similar to the heat transfer through a solid
–
gasemulsion [19].These considerations point out that the evaluation of particleconvective heat transfer rate requires the estimation of particlemotion in proximity of the immersed surface [25,37]. Particlemotion can be analyzed using direct methods, as opticalinvestigations [3,30] and Positron Emission Particle Tracking(PEPT) techniques [25], or indirect ones, such as numericalsimulations [4,5,27
–
29], abrasion [2] or heat and mass transfer
measurements (as the classical work of Mickley and Fairbanks[6]). The characteristics of this motion can be resumed by meansof a particle-to-surface residence time distribution and a surface bed voidage. However, while the particle residence timedistribution near immersed surface has been analysed in details[6,21,25,31
–
39],thesurfacevoidfractionhasbeenestimatedinaverylimitednumberofexperimentalandnumericalstudies[3
–
5].At the moment, the lack in information on surface voidfraction can be considered as the main responsible of theabsence of a definite correlation between heat transfer and bedstructure near the immersed surface.This work discusses a method for the estimation of particleconvective heat transfer coefficient in fluidized beds to describethe effective heat transfer rate for submerged surfaces in a broadrange of operating conditions and provide a correlation betweenheat transfer coefficient and surface void fraction. For thesereasons, the model has been tested on numerous experimentaldata obtained in different operative conditions by varying pressure, bed material properties and temperatures.
2. Particle convective heat transfer model
The particle convective heat transfer coefficient can bedescribed starting from the analysis of the thermal transient of the particles in contact with the exchange surface. In this sense,the proposed model has been developed under the followinghypotheses:
–
Validity of the approach of single particle heat transfer mechanism [19].
–
The solid particles are considered as monosized sphereswhose diameter is equal to the averaged Sauter diameter of the bed material.
–
Negligible thermal gradient within the particle (i.e. uniform particletemperature).Atambientconditionsthisassumptionisverified for a ratio of particle and gas thermal conductivitieshigher than 30 [26].
–
The main wall-to-particle heat transfer resistance,
R
w
, isrepresented by a contact resistance due to the gas gap betweenthe surface and the particle,
δ
.
R
w
is expressed as the ratio between
δ
and the gas thermal conductivity
K
g
. For spherical particles, the thickness of this gas layer is usually assumedequal to
d
p
/10 (e.g. [8]), while Botterill [18] reported that a
valueoftheorderof
d
p
/24wouldbemorerealisticforirregular shaped particles.Although in real conditions the particles touch the surface in alimited number of contact points, Botterill [18] has verified that these points can be considered as surface discontinuities whosecontribution on the overall heat flow is negligible. Moreover,Molerusetal.[21]haveshowntheexistenceofradialandrotationalcomponentsinthemotionofparticlesnearthesurfaceandthus,the particle-to-surface contact points are not constant in time.According to these considerations, the model assumes a uniform boundary condition on the whole particle surface, so that the particle-to-surface contact area coincides with the whole particlesurface.Under the hypotheses of single particle approach, the surface-to-bedheatflowduetoparticleconvectioncanbemodelledasthesum of the heat flows between the surface and each particle,
q
(
t
),weighted by means of the particle residence time distribution,
E
(
t
), and multiplied by the number of particles simultaneously incontact with the surface,
n
. Therefore,
h
pc
is calculated as:
h
pc
¼
:
Q A
s
T
w
T
l
ð Þ ¼
1
A
s
T
w
T
l
ð Þ
Z
þ
l
0
q t
ð Þ
d
E t
ð Þ
d
n dt
ð
2
Þ
According to the hypotheses of particle sphericity andnegligible particle thermal gradients, the thermal behaviour of aspherical particle at initial temperature
T
∞
that contacts theexchange surface at temperature
T
w
, is governed by a first order differential equation [7,16]:
q
p
c
p
V
p
dT dt
¼
A
p
R
w
T
w
T
ð Þ
;
t
¼
0
;
T
¼
T
l
;
ð
3
Þ
70
F. Di Natale et al. / Powder Technology 187 (2008) 68
–
78
Where
V
p
and
A
p
are the volume and the lateral surface of the sphere [21].The surface-to-particle heat flow as derived from Eq. (3) is:
q t
ð Þ ¼
q
p
c
p
V
p
dT dt
¼
q
p
c
p
V
p
s
d
T
w
T
l
d
exp
t
s
;
ð
4
Þ
where
τ
is the characteristic particle heating time defined as:
s
¼
q
p
c
p
d
p
6
d
K
g
ð
5
Þ
The residence time distribution for the contact of particleswith the exchange surface,
E
(
t
), as described by a Gammafunction with shape factor
α
equal to 1 or 0 [21,33,34]:
E t
ð Þ ¼
1
a
!
b
t
N
a
þ
1
d
t
a
d
exp
t
b
t
N
;
ð
6
Þ
where
b
t
N
is the mean residence time and its inverse,
f
, isusually defined as surface renewal frequency. The particleresidence time distribution is the result of different scales of motion that include the mixing induced by bubbles motion, theBrownian-like microscopic motion typical of a granular flowand the particle displacement due to the peculiar fluid dynamicfield near the surface. As previously reported, this last consistsin the formation of a layer of higher gas velocity and bed porosity, which may eventually give rise to the formation of superficial bubbles. Typical values of particle motion frequen-cies induced by this surface layer are around 5
–
20 Hz[25,31,38], while its average thickness is of 6
–
10 mm [3
–
5] avalue close to Phillips' [19] estimation of the Kolmogorovlength scale in a fluidized bed. Hence, its effect on particlemotion can be considered more like a rigid high frequencydisplacement than a mixing phenomenon. For exchangesurfaces with characteristic dimension of a few millimetres,the surface renewal mechanism is mainly related to the surfacelayer, as the particle displacement induced by this phenomenonis higher than the surface length. Hence, the particles in contact with the surface at a given time can be almost completelyreplaced by fresh particles due to the action of the surface layer flow field. This result has been clearly pointed out by the worksof Boerefijn et al. [39] and Pence et al. [38] starting from the
measures of transient heat transfer coefficients for probes withvery small exchange surface area.For exchange surfaces with characteristic dimensions of theorder of some centimetres, as those typically involved in heat exchange processes of industrial interest, this rapid displace-ment does not give rise to a complete renewal of particulate phase near the surface due to the presence of multiple particle-to-surface contacts that happen at different surface positions[21,33,34]. In these cases, lower values of surface renewalfrequency are obtained (typically 0.7
–
2 Hz) and the leading phenomenon for particle mixing appears to be that induced bythe bubbles motion. Accordingly, data on temperature
–
timesignals (e.g. [6]) reveal that the surface temperature mainlychanges at the passage of a rising bubble while only a limitedoscillation is observed during the contact with the emulsion phase, due to the effect of surface gas layer. For this reason, in afirst analysis, for a fully bubbling or a slugging regime, thesurface renewal frequency almost coincides with the bubblefrequency.Detailed values of surface renewal frequency are available inthe literature [6,21,33,34,36,37] highlighting that it results to bea function of several parameters as physical properties of solidand gas phases, fluidization velocity and particle diameter. In particular, Mickley and Fairbanks [6] report the effect of gasvelocity and particle diameter at 100 kPa on the surface renewalfrequency (Fig. 1A), while Molerus et al. [21] reported the value
of the renewal frequency in a bubbling fluidized bed of 250 µm ballotini as a function of superficial velocity and gas pressure(Fig. 1B). The renewal frequency increases by increasing pressure and superficial velocity. For the case of silica sand and ballotini, a power law regression function for the description of
Fig. 1. Surfacerenewalfrequencyinfunctiontoexcessgasvelocity,pressureandparticlediameter.A:DatafromMickleyandFairbanks[6];B:DatafromMolerus[21].
71
F. Di Natale et al. / Powder Technology 187 (2008) 68
–
78
experimental data available in the literature [6,21,33,34,36,37]gives the following expression for
f
:
f
¼
k
d
Re
0
:
287 p
d
d
1 p
R
2
¼
0
:
94
ð
7
Þ
Thecoefficientofproportionality
k
isequalto2.045·10
−
4
m/s.For the evaluation of the number of particles simultaneouslyin contact with the surface, a simple geometrical relation can beconsidered:
n
¼
A
s
k
1
e
w
ð Þ
V
p
ð
8
Þ
In Eq. (8)
λ
is the thickness of the thermal gradient layer,
ε
w
is the averaged volumetric void fraction near the surface,
A
s
λ
(1
−
ε
w
) is the solid volume fraction in a volumetric shell of depth
λ
around the exchange surface of area
A
s
and
V
p
is thevolume of a single particle. The value of
λ
can be related to theratio between the particle residence time (
b
t
N
=1/
f
) and thecharacteristic time for the complete particle heating/cooling,
t
H
,which, according to the first order mechanism of heat transfer (Eq. (3)) is about five times the characteristic particle heatingtime,
τ
[41]. Both the times depend on particle and gas physical properties and on particle diameters.Former experimental and theoretical evidences pointed out that for glass or silica sand particles coarser than 300
μ
m andfluidized with air,
λ
coincides with the particle diameter [18,19,40], while, for finer particles,
λ
is assumed, at first approximation, to be around 500
μ
m [17
–
19,32]. For thissystem, at relative gas velocity
U
−
U
mf
=0.2 m/s,
T
=25 °C and
P
=100 kPa, the ratio
b
t
N
/
t
H
is equal to 1 for
d
p
=500 µm and it is around 2 for
d
p
=270 µm.Finally, for the estimation of surface void fraction only a fewworks can be considered. In particular,numerical estimations on2-D systems show that the value of
ε
w
is between 0.7 and 0.8 for a 600
–
700 µm ballotini fluidized bed at minimum fluidizationvelocity in which a plane slab [20] or an horizontal tubes [5] are
immersed.Similarresultshavebeenobtainedfromexperimentaltests [3] that also show a certain increase of
ε
w
with gas velocity.Quantitative estimations of
ε
w
derive from semi empiricalcorrelations proposed by Buyevich et al. [3], Ganzha et al. [4]
and Kubie and Broughton [43] and are reported in Table 1.
These equations show that the surface void fraction is afunction of the dimensionless distance from the surface referredto particle diameter,
x
/
d
p
, the dimensionless position of thesurface itself,
y
/
L
, the void fraction at minimum fluidizationvelocity,
ε
mf
, and the fluidization velocity,
U
/
U
mf
. In particular,surface void fraction increases with particle diameter and gasvelocity.The substitution of the Eqs. (4), (6) and (8) in Eq. (2) and theresolution of the integral, gives the following expression for the particle convective heat transfer coefficient:
h
pc
¼
q
p
c
p
k
1
e
w
ð Þ
f
1
þ
f
s
;
ð
9
Þ
It is worth noting that, due to the exponential expression of single particle heat flow,
q
(
t
), the value of
α
which appears inEq. (6) is not significant for the solution of the integral of Eq. (2).
3. Results and discussion
3.1. Estimation of averaged surface void fraction fromexperimental results
Under the hypothesis of complete availability of experi-mental values on
f
,
λ
and
ε
w
, the proposed model can bevirtually considered asapredictive one.Nevertheless, itisworthnoting that while the estimation of
f
and
λ
has been verified byseveral experimental and theoretical analyses, data on
ε
w
have been obtained only in a few experimental or computationalstudies [3
–
5,25,42,43] in a small range of operational condi-tions. The difficulties in the estimation of surface void fractionare due to its dependence on particle motion near the surfacewhich is strongly connected to the surface shape and whose properties are still a matter of intensive research [25].However, the proposed model, with its direct correlation between heat transfer coefficient and surface void fraction, can be used for the evaluation of
ε
w
in a broad range of experimentalconditions, by using the wide number of experimental studieson heat transfer coefficient available in literature. Typically,heat transfer experiments consist in a complete
h
exp
=
h
exp
(
U
) curvecorresponding to a given pressure, temperature and bed
Table 1Available estimations for surface void fractionAuthors Expression Surface NoteBuyevich et al. [3] 1
e
w
1
e
mf
¼
2
:
4
U
=
U
mf
ð Þ
0
:
347
Ar
0
:
077
y
=
L
ð Þ
0
:
83
;Plane slabKubie and Broughton [43] 1
e
w
1
e
mf
¼
2
:
04
d
x
=
d
p
d
1
0
:
51
d
x
=
d
p
;Plane slab
x
/
d
p
b
1Ganzha et al. [4] 1
e
w
1
e
mf
¼
0
:
7293
þ
0
:
5139
d
d
p
=
D
1
þ
d
p
=
D
;Vertical cylinder 72
F. Di Natale et al. / Powder Technology 187 (2008) 68
–
78

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