Research ArticleWall-to-Bed Heat Transfer at Minimum Gas-Solid Fluidization
Huili Zhang,1 Jan Degrève,1 Jan Baeyens,2 and Raf Dewil3
1 Department of Chemical Engineering, KU Leuven, Chemical and Biochemical Process Technology and Control Section,de Croylaan 46, 3001 Heverlee, Belgium
2 School of Engineering, University of Warwick, Coventry, UK3Department of Chemical Engineering, KU Leuven, Process and Environmental Technology Lab, J.P. De Nayerlaan 5,2860 Sint-Katelijne-Waver, Belgium
Correspondence should be addressed to Jan Baeyens; [email protected]
Received 7 May 2014; Accepted 3 September 2014; Published 17 September 2014
The heat transfer from a fluidized bed to the cooling jacket of the vessel has been studied for various powders at minimumfluidization conditions, by both convection and conduction approaches. These heat transfer characteristics are important as thepoint of transition between packed and fluidized bed operations and are needed in designing heat transfer operations where bubble-flow is not permitted.The effective thermal conductivity of the emulsion moreover determines the contact resistance at the heatingor cooling surface, as used in packet renewal models to predict the wall-to-bed heat transfer. In expressing the overall heat transferphenomenon as a convective heat transfer coefficient, it was found that the results could be fitted by Numf,𝑗 = 0.01Ar
0.42.
1. Introduction
A powder is a heterogeneous system in which solid particlesare surrounded by gas. There are an unlimited number ofsolid-gas systems possible ranging from the single solid insingle gas system to the more complex fluidized bed.
The specific reasons for investigating the heat transferat minimum fluidization are fourfold: (i) it is an importantdesign value for operations where bubble-flow is not permit-ted, for example, cooling of safety glass or slow and controlledcooling/hardening of metal-alloy wire; (ii) it is the point oftransition between packed and fluidized bed operations; (iii)it defines the extent of the thermal gradient within the bedclose to the heat exchanging wall; and (iv) it provides dataof the effective thermal conductivity of the bed at minimumfluidization: data on the effective thermal conductivity areessential to the estimation of the contact resistance at theheating or cooling surface, as used in packet renewal modelsto predict the wall-to-bed heat transfer and further discussedin Section 3.4.
Attempts to understand how heat is transferred throughthe system usually devolve into attempts to determine its
“convective heat transfer coefficient,” ℎ (W/m2K), as definedin the standard equation for heat transfer by convection:
𝑄 = ℎ𝐴exΔ𝑇. (1)
Experiments allow the determination of the temperaturedifference (Δ𝑇, in 𝐾) for a known heat flow rate, 𝑄, (𝑊)
and a known surface area of the heat exchanger 𝐴ex (m2).The heat exchange surfaces are either an “outside-wall,” thatis, a heat transfer jacket, or an “internal” surface with animmersed tubular heat exchanger. Both experimental andindustrial equipment commonly use cylindrical configura-tions to contain the bed, often with immersed tubular heatexchanger and/or heat transfer jacket.
The heat transfer coefficient in bubbling fluidized bedshas been extensively investigated (e.g., [1–4]): the wall-to-bed heat transfer rate increases with increasing gas flowrate because of the more vigorous bubble-induced particlemixing and hence a faster renewal of the emulsion phasein contact with the heat exchange surface (Figure 1). Theincrease is however limited, since higher gas flow rates causea longer time fraction of contact between the gas bubbles andthe surface. This phenomenon will even dominate at very
Hindawi Publishing CorporationJournal of Powder TechnologyVolume 2014, Article ID 163469, 8 pageshttp://dx.doi.org/10.1155/2014/163469
2 Journal of Powder Technology
high gas flow rates, where the heat transfer coefficient willgradually decrease [5, 6]. In deep fluidized beds of smallerinternal diameter, the freely bubbling mode can transforminto slugging [7, 8], where again poorer mixing and long slugcontact with the heat transfer surface will reduce the heattransfer coefficient [6].
This bubble-induced heat transfer mechanism is illus-trated in Figure 1, where packets of particles are broughtinto contacting the transfer surface, absorbing heat duringtheir contact by unsteady state conduction and dissipating thecaptured heat into the bulk of the bed by the bubble-inducedmixing. This packet renewal mechanism will be discussed inSection 3.4 of the paper.
A different approach to describing the heat transferby convection considers conduction through the emulsionphase in contact with the heat transfer surface.This approachresults in the determination of the thermal conductivity,𝑘 (W/mK). This property of the system determines thetemperature gradient under a fixed heat flow. The definitionof 𝑘 is obtained from Fourier’s law
𝑄 = −𝑘𝐴exΔ𝑇
Δ𝑥
, (2)
where the heat flow rate, 𝑄, (W) and the temperaturegradient, Δ𝑇/Δ𝑥, (K/m) are perpendicular to the exchangearea, 𝐴ex (m
2).Integration of (2) for a cylindrical space with height 𝐿 (m)
and for radii 𝑟1
and 𝑟2
(m) leads to expressions for the radialthermal conductivity as illustrated in Figure 2.
The radial thermal conductivity, 𝑘𝑟
(W/mK), is deter-mined by the geometry, the radial heat flow,𝑄
𝑟
, (W), and thetemperature difference Δ𝑇 (K), according to
𝑘𝑟
=
𝑄𝑟
2𝜋𝐿Δ𝑇
ln(𝑟2𝑟1
) . (3)
Thermal conductivities may be measured by either staticor dynamic methods. The former involves temperature mea-surements under steady state operation whereas in the lattercase the temperature change with time is measured at one ormore positions. Since confidence in unsteady state methodsis usually based on agreement with steady state methods, theuse of the former seems desirable until a fairly comprehensivecollection of reliable data has been acquired. The specificstudy of thermal conductivities in fluidized bed is uncommonsince it is often impossible to measure a thermal gradient inthe bulk of the bed, due to the high thermal conductivity, dueto the relatively small heat fluxes used in the experiments, anddue to the temperature fluctuations in the bed [6]. It is alsoknown from experiments that the axial conductivity is about10 times greater than in the radial direction, due to the betteraxial mixing in the wake of a bubble, as mentioned by Roweet al. [9].
The experimental heat transfer surface could be a planeslab, a hollow sphere, or a cylinder but since the constructionof uniformhollow spheres or slabs is very difficult, cylindricalarrangements are very common and have been used inmost of the previous experiments, as reviewed by Grace andBaeyens [10].
Heat transfersurface
Direct contactof particle with
surface
Next packet ofemulsion to contact
the surface
Equivalent thicknessof emulsion layer, P0Thickness of gas
film, 𝛿
Figure 1: General mechanism of bed-to-wall heat transfer.
T1T2
r2
r1L
Figure 2: Geometry for the radial heat flow in a cylindrical unit.
Ideally, only a heat flow perpendicular to the surfaceshould occur. For the study of radial heat transfer, the flatend surfaces of the cylinder ought to be covered with anonconductor of heat. As there is no such perfect insulator,two techniques are frequently used: heat guards are used inorder to keep the temperature constant over the whole lengthof the test section, or results are only taken in the centresection of a test specimen which is long in comparison toits diameter and where the isothermal surfaces are essentiallycylindrical over the centre section. In the present study theguard-heater approach will be used, as described further inthe text.
2. Experimental Layout and Procedure
The experiments involved the use of a mild steel fluidizationcolumn of 21.6 cm I.D., whereby the fluidized bed was heatedwith an immersed electrical heater and cooled by watercirculating in the jacket of the column. The equipment isschematically represented in Figure 3.
The water jacket is the annular space between the 21.6 cmbed and the outer 25 cm I.D. pipe. Coldwater from the servicepipe (T ≈ 20∘C) was fed to a constant head tank, and a needlevalve in the underflow controlled the flow rate. The waterflow meter was calibrated by measuring the volume of watercollected over a measured time. The central heater consisted
Journal of Powder Technology 3
23
2210
99
8
21
1620
21 15
1819
18
11713
1817
18
206
14
1 2
3
5
4
T
T
T
T
T
T
T
T
ΔP
P
(1) Compressor
(2) Shut off valve and water trap
(3) Pressure gauge
(4) Rotameters for air flow
(5), (10) Thermometers ±0.5 ∘C
(6) Distributor
(7) Fluidized bed
(8) Expansion chamber
(9) Wire mesh filters (aperture 150𝜇m)
when fluidizing fine powder, these filters were removed
and the outlets were connected to cyclone 22, followed
by filter 23.
(12) Constant head tank
(13) Rotameter for water flow
(14), (16) Thermometers ±0.2 ∘C
(17) Thermocouples welled onto the wall, connected to
data logger
(18) Moveable thermocouples, connected to data logger
(19) Heater, consisting of 3 elements (1.5 cm O.D.)
(20) Supports for the heater
(21) Variac voltage regulators
(11) H2O manometer
Figure 3: Layout of the experimental rig.
4 Journal of Powder Technology
of three heater elements, manufactured by Masser [11]: themiddle heater was 300mm long and the guard heaters wereeach 100mm. These elements were incorporated in a coppertube (≈10mm I.D. −15mm O.D.) and the complete heaterwas positioned centrally in the column by means of a brasstube fixed onto the distributor at the bottom of the heaterand by means of 2 radial rods supporting a central brass tubefixed to the top of the heater. By using the guard heatersa nearly constant temperature along the full length of themiddle heating element was obtained.
Thermocouples were imbedded in the heater wall andbrazed into position. The bed temperature was measuredwith moveable sheathed thermocouples at different radialpositions and four different heights. These thermocoupleswith stainless steel sheaths were fixed with Araldite into a2mm O.D. copper tube to prevent them bending at the tip.The thermocouples protruded 10mm beyond the supportingtubes. The thermocouples were moved in steps of ∼1mm inthe vicinity of the heater and 5mm in the bulk of the bed.Theposition of the thermocouples was defined with an accuracyof 0.5mm by means of a mark on the copper sheath.
The thermocouple outputs, with 0.5∘C accuracy, wereconnected to a data logger. The experimental work involvedthe use of 21 powders and the static bed height was about50 cm for all powders tested. The minimum fluidizationvelocity was determined from the commonly used mea-surement of the pressure drop across the bed, Δ𝑃 (Pa), atincreasing superficial gas velocity, 𝑈 (m/s) [12]. The air flowrate was adjusted to the superficial gas velocity at minimumfluidization,𝑈mf (m/s), where the first small bubbles occurredat the bed surface. Almost all of the powders tested are ofthe Geldart B-type [12], where the onset of bubbling andthe condition of minimum fluidization coincide. The onsetof bubbling occurs at a higher gas velocity than 𝑈mf onlyfor A-type powders. The heater was switched on and thebed was allowed to reach steady state conditions, resultingin a constant temperature at the heater and in the bed. Thetemperatures were recorded over 30 minutes and the voltageacross the heater was measured with an AVO-meter. Theexperimental results will be described hereafter.
The gas distributor was a sandwich of filter paper betweentwo perforated metal plates, with 8mm I.D. holes at a25.4mm pitch, as illustrated in Figure 4.
ThreeVariac voltage regulatorswere used.While centrallyheating and simultaneously cooling (wall), local temperaturesin the bed and at the inner and outer walls were measured.The details of thermocouple locations on both surfaces andin the bed are given in Figure 3. The powder sphericityand solids’ thermal conductivity, 𝑘
𝑠
(W/mK), are given inTable 1.
3. Experimental Results and Discussion
3.1. Temperature Profiles and Heat Transfer Properties. Fromthe experiments, graphs similar to Figure 5 were obtainedwhilst the use of guard heaters maintained the temperatureat the heater wall to within 2∘C over the entire height.The temperature of the jacket was also nearly independent
Steel plate, 1.6 mm thick
= 25.4 mmpitch
330.2mm
R
++++
++
++
+++++++
++
++
+++++++
++
+
+
+
Holes of 8mm
Figure 4: Perforated plate distributor used for testing.
70
60
50
40
30
20
0.75 0.8 1.0 1.2 1.4 1.6 8 10
Jacket surface
Heater surface
r(cm)
Ph
Pj
T(∘
C)
Figure 5: Semilogarithmic plot of the average temperature 𝑇 versusradial distance 𝑟.
Table 1: Properties of the components of the powder gas systemused in calculations.
of height in the bed (within 1.5∘C). ΔT could be readilycalculated from arithmetic means.
The heater output was typically 25.2–33.6W. The coolingoutput from the bed to the jacket was calculated fromthe known water flow rate and the measured temperaturedifference of thewater between input and output to the jacket.The surface area of the jacket was calculated as 𝜋 ∗ 21.6 ∗
𝐻mf cm2, where𝐻mf is the bed height at incipient fluidization
(close to 50 cm in all runs).
Journal of Powder Technology 5
Table 2: Experimental results in the 21.6 cm I.D bed using a 1.0 cm O.D. heater.
Note: propertiesmarked∗were experimentally determined. Average particle sizes weremeasured byMalvern laser diffractometry.The absolute particle densitywas taken from suppliers’ data.
Experiments were repeated 3 times for each powder, andthe respective calculated heat transfer coefficients were allwithin 7% of the calculated (and further reported) averagevalues. The experimental temperature profiles, as illustratedin Figure 5, demonstrate the existence of a distinct zone nearthe central heater or near the outside-wall where a majortemperature gradient is observed. These thicknesses of thesezones are given in Figure 5 as 𝑃
ℎ
and 𝑃𝑗
for the heaterand wall zones, respectively. The temperature gradients inthese contact layers are significant. The temperature remainshowever nearly constant outside these layers, that is, inthe central part of the bed. These zones correspond to thecontact transfer layers in the mechanistic surface renewalmodels (e.g., [13–16]). This differs from a packed bed, wherethe thermal gradient extends from centre to wall [17]. Theoverall properties of such layers can be derived if theircylindrical symmetry is taken into account. This has beenperformed graphically on the semilogarithmic plots of thegradient thicknesses of the layer near the outer wall. Withdetermined heat transfer coefficients and layer thicknesses,the effective thermal conductivity of the layers near the jacketcan be calculated and values of the thermal conductivityat minimum fluidization near the jacket, 𝑘mf,𝑗 (W/mK), areincluded in Table 2.
3.2. The Convective Bed-to-Wall Heat Transfer Coefficientsat Minimum Fluidization. All wall-to-bed convectionheat transfer coefficients at minimum fluidization, ℎmf,𝑗,were transformed into their respective Nusselt-numbers(ℎmf,𝑗𝑑sv/𝑘𝑔), with 𝑘
𝑔
as thermal conductivity of thefluidization gas (W/mK).
The particle size, 𝑑sv (m), absolute particle density, 𝜌𝑝
(kg/m3), gas density, 𝜌𝑔
(kg/m3), gas viscosity, 𝜇 (kg/ms), andthe gravitational constant, 𝑔 (m/s2), were grouped into theArchimedes number, defined as
Ar =𝑑
3
sv (𝜌𝑝 − 𝜌𝑔) 𝜌𝑔g𝜇
2
.(4)
The increase of the minimum fluidization heat transfercoefficient with increasing 𝑑sv is due both to a higher valueof the gas flow, hence increasing the effect of forced convec-tion, and to a decreased voidage. A log-log plot of Nusseltnumber versus Archimedes number for all powders testedyields a straight line, as illustrated in Figure 6, suggesting acorrelation of the form Numf,𝑗 = 𝐾Ar𝑐, where 𝐾 and 𝑐 havevalues as given in (6), for 10 ≤ Ar ≤ 2∗103:
Numf,𝑗 = 0.01Ar0.42
. (5)
6 Journal of Powder Technology
1
0.1
0.0110 100 1000
Ar (—)
(—)
Nu m
f,j
Figure 6: Nusselt number versus Archimedes numbers for all 21powders tested.
A comparison of our results with former investigationsis very difficult because these researchers either only repre-sented their data graphically, or included no precise data inthe vicinity of 𝑈mf.
3.3. The Effective Thermal Conductivity. As already men-tioned, Figure 5 suggests that heat is being conducted awayfrom the heater or jacket through a distance of 𝑃
0
cm,characterizing the contact thermal layer, and thereafter near-isothermal conditions prevail in the bed. Values are presentedin Figure 7.
It can be seen that the thickness 𝑃0
has a higher value forfine particles than for coarser ones and the same trend existsfor spherical/rounded materials versus angular powders. Ifthe heat transfer were purely conductive, one would expectthe opposite trend. It should be remembered however thatthe mechanism of heat transfer includes a transfer to thegas. Due to the deflection of the gas stream by the particles,a complex gas flow pattern (counter current, concurrent,and perpendicular to the heat flow) is responsible for theradial dispersion of the heat. Since the interstitial gas velocityin any direction increases as the bed voidage at minimumfluidization (𝜀mf) decreases and the minimum fluidizationvelocity (𝑢mf,m/s) increases, it is clear that with increasing𝑑svthe convective radial heat interchange will increase and thatthe thermal gradient will be limited to a smaller zone nearthe wall only. The proposed thermal conductivity defines theradial temperature gradient and was introduced only becauseit was found impossible to describe the separate contributionof forced convection to the overall heat transfer.
Using the values of 𝑘mf,𝑗, ℎmf,𝑗 can be calculated for aparticular solid-gas system from the use of the value of 𝑃
𝑗
asgiven in Figure 7:
ℎmf,𝑗 =𝑘mf,𝑗
(𝑅 − 𝑃𝑗
) ln (𝑅/ (𝑅 − 𝑃𝑗
))
, (6)
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0100 1000
Spherical glassRounded sand
Angular sandAngular catalyst
Laye
r thi
ckne
ssPj
(cm
)
dsv (𝜇m)
Figure 7: Average values of the layer thickness, 𝑃𝑗
, versus particlesize, 𝑑sv.
where 𝑅 is the radius of the column. Smaller column diam-eters will hence see an increased value of the heat transfercoefficient.
3.4. The Contact Resistance in Heat Transfer Modelling. Theheat transfer in gas-solid fluidized beds has been modelledby different approaches.
In single or multiple particle heat transfer models (e.g.,[18–21]), the contact heat transfer is considered as a thin gasfilm adjacent to the heat transfer surface and of thickness 𝛿(m).The thermal conductivity of the gas, 𝑘
𝑔
(W/mK), and the(unknown) thickness of the gas film (expressed as a fractionof the particle size, 𝑑sv) define the contact resistance as
ℎ𝑐
=
𝑘𝑔
𝛿
,(7)
with 𝛿 equal to 0.1 to 0.2𝑑sv.In the packet renewal models (e.g., [6, 22, 23]), the
thermal conductivity at minimum fluidization is recognizedas basis of the heat transfer resistance.
Baeyens and Geldart [6] adapted the packet renewalmodel by adding a time-independent contact resistance, ℎ
𝑐
(W/m2K), between the packet and the surface, and deter-mined this contact resistance as
ℎ𝑐
=
𝑚 ⋅ 𝑘𝑔
𝑑sv∼
𝑘mf𝑑sv
. (8)
Different values of 𝑚 have been given in literature, being3.2 for nonmetallic powders [6], about 7.2 [24], to 8 [13], and
Journal of Powder Technology 7
even 10 [25]. The wide range of 𝑚-values cited in literaturedoes not provide an unambiguous relationship between 𝑘mfand 𝑘𝑔
.Having determined the thermal conductivity at mini-
mum fluidization, 𝑘mf, the use of packet renewal models ishowever facilitated, since its value can be directly used in theexpressions of the heat transfer coefficient, ℎ (W/m2K). Thisis illustrated for the expressions of two model approaches, asfollows.
Mickley and Fairbanks [22]
ℎ =√
𝑘mf𝜌mf𝑐𝑝,mf
𝜋𝜃
,(9)
Baeyens and Geldart [6]
ℎ =
𝜋ℎ𝑐
1 + (6ℎ𝑐
/𝜌𝑝
𝑐𝑝
𝑑sv) 𝜃with ℎ
𝑐
∼
𝑘mf𝑑sv
. (10)
The bubble-induced particle mixing determines the timeof contact, 𝜃(s), between the particles and the heat transfersurface [6]. Introducing the relevant particle properties and𝑘mf allows us to predict the heat transfer coefficient, ℎ, at agiven gas velocity and the packet contact time at the heatexchange surface, 𝜃(s).
4. Conclusions
The heat transfer from a fluidized bed to the outside wall,at the onset of fluidization, has been studied for variouspowders.
In expressing the overall heat transfer phenomenon asa convective heat transfer coefficient, it was found thatthe results can be calculated from the equation Numf =
0.01Ar0.42.Effective thermal conductivities were also predicted. A
combination of the data of both the thicknesses of the thermallayer and the effective thermal conductivity allows predictionof the heat transfer coefficient for any geometry of the heattransfer surfaces, expressed by their respective radius, 𝑅. Themeasured thermal conductivity can moreover be applied influidized bed heat transfer modelling.
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
Acknowledgment
The authors acknowledge the European Commission forcofunding the “CSP2” Project—Concentrated Solar Power inParticles (FP7, Project no. 282 932).
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