Heat transfer in fluidized beds: design methods

Heat transfer in fluidized beds: design methods

Powder Technology 150 (2005) 123 – 132 www.elsevier.com/locate/powtec Heat transfer in fluidized beds: design methods John C. Chena,*, John R. Graceb...

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Powder Technology 150 (2005) 123 – 132 www.elsevier.com/locate/powtec

Heat transfer in fluidized beds: design methods John C. Chena,*, John R. Graceb, Mohammad R. Golrizc a

b

Department of Chemical Engineering, Lehigh University, Bethlehem, PA 18015-4791, USA Department of Chemical and Biological Engineering, University of British Columbia, Vancouver, Canada V6T 1Z4 c Department of Applied Physics and Electronics, Umea University, Umea, SE-90187, Sweden Available online 26 January 2005

Abstract Large-scale fluidized beds for commercial processes commonly require heat transfer surfaces. Design then demands that heat transfer coefficients be specified. Empirical correlations are unable to cover the wide range of variables and conditions encountered. Mechanistic models are more reliable, but must be chosen carefully. For bubbling beds, the packet model approach gives reasonable predictions for the convective component of transfer, but further work is required to provide reliable estimates of two required time constants, dependent on the hydrodynamics. For industrial-scale circulating beds, a mechanistic model that incorporates the key factors influencing heat transfer, assumes fully developed transfer, and utilizes results from large-scale units is recommended. D 2004 Elsevier B.V. All rights reserved. Keywords: Heat transfer; Fluidized bed; Design method

1. Introduction

2. Bubbling fluidized beds (BFBs)

To control temperature in many applications of fluidized beds, it is necessary to add or extract thermal energy. Often, this requirement is associated with endothermic or exothermic chemical reactions. For example, fluidized bed combustors require the removal heat of combustion in order to maintain operating temperature. Fluidized beds that require such heat transfer are designed to contact the fluidized medium with heat transfer surfaces—either immersed tube bundles or membrane waterwalls. Design and scale-up of these surfaces require knowledge of the heat transfer coefficient at the tube or wall surfaces in contact with the fluidized medium. This paper discusses methods for estimating the heat transfer coefficients for two important types of fluidized beds, namely bubbling dense beds and circulating beds operated in the fast fluidization flow regime. In both cases, the need is for models that can predict the effective heat transfer coefficient for submerged surfaces and are applicable over broad ranges of operating conditions and system sizes.

Most applications of bubbling fluidized beds that require heat transfer utilize bundles of heat transfer tubes, which may be horizontal or vertical. Other orientations are not recommended. Hence the requirement here is to develop engineering methods for predicting the heat transfer coefficient on tubes submerged in bubbling fluidized media. It is generally recognized that heat transfer occurs by gaseous convection during brief periods of contact with a lean gas phase, by particle convection/conduction during times of contact with a dense particle phase, and by radiation in the case of high temperature operations [1]. Thus:

* Corresponding author. Tel.: +1 610 758 4260; fax: +1 610 758 5057. E-mail address: [email protected] (J.C. Chen). 0032-5910/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2004.11.035

htot ¼ fd hd þ ð1  fd Þhl þ hr

ð1Þ

where h tot, h d, h l, and h r are the total effective heat transfer coefficient, heat transfer coefficient during bdenseQ (particle) phase contact, heat transfer coefficient during bleanQ gas phase contact, and heat transfer coefficient for radiation, respectively, while f d is the time fraction of contact by dense phase. It is usually assumed that the three heat transfer coefficients on the right-hand side above are independent. This approach is also followed here.

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2.1. Convection Often the two convective terms are lumped into a single convection coefficient, i.e., hc ¼ fd hd þ ð1  fd Þhl :

ð2Þ

Many attempts have been made to correlate the convective heat transfer coefficient with gas flow rate and properties of the fluid and solid particles. Most often, such correlations seek empirical relationships between a Nusselt number (Nu) with the fluid Prandtl number (Pr) and a Reynolds number (Re). A typical example is a correlation presented by Vreedenberg [2] for submerged horizontal tubes: !0:3 !0:3 l2g  0:3 qp hc Dt N ut u ¼ 420 Ret Prg 2 3 kg qg gqp dp ! qp for Ret N2250 ð3Þ qg where Re t is the Reynolds number based on tube diameter, D t. while k g, q p, q g, l g, and d p are the gas thermal conductivity, particle density, gas density, gas viscosity, and mean particle diameter, respectively. Another example is the correlation of Borodulya et al. [3] for vertical tubes: !0:14   hc dp cpp 0:24 0:1 qp Nup u ¼ 0:74ð ArÞ ð1  eÞ0:67 kg qg cpg   ð1  eÞ0:67 þ 0:46 Rep Prg e

ð4Þ

where Nu p, Ar, and Re p are the Nusselt number, Archimedes number, and Reynolds number, all based on the particle diameter, while c pp, c pg, and e are the specific heat of particles, specific heat of gas, and overall void fraction, respectively.

Recently Chen [4] compared several different correlations against a set of experimental data for heat transfer in a fluidized bed of 240 Am glass spheres. Fig. 1 (from [4]) shows that in fact there is very little agreement among the various correlations, or between the correlations and experimental data—with deviations on the order of 100%. While empirical, convective correlations are easy to use and may be appropriate for specific scale-up applications, their generality is questionable. The number of variables is so large and the range of data used to generate the correlations is so limited that empirical correlations are unable to provide reliable predictions. In the opinion of the authors, such empirical correlations for the convective heat transfer coefficient in bubbling fluidized beds for scale-up purposes can only be used with considerable caution. Most design correlations (as discussed above) represent convective heat transfer as a steady state process driven by gas flow. In contrast, visual observations in actively bubbling beds indicate that the particle emulsion actually remains relatively static, only being periodically disturbed by the passage of discreet gas bubbles. Mechanistically, the most representative model for fluidized bed heat transfer may well be the packet model originally suggested by Mickley and Fairbanks in 1955 [5]. These authors considered the heat transfer surface to be alternately contacted by gas bubbles and closely packed particle emulsion (packets). Thus the transport process becomes a surface renewal phenomenon where heat transfer occurs primarily by transient conduction between the particle packets and the surface during periods when the packets reside at the heat transfer surface. In recent years, this basic concept has been experimentally confirmed by researchers who were able to measure transient variations of particle concentration at the heat transfer surfaces. For example, Ozkaynak and Chen [6] and Chandran and Chen [7] used flush-mounted capacitance electrodes to obtain real time measurements of fluctuating particle concentration at surfaces submerged in bubbling beds. Results indicated that the particle concentration is highly transient, with instantaneous

Fig. 1. Comparison of correlations for convective heat transfer with experimental data of Ozkaynak and Chen [6] for vertical tube in bubbling beds.

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concentrations alternating between almost pure gas (bubble phase) and that of loosely packed beds. Such evidence of highly transient contacting lends support to the surface renewal model. The convective heat transfer coefficient can then include bubble phase convection and packet phase convection, as expressed in Eq. (2). Following the approach of Mickley and Fairbanks, the dense phase is equated to particle bpackets,Q where f d is the fraction of time during which surfaces are in contact with the dense (packet) phase. If one approximates the particle packet as a pseudohomogeneous medium of solid volume fraction (1e Pa), transient conduction analysis results in the average densephase coefficient as:   kPa qp cpp ð1  ePa Þ 1=2 : ð5Þ hd ¼ 2 psPa Here k Pa is packet thermal conductivity and e Pa is void fraction in the packets, while s Pa is the root mean residence time of packets at the heat transfer surface, given by: 32 2 i¼n X ðsPa Þi 7 6 7 6 i¼1 7 : sPa u6 ð6Þ 6X i¼n qffiffiffiffiffiffiffiffiffiffiffi 7 5 4 ðsPa Þi i¼1

In most bubbling fluidized beds, the bubble phase contribution is small compared to the packet phase contribution so that the total convective heat transfer coefficient can be approximated as:   kPa qp cpp ð1  ePa Þ 1=2 hc gf d hd ¼ 2fd : ð7Þ psPa All parameters in Eq. (7) can be evaluated from properties of the solid particles and gas, with the exception of the two time parameters f d and s Pa. Where direct measurements of transient solid concentrations have been obtained at the heat transfer surface, these two parameters can be evaluated directly. Chen [4] used this concept for the case of a submerged vertical tube in bubbling fluidized beds of glass

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beads. Based on the measurements of Ozkaynak and Chen [6], empirical curves for the mean residence time (s Pa) and time fraction ( f d) were obtained as functions of the excess gas velocity (UU mf). Figs. 2 and 3, taken from Chen [4], illustrate the experimental data for these two time parameters and their corresponding empirical curves. It is seen that both the mean residence time and the time fraction decrease with increasing excess gas velocity for particles of a given size, and increase with increasing particle size. For vigorously bubbling beds the residence time of particle packets on a heat transfer surface tends to be on the order of less than 1 s. The Fourier modulus for transient conduction in the particle packets is then of order: kPa sPa Fou VO½10: ð8Þ qp cpp ð1  ePa Þdp2 This magnitude of the Fourier modulus implies that conduction waves in the emulsion packets are able to penetrate distances of only a few particle diameters during the packet residence period. It should be noted that since the particle packing density is reduced in the first layer of particles at a solid surface, one should account for the corresponding reduction of effective thermal conductivity in this near-surface region. From the solution of the transient conduction equation, the penetration depth, (n), defined as the distance into the emulsion packet where 90% of the temperature change is obtained in residence time, s Pa, is given [4] by:   n pffiffiffiffiffiffiffiffiffiffiffiffi aPa sPa ¼ 0:9 erf 2 pffiffiffiffiffiffiffiffiffiffiffiffi ð9Þ ni2:32 aPa sPa where aPa u

kPa : qp cpp ð1  ePa Þ

ð10Þ

For distances (x) within one particle diameter of the heat transfer surface, an approximation of the local void fraction

Fig. 2. Example of residence times for particle packets on vertical heat transfer surface in bubbling beds, from [4].

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Fig. 3. Example of time fraction for packets on vertical heat transfer in surface in bubbling beds, from [4].

can be obtained by fitting results of Kubie and Broughton [8]:     x x ex ¼ 1  2:04ð1  ePa Þ 1  0:51 : ð11Þ dp dp At positions beyond one particle diameter, the local void fraction and packet properties are approximately equal to the values at bulk emulsion voidage, e Pa. Chen [4] suggested that the packet model of Mickley and Fairbanks be modified for cases where n is less than one particle diameter by taking the effective thermal conductivity of the emulsion packet at x=n/2. Results of Kunii and Smith [9] for packed media in stationary fluid can be represented by the following expressions for the effective packet thermal conductivity:  b kp kPa ¼ akg ; a ¼ 3:5  2:5e; b ¼ 0:46  0:46e kg ð12Þ where, for n/2zd p: e¼ePa ; the bulk void fraction in the emulsion packets;cemf ð13Þ while, for n/2bd p:      n=2 n=2 e ¼ 1  2:04 1  ep 1  0:51 : dp dp 

ð14Þ

To test this mechanistic approach, Chen [4] used the above equations with empirical correlations for f d and s Pa to predict convective heat transfer coefficients for a submerged vertical tube and compared the predictions to independently measured heat transfer coefficients at the same operating conditions. Fig. 4 compares the model predictions to experimental measurements for bubbling fluidized beds for two different particle sizes. It is seen that the modified packet model based on a surface renewal mechanism gives good agreement with the experimentally measured heat

transfer coefficients. The model not only predicts the magnitude of the heat transfer coefficient, but also correctly duplicates the trends for parametric variations with particle size and gas velocity. The conclusion is that for scale-up of heat transfer surfaces submerged in bubbling fluidized beds, the mechanistic surface renewal model can give reliable predictions. A problem in using this mechanistic approach is the lack of reliable information on the two time parameters s Pa and f d. Chandran and Chen [7] noted that in the case of horizontal tubes, large variations in both the residence time (s Pa) and time fraction ( f d) occur around the circumference, resulting in wide variations in the corresponding local heat transfer coefficients. These authors presented correlations for s Pa and f d, but recognized that many more data are required for any general correlation. Given this situation where the mechanistic surface renewal model appears to be realistic, but information on the time parameters of residence time and fractional coverage are lacking, what is the promise for scale-up and new designs? Clearly, the technical community needs to collect experimental measurements of these time parameters for broad ranges of particles, operating conditions, and bed sizes. Empirical correlations or mechanistic predictive methods can then be developed for these two time parameters, allowing the modified packet model to calculate convective heat transfer coefficients. Alternatively, simple mechanistic approaches used to describe and predict the hydrodynamics of bubbling beds could be adapted to estimate these two parameters. One hopes that such developments will be attained in the near future. Another promising approach is to develop deterministic models for the dynamics of particle movement in fluidized beds by numerical simulation. In recent years, progress has been made with both Eulerian models using local averaging at the scale of the computational cell and discreet particle models using Lagrangian formulations to track particle motion. Especially promising results have been reported by

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Fig. 4. Agreement between modified packet model and data for heat transfer to vertical tube in bubbling beds, from [4].

researchers who have combined the Eulerian and Lagrangian models to simulate bubbling fluidization. Fig. 5, taken from Xu et al. [10], shows the results of simulation for a two-dimensional bed of air-fluidized 550-Am-diameter particles. This figure presents snapshots of the simulation results at three specific times indicating the transient void fractions in the bubbling bed. The gradations of shade indicate instantaneous voidages in the bed. One can expect that if a heat transfer tube were submerged in this bed, the simulation would predict the dynamics of alternating contact by gas bubbles and particle packets. It is clear that a simulation covering a sufficient time duration should give statistical information on the time parameters of interest, s Pa and f d. Thus the combination of experimental measure-

ments, empirical correlations, and numerical CFD simulation should in the future provide the necessary time-domain information to enable direct application of the surface renewal model for predicting convective heat transfer to submerged surfaces in bubbling fluidized beds. 2.2. Radiative heat transfer For bubbling fluidized beds operating at temperatures greater than 500 8C, thermal radiation contributes significantly to the overall heat transfer. Models to estimate the radiative heat transfer flux can be grouped into two classes. The more physically based approach attempts to calculate radiant emission, absorption, and scattering within the

Fig. 5. Transient void fractions for a bubbling bed from numerical simulation of Xu et al. [10]. Air-fluidized bed of particles with d p=550 Am, q p=2500 kg/m3.

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fluidized media. Examples of this approach include the works of Bhattacharya and Harrison [11], and Chen and Chen [12]. This first type of model is mechanistically sound but complicated to apply, often involving the solution of nonlinear integrodifferential equations for photon transport. The second, simpler approach is to treat the radiation transfer as exchange between opaque gray bodies separated by a nonparticipating medium, so that:   r Tb4  Tw4     qr ¼ ð15Þ A w 1  eb 1 1  ew þ þ F Ab eb ew where T b and Tw are the bulk bed temperature and wall/ surface temperature, respectively. Unless the heat transfer surfaces are tightly packed within the fluidized bed or finned, the view factor ( F) approaches unity and Aw=A b. Emissivity of the heat transfer surface (e w) is a property of the surface. For most metallic surfaces operating at high temperatures, e w tends to be in the range of 0.7–1.0. The parameter of greatest uncertainty in Eq. (15) is the effective bed emissivity (e b). Ozkaynak and Chen [13] obtained measurements of effective bed emissivity which showed that e b also ranged from 0.7 to 1.0 in fluidized beds of sand. Thus, a simple engineering estimate of the radiative heat transfer coefficient can be obtained from:   r Tb4  Tw4 qr  : hr ¼ ¼ 1  eb 1  ew Tb  Tw ð Tb  T w Þ þ1þ eb ew ð16Þ

3. Circulating fluidized beds (CFBs) Circulating fluidized beds are used in a number of processes, most notably combustion and catalytic reactions. From a heat transfer point of view, the major interest is in

circulating fluidized bed combustors, where heat is usually extracted from membrane wall surfaces around the periphery of the reactor, sometimes supplemented by bubbling bed transfer in the return portion of the external circulation loop. Given the wear caused by particles moving rapidly upwards and downwards, CFB reactors seldom employ internal surfaces. Flow conditions in the reactor usually correspond to the fast fluidization flow regime. Hence the design problem of greatest interest is how to predict the heat transfer coefficient for membrane surfaces at the wall of a fast-fluidized bed. In almost all cases of practical importance, the membrane surface is disposed in such a manner that the tubes are oriented vertically. Heat transfer to the wall of a CFB riser can again be assumed to involve additive components due to conduction, convection, and radiation. Many workers have begun with a two-phase structure, somewhat similar to that described above for bubbling beds. However, the two phases are different than those in bubbling beds, with the flow at the wall dominated by streamers or clusters traveling mostly downward, interspersed with periods where there is upwards flow of a dilute suspension. Complementing similar experiments in the bubbling bed case referred to above, CFB experiments by Wu et al. [14] with small flush-mounted heat transfer surfaces have shown rapidly varying local instantaneous heat transfer coefficients, with the fluctuations corresponding to the arrival of streamers at the heat transfer surface. Hence the processes governing heat transfer are somewhat similar, with packets of particles traveling primarily vertically interspersed with periods of relatively dilute conditions. Complicating the situation, however, are several factors: (a)

Whereas the voidage distribution in bubbling beds is essentially binary, varying between a discreet dense phase voidage and nearly pure voids inside bubbles, the voidage distribution in fast fluidization tends to be continuous and widely dispersed.

Fig. 6. Heat transfer coefficients for large circulating fluidized bed combustors. Sources of data are fully specified in the original paper [15].

J.C. Chen et al. / Powder Technology 150 (2005) 123–132

(b)

Streamers in circulating fluidized beds can reverse directions, moving both upwards and downwards.

Boiler makers and other manufacturers of large CFB units typically base their estimates of heat transfer coefficient on experience from previous units, with adjustments to account for such factors as changes in mean particle diameter, suspension density, and membrane surface geometry. They can be said to use binternal correlationsQ based on confidential data. Published correlations, which are purely empirical, are not common for circulating fluidized beds. Instead, a number of semiempirical models have been proposed in the literature based on periodic renewal of particle clusters at the heat transfer surface. Analogous to Eq. (1), one treats the heat transfer as being composed of superimposed conduction/ convection and radiation, assumed to be additive. The experimental data available for developing hydrodynamic predictions and for testing heat transfer predictions for CFBs are mostly from small-scale vessels. The few cases reported for larger CFBs, with cross-sectional areas varying from 2.4 to 88 m2, have been summarized by Golriz and Grace [15]. Fig. 6 (from [15]) plots the heat transfer coefficients versus suspension density for these large-scale units. While the limited number of data from large units is a definite disadvantage, there is one significant advantage of larger scale equipment. This is that large risers are tall enough that both the flow and the heat transfer can be assumed to be fully developed [15], whereas in small scale vessels, both the flow and the heat transfer are always likely to be developing, meaning that heat transfer changes significantly with the vertical length of the heat transfer surface [16]. Golriz and Grace [15] devised a model for large units based on the assumptions of fully developed conditions and radially uniform clusters at the wall. Fig. 7 portrays this model in electrical network analogy form, with six resistances. At any instant, some portions of the surface are bare, while other portions of the surface are covered by clusters, each separated from the wall by a thin gap of thickness d g. Different heat transfer mechanisms are assumed for the bare and covered portions. For the bare

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sections, transfer is by gas convection (denoted by subscript bgcQ) and by radiation from the suspension to the wall (subscript bsrQ). The rest of the wall is covered by clusters/ streamers, providing a parallel transfer path. The transfer rate is then assumed to be controlled by a particle horizontal exchange flux, G sh, in parallel with radiation, these two being in series with transfer from the streamer/cluster to the wall by parallel conduction and radiation across a gas gap at the wall. The overall heat transfer is then:   htot ¼ hgc þ hsr ð1  f Þ þ

f : 1 1   þ Gsh cpp þ hrad kg =dg þ hrad ew be ð17Þ

The above quantities are then estimated as follows. A new correlation was given for the fractional coverage, f, accounting for the scale of the unit:     2 f ¼ 1  exp  25; 000 1  0:5D ð1  esus Þ e þ e0:5D ð18Þ where D is the riser equivalent diameter (4 cross-sectional area/perimeter) in metres. For large units, f approaches unity, meaning that the entire wall becomes covered by clusters. An alternative relationship giving somewhat lower values of f as D increases has recently been suggested by Dutta and Basu [17]. The gas convective transfer coefficient, h gc, was obtained from the well-known Dittus– Boelter correlation [18]. Radiation between the suspension and the bare wall was estimated from the parallel surface expression, similar to Eq. (16) above:   r Tb4  Tw4   hsr ¼ ð19Þ 1 1 ðTb  Tw Þ þ 1 esus ew where T b and Tw are the bulk and wall temperatures, while e w and e sus are the wall and suspension emissivities. The latter is estimated from the Brewster [19] correlation. The

Fig. 7. Golriz and Grace [15] model in network analogy form.

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two remaining radiation heat transfer coefficients are given by:    4  1 1 4 hrad ¼ 4r T  T þ  1 ðTe  Tw Þ ð20Þ ew e w esus ew

=

hrad be

  ¼ 2r Tb4  Te4

=



2 esus

  1 ðTb  Te Þ

ð21Þ

with Te being the emulsion temperature at the edge of the gas gap given by: Tb R56 þ Tw R34 Te ¼ R34 þ R56

R3 R4 with R34 ¼ R3 þ R4 R5 R6 and R56 ¼ R5 þ R6

ð22Þ

where the resistances R 3, R 4, R 5, and R 6 are all defined in Fig. 7. Eq. (21) allows for some radiation shielding by the intervening particles. The gas gap thickness is estimated [20] from: dg ¼ 0:0282dp ð1  esus Þ0:59

ð23Þ

where d p is the average particle diameter and e sus is the crosssectional average suspension void fraction. An expression for the lateral solids flux was obtained by fitting all heat transfer data for units of hydraulic diameter 1 m or larger where the suspension densities, q sus=q p(1e sus)+q ge sus, were greater than or equal to 5 kg/m3, leading to: Gsh ¼ 0:0225lnðqsus Þ þ 0:1093:

ð24Þ

As shown in Fig. 8, predictions of this method are in good agreement with available large-scale heat transfer data, with few points lying outside the F25% bands. This approach is rational and appears to provide the best quantitative means of estimating heat transfer coefficients

for design of large-scale CFB units. However, it is simplified in not considering such factors as the interaction between radiation and convection, the coupling of heat transfer resistance in the tube wall and on the steam side, reversing motion of clusters at the wall, and the highly three-dimensional nature of the membrane wall surface. For a more advanced model that accounts for these factors, see recent papers by Xie et al. [21,22].

4. Conclusion Design or scale-up of heat transfer systems in fluidized beds requires estimation of the effective heat transfer coefficient on tube or wall surfaces in contact with the fluidized medium. Manufacturers typically base their estimates on experience obtained in previous units, utilizing empirical correlations to adjust for changes in particle size and operating conditions. Such empirical correlations must be used with considerable caution. In this paper, it is suggested that mechanistic models based on the surface renewal concept hold promise for design and scale-up of heat transfer systems for both bubbling dense beds and fast circulating fluidized beds.

Nomenclature Ab Projected area of bulk fluidized media as seen by Aw, m2 Aw Area of heat transfer surface, m2 a Constant in Eq. (12), – Ar Archimedes number, (gd p3(q pq g)/q gm g2), – b Constant in Eq. (12), – c pg Specific heat of gas, J/kg K c pp Specific heat of particles, J/kg K D Riser equivalent diameter, m

Fig. 8. Heat transfer coefficients predicted by Golriz and Grace [15] model compared with experimental data. Dashed lines indicate F25% deviations. CUT indicates Chalmers University of Technology.

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Dt dp eb ep e sus ew F f

Diameter of heat transfer tube, m Mean particle diameter, m Effective bed emissivity, – Particle emissivity, – Suspension emissivity, – Wall emissivity, – Radiative view factor between Aw and A b, – Fraction of total time during which wall surface is covered by clusters, – Fo Fourier modulus (Eq. (8)), – fd Time fraction of contact by dense phase, – g Gravitational acceleration, m2/s G sh Lateral particle flux, kg/m2 s h rad Heat transfer coefficient for bulk-to-emulsion be radiative transfer, W/m2 K hc Convective heat transfer coefficient, W/m2K hd Heat transfer coefficient for bdenseQ (particle) phase contact, W/m2 K rad h ew Heat transfer coefficient for radiation transfer across gas gap, W/m2 K h gc Gas convective heat transfer coefficient for uncovered portions of wall, W/m2 K hl Heat transfer coefficient for bleanQ gas phase contact, W/m2 K hr Heat transfer coefficient for radiation, W/m2 K h sr Radiative heat transfer coefficient from bulk suspension to bare wall, W/m2 K h tot Total heat transfer coefficient, W/m2 K kg Gas thermal conductivity, W/m K kp Particle thermal conductivity, W/m K k Pa Thermal conductivity of particle packet, W/m K Nu p Nusselt number based on mean particle diameter, (h cd p/k g), – Nu t Nusselt number based on tube diameter, (h cD t/k g), – Nu Nusselt number based on riser equivalent diameter, (h cD/k g), – Pr g Prandtl number of gas, (c pgl g/k g), – qr Radiative heat flux, W/m2 R 1R 6 Heat transfer resistances defined in Fig. 7, m2 K/W Re D Reynolds number based on the riser equivalent diameter, (Dq gU/l g), – Re p Reynolds number based on the particle diameter, (d pq gU/l g), – Re t Reynolds number based on tube diameter (D tq gU/l g), – Tb Bulk suspension temperature, K Te Emulsion temperature, K Tw Wall temperature, K U Superficial gas velocity, m/s U mf Superficial gas velocity at minimum fluidization, m/s Greek a Pa dg e e mf

letters Thermal diffusivity of particle packets (Eq. (10)), m2/s Gas gap thickness, m Volume fraction gas (void fraction), – Void fraction at minimum fluidization, –

e Pa e sus ex lg mg qg qp q sus r s Pa n

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Void fraction in the emulsion packet, – Cross-sectional average voidage, – Void fraction at distance x from heat transfer surface, – Gas viscosity, kg/m s Gas kinematic viscosity, m2/s Gas density, kg/m3 Particle density, kg/m3 Cross-sectional average suspension density, kg/m3 Stefan–Boltzmann constant, 5.67 108 W/m2 K4 Root mean residence time of packet at heat transfer surface, s Penetration distance for conduction wave, m

Subscripts b Bulk be Bulk-to-emulsion c Convective cov Covered portion d Dense phase ew Emulsion-to-wall g Gas gc Gas convection l Lean phase mf Minimum fluidization p Particle Pa Packet r Radiation sr Suspension radiation sus Suspension t Tube tot Total w Wall or heat transfer surface x Distance from heat transfer surface into packet

References [1] J.C. Chen, Fluidization solids handling and processing, in: W.C. Yang (Ed.), Noyes Publ., 1999, pp. 153 – 208. [2] H.A. Vreedenberg, Chem. Eng. Sci. 9 (1958) 52 – 60. [3] V.A. Borodulya, A.P. Teplitsky, V.V. Sorokin, I.I. Markevich, A.F. Hassan, T.P. Yeryomenko, Int. J. Heat Mass Transfer 34 (1991) 47 – 53. [4] J.C. Chen, Max Jakob award lecture, J. Heat Transfer 125 (2003) 549 – 566. [5] H.S. Mickley, D.F. Fairbanks, AIChE J. 1 (3) (1955) 374 – 384. [6] T.F. Ozkaynak, J.C. Chen, AIChE J. 26 (4) (1980) 544 – 550. [7] R. Chandran, J.C. Chen, AIChE J. 29 (6) (1982) 907 – 913. [8] J. Kubie, J. Broughton, Int. J. Heat Mass Transfer 18 (1975) 289. [9] D. Kunii, J.M. Smith, AIChE J. 6 (1960) 71 – 78. [10] B.H. Xu, A.B. Yu, S.J. Chew, P. Zulli, Powder Technol. 109 (2000) 13 – 26. [11] S.C. Bhattacharya, D. Harrison, Proc. Euro. Conf. Particle Tech. K7 (1976) 12 – 16. [12] J.C. Chen, K.L. Chen, Chem. Eng. Commun. 9 (1981) 255 – 271. [13] T.F. Ozkaynak, J.C. Chen, Fluidization, in: D. Kunii, R. Toei (Eds.), Eng. Foundation Pub., 1983, pp. 371 – 378.

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[14] R.L. Wu, C.J. Lim, J.R. Grace, Can. J. Chem. Eng. 67 (1989) 301 – 307. [15] M.R. Golriz, J.R. Grace, in: J.R. Grace, J. Zhu, H. deLasa (Eds.), Circulating Fluidized Bed Technology VII, CSChE, Ottawa, 2002, pp. 121 – 128. [16] R.L. Wu, J.R. Grace, C.J. Lim, Chem. Eng. Sci. 45 (1990) 3389 – 3398. [17] A. Dutta, P. Basu, Int. fluid bed combustion conf., Jacksonville, Florida FBC 2003-042, 2003. [18] J.R. Welty, Engineering Heat Transfer, John Wiley and Sons Publ., 1974, p. 262. [19] M.Q. Brewster, Trans. ASME 108 (1986) 710 – 713. [20] T. Lints, L.R. Glicksman, AIChE Symp. Ser. 89 (296) (1993) 35 – 52.

[21] D. Xie, B.D. Bowen, J.R. Grace, C.J. Lim, Int. J. Heat Mass Transfer 46 (2003) 2179 – 2205. [22] D. Xie, B.D. Bowen, J.R. Grace, C.J. Lim, Chem. Eng. Sci. (2003) 4247 – 4258.

Further reading [1] H.A. Vreedenberg, Chem. Eng. Sci. 11 (1960) 274 – 285. [2] A. Baerg, J. Klasson, P.E. Gishler, Can. J. Res. (1950) 287. [3] L. Wender, G.T. Cooper, AIChE J. 4 (1958) 15 – 23.