Heat Transfer to a Single Sphere Immersed in ... - ACS Publications

to a gas from densely packed beds of monodisperse spherical particles. Arpit Singhal , Schalk Cloete , Stefan Radl , Rosa Quinta-Ferreira , Shahri...
0 downloads 0 Views 242KB Size
Download PDF
5632

Ind. Eng. Chem. Res. 2004, 43, 5632-5644

Heat Transfer to a Single Sphere Immersed in Beds of Particles Supplied by Gas at Rates above and below Minimum Fluidization Stuart A. Scott,* John F. Davidson, John S. Dennis, and Allan N. Hayhurst Department of Chemical Engineering, University of Cambridge, Pembroke Street, Cambridge, U.K. CB2 3RA

The coefficient of heat transfer to a mobile sphere in a fluidized bed of relatively large particles has been measured. Measurements were also made for U < Umf, where the solids are stationary. One feature of these experiments is that the heat-transfer sphere was of a size comparable to that of the bed particles. Two methods of measuring heat-transfer coefficients in the bed were employed: (i) The rate of cooling of a sphere, initially hot, was measured by means of a tiny inserted thermocouple with fine lead wires connected to a temperature recorder external to the bed, and (ii) a small number of spherical CO2 particles (dry ice) were put into an air-fluidized bed of inert particles. The subsequent concentration of CO2 in the off-gas from the bed provided a measure of the evaporation rate of the dry ice particles and, hence, of the heat-transfer rate to each dry ice particle. Good agreement between values of h from the two methods implies that the thermocouple lead wires of method (i) did not restrain the free motion of the heat-transfer sphere. The meaning of the heat-transfer coefficient is discussed with regard to its applicability to a transient experiment, and a criterion is suggested for determining when it is valid to derive a heat-transfer coefficientsa steady-state parametersfrom a transient experiment. At high Reynolds numbers (100 < Remfs < 830) and with small spheres in a bed of large particles (0.2 < ds/dp < 2.75), the dominant mechanism of heat transfer is found to be to the flowing gas. In this case, Nu ) 2 + 1.0Remfs0.6(ds/dp)0.26. 1. Introduction A major advantage of a gas-solid fluidized bed is the high rate of heat transfer, either to fixed surfaces or to particles reacting in the bed. There have been numerous studies of heat transfer to fixed surfaces in fluidized beds, reviewed, e.g., by Botterill1 and Zhu and Fan.2 Heat transfer to a freely moving sphere in a bed fluidized by gas has also been studied,3-6 although mainly for the case of a large mobile sphere in a bed of smaller particles. Typically, the ratio of the diameter of the heat-transfer sphere to that of the fluidized particles, ds/dp, has been greater than 3. Much less work has been done on heat transfer to a mobile sphere similar in size to, or smaller than, the fluidized particles. This regime has been examined experimentally7,8 using spheres with embedded thermocouples in beds of relatively large fluidized particles, covering the range 0.2 < ds/dp < 1.8. Baskakov et al.9 analyzed published data and, by interpolating between various regimes of heat transfer, suggested correlations for a wide range of ds/ dp values. Heat transfer to a mobile sphere in a bed of fluidized particles of comparable or larger size is the focus of this paper. Attention has also been paid to heat transfer to such a sphere when the superficial velocity is reduced below that for incipient fluidization, so the bed changes from being fluidized to being packed. Much of the initial work on heat transfer to a freely moving sphere in a fluidized bed arose from work on the combustion of carbon particles, especially coal. Here, the temperature of a burning particle is crucial in determining its rate of reaction. Tamarin et al.10 and Ross et al.11 measured * To whom correspondence should be addressed. Tel.: +44 (0) 1223 330134. Fax: +44 (0) 1223 334796. E-mail: [email protected].

the coefficient of heat transfer to particles of coke burning in a bubbling fluidized bed of sand. Tamarin et al.10 embedded a thin thermocouple in a particle of coke to record its temperature during combustion. Ross et al.11 photographed the surface of a fluidized bed and estimated the temperature of the burning particles from the color and intensity of the emitted light. In both of these studies, the heat-transfer coefficient was derived from the measured rate of combustion, together with the measured temperature of the particles. However, assumptions had to be made about the location of the combustion reactions, specifically whether the carbon burns to give CO2 directly at the surface of a particle or whether CO2 is formed from CO burning away from the surface. Others have embedded a thin, flexible thermocouple in an inert particle which was then either heated or cooled by immersion in a fluidized bed.3-7 The heat-transfer coefficient was then obtained from the dynamic temperature response of the mobile sphere. One concern with this type of experiment is whether the thermocouple lead impedes the motion of the sphere. Hayhurst and Parmar12 compared the burnout times of graphite spheres both with and without an embedded thermocouple; they found that the presence of the thermocouple did not alter the burnout time if the external diameter of the thermocouple was ∼0.20 mm or less. Rios and Gilbert13 overcame concerns about mobility by introducing a large plastic sphere into a fluidized bed and allowing it to circulate for a specified time before removing it and measuring its heat content in a calorimeter. The large sphere had a tendency to float on the surface of the bed. The heat-transfer coefficient increased when the superficial velocity was sufficient to cause the sphere to be dragged down into the bulk of the bed. In the present study, measurements of heat transfer from a mobile sphere were performed using a thermo-

10.1021/ie0307380 CCC: $27.50 © 2004 American Chemical Society Published on Web 07/15/2004

Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004 5633

couple embedded in the sphere; the thermocouple was small, and its lead wires were thin and flexible. In addition, heat-transfer coefficients were determined by measuring the rate of production of gaseous CO2 from subliming spheres of solid CO2 particles added to a fluidized bed. Embedded thermocouples were not needed, so the spheres had completely unrestricted movement within the fluidized bed. 2. Experimental Method and Analysis 2.1. Experiments with an Embedded Thermocouple. Phosphor bronze (92 vol % Cu, 7.7 vol % Sn, and 0.3 vol % P) and graphite spheres (2-6-mm o.d.) were each attached to a thin thermocouple, heated in an oven to ∼573 K, and subsequently dropped into a bed of particles fluidized by air at room temperature. The thermal conductivity, λs, of each sphere was high enough to ensure that the Biot number, Bi ) hds/λs, was much less than unity. Here, h is the heat-transfer coefficient from the surface of the sphere, and ds is its diameter; λs ) 2000 and 404 W m-1 K-1 at 300 K for graphite parallel to the basal plane14 and for phosphor bronze,5 respectively. The largest value of Bi was 0.004, ensuring negligible temperature gradients within a sphere. The graphite and phosphor bronze spheres differed significantly in density; the graphite spheres had a specific gravity of 1.8 (i.e., much less than the value of 8.7 for phosphor bronze) and so were closer in density to that of the emulsion phase (specific gravity ≈ 0.5-1.7) of the fluidized bed. The spheres of phosphor bronze were brazed to inconel-sheathed, K-type thermocouples (TC Ltd.; 0.25-mm sheath diameter); each sphere had been drilled to accept the sheath. Figure 1a shows one such sphere and its thermocouple leads after being withdrawn from a fluidized bed. Alternatively, heat-transfer spheres made from graphite were attached to either a sheathed K-type (0.25-mm o.d.) thermocouple or to a more flexible, unsheathed, thermocouple using AutoStic, a ceramic cement with a thermal conductivity of 0.65 W m-1 K-1. The unsheathed thermocouples were made from Alumel and Chromel (K-type) wires (0.1- and 0.2-mm o.d.) joined to give a thermocouple junction at the tip; one such sphere and its two thermocouple wires are shown in Figure 1b. In both images a and b of Figure 1, the thermocouple wires were almost tied in knots following immersion in the fluidized bed for the duration of an experiment, indicating that the wires were flexible enough not to affect the motion of the heat-transfer sphere. In an experiment, a single heat-transfer sphere was heated to ∼573 K in an oven and plunged into a fluidized bed at room temperature. The temperature, T, of the heat-transfer sphere was recorded as a function of time, t, using either a chart recorder or a computer. The bed was contained in a Perspex tube (either 90- or 140-mm i.d.). A layer of steel balls (3-mm o.d.), supported on a steel mesh, was used as the distributor. The height of the layer of balls was adjusted to give even fluidization; typically, the pressure drop across the distributor was 0.2-0.5 times that across the bed at incipient fluidization. Details of all of the fluidized particles and the experimental configurations used in this study are given in Table 1. The fluidized particles were of various materials with average sizes ranging from 0.78 to 9 mm, corresponding to 0.2 < ds/dp < 7.6. Relatively shallow beds (unfluidized depth of 70-100 mm) were used to avoid slugging. In some cases, large bubbles were

Figure 1. Photographs of heat-transfer spheres on a thermocouple. The lead wires were initially straight; here, they are shown following an experiment. (a) Sphere of phosphor bronze (diameter ) 2 mm) on its sheathed thermocouple (0.25-mm o.d.). (b) Sphere of graphite (diameter ) 6 mm) with its unsheathed thermocouple (wire diameter ) 0.1 and 0.2 mm).

formed periodically (a characteristic of slugging), but no well-defined slugs were observed. Large bubbles were most apparent with larger fluidized particles (dp > 2 mm) at U/Umf ≈ 1.5; higher values of U/Umf resulted in more vigorous mixing in the bed and disrupted the formation of big bubbles. Flow rates of air to the bed for dp > 0.78 mm were derived by measuring the gas velocity in the freeboard using a hot wire anemometer. In fact, the gas velocity was measured about 1 m above the center of the bed surface and reduced by a factor of 1.2 to allow for the fact that, with a turbulent velocity profile, the maximum velocity is 1.2 times the mean. The Reynolds number of the gas flow in the freeboard always exceeded 5000 at Umf. At lower flow rates, but still turbulent, the method gave flow rates within 5% of those measured from the rate of rise of a soap film in a vertical tube. For the smaller fluidized particles (dp e 0.78 mm), a rotameter which had been calibrated using the same anemometer in the freeboard was used. The measured value of Umf for the silica sand (+710, -850 µm) was within 5% of the value given by the correlation of Wen and Yu.15 Some experiments were conducted with the superficial velocity, U, less than that needed to fluidize the bed, Umf. To accomplish this, the hot sphere was first introduced into a bed with U > Umf, and then the gas velocity was immediately reduced to slump the bed at the required U < Umf. The valves metering the gas flow were such that this took ∼3 s, by which time the hot sphere had cooled somewhat. Care was taken to note on a plot of T against time, t, when the bed became

5634 Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004 Table 1. Properties of the Solids Fluidized, Including τp, the Time Constant for the Heating of a Fluidized Particle fluidized solids τp ) FpCpds2/λp (s)

Fp (kg m-3)

Umf at 290 K (m s-1)a

2

1650

0.85

3.2

1500

1.13

0.84

743

1.5

2600

0.85

1.3

743

0.24

0.71-0.85 9

2600 900

0.40 2.07

1.3 0.678

743 1170

0.88 125.8

dp (mm)

material alumina catalyst supportb silica catalyst support glass ballotinic silica sandc polystyrene spheres

bed

a

λp (W m-1 K-1)

Cp (J kg-1 K-1)

14.9

765

0.34 14.0

diameter (mm)

sphere

height, unfluidized (mm)

90

70

90

70

90

70

90, 140 140

70, 90 100

graphite phosphor bronze phosphor bronze graphite graphite graphite phosphor bronze graphite

Yu.15

ds (mm)

material

2, 4.6, 6 3, 5.5 2, 3 6 6 2, 4.6, 6 2 6

b

Calculated from the correlation of Wen and Values shown were in agreement with observations of the bed. Thermal conductivity values have been estimated by taking the value of solid alumina (∼36 W m-1 K-1) and multiplying by the particle porosity. c Values taken to be those of silica given by Perry and Green.14

slumped, so that only subsequent measurements were interpreted. For a sphere with Bi , 1, its temperature, T, after being plunged into a fluidized bed, is governed by

VsFsCps

dT ) -hAs(T - T∞) dt

(1)

Here, As and Vs are the surface area and volume of the sphere, respectively, and Fs and Cps are the density and the specific heat capacity of the material of the sphere, respectively. At the temperatures used (T < 573 K), radiation was negligible. The temperature of the bed, T∞, was assumed to be constant. The heat capacity of the phosphor bronze spheres was taken to be constant (382 J kg-1 K-1) over the temperature range of 293573 K used.5 Integrating eq 1 for a solid with constant heat capacity gives

ln

(

)

-hAs T - T∞ -6h -t ) t) t) To - T ∞ VsCpsFs dsCpsFs τsphere

(2)

When the chart recorder was used, the time constant, τsphere, was derived by reading from the trace the time at which the signal dropped to 1/e2 of the original value, i.e., at t ) 2τsphere. When the temperatures of the heattransfer sphere were logged by a computer, the heattransfer coefficient was found from the gradient of a plot of ln(T - T∞) against time, which, by eq 2, should be linear and have a gradient of -1/τsphere. The two techniques gave very similar results (typically to within better than 3%). For graphite, the heat capacity varies considerably from 293 to 573 K. Therefore, the molar heat capacity as a function of temperature given by Chase et al.16 was fitted to the expression

Cpms(T) ) Co + bT kmol-1

(3)

K-1,

where Co ) 6.727 × J b ) 3.207 J kmol-1 K-2, and T is in Kelvin. Equation 3 gives a maximum error of 0.1% over the relevant temperature range. Combining eqs 1 and 3 and integrating gives 103

Thus, it is possible to obtain the heat-transfer coefficient from the slope of a plot of the left-hand side of eq 4 against t. 2.2. Experiments with Spheres of Dry Ice. Spheres of solid carbon dioxide (initial diameter ≈ 5 mm) were prepared using a press. A batch of spheres (between 10 and 20 in number) was first weighed and then thrown into a bed of fluidized sand; the temperature of the bed (initially that of the room) was measured with a thermocouple. The sublimation of a sphere of dry ice is controlled by the coupled transport of heat to its surface and of gaseous CO2 away from the surface, whose temperature is 194.7 K17 for a vapor pressure of 101.32 kPa. The density of dry ice (1400-1500 kg m-3) is similar to the density of the emulsion phase of the fluidized bed. Of course, the motion of these spheres is not impeded by an attached thermocouple. The concentration of CO2 in the off-gas from the bed was monitored continuously during an experiment, by sampling into an infrared analyzer (ABB Easyline IR) and a mass spectrometer (Hiden Analytical, HPR-20). Plots of the measured concentration of CO2 ([CO2], kmol m-3) against time were deconvoluted12 to allow for mixing in the sampling system. In these experiments, the Perspex bed with a diameter of 90 mm was used; the Perspex tube was lined with aluminum foil to prevent the accumulation of static charges and to protect the Perspex from direct contact with the dry ice. The catalyst support particles (details are given in Table 1) were found to adsorb carbon dioxide; hence, the experiments with dry ice were limited to beds of sand. The rate of sublimation from a single sphere of dry ice, Q (in kmol s-1), can be calculated from the measured CO2 concentration in the off-gas, the cross-sectional area of the bed, Abed, the number of spheres of dry ice, N, and the superficial velocity, U, using

dsFms [bT + (Co + bT∞) ln(T - T∞)] ) 6 dsFms -ht + [bTo + (Co + bT∞) ln(To - T∞)] (4) 6

Q)

(

)

3 AbedU[CO2] d πds Fms )N dt 6

(5)

where Fms is the molar density of dry ice. Thus, the diameter, ds, of the subliming sphere at time t was calculated from the area under a plot of the concentration of CO2 against time, because eq 5 gives πFms(dso3 ds3)/6 ) ∫0tQ dt, where dso is the initial diameter of the sphere of solid CO2. Also, Q is related to the heattransfer coefficient, hsub, and the latent heat of sublima-

Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004 5635

tion, ∆H (26.1 MJ kmol-1,18 evaluated at Ts, by a heat balance on the subliming particle’s exterior

∆HQ ) πds2hsub(T∞ - Ts)

(6)

Hence, a plot of the measured CO2 concentration in the off-gas versus time can be converted to a plot of the heattransfer coefficient versus the diameter of the sphere. The temperature of the bed, T∞, was measured, but did not decrease by more than 2 K, so it was taken to be constant; this approximation gave errors below 1%. Although the heat-transfer coefficient, hsub, defined in eq 6 is consistent with the values reported in the general literature (e.g., Bird, Stewart, and Lightfoot19), it does differ from the heat-transfer coefficient, h, measured in an experiment with an inert sphere of the same size. This is because the enthalpy needed to heat the gaseous CO2 while it diffuses away from the sphere must be taken into account. The effect can be estimated by considering the film model of heat transfer, in which the main resistance to heat transfer is across a boundary layer of thickness δ and thermal conductivity λ. In the steady state, the flow rate of enthalpy across a spherical control surface at a radius r is independent of r and equals

dT dr

-4πr2λ

|

r

+ QCpCO2T ) -∆HQ + QCpCO2Ts (7)

where the left-hand side represents the value at radius r and the right-hand side gives the value at the surface of the sphere. In eq 7, the reference enthalpies do not appear because they cancel. Integrating eq 7 between r ) ds/2 and r ) (ds/2 + δ) gives

(

ln 1 +

)

(

)

QCpCO2 1 T∞ - Ts 1 ) ∆H/CpCO2 4πλ ds/2 ds/2 + δ

(8)

Using eq 6 to eliminate Q from eq 8 and rearranging yields hsub )

{ ( )} [ ds λ 1+ 2δ (ds/2)

×

(

∆H/CpCO2 T∞ - Ts ln 1 + ∆H/CpCO2 (T∞ - Ts)

)]

(9)

where the first term in brackets represents h and the second term is a correction factor. Thus, the coefficient of heat transfer to the subliming sphere, hsub, can be written as the coefficient of heat transfer to an inert sphere, h (first term in eq 9, Paterson and Hayhurst20), multiplied by a correction factor that accounts for the flux of CO2 away from the subliming sphere of dry ice. In this study, the correction factor in eq 9, 0.933, was used to convert the heat-transfer coefficients measured for the sublimation of a dry ice sphere to that of an equivalent inert sphere. 3. Results 3.1. Experiments with an Embedded Thermocouple. Figure 2a shows plots of the temperature against time for a graphite sphere (ds ) 6 mm) cooling in a bed of alumina catalyst support pellets (dp ) 2 mm); the results shown are for 0 < U/Umf < 1.0. Figure 2b shows the plots derived from eq 4; the slope should be -h, so for a constant h, this plot should be a straight line. For U > Umf, all of the plots similar to Figure 2b were linear, but for U < 0.5Umf (although the exact value varied between beds), there was some curvature,

Figure 2. Plots of (a) the temperature and (b) the left-hand side of eq 4 versus time during experiments in which a graphite sphere (diameter ) 6 mm) was cooled in a packed bed of alumina catalyst pellets (dp ) 2 mm). Values of U are given.

as shown in Figure 2b. In such cases, a mean slope was taken, although h decreased with time. It is shown in section 4.3 that the curvature can be attributed to the effects of the unsteady-state conduction of heat from the hot sphere to the unfluidized particles. The mobility of a heat-transfer sphere within the fluidized bed is likely to depend on (i) the type of thermocouple lead, (ii) the value of U/Umf, and (iii) the sizes and densities of both the heat-transfer sphere and the fluidized particles. The phosphor bronze spheres (specific gravity ) 8.7) sank in beds of solids with a particle diameter of less than 2 mm, but were observed to move freely in the beds of 2- and 3.2-mm catalyst support particles and 9-mm polystyrene spheres. Figure 1a shows that a bronze sphere (diameter ) 2 mm) on a sheathed thermocouple (o.d. ) 0.25 mm) moved freely around a bed of silica catalyst support particles (diameter ) 3.2 mm) at U/Umf ≈ 3. Similarly, a graphite sphere (specific gravity ) 1.8) was not constrained by the unsheathed thermocouple shown in Figure 1b.

5636 Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004

Figure 3. Measured heat-transfer coefficients for graphite spheres with diameters of (a) 6, (b) 4.6, and (c) 2 mm in a bed of sand (+710, -850 µm) at 293 K for different values of U/Umf.

These graphite spheres did not sink, but moved freely around the bed. It is thus concluded that the measurements of h are unaffected by the presence of the thermocouple. This conclusion is confirmed below by good agreement between measurements of h using thermocouples and measurements with spheres of dry ice. Figures 3 and 4 are plots of h against U/Umf and illustrate two limiting types of behavior. Typically, each experiment was repeated 5 times with U greater than Umf; the error bars shown on these figures indicate the 95% confidence intervals for the mean heat-transfer coefficient. Figure 3 shows plots of the heat-transfer coefficients for graphite spheres of various diameters in a bed of silica sand (+710, -850 µm) fluidized by air. For the largest sphere (diameter ) 6 mm, Figure 3a), there is a discontinuity at U ≈ Umf due, no doubt, to the onset of particle convection when the sand became fluidized. With a smaller graphite sphere (diameter ) 4.6 mm, Figure 3b), this discontinuity is less apparent. However, for the smallest graphite sphere (diameter ) 2 mm, Figure 3c), the measured values of h show no discontinuity at U ≈ Umf. Such behavior suggests that heat transfer by convection of moving sand particles becomes less important as ds/dp is reduced. For Figure 3a, with the 6-mm graphite sphere, both the sheathed and unsheathed thermocouples were used. The difference in measured h values was noticeable only near U ) Umf; for U > 1.5Umf, the difference was within the variability of the results. Near U ) Umf, the difference was probably due to a slight maldistribution of the fluidizing air. Both the Reynolds number of the gas flow around the sphere and ds/dp are likely to determine whether a discontinuity occurs near U ) Umf. Discontinuities were

Figure 4. Measured heat-transfer coefficients for various spheres to a bed of alumina particles (dp ) 2 mm) fluidized by air at 293 K. (a) Phosphor bronze: ds ) (O) 3 and (4) 5.5 mm. (b) Graphite: ds ) (2) 2, (b) 4.6, and (×) 6 mm.

seen consistently in this study only when ds/dp > 3, as in Figure 3a and b. In Figure 4, for alumina catalyst support particles (dp ) 2 mm), there is no obvious discontinuity in the magnitude of the heat-transfer coefficient at U/Umf ) 1. Also, given the variability in the experimental results, it is difficult to identify a maximum in these plots of h versus U/Umf for U/Umf > 1 in Figure 4. This behavior was seen in all of the experiments, excluding those in Figures 3a and 3b and experiments with a large sphere of graphite (diameter ) 6 mm) in glass ballotini (dp ) 1.5 mm), which showed a small discontinuity. 3.2. Heat Transfer to Subliming Spheres of Dry Ice. Figure 5 shows the results from experiments using a batch of spherical pellets of solid carbon dioxide added to a bed of silica sand (+710, -850 µm), fluidized by air at 293 K for different values of U/Umf. The values of h were calculated from hsub using eq 9. For these experiments, the total yield of CO2, obtained by integrating the measured plot of the CO2 concentration against time, was within 8% of the amount added to the fluidized bed. Once the spheres of dry ice had shrunk to ∼0.5 mm, the error in the estimated h values became very large; hence, only results for ds > 1 mm are reported. The spheres of solid CO2 first sank into the bed and were then seen to circulate, occasionally coming to the surface before being dragged down into the bed. The velocity of gaseous CO2 leaving the surface of each sphere of dry ice was calculated to be ∼0.01 m s-1; this is only 2.5% of Umf (given in Table 1) and, hence, is unlikely to affect the flow patterns around the heattransfer spheres significantly. These results can be compared directly with those in Figure 3 for spheres of

Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004 5637

Figure 5. Coefficient of heat transfer to spheres of dry ice (initial diameter ≈ 5 mm) subliming in a bed of silica sand (+710, -850 µm) fluidized by air at 293 K: U/Umf ) (b) 1.5, (2) 1.8, (]) 2 , (solid line) 2.3. Values of h from Figure 3 are given for graphite spheres, each with an embedded thermocouple (o.d. ) 2, 4.6, and 6 mm) in the same bed at U/Umf ) 2.1.

Figure 6. Coefficient of heat transfer to spheres of dry ice (initial diameter ≈ 5 mm) subliming in a bed of silica sand (+500, -600 µm) fluidized by air at 293 K: U/Umf ) (0) 1, (dashed line) 1.4, (solid line) 1.59, (solid gray line) 1.9, (∆) 2.35, and (*) 2.8.

similar diameters but containing an embedded thermocouple. Figures 3 and 5 show that, for both types of experiment, the heat-transfer coefficient reaches a plateau for U > 1.7Umf. For comparison, Figure 5 shows values of h taken from Figure 3, for spheres of graphite, each with an embedded thermocouple at U ) 2.1Umf: the heat-transfer coefficient at U ≈ 2.1Umf increases from ∼300 W m-2 K-1 for a 6-mm sphere to ∼375 W m-2 K-1 for a 2-mm sphere. This is comparable to the increase from 250 W m-2 K-1 for 5-mm dry ice spheres to ∼350 W m-2 K-1 for spheres of dry ice of diameter 2 mm. The differences in the absolute values of the heattransfer coefficient between the two types of experiment (at worst ∼20%) are within experimental error. This confirms that the embedded thermocouples did not impede the motion of the sphere. The results in Figure 5 provide some evidence that attrition of the spheres of dry ice was negligible: (i) Attrition would give a greater value of h from the experiments with dry ice than from the experiments with inert mobile spheres. (ii) For values of U near Umf, attrition must be negligible, because the bed is nearly motionless. Attrition should give h increasing with

U/Umf. The fact that the values of h in Figure 5 deduced from the dry ice experiments are almost independent of U/Umf indicates that attrition is negligible. Figure 6 shows heat-transfer coefficients measured in experiments with batches of dry ice spheres in a bed of smaller sand (+500, -600 µm) fluidized by air at 293 K. In this case, the heat-transfer coefficient remained fairly constant throughout sublimation, but increased monotonically with U/Umf over the range studied. It is shown below that, if heat transfer is dominated by convection of the sand particles, h is only a weak function of the diameter of the heat-transfer sphere. This is apparent from Figure 6: during the sublimation of CO2, ds decreases and with the small sand particles used, ds/dp varies in the range 2 < ds/dp < 8, but h is almost constant while the CO2 spheres shrink. This indicates that heat transfer is dominated by the convection of sand5 for the small value of Remfs in these experiments. 4. Discussion 4.1. Transient Heat Conduction and the Concept of a Heat-Transfer Coefficient. For unsteady-state

5638 Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004

experiments in which a hot sphere is immersed in a cold medium, it is important to determine when it is meaningful to assign a heat-transfer coefficient to the cooling. A heat-transfer coefficient is usually, and ideally, measured in a steady-state experiment. For example, Gabor21 and Denton22 measured the rate of heat transfer from heated objects (cylinders and spheres, respectively) to a packed bed when the object and surrounding bed reached a steady state; the rate of heat loss from the immersed object was then equal to the rate of energy supplied. Another important feature of the steady state is that the temperature profile around the hot body does not change with time; thus, the total enthalpy flow at a radius of r is independent of r. A temperature profile fulfilling these criteria is henceforth referred to as a “steady-state” profile. In a transient experiment, it is, strictly speaking, impossible to form a steady-state temperature profile; however, it is possible to find conditions giving a profile closely approximating that of the steady state. Only under these conditions, when there is a “pseudo-steady-state” temperature profile, does h have its usual meaning. This section explores the conditions required to give a pseudo-steady state for a sphere cooling in a bed of fluidized particles. The effect of transient heat transfer on the cooling of a sphere immersed in the emulsion phase of a fluidized bed is best understood by analogy with the case of a sphere cooling in a stagnant fluid of thermal conductivity λ, heat capacity C, and density F. Conduction in the fluid is governed by the equation

[ ( )]

∂T λ ∂ 2∂T ) 2 r ∂t r ∂r ∂r

FC

(10)

where r is the radial coordinate and T(r,t) is the temperature of the fluid. The boundary conditions are

T f T∞ as r f ∞

(11a)

and for a heat-transfer sphere (radius ) a) with uniform internal temperature Ts

dTs 3λ ∂T ) dt FsCpsa ∂r r)a

( )

(11b)

Boundary condition 11b assumes that the sphere and adjacent fluid at r ) a are at the same temperature and ensures that the sphere cools at a rate determined by the enthalpy flow away from the surface of the sphere. Here, the subscript s refers to the material of the sphere. The initial conditions are that (i) T(a < r e ∞) ) T∞ and (ii) the temperature of the fluid at the surface of the sphere, T(r ) a), is equal to the initial temperature of the sphere, Tso. It is possible to assign a heat-transfer coefficient, h, to describe such a cooling of the sphere; however, the value of h will be independent of t only if the surrounding fluid can quickly establish a temperature profile which approximates to the steady-state temperature profile faster than any changes in the temperature of the sphere. This would be the case if, for example, the thermal capacity of the heat-transfer sphere were much higher than that of the surrounding fluid. In this case, eq 10 can readily be solved, provided that the heattransfer sphere has a constant temperature, Ts, giving

{ [( ) ]}

T - T∞ a ) 1 - erf Ts - T ∞ r

r -1 a 4λt 1/2 a2FC

(12)

The resulting time constant for approach to the steadystate temperature profile is then τfluid ) a2FC/4λ. This can be compared with the time constant,τsphere, for cooling the heat-transfer sphere in a stagnant medium, which from eq 2 with Nu ) 2 and therefore h ) λ/a, is τsphere ) a2FsCps/3λ. Here, Fs and Cps are now the density and specific heat capacity, respectively, of the heattransfer sphere. By assuming Nu ) 2, an implicit assumption is made that the temperature profile in the surrounding medium adjusts quickly to that of a steadystate condition with a fixed sphere temperature. The ratio of these two time constants

Γ)

τsphere FsCps ∼ τfluid FC

(13)

is a measure of the relative thermal capacities of the surrounding medium and of the heat-transfer sphere. When Γ . 1, a steady-state temperature profile will be established more quickly in the surrounding fluid than the temperature of the heat-transfer sphere will decrease. Only then is it, strictly speaking, valid to define a conventional heat-transfer coefficient. In the experiments discussed here, Γ, as defined by eq 13, is approximately equal to 1. The argument so far has assumed that the particulate phase of a fluidized or packed bed behaves as a stagnant continuum. Hence, the thermal boundary layer extends from the surface of the heat-transfer sphere to infinity. This is a gross simplification, although many models of heat transfer in a fluidized bed do treat the emulsion phase as a continuum.2 When there is a thermal boundary layer of finite thickness, eq 13 must be modified. For a pseudo-steady temperature profile penetrating a distance δ into the medium, the time constant for changes in the temperature profile within δ is τfluid ≈ δ2FC/λ. Also, eq 2 gives τsphere ) aFsCps/3h, so the ratio of the two time constants, Γ, then becomes Γ ) τsphere/τfluid ∼ aFsCpsλ/CFhδ2. Now, F and C should be interpreted as the effective density and heat capacity, respectively, of the fluid in the “boundary layer” surrounding the sphere. If the film model is used to evaluate h, the Nusselt number is given by20

a 2ha )21+ λ δ

(14)

τsphere FsCps 1 ∼ τfluid FC (δ2/a2 + δ/a)

(15)

Nu )

(

)

so that Γ becomes

Γ)

If the thickness of the boundary layer is much smaller than the radius of the heat-transfer sphere (δ/a , 1), which, for example, would be the case for turbulent flow around a sphere, then eq 15 indicates Γ . 1. It is then valid to define a heat-transfer coefficient. The presence of a thin thermal boundary layer implies that the temperature profile is confined to a small region near the sphere. This could occur in a fluidized bed if the particles in the emulsion phase convected heat away from the heat-transfer sphere sufficiently quickly to

Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004 5639

Figure 7. Plots of (a) the temperature and (b) the left-hand side of eq 4 versus time (plot b is linear for h ) constant) for a graphite sphere (ds ) 6 mm) cooling in a bed of alumina (dp ) 2 mm). Initial sphere temperature ) 343 K; bed temperature ) 293 K. Measurements with U ) 0, solid line. The circles are the results of theoretical calculations for unsteady-state heat conduction in a medium of effective thermal conductivity of 0.103 W m-1 K-1.

ensure that the temperature profile did not penetrate far into the emulsion phase. A second possibility is that all of the resistance to heat transfer is due to a gaseous boundary layer around the sphere. This might occur in the systems examined in this paper either because (i) the velocity of gas is locally high or (ii) the heat-transfer sphere is relatively small, so that only a few fluidized particles touch it. 4.2. Solution of the Unsteady-State Conduction Equation: Comparison of Theory and Experiment for U < Umf. The transient heat conduction eq 10, together with the boundary conditions (eqs 11a and b), which allow for the changing temperature of the heattransfer sphere, were solved using the method of finite volumes. Figure 7 shows (a) the measured temperature of the heat-transfer sphere, along with the computed values (using eqs 10 and 11) for an experiment in which a hot graphite sphere (ds ) 6 mm) was immersed in a bed of alumina particles (dp ) 2 mm) initially at room temperature, with U ) 0, and (b) the equivalent plots of the left-hand side of eq 4 against t (as in Figure 2b). The effective thermal conductivity of the surrounding packed bed, λ ) 0.103 W m-1 K-1, was determined by fitting the theoretical solution to the experimental curve using a nonlinear least-squares method. For a constant value of h, eq 4 predicts that a plot with the coordinates of Figure 7b should be linear. However, both experimental and theoretical predictions (eqs 10 and 11) deviate from linearity. The heat-transfer coefficient (indicated by the gradient of the plot in Figure 7b) decreases while the heat-transfer sphere cools, indicating that transient effects are important. Figure 8 shows the radial temperature profiles calculated for both pseudo-steady and transient heat conduction during the cooling of the 6-mm graphite sphere with the value of λ deduced from Figure 7b. Figure 8a, which shows the temperature profiles in the surrounding fluid given by steady-state conduction, differs significantly from Figure 8b, showing the temperature profile given by unsteady-state conduction. The thermal capacity of the sphere is insufficient to allow a

Figure 8. Computed radial temperature profiles around a graphite sphere (ds ) 6 mm), initially at 343 K, cooling in a medium of effective thermal conductivity of 0.103 W m-1 K-1 and bulk temperature 293 K: (a) pseudo-steady-state conduction; (b) transient heat conduction, eqs 10 and 11.

steady-state temperature profile to form without changing the temperature of the sphere. When the effects of transient heat conduction are taken into account, the material around the sphere is initially cold and acts as a heat sink. Thus, for transient conduction, the temperature gradient at the surface of the sphere is greater than for steady-state conduction. Therefore, the nominal heat-transfer coefficient will, initially, be higher than expected and will decrease while the surrounding material heats up. An important feature of Figure 8 is that, during the first 50 s of cooling, the depth of penetration of the unsteady temperature profile into the “fluid” is ∼5 mm, i.e., less than 3 times the diameter (2 mm) of a single fluidized particle. Consequently, the deviation from the theory of unsteady conduction shown in Figure 7 is likely to be due to the breakdown of the continuum approximation for the particulate phase. Nevertheless, these calculations show that the transient nature of heat conduction dominated the experiment reported in Figure 7. These calculations also cast doubt on the validity of using a constant heat-transfer coefficient for transient experiments with U < Umf. Indeed, this explains why the plot in Figure 7b is not linear; the measurements would lie on a straight line only if h were constant. The unsteady nature of the experiments in this study makes them unsuitable for quantitative evaluation of the thermal conductivity of a packed bed. Yagi and Kunii23 showed that the effective thermal conductivity of a packed bed is

λ(U) ) λ(U ) 0) + ξFgCpgdpU

(16)

where Fg and Cpg are the density and heat capacity, respectively, of the fluidizing gas and ξ is a constant that includes both a shape factor and a correction for the proportion of fluid flowing in the direction of heat transport. From eq 16, the heat-transfer coefficient should vary linearly with U, as in Figure 4, for U < Umf. However, it would be dangerous to calculate an effective thermal conductivity of the bed from our experiments;

5640 Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004

Figure 9. Typical dependence of the heat transfer to fixed surfaces and mobiles spheres in fluidized beds of fine particles.

unsteady-state simulations of experiments conducted in this work have shown that the derived values of thermal conductivity vary by a factor of 2 depending on whether transient or steady-state temperature profiles were used in the simulations. 4.3. Effect of Transient Heat Conduction on the Present Results. The curvature of the plots in Figure 2b, used to determine the heat-transfer coefficient, was found to be largest at low gas flow rates and U < Umf. For both U > Umf and packed beds of large particles with U greater than ∼0.5 Umf (although this depended on the solid being fluidized), the results gave linear plots. Such behavior can be seen in Figure 2b: the fact that the plots became linear as U increased indicates that a near-steady-state temperature profile is quickly established in a small region near the surface of the heat-transfer sphere; the effects of transient conduction are then negligible. When U/Umf . 1, the linearity could be due to the rapid renewal of particles at the surface of the heat-transfer sphere, or alternatively, it could be caused by a boundary layer of gas near the exterior of the heat-transfer sphere. In a fixed bed, there is no renewal of particles at the surface of the sphere; this suggests that the transition from experiments showing nonlinearity (in Figure 2b) at low flow rates, through to linearity nearer U ) Umf and for U > Umf, is likely to be due to the formation of a thin boundary layer of gas at the surface of the heat-transfer sphere. 4.4. Heat-Transfer Mechanisms in a Fluidized Bed. In general, a plot of the heat-transfer coefficient against the superficial velocity for a fixed surface in a fluidized bed of fine particles has the qualitative shape shown by curve a in Figure 9.1 The heat-transfer coefficient is small for a packed bed, but increases sharply with fluidizing velocity when U becomes larger than Umf. This sharp increase is caused by the onset of particle convection, i.e., heat is removed from the stationary surface by packets of fluidized particles coming to the surface, where they heat up before being displaced by other such packets. When the superficial velocity is increased further, a maximum is often seen. This is considered to be caused by the heat-transfer surface being covered less by particles and progressively more by bubbles, whose volume fraction increases with U.6 Similar curves have been obtained for a mobile sphere in a fluidized bed of small particles6 with 3 < ds/dp < 200. In contrast, most of the experiments

presented in this paper, for 0.2 < ds/dp < 3, did not show a discontinuity at U ) Umf and exhibited no obvious maximum for U > Umf. The similarity of the shapes of the curves for fixed surfaces and some mobile spheres has prompted workers to extend the mechanistic models of heat transfer to a fixed surface to the case of a mobile sphere.3-5 This approach has been most successful for large, mobile spheres in a fluidized bed of finer particles with 3 < ds/dp < 200. Others have correlated the Nusselt number with the Reynolds number, both being based on the properties of the fluidizing gas.6,9 Correlating Nu with Re neglects the properties of the fluidized particles and hence is valid only when applied to beds that have appropriate similarity or when heat transfer to the gas dominates. The prevailing view in the literature is that heat transfer to a surface in a fluidized bed can be rationalized by separating the observed heat-transfer coefficient, h, into the contributions from gas convection, hg, and from particle convection, hp,3,5,24 so that h ) hp + hg. Mickley and Fairbanks25 suggested that the high rates of heat transfer in a fluidized bed are caused by continual replacement of “packets” of particles at the heat-transfer surface with fresh particles from the bulk particulate phase. By treating the particulate phase as a continuum, they were able to use the theory of unsteady-state conduction to derive an expression for hp to a planar surface. Their analysis gave a heattransfer coefficient which is a function of only one parameter, namely, tc, the time a packet of emulsion phase is in contact with the surface. The analogous result for a fixed sphere immersed in the emulsion phase is found by considering the transport of heat from a sphere at constant temperature to the surrounding medium which is initially at the temperature of the bulk of the bed. The surrounding material heats up for a time, tc, the contact time, when the medium, now hot, is replaced by a fresh batch of particles from the bulk of the bed. Between each “surface renewal”, the temperature profile in the emulsion phase is given by eq 12. Differentiating eq 12 and then integrating from t ) 0 to t ) tc gives the total heat loss, from which the heattransfer coefficient averaged over the time tc is found to be

hp )

x

λem +2 a

FemCpemλem πtc

(17)

Here, Fem, Cpem, and λem are the effective density, specific heat capacity, and thermal conductivity, respectively, of the emulsion phase; λem is often calculated using Yagi and Kunii’s23 expression, eq 16. The first term in eq 17, which represents the heat transfer coefficient at steadystate to a sphere in a stagnant medium, is often neglected in mechanistic models of heat transfer to freely moving spheres. The particle convective heattransfer coefficient, hp, often includes a gas film resistance in series with Mickley and Fairbanks’ surface renewal model25 to take into account the higher voidage fraction of the bed near the heat-transfer surface.4,24 Despite the difficulty in evaluating tc in eq 17, these models have been successfully applied to a large sphere in a fluidized bed of much smaller particles.3-5 There are some limitations to the particle convection model when applied to the transient experiments of this study: (i) The constant-temperature boundary condition used to derive the governing equations (eqs 12 and 17)

Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004 5641

is valid only if the time constant for the sphere cooling is longer than the time constant for the renewal of particles. This is the case if the penetration depth of the temperature profile, δ, is small, i.e., Γ . 1. (ii) The continuum approximation, when applied to the emulsion phase of the bed, breaks down if the time constant for heating a single bed particle, τp, is much greater than the contact time, i.e., τp . tc. Then the temperature profile does not extend appreciably beyond the single layer of fluidized particles adjacent to the heat-transfer sphere. This last point might explain the lack of discontinuity in the plots of h versus U/Umf (e.g., in Figure 4) near Umf for the smaller values of ds/dp. If τp . tc, or τp is longer than the duration of the transient experiment, the particles near the heat-transfer sphere will have a constant temperature irrespective of the rate of refreshment of material at the surface. Table 1 includes the time constant for heating each fluidized particle. For the larger particles, such as silica and polystyrene, τp is large, and in the case of polystyrene spheres, τp (∼126 s) exceeds the duration of the experiment. However, for the alumina particles, the glass ballotini, and the sand, τp is low ( Umf and where h is approximately constant. Richardson and Collier,7 noting that there was no discontinuity on either side of U ) Umf, suggested that the Nusselt number could be related to the Reynolds number based on U for 0.5 < U/Umf < 1.3. Their correlation showed scatter, but the implicit assumption that the dominant heat-transfer resistance is essentially the same in the packed bed when U is near Umf and when the bed becomes fluidized was insightful. This is consistent with heat transfer being dominated by a thin boundary layer of gas. A careful analysis of the measurements of Collier et al.8 indicated that Nu ) 2 + χRemfsR, or

ln(Nu - 2) ) ln(χ) + R ln(Remfs)

(18)

In particular, it is clear that, for ds/dp e 1, their values of Nu for U e Umf extrapolate to Nu ) 2 for Re f 0. Such a value of Nu ) 2 when Re ) 0 has been questioned, e.g., by Baskakov et al.,9 who concluded

Nu ) 2 + 8(ds/dp) when Re tends to zero for ds e dp. Such an expression gives Nu ) 2 when (ds/dp) f 0, but Nu ) 10 for ds ) dp. This value of Nu ) 10 arose from a simple geometrical argument by Zabrodsky27 that there is an upper limit to δ, the effective thickness of the gaseous boundary layer, determined by the proximity of the surrounding particles. For low Reynolds numbers, δ, the thickness of the boundary layer becomes as large as dp; conduction of heat through the particles then becomes important, so the Nu value based on the conductivity of the gas can never be as low as 2. However, Collier et al.8 made measurements when the interstitial gas was turbulent for U e Umf. It might be that the limiting film thickness, which gives rise to Nu f 10 for Re f 0 for ds ) dp, manifests itself only when eq 18 breaks down. This could occur if Re were reduced low enough for there to be laminar flow around the heat-transfer sphere, so that its thermal boundary layer would become thicker than some limiting value of film thickness. In fact, Collier et al.8 gave

Nu ) 2 + 0.90Res0.62(ds/dp)0.2

(19)

for turbulent flow around the heat-transfer sphere. When the minimization of least squares was used to fit our experimental measurements to eq 18, the residuals (i.e., actual minus predicted values) were found to correlate linearly with ln(ds/dp). It is noteworthy that there was no correlation between the residual errors and the thermal diffusivity of the fluidized particles; such a correlation would be expected if particle convection were dominating heat transfer. The linear correlation between the residual errors and ln(ds/dp) suggests that eq 18 should be modified to include the ratio of the particle diameters; thus, Nu ) 2 + χRemfsR(ds/dp)β and a leastsquares fit of the values of the Nusselt number gave

Nu ) 2 + 1.0Remfs0.6(ds/dp)0.26

(20)

Any results from this study with a discontinuity in the plots of h against U/Umf were not included in the correlation of Nu with Remfs leading to eq 20; this is

5642 Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004 Table 2. Comparison of Nu (for U > Umf) Predicted According to Eq 20, with the Experimentally Determined Value of Nu,a and Thickness of the Gaseous Boundary Layer Estimated Using Eq 14 dp (mm)

ds (mm)

Umf (m/s)

ds/dp

Remf (bed)

Remfs (sphere)

Nu

predicted Nu (eq 20)

error (%)

δ (mm) (eq 14)

0.78 9 9 3.24 3.24 3.24 2 2 2 2 2

2 2 6 3 2 6 3 5.5 2 4.6 6

0.40 2.07 2.07 1.13 1.13 1.13 0.85 0.85 0.85 0.85 0.85

2.56 0.22 0.67 0.93 0.62 1.85 1.50 2.75 1.00 2.30 3.00

24.1 1250 1250 246 246 246 114 114 114 114 114

53.7 278 834 228 152 455 171 314 114 262 342

28 19 48 27 18 46 30 52 19 37 45

16 22 52 27 20 48 26 43 19 37 46

-43 +11 +9 +2 +10 +5 -14 -18 -2 -1 +2

0.08 0.11 0.13 0.12 0.12 0.14 0.11 0.11 0.11 0.13 0.14

2 3 5.5 2 3 5.5

1.07 1.07 1.07 0.7 0.7 0.7

Results from Richardson and Collier7,8 0.20 717 143 16 0.30 717 215 26 0.55 717 394 40 0.67 148 98.7 14 1.00 148 148 22 1.83 148 271 32

15 20 33 16 22 36

-8 -23 -18 +14 0 +11

0.14 0.13 0.14 0.17 0.15 0.18

10 10 10 3 3 3

a Experiments showing a significant jump in heat-transfer coefficient when the fluidizing velocity was increased beyond U mf are not included.

because heat transfer in these experiments (e.g., Figure 3a) will be due, in part, to the convection of fluidized particles. Table 2 compares the measured values of the average Nusselt number with those given by eq 20. The first entry in Table 2 was not included in the correlation because it appeared to be an outlier: otherwise, the error was typically less than 20%. The correlation coefficient, R2, was found to be 0.89. The similarity between eq 19,8 derived using measurements of h when U e Umf, and eq 20, derived from measurements with U g Umf, is remarkable. Equation 20 has a form similar to the expression for heat transfer from a heated sphere at steady state to a packed bed of spheres of equal size at high Reynolds numbers, given by Denton22 as

Nu ) 0.72PrairRe0.7

(21)

Assuming that the Reynolds number used by Denton22 was defined in the same way as Remfs, eq 21 gives values of Nu within 30% of those predicted by eq 20 over the range of conditions considered here. Because Denton22 subtracted the heat flow at U ) 0, there might be an argument for adding 2 to eq 21. This would reduce the difference between eq 21 and eq 20 to within 20%. Thus, the majority of the experiments presented in this study with relatively small values of ds/dp have been shown to be dominated by a thin boundary layer of gas around a sphere immersed in a fluidized bed. The effect of the surrounding particulate phase is two-fold: first, the presence of the fluidized particles increases the local gas velocity experienced by the heat-transfer sphere; second, contacts with the fluidized particles disrupt the gaseous boundary layer. For heat transfer to be dominated by such a gaseous boundary layer, its thickness should be smaller than, or comparable to, the diameters of the surrounding fluidized particles. An estimate of the thickness of the gaseous boundary layer, δ, calculated using eq 14 is given in Table 2; in all cases, this gas film thickness is much less than the diameter of the fluidized particles. Furthermore, the number of contacts between the fluidized particles and the heat-transfer sphere should also be small. The discontinuity in the plots of h versus U/Umf (e.g., as in Figure 3a) has been associated with

the onset of particle convection at U ) Umf. Insufficient experiments were performed to identify precisely the region in which the resistance of the gas film dominates heat transfer; however, the results indicate that, for 53 < Remfs < 830 and 0.2 < ds/dp < 2.75, the gas film will be the main resistance to heat transfer. The gas film will also dominate at Reynolds numbers higher than those examined in this study and for smaller ratios of ds/dp. 5. Conclusions The heat-transfer coefficient, h, to a mobile sphere has been measured in fluidized and packed beds of relatively large particles. These measurements revealed the following: (1) Good agreement was obtained between the values of h deduced from (a) the rate of cooling of a single mobile sphere as measured by an embedded thermocouple and (b) the rate of sublimation of a small batch of spheres of dry ice (solid CO2) placed in the bed. This implies that the effects of the trailing wires from the thermocouple in the mobile sphere and the attrition of the dry ice were both negligible. (2) The heat-transfer coefficient, strictly speaking a steady-state parameter, cannot generally be measured in an unsteady-state experiment. The condition Γ ) τsphere/τfluid . 1 must be fulfilled to define a heat-transfer coefficient from a transient experiment. Here, τsphere and τfluid are the time constants for the cooling of the heattransfer sphere and the formation of the steady-state temperature profile around the sphere, respectively. (3) There was a progressive reduction in the effects of transient conduction as the gas velocity was increased from U ) 0 to U ) Umf before the bed was fully fluidized. This suggested that the temperature profile around the heat-transfer sphere was confined to a small region near its surface. Because particle convection was absent for U < Umf, this was in all probability caused by the formation of a gaseous boundary layer around the sphere. Under these conditions, Γ ) τsphere/τfluid . 1, and it is possible to measure a heat-transfer coefficient from a transient experiment. (4) When U is progressively increased through Umf, the bed becomes fully fluidized, and there can be a sharp

Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004 5643

rise in h on passing from U < Umf to U > Umf. The sharp rise is observed for (a) large values of ds/dp (ds/dp > 3) and (b) low values of Remfs, the Reynolds number at minimum fluidization, based on ds. These observations imply a sharp rise in h when particle convection is large, as for a fixed heat-transfer surface in a bed of small particles. In contrast, experiments with 53 < Remfs < 830 and 0.2 < ds/dp < 2.75 showed that h (a) remained constant, when U/Umf was increased above unity and (b) was continuous through the onset of fluidization, i.e., there was no step increment in h at U ) Umf. This implies that heat transfer to the mobile sphere is governed by behavior in the particulate phase, which is incipiently fluidized. In this case, bubbles have only a small effect on the mobile sphere, which is influenced only by its adjacent environment of incipiently fluidized particles. (5) The experiments indicated that, for ds/dp < 3 and Remfs > 60, the dominant heat-transfer resistance is that of a gas film around the sphere. Thus, it was possible to correlate the Nusselt number against the Reynolds number on the basis of both the fluidizing velocity at minimum fluidization and the sphere’s diameter, as well as the ratio ds/dp. A least-squares fit to the measurements gave Nu ) 2 + 1.0Remfs0.6(ds/dp)0.26 for 100 < Remfs < 830 and 0.2 < ds/dp < 2.75. Acknowledgment The authors thank Dr. Martin Hayes at Johnson Matthey Chemicals for help in acquiring materials for this study and Mr. David Wilson, who assisted with the experiments. The authors are grateful for the support of the Engineering and Physical Sciences Research Council (GR/R21547), Anglian Water plc, and EMC Environmental Engineering Ltd. Nomenclature a ) radius of the heat-transfer sphere (m) Abed ) cross-sectional area of the fluidized bed (m2) As ) surface area of the heat-transfer sphere (m2) b ) rate of change of the heat capacity of graphite with temperature (J kmol-1 K-2) C ) heat capacity of fluid (J kg-1 K-1) Co ) heat capacity of graphite, extrapolated to 0 K (J kmol-1 K-1) Cp ) heat capacity (mass basis) (J kg-1 K-1) CpCO2 ) heat capacity of gaseous CO2 (molar basis) (J kmol-1 K-1) Cpm ) heat capacity (molar basis) (J kmol-1 K-1) [CO2] ) concentration of CO2 (kmol m-3) dp ) bed particle diameter (m) ds ) sphere diameter (m) h ) heat-transfer coefficient (W m-2 K-1) h h ) mean heat-transfer coefficient from n repeats (W m-2 K-1) hsub ) coefficient for heat transfer from a subliming particle (W m-2 K-1) ∆H ) heat of sublimation of dry ice (J kmol-1) n ) number of repeats of experiment N ) number of spheres of dry ice added to the fluidized bed Q ) rate of sublimation of dry ice (kmol s-1) r ) radial coordinate (m) T ) temperature (K) t ) time (s) tc ) contact time (s) U ) superficial velocity of gas in the bed (m s-1) Umf ) superficial velocity at incipient fluidization (m s-1)

Vs ) volume of heat-transfer sphere (m3) Greek Letters R ) parameter used in correlation β ) parameter used in correlation χ ) parameter used in correlation δ ) boundary layer thickness (m) mf ) void fraction at incipient fluidization ξ ) parameter in eq 16 λ ) thermal conductivity (W m-1 K-1) µ ) gas viscosity (Pa s) F ) density (mass basis) (kg m-3) Fm ) density (molar basis) (kmol m-3) σn-1 ) sample standard deviation of h from n repeats (W m-2 K-1) τfluid ) time constant to establish a steady-state temperature profile in surrounding fluid (s) τp ) time constant for the heating of a single fluidized particle (s) τsphere ) time constant for the cooling of the sphere (s) Dimensionless Variables Remf ) UmfdpFg/µg ) Reynolds number at minimum fluidization based on bed material size Remfs ) UmfdsFg/µg ) Reynolds number at minimum fluidization based on the heat-transfer sphere diameter Res ) UdsFg/µg ) Reynolds number based on the heattransfer sphere diameter Nu ) hds/λg ) Nusselt number (based on gas thermal conductivity) ∆Nu ) rise in Nu at U ) Umf Bi ) hds/λs ) Biot number Γ ) τsphere/τfluid ) ratio of time constants Pr ) µgCp/λg ) Prandtl number (based on gas properties) Subscripts ∞ ) conditions a long way from particle em ) emulsion phase g ) gas m ) molar basis mf ) conditions at incipient fluidization o ) initial value p ) bed particles s ) surface or solid sphere

Literature Cited (1) Botterill, J. S. M. Fluid Bed Heat Transfer; Academic Press: New York, 1975. (2) Zhu, C.; Fan, L. S. Hydrodynamics of Heat Transfer in Fluidised Beds; Cambridge University Press: Cambridge, U.K., 1998. (3) Agarwal, P. K. Transport phenomena in multi-particle systems-IV. Heat transfer to a large freely moving particle in gas fluidised bed of smaller particles. Chem. Eng. Sci. 1991, 46, 1115. (4) Linjewile, T. M.; Hull, A. S.; Agarwal, P. K. Heat transfer to a large mobile particle in gas fluidised beds of smaller particles. Chem. Eng. Sci. 1993, 48, 3671. (5) Parmar, M. S.; Hayhurst, A. N. The heat transfer coefficient for a freely moving sphere in a bubbling fluidised bed. Chem. Eng. Sci. 2002, 57, 3485. (6) Prins, W. Fluidised bed combustion of a single carbon particle. Ph.D. Thesis. University of Twente, Twente, The Netherlands, 1987. (7) Richardson, J. L.; Collier, A. Heat Transfer from Small Particles in a Fluidised Bed; Part IIB Research Project Report; Department Chemical Engineering, University of Cambridge: Cambridge, U.K., 1993. (8) Collier, A.; Hayhurst, A. N.; Richardson, J. L.; Scott, S. A. The heat transfer coefficient between a particle and a bed (packed or fluidised) of much larger particles. Chem. Eng. Sci., In press.

5644 Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004 (9) Baskakov, A. P.; Filippovskii, N. F.; Munts, V. A.; Ashikhmin, A. A. Temperature of particles heated in a fluidised bed of inert material. J. Eng. Phys. 1987, 52, 788. (10) Tamarin, A. I.; Galershtein, D. M.; Shuklina, V. M. Heat transfer and the combustion of coke particles in a fluidised bed. J. Eng. Phys. 1982, 41, 21. (11) Ross, I. B.; Patel, M. S.; Davidson, J. F. The temperature of burning carbon particles in fluidised beds. Trans. Inst. Chem. Eng. 1981, 59, 83. (12) Hayhurst, A. N.; Parmar, M. S. Does solid carbon burn in oxygen to give the gaseous intermediate CO or produce CO2 directly? Some experiments in a hot bed of sand fluidized by air. Chem. Eng. Sci. 1998, 53, 427. (13) Rios, G. M.; Gilbert, H. Heat transfer between a gas fluidized bed and big bodies: analysis and explanation of the big body mobility effects. In Fourth International Engineering Foundation Conference on Fluidisation; Kunii, D., Toei, R., Eds.; AIChE: New York, 1983; pp 363-371. (14) Perry, R. H.; Green, W. H. Perry’s Chemical Engineer’s Handbook, 7th ed.; McGraw-Hill: New York, 1998. (15) Wen, C. Y.; Yu, Y. H. A generalised method for predicting the minimum fluidization velocity. AIChE J. 1966, 12, 610. (16) Chase, M. W.; Davies, C. A.; Downey, J. R.; Frurip, D. J.; McDonald, R. A.; Syverud A. N. JANAF Thermochemical Tables (3rd ed.). J. Phys. Chem. Ref. Data 1985, 14 (Suppl 1). (17) Lide, R. D.; Frederikse, H. P. R. CRC Handbook of Chemistry and Physics, 77th ed.; CRC Press: Boca Raton, FL, 1996.

(18) www.webbook.nist.gov/chemistry/ (accessed June 2004). (19) Bird, B. R.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena, 2nd ed.; Wiley: New York, 2002. (20) Paterson, W. R.; Hayhurst, A. N. Mass or heat transfer from a sphere to a flowing fluid. Chem. Eng. Sci. 2000, 55, 1925. (21) Gabor J. D. Heat transfer to particle beds with gas flows less than or equal to that required for incipient fluidisation. Chem. Eng. Sci. 1970, 25, 979. (22) Denton, W. H. The Heat Transfer and Flow Resistance for Fluid Flow Through Randomly Packed Spheres. In Proceedings of the General Discussion on Heat Transfer, Institute of Mechical Engineers and ASME: London, 1951; pp 370-373. (23) Yagi, S., Kunii, D. Studies on effective thermal conductivities in packed beds. AIChE J. 1957, 3, 375. (24) Xavier, A. M.; Davidson, J. F. Heat transfer to surfaces in fluidised beds and in the freeboard region. AIChE Symp. Ser. 1981, 77 (208), 368. (25) Mickley, H. S.; Fairbanks, D. F. Mechanism of heat transfer to fluidised beds. AIChE J. 1955, 1, 374. (26) Dean, W. M. Analysis of Transport Phenomena; Oxford University Press: Oxford, U.K., 1998. (27) Zabrodsky, S. S. Hydrodynamics and Heat Transfer in Fluidized Beds; MIT Press: Cambridge, MA, 1966; p 169.

Received for review September 25, 2003 Revised manuscript received April 2, 2004 Accepted April 7, 2004 IE0307380