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Effects of micron-sized particles on hydrodynamics and local heat transfer in a slurry bubble column
Powder Technology 133 (2003) 171 – 184 www.elsevier.com/locate/powtec
Effects of micron-sized particles on hydrodynamics and local heat transfer in a slurry bubble column H. Li, A. Prakash *, A. Margaritis, M.A. Bergougnou Department of Chemical and Biochemical Engineering, The University of Western Ontario, London, ON, Canada N6A 5B9 Received 10 July 2002; accepted 15 April 2003
Abstract Hydrodynamics and heat transfer in an air – water – glass bead system are investigated to study the effects of particle size (11 – 93 Am) and slurry concentration (up to 40 vol.%). The effect of particle size on gas holdup is only slight over the range of particle size investigated. This can be attributed to differences in rise velocities of different bubble fractions. Small bubble rise velocities are systematically lower for a particle size of 93 Am compared to 11- and 35-Am particles. Larger bubble rise velocities decreased with increasing particle size. Heat transfer coefficients decreased with increasing particle size at the column center, but no significant difference is observed at the column wall. At the column bottom, the heat transfer behaviour of the 11-Am particles is distinctly different from the 35- and 93-Am particles. With the 11-Am particles, there is a smaller effect of slurry concentration on local heat transfer coefficient. Estimated interfacial area results show a sharp decrease due to addition of fine particles into the liquid. The decrease is more dramatic as the slurry concentration is increased. D 2003 Elsevier B.V. All rights reserved. Keywords: Slurry bubble column; Hydrodynamics; Local heat transfer; Interfacial area
1. Introduction Slurry bubble column reactors provide benefits for a number of industrial processes in the areas of heavy oil upgrading, Fischer –Tropsch synthesis, environmental pollution control and biotechnology [1 –5]. The advantages of these reactors include high heat and mass transfer rates, isothermal conditions and on-line catalyst addition and withdrawal. Also, there are low maintenance requirements due to simple construction and absence of any moving parts. However, proper design and scale-up of slurry bubble columns require a thorough understanding of hydrodynamic and associated heat and mass transfer properties. In recent years, gas holdup, bubble rise velocity and heat transfer have been investigated under various gas velocities, slurry concentrations and operating pressures [6 –14]. The effects of particle size on gas holdup have also been reported [12,15]. Typical variations in gas holdups with particle diameter have been discussed by Khare and Joshi [15]. It is pointed out that up to a particle size of 100 Am, there is a slight increase in gas holdup with particle diameter. This is followed by a decrease up to a
particle diameter of about 600 Am. Gas holdups have usually been found to decrease with increasing slurry concentrations except at very low slurry concentrations [12,15]. Saxena et al. [12] pointed out that the effect of slurry concentration becomes negligible at higher slurry concentrations. The particle size range covered in this study includes applications of the slurry reactor in catalytic chemical reactors as well as bioreactors. While most literature studies have used slurry concentrations less than 10 vol.%, this study investigates slurry concentrations up to 40 vol.%. There is also a lack of information on the effects of particle size on local heat transfer. The present work deals with the effect of particle size on local heat transfer, particularly the distributor and bulk regions of the column in more detail. The local heat transfer coefficients show the relationship between local hydrodynamics and mixing in the column. Moreover, bubble population measurements provided insights into the role of particle size and slurry concentration on rise velocities of different bubble size fractions.
2. Experimental * Corresponding author. E-mail address: [email protected] (A. Prakash). 0032-5910/03/$ - see front matter D 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0032-5910(03)00118-9
Experiments were conducted in a Plexiglas column of 0.28 m inside diameter and 2.4-m height. A detailed
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schematic of the column is given elsewhere [5,6]. It consisted of 0.6 m high top and bottom sections and a 1.2 m high middle section. A six-arm gas distributor introduced air into the column bottom. The gas distributor arms were 5 mm in diameter and 0.14 m long. Each arm had four holes of 1.5-mm diameter facing downwards. An electric heater was located near the bottom to maintain a constant slurry temperature. Tap water was the liquid phase, while compressed air and glass beads (11, 35 and 93 Am) constituted gas and solid phases, respectively. Gas flow was measured by a rotameter (Omega FL-1660) and superficial gas velocity was varied from 0.05 to 0.30 m/s. The unaerated liquid height was 1.4 m and the temperature was maintained at 23 jC in the column. Two pressure transducers, supplied by OMEGA (PX541), were flush mounted on the wall of the column at 0.07 and 1.33 m above the bottom. The pressure transducers provided fast response (2 ms) to pressure changes and pressure signals were recorded at a rate of 60 Hz. Pressure signals were also recorded after switching off gas by a quick closing ball valve located at the gas inlet into the column. The signals provided dynamic variations in pressure during gas disengagement. The data acquisition for measuring pressure started about 10 s before gas flow was stopped, so as to record the whole dynamic process of pressure variation. The whole process consisted of three parts: steady state operation, gas disengagement after stoppage of gas flow and gas free suspension. As the gas inlet valve was closed, the instantaneous gas holdup was monitored by two pressure transducers and was defined as: eg ðtÞ ¼ 1
1 DPðtÞ ðql /l þ qs /s Þg DH
ð1Þ
This technique is based on the principle that different bubble classes in a dispersion can be distinguished if there are significant differences between their rise velocities. The rate at which the instantaneous gas holdup drops would depend on the fraction and rise velocities of different bubble classes. The procedure to obtain rise velocities and holdups due to different bubble fractions was presented in an earlier study by Li and Prakash [7]. The instantaneous heat transfer flux was measured by means of a fast response (0.02 s) heat flow sensor (RDF corporation, microfoil heat flow sensor No. 20453-1) mounted on a cylindrical probe. The sensor provided an output voltage proportional to the local heat flux and also measured the probe surface temperature. The signals from the heat flux sensor were collected at 60 Hz for 35 s. The microvolt signals from the heat flux sensor were amplified to millivolts before collection by the data acquisition system. The details of probe design and construction are described elsewhere [6,8,9]. Three heat transfer probes were located at 0.07, 0.93 and 1.28 m from the column bottom. The probes were designed to be movable along radial direction. The bed temperature was measured by a different probe equipped with copper-constantan thermocouples
placed inside stainless steel tubes (6.4 mm in diameter) at various radial locations. The time and space average bed temperature was monitored by a digital thermometer. The measurements were started after steady state bed temperature was reached. The local instantaneous heat transfer coefficient (hi) was estimated from heat flux ( Q) and the temperature difference between the probe surface and the bed (DTsi). The time-averaged heat transfer coefficient at a given location was obtained by averaging the instantaneous heat transfer data:
h¼
N 1X Q N i¼1 DTsi
ð2Þ
3. Results and discussion 3.1. Hydrodynamics Fig. 1 presents gas holdups obtained in different systems with increasing gas velocities. Addition of particles in liquid decreased gas holdups and the difference increased with gas velocity. It can also be seen from Fig. 1 that gas holdups in suspensions of 11-Am particles were slightly lower compared to larger particles which had similar holdups. The effects of increasing slurry concentration on gas holdups are presented in Fig. 2 for different particle sizes. It can be observed that gas holdups generally decreased with increasing slurry concentration. However, the rate of decrease is low at concentrations above 20 vol.%. It is also observed that gas holdups in slurries of 93-Am particles are slightly higher compared to smaller particles. The effects of particle size and slurry concentration were further explored based on
Fig. 1. Variation of gas holdup with gas velocity (air – water; slurry concentration: 10 vol.%).
H. Li et al. / Powder Technology 133 (2003) 171–184
Fig. 2. Variation of gas holdup with slurry concentration and different particle sizes.
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the rise velocities of different fractions of bubbles obtained with the dynamic disengagement technique. It is seen from Fig. 3 that large bubble rise velocities increased with slurry concentration up to about 20 vol.%. At higher slurry concentrations, there is no significant effect on large bubble rise velocity. It is also noted that rate of increase in bubble rise velocities is higher with decreasing particle size. This behavior could be attributed to higher stable bubble size in suspensions of smaller particles. Small particles can penetrate the boundary layer surrounding the bubbles, thereby dampening the turbulence effect on bubble break-up. It is interesting to compare these results with the large bubble rise velocities reported by Prakash et al. [5] in suspensions of yeast particles. The large bubble rise velocities reported in suspensions of yeast particles are higher than those in suspensions of the 11-Am particles in the present study, although yeast cells concentrations used by Prakash et al. [5] were much lower. This observation confirms the stabilizing effect of small particles on bubble surface since yeast particles are about 8 Am in diameter. Increase in rise velocity of large bubbles with increasing slurry concentration leads to a corresponding drop in gas holdup due to large bubble fraction (Fig. 4). The effects of slurry concentration and particle size on rise velocities of small bubbles fraction are presented in Fig. 5. The data can be divided into two main groups, one for particle sizes of 11 and 35 Am and the other for 93-Am
Fig. 3. Large bubble rise velocity as a function of slurry concentration.
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Fig. 4. Large bubble holdup as a function of slurry concentration.
particles. It can be seen that rise velocities of small bubbles for particle size of 93 Am are generally lower than for particle sizes of 11 and 35 Am. The differences between
these two sets are initially low but increase with slurry concentration of up to about 25 vol.%. The difference again decreases at higher slurry concentrations. This behavior
Fig. 5. Small bubble rise velocity as a function of slurry concentration.
H. Li et al. / Powder Technology 133 (2003) 171–184
cannot be explained due to differences in particle terminal velocities alone. The rise velocities of small bubble fraction are plotted as a function of gas velocity in Fig. 6 for two slurry concentrations. It can be seen that bubble rise velocities decrease with increasing gas velocity in all systems except with the 11-Am particles, where it shows a tendency to increase with gas velocity. The decrease in rise velocity of small bubble fraction can be attributed to increases in bubble break-up rate with increasing gas velocity. Both bubble break-up and bubble coalescence rates are known to increase with increasing gas velocity [16]. However, the rate of increase of bubble break-up with gas velocity is usually higher than increase in bubble coalescence rate [16]. This does not seem to be the case in suspensions of the 11-Am particles, where the bubble break-up rate seems to decrease with increasing gas velocity, resulting in larger bubble size. This behavior could be due to slurry rheology similar to that of a dilatant fluid. 3.2. Local heat transfer Measurements of local heat transfer coefficients provided further insights into local hydrodynamics. It can be seen from Fig. 7 that heat transfer coefficients are lower in slurries of fine glass particles compared to the air –water system. This behavior has been attributed to reduced turbulence intensity in bubble wake due to increase in apparent suspension viscosity caused by addition of particles [6,17]. Although bubble size increases in slurry systems resulting in
175
larger bubble wake, the lower turbulence intensity in the wake region seems to negate the effects of the larger wake region. The heat transfer coefficients increased with gas velocities in both the central and wall regions of the column in slurries of different particle sizes. However, in the center, the heat transfer coefficients are higher in the 11-Am particle slurry and decrease with increasing particle size. On the other hand, in the wall region, there is no effect of particle size on heat transfer coefficient. The effect of particle size in the center can be attributed to large bubbles passing through the region [9]. As observed earlier, large bubble rise velocities increased with a decrease in particle size (Fig. 3). Change in bubble size distribution apparently has little effect on wall region heat transfer coefficient. In this region, heat transfer is dominated by back flow circulation created by radial gradient in gas holdup and not directly by bubble wake phenomena. Fig. 8 presents the effects of particle size and slurry concentration on heat transfer at the center in the bulk region. It is seen that heat transfer coefficients generally decrease with slurry concentration. This can be attributed to an increase in the apparent slurry viscosity and corresponding increase in average boundary layer thickness with increasing slurry concentration [9]. It is shown that the boundary layer thickness for heat transfer increases with slurry concentration, but decreases with increasing bubble size. But calculated boundary layer thickness is higher in slurry systems compared to air – water system. It is also observed from Fig. 8 that the rate of decrease in heat
Fig. 6. Variation of small bubble rise velocity with gas velocity in low and high slurry concentrations.
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Fig. 7. Comparison of heat transfer at the center and wall area of bulk region (slurry concentration: 10 vol.%; z = 1.28 m).
transfer is slower in slurries of smaller particles. This effect becomes more significant at higher superficial gas velocities. This can again be attributed to the formation of large bubbles in slurries of smaller particles. The faster rising large bubbles can enhance the heat transfer rate in their wake region [17,18]. Fig. 9 shows that the heat transfer rate decreased with increasing slurry concentration in the wall region as well. However, as noted earlier, there is no effect of particle size on the heat transfer process in this region. This indicates different levels of turbulence and resulting in heat transfer mechanisms in the two regions. The turbulence near the wall would be more homogeneous due to smaller bubbles and their smaller wake effects in the region. Complete radial profiles of heat transfer coefficients are presented in Figs. 10 and 11 for two slurry concentrations and different particle sizes. As noticed above, differences in slurries of different particle sizes are not significant in the wall region but differences increase toward the column center. As the particle size is reduced, the size of large bubbles is increased and the central region affected by the large bubbles becomes wider. This is confirmed from the radial profile of heat transfer coefficients, where reduced particle size increases the heat transfer and widens the central section. At slurry concentration of 20 vol.% and gas velocity of 0.3 m/s, the radial profile is nearly flat up to about half of column diameter (Fig. 11), indicating that large bubbles occupied nearly half of the column
diameter. At low gas velocity, bubble distribution is more uniform and the effect of particle size on radial profile becomes small. Li and Prakash [9] developed a correlation to estimate the radial profile of heat transfer coefficients in the bulk region and pointed out that the wall region heat transfer coefficients could be well predicted by the model presented by Deckwer et al. [19]. The wall region heat transfer coefficients obtained in the present study were also well predicted by the model of Deckwer et al. [19]. For other radial locations, the following equation developed by Li and Prakash [9] was applied. r 0:2 h hw ¼ 0:217 1:0 R hw
ð3Þ
Eq. (3) predicted well ( F 8%) the radial heat transfer coefficients up to a slurry concentration of 10 vol.% for all particle sizes. The errors, however, increased for higher slurry concentrations—the predictions were generally lower for the 11-Am particle slurries and higher for 93-Am particle slurries. It may be pointed out that large bubble rise velocities in slurries up to 10 vol.% were close for different particle sizes (Fig. 3). For higher slurry concentrations, however, the differences in large bubbles rise velocities increased in slurries of different particle sizes. This indicates that coefficients in Eq. (3) will vary with particle size.
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Fig. 8. Heat transfer coefficients at the center of the bulk region for different particle sizes and slurry concentrations (z = 1.28 m).
Heat transfer coefficients in the center of the bulk region were analyzed based on consecutive and surface-renewal theory proposed by Wasan and Ahluwalia [20]. The application of this model was investigated in the 35-Am particle slurry system by Li and Prakash [9]. This work extends its application in slurries of other particle sizes. In the model, the time average heat transfer coefficient from heating surface to the bed is expressed by physical properties of slurry, the film thickness (d) and the contact time between the liquid elements and the film (hc). 2ksl ksl d h ¼ pffiffiffiffiffiffiffiffiffiffi þ pahc ahc
pffiffiffiffiffiffiffi
ahc ahc 1 erf exp 1 ð4Þ d d2
pffiffiffiffiffiffiffi the term ð ahc =dÞ accounts for the contribution of film resistance to heat transfer. The contact time (hc) at the center and wall is estimated based on the approach presented by Li and Prakash [9]. The contact time (hc) at the center can be described using large bubble rise velocity (Ub,L) and probe diameter (dp): hc ¼
dp Ub;L
ð5Þ
The film thickness of thermal conduction d is related to thickness of the laminar viscous sublayer (do) as: d¼
do Pr1=3
ð6Þ
The thickness of the laminar viscous sublayer depends on the geography of the surface of the heat transfer source [21]. For the cylindrical probe used in this study, the heat flux sensor was located at an angle of 90j in front of liquid flow. Thus, the thickness of the laminar viscous sublayer can be expressed as [9]: do ¼
2:5dp 1=2
2Rep
ð7Þ
here, dp is the diameter of the cylindrical probe. Rep is the Reynolds number based on probe diameter and bubble rise velocity ( = qsl Ub,L dp/lsl). Thus, the film thickness can be calculated by combining Eqs. (6) and (7), expressed as: d¼
1:25dp 1=2
Rep Pr1=3
ð8Þ
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Fig. 9. Heat transfer coefficients at the wall of the bulk region for different particle sizes and slurry concentrations (z = 1.28 m).
The effective densities and heat capacities of suspensions were estimated from liquid and solid properties and their relative fractions:
Fig. 12 shows that there is good agreement between the predicted and experimental values of heat transfer coefficients.
qsl ¼ /s qs þ ð1 /s Þql
3.3. Interfacial area
ð9Þ
Cp;sl ¼ ws Cps þ ð1 ws ÞCpl
ð10Þ
Thermal conductivities of suspensions required in the above equations were estimated by the equation proposed by Tareef [22]: ksl ¼ kl
2kl þ ks 2/s ðkl ks Þ 2kl þ ks /s ðkl ks Þ
ð11Þ
Slurry viscosities were estimated by the correlation proposed by Vand [23] based on recommendation in literature studies [9]: lsl ¼ lL exp
2:5/s 1 0:609/s
ð12Þ
Based on the measurements of this study, an approximate method was developed to estimate the interfacial area to study the effects of particle size and concentration. Interfacial area is a good indicator of gas – liquid mass transfer capabilities of a system. It can be estimated from gas phase holdups due to two bubble populations and the average diameter of the population—assuming spherical bubbles. Appropriate literature correlations were selected to relate bubble rise velocity of each bubble fraction to average bubble diameter of the population.
a¼
6eb;L 6eb;sm þ db;L db;sm
ð13Þ
H. Li et al. / Powder Technology 133 (2003) 171–184
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Fig. 10. Radial profile of heat transfer coefficient in slurry concentration of 10 vol.% of different particle sizes.
We can write the total gas holdup as a sum of gas holdups due to two bubble fractions. Vg;sm˜ Vg;L þ Ub;L Ub;sm˜
ð14Þ
Vg ¼ Vg;L þ Vg;sm˜
ð15Þ
eg ¼
Eqs. (14) and (15) can be solved for gas velocities associated with individual fractions and the gas holdups due to individual fractions can be calculated as below: eb;L ¼
Vg;sm˜ Vg;L ; eb;S ¼ Ub;L Ub;sm˜
ð16Þ
For the estimation of the average bubble diameter of each bubble population, literature correlations relating bubble rise velocity to bubble diameter were tested [24 –26]. For large bubble fraction, the correlation proposed by Kim et al. [25] was found to be reasonable since other literature correlations [24,26] predicted large bubble diameters to be larger than the column diameter. Moreover, the correlation of Kim et al. [25] provided estimates of large bubble diameter which can
be supported by observations in this study. In the slurry of the 11-Am particles, this correlation predicted the large bubble diameter to be about half the column diameter at slurry concentration of 20 vol.% and gas velocity of 0.3 m/s. This coincides with the observations from radial profiles of heat transfer coefficients. As observed in Fig. 11, the radial profile of heat transfer coefficient became nearly flat, extending up to half of the column diameter as slurry concentration increased. db;L ¼ ðUb;L =17:1Þ1:011
ð17Þ
For small bubble fraction, the generalized form of the equation by Fan and Tsuchiya [26] as presented in Fan et al. [27] was selected to predict variations of bubble rise velocity with bubble diameter. This equation can account for the influence of the physical properties of the phases on bubble rise velocity. (" U Vb ¼
#n n=2)1=n Mo1=4 Dq 5=4 2 2c Dq d bV d Vb þ þ qsl qsl 2 Kb d Vb ð18Þ
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Fig. 11. Radial profile of heat transfer coefficient in slurry concentration of 20 vol.% and different particle sizes.
The dimensionless velocity (UbV) and the dimensionless bubble diameter (d bV) are defined as: UV b ¼ Ub
qsl rg
1=4
; dV b ¼ db
q g 1=2 sl r
ð18aÞ
The first and second terms in Eq. (18) represent the terminal rise velocities for the fine and small bubbles and for intermediate to large size bubbles, respectively. Li and Prakash [7] observed that this equation predicted well the influence of slurry concentration on small bubble rise velocities in a suspension of fine particles. The rise velocities of large bubbles predicted by the above equation were generally unrealistic since the predicted values were higher than the column diameter. For monocomponent contaminated liquids, constants n, c and Kb are given by Fan and Tsuchiya [26]: n ¼ 0:8; c ¼ 1:2; Kb ¼ maxð14:7Mo0:038 ; 12Þ
ð18bÞ
For the fine particles used in this study, the slurry phase was assumed to be homogeneous and its viscosity was estimated by the correlation proposed by Vand [23]. Contribution to total interfacial area due to large bubbles was on
average around 10% only—thus, the main contribution to interfacial area is due to small bubble fraction. Any errors in estimating large bubble size would have a small effect on estimated interfacial area. The interfacial areas obtained by this approximate procedure were found to be generally within 20% of the measured values reported in literature. Estimated interfacial areas in gas –liquid system were comparable with the results reported by Buchholz et al. [28] and in three-phase system with the results of Fukuma et al. [29]. Fig. 13 shows that there is a significant drop in gas –liquid interfacial area due to presence of fine particles. The drop increases with increasing slurry concentration and decreasing particle size. The data in 20 vol.% slurries are replotted in Fig. 14 for clarity. The plots for the three-particle sizes seem to follow the rheological behavior of non-Newtonian fluids. For the 93-Am particle slurry, the behavior is close to the shear diagram for Bingham plastic; for the 35-Am slurry, the shape is closer to pseudoplastic fluid while the 11-Am particle curve exhibits characteristics of a dilatant fluid. Fig. 15 shows the effect of particle size and slurry concentrations on heat transfer at the column bottom. It can be seen that heat transfer coefficients decrease with slurry concentration, but the effect is generally small above
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Fig. 12. Comparison of measured heat transfer coefficients in the central region with the model predictions.
10 vol.%. It is also seen that heat transfer coefficients in slurries of the 11-Am particles are higher compared to larger particles. It may be pointed out that gas jet effects are expected to dominate the region with increasing gas velocity
[9]. In slurry systems, the entrainment by gas jets and turbulence generated in the surrounding medium are expected to be a function of particle diameter and particle density. Turbulence dissipation in the vicinity of the jets is
Fig. 13. Variation of the estimated interfacial area with gas velocity and particle size in high and low slurry concentrations.
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Fig. 14. Variation of interfacial area with gas velocity in concentrated slurry (slurry concentration: 20 vol.%).
Fig. 15. Heat transfer coefficients in the wall region of the column bottom in slurries of different particle sizes.
H. Li et al. / Powder Technology 133 (2003) 171–184
expected to increase with increasing particle size. The following can be concluded based on a comparison of heat transfer coefficients in different regions of the column presented in Fig. 16. – In the bulk region of the column, there is an effect of particle size on local heat transfer coefficient at the center but not at the wall. In this region, heat transfer coefficients are higher at the center compared to wall area. – Heat transfer coefficients are significantly higher in the bulk region compared to the distributor region of the column. – In the distributor region, there is a large difference between heat transfer coefficients in the 11-Am particle slurries and other particle sizes. This behavior can be related to gas jet effects which dominate the region. 3.4. Concluding remarks Change in particle size affects bubble size distribution in a slurry bubble column but has a small effect on overall gas holdup. Large bubble rise velocity is increased with decreasing particle size. The increase in rise velocity of large
183
bubble fraction is supported by local heat transfer measurements at the column center. Different heat transfer mechanisms seem to prevail at the wall and at the center in the bulk region. The central region of the column is dominated by large bubbles rising through the region. Gas – liquid interfacial area decreases very quickly with increase in slurry concentration of fine particles. This would have a profound negative effect on gas – liquid mass transfer. Significantly lower heat transfer coefficients in the distributor region are attributed to lower turbulence in the region dominated mainly by gas jets. Nomenclature a interfacial area for gas –liquid mass transfer (m 1) c constant defined in Eq. (18b) Cp heat capacity (J/kg jC) db bubble diameter (m) dbV dimensionless bubble diameter (Eq. (18a)) dp probe diameter (m) h local average heat transfer coefficient (kW/m2 jC) hw average heat transfer coefficient at column wall (kW/m2 jC) DH distance between pressure transducer (m)
Fig. 16. Comparison of local heat transfer coefficients in different regions of the column (Vg = 0.3 m/s).
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k Kb Mo n N Pr DP Q r R Rep t DTsi Ub UbV Vg W z
H. Li et al. / Powder Technology 133 (2003) 171–184
thermal conductivity (kW/m jC) constant defined in Eq. (18b) Morton number [ gDql4sl/(q2sl r3] exponent defined in Eq. (18b) number of data points Prandtl number for suspension, (Cpl/k) pressure difference between the top and bottom transducers (Pa) heat flux (kW/m2) radial distance (m) column radius (m) Reynolds number based on probe diameter, (qUbdp/l) time (s) temperature difference between sensor surface and bed (jC) bubble rise velocity (m/s) dimensionless bubble rise velocity (Eq. (18a)) superficial gas velocity (m/s) weight fraction axial distance (m)
Greek letters a thermal diffusivity, k/qCp (m2/s) do laminar boundary layer (m) d thermal boundary layer (m) e phase holdup eb bubble fraction holdup hc contact time(s) l viscosity (Ps s) / volume fraction of solids in slurry phase q density (kg/m3) Subscripts g gas l liquid L large s solids sm small ˜ sl slurry
Acknowledgements This research project was supported by the Natural Science and Engineering Research Council of Canada
(NSERC) through individual research grants awarded to Dr. A. Prakash, Dr. A. Margaritis and Dr. M.A. Bergougnou.
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Effects of micron-sized particles on hydrodynamics and local heat transfer in a slurry bubble column H. Li, A. Prakash *, A. Margaritis, M.A. Bergougnou Department of Chemical and Biochemical Engineering, The University of Western Ontario, London, ON, Canada N6A 5B9 Received 10 July 2002; accepted 15 April 2003
Abstract Hydrodynamics and heat transfer in an air – water – glass bead system are investigated to study the effects of particle size (11 – 93 Am) and slurry concentration (up to 40 vol.%). The effect of particle size on gas holdup is only slight over the range of particle size investigated. This can be attributed to differences in rise velocities of different bubble fractions. Small bubble rise velocities are systematically lower for a particle size of 93 Am compared to 11- and 35-Am particles. Larger bubble rise velocities decreased with increasing particle size. Heat transfer coefficients decreased with increasing particle size at the column center, but no significant difference is observed at the column wall. At the column bottom, the heat transfer behaviour of the 11-Am particles is distinctly different from the 35- and 93-Am particles. With the 11-Am particles, there is a smaller effect of slurry concentration on local heat transfer coefficient. Estimated interfacial area results show a sharp decrease due to addition of fine particles into the liquid. The decrease is more dramatic as the slurry concentration is increased. D 2003 Elsevier B.V. All rights reserved. Keywords: Slurry bubble column; Hydrodynamics; Local heat transfer; Interfacial area
1. Introduction Slurry bubble column reactors provide benefits for a number of industrial processes in the areas of heavy oil upgrading, Fischer –Tropsch synthesis, environmental pollution control and biotechnology [1 –5]. The advantages of these reactors include high heat and mass transfer rates, isothermal conditions and on-line catalyst addition and withdrawal. Also, there are low maintenance requirements due to simple construction and absence of any moving parts. However, proper design and scale-up of slurry bubble columns require a thorough understanding of hydrodynamic and associated heat and mass transfer properties. In recent years, gas holdup, bubble rise velocity and heat transfer have been investigated under various gas velocities, slurry concentrations and operating pressures [6 –14]. The effects of particle size on gas holdup have also been reported [12,15]. Typical variations in gas holdups with particle diameter have been discussed by Khare and Joshi [15]. It is pointed out that up to a particle size of 100 Am, there is a slight increase in gas holdup with particle diameter. This is followed by a decrease up to a
particle diameter of about 600 Am. Gas holdups have usually been found to decrease with increasing slurry concentrations except at very low slurry concentrations [12,15]. Saxena et al. [12] pointed out that the effect of slurry concentration becomes negligible at higher slurry concentrations. The particle size range covered in this study includes applications of the slurry reactor in catalytic chemical reactors as well as bioreactors. While most literature studies have used slurry concentrations less than 10 vol.%, this study investigates slurry concentrations up to 40 vol.%. There is also a lack of information on the effects of particle size on local heat transfer. The present work deals with the effect of particle size on local heat transfer, particularly the distributor and bulk regions of the column in more detail. The local heat transfer coefficients show the relationship between local hydrodynamics and mixing in the column. Moreover, bubble population measurements provided insights into the role of particle size and slurry concentration on rise velocities of different bubble size fractions.
2. Experimental * Corresponding author. E-mail address: [email protected] (A. Prakash). 0032-5910/03/$ - see front matter D 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0032-5910(03)00118-9
Experiments were conducted in a Plexiglas column of 0.28 m inside diameter and 2.4-m height. A detailed
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schematic of the column is given elsewhere [5,6]. It consisted of 0.6 m high top and bottom sections and a 1.2 m high middle section. A six-arm gas distributor introduced air into the column bottom. The gas distributor arms were 5 mm in diameter and 0.14 m long. Each arm had four holes of 1.5-mm diameter facing downwards. An electric heater was located near the bottom to maintain a constant slurry temperature. Tap water was the liquid phase, while compressed air and glass beads (11, 35 and 93 Am) constituted gas and solid phases, respectively. Gas flow was measured by a rotameter (Omega FL-1660) and superficial gas velocity was varied from 0.05 to 0.30 m/s. The unaerated liquid height was 1.4 m and the temperature was maintained at 23 jC in the column. Two pressure transducers, supplied by OMEGA (PX541), were flush mounted on the wall of the column at 0.07 and 1.33 m above the bottom. The pressure transducers provided fast response (2 ms) to pressure changes and pressure signals were recorded at a rate of 60 Hz. Pressure signals were also recorded after switching off gas by a quick closing ball valve located at the gas inlet into the column. The signals provided dynamic variations in pressure during gas disengagement. The data acquisition for measuring pressure started about 10 s before gas flow was stopped, so as to record the whole dynamic process of pressure variation. The whole process consisted of three parts: steady state operation, gas disengagement after stoppage of gas flow and gas free suspension. As the gas inlet valve was closed, the instantaneous gas holdup was monitored by two pressure transducers and was defined as: eg ðtÞ ¼ 1
1 DPðtÞ ðql /l þ qs /s Þg DH
ð1Þ
This technique is based on the principle that different bubble classes in a dispersion can be distinguished if there are significant differences between their rise velocities. The rate at which the instantaneous gas holdup drops would depend on the fraction and rise velocities of different bubble classes. The procedure to obtain rise velocities and holdups due to different bubble fractions was presented in an earlier study by Li and Prakash [7]. The instantaneous heat transfer flux was measured by means of a fast response (0.02 s) heat flow sensor (RDF corporation, microfoil heat flow sensor No. 20453-1) mounted on a cylindrical probe. The sensor provided an output voltage proportional to the local heat flux and also measured the probe surface temperature. The signals from the heat flux sensor were collected at 60 Hz for 35 s. The microvolt signals from the heat flux sensor were amplified to millivolts before collection by the data acquisition system. The details of probe design and construction are described elsewhere [6,8,9]. Three heat transfer probes were located at 0.07, 0.93 and 1.28 m from the column bottom. The probes were designed to be movable along radial direction. The bed temperature was measured by a different probe equipped with copper-constantan thermocouples
placed inside stainless steel tubes (6.4 mm in diameter) at various radial locations. The time and space average bed temperature was monitored by a digital thermometer. The measurements were started after steady state bed temperature was reached. The local instantaneous heat transfer coefficient (hi) was estimated from heat flux ( Q) and the temperature difference between the probe surface and the bed (DTsi). The time-averaged heat transfer coefficient at a given location was obtained by averaging the instantaneous heat transfer data:
h¼
N 1X Q N i¼1 DTsi
ð2Þ
3. Results and discussion 3.1. Hydrodynamics Fig. 1 presents gas holdups obtained in different systems with increasing gas velocities. Addition of particles in liquid decreased gas holdups and the difference increased with gas velocity. It can also be seen from Fig. 1 that gas holdups in suspensions of 11-Am particles were slightly lower compared to larger particles which had similar holdups. The effects of increasing slurry concentration on gas holdups are presented in Fig. 2 for different particle sizes. It can be observed that gas holdups generally decreased with increasing slurry concentration. However, the rate of decrease is low at concentrations above 20 vol.%. It is also observed that gas holdups in slurries of 93-Am particles are slightly higher compared to smaller particles. The effects of particle size and slurry concentration were further explored based on
Fig. 1. Variation of gas holdup with gas velocity (air – water; slurry concentration: 10 vol.%).
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Fig. 2. Variation of gas holdup with slurry concentration and different particle sizes.
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the rise velocities of different fractions of bubbles obtained with the dynamic disengagement technique. It is seen from Fig. 3 that large bubble rise velocities increased with slurry concentration up to about 20 vol.%. At higher slurry concentrations, there is no significant effect on large bubble rise velocity. It is also noted that rate of increase in bubble rise velocities is higher with decreasing particle size. This behavior could be attributed to higher stable bubble size in suspensions of smaller particles. Small particles can penetrate the boundary layer surrounding the bubbles, thereby dampening the turbulence effect on bubble break-up. It is interesting to compare these results with the large bubble rise velocities reported by Prakash et al. [5] in suspensions of yeast particles. The large bubble rise velocities reported in suspensions of yeast particles are higher than those in suspensions of the 11-Am particles in the present study, although yeast cells concentrations used by Prakash et al. [5] were much lower. This observation confirms the stabilizing effect of small particles on bubble surface since yeast particles are about 8 Am in diameter. Increase in rise velocity of large bubbles with increasing slurry concentration leads to a corresponding drop in gas holdup due to large bubble fraction (Fig. 4). The effects of slurry concentration and particle size on rise velocities of small bubbles fraction are presented in Fig. 5. The data can be divided into two main groups, one for particle sizes of 11 and 35 Am and the other for 93-Am
Fig. 3. Large bubble rise velocity as a function of slurry concentration.
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Fig. 4. Large bubble holdup as a function of slurry concentration.
particles. It can be seen that rise velocities of small bubbles for particle size of 93 Am are generally lower than for particle sizes of 11 and 35 Am. The differences between
these two sets are initially low but increase with slurry concentration of up to about 25 vol.%. The difference again decreases at higher slurry concentrations. This behavior
Fig. 5. Small bubble rise velocity as a function of slurry concentration.
H. Li et al. / Powder Technology 133 (2003) 171–184
cannot be explained due to differences in particle terminal velocities alone. The rise velocities of small bubble fraction are plotted as a function of gas velocity in Fig. 6 for two slurry concentrations. It can be seen that bubble rise velocities decrease with increasing gas velocity in all systems except with the 11-Am particles, where it shows a tendency to increase with gas velocity. The decrease in rise velocity of small bubble fraction can be attributed to increases in bubble break-up rate with increasing gas velocity. Both bubble break-up and bubble coalescence rates are known to increase with increasing gas velocity [16]. However, the rate of increase of bubble break-up with gas velocity is usually higher than increase in bubble coalescence rate [16]. This does not seem to be the case in suspensions of the 11-Am particles, where the bubble break-up rate seems to decrease with increasing gas velocity, resulting in larger bubble size. This behavior could be due to slurry rheology similar to that of a dilatant fluid. 3.2. Local heat transfer Measurements of local heat transfer coefficients provided further insights into local hydrodynamics. It can be seen from Fig. 7 that heat transfer coefficients are lower in slurries of fine glass particles compared to the air –water system. This behavior has been attributed to reduced turbulence intensity in bubble wake due to increase in apparent suspension viscosity caused by addition of particles [6,17]. Although bubble size increases in slurry systems resulting in
175
larger bubble wake, the lower turbulence intensity in the wake region seems to negate the effects of the larger wake region. The heat transfer coefficients increased with gas velocities in both the central and wall regions of the column in slurries of different particle sizes. However, in the center, the heat transfer coefficients are higher in the 11-Am particle slurry and decrease with increasing particle size. On the other hand, in the wall region, there is no effect of particle size on heat transfer coefficient. The effect of particle size in the center can be attributed to large bubbles passing through the region [9]. As observed earlier, large bubble rise velocities increased with a decrease in particle size (Fig. 3). Change in bubble size distribution apparently has little effect on wall region heat transfer coefficient. In this region, heat transfer is dominated by back flow circulation created by radial gradient in gas holdup and not directly by bubble wake phenomena. Fig. 8 presents the effects of particle size and slurry concentration on heat transfer at the center in the bulk region. It is seen that heat transfer coefficients generally decrease with slurry concentration. This can be attributed to an increase in the apparent slurry viscosity and corresponding increase in average boundary layer thickness with increasing slurry concentration [9]. It is shown that the boundary layer thickness for heat transfer increases with slurry concentration, but decreases with increasing bubble size. But calculated boundary layer thickness is higher in slurry systems compared to air – water system. It is also observed from Fig. 8 that the rate of decrease in heat
Fig. 6. Variation of small bubble rise velocity with gas velocity in low and high slurry concentrations.
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Fig. 7. Comparison of heat transfer at the center and wall area of bulk region (slurry concentration: 10 vol.%; z = 1.28 m).
transfer is slower in slurries of smaller particles. This effect becomes more significant at higher superficial gas velocities. This can again be attributed to the formation of large bubbles in slurries of smaller particles. The faster rising large bubbles can enhance the heat transfer rate in their wake region [17,18]. Fig. 9 shows that the heat transfer rate decreased with increasing slurry concentration in the wall region as well. However, as noted earlier, there is no effect of particle size on the heat transfer process in this region. This indicates different levels of turbulence and resulting in heat transfer mechanisms in the two regions. The turbulence near the wall would be more homogeneous due to smaller bubbles and their smaller wake effects in the region. Complete radial profiles of heat transfer coefficients are presented in Figs. 10 and 11 for two slurry concentrations and different particle sizes. As noticed above, differences in slurries of different particle sizes are not significant in the wall region but differences increase toward the column center. As the particle size is reduced, the size of large bubbles is increased and the central region affected by the large bubbles becomes wider. This is confirmed from the radial profile of heat transfer coefficients, where reduced particle size increases the heat transfer and widens the central section. At slurry concentration of 20 vol.% and gas velocity of 0.3 m/s, the radial profile is nearly flat up to about half of column diameter (Fig. 11), indicating that large bubbles occupied nearly half of the column
diameter. At low gas velocity, bubble distribution is more uniform and the effect of particle size on radial profile becomes small. Li and Prakash [9] developed a correlation to estimate the radial profile of heat transfer coefficients in the bulk region and pointed out that the wall region heat transfer coefficients could be well predicted by the model presented by Deckwer et al. [19]. The wall region heat transfer coefficients obtained in the present study were also well predicted by the model of Deckwer et al. [19]. For other radial locations, the following equation developed by Li and Prakash [9] was applied. r 0:2 h hw ¼ 0:217 1:0 R hw
ð3Þ
Eq. (3) predicted well ( F 8%) the radial heat transfer coefficients up to a slurry concentration of 10 vol.% for all particle sizes. The errors, however, increased for higher slurry concentrations—the predictions were generally lower for the 11-Am particle slurries and higher for 93-Am particle slurries. It may be pointed out that large bubble rise velocities in slurries up to 10 vol.% were close for different particle sizes (Fig. 3). For higher slurry concentrations, however, the differences in large bubbles rise velocities increased in slurries of different particle sizes. This indicates that coefficients in Eq. (3) will vary with particle size.
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Fig. 8. Heat transfer coefficients at the center of the bulk region for different particle sizes and slurry concentrations (z = 1.28 m).
Heat transfer coefficients in the center of the bulk region were analyzed based on consecutive and surface-renewal theory proposed by Wasan and Ahluwalia [20]. The application of this model was investigated in the 35-Am particle slurry system by Li and Prakash [9]. This work extends its application in slurries of other particle sizes. In the model, the time average heat transfer coefficient from heating surface to the bed is expressed by physical properties of slurry, the film thickness (d) and the contact time between the liquid elements and the film (hc). 2ksl ksl d h ¼ pffiffiffiffiffiffiffiffiffiffi þ pahc ahc
pffiffiffiffiffiffiffi
ahc ahc 1 erf exp 1 ð4Þ d d2
pffiffiffiffiffiffiffi the term ð ahc =dÞ accounts for the contribution of film resistance to heat transfer. The contact time (hc) at the center and wall is estimated based on the approach presented by Li and Prakash [9]. The contact time (hc) at the center can be described using large bubble rise velocity (Ub,L) and probe diameter (dp): hc ¼
dp Ub;L
ð5Þ
The film thickness of thermal conduction d is related to thickness of the laminar viscous sublayer (do) as: d¼
do Pr1=3
ð6Þ
The thickness of the laminar viscous sublayer depends on the geography of the surface of the heat transfer source [21]. For the cylindrical probe used in this study, the heat flux sensor was located at an angle of 90j in front of liquid flow. Thus, the thickness of the laminar viscous sublayer can be expressed as [9]: do ¼
2:5dp 1=2
2Rep
ð7Þ
here, dp is the diameter of the cylindrical probe. Rep is the Reynolds number based on probe diameter and bubble rise velocity ( = qsl Ub,L dp/lsl). Thus, the film thickness can be calculated by combining Eqs. (6) and (7), expressed as: d¼
1:25dp 1=2
Rep Pr1=3
ð8Þ
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Fig. 9. Heat transfer coefficients at the wall of the bulk region for different particle sizes and slurry concentrations (z = 1.28 m).
The effective densities and heat capacities of suspensions were estimated from liquid and solid properties and their relative fractions:
Fig. 12 shows that there is good agreement between the predicted and experimental values of heat transfer coefficients.
qsl ¼ /s qs þ ð1 /s Þql
3.3. Interfacial area
ð9Þ
Cp;sl ¼ ws Cps þ ð1 ws ÞCpl
ð10Þ
Thermal conductivities of suspensions required in the above equations were estimated by the equation proposed by Tareef [22]: ksl ¼ kl
2kl þ ks 2/s ðkl ks Þ 2kl þ ks /s ðkl ks Þ
ð11Þ
Slurry viscosities were estimated by the correlation proposed by Vand [23] based on recommendation in literature studies [9]: lsl ¼ lL exp
2:5/s 1 0:609/s
ð12Þ
Based on the measurements of this study, an approximate method was developed to estimate the interfacial area to study the effects of particle size and concentration. Interfacial area is a good indicator of gas – liquid mass transfer capabilities of a system. It can be estimated from gas phase holdups due to two bubble populations and the average diameter of the population—assuming spherical bubbles. Appropriate literature correlations were selected to relate bubble rise velocity of each bubble fraction to average bubble diameter of the population.
a¼
6eb;L 6eb;sm þ db;L db;sm
ð13Þ
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Fig. 10. Radial profile of heat transfer coefficient in slurry concentration of 10 vol.% of different particle sizes.
We can write the total gas holdup as a sum of gas holdups due to two bubble fractions. Vg;sm˜ Vg;L þ Ub;L Ub;sm˜
ð14Þ
Vg ¼ Vg;L þ Vg;sm˜
ð15Þ
eg ¼
Eqs. (14) and (15) can be solved for gas velocities associated with individual fractions and the gas holdups due to individual fractions can be calculated as below: eb;L ¼
Vg;sm˜ Vg;L ; eb;S ¼ Ub;L Ub;sm˜
ð16Þ
For the estimation of the average bubble diameter of each bubble population, literature correlations relating bubble rise velocity to bubble diameter were tested [24 –26]. For large bubble fraction, the correlation proposed by Kim et al. [25] was found to be reasonable since other literature correlations [24,26] predicted large bubble diameters to be larger than the column diameter. Moreover, the correlation of Kim et al. [25] provided estimates of large bubble diameter which can
be supported by observations in this study. In the slurry of the 11-Am particles, this correlation predicted the large bubble diameter to be about half the column diameter at slurry concentration of 20 vol.% and gas velocity of 0.3 m/s. This coincides with the observations from radial profiles of heat transfer coefficients. As observed in Fig. 11, the radial profile of heat transfer coefficient became nearly flat, extending up to half of the column diameter as slurry concentration increased. db;L ¼ ðUb;L =17:1Þ1:011
ð17Þ
For small bubble fraction, the generalized form of the equation by Fan and Tsuchiya [26] as presented in Fan et al. [27] was selected to predict variations of bubble rise velocity with bubble diameter. This equation can account for the influence of the physical properties of the phases on bubble rise velocity. (" U Vb ¼
#n n=2)1=n Mo1=4 Dq 5=4 2 2c Dq d bV d Vb þ þ qsl qsl 2 Kb d Vb ð18Þ
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Fig. 11. Radial profile of heat transfer coefficient in slurry concentration of 20 vol.% and different particle sizes.
The dimensionless velocity (UbV) and the dimensionless bubble diameter (d bV) are defined as: UV b ¼ Ub
qsl rg
1=4
; dV b ¼ db
q g 1=2 sl r
ð18aÞ
The first and second terms in Eq. (18) represent the terminal rise velocities for the fine and small bubbles and for intermediate to large size bubbles, respectively. Li and Prakash [7] observed that this equation predicted well the influence of slurry concentration on small bubble rise velocities in a suspension of fine particles. The rise velocities of large bubbles predicted by the above equation were generally unrealistic since the predicted values were higher than the column diameter. For monocomponent contaminated liquids, constants n, c and Kb are given by Fan and Tsuchiya [26]: n ¼ 0:8; c ¼ 1:2; Kb ¼ maxð14:7Mo0:038 ; 12Þ
ð18bÞ
For the fine particles used in this study, the slurry phase was assumed to be homogeneous and its viscosity was estimated by the correlation proposed by Vand [23]. Contribution to total interfacial area due to large bubbles was on
average around 10% only—thus, the main contribution to interfacial area is due to small bubble fraction. Any errors in estimating large bubble size would have a small effect on estimated interfacial area. The interfacial areas obtained by this approximate procedure were found to be generally within 20% of the measured values reported in literature. Estimated interfacial areas in gas –liquid system were comparable with the results reported by Buchholz et al. [28] and in three-phase system with the results of Fukuma et al. [29]. Fig. 13 shows that there is a significant drop in gas –liquid interfacial area due to presence of fine particles. The drop increases with increasing slurry concentration and decreasing particle size. The data in 20 vol.% slurries are replotted in Fig. 14 for clarity. The plots for the three-particle sizes seem to follow the rheological behavior of non-Newtonian fluids. For the 93-Am particle slurry, the behavior is close to the shear diagram for Bingham plastic; for the 35-Am slurry, the shape is closer to pseudoplastic fluid while the 11-Am particle curve exhibits characteristics of a dilatant fluid. Fig. 15 shows the effect of particle size and slurry concentrations on heat transfer at the column bottom. It can be seen that heat transfer coefficients decrease with slurry concentration, but the effect is generally small above
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Fig. 12. Comparison of measured heat transfer coefficients in the central region with the model predictions.
10 vol.%. It is also seen that heat transfer coefficients in slurries of the 11-Am particles are higher compared to larger particles. It may be pointed out that gas jet effects are expected to dominate the region with increasing gas velocity
[9]. In slurry systems, the entrainment by gas jets and turbulence generated in the surrounding medium are expected to be a function of particle diameter and particle density. Turbulence dissipation in the vicinity of the jets is
Fig. 13. Variation of the estimated interfacial area with gas velocity and particle size in high and low slurry concentrations.
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Fig. 14. Variation of interfacial area with gas velocity in concentrated slurry (slurry concentration: 20 vol.%).
Fig. 15. Heat transfer coefficients in the wall region of the column bottom in slurries of different particle sizes.
H. Li et al. / Powder Technology 133 (2003) 171–184
expected to increase with increasing particle size. The following can be concluded based on a comparison of heat transfer coefficients in different regions of the column presented in Fig. 16. – In the bulk region of the column, there is an effect of particle size on local heat transfer coefficient at the center but not at the wall. In this region, heat transfer coefficients are higher at the center compared to wall area. – Heat transfer coefficients are significantly higher in the bulk region compared to the distributor region of the column. – In the distributor region, there is a large difference between heat transfer coefficients in the 11-Am particle slurries and other particle sizes. This behavior can be related to gas jet effects which dominate the region. 3.4. Concluding remarks Change in particle size affects bubble size distribution in a slurry bubble column but has a small effect on overall gas holdup. Large bubble rise velocity is increased with decreasing particle size. The increase in rise velocity of large
183
bubble fraction is supported by local heat transfer measurements at the column center. Different heat transfer mechanisms seem to prevail at the wall and at the center in the bulk region. The central region of the column is dominated by large bubbles rising through the region. Gas – liquid interfacial area decreases very quickly with increase in slurry concentration of fine particles. This would have a profound negative effect on gas – liquid mass transfer. Significantly lower heat transfer coefficients in the distributor region are attributed to lower turbulence in the region dominated mainly by gas jets. Nomenclature a interfacial area for gas –liquid mass transfer (m 1) c constant defined in Eq. (18b) Cp heat capacity (J/kg jC) db bubble diameter (m) dbV dimensionless bubble diameter (Eq. (18a)) dp probe diameter (m) h local average heat transfer coefficient (kW/m2 jC) hw average heat transfer coefficient at column wall (kW/m2 jC) DH distance between pressure transducer (m)
Fig. 16. Comparison of local heat transfer coefficients in different regions of the column (Vg = 0.3 m/s).
184
k Kb Mo n N Pr DP Q r R Rep t DTsi Ub UbV Vg W z
H. Li et al. / Powder Technology 133 (2003) 171–184
thermal conductivity (kW/m jC) constant defined in Eq. (18b) Morton number [ gDql4sl/(q2sl r3] exponent defined in Eq. (18b) number of data points Prandtl number for suspension, (Cpl/k) pressure difference between the top and bottom transducers (Pa) heat flux (kW/m2) radial distance (m) column radius (m) Reynolds number based on probe diameter, (qUbdp/l) time (s) temperature difference between sensor surface and bed (jC) bubble rise velocity (m/s) dimensionless bubble rise velocity (Eq. (18a)) superficial gas velocity (m/s) weight fraction axial distance (m)
Greek letters a thermal diffusivity, k/qCp (m2/s) do laminar boundary layer (m) d thermal boundary layer (m) e phase holdup eb bubble fraction holdup hc contact time(s) l viscosity (Ps s) / volume fraction of solids in slurry phase q density (kg/m3) Subscripts g gas l liquid L large s solids sm small ˜ sl slurry
Acknowledgements This research project was supported by the Natural Science and Engineering Research Council of Canada
(NSERC) through individual research grants awarded to Dr. A. Prakash, Dr. A. Margaritis and Dr. M.A. Bergougnou.
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