Ind. Eng. Chem. Res. 2001, 40, 457-464
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Pressure Drop and Heat Transfer for Cocurrent Upflow of Dilute Gas-Coal Particle Suspensions in a Circular Tube R. A. Sorensen, J. D. Seader,* and B. S. Brewster Department of Chemical and Fuels Engineering, University of Utah, Salt Lake City, Utah 84112
Pressure drop and heat transfer for dilute gas-coal particle suspensions flowing cocurrently upward in a 0.498-in.-i.d. tube were measured and compared to previously published results for cocurrent downflow. A 4-ft-long electrically heated test section followed a 26-ft-long pressuredrop test section. Room air was employed as the carrier gas for 300-µm (average screen size) suspensions of coal particles. The gas Reynolds number was varied from 10 000 to 30 000, with solids loading ratios up to 15 lb of solids/lb of gas. Frictional pressure drop for upflow was found to be greater than that for cocurrent downflow. The wall Nusselt number was slightly less than that for air alone but somewhat higher than that for cocurrent downflow. Particle Nusselt numbers were found to depend on the gas Reynolds number and the solids loading ratio and were almost identical with those for cocurrent downflow. Introduction Interest in transport phenomena for particulate suspensions has increased during the past 50 years because of their frequent occurrence in industrial applications, including pneumatic conveyance, catalytic cracking, nuclear cooling systems, and coal combustion. Although numerous investigations have been reported in the past (Depew and Kramer1), general correlations of particle motion, pressure drop, and heat transfer for particulate suspensions in equipment configurations of interest do not exist. Furthermore, available experimental data frequently scatter widely, and results of different investigators are often contradictory. If a correlation is proposed, its applicable range is limited. Obvious suspension flow configurations of interest are vertical upflow, vertical downflow, and horizontal and inclined arrangements. The major effort of past research has been on the vertical upflow and horizontal and inclined arrangements, with only the investigations of Kim and Seader2,3 and Brewster and Seader4,5 dealing with heat transfer and pressure drop for gas-solids suspensions in cocurrent vertical downflow. Kim and Seader measured pressure-drop and heat-transfer rates for suspensions of 329-µm glass spheres in air in vertical downflow. Brewster and Seader5 compared their data for vertical downflow of air-coal particle suspensions with those of Kim and Seader to discern the effects of particle properties, shape, and size distribution. The experiments of Brewster and Seader were conducted with 100- and 300-µm (average screen size) coal particles flowing at superficial gas Reynolds numbers of 10 000-30 000 with solids loading ratios of 0-18 lb of solids/lb of gas. The present study deals with pressure drop and heat transfer for a suspension of 300-µm coal particles in air for vertical upflow over a superficial gas Reynolds number range of 10 000-30 000 and solids loading ratios of up to 15. Pressure-drop profiles, as well as the pressure drop for fully developed flow, were measured at various solids loading ratios. Pressure drop * To whom all correspondence should be addressed. Phone: 801-581-6916. Fax: 801-585-9291. E-mail: J.Seader@ m.cc.utah.edu.
was analyzed in terms of both a solids friction factor and a suspension-flow friction factor. Rates of heat transfer between the wall and the gas and between the gas and the particles are reported in terms of wall and particle Nusselt numbers, respectively. The results of these experiments are compared with the results of Brewster and Seader. The data are intended to help clarify the differences in suspension transport phenomena between upflow and downflow configurations. Theoretical Considerations This section summarizes the theoretical basis for processing and analyzing the experimental results. First, procedures used to account for the irregular shape of conveyed coal particles are presented. These procedures result in an expression for the equivalent particle diameter. The macroscopic continuity relations are then described and used to obtain a relationship between the holdup and slip velocity of the phases. An estimation for the slip velocity is obtained from a terminal velocity analysis involving the one-dimensional equation of motion for a particle in an infinite fluid. Finally, equations for the pressure drop and the one-dimensional heat balance governing the gas and particle temperatures are presented. Particle-Shape Effects. Volumes of coal particles with diameters as small as 60 µm were measured by Needham and Hill,6 who obtained the following empirical correlation for the particle volume, Vp, in cubic microns, in terms of the diameter, da, in microns, of a sphere of equivalent cross-sectional area, and davg, the average screen size in microns:
Vp ) 0.073davg0.15da3
(1)
The particle cross-sectional (projected) area ) Ap ) πda2/4, which combined with eq 1 gives
Ap π ) Vp 4(0.073d
0.15
avg
)da
(2)
Later, Needham and Hill7 reported settling velocities
10.1021/ie000531w CCC: $20.00 © 2001 American Chemical Society Published on Web 12/01/2000
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for coal and other mineral particles in water. From these data, the following more useful correlation, in terms of the average screen size, was developed for application to the pressure drop measurements of this study:
Ap ) Vp 4(0.073d
0.15 avg
π (3) )(0.67 + 0.2 log davg)davg
The external surface area of a particle plays an important role in heat transfer to the particle. Austin et al.8 measured the external surface area of coal particles with diameters of 40-1000 µm by a permeability method and reported the following correlation for the ratio of external area to volume
Sp 6 ) Vp ψdavg
(4)
where ψ is the sphericity shape factor, defined as the ratio of the surface area of a sphere of the same volume as the coal particle to the surface area of the coal particle. Typical values of ψ for coal range from 0.7 to 0.8, with 0.75 as an average value. As discussed by Kreith,9 convective heat-transfer correlations for spheres satisfactorily predict heattransfer coefficients for particles of irregular shape if the sphere diameter is replaced by ds, the diameter of a spherical particle having the same external surface area as the irregular particle. Thus, 1/2
ds ) (Sp/π)
(5)
Equation of Motion for a Single Particle. The acceleration of a single particle in an infinite fluid was examined analytically by solving the one-dimensional particle-motion equation. Cocurrent-countergravity and cocurrent-cogravity configurations were considered along with particle size effects. Boothroyd10 discusses the importance of particle-particle effects in polydisperse systems and states that the only significant fluidsolid interaction for a high-density solid system is the drag force. With this assumption, the equation of motion for a single particle in a flowing fluid is
Fg Ap (Ug - Up)|Ug - Up| dUp ) CD -g dt Fs V p 2
log CD ) 0.0958(log Rep)2 - 0.891(log Rep) + 1.43 (8) where
Rep )
(7)
However, for the irregularly shaped particles encountered here, this ratio is given by eq 3. Robertson and Crowe11 give the drag coefficient for a sphere, with the
Fg|Us|da µg
(9)
with, from eq 3, da ) davg(0.67 + 0.2 log davg). For davg ) 300 µm, da ) 350 µm. Particle Velocity. Particle velocity is an important and difficult parameter to determine accurately. In the absence of experimental values, a reasonable estimation is required. Klinzing12 discusses the applicability of the terminal velocity analysis to suspensions of fine particles in the dilute-phase regime. For upflow, the particle phase is assumed to lag behind the gas phase by an amount equal to the terminal velocity as obtained from the one-dimensional equation of motion for the particle in an infinite gas flow field. This model neglects any particle-particle or particle-wall effects that may be present in the flowing suspension. Hinkle,13 in extensive experiments, correlated the particle velocity, with a number of system parameters, as
Up ) Ug(1 - 1.53dp0.92 Fs0.5 Fg-0.2DT-0.54)
(10)
where dp is the diameter of the particles in feet, Fs and Fg are the densities in pounds per cubic foot of particles and fluid, respectively, and DT is the inside diameter in feet of the conveying duct. A comparison of these estimation methods for the particle velocity, as applied to the present study, showed the two to agree within approximately 15%. Macroscopic Continuity Equations. Steady-state macroscopic equations are used to relate actual and superficial gas velocities. The continuity equations for gas and particle phases respectively are
(6)
where the first term on the right-hand side is the viscous drag force exerted by the gas on the particle and the second term is the gravity force acting on the particle. The upward direction is positive and, because velocity is a vector, the absolute value factor is required to give that term the correct sign. Values of the particle projected cross-sectional areato-volume ratio, Ap/Vp, and the drag coefficient, CD, must be known to solve eq 6. For a sphere, the crosssectional area-to-volume ratio is
Ap 3 ) Vp 2dp
following approximation accurate to within 2% over a range of particle Reynolds numbers, Rep, from 1 to 250:
Fg(1 - E)Ug ) FgUsg
(11)
FsEUp ) FsUsp
(12)
where E is the volume fraction of solids, Usg is the superficial gas velocity, and Usp is the superficial solids velocity. The slip velocity, Vs, is defined as the difference between velocities of the gas and particle phases, where, using eqs 11 and 12,
Vs ) Us - Up )
Up Usg 1-E E
(13)
The volume fraction of solids (holdup), in terms of the slip velocity, can be expressed by
(
E)-
) [(
Um - Vs ( 2Vs
)
Um - Vs 2Vs
2
+
(
)]
Um Qs 2Vs Qs + Qg
1/2
(14) where Qs and Qg are volumetric flow rates of solid and
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gas, respectively, and
as constants given by
Um ) Usg + Usp
(15)
The sign of the second term of eq 14 is selected so that 0 e E e 1. Pressure Drop. The method of Soo14 for analyzing the pressure drop for a gas-solid, two-phase flow was applied, where a macroscopic mechanical energy balance includes pressure drops due to acceleration, friction, and static head, of both the gas and particle phases, respectively, as given by
∆PT ) ∆Pag + ∆Pas + ∆Pfg + ∆Pfs + ∆PFg + ∆PFs (16) In the experiments reported here, the two acceleration terms in eq 16 were negligible. The frictional pressure drop for the gas phase is
FgUg2 L 2 D
∆Pfg ) 4f
(17)
where f is the Fanning friction factor, correlated in terms of the Reynolds number of the gas stream, Reg, and the pipe inside-wall roughness. Govier and Aziz15 thoroughly discuss the static solids head, which is the force required to support the solids against the force of gravity
∆PFs ) EFsL
(18)
If it is assumed that the frictional pressure drop due to the gas in the suspension flow is the same as that for the flow of gas alone at the same gas superficial mass velocity, the solids frictional pressure drop represents the increased frictional pressure drop due to introduction of particles into the conveying stream. From experimental data, eqs 17 and 18 are used to calculate the solids frictional pressure drop from the manometer reading, which with lines full of liquid is
∆Pm ) ∆Pf + ∆PFs ) ∆Pfg + ∆Pfs + ∆PFs
(19)
From the solids frictional pressure drop, a solids friction factor is defined as
fs )
2D∆Pfs 4FsEUp2L
(20)
Pfeffer et al.16 discuss the expected dependence of the solids friction factor on the superficial gas Reynolds number, Resg, solids loading ratio, and particle diameter. Heat-Transfer Analysis. One-dimensional macroscopic equations governing bulk gas and particle temperatures are based on the work of Lempel,17 neglecting radiative fluxes to and internal temperature gradients in the particles. For the gas phase and particle phase, respectively,
dTg ) β - γ(Tg - Tp) dz
(21)
dTp ) R(Tg - Tp) dz
(22)
where for given run conditions R, β, and γ can be taken
Sp kg Nu Vp dpFsUpCps p
(23)
β ) 4qwi/kgPe
(24)
γ ) RLR(Cps/Cpg)
(25)
R)
where kg is the gas thermal conductivity, Cps and Cpg are the specific heats of the solid and gas, respectively, Nup is the particle Nusselt number, qwi is the inside wall heat flux, Pe is the gas Peclet number, and LR is the solids loading ratio. Assuming that physical properties of the flowing gas and solid particles are constant for the temperature range of the experiments and that gas and particle temperatures are equal at the inlet to the heat-transfer test section, then R, β, and γ are constant for a given experimental run with a constant wall heat-flux boundary condition. For these conditions, eqs 21 and 22 are readily integrated for an inlet gas temperature of Tg0 to
Tg(z) ) Tg0 +
( ) [
]
Tp(z) ) Tg0 +
( ) [
]
Rβ βγ z+ × R+γ (R + γ)2 {1 - exp[-(R + γ)z]} (26) Rβ Rβ z+ × R+γ (R + γ)2 {1 - exp[-(R + γ)z]} (27)
Over the bulk gas and solids temperature ranges of the experiments, the specific heats varied by less than 1%, while the gas thermal conductivity varied by less than 3.5% from the average value. Experimental Apparatus and Procedure Experimental difficulties involved in the measurement of flowing gas-particle systems are substantial. The abrasive nature of the suspension and the tendency for particles to destroy or clog any probe or tap are soon encountered. The experimental apparatus, depicted in Figure 1, was comprised of components to admit a regulated supply of compressed air and coal particles into a smooth-wall transport line, establish a developed cocurrent upward suspension flow, and transfer heat to the suspension. Air flowed through the apparatus on a once-through basis, but coal particles were recycled. Metered air was obtained from a gas supply and metering system. Compressed air at 90 psig passed through a filter, a primary gas regulator to reduce the pressure to 60 psig, another regulator to deliver the air at a pressure of 20-25 psig, and a calibrated rotameter. With this system, no appreciable fluctuations were observed in the gas supply pressure to the rotameter under operating conditions. Coal particles were metered into the air stream with a calibrated electromagnetic vibratory feeder, which controlled the feed rate by varying the dc power input to the magnetic coil of the feeder. After passage through the pressure drop and heat-transfer test sections, a heat exchanger, and a return line, separation of the coal
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Figure 1. Experimental apparatus.
particles from the conveying air was performed by a collector hopper, which acted as a gravity-settling chamber. The entire flow system was grounded to avoid buildup of static electric charge, which was readily detectable in a flexible polyethylene-diverting tube located in the uppermost region above the heat-transfer section. The coal-particle feed was prepared by screening with standard woven wire mesh screens. The average particle size was 300 µm, with a size range of 100-500 µm. Approximately 80% of the particle mass was in the range of 250-400 µm. With time, larger particles underwent unavoidable mechanical breakage. Particles of diameter of less than 60 µm were continuously removed by passing through the collector hopper. Sizedistribution sieve analyses were performed periodically during the course of the experiments to maintain an average particle size of 300 µm. Pressure-Drop and Temperature Measurements. The axial distribution of the pressure drop was measured in a 25-ft-high test section so that both acceleration and fully developed flow regimes could be observed for the upward-flowing, coal-air suspensions. The test section was constructed from smooth-walled, seamless, stainless steel tubing (type 304) of 5/8 in. o.d. × 0.065 in. wall. Wall pressure taps were located at 14 different axial positions spaced from 1 to 2 ft apart. All pressure lead lines were of 1/4-in. polyethylene tubing. Each line contained a particle trap located near the tube-wall tap. These traps removed entrained solids in the pressure lines and damped any of the smaller pressure fluctuations of the flowing suspension. Pressure readings were obtained by sequentially selecting pressure lead lines through a valve-switching bank connected to various manometers. Air leakage from the system was found to be insignificant. Pressure-drop profiles for the continuous, steady-state flow of air alone were measured
for Reynolds numbers between 10 000 and 30 000 and found to agree closely with accepted friction-factor correlations. Profiles were then measured for the suspension flow over the same range of Reynolds numbers for a range of loading ratios. The long pressure-drop test section provided a fully developed suspension flow upon entering the heat-transfer test section. Radial temperature profiles for suspensions leaving the heat-transfer test section were measured for all runs, including those with air alone. Frequent removal of coal from the heattransfer surface area of the test section kept fouling effects to a minimum. The heat-transfer test section was constructed from 50 in. of the same straight tubing as that used for the pressure-drop test section. The two test sections were electrically isolated from each other by placing 2-in.-thick phenolic insulators between copper connector flanges that had been silver-soldered to the stainless steel transport line. The tubing of the heattransfer test section was heated by electrical resistance heating provided by a dc rectifier-type welding unit capable of supplying up to 300 A. Heat losses from the heat-transfer test section were minimized by use of insulation. With electrical heating, the axial wall heat flux was essentially uniform and was determined from measurements of voltage and current. The outside axial tube-wall surface temperature profile was measured by 21 30-gauge thermocouples (type K), which were electrically isolated from the tube wall by 0.001-in.-thick mica sheets. The radial temperature profile of the suspension leaving the heat-transfer test section was obtained from a thermocouple probe, which traversed the tube diameter. The probe was constructed from a 20-gauge stainless steel hypodermic needle of 0.036-in.o.d. × 0.006-in. wall thickness. Located within the needle was a 0.020-in.-o.d. type K thermocouple that extended 1/8 in. past the needle top. The needle was bent so that the enclosed thermocouple protruded 1-1/2 in. into the oncoming flow to minimize heat conduction loss. Data for heat transfer to air alone were obtained for average gas Reynolds numbers between 10 000 and 30 000. Measured Nusselt numbers agreed to within (8% of the correlation of Kays.18 Heat-transfer rates to upward-flowing, coal-air suspensions were measured for three gas flow rates at varying solids loading ratios. Particle impact heating of the radial exit-gas temperature thermocouple was shown to be negligible in isothermal experimental runs. Solids loading ratios of up to 8 were examined under ambient conditions, and no appreciable frictional heating of the radial thermocouple was observed. Experimental Results and Discussion In this section, results are presented for axial pressure drop profiles, fully developed pressure drop, radial temperature profiles, and fully developed heat transfer under a constant wall heat-flux boundary condition for the upward flow of air-coal particle suspensions. The results are compared to the flow of air alone and to the downward flow of air-coal particle suspensions. Axial Pressure-Drop Profiles. Typical profiles of measured pressure-drop (∆Pm) gradients for air alone and suspension flows for loading ratios of up to 11.0 lb of solids/lb of air are shown in Figure 2, where the superficial gas Reynolds number is 10 000, the temperature is 81 °F, and the pressure range is 12.6-13.1 psia. The maximum distance from the tube entry to a pressure tap was 20.89 ft. With a tube inside diameter
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Figure 2. Typical suspension pressure-drop profiles for Resg ) 10 000.
Figure 3. Fully developed pressure drop.
of 0.041 25 ft, this corresponds to an entry length-todiameter ratio, L/D, of approximately 500. For the flow of air alone, the pressure-drop profile is established by an L/D of less than 50, which is in agreement with the literature. For suspension flow, large pressure gradients are required to accelerate the particles, and the entry length is increased significantly in the solids loading range of 4.4-11.0. For the highest solids loading, the entry length is approximately 200 tube diameters, and the fully developed pressure gradient is almost twice that for air alone. Similar experimental results for cocurrent downflow showed approximately the same increased entry length; however, the fully developed pressure gradient was almost identical with that for air alone. At a superficial gas Reynolds number of almost 20 000 for cocurrent upflow, entry lengths were approximately the same as those for a Reynolds number of 10 000, and fully developed pressure gradients with solids present were greater than those for air alone. Fully Developed Pressure-Drop Gradient. As shown in Figure 3, measurements of the fully developed pressure-drop gradient for cocurrent upflow with superficial gas Reynolds numbers in the range of 10 000-
Figure 4. Measured and frictional components of suspension pressure drop.
20 000 show that the pressure-drop gradient is a linear function of solids flow rates as high as 166 lb/h, for a fixed Reynolds number. At a Reynolds number of 10 000, this solids flow rate corresponds to a solids loading ratio of 11.0. At this condition, the fractional volumetric holdup of solids, computed from eq 14, is 0.009 or 0.9%. The linear relationship between the solids mass flow rate and the fully developed pressure-drop gradient is characteristic of dilute phase, coarse-particle systems and was observed throughout the duration of the study. Solids Friction Factor. Using eqs 3, 6, and 8-15, the total frictional component of the pressure-drop gradient, ∆Pf/L, was computed by eq 19 from the measured pressure-drop gradient and the computed solids static head from eq 18. The results are shown in Figure 4 for superficial gas Reynolds numbers of approximately 10 000 and 20 000, where the frictional component increases with an increase in the coal mass flow rate. Using eqs 17, 19, and 20, solids friction factors were computed. These are shown in Figure 5 as a function of the solids loading ratio for four Reynolds numbers in the range of 10 000-20 000, where in all four cases the solids friction factor decreases with an increase in the solids loading factor. The rate of decrease increases with increasing Reynolds number. Included in Figure 5 is a line that represents the correlation of the data of Brewster19 for downflow of 300-µm particles of coal in air at Reynolds numbers of 10 000-20 000. At the lower solids loadings ratios, the values of the solids friction factor for downflow are somewhat less than 50% of the values for upflow. Computed solids friction factors and corresponding particle velocities, assumed here to be the difference between the gas and particle terminal velocities, are compared to literature correlations in Figure 6. For the range of Reynolds numbers covered in the experiments, the values of the friction factors lie mostly within the range of the correlations, although at the lower limit of the correlations. Solids friction factors determined for
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Figure 5. Effect of the solids loading ratio on the solids friction factor.
Figure 7. Typical axial tube-wall temperature profile.
Figure 8. Typical radial temperature profile.
Figure 6. Comparison of the solids friction factors to correlations and downflow data.
cocurrent downflow by Brewster19 over the same ranges of superficial gas Reynolds number and solids loading ratio, shown by an envelope in Figure 6, are approximately 50% of the values for cocurrent upflow. This may be due to a higher degree of turbulence in upflow compared to downflow. Axial and Radial Temperature Profiles. Complete axial and near-exit radial gas-temperature profiles were measured in all experimental heat-transfer runs with both air alone and flowing suspensions. Figure 7 shows the effect of solids loading on the axial walltemperature profile for an average superficial gas Reynolds number of 10 000 for solids loading ratios of up to 4.8. For air alone, the thermal entry region extends to an entry length of approximately 15 tube diameters. From that location, the temperature increases linearly for the constant wall heat-flux boundary condition. In the presence of solids, the thermal entry length is increased considerably, because of the increased thermal capacity of the suspension and the time lag for the transfer of energy from the gas to the solids. Typical radial temperature profiles are shown in Figure
8 at an entry length of 80 tube diameters for a superficial gas Reynolds number of approximately 10 000 and solids loading ratios up to 6.7. For all four runs, the wall temperature at that location was 185 ( 2 °F. As the solids loading ratio is increased, the temperature profile becomes less flat. For the highest solids loading ratio, the difference between the centerline temperature and the wall temperature is almost 100 °F, while for air alone, the difference is less than 50 °F. Similar increasing temperature differences were observed as the solids loading ratio was increased for cocurrent downflow. As mentioned earlier, no particle-impact heating of the radial thermocouple probe was observed, and, therefore, no correction was applied to the measured thermocouple reading. No erosion of the thermocouple probe was noted even after significant accumulated run time, although initial pitting of the stainless steel thermocouple containment sheath was observed during the final suspension experiments. Nusselt Numbers for Heat Transfer. The uniform rate of electrical heat generation along the length of the heat-transfer test section was computed from the measured current and voltage drop. Most of this heat was transferred to the suspension flowing through the tube, but up to 10% of the heat generated was lost to the surroundings. The magnitude of the heat loss was determined by measurements with air alone, where the exit gas profile was used to determine the average exiting gas temperature, from which the energy transfer to the gas could be determined and compared with the
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Figure 9. Effect of the solids loading ratio on the particle Nusselt number.
rate of electrically generated heat. The rate of heat loss was correlated with the measured tube-outside wall temperature. The correlation was used to correct the heat-transfer measurements with the flow of air-coal particle suspensions. As mentioned above, fully developed Nusselt numbers for air alone were in excellent agreement with literature correlations. The experimental temperature measurements were used to compute both wall-to-gas and gas-to-particle Nusselt numbers, given by Nuw ) hwgD/kg and Nup ) hgsdp/kg, respectively, where hwg was based on the difference between the inside wall and bulk gas temperatures. For the former, inside wall temperatures were computed from measured outside wall temperatures using the method described by Massier.20 The inside and outside wall temperatures generally differed by only about 1 °F. Average values of Nup were calculated to force exit bulk-gas temperatures predicted by eq 26 to agree with those calculated by integrating measured exit gas radial temperature profiles. Calculated values are shown in Figure 9 as a function of solids loading ratios of up to 4.8 and superficial gas Reynolds numbers from 10 000 to 20 000. The values of the particle Nusselt numbers are in the range of 5-9, which is higher than the limiting value of 2 for single particles in infinite quiescent surroundings. The particle Nusselt numbers decrease with an increase in the solids loading ratio and increase with an increase in the Reynolds number. Almost identical particle Nusselt numbers were observed for cocurrent downflow of gas-coal particle suspensions. Using eq 26, local wall Nusselt numbers, Nuw, were determined. Values for a superficial Reynolds number of approximately 10 000 are plotted in Figure 10 as a function of the dimensionless axial distance for air alone and two solids loading ratios. The effect of solids on the thermal entry length is substantial. However, for Reynolds numbers of 15 000 and 20 000, the effect of solids was much less. Similar trends were observed for the cocurrent downward flow. For suspension flow, asymptotic wall Nusselt numbers were usually achieved within 40 tube diameters. Results for three superficial gas Reynolds numbers as a function of solids loading are shown in Figure 11. Little effect of the solids loading ratio is observed above a value of 1. At the lowest Reynolds number, the presence of the particles has no effect on the wall Nusselt number.
Figure 10. Typical axial wall Nusselt number profile.
Figure 11. Effect of the solids loading ratio on the asymptotic wall Nusselt number.
For the two higher Reynolds numbers, the wall Nusselt number is reduced by not more than 10%. At a Reynolds number of 15 000, asymptotic wall Nusselt numbers for cocurrent downflow were observed to be approximately 10% lower than those for cocurrent upflow, probably because of the increased frictional effect of the particles and wall in the upflow case. For cocurrent upflow, the trends of the data in Figure 11 are in general agreement with the theory of Louge et al.21 and the experimental data of Jepson et al.,22 in that at low solids loading ratios the wall Nusselt number decreases with an increase in the solids loading ratio. Conclusions Based on pressure-drop and heat-transfer measurements for cocurrent upflow of air-coal particle suspensions and earlier work for cocurrent downflow, the following conclusions are presented for dilute suspensions: 1. The hydraulic and thermal entry length regions for suspension flow are extended over those for air alone. The entry regions are weakly dependent on both the solids loading ratio and the superficial gas Reynolds number. 2. Measured and frictional pressure drops increase linearly with solids loading. Frictional pressure drops for cocurrent upflow are larger than those for cocurrent downflow. The frictional pressure drop obtained by
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subtracting the solids static head from the measured pressure drop appears to be independent of the gas Reynolds number but is a stronger function of the solids loading ratio than is the corresponding cocurrent downflow frictional pressure drop. 3. A solids friction factor, defined analogously to the Fanning friction factor, correlates the solids frictional pressure drop in terms of the loading ratio and the superficial gas Reynolds number, if the particle velocity is known. 4. The presence of solids can substantially increase the difference between the centerline temperature and the wall temperature for both cocurrent upflow and cocurrent downflow. 5. Particle Nusselt numbers are higher than the lower limiting value of 2 and increase with an increase in the Reynolds number but decrease with an increase in the solids loading ratio. Particle Nusselt numbers for cocurrent downflow are almost identical with those for cocurrent upflow. The wall Nusselt number decreases somewhat with the solids loading ratio and is somewhat higher than that for cocurrent downflow. Literature Cited (1) Depew, C. A.; Kramer, T. J. Heat Transfer to Flowing GasSolid Mixtures. Advances in Heat Transfer; Academic Press: New York, 1973. (2) Kim, J. M.; Seader, J. D. Heat Transfer to Gas-Solids Suspensions Flowing Cocurrently Downward in a Circular Tube. AIChE J. 1983, 29, 306. (3) Kim, J. M.; Seader, J. D. Pressure Drop for Cocurrent Downflow of Gas-Solids Suspensions. AIChE J. 1983, 29, 353. (4) Brewster, B. S.; Seader, J. D. Measuring Temperature in a Flowing Gas-Solids Suspension with a Thermocouple. AIChE J. 1984, 30, 676. (5) Brewster, B. S.; Seader, J. D. Coal Particle Suspensions in Vertical Downflow. AIChE J. 1984, 30, 996. (6) Needham, L. W.; Hill, N. W. The Shape and Specific Surface of Coal Particles. Fuel Sci. Pract. 1935, 14 (8), 222. (7) Needham, L. W.; Hill, N. W. The Settling of Mineral Particles in Water. Fuel Sci. Pract. 1947, 26 (4), 101. (8) Austin, L. G.; Gardner, R. P.; Walker, P. L., Jr. The Shape Factors of Coals Ground in a Standard Hardgrove Mill. Fuel 1963, 42, 319. (9) Kreith, F. Principles of Heat Transfer, 3rd ed.; Intext Press: New York, 1973. (10) Boothroyd, R. G. Flowing Gas-Solids Suspensions; Chapman and Hall: London, 1971.
(11) Robertson, J. Q.; Crowe, C. T. Engineering Fluid Mechanics; Houghton Mifflin Co.: Boston, 1975. (12) Klinzing, G. E. Gas-Solid Transport; McGraw-Hill: New York, 1981. (13) Hinkle, B. L. Ph.D. Dissertation, Georgia Institute of Technology, Atlanta, GA, 1953. (14) Soo, S. L. Fluid Dynamics of Multiphase Systems; Ginn and Blaisdell Publishing Co.: Walthan, MA, 1967. (15) Govier, G. W.; Aziz, K. The Flow of Complex Mixtures in Pipes; Van Nostrand Reinhold: New York, 1972. (16) Pfeffer, R.; Rosetti, S.; Lieblein, S. Analysis and Correlation of Heat Transfer Coefficients and Friction Factor Data for Dilute Gas-Solid Suspensions; NASA Technical Note D-3603; 1962. (17) Lempel, M. J. A Numerical Method for Determining HeatTransfer Characteristics for a Dilute Gas-Solids Mixture in an Externally Heated Tube. Bur. Mines Inf. Circ. 1967, 8343. (18) Kays, W. M. Convective Heat and Mass Transfer; McGrawHill: New York, 1980. (19) Brewster, B. S. Heat Transfer and Pressure Drop in CoalAir Suspensions Flowing Downward Through a Vertical Tube. Ph.D. Dissertation, Department of Chemical Engineering, University of Utah, Salt Lake City, UT, 1979. (20) Massier, P. F. A Forced-Convection and Nucleate-Boiling Heat-Transfer Test Apparatus; Technical Report No. 32-47; Jet Propulsion Laboratory: Pasadena, CA, Mar 3, 1961. (21) Louge, M.; Yusof, J. M.; Jenkins, J. T. Heat Transfer in the Pneumatic Transport of Massive Particles. Int. J. Heat Mass Transfer 1993, 2, 265. (22) Jepson, G.; Poll, A.; Smith, W. Heat Transfer from Gas to Wall in a Gas/Solids Transport Line. Trans. Inst. Chem. Eng. 1963, 41, 207. (23) van Swaaij, W. P. M.; Buurman, C.; van Breugel, J. W. Shear Stresses on the Wall of a Dense Gas-Solids Riser. Chem. Eng. Sci. 1970, 25, 1818. (24) Reddy, K. V. S.; Pei, D. C. T. Particle Dynamics in SolidsGas Flow in a Vertical Pipe. Ind. Eng. Chem. Fundam. 1969, 8, 490. (25) Capes, C. E.; Nakamura, K. Vertical Pneumatic Conveying: An Experimental Study with Particles in the Intermediate and Turbulent Flow Regimes. Can. J. Chem. Eng. 1973, 51, 31. (26) Konno, H.; Saito, S. Pneumatic Conveying of Solids Through Straight Pipes. J. Chem. Eng. Jpn. 1969, 2, 211. (27) Stemerding, S. Pneumatic Transport of Cracking Catalyst in Vertical Risers. Chem. Eng. Sci. 1962, 17, 599.
Received for review May 31, 2000 Revised manuscript received September 22, 2000 Accepted October 6, 2000 IE000531W
