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LDV measurements and analysis of gas and particulate phase velocity profiles in a vertical jet plume in a 2D bubbling fluidized bed Part II: Mass and momentum transport
Powder Technology 220 (2012) 47–54
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LDV measurements and analysis of gas and particulate phase velocity profiles in a vertical jet plume in a 2D bubbling fluidized bed Part II: Mass and momentum transport Alexander G. Mychkovsky ⁎, Steven L. Ceccio University of Michigan, Department of Mechanical Engineering, Ann Arbor, MI, 48109, USA
a r t i c l e
i n f o
Available online 19 September 2011 Keywords: Fluidization Convective transport Momentum transfer Multiphase flow Laser velocimetry Two-phase jet
a b s t r a c t A Laser Doppler Velocimetry (LDV) technique was implemented to simultaneously measure the gas and particulate phase velocities in a high-speed jet plume in a two-dimensional (2D) bubbling fluidized bed. The gas and particulate phase velocity profiles are presented and analyzed. This includes similarity profile scaling as well as volume fraction, mass flow, and momentum transport calculations for the two phases. Furthermore, applying the Eulerian equation of motion to the particulate phase with the measured velocity profiles, the bed particle drag coefficient is recovered and is found to be consistent with the established empirical value. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Fluidized beds are often employed as chemical reactors and solid fuel combustors due to the rapid mixing of their contents. In many configurations, high-speed gas jets are employed to introduce reactants into a bubbling particulate emulsion. When a gas jet is injected into a bubbling bed, particles and interstitial gas in the emulsion are vigorously advected in the jet plume region. Quantifying the mass and momentum transport of the jet is necessary to predict and optimize the efficiency of the reactor. The mass entrainment and subsequent momentum transfer between the gas jet and the emulsion are still not completely understood due, in part, to lack of experimental data. This is because the gas-particle flow is largely opaque in the bubbling emulsion and extremely harsh and abrasive in the jet plume. As a result, semi-empirical relations are often used to describe the jet dynamics [1–3]. In order to quantify the mass and momentum transport in the jet, the velocity profiles of both phases at various axial locations must be known. An overview of transverse and axial velocity profile measurements in bubbling bed gas jet plumes is given by [4]. Typically, gas velocities are obtained with Pitot tubes and high speed video is used to determine the particle velocities. The experimental data gathered with these techniques indicates that the gas and particulate phase velocity profiles appear approximately self-similar [5,6]. However, integration of these profiles to quantify the mass flow and momentum transport of the two phases
⁎ Corresponding author. E-mail address: [email protected] (A.G. Mychkovsky). 0032-5910/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2011.09.028
has not led to consistent results. [7] noted that the volumetric flow of air that was determined by integrating the gas velocity profiles far exceeded the value recorded by the air flow meters, and that estimated air entrainment was not sufficient to account for this discrepancy. Consequently, analysis of the velocity profiles reported in the literature has been limited to describing the general shapes and development of the velocity profiles. Presented here are simultaneously measured gas and particulate phase velocity profiles in a high-speed jet plume in a twodimensional (2D) bubbling fluidized bed. These gas and particulate phase velocity profiles are presented and analyzed, including similarity profile scaling as well as volume fraction, mass flow, and momentum transport calculations for the two phases. A momentum balance for the two phases is applied to determine the void fraction and these data are also used to recover the gas-particle drag coefficient in the high Reynolds number gas jet plume.
2. Experimental setup Experiments were conducted in a 2D bubbling fluidized bed, which is shown in Fig. 1. The bed dimensions are 457 mm wide by 1 m tall with a 12.7 mm gap (w). The walls are transparent acrylic with 102 mm by 153 mm by 5 mm thick quartz viewing windows inserted 50 mm above the vertical jet inlet orifice, which is 9.2 mm in diameter (Dj). The vertical jet is located midway across the porous polyethylene fluidization distributor and is set flush with its surface. The axial coordinate y is referenced with respect to the jet inlet and the transverse coordinate x is referenced with respect to the jet centerline.
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Fig. 1. Vertical gas jet in the laboratory 2D bubbling fluidized bed. The bed dimensions are 457 mm wide by 1 m tall with a 12.7 mm gap. The walls are transparent acrylic with 102 mm by 153 mm by 5 mm thick quartz viewing windows inserted 50 mm above the vertical jet inlet orifice, which is 9.2 mm in diameter. The square quartz windows may be used for horizontal jet experiments in the future. The vertical jet is located midway across the porous polyethylene fluidization distributor. The particle emulsion is white and the jet plume is black.
The particles used in the emulsion are 838 μm Sauter mean diameter (910 μm volume mean diameter) high-density polyethylene (HDPE) micropellets, which have a density of 900 kg/m 3. These particles are considered Geldart Group B particles, which bubble immediately beyond minimum fluidization. The minimum fluidization velocity (vmf) for the emulsion was experimentally determined to be 29 cm/s. The velocity profiles presented in this paper were obtained with a jet inlet velocity (Vj) of 92 m/s and a distributor fluidization velocity (vfl) 15% greater than the minimum fluidization value. The jet the inlet mass flow rate (mj) was 8 g/s and the inlet momentum (Jj) 0.73 kg.m/s 2. The temperature of the jet (Tj) was maintained at − 5 °C with an air density (ρg) of 1.3 kg/m 3. In order to minimize static charge effects, a pinch of Larostat 519 powder was added to the emulsion and the fluidization air was humidified with a room temperature bubbler. Bed particle and gas phase velocities were obtained with a twocomponent LDV system. In order to avoid damaging the optical access windows, the high-speed jet gas was seeded with ice crystals, which were formed by rapidly condensing and freezing the moisture in the jet air just prior to injection via a dry ice heat exchanger. LDV bursts from the bed particles and gas tracer ice crystals were simultaneously recorded. These Doppler signals were differentiated based on their burst intensity and temporal coincidence to yield the particulate and gas phase velocities at a given location within the jet plume. The uncertainty for these measurements is approximately ±0.5 m/s along the centerline, ±1 m/s within the half-velocity point core region, and ±2 m/s near the edges along the plume boundary. A detailed description of the measurement technique is given in [8]. 3. Gas jet into the empty bed 3.1. Empty bed velocity profiles Before attempting to analyze the two-phase measurements in the bubbling bed jet plume, the behavior of the single-phase gas jet in the empty 2D bed was examined. Jet cross-sectional velocity profiles were measured at various axial locations at the same inlet conditions as described in the previous section. The jet transverse velocity profiles taken at axial locations of y = 70 mm, 100 mm, and 130 mm are shown in Fig. 2. Velocity profiles could not be recorded at
Fig. 2. Gas phase velocity profiles in the empty bed. Vj = 92 m/s.
y = 60 mm for the empty bed configuration (as they were in the bubbling bed) because the gas velocities near the centerline at this near axial location exceeded the dynamic range of the LDV system. There are several ways to describe the bell-curve shape of the velocity profiles at a given axial location. One of the most common profiles referenced for the jet plume transverse velocity distribution in a fluidized bed is the Schlichting profile [1,9]. However, for the sake of computational ease, a Gaussian profile is often used (Ounnar, 2009). h i vg 2 ¼ exp − lnð2Þξg vg;m
ð1Þ
where the similarity variable is scaled with the half-velocity point ξg ¼
x xg;1=2
ð2Þ
and vm is the centerline, maximum velocity at a given axial location and the subscript ‘g’ denotes the gas phase. The normalized velocity profiles are scaled with their respective half-velocity points and compared with the Gaussian profile in Fig. 3. The values of the half velocity points are determined by the slope of the data when plotted on a semi-log plot. The empty bed gas jet velocity profiles appear selfsimilar when presented in this manner. 3.2. Empty bed transport It should be noted that a classic 2D jet is formed with a narrow injection slot that is very long in the third dimension. However, this configuration is not practical in a fluidized bed, for optical access reasons, as it would require a very wide bed gap. Therefore, instead of being infinitely long in the third dimension, a 2D fluidized bed simply restricts the bed particle and gas movement within a plane formed by a narrow gap between two bounding walls. At the axial distances of the velocity profile measurements, the gas jet exhibits plume spreading and axial velocity decay behavior consistent with a 2D jet [10]. Though the jet can only spread in two dimensions, the bounding walls create a velocity gradient across the bed gap, as shown in Fig. 4. Note that the no-slip condition has been applied at the wall boundaries since LDV measurements near the windows were not possible, as slight window blemishes are recorded as stationary Doppler bursts amongst other optical reasons [11]. This situation is analogous to a Hele-Shaw flow, where the velocity profile in the z direction is parabolic. This permits integration of the velocity profile with regard
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49
where C1 = 0.7 and C2 = 0.55. From this point forward, v = vavg and v 2 = v 2avg as defined by Eqs. (3) and (4). Rather than overestimate measured values, the gas mass flow rate at any axial location in the jet plume should be calculated as b
m˙ g ¼ C1 ρg w ∫ vg dx
ð5Þ
−b
and the jet momentum rate as b
2 J˙ g ¼ C2 ρg w ∫ vg dx
ð6Þ
−b
Fig. 3. Normalized gas phase velocity profiles in the empty bed scaled with respect to the half-velocity points and compared to the Gaussian profile. Vj = 92 m/s.
to z and thus consider an effective velocity field in only the x, y plane. Since the LDV measurement volume is located in the middle of the bed gap, the velocity values reported in the prior velocity profiles correspond to peak values rather than average values across the bed gap. Therefore, in order to use the integrated velocity profiles in the jet plume to calculate the mass and momentum transport, relationships between the average value and the peak values across the bed gap must be used to correct for overestimations. (Note that vg,peak is used with respect to the third dimension, whereas vg,m is used with respect to the transverse direction.) The average value of the velocity across the gap is approximately 70% of the peak value and the average value of the velocity squared is about 55% of the peak value squared, so that vg;avg ¼
2
vg;avg
1 ∫ vðzÞdz ≈ C1 vg;peak ww
1 2 2 ¼ ∫ ½vðzÞ dz ≈ C2 vg;peak ww
ð3Þ
ð4Þ
where b is the jet plume width. This situation has been observed but not properly accounted for in the literature. [7] noted that the volumetric flow of air calculated by integrating jet gas velocity profiles obtained via pitot tube measurements in their 2D fluidized bed greatly exceeded the amount of air injected and that entrained air could not account for this excess. Most likely, the reason for this overestimation of gas flow is because the three dimensional nature of the gas velocity profile was not taken into account and the measured peak velocity values were used rather than an average value across the bed gap. In addition to overestimating the jet mass flow rate at various downstream locations, [7] also overestimated the momentum rate. In order to compensate for this, the momentum transport at the various axial locations were normalized with ∫
v2g Vj2
ð7Þ
dξ
This is not the same as normalizing with respect to the jet inlet momentum since the inlet has a smaller, circular cross-sectional area. Therefore, this scaling omits a factor of (π/4)2 and overestimates the actual inlet momentum rate but was necessary for normalization so that the momentum transport reported downstream did not exceed unity. One of the defining characteristics of a free jet is that its axial momentum is conserved. The vertical jet gas momentum transport can be measured at an axial location by integrating measured velocity profiles according to Eq. (6). Written in terms of a similarity velocity profile, vg ¼ vg;m ðyÞf
x
! ð8Þ
xg;1=2 ðyÞ
the jet momentum transport is " b ˙J ¼ C ρ w ∫ v2 f g;m g 2 g −b
x xg;1=2
!#2 dx
ð9Þ
Changing the variable of integration to ξ = x/xg,1/2 yields 2 2 2 J˙ g ¼ C2 ρg w vg;m xg;1=2 ∫ ½ f ðξÞ dξ
ð10Þ
−2
Only the maximum velocity and half-velocity width are functions of axial distance while the other terms are constant. The finite integral can be evaluated for a given similarity velocity profile shape. For the sake of computational ease, the Gaussian profile is used and the integral term can be evaluated with the error function yielding a value of 1.50. Therefore, the momentum rate can be simply calculated as Fig. 4. Bed gap velocity measurements with a parabolic velocity profile. vg,avg / vg,peak ~ 0.7 2 2 and vg,avg / vg,peak ~ 0.55.
2 J˙ g ¼ 1:5 C2 ρg w vg;m xg;1=2
ð11Þ
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Using experimental values for vg,m and xg,1/2 at the various axial locations the calculated momentum rates based on the integration of the Gaussian similarity velocity profile are reported in Table 1. Similarly, the mass flow rate, expressed in terms of the similarity velocity profile is 2 m˙ g ¼ C1 ρg w vg;m xg;1=2 ∫ f ðξÞ dξ
Table 2 Mass flow rates in the jet plume in the empty bed determined from the Gaussian profiles at downstream locations. The jet inlet mass flow is mj = 8 g/s.
ð12Þ
Physical Location, y (mm)
mg/mj
70 100 130
1.7 2.1 2.6
−2
Again, using the error function to integrate the Gaussian profile, the value of the finite integral is 2.09, so that the mass flow rate can be simply calculated as m˙ g ¼ 2:09 C1 ρg w vg;m xg;1=2
ð13Þ
reveals the presence of particles significantly contributes to the decay of the gas phase velocity. The slight decline in the measured particulate phase Eulerian velocity as a function of axial distance is due to the entrainment of relatively stationary particles. 4.2. Void Fraction in the two-phase jet plume
The jet mass flow rate calculated at the three downstream axial locations using experimental values for vg,m and xg,1/2 are reported in Table 2. Note that in this transport analysis of the gas jet plume heat exchange is neglected and thus the gas phase density is considered to be constant in the measurement region.
In the bubbling bed, an average void fraction at various axial locations in the jet plume can be determined by a momentum balance with the jet inlet. Writing the momentum transport in terms of the velocity profiles and volumetric void fraction, ε " # b b 2 2 J˙ j ¼ wC2 ∫ ερg vg dx þ ∫ ð1−εÞρp vp dx
4. Gas jet into the bubbling bed
−b
4.1. Bubbling bed velocity profiles
ð16Þ
−b
Once the vertical jet in the empty bed had been characterized, the bed was filled with HDPE microspheres and fluidized 15% beyond minimum fluidization. The gas jet was run at the same inlet conditions as before. The velocity profiles of both the gas and particulate phases taken at axial locations of y = 60 mm, 70 mm, 100 mm, and 130 mm are shown in Figs. 5 and 6. From this data, the slip velocity, defined as
The densities of the gas and particulate phases are considered to be constant. The phase concentration is considered to be uniform across the jet so that ε = ε(y) as particles are entrained and the jet expands. This is consistent with the hydrodynamic model of Massimilla in Davidson et al. (1985).
vs ¼ vg −vp
ε¼
ð14Þ
can be calculated. Likewise, the particle Reynolds number in the jet plume, defined as
Rep ¼
ρg vg −vp g dp
ð15Þ
μg
b
J˙ j −wC2 ρp ∫ v2p dx "
−b b
wC2 ρg ∫ −b
b
v2g dx−ρp
∫
#
ð17Þ
v2p dx
−b
The momentum averaged volumetric void fraction at each axial location is shown in Table 3. As expected, the jet plume is very dilute with an average gas volume fraction of 95 to 96%. Since the average value of the void fraction is high and experiences only a small change,
is plotted in Fig. 7. The normalized velocity profiles, scaled with their respective half-velocity points, are shown with the Gaussian profile in Figs. 8–10. The gas, particulate, and slip velocity profiles appear to be self-similar when presented in this manner. The nature of the jet plume spreading and centerline velocity decay is shown in Figs. 11 and 12. These data are juxtaposed with gas phase measurements obtained in the empty bed at the same jet inlet conditions for the sake of comparison. Fig. 11 indicates that particulate phase velocity profile is slightly wider that the gas phase velocity profile. This is reasonable since all of the particles in the jet plume are entrained from rest along the emulsion boundary whereas the majority of the gas in the plume originates from the jet orifice. However, the half-velocity point growth for the gas phase in the bubbling and empty bed is remarkably similar. As expected, Fig. 12
Table 1 Momentum rates in the jet plume in the empty bed determined from the Gaussian profiles at downstream locations. The jet momentum is Jj = 0.73 kg.m/s2. Physical Location, y (mm)
Jg/Jj
70 100 130
0.97 0.99 1.04
Fig. 5. Gas phase velocity profiles for the jet in the bubbling bed of 838 μm HDPE particles. Vj = 92 m/s. Vfl/Vmf = 1.15.
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Fig. 6. Particulate phase velocity profiles for the jet in the bubbling bed of 838 μm HDPE particles. Vj = 92 m/s. Vfl/Vmf = 1.15.
ε could be considered constant for the gas phase mass and momentum calculations in this particular case. On the other hand, the volumetric solids fraction (1 − ε) cannot be considered constant for the particulate transport calculations. This is because the value of (1 − ε) is very small, and therefore even the small change in solids fraction in the streamwise direction is significant. Furthermore, the density of the particles is nearly three orders of magnitude greater than the gas. The calculated values suggest that, perhaps contrary to intuition, the solids fraction actually decreases in the streamwise direction shortly beyond the jet orifice. However, this is consistent with the work of [1,12], who defined an initial entrainment zone followed by a linear expansion region where the particle entrainment rate gradually decreases.
Fig. 8. Normalized gas phase velocity profiles scaled with respect to the half velocity point for the jet into the bubbling bed of 838 μm HDPE particles. Vj = 92 m/s. Vfl/Vmf = 1.15.
void fraction are constant across the profile. As before, using the error function to analytically integrate the self-similar Gaussian velocity profiles, the mass flow and momentum rates can be obtained with knowledge of the respective maximum velocity, half-velocity point, and volumetric void and solids fractions as a function of axial location. m˙ g ¼ 2:09 C1 ερg w vg;m xg;1=2
ð18Þ
m˙ p ¼ 2:09 C1 ð1−εÞρp w vp;m xp;1=2
ð19Þ
4.3. Bubbling bed mass and momentum transport The mass flow and momentum rates for the two phases can be calculated at each downstream location assuming that the densities and
Fig. 7. Particle Reynolds number profiles for the jet in the bubbling bed of 838 μm HDPE particles. Vj = 92 m/s. Vfl/Vmf = 1.15.
Fig. 9. Normalized particulate phase velocity profiles scaled with respect to the half velocity point for the jet into the bubbling bed of 838 μm HDPE particles. Vj = 92 m/s. Vfl/Vmf = 1.15.
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A.G. Mychkovsky, S.L. Ceccio / Powder Technology 220 (2012) 47–54
Fig. 12. Maximum velocity values for the gas, particulate, and slip velocity profiles in the bubbling bed of 838 μm HDPE particles compared to the empty bed gas phase data. Vj = 92 m/s. Vfl/Vmf = 1.15. Fig. 10. Normalized slip velocity profiles scaled with respect to the half velocity point for the jet into the bubbling bed of 838 μm HDPE particles. Vj = 92 m/s. Vfl/Vmf = 1.15.
2 J˙ g ¼ 1:5 C2 ερg w vg;m xg;1=2
ð20Þ
2 J˙ p ¼ 1:5 C2 ð1−εÞρp w vp;m xp;1=2
ð21Þ
nearly constant at the inlet value. The momentum data indicates that more than half of the initial jet gas momentum is already transferred to the particulate phase at first axial measurement location. Significant initial momentum transfer is to be expected due to high particle entrainment and large relative slip velocity values near the jet inlet. 5. Particle drag coefficient in the jet plume
Using experimental values for these variables, the calculated mass flow and momentum rates for both phases are plotted with respect to the physical origin in Figs. 13 and 14. The mass flow rates are juxtaposed with the empty bed gas phase data. Recall that the momentum rate for the gas phase in the empty bed remains constant at the inlet value, and is therefore not shown in the figures. The mass flow data indicates that particle entrainment near the jet inlet is significant and that the rate of entrainment into the jet plume decreases with streamwise distance. Once again, this is consistent with an initial entrainment zone followed by a linear expansion region where the particle entrainment rate gradually decreases. On the other hand, the gas phase mass flow in the jet plume remains
Momentum is transferred from the jet gas to the entrained particles via drag force. The drag force on a particle in a gas flow is FD ¼
2 1 C A ρ v −vp 2 D p g g
ð22Þ
which is positive in the upwards vertical direction since the coefficient of drag (CD) is a positive value and vg N vp everywhere in the jet plume. Note that AP is the projected area (circular cross-section) of the particle, as opposed to the surface area of the particle. The equation of motion for an individual particle is FD −Wp ¼ mP vp
dvp dy
ð23Þ
Since ρp ≫ ρg, acceleration terms (added mass and Basset force) have been neglected. Eq. (23) indicates that as long as FD N Wp, an individual particle's velocity will increase in the streamwise direction. Fig. 4 indicates that the particle Reynolds number in the plume is greater than 1,000 for all locations near the jet centerline. At these high Reynolds numbers, the coefficient of drag on a sphere has a nearly constant value of 0.4. Therefore, the drag force on an individual
Table 3 Void fraction values in the jet plume in the bubbling bed determined from a momentum balance at each axial location with the jet inlet using Eq. (17).
Fig. 11. Half-velocity point values for the gas, particulate, and slip velocity profiles in the bubbling bed of 838 μm HDPE particles compared to the empty bed gas phase data. Vj = 92 m/s. Vfl/Vmf = 1.15.
Physical Location, y (mm)
ε (%)
60 70 100 130
95 95 96 96
A.G. Mychkovsky, S.L. Ceccio / Powder Technology 220 (2012) 47–54
53
where the drag and weight forces are fD ¼ Np FD ¼ ð1−εÞ
ΔxΔy F Vp D
ð25Þ
wp ¼ ð1−εÞρp gΔxΔy
ð26Þ
where Np is the number of particles in the control volume and Vp is the volume of an individual particle. The momentum transport is 2 J˙ p ¼ ð1−εÞρp vp Δx
ð27Þ
Substituting Eqs. (25)–(27), (22), and (14) into (24) and taking the limit as Δx and Δy go to zero ð1−εÞ Fig. 13. Gas and particulate phase mass flow rates in the jet plume determined from the Gaussian profiles and calculated void fractions. The two-phase data in the bubbling bed of 838 μm HDPE particles is compared to the empty bed gas phase data. Vj = 92 m/s. Vfl/Vmf = 1.15.
838 μm HDPE particle in the jet core is about 50,000 times greater than the weight force due to gravity. However, Fig. 8 indicates that vp does not increase in the streamwise direction. The reason for this apparent inconsistency is due to the different frames of reference. The particle velocity in the equation of motion for an individual particle is taken in the Lagrangian frame of reference. On the other hand, the LDV system records the velocity of particles passing though a specified location, rather than tracking the velocity of a specific particle. These velocity measurement values are with respect to the Eulerian frame of reference, which treats the particulate phase as a continuum. To clarify, the velocity of an individual particle does increase as it is accelerated downstream by the vertical drag force of the jet gas (Lagrangian frame). However, the average particle velocity decreases with downstream distance since several slow moving particles are entrained along the jet boundaries (Eulerian frame). Considering the jet to be steady state, the Eulerian equation of motion per unit bed width for a differential area (Δx by Δy) of the particulate phase is J˙ p y − J˙ p yþΔy þ fD −wp ¼ 0
! i CD Ap ρg 2 d h 2 ð1−εÞvp dx vs dx−ð1−εÞρp gdx ¼ ρp dy 2Vp
ð28Þ
Integrating across the region of interest in the jet plume and considering (1 − ε) to be a function of y only yields # ! " y2 b CD Ap ρg y2 2 ∫ ð1−εÞ ∫ vs dx dy−2ρp g∫ ð1−εÞbdy 2Vp y1 y1 −b b nh i h i o 2 2 ¼ ρp ∫ ð1−εÞvp y2 − ð1−εÞvp y1 dx
ð29Þ
−b
Expressing the velocity profiles in the similarity form of v(x, y) = vm(y)f(ξ), where ξ = x/x1/2, # !y " y2 b CD Ap ρg 2 2 2 ∫ ð1−εÞvs;m ∫ fs ðξÞdx dy−2g∫ ð1−εÞbdy 2Vp ρp y y ¼
h
−b
1
ð1−εÞv2p;m
1
i b h h i b h i i 2 2 2 ∫ fp ðξÞ dx− ð1−εÞvp;m ∫ fp ðξÞ dx y2
−b
y2
y1
−b
ð30Þ
y1
Changing variables in order to carry out the Gaussian velocity profile integrations b
2
∫ f ðξÞdx ¼ 1:5 x1=2
ð31Þ
−b
ð24Þ
If the weight of the particle is negligible, knowledge of the shape of the self-similar profile is not necessary, only the fact that b
2
2
2
∫ f ðξÞdx ¼ x1=2 ∫ f ðξÞdξ ¼ ðconst Þx1=2
ð32Þ
−2
−b
is needed as the constant would cancel from both sides of Eq. (30) if gravity is neglected. However, the particle weight is considered in this general formulation. Also expressing b = 2x1/2, the coefficient of drag can be solved by re-arranging Eq. (30) y2
CD ¼
2Vp ρp Ap ρg
! 4g ∫ ð1−εÞxp;1=2 dy þ 1:5
h
i h i 2 2 ð1−εÞvp;m xp;1=2 − ð1−εÞvp;m xp;1=2 y2
y1
y1
y2
1:5 ∫ ð1−εÞv2s;m xs;1=2 dy y1
ð33Þ
Fig. 14. Gas and particulate phase momentum rates in the jet plume in the bubbling bed of 838 μm HDPE particles determined from the Gaussian profiles and calculated void fractions. The empty bed gas phase momentum rate remains constant at the inlet value. Vj = 92 m/s. Vfl/Vmf = 1.15.
The integrals in Eq. (33) can either be carried out numerically with 2 andx1/2for the particulate and slip velocity experimental values of vm profiles at the various axial locations or by analytically integrating the 2 expressions for vm and x1/2 as a function of y if they are known. The former was chosen and the results are shown in Table 4. The reasonable agreement between these values and the established empirical
54
A.G. Mychkovsky, S.L. Ceccio / Powder Technology 220 (2012) 47–54 Table 4 Particle drag coefficients calculated from Eq. (33). The established empirical value is approximately 0.4. The HDPE micropellets are treated as spheres and the particulate phase and slip velocity Gaussian profiles as well as the calculated solids fraction values are used. The major source of error is the uncertainty of the solids volume fraction at each axial location. Physical Distance, y2-y1 (mm)
CD
60–70 70–100 100–130
0.47 0.29 0.40
value of 0.4 indicates that gas-particle drag is the dominant mechanism of momentum transfer and that particle-particle collisions are not significant in the dilute jet plume. The discrepancy between the values in Table 4 is primarily due to the uncertainty in (1 − ε). Furthermore, the bed particle microspheres are slightly cylindrical in shape, and therefore this analysis can only serve as a first order approximation. 6. A discussion of uncertainty estimates The uncertainty for velocity measurements is approximately ±0.5 m/s along the centerline, ±1 m/s within the half-velocity point core region, and ±2 m/s near the edges along the plume boundary and has been addressed in [8]. In order to determine the repeatability of the jet plume transport analysis procedure, the mass flow and momentum rates at each axial location were determined by 1) numerically integrating the LDV data points and by 2) using Gaussian velocity profiles for the respective phases. The resulting mass and momentum transport values are within +/−5% [10]. As expected, the data shows little deviation at the y = 60 and 70 mm locations but increases downstream due to the jet fluctuations. The void fractions and momentum rates calculated using the two techniques are nearly identical. The mass flow rates calculated using the Gaussian profile technique are slightly higher. This is because the momentum, and thus void fraction, calculations involve the square of the velocity profile, which is steeper than the velocity profile used for the mass flow calculations. Therefore, nearly all of the jet momentum is captured by the LDV data points whereas some of the jet mass flow lies beyond these measurements. 7. Conclusions Gas and particulate phase velocity profiles of a high-speed gas jet in a bubbling 2D fluidized bed have been successfully measured and analyzed. First, the high-speed gas jet in the empty 2D bed was characterized. In order to accurately quantify the mass and momentum transport in the jet plume, the 3D velocity gradients due to the bounding walls must be considered. As verification, the gas jet in the empty bed was found to conserve momentum at the downstream measurement locations. It should be noted that the 3D correction coefficients in presented in this work are not universal but rather specific to the geometry of the fluidized bed. Next, gas and particulate phase velocity profiles in a gas jet plume in a bubbling 2D fluidized bed were obtained simultaneously via LDV burst intensity and coincidence subranging. The emulsion of Geldart Group B particles was fluidized approximately 15% above minimum fluidization. The gas, particulate, and slip velocity profiles appear to
be self-similar and Gaussian. When compared to the empty bed gas jet plume at the same inlet conditions, the gas phase velocity is significantly reduced though the plume width growth is similar. The particulate phase centerline velocity decreases downstream due to the entrainment of slow moving particles from the emulsion along the plume boundary. An average void fraction was calculated at each axial measurement location from a momentum balance with the jet inlet and was found to be 95 to 96%. The mass and momentum transport of the gas and particulate phases was calculated from the measured velocity profiles. The results reveal significant momentum transfer from the jet gas to the entrained particles. Mass flow calculations show considerable particle entrainment near the jet inlet while the amount of gas in the jet plume remains nearly constant at these bed conditions. Using the particulate and slip velocity profile data along with the particulate phase equation of motion, the particle drag coefficient was calculated and is found to be consistent with the established empirical value of 0.4. This indicates that high Reynolds number gasparticle drag is the dominant mechanism of momentum transfer and that particle-particle collisions are not significant in the dilute jet plume. Acknowledgements This project is sponsored by the Department of Energy's Office of Fossil Energy's University Research Program under project number DE-NT0007649. The research is also performed under appointment to the Rickover Graduate Fellowship Program sponsored by Naval Reactors Division of the U.S. Department of Energy (A. Mychkovsky). References [1] J.M.D. Merry, Penetration of a horizontal gas jet into a fluidized bed, Transactions of the Institution of Chemical Engineers 49 (1971) 189–195. [2] J.M.D. Merry, Penetration of a vertical jets into fluidized beds, American Institute of Chemical Engineers 21 (1975) 507–510. [3] P.E. Roach, The penetration of jets into fluidized beds, Fluid Dynamics Research 11 (1993) 197–216. [4] L. Massimilla, Gas Jets in Fluidized Beds, in: J.F. Davidson, R. Clift, D. Harrison (Eds.), Fluidization, second ed., Academic Press, New York, 1985, pp. 133–171. [5] M. Filla, L. Massimilla, S. Vaccaro, Gas Jets in Fluidized Beds and Spouts: A Comparison of Experimental Behavior and Models, Canadian Journal of Chemical Engineering 61 (1983) 370–376. [6] A. Ounnar, J. Arrar, F. Bentahar, Hydrodynamic behaviour of upflowing jet in fluidized bed: velocity profiles of sand particles, Chemical Engineering and Processing 48 (2009) 617–622. [7] C. Xuereb, C. Laguerie, T. Baron, Etude du comportement de jets continues horizontaux ou inclines introduits dans un lit fluidise par un gaz Deuxieme partie: profiles de vitesse du gaz dans les jets horizontaux, Powder Technology 64 (1991) 271–283. [8] A.G. Mychkovsky, D. Rangarajan, S.L. Ceccio, LDV measurements and analysis of gas and particulate phase velocity profiles in a vertical jet plume in a 2D bubbling fluidized bed Part I: Two phase LDV measurement technique. Submitted for publication to the 2010 NETL Multiphase Flow Workshop Special Issue of Powder Technology. [9] N.A. Shakhova, G.A. Minaev, An engineering method of calculating a jet in a fluidized bed,, Inzhenerno-Fizicheskii Zhurnal 19 (1970) 1002–1011. [10] A.G. Mychkovsky, 2010. LDV measurements and analysis of gas and particulate phase velocity profiles in a vertical jet plume in a 2D bubbling fluidized bed. Dissertation, University of Michigan, Ann Arbor. [11] A.G. Mychkovsky, N.A. Chang, S.L. Ceccio, Bragg cell laser intensity modulation: effect on laser Doppler velocimetry measurements, Applied Optics 48 (2009) 3468–3474. [12] C. Xuereb, C. Laguerie, T. Baron, Etude du comportement de jets continues horizontaux ou inclines introduits dans un lit fluidise par un gaz I: Morphologie des jets, Powder Technology 67 (1991) 46–56.
Contents lists available at SciVerse ScienceDirect
Powder Technology journal homepage: www.elsevier.com/locate/powtec
LDV measurements and analysis of gas and particulate phase velocity profiles in a vertical jet plume in a 2D bubbling fluidized bed Part II: Mass and momentum transport Alexander G. Mychkovsky ⁎, Steven L. Ceccio University of Michigan, Department of Mechanical Engineering, Ann Arbor, MI, 48109, USA
a r t i c l e
i n f o
Available online 19 September 2011 Keywords: Fluidization Convective transport Momentum transfer Multiphase flow Laser velocimetry Two-phase jet
a b s t r a c t A Laser Doppler Velocimetry (LDV) technique was implemented to simultaneously measure the gas and particulate phase velocities in a high-speed jet plume in a two-dimensional (2D) bubbling fluidized bed. The gas and particulate phase velocity profiles are presented and analyzed. This includes similarity profile scaling as well as volume fraction, mass flow, and momentum transport calculations for the two phases. Furthermore, applying the Eulerian equation of motion to the particulate phase with the measured velocity profiles, the bed particle drag coefficient is recovered and is found to be consistent with the established empirical value. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Fluidized beds are often employed as chemical reactors and solid fuel combustors due to the rapid mixing of their contents. In many configurations, high-speed gas jets are employed to introduce reactants into a bubbling particulate emulsion. When a gas jet is injected into a bubbling bed, particles and interstitial gas in the emulsion are vigorously advected in the jet plume region. Quantifying the mass and momentum transport of the jet is necessary to predict and optimize the efficiency of the reactor. The mass entrainment and subsequent momentum transfer between the gas jet and the emulsion are still not completely understood due, in part, to lack of experimental data. This is because the gas-particle flow is largely opaque in the bubbling emulsion and extremely harsh and abrasive in the jet plume. As a result, semi-empirical relations are often used to describe the jet dynamics [1–3]. In order to quantify the mass and momentum transport in the jet, the velocity profiles of both phases at various axial locations must be known. An overview of transverse and axial velocity profile measurements in bubbling bed gas jet plumes is given by [4]. Typically, gas velocities are obtained with Pitot tubes and high speed video is used to determine the particle velocities. The experimental data gathered with these techniques indicates that the gas and particulate phase velocity profiles appear approximately self-similar [5,6]. However, integration of these profiles to quantify the mass flow and momentum transport of the two phases
⁎ Corresponding author. E-mail address: [email protected] (A.G. Mychkovsky). 0032-5910/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2011.09.028
has not led to consistent results. [7] noted that the volumetric flow of air that was determined by integrating the gas velocity profiles far exceeded the value recorded by the air flow meters, and that estimated air entrainment was not sufficient to account for this discrepancy. Consequently, analysis of the velocity profiles reported in the literature has been limited to describing the general shapes and development of the velocity profiles. Presented here are simultaneously measured gas and particulate phase velocity profiles in a high-speed jet plume in a twodimensional (2D) bubbling fluidized bed. These gas and particulate phase velocity profiles are presented and analyzed, including similarity profile scaling as well as volume fraction, mass flow, and momentum transport calculations for the two phases. A momentum balance for the two phases is applied to determine the void fraction and these data are also used to recover the gas-particle drag coefficient in the high Reynolds number gas jet plume.
2. Experimental setup Experiments were conducted in a 2D bubbling fluidized bed, which is shown in Fig. 1. The bed dimensions are 457 mm wide by 1 m tall with a 12.7 mm gap (w). The walls are transparent acrylic with 102 mm by 153 mm by 5 mm thick quartz viewing windows inserted 50 mm above the vertical jet inlet orifice, which is 9.2 mm in diameter (Dj). The vertical jet is located midway across the porous polyethylene fluidization distributor and is set flush with its surface. The axial coordinate y is referenced with respect to the jet inlet and the transverse coordinate x is referenced with respect to the jet centerline.
48
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Fig. 1. Vertical gas jet in the laboratory 2D bubbling fluidized bed. The bed dimensions are 457 mm wide by 1 m tall with a 12.7 mm gap. The walls are transparent acrylic with 102 mm by 153 mm by 5 mm thick quartz viewing windows inserted 50 mm above the vertical jet inlet orifice, which is 9.2 mm in diameter. The square quartz windows may be used for horizontal jet experiments in the future. The vertical jet is located midway across the porous polyethylene fluidization distributor. The particle emulsion is white and the jet plume is black.
The particles used in the emulsion are 838 μm Sauter mean diameter (910 μm volume mean diameter) high-density polyethylene (HDPE) micropellets, which have a density of 900 kg/m 3. These particles are considered Geldart Group B particles, which bubble immediately beyond minimum fluidization. The minimum fluidization velocity (vmf) for the emulsion was experimentally determined to be 29 cm/s. The velocity profiles presented in this paper were obtained with a jet inlet velocity (Vj) of 92 m/s and a distributor fluidization velocity (vfl) 15% greater than the minimum fluidization value. The jet the inlet mass flow rate (mj) was 8 g/s and the inlet momentum (Jj) 0.73 kg.m/s 2. The temperature of the jet (Tj) was maintained at − 5 °C with an air density (ρg) of 1.3 kg/m 3. In order to minimize static charge effects, a pinch of Larostat 519 powder was added to the emulsion and the fluidization air was humidified with a room temperature bubbler. Bed particle and gas phase velocities were obtained with a twocomponent LDV system. In order to avoid damaging the optical access windows, the high-speed jet gas was seeded with ice crystals, which were formed by rapidly condensing and freezing the moisture in the jet air just prior to injection via a dry ice heat exchanger. LDV bursts from the bed particles and gas tracer ice crystals were simultaneously recorded. These Doppler signals were differentiated based on their burst intensity and temporal coincidence to yield the particulate and gas phase velocities at a given location within the jet plume. The uncertainty for these measurements is approximately ±0.5 m/s along the centerline, ±1 m/s within the half-velocity point core region, and ±2 m/s near the edges along the plume boundary. A detailed description of the measurement technique is given in [8]. 3. Gas jet into the empty bed 3.1. Empty bed velocity profiles Before attempting to analyze the two-phase measurements in the bubbling bed jet plume, the behavior of the single-phase gas jet in the empty 2D bed was examined. Jet cross-sectional velocity profiles were measured at various axial locations at the same inlet conditions as described in the previous section. The jet transverse velocity profiles taken at axial locations of y = 70 mm, 100 mm, and 130 mm are shown in Fig. 2. Velocity profiles could not be recorded at
Fig. 2. Gas phase velocity profiles in the empty bed. Vj = 92 m/s.
y = 60 mm for the empty bed configuration (as they were in the bubbling bed) because the gas velocities near the centerline at this near axial location exceeded the dynamic range of the LDV system. There are several ways to describe the bell-curve shape of the velocity profiles at a given axial location. One of the most common profiles referenced for the jet plume transverse velocity distribution in a fluidized bed is the Schlichting profile [1,9]. However, for the sake of computational ease, a Gaussian profile is often used (Ounnar, 2009). h i vg 2 ¼ exp − lnð2Þξg vg;m
ð1Þ
where the similarity variable is scaled with the half-velocity point ξg ¼
x xg;1=2
ð2Þ
and vm is the centerline, maximum velocity at a given axial location and the subscript ‘g’ denotes the gas phase. The normalized velocity profiles are scaled with their respective half-velocity points and compared with the Gaussian profile in Fig. 3. The values of the half velocity points are determined by the slope of the data when plotted on a semi-log plot. The empty bed gas jet velocity profiles appear selfsimilar when presented in this manner. 3.2. Empty bed transport It should be noted that a classic 2D jet is formed with a narrow injection slot that is very long in the third dimension. However, this configuration is not practical in a fluidized bed, for optical access reasons, as it would require a very wide bed gap. Therefore, instead of being infinitely long in the third dimension, a 2D fluidized bed simply restricts the bed particle and gas movement within a plane formed by a narrow gap between two bounding walls. At the axial distances of the velocity profile measurements, the gas jet exhibits plume spreading and axial velocity decay behavior consistent with a 2D jet [10]. Though the jet can only spread in two dimensions, the bounding walls create a velocity gradient across the bed gap, as shown in Fig. 4. Note that the no-slip condition has been applied at the wall boundaries since LDV measurements near the windows were not possible, as slight window blemishes are recorded as stationary Doppler bursts amongst other optical reasons [11]. This situation is analogous to a Hele-Shaw flow, where the velocity profile in the z direction is parabolic. This permits integration of the velocity profile with regard
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49
where C1 = 0.7 and C2 = 0.55. From this point forward, v = vavg and v 2 = v 2avg as defined by Eqs. (3) and (4). Rather than overestimate measured values, the gas mass flow rate at any axial location in the jet plume should be calculated as b
m˙ g ¼ C1 ρg w ∫ vg dx
ð5Þ
−b
and the jet momentum rate as b
2 J˙ g ¼ C2 ρg w ∫ vg dx
ð6Þ
−b
Fig. 3. Normalized gas phase velocity profiles in the empty bed scaled with respect to the half-velocity points and compared to the Gaussian profile. Vj = 92 m/s.
to z and thus consider an effective velocity field in only the x, y plane. Since the LDV measurement volume is located in the middle of the bed gap, the velocity values reported in the prior velocity profiles correspond to peak values rather than average values across the bed gap. Therefore, in order to use the integrated velocity profiles in the jet plume to calculate the mass and momentum transport, relationships between the average value and the peak values across the bed gap must be used to correct for overestimations. (Note that vg,peak is used with respect to the third dimension, whereas vg,m is used with respect to the transverse direction.) The average value of the velocity across the gap is approximately 70% of the peak value and the average value of the velocity squared is about 55% of the peak value squared, so that vg;avg ¼
2
vg;avg
1 ∫ vðzÞdz ≈ C1 vg;peak ww
1 2 2 ¼ ∫ ½vðzÞ dz ≈ C2 vg;peak ww
ð3Þ
ð4Þ
where b is the jet plume width. This situation has been observed but not properly accounted for in the literature. [7] noted that the volumetric flow of air calculated by integrating jet gas velocity profiles obtained via pitot tube measurements in their 2D fluidized bed greatly exceeded the amount of air injected and that entrained air could not account for this excess. Most likely, the reason for this overestimation of gas flow is because the three dimensional nature of the gas velocity profile was not taken into account and the measured peak velocity values were used rather than an average value across the bed gap. In addition to overestimating the jet mass flow rate at various downstream locations, [7] also overestimated the momentum rate. In order to compensate for this, the momentum transport at the various axial locations were normalized with ∫
v2g Vj2
ð7Þ
dξ
This is not the same as normalizing with respect to the jet inlet momentum since the inlet has a smaller, circular cross-sectional area. Therefore, this scaling omits a factor of (π/4)2 and overestimates the actual inlet momentum rate but was necessary for normalization so that the momentum transport reported downstream did not exceed unity. One of the defining characteristics of a free jet is that its axial momentum is conserved. The vertical jet gas momentum transport can be measured at an axial location by integrating measured velocity profiles according to Eq. (6). Written in terms of a similarity velocity profile, vg ¼ vg;m ðyÞf
x
! ð8Þ
xg;1=2 ðyÞ
the jet momentum transport is " b ˙J ¼ C ρ w ∫ v2 f g;m g 2 g −b
x xg;1=2
!#2 dx
ð9Þ
Changing the variable of integration to ξ = x/xg,1/2 yields 2 2 2 J˙ g ¼ C2 ρg w vg;m xg;1=2 ∫ ½ f ðξÞ dξ
ð10Þ
−2
Only the maximum velocity and half-velocity width are functions of axial distance while the other terms are constant. The finite integral can be evaluated for a given similarity velocity profile shape. For the sake of computational ease, the Gaussian profile is used and the integral term can be evaluated with the error function yielding a value of 1.50. Therefore, the momentum rate can be simply calculated as Fig. 4. Bed gap velocity measurements with a parabolic velocity profile. vg,avg / vg,peak ~ 0.7 2 2 and vg,avg / vg,peak ~ 0.55.
2 J˙ g ¼ 1:5 C2 ρg w vg;m xg;1=2
ð11Þ
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Using experimental values for vg,m and xg,1/2 at the various axial locations the calculated momentum rates based on the integration of the Gaussian similarity velocity profile are reported in Table 1. Similarly, the mass flow rate, expressed in terms of the similarity velocity profile is 2 m˙ g ¼ C1 ρg w vg;m xg;1=2 ∫ f ðξÞ dξ
Table 2 Mass flow rates in the jet plume in the empty bed determined from the Gaussian profiles at downstream locations. The jet inlet mass flow is mj = 8 g/s.
ð12Þ
Physical Location, y (mm)
mg/mj
70 100 130
1.7 2.1 2.6
−2
Again, using the error function to integrate the Gaussian profile, the value of the finite integral is 2.09, so that the mass flow rate can be simply calculated as m˙ g ¼ 2:09 C1 ρg w vg;m xg;1=2
ð13Þ
reveals the presence of particles significantly contributes to the decay of the gas phase velocity. The slight decline in the measured particulate phase Eulerian velocity as a function of axial distance is due to the entrainment of relatively stationary particles. 4.2. Void Fraction in the two-phase jet plume
The jet mass flow rate calculated at the three downstream axial locations using experimental values for vg,m and xg,1/2 are reported in Table 2. Note that in this transport analysis of the gas jet plume heat exchange is neglected and thus the gas phase density is considered to be constant in the measurement region.
In the bubbling bed, an average void fraction at various axial locations in the jet plume can be determined by a momentum balance with the jet inlet. Writing the momentum transport in terms of the velocity profiles and volumetric void fraction, ε " # b b 2 2 J˙ j ¼ wC2 ∫ ερg vg dx þ ∫ ð1−εÞρp vp dx
4. Gas jet into the bubbling bed
−b
4.1. Bubbling bed velocity profiles
ð16Þ
−b
Once the vertical jet in the empty bed had been characterized, the bed was filled with HDPE microspheres and fluidized 15% beyond minimum fluidization. The gas jet was run at the same inlet conditions as before. The velocity profiles of both the gas and particulate phases taken at axial locations of y = 60 mm, 70 mm, 100 mm, and 130 mm are shown in Figs. 5 and 6. From this data, the slip velocity, defined as
The densities of the gas and particulate phases are considered to be constant. The phase concentration is considered to be uniform across the jet so that ε = ε(y) as particles are entrained and the jet expands. This is consistent with the hydrodynamic model of Massimilla in Davidson et al. (1985).
vs ¼ vg −vp
ε¼
ð14Þ
can be calculated. Likewise, the particle Reynolds number in the jet plume, defined as
Rep ¼
ρg vg −vp g dp
ð15Þ
μg
b
J˙ j −wC2 ρp ∫ v2p dx "
−b b
wC2 ρg ∫ −b
b
v2g dx−ρp
∫
#
ð17Þ
v2p dx
−b
The momentum averaged volumetric void fraction at each axial location is shown in Table 3. As expected, the jet plume is very dilute with an average gas volume fraction of 95 to 96%. Since the average value of the void fraction is high and experiences only a small change,
is plotted in Fig. 7. The normalized velocity profiles, scaled with their respective half-velocity points, are shown with the Gaussian profile in Figs. 8–10. The gas, particulate, and slip velocity profiles appear to be self-similar when presented in this manner. The nature of the jet plume spreading and centerline velocity decay is shown in Figs. 11 and 12. These data are juxtaposed with gas phase measurements obtained in the empty bed at the same jet inlet conditions for the sake of comparison. Fig. 11 indicates that particulate phase velocity profile is slightly wider that the gas phase velocity profile. This is reasonable since all of the particles in the jet plume are entrained from rest along the emulsion boundary whereas the majority of the gas in the plume originates from the jet orifice. However, the half-velocity point growth for the gas phase in the bubbling and empty bed is remarkably similar. As expected, Fig. 12
Table 1 Momentum rates in the jet plume in the empty bed determined from the Gaussian profiles at downstream locations. The jet momentum is Jj = 0.73 kg.m/s2. Physical Location, y (mm)
Jg/Jj
70 100 130
0.97 0.99 1.04
Fig. 5. Gas phase velocity profiles for the jet in the bubbling bed of 838 μm HDPE particles. Vj = 92 m/s. Vfl/Vmf = 1.15.
A.G. Mychkovsky, S.L. Ceccio / Powder Technology 220 (2012) 47–54
51
Fig. 6. Particulate phase velocity profiles for the jet in the bubbling bed of 838 μm HDPE particles. Vj = 92 m/s. Vfl/Vmf = 1.15.
ε could be considered constant for the gas phase mass and momentum calculations in this particular case. On the other hand, the volumetric solids fraction (1 − ε) cannot be considered constant for the particulate transport calculations. This is because the value of (1 − ε) is very small, and therefore even the small change in solids fraction in the streamwise direction is significant. Furthermore, the density of the particles is nearly three orders of magnitude greater than the gas. The calculated values suggest that, perhaps contrary to intuition, the solids fraction actually decreases in the streamwise direction shortly beyond the jet orifice. However, this is consistent with the work of [1,12], who defined an initial entrainment zone followed by a linear expansion region where the particle entrainment rate gradually decreases.
Fig. 8. Normalized gas phase velocity profiles scaled with respect to the half velocity point for the jet into the bubbling bed of 838 μm HDPE particles. Vj = 92 m/s. Vfl/Vmf = 1.15.
void fraction are constant across the profile. As before, using the error function to analytically integrate the self-similar Gaussian velocity profiles, the mass flow and momentum rates can be obtained with knowledge of the respective maximum velocity, half-velocity point, and volumetric void and solids fractions as a function of axial location. m˙ g ¼ 2:09 C1 ερg w vg;m xg;1=2
ð18Þ
m˙ p ¼ 2:09 C1 ð1−εÞρp w vp;m xp;1=2
ð19Þ
4.3. Bubbling bed mass and momentum transport The mass flow and momentum rates for the two phases can be calculated at each downstream location assuming that the densities and
Fig. 7. Particle Reynolds number profiles for the jet in the bubbling bed of 838 μm HDPE particles. Vj = 92 m/s. Vfl/Vmf = 1.15.
Fig. 9. Normalized particulate phase velocity profiles scaled with respect to the half velocity point for the jet into the bubbling bed of 838 μm HDPE particles. Vj = 92 m/s. Vfl/Vmf = 1.15.
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A.G. Mychkovsky, S.L. Ceccio / Powder Technology 220 (2012) 47–54
Fig. 12. Maximum velocity values for the gas, particulate, and slip velocity profiles in the bubbling bed of 838 μm HDPE particles compared to the empty bed gas phase data. Vj = 92 m/s. Vfl/Vmf = 1.15. Fig. 10. Normalized slip velocity profiles scaled with respect to the half velocity point for the jet into the bubbling bed of 838 μm HDPE particles. Vj = 92 m/s. Vfl/Vmf = 1.15.
2 J˙ g ¼ 1:5 C2 ερg w vg;m xg;1=2
ð20Þ
2 J˙ p ¼ 1:5 C2 ð1−εÞρp w vp;m xp;1=2
ð21Þ
nearly constant at the inlet value. The momentum data indicates that more than half of the initial jet gas momentum is already transferred to the particulate phase at first axial measurement location. Significant initial momentum transfer is to be expected due to high particle entrainment and large relative slip velocity values near the jet inlet. 5. Particle drag coefficient in the jet plume
Using experimental values for these variables, the calculated mass flow and momentum rates for both phases are plotted with respect to the physical origin in Figs. 13 and 14. The mass flow rates are juxtaposed with the empty bed gas phase data. Recall that the momentum rate for the gas phase in the empty bed remains constant at the inlet value, and is therefore not shown in the figures. The mass flow data indicates that particle entrainment near the jet inlet is significant and that the rate of entrainment into the jet plume decreases with streamwise distance. Once again, this is consistent with an initial entrainment zone followed by a linear expansion region where the particle entrainment rate gradually decreases. On the other hand, the gas phase mass flow in the jet plume remains
Momentum is transferred from the jet gas to the entrained particles via drag force. The drag force on a particle in a gas flow is FD ¼
2 1 C A ρ v −vp 2 D p g g
ð22Þ
which is positive in the upwards vertical direction since the coefficient of drag (CD) is a positive value and vg N vp everywhere in the jet plume. Note that AP is the projected area (circular cross-section) of the particle, as opposed to the surface area of the particle. The equation of motion for an individual particle is FD −Wp ¼ mP vp
dvp dy
ð23Þ
Since ρp ≫ ρg, acceleration terms (added mass and Basset force) have been neglected. Eq. (23) indicates that as long as FD N Wp, an individual particle's velocity will increase in the streamwise direction. Fig. 4 indicates that the particle Reynolds number in the plume is greater than 1,000 for all locations near the jet centerline. At these high Reynolds numbers, the coefficient of drag on a sphere has a nearly constant value of 0.4. Therefore, the drag force on an individual
Table 3 Void fraction values in the jet plume in the bubbling bed determined from a momentum balance at each axial location with the jet inlet using Eq. (17).
Fig. 11. Half-velocity point values for the gas, particulate, and slip velocity profiles in the bubbling bed of 838 μm HDPE particles compared to the empty bed gas phase data. Vj = 92 m/s. Vfl/Vmf = 1.15.
Physical Location, y (mm)
ε (%)
60 70 100 130
95 95 96 96
A.G. Mychkovsky, S.L. Ceccio / Powder Technology 220 (2012) 47–54
53
where the drag and weight forces are fD ¼ Np FD ¼ ð1−εÞ
ΔxΔy F Vp D
ð25Þ
wp ¼ ð1−εÞρp gΔxΔy
ð26Þ
where Np is the number of particles in the control volume and Vp is the volume of an individual particle. The momentum transport is 2 J˙ p ¼ ð1−εÞρp vp Δx
ð27Þ
Substituting Eqs. (25)–(27), (22), and (14) into (24) and taking the limit as Δx and Δy go to zero ð1−εÞ Fig. 13. Gas and particulate phase mass flow rates in the jet plume determined from the Gaussian profiles and calculated void fractions. The two-phase data in the bubbling bed of 838 μm HDPE particles is compared to the empty bed gas phase data. Vj = 92 m/s. Vfl/Vmf = 1.15.
838 μm HDPE particle in the jet core is about 50,000 times greater than the weight force due to gravity. However, Fig. 8 indicates that vp does not increase in the streamwise direction. The reason for this apparent inconsistency is due to the different frames of reference. The particle velocity in the equation of motion for an individual particle is taken in the Lagrangian frame of reference. On the other hand, the LDV system records the velocity of particles passing though a specified location, rather than tracking the velocity of a specific particle. These velocity measurement values are with respect to the Eulerian frame of reference, which treats the particulate phase as a continuum. To clarify, the velocity of an individual particle does increase as it is accelerated downstream by the vertical drag force of the jet gas (Lagrangian frame). However, the average particle velocity decreases with downstream distance since several slow moving particles are entrained along the jet boundaries (Eulerian frame). Considering the jet to be steady state, the Eulerian equation of motion per unit bed width for a differential area (Δx by Δy) of the particulate phase is J˙ p y − J˙ p yþΔy þ fD −wp ¼ 0
! i CD Ap ρg 2 d h 2 ð1−εÞvp dx vs dx−ð1−εÞρp gdx ¼ ρp dy 2Vp
ð28Þ
Integrating across the region of interest in the jet plume and considering (1 − ε) to be a function of y only yields # ! " y2 b CD Ap ρg y2 2 ∫ ð1−εÞ ∫ vs dx dy−2ρp g∫ ð1−εÞbdy 2Vp y1 y1 −b b nh i h i o 2 2 ¼ ρp ∫ ð1−εÞvp y2 − ð1−εÞvp y1 dx
ð29Þ
−b
Expressing the velocity profiles in the similarity form of v(x, y) = vm(y)f(ξ), where ξ = x/x1/2, # !y " y2 b CD Ap ρg 2 2 2 ∫ ð1−εÞvs;m ∫ fs ðξÞdx dy−2g∫ ð1−εÞbdy 2Vp ρp y y ¼
h
−b
1
ð1−εÞv2p;m
1
i b h h i b h i i 2 2 2 ∫ fp ðξÞ dx− ð1−εÞvp;m ∫ fp ðξÞ dx y2
−b
y2
y1
−b
ð30Þ
y1
Changing variables in order to carry out the Gaussian velocity profile integrations b
2
∫ f ðξÞdx ¼ 1:5 x1=2
ð31Þ
−b
ð24Þ
If the weight of the particle is negligible, knowledge of the shape of the self-similar profile is not necessary, only the fact that b
2
2
2
∫ f ðξÞdx ¼ x1=2 ∫ f ðξÞdξ ¼ ðconst Þx1=2
ð32Þ
−2
−b
is needed as the constant would cancel from both sides of Eq. (30) if gravity is neglected. However, the particle weight is considered in this general formulation. Also expressing b = 2x1/2, the coefficient of drag can be solved by re-arranging Eq. (30) y2
CD ¼
2Vp ρp Ap ρg
! 4g ∫ ð1−εÞxp;1=2 dy þ 1:5
h
i h i 2 2 ð1−εÞvp;m xp;1=2 − ð1−εÞvp;m xp;1=2 y2
y1
y1
y2
1:5 ∫ ð1−εÞv2s;m xs;1=2 dy y1
ð33Þ
Fig. 14. Gas and particulate phase momentum rates in the jet plume in the bubbling bed of 838 μm HDPE particles determined from the Gaussian profiles and calculated void fractions. The empty bed gas phase momentum rate remains constant at the inlet value. Vj = 92 m/s. Vfl/Vmf = 1.15.
The integrals in Eq. (33) can either be carried out numerically with 2 andx1/2for the particulate and slip velocity experimental values of vm profiles at the various axial locations or by analytically integrating the 2 expressions for vm and x1/2 as a function of y if they are known. The former was chosen and the results are shown in Table 4. The reasonable agreement between these values and the established empirical
54
A.G. Mychkovsky, S.L. Ceccio / Powder Technology 220 (2012) 47–54 Table 4 Particle drag coefficients calculated from Eq. (33). The established empirical value is approximately 0.4. The HDPE micropellets are treated as spheres and the particulate phase and slip velocity Gaussian profiles as well as the calculated solids fraction values are used. The major source of error is the uncertainty of the solids volume fraction at each axial location. Physical Distance, y2-y1 (mm)
CD
60–70 70–100 100–130
0.47 0.29 0.40
value of 0.4 indicates that gas-particle drag is the dominant mechanism of momentum transfer and that particle-particle collisions are not significant in the dilute jet plume. The discrepancy between the values in Table 4 is primarily due to the uncertainty in (1 − ε). Furthermore, the bed particle microspheres are slightly cylindrical in shape, and therefore this analysis can only serve as a first order approximation. 6. A discussion of uncertainty estimates The uncertainty for velocity measurements is approximately ±0.5 m/s along the centerline, ±1 m/s within the half-velocity point core region, and ±2 m/s near the edges along the plume boundary and has been addressed in [8]. In order to determine the repeatability of the jet plume transport analysis procedure, the mass flow and momentum rates at each axial location were determined by 1) numerically integrating the LDV data points and by 2) using Gaussian velocity profiles for the respective phases. The resulting mass and momentum transport values are within +/−5% [10]. As expected, the data shows little deviation at the y = 60 and 70 mm locations but increases downstream due to the jet fluctuations. The void fractions and momentum rates calculated using the two techniques are nearly identical. The mass flow rates calculated using the Gaussian profile technique are slightly higher. This is because the momentum, and thus void fraction, calculations involve the square of the velocity profile, which is steeper than the velocity profile used for the mass flow calculations. Therefore, nearly all of the jet momentum is captured by the LDV data points whereas some of the jet mass flow lies beyond these measurements. 7. Conclusions Gas and particulate phase velocity profiles of a high-speed gas jet in a bubbling 2D fluidized bed have been successfully measured and analyzed. First, the high-speed gas jet in the empty 2D bed was characterized. In order to accurately quantify the mass and momentum transport in the jet plume, the 3D velocity gradients due to the bounding walls must be considered. As verification, the gas jet in the empty bed was found to conserve momentum at the downstream measurement locations. It should be noted that the 3D correction coefficients in presented in this work are not universal but rather specific to the geometry of the fluidized bed. Next, gas and particulate phase velocity profiles in a gas jet plume in a bubbling 2D fluidized bed were obtained simultaneously via LDV burst intensity and coincidence subranging. The emulsion of Geldart Group B particles was fluidized approximately 15% above minimum fluidization. The gas, particulate, and slip velocity profiles appear to
be self-similar and Gaussian. When compared to the empty bed gas jet plume at the same inlet conditions, the gas phase velocity is significantly reduced though the plume width growth is similar. The particulate phase centerline velocity decreases downstream due to the entrainment of slow moving particles from the emulsion along the plume boundary. An average void fraction was calculated at each axial measurement location from a momentum balance with the jet inlet and was found to be 95 to 96%. The mass and momentum transport of the gas and particulate phases was calculated from the measured velocity profiles. The results reveal significant momentum transfer from the jet gas to the entrained particles. Mass flow calculations show considerable particle entrainment near the jet inlet while the amount of gas in the jet plume remains nearly constant at these bed conditions. Using the particulate and slip velocity profile data along with the particulate phase equation of motion, the particle drag coefficient was calculated and is found to be consistent with the established empirical value of 0.4. This indicates that high Reynolds number gasparticle drag is the dominant mechanism of momentum transfer and that particle-particle collisions are not significant in the dilute jet plume. Acknowledgements This project is sponsored by the Department of Energy's Office of Fossil Energy's University Research Program under project number DE-NT0007649. The research is also performed under appointment to the Rickover Graduate Fellowship Program sponsored by Naval Reactors Division of the U.S. Department of Energy (A. Mychkovsky). References [1] J.M.D. Merry, Penetration of a horizontal gas jet into a fluidized bed, Transactions of the Institution of Chemical Engineers 49 (1971) 189–195. [2] J.M.D. Merry, Penetration of a vertical jets into fluidized beds, American Institute of Chemical Engineers 21 (1975) 507–510. [3] P.E. Roach, The penetration of jets into fluidized beds, Fluid Dynamics Research 11 (1993) 197–216. [4] L. Massimilla, Gas Jets in Fluidized Beds, in: J.F. Davidson, R. Clift, D. Harrison (Eds.), Fluidization, second ed., Academic Press, New York, 1985, pp. 133–171. [5] M. Filla, L. Massimilla, S. Vaccaro, Gas Jets in Fluidized Beds and Spouts: A Comparison of Experimental Behavior and Models, Canadian Journal of Chemical Engineering 61 (1983) 370–376. [6] A. Ounnar, J. Arrar, F. Bentahar, Hydrodynamic behaviour of upflowing jet in fluidized bed: velocity profiles of sand particles, Chemical Engineering and Processing 48 (2009) 617–622. [7] C. Xuereb, C. Laguerie, T. Baron, Etude du comportement de jets continues horizontaux ou inclines introduits dans un lit fluidise par un gaz Deuxieme partie: profiles de vitesse du gaz dans les jets horizontaux, Powder Technology 64 (1991) 271–283. [8] A.G. Mychkovsky, D. Rangarajan, S.L. Ceccio, LDV measurements and analysis of gas and particulate phase velocity profiles in a vertical jet plume in a 2D bubbling fluidized bed Part I: Two phase LDV measurement technique. Submitted for publication to the 2010 NETL Multiphase Flow Workshop Special Issue of Powder Technology. [9] N.A. Shakhova, G.A. Minaev, An engineering method of calculating a jet in a fluidized bed,, Inzhenerno-Fizicheskii Zhurnal 19 (1970) 1002–1011. [10] A.G. Mychkovsky, 2010. LDV measurements and analysis of gas and particulate phase velocity profiles in a vertical jet plume in a 2D bubbling fluidized bed. Dissertation, University of Michigan, Ann Arbor. [11] A.G. Mychkovsky, N.A. Chang, S.L. Ceccio, Bragg cell laser intensity modulation: effect on laser Doppler velocimetry measurements, Applied Optics 48 (2009) 3468–3474. [12] C. Xuereb, C. Laguerie, T. Baron, Etude du comportement de jets continues horizontaux ou inclines introduits dans un lit fluidise par un gaz I: Morphologie des jets, Powder Technology 67 (1991) 46–56.