(159h) Application of Drift-Flux Model to Phase Holdup in Liquid-Solids Circulating Fluidized Bed

Solids holdup and solids velocity in the riser of a liquid-solid circulating fluidized bed (LSCFB) are determined covering a wide range in experimental conditions. The data along with the data reported in literature are analyzed using the drift-flux model, and the model parameters are related to the operating conditions and particle characteristics.

Phase holdup in two-phase flow is due to (i) the velocity and concentration profiles across the cross-section and (ii) the local relative velocity between phases caused by gravitational effects. Drift-flux model takes into account the first effect by a distribution parameter and the second effect by the weighted average drift velocity. Though the model is satisfactorily validated to the flow of gas-liquid systems (Zuber and Findlay), its application to liquid-solid fluidized beds, especially to liquid-solid circulating fluidized beds, has not been investigated, an aspect which forms the subject matter of this paper.

Liquid-solid circulating fluidized beds offer effective contact between the phases giving enhanced heat and mass transfer rates, high liquid throughput per unit cross-section, a facility for continuous operation, and control of solids velocity and solids volume fraction through a choice of solids and liquid feed rates. They find wide application in biochemical processes, protein purification processes, and non-catalytic and catalytic liquid-solid reactions and in solids mixing and ore-dressing.

When liquid flows through a bed of solids at velocities exceeding the terminal velocity of the particle, particle entrainment begins, necessitating the feeding of fresh solids or recirculation of entrained solids to the bottom of the bed to maintain the solids inventory within the bed. This kind of operation is known as liquid-solid circulating fluidized bed (LSCFB). The permissible liquid velocities in LSCFB range from terminal velocity of the particles to the transport velocity. The flow structure is particulate fluidization with no evidence of formation of particle aggregates and axial bed voidage non-uniformities.

Significant earlier literature relating to the solids holdup and slip velocity is due to Kuramoto et al(1998), Liang and Zhu (1997), Linag et al(1997), Zheng et al (1999), Zheng and Zhu (2000). Zheng et. al. (2002).

Experimental:

Experiments are conducted using a Plexiglas riser column of 94mm i.d. and 2400 mm high (Figure 1). The base of the riser has two distributors, one each for the primary and auxiliary liquid flows into the riser. Primary liquid flows through a multi-tubular distributor, and the auxiliary liquid flows through a perforated plate ensuring uniform liquid distribution across the riser cross-section. Auxiliary liquid flow serves to fluidize the particles at the base of the riser, facilitates and regulates the solids flow into the riser. The combined flow of the primary and auxiliary liquid enables the particles to move cocurrently to the top of the riser, where they are separated from the liquid, and returned to the particle storage vessel via the particle flow-rate measuring device. The provision of the dual liquid flows in the distributor enables control of the liquid flow rate and the solids circulation rate independently by adjusting the auxiliary and primary liquid flows. Sand and cation-exchange resin of different sizes are used with water as the fluidizing medium. The average solids holdup is determined from the pressure gradient measured for each section along the riser length.  The experiments are conducted for different combinations of auxiliary and primary liquid flow rates.

Results and Discussion:

The experimental data show that U_s increases rapidly with U_f initially and approaches an asymptotic value, U_sm at high liquid velocities. U_s and U_sm are found to depend on the auxiliary liquid rate and particle characteristics. The solids holdup, e_s decreases with an increase in U_f and is higher for higher values of U_a. e_s increases with increase in U_s, for a given total liquid flow rate, U_l ; the increase is faster for particles of smaller size or of lower density. For a given U_a, however, e_s decreases with increase in U_s. The variation in e_s with a variation in U_s is higher for higher values of either U_a or U_l. Based on the observations, the data are correlated as

   σ = 0.12                       (1)

and in terms of solids circulation rate as

             σ = 0.15                       (2)

The Drift-flux Model:

The drift-flux model relates the true velocity of particles,  to the cross-sectional average volumetric flux density of the mixture, <j> as (Zuber and Findlay 1965).

                                                   (3)

where C_o is the distribution parameter and  is the weighted average drift velocity. C_o and  are evaluated using the experimental data of the present study and that of literature.   is found to range from 5x10^-3 to 8x10^-2 for the range of data (93 ≤ d_p ≤ 655 μ_m; 1100 ≤ ρ_s ≤ 2700 Kg/m³;   6.4x10^-3 ≤ U_t ≤ 8x10^-2 m/s).  The very small values for  are obtained with solids of small size or low density suggesting that the particles and the liquid have the same velocity at any radial position.  The average velocity of the particles is smaller than that of the liquid phase because of the particles are concentrated in the region of lower liquid velocity i.e. the wall region.  It is further expected that the local drift velocity depends on particle concentration as the presence of other particles affects the motion of a given particle.   is found to relate as 

                                                     (4)

          For the aforementioned range, C_o varies from 0.82 to 0.99, except for silica gel and plastic beads where C_o = 0.45.  C_o is less than one suggests that the solids concentration near the wall region is higher than the average holdup which is in conformity with the measured radial distributions. (Liang et al (1997); Zheng et al, 2002) presented measured radial concentration profiles in LSCFB.  Approximating the velocity concentration distributions by radial power laws. 

                                                                    (5a)

                                                     (5b)

the experimental data of the authors are satisfactorily compared (s < 0.03) with the model equations; n rages from 2 to 7, and directly relates to e_s.  Radial non-uniformity in solids holdup decreases with increase in U_l or a decrease in U_s; radial profile is unaltered with the axial position in a fully developed LSCFB.   m varies over small ranges from 2 to 3; m decrease with increase in U_l and does not significantly vary with U_s.  The foregoing analysis proves the applicability of drift-flux model to LSCFB to evaluate the relative magnitudes of the influence of radial non-uniform distributions and the local relative velocity between the phases.

Slip velocity for the different experimental conditions covering the present data and the data of Liang et al (1997), Kuramota et al (1998) and Zheng et al(1999) is estimated using the formula

                                                         (6)

U_R is satisfactorily related for the range 11.8  Ar  11484 as

U_R (m/s)  = 3x10^-3Ar ^0.466  ε^3.5                                                        (7)

Notations

Ar        Archimedes number,     C_o        Distribution Parameter,  d_p particle diameter,    j            liquid velocity at the center of the riser,  <j>     Average volumetric flux density, m, n   exponents in equations 5,  R   Radius of the riser,    U_a           auxiliary liquid velocity, m/s,   U_f                primary liquid velocity, m/s, U_l total liquid velocity, m/s,   U_S   solids circulation rate m/s,   U_sm maximum possible solids circulation rate, m/s  Ut  particle terminal velocity, m/s,   Weighted average drift flux velocity.

Greek letters

ε_l          bed voidage (-), ε_s        solids holdup (-),ε_cs solids holdup at center, ε_sw  solids holdup nearwall, σ          Root Mean Square deviation, ρ_l  liquid density (kg/m³), ρ_s    particle density (kg/m³)

References:

1.      Koji Kuramoto, et. al. (1998), ?Macroscopic flow structure of Solid particles in circulating liquid solid fluidized bed riser?, Journal of Chemical Engineering of Japan, 31, p.258-265.

2.      Wu-Geng Liang, et. al. (1997a), ?Flow characteristics of the liquid solid circulating fluidized bed?, Powder Technology, 90 p.95-102.

3.      Wu-Geng Liang, Jing-Xu Zhu (1997b), ?A core-annulus model for the radial flow structure in a liquid solid circulating fluidized bed (LSCFB)? Chemical Engineering Journal, 68, p.51-62.

4.      Zheng .Y. J-X Zhu et. al., Radial solids flow structure in a Liquid Solids Circulating fluidized bed, The Chemical Engineering Journal, 2002, 141-150.

5.      Zheng Y., Jing-Xu (Jesse) Zhu, et al(2000a), ?(Gas-) Liquid-Solid circulating fluidized beds and their potential applications to Bioreactor engineering?, The Canadian Journal of Chemical Engineering, 78, p.82-94.

6.      Zheng Y. et. al. (1999), ?The Axial Hydrodynamic Behavior in  a Liquid-solid circulating fluidized bed?, The Canadian Journal of Chemical Engg., 77, p.284-290.

7.      Zuber. N. and Findlay,  Average volumetric concentration in two-phase flow system, Journel of Heat Transfer, 1965, 453-468.