(260d) Dependence of Particle Fluctuation Velocity on Gas Flow, Particle Diameter, and Density in Gas Fluidized Beds for Polymer, Glass and Metal Spheres in the Geldart B Fluidization Regime

This paper discusses the experimental and theoretical implications of new experimental data[1] on the steady state, mean squared fluctuation velocity, or granular temperature, of Geldart B (bubble at fluidization), monodispersed, polymer, glass, nickel, and stainless steel spheres, as a function of superficial gas velocity in a fluidized bed. The data was obtained by power spectral analysis of the wall vibrational energy in a frequency range, 10-20 kHz, dominated by the Acoustic Shot Noise (ASN) of random sphere impact of the wall, and calibrated by hammer excitation over the wall, of the fluidized bed. All of the data is consistent with the conclusion of an earlier paper[2] that the ratio of the fluctuation velocity of Geldart B glass spheres to the gas superficial velocity, is inversely proportional to sphere diameter; and defines a fundamental length scale, D_o^B, for the particle diameter in gas fluidized beds. We show that independent data on the fluctuation velocity of gas fluidized glass spheres, derived from diffusing wave spectroscopy (DWS) analysis of the interference of reflected laser light scattered by random particle motion near the wall by Menon and Durian [3], and from measurements of the granular pressure exerted by random sphere impact on a porous membrane by Campbell and Wang [4], are consistent with the ASN data[1]. Clearly the length scale, D_o^B, is a fundamental feature of steady state random particle motion in a gas fluidized beds within, near and at the wall, and is not an experimental artifact. The inverse scaling to sphere diameter, of the dimensionless ratio of the particle fluctuation velocity to gas superficial velocity, and the consequent, necessary length scale, D_o^B, was not a feature of any published theoretical model on the steady state particle motion in gas fluidized beds, prior to 1999. Remarkably, it is an implicit feature of two recent dense kinetic theories of gas fluidized beds recently published by Buyevich and Kapbasov[5, 6] and Koch and Sangani[7]. Under the critical condition of small, but finite, collisional energy loss, both theories define a fundamental length scale, D_o^B = CD* = C(μ_f²/ ρ_f²g)^1/3 where: ρ_p is the sphere density; μ_f is the gas viscosity; g is the gravitational field in the fluidized bed. The constant C depends on the parameters of the specific theory. It is somewhat surprising that D* had not been previously identified as a significant sphere diameter length scale for gas fluidized beds, since many of the dimensionless constants that are used to characterize the gas fluidized bed in theoretical models[8] and CFD calculations[9] such as the Stokes, Reynolds, Froude, Galileo, and Archimedes numbers, are proportional to (D/D*)³. From the experimental data on the fluctuation velocity for polymer, glass, nickel and stainless steel spheres D_o^B =(56±2) D*, where for glass spheres, argon fluidization, and the earth gravitational field, D*=1.96 microns. The only free parameter of the Buyevich and Kapbasov model[5, 6] is the coefficient of restitution, and with e=0.99990 , D_o^B = 70 D*, in good agreement with the experimental data for Geldart B glass spheres and argon fluidization[1]. Under the same fluidization conditions, and the parameters of the Koch and Sangani model[7], D_o^B =10 D*, or a granular temperature and pressure that is 50 times less than that derived from the experimental data[1]. From the experimental data for the particle fluctuation velocity we can define the granular pressure, and use the Anderson-Jackson stability model[8, 10] for the initial, one dimensional, stability of a uniform gas fluidized bed against perturbations in particle concentration, to define a critical sphere diameter, D_G , such that for DG, the initial fluidized bed is stable, and for D> D_G, the initial fluidized bed is unstable. From the initial stability theory at minimum fluidization velocity[1], D_G is a linear function of D_o^B, and from the experimental fluctuation velocity data on monodispersed spheres, D_G = D_o^B =60 D* = 60(μ_f²/ ρ_f²g)^1/3. This first principles definition of D_G may be compared with the empirical definition of D_B/A, due to Geldart[11], where the condition D B/A, defines the Geldart A fluidization regime, which "fluidizes before bubbling", and the condition D>D_B/A, defines the Geldart B fluidization regime, which "bubbles at fluidization"[1]. While D_G=D_B/A=120 microns for glass, D_G=2D_B/A, for nickel/stainless steel densities, and D_G =0.5D_B/A, for polymer densities, since D_B/A≈(1/ρ_p)^1.2 and D_G≈(1/ρ_p)^0.67. These differences offer significant opportunities for experimental validation of the Anderson-Jackson stability model as a function of density. Moreover the inverse power law dependence of D_G ≈ (1/g)^0.33 on laboratory gravitational constant, g, is a factor of four smaller than that defined by the empirical Geldart stability scale, where D_B/A ≈(1/g)^1.2. The consequent factor of two to three difference on the power law dependence D_B/A and D_G on the laboratory gravitational field, g, should be a significant factor in designing, and interpreting, experiments on the Geldart fluidization regimes in a centrifugal gravitational field[12]. Finally the quantitative agreement between the recent experimental data for glass spheres in gas fluidized beds and the dense kinetic theory of Buyevich and Kapbasov[5,6], for the magnitude and functional dependence of the particle fluctuation velocity on gas flow, particle diameter, and density, suggests many theoretical, and CFD, opportunities to explore the limits of fundamental dense kinetic models for the hydrodynamics of fluidization. [8, 13].

References

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