Distended Beds Research Articles

Power Law Fluid Flow Through Beds of Spheres at Intermediate Reynolds Numbers: Pressure in Fixed and Distended Beds

Extensive numerical predictions of friction, pressure and total drag coefficients for the steady incompressible flow of power law liquids through assemblages of spherical particles are presented in this paper. The governing equations (continuity and momentum) have been solved numerically for the unknown pressure and velocity components which, in turn, have been used to infer the values of the derived variables like vorticity, stream function, pressure and frictional components of the overall drag coefficient. The inter-particle hydrodynamic interactions have been mimicked by the two commonly used concentric spherein-sphere cell models. The numerical results presented herein span the following ranges of kinematic and physical conditions: power law index, 0.5, 0.6, 0.8, 1, 1.2, 1.4, 1.6, 1.8, thereby including both shear-thinning and shear-thickening flow behaviour; the particle Reynolds number, 1–500; and bed voidages of 0.4, 0.5 and 0.6, thereby covering packed and distended bed conditions. The accuracy of the numerical solution procedure has been validated by carrying out extensive comparisons with the previously available analytical and scant numerical results, mainly limited to the creeping flow conditions. The utility of such a simple approach is demonstrated by presenting detailed comparisons with the experimental results available in the literature.

Hydrodynamics of creeping flow of power law fluids through particle assemblages

The equations of motion describing the creeping flow of power law liquids past particle assemblages have been solved numerically using the finite element method. The free surface cell model has been employed to analyse the hydrodynamic interactions between the particles. Detailed flow field features such as stress profiles on the particles and macroscopic quantities such as drag coefficient have been predicted for a wide range of fluid behaviour ( 1 ≥ n ≥ 0.2) and bed voidage, ( 0.3 ≤ ε ≤ 0.9999) thereby covering the complete spectrum of conditions encountered during the fluid flow through fixed, fluidized and distended beds. The observed dependences of the stress profiles and the drag coefficients on the power law index and bed voidage have been explained qualitatively with the aid of physical reasoning and order of magnitude considerations. Finally, the numerical results reported herein are compared with the appropriate experimental results available in the literature.

A GENERALIZED APPROACH TO FLOW THROUGH FIXED BEDS FLUIDIZATION AND HINDERED SEDIMENTATION

A generalized approach for a unified description of flow through mono and polydisperse uniform fixed and distended beds, particulate fluidization and hindered sedimentation is presented. Drag forces exerted by fluids on individual particles are used to evaluate friction factors and pressure drops across fixed beds. Extended definitions of drag coefficient and of its correlation with Reynolds number are evaluated and compared with literature data. It is shown that the new approach can be used reliably for creeping and turbulent flows, and also that it applies to the full range of packing densities which characterize dispersions and different types of beds comprising spherical particles. Finally effect of size distribution, bed structure, and particle segregation, on pressure drop, and implication of the new model for analysis of onset of fluidization are considered.

Transport phenomena in multi-particle systems—I. Pressure drop and friction factors: Unifying the hydraulic-radius and submerged-object approaches

A submerged-object model which accounts for the presence of stagnant zones is developed for predicting the pressure drop in random multi-particle systems. This model is then combined with the hydraulic-radius approach. With appropriate definition of the parameters involved, these approaches are found to yield complementary results. The tortuosity factor is considered as a velocity enhancement coefficient and is then related to the stagnant bed voidage. An expression for the tortuosity factor in random multi-particle systems is obtained. A definition for the multi-particle Reynolds number is developed which is applicable to the entire range of void fractions. A generalized friction factor is also developed from the definition of the friction factor for flow in a conduit and is related to the porosity-dependent drag coefficient for individual spheres. The friction factor reduces to the single-sphere drag coefficient in the limit ε→1. The approach, in principle, permits the collapsing of the entire momentum transfer data for multi-particle systems to a single curve coincident with the single-sphere drag coefficient relation. Friction factors calculated from the experimental data reported in the literature on packed, fluidized and distended beds, as well as sedimenting systems, are in good agreement with the theory for laminar as well as turbulent regimes. An expression for the relative velocity of sedimenting suspensions, applicable to a wide range of Reynolds number and void fractions, is obtained. An approximation for the estimation of the minimum fluidizing velocity is also developed and compared with the data available in the literature.

An analysis of transport phenomena for multi-solid particle systems at higher Reynolds numbers by a 5th order polynomial Karman-Pohlhausen method

Using the concentric spheres free surface model and a 5th order polynomial Karman-Pohlhausen method of the laminar boundary layer theory, the dimensionless tangential stress distributions, the dimensionless pressure distributions around a solid sphere in a swarm and the viscous, form and total drag coefficients for multi-solid sphere systems were numerically computed at higher Reynolds numbers, based on the first assumption that the pressure distribution equals that of potential flow between concentric spheres up to the separation point, and behind the separation point in the wake region the pressure does not recover and keeps constant, and on the second assumption that the pressure distribution varies according to the measurement of Flachsbart. The theoretical drag coefficient of single solid spheres in an infinite medium based on the second assumption agreed with the experimental data in the range of the Reynolds numbers from 3 × 10 2 to 10 5 . The friction factor for multi-solid particle systems based on the first assumption is almost the same as that on the 4th order polynomial and agreed with the experimental data of packed and distended beds. The void functions obtained from the drag coefficients for multi-solid particle systems based on both first and second assumptions were almost the same as the one on the 4th order polynomial. Using the velocity profiles based on concentric spheres free surface model and a 5th order polynomial Karman-Pohlhausen method of the laminar boundary layer obtained previously, the diffusion equation was solved numerically at higher Reynolds numbers on the first assumption of the pressure distribution around a solid sphere in a swarm equals that of potential flow between concentric spheres from the frontal stagnation point to the separation one, and the pressure does not recover, but keeps constant behind the separation point in the wake region. The mass transfer rate for multi-solid particle systems so computed was almost the same as that on the 4th order polynominal and agreed with the experimental data of single Solid spheres, and packed and particulate fluidized beds at higher Reynolds numbers.

Mass transfer in liquid fluidized beds of ion exchange particles

Mass transfer rate data in a shallow liquid fluidized bed of ion exchange resin particles have been obtained in the range 2 < Re s, < 25. At low voidage, ε < 0.55–0.60, the rate of mass transfer is reduced and it is inferred that a fluidized bed tends to maintain an ordered axial structure. This is lost at high voidage due to increased flow perturbation caused by the distributor plate. A generalised correlation is given for fluidized, fixed and distended beds in the range 2 < Re s, < 25.

An analysis of transport phenomena for multi-solid particle systems at higher reynolds numbers by a standard karman—pohlhausen method—I Momentum transfer

Using the concentric spheres free surface model and a standard 4th polynomial Karman—Pohlansen method, the dimensionless tangential stress distribution, and the dimensionless pressure distribution around a solid sphere in a swarm, and the viscous, form and total drag coefficient for multi-solid particle systems were computed numerically at higher Reynolds numbers. The void function obtained from the drag coefficient for multiparticle system was comparative to the experimental data. The friction factor for multi-particle systems so computed agreed with the experimental data of packed and distended beds.

A generalized approach to the fluid dynamics of particulate systems part III: General correlation for the pressure drop through fixed beds of spherical particles

A general correlation for pressure drop through fixed beds of spherical particles, based upon a discrete particle model corrected for particle interaction, is presented in this paper. The standard C D versus Re curve for single spheres is extended to multiparticle systems by incorporating proper functions of the volumetric particle concentration. The resulting correlation is compared with others and with data from eleven independent sources. A good fit is obtained in the creeping flow and intermediate regions as well as for distended bed data in the highly turbulent range. However, a certain deviation is noticed in the turbulent range with normal packed bed data. This paper, together with the others in this series, allows the prediction of pressure drop or drag in single particle and multiparticle systems by a single curve. Systems correlated with this curve include fluidization and sedimentation with either solid or fluid particles, minimum fluidization and, now, fixed beds.

A Generalized approach to the fluid Dynamics of particulate systems part III: General correlation for the pressure drop through fixed beds of spherical particles

A general correlation for pressure drop through fixed beds of spherical particles, based upon a discrete particle model corrected for particle interaction, is presented in this paper. The standard C D versus Re curve for single spheres is extended to multiparticle systems by incorporating proper functions of the volumetric particle concentration. The resulting correlation is compared with others and with data from eleven independent sources. A good fit is obtained in the creeping flow and intermediate regions as well as for distended bed data in the highly turbulent range. However, a certain deviation is noticed in the turbulent range with normal packed bed data. This paper, together with the others in this series, allows the prediction of pressure drop or drag in single particle and multiparticle systems by a single curve. Systems correlated with this curve include fluidization and sedimentation with either solid or fluid particles, minimum fluidization and, now, fixed beds.

Creeping flow mass transfer to a single active sphere in a random spherical inactive particle cloud at high schmidt numbers

Using the solution by Tam of Navier-Stokes equations for creeping flow around an active sphere surrounded by a random cloud of inactive spheres, an asymptotic solution of the convective diffusion equation is obtained for high Schmidt numbers. The Sherwood number for the overall mass transfer coefficient to the active sphere has been analytically related to the Peclet number as ▪ It agrees very well with the experimental mass transfer data on single active spheres for σ = 0476, R e < 10 and large Sc. This analytical result becomes invalid as σ decreases to 0.33. Pfeffer's model for the same problem has excellent agreement with the mass transfer data on single active spheres for σ = 026, R e < 10 and Sc = 1600. Pfeffer's model seems to be quite satisfactory for the usual range of void volume fractions in packed beds. The present model seems to be more accurate at higher values of void volume fractions in packed and distended beds.

Effects of vasoconstriction on blood viscosity in vivo.

AbstractThe influence of vascular dimensional changes on blood viscosity in vivo (“effective” viscosity) was analyzed in the calf muscle vascular bed of cats. Values for effective blood viscosity over an extensive flow range were obtained from comparisons of pressure‐flow curves for blood and a Newtonian fluid (Dextran‐Tyrode) with known viscosity. Such comparisons were made both at high and low flows during maximal vasodilataion and at low flows during vascular narrowing, caused either by vessel collapse or by intense, stable vasoconstriction as a result of barium chloride infusion. At indentical low flows effective blood viscosity was substantially lower in the constricted or collapsed bed than in the dilated and distended bed. With reduction of flow from maximal to very low values in the completely dilated, non‐collapsed bed effective blood viscosity increased from about 2.35 cP by 200–300 per cent. If the same low flows occurred as a result of intense vasoconstriction, implying a more than 5000 per cent increase of flow resistance, the viscosity increased by about 30 per cent.Apparently, effective blood viscosity is normally low and varies very little within the extensive flow range met with in vivo as a consequence of alterations of vascular tone. Therefore, even at extremely low flows in the constricted bed, as after severe blood loss, effective blood viscosity is hardly a significant determinant of flow, as long as blood composition is not changed.

Total and form drag friction factors for the turbulent flow of air through packed and distended beds of spheres

Total and form drag friction factors for the turbulent flow of air through packed and distended beds of spheres

Pressure drops in the flow of gases through packed and distended beds of spherical particles

AbstractPressure drop measurements were made across packed and distended beds having five layers of smooth plastic spheres, 1.23 in. in diamter, arranged in cubic, body‐centered cubic, and face‐centered cubic orientations of void fractions varying from 0.354 to 0.882. The distended beds were prepared by separating the spheres with permanently attached short lengths of fine rigid wire. Friction factors were calculated from the overall pressure drop measurements with the Ergun equation and were plotted against the corresponding modified Reynolds number which ranged from 2,550 to 64,900. A single relationship resulted for both types of beds which is independent of the geometric orientation and void fraction of the spheres of the bed.To eliminate entrance and exit effects of the air flowing through the bed pressure drop measurements were also made across the middle layer of each distended bed. Again a single relationship between the corresponding friction factors and modified Reynolds number was obtained which is independent of the geometric orientation and void fraction of the bed.From the two relationships between friction factors and modified Reynolds number the ratio Pk/fk for a packed or distended bed having five layers of spheres is 1.13. This ratio should decrease with increasing number of layers of spheres and approach the limiting value of 1 when the number of layers becomes very large.

Transfer processes in the flow of gases through packed and distended beds of spheres

AbstractSimultaneous mass, heat, and momentum transfer was experimentally investigated for the flow of air through packed and distended beds of spheres saturated with water. The porous spheres were approximately ⅝ in. in diameter. The void fraction of the packed bed was ϵ = 0.416, and the distended beds, whose spheres were positioned in a body‐centered cubic arrangement, had respective void fractions of ϵ = 0.576 and 0.778. The distended beds were formed by separating the spheres and connecting them with short lengths of fine rigid wire.Similar relationships for the packed and the distended fixed beds were obtained when the transfer factors were related to the modified Reynolds number. The relationship obtained for mass transfer in packed and distended beds was found to apply to data reported for fluidized beds of spheres.Pressure drop measurements were made which were not exacting but which indicate that the Ergun relationship for packed beds can be used to relate the friction factor to the modified Reynolds number for both the packed and distended beds investigated.