A theoretical study of pressure drop for non‐Newtonian creeping flow past an assemblage of spheres

Abstract

A combination of Happel's free surface model and variational principles is used to obtain bounds on the drag offered by the creeping flow of a power law fluid past an assemblage of solid spheres. The theoretical predictions of the product of the Fanning friction factor f and Reynolds number Re, are in close agreement with available experimental data on non-Newtonian flow through porous media. The product (f Re,) reduces for the Newtonian case to that of Happel and Brenner. SCOPEThe flow of a Newtonian fluid through packed and fluidized beds has received considerable attention in the chemical engineering literature. A free surface model developed by Happel is widely used to predict the friction factor for ~~~~~~i~~ fluids. H~~~~~~, the analogou9 problem of flow of a non-Newtonian fluid through packed and fluidized beds has not been analyzed so far with this model. This paper extends the Happel's free surface model to the flow of power law fluids by making use of variational principles and presents an analysis for the friction factor in packed and fluidized beds. CONCLUSIONS A N D SIGNIFICANCE A combination of Happel's free surface model and variational principles yields numerical values for the product of the Fanning friction factor f and the Reynolds number Re, for various values of bed porosities and flow behavior indexes. An expression for the product (f Re,) in terms of the porosity and flow behavior index is developed which well predicts experimental values of friction factor for power law flow through packed and fluidized beds. A correlating equation was developed which provides a continuous STUART W. CHURCHILL representation f<r d l Pr and Re. Different constants are suggested for the local and mean Nusselt numbers and for uniform wall temperature and heating. These constants are based on the best available theoretical

Drag and Mass Transfer in Slow Non-Newtonian Flows Over an Ensemble of Newtonian Spherical Drops or Bubbles

“…Due to the lack of both experimental data for swarms of drops moving in power law liquids, and also of the other theoretical analyses of the same problem, the reliability of the predictions presented can not be assessed. By analogy with cases of a similar physical nature analysed and verified by Mohan and Raghuraman (1976b) and Kawase and Ulbrecht (1981a, b) one may expect these predictions to be similarly reliable, especially if deviations from Newtonian behavior are not large, 0 . 8 < n < 1, and if the liquids are not contaminated by surface active impurities.…”

“…Unlike the case of a single particle or a single bubble, the correction factors for drag coefficient and mass transfer for the motion of an assemblage of solid spheres or bubbles were found to decrease with increasing pseudoplasticity (Mohan and Raghuraman, 1976b, Bhavaraju er al., 1978, Kawase and Ulbrecht, 1981b. These theoretical predictions were later validated by Kawase and Ulbrecht (1981b) by comparing experimental data on pressure drop for nonNewtonian flows in fixed and fluidized beds reported by Yu el al.…”

Hydrodynamics of the motion of spherical particles, droplets, and bubbles in a non-Newtonian liquid: Analytical methods of investigation

“…Coefficients A 1 , A 3 , A 4 in (36) are the functions of flow index n and gas content ϕ in the power law liquid as follows [16]: At ϕ = 0, expression (36) is identical to the param eter Y A-R , which was previously obtained in [9] and is a part of formula (24). The values of correcting param eters Y St , Y StSw of the drag coefficients of a single sphere (a bubble with immobilized interface) and the ensem ble of these spheres were given in [16] with reference to [2] and [19] (Table 2). In this case, the upper boundary of Y St values apparently was used (Fig.…”

“…While the values of the correcting param eter Y and drag coefficient C D for a single bubble increased with decreasing flow index n of a power law liquid (both in the Stokes and Adamar-Rybczynski modes), a converse effect took place for a bubble ensemble: Y and C D values decreased with decreasing n in both hydrodynamic modes. These calculations (as [16]) were proved by experimental stud ies from an ensemble of solid spheres [19].…”

“…For the Levich regime, which is characterized by the free surface of a spherical bubble and values of the Reynolds number of ~100-1000, the drag coefficient C DL appears as (19) Parameter Y L (designated K n in the study) is deter mined by the equation…”