Drag force/correlations

Note that the validity of both the Ergun and Wen and Yu equations has recently been questioned on the basis of LB data, and alternative drag-force correlations have been proposed. From LB simulations, Hill et al. (2001a, b) suggest the following relation for Stokes flow (lim Rea->0) ... 

Beetstra, R., van der Hoef, M. A., and Kuipers, J. A. M. Numerical study of segregation using a new drag force correlation for polydisperse systems derived from lattice Boltzman simulations, Manuscript submitted the Chem. Eng. Sci. (2006, in press). 

Fig. 10. Normalized drag force at arbitrary Reynolds numbers and gas fractions. The symbols represent the simulation data, the solid line the Ergun correlation Eq. (18), the dashed line the Wen-Yu correlation Eq. (46) for e = 0.8, and the grey line the correlation by Hill et al. (2001a,b) Eq. (47) and the long-dashed line Eq. (19), both for e = 0.5.

Expression in Eq. (19) is within 8% of all simulation data up to Re — 1000. Since this relation has been derived very recently (Beetstra et al., 2006), it has not been applied yet in the higher scale models discussed in Sections III and IV. However, the expression by Hill et al. in Eq. (47) derived from similar type of LBM simulations is consistent with our data, in particular when compared to the large deviations with the Ergun and Wen and Yu equations. So, we expect that the simulation results presented in Section IV.F using the Hill et al. correlation will not be very different from the results that would be obtained with expression in Eq. (19). A more detailed account of the derivation of expression in Eq. (19) and a comparison with other drag-force relations can be found in Ref. Beetstra et al. (2006). 

The most satisfactory way of representing the relation between drag force and velocity involves the use of two dimensionless groups, similar to those used for correlating information on the pressure drop for flow of fluids in pipes. 

The pair correlation function g describes the distribution of molecular centers in the solution. In concentrated systems at rest, gxl. Flow alters g, and it is this change which gives rise to the drag forces. For sufficiently slow shearing flows,... 

The drag force gives a velocity component l/p that must be obtained from experimental correlations, and combining these components gives 1/pi. 

Allowable flows of vapor through the froth are correlated on the basis of liquid entrainment. Balancing of the drag force of the vapor on a representative drop of liquid against the gravitation force on the drop has led to the relationship. 

Equally tentative correlations are presented by them for the drag force and angular velocity. 

Because of the complexity of the fluid dynamics, correlations are used to estimate the drag coefficient, Cn, from which the drag force is calculated. The drag coefficient is defined as the ratio of the force per area (Fo/Ap b, perpendicular to the fluid flow and where Ap t = is the projected surface area of the bob) to the product of the fluid density and the velocity head ... 

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