Applications of the Ceriiansky and Siroky Model
The objectives of the experimental studies were threefold:
1. Determine an empirical relationship between permeability and porosity in the form of Eq. 10-92.
2. Determine the values of the deposition and entrainment rate constants, kp and k'e.
3. Study the effects of the length of porous media and the rate and concentration of the particle suspension injected into the porous media.
The pressure difference across the porous media and the particle concentration of the effluent were measured as functions of time during the injection of a suspension of finely ground limestone particles at a given concentration and rate. The porous material was prepared by using nonwoven felt of filaments of polypropylene. The porous material samples of 4 cm diameter and 0.5,1.0, 1.5, and 2.0 cm lengths were used. The particle suspension was prepared using finely ground limestone of 2,825 kg/m3 density in water. The porosity was determined by the weighting method. The discrete times at which measurements are taken are denoted by the subscript i = 2,3,...,N and the initial time is denoted by / = 1. The permeability was determined by Darcy's equation by neglecting the effect of gravity for short samples:
The volume of particles deposited per unit volume of porous media was calculated, by integrating Eq. 10-84 and applying the mean value theorem:
where £0 =0 for an initially particle-free porous material. Eq. 10-172 is evaluated numerically by applying the trapezoidal rule of integration as, for a constant injection suspension particle concentration:
Eqs. 10-170 and 173 were applied at different times and the data were plotted in Figure 10-16. As shown in Figure 10-16, virtually the same results were obtained for different injection velocities of u = 0.5 and 1.0 cm/s. The E and G parameter values were determined as a function of the length of porous media by nonlinear regression of Eq. 10-92 to the data given in Figure 10-16. Plot of E and G vs. the length are given in this article exponential regressions of these data indicate that E = 0.14 and G = 5.3 as the length approaches zero, although the data are of low
quality, as indicated by the coefficients of regressions R2 = 0.78 and R2 = 0.18, respectively. The mean diameter of particles is 20.2jam and the estimated dimension of pores 50jom. The porosity is (j>0=0.31. The concentration of the injected suspension is cin =0.lkg/m3 or <5in = 0.1&g/ra3)/(2,825£g/m3) = 3.54xlO-5m3/m To determine the deposition and entrainment rate constants, in Eq. 10-91 i'cr - 0 was substituted and the derivative with respect to time was evaluated numerically using the central and backward finite difference approximations given below, respectively, for the interior and the final points:
The average concentration was estimated as the logarithmic mean value of the injection and the effluent suspension concentrations according to:
The rate parameters, kp and k'e, in Eq. 10-91 were determined for different injection velocities using the method of least squares with the values calculated by Eqs. 10-170 and 174-176. The results presented in this article indicate that the retention rate coefficient, kp, decreases and the entrainment rate coefficient, ke, increases with the injection velocity. These calculations were repeated for different length porous media and the results are summarized. These values can be extrapolated to zero core length, however, again the quality of data is not good.
The effect of the suspension particle concentration and particle size on the pressure drop. For a given injection suspension particle concentration and rate, at equilibrium,
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