A well-dispersed mixture of solids and liquid is to be concentrated by being pumped through a tank with a porous plate at the bottom through which pure liquid may pass, as shown schematically in Figure 4P.1. Because the dispersion is very good you may consider the mixture to be homogeneous liquid with a solids concentration c.

a. Show that the mixture density is a linear function of c. The total volume V is the sum of the solids volume V_S and the liquid volume V_L.
b. It is observed experimentally that the logarithm of height varies linearly with time when pure water flows through the porous plate. Assume that the volumetric flow rate q₃ depends on the liquid density and viscosity, ρ_L and η_L, respectively; the plate area, A; and the pressure change per unit thickness across the plate, ∆p/δ. Obtain a constitutive equation for q₃.
c. Obtain a steady-state design equation relating the tank dimensions to the liquid throughput and the desired degree of concentration.

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Porous plate thickness d area A Mixture Solids conc. C₁ Volumetric flow rate q₁ Pure liquid Volumetric flow rate 93 Mixture Solids conc. C₂ Volumetric flow rate 92