The Ernst Ratz Analytical Solution in a High Speed Rotating Cylinder Revisited

In this study the Ernst Ratz differential equation governing the concentration field in a high speed rotating cylinder with and without axial back diffusion was revisited, and the solution of the differential equation was developed to study the effect of product baffle opening on the optimum feed flow rate and on the optimum enrichment for a wide range of normalized counter-current (L/F in the range 1.2 to 8) with unit cut (P/F) equal to 0.4, 0.5 and 0.6. Here, L is the counter-current circulation rate, F is the feed flow rate, and P is the product flow rate. The analysis shows that at a given unit cut and normalized counter-current, the optimum feed flow rate can be reduced by lowering the product baffle opening, and the effect is significant for normalized counter-current L/F up to 5, and beyond that point the influence is not as much of important, whereas in the case of optimum enrichment (N_P – N_W)_opt , as the product baffle opening is lowered, the optimum enrichment increases monotonically with normalized counter-current. Here, N_P and N_W are the concentration of the product and waste stream respectively. Next, the flow profile efficiency (E_F) and mass flow efficiency (E_M) have been studied for wall pressure in the range 20 to 100 m-bar based on the mass flow rate in the inner stream (m_i), and the analysis indicates that the product of flow profile and mass flow efficiency (E_F x E_M ) has an optimum for each wall pressure, and the separative power (δu) attains its maximum value at that m_i value. The comparison between axial back diffusion and without axial back diffusion reveals that at a given unit cut, the rectifier has an additional length due to axial back diffusion effect, and the influence can be reduced by increasing the aspect ratio (Z/R_w) of the cylinder ((Pradhan & Kumaran, J. Fluid Mech., vol. 686, 2011, pp. 109-159); (Kumaran & Pradhan, J. Fluid Mech., vol. 753, 2014, pp. 307-359)). Here, Z is the effective length of axial counter-current, and R_w is the radius of the cylinder. An important finding is that there is a cross over point for normalized counter-current beyond which the optimum feed flow rate is higher with axial back diffusion compared to without axial back diffusion, and the cross over point shifted to smaller values with the lowering of product baffle opening.