Analytical Solution for Real Cascade Equation in Steady State

Abstract

Steady state real cascades are analyzed. The method adopted is based on two difference equation representing the conservation of the total flow and the desired material flow, respectively. Introducing what is termed the "desired material flow cut" as a stage parameter corresponding to the "total flow cut" or "cut", the flow relations of the desired material flow are reduced to the same form as that of the total flow rate. The flow rate is not subject to any restrictive assumption except the conservation of mass flow, so that the method can be applied to almost any type of real cascade with side flows and losses. Adoption of a suitable iterative calculation will permit this treatment to deal with cascades constituted of separators with large values of separa-tion factor without resorting to approximation, because the "desired material flow cut" of each stage is described exactly by the separation factors of each stage.
The difference equations are solved analytically, for which the determinants B(i, j) and D(i, j) are introduced, which are the functions of stage parameters from the i-th to the j-th stage. With these determinants, explicit relations are obtained between the flow rate of each stage and the stage parameters of the whole cascade. The determi-nants are calculated by the recurrence formula.