5.63. Acetone can be removed from acetone-air mixtures using simple counter- current cascades, by adsorption onto charcoal (Foust et al. 1980). We wish to find the required number of equilibrium stages to reduce a gas stream carrying 0.222 kg acetone per kg air to a value 0.0202 kg acetone per kg air. Clean charcoal (X = 0) enters the system at 2.5 kg/sec, and the air rate is constant at 3.5 kg/sec. Equilibrium between the solid and gas can be taken to obey the Langmuir-type relationship where Y = KXn 1+KX K = 0.5 Y = kg acetone/kg air = X, kg acetone/kg charcoal (a) Write the material balance between the first stage (where X, enters) and the nth stage, and use the equilibrium relationship to derive the finite difference equation for X. (b) Rearrange the expression in part (a) to show that the Riccati equation arises Xxn+1+AX+1 + BX + C = 0 What are the appropriate values for A, B, and C? (c) Use the condition X = 0 (clean entering charcoal) to evaluate the arbitrary constant in the general solution from part (b) and thus obtain a relationship to predict X,, = f(n). (d) Use an overall material balance to find X, and use this to calculate the number of equilibrium stages (N) required. Ans: (d) 3.5 stages