Question - A fluidised bed reactor is used to produce zinc oxide from spherical zinc sulphide particles in a reaction proceeding according to ZnS(s) + b O₂(g) → products where b is a stoichiometric coefficient. A shrinking core model is appropriate for analysis of the reaction, with chemical reaction being the controlling resistance. The rate of reaction is first order with respect to the concentration of O₂ and zeroth order with respect to ZnS. There is no significant change in particle size during the reaction.

(a) Sketch the radial profile of the concentration of O₂ through a particle at an arbitrary time during the reaction before complete conversion.

(b) Derive an expression for the time τ_c for complete conversion of ZnS in a particle of radius R. Hence show that the fractional conversion X of ZnS in a particle, up until time t = τc, obeys

X = 1 - (1- t/τ_c)³

(c) The fluidised bed reactor can be modelled as a gas-solid reactor with a uniform gas concentration and a particle residence time distribution obeying

E(t) = exp(-t/τ_m)/τ_m, where τ_m is the mean residence time.

Explain why the average fractional conversion X^- of ZnS at the reactor outlet is

X^- = 1 - ₀∫^τ_c (1 - t/τ_c)³(exp(-t/τ_m)/τ_m) dt

When τ_m >> τ_c then exp(-t/τ_m) ≈               1 - t/τ_m for t ≤ τ_c. Show that, under conditions when the conversion is high, the average fractional conversion at the reactor outlet obeys

X^- ≈ 1 - ¼ τ_c/τ_m + 1/20(τ_c/τ_m)²

The solid feed to the reactor consists of a mixture of 30 wt% ZnS particles with diameter 50 μm and τ_c = 1 min, and 70 wt% ZnS particles with diameter 100 μm. The mean residence time is 10 min for both sizes of particles in the reactor. Calculate the overall average conversion of ZnS at the reactor outlet.