Answer Happy

Consider a sphere of radius R immersed in a large body of fluid. The sphere has an excess of mass of species 1 on its surface, but it is sufficiently slight that there is no buoyancy-induced motion in the fluid at a partial pressure of P₁0 and corresponding mass fraction of 10- Far away from the sphere, species 1 partial pressure is p₁ and its corresponding mass fraction is w₁f. The rate at which moles of species 1 are transported outward is equal and opposite to the rate at which moles of species 2 are transported inward. This is equimolar counter- diffusion from a sphere. (a) Show that the pertinent describing differential equations are Continuity Diffusion pu, d(r²pu,) dr m1, total dw₁ dr with boundary conditions of w₁(r = R) = w 10 w₁(r = ∞0) = ₁/ (b) Show that the equimolar counterdiffusion requirement m,(r) M₁ leads to the relationship that 0 4TD 12 PM₂R RT m1, total D12 d(pr² dw₁/dr) dr or or P₁(r = R) = P10 P₁(r = ∞0) = P₁ m₂ (r) M₂ (w₁0 - wis) X [1 + w ₁0(M₂/M₁ − 1)][1 + w₁ƒ(M₂/M₁ − 1)] - (c) Show from the result of part b that defining a mass-transfer coefficient hp as Apho(w10-wis) [1 + w₁(M₂/M₁ - 1)][1 + w₁/(M₂/M₁ =¯1)} leads to the Sherwood number Sh being hpD Sh= D12 = 2 ||| = 3