2. 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

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2 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 1 (94.55 KiB) Viewed 8 times

2. 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 w₁. 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 d(r²pu,) dr m₁ ¹1, total dw₁ pur dr <<=0 with boundary conditions of <=310 or w₁(r =R) P₁(r = R) = P10 w₁(r = ∞0) = W₁s or P₁(r = ∞0) = P₁S (b) Show that the equimolar counterdiffusion requirement m,(r) M₁ leads to the relationship that 4TD12PM₂R RT = m1, total D12 d(pr² dw,/dr) dr m₂(r) M₂ (W₁0 - wis) X [1 + ₁0(M₂/M₁ − 1)][1 + w₁(M₂/M₁ − 1)] ಅ - - (c) Show from the result of part b that defining a mass-transfer coefficient ho as Aphp(wo-wis) = [1 + w ₁0(M₂/M₁ − 1)][1 + w₁/(M₂/M₁ =1)]* leads to the Sherwood number Sh being Sh= hpD D12 = 2