Assume that a spherical liquid drop contains a supersaturated solution of a give

Question

Assume that a spherical liquid drop contains a supersaturated solution of a given species A with a concentration C_A, sat. The solute species A is volatile and it has vapor pressure P-A, sat. The drop is surrounded by a large body of stagnant gas. (a) Derive a model (an equation) to estimate the steady diffusion of the species A into the gas. (b) Now consider that the release of the species A is into a well-stirred region of fluid surrounding the droplet in which the species is transferred by convection. To model the problem, assume that the transfer of the species A is equivalent to the diffusion through a boundary layer of a known thickness delta. Derive a model (an equation) to estimate the steady diffusion of the species A into the gas.

Explanation / Answer

4 r2 NAr (r) - 4 (r + r)2 NAr (r + r) = 0

(r + r)2 NAr (r + r) - r2NAr (r) = 0 / r

NAr = xA (NAr + NBr) - c DAB dxA/dr

dxA/dr = - ( C1/cDAB) (1/r2)

xA (a) = xA1

xA () = 0

C1 = cDABaxA1 ; C2 = 0

xA (r) / xA1 = a/r

NAr (r) = cDAB axA1 1/r2

WA = 4 a2 NAr (a) = 4cDAB axA1 = 4 DAB acA1

d/dt (4/3 a3) = 4a2 da/dt = - 4 DAB MA acA1

a da/dt = -DAB MA cA1 /

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