An airborne spherical cellular organism, 0.015 cm in diameter, utilizes 4.5 gmol O2/(hour kg of cell mass). Assume Sh = 4 for external convective resistance to O2 transfer to the cell. (Sh = kd/D is based on diffusivity in the gas phase). Assume zero-order kinetics for respiration.
What is the concentration of O2 at the center of the cell? Use a diffusion coefficient for O2 through the cellular materials of 10^(-5) cm^2/second. Take the solubility of oxygen in the cellular material to be 1.4 x 10^(-6) mol/cm^3, in equilibrium with air at 25 degrees C and one atmosphere total pressure.
Equation of continuity in spherical coordinates.
The Attempt at a Solution
There's no time dependency or velocities, and if it's assumed symmetric then everything will be happening in the r direction. The equation of continuity is reduced to:
0 = D(1/r^2) d/dr (r^2 dCa/dr) + Ra
Where Ra is the reaction, which in zero-order kinetics reduces to Ra = -k
After solving the differential equation, you're left with:
Ca = (-k/6D)r^2 - C1/r + C2
Where C1 and C2 are constants. Here is where I am failing to understand the problem: the boundary conditions
Boundary Condition #1:
flux through object to surface = flux through boundary layer from surface
-D* dCa/dr = k(Ca(fluid) - Ca(bulk stream)) at r=R
Where I guess I can solve for k through the Sherwood number given, in terms of D and the diameter since I'm not given k.
**edit: this is not the k from the reaction, unrelated
Boundary condition #2
My brain tells me it should be one of the following and I'm not sure which one to use (or if it matters which I choose)
a) dCa/dr = 0 in the middle of the cell (symmetry). This would give rise to C1 = 0 in order to keep the solution finite in the middle.
b) The concentration at the surface = the concentration in equilibrium of the outside, namely
Ca = αCa(gas) at r = R, where α is an equilibrium relationship I am able to find.
Any help would greatly be appreciated. I am getting confused also at subscripts in this problem: there are two diffusivity constants? We're given the one through the cell fluid, but the one in the boundary condition is defined for the gas phase through the Sherwood number.