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Mass Transfer - Boundary Conditions

  1. Apr 5, 2013 #1
    1. The problem statement, all variables and given/known data

    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.

    2. Relevant equations

    Equation of continuity in spherical coordinates.

    3. 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.
     
    Last edited: Apr 5, 2013
  2. jcsd
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