Mass Transfer - Boundary Conditions

In summary, the problem involves an airborne spherical cellular organism with a diameter of 0.015 cm and a rate of oxygen utilization of 4.5 gmol O2/(hour kg of cell mass). The external convective resistance to oxygen transfer to the cell is assumed to be 4. Using a diffusion coefficient for oxygen through the cellular materials of 10^(-5) cm^2/second and a solubility of 1.4 x 10^(-6) mol/cm^3 at equilibrium with air at 25 degrees C and one atmosphere total pressure, the concentration of oxygen at the center of the cell is to be determined. The equation of continuity in spherical coordinates is used and the resulting differential equation is solved
  • #1
dweeegs
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Homework Statement



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.

Homework Equations



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.
 
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  • #2
Also, the concentration at the surface of the cell would not be the concentration in the gas phase.
 

FAQ: Mass Transfer - Boundary Conditions

What is mass transfer?

Mass transfer is the movement of mass from one location to another. In science, it refers to the transfer of molecules or particles between phases, such as gas, liquid, or solid.

What are boundary conditions in mass transfer?

Boundary conditions in mass transfer refer to the conditions at the interface between two phases, such as the gas-liquid interface. These conditions can include temperature, pressure, and concentration gradients that affect the transfer of mass.

Why are boundary conditions important in mass transfer?

Boundary conditions are important in mass transfer because they determine the rate and direction of mass transfer between phases. They also play a crucial role in predicting and controlling the behavior of systems that involve mass transfer.

What are some common boundary conditions in mass transfer?

Some common boundary conditions in mass transfer include constant concentration, constant flux, and no mass transfer. These conditions can be applied to different types of interfaces, such as gas-liquid or liquid-solid interfaces.

How do boundary conditions affect mass transfer?

Boundary conditions affect mass transfer by creating a gradient that drives the transfer of mass between phases. They can also affect the direction and rate of mass transfer, as well as the overall behavior of the system. In some cases, boundary conditions can be manipulated to optimize or control mass transfer processes.

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