[Anderson2019]

Hi,

I have a problem with a boundary condition and would appreciate any help.

I've been using Python and the Finite Volume Method. I've been writing my own code for this specific problem.

The situation is the following: there are two fluids, blood and almost pure O2 gas, flowing side-by-side separated by a thin membrane, which is permeable to CO2. Thus, CO2 may leave the blood through the membrane and into the gas current.

According to the literature, the main resistance in the process of CO2 removal from blood is in the blood side. Therefore I do not include in the model the membrane itself. I consider the CO2 to be removed convectively by the gas current (as if there was no membrane in between).

So, my boundary condition (BC) for CO2 at the wall where CO2 leaves the domain is:
-Dco2.d[CO2]/dx = km (kp[CO2] - [CO2]g)

where Dco2 is the diffusion coefficient in blood; km is the convective mass transfer coefficient in the gas phase; kp is the partition coefficient; [CO2]g is the constant CO2 concentration in the gas current.

Based on the results I have so far, the previous BC works fine.

The problem arises when I try to include an heterogeneous reaction by an enzyme immobilized on the interface bewteen blood and gas (where there should be a membrane, which I haven't included in the model).

In this situation I have to include diffusion through blood, reaction on the surface, and CO2 removal by convection. The reversible reaction catalised by the enzyme is HCO3 + H <-> CO2 + H2O. After doing some research, I found one reasonable way to express this BC, which is called a discontinuity of flux boundary (from here).

I understood this BC as: the flux of CO2 leaving the blood convectively equals the CO2 flux that arrives at the interface diffusively plus (or minus) the amount of CO2 generated (or consumed) by the reversible reaction.

Equation: arrive - leave = reaction
-D*dc/dx - km*(kp*c - Cg) = kcat*E*(c - w*h/k1)/(M + c - w*h/k1) #for CO2; Only CO2 leaves the blood, that's why this is the only BC with a convection term.
-D*dw/dx = -kcat*E*(c - w*h/k1)/(M + c - w*h/k1) #for HCO3
-D*dh/dx = -kcat*E*(c - w*h/k1)/(M + c - w*h/k1) #for H

where: c = CO2 concentration, w = HCO3 concentration, h = H concentration, km = convective mass transfer coefficient, Cg = CO2 concentration in the gas, kcat = turnover number, E = enzyme concentration, k1 = equilibrium constante, M = Michaelis-Menten constant. Note: there is a minus sign in front of each diffusion term so that it becomes a positive flux.

I've run some simulations with these BCs and they seemed to be working fine since the CO2 partial pressure in the blood reduced faster for the case with enzymes, as expected. The problem is that the CO2 was reducing faster bacause it was being converted to HCO3 instead of leaving the blood. That's not what experimental works in the literature have shown. CO2 partial pressure should decrease faster because of a higher CO2 removal when the enzymes are present.

I've done some reserach on how this kind of problem (immobilized enzyme) is modeled but either those works had much more complex problems or were purely experimental.

After all this, my question is: is this 'flux discontinuity boundary condition' a good way to express the physics of my problem? If not, what are other possible ways?

Thank you in advance.