I’m working on simulating a system where a rough vacuum is used to evaporate water. To do this, I need to perform a mass and species balance on the vacuum side of the system, but I’ve been having some trouble.
The system consists of a water reservoir and a chamber where the vacuum is applied, separated by a membrane.
Ideally, only water vapor can pass through the membrane and enter the chamber. The mass flux (water vapor) entering the vacuum chamber is given by the following equation:
$$j_w=K(p_{sat}-p_{w,vac}),$$
where $K$, $p_{sat}$ and $p_{w,vac}$ are the membrane mass transfer coefficient ($kg\,m^{-2}\,s^{-1}\,Pa^{-1}$), saturation pressure of water in the water reservoir and the partial pressure of water vapor in the vacuum chamber. So far, things are fine, because both $K$ and $p_{sat}$ are known, and the amount of mass entering the system per unit time is
$$\dot{M}_{in}=j_w\times A,$$
where $A$ is the membrane surface. A vacuum pump is used to keep the absolute pressure low. I know what the absolute pressure $p_{vac}$ in the vacuum side is, and from this, I can infer two things:
Since $p_{vac}>p_{sat}$, there is some air inside the vacuum chamber.
I also know what the suction flow rate $Q_{suc}\,(m^3\,s^{-1})$ is: I can calculate it from the characteristic curve of the pump ($p_{vac}\, \text{vs}\,Q_{suc}$).
As the system reaches steady state, there can be no mass accumulation in the vacuum chamber, so
$$\dot{M}_{in}-\dot{M}_{out}=0$$
My problem arises when calculating $\dot{M}_{out}$. One possible approach I’ve considered is
$$\dot{M}_{out}=\rho_{vac}Q_{suc},$$
but I have no way of calculating $\rho_{vac}$ without knowing the composition (mass fraction $\omega$ of air and water) inside the vacuum chamber.
Do you have any idea on how can I derive an equation for $\dot{M}_{out}$?
P.S.: the vacuum pump used is a Steam Ejector
