Evaporation of water using a vacuum pump - COMSOL

Question

I'm using COMSOL to simulate a system where liquid water is evaporated using a vacuum pump, but I am struggling to understand the mass transport process of vapor from the liquid-air interphase to the outlet connected to the pump. The pump reduces the absolute pressure in the chamber, extracts vapor from a chamber, which reduces the vapor concentration, and increases the evaporation rate.

My system consists on a channel where liquid water flows (rectangle 1) and a chamber where the vacuum is applied (rectangle 2). The red line on the top represents the outlet connected to the vacuum pump.

The green line represents a hydrophobic membrane that prevents the liquid water from going up when the absolute pressure on the top is low. However, the water vapor can cross the membrane.

From what I know, in steady state, depending on the vacuum level created by the pump, two situations can happen:

• If the absolute pressure is lower than the saturation pressure of the liquid water, the chamber will contain only vapor and the air content can be neglected. Simulating such a system should be easy. The flow of vapor would be laminar and two boundary conditions would be needed, $p_{sat}$ in the gas-liquid interphase and $p_{vacuum}$ at the outlet.
• However, if the absolute pressure set by the pump ($p_{vacuum}$) is greater than the saturation pressure of the liquid water, some traces of air will remain in the chamber. In this situation, I don't know whether the vapor transport is driven by diffusion only or by both diffusion and convection.

My problem arises in this second situation because I have been unable to find out if both diffusion/convection contribute to the transport of vapor and to find the correct boundary conditions for each of them.

The only B.C. I'm sure of is the concentration B.C. in the liquid-gas interphase, where the concentration equals the saturation concentration.

Any suggestion on how to solve this problem?

Answer 1

The transport rate through the hydrophobic membrane is driven by the permeation flux. The permeation flux rate for water through the membrane $J_{w,M}$ (mols / m$^2$ s) is proportional to the difference in pressures of the component across the membrane as

$$J_{w,M} = P_M(p_{w,vap} - p_{w,M}) $$

where $p_{w,vap}$ is the vapor pressure of the liquid water (Pa), $p_{w,M}$ is the partial pressure of water vapor a the immediate membrane surface on the side in the chamber that is being pumped, and $P_M$ is the membrane permeance (with associated units), dependent on material (permeation diffusion constant) and thickness to first order.

Convection does not occur on the liquid side when the membrane is in direct contact with the water. Convection could be included on the vapor chamber side. The governing flux equation is

$$J_{w,c} = k_w(p_{w,M} - p_{w,B}) $$

where $k_w$ is the convection coefficient for water in air at the total pressure and $p_{w,B}$ is the partial pressure for water in the ``bulk'' (far removed from the membrane surface).

The total flux will be controlled by the lower of the two factors, $J_{w,M}$ versus $J_{w,c}$. The controlling factor is not the total pressure in the pumped chamber, it is the partial pressure of water vapor in that chamber. When $p_{w,B} > p_{w,vap}$, no transport occurs. The operation of the system must be defined accordingly. To do so, you must have the governing transport parameters ($P_{w,M}, k_{w}$), the vapor pressure $p_{w,vap}$ at the given system temperature, and $p_{w,B}$. The unknown $p_{w,M}$ can be eliminated either by assuming that $p_{w,M} = p_{w,B}$ (a well mixed system with no convection) or by setting the two flux rates equal and solving for $p_{w,M}$ as an unknown.

Finally, in practice, all vacuum pumps have a base pump rate and lowest attainable pressure as their operating specifications. Also, the pumped chamber will always contain a variation of components between air (oxygen, nitrogen, argon) and water vapor dependent on the type of pump (rotary, piston, turbo-molecular, diffusion, ...), as each type of pump is more or less effective at pumping various species. These three factors (pump rate, base pressure, and effectiveness for each component) must be considered to continue the operating materials balance equations between the chamber and pump itself. The three factors are not amenable to a theoretical model. They depend on pump type, chamber size and wall material, and temperature.