Assumptions
Water vapor condensation phenomenon is fundamentally limiting the application of membranes for oxygenation as explainedhere. Modeling water vapor transfer through hollow fiber membrane can be considerably complicated. But, it can be performed fairly simply by assuming the following two.
 No gas transfer across the membrane except water vapor.
 No humidity gradient in the radial direction of hollow fiber
The first assumption is true only when the gas is saturated in water phase and no gas consumption occurs, e.g. air passing through hollow fiber membranes immersed in clean water. The second assumption is truer for smaller fibers, where radial mixing is easy. Although the derived equation will not be rigorous, it provides us an idea on how quick the water vapor saturation is in typical conditions.
Modeling
Symbols
: Thickness of nonporous layer (m) : Relative humidity (=) : Initial relative humidity (=) : Fiber inner diameter (m) : Permeability of nonporous layer (mol^{.}m/m^{2}/s/Pa) L: Effective membrane length ( m ) : Number of fibers () : Gas flow rate, total (m^{3}/s) X : Distance from the entrance (m) : Vapor pressure in lumen (Pa) : Vapor pressure, saturated (Pa)

The driving force of water vapor transfer is the difference of water vapor pressures across the membrane wall. Therefore, the water vapor transfer rate can be assumed proportional to the vapor pressure difference between lumen (gas phase) and shell (liquid phase). Under a steady state, the number of moles of water vapor added to the sweeping gas flow () in the small segment of the hollow fiber membrane shown in Fig. 1 can be calculated by multiplying the gas flow rate by water vapor differential (dP) as shown by the term in the left of the equation. The amount of water vapor added to the same segment of the hollow fiber can be described by the right term of the equation, where permeability () is multiplied by driving force (–) and membrane surface area (dA) and divided by the thickness of nonporous layer ().
Â Â Â Â Â Â Â Â Â Â Â Â Â Â —————————— (1)
Fig. 1. Water vapor intrusion to hollow fiber lumen while gas flows.
In the meantime, membrane surface area of the small segment of interest can be calculated by Eq. (2).
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â —————————— (2)
By combining above two equations, following equation is obtained.
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â —————————— (3)
The above equation can be integrated with two boundary conditions as follow : (,) and (,).
Â Â Â Â Â Â Â —————————— (4)
Finally Eq. (5) is obtained, which shows the relative humidity () along the fiber as a function of mass transfer coefficient (). This equation can be further simplified by dividing byÂ
Â Â Â Â Â —————————— (5)
Â Â Â Â Â Â Â Â Â Â Â Â Â Â —————————– (5â€™)
Above equation can be rearranged against permeability,, which is effective only when relative humidity remains less than unit. This equation is used to calculate permeability from experimental data.
Â Â Â Â Â Â Â Â —————————— (6)
Result
The permeabilities of silicon coated nonporous membrane for various gases are found here. One thing noticeable is that water vapor has 60 folds higher permeability than oxygen (36,000 Barrer vs 600 Barrer) for silicon. Therefore, when the silicone coated membranes are used to dissolve oxygen into water, it is not hard to imagine air will be saturated with water vapor when only a fraction of oxygen is transferred/dissolved into water. The units (Barrer) of the permeability in the reference can be converted to proper ones (mol^{.}m/m^{2}/s/Pa) to be used with above equations by multiplying a conversion factor, e.g. 3.348 x 10^{16}_{.}
The graph in the calculator in the bottom shows the relative humidity profile inside a hollow fiber membrane for a likely condition the gas transfer membrane is used, e.g. 20 ^{o}C and 10 micron coating thickness.Â Although the membrane length required to saturate water vapor varies depending on all the parameters in the calculator, it takes only ~5 cm in the example given in the following. According to a literature (Fang, 2004), where similar modeling were performed with slightly different assumptions, air saturated within 1.5 cm – 3.0 cm from the hollow fiber entrance depending on condition.
It is noteworthy that the curve in the following graph is for the most optimistic case. Unlike the assumption made in the equation derivation, a portion of gas volume is lost continuously while the inlet gas proceeds in the lumen. This is an extra mechanism that increases water vapor pressure in addition to the water vapor transfer to lumen. Therefore, water vapor saturation should occur faster than predicted by this model.
In theory, water vapor saturation and subsequent lumen blockage by the condensed water can be prevented by reducing membrane length, increasing gas flow rate, increasing wall thickness, increasing selectivity of membrane in favor of gas component that needs to be transferred, etc. However, it is not straightforward how to optimize all these parameters since all such solutions cannot be imposed without losing either performance or cost competitiveness. More studies are required to overcome the intrinsic technical issue of lumen condensation.
Â© Seong Hoon Yoon