>⁢
9
⁢
8

Eq
.

5

Thus, Eq. 3 can be re-written once more to provide a differential equation completely in terms of the humidity ratio.

$\begin{matrix} ρ_{a i r} V_{e} \frac{d ω_{e}}{dt} = {\dot{m}}_{i n} - β A_{m} (\frac{P_{e} ω_{e}}{ω_{e} + 0.6 2 1 9 8} - \frac{P_{a m b} ω_{a m b}}{ω_{a m b} + 0.6 2 1 9 8}) & Eq . 6 \end{matrix}$

Eq. represents the differential equation that will be solved to calculate the humidity immediately next to the membrane as a function of time. For the Joule Heating analysis, P_eis increased according to the assumed temperature increase, described by Eq. 7 from the ideal gas law.

$\begin{matrix} P_{e} = \frac{T_{e}}{T_{a m b}} P_{a m b} & Eq . 7 \end{matrix}$

Similarly, in the vacuum-driven analysis, P_ambis replaced by a vacuum pressure (P_vac). This differential equation only captures the humidity removal at the membrane interface. However, we also need to account for the passive diffusion of water vapor throughout the enclosure (also referred to as the effect of concentration polarization).

As mentioned above, it is helpful to account for an affect referred to as concentration polarization. Because we have a stagnant volume of air inside of the enclosure, when humidity immediately adjacent to the membrane is removed, the humidity in the rest of enclosure will migrate towards the membrane to be removed by the membrane. This tradeoff between membrane permeation and water vapor diffusion in air is a common point of analysis in related HVAC research. FIG. 5 shows a simple representation of this concentration polarization effect.

The differential equation presented in Eq. 6 is solved numerically, thus we can easily implement Fick's Law of Diffusion to account for the diffusion of water vapor throughout the volume. To achieve this, the 1D mass transfer model is discretized between the membrane surface and the opposite surface of the enclosure. Time is also discretized. A simple first-order Euler approximation scheme is applied to solve the ODE.

For any given scenario, the initial humidity conditions are known. At each time step, the humidity adjacent to the membrane (i.e. the first discrete point in space) is calculated from Eq. 6. Then, the remaining humidity values in the 1D enclosure are estimated based on Fick's Law (mass balance).

$\begin{matrix} d m_{x = i, t = j} = \frac{D_{A B} A_{m} Δ t}{Δ x} [(A_{H, i + 1, j} - A_{H, i, j}) - (A_{H, i, j} - A_{H, i - 1, j}) & Eq . 8 \end{matrix}$

Here, D_ABis the diffusion coefficient for water vapor in air, Δt represents the time between discrete time points and Δx represents the distance between discrete nodes between the membrane and opposite enclosure surface. Eq. 8 calculates the change in water vapor mass within each discrete element, which can then be converted to a change in absolute humidity and thus a change in humidity ratio.

With Eq. 8 and Eq. 6 nestled inside the same loops, the equations march through space and time, calculating the humidity ratio at each discrete location node at each time in the given analysis. In the end, the humidity ratio can be plotted for the enclosure and converted to dew point to present the same information in a different manner. While this explanation of the approach is brief, it should give some insight into the level of detail applied to the model.

For the purpose of showing the capability of the current model, several analyses were conducted. For now, the relevant dimensions of the enclosure are described in

Table 2. Additionally, it is assumed that one of the “large” faces on the enclosure would be a selective membrane. It is assumed that the enclosure is ‘leaky’ and not airtight or otherwise sealed. All of the calculations and plotting may be done automatically, so if any of these dimensions are deemed unrealistic, or if we simply want to test different dimensions, the entire analysis, including plotting, can be re-done in a matter of minutes.

TABLE 2

Summary of the enclosure dimensions

used in the following sample results.

Variable
Symbol
Value

Width
W
3.65 cm

Height
H
2.35 cm

Length
L
5.8 cm

Membrane Area
A_m
21.17 cm²

Membrane
β
2300 GPU

Permeance

Next, we present some results for the vacuum-driven operation mode, as this is the more practical, near-market approach that could be easily implemented in real applications. FIG. 6 shows the types of plots that can currently be produced by the code, all of which produce some metric relating to the humidity inside the enclosure as a function of time. This plot shows the distribution of humidity in the single dimension as a function of time. So, what you notice is that at a distance of x/L=0 (which is adjacent the membrane) the humidity drops to a final value rapidly. Meanwhile, on the opposite side of the enclosure at x/L=1, the humidity drops more gradually, relying on water vapor diffusion towards the membrane surface for removal from the enclosure volume. The model can also be quickly re-run to produce comparisons between different ambient conditions.

Overall, the Joule heating portion of the code uses nearly identical code, with the major difference being that now the pressure inside the enclosure is elevated due to the increased temperature. It should be noted that currently localized heating is not specifically accounted for. In other words, an enclosure temperature is set, and it is assumed the entire enclosure volume is quickly heated to that temperature. This allows a simple analysis on the concept of heating the volume to produce the required pressure gradient, giving insight into what temperatures might be required and what kind of humidity reduction is ultimately achievable. Of course, when adding even more detail, we can account for heating convection (which will delay the time to remove humidity and could induce convective air currents to help mix the volume and reduce the effect of concentration polarization). This level of detail could be useful, though it is approaching the limits of a simple numerical framework and could require commercial simulations using a program. Either way, this portion of the code can produce the same type of results as the vacuum-driven operation code, so an abbreviated sample of results are shown below.

Some interesting takeaways from these sample plots is that raising the temperature of the enclosure to approximately 128° C. will yield similar humidity reduction. What we have to consider is which scenario proves more energy efficient—pulling an 80 kPa absolute pressure vacuum (i.e., 20 kPa below atmosphere) or heating the enclosure to 128° C. Dealing with such high temperatures could be damaging to the membranes or to the electronics. If high temperature operation is not a concern, this Joule heating approach could still be beneficial due to the small scale and simplicity of using a resistive heating element compared to requiring a dedicated pump or fan to pull a negative pressure. Likely, both operational modes will have unique applications where they are more favorable, which is why it is important to consider both options and/or a hybrid of the two.

The primary performance metric of the preliminary experiments explores the humidity ratio or dew point temperature versus time, much like the results in the modeling section of this document. We want to understand how quickly the humidity level is reduced inside the control volume, and how the vacuum levels or elevated volume temperatures impact the speed of dehumidification and final humidity. To monitor these parameters, several components/sensors shown in Table 3 are used to control the device and acquire data.