Performance evaluation and model of spacesuit cooling by hydrophobic hollow fiber-membrane based water evaporation through pores

Abstract

A comprehensive mathematical model is presented that accurately estimates and predicts failure modes through the computations of heat rejection, temperature drop and lumen side pressure drop of the hollow fiber (HF) membrane-based NASA Spacesuit Water Membrane Evaporator (SWME). The model is based on mass and energy balances in terms of the physical properties of water and membrane transport properties. The mass flux of water vapor through the pores is calculated based on Knudsen diffusion with a membrane structure parameter that accounts for effective mean pore diameter, porosity, thickness, and tortuosity. Lumen-side convective heat transfer coefficients are calculated from laminar flow boundary layer theory using the Nusselt correlation. Lumen side pressure drop is estimated using the Hagen-Poiseuille equation. The coupled ordinary differential equations for mass flow rate, water temperature and lumen side pressure are solved simultaneously with the equations for mass flux and convective heat transfer to determine overall heat rejection, water temperature and lumen side pressure drop. A sensitivity analysis is performed to quantify the effect of input variability on SWME response and identify critical failure modes. The analysis includes the potential effect of organic and/or inorganic contaminants and foulants, partial pore entry due to hydrophilization, and other unexpected operational failures such as bursting or fiber damage. The model can be applied to other hollow fiber membrane-based applications such as low temperature separation and concentration of valuable biomolecules from solution.

Graphical Abstract

Introduction

Polymeric hollow fiber membranes are ubiquitous in membrane-based separations ranging from membrane distillation to pervaporation for water purification, desalination, gas separation, liquid-liquid extraction etc. [1–5]. Vacuum membrane distillation (VMD) in particular is used for water purification and treatment, desalination, liquid-liquid separation, removal of organics and recovery of solutes [6–10].

The vast amount of literature published on VMD, and other membrane distillation applications using hollow fiber membranes, is relevant to the design of innovative devices like NASA’s hollow fiber-based Spacesuit Water Membrane Evaporator (SWME) [11–14], The SWME is a critical component of the portable life support system in spacesuits used for extravehicular activities. It is designed to reject the heat generated by the crew member wearing the suit and the electrical components of the suit’s portable life support subsystem. Heat is removed by circulating a heat transfer fluid (water) from a reservoir through a heat exchange network in the suit. The heated water is cooled by evaporating a portion of the water to space in the SWME and recirculated for thermal control. The rate of water evaporation and associated exiting liquid water temperature are controlled by adjusting the water vapor pressure in the shell of the hollow fiber module with a backpressure valve as illustrated in Figure 1. The water reservoir is refilled as needed.

Fig. 1.

(a) Diagram of the full SWME module integrated in spacesuit, schematic of a SWME module and an experimental unit [26] (included with permission from NASA) (b) Simplified conceptual diagram of a single fiber in the NASA SWME module and (c) differential control volume for mass and energy balance.

SWME operation is similar to VMD. A hot water stream is passed through a membrane module where evaporation and cooling occur. However, operational objectives are significantly different. VMD seeks to minimize heat loss due to evaporation while SWME seeks to maximize it. In contrast to VMD, SWME operations is at a lower inlet temperature and the main goal is to cool the water in the lumen side rather than producing a particular quality distillate.

VMD modeling [10, 15–17] may be adapted for the SWME unit to reduce expensive experimental testing and performance evaluation and enables early detection of degradation that could lead to critical failures. This provides an added layer of operational protection for astronauts during spacewalks.

Most past studies of VMD focus on the production of potable water from saline solutions, which is usually performed at relatively high temperatures (30–80 °C) [17]. Energy efficiency is a key issue in VMD [18] but past work has not focused on thermal performance and characterizing the dependence of performance on membrane properties and operating conditions, which is key in design and operation of SWME operation. A comprehensive thermal performance model also is needed to analyze other advanced hollow fiber membrane applications like low temperature concentration of expensive biomolecules or foodstuffs where lyophilization is not suitable [19].

Performance metrics critical to SWME application include outlet water temperature, heat rejection, and rate of water loss per unit heat rejection. Lumen-side pressure drop and vapor flow rate across the membrane are important as they provide insight into undesirable scenarios such as bursting and water leakage or weeping through the pores. A comprehensive mathematical model that can predict thermal performance as a function of water inlet conditions and membrane parameters (pore size and shape, porosity, tortuosity etc.) is of utmost importance. However, modeling is complicated by several factors (e.g., low pressure evaporation of water, pore non-uniformity, difficulties in measuring tortuosity, maintaining sufficient porosity without compromising structural integrity etc.) and an unavoidable long term performance deterioration due to particulate, contaminant, and foulant build-up in the system [20–22].

The VMD modeling literature has relied predominantly on arithmetic approaches with algebraic equations [16, 23]. Recently, Kim et al. [17] reported a model for VMD with differential mass and energy balances, in which they estimated the vapor flux by fitting the model with experimental data. Zhang et al. [16] used the Knudsen diffusion flux equation with a single membrane transport parameter (heretofore referred to as the “membrane structure parameter”) and used gas permeation to estimate the parameter experimentally. However, the prediction of thermal performance (heat rejection and temperature) by the VMD model was not explored in both studies.

Here, we develop a model for the NASA SWME unit using a differential mass and energy balance approach. The model explicitly includes the temperature dependence of all water physical properties (water latent heat of vaporization, heat capacity, vapor pressure, density, and viscosity). Membrane vapor flux is calculated using Knudsen diffusion (based on Knudsen number) and the lumen-side convective heat transfer is calculated from the Nusselt correlation for laminar flow in circular flow channels. The lumen-side pressure drop is estimated from the Hagen-Poiseuille equation. The membrane structure parameter is determined by fitting predicted heat rejection values to the experimental SWME values reported by Bue et al. [24].

In contrast to various VMD publications, which are mainly explored for water desalination, our paper advances the science of membrane-based cooling of water. Quantification of the role of input process and driving force variables are performed, as well as membrane properties through fundamental mass and heat transfer equations. In addition to the elegant spacesuit applications, one can apply this concept for low temperature separation and concentration of heat sensitive materials. The model is used to evaluate the effect of membrane structure parameter, porosity, pore diameter and tortuosity on heat rejection, temperature drop and lumen side pressure drop. Additionally, the dependence of thermal performance on pore size distribution, hydrophobicity change, and the role of contaminant fouling during continuous SWME operation is examined as part of a failure mode analysis. The specific objectives are: (i) developing a comprehensive mathematical model to accurately predict reported experimental thermal performances and estimating membrane structure parameter for SWME, (ii) performing a sensitivity analysis to understand the effect of input process variables on output thermal performance, (iii) understanding the role of contaminants and organic foulants on long-term thermal performance degradation and (iv) evaluating the conditions that can lead to critical failure modes.

Materials and Methods

Pore structure and morphology of the hollow fibers for SWME module (HF-SWME) were examined using a FEI Helios Nanolab 660 Focused Ion Beam/Scanning Electron Microscope (SEM) at the University of Kentucky Electron Microscopy Center (EMC). Samples were prepared for scanning electron microscopy (SEM) imaging by mounting on a 90° sample holder (aluminum stub) with double sided carbon tape (Nisshin). For lumen surface views, a small fiber piece was cut axially with scissors to expose the lumen. For cross-sectional views, fibers were cut using a sharp razor blade at room temperature and mounted on a silicon wafer with copper tape for polishing using an argon ion-beam polisher (JEOL cooling cross section polisher IB-19520CCP) with settings of 4.0 kV, 5.0 Ar gas flow setting at −120 °C with 40s/20s (on/off) for 8 h and stage swing enabled) [25]. The polished samples were mounted on a 90° sample holder (aluminum stub) and sputter coated with 5 nm Pt layer (using a Leica EM ACE600 instrument) prior to SEM imaging. Surface characterization of the hollow fibers was performed using N₂ sorption experiments conducted at 77 K with a Micromeritics TriStar 3000 instrument. Specific surface area was determined using the Brunauer Emmett-Teller (BET) isotherm method and pore volume, average pore size and pore size distribution were calculated using the method of Barrett, Joyner and Halenda (BJH).

Model Development

The model equations were developed assuming all fibers are identical and fluid distribution into the fibers from the distribution manifolds is uniform. Fig. 1 illustrates a single representative fiber and the operational variables: inlet water flow rate, ${\dot{m}}_{in}$; inlet water temperature, $T_{in}$; and the driving force for water vapor transport from lumen to shell side, $\text{Δ}P=P_{vapor}-P_{shell}$, where $P_{vapor}$ is the water vapor pressure at the water temperature adjacent to the membrane and $P_{shell}$ is the shell side pressure. Fig. 1 also illustrates the SWME module and the backpressure valve on the case used to control pressure in the shell-side space outside the fiber; the valve opens to reduce shell-side pressure if the target outlet water temperature is too high. Detailed model development from mass and energy balances for a control volume surrounding a differential fiber length (see Fig. 1) is provided in the Supplementary Information.

Overall model to predict SWME Performance:

The variation of water flow rate in the lumen is given by:

$$\frac{d\dot{m}}{d\xi }=-J_k\pi d_{l}L$$ (1a)

where, $\dot{m}$ is the water mass flow rate (kg/s), $J_k$ is the vapor mass flux across the membrane (kg/m²/s), $d_l$ is the lumen diameter, L is total fiber length, and $\xi$ is the normalized dimensionless length. Note that the vapor flux depends on temperature and thus is a function of axial position. The initial condition for Eq. (1a) is:

$$\dot{m}={\dot{m}}_{\text{in}}\text{at}\xi =0$$ (1b)

where ${\dot{m}}_{in}$ is the inlet mass flow rate per fiber.

Similarly, the variation of water temperature in the lumen is given by Eq. (2a) (see Supplementary Information for detail):

$$\frac{dT}{d\xi }=\frac{-{\lambda }_v}{\dot{m}{c}_p}\left(J_k\pi d_{l}L\right)$$ (2a)

where, $C_p$ is the specific heat capacity of water (Joules/kg/K), a function of temperature and ${\lambda }_v$ is the water latent heat of vaporization (Joules/kg), also a function of temperature. Initial condition for Eq. (2a) is:

$$T=T_{\text{in}}\text{at}\xi =0$$ (2b)

where $T_{in}$ is the inlet temperature.

Since water vapor fluxes are small compared to the lumen flow rate, the lumen pressure drop is calculated from the differential Hagen–Poiseuille Equation:

$$\frac{d{P}_{lumen}}{d\xi }=-\frac{128\mu \dot{m}L}{\pi \rho d_l^4}$$ (3a)

where, $\mu$ is the dynamic viscosity of water (Pa.s) and $\rho$ is the density (kg/m³). Note both are functions of lumen temperature. The initial condition for Eq. (3a) is:

$$P_{lumen}={\left(P_{lumen}\right)}_{in}\text{at}\xi =0$$ (3b)

where ${\left(P_{lumen}\right)}_{in}$ is the inlet lumen water pressure.

The equation for convective heat transfer in lumen side boundary layer can be expressed as:

$$h_c\left(T-T_p\right)=-{\lambda }_v{J}_k$$ (4)

where, $h_c$ is the convective heat transfer coefficient (W/m²/K) and $T_p$ is the pore mouth temperature (K), a function of position along the fiber (can be assumed equal to $T_m$, liquid-vapor interface temperature when the interface is at the pore mouth).

The convective heat transfer coefficient is estimated from the Nusselt correlation for laminar flow in tubes. In the entry mass transfer limit, $Nu=\frac{h_c{d}_l}{{\kappa }_w}=1.86{\left(\frac{RePr{d}_l}{L}\right)}^{0.33}$, where $Re=\frac{v_w{d}_l\rho }{\mu }$ is the Reynolds number and $Pr=\frac{C_p\mu }{{\kappa }_w}$ is the Prandtl number. The thermal entry length is 1/100th of the fiber length in the absence of evaporation and the flow in the lumen can be considered fully developed laminar flow. For fully developed laminar flow in the well-developed heat transfer limit, $Nu=4.36$. Since the shell is under high vacuum, heat conduction in the shell is negligible in VMD [15]. While the fluid thermal boundary layer is indeed included in the heat transfer calculations, the effect is small. The difference between the bulk and inner surface temperature is less than 1 °C for the results presented here. Also note that in our supplement we do discuss (Eq S7 and S8, and Eq 4 above)) and account for pore temp being not same as bulk lumen side water.

The calculation of water vapor flux through the pores is critical to modeling SWME performance. The flux should be calculated using either Knudsen diffusion or molecular diffusion based on the Knudsen number, $K_n=\frac{l}{d_p}$, where $l$ is the mean free path and $d_p$ is the mean pore diameter.

The mean free path can be calculated from:[27]

$$l=\frac{k_B{T}_p}{\sqrt{2}\pi P{\sigma }^2}$$ (5)

where, $k_B$ is the Boltzmann constant ($k_B=1.381\times {10}^{-23}\text{Joules}/\text{K}$), $T_p$ is the pore temperature, $P$ is the mean pore pressure at pore opening $\left(P=P_{vapor}\right)$ [28], and $\sigma$ is the collision diameter (2.641 × 10⁻¹⁰ m for water).

When water permeation occurs by Knudsen diffusion, the water flux is given by:

$${\text{J}}_{\text{k}}=(0.0248)S_P\sqrt{\frac{1}{T_p}}(\text{Δ}P)$$ (6)

where $S_p$ is the overall membrane structure parameter $\left(S_P=\frac{d_p{\varphi }_p}{\delta \tau }\right)$, $d_p$ is the mean pore diameter, ${\phi }_p$ is membrane porosity, $\delta$ is membrane thickness, and $\tau$ is membrane tortuosity.

Differential equations (1)–(3) are solved in MATLAB^® using an adaptive Runge-Kutta method (“ode45 solver”). Values for the temperature dependent material properties are provided in the Supplemental Information (Tables S1). Ranges of operational variables (${\dot{m}}_{in}$, $T_{in}$, ${\left(P_{lumen}\right)}_{in}$ and $P_{shell}$) and membrane parameters used in the simulations are provided in Tables S2 in the Supplementary Information.

Model outputs are temperature and mass flow rate as a function of position. These are used to calculate heat rejection, total mass of water vaporized and lumen side pressure drop from Eqs. (7)–(9), respectively:

$$\text{Heat}\text{rejection},q={\int }_0^L{\lambda }_v{\dot{m}}_{v}dz$$ (7)

$$\text{Total}\text{mass}\text{of}\text{water}\text{vaporized},{\left({\dot{m}}_v\right)}_{tot}={\dot{m}}_{in}-{\dot{m}}_{out}$$ (8)

$$\text{Pressure}\text{drop}\text{across}\text{lumen}\text{side},\text{Δ}{P}_{lumen}={\left(P_{lumen}\right)}_{in}-{\left(P_{lumen}\right)}_{out}$$ (9)

In Eq. (7), ${\lambda }_v$ and ${\dot{m}}_v$ are temperature dependent and therefore vary with position.

For certain cases (low inlet water temperature, < 10 °C), one can convert the above equations into an algebraic form by considering the average properties of water at $T_{avg}=\left(T_{in}+T_{out}\right)/2$. The result is given by (see Supporting Information for derivation):

$${\left(h_c\right)}_{avg}\left(T_{in}-\frac{{\left({\lambda }_v\right)}_{avg}}{2{\left(C_p\right)}_{avg}}\ln \left(\frac{{\dot{m}}_{in}}{{\dot{m}}_{in}-{\left(J_k\right)}_{avg}\pi d_{l}L}\right)-{\left(T_p\right)}_{avg}\right)=+{\left({\lambda }_v\right)}_{avg}{\left(J_k\right)}_{avg}$$ (10)

where, ${\left(J_k\right)}_{avg}$, ${\left({\lambda }_v\right)}_{avg}$, ${\left(C_p\right)}_{avg}$ and ${\left(h_c\right)}_{avg}$ are the vapor mass flux, latent heat of vaporization, water heat capacity, and convective heat transport coefficient evaluated at the average pore temperature ${\left(T_p\right)}_{avg}$.

Eq. (10) is solved along with Eq. 6 to find ${\left(T_p\right)}_{avg}$. The value is used to calculate ${\dot{m}}_{out}$ and $T_{out}$ (see Supporting Information). Finally, heat rejection, mass of water vaporized, and lumen side pressure drop are calculated using Eqs. (7), (8) and (11) respectively.

$$\text{Δ}{P}_{lumen}=\frac{128{\mu }_{avg}L{(\dot{m})}_{avg}}{\pi {\rho }_{avg}{d}_l^4}$$ (11)

where, ${\mu }_{avg}$ and ${\rho }_{avg}$ are the viscosity and density of water evaluated at $T_{avg}$ and ${(\dot{m})}_{avg}=\frac{{\dot{m}}_{in}+{\dot{m}}_{out}}{2}$ is the average mass flow rate of water.

Liquid entry into the pores:

Liquid entry into the pores is analyzed by Laplace equation as shown in Eq. (11), which relates liquid entry pressure ($LEP$) to membrane and liquid properties.

$$LEP=-\frac{4B_g{\gamma }_{L}cos{\theta }_o}{d_P}$$ (12)

where, ${\gamma }_L$ and ${\theta }_o$ are the surface tension of the liquid and initial contact angle of the liquid with polymeric membrane, respectively, and $B_g$ is the geometric factor ($B_g=1$, for cylindrical pores). Now if a portion of the pore in contact with liquid (lumen side) gets hydrophilized enough due to fouling such that the lumen side pressure is greater than LEP, liquid will enter the pores up to the end of hydrophilized portion (Fig. S1 in Supplementary Information). For a single pore with partial liquid entry up to the length of ${\delta }_{\text{L}}$, if the vapor traveling path length is ${\delta }_{\text{V}}$, the water vapor flux in that case is given by:

$${\left(J_k\right)}_{vap}=(0.0248)\frac{\delta }{{\delta }_V}{S}_P\sqrt{\frac{1}{T_s}}(\text{Δ}P)$$ (13)

where $T_s$ is the vapor liquid interface temperature. In the absence of liquid entry, $T_s$ is assumed to be equal to $T_p$ as discussed previously. However, in the case of partial liquid entry $T_s\ne T_p$ and $P_{vapor}$ in Eq. (12) is a function of $T_s$, which also is unknown.

If water wets the pores, the liquid mass flux can be estimated from the Hagen-Poiseulle equation for flow through the pores [29]:

$${\left(J_k\right)}_{liq}=\frac{\pi \rho d_P^4}{128\mu {\delta }_L}\left(P_{lumen}-P_{vapor}+P_C\right)$$ (14)

where, $P_C=\frac{4{\gamma }_{L}cos\theta }{d_P}$ is the capillary pressure and ${\gamma }_L$ and $\theta$ are the surface tension of the liquid and changed contact angle of the fiber in the portion of the membrane that is wetted, respectively. Physical properties $\mu$ and $\rho$ also are temperature dependent and can be evaluated at an average temperature of $\frac{T_P+T_S}{2}$.

Results and Discussion

The system of ordinary differential equations (ODEs) that comprise the conservation of mass, momentum, and energy equations for a single SWME fiber (Eqs. (1)–(3)) was solved with MATLAB^® ode45 solver along with the equations for convective heat transfer coefficient in the lumen side and diffusion of vapor through the pores to obtain liquid water mass flow rate through the lumen, liquid water temperature, pore temperature and lumen side pressure profiles as a function of axial position. These results are then used to calculate the values of heat rejection, total mass of water loss, temperature drop and lumen side pressure drop that are critical to performance of the spacesuit SWME.

Hollow fiber characterization:

NASA experimental thermal performance measurements for the HF-SWME were used to validate the model and estimate the membrane structural parameter $S_p$. Scanning electron microscopic (SEM) images of the lumen side surface, shell side surface and cross section of the HF-SWME membranes are shown in Fig. 2. Images show more uniform pore structure in both lumen and shell side with somewhat ellipsoidal pore opening shape, whereas the internal pore structure is more fibrous than spongy (i.e. bounded by polymer strands (not ellipsoidal cavities) as observed in polyvinylidene fluoride or PVDF membranes) as seen in the cross-sectional image [25]. As a result, SWME fibers have less variability in pore size compared to more commonly used PVDF or polyether sulfone (PES) membranes. The lumen-side pore size distribution (major axis, minor axis and circular equivalent diameter, $\sqrt{\mathrm{major}\mathrm{axis}\ast \mathrm{minor}\mathrm{axis}}$ is provided in Supplementary Information Fig. S3 based on measurements of 120 random pores from SEM images like the one provided in Fig. 2a. The major axis has wide pore distribution, whereas minor axis (which is more important for the diffusion of water vapor) has narrower pore distribution with some pores skewed towards the 100 nm size. The equivalent circular diameter (Fig. S3c) shows ~6% of the pores are larger than 100 nm.

Fig. 2.

SEM images of hollow fiber for SWME module (HF-SWME) with non-circular ellipsoidal pore structures: (a) lumen side surface, (b) shell side surface and (c) cross-sectional view. Scale bar for all images 1.0 μm.

Sorption isotherms and the corresponding BJH pore size distribution obtained from the desorption branch are provided in the Supplementary Information Fig. S4. The isotherms showed capillary condensation with a hysteresis loop consistent with a pore size around 50 nm [25]. The BJH pore size distribution showed a peak at 34 nm, which is consistent with minor axis pore size distribution of SEM images, with 5% of pores >100 nm based on the cumulative pore volume. Volumetric porosity calculated from the pore volume of nitrogen sorption are 24–28% for different experiments, which is much smaller than the 40% manufacturer reported porosity.

Estimation of structure parameter and model validation:

Only Knudsen diffusion through the pores is considered to model vapor transport through the membrane pores based on the calculated Knudsen number as a function of pore temperature for different pore diameters (see Fig. S5 in Supplementary Information). For pore diameters up to 400 nm and pore temperatures of 40 °C, the Knudsen number is well above the threshold for Knudsen diffusion (Kn > 1.0), so the vapor mass flux is estimated using the Knudsen diffusion equation (Eq. (5)).

Pore diameter, porosity, membrane thickness and tortuosity are lumped together in the overall membrane structure parameter $\left(S_P=\frac{d_p{\varphi }_p}{\delta \tau }\right)$ in the flux equation. Using the manufacturer reported values for the HF-SWME ($d_P=40\text{nm}$, ${\varphi }_P=40\text{\%}$ with $\delta =40\mu \text{m}$ and $\tau =1.0$ (if pores are straight)), ${\text{S}}_{\text{P}}$ is 4.0 × 10^-4. Although pore diameter of the membrane (HF-SWME) was measured and found close to manufacturer reported values (both lumen and shell side) [25], nitrogen adsorption gives a porosity of 24% and tortuosity was not verified. However, model predictions can be compared to experimental thermal performance data to obtain the best fit value of $S_P$. Heat rejection values over a range of inlet temperatures reported by NASA for a hollow fiber module of 18 cm length, 14,300 fibers with fully open backpressure valve (Bue et al. [24]) are used for the comparison and model validation. Using a $S_p$ value of 4.0 × 10⁻⁴, the model overpredicts heat rejection by a factor of two (Fig. 3a). The value of $S_p$ that best fits the data is 1.04 × 10⁻⁴ with a R² value of 0.9965. $S_P=1.04\times {10}^{-4}$ provides the best fit and the error in heat rejection is only 0.3%. Using ${\text{d}}_{\text{p}}=42\text{nm}$ (shell surface) and 46 nm (lumen surface), ${\varphi }_{\text{P}}=24\text{\%}$ (measured by nitrogen adsorption [25]), and $\delta =40\mu \text{m}$ provides $\tau =2.44$ and 2.67, respectively, for $S_P=1.04\times {10}^{-4}$. Note that both values are very close to the previously reported $\tau =2.5$ for polypropylene membrane hollow fibers [5, 30]. For $S_P=1.04\times {10}^{-4}$, model predictions of heat rejection are in excellent agreement with experimental data for the range of feed temperatures and shell pressures reported indicating robustness of the model in hollow fiber membrane applications.

Fig. 3.

(a) Model fitting with NASA reported experimental heat rejection (experimental data taken from Bue et al. [24]) as a function of inlet water temperature with fully open backpressure valve to optimize overall membrane structure parameter (${\text{S}}_{\text{P}}$) and (b) model predicted heat rejection with optimized ${\text{S}}_{\text{P}}$ along with experimental heat rejection as a function of shell side pressure. HF-SWME membranes, number of fibers 14,300 and fiber length 18 cm with inlet liquid water mass flow rate of 91 kg/h.

The model was also validated by the experimental vapor flux reported by Sun et al. (2014) [31] for PVDF membranes with the reported membrane properties: average pore size, porosity and membrane permeability (Fig. S6a in the Supplementary Materials). Model predicted vapor flux values are similar to the experimental flux values except for $d_P=0.282\mu \text{m}$ and inlet water temperature of 80 °C. The agreement is remarkable despite a very non-uniform pore structure with large macrovoids introduced by the phase inversion process used to form the membranes. Model estimated temperature drop is shown in Fig. S6b, which shows a low temperature drop due to high water flow rate in the lumen side.

Model prediction results for new generation SWME module:

Using the best fit value for the HF-SWME membrane structure factor, the model was used to evaluate the current SWME module design containing 27,900 fibers and an active fiber length of 11.9 cm (see Supplementary Information for other membrane input parameters). Model predicted liquid water temperature as a function of dimensionless length is shown in Fig. 4a and b for shell side pressure of 0.5 torr and 7 torr (Mars atmospheric pressure), respectively. For 0.5 torr shell side pressure, at high $S_P\left(4.0\times {10}^{-4}\right)$, the temperature approaches 0 °C at the midpoint of the fiber length whereas the temperatures remain above zero for $S_P=1.04\times {10}^{-4}$. Although a negative water temperature can occur for water evaporation at low pressure [32], the lumen side pressure is >1 bar and a negative temperature will form ice which should be avoided in the SWME application. Note that calculations were stopped when the temperature dropped to zero. For 7 torr shell side pressure (Fig. 4b), the outlet water temperature was 7.2 °C and 12.9 °C for $S_P=4.0\times {10}^{-4}$ and 1.04 × 10⁻⁴, respectively.

Fig. 4.

Liquid water temperature as a function of dimensionless length for $\text{Sp}=4.0\times {10}^{-4}$ and 1.04 ×10⁻⁴ with shell side pressure (a) ${\text{P}}_{\text{shell}}=0.5\text{torr}$ and (b) ${\text{P}}_{\text{shell}}=7\text{torr}$. ${\text{T}}_{\text{in}}={20}^{\circ }\text{C}$ and ${\dot{m}}_{in}=91\text{kg}/\text{h}$.

For 0.5 torr shell side pressure and $S_P=4.0\times {10}^{-4}$, the water vapor flux is very high and water temperature drops rapidly as a result. Fig. S7 in Supporting Information shows corresponding water vapor mass flow rate through the pores for $T_{in}={20}^{\circ }\text{C}$. For $P_{shell}=0.5\text{torr}$ (Fig. S7a) and high $S_P$, water vapor mass flow rate is very high near the entrance but drops rapidly as the water cools. Conversely, for $P_{shell}=7\text{torr}$ (Fig. S7b), the vapor mass flow rate is high near the entrance for high $S_P$ but drops below the value for $S_P=1.04\times {10}^{-4}$ around $\xi =0.67$.

Fig. 5 shows heat rejection, total mass of water vaporized, liquid water temperature at the outlet and lumen side pressure drops as a function of overall membrane structure parameter for different shell side pressure at ${\text{T}}_{\text{in}}={20}^{\circ }\text{C}$. Heat rejection and mass of water vaporized increase as either the structural parameter increases or shell pressure decreases (Fig. 5a and 5b). However, for ${\text{P}}_{\text{shell}}=0.5\text{torr}$ and 3 torr, heat rejection reaches a maximum at a certain ${\text{S}}_{\text{P}}$, after which there is no increase in heat rejection as well as mass of water vaporized. Also, at low shell side pressure outlet temperature approaches 0 °C for higher values of ${\text{S}}_{\text{P}}$ (Fig. 5c). For lower shell side pressure and higher values of S_P, liquid water temperature approaches 0 °C before the end of the fiber and no heat rejection can be obtained downstream of that point (calculation stopped), which can help to estimate fiber length necessary for low temperature hollow fiber applications.

Fig. 5.

(a) Heat rejection, (b) total mass of water vaporized, (c) outlet liquid water temperature and (d) lumen side pressure drop as a function of overall membrane structure parameter ($\text{Sp}$) for different shell side pressure $\left({\text{P}}_{\text{shell}}\right)$. ${\text{T}}_{\text{in}}={20}^{\circ }\text{C}$ and ${\dot{m}}_{in}=91\text{kg}/\text{h}$.

In addition to membrane structure parameter and shell side pressure, inlet water temperature strongly influences thermal performance. Heat rejection, total mass of water vaporized, outlet water temperature and lumen side pressure drop as a function of inlet water temperature for ${\text{P}}_{\text{shell}}=7\text{torr}$ are shown in Fig. 6. As expected, heat rejection and mass of water vaporized increase with inlet water temperature for all values of $S_P$ due to increased driving force for vapor flux. Outlet water temperature also increases with inlet water temperature, however for high value of ${\text{S}}_{\text{P}}\left(=6\times {10}^{-4}\right)$ the effect is minimal. Under these conditions, outlet temperature never decreases below 6 °C, at which temperature the vapor pressure of water is 7 torr. For ${\text{P}}_{\text{shell}}=7\text{torr}$ (atmospheric pressure at Mars), inlet temperatures below 6 °C will not produce any cooling. However, in the moon, where atmospheric pressure is 0 torr, very low inlet water temperature is also feasible. The lumen side pressure drop is very small compared to the inlet water pressure (1.2 bar). At 0.5 torr (data not shown), heat rejection and mass of water vaporized increase with increasing temperature for all values of $S_P$. However, for higher values of $Sp$ there is no increase in heat rejection or mass of water vaporized with increasing ${\text{S}}_{\text{P}}$, consistent with Fig. 5, and the outlet temperature remains close to 0 °C for higher ${\text{S}}_{\text{P}}$ values.

Fig. 6.

(a) Heat rejection, (b) total mass of water vaporized, (c) outlet liquid water temperature and (d) lumen side pressure drop as a function of inlet water temperature for different overall membrane structure parameter ($S_p$). $P_{shell}=7\text{torr}$ and ${\dot{m}}_{in}=91\text{kg}/\text{h}$.

Another important inlet variable is the liquid water mass flow rate, which gives us the indication of blockage or bursting in the lumen side. Fig. S8 in the Supplementary Information shows heat rejection and outlet water temperature as a function of inlet water flow rate at 25 °C and $S_P=1.04\times {10}^{-4}$ for different shell side pressure. Outlet water temperature and heat rejection both decrease with decreasing water flow rate as there is less water to cool and temperature drops rapidly.

Evaluation of Conditions for finding fixed lumen outlet water temperature:

For nominal conditions, the anticipated heat load for a spacewalk is around 350 watts (300 watts from the astronaut’s body and 50 watts from the electronics) [33]. The nominal inlet water temperature is around 17 °C and targeted outlet temperature is 10 °C. From Fig. 5, when $T_{in}={20}^{\circ }\text{C}$, the adjusted membrane parameter ($S_p=1.04\times {10}^{-4}$, obtained from fitting SWME experimental thermal performance data) results in $T_{out}={13}^{\circ }\text{C}$ for $P_{shell}=7\text{torr}$, which is higher than targeted 10 °C. Outlet temperature and heat rejection plots for lower $T_{in}$ are shown in the Supplementary Information (Fig. S9). When $T_{in}={17}^{\circ }\text{C}$, $S_P=1.04\times {10}^{-4}$ results in $T_{out}={11.7}^{\circ }\text{C}$ for $P_{shell}=7\text{torr}$, which is higher than targeted 10 °C. However, heat rejection is 556 watts for $P_{shell}=7\text{torr}$, which is higher than nominal heat load (350 watts), and a reduction of 44% ${\text{S}}_{\text{P}}$ can be tolerated to maintain 360 watts heat rejection. When $T_{in}={13.4}^{\circ }\text{C}$, $S_P=1.04\times {10}^{-4}$ results in $T_{out}={10}^{\circ }\text{C}$ for $P_{shell}=7\text{torr}$, which is exactly the outlet temperature desired. For $T_{in}={13.4}^{\circ }\text{C}$, heat rejection is also 355 watts for $T_{in}={13.4}^{\circ }\text{C}$, at that point any reduction in ${\text{S}}_{\text{P}}$ will results in increase in outlet temperature due to low heat rejection.

Fig. 7a illustrates the dependence of outlet water temperature ($T_{out}$) on inlet water temperature and membrane structure parameter ($S_P$) as a color heat map for a shell pressure of 7 torr. Clearly, outlet water temperature increases with both inlet water temperature and reduction of $S_P$. The corresponding contour plot is provided in Fig. 7b. Using the contour plots, combination of inlet water temperature, $S_P$ and $P_{shell}$ can be chosen for a targeted outlet water temperature (for example 10 °C).

Fig. 7.

(a) 3-dimensional (3D) surface plot of SWME outlet water temperature as a function of inlet water temperature and membrane structure parameter (${\text{S}}_{\text{P}}$) and (b) corresponding contour plot showing different outlet temperature line for inlet water temperature and ${\text{S}}_{\text{P}}$. For current SWME module: number of fibers is 27,900, inlet water mass flow rate is 91 kg/h.

Evaluation of Approximate Algebraic Solution:

An approximate algebraic solution of the comprehensive mathematical model (ODE) can be developed to predict the heat rejection and temperature drop by considering the physical properties as constant (at an average temperature between inlet and outlet) to simplify the equations. The changes in all water physical properties (density, heat capacity, and thermal conductivity of water) within the SWME operating range (5–30 °C) are small. Although water vapor pressure is a strong function of temperature (Fig. S2), the analytical algebraic solution assumes the vapor pressure can be evaluated at the average temperature along SWME fiber, $\left(T_{in}+T_{out}\right)/2$. The algebraic solution thus obtained is expected to possess greater error at higher inlet temperatures and membrane structure parameters, but it can provide a quick tool to predict the thermal performance of the SWME module without the computational rigour of the more accurate ODE model.

Fig. S10 in Supplementary Information illustrates heat rejection, mass of water vaporized, outlet temperature and lumen side pressure drop as a function of membrane structure parameter, $Sp$ for shell side pressre of 7 torr and inlet water temperature of 20 °C. Heat rejection increases with increasing $Sp$, but the algebraic model overpredicts the values for high $Sp$.

The temperature distribution along the fiber is plotted in Fig. S11 which shows $T_{avg}$ is higher (12.2 °C) for the algebraic model than the ODE model (11.0 °C). This leads to overprediction of water evaporation and heat rejection. Heat rejection and temperature drop along the fiber as a function of inlet water temperature also show that model prediction error increases with increasing inlet temperature (Fig. 8). Nonetheless, the error in predicting outlet temperature, $\left({\text{Tout}}_{\text{ODE}}-{\text{Tout}}_{\text{algebraic}}\right)/{\text{Tout}}_{\text{ODE}}$, is < 10 % for inlet temperatures less than 8 °C.

Fig. 8.

Comparison between heat rejection and temperature drop predicted by ODE and algebraic model as a function of inlet water temperature. $\text{Sp}=1.04\times {10}^{-4}$, ${\text{P}}_{\text{shell}}=7\text{torr}$.

Role of particulates, contaminants and foulants in lumen water:

Fouling by scaling or particulate deposition is one of the major reasons for performance degradation during membrane distillation. Fouling also may be an issue for the SWME. Polypropylene (PP) hollow fiber membranes have shown ${\text{CaCO}}_3$ scaling and significant water flux reduction with tap water in membrane distillation process [34, 35]. Similar contaminant precipitation and scaling may occur in the SWME loop as metal ions and other contaminants in the feed are concentrated during operation. Adsorption of bovine serum albumin (BSA) on silica particulate aggregates was also reported to reduce vapor flux sharply [36]. The effects of particulates in the membrane pore can be multifaceted, from pore blocking to reduction in pore size. Fig. 9 illustrates the effect of reducing pore size by a factor of two on heat rejection. A dramatic decrease occurs over the range of inlet temperatures considered. However, the dependence of heat rejection on changing pore size is approximately linear. Fig. 9b also suggests the “reduction in heat flux” (heat flux ratio) is a linear function of inlet T.

Fig. 9.

Effect of pore size reduction by scaling or fouling on SWME heat rejection as a function of inlet water temperature: (a) heat rejection when pore size is reduced by half compared to regular pore size and (b) ratio of heat rejection when pore size is halved versus regular pore size. ${\text{P}}_{\text{shell}}=7\text{torr}$ and ${\text{S}}_{\text{P}}=1.04\times {10}^{-4}$ for regular pore size.

Physical processes such as fouling can lead to $S_P$ reduction and a concomitant reduction in thermal performance that can cause system failure. For example, a change is ±5% is the ${\text{S}}_{\text{P}}$ value cause a ±3% change in the temperature drop of SWME module. If only pore diameter reduces due to deposition (most probably at the pore openings), $S_P$ should decrease linearly. However, a reduction of pore size will also lower porosity. For example, a 10% reduction in $d_P$ will cause 19% reduction in surface porosity and a net reduction of 27% in S_P. However, if some small pores are blocked completely only ${\varphi }_{\text{P}}$ will be reduced (and corresponding linear reduction in ${\text{S}}_{\text{P}}$ with respect of porosity) without significant $d_P$ changes. The combination of small pore blockage and reduction of pore opening due to deposition and fouling will be much more complicated. We do not expect significant change in membrane thickness or tortuosity of the pores resulting from contamination, deposition, and fouling.

Complete and partial pore entry by water:

The literature suggests polypropylene hollow fiber membranes may experience water intrusion and reduction in surface contact angle when submerged in deionized water from ${\theta }_o>{115}^{\circ }$ to 100° for 90 days at room temperature [37]. Although the membrane was from a different supplier, contact angle reduction is likely after prolonged exposure to water. Particulates and scaling also can increase the hydrophilicity of the pore inner surface. Although a surfactant can enhance water evaporation rate from oil-in-water emulsion [38], it severely reduces water surface tension [39] and liquid entry pressure (LEP). Fig. S12 in the Supplementary Information shows SWME heat rejection when some portions of the pores are larger size (0–10% pores with 200 nm average diameter, which is consistent with the SEM images and pore size distribution in Fig. S3). As seen, presence of larger pore increases the heat rejection because of higher ${\text{S}}_{\text{P}}$ values. However, liquid water intrusion, if it occurs, can have serious consequences. Pore intrusion of liquid water can be analyzed using Laplace-Young equation (Eq. (11)), which relates liquid intrusion pressure as a function of four parameters: geometric factor (=1 for cylindrical pores), water surface tension, contact angle between the liquid and surface and average pore diameter.

The geometric factor (${\text{B}}_{\text{g}}$) is less than 1.0 for non-cylindrical pores, usually 0.4–0.6 for the pores in stretched membranes [40]. As a result, liquid entry pressure can be half for stretched membrane pores, based on the geometric factor for similar sized cylindrical pores. At the operating lumen pressure (1.2 bar), liquid water may enter >100 nm pores for a contact angle of 95 degrees and >200 nm pores for a contact angle of 100 degrees. Presence of organics can also affect the surface tension of water, however the effect should be negligible (as a reference, presence of 1mM SDS (288 ppm) reduced water surface tension by 3%, whereas 2 mM SDS (576 ppm) reduces water surface tension by 9–10% [39]. Fig. 10a illustrates LEP as a function of average pore diameter and contact angle for different liquid water surface tension (40–70 mN/m). The corresponding contour plots are provided in Fig. 10b–e, where each line represents a fixed LEP for a combination of average pore diameter and contact angle. If LEP = 1 bar, significant water entry in the pores will occur and potentially lead to system failure.

Fig. 10.

(a) 3D surface plots of liquid entry pressure (LEP) as a function of the contact angle and average pore diameter at different water surface tension (${\gamma }_L=40-70\text{mN}/\text{m}$) and contour plots showing different LEP lines for (b) ${\gamma }_I=70\text{mN}/\text{m}$, (c) ${\gamma }_L=60\text{mN}/\text{m}$, (d) ${\gamma }_I=50\text{mN}/\text{m}$, and (e) ${\gamma }_L=40\text{mN}/\text{m}$. Geometric factor was considered 1.0 in these figures (cylindrical pores).

If partial liquid entry happens due to hydrophilization, the vapor path length is reduced which will increase $S_P$, and the flux in such case can be found by using Eq. (12). While the effect of water entry on $S_P$ may not be significant, the presence of a stagnant liquid layer also will affect the interface temperature between the liquid and vapor (not the temperature at pore entry) and the associated water vapor pressure driving evaporation as it introduces an additional thermal resistance.

Chamani et al. [29] have modeled partial and full pore entry by the liquid due to hydrophobicity change, which will be important during SWME operation in case of hydrophobicity change. Their results suggest the net effect of partial liquid entry will be small as the increase in vapor flux from reduced vapor path length offsets the increase in thermal resistance due to the stagnant liquid layer.

Membrane thickness has profound impact on permeant flux during vacuum membrane distillation, as well as heat transfer increase for low membrane thickness [41]. Hence, thermal performance should increase due to apparent vapor path reduction as a result of partial liquid entry. However, due to the presence of stagnant liquid layer things is much complicated. But the net effects should not be detrimental in terms of ${\text{S}}_{\text{P}}$.

It should be noted that a non-equilibrium evaporation layer exists at the interface between liquid and vapor. The thickness of this layer is given by Eq. (S23) in the Supplementary Information [42]. Fig. S13 shows the calculated thickness of Knudsen layer as a function of pore temperature (interface temperature for partial pore entry). If the length of vapor path (${\delta }_{\text{V}}$) is less than Knudsen layer flux equation will not be valid to describe the system.

Analysis of critical failure modes for SWME module:

Based on the model results and subsequent analysis, several critical failure modes can be deduced for NASA SWME module. If liquid water penetrates completely through the fraction of the pores (due to hydrophilization), it will vaporize in the shell side (For example 1% of pore leakage leads to 3.3 kg/h of additional water loss from the system according to Hagen-Poiseuille equation). Since the reservoir capacity is around 3.6 kg of water, this water loss due to pore leakage will sustain the spacewalk activity only for 1-2 h.

Burst pressure of different polymeric fibers were found to be more than 30 bars from temperature −40 to 160 °C [43]. Hence, high lumen water pressure induced fiber bursting is improbable based on an intended operational pressure of SWME water less than 1.2 bar in space. However, if any fiber detachment happens from epoxy casing, liquid water loss will be significant and reduce spacesuit operational time.

The presence of surfactant can reduce vapor pressure [44] and the driving force for evaporation significantly. This detrimental effect on thermal performance occurs even if water does not enter the pore due to a reduction in LEP; water passage through the pores (weeping) will just exacerbate the detrimental effects. Surfactant will not be introduced intentionally but potential sources of accidental introduction should be mitigated.

Another failure mode is associated with reduction of $S_P$ due to pore blockage, contamination, precipitation of particulates, and/or organic fouling. As shown in Fig. S9 in the Supplementary Information, a reduction in $S_P$ results in reduced thermal performance and a corresponding increase in the outlet temperatures. If ${\text{T}}_{\text{out}}$ is more than the desired value (say 10 °C), the shell side pressure control valve in SWME will open more to bring it down. However, when the vacuum reaches maximum (surface pressure of Mars is 7 torr), further opening of pressure control valve will not produce desired outlet temperature and target heat rejection cannot be achieved.

Conclusions

This work develops a comprehensive predictive mathematical model to estimate the thermal performance of the NASA spacesuit water membrane evaporator (SWME). The model was developed by considering mass and energy balance during the evaporation of water through the membrane pores along with diffusive transport of water vapor through the pores and differential Hagen-Poiseuille equation for lumen side water pressure drop. The model correlates critical input variables: inlet water temperature, inlet water mass flow rate, shell side pressure and overall membrane structure parameter ($S_P$) to thermal performance output parameters: heat rejections, outlet water temperature, temperature drop, and lumen side pressure drop. NASA reported experimental thermal performance data was used to fit the model and estimate $S_P$ to be 1.04 ×10⁻⁴, which, based on the experimentally measured average pore diameter (using SEM) and volumetric porosity (by nitrogen adsorption) provides a tortuosity value between 2.44 and 2.67. After validation, the model was used to perform a detailed sensitivity analysis to study the effect of input variables changes on the output thermal performance parameters. Heat rejection increases (and outlet water temperature decreases) with increasing membrane parameter and/or inlet water temperature but decreases with higher shell side pressure, as expected. An approximate algebraic solution for the model was also developed by considering average physical properties, which was found to be applicable for ${\text{T}}_{\text{in}}<8^{\circ }\text{C}$. The role of particulates, contaminants and foulants on thermal performance of SWME unit was analyzed and discussed as well as partial pore entry caused by hydrophobicity change in the entrance portion of the pores (lumen side). For example, a 45% reduction in ${\text{S}}_{\text{P}}$ (due to pore blockage and/or pore size reduction) is tolerable to maintain nominal heat load at 7 torr (Martian surface atmospheric pressure). The critical failure modes are discussed based on the model analysis and results. As an example, 1% of fiber bursting/leakage will lead to an additional 3.6 kg water loss from the SWME system, which will reduce spacewalk activity to only 1–2 h. The work delivers tremendous advancement in the field of hollow fiber-based membrane applications by providing a mathematical framework to analyze and predict critical process parameters and the effects of contaminants and foulants on thermal performances. A future application of the model and analysis will be to predict hollow fiber-based membrane separation performance following concentration and purification of biomolecules at lower temperature from dilute aqueous solution.

Highlights:

• Hollow fiber membranes for spacewalk heat exchange application

• Detailed math model for NASA Spacesuit Water Membrane Evaporator (SWME) module

• Predicting thermal performance and explaining critical SWME failure modes

• Role of contaminants and organic foulants on SWME performance discussed

Nomenclature

$A$, $B$, $C$

Antoine constants

$A_p$

Area of one pore opening (m²)

$B_g$

Geometric factor

$C_p$

Water heat capacity (Joules/kg/K)

$D_k$

Knudsen diffusivity (m²/s)

$d_l$

Lumen diameter (m)

$d_p$

Mean pore diameter (m)

$h_c$

Convective heat transfer coefficient (W/m²/K)

$J_k$

Mass flux of vapor through the pores (kg/m²/s)

$k_B$

Boltzmann constant (Joules/K)

$K_n$

Knudsen number

$l$

Mean free path (m)

$L$

Length of the fiber (m)

$LEP$

Liquid entry pressure (Pa)

$M_w$

Molecular weight of water (kg/mol)

$\dot{m}$

Mass flow rate of water in lumen side (kg/s)

${\dot{m}}_{\text{in}}$

Inlet mass flow rate of water in lumen side (kg/s)

$n$

Number of pores per unit length of fiber

$Nu$

Nusselt number

$N_k$

molar flux of vapor (mol/m²/s)

$Pr$

Prandtl number

$P_{lumen}$

Lumen side liquid water pressure (Pa)

$P_m$

Mean pore pressure at pore opening (Pa)

$P_{shell}$

Shell side pressure (Pa)

$P_{vapor}$

Water vapor pressure (Pa)

$q$

Heat rejection (watts)

$Re$

Reynolds number

$R_g$

Gas constant (Joules/mol/K)

$r_l$

Radius of the lumen (m)

$r_p$

Pore radius (m)

$S_P$

Overall membrane structure parameter

$T$

Liquid water temperature (K)

$T_p$

Pore temperature (K)

$v_w$

Water velocity (m/s)

$T_{shell}$

Shell side temperature (K)

$T_{in}$

Inlet water temperature (K)

$T_m$

Liquid-vapor interface temperature (K)

$T_p$

Pore temperature (K)

$z$

Distance along the fiber length (m)

Greek Letters:

${\gamma }_L$

Water surface tension (N/m)

$\delta$

Membrane thickness (m)

${\delta }_L$

Liquid entry length (m)

${\delta }_V$

Vapor path length (m)

$\text{Δ}P$

Pressure drop across membrane (Pa)

${(\text{Δ}P)}_{lumen}$

Lumen side liquid water pressure drop (Pa)

$\theta$

Water-membrane contact angle (degree)

${\theta }_o$

Initial water-membrane contact angle (degree)

$\kappa$

Membrane thermal conductivity (W/m/K)

${\kappa }_{pp}$

Polypropylene thermal conductivity (W/m/K)

${\kappa }_w$

Water thermal conductivity (W/m/K)

${\lambda }_v$

Water latent heat of vaporization (Joules/kg)

$\mu$

Dynamic viscosity of water (Pa.s)

$\xi$

Dimensionless distance along fiber length

$\rho$

Density of liquid water (kg/m³)

$\sigma$

Collision diameter (m)

$\tau$

Membrane tortuosity

${\varphi }_p$

Membrane porosity