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Research Papers: Evaporation, Boiling, and Condensation

# A New Approach to Delay or Prevent Frost Formation in MembranesPUBLIC ACCESS

[+] Author and Article Information
Pooya Navid

Mechanical Engineering Department,
57 Campus Dr,

Shirin Niroomand

Mechanical Engineering Department,
57 Campus Dr,

Carey J. Simonson

Mechanical Engineering Department,
57 Campus Dr,

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received March 2, 2018; final manuscript received September 15, 2018; published online November 16, 2018. Assoc. Editor: Zhixiong Guo.

J. Heat Transfer 141(1), 011503 (Nov 16, 2018) (11 pages) Paper No: HT-18-1122; doi: 10.1115/1.4041557 History: Received March 02, 2018; Revised September 15, 2018

## Abstract

Saturation of the water vapor is essential to form frost inside a permeable membrane. The main goal of this paper is to develop a numerical model that can predict temperature and humidity inside a membrane in order to show the location and time of saturation. This numerical model for heat and mass transfer is developed to show that frost formation may be prevented or delayed by controlling the moisture transfer through the membrane, which is the new approach in this paper. The idea is to simultaneously dry and cool air to avoid saturation conditions and thereby eliminate condensation and frosting in the membrane. Results show that saturation usually occurs on side of the membrane with the highest temperature and humidity. The numerical model is verified with experimental data and used to show that moisture transfer through the membrane can delay or prevent frost formation.

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## Introduction

In cold climates like Canada and Northern Europe, there is a high risk of frost formation in energy exchangers. Frosting occurs when the warm and humid air flowing through the exchangers cools below the frosting point and the water vapor in the air changes phase and forms crystals on the surface that separates the two airstreams. It is important to note that saturation is a prerequisite for phase change (frosting or condensation) [1]. The frost layer acts like an insulation layer that reduces heat and moisture transfer between the two streams. Frost formation significantly reduces the performance of the exchangers and heating, ventilation, and air conditioning (HVAC) systems [24]. Frost formation has negative effects in many industrial applications such as refrigeration, compressor blades, aircraft wings, wind turbines, heat pumps, and heat recovery systems, which are studied in the literature [2,57].

There are many studies on frost formation on impermeable surfaces. Many studies focus on reducing or removing frost through heating, anti-icing chemicals and surface modification especially hydrophobic surfaces [814]. Other studies have shown that moisture transfer reduces the risk of frosting in energy wheels, which are rotating energy exchangers [15,16]. Additionally, Sun and Rykaczewski [17] use bilayer coatings to nanoengineer the integral humidity sink effect to provide extreme antifrosting performance. Moreover, experimental studies have shown that the performance of membrane energy exchangers under frosting operating conditions is generally better than the performance of sensible heat exchangers [3,18,19]. However, these experimental studies were on an entire exchanger and therefore the conditions within the membrane were not measured. Membranes have been used widely in industries such as food and beverage [20], water treatment [21], pharmaceutical [22], gas separation [23], and HVAC systems [19,24]. Detailed studies of a membrane under frosting operating conditions have not been presented in the literature. Furthermore, most numerical studies about frost are related to frost growth on impermeable plates [4,25,26]. This paper contributes to the literature by using a numerical study to provide understanding of frosting in membranes for energy exchangers.

The main objective of this paper is to investigate the possibility of delaying or preventing saturation (which is required for frost formation) in membrane exchangers by controlling moisture transfer through the membrane. The idea is to simultaneously dry and cool the air in contact with the membrane to avoid saturation conditions and thereby eliminate condensation and frosting. Therefore, this new approach to avoiding frosting can prevent frost formation in permeable membranes not only for HVAC applications, but also any other applications which use vapor permeable membranes.

## Physical Model

To describe the physical problem, a cross section of a membrane is shown in Fig. 1. The properties of the membrane (Propore™) and impermeable plate studied in this paper are presented in Table 1. Warm and humid air flows over the top of the membrane/plate and cold liquid desiccant passes under the bottom of the membrane/plate. The heat and mass transfer are assumed to be one-dimensional (1D) because the length and the width of the membrane/plate are much greater than the thickness. It should be mentioned that the purpose of this model is to show that frost formation may be prevented or delayed by controlling the moisture transfer through the membrane and that is why one-dimensional is chosen for simulation. For applications which will not be one-dimensional, the model should be developed.

Three cases are investigated in this paper:

1. (1)Permeable membrane
2. (2)Impermeable membrane
3. (3)Impermeable plate

In this case of the permeable membrane, convection heat and mass transfer occurs between the permeable membrane and the fluids flowing over the top and bottom of the permeable membrane. There is pure heat conduction and diffusion of water vapor through the permeable membrane. In this case, the impermeable membrane is as same as the permeable membrane excepting that the bottom surface of the membrane is impermeable to moisture transfer. Therefore, there is diffusion of water vapor through the impermeable membrane from the top surface, but water vapor cannot pass from the bottom surface. This second case is selected to investigate the effect of sorption on the initiation of saturation (or frost). The case of the impermeable plate is an impermeable plate with the same thickness of the membrane. In this case, there is only heat transfer between the plate and the fluids flowing over the top and bottom of the plate and one-dimensional heat conduction through the plate. For all cases, the sides of the membrane/plate are assumed impermeable and adiabatic.

## Numerical Model

To find the location of saturation inside the membrane, a porous media model based on the theory of local volume averaging and local thermal equilibrium is used. The model solves the temperature and vapor density profiles in a membrane to determine when and where saturation conditions occur. The model is used to determine saturation conditions within the membrane because saturation conditions are a prerequisite for frosting to occur. Therefore, if moisture transfer avoids saturation condition throughout the membrane, frosting (phase change) can be avoided. Some assumptions were made to simplify the calculations.

1. (1)The total gas pressure is constant.
2. (2)The mass transfer through the membrane is by diffusion only.
3. (3)The air–water vapor moisture behaves like a perfect gas.
4. (4)The porous membrane is homogenous.
5. (5)No chemical reactions occur in the porous membrane other than the phase change due to adsorption.
6. (6)The thermophysical properties of the fluid and air are assumed to be constant.

###### Governing Equations.

The governing equations needed to calculate the temperature and relative humidity in the membrane are presented in this section. The control volume is defined around the membrane as shown in Fig. 1. Fundamental equations of conservation of energy and mass are given below [29].

• Energy Equation Display Formula

(1)$ρcpeff∂T∂t+m˙hfg=∂∂xkeff∂T∂x$

• Water vapor diffusion equation Display Formula

(2)$∂εgρv∂t−m˙=∂∂xDeff∂ρv∂x$

where $Deff$ is the effective diffusion coefficient of the membrane and is related to binary diffusion coefficient for water vapor in air and the tortuosity $τ$ of the membrane, as follows: Display Formula

(3)$Deff=εgDABτ$

• Continuity equation Display Formula

(4)$∂εl∂t+m˙ρl=0$

The phase change rate $(m˙)$ is calculated from Eq. (5) and is a function of the moisture content $u$ of the membrane, which is presentenced in Eq. (6). Display Formula

(5)$m˙=∂u∂tρeff,dry$

Where the sorption isotherm is [30]: Display Formula

(6)$u=0.0303RH3−0.02938RH2+0.01629RH$

In Eq. (2.6), the total uncertainty for the moisture content $(u)$ is equal to ±0.001 kg/kg [30].

• Where the sorption isotherm volumetric constraint equation Display Formula

(7)$εg+εl+εmem=1$

• Physical properties

The physical properties of the membrane change due to changes in the moisture content and temperature and are calculated as follows: Display Formula

(8)$ρeff=εgρv+ρAIR+εlρl+εmemρmem$
Display Formula
(9)$cpeff=εgρvcp,v+ρAIRcp,AIR+εlρlcp,l+εmemρmemcp,memρeff$
Display Formula
(10)$keff=εgρvkv+ρAIRkAIRρv+ρAIR+εlkl+εmemkmem$

Dimensionless properties, $cp*$, $ρ*$, $u*$, $k*$, $L*$, and $D*$, are defined relative to the base case properties in Table 1 and the following equation: Display Formula

(11)$α*=φsensφbase$

where $α*$ is the general properties ($cp*$, $ρ*$, $u*$, $k*$, $L*$, and $D*$) and $φsens$ is the value of the properties in a sensitivity study, and $φbase$ is the value of properties in Table 1 and Eq. (11)

• Thermodynamic relations

The water vapor density in Eq. (2) can be converted to relative humidity by using the following thermodynamic relations: Display Formula

(12)$pv=RvρvT$
Display Formula
(13)$pAIR=RAIRρAIRT$
Display Formula
(14)$pAIR=pg−pv$
Display Formula
(15)$RH=pvpsat$

• Boundary conditions

As shown in Fig. 1, the top and bottom of the control volume are in the contact with the air and the liquid desiccant, respectively. The sides of the control volume are adiabatic and impermeable. The top $x=L$ and bottom $x=0$ boundary conditions are: Display Formula

(16)$hh,LDTLD−T0,t=−keff∂T∂xx=0$
Display Formula
(17)$ρ0,t=ρLD=fCLD,TLD$
Display Formula
(18)$∂ρv∂xx=0=0$
Display Formula
(19)$hh,AIRTL,t−TAIR=−keff∂T∂xx=L$
Display Formula
(20)$hm,AIRρL,t−ρAIR=−Deff∂ρv∂xx=L$

The combined natural convection and radiation heat transfer are considered to calculate heat transfer coefficient between the air and the membrane[31]. The heat transfer coefficient for natural convection on a horizontal plate for the air side is calculated from the following equations: Display Formula

(21)$NuAIR=0.52Ra0.2$
Display Formula
(22)$hh,AIR=kAIRNuAIRLT$

For the bottom of the membrane, the heat transfer coefficient is calculated for fully developed laminar $(ReLD≈4)$ flow in a duct $(NuLD=3.287)$ [32] as follows: Display Formula

(23)$hh,LD=kLDNuLDLT$

The Chilton–Colburn analogy [33] for heat and mass transfer is used to determine the convection mass transfer coefficients between the membrane and the air and liquid desiccant. Display Formula

(24)$hhhm=cpLe23$

For the case of an impermeable membrane with no moisture transfer from the bottom surface (case 2), Eq. (18) is used. For case 3 (impermeable plate) with no moisture transfer from the bottom and top surface of the surface to the liquid desiccant and the air, Eq. (18) is used and $hm,AIR=0$ in Eq. (20), and $Deff=0$ in Eq. (2).

The properties of the membrane and impermeable plate in Table 1 and the temperature, the humidity, and the convective transfer coefficients in Table 2 are measured and calculated based on the experimental tests conducted on the Propore™ membrane and impermeable plate. More details on experiment will be explained in experiment section (Sec. 5).

• Initial conditions

Initially, the membrane and impermeable plate temperature, humidity ratio, and moisture content are assumed to be uniform and in equilibrium with the temperature and humidity ratio of the liquid desiccant.

• Solution scheme

The governing equations are discretized using the finite difference method. A matlab code has been developed to solve the governing equations (energy equation, water vapor diffusion equation, continuity equation, moisture adsorption equation). The algorithm used to calculate the temperature and relative humidity can be summarized as follows:

1. (1)Input physical data, including physical properties (thickness, thermal conductivity, density, heat capacity, diffusion coefficient of the selected membrane), and operating conditions (temperature, relative humidity, heat and mass transfer coefficients on both sides of the membrane).
2. (2)Calculate and update the effective properties of the membrane.
3. (3)Calculate the temperature at each node from the energy equation (Eq. (1)).
4. (4)Calculate the humidity at each node from the diffusion equation (Eq. (2)).
5. (5)Calculate the phase change rate at each node from the moisture adsorption equation (Eq. (5)).
6. (6)Calculate the gas volume fraction at each node from the volumetric constraint equation (Eq. (7)).
7. (7)Repeat steps 2–6 until a converged solution field is obtained which satisfies both temperature and vapor density.
8. (8)Repeat steps 2–7 for the next time step.

In this paper, a time-step of 0.01 s, a grid size of 2.5 μm, a normalized convergence criteria of 10−8, and a relaxation factor of 0.001 are used. These values were selected based on sensitivity studies; in order to ensure numerical accuracy with these value, the numerical results satisfy energy and mass balances within ±1%.

## Experiment

An experiment was conducted to verify the numerical model by detecting the onset of frosting on the impermeable plate and permeable membrane described in Table 1. The plate/membrane (5 cm × 5 cm) is glued on the test section of the test facility shown in Fig. 2. The test in this paper has natural convection conditions for the air above the plate/membrane and forced convection for the liquid desiccant below the plate/membrane (Table 2). Cold liquid desiccant $LiCl$ provided by a thermal bath, flows into the test section to cool the temperature of the plate/membrane aerating a forced convection heat and moisture transfer boundary condition on the bottom of the plate/membrane (Table 2). T-Type thermocouples are used to measure the temperature of the liquid desiccant at the inlet and outlet of the test section and also the ambient air temperature. A thermocouple is attached on the upper surface of the membrane/plate in the center of the test section with using thermal paste to measure the surface temperature of the plate/membrane. The humidity ratio of the ambient air is measured during experiments using a capacitance-based RH sensor.

The thermocouples and RH sensors are calibrated before the experiments with a Hart scientific 9107 Dry Well Calibrator and a Thunder Scientific model 1200 Mini Humidity Generator respectively giving systematic uncertainties $B$ of ±0.1 °C and ±0.5% RH. The random uncertainties $P$ for thermocouple and RH sensors are equal to ±0.1 °C and ±0.4% RH. The total uncertainty $U$ in each thermocouple is calculated to be ±0.14 °C and ±0.64% RH using the following equation [34]: Display Formula

(25)$U=B2+P2$

## Results and Discussion

In this section, numerical results showing the temperature and relative humidity profiles in the membrane will be presented to show when and where saturation conditions occur in the membrane for the conditions specified in Table 2. In addition, the model will be verified with experimental data, and sensitivity studies will be presented to investigate the effect of each parameters on the onset of the saturation.

###### Temperature Profile.

In order to find when and where saturation occurs in the membrane, the model must determine the temperature at different locations in the membrane as a function of time. Figure 3 presents the temperature as a function of time for different positions within the membrane when the liquid desiccant temperature $TLD$ is −10 °C and −15 °C (other boundary conditions are in Table 2).

As can be seen in Fig. 3, the transient period is very short and the temperatures reach the steady-state within a few minutes. At the steady-state, the temperature difference across the thin membrane ($200$μm) is small ($0.1$° C), which means the thermal resistance of the membrane is small compared to the convective heat transfer resistances. Therefore, the steady-state temperature and heat transfer rate through the membrane mainly depend on the temperature of the air and the liquid desiccant, and the convective heat transfer coefficients. Results showed that changing the air relative humidity from $RHAIR=12%$ to $RHAIR=20%$ does not affect the temperature response and is not presented here.

The temperature profile inside the membrane at different time and steady-state conditions is shown Fig. 4. As it can be seen, the temperature profile through the membrane is linear at steady-state (Fig. 4(b)). It should also be noted that the temperature profile is also quite linear during the transient period (Fig. 4(a)).

In order to show that the numerical model calculates temperature accurately, numerical results for top surface temperature of the membrane at steady-state conditions are verified with experimental results. Figure 5 shows the numerical and experimental data for the temperature at the top surface of the membrane $Tx=L$ at steady-state conditions at different liquid desiccant temperatures. The temperature at top surface of the membrane $Tx=L$ decreases as the liquid desiccant temperature $TLD$ decreases, which is expected. The air relative humidity is constant at $RHAIR=12%$ in Fig. 5. There is good agreement between the numerical and experimental data within the measurement uncertainty. The maximum difference between the numerical and experimental data is 0.8% based on Eq. (26).

To further verify the model, the results of the numerical model are compared with the analytical solution [33] for the case of heat transfer only (no moisture transfer). The detailed results are not presented here, but the average and maximum errors between numerical and analytical data are 0.3% and 1.1% based on the following equation: Display Formula

(26)$err=Tnum−Texp/anlTAIR−TLD×100$

###### Relative Humidity Profile.

The relative humidity of the air inside the permeable membrane is presented as function of time in Fig. 6 for the case of $TLD=−15$° C and $RHAIR=12%$. It should be noted that relative humidity at the top surface of the membrane $x=L$ is the highest and is the first location to reach saturation conditions. Thus, the top surface is the most critical point of the membrane and should be monitored for saturation conditions. It is reminded that the top surface of the membrane is the warm side of the membrane. As shown in Fig. 6, the relative humidity at the bottom surface of the membrane is close to the equivalent relative humidity of the liquid desiccant. That is because the mass transfer resistance at the bottom surface of the membrane is much lower than the moisture resistance of the membrane and the mass transfer resistance at the top surface of the membrane.

The transient relative humidity values of the air at the most critical point in the membrane (i.e., at the top surface) are presented for the permeable and impermeable membrane in Fig. 7 for the case of $TLD=−15$ °C and $RHAIR=12%$ and $RHAIR=20%$. Numerical results show that the air within both the permeable and impermeable membranes becomes saturated when $RHAIR=20%$. If the humidity of the air above the permeable membrane is $20%$ RH $RHAIR=20%$, the humidity inside the permeable membrane reaches the saturation line $(100%)$ in about $2$ min. Therefore, phase change and frosting are expected in this case. However, if $RHAIR=20%$ for the impermeable membrane, the humidity inside the impermeable membrane reaches the saturation line $100%$ in less than 1 min (Fig. 7(a)). Therefore, moisture transfer through the permeable membrane delays frosting.

Figure 7(b) shows that if $RHAIR=12%$, the humidity inside the permeable membrane does not reach the saturation line $(100%)$ while the impermeable membrane reaches saturation quite quickly (within about 1 min). This means that at these conditions, frost is expected in the impermeable membrane, while no frost is expected in the permeable membrane. Therefore, moisture transfer through the permeable membrane is able to eliminate frosting for these conditions. It should be mentioned that experiments were conducted at $TLD=−15$ °C and $RHAIR=12%$ (Fig. 7(b)) for permeable membrane and impermeable plate which will be presented in Sec. 6.3 and experimental results agree with the numerical model.

Figure 8 shows the humidity ratio inside the permeable membrane at various times. In Fig. 8, the humidity ratio has a linear profile with position at steady-state conditions.

The numerical model will be verified with experimental data in Sec. 5.3. Here, the numerical model is verified with analytical data [35] for the case of mass transfer only (no heat transfer). The detailed results are not presented here, but the average and maximum differences between the numerical and analytical results are 0.4% and 1.3% based on the following equation: Display Formula

(27)$err=ρnum−ρanlρAIR−ρLD×100$

Figures 4 and 8 show that the temperature and humidity profiles inside the membrane are linear at steady-state. Therefore, it may be possible to develop an analytical model based on the thermal and mass resistance that is able to accurately predict the temperature and relative humidity at the top and bottom surfaces of the membrane under steady-state conditions. This is left for future work.

###### Detection of Frost at Steady-State Conditions.

This section will present the effect of $TLD$ and $RHAIR$ on the relative humidity of the air at the top surface of the membrane/plate. In each case, pictures from experimental will presented to verify the predictions of the numerical model.

• Effect of liquid desiccant temperature

Figures 9 and 10 present numerical results for the steady-state relative humidity at top surface of the membrane/plate ($RHx=L$) as a function of liquid desiccant temperature for permeable membrane and impermeable plate, respectively. Figures 9 and 10 also contain pictures of the membrane/plate for experiments with and without frosting. The air relative humidity is $12%$ in Figs. 9 and 10. Figure 9 shows that the models predicts that $RHx=L$ will increase from 78% RH to 100% RH as $TLD$ decreases from −14 °C to −17 °C for the permeable membrane. Thus, saturation and frosting are expected for $TLD≤−17$° C. The picture from the experiment confirm the frost limit of $TLD=−17$° C predicted by the model as the photograph from the test with $TLD=−16$° C shows no frost while the photograph at $TLD=−17$° C shows a distinct frost layer on the membrane. Similar results are seen in Fig. 10 for the impermeable plate, except frost is predicted and confirmed for $TLD≤−15$° C. Both the numerical and experimental data show that moisture transfer through the membrane delays frosting.

• Effect of air relative humidity

If air relative humidity increases, there is a greater risk of frosting in the membrane. Figure 11 presents the numerical results for the steady-state values of the relative humidity at top surface of the membrane ($RHx=L$) as a function of the air relative humidity ($RHAIR$) for the permeable membrane and impermeable plate. Once again, the top surface of the membrane is presented because it is the most critical location and reaches saturation conditions first (Fig. 7). The liquid desiccant temperature is $−15$ °C in Fig. 11.

As $RHAIR$ increases, $RHx=L$ increases for both the membrane and plate. Saturation conditions exist (numerical model) when $RHAIR≥11%$ for the impermeable plate and for $RHAIR≥16%$ for the permeable membrane. The model is verified by pictures from experiments with $TLD=−15$ °C and $RHAIR=12%$. The pictures in Fig. 11 show that there is frost on impermeable plate at $RHAIR=12%$ but no frost on the permeable membrane at $RHAIR=12%$ (Fig. 11). Comparing the results for the permeable membrane and impermeable plate shows that moisture transfer through the membrane delays frosting. It should be mentioned that numerical results for the impermeable membrane are the same as the impermeable plate.

• Sensitivity studies

Sensitivity studies can show how different values of each of the properties impact the time at which saturation (frosting) occurs (i.e., the saturation time) when all other properties are constant. Saturation time is defined as the time when relative humidity inside the membrane reaches saturation conditions. Figure 12 shows the impact of different membrane properties on the saturation time when liquid desiccant temperature is −15 °C and air relative humidity is 20%.

With $TLD=−15$° C and $RHAIR=20%$, the saturation time is two minutes for the base membrane properties. As shown in Fig. 12, changing the density, moisture content, thickness, and diffusion coefficient of the membrane have a noticeable effect on saturation time. These parameters affect moisture transfer and thus influence the saturation time. On the other hand, the heat capacity ($cp$) and thermal conductivity ($k$) have a negligible effect on saturation time because heat capacity and thermal conductivity do not affect moisture transfer. These results confirm the hypothesis that moisture transfer affects saturation. It should be noted that changing heat capacity, density, moisture content, and the diffusion coefficient do not change surface temperature (Fig. 12(a)); meanwhile, changes in thermal conductivity and the thickness of the membrane change surface temperature (Fig. 12(b)). As shown in Fig. 12, changing the membrane properties can delay the onset of saturation, which would also delay the onset of frosting. However, the maximum delay is less than five minutes in all the cases tested.

## Conclusion

In this paper, a numerical model that predicts the onset of saturation conditions in permeable membranes has been developed and verified with experiment data. Saturation conditions are required for frosting to occur. Numerical and experimental data show that moisture transfer through the membrane can delay frost formation and even prevent frost formation in some conditions. Therefore, the results in this paper confirm that it is possible to achieve a frost-free exchanger with a vapor permeable membrane. Numerical results show that changing the moisture properties of the membrane by a factor of two can delay the onset of saturation but for less than 5 min. Therefore, in practice, delaying saturation is not as promising as avoiding saturation condition to achieve a frost-free exchanger with a vapor permeable membrane. Based on these findings, it is important to investigate the effects of different parameters, which influence frost formation at the steady-state condition in order to prevent frosting. The results in this paper also show that the temperature and relative humidity at steady-state conditions can possibly be calculated using an analytical model based on thermal and mass resistance.

## Acknowledgements

This research was financially supported by the Natural Science and Engineering Research Council of Canada (NSERC) and the College of Graduate and Postdoctoral Studies at University of Saskatchewan.

## Nomenclature

Acronyms
• HVAC =

heating, ventilation, and air conditioning

• LAMEE =

liquid-to-air membrane energy exchanger

• TC =

thermocouple

English Symbols
• $B$ =

systematic uncertainty

• $C$ =

concentration, $%$

• $cp$ =

specific heat, J/(kg·K)

• $d$ =

diameter of membrane fibers, m

• $D$ =

diffusion coefficient for water vapor in air, m2/s

• $DAB$ =

binary diffusion coefficient for water vapor in air, m2/s

• $err$ =

error

• $eqv$ =

equivalent

• $hfg$ =

evaporation specific enthalpy of saturated liquid water, J/kg

• $hm$ =

local convective mass transfer coefficient, m/s

• $hh$ =

heat transfer coefficient, W/(m2·K)

• $k$ =

thermal conductivity, W/(m·K)

• $l$ =

representative elementary volume of characteristic length, m

• $L$ =

thickness of membrane, m

• $LT$ =

characteristic length, m

• $Le$ =

Lewis Number

• $m˙$ =

phase change rate, kg/(s·m3)

• $Nu$ =

Nusselt number

• $P$ =

pressure or random uncertainty, Pa

• $R$ =

specific gas constant, J/(kg·K)

• $Ra$ =

Rayleigh number

• $RH$ =

relative humidity, $%$

• $t$ =

time, s

• $T$ =

temperature, °C

• $u$ =

mass of moisture adsorbed per kg of dry membrane, kg/kg

• $U$ =

total uncertainty

• $W$ =

humidity ratio of air, gv/kga

• $x$ =

length direction perpendicular to the membrane, m

Greek Symbols
• $α$ =

dimensionless properties

• $ε$ =

volume fraction

• $ρ$ =

density, kg/m3

• $τ$ =

tortuosity coefficient

• $φ$ =

different properties

Subscripts
• $anl$ =

analytical

• $eff$ =

effective

• $err$ =

error

• $exp$ =

experimental

• $g$ =

gas

• $h$ =

heat

• $l$ =

liquid

• $LD$ =

liquid desiccant

• $m$ =

moisture

• $mem$ =

membrane

• $num$ =

numerical

• $sat$ =

saturation

• $sens$ =

sensitivity

• $tot$ =

total

• $v$ =

water vapor

Superscript
• $*$ =

dimensionless quantity

Chemical Symbols
• $LiCl$ =

lithium chloride

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Parent, O. , and Ilinca, A. , 2011, “ Anti-Icing and de-Icing Techniques for Wind Turbines: Critical Review,” Cold Regions Sci. Technol., 65(1), pp. 88–96.
Shang, W. , Chen, H. , and Besant, R. W. , 2005, “ Frost Growth in Regenerative Wheels,” Heat Transfer, 127(9), pp. 1015–1026.
Mahmood, G. , and., and Simonson, C. J. , 2012, “ Frosting Conditions for an Energy Wheel in Laboratory Simulated Extreme Cold Weather,” Seventh International Cold Climate HVAC Conference, Calgary, AB, Canada, Nov. 12–14, p. 9.
Sun, X. S. , and Rykaczewski, K. , 2017, “ Suppression of Frost Nucleation Achieved Using the Nanoengineered Integral Humidity Sink Effect,” ACS Nano, 11(1), pp. 906–917. [PubMed]
Alonso, M. J. , Mathisen, H. M. , Aarnes, S. , and Liu, P. , 2017, “ Performance of a Lab-Scale Membrane-Based Energy Exchanger,” Appl. Therm. Eng., 111, pp. 1244–1254.
Rafati Nasr, M. , Kassai, M. , Ge, G. , and Simonson, C. J. , 2015, “ Evaluation of Defrosting Methods for Air-to-Air Heat/Energy Exchangers on Energy Consumption of Ventilation,” Appl. Energy, 151, pp. 32–40.
Dalena, F. , Basile, A. , and Rossi, C. , 2017, “ Integration of Membrane Technologies Into Conventional Existing Systems in the Food Industry,” Bioenergy Systems for the Future: Prospects for Biofuels and Biohydrogen, Elsevier Science & Technology, Sawston, England.
Alzahrani, S. , and Mohammad, A. W. , 2014, “ Challenges and Trends in Membrane Technology Implementation for Produced Water Treatment: A Review,” J. Water Process Eng., 4, pp. 107–133.
Yu, C. , Liu, Y. , Chen, G. , Gu, X. , and Xing, W. , 2011, “ Pretreatment of Isopropanol Solution From Pharmaceutical Industry and Pervaporation Dehydration by NaA Zeolite Membranes,” Chin. J. Chem. Eng., 19(6), pp. 904–910.
Hamm, J. B. S. , Ambrosi, A. , Griebeler, J. G. , Marcilio, N. R. , Tessaro, I. C. , and Pollo, L. D. , 2017, “ Recent Advances in the Development of Supported Carbon Membranes for Gas Separation,” Int. J. Hydrogen Energy, 42(39), pp. 24830–24845.
Abdel-Salam, M. R. , Ge, G. , Fauchoux, M. , Besant, R. W. , and Simonson, C. J. , 2014, “ State-of-the-Art in Liquid-to-Air Membrane Energy Exchangers (LAMEEs): A Comprehensive Review,” Renewable Sustainable Energy Rev., 39, pp. 700–728.
Wu, X. , Ma, Q. , Chua, F. , and Hu, S. , 2016, “ Phase Change Mass Transfer Model for Frost Growth and Densification,” Int. J. Heat Mass Transfer, 96, pp. 11–19.
Zhuang, D. , Ding, G. , Hu, H. , Fujino, H. , and Inoue, S. , 2015, “ Condensing Droplet Behaviors on Fin Surface Under Dehumidifying Condition—Part I: Numerical Model,” Appl. Therm. Eng., 105, pp. 336–344.
Ge, G. , Mahmood, G. I. , Ghadiri Moghaddam, D. , Simonson, C. J. , Besant, R. W. , Hanson, S. , Erb, B. , and Gibson, P. W. , 2014, “ Material Properties and Measurements for Semi-Permeable Membranes Used in Energy Exchangers,” J. Membr. Sci., 453, pp. 328–336.
Hemingson, H. , 2010, “ The Impacts of Outdoor Air Conditions and Non-Uniform Exchanger Channels on a Run-Around Membrane Energy Exchanger,” M.Sc. thesis, University of Saskatchewan, Saskatoon, SK, Canada.
Talukdar, P. , Iskra, C. R. , and Simonson, C. J. , 2008, “ Combined Heat and Mass Transfer for Laminar Flow of Moist Air in a 3D Rectangular Duct: CFD Simulation and Validation With Experimental Data,” Int. J. Heat Mass Transfer, 51(11–12), pp. 3091–3102.
Iskra, C. R. , 2007, “ Convective Mass Transfer Between a Hydrodynamically Developed Airflow and Liquid Water With and Without a Vapor Permeable Membrane,” M.Sc. thesis, University of Saskatchewan, SK, Canada.
Fauchoux, M. , 2012, “ Design and Performance Testing of a Novel Ceiling Panel for Simultaneous Heat and Moisture Transfer to Moderate Indoor Temperature and Relative Humidity,” Ph.D. thesis, University of Saskatchewan, Saskatoon, SK, Canada.
Rohsenow, W. M. , Hartnett, J. P. , and Cho Young, I. , 2007, Handbook of Heat Transfer, McGraw-Hill, New York.
Bergman, T. L. , Lavine, A. S. , Incropera, F. P. , and Dewitt, D. P. , 2011, Fundamentals of Heat and Mass Transfer, 7th ed., Wiley, Hoboken, NJ.
ASME, 2013, “Test Uncertainty,” American Society of Mechanical Engineers, New York, Standard No. PTC 19.1.
Talukdar, P. , Osanyintola, O. F. , Olutimayin, S. O. , and Simonson, C. J. , 2007, “ An experimental data set for benchmarking 1-D, transient heat and moisture transfer models of hygroscopic building materials. Part I: Experimental facility and material property data,” Int. J. Heat Mass Transfer, 50(23–24), pp. 4527–4539.
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## References

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Amer, M. , and Wang, C.-C. , 2017, “ Review of Defrosting Methods,” Renewable Sustainable Energy Rev., 73, pp. 53–74.
Liu, P. , Rafati Nasr, M. , Ge, G. , Alonso, M. J. , Mathisen, H. M. , Fathieh, F. , and Simonson, C. , 2016, “ A Theoretical Model to Predict Frosting Limits in Cross-Flow Air-to-Air Flat Plate Heat/Energy Exchangers,” Energy Build., 110, pp. 404–414.
Léoni, A. , Mondot, M. , Durier, F. , Revellin, R. , and Haberschill, P. , 2016, “ State-of-the-Art Review of Frost Deposition on Flat Surfaces,” Int. J. Refrig., 68, pp. 198–217.
Kim, P. , Wong, T.-S. , Alvarenga, J. , Kreder, M. J. , Adorno-Martinez, W. E. , and Aizenberg, J. , 2012, “ Liquid-Infused Nanostructured Surfaces With Extreme Anti-Ice and Anti-Frost Performance,” ACS Nano, 6(8), pp. 6569–6577. [PubMed]
Hong, S. J. , Lear, W. E. , and Kim, M. S. , 2014, “ Physical Characteristics of Frost Formation in Semi-Closed Cycle Turbine Engines,” J. Mech. Sci. Technol., 28(4), pp. 1581–1588.
Li, Y. , Li, W. , Liu, Z. , Lu, J. , Zeng, L. , Yang, L. , and Xie, L. , 2017, “ Theoretical and Numerical Study on Performance of the Air-Source Heat Pump System in Tibet,” Renewable Energy, 114, pp. 489–501.
Jeong, C. H. , Lee, J. B. , Lee, S. H. , Lee, J. , You, S. M. , and Choi, C. K. , 2016, “ Frosting Characteristics on Hydrophilic and Superhydrophobic Copper Surfaces,” ASME J. Heat Transfer, 138(2), p. 020913.
Kim, H. , Kim, D. , Jang, H. , Kim, D. R. , and Lee, K.-S. , 2016, “ Microscopic Observation of Frost Behaviors at the Early Stage of Frost Formation on Hydrophobic Surfaces,” Int. J. Heat Mass Transfer, 97, pp. 861–867.
Sommers, A. D. , Truster, N. L. , Napora, A. C. , Riechman, A. C. , and Caraballo, E. J. , 2016, “ Densification of Frost on Hydrophilic and Hydrophobic Substrates—Examining the Effect of Surface Wettability,” Exp. Therm. Fluid Sci., 75, pp. 25–34.
Rahimi, M. , Afshari, A. , Fojan, P. , and Gurevich, L. , 2015, “ The Effect of Surface Modification on Initial Ice Formation on Aluminum Surfaces,” Appl. Surf. Sci., 355, pp. 327–333.
Oberli, L. , Caruso, D. , Hall, C. , Fabretto, M. , Murphy, P. J. , and Evans, D. , 2014, “ Condensation and Freezing of Droplets on Superhydrophobic Surfaces,” Adv. Colloid Interface Sci., 210, pp. 47–57. [PubMed]
Na, B. , and Webb, R. L. , 2003, “ A Fundamental Understanding of Factors Affecting Frost Nucleation,” Int. J. Heat Mass Transfer, 46(20), pp. 3797–3808.
Parent, O. , and Ilinca, A. , 2011, “ Anti-Icing and de-Icing Techniques for Wind Turbines: Critical Review,” Cold Regions Sci. Technol., 65(1), pp. 88–96.
Shang, W. , Chen, H. , and Besant, R. W. , 2005, “ Frost Growth in Regenerative Wheels,” Heat Transfer, 127(9), pp. 1015–1026.
Mahmood, G. , and., and Simonson, C. J. , 2012, “ Frosting Conditions for an Energy Wheel in Laboratory Simulated Extreme Cold Weather,” Seventh International Cold Climate HVAC Conference, Calgary, AB, Canada, Nov. 12–14, p. 9.
Sun, X. S. , and Rykaczewski, K. , 2017, “ Suppression of Frost Nucleation Achieved Using the Nanoengineered Integral Humidity Sink Effect,” ACS Nano, 11(1), pp. 906–917. [PubMed]
Alonso, M. J. , Mathisen, H. M. , Aarnes, S. , and Liu, P. , 2017, “ Performance of a Lab-Scale Membrane-Based Energy Exchanger,” Appl. Therm. Eng., 111, pp. 1244–1254.
Rafati Nasr, M. , Kassai, M. , Ge, G. , and Simonson, C. J. , 2015, “ Evaluation of Defrosting Methods for Air-to-Air Heat/Energy Exchangers on Energy Consumption of Ventilation,” Appl. Energy, 151, pp. 32–40.
Dalena, F. , Basile, A. , and Rossi, C. , 2017, “ Integration of Membrane Technologies Into Conventional Existing Systems in the Food Industry,” Bioenergy Systems for the Future: Prospects for Biofuels and Biohydrogen, Elsevier Science & Technology, Sawston, England.
Alzahrani, S. , and Mohammad, A. W. , 2014, “ Challenges and Trends in Membrane Technology Implementation for Produced Water Treatment: A Review,” J. Water Process Eng., 4, pp. 107–133.
Yu, C. , Liu, Y. , Chen, G. , Gu, X. , and Xing, W. , 2011, “ Pretreatment of Isopropanol Solution From Pharmaceutical Industry and Pervaporation Dehydration by NaA Zeolite Membranes,” Chin. J. Chem. Eng., 19(6), pp. 904–910.
Hamm, J. B. S. , Ambrosi, A. , Griebeler, J. G. , Marcilio, N. R. , Tessaro, I. C. , and Pollo, L. D. , 2017, “ Recent Advances in the Development of Supported Carbon Membranes for Gas Separation,” Int. J. Hydrogen Energy, 42(39), pp. 24830–24845.
Abdel-Salam, M. R. , Ge, G. , Fauchoux, M. , Besant, R. W. , and Simonson, C. J. , 2014, “ State-of-the-Art in Liquid-to-Air Membrane Energy Exchangers (LAMEEs): A Comprehensive Review,” Renewable Sustainable Energy Rev., 39, pp. 700–728.
Wu, X. , Ma, Q. , Chua, F. , and Hu, S. , 2016, “ Phase Change Mass Transfer Model for Frost Growth and Densification,” Int. J. Heat Mass Transfer, 96, pp. 11–19.
Zhuang, D. , Ding, G. , Hu, H. , Fujino, H. , and Inoue, S. , 2015, “ Condensing Droplet Behaviors on Fin Surface Under Dehumidifying Condition—Part I: Numerical Model,” Appl. Therm. Eng., 105, pp. 336–344.
Ge, G. , Mahmood, G. I. , Ghadiri Moghaddam, D. , Simonson, C. J. , Besant, R. W. , Hanson, S. , Erb, B. , and Gibson, P. W. , 2014, “ Material Properties and Measurements for Semi-Permeable Membranes Used in Energy Exchangers,” J. Membr. Sci., 453, pp. 328–336.
Hemingson, H. , 2010, “ The Impacts of Outdoor Air Conditions and Non-Uniform Exchanger Channels on a Run-Around Membrane Energy Exchanger,” M.Sc. thesis, University of Saskatchewan, Saskatoon, SK, Canada.
Talukdar, P. , Iskra, C. R. , and Simonson, C. J. , 2008, “ Combined Heat and Mass Transfer for Laminar Flow of Moist Air in a 3D Rectangular Duct: CFD Simulation and Validation With Experimental Data,” Int. J. Heat Mass Transfer, 51(11–12), pp. 3091–3102.
Iskra, C. R. , 2007, “ Convective Mass Transfer Between a Hydrodynamically Developed Airflow and Liquid Water With and Without a Vapor Permeable Membrane,” M.Sc. thesis, University of Saskatchewan, SK, Canada.
Fauchoux, M. , 2012, “ Design and Performance Testing of a Novel Ceiling Panel for Simultaneous Heat and Moisture Transfer to Moderate Indoor Temperature and Relative Humidity,” Ph.D. thesis, University of Saskatchewan, Saskatoon, SK, Canada.
Rohsenow, W. M. , Hartnett, J. P. , and Cho Young, I. , 2007, Handbook of Heat Transfer, McGraw-Hill, New York.
Bergman, T. L. , Lavine, A. S. , Incropera, F. P. , and Dewitt, D. P. , 2011, Fundamentals of Heat and Mass Transfer, 7th ed., Wiley, Hoboken, NJ.
ASME, 2013, “Test Uncertainty,” American Society of Mechanical Engineers, New York, Standard No. PTC 19.1.
Talukdar, P. , Osanyintola, O. F. , Olutimayin, S. O. , and Simonson, C. J. , 2007, “ An experimental data set for benchmarking 1-D, transient heat and moisture transfer models of hygroscopic building materials. Part I: Experimental facility and material property data,” Int. J. Heat Mass Transfer, 50(23–24), pp. 4527–4539.

## Figures

Fig. 1

Schematic of the problem of heat and moisture transfer through a membrane separating air and liquid desiccant

Fig. 2

Schematic of the experimental facility used to detect the onset of frost on a vapor permeable membrane or impermeable plate

Fig. 3

Predicted transient temperature on the top x=L and bottom (x=0) surfaces and the middle (x=1/2) of the membrane as a function of time for (a) TLD=−10 °C and (b) TLD=−15 °C

Fig. 4

Temperature profile within the membrane for TLD=−15 °C at (a) different time and (b) steady-state condition

Fig. 5

Comparison between numerical and experimental temperature of the top surface of the membrane under steady-state conditions when RHAIR=12%. The error bars indicate the 95% uncertainty bounds in the measured temperature.

Fig. 6

Predicted humidity inside the permeable membrane as a function of time when TLD=−15 °C and RHAIR=12%

Fig. 7

Predicted humidity on the top surface of the permeable membrane and impermeable membrane as a function of time for TLD=−15 °C: (a) RHAIR=20% and (b) RHAIR=12%

Fig. 8

Humidity ratio change as a function of time and location within the membrane for TLD=−15 °C and RHAIR=12%

Fig. 9

Simulated relative humidity on the top surface of the permeable membrane as a function of the liquid desiccant temperature at steady-state when RHAIR=12%. Pictures of the top surface of the permeable membrane are presented for TLD=−17 °C and TLD=−16 °C. (There is white layer of frost on the permeable membrane when TLD=−17 °C.)

Fig. 10

Simulated relative humidity on the top surface of the impermeable plate as a function of the liquid desiccant temperature at steady-state when RHAIR=12%. Pictures of the top surface of the impermeable plate are presented for TLD=−15 °C and TLD=−14 °C. (The white spots in the images are frost crystals and are present for TLD=−15 °C but are absent for TLD=−14 °C.)

Fig. 11

Simulated relative humidity on the top surface of the membrane as a function of air relative humidity for steady-state conditions when TLD=−15 °C. Picture of the top surface of the membrane is presented for TLD=−15 °C and RHAIR=12%. (The white points on the impermeable plate are frost crystals, meanwhile there is not any frost on the permeable membrane.)

Fig. 12

Impact of dimensionless coefficients on dimensionless saturation time when TLD=−15 °C and RHAIR=20% for different dimensionless, (a) heat capacity, density, moisture content, and diffusion coefficient (b) thermal conductivity, and thickness of the membrane.

## Tables

Table 1 Base properties of the permeable and impermeable membrane, and impermeable plate used in the numerical simulation.
Table 2 Boundary conditions

## Errata

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