Improved thermal performance of face mask using phase change material

Abstract

The purpose of this work is to develop a mathematical model to study the thermal performance of a cloth face mask in which phase change material (PCM) is encapsulated as compared to a cloth mask alone. The thermal performance is assessed by the ability to reduce heat loss and water vapor transfer from the exhaled air to inhaled air as well as to reduce the potential for condensation in the mask fabric layers. The mathematical model incorporated all the complex transport mechanisms of heat and moisture, including the realistic breathing pattern for the air passing through the fabric. The core assumption of the model of non-local-thermo-equilibrium was validated by performing experiments on a common cotton cloth mask. The model predicted well the experimentally measured air temperature through a cotton cloth mask during sudden changes in thermal inflow conditions. The validated model was used to perform a parametric study to determine environmental conditions and PCM material characteristics for the enhanced performance of the cloth mask. It was found that a considerable increase in the reclaiming of sensible and latent energy was reached for the same ambient conditions by the introduction of PCM with melting temperature of 22℃ and PCM mass fraction of 20%. Furthermore, the incorporation of PCM allows inhalation of colder outdoor conditions with a high range of relative humidity without condensation at ambient temperatures lower than 12℃, which is not possible without PCM. For higher temperatures than 12℃, the introduction of PCM enlarged the possible range of relative humidity reached without condensation.

Keywords

Phase change material heat and moisture transport in a face mask face mask effectiveness

People are frequently exposed to cold and dry environments when living in regions at subzero temperatures or at high altitude mountains. These hard conditions affect significantly the physiology of the human body during inspiration. In fact, inspired air is warmed up to the body temperature and humidified to the saturation state by the upper airways before reaching the trachea.¹ During expiration much water and body heat are lost which causes dehydration and heat loss of the body,^2,3 decreasing athletic performance and augmenting the probability of hypothermia.⁴ Furthermore, on the economic level, recovering this water and heat using stocks of water and energy sources is costly, particularly during long trips to very cold environments.⁵ Offhand solutions have been practiced, such as wearing a scarf covering the nose and mouth.⁶ Recent research investigated by experiments the use of a face mask as a heat exchanger between the inspired and expired air in order to retain heat and moisture from the exhaled air and deliver them to the inhaled air.^4,5 The more heat and water vapor recovery is achieved from the expired air, the less are the strains imposed on the upper airways and the losses by the human body due to respiration. Rosen and Rosen reported that in severe cold conditions people sleeping in a tent and wearing a cheap and lightweight heat and moisture retaining mask can significantly decrease heat and water losses compared with non-wearers.⁵ The facial mask used in their experiments was the Air Warming Mask manufactured by the 3M company (3M, Minneapolis, MN) and composed of non-woven, synthetic fibers.⁵ They reported a higher weight loss of 0.13 kg during eight hours of sleep for non-wearers compared to mask wearers, which is a considerable amount. However, when large differences between thermal conditions of inspired and expired air, condensation might take place on the fiber surfaces through the mask between the mouth and the ambient side. It is not desirable to have condensation of water on the fiber surfaces inside the mask or to have any movement of liquid condensates through the mask. High relative humidity favors the growth of viruses and bacteria⁷ and liquid condensates is the main path of their diffusion toward the mouth.^8,9 Furthermore, liquid condensates reduce the permeability of the face mask by decreasing the void space through which the air passes, making the respiration process more difficult and requiring more effort from the wearer to breathe, leading to discomfort. Consequently, an exchanger mask in which condensation is prevented or controlled would enhance the comfort of the wearer.

Carnevale and Ducharme⁶ conducted a literature review on heat and moisture exchanger masks and concluded that the simple winter scarf has a comparable performance in heat and moisture retaining compared to the majority of available commercial face masks and presents the advantages of being simple, cheap, and widely available and acceptable by the wearer. However, cloth mask efficacy as a heat and moisture exchanger can be improved if innovative means are provided to control the mask outer surface temperature so that it exchanges more heat and moisture with the flowing air and does not reach low values that cause a drop in the flowing air temperature below its dew point, which leads to condensation. One of the innovative means is to investigate the use of phase change material (PCM) as a buffer for control of temperature during expiration and inspiration. The incorporation of PCM to the cloth material is expected to enhance the heat and moisture retaining from the exhaled air and to lower the potential for condensation, which favors virus and bacteria diffusion toward the mouth and reduces the comfort of the wearer.

PCMs are used to improve the thermal performance of clothing during transient periods when human environment changes from warm to cold and vice versa. This is achieved by absorbing or releasing heat when subjected to heating or cooling during a phase change and thus reducing the effect of temperature swings on human body heat loss.^10,11 PCM has been used widely in cloth material for the sake of minimizing sudden temperature changes at the body interface and providing thermal comfort.¹¹ Ghali et al.¹² investigated the use of PCM in fabric when subject to periodic ventilation induced by walking or motion. They concluded that the PCM was not able to regenerate due to the relatively low airflow rate passing through the PCM fabric.¹² The PCM effect lasted for a short period of time because of the lack of the ability to regenerate it.¹² However, the use of PCM in cloth mask might be promising due to the relatively high flow rates of air enhancing heat exchange and also due to the periodicity of alternating inhalation /exhalation streams which could regenerate the PCM material. To our knowledge, there is no published work that has investigated the performance of a PCM face mask. The PCM is characterized by its melting/crystallization temperature at which PCM delivers heat by changing from liquid to solid state or absorbs heat by moving from solid to liquid state. The potential for condensation may decrease by adding a PCM layer which can block the rapid fall of the air temperature below its dew point during exhalation through the mask. The proper choice of the PCM melting temperature may decrease the condensation rate leading to a better performance of the face mask by minimizing virus and bacteria penetration while enhancing heat and moisture exchange and wearer comfort. As for the PCM quantity, it is not recommended by the textile industry to increase the percentage of PCM because it will increase the cost of the fabric as well as its weight. A 20% fraction of PCM is representative of commonly used values by the industrial manufacturers and is adopted in this work.¹²

In order to study the effect of incorporating PCM material on the performance of the face mask modeling of the physical processes taking place through the mask is necessary. The mask can simply be treated as a porous media in which heat and moisture transport is coupled. The modeling of heat and moisture transfer in textile materials and face masks present many similarities, however, the transient periodic airflow due to the breathing cycles adds complexity. In the modeling of thin fabric material subject to normal flow, many researchers assumed local thermal equilibrium (LTE) between the air and the fibers due to the relatively low flow rates through the fabric and its small dimensions.^13,14 Amiri and Vafai¹⁵ stated that the LTE must be relaxed in the cases where the temperature difference between the fluid and the solid fabric were significant. Ghali et al.¹⁶ were the first to study the effect of periodic ventilation on heat and mass transport through fibrous material by developing a three-node fabric thermal model that did not assume LTE in the fiber. They modeled the heat and moisture transfer in thin porous textile material in the case of vigorous motion of the human body and conducted experiments and reported heat and mass transfer coefficients between the passing air and the fabric nodes.¹⁷ Similarly, due to the alternating inspiration and expiration cycles with different conditions through the face mask, the LTE assumption may not be valid although it was adopted by Li et al.⁷ in their numerical modeling of diffusion of viruses by movement of liquid condensates through a face mask. The main physical processes involved in the textile material which can be extended to the face mask application are the condensation/evaporation, sorption/desorption, conduction, convection, and diffusion, in addition to melting/crystallization processes when PCM is incorporated into the fabric material. Owing to the common features in physical processes in cloth material and face mask application, it is reasonable to investigate the efficiency of using a cloth material as a cheap and affordable face mask. Modeling of heat and moisture transfer in cloth face masks, with all their associated complex physical processes, is not widely developed in the literature.

The aim of this work is to study by mathematical modeling the thermal performance of a PCM face mask during breathing in a cold environment and compare it to the performance of a simple cloth mask of the same material without the encapsulated PCM. The mathematical model will predict the enhancement in thermal performance of the PCM mask for different inspiration air conditions, and its ability to minimize the potential for condensation. The mathematical model will incorporate all the complex transport mechanisms of heat and moisture, including the incorporation of a realistic breathing pattern for the air passing through the fabric. The model core assumption of the non-local thermal equilibrium (NLTE) will be validated by performing experiments on a common cotton cloth mask using a built experimental set up. The validated model will be used to perform a parametric study to determine environmental conditions and PCM material characteristics for enhanced performance of the cloth mask.

Mathematical formulation

Figure 1 presents the mouth–fabric environmental system and the fabric–PCM three nodes. The air–fiber model is best represented by a flow of air around cylinders in cross flow in multilayers, where the air voids are connected through the cylinders (yarns) as shown in Figure 1. The mask area is A_f and the fabric thickness is e_f. The airflow is normal to the fabric plane. The mask fabric model is based on a multilayer fabric model that uses, for each layer, the three-node model of Ghali et al.¹⁶ in which the fabric is represented by an outer node, inner node, and air-void node. The outer node represents the exposed surface of the yarns (yarns that are woven into a fabric), which is in direct contact with the penetrating air in the void space between the yarns (the air-void node). The inner node represents the inner portion of the “solid” yarn (fibers on the interior of the yarn), which is completely surrounded by the outer node. The outer node exchanges heat and moisture transfer with the flowing air and with the inner node, while the inner node exchanges heat and moisture by diffusion only with the outer node. The moisture uptake in the fabric is initially very fast due to the convection effect at the yarns surface, followed by diffusion to the yarn interior.¹⁷ The PCM is assumed to be encapsulated inside the hollow fibers and distributed evenly within the fabric inner and outer nodes. However, the percentage of phase of change in each node may differ depending on the rates of energy transfer.

Figure 1.

Schematics of the mouth–fabric environmental system and the fabric–PCM three-node.

It is important to keep the model simple while capturing the complex interacting physical processes during respiration through the face mask. Several simplifying assumptions are adopted. The thermophysical properties of the dry air and the fibrous material are assumed constant. The pressure and velocity losses of the air stream in the axial direction (along E_f) are negligible for both the inspired and expired air. The heat of adsorption is assumed to be entirely delivered to the fabric material and a NLTE is assumed to exist between the flow of air and outer node and between the inner and outer nodes.¹⁶ The heat and mass-transfer coefficients between the air/outer nodes and outer/inner nodes are a function of the air cross-flow velocity and hence they vary in a sinusoidal way with time along the channel.¹⁷ In addition, the PCM is assumed homogeneous and isotropic at thermophysical properties of the fiber node in which it is incorporated in each phase while the phase change is assumed to occur at a single temperature and the difference in density between solid and liquid phases is negligible.

To accurately model the breathing process through the face mask, conduction, convection, diffusion, melting/solidification, and adsorption/desorption phenomena must be incorporated. Given the large difference in thermal conditions between the inspired and expired air, condensation may occur on the outer cold surface of the fibers. However, the aim of this work is to investigate cases for which the introduction of PCM prevents condensation by controlling the mask temperature such that the air temperature remains above its dew point. For this reason the condensation/evaporation processes will not be included in our mask model.

Mask–air layer breathing model

It is important to establish thermal conditions and flow rate of the air penetrating the mask during inspiration and expiration. These conditions are used as input to the fabric three-node transient model.

The human breathing pattern is nearly sinusoidal.¹⁸ Chen et al.¹⁹ conducted experimental work to determine the respiration amplitude and frequency for subjects of different sex and size (height, weight, body surface area). They concluded that the period of inhalation was slightly lower than the exhalation one and reported the following correlations of the inspiration and expiration flow rates as a function of time and physical properties of the human:

q_{i} (t) = a_{i} \sin (b_{i} t) \times 10^{- 3}

(1{\rm a})

where the general expression q (m³/s) for the sinusoidal behavior of flow rate during inhalation and exhalation is characterized by amplitude a (L/s). The index i represents alternatively the inspiration (in) or the expiration (ex). The values of a and b differ slightly between inspiration and expiration and are related by the following expression for each of the periods:¹⁹

a_{i} = \frac{b_{i} TV}{2}

(1{\rm b})

b_{i} = \frac{π {RF}_{i}}{30}

(1{\rm c})

The tidal volume (TV) which is the volume exhaled during each breath is given by:¹⁹

TV = \frac{MV ({RF}_{in} + {RF}_{ex})}{2 {RF}_{in} {RF}_{ex}}

(1{\rm d})

where MV is the volume expired in one minute and RF_in and RF_ex are the number of breaths per minute during inspiration and expiration, respectively. The TV, MV, and RF factors have been widely studied in the literature²⁰ and were shown to depend on several physiological parameters such as the age, sex, height, weight, body surface area, posture, and activity level.²¹ For instance, the experimentation performed by Chen et al.²² led to the following correlations for females:

MV = 4.634 \times S

(1{\rm e})

{RF}_{in} = 46.43 - 18.85 \times H

(1{\rm f})

{RF}_{ex} = 54.47 - 25.48 \times H

(1{\rm g})

MOA = 1.16 \pm 0.67

(1{\rm h})

where H is the height in meters, MOA is the mouth opening (cm²), and S is the body surface area (m²). The mean values of all the mentioned variables were used to calculate the periods of inspiration and expiration for normal breathing and the airflow rates and cross-sectional area A_f through the mask in m² for women¹⁷ as follows:

t_{in} = \frac{30}{{RF}_{in}} = 1.96 s

(2{\rm a})

t_{ex} = \frac{30}{{RF}_{ex}} = 2.41 s

(2{\rm b})

q_{in} (t) = 4.308 \times 10^{- 4} \sin (1.6051 t)

(2{\rm c})

q_{out} (t) = 3.493 \times 10^{- 4} \sin (1.3014 t)

(2{\rm d})

A_{f} = MOA \times k \times 10^{- 4}

(2{\rm e})

where k is the expansion factor of the airflow limiting area from the mouth to the face mask and was set to 7.0 resulting in a cross flow-section of approximately 10^–3 m² through the mask.

Boundary conditions

The breathing pattern provided in the above sinusoidal expression of the airflow of the air through the mask was used as the air normal flow in our model. The associated airflow thermal conditions change periodically. Experimental values on relative humidity of expired air have been reported in the literature to vary in the range of 80%^22–24 to nearly 100%.^25,26 The discrepancy in the obtained results was related to their variations with external factors, such as the variation of ambient conditions, activity level, and measurement errors due to the slow response time of humidity and temperature sensors. Cain et al.²⁶ studied the effect of activity level and ambient temperature on the expired conditions and came up with the conclusion that the humidity level was higher than 90% in the majority of the cases. The expired air temperature was reported not to vary significantly with the work rate but was affected by the outdoor conditions, varying in average from 32℃ to 34℃ at ambient temperatures from 0℃ to 20℃, and was expected to reach 37℃ as the ambient temperature approached 37℃.²⁶ In this work, the conditions of the exhaled air were varied according to the literature findings with the ambient conditions using the linear interpolation method. As the variables are functions of position and time, initial values of temperature and water vapor pressure for airflow, inner, and outer nodes as well as regains of the outer and inner nodes should be specified however the steady-state solution is reached and is independent of the initialized state. During expiration, the temperature and water vapor pressure of the expired air were imposed at the mouth side of the face mask while a zero flux condition (Neumann boundary condition type²⁷) was imposed at the ambient side. On the other hand, during inspiration, temperature and water vapor pressure of the ambient air were imposed at the ambient side of the face mask while a zero flux condition was imposed at the mouth side. The boundary conditions at the internal and external side of the mask during the respiration period are presented below.

During the inspiration period:

T_{a} (x = e_{f}, t) = T_{in}

(2{\rm f})

(\frac{\partial T a}{\partial x}) x = 0 = 0

(2{\rm g})

P a (x = e f, t) = P in

(2{\rm h})

(\frac{\partial P a}{\partial x}) x = 0 = 0

(2{\rm i})

where e_f is the fabric thickness, x is the axial coordinate (x = 0 at the mouth side; x = e_f at the ambient side), t is the time, T_a and P_a are, respectively, the temperature and water vapor pressure of the airflow through the mask as a function of position and time, T_in and P_in are the temperature and water vapor pressure imposed at the ambient side during the inspiration period.

During the expiration period:

T_{a} (x = 0, t) = T_{ex}

(2{\rm j})

(\frac{\partial T a}{\partial x}) x = e f = 0

(2{\rm k})

P a (x = 0, t) = P ex

(2{\rm l})

(\frac{\partial T a}{\partial x}) x = e f = 0

(2{\rm m})

T_ex and P_ex are the temperature and water vapor pressure imposed at the mouth side during the expiration period.

Mask fabric model

Nodal mass balances for each fabric layer of thickness dx

In the derivation of the mass balances of the air-void node, the air–water vapor mixture is assumed dilute and the bulk velocity of the mixture is very close to the velocity of the void air. This assumption allows simplification of the mass balances by ignoring the effect of counter transfer of the air. The thermophysical properties of the dry air and the fibrous material are assumed to be constant and are evaluated at the mid temperature range, since the range of involved temperatures is limited to the interval (5℃ to 37℃). The pressure and velocity losses of the air stream in the axial direction were negligible for both the inspired and expired air due to the small thickness of the fabric material. The PCM microcapsules do not play any role in the water vapor mass transport since they are impermeable and do not affect the permeability of the fabric material as they are encapsulated inside the fibers. So the water vapor mass balances for the outer node and the inner node are given in equations (3) and (4), respectively, following the model of Ghali et al.¹²:

γ ɛ_{f} ρ_{f} dx \frac{\partial R_{o}}{\partial t} = H'_{mo} (P_{a} - P_{o}) + H'_{mi} (P_{i} - P_{o})

(3)

(1 - γ) ɛ_{f} ρ_{f} dx \frac{\partial R_{i}}{\partial t} = H'_{mi} (P_{o} - P_{i})

(4)

where R_o is the regain of the outer node, R_i is the regain of the inner node, ρ_f is the density of the fabric, γ is the fraction of the outer node volume to the total fiber one¹⁷ reported equal to 0.6, ɛ_f is the volume fraction of the fibers, P_o is the local equilibrium vapor pressure in the outer node that does reflect the moisture content and temperature, P_i is the local equilibrium pressure in the inner node, P_a is the local pressure in the air void node, H′_mo and H′_mi are the normalized mass transfer coefficients reported by Ghali et al.¹⁷ between the outer node and the penetrating air node and the outer node and the inner node, respectively.

The term on the left-hand side (LHS) of outer node mass balance of equation (3) represents the water vapor mass storage in the outer node, the first term on the right-hand side (RHS) of equation (3) represents the convective mass transfer between the outer node and the air, and the second term on the RHS represents the diffusion of water vapor between the outer and the inner nodes lumped by a convective term. Whereas the term on the LHS of the inner node mass balance in equation (4) represents the water vapor mass storage in the inner node, and the term on the RHS represents the diffusion of water vapor between the outer and the inner nodes lumped by a convective term. The total fabric regain R (kg of adsorbed H₂O/kg dry fiber) can simply be calculated as R = γR_o + (1 – γ)R_i. The regain values are easily obtained from the sorption curves of the fibers’ material relating the regain to the relative humidity of the air in equilibrium with the fibers.²⁸

The water vapor mass balance in the air void node is given in equation (5):

ρ_{a} ɛ_{a} \frac{\partial Y_{a}}{\partial t} = \frac{\partial}{\partial x} (D_{a} ɛ_{a} \frac{\partial Y_{a}}{\partial x}) + ρ_{a} \frac{q_{i} (t)}{A_{\binom{f}{}}} \frac{\partial Y_{a}}{\partial x} + H'_{mo} (P_{o} - P_{a})

(5)

where ρ_a is the density of the air, ɛ_a is the volume fraction of the air void, D_a is the diffusion coefficient of water vapor in the air, A_f is the cross-sectional area of the fabric, q_i(t) is the instantaneous airflow rate through the mask, and Y_a is the water vapor content in kg of H₂O per kg of dry air.

Nodal energy balances for each fabric layer of thickness dx

A NLTE exists between the flow of air and the inner and outer node due to the relatively quick heat and moisture exchange caused by the periodic breathing and the significant change in temperature and relative humidity that takes place between the inspired and expired air conditions. This fundamental assumption was validated experimentally as presented in the experimental section. The energy balance on the outer and inner nodes depends on the PCM (paraffin) state. The temperatures of the inner and outer nodes can vary only if they are different from the melting/crystallization point of the PCM. Once this point is reached, the temperature of the nodes remains constant until the phase change is completed from liquid to solid (when heat is released by the PCM) or from solid to liquid (when heat is absorbed by the PCM). The vapor enters the pores, diffuses in the pores, and meanwhile is adsorbed. Hence, the heat of adsorption is assumed to be totally delivered to the fabric material immediately when water vapor enters the porous layer. The energy equation of the outer node, when the outer node temperature is not equal to the melting temperature of the paraffin (no phase change takes place), is written as:

γ ɛ_{f} ρ_{eq} c_{eq} dx \frac{\partial T_{o}}{\partial t} = H'_{co} (T_{a} - T_{o}) + H'_{ci} (T_{i} - T_{o}) + h_{ads} γ ɛ_{f} ρ_{f} dx \frac{\partial R_{o}}{\partial t}

(6{\rm a})

where γ is the volume fraction of the outer node in the fabric material, ɛ_f the volume fraction of the fibers, c_eq is the equivalent specific heat of the fibers incorporated with PCM and ρ_eq is their density, T_a_, T_o, and T_i are, respectively, the temperatures of the air void, outer and inner nodes, H′_co is the convection heat-transfer coefficient between the air and outer nodes, H′_ci is the conduction heat-transfer coefficient between the inner and outer nodes, h_ads is the adsorption heat of water vapor by the fabric material, and R_o is the regain of the outer node.

The term on the LHS of equation (6a) represents the energy storage in sensible heat form of the outer node, the first and second terms of the RHS represent the convective and conduction heat transfer between the outer node and the air and inner node, respectively. The last term illustrate the adsorption/desorption heat delivered/extracted by water vapor adsorbed/desorbed by the outer node. When the temperature of the outer node is equal to the melting temperature of the PCM (∂T_o/∂t = 0), the outer node energy balance is given by:

γ ɛ_{f} ρ_{f} h_{f} dx \frac{\partial α_{o}}{\partial t} = H'_{co} (T_{a} - T_{m}) + H'_{ci} (T_{i} - T_{m}) + h_{ads} γ ɛ_{f} ρ_{f} dx \frac{\partial R_{o}}{\partial t}

(6{\rm b})

where h_sf is the melting heat of the PCM material and α_o is the liquid fraction of the PCM of the outer node, and T_m is the melting temperature of the PCM. The term on the LHS of equation (6b) represents the energy storage in latent heat form by the PCM of the outer node.

The energy equation of the inner node when the temperature of the inner node is different from the melting temperature of the PCM and is given by:

(1 - γ) ɛ_{f} ρ_{eq} c_{eq} dx \frac{\partial T_{i}}{\partial t} = H'_{ci} (T_{o} - T_{i}) + h_{ads} (1 - γ) ɛ_{f} ρ_{f} dx \frac{\partial R_{i}}{\partial t}

(7{\rm a})

where R_i is the regain of the inner node. The term on the LHS of equation (7a) represents the energy storage in sensible heat form of the inner node; the first term on the RHS represents the conduction heat transfer between the inner node and the outer node. The last term illustrates the adsorption/desorption heat delivered/extracted by water vapor adsorbed/desorbed by the inner node. When the temperature of the inner node is equal to the melting temperature of the PCM, (∂T_i/∂t = 0), the inner node energy balance is given by

(1 - γ) ɛ_{f} ρ_{f} h_{sf} dx \frac{\partial α_{i}}{\partial t} = H'_{ci} (T_{o} - T_{m}) + h_{ads} (1 - γ) ɛ_{f} ρ_{f} dx \frac{\partial R_{i}}{\partial t}

(7{\rm b})

where α_i is the liquid fraction of the PCM of the inner node. The term on the LHS of equation (7b) represents the energy storage in latent heat form by the PCM of the inner node.

The energy equation of the air void node is given by:

ρ_{a} c_{a} ɛ_{a} \frac{\partial T_{a}}{\partial t} = \frac{\partial}{\partial x} (k_{a} ɛ_{a} \frac{\partial T_{a}}{\partial x}) + ρ_{a} \frac{q_{i} (t)}{A_{f}} \frac{\partial T_{a}}{\partial x} + H'_{co} (T_{o} - T_{a})

(8)

where ɛ_a is the volume fraction of the void space, c_a_, ρ_a, and k_a are, respectively, the specific heat, the density, and the conductivity of the air. The term on the LHS of equation (8) represents the energy storage in sensible heat form of the air, the first term on the RHS represents the heat conduction in the air in the direction of the flow, the second term on the RHS represents the convection in the air in the direction of the flow, and the last term represents the convective heat transfer between the air and the outer node.

Thermophysical-property equations

Interrelationships between some of the parameters of the equations are needed.to solve for the different variables predicted by the mathematical model:

{RH}_{j} = - 6.33 R_{j}^{3} - 23.7337 R_{j}^{2} + 10.42 R_{j} - 0.05

(9{\rm a})

P_{sat, j} = \exp (23.196 - \frac{3816.44}{T_{j} - 48.13})

(9{\rm b})

P_{j} = {RH}_{j} P_{sat, j}

(9{\rm c})

P_{a} = \frac{P_{atm} Y_{a}}{0.62188 + Y_{a}}

(9{\rm d})

where the index j refers to the inner or outer node, the index a refers to air, RH is the relative humidity, R is the regain, Y is the absolute humidity, P is the water vapor pressure, P_sat the saturation pressure, P_atm the atmospheric pressure.

Numerical modeling method

Non-linear partial differential equations can be solved using different methods: analytical techniques such as the homotopy method^27,28 or numerical methods.^12,29 The complexity of the current physical model, the large number of physical parameters and properties that depend on environment and fabric temperature, and the presence of PCM which requires a separate energy balance equation due to melting/solidification presented a challenge to use analytical methods. In this work, a numerical approach for solving similar equations has been used.¹²

The mask domain is divided into 40 control volumes of length dx = 5 × 10^–5 m in the direction of the flow. The coupled mass- and heat-transport equations of the outer and inner nodes of the fabric, and the air void are discretized spatially and temporally with a time step of 0.005 into algebraic equations using a semi-implicit finite volume method developed by Patankar.²⁹ A three-diagonal matrix algorithm is adopted for solving the time-dependent mass and energy balances, derived from conservation laws and applied on each control volume representing a layer that includes a set of fiber layers separated by voids through which air penetrates. The simulation period is considered long enough to ensure that steady periodic solution is obtained and that the initial conditions no longer affect results. The vapor pressure of the flowing air in the air spacing layer or in the fabric voids is related to the air relative humidity, and temperature and is calculated using the psychrometric formulas of Hyland and Wexler³⁰ to predict the saturation water-vapor pressure and hence the vapor pressure at the specified relative humidity. Cotton was selected as the mask material because it is a common cloth component. The regain of cotton has a definite relation to the relative humidity of the water vapor through a property curve of regain versus relative humidity and an interpolative correlation has been used.³¹ The correlations are used in the simulation to calculate the inner and outer nodes’ relative humidities corresponding to the values obtained for inner and outer regains, respectively. At every time step, the air mass flow rate and the total regain are updated and the inner and outer nodes temperatures are compared with the melting temperature of the PCM paraffin to determine the time when phase changes takes place to the time when the paraffin completely solidifies or melts. The small spatial and time steps are used to ensure the accuracy and stability of the solution with convergence criteria of the residuals of the equations for all the predicted variables set to 10^–7.

The developed simulation model solves for the following variables with respect to time and position in the direction of the respiration: (1) the temperature of the outer node, inner node, and air void, (2) the vapor pressure at the surfaces of the outer and inner nodes and of the air in the voids, (3) the regains of the outer and inner nodes, and (4) the fractions of melted PCM in the inner and outer nodes.

Experimental details

A major physical characteristic of the flow in the mask is the non-existence of thermal equilibrium between the penetrating air and the fiber. The ability of the model to capture this characteristic over a short transient period needs to be tested. The objective of the experiment is to check the validity of the NLTE modeling approach in the proposed model through testing the ability of the model to predict transient effects on air temperature on both sides of the mask when a sudden change in flow temperature through the mask takes place. To meet this objective, we conducted the experiment on a simple highly permeable cloth material that does not contain PCM. Two air streams were drawn from two separated climatic chambers of different environmental conditions. The airflow through the cloth was abruptly changed in flow direction and thermal conditions (temperature and moisture content) to test the NLTE of cloth material. The experimental setup is designed using fast responsive switching mechanisms of the two airflows from different chambers without complex thermal control of the conditions of the two streams.

Description of the experimental setup

An experimental setup was constructed as shown in Figure 2 to reproduce a sudden change in airflow temperature in which the flow through the cloth mask temperature suddenly changes from a value at the cool environment to a value at the average temperature of exhaled air. This is achieved by subjecting the mask cloth material to a set of conditions for a long period of time in order to reach a steady periodic state followed by an abrupt change of conditions and report the temperatures in front and behind the fabric. Untreated cotton was chosen as a representative of a most common worn fabric to use in the ventilation tests. The cotton was obtained from Test Fabrics Inc. (Middlesex, NJ08846), and is made of un-mercerized cotton duck, style #466 of thickness 1 mm. In order to avoid the mixing of warm and cold air, two different air passages were controlled by two separate fans drawing air from two separate environmental chambers in which temperature and humidity were controlled. Two check and on–off valves were used to insure a unidirectional airflow and to avoid back flows and air leakages. Two thermocouples of type T allowed the measurement of the temperature of the flows in both directions before and after reaching the fabric. The accuracy of the temperature readings was ± 0.5℃ and response time was 0.1 s.

Figure 2.

A schematic of the experimental setup.

The built system consisted of the following components: (1) two electric fans, (2) two flow meters, (3) two check valves, with two on–off valves, (4) four aluminum pipes, (5) two thermocouples, (6) a cotton fabric, and (7) two couplers. The specifications of the various components are summarized in Table 1.

Table 1.

Parameters of the components of the experimental setup

Item	Description
Aluminum pipe	Diameter = 2.3 cm
Length = 0.4–0.5 m
Cotton fabric	Thickness = 1 mm
Density = 1540 kg/m³
Couplers	Manufactured from polyamide with high insulation properties
Fan	Power = 2000 W
Humidifier	VapacMicrovap
Humidity sensor	HX71 series
Mass flow meter	Code: 4043
Range of measurement: [0; 200 L/min]
Thermocouple	GG-T-20-SLE
Accuracy: ± 0.5℃
Response time: 0.1 s

Since the model does not account for condensation, high relative humidities through the mask can cause condensation on the thermocouple and affect the accuracy of the results, low relative humidity (50%) was used in both chambers. However, a large difference of temperature of 20℃ was established between the two climatic chambers (T = 35℃ in Chamber 1 and T = 15℃ in Chamber 2) to reveal the thermal performance of the cotton fabric by retaining heat from a hot flow and delivering heat for a cold one. The precision in the set conditions of the climatic chamber temperature was ± 0.5℃ and chamber relative humidity was ± 2%.

The experimental setup was used to investigate the transient effect of the mask. A constant airflow rate of 20 L/min amplitude was used because this value is close to the mean flow rate of the human respiration.¹⁹ The experiment started by operating Fan 1 drawing air at 35℃ through the mask from Chamber 1for a period of one hour to insure that steady state is reached, meanwhile the on–off Valve 1 was closed and the on–off Valve 2 was opened to allow the flow to go in one direction, then the operation of Fan 1 was interrupted, the on–off Valve 1 was opened, and the on–off Valve 2 was closed, and Fan 2 was operated in order to reverse the airflow so that air at 15℃ flows from Chamber 2 to 1 passing by the cotton-cloth mask. The switching process took approximately 2 s. The temperatures before and after the fabric were recorded every 0.1 second with a relative accuracy of the thermocouple of ± 0.5℃. Once the steady-state condition was reached the flow was reversed again and air at 35℃ flows from Chamber 1 to 2 passing by the fabric material using the same procedure. The whole experiment took about two-and-a-half hours and was repeated several times to minimize experimental errors.

Results and discussion

Model validation

The developed simulation model predicted the temperature of the outer node, inner node, and air void temperature of the fabric. Measurements were taken of the air temperature exiting the fabric air layer at air void temperature for the two opposite directions of the flow. In the absence of the mask, the temperature of the air stream dropped instantaneously from 35 ± 0 .5℃ to 15 ± 0.5℃ following the temperature of the source chamber.

Figure 3 presents a plot of measured and predicted values of air temperature at airflow rate of 20 L/min exiting the cloth mask. There is good agreement between measured and predicted values with the error not exceeding ± 0.5℃. The results clearly show that the stored heat by the mask material from the hot airflow prevented the sharp decrease in air temperature and in the initial period of one or more minutes, the air temperature was above 15℃ (Figure 3 after switch 1). In addition, the heat extracted from the air by the mask when switching to the warm flow postponed the spontaneous increase in the air temperature (Figure 3 after switch 2). The transient effect for both cases of cold and hot stream lasted for about 70s.

Figure 3.

Comparison of the experimentally measured and model predicted air temperature.

Since the NLTE assumption is validated experimentally, the mathematical model is then used to simulate both cloth and PCM masks subjected to real breathing cycles and the results are presented in the next section.

Parametric study of PCM-Mask for effective operation

Incorporating PCM in a cloth mask is expected to present an improvement in cloth mask performance resulting in minimizing the occurrence of condensation and in retaining heat and moisture from the exhaled air to be captured by the inhaled air. In fact, the regeneration of the PCM material in the cloth mask by the periodic breathing flow is a promising factor that could enhance significantly the heat and mass transfer performance of the mask and regulate the variation of the outer node temperature such that the air temperature remains above its dew point. By proper selection of the melting temperature, the incorporation of PCM in cloth material may reduce the potential for condensation which favors infected particle diffusion.

In order to illustrate the improvement of the heat and mass transfer provided by the incorporation of PCM, the case of ambient conditions of T_∞ = 12℃, RH = 30%, and the corresponding exhaled conditions of T = 33℃, RH = 90% was simulated for cotton cloth mask of 2 mm thickness and cross-sectional area of 10 cm² with and without PCM and for a breathing period of 4.37 s (1.96 s of inspiration, 2.41 s of expiration). The selected common PCM is paraffin, incorporated in the fiber material in an encapsulated form which will prevent the generation of any paraffin particles that may be inhaled during breathing and risk the health of the wearer, and is characterized by a latent melting heat of 200 kJ/kg, a melting temperature of 22℃, and a PCM mass fraction of 20%.³² As reported by Ghali et al.¹², a paraffin PCM material can be prepared as a combination of different kinds of paraffin (octadecane, nonadecane, hexadecane, etc.) each of which is characterized by its proper melting/solidification temperature. By using the appropriate proportion of each paraffin type, the desired crystallization temperature of the PCM could be obtained. Different properties used in the current application of the mathematical model are summarized in Table 2. Figure 4 shows a schematic of the simulated mask–air system with selected environmental boundary conditions for the simulations. The variation of different variables is presented for two normal breathing cycles.

Figure 4.

Representation of one of the simulated boundary conditions through the mask.

Table 2.

Properties used in the simulation model

Physical parameter	Value
Density of the cotton material (kg/m³)	1540
Density of the airflow (kg/m³)	1.25
Specific heat of the cotton material (J/kg K)	1540
Specific heat of the airflow (J/kg K)	1000
Specific heat of the PCM (J/kg K)	2000
Porosity of the fabric	0.8
Adsorption heat of the water vapor (J/kg)	2.522 × 10⁶
Heat conduction coefficient of the air (W/m K)	0.0245
Diffusion coefficient of water vapor in the air (m²/s)	2.4 × 10^–5

The differences in temperature and water vapor pressure between the air and outer nodes are the driving factors for heat and mass transfer, respectively. As these differences are reduced during inspiration and expiration phases, the potentials for energy and moisture exchanges decrease with time. Hence, there is a need to maintain continuous potentials for heat and mass transfer. Figure 5a shows that the average water vapor regain of the outer node oscillates between 0.103 and 0.112 kg/kg of dry fiber without PCM while it varies with an approximately 35% increase in amplitude between 0.109 and 0.121 with PCM. Figure 5b shows the variation of the PCM fraction of the outer nodes at different locations through the mask. The periodicity of the breathing pattern not only allows the regeneration of the adsorbed moisture content of the fabric material (Figure 5a) during inspiration so that it could adsorb water vapor again during expiration, but also is able to regenerate the PCM at different positions through the mask (Figure 5b). Figure 5b reveals that the PCM liquid fraction varies with the position through the mask due to the variable energy rate transfer, which is maximal at the external side of the mask where the PCM liquid fraction drops from 100% to 0% during inspiration and was able to completely regenerate during expiration. On average, 70% of the PCM liquid fraction is solidified during inspiration and melted again during expiration, showing that the choice of PCM mass fraction of 20% is convenient. Because of the alteration between inspiration and expiration, continuous heat and mass transfer are possible through the mask.

Figure 5.

The variation in time of (a) the average regain of the outer nodes and (b) the PCM fraction of the outer nodes at different locations through the mask at PCM melting temperature of 22℃.

The enhanced performance of the incorporated face mask is mainly due to the phase change period of the PCM stabilizing the temperature of the outer nodes at the melting/crystallization temperature, which is 22℃ in the current study (Figure 6(a) and (b)). Without PCM the outer node temperature is not stabilized, which increases the rate of change of the temperature of the outer nodes and consequently the variation in their water vapor pressure decreases the driving factors for heat and mass exchanges. For instance, as shown in Figure 6(a) and (b), the drop of the temperature at the external side of the mask without PCM during inspiration is ΔT_o = 21.5℃ (from 33℃ to 11.5℃), while it is reduced for the PCM mask to ΔT_o = 12.5℃ (from 26℃ to 13.5℃).

Figure 6.

The variation in time of the temperature of the outer node for the face mask (a) without PCM and (b) with PCM at melting temperature of 22℃.

The heat and mass transfer are strongly coupled. In fact, the heat transfer is reduced by the desorption mechanism, which extracts heat from the outer node, lowering its temperature and its water vapor pressure. This decreases the heat and mass transfer driving factors and may reverse the warming effect during inhalation to a cooling one. Similar heat-transfer reduction happens during exhalation due to the adsorption process. This is what causes the drop of temperature of the outer nodes at the external side of the mask below 12 ℃ (Figure 6(a)) during the inspiration period which is an undesirable value as it enhances the risk of condensation. The incorporation of PCM played a positive role by increasing the temperature of the outer nodes through the mask at the beginning of the expiration period, reducing the undesired temperature drop resulting from the desorption process which might cause condensation. To illustrate the improved performance of the mask with and without PCM, the outer node temperatures at the beginning of the expiration period for both cases (see Figure 6(a) and (b)) revealed an increase of 2℃ (from 11.5℃ to 13.5℃) at the external side of the mask and an increase of 6℃ (from 16℃ to 22℃) at its internal side with the introduction of PCM. In addition, Figure 6(a) and (b) show that without PCM the maximum difference in temperature of the outer nodes between the external and internal mask sides is 4℃ but 10℃ with PCM, making the mouth side of the face mask significantly warmer than the ambient one, which is desired. In fact, the incorporation of PCM in the cotton cloth mask was not only capable of enhancing the heat- and mass-transfer processes, but also of decreasing the potential for condensation by lowering the maximum relative humidity through the fabric. As the expired air is nearly saturated condensation might occur. The main cause of condensation is the high decrease of the saturation water vapor pressure due to the relatively large fall of air temperature through the mask. Since the variation of the outer node temperature of the fibers plays a major role in the variation of the temperature of the expired air, setting that temperature at a certain value during the phase change of the PCM is an important design intervention. This is necessary in order to avoid very low temperatures of the outer node at the beginning of the expiration period and avoid the high drop of the air temperature through the mask.

Figure 7 shows the variation in time of (a) the air temperature and (b) the air water-vapor content for the cases of breathing without mask, with cloth mask, and with PCM mask. It is clear that in the studied case, the drop of the leaving air temperature at the beginning of the expiration period was 14℃ (from 33℃ to 19℃) without PCM, as shown in Figure 7(a), while it was 10℃ (from 33℃ to 23℃) with the PCM mask. Figure 8 compares, for cloth mask with and without PCM, the variation in time of the relative humidity at the proximity of the internal side of the mask at x = 0.25 mm, the critical position for which maximum relative humidity through the mask is reached at the beginning of the expiration period. The PCM mask decreased the maximum relative humidity through the mask from 99.5% to 93%.

Figure 7.

The variation in time of (a) the air temperature and (b) the air water vapor content for the cases of breathing without mask, with cloth mask, and with PCM mask at the PCM melting temperature of 22℃.

Figure 8.

The variation in time of the relative humidity at the proximity of the internal side of the mask (x = 0.25 mm).

Examination of the temperature and water vapor content of air leaving the mask during the respiration cycle revealed that wearing a mask creates new respiration profiles for the temperature and relative humidity of air compared to the normal step profile without a mask (Figure 7(a) and (b)). The comparison of modified respiration patterns caused by masks incorporated with or without PCM shows clearly the enhancement of the mask performance due to the incorporation of PCM. For both cases the air is warmed and humidified progressively while passing through the mask from the ambient to the mouth side by the fabric material during inhalation and cooled and dried when flowing in the opposite direction during exhalation. However, the introduction of PCM allows the mask to reclaim more heat and moisture from the expired air, establishing a higher difference in temperature and water vapor content between the airflows entering and leaving the mask for the inhaled and exhaled air.

In order to show the efficiency of the mask in retaining heat and moisture during inspiration, latent heat (Q_L) and sensible heat (Q_H) recovering were calculated for the different simulated cases according to the following definitions:

Q_{L} = \int_{0}^{t_{in}} \frac{ρ_{a} q_{in} (t) h_{fg} (Y_{ms} (t) - Y_{amb}) dt}{t_{\binom{in}{}}}

(10{\rm a})

Q_{H} = \int_{0}^{t_{in}} \frac{ρ_{a} q_{in} (t) c_{pa} (T_{ms} (t) - T_{amb}) dt}{t_{\binom{in}{}}}

(10{\rm b})

where t_in is the inspiration period of the breathing cycle, Y_amb, T_amb _, are, respectively, the water vapor content and temperature of the air imposed at the ambient side during the inspiration period, q_in(t), T_ms(t), and Y_ms(t) are, respectively, the instantaneous flow rate, temperature, and water vapor of the air leaving the mask during the inspiration period. The simulations of variable ambient conditions are summarized in Table 3.

Table 3.

Heat and moisture retention and recovery for different cold conditions for mask of area 10 cm² with and without PCM.

Ambient conditions	Performance parameter	Without PCM	With PCM
12℃, 30%	Maximum relative humidity (%)	99.5	93
	Average sensible heat recovery through the mask area, Q_H (W/m²)	1847	2683
	Average latent heat recovery through the mask area, Q_L (W/m²)	5225	6781
15℃, 50%	Maximum relative humidity (%)	96	92
	Average sensible heat recovery through the mask area, Q_H (W/m²)	1578	2086
	Average latent heat recovery through the mask area, Q_L (W/m²)	4534	5527

It was found from the simulations at different ambient conditions that the higher the differences between the ambient and exhaled conditions, the larger the heat and mass recovery by the face mask. In addition, a considerable increase of sensible and latent energy reclaiming is reached for the same ambient conditions by the introduction of PCM of common characteristics (melting heat of 200 kJ/kg, melting temperature of 22 ℃, and PCM mass fraction of 20%) as shown in Table 3.

In conclusion, the energy efficiency of PCM cloth mask is increased by colder and dryer conditions in dry winters. Furthermore, the incorporation of PCM allows inhalation of colder outdoor conditions at higher relative humidity without condensation, as illustrated in Table 4. It is expected that the occurrence of condensation would increase when ambient temperature conditions decrease. The model simulations have shown that for a given outdoor cold temperature, there is a limiting threshold outdoor relative humidity above which condensation occurs. The incorporation of PCM increases this limiting threshold relative humidity for the same outdoor temperature. For example, from the results summarized in Table 4, it is clear that temperatures of 5℃ and 10℃ resulted in condensation when a face cloth mask is used without PCM. However, using the same cloth material for the mask with PCM incorporated, condensation does not occur at the same temperatures of 5℃ and 10℃. This means that with proper use of PCM melting temperature in the mask, it can become functional at relative humidities of up to 30 and 40%, respectively. For higher temperatures as 12℃ and 15℃, the introduction of PCM enlarged the possible range of relative humidity reached without condensation. Hence, the role played by the incorporation of PCM in cloth mask in reducing the potential for condensation is viable in winters even if humidity is moderate.

Table 4.

Range of relative humidity reached without condensation for different ambient temperatures

Ambient temperature (℃)	5	10	12	15
Range of relative humidity (minimum; maximum) (%)	Without PCM	Condensation occurred, 100%	Condensation occurred, 100%	(0; 30)	(0; 50)
Range of relative humidity (minimum; maximum) (%)	With PCM	(0; 30)	(0; 40)	(0; 55)	(0; 70)

Conclusions

When exposed to severe cold conditions, humans tend to use cloth material or their hands as a protection from inhaling cold dry air in order to retain warmer and more humid air for inhalation. Ordinary cloth can serve as a heat and moisture exchanger between the inhaled and exhaled air and hence decrease the heat and water losses from the human body, which may be significant in severe cold conditions. The higher the difference in temperature between the inspired and expired conditions, the larger are the body energy savings by the face mask but also the potential for condensation which favors diffusion of viruses. Incorporating PCM in the fabric is found to interrupt the variation in the temperature of the mask due to the possible regeneration of the PCM by periodic breathing retarding the condensation process or eliminating it.

The developed mathematical model incorporated all the complex transport mechanisms of heat and moisture, including the incorporation of a realistic breathing pattern for the air passing through the fabric. The core assumption of the model of NLTE was validated by performing experiments on a common cotton cloth mask that showed good agreement between predicted and measured air temperatures through the PCM during inspiration and expiration inflow conditions. A parametric study using the developed model showed that the introduction of the PCM in the fabric material can prevent condensation by controlling the mask temperature such that it is above the airflow dew point. This temperature is strongly dependent on the melting point of the PCM and on air ambient temperature and relative humidity.

Footnotes

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Acknowledgments

The authors acknowledge the help of G. Salame, A. Frayha, J. Khashab, W. Shaar, and H. Sadek in designing and building the experimental breathing machine.

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