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Heat and Moisture Transfer with Sorption and Phase Change Through Clothing Assemblies Part II: Theoretical Modeling, Simulation, and Comparison with Experimental Results JINTU FAN1

AND

XIAO-YIN CHENG

Institute of Textiles and Clothing, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong ABSTRACT Part II of this series reports on a theoretical model and simulation results of coupled heat and moisture transfer with phase change and mobile condensates in clothing assemblies consisting of porous fibrous battings sandwiched by inner and outer layers of a thin covering fabric. The model considers moisture movement induced by partial water vapor pressure, a super saturation state in the condensing region, dynamic moisture absorption of fibrous materials, and the movement of liquid condensates. The theoretical results of the model are compared and agree well with the experimental ones. A numerical simulation using the model shows that inner fibrous battings with higher fiber contents, finer fibers, greater fiber emissivity, higher air permeability, a lower disperse coefficient of surface free water, and a lower moisture absorption rate cause less condensation and moisture absorption, which is beneficial to thermal comfort during and after exercising in cold weather conditions.

Understanding coupled heat and moisture transfer with phase change and mobile condensate is not only important to clothing comfort but also to building engineering and energy conservation. Since the 1980s, a number of theoretical models have been proposed in this field. Ogniewicz and Tien [11] proposed a model that assumed heat is transported by conduction and convection and the condensate is in a pendular state. The analysis was limited to a quasi-steady state, that is, temperature and vapor concentration remain unchanged with time before the condensates become mobile. Motakef and El-Masri [9] first considered the quasi-steady state corresponding to mobile condensate, where the condensates diffuse toward the wet zone’s boundaries as liquid and re-evaporate at these boundaries, leaving the timeinvariant temperature, vapor concentration, and liquid content profiles. This theoretical model was later extended by Shapiro and Motakef [12], who analyzed unsteady heat and moisture transport processes and compared the analytical results with experimental ones under some very limited circumstances. This analysis is only valid when the time scale for the motion of the dry-wet boundary in porous media is much larger than the ther1

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Textile Res. J. 75(3), 187–196 (2005)

mal diffusion time scale, which may, however, not be the case with frost and low moisture accumulation [13]. Farnworth [7] presented the first dynamic model of coupled heat and moisture transfer with sorption and condensation. This model was rather simplified and only appropriate for multi-layered clothing, because Farnworth assumed that the temperature and moisture content in each clothing layer were uniform. Vafai and Sarkar [15] rigorously modeled transient heat and moisture transfer with condensation. For the first time, they found the interface of the dry and wet zones directly from the solution of the transient governing equations. In that work, they numerically analyzed the effects of boundary conditions, the Peclet and Lewis number, on the condensation process. Later Vafai and Tien [16] extended the analysis to two-dimensional heat and mass transport accounting for phase changes, in a porous matrix. Tao et al. [13] first analyzed the frost effect in an insulation slab by applying Vafai and Sarker’s model to the case of temperatures below the triple point of water. Tao, Besant, and Rezkallah [14] also for the first time considered the hygroscopic effects of insulation materials in their model. Murata [10] first considered the falling of condensate under gravity and built this phenomenon into his steady-state model.

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We (Fan and co-workers [4, 5]) first introduced the dynamic moisture absorption process and radiative heat transfer as well as the movement of liquid condensates [5] in their transient models. We now improve our model further by considering moisture bulk flow induced by the vapor pressure gradients and super saturation state. In this paper, we describe the improved model, compare its results with the experimental ones, and analyze the effects of various material parameters on the accumulation of water in clothing as a result of condensation or moisture absorption based on computed numerical results.

u⫽ ⫺

,

(1)

where p is the water vapor pressure in the interfiber void, calculated by p ⫽ psat ⫻ Rhf. Based on the conservation of heat energy and applying the two-flux model of radiative heat transfer, at position x and time t, we obtain the heat transfer equation: C v 共 x,t兲

冉

⭸T ⭸T ⭸T ⭸ k共 x,t兲 ⫽ ⫺ ␧uC va 共 x,t兲 ⫹ ⭸t ⭸x ⭸x ⭸x ⫹

Model Formulation The model considers a clothing assembly consisting of a thick porous fibrous batting (⬃10 mm) sandwiched between one thin inner fabric (⬃0.1 mm) next to the skin and the other fabric layer (⬃0.1 mm) next to the cold environment. The schematic diagram is shown in Figure 1. Since the fibrous batting is highly porous and the temperature difference between the skin and the environment is great, radiative heat transfer within the fibrous batting is considered very important. In forming the mathematical model, we assume that the porous fibrous batting is isotropic in fiber arrangement and material properties; volume changes of the fibers due to changing moisture and water content are neglected; local thermal equilibrium exists among all phases, and as a consequence, only sublimation or ablimation is considered in the freezing region; and the moisture content at the fiber surface is in sorptive equilibrium with that of the surrounding air.

K x ⭸p ␮ ⭸x

⭸F L ⭸F R ⫺ ⫹ ␭ 共 x,t兲⌫共 x,t兲 ⭸x ⭸x

冊 ,

(2)

where ⭸F L ⫽ ␤ 共 x兲 F L ⫺ ␤ 共 x兲 ␴ T 4共 x,t兲 ⭸x

(3)

and ⭸F R ⫽ ⫺ ␤ 共 x兲 F R ⫹ ␤ 共 x兲 ␴ T 4共 x,t兲 ⭸x

,

(4)

where the effective thermal conductivity k(x,t) is a volumetric average calculated by k(x,t) ⫽ ␧ka ⫹ (1 ⫺ ␧)(kf ⫹ ␳Wkw), the effective volumetric heat capacity of the fibrous batting is calculated by Cv ⫽ ␧Cva ⫹ (1 ⫺ ␧)(Cvf ⫹ ␳WCvw), and the porosity of fiber plus condensates (liquid water or ice) is calculated by ␧ ⫽ ␧⬘ ⫺ (␳ ⫺ ␳ice)W(1 ⫺ ␧⬘). According to mass conservation, water vapor transfer in the interfiber void is controlled by the moisture transfer equation: ␧

⭸C a ⭸C a D a␧ ⭸ 2C a ⫽ ⫺ ␧u ⫹ ⫺ ⌫共 x,t兲 ⭸t ⭸x ␶ ⭸ x2

.

(5)

Even when there is no condensation on the surface of a fiber in the porous batting (i.e., the relative humidity is less than 100%), fibers absorb or desorb moisture, and the absorption or desorption rate is of the form ⌫ s 共 x,t兲 ⫽ ␳ 共1 ⫺ ␧兲

⭸C f 共 x,t兲 ⭸t

,

(6)

where Cf(x,t) is the moisture content within the fiber, which can be integrated by [4] FIGURE 1. Schematic diagram of the porous clothing ensemble.

C f 共 x,t兲 ⫽ 兵2/共 ␳ R f2兲其 In this work, we believe that moisture bulk flow is induced as a result of the gradient of partial water vapor pressure, as in the case of wood drying [1]. The speed of the movement of moist air is modeled by Darcy’s law:

冕

Rf

C f⬘rdr

,

(7)

0

where Rf is the radius of the fiber. Cf⬘ is the volumetric moisture concentration in the fiber, which can be determined by the Fickian diffusion law [4]:

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冉

⭸C f⬘ 1 ⭸ ⭸C f⬘ df ⫽ ⭸t r ⭸r ⭸r

冊

.

(8)

When the relative humidity reaches 100%, condensation or freezing occurs in addition to absorption. Many previous models [5, 9, 13, 14] assumed that extra moisture in the air condenses instantaneously, so that the maximum relative humidity in the air is 100%. This was considered less appropriate and the cause of some discrepancies between the numerical results of the previous models and experimental results. We now believe that there is a temporary super-saturation state (or Ca ⬎ Ca* or Rhf ⱖ 1.0). In other words, the moisture concentration in the air exceeds the saturated moisture concentration, so time is required for condensation to take place. Given sufficient time, however, the extra moisture in the air will condense until the moisture concentration in the air decreases to the saturated moisture concentration. On the other hand, when the humidity of the surrounding air is below 100%, evaporation or sublimation occurs if there is free water or ice on the fiber surface. Water condensation and evaporation are modeled using the Hertz-Knudsen equation [8]. The condensation or evaporation rate per unit surface area of fiber covered with condensates (liquid water or ice) is ⌫ sce 共 x,t兲 ⫽ ⫺ E 冑M/ 2 ␲ R共P sat / 冑T s ⫺ P v / 冑T v兲

.

(9)

From Equation 9, we get [5] ⌫ sce 共 x,t兲 ⫽ ⫺ E 冑M/ 2 ␲ R共1 ⫺ Rhf 兲 P sat / 冑T

.

(10)

Since the surface area of the fiber covered by condensates in the control volume is 2 冑共1 ⫺ ␧⬘兲共1 ⫺ ␧兲 Rf

,

the condensation or evaporation rate per unit volume is ⌫ ce 共 x,t兲 ⫽ ⫺

2E 冑共1 ⫺ ␧⬘兲共1 ⫺ ␧兲 Rf ⫻ 冑M/ 2 ␲ R共1 ⫺ Rhf 兲 P sat / 冑T

.

(11)

.

(12)

The free water, i.e., the water on the fiber surface, may diffuse when it is in liquid form and its content exceeds a critical value. According to mass conservation, we have

␳ 共1 ⫺ ␧兲

˜ ˜ ⭸ 2W ⭸W ⫹ ⌫ ce 共 x,t兲 ⫽ ␳ 共1 ⫺ ␧兲d l ⭸t ⭸ x2

k共0,t兲

⭸T ⭸x

k共L,t兲

冏

⫽ x⫽0

⭸T ⭸x

D a␧ ⭸Ca ␶ ⭸x

冏

D a␧ ⭸Ca ␶ ⭸x

冏

冏

1 共T兩 x⫽0 兲 ⫺ T 0 r0 ⫽

x⫽L

,

(14)

,

(15)

⫽

Ca兩 x⫽0 ⫺ Ca 0 w0

,

(16)

⫽

Ca 1 ⫺ Ca兩 x⫽L w 1 ⫹ 共1/h c 兲

.

(17)

x⫽0

x⫽L

T 1 ⫺ T兩 x⫽L r 1 ⫹ 共1/h t 兲

冊

Considering the radiative heat transfer at the interface of the inner thin fabric and the fibrous batting and that of the outer thin fabric and the fibrous batting, we have initial conditions for Equations 3 and 4 as follows: 共1 ⫺ ␨ 1 兲 F L 共0,t兲 ⫹ ␨ 1␴ T 4共0,t兲 ⫽ F R 共0,t兲

,

(18)

共1 ⫺ ␨ 2 兲 F R 共L,t兲 ⫹ ␨ 2␴ T 4共L,t兲 ⫽ F L 共L,t兲

.

(19)

These equations and boundary conditions are solved using the finite difference method.

Therefore, the total water accumulation rate ⌫(x,t) is ⌫ ⫽ ⌫ s ⫹ ⌫ ce

˜ ⫽ W(x,t) ⫺ Wf (x,t) is the free water content, where W Wf (x,t) ⫽ Cf (x,t)/␳ is the water absorbed within the fiber, 1 t and W共x,t兲 ⫽ 冕 ⌫共x,t兲dt is the total water content, ␳ 0 including that absorbed by the fibers and on the fiber surface. We define dl phenomenologically, and it depends on the water content, temperature, and properties of the fiber batting; dl ⫽ 0 when the condensate is immobile, which is the case when the water content is less than a critical value Wc, or when the free water is frozen. The boundary conditions to the main differential equations (2 and 5) are the same as those reported previously [4, 5]. Since the conductive heat transfer and moisture transport at the interfaces between the inner covering fabric and the batting as well as between the batting and the outer covering fabric should be continuous, we have

,

(13)

Theoretical Results and Comparison with Experimental Ones In the numerical computation, the initial conditions are 20°C and 65% RH when the fibrous battings are conditioned before testing. In addition to the standard parameters, which can be found in the handbooks, we used actually measured values of the parameters of fibrous battings and covering fabrics in the numerical computation except for the diffusion coefficient of moisture in the fiber Df and the diffusion coefficient of free water on the fiber surface Dl. We found that the water content in the

190 batting next to the skin is mostly determined by the diffusion coefficient of moisture in the fiber Df. The most appropriate diffusion coefficients of moisture in the fiber Df can therefore be determined from experimental data. Df ⫽ 1.512e-16 m2s for viscose batting and Df ⫽ 1.0e-16 m2/s for polyester batting. The diffusion coefficient of free water on the fiber surface Dl and the critical water content (Wc), where there is no liquid water diffusion, depends on the porosity and the surface tension of the fibers. Because no direct measurements of Dl and Wc were possible, we determined them by fitting the experimental data: Dl ⫽ 5.4e ⫺ 11 m2/s for viscose batting and Dl ⫽ 1.35e ⫺ 13 m2/s for polyester batting, with Wc assumed to be 20%. Figures 2–5 compare the numerical and experimental results of water content distribution in fibrous battings. Here, we see generally good agreement between the numerical results of the model and the experimentally measured water content distributions, except for the outer regions of the battings. The model predicts that the greatest condensation takes place at the outermost layer of the battings; however, in a few cases (one out of a total of eight cases) the highest water content appeared at the second or third outermost layer of the battings. One possible explanation is that some ice stuck to the outer covering fabric, as we observed during the experiments.

FIGURE 2. Comparison of water content distribution between simulation and experiment (fifteen plies of viscose batting sandwiched by two layers of a nylon fabric).

TEXTILE RESEARCH JOURNAL

FIGURE 3. Comparison of water content distribution between simulation and experiment (fifteen plies of viscose batting sandwiched by two layers of a laminated fabric).

FIGURE 4. Comparison of water content distribution between simulation and experiment (six plies of polyester batting sandwiched by two layers of a nylon fabric).

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191 As we see from Figure 6, most of the changes in temperature distribution take place within 1/2 hour of exposure to the cold environment. After the clothing assembly is placed on the sweating, guarded hot plate, the inner region is quickly heated up by the heat from the warm skin, and the outer region quickly cools down due to heat loss to the environment. After stabilization, the temperature reduces from the inner region to the outer region. This reduction is gradual in the inner region, but becomes steeper toward the outer region. This is a result of the combined heat transfer of convection, conduction, and radiation as well as the influence of the heat of moisture absorption and/or condensation. The trends of temperature distribution are in good agreement with the experimental findings reported in Part I of this series [3].

FIGURE 5. Comparison of water content distribution between simulation and experiment (six plies of polyester batting sandwiched by two layers of a laminated fabric).

It may also be possible that some of the condensates at the outermost layer might drop to the layers underneath. With regard to polyester battings, we can see that there is a relatively high discrepancy between the theoretical and experimental results for the water content distribution after 8 hours, but not for the water content distribution after 24 hours. This might be because the polyester batting was non-absorbent and highly porous, the water condensates were on the fiber surface, and before they were frozen into ice within the first few hours of the experiments, the liquid water tended to drop to the lower layers, resulting in less sharp increases in water content in the outer layers and higher water content in the lower layers.

FIGURE 6. Distribution of temperature.

From Figure 7, we see that the gradient of moisture concentration in the batting is small, especially in the

Mechanisms of Condensation in Clothing Assemblies From the generally good agreement between the theoretical and experimental results, we believe that our model provides a good mapping of the mechanisms of heat and moisture transfer with absorption and phase change within clothing assemblies. In order to gain a better understanding of the interactions of the different mechanisms involved, we have plotted the distribution of temperature, moisture concentration, and relative humidity as well as the accumulation of water content within the battings in Figures 6 –9.

FIGURE 7. Distribution of moisture concentration.

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inner region, so moisture transfer by diffusion is rather small. Much of the moisture transfer is caused by moisture bulk flow induced by the gradient of partial water vapor pressure. Because the temperature at the outer regions is low, the saturated moisture concentration is low, and the moisture concentration in the outer regions creates a temporary super saturation state (i.e., relative humidity exceeds 100%), causing condensation. The distribution of relative humidity can be seen in Figure 8.

optimizing clothing design for thermal comfort. In each simulation, we have changed one parameter, but the rest are constant as listed in Tables I and II in Part I of our series [3]. EFFECT OF RADIATIVE SORPTION CONSTANT FIBERS 〉

OF THE

The effect of the radiative sorption constant of the fibers ␤ on the water content distribution is shown in Figure 10. The water content reduces with the increase of ␤, but the reduction at the outermost region is less than that in the inner and middle regions.

FIGURE 8. Distribution of relative humidity.

The water content in the batting, including moisture absorption by the fibers and condensation, is cumulative, as shown in Figure 9. The condensed water in liquid form may wick to inner regions where the water content is lower. FIGURE 10. Effect of ␤ (beta) on water content distribution.

From the view of thermal comfort of clothing for wearers, sweating occurs when the human body cannot release its heat through dry heat loss alone. During sweating, less accumulated water in clothing is desirable, and hence a greater value of ␤ is preferred. Since ␤ is related to the fractional fiber volume, fiber emissivity, and fiber radius [7],

␤⫽ FIGURE 9. Accumulation of water content within battings.

Effects of Various Material Parameters Investigating the effects of various material parameters through numerical simulation is important in view of

共1 ⫺ ␧兲 ␨ f Rf

,

(20)

where Rf is the fiber radius, ␧ is the porosity of the batting, and ␨f is the emissivity of the fiber. Based on this relationship, higher fiber content, finer fibers, and greater emissivity of the fibers are preferred for less condensation within the fibrous batting.

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EFFECT OF DIFFUSION COEFFICIENT FIBER df

OF

MOISTURE

IN THE

The effect of the diffusion coefficient of moisture in the fiber df on the water content distribution is shown in Figure 11. As we can see, the water content increases with increased df , particularly in the inner regions of the batting where water accumulation is mostly due to moisture absorption. With a higher diffusion coefficient (df), the average water content increases much quicker in the first hour, then gradually as in the case of a low diffusion coefficient. These results indicate that hygroscopic fibers with a high df value may be disadvantageous for thermal comfort in cold conditions, since when exercising and sweating, there is a larger and quicker accumulation of water in clothing. The accumulated water content may be a source of “after-chill” discomfort when the wearer stops exercising.

FIGURE 12. Effect of dl on water content distribution.

bution of water content is almost even. A small dl with less liquid movement is preferred, as water accumulating at the outer region should create less discomfort. EFFECT

FIGURE 11. Effect of df on water content distribution.

EFFECT OF DISPERSE COEFFICIENT THE FIBROUS BATTING dl

OF

FREE WATER

IN

The effect of the disperse coefficient of free water in the fibrous batting dl on the water content distribution is shown in Figure 12. When dl ⫽ 0.0, there is no movement of liquid water on the fiber surface, the curve of water content distribution is concave, and the peak appears at the outermost side of the batting. With the increase in dl, when the amount of liquid condensate exceeds a certain value, the liquid water overcomes the surface tension and moves to the region with a lower water content. When dl ⫽ 5.4 ⫻ 10⫺8m2s⫺1, the distri-

OF

AIR PERMEABILITY

OF

FIBROUS BATTING

The effect of the air permeability of fibrous batting (i.e., Kx coefficient of Darcy’s Law kx ⫽ ) on the water content ␮ distribution is shown in Figure 13, assuming the porosity of the batting is not changed. When kx⫽0.0 (i.e., there is no flow of moist air within the fibrous batting, which may be achieved by interlacing the layers of battings with airimpermeable fabrics), the distribution of water content is convex. With the increase in kx, the shape of the distribution gradually changes to concave. It is clear that a higher kx causes a greater difference between the water content at the inner region and that at the outer region of the batting. The reduction in air permeability kx, probably by interlacing the layers of the battings with less permeable fabrics, can create a more even distribution of water content. EFFECT OF MOISTURE VAPOR RESISTANCE INNER COVERING

OF

Figure 14 shows the effect of the water content distribution within the fibrous batting for the inner covering fabric with a moisture vapor resistance ranging from 64.7 to 2087.0 sm⫺1, when the outer covering fabric is the woven nylon with the properties specified in Table I of

194

TEXTILE RESEARCH JOURNAL the moisture concentration within the batting is very close to the moisture concentration next to the skin because of the thick batting and the outer covering fabric. EFFECT OF MOISTURE VAPOR RESISTANCE OUTER COVERING

OF

Water content distribution within the batting is significantly affected by the moisture vapor resistance of the outer covering fabric, as shown in Figure 15. With an outer covering fabric of lower moisture vapor resistance, less condensation will take place within the batting, particularly at its outer regions. This is because more moisture will be transmitted by convection or diffusion into the environment through a more permeable outer covering fabric. The vapor permeability of the outer fabric is critical and should be minimal while also meeting the requirements of protection.

FIGURE 13. Effect of kx on water content distribution.

FIGURE 15. Effect of moisture vapor resistance of outer covering fabric on water content distribution. FIGURE 14. Effect of moisture vapor resistance of inner covering fabric.

Conclusions Part I. As we can see, the effect of the moisture vapor resistance of the inner covering fabric is very small within the commercial range (from a permeable woven fabric to a breathable one). The condensation within the battings can only be reduced when the inner covering fabric is almost impermeable. This is probably because

In this paper, we have presented a theoretical model of coupled heat and moisture transfer within clothing assemblies with moisture absorption, phase change, and mobile condensates. We compare the results of this model with the experimental ones and find good agreement. Based on our theoretical analysis, we can

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better understand the mechanism of condensation within clothing. We report numerical simulation results on the effects of various clothing parameters. Based on this analysis, we believe that for clothing consisting of fibrous battings sandwiched by two layers of thin fabrics, inner fibrous battings with a higher fiber content, finer fibers, greater fiber emissivity, higher air permeability, a lower disperse coefficient of surface free water, and a lower moisture absorption rate cause less condensation and moisture absorption, which is beneficial to thermal comfort during and after excising in cold weather conditions. With regard to the inner and outer covering fabrics, the inner one is probably less critical, but the outer one should be as permeable as possible in order to minimize water condensation within the battings. ACKNOWLEDGEMENT We would like to thank the Research Grant Committee of the Hong Kong University Grant Council for funding this project (PolyU 5142/00E).

Appendix NOMENCLATURE

Ca Cai

Ca* Cf Cf⬘ Cv Cva Cvf Cvw Da df dl

water vapor concentration in the interfiber void space, kg m⫺3 moisture concentration at the boundaries, K (i.e., i ⫽ 0, surface next to human body; i ⫽ 1, surrounding air) saturated water vapor concentration in the interfiber void space, kg m⫺3 mean water vapor concentration in the fiber, kg m⫺3 volumetric moisture concentration in the fiber (it varies over the radius of the fibers), kg m⫺3 effective volumetric heat capacity of the fibrous batting, kJ m⫺3 K⫺1 volumetric heat capacity of the dry air, kJ m⫺3 K⫺1 effective volumetric heat capacity of the fiber, kJ m⫺3 K⫺1 volumetric heat capacity of water, kJ m⫺3 K⫺1 diffusion coefficient of water vapor in the air, m2s⫺1 diffusion coefficient of moisture in the fiber, m2 s⫺1 disperse coefficient of free water in the fibrous batting, m2 s⫺1

E FL FR hc ht k ka kf kw Kx kx L Li M p psat pv R Rf r ri

RHi

Rhf T Ti

Ts Tv t u ui

W

condensation or evaporation coefficient, dimensionless total incident thermal radiation traveling to the left, W total incident thermal radiation traveling to the right, W convective mass transfer coefficient, m s⫺1 convective thermal transfer coefficient, W m⫺2 K⫺1 effective thermal conductivity of the fibrous batting, W m⫺1 K⫺1 thermal conductivity of air, W m⫺1 K⫺1 thermal conductivity of fiber, W m⫺1 K⫺1 thermal conductivity of water in the fibrous batting, W m⫺1 K⫺1 permeability of porous batting, m2 coefficient of Darcy’s law, m2 (Pa.s)⫺1 thickness of the fibrous batting, m thickness of the inner and outer covering fabrics, m (i ⫽ 0, inner fabric; i ⫽ 1, outer fabric) molecular weight of the evaporating substance, M ⫽ 18.0152, (g.mol⫺1, for water pressure of water vapor in the interfiber void, Pa saturated water vapor pressure at temperature Ts, Pa vapor pressure in vapor region at Tv, Pa the universal gas constant, R ⫽ 8.314471, J K⫺1 mol⫺1 radius of fibers, m radial distance, m resistance to heat transfer of inner or outer covering fabric, Km2 W⫺1 (i.e. i ⫽ 0, inner fabric; i ⫽ 1, outer fabric) relative humidity of the surroundings, % (i.e., i ⫽ 0, surface next to human body; i ⫽ 1, surrounding air) relative humidity of the air space within the porous batting, % temperature, K temperature of the boundaries, K (i.e., i ⫽ 0, surface next to human body; i ⫽ 1, surrounding air) temperature at the interface of condensates and vapor, K temperature in the vapor region, K time, seconds speed of moisture vapor within fibrous batting, m s⫺1 speed of moisture vapor through the covering fabric (i.e., i ⫽ 0, through inner covering fabric; i ⫽ 1, through outer covering fabric) water content of the fibrous batting, %

196 Wc Wf WCi wi x x⬘

␤ ␧ ␧⬘

␭ ␮ ␳ ␳ice ␴ ␶

␨f ␨i ⌫ ⌫ce ⌫sce ⌫s

TEXTILE RESEARCH JOURNAL critical level of water content above which liquid water becomes mobile, % water content of the fibers in the porous batting, % water content of the ith layer of the batting, % resistance to water vapor (i.e., i ⫽ 0, inner fabric; i ⫽ 1, outer fabric), s m⫺1 distance from the inner covering fabric, m dimensionless distance from the inner covering fabric, x⬘ ⫽ x/L radiative sorption constant of the fibers, m⫺1 porosity of fiber plus condensates (liquid water or ices) porosity of the fibrous batting, ␧ ⫽ cubic volume of interfiber space/total cubic volume of batting space latent heat of (de)sorption of fibers or condensation of water vapor, kJ kg⫺1 dynamic viscosity of dry water vapor, kg m⫺1 s⫺1 density of the fibers, kg m⫺3 density of ice, kg m⫺3 Boltzmann constant ␴ ⫽ 5.6705 ⫻ 10⫺8, WK⫺4 m⫺2 effective tortuosity of the fibrous batting; the degree of bending or twist of the passage of moisture diffusion due to the bending or twist of fibers in the fibrous insulation, which normally changes between 1.0 and 1.2, depending on the fiber arrangements fiber emissivity surface emissivity of the inner and outer covering fabrics (i ⫽ 1, inner fabric; i ⫽ 2, outer fabric) total rate of (de)sorption, condensation, freezing, and/or evaporation, kg s⫺1 m⫺3 rate of condensation, freezing, and/or evaporation, kg s⫺1 m⫺3 condensation or evaporation rate per unit surface area of fiber covered with condensates, kg s⫺1 m⫺2 rate of (de)sorption, kg s⫺1 m⫺3

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