Statistical modeling of water vapor transport through woven fabrics

Water vapor transport plays a significant role in determining the thermal comfort of a fabric since it represents the ability to transfer perspiration from the body. It becomes extremely important in determining thermal comfort in hot environments in which sweat production and evaporation from the human body are the major cooling mechanisms for maintaining thermal comfort and avoiding heat stress. Free movement of moisture vapor to the fabric surface is essential in cold weather as well, since perspiration can cause fabric wetness, resulting in freezing in winter. ¹

Water vapor transport through textile structures is a complex phenomenon determined by various parameters. The mechanisms of water vapor transmission (WVT) through porous textiles involve diffusion of water vapor through voids (openings) and diffusion through individual fibers. ² Diffusion of water vapor through the void spaces is controlled by the water vapor pressure gradient across the two sides of the fabric. Pores in woven fabrics can be interfiber and interyarn; thus, the rate of diffusion is governed by the yarn/fabric structure, including the size and concentration of intra/interyarn pores and the fabric thickness. Transport of water vapor through the fiber can be either adsorption or absorption, depending upon the fiber morphology. Absorption involves the diffusion of water vapor through the polymer structure of the fiber by permeation or solution and its desorption at the outer surface. Thus, the ability of the fibers to undergo diffusion depends on the hydrophilic or hydrophobic nature of the fiber. Adsorption is a surface phenomenon with the adhesion of molecules of gas, liquid, or solids to a solid surface that can result in a liquid wicking through a fibrous assembly. There has been a great deal of research on the diverse aspects of moisture vapor transport through woven textiles, both experimentally and theoretically. ^{3 – 12} Fourt and Harris ³ studied the diffusion of water vapor through woven fabrics, and reported that the resistance of a woven fabric to the transport of moisture vapor depends on the type of fiber, fabric thickness, and the tightness of the weave. They also noted that water vapor transport occurs primarily through the voids in the material due to the readily available, low-resistance pathway; thus, the nature of fibers itself only contributes to vapor transmission when the volume occupied by the fibers is considerably larger. Rainard ⁴ showed experimentally and theoretically that the intrinsic permeability of a fabric was related to the amount of voids present in the fabric and the void geometry. Backer ⁵ reported that air/gas permeability of a fabric was dependent on the fabric thickness, pore size, and the number of pores in a given area. In a study of the geometry of fabric interstices as related to gas transport, Backer ⁶ observed that the different behavior of various textile structures possessing equal projected pore areas may be attributed to the nature of the interlacing of warp and filling yarns. He also suggested that where fabrics are woven with maximum tightness, it is likely that interfiber spacings within yarns also contribute to transport. However, if a reasonably circular yarn section is maintained in moderately tight fabrics, the relative flow- resistance of the interyarn pores should dominate the flow through the fabric.

Whelan et al. ⁷ developed an analytical and experimental model to predict WVT of textile fabrics based on vapor flow through the uniform holes of a perforated metal plate. The model showed that diffusion of water vapor through a perforated metal plate is a function of the thickness of the plate, the fractional pore area of the plate, and the diameter of the individual holes, which stresses the importance of fabric thickness and porosity in WVT through woven fabrics. This plate model, however, assumes that pore diameters are constant throughout the thickness of the material and that transmission through the solid areas of the plate, corresponding to the contribution of fabric yarns and fibers, is zero. Weiner ⁸ proposed two mathematical models to predict moisture vapor transmission in terms of fabric geometry. One model was based on a perforated plate theory in which moisture vapor diffuses partly through the pores and through the non-perforated areas of the fabric. In the other model, he considers the fabric as a relatively uniform mixture of fiber and air and calculates the resistance to diffusion as the sum of the components derived from fiber-space and air-space resistances. Woo et al. ⁹ presented an analytical model to describe water vapor diffusion through non-woven fabrics composed of non-hydrophilic fibers. They found that fiber volume fraction and shape coefficient are the most influential structural parameters affecting water vapor diffusivity through non-hydrophilic nonwoven fabrics.

Although theoretical models provide meaningful insight into the relationship between the various parameters and moisture vapor transport of porous textiles, assumptions underlying their calculations are often violated due to the complexity of a textile structure. As described by previous researchers, ⁶ the yarns of open-weave fabrics are often distorted both in and out of the plane of the fabric to achieve static equilibrium within the weave structure. In tightly woven fabrics, there is an increased tendency toward flattening of the yarns when they are bent over cross yarns, which violates the assumption in most theoretical models that textile yarns act as inextensible, circular cylinders. Furthermore, the pore structure in fibrous materials is complex in that multiple pores have different shapes and size distributions. Pore size may vary even along a single pore path, and the material may contain closed or blind pores that terminate inside the material, which do not contribute to fluid flow through. The numerous interrelated structural features of textiles make it hard to generalize the analytical aspects of the mechanism of flow-through fabrics.

From the aspect of providing information for practical situations, a statistical model would be useful as a supplement to theoretical modeling. Statistical analyses would further clarify the governing factors that impact the transmission of water vapor through woven textiles, reveal the relative significance of different variables, and improve the understanding of the complex behavior of moisture vapor transport through textile structures. Although previous studies elucidated influential factors in moisture movement through woven fabrics, much research has focused on parts of the elements contributing to moisture vapor transport, rather than covering the contribution of all the elements. To have a comprehensive understanding, a systematic approach based on the mechanisms of moisture movement through porous textiles is needed.

Our objectives in this study are to understand parameters that are critical in the moisture vapor transport through woven fabrics, to quantify the individual contributions of those parameters to water vapor transport, and to develop a statistical model to predict WVT through woven fabrics.

Materials and methods

Fifteen woven fabrics with various fabric thickness, weight, fabric construction, and staple fiber type were selected. Fiber types included cotton, lyocell, nylon, polyester, and polypropylene to represent different levels of hydrophilic nature of fibers. For cotton fabrics, which are hydrophilic, fabrics with a range of thickness, weight, and fabric count were chosen, so that cotton fabrics from thin, open structures to heavy, compact structures were included to assess the influence of structural characteristics on moisture vapor transport when intrinsic fiber properties are controlled. Fabrics had no finishing treatments. Fabric descriptions are given in Table 1.

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Table 1.

Fabric descriptions

Fabric code	Fiber	Fabric count, yarns/cm; W × F	Fabric weight, g/m2	Fabric thickness, mm	Construction
W1	Cotton	34 × 28	108 (1)	0.206 (0.005)	Plain
		(0.2) (0.2)
W2	Cotton	13 × 11	540 (4)	1.000 (0.022)	Plain
		(0) (0.2)
W3	Nylon 6,6	20 × 18	125 (0.9)	0.323 (0.011)	Plain
		(0) (0.2)
W4	Polyester	20 × 20	126 (2)	0.301 (0.006)	Plain
		(0.2) (0.2)
W5	Polyester	16 × 13	179 (2)	0.402 (0.006)	Plain
		(0.2) (0)
W6	Polypropylene	16 × 13	189 (3)	0.552 (0.007)	Plain
		(0.2) (0)
W7	Cotton	31 × 11	316 (2)	0.615 (0.008)	Rib
		(0.2) (0.2)
W8	Cotton	35 × 11	209 (2)	0.474 (0.008)	Rib
		(0.2) (0)
W9	Cotton	22 × 16	359 (3)	0.601 (0.010)	Plain
		(0) (0)
W10	Cotton	45 × 20	278 (2)	0.480 (0.008)	Twill
		(0.2) (0.2)
W11	Lyocell	36 × 27	148 (3)	0.254 (0.014)	Plain
		(0.2) (0.3)
W12	Lyocell	28 × 20	205 (1)	0.421 (0.008)	Plain
		(0.2) (0.2)
W13	Cotton	29 × 19	358 (2)	0.683 (0.008)	Twill
		(0) (0.3)
W14	Cotton	34 × 11	445 (8)	0.813 (0.036)	Rib
		(0.2) (0.2)
W15	Cotton	18 × 18	474 (2)	0.972 (0.008)	Twill
		(0) (0)

Standard deviations in parentheses.

Textile parameters

Fabric thickness

was measured according to ASTM D 1777-96. ¹³ Measurements were taken with a Frazier Compressometer using a 0.95 cm diameter presser foot at 6.9 kPa of pressure at 10 randomly selected areas.

Mass per unit area

Mass per unit area (weight) was measured according to ASTM D 3776-96. ¹⁴ Measurements were taken for four randomly selected samples.

Fabric count

Fabric count, that is, number of yarns per unit distance, was measured according to ASTM D 3775-03. ¹⁵ Measurements were taken at five randomly selected areas.

Yarn number

Yarn number was measured according to ASTM D 1059-01. ¹⁶ Measurements were taken from 10 warp samples and 10 filling samples for each fabric.

Yarn twist

Yarn twist was measured according to ASTM D 1423-99. ¹⁷ Measurements were taken from 25 warp samples and 25 filling samples for each fabric.

Yarn diameter

Yarn diameter was measured using a microscope equipped with a calibrated micrometer (Continuµm™, Thermo Nicolet Co., Madison, WI). Measurements were taken from 50 warp samples and 50 filling samples for each fabric.

Pore size distribution

Pore size distribution was measured by a Capillary Flow Porometer (Model CFP-1200-AEX, Porous Materials, Inc., Ithaca, NY). Gas flow rates through wet and dry samples versus differential pressure were measured, and pore diameters of through-pores at the most constricted part of the pore were determined. ¹⁸

Water vapor transmission

WVT rate was measured according to ASTM E 96-00 ¹⁹ using a dish assembly (Vapometer, Thwing-Albert Instrument Company, Philadelphia, PA) for three samples.

Statistical analyses

Statistical analyses were performed on textile measurements and water vapor transport data using the SAS® system (SAS Institute Inc., Cary, NC). Multivariate statistical methods, including multiple linear and multiple polynomial regressions, all-subsets regression, and stepwise regression were performed. Before the analyses, data were centered by subtracting the mean foreach predictor variable to reduce the degree of multicollinearity.

Results and discussion

Selection of textile parameters

Fabric weight and thickness were measured for each specimen. Fabrics ranged in mass per unit area from 108 to 540 g/m², and varied in thickness from 0.206 to1.000 mm. Weight and thickness are presented in Table 1.

Based on the mechanisms of water vapor transport through porous textile fabrics, water vapor may diffuse: (1) through the interyarn spaces; (2) through the interfiber spaces; (3) through the fiber substance itself. In order to have a comprehensive understanding of WVT through a textile fabric and to obtain an expression for the moisture vapor transport, the contribution of each of these parts needs to be taken into account.

Woven textiles have complex geometric configurations because of the fabric weave and yarn structure. In woven structures, pores can be interfiber and interyarn. The structure and dimension of intra- and interyarn pores strongly depend on hardness of twist in theyarn structure and compactness in the woven structure. ²⁰ Thus, textile structural parameters that could describe the relative magnitude of interyarn space and interfiber space should be included in the modeling. In order to be used in modeling, the concepts described in the mechanisms of moisture vapor movement through porous textiles should be expressed in appropriate variables that are quantifiable.

In this study, the fabric cover factor was selected as a possible predictor variable in the modeling to approximate the contribution of interyarn spaces. The cover factor is a measure of fabric tightness and describes the compactness of the weaving of a given yarn system, which could represent the relative magnitude of interyarn space of a given fabric. The cover factor was calculated from fabric counts and yarn diameters using the following equation:

(1)

where C represents cover factor of the fabric; e is the number of warp yarns over 1 cm of fabric width; p is the number of filling yarns over 1 cm of fabric width; d₁ is the diameter of the warp yarn (cm); d₂ is the diameter of the filling yarn (cm). Cover factors for the specimens, which ranged from 0.71 to 1.00, are shown in Table 2.

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Table 2.

Fabric parameters

Fabric code	Fabric cover factor	Yarn twist factor	Yarn packing factor	Solid volume fraction	Moisture regain, %21,22
W1	0.71	3.65	0.62	0.345	7.5
W2	0.96	5.30	0.41	0.355	7.5
W3	0.78	2.92	0.46	0.340	4.5
W4	0.87	3.08	0.30	0.303	0.4
W5	0.88	3.83	0.26	0.322	0.4
W6	0.91	2.53	0.37	0.369	0.1
W7	1.00	4.12	0.53	0.338	7.5
W8	0.90	3.61	0.55	0.290	7.5
W9	0.88	5.10	0.66	0.394	7.5
W10	1.00	3.86	0.64	0.381	7.5
W11	0.78	3.64	0.70	0.386	12
W12	0.86	3.67	0.50	0.322	12
W13	0.94	4.11	0.56	0.345	7.5
W14	1.00	4.37	0.55	0.360	7.5
W15	0.94	4.38	0.48	0.321	7.5

The relative magnitude of interfiber space could be quantified by textile parameters such as yarn twist factor or yarn packing factor. The twist factor is a measure of ‘twist hardness’ of yarn and describes compactness in yarns of the same size. The twist factor was calculated using the following equation:

(2)

where t_w represents the twist factor of the yarn; tpi is the twist in turns per inch; N_e is the yarn number in the cotton system. Average twist factors of warp and filling yarns for each specimen, which ranged from 2.53 to 5.30, are presented in Table 2.

The yarn packing factor, which describes the ratio of the total fiber area to actual yarn area in the cross-section of a multifilament yarn, was calculated as follows:

(3)

where ϕ represents the packing factor of the yarn; d is the diameter of the yarn (cm); ρ is the density of the fiber (g/cm³); N_t is the linear density of the yarn (tex); S is the constant, 10⁵. Densities for cellulose, nylon, polypropylene, and polyester were obtained from Morton and Hearle. ²¹ Average packing factors of warp and filling yarns for each specimen, which ranged from 0.26 to 0.70, are presented in Table 2.

The solid volume fraction of fabric, the ratio of solid parts to the total volume of the material, which has shown to be an influential structural parameter affecting water vapor diffusivity through non-hydrophilic nonwoven fabrics, ⁹ was calculated as follows:

(4)

where S represents the solid volume fraction of the fabric; m is the mass of the specimen at standard condition (g); ρ is the density of the fiber (g/cm³); a is the area of the specimen (cm²); t is the thickness of the specimen (cm). Solid volume fractions for the specimens, which ranged from 0.290 to 0.394, are shown in Table 2.

Since water vapor may diffuse not only through the void spaces but also through the fiber substance, the nature of fibers may contribute to moisture vapor transport. Moisture regain is a property that reflects the hydrophilic or hydrophobic nature of fibers, and such inherent fiber characteristics may affect moisture vapor transport through a textile fabric. Absorption of moisture can result in fiber swelling, particularly for hydrophilic fibers with high moisture regain. This moisture absorption is a dynamic factor until a steady state is reached, which is the condition at which WVT was measured. In this study, moisture regain of fiber was selected as a possible predictor variable to represent the hydrophilic or hydrophobic nature of fibers and possible changes in porosity due to fiber swelling. Moisture regains for cotton, lyocell, nylon, polyester, and polypropylene were obtained from literature^21,22 and are presented in Table 2.

The fabric weave and nature of the yarn interlacing influence the amount and size of voids and the pore geometry, and therefore vapor transport through fabric. To represent the interlacing pattern between warp and filling yarns we chose variables that could reflect the effect of fabric weave and that are quantifiable to be used in the modeling. The fabric cover factor was selected to describe fabric tightness and solid volume fraction to quantify the amount of voids.

Pore size and shape are also affected by fabric and yarn structure, as well as fiber size. Pore size was reported as an important parameter affecting WVT through woven fabrics.^5,7 Although woven textiles have complex geometric configurations with a range of pore diameters, a descriptor that could represent the pore dimension of woven fabric needs to be explored in order to be used as a predictor variable in modeling and to examine the contribution of pore size to moisture vapor movement through the material. In this study, capillary flow porometry was used to measure pore diameters of only through-pores at the most constricted part of the pore for each fabric. From the through-pore size distribution of each specimen, we computed two different types of pore size that might be used as a possible predictor variable in modeling: mean flow pore diameter and diameter at maximum pore size distribution. Mean flow pore diameters for the specimens ranged from 4.89 to 49.59 µm, and diameters at maximum pore size distribution ranged from 0.60 to 7.50 µm. Pore size data for each specimen are given in Table 3.

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Table 3.

Pore size measurements and water vapor transmission rate of woven fabrics

Fabric code	Pore size		Water vapor transmission rate, g/h/m2
	Mean flow pore diameter µm	Diameter at maximum pore size distribution, µm
W1	28.73	7.18	19.70
W2	14.07	7.21	18.48
W3	32.65	7.50	19.54
W4	36.31	7.34	20.31
W5	42.56	7.49	19.67
W6	49.59	6.34	19.24
W7	9.71	5.45	18.96
W8	13.04	7.06	19.08
W9	8.21	5.55	18.76
W10	9.35	6.29	18.87
W11	34.74	7.26	19.71
W12	18.58	5.70	19.72
W13	4.89	0.60	18.36
W14	6.52	1.36	18.64
W15	7.70	7.04	18.36

Based on the mechanisms of WVT through woven fabrics and previous studies, we selected seven factors for statistical analyses: fabric thickness, cover factor, solid volume fraction, yarn twist factor, yarn packing factor, pore diameter, and moisture regain of fiber. The selection process is summarized in Figure 1. OPEN IN VIEWER

Figure 1.

Influential factors involved in water vapor transmission through woven fabrics.

Parameters influencing water vapor transport

WVT rates for each specimen are shown in Table 3. Statistical analyses were performed to determine the characteristics for each parameter that influences the WVT. Table 4 shows the correlation coefficients between WVT and textile parameters. Fabric thickness has the highest correlation coefficient, followed by the mean flow pore diameter of fabric, fabric cover factor, yarn twist factor, diameter at maximum pore size distribution, yarn packing factor, moisture regain of fiber, and solid volume fraction of fabric. It shows that the fabric thickness, fabric cover factor, yarn twist factor, yarn packing factor, moisture regain of fiber, and solid volume fraction of fabric are negatively related to WVT, whereas pore diameters of fabric are positively related to moisture vapor movement through the material. In other words, WVT decreases with increased fabric thickness, fabric cover factor, yarn twist factor, yarn packing factor, moisture regain of fiber, and solid volume fraction, whereas WVT increases with increased pore diameter of fabric.

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Table 4.

Correlation coefficients between water vapor transmission and fabric parameters

Fabric parameters	Correlation coefficient
Fabric thickness	−0.86
Mean flow pore diameter of fabric	0.75
Fabric cover factor	−0.67
Yarn twist factor	−0.66
Diameter at maximum pore size distribution	0.52
Yarn packing factor	−0.29
Moisture regain of fiber	−0.28
Solid volume fraction of fabric	−0.28

To develop a statistical model predicting WVT through woven fabrics, regression analyses were performed using those textile parameters. Since some of the parameters, such as mean flow pore diameter and diameter at maximum pore size distribution, are highly interrelated, all-subsets regression and stepwise regression were performed first to screen the variables that have little influence in predicting the response variable when other predictor variables are present in the model. We found that diameter at maximum pore size distribution does not contribute much to the model when the mean flow pore diameter is present in the model. Wealso found that solid volume fraction does not contribute to the model when the fabric cover factor is present in the model. Thus, the two variables, diameter at maximum pore size distribution and solid volume fraction of fabric, which were not significant in the presence of other predictor variables, were eliminated in further modeling efforts.

We started with a multiple linear regression model, and the regression equation resulted in an adjusted R² value of 0.87 (Table 5, Model 1). Table 5 shows the coefficients for textiles parameters with adjusted R² values for models developed. Since partial regression leverage plots of the dependent variables versus independent variables showed quadratic relationships, a polynomial multiple regression model with linear and quadratic terms was refitted to the data. When the quadratic terms, as well as linear terms, were entered in the model, the adjusted R² value increased to 0.99 (Table 5, Model 2). For the polynomial model, the linear terms of fabric thickness and moisture regain were found to be significant factors among the six linear terms, and the quadratic terms of cover factor, mean flow pore diameter, and moisture regain were found to be significant factors among the six quadratic terms. The variance inflation factors were much higher than the critical value, 10, which indicates high multicollinearity among the independent variables. Since the four other linear terms and three other quadratic terms did not appear to be influential for prediction of WVT, those terms were dropped at the 5% significance level. The polynomial model with linear terms of thickness and moisture regain and quadratic terms of cover factor, mean flow pore diameter, and moisture regain resulted in an adjusted R² value of 0.97, keeping only significant terms at a 5% significance level (Table 5, Model 3). Since it is a rule of thumb to keep lower order terms of predictor variables even if they are non-significant in polynomial models, the linear terms of cover factor and mean flow pore diameter were kept in the model. The final model selected to predict WVT through woven fabrics was a polynomial model with linear terms of fabric thickness, cover factor, moisture regain of fiber, and mean flow pore diameter of fabric, and quadratic terms of cover factor, moisture regain of fiber, and mean flow pore diameter of fabric (Table 5, Model 4). Thus, a regression equation to predict WVT through woven fabrics is as follows:

(5)

where T is fabric thickness (mm); C is the fabric cover factor; M is the moisture regain of fiber (%); P is the mean flow pore diameter of fabric (µm). The adjusted R² value is 0.97, which indicates that the four textile parameters, fabric thickness, cover factor, moisture regain of fiber, and mean flow pore diameter of fabric, accounted for significant amounts of the differences observed in WVT of woven fabrics among the fabrics examined. Influence of the yarn twist and yarn packing factor was shown to be insignificant at the 5% significance level for these experimental conditions for models that also include fabric thickness and cover factor. This may indicate that, for woven fabrics, interyarn spacings may have greater influence on moisture vapor transport than interfiber spacings within yarns. Fabric thickness was one of the highly influential factors affecting water vapor transport of woven fabrics as a singular parameter with a negative relationship between fabric thickness and WVT. Moisture regain of fiber was shown to be another important factor in moisture vapor transport. Previous researchers ³ noted that significant amounts of moisture vapor travel through the fiber substance for hydrophilic fibers, andthis effect of fiber type was observed when the volume occupied by the fibers exceeded 30–40%. The majority of our specimens ranged from 0.30 to 0.40 in solid volume fraction; thus, wesuppose that the contribution of the intrinsic fiber properties was revealed in the model. There was no significant difference between Models 2, 3, and 4 in terms of R² at the 5% significance level. Thus, the final model selected is the one showing higher R² with the fewest number of significant variables.

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Table 5.

Summary of coefficients and R² values for regression models for water vapor transmission

	Regression coefficients a													Adjusted R2
	Intercept	T	C	tW	ϕ	M	P	T 2	C 2	tW 2	ϕ2	M 2	P 2
Model 1	0	−2.2	0.2	0.07	−2.5	0.05	0.008	-	-	-	-	-	-	0.87
		(0.5)	(1)	(0.1)	(0.8)	(0.03)	(0.009)
Model 2	0.6	2	−1	0.07	4	−0.19	0.03	−3	14	−0.20	−8	0.045	−0.0041	0.99
	(0.2)	(1)	(1)	(0.2)	(1)	(0.04)	(0.01)	(2)	(10)	(0.09)	(9)	(0.005)	(0.0007)
Model 3	−0.08	−1.0	-	-	-	−0.06	-	-	12	-	-	0.026	−0.0017	0.97
	(0.06)	(0.2)				(0.01)			(4)			(0.003)	(0.0002)
Model 4	−0.05	−1.0	−0.2	-	-	−0.06	0.002	-	11	-	-	0.025	−0.0017	0.97
	(0.1)	(0.2)	(0.7)			(0.02)	(0.008)		(6)			(0.005)	(0.0002)

a

T (fabric thickness); C (fabric cover factor); t_W (yarn twist factor); ϕ (yarn packing factor); M (moisture regain); P (Mean flow pore diameter).

Standard errors of coefficients in parentheses.