Textile Research Journal Article Dynamic Moisture Absorption Behavior of Polyester Cotton Fabric and Mathematical Model Abstract We investigate the dynamic moisture absorption behavior of polyester cotton fabrics of different warp and weft densities, survey the drybasis moisture contents of these various fabrics under variant temperature and relative humidity, and acquire a dynamic moisture absorption curve and balanced moisture content of these fabrics. The results show that the fabric moisture absorption velocity is in reverse relation with its warp and weft densities. By applying the Fick s Law for diffusion in porous media, a mathematical model for the fabric s dynamic moisture absorption is established, which is represented in the form of a non-linear partial differential equation boundary value problem; in order to simplify the calculation of the solution, a model parameter a is put forward, defined as the structural factor, relating to the fabric structure. After the model is solved under six-point format finite difference method, the distribution of the dry-basis moisture content in the diffusion direction at different times can be obtained. Based on the experimental data, the golden section method is utilized to estimate the model parameter a, and the linear relation between the parameter a and the product of warp and weft densities j is obtained by mathematical software Mathematica. The simulation value of dry-basis moisture content from the model matches the experimental observations. Yingchun Du 1 and Jin Li Beijing Institute of Fashion Technology, North Exit of Heping Street, Beijing 100029, China Key words fabric structure, moisture absorption behavior, dynamic, golden section method, mathematical model, warp and weft densities The hot wet comfort of clothing an important factor in clothing s wear comfort. At present, comfort studies mainly focus on heat moisture comfort, including research into the heat moisture conductivity of fabric, the formulation of reasonable and applicable assessment methods of clothing heat moisture comfort, and so on [1 4]. 1 The setup of a mathematical model describing the heat moisture transmission property of fabric is significant in the exploration of the heat moisture transfer mechanism in fabric made of porous organic materials and in the quantitative research into the hot wet comfort of clothing [5 9]. Henry 1 Corresponding author: e-mail: clydych @bift.edu.cn Textile Research Journal Vol 80(17): 1793 1802 DOI: 10.1177/0040517510365950 Figures 1 6, 8 appear in color online: http://trj.sagepub.com The Author(s), 2010. Reprints and permissions: http://www.sagepub.co.uk/journalspermissions.nav
TRJ 1794 Textile Research Journal 80(17) has brought forward a mathematical model describing the process of thermal wet transfer in fabrics; the model is analyzed by assuming that the moisture absorption quantity in fiber depends only on temperature and air humidity, and the equilibrium between fiber and neighboring air is instantaneous [10]. However, this assumption is too far removed from the actual fiber adsorption process, and so limits its applications. To improve the abovementioned model, Nordon and David have developed experimental expressions about the moisture content in fiber and surrounding relative humidity (RH), and presented the numerical solution of the equation by combining with several factors neglected by Henry [11], but these expressions lose sight of the adsorption dynamics theory of fabrics. The adsorption rate equation brought forward by Li and Holcombe considers the two-stage adsorption properties of wool fiber, and simulates the adsorption behavior of woven fabrics by combining with more practical boundary conditions [12]. Li and Luo have improved the mathematical simulation method of moisture absorption process in fiber [13], and simulate the two-stage adsorption process of woven fiber through one uniform diffusion equation and two sets of different diffusion coefficients. Liu et al. [14] have conducted a numerical simulation on the dynamic thermal wet transfer of human body, clothing, and environmental system. So far, the study is small, and the balanced moisture regain, used to evaluate the moisture absorption of fabric, indicates the moisture content once the fabric finally reaches the balance of moisture absorption, but cannot observe or express the process of fabric s dynamic moisture absorption. This paper deals with the moisture absorption behavior of polyester cotton fabrics of different warp and weft densities, determines the regular pattern of influence on the moisture absorption performance of fabric caused by the fabric density and humidity, sets up a mathematical model able to describe the dynamic process of fabric moisture absorption, and puts forward the structural factor a, the model parameter depicting the relationship between the fabric structure and moisture absorption performance. The simulation value of dry-basis moisture content from the model is consistent with that obtained experimentally. Experimental Details Experiment Material 2-plied polyester-cotton blended yarn, containing cotton 35% and polyester 65% and with fineness 13 te was used (Baoding Yimian Group). Equipment and Instrumentation The following equipment was used: SL7900 fully automated sampling weaving loom (made in Taiwan); Adventure TMAR2140 electronic analytical balance; drying oven; constant temperature and constant humidity room (20 C, RH 65%); SDH0501 adjustable temperature moisture test box (RH 20% 98%, and temperature 5 100 C). Fabrication of Sample Cloth The polyester cotton blended yarns were fabricated to the plain cloth in different warp and weft densities, under the tatting method. The samples have the warp density and weft density, respectively, in the unit of yarns/dm (as same in hereinafter), 200 (110, 140, 170, 200), and 280 (140, 180, 220, 260). The yarns were subject to heat setting treatment before fabrication, and after weaving the cloth was not subjected to any post-processing. Tests of Moisture Absorption Test 1 The electrical analytical balance was placed in the adjustable temperature moisture test box, temperatures and relative humidity were set to reach the moisture heat balance in the box. The sample cloth was baked for 2.5 h at 105 C, its dry mass was weighed; the cloth was then placed on the analytical balance, and readings were obtained every 5 min. Four fabrics were tested with warp density at 200 and weft density at 110, 140, 170, 200, respectively, within 20% 65% RH and temperature at 20 45 C, and the process of dynamic moisture absorption was observed. Test 2 To enable accurate observation of the dynamic moisture absorption of fabric, the test was conducted in a constanttemperature constant-humidity room. The analytical balance was placed in the room at 20 C and at 65% RH. The sample cloth was baked for 2.5 hours at 105 C, and its dry mass was weighed. The cloth was then placed on the analytical balance and readings were taken every 2 min. Four fabrics, with warp density at 280 and weft density at 140, 180, 220, 260, respectively, were tested for dynamic moisture absorption. Results and Discussion The influence of warp and weft densities on dynamic moisture absorption The changes in the dry-basis moisture content of fabrics at different warp and weft densities, under the conditions of RH at 65% and temperature ranging from 20 40 C, are shown in Figures 1 3. It is found that the content increases as the time prolongs. The general trend of each curve is
Dynamic Moisture Absorption Behavior of Polyester Cotton Fabric Y. Du and J. Li 1795 TRJ Figure 1 Moisture absorption curve (20 C, relative humidity 65%, warp density 200/dm). Figure 2 Moisture absorption curve (30 C, relative humidity 65%, warp density 200/dm). similar. However, the four kinds of fabrics do not have same moisture absorption velocity at the beginning. As the weft density rises, the velocity decreases gradually. After a certain period of time, the dynamic balance is reached when the balanced dry-basis moisture content is similar and even equal. In the initial phase, the moisture absorption velocities for the four fabrics are quite high. This is mainly because there are a number of free hydrophilic radicals in the cotton fiber of polyester cotton fabric, which combine water through the hydroxyl. The polyester, possessing no hydrophilic radicals, depends on the water adsorption on the surface of a few of unformed areas and crystal areas. The porous structure of fabrics causes an important influence on the moisture absorption performance, which will be discussed later. After a certain period of time, the rate of moisture absorption is diminished, and finally the balance is reached. The density of the fabric only determines additional hindering when the water molecules enter into the fabrics. Fabric of lesser density reaches balance status in a shorter time, while a fabric with greater density takes more time to reach the balance status. The final moisture regain, however, is equal.
TRJ 1796 Textile Research Journal 80(17) Figure 3 Moisture absorption curve (40 C, relative humidity 65%, warp density 200/dm). Influence of relative humidity on dynamic moisture absorption of fabric Figures 4 6 show the moisture absorption curves under different conditions of RH (25%, 65% and 95%), at temperatures of 25 C, 30 C, and 35 C. It is can be seen that, as the RH rises, the dry-basis content in same absorbing period increases obviously. When the RH reaches 95%, the curve of moisture absorption gives a velocity clearly larger than those at 20% and 65%. This is because cotton, in high humidity, swells by absorption of moisture, increases its bulk, and adsorbs more water molecules. In addition, in conditions of high humidity the water vapor has a larger partial pressure outside the fiber, so that the drive to absorb is increased. In high RH, the fiber s moisture absorption takes a longer time to reach balance status, because there are more water molecules adsorbed on the fiber surface when the vapor s partial pressure is larger. These water molecules are moving sideways, and they spread at the surface of fiber, moisturizing the fiber. When the moisturized surface continues to adsorb the water molecules, the molecules are condensed. Figure 4 Moisture absorption curve (25 C, warp and weft densities 200/ dm 170/dm).
Dynamic Moisture Absorption Behavior of Polyester Cotton Fabric Y. Du and J. Li 1797 TRJ Figure 5 Moisture absorption curve (30 C, warp and weft densities 200/ dm 170/dm). Figure 6 Moisture absorption curve (35 C, warp and weft densities 200/ dm 170/dm). Mathematical Model and Parameter Estimation of Fabric Moisture Absorption Set-up of Model The fabric contains multiple apertures or gaps. There are three kinds of apertures or gaps, as assessed by direction of water transmission [5]. First, there are cavities in the fiber (such as the middle cavities of cotton or hemp, the marrow of crude fiber, and intermediate cavities of hollow fiber) and the aperture of various fibrils (elementary fibril, microfibril, fibril, macrofibril). The former has large size (widthwise size is 0.05 0.6 10 µm), but a considerable portion is nonthrough gaps. The latter has smaller size (often, 1 100 nm), but is through gaps. Second, there are gaps between the fibers within the yarn, with sizes ranging between 0.2 and 200 µm, and mostly from 1 60 µm; and they are through gaps. Third, there are gaps between yarns in a fabric. The size is from 20 1000 µm, and generally through gaps. The moisture absorption process of fabric can be seen as water molecules physically diffusing in series within the gaps. The experiment adopts a single-layer plain fabric, with a thickness of 0.5 mm only. The gap between yarns is large. The above-mentioned third kind of gap gives very small
TRJ 1798 Textile Research Journal 80(17) Figure 7 Differential element sketch map of dynamic mathematical model. resistance, which can be ignored. The steam is easy to diffuse from the outside to the yarn surface of fabric, and this can be considered as the same as diffusion in air. The second gap described above has a diameter much larger than the molecular diameter of water, which diffuses, along the yarn surface and through the gaps, into the internal fiber surface. In the process, it is not likely that the vapor molecule collides with the aperture wall. Such diffusion abides by the Fick s Law [15]. As the resistance is very small, this too can be neglected. As the first kind of aperture is smaller in diameter, the resistance, caused by collisions with the wall of aperture (called Knudsen diffusion), becomes larger as the vapor diffuses from the fiber surface to the fiber s internal. This procedure is key to influencing moisture absorption at quick or slow velocity. Based on above analysis, this paper adopts the microelement method, applies the law of conservation of mass and Fick s Law, and establishes a mathematical model to describe the moisture transfer in the first kind of aperture. As Figure 7 shows, the fiber can be deemed as a column with infinite length and the moisture transfer, within the column, moves along the x direction. The moisture distribution of points on the column surface is symmetric alongside the axis. A thin-shell cylinder, thickness of which is dr, radius is r, length is l and x-away to the surface, is taken for the differential moisture calculation. Here, R is the radius of fiber, and x + r = R. According to Fick s Law and the mass balance equation, i.e. the transfer rate of the moisture inputting from the external surface of micro-cylinder minus that outputting from the micro-cylinder s internal surface equals that growing within the micro-cylinder, we can obtain following equation: Where ρ is the dry-basis density of fiber, C is the dry-basis moisture content within the differential cylinder, t is the time, and D is the diffusion coefficient of water in fiber. As the densities of warp and weft increase, the first kind of aperture may be squeezed so as to decrease the aperture and influence the moisture transmission velocity. A model parameter a is introduced to simplify the solution of equation 1.1. This relates to the fabric structure, is defined as fabric structural factor, and amends the diffusion coefficients of fabrics with different warp and weft densities, i.e. the water diffusion coefficient in fabric fiber is the product of a and D. Due to of polyester s very poor moisture absorption, the polyester cotton fabric utilized in this paper makes its absorption mainly by cotton fiber. Thus, the D is obtained from Liu et al. [14]. D = (0.85 + 50.6C 1100C) 10 14, D = 2.5{1 exp[ 3.5 exp( 45C)]} 10 14, t < 540 s Thus the initial condition of Equation 1-1 is that C t = 0 = 0 t < 540 s (1.2) As the distribution of dry-basis moisture content is symmetric about the column (fiber) s central axis and the introduction of fabric structural factor a, boundary conditions of differential equations can be simplified as follows: C x 0 C = = C 0, ------ = 0 x x = R (1.3) ad 2 C 1 -------- -- C ------ + a D ------- C ------ x 2 r x x x = C ------ t (1.1)
Dynamic Moisture Absorption Behavior of Polyester Cotton Fabric Y. Du and J. Li 1799 TRJ Figure 8 The dynamic distribution of moisture content within the fibers of fabric. Solution of Model With a finite difference method, the non-linear partial differential equation (PDE) boundary value problems in equations 1.1 1.3 are solved. The six-point format is used for discrete difference and the limit is acquired through L Hospital s Rule at the position of x = R; in this way, the discrete difference could be implemented smoothly [16]. A row of points are virtualized out of the central axis and the central difference quotient is used as of the discrete form of derivative in internal boundary conditions. As the equation contains the undetermined parameter a, the iteration method is adopted for solution. A value of D is selected within the proper range to obtain the iterative initial values of C on every layer, and then the Gauss-Seidel iteration is implemented to obtain the iterative results, i.e., the distribution of dry-basis moisture content in x direction in different moments. Figure 8 shows the distribution of moisture content within the fiber along the time under conditions including temperature of 20 C, RH of 65%, a = 1, and fabric with warp density of 280 yarns/dm and weft density of 180 yarns/dm. The positive and negative signs indicate symmetry alongside the center. Figure 9 shows the distribution of the moisture content along the radius of the fiber. Figure 10 gives the simulation value of the moisture content with time. Figure 9 Stimulation diagram of the moisture content at every point in fiber with radius. Figure 10 Stimulation diagram of the moisture content at every layer in the fiber with time.
TRJ 1800 Textile Research Journal 80(17) Parameter estimate and verification of model Estimation of model parameter a The golden section method is a direct method in onedimension optimization, and it is easy because it does not acquire the derivative of object function [17]. By applying the golden section method, basing on the observed data from Test 2, and relying on the Visual c++ environment, a program for parameter estimate of non-linear partially differential equation is prepared. Figure 11 shows the computer diagram for parameter estimation and the model s simulation calculation, where S 0 and S 1 are values of the objective function when a 0 and a 1, respectively, and can be expressed with the following: Where and X i are the calculated and experimental values of total dry-basis moisture content at a moment, respectively, and n is the number of experimental points. The acquired estimation a is shown in Table 1. Using mathematical software Mathematica, the relation between a and j, the product of warp and weft densities, is obtained, and can be expressed as follows: X i * Verification of Model n * S = ( X i ) 2 i = 1 For fabric of certain warp and weft densities, the model parameter a can be obtained from equation (3). Then C at X i a = 2.37925 0.223839j (2) (3) Figure 11 Diagram of parameter estimation and model solution algorithm.
Dynamic Moisture Absorption Behavior of Polyester Cotton Fabric Y. Du and J. Li 1801 TRJ Table 1 Estimated a values of fabrics of different warp and weft densities. j/(100/dm) 2 2.8 1.4 = 3.92 2.8 1.8 = 5.04 2.8 2.2 = 6.16 2.8 2.6 = 7.28 a 1.549 1.166 1.029 0.759 a certain radial position at moment of t can be determined by solving equation (1), and the simulation value of total dry-basis moisture content X at the certain time t is acquired by numerical integration under the Romberg method. Figure 13 gives the simulation value and the observed value at 20 C, RH 65%. We see that the simulation value matches the observed value, meaning that it is practical to consider only the diffusion resistance in the Figure 12 The relation between structural factor a and product of warp and weft densities. Figure 13 Comparison between total moisture absorption simulation value and observed value of fabrics of different warp and weft densities.
TRJ 1802 Textile Research Journal 80(17) fiber, and ignore that in the apertures and gaps between the fibers and the yarns. The rise of warp and weft densities can cause the extrusion of the aperture passage, and fibers deviate further from the assumed cylindrical shape, the influence of which can be compensated by the structural factor a. Conclusions The dry-basis moisture content of polyester cotton fabrics made with the same kind of yarn will increase as the duration of moisture absorption is prolonged, but the balanced moisture content is almost same. The moisture absorption velocity will be slower as the warp and weft densities of fabric increase. By applying Fick s Law for diffusion in porous media, we set up a model for a fabric s dynamic moisture absorption, which is represented in the form of a non-linear PDE boundary value problem. In order to simplify calculation of the solution, a model parameter a is put forward, named structural factor, and the linear relation between a and j the product of warp and weft densities is obtained under conditions including temperature of 20, RH of 65%, and fabric warp and weft density ranging from 280 140 to 280 260 yarns/dm. The simulation value of the moisture content matches the observed value made under the above conditions, meaning that it is practical to consider only the diffusion resistance in the fiber, and ignore that in the apertures and gaps between the fibers and the yarns. The rise of warp and weft densities can cause the extrusion of the aperture passage, and fibers deviate further from the assumed cylindrical shape, the influence of which can be compensated by the structural factor a. If, beyond the scope of the above experimental conditions, the situation should be more complex, the external diffusion resistance caused by gaps between fibers and yarn could be given by the ratio of the average gap thickness and the diffusion coefficient. References 1. Tao, H. M., and Sun, L., Analysis on clothing comfort. Foreign Silk (Chinese), 10, 28 29, (2004). 2. Chen, W. T., Fu, J. Q., Li, W. Z., et al., Moisture absorption and release performance of fabrics. J. Beijing Inst. Cloth. Technol. (China), 25(4), 48 55, (2005). 3. Inst. Text. Cloth., Hong Kong Polytechnic Univers., Hong Kong Cloth. Prod. Devel. & Market. Research Cent. Clothing comfort and product development (Chinese). Beijing: Textile Press, 2002. 4. Li, R. Q., Experiment and study on moisture absorption rate of textile materials. J. Text. (China), 6, 35 37, (1991). 5. Zheng, L. F., Study for the effect of fabric parameters and structure on the heat and moisture comfort, Master s thesis, Qingdao University, China, 2004. 6. Ogniewicz, Y., and Tien, C. L., Analysis of condensation in porous insulation. Int. J. Heat Mass Transfer, 24, 421 429, (1981). 7. Bouddour, A., Uriault, J. L., Mhamdi, A. M., Heat and mass transfer in wet porous media in presence of evaporation condensation. Int. J. Heat Mass Transfer, 41(15), 2263 2277, (1998). 8. Hsieh, W. H., Lu, S. F., Heat transfer analysis and thermal dispersion in thermally developing region of a sintered porous metal channel. Int. J. Heat Mass Transfer, 43, 3001 3011, (2000). 9. Zhu, Q. Y., and Li, Y., Mathematical model of the heat and moisture transfer in porous organic textiles. Chin. J. Computat. Mech. (China), 20(6), 641 648, (2003). 10. Henry, P. S. H., The diffusion of moisture and heat through textiles. Discuss. Faraday Soc., 3, 243 257, (1948). 11. Nordon, P., and David, H. G., Coupled diffusion of moisture and heat in hygroscopic textile materials. Int. J. Heat Mass Transfer, 10(2), 853 866, (1967). 12. Li, Y., and Holcombe, B. V., A two stage sorption model of the coupled diffusion of moisture and heat in wool fabric. Textile Res. J., 64(4), 211 217, (1992). 13. Li, Y., and Luo, Z. X., An improved mathematical simulation of the coupled diffusion of moisture and heat in wool fabric. Textile Res. J., 69(10), 760 768, (1999). 14. Liu, Y. X., Li, F. Z., Luo, Z. X., et al. Numerical simulation on the dynamic thermal-wet transfer of human body, clothing, and environmental system. J. Text. (China), 25(5), 24 27, (2004). 15. Wang, S. T., and Chen, T., Basis of transfer processes in chemical engineering (Chinese), Beijing: Chem. Ind. Press, 220 230, 1998. 16. Zhang, J. R., Numerical method in chemical engineering (Chinese), Bejing: Petrochem. Press, 1995. 17. Shi, W. P., Application of golden section method in solving unconstrained multiple variable optimization. J. Northeast Normal Univers. (China), 35(2), 11 16, (2003).