Modeling wicking in textiles using the dual porosity approach

Abstract

We develop a dual porosity diffusivity model to simulate the complex dynamic wicking behavior in textiles: wicking inside yarns coupled with wicking in the voids in between the yarns. The model expands the Richards equation to account for mass exchange between the two pore systems. This exchange, however, appears to be very small for cotton textiles and the system appears to behave as two parallel pore systems. The water uptake in the yarn pore system is mostly affected by the textile structure (woven versus knit), while the void pore system differs in the maximum moisture content that can be achieved during uptake. Gravity is shown to play an important role, especially for the coarser void pore system.

Keywords

dual porosity diffusivity model dynamic wicking behavior gravity effect

Wicking, that is, the spontaneous liquid imbibition due to capillary forces in textiles, is an important phenomenon to understand since wrong moisture management can lead to injuries such as steam burns for firefighters¹ and decubitus for immobile patients.^2–4 Wicking occurs in fabrics with wettable fibers and whose geometry allows the formation of a meniscus.⁵ A textile fabric is a multi-scale porous material: fibers themselves can be porous, forming the fiber pore system, for example cotton fibers, where the space between the fiber forming yarns is the second pore system, called the yarn pore system, and the space between yarns in a fabric structure forms the third pore system, called the void pore system. The fiber pore system depends only on fiber material and production. The yarn pore space is defined by the fiber geometry, number of fibers and yarn twisting level, while the void pore system is defined by the fabric structure. Some examples of cotton fabric structures are knit and woven. In this study we will focus on the water uptake in the yarn and void pore systems of cotton textiles.

To understand wicking, it is necessary first to measure this process accurately. Many experimental techniques have been applied to wicking, such as visible light,⁶ infrared light,⁷ electrical contacts⁸ and thermo-couples.⁹ All of these cannot quantify moisture transfer, that is, moisture content distribution, throughout the whole fabric, as they rely either on the sharp-front approximation or point measures along the textile. X-ray computed tomography (CT) techniques were applied to moisture quantification in wicking,^10,11 but water does not interact strongly with X-rays, leading to a reduced signal. Neutron radiography, which is sensitive to hydrogen molecules and therefore is able to quantify water,¹² has been applied to wicking in a previous study.¹³ The neutron radiography technique is a non-destructive imaging technique, which enables high-resolution moisture content identification. The experimental method is described in the Experimental setup and neutron radiography section but further details, including the image analysis procedure, are described elsewhere.¹³ The cotton samples showed wicking in distinctively two regions: in the yarn pore space and in the spaces between the yarns, the void pore space.

The wicking process in textiles is commonly modeled using capillary models, such as the models of Lucas¹⁴ and Washburn,¹⁵ where no gravity or inertial effects are considered. This model is based on the assumption of a moving sharp wetting front followed by a uniform moisture distribution in the wetted zone. Some work was done to address the gravity and inertial effects,^16,17 still giving no further information regarding the moisture distribution. Other enhancements of the model include the addition of the swelling effects of the fibers.¹⁸ Another approach of modeling wicking in textiles is based on pore networks.¹⁹ A pore network model, however, requires knowledge of the distribution of pore sizes and their connectivity, which requires high-resolution three-dimensional (3D) information on the porous structure of the material. Finally, it is possible to model wicking through a continuum approach to porous media based on the work of Darcy²⁰ and Richards.²¹ These approaches have been applied to fields such as gas diffusion layers in fuel cells²² and geothermal energy.²³ Dual pore systems have been investigated using the continuum approach for fractured²⁴ and highly heterogeneous²⁵ materials. The advantage of the continuum approach is that the moisture transport properties, such as diffusivity, can be more easily obtained from macroscopic measurements, such as transient moisture content profiles measured with non-destructive measurement techniques, for example, X-ray or neutron radiography experiments.²⁶

In this paper, we aim to model the wicking behavior in cotton by means of a dual porosity diffusivity model. We first describe the materials and the neutron radiography experiment. Then we build our diffusivity model from the Richards equation,²¹ adding the exchange terms between the two pore systems. Finally, we compare the simulated results to the measured data and interpret the obtained results.

Experimental materials and methods

Materials

We use two different cotton yarns (two different breeds: natural white and natural brown) both with the same twisting level (800 tpm). They are stapled yarns, that is, built from staple ribbon-like fibers of short (1–2 cm) length (see Figure 1(a)). The yarn cross-section is of the order of 200 µm, while the fiber cross-section is of the order of 20 µm. The contact angles are measured from digital photographs taken with the yarns perpendicularly dipped in a de-ionized water reservoir. The angles are measured using the tangent method on both sides of the yarn using the Fiji distribution of ImageJ.²⁷ At least three images of each yarn were captured and measured. See Figure 2 for a sample of the images used for contact angle measurement. The yarn characteristics are listed in Table 1.

Figure 1.

(a) Electron microscopy image of cotton yarn. (b) Simple jersey knit textile. (c) Plain woven textile.

Figure 2.

Contact angle measurement.

Table 1.

Yarn characteristics

Material	Yarn linear density (dtex = g/10 km)	Twisting levels (tpm^a)	Contact angle (°)
Cotton brown	197 (Ne^b 30)	800	38 ± 12
Cotton white	148 (Ne 40)	800	21 ± 7

tpm: number of twists per meter length of yarn.

Ne: English cotton number. Equivalent to number of hanks (770 m) of yarn per pound.

The yarns are either knitted in a circular knitting machine in the simple jersey pattern or woven in a manual loom mounted with a polyester yarn as the warp thread. Sample knit and woven structures can be seen in Figure 1(b) and (c). Before assembly in the sample holder, they are washed three times at 40 ℃ to remove any trace of fabrication residuals. The textiles are assembled in an aluminum frame that keeps the samples flat and perpendicular to the beam during the experiment. The textiles are first cut to 70 × 100 mm², which is the size of the sample holder. They are assembled in a sandwich between two sample holder frames (Figure 3(a)). The knit textiles are assembled such that the course direction is vertical, and the woven textile is assembled with the weft direction in the vertical position.

Figure 3.

(a) Samples assembled in the sample holder. (b) Example of a raw image of neutron radiography during wicking. (c) Setup schematic showing the sample, reservoir, reservoir electric lift, base structure and balance. The reservoir and its lift are supported by two pillars (on both sides of the lift) attached to the base, which rests on the balance casing (but not on the measuring plate). The sample is supported by two pillars that reach the measuring plate of the balance through two access holes in the base. (d) Photo of the setup inside the NEUTRA beamline, with boronated polyethylene blocks on its left to protect the electronics from neutron radiation, and with the scintillator on its right. The neutron beam crosses from left to right.

Experimental setup and neutron radiography

The setup (Figure 3(c) and (d)) consists of a balance, a support for the sample that rests on the measuring plate of the balance and another support that carries the reservoir, which is automatically elevated by an electric lift. The imaging procedure consists of first imaging the sample in the dry state, with the reservoir at its lowest height. Second, the reservoir is brought up to the sample by the electric lift (activated remotely). Wicking starts and the images are taken every 6 seconds. Once the water front reaches the top of the sample or does not perceptually move anymore, the reservoir is lowered and images are taken of the final wet sample. This last step is important to have images of the full sample without the reservoir in the field of view (as further explained below). The whole experiment is performed with the sample resting on the balance and its increasing mass digitally monitored. Although uncontrolled, for monitoring purposes, the air temperature and relative humidity (RH) are continuously logged in the experimental room and their values are 26.2 ± 0.4 ℃ and 48 ± 6% RH, respectively.

Our measurements were performed at the NEUtron Transmission RAdiography (NEUTRA) beamline at the Swiss Spallation Neutron Source (SINQ) of the Paul Scherrer Institute (PSI) in Villigen, Switzerland. The conical neutron beam is controlled by a collimator and the sample is situated at a distance from the collimator, yielding a final field of view of 107 × 133 mm². The beam traverses the sample and, as it meets the 100 µm thick zinc sulfide scintillator doped with ⁶Li as the neutron absorbing agent, it is transformed in visible light, which is photographed with a sCMOS (Andor Neo) detector system of 2160 × 2560 pixels. The images are binned 2 × 2, resulting in 1080 × 1280 pixels. The system yields a pixel size of 104 × 104 µm² at a 16-bit dynamic range. More details of the facility are available in Lehmann et al.²⁸ The beam is polychromatic within the thermal spectrum²⁹ and the treatment of information is spectral.

Experimental results

The raw intensity images are converted to yield an equivalent water mass at each pixel. First the images are converted from intensity to water thickness based on the Beer–Lambert law³⁰ using the Quantitative Neutron Imaging (QNI) software tool,³¹ which also corrects for the polychromatic nature of the beam, dark current, flat field and noise. Next, the neutron scattering effect of the reservoir is removed. Finally, the dry images are used as references and subtracted from the whole dataset to obtain moisture distributions. Details of the image processing for moisture quantification can be found in Parada et al.¹³ These processed images provide the moisture mass per pixel. Figure 4 shows the time evolution of moisture content within a cotton knit sample.

Figure 4.

Moisture content evolution for cotton brown knit sample during wicking over 275 seconds. The red bar at the bottom is the water reservoir. (Color online only.)

In wicking, as can be seen from early on (under 25 seconds), there is always a blue region that precedes the orange region. Close examination of the moisture content distribution reveals that only the yarn pore space is filled in the region with lower moisture content (blue regions, around 2 × 10⁻⁶ g/pixel), while in regions with higher moisture content (yellow and orange regions, above 3 × 10⁻⁶ g/pixel) both yarn and voids pore space are filled. The stagnation of the height of the region with higher moisture content indicates the effect of gravity. Indeed, the water front in the void pore space reaches its maximum height quickly due to gravity, while wicking in the yarn pore space continues until the top of the sample is reached.

From this pixel-level data, we can obtain moisture content profiles by averaging the moisture content of the central part of the sample (20 mm) along the width for each time step. The average thickness of the textile is around 0.76 mm and was used to determine the moisture content per volume (kg/m³). We remark that the exact thickness value is less important and is merely used as a scaling factor. Note, however, that for very thick samples the side effects can be non-negligible. Furthermore, the thickness of a textile is not well defined and, consequently, difficult to measure. For comparison's sake, we use the same thickness for all specimens. Figure 5 shows the measured moisture content profiles for different time steps alongside our simulated results.

Figure 5.

Comparison of measured and simulated moisture contents at different time steps. (Color online only.)

Steady-state model: film hanging between parallel cylinders

As supplementary analysis of the water uptake in cotton, we imaged the water configuration using a laboratory X-ray computer tomography scanner (CT) with an anode voltage of 50 kV.¹³ This analysis allowed us to approximate the uptake in the void pore space as capillary uptake between two parallel cylinders, where each cylinder represents a yarn of cotton fibers. This analytical model predicts the height of the water uptake in the void pore space, but requires that the radius of the meniscus of the film between the cylinders is known, which is difficult to obtain by, for example, microscope or camera measurement. This previous model did not predict the dynamic wicking process either in the yarns and in the void pore space, and we therefore propose in this paper a dual porosity continuum model in order to encompass better these two systems.

Model description

The water transport in a porous medium taking into account gravity can be described as²¹

\frac{\partial w}{\partial t} = \nabla [K \cdot \nabla (p_{c} + ρ gz)]

(1)

where w is the moisture content (kg/m³), K is the permeability (s),

p_{c}

is the capillary pressure (Pa),

ρ

is the liquid density (kg/m³), g is gravity acceleration (m/s²) and z is the height (m)

For one-dimensional (vertical) flow, we can rewrite the equation as

\begin{matrix} \frac{\partial w}{\partial t} = \frac{\partial}{\partial z} [K \cdot \frac{\partial}{\partial z} (p_{c} + ρ gz)] = \frac{\partial}{\partial z} [K \cdot \frac{\partial p_{c}}{\partial z} + K ρ g] \\ = \frac{\partial}{\partial z} (K \cdot \frac{\partial p_{c}}{\partial z}) + ρ g \frac{\partial K}{\partial z} \end{matrix}

(2)

Equation (2) requires knowledge of the capillary pressure curve w(p_c) and permeability K of the textile as a function of the capillary pressure. Since both are difficult to measure independently for textiles, we use a diffusivity approach combining both in a single moisture transport property, the diffusivity, which can be determined more easily and obtained from dynamic moisture content distributions measured in experiments.

Permeability K can be calculated into diffusivity D (m²/s) using the capillary pressure curve $w (p_{c})$ . Diffusivity is the gradient of concentration divided by the diffusion flux. Permeability is a measure of the rate at which moisture can spread under the gradient of capillary pressure, whereas diffusivity is a measure of the rate at which moisture can spread under the gradient of moisture content. The relation between diffusivity and permeability is

D = \frac{K}{C}

(3)

where C is the moisture capacity (s²/m²), defined as the slope of the capillary pressure curve at a particular value of moisture content w

C = \frac{\partial w (p_{c})}{\partial p_{c}}

(4)

Equation (2) can be rewritten as

\frac{\partial w}{\partial t} = \frac{\partial}{\partial z} (D \cdot \frac{\partial w}{\partial z}) + ρ g \frac{\partial K}{\partial w} \frac{\partial w}{\partial z}

(5)

We remark that this equation needs knowledge of the diffusivity and the derivative of the permeability to the moisture content.

A dual porosity model is used to describe water transport in the textile. It is assumed that the textile consists of two pore systems, a yarn pore system and a void pore system. We add a term to equation (5) to account for the mass exchange between the two pore systems

\frac{\partial w_{y}}{\partial t} = \frac{\partial}{\partial z} (D_{y} \cdot \frac{\partial w_{y}}{\partial z}) + ρ g \frac{\partial K_{y}}{\partial w_{y}} \frac{\partial w_{y}}{\partial z} + α (w_{v} - w_{y})

(6)

\frac{\partial w_{v}}{\partial t} = \frac{\partial}{\partial z} (D_{v} \cdot \frac{\partial w_{v}}{\partial z}) + ρ g \frac{\partial K_{v}}{\partial w_{v}} \frac{\partial w_{v}}{\partial z} + α (w_{y} - w_{v})

(7)

where the subscripts y and v represent the yarn and void pore systems, respectively;

α

is the exchange coefficient between the two systems (s⁻¹). The terms

\frac{\partial K_{y}}{\partial w_{y}}

and

\frac{\partial K_{v}}{\partial w_{v}}

are in general a function of moisture content, but in a first approach we will consider them as constants. The exchange coefficient α is also considered to be a constant.

The moisture diffusivity is parametrically described with the following equation²⁶

D = D_{cap} \cdot exp [F \cdot \frac{w - w_{cap}}{w_{cap}}]

(8)

where

D_{cap}

is the moisture diffusivity at capillary moisture content (m²s⁻¹);

w_{cap}

is the maximum moisture content (kg m⁻³) attained during the capillary uptake process. The parameter F describes the slope of the diffusivity curve in a semi-log plot.

The model is built and run in COMSOL 5.2a. The computational domain in two dimensions consists of a rectangle of 20 mm width and 80 mm height. The vertical domain is discretized into 100 uniform elements based on a mesh refinement study. All boundaries, except the bottom, are no flux boundaries. The bottom is a Dirichlet boundary with the moisture content set at capillary moisture content. Initially, all elements have zero moisture content. The time-dependent problem is solved by integration of the partial differential equations in time according to an implicit backwards differentiated formula (BDF) method. The COMSOL solver used is the direct solver MUMPS (MUltifrontal Massively Parallel Sparse direct Solver) with its default solver options. Numerical time steps are automatically selected by the COMSOL solver, depending on relative or absolute tolerance for the accuracy of the integration.

Comparison of experimental results and model

The moisture transport properties, that is, the diffusivity parameters, derivative of permeability to moisture content and exchange parameters of the two pore systems, are obtained by comparing simulated moisture content profiles and accumulated moisture uptakes at different time steps against measurements by the trial and error method (Figures 5 and 6). Trial and error calibration consists of adjusting model parameters manually and evaluating the agreement between simulation and measurement results. Many trial runs are needed to improve the agreement between simulation and measurement results. In general, the simulation results agree well with measurement results (Figures 5 and 6). For example, the root mean square error for measured and simulated accumulated moisture uptake is 5.79 × 10⁻⁶, 1.28 × 10⁻⁵, 5.78 × 10⁻⁶ kg for cotton brown knit, cotton white knit and cotton brown woven, respectively. The comparison between measurement and simulation shows that the dual porosity model is capable of simulating the wicking process in voids and yarn pore spaces.

Figure 6.

Comparison of measured and simulated accumulated moisture uptake.

Table 2 summarizes the obtained moisture transport properties. Note the large difference in the diffusivity parameters (especially

D_{cap}

and F) between the yarn and void pore systems. Figure 7 shows the obtained moisture diffusivity of the two pore systems for a single sample. Due to the very small value of parameter F for the void pore system, the diffusivity almost does not vary with moisture content. By comparison, the diffusivity for the yarn pore system increases with moisture content. The diffusivity for the yarn pore system is smaller than that of the void pore system when the moisture content is lower than 50 kg/m³. When the moisture content is above 50 kg/m³, the diffusivity for the yarn pore system becomes larger.

Table 2.

Moisture transport properties for the two pore systems

	Yarn pore system				Void pore system
Sample	$D_{cap}$	$w_{cap}$	$F$	$\frac{\partial K}{\partial w}$	$D_{cap}$	$w_{cap}$	$F$	$\frac{\partial K}{\partial w}$	$α$
Cotton brown, 800 tpm, woven	1.07 × 10⁻⁴	100	6.1	1.08 × 10⁻⁴	4.2 × 10⁻⁶	130	8.46 × 10⁻⁴	3.5 × 10⁻⁴	0
Cotton brown, 800 tpm, knit	1.02 × 10⁻⁴	280	6.1	2.5 × 10⁻⁴	4.2 × 10⁻⁶	500	8.46 × 10⁻⁴	3.5 × 10⁻⁴	0
Cotton white, 800 tpm, knit	1.02 × 10⁻⁴	280	6.1	2.5 × 10⁻⁴	4.2 × 10⁻⁶	300	8.46 × 10⁻⁴	3.5 × 10⁻⁴	0

Figure 7.

Moisture diffusivity of the void and yarn pore systems for the cotton brown knit sample.

Table 2 also shows that the difference between the two knits of different cotton yarns is not very large. The difference between knit and woven structures, however, is quite large for the capillary moisture content $w_{cap}$ in both yarn and void pore systems. The difference comes from the different textile structure (resulting in different loop sizes) and the presence of polyethylene terephthalate (PET) yarn in the woven fabric (which has a much lower moisture absorption than cotton). An exchange parameter α equal to zero is found to give the best agreement between measurement and simulation.

Discussion

This paper shows that the dual porosity model is able to simulate the wicking process in cotton. The physical understanding is that there are essentially two pore systems in parallel: one inside the yarns and another between the yarns (voids). Although the model uses simplifying assumptions, for example, that the terms $\frac{\partial K_{y}}{\partial w_{y}}$ and $\frac{\partial K_{v}}{\partial w_{v}}$ are constant, it manages to simulate the whole dynamic behavior of wicking adequately, matching the experimental results at all time steps. The exchange coefficient α in all our samples was shown to be 0, indicating that there is no mass exchange between the two pore systems.

The main differences found between the yarn and void pore systems is the order of magnitude of the parameters $D_{cap}$ and F, while the parameters $\frac{\partial K}{\partial w}$ are of the same order of magnitude. If we compare the terms in diffusivity and the parameter $\frac{\partial K}{\partial w}$ in equations (6) and (7), we can see that the void pore system is much more impacted by the term $\frac{\partial K}{\partial w}$ than the yarn pore system, showing that the effect of gravity is much more pronounced on the coarse void pore system than on the finer yarn pore system.

The influence of gravity on moisture transport is illustrated in Figures 8 and 9. When gravity is not considered, moisture is transported much faster and more moisture is accumulated in the sample. The simulation results for the case without gravity are very different from the measured ones, showing that gravity plays an important role in the wicking process in cotton textiles.

Figure 8.

Influence of gravity on moisture transport for the cotton brown knit sample. (Color lines online only.)

Figure 9.

Influence of gravity on moisture transport in the void pore system for the cotton brown knit sample.

Figure 10 compares the measured accumulated moisture uptake in the textile versus time with the simulated ones with and without gravity in a log–log plot. We clearly see that without gravity the textile would follow a square root of time behavior until reaching the top, as predicted by the Washburn law. However, in reality, due to the influence of gravity, the accumulated curve deviates from this behavior, as gravity plays a greater role with absorption height.

Figure 10.

Simulated accumulated moisture content versus time with and without gravity (lines). Measured accumulated moisture content versus time (crosses). (Color lines online only.)

For the void pore system, the only parameter that varies across the different samples is the capillary moisture content $w_{cap}$ , showing the difference in maximum volume of moisture stored in the pore system. For the yarn pore system, $w_{cap}$ , $D_{cap}$ and $\frac{\partial K_{y}}{\partial w_{y}}$ vary between the woven and knit structures. This may be an indication of the influence of the polyester yarn at the warp (horizontal) threads or the different yarn paths (tortuosity and contact to other yarns) inherent in the different fabric structures.

It is important to note that the diffusivity coefficient and the $\frac{\partial K}{\partial w}$ terms for each system are aggregated characteristics of the system. Such terms include all possible transport mechanisms, such as capillary uptake in pores as well as film flow. From our previous work,¹³ we know the moisture present in the fabric voids is more closely related to a film hanging between two cylinders than a classical flow in a porous medium. The film model, as developed in our previous paper, however, only gives us equilibrium insights (such as the maximum height at equilibrium) and not the dynamics of the wicking process. This study here shows that the diffusivity approach is capable of modeling the dynamics of wicking in fabrics even when there are two wicking regions: inside the yarns and between them.

Conclusion

In this study, a dual porosity diffusivity model is developed to simulate the complex dynamic wicking behavior in yarn and void pore systems in textiles. The proposed methodology relies on experimental data. The use of neutron imaging allows the complete documentation of the moisture content distribution versus time, and such information is mandatory for the model developed. A good agreement is obtained between measured and simulated moisture content profiles. The moisture exchange between yarn and pore void systems is very small for cotton textiles. Gravity plays an important role in wicking, especially for the coarser void pore system. Although neutron measurements are not mainstream, such measurements are becoming more accessible to the scientific community. Once typical textiles are characterized and the proposed dual modeling applied on a sufficient wide array of samples, the typical exchange coefficient or other model inputs could be identified for general use. Although the current dual porosity model could well predict the macroscopic dynamic wicking behavior, it does not provide any information about wicking behavior at the pore scale. Pore scale models are still needed to simulate accurately fluid flow in the yarn and void pore systems.

Footnotes

Acknowledgements

The authors would like to thank CWC Textil AG and LEGS for providing the yarns. We also thank Stefan Carl, from Empa, for support in developing the different experimental devices and Jan Hovind and Peter Vontobel, from PSI, for support during neutron radiography experiments. We thank Lukasz Pawlicki for the production of .

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by an ETH Grant (# 0-20909-13).

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