Steam impinging and heat and water spreading in fabrics

Abstract

This study is about the heat and mass transport phenomena in a system with steam jet flow to eliminate/alleviate cloth wrinkles. We first adopted a theoretical approach to derive the mean capillary radii so that a fabric can be characterized as an assembly of capillary tubes with varying diameters. We then analyzed the processes as a heat transfer via the fibers and water via the pores in fabrics of different anisotropies. During water movement, the water weight actually intensifies the inherent anisotropy of the fabric in the water flow pattern. For heat transfer, the water weight becomes irrelevant and both convection and radiation are shown to be too trivial to include. Corresponding experiments are also conducted, using infrared and visible light cameras to record the heat and water flow processes, respectively. The results are compared with the theoretical predictions and the discrepancies are explored and explained.

Keywords

fabrics steam impinging capillary flow heat and water transfer anisotropy

Our overall research objective concerns new ironing devices with a steam jet flow to eliminate/alleviate cloth wrinkles. Ironing is a process for fabrics that uses high temperature and moisture (steam) and has been a dominant method in cloth refurbishment. The problems with the conventional household ironing tool are evident, such as tediousness in assembling/disassembling the system, clumsiness in operation, and lacking safety protections for both the cloth and the user from scorching, and together have prompted some improved designs. More recently, vertical mini cloth steamers have been developed for quickly removing garment wrinkles without the pressure from a heavy iron, thus termed as vertical pressure-free ironing.^1–3 The new steamers are easier to use and hence beneficial to today’s fast pace in daily life. They relax the fabrics, rather than flatten them as in normal ironing, and hence are gentler on clothing and more applicable to delicate fabrics such as silk. Figure 1 is a sketch of a vertical mini cloth steamer where steam is produced from the generator, flowing through the pipe, and projected toward the suspended cloth. Once it touches the cloth, the steam condenses into liquid water with the same high temperature. Both heat and water will then spread outward through the cloth. Cameras were placed in the back of the fabric surface to record the temperature (via an infrared camera) and liquid (a normal camera) changes and distributions, as illustrated in Figure 1.

Figure 1.

The test platform and physical process.

A logical question about pressure-free ironing is what factors influence the ironing effect and to what extent. A thorough literature review suggests that there is not much work carried out on fabric response during the new pressure-free ironing. In fact, there is barely any public information available on the effects of heat, moisture, ironing time, and cooling wind for fabric vertical steam ironing. Our group has initiated research in this area since 2015 and mainly published our experimental results.^4–6 This paper represents an attempt at a theoretical treatment of the key issues involved.

Apparently, this new cloth ironing method still involves a complex physical process, and our analysis focuses on steps such as the steam jet impingement on fibrous materials, steam condensing into water, water and heat transfer through a porous medium, and the changes caused in the physical and surface properties of the fabric during the process. Although, at the end of the ironing process, the three-phase fabric system (solid fibers, steam vapor, and liquid water droplets) is in a thermodynamic equilibrium, theoretical analysis of the entire process is still lacking.

From a material science perspective, a fabric is a multiphase, porous and flexible system of fiber (mostly polymeric), air, and moisture, balanced with the ambient conditions, chiefly temperature and relative humidity. Consequently, any fabric property is the collective contribution of all three components; yet in different extent because of the distinctive properties each phase contributes to the system behavior.^7,8 For instance, fibers impart structural integrity and strength; air and moisture contribute to the system flexibility and deformation, thermal and moisture transfer, and antistatic properties, and air also reduces the system weight. Water or water steam, vapor at the boiling point of water,^9,10 have properties highly susceptible to the ambient conditions, especially the temperature. Steam acts as a source of both heat and moisture in the new ironing process. Research on fabric mechanics, both theoretical^11–13 and experimental,^14,15 is still too sporadic for use in such a modeling effort.

One issue in dealing with fabrics has to do with the multi-scale nature of the material. The fabric is formed by yarns that, in turn, are made of polymeric fibers; likewise, a fiber consists of individual polymer molecules. The constituents at each structure level have their own properties.^7,8 For instance, the molecules, fibers, yarns, and the fabric all have densities that, in general, possess different values – a distinctive characteristic of multiphasic porous materials. In practice, however, it is often easy to blur the difference between, say, fiber porosity and fabric porosity.

In a slender form like a tree, a single fiber is clearly not isotropic. To simplify the analysis, however, a fiber is usually treated as isotropic so that all its properties are direction-independent. There are cases, however, where such simplification can lead to gross errors. For instance, almost all the properties along the fiber length and across it are significantly different. In the case of heat conduction, research has demonstrated¹⁶ that the axial thermal conductivity is nearly 10 times greater than the lateral value. Similarly, one important peculiarity about textile fabric is its anisotropy, an indispensable attribute for clothing purposes. Although remotely related to the fiber anisotropy, fabric anisotropy is a reflection of the irregularity of fiber density distribution along different directions, caused by^7,17 two variables: the fiber amount and fiber orientation. Most research about textile anisotropy is on the mechanics such as tensile behavior, wrinkle recovery etc.; much less on other aspects. Also, textile fabric is a very thin material; thus allowing, in many cases, the analyses to be limited to the two dimensional range, thus significantly simplifying the questions.

As for cloth wrinkling, material crease or buckling is usually a destructive phenomenon and an indication of structural failure in many engineering applications. However, fabric wrinkling allows multi-curvature bending, clearly a distinctive attribute in fabrics and critical for its formability, as summarized by Hearle et al.¹² and Postle et al.¹³, to fit on a human body elegantly. For fabrics, wrinkling is a macroscopic reflection of fiber deformation. In a deformed fabric, once the load is released, the bent fibers will bounce back to their original state and the wrinkles disappear. However, over time and under repeated actions, some plasticity-associated wrinkles will stay even in the absence of external stress, thus altering the appearance of the cloth. In ironing, by applying a temperature exceeding the glass transition temperature of the polymeric molecules, the deformed fibers acquired the energy to relax and return to their original positions and the wrinkles recover. The new state is fixed once the temperature has cooled down. Other factors, including moisture and pressure, also facilitate the process.^18–24

To facilitate our analysis, we summarize our major assumptions, which may or may not be stated explicitly during the related analysis, as below:

A local equilibrium between the fluid (vapor and liquid) and solid;

As the steam tube is very close to the fabric, the steam impinged to the fabric keeps the same temperature of 100℃ as in the tube;

The wet steam is taken as an ideal gas;

The radiation is neglected due to the porous medium temperature not being sufficiently high.²⁵ We also validated this in our calculations below.

Statistical variations of all material properties and system structure are neglected.

The properties of the fibers relevant to this study are assembled in Table 1.^26–28 The data are collected from various sources after careful examination. Table 1 takes the properties and values most relevant to the present work. Among them, the specific heat capacity of wood fibers was not needed in this study, and is hence marked as N/A. The properties of water at 25℃ are listed in Table 2, collected from several sources^10,29 as well.

Table 1.

Fiber properties

Fiber type	Cotton²⁷	Polyester²⁷	Wood fiber^26,28
Fiber density (g/cm³)	1.52	1.39	0.6
Fiber radius (m) 10⁻⁶	6.65	20	5 ∼ 10
Specific heat capacity (J/g·K)	1.25	1.41	N/A
Fiber axial thermal conductivity (W/m·K)	0.04	0.10	0.3

Table 2.

Water properties at 25℃

Surface tension (mN/m)	γ	71.97
Thermal conductivity (W/m·K)	k	0.608
Contact angle against the capillary tube wall	θ	41.5
Viscosity (Pa·s)	μ	0.89 × 10⁻³

Two fabric samples, a white cotton fabric of plain weave and a polyester nonwoven fabric, were used in this study, as specified in Table 3, and the properties of the cotton and polyester were measured for the samples used in the research. However, the wetting process in the nonwoven fabric recorded by camera images doesn’t show well so we added another paper sample in Table 3 that has a random fiber orientation just like the nonwoven fabric but generates much better photos as a cross reference. The average porosities of these three kinds of samples are calculated based on the fiber density and total volume and weight. Furthermore, fabrics are, in general, anisotropic, with properties that are direction dependent. For woven fabrics, differences in warp and weft will lead to fabric structural anisotropy, so we need to specify the fiber volume fractions in both directions; whereas for random orientation nonwoven and paper we can use the mean fiber volume fraction.

Table 3.

Fabric properties

	Woven fabric	Nonwoven	Paper
Fiber type	Cotton	Polyester	Wood fiber
Fabric thickness (cm)	0.04	0.10	0.03
Fabric width (cm)	35.00	33.00	21.00
Fabric length (cm)	38.00	35.00	21.00
Fabric weight (g)	17.66	19.97	2.80
Fabric volume (cm³)	46.20	115.5	13.23
Fabric density (g/cm³)	0.38	0.17	0.21
Fabric thermal conductivity k (W/m·K)	0.16¹⁵	0.04³⁰	N/A
Fabric specific heat capacity C_p (J/g·K)	1.21¹⁵	1.34³¹	N/A
Fabric thermal diffusivity α (J/g·K) 10⁻⁷	3.48	1.32	N/A
Fabric count (cm⁻¹, warp/weft)	60.00/30.00	N/A	N/A
Fabric fiber volume fraction (warp/weft)	0.30/0.15	0.12	0.52
Fabric mean porosity (warp/weft)	0.70/0.85	0.88	0.48
Mean capillary radius (warp/weft) 10⁻⁵ m	5.15/7.25	0.17	4.62

Sample structure characterization

Fiber properties versus the fabric properties

As stated before, a fabric is a composite of fiber, air and moisture and its properties are the collective contributions of all three components. However, in practice, air and moisture are frequently neglected (often rightly so), and a fabric is considered a mere collection of dry fibers. That is why in many research papers the difference between the properties of a fiber and a fabric made of this fiber is blurry, thus the two share the same properties. In some cases, the discrepancy can be small. For instance, the weight of a fabric is very close to the weight of the total fibers. Yet, in other cases, e.g. in heat transfer, the thermal conductivity of a fabric is nearly independent of the thermal conductivity of its constituent fibers.¹⁶ That is why, using the same type of fiber, we can produce summer cooler T-shirts or a winter long coat with drastically diverse thermal performance. A similar result is readily observable by comparing the fiber properties in Table 1 and those of the fabrics in Table 3.

For this study, we carefully examined the existing literature and collected the thermal properties for fibers: fiber specific heat capacity (J/g·K) for cotton = 1.25,³¹ polyester = 1.41;³¹ fiber axial thermal conductivity (W/m·K) for cotton = 0.04,¹⁶ polyester = 0.10;¹⁶ and for fabrics: fabric specific heat capacity (J/g·K) for cotton woven = 1.21,¹⁵ polyester nonwoven = 1.34;³¹ fabric transverse thermal conductivity (W/m·K) for cotton woven = 0.16,¹⁵ polyester nonwoven = 0.04.³⁰ Other values were measured and/or calculated in our lab, as described when they first appear.

There are still more complex cases, as in all surficial and interfacial problems. Since more than one type of material constituent is involved, for instance in liquid spreading in fibrous media, only a parameter describing the properties of all those involved is meaningful – e.g. there have been suggestions that a compound term $g_{LV} cos θ$ could be derived – a system “specific wettability” – that could be used to characterize the interactions at the fiber – air-liquid interface.^32–36

Seeking the connection between the behaviors of the constituents and the global system is the eternal goal of the major objectives in science and, in particular, in statistical physics. The simplest solution for such problems is to use the often-called Role of Mixtures,⁷ i.e.

P_{s} = P_{1} \cdot V_{f 1} + P_{2} \cdot V_{f 2} + \dots P_{n} \cdot V_{fn} = \sum_{i = 1}^{n} P_{i} \cdot V_{fi}

(1)

That is, the system property P_s is the average of the corresponding values P_i of all n constituents, weighed by its associated volume fraction V_fi. This relationship, of course, only holds in cases where the coupling interactions between the constituents are negligible, i.e. in determining the system weight. For other properties, e.g. the thermal ones discussed here, using this equation can only yield miserable results. In other words, we need the term that relates P_i with P_j to reflect the interactions between the two constituents.

Our analysis below therefore can be rephrased as an attempt to analyze the heat and liquid flow behaviors given the fiber thermal and wetting properties and the system structures.

The porosity and fiber volume fraction in fabric

As the essential parameter for a fibrous medium, the average fiber volume fraction V_f of the fabrics is calculated based on the fiber density, fabric volume, weight and weight fraction W_f as

V_{f} = \frac{ρ_{t}}{ρ_{f}} W_{f} = \frac{{\bar{W}}_{t} W_{f}}{\bar{V_{t}} ρ_{f}}

(2)

where

\bar{W}

\bar{V}

and ρ are the weight, volume and density, respectively. The subscript f represents the fiber and t is for the fabric, and when the air weight is negligible, there is

W_{f} = \frac{\bar{W_{t}}}{\bar{W_{f}}} \approx 1

. The corresponding fabric porosity follows as

φ = 1 - V_{f}

(3)

Since more than one type of material constituent is involved, for instance in liquid spreading in fibrous media, only a parameter describing the properties of all those involved is meaningful – e.g. there have been suggestions that a compound term $g_{LV} cos$ could be derived – a system “specific wettability” – that could be used to characterize the interactions at the fiber – air-liquid interface. To reflect the structural anisotropy of the woven sample, we calculated the fabric structure parameters in both the warp and weft directions listed in Table 3. Noting that the fabric count in the warp direction is 60 yarns/cm and 30 yarns/com in the weft, since all the other parameters remain the same, the fiber volume fraction in the warp direction is twice the value of that in weft.

To describe the anisotropy in fabrics, we can define a porosity disparity indicator (PDI) ɛ to reflect the fabric anisotropy in porosity

ɛ = \frac{φ_{1}}{φ_{2}}

(4)

where

φ_{1}

and

φ_{2}

are the porosities in the warp and weft directions, calculated respectively based on equation (3). In fabric, different porosity directly impacts heat and liquid flow in fabric when other parameters are fixed according to the analysis below. For our woven specimen in Table 3,

φ_{1} = 0.70

while

φ_{1} = 0.85

, thus

ɛ = 0.82

, indicating an anisotropic structure, in comparison with the nonwoven where

φ_{1} = φ_{2}

so that

ɛ = 1

The radii distribution of capillary pores in fabric

Fabrics can be considered as made of solid fibers and the capillary pores in between the fibers. Clearly water can only transfer via the capillaries and heat via the fibers. The size (radius and tortuosity) of the capillary pores varies over a wide range. In this study, we assume a constant radius for each capillary tube, and the effect of tortuosity can be ignored. As our major concern is the system behavior at steady state, an effective average radius is sufficient for our analysis. For a porous fibrous structure with totally random fiber orientation distribution, Pan^37,38 derived the average effective radius $\bar{r} = r_{c}$ of all the pores as

r_{c} = \frac{π r_{f}}{\sqrt{V_{f}}} e^{\frac{V_{f}}{2 π^{2}}}

(5a)

\frac{r_{c}}{r_{f}} = \frac{π}{\sqrt{V_{f}}} e^{\frac{V_{f}}{2 π^{2}}}

(5b)

From equation (5b), the right-hand side is always greater than one, meaning the average capillary radius cannot be smaller than the fiber radius, and the only variable that can alter the radii is the fiber volume faction V_f. Plotting in relative terms, $r_{c} / r_{f} \to V_{f}$ in Figure 2, when the fiber amount (V_f) increases, r_c reduces but with r_f as the asymptote, for in the extreme case $V_{f} \to 1$ , there is still $r_{c} / r_{f} \to 1.65$ . That is, the capillary radius is always greater than the fiber radius. In deriving equation (4), the only assumption used is random fiber orientation distribution, so the conclusions drawn here are valid to any fibrous media with random fiber orientation distribution; one cannot generate pores with diameters smaller than the constituent fibers. So, for nonwoven fabrics where fibers are randomly distributed, by using the effective mean porous radius alone $\bar{r} = r_{c}$ , the whole structure is transformed into a macroscopically isotropic one.

Figure 2.

The mean capillary radius as a function of fiber volume fraction.

For other fibrous sheets with non-random fiber orientation distribution like woven fabrics, an equation like equation (5) is not available, for there has been no fiber orientation density function derived for them. However, from equation (5), we can still calculate the mean porous radius for any given local regions so long as the corresponding fiber volume fraction is known. For example, in the present woven fabric case, we know from Table 3 that the fiber amount in the warp direction is twice the value in the weft direction. For the same piece of fabric where the effect of fiber orientation is nulled, we can find the fiber volume fractions in both the warp and weft directions $V_{f 1}$ and $V_{f 2}$ , respectively. The corresponding mean capillary radii $r_{c 1}$ and $r_{c 2}$ can then be obtained, as shown in Table 3. Bringing this direction dependence of the mean capillary radii ( $r_{c 1}$ , warp; $r_{c 2}$ , weft) into calculations, various behavior anisotropies can be predicted.

It is often said³⁹ that a porous medium can be viewed as an assembly of capillary tubes. Knowing the capillary radius distribution is then clearly a premise for any theoretical analysis of the system.

Theoretical analysis of water spreading in fabrics

Assume the steam remains the same temperature at the tube outlet and when contacting the fabric, because of the very close distance between the two. The steam condition: T_g = 100℃, RH_g = 100%, and the ambient condition: T_a = 30℃, RH_a = 65%. As explained in Figure 1, the entire process, from steam jet impinging the fabric target, steam condensing into water, to water spreading outwards over the fabric, is a rather complex theoretical problem. With certain assumptions and simplifications, the physics of the process is analyzed below in the sequence of occurrence.

Heat produced from steam to liquid water under ambient condition

The source of the energy and mass is the steam gas and the total heat energy involved consists of two parts: the heat q₁ when the steam condensed into liquid water, and the hot water cooled down to the room temperature q₂. Let

Δ \overset{\cdot}{m}

be the mass rate of steam, the heat involved

q_{1} = H_{v} Δ \overset{\cdot}{m}

(6a)

and

q_{2} = S_{e} (T_{g} - T_{a}) Δ \overset{\cdot}{m}

(6b)

where H_v = 2256.47 J/g is the enthalpy of moisture condensation and S_e = 1.307 J/(g·K) is the specific entropy of water at the temperature difference T_g − T_a,¹⁰ and T_g = 373 K, T_a = 303 K are temperatures of the initial moisture and the room, respectively. Table 4 shows the calculated results for both fabrics at different steam flow rates and effective water flow speeds. q₂ is the heat consumed during water and heat transfer in the fabrics when the hot water cooled down to room temperature, while q₁ is the heat dissipated when the hot steam condensed into water of the same temperature. According to the data in Table 4, this set up is not energy efficient at all since the dominant amount of energy q₁ is dissipated during condensation.

Table 4.

Energies dissipated in different steam flow rate

	Flow rate (g/s)	Δw (g)	q₁ (J)	q₂ (J)
Woven fabric	0.5	0.92	2066.93	91.69
	1	1.67	3770.56	167.27
	1.5	2.64	5948.05	263.86
Nonwoven	0.5	0.49	1114.70	49.45
	1	1.32	2974.03	131.93
	1.5	1.75	3937.54	174.67

Two kinds of transfers occur once the hot steam condenses into water of the same temperature in the sample, i.e. water flowing outward via the capillary pores in the sample and heat transferring along the fibers from T_g down to T_a, and both are analyzed in the following sections.

The initial spreading speeds

Condensed water in the center of the fabric will spread outward with speed under the action of several pressures. The dynamic pressure p_d is formed by the so-called effective steam,⁴⁰ i.e. the steam condensed in the fabric.

p_{d} = \frac{1}{2} ρ u^{2}

(7)

where u is the initial speed of condensed water flowing outward in the fabric, ρ is the density of the water. For a given water flow mass rate

Δ \overset{\cdot}{m}

Δ \overset{\cdot}{m} = ρ uA

(8)

and

A = δ π D

(9)

Where A is the wall area of the steam cylinder, δ is the thickness of fabric, D is the diameter of the tube shown Figure 1. Combining the equations yields the initial water spreading speed

u = \frac{Δ \overset{\cdot}{m}}{ρ δ π D}

(10)

Δ \overset{\cdot}{m}

is given in the experiments. After plugging in the water density ρ = 0.997 (g/cm³), the diameter of the tube D = 5 mm, fabric thickness δ = 0.40 × 10⁻³ mm, the values of u at different

Δ \overset{\cdot}{m}

levels are calculated in Table 5, along with the measured total flow time. From Table 5, the effect of u is clear, hence the dynamic pressure p_d is negligible.

Table 5.

The inertial flow speed and its effect on total flow time

$Δ \overset{\cdot}{m}$ (s) 10⁻³	u (m/s) 10⁻³	Time (s)
15.33	2.44	80.00
27.83	4.43	79.99
44.00	7.00	79.99

Liquid water spreading outward in the suspended fabric sample

Once condensed to liquid in the sample, water will start wicking outward to the surrounding fabric substrate, driven by capillary forces^41,42 p_c and the dynamic pressure p_d. Although there are, in general, two mechanisms in water spreading, i.e. wetting and wicking, occurring in sequence, we will not differentiate them, mainly because both are driven by the capillary forces and described by the same fundamental equations.^33,40 As the specimen is suspended vertically, the gravity related term p_g will also influence the capillary flow in the fabric along the vertical directions.

The liquid spreading in a porous medium can be considered as flow through a network of interconnected capillaries. The Lucas–Washburn equation is widely used to describe such flow, which is actually a special form of the Hagen–Poiseuille law.^42,43 In this case, the flow volumetric speed:

\frac{d V}{dt} = \frac{π r_{c}^{4} \sum p}{8 μ l}

(11a)

where

\sum p = p_{c} + p_{d} + p_{g}

(11b)

where r_c is the capillary radius, V is the mass volume and μ the viscosity of the liquid; l is the flow length. The capillary pressure p_c is defined as⁴⁰

p_{c} = \frac{2 γ cos θ}{r_{c}}

(12)

while the dynamic pressure p_d is given in equation (7), the liquid weight caused pressure p_g⁴³ is

p_{g} = - l ρ g sin β

(13)

where β is the angle between the flow and the horizontal directions, as in Figure 1. Plugging in all the relations, equation (11) becomes

\frac{d V}{dt} = \frac{π r_{c}^{4}}{8 μ l} (\frac{2 γ cos θ}{r_{c}} + \frac{1}{2} ρ u^{2} - l ρ g sin β)

(14)

In capillary flow, the liquid volume V is expressed as $V = π r_{c}^{2} l$ and then equation (14) can be rewritten as

\frac{dl}{dt} = \frac{r_{c}^{2}}{8 μ l} (\frac{2 γ cos θ}{r_{c}} + \frac{1}{2} ρ u^{2} - l ρ g sin β)

(15a)

Flowing in the horizontal (weft) direction, β = 0 and π, then sin β = 0 and equation (15a) becomes

\frac{dl}{dt} = \frac{r_{c}^{2}}{8 μ l} (\frac{2 γ cos θ}{r_{c}} + \frac{1}{2} ρ u^{2})

(15b)

By setting β = 90° and 270°, flowing in the vertical direction along and against the gravitational effect. Equation (15a) holds, but $sin 90^{\circ} = 1, sin 270^{\circ} = - 1$ , respectively.

Still note that $u = dl / dt$ , equation (15a) is a nonlinear differential equation whose solution in a closed form may not exist. To overcome this, in comparison between the other two pressure terms p_c and p_g, the dynamic pressure p_d with the term $u^{2} = (dl / dt)^{2}$ is a higher order of a small differential $dl / dt$ , and hence negligible p_d, as already verified in Table 5. Equation (15a) can thus be simplified into

\frac{dl}{dt} = \frac{r_{c}^{2}}{8 μ l} (\frac{2 γ cos θ}{r_{c}} - l ρ g sin β)

(15c)

Bringing r_c in equation (5) into equation (15c), we obtain the final expression in equation (15d) for the flow speed

\frac{dl}{dt} = \frac{r_{f}}{8 μ l \sqrt{V_{f}}} e^{\frac{V_{f}}{2 π^{2}}} (2 γ cos θ - \frac{r_{f} π}{\sqrt{V_{f}}} e^{\frac{V_{f}}{2 π^{2}}} l ρ g sin β)

(15d)

Equation (15d), for the first time, specifies how the fiber size r_f, and amount V_f; the liquid properties viscosity μ, contact angle θ, fluid density ρ and surface tension γ all affect the flow speed in a rather complex way. We plotted equation (15d) in Figure 3, where the flow speed dl/dt (m/s) is against the fiber volume fraction V_f. First of all, as dl/dt in the figure is always positive, so the gravitational term is always smaller than the capillary suction force. Also, the fiber volume fraction has to be greater than a critical value V_fc, below which equation (15d) will not offer a meaningful result, as shown in both the equation and the figure. When other parameters remain constant, i.e. for a given curve in Figure 3, a larger V_f and hence a smaller mean capillary radius r_c will result in a higher capillary flow speed. At a given V_f, the flow speed acquires different values along different directions (i.e. the flow anisotropy) and the difference is caused by two factors, the fabric structural anisotropy and the effect of the liquid weight as the fabric specimen is vertically suspended. Therefore, at a given V_f, the speed reaches its highest in the direction β = 3π/2, where the fiber volume fraction and liquid weight both facilitate the liquid spreading; whereas, in the direction β = π/2, the liquid weight is against the flow, leading to the lowest flow rate, and the flow speeds are in between the two extremes at β = 0,π and other directions. Of course, as equation (15d) is available now, we can even separate and compare in more detail the terms due to liquid weight and fabric type.

Figure 3.

V_f influence on the dl/dt at different flow directions.

According to reference 40, equation (15c) can be integrated to the flow time as a function of flow distance l:

t (l) = - \frac{8 η l_{m}}{ρ {gr}_{c}^{2}} [\frac{l}{l_{m}} + \ln (1 - \frac{l}{l_{m}})]

(16)

where l_m is the terminal (final) flow distance.

In other words, using the β value, the angle between the flow and the horizontal direction in Figure 1, we can conveniently deal with both sample structural anisotropy and the effect of liquid weight.

When β = 0 and 180°, flow is along the horizontal direction and the liquid weight is inapt, equation (15d) then reduces to Washburn’s law and a more direct relation between the flow distance l and time t can be derived.⁴²

To verify the inertial effect, the capillary flow distance 4.66 cm in β = 0 and the values of u in Table 5, also applied in equation (15c), and the corresponding time were calculated via equation (16). This is how the last column in Table 5 was acquired, clearly showing that the dynamic effect or the flow speed u has a very minor effect on the result.

Theoretical analysis of the heat transfer in fabrics

Heat is transmitted largely through solid fibers, for air is virtually a heat insulator. In general, the spaces between the fibers and yarns are so small (∼10⁻⁶ m) that the effect of heat convection is negligible. The heat transfer via radiation can be calculated below. Assuming the steam tube is an ideal radiator (blackbody, no energy dissipation, the most conservative case), then according to the Stefan–Boltzmann law, for example in reference 44

E_{b} = σ T_{s}^{4}

(17)

T_s = T_g is the steam temperature on the absolute scale (373 K), σ is the Stefan–Boltzmann constant, E_b is the power released per unit area (w/m²). Then the radiation power in this case is

w = E_{b} A

(18)

The area A is given in equation (9). Then the radiation heat per minute is

q = w \times 60 s

(19)

However, our calculation in Table 4 shows that the radiation heat q₂ is much smaller than the steam condensation heat q₁. Clearly the effect of radiation can be neglected. Furthermore, as the gravitational force has no effect on heat transfer, so the anisotropy in heat transfer is not as significant as in the water spreading case. Our question thus reduces to a steady axisymmetric heat conduction, and the temperature distribution depends only on a single coordinate l⁴⁵

\frac{d}{dl} (l \frac{d T}{dl}) = 0

(20)

For steady-flow, it simplifies into

l \frac{d T}{dl} = a

(21)

Where a is an integration constant. For an independent l, the final solution is

T = a \ln l + b

(22)

To find the integration constants a and b, we have the boundary conditions T = T_g at l = 0 and

T = T_i at l = l_i. Applying them yields the final solution on temperature fading with flow path l:

T = (T_{i} - T_{g}) \ln l + T_{g}

(23)

Note that, for different samples, the l and T values at the end will be different. What is interesting here is that, according to equation (23), the temperature distribution is determined by the initial temperature and the distance l and appears independent of the fiber properties! For a heterogeneous and anisotropic sample where the path l is a function of both location and direction, however, the solution for equation (21) will be more complex if the expression of l is known. That solution nevertheless is able to predict the influences of the sample heterogeneity and anisotropy on the heat transfer.

Experimental verification and discussion

The experimental platform and process

Before the calculation, all the parameters were listed in Tables 1 through 3, including for the woven sample the volume fractions in the warp and weft direction V_f1 = 0.30 and V_f2 = 0.15 and the corresponding mean capillary radii r_c1 = 5.15 × 10⁻⁵ m and r_c2 = 7.25 × 10⁻⁵ m.

Both the experiments and numerical analysis were conducted at three steam flow rate levels $Δ \overset{\cdot}{m}$ : 0.5, 1 and 1.5 g/s when steam sprays the fabric. For simplicity, the deformation of the fabric is ignored. The specimen mass change Δw was measured along with the calculated heat dissipation during steam condensation, as shown in Table 4.

Water spreading photos – data and curves

Influence of the inertial dynamic flow speed

As discussed before, by inserting the u values from Table 5 into equations (15a) and (16), the flowing times are calculated as the last column in Table 5. It is clear that the impact of the initial flow speed u is indeed negligible.

Influence of the sample anisotropy and water weight

Figures 4(a) and (b) show the patterns of water wetting in the woven fabric in comparison with the nonwoven (paper) sample as time goes from the 1st second to the 30th second. We need to point out that all the cotton fabric samples are plain fabric. In Figure 4(b), especially the 1st second, 20th second and 30th second, the sample appears like twill, but this is caused by the camera’s Moiré pattern effect (in television and digital photography, a pattern on an object being photographed can interfere with the shape of the light sensors to generate unwanted artifacts).

Figure 4.

The capillary flow in the samples. (a) Water wetting in the nonwoven fabric. (b) Water wetting in the woven fabric. (c) Flow pattern in the woven fabric. (d) Flow pattern in the nonwoven fabric

First, as the nonwoven sample possesses a more uniform structure with randomly distributed fibers, it indeed demonstrates a more uniform flowing profile than the woven sample, which shows a sharp angle in both the warp and weft directions. Nonetheless, both samples exemplify a typical anisotropic behavior: water flows faster along the vertical direction than the horizontal. This is easy to understand in the case of the woven fabric because of a higher fiber volume fraction, hence a greater flowing speed along the vertical (warp) direction, as we have calculated and shown in Table 3.

Of course, another significant factor responsible for the flowing profile is the weight of the liquid as the samples are suspended. A closer look at the flow pattern in the woven sample in Figure 4(c) reveals that the up and down halves are not symmetrical, and the flow distance in the lower half (5.74 cm) is visibly longer than that in the top half (4.66 cm), due apparently, to the effect of the liquid water weight. This appears more pronounced in the nonwoven sample in Figure 4(d), where a more significant difference is exhibited between the top (1.80 cm) and the lower halves (3.29 cm). So we can conclude that, even though the nonwoven sample is structurally isotropic, the liquid weight still leads to an anisotropic flow pattern due to gravity.

We then turned the samples by 90° so that the structural influence was altered as the horizontal direction now aligned with the warp, yet the liquid weight effect remained the same. Figure 5 shows the results after turning. Overall, the flowing profiles for the nonwoven sample in Figure 5(a) remain largely unchanged from Figure 4(a), as there is little structure anisotropy in the nonwoven sample but the liquid weight is still acting. So, turning the sample direction has little effect on an isotropic sample, as it is structurally independent of direction. In contrast, the woven sample in Figure 5(b) looks less elliptic than that in Figure 4(b) because structure anisotropy in the woven sample is now offset by the effect of water weight along the vertical (now the weft) direction.

Figure 5.

The capillary flow inside the samples after 90° rotation.

The flow anisotropy in the woven sample can be predicted using equation (15d) as well, by assuming that the samples are simply assemblies of capillary tubes with various diameters. As shown in equation (5), the mean capillary radius r_c is a function of the fiber volume fraction.

Flows along different directions are reflected by the sample orientation angle β. For the horizontal (weft) direction, β = 0,π, whereas β = π/2 and 3π/2 in the vertical (warp) direction. Furthermore, at the top direction β = π/2, the liquid weight is against the liquid flow; yet, for the downward direction β = 3π/2, the weight acts along with the flow.

Let’s take the woven sample as an example. The fiber volume fractions in equation (2) for the warp and weft directions are V_f1 and V_f2, respectively, and the corresponding mean capillary radii r_c1 and r_c2 are calculated using equation (5), all in Table 3. According to equation (15d), when other parameters remain constant, a smaller r_c will cause a faster capillary flow travel speed.

Plugging in all the data from Table 3 for the horizontal direction β = 0,π; r_c = r_c2 and there is no gravitational effect, equation 15(d) is reduced to

t = 235.59 l^{2}

(24a)

Whereas for the vertical direction, when β = π/2, r_c = r_c1 equation (15d) turns into

t = - 407.16 [0.38 l + 0.09 \ln (1 - 4.29 l)]

(24b)

and for β = 3π/2,

r_{c} = r_{c 1}

there is

t = 407.17 [0.38 l - 0.09 \ln (1 + 4.29)]

(24c)

where t is the flow time and l is the capillary flow distance. By plotting the three equations, we can compare the flow time–distance connection for a given fabric; then, by rotating the β value and using the corresponding mean effective pore radius r_c, we can predict the flow time–distance for the sample rotated by π/2 to reveal the liquid weight influence, all as illustrated in Figure 6.

Figure 6.

The capillary flow in the samples. (a) Woven sample flow prediction. (b) Nonwoven sample flow prediction.

Figure 6(a) is for the woven sample. Curves 2, 3 and 5 are the predictions for the horizontal β = 0,π, the vertical up β = π/2 and the vertical down β = 3π/2 directions, respectively, whereas Curves 1, 4 and 6 are the corresponding results after rotating the sample by π/2.

The differences between Curves 2, 3 and 5 in Figure 6(a) reflect the flow behaviors in the different directions – the flow anisotropy of the sample. For instance, at the same time t = 7 s, Curve 2 is along the warp but at the vertical downward direction with the help of liquid weight. The experimental value, l_m = 5.74 cm is applied in Figure 4(c). Still warp but now the liquid is against the flow in the vertical upward direction β = π/2, Curve 5 shows the lowest flow, l_m = 4.66 cm. Curve 3 is the prediction for the horizontal (weft) direction where liquid weight becomes irrelevant; the flow therefore is in the middle, l_m = 3.92 cm in Figure 4(c), but very close to Curve 2.

Once the samples are turned by 90°, the warp and weft switch places for the woven sample, but the liquid weight remains in the vertical direction. Now we have the Curves 1, 4 and 6. Curve 1 is the weft but now in the vertical upward direction, it is Curve 3 but with the help of liquid weight. Curve 1 is the highest one of the six curves; Curve 4 is the warp direction but now aligned horizontally with no influence from the liquid weight; Curve 6 is the weft, but now along the vertical upward direction against the liquid weight, it is the lowest of all.

The same comparison is also made for the nonwoven sample without the ones after the 90° rotation in Figure 6(b). Although the three curves are not overlapping each other, the differences between them are much smaller – due to a less anisotropic structure. The vertical downward direction β = 3π/2 shows the highest curve attributed to the liquid weight, for without it, β = π/2, corresponds to the lowest curve. When the liquid weight disappears, the resultant curve is again in between, similar to the woven sample in terms of liquid weight effect.

Comparisons are also made in Table 6 and Figure 7 between the predicted results from equations (24) against the experimental data measured from Figures 4 and 5, for both woven and nonwoven samples. For more generality and easy viewing, we normalized all the data, dividing the distance value l by its corresponding maximum value l_max, so that the numbers are within the range of zero to one. Note that, at a given time, a smaller number indicates a longer way remains to go, therefore a higher curve. So, at time t = 10 s, both experimental and theoretical results rank {β = 3π/2, r_c1} = the number one; {β = 0, r_c2} the number two and {β = π/2, r_c1} the last one; not only consistent between the experiments and the predictions but also the same ranking Curve 2 > Curve 3 > Curve 5 as in Figure 6(a). Likewise, the comparison for the nonwoven sample is in Table 7 and Figure 8. The data show smaller variations along different directions. In all cases, a good agreement is seen between the experimental and theoretical results.

Table 6.

Experiments and theoretical predictions along warp and weft

	Experimental results (l/l_max)			Theoretical predictions (l/l_max)
	β = 0 and π	β = π/2	β = 3π/2	β = 0 and π	β = π/2	β = 3π/2
Time (s)	r_c2	r_c1	r_c1	r_c2	r_c1	r_c1
1	0.22	0.21	0.16	0.18	0.34	0.14
5	0.52	0.64	0.38	0.41	0.64	0.30
10	0.62	0.78	0.53	0.58	0.80	0.47
15	0.68	0.83	0.80	0.71	0.89	0.61
20	0.73	0.89	0.81	0.82	0.94	0.75
30	1.00	1.00	1.00	1.00	1.00	1.00

Figure 7.

The experiments and theoretical prediction comparison for the woven sample along different fabric directions. (a) β = 0 and π(b) β = π/2 (c) β = 3π/2.

Figure 8.

The capillary flow in the nonwoven sample along different directions. (a) β = 0 and π (b) β = π/2 (c) β = 3π/2.

Table 7.

Nonwoven material: comparison between experiments and theoretical predictions

	Theoretical predictions (l/l_max)			Experimental results (l/l_max)
Time (s)	β = 0 and π	β = π/2	β = 3π/2	β = 0 and π	β = π/2	β = 3π/2
1	0.20	0.26	0.12	0.26	0.30	0.36
3	0.35	0.46	0.23	0.40	0.44	0.50
5	0.45	0.60	0.31	0.56	0.61	0.70
7	0.53	0.68	0.38	0.59	0.70	0.80
9	0.60	0.74	0.44	0.64	0.77	0.86
30	1.00	1.00	1.00	1.00	1.00	1.00

In summary, the shape variations of the flow profiles are the cumulated results of the irregularities in all related parameters characterized by the porosity or fiber volume fraction and the liquid weight. As stated above, if the fiber volume fraction can be given as a function of location and direction, the solution can predict the influences of the sample heterogeneity and anisotropy on the flow behaviors.

Heat transfer IR photos – data and curves

For the heat transfer process recorded by the infrared camera, Figure 9(a) is the results for both cotton and nonwoven samples. As the liquid weight is not pertinent to heat flow, the heat images are not as extreme as the water flow patterns, and the irregularities in the profiles are entirely a reflection of variations in the fiber volume fraction of the samples. Therefore, the heat transfer patterns between the woven and nonwoven samples exhibit similar shapes, but the one for the woven sample is larger in size. Then, the water flow and heat transfer behaviors of the woven sample are compared side by side in Figure 9(b); the heat transfer patterns are rounder than those in the case of water flow, closer to the prediction by equation (23) of coaxial rings at different coordinate l.

Figure 9.

(a) Heat transferring in samples: woven versus nonwoven at 60 g/min. (b) Comparison between water flow and temperature transfer in the woven sample.

The best parameter to measure the heat transfer rate of a material from the hot side to the cold side is to use the thermal diffusivity α of the sample, defined as:⁴⁵

α = \frac{k}{ρ c_{p}}

Where k, ρ and c_p are the thermal conductivity, the density and the specific heat capacity of the material. That is, even though the α value is proportional to the thermal conductivity k, the material density and heat capacity are just as important. The thermal diffusivities for the samples are calculated and provided in Table 3 as well. The thermal diffusivity for the cotton fabric 3.48 (J/g·K) 10⁻⁷ is greater than that for the polyester fabric 1.32 (J/g·K) 10⁻⁷, leading to a larger heat spreading area.

Equation (23) predicts a similar heat flow behavior for both samples, which is more or less consistent with the experiments in Figure 9(a). First, the temperature is fading with time. The irregularities in the beginning were intense but somehow slowed down gradually. Even though the nonwoven sample is much more uniform than the woven structurally, that difference doesn’t seem to be reflected in the heat images. Overall, the heat in the woven sample does spread wider than in the nonwoven, as explained by the fabric thermal diffusivities above. As heat transfers virtually through fibers only, it moves further in the woven sample with an overall density 0.38 (g/cm³) than in the nonwoven with a density 0.17 (g/cm³), i.e. the nonwoven sample is much more porous (with less fibers).

A more careful examination reveals further differences. For the woven sample, the pictures, especially in the end, show an outline close to elliptic shape with the major axis aligned in the warp direction with V_f1 > V_f2 of the weft direction, again leading to a higher heat transfer rate in the warp direction.

In fact, when applying equation (23) to predict temperature changes with time, for the woven sample for example, since the constants T and l possess different values at different directions determined based on the experiment results from the infrared images, equation (23) can be used to predict the temperature distributions at different directions.

To further examine the process, the theoretical prediction and experimental data, all normalized, are plotted in Figure 10 for the woven sample. Figure 10(a) shows the experimental results from Figure 9(a) for the woven fabric, clearly demonstrating the heat transfer anisotropy where the heat flow is furthest in the warp downward direction β = 3π/2, followed by that in the warp upward direction β = π/2 and then finally the horizontal weft direction β = 0. The theoretical predictions using equation (24) are plotted in Figure 10(b). First, the predictions and test data exhibit a similar trend. The temperature declines from the center to the edge in a logarithmic manner. Also, the predicted results offer a ranking in the flow range along the three directions of $β = \frac{3 π}{2} > β = \frac{π}{2} > β = 0$ , identical to that given by the experimental data. Yet there are some discrepancies between Figures 10(a) and (b). At the same distance, r = 0.1 m for example, the temperature coverage based on the experimental result is 0.50 ∼ 0.66 in Figure 10(b), whereas in Figure 10(a), the range is 0.62 ∼ 0.78. That is, the theoretical predictions depict a slower temperature fading rate. The discrepancies between the prediction and experimental data may be caused by the external steam flow, i.e. the steam flow along the fabric surface. When we conducted the experiment, it was not possible to ensure that the orifice was 100% sealed and attached directly to the fabric. Therefore, some steam will leak and form an external flow, which is complex. The cross section of the orifice may also influence the external effect, which needs further study.

Figure 10.

Fabric temperature: prediction and real data. (a) Experiments. (b) Predictions.

Again, since the liquid weight effect is now irrelevant, there is much less directional impact and the figures, including those in Figure 9(a), are less dramatic than those in the water spreading process.

Conclusions

The heat and mass transport processes, including steam condensation, heat and water spreading in a suspended fabric sample in a new ironing device are presented. The steam flow rate Δn& is the sole source for both heat and liquid, and the specimen mass change Δw represents the liquid spreading into the fabric. The device is found to be energy inefficient due to the disproportionally large amount of heat dissipated during steam condensation once impinging the fabric. The heat flow is largely done through fiber conduction whereas the water spreading is via the capillary pores. Therefore for a given system with a known fiber orientation function, the fiber volume fraction V_f of the sample is the prevailing factor for it determines the porosity of the sample and the amount and diameter of the capillary pores.

A fabric can be treated as a system of capillary tubes with diverse diameters determined by the corresponding fiber volume fractions. Once the fiber volume fraction distribution along different directions in the fabric is known, the anisotropy in both heat and water flow can be characterized and studied. A higher fiber volume fraction leads to a smaller capillary diameter and hence a fast liquid flow speed – consistent with existing theories. In a system with random fiber orientation, the diameters of the capillary pores generated cannot be smaller than the constituent fibers.

Structural anisotropy in fabrics is a measurable feature. The water weight results in an enhanced behavior anisotropy in water flow but is irrelevant in the heat spreading process.

Footnotes

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: Liang S.T. would like to acknowledge the financial support from CSC, China for his visit to UC Davis where the manuscript is completed and his gratitude for financial support from Shanghai Science and Technology Committee (Grant Number 17DZ2202900), National Natural Science Foundation (Grant Number 71373041), Shanghai Summit Discipline in Design through Project on Fashion Technology Innovation, Donghua University Institute for Nonlinear Sciences (Grant Number 231-08-0001) and the Fundamental Research Funds for the Central Universities (Grant Number 17D310702).

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