Prediction of air permeability coefficient and water-vapor resistance of 3D textile layer

Figures & data

Figure 1. The sample of 3D textile.

Table 1. Measured construction parameters and air permeability of samples (Zupin et al., 2012).

Sample number	Measured warp density (ends/cm)	Measured weft density (picks/cm)	Thickness (mm)	Measured air permeability (mm/s)	Theoretical air permeability (mm/s) (Pezzin, 2015)
1	21	15	0.439	2391.67	2316
2	21	20	0.438	1571.67	1526

Figure 2. Scheme of the model simulating A Sweating Guarded Hot Plate M259B (Neves et al., 2015) experiment.

Table 2. Thermal-physiological comfort rating system in accordance with R_et coefficient values (Relji et al., 2016).

Rating	Ret value	Description
Very good	0–6	Extremely breathable and comfortable at a higher level of activity.
Good	7–13	Very breathable and comfortable at a moderate rate of activity.
Satisfactory	14–20	Breathable, but uncomfortable at a higher rate of activity.
Unsatisfactory	21–30	Slightly breathable, giving moderate comfort at a low rate of activity.
Very unsatisfactory	31+	Not breathable and uncomfortable, with a short tolerance time.

Figure 3. Geometry of Model 1, Model 2, Model 3, Model 4, Model 5, Model 6.

Figure 4. Model 7 geometry with boundary conditions for air permeability simulation.

Table 3. Air domain and textile layer parameters.

Model number	Length of x direction (mm)	Width of y direction (mm)	Height of z direction (mm)	Thickness of textile layer (mm)	Distance between two layers of textile (mm)	Number of layers
1, 3	1.4	1	10	0.439	–	1
5, 6, 7	1.4	1	10	0.439	3.06	2
2, 4	1.05	1	10	0.438	–	1

Table 4. Statistics of completed mesh.

Sample	Number of elements (fabric domain)	Total number of elements	Number of mesh vertices
Model 1	2,35,566	2,94,324	56,890
Model 2	2,24,629	2,88,753	55,489
Model 3	7394	59,191	11,805
Model 4	7668	57,012	11,561
Model 5	4,71,396	5,86,144	1,12,959
Model 6	26,861	1,41,248	28,062
Model 7	34,500	1,73,561	33,430

Table 5. Boundary conditions for the simulation of air permeability.

Surface	Boundary conditions	Settings and assumptions
On ∂Ωt	$\mathbf{u}=0$	Wall condition: No slip. No slip condition is assumed on 3D textile surface. The fluid (air) velocity on this surface is zero.
On ∂Ωair	$\mathbf{u}\cdot \mathbf{n}=0{\mathbf{K}}_{\mathbf{n}}-({\mathbf{K}}_{\mathbf{n}}\cdot \mathbf{n})\mathbf{n}=0,{\mathbf{K}}_{\mathbf{n}}=\mathbf{Kn}$ In case of air domain: $\mathbf{K}=\mu (\nabla \mathbf{u}+{(\nabla \mathbf{u})}^{\mathrm{T}})$ In case of porous media domain: $\mathbf{K}=\frac{\mu }{{\epsilon }_p}(\nabla \mathbf{u}+{(\nabla \mathbf{u})}^{\mathrm{T}})-\frac{2}{3}\frac{\mu }{{\epsilon }_p}(\nabla \cdot \mathbf{u})\mathbf{I}$	Wall condition: slip. A slip condition is on air surface that has no contact with 3D textile layer.
On ∂Ωinlet	$${\mathbf{n}}^{\mathrm{T}}\left[-\hat{p}\ge +\mathbf{K}\right]\mathbf{n}=-{\hat{p}}_0{\hat{p}}_0\ge p_0,\mathbf{u}\cdot \mathbf{t}=0p_0=200\text{Pa}$$	Inlet (Suppress backflow). According to experiment the pressure difference is 200 Pa. Pressure condition $p_0=200\text{Pa}$ on ∂Ωinlet surface was set on. In this simulation two pressure boundary conditions on ∂Ωinlet and on ∂Ωoutlet surface were set on. For solver it takes extra time to recalculate velocity ($\mathbf{u}\cdot \mathbf{t}=0$).
On ∂Ωoutlet	$\left[-p\mathbf{I}+\mathbf{K}\right]\mathbf{n}=-{\hat{p}}_0\mathbf{n}{\hat{p}}_0\le p_0{p}_0=0\text{Pa}$	Outlet. Relative pressure $p_0=0\text{Pa}$ on ∂Ωoutlet surface was set on. Physical meaning that absolute pressure is equal to reference pressure (1 [atm]). It can be expressed as $p_A=p_{\mathit{ref}}+p_0.$

Table 6. Porous matrix properties (Pezzin, 2015).

All models	Porosity ϵp	0.58
Model 2, Model 4	Hexagonal array permeability k	1.35 × 10^−11 [m^2]
Model 1, Model 3, Model 5–7	Square array permeability k	1.64 × 10^−11 [m^2]

Table 7. Material properties.

	Dry air at T = 20 °C	Water-vapor at T = 35 °C, ϕ = 0.4
Dynamic viscosity µ	1.8139 × 10^−5 [Pa·s]	2.0386·10^−4 [Pa·s]
Density ρ	1.2043 [kg/m^3]	0.0158 [kg/m^3] (vapor content) $\rho =\frac{p_v}{R_v\cdot t}=\frac{2250Pa}{461.5J/\mathit{kgK}\cdot 308.15K}$

Table 8. Boundary conditions for the simulation of water-vapor resistance coefficient.

Surface	Boundary conditions	Settings and assumptions
On ∂Ωinlet	$\mathbf{u}=-U_0\mathbf{n}{U}_0=1\mathrm{m}/\mathrm{s}$	Inlet (Suppress backflow). According to experiment the normal inflow velocity is 1 m/s.

Table 9. Calculated air permeability and water-vapor resistance values.

Numerical model	Layer/density (warp/weft)	Air permeability (mm/s)	Air permeability error	Water-vapor transmission coefficient ${\delta }_p$(s/m)	Water-vapor resistance Ret (s/m)
Model 1	1 layer 21/15	2387.7	0.17 %	4.7159E − 5	4.4675
Model 2	1 layer 21/20	1534.4	2.37%	2.5860E − 5	8.1470
Model 3	1 layer approximated 21/15	2386.6	0.21%	6.0892E − 5	3.4599
Model 4	1 layer approximated 21/20	1556.3	0.98%	2.3649E − 5	8.9085
Model 5	2 layer 21/15	1462.5	–	2.6894E − 5	7.8337
Model 6	2 layer approximated 21/15	1493.2	–	3.3128E − 5	6.3595
Model 7 (3D textile model)	2 layer approximated 21/15 with spacer	1525.9	–	3.4670E − 5	6.0769

Figure 5. Simulation results of Model 7. Distribution of the velocity flow.

Figure 6. Dependence of calculated air permeability coefficient values against the number of elements.

Table

ϵp	Porosity of the textile	–	0.58
κ	Intrinsic permeability of yarns	m^2	1.64 × 10^−11
p0	Inlet pressure	Pa	200
p0	Outlet pressure	Pa	0
$p_A$	Absolute pressure	Pa
u	Velocity flow	m/s
Uo	Normal inflow velocity	m/s	1
psat	Saturation water-vapor partial pressure at 35 °C	Pa	5620
pv	Water-vapor partial pressure at relative humidity ϕ = 0.4	Pa	2250
${\delta }_p$	Water-vapor transmission coefficient	s/m	$\frac{w_v}{A\Delta p}$
A	Area	mm^2	1.4
$w_v$	Mass flow rate of water-vapor	kg/s
Ret	Water-vapor resistance	s/m
H	Heating power supplied to the measuring unit	W
R	Universal gas constant	J/(mol·K)	8.314
$R_v$	Gas constant for water vapor	J/(kg·K)	$\frac{R}{0.018(\text{kg}/\text{mol})}=461.5$
ϕ	Relative humidity	–	0.4
t	Temperature	K

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