Simultaneous Determination of Thickness and Heat Conductivity in Fabric Design: Steady-state Modeling and PSO Algorithms

Abstract

Fabric thickness and texture directly affect heat and moisture transfer characteristics in the human body–fabric–environment system, which determine the heat-moisture comfort level of the human body. Based on the model of steady-state heat and moisture transfer from skin to environment, we put forward an Inverse Problem of Thickness-texture Determination (IPTTD). Adopting the idea of the least-squares method, we formulate IPTTD into a function minimization problem. We employ Particle Swarm Optimization (PSO) method to directly search optimal solution of the objective function. The results of numerical simulation show the effectiveness of the presented algorithms and validity of the proposed IPTTD.

1. Introduction

In recent years, functional textile material design has been paid more and more attention due to the human increasing requirements on daily life and some special applications. Some mathematical models of heat and moisture transfer in the body-fabric-environment system have been formulated [1,2,3,4,5,6,7,8]. From these models, we can find some main factors which affect human body comfort and find out these factors’ intrinsic rules. Thereby we usually call these models Direct Problems (DP). Now, some numerical methods have been put forward to solve the DP such as finite-difference method, finite-volume method, finite-element method and controllability volumetime- domain recursive method [9,10,11].

Although the DP helps us to know about human body heat-moisture comfort level for given textile materials and environmental conditions, we don’t know how to determine the textile texture and thickness mathematically in order to make one feel more comfortable when the textile materials are not given. So the study of inverse problems of textile material design is fundamentally important for us to study human body comfort. Xu proposed inverse problems of textile thickness or type design for a single layer or bilayer textile material based on the model of steady-state or dynamic-state heat and moisture transfer through textile and presented effective numerical algorithms to solve the inverse problems [12,13,14,15].

In this paper, we formulate an Inverse Problem of fabric Thickness-texture Determination (IPTTD) which is based on the steady model of heat and moisture transfer from skin to environment through fabric [16] and make numerical calculation to IPTTD under the given environmental conditions.

2. Mathematical Formulation of Heat and Mass Transfer in the Body-Fabric-Environment System

2.1. The Schematic Diagram and Assumptions

The body-fabric-environment system discussed in this paper consists of human skin, an air layer between the skin and fabric (microclimate area), fabric and environment as shown in Figure 1. The outer surface of fabric is exposed to the natural environment. The fabric and the skin are placed perpendicular to the direction of gravity, and this lay-out is called a horizontal system. Radiation and surface diffusion are considered in addition to conduction and convection [16]. The assumptions are described same as that in reference [17].

Figure 1. Pathways of the heat and moisture transfer from skin to environment through fabric: horizontal system.

2.2. The Direct Problems (DP)

Based on the above system and the model of heat and moisture transfer from skin to environment through fabrics, we attribute the model to the boundary problem of complicated nonlinear equations. The temperature T(x) and relative humidity w(x) in the horizontal system satisfy

(1)

(2)

According to energy conservation theory and mass conservation theory, the following relationships are satisfied at the steady state

(3)

2.3. Numerical Solution to the DP

According to the equations in (1), we can see that T(x) and w(x) are piecewise functions

(4)

(5)

where x_i and y_i (i = 1,⋯,4) are coefficients which will be derived by solving the DP (1)–(3).

By computation we turn to solve following nonlinear equation with respect to the variable x₁

(6)

where

2.4. Numerical Experiments

We illustrate representative fabrics such as cotton, wool and PET (Polyethylene Terephthalate). The parameters used in the computation procedure, structural and physical parameters of three fabric materials are listed in Table 1 and Table 2 respectively.

Table 1. The parameters used in the computational process.

Table 2. Structural and physical parameters of three fabric materials.

Therefore the distributions of the temperature T(x) and relative humidity w(x) are described in Figure 2 for three fabric materials.

Figure 2. Distribution of temperature T(x) and relative humidity w(x).

3. An Inverse Problem of Thickness-texture Determination (IPTTD)

3.1. Mathematical Formulation of the IPTTD

We consider the IPTTD in the body-fabric-environment system.

IPTTD: In the body-fabric-environment system, we assume that

(1) Environment: (T^e,RH^e) ∈ [T_min,T_max] × [RH_min,RH_max], where T_min and T_max are minimum average temperature and maximum average temperature at a specific place during a specific time period respectively; Similarly RH_min and RH_max are minimum relative humidity and maximum average relative humidity respectively.

(2) Fabric: The structural parameters of fabric such as the radius of pore, porosity of fabric surface and effective tortuosity are all known.

(3) Body heat-moisture-comfort indexes: The literatures on clothing thermal comfort have indicated that the comfort indexes in the microclimate area are given as follows [18]: temperature (305.16 ± 1)K, relative humidity (50% ± 10%), wind speed (0.25 ± 0.15)m/s.

Given the environmental temperature and relative humidity together with the comfort indexes, according to the boundary value conditions

(7)

we shall determine the texture and its thickness by the DP (1)–(3) to make sure that the temperature and relative humidity in microclimate area approach the comfort indexes as much as possible.

3.2. Weighted Least-squares (WLS) Solution to the IPTTD

With regard to IPTTD, we construct a numerical algorithm to implement numerical simulation. We discretize the combination of environmental temperature and humidity as (T_i^e,RH_j^e)(i = 1,…,n; j = 1,…,m) in order to obtain the WLS solution. Suppose that T^∗,RH^∗ is the experience value of temperature and relative humidity in comfortable state, for example, we choose T^∗ = 305.16K, RH^∗ = 50%.

We can attribute the IPTTD to the following weighted least squares problem:

(8)

where W₁,W₂ are different weights satisfying W₁ + W₂ = 1.

The objective of the weighted least-squares method is to find the minimizer (L_wls,k_wls) such that the least-squares function J(L,k) attains the minimal value

(9)

where (L_wls,k_wls) is called the weighted least-squares solution to the IPTTD.

3.3. The Description of Particle Swarm Optimization (PSO)

Since the objective function is complicated and nonlinear, we employ particle swarm optimization to directly search the WLS solution to the IPTTD.

We assume the search space is D-dimensional and the number of particle is n. Vector X_i = (x_i1,x_i2,…,x_iD) and P_i = (p_i1,p_i2,…,p_iD) are the present position and the past optimal position in history flight records of the i particle. Pg is the optimal position of all particles’ past optimal position P_i(i = 1,2,…,n). V_i = (v_i1,v_i2,…,v_iD) is the i particle’s position change rate(speed). The position of each particle changes according to the following formula:

where the positive constants ac₁,ac₂ are called acceleration factors. rand() is random number belong to [0, 1]. in is called inertia factor. There are a positive correlation between the value of in and the size of solution space. j is the current iteration number and m is maximum iteration number. Now the POS steps are concluded as follows:

Step 1. Initialize. (a) Set particle number n. (b) Set particle initial position X_i(0) and initial velocity V_i(0),i = 1,2,…,n. (c) Set particle initial personal optimal position P_i(0) and find global optimal position Pg(0).

Step 2. Generate random number and calculate velocity and update position to each particle. Calculate fitness value f(x_i(j)) to updated position of particle, compared and find personal optimal value and global optimal value. (a) Personal optimal value: if f(x_i(j)) < f(P_i(j)), then P_i(j + 1) = x_i(j), else P_i(j + 1) = P_i(j). (b) Global optima value: if f(P_i(j)) < f(Pg(j)) then Pg(j + 1) = P_i(j), else P_i(j + 1) = Pg(j).

Step 3. Judge iteration number. Compute output optimal solution if it gets to maximum iteration; repeat the step 2 if it doesn’t get to maximum iteration.

3.4. Numerical Simulation for the IPTTD

We choose five different environmental conditions (five cases, here environmental temperature keep changeable and relative humidity is always kept between 30% and 70%) to do numerical simulation and determine thickness and thermal conductivity simultaneously by PSO algorithms. Compared with the heat conductivity table in [17], we can predict suggested fabric materials. The detailed information is listed in Table 3. We select representative cases such as case 1, 2 and 5 to draw graphs of function J(L,k_fabric). The following pictures (Figure 3, Figure 4 and Figure 5) are the graphs of function J(L,k_fabric) for the case 1, 2 and 5.

Table 3. The results of numerical simulation for the IPTTD.

Figure 3. Function J(L,k_fabric) in case 1.

Figure 4. Function J(L,k_fabric) in case 2.

Figure 5. Function J(L,k_fabric) in case 5.

4. Validation of the IPTTD

We can not only make sure that the IPTTD has unique solution from the Table 3 and Figure 3, Figure 4 and Figure 5, we also find the uniqueness of J(L,k_fabric) by its graphs as L or k_fabric is fixed. The graphs are given in Figure 6 and Figure 7 for the case 1, 2 and 5.

Figure 6. Graph of function J(L,k_fabric) when k_fabric is fixed.

Figure 7. Graph of function J(L,k_fabric) when L is fixed.

5. Conclusion

According to the numerical results, we make some remarks. When the human body is situated in lower temperature than the skin temperature, we make sure that there is a positive correlation between the fabric thickness and fabric heat conductivity. From the computational results and graphs we can find out that the objective function reaches much smaller value than the results of IPTD and IPMD in [17]. It shows that IPTTD is more reasonable than the IPTD and IPMD. Hence we can simultaneously determine fabric thickness and heat conductivity to make human body more comfortable.