Mathematical Modeling Of Heat-moisture Transfer Within Textiles And The Corresponding Inverse Problems

Textile material design aims at getting insight into the law of thermodynamics within textiles by mathematical or scientific methods, and hence providing theoretical support for textiles, especially for functional fabric materials. The study tends to focus eyesight on heat-moisture transfer models and the last few years have witnessed a remarkable development in this area. However, these models may not be applicable in some cases where the classical Fourier’s law or Fick’s law is no longer valid. Thermal protective fabric can be treated as a porous medium with tinier pore diameter and the garment consists of multilayered fabric, designed for firefighters or people working under high temperature to avoid thermal damage as a result of exposure to a flash fire. Above factors make the heat transfer process within textiles more complicated, and hence the classical Fourier’s law or Fick’s law is surely invalid under this circumstance. In recent years, anomalous diffusion has gained unprecedented attention of scholars. The reason lies in the increasing anomalous phenomena observed in porous media or complex media.This paper is organized as follows.In chapter 1, we introduce the background and significance of studying textiles or ther-mal protective clothing. The previous research results and the content of this paper are briefly described as well.The whole summary of textile material design in the case of classical model is presented in chapter 2, including direct problem for temperature, inverse problem of boundary value determination or sideways problem for moisture and inverse problem of textile material design. Similar to the energy estimates of continuous problem, discrete variational method is applied to prove the stability of numerical scheme of bilayer model. The corresponding convergence rate is also presented. As for sideways problem, we discuss some characteristics of finite difference method without mollifying. The above two problems are combined in the inverse problem of textile material design. We numerically solve the porosity determination problem of bilayer model and provide the modified algorithm by using BP neural network.In view of the importance of sideways problem, in chapter 3, a novel forward colloca-tion method (FCM) is investigated. The FCM approximates the boundary value by finite dimensional combination in the form of interpolation, which in turn transforms the origi-nal problem into a system of well-posed direct problems and an ill-posed algebraic system. Therefore, the errors are mainly determined by the numerical accuracy of the direct prob- lems. The result indicates that the error is bounded by the numerical error of approximated quasi-solution and confirms the previous claim.Based on super-diffusion law, in chapter 4, a space fractional heat transfer model is proposed to describe the faster transmission under high temperature. Similar to the process of establishing weak solution for classical problem, we put forward a new definition of weak solution for fractional equation and prove the conditional well-posedness by using Galerkin approximation method.As for the thermal protective clothing, we aim at minimizing the thermal damage so as to determine some physical parameters including thickness, thermal conductivity and poros-ity, etc. For simplicity, in chapter 5, we require that temperature at the interface between the epidermis and dermis in the human skin is lower than the critical temperature of thermal damage; hence a reasonable definition of the inverse problem of textile material design is put forward. As examples, we numerically solve the corresponding inverse problems. As an early start of inverse problems modeling for thermal protective clothing design, we should continue giving much more research achievements in the very near future.