## Abstract

The paper is concerned with the heat and sweat transport in porous textile media with complex phase changes, which is described as a non-isothermal, multi-phase and multi-component fluid flow and governed by a nonlinear, degenerate and strongly coupled parabolic system. The phase change, condensation/evaporation and fibre absorption, play an important role in the design of functional clothing. In this paper, we present some more precise formulations on condensation/evaporation, fibre absorption and heat capacity to maintain the physical conservation of mass and energy. A typical clothing assembly with a hydrophobic batting material and two different types of cover is investigated numerically. Numerical results show that for the hydrophobic material, the evaporation and condensation zones arise simultaneously in the batting area and that the proposed formulations are more realistic to describe the phase change. Existence and uniqueness of a classical positive solution for the incompressible model are also proved.

## 1. Introduction

Mathematical modelling for heat and moisture transfer in textile materials has been studied by many authors, which is often described as a multi-component compressible (Fan *et al.* 2004; Li *et al.* 2005; Huang *et al.* 2008; Henrique *et al.* 2009) or incompressible (Wang & Catton 2001; Cimolin 2008; Canuto & Cimolin 2010) fluid flow through porous textile media. Some earlier works can be found in David & Nordon (1939), Ogniewicz & Tien (1981) and Farnworth (1986) with relatively simple models. Recently, Li & Zhu (2003) presented a model including both the phase change and fibre absorption. Fan and his co-workers studied a more general dynamic model for heat and moisture transfer in textile materials. A typical application of their model is a clothing assembly consisting of a thick porous fibrous batting sandwiched between two thin cover fabrics. In the models, only a single-component (vapour) flow was concerned and the air motion was ignored. In addition, the condensation/evaporation was introduced in terms of the Hertz–Knudsen equation, which is true only for a single-zone case (wet or dry) in general. Numerical results (Fan *et al.* 2004) show that the vapour velocity in the single-component model is larger than expected. A multi-phase and multi-component flow model for textile materials with phase change was proposed by Huang *et al.* (2008). The model is a generalization of a single-component model used in the previous study by taking both air motion and vapour motion into account as both air and vapour behave differently and air could provide extra resistance to vapour movement. Ye *et al.* (2008) made several further modifications on this model. As the volume fraction of water is relatively small, a simplified water equation was introduced by neglecting convection and diffusion (capillary effect) of the liquid water. Therefore, the liquid water is immobile and stays at the condensation site. The condensation/evaporation in these multi-component models was also formulated by the Hertz–Knudsen equation, the same as in those single-component models. Owing to the source free of the air equation, the air concentration reached a fixed profile in a very short period. A quasi-steady-state model was proposed in our recent work (Ye *et al.* 2010), which consists of a steady-state air equation and dynamic state equations for other components. Under certain conditions, an analytical formula of the air concentration was given in terms of the mixture gas (air and vapour) concentration and temperature (or mixture pressure). With the analytical formula, the multi-component model reduces to a new single-component model, which is described by a system of nonlinear parabolic equations involving only the vapour concentration (or pressure), temperature and water content. Moreover, the system can be written in the same form as the single-component model studied by Le *et al.* (1995), Cheng & Wang (2008), Choudhary *et al.* (2004) and Fan *et al.* (2004) with an additional (air) permeability *K*, which, in other words, represents the air resistance to the vapour motion. As the global velocity in textile materials is usually small, the problem in some other applications was considered approximately as an incompressible fluid flow (Wang & Catton 2001; Cimolin 2008; Canuto & Cimolin 2010).

Applications of single- or multi-component and multi-phase fluid flow in porous media can be found in many other areas, such as the petroleum engineering and the groundwater hydrology (Ewing 1983; Mikelic 1991; Feng 1995; Wang 2008), food industry (Huang *et al.* 2007) and building materials (Ogniewicz & Tien 1981; Choudhary *et al.* 2004; Henrique *et al.* 2009). Numerical methods and their analysis can be found in Ewing *et al.* (1984), Wang *et al.* (2006), Wang (2008) and Jia *et al.* (2011). Although the physical models are similar in these applications, unique characteristics exist for each problem. The most important feature in the underlying model is its complex phase changes, condensation/evaporation and fibre absorption. Moreover, the water content distribution, which is mainly determined by the phase changes, plays an important role in the study of functional clothing, although the water equation is simple. Both single- and multi-component models are governed mathematically by a system of nonlinear, coupled and degenerate parabolic partial differential equations (PDEs). Mathematical analysis for these models is very limited. Some simplified models simulate the physical processes well in certain normal circumstances (Fan *et al.* 2000, 2004; Li & Zhu 2003). However, the simplification may result in physical inconservation in some critical circumstances and difficulty in mathematical analysis. In recent works (Li & Sun 2010; Ye *et al.* 2010), the existence of a solution for the steady-state case and the dynamic case of both single- and two-component models was proved, respectively. It was assumed in these works that the batting zone is wet everywhere, materials are non-hydrophobic (i.e. no fibre absorption) and the water equation is not involved in the system. To take the water content into account, the phase change should be defined more carefully.

In this paper, we first present more realistic formulations for the multi-component (air–vapour–water–heat) and multi-phase flow in porous textile media based on those previous works, which include the following two aspects: (i) A truncated Hertz–Knudsen equation is introduced to describe the condensation/evaporation for the general dry–wet case, which is more realistic in clothing assembly. (ii) In all previous works (Li & Zhu 2003; Fan *et al.* 2004; Li *et al.* 2005; Huang *et al.* 2008), the fibre absorption was described by an evolutionary equation along the fibre radius with a saturated Dirichlet boundary condition on the surface of a single fibre. This implies that the absorption on the fibre surface is enforced to be in saturation immediately for a given relative humidity. This is not realistic in physics as the absorption is a time-dependent process and also, this results in difficulty in mathematical analysis. Here, we introduce a Robin-type boundary condition, with which the absorption on the fibre surface is always smaller than the saturation absorption before the steady state is reached. Numerical simulations for a typical clothing assembly with a hydrophobic batting material are presented. Our numerical results show that the evaporation and condensation zones arise simultaneously in the batting area during almost the whole period, whereas in the same physical process, a clothing assembly with a non-hydrophobic batting material is in condensation everywhere after several minutes (Fan *et al.* 2004; Ye *et al.* 2010). Clearly, a suitable choice of both batting material and cover will influence the water content distribution and the evaporation/condensation interface significantly. The latter is the main factor to be considered in the design of functional clothing. Numerical results also show that the fibre-absorption rate with the new formulation is more reasonable, compared with the rate from previous works. Secondly, with the new setting, we present theoretical analysis for an incompressible case in the presence of the complex phase changes and liquid water. We prove the existence and uniqueness of the classical and positive solution for the system with a class of commonly used Robin-type boundary conditions under more general assumptions for the saturation pressure function and the absorption rate.

The rest of the paper is organized as follows. In §2, we present a review on multi-phase and multi-component models of heat and moisture transfer in textile materials with some modification to previous works. A more precise mathematical formulation with condensation/evaporation and fibre absorption is introduced by a system of nonlinear, coupled and degenerate parabolic PDEs. In §3, we present our numerical simulations on clothing assemblies consisting of viscose batting with two different covers: laminated and nylon, respectively, in comparison with experimental data given in Fan *et al.* (2000). A human sweating model is also simulated with a normal sweating rate and a sweating rate during exercise, respectively. In §4, we present our theoretical analysis for an incompressible case.

## 2. The mathematical models

Here, we consider a heat and moisture transport system only in a one-dimensional setting as the thickness of clothing assemblies is often smaller compared with sizes of the other two dimensions. We present a modified multi-phase and multi-component flow model mainly based on the work of Ye *et al.* (2008), which can also be viewed as a generalization of models developed earlier by Le *et al.* (1995), Fan *et al.* (2000, 2004), Li & Zhu (2003) and Choudhary *et al.* (2004).

### (a) A two-component and multi-phase flow model

From the conservation of mass and energy, the physical process can be described by
2.1
2.2
2.3
and
2.4where *ϵ* is the porosity of the media, *C*_{v}, *C*_{a} and *C*=*C*_{a}+*C*_{v} denote the vapour density, the air density and the density of the gas mixture (mol m^{−3}), respectively, *T* the absolute temperature (K), *W* the relative liquid water content, *C*_{f} the fibre absorption, *u*_{g} the velocity of the gas mixture, and the total heat capacity of the mixture and the volumetric heat capacity of gas, respectively, and *λ* the latent heat of evaporation/condensation (J mol^{−1}). The generalized Fick’s law has been used for the relative diffusion, where *D*_{g} is the molecular diffusion coefficient for air and vapour.

The velocity *u*_{g} of the gas mixture is given by Darcy’s law
2.5
where *k* and *k*_{rg} denote the permeability and relative permeability, respectively, and *μ*_{g} the dynamic viscosity, which usually is density-dependent for the compressible case (Mikelic 1991; Feng 1995). The pressure *P* is given by the ideal gas law *P*=*RCT*, with *R* being the universal gas constant.

### (b) Heat capacity and thermal conductivity

In most existing engineering models (see Le et al. 1995; Fan *et al.* 2000, 2004; Li & Zhu 2003; Choudhary *et al.* 2004), the total heat capacity was assumed to be a constant. In fact, the total heat flux is the sum of the heat flux in solid (fibre) and the heat flux in free space. The convective part in the latter is determined by the mass convective flux. The effective heat capacity is defined by
where and are the volumetric heat capacity of the gas mixture and the molar heat capacity of fibre/liquid water, respectively. More precisely, the volumetric heat capacity of gas depends upon the density of the gas, and usually
2.6
with , the molar heat capacity of gas, being a constant. Basically, for the underlying clothing assembly models, the difference between a constant used in those previous works and a variable in equation (2.6) does not influence the reality or results of these models in normal circumstances. However, the formulae above produce the more precise conservations such that the theoretical analysis becomes possible (Li & Sun 2010).

The effective heat conductivity *κ* is given by
2.7
where *κ*_{g} and *κ*_{s} are the thermal conductivities of the gas mixture and fibre liquid water.

### (c) The phase-change rate

The phase-change rate consists of two parts: the condensation/evaporation rate *Γ*_{ce} and the fibre-absorption rate . In most previous works (Fan *et al.* 2000, 2004; Li & Zhu 2003; Canuto & Cimolin 2010), the former was defined through the Hertz–Knudsen equation (Jones 1992)
2.8
where *P*_{v}=*RC*_{v}*T* is the partial vapour pressure and *P*_{sat} is the saturation pressure determined from experimental measurements (Fan *et al.* 2002, 2004).

Physically, the condensation occurs when *H*>0 (i.e. *P*_{v}>*P*_{sat}), the evaporation occurs only when *H*<0 and simultaneously the amount of liquid water in the core void is positive. For the clothing assembly cases presented in Fan *et al.* (2000, 2004) and Li & Zhu (2003), numerical simulations show that the physical process in the whole batting area is in condensation and no dry zone exists after a short period. In these models, the condensation/evaporation rate was defined directly by
2.9
A pure evaporation process was considered in Cimolin (2008) and Canuto & Cimolin (2010) for a motorcycle helmet model, in which
2.10
where *h*(⋅) is the Heaviside function. However, in many other cases, condensation and evaporation occur simultaneously in the batting area. Here, we define the condensation/evaporation rate by a general form
2.11
where
and *a*(*W*) is a monotonically increasing function of the water content with *a*(0)=0 and 0≤*a*(*W*)≤1. Formula (2.11) reduces to formulae (2.9) and (2.10), respectively, for the pure condensation case and the pure evaporation case. This also implies that the evaporation rate depends upon the liquid water left and no evaporation occurs when *W*=0.

On the other hand, the absorption process of fibre in a wet environment is described by an evolutionary equation (Haghi 2003; Li & Zhu 2003; Fan *et al.* 2004; Li *et al.* 2005; Huang *et al.* 2008)
2.12
where *R*_{f} is the radius of fibre, and *D*_{f}, the diffusion coefficient, may depend upon the water content (Haghi 2003; Li *et al.* 2005) and
2.13

Usually, the saturation (maximal) absorption of a fibre material, denoted by *W*_{f}′(*RH*), is dependent upon the relative humidity, *RH*:=*P*_{v}/*P*_{sat}, and is available via experimental measurements (Fan *et al.* 2000, 2004). In all the previous works (Fan *et al.* 2000, 2004; Li & Zhu 2003; Li *et al.* 2005; Huang *et al.* 2008), the absorption by fibre is described by the evolutionary equation (2.12) with the Dirichlet boundary condition
2.14
at *r*=*R*_{f} and the Neumann boundary condition
2.15
at *r*=0. The Dirichlet boundary condition (2.14) implies that the absorption on the fibre surface (*r*=*R*_{f}) is enforced to be in saturation immediately for a given relative humidity. In reality, even for a constant relative humidity, the absorption on the fibre surface is always smaller than the saturation absorption before the steady state is reached. Here, we introduce the following Robin-type boundary condition
2.16
replacing the Dirichlet boundary condition (2.14) on the fibre surface, where *h*_{f} denotes the corresponding resistance coefficient. A commonly used initial condition is
2.17
Obviously, when a constant relative humidity is given, the boundary value problems with the Dirichlet boundary condition (2.14) and the Robin-type boundary condition (2.16) have the same steady-state solution, *C*_{f}′=(*ρ*_{w}/*M*)*W*_{f}′(*RH*).

With the above initial and boundary conditions, the equation (2.12) has a unique classical solution for any given *RH*(*t*) at *r*=*R*_{f}. In addition, the solution can be expressed, in terms of Green’s function, as
2.18
where *G*(*r*,*t*;*R*_{f},*τ*) is Green’s function of equation (2.12) with the boundary and initial conditions (2.15)–(2.17). For *D*_{f} being a constant, we have further
where *J*_{0} is the Bessel function of order zero, *β*_{n}, *n*=1,2,…, are the roots of the equation
and *α*_{n} is given by

In general, we denote by **G**, the linear operator, from *W*_{f}′(*RH*) to *C*_{f}′(*R*_{f},*t*), i.e.
2.19
Then by equations (2.12) and (2.13),

Note that **G** is a linear operator and, by using the maximum principle for the parabolic equation (2.12) with the boundary and initial conditions (2.15)–(2.17), we have the following inequality:
2.20

### (d) Boundary–initial–interface conditions

As the thickness of cover layers is much smaller than that of the batting layer, the properties of heat and moisture transfer in the covers were often described by simple resistances to heat, vapour and air transfer. A class of commonly used Robin-type boundary conditions were introduced by Fan *et al.* (2004) to approximate the experimental setup in Fan *et al.* (2002), in which boundary conditions are defined by a combined simulation of cover layers and ambient environment. We use these Robin-type boundary conditions in this paper.

At the outer boundary,
2.21
where , *R*^{o}_{g} and *R*^{o}_{t} are the resistances of the outer cover to vapour, gas mixture and heat. *H*^{o}_{v} and *H*^{o}_{g} are the mass transfer coefficients in the outer environment for vapour and gas mixture, respectively, and *H*^{o}_{t} is the heat transfer coefficient in the outer environment for heat.

We assume that the background temperature is fixed at *T*^{o} (e.g. −20^{°}C) with a relative humidity of RH^{o} (e.g. 70%). The background vapour and air concentrations can be calculated by
2.22
respectively, where *P*_{atm} is the atmospheric pressure.

Similar to the outer boundary, we have the following boundary conditions on the inner boundary
2.23
where *R*^{i}_{v}, *R*^{i}_{g}, , *H*^{i}_{v}, *H*^{i}_{g}, *H*^{i}_{v} and *H*^{i}_{t} are defined analogously. In addition, we assume a relative humidity at the inner environment as 100 per cent owing to a constant evaporation, i.e. *RH*^{i}=1. The vapour concentration can be computed by
2.24
where *T*^{i} is the inner background temperature.

The initial conditions are given by 2.25

When the outer cover of the assembly is exposed to a cold environment under the freezing point and the inner cover is exposed to a higher temperature (e.g. human skin), there are a wet zone and a frozen zone in the batting area. A moving interface occurs at the position of the freezing temperature, *T*=273 *K*. As the amount of the water content (sweat) in assemblies is relatively small, the difference between the densities of water and ice will be neglected. Therefore, the interface arises mainly owing to the jump of the heat capacity (more precisely ), the heat conductivity *κ* and latent heat *λ* at the freezing point. By classical formulae of two-phase flow (Gupta 2003), the interface location *α*(*t*) satisfies the following interface equation
2.26
where [*u*] denotes the jump of *u* at the interface, *ρ*_{f} denotes the density of fibre and *λ*_{wi} the phase change latent heat from water to ice.

## 3. Numerical results and discussion

In this section, computational results are presented for the heat/moisture transfer in a clothing assembly with a 15-pile viscose batting sandwiched with two covering layers: laminated cover and nylon cover. Physical parameter values can be found in Fan *et al.* (2004) and Huang *et al.* (2008).

### Example 3.1

To compare with experimental measurements performed in Fan *et al.* (2002), we solve systems (2.1)–(2.4) with the boundary and initial conditions given in equations (2.21)–(2.25). Here, all numerical results are obtained by using the finite volume method presented in Ye *et al.* (2008) with Δ*t*=10s, Δ*x*=*L*/100 (1% of the batting length) and Δ*r*=*R*_{f}/50.

We present the numerical interfaces (figure 1) between evaporation and condensation zones (*Γ*_{ce}=0) and between the water and ice zones (*T*=0^{°}C (273 K)), respectively, for viscose batting with nylon cover and laminated cover. One can see clearly from the left subfigure that for the viscose material, the batting area is divided into two zones: evaporation zone and condensation zone, although the dry initial condition is enforced. Thus, the truncated Hertz–Knudsen formulation (2.8) is necessary. Numerical results for polyester material were presented in Fan *et al.* (2004) and Ye *et al.* (2010), in which the whole batting area quickly becomes wet from the dry initial condition after several minutes. Compared with the nylon cover, the clothing assembly with laminated cover contains a larger evaporation zone near the inner cover, which results in less water content being distributed in this area (as observed from figures 3 and 4). The latter is the main factor to be considered in the design of functional clothing. As the outer environmental temperature (*T*^{o}=−20^{°}C) is lower than the freezing point, there is a moving interface arising at *T*=0^{°}C (273 K). The interface for the problem with the laminated cover goes faster than that with the nylon cover, which implies that the latter keeps the human body warmer.

In figure 2, we present numerical results of the fibre absorption *C*_{f} at *x*=0.5*L* obtained by solving the evolutionary equation (2.12) with the saturated Dirichlet boundary conditions (2.14) used in previous works (Fan *et al.* 2004; Ye *et al.* 2008) and with the proposed Robin-type boundary condition (2.16), respectively. Clearly, the absorption rate from those previous works is much higher than that from our proposed formulation as the absorption process is time-dependent and enforcing the saturated absorption on the surface of fibre may not be realistic.

For completeness, we present numerical results at 8 and 24 h (figures 3 and 4), respectively, in which *C*_{v}, *C*_{a}, *T* and *W* are the vapour concentration, air concentration, temperature and water content. The comparisons with experimental measurement of water content performed in Fan *et al.* (2002) are given in the last two subfigures. The total water content is positive as the fibre absorption occurs everywhere.

### Example 3.2

Finally, we test a clothing assembly with certain human sweating system. In this case, the vapour flux at the inner cover is defined by the amount of sweat provided by the human body. The boundary condition at the inner cover now is given by
3.1
3.2
and
3.3
where *s*_{w} denotes the local sweating rate, which is a function of the local skin temperature, the average of skin temperature and body temperature in general (Nadela 1983). We also assume that the inner side of the clothing assembly is sealed off from the air supplies.

Here, we consider two different cases: normal human sweating case with sweating rate 3.4 and an extremal case during exercise with the rate 3.5 We present numerical results of the clothing assembly in the normal case at 24 h (figure 5) and numerical results in the extremal case at 1 h (figure 6).

In the normal case, as the sweating rate is relatively low and the vapour density is small, the condensation (*Γ*_{ce}≥0) occurs only near the outer cover, whereas the evaporation zone (*Γ*_{ce}≤0) takes almost the whole batting area. The water content, mainly owing to the fibre absorption, is distributed increasingly in the batting area with a high accumulation near the outer cover. In the exercise case, owing to the increase in sweating rate, the water content is much larger than in the normal case and the water vapour has a strong condensation near the inner cover.

## 4. Incompressible case

When the global velocity is small, the problem can be considered approximately as an incompressible flow as in some other applications (Wang & Catton 2001; Cimolin 2008; Canuto & Cimolin 2010). In this case, the system with non-dimensionalization reduces to
4.1
4.2
4.3
4.4
with the boundary conditions
4.5
and the initial conditions
4.6
where *c*=*C*_{v}/*C* is the vapour concentration and for *l*=0,*L* and for *l*=0,*L*, *i*=*P*,*c*,*T* are positive constants which satisfy
4.7
The source is given by *Γ*=*Γ*_{ab}+*Γ*_{ce} and the absorption *Γ*_{ab} and the evaporation/condensation *Γ*_{ce} are defined by
4.8
and
4.9
respectively, where **G** is a linear operator that satisfies equation (2.20) and is the relative humidity with .

In this section, we present theoretical analysis for systems (4.1)–(4.6) with the complex phase change (4.8) and (4.9). First, we re-define *H* by
4.10
where *χ*(⋅) is a cut-off function defined by
Later, we will show that 0≤*c*≤1 from which we see *χ*(*c*)=*c*.

We assume that the resistance coefficient *η*(*T*) is a bounded and non-negative smooth function of *T*≥0 with *η*(0)=0. Based on the physical data, we also assume that the functions *g*(⋅) and *P*_{s}(⋅) are smooth, non-decreasing and non-negative functions in [0,1] and , respectively, such that *g*(0)=*P*_{s}(0)=0 and
4.11
For completeness, we define *P*_{s}(*T*)=0 for *T*<0; *g*(*R*_{H})=0 for *R*_{H}<0 and *g*(*R*_{H})=*g*(1) for *R*_{H}>1. Therefore, *g*(*R*_{H}) and **G**[*g*(*R*_{H})] are both bounded by *g*(1).

The physical parameters *κ*, *d*_{g}, *λ* and *β* are assumed to be positive constants. To simplify the notations, we denote by *E*_{0} a generic positive constant which depends solely upon the physical parameters involved in the equation as well as in the initial and boundary conditions. We present *a priori* estimates below for the solution of systems (4.1)–(4.6) in the domain *Ω*_{τ}=*Ω*×(0,*τ*) under the physical assumptions (4.7)–(4.11).

### (a) Non-negativity of solution

We multiply equation (4.1) by *P*^{−} and integrate the result to obtain
By the definition of *H* and *g*(*R*_{H}), we have *H*^{+}*P*^{−}≡*g*(*R*_{H})*P*^{−}≡0. Moreover, by noting the fact (*P*−*P*^{0})*P*^{−}|_{x=0}≤0 and (*P*−*P*^{L})*P*^{−}|_{x=L}≤0, ∥*P*^{−}_{x}∥_{L2(Ω)}=0 and so *P*≥0 in *Ω*_{τ}.

Similarly, multiplying equation (4.2) by *c*^{−} and integrating the result leads to
Owing to the presence of the cut-off function *χ*, we have and *g*(*R*_{H})*c*^{−}≡0. Again, as and , the above inequality reduces to
Using Gronwall’s inequality, we derive that *c*^{−}=0 and so *c*≥0 in *Ω*_{τ}. To prove *c*≤1, we let . Equation (4.1) minus equation (4.2) gives
The corresponding boundary conditions for *P* and *c* yield
and
The initial condition for is . By assumption (4.7), the above equation for has a structure similar to the equation for *c*. With the same argument, we can derive that i.e. in *Ω*_{τ}, which implies that 0≤*c*≤1. Thus *χ*(*c*)=*c*, i.e. the cut-off function *χ* in equation (4.10) can be removed.

To prove the positivity of temperature, we multiply equation (4.1) by *T* and subtract the result from equation (4.3) to get
4.12
Let *T*=e^{−γt}(*θ*+*δ*) with *γ* and *δ* being positive constants, then *θ* is the solution of the equation
4.13
with the boundary and initial conditions
4.14
where
We can choose such that the right-hand sides in equation (4.14) are non-negative. As
at those points where *θ*≤0, we can also choose
such that *S*≥0 at the points where *θ*≤0. Integrating equation (4.13) times *θ*^{−}, we obtain with the *δ* and *γ*,
4.15
Using Gronwall’s inequality, we find that *θ*^{−}≤0. In other words, *T*≥e^{−γτ}*δ* i.e. in *Ω*×(0,*τ*).

The non-negativity of the liquid water *W* can be proved analogously by multiplying equation (4.4) with *W*^{−} and integrating the result.

### (b) Energy estimates

We integrate equation (4.1) times *P* to get
which together with the inequalities and *η*(*T*)**G**[*g*(*R*_{H})]≤*E*_{0}*g*(1) leads to
4.16
Let
4.17
Again, multiplying equations (4.1) and (4.3) by −*h*_{1}(*T*) and *h*_{2}(*T*), respectively, and integrating the their sum give
4.18
where *h*_{4}(*T*)=*λh*_{2}(*T*)+*h*_{1}(*T*) and
For any positive integer
we have
We see from equation (4.18) that
By the definition of *Γ*, the above inequality reduces to
By inequality (2.20) and equation (4.16), we have further
4.19
To estimate the last term in equation (4.19), we introduce two Young dual functions. Let
for *y*,*z*≥0, where *P*^{−1}_{s}(⋅) defines the inverse function of *P*_{s}(⋅) and *ε*_{0} is a small positive constant. As
and *Φ*(*y*) is monotonically increasing, the function *Ψ* is well defined. Clearly,
Taking *y*=*a*(*W*)*H*^{−} and *z*=*h*_{4}(*T*)*T*^{−1/2} in the above inequality, we can see that
As *a*(*W*)*H*^{−}≤*P*_{s}(*T*), we have *Φ*(*a*(*W*)*H*^{−})≤*ε*_{0}*h*_{4}(*T*)*a*(*W*)*H*^{−}, then it is not difficult to prove that
The inequality (4.19) now reduces to
With a fixed *m*=*m*_{0} and , we arrive at
which together with Gronwall’s inequality produces
In particular, the above inequality with equation (4.16) implies that
4.20

Now, we rewrite equation (4.19) by
With Gronwall’s inequality and equation (4.20), we get
Integrating the above inequality with respect to *t*, we derive that ∥*T*∥_{Lm+1(Ω×(0,τ))}≤*E*_{0}, where *E*_{0} is independent of *m*. Letting , we obtain
4.21
The estimates (4.16) read
4.22

### (c) Regularity of solution

From the definition of the source *Γ* and the upper and lower bounds of *T*, we immediately get
4.23
From equation (4.1), we derive that . With the regularity of *P*_{x} and *Γ*, we can apply the estimates to equations (4.2) and (4.3) to obtain , for some large positive *q*. Moreover, we can see that , , . With the *a priori* estimates presented above, existence and uniqueness of classical solution can be obtained by a routine argument with the Leray–Schauder fixed-point theorem. We summarize the main result of this section in the following theorem.

### Theorem 4.1

*The system of equations (4.1)–(4.6) admits a unique classical solution (P,c,T,W) which satisfies
*
4.24
*and
*
4.25
*for some α∈(0,1).*

## 5. Conclusion

We have reformulated the heat–air–vapour–water transport model with a more precise description on the complex phase changes, condensation/evaporation and the fibre absorption. The new formulation maintains more physical conservation which makes the model more realistic and the mathematical analysis possible. In the present paper, we present only a one-dimensional model with theoretical analysis for the incompressible case. Existence for the non-isothermal, multi-phase and multi-component compressible flow in three-dimensional porous textile media requires more rigorous analysis and we present it in a subsequent work. In addition, for a full three-dimensional model with a high water content, gravitational effects should be considered.

## Acknowledgements

The authors would like to thank the anonymous referees for useful suggestions. The work of authors was supported in part by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 102409).

- Received February 22, 2011.
- Accepted June 23, 2011.

- This journal is © 2011 The Royal Society