## Abstract

Heat and moisture transport plays an important role in many engineering areas. In many sweat transport systems, such as clothing assembly, the moisture concentration (or sweat) is relative small and the air concentration reaches a steady state quickly. In this paper, a quasi-steady-state multi-component and multi-phase model for heat and moisture transport in porous textile materials with phase change is proposed. An analytic form of the air concentration is obtained in terms of the mixture gas (vapour and air) concentration and temperature. The new model is described in the form of a single-component flow with an extra air resistance (permeability), involving only the vapour concentration (or pressure), temperature and water content. The existence of the classical positive solutions of the corresponding steady-state model is proved. Two types of clothing assemblies are investigated numerically. The comparisons among the experimental measurements and numerical results of the fully dynamic model, the proposed quasi-steady-state model and steady-state model are also presented. Numerical results show that the proposed quasi-steady-state model is realistic and less complicated.

## 1. Introduction

Simultaneous heat and moisture transfer in porous media attracts considerable attention as it can be found in a wide range of industrial and engineering domains, such as food industry (Huang *et al.* 2007), building materials (Ogniewicz & Tien 1981; Choudhary *et al.* 2004) and more recently, textile materials (Smith & Twizell 1985; Fan *et al.* 2000; Li & Zhu 2003; Cheng & Wang 2008; Huang *et al.* 2008). Here we focus our attention on clothing assemblies. In this application, heat and moisture transfers are coupled in rather complicated mechanisms. Water vapour moves through the clothing assembly by convection and diffusion, which are induced by the pressure gradient and concentration gradient. The heat is transferred by conduction in all phases (liquid, fibre and gas) and convection in gas. Phase changes occur in the form of evaporation/condensation and/or sublimation and fibres absorb the water vapour owing to their chemical/physical nature.

There are two types of approaches to modelling the heat and moisture transport in porous textile materials in general, a single component (vapour) model (Ogniewicz & Tien 1981; Le *et al.* 1995; Fan *et al.* 2000, 2004; Li & Zhu 2003) and a multi-component (vapour and air) model (Huang *et al.* 2008; Ye *et al.* 2008; Henrique *et al.* 2009). Ogniewicz & Tien (1981) studied a single-component model of heat and vapour transport in fibrous insulation, where the condensation in wet zones was a given function of temperature related to the saturated vapour. In their work (Ogniewicz & Tien 1981), steady-state solutions of temperature and vapour concentration were obtained numerically while convection with a constant vapour velocity and a linear conduction/diffusion process were included. In practice, it is difficult to estimate the vapour velocity precisely. Motakef & El-Masri (1986) took the same condensation approach as in Ogniewicz & Tien (1981) and an extra water diffusion was added in their model when the water content exceeded a critical liquid content. Recently, Li & Zhu (2003) studied the heat and moisture transport in textile materials and presented a model including both phase change and fibre absorption. Fan and his co-workers studied more general dynamic models in textile materials. A typical application of their models is a clothing assembly consisting of a thick porous fibrous batting sandwiched by two thin cover fabrics. A further improved model of coupled heat and moisture transfer with phase change, mobile condensates and thermal radiation in clothing assemblies was presented in Fan *et al.* (2004). The moisture movement was introduced with Darcy’s velocity, which was proportional to the pressure gradient. The thermal radiation equation was solved analytically, in terms of temperature. However, in all these models only a single-component (vapour) fluid flow was concerned and the air motion was ignored. Numerical results (Fan *et al.* 2004) showed that the vapour velocity in the single-component model was larger than expected. More recently, a multi-phase and multi-component flow model for textile materials with phase change was proposed in Huang *et al.* (2008). The model was a generalization of a single-component model used in the previous study, taking both air motion and vapour motion into account, since air and vapour behaved differently and the air could provide extra resistance to the vapour movement. Ye *et al.* (2008) made several further modifications to this model. The movement of the water, convection and diffusion (capillary effect) of the liquid water, was neglected in the model, since the water content of the human sweating system in normal circumstances is very low and the water stays at the condensation site in general. To describe the air motion, an extra air equation was introduced in these multi-component models, which were a little more complicated compared with single-component models.

Numerical simulations done in Huang *et al.* (2008) and Ye *et al.* (2008) for a class of clothing assemblies showed that there were different time scales for different mechanisms in the multi-phase and multi-component heat and moisture transfer model. Since phase changes occur between vapour and water, the air equation is source-free. Therefore, the air concentration reached a fixed profile in a very short period. This motivated us to study a quasi-steady state of existing models. The purpose of this paper is twofold. Firstly, we propose a quasi-steady-state model that consists of a steady-state air equation and dynamic-state equations for other components. An analytic formula of the air concentration is given in terms of the mixture gas (air and vapour) concentration and temperature (or mixture pressure). The new model is described by a system of nonlinear parabolic equations only involving the vapour concentration (or pressure), temperature and water content. Moreover, the system can be written in a form of the single-component model as studied in Choudhary *et al.* (2004), Fan *et al.* (2004), Le & Ly (1995), Le *et al.* (1995) and Ogniewicz & Tien (1981) with an additional (air) permeability *K*, which, in other words, represents the air resistance to the vapour motion. For the clothing assembly case, we can show that *K*≈1/*P*_{a} by an asymptotic analysis, where *P*_{a} is the air pressure. Numerical results illustrate the difference between the fully dynamic model and the proposed quasi-steady-state model on the air concentration and temperature vanishes after a short period, around 10 min. The difference in vapour concentration is less than 2 per cent after 30 min. Secondly, we study the corresponding steady-state solution with a class of physical boundary conditions and prove the existence of classical positive solutions under a more general assumption for the saturation pressure function. The existence of a weak solution of a single-component model was presented in Li *et al.* (submitted) and the existence of a weak solution of the proposed quasi-steady-state model may be obtained along the line of argument in Li *et al.* (submitted). Theoretical analysis for the fully dynamic multi-phase and multi-component flow model has not been explored, although some isothermal models and certain different non-isothermal models from other applications were studied (Mikelic 1991; Feng 1995; Straughan 2006; Cheng & Wang 2008).

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. In §3, we derive an analytic formula of the air concentration from the steady-state air equation with appropriate physical boundary conditions and present a new quasi-steady state model. In §4, we present our theoretical analysis of the steady-state model by the Leray–Schauder fixed point theorem. Numerical experiments are conducted in §5, comparing numerical results from the fully dynamic model, the proposed quasi-steady-state model, the steady-state model and experimental data in Fan *et al.* (2002). Numerical results show that the proposed model is realistic and less complicated.

## 2. A multi-phase and multi-component model

We consider a typical clothing assembly of three layers, a porous batting sandwiched by an inner layer and an outer layer of thin covering fabrics, see figure 1 for the schematic diagram. The outer cover of the assembly is exposed to a cold environment with fixed temperature and relative humidity while the inner cover is exposed to a mixture of air and vapour at higher temperature and higher relative humidity. The mathematical model for the clothing assembly has been studied by many authors. The multi-phase and multi-component flow model studied here is mainly based on the work in Ye *et al.* (2008), which can be viewed as a generalization of models developed earlier in Choudhary *et al.* (2004), Fan *et al.* (2000, 2004), Le & Ly (1995) and Li & Zhu (2003).

From the conservation of mass and energy, the physical process can be described by
2.1
2.2
2.3and
2.4
Here *C*_{v} is the vapour concentration (mol m^{−3}), *C*_{a} the air concentration (mol m^{−3}), *T* the temperature (K) and *W* the relative liquid water content (%). The generalized Fick’s law has been used for the binary multi-component gas mixture (vapour and air). *C*=*C*_{v}+*C*_{a} is the mixture gas (molar) concentration, *τ*_{c} the tortuosity of the assembly for the air–vapour diffusion, *D*_{g} the molecular diffusion coefficient for air and vapour, *ρ* the density of fibre and *M* the molecular weight of water. *λ* is the latent heat of evaporation/condensation in the wet zone while in the frozen zone, it represents the latent heat of sublimation.

The porosities with liquid water content (*ϵ*) and without liquid water content (*ϵ*′) are related by
2.5
where *ρ*_{w} is the density of water.

The vapour–air mixture velocity (volumetric discharge) is given by Darcy’s law
2.6
where *k* is the permeability, *k*_{rg} and *μ*_{g} are the relative permeability and the viscosity of the gas mixture, respectively. *P*=*P*_{v}+*P*_{a} is the mixture pressure. From the ideal gas assumption, *P*_{α}=*RC*_{α}*T*, *α*=*a*,*v*.

The effective heat conductivity is defined in terms of the mixture gas conductivity *κ*_{g} and mixture solid/liquid (water, ice and fibre) conductivity *κ*_{s} by
where the mixture solid/liquid conductivity *κ*_{s} is a combination of the water/ice conductivity *κ*_{w} and fibre conductivity *κ*_{f},
2.7
In the same way, the effective heat capacity *C*_{vt} is defined by
where *C*_{vg} is the volumetric heat capacity of the mixture gas, *C*_{vs} the specific heat capacity of mixture solid/liquid and
2.8
where *C*_{vf} and *C*_{vw} are the heat capacity of fibre and water/ice, respectively.

The (molar) rate of phase change per unit volume *Γ* is defined by the Hertz–Knudsen equation (Jones 1992)
2.9
for condensation and evaporation (molar rate), where *R* is the universal gas constant, *R*_{f} the radius of fibre and *E* the non-dimensional phase change coefficient. The saturation pressure *P*_{sat} is determined from experimental measurements (figure 2).

For many applications in textile industries, vapour concentration is one of the most important factors while it is relative small, compared with the air concentration. A single-component model, which concerned the vapour transport in the void only, has been investigated by many authors (Le & Ly 1995; Le *et al.* 1995; Fan *et al.* 2000, 2004; Li & Zhu 2003; Choudhary *et al.* 2004). The vapour equation involving diffusion and convection processes was given by
2.10
The equations (2.3) and (2.4) together with the above equation describe a multi-phase and single-component flow in porous textile materials, as studied in Choudhary *et al.* (2004), Fan *et al.* (2004), Le & Ly (1995) and Le *et al.* (1995). Moreover, when the convection is dominant, the vapour equation reduces to
2.11
where *u*_{v}=(*k*/*μ*)(∂*P*_{v}/∂*x*). The single-component flow model is simpler than the one given in equations (2.1)–(2.4) since only the vapour, temperature and water content are involved in the system. However, the vapour velocity in the single-component model is larger than expected (Fan *et al.* 2004), owing to the neglect of the air motion. A more realistic single-component model will be derived in §3.

## 3. Quasi-steady-state solutions

Numerical results presented in Huang *et al.* (2008) for the multi-phase and multi-component flow model defined in equations (2.1)–(2.4) showed that there were several separations of time scales for the physical processes involved. The convection time scale for vapour and air was comparable to the diffusive time scale. Phase change was the most dominant process that occurs at a much faster time scale. The air concentration quickly reached a steady state, particularly in a cold environment, where low or no air ventilation was assumed at the inner side of the clothes to reduce heat loss. In this case, the air concentration reached a fixed profile after several minutes. Here, we propose a quasi-steady-state model, in which the air concentration is described by a steady-state equation and the vapour concentration (or pressure), temperature and water content are in unsteady states.

Based on the above description, the air equation in steady state is given by
3.1
with the boundary conditions
and
where *V* _{G}=*kk*_{rg}/*μ*_{g}, *δ*_{G}=*D*_{g}/*τ*_{c}, *R*^{o}_{a} the resistance of outer cover to heat, *H*^{o}_{a} the mass transfer coefficients of outer cover for air and *C*^{o}_{ar} the ratio of air concentration to the mixture gas concentration in a cold environment. Since the amount of water in clothing assemblies is very small, we assume that these physical parameters involved in the air equation of steady state are constants, independent of the amount of water.

Integrating the air equation (3.1) from 0 to *L*, we have
By the boundary conditions of the air equation,
which implies the Dirichlet boundary condition
For simplicity, we denote *P*|_{x=L}, *C*|_{x=L} and *C*_{a}|_{x=L} by *P*_{L}, *C*_{L} and *C*_{aL} in the following.

Integrating the air equation and noting the flux boundary condition, we get
and moreover,
Integrating it again gives
and finally, we obtain
3.2
or
which in turn leads to
3.3
In terms of the above formula, a single-component system (mixed gas concentration–temperature) is described by
3.4and
3.5
with Darcy’s law (2.6) and *C*_{a}=*Cg*(*P*), where

In many physical problems, one often chooses the mixture pressure *P* as a main variable instead of *C*. A system of pressure–temperature–water equations is defined by
3.6and
3.7
and the water equation (2.4), where . The corresponding boundary conditions are given by
where are the heat transfer and moisture transfer coefficients of inner cover, respectively, the resistances of inner cover to heat and vapour, *T*^{o} and *P*^{o} the temperature and pressure in cold environments and *T*^{i} the temperature near the human body.

On the other hand, by equation (3.3),
and the mixed gas velocity is written by
3.8
where
3.9
Note that *u*_{v}=(*k*/*μ*)(∂*P*_{v}/∂*x*) represents the vapour velocity with the neglect of air in the void and the *u*_{g} represents the mixed gas velocity. Equation (3.8) shows that the additional air permeability *K* should be included when air exists in the void. Since *V* _{G}/*δ*_{G}∼*O*(1) and *P*∼10^{5}>>1,
i.e. air permeability is proportional to the reciprocal of the air pressure *P*_{a}. The single-component model in Choudhary *et al.* (2004), Fan *et al.* (2004) and Le & Ly (1995) takes *K*=1. However, for clothing assembling problems, the amount of the vapour (sweat) is relatively small compared with the air. In this case, *K*≈1/*P*_{a}∼10^{−5}.

Theoretical analysis for a single-component model (vapour–temperature) was given in a recent work (Li *et al.* submitted), in which the existence of a weak solution was obtained. The model studied in Li *et al.* (submitted) can be viewed as a special case of the proposed quasi-steady-state system with *g*(*P*)=0. The existence of weak solution of the quasi-steady-state system can be obtained by an analogous approach. However, the existence of a classical solution is unknown. Owing to the practical interest in a long time period (8–24 h), we study the existence of a classical positive solution of the corresponding steady-state system in the following section. Numerical results presented in Fan *et al.* (2004) and Ye *et al.* (2008) and in §5 show that both temperature and pressure (or mixed gas concentration) approximately reach a steady state quickly.

## 4. Steady-state equations

We assume that all the physical parameters, the heat conductivity, heat capacity and permeability, are positive constants. With non-dimensionalization and the neglect of the water equation, the steady-state system of the proposed model in equations (3.4)–(3.5) is described by
4.1and
4.2
where *P*=*CT* and is defined by
*g*(*P*) is defined in equation (3.2) and . The corresponding boundary conditions in the last section reduce to
4.3
For the clothing assembling,
Since *C*^{o}_{ar} is the ratio of the air concentration and mixed gas concentration in the cold environment, *C*^{o}_{ar}<1. Based on the experimental data in figure 2, we assume further that *p*_{s}(*T*) is a smooth and increasing function of *T* such that
4.4

### (a) Existence of a classical positive solution

We prove the existence of classical solutions to the systems (4.1) and (4.2) by using the Leray–Schauder fixed point theorem. We present the theorem below. The proof is a only a slight variation of the proof given in Gilbarg & Trudinger (1998) and we omit it.

### Lemma 4.1 (Leray–Schauder)

*Let X be a Banach space. Let* *be a continuous and compact map. We assume that there exists a positive constant M such that ∥**A**(z,0)∥≤∥z∥ for ∥z∥≥M. If the subset*
*is bounded in X, then the mapping* *A**(⋅,1) has a fixed point z∈X, i.e.* *A**(z,1)=z.*

To construct the map **A**, we rewrite the systems (4.1) and (4.2) by
4.5
Obviously, if (*P*,*T*) is a classical solution of the above system, then it is a classical solution of the systems (4.1) and (4.2).

Let *X*=*C*[0,*L*]. For any given (*u*_{0},*v*_{0})∈*X*^{2} and *σ*∈[0,1], we let *P*_{0}=*e*^{u0} and *T*_{0}=*e*^{v0}. Then,
We define *P* to be the solution of the following quasi-linear elliptic equation
4.6
Where
With *P* in hand, we define *T* to be the solution of the equation
4.7

By the definition, *f*(*P*) is an increasing function of *P* for *P*≥*P*_{m}, where *g*(*P*_{m})=1. Let , ,
and
By the strictly increasing of *f*(*P*) for *P*≥*P*_{m}, we have *γ*_{1}>*P*_{m} and *γ*_{2}>0.

To prove the uniform lower boundedness of *T* and *P*, we consider the modified equation
4.8
with
From the definition of *γ*_{1}, the right-hand side of equation (4.8) is non-negative. By the theory of quasi-linear elliptic equations (Ladyzhenskaya & Ural’tseva 1968), the equation (4.8) has a unique solution *P*∈*H*^{1}(0,*L*). By multiplying equation (4.8) by (*P*−*γ*_{1})^{−} and integrating the resulting equation, we obtain
which implies (*P*−*γ*_{1})^{−}=0, hence *P*≥*γ*_{1}. It follows that equation (4.8) reduces to equation (4.6). Similar argument can be used to derive *T*≥*γ*_{2}.

Thus *P* and *T* are uniformly bounded in *H*^{1}(0,*L*)↪↪*C*[0,*L*] when *u*_{0},*v*_{0} are uniformly bounded in *C*[0,*L*].

Let and . We denote the mapping from (*u*_{0},*u*_{0},*σ*) to (*u*,*v*) by **A**. It is obvious the mapping is continuous and compact. When *σ*=0, from the maximum principle, *P*^{o}≤*P*≤*P*^{i} and *T*^{o}≤*T*≤*T*^{i}, i.e. the bounds of *u*, *v* are independent of *u*_{0},*v*_{0}. This implies ∥**A**(*u*_{0},*v*_{0},0)∥_{X2}≤∥(*u*_{0},*v*_{0})∥_{X2} when ∥(*u*_{0},*v*_{0})∥_{X2} is large enough.

### Theorem 4.2.

*There exists a positive classical solution ( C,P) for the system (4.1)–(4.3), which satisfies* 1−

*g*(

*P*)>0.

By lemma 4.1, it suffices to prove that all the functions (*u*,*v*)∈*X*^{2} satisfying
4.9
are uniformly bounded from above in *X*^{2}.

### (b) A priori estimate

Let *P*=*e*^{u}, *T*=*e*^{v} and *C*=*P*/*T*. Firstly, we present an a priori estimate for *P*, *T* and *C*. In the remainder of this section, we denote by *E*_{c} a generic positive constant independent of *σ*. From equations (4.6) and (4.7), we see that *P*,*T*,*C*∈*C*^{2}[0,*L*] satisfy the following boundary value problems
4.10
and
4.11
Adding equation (4.10) times (*λ*_{M}+*T*) to equation (4.11), we have
Integrating the above equation from 0 to *x* gives
from which
4.12
Integrating the above equation from *x* to *L* again, we obtain
4.13

Let . If *x*_{M} is one of the endpoints, from the boundary conditions, . If *x*_{M}∈(0,*L*), by noting the fact that *T*′(*x*_{M})=0 and *T*′′(*x*_{M})≤0, from equation (4.11), we derive that
Moreover, by the assumption (4.4),
which, together with equation (4.13), leads to
Since is strictly increasing, we obtain *T*(*x*_{M})≤*E*_{c} from the above inequality, i.e. ∥*T*∥_{C[0,L]} is uniformly bounded with respect to *σ*∈[0,1].

Applying the maximum principle to equation (4.10), we observe that *P* is uniformly bounded in *C*[0,*L*].

Secondly, we prove the uniform boundedness of (*u*,*v*). Clearly, we only need to prove that *P*,*T* are both uniformly bounded from below at 0. Let
We consider the following two different cases.

*Case* 1. *x*_{T},*x*_{P}∈(0,*L*). Noting the fact that at a minimum point of a smooth function, its first-order derivative is zero and its second-order derivative is non-negative, we see from equations (4.10) and (4.11) that
4.14
Since *f*(*P*) and are increasing functions for *P*>*P*_{m} and *T*>0, respectively, we have
The *P* and *T* achieve their minimum at the same point, say *x*_{m}. If , from equation (4.12), the function is increasing and therefore,
which shows *P*(*x*_{m})=*P*(0) and *T*(*x*_{m})=*T*(0). Similarly, if , then
which implies that either
or
i.e. one of *T* and *P* is uniformly larger than a positive constant. From equation (4.14), we observe that the other one is also uniformly larger than a positive constant.

*Case* 2. One of *x*_{T} and *x*_{P} is an endpoint. We only consider the case *x*_{T}=0. Noting the fact *T*′(*x*_{T})≥0 and the boundary condition in equation (4.11), we get *T*(*x*)≥*T*(*x*_{T})≥*T*^{i}. In this case, if *x*_{P}∈(0,*L*), we can see from equation (4.14) that *f*(*P*(*x*_{P}))≥*E*_{c}; otherwise *x*_{P} is also an endpoint, a similar argument shows that *P*(*x*_{p})≥*P*^{o}. Moreover, by the analysis of §4*a*, we have *P*≥*γ*_{1}>*P*_{m}.

Now, we conclude that *P*, *T* are uniformly bounded both from above and from below at 0, which implies that and are uniformly bounded in *C*[0,*L*]. By lemma 4.1, there exists a fixed point (*u*,*v*) for mapping **A**(⋅,1). Thus, there exists a positive classical solution (*P*,*T*) for the systems (4.1)–(4.3) and *P*≥*γ*_{1}>*P*_{m}, which implies *f*(*P*)=*P*(1−*g*(*P*))>0. The proof of theorem 4.2 is complete.

## 5. Numerical experiments

In this section, we present our numerical results for a textile assembly with a porous batting sandwiched by two covering layers, which was investigated in Ye *et al.* (2008) for the fully dynamic multi-phase and multi-component model. A polyester batting and two cover materials, laminated and nylon, are tested here. The values of these physical parameters are presented in table 1 for these two covers and in table 2 for the polyester batting. Other parameter values can be found in Fan *et al.* (2004) and Huang *et al.* (2008). To compare with the experimental data, the vapour flux type boundary conditions are used in our numerical test. The initial conditions are given by
The dynamic multi-phase and multi-component model in equations (2.1)–(2.4) and the quasi-steady-state model in equations (3.6) and (3.7) and equation (2.4) are solved by a splitting finite-volume method presented in Ye *et al.* (2008) with Δ*t*=1*s*, Δ*x*=*L*/100 (1% of the batting length). The finite-volume method with a smaller time step, Δ*t*=0.01 s, and spatial step, Δ*x*=*L*/500, is further tested to confirm our numerical results. The computation is performed on a Blade 1000 Sun-workstation in double precision.

We present in figure 3 numerical results of the fully dynamic multi-phase and multi-component model with the laminated cover at 8 h and 24 h, in which *C*_{v}, *C*_{a}, *T*, and *W* are the vapour concentration, air concentration, temperature, source term and water content, respectively. The comparisons with experimental measurements of water content done in Fan *et al.* (2002) are given in the last two subfigures. Numerical results of the quasi-steady-state model and steady-state model with the laminated cover are presented in figure 4, in which the comparisons with experimental measurement of water content are also included. To see the difference between solutions of the fully dynamic model and the proposed quasi-steady-state model clearly, we present in figure 5 the comparisons between numerical results of these two models at *t*=5, 10, 30 and 60 mins, where *C**_{v}, *C**_{a} and *T** denote the vapour concentration, air concentration and temperature from the quasi-steady-state model. Numerical results for the assembly with the nylon cover are presented in figures 6–8.

In both cases, the vapour concentration and temperature are monotonically decreasing from the inner cover to the outer cover owing to higher vapour concentration in the warm environment while the air concentration is monotonically increasing. This validates that the air flow could provide extra resistance to the vapour flux. We can see from figure 5 that for the solutions the dynamic model and the quasi-steady-state model, reach the same steady state while air concentration and temperature reach the state more quickly. The relative difference in air concentration and temperature between these two models is less than 2 per cent after only 5 min. This shows that the proposed quasi-steady-state model is a good approximation to the fully dynamic multi-phase and multi-component model, while the solutions in the first 5 min are less important owing to some uncertainty of initial condition in practical cases. The relative difference in vapour concentration between these two models is less than 2 per cent after 30 min. The profiles of the water contents from these two models are close enough although there is some difference owing to the accumulation of the water content.

## 6. Conclusions

In this paper, we have studied the heat and moisture transfer in porous textile materials. Based on our numerical observations and the nature of practical textile assemblies, the dynamic air equation in existing models (Huang *et al.* 2008; Ye *et al.* 2008) is replaced with a steady-state air equation from which we derive an analytic form of the air concentration in terms of the temperature and mixture gas concentration. Then a new temperature–pressure–water system is proposed. Compared with existing single-component temperature–vapour–water models in Choudhary *et al.* (2004), Fan *et al.* (2004), Le & Ly (1995), Le *et al.* (1995), Ogniewicz & Tien (1981), our quasi-steady-state model is more realistic. Numerical results show that the difference between the solution of the fully dynamic multi-phase and multi-component model and the solution of the proposed quasi-steady-state model is negligible after a short period. Since the influx and outflux of vapour and heat depend upon the values of vapour and temperature at the inner and outer boundaries, a balance among the vapour/heat influx, the vapour/heat generated by phase changes and the vapour/heat loss is reached quickly. Moreover, we have presented a theoretical analysis for the corresponding steady-state model and showed the existence of a classical positive solution.

Multi-component flow models can also be found in many other areas, such as the oil industry (Chen *et al.* 2000) and food industry (Huang *et al.* 2007). Numerical studies in these areas have been done extensively (Douglas *et al.* 1997; Wang *et al.* 2002, 2006; Wang 2008). There may be certain different time scales for different components in these models. It is possible to extend the quasi-steady-state model to these applications. Since in many practical cases, the media may not be distributed uniformly and gravitational effect does exist, one future work is to study the heat and moisture transport in a three-dimensional porous textile medium.

## Acknowledgements

The work of the authors was supported in part by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project no. CityU 101906).

The authors would like to thank Professors J. Fan and H. Huang for helpful discussions, and the referees for their useful suggestions.

## Footnotes

- Received January 9, 2010.
- Accepted March 5, 2010.

- © 2010 The Royal Society