Modeling of Annual Heat and Moisture Diffusion in a Multilayer Wall

Abstract

We present an 1D numerical model of heat, steam, and water transfer across a wall consisting of several layers of different materials. The model is the system of coupled diffusion equations for wall temperature; vapor pressure, and water concentration in material pores, with account of vapor condensation and water evaporation. The system of nonlinear PDEs is solved numerically using the finite difference method. The main objective of modeling is simulation of long-term behavior of building wall moisture distribution under influence of seasonal variations in atmospheric air temperature and humidity.

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1 Introduction

Simultaneous heat, steam, and moisture transfer with condensation and evaporation in porous materials is of practical importance in applications in civil engineering. The transport of water vapor across building walls and its possible condensation increases the thermal conductivity of the porous materials and may cause structural damage. Complexity of the phenomenon and the large variety of conditions under which it can proceed cause the development of new models and approaches to the problem in a lot of research works. In particular, we can refer the works [1–9].

The main goal of this paper is not the development of original model of the phenomenon, but the presentation of a sufficiently precious and stable numerical algorithm for solution of the model of coupled heat, steam, and moisture diffusion, invented by Fokin K.F. in his book [10]. The Fokin model is widely used in civil engineering practice in Russia for estimation of long term thermal and moisture behavior in building envelopes [11]. The model is phenomenological model based on simple, intuitive physical principles. The diffusion flux of each component is proportional to gradient of corresponding variable: temperature, partial pressure of steam, and water concentration. All parameters of the model can be measured experimentally. Evolution equation for each component is the diffusion equation. Mutual coupling between equations is performed by the dependence of diffusion coefficients on moisture, and trough the source terms related to the phase transitions. Main difficult in numerical solution of the model is connected with variability in time of regions of vapor condensation.

2 Mathematical Formulation of the Model

Model parameters. For definiteness, in this paper we will consider three-ply wall. The thickness of wall layers is denoted as $d_1$, $d_2$, and $d_3$ [m] respectively. Let the first layer is the external (outward building) and the third layer is the internal (inward building) layer of the wall. We will consider the fluxes of heat and moisture in the direction traversal to wall only, – the direction along x axis. The point $L_0=0$ is the outer edge of external layer, the points $L_1 = d_1$ and $L_2=L_1+d_2$ are the interface points between layers, and the point $L_3=L_2+d_3$ is the outer edge of internal layer.

The material of each layer is characterized by following parameters: density – $\rho$ [kg/m$^3$]; heat capacity – c [kJ/(kg$\cdot$K)]; coefficients of thermal – $\lambda$ [kJ/(h$\cdot$m$\cdot$K)]; vapor – $\mu$ [g/(h$\cdot$m$\cdot$Pa)]; and hydraulic – $\beta$ [g/(h$\cdot$m$\cdot$%)], conductivity. The parameters $\rho$, c, and $\mu$ are supposed to be constant in considered range of temperature T [K] and moisture volumetric concentration $\omega$ [%], while other parameters can depend on $\omega$. For given material the dependencies $\lambda (\omega )$ and $\beta (\omega )$ used in calculations are polynomial interpolations of experimental measurements. Under equilibrium conditions at constant temperature the relation between air humidity $\varphi$ and moisture $\omega$ is determined from experimentally measured sorption isotherm $\omega = o(\varphi )$.

The external layer of the wall contacts with atmospheric air, and the internal layer contacts with building interior air. The air temperature $T_\mathrm {ex}(t)$, $T_\mathrm {in}(t)$, and humidity $\varphi _\mathrm {ex}(t)$, $\varphi _\mathrm {in}(t)$ are given function of time t [h]. Heat exchange between wall and air is determined by coefficients $\alpha _\mathrm {ex}$ and $\alpha _\mathrm {in}$ [kJ/(h$\cdot$m$^2\cdot$K)]. Steam exchange at the wall borders is determined by coefficients $\gamma _\mathrm {ex}$ and $\gamma _\mathrm {in}$ [g/(h$\cdot$m$^2\cdot$Pa)].

Heat conductivity. The transversely heat transfer in each wall layer is described by the heat conduction equation for the material temperature T(t, x):

$$\begin{aligned} c\rho \frac{\partial T(t,x)}{\partial t}=\frac{\partial }{\partial x}\left( \lambda (\omega )\frac{\partial T(t,x)}{\partial x}\right) +Q(T,\omega ). \end{aligned}$$

(1)

The source term Q takes into consideration the latent heat of vapor condensation and water evaporation (the water-ice transitions are not included in the model).

At the wall borders and layer interfaces there are imposed boundary and conjugation conditions:

$$\begin{aligned} -\lambda (\omega )\frac{\partial T(t,x)}{\partial x}{\bigg \vert } _{x=L_0}=\alpha _\mathrm {ex}\left( T_\mathrm {ex}(t)-T(t,x)\right) {\bigg \vert }_{x=L_0}; \end{aligned}$$

(2)

$$\begin{aligned}&T(t,L_i-0)=T(t,L_i+0),\nonumber \\&\lambda (\omega )\frac{\partial T(t,x)}{\partial x}{\bigg \vert }_{x=L_i-0}=\lambda (\omega )\frac{\partial T(t,x)}{\partial x}{\bigg \vert }_{x=L_i+0},\ i=1,2; \end{aligned}$$

(3)

$$\begin{aligned} -\lambda (\omega )\frac{\partial T(t,x)}{\partial x}{\bigg \vert }_{x=L_3}=\alpha _\mathrm {in}\left( T(t,x)-T_\mathrm {in}(t)\right) {\bigg \vert }_{x=L_3}. \end{aligned}$$

(4)

Vapor and liquid water conductivity. It is supposed that the water in wall material can be in three forms: water vapor (steam), and mobile liquid water in material pores, and immobile absorbed water rigidly connected with material skeleton. The concentration of absorbed water $\omega$ depends upon local air humidity in pores $\varphi (t,x)$ and assumed to be equal to equilibrium concentration $o(\varphi )$. If $\varphi < 1$, the material contains only steam and absorbed water, the mass of absorbed water in unit volume of material is equal to $0.01\omega \rho$.

The air humidity $\varphi$ is defined as ratio $\varphi =e/E(T)$, where e [Pa] is partial pressure of water vapor, and E(T) is pressure of saturated vapor at air temperature T. For calculation of E we use the approximation formula:

$$\begin{aligned} E(T)={\left\{ \begin{array}{ll} 4.688(1.486+T/100)^{12.3}, &{} T<0,\\ 288.58(1.098+T/100)^{8.02}, &{} T\ge 0. \end{array}\right. } \end{aligned}$$

(5)

The function $\varphi (t,x)$ is supposed to be continuous function of both variables. So, at every t one can define the subset $V_t\subseteq (L_0,L_3)$, $V_t = \{x: \varphi (t,x)<1\}$, and the complementary subset $W_t=(L_0,L_1)\setminus V_t$.

In subset $V_t$ the moisture moves in the form of steam only, and its motion is governed by the vapor conduction equation

$$\begin{aligned} \xi (\omega )\rho \frac{\partial e(t,x)}{\partial t}=\mu \frac{\partial ^2e(t,x)}{\partial x^2}. \end{aligned}$$

(6)

(Thermo-diffusion of steam, that is the diffusion induced by temperature gradient, does not included in the model.) Here the parameter $\xi$ defines the ‘vapor capacity’ of material and can be estimated by the equation

$$\begin{aligned} \xi (\omega )=\frac{do(\varphi )}{d\varphi }. \end{aligned}$$

(7)

As noted above, Eq. (6) is defined in the subset $V_t$. If the point $L_0$ and/or $L_3$ are the boundary points of $V_t$, then the boundary conditions of convective exchange of steam between air in material pores and surrounding air are imposed similar to (2), (4) with replaced T by e, $\lambda$ by $\mu$, and $\alpha$ by $\gamma$. If any of interface points $L_1$, $L_2$ belongs to $V_t$, then the conjugation condition assuming continuity of vapor pressure and flux is imposed similar to (3).

In subset $W_t$ the material pores contain liquid water together with water vapor, and it is supposed that between water and vapor there keeps up the dynamic equilibrium. That is the partial pressure of vapor in $W_t$ equals to pressure of saturated vapor, $e(t,x)\equiv E(T(t,x))$ for all $x\in W_t$. To estimate the volumetric concentration of liquid water w [%] we propose the following formula

$$\begin{aligned} w(t,x)=\omega (t,x)-o(1), \end{aligned}$$

(8)

where o(1) is maximal concentration of absorbed water, corresponding to $\varphi =1$.

Diffusive motion of liquid water is described by equation

$$\begin{aligned} 10\rho \frac{\partial w(t,x)}{\partial t}=\frac{\partial }{\partial x}\left( \beta (\omega )\frac{\partial w(t,x)}{\partial x}\right) +\mu \frac{\partial ^2E(T)}{\partial x^2}. \end{aligned}$$

(9)

Here numerical coefficient ‘10’ appears due to different dimensions of parameters (kg in $\rho$, g in $\mu$ and $\beta$, and % in w). Second term in the right hand side corresponds to vapor condensation or water evaporation depending on its sign. To avoid extra model complication the condition of water impermeability at border points of $W_t$ is accepted

$$\begin{aligned} \frac{\partial w}{\partial x}{\bigg \vert }_{\partial W_t}=0. \end{aligned}$$

(10)

If any of points $L_1$, $L_2$ is inner point of $W_t$, then there is imposed the condition of continuity of water concentration and water flux, similar to (3). In the points, separating subsets $V_t$ and $W_t$, we suppose continuity of pressure e(t, x).

Latent heat of phase transition. The term $\nu =\mu (\partial ^2E(T)/\partial x^2)$ [g/(h$\cdot$m$^3)$] in Eq. (9) represents the rate of change of water concentration due to water evaporation or vapor condensation, that is in result of phase transition. The phase transitions are accompanied by release or absorption of latent heat, depending on the sign of $\nu$. The latent heat of phase transition is accounted in Eq. (1) via the source term Q, numerical value of which is calculated with the formula

$$\begin{aligned} Q = {\left\{ \begin{array}{ll} q_\mathrm {L}\nu , &{} x \in W_t, \\ 0, &{} x\in V_t, \end{array}\right. } \end{aligned}$$

(11)

where $q_\mathrm {L}$ is the specific heat of water vaporization (= 2.26 kJ/g).

3 Numerical Scheme

The system of Eqs. (1), (6), and (9), with all additional conditions listed above is solved numerically with the help of the finite differences method. To approximate the space derivatives with finite differences we define the uniform grid of N nodes $x_k$ on the interval $(L_0,\,L_3)$: $x_k=L_0+(k-0.5)h$, $k=1,2,\dots ,N$, $h=(L_3-L_0)/N$. The time derivative is approximated on the time grid $t_0=0,t_1,t_2,\dots ,t_n,\dots$, with variable time step $\tau _n=t_{n+1}-t_n$, $n\ge 0$. Each function f(t, x) is replaced its grid approximation $f_k^n=f(t_n,x_k)$. Below, for brevity, we will drop the upper index denoting $f_k^n$ as $f_k$, and $f_k^{n+1}$ as $\hat{f}_k$.

Finite difference equations. To derive the finite difference equations we use implicit scheme and heat and mass balance method. The finite-difference counterparts of Eqs. (1), (6), and (9) are the following

$$\begin{aligned} hc_k\rho _k\frac{\hat{T}_k-T_k}{\tau }=\hat{\varLambda }_k^-(\hat{T}_{k-1}-\hat{T}_k) - \hat{\varLambda }_k^+(\hat{T}_k-\hat{T}_{k+1}) + \hat{Q}_k,\ k=1,\dots ,N. \end{aligned}$$

(12)

$$\begin{aligned} h\hat{\xi }_k\rho _k\frac{\hat{e}_k-e_k}{\tau }=M_k^-(\hat{e}_{k-1}-\hat{e}_k) - M_k^+(\hat{e}_k-\hat{e}_{k+1}),\ x_k \in V_n. \end{aligned}$$

(13)

$$\begin{aligned} 10h\rho _k\frac{\hat{w}_k-w_k}{\tau }=\hat{B}_k^-(\hat{w}_{k-1}-\hat{w}_k) - \hat{B}_k^+(\hat{w}_k-\hat{w}_{k+1}) + \hat{\nu }_k,\ x_k \in W_n. \end{aligned}$$

(14)

The coefficients $\varLambda$ are calculated as follows

$$\begin{aligned}&\varLambda _1^-=\frac{2\alpha _\mathrm {ex}\lambda _1}{\alpha _\mathrm {ex}h+2\lambda _1},\ \varLambda _k^-=\frac{2\lambda _{k-1}\lambda _k}{h(\lambda _{k-1}+\lambda _k)},\ k=2,\dots ,N;\nonumber \\&\varLambda _k^+=\frac{2\lambda _k\lambda _{k+1}}{h(\lambda _k+\lambda _{k+1})},\ k=1,\dots ,N-1,\ \varLambda _N^+=\frac{2\lambda _N\alpha _\mathrm {in}}{2\lambda _N+\alpha _\mathrm {in}h}. \end{aligned}$$

(15)

The coefficients M and B are calculated similar to $\varLambda$. The term $\nu$ is approximated by second difference

$$\begin{aligned} \nu _k=M_k^-(E_{k-1}-E_k)-M_k^+(E_k-E_{k+1}), \end{aligned}$$

(16)

where $E_k=E(T_k)$, if $x \in W_n$; $E_k=e_k$, if $x \in V_n$; $E_0=e_\mathrm {ex}(t_n)$, $E_{N+1}=e_\mathrm {in}(t_n)$.

Assuming that $T_0=T_\mathrm {ex}(t_n)$, $T_{N+1}=T_\mathrm {in}(t_n)$; $e_0=e_\mathrm {ex}(t_n)$, $e_{N+1}=e_\mathrm {in}(t_n)$, one obtains the closed system of 2N algebraic equations with 2N unknowns. First N unknowns are the values of temperature $\hat{T}_k$ at grid nodes $x_k$ in time moment $t_{n+1}$. Other N unknowns are the values of steam partial pressure $\hat{e}_k$ at nodes $x_k\in V_n$ or the values of water concentration $\hat{w}_k$ at nodes $x_k\in W_n$.

Iterative solution. The system of finite-difference Eqs. (12), (13), (14) is the non-linear system because of the scheme is implicit and the coefficients $\lambda$ and $\beta$ depend on the system solution. To solve the non-linear system we use an iterative procedure.

Using known solution at time $t_n$ we calculate the coefficients $\varLambda$ and B (coefficients M are constant) and substitute them in system (12), (13), (14). Solving the resulting linear system we obtain first approximation to solution at moment $t_{n+1}$, $\{T_k^{(1)},\;e_k^{(1)},\;w_k^{(1)}\}$. Then the first approximation is used in the same way to obtain second approximation $\{T_k^{(2)},\;e_k^{(2)},\;w_k^{(2)}\}$, and so on. The iterations are ended when the relative difference between two successive approximations becomes sufficiently small.

Designation of subset $W_t$. At initial time moment $t_0$ all nodes $x_k$, where initial moisture concentration $\omega _k$ is not less than maximal adsorption concentration, $\omega _k\ge o(1)$, are included in subset $W_0$. Further, the domain W is re-designated after each successful time step. The nodes $x_k$ from $V_n$, where the vapor pressure $e_k\ge E_k$ pass from $V_n$ in $W_{n+1}$, while the nodes $x_k$ from $W_n$, where $w_k < 0$, pass from $W_n$ in $V_{n+1}$. There are no restrictions on shape and size of subset V (and, therefore, of W).

Choice the value of time step $\tau$. To avoid possible numerical instability time step $\tau$ is limited by some empirical maximal value $\tau _\mathrm {max}$. Initial time step is assigned less than $\tau _\mathrm {max}$ in several orders of magnitude. Time step with assigned value of $\tau$ is considered as successful, if (i) the number of iterations in solution of system (12), (13), (14) is not exceed assigned maximal number of iterations, and (ii) relative difference between solution at current time step and solution at previous time step is sufficiently small. If current time step is not successful, the value of $\tau$ is diminished by definite factor, and step is repeated until the time step becomes successful. If the number of successive successful steps exceeds assigned number, the value of $\tau$ increases by definite factor.

4 Numerical Example

As an example we consider a three-ply wall. The external layer of wall is a thin stucco, and other two layers of equal thickness are concrete and mineral wool as heat insulator. The main goal of our example is to demonstrate the difference (well-known in construction practice) in heat and moisture behavior in two walls consisting of these layers. First wall is ‘correct’ wall with layers order: stucco - heat insulator - concrete; and second wall is ‘incorrect’ with layers order: stucco - concrete - heat insulator.

Fig. 1.

(a) – sorption isotherm $o(\varphi )$, (b) – coefficient of water conductivity $\beta (\omega )$.

In this paper we use some ‘typical’ values of material parameters, which can be found in numerous literature. The accepted values of constant material parameters are presented in Table 1 (dimensions were specified in text above). Two additional parameters k and $\lambda _0$ are used for calculation of material coefficient of thermal conductivity $\lambda$, $\lambda =\lambda _0+k\omega$.

Table 1. Material parameters.

Fig. 2.

Space-time plot of temperature in ‘correct’ (a), and ‘incorrect’ (b) wall.

Fig. 3.

Space-time plot of water concentration in ‘correct’ (a), and ‘incorrect’ (b) wall.

The graphs of the sorption isotherms and the coefficient $\beta$ in dependence on $\omega$ are shown in Fig. 1, where the experimental measurements, depicted by markers, are connected via polynomial interpolations. In the building interior the constant air temperature and humidity are supposed, $T_{in}=20\,^\circ$C, $\varphi _{in}=0.6$. The temperature and humidity of external air simulate their seasonal variations for Moscow region. The heat exchange coefficients $\alpha _{in}=31.4$, $\alpha _{ex}=85.8$, and the vapor exchange coefficients $\gamma _{in}=0.075$, $\gamma _{ex}=0.750$.

The initial time moment corresponds to the middle of July with average daily air temperature of 19.3 $^\circ$C. The initial temperature distribution in wall is linear between 19.3 $^\circ$C at outer and 20 $^\circ$C at inner wall surface. The initial moisture concentration is assigned for every wall layer. In our example it was $0.9\times o(1)=3.597$ for stucco, $1.5\times o(1)=1.775$ for concrete, and $1.5\times o(1)=2.813$ for mineral wool. That is, the wall is waterlogged and have to dry.

Figure 2 shows, that (as it should be expected) the fall of temperature in winter time occurs mostly in the heat insulator layer. Hence, during winter the concrete temperature is high in ‘correct’ wall, and low in ‘incorrect’ wall. The vapor condensation proceeds in that regions, where the vapor pressure e becomes higher than the pressure of saturated vapor E. Such a condition is mainly created at low temperatures. Therefore, there are much more possibilities for condensation in ‘incorrect’ wall than in ‘correct’ wall. This is confirmed by Fig. 3.