# Model

## 1. Assumptions

The hygrothermal equations specified in the following clauses contain the following assumptions:

• constant geometry, no swelling and shrinkage;

• no chemical reactions are occurring;

• latent heat of sorption is equal to latent heat of condensation/evaporation;

• no change in material properties by damage or ageing;

• local equilibrium between liquid and vapour without hysteresis;

• moisture storage function is not dependent on temperature

• temperature and barometric pressure gradients do not affect vapour diffusion.

The development of the equations is based on the conservation of energy and moisture. The mathematical expression of the conservation laws are the balance equations. The conserved quantity changes in time, only if it is transported between neighbouring control volumes.

Heat conservation shall be expressed by

$\left(c_{\mathrm{m}} \cdot \rho_{\mathrm{m}}+c_{\mathrm{w}} \cdot w\right) \cdot \frac{\partial T}{\partial t}=-\frac{\partial\left(q_{\mathrm{sens}}+q_{\mathrm{lat}}\right)}{\partial x}$

The increase of the moisture content of a control volume shall be determined by the net inflow of moisture. The moisture flow rate equals the sum of the vapour flow rate and the flow rate of liquid water.

$\begin{array}{l} \frac{\partial w}{\partial t}=-\frac{\partial g}{\partial x} \\ g=g_{v}+g_{1} \end{array}$

The relative humidity shall be defined by the following equation:

$\varphi=\frac{p_{\mathrm{v}}}{p_{\mathrm{v}, \text { sat }}(T)}$

The pressure acting on the water inside a building material due to the capillary forces is different from the pressure of the surrounding air. The difference is called suction.

$p_{\mathrm{suc}}=p_{\mathrm{a}}-p_{\mathrm{w}}$

The suction of the pore water is related to the relative humidity of the surrounding air by the Kelvin equation:

$p_{\mathrm{suc}}=-\rho_{\mathrm{w}} R_{\mathrm{H} 2 \mathrm{O}} T \ln \varphi$

The relation between the state variables $\varphi, p_{\mathrm{v}}, p_{\mathrm{suc}}, T$ and the moisture content of a building material is defined by the moisture storage function. The moisture storage function of a building material shall be expressed either as the moisture content as a function of suction (suction curve), $w\left(p_{\text {suc }}\right),$ or as the moisture content as a function of the relative humidity (sorption curve), $w(\varphi)$

## 2. Transport of heat and moisture

### 2.1. Heat transport

#### 2.1.1. Heat transport inside materials

Heat transport shall be composed of sensible and latent components. Sensible heat transport shall be calculated with Fourier’s law with a thermal conductivity which depends on moisture content. $q_{\text {sens }}=-\lambda(w) \cdot \frac{\partial T}{\partial x}$

#### 2.1.2. Heat transport across boundaries

The heat flow from the surrounding environment into the construction consists of convection, shortwave radiation from the sun and longwave radiation exchange with sky and surrounding surfaces.

Sensible heat flow from each surrounding environment to the building envelope shall be given by:

$q_{\text {sens }}=h\left(T_{\text {eq }}-T_{\text {surf }}\right)$

The heat transfer coefficient and the equivalent temperature are:

$\begin{array}{l} h=h_{\mathrm{C}}+h_{\mathrm{r}} \\ T_{\mathrm{eq}}=T_{\mathrm{a}}+\frac{1}{h}\left(E_{\mathrm{sol}} \alpha_{\mathrm{sol}}+\left(T_{\mathrm{r}}-T_{\mathrm{a}}\right) h_{\mathrm{r}}\right) \end{array}$

The radiative and convective heat exchanges are represented by an equivalent temperature. Other means of accounting for these effects may be used.

If the surface temperature is known it can be used as a boundary condition.

Latent heat flow to and from the boundaries is proportional to the vapour flow rate at the surfaces (see 4.2.2 ).

### 2.2. Moisture transport

#### 2.2.1. Moisture transport inside materials

Moisture is transported by capillary forces and diffusion. The transport equations shall be formulated using the partial vapour pressure and the suction as the driving potentials.

$\begin{array}{l} g=g_{v}+g_{w} \\ g_{v}=\frac{1}{\mu(\varphi)} \delta_{0} \frac{\partial p_{v}}{\partial x} \\ g_{w}=K\left(p_{s u c}\right) \frac{\partial p_{s u c}}{\partial x} \end{array}$

The temperature dependence of the liquid conductivity may be neglected.

 For the liquid transport alternative potentials such as relative humidity, moisture content and temperature may be used, if the transport coefficients are transformed and the interfaces between two materials are handled in such a way that the suction and the partial vapour pressure are still continuous functions across the interface.

#### 2.2.2. Moisture transport across material interfaces

Internal interfaces

The details of the contact between two layers of building materials can have a large influence on the liquid moisture transport. Additional coatings, such as adhesives, can also modify the diffusive moisture transport Small air gaps between materials and the modification of pore structure at material interfaces, because of chemical reaction products, reduce the capillary water transport across the interface. The influence of the interface on the liquid moisture flow may be described by a moisture resistance, $R_{w},$ which is defined by:

$g_{w}=\frac{\Delta p_{\text {suc }}}{R_{w}}$
External interfaces

Coatings and paints can cause additional resistance for water uptake and drying. The impact on diffusion can be described by an additional moisture resistance at the surface, $s_{\mathrm{d}, \mathrm{s}} / \delta_{0},$ defined by:

$g_{\mathrm{v}}=\frac{\delta_{0}}{s_{\mathrm{d}, \mathrm{s}}}\left(p_{\mathrm{v}, \mathrm{a}}-p_{\mathrm{v}, \mathrm{s}}\right)$

where $s_{\mathrm{d}, \mathrm{s}}$ is the equivalent vapour diffusion thickness of the interface, in $\mathrm{m}$ The uptake of driving rain is limited by the amount of water which can be absorbed by the material at the surface:

$g_{\mathrm{w}, \mathrm{max}}=\left(K \frac{\partial p_{\mathrm{suc}}}{\partial x}\right)$

so that:

$g_{\mathrm{w}}=\min \left(g_{\mathrm{p}}, g_{\mathrm{w}, \max }\right)$

where $g_{\mathrm{p}}$ is the water available for absorption from precipitation.