# Heat and Moisture Transfer in Building Modeling

 COMSOL uses the models described below in [CEN2007].

## 1. Heat transfer

[CEN2007] proposes the following equations for heat transfert in building materials

$\begin{array}{c} \left(\rho C_{p}\right)_{\mathrm{eff}} \frac{\partial T}{\partial t}+\nabla \cdot \mathbf{q}=Q \\ \mathbf{q}=-\left(k_{\mathrm{eff}} \nabla T+L_{\mathrm{v}} \delta_{\mathrm{p}} \nabla\left(\phi p_{\mathrm{sat}}\right)\right) \end{array}$

It considers the building material as a porous medium in local thermal equilibrium in which the following mixing rules apply:

$\left(\rho C_{p}\right)_{\mathrm{eff}}\left(\text { SI unit: } J /\left(\mathrm{m}^{3} \cdot \mathrm{K}\right)\right)$ is the effective volumetric heat capacity at constant pressure, defined to account for both solid matrix and moisture properties:

$\left(\rho C_{p}\right)_{\mathrm{eff}}=\rho_{\mathrm{s}} C_{p, \mathrm{s}}+w C_{p, \mathrm{w}}$

where

• $\rho_{\mathrm{s}}$ (SI unit: $\mathrm{kg} / \mathrm{m}^{3}$ ) is the dry solid density,

• $C_{p, \mathrm{s}}$ (SI unit: $\mathrm{J} /(\mathrm{kg} \cdot \mathrm{K})$ ) is the dry solid specific heat capacity,

• $w$ (SI unit: $\mathrm{kg} / \mathrm{m}^{3}$ ) is the water content given by a moisture storage function, and

• $C_{p, \mathrm{w}}(\text { SI unit: } \mathrm{J} /(\mathrm{kg} \cdot \mathrm{K}))$ is the water heat capacity at constant pressure.

$k_{\mathrm{eff}}(\text { SI unit: } \mathrm{W} /(\mathrm{m} \cdot \mathrm{K}))$ is the effective thermal conductivity, defined as a function of the solid matrix and moisture properties:

$k_{\mathrm{eff}}=k_{\mathrm{s}}\left(1+\frac{b w}{\rho_{\mathrm{s}}}\right)$

where

• $\left.k_{\mathrm{s}} \text { (SI unit: } \mathrm{W} /(\mathrm{m} \cdot \mathrm{K})\right)$ is the dry solid thermal conductivity and

• $b$ (dimensionless) is the thermal conductivity supplement.

This definition neglects the contribution due to the volume fraction change of the moist air.

The heat source due to moisture content variation is expressed as the vapor diffusion flow multiplied by latent heat of evaporation:

$L_{\mathrm{v}} \delta_{\mathrm{p}} \nabla\left(\phi p_{\mathrm{sat}}\right)$

where

• $L_{\mathrm{v}}$ (SI unit: $\mathrm{J} / \mathrm{kg}$ ) is the latent heat of evaporation,

• $\delta_{\mathrm{p}}$ (SI unit: $\mathrm{s}$ ) is the vapor permeability,

• $\phi$ (dimensionless) is the relative humidity, and

• $p_{\text {sat }}$ (SI unit: $\mathrm{Pa}$ ) is the vapor saturation pressure.

## 2. Moisture transport in Building

[CEN2007] proposes the following moisture transport in building materials

$\xi \frac{\partial \phi}{\partial t}+\nabla \cdot\left(-\xi D_{\mathbf{w}} \nabla \phi-\delta_{\mathbf{p}} \nabla\left(\phi p_{\text {sat }}(T)\right)\right)=G$

This equation models the moisture transfer as the sum of the capillary moisture flux:

$-D_{\mathbf{w}} \nabla(w(\phi))=-D_{\mathbf{w}} \frac{\partial w}{\partial \phi} \nabla \phi=-\xi D_{\mathbf{w}} \nabla \phi$

and the vapor diffusion flux:

$\delta_{\mathrm{p}} \nabla p_{\mathrm{v}}(T)=\delta_{\mathrm{p}} \nabla\left(\phi p_{\mathrm{sat}}(T)\right)$

with the following material properties, fields, and source:

• $\xi\left(\text { SI unit: } \mathrm{kg} / \mathrm{m}^{3}\right)$ is the moisture storage capacity.

• $\cdot \delta_{\mathrm{p}}(\text { SI unit: } s)$ is the vapor permeability.

• $\phi$ (dimensionless) is the relative humidity.

• $p_{\text {sat }}(\text { SI unit: } \mathrm{Pa})$ is the vapor saturation pressure.

• $\cdot T(\text { SI unit: } K)$ is the temperature.

• $D_{\mathrm{w}}\left(\text { SI unit: } \mathrm{m}^{2} / \mathrm{s}\right)$ is the moisture diffusivity.

• $G\left(\text { SI unit: } \mathrm{kg} / \mathrm{m}^{3} \cdot \mathrm{s}\right)$ is the moisture source.

## 3. Theory for Moisture Transport in Air

For the moment, we do not consider moisture transport in air but models do exist.

## 4. Heat and Moisture transfert coupling

We have the following equations for heat and moisture transport, see [CEN2007]:

$\begin{array}{l} \left(\rho C_{p}\right)_{\mathrm{eff}} \frac{\partial T}{\partial t}+\nabla \cdot\left(-k_{\mathrm{eff}} \nabla T-L_{\mathrm{v}} \delta_{\mathrm{p}} \nabla\left(\phi p_{\mathrm{sat}}\right)\right)=Q \\ \quad \xi \frac{\partial \phi}{\partial t}+\nabla \cdot\left(-\xi D_{\mathrm{w}} \nabla \phi-\delta_{\mathrm{p}} \nabla\left(\phi p_{\mathrm{sat}}\right)\right)=G \end{array}$

where:

• $\left(\rho C_{p}\right)_{\text {eff }}\left(\text { SI unit: } J /\left(\mathrm{m}^{3} \cdot \mathrm{K}\right)\right)$ is the effective volumetric heat capacity at constant pressure.

• $T(\text { SI unit: } K)$ is the temperature.

• $k_{\mathrm{eff}}(\text { SI unit: } \mathrm{W} /(\mathrm{m} \cdot \mathrm{K}))$ is the effective thermal conductivity.

• $L_{\mathrm{v}}(\text { SI unit: } \mathrm{J} / \mathrm{kg})$ is the latent heat of evaporation.

• $\delta_{\mathrm{p}}(\text { SI unit: } s)$ is the vapor permeability

• $\phi(\text { dimensionless })$ is the relative humidity.

• $p_{\text {sat }}(\text { SI unit: } \mathrm{Pa})$ is the vapor saturation pressure.

• $Q\left(\text { SI unit: } W / m^{3}\right)$ is the heat source.

• $\xi\left(\text { SI unit: } \mathrm{kg} / \mathrm{m}^{3}\right)$ is the moisture storage capacity.

• $D_{\mathrm{w}}\left(\text { SI unit: } \mathrm{m}^{2} / \mathrm{s}\right)$ is the moisture diffusivity.

• $G\left(\text { SI unit: } \mathrm{W} / \mathrm{m}^{3}\right)$ is the moisture source.

## 5. Required data

The used model requires the following hygrothermal properties:

Sorption isotherm

A function that links the water content $w$ to the relative humidity $\phi$. The sorption isotherm is unique for each material. It indicates, for a relative humidity value, the corresponding water content at a given constant temperature:

$w = f(\phi)\\ \phi = f(w)$
Water retention curve

A function that links the water content $w$ to the suction pressure $p_\text{suc}$:

$w = f(p_\text{suc})\\ p_\text{suc} = f(w)$

The sorption isotherm and the water retention curve are determined experimentally.

Water storage capacity

The water storage capacity can be determined by deriving the sorption function $w=f(\phi)$:

$\xi = \frac{dw}{d\phi}$
Vapor permeability

For each material, the vapor permeability is given as a function of the temperature and the relative humidity:

$\delta_p = f(T, \phi)$
Moisture diffusivity

The moisture diffucivity $D_w$ can be determined using the following formula [CEN2007]:

$D_w (w) = - \frac{K\left(p_\text{suc}\right)} { \frac{\partial w}{\partial p_\text{suc}}}$

where

$K\left(p_\text{suc}\right)$ $[s/m$] is the material liquid water permeability.

Effective heat capacity
$\left(\rho C_{p}\right)_{\mathrm{eff}}=\rho_{\mathrm{s}} C_{p, \mathrm{s}}+w C_{p, \mathrm{w}}$

where

• $\rho_{\mathrm{s}}$ $[\mathrm{kg} / \mathrm{m}^{3}$] is the dry solid density,

• $C_{p, \mathrm{s}}$ $[\mathrm{J} /(\mathrm{kg} \cdot \mathrm{K})$] is the dry solid specific heat capacity,

• $w$ $[\mathrm{kg} / \mathrm{m}^{3}$] is the water content given by a moisture storage function, and

• $C_{p, \mathrm{w}}$ [$\mathrm{J} /\left(\mathrm{kg} \cdot \mathrm{K}\right)$] is the water heat capacity at constant pressure.

Effective thermal conductivity
$k_{\mathrm{eff}}=k_{\mathrm{s}}\left(1+\frac{b w}{\rho_{\mathrm{s}}}\right)$

where

• $k_{\mathrm{s}}$ [$\mathrm{W} /(\mathrm{m} \cdot \mathrm{K})$] is the dry solid thermal conductivity

• $\rho_{\mathrm{s}}$ $[\mathrm{kg} / \mathrm{m}^{3}$] is the dry solid density

• $b$ [-] is the thermal conductivity supplement.It indicates by how many percent the thermal conductivity increases per mass percent of moisture. Examples are available in [KUN1995] (Table 4, p.26).

Vapor saturation pressure

[KUN1995] (equation 50) gives an empirical value of $p_\text{sat}$, as function of the temperature:

$p_\text{sat} = 611\cdot \exp\left(\dfrac{a \cdot T}{T_0+T}\right)$

with :

• $a=22.44$ and $T_0=272.44$°C if $T<0$°C

• $a=17.08$ and $T_0=234.18$°C if $T\geqslant 0$°C

## 6. Solution Strategy

### 6.1. Variationnal problem

Fisrt of all, we focus on the first equation

$\left(\rho C_{p}\right)^n_{\mathrm{eff}} \frac{T^{n+1}-T^n}{\Delta t}+ \nabla \cdot\left(-k^n_{\mathrm{eff}} \nabla T^{n+1}-L^n_{\mathrm{v}} \delta^n_{\mathrm{p}} \left(\nabla\phi\right) p^n_{\mathrm{sat}} + \phi^{n+1} \left( \nabla p^n_{\mathrm{sat}}\right) \right)=Q^{n+1}$

We multiply by a test function $v$ of the temperature space $X_h$

$\int_\Omega \left(\rho C_{p}\right)^n_{\mathrm{eff}} \frac{T^{n+1}}{\Delta t} v+ \int_\Omega \nabla \cdot\left(-k^n_{\mathrm{eff}} \nabla T^{n+1} \right) v + \int_\Omega \nabla \cdot\left(-L^n_{\mathrm{v}} \delta^n_{\mathrm{p}} p^n_{\mathrm{sat}} \left(\nabla\phi^{n+1}\right) - L^n_{\mathrm{v}} \delta^n_{\mathrm{p}} \left( \nabla p^n_{\mathrm{sat}}\right) \phi^{n+1} \right) v = \int_\Omega \left( \left(\rho C_{p}\right)^n_{\mathrm{eff}} \frac{T^n}{\Delta t} + Q^{n+1} \right) v \\$
$\int_\Omega \left(\rho C_{p} \right)^n_{\mathrm{eff}} \frac{T^{n+1}}{\Delta t} v+ \int_\Omega \left(k^n_{\mathrm{eff}} \nabla T^{n+1} \right) \cdot \nabla v +\int_{\partial\Omega} \left(-k^n_{\mathrm{eff}} \nabla T^{n+1} \cdot n \right) v +\int_\Omega L^n_{\mathrm{v}} \delta^n_{\mathrm{p}} p^n_{\mathrm{sat}} \left(\nabla\phi^{n+1} \right) \cdot \nabla v +\int_{\partial\Omega} \left(-L^n_{\mathrm{v}} \delta^n_{\mathrm{p}} p^n_{\mathrm{sat}} \left(\nabla\phi^{n+1} \right) \cdot n \right) v +\int_\Omega \left(L^n_{\mathrm{v}} \delta^n_{\mathrm{p}} \left( \nabla p^n_{\mathrm{sat}}\right) \phi^{n+1} \right) \cdot \nabla v +\int_{\partial\Omega} \left( -L^n_{\mathrm{v}} \delta^n_{\mathrm{p}} \left( \nabla p^n_{\mathrm{sat}}\right) \phi^{n+1} \cdot n \right) v = \int_\Omega \left( \left(\rho C_{p}\right)^n_{\mathrm{eff}} \frac{T^n}{\Delta t} + Q^{n+1} \right) v \\$

Now the second one :

$\xi^n \frac{\phi^{n+1}-\phi^n}{\Delta t}+\nabla \cdot\left(-\xi^n D^n_{\mathrm{w}} \nabla \phi^{n+1}-\delta^n_{\mathrm{p}} (\nabla\phi^{n+1}) p^n_{\mathrm{sat}} - \delta_p\phi^{n+1}\nabla p^n_\mathrm{sat}\right)=G^{n+1}$

We multiply by a test function $q$ of the humidity space $M_h$ :

$\int_\Omega \left(\xi^n \frac{\phi^{n+1}}{\Delta t}\right)q +\int_\Omega\nabla \cdot\left(-\xi^n D^n_{\mathrm{w}} \nabla \phi^{n+1}\right)q +\int_\Omega \nabla \cdot\left(-\delta^n_{\mathrm{p}} (\nabla\phi^{n+1}) p^n_{\mathrm{sat}}\right)q +\int_\Omega \nabla \cdot\left(-\delta_p\phi^{n+1}\nabla p^n_\mathrm{sat}\right)q =\int_\Omega \left(\xi^n\frac{\phi^n}{\Delta t}+G^{n+1}\right) q$
$\int_\Omega \left(\xi^n \frac{\phi^{n+1}}{\Delta t}\right)q +\int_\Omega\left(\xi^n D^n_{\mathrm{w}} \nabla \phi^{n+1}\right)\cdot\nabla q +\int_{\partial\Omega}\left(-\xi^n D^n_{\mathrm{w}} \nabla \phi^{n+1}\cdot n\right)q +\int_\Omega \left(\delta^n_{\mathrm{p}} (\nabla\phi^{n+1}) p^n_{\mathrm{sat}}\right)\nabla q +\int_{\partial\Omega} \left(-\delta^n_{\mathrm{p}} (\nabla\phi^{n+1}) p^n_{\mathrm{sat}}\cdot n\right)q +\int_\Omega \left(\delta_p\phi^{n+1}\nabla p^n_\mathrm{sat}\right)\nabla q +\int_{\partial\Omega} \left(-\delta_p\phi^{n+1}\nabla p^n_\mathrm{sat}\cdot n\right)q =\int_\Omega \left(\xi^n\frac{\phi^n}{\Delta t}+G^{n+1}\right) q$

At time $n+1$, the variationnal problem is : Find $(T^{n+1},\phi^{n+1})\in X\times M$ such as

$\begin{array}{l} a(T^{n+1},v) + b(\phi^{n+1},v) = f(v) & \forall v\in X\\ c(T^{n+1},q) + d(\phi^{n+1},q) = g(q) & \forall q\in M \end{array}$

with :

• $a(T,v) = \displaystyle\int_\Omega\left(\rho C_{p}\right)^n_{\mathrm{eff}} \frac{T^{n+1}}{\Delta t} v + \int_\Omega \left(k^n_{\mathrm{eff}} \nabla T^{n+1} \right) \cdot \nabla v - \int_{\partial\Omega} \left(k^n_{\mathrm{eff}} \nabla T^{n+1} \cdot n \right) v$

• $b(T,v) = \displaystyle \int_\Omega L^n_{\mathrm{v}} \delta^n_{\mathrm{p}} p^n_{\mathrm{sat}} \left(\nabla\phi^{n+1}\right) \cdot \nabla v - \int_{\partial\Omega} \left(L^n_{\mathrm{v}} \delta^n_{\mathrm{p}} p^n_{\mathrm{sat}} \left(\nabla\phi^{n+1}\right) \cdot n \right) v + \int_\Omega \left(L^n_{\mathrm{v}} \delta^n_{\mathrm{p}} \left( \nabla p^n_{\mathrm{sat}}\right) \phi^{n+1} \right) \cdot \nabla v - \int_{\partial\Omega} \left(L^n_{\mathrm{v}} \delta^n_{\mathrm{p}} \left( \nabla p^n_{\mathrm{sat}}\right) \phi^{n+1} \cdot n \right) v$

• $c(T,q) = 0$

• $d(T,v) = \displaystyle\int_\Omega \left(\xi^n \frac{\phi^{n+1}}{\Delta t}\right)q + \int_\Omega\left(\xi^n D^n_{\mathrm{w}} \nabla \phi^{n+1}\right)\cdot\nabla q - \int_{\partial\Omega}\left(\xi^n D^n_{\mathrm{w}} \nabla \phi^{n+1}\cdot n\right)q + \int_\Omega \left(\delta^n_{\mathrm{p}} (\nabla\phi^{n+1}) p^n_{\mathrm{sat}}\right)\nabla q - \int_{\partial\Omega} \left(\delta^n_{\mathrm{p}} (\nabla\phi^{n+1}) p^n_{\mathrm{sat}}\cdot n\right)q + \int_\Omega \left(\delta_p\phi^{n+1}\nabla p^n_\mathrm{sat}\right)\nabla q - \int_{\partial\Omega} \left(\delta_p\phi^{n+1}\nabla p^n_\mathrm{sat}\cdot n\right)q$

• $f(v) = \displaystyle\int_\Omega\left( \left(\rho C_{p}\right)^n_{\mathrm{eff}} \frac{T^n}{\Delta t} + Q^{n+1} \right) v$

• $g(q) = \displaystyle\int_\Omega \left(\xi^n\frac{\phi^n}{\Delta t}+G^{n+1}\right) q$

The discret problem is :

$\begin{bmatrix} A & B \\ 0 & D \end{bmatrix}\begin{bmatrix} T^{n+1} \\ \phi^{n+1} \end{bmatrix} = \begin{bmatrix} F \\ G \end{bmatrix}$

## 7. Benchmark examples

Two benchmark examples are studied:

## References

• [CEN2007] EN 15026, Hygrothermal performance of building components and building elements - Assessment of moisture transfer by numerical simulation, CEN, 2007.

• [HAM2002] C.-E. Hagentoft, HAMSTAD – Final report: methodology of HAM-modeling, Report R-02:8, Gothenburg, Department of Building Physics, Chalmers University of Technology, 2002.

• [Kumaran1994] Kumaran, M. K., Mitalas, G. P. and Bomberg, M. T. (1994), 'Fundamentals of transport andstorage of moisture in building materials and components', Trechsel, H. R. (ed), Moisture control in buildings.

• [KUN1995] Künzel H, Simultaneous Heat and Moisture Transport in Building Components, PhD thesis, Fraunhofer Institute of Building Physics, 1995.

• [Kunzel2004] Kunzel, H.M., Holm, A., Zirkelbach, D. and Karagiozis, A.N., Simulation of indoor temperature and humidity conditions including hygrothermal interactions with building envelope Solar Energy 78 (2005) 554–561.

• [Mendes2019] Mendes, N., Chhay, M., Berger, J. and Dutykh, D., Numerical methods for diffusion phenomena in building physics, Springer Nature Switzerland AG 2019.

• [Neymark2002] Neymark, J. and Judkoff, R. International energy agency building simulation test and diagnostic method for heating, ventilation, and air-conditioning equioement models (HVAC BESTEST). Volume 1: Cases E100-E200 Technical Report NREL/TP-550-30152.

• [Škerget2014] Škerget, L. and Tadeu, A. BEM numerical simulation of coupled heat and moisture flow through a porous solid Engineering Analysis with Boundary Elements, 2014.

• [Straube2002] Straube J. F., Moisture in Buildings, ASHRAE Jornal, January 2002.