Coupled heat and mass transport material model - HeMoKunzel

The presented formulation is based on the work of Kuenzel [Kuenzel]. The model is suitable for problems with dominating water diffusion and negligible water convection. The governing equations for temperature and humidity reads

(127)\[\begin{split}\frac{\partial Q}{\partial t} &=& \frac{\partial Q}{\partial T} \frac{\partial T}{\partial t} = C_v \frac{\partial T}{\partial t} = -\nabla q_T = \nabla \left( \lambda \nabla T \right ) + h_v \nabla \left( \delta_p \nabla (H p_{sat}) \right )\\ \frac{\partial w}{\partial t} &=& \frac{\partial w}{\partial H} \frac{\partial H}{\partial t} = -\nabla q_H = \nabla \left( D_H \nabla H + \delta_p \nabla (H p_{sat}) \right )\end{split}\]

Table 67 Parameters from Kunzel’s model.

Symbol	Unit	Description
\(T\) \(H\)	(-)	Temperature Relative humidity 0-1
\(\frac{\partial Q}{\partial T} \approx C_v\) \(\frac{\partial w}{\partial H}\) \(Q\) \(q_T\) \(\lambda\) \(h_v\) \(\delta_p\) \(p_{sat}\) \(w\) \(D_H\) \(D_w\)	J/K/m\(^3\) kg/m\(^3\) J/m\(^3\) W/m\(^2\) W/m/K J/kg kg/m/s/Pa Pa kg/m\(^3\) kg/m/s m\(^2\)/s	Heat storage capacity per volume Moisture storage capacity - sorption isotherm Total amount of heat in unit volume Heat flux Thermal conductivity Evaporation enthalpy of water Water vapour permeability Water vapour saturation pressure Moisture content Liquid conduction coefficient Water diffusivity

Numerical solution leads to the system of equations

(128)\[\begin{split}\left[ \begin{array}{cc} \tenss{K}_{TT} & \tenss{K}_{TH} \\ \tenss{K}_{HT} & \tenss{K}_{HH} \end{array} \right] \left\{ \begin{array}{c} \tenss{r}_T \\ \tenss{r}_H \end{array} \right\} + \left[ \begin{array}{cc} \tenss{C}_{TT} & \tenss{C}_{TH} \\ \tenss{C}_{HT} & \tenss{C}_{HH} \end{array} \right] \left\{ \begin{array}{c} \dot{\tenss{r}}_T \\ \dot{\tenss{r}}_H \end{array} \right\} = \left\{ \begin{array}{c} \tenss{q}_{T} \\ \tenss{q}_{H} \end{array} \right\},\end{split}\]

where

\[\tenss{K}_{TT} = \int_{\Omega} \tenss{B}^T k_{TT} \tenss{B}{\rm d}\Omega,\qquad \tenss{K}_{TH} = \int_{\Omega} \tenss{B}^T k_{Th} \tenss{B}{\rm d}\Omega,\qquad \tenss{K}_{HT} = \int_{\Omega} \tenss{B}^T k_{hT} \tenss{B}{\rm d}\Omega,\qquad \tenss{K}_{HH} = \int_{\Omega} \tenss{B}^T k_{hh} \tenss{B}{\rm d}\Omega,\qquad\]

\[\tenss{C}_{TT} = \int_{\Omega} \tenss{N}^T c_{TT} \tenss{N} {\rm d}\Omega,\qquad \tenss{C}_{TH} = \int_{\Omega} \tenss{N}^T c_{Th} \tenss{N} {\rm d}\Omega,\qquad \tenss{C}_{HT} = \int_{\Omega} \tenss{N}^T c_{hT} \tenss{N} {\rm d}\Omega,\qquad \tenss{C}_{HH} = \int_{\Omega} \tenss{N}^T c_{hh} \tenss{N} {\rm d}\Omega,\qquad\]

\[\tenss{q}_T = \int_{\Gamma_2} \tenss{N}^T \overline{q}_{T}{\rm d}\Gamma,\qquad \tenss{q}_H = \int_{\Gamma_2} \tenss{N}^T \overline{q}^{h}{\rm d}\Gamma,\qquad\]

where

(129)\[\begin{split}k_{TT} &=& \lambda(w) + h_v \cdot \delta_p(T) \cdot H \cdot \frac{\Delta p_{sat}}{\Delta T}(T),\\ k_{TH} &=& h_v \cdot \delta_p(T) \cdot p_{sat}(T),\\ k_{HT} &=& \delta_p(T) \cdot H \cdot \frac{\Delta p_{sat}}{\Delta T}(T),\\ k_{HH} &=& D_w(H) \cdot \frac{\Delta w}{\Delta H}(H) + \delta_p(T) \cdot p_{sat}(T),\\ c_{TT} &=& C_s \cdot \rho + C_w \cdot w,\\ c_{TH} &=& 0,\\ c_{HT} &=& 0,\\ c_{HH} &=& \frac{\Delta w}{\Delta H}(H).\end{split}\]

Note, that conductivity matrix tenss{K} is unsymmetric hence unsymmetric matrix storage needs to be used (smtype).

The model parameters are summarized in Table 68.

Table 68 Coupled heat and mass transfer material model Kunzel - summary.

Description	Coupled heat and mass transfer material model
Record Format	HeMoKunzel num (in) d (rn) iso_type (in) iso_wh (rn) mu (rn) permeability_type (in) A (rn) lambda0 (rn) b (rn) cs (rn) [pl (rn)] [rhoH2O (rn)] [cw (rn)] [hv (rn)]
Parameters	num material model number d bulk density of dry building material [kg/m\(^3\)] iso_type=0 is isotherm from Hansen needing iso_n, iso_a, =1 is Kunzel which needs iso_b iso_wh maximum adsorbed water content [kg/m\(^3\)] mu water vapor diffusion resistance [-] permeability_type =0 is Multilin_h needing perm_h, perm_Dw(h), =1 is Multilin_wV needs perm_wV, perm_DwwV, =2 is Kunzelperm needs A as water absorption coefficient [kg/m/s\(^0.5\)] lambda0rn thermal conductivity [W/m/K] b thermal conductivity supplement [-] cs specific heat capacity of the building material [J/kg/K] [pl] ambient air pressure [Pa], default = 101325 [rhoH2O] water density [kg/m3], default = 1000 [cw] specific heat capacity of liquid water, default = 4183 [hv] latent heat of water phase change [J/kg], default = 2.5e+6
Supported modes	_2dHeMo, _3dHeMo
Tests/Examples	tm/HeMoKunzel_1.in