Multiphase model for hygrothermal analysis of porous media with salt crystallization and hydration

Abstract

A new fully coupled multiphase model for hygrothermal analysis and prediction of salt diffusion and crystallization in porous building materials is presented. The relative humidity, the temperature, the concentration of the dissolved salt and the concentration of precipitated salts are assumed as independent variables. A suitable modelling of the crystallization/dissolution and hydration/dehydration processes allows considering salts with hydrous and anhydrous crystals. Some numerical applications on fired-clay bricks show the effectiveness of the proposed approach.

Appendices

Appendix 1

Some useful data are reported in this section.

1.1 Thermal conductivity of the liquid water function of temperature [29]

Temperature (K)	Thermal conductivity of liquid water (W/m/K)
275	0.5606
280	0.5715
285	0.5818
290	0.5917
295	0.6009
300	0.6096
305	0.6176
310	0.6252
315	0.6322
320	0.6387
325	0.6445
330	0.6499
335	0.6546

1.2 Molar dissolution enthalpy at infinite dilution in J mol⁻¹ ($273,15 < T < 373,15\;{\text{K}}$):$\overline{{\Delta_{\text{sol}} H^{\infty } }} = a_{\infty } + b_{\infty } (T - T_{0} )\quad {\text{with}}\quad T_{0} = 298,15\,{\text{K}}$ [22]

Solid species	$a_{\infty }$ (J/mol)	$b_{\infty }$ (J/mol/K)
NaCl	3799.7	−115.80
Na2SO4	−1380.4	−302.09

1.3 Molar latent heat of hydration/dehydration for H₂O–Na₂SO₄ solutions function temperature ($291,15 < T < 305,15\;{\text{K}}$) $h_{\text{hyd}} = h_{\text{hyd}} (T_{0} ) + \Delta_{\text{hyd}} C_{\text{p}} (T - T_{0} )$ with $T_{0} = 298,15\,{\text{K}}$ [22]

Solid species	$h_{\text{hyd}} (T_{0} )$ (J/mol)	$\Delta_{\text{hyd}} C_{\text{p}}$ (J/mol/K)
Na2SO4	−81 556	−687.91

Appendix 2

Some useful expressions are derived in this section:

2.1 Latent heat of crystallization for an anhydrous salt $H_{{{\text{cry}}_{j}}}$ [22]

$$ H_{{{\text{cry}}_j}} = \frac{{{m_{\text{w}}^{\text{l}}}\!^{\prime}\Delta_{\text{sol}} \hat{H}(m^{\prime}) - m_{\text{w}}^{\text{l}} \Delta_{\text{sol}} \hat{H}(m)}}{{x\;M_{{{\text{s}}_{j} }}^{\text{s}} }} $$

where:

$$ \Delta_{\text{sol}} \hat{H}(m^{\prime}) = \hat{L}(m^{\prime}) + m^{\prime}\,{\kern 1pt} \overline{{\Delta_{\text{sol}} H^{\infty } }} \,\,\Delta_{\text{sol}} \hat{H}(m) = \hat{L}(m) + m\,{\kern 1pt} \overline{{\Delta_{\text{sol}} H^{\infty } }} $$

$m^{\prime}$ is the molality of the electrolytic solution at the end of the crystallization process, $m$ is the molality of the electrolytic solution at the beginning of the crystallization process, $\hat{L}(m^{\prime})$ and $\hat{L}(m)$ is the excess enthalpy of a electrolytic solution containing 1 kg of water, calculated through the Pitzer’s model as reported in [22], $x$ is the number of moles of salt formed during the crystallization process

$$ \begin{aligned} H_{{{\text{cry}}_{j}}} \,&= \,\frac{{{m_{\text{w}}^{\text{l}}}\!^{{\prime} }\left( {\hat{L}(m^{{\prime }}) + m^{{\prime }}\,{\kern 1pt} \overline{{\Delta_{\text{sol}} H^{\infty } }} } \right) - m_{w}^{l} \left( {\hat{L}(m) + m\,{\kern 1pt} \overline{{\Delta_{\text{sol}} H^{\infty } }} } \right)}}{{x\;M_{{{\text{s}}_{j} }}^{\text{s}} }} \hfill \\ \,&=\, \frac{1}{{M_{{{\text{s}}_{j} }}^{\text{s}} }}\left[ {\frac{{{\kern 1pt} \overline{{\Delta_{\text{sol}} H^{\infty } }} }}{x\;}\left( {{m_{\text{w}}^{\text{l}}}\!^{{\prime }}m^{\prime } - m_{\text{w}}^{\text{l}} \,m} \right) + \frac{{{m_{\text{w}}^{\text{l}}}\!^{{\prime} }\,\hat{L}(m') - m_{\text{w}}^{\text{l}} \,\hat{L}(m)}}{x}} \right] \hfill \\ \,&= \,\frac{1}{{M_{{{\text{s}}_{j} }}^{\text{s}} }}\left( { - {\kern 1pt} \overline{{\Delta_{\text{sol}} H^{\infty } }} + \Delta H_{L} } \right). \hfill \\ \end{aligned} $$

Taking into account that $\frac{{{m_{\text{w}}^{\text{l}}}\!^{{\prime }}m^{\prime } - m_{\text{w}}^{\text{l}} \,m}}{x} = - 1$ and defining $\Delta H_{L}$ as:

$$ \Delta H_{L} = \frac{{{m_{\text{w}}^{\text{l}}}\!^{{\prime }}\,\hat{L}(m^{\prime}) - m_{\text{w}}^{\text{l}} \,\hat{L}(m)}}{x} .$$

Kinetic hydration/dehydration parameter for sulphate solutions

$$ K_{H_{ij}} = \frac{{K_{\text{esp}} \,M_{{{\text{s}}_{i} }}^{\text{s}} }}{{10\,M_{{{\text{H}}_{ 2} {\text{O}}}} {\kern 1pt} \rho_{{{\text{s}}_{i} }}^{\text{s}} \pi {\kern 1pt} r_{p}^{2} }} = \frac{{K_{\text{esp}} \,\left( {V_{\text{m}} } \right)_{{{\text{s}}_{i} }}^{\text{s}} }}{{10\,M_{{{\text{H}}_{ 2} {\text{O}}}} {\kern 1pt} \pi {\kern 1pt} r_{\text{p}}^{2} }}\,\,K_{H_{ji}} = \frac{{K_{\text{esp}} \,M_{{{\text{s}}_{j} }}^{\text{s}} }}{{10\,M_{{{\text{H}}_{ 2} {\text{O}}}} {\kern 1pt} \rho_{{{\text{s}}_{j} }}^{\text{s}} \pi {\kern 1pt} r_{\text{p}}^{2} }} = \frac{{K_{\text{esp}} \,\left( {V_{\text{m}} } \right)_{{{\text{s}}_{j} }}^{\text{s}} }}{{10\,M_{{{\text{H}}_{ 2} {\text{O}}}} {\kern 1pt} \pi {\kern 1pt} r_{\text{p}}^{2} }} $$

$i$ = hydrated crystallized form

$j$ = dehydrated crystallized form

$K_{\text{esp}}$ = 0.45–0.50 mg/min experimental water absorption coefficient for sodium sulphate solutions [15].

The coefficient of the equations of the model are reported in the following.

In particular those related to Eq. (66) are:

$$ \phi_{h} = \frac{{\partial c_{\text{w}} }}{\partial h},\,\,\phi_{h\omega } = \frac{{\partial c_{\text{w}} }}{\partial \omega },\,\,\phi_{hT} = \frac{{\partial c_{\text{w}} }}{\partial T},\,\,\phi_{{hs_{i} }} = \frac{{\partial c_{\text{w}} }}{{\partial c_{{{\text{s}}_{i} }}^{\text{s}} }}, $$

$$ C_{hh} = \frac{{D_{\text{v}} }}{{R_{\text{v}} T}}p_{\text{v,sat}} + (1 - \omega )\frac{{\rho_{\text{w}}^{\text{l}} R_{\text{v}} TK_{\text{l}} }}{h},\,\,C_{hT} = \frac{{D_{\text{v}} h}}{{R_{\text{v}} T}}\frac{{\partial p_{\text{v,sat}} }}{\partial T} + (1 - \omega )\rho_{\text{w}}^{\text{l}} R_{\text{v}} K_{\text{l}} \ln (h), $$

$$ C_{h\omega } = \frac{{D_{\text{v}} h}}{{R_{\text{v}} T}}\frac{{\partial p_{\text{v,sat}} }}{\partial \omega } - \rho_{\text{ws}}^{\text{l}} K_{\text{s}} . $$

The coefficients related to Eq. (67) are:

$$ \phi _{\omega } = \frac{{c_{{\text{w}}} }}{{(1 - \omega )^{2} }} + \frac{\omega }{{1 - \omega }}\phi _{{h\omega }} ,{\mkern 1mu} {\mkern 1mu} \phi _{{\omega h}} = \frac{\omega }{{1 - \omega }}\phi _{h} ,{\mkern 1mu} {\mkern 1mu} \phi _{{\omega T}} = \frac{\omega }{{1 - \omega }}\phi _{{hT}} ,{\mkern 1mu} {\mkern 1mu} \phi _{{{\text{s}}_{i} }} = 1 + \frac{\omega }{{1 - \omega }}\phi _{{h{\text{s}}_{i} }} , $$

$$ C_{\omega \omega } = \rho_{\text{ws}}^{\text{l}} K_{\text{s}} ,\,\,C_{\omega h} = \omega \frac{{\rho_{\text{w}}^{\text{l}} R_{\text{v}} TK_{\text{l}} }}{h},\,\,C_{\omega T} = \omega \rho_{\text{w}}^{\text{l}} R_{\text{v}} TK_{\text{l}} \ln (h). $$

The coefficients related to the energy balance Eq. (68) are:

$$ C_{TT} = \lambda_{\text{eff}} + \left( {\beta_{\text{w}}^{\text{g}} T + H_{\text{eva}} } \right)\frac{{D_{\text{v}} h}}{{R_{\text{v}} T}}\frac{{\partial p_{\text{v,sat}} }}{\partial T} + \left( {\beta_{\text{w}}^{\text{l}} (1 - \omega ) + \beta_{\text{s}}^{\text{l}} \omega } \right)\rho_{\text{w}}^{\text{l}} R_{\text{v}} K_{\text{l}} \ln (h)T, $$

$$ C_{Th} = \left( {\beta_{\text{w}}^{\text{g}} T + H_{\text{eva}} } \right)\frac{{D_{\text{v}} }}{{R_{\text{v}} T}}p_{\text{v,sat}} + \left( {\beta_{\text{w}}^{\text{l}} (1 - \omega ) + \beta_{\text{s}}^{\text{l}} \omega } \right)\frac{{\rho_{\text{w}}^{\text{l}} R_{\text{v}} T^{2} K_{\text{l}} }}{h}, $$

$$ C_{T\omega } = \left( {\beta_{\text{w}}^{\text{g}} T + H_{\text{eva}} } \right)\frac{{D_{\text{v}} h}}{{R_{\text{v}} T}}\frac{{\partial p_{\text{v,sat}} }}{\partial \omega } + \left( {\beta_{\text{s}}^{\text{l}} - \beta_{\text{w}}^{\text{l}} } \right)\rho_{\text{ws}}^{\text{l}} K_{\text{s}} T, $$

$$ B_{\text{w}} = \left( {\beta_{\text{w}}^{\text{g}} - \beta_{\text{w}}^{\text{l}} } \right)T + H_{\text{eva}} ,\,\,B_{\text{w}}^{*} = \beta_{\text{w}}^{\text{l}} T, $$

$$ \left( {\varphi_{{T{\text{s}}_{i} }} } \right)_{\text{cry}} = \left( {\beta_{{{\text{s}}_{i} }}^{\text{s}} - \beta_{\text{s}}^{\text{l}} } \right)T + H_{{{\text{cry}}_{i} }} ,\,\,\left( {\varphi_{{T{\text{s}}_{i} }} } \right)_{\text{hyd}} = \left( {\beta_{{{\text{s}}_{i} }}^{\text{s}} - \beta_{\text{s}}^{\text{l}} } \right)T + H_{{{\text{hyd}}_{ij} }} , $$

$$ \varphi_{T} = \beta_{\text{eff}} \rho_{\text{eff}} . $$

Finally, coefficients of the kinetic Eq. (69) are:

$$ C_{{{\text{ss}}_{i} }} = S_{\text{ws}}^{\text{l}} \left( {n_{i} \phi_{0} \rho_{{{\text{s}}_{i} }}^{\text{s}} \pi r_{\text{p}}^{2} } \right)K_{{C_{i} }} ,\,\,C_{{{\text{s}}h_{i} }} = \frac{{\pi {\kern 1pt} r_{\text{p}}^{2} \rho_{{{\text{s}}_{i} }}^{\text{s}} }}{{V_{\text{tot}} }}K_{H_{ij}} . $$