Difference between revisions of "Modeling of Salt and Humidity Transport"

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{{EngVersion|text=[[CGerdwilker|Christa Gerdwilker]]}}
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<!--The English version of the article is edited by CGerdwilker|Christa Gerdwilker]]}}-->
  
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''Author: [[User:ANicolai|Dr. Andreas Nicolai ]]''<br>
 
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English version by Christa Gerdwilker
''Author: [[User:ANicolai|Dr. Andreas Nicolai ]]''
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<br>
<br><br>
 
 
back to [[Saltwiki:Community portal]]
 
back to [[Saltwiki:Community portal]]
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'''Linked Article '''
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'''Linked Articles '''
 
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   category = ANicolai&Salt transport modeling
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   category = Nicolai,Andreas&Salt Moisture Transport
 
   format  = ,\n* [[%PAGE%|%TITLE%]],,
 
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__TOC__
 
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= Introduction =
 
= Introduction =
  
The transport of salts in porous materials (building materials, stone, flooring etc.)is dependent on many factors. The types of salts, their constituents and distribution, the extent of dampness, temperature, environmental conditions in general and also the  microscopic structure of the material all influence the movement of salts and the appearance of decay over time.
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Salt transport in porous materials (building materials, stone, floors, etc.) depends on many factors, such as: the type of salt(s) present, their composition and distribution in the porous material. Furthermore, the microscopic structure of the materials, including pore types and pore size distribution, the presence of moisture in it, as well as environmental conditions, such as temperature and RH, will also influence the movement of salts and the appearance of deterioration over time.  
  
Usually, only specific combinations of materials, salt mixtures and environmental conditions can be measured under laboratory conditions (common laboratory experiments for the determination of salt transport properties can be found in the article "[[Experimental calibration of salt transport parameters]]". Due to diverse parameters and decay scenarios it is rarely possible to directly apply such imperial measurements in the practical environment.  
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In general, only specific combinations of materials, salt mixtures and environmental conditions can be measured under laboratory conditions (common laboratory experiments for the determination of salt transport properties can be found in the article "[[Experimental calibration of salt transport parameters]]". Due to diverse parameters and decay scenarios on actual buildings and monuments it is rarely possible to directly apply such empirical measurements.  
  
Alternatively, transport models can be used which realistically simulate the physical and chemical context and thus allow mathematical predictions regarding the dispersion and accumulation of salts and material decay over time. The aim of this article and linked reference material is to provide a summary of current research in the modeling of salt transport.  
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Alternatively, transport models can be used which realistically simulate the physical and chemical context and thus allow mathematical predictions regarding the distribution and accumulation of salts and the consequent material decay over time. The aim of this article and linked reference material is to provide a summary of current research in mathematical modeling of salt transport.  
  
Due to insufficient nomenclature in relevant norms and literature, a bespoke form of annotation and symbols for salt transport models [[Annotation and list of symbols used for salt transport models]] is used for the models below.
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Due to an insufficiency in nomenclature in relevant norms and literature, a custom  form of annotation and symbols for salt transport models (Annotation and list of symbols used for salt transport models) is used for the models below.
  
 
= Fundamental aspects of modeling salt transport =
 
= Fundamental aspects of modeling salt transport =
  
 
Due to the complexity of the involved processes, salt transport models must be able to describe a number of effects, amongst others:  
 
Due to the complexity of the involved processes, salt transport models must be able to describe a number of effects, amongst others:  
* Moisture transport and moisture retention  
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* Moisture transport and moisture retention.
* Thermal transfer through thermal conductivity and radiation  
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* Thermal transfer via thermal conductivity and radiation.
* Enthalpy transport, e.g. latent heat to describe cooling during evaporation  
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* Enthalpy transport, e.g., latent heat to describe cooling during evaporation.
* Balanced salt phase change constituents and phase change kinetics
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* Balance between different salt phases and kinetics of phase changes* Salt diffusion and distribution.
* Salt diffusion and dispersion
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* Efflorescence (removal of salts from the calculated domain).
* Efflorescence (removal of salts from the calculated domain)
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* Change of pore space through crystallization and resultant effect on moisture and salt transport.
* Change of pore space through crystallization and resultant effect on moisture and salt transport  
+
 
 +
Because the linked thermal and moisture transport models form the basis for salt transport modeling, a summary of the current state of research in moisture transport modeling is provided.
  
Because the linked thermal and moisture transport models form the basis for salt transport modeling, a summary of the current state of research in moisture transport modeling is provided. Generally, the presented model corresponds to the jointly defined transport model <bib id="Hagentoft.etal:2004"/> of the HAMSTAD Projekt.  
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Generally, the presented model corresponds to the jointly defined transport model <bib id="Hagentoft.etal:2004"/> of the HAMSTAD Projekt.
  
 
= Moisture transport modeling =
 
= Moisture transport modeling =
  
Water is the transport medium for salts. In dry material salt is immobile. Only the mobilization of salts through penetrating moisture and enrichment of salts during evaporation of water in a different area of porous masonry results in decay. Subsequently a detailed moisture transport model is a primary pre-requisite for every salt transport model.
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Water is the transport medium for salts. In dry material salt remains immobile. Decay only results through the mobilization of salts by penetrating moisture and enrichment of salts as evaporation of water takes place in different area(s)of the porous masonry. Therefore, a detailed moisture transport model is a primary pre-requisite for every salt transport model.
  
Moisture transport models describe the different transport processes of moisture in a porous medium as well as the retention of moisture and subsequently the interrelation between the material conditions (moisture content and mass) and the intrinsically thermo-dynamic environmental conditions (capillary pressure, relative humidity, etc.).  
+
Moisture transport models describe the different transport processes of moisture in a porous medium as well as the retention of moisture and subsequently the interrelation between the material conditions (moisture content and mass) and the intrinsically thermo-dynamic environmental conditions (capillary pressure, relative humidity, etc.).  
  
 
== Fundamental moisture transport mechanisms in damp masonry ==
 
== Fundamental moisture transport mechanisms in damp masonry ==
  
Building materials, stone and generally porous materials can absorb moisture in vapor and liquid form. Accordingly the transport mechanisms for liquid water and vapor are differentiated:  
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Building materials, stone and generally porous materials can absorb moisture in both vapor and liquid form. Accordingly, the transport mechanisms for liquid water and vapor are differentiated:  
* Water vapor diffusion  
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* Water vapor diffusion;
* Convection of vapor in air currents
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* Convection of vapor in air currents;
* Liquid water flow induced by differential water pressure  
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* Liquid water flow induced by differential water pressure.
  
 
These transport mechanisms are described in the article "[[Moisture transport mechanisms]]".  
 
These transport mechanisms are described in the article "[[Moisture transport mechanisms]]".  
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== Correlation between driving forces and moisture content ==
 
== Correlation between driving forces and moisture content ==
  
The correlation between a volume or mass based amount of e.g. moisture content and the intrinsic (and therefore volume independent) amount of relative humidity or capillary pressure are created by the moisture retention capacity. This is differentiated into  
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The correlation between a volume or mass based amount of e.g., moisture content and the intrinsic (and therefore volume independent) amount of relative humidity or capillary pressure are created by the moisture retention capacity. This is differentiated into:
* The sorption isotherm and
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* The sorption isotherm, and
 
* The moisture retention capacity (MRC)  
 
* The moisture retention capacity (MRC)  
  
The sorption isotherm is commonly defined for standard conditions and with that, constant temperature (hence the term isotherm) and establishes the correlation between moisture content and relative humidity <math>\phi</math>. The moisture retention capacity establishes the correlation between the moisture content and capillary pressure <math>p_c</math>.  
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The sorption isotherm is commonly defined for standard conditions, i.e., at constant temperature (hence the term isotherm) and establishes the correlation between moisture content and relative humidity <math>\phi</math>. The moisture retention capacity establishes the correlation between the moisture content and capillary pressure <math>p_c</math>.  
  
The moisture content can be measured as moisture mass over volume of material i.e. as moisture mass density <math>\rho^{m_{w+v}}</math> or as water volume over material volume i.e. the moisture content <math>\theta_\ell</math>.  
+
The moisture content can be measured as the moisture mass per volume of material, i.e., as moisture-mass density    
 +
<math>\rho^{m_{w+v}}</math> or as the ratio of water volume per material volume, i.e., the moisture content <math>\theta_\ell</math>.  
  
Relative humidity and capillary pressure are interrelated (in ''saltfree!'' material) which is expressed in the Kelvin equation  
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Relative humidity and capillary pressure are interrelated (in salt free material) and are expressed by the Kelvin equation:
 
  <math>
 
  <math>
 
\ln  \phi = \frac{p_c}{\rho_w R_v T}
 
\ln  \phi = \frac{p_c}{\rho_w R_v T}
 
</math>  
 
</math>  
The capillary pressure is defined as negative pressure in contrast to tension caused by suction which is defined as negative capillary pressure.  
+
The capillary pressure is defined as the inverse of the capillary tension that gives rise to capillarity.  
  
 
The article "[[Moisture retention in porous materials]]" further elaborates on this interrelation.
 
The article "[[Moisture retention in porous materials]]" further elaborates on this interrelation.
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== Phase-changing processes (without salt) ==
 
== Phase-changing processes (without salt) ==
  
Even without the presence of salt in porous materials the following phase changes within the pore system need to be considered:  
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The following phase changes within the pore system — even in the absence of salt in them — need to be considered:
* Evaporation
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* Evaporation;
* Condensation,
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* Condensation;
* Freeze
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* Freezing;
* Thaw
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* Thawing;
The related phase change enthalpies are always of critical issue here. <!--Included is the description of the ice formation processes. -->
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The corresponding phase change enthalpies are always a critical issue. <!--Included is the description of the ice formation processes. -->
  
 
The article "[[Modeling of the phase changes between ice, water and vapor]]" discusses the common approaches to linked hygro-thermal transport models.  
 
The article "[[Modeling of the phase changes between ice, water and vapor]]" discusses the common approaches to linked hygro-thermal transport models.  
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The general equation for moisture and ice mass (for the kinetic description of ice formation) is  
 
The general equation for moisture and ice mass (for the kinetic description of ice formation) is  
  
<math>\frac{\partial \rho^{m_{w+v}}}{\partial t} &= - \nabla \left( j^{m_{w}} + j^{m_{v}}_{dif\!f} + j^{m_{v}}_{conv} \right) - \sigma_{w \rightarrow \text{ice}}</math>
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<math>\frac{\partial \rho^{m_{w+v}}}{\partial t} &= - \nabla \left( j^{m_{w}} + j^{m_{v}}_{dif\!f} + j^{m_{v}}_{conv} \right) - \sigma_{w \rightarrow \text{ice}}</math>
 +
 
 +
<math>\frac{\partial \rho^{m_{\text{ice}}}}{\partial t} &=  \sigma_{w \rightarrow \text{ice}}</math>
 
   
 
   
 
+
The second part of the balance equation and phase change term <math>\sigma_{w \rightarrow \text{ice}}</math> can be ignored if the crystallization of ice is not of interest. The term ‘isothermal process’ to describe moisture transport is rarely appropriate. In practice, it is not applicable when either cooling (during evaporation) or warming (during condensation) processes are the most important. Therefore, the moisture mass equation needs to be complemented by the energy equation.
 
 
 
<math>\frac{\partial \rho^{m_{\text{ice}}}}{\partial t} &=  \sigma_{w \rightarrow \text{ice}}</math>
 
 
 
 
 
 
The second part of the balance equation and phase change term <math>\sigma_{w \rightarrow \text{ice}}</math> can be ignored if the crystallization of ice is not of interest. The use of the term ‘isothermal process’ to describe moisture transport is rarely sufficient. It is not applicable where cooling during evaporation and warming during condensation form substantial processes in practice. Therefore, the moisture mass balance equation is complemented by the energy balance equation.
 
  
 
= Thermal conductivity, thermal transfer and energy equation =
 
= Thermal conductivity, thermal transfer and energy equation =
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Heat retention is differentiated into:  
 
Heat retention is differentiated into:  
* Sensible heat i.e. heat which is stored in the kinetic energy caused by atom vibration or the movement energy of molecules<br />Every material has a specific thermal capacity, which together with density and temperature changes, result in changes to the stored sensitive heat: <math> \Delta U = c_T \rho \Delta T</math>
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* Sensible heat, i.e., heat which is stored in the kinetic energy caused by atom vibration or the movement energy of molecules<br />Every material has a specific thermal capacity, which together with density and temperature changes, result in changes to the stored sensitive heat:
* latent Wärme, i.e. phase change enthalpies <br />Latent heat is commonly defined in relation to the state of aggregate e.g. the liquid aggregate state of water. During heating above boiling point steam absorbs latent heat which is released again during condensation. Analogously, latent heat is released during freezing which has to be re-introduced during the melting of ice. Latent heat is several magnitudes bigger than sensible heat.  
+
<math> \Delta U = c_T \rho \Delta T</math>
* Chemically bound heat <br />This energy is balanced in reaction equations e.g. during the crystallization or solution of salts.  
+
* Latent heat, i.e., phase change enthalpies <br />Latent heat is commonly defined in relation to the state of aggregation, e.g., the liquid aggregate state of water. During heating above boiling point, steam absorbs latent heat that is then released during condensation. Analogously, latent heat is released during freezing that needs to be re-introduced during the melting of ice. Latent heat is several magnitudes larger than sensible heat.
 +
* Chemically bound heat <br />This energy is balanced in reaction equations, e.g., during crystallization or dissolution of salts.  
  
The stored energies can be summarized in relation to the reference temperature <math>T_{Ref}</math> and produce, in the case of a salt free material, the energy density difference <math>\rho^U - \rho^{U_{Ref}}</math>:   
+
The stored energies can be summarized in relation to the reference temperature <math>T_{Ref}</math> and produce, in the case of a salt free material, the energy density difference
 +
<math>\rho^U - \rho^{U_{Ref}}</math>:   
  
<math>
+
<math>
 
\rho^U - \rho^{U_{Ref}} = \rho_b c_T \left(T - T_{Ref} \right) \\     
 
\rho^U - \rho^{U_{Ref}} = \rho_b c_T \left(T - T_{Ref} \right) \\     
 
+ \quad \quad \rho^{m_v}  \left[  c_{T,v}  \left(T - T_{Ref} \right) + h_v\right]  \\
 
+ \quad \quad \rho^{m_v}  \left[  c_{T,v}  \left(T - T_{Ref} \right) + h_v\right]  \\
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==== Illustrative example ====
 
==== Illustrative example ====
The following mental experiment can be conducted during consideration of the equation for energy density:  
+
The following example serves to illustrate the equations that follow the changes for energy density during the evaporation of water:  
 
* The control volume is adiabatic so that the energy density is constant  
 
* The control volume is adiabatic so that the energy density is constant  
* Water is to evaporate so that <math>\Delta \rho^{m_v} = - \Delta \rho^{m_w}</math>, which results in a smaller liquid density and a larger vapor mass density.  
+
* As water evaporates so that <math>\Delta \rho^{m_v} = - \Delta \rho^{m_w}</math>, this results in a smaller liquid density and a larger vapor mass density.  
* If the energy density equation is now adjusted for temperature, a lower temperature than before is obtained; this cooling through evaporation corresponds with our expectations.  
+
* If the energy density equation is now adjusted for temperature, a lower temperature than before is obtained; this cooling through evaporation corresponds to that observed in real life.  
  
 
== Thermal transfer mechanisms ==
 
== Thermal transfer mechanisms ==
 
Different mechanisms of thermal transfer are considered:
 
Different mechanisms of thermal transfer are considered:
* Thermal conductivity
+
* Thermal conductivity.
* Thermal transfer through short-/ long-wave radiation (short-wave radiation is only of significance during the description of frame conditions whereas long wave radiation is important within constructions e.g. inside buildings).  
+
* Thermal transfer through short-/ long-wave radiation (short-wave radiation is only of significance during the description of frame conditions whereas long wave radiation is important within constructions, e.g., inside buildings).  
* Convection of latent and sensible heat  
+
* Convection of latent and sensible heat.
  
The article "[[Mechanisms of thermal transfer]]" discusses models for the different thermal transfer mechanisms, in particular thermal conductivity <math>j^Q</math> and enthalpy transport <math>h j^m</math> of the different components.  
+
The article "[[Mechanisms of thermal transfer]]" discusses models for the different thermal transfer mechanisms, in particular thermal conductivity <math>j^Q</math> and enthalpy transport <math>h j^m</math> of the different components.
  
 
== Energy balance equation ==
 
== Energy balance equation ==
 
The equation for the energy stored in a salt-free porous materials is  
 
The equation for the energy stored in a salt-free porous materials is  
<math>\frac{\partial \rho^U }{\partial t} &= - \nabla \left( j^Q + h_w j^{m_{w}} + h_v j^{m_{v}}_{dif\!f} + h_v j^{m_{v}}_{conv} \right)</math>
+
<math>\frac{\partial \rho^U }{\partial t} &= - \nabla \left( j^Q + h_w j^{m_{w}} + h_v j^{m_{v}}_{dif\!f} + h_v j^{m_{v}}_{conv} \right)</math>
  
No expanding or reducing terms for the phase change enthalpy are given in the energy equation. The change enthalpies are considered in the energy density (see paragraph above).  
+
No expanding or reducing terms for the phase change enthalpy are given in the energy equation. The enthalpy changes are considered as energy density (see paragraph above).  
  
The enthalpy of dry air <math>h_a \rho^{m_a}</math> (dry air = all gas phase components except water vapor) is much smaller than the enthalpy of water vapor so that it is commonly neglected. The energy equation can be expanded accordingly in specific applications (e.g. compressed air drying of porous materials). In typical application scenarios the speed of air currents in porous substances can be neglected so that <math>h_v j^{m_{v}}_{conv} </math> can also be disregarded.
+
The enthalpy of dry air <math>h_a \rho^{m_a}</math> (dry air = all gas phase components except water vapor) is much smaller than the enthalpy of water vapor so that it is commonly neglected. The energy equation can be expanded accordingly in specific applications(e.g., drying of porous materials by compressed air). In typical application scenarios the speed of air currents in porous substances can be neglected so that <math>h_v j^{m_{v}}_{conv} </math> can also be disregarded.
  
 
= Modeling of salt transport =
 
= Modeling of salt transport =
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== Transport mechanisms for salts/ions and modeling approaches ==
 
== Transport mechanisms for salts/ions and modeling approaches ==
Generally, salts can only be transported within building materials in the presence of liquid phase i.e. capillary water. This transport happens through ''diffusion'' or ''convection''.   
+
Generally, salts can only be transported within building materials in the presence of a liquid phase, e.g., capillary water. This transport happens through ''diffusion'' or ''convection''.   
* Diffusion describes the (mass-centric) exchange of ions  
+
* Diffusion describes the (mass-centric) exchange of ions.
* Convection describes the transport of dissolved salts together with the liquid phase  
+
* Convection describes the transport of dissolved salts together with the liquid phase.
  
During the convection of a salt solution through porous material, different current paths through the pore network can result in higher salt concentrations fronts "forging ahead" or "lagging behind" in some areas.  The observed spread of a concentration front, similar to a diffusion process, is termed ''dispersion''. The effect of dispersion and diffusion is similar and difficult to differentiate in the presence of convection. In still liquids only diffusion will take place.  
+
During the convection of a salt solution through a porous material, different current paths will develop in the pore network resulting in higher salt concentrations fronts "forging ahead" or "lagging behind" in some areas.  The observed spread of a concentration front, similar to a diffusion process, is termed ''dispersion''. The effect of dispersion and diffusion is similar and difficult to differentiate in the presence of convection. In still liquids only diffusion will take place.
  
 
== Phase changes ==
 
== Phase changes ==
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To describe:
 
To describe:
* Increased hygroscopic moisture absorption in the presence of salt loads  
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* Increased hygroscopic moisture absorption in the presence of salt loads;
* Reduction in vapor pressure, water activity  
+
* Reduction in vapor pressure, water activity;
* Surface tension + Kelvin-equation
+
* Surface tension + Kelvin-equation.
  
 
== Influence of salts on moisture transport ==
 
== Influence of salts on moisture transport ==
* Viscosity
+
* Viscosity changes.
* Influence on density/gravitational forces
+
* Influence on density/gravitational forces.
  
== Crystallization within and outwith the pore network ==
+
== Crystallization within and outside of the pore network ==
* Decrease in pore space due to crystallization, resultant effect on reservoir capacity and transport mechanisms  
+
* Decrease in pore space due to crystallization with the resultant effect on reservoir capacity and transport mechanisms.
* Efflorescence-Models
+
* Efflorescence-Models.
  
 
= Literature =
 
= Literature =
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Note: The listed literature is still to be shown and reviewed  
 
Note: The listed literature is still to be shown and reviewed  
  
<bib id=Koniorczyk.etal:2008/>
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<bibprint filter="year:2008, author:%Koniorczyk%"/>
 
 
  
<bibprint />
+
<biblist />
  
[[Category:Salt Moisture Transport]] [[Category:R-ANicolai]] [[Category:ANicolai]] [[Category:R-CBlaeuer]] [[Category:Editing]]
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[[Category:Salt Moisture Transport]] [[Category:R-ANicolai]] [[Category:Nicolai,Andreas]] [[Category:R-CBlaeuer]] [[Category:approved]] [[Category:Modelling]]

Latest revision as of 12:42, 20 September 2013


Author: Dr. Andreas Nicolai
English version by Christa Gerdwilker
back to Saltwiki:Community portal
Linked Articles

Introduction

Salt transport in porous materials (building materials, stone, floors, etc.) depends on many factors, such as: the type of salt(s) present, their composition and distribution in the porous material. Furthermore, the microscopic structure of the materials, including pore types and pore size distribution, the presence of moisture in it, as well as environmental conditions, such as temperature and RH, will also influence the movement of salts and the appearance of deterioration over time.

In general, only specific combinations of materials, salt mixtures and environmental conditions can be measured under laboratory conditions (common laboratory experiments for the determination of salt transport properties can be found in the article "Experimental calibration of salt transport parameters". Due to diverse parameters and decay scenarios on actual buildings and monuments it is rarely possible to directly apply such empirical measurements.

Alternatively, transport models can be used which realistically simulate the physical and chemical context and thus allow mathematical predictions regarding the distribution and accumulation of salts and the consequent material decay over time. The aim of this article and linked reference material is to provide a summary of current research in mathematical modeling of salt transport.

Due to an insufficiency in nomenclature in relevant norms and literature, a custom form of annotation and symbols for salt transport models (Annotation and list of symbols used for salt transport models) is used for the models below.

Fundamental aspects of modeling salt transport

Due to the complexity of the involved processes, salt transport models must be able to describe a number of effects, amongst others:

  • Moisture transport and moisture retention.
  • Thermal transfer via thermal conductivity and radiation.
  • Enthalpy transport, e.g., latent heat to describe cooling during evaporation.
  • Balance between different salt phases and kinetics of phase changes* Salt diffusion and distribution.
  • Efflorescence (removal of salts from the calculated domain).
  • Change of pore space through crystallization and resultant effect on moisture and salt transport.

Because the linked thermal and moisture transport models form the basis for salt transport modeling, a summary of the current state of research in moisture transport modeling is provided.

Generally, the presented model corresponds to the jointly defined transport model [Hagentoft.etal:2004]Title: Assessment Method of Numerical Prediction Models for Combined Heat, Air and Moisture Transfer in Building Components: Benchmarks for One-dimensional Cases
Author: Hagentoft, Carl-Eric; Kalagasidis, Angela Sasic; Adl-Zarrabi, Bijan; Roels, Staf; Carmeliet, Jan; Hens ,Hugo; Grunewald, John; Max Funk; Rachel Becker; Dina Shamir; Olaf Adan; Harold Brocken; Kumar Kumaran; Reda Djebbar
Link to Google Scholar
of the HAMSTAD Projekt.

Moisture transport modeling

Water is the transport medium for salts. In dry material salt remains immobile. Decay only results through the mobilization of salts by penetrating moisture and enrichment of salts as evaporation of water takes place in different area(s)of the porous masonry. Therefore, a detailed moisture transport model is a primary pre-requisite for every salt transport model.

Moisture transport models describe the different transport processes of moisture in a porous medium as well as the retention of moisture and subsequently the interrelation between the material conditions (moisture content and mass) and the intrinsically thermo-dynamic environmental conditions (capillary pressure, relative humidity, etc.).

Fundamental moisture transport mechanisms in damp masonry

Building materials, stone and generally porous materials can absorb moisture in both vapor and liquid form. Accordingly, the transport mechanisms for liquid water and vapor are differentiated:

  • Water vapor diffusion;
  • Convection of vapor in air currents;
  • Liquid water flow induced by differential water pressure.

These transport mechanisms are described in the article "Moisture transport mechanisms".

Correlation between driving forces and moisture content

The correlation between a volume or mass based amount of e.g., moisture content and the intrinsic (and therefore volume independent) amount of relative humidity or capillary pressure are created by the moisture retention capacity. This is differentiated into:

  • The sorption isotherm, and
  • The moisture retention capacity (MRC)

The sorption isotherm is commonly defined for standard conditions, i.e., at constant temperature (hence the term isotherm) and establishes the correlation between moisture content and relative humidity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} . The moisture retention capacity establishes the correlation between the moisture content and capillary pressure Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_c} .

The moisture content can be measured as the moisture mass per volume of material, i.e., as moisture-mass density

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho^{m_{w+v}}}
 or as the ratio of water volume per material volume, i.e., the moisture content Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta_\ell}
. 

Relative humidity and capillary pressure are interrelated (in salt free material) and are expressed by the Kelvin equation:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle  \ln  \phi = \frac{p_c}{\rho_w R_v T} }
 

The capillary pressure is defined as the inverse of the capillary tension that gives rise to capillarity.

The article "Moisture retention in porous materials" further elaborates on this interrelation.

Phase-changing processes (without salt)

The following phase changes within the pore system — even in the absence of salt in them — need to be considered:

  • Evaporation;
  • Condensation;
  • Freezing;
  • Thawing;

The corresponding phase change enthalpies are always a critical issue.

The article "Modeling of the phase changes between ice, water and vapor" discusses the common approaches to linked hygro-thermal transport models.

Balance equations

Following the discussion and illustration of the individual processes involved in moisture transport and retention, these can be summarized in the equations below.

The general equation for moisture and ice mass (for the kinetic description of ice formation) is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial \rho^{m_{w+v}}}{\partial t} &= - \nabla \left( j^{m_{w}} + j^{m_{v}}_{dif\!f} + j^{m_{v}}_{conv} \right) - \sigma_{w \rightarrow \text{ice}}}

 
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial \rho^{m_{\text{ice}}}}{\partial t} &=  \sigma_{w \rightarrow \text{ice}}}


The second part of the balance equation and phase change term Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_{w \rightarrow \text{ice}}} can be ignored if the crystallization of ice is not of interest. The term ‘isothermal process’ to describe moisture transport is rarely appropriate. In practice, it is not applicable when either cooling (during evaporation) or warming (during condensation) processes are the most important. Therefore, the moisture mass equation needs to be complemented by the energy equation.

Thermal conductivity, thermal transfer and energy equation

Heat retention

Heat retention is differentiated into:

  • Sensible heat, i.e., heat which is stored in the kinetic energy caused by atom vibration or the movement energy of molecules
    Every material has a specific thermal capacity, which together with density and temperature changes, result in changes to the stored sensitive heat:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle  \Delta U = c_T \rho \Delta T}

  • Latent heat, i.e., phase change enthalpies
    Latent heat is commonly defined in relation to the state of aggregation, e.g., the liquid aggregate state of water. During heating above boiling point, steam absorbs latent heat that is then released during condensation. Analogously, latent heat is released during freezing that needs to be re-introduced during the melting of ice. Latent heat is several magnitudes larger than sensible heat.
  • Chemically bound heat
    This energy is balanced in reaction equations, e.g., during crystallization or dissolution of salts.

The stored energies can be summarized in relation to the reference temperature Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{Ref}} and produce, in the case of a salt free material, the energy density difference

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho^U - \rho^{U_{Ref}}}
:  
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle  \rho^U - \rho^{U_{Ref}} = \rho_b c_T \left(T - T_{Ref} \right) \\     + \quad \quad \rho^{m_v}   \left[  c_{T,v}  \left(T - T_{Ref} \right) + h_v\right]  \\ + \quad \quad \rho^{m_w} c_{T,w}  \left(T - T_{Ref} \right)  \\ + \quad \quad \rho^{m_\text{ice}}  \left[  c_{T,\text{ice}}  \left(T - T_{Ref}  \right) - h_{\text{ice}}  \right]  }

Simplified, the energy density Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho^U} can be seen as a differential value and, therefore, an explicit specification of the reference temperature can be omitted.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle  \rho^U = \rho_b c_T T + \rho^{m_v}   \left(  c_{T,v}  T + h_v \right)  + \rho^{m_w} c_{T,w}  T + \rho^{m_\text{ice}}  \left(  c_{T,\text{ice}}  T - h_{\text{ice}}  \right)  }

The reference phase here is the liquid phase, so that phase change enthalpy is added to the steam and the relevant enthalpy is deducted from ice. The first term of the added energy density is the energy stored within the material matrix.

Illustrative example

The following example serves to illustrate the equations that follow the changes for energy density during the evaporation of water:

  • The control volume is adiabatic so that the energy density is constant
  • As water evaporates so that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta \rho^{m_v} = - \Delta \rho^{m_w}} , this results in a smaller liquid density and a larger vapor mass density.
  • If the energy density equation is now adjusted for temperature, a lower temperature than before is obtained; this cooling through evaporation corresponds to that observed in real life.

Thermal transfer mechanisms

Different mechanisms of thermal transfer are considered:

  • Thermal conductivity.
  • Thermal transfer through short-/ long-wave radiation (short-wave radiation is only of significance during the description of frame conditions whereas long wave radiation is important within constructions, e.g., inside buildings).
  • Convection of latent and sensible heat.

The article "Mechanisms of thermal transfer" discusses models for the different thermal transfer mechanisms, in particular thermal conductivity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j^Q} and enthalpy transport Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h j^m} of the different components.

Energy balance equation

The equation for the energy stored in a salt-free porous materials is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial \rho^U }{\partial t} &= - \nabla \left( j^Q + h_w j^{m_{w}} + h_v j^{m_{v}}_{dif\!f} + h_v j^{m_{v}}_{conv} \right)}

No expanding or reducing terms for the phase change enthalpy are given in the energy equation. The enthalpy changes are considered as energy density (see paragraph above).

The enthalpy of dry air Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_a \rho^{m_a}} (dry air = all gas phase components except water vapor) is much smaller than the enthalpy of water vapor so that it is commonly neglected. The energy equation can be expanded accordingly in specific applications(e.g., drying of porous materials by compressed air). In typical application scenarios the speed of air currents in porous substances can be neglected so that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_v j^{m_{v}}_{conv} } can also be disregarded.

Modeling of salt transport

The previously described modeling approaches and equations initially apply to salt-free materials with the added restriction of a non-variable material structure. If salts are also to be considered, several influential factors will need to be integrated into the model.

Transport mechanisms for salts/ions and modeling approaches

Generally, salts can only be transported within building materials in the presence of a liquid phase, e.g., capillary water. This transport happens through diffusion or convection.

  • Diffusion describes the (mass-centric) exchange of ions.
  • Convection describes the transport of dissolved salts together with the liquid phase.

During the convection of a salt solution through a porous material, different current paths will develop in the pore network resulting in higher salt concentrations fronts "forging ahead" or "lagging behind" in some areas. The observed spread of a concentration front, similar to a diffusion process, is termed dispersion. The effect of dispersion and diffusion is similar and difficult to differentiate in the presence of convection. In still liquids only diffusion will take place.

Phase changes

  • Crystallization Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_{s\rightarrow p}} and solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_{p\rightarrow s}}
  • Hydration Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_{p \rightarrow p,\text{hyd}}} and dehydration Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_{p,\text{hyd} \rightarrow p}}
  • Deliquescence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_{p\rightarrow s,\text{del}}}

The detailed description of the phase changing models is shown in the article "Modeling of the phase change reaction of salts ".

Influence of salts on moisture retention

To describe:

  • Increased hygroscopic moisture absorption in the presence of salt loads;
  • Reduction in vapor pressure, water activity;
  • Surface tension + Kelvin-equation.

Influence of salts on moisture transport

  • Viscosity changes.
  • Influence on density/gravitational forces.

Crystallization within and outside of the pore network

  • Decrease in pore space due to crystallization with the resultant effect on reservoir capacity and transport mechanisms.
  • Efflorescence-Models.

Literature

Note: The listed literature is still to be shown and reviewed

[Koniorczyk.etal:2008]Koniorczyk, Marcin; Gawin, Dariusz (2008): Heat and Moisture Transport in Porous Building Materials Containing Salt. Journal of Building Physics, 31 (4), 279-300Link to Google Scholar
[Hagentoft.etal:2004]Hagentoft, Carl-Eric; Kalagasidis, Angela Sasic; Adl-Zarrabi, Bijan; Roels, Staf; Carmeliet, Jan; Hens ,Hugo; Grunewald, John; Max Funk; Rachel Becker; Dina Shamir; Olaf Adan; Harold Brocken; Kumar Kumaran; Reda Djebbar (2004): Assessment Method of Numerical Prediction Models for Combined Heat, Air and Moisture Transfer in Building Components: Benchmarks for One-dimensional Cases. Journal of Thermal Envelope and Building Science, 27 (4), 327-352Link to Google Scholar