# Modeling of Salt and Humidity Transport

<bibimport/>
*Author: Dr. Andreas Nicolai *

English version by Christa Gerdwilker

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## Contents

# Introduction

Salt transport in porous materials (building materials, stone, floors, etc.) is dependent on many factors, such as: the type of salt, its composition and distribution in the porous material. Furthermore, the microscopic structure of the materials, including pore types and pore size distribution, and the presence of moisture in it, as well as the 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 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 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 through 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 dispersion.
- 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*

of the HAMSTAD Projekt.

# 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.

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 contentFailed 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 of 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,. It is not applicable in practice when cooling during evaporation, and 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

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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]**Title: ** *Heat and Moisture Transport in Porous Building Materials Containing Salt***Author: ** *Koniorczyk, Marcin; Gawin, Dariusz*

[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), 327-352 |