Moisture transport mechanisms

Author: Dr. Andreas Nicolai

English version by Sandra Leithäuser

back to Saltwiki:Community portal

Linked Articles

• Modeling of Salt and Humidity Transport
• Moisture retention in porous materials

Abstract

This article describes the different moisture transport mechanisms and common model approaches for the mathematical description of moisture flows.

back to Modeling of salt- and humidity transport

Vapor diffusion

The vapor diffusion process is an exchange between water vapor molecules and other molecules of the "dry air". Dry air in this context means: all molecules that constitute the air, apart from water vapour. According to the thermodynamic definition, the vapor diffusion process is a "mass-centric??" exchange process. The thermodynamic driving force (thrust?) for diffusion is entropy production, or in the transferred sense, the "gradient of the chemical potential??". In the field of building physics the use of this gradient can, under the usual assumptions and limitations, be transferred to the gradient of vapour pressure. Under (relatively unrealistic) isobaric conditions, the gradient of vapour concentration can also be used.

The diffusive vapour flow can now be described as a product of the vapour pressure gradient and the conductivity of vapour Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle delta_v} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K_v}  :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j^{v}_{dif\!f} = - K_v \frac{\partial p_v}{\partial x} = - \delta_v \frac{\partial p_v}{\partial x} }

Here Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_v} is vapor pressure and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j^{m_v}_{dif\!f}} is vapour current density as the mass flow density. The vapour conductivity, i.e. the transport coefficient, can be described in different ways in porous materials. Common formulations are:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j^{v}_{dif\!f} = - \frac{D_v(\theta_\ell)}{R_v T} \frac{\partial p_v}{\partial x}\\ }

with the help of the moisture content dependant vapor diffusivity function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_v} of the material. Alternatively, the vapour diffusivity of the air Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_{v,air} } in combination with the water vapor diffusion resistance factor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} can be used. However, in this approach the reduction of the vapor diffusion in the available air pore cross-section through the ratio Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\theta_\ell}{\theta_{ef\!f}}} has to be explicitly taken into account.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j^{v}_{dif\!f} = - \frac{D_{v,air}}{\mu R_v T} \frac{\theta_\ell}{\theta_{ef\!f}} \frac{\partial p_v}{\partial x} }

Vapor convection

If , due to a pressure gradient, air or pore gas flows through the pore system, all components of the gas phase are transported by convection. The air flow in a porous system can be described by Darcy’s law (laminar, not turbulent). The driving force for the air flow is the gradient of the air-pressure, i.e. the gas-pressure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_g} . The distinction between dry air and gas (gas=dry air + water vapor) is not absolutely necessary in low temperature and relative humidity conditions, because the partial pressure of water vapor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_v} (in the range of 1-3kPa) is much lower than the air pressure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_a} (in the range of 100 kPa). If the model, however, is used at higher temperatures this distinction is important.

The gas mass flow density is given with:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j^{g} = -K_g \frac{\partial p_g}{\partial x} }

with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_g = p_a + p_v } ,

at moderate temperatures applies:Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_g \simeq p_a} .

The proportion of water vapour transported by convection can easily be calculated by multiplying the concentration of water vapour Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_v} (thermodynamic definition, mass component per mass of phase).

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j^{v}_{conv} = c_v \; j^{g} }

Liquid water conduction/capillary conduction

The mechanism of liquid water conduction/capillary conduction within porous materials is best explained by looking at the effect a drinking straw causes inside a glass of water. Due to surface tension water rises inside a thin tube (capillary) against gravity. The thinner the capillary, the higher the liquid rises in comparison to larger capillaries.

Now a pore system may be a bundle of different capillary tubes with a diverse range of diameters. Brickwork, for instance, has many capillaries with a diameter ranging between 1-10 µm. The level of capillary rise is higher in thin capillaries than in large capillaries. Even when the moisture is introduced horizontally (without the counteraction of gravity) higher tension in the thin capillaries increases the moisture penetration into the building material.

However, because friction loss is higher in thin pores, the velocity of the water flow is higher in large pores. Thus, the transport rate does not only depend on the presence of thin pores and high " suction ?" tension, but also on the diameter of the available tubes. This effect has to be taken into account when making moisture transport models.

Under microscopic magnification porous materials often resemble the idealized bundle of tubes, yet the capillary effect is similar with other geometries. Therefore the capillary tube model is fairly representative for the application on any possible microscopic geometry.

Diffusivity approach

The capillary transport generally leads to capillary water from areas of high humidity to re-distribute to areas of low humidity. This idea is based on the diffusivity approach. The driving force of moisture transport is therefore used as the gradient of moisture content. The "relationship (function, dependence??) " of "redistribution speed?(velocity?) " is described by the transport coefficient of moisture diffusivity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_\ell} . (Die Abhängigkeit der Umverteilungsgeschwindigkeit wird durch den Transportkoeffizienten, der Feuchtediffusivität Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_\ell} beschrieben.)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j^{w} = - \rho_\ell D_\ell(\theta_\ell) \frac{\partial \theta_\ell}{\partial x} }

The transport coefficient depends on moisture. Conventional approaches to models are based on the idea of a steadily increasing diffusivity as a function of increasing moisture content.

The diffusivity approach has an advantage: it is easy to define parameters and the diffusivity function has a low non-linearity. (Der Diffusivitätsansatz hat den Vorteil einer einfachen Parametrisierung und geringen Nichtlinearität der Diffusivitätsfunktion) On the downside, however, this approach does not allow for the description of liquid water transport in conjunction with positive pressures, e.g. due to groundwater, because, when total moisture saturation occurs, the driving potential is dropped (the moisture content gradient is zero) and therefore the model shows no moisture content at all.

(Hydraulic?) Conductivity approach/ Darcy-flow model

The limitations of the diffusivity model are compensated through the use of a pressure gradient. Analogously to air flow (through a porous material?) , the Darcy- flow is thought to run through the porous material, driven by the gradient of liquid water pressure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_\ell} .

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j^{w} = - K_\ell(\theta_\ell) \frac{\partial p_\ell}{\partial x} }

(Die Einschränkungen des Diffusivitätsmodells werden durch die Verwendung eines Druckgradienten aufgehoben. Analog zur Luftdurchströmung wird eine Darcy-Strömung durch das poröse Material angenommen, angetrieben vom Gradienten des Flüssigwasserdrucks Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_\ell} .)

In moisture saturated materials the impressed? hydrostatic water pressure can simply be used as a driving force. (In feuchtegesättigten Materialien kann einfach der aufgeprägte hydrostatische Wasserdruck als treibende Kraft verwendet werden.) In partially saturated materials the liquid water pressure is < 0 and corresponds to the isobaric conditions of capillary pressure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_c} . In the case of air pressure differences the equilibrium applies:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_\ell = p_c + p_g }

Yet, when looking at the degree of normal air pressure differences (10- 100 Pa) and the differences in capillary pressures (0-10⁷ Pa), it becomes apparent that, for practical problems, isobaric conditions can be expected. At this point the relationship between capillary pressure and moisture content is crucial. The article Moisture retention in porous materials describes these relationships in detail.

The conductivity of liquid water Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K_\ell} is a highly non-linear function and therefore the parameterization of the model is accordingly challenging. The advantage of the conductivity model is the continuity of the driving force (thrust?) at material boundaries. In the equilibrium, the liquid water pressure at a layer boundary of a material is identical on both sides, whereas the equilibrium moisture content of two materials can be greatly different. The resulting leap in moisture content has mainly consequences on the numerical solution methods, when using the diffusivity model.

Kirchhoff- potential formula Kirchhoff-Potential-Formulierung

The high non-linearity of the moisture transport functions leads to difficulties for numerical solutions (in particular interrelationships between results of averaging methods and calculation grids). The introduction of mathematical transformation allows a generalization of the moisture transport description and the reduction of numerical problems. The Kirchhoff- potential function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Psi_\ell(\theta_\ell)} is introduced and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j^{w} = - \frac{\partial \Psi_\ell}{\partial x} } applies.

A comparison of the coefficients of the previously shown model demonstrates:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial \Psi_\ell}{\partial x} = K_\ell(\theta_\ell) \frac{\partial p_\ell}{\partial x} = \rho_\ell D_\ell(\theta_\ell) \frac{\partial \theta_\ell}{\partial x} }

and therefore

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K_\ell = \frac{\partial \Psi_\ell}{\partial p_\ell} }

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_\ell = \frac{1}{\rho_\ell } \frac{\partial \Psi_\ell}{\partial \theta_\ell} }

applies. At a given diffusivity function or respectively conductivity function the Kirchhoff- potential function can be determined by integration, making the numerical solution independent of the actual physical modeling.

Weblinks

http://web.byv.kth.se/bphys/pdf/art_0299.pdf

Literature

http://jmst.ntou.edu.tw/marine/7-2/125-131.pdf http://www.elsevierdirect.com/companions/9780122578557/PDFs/Ch_3.pdf