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Water Vapor Diffusion

This help topic discusses several physical quantities used to describe water vapor diffusion, since these quantities and their interrelationships may not be familiar to all WUFI users.

Water vapor diffusion resistance factor, µ-value

Diffusion of water vapor in air can be described by the equation

gv = -δ * dp/dx (in air),
gv[kg/m²s]:water vapor flux density
p[Pa]:water vapor partial pressure
δ [kg/(msPa)]:water vapor diffusion coefficient in air,
δ = 2.0·10-7 T0.81 / PL [1]
T[K]:absolute ambient temperature
PL[Pa]:ambient atmospheric pressure.

In porous building materials, diffusion likewise takes place in air (in the pore spaces), but it is impeded by the reduction of the accessible cross-section, adsorption effects at the pore walls and the tortuosity of the pore paths. In the context of building physics, it is admissible to allow for this by simply introducing a diffusion resistance factor µ:

gv = -(d/µ) * dp/dx (in porous material),
µ[-]:water vapor diffusion resistance factor.

Retaining δ as a separate coefficient in the above equation has the advantage that it already describes the temperature and pressure dependence of water vapor diffusion and µ is therefore practically independent of temperature and pressure, i.e. it is a constant which only depends on the material in question.

However, measurements of µ which are performed at different levels of relative humidity (dry-cup and wet-cup) may result in different values for one and the same material. This is due to surface diffusion which becomes noticeable at higher humidities but would be more properly treated as liquid transport [2]. This additional moisture transport is usually not separated out in the analysis of the measurements and, lumped together with vapor diffusion, reduces the apparent diffusion resistance, resulting in a lower µ-value. In these cases, it is more appropriate to use a constant µ-value and to adjust the liquid transport coefficients to include surface diffusion. However, WUFI also allows to use a moisture-dependent µ-value to simplify the treatment of cases where this distinction can be neglected.

The µ-value represents the ratio of the diffusion coefficients of water vapor in air and in the building material and has therefore a simple interpretation: it is the factor by which the vapor diffusion in the material is impeded, as compared to diffusion in stagnant air. For very permeable materials, such as mineral wool, the µ-value is thus close to 1, whereas it increases for materials with greater diffusion resistance.

The following table lists µ-values for some common materials:

(3% - 50% RH)
(50%-93% RH)
cellular concrete 7.77.1
lime silica brick 27 18
solid brick 9.58.0
gypsum board 8.37.3
concrete (B25) 110150 (*)
cement-lime plaster 19 18
lime plaster 7.36.4
Saaler sandstone 60 28
Baumberger sandstone 20 17
Worzeldorfer sandstone38 22

(*) the increase of the µ-value in the wet-cup measurement of concrete is probably due to swelling effects [3].

In WUFI, the µ-value is one of the basic material properties which needs to be specified for each material.


Vapor diffusion thickness, sd-value

Assuming constant temperature and µ-value, the diffusion flow through a material layer with thickness Dx=s is

gv_mat = -δ/µ * Δp/Δx = -δ/(µ*s) * Δp,

whereas the diffusion flow through a stagnant air layer (µ=1) with thickness sd is

gv_air = -δ/µ * Δp/Δx = -δ/(1*sd) * Δp = -δ / sd * Δp.

Dividing the former equation by the latter yields

gv_mat / gv_air = sd / (µ*s).

If the air layer is chosen so that its vapor diffusion resistance is the same as that of the material layer of thickness s (and therefore gv_mat = gv_air), then its thickness sd must be

sd = µ*s.

For a material layer with diffusion resistance factor µ and thickness s, the product µ*s thus gives the thickness which a stagnant air layer would need in order to have the same diffusion resistance. This "sd-value" or "vapor diffusion thickness" expresses the diffusion resistance of a layer in a form which is easily understood and applied.

Since the definition of the sd-value contains the the thickness Δx of the layer, the sd-value is a property of the given layer, not of the material itself. Two layers made from the same material but with different thicknesses will have different sd-values, but in both cases the material will have the same µ-value.

For some building components, only their total diffusion resistance is of importance but not their µ-value and their thickness separately. They may then simply be specified in terms of their sd-value. Also, measuring the sd-value does not require to determine the thickness of the sample.
In particular, the sd-value is used to characterise vapor retarders (sd >= 10 m), vapor barriers (sd >= 1000 m) and surface coatings (mineral paints: sd ~ 0.04 m, oil paints: sd = 1.0 .. 2.6 m), where it can be difficult to determine their thickness properly.

In WUFI, surface coatings on the component need not be explicitly modelled in the component assembly; their diffusion resistance may instead be taken into account by simply specifying their sd-value in the dialog Edit Surface Coefficients.


Permeance, Permeability

In the formula for water vapor diffusion in a porous material

gv = -δ/µ * Δp/Δx = - δ/sd * Δp =: - Δ * Δp

the so-called permeance Δ may be introduced by defining

Δ := δ/sd [kg/(m²sPa)].

Since the definition of the permeance contains the thickness Δx of the layer, the permeance is a property of the given layer, not of the material itself. Two layers made from the same material but with different thicknesses will have different permeances.

In Inch-Pound units the permeance is measured in perm. One perm is one grain (avoirdupois) of water vapor per hour flowing through one square foot of a layer, induced by a vapor pressure difference of one inch of mercury across the two surfaces. In SI units, this corresponds to 57.45e-12 kg/(m²sPa) [4]. Therefore, with Δ expressed in perm:

57.45e-12*Δ = δ/sd [kg/(m²sPa)], or
sd = δ/(57.45e-12*Δ) [m].

For a reference temperature of 5°C and a barometric pressure of 1013.25 hPa, δ has the value 1.884e-10 kg/(msPa) (see above), and we obtain

sd = 3.28/Δ,

where Δ is expressed in perm and sd in m.

The permeance describes a property of a specific construction layer with a given thickness. Multiplying the permeance by the layer thickness Δx yields the permeability Π [perm inch] of the layer material:

Δ * Δx = δ/sd * Δx = δ/µ =: Π , so that
gv = Π Δp/Δx.

The conversion factor for Δ*Δx is 57.45e-12 [kg/(m²sPa)]/[perm] * 0.0254 [m]/[in] = 1.459e-12 [kg/(msPa)]/[perm in]:

1.459e-12*Π = δ/µ [kg/(msPa)],

and with the same reference value for δ as above:

7.744e-3*Π = 1/µ [-], or

µ = 129 / Π [-],

where Π is expressed in perm inch and µ is dimensionless.


[1]Schirmer, R.: Die Diffusionszahl von Wasserdampf-Luft-Gemischen und die Verdampfungsgeschwindigkeit,
Beiheft VDI-Zeitschrift, Verfahrenstechnik (1938), H. 6, p. 170-177.
[2]Krus, M.: Moisture Transport and Storage Coefficients of Porous Mineral Building Materials. Theoretical Principles and New Test Methods
Fraunhofer IRB Verlag, 1996
[3]Holm, A., Krus, M., Künzel, H.M.: Concrete from the viewpoint of moisture technology: Parameters and mathematical approaches to the evaluation of climatic effects in external structural elements made of concrete.
To be published in Concrete Science and Engineering
[4]ASHRAE Terminology of Heating, Ventilation, Air Conditioning, & Refrigeration, 2nd ed. 1991