Details:LiquidTransportCoefficients

Liquid Transport Coefficients

The predominant moisture transport mechanism in capillary porous materials is the capillary liquid transport. Although it is basically a convective phenomenon, in the context of building physics it is sufficiently accurate to regard the liquid transport in the pore spaces as a diffusion phenomenon:

gw = -Dw(w) · grad w
gw	[kg/m²s]:	liquid transport flux density
w	[kg/m²]:	moisture content
Dw	[m²/s]:	liquid transport coefficient,

where the liquid transport coefficient Dw is generally strongly dependent on the moisture content. The main reason why the liquid transport can adequately be described by a diffusion formula is that it correctly reproduces the linear increase of the imbibed amount of liquid over the square root of time.

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The liquid transport coefficient is not a pure material property, however - it depends on the material as well as the boundary conditions [1].

The liquid transport coefficient for suction Dws describes the capillary uptake of water when the imbibing surface is fully wetted. In the context of building physics this corresponds to rain on a facade or an imbibition experiment. The suction transport is dominated by the larger capillaries, since their lower capillary tension is more than compensated by their markedly lower flow resistance.

The liquid transport coefficient for redistribution Dww describes the spreading of the imbibed water when the wetting is finished, no new water is taken up any more and the water present in the material begins to redistribute. In a building component, this corresponds to the moisture migration in the absence of rain. The redistribution is dominated by the smaller capillaries since their higher capillary tension draws the water out of the larger capillaries.

Since the redistribution is a slower process (taking place in the small capillaries with their high flow resistance), the corresponding liquid transport coefficient is generally markedly less than the coefficient for suction.

WUFI therefore uses two distinct liquid transport coefficients for each capillary-active material which are used depending on the boundary conditions (rain / no rain). Each coefficient is entered in a separate table.

As a rough approximation, the liquid transport coefficients show a more or less exponential dependence on the moisture content. That is why WUFI uses a logarithmic interpolation between the table entries (i.e., linear in a semilogarithmic diagram).

The first entry in a table should be the pair (0 ; 0); the next entry then corresponds to the moisture content below which the liquid transport is negligible, i.e., the transport coefficients are zero. The logarithmic interpolation automatically results in interpolated values of zero for the coefficients below that threshold.

The last entry is also used for all higher moisture contents up to wmax. Although there is little capillary conduction above free saturation [2], there might be other transport mechanisms which could approximately be described by finite liquid transport coefficients (convection due to gravitation or pressure differentials etc.).
To allow for this kind of transport processes (if desired), WUFI accepts arbitrary liquid transport coefficients for this moisture region, too. In general, you will rarely encounter moisture contents in this region which is difficult to treat in calculation. By setting the last entry to zero, the capillary conduction in the moisture region above the last entry may be suppressed.

Example: Baumberger sandstone

w		Dws
[kg/m³]		[m²/s]
0		0
31		2.5E-10
52		3.9E-9
84		5.2E-9
126		1.0E-8
168		2.8E-8
189		5.0E-8
200		1.0E-7
210		3.0E-7

This table states, among other things, that

• there is no capillary conduction at a moisture content below 31 kg/m³,
• at a moisture content of 40 kg/m³, the liquid transport coefficient for suction is 8.1E-10 m²/s (interpolated), and
• for all moisture contents above 210 kg/m³, a liquid transport coefficient of 3.0E-7 m²/s is being used.

During the calculation WUFI performs iterations where it samples small regions of the tabulated curve. Very sharp bends in the curve may throw the iteration off and thus impede its convergence. In such a case, the curve should be smoothed by inserting additional points. A large number of entries may slow the search in the table, however.

 

Unfortunately, measured liquid transport coefficients are available only for a relatively small number of materials. The ability to at least estimate them from standard material data is therefore desirable.

In many cases, the increase of Dws with increasing moisture content can be approximately described by an exponential function which, for most mineral building materials, spans about three powers of ten. Under these conditions, there is the following approximate relation between Dws and the A-value A:

Dws(w) = 3.8 · (A / wf)² · 1000(w / wf) - 1
Dws	[m²/s]:	liquid transport coefficient for suction
A	[kg/m²s1/2]:	water absorption coefficient
w	[kg/m³]:	moisture content
wf	[kg/m³]:	free water saturation

In this way, WUFI can automatically generate a table with estimated liquid transport coefficients for suction. Only the water absorption coefficient needs to be entered, and WUFI will use it, the moisture storage function and the above formula to generate a table with the following entries:

w	Dws
[kg/m³]	[m²/s]
0	0
w80	Dws(w80)
wf	Dws(wf)

In the same way, WUFI can also generate an estimated table with liquid transport coefficients for redistribution. WUFI will create this:

w	Dww
[kg/m³]	[m²/s]
0	0
w80	Dws(w80)
wf	Dws(wf)/10

Please note that this method is just a rough estimate that proves successful in many cases, but which is not necessarily useful for all materials. Especially, there may be inaccuracies in the shape of the suction profiles. A generated table is just meant to be some assistance; you should not blindly rely on it. Future WUFI versions are planned to offer more sophisticated methods.

Please note that the water absorption coefficient here has SI units [kg/m²s^1/2], whereas the relevant German standard uses [kg/m²h^1/2]. Divide the latter values (e.g. 2.6 kg/m²h^1/2 for Baumberger sandstone) by 60 to get SI units (0.043 kg/m²s^1/2).

Literature:

[1]	Krus, M.: Moisture Transport and Storage Coefficients of Porous Mineral Building Materials. Theoretical Principles and New Test Methods Fraunhofer IRB Verlag, 1996
[2]	Krus, M., Künzel H.M.: Flüssigtransport im Übersättigungsbereich. IBP-Mitteilung 22 (1995), Nr. 270.