Quelltext der Seite 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:

<TABLE>
<TR><TD COLSPAN="3">g<small>w</small> = -D<small>w</small>(w)
· grad w</TD></TR>

<TR><TD>g<small>w</small></TD><TD ALIGN="RIGHT">[kg/m&sup2;s]:</TD><TD>liquid transport flux density</TD></TR>
<TR><TD>w</TD><TD ALIGN="RIGHT">[kg/m&sup2;]:</TD><TD>moisture content</TD></TR>
<TR><TD>D<small>w</small></TD><TD ALIGN="RIGHT">[m&sup2;/s]:</TD><TD>liquid transport coefficient,</TD></TR>
</TABLE>


where the liquid transport coefficient D<small>w</small> 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.


<IMG SRC="pix/e_Dw.gif" WIDTH="197" HEIGHT="152" VSPACE="0" HSPACE="0" ALT="">

<P>
The liquid transport coefficient is not a pure material property,
however - it depends on the material as well as the boundary
conditions [1].
</P>
<P>
The <B>liquid transport coefficient for suction D<small>ws</small></B>
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.
</P>
<P>
The <B>liquid transport coefficient for redistribution
D<small>ww</small></B> 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.
</P>
<P>
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.
</P>
<P>
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.
</P>
<P>
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<!--; cf.
the printable material data sheets //-->).
</P>
<P>
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.
</P>
<P>
The last entry is also used for all higher moisture contents up
to w<small>max</small>. 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.).<BR>
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.
</P>
<P>
Example: Baumberger sandstone
</P>

<CENTER>
<TABLE>
<TR ALIGN="CENTER"><TD>w</TD><TD>&nbsp;</TD><TD>D<small>ws</small></TD></TR>
<TR ALIGN="CENTER"><TD>[kg/m&sup3;]</TD><TD>&nbsp;</TD><TD>[m&sup2;/s]</TD></TR>
<TR><TD ALIGN="RIGHT">0  </TD><TD>&nbsp;</TD><TD>0</TD></TR>
<TR><TD ALIGN="RIGHT">31 </TD><TD>&nbsp;</TD><TD>2.5E-10</TD></TR>
<TR><TD ALIGN="RIGHT">52 </TD><TD>&nbsp;</TD><TD>3.9E-9 </TD></TR>
<TR><TD ALIGN="RIGHT">84 </TD><TD>&nbsp;</TD><TD>5.2E-9 </TD></TR>
<TR><TD ALIGN="RIGHT">126</TD><TD>&nbsp;</TD><TD>1.0E-8 </TD></TR>
<TR><TD ALIGN="RIGHT">168</TD><TD>&nbsp;</TD><TD>2.8E-8 </TD></TR>
<TR><TD ALIGN="RIGHT">189</TD><TD>&nbsp;</TD><TD>5.0E-8 </TD></TR>
<TR><TD ALIGN="RIGHT">200</TD><TD>&nbsp;</TD><TD>1.0E-7 </TD></TR>
<TR><TD ALIGN="RIGHT">210</TD><TD>&nbsp;</TD><TD>3.0E-7 </TD></TR>
</TABLE>
</CENTER>

<P>
This table states, among other things, that
</P>
<UL>
<LI>there is no capillary conduction at a moisture content
    below 31 kg/m&sup3;,</LI> 
<LI>at a moisture content of 40 kg/m&sup3;, the liquid transport
    coefficient for suction is 8.1E-10 m&sup2;/s (interpolated), and</LI>
<LI>for all moisture contents above 210 kg/m&sup3;, a liquid transport
    coefficient of 3.0E-7 m&sup2;/s is being used.</LI>
</UL>


<P>
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.
</P>
<P>
&nbsp;
</P>
<P>
Unfortunately, measured liquid transport coefficients are available only
for a relatively small number of materials. The ability to at least
<B>estimate</B> them from standard material data is therefore desirable.
</P>
<P>
In many cases, the increase of D<small>ws</small> 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 D<small>ws</small> and the A-value A:
</P>

<TABLE>
<TR><TD COLSPAN="3">D<small>ws</small>(w) = 3.8 · (A / wf)&sup2; · 1000<sup>(w / wf) - 1</sup></TD></TR>
<TR><TD>D<small>ws</small></TD><TD ALIGN="RIGHT">[m&sup2;/s]:</TD><TD>liquid transport coefficient for suction</TD></TR>
<TR><TD>A</TD><TD ALIGN="RIGHT">[kg/m&sup2;s<sup>1/2</sup>]:</TD><TD>water absorption coefficient</TD></TR>
<TR><TD>w</TD><TD ALIGN="RIGHT">[kg/m&sup3;]:</TD><TD>moisture content</TD></TR>
<TR><TD>w<small>f</small></TD><TD ALIGN="RIGHT">[kg/m&sup3;]:</TD><TD>free water saturation</TD></TR>
</TABLE>

<P>
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:
</P>

<CENTER>
<TABLE>
<TR ALIGN="CENTER"><TD>w</TD><TD>D<small>ws</small></TD></TR>
<TR ALIGN="CENTER"><TD>[kg/m&sup3;]</TD><TD>[m&sup2;/s]</TD></TR>
<TR><TD>0</TD><TD>0</TD></TR>
<TR><TD>w<small>80</small></TD><TD>D<small>ws</small>(w<small>80</small>)</TD></TR>
<TR><TD>w<small>f</small></TD><TD>D<small>ws</small>(w<small>f</small>)</TD></TR>
</TABLE>
</CENTER>

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

<CENTER>
<TABLE>
<TR ALIGN="CENTER"><TD>w</TD><TD>D<small>ww</small></TD></TR>
<TR ALIGN="CENTER"><TD>[kg/m&sup3;]</TD><TD>[m&sup2;/s]</TD></TR>
<TR><TD>0</TD><TD>0</TD></TR>
<TR><TD>w<small>80</small></TD><TD>D<small>ws</small>(w<small>80</small>)</TD></TR>
<TR><TD>w<small>f</small></TD><TD>D<small>ws</small>(w<small>f</small>)/10</TD></TR>
</TABLE>
</CENTER>

<P>
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 <I>shape</I> 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.
</P>
<P>
Please note that the water absorption coefficient here has SI units
[kg/m&sup2;s<sup>1/2</sup>], whereas the relevant German standard uses
[kg/m&sup2;h<sup>1/2</sup>]. Divide the latter values (e.g. 2.6
kg/m&sup2;h<sup>1/2</sup> for Baumberger sandstone) by 60 to
get SI units (0.043 kg/m&sup2;s<sup>1/2</sup>).
</P>
<P>
Literature:
</P>
<TABLE>
<TR><TD VALIGN="TOP">[1]</TD><TD>Krus, M.: Moisture Transport and
Storage Coefficients of Porous Mineral Building Materials. Theoretical
Principles and New Test Methods<BR>
Fraunhofer IRB Verlag, 1996</TD></TR>

<TR><TD VALIGN="TOP">[2]</TD><TD>Krus, M., K&uuml;nzel H.M.:
Fl&uuml;ssigtransport im Übersättigungsbereich.<BR>
IBP-Mitteilung 22 (1995), Nr. 270.</TD></TR>
</TABLE>