A MODEL FOR ADVECTION HEAT AND MOISTURE FLOWS IMPLEMENTED IN
A PROGRAM FOR WHOLE-BUILDING HYGROTHERMAL SIMULATION
Karl Grau1 and Carsten Rode 2
1
Danish Building Research Institute, Aalborg University
2
Technical University of Denmark
wind direction, as well as the surroundings of the
ABSTRACT building.
A model for calculating exfiltration &infiltration air
flows in exterior building envelope constructions has Filtration through small cavities in a construction can
been implemented in the whole-building hygrother- be expressed according to the Classic Orifice method
mal simulation tool BSim. The tool is able to predict (Jensen, 2005) as:
indoor humidity conditions using a transient model q = k ⋅ A ⋅ 2 ⋅ ΔP / ρ
inf
(1)
for the moisture conditions in the building envelope.
where
qinf Air volume, m3/s
INTRODUCTION
When air flow passes through the building envelope, k Constant taking into account the fric-
it has some notable influence on the heat flow tion of the cavity, -
through the wall. However, the impact on the mois- A Area of cavity, m2
ture conditions can, even for small air flows, com- ΔP Pressure difference, Pa
pletely change the situation, e.g. from a healthy con- ρ Air density, kg/m3
struction to a disaster. Air flows from warm/humid
environment towards colder climates should gener- ΔP = pn − pi (2)
ally be avoided, whereas air flows in the opposite
direction could have a drying effect. where
pn Air pressure on the construction facing
When infiltrating air flows can be seen to provide outside, Pa
some of the required fresh air supply to the indoor pi Air pressure on inside, Pa
environment, then the so-called dynamic wall princi-
ples applies, and the wall itself works as a heat ex- The air pressure can be expressed as
changer where some of the heat lost by conduction is pn = ½ ⋅ cpn ⋅ ρ ⋅ v 2 (3)
recovered by the ingoing air. where
For these reasons there is a desire to implement a cpn Pressure coefficient on outside, -
model for advective heat and moisture flow in a ρ Air density, kg/m3
whole building simulation tool. BSim from the Dan- v Air velocity in reference height, m/s
ish Building Research Institute has been chosen as and
one such tool. BSim has been presented previously to
IEA Annex 41 (Rode & Grau, 2003 and 2004) and in pi = ½ ⋅ cpi ⋅ ρ ⋅ v 2 (4)
Grau & Rode, 2005. A multizone model for air flow where
between zones (rooms) within a building has been cpi Pressure coefficient for the zone, -
implemented into BSim as described previously by
Because of continuity, the sum of infiltration and
Grau & Rode, 2005. For more information about
exfiltration through constructions facing the zone
BSim, see www.bsim.dk.
must be zero, the pressure coefficient can be calcu-
lated.
MODEL FOR INFILTRATION &
∑ ∑
n n
EXFILTRTAION AIR FLOWS cp =i 1
A ⋅ cp /
k k A
1 k
(5)
The infiltration and exfiltration air flows in exterior where
constructions is dependent on the wind speed and Ak Area of construction k facing outside,
m2
cpk Pressure coefficient for construction k, - where:
ρ Density of the material kg/m
3
The filtration through a construction then can be
expressed as cp Specific heat J/(kg K)
Δx Width of the control volume m
qinf, m = fac ⋅ ½ ⋅ cp n −cpi ⋅ v 2 / Ac (6)
T Temperature K
where Δt Time step s
qinf,m Filtration, m3/s/m2 λ Thermal conductivity W/(m K)
R Possible interface resistance 2
m K/W
fac A user given factor taking into account between control volumes
the area and friction of the cavities in the i Index for control volume number -
construction, - j Index for time step -
Ac Area of the construction, m2
With air flow, equation (7) is modified as follows:
cp n −cpi Infiltration
T j +1 − Ti j
>0 ( )
ρ c p i Δxi i
Δt
=
cp n −cpi Exfiltration j+
Ti +1 1 − Ti j +1 j+
Ti j +1 − Ti −1 1
<0 + + (8)
Δxi Δxi +1 Δxi −1 Δxi
The pressure coefficients are dependant of the orien- + + Ri +½ + + Ri −½
tation and slope of the construction, the wind direc- 2 λi 2 λi +1 2 λi −1 2 λi
tion, and of the surroundings, sheltered or exposed.
In BSim there is tables of typical pressure coeffi-
(
qair ρ c p Tupstream − Ti j +1
j +1
)
cients that will be used. The wind speed is dependent where:
on the height of the building, and of the terrain type, Tupstream Temperature in the adjacent control K
e.g. opens flat country, or located in a city. volume from which the air comes
It is realized that there could be a need for implemen- For vapour diffusion, the finite difference form of
tation of another air flow model, such the air flow is the moisture balance equation looks:
j+1 j
governed by Darcy flow through porous wall materi- pi p
j+1
- ji
als. This would give some other air flow rates, where p s ,i p s , i
( ρ ξ )i Δ x i =
the rate is proportional to the pressure difference Δt (9)
across the building envelope, as opposed to in the j+1
p i-1 - p i
j+1 j+1
p i+1 - p i
j+1
here implemented model where it is proportional to +
Δxi −1 Δxi Δxi Δxi +1
the square root of the pressure difference. + + Z i −½ + + Z i+½
2 δ i-1 2 δ i 2 δ i 2 δ i+1
PROCEDURES TO CALCULATE HEAT where:
AND MOISTURE TRANSPORT ξ Moisture capacity kg/kg
(slope of sorption isotherm)
THROUGH POROUS WALLS WITH A p Vapour pressure Pa
FILTRATING AIR FLOW ps Saturation vapour pressure Pa
δ Water vapour permeability kg/(m s Pa)
The filtration air flow qinf,m is calculated accord- Z Possible interface water va- 2
Pa m s/kg
ing to the procedures described in the previous pour resistance between con-
section. trol volumes
Both temperature and vapour pressures are With air flow, equation (9) is modified as follows:
calculated in BSim using an implicit finite control pi
j+1
p
j
volume scheme j+1
- ji
p s ,i p s , i
( ρ ξ )i Δ x i =
For heat conduction, the finite difference form of Δt
j+1 j+1 j+1 j+1
the heat balance equation form for one control p i-1 - p i p i+1 - p i
+ + (10)
volume looks: Δxi −1 Δxi Δxi Δxi +1
+ + Z i −½ + + Z i+½
T j +1 − Ti j 2 δ i-1 2 δ i 2 δ i 2 δ i+1
( )
ρ c p Δxi i =
i Δt
qair
(p j +1
upstream − pij +1 )
j+
Ti +1 1 − Ti j +1 j+
Ti j +1 − Ti −1 1 (7)
RvTi
+
Δxi Δxi +1 Δxi −1 Δxi where:
+ + Ri +½ + + Ri −½
2 λi 2 λi +1 2 λi −1 2 λi pupstream Vapour pressure in the adja- K
cent control volume from
which the air comes u Moisture content kg/kg
Rv Gas constant for water vapour J/(kg K
= 461.5 )
T Absolute temperature K The equilibrium relative humidity is determined for
each control volume by taking he calculated moisture
A “new” vapour pressure distribution has been de- contents and using the sorption curve for the relevant
termined. However, since the whole problem is non- material (that is, an inverse expression for the sorp-
linear (because the moisture capacity is not a con- tion curve will be used).
stant value), a special procedure must be followed to
ensure that the mass balance will be correct. The EXAMPLE 1 - VALIDATION
vapour pressures just found must be regarded as An existing BSim model from one of the variations of
preliminary indications of the new values. It could Common Exercise 1 (the BESTEST building) has
even be that some of the preliminary vapour pres- been taken as starting point for illustrating the new
sures have values that exceed the values of ps at the model with filtration air flow in the building enve-
same location, or they may be negative. This is either lope. The building consists of 150 mm solid aerated
neglected at first (!), or the time step is repeated with concrete walls, and the walls have no surface coat-
a smaller Δt. The found vapour pressures are used in ing.
the moisture balance of each control volume as fol- One of the exterior walls of the building is split up
lows. such that half of the wall has a filtration air flow
going through it, while the other is calculated as
The vapour flux across the interface between being air tight. See Figure 1. The building is exposed
control volumes, i and i+1, for the time step to an outdoor climate which artificially was constant
from j to j+1 is: at 20ºC and 30% RH, while the indoor climate was
j+1
pi
j+1
- p i-1 assumed operated so it was constantly 30ºC and 50%
j+1
g i −½ = - RH. Outside, the wind was assumed blowing from
Δ x i-1 Δ (11)
+ x i + Z i −½ west at 5 m/s, so by turning the building so the fa-
2 δ i-1 2δ i çade with filtration air flow either faces the wind, or
where: is on the leeward side, then the heat and moisture
g Water vapour flux kg/m2s flows in the building envelope could be investigated
with the advection phenomena. With constant
boundary conditions and constant air flows, the re-
The vapour flux is calculated from the vapour pres- sults can be compared with results of analytical cal-
sures at the end of the time step, as predicted by culations. The air flows were adjusted to always give
Equation (10). A similar equation to Equation (11) is 2
a mass flow rate of 0.0005 kg/(m s), corresponding
set up for gi+½ and the increase in moisture content 3 2
to 1.57 m /(m h) – either infiltrating or exfiltrating.
control volume i is found as follows:
( )
j+1
- uij The mesh had control volumes of thickness 1.7 mm
ρ i Δ xi ui +
= - g ij+11 - g ij+1 (12) closest to the indoor climate, increasing to 10 mm
Δt through most of the wall, so there were a total of 17
where: control volumes to represent the wall.
Figure 1 BESTEST building modelled in BSim. The wall that that is split in two halves has one half calcu-
lated with the filtration model, whereas the other half is calculated as being air tight.
2
The analytical solution for the temperature is given M air Mass flow rate of air kg/(m s)
by (Hagentoft, 2001):
Figure 2 shows the comparison between the tem-
x peratures calculated numerically using BSim and
e −1 the analytical solutions for exfiltration, no air flow,
T ( x) = Tout + (Tin − Tout ) L
(13)
and infiltration, respectively.
e −1
where The table below gives a comparison between the
analytical and numerically calculated temperatures
L Thickness of the wall m
in the middle of the wall.
A quantity of dimension length that m
characterises the interaction between Analytical Numerical
conductive and convective heat flows,
such that: Exfiltration 25.52 25.55
λ No air flow 25.00 25.04
= (14)
cair M air Infiltration 24.48 24.50
where
cair Heat capacity of air = 1005 J/(kg K)
Temperature
30
Infiltration
air flow,
kg/m²s
-0.0005
Temperature, °C
0.000
25
0.0005
20
0.000 0.050 0.100 0.150
x, m
Figure 2 Analytical (solid lines) and numerical calculation (symbols) of temperatures through the wall with
exfiltration (red), no air flow (bright green) and infiltration (blue) air flow through the outer wall.
The temperature determines the saturation vapour Vapour Pressure
pressure. An approximation of the analytical va- 2100.00
pour pressure is found as: 1900.00
x 1700.00 Infiltration
e v −1
air flow,
Vapour Pressure, Pa
kg/m²s
p( x) = pout + ( pin − pout ) L
(15) 1500.00 -0.0005
0.0000
0.0005
e v −1
1300.00
1100.00
where
900.00
v
A quantity of dimension length that m 700.00
0.00 0.05 0.10 0.15
characterises the interaction between x, m
conductive and convective water vapour
flows, such that: Figure 3 Vapour pressure calculated according
to Equation (15).
δ
v = (16)
cv M air Finally, RH can be determined as p/ps. Figure 4
where shows the comparison between the temperatures
calculated numerically using BSim and the analyti-
cv Ratio between humidity of the Pa-1 cal solutions for exfiltration, no air flow, and infil-
air (x) and the vapour pres- tration, respectively.
sure = 0.622/(101.325 Pa – p)
cv is not quite constant since p varies with location The table below gives a comparison between the
in the construction, and therefore, the analytical analytical and numerically calculated relative hu-
solution is only approximate (with a few percents’ midity in the middle of the wall.
deviation). Analytical Numerical
Exfiltration 66.8 64.9
No air flow 44.7 44.7
Infiltration 22.5 23.4
Relative Humidity
100
90
80
Infiltration
70 air flow,
kg/m²s
Relative Humidity, %
60 -0.0005
0.0000
50
0.0005
40
30
20
10
0
0.00 0.05 0.10 0.15
x, m
Figure 4 Analytical (solid lines) and numerical calculation (symbols) of relative humidity through the wall
with exfiltration (red), no air flow (bright green) and infiltration (blue) air flow through the outer
wall.
It can be concluded that the advection model in the results of a similar calculation without filtration
BSim predicts the analytical temperature distribu- air flow. The building now has a heating system
tion with a good satisfaction, while also the relative with a set-point of 20ºC, and a cooling system
humidity distribution is predicted with a fair 3
(26ºC). The 129.6 m big room has for 8 hours per
amount of accuracy. day a moisture load of 500 g/h in (and 50 g/h the
rest of the day). The air change rate is constant at
EXAMPLE 2 – IMPACT ON ANNUAL -1
0.5 h . The building is simulated without windows.
CONDITIONS
Figure 5 shows the monthly values of air exfiltra-
In this example, all walls of the building were cal- tion/infiltration for all four wall orientations. It is
culated as if they allowed for air infiltration and sen that exfiltration dominates except in the west
exfiltration, but now the building was exposed to wall. In February and March, however it is oppo-
the outdoor climate of Denmark, as it is described site, then infiltration dominates in the east wall,
in the Design Reference Year for Copenhagen. A while there is exfiltration in the west wall. This
full year simulation was carried out, and the results corresponds with the average wind directions that
of heat and moisture flows have been compared to can be seen in Figure 6.
2007
1.0
0.8
0.6
0.4
Filtration(Const20)m³/h
Filtration(Const31)m³/h
0.2
Filtration(Const40)m³/h
Filtration(Const49)m³/h
-0.0
-0.2
-0.4
1 2 3 4 5 6 7 8 9 10 11 12 13
Month
\My Documents\PROJEKTR\IEA Annex 41\Florianapolis\Subtask 1\BSim\CE1A open filtration house w DRY 080407.d
3 2
Figure 5 Monthly average values over the year of the air flow (m /(m h) - positive values indicate exfiltra-
tion, negative are for infiltration) for the four facades of the BESTEST building (edition made of
solid aerated concrete) exposed to the Danish climate. Const20 (red) faces south, const 31 (blue)
faces east, const 40 (pink) faces west, and const 49 (dark grey) faces west.
2007 Figure 7 shows the monthly annual moisture con-
220 tent in the exterior part of the walls when there is
210
no air flow through the walls. It is seen how the
200
moisture content is generally higher in winter than
190
180
in summer, and that the north wall has the highest
170
moisture content, and the south wall the lowest.
160
WindDir([Outdoor])deg
The east and west walls exhibit almost the same
150
moisture content and annual variation, and their
140
level lies between the values for the north and south
130
120
walls.
1 2 3 4 5 6 7 8 9 10 11 12 13
Month
\My Documents\PROJEKTR\IEA Annex 41\Florianapolis\Subtask 1\BSim\CE1A open filtration house w DRY 080407.d
Figure 6 Monthly average wind direction in
the Danish DRY, North = 0º, east =
90º.
2007
0.055
0.050
0.045
Mc18[Aerated](Const20)kg/kg
Mc18[Aerated](Const31)kg/kg
0.040 Mc18[Aerated](Const40)kg/kg
Mc18[Aerated](Const49)kg/kg
0.035
0.030
1 2 3 4 5 6 7 8 9 10 11 12 13
Month
My Documents\PROJEKTR\IEA Annex 41\Florianapolis\Subtask 1\BSim\CE1A open nofiltration house w DRY 080407
Figure 7 Moisture content (kg/kg) calculated without filtration in the wall 1.5 cm behind its exterior surface
for each of the four orientations: Const20 (red) south, const 31 (blue) east, const 40 (pink) west,
and const 49 (dark grey) west.
With air flow, the similar graph for moisture con- from one another, such the east wall is the driest of
tent in the exterior wall layers can be seen in Figure the two in the first months of the year, and the west
8. The general pattern is the same: The north wall is the driest in the fall. It seems, once again, that
has the highest moisture content, and the south wall weather/wall combinations where infiltration domi-
has the lowest by the end of the year. However, it nates are the incidents that cause the lowest mois-
can now be seen how the east and west walls differ ture content in the walls.
2007
0.060
0.055
0.050
Mc19[Aerated](Const20)kg/kg
0.045
Mc18[Aerated](Const31)kg/kg
Mc18[Aerated](Const40)kg/kg
0.040 Mc18[Aerated](Const49)kg/kg
0.035
0.030
1 2 3 4 5 6 7 8 9 10 11 12 13
Month
\My Documents\PROJEKTR\IEA Annex 41\Florianapolis\Subtask 1\BSim\CE1A open filtration house w DRY 080407.d
Figure 8 Moisture content (kg/kg) calculated with filtration in the wall 1.5 cm behind its exterior surface for
each of the four orientations: Const20 (red) south, const 31 (blue) east, const 40 (pink) west, and
const 49 (dark grey) west.
whole-building hygrothermal analysis. IEA
ECBCS Annex 41, Working meeting, Lyon,
FURTHER WORK October 2006. Paper A41-T1-DK-06-
- Further checking and validation of the Hagentoft, C.-E., Introduction to Building Physics,
model… Studentlitteratur, 2001.
- Implementation of Darcy flow model for walls Jensen, Rasmus Lund, 2005, Modelling of Natural
instead/as a supplement to orifice flow model Ventilation and Night Cooling - by the Loop
- Adjustment of indoor sir infiltration Equation Method. Ph.D. Thesis. Department of
Building Technology and Structural Engineer-
- Energy impacts should be analysed, e.g. the ing, Aalborg University, ISSN 1397-7953
dynamic wall principle R0402. In Danish
CONCLUSION Rode, C. and Grau, K. 2003. Whole Building Hy-
grothermal Simulation Model, American
A preliminary model for filtration air flow through Society of Heating, Refrigriation and Air-
wall has been implemented to investigate its effect conditioning Engineers. Recent Advances in
on heat and moisture conditions in the wall.
Energy Simulation: Building Loads,
Symposium CH-03-09, Chicago, USA.
LITERATURE
Rode, C. and Grau, K. 2004. Calculation Tool for
Grau, K. and Rode, C. 2005. Whole building HAM Whole Building Hygrothermal Analysis. IEA
simulation with a multizone air flow model. ECBCS Annex 41, Working meeting, Zürich,
IEA ECBCS Annex 41, Working meeting, May 2004. Paper A41-T1-DK-04-1
Trondheim, October 2005. Paper A41-T1-DK-
05-6.
Grau, K. and Rode, C. 2005. A model for air flow
in ventilated cavities implemented in a tool for