FINITE ELEMENT MODELLING OF COUPLED HEAT AND MOISTURE TRANSFER IN

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Eleventh International IBPSA Conference
Glasgow, Scotland
July 27-30, 2009

FINITE ELEMENT MODELLING OF COUPLED HEAT AND MOISTURE
TRANSFER IN TYPICAL EARTH-SHELTERED BUILDING ENVELOPE

Xibin Ma1, Baoyi Cheng1, Jinfeng Mao1, Wenjie Liu2, Dongyi Zi1
1
Engineering Institution of Engineering Corps, PLA University of Science and Technology
Nanjing, 210007, P.R. China, xibinma@gmail.com
2
Investigation and Design Institute of Guangzhou Military Region Air Force
Guangzhou, 510052, P.R. China, newlewin@gmail.com

irreversible thermodynamics to describe the
ABSTRACT interactions of the forces and fluxes involved. A
It is necessary to do research on the heat and mechanistic approach based on physical models of
moisture transfer in the earth-sheltered building for the phenomenological processes that occur in the soil
getting a better underground environment. Based on was established by Philips and De Vries (Philip et al.,
Philips and De Vries’ and Zhang’s theory (Philip et 1957; De Vries, 1958). Whitaker (Whitaker, 1973;
al., 1957; Zhang et al., 2006), this paper establishes a Whitaker, 1977; Whitaker, 1980) averaged the
detailed 2-D transient mathematical model to transport equation on a representative elementary
understand the heat and moisture transfer exactly in volume (REV) at the continuum level and obtained
the earth-sheltered building envelope and the the governing equations in a higher level. This
surrounding soils. A general FEM (Finite Element modelling method overcomes the modelling
Method) software is adopted to solve the highly difficulty that porous media are heterogeneous. But
nonlinear governing equations. The numerical the transport coefficients of the model can not be get
simulation showed that the existence of waterproof easily without the complex experiments. These
layer makes the temperature field continuous and the models have been adopted widely by many
moisture field discontinuous. The results may afford researchers (Deru, 2001; Lu, 2002; Lu, 2002; Janssen
the basis of heat and moisture load calculation for the et al., 2004; Qin et al., 2006). According to the
earth-sheltered building. models of Philip, De Vries, Luikov and Whitaker,
Liu (Liu et al., 1995; Liu et al., 1998) developed the
INTRODUCTION multiphysics-phase change-diffuse model recently
With the development of economy and society, the which based on the Navier-Stokes equation. This
underground space is explored widely in China to model has seven field variables. Besides, there are
enlarge the urban space (Carmody et al., 1982; Tong, more models established by the researchers
2005; Qian et al., 2007). However, moisture is one of worldwide (Thomas, 1987; Pedersen, 1990;
the most important factors limiting an earth-sheltered Matsumoto et al., 1997; Mendes et al., 2002; dos
building’s service life. Moisture damage has been Santos et al., 2006; Tariku et al., 2006).
identified as one of the main reasons for building Compared to the heat and moisture research in the
envelope deterioration. More recently recognized are ground buildings, the literature referred to earth-
the potential serious health hazards of mould and sheltered building is limited. Ogura analysed the heat
other organisms which flourish in buildings and and moisture behaviour in underground space by
constructions with excessive moisture (Geving, quasilinearized method (Ogura et al., 1999). Zhang
1997). Therefore, it is necessary to do research on the had presented a mathematical model which described
heat and moisture transfer in the earth-sheltered with temperature and relative humidity as driving
buildings for getting a better underground potential for coupled heat and moisture transfer in
environment. porous wall materials. This model could make the
For decades, many researchers have devoted to the discontinuity on the moisture content profile
work of modelling of heat and moisture transfer in continuous. However, both the models did not take
ground building envelope (Straube et al., 2001). As the influence of the waterproof layer into
early as 1960s, Luikov (Luikov, 1966) has firstly consideration and the quasilinearized method could
proposed a mathematical model for simultaneous not reflect the heat and moisture variations actually.
heat and mass (moisture) transfer in building porous Then the purpose of this paper is to clarify the heat
materials. The conservation equations include the and moisture behaviour in the earth-sheltered
mass, the momentum and the energy conservation building envelope under the natural condition.
equations. The constitutive equations are described
by Darcy’s law, Fick’s law and Fourier’s law. PHYSICAL AND MATHEMATICAL
However, the solutions are either numerical or MODEL
complicated involving complex eigenvalues. Cary
To analyse the coupled heat and moisture transfer in
and Taylor (Cary et al., 1962; Cary et al., 1962) uses
the envelope of the typical earth-sheltered building,

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this paper takes an underground building in Nanjing Zhang’s model (Zhang et al., 2006), the coupled heat
(China) as an example, as depicted in Figure 1. The and moisture transfer model in this paper is derived
length (L), with (W), height (h) and the arc height (h1) according to the two. Based on the assumption above,
is 50m, 6m, 5.3m and 1.5m, separately. The distance the governing equations describing the heat and
between the soil surface and the top of the building is moisture transfer in the envelope and the surrounding
1.5m and the thickness of the envelope is 0.5m soils are as follows.
(Suppose the thickness in the different area of the T
envelope is same).  m cm    keff T  (1)

Y 
    D   DT T  (2)
l 
X
0
Where
d+h2
L T  Dv pv , sat   L T  
keff  km    1 (3)
Rv T 2  Rv 
h1
Ω1 Dv pv , sat Dl l Rv T
D   (4)
Ω2 Rv T 
H
Dv pv , sat   L T  
h
W DT  Dl l Rv  ln      1 (5)
Rv T 2  Rv 
h2 Dl in Equation (4) is determined by Equation (6)
(Valen, 1998),
Dv  v , sat
Dl  (6)
Figure 1 Cross-section of typical shallow buried Rv T l
underground engineering
The meaning of the symbols and notations used in
Before the mathematical modelling, there are some the equations are given in the Nomenclature.
assumptions. The upper surface of the physical domain is exposed
 Homogeneous and isotropic porous media to solar and long-wave radiations, convection heat
(envelope and the surrounding soil) with no and moisture transfer. This way, for Y  0 , the
distension and contraction is considered. boundary condition becomes
 Local thermal equilibrium is satisfied keff T Y  0  hhs To  T Y  0    I  

 
throughout the porous media. (7)
 T Y  0  Tsky 4  L T  hms  o   Y  0 
4

 Capillary pressure on the interfaces of the
different material layers is equal, i.e. the and the mass balance is written as
 D   DT T  hms   o   Y  0 
hydraulic continuous.
(8)
 The moisture absorption and desorption
progress is isothermal without considering The heat and moisture balance on  2 is described as
the influence of the capillary hysteresis.
 The length of the building is relative long keff T Y  0  hhi Ti  T  2  (9)
compared to the with and height, the two
dimensional is adopted.
 L T  hmi i      2

 The drive potential for the liquid water in the
porous material and the water vapor is
 D   DT T  hmi    
i 2
(10)
density gradient and capillary pressure The interface 1 represents the waterproof layer, so
gradient. the heat is continuous while the moisture could not
 The waterproof layre can stop the mositure transfer from it. The boundary is
transfer completely.
 The thermal parameters are constants for
  D   D  T 
 T soil 1
0 (11)
given material.
 There is no heat or moisture source in the
  D   D  T 
 T envelope 
1
0 (12)
envelope and the surrounding soils.
Tsoil 1
 Tenvelope (13)
As the Philips and De Vries’ model (Philip et al., 1

1957) was adopted worldwide and the merit of

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Through the model of natural soil temperature field,
the depth formula of the constant temperature layer 40
was deduced by Liu et al. (Liu et al., 2007). So in the

Outdoor temperature /C
depth of H  20 m, it is supposed that the water table 30
is there and the temperature is constant. As a result,
the heat and moisture condition in this layer in 20
Nanjing (China) is written as (Liu et al., 2007)
T Y  20  17.5  273.15  K (14) 10

 Y  20  90% (15) 0
If it is sufficient far away from the envelope, the heat
and moisture transfer can no longer influence the soil -10
0 2000 4000 6000 8000
temperature profile, where is defined as the adiabatic
and impermeable. When it refers to the determination Time /h
of l , it is another problem. It comes true only l   Figure 2 Annual variation of the dry bulb
in theory. However, it is unrealistic in the numerical temperature Nanjing (China) form CSWD
simulation. 0.5l  15  20 m in many simulations
(Adjali et al., 1998; dos Santos et al., 2003) is 1.0
adopted here.
Symmetry reduces the modelling domain to half of
the model. It gets a symmetry boundary on X  0 . Relative humidity /% 0.8

SIMULATION 0.6
The governing equations were solved using
COMSOL Multiphysics (COMSOL AB, 2008).
COMSOL Multiphysics is a powerful interactive 0.4
environment for modelling and solving all kinds of
scientific and engineering problems based on partial 0.2
differential equations (PDEs). With this software you
can easily extend conventional models for one type 0 2000 4000 6000 8000
of physics into multiphysics models that solve
coupled physics phenomena—and do so Time /h
simultaneously. The software runs the finite element Figure 3 Annual variation of relative humidity
analysis together with adaptive meshing and error Nanjing (China) form CSWD
control using a variety of numerical solvers.
1000
horizontal global irradiation /(W m )

Suppose the material of the envelope consists of
2

concrete, and the soils surrounding the underground
building is made of lime rock. The thermophysical 800
parameters of them are listed in Table 1.

Table 1
600
Thermophysical parameters of the building materials
and the soils (Ma et al., 1986) 400
MATERIAL LIME ROCK CONCRETE
 [kg/m3] 1700 2200 200
c [J/(kg K)] 930 840
K [W/(m K)] 0.93 1.28
0
Dv [m2/s] 6.9  106 1.4  105 0 2000 4000 6000 8000
The CSWD, Chinese Standard Weather Data, (China Time /h
Meteorological Bureau Climate Information Center Figure 4 Annual variation of horizontal global
Climate Data Office et al., 2005) for the city of irradiance Nanjing (China) form CSWD
Nanjing was used, with a constant convection heat
transfer coefficient of 15 and 8.14 W/(m2 K) for hhs The approximate expression of mass convection
and hhi (Yuan, 2005), an absorptivity of 0.5 (dos coefficient is determined by (Galbraith, 1992)
Santos et al., 2003) and emissivity of 0.9 (Liu et al., hm  9.28  104  hh (16)
1983). Figure 2-Figure 5 present the annual variation
of temperature, relative humidity, horizontal global
irradiation and equivalent sky temperature in Nanjing.

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The moisture absorption and desorption process lines Figure 6 Temperature profile of the earth-sheltered
for concrete and lime rock are provided with the building envelope and the surrounding soils at the
following equations (Ma et al., 1986; Pei et al., 1999), time of 2 years
0.000392
w (17)
1  1.8504  0.9874 2
w  0.014 4  0.021 3  0.014 2
(18)
0.0014  0.00042

320

300
Sky temperature /K

280

260
Figure 7 Relative humidity profile of the earth-
240 sheltered building envelope and the surrounding
soils at the time of 2 years
0 2000 4000 6000 8000
Figure 6 and Figure 7 present the temperature and
Time /h
relative humidity spatial distributions at 14:00 on
Figure 5 Annual variation of the equivalence sky July 20th. It is possible to notice in Figure 6 that the
temperature Nanjing (China) form CSWD existance of the earth-sheltered building had
influenced the soil’s temperature distribution a lot.
RESULTS AND DISCUSSIONS For the indoor air have a constant temperature of
In this numerical simulation, the finite element 26°C, the heat is tranfered from the envelope to the
method is triangle and 945 mesh points, 1760 soil in summer. While the waterproof layer had not
elements were used. Take 15:00 on July 20th, which affect the heat transfer progess on the interface of the
is the time of outdoor maximal temperature (37.2°C), envelope and the rock, it did effect the moisture
as the initial time for the simulation. For initial transfer (see in Figure 7). Note that in Figure 7, the
condition for the soil, a temperature of 17.5°C and a left colorbar is for the rock and the right one for the
relative humidity of 50% were utilized. While for the envelope. After a whole year, the moisture had
envelope, temperature of 17.5°C and a relative absorped by the upper lay of soil in this simulation.
humidity of 90%. Suppose the underground space is Due to the waterproof layer, the moisture in the soils
air-conditioned by the HVAC system, and the didn’t transfer to the envelope, and the moisture in
temperature and relative humity maintains 26°C and the envelope had been transferred into the indoor
60%, separately. The simulation period is 2 year environment afer construction.
(17520 hours) and time step is 1 hour (Wang, 2007). 6 0.5

5
0.4
Moisture flux

Heat flux
m2)

4
0.3
Heat flux q/(W

3
/(g m2 h )

0.2
2
Moisture flux
1

1 0.1
0
00
00
00
00

0
0
0
0
00
00
00
00
20
40
60
80
10
12
14
16

Time  /h
Figure 8 Variation of the heat and moisture flux on
the wall of the earth-sheltered building

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Figure 8 shows the variation of the heat and moisture coupled heat and moisture flow in soils (Shen, 1986),
flux on the wall of the earth-sheltered building. The results from the coupled model were compared to
moisture flux varies similarly with the heat flux. The results obtained when the governing equations were
maximum moisture flux in this simulation is decoupled. For the sand simulation, a 9% greater
3.03g/(m2 h) and it fluctuates between 0.1-0.3 g/(m2 wall heat loss during the winter and almost a 50%
h), Which meets the experimental results of 1.7g/(m2 increase in summer heat gain occurred due to
h) and 0.12 g/(m2 h) in China (Editorial Group, 1983). coupling. Due to the lack of thermophysical
So the models in this paper is validated based on the parameters of the building materials, only the lime
assumption that the waterproof layre can stop the rock and concrete were studied in this paper.
mositure transfer completely. However, both the results indicates that the couple
heat and moisture transfer model should be
3.0 0.7 considered in some cases.
2.8
2.5
0.6
CONCLUSION
/(g m2 h )
1

2.3 0.5
Based on the theory of Philips and De Vries’ and
2.0 0.4
Zhang’s model, this paper built a mathematical
1.8 0.3 model to analyze the coupled heat and moisture
1.5 Ceiling transfer in the envelope of earth-sheltered building
Moisture flux

0.2
1.3 Wall and the surrounding soils. The COMSOL
1.0 Floor 0.1
Mutiphysics was utilized to solve the governing
0.8 0.0
equations in the porous materials. The software could
0.5 couple the temperature field with the moisture field
0.3
and solve the nonlinear problem effectively. It may
0.0
be conclude from the results that,
0
00
00
00
00

0
0
0
0
00
00
00
00

1. The waterproof layer could prevent the moisture
20
40
60
80
10
12
14
16

tranfering between the envelope and the
Time  /h
surrounding soils, while the temperature field is
Figure 9 Variation of the moisture flux in the ceiling, still coutinuous. The mathematical model could
floor and wall of the earth-sheltered building describe the coupled heat and moisture transfer
in the different porous material simultaneously
40 and independently.
2. The moisture flux in the wall, ceiling and floor of
q/(W m2)

the earth-sheltered building is different. Of the
30 three, moisture flux in the ceiling is the biggest.
Coupled heat and moisture
Heat only Becasuse the heat transfer in the ceiling is
20 affected by the ourdoor meteorologic paratmeters
Heat flux

more. Accordingly, the moisture transfer is
infuenced by the temprature.
10
3. The couple heat and moisture transfer model
should be considered in some cases when do
0 investigation of heat transfer in the porous
materials.
0
00
00
00
00

0
0
0
0
00
00
00
00
20
40
60
80
10
12
14
16

The research may reflect the heat and moisture
Time  /h transfer in the underground engineering envelope and
Figure 10 Comparison of the heat flux by the model the surrounding soils objectively and the results
in this paper and the heat transfer only model could afford the basis of heat and moisture load
calculation for underground engineering. As the
The difference of moisture flux in the ceiling, floor results were obtained by the numerical simulation, it
and wall of the earth-sheltered building is depicted in is necessary to do verification and analysis by both
Figure 9. The moisture flux on the ceiling fluctuated the experiments in the lab and field test further.
more than the one on the floor or wall. It may due to
the influence of the vaiation of outdoor meteorologic ACKNOWLEDGEMENT
paratmeters. Figure 10 compares the heat flux on the The ﬁnancial support from China National Civil Air
surface of the underground engineering envelope Defense Ofﬁce (Project NO. ZC1302K30) is
based on the coupled heat and moisture transfer gratefully acknowledged.
model and heat transfer model, separately. The mean
relative error during the simulation is 11.4%. Shen REFERENCES
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