# FINITE ELEMENT MODELLING OF COUPLED HEAT AND MOISTURE TRANSFER IN by liaoqinmei

VIEWS: 29 PAGES: 7

• pg 1
```									                                                                             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,

- 1850 -
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
 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

- 1851 -
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
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.

- 1852 -
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

- 1853 -
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
did an an investigation of transient, two-dimensional            Adjali, M. H., M. Davies and J. Littler.1998. Three-
dimensional earth-contact heat flows: A

- 1854 -
comparison of simulated and measured data for a           heat loss via the ground. Building and
buried structure. Renewable Energy, 15 (1-4):             Environment, 39 (7): 825-836.
356-359.                                             Liu, S. and Y. Huang. 1983. Discussion on effective
Carmody, J. and D. Derr.1982. The use of                      sky temperature. ACTA ENERGIAE SOLARIS
underground space in the People's Republic of             SINIC, 4 (1): 63-68. [In Chinese]
China. Underground Space, 7 (1): 7-11.               Liu, W., S. Peng and K. Mizukami.1995. A general
Cary, J. W. and S. A. Taylor.1962. The interaction of         mathematical modelling for heat and mass
the simultaneous diffusions of heat and water             transfer in unsaturated porous media: an
vapor. Soil Science Society of America Journal,           application to free evaporative cooling. Heat and
26 (5): 413-416.                                          Mass Transfer, 31 (1): 49-55.
Cary, J. W. and S. A. Taylor.1962. Thermally driven      Liu, W., X. X. Zhao and K. Mizukami.1998. 2D
liquid and vapor phase transfer of water and              numerical      simulation    for     simultaneous
energy in soil. Soil Science Society of America           heat,water and gas migration in soil bed under
Journal, 26 (5): 417-420.                                 different environmental conditions. Heat and
China Meteorological Bureau Climate Information               Mass Transfer, 34 (4): 307-316.
Center Climate Data Office and Tsinghua              Liu, X., J. Zhao and C. Shi, et al.2007. Study on soil
University Department Of Building Science And             layer of constant temperature. ACTA
Technology ,2005. China Standard Weather                  ENERGIAE SOLARIS SINIC, 28 (5): 494-498.
Data for Analyzing Building Thermal                       [In Chinese]
Conditions. Beijing, China Building Industry         Lu, X.2002. Modelling of heat and moisture transfer
Publishing House. [In Chinese]                            in buildings Ⅰ . Model program. Energy and
COMSOL AB ,2008. COMSOL Multiphysics User's                   Buildings, 34 (10): 1033-1043.
Guide. Los Angeles, COMSOL, Inc.                     Lu, X.2002. Modelling of heat and moisture transfer
De Vries, D.1958. Simultaneous transfer of heat and           in buildings II. Applications to indoor thermal
moisture in porous media. Transactions of the             and moisture control. Energy and Buildings, 34
American Geophysical Union, 39 (5): 909-916.              (10): 1045-1054.
Deru, M. P. ,2001. Ground-coupled heat and               Luikov, A. V. ,1966. Heat and mass transfer in
moisture transfer from buildings. Fort Collins,           capillary porous-bodies. Oxford, Pergamon Press.
CO, Colorado State University.                       Ma, Q., R. Fang and L. Xiang, et al. ,1986. Practical
dos Santos, G. H. and N. Mendes.2003. The SOLUM               hand book of thermophysical properties. Beijing,
program for predicting temperature profiles in            China Agriculture and Machine Press. [In
soils: Mathematical models and boundary                   Chinese]
conditions analyses. 8th International IBPSA         Matsumoto, M., S. Hokoib and E. Ka, Ed. (1997).
Conference,        Eindhoven,       Netherlands,          Analysis of coupled heat and moisture transfer in
International Building Performance Simulation             buildings considering the influence of radiant
Association.                                              heat transfer. ASHRAE Transactions.
dos Santos, G. H. and N. Mendes.2006. Simultaneous       Mendes, N., P. C. Philippi and R. Lamberts.2002. A
heat and moisture transfer in soils combined              new mathematical method to solve highly
with building simulation. Energy and Buildings,           coupled equations of heat and mass transfer in
38 (4): 303-314.                                          porous media. International Journal of Heat and
Editorial Group of Design handbook for the heating,           Mass Transfer, 45 (3): 509-518.
ventilation and air-conditioning of the              Ogura, D., T. Matsushita and M. Matsumoto.1999.
underground engineering. Beijing: China                   Analysis of heat and moisture behavior in
Architecture and Building Press, 1983.                    underground space by quasilinearized method.
Galbraith, G. H. ,1992. Heat and mass transfer within         6th International IBPSA Conference, Kyoto,
porous building materials. Glasgow, University            Japan, International Building Performance
of Strathclyde.                                           Simulation Association.
Geving, S..1997. A systematic method for                 Pedersen, C. R. ,1990. Combined heat and moisture
hygrothermal analysis of building constructions           transfer in building constructions. Lyngby,
using computer models. 6th IBPSA Conference,              Technical University of Denmark.
Prague, Czech Republic, International Building       Pei, Q. and Z. Pei.1999. An experimental study on
Performance Simulation Association.                       isothermal moisture absorption and desorption
Janssen, H., J. Carmeliet and H. Hens.2004. The               process of some common-used building
influence of soil moisture transfer on building           materials. Journal Of Hunan University (Natural
Sciences Edition), 26 (4): 96-99. [In Chinese]

- 1855 -
Philip, J. R. and D. De Vries.1957. Moisture                 Nanjing, PLA University of Science and
movement in porous material under temperature           Technology. [In Chinese]
gradients. Transactions of the American              Zhang, H., C. Liu and F. Liu, et al.2006. Numerical
Geophysical Union, 38 (2): 222-232.                     simulation of heat and moisture transfer in
Qian, Q., Z. Chen and Y. Wang, et al. ,2007.                 porous wall in underground caves. HV&AC, 36
Developing and making use of underground                (12): 9-13. [In Chinese]
space properly. Nanjing, Phoenix Science Press.
[In Chinese]                                         NOMENCLATURE
Qin, M., R. Belarbi and A. Ait-Mokhtar, et al.2006.        cm         -   Specific heat of material [J/(kg K)]
An analytical method to calculate the coupled         Dl         -   Permeability coefficient [kg/(m s Pa)]
heat and moisture transfer in building materials.     Dv         -   Diffusion coefficient of water vapor
International Communications in Heat and Mass
[m2/s]
Transfer, 33 (1): 39-48.
hh         -   Convection coefficient [W/m2 K]
Shen L S.1986. An investigation of transient, two-
dimensional coupled heat and moisture flow in         hm         -   Mass convection coefficient [m/s]
soils. Minneapolis, MN: University of                 keff       -   Effective thermal conductivity [W/(m
Minnesota.                                                           K)]
Straube, J. and E. Burnett.2001. Overview of               km         -   Thermal conductivity of material [W/(m
hygrothermal (HAM) analysis methods. 2009-                           K)]
02-02,                                                L T      -   Latent heat of vaporization [J/kg]
http://www.astm.org/BOOKSTORE/MNL40PD                 pv , sat   -   Saturation pressure of water vapour [Pa]
F/Ch5.pdf.                                            Rv         -   Gas constant of water vapour [J/(kg K)]
Tariku, F. and M. K. Kumaran.2006. Hygrothermal            T          -   Temperature [K]
modeling of aerated concrete wall and                 w          -   Moisture content [g/kg]
comparison with field experiment. 3rd                           -   Solar absorptivity [-]
International Building Physics Conference,                      -   Radiation emissivity
Montreal, QC.                                                   -   Relative humidity [-]
Thomas, H. R.1987. Nonlinear analysis of heat and          l         -   Water density [kg/m3]
moisture transfer in unsaturated soil. ASCE           m         -   Material density [kg/m3]
Journal of Engineering Mechanics, 113 (8):
v , sat   -   Density of saturated water vapour
1163-1180.                                                           [kg/m3]]
Tong, L. ,2005. Underground space and development                    -   Stefan-Bolzmann’s constant [W/(m2 K4)]
of urban modernization. Beijing, China                          -   Time [s]
Architecture and Building Press. [In Chinese]                   -   slope of the moisture equilibrium
Valen, M. S. ,1998. Moisture transfer in organic                          curve,   w / 
coatings on porous materials: The influence of
varying environmental conditions. Trondheim,
Norwegian University of Science and
Technology.
Wang, Q.2007. Three dimensional transient coupled
conduction and convection heat transfer in the
underground engineering. Nanjing, PLA
University of Science and Technology. [In
Chinese]
Whitaker, S.1973. The transport equations for multi-
phase systems. Chemical Engineering Science,
28 (1): 139-147.
Whitaker, S.1977. Simultaneous heat mass and
momentum transfer in porous media: a theory of
drying. Advances in Heat Transfer, 13 : 119-203.
Whitaker, S.1980. Heat and mass transfer in granular
porous media. Advances in Drying, 1 : 23-61.
Yuan, Y.2005. Research on the coupled heat transfer
in the soils in the underground engineering.

- 1856 -

```
To top