Energy and Buildings 40 (2008) 1790–1798
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Energy and Buildings
journal homepage: www.elsevier.com/locate/enbuild
Optimization of the configuration of 290 140 90 hollow clay bricks
with 3-D numerical simulation by finite volume method
L.P. Li a, Z.G. Wu a, Y.L. He a, G. Lauriat b, W.Q. Tao a,*
a
b
School of Energy & Power Engineering, Xi’an Jiaotong University, Xi’an, China
LETEM, Universite de Marne-la-Valle, France
A R T I C L E I N F O
A B S T R A C T
Article history:
Received 27 January 2008
Received in revised form 18 March 2008
Accepted 18 March 2008
This paper is aimed at finding the optimum configuration of the number of holes and their arrangement
for the 290 140 90 hollow clay bricks with 3-D numerical simulation by a home-made code with
finite volume method. Seventy-two kinds of configurations with different hole number and arrays are
chosen elaborately and their equivalent thermal conductivities are numerically predicted. In addition,
the effects of the hole surface radiation and the indoor–outdoor temperature difference on the equivalent
thermal conductivity are also investigated. The major findings are as follows. The radiation of the hole
surfaces makes heat transfer enhanced and the equivalent thermal conductivity enlarged in some extent,
ranging from 25.8% to 4.6%. The optimum configuration has eight holes in length, four holes in width and
one holes in height, whose equivalent thermal conductivity is the lowest and of 0.400 W/(m K),which is
only 59% of the highest thermal conductivity of the all cases studied. When the indoor–outdoor
temperature difference varies from 50 8C to 20 8C, the equivalent thermal conductivity of the 72 kinds of
hollow bricks does not vary too much, usually within 5%. Especially, the equivalent thermal conductivity
of the optimum configuration holds no change within this variation range of indoor–outdoor temperature
difference.
ß 2008 Elsevier B.V. All rights reserved.
Keywords:
Hollow clay bricks
Numerical simulation
Equivalent thermal conductivity
Natural convection
Surface radiation
1. Introduction
Hollow clay bricks used as building materials have obvious
advantages in improving thermal insulation performance of
building walls, separating sound as well as reducing building wall
loads. Thermal insulation performance of the hollow clay brick
mainly depends on the hole configuration, that is, on the number of
holes, their arrangement and the void fraction, and investigation
on the hole configuration effect is of great significance to improve
its thermal insulation performance. It is well-known that the heat
transfer and fluid flow process within a hollow clay brick is a
typical complicated combined mode problem, with conduction,
convection and surface radiation being all involved. Theoretically
speaking, both experimental and numerical methods can be used
to find an optimum configuration of a hollow brick under certain
condition. However, because of the large number of the possible
variants of the hole configurations experimental study is very costexpensive. Meanwhile apart from surface radiation the transport
process in the hollow brick is conduction and laminar flow in
nature, and this is a typical case that numerical simulation can play
* Corresponding author.
E-mail address: [email protected] (W.Q. Tao).
0378-7788/$ – see front matter ß 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.enbuild.2008.03.010
a great role. This is probably the main reason that most related
literatures are of numerical study. In the following, a brief review
of literatures in recent 10 years is presented. In [1] 2-D simulation
was conducted for enclosures similar to the hollow clay bricks with
single or double holes. Surface radiation was taken into account by
treating the radiation energy as the additional source term of the
control volume bounded with the surface. The results emphasize
the effect of the thermal boundary condition on the total heat
transfer. Lorente et al. [2] studied the heat flux and thermal
resistance of a hollow brick with a single vertical hole by using
simplified analytical model. They also studied the influence of
environment temperatures on thermal resistance of the walls built
with some shapes of vertical hollow bricks [3]. When time entered
into 21th century, study on the hollow brick related problems
receives more interests of researchers because the world-wide
energy shortage. Castro Cadoso et al. [4] numerically studied a
structure with 12 rectangular holes for which the horizontal top
and bottom surfaces were adiabatic and vertical left and right
surfaces maintained at constant but different temperatures. The
effect of the hole surface radiation was examined. Hinojosa et al.
[5] predicted the Nusselt number for the natural convection and
surface thermal radiation in a square tilted open cavity. They found
that the heat transfer via radiation is in the same order of natural
convection. In [6], indoor thermal environment of office space
L.P. Li et al. / Energy and Buildings 40 (2008) 1790–1798
Nomenclature
area of inner surface of calculation
area of outer surface of calculation
specific heat of air (k J/(kg K))
nominal flow rate of cross section in holes
R
(G = rju jdydz, kg/m3)
convection heat transfer coefficient of inner surface
h1
of wall (W/(m2 K))
convection heat transfer coefficient of outer surh2
face of wall (W/(m2 K))
J
radiosity (W/m2)
L1, M1, N1 the grid number in x, y and z direction,
respectively
p
Pressure (Pa)
effective pressure (Pa)
Peff
Pr
Prandtl number
net radiant heat flux (W/m2)
qr
total heat transfer rate cross inner surface of wall
Qinwall
(W)
Qoutwall total heat transfer rate cross outer surface of wall
(W)
the largest mass residual in whole computation
Rmax
field
Sc,ad,air additional source term of control volume at air side
Sc,ad,clay additional source term of control volume at clay
side
T
temperature (K)
reference temperature (K)
Tc
indoor temperature (K)
Tf1
environment temperature (K)
Tf2
average temperature of inner surface of wall (K)
Tw1
average temperature of outer surface of wall (K)
Tw2
u
velocity component in x direction (m/s)
v
velocity component in y direction (m/s)
!
velocity vector in x–y plane
V
w
velocity component in z direction (m/s)
X1, Y1, Z1 the length of calculation unit in x, y and z
direction, respectively (m)
view factor
XIJ
x, y, z
coordinate (m)
Aw1
Aw2
cpf
G
Greek letters
b
volumetric thermal expansion coefficient (K1)
e
h
lf
lnat
lrad
ls
leq
r
rc
rf
surface emissivity
viscosity (kg/(m s))
thermal conductivity of air (W/(m K))
equivalent thermal conductivity when radiation
neglected (W/(m K))
equivalent thermal conductivity when radiation
considered (W/(m K))
thermal conductivity of clay (W/(m K))
equivalent thermal conductivity (W/(m K))
mass density (kg/m3)
reference mass density (kg/m3)
mass density of air (kg/m3)
1791
controlled by cooling panel system was investigated by numerical
simulation and field measurement, and the results showed the
importance of radiation in the thermal environment simulation. In
[7], natural convection and radiation heat transfer in a cavity
having a square body at its center had been simulated. The results
indicate that the heat transfer by convection and radiation are
usually of the same order of magnitude and the radiation
homogenizes the temperature inside the cavity. In [8] coupled
natural convection–conduction effects on the heat transfer
through three kinds of hollow clay bricks had been studied using
commercial software FLUENT, in which the first one was a typical
hollow brick with three identical holes, the second was filled with
ordinary polystyrene bars and the third filled with hollow
polystyrene bars. The results show that, compared with the first,
the second can reduce the heat transfer rate by 36% while the third
can reduce only by 6% due to the air motion inside the holes in the
polystyrene bars. All of above studies are of 2-D simulation. In
[9,10] 3-D numerical thermal analysis of some light concrete
hollow brick walls was conducted by the finite element method in
which the parameter of mass overall thermal efficiency was used
to evaluate the economic efficiency of light concrete hollow brick.
Lauriat and Desrayaud [11] simulated the conjugate natural
convection in partially open enclosures with surface radiation
being taken into account. In [12] detailed 3-D numerical
simulation about 240 115 90 hollow clay bricks was conducted for 50 kinds of different hole configuration and the
optimum configuration was found which is wildly used to
construct 240 mm walls in China. In addition, the parameter of
mass overall thermal efficiency was adopted in [12] and found that
it is not suitable to characterize the hollow clay brick performance.
The major purpose of this paper is to find out the optimum
configuration of the 290 140 90 hollow clay bricks. By optimum
configuration we mean that the heat transfer rate through the wall of
the configuration under given conditions is the least. This is an
important way to save building energy either in summer or in winter
seasons. The overall heat transfer process from the inner side of a
room to the environment is consisted of three steps: from inner side
to the inner surface of the wall via convection and radiation heat
transfer, from the inner surface of the wall to the outside surface of
the wall through heat conduction and from the outside surface of the
wall to the environment via convection and radiation. The thermal
resistance of the conduction process can be represented by d/leq,
where leq is the equivalent thermal conductivity of the brick
structure [13,14]. As indicated above, within the hollow brick, heat is
transferred via conduction, convection and radiation, and different
configuration of the hollow brick will affect the relative importance
of conduction, convection and surface radiation in the heat transfer
process from inner surface to outer surface. However, the overall
thermal insulation performance of the hollow brick can be
represented by the equivalent thermal conductivity. The equivalent
thermal conductivity is the thermal conductivity of an equivalent
solid brick which can conduct the same heat under the same indoor–
outdoor conditions and the same brick thickness. Strictly speaking,
the heat transfer process through the wall is unsteady in nature,
because either the indoor or the outdoor conditions are actually
changing from time to time. However, from engineering point of
view, thermal design of the building can be conducted for some
typical situations of indoor and outdoor conditions with a steady
state assumption. In this study numerical simulation of the steady
heat transfer process through the hollow brick will be conducted for
some typical combinations of indoor and outdoor conditions. The
major purpose of the study is to find out the hole configuration with
which the brick has the lowest equivalent thermal conductivity. In
order to do that, 72 kinds of different hole configurations are
designed.
L.P. Li et al. / Energy and Buildings 40 (2008) 1790–1798
1792
2. Mathematical formulation
The 290 140 90 hollow clay bricks is one type of bricks
commonly used in China to construct building walls with 300 mm
thickness. Their exterior sizes are 290 mm in length, 140 mm in
width and 90 mm in height. For simplicity, the denotation of
Lm1Wm2Hm3 is used to represent the brick with m1 holes in
length, m2 holes in width and m3 holes in height such as
denotation L8W4H1 represents the brick having eight holes in
length, four holes in width and one hole in height. The denotations
of some selected hollow bricks from all 72 kinds are listed in
Table 1.
From the building wall structure the smallest computational
domain (unit) consists of four blocks of the bricks that have
identical configuration and is conglutinated by mortar according
to the manner of putting bricks. It is considered that its
boundaries are repeated periodically in both up/down and left/
right directions. Fig. 1 shows one of such 72 computational
domains consisting of four blocks of L8W4H1. It should be noted
that all the holes are rectangular in cross section as shown in
Fig. 1. It is obvious that the heat transfer phenomenon is threedimension in nature, i.e., both the temperature (of the solid part
and fluid part) and the fluid flow velocity in the holes vary with
the three coordinates. Hence, their equivalent thermal conductivities are numerically calculated with 3-D numerical
simulation of the temperature and air velocity in Cartesian
coordinates by using a home-made code of finite volume
method.
As indicated above the equivalent thermal conductivity of the
hollow clay bricks depends on the hole number, their arrangement
(arrays) and void fraction. The 72 kinds have two things in
common: there is only one hole in the height and in length and
width directions the holes are uniformly distributed. The hole
number varies from 1 to 14 in length and 1 to 6 in width. All of
them have the largest void fraction at given hole number and
arrays. Here, the void fraction means the percentage of the volume
filled by air divided by the total volume of the hollow clay brick.
The largest void fraction means that for given hole number and
array, the hull of the brick is 10 mm, the ribs parting the holes are
7.5–10 mm, which are the essential need for hollow clay bricks not
to be split.
Based on the fundamental physical process, the following
approximations can be adopted in our numerical model: (1) The
hollow clay bricks and the air in the holes are of constant thermal
physical properties; (2) Air flow in the holes of the hollow clay
bricks is incompressible, steady state and laminar; (3) When
radiation is considered air in the holes is a non-participating
medium; (4) The Boussinesq assumption is adopted for the natural
convection in the holes [15]; (5) the convective boundary
conditions are taken at the indoor and outdoor surfaces of the
computation domain while the other four surfaces are considered
as adiabatic (Fig. 1(a)); (6) The mortar is assumed to have the same
conductivity as the clay brick; (7) The hole surface is gray and
diffusive.
The numerical prediction of the equivalent thermal conductivity will be performed under given third kind of boundary
conditions for both indoor and outdoor sides. That is the fluid
temperatures and the total heat transfer coefficients of the inner
and outer surfaces of the wall are prespecified. Then in order to
predict the equivalent thermal conductivity of the brick
structure, its two surfaces temperatures have to be known. In
order to obtain these two surface temperatures the partial
differential equation of the temperature in the brick structure
should be solved under the given boundary conditions. Since
within the brick holes air flows exist and the fluid velocity will
affect the heat transfer process, the momentum equations of the
fluid in the holes should be simultaneously solved. Thus, to
determine the equivalent thermal conductivity, the governing
equations for the temperature and fluid velocity in the brick
should be simultaneously solved.
Table 1
Comparison lnat with lrad for 290 140 90 bricks
Kinds
Holes number
Void fraction (%)
lnat (W/(m K))
lrad (W/(m K))
(lrad lnat)/lrad (%)
L01W1H1
L04W1H1
L07W1H1
L10W1H1
L14W1H1
L01W2H1
L04W2H1
L07W2H1
L10W2H1
L14W2H1
L01W3H1
L04W3H1
L07W3H1
L10W3H1
L14W3H1
L01W4H1
L04W4H1
L08W4H1
L10W4H1
L14W4H1
L01W5h1
L04W5H1
L07W5H1
L10W5H1
L14W5H1
L01W6H1
L04W6H1
L07W6H1
L10W6H1
L14W6H1
1
4
7
10
14
2
8
14
20
28
3
12
21
30
42
4
16
32
40
56
5
20
35
50
70
6
24
42
60
84
79.8
71.0
65.0
59.1
50.2
73.2
65.0
59.6
54.2
46.1
66.5
59.1
54.2
49.3
41.9
63.2
56.1
50.3
46.8
39.8
59.9
53.2
48.8
44.3
37.7
53.2
47.3
43.3
39.4
33.5
0.680
0.426
0.423
0.429
0.433
0.628
0.356
0.351
0.361
0.384
0.631
0.355
0.351
0.364
0.393
0.626
0.355
0.355
0.369
0.402
0.608
0.358
0.358
0.377
0.414
0.566
0.374
0.379
0.402
0.440
0.905
0.554
0.522
0.511
0.498
0.807
0.450
0.422
0.419
0.429
0.781
0.431
0.408
0.410
0.428
0.758
0.421
0.400
0.408
0.431
0.726
0.417
0.401
0.411
0.439
0.670
0.424
0.415
0.430
0.461
24.9
23.1
19.0
16.0
13.1
22.2
20.9
16.8
13.8
10.5
19.2
17.6
14.0
11.2
8.2
17.4
15.7
11.3
9.6
6.7
16.3
14.1
10.7
8.3
5.7
15.5
11.8
8.7
6.5
4.6
L.P. Li et al. / Energy and Buildings 40 (2008) 1790–1798
@T
¼0
@z
@T
z ¼ Z1 u ¼ v ¼ w ¼ 0
¼0
@z
z¼0
u¼v¼w¼0
1793
(2c)
where X1, Y1 and Z1 express the size of the computation domain in
x, y and z direction, respectively. Eq. (2a) expresses the no-slip
boundary condition of the three velocity components and the third
kind boundary condition of the temperature at the inner and outer
surfaces. Eqs. (2b) and (2c) represent the symmetric boundary
condition, i.e., adiabatic, at the boundaries of y and z coordinates
for the temperature. It is interesting to note that the above
governing equations are valid for both solid part of the brick and
the air in the hole. Here, an idea from [17] is adopted, i.e., the solid
is regarded as a special fluid with very large viscosity. This is a very
successful numerical technique to treat the complicated fluid flow
problems where both the fluid and solid temperature should be
simultaneously predicted, and has been widely adopted in many
numerical simulation [16]. With such a technique, the surfaces of
the holes are no longer the computational boundaries, rather they
are the interfaces within the computational domain, and no any
special treatment is needed for such interfaces when the governing
equations are solved. The details of the numerical technique will be
presented below.
According to general building construction engineering practice, following thermal physical properties are used in simulation:
Tf1 = 293 K; Tf2 = 253 K; h1 = 8.72 W/(m2 K); h2 = 23.26 W/
2
(m K); Pr = 0.707; cpf = 1005 J/(kg K), lf = 0.0244 W/(m K), h =
17.2 106 kg/(m s), rf = 1.293 kg/m3, ls = 0.755 W/(m K), e = 0.85.
3. Numerical methods
Fig. 1. Computational domain.
The governing equations for the temperature and velocity are as
follows [16]:
The governing equations are discretized with the finite volume
method [16,17]. Discretization of convection term uses the SGSD
scheme [18], which is a stability-guaranteed second-order
difference scheme, and the SIMPLE is adopted as pressure–velocity
solution algorithm. Discretized algebraic equations are solved in
the whole computational domain by TDMA + ADI method [16,17].
As indicated above the surfaces parting solid and fluid areas
become inner parts of the simulation domain. To guarantee the
success of such simple and efficient numerical treatment, the
equivalent diffusion coefficient at the separating surfaces should
be calculated by the harmonic mean [16,17,19]. The thermal
!
@ðru2 Þ @ðruvÞ @ðruwÞ
@p
@2 u @2 u @2 u
þ
þ
¼ eff þh
þ
þ
@x
@y
@z
@x
@x2 @y2 @z2
!
@ðruvÞ @ðrv2 Þ @ðrvwÞ
@p
@2 v @2 v @2 v
þ
þ
¼ eff þh
þ
þ
þ rc g bðT T cÞ
@x
@y
@z
@y
@x2 @y2 @z2
!
@ðruwÞ @ðrvwÞ @ðrw2 Þ
@p
@2 w @2 w @2 w
þ
þ
¼ eff þh
þ
þ
@x
@y
@z
@z
@x2 @y2 @z2
!
2
2
@ðruTÞ @ðrvTÞ @ðrwTÞ l @ T @ T @2 T
þ
þ
¼
þ
þ
@x
@y
@z
cp @x2 @y2 @z2
where, peff = p rcgy.
The boundary conditions of the governing equations are
@T
@x
@T
x ¼ X1 u ¼ v ¼ w ¼ 0 h1 ðT w1 T f1 Þ ¼ ls
@x
(2a)
@T
¼0
@y
@T
¼0
y ¼ Y1 u ¼ v ¼ w ¼ 0
@y
(2b)
x¼0
y¼0
u¼v¼w¼0
h2 ðT f2 T w2 Þ ¼ ls
u¼v¼w¼0
(1)
conductivity of solid and fluid regions are adopted individually
while the specific heat for the solid area should be replaced by the
value of fluid area in order to guarantee the heat flux continuity at
the separating surfaces [16,19]. By adopting a very large value of
viscosity amounting to1030 in the momentum equation for the
solid part of the hollow brick, the zero velocity of solid area can be
actually gained.
From the simulated temperature field the equivalent thermal
conductivity can be determined. The relationship of the equivalent
thermal conductivity and the predicted temperature field can be
found as follows.
1794
L.P. Li et al. / Energy and Buildings 40 (2008) 1790–1798
3.1. Formulae for calculation of equivalent thermal conductivity
transfer theory [13,14]:
The equivalent thermal conductivity can be calculated according to the following relationship of heat flux equilibrium
h1 ðT f1 T W1 Þ ¼
leq
X1
ðT W1 T W2 Þ ¼ h2 ðT W2 T f2 Þ
"
qr ðIÞ ¼
(3)
T f1 T w1
T w1 T w2
(4a)
T w2 T f2
T w1 T w2
(4b)
or
leq ¼ X 1 h2
where
T w1
T w2
2
3
1 4 X
¼
AðiÞT w ðiÞ5
Aw1 i 2 A
2 w1
3
1 4 X
¼
AðiÞT w ðiÞ5
Aw2 i 2 A
(5a)
(5b)
4
#
JðIÞ ;
X
TðIÞ 4
JðIÞ ¼ C 0 eðIÞ
þ ð1 eðIÞÞ
X IJ JðJÞ
100
J
then
leq ¼ X 1 h1
eðIÞ
TðIÞ
C0
1 eðIÞ
100
(7)
where, I = 1, 2, 3, 4, 5, 6 and J = 1, 2, 3, 4, 5, 6; C0 = 5.67; T (I) is the
mean temperature of the surface (I) of the six-surface enclosure.
It is to be noted that for each surface of a hole it may be discretized
by several smaller surfaces depending on the generated grid. Thus,
the mean temperature T (I) should be calculated as follows
"
#
1 X
TðIÞ4 ¼
(8)
AðiÞTðiÞ4
AðIÞ i 2 I
or
"
#
1 X
AðiÞTðiÞ
TðIÞ ¼
AðIÞ i 2 I
(9)
w2
where Tw(i) is the temperature of the grid points at the inner
or outer surfaces of the computational domain and h1,h2 are
the specified heat transfer coefficients at the inner and out
surfaces.
3.2. Radiation between hole inner surfaces
The radiant heat transfer between the internal surfaces of the
holes in hollow clay bricks should not be neglected [1,4,11,12]. The
radiant heat flux at inner surfaces can be treated as additional
source terms in the control volumes close to the inner surfaces
[1,12], which can be calculated as follows:
For the control volumes at the air side (see Fig. 2)
Sc;ad;air
q dx =ls
1
¼ r e þ
dxe =ls þ dxe =lf Dxþ
(6a)
For the control volumes at the clay side
Sc;ad;clay ¼ d
qr xþ
e= f
þ
x
=
s þ xe = f
e
d
l
l
1
d l Dx where T(i) is a local temperature of the interface I, which can be
readily obtained during the iterative solution process by interpolation from the temperatures of the neighboring grid points
shown below.
In Eqs. (8) and (9) the expression i 2 I stands for all the grids
belonging to the I surface of the enclosure. Our numerical practices
show that the final results obtained from Eqs. (8) and (9) are almost
the same while Eq. (9) can get a bit faster convergence because no
fourth power calculation is needed in this definition.
The interface temperature T(i) can be interpolated from
predicted neighboring grid points according to the heat flux
balance. For example, the temperature of e-interface in Fig. 2 can
be calculated with the following equation
TðiÞ ¼ T e ¼
þ
T P ls =dx
e þ T E lf =dxe qr ðeÞ
ls =dxe þ lf =dxþe
(10)
3.3. View factor calculation
(6b)
where, qr is the net radiant heat flux at the inner surfaces of the
holes in the hollow clay bricks. As can be seen from Fig. 1 each
hole is composed of six surfaces: top, bottom, and four lateral
surfaces. These six surfaces are further assumed to be at
individual constant temperatures and taken as the computational radiation surfaces, for each of which the radiosity J(I), and
the heat flux qr(I) can be determined according to radiative heat
It can be seen that in order to determine the interface
temperature the surface radiative flux is required (Eq. (10)). To
determine the surface heat flux from Eq. (7) the view factor data for
each surface of the six-surface enclosure should be supplied. The
relationship between every two surfaces of the enclosure is either
two aligned parallel rectangular planes or two perpendicular
rectangular planes. For such combinations, the view factor, XIJ, can
be calculated according to the following equation [13,14]:
For aligned parallel rectangles
8 2
9
2
2 31=2
>
>
=
1þZ
2 < 4 1þX
X
Z
2 1=2
2 1=2
1
1
1
1
5
X IJ ¼
þ
X
1
þ
Z
tan
þ
Z
1
þ
X
tan
Xtan
X
Ztan
Z
ln
2
2
1=2
1=2
>
2
2
pXZ >
:
;
1þX þZ
1þZ
1þX
(11)
For perpendicular rectangles with a common edge
0
1
2
1
2 1=2
1 1
1 1
1
Xtan
Y
þ
X
tan
þYtan
B
C
2
2 1=2
X
Y
B
C
Y þX
B
C
9
8
B
C
2
2
1 B
C
2
3
2
3
X
Y
>
>
X IJ ¼
>
>
B
C
2
2
2
2
2
2
2
2
>
>
=
<
C
1
þ
1
þ
X
Y
X
1
þ
X
þ
Y
Y
1
þ
Y
þ
X
pX B
6
7 6
7
B þ 1 ln
C
4
5
4
5
B
C
2
2
2
2
2
2
2
2
>
@ 4 >
A
>
>
X
þ
Y
X
X
þ
Y
Y
Y
þ
X
1
þ
1
þ
1
þ
>
>
;
:
(12)
L.P. Li et al. / Energy and Buildings 40 (2008) 1790–1798
1795
1.3%. So the grid system of case 1, which is L1 = 122, M1 = 82 and
N1 = 122, has been chosen for all kinds of 290 140 90 hollow
clay bricks to keep a balance between numerical accuracy and
computational cost. This set of grid system can ensure that there is
at least 720 control volumes in the smallest holes. All grids are
distributed evenly in the computation domain.
4. Simulation results and discussion
Fig. 2. The control volumes at the sides of e-interface.
Obviously, the solution process is iterative in nature. Both T(I)
and qr(I) should be updated during iteration procedure. Besides, in
order to ensure the convergence of iteration, the relaxation for
velocity components and pressure is absolutely necessary for all
cases.
In the following presentation, the effects of the hole configuration (without surface radiation), the surface radiation and the
indoor–outdoor temperature difference on the equivalent thermal
conductivity will be discussed in order, followed by some typical
velocity and temperature fields. Simulations are conducted for two
series of conditions. In the first series the surface radiation is not
considered, and the heat is transferred only by conduction and
natural convection. The resulting equivalent thermal conductivity
is denoted by lnat. In the second series, radiation is taken into
account and the resulting equivalent thermal conductivity is
expressed by lrad.
4.1. The influence of the hole number on the equivalent thermal
conductivity
In the preliminary computations, the test of grid independence
of the solution has been carried on for the most complicated
L14W6H1 hollow clay bricks. The results are drawn in Fig. 3.
Compared case 1 with case 4, when the total grid number increases
from 122 104 to 949 104, lrad has an increment no more than
Simulation started on the cases in which surface radiation in the
holes was neglected. The results of lnat for 72 kinds of
290 140 90 hollow clay bricks are drawn in Fig. 4 and some
selected values are listed in Table 1.
From this graph we can see that when the holes in width
direction are 1–6, with the increase of the hole number in length
from one to two holes lnat decreases sharply, it further drops off
with the increase of lengthwise holes from 2 to 4,and reaches its
minimum at holes of 4–6. Then it increases gradually. When the
lengthwise hole number is greater than 4, widthwise hole
number shows some effect on lnat: the more the widthwise
holes, the larger the value of lnat. This variation tendency can be
understood as follows. If surface radiation in the holes is
neglected, the main influences on the equivalent thermal
conductivity are the heat conduction through the clay and
natural convection in the holes. When the hole number is not
much, the increase of the hole number leads to such a situation
that the deterioration of natural convection is larger than the
enhancement in heat conduction through the increased rib,
which makes lnat smaller. Otherwise, the increase of the hole
number leads to an opposite results and makes lnat larger.
Therefore, whether lnat increases or decreases with hole
number depends on which factor is dominant.
Fig. 3. The validation of grid independence.
Fig. 4. lnat for the 290 140 90 hollow clay bricks.
3.4. Convergence criteria
Convergence criteria of the iteration procedure for the resulted
algebraic equations discretized from the governing equation,
Eq. (1), are taken as follows:
Rmax/G < 106; jQinwall Qoutwallj/min (Qinwall, Qoutwall)<102;
and
kþ300
k
l
l 6
10
kþ300
l
for all cases to be simulated.
3.5. Grid-independence examination
L.P. Li et al. / Energy and Buildings 40 (2008) 1790–1798
1796
Fig. 5. lrad for the 290 140 90 hollow clay bricks.
4.2. The influence of the surface radiation on the equivalent thermal
conductivity
The equivalent thermal conductivities of 72 kinds of
290 140 90 hollow clay bricks with the effects of conduction,
convection and surface radiation all being considered simultaneously are drawn in Fig. 5 and some selected values are listed in
Table 1. Compared with lnat in Fig. 4, it can be observed that both lnat
and lrad have the same variation tendency: with the increase in hole
number the equivalent thermal conductivity decreases first, reaches
the minimum and then increases gradually in some extent.
However, the surface radiation enhances the equivalent thermal
conductivity in some extent. The L2W1H1 hollow clay brick has the
largest increment of 25.8% while the L14W6H1 hollow clay brick has
the lowest increment of 4.6%. Generally speaking, the more the hole
number, the less the surface radiation effect. The detailed
comparison between lrad and lnat can be found in Table 1.
From Fig. 5, following features may be noted. First, the brick
having only one hole in both width and length has significantly
higher equivalent thermal conductivity than all the other cases.
Second, for the case with at least two holes in width, the effect of
hole number in width becomes much weaker, as all the curves with
2–6 holes in width direction being more or less compacted. And for
the cases with 37 holes in length direction, the equivalent
thermal conductivity of 5 holes in width is the least. The character
of the hole number effect is due to the complicated heat transfer
process caused by conduction, convection and surface radiation
taking place in the hollow clay bricks simultaneously. With the
increase of the hole number convection and surface radiation in the
holes are deteriorated while conduction through the clay may be
enhanced. Therefore, the equivalent thermal conductivity depends
on the total effects of conduction, convection and surface radiation.
According to the simulation results, the optimum configuration of
the 290 140 90 hollow clay bricks is the L08W4H1 brick, which
has the lowest lrad of 0.400 W/(m K). Compared with the lowest lnat
of 0.350 W/(m K), there is an increment of 12.5%. Therefore, the effect
of surface radiation in the holes cannot be neglected. It is interesting
to note that this lowest value is quite close to the one of
240 115 90 brick, which is 0.419 W/(m K), and the related void
fraction is not different too much either (from 46.9% to 50.3%)
4.3. The effect of temperature difference between indoor and outdoor
on the equivalent thermal conductivity
By keeping the indoor temperature at 20 8C while changing
outdoor temperatures from 30 8C to 0 8C, the effects of the
Table 2
The temperature influence on lrad for 290 140 90 bricks
Kinds
Tf1 Tf2 = 50 8C lrad
(W/(m K))
Tf1 Tf2 = 40 8C lrad
(W/(m K))
Tf1 Tf2 = 30 8C lrad
(W/(m K))
Tf1 Tf2 = 20 8C lrad
(W/(m K))
Maximum difference
(compared with Tf1 Tf2 = 40 8C) (%)
L01W1H1
L03W1H1
L07W1H1
L10W1H1
L14W1H1
L01W2H1
L02W2H1
L07W2H1
L09W2H1
L14W2H1
L01W3H1
L02W3H1
L07W3H1
L09W3H1
L14W3H1
L01W4H1
L02W4H1
L07W4H1
L09W4H1
L14W4H1
L01W5H1
L02W5H1
L07W5H1
L09W5H1
L14W5H1
L01W6H1
L03W6H1
L08W6H1
L10W6H1
L14W6H1
0.922
0.594
0.530
0.521
0.510
0.822
0.569
0.424
0.419
0.432
0.799
0.545
0.409
0.406
0.428
0.781
0.533
0.402
0.401
0.431
0.755
0.524
0.402
0.403
0.438
0.703
0.449
0.417
0.429
0.460
0.905
0.586
0.522
0.511
0.498
0.807
0.561
0.422
0.417
0.429
0.781
0.537
0.408
0.405
0.428
0.758
0.523
0.401
0.401
0.431
0.726
0.512
0.401
0.403
0.439
0.670
0.444
0.417
0.430
0.461
0.885
0.577
0.513
0.499
0.485
0.787
0.552
0.419
0.414
0.427
0.758
0.527
0.407
0.404
0.429
0.729
0.510
0.401
0.401
0.433
0.691
0.499
0.401
0.404
0.441
0.631
0.438
0.418
0.432
0.463
0.857
0.563
0.499
0.484
0.468
0.760
0.538
0.416
0.410
0.423
0.725
0.512
0.405
0.403
0.429
0.688
0.494
0.400
0.401
0.435
0.644
0.482
0.402
0.405
0.443
0.584
0.430
0.420
0.434
0.464
1.9 to 5.3
1.4 to 3.9
1.5 to 4.4
2.0 to 5.3
2.4 to 6.0
1.9 to 5.8
1.4 to 4.1
0.5 to 1.4
0.5 to 1.7
0.7 to 1.4
2.3 to 7.2
1.5 to 4.7
0.2 to 0.7
0.2 to 0.5
0.0 to 0.2
3.0 to 9.2
1.9 to 5.5
0.2 to 0.2
0.0 to 0.0
0.0 to 0.9
4.0 to 11.3
2.3 to 5.9
0.2 to 0.2
0.0 to 0.5
0.2 to 0.9
4.9 to 12.8
1.1 to 3.2
0.0 to 0.7
0.2 to 0.9
0.2 to 0.7
L.P. Li et al. / Energy and Buildings 40 (2008) 1790–1798
1797
Fig. 6. Isotherms for the L1W1H1 hollow clay brick (z = 0.02).
Fig. 8. V vectors for the L1W1H1 hollow clay brick (z = 0.02 and z = 0.21).
indoor–outdoor temperature difference on lrad were simulated.
Some simulation results are listed in Table 2.
We can see from this table that the effect of temperature
difference on lrad with different hole number and arrays is not the
same. The values of lrad for some kinds of bricks decrease when the
temperature differences decrease while for other kinds lrad
increase in some extent. Except L1W6H1 hollow clay brick for
which the maximum difference percentage is 12.8% between the
case of Tf1 Tf2 = 20 8C and Tf1 Tf2 = 40 8C, most kinds of bricks
have the percentage difference no more than 5%. It is wonderful to
note that the hollow clay brick with the optimum configuration
(L8W4H1) has the lrad almost constant when outdoor temperatures changes. Besides, the nearer to the optimum configuration
the smaller the lrad change with the temperature difference. This
implies that the optimum configuration does not change with the
indoor–outdoor temperature difference. In [12] the similar result
was obtained for the 240 115 90 hollow clay bricks.
4.4. Isothermals and velocity vectors for some representative hollow
clay bricks
Three representative kinds of 290 140 90 hollow clay
bricks are selected to present flow and temperature fields. These
are L01W1H1, L08W4H1 and L14W6H1: L01W1H1 has the
smallest hole number (one hole) and the largest void fraction
(79.8%), L08W4H1 is the optimum configuration (void fraction of
50.3%) and L14W6H1has the most holes number (84) and the
Fig. 7. Isotherms for the L1W1H1 hollow clay brick (z = 0.21).
smallest void fraction(33.5%). The isothermals and velocity vectors
for these three kinds of hollow clay bricks at some cross sections
(z = 0.02 and z = 0.21) are shown in Figs. 6–11. It should be noted
that these figures are drawn for the entire computational domain
which is composed of four identical bricks with two orientations.
From these figures we can note the following features. First
from the isotherms with the increase in hole number the natural
convection in the holes becomes more and more conductionpredominated. The isotherms in Figs. 6 and 7 show very strong
convection, characterized by the horizontal part of the isotherms in
the lower part of the enclosure. While in Fig. 9 isotherms of
L8W4H1 hollow clay brick are more or less parallel to each other
and finally almost all the isotherms of L14W6H1 hollow clay brick,
either in the solid clay or in the holes, exhibit quite well parallel
character (Fig. 11), characterized by pure heat conduction. Second
from the velocity fields, the flow in L01W1H1 is highly 3-D in
nature. This can be witnessed by the velocity fields in two cross
sections: at z = 0.02, the flow field belongs to conductionpredominated type, while the flow in z = 0.21 exhibits strong
convection, and boundary layer type flow can be observed along
the two vertical surfaces of the hole (Fig. 8). With the increase in
the hole number convection becomes weaker and weaker, and for
the L14W6H1 the magnitude of velocity has a two-order difference
compared with that of L01W1H1. Only the V vectors for the
L8W4H1 brick are shown in Fig. 10, and those for L14W6H1 brick
Fig. 9. Isotherms for the L8W4H1 hollow clay brick (z = 0.21).
L.P. Li et al. / Energy and Buildings 40 (2008) 1790–1798
1798
Fig. 10. V vectors for the L8W4H1 hollow clay brick (z = 0.02 and z = 0.21).
conductivity with the length direction hole number are quite
close to each other. The reason is explained in the context.
3. Within the range of 20–50 8C of the indoor–outdoor temperature difference, its effect on the equivalent thermal conductivity
is not very significant, with most percentage difference being
within 5%. Interestingly, for the optimum configuration
L8W4H1, its equivalent thermal conductivity almost does not
change with the temperature difference. Moreover, the nearer to
the optimum configuration, the smaller the change of their
equivalent thermal conductivity with the temperature difference.
4. Compared with the results in [12], it is found that the two kinds
of hollow clay bricks (290 140 90 vs. 240 115 90) have
much similar variation trends about the influence of the hole
number and arrays, surface radiation and indoor–outdoor
temperature difference. Therefore, it can be anticipated that
above variation trends may also be suitable to other types of
hollow clay bricks. However, to optimize the configuration of
each type of hollow clay bricks individual simulation is required.
Acknowledgments
This work is supported by the National Natural Science
Foundation of China (50636050) and the National Fundamental
Projects of R&D of China (2007CB206902).
References
Fig. 11. Isotherms for the L14W6H1 hollow clay brick (z = 0.21).
are so small that they can hardly be spotted with the same printed
scale and are not shown by figures.
5. Conclusions
Through an extensively numerical study on the influence of the
hole number and arrays, with and without surface radiation as well
as variable indoor–outdoor temperature difference on the
equivalent thermal conductivity of 290 140 90 hollow clay
bricks, following conclusions can be made.
1. The surface radiation in the holes makes the equivalent thermal
conductivity augmented in some extent, with the maximum
increment being 25.8% for L2W1H1 brick and the minimum
increment being 4.6% for L14W6H1. With the increase in hole
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2. The hole number and their arrays have crucial influence on the
equivalent thermal conductivity. The brick with only one hole in
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equivalent thermal conductivity. The L8W4H1 has the smallest
equivalent thermal conductivity of 0.400 W/(m K), which is the
optimum configuration of the 290 140 90 hollow clay
bricks. The effect of hole number in length direction is larger
than that in width direction. For the hole number of 2–6 in width
direction their variation curves of the equivalent thermal
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