EPJ Web of Conferences 143, 02007 (2017 )
DOI: 10.1051/ epjconf/201714302007
EFM 2016
New methodology for the walls design in buildings by
numerical simulation of the thermal convection
Elhadj Benachour1,2, a, Belkacem Draoui1, Bachir Imine2, Khadidja Asnoune1, Elmir Mohamed1
1
Laboratory of Energy in Arid Regions, University of Tahri Mohamed - Bechar, P.O. P 417. 08000 , Road
Kenedza, Bechar, Algeria
2
Department of Mechanical Engineering, University of Sciences and Technology of Oran-Mohamed Boudiaf,
P.O.1505 El-Mnaouar, Oran, Algeria
Abstract. Buildings are complex systems composed of several elements, which are assembled to respond to a
number of needs functional and symbolic according to set of legal and environmental requirements and
potentially accommodate users with different levels of demand. Predicting the conception of the external wall
is beneficial in the design of house and building structures.in this study, an analogy was used for the functions
which are discretized by the finite difference method and integrated in the CFD code which is based on the
finite volume method. The CFD software is used as a technique to modelling the behaviour of fluid and the
thermal convection in the external wall of the house with different Rayleigh numbers [103 Ra 105]. In the
second phase, we change the thickness of the wall several times and calculate the Nusselt number and
exchange coefficient of heat transfer aims to find a cloud point respectively for the thicknesses e = 0, L /40,
L /20 and L /10. After, we developed a relationship that helps us to know the exchange ratio for each thickness
( e ) belongs to the interval [0, L /10] by the Lagrange polynomial interpolation method for Rayleigh number
equal 104 , and then we developed a FORTRAN program to control the nonlinear equation of order three. This
method for predicting exchange coefficient of convection for to optimize the design of walls in buildings.
1 Introduction
The thermal behavior of buildings is a current problem
Actually, In Algeria, The construction industry must start
to take into account the changing needs of the population
in its projects. It’s not just homes that need to be adapted
to ensure a safe and inclusive environment, but also the
wider surroundings. Local shops and town centres have
to be accessible and to accommodate the needs of people.
This survey indicates that the biggest improvements
could be made to the building sector, which must be
properly adapted to ensure a safe environment for a
workforce .The building sector today is known to be
consuming 40% of the world energy [1,2], and in turn,
supports 23–40% of the world's greenhouse gas emission,
particularly CO2 [3]. It has become common in recent
years to gather data on human attitudes and behaviour in
building energy research through interviews with
building occupants and other relevant actors. Examples of
this kind of research are found in the main journals which
deal with technical aspects of energy in buildings,
including
Building
and
Environment
[4–12],
Building and Environment [13-15], Energy Storage [1620], Energy and Buildings [21–33]. In most situations,
the mechanical cooling devices offer solutions that are
a
neither environment friendly nor energy sustainable. The
mechanical devices are non-functional and cannot offer
thermal comfort without energy input. Hence utilization
of advanced building materials and passive technologies
in buildings may offer the solution for thermal comfort
demands, substantially reduce the energy demand, impact
on the environment and carbon footprint of building stock
worldwide [34].Thermal comfort is dependent and
influenced by a range of environmental factors viz. air
temperature, radiant temperature, humidity, air
movement, metabolic rate or human activity [35,36].
Also, numerous studies across the world have shown the
impacts of hot working environments on the working
population [37–43]. The choice of which material or
combination of materials is used depends on a wide
variety of factors. Some insulation materials have health
risks, some so significant the materials are no longer
allowed to be used but remain in use in some older
buildings such as asbestos fibers and urea. Also, the cost
can be high compared to the traditional insulation. The
interior or external insulation does not often become the
only possible solution in particular with the old buildings
in the Sahara of Algeria. Hygrothermal parameters of the
existing wall material should be reported and well known
for designing powerful and durable walls in the time
Corresponding author: benachour_elhadj@yahoo.fr
© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution
License 4.0 (http://creativecommons.org/licenses/by/4.0/).
EPJ Web of Conferences 143, 02007 (2017 )
DOI: 10.1051/ epjconf/201714302007
EFM 2016
x
study. First, in this paper, CFD software is used as a
technique to modelling the behaviour of fluid and the
thermal convection in the external wall of the house with
different Rayleigh numbers [103 Ra105], In this
context, an analogy was used for the functions which are
discretized by the finite difference method and integrated
in the CFD code which is based on the finite volume
method. Secondly, The most important part in this work
is to vary the thickness of the building material of the
outer wall four times and calculate the Nusselt number
and exchange coefficient of heat transfer aims to find a
cloud point respectively for the thicknesses e = 0, L / 40,
L / 20 and L / 10. After, we developed a relationship that
helps us to know the exchange ratio for each thickness e
belongs to the interval [0, L / 10] by the Lagrange
polynomial interpolation method, and then we developed
a FORTRAN program to control the nonlinear equation
of order three. This method for predicting exchange
coefficient of convection to optimize the design of walls
in buildings before starting the wall construction for
Rayleigh number equal 104 .
Study the effect of the Rayleigh number on the
convection in building.
4 Mathematical model
The fluid is assumed incompressible and obeys the
Boussinesq approximation. In these cases, continuity in
two dimensions and the equations governing the flow and
energy is given by:
Continuity.
wU wV
wX wY
0
(1)
X-momentum.
wU
wU
wU
U
V
wt
wY
wX
1 wP
X 2U (2)
U wX
Y-momentum.
2 Geometric configuration
wV
wV
wV
U
V
wY
wX
wt
1 wP
U wX
X 2V gE
wT
wX
(3)
Energy.
O
1
2T
M (4)
U cp
U cp
wT
wT
wT
V
U
wY
wX
wt
The derived equation of motion Eq. (2) over Y and the
equation of motion Eq. (3) by contributing to x, then, after
subtracting the two equations obtained, we obtain the
equations dimensionless variables in writing Helmotz in
terms of vorticity and stream function formulation are as
follows:
Fig. 1. Schematic of the studied configuration
wZ
wZ
wZ
V
U
wY
wX
wt
To simplify the problem, assume that:
• The fluid is Newtonian and incompressible.
• The heat dissipation by viscous friction is neglected.
• The Boussinesq approximation is considered
wT
wT
wT
V
U
wt
wY
wX
w 2\
w 2\
wX 2 wY 2
3 Objectives
We can express our aim for this study in the
following
points:
x Prediction of the wall design through numerical
evaluation of the convection in buildings by the
Lagrange polynomial interpolation method.
x Modelling Study of the convection – conduction
coupling.
x Study of the effect of the distribution of the heat
inside buildings on the convection.
x Comparative study between the construction of
buildings with different thicknesses and without
insulation.
wT
wX
(5)
M
L2
w 2T w 2T
2
2
U Cp a 'T
wY
wX
(6)
Pr 2 Z Ra Pr
Z
(7)
The stream function and vorticity are related to the
velocity components by the following expressions:
U
w\
,V
wY
w\
and Z
wX
wV wU
wX wY
(8)
The dimensionless parameters in the equations above are
defined as follows:
2
EPJ Web of Conferences 143, 02007 (2017 )
DOI: 10.1051/ epjconf/201714302007
EFM 2016
°X
°°
®
°
°V
°¯
x
,
L
Q
ui
,P
Y
y
,
L
P
,T *
U ui 2
U
u
ui
(9)
Nu
(average
number)
(hot wall)
T Ti
t
,t *
Tc Ti
L2 / a
De Vahl
Davis
[45]
De Vahl
Davis
[45]:
De Vahl
Davis
[45] :
1.12
2.243
4.52
For solid (Concrete), we are interested only in the
following heat equation:
wT
wt
1 w 2T w 2T
(
)
a w X 2 wY 2
7 Mathematical model for the lagrange
polynomial interpolation method
(10)
The
Lagrange
interpolating
polynomial
is
the polynomial of degree (n-1) that passes through
the n points (x1, y1=f(x1)) , (x2, y2=f(x2)) , ..., (xn,
5 Procedure of simulation
yn=f(xn)), the interpolation polynomial in the Lagrange form is
a linear combination and is given by :
First, The numerical calculation was conducted using the
GAMBIT and FLUENT software. The numerical
procedure used in this work is that of finite volume. It
involves the integration of differential equations of
mathematical model on finite control volumes for the
corresponding algebraic equations. In this context, an
analogy was used for the functions which are discretized
by the finite difference method and integrated in the CFD
code which is based on the finite volume method. The
SIMPLE algorithm [44] was chosen for the coupling
speed pressure in the Navier-Stokes equations on a
staggered grid. The convective terms in all equations are
evaluated using the schema apwind first order. The
discretization of the time term is made in a totally implicit
scheme. The convergence of the solution is considered
reached when the maximum relative change of all
variables (u, v, w, p, t) between two successive time is not
less than 104. With an aim of following well any variation
of the fields thermal and hydrodynamic, we used a
uniform grid of 14241 nodes and 14480 Elements in
non-stationary mode. Secondly, we developed a
FORTRAN program to control the nonlinear equation of
order three. This method of interpolation was used for
predicting exchange coefficient of convection to optimize
the design of walls in buildings
n
Pn (x)
¦y
Li (x)
i
(11)
i 0
For Lagrange basis polynomials
n
(x x )
j
Li (x)
j 0
j zi
n
n
(x
i
xj )
(x x j )
(x
j 0
j zi
i
xj )
(12)
j 0
j zi
Where
°Li (xj ) 0 if i z j
®
°̄Li (xj ) 1 if i j
(13)
7.1 The polynomial interpolation points
We wish to find the polynomial interpolating the points
Table2 The sets of polynomial interpolation points.
B=e/L
Nua
0
2.24
1/40
1.59
1/20
1.35
1/10
0.98
6 Validation of the Code of Calculation
Where, the points of Nu (e) are obtained by numerical
simulation of the FLUENT software, for the Rayleigh
number equal 104.
6.1 Validation of the model
8 Results and discussion
The computer code was validated on a natural convection
problem of stale air in a square cavity with vertical walls
differentially heated and adiabatic horizontal walls. Our
results were compared with those obtained by De Vahl
Davis (1983) [45] .The latter dealt with the same problem
by adopting the finite difference method with the
vorticity-stream function formulation (see Table 1).
The boundary conditions have been established to
simulate a geometric configuration used frequently in
two-dimensional approximation .The structure of the
flow, the temperature field and heat transfer through the
hot wall are discussed in this section.In this study, to
target the most important goal, we will show and studied
the dynamic and thermal behavior of the fluid in the
cavity.
Table 1 Comparisons of the results of validation
Nu
(average
number)
(hot wall)
Ra = 103
Ra = 104
Ra = 105
8.1 Isotherms
present
study:
present
study :
Present
study :
1.2
2.257
4.64
The isotherms are shown in Figures [2]. The heat
distribution in the cavity is in accordance with the fluid
circulation revealed by isotherms and iso-currents. Indeed
3
EPJ Web of Conferences 143, 02007 (2017 )
DOI: 10.1051/ epjconf/201714302007
EFM 2016
we find a heating fluid from the interface, if it causes the
change of the heat distribution in the cavity (see figure)
for different numbers of Rayleigh For a fixed value of the
number of Prandtl equal 0.71 (air fluid). Gradually, as the
Rayleigh number has increased, the isotherms become
increasingly wavy and heat transfer increases, so the flow
intensifies and natural convection is expanding and
predominates (natural convection is predominant).
l3
(17)
We can see that each term is of degree three, so the entire sum
has degree at most three.
The polynomial P(x) given by the above formula is called
Lagrange’s interpolating polynomial and the functions Eq. (l4,
Eq). (l5), Eq). (l6), Eq.( l7) are called Lagrange’s interpolating
basis functions.
0.8
8.3 Nusselt number correlations
R a= 1 .e 3
R a= 1 .e 4
R a= 1 .e 5
0.6
Temperature
L
L
) (x
)
40
20
L
L
L
L
L
0) (
(
) (
)
10
10
40
10
20
(x 0) (x
It should be noted that the numerical result given by the
equation Eq.(20). Remarkably similar to the estimate of
the average Nusselt number by the Lagrange interpolation
method for each wall thickness between [0, L / 10] but
only for Ra=104.
0.4
0.2
P3 ( x) y0l0 y1 l1 y2 l2 y3 l3
0
0.2
0.4
0.6
0.8
1
X
P3 (x) 2.24l0 1.59 l1 1.35 l2 0.98 l3 (19)
Fig. 2. The Isotherms for wall thickness e = L/20, and for
different Rayleigh Numbers, Pr=0.71.
P3 ( x )
The points of
Nu (e) are obtained by numerical
simulation of the FLUENT software, for the Rayleigh
number equal 104 . (See table 3)
Nu(e)
Xi=e
0
0
2.24
4
1
e=L/40
1.59
4
2
e=L/20
1.35
4
3
e=L/10
0.98
10
10
Nu
l1
l2
L
)
10
L
)
10
L
)
10
L
)
10
L
)
10
L
)
10
(22)
In heat transfer within a fluid, the Nusselt number (Nu) is
the ratio of convective to conductive heat transfer. The
average value of the heat transfer coefficient is often
needed in process engineering and design applications. In
this context, the average Nusselt number is given by:
Nu
y 0 l 0 y1 l1 y 2 l 2 y 3 l 3
L
L
) (x
) (x
40
20
L
L
(0
) (0
) (0
40
20
L
(x 0) (x
) (x
20
L
L
L
L
(
) (
0) (
40
40
20
40
L
(x 0) (x
) (x
40
L
L
L
L
(
) (
0) (
20
20
40
20
(x
2986.6 B3 552 B2 37.93 B 2.24
Where: B: e/L
hH
O
h
O Nu
H
(23)
Where h is the convective heat transfer coefficient of the
flow, H is the characteristic length, is the thermal
conductivity of the fluid.
We would have the four basis polynomials
l0
2986.6 3 552 2 37.93
e 2 e
e 2.24 (21)
L3
L
L
e : Wall thickness.
Also, we can write:
If (0, 2.24),(L/40, 1.59), (L/20, 1.35), (L/10, 0.98), are
given data points, then the cubic polynomial passing
through these points can be expressed as,
P3 ( x)
(20)
Nu : Average Nusselt number.
Yi=Nu (e)
Average Nusselt
number
104
10
8960 3 1656 2 113 . 8
x
x
x 2 .24
3 L3
3 L2
3L
Where:
Table3 Nusselt number for different values of thickness.
and for Ra=104
i
Therefore, we can write,
8.2 Lagrange polynomials
Ra
(18)
(14)
8.4 Average Nusselt numbers
The influence of the Rayleigh number and Wall thickness
on the average Nusselt number is shown in Fig. 5 and fig
6. The Nusselt number decreases with increasing of the
wall thickness. Therefore, the heat transfer also decreases
because the inertia and the thermal resistance of the wall
increases. The average value of the Nusselt number
(15)
(16)
4
EPJ Web of Conferences 143, 02007 (2017 )
DOI: 10.1051/ epjconf/201714302007
EFM 2016
belongs to the interval [0, L / 10] by the Lagrange
polynomial interpolation method, and then we developed
a FORTRAN program to control the nonlinear equation of
order three (equation Eq.(21).or. Eq.(22). This method for
predicting exchange coefficient of convection for to
optimize the design of walls in buildings. Since,
Interpolation is the process of defining a function that
takes on specified values at specified points.
Predictive simple correlation was developed to estimate
the value of the average Nusselt number and the
coefficient of heat transfer for exchange any thickness
ranging from 0 to L/10 .it just enough to replace the
thickness value in equation Eq.(21), to calculate the
Nusselt number planned before the design and
construction of walls.
increases as the value of the Rayleigh number (Ra)
increases. Broadly, advection becomes stronger and thus
heat transfer increases. This trend is qualitatively
consistent with that seen in the convection for important
values of Ra.
5
Ra = 1.E3
4.5
Ra = 1.E4
Ra = 1.E5
4
Nu(Average number)
3.5
3
2.5
2
1.5
1
0.5
0
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Acknowledgment
0.1
e/l (wall thickness)
Fig. 5. Profiles of average Nusselt number for different wall
thickness and for different Rayleigh number, Ra=103,104,105, Pr
= 0.71.
This work supported by the Research program sponsored
by the Faculty of Science and Technology, Department
of
technology, ENERGARID laboratory,
Tahri
Mohamed University, Bechar, Algeria.
I would like to thank all reviewers for taking the time and
energy to review our work.
4.5
Wall thickness e=0
4
Wall thickness e=L/40
Wall thickness e=L/20
Nu(average number)
3.5
Wall thickness e=L/10
References
3
2.5
1.
2
1.5
2.
1
0.5 3
10
4
10
5
10
Ra
3.
Fig. 6. Profiles of average Nusselt number for different Rayleigh
number, Ra=103,104,105, Pr = 0.71.
4.
9 Conclusion
Extensive numerical results for the laminar natural
convection heat transfer from a heated wall are obtained
for the range of conditions as: Rayleigh number, 103 Ra
105 and Prandtl number Pr = 0.71, Concrete wall for
different thicknesses is viewed with 0 e L/10. We are
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software is used as a technique to modelling the
behaviour of fluid and the thermal convection in the
external wall of the house, in this context, an analogy was
used for the functions which are discretized by the finite
difference method and integrated in the CFD code which
is based on the finite volume method. Secondly, The most
important part in this work is to vary the thickness of the
building material of the outer wall four times and
calculate the Nusselt number and exchange coefficient of
heat transfer aims to find a cloud point respectively for
the thicknesses e = 0, L / 40, L / 20 and L / 10. After, we
developed a relationship that helps us to know the Nusselt
number and exchange ratio for each thickness ( e )
5.
6.
7.
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