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Abstract

A two-dimensional model is proposed for energy efficiency assessment through the simulation of heat transfer in building envelopes, considering the influence of the surrounding environment. The model is based on the Du Fort–Frankel approach that provides an explicit scheme with a relaxed stability condition. The model is first validated using an analytical solution and then compared to three other standard schemes. Results show that the proposed model offers a good compromise in terms of high accuracy and reduced computational efforts. Then, a more complex case study is investigated, considering non-uniform shading effects due to the neighboring buildings. In addition, the surface heat transfer coefficient varies with wind velocity and height, which imposes an addition non-uniform boundary condition. After showing the reliability of the model prediction, a comparison over almost 120 cities in France is carried out between the two- and the one-dimensional approaches of the current building simulation programs. Important discrepancies are observed for regions with high magnitudes of solar radiation and wind velocity. Last, a sensitivity analysis is carried out using a derivative-based approach. It enables to assess the variability of the solution according to the modeling of the two-dimensional boundary conditions. Moreover, the proposed model computes efficiently the solution and its sensitivity to the modeling of the urban environment.
arXiv:2111.09131v1 [cs.CE] 17 Nov 2021
An efficient two-dimensional heat transfer model for building envelopes
Julien Berger a, Suelen Gasparina, Walter Mazuroskib, Nathan Mendesc
November 18, 2021
aLaboratoire des Sciences de l’Ingénieur pour l’Environnement (LaSIE), UMR 7356 CNRS, La Rochelle Université,
CNRS, 17000, La Rochelle, France
bUniv. Savoie Mont Blanc, LOCIE, 73000 Chambéry, France
cThermal Systems Laboratory, Mechanical Engineering Graduate Program,
Pontifícia Universidade Católica do Paraná, Rua Imaculada Conceição, 1155, CEP : 80215-901, Curitiba, Brazil
Abstract
A two-dimensional model is proposed for energy efficiency assessment through the simulation of heat
transfer in building envelopes, considering the influence of the surrounding environment. The model
is based on the Du FortFrankel approach that provides an explicit scheme with a relaxed stability
condition. The model is first validated using an analytical solution and then compared to three other
standard schemes. Results show that the proposed model offers a good compromise in terms of high
accuracy and reduced computational efforts. Then, a more complex case study is investigated, consider-
ing non-uniform shading effects due to the neighboring buildings. In addition, the surface heat transfer
coefficient varies with wind velocity and height, which imposes an addition non-uniform boundary condi-
tion. After showing the reliability of the model prediction, a comparison over almost 120 cities in France
is carried out between the two- and the one-dimensional approaches of the current building simulation
programs. Important discrepancies are observed for regions with high magnitudes of solar radiation and
wind velocity. Last, a sensitivity analysis is carried out using a derivative-based approach. It enables to
assess the variability of the solution according to the modeling of the two-dimensional boundary condi-
tions. Moreover, the proposed model computes efficiently the solution and its sensitivity to the modeling
of the urban environment.
Key words: Du FortFrankel method; Two-dimensional heat transfer; Numerical model; Building
energy efficiency; Non-uniform boundary conditions; Surface solar fraction.
1 Introduction
The building sector is responsible for almost 33% of the world global energy consumption and the current
environmental context imposes an improvement of the energy efficiency of building envelopes [1]. For this,
several tools, called building simulation programs, have been developed over the last 50 years to assess
building energy performance. A review of such models has been proposed in [2] with a recent update in [3].
Among the most contemporary, one can cite Domus [4] or EnergyPlus [5] as examples that employ modern
techniques of shading assessment, for instance, but have building envelope engines limited to one-dimensional
heat transfer modeling.
Among all the phenomena involved in building physics, the heat transfer process through the envelope
is one of the most important since it represents a major part of the energy consumption. The conduction
loads through the envelope require fine and accurate modeling to guarantee the reliability of the building
simulation programs. However, several drawbacks can be outlined.
First, generally, the building simulation programs mentioned in [2,3] model the heat transfer process
through the building envelope in one-dimension, as mentioned above. Indeed, for simulation at large scales
(district or urbanity), the reliability of one-dimensional envelope models is reduced. Furthermore, most
common approaches are the resistance-capacitance model or response-factor method to simulate the heat
transfer through the building envelope [6]. As reported in [7,8], detailed models based on two- or three-
dimensional approaches are required to increase the accuracy of the predictions. Some attempts have been
made to include two-dimensional modeling in [9] considering both heat and mass transfer. In [10,11], a
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An efficient two-dimensional heat transfer model for building envelopes
two-dimensional model has been proposed based on an intelligent co-simulation approach. However, those
works assume only simple time-varying boundary conditions.
The second drawback arises from the modeling of the outside boundary conditions. Those are given by
time-varying climatic data with a time step of one hour, which may increase the inaccuracy particularly for
the modeling of the outside incident radiation flux [7]. Moreover, as reported in [12], many tools use simple
trigonometric methods for shading assessment. Some alternative techniques have been proposed to increase
the accuracy of the methods. Particularly, in [12], a pixel counting technique is developed and validated
using experimental data. Even if this approach has been integrated into the Domus building simulation
program, the simulation still considers one-dimensional transfer through the envelope. Similarly, the heat
transfer coefficient at the interface between the wall and the outside environment is modeled using empirical
models. An extensive literature review is given in [13]. One can note in Table 1 of the mentioned reference
[13] that most programs assume constant values. In building simulation programs such as EnergyPlus, the
coefficient may vary according to the wind speed velocity and/or the height. Nevertheless, as mentioned in
[14], the building simulation programs cannot handle spatially variable boundary conditions.
Even with the drawbacks identified, the development of two- or three-dimensional heat transfer model
in building envelope is still a difficult task. Indeed, the physical phenomena in buildings are generally
observed over (at least) one year. Besides, building physical domains scale with several meters. Thus, the
characteristic time and space lengths may induce significant computational cost. Thus, efficient numerical
models are worth of investigation. In this paper, an innovative numerical model based on the Du Fort
Frankel scheme is studied, which has already demonstrated a promising efficiency in [1517] for the
simulation of one-dimensional heat and mass transfer through porous materials in building envelopes. Here,
the model is extended to simulate two-dimensional heat transfer in a building facade over one year. It
considers time and space varying convective and radiative boundary conditions at the external surface. A
comparison is performed to analyze the influence of two-dimensional modeling to predict building energy
efficiency. For this, a derivative-based approach is used to compute efficiently the time-varying sensitivity
of the critical outputs.
To assess this study, the article is organized as follows. The mathematical model for the solution and
its sensitivity is described in Section 2. Then, the Du FortFrankel numerical scheme is presented in
Section 3. A validation procedure is carried out using an analytical solution in Section 4. Then, a more
realistic case study of a building within aurban environment is treated in Section 5. To conclude, final
remarks are addressed in Section 6.
2 Description of the mathematical model
The problem involves heat transfer through the facade of a building located in an urban area where the
studied building faces another one (Figure 1). The front building is located at a distance Dmand it has
a height Fmand induces a time-varying shadow on the studied building.
2.1 Governing equations
The two-dimensional heat diffusion transfer is considered in a facade composed of a multi-layered wall.
The process occurs over the time domain Ωt=0, t f. The space domain is illustrated in Figure 1, where
L,Hand Ware the length, height and width of the wall. The space coordinates xand ybelong to the
domains:
x
def
:= xx0, L and Ω y
def
:= yy0, H .
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An efficient two-dimensional heat transfer model for building envelopes
outside inside
front buildings
wall facade
shadow sunlit
Figure 1. Illustration of the problem considered and the physical domain of heat transfer in the building
facade.
Thus, the spatial domain of the wall is Ω def
:= xy. The four boundaries of the domains are defined
such as:
Γ1
def
:= nx , y yy, x = 0 o,Γ2
def
:= nx , y xx, y =Ho,
Γ3
def
:= nx , y yy, x =Lo,Γ4
def
:= nx , y xx, y = 0 o.
Thus, the whole boundary of the spatial domain is Γ def
:=
4
[
i=1
Γi. The governing equation of heat transfer is:
c(x , y )·T
∂t
∂x k(x , y )·∂T
∂x
∂y k(x , y )·T
∂y = 0 ,(1)
where cJ.m3.K1and kW.m1.K1are the volumetric heat capacity and the thermal conductiv-
ity. The wall is composed of Nmaterials and Ω =
N
[
i=1
i, with ibeing the space domain of the material
i. Thus, both kand cdepend on space coordinates:
k(x , y ) =
N
X
i= 1
ki·φi(x , y ), c (x , y ) =
N
X
i= 1
ci·φi(x , y ),
where kiand ciare the thermal conductivity and the heat capacity of the material iassumed as constant.
The function φicorresponds to a piece wise function basis:
φi(x , y ) =
1,(x , y )i,
0,(x , y )/i.
Initially, the wall is assumed in steady-state condition:
T=T0(x , y ),(x , y ), t = 0 ,
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An efficient two-dimensional heat transfer model for building envelopes
where T0is a given function dependent on space coordinates. The latter requires to be consistent with the
boundary conditions. One important output is the heat flux jW.m2defined by:
j(x , y , t ) = k(x , y )·T·nx , y ,
with nxand nybeing the unitary normal vector of xand yaxis, respectively. The total heat flux
JW.m2impacting at the inside of the ambient zone is computed by:
J(t) = 1
HZΓ3k(x=L , y )·T
∂x x=L
dy . (2)
The last interesting output is the integrated thermal gain (also called thermal or conduction loads):
E(t) = ZtJ(t) dt , (3)
where ttis a time interval generally defined as one month. The thermal loads indicate the amount
of thermal energy transferred through the wall.
2.2 Boundary conditions
At the interface between two materials, the continuity of the heat flux and temperature field are assumed.
At the interface between the wall and the air, the diffusive heat flux entering is equal to the convective and
radiative ones. Thus, a Robin type condition are assumed at the boundary Γi:
ki
∂T
∂n i
+h, i ·T=h, i ·T, i +q, i ,(x , y )Γi,i1, . . . , 4,
where ∂T
∂n
def
:= ∂T
∂x ·n·nxor T
∂n
def
:= ∂T
∂y ·n·nywith nbeing the outward normal of the considered boundary,
h, i W.m2.K1is the surface heat transfer coefficient between the material and the surrounding
ambient air and q, i W.m2is the incident short-wave radiation flux. The air temperature T, i
depends on time:
T, i :t7−T, i (t).
For the external boundary Γ 1, the surface heat transfer coefficient and the incident short-wave radiation
flux depend on both time and space:
h,1: ( y , t )7−h,1(y , t ), q ,1: ( y , t )7−q,1(y , t ).
The coefficient h,1depends on height yand time according to the wind velocity vm.s1[18]:
h,1(y , t ) = h10 +h11 ·v
v0·y
y0β
,(4)
where
v:t7−v(t),
and h10 , h 11 W.m2.K1and βare given coefficients. The reference quantities v0and y0are
set to v0= 1 m.s1and y0= 1 m. The mean value of the surface heat transfer coefficient is defined by:
h,1
def
:= 1
H·tfZtZy
h,1(x , t ) dxdt .
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An efficient two-dimensional heat transfer model for building envelopes
shadowshadow
sunlit
Figure 2. Illustration of the indicator function χ, using the sunlit area ratio computed by Domus.
The incident short-wave radiation flux q,1also depends on space and time according to the variation of
the sunlit on the facade. It is constituted with the direct qdr
, diffuse qdf
and reflected qrf
components. The
direct flux qdr
(t) depends on the total direct solar radiation IW.m2:
qdr
(t) = I(t)·cosθ(t),
where θis the angle between the wall normal and the solar beam. The magnitude of the direct heat
flux depends on the position of the shadow on the facade. The latter can be the consequence of different
shading elements such as screens, trees or other buildings. Thus, the incident radiation flux is decomposed
as:
q,1(y , t ) = α·qdr
(t)·χ(y , t ) + qdf
(t) + qrf
(t),(5)
where αis the wall absorptivity and χ(y , t ) is an indicator function which feature is illustrated in
Figure 2. It returns 1 if yis out of the shadow and 0 if yis in the shadow. Thus, the indicator function is
defined as:
χ(y , t ) =
0, y 6h(t),
1, y > h(t),
(6)
where h(t)mis the height of the shadow on the outside wall facade. It is computed according to:
h(t) = H·1S(t),
where S(t)is the sunlit area ratio perpendicular to the ground. It corresponds to the ratio between
the sunlit area Asm2and the total area Aof the wall facade. Assuming that the frontier between the
sunlit and shadow area as a straight line, the sunlit area ratio is given by:
S(t) = Hh(t)
H.(7)
It is calculated using the pixel counting technique described in [19] and implemented in the Domus building
simulation program [4,12].
2.3 Sensitivity analysis of the two dimensional aspect of the boundary conditions
The outside boundary conditions are modeled in two dimensions, i.e., varying according to the time t
and the height y. To evaluate the influence of such modeling on the assessment of the energy efficiency of
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An efficient two-dimensional heat transfer model for building envelopes
the facade, a derivative-based sensitivity analysis is carried out [2023]. The four essential parameters in
modeling the outside boundary conditions are the height of the front building F, the distance of the front
building D, the first-order coefficient h11 and the coefficient βof the power law described in Eq. (4). A
Taylor development of the temperature is expressed:
T(x , t , h 11 , β , F , D ) = T(x , t , h
11 , β , F , D )
+∂T
∂h 11 h11 =h
11 ·δh 11 +T
∂β β=β·δβ +T
∂F F=F·δF +T
∂D D=D·δD
+Oδh 2
11 , δβ 2, δF 2, δD 2,(8)
where
δh 11
def
:= h11 h
11 , δβ def
:= ββ, δF def
:= FF, δD def
:= DD.
This development enables to assess the variability of the temperature for any value of parameters β , h 11 , F , D
around the given ones β, h
11 , F , D . Note that the Taylor development -Eq. (8)- can be written
for any other chosen output such as, for instance, the total heat flux Jfrom Eq. (2) or the thermal loads E
from Eq. (3).
Instead of performing costly discrete sampling to assess the partial derivative relative to each of the
four parameters β , h 11 , F , D , the governing equation (1) is directly differentiated with respect to the
selected parameter. For this, we denote the four sensitivity coefficients by:
Θ1=∂T
∂h 11
,Θ2=∂T
∂β ,Θ3=T
∂F ,Θ4=T
∂D .
Each of them is the solution to the following partial differential equations:
c·Θi
∂t
∂x k·Θi
∂x
∂y k·Θi
∂y = 0 ,i1, . . . , 4.(9)
The initial condition is θi= 0 ,(x , y ), t = 0 ,i1, . . . , 4. The differences in the
computation of the sensitivity coefficients Θiarise in the boundary conditions. For h11 and β, the boundary
conditions are:
ki·Θ1
∂n i
+h, i ·Θ1=∂h , i
∂h 11 ·T, i T,
ki·Θ2
∂n i
+h, i ·Θ2=∂h , i
∂β ·T, i T,(x , y )Γi,i1, . . . , 4,
with
∂h , i
∂h 11
(y , t ) =
0,i2,3,4,
v
v0·y
y0β
, i = 1 .
and
∂h , i
∂β (y , t ) =
0,i2,3,4,
h11 ·v
v0·lny
y0·y
y0β
, i = 1 .
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An efficient two-dimensional heat transfer model for building envelopes
For Θ 3and Θ 4, the boundary conditions are:
ki·Θ3
∂n i
+h, i ·Θ3=∂q , i
∂F ,
ki·Θ4
∂n i
+h, i ·Θ4=∂q , i
∂D ,(x , y )Γi,i1, . . . , 4.
For those two sensitivity coefficients, the purpose is to obtain the derivative of q, i according to For D.
First, it should be noted that the incident flux on the boundaries Γ 2, Γ3and Γ 4do not vary with those
parameters:
∂q , i
∂F =∂ q , i
∂D = 0 ,i2,3,4.
Then, from Eq.(5), we have:
∂q ,1
∂F =α·qdr
·∂χ
∂F ,
and similarly
∂q ,1
∂D =α·qdr
·∂χ
∂D .
Since there is no direct analytical relation between the indicator function χand the geometric parameters
Fand D, the partial derivatives ∂χ
∂F and χ
∂D are obtained using a discrete modeling and geometric con-
siderations. As illustrated in Figures 3(a) and 3(b), the increase of the building front height and distance is
denoted by δF and δD , respectively. Thus, we have
∂χ
∂F χ(F+δF , t )χ(F)
δF +OδF ,χ
∂D χ(D+δD , t )χ(D)
δD +OδD .
Note that discrete derivative of higher-order accuracy can be defined if required. The derivative according
to Fis first treated. An increase of the front building height δF induces an increase δhof the shadow height.
Thus, the new shadow height is given by:
˜
h=h+δh= min h+δF , H .
For an increase of the front building height, the new shadow height verifies h<˜
h< H . Thus, the indicator
function at F+δF can be evaluated by:
χ(F+δF , t ) =
0, y 6˜
h(t),
1, y > ˜
h(t),
to obtain the discrete derivative of the indicator function:
∂χ
∂F 1
δF
0, y 6h(t),
1,h(t)< y 6˜
h(t),
0, y > ˜
h(t).
Similarly, the derivative according to Dis assessed. Using geometrical consideration from Figure 3(b), an
increase of the front building distance δD implies a decrease of the shadow height:
˜
h= max hδD ·tan θ , 0.
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An efficient two-dimensional heat transfer model for building envelopes
front buildings
wall facade
(a)
front buildings
wall facade
(b)
Figure 3. Variation of the shadow height according to a slight variation of the front building height (a) or
distance (b).
The new shadow height verified 0 <˜
h<h. The discrete derivative of the indicator function according to
parameter Dis:
∂χ
∂D 1
δD
0, y 6˜
h,
1,˜
h< y 6h(t),
0, y > h(t).
A similar development can be done for a decrease in the height or distance of the front building. Using the
governing equation combined with the initial and boundary conditions, the four sensitivity coefficients can
be computed to perform a Taylor development of the interesting output. The computation is carried out
with the governing equation of heat transfer (1). Note that Eq. (9) and (1) are equal from a mathematical
point of view. The same efficient numerical model can be employed to compute the solution. Within an
explicit time scheme, the total cost to assess the sensitivity of the output scales with 5 times the cost of the
direct problem (4 sensitivity coefficient plus the equation of heat transfer). This cost is strongly reduced
compared to sampling approaches.
2.4 Dimensionless formulation
To perform efficient numerical computations, it is of major importance to elaborate a dimensionless
formulation of the problem. For this, the temperature is transformed into the dimensionless variable u:
udef
:= TT0
δT ,(10)
where T0and δT are chosen reference temperatures. This transformation is also applied to the initial
condition T0and to the boundary condition T. The space and time coordinates are also changed:
tdef
:= t
t0
, x def
:= x
Lx , 0
, y def
:= x
Ly , 0
.
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An efficient two-dimensional heat transfer model for building envelopes
where t0,Lx , 0=Land Ly , 0=Hare reference time and length quantities. A different reference length
is chosen for xand ycoordinates to re-scale the dimensionless problem on the plate 0,1×0,1. The
material properties are converted to:
kdef
:= k
k0
, c def
:= c
c0
,
where k0and c0are reference thermal conductivity and volumetric heat capacity. The coefficients c
and kare called distortion ones according to the reference conditions. Through these transformations,
dimensionless numbers are enhanced. Namely, the Fourier number characterizes the diffusion process
through the xor ydirections. The Biot number translates the intensity of the heat penetration at the
interface between the air and the material. Both are defined such as:
Fo x
def
:= k0t0
c0L2
x , 0
,Fo y
def
:= k0t0
c0L2
y , 0
,Bi x
def
:= h L x , 0
k0
,Bi y
def
:= h L y , 0
k0
.
At the boundaries, the heat flux is changed such as:
q
, x
def
:= qLx , 0
k0δT , q
, y
def
:= qLy , 0
k0δT .
In the end, the dimensionless problem is defined as:
c·∂u
∂t Fo x·
∂x k·u
∂x Fo y·
∂y k·u
∂y = 0 ,(11)
with the boundary conditions:
k
i·∂u
∂n i
+ Bi i·u= Bi i·u, i +q
, i ,
and the initial condition u=u0.
3 Direct numerical model
3.1 The Du Fort–Frankel numerical method
3.1.1 Numerical scheme
A uniform discretisation is considered for space and time lines. For the sake of clarity, the super-script
is removed in this section for the description of the numerical method. The discretisation parameters
are denoted using ∆tfor the time, ∆xfor the xspace and ∆yfor the yone. The discrete values of the
function u(x , y , t ) are written as un
j i
def
:= u(xj, y i, t n) with i=1, . . . , N y,j=1,... ,Nx
and n=1,... ,Nt.
The Du FortFrankel scheme is employed to build an efficient numerical model for the two-dimensional
heat diffusion equation. For the sake of simplicity, to explain the numerical scheme the latter is written as:
c·∂u
∂t =
∂x kx·∂u
∂x +
∂y ky·u
∂y .(12)
According to Eq. (11), we have kx
def
:= Fo x·kand ky
def
:= Fo x·k. The coefficients c,kxand kyare assumed
as constant, independent on time or space. First, Eq. (12) is discretized using finite central differences and
forward Euler approach:
c
t·un+1
j i un
j i =kx
x2·un
j1i2un
j i +un
j+1 i+ky
y2·un
j i12un
ji +un
j i+1 .(13)
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An efficient two-dimensional heat transfer model for building envelopes
Figure 4. Du FortFrankel numerical scheme stencil.
Then, to obtain the Du FortFrankel scheme, the term un
j i is replaced by 1
2un+1
j i +un1
j i in Eq. (13).
It yields to the following explicit expression:
un+1
j i = Σ x·un
j1i+un
j+1 i+ Σ y·un
j i1+un
j i+1 + Σ xy ·un1
j i ,(14)
where the coefficient Σ x, Σ yand Σ xy are given by:
Σx
def
:= λx
1 + λx+λy
,Σy
def
:= λy
1 + λx+λy
,Σxy
def
:= 1λxλy
1 + λx+λy
,
λx
def
:= 2 ∆t
x2
kx
c, λ y
def
:= 2 ∆t
y2
ky
c.
The stencil of the scheme is illustrated in Figure 4. The scheme is explicit expressed so no costly inversion
of matrix is required, as in implicit approaches. Furthermore, as demonstrated in next section, it has
an extended stability region, so the so-called Courant-Friedrichs-Lewy (CFL) restriction [24] is relaxed.
Interested readers may consult [15,16,25] for example of its applications for one-dimensional heat and
moisture transfer in building porous materials.
3.1.2 Stability
To proof the unconditional stability of the numerical scheme, a standard von Neumann analysis is
carried out. Assuming constant diffusion coefficient, the solution is decomposed according to:
un
ji =ρn·expiβ x j)·expiγ y i),(15)
where i = 1 , γand βare real numbers and ρis a complex one. Substituting Eq. (15) into Eq. (14),
one obtains:
ρ= Σ x·expiβx) + expiβx)+ Σ y·expiγy) + expiγy)+Σxy
ρ.
It leads to the following second-order polynomials in ρequation:
ρ2B ρ +C= 0 ,(16)
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An efficient two-dimensional heat transfer model for building envelopes
with
Bdef
:= 2 Σxcosβx) + Σ ycosγy), C def
:= Σxy .
The general solution of Eq. (16) is:
ρ±=1
2B±D, D def
:= B24C .
The modulus ρ±verifies:
ρ±61
2B±D.
It is straightforward that 1
2B6Σx+ Σ y. Given the expression of D, we have:
1
2D=1
1 + λx+λy· λxcosβx) + λycosγy)2+ 1 λx+λy2!
1
2
.
Thus,
1
2D61
1 + λx+λy
,
and
ρ±6Σx+ Σ y+1
1 + λx+λy
.
One can note that the right hand side is equal to
Σx+ Σ y+1
1 + λx+λy
= 1 .
Therefore, the ρ±61 always holds and the scheme is unconditionally stable.
3.1.3 Accuracy
The consistence analysis of the scheme (14), using Taylor expansion, gives the following result:
un+1
j i Σx·un
j1i+un
j+1 iΣy·un
j i1+un
j i+1 Σxy ·un1
j i
=c·∂u
∂t
∂x kx·∂u
∂x
∂y ky·u
∂y + kx
x2+ky
y2·2u
∂t 2+1
6
3u
∂t 3!·t2
+Ox2+ ∆y2+ ∆t4.
Thus, the scheme is second-order accurate in space Ox2+ ∆y2. However, the accuracy in time
depends on the quantity τdefined as:
τdef
:= kx
x2+ky
y2·t2.
If τ1 , then the scheme is second-order accurate in time Ot2. If the condition τ1 is not
respected, then the scheme is not consistent with the discretized equation. For practical applications, in the
case ∆xy, then the second-order accuracy is obtained when ∆t=Ox2.
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An efficient two-dimensional heat transfer model for building envelopes
3.2 Metrics of efficiency and reliability of a model
To evaluate the efficiency of a numerical model, one criterion is the computational (CPU) run time
required to compute the solution. It is measured using the Matlab environment with a computer equipped
with Intel i7 CPU (2.7GHz 6th generation) and 32 GB of RAM. Hence the following ratio is defined:
Rcpu
def
:= tcpu
t0
,
where tcpu sis the measured CPU time and t0is a chosen reference time.
The accuracy of the numerical model is assessed by comparing the results to a reference solution denoted
by the superscript ref. The error can be applied to a certain time varying output Φ, as for instance the
temperature, the flux or the thermal loads, that may depend on the space coordinates xand y. Then, the
error for this output is defined by the compound function:
ε2Φ(t)def
:= 1
L H ZxZyΦ( x , y , t )Φref (x , y , t )2dxdy!
1
2
.
A normalized version of the error is also used:
ε2Φ(t)def
:= 1
L H ZxZyΦ( x , y , t )Φref (x , y , t )
max
x y Φref (x , y , t )min
x y Φref (x , y , t )2
dxdy!
1
2
.
For investigations of the physical phenomena, the relative error can also be relevant:
εrΦ(t)def
:= 1
L H ZxZyΦ( x , y , t )Φref (x , y , t )
Φref (x , y , t )2
dxdy!
1
2
.
4 Validation of the numerical model
4.1 Case study
To validate the implementation and verify the theoretical features, the model results are compared with
an analytical solution from the EXACT1toolbox, namely X33B00Y33B00Tx5y5 [26]. Since the objective
of this section is the validation, the problem is described in dimensionless formulation. The domain is
defined for ( x, y )0,1×0,1. The initial condition is piece-wise defined on the sub-domain
ab
def
:= 0, L
a×0, L
b:
u0(x, y ) =
1,(x, y )ab ,
0,(x, y )/ab ,
where L
a= 0.6 and L
b= 0.5 . The Fourier numbers are s et to unity Fo x= Fo y= 1 . The plate
is composed of one material so the distortion coefficients are equal to the unity k=c= 1 . At the
boundaries, the imposed Robin condition is homogeneous, so u, i =q, i = 0 ,i1, . . . , 4.
The Biot numbers are equal to:
Bi 1= 3 ,Bi 2= 0.5,Bi 3= 1.5,Bi4= 4 .
The time horizon is t
f= 0.04 .
1http://exact.unl.edu/exact/home/home.php
12 /37
An efficient two-dimensional heat transfer model for building envelopes
Table 1. Efficiency of the models for the validation case.
Model Time step Space mesh Error Computational time
tx= ∆yε2u(t
f)Rcpu
Euler implicit 10 410 24.63 ·10 31
Euler explicit 10 510 24.63 ·10 30.11
Alternating Direction Implicit 10 410 24.63 ·10 30.05
Du FortFrankel 10 410 24.64 ·10 30.007
the reference computational time is t0= 773 s,the one of the Euler implicit model.
4.2 Results
The solution is computed using four numerical models, namely the Du FortFrankel (denoted DF),
the implicit Euler (denoted IM), the explicit Euler (denoted EX) and the Alternating Direction Im-
plicit (denoted ADI). The second and third models use central finite difference approaches for the space
discretisation. The ADI method is described in [27] with details in Appendix A. First, all models except
Euler explicit considers the same space and time discretisations ∆t= 104and ∆x= ∆y= 10 2.
For the Euler explicit approach, the discretization parameters are required to satisfy the following stability
condition:
t6xy2
2 Fo x2+ ∆y2,(17)
which corresponds to ∆t62.5·10 5for this case study. Thus, a smaller time step is used ∆t= 10 5
for this model. Figures 5(a) to 5(c) compare the solutions. All of them are overlapped, highlighting the
validation of the numerical models compared to the analytical solution. Figure 5(d) enhances the two-
dimensional aspect of the heat transfer through the domain. Table 1provides a synthesis of the efficiency of
the numerical models. All models have a satisfying error with the same order of accuracy ε2=O( 10 3) .
It also justifies why the solutions are overlapped in Figures 5(a) to 5(c). However, the computational time
ratio is very different among the models. The Euler explicit requires only 10% of the computational time of
the Euler implicit, even for a time step one order lower. However, this model is not reliable for predicting
the phenomena in building materials due to its conditional stability Eq. (17). The Alternating Direction
Implicit and Du FortFrankel approaches require only 5% and 0.7% of the implicit computational time.
The important differences are due to the computational efforts to inverse the matrix in the implicit method.
It represents 99.5% of the total computational time. Note that the problem is linear in parameters. The dif-
ferences in computational time should increase when considering nonlinear problems due to the requirement
of subiterations to treat the nonlinearities.
Further investigations are carried out by setting the space mesh to ∆x= ∆y= 102and performing
computations for several values of time discretisation ∆t. For each computation, the error with the
analytical solution and the computational time of the four numerical models are evaluated. Figure 6(a) shows
the variation of the error according to the time discretisation. Several theoretical results can be confirmed.
First, the Euler explicit scheme enables to compute the solution only until the CFL restriction ∆t6
2.5·105. Then, it can be remarked that the Du FortFrankel, the Euler implicit and the Alternating
Direction Implicit approaches are unconditionally stable, as proven theoretically in Section 3.1.2 for the
primer. However, some differences are observed between those models. The Du FortFrankel scheme is
second-order accurate in time while the two others are only first order. Figure 6(b) gives the variation of
the accuracy with the computational ratio. The Du FortFrankel model is always faster than the other
approaches. For ∆t= 104, it can be remarked that the Du FortFrankel approach is as accurate as
the others. However, it computes more than a thousand times faster than the Euler implicit model.
13 /37
An efficient two-dimensional heat transfer model for building envelopes
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
1.1
(a)
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
1.1
(b)
0 0.1 0.2 0.3 0.4
0
0.2
0.4
0.6
0.8
1
(c)
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
0.1
0.2
0.3
0.4
0.5
0.6
(d)
Figure 5. Variation of the field according to xat y= 0.5 (a), according to yat x= 0.4 (b) and
according to t(c). Slice of the solution from the Du FortFrankel model at t= 0.04 (d).
14 /37
An efficient two-dimensional heat transfer model for building envelopes
10 -5 10 -4 10 -3 10 -2 10 -1
10 -3
10 -2
10 -1
10 0
(a)
10 -4 10 -3 10 -2 10 -1
10 -3
10 -2
10 -1
(b)
Figure 6. Variation of the numerical models errors according to the time discretisation (a). Variation of
the numerical models errors according to the computational effort ( b). For the last one, the reference
computational time is t0= 294 s, the one of the Euler implicit model.
15 /37
An efficient two-dimensional heat transfer model for building envelopes
Table 2. Thermal properties of the material composing the facade.
Layer Thermal conductivity Volumetric heat capacity length
W.m1.K1 MJ .m3.K1 m
concrete 1.4 2 0.2
wood fiber 0.05 0.85 0.15
gypsum 0.25 0.85 0.02
5 Real case study
5.1 Description
The case study considers a south-oriented facade of a house located in Paris, France. The wall is composed
of three layers: concrete (outside part), wood fiber insulation and gypsum board (inside part). The material
properties are taken from the French standards [28] and given in Table 2. This configuration corresponds
to a building with improved energy efficiency. As illustrated in Figure 1, the height and width of the wall
are H= 3 mand L= 37 cm . The facade is located in an urban area so it is facing other buildings. The
latter is located at a distance D= 5 mand has a height F= 3 m, which induces a shadow on the studied
facade. The height of the shadow varies according to time. As a consequence, the outside incident radiation
flux q,1varies according to height and time. The diffusivity is set as α= 0.6 . The outside surface
heat transfer coefficient h1depends on height position yand time varying climate wind velocity as defined
in Eq. (4). The following parameters are used h10 = 5.82 W.m2.K1,h11 = 3.96 W.m2.K1,
v0= 1 m.s1,y0= 1 mand β= 0.32 . The outside wind velocity is shown in Figure 7(a). It varies
around a mean of 4 m.s1. The occurrences of the surface heat transfer coefficient h,1are shown in
Figure 7(b). The surface heat transfer coefficient increases according to the height, with a mean around
8.6W.m2.K1at y= 0.3m, 10.5W.m2.K1at y= 1.5mand 11.5W.m2.K1at y= 2.7m.
The mean over the whole year and height gives a coefficient of h,1= 10.28 W.m2.K1. Figures 7(c)
and 7(d) enable to compare the time variation of the coefficient between the bottom and the top of the
facade. Higher magnitudes of variation are observed at the top of the facade. Moreover, the discrepancy
with the value of 12 W.m2.K1used for standard computation is locally important.
The outside temperature is also given by weather data file. The inside temperature is controlled and
defined according to sinusoidal variations depending on the winter and summer seasons. The time variation
of inside and outside temperature is shown in Figure 8. The inside surface transfer coefficient is set as
constant to h3= 10 W.m2.K1. The top and bottom boundaries of the facade Γ 2and Γ4are set
as adiabatic. Indeed, the investigations focus on the influence of the space and time variations of outside
boundary conditions on the thermal efficiency of the facade. The simulation horizon is of one year so
tf= 365 d.
5.2 Generating the incident radiation flux
The pixel counting technique is employed to determine the outside incident radiation heat flux and the
variation of the shadow height. A time step of 6 min is used to provide the data. Figure 9(a) compares the
results of the pixel counting technique with the shadow height. From these results, the indicator function
χdefined in Eq. (6) can be computed as illustrated in Figure 9(b). For this winter day, the bottom of the
facade remains in the shadow. Around midday, the top of the facade receives the direct sun. During this
time, the indicator function is equal to 1 and thus this part of the facade receives the direct heat flux added
to the reflected and diffuse ones. As remarked in Figure 9(c), the magnitude of the flux is three times higher
on the top of facade around midday.
Figures 10(c) and 10(d) illustrates the variation of the incident shot-wave radiation flux for two different
weeks. In summer, the sunlit area covers the whole facade, as shown in Figures 10(a). Thus, there is
16 /37
An efficient two-dimensional heat transfer model for building envelopes
J F M A M J J A S O N D
0
5
10
15
20
(a)
(b)
J F M A M J J A S O N D
4
8
12
16
20
24
28
32
(c)
J F M A M J J A S O N D
4
8
12
16
20
24
28
32
(d)
Figure 7. Variation of the outside velocity (a). Probability density function of the outside surface heat
transfer coefficient h1(b). Variation of the outside surface heat transfer coefficient h1at y= 0.3m(c)
and y= 2.7m(d).
J F M A M J J A S O N D
0
10
20
30
Figure 8. Variation of the inside and outside temperatures.
17 /37
An efficient two-dimensional heat transfer model for building envelopes
no significant difference in terms of incident flux on the top and bottom surfaces. However, in winter the
contrast is more noticeable since the height of the shadow reaches almost y= 1.5meach day at midday.
As a consequence, the bottom of the facade receives less flux. Note that when the direct flux is negligible
compared to the diffuse one, the magnitude of the total incident flux is homogeneous over the whole facade.
This can be remarked on December 6 th in Figure 10(d).
The incident flux on the facade varies according to the height. The ratio between the effective and total
incident radiation flux on the facade is illustrated for two different heights in Figures 11(a) and 11(b). For
the top of the facade, the ratio is almost always equal to 1 . In other words, the top of the facade is not
affected by the shadow and it receives the total incident radiation flux. However, at the bottom, the ratio can
reach 20%. During the winter period, the effective incident radiation flux is particularly reduced compared
to the total one. Note that the sunlight exposure of the facade is shorter in winter than in summer. The
average time of daily sunlight exposure is 7.1hin December and 8.9hn August.
5.3 Assessing the thermal efficiency
After generating the variation of the boundary conditions according to space and time, the numerical
model is used to evaluate the thermal performance of the wall. The discretisation parameters are ∆t=
36 sand ∆x= ∆y= 3.7mm . The computational time of this simulation is of tcpu = 5.6min ,
corresponding to a ratio Rcpu = 0.93 s/days of physical simulation. Comparatively, the same simulation
with the one-dimensional Du FortFrankel model and same discretisation parameters has a ratio of
Rcpu = 0.54 s/days of physical simulation. The increase of computational time is moderate compared to
the one-dimensional approach. Considering the CFL stability condition from Eq. (17) and the parameters
of the problem, the computational ratio with the Euler explicit approach is estimated to Rcpu = 13 % .
The Du FortFrankel numerical model enables to save significant computational efforts compared to
standard approaches.
The temperature variation according to xand yis provided in Figures 12(a) to 12(f) for summer and
winter periods. Since the incident heat flux is more homogeneous on the facade in summer, there are
no important differences between the temperature at the bottom and top. As remarked in Figure 12(e),
the differences scale with 2 C. Those small differences are mainly due to the variation of the surface
heat transfer coefficient according to the height y. In winter, there are more discrepancies in the surface
temperature between the top and bottom of the facade. Figure 12(f) highlights those contrasts. At 06:00,
the temperature is relatively homogeneous along the facade since there is no incident flux. However, at
12:00 the incident flux induces a variation of almost 7 Con the temperature between the top and bottom.
Note that on the inside surface at x= 0.37 m, the temperature does not vary with the height of the
facade. Those results are confirmed by the sections of temperature illustrated in Figure 13. The influence of
the boundary conditions is mostly remarkable in the first concrete layer of the facade. The insulation layer
reduces significantly the temperature gradients along with the height.
Even if the variation of temperature is small at the inner surface, it still induces a variation of the heat
flux along yas remarked in Figure 14. It is important to note that in summer the flux changes of sign
between the bottom and top of the wall. At 18:00, below y= 1.25 mthe flux is positive so the bottom
of the wall is heating the inside zone. Inversely, the top of the wall is cooling the inside zone. This effect
is not due to the incident radiation but to the variation of the heat surface coefficient with the height. The
latter is higher at the top of the facade. It increases the heat transfer at the top surface. In winter, the flux
on the inside surface is entirely negative. Thus, the inside zones are losing energy through the wall.
The time and space variation of the climatic boundary conditions induces two-dimensional heat transfer
through the facade. It is important to evaluate the thermal efficiency of the wall compared to standard
approaches. The building simulation program Domus is used to assess standard building energy efficiency.
Within this approach, the heat transfer is modeled in one-dimension and the outside surface heat transfer
coefficient is considered as constant. The latter is set to the mean h,1= 10.28 W.m2.K1. Moreover,
the outside incident short-wave radiation flux includes the shading effects by evaluating the sunlit area ratio
18 /37
An efficient two-dimensional heat transfer model for building envelopes
01:00 03:00 05:00 07:00 09:00 11:00 13:00 15:00 17:00 19:00 21:00 23:00
0
0.5
1
1.5
2
2.5
3
h
(a)
0
0.5
05/12 01:00
1
05/12 05:00
1.5
2
05/12 09:00
2.5
0
3
05/12 13:00
05/12 17:00
05/12 21:00
1
(b)
0
0.5
05/12 01:00
1
05/12 05:00
1.5
2
05/12 09:00
2.5
0
3
05/12 13:00 40
05/12 17:00 80
120
05/12 21:00 160
200
(c)
Figure 9. Results of the pixel counting technique compared with the height of the shadow on the facade for
December 5th (a). Variation of the indicator function (b) and the corresponding incident radiation heat
flux (c)
19 /37
An efficient two-dimensional heat transfer model for building envelopes
01/08 03/08 05/08 07/08
0
0.5
1
1.5
2
2.5
3
h
(a) summer
01/12 03/12 05/12 07/12
0
0.5
1
1.5
2
2.5
3
h
(b) winter
(c) summer (d) winter
Figure 10. Variation of the shadow height (a,b) and incident short-wave radiation heat flux (c,d).
J F M A M J J A S O N D
0
0.2
0.4
0.6
0.8
1
(a) y= 0.3m
J F M A M J J A S O N D
0
0.2
0.4
0.6
0.8
1
(b) y= 2.7m
Figure 11. Ratio between the effective and total incident radiation heat flux at two different heights.
20 /37
An efficient two-dimensional heat transfer model for building envelopes
Son the facade using the pixel counting technique:
qst
,1(t) = α·qdr
(t)·S(t) + qdf
(t) + qrf
(t).(18)
The results from the standard approach are compared to the one obtained with the two-dimensional mod-
eling. Figure 15(a) compares the heat flux at the inside surface. Small discrepancies are noted in both
winter and summer periods. The magnitude of the flux is higher with the two-dimensional modeling. In the
mid-season, the two approaches have similar predictions. Table 3gives the thermal loads per month for the
two-dimension model. The relative error with the one-dimensional modeling is also presented. The error
is higher for the summer period. In July, the one-dimensional model underestimated by 80% the thermal
loads. In winter the error is lower by around 1% .
A parametric comparison is carried out to analyze the influence of the shadow on the incident short-wave
radiation flux and the variation of the surface heat transfer coefficient with the height and the wind speed.
Three additional simulations are performed with the two-dimensional model. The first one considers both
a constant outside surface heat transfer coefficient h,1= 10.28 W.m2.K1and no shadow modeling
on the facade. Thus, the sunlit area is set to unity S= 1 . The second computation deals only with a
constant coefficient while the third one only does not take into account the shadow modeling. Results are
shown in Table 3. The shadow modeling does not impact the prediction of thermal loads during the summer
period (April to September). It is consistent with the analysis of Figures 11(a) and 11(b) since the wall is
always exposed to the sunlited. However, the modeling of the heat surface coefficient with height and wind
velocity significantly influence the predictions during this period. In July, the relative error reaches 80%
when considering a constant surface transfer coefficient. In the winter period (October to February), both
shadow modeling and varying surface transfer coefficients have remarkable effects on the predictions. For
the month of December, the effects are counter-balanced. Not including the shadow modeling induces a
relative error of 3.65%. On the contrary, not taking into account the time variation of the surface transfer
coefficient implies a relative error of 1% .
A similar study is performed by computing the relative error on the thermal loads with the one-
dimensional model. The first computation is the standard one described above. It considers a constant
surface transfer coefficient and a direct radiation flux computed according to the sunlit area ratio. The
second simulation combines the same approach for the direct radiation flux and a time varying surface
transfer coefficient compute using Eq. (4), for the middle height y= 1.5m. The last simulation includes
a constant surface transfer coefficient. For the radiation, it is assumed that the facade is always exposed to
the sunlit. The results are presented in Table 3. For the summer period, the 1D model with ratio radiation
flux and time varying coefficient has a lower error compared to the 2D modeling. For the winter, there is no
particular tendency. It can be highlighted that the 1D model with no sunlit ratio and constant coefficient
lacks of accuracy to predict the thermal loads in this case.
Those results highlight the importance of modeling the two-dimensional transfer induced by time and
space variations of the incident short-wave radiation heat flux and of the heat surface coefficient to accurately
predict the wall energy efficiency. Both the shadow and the surface transfer coefficient modelings have an
important effect on the thermal loads. The one-dimensional model cannot predict the phenomena with a
reduced relative error over the whole year.
5.4 Results for other cities
Further investigations are carried for 119 cities in France, according to weather data filed taken from
Meteonorm [29]. The purpose is to evaluate the difference between the two- and the one-dimensional modeling
for the given case study. For that, the procedure is as follows: (i) compute the incident radiation and shadow
height using Domus software, (ii) compute the fields in the facade using the proposed two-dimensional model,
(iii) compute the fields in the facade using the one-dimensional model considering the mean value h,1
and the average value of radiation flux given by Eq. (18)(iv) compute the errors on the inside heat flux J
and the thermal loads E. The discretisation parameters and the inside boundary conditions are equal to
the one described in Section 5.1. The results are then projected on the France map using the geographic
21 /37
An efficient two-dimensional heat transfer model for building envelopes
0 0.1 0.2 0.3 0.37
16
18
20
22
24
26
28
30
32
(a) summer, y= 2.7m
0 0.1 0.2 0.3 0.37
4
6
8
10
12
14
16
18
20
(b) winter, y= 2.7m
0 0.1 0.2 0.3 0.37
16
18
20
22
24
26
28
30
32
(c) summer, y= 0.3m
0 0.1 0.2 0.3 0.37
4
6
8
10
12
14
16
18
20
(d) winter, y= 0.3m
16 18 20 22 24 26 28 30 32
0
0.5
1
1.5
2
2.5
3
(e) summer, x= 0 m
6 7 8 9 10 11 12 13 14 15 16
0
0.5
1
1.5
2
2.5
3
(f) winter, x= 0 m
Figure 12. Temperature variation according to the height (a,b,c,d) and length (e,f) of the wall.
22 /37
An efficient two-dimensional heat transfer model for building envelopes
0 0.1 0.2 0.3 0.37
0
0.5
1
1.5
2
2.5
3
24
25
26
27
28
29
30
31
(a) summer, 03/08
0 0.1 0.2 0.3 0.37
0
0.5
1
1.5
2
2.5
3
8
9
10
11
12
13
14
15
16
17
18
19
(b) winter, 01/12
Figure 13. Section of the temperature according to xand yat 13 : 00 in summer (a) and winter (b).
-0.7 -0.5 -0.3 -0.1 0.1 0.3 0.5 0.7
0
0.5
1
1.5
2
2.5
3
(a) summer
-4.1 -3.9 -3.7 -3.5 -3.3
0
0.5
1
1.5
2
2.5
3
(b) winter
Figure 14. Variation of the heat flux on the inside boundary x= 0 in summer (a) and winter (b).
Table 3. Influence of several hypotheses on the thermal loads of the wall.
Hypothesis Months
Model Shad. Conv. Output J F M A M J J A S O N D
2D Yes f(y , t )E[MJ]9.68 7.99 7.06 4.98 2.41 1.30 0.67 1.52 1.99 4.36 7.92 9.69
2D No f(y , t )εrE[%] 3.84 4.62 0.08 0.01 0.05 0.16 0.39 0.14 0.02 2.52 5.87 3.65
2D Yes Const. εrE[%] 0.69 0.45 0.13 4.61 5.52 15.280.531.81.64 8.48 2.24 1
2D No Const. εrE[%] 2.61 4.5 0.19 4.62 5.46 15.180.231.71.62 6.34 2.74 2.32
1D Ratio Const. εrE[%] 0.38 0.44 0.09 4.56 5.62 15.380.631.71.54 8.43 2.21 0.98
1D Ratio f(t)εrE[%] 0.11 0.42 1.41 2.66 5.61 10.518.49.22 6.08 1.97 0.19 0.21
1D No Const εrE[%] 2.92 4.49 0.16 4.57 5.56 15.280.331.61.51 6.28 2.77 2.34
23 /37
An efficient two-dimensional heat transfer model for building envelopes
J F M A M J J A S O N D
-6
-5
-4
-3
-2
-1
0
1
2
(a)
Figure 15. Variation of the total heat flux on the inside boundary computed with the standard one
dimensional approach and with the two-dimensional modeling.
information system QGIS [30]. The interpolation between the results is carried using the inverse distance
weight method.
Figures 16(a) and 16(b) show the error between the two modeling approaches for the inside heat flux
and for the thermal loads, respectively. The maximal errors ε2nare 0.069 and 0.088 for the inside flux and
thermal loads, respectively. For the inside heat flux, the South-East and North-West regions of France are
zones of higher errors. Indeed, as remarked in Figures 17(a) and 17(b), it corresponds to a geographic area
of high radiation flux and wind velocity. Thus, the accurate description of the outside boundary condition in
the two-dimensional model is essential to predict accurately the inside heat flux. The diagonal South-West
North-East is a region of a relatively small error. It corresponds to the same diagonal region with lower
wind velocity. For the thermal loads, the error is maximal in the North-West region. On the contrary to
the inside thermal flux, the error has a low magnitude in the South-East region. This is probably due to
the fact that the magnitude of the thermal loads is lower.
5.5 Local sensitivity analysis of the two-dimensional boundary conditions
As remarked with the previous results, the modeling of the boundary conditions outside the facade
can impact significantly the assessment of the energy efficiency. To investigate this influence, the Taylor
expansion (8) is used. It enables to assess the sensitivity of the important fields, i.e. the temperature, the
total heat flux on the inside part and the thermal loads according to variation of the input parameters. The
latter includes the first order surface heat transfer coefficient h11 and the coefficient βused in Eq. (4). Both
parameters influence the empirical model of the outside surface heat transfer coefficient and its modification
with the height yand the wind velocity. Two additional parameters are considered, the front building height
position Fand distance D. Those two parameters are related to the height of the shadow on the outside
facade and thus, the incident direct flux. Those four parameters enable to analyze the uncertainty in the
two-dimensional modeling, i.e. in space height yand in time t, boundary conditions. The variations are
δh 11 =±0.198 W.m2.K1,δβ =±0.016 , δF =±0.15 mand δD =±0.15 m. It corresponds to
an equal relative variation for each parameter:
δh 11
h11
=δβ
β=δF
H=δD
H= 0.05 .
For the sake of clarity, we denote by:
δp def
:= δh 11 ,δβ , δF , δD ,
24 /37
An efficient two-dimensional heat transfer model for building envelopes
<= 0.030
0.030 - 0.035
0.035 - 0.040
0.040 - 0.045
0.045 - 0.050
0.050 - 0.055
0.055 - 0.060
0.060 - 0.065
> 0.065
(a)
<= 0.025
0.025 - 0.030
0.030 - 0.035
0.035 - 0.040
0.040 - 0.050
0.050 - 0.060
0.060 - 0.070
0.070 - 0.080
> 0.080
(b)
Figure 16. Error between one- and two-dimensional modeling for the inside heat flux (a) and the thermal
loads (b) across France.
25 /37
An efficient two-dimensional heat transfer model for building envelopes
<= 52.00
52.00 - 56.00
56.00 - 60.00
60.00 - 64.00
64.00 - 68.00
68.00 - 72.00
72.00 - 76.00
76.00 - 80.00
> 80.00
(a)
<= 2.30
2.30 - 3.00
3.00 - 3.70
3.70 - 4.40
4.40 - 5.10
5.10 - 5.80
5.80 - 6.40
6.40 - 7.00
> 7.00
(b)
Figure 17. Time mean total incident radiation flux (a) and air velocity (b) across France.
26 /37
An efficient two-dimensional heat transfer model for building envelopes
which by convention induces a decrease of entering heat flux on the outside surface of the facade. The
numerical model composed of the governing and the four sensitivity equations is computed with the following
discretisation parameters: t= 36 sand ∆x= ∆y= 3.7mm . The climatic data from Marseille is
selected. The computational ratio is Rcpu = 3.77 s/days . As expected, the computational time increases
compared to the one in Section 5.3 since there are additional equations to solve.
First, the uncertainties on the boundary condition is presented. The modification of the surface transfer
coefficient of the facade is illustrated in Figure 18(b) according to slight variations of both parameters h11
and β. The parameters variation have a higher impact on the modeling of the heat transfer coefficient
between 12 : 00 and 16 : 00 due to higher wind velocity, as noted in Figure 18(a). Then, Figures 19(a)
and 19(b) show the variation of shadow height on the facade during one day in winter according to slight
variations of the front building facade height and distance, respectively. An increase of the height or a
decrease of the distance induces a larger shadow on the building facade. Around noon, the shadow increases
scales with 15 cm and 3.4cm due to facade height and distance variations, respectively. As a consequence,
the incident direct radiation flux on the facade is altered. Figure 19(c) gives the variation of the flux for
a point located at y= 0.6maccording to a slight variation of the front building facade height. At this
specific height, the influence only occurs in the morning and afternoon, when the sun is rising or decreasing.
At noon, there is no variation since the point y= 0.6mis in the sunlit with or without variations of
the front building height. Similar results are obtained for the variation of the incident flux according to a
variation of the front building distance. One can note that Fand Dhave opposite effects. An increase of
the height and a decrease of the distance leads to an increase of the shadow height, respectively.
Figures 18 and 19 highlight the time and space changes of the boundary conditions modeling due to
uncertainties in the input parameters. Those modifications impact the heat transfer process through the
whole facade. Using the Taylor expansion and the sensitivity coefficients of each parameters, it is possible
to evaluate the impact of those variations on the fields. Figure 20(a) shows the temperature variation during
one week according to a modification on the all parameters. One can observe that using this approach a
time varying temperature sensitivity is computed. A detailed analysis is carried in Figure 20(b) to evaluate
the impact of the change of each parameter on the temperature. For this day and this point, the change
of the facade height contributes to 5 Cof variation. This parameter has a strong impact compared to the
three others. Similarly, the extension can be carried for the total flux Jon the inside part of the facade.
Figures 20(c) and 20(d) give the variation according to change of all parameters for winter and summer
weeks. The sensitivity of the flux can reach 0.5Wfor this period. It varies according to time since it depends
on the magnitude of the radiation flux and wind velocity. Similarly, the influence of the boundary conditions
uncertainties on the thermal loads of the facade can be investigated in Figure 21(a). The variability of the
facade energy efficiency depending on the modeling of the boundary conditions is presented. The maximum
variation of Eoccurs in September and reaches 0.91 MJ .m2. The variation + δp has a higher impact on
the thermal loads. It is probably due to the decrease of the incident radiation flux in such configuration,
which strongly affects the heat transfer process through the facade. A detailed investigation is presented
in Figures 21(b) and 21(c). The relative variation of the thermal loads is given according to positive or
negative change of each parameter. The sensitivity varies according to the month. In Figure 21(b), the
parameters h11 and βhave a higher impact than other parameters, particularly in April, May and October.
It corresponds to months with important wind velocity values. Furthermore, the thermal loads are almost
not sensible to a decrease and increase of the building front height and distance, respectively. Indeed, those
changes only modify the incident radiation flux in the morning as noted in Figures 19(c) and 19(d). In
Figure 21(c), those two parameters have more impact when the height and distance increases and decreases,
respectively. Particularly in October, the relative variation can reach 70 % due to a reduction of the
building front distance. Except for the months of high wind velocity, the parameters h11 ,βand Fhave a
comparable sensitivity on the thermal loads.
27 /37
An efficient two-dimensional heat transfer model for building envelopes
00:00 04:00 08:00 12:00 16:00 20:00 00:00
0
2
4
6
8
10
12
14
16
(a)
00:00 04:00 08:00 12:00 16:00 20:00 00:00
4
6
8
10
12
14
16
18
20
22
24
26
28
30
(b)
Figure 18. Time evolution of the wind velocity (a). Taylor extension of the outside surface transfer
coefficient at y= 0.6maccording to a slight variation of parameters h
11 and β(b). All results are
plotted for February 7th .
6 Conclusion
The development of more accurate numerical models is essential to assess the energy efficiency of buildings
considering the influence of urban environment. Due to computational issues, most of the today’s building
simulation programs proposes a one-dimensional approach to predict the phenomena of heat transfer within
the building envelope. It is worth of investigation to propose innovative numerical methods to build a
reliable model with high reliability. This paper presented an efficient numerical model for two–dimensional
heat transfer in building facade, considering complex outside boundary conditions with shading effects and
varying surface heat transfer coefficient. The main advantage is the fast computation of the solution, i.e. the
temperature field, and its sensitivity on the modeling of the boundary conditions. Using a Taylor expansion
of the solution and the sensitivity functions, it is possible to evaluate the two-dimensional modeling of the
boundary conditions on the energy efficiency.
The numerical models are described in Section 3. It is based on the Du FortFrankel scheme. It
provides an explicit formulation, which enables a more direct treatment of the nonlinearities. An important
advantage is the relaxed stability condition of the scheme compared to the traditional Euler explicit
approach. A first case is considered with an analytical solution in Section 4. It validates the theoretical
results and highlights the efficiency of the proposed numerical model. A perfect agreement is remarked with
the analytical solution and the three other schemes: ADI, Euler implicit and the Euler explicit. The
Du FortFrankel model is the one proposing the best compromise between high accuracy of the solution
and reduced computational efforts.
In Section 5, a more realistic case study was presented and investigated. The heat transfer occurs in a
whole building facade. Shading effects are induced by the facing buildings of a urban environment. Thus,
the incident radiation flux on the facade varies according to the height of the shadow. Both are computed
using the pixel counting technique implemented in the Domus building simulation program. The outside
heat transfer coefficient is varying according to the wind velocity and to the height of the facade using an
empirical correlation obtained from the literature. In this way, the boundary conditions are modeled in
two-dimensions, i.e. depending on time and height. The model enables to compute accurately the two-
dimensional fields with a reduced computational effort. Then, a comparison is carried for almost 120 cities
in France between the two-dimensional approach and the traditional one-dimensional one. The highest error
on the prediction of the physical phenomena occurs in regions with high magnitude of wind and high short–
wave radiation flux. Last, a sensitivity analysis is carried out using a derivative-based approach to highlight
28 /37
An efficient two-dimensional heat transfer model for building envelopes
06:00 10:00 14:00 18:00 22:00 02:00 06:00
0
0.5
1
1.5
2
2.5
3
h
h
h
h
(a)
06:00 10:00 14:00 18:00 22:00 02:00 06:00
0
0.5
1
1.5
2
2.5
3
h
(b)
06:00 10:00 14:00 18:00 22:00 02:00 06:00
0
50
100
150
200
250
300
350
(c)
06:00 10:00 14:00 18:00 22:00 02:00 06:00
0
50
100
150
200
250
300
350
(d)
Figure 19. Variation of the shadow height according to a slight variation of the front building facade height
(a) and distance (b). Taylor extension of the direct flux incident on the facade at y= 0.6maccording to
a slight variation of the front building facade height (c) and distance (d). All results are plotted for
February 7th .
29 /37
An efficient two-dimensional heat transfer model for building envelopes
07/02 08/02 09/02 10/02 11/02 12/02 13/02 14/02
5
10
15
20
25
30
35
40
(a)
12:00 14:00 16:00 18:00 20:00
10
15
20
25
30
(b)
07/02 08/02 09/02 10/02 11/02 12/02 13/02 14/02
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
(c)
01/08 02/08 03/08 04/08 05/08 06/08 07/08
0
0.5
1
1.5
2
2.5
3
3.5
(d)
Figure 20. Taylor extension of the temperature on the facade at (x , y ) = ( 0 ,0.6 ) maccording to a
slight variation of the four parameters of interests (a). Detailed contribution of the sensitivities to each of
the four parameters on the temperature on the facade at (x , y ) = ( 0 ,0.6 ) mfor February 7th (b).
Taylor extension of the total heat flux on the inside boundary for a slight variation of the four parameters
of interests in winter (c) and summer (d) periods.
30 /37
An efficient two-dimensional heat transfer model for building envelopes
(a)
J F M A M J J A S O N D
-0.05
0.05
0.15
0.25
0.35
(b)
J F M A M J J A S O N D
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
(c)
Figure 21. Taylor extension of the thermal loads of the facade at according to a slight variation of the four
parameters of interests (a). Detailed contribution of the relative variation of each parameters on the
thermal loads of the facade(b,c).
31 /37
An efficient two-dimensional heat transfer model for building envelopes
the most influencing parameters in the modeling of the two-dimensional boundary conditions. The influence
of each parameter can be analyzed according to the time line. The model of the surface convective heat
transfer coefficient has a significant effect on the solution for months with high wind velocity. A combined
increase and decrease of the height and distance front building can induce a relative variation of 70 % on
the prediction of the thermal loads.
Further work should be dedicated to implement such efficient numerical models in building simulation
programs to simulate in a city scale. Particularly, the explicit formulation of the proposed model is a
promising feature for future implementation and coupling with other numerical tools.
Acknowledgments
The authors acknowledge the French and Brazilian agencies for their financial support through the project
CAPES–COFECUB, as well as the CNPQ of the Brazilian Ministry of Science, Technology and Innovation,
for co-funding.
32 /37
An efficient two-dimensional heat transfer model for building envelopes
Nomenclature and symbols
Physical parameters
Latin letters
cvolumetric heat capacity J.m3.K1
Dfront building distance m
Ethermal loads J.m2
Ffront building height m
hshadow height m
h , h , h 10 , h 11 surface heat transfer coefficient W.m2.K1
Hbuilding facade height m
Itotal direct solar radiation W.m2
j , Jheat flux W.m2
kthermal conductivity W.m1.K1
Lwall length m
qradiation flux W.m2
Ssunlit area ratio
t , t ftime s
Ttemperature K
xhorizontal space coordinate m
yvertical space coordinate m
vair velocity m.s1
Greek letters
βsurface transfer coefficient
χIndicator function of sunlit
t,ttime domain s
x,yspace domain m
Γ1,Γ2,Γ3,Γ4Spatial boundary m
Φ interesting output variable
θangle between wall normal and sun position
Θ sensitivity function variable
ε2error unit of Φ
εrrelative error
33 /37
An efficient two-dimensional heat transfer model for building envelopes
Mathematical notations
Latin letters
Bi Biot number
Fo Fourier number
Nx, N y, N tnumber of elements
Rcpu CPU time ratio
udimensionless temperature
Greek letters
δslight variation
xspace mesh
ttime step
λx, λ yDu FortFrankel scheme coefficients
φpiece wise function
Σx,Σy,Σxy Du FortFrankel scheme coefficients
τaccuracy Du FortFrankel scheme coefficients
Subscripts and superscripts
0 reference value or initial condition
dr direct flux component
df diffuse flux component
rf reflective flux component
dimensionless value
boundary
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36 /37
An efficient two-dimensional heat transfer model for building envelopes
A Alternating Direction Implicit numerical scheme
The idea of the Alternating Direction Implicit (ADI) numerical scheme is to split the time step into two
intermediate stages. For the first stage tn+1
2=tn+1
2t, the scheme considers an implicit formulation
in the xdirection and an explicit one in the ydirection. Using central finite differences for the space
discretisation of Eq. (12), it yields to:
un+1
2
j i un
j i = Λ xun+1
2
j+1 i2un+1
2
j i +un+1
2
j1i+ Λyun
j i+1 2un
j i +un
j i1,(19)
with
Λx
def
:= t
2 ∆x2
kx
c,Λy
def
:= t
2 ∆y2
ky
c.
Thus, Eq. (19) gives an implicit formulation to compute un+1
2
j i :
1 + 2 Λ xun+1
2
j i Λxun+1
2
j+1 iΛxun+1
2
j1i=un
j i + Λ yun
j i+1 2un
j i +un
j i1.
This system can be written in a matrix formulation:
Axun+1
2
i=bi
with
un+1
2
i=
un+1
2
1i
.
.
.
un+1
2
Nxi
.
The system is then solved for i1, . . . , Ny. The second stage enables to compute un+1
j i from un+1
2
j i .
For this, it assumes an implicit formulation on the ydirection and an explicit one on the xdirection:
un+1
j i un+1
2
j i = Λ xun+1
2
j+1 i2un+1
2
j i +un+1
2
j1i+ Λyun+1
j i+1 2un+1
j i +un+1
j i1,
which can be formulated into an implicit expression:
1 + 2 Λ yun+1
j i Λyun+1
j+1 iΛyun+1
j1i=un+1
2
j i + Λxun+1
2
j i+1 2un+1
2
j i +un+1
2
j i1.
Again the system is formulated as:
Ayun+1
j=bj
with
un+1
j=
un+1
j1
.
.
.
un+1
j N y
,
and solved for j1, . . . , N x. The accuracy of the scheme is second order Ox2,y2,t2. The
scheme is unconditionally stable.
37 /37
ResearchGate has not been able to resolve any citations for this publication.
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