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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 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, 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 Fort–Frankel 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 [15–17] 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 Fort–Frankel 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:

Ωx

def

:= xx∈0, L and Ω y

def

:= yy∈0, 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

:= Ω x∪Ωy. The four boundaries of the domains are deﬁned

such as:

Γ1

def

:= nx , y y∈Ωy, x = 0 o,Γ2

def

:= nx , y x∈Ωx, y =Ho,

Γ3

def

:= nx , y y∈Ωy, x =Lo,Γ4

def

:= nx , y x∈Ωx, 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.m−3.K−1and kW.m−1.K−1are the volumetric heat capacity and the thermal conductiv-

ity. The wall is composed of Nmaterials and Ω =

N

[

i=1

Ωi, with Ω 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 )∈Ωi,

0,(x , y )/∈Ωi.

Initially, the wall is assumed in steady-state condition:

T=T0(x , y ),∀(x , y )∈Ω, 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.m−2deﬁned 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.m−2impacting at the inside of the ambient zone is computed by:

J(t) = 1

HZΓ3−k(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) = ZΩtJ(t) dt , (3)

where Ωt⊂Ωtis 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,∀i∈1, . . . , 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.m−2.K−1is the surface heat transfer coeﬃcient between the material and the surrounding

ambient air and q∞, i W.m−2is 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 v∞m.s−1[18]:

h∞,1(y , t ) = h10 +h11 ·v∞

v0·y

y0β

,(4)

where

v∞:t7−→ v∞(t),

and h10 , h 11 W.m−2.K−1and β−are given coeﬃcients. The reference quantities v0and y0are

set to v0= 1 m.s−1and y0= 1 m. The mean value of the surface heat transfer coeﬃcient is deﬁned by:

h∞,1

def

:= 1

H·tfZΩtZΩy

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.m−2:

qdr

∞(t) = I(t)·cosθ(t),

where θ−is 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 α−is 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·1−S(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) = H−h(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 [20–23]. 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

:= β−β◦, δF def

:= F−F◦, δD def

:= D−D◦.

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 ,∀i∈1, . . . , 4.(9)

The initial condition is θi= 0 ,∀(x , y )∈Ω, t = 0 ,∀i∈1, . . . , 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,∀i∈1, . . . , 4,

with

∂h ∞, i

∂h 11

(y , t ) =

0,∀i∈2,3,4,

v∞

v0·y

y0β

, i = 1 .

and

∂h ∞, i

∂β (y , t ) =

0,∀i∈2,3,4,

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,∀i∈1, . . . , 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 ,∀i∈2,3,4.

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 h−δD ·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

:= T−T0

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:

t⋆def

:= 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:

k⋆def

:= k

k0

, c ⋆def

:= c

c0

,

where k0and c0are reference thermal conductivity and volumetric heat capacity. The coeﬃcients c⋆

and k⋆are 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

:= q∞Lx , 0

k0δT , q ⋆

∞, y

def

:= q∞Ly , 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 Fort–Frankel 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

j−1i−2un

j i +un

j+1 i+ky

∆y2·un

j i−1−2un

ji +un

j i+1 .(13)

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An eﬃcient two-dimensional heat transfer model for building envelopes

Figure 4. Du Fort–Frankel numerical scheme stencil.

Then, to obtain the Du Fort–Frankel scheme, the term un

j i is replaced by 1

2un+1

j i +un−1

j i in Eq. (13).

It yields to the following explicit expression:

un+1

j i = Σ x·un

j−1i+un

j+1 i+ Σ y·un

j i−1+un

j i+1 + Σ xy ·un−1

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

:= 1−λx−λy

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·exp−iβ x j)·exp−iγ 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·expiβ∆x) + exp−iβ∆x)+ Σ y·expiγ∆y) + exp−iγ∆y)+Σxy

ρ.

It leads to the following second-order polynomials in ρequation:

ρ2−B ρ +C= 0 ,(16)

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An eﬃcient two-dimensional heat transfer model for building envelopes

with

Bdef

:= 2 Σxcosβ∆x) + Σ ycosγ∆y), C def

:= −Σxy .

The general solution of Eq. (16) is:

ρ±=−1

2B±√D, D def

:= B2−4C .

The modulus ρ±veriﬁes:

ρ±61

2B±√D.

It is straightforward that 1

2B6Σx+ Σ y. Given the expression of D, we have:

1

2√D=1

1 + λx+λy· λxcosβ∆x) + λycosγ∆y)2+ 1 −λx+λy2!

1

2

.

Thus,

1

2√D61

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

j−1i+un

j+1 i−Σy·un

j i−1+un

j i+1 −Σxy ·un−1

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

+O∆x2+ ∆y2+ ∆t4.

Thus, the scheme is second-order accurate in space O∆x2+ ∆y2. However, the accuracy in time

depends on the quantity τdeﬁned as:

τdef

:= kx

∆x2+ky

∆y2·∆t2.

If τ≪1 , then the scheme is second-order accurate in time O∆t2. If the condition τ≪1 is not

respected, then the scheme is not consistent with the discretized equation. For practical applications, in the

case ∆x≈∆y, then the second-order accuracy is obtained when ∆t=O∆x2.

11 /37

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:

ε2◦Φ(t)def

:= 1

L H ZΩxZΩyΦ( x , y , t )−Φref (x , y , t )2dxdy!

1

2

.

A normalized version of the error is also used:

ε2◦Φ(t)def

:= 1

L H ZΩxZΩyΦ( 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:

εr◦Φ(t)def

:= 1

L H ZΩxZΩyΦ( 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

Ωab

def

:= 0, L ⋆

a×0, L ⋆

b:

u0(x⋆, y ⋆) =

1,(x⋆, y ⋆)∈Ωab ,

0,(x⋆, y ⋆)/∈Ωab ,

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 ,∀i∈1, . . . , 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 †

∆t⋆∆x⋆= ∆y⋆ε2◦u(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 Fort–Frankel 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 Fort–Frankel (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⋆= 10−4and ∆x⋆= ∆y⋆= 10 −2.

For the Euler explicit approach, the discretization parameters are required to satisfy the following stability

condition:

∆t⋆6∆x⋆∆y⋆2

2 Fo ∆x⋆2+ ∆y⋆2,(17)

which corresponds to ∆t⋆62.5·10 −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 Fort–Frankel 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⋆= 10−2and 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 ∆t⋆6

2.5·10−5. Then, it can be remarked that the Du Fort–Frankel, 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 Fort–Frankel 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 Fort–Frankel model is always faster than the other

approaches. For ∆t⋆= 10−4, it can be remarked that the Du Fort–Frankel 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 x⋆at y⋆= 0.5 (a), according to y⋆at x⋆= 0.4 (b) and

according to t⋆(c). Slice of the solution from the Du Fort–Frankel 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.m−1.K−1 MJ .m−3.K−1 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.m−2.K−1,h11 = 3.96 W.m−2.K−1,

v0= 1 m.s−1,y0= 1 mand β= 0.32 . The outside wind velocity is shown in Figure 7(a). It varies

around a mean of 4 m.s−1. 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.m−2.K−1at y= 0.3m, 10.5W.m−2.K−1at y= 1.5mand 11.5W.m−2.K−1at y= 2.7m.

The mean over the whole year and height gives a coeﬃcient of h∞,1= 10.28 W.m−2.K−1. 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.m−2.K−1used 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.m−2.K−1. 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)

5 7.5 10 12.5 15 17.5 20

0

0.2

0.4

0.6

0.8

1

(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 Fort–Frankel 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 Fort–Frankel 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.m−2.K−1. 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.m−2.K−1and 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 )εr◦E[%] 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. εr◦E[%] −0.69 0.45 0.13 4.61 −5.52 −15.2−80.5−31.8−1.64 −8.48 −2.24 −1

2D No Const. εr◦E[%] 2.61 4.5 0.19 4.62 −5.46 −15.1−80.2−31.7−1.62 −6.34 2.74 2.32

1D Ratio Const. εr◦E[%] −0.38 0.44 0.09 4.56 −5.62 −15.3−80.6−31.7−1.54 −8.43 −2.21 −0.98

1D Ratio f(t)εr◦E[%] 0.11 −0.42 −1.41 −2.66 −5.61 −10.5−18.4−9.22 −6.08 −1.97 −0.19 −0.21

1D No Const εr◦E[%] 2.92 4.49 0.16 4.57 −5.56 −15.2−80.3−31.6−1.51 −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.m−2.K−1,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 .m−2. 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 Fort–Frankel 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 Fort–Frankel 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.m−3.K−1

Dfront building distance m

Ethermal loads J.m−2

Ffront building height m

hshadow height m

h , h , h 10 , h 11 surface heat transfer coeﬃcient W.m−2.K−1

Hbuilding facade height m

Itotal direct solar radiation W.m−2

j , Jheat ﬂux W.m−2

kthermal conductivity W.m−1.K−1

Lwall length m

qradiation ﬂux W.m−2

Ssunlit area ratio −

t , t ftime s

Ttemperature K

xhorizontal space coordinate m

yvertical space coordinate m

vair velocity m.s−1

Greek letters

βsurface transfer coeﬃcient −

χIndicator function of sunlit −

Ωt,Ωttime domain s

Ωx,Ωyspace 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 Fort–Frankel scheme coeﬃcients

φpiece wise function

Σx,Σy,Σxy Du Fort–Frankel scheme coeﬃcients

τaccuracy Du Fort–Frankel 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

2∆t, 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 i−2un+1

2

j i +un+1

2

j−1i+ Λyun

j i+1 −2un

j i +un

j i−1,(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 i−Λxun+1

2

j−1i=un

j i + Λ yun

j i+1 −2un

j i +un

j i−1.

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 i∈1, . . . , 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 i−2un+1

2

j i +un+1

2

j−1i+ Λyun+1

j i+1 −2un+1

j i +un+1

j i−1,

which can be formulated into an implicit expression:

1 + 2 Λ yun+1

j i −Λyun+1

j+1 i−Λyun+1

j−1i=un+1

2

j i + Λxun+1

2

j i+1 −2un+1

2

j i +un+1

2

j i−1.

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 j∈1, . . . , N x. The accuracy of the scheme is second order O∆x2,∆y2,∆t2. The

scheme is unconditionally stable.

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