Energy and Buildings 41 (2009) 1031–1036

Thermal optimization of multilayered walls using genetic algorithms

V. Sambou a,b, B. Lartigue a,*, F. Monchoux a, M. Adj b

a P.H.A.S.E., Universite de Toulouse - UPS, 118 route de Narbonne, 31062 Toulouse Cedex 9, Franceb L.E.A., Ecole Superieure Polytechnique, BP 5085 Dakar Fann, Senegal

A R T I C L E I N F O

Article history:

Received 3 January 2009

Accepted 16 May 2009

Keywords:

Building wall

Optimization

Genetic algorithm

Thermal resistance

Thermal capacitance

A B S T R A C T

This paper reports on an optimization of a building wall using genetic algorithms. The double objective of

optimization is maximizing thermal insulation and maximizing thermal inertia. If insulation is easily

quantified with thermal resistance, inertia is a criterion not characterized with a single parameter. Using

the quadrupoles method, we propose the thermal capacitance as a way to quantify the inertia of the wall.

Walls realizing the best trade-off between the two conflicting objectives are presented in a Pareto front.

Optimal walls composition shows that the best layers disposition is massive layer at indoor side and

insulating layer at outdoor side. An important and new result obtained in this study is that the optimal

thickness of the indoor side massive layer is L/4 where the thermal wavelength L is an intrinsic

parameter of the layer material depending on the period of oscillations.

� 2009 Elsevier B.V. All rights reserved.

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1. Introduction

Industrialized nations need to reduce their consumption ofenergy and emissions by a factor of four to five within the next 50years. France has already adopted this objective as a part of itsNational Strategy for Sustainable Development, introduced in June2003, and its Climate Plan, introduced in July 2004.

In France, the building industry consumes 43% of the totalnational energy, making it the biggest consumer across all sectorsof the economy. The corresponding greenhouse gas emissionsrepresent 25% of the total emissions. Therefore, reducing theenergy consumption in the building sector is critical to meeting theFrench national commitments of reducing greenhouse gasesemission by 75% by 2050. The reinforced requirements of thethermal standard RT 2005 are part of the French response. Amongthese requirements, we find:

� the reinforcement of thermal insulation;� the valorization of bioclimatic conception;� the limitation of the use of air-conditioning.

Yet, high insulation which decreases heating energy duringwinter period leads to overheating during hot days. To overcome thisphenomenon, various methods are possible, as free cooling [1,2],architectural design [3], or thermal inertia [4,5]. Thermal inertia hasa positive effect on the indoor temperature since the solar heat gainduring the day is stocked and later discharged indoor.

* Corresponding author. Tel.: +33 561 55 68 97; fax: +33 561 55 81 54.

E-mail address: lartigue@cict.fr (B. Lartigue).

0378-7788/$ – see front matter � 2009 Elsevier B.V. All rights reserved.

doi:10.1016/j.enbuild.2009.05.007

Studies aiming to optimize building walls thermal insulationcan be found in literature. Comakli and Yuksel [6] have determinedthe optimal insulation thickness of an external wall using the lifecycle of buildings in the coldest cities of Turkey. Al-Khawaja [7]determined the optimal thickness of any insulation with the totalcost of insulation and energy in hot countries as criteria ofoptimization. Lollini et al. [8] undertook a study to determine thebest insulation level of the new buildings focusing on energy,economic and environmental issues. All above studies focus onlyon thermal insulation and do not consider thermal inertia.

The role of thermal inertia in buildings is a topic widely studiedin the literature. Balaras [9] emphasized the impact of thermalmass on the cooling load of buildings. Some studies [10,11] showedthat the wall thermophysical properties have great effects on timelag and decrement factor. Ulgen [11] suggested to use multilayeredand insulated wall formations for a continuously occupied buildingand single-layered wall for buildings used for specific timeintervals.

In order to characterize the dynamics of the building,Antonopoulos and Koronaki [12] defined apparent and effectivethermal capacitance of buildings. The apparent thermal capaci-tance is obtained by adding the distributed specific heats of allbuilding elements. They proposed a simple model to determine theindoor temperature using building heat-loss coefficient andefficient thermal capacitance as parameters. The same authorsin [13] developed correlations giving the dynamic thermalparameters of buildings as effective thermal capacitance, thermaltime constant and thermal delay.

Tsilingiris [14] studied thermal time constants of a wall,defining two time constants – the forward and reverse – whichconventionally correspond to indoor and outdoor wall sides

Nomenclature

a thermal diffusivity (m s�2)

c specific heat (J kg�1 K�1)

Cth thermal capacitance (J m�2 K�1)

k thermal conductivity (W m�1 K�1)

L thickness (m)

p Laplace parameter

q unit surface heat flux (W m�2)

R thermal resistance (m2 K W�1)

t time (s)

T temperature (C)

Z impedance (m2 K W�1)

Symbols

L thermal wave length (m)

f complex heat flux (W m�2)

u complex temperature (C)

r density (kg m�3)

v angular frequency (s�1)

Subscripts

in indoor side

out outdoor side

w wall

V. Sambou et al. / Energy and Buildings 41 (2009) 1031–10361032

respectively. He emphasized the impact of the insulation positionon the two time constants and showed the advantage of employinglow thermal forward time constant walls in intermittently heatedor cooled buildings. The same author analyzed wall heat loss fromintermittently conditioned spaces [15]. The effect of varioussignificant design and system operational parameters such as wallheat capacity, thermal time constant and insulation layer positionare investigated on the quasi steady-state energy loss and dailyindoor temperature swing under typical winter conditions.Tsilingiris [16] developed a parametric study to analyze theinfluence of the distribution of wall heat capacity and thermalresistance on the dynamic behavior of walls.

Other studies showed the impact of the insulation position inthe wall on the dynamic behavior of the buildings [17–19].Kossecka and Kosny [20] analyzed the influence of insulationconfiguration in a continuously used building on the total heatingand cooling load. This analysis made for six different US climatesshowed that the best thermal performance is obtained whenmassive material layers are located at the inner side and directlyexposed to the interior space.

The contribution of our paper is to define the optimalcomposition of a wall building by optimizing simultaneouslythermal insulation and thermal inertia of the wall withoutdiscriminating any parameter. Yet, if insulation can be easilyquantified with thermal resistance, thermal inertia is a criterionthat cannot be simply characterized with a single parameter. Weuse the quadrupoles method to introduce, by electrical analogy,the thermal capacitance and propose it as the parametercharacterizing thermal inertia [21].

2. Quadrupoles method

The method to determine the transmission matrix of heattransfer for one-dimensional multilayer plane walls in Laplacedomain is known and can be found for example in [22]. The heattransfer equation in a layer of building homogeneous material is

represented by:

@Tðx; tÞ@t

¼ a � @2Tðx; tÞ@x2

(1)

the heat flux q for a unity surface at arbitrary time t and location x

in the wall is represented by:

qðx; tÞ ¼ �k � @Tðx; tÞ@x

(2)

applying Laplace transform to Eqs. (1) and (2) leads to a matrixrelationbetweentemperatures and heatflux at both sides of the wall:

Toutð pÞqoutðpÞ

� �¼ AðpÞ Bð pÞ

CðpÞ Dð pÞ

� �T inð pÞqinðpÞ

� �¼ MðpÞ T inð pÞ

qinð pÞ

� �(3)

where p is the Laplace variable.The total transmission matrix of the entire wall M(p) is obtained

by multiplying individual layer transmission matrices includingthe surface films at both sides as follows:

Mð pÞ ¼ Mout �MNðpÞ � . . . �M2ð pÞ �M1ð pÞ �Min (4)

where

M j ¼1 R j

0 0

� �; j ¼ out or in (5)

� Rj is film thermal resistance

� MiðpÞ ¼Aið pÞ Bið pÞCið pÞ Dið pÞ

� �; i ¼ 1;2; . . . ;N (6)

Transmission matrix elements of a layer i can be given inhyperbolic function forms:

Aið pÞ ¼ DiðpÞ ¼ cos h bið pÞ � Li½ � (7)

Bið pÞ ¼sin h bið pÞ � Li½ �

ki � bið pÞ(8)

Cið pÞ ¼ ki � bið pÞ � sin h bið pÞ � Li½ � (9)

with biðpÞ ¼ffiffiffipai

qIn the case of sinusoidal heat transfer, the elements of each

layer transmission matrix are obtained by substituting jv( j ¼

ffiffiffiffiffiffiffi�1p

) to p.

biðvÞ ¼

ffiffiffiffiffiffiffiffiffiffij �vai

s¼ ð1þ jÞ � 2 � p

LiðvÞ(10)

Li(v) can be defined as the thermal wavelength of the material oflayer i at an angular frequency v. This parameter is given by:

LiðvÞ ¼ 2 � p �ffiffiffiffiffiffiffiffiffiffiffi2 � ai

v

r(11)

the complex elements of each layer transmission matrix are:

AiðvÞ ¼ DiðvÞ ¼ cos h 2 � p � ð1þ jÞ � Li

LiðvÞ

� �(12)

BiðvÞ ¼Ri

2 � p � ð1þ jÞ � ðLi=LiðvÞÞ

� sin h 2 � p � ð1þ jÞ � Li

LiðvÞ

� �(13)

CiðvÞ ¼2 � p � ð1þ jÞ � ðLi=LiðvÞÞ

Ri

� sin h 2 � p � ð1þ jÞ � Li

LiðvÞ

� �(14)

where Ri is thermal resistance of the layer i (Ri = Li/ki).

Fig. 1. Thermal quadrupole representation in «P» scheme.

Table 1Boundary values of variables for case 1 study.

Min value Variable Max value

0 L (m) 0.4

0.043 k (W m�1 K�1) 1.731

32 r (kg m�3) 2243

840 c (J kg�1 K�1) 2510

V. Sambou et al. / Energy and Buildings 41 (2009) 1031–1036 1033

Note that the transmission matrix elements of each layer onlydepend on two parameters: R and L/L.

The quadrupole Eq. (3) tying complex temperatures and heatflux at both sides of the wall becomes:

uoutðvÞfoutðvÞ

� �¼ AðvÞ BðvÞ

CðvÞ DðvÞ

� �uinðvÞfinðvÞ

� �(15)

the multilayered wall can be represented by three impedancesassociated in a « P » scheme as showed in Fig. 1. The threeimpedances can be expressed as:

ZoutðvÞ ¼BðvÞ

DðvÞ � 1(16)

ZtðvÞ ¼ BðvÞ (17)

ZinðvÞ ¼BðvÞ

AðvÞ � 1(18)

The representation in Fig. 1 shows the advantage of putting outthree supplementary heat flux. f’out and f’in are flux absorbed bythe wall at both sides. Impedances Zout and Zin play then the role ofheat accumulator. The thermal capacitances related to thoseimpedances are expressed as:

CthoutðvÞ ¼ �1

v � I ZoutðvÞ½ � (19)

CthinðvÞ ¼ �1

v � I ZinðvÞ½ � (20)

The thermal capacitance Cthin related to the interior side of thewall characterizes the wall thermal inertia as recommended byinternational standard ISO 13786 [23] since it attenuates theinternal temperature fluctuations.

Since the outside temperature changes diurnally, a period of24 h is considered here.

3. Multi-objective optimization

Our aim is to optimize both thermal insulation and thermalinertia of a wall by maximizing the two objectives: thermalresistance and indoor side thermal capacitance.

The optimization tool is a genetic algorithm code. Geneticalgorithms are stochastic search methods that mimic naturalbiological evolution. They operate on a population of candidatesolutions applying the principle of survival of the fittest to produceincreasingly improved approximations of a solution.

The genetic algorithm code used in this work has been developedby Leyland [24] and Molyneaux [25] from the Ecole PolytechniqueFederale de Lausanne, Switzerland. Originally, this multi-objectivecode was developed for energy systems with conflicting optimiza-tion criteria. In contrast with single objective optimization with onesingle solution, multi-objective optimization aims at finding a set ofPareto solutions. A solution is said to be Pareto optimal if and only if

it is not dominated by any other solution in the decision variablespace. If solution X1 dominates another solution X2, it implies that X1

is non-inferior to X2 for all the considered performance criteria but itis better than X2 for at least one criterion. All the points in theobjective function space corresponding to optimal solutions form aPareto front, which is useful to understand the trade-off between theperformance criteria.

4. Results and analysis

Walls are made of N parallel layers of homogeneous andisotropic materials. Walls’ thickness is fixed to Lw = 0.4 m.

4.1. Case study 1: optimal walls made of possibly fictitious materials

This case study consists of determining the thickness andthermophysical properties of the wall layers that realize the besttrade-off between high thermal insulation and high thermalinertia. The problem variables are thickness L, thermal conductiv-ity k, density r and specific heat c of each layer. Then, the followingvector constitutes the chromosome of an individual (a wall):

x ¼ ðL1; k1;r1; c1; L2; k2;r2; c2; . . . ; LN; kN;rN; cNÞ (21)

The problem constraints are:

0 � Li � Lw

kmin � ki � kmax

rmin � ri � rmax

cmin � ci � cmax

8>><>>: (22)

The boundary values of the thermophysical parameters arebased on ASHRAE database of walls materials [26] and aresummarized in Table 1.

In order to help interpret the results, the layers number N isfixed to 3 since it has been shown that for N higher or equal to 3, thePareto optimal front of solutions is the same.

Pareto optimal walls front is presented in Fig. 2. Two distinctiveparts can be identified: the first (part I) corresponds to a very slowdecrease in the thermal capacitance as the thermal resistanceincreases, and the second (part II) where the capacitance decreasesdrastically.

In part I, optimal walls are characterized by a massive indoor sidelayer with a maximal thermal effusivity i.e. (k�r�c)1/2, meaning thatits thermophysical properties k, r and c reach their maximal value.Another important result is that walls have indoor massive layerwith thickness closed to L/4. The thermal wavelength L for themassive material is equal to 0.144 m. Then, one can consider that theoptimal thickness of a massive layer is L/4. The two other layers areinsulating with a random volumetric heat capacity (r�c). These twolayers contribute only to the thermal resistance and not capacitance.

In part II, the indoor massive layer of each wall has a thicknessless than L/4. In this case, the second (and central) layer is verythin and contributes to the thermal capacitance since itsvolumetric heat capacity (r�c) is maximal. However, its thermalconductivity is minimal in order to ensure insulation. The last layeris insulated with a random volumetric heat capacity.

Solutions found in this optimization are not realistic. Indeed,the massive layer would be a heavy concrete with wood specific

Fig. 2. Pareto front for case 1 solutions.Fig. 3. Pareto optimal walls front of case 2 and ASHRAE walls.

Table 2Selected Pareto optimal walls composition.

R (m2 K/W) Cthin (kJ/(m2K)) Wall layers (in! out) Lw (m)

Wall 1 2.2 331.3 E0 C5 C13 B23 B10 A1 A0 0.388

Wall 2 3.8 324.5 E0 C5 C13 B21 B27 A0 0.400

Wall 3 5.9 278.6 E0 C13 E3 B13 B27 B5 A0 0.400

Wall 4 7.21 190.76 E0 C5 B12 B13 B23 B23 A0 0.400

A0: outside surface film resistance.

A1: 25 mm stucco.

B10: 50 mm wood.

B5/B12/B13/B23/B27: 25 mm/75 mm/100 mm/60 mm/115 mm insulation.

C5/C13: 100 mm/150 mm heavyweight concrete.

E0: inside surface film resistance.

E3: 10 mm felt and membrane.

V. Sambou et al. / Energy and Buildings 41 (2009) 1031–10361034

heat and the second layer of walls in part II would be insulated withheavy concrete density and wood specific heat. To our knowledge,there is no material having such thermophysical properties.

A second case study is undertaken in order to optimize realbuilding walls.

4.2. Case study 2: optimal walls made of fixed-thickness

ASHRAE layers

The aim of the second case study is to determine the optimalcomposition of a wall made of ASHRAE material layers (definedwith fixed-thickness) [26]. These layers are stocked in a matrixwhose length is the number of layer types i.e. 55 excluding steelsiding (code: A3) and air space (codes: B1 and E4). The number oflayers is fixed to a maximum of 5 (the maximum ASHRAE walllayers number). Problem variables are the identification number ofeach layer in the matrix. The chromosome of an individual (a wall)is then a vector of 5 integers ni:

x ¼ ðn1;n2;n3;n4;n5Þ (23)

The problem constraints are:

0 � ni � 55Lw � 0:40 m

�(24)

The number 0 corresponds to an absence of layer at thecorresponding position in the vector. Thus, the number of wall layerscorresponding to the vector is decreased by one unit, and so on.

Fig. 3 represents the Pareto front of optimal walls realizing thebest trade-off between high thermal resistance and high indoorside thermal capacitance. The shape of this curve is the same as theone of Fig. 2, auguring that the same remarks apply. However, onecan see that the thermal capacitance of walls in part I of the Paretofront is far lower than in Fig. 2.

Fig. 3 also presents ASHRAE walls in order to compare them tooptimal walls. Although the thickness of some ASHRAE walls issuperior to Lw, it can be seen that none of these walls is Pareto-optimal.

The composition of some selected walls in the Pareto front isdetailed in Table 2.

Several conclusions can be drawn from the results of thisoptimization:

* The composition of optimal walls shows that the optimalposition of insulation is at the wall outdoor side. This result has

been reported by many authors such as [20] for continuous-usebuildings.

* Walls located in part I of the front have massive layers at indoorside. These layers are 100 mm and 150 mm heavyweightconcrete. Total thickness of the massive layer is 0.25 mcorresponding to L/4 for the heavyweight concrete. Thisinteresting result means that it is not necessary to have massivelayer thicker than L/4 to improve the wall thermal inertia.

The Pareto optimal walls given in this case study are composedof fixed-thickness layers. The objective of the next case is todetermine optimal thickness of layers composing a Pareto optimalwall.

4.3. Case study 3: optimal walls made of ASHRAE materials

The problem at hand consists of determining the composition ofoptimal walls and their layers thickness. The problem variables arevectors of 2N components, N being the maximal layers number fora wall. N is fixed to 5 (the maximum ASHRAE wall layers number).Then, a wall chromosome is given by:

x ¼ ðn1; L1;n2; L2;n3; L3;n4; L4;n5; L5Þ (25)

The odd components ni of a variable x are the identificationnumber of the material of each wall layer given in Table 3. The evencomponents Li are the thickness of each wall layer.

The constraints of the problem are:

0 � Li � Lw

1 � ni � 22

�(26)

Table 3ASHRAE construction materials (optimal materials in italic).

N Material k (W m�1 K�1) r (kg m�3) c (J kg�1 K�1)

1 Stucco 0.692 1858 840

2 Face brick 1.333 2002 920

3 Slag 0.19 1121 1670

4 Finish 0.415 1249 1090

5 Insulation I 0.043 32 840

6 Insulation II 0.043 91 840

7 Wood 0.121 593 2510

8 Clay tile 0.571 1121 840

9 Lightweight concrete block 0.381 609 840

10 Heavyweight concrete block 0.813 977 840

11 Common brick 0.727 1922 840

12 Heavyweight concrete 1.731 2243 840

13 Lightweight concrete block 0.571 609 840

14 Heavyweight concrete block 1.038 977 840

15 Lightweight concrete 0.173 641 840

16 Lightweight concrete block (filled) 0.138 288 840

17 Heavyweight concrete block (filled) 0.588 849 840

18 Lightweight concrete block (filled) 0.138 304 840

19 Heavyweight concrete block (filled) 0.675 897 840

20 Plaster or gypsum 0.727 1602 840

21 Slag or stone 1.436 881 1670

22 Acoustic tile 0.061 481 840

Fig. 4. Pareto optimal walls front of case 3.

V. Sambou et al. / Energy and Buildings 41 (2009) 1031–1036 1035

Fig. 4 presents the Pareto optimal walls front for Lw = 0.40 m.This front is identical to the one of Fig. 3 but with more regularity.

Composition of selected walls is given in detail in Table 4.Several conclusions can be drawn from detailing the composi-

tion of Pareto optimal walls:

* Only 6 materials among the 22 contained in Table 3 enter in thecomposition of the optimal walls. Those materials are heavy-weight concrete, the two insulation types, wood, face brick andslag.

* Although the number of layers is fixed to 5, most Pareto optimalwalls only have two or three layers.

* Walls in the first step of the front (where Cthin varies slowlywith thermal resistance) are composed only of three materials:heavyweight concrete with a thickness of L/4 at indoor side,insulation and wood. The order of the last two layers (insulationand wood) seems to have no influence on the thermalcapacitance. This result is interesting since it shows thatinsulation in ‘‘sandwich’’ configuration does not alter thermalinertia of a wall with internal side massive layer of thicknessabout L/4.

Table 4Characteristics of selected optimal walls (layers thicknesses are in millimeters).

* In the second step of the Pareto front, Cthin decreases drasticallywhen thermal resistance increases linearly with the insulationthickness.

5. Conclusion

Thermal inertia is important to summer thermal comfort andbioclimatic building conception. Thermal capacitance based onwall quadrupole representation by electrical analogy is a pertinentparameter to characterize thermal inertia.

The optimization of a building wall to limit winter energyconsumption and to improve summer thermal comfort is based onthe simultaneous evaluation of wall thermal resistance andthermal capacitance that must have the highest values as possible.Wall thickness and wall composition must be realistic.

Multilayered wall optimization determines the optimal layersdisposition. The best configuration consists of a massive layerinside and insulation outside. An important and new finding is theoptimal thickness of the massive layer L/4. The thermalwavelength L is an intrinsic parameter of the layer materialdepending on the period of oscillations. Insulating layer thicknessdepends directly on the target insulation level.

Results from this optimization can be used in studies dealingwith buildings’ energy consumption and comfort. Future researchwill study the impact of such optimized walls on the behavior ofbuildings. It is likely that some supplementary optimizationcriteria depending on climate or occupation schedule should beintroduced in order to choose among multiple Pareto optimalsolutions.

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