A building thermal bridges sensitivity analysis

Applied Energy 107 (2013) 229–243

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A building thermal bridges sensitivity analysis Alfonso Capozzoli, Alice Gorrino, Vincenzo Corrado ⇑ Department of Energy, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy

h i g h l i g h t s " Heat transfer through the most typical thermal bridges is analyzed through a finite difference method. " The linear thermal transmittance for a large number of design parameters is catalogued. " Non-linear regression models of the most typical thermal bridges are identified. " A sensitivity analysis of the most relevant design parameters is carried out.

a r t i c l e

i n f o

Article history: Received 26 July 2012 Received in revised form 20 January 2013 Accepted 15 February 2013 Available online 16 March 2013 Keywords: Thermal bridges Transmission heat losses Parametric analysis Regression models Sensitivity analysis Analysis of variance

a b s t r a c t Along with the entry into force of the new European Directive 2010/31/EU on the Energy Performance of Buildings (EPBD recast), each Member State has the responsibility of supporting activities for the construction of nearly zero energy buildings with a very high energy performance. In order to achieve the new EU directive targets, designers, in addition to having to use innovative building components, also have to pay more attention to the construction details which mostly affect building envelope heat losses. It is therefore necessary not only to properly design structural nodes, in order to minimize such energy losses, but also to identify accurate numerical methods in order to appreciate the benefits of a proper design. A sensitivity analysis based on an extensive study of the linear thermal transmittance value of many types of thermal bridge, based on the methodology specified in EN ISO 10211, has been carried out in the presented work. After having defined the input design variables and considering a range of variation for each of them for the linear thermal transmittance evaluation, a non-linear regression model has been specifically developed for each analyzed thermal bridge, considering the output values of a numerical code as data set. In order to perform the sensitivity analysis a significant and representative number of cases have been generated, using a sampling technique. The ANOVA–FAST method has been performed, on the basis of the obtained results, in order to assess the contribution of each input design variable to the deviation of the linear thermal transmittance for each kind of thermal bridge. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction 1.1. Influence of thermal bridges on building energy performance With the entry into force of European Directive 2010/31 on the Energy Performance of Buildings (EPBD recast) [1], each Member State is required to draw up national plans to increase the number of nearly zero-energy buildings. Moreover, starting from European Directive 2002/91/CE, the Italian national legislation on building energy efficiency (Legislative Decree no. 192/2005, Legislative Decree no. 311/2006) has led to an improvement in opaque and transparent envelope performance. In order to achieve the objectives of the EPBD recast, the designer is required on one hand to use innovative envelope ⇑ Corresponding author. Tel.: +39 011 090 4456; fax: +39 011 090 4499. E-mail address: [email protected] (V. Corrado). 0306-2619/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.apenergy.2013.02.045

components, while on the other hand to pay greater attention to construction details. In fact, the mere addition of an insulation layer only reduces the one-dimensional heat flow, but does not significantly decrease the multi-dimensional one, if no attention has been paid to the heat flow through thermal bridges. Although it is not possible to obtain general results concerning the weight of thermal bridges on the energy needs of buildings, several studies have presented numerical results for different cases. A study was conducted in Greece on a typical three-storey apartment building with an open ground-floor space (pilotis) and a flat roof; the façades are composed of two brick layers with interposed insulation [2]. The study shows that the double brick wall construction widely used in Greece is affected to a great extent by thermal bridges. Even if the actual construction presents high insulation levels, the heating need can be 30% higher than the one calculated without taking into account the thermal bridge effects. Cappelletti et al. [3] have shown that the weight of thermal

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Nomenclature Symbols B0 btr e H l L2D q R S SI U

characteristic dimension of the floor (m) temperature reduction factor (–) width (m) overall heat transfer coefficient (W K1) length (m) thermal coupling coefficient from bi-dimensional calculation (W m1 K1) heat flow per meter length (W m1) thermal resistance (m2 K W1) thickness (m) sensitivity index thermal transmittance of a building component (W m2 K1)

Greek symbols h air temperature (°C) k thermal conductivity (W m1 K1) w linear thermal transmittance (W m1 K1)

b calc e fl fr gr i int ins m mas M1 M2 p pred r se si tr w

balcony calculated external floor frame ground internal intermediate insulation layer mean masonry internal masonry external masonry pillar predicted roof external surface internal surface transmission wall

Subscripts adj adjacent, adjusted

bridges on building energy needs for space heating can reach 67% for a building with a double brick wall (U = 0,15 W/(m2 K)) in a typical Italian climatic zone. A collection of interesting studies has been reported in [4]. The impact of thermal bridges on the energy quality of a building has also emerged from a study carried out in the Czech Republic: the case study was a residential building with brick walls and wooden frame windows. From the same study it is highlighted that the relative impact of the thermal bridges on the annual energy needs varies from 7%, for typical houses of the seventies to 28%, for modern high-quality houses. The impact of thermal bridges on the heating energy needs of different European Member States is generally as high as 30% [4]. Moreover, the more a building envelope is insulated, the more thermal bridges play a relatively increasing role in the global heat losses and consequently in the energy needs for space heating. A study on the effects of thermal bridges in an Italian climate has been carried out on two building types (terraced houses and semi-detached houses) and three envelope configurations by Evola et al. [5]. The results show that the correcting thermal bridges is an effective way of reducing the primary energy heating demand (25% for terraced houses, 17.5% for semi-detached house), but only a slight improvement – about 3.5% – can be achieved in the cooling performance of the building. The overall annual energy savings is about 8.5%, but a cost analysis has shown that the savings determined by correcting thermal bridges are not sufficient to recover the additional construction and refurbishment costs. Some studies focus on window thermal bridges giving practical and technical solutions to minimize their effect on building thermal losses. Thermal bridging has been evaluated through three different window systems commonly used in buildings in hot regions, by Ben-Nakhi [6] by means of the linear thermal transmittance (w) approach. The results show that the classical window system, which is the most common in Kuwait, is affected by significant thermal bridging and that this effect should be considered in building design. Some practical methods used to reduce the w value for the classical window system were also evaluated in this paper. The influence of thermal bridges on the performance of windows has also been analyzed by Cappelletti et al. [7] for the case of clay walls with external and cavity insulation. The results pointed out a

consistent reduction of up to 70–75% in linear thermal transmittances when the window is moved from the internal to the external position. This decrease mainly depends on the position of the insulating layer installed in the window hole. Hence, in order to minimize thermal bridge energy losses, it is necessary not only to design the structural nodes properly, but also to identify accurate methods to calculate heat losses, in order to make it possible to appreciate the benefits induced by a correct design. 1.2. Existing methods for the calculation of energy losses through thermal bridges In this paper the thermal bridging effect is evaluated by means of the linear thermal transmittance approach. Since this parameter is calculated under stationary conditions, it is generally applied for building energy need assessment through quasi-steady state methods in the framework of energy performance technical standards and regulations. The use of linear thermal transmittance approach presents some limitations for building energy simulations (BESs); therefore the aim of this paper is not to implement it in dynamic energy simulation codes. The heat exchange through thermal bridges can be calculated using the different methodologies specified in the relevant technical standards [8,9], which present both simplified and detailed methods to calculate thermal losses through thermal bridges under steady-state conditions. Italian Technical Specification UNI/TS 11300-1 [10], for existing buildings, in the absence of reliable project data or more accurate information, for some building types, considers a percentage increase of the overall heat transfer coefficient by transmission (Htr,adj), while European Standard EN 12831 [11] introduces a corrected thermal transmittance of building element, taking into account linear thermal bridges through an increase of the thermal transmittance of the element. Both documents mention EN ISO 14683 standard as the main reference for the calculation of heat transfer through the thermal bridges in all the remaining cases. In the cases of linear thermal bridges, and in the absence of specific data, EN ISO 14683 standard [8] provides the use of default

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linear thermal transmittance values with reference to typical twodimensional thermal bridges, and reports these values in Annex A, with a cautious overestimation. The standard also specifies that these values can only be used if no details are available on the particular thermal bridge or in the case in which an approximate value of w is appropriate in relation to the accuracy required to determine the overall heat transfer. As far as numerical calculations are concerned, EN ISO 14683 refers to EN ISO 10211, which established the specifications for a threedimensional and a two-dimensional geometric model of a thermal bridge for the numerical calculation of heat flows, in order to assess the overall heat loss from a building or part of it, and the minimum surface temperatures in order to assess the risk of surface condensation. EN ISO 14683 [8] suggest the following calculation methods, in descending order of accuracy: – Numerical calculations according to EN ISO 10211 [9] (typical accuracy ±5%). – Thermal bridge catalogues (typical accuracy of ±20%). – Manual calculations (typical accuracy ±20%). – Default values provided by EN ISO 14683 itself (typical accuracy ±50%). The default w-values provide the most inaccurate results above all because the parameters used to calculate the linear thermal transmittance (boundary conditions, geometric parameters and material thermal properties) are set at default values that are close to the maximum which is likely to occur in practice and are thus cautious overestimates of the thermal bridging effects. Thermal bridges catalogues are much more accurate when the parameters used to calculate the w values correspond to the real construction characteristics. In this case, the accuracy can be comparable to that of a numerical calculation [8,9]. As far as the numerical calculations according to EN ISO 10211, which provides the most accurate results, are concerned, many simulation codes based on finite difference, finite element or finite volume calculations have been developed. Current studies are focusing on the development of a thermal parameter characterizing thermal bridges under dynamic conditions [12] and its implementation in dynamic energy simulation codes [13–17], although some critical aspects still remain regarding the inertia of thermal bridges, as can be seen in Martin et al. [18]. Starting from numerical simulations, Dilmac et al. [19] have proposed a new method to calculate the parameters cited in ISO 9164 for the floor/beam-wall intersections used in the equations for the calculation of the energy transmission heat loss coefficient, HT, especially in the evaluation of thermal bridges. The proposed method is simpler than that specified in EN ISO 10211 and yields more accurate results for floors with web beams than EN ISO 14683. Another simple way to evaluate the effect of thermal bridges on the global building heat losses is defined in Asdrubali et al. [20] starting from results acquired by the thermographic investigations and elaborated through an analytical process. In order to provide linear thermal transmittance correlations of some thermal bridges, Ben Larbi [21] has calculated different w values by varying the thermal bridge properties, and has developed a simple statistical model from results obtained through a numerical resolution based on EN ISO 10211 in order to obtain w values through non-linear regression equations. Despite the intense research activity on this topic, only few investigations concern the global influence of the envelope design variables on thermal bridges linear thermal transmittance and on the relevant heat losses. The purpose of this study is to evaluate the impact of the main input design variables on the variance of the linear thermal transmittance (w value) for several types of thermal bridge. A sensitivity

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analysis, based on an extensive study carried out to calculate the linear thermal transmittance of many types of thermal bridges by means of detailed numerical methods, is presented. The main design variables of interest for the creation of thermal bridge catalogues were investigated, using the finite differences calculation methodology, according to EN ISO 10211. Using the abovementioned methodology, many thermal bridges catalogues have been published till now. For the present study 36 thermal bridges, including the most typical junctions of the building envelope, have been selected from a catalogue of about 100 thermal bridges drawn by the Authors [22]. Having defined the main input variables pertaining to thermal bridges and considering a range of variation that covers most of the current building envelope design solutions for each of them, about 190 different calculated configurations have been considered for each thermal bridge, varying the values of the design variables. An example of a thermal bridge form containing the considered configurations is presented in Fig. 1. From these configurations, for each thermal bridge a multiple non-linear regression model has been identified, considering the output values of the numerical code as the data set for the regression function. In order to perform a sensitivity analysis, a significant number of representative cases (about 1000) have been generated for each thermal bridge using a sampling technique. Each case consists of a set of input variables generated according to a uniform probability distribution. The linear thermal transmittance values of each thermal bridge have been predicted for each case by means of the identified regression equations. A sensitivity analysis of the results has been performed by means of the ANOVA (Analysis of Variance) technique [23], with the aim of assessing the contribution of each input variable to the deviation of the linear thermal transmittance for each analyzed thermal bridge. The aim of the sensitivity analysis is to provide designers and policymakers with a simple and practical way to identify the main important design variables affecting the heat losses through each thermal bridge and to address a more correct application of technical standards and regulations. 2. Calculation of linear thermal transmittance The sensitivity analysis in this paper starts with an extensive study which was carried out to evaluate the linear thermal transmittance of many different types of thermal bridge, with a two dimensional geometric model under steady state conditions, according to EN ISO 10211. The procedure starts with the calculation of the thermal coupling coefficient (L2D), that is the two-dimensional heat transfer coefficient, expressed in W m1 K1, obtained by means of finite difference methods or finite element analysis:

L2D ¼

q ðhi  he Þ

ð1Þ

where q is the heat flow per meter length, while hi and he are the internal and external temperatures, respectively. The linear thermal transmittance of the linear thermal bridge separating the two environments, w, is calculated through the following equation:

w ¼ L2D 

Nj X U j  lj

ð2Þ

j¼1

where Uj is the thermal transmittance of the 1-D component j that separates the considered environments, lj, is the length in the 2-D geometrical model over which the Uj value applies. The portion of the envelope taken into consideration in the calculation of L2D is identified according to the specifications of the standard, and it is extended to a suitable distance from the node under consideration,

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Fig. 1. Example of a thermal bridge form including the considered configurations.

in order to include the entire region in which the heat flow is disrupted from the one-dimensional conditions. The geometric model for the calculation of the heat flow is delimited by shear plains and it is divided into cells by auxiliary plains (Fig. 2). In the present study the system based on external dimensions, measured between the finished external faces of the external elements of the building, has been assumed [8]. 3. Analyzed thermal bridges Different junctions were chosen from the thermal bridge catalogue developed by the Authors [22]. Thirty-six thermal bridges were considered in order to analyze the most typical junctions of a building envelope. The following categories were considered:

wall–roof junctions, wall–balcony junctions, convex and concave corners between walls with and without pillars, wall – intermediate floor junctions, wall – slab on ground floor junctions, wall – window frame junctions, wall – pillar junctions. Three different types of wall insulation were analyzed for each category of thermal bridge: external, intermediate and internal (in the ‘‘Description’’ column of Table 1 a simplified scheme of each analyzed thermal bridge is reported). Since a building could have a load-bearing wall structure rather than a concrete structure, convex and concave corners were considered with and without pillars. Only corners with pillars were chosen for the double brick wall with intermediate insulation, which cannot be a load-bearing wall, considering both insulated (C11 and C19) and non-insulated junctions (C10 and C18).

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been verified through different calculation tests. For example it has been observed that frame U-values ranging from 2.0 to 3.0 W/(m2 K) do not give relevant variations of linear thermal transmittance. 4. Identification of regression models

Fig. 2. Representation of the geometric model by means of shear plains (that delimitate the model) and auxiliary plains (that divide the model into cells).

Three different types of configuration were also considered for double brick wall – pillar junctions: without an insulation layer (P2), with an insulation panel on the external surface (P7) and with insulation surrounding the pillar (P9). Another typical configuration was analyzed for a double brick wall-intermediate floor junction: with insulation on the external part of the floor (IF5) and without insulation (IF2). Two different types of configuration were chosen for the three types of wall – window frame junctions: with an interrupted insulation layer, which means the insulation does not cover the window frame (W7, W8 and W9) and with an insulation layer up to window frame (W12a, W11 and W12). About 190 configurations were calculated under steady-state conditions for each thermal bridge by means of TRISCO simulation code [24], varying the main design variables that influence the linear thermal transmittance value. This code applies the finite difference method to calculate the global heat flows through the thermal bridge and the adjacent walls, according to the procedure specified in EN ISO 10211. About 190 values of linear thermal transmittance were therefore calculated for each selected thermal bridge as a function of the input variables (which differ for each thermal bridge) and of some fixed input parameters and boundary conditions, as reported in Tables 2 and 3. Each input variable was set according to discrete steps within a variation range that includes most of the current building envelope design solutions. The values considered for each input variable in order to evaluate the linear thermal transmittance are reported in the ‘‘Values’’ column of Table 4. For each thermal bridge all the possible combinations of these values for the different variables have been considered. As far as the wall–roof junction is concerned, the temperature reduction factor (btr) was also identified as a design parameter in order to take into account heat losses through unconditioned spaces, according to EN ISO 13789 [25]. In general the design parameters were considered as input variables if their values significantly influence the variation of w values or if their range of variation in current design solutions is relevant. This is for example the case of thermal conductivity of masonry wall, whose value considerably varies from clay blocks to lightweight masonry. Otherwise, other input parameters (e.g. the thermal conductivity of the insulation layer and of the ground, the window frame U-value) were considered fixed as their ranges of variation are very strict and/or their variation has a very low influence on the deviation of the w values. These assumptions have

A non-linear regression model was developed from the linear thermal transmittance values calculated by means of the TRISCO simulation code, for each identified thermal bridge. The statistical model used to fit the linear thermal transmittance values obtained from the computer simulation is based on the extra heat flow with respect to 1-D one, per meter and per temperature difference between the inside and outside environment through the part of the envelope components that constitute the thermal bridge (Fig. 3). The shape of the identified equations was adapted to each thermal bridge on the basis of the thermal characteristics of the bridge (heat flow lines) and the dependent variables. The general form of the equation is:

w¼

c1  e þ c2

Rðbi  Ri Þ þ c3

þ f ðRadj ; eadj Þ

ð3Þ

where c1, c2, c3 and bi are the parameters estimated by the regression analysis. This equation states that the linear thermal transmittance value of each thermal bridge is equal to the ratio between the width of the thermal bridge and the thermal resistance, to which a function of the adjacent length and thermal resistances of the adjacent components is added. The calculations have been limited to adjacent walls according to European Standard EN ISO 10211. A similar approach to identify the shape of equation was adopted by Ben Larbi [21], in which the linear thermal transmittance was expressed as the sum of two terms: the ratio of the thermal bridge width to the total thermal resistance of the bridge (related to the main thermal flow through the thermal bridge) and a linear function of the adjacent wall thermal resistance. In the present study an improved formulation has been proposed, deducing each equation through the observation of the thermal field distortion phenomenon for each node and the associated heat flow lines. Let us examine, for example, the thermal bridge B1, shown in   Fig. 4. The first part of the equation w01 describes the heat flow per unit length that affect the node (in this case, the thermal bridge is represented by balcony slab), which is a function of the thickness of the slab (Sb) and of the relative thermal resistance (Smas + Sins,w)/   kb. The second part of the equation w02 describes the heat flow per unit length in proximity of the node. This flow is a function of the thickness of the wall and the slab and their thermal resistances. In   analogy with the first two parts of the equation, the third part w03 is the heat flow that only affects the wall. Fig. 5 shows the correlation between the linear thermal transmittance values obtained using the detailed method and the values predicted by the identified inverse model for the thermal bridge B1. A non-linear regression model was specifically developed for each analyzed thermal bridge using the above described general equation shape on the basis of the flux flow lines and the dependent variables. The regression equations and the estimated parameters for all the analyzed thermal bridges are reported in Table 1. In the same table the fixed input data are also reported. Table 5 reports the standard deviation and the average values of the linear thermal transmittance calculated using the method adopted in EN ISO 10211 (wcalc) and the expected linear thermal transmittance values obtained from the regression models (wpred),

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Table 1 Identification of analyzed thermal bridges and regression equations. Description

Regression equations

(a) Externally insulated wall

Sins,r

λmas

Sr

wR5 ¼ Smas kr

Sr Sins;w

þ

kins

0:153Smas 0:048Sins;w

þ Smas

Sins;w kmas 0:008 kins þ0:006

þ0:298

mas 2:644Sr Þ 1 mas þ0:183 þ btr þ1:795  ð2:465S þ 0:201S  0:077 Sins;r Sins;r Sr Sr kins

0:068kr þ0:178

kins þ kmas þ1:089

Fixed input data for R5: kr, kins

Smas Sins,w

R5 - Wall – roof junction λ mas Smas S b

wB1 ¼ Smas SbSþ0:130 ins;w kb

þ

kb

mas 1:007Sb Þ þ ð0:785S þ Smas Smas Sins;w

1:368kmas þ

þ0:217

kins

0:504Smas Sins;w kmas þ2:876 kb þ0:046

0:044

 0:189

Fixed input data for B1: kb, kins

Sins,w

B1 - Wall – balcony junction Sins,w

0:327Smur Sins;w kmas þ0:287 kins 0:080

mas wC1 ¼ Smas1:381S Sins;w kmas þ kins

wC14 ¼ Smas

λ mas

kpil

þ Smas

0:097

0:332Smas Sins;w

þ0:188

kins

þ0:108

þ Smas

 0:051

ð2:055Smas Þ

Sins;w kmas þ1:273 kins þ0:305

 0:060

Fixed input data for C1 and C14: kpil, kins

Smas

C1 – C14 - Convex corner between walls / Convex corner between walls with pillar Sins,w

λ mas

Smas

0:071 mas wC5 ¼ 0:0910:697S þ Smas0:705SmasSins;w Smas Sins;w kmas þ kins

0:40

mas wC17 ¼ Smas0:399S Sins;w kpil

þ

kins

kmas þ0:710 kins

þ0:111

0:324

þ 0:019

ð0:105Smas Þ

þ Smas

Sins;w kmas þ0:261 kins þ0:169

þ 0:017

Fixed input data for C5 and C17: kpil, kins

C5 – C17 – Concave corner between walls / Concave corner between walls with pillar λ mas Smas

wIF1 ¼ Smas kfl

Sfl

Sfl þ0:589

0:149Smas 1:850Sfl

þ

Sins;w kins þ0:101

Smas Sins;w kmas þ kins þ0:267

Fixed input data for IF1: kfl, kins

Sins,w

IF1 - Wall – intermediate floor junction Sins,w Sfl

λ mas Smas Sins,fl

wGF5 ¼

S

S

3:056Smas

1:811 kins;fl þ fl þ3:561 mas k fl

Sins;w Smas kgr þ3:118kmas þ0:504

þ

ð0:296Þ S

fl

0

Fixed input data for GF5: kfl, kins,,kgr, B

GF5 - Junction wall – slab-onground floor Sins,w

λ mas

Smas

wW7 ¼ SmasSmas

kmas þ0:167

wW12A ¼

þ

ð0:507Smas 0:134Þ Sins;w kins 0:018

1:546Skmas þ mas

0:445Smas þ0:030 Smas kmas þ0:270

þ

ð0:271Smas 0:241Þ 1:330Skmas þ mas

Sins;w kins þ1:025

þ 0:059

Fixed input data for W7 and W12a:kins, Ufr

W7 - W12a - Junction wall – window frame with continuous insulation Sins,w Smas

λ mas Spil

P5 - Wall –pillar junction

wP5 ¼ Smas kpil

Spil þ0:864

Sins;w kins þ0:166

þ Smas

ð0:838Spil Þ

Sins;w kmas þ0:746 kins þ0:049

Fixed input data for P5:kpil, kins

Sins;fl

0:159kfl þ

kins

1:783B0 þ0:223B02 þ3:052

þ

ð1:007Sfl 2:357Smas þ0:341Þ Sins;w

þ 2:729kSmas mas

kins

þ0:090

 0:048

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A. Capozzoli et al. / Applied Energy 107 (2013) 229–243 Table 1 (continued) Description

Regression equations

(b) Wall with intermediate insulation

Sins,r

λ mas

wR6 ¼

Sr

SM1 S

S

M1 þ ins;r þ2:329 0:589Skrr þ0:553kmas kmas

þ

SM1 kr

8:306Sins;r 14:444Sins;w 7:035SM1 5:275SM2 þ1:72Sr þ3:941 S

S

M1 þ M2 þ 0:415kmas kmas

Sins;w kins þ0:461

1  btr þ3:719 þ Sins;r kins

ð2:676Þ 0:652kSrr þ0:469

1  btr þ1:821

Fixed input data for R6: kr, kins

SM1 SM2 Sins,w

R6 - Wall – roof junction λ mas

b wB2 ¼ SM1 þSins;w þSM2Sþ0:785S b kb

Sins,w S b

þ0:048

0:334SM2 1:035Sb Þ þ ð0:333SM1 þ Sins;w S M1 þSM2 þ

kins

0:339SM1 þ0:339SM2 0:161Sins;w þ0:199Sb SM1 þSM2 Sins;w 0:186Sb þ k þ k þ0:1 kmas b b

kmas

Fixed input data for B2: kb, kins

SM1 SM2

B2 - Wall – balcony junction Sins,w

wC10 ¼ wC11 ¼

λ mas SM1 SM2

6:061SM1 Sins;w SM1 SM2 SM2 kpil þ3:274 kpil þ kpil þ0:795kmas þ0:445

S

M1 þ 3:451kmas

SM1 Sins;w S SM1 SM2 SM2 M1 kmas þ0:31 kpil þ kpil 0:31 kpil þ0:205kmas þ0:057

þ

M1 1:388SM2 0:114Þ þ ð2:182S þ 0:158 Sins;w SM1 þSM2 kins

þ1:117

kmas

0:089

ð2:279SM1 1:771SM2 2:497Sins;w 0:321Þ Sins;w S SM2 M1 kins þkmas þkmas þ0:28

þ 0:265

Fixed input data for C10 and C11: kpil, kins

C10 – C11 - Convex corner between walls with pillar Sins,w

wC18 ¼

λ mas

SM1 SM2

wC19 ¼

1:964SM1 þ0:882SM2 Sins;w S SM2 SM2 M1 kpil þ kpil þ1:725 kpil þ1:108kmas þ0:087

S

M1 þ 0:814kmas

SM1 Sins;w S SM1 SM2 SM2 M1 kmas þ0:34 kpil þ kpil 0:265 kpil þ0:082kmas þ0:057

0:74SM1 0:215 þ Sins;w0:099SM2SM1 SM2 kins

þ

þ0:569kmas þ2:622kmas þ0:196

þ 0:324

0:24SM2 0:051SM1 0:889Sins;w 0:637 Sins;w S SM2 M1 kins þkmas þkmas þ1:034

þ 0:334

Fixed input data for C18 and C19: kpil, kins

C18 – C19 - Concave corner between walls with pillar λ mas

S þ0:075

wIF2 ¼

Sins,w Sfl

0:847

fl SM1 þSM2 Sins;w þ k þ0:137 kfl fl

wIF5 ¼ SM1 1:016SM2 kfl

ð2:396Sfl 0:214Þ

þ Sins;w kins

SM1 þSM2 þ0:679 kmas

þ

Sfl Sins;w SM2 kfl þ0:179kmas þ0:564

þ

þ Sins;w kins

M1 þ0:311SM2 þ0:019 þ 0:368S  0:166 Sins;w SM1 þSM2

þ2:174

kmas

ð1:353Sfl Þ SM1 þSM2 0:168 kmas

þ

þ0:043

kfl

0:151SM1 þ0:086SM2þ0:054

þ

SM2 kmas Sins;w þ kfl þ0:037

S

M1 þ0:354 0:63kmas

 0:004

Fixed input data for IF2 and IF5: kfl, kins

SM1 SM2

IF2 – IF5 - Wall – intermediate floor junction λ mas

Sins,w SM2 Sfl

wGF6 ¼

SM1 Sins,fl

SM1 SM1 SM1 þSins;w þSM2 kfl Sfl kmas þ þ kfl þ0:318

þ

ð1:253SM1 0:693SM2Þ SM1 SM2 kmas Sins;w kins 0:03 kmas þ þ

þ Sins;fl kins

0:6440:016B0 S

1:2kfl 1:069B0 þ11:573

0:373Sfl þ2:938 SM1 þ13:652SM2þS kmas ins;fl Sfl ins;w þ15:306 4:153  kins þkfl þ

þS

fl

 0:196

0

Fixed input data for GF6: kfl, kins, kgr, B

GF6 - Junction wall – slab-onground floor SM2 Sins,w

wW8 ¼

0:523SM1 þ0:602SM2 þ0:181Sins;w þ0:057 S SM1 þSM2 þ0:015 ins;w kmas kins

wW11 ¼

λ mas

þ 0:116SM10:012 þSins;w þ0:054SM2  0:012 kmas

0:178Sins;w 0:137SM1 0:142SM2 0:058 S SM1 þSM2 þ0:174 ins;w kins þ1:169 kmas

þ

0:024Sins;w þ0:427SM1 þ0:441SM2 1:693

Sins;w kins

SM1 þSM2 kmas

þ

þ 0:022

Fixed input data for W8 and W11: kins, Ufr

SM1

W8 - W11 - Junction wall – window frame with interrupted insulation/ with continuous insulation S

λ mas Spil

Sins,w SM2 SM1

P2 - P7 - P9 - Wall –pillar junction

pil wP2 ¼ SM1 þSM2 þSins;w kpil

þ0:158

þ

ð1:028Spil 0:383SM1 0:299SM2 0:396Sins;w Þ SM1 þSM2 Sins;w þk kmas ins

Spil

wP7 ¼ SM1 1:019SM2 þSins;w kpil

wP9 ¼

ð1:457Spil Þ

S

M2 þ0:713 þ0:148kmas

ð1:324Spil 0:494SM2 Þ SM1 þSM2 Sins;w þ k þ0:261 kmas ins

þ 1:348SM1 þ0:809SM2 kmas

þ

Sins;w kins

þ

0:49SM1 þ0:285SM2 0:16Sins;w þ0:058 SM1 þSM2 Sins;w þ k þ0:157 kmas pil

0:321SM1 þ0:177SM2 þ0:075 þ 0:96S  0:031 Sins;w M1 þ0:369SM2 kmas

þ

kpil

þ0:068

Spil þ0:713SM2

þ SM1 þSins;w kpil

S

M2 þ0:85 þ0:31kmas

Fixed input data for P2, P7 and P9: kins, kpil (continued on next page)

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A. Capozzoli et al. / Applied Energy 107 (2013) 229–243

Table 1 (continued) Description

Regression equations

(c) Internally insulated wall

Sins,r

λ mas

wR7 ¼

Sr

Sr Sins;w Smas kr þ0:892 kr þ0:086

0:712kSrr þ

0:12Smas 0:021Sins;w

þ Smas

Sins;w kmas þ0:006 kins þ0:018

mas 0:336Sr Þ þ b1tr  ð0:123S þ btr  Sins;r Sr kr þ0:324 kins

Fixed input data for R7: kr, kins

Smas Sins,w

R7 - Wall – roof junction λ mas

wB3 ¼ Smas SbSþ0:136 ins;w þ

kb

Smas S b

þ0:233

kb

0:413Smas Sins;w kmas þ0:056 kins þ0:035

þ Smas

þ

ð0:014Smas 0:02Sb Þ Sins;w kb 0:188

0:029kSmas þ mas

Fixed input data for B3: kb, kins

Sins,w

B3 - Wall – balcony junction Sins,w

wC3 ¼

1:539Smas 0:954Sins;w Sins;w Smas kmas þ0:713 kins þ0:167

1:93Smas Sins;w kmas þ0:75 kins þ0:145

wC9 ¼ Smas

λ mas

þ

0:216Smas 0:554Sins;w 0:421

Sins;w S mas kins þ kpil þ0:038

Fixed input data for C3 and C9: kpil, kins

Smas

C3 – C9 - Convex corner between walls / Convex corner between walls with pillar Sins,w

wC7 ¼ Smas

Smas

wC22 ¼ Smas

0:458Smas Sins;w kmas þ0:234 kins þ0:02

λ mas

0:54Smas Sins;w kmas þ0:581 kins þ0:175

þ

0:237Smas þ0:114Sins;w 0:215

Sins;w S mas kins þ kpil þ0:05

Fixed input data for C7 and C22: kpil, kins

C7 – C22 Concave corner between walls / Concave corner between walls with pillar λ mas Smas

wIF3 ¼ Smas kfl

Sfl

Sfl þ0:086 þ

Sins;w kfl þ0:184

þ

0:27Smas 0:84Sfl Smas Sins;w kmas þ kins 0:082

þ Smas0:263SmasSins;w  0:127 kmas þ1:825 kfl

Fixed input data for IF3: kfl, kins

Sins,w

IF3 - Wall – intermediate floor junction Smas Sfl

λ mas Sins,w Sins,fl

wGF7 ¼

1:008Smas 0:972Sfl S

Sins;fl kins þ0:664

1:594kfl þ fl

0:113Smas þ0:082Sfl

þ Sins;w kins

0:057Skmas þ0:563 mas

 0:055

Fixed input data for GF7: kfl, kins, kgr, B0

GF7 - Junction wall – slab-on-ground floor λ mas

Smas

wW9 ¼

0:475Smas 0:134

Sins;w kins þ0:478

mas þ 1:219Skmas

kmas þ0:292

mas 0:241 mas þ0:024 wW12 ¼ 0:127S þ 0:414S þ 0:055 Smas Smas Sins;w kmas þ kins

Sins,w

þ SmasSmas

kmas þ0:448

þ1:412

Fixed input data for W9 and W12:kins, Ufr

W9 - W12 - Junction wall – window frame with interrupted insulation/ Junction wall – window frame with continuous insulation λ mas Spil

Sins,w Smas

P6 - Wall –pillar junction

wP6 ¼ Smas kpil

Spil þ0:04 þ

Sins;w kins þ0:202

0:706Spil

þ Smas

Sins;w kmas þ0:655 kins

Fixed input data for P6:kpil, kins

þ0:029

ð0:036Sr Þ Sins;w Sr kr 0:007kr

0:026Skmas þ mas

 0:063

237

A. Capozzoli et al. / Applied Energy 107 (2013) 229–243 Table 2 Fixed input data used for the calculation of the linear thermal transmittance. Parameter

Symbol

Value

Unit

Insulation thermal conductivity Concrete thermal conductivity of the floor, roof, balcony or pillar Ground thermal conductivity Thermal transmittance of the window frame Thickness of the window frame Characteristic dimension of the floor

kins kfl, kr, kb, kpil kgr Ufr Sfr B0

0.04 2.0 2.0 2.0 0.06 2.0

W m1 K1 W m1 K1 W m1 K1 W m2 K1 m m

Table 3 Fixed boundary conditions used to calculate the linear thermal transmittance. Parameter

Symbol

Value

Unit

External surface thermal resistance Internal surface thermal resistance Indoor temperature Outdoor temperature

Rse Rsi hi he

0.04 0.13 20.0 0.0

m2 K W1 m2 K W1 °C °C

as well as of the residual values of all the considered thermal bridges. The relevant correlation coefficient adjusted R-squared is also reported for each thermal bridge. Examining these indices, it is possible to appreciate that the models developed are well able to represent the 2D-w values. Only the thermal bridges in the case of externally insulated wall are reported in Table 5. However, similar results were obtained for the inverse models developed to fit the 2D-w values for the other thermal bridges considered in this paper (intermediate and internal insulation). The adjusted R-squared of the identified regression models are on average very close to 100% for all the analyzed thermal bridges. The proposed models are a simple way to obtain w values with relative errors very similar to those obtained by numerical calculations.

5. Sensitivity analysis A sensitivity analysis was conducted by generating a random sample of the possible combinations of physical parameters that determine the linear thermal transmittance value for each analyzed thermal bridge. The sample was generated and the sensitivity analysis was conducted using the ANOVA–FAST technique [23,26,27] by means of SimLab 2.2 code [28]. The FAST method is based on a transformation that converts the variance of a variable Y, which is a k-dimensional integral, to a single dimensional integral with respect to a scalar variable s, by transforming each input factor Xi into the form Xi = Gi(sin (xis)). The different sensitivity indexes can be evaluated for an appropriate

set of transformations functions Gi and integer frequencies xi using the Monte Carlo method [27]. About 1000 cases were generated for each thermal bridge using a combination of input variables and considering a variation for each of them within the range specified in the ‘‘Range’’ column of Table 4. The linear thermal transmittance values of the thermal bridges were identified for each case through the developed regression models. All the combinations of the input variables were considered excluding only the cases for which the resulting envelope thermal transmittance was out of the validity range. The ANOVA technique is used to split the variance of the output variable between the different input variables. The variance is a measure of the dispersion of the output variable, which in our case, is represented by the linear thermal transmittance of each thermal bridge. Using this technique, it is then possible to evaluate the contribution of each input variable to the variability of the linear thermal transmittance from a quantitative point of view. Furthermore, using the ANOVA method, it is possible to evaluate the effect of the interactions between the input variables on the variability of the output. Consequently, adopting this methodology, it has been possible to carry out an analysis with the aim of assessing the relevance of each input design variable. The technique is based on the decomposition of the model variance into:  A first order sensitivity index, SIi, caused by each input factor.  A second order sensitivity index, SIi,j caused by the interactions between any pair of input factors i and j (–i), which cannot be explained by the sum of the individual effects due to the two factors.  Analogously, higher order sensitivity indexes (SI1. . .i. . .k) caused by the interactions between more than two factors. A total sensitivity index, SITi, can be defined for each factor, which is the sum of the first order sensitivity index of the factor under investigation and the higher sensitivity indices involving this factor. The first order sensitivity index related to each input factor represents the impact of the variation of that factor on the variance of the linear thermal transmittance. Some of the effects

Table 4 Input design variables: values considered to calculate the linear thermal transmittance and ranges considered to generate the sample. Variable Thickness Thickness Thickness Thickness

of of of of

the the the the

intermediate floor, roof or balcony pillar masonry masonries for a double brick wall with intermediate insulation

Thickness of the floor or roof insulation ? Thermal transmittance of the floor or roof Wall insulation thickness ? Wall thermal transmittance Thermal conductivity of the masonry Temperature reduction factor

Symbol

Values

Range

Unit

Sfl = Sr = Sb Spil Smas SM1 SM2 Sins,fl, Sins,r Ufl, Ur Sins,w Uw kmas btr

[0.15, 0.20, 0.25] [0.30, 0.40] [0.20, 0.30, 0.40] [0.10, 0.20, 0.25, 0.30] [0.10, 0.20, 0.25, 0.30]

[0.15–0.25] [0.30–0.40] [0.20–0.40] [0.1–0.3] [0.1–0.3] [0.0453–0.3902] ? [0.10–0.70] [0.0023–0.3843] ? [0.10–0.70] [0.25–0.90] [0.50–1.0]

m m m m

[0.10, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70] [0.10, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70] [0.25, 0.50, 0.90] [0.50, 0.65, 0.80, 1.0]

m W m2 K1 m W m2 K1 W m1 K1 –

238

A. Capozzoli et al. / Applied Energy 107 (2013) 229–243

Fig. 3. Geometrical model of a 2D thermal bridge.

Fig. 4. Geometrical model of thermal bridge B1 and explanation of the regression equation.

Table 5 Statistical indices for the calculated and predicted linear thermal transmittance referring to the thermal bridges in the case of the externally insulated wall. Thermal bridge code

B1 C1 C5 W7 R5 GF5 C14 C17 W12A P5

Mean value (W/(m K))

Standard deviation (W/(m K))

Adjusted R2

wcalc

wpred

Residuals

wcalc

wpred

Residuals

0.637 0.147 0.040 0.301 0.063 0.009 0.090 0.067 0.166 0.077

0.630 0.147 0.039 0.300 0.063 0.008 0.090 0.067 0.167 0.077

0.000 0.000 0.000 0.001 0.000 0.000 0.000 0.000 0.000 0.000

0.136 0.078 0.020 0.154 0.068 0.108 0.042 0.065 0.069 0.128

0.137 0.078 0.021 0.154 0.067 0.108 0.041 0.065 0.069 0.128

0.008 0.005 0.002 0.008 0.006 0.008 0.003 0.004 0.006 0.005

on the output variable are due to the interactions between the input variables and are therefore not explained by the first order index. The share allocated to the interaction item represents the portion of the variance of the linear thermal transmittance that

0.997 0.997 0.989 0.998 0.994 0.995 0.995 0.995 0.992 0.998

is not due to linear effects and which can therefore be interpreted as a measure of the non-linearity of the model. The sensitivity analysis was performed for each thermal bridge according to the following phases:

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A. Capozzoli et al. / Applied Energy 107 (2013) 229–243

1.00

(W m -1 K-1)

Calculated linear thermal transmittance

2

R = 0.997 1.20

0.80 0.60 0.40 0.20 0.00

0

0.2

0.4

0.6

0.8

1 -1

1.2

-1

Predicted linear thermal transmittance (W m K ) Fig. 5. Correlation between the calculated linear thermal transmittance and that predicted by the regression model. Thermal bridge B1.

Table 6 First-order sensitivity indices – linear thermal transmittance.

 Definition of the range of each variable to cover most of the current building envelope design solutions, considering the same probability of occurrence for each selected amplitude.  Definition of a probability density function for the input variables, choosing a uniform function in the range of variability considered in the ‘‘Range’’ column of Table 4 for which the

validity of each model regression was analyzed; the uniform distribution is a bounded continuous distribution over an interval [a, b] so that each variable, with a value between the bounds, has an equal probability.  Definition of fixed input parameters and boundary conditions during the simulations (see Tables 2 and 3).

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 Generation of a matrix whose number of rows (n) corresponds the number of cases generated while the number of columns (k) corresponds to the number of input variables.  Application of the developed regression model (for each thermal bridge) to generate an output vector of the linear thermal transmittance values with a number of rows equal to n.  Evaluation of the first order sensitivity indices for each input variable using the ANOVA–FAST analysis.  Evaluation of the total order sensitivity indices for each input variable using the ANOVA–FAST analysis. 6. Results and discussion The results relative to the sensitivity analysis are presented in this section. The weight that the uncertainty of each input variable has on the uncertainty of the output variable (linear thermal transmittance w) has been evaluated for each type of thermal bridge. The first and total order indices of each variable are reported in Tables 6 and 7 for all the thermal bridges analyzed with the ANOVA–FAST technique using the developed inverse models to generate the output vector. In Table 6 the first-order sensitivity indices related to each thermal bridge higher than 0.20 have been highlighted, while in Table 7 total order sensitivity indices related to each thermal bridge higher than 0.25 have been highlighted. These values have been conventionally selected to exclude those

Table 7 Total sensitivity indices – linear thermal transmittance.

variables that can be considered irrelevant according to literature indications [27]. For the same thermal bridges, the most effective variables are also illustrated in Table 8 with different colors in order to better identify the ranking evaluated by the total order sensitivity indices analysis. This Table can be useful for the designers and policymakers, because it identifies for each thermal bridge the design variables that need more attention for their effect on thermal bridging and on building heat losses and energy performance. As far as the junction between the wall and roof, with both an external and intermediate insulated wall (R5 and R6) is concerned, the thermal insulation thickness of the roof and the wall, and hence their relative thermal resistance, have the most effect on the variance of the linear thermal transmittance. In the case of internal insulation (R7), the thermal insulation roof and the roof thickness have an important impact. The variables that have the greatest weight in the wall – balcony junction (B1, B2 and B3) are the balcony thickness and subsequently the thermal resistance of the massive part of the floor, as well as the thermal insulation thickness of the wall. Moreover the internal masonry part of the wall (SM1) has an important impact for B2. The most important variable for the convex and concave corners between the walls, with and without pillar (C1, C5, C14, C17, C10, C11, C3, C9, C7, and C22) is the insulation thickness of the wall and hence the thermal resistance insulation layer. The

Table 8 Identification of thermal bridges most effective variables (in red, orange and yellow colors are shown the first, the second and the third most important variables respectively, with relation to total sensitivity indices). Wall–roof junction

Sins,fl

Sins,fl

Sins,fl

Sins,w

Sins,w

Sfl

R6

R7

R5

SM1

Wall – balcony junction

Sfl

Sins,w

Sfl

Sins,w

Sfl

Sins,w B2

B3

Sins,w

Sins,w

Sins,w

Sins,w

Sins,w

Sins,w

Sins,w

C7 – C22

C18 – C19

Sins,w

Sins,w

Sins,w

SM1

λ mas

Sfl

IF2 – IF5

IF1 Junction wall – slab-on-ground floor

Sins,w

Sins,w

C5 – C17 Wall – intermediate floor junction

C3 – C9

λ mas

λ mas

IF3

λ mas Sins,w

Sins,fl

λmas GF5

Sins,w

Sins,w

C10 – C11

C1 – C14 Concave corner between walls/concave corner between walls with pillar

Sins,w

Sins,w

A. Capozzoli et al. / Applied Energy 107 (2013) 229–243

B1 Convex corner between walls/convex corner between walls with pillar

GF6

GF7 (continued on next page) 241

A. Capozzoli et al. / Applied Energy 107 (2013) 229–243

P2 - P7 - P9

Sins,w

P5

λ mas Sins,w

P6

Sins,w Sins,w

W8 - W11 Spil Sins,w Wall –pillar junction

W7 - W12a

λ mas λ mas

Sins,w

Sins,w Sins,w Junction wall – window frame with interrupted insulation/junction wall – window frame with continuous insulation

Table 8 (continued)

λ mas

Sins,w λ mas

W9 - W12

Sins,w λ mas

242

thermal conductivity of the masonry has the most important impact on the variance of linear thermal transmittance in the concave corner between the walls with a pillar (C18 and C19). The thickness of insulation layer is also non-negligible for thermal bridge C19. The most important variable for the junction wall – insulated pillar with an externally and internally insulated wall (P5 and P6) is the insulation thickness of the wall and hence the thermal resistance insulation layer. The most significant variable for the case of a double brick wall with intermediate insulation in the thermal bridge P2 is the thickness of the pillar and of the insulation layer. When the same junction is thermally corrected with an external insulation panel on the pillar (P7), the thermal conductivity of the masonry becomes the most important variable followed by the insulation layer. The thermal bridge P9 characterized by a correction all around the pillar with a continuous insulation layer is almost exclusively influenced by the wall insulation layer. The most important variable for the wall – intermediate floor junction in the case of an externally insulated wall (IF1) is the insulation thickness of the wall. The thickness of the internal masonry part of the wall (SM1) also has a large impact on the variance of linear thermal transmittance for thermal bridge IF2 as well as the wall insulation layer; in the case of the junction between the wall and the intermediate floor with a correction (IF5), the most significant variables are the conductivity of the masonry and the wall insulation layer. In the case of an internally insulated wall (IF3), the variables which show the greatest impact are the thickness of the insulation layer and the thickness of the floor. The variables that have the greatest impact on the variance of the linear thermal transmittance for the wall–slab-on-the ground floor thermal bridge, in the case of an externally insulated wall and double brick masonry, with intermediate insulation (GF5 and GF6), are the insulation thickness of the wall and the thermal conductivity of the masonry. In the case of an internally insulated wall (GF7), the most important variable is the insulation thickness of the slab on the ground floor. The variables that have most impact on the junction wall – window frame, in the case of an externally and intermediate insulated wall (W9, W12, W12a, and W7) are the insulation thickness of the wall and the thermal conductivity of the masonry. In thermal bridge W8 (wall – window frame with intermediate interrupted insulation), the most significant variable is the thermal conductivity of the masonry; the only significant variable in the case of intermediate continuous insulation (W11) is the insulation thickness of the wall. In the analysis carried out the thermal transmittance of the window frame has been considered constant as it has been found that it plays a negligible role on the linear thermal transmittance variation. For this reason the regression equations to calculate the linear thermal transmittance of the junction wall–window frame would be also valid for any frame in the range from 2 to 3 W m2 K1. The results in general show that in the case of an externally and internally insulated wall, when the insulation layer is continuous at the junction, the most important design variable is the insulation thickness of the wall, while, in the case of interrupted insulation, the variables that affect the thermal resistance of a non-insulated junction (thermal conductivity of masonry, floor or roof thickness, wall thickness) are also important. Moreover, among the variables that affect the thermal resistance of a wall junction, masonry conductivity has more effect than thickness on the linear thermal transmittance variation. When the node is represented by a floor or a roof, the thickness is more important, because the concrete thermal conductivity is constant. In the case of cavity insulation, since there is no continuous insulation, the variance of the linear thermal transmittance is also influenced by the thermal resistance of the junction and, above all, by the thermal conductivity of the wall masonry.

A. Capozzoli et al. / Applied Energy 107 (2013) 229–243

A thermal correction in a junction generally induces a change in the most important variables affecting the variance of linear thermal transmittance. In fact for thermal bridges C18–C19, IF2– IF5, P2–P7–P9 the thermal correction influences the thermal field distortion phenomenon and the associated heat flow lines. It is important to underline that the results of the sensitivity analysis are affected by the relative amplitude of the selected ranges for each input variable. For this reason the set of the most typical design envelope solutions have been taken into account to select the input variable range. 7. Conclusion The paper presents a sensitivity analysis carried out on thirtysix thermal bridges, including the most typical junctions of the building envelope. The adopted methodology has allowed non-linear regression models to be identified as a function of the design variables of each thermal bridge, on the basis of linear thermal transmittance values calculated using a numerical method according to EN ISO 10211. The results show that the statistics of the identified regression models are very good with an adjusted Rsquared on average very close to 100% for all the analyzed thermal bridges. Under the validity range of each variable, all the proposed models generally show relative errors very similar to those obtained by numerical calculations. The equations can be used by designers and practitioners for their reliability and simplicity compared to numerical methods with a calculation error lower than using default w values provided by EN ISO 14683. The variance of the linear thermal transmittance has been decomposed through the ANOVA–FAST analysis into the input variables considered for each type of thermal bridge. The impact of each design variable on the variance of the linear thermal transmittance through the calculation of the first and total order indices has been evaluated. It has been observed in a large number of simulations that the linear thermal transmittance values vary significantly for changes in the considered design variables. For almost each analyzed thermal bridge, the thickness of the insulation layer is one of the most important variables affecting the deviation of linear thermal transmittance. Whenever the insulation layer does not cover the junction, the thermal resistance of the non-insulated junction is also significant. Moreover, among the variables affecting the thermal resistance of a wall junction, the masonry thermal conductivity has a greater effect on the linear thermal transmittance variation than its thickness. A thermal correction of a junction generally determines a change in the most w value influencing design variables. Since a variation of the linear thermal transmittance affects the building heating energy need, the performed sensitivity analysis has allowed the design variables that have the highest influence on building energy performance to be identified. Moreover, to this purpose a Table has been presented as a practical and useful tool for designers and policymakers. Acknowledgement This work has been carried out as part of a research activity supported by Edilclima S.p.A. concerning the ‘‘Development and

243

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