Statistical Modelling Of Heat Transfer For Thermal Bridges

Energy and Buildings 37 (2005) 945–951 www.elsevier.com/locate/enbuild

Statistical modelling of heat transfer for thermal bridges of buildings A. Ben Larbi * Centre Scientifique et Technique du Baˆtiment, 4 avenue du Recteur Poincare´, F-75782 Paris Cedex 16, France Received 10 June 2004; received in revised form 19 September 2004; accepted 2 December 2004

Abstract In this paper, we develop statistical models of the thermal transmittance of 2D thermal bridges (c). We aim to give building designers plain and practical tools for the evaluation of the most common 2D thermal bridges. Three examples are considered: a slab-on-grade floor–wall junction, a floor–wall junction and a roof–wall junction. For each case, we perform computer simulations of the thermal transmittance, c, for several values of the most important variables. Then, we fit these numerical results to a nonlinear regression model. The results appear to be good with relative errors less than 5%. # 2005 Elsevier B.V. All rights reserved. Keywords: Heat transfer; Thermal bridge; Building; Computer simulation; Statistical modelling

1. Introduction Thermal bridges are parts of the building envelope where the otherwise uniform thermal resistance are significantly changed (e.g. at structural joints with roofs, floors, ceilings, and other walls, or other building envelope details such as corners, window or door openings) [1], resulting in a multidimensional heat flow. They have a major effect on the thermal performance of the building envelope, significantly increasing winter heat loss and summer heat gain. The better walls are insulated, the more thermal bridges amount in the overall envelope heat losses. The temperature of the inside surface over a thermal bridge is lower than that of the adjacent construction during the heating season. The difference in the temperature gradient may cause condensation and mould growth (so reducing the indoor air quality). There are two types of thermal bridges. The linear or 2D ones are situated at the junction of two or more building elements and they are characterised by a linear thermal transmittance (or c-value in W/m K). The point or 3D ones lie where an insulated wall is perforated by an element with high thermal conductivity or where there are threedimensional corners and they are characterised by a point * Tel.: +33 1 30 85 20 93; fax: +33 1 30 85 21 34. E-mail address: [email protected] 0378-7788/$ – see front matter # 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.enbuild.2004.12.013

thermal transmittance (or x-value in W/K). In most cases evaluations will be limited to linear thermal bridges, most common; 3D calculations will be exceptional. The evaluation of thermal bridges can be done experimentally by using standardised test methods on two identical building elements, the first one with and the second one without a thermal bridge. This method is limited to those building elements that can be so tested. Thus, the accuracy of the assessment is rather uncertain, it is time-consuming, expensive and laborious, only applicable for important projects or for checking the reliability of simulation calculations [2]. Thermal bridges can therefore be evaluated by using numerical methods. A lot of software helps for it [3–5]. However they need minimum skills and some care for defining the boundary conditions. Catalogues [6] give several examples of thermal bridges for fixed parameters (e.g. dimensions and kind of material). So they are less flexible than calculations. In France, like in many European countries, tabulated values of c are given for typical cases [7]. But they often do not match with the actual details of any building project. In this paper, we develop analytical formulas for the thermal transmittance of three different 2D thermal bridges. We begin with a numerical computerization of the thermal transmittance for a limited number of values for the most important variables that influence the c-value. Then we fit the numerical results to a regression model that

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A.B. Larbi / Energy and Buildings 37 (2005) 945–951

Nomenclature adj e em ep In. L Out. R Riv Rsp DT U

adjacent width of a thermal bridge (m) width of a wall (cm) width of a roof (cm) inside length of an adjacent wall (m) outside thermal resistance (m2 K/W) thermal resistance of wall insulation (m2 K/ W) thermal resistance of roof insulation (m2 K/ W) temperature difference between the inside and outside environment (K) thermal transmittance (W/m2 K)

Fig. 1. Geometrical model of a 2D thermal bridge.

The linear thermal transmittance is given by c¼

Greek letters x point thermal transmittance (W/K) Fg global heat flow per metre length (W/m) c linear thermal transmittance (W/m K)

N Fg X  U i Li DT i¼1

ðW=mKÞ

(1)

where Fg is the global heat flow per unit of length (W/m), Ui the thermal transmittance of the 1D component i separating the two environments (W/m2 K), Li the length within the 2D geometrical model over which the value Ui applies (m), DT the temperature difference between the inside and outside environment (K) and N the number of 1D components.

Subscripts adj adjacent

2.1. Hypotheses of calculation

gives an analytical formula for a wider range of configurations. By doing so, we aim to give the building actors (project manager, designer, engineer, etc.) plain and practical tools for evaluating the most important 2D thermal bridges.

The calculation hypotheses are given in the following tables for the boundary conditions (Table 1) and the thermal conductivities (Table 2).

3. Statistical modelling of the heat transfer through a thermal bridge

2. Numerical calculations of the heat transfer through a 2D thermal bridge

The statistical model used to fit the values of a linear thermal transmittance obtained from the computer simulation is based on the heat flow per metre through the part of the wall that constitutes the thermal bridge (Fig. 2), given by the relation:

Numerical calculations of the heat transfer through a thermal bridge require the use of methods with numerical resolution like finite element or finite difference methods. The European Standard EN ISO 10211-2 [8] describes the calculus method for linear thermal bridges and superficial temperatures. The software used here, BISCO [4], refers to the finite difference method in order to calculate the global heat flows through the thermal bridge and the adjacent walls under steady-state conditions. Fig. 1 shows a geometrical model of 2D thermal bridge.

F ¼ eUDT

ðW=mÞ

(2)

where e is the width of the thermal bridge (m), U the thermal transmittance of the portion of the wall that constitutes the thermal bridge (W/m2 K), and DT the temperature difference between the inside and outside environment (K).

Table 1 Boundary conditions Title

Internal External

Surface heat transfer coefficient (W/m2 K)

Surface thermal resistance (m2 K/W)

Horizontal flow

Upward flow

Downward flow

Horizontal flow

Upward flow

Downward flow

7.69 25

10 25

5.88 25

0.13 0.04

0.10 0.04

0.17 0.04

Temperature (8C)

20 0

A.B. Larbi / Energy and Buildings 37 (2005) 945–951 Table 2 Thermal conductivity of materials Material

Conductivity (W/m K)

Insulation Concrete Masonry

0.04 2.00 0.70

Eq. (5) is adapted to each case according to the thermal characteristics of the bridge (flow lines) and the dependent variables.

4. Results and discussion

However, the heat flow through the thermal bridge also depends on the insulation of the building envelope. The more the envelope is insulated, the higher the heat flows through the thermal bridge are. Following the European Standard EN ISO 10211-1 [1], we limit the calculations to adjacent walls. In order to take into account the effect of the adjacent walls, we add a function of the thermal resistance of the adjacent walls (Radj) in Eq. (2): F ¼ eUDT þ f ðRadj Þ ðW=mÞ

f ðRadj Þ F ¼ eU þ DT DT

ðW=mKÞ

(4)

By assuming f (Radj) as a linear function of the adjacent walls thermal resistance: c¼P

c1  e þ c3 Radj1 þ c4 Radj2 þ cn c i 2  Ri þ cd

4.1. Computer simulations This section presents numerical results for three thermal bridges, which amounts for the most of the overall envelope heat loss. For each one, we give on one side the thermal characteristics (flow lines) and on the other the c-value (W/ m K) resulting from the numerical calculations. For each thermal bridge two situations have been simulated; the exterior wall, with internal insulation, is made of either concrete or masonry. In all cases the floor is made of concrete.

(3)

The linear thermal transmittance is obtained by dividing the heat flow by the temperature difference between the inside and outside environment (DT): c¼

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4.1.1. Slab-on-grade floor–wall junction Figs. 3 and 4 show the cross section of the 2D thermal bridge and the flow lines, respectively. In this case the cvalue depend on the floor thickness (ep), the wall and floor insulations (Riv and Rsp). As shown in Fig. 4 the heat flow lines concentrate on the floor width, thus the thermal bridge width is proportional to the wall width. The computerized c-values for a junction with a concrete wall (resp. a masonry wall) are given in Table 3 (resp. Table 4). The foundation is made of concrete.

(5)

P where e is the thermal bridge width, i Ri the total thermal resistance of the bridge, Radj1, Radj2 are the insulation resistances of the adjacent walls and c1, c2, c3, c4, cd, cn are the estimated coefficients.

4.1.2. Floor–wall junction Figs. 5 and 6 show the cross section of the 2D thermal bridge and the flow lines, respectively. The c-value depends on the floor thickness (ep), the wall width (em) and the wall insulation (Riv). The thermal bridge width is proportional to the floor thickness where the flux lines concentrate (Fig. 6).

Fig. 2. Modelling of heat transfer through a 2D thermal bridge.

Fig. 3. Cross-section.

ðW=mKÞ

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A.B. Larbi / Energy and Buildings 37 (2005) 945–951

Fig. 4. Flow lines.

Fig. 5. Cross-section.

Table 3 The c-values for a slab-on-grade floor–wall junction (exterior wall made in concrete) Riv

Rsp (m2 K/W) 1.5

1.5 2.5 3.5

2.5

3.5

15

20

25

15

20

25

15

20

25

0.54 0.51 0.48

0.62 0.59 0.56

0.69 0.66 0.63

0.57 0.54 0.50

0.65 0.62 0.58

0.72 0.69 0.65

0.59 0.56 0.52

0.67 0.64 0.60

0.74 0.71 0.67

Floor thickness, ep = 15, 20 and 25 cm.

Table 4 The c-values for a slab-on-grade floor–wall junction (exterior wall made in masonry) Riv

Rsp (m2 K/W) 1.5

1.5 2.5 3.5

2.5

3.5

15

20

25

15

20

25

15

20

25

0.45 0.43 0.41

0.52 0.50 0.48

0.59 0.57 0.55

0.48 0.46 0.44

0.55 0.53 0.51

0.62 0.60 0.58

0.50 0.48 0.46

0.57 0.55 0.53

0.64 0.62 0.60

Floor thickness, ep = 15, 20 and 25 cm. Fig. 6. Flow lines.

Table 5 The c-values for a floor–wall junction (exterior wall made in concrete) em

Riv (m2 K/W)

em

1.5

17.5 22.5 27.5

Table 6 The c-values for a floor–wall junction (exterior wall made in masonry)

2.5

3.5

1.5

15

20

25

15

20

25

15

20

25

0.91 0.87 0.83

1.08 1.03 0.98

1.23 1.17 1.11

0.83 0.80 0.77

0.99 0.95 0.92

1.14 1.09 1.05

0.76 0.73 0.71

0.91 0.88 0.85

1.05 1.01 0.98

Floor thickness, ep = 15, 20 and 25 cm.

Riv (m2 K/W)

22.5 25.0 27.5

2.5

3.5

15

20

25

15

20

25

15

20

25

0.72 0.70 0.67

0.87 0.85 0.82

1.02 0.99 0.95

0.67 0.65 0.63

0.82 0.80 0.77

0.96 0.93 0.90

0.62 0.61 0.59

0.77 0.75 0.73

0.90 0.88 0.85

Floor thickness, ep = 15, 20 and 25 cm.

A.B. Larbi / Energy and Buildings 37 (2005) 945–951

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The computerized c-values for a junction with a concrete wall (resp. a masonry wall) are given in Table 5 (resp. Table 6). 4.1.3. Roof–wall junction Figs. 7 and 8 show the cross section of the 2D thermal bridge and the flow lines, respectively. The c-value depends on the roof width (ep), the wall width (em), and the wall and floor insulations (Riv and Rsp). The thermal bridge width is proportional to the roof width where the flux lines concentrate (Fig. 8). The computerized c-values for a junction with a concrete (resp. a masonry wall) are given in Table 7 (resp. Table 8). 4.2. Regression modelling Fig. 7. Cross-section.

The analytical formulas for 2D thermal bridges are intrinsically non linear in their parameters. So we use nonlinear regression models, which allow nonlinearity in the parameters of the regression. The estimation method is the nonlinear ordinary least squares estimation: the values of the parameters minimize the sum of squared deviations [9]. For every thermal bridge analyzed, we give on the one hand the regression model and figure showing the model fitted to the numerical simulation. In the regression model, the value of the parameter for Riv (at the denominator) is chosen as 0.02, equal to the insulation thermal conductivity divided by the floor thermal conductivity. The goodness of fit can be appreciated by considering the adjusted R-squared (R¯ 2 ) and the t-statistic for each coefficient estimate. 4.2.1. Slab-on-grade floor–wall junction As shown in Fig. 9, the regression model for the linear thermal transmittance of slab-on-grade floor–wall junction is close to the numerical simulation. For concrete wall, the absolute error reaches a maximum at 0.02 and the relative error is less than 3%. For masonry wall, the absolute error reaches a maximum at 0.01 and the relative error is less than 2%.

Fig. 8. Flow lines.

Table 7 The c-values for a roof–wall junction (exterior wall made in concrete) Rsp (m2 K/W)

em

Riv (m2 K/W) 2

3

4

15

20

25

15

20

25

15

20

25

1.5

17.5 22.5 27.5

0.72 0.70 0.68

0.83 0.80 0.78

0.92 0.89 0.86

0.66 0.65 0.63

0.77 0.75 0.73

0.86 0.83 0.81

0.61 0.60 0.59

0.72 0.70 0.69

0.80 0.78 0.77

2.5

17.5 22.5 27.5

0.74 0.72 0.70

0.86 0.83 0.81

0.96 0.93 0.90

0.67 0.66 0.65

0.79 0.77 0.75

0.89 0.86 0.84

0.62 0.61 0.60

0.74 0.72 0.70

0.83 0.81 0.79

3.5

17.5 22.5 27.5

0.75 0.73 0.71

0.88 0.85 0.83

0.98 0.95 0.92

0.68 0.67 0.66

0.81 0.78 0.76

0.91 0.88 0.86

0.63 0.62 0.61

0.75 0.73 0.71

0.85 0.82 0.80

Floor thickness, ep = 15, 20 and 25 cm.

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A.B. Larbi / Energy and Buildings 37 (2005) 945–951

Table 8 The c-values for a roof–wall junction (exterior wall made in masonry) Rsp (m2 K/W)

em

Riv (m2 K/W) 2

3

4

15

20

25

15

20

25

15

20

25

1.5

22.5 25 27.5

0.66 0.65 0.64

0.76 0.74 0.73

0.85 0.83 0.82

0.62 0.61 0.60

0.71 0.70 0.69

0.80 0.78 0.77

0.57 0.56 0.56

0.67 0.66 0.65

0.75 0.74 0.73

2.5

22.5 25 27.5

0.68 0.66 0.65

0.78 0.77 0.76

0.88 0.86 0.85

0.63 0.62 0.61

0.73 0.72 0.71

0.82 0.80 0.79

0.58 0.57 0.57

0.68 0.67 0.67

0.78 0.76 0.75

3.5

22.5 25 27.5

0.69 0.67 0.66

0.80 0.78 0.77

0.90 0.88 0.86

0.63 0.62 0.61

0.74 0.73 0.72

0.84 0.82 0.81

0.59 0.58 0.57

0.70 0.69 0.68

0.80 0.78 0.77

Floor thickness, ep = 15, 20 and 25 cm.

masonry wall; c ¼

0:409  ep þ 0:025Rsp þ 0:184 0:02Riv þ 0:243 (7)

4.2.2. Floor–wall junction For the floor–wall junction, the absolute error reaches a maximum at 0.02, and the relative error is less than 2% with a concrete wall. The absolute error reaches a maximum at 0.01 and the relative error is less than 2% with a masonry wall (Fig. 10). Regression model with a ep concrete wall; c ¼ 0:02Riv þ 0:433  em þ 0:195  0:04Riv þ 0:462

Fig. 9. Fitted model for a slab-on-grade floor–wall junction (concrete and masonry wall).

masonry wall; c ¼ Regression model with a concrete wall; c ¼

0:285  ep þ 0:023Rsp þ 0:252 0:02Riv þ 0:141 (6)

Fig. 10. Fitted model for a floor–wall junction (concrete and masonry wall).

(8)

ep 0:02Riv þ 0:613  em þ 0:153  0:018Riv þ 0:276

(9)

Fig. 11. Fitted model for a roof–wall junction (concrete and masonry wall).

A.B. Larbi / Energy and Buildings 37 (2005) 945–951

4.2.3. Roof–wall junction In this case the absolute error reaches a maximum at 0.04 and the relative error is less than 5% for concrete or masonry wall (Fig. 11). Regression model with a 0:287  ep 0:02Riv þ 0:155  em þ 0:06

concrete wall; c ¼

þ 0:02Rsp þ 0:34 masonry wall; c ¼

0:31  ep 0:02Riv þ 0:213  em þ 0:065 þ 0:017Rsp þ 0:312

(11)

5. Conclusion This paper presents regression models of the thermal transmittance for three examples of 2D thermal bridges (cvalues): slab-on-grade floor–wall junction, floor–wall junction and roof–wall junction. For slab-on-grade floor–wall junction, the results show that the models, for concrete or masonry wall, give results with relative errors less than 3%. The statistics of this model are also good. The adjusted R-squared (R¯ 2 ) is about 0.99 for concrete wall and is close to 1 for masonry wall. For all the coefficients in the two cases, the t-statistics are higher than 10, which mean that all the coefficients are significant. For floor–wall junction, the relative errors are less than 2% for the two cases. The R¯ 2 are close to 1 and the t-statistics are higher then 20. For roof–wall junction, the relative errors are less than 5% and R¯ 2 is about 0.98 for concrete or masonry wall. The t-statistics are sufficiently higher to justify the choice of the variables in the models (Tables 9–11). For all the models, the relative errors are less than 5%. If we added these errors to those obtained generally by numerical calculations, which are about 5%, the global Table 9 Statistics for a slab-on-grade floor–wall junction model Variable

ep cd Rsp cn a b

Concrete walla

Masonry wallb

Coefficient

t-Statistic

Coefficient

t-Statistic

0.285 0.141 0.023 0.252

13.19 11.97 12.60 28.47

0.409 0.243 0.025 0.184

16.38 15.08 20.01 40.96

No. of observations: 27; R¯ 2 : 0.988. No. of observations: 27; R¯ 2 : 0.995.

Table 10 Statistics for a floor–wall junction model Variable

Concrete walla Coefficient

t-Statistic

Coefficient

t-Statistic

em cd Riv cn

0.433 0.195 0.04 0.462

22.37 47.54 32.00 55.03

0.613 0.153 0.018 0.276

22.20 23.58 21.35 47.31

a

(10)

951

b

Masonry wallb

No. of observations: 27; R¯ 2 : 0.988. No. of observations: 27; R¯ 2 : 0.999.

Table 11 Statistics for roof–wall junction model Variable

Concrete walla Coefficient

t-Statistic

Coefficient

t-Statistic

ep em cd Rsp cn

0.287 0.155 0.06 0.02 0.34

28.34 12.86 14.95 11.12 39.68

0.31 0.213 0.065 0.017 0.312

25.11 8.49 8.89 11.19 41.83

a b

Masonry wallb

No. of observations: 81; R¯ 2 : 0.981. No. of observations: 81; R¯ 2 : 0.982.

relative errors of the proposed models will be about 10% which are less than errors generally obtained by calculation formulas (about 20%) [6]. The presented models can be used by practitioners, provided that both boundary conditions and material characteristics are similar to those considered above.

References [1] EN ISO 10211-1, Thermal bridges in building construction: heat flows and surface temperatures. Part 1. General Calculation Methods, 1995. [2] A.G. Mcgowan, A.O. Desjarlais, Investigation of common thermal bridges in walls, ASHRAE Transactions 103 (1997) 509– 517. [3] A.W.M. van Schijndel, Modelling and solving physics problems with FemLab, Building and Environment 38 (2003) 319–327. [4] BISCO, 2D Steady-state Heat Transfer Free Form, Physibel, Belgium. [5] KOBRA. Computer Program to Query an Atlas of Building Details on their Thermal Behaviour (Two-dimensional Steady State), Physibel, Belgium, 2002. [6] EN ISO 14683, Thermal bridges in building construction: linear thermal transmittance, Simplified methods and default values, 2000. [7] Ministe`re de l’e´ quipement, Guide de la re´ glementation thermique, France, 2000. [8] EN ISO 10211-2, Thermal bridges in building construction: heat flows and surface temperatures. Part 2. Linear thermal bridges, 2001. [9] W.H. Greene, Econometric Analysis, 5th ed. Prentice-Hall, 2003.