Effect Of 2d Modelling Of Thermal Bridges On The

Energy and Buildings 33 (2001) 583±587

Effect of 2D modelling of thermal bridges on the energy performance of buildings Numerical application on the Matisse apartment F. DeÂqueÂa,*, F. Ollivierb, J.J. Rouxc a

Electricite de France, Direction Recherches et DeÂveloppements, DeÂpartement ADE B Les RenardieÁres, Route de Sens, 77818 Moret sur Loing, France b ALTRAN Technologies, 58 Boulevard Gouvion-saint-cyr, 75 858 Paris, France c Centre de Thermique de Lyon (CETHIL), UMR CNRS 5008, BaÃtiment 307, INSA Lyon, 20 Av. Albert Einstein, 69621 Villerbanne Cedex, France Received 3 September 2000; accepted 30 September 2000

Abstract This paper describes the modelling approach used to accurately evaluate the effect of thermal bridges on the energy performance of buildings. The heat transfers in the intersections of walls were initially modelled in Sisley software. These models were then reduced and integrated in Clim 2000. The simulation results were compared against the models obtained from thermal regulation values. For standard wall con®gurations, it was seen that the detailed modelling of heat transfers provides an additional accuracy of about 5% in terms of the evaluation of the building's heat losses. # 2001 Published by Elsevier Science B.V. Keywords: Computer simulation; Energy consumption; Building envelope; Thermal bridges

1. Introduction Insulating walls represent one of the simplest solutions for decreasing the building's heat losses. Adding a layer of insulation is not, however, necessarily a suf®cient solution to reduce these losses. Indeed, thermally low points, commonly known as thermal bridges appear at the intersection of the walls of homes and decrease the effectiveness of insulations. These preferential heat loss areas are most often caused by a discontinuity in the building's insulation. In the remainder of this paper, these wall intersections, or the models that represent them, will be referred to as ``thermal bridges'' or ``thermal bridge models''. The increase in the level of insulation of buildings thus heightens the weight of thermal bridges in the overall energy consumption. The evaluation of these heat losses and their effect on the overall building performance represents a dif®culty frequently encountered by building professionals. As standard con®gurations and statutory calculation modes go back to 1977, they are often not suf®cient to classify, calculate or take into account innovative building and insulation tech* Corresponding author. E-mail address: [email protected] (F. DeÂqueÂ).

0378-7788/01/$ ± see front matter # 2001 Published by Elsevier Science B.V. PII: S 0 3 7 8 - 7 7 8 8 ( 0 0 ) 0 0 1 2 8 - 6

niques. Indeed, the use of ®xed tabulated values may lead to an under-estimation of the heat losses caused by thermal bridges. Numerical simulation then provides an ef®cient response to this problem. Two categories of tools are available in this ®eld. The ®rst category is simulation tools for evaluating the overall building energy performance. On the basis of a description of the whole envelope Ð opaque and glass walls Ð the heating systems and their regulation, modular software such as TRNSYS [1], SPARK [2], IDA [3] or Clim 2000 [4] simulate the thermal phenomena of building through a modular approach. The calculation of thermal bridges is often very approximate in these programs. The second category of tools includes simulation software such as THERM 2.0 [5] or TB3D/FMD [6]. These types of software are the calculation programs speci®c to the heat transfers in the walls. They are primarily based on entering the geometry in two dimensions (2D) or three dimensions (3D) of the thermal bridges through a graphic interface. Then they calculate the distribution of ¯ows and temperatures under steady-state conditions via a meshing, which may not be automatic. These tools do not, however, provide the possibility of evaluating the effect of thermal bridges on the overall performance of the building.

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In order to provide an accurate assessment of the effect on the building performance, we developed a two-stage modelling approach. The ®rst stage involves accurately modelling the heat transfers in thermal bridges through the Sisley software. On the basis of 2D modelling of the heat transfers in walls, this program automatically generates the meshing, the state model and the reduced dynamic model of thermal bridges. These reduced models are then integrated in the other components of the envelope to be simulated in Clim 2000. The ®nal paragraph in this paper illustrates this approach through an application example. Consideration will be given to the advantage of accurately taking thermal bridges into account over a conventional approach using tabulated values derived from thermal regulations. 2. 2D dynamic models of thermal bridges This ®rst stage in the modelling approach involves obtaining detailed models of thermal bridges through the Sisley software. Sisley is based on the ®nite-volume method and models heat transfers in 2D. This user-friendly tool was developed for this purpose by EDF's Research and Development Division in partnership with CETHIL (Lyon University). This is a tool, which is particularly well suited to the study of thermal bridges, of heated ¯oors and/or ceilings and windows. The main phases involved in obtaining a reduced dynamic model of thermal bridges are the followings ones:  Graphic description of walls. The user draws the configuration of the walls using a library of graphic components (see Fig. 1). This library may be shared by several users and enhanced as one wishes. Users then graphically define the inputs and outputs of their model in terms of temperature and/or heat flow.  Automatic meshing. Using this graphic description, Sisley automatically creates the whole meshing used to model the thermal bridge. The calculation algorithm is based on an irregular meshing of the ``Tchebycheff'' [7] type. All

Fig. 1. Thermal bridge description: choice of materials.

Fig. 2. ``Tchebycheff'' type meshing.

the meshing points are distributed over a half circle. The positions of these points are projected onto the segment (see Fig. 2).  Calculation of isothermal lines under steady-state conditions. To ensure that data are correctly entered, the user can perform a calculation under steady-state conditions. After defining the numerical values at the model inputs, he displays the results in tabular form Ð heat loss calculation in W/m Ð or graphic form (see Fig. 3).  Automatic generation of the physical model. Sisley then generates the equations of the thermal bridge dynamic model in the state space form (see Eq. (1)). In the case of thermal bridges, the model inputs are the temperatures outside and inside the home and the outputs are the heat flows corresponding to the heat losses. T represents the field of temperatures inside the walls, Text the outside temperature, Tint the temperature inside the home and f the heat losses in the wall. Eq. (1) shows the representation of the model in the state space form. T_ ˆ AT ‡ B1  Text ‡ B2  Tint ; f ˆ HT ‡ D1  Text ‡ D2  Tint

(1)

 Automatic generation of the reduced model. Sisley finally generates the equations of the reduced model. The following section describes the technique used to reduce models.

Fig. 3. Graphical display of isothermal lines.

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3. Reducing physical models by the Moore method The modelling of complete buildings including several rooms each containing 10 or so thermal bridges calls for the availability of models adapted to numerical simulation. For this reason, the physical models of thermal bridges generated by Sisley, including a large number of equations Ð between 300 and 1000 equations Ð are reduced. To reduce these models, we use Moore's reduction method [8,9]. This is a truncating technique, which eliminates a part of the model so that only the dominant components of the model are conserved. These dominant components are enhanced by a base change. The truncation model, already used in building thermal response [10], involves retaining the largest time constants in the reduced model. In this method, only small time constants are eliminated. Moore's reduction is similar, in principle, to modal truncation. Moore's truncation initially involves passing from the thermal base (Eq. (1)) to the balanced base. This base is based on the controllability and observability concepts used in automatic control. A strict de®nition of controllability and observability is given in [8], together with a description of the automatic concepts introduced in this section. To make it simple, controllability is the possibility, to obtain and vary the model states using the system inputs. Observability is the possibility to determine the model states from the outputs. The controllable and observable state components are de®ned by their controllability matrix Wc and observability matrix Wo, obtained by a Lyapounov type resolution: A  Wc ‡ Wc  AT ˆ B  BT ; AT  Wo ‡ Wo  A ˆ H T  H The ``balanced'' base, is constructed so that the controllability and observability matrices are identical and equal to a diagonal matrix W. The term ``balanced'' means that the system's controllability and observability have an equally important role in the construction of this base. Moore's truncation involves retaining the most controllable and observable state components of the system. The states retained by the truncation are those which:  are the easiest to excite for an energy signal at the model's input;  has the greatest effect on the observed outputs. The models obtained after reduction are third or ®fth order dynamic models. The reduced model faithfully reproduces the physical model in the stationary and transient state. Fig. 4 shows the internal heat ¯ow response of the model in Fig. 1 over a variation of 18C in the outside temperature. The two responses Ð 810-equations model and 3-equations reduced model Ð are similar.

Fig. 4. Heat flow response to a 18C step change in the outside temperature.

4. Reduced models of thermal bridges in Clim 2000 The thermal models are then stored in a Unix tree structure for simulation. Generic models of thermal bridges are then created in Clim 2000. Each of these models corresponds to a particular structure of thermal bridges. This paper presents the two structures most commonly used. These are the L and T structures (see Figs. 5 and 6). 5. Application: evaluation of the energy consumption of the Matisse apartment Ð effect of taking thermal bridges into account accurately and in detail The objective of this application is to evaluate the effect on the energy consumption of an accurate, detailed model of thermal bridges. We will compare the results obtained against a simpli®ed model based on statutory tabulated values [11].

Fig. 5. L shaped thermal bridges.

Fig. 6. T shaped thermal bridges.

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Fig. 7. Matisse home layout.

5.1. Description of the Matisse home Matisse is a building front apartment with a living area of 65.8 m2 including three main rooms …living room‡ two bedrooms†. It is located on the top ¯oor, under a terraced roof (see Fig. 7) [12]. 5.2. Modelling of thermal bridges

Fig. 9. T shaped ``thermal bridge'' model.

In this application, the model only requires the T and L structures. Figs. 8 and 9 illustrate two con®gurations representative of this model. In a conventional approach, the heat transfers in the walls are modelled by a ``dummy'' wall the heat losses of which correspond to the statutory tabulated values. In our approach, all the wall intersections giving rise to thermal bridges are modelled in Sisley and then introduced in a Clim 2000 study (see Fig. 10). 5.3. Simulation results The simulations are carried out with three different meteorological scenarios, representing cold, medium and mild climates (Paris in 1970±1971, Perpignan in 1989±1990 and Nancy in 1986±1987). The simulations are performed Fig. 10. Clim 2000 graph.

Fig. 8. L shaped ``thermal bridge'' model.

Fig. 11. Heating power: solid curve Ð reduced 2D model; dashed curve Ð statutory model.

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over a heating season from 1 October to 21 May. The heating setting is 198C. The results obtained Ð zoom on heating power (Fig. 11) Ð show that ``statutory'' models under-evaluate heat losses. According to the meteorological scenario, differences of 5±7% in the overall building consumption are obtained. 6. Conclusion This paper presents the two-stage approach used to accurately model heat losses in buildings due to thermal bridges. This approach is based on the one hand on Sisley tool for 2D modelling and the reduction of wall models and on the other hand on Clim 2000 for the simulation of the building energy performance. The modelling of the Matisse apartment illustrates this approach. Although the geometrical con®gurations of the wall intersections are relatively conventional in our study (T and L shaped structures), the accuracy of heat losses is increased by about 5±7% by taking 2D models of thermal bridges into account. It is then considered that these differences increase considerably in the case of more complex con®gurations differing signi®cantly from statutory references. The modelling approach adopted and the simulation tools used then provide essential supports for evaluating these complex con®gurations. References [1] S.A. Klein (Ed.), TRNSYS, a transient system simulation program, Engineering Experiment Station Report No. 38-13, University of Wisconsin, Madison, WI, 1990.

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