Energy and Buildings

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1 Energy and Buildings 41 (2009) Contents lists available at ScienceDirect Energy and Buildings journal homepage: Numerical study on the heat storing capacity of concrete walls with air cavities Z.L. Zhang a, *, B.J. Wachenfeldt b a Dept. of Structural Engineering, Norwegian University of Science and Technology (NTNU), Trondheim, Norway b SINTEF Technology and Society, Institute for Architecture and Building Technology, Trondheim, Norway ARTICLE INFO ABSTRACT Article history: Received 29 September 2008 Received in revised form 9 February 2009 Accepted 22 February 2009 Keywords: Heat storing capacity Concrete walls Air cavities Building energy simulations Finite element analyses using COMSOL have been carried out to study the heat transfer behavior and storing capacity of concrete walls with air cavities, and to explore the possibility of using one-layer and two-layer one-dimensional models with equivalent thermal conductivity and mass density to represent the effect of concrete walls with air cavities in a building energy simulation. Three typical wall geometries were chosen and both stationary and transient analyses have been carried out. The stationary analyses were performed first to find the equivalent thermal conductivity which was further used in the transient analyses to fit the equivalent mass density. Because of the presence of air cavities the equivalent thermal conductivities are always smaller than the bulk thermal conductivity. However, for the one-layer model an exaggerated equivalent mass density as high as two times the bulk density should be used in order to simulate the heat storing capacity of the concrete walls with air cavities. The values of the fitted equivalent mass density are strongly dependent on the wall thickness. ß 2009 Elsevier B.V. All rights reserved. 1. Introduction Building simulation is considered to be one of the most important efforts in increasing buildings efficiency. Though rarely using full-scale building models, it has become increasingly common to apply building simulation tools to predict the thermal environment, in particular in extreme rooms, to identify the need for cooling. Correct simulation can realize the potential benefits for the thermal environment such as reduction of temperature peaks, temperature swings and number of hours with excessive operative temperatures [1]. In reality both the energy performance and the thermal environment is strongly dependent on the thermophysical properties of the materials. The stationary as well as transient behavior of structural wall elements is therefore of fundamental importance in building energy simulation [1]. Thermophysical properties of concrete structures are usually given for bulk concrete without air cavities. The existence of air cavities in the concrete construction will influence both the heat transfer and heat storage capacity. In building simulation a concrete construction is commonly treated as a combination of one-dimensional homogenous isotropic elements, see Fig. 1 [2]. The construction thickness can be divided into one or several elements. For constructions without air cavities, the standard bulk values of the thermophysical properties can be used for these 1D elements. If a construction with air cavities is to be modelled as a 1D element, the thermophysical properties should be modified to account for the effect of the cavities. Several researchers have studied the thermal behaviour of hollow-cored concrete slabs. A theoretical study of a hollow-cored concrete slab exposed to the ambient condition, concluded that the location of the air cavity did not affect the thermal resistance of the slab [3]. However, this study was based on a one-dimensional * Corresponding author. Tel.: address: zhiliang.zhang@ntnu.no (Z.L. Zhang). Fig. 1. 1D Homogenous isotropic one-dimensional wall elements in building simulation /$ see front matter ß 2009 Elsevier B.V. All rights reserved. doi: /j.enbuild

2 770 Z.L. Zhang, B.J. Wachenfeldt / Energy and Buildings 41 (2009) Fig. 2. (a) Cross-section of the concrete walls with air cavities and (b) 2D unit cell. Table 1 Geometry for the three hollow-cored concrete slabs. Cases H [m] W [m] R [m] Cavity area fraction Case % Case % Case % analysis of the heat transfer through the slab assuming a uniform cavity separating the outer and inner layer of concrete, and thus ignored the geometrical complexity of the thermal bridges through the air cavity, i.e. in slabs with air cavities. Simplified, onedimensional theoretical models to estimate the thermal conductance of hollow cored slabs have also been developed [4]. However, none of these models are suitable to take the real geometry of hollow-cored concrete slabs into account to make an analogous 1D model suitable to be implemented in building simulation models. The concrete wall geometry considered in this study is shown in Fig. 2. The cavities are assumed to be cylindrical. The top and bottom surfaces of the walls are assumed to possess identical thermal conditions. Because of the periodicity, a 2D unit cell model for each wall geometry has been utilized for the finite element analysis. Due to the symmetry, a quarter-model is finally used in finite element analyses. Three concrete slab geometries with cylindrical air cavities commonly used in Norwegian building industry, shown in Table 1, have been selected [5]. The objective of the study is to investigate the effect of air cavities on the heat transfer behavior and diurnal heat storing capacity of the concrete slabs, and to use this information to create an analogous, generic 1D model for this type of constructions. The approach is to use the finite element results of the 2D unit cell model for fitting the equivalent thermal conductivities and mass densities for the 1D elements used in building energy simulation. The commercial program COMSOL 1 has been used for the analyses. Both stationary and transient analyses have been performed. The paper is organized as follows: The material data for the bulk concrete walls without air cavities as well as the finite models are described in the following section. In Section 3 the main results will be presented for the 2D models first. The approach we adopted here is to fit the equivalent thermal conductivities first from stationary analyses. The fitted equivalent thermal conductivity will be further used to fit the equivalent mass densities in the transient analyses. The paper is closed with discussions and concluding remarks. element models. The heat storing capacity of the air in the cavities has been neglected and a thermal insulation boundary condition is applied to the cavities. A sinus temperature profile fluctuating between 20 and 30 8C within a period of 5 days has been applied. The initial temperature for all the models was assumed to be 20 8C. Fig. 3 shows the 2D finite element mesh used in both the stationary and transient analyses. A mesh convergence study has been carried out to ensure the accuracy of the finite element results. In general, the results are not particularly sensitive to the mesh size. In the 1D models, more than 30 elements have been used. Both a one-layer and bi-layer 1D models have been considered. For the one-layer model with length H, the equivalent thermal conductivity ðk e 1Þ can be directly calculated from the finite element results of the 2D unit cell model. For a bi-layer model, the length of the outer layer and inner layer are calculated as L 1 = H R and L 2 = R, respectively. The thermal conductivity of the outer later is taken as the value of the bulk concrete, k 1 = k c. The equivalent thermal conductivity of the inner layer can be calculated from: k e 2 ¼ L L 2 a 2a 2 ¼ (1) 1 ða=a 1 Þ where a = k e /L anda 1 = k c /L 1, a 2 ¼ k e 2 =L 2, L = H, k e is the equivalent thermal conductivity for the one-layer model which is readily calculated form the finite element analyses. In the above calculation, equivalent heat flux conditions a 1 DT 1 = a 2 DT 2 = adt = a(dt 1 + DT 2 ) have been used. 3. Results 3.1. Effect of air cavities on the thermal conductivity stationary analyses A parameter study on the effect of air cavities on the thermal conductivity of a square unit cell model with H = W = 0.1 m, is 2. Material data, finite element models, procedures Based on tabulated data in [6], the thermal conductivity k c,mass density, r c and specific heat, c c of the bulk concrete is assumed to be 1.8 W/mK, 2350 kg/m 3 and 1050 J/kg K, respectively. The heat transfer coefficient between the wall surface and the media temperature has been calculated according to the standard NS-EN 15265:2007 as 8 W/m 3 K. Because of the symmetry of the wall, a quarter unit cell 2D model has been used. Fig. 3 shows the finite 1 Fig. 3. 2D and 1D finite element models.

3 Z.L. Zhang, B.J. Wachenfeldt / Energy and Buildings 41 (2009) Fig. 4. 2D cell model finite element results on the effect of cavity area fraction on the reduction of thermal conductivity. The square unit cell has dimensions H = W = 0.1 m. performed first. Fig. 4 shows the thermal conductivity ratio of the unit cell to the bulk concrete versus the cavity area fraction. The cavity area fraction is calculated as the ratio of the cavity area to the unit cell area. In order to facilitate the practical application, the effect of cavity area fraction on the thermal conductivity of the square unit cell has been numerically fitted by the following polynomial function: k e k c ¼ 0:9848 f 3 þ 1:7475 f 2 1:9769 f þ 0:9997 (2) where k e is the thermal conductivity of the unit cell and f is the cavity area fraction. The thermal conductivities of the unit cell models for the three cases considered have been found to be 0.554, and W/ mk, respectively. The results have been normalized by the bulk concrete thermal conductivity and are also plotted in Fig. 4. It can be seen that the reduction of the thermal conductivity is not solely determined by the cavity area fraction. It is therefore not possible to derive a general relation for non-square unit cell model. Nevertheless, Eq. (2) obtained from a square unit cell is a good approximation for the three cases and can be used as a rough estimation for the equivalent thermal conductivity Equivalent thermal conductivity for the 1D models The finite element results of the three cases have been used to calculate the equivalent thermal conductivity for the 1D models. For the one-layer model, the corresponding equivalent thermal conductivities for the three cases are exactly the same the as the unit cell model results. For the bi-layer model, the thermal conductivity for the outer layer is taken as the value of bulk concrete, while the equivalent thermal conductivities for the inner layer have been calculated by Eq. (1). The results for the three cases are presented in Table 2. It can be seen from Table 2 that the equivalent thermal conductivities for the inner layer varied from for case 2 to Fig. 5. Typical heat flux versus time for a 5-day period from 2D model of case for case 3. The values of the equivalent thermal conductivity are dependent on the absolute wall thickness as well as the cavity area fraction Equivalent mass density transient analysis Having found the equivalent thermal conductivities for the 1D one layer and bi-layer models, the thermal resistance of the real hollow-cored concrete slabs are well represented by the 1D models. The next step is to adjust the mass density and specific heat capacity of the different layers so that the dynamic effects of the slab are also well represented by the 1D models. The roles of mass density and specific heat capacity with respect to the heat storing capacity of thermal mass are identical and only the mass density will be focused here. A typical transient averaged heat flux versus time curve is shown in Fig. 5. In the transient analysis, the heat stored for all the models in the last half day with positive heat flux (shadowed area) has been integrated and compared. Fig. 6 compares the results of the 1D one-layer model with the results of the 2D unit cell model with H = W = 0.1 m. The mass densities for the 1D and 2D models are the same. However, the equivalent thermal conductivity shown in Table 2 has been used for the one layer model. Fig. 6 shows that when an equivalent thermal conductivity was used the one-layer model with the bulk mass density always underestimates the heat storing capacity of the unit cell model. In order to correctly simulate the heat storing capacity an artificially enlarged mass density should be used for the one-layer model. In the following the equivalent thermal conductivity shown in Table 2 will be applied to determine the equivalent mass density for both the one-layer and bi-layer models. Because there is no Table 2 Fitted thermal conductivities for the three concrete walls with air cavities. Cases Thermal conductivity without air cavities [W/mK] Thermal conductivity unit cell (FEM) [W/mK] One-layer Two-layer model L 1 [m] K 1 [W/mK] L 1 [m] K 1 [W/mK] L 2 [m] K 2 [W/mK] Case Case Case

4 772 Z.L. Zhang, B.J. Wachenfeldt / Energy and Buildings 41 (2009) Fig. 6. Comparison of the one-layer model prediction with the 2D FEM results. The equivalent thermal conductivity for the one-layer model was taken as the same as the 2D FEM analysis. The bulk mass density was used in both models. Fig. 8. Fitting of the equivalent mass density for case 2. analytical solution for solving the equivalent mass density, a wide range of the mass density has to be considered. Fig. 7 compares the integrated heat flux as a function of mass density in the last half cycle of a period of 5 days of both the onelayer and the bi-layer models with the 2D unit cell results for case 1. The 2D unit cell result has been plotted as a horizontal curve. It should be noted that the horizontal curve represents one value not a function of the mass density. It can be clearly seen from Fig. 7 that when the equivalent thermal conductivity fitted from the stationary analysis is used, an equivalent mass density as large as 4000 kg/m 3 should be used in order to achieve the same integrated heat flux as that of the 2D unit cell model. For the bilayer model, an equivalent mass density about 1400 kg/m 3 which is smaller than the bulk concrete density 2050 kg/m 3, should be applied. It is interesting to observe that the integrated heat flux for the one-layer model is clearly a linear function of the mass density while for the bi-layer model it is a non-linear relation. Similar results for case 2 are shown in Fig. 8. For the one-layer model the equivalent mass density should be around 5000 kg/m 3, which is 2.4 times that of the bulk concrete density. The fitted equivalent mass density for the bi-layer model is even larger, 5400 kg/m 3. It is interesting to note that when the mass density is about 2800 kg/m 3 the integrated heat flux for the bi-layer model reaches a minimum. The integrated heat flux will be increased when the mass density is either larger or smaller than this value. Similar observations can be made for case 3 in Fig. 9. The fitted equivalent mass densities for the one-layer and bi-layer models are around 3800 and 4200 kg/m 3, respectively. A minimum occurred at a mass density around 1600 kg/m 3. It is also interesting to note that the effect of mass density on the integrated heat flux for the one-layer model is consistent for the three cases a linear relation. However, the behavior for the 2D model is different in case 1 and cases 2 and 3. This probably can be explained by the wall thickness. The wall thickness for case 1 (200 mm) is the smallest among the three cases and close to the critical wall thickness for maximum diurnal heat capacity for concrete materials [7]. When the thickness is larger than this critical thickness, the diurnal heat capacity will decrease and the effect of inner layer mass density in a bi-layer model will be reduced. Thus an unrealistically large mass density should be used to achieve the same integrated heat flux as the unit cell model. The total amount of heat a building component can absorb is dependent on the exposed surface area and the thickness of the material. During a diurnal cycle, most of the recoverable heat is contained in the first 5 cm layer, and thicknesses above 10 cm provide little additional effect, assuming the properties equal to heavyweight concrete [7]. Fig. 7. Fitting of the equivalent mass density for case 1. Fig. 9. Fitting of the equivalent mass density for case 3.

5 Z.L. Zhang, B.J. Wachenfeldt / Energy and Buildings 41 (2009) Conclusions Fig. 10. Comparison of the heat flux versus day curves during the last half positive cycle for case Implementation in building simulation models The results can now be applied in building simulation models with hollow-core concrete slabs of the type shown in Fig. 2. For the three cases investigated, the slabs can be modelled as 1D elements as follows: The slab can be modelled as a 1D construction component where its total thickness equals 2L = 2(L1 + L2) (see Fig. 3 which can be seen as representing the bottom half of the slab). The slab can be split up into two elements with thickness L (one layer model), or four elements where the outer elements has thickness L1, and the inner elements has thickness L2 (the bilayer model). If the one layer model is chosen: Set the thermal conductivity of the layers according to Table 2, i.e W/mK if the geometry of the slab is according to case 1, W/mK for case 2 or W/mK for case 3. Set the equivalent mass density according to Figs. 7, 8 and 9, i.e. to 4000, 5000 or 3800 kg/m 3 for cases 1, 2 and 3, respectively (NB: the specific heat capacity is not altered). If the bi-layer model is chosen: Set the thermal conductivity and mass density of the outer layers (with thickness L1) equal to that of bulk concrete, i.e. to 1.8 W/ mk and 2050 kg/m 3, respectively. Set the thermal conductivity of the inner layers (with thickness L2) according to Table 2, i.e W/mK if the geometry of the slab is according to case 1, W/mK for case 2 or W/mK for case 3. Set the equivalent mass density of the inner layers (with thickness L2) according to Figs. 7, 8 and 9, i.e. to 1400, 5400 or 4200 kg/m 3 for cases 1, 2 and 3, respectively. The thermal behaviour of the 1D model should then closely resemble the behaviour of the real hollow-cored slabs. When steady state heat transfer in hollow-cored concrete slabs is concerned, the thermal conductivity is the only thermophysical parameter. The equivalent thermal conductivity for either the one one-layer or bi-layer analogous models established here can be directly determined from the 2D unit cell finite element analyses. For this purpose, the thermal conductivity of a square unit cell with a spherical cavity has been numerically fitted with a polynomial equation. The result is given as the thermal conductivity ratio as a function of the cavity area fraction. The thermal conductivity of a unit cell, k e 1, depends on its geometry as well as the cavity area fraction. Nevertheless, the proposed equation gives a good approximation of the thermal conductivity of the unit cell model. The equivalent thermal conductivity for the one-layer model equals k e 1, while the equivalent thermal conductivity for inner layer of the bi-layer model can be calculated from Eq. (1), setting the thermal conductivity of the outer layer equal to the bulk concrete, k 1 ¼ k c. When the heat storage capacity is concerned a transient analysis of a period of a couple of days should be carried out and the mass density and specific heat capacity have to be determined. Because the mass density and specific heat capacity play the same role in the governing heat transfer equation the focus has been put in this study on the mass density. For the one-layer models the integrated heat flux is a linear function of the mass density and the fitted equivalent mass density, which represents the heat flux behaviour of the 2D unit cell model, can be more than two times that of the bulk concrete mass density. For case 1 with a wall thickness 200 mm, the fitted equivalent mass density for the inner layer is less than the bulk density of concrete. The equivalent mass densities for cases 2 and 3 are unrealistically large and more than two times that of the bulk concrete value. This could be explained by the fact that the thickness of cases 2 and 3 are larger than the critical thickness at which the maximum diurnal heat capacity reaches the maximum. By using the equivalent thermal conductivity and mass density fitted in this paper, both the stationary and transient diurnal heat flux of the 2D unit cell models can be correctly represented the one-dimensional one-layer or bi-layer models. It should be noted that in the transient analysis only the integrated heat flux has been used as an indicator for the fitting of the equivalent mass. The maximum peak temperature as well as the peak heat flux, and the time it occurs, is not necessarily the same, see Fig. 10. References [1] Rasmus Z. Høseggen, Dynamic use of the building structure energy performance and thermal environment, Thesis at Norwegian University of Science and Technology (2008) 51. [2] J.A. Clarke, Energy Simulation in Building Design, 2nd ed., Butterworth-Heinemann, Oxford, UK, [3] P. Gandhidasan, K.N. Ramamurthy, Thermal behaviour of hollow-cored concrete slabs, Applied Energy 19 (1985) 41. [4] K.P. Rao, P. Chandra, A study of the thermal performance of concrete hollow blocks by an electric analogue method, Building Science 5 (1970) 31. [5] Spenncon, Spenndekk - raskt, sikkert og fleksibelt, Brochure with product information, 2008, [6] J.A. Clarke, Energy Simulation in Building Design, Butterworth-Heinemann, [7] J.D. Balcomb (Ed.), Passive Solar Buildings, MIT, USA, 1992.