Conduction / Natural convection analysis of heat transfer across multi-layer building blocks

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1 Conduction / Natural convection analysis of heat transfer across multi-layer building blocks Abstract H. Baig and M. A. Antar KFUPM, Dhahran, 31261, Saudi Arabia King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia Heat loss across a multi-layer wall that includes hollow blocks is studied numerically. Conjugate conduction-natural convection heat transfer is carried out to compute the heat leak/ R-Value for different numbers of air-filled cavities and for different cavity layouts. Interaction between conduction and convection at the boundaries is considered rather than the classical isothermal and adiabatic boundary conditions reported in open literature. This is believed to provide more accurate estimation of the heat transfer rate across the walls. Therefore, boundary conditions include a heat balance at the boundaries between the two heat transfer modes. Results show that increasing the number of cavities can be effective in reducing the heat flux without compromising the building blocks strength. Nevertheless, a considerable reduction in the heat flux or increase in the R-value can be achieved through changing the layout of the cavities within the wall. The effect of both techniques can compensate for removing the insulation layer in the cavity used to increase the wall thermal resistance. This is due to decreasing the cavities width that contributes significantly in suppressing the convection heat transfer effects (reduced value of Ra). 1. Introduction Heat leak through building walls and ceilings consume a substantial amount of energy. Since climate control units require a significant amount of electric energy, studies of heat leak has received considerable attention in the past decades. An accurate estimate of the heat leak through the composite, multi-layered walls accompanied with practical low cost methods for reducing the heat leaks is an effective way of reducing energy consumption. Hollow blocks are major wall elements where the modes by which heat transfer occurs are heat conduction in the solid sections and natural convection within the cavities. Investigations of the heat transfer through the hollow spaces have been treated both experimentally and theoretically by several researchers Lacarrier et al. (2003) analyzed numerically the vertically perforated bricks. They reported that walls can be constructed without any other materials than clay and mortar. They reported that heat transfer in these assemblies is not totally understood. For perforated brick construction, it is indicated that convection heat transfer is negligible in the perforations. Therefore, the thermal resistance of the brick increases. In a particular study of the ruptures it is concluded that the convection present in these regions is a local phenomenon since it breaks the thermal bridges created by the mortar fill. Bajorek and Lloyd (1982) carried out an experimental study to investigate the natural convection heat transfer within a two dimensional partitioned enclosure of unit aspect ratio using an interferometer. They reported that dividing the cavity along its

2 vertical axis results in a reduction in the heat transfer by approximately 15%. Nishimura et al. (1988) reported that Nusselt number is inversely proportional to the number of partitions which was also confirmed by experiments. They also observed that effective heat leak reduction is attained using 2-5 partitions. Aviram et al. (1988) investigated experimentally variable aspect ratio cavity and reported that increasing aspect ratio decreases flow magnitude, reduces circulation intensity and increases the cavity thermal resistance. Nusselt number diminished with reduced cavity depth. Recently, del Coz Diaz et al (2006) carried out an experimental and numerical study to investigate the thermal transmittance coefficient, U, of a wall made of Arliblock bricks. They observed that wall insulation decreases with the increase in the mortar and material conductivities. They also noticed that changing the profiles of the holes alters the rate of the heat transfer through the hollow blocks. Then, they (2007) studied major variables influencing the thermal conductivity of masonry materials and carried out an optimization study for different brick geometries based on both thermal resistance and weight. The minimum web thickness for safe construction was reported by Kumar (2003). Ciofalo and Karayiannis (1991) reported that the mechanism responsible for the large reduction in heat transfer in partitioned enclosures was because of the breaking down of the unicellular circulation near the regions. Manz (2003) studied natural convection heat transfer in rectangular, gas-filled tall cavities in building elements such as insulating glazing units, double-skin facades and others. He reported that flow regimes depend on Ra and the aspect ratio. A linear temperature profile exists as a function of the x-position within the so-called conduction regime. Al-Hazmy (2006) investigated the heat transfer through a common hollow building brick. Insulation assessment of the building blocks was examined based upon the heat transfer rate. Three different configurations for building bricks were studied including a gas-filled and insulation-filled cavity. Results show that the cellular air motion inside blocks cavities contributes significantly to the heat loads. The insertion of polystyrene bars reduced the heat rate by a maximum of 36%. Lee and Pessiki (2006) carried out a study to investigate the performance characteristics of precast concrete sandwich wall panels with two or three wythes separated by air layers. It was found that, in general, the thermal performance of three-wythe panels is better than that of two-wythe panels due to the increased thermal path length. Ho and Yih (1987) analyzed conjugate natural convection and conduction in a multi-layer wall. They considered isothermal left and right sides of the wall and adiabatic boundary condition in both top and bottom surfaces. Tong and Gerner (1986) analyzed natural convection in partitioned air-filled rectangular enclosures and reported that placing a partition midway between the vertical walls results in the greatest reduction in heat transfer. Kangni et al. (1991) investigated natural convection in partitioned walls for various aspect ratios and for a wide range of Ra and wall thicknesses. Turkoglu and Yucel (1996) investigated numerically natural convection heat transfer in enclosures with conducting multiple partitions and side walls. However, in their analysis the sidewalls were assumed to be isothermal, thus eliminating the temperature gradient in the y-direction within the solid. They also kept the top and bottom surfaces perfectly insulated. They reported that Nusselt number

3 decreases as the number of partitions is increased up to 4. They also reported that the cavity aspect ratio had an insignificant effect on their calculations Lorente (2002) published a review article to illustrate the heat flow through walls with relatively complicated internal structure. He reported the effect of Rayleigh number and aspect ratio showing that for Re= 3550, no fluctuations were observed and a unicellular flow was observed. As the Rayleigh number increases, the flow becomes multi-cellular. Antar and Thomas (2001) addressed the heat transfer across a hollow building block and estimated two dimensional effects of the heat transfer across the block. In another study they (2004) developed a numerical finite-difference analysis for steadystate heat transfer in a composite wall with a two-dimensional rectangular gray body radiating cavity with and without natural convection circulation of air. The purpose of their analysis was to provide a basis for evaluating the accuracy of the first-order twodimensional model. Recently, Antar (2006) investigated the significance of multidimensional effects in estimating the rate of heat loss and identified the cases where simple one-dimensional convection/radiation analysis may be considered a good approximation for heat transfer rate calculations. It was reported by Antar and Thomas (2001, 2004) and Antar (2006) that the approximate simple one dimensional analysis for the problem under investigation has two alternative thermal circuits, an upper bound thermal circuit and a lower bound one. Calculations show that the percentage difference in estimating the upper bound and lower bound heat transfer rate reaches 39 %. This indicates significant two dimensional effects. Neither the upper bound nor the lower bound solution provides a reliable value for the heat leak, and a two dimensional model is needed to estimate the heat transfer rate accurately. Therefore, this work is aimed at developing and using a two-dimensional model for investigating the effect of cavities layout on the thermal resistance of the block. A block of 5 cavities in the heat flow direction was considered with the objective of increasing the thermal resistance. The previous literature search considered the number of cavities as a major factor for reducing the heat leak.this study shows that the shape and distribution of the cavity plays an equally important role in this regard. More practical boundary conditions will be used such as nonuniform vertical walls temperatures and non-adiabatic horizontal surfaces of the cavities. 2. Formulation The basic geometry of the problem under investigation is shown in Figure 1. Heat transfer by conduction occurs in the solid part whereas natural convection heat transfer occurs within the cavities. Other layouts may have different cavity size or structure, see Figure 4, but the modes are unchanged.

4 w 1 w 2 w Conduction heat transfer L 1 L 2 L 1 L 2 L 1 L 2 L 1 L 2 L 1 L 2 L 1 Figure 1: Hollow brick geometry, case 1 Steady two dimensional conduction heat transfer in the block solid material is governed by the following equation: 2T 2T + = 0 x 2 2 (1) T = T i at x = 0 T = T o at x = L (2) T T = 0 at y = 0 = 0 at y = w (3) T k = qs,x " x at x = L 1 and at x = L 1 + L 2 for w 1 y w 1 + w 2 (4) T k = qs,y " y at y = w 1 and at y = w 1 + w 2 for L 1 x L 1 + L 2 (5) where L = (L 1 + L 2 + L 1 ), w = (w 1 + w 2 + w 1 ) q s,x " and q s,y "are the convection heat transfer flux at the interface in both x and y directions. The same boundary conditions are applied to all cavities. 2.2 Convection heat transfer The governing differential equations for the free convection within each of the enclosures are given for constant properties, no heat dissipation, applying Boussinesq approximation - as: u v + = 0 (6) x 2 2 u u p u u ρ u v μ + = (7) x x x 2 2 v v p v v ρ u + v = + μ + ρ gβ( T T ) 2 2 x x (8)

5 2 2 T T k T T u + v = + (9) 2 2 x ρ C x At the inner walls of the gaps, the no slip condition applies (U=V=0) (10) Note that air properties are considered as: ρ = kg/m 3, c P = kj/kg K and k = W/mK. The boundary conditions at the interface of the air gap include continuity of temperature and heat flux. However, temperatures are unknown, and hence an iterative solution will be adopted to guess the temperatures at the interface and apply the heat balance equation to calculate them (refer to equations 4 and 5) " " q conduction = q convection (11) Note that the boundary conditions are applied for all the air gaps in the material, i.e. once for the arrangement given in Fig. 1, and as many as the number of gaps 3. Numerical Solution and Grid independence The grid used in both solid and fluid phases is designed such that more nodes are concentrated at all the corners of both solid and gas phases where higher gradients exist. Other layouts follow the same trend, see figure 2 for a sample of the numerical grid. The Control volume approach is employed in the numerical scheme. Variables are computed at ordinary nodal points, except the velocities, which are determined at staggered grid centered around the faces of the cells. More details of the numerical scheme are given by Patankar (1980). Grid independent tests are conducted as shown in Fig. 3 where a grid of nodes is considered on the basis of less computation time without compromising the solution accuracy in the case of a 5 cavities. The grid independence study was repeated for all other configurations. The number of nodes increases as we use more cavities within the block to take care of the boundary layer growth within each cavity. Figure 2: Numerical grid Figure 3: Grid independence 4. Discussion of Results Since the current layouts are not reported in the open literature, the numerical results are validated by calculating the Nusselt number using the layouts employed by other investigators. The percentage difference in calculating the Nusselt number (at Ra = 10 5 and H/W = 1) compared with Ho and Yih (1987) was 2%. Moreover, Nusselt

6 number calculations compared with Kangni et al. (1991) for a different case (Ra = 10 6 and H/W = 5) showed a deviation of 0.2%. The R-value for the layout shown in Figure 1 (based on one third of the block due to symmetry) is 0.47 K m 2 /W (case 1). Changing the layout to the one shown in Figure 4a (case 2) results in an increase in the R-value (decrease in the heat transfer rate) by 13.5 %. This is due to narrowing the path of conduction heat transfer in the solid material above and below the cavities. A further change in the layout to the case 3, shown in Figure 4b results in a further increase in the R-value by % (R-0.56 K.m 2 /W) compared to the basic case. This layout is better than the previous one (case 2, Fig 4a) since it does not have small aspect ratio cavities and therefore has less convection effects. It is interesting to mention that the layout of case 4, Figure 4c did not show an improvement in the heat leak, R-0.51 K.m 2 /W. This is due to pronounced role of conduction heat transfer in the solid surrounding the cavities thus providing unnecessary thermal bridges. In addition, more cavities in this case with lower aspect ratio results in a more air circulation that promotes the convection heat transfer rate. Therefore, changing the layout of the brick can play a substantial role in increasing its thermal resistance. The layout shown in case 5, Figure 4d is also tested and it results R-value 0f 0.43 K.m 2 /W, i.e. an 8.8 % increase in the heat transfer rate/reduction in R-value. It is fairly obvious that this layout is not a preferred alternative. This is due to higher convection effects and less conduction resistance (because of more thermal bridges) layout compared with the other layouts. (a) case 2 (b) case 3 (c) case 4 (d) case 5 Figure 4: Stream function in the alternative brick layouts

7 Figure 5 shows the mid-plane (y=w/2) temperature distribution as it drops from T o at the right exterior wall to T i at the left exterior one for the two layouts case 2 and case 3. These two layouts were selected since they have the lowest heat transfer rates (highest R-value). Case 3, shown in Figure 4b has a higher resistance compared to case 2 shown in Fig. 4a. This can be attributed to more homogeneous and steeper temperature gradient established across the width of the whole block (20 cm in this study) leading to better temperature distribution and higher thermal resistance across the block. Case 2 Case 3 Figure 5: Centerline temperature profile for cases 2 and 3 5. Conclusion Conduction / Natural convection conjugate heat transfer within masonry blocks with hollow cavities is investigated numerically. Changing the layout of the cavities can play a substantial role in increasing the R-value (decreasing the heat leak) and hence enhances thermal insulation through increasing the R-value by 13.5% (case 2 compared to case 1) and up to 17.65% (case 3 compared to case 1) without affecting the structural characteristics of the blocks. Decreasing the thickness of the solid material used (within the safe limits) decreases the thermal bridges and thus avoids unnecessary conduction heat leak. In addition, using cavities of high aspect ratio (case 3 instead of case 2) and less width decreases the convection heat transfer effects since decreasing the cavity width reduces the Rayleigh number and hence reduces the convection heat transfer coefficient. It is believed that these results will provide guidelines to masonry block manufacturers for better energy conservation. Acknowledgement The support provided by King Fahd University of Petroleum and Minerals, KFUPM to carry out this investigation through project number IN is gratefully acknowledged. 6. References [1] Lacarrie`re, B., Lartigue, B. and Monchoux, F., 2003, Numerical study of heat transfer in a wall of vertically perforated bricks: influence of assembly method, Energy and Buildings, 35, [2] Kumar, S., 2003, Fly ash lime phosphogypsum hollow blocks for walls and partitions, Building and Environment, 38, [3] Bajorek, S. M. and Lloyd, J. R., 1982, Experimental Investigation of Natural Convection in partitioned Enclosures, Journal of Heat Transfer, 104,

8 [4] Nishimura, T., Shiraishi, M., Nagasawa, F. and Kawamura, Y., 1988, Natural convection heat transfer in enclosures with multiple vertical partitions, International Journal of Heat and Mass Transfer, 31 (8), [5] Aviram, D P., Fried A. N., and Roberts, J. J., 2001, Thermal properties of a variable cavity wall, Building and Environment, 36, [6] del Coz Diaz, J. J., Garcia Nieto, P. J., Martin Rodriguez, A., Lozano Martinez- L uengas, A., Betegon Biempica, C., 2006, Non-linear thermal analysis of light concrete hollow brick walls by the finite element method and experimental validation, Applied Thermal Engineering, 26, [7] del Coz Dı az a, J. J., Garcı a Nieto, P. J., Betego n Biempica, C., Prendes Gero, M. P., 2007, Analysis and optimization of the heat-insulating light concrete hollow brick walls design by the finite element method, Applied Thermal Engineering, 27, [8] Ciofalo, M. and Karayiannis, T. G., 1991, Natural convection heat transfer in a partially or completely partitioned vertical rectangular enclosure, International Journal of Heat and Mass Transfer, 34 (1), [9] Manz, H., 2003, Numerical simulation of heat transfer by natural convection in cavities of facade elements, Energy and Buildings, 35, [10] Al-Hazmy, M. M., 2006, Analysis of coupled natural convection conduction effects on the heat transport through hollow building blocks, Energy and Buildings, 38, [11] Lee, B. J., Pessiki, S., 2006, Thermal performance evaluation of precast concrete three-wythe sandwich wall panels, Energy and Buildings, 38, [12] Ho, C. J. and Yih, Y. L., 1987, Conjugate natural convection heat transfer in an air-fileld rectangular cavity, International Communication in Heat and Mass Transfer, 14, [13] Tong, T. W. and Gerner, F. M., 1986, Natural convection in partitioned airfilled rectangular enclosures, International Communication in Heat and Mass Transfer, 13, [14] Kangni, A., Yedder B., and Bilgen, E., 1991, Natural convection and conduction in enclosures with multiple vertical partitions, Int. J. Heat and Mass Transfer, 34, [15] Torkoglu, H. and Yucel, N., 1996, Natural convection heat transfer in enclosures with conducting multi le partitions and side walls, Heat and Mass Transfer, 2, 1-8. [16] Lorente, S., 2002, Heat losses through building walls with closed, open and deformable cavities, International Journal of Energy Research, 26, [17] Antar, M. A. and Thomas, L. C., 2001, Heat transfer through a composite wall with enclosed spaces: A practical two-dimensional analysis approach, ASHRAE Transactions, 106, [18] Antar, M. A. and Thomas, L. C., 2004, Heat Transfer Through a Composite Wall with an Evacuated Rectangular Gray body Radiating Space: A Numerical Solution, ASHRAE Transactions, 110 (2), [19] Antar, M. A., 2006, Multi-Dimensional Effects in Estimating the Heat Loss Across Building Envelopes Proceedings of the Second International Conference on Thermal Engineering Theory and Applications, Al-Ain, UAE 3-6 Jan. [20] Patankar, S. V., 1980, Numerical Heat Transfer, McGraw-Hill, New York.