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5COUPLED HEAT TRANSFERS THROUGH BUILDING ROOFS FORMED BY HOLLOW CONCRETE BLOCKS
T. Ait-Taleb, A. Abdelbaki, Z. Zrikem
LMFE, Department of Physics Cadi Ayyad University Faculty of Sciences Semlalia, BP 2390, Marrakesh, Morocco Tel.: +212-24-43-46-49 (post 489); Fax: +212-24-43-74-10 E-mail: zrikem@gmail.com, abdelbaki@ucam.ac.ma
Received: 23 Sept 2007; accepted: 31 Oct 2007
The aim of this work is to study numerically the coupled heat transfers by conduction, natural convection and radiation through building roofs constructed by hollow concrete blocks. Heat transfer is assumed to be two-dimensional and the fluid flow in the different air cells of the system under investigation is laminar. Equations governing heat transfer are discretized using the control volumes approach and are solved by the SIMPLE algorithm. Streamlines, isotherms and overall heat fluxes are presented for building roofs heated from below and formed by different types of hollow concrete blocks. Analysis of results obtained for different numbers of air cells in the horizontal direction shows that the study of the entire system can be reduced, with a very good approximation, to the study of a single hollow concrete block. This permits a very important reduction of the computational time.
Keywords: building roof, hollow concrete blocks, conduction, convection, radiation, numerical simulation
Doctorat student at the Cadi Ayyad University, Faculty of Sciences Semlalia, Department of Physics, Fluid Mechanics and Energetics Laboratory, Marrakech, Morocco. He received the Licence (1997) and the DESA (2001) in physics from Cadi Ayyad University, Faculty of Sciences Semlalia, Marrakech, Morocco. It's research field is the study of coupled heat transfer by natural convection, conduction and radiation in building elements and the development of the heat transfer functions for these elements. About ten papers in international scientific journals and congresses was published.
T. Ait-Taleb
Professor at the Cadi Ayyad University, Faculty of Sciences Semlalia, Department of Physics, Fluid Mechanics and Energetics Laboratory, Marrakech, Morocco. He received the DES (1993) and the Doctorat d'Etat (2000) in physics from the Cadi Ayyad University, Faculty of Sciences Semlalia, Marrakech, Morocco. The main range of scientific interests are the study of the coupled heat transfer by natural convection, conduction and radiation in different configurations; the heat transfer between soil and buildings, solar systems, the development of the heat transfer functions for the building elements. About seventy papers in international scientific journals and congresses was published.
A. Abdelbaki
li
Z. Zrikem
Professor at the Cadi Ayyad University, Faculty of Sciences Semlalia, Department of Physics, Fluid Mechanics and Energetic Laboratory, Marrakech, Morocco. He is Engineer (1981) in Mechanical Engineering from Ecole Mohammadia d'Ingénieurs, Rabat, Morocco and he received the Ph.D. (1988) in Mechanical Engineering from Ecole Polytechnique, Montreal, Canada. The main range of scientific interests are the study of the coupled heat transfer by natural convection, conduction and radiation in different configurations; the heat transfer between soil and buildings, solar systems, the development of the heat transfer functions for the building elements. About a hundred of papers in international scientific journals and congresses was published.
30
International Scientific Journal for Alternative Energy and Ecology № 6 (62) 2008
© Scientific Technical Centre «TATA», 2008
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Introduction
Hollow concrete blocks are often used in the construction of building roofs. In general, each hollow block is formed by three cavities surrounded by solid concrete partitions. Thus, the heat transfer within such structures is done simultaneously by conduction in the solid partitions, natural convection inside the cavities and radiation between the internal faces of these cavities. Furthermore, the three heat transfer processes are intimately bound. Therefore, the fine study of the thermal behavior of the hollow blocks needs a simultaneous solving of the complex and sometimes non linear equations modeling the different thermal transfer mechanisms.
The coupling problem between the three modes of heat transfer was the subject of numerous investigations. The bibliographic studies presented in the works [1-6] show that the most of these investigations are generally limited to simple configurations constituted by rectangular cavities with one or several conducting walls and were especially has been interested to the study of the effects of conduction and/or radiation on natural convection in the cavities.
The coupled heat transfers in vertical alveolar building envelopes formed by hollow clay tiles have been studied by Abdelbaki and Zrikem [5]. Based on a detailed simulation model, the authors determined characteristic coefficients that permit fast and accurate estimation of heat exchanges through the alveolar walls without solving the complex coupled equations governing the different thermal mechanisms [6]. These characteristic coefficients are the overall heat exchange coefficients in the steady state regime and the transfer function coefficients in time varying regime. Recently, the previous studies are extended to the case of hollow concrete blocks used in the construction of
building roofs by Ait-Taleb et al. [7, 8]. In this case, the considered alveolar structures are horizontal but are subjected to a vertical gradient temperature. The two situations of heating have been considered: heating from below and from above.
In the practice, a building roof having, for example, 4 meters length contains about 8 units of hollow blocks. In construction, the hollow blocks are placed in such a way that each hollow blocks possesses 3 cavities (Nx = 3) in the horizontal direction separated by supporting girders. Therefore, the total number of alveolar in the considered building roof is about 24 cavities in the horizontal direction (Nx = 24). Consequently, the simulation of the global system requires an important computational time. Then, it is more interesting to reduce the size of the system under investigation. However, this reduction is only efficient if its influence on the global thermal behavior of the system is limited. To examine this solution, this paper presents a numerical study of the vertical coupled heat transfers by conduction, convection and radiation through horizontal hollow structures that differ by the alveolar number (Nx) in the horizontal direction and by the aspect ratio of the internal cells.
Mathematical formulation
The studied alveolar system is sketched in Fig. 1 and is formed by a range of Nx rectangular cavities of width l and height h surrounded by vertical solid partitions of thickness ext (1< i < Nx + 1) and horizontal partitions of thickness eyj (1< j < 2). The top and bottom sides of the hollow structure are considered isothermal and are maintained at constant temperatures T0 and T respectively. The structure vertical sides are considered adiabatic or submitted to periodicity conditions.
concrete joints
Fig. 1. Schematic diagram of disposition of hollow blocks in building roofs
In formulating governing equations, the fluid motion and the heat transfer are considered two-dimensional and the fluid flow is laminar. The solid and fluid properties are assumed to be constant except for the density in the buoyancy term where the Boussinesq approximation is utilized. Viscous heat dissipation in the fluid is neglected. The fluid is assumed to be non-participating to radiation and the inside surfaces of all cavities are considered diffuse-grey. The governing equations are written in dimensionless form as:
dU dV
IX +1Y = 0
dU dU dU dU + U — + V — дт dX dY
dP + Pr № + dU dX IdX2 dY2 ,
(1) (2)
dV dV dV dP 1Г + U dX + V ~dY =~dY + Pr
f d2V
d2V ^
dX2 + dY2
+ RaPrQ
f
(3)
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у
x
de
def
def
dT
f + U—t + F- f
d2ef d2e
f
dX
dY 3X2
+
f
dY2
(4)
Of de^ = d^ + d 2es
a dT dX2 dY2
(5)
where as is the solid thermal diffusivity and 6s is the dimensionless solid temperature. The boundary conditions of the problem are:
* U = V = 0 on the inner sides of each cavity.
* es (x,0) = 1 and es (x,l) = 0 (0< X < L/H)
'es (0, Y) = es (L/H, Y) (0 < Y < 1) for the periodicity condition
dX
— 0(0 ^ Y ^ 1) for the adiabaticity condition
'X — 0; L/H
The continuity of the temperature and the heat flux at the fluid-wall interfaces gives:
es (x ,y) —e f (x ,y),
de
de f
dn Nk dn
•+ NrQr
(6)
(7)
qr,k (rk) — Jk (rk) - Ek (rk)
(8)
Jk (rk ) — e k a(Tk (rk ))4 + (1 -e k )Ek (rk ):
(9)
Z I
Ek (rk ) — ZJ J, (r, )dFdSi -
J=1 A,
k -dSj (rk ,rj) :
(10)
where U and V are the dimensionless velocity components in X and Y directions respectively, P is the pressure, 6/ is the fluid temperature, Pr is the Prandtl number and Ra is the Rayleigh number given by:
g 3AT v Ra =-2-Pr , Pr = —, where v and af are
v af f respectively the fluid kinematic viscosity and the thermal diffusivity.
The dimensionless equation of heat conduction in the solid walls is:
where ek is the emissivity of the surface k and dF(Sk _ dsj is the view factor between the finite surfaces dSk and dSj located at rk and rj respectively. Taking into account equations (8) to (10), the dimensionless radiative heat flux can be expressed as:
Qr.
rk — e
«1 - G
% (ri.
+^
-e,
:t J
j—1
SJ'j (r( WdSk-dSj
(11)
where G is the temperature ratio T0/Ti, J' (j is the
dimensionless radiosity at the position rj on surface j. By dividing the walls into finite isothermal surfaces, equation (11) leads to a set of linear equation where the unknowns are the dimensionless radiosities J' (rj).
The dimensionless average heat flux across the structure is given by:
Qa — -
hl f des Hl f deJ
H J # ^—-t J lYl d
L 0 01 Y—0 L 0 01 Y—1
dX .(12)
where n represents the dimensionless coordinate normal to the wall, Nk is the thermal conductivity ratio Kf/Ks, Qr is the dimensionless radiative heat flux and Nr is the dimensionless radiation to conduction parameter defined by:
N =<H r ksAT ■
The dimensionless radiative heat flux Qr is related to the radiative heat flux qr by:
q =
Qr ^T04.
The net radiative heat flux qrk(rk) exchanged by the finite area dSk, located at a position rk on the surface k, is given by:
where Jk(rk) is the radiosity and Ek(rk) is the incident radiative heat flux on the surface dSk given respectively by:
The previous equations are discretized using the finites differences method based on the control volumes approach with a power law scheme and are solved by the SIMPLE algorithm. The resulting system of algebraic equations is solved by the Tri-Diagonal-Matrix-Algorithm. To accelerate the convergence of solutions, the governing equations are solved in their instationary form. The numerical code has been validated by comparing its results with those reported in reliable works in literature [4]. To realize a compromise between accuracy and computation time, a study on the effects of both grid spacing and time step on the simulation results has been conducted. It was found that a no uniform grid size of 185x20 is sufficient to model accurately the heat transfer and fluid flow inside the hollow block. The dimensionless time used is 10-4. The convergence criterion is 10-4.
Results and discussion
Results presented in this study (Table 1) are obtained for the hollow blocks mostly used in Morocco which are made in light concrete and characterized by a thermal conductivity Ks = 0.5 W/m-K and emissivity e = 0.9. The geometrical parameters for the different hollow blocks considered are given in table 1 where l is the length and h is the height of the internal cavities, Ac is the aspect ratio of these cavities (Ac = h/l), ex and ey are respectively the thickness of the vertical and horizontal solid partitions. The Prandtl number is Pr = 0.71. The temperature difference AT = (T0 - T) values vary between 1 °C and 30 °C in accordance with the practical conditions.
32
International Scientific Journal for Alternative Energy and Ecology № 6 (62) 2008
© Scientific Technical Centre «TATA», 2008
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Table 1
Geometrical dimensions of the different studied hollow blocks
Aspect ratio l (m) h (m) e* (m) (m) Ac = h/l
Ac = Ad « 1/4 0.13 0.035 0.025 0.02 0.26
Ac = Ac2 « 1/2 0.13 0.07 0.025 0.02 0.53
Ac = Ac3 « 1 0.13 0.1 0.025 0.02 0.77
Streamlines and isotherms
Fig. 2 presents the streamlines contours and the isotherms obtained for Nx = 12, AT = 20 °C, G = 1.035 and for the two considered types of hollow blocks characterized by Ac ~ 1/2 and Ac ~ 1. The other dimensionless parameters corresponding to these data are RaH = 2.5-106 and Nr = 5.850 for Ac = 1/2, RaH = = 5.169-106 and Nr = 4.596 for Ac = 1. It should be mentioned that the effective Rayleigh numbers in the different internal cavities are Rah = 5.13 105 for Ac ~ 1/2 and Rah = 1.53-106 for Ac~ 1 in such way the air flow remains laminar. Results of the Fig. 2 show that the flow structures and the temperature fields in the different cells of the same building roof are practically similar. Indeed, it is shown a nearly repetitivity of the flow structures and the temperature profiles in the different hollow blocks. It can be seen also that, the flow nature and temperature distribution are significantly affected by the aspect ratio of the alveolar. In fact, natural convection in the hollow block of aspect ratio Ac ~ 1/2 is characterized by the formation of two adjacent cells circulating in opposite directions and symmetrical with respect the central vertical axis of the cavity. However, for the same temperature difference, the flow structure in the hollow block of aspect ratio Ac ~ 1, is characterized by a single cell. The concentration of the isotherms near the superior horizontal surfaces of the alveolar indicates an important gradient of temperature in these regions. This is due essentially to the motion of the fluid going up following the alveolar vertical axis. Toward the central and low parts of the cavities, the heat transfer is less important but has a very marked two dimensional character.
ев 'OB1 ос» во во (вв тети вввввв
b
Fig. 2. Streamlines (at the top) and isotherms (at the bottom) obtained for AT = 20 °C and for the aspect ratios: (a) Ac ~ 1/2; (b) Ac~ 1
As expected, in the solid partitions separating the hollow blocks (concrete joints). the stratification of the isotherms indicates the linear character of the conductive heat transfer through these partitions.
Heat transfer
In order to show the effect of the alveolar number Nx on the global heat transfer through the system, Fig. 3 presents the variation of the dimensional heat flux Q (W/m2) crossing the structure as a function of the temperature difference AT for Ac ~ 1/2 and different values of Nx. The heat fluxes obtained for Nx = 3, 6, 9 and 12 by applying adiabatic boundary condition imposed on the vertical limits of the system (building floor) are compared to those calculated for Nx = 3 using the periodicity condition at the system vertical limits. Fig. 3 shows that the results obtained in the five considered situations are in very good agreement. Indeed, the observed difference on heat fluxes calculated for different values of AT and Nx are lower than 0,5 %. This result confirms the observations made in the previous paragraph when the analysis of the structures flow and the temperature fields has indicated a repetitively physical phenomenon in the different hollow blocks of the same building roofs.
T0 -Ti (°C)
Fig. 3. Effects of both alveolar number in the horizontal direction Nx and boundary conditions on the average heat transfer through the hollow block of aspect ratio Ac ~ 1/2
The results established above are observed also for other alveolar aspect ratios. In fact, Fig. 4 presents the variations of the mean vertical heat flux Q (W/m2) as a function of the temperature difference AT (°C) between the horizontal surfaces for Ac ~ 1/4, Ac ~ 1/2 and Ac ~ 1. This figure gives the results obtained for different values of Nx. It can be noted that, for each type of hollow block, the heat fluxes simulated in the different considered cases (Nx = 3, 6, 9 and 12) are in very good agreement. The maximal relative difference between the results obtained for the different values of Nx is lower than 0,7 %. Therefore the study of the entire system formed by Nx alveolar can be reduced with a very good approximation to that of hollow block with three alveolar
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(Nx = 3). This approximation permits significant reduction in computational time especially in transient conditions where the latter becomes extremely high.
To - T (°C)
Fig. 4. Variations of the global heat flux as function of the temperature differences AT for the three types of hollow blocks of aspect ratios Ac~ 1/4; Ac~ 1/2; Ac~ 1/4 and different values of Nx
Conclusion
The simulation results showed that the flow structures and the thermal behaviors in the different hollow blocks of a alveolar building roof are similar. For the different types of hollow block considered, the average heat flux crossing roofs that differ by the alveolar number Nx in the horizontal direction are practically the same. Therefore, the numerical simulation of the thermal behavior of the building roof can be reduced with a very good approximation to that of hollow block with three cells in the horizontal direction. This permits a considerable reduction of the computation time, especially in transient conditions where the latter becomes extremely important.
References
1. Kim D.M., Viskanta R. Study of the effects of wall conductance on natural convection in differentially oriented square cavities // J. Fluid Mech. 1984, Vol. 144. P.153-176.
2. Balaji C., Venkateshan S.P. Interaction of surface radiation with free convection in a square cavity // Int. J. Heat Fluid Flow. 1993, Vol. 14, P. 260-267.
3. Akiyama M., Chong O.P., Numerical analysis of natural convection with surface radiation in a square enclosure// Numer. Heat Transfer. 1997, Vol. 31, Part. A, P. 419-433.
4. Ramesh N., Venkateshan S.P., Effect of surface radiation on natural convection in a square enclosure // J. Thermophys. Heat Transfer. 1999, Vol. 13, P. 299-301.
5. Abdelbaki A., and Zrikem Z., Simulation numérique des transferts thermiques couplés à travers les parois alvéolaires des bâtiments // Int. J. Therm. Sci. 1999. Vol. 38, P. 719-730.
6. Abdelbaki A., Zrikem Z., Haghighat F., Identification of empirical transfer function coefficients for a hollow tile based on detailed models of coupled heat transfers // Building and Environment. 2001, Vol. 36, P. 139-148.
7. Ait-Taleb T., Abdelbaki A., and Zrikem Z., Etude numérique de l'interaction entre les transferts thermiques conductifs, convectifs et radiatifs dans une structure alvéolaire chauffée par le bas // Phys. and Chem. News, 2006, Vol. 30, P. 20-24.
8. Ait-Taleb T., Abdelbaki A., and Zrikem Z., Numerical simulation of coupled heat transfers by conduction, natural convection and radiation in hollow structures heated from below or above // Int. J. Therm. Sci. In press.
International Scientific Journal for Alternative Energy and Ecology № 6 (62) 2008
© Scientific Technical Centre «TATA», 2008
