Abstract. 1 Introduction

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1 Combined natural convection, conduction and radiation heat transfer in a cavity with solar control coating deposited to a vertical semitransparent wall G. Alvarez," C.A. Estrada'' CENIDET-DGIT-SEP, A.P , Cuernavaca, Morelos CP 62050, Mexico >>Laboratorio de Energia Solar-IIM-Unam, A.P. 34, Temixco CP 62581, Temixco, Morelos, Mexico Abstract This study presents a transient two-dimensional computational model of a combined natural convection, conduction, and radiation in a cavity with an aspect ratio of one containing air as a non participant fluid. The cavity has two opaque adiabatic horizontal walls, one opaque isothermal vertical wall The opposite vertical wall is a semitransparent wall that consists of a 6 mm glass sheet with a solar control coating of SnS-Cu*S facing the inside of the cavity permitting solar radiation exchanges between the cavity and its surroundings' The momentum and energy equations in transient state were solved by finite differences using the ADI technique. Coupled to this equations are the transient conduction equation and the radiative energy flux boundary conditions The results in this paper are limited to the range of 10*<Gr<10' with an isothermal vertical cold wall of 21 C, outside air temperatures (T.) of 35 C and incident solar radiation of AM2 (750 W/m') normal to the semitransparent wall A Nusselt number correlation as a function of the Rayleigh number is presented. 1 Introduction During the last two decades new technologies have been proposed and developed to control the thermal gains through the large window panes of tall tower type buildings and hence, to reduce their energy consumption The technology of the solar control coatings is still in progress and is part of this new technology. The solar control coatings are used in locations with warm climates to control in a spectrally selective mode the entrance of sunlight to the rooms and thus to reduce the thermal gains inside the building. An important optical characteristic of this kind of material is its high absorptance in the solar spectrum which increases the temperature of the solar control coating and glass

2 176 Advanced Computational Methods in Heat Transfer This increment of the film and glass temperatures may cause additional thermal gains inside the room instead of reduce them. Several models has been developed to get a better understanding of the thermal performance in a room, among them, combined natural convection conduction and radiation in a cavity with opaque walls and non participant media appears with Larson in 1972 [4]. Webb and Viskanta [7] presented experimental and numerical steady state results of natural convective flow movement in a cavity with a semitransparent vertical wall and a participant media, they considered incident solar radiation. Behnia et al. [2] presented the same case as before but with a non participant media and no incident radiation. None of those studies have considered the transient process of the heat transfer in a cavity with a solar control coating on a semitransparent wall. As a first approach, to understand the thermal performance of a room with a glass window with solar control coating, a theoretical study of a cavity with a semitransparent wall is presented. This study considers direct insulation, conduction of energy through the semitransparent wall, radiative exchange between the walls and a radiatively non participant fluid. 2 Physical Model The air filled cavity with an aspect ratio of one is shown in figure 1. The assumption of a two-dimensional model for the cavity is afirstapproach to the problem. All the walls are assumed with non slip conditions. The left face of the cavity is an isothermal opaque cold wall at 21 C; the right face is the semitransparent conductive hot wall with a solar control coating where the solar isothermal j j_ wall \ adiabatic wall ^ ^' 3 2 Air 4 ^ solar sole radiation G 1 %2g%%%%@%%%%%%%%%%%%y 2 7 \ X adiabatic wall Figure 1. Physical model and the co-ordinate system. radiation flux is transmitted through it. The conduction heat transfer process in this wall is considered in one-dimension. The governing equations assume the Boussinesq approximation. The opaque and the semitransparent walls are considered grey, diffuse reflectors and emitters of radiation. The incident solar radiation flux in the solar control semitransparent wall is assumed normal with a constant value of AM2 (750 W/nf). The optical and thermophysical properties

3 Advanced Computational Methods in Heat Transfer 177 are considered constant. Although, the solar control coating transmittance and reflectance do depend on the wavelength, in the visible region increases up to 40% but in the near infrared region remains constant and its value is less than 20%, the optical properties considered here are the integrated spectrum values The air inside the cavity is assumed to be radiatively non-participant for a low content of water vapour. The thickness of the glass is 6 mm with a solar control coating of SnS-Cu.S of negligible thickness (<6 urn). The properties are given in Table 1. The initial temperature is 21 C (294 K). Table 1. Optical and thermophysical properties of glass and SnS-Cu*S solar Glass (6 mm) a, =0.14 %e 0.78 Pa E%= 0.85 kg = 1.12 W/m K Glass (3 mm) a* =0.06 Tg Pe densg-2500 kg/iif film (SnS-CuxS) A*soi = 064 T*soi = 0.15 R*,o, = f = 0.40 Cpf=0.750kJ/kgK Air ka = 26.3x10-3 VV/mK v= 15.89x10-6 mvs Cpa=1.012kJ/kgK Densa=1.204kg/nr' 3 Mathematical Model The governing equations are those of the conservation of mass, momentum and energy equations for a cavity with radiative exchange at the boundaries and conduction in the solar control semitransparent wall. The stream functionvorticity formulation is used. The equations are expressed in a dimensionless form as: -+ V = F^r ^ di 5Y Gr^L9X 1; v ax* a^ i r ^ + u + v = at ax ay the velocities are related to the streamfunction by J^ae AT ax Co (2) (3) ax (4) JThe dimensionless variables were chosen according to reference [5] as t=tu,/l, G=T/TH, where The initial conditions in the cavity are those of a specified uniform temperature and a stagnant gas. The hydrodynamic boundary conditions are in terms or the streamrunction as: V(0,Y,T) = (j/(l,y,t) = V(X,0,i) = M/(X,1,T) = 0 (5) The boundary conditions on vorticity are not known explicitly but they are determined by the Taylor series expansion of the stream function in the vicinity

4 178 Advanced Computational Methods in Heat Transfer of the wall. The temperature boundary conditions on the top, bottom and semitransparent wall are given by the following energy balances: Energy balance on the bottom insulated wall 1: q% = q% +q^ =0,or The boundary condition for the isothermal wall 2 T_ T A 2_ / 02 ~ ~^~ I 'H The boundary condition for the insulated wall 3, w Q,,= (8) The temperature boundary condition of the vertical semitransparent wall 4 must have careful consideration. The conduction equation for this wall is coupled to the fluid energy equation by its interior boundary condition. Thus the conduction equation in dimensionless form of wall 4 is *>.- N. ph),,df] 5t N^PrVGrlciX'' dx'j with F'(X') = N^G'exp[-s^L'(l-X')], where Sg is the glass extinction T 3j I s~~\ coefficient, N = - k» and G'= - - GT, with the following boundary conditions: 1) energy balance between the inside air and the solar control coating q absorbed = q k ~ q ka + ^. dimensionless form p \ -. r.-y^ ^ ^^a,t.n,g--^-u.,-n,.q,=0 (10) 2) energy balance between the glass and ambient air ax' 0 (11) L'h with N, = * k g The local Nusselt number in dimensionless form is given by Nu =. 8,-8, 5X The dimensionless temperature of the vertical wall opposite to the vertical semitransparent wall is specified as 6(0,Y)=0.84 which corresponds to 21 C (294 K) for a TH of 1TC (350 K). The terms Q,, Q^, Q^ yq, are the dimensionless radiative fluxes inside the walls of the cavity given by the radiative transfer equations. For a grey diffusive walls with an arbitrary temperature distribution, the radiation configuration factors between any pair of

5 Advanced Computational Methods in Heat Transfer 179 elements of the boundary must be determined. The radiative heat flux alone the wall us given by q, = J,(x,)-q.(x,) f^ with J,(x,) = 8,oT/+p,q,(x.) and q,(x,) = Jj,(r,)dF_, (14) J=> Aj where q,(x,) is the incoming radiative heat flux of the wall i and J is the radiosity. The configuration factors Fy are already given in reference [51 Solving the radiosity equations, dimensional values are used in the calculation of the radiative fluxes on the boundaries and the result is non-dimensionalized. 4 Method of Solution Equations (l)-(4), conduction equation (9) along with the radiative flux equations (13)-(14) and their initial and boundary conditions (5)-(l 1) define the problem completely. The derivatives of the partial differential equations (1) and (3) were approximated by forward time and central space differences, the ADI (Alternatmg-Direction-Implicit) finite differences method was used The streamline equation (2) was solved by the method of Paceman and Rachford I he conduction differential equation was solved numerically by using a explicit finite differences formulation. The net radiative heat fluxes were calculated from the radiative net heat flux equation (13); the radiosity equations were solved by the method of successive approximation. The integrals were evaluated by the Simpson's rule. The solution procedure goes in the following order. Knowing the initial and boundary values, the radiative net heat flux is computed. As solar energy radiation strikes the glass and film, the energy conduction equation (9) for the glass and their boundary conditions (10) and (11) are solved to update the temperature distribution in the glass. The normal temperature gradients are ca culated from equations (6) and (8). The partial differential equation (3) is solved to obtain the inside air temperature patterns, then the vorticity-transport equation (1) and the stream function equation (2). After that, the velocity field is calculated from equations (4). The procedure is repeated until steady state is reached. 5 Results Several numerical experiments were tested to determine the time step grid size and convergence criteria. For Or of 10* and 10*, a mesh of 31x31 nodal points and a time increment of were adequate in terms of accuracy and computed economy. For Gr=10«a mesh of 51x51 and a time increment of were satisfactory but the computer time increases appreciably Also to validate the model, a comparison with the classical problem of natural convection m a cavity with isothermal vertical walls was carried out The comparison was satisfactory [1]. The parameter values used were for a square cavity containing air with an ambient exterior temperature of 35 C, an external

6 180 Advanced Computational Methods in Heat Transfer convective heat transfer of 6.2 W/m^ C. All the surface walls were assumed to have an emittance of 0.9 except for the semitransparent wall, which has an inside film emittance of 0.4 and an exterior glass emittance of To examine the effects of the use of the solar control coating on the semitransparent wall, two cases were considered, case A; a cavity with a semitransparent wall of glass and film and case B; the same as case A except that the semitransparent wall is a clear glass. The semitransparent wall is located on the right side of the cavity. Figure 2 and 3 show the transient isothermal and streamline plots for case A for Gr of 10*, 10^ and 10*. The temperature of semitransparent wall increases very rapidly due to the solar radiation absorbed by the solar control coating and the glass. In case A, for 1=6 the mean temperature of the film reaches (60.4 C) having little increase for later times; in T=60, 0=0.955 (61.1 T) and in case B, for 1=6 the film reaches (37.9 C) having also little increase; in x=60, 0=0.889 (38.2 C). The fluid circulates in counterclockwise direction. As the Grashof increases, temperature gradients become more severe near the vertical walls. For Gr=10* some stratification is presented at the center of the Figure 2. Isotherms for case A, for Gr=10\ 10* and \(f and dimensionless time from left to right of 6, 12, 20 and 60. cavity at later times. Figure 4 and 5 are the same asfigures2 and 3 but for case B. Although the shapes of the isotherms look similar infigures2 and 4 there are differences; the isotherms are curveless in case B compared with case A, indicating that the convection regime is more pronounced for case A. That happen because part of the energy that goes through the glass+film is absorbed (a=0.61), increasing the temperature of the coating and glass. The heat absorbed is transmitted to the fluid. The clear glass absorbs less energy

7 Advanced Computational Methods in Heat Transfer * 10* Figure 3. Streamlines for case A, for Gr=10\ 10* and 10* and dimensionless time from left to right of 6, 12, 20 and 60. (a-0.14). The thermal boundary layer thickness of cases A and B for Gr=10* and 10* in the lower side of the cavity is thinner for the case A than for the case B, but the isotherms in the upper side are curveless in case B compared with A, indicating that the diffusion is slower towards the core of the cavity. 10' 10" Figure 4. Isotherms for case B, for Gr=10*, 10* and 10* and dimensionless time from left to right of 6, 12, 20 and 60.

8 182 Advanced Computational Methods in Heat Transfer Figure 5. Streamlines for case B, for Gr=lO\ 10^ and 10* and dimensionless time from left to right of 6, 12, 20 and 60. Looking at figures 3 and 5, one vortex is presented in most of the cases except for Gr= 10* for times of 20 and 60 where two vortex are appreciated. Initially the vortex is located towards the upper right side of the cavity then it moves to the center of the cavity. For Gr=10^ the central streamline is elongated and distorted in an elliptic shape. The effect of convection is more pronounced in this case than in Gr=10*. The streamline gradients are stronger in case A than in case B indicating the influence of the radiative heat flux which makes the fluid moves slower. For Gr=10*, the streamline patterns for cases A and B, the vortices move closer to the walls and are convected downstream. The boundary layers adjacent to the vertical walls become thin and fast. In figure 3, for Gr=10* and 1=6 the flow spreads out from a thin layer in the vertical wall and turns over the corner, as the flow turns over, it starts to separate from the wall and then reattached to it, this effect is not shown for case B for x=6 but for the next times the effect is more pronounced for the two cases. This oscillation has been already reported in natural convection cavities [3,6] and not in combined natural convection and radiation. A correlation for the average Nusselt number were derived using least square linear regression for 1x10* < Gr < 5 x 10* and Pr=0.73. The correlation is: Nue = (Pr Gr)^" for Pr=0.73 and 1x10* < Gr < 5 x 10* Using arguments of energy balance in the cavity, it was found that, the percentage difference was less than 4% (3.75%), showing a possible total numerical error less than this number

9 Advanced Computational Methods in Heat Transfer 183 The shading coefficient (SC) is a standard measure of the efficiency of the thermal heat gain of the glass and the solar control coating and represents the ratio of the heat that goes to the interior of the cavity from the semitransparent wall to that for a clear 3 mm glass (case B) Figure 6 shows the curves for the shading coefficient vs ambient temperature. The upper curve is for a cavity corresponding to case B and the lower curve for a cavity corresponding to case A. It is observed that for an ambient temperature of 35 C the shading coefficient SC * CCO-1E5 VSC-1E5 * Figure 6. Shading coefficient vs ambient temperature for cases A and B. for the cavity using the solar control coating was of 42%, meanwhile for the cavity with clear glass was of 92.2%, indicating that, from the 100% of energy that would go through if we use a clear 3 mm thick glass, just the 42% goes through because of the use of a 6 mm glass with a solar control coating attached to it. If a clear glass of 6 mm is used, the heat gain to the inside of the cavity would be of 92.2%. 7 Conclusions A transient two dimensional mathematical model of a combined natural convection, conduction and radiation in a square cavity containing air as non participant fluid has been presented. A comparison between cavities with a vertical semitransparent wall with and without solar control coating was presented. It was observed the influence of the radiative exchange on the flow and temperature patterns. They were very similar qualitatively but not quantitatively. A decrease in the convective regime was observed. For Gr=10^, in both cases A and B, a wave effect appeared on the upper side of the cavity. A correlation of the convective Nusselt number as a function of the Rayleigh number and the shading coefficient as a function of the ambient temperature were presented. Those results indicated that, in accordance to a one dimensional model reported by Alvarez et al. [1], even though the glass and the film are warmer than the clear glass, the energy entering through the glass-film is lower than that of the clear glass. The difference was of 50.2% demonstrating the

10 184 Advanced Computational Methods in Heat Transfer clear advantage, from the thermal point of view, of the use of the solar control coating on glass windows. Nomenclature Latins A* G Or h«k NO. Nk Nu Pr R*soi T t X,Y Greeks a Index f,4 g Solar absorptancc, % Cp Incident Solar radiation, W/m^ g Grashof number, gpatl%* H Heat transfer coefficient, W/rn^ K Jj thermal conductivity, W/m K L, L' Dimensionless param. otg/ a NH Dimensionless param. kw/k, NL Nusselt number, hl/k, q Prandtl number, u/a R*,<>i Integrated Reflectance, % s, Temperature, K T *,«! time, s II, V Dimensionless coord, directions Air diffiisivity, mvs a, Thermal expansion coefficient, 1/K 8 Kinematic viscosity, mvs PJ Dimensionless temperature, T/Tn vy Vorticity tg Dimensionless time, tu^/l a incident energy on wall i film glass References a H I Specific heat Gravity, m/s^ Height of cavity, m Radiosity of wall i, Wide of the cavity, Thickness of glass, m Dimensionless param. hl'/kg Dimensionless parameter, LVL Heat, W/m* Integrated Reflectance, % Extinction coefficient of glass, 1/m Integrated transmittance, % Dimensionless velocity,u/u<, Glass absorptance Emittance Reflectance of wall i Streamlines Glass transmissivity Stefan Boltzm. const., air highest temperature inside W/m* net radiation on wall i conduction outside 1. Alvarez, G. Transferencia de Calor en una Cavidad con Interaction Termica a Traves de una Cara Semitransparente con Controlador Optico, Doctor in Engineering Thesis, National University of Mexico, Behnia, M., J A Rizes & G de Vahl Davis. Combined Radiation and Natural Convection in a Cavity with a Semitransparent Wall and Containing a Non-participant Fluid, Int. J. for Numerical methods in Fluids, 1990, 10, Ivey, G.N. Experiments on Transient Natural Convection in a cavity, J. of Fluid Mech. 1984, 144, Larson, D.W. Analytical Study of Heat Transfer in an Enclosure, Ph.D. Thesis, Purdue University, Larson, D.W & R. Viskanta. Transient Combined Laminar Free Convection and Radiation in a Rectangular Enclosure, J. Fluid Mechanics, 1976, 78, Ravi, MR, RAW Henkes & J. Hoogendoon. On the High Rayleigh Number Structure of Steady Laminar Natural Convection Flow in Square Enclosure, J. FluidMech., 1994, 262, Webb, B.W. & R Viskanta. Radiation Induced Buoyancy Driven Flow in Rectangular Enclosures: Experiment and analysis, J. of Heat Transfer, 1987, 109,