Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 53, 3, pp. 643-652, Warsaw 2015 DOI: 10.15632/jtam-pl.53.3.643 THE COMBINED HEAT TRANSFER OF RADIATION AND MIXED CONVECTION ANALYSIS IN A LID-DRIVEN TRAPEZOIDAL CAVITY Malihe Mohammadi, Seyyed Abdolreza Gandjalikhan Nassab Department of Mechanical Engineering, School of Engineering, Shahid Bahonar University, Kerman, Iran e-mail: mohammadi2345@yahoo.com In this paper, the effect of radiation heat transfer onmixed convection in a lid-driven trape- zoidal cavity is studied numerically. The governing equations ofmixed convection are solved based on the SIMPLE algorithm and the solution of the radiative transfer equation inside the participatingmedium is carried out using the discrete ordinatesmethod, simultaneously. To study the heat transfer and flow characteristics, sensitivity analysis is carried out based on the two parameters including the optical thickness and radiation-conduction parameter. Several interesting results are obtained such as sweepbehavior on the isotherms, streamlines and convective Nusselt number with optical thicknesses. Keywords: mixed convection, radiation, lid-driven trapezoidal cavity 1. Introduction Themixed convection process inside a lid-driven cavity is found inmanyengineering applications such as solar collectors, cooling of electronic devices, heat exchangers (Cheng and Liu, 2010). Study of the literature shows that the problem is considered along two different parts, those are square or rectangular geometry and irregular geometries. The majority of previous works investigated the effect of several boundary conditions on flow and temperature characteristics inside a lid-driven square or rectangular cavity (Moallemi and Jang, 1992; Prasad and Koseff, 1996; Shankar et al., 2002; Oztop, 2006). Chen and Cheng (2004, 2005) studied the flow and temperature characteristics formixed convective inside an arc-shaped lid-driven cavity by consi- dering variable inclination angle. Recently, few literatures items have been reported on topic of mixed convective flow insidea lid-driven trapezoidal cavity.Theanalysis of flowand temperature characteristics inside irregular geometries is very important for advance of heating and cooling arrangements ormaterial processing systems (Bhattacharya et al., 2013). Inmost of the studies on themixed convection in lid-driven cavities, the role of the radiative heat transfer is neglected. Therefore, the objective of this work is to study the effect of radiative heat transfer on mixed convection in a lid-driven trapezoidal cavity numerically. To achieve this purpose, the flow and heat transfer characteristics is investigated for a broad range of optical thickness (0<τ < 100) and radiation-conduction parameters (0.5 0 (2.4) The discrete ordinates method is used to solve the equation of radiative transfer equation (Eq. (2.3)). In this method, Eq. (2.3) is replaced by a set of n equations for a finite number of n different directions si, and each integral is replaced by a quadrature series of the following form (si·∇)I(r,si)=−βI(r,si)+σaIb(r)+ σs 4π n ∑ j=1 wjI(r,sj)Φ(sj,si) i=1,2,3, . . . ,n (2.5) with the boundary conditions I(rw,si)= εwIb(rw)+ 1−εw π ∑ |nw|,|sj|<0 )wjI(rw,sj)|nw,sj| (2.6) 646 M.Mohammadi, S.A. Gandjalikhan Nassab In the above equations, wk is the weighting factor for each discrete ordinate. The original equ- ation by this angular approximation is transformed to a set of coupled differential equations. Equation (2.5) in Cartesian coordinates becomes ξi ∂Ii ∂x +νi ∂Ii ∂y +µi ∂Ii ∂z +βIi =βSi Si =(1−ω)Ib+ ω 4π n ∑ j=1 wjI(r,sj)Φ(sj,si) i=1,2,3, . . . ,n (2.7) in which ξm, νm, µm are the directional cosines of si. Non-dimensional Eqs. (2.1)-(2.3) are obtained using the following dimensionless variables X = x L Y = y L U = u U0 V = v U0 θ= T −Tc Th−Tc P = p ρU20 Re= ρU0L µ θ1 = Tc Th−Tc θ2 = Th Tc I∗ = I σT4 h S∗ = S σT4 h τ =βL 1−ω= σa β Pr= ϑ α Pe=RePr Gr= gβ(Th−Tc)L 3 ϑ2 RC = σT3 h L k q∗r = qr σT4 h (2.8) The non-dimensional form of Eqs. (2.1)-(2.3) can be written as ∂U ∂X + ∂V ∂Y =0 U ∂U ∂X +V ∂U ∂Y =− ∂P ∂X + 1 Re (∂2U ∂X2 + ∂2U ∂Y 2 ) U ∂V ∂X +V ∂V ∂Y =− ∂P ∂Y + 1 Re (∂2V ∂X2 + ∂2V ∂Y 2 ) + Gr Re2 θ ∂ ∂X ( Uθ− 1 Pe ∂θ ∂X ) + ∂ ∂Y ( Vθ− 1 Pe ∂θ ∂Y ) + τ(1−ω)RCθ1θ2 Pe [ 4 θ42 ( θ θ1 +1 )4 − n ∑ i=1 I∗iwi ] =0 (2.9) Themodified boundary conditions are: — on the upper wall U =1 V =0 θ=0 (2.10) — on the lower wall U =0 V =0 θ=1 (2.11) — on the inclined walls U =0 V =0 θ=1−Y (2.12) The convective, radiative and total Nusselt number at the walls are determined using the heat fluxes as Nuc = qcL k∆T Nur = qrL k∆T Nut =Nuc+Nur (2.13) The combined heat transfer of radiation and mixed convection... 647 3. Numerical solution procedure The applied numerical method to solve the governing equations (Eqs. (2.9)) is a line-by-line iteratively method based on the SIMPLE algorithm. The equations are discretized using the finite control volume technique on staggered control volumes for the x and y velocity compo- nents. The RTE is solved using the DOMmethod to calculate the radiation source term in the energy equation (Modest, 2003). This calculations are written in a FORTRAN program. The convergence criterion for solving the governing equations is assumed that the amount of residual terms between two consecutive iteration is less than 10−6, and the convergence criterion for the RTE is max ∣ ∣ ∣ I∗np − I ∗n−1 p I∗np ∣ ∣ ∣ ¬ 10−5 (3.1) 4. Validation of the code To validate the puremixed convection case, comparison is performedwith the studied problem by Bhattacharya et al. (2013). The schematic of this problem is a two-dimensional trapezoidal enclosurewith cold topwall (lid), hot bottomwall and adiabatic inclinedwalls. Thedistribution of the Nusselt number at the bottom wall for Re = 100, Gr = 105 and Pr = 0.7 is shown in Fig. 2.We can conclude that there is a good consistency between the present numerical results and those reported by Bhattacharya et al. (2013). Fig. 2. Nusselt number profile at the bottomwall,RC =10, Re=100, Gr=105 and Pr=0.7 To investigate validity of thediscreteordinatesmethod, sinceno studyexists on thecombined heat transfer of mixed convection and radiation in a lid-driven cavity, a combined conductive- radiative heat transfer problem is considered. This problem was studied by Mahapatra et al. (2006). The schematic is a square enclosure of length L, containing an absorbing, emitting, and scattering medium. A plot of the mid-plane temperature inside the medium for RC = 10, ε=1.0, τ = 1.0 and ω =0.5 is shown in Fig. 3. It is shown that the present numerical results are in accordance with the theoretical ones reported by Mahapatra et al. (2006). Also, to get grid-independent solutions, the maximum total Nusselt number at the bottom wall for three different radiation-conduction parameters and optical thicknesses for three cases are shown in Table 1. As it is seen, the amount of the total Nusselt number is different only in the second or third decimal places. Therefore, we used a uniform mesh with 90× 80 grid numbers for all cases. Also, themaximumtotal Nusselt number at the bottomwall withS4 andS6 approximations at Re = 100, Gr = 105, Pr = 0.7 is shown in Table 2. These results demonstrate that there is very little difference between S4 and S6 approximations. Thus, to reduce computational effort, S4 approximation is used for all cases. 648 M.Mohammadi, S.A. Gandjalikhan Nassab Fig. 3. Variation of mid-plane temperature, ε=1.0 and τ =1.0 Table 1.Grid independence study in terms of the total Nusselt number at Re=100, Gr=105 and Pr=0.7 Grid Size τ =0 τ =0.5 τ =1 RC =0.5 RC =2 RC =5 80×70 13.325 11.799 10.927 7.169 2.432 13.916 90×80 13.310 11.781 10.916 7.157 9.418 13.903 100×90 13.313 11.781 10.907 7.148 9.408 13.895 Table 2.Angular quadrature study in terms of the total Nusselt number atRe=100,Gr=105 and Pr=0.7 τ =0 τ =0.5 τ =1 RC =0.5 RC =2 RC =5 S4 13.310 11.781 10.916 7.157 9.418 13.903 S6 13.380 11.836 10.944 7.160 9.436 13.951 5. Results and discussion In this study, the effect of radiative heat transfer on the heat transfer and flow characteristics of the mixed convection in a lid-driven trapezoidal cavity is investigated numerically. Figures 4 and 5 indicate the isotherms and streamlines atRi= 10 for different optical thicknesses and also for the pure mixed convection case. Optical thickness τ = 0 shows the results for a radiatively transparent medium. These figures demonstrate the sweep behavior on the isotherms and stre- amlines. An increase in the optical thickness from τ =0 to τ =2makes the difference between isotherms for the puremixed convection and the radiative case increases as well. But when the optical thickness increases from τ =2 to τ =100, the isotherms of pure mixed convection case will be obtained. In fact, for high values of optical thickness, the radiation intensity cannot permeate the cavity, and it is absorbed near the walls. This phenomenon is also seen for the streamlines along the optical thickness, such that by increasing the optical thickness from τ =0 to τ =10 the difference between streamlines for the puremixed convection and the radiative case increases. But by further increasing of the optical thickness the reverse behavior is observed. In Fig. 6, the effect of radiation-conduction parameter (RC) on the thermal behavior of the system at Ri = 10 is shown. This parameter shows the relative importance of the radiative heat transfer mechanism compared with its conduction counterpart. As can be seen in Fig. 6, as the RC parameter increases due to changing the dominant heat transfer mechanism from conduction to radiation, the difference between the isotherms for the radiative and pure mixed convection case increases. The combined heat transfer of radiation and mixed convection... 649 Fig. 4. Isotherms for different optical thicknesses and the pure mixed convection case at Ri= 10 Fig. 5. Streamlines for different optical thicknesses and pure mixed convection at Ri= 10 650 M.Mohammadi, S.A. Gandjalikhan Nassab Fig. 6. Isotherms for different radiation-conduction parameters and pure mixed convection at Ri=10 Fig. 7. The effect of optical thicknesses on the Nusselt number distribution along the bottomwall; (a) total Nusselt number, (b) radiative Nusselt number, (c) convective Nusselt number The radiative, convective and total Nusselt numbers along the bottom wall for different optical thicknesses at Ri = 10 are shown in Figs. 7a-7c. As it is seen in Fig. 7a, as optical thickness increases, the total Nusselt number decreases. This figure shows that the radiatively transparent medium (τ = 0) has the maximum amount of the total Nusselt number and the puremixed convection case has the smallest total Nusselt number.Therefore, it can bededucted that the radiatively transparent medium has the maximum amount of thermal exchange with the environment. Figure 7b shows that as the optical thickness increases, the radiative Nusselt The combined heat transfer of radiation and mixed convection... 651 number decreases. This is because of the increasing gas absorption, hence the radiative heat flux at the bottomwall decreases. Figure 7c indicates that there is sweep behavior in the convective Nusselt number. By increasing the optical thickness from τ =0 to τ =5, the convective Nusselt number decreases at the core of the cavity (0.5