Present research has been financially supported by the National Key Research and Development Program of the Ministry of Science and Technology of China (Grant No. 2018YFC0705201, Grant No. 2018YFB0904200), National Natural Science Foundation of China (Grant No. 51778504, Grant No. U1867221), Beijing Institute of Satellite Environmental Engineering (CAST-BISEE Grant No. CAST-BISEE2019-025), Joint Zhuzhou-Hunan Provincial Natural Science Foundation (Grant No. 2018JJ4064), National Defense Research Funds for the Central Universities (Grant No. 2042018gf0031, Wuhan University), and Teaching Research Program (Grant NO. 2019JG030, Wuhan University), and Shandong Provincial Natural Science Foundation (Grant No. ZR2018MEE035).
MHD convection of CuO-water nanofluid inside a differentially heated enclosure with porous fins placed at the hot sidewall is analyzed in the present study. Empirical correlating formulas are used to predict the effective thermal conductivity and dynamic viscosity of nanofluid. Results are presented in terms of streamlines, heatlines, isotherms and the average Nusselt number. These outcomes are outlined as follows: For high Darcy number and thermal conductivities ratio of porous medium, heat transfer rate increases by using more porous fins while this effect is opposite in tiny Darcy number and low thermal conductivities ratio of porous materials. This phenomenon occurs due to the dual effects of porous fins on heat conduction and convection. For high Darcy and Rayleigh numbers, case N sf shows the most effective performance of heat transfer, which could report 75.62 per cent heat transfer augmentation over the reference case. Meanwhile, case with eight porous fins has the best performance of heat transport for intermediate Darcy number and large thermal conductivities ratio of porous materials. The prediction of average Nusselt number decreases with an increase in Hartmann number. There also exist the optimal designs of porous fins and corresponding nanoparticles concentrations to maximize the heat transfer if some conditions were satisfied and Correlations of average Nusselt number have been established for these governing parameters and they will be beneficial for thermal design applications, such as the effective cooling methodology for future electronics simultaneously of size-minimization and energy-maximization. Present research has been financially supported by the National Key Research and Development Program of the Ministry of Science and Technology of China (Grant No. 2018YFC0705201, Grant No. 2018YFB0904200), National Natural Science Foundation of China (Grant No. 51778504, Grant No. U1867221), Beijing Institute of Satellite Environmental Engineering (CAST-BISEE Grant No. CAST-BISEE2019-025), Joint Zhuzhou-Hunan Provincial Natural Science Foundation (Grant No. 2018JJ4064), National Defense Research Funds for the Central Universities (Grant No. 2042018gf0031, Wuhan University), and Teaching Research Program (Grant NO. 2019JG030, Wuhan University), and Shandong Provincial Natural Science Foundation (Grant No. ZR2018MEE035). Figure 1. Schematic diagram of physical model and coordinate system Figure 2. Schematic diagram of special structure fin case N sf ( W fin = 1.0) Figure 3. The validation of streamlines and isotherms of natural convection heat transfer within a square enclosure filled with air at Ra = 10 7 and Pr = 0.71 Figure 4. The validation of streamlines and isotherms of natural convection heat transfer within a square enclosure with a solid fin centrally mounted at the heated wall at Ra = 10 5 , Pr = 0.71, ε = 0, λ eff = 10 4 , N = 1, L fin = 0.5 and W fin = 0.01 Figure 5. Comparison of the heat transfer enhancement ratio between present results and those presented by Cianfrini et al. (2015) at Ra = 10 5 , Pr = 7.0, d P = 25 nm and T h = 323 K Figure 6. Streamlines, isotherms and heatlines plots (from left to right) with different Rayleigh number for case N sf at λ s = 10 3 , Ha = 0, Da = 10 −1 and φ = 0 Figure 7. Streamlines, isotherms and heatlines plots (from left to right) with different Rayleigh number for case with eight porous fins at λ s = 10 3 , Ha = 0, Da = 10 −1 and φ = 0 Figure 8. Variations of average Nusselt number with different Rayleigh numbers and designs of porous fins at Ha = 0, Da = 10 −1 , λ s = 10 3 and φ = 0 Figure 9. Streamlines, isotherms and heatlines plots (from left to right) with different Darcy number for case N sf at Ra = 10 7 , Ha = 10, λ s = 10 3 , and φ = 0 Figure 10. Streamlines, isotherms and heatlines plots (from left to right) with different Darcy number for case with eight porous fins at Ra = 10 7 , Ha = 10, λ s = 10 3 and φ = 0 Figure 11. Variations of the average Nusselt number with Darcy number and designs of porous fins for different λ s at Ra = 10 7 , Ha = 0 and φ = 0 Figure 12. Streamlines, isotherms and heatlines plots (from left to right) with varying thermal conductivities of porous matrix for case N sf at Ra = 10 7 , Ha = 0, Da = 10 −1 and φ = 0 Figure 13. Streamlines, isotherms and heatlines plots (from left to right) with varying thermal conductivity of porous matrix for case with eight porous fins at Ra = 10 7 , Ha = 0, Da = 10 −1 and φ = 0 Figure 14. Variations of average Nusselt numbers with λ s and designs of porous for different Darcy number at Ra = 10 7 , Ha = 0 and φ = 0 Figure 15. Effect of Hartmann number on streamlines, isotherms and heatlines plots (from left to right) for case N sf at Ra = 10 7 , λ s = 10 3 , Da = 10 −2 and φ = 0.04 (Ha = 0 (…)) Figure 16. Effect of Hartmann number on streamlines, isotherms and heatlines plots (from left to right) for case with eight porous fins at Ra = 10 7 , λ s = 10 3 , Da = 10 −2 and φ = 0.04 (Ha = 0 (…)) Figure 17. Variations of average Nusselt numbers with Ha and designs of fins for various Darcy number at Ra = 10 7 , λ s = 10 3 and φ = 0.04 Figure 18. Effect of nanoparticles concentration on streamlines, isotherms and heatlines plots (from left to right) for case N sf at Ra = 10 7 , Ha = 10.0, λ s = 10 3 and Da = 10 −1 (nanofluid (—) and pure fluid (φ = 0) (…)) Figure 19. Effect of nanoparticles concentration on streamlines, isotherms and heatlines plots (from left to right) for case with eight porous fins at Ra = 10 7 , Ha = 10.0, λ s = 10 3 and Da = 10 −1 (nanofluid (—) and pure fluid (φ = 0) (…)) Figure 20. Variations of average Nusselt numbers with φ and designs of porous fins for different λ s at Ha = 0, Ra = 10 7 , Da = 10 −1 Figure 21. Average Nusselt number correlations of discrete heat transfer rates versus the equation (50) Table I. Thermo-physical properties of water and nanoparticles ρ (kg/m 3 ) C P (J/kgK) k (W/mK) β (K −1 ) d p (nm) σ (Ω·m) −1 water 997.1 4179 0.613 0.00021 − 0.05 CuO 6500 535.6 20 0.000051 25 10 −10 Table II. Grid independency for nanofluid at Ra =10 7 , Da = 10 −6 , λ s = 10 3 , Ha= 50 and φ = 0.04 Nusselt numbers for different grids and fins' design Case 102 × 102 202 × 202 302 × 302 402 × 402 N sf 14.371 14.203 14.132 14.087 N = 1 15.783 15.511 15.458 15.490 N = 2 15.365 15.147 15.086 15.072 N = 4 14.251 14.067 14.025 13.990 N = 8 11.029 10.919 10.858 10.828 Table III. Comparison of average nusselt numbers with benchmark results Ra φ Nu Ogut (2009) Present Case 1 Natural convection of CuO-water in a square enclosure 10 3 0 1.948 1.962 0.05 2.157 2.168 0.1 2.368 2.394 0.15 2.585 2.618 10 4 0 3.534 3.486 0.05 3.902 3.924 0.1 4.302 4.328 0.15 4.714 4.745 10 5 0 6.209 6.201 0.05 6.921 6.919 0.1 7.653 7.649 0.15 8.409 8.404 Case 2 Natural convection in a square cavity filled with air Ra = 10 3 Ra = 10 4 Ra = 10 5 Ra = 10 6 Present 1.117 2.246 4.536 8.922 Davis (1983) 1.118 2.243 4.519 8.799 Fusegi et al. (1991) 1.105 2.302 4.646 9.012 Barakos and Mitsoulis (1994) 1.108 2.201 4.430 8.754 Khanafer et al. (2015) 1.115 2.226 4.505 8.778 Table IV. Validation of Darcy–Forchheimer model Da Ra Nu ε = 0.4 ε = 0.6 Nithiarasu et al. (1997) Present Nithiarasu et al. (1997) Present 10 −4 10 5 1.067 1.064 1.071 1.067 10 6 2.550 2.606 2.725 2.719 10 7 7.810 7.895 8.183 8.235 5 × 10 7 13.820 14.055 15.567 15.851 10 −2 10 3 1.010 1.007 1.015 1.012 10 4 1.408 1.361 1.530 1.492 10 5 2.983 2.998 3.555 3.446 5 × 10 5 4.990 5.010 5.740 5.798 Table V. Comparison of average nusselt numbers between ours and those published Nu Da = 10 −6 Da = 10 −5 Da = 10 −4 Da = 10 −3 Da = 10 −2 Khanafer et al. (2015) 4.327 4.682 5.474 5.880 5.946 Present Work 4.325 4.587 5.489 5.920 5.988 Notes: ( Ra = 10 5 , ε = 0.9 and λ eff = 10 2 )
