A Mixed Convection Study in Inclined Channels with Discrete Heat Sources 19 plays a considerable role on the second module heating. For instance, what has just been said happens in the case where Re = 1000 and d = 1, 2, and 3. Gr =10 3 γ (degree) 0 20406080 Nu 1 2 3 4 5 6 7 Re=1 Re=5 Re=10 Re=50 Re=100 (a) Gr = 10 4 γ (degree) 020406080 N u 2 3 4 5 6 7 8 9 Re=1 Re=5 Re=10 Re=50 Re=100 Re=200 (b) Convection and Conduction Heat Transfer 20 Gr = 10 5 γ(degree) 0 20406080 Nu 4 5 6 7 8 9 10 11 12 Re = 1 Re = 5 Re = 10 Re = 50 Re =100 Re = 200 Re = 500 (c) Fig. 12. Average Nusselt vs γ: 1 ≤ Re ≤ 500 and (a) Gr = 10 3 , (b) Gr = 10 4 , and (c) Gr = 10 5 Re=1 d=1 Re = 10 d = 1 Re=100 d=1 Re=1 d=2 Re=10 d=2 Re=100 d=2 Re=1 d=3 Re=10 d=3 Re=100 d=3 Fig. 13. Velocity vector for Gr = 10 5 , Re = 1, 10, 100, and d = 1, 2, 3 A Mixed Convection Study in Inclined Channels with Discrete Heat Sources 21 0 . 0 2 0 0 . 0 2 0 0 . 0 2 0 0 . 0 4 0 0 . 0 4 0 0 . 0 4 0 0 . 0 6 1 0 . 0 6 1 0 . 0 6 1 0 . 0 6 1 0 . 0 8 1 0 . 0 8 1 0 . 0 8 1 0 . 0 8 1 0 . 1 0 1 0 . 1 0 1 0 . 1 2 1 0 . 1 2 1 0 . 1 2 1 0 . 1 4 1 0 . 1 4 1 0 . 1 6 2 0.024 0 . 02 4 0 . 0 2 4 0 . 0 4 7 0 . 0 4 7 0 . 0 7 1 0 . 0 7 1 0 . 09 5 0 . 0 9 5 0 . 1 1 8 0 . 1 1 8 0 . 1 4 2 0 . 1 4 2 0 . 1 6 6 0 . 1 8 9 0 . 2 3 7 0 . 0 1 9 0 . 0 1 9 0 . 0 3 8 0 . 0 3 8 0 . 0 5 7 0 . 0 7 7 0 .0 9 6 0 . 1 3 4 0 . 1 3 4 0 . 1 5 3 0 . 1 7 2 0 . 1 9 1 0 . 1 9 1 0 . 2 1 0 0 . 0 1 6 0 . 0 1 6 0 . 0 3 2 0 . 0 4 8 0 . 0 6 5 0 . 0 9 7 0 . 1 1 3 0 . 1 2 9 0 . 1 6 1 0.007 0.007 0.007 0.015 0.015 0.015 0.022 0.022 0.022 0.029 0.029 0.029 0.029 0.036 0.036 0.044 0.0440.051 0.058 0.058 0.065 0.073 0 . 0 2 0 . 0 2 0 . 0 2 0.02 0 . 0 4 0 . 0 4 0 . 0 4 0 . 0 4 0 . 0 4 0 . 0 4 0 . 0 6 0 . 0 6 0 . 0 6 0 . 0 6 0 . 0 6 0 . 0 8 0 . 0 8 0 . 0 8 0 . 0 8 0 . 0 8 0 . 0 8 0 . 1 0 . 1 0 . 1 0. 1 0 . 1 0 . 1 2 0 . 1 2 0 . 1 2 0 . 1 2 0 . 1 4 0 . 1 4 0 . 1 6 0 . 1 8 0 . 1 8 0 . 2 2 0 . 2 4 0 . 0 2 4 0 . 0 2 4 0 . 0 2 4 0 . 0 2 4 0 . 0 4 7 0 . 0 4 7 0 . 0 4 7 0.047 0 . 0 4 7 0 . 0 7 1 0 . 0 7 1 0 . 0 7 1 0 . 0 7 1 0 . 0 7 1 0 . 0 9 4 0 . 0 9 4 0 . 0 9 4 0 . 0 9 4 0 . 1 1 8 0 . 1 1 8 0 . 1 1 8 0 . 1 1 8 0 . 1 41 0 . 1 4 1 0 . 1 6 5 0 . 2 1 2 0 . 0 2 0 0 . 0 4 0 0 . 0 4 0 0 . 0 6 0 0 . 0 6 0 0 . 0 8 0 0 .1 0 0 0 . 1 2 0 0 . 1 6 0 0 . 0 1 7 0 . 0 3 3 0 . 0 5 0 0 .0 5 0 0.067 0.084 0 . 1 0 0 0 . 1 1 7 0 . 1 5 0 0 . 2 0 1 0 . 0 0 8 0 . 0 3 0 0 . 0 3 8 0 . 0 4 5 0 . 0 9 1 0 . 0 2 1 0 . 0 2 1 0 . 0 2 1 0 . 0 2 1 0 . 0 2 1 0 .0 2 1 0 . 0 4 2 0 . 0 42 0 . 0 4 2 0 . 0 4 2 0 . 0 4 2 0 . 0 4 2 0 . 0 6 3 0 . 0 6 3 0 . 0 6 3 0 . 0 6 3 0 . 0 6 3 0 . 0 8 4 0 . 0 8 4 0 . 0 8 4 0 . 0 8 4 0 . 0 8 4 0. 08 4 0 . 1 0 5 0 . 1 0 5 0 . 1 0 5 0 . 1 0 5 0 . 1 2 6 0 . 1 2 6 0 . 1 2 6 0 . 1 4 7 0 . 1 4 7 0 . 1 6 8 0 . 1 8 9 0 . 1 8 9 0 . 2 1 0 0 . 2 3 1 0 . 0 2 2 0 . 0 2 2 0 . 0 2 2 0 . 0 2 2 0 . 0 4 5 0 . 0 4 5 0 . 0 4 5 0 . 0 4 5 0.045 0 . 0 6 7 0 .0 6 7 0 . 0 6 7 0.067 0 . 0 6 7 0 . 0 8 9 0 . 0 8 9 0 . 0 8 9 0 . 0 8 9 0 . 1 1 2 0 . 1 3 4 0 . 1 5 7 0 . 1 5 7 0 . 1 7 9 0 . 1 7 9 0 . 2 0 1 0 . 2 2 4 0 . 2 4 6 0 . 0 2 0 0 . 0 3 9 0.059 0 . 0 5 9 0.079 0 . 0 9 8 0 . 1 1 8 0 . 1 5 7 0 . 1 7 7 0 . 2 1 6 0 . 2 5 5 0 . 0 1 6 0 . 0 3 3 0.049 0.049 0 . 0 6 5 0 .0 6 5 0 . 0 8 2 0 . 0 9 8 0 . 1 1 5 0 . 1 1 5 0 . 1 3 1 0 . 1 8 0 0 . 1 9 6 0 . 2 1 3 0 . 0 0 8 0 . 0 1 5 0.030 0 . 0 3 8 0 . 0 5 3 Fig. 14. Isotherms for Re = 1, 10, 100, 1000, γ = 0° , and Gr = 10 5 , Δθ = 0.02 Re = 1 d = 1 Re = 10 d = 1 Re = 50 d = 1 Re = 1 d = 2 Re = 1000 d = 1 Re = 100 d = 1 Re = 100 d = 2 Re = 50 d = 2 Re = 10 d = 2 Re = 10 d = 3 Re = 1 d = 3 Re = 1000 d = 2 Re = 50 d = 3 Re = 100 d = 3 Re = 1000 d = 3 Convection and Conduction Heat Transfer 22 Gr=10 3 -Heater1 1 10 100 1000 0 2 4 6 8 10 12 14 d=1 d=2 d=3 Gr=10 3 - Heater 2 1 10 100 1000 Nu 0 2 4 6 8 10 12 d=1 d=2 d=3 Gr=10 4 - Heater 1 1 10 100 1000 0 2 4 6 8 10 12 14 d=1 d=2 d=3 Gr=10 4 - Heater 2 1 10 100 1000 NU 0 2 4 6 8 10 12 d=1 d=2 d=3 Gr = 10 5 - Heater 1 Re 1 10 100 1000 0 2 4 6 8 10 12 14 d=1 d=2 d=3 Gr=10 5 - Heater 2 Re 1 10 100 1000 NU 0 2 4 6 8 10 12 d=1 d=2 d=3 Fig. 15. Nu for Re = 1, 10, 10 2 , 10 3 , d = 1, 2, 3, Gr = 10 5 , 10 4 , and 10 5 on Heater 1 and 2 Figure (15) depicts the effect of the Reynolds number on heat transfer for Gr = 10 3 , 10 4 , and 10 5 , Re = 1, 10, 100, and 1000, and finally, d = 1, 2, 3, in the pair of heat sources. This picture shows some points already discussed previously such as the module distance effect which is almost negligible on Heater 1 and moderate on Heater 2. It can be clearly seen the balance between forced and natural convections. In a general way, the distance d = 3 is the one which offers better work conditions since the temperatures are lower. Figure (16) presents the temperature distributions θ on Heater 1 and 2 for Re = 100 and 1000 and d = 1, 2, and 3. The distance between the modules does not affect the temperature on Heater 1 whereas this effect can be distinctively seen on Heater 2. It is interesting noticing in Heater 2, that for Re = 100 and 1000, distances d = 2 and 3 do not present significant changes, but d = 1. Then, there is an optimum distance in which two heat sources can be placed apart to have lower temperatures and this is the case of d = 3 here, although d = 2 does not present a meaningful change in temperature either. This, in a certain way, can lead us to a better layout of the heat sources in an array. Of course, the presence of more heat sources and the geometry of the channel must be taken into account. Anyway, this behavior is food for thought for future studies. Finally, the time distribution of the Nusselt number along Heater 1 and 2 for Gr = 10 5 , Re = 10, 100, and 1000, d = 1, 2, and 3, is shown in Fig. (17). In all cases, as expected, the first A Mixed Convection Study in Inclined Channels with Discrete Heat Sources 23 Re = 100, Gr = 10 5 Heater 1 θ 0,00 0,05 0,10 0,15 0,20 0,25 d = 1 d = 2 d = 3 Re = 1000, Gr = 10 5 Heater 1 θ 0,00 0,05 0,10 0,15 0,20 0,25 d = 1 d = 2 d = 3 Re = 100, Gr = 10 5 Heater 2 θ 0,06 0,08 0,10 0,12 0,14 0,16 0,18 0,20 0,22 0,24 d = 1 d = 2 d = 3 Re = 1000, Gr = 10 5 Heater 2 θ 0,00 0,05 0,10 0,15 0,20 0,25 d = 1 d = 2 d = 3 Fig. 16. Temperature on modules 1 and 2 for d = 1, 2, 3; Re = 100, 1000, and Gr = 10 5 module is submitted to higher heat transfer since it is constantly been bombarded with cold fluid from the forced convection. On the other hand, it can be seen again that a flow wake from the first source reaches the second one and this is responsible for the bifurcation of the Nusselt number curves. Here, one can note the time spent by the hot fluid coming from the first source and traveling to the second one. For example, for Re = 100 and d = 1, 2, and 3, the time shots are, respectively, around t = 1.4, 3.0, and 4.0. However, the converged values for these last cases are almost the same. As seen earlier, periodic oscillations appear for Re = 10. 5.3 Case with three heat sources The results presented here are obtained using the finite element method (FEM) and a structured mesh with rectangular isoparametric four-node elements in which ΔX = 0.1 and ΔY = 0.05. A mesh sensibility analysis was carried out (Guimaraes, 2008). The temperature distributions for Reynolds numbers Re = 1, 10, 50, and 100, Grashof number Gr = 10 5 , and inclination angles γ = 0° (horizontal), 45°, and 90° (vertical) are available in Fig. (18). For Re = 1 and γ = 0° and 45°, there is a formation of thermal cells which are localized in regions close to the modules. When Re = 1, the flow is predominantly due to natural convection. As Re is increased, these cells are stretched and hence forced convection starts to be characterized. By keeping Re constant, the inclination angle variation plays an important role on the temperature distribution. The effect of γ on temperature is stronger when low velocities are present. For example, when Re = 10 and γ = 0°, 45° and 90°, this behavior is noted, that is, for γ = 0° and Re = 10, a thermal cell is almost present, however, for γ = 45°and Re = 10, those cells vanish. This is more evident when Re =1 and γ = 45° and 90°. Convection and Conduction Heat Transfer 24 Re = 10, d = 1 t 2468101214 NU 5 10 15 20 25 30 Heater 1 Heater 2 Re = 100, d = 1 t 2468101214 NU 5 10 15 20 25 30 Heater 1 Heater 2 Re = 1000, d = 1 t 2468101214 NU 5 10 15 20 25 30 Heater 1 Heater 2 Re = 10, d = 2 t 5101520 NU 5 10 15 20 25 30 Heater 1 Heater 2 Re = 100, d = 2 t 5101520 NU 5 10 15 20 25 30 Heater 1 Heater 2 Re = 1000, d = 2 t 5101520 NU 5 10 15 20 25 30 Heater 1 Heater 2 Re = 10, d = 3 t 5101520 NU 5 10 15 20 25 30 Heater 1 Heater 2 Re = 100, d = 3 t 5101520 NU 5 10 15 20 25 30 Hea ter 1 Hea ter 2 Re = 1000, d = 3 t 5101520 NU 5 10 15 20 25 30 Heater 1 Heater 2 Fig. 17. Average Nusselt number vs Time: Gr = 10 5 , Re = 10, 10 2 , 10 3 , d =1, 2 , 3, Heater 1, 2 A Mixed Convection Study in Inclined Channels with Discrete Heat Sources 25 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 0.5 1 0.02 0.04 0 . 0 6 0 . 0 8 0.02 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0 . 1 0.02 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0 0.5 1 0 . 0 2 0 . 0 4 0 . 0 4 0 . 0 4 0 . 0 6 0 . 0 6 0.06 0 . 0 8 0.08 0 . 0 8 0 0.5 1 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 4 0 . 0 6 0 . 0 8 0 . 0 4 0 . 0 6 0 . 0 8 0 0.5 1 0 . 0 2 0 . 0 2 0 .0 2 0 . 0 4 0 . 0 4 0 . 0 4 0 . 0 6 0 . 0 6 0 . 0 6 0 0.5 1 0.08 0.10 0.10 0 . 0 6 0.06 0.04 0.02 0 . 0 4 0 . 0 4 0 . 0 6 0 . 0 8 0 . 0 8 0 . 0 2 0 . 0 2 0 0.5 1 0.02 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0 0 . 0 6 0 . 0 8 0 . 1 0 0 . 0 4 0 . 0 6 0 . 0 8 0 0.5 1 0.02 0.02 0.04 0.06 0 . 0 8 0.04 0.06 0.08 0.02 0.04 0 . 0 6 0 0.5 1 0.02 0 . 0 4 0 . 0 6 0.08 0 . 0 6 0 . 0 8 0 . 1 0 0 . 1 2 0 . 1 0 0 . 0 8 0 . 1 0 0 0.5 1 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 6 0 . 0 8 0 . 1 0 0 . 0 8 0 . 0 6 0 . 0 8 0 . 1 0 0 0.5 1 0.02 0 . 0 4 0 . 0 4 0.06 0 . 0 6 0 . 0 8 0.10 0 . 0 8 0 . 0 6 0 . 0 8 0 0.5 1 0.02 0 . 0 4 0.04 0.04 0.06 0 . 0 6 0 . 0 6 0 . 0 8 0 . 0 8 0 0.5 1 Re = 1 γ = 0 0 Re = 10 γ = 0 0 Re = 50 γ = 0 0 Re = 100 γ = 0 0 Re = 1 γ = 45 0 Re = 10 γ = 45 0 Re = 50 γ = 45 0 Re = 100 γ = 45 0 Re = 1 γ = 90 0 Re = 10 γ = 90 0 Re = 50 γ = 90 0 Re = 100 γ = 90 0 Fig. 18. Isotherms for Gr = 10 5 , Re = 1, 10, 50, 100 and γ = 0°, 45°, 90° Convection and Conduction Heat Transfer 26 It is worth observing that, the fluid heated in the first heater reaches the second one, and then the third one. Thus, this process of increasing temperature provides undesirable situations when cooling is aimed. Re = 10, γ = 0° Re = 100, γ = 0° Re = 10, γ = 45° Re = 100, γ = 45° Re = 10, γ = 90° Re = 100, γ = 90° Fig. 19. Velocity vectors for Gr = 105, Re = 10 and 100, and γ = 0°, 45°and 90° Figure (19) depicts the velocity vectors for Re = 10 and 100 and Gr = 10 5 for γ = 0°, 45°, and 90°. It can be noted that for Re = 10 and γ = 0°, 45°, and 90°, recirculations are generated by the fluid heated on the sources. For Re = 10 and γ = 0°, three independent recirculations appear. The distance among the heat sources enables the reorganization of the velocity profile until the fluid reaches the next source and then the recirculation process starts all over again. Now, concerning the cases where Re = 10 and γ = 45° and 90°, there are two kinds of recirculations, that is, a primary recirculation along all channel that encompasses A Mixed Convection Study in Inclined Channels with Discrete Heat Sources 27 another two secondary recirculations localized just after the sources. Moreover, for these later cases, a reversal fluid flow is present at the outlet. As Re is increased by keeping γ constant, these recirculations get weaker until they disappear for high Re. Clearly, one can note the effect of the inclination on the velocity vectors when Re = 10. The strongest inclination influence takes place when it is between 0° and 45°. Figure (20) presents the average Nusselt number distributions on the heat sources, NU H1 , NU H2 , and NU H3 for Reynolds numbers Re = 1, 10, 50, 100, and 1000, Grashof numbers Gr = 10 3 , 10 4 , and 10 5 , and inclination angles γ = 0°, 45°, and 90°. In general, the average Nusselt number for each source increases as the Reynolds number is increased. By analyzing each graphic separately, it can be observed that NU H1 tends to become more distant from NU H2 and NU H3 as Reynolds number is increased, starting from an initial value for Re = 1 which is almost equal to NU H2 and NU H3 . This agreement at the beginning means that the heaters are not affecting one another. Here, it can be better perceived that behavior found in Fig. (13), where a heater is affected by an upstream one. That is the reason why NU H1 shows higher values. The only case in which the heaters show different values for Re = 1 is when Gr = 10 5 and γ = 90°. Overall, the strongest average Nusselt number variation is between 0° and 45°. Practically in all cases, NU H1 , NU H2 ,and NU H3 increase in this angle range, 0° and 45°, while for Gr = 10 5 and Re = 1000, NU H2 and NU H3 decrease. When electronic circuits are concerned, the ideal case is the one which has the highest Nusselt number. Thus, angles 45° and 90° are the most suitable ones with not so Re 10 0 10 1 10 2 10 3 NU 0 2 4 6 8 10 12 14 16 NUH1 NUH2 NUH3 Gr=10 3 , γ = 0° Re 10 0 10 1 10 2 10 3 NU 0 2 4 6 8 10 12 14 16 NUH! NUH2 NUH3 Gr=10 3 , γ = 45° Re 10 0 10 1 10 2 10 3 Nu 0 2 4 6 8 10 12 14 16 NUH1 NUH2 NUH3 Gr=10 3 , γ = 90° Re 10 0 10 1 10 2 10 3 NU 0 2 4 6 8 10 12 14 16 NUH1 NUH2 NUH3 Gr=10 4 , γ = 0° Re 10 0 10 1 10 2 10 3 NU 0 2 4 6 8 10 12 14 16 NUH1 NUH2 NUH3 Gr=10 4 , γ = 45° Re 10 0 10 1 10 2 10 3 NU 0 2 4 6 8 10 12 14 16 NUH1 NUH2 NUH3 Re 10 0 10 1 10 2 10 3 NU 0 2 4 6 8 10 12 14 16 NUH1 NUH2 NUH3 Gr=10 5 , γ = 0° Gr=10 4 , γ = 90° Re 10 0 10 1 10 2 10 3 NU 0 2 4 6 8 10 12 14 16 NUH1 NUH2 NUH3 Gr=10 5 , γ = 45° Re 10 0 10 1 10 2 10 3 NU 0 2 4 6 8 10 12 14 16 NUH1 NUH2 NUH3 Gr=10 5 , γ = 90° Fig. 20. Average Nusselt number vs Reynolds number for Gr = 10 3 , 10 4 , 10 5 , γ = 0°, 45°, 90° Convection and Conduction Heat Transfer 28 much difference between them. An exception would be the case where Gr = 10 5 , Re = 1000, and γ = 0°. Figure (21) presents the local dimensionless temperature distributions on the three heat sources for Re = 10, 100, 1000, Gr = 10 5 , γ = 0°, 45°, and 90°. Again, the cases where Re = 10 and 100 show the lowest temperatures when γ = 90°. On the other hand, this does not happen when Re = 1000, where the horizontal position shows the lowest temperatures along the modules. All cases in which γ = 0°, the second and third sources have equal temperatures. However, the first module shows lower temperatures. As mentioned before, this characterizes the fluid being heated by a previous heat source, thus, not contributing to the cooling of an upstream one. Figure (22) presents the average Nusselt number variation on H 1 , H 2 , and H 3 against the dimensionless time t considering Re = 10, 100, Gr = 10 3 , 10 4 , 10 5 and γ = 90°. In the beginning, all three Nusselt numbers on H 1 , H 2 , and H 3 have the same behavior and value. These numbers tend to converge to different values as time goes on. However, before they do so, they bifurcate at a certain point. This denotes the moment when a heated fluid wake from a previous source reaches a downstream one. Re=10, Gr=10 5 , γ=0° θ 0,00 0,05 0,10 0,15 0,20 0,25 0,30 Sour ce 1 Sour ce 2 Sour ce 3 Re=10, Gr=10 5 , γ=45° Sou rce θ 0,05 0,10 0,15 0,20 0,25 0 , 30 Sour ce 1 Sour ce 2 Sour ce 3 Re=10, Gr=10 5 , γ=90° Source θ 0,05 0,10 0,15 0,20 0,25 0 , 30 Sour ce 1 Sour ce 2 Sour ce 3 Re=100, Gr=10 5 , γ=0° Sour ce θ 0,00 0,05 0,10 0,15 0,20 0,25 0,30 Sour ce1 Sour ce2 Sour ce3 Re=100, Gr=10 5 , γ=45° Sour ce θ 0,00 0,05 0,10 0,15 0,20 0,25 0,30 Source1 Source2 Source3 Re=100, Gr=10 5 , γ=90° Sour ce θ 0,00 0,05 0,10 0,15 0,20 0,25 0,30 Sour ce1 Sour ce2 Sour ce3 Re=1000, Gr=10 5 , γ=0° Sour ce θ 0,00 0,02 0,04 0,06 0,08 0,10 0,12 Sour ce 1 Sour ce 2 Sour ce 3 Re=1000, Gr=10 5 , γ=45° Sour ce θ 0,00 0,02 0,04 0,06 0,08 0,10 0,12 Source1 Source2 Source3 Re=1000, Gr=10 5 , γ=90° Sou rce θ 0,00 0,02 0,04 0,06 0,08 0,10 0,12 Sour ce 1 Sour ce 2 Sour ce 3 Source Fig. 21. Module temperatures for Re = 10, 100, 1000; Gr = 105, γ = 0°, 45°, 90° [...]... Transfer, Vol 32, pp 124 4– 125 2 Binet, B & Lacroix, M (20 00) Melting from heat sources flush mounted on a conducting vertical wall, Int J of Numerical Methods for Heat and Fluid Flow, Vol 10, pp 28 6–306 Baskaya, S.; Erturhan, U.; Sivrioglu, M (20 05) An experimental study on convection heat transfer from an array of discrete heat sources, Int Comm in Heat and Mass Transfer, Vol 32, pp 24 8 25 7 Da Silva,.. .29 A Mixed Convection Study in Inclined Channels with Discrete Heat Sources 10 10 8 NUM NUM 8 H1 H2 H3 , R e = 10, G r = 103 , γ = 90° 6 4 2 4 0 5 10 15 20 25 30 2 10 20 30 40 6 60 t 6 4 2 0 5 10 15 20 25 2 t 10 H1 H2 H3 4 R e = 100, G r = 10 , γ = 90° 0 10 20 30 40 50 60 70 t 10 H1 H2 H3 R e = 1 0 , G r = 1 0 5, γ = 90° 8 8 NUM 6 6 H1 H2 H3 5 R e = 100, G r = 10 , γ = 90° 4 2 50 8 H1 H2 H3 4... ρ ∂y 2 u 2 u + 2 2 ∂x ∂y 2 v 2 v + 2 2 ∂x ∂y ∂T ∂T ∂T +u +v =κ ∂t ∂x ∂y + gβ ( T − T0 ) 2 T 2 T + 2 2 ∂x ∂y (2) (3) (4) where u and v are the velocity components along x − and y−directions, t is the time, p is the pressure, ν, ρ, β and κ are kinematic viscosity, density of the fluid, coefficient of thermal expansion and thermal diffusivity respectively, g is the acceleration due to gravity and T... weak heat transfer and a period with intensive heat transfer for each aspect ratio The weak heat transfer corresponds to the day time condition when the heat transfer is dominated by conduction and the strong heat transfer corresponds to the night-time condition when convection dominates the flows and the instabilities occur in the form of rising and sinking plumes During the day time the heat transfer. .. conjugate mixed convection in a vertical channel with heat generating components, International Journal of Heat and Mass Transfer, Vol 50, pp 3561–3574 Muftuoglu, A & Bilgen E (20 07) Conjugate heat transfer in open cavities with a discrete heater at its optimized position, International Journal of Heat and Mass Transfer, doi:10.1016/j.ijheatmasstransfer .20 07.04.017 Premachandran, B & Balaji C (20 06) Conjugate... that there is not much difference in heat transfer for 48 16 Convection and Conduction Heat Transfer Will-be-set-by-IN-TECH the aspect ratios A = 0.5 and 0 .2 The highest average Nusselt numbers for A = 1.0, 0.5 and 0 .2 are 6.55, 8. 72 and 8.76 respectively 5.4 Effects of Rayleigh number on the flow response and heat transfer Fig.10 shows snapshots of stream function and temperature contours for the aspect... mixed convection with surface radiation from a horizontal channel with protruding heat sources, International Journal of Heat and Mass Transfer, Vol 49, pp 3568–35 82 Dogan, A.; Sivrioglu, M.; Baskaya S (20 05) Experimental investigation of mixed convection heat transfer in a rectangular channel with discrete heat sources at the top and at the bottom, International Communications in Heat and Mass Transfer, ... Table 2 shows the scaling values of the filling-up times for sudden and ramp heating/cooling boundary conditions for different A and Ra It is seen that the heating-up or cooling-down times for the sudden heating/cooling boundary condition for A = 0.5 and Ra = 1.5 × 106 is 42. 39s and for Ra = 7 .2 × 103 and the same aspect ratio is 159.01s For aspect ratios 0 .2 and 1.0 the filling-up times are 83 .24 s and. .. al (20 10a) is tsr = tfr = h(1 + A2 )1 /2 − Ax1 2 A1 /2 κRa1/4 (1 + A2 )5/4 , (8) where x1 is given by ⎡ x 1 ∼ L ⎣1 − κA1 /2 Ra1/4 (1 + A2 )1/4 tp 1− h2 1 /2 ⎤ ⎦, (9) However, if the cavity is filled with cold fluid before the ramp is finished then the filling up time is given in Saha et al (20 10a) as tfq ∼ h8/7 t3/7 p κ4/7 Ra1/7 A2/7 (1 + A2 )1/7 , (10) Table 1 presents the scaling values of the steady and. .. by conduction and the strong heat transfer corresponds to the night-time condition At night, the boundary layers adjacent to the inclined walls and the bottom are unstable Therefore, sinking and rising plumes are formed in the inclined and horizontal boundary layers These plumes dominate 44 12 Convection and Conduction Heat Transfer Will-be-set-by-IN-TECH Fig 5 Average Nusselt number on the top and . 1 Heater 2 Re = 1000, d = 1 t 24 6810 121 4 NU 5 10 15 20 25 30 Heater 1 Heater 2 Re = 10, d = 2 t 5101 520 NU 5 10 15 20 25 30 Heater 1 Heater 2 Re = 100, d = 2 t 5101 520 NU 5 10 15 20 25 30 Heater. and γ = 45° and 90°. Convection and Conduction Heat Transfer 24 Re = 10, d = 1 t 24 6810 121 4 NU 5 10 15 20 25 30 Heater 1 Heater 2 Re = 100, d = 1 t 24 6810 121 4 NU 5 10 15 20 25 30 Heater. 2 t 5101 520 NU 5 10 15 20 25 30 Heater 1 Heater 2 Re = 1000, d = 2 t 5101 520 NU 5 10 15 20 25 30 Heater 1 Heater 2 Re = 10, d = 3 t 5101 520 NU 5 10 15 20 25 30 Heater 1 Heater 2 Re = 100, d = 3 t 5101 520 NU 5 10 15 20 25 30 Hea