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In this work, numerical simulation of natural convection of nanofluids within a square enclosure were investigated by using the non-dimensional lattice Boltzmann method (NDLBM). The effects of key governing parameters Knudsen number ($10^{-6} \leqslant Kn_{{\rm{f}},{\rm{s}}} \leqslant 10^4$), Rayleigh number ($10^3 \leqslant Ra_{{\rm{f}},{\rm{L}}} \leqslant 10^6$), and nanoparticle volume fraction ($10^{-2} \leqslant \phi_{\rm{s}} \leqslant 10^{-1}$) on the heat and mass transfer of nanofluids were discussed. The results show that in the low $Ra_{{\rm{f}},{\rm{L}}}$ conductive dominated regime, the nanoparticle size has little effect on heat transfer, whereas in the high $Ra_{{\rm{f}},{\rm{L}}}$ convective dominated regime, larger nanoparticle sizes significantly enhance flow intensity and heat transfer efficiency. As fixed $Ra_{{\rm{f}},{\rm{L}}}$ and $\phi_{\rm{s}}$, the heat transfer patterns change from conduction to convection dominated regimes with increasing $Kn_{{\rm{f}},{\rm{s}}}$. The influence of nanoparticle volume fraction was also investigated, and in convection dominated regime, the maximum heat transfer efficiency was achieved when $\phi_{\rm{s}} = 8 {\text{%}}$, balancing both thermal conduction and drag fore of nanofluids. Additionally, by analyzing the full maps of mean Nusselt number ($\overline {Nu}_{{\rm{f}},{\rm{L}}}$) and the enhancement ratio related to the base fluid ($Re_{{\rm{n}},{\rm{f}}}$), the maximum values of $\overline {Nu}_{{\rm{f}},{\rm{L}}}$ and $Re_{{\rm{n}},{\rm{f}}}$ occur when the nanoparticle size is $Kn_{{\rm{f}},{\rm{s}}} = 10^{-1}$ for both conductive and convective dominated regimes. To capture the effects of all key governing parameters on $\overline {Nu}_{{\rm{f}},{\rm{L}}}$, a new empirical correlation has been derived from the numerical results, providing deeper insights into how these parameters influence heat transfer performance.
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Keywords:
- Knudsen number /
- Nanofluid /
- Natural convection /
- lattice Boltzmann method
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图 1 物理模型和边界条件示意图
Figure 1. Sketch of the problem and boundary conditions.
图 2 不同网格数对平均努塞尔数$ \overline{Nu}_{{{\rm{f}}, {\rm{L}}}} $的影响
Figure 2. The influence of different mesh grid on the mean Nusselt number $ \overline{Nu}_{{{\rm{f}}, {\rm{L}}}} $.
图 4 在固定颗粒体积分数$ \phi_{\rm{s}} = 8 {\text{%}} $, 不同颗粒粒径$ 1 \times 10^{-4} \leqslant Kn_{{\rm{f}}, {\rm{s}}} \leqslant 1 \times 10^2 $和瑞利数$ 1 \times 10^{3} \leqslant Ra_{{{\rm{f}}, {\rm{L}}}} \leqslant 1 \times 10^6 $下的无量纲速度场流线分布图像 (a) $ Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^{3} $; (b) $ Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^{4} $; (c) $ Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^{5} $; (d) $ Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^{6} $
Figure 4. Dimensionless streamlines for different nanoparticle size $ 1 \times 10^{-4} \leqslant Kn_{{\rm{f}}, {\rm{s}}} \leqslant 1 \times 10^2 $ and Rayleigh number $ 1 \times $$ 10^{3} Ra_{{{\rm{f}}, {\rm{L}}}} \leqslant 1 \times 10^6 $ with fixed $ \phi_s = 8 {\text{%}} $: (a) $ Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^{3} $; (b) $ Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^{4} $; (c) $ Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^{5} $; (d) $ Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^{6} $
图 5 在固定颗粒体积分数$ \phi_{\rm{s}} = 8 {\text{%}} $, 不同颗粒粒径$ 1 \times 10^{-4} \leqslant Kn_{{\rm{f}}, {\rm{s}}} \leqslant 1 \times 10^2 $和瑞利数$ 1 \times 10^{3} \leqslant Ra_{{{\rm{f}}, {\rm{L}}}} \leqslant 1 \times 10^6 $下的无量纲温度场等温线分布图像 (a) $ Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^{3} $; (b) $ Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^{4} $; (c) $ Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^{5} $; (d) $ Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^{6} $
Figure 5. Dimensionless isotherms for different nanoparticle size $ 1 \times 10^{-4} \leqslant Kn_{{\rm{f}}, {\rm{s}}} \leqslant 1 \times 10^2 $ and Rayleigh number $ 1 \times $$ 10^{3} Ra_{{{\rm{f}}, {\rm{L}}}} \leqslant 1 \times 10^6 $ with fixed $ \phi_s = 8 {\text{%}} $: (a) $ Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^{3} $; (b) $ Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^{4} $; (c) $ Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^{5} $; (d) $ Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^{6} $
图 6 不同颗粒体积分数和瑞利数下, 平均努塞尔数$ \overline {Nu}_{{{\rm{f}}, {\rm{L}}}} $与克努森数$ 1 \times 10^{-6} \leqslant Kn_{{\rm{f}}, {\rm{s}}} \leqslant 1 \times 10^4 $关系 (a) $ Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^3 $; (b) $ Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^4 $; (c) $ Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^5 $; (d) $ Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^6 $
Figure 6. The mean Nusselt number $ \overline {Nu}_{{{\rm{f}}, {\rm{L}}}} $ versus Knudsen number $ 1 \times 10^{-6} \leqslant Kn_{{\rm{f}}, {\rm{s}}} \leqslant 1 \times 10^4 $ with different volume fraction and Rayleigh number: (a) $ Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^3 $; (b) $ Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^4 $; (c) $ Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^5 $; (d) $ Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^6 $.
图 7 不同瑞利数$ 1 \times 10^{3} \leqslant Ra_{{{\rm{f}}, {\rm{L}}}} \leqslant 1 \times 10^6 $和固定颗粒体积分数$ \phi_{\rm{s}} = 8{\text{%}} $下, 平均努塞尔数$ \overline {Nu}_{{{\rm{f}}, {\rm{L}}}} $与克努森数关系
Figure 7. The mean Nusselt number $ \overline {Nu}_{{{\rm{f}}, {\rm{L}}}} $ versus Knudsen number with different Rayleigh number $ 1 \times 10^{3} \leqslant $$ Ra_{{{\rm{f}}, {\rm{L}}}} \leqslant 1 \times 10^6 $ and fixed nanoparticle volume fraction $ \phi_{\rm{s}} = 8{\text{%}} $.
图 8 不同瑞利数$ 1 \times 10^{3} \leqslant Ra_{{{\rm{f}}, {\rm{L}}}} \leqslant 1 \times 10^6 $和固定颗粒体积分数$ \phi_{\rm{s}} = 8{\text{%}} $下, 纳米流体相较于基液传热增加率$ Re_{{\rm{n}}, {\rm{f}}} $与克努森数关系
Figure 8. The enhancement ratio $ Re_{{\rm{n}}, {\rm{f}}} $ versus Knudsen number with different Rayleigh number $ 1 \times 10^{3} \leqslant Ra_{{{\rm{f}}, {\rm{L}}}} \leqslant $$ 1 \times 10^6 $ and fixed nanoparticle volume fraction $ \phi_{\rm{s}} = 8{\text{%}} $.
图 9 不同瑞利数$ 1 \times 10^{3} \leqslant Ra_{{{\rm{f}}, {\rm{L}}}} \leqslant 1 \times 10^6 $和固定颗粒粒径$ Kn_{{\rm{f}}, {\rm{s}}} = 10^{-1} $下, 平均努塞尔数$ \overline {Nu}_{{{\rm{f}}, {\rm{L}}}} $与颗粒体积分数关系
Figure 9. The mean Nusselt number $ \overline {Nu}_{{{\rm{f}}, {\rm{L}}}} $ versus nanoparticle volume fraction with different Rayleigh number $ 1 \times 10^{3} \leqslant Ra_{{{\rm{f}}, {\rm{L}}}} \leqslant 1 \times 10^6 $ and fixed nanoparticle size $ Kn_{{\rm{f}}, {\rm{s}}} = 10^{-1} $.
图 10 不同瑞利数$ 1 \times 10^{3} \leqslant Ra_{{{\rm{f}}, {\rm{L}}}} \leqslant 1 \times 10^6 $和固定颗粒粒径$ Kn_{{\rm{f}}, {\rm{s}}} = 10^{-1} $下, 纳米流体相较于基液传热增加率$ Re_{{\rm{n}}, {\rm{f}}} $与颗粒体积分数关系
Figure 10. The enhancement ratio $ Re_{{\rm{n}}, {\rm{f}}} $ versus nanoparticle volume fraction with different Rayleigh number $ 1 \times $$ 10^{3} \leqslant Ra_{{{\rm{f}}, {\rm{L}}}} \leqslant 1 \times 10^6 $ and fixed nanoparticle size $ Kn_{{\rm{f}}, {\rm{s}}} = 10^{-1} $.
图 11 不同克努森数$ 1 \times 10^{-3} \leqslant Kn_{{\rm{f}}, {\rm{s}}} \leqslant 1 \times 10^2 $和固定颗粒体积分数$ \phi_{\rm{s}} = 8 {\text{%}} $下, 平均努塞尔数$ \overline {Nu}_{{{\rm{f}}, {\rm{L}}}} $与瑞利数关系
Figure 11. The mean Nusselt number $ \overline {Nu}_{{{\rm{f}}, {\rm{L}}}} $ versus Rayleigh number with different Knudsen number $ 1 \times 10^{-3} \leqslant Kn_{{\rm{f}}, {\rm{s}}} \leqslant $$ 1 \times 10^2 $ and fixed nanoparticle volume fraction $ \phi_{\rm{s}} = 8 {\text{%}} $.
图 12 不同克努森数$ 1 \times 10^{-3} \leqslant Kn_{{\rm{f}}, {\rm{s}}} \leqslant 1 \times 10^2 $和固定颗粒体积分数$ \phi_{\rm{s}} = 8 {\text{%}} $下, 纳米流体相较于基液传热增加率$ Re_{{\rm{n}}, {\rm{f}}} $与瑞利数关系
Figure 12. The enhancement ratio $ Re_{{\rm{n}}, {\rm{f}}} $ versus Rayleigh number with different Knudsen number $ 1 \times 10^{-3} \leqslant Kn_{{\rm{f}}, {\rm{s}}} \leqslant $$ 1 \times 10^2 $ and fixed nanoparticle volume fraction $ \phi_{\rm{s}} = 8 {\text{%}} $.
图 13 全参数范围下平均努塞尔数的对数函数$ \lg (\overline {Nu}_{{{\rm{f}}, {\rm{L}}}}) = $$ l \lg (\dfrac{{{k_{\rm{n}}}}}{{{k_{\rm{f}}}}}\overline {Nu}_{{{\rm{n}}, {\rm{L}}}}) $ (a) 和纳米流体相较基液传热增加率$ Re_{{\rm{n}}, {\rm{f}}} $ (b) 随不同克努森数$ Kn_{{\rm{f}}, {\rm{s}}} $、瑞利数$ Ra_{{{\rm{f}}, {\rm{L}}}} $、颗粒体积分数$ \phi_{\rm{s}} $变化的三维等值面图
Figure 13. The three dimensional isosurfaces of logarithmic function of mean Nusselt number $ \lg (\overline {Nu}_{{{\rm{f}}, {\rm{L}}}}) = \lg (\dfrac{{{k_{\rm{n}}}}}{{{k_{\rm{f}}}}}\overline {Nu}_{{{\rm{n}}, {\rm{L}}}}) $ (a) and enhancement ratio $ Re_{{\rm{n}}, {\rm{f}}} $ (b) over the full parameter range as a function of Knudsen number $ Kn_{{\rm{f}}, {\rm{s}}} $, Rayleigh number $ Ra_{{{\rm{f}}, {\rm{L}}}} $, and nanoparticle volume fraction $ \phi_{\rm{s}} $.
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