<p>This study dives into the natural convection (NC) of a Herschel–Bulkley fluid (HBF) that contains nano-encapsulated phase change materials (NEPCMs) within a cavity featuring three vertically heated indented plates. To carry out the analysis, we use a numerically stable cascaded lattice Boltzmann method (CLBM) based on central moments, which is enhanced by GPU computing. The indented plates are kept at a heated temperature, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$T_h$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>h</mi> </msub> </math></EquationSource> </InlineEquation>, while the left and right walls of the cavity are maintained at a cooler temperature, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$T_c$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mi>c</mi> </msub> </math></EquationSource> </InlineEquation>. The NEPCM nanoparticles have a core–shell structure, with the phase change material (PCM) concentrated closer to the core. This research aims to explore heat transfer mechanisms, phase change behavior, and the overall thermal performance. We take into account several key parameters, including the Bingham number (<InlineEquation ID="IEq3"> <EquationSource Format="TEX">$0 \le \text {Bn} \le 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>≤</mo> <mtext>Bn</mtext> <mo>≤</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>), power-law index (<InlineEquation ID="IEq4"> <EquationSource Format="TEX">$n=0.85$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>0.85</mn> </mrow> </math></EquationSource> </InlineEquation>), volume fraction (<InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\phi = 0.04$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϕ</mi> <mo>=</mo> <mn>0.04</mn> </mrow> </math></EquationSource> </InlineEquation>), Prandtl number (<InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\text {Pr}=16.6$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>Pr</mtext> <mo>=</mo> <mn>16.6</mn> </mrow> </math></EquationSource> </InlineEquation>), fusion temperature (<InlineEquation ID="IEq7"> <EquationSource Format="TEX">$0.2 \le \theta _f \le 0.8$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0.2</mn> <mo>≤</mo> <msub> <mi>θ</mi> <mi>f</mi> </msub> <mo>≤</mo> <mn>0.8</mn> </mrow> </math></EquationSource> </InlineEquation>), Rayleigh number (<InlineEquation ID="IEq8"> <EquationSource Format="TEX">$10^4 \le \text {Ra} \le 10^6$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mn>10</mn> <mn>4</mn> </msup> <mo>≤</mo> <mtext>Ra</mtext> <mo>≤</mo> <msup> <mn>10</mn> <mn>6</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>), and Stefan number (<InlineEquation ID="IEq9"> <EquationSource Format="TEX">$0.2 \le \text {Ste} \le 0.8$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0.2</mn> <mo>≤</mo> <mtext>Ste</mtext> <mo>≤</mo> <mn>0.8</mn> </mrow> </math></EquationSource> </InlineEquation>). Our results show that as Ra and fusion temperatures (<InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\theta _{f}=0.2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>θ</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>0.2</mn> </mrow> </math></EquationSource> </InlineEquation>, 0.6, 0.8) increase, the streamline patterns near the heated walls expand, while their intensity decreases near the cold walls. For <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\text {Ra}=10^6$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>Ra</mtext> <mo>=</mo> <msup> <mn>10</mn> <mn>6</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$0.2\le \text {Ste} \le 0.8$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0.2</mn> <mo>≤</mo> <mtext>Ste</mtext> <mo>≤</mo> <mn>0.8</mn> </mrow> </math></EquationSource> </InlineEquation>, the isothermal lines flatten out uniformly, which enhances heat transfer. With a constant <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\text {Bn}$</EquationSource> <EquationSource Format="MATHML"><math> <mtext>Bn</mtext> </math></EquationSource> </InlineEquation>, an increase in <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\text {Ra}$</EquationSource> <EquationSource Format="MATHML"><math> <mtext>Ra</mtext> </math></EquationSource> </InlineEquation> has a significant impact on heat capacity. The peak average Nusselt number (<InlineEquation ID="IEq15"> <EquationSource Format="TEX">$\overline{Nu}$</EquationSource> <EquationSource Format="MATHML"><math> <mover> <mrow> <mi mathvariant="italic">Nu</mi> </mrow> <mo>¯</mo> </mover> </math></EquationSource> </InlineEquation>) improves by 72.30% at the highest <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$\text {Ra}$</EquationSource> <EquationSource Format="MATHML"><math> <mtext>Ra</mtext> </math></EquationSource> </InlineEquation> and lowest <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$\text {Bn}$</EquationSource> <EquationSource Format="MATHML"><math> <mtext>Bn</mtext> </math></EquationSource> </InlineEquation>. We also developed a mathematical correlation for <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$\overline{Nu}$</EquationSource> <EquationSource Format="MATHML"><math> <mover> <mrow> <mi mathvariant="italic">Nu</mi> </mrow> <mo>¯</mo> </mover> </math></EquationSource> </InlineEquation>, which demonstrates excellent predictive accuracy.</p>

Simulation of Herschel–Bulkley Fluid Dispersed with Nano-Encapsulated Phase Change Materials for Prototyping Energy Storage with Three Vertical Heated Plates Using Cascaded Lattice Boltzmann Method

期刊论文
发表日期：

2025年10月9日

查看全文

Md. Mamun Molla,
Khairunnahar Suchana,
Azad Rahman,
Sadia Siddiqa

摘要

This study dives into the natural convection (NC) of a Herschel–Bulkley fluid (HBF) that contains nano-encapsulated phase change materials (NEPCMs) within a cavity featuring three vertically heated indented plates. To carry out the analysis, we use a numerically stable cascaded lattice Boltzmann method (CLBM) based on central moments, which is enhanced by GPU computing. The indented plates are kept at a heated temperature, $T_h$ $T_{h}$ , while the left and right walls of the cavity are maintained at a cooler temperature, $T_c$ $T_{c}$ . The NEPCM nanoparticles have a core–shell structure, with the phase change material (PCM) concentrated closer to the core. This research aims to explore heat transfer mechanisms, phase change behavior, and the overall thermal performance. We take into account several key parameters, including the Bingham number ( $0 \le \text {Bn} \le 1$ $0 \leq Bn \leq 1$ ), power-law index ( $n=0.85$ $n = 0.85$ ), volume fraction ( $\phi = 0.04$ $ϕ = 0.04$ ), Prandtl number ( $\text {Pr}=16.6$ $Pr = 16.6$ ), fusion temperature ( $0.2 \le \theta _f \le 0.8$ $0.2 \leq θ_{f} \leq 0.8$ ), Rayleigh number ( $10^4 \le \text {Ra} \le 10^6$ $10^{4} \leq Ra \leq 10^{6}$ ), and Stefan number ( $0.2 \le \text {Ste} \le 0.8$ $0.2 \leq Ste \leq 0.8$ ). Our results show that as Ra and fusion temperatures ( $\theta _{f}=0.2$ $θ_{f} = 0.2$ , 0.6, 0.8) increase, the streamline patterns near the heated walls expand, while their intensity decreases near the cold walls. For $\text {Ra}=10^6$ $Ra = 10^{6}$ and $0.2\le \text {Ste} \le 0.8$ $0.2 \leq Ste \leq 0.8$ , the isothermal lines flatten out uniformly, which enhances heat transfer. With a constant $\text {Bn}$ $Bn$ , an increase in $\text {Ra}$ $Ra$ has a significant impact on heat capacity. The peak average Nusselt number ( $\overline{Nu}$ $\bar{Nu}$ ) improves by 72.30% at the highest $\text {Ra}$ $Ra$ and lowest $\text {Bn}$ $Bn$ . We also developed a mathematical correlation for $\overline{Nu}$ $\bar{Nu}$ , which demonstrates excellent predictive accuracy.