Purpose

This study aims to analyze the impact of fractional derivatives on heat transfer and entropy generation during transient free convection inside various complex porous enclosures, such as triangle, L-shape and square-containing wavy surfaces. These porous enclosures are saturated with Cu-water nanofluid and subjected to the influence of a uniform magnetic field.

Design/methodology/approach

In the present study, Darcy’s model is used for the momentum transport equation in the porous matrix. Additionally, the Caputo time fractional derivative is introduced in the energy equation to assess the heat transfer phenomenon. Furthermore, the total entropy generation has been computed by combining the entropy generation due to fluid friction (S_ff), heat transfer (S_ht) and magnetic field (S_mf). The complete mathematical model is further simulated using the penalty finite element method, and the Caputo time derivative term is approximated using the L1 scheme. The study is conducted for various ranges of the Rayleigh number (102≤Ra≤104), Hartmann number (0≤Ha≤20) and fractional order parameter (0<α<1) with respect to time.

Findings

It has been observed that the fractional order parameter α governs the characteristics of entropy generation and heat transfer within the selected range of parameters. The Bejan number associated with heat transfer (Be_ht), fluid friction (Be_ff) and magnetic field (Be_mf) further demonstrate the dominance of flow irreversibilities. It becomes evident that the initial evolution state of streamlines, isotherms and local entropy varies according to the choice of α. Additionally, increasing Ra values from 10² to 10⁴ shows that the heat transfer rate increases by 123.8% for a square wavy enclosure, 7.4% for a triangle enclosure and 69.6% for an L-shape enclosure. Moreover, an increase in the value of Ha leads to a reduction in heat transfer rates and entropy generation. In this case, Bemf→1 shows the dominance of the magnetic field irreversibility in the total entropy generation.

Practical implications

Recently, fractional-order models have been widely used to express numerous physical phenomena, such as anomalous diffusion and dispersion in complex viscoelastic porous media. These models offer a more accurate representation of physical reality that classical models fail to capture; this is why they find a broad range of applications in science and engineering.

Originality/value

The fractional derivative model is used to illustrate the flow pattern, heat transfer and entropy-generating characteristics under the influence of a magnetic field. Furthermore, to the best of the author’s knowledge, a fractional-derivative-based mathematical model for the entropy generation phenomenon in complex porous enclosures has not been previously developed or studied.

中文翻译：

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复杂多孔外壳中热磁分数阶对流的熵生成：数值研究

目的

本研究旨在分析分数阶导数对各种复杂多孔外壳（例如三角形、L 形和方形波状表面）内瞬态自由对流期间传热和熵产生的影响。这些多孔外壳充满了铜水纳米流体，并受到均匀磁场的影响。

设计/方法论/途径

在本研究中，达西模型用于多孔基体中的动量传递方程。此外，能量方程中引入了卡普托时间分数阶导数来评估传热现象。此外，通过结合流体摩擦 ( S _ff )、传热 ( S _ht ) 和磁场 ( S _mf ) 产生的熵来计算总熵产生。使用惩罚有限元法进一步模拟完整的数学模型，并使用L1格式逼近Caputo时间导数项。该研究针对不同范围的瑞利数进行 （102≤右A≤104）, 哈特曼数 （0≤HA≤20）和分数阶参数 （0<α<1）就时间而言。

发现

据观察，分数阶参数 α 控制着选定参数范围内的熵产生和传热特性。与传热 ( Be _ht )、流体摩擦 ( Be _ff ) 和磁场 ( Be _mf ) 相关的贝让数进一步证明了流动不可逆性的主导地位。很明显，流线、等温线和局部熵的初始演化状态根据 α 的选择而变化。此外，将Ra值从10 ²增加到10 ⁴表明，方形波状外壳的传热率增加了123.8%，三角形外壳的传热率增加了7.4%，L形外壳的传热率增加了69.6%。此外，Ha值的增加会导致传热率和熵产生的降低。在这种情况下， 乙e米F→1显示了磁场不可逆性在总熵产生中的主导地位。

实际影响

近年来，分数阶模型已被广泛用于表达许多物理现象，例如复杂粘弹性多孔介质中的反常扩散和色散。这些模型提供了经典模型无法捕捉到的更准确的物理现实表示；这就是为什么它们在科学和工程领域有着广泛的应用。

原创性/价值

分数阶导数模型用于说明磁场影响下的流型、传热和熵产生特性。此外，据作者所知，之前尚未开发或研究过用于复杂多孔外壳中熵生成现象的基于分数阶导数的数学模型。