# A Significant Role of Activation Energy and Fourier Flux on the Quadratically Radiated Sphere in Low and High Conductivity of Hybrid Nanoparticles

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## Abstract

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## 1. Introduction

_{2}) across a sphere with Arrhenius activation energy and quadratic thermal radiation. The results are presented in a graphical format using bvp4c (MATLAB in-built function). In order to determine the influence that the parameters have on the heat transfer rate, the mass transfer rate, and the skin friction coefficient, a statistical method known as multiple linear regression is utilized.

## 2. Formulation

_{2}) flow through a convectively heated sphere with viscous dissipation and quadratic thermal convection parameters. The current flow pattern of a fluid over a sphere is depicted in Figure 1. Table 1 presents the thermo-physical property values of the base fluid and the nanomaterials utilized in the production of the present hybrid nanofluid. It is assumed that $x$ is the coordinate measured along the surface of the sphere starting from the lower stagnation point and $y$ is the coordinate measured in the direction normal to the surface of the sphere. Surface temperature and free stream temperature are denoted by ${T}_{w}$ and ${T}_{\infty}$, respectively. The heat flux and mass flux of the sphere are indicated by ${q}_{w}$ and ${s}_{w}$, respectively. $a$ is the radius of the sphere; $r=r\left(x\right)=a\mathrm{sin}\left(\frac{x}{a}\right)$ is the radial distance between the symmetrical axis and the sphere’s surface, and $g$ is the acceleration due to gravity. ${B}_{0}$ is the uniform magnetic field strength applied in the direction perpendicular to the surface. Fluid flow is considered near the stagnation point only. In addition to this, it is presumed that the induced magnetic field is not taken into account.

**Table 1.**The values of base fluid and nanomaterial thermophysical properties (Jayadevamurthy et al. [47]).

S. No. | $\mathbf{Water}\text{}\left(\mathit{f}\right)$ | $\mathit{C}\mathit{u}$ $\left({\mathit{s}}_{1}\right)$ | $\mathit{S}\mathit{i}{\mathit{O}}_{2}$ $\left({\mathit{s}}_{2}\right)$ | |
---|---|---|---|---|

1 | $\rho \left(\mathrm{Kg}/{\mathrm{m}}^{3}\right)$ | 997.1 | 8933 | 2200 |

2 | ${C}_{p}\left(\mathrm{J}/\mathrm{Kg}\text{}\mathrm{K}\right)$ | 4179 | 385 | 745 |

3 | $k\left(\mathrm{W}/\mathrm{m}\text{}\mathrm{K}\right)$ | 0.613 | 400 | 1.4 |

4 | $\sigma \left(\mathrm{S}{\mathrm{m}}^{-1}\right)$ | 0.05 | $5.96\times {10}^{7}$ | 0.000975 |

#### Engineering Parameters of Concern

## 3. Validation of Current Results

## 4. Discussion of the Outcomes

## 5. Conclusions

- The magnetic field parameter can be used to influence the fluid flow;
- A bigger buoyancy ratio parameter escalates the fluid velocity;
- An increase in the Biot number leads to the rise in the fluid temperature;
- As the Eckert number improves, the Nusselt number decreases;
- The concentration of the fluid enriches with the raise in the activation energy;
- The increase in the Sherwood number is caused by a rise in both ${S}_{c},\text{}\alpha $;
- The radiation parameter and the heat source parameter have a negative influence on the Nusselt number;
- The friction factor increases when the volume percentage of nanoparticles gets larger, and the buoyancy ratio parameter also increases;
- In the future, this work may serve as a good reference case for extension to more complicated situations, involving unsteady flows, non-uniform flows, and/or non-spherical bodies.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$a$ | Radius of the sphere |

${G}_{r}$ | Local thermal Grashoff number |

$m$ | Fitted rate constant |

$k$ | Thermal conductivity (W·m^{−1}·K^{−1}) |

$g$ | Acceleration due to gravity |

${f}^{\prime}$ | Dimensionless velocity |

${D}_{m}$ | Molecular diffusivity (m^{2}·s^{−1}) |

$T$ | Dimensional temperature of fluid (K) |

${h}_{f}$ | Convective heat transfer coefficient |

$\theta $ | Dimensionless temperature of fluid |

${C}_{p}$ | Heat capacity |

$C$ | Dimensional concentration of fluid (mol·m^{−3}) |

$u,v$ | Velocity components in $x,y$ directions (m·s^{−1}) |

${E}_{ck}$ | Eckert number |

${S}_{c}$ | Schmidt number |

$k*$ | Mean absorption coefficient |

${E}_{n}$ | Activation energy parameter |

${Q}_{0}$ | Volumetric rate of heat source parameter |

$M$ | Magnetic field parameter |

${Q}_{t}$ | Heat source parameter |

${k}_{1}$ | Boltzmann constant—8.314 J·mol^{−1}·K |

${E}_{0}$ | Dimensional activation energy parameter |

${G}_{c}$ | Local diffusion Grashoff number |

${\mathrm{P}}_{r}$ | Prandtl number |

${k}_{0}$ | Chemical reaction rate |

$Bi$ | Biot number |

${K}_{r}$ | Dimensional reaction rate parameter |

Greek Letters | |

$\mu $ | Dynamic viscosity of fluid (kg·m^{−1}·s^{−1}) |

$\sigma *$ | Stefan–Boltzmann constant |

$\delta $ | Buoyancy ratio parameter |

$\xi $ | Dimensionless coordinate |

$\delta *$ | Nonlinear thermal convection parameter |

$\upsilon $ | Kinematic viscosity (m^{2}·s^{−1}) |

${\beta}_{T}$ | Volumetric coefficient of thermal expansion |

$\alpha $ | Reaction rate constant |

$\psi $ | Stream function |

$\rho $ | Density of fluid (kg·m^{−3}) |

$\eta $ | Similarity variable |

${\beta}_{C}$ | Volumetric coefficient of diffusion expansion |

${\beta}_{1T}$ | Nonlinear thermal convection parameter |

${\beta}_{1C}$ | Nonlinear diffusion convection parameter |

$\sigma $ | Electrical conductivity |

${\delta}_{1}$ | Dimensionless nonlinear convection parameter related to species concentration parameter |

$\mathsf{\Phi}$ | Dimensionless concentration of fluid |

${\theta}_{r}$ | Temperature ratio parameter |

Subscripts | |

$f$ | Fluid |

$nf$ | Nanofluid |

$hnf$ | Hybrid nanofluid |

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**Table 2.**Comparison of current findings to previous results under specific circumstances such as $\mathrm{Pr}=7$.

$\mathit{\xi}$ | $\mathit{N}\mathit{u}$ | |
---|---|---|

Alkasasbeh [48] | Current Results | |

0.959481 | 0.959467 | |

${10}^{0}$ | 0.957203 | 0.957216 |

${20}^{0}$ | 0.950561 | 0.950569 |

${30}^{0}$ | 0.939668 | 0.939648 |

${40}^{0}$ | 0.924310 | 0.924301 |

${50}^{0}$ | 0.904501 | 0.904499 |

${60}^{0}$ | 0.880058 | 0.880050 |

${70}^{0}$ | 0.851032 | 0.851037 |

${80}^{0}$ | 0.816761 | 0.816799 |

${90}^{0}$ | 0.779178 | 0.779172 |

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**MDPI and ACS Style**

Venkateswarlu, A.; Murshid, N.; Mulki, H.; Abu-samha, M.; Suneetha, S.; Babu, M.J.; Raju, C.S.K.; Homod, R.Z.; Al-Kouz, W.
A Significant Role of Activation Energy and Fourier Flux on the Quadratically Radiated Sphere in Low and High Conductivity of Hybrid Nanoparticles. *Symmetry* **2022**, *14*, 2335.
https://doi.org/10.3390/sym14112335

**AMA Style**

Venkateswarlu A, Murshid N, Mulki H, Abu-samha M, Suneetha S, Babu MJ, Raju CSK, Homod RZ, Al-Kouz W.
A Significant Role of Activation Energy and Fourier Flux on the Quadratically Radiated Sphere in Low and High Conductivity of Hybrid Nanoparticles. *Symmetry*. 2022; 14(11):2335.
https://doi.org/10.3390/sym14112335

**Chicago/Turabian Style**

Venkateswarlu, Avula, Nimer Murshid, Hasan Mulki, Mahmoud Abu-samha, Sangapatnam Suneetha, Macherla Jayachandra Babu, Chakravarthula Siva Krishnam Raju, Raad Z. Homod, and Wael Al-Kouz.
2022. "A Significant Role of Activation Energy and Fourier Flux on the Quadratically Radiated Sphere in Low and High Conductivity of Hybrid Nanoparticles" *Symmetry* 14, no. 11: 2335.
https://doi.org/10.3390/sym14112335