# Dynamics of Heat Transfer Analysis of Convective-Radiative Fins with Variable Thermal Conductivity and Heat Generation: Differential Transformation Method

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## Abstract

**:**

## 1. Introduction

## 2. Fundamental Operations of DTM

## 3. Mathematical Formulation

- The temperature is a function of $x$ and remains constant over time.
- The temperature variance due to fin thickness is neglected.
- The fin bed is kept at a steady temperature.
- Solid matrix and fluid are in a dynamic state of equilibrium.
- Fin is considered to be in a steady state.

- For quadrilateral fin

- For exponential fin

- For convex fin$$\mathsf{\Gamma}\left(x\right)={\mathsf{\Gamma}}_{b}{\left(\frac{x}{L}\right)}^{0.5}$$$$\theta =\frac{T}{{T}_{b}},{\theta}_{a}=\frac{{T}_{a}}{{T}_{b}}X=\frac{x}{L},{N}^{2}=\left(\frac{h{L}^{2}}{{k}_{b}{A}_{b}}\right)Nr=\frac{\epsilon \sigma {L}^{2}{T}_{b}^{3}}{{A}_{b}{k}_{a}}G=\frac{{L}^{2}{q}^{\ast}}{{A}_{b}{k}_{a}{T}_{b}}$$

- For rectangular profile

- For exponential profile

- For convex profile$$\beta {\left(\frac{d\theta}{dy}\right)}^{2}+[1+\beta (\theta -{\theta}_{a})]\frac{{d}^{2}\theta}{d{y}^{2}}-Nr4y\left({\theta}^{4}-{\theta}_{a}^{4}\right)-{N}^{2}\left(\theta -{\theta}_{a}\right)4y+G4y=0$$$$\frac{d\theta \left(0\right)}{dX}=0,\theta \left(1\right)=1$$

## 4. Solution Method with DTM

- For rectangular profile

- For exponential profile

- For convex profile

- For rectangular profile$$\begin{array}{l}Q\left[2\right]=\frac{-G+a{N}^{2}+{a}^{4}Nr-{N}^{2}{\theta}_{a}-Nr{\theta}_{a}^{4}}{2\left(1+a\beta -\beta {\theta}_{a}\right)}\\ Q\left[3\right]=0\\ Q\left[4\right]=\frac{{N}^{2}Q\left[2\right]-6\beta Q{\left[2\right]}^{2}}{12\left(1+a\beta -\beta {\theta}_{a}\right)}\\ Q\left[5\right]=0\\ Q\left[6\right]=\frac{{N}^{2}Q\left[4\right]-30\beta Q\left[2\right]Q\left[4\right]}{30\left(1+a\beta -\beta {\theta}_{a}\right)}\\ Q\left[7\right]=0\end{array}$$

- For exponential profile$$\begin{array}{l}Q\left[2\right]=\frac{-G+a{N}^{2}+{a}^{4}Nr-{N}^{2}{\theta}_{a}-Nr{\theta}_{a}^{4}}{2\left(1+a\beta -\beta {\theta}_{a}\right)}\\ Q\left[3\right]=\frac{-2aQ\left[2\right]-a\beta Q\left[2\right]-{a}^{2}\beta Q\left[2\right]+2a\beta {\theta}_{a}Q\left[2\right]}{3\left(1+a\beta -\beta {\theta}_{a}\right)}\\ Q\left[4\right]=\frac{\begin{array}{l}-3{a}^{2}Q\left[2\right]+{N}^{2}Q\left[2\right]-2a\beta Q\left[2\right]-2{a}^{2}\beta Q\left[2\right]+3{a}^{2}\beta {\theta}_{a}Q\left[2\right]-4a\beta Q{\left[2\right]}^{2}-{a}^{2}\beta Q{\left[2\right]}^{2}-9aQ\left[3\right]-6a\beta Q\left[3\right]\\ -3{a}^{2}\beta Q\left[3\right]+9a\beta {\theta}_{a}Q\left[3\right]\end{array}}{12\left(1+a\beta -\beta {\theta}_{a}\right)}\end{array}$$

- For convex profile$$\begin{array}{l}Q\left[2\right]=0\\ Q\left[3\right]=-\frac{2\left(G-a{N}^{2}+a{N}^{2}{\theta}_{a}+aNr{\theta}_{a}^{4}\right)}{3\left(1+a\beta -\beta {\theta}_{a}\right)}\\ Q\left[4\right]=0\\ Q\left[5\right]=0\\ Q\left[6\right]=\frac{4{N}^{2}Q\left[3\right]-4{N}^{2}{\theta}_{a}Q\left[3\right]-4Nr{\theta}_{a}^{4}Q\left[3\right]-15\beta Q{\left[3\right]}^{2}}{30\left(1+a\beta -\beta {\theta}_{a}\right)}\\ Q\left[7\right]=0\\ Q\left[8\right]=0\end{array}$$

- For rectangular profile

#### Fin Efficiency

## 5. Results

## 6. Thermal Analysis

- Aluminum alloy (AA6061) is considered a fin material as it is a good thermal and electrical conductor with heat conduction of 300 W/m K.
- Heat conduction is considered 1D and longitudinal.
- h is considered to be 39.9 W/m
^{2}K above the fin surface. - The fin base is kept at 550 K, and 283 is the ambient temperature.

## 7. Conclusions

- Upon enhancing the convection–conduction parameter, the thermal dispersal in the fin lowers.
- A strengthened heat transfer fine is observed for the radiative-conduction constant.
- The thermal rate of the fin improves with an augmented change in a heat-generating parameter.
- This scrutiny convinces us that DTM algorithms are efficient and convenient methods for nonlinear differential systems.
- Thermal radiation and natural convection have a significant influence on the cooling of a fin.
- In the steady state, fins dissipate heat to the environment because heat production within a fin surges the temperature of the fins.
- The temperature scatters of a fin for different profiles are calculated using the ANSYS software, considering aluminum alloy (AA6061) as the fin body material. The fin base has a higher temperature and reduces drastically toward the fin tip.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$P$ | fin cross-section (m^{2}) |

$a$ | exponential parameter |

$h$ | heat transfer coefficient (wm^{−1}k^{−1}) |

$k$ | heat conduction (wm^{−1}k^{−1}) |

$Nr$ | radiative parameter |

$G$ | heat generation parameter |

$L$ | fin length (m) |

$N$ | convective parameter |

$T$ | temperature (k) |

$\phi $ | transformed function |

$\varphi $ | original analytic function |

$a$ | fin base temperature |

$\beta $ | thermal expansion coefficient (K^{−1}) |

$\zeta $ | dimensional constant (K^{−1}) |

$\eta $ | efficiency of the fin |

$U$ | transformed equation |

$\theta $ | dimensionless temperature |

$a$ | ambient temperature |

$b$ | base of the fin |

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**Figure 8.**(

**a**) Variation of ${Q}_{b}$ with $N$ for several assigned values of $\beta $. (

**b**) Variation of ${Q}_{b}$ with $N$ for several assigned values of ${\theta}_{a}$.

**Figure 9.**Temperature distribution of (

**a**) rectangular; (

**b**) exponential; (

**c**) convex profile for aluminum alloy (AA6061).

**Figure 10.**Influence of: (

**a**) G and $\beta $ on fin efficiency; (

**b**) Nr and ${\theta}_{a}$ on fin efficiency.

Initial Function | Converted Function |
---|---|

$\varphi \left(r\right)=\frac{dg\left(r\right)}{dx}$ | $\phi \left(v\right)=\left(v+1\right)G\left(v\right)$ |

$\varphi \left(r\right)=\frac{{d}^{2}g\left(r\right)}{d{x}^{2}}$ | $\phi \left(v\right)=\left(v+1\right)\left(v+2\right)G\left(v+1\right)$ |

$\varphi \left(r\right)=1$ | $\phi \left(v\right)=\delta \left(v\right)$ |

$\varphi \left(r\right)=t$ | $\phi \left(v\right)=\delta \left(v-1\right)$ |

$\varphi \left(r\right)={r}^{m}$ | $\phi \left(v\right)=\delta \left(v-w\right)=\{\begin{array}{c}1ifv=w\\ 0ifv\ne w\end{array}$ |

$\varphi \left(r\right)=g\left(r\right)h\left(r\right)$ | $\phi \left(v\right)={\displaystyle \sum _{w=0}^{v}H\left(v\right)G\left(v-w\right)}$ |

$\varphi \left(r\right)={e}^{ar}$ | $\phi \left(v\right)=\frac{{a}^{v}}{v!}$ |

**Table 2.**Comparison of $\theta \left(X\right)$ obtained by different studies for rectangular fins by considering $\beta =0,G=0,Nr=0,{\theta}_{a}=0$ and $N=0.5$.

$\mathit{X}$ | HPM (Languri et al. [42]) | ADM (Arslanturk [43]) | VIM (Languri et al. [42]) | DTM (Current Study) |
---|---|---|---|---|

$\theta \left(X\right)$ | ||||

0 | 0.886819 | 0.886819 | 0.886819 | 0.886818 |

0.2 | 0.891257 | 0.891257 | 0.891257 | 0.8912567 |

0.4 | 0.904614 | 0.904615 | 0.904614 | 0.940614 |

0.6 | 0.927026 | 0.927026 | 0.927026 | 0.927027 |

0.8 | 0.958715 | 0.958716 | 0.958715 | 0.958715 |

1 | 1.000000 | 1.000000 | 1.000000 | 1.000000 |

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**MDPI and ACS Style**

Ananth Subray, P.V.; Hanumagowda, B.N.; Varma, S.V.K.; Zidan, A.M.; Kbiri Alaoui, M.; Raju, C.S.K.; Shah, N.A.; Junsawang, P.
Dynamics of Heat Transfer Analysis of Convective-Radiative Fins with Variable Thermal Conductivity and Heat Generation: Differential Transformation Method. *Mathematics* **2022**, *10*, 3814.
https://doi.org/10.3390/math10203814

**AMA Style**

Ananth Subray PV, Hanumagowda BN, Varma SVK, Zidan AM, Kbiri Alaoui M, Raju CSK, Shah NA, Junsawang P.
Dynamics of Heat Transfer Analysis of Convective-Radiative Fins with Variable Thermal Conductivity and Heat Generation: Differential Transformation Method. *Mathematics*. 2022; 10(20):3814.
https://doi.org/10.3390/math10203814

**Chicago/Turabian Style**

Ananth Subray, P. V., B. N. Hanumagowda, S. V. K. Varma, A. M. Zidan, Mohammed Kbiri Alaoui, C. S. K. Raju, Nehad Ali Shah, and Prem Junsawang.
2022. "Dynamics of Heat Transfer Analysis of Convective-Radiative Fins with Variable Thermal Conductivity and Heat Generation: Differential Transformation Method" *Mathematics* 10, no. 20: 3814.
https://doi.org/10.3390/math10203814