Journal of Thermal Analysis and Calorimetry

, Volume 138, Issue 4, pp 2913–2921

# Analysis of transient heat transfer in radial moving fins with temperature-dependent thermal properties

Article

## Abstract

In this article, thermal analysis of radial moving fins of hyperbolic and rectangular profiles is performed. A time-dependent nonlinear differential equation modeling heat transfer in a radial moving fin is considered. The thermal conductivity and heat transfer coefficient are modeled as linear and power-law functions of temperature, respectively. The analytical solutions are generated using the differential transform method which is an analytical solution technique that can be applied to various types of differential equations. The accuracy of the analytical solutions is validated by benchmarking against the numerical solutions obtained by applying the inbuilt numerical solver in Maple (pdsolve). A good agreement is observed between the analytical and numerical solutions. The effects of embedding parameters on the dimensionless temperature and heat transfer rate are analyzed. The results show that the fin rapidly dissipates heat to the surrounding fluid with an increase in the speed of the moving fin, that is, an increase in the values of the Peclet number. The results also show that the heat transfer through a rectangular fin is more efficient when compared to a fin of hyperbolic profile.

## Keywords

Moving fin Differential transform method Thermal conductivity Heat transfer Fin tip temperature

## List of symbols

c

Specific heat ($${\text{J}\,\text{kg}}^{-1}\,{\text{K}}$$)

H

Heat transfer coefficient ($${\text{W}\,\text{m}}^{-2}\,\text{K}$$)

$$h_\mathrm{b}$$

Heat transfer coefficient at the fin base ($${\text{W}\,\text{m}}^{-2}\,{\text{K}}$$)

K

Thermal conductivity ($${\text{W}\,\text{m}^{-1}}\,{\text{K}}$$)

$$k_\mathrm{c}$$

Thermal conductivity at ambient temperature ($${\text{W}\,\text{m}^{-1}}\,{\text{K}}$$)

t

Time (s)

T

Temperature (K)

U

Velocity of the fin (m s$$^{-1}$$)

R

r

M

Thermo-geometric fin parameter

$$A_\mathrm{c}$$

Cross-sectional area ($${\text{m}}^2$$)

$$r_{\mathrm{b}}, r_{\mathrm{t}}$$

Inner and outer radii, respectively (m)

L

Fin length (m)

F(R)

Fin profile

$$T_ \mathrm{a}$$

Ambient temperature (K)

$$T_\mathrm{b}$$

Fin base temperature (K)

P

Fin perimeter (m)

Pe

Peclet number

$$\varPhi (t,x)$$

Transformed analytical function

$$\phi (t,x)$$

Original analytical function

## Greek symbols

$$\sigma$$

Boltzmann constant ($${\text{W}\,\text{m}}^{-2}\,{\text{K}}^4$$)

$$\delta _\mathrm{b}$$

Base fin width (m)

$$\beta$$

$$\rho$$

Density ($${\text{kg}\,\text{m}}^{-3}$$)

$$\tau$$

Dimensionless time

$$\alpha$$

Thermal diffusivity ($${\text{m}}^2\,{\text{s}^{-1}}$$)

$$\theta$$

Dimensionless temperature

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## Authors and Affiliations

• Partner Luyanda Ndlovu
• 1
• 2
• Raseelo Joel Moitsheki
• 1
1. 1.School of Computer Science and Applied MathematicsUniversity of the Witwatersrand, JohannesburgWitsSouth Africa
2. 2.Standard Bank of South AfricaJohannesburgSouth Africa