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Thermal analysis of a convective–conductive–radiative annular porous fin with variable thermal parameters and internal heat generation

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Abstract

The thermal characteristics of annular porous fins with rectangular and hyperbolic cross-sections and internal heat generation were comprehensively studied by the homotopy perturbation method (HPM). The convective–conductive–radiative mode of heat transfer was considered in the current analysis. All thermal parameters were considered as a function of temperature. An approximate closed-form solution was obtained by solving the nonlinear heat transfer equation using HPM. Darcy’s model was employed to formulate the governing equation of heat transfer through porous media. Unknown constants were the initial approximations of the solution and were evaluated based on the boundary and initial conditions of the problem. The effects of pores and different thermal parameters on the dimensionless temperature distribution and the fin efficiency were graphically presented. In order to evaluate the accuracy of the closed-form solution, the obtained results (for both dimensional and non-dimensional forms) were validated by numerical solutions.

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Abbreviations

A c :

Cross-sectional area of fin (m2)

C p :

Specific heat of fluid (J g−1 K−1)

D a :

Darcy Number

e :

Co-efficient of linear variation of heat generation per unit volume (K−1)

e o :

Dimensionless co-efficient for linear variation of heat generation per unit volume

Gr:

Grashof Number

g :

Gravitational constant (9.81 m s−2)

h :

Convective heat transfer coefficient (W m−2 K−1)

h o :

Convective heat transfer coefficient at fin base temperature Tb (W m−2 K−1)

k s, k f :

Thermal conductivity of solid, fluid (W m−1 K−1)

k eff :

Effective thermal conductivity of porous matrix (W m−1 K−1)

k r :

Ratio of thermal conductivity of solid to that of fluid at ambient temperature Ta

\(k_{{\text{s}}}^\text{o} ,\;k_{{\text{f}}}^\text{o}\) :

Thermal conductivity of solid and fluid at temperature Ta (W m−1 K−1)

\(k_{{{\text{eff}}}}^\text{o}\) :

Effective thermal conductivity of porous matrix at temperature Ta (W m−1 K−1)

m :

Exponent for variation of thickness of fin

ṁ :

Mass flow rate of fluid (kg s–1)

n :

Exponent of variable convective heat transfer coefficient

N c :

Non-dimensional convection parameter

N cc :

Non-dimensional conduction–convection parameter

N r :

Non-dimensional radiation parameter

P r :

Prandtl Number

P :

Perimeter enclosing the cross-sectional area of the fin (m)

q :

Heat flow from base to tip of the fin (W)

q o :

Internal heat generation at Ta (W m−3)

Q * :

Non-dimensional heat generation parameter

R a :

Rayleigh Number

R :

Dimensionless radius

r i, r o, t b :

Inner radius, outer radius and thickness of the fin at the base (m)

T b, T a and T s :

Base temperature of fin, ambient temperature and radiation sink temperature (K)

α f :

Thermal diffusivity (m2 s–1)

β :

Co-efficient of volumetric thermal expansion of fin (K−1)

ε :

Surface emissivity of the fin

ε o :

Surface emissivity at the sink temperature Ts

η :

Efficiency of the fin

θ :

Dimensionless temperature

θ a :

Dimensionless ambient temperature

θ s :

Dimensionless radiation sink temperature

κ :

Permeability of porous media (m2)

κ 1, κ 2 :

Co-efficient of thermal conductivity of solid and fluid (K−1)

λ :

Co-efficient of linear variation of emissivity (K−1)

λ o :

Dimensionless co-efficient for linear variation of emissivity

ν f :

Kinematic viscosity of fluid (m2s–1)

ρ :

Density of fluid (kg m−3)

σ :

Stefan–Boltzmann Constant (5.67 × 10–8 W m–2 K–4)

ϕ :

Porosity of the fin

τ w :

Variable thermal conductivity parameter for solid–fluid interaction

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Acknowledgements

The authors would like to thank Dr. Rajiv Ranjan for providing the preliminary ideas of HPM to Mr. Venkitesh (First author). The authors are grateful to Dr. Satyabratta Sahoo, Dept. of Mechanical Engineering, IIT(ISM) Dhanbad and the reviewers for their critical comments and suggestions that greatly improved the manuscript.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Indian Institute of Technology (ISM) Dhanbad, Dhanbad, India

    V. Venkitesh & Ashis Mallick

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  1. V. Venkitesh
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Correspondence to Ashis Mallick.

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Appendix

Appendix

The solution of the governing equation is assumed to be of the form

$$ \theta = \theta_{0} + p\theta_{1} + p^{2} \theta_{2} $$

It is assumed that θ0 = 1.

The co-efficient of p is obtained by solving the differential equation

$$ \theta_{1}^{^{\prime\prime}} - \left\{ {N_{{{\text{CC}}}} (\theta_{0} - \theta_{{\text{a}}} )^{2} - N_{{\text{C}}} \frac{{(\theta_{0} - \theta_{{\text{a}}} )^{\text{n} + 1} }}{{(1 - \theta_{{\text{a}}} )^\text{n} }} - N_{{\text{R}}} (1 + \lambda_{0} (\theta_{0} - \theta_{{\text{s}}} ))\left( {\theta_{0}^{4} - \theta_{{\text{s}}}^{4} } \right)} \right\}R^{ - \text{m}} + Q(1 + e_{0} (\theta_{0} - \theta_{{\text{a}}} )) = 0 $$

which is solved subjected to boundary conditions

θ1 = 0 at R = 1 and θ1′ = 0 at R = RT.

$$ \theta_{1} = \left\{ {N_{{{\text{CC}}}} (1 - \theta_{{\text{a}}} )^{2} + N_{{\text{C}}} (1 - \theta_{{\text{a}}} ) + N_{{\text{R}}} (1 + \lambda_{0} (1 - \theta_{{\text{s}}} ))(1 - \theta_{{\text{s}}}^{4} )} \right\}\frac{{R^{ - \text{m} + 2} }}{(1 - m)(2 - m)} - Q(1 + e_{0} (\theta_{0} - \theta_{{\text{a}}} ))\frac{{R^{2} }}{2} + C_{1} R + C_{2} $$

where

$$ \begin{aligned} C_{1} = & Q(1 + e_{0} (\theta_{0} - \theta_{{\text{a}}} ))R - \left\{ {N_{{{\text{CC}}}} (1 - \theta_{{\text{a}}} )^{2} + N_{{\text{C}}} (1 - \theta_{{\text{a}}} ) + N_{{\text{R}}} (1 + \lambda_{0} (1 - \theta_{{\text{s}}} ))\left( {1 - \theta_{{\text{s}}}^{4} } \right)} \right\}\frac{{R^{ - m + 1} }}{(1 - m)} \\ C_{2} = & \frac{Q}{2}(1 + e_{0} (\theta_{0} - \theta_{{\text{a}}} )) - \left\{ {N_{{{\text{CC}}}} (1 - \theta_{{\text{a}}} )^{2} + N_{{\text{C}}} (1 - \theta_{{\text{a}}} ) + N_{{\text{R}}} (1 + \lambda_{0} (1 - \theta_{{\text{s}}} ))\left( {1 - \theta_{{\text{s}}}^{4} } \right)} \right\}\frac{1}{(1 - m)(2 - m)} - C_{1} \\ \end{aligned} $$

Similarly, the co-efficient of p2 is obtained by solving the differential equation

$$ \theta_{2}^{^{\prime\prime}} + \tau_{{\text{W}}} (1 - \theta_{{\text{a}}} )\theta_{1}^{^{\prime\prime}} + (\tau_{{\text{W}}} (1 - \theta_{{\text{a}}} ) + 1)\frac{m + 1}{R}\theta_{1}^{^{\prime}} - [N_{{\text{C}}} (1 + n) + N_{{\text{R}}} \left\{ {\lambda_{0} (1 - \theta_{{\text{s}}}^{4} ) + 4(1 + \lambda_{0} (1 - \theta_{{\text{s}}} ))} \right\} + 2N_{{{\text{CC}}}} (1 - \theta_{{\text{a}}} )]\frac{{\theta_{1} }}{{R^{m} }} + Qe_{0} \theta_{1} = 0 $$

Subjected to boundary conditions

θ2 = 0 at R = 1 and θ2′ = 0 at R = RT

$$ \begin{aligned} \theta_{2} = & A_{2} \left[ {\frac{{A_{1} R^{ - 2\text{m} + 4} }}{(4 - 2m)(3 - 2m)(1 - m)(2 - m)} - \frac{{Q_{1} R^{ - \text{m} + 4} }}{2(3 - m)(4 - m)} + \frac{{C_{1} R^{ - \text{m} + 3} }}{(3 - m)(2 - m)} + \frac{{C_{2} R^{ - \text{m} + 2} }}{(2 - m)(1 - m)}} \right] \\ & - Q_{2} \left[ {\frac{{A_{1} R^{ - \text{m} + 4} }}{(4 - m)(3 - m)(1 - m)(2 - m)} - \frac{{Q_{1} R^{4} }}{24} + \frac{{C_{1} R^{3} }}{6} + \frac{{C_{2} R^{2} }}{2}} \right] - \tau_{{\text{W}}} (1 - \theta_{a} )\left[ {\frac{{A_{1} R^{ - \text{m} + 2} }}{(1 - m)(2 - m)} - \frac{{Q_{1} R^{2} }}{2}} \right] \\ & - B_{1} (m + 1)\left[ {\frac{{A_{1} R^{ - \text{m} + 2} }}{(1 - m)(2 - m)} - \frac{{Q_{1} R^{2} }}{2}} \right] + C_{1} \left[ {R\ln R - R} \right] + C_{3} R + C_{4} \\ \end{aligned} $$

where

$$ \begin{aligned} A_{1} = & N_{{{\text{CC}}}} (1 - \theta_{{\text{a}}} )^{2} + N_{{\text{C}}} (1 - \theta_{{\text{a}}} ) + N_{{\text{R}}} (1 + \lambda_{0} (1 - \theta_{{\text{s}}} ))\left( {1 - \theta_{{\text{s}}}^{4} } \right) \\ Q_{1} = & Q(1 + e_{0} (\theta_{0} - \theta_{{\text{a}}} )) \\ B_{1} = & (\tau_{{\text{W}}} (1 - \theta_{{\text{a}}} ) + 1) \\ A_{2} = & N_{{\text{C}}} (1 + n) + N_{{\text{R}}} \left\{ {\lambda_{0} \left( {1 - \theta_{{\text{s}}}^{4} } \right) + 4(1 + \lambda_{0} (1 - \theta_{{\text{s}}} ))} \right\} + 2N_{{{\text{CC}}}} (1 - \theta_{{\text{a}}} ) \\ Q_{2} = & Qe_{0} \\ \end{aligned} $$
$$ \begin{aligned} C_{3} = & \tau_{{\text{W}}} (1 - \theta_{{\text{a}}} )\left[ {\frac{{A_{1} R^{ - \text{m} + 1} }}{(1 - m)} - Q_{1} R} \right] + B_{1} (m + 1)\left[ {\frac{{A_{1} R^{ - \text{m} + 1} }}{{(1 - m)^{2} }} - Q_{1} R + C_{1} \ln R} \right] \\ & + Q_{2} \left[ {\frac{{A_{1} R^{ - \text{m} + 3} }}{(3 - m)(1 - m)(2 - m)} - \frac{{Q_{1} R^{3} }}{6} + \frac{{C_{1} R^{2} }}{2} + C_{2} R} \right] - A_{2} \left[ {\frac{{A_{1} R^{ - 2\text{m} + 3} }}{(3 - 2m)(1 - m)(2 - m)} - \frac{{Q_{1} R^{ - \text{m} + 3} }}{2(3 - m)} + \frac{{C_{1} R^{ - \text{m} + 2} }}{(2 - m)} + \frac{{C_{2} R^{ - \text{m} + 1} }}{(1 - m)}} \right] \\ \end{aligned} $$
$$ \begin{aligned} C_{4} = & \tau_{{\text{W}}} (1 - \theta_{{\text{a}}} )\left[ {\frac{{A_{1} }}{(2 - m)(1 - m)} - \frac{{Q_{1} }}{2}} \right] + B_{1} \left[ {\frac{{A_{1} }}{{(2 - m)(1 - m)^{2} }} - \frac{{Q_{1} }}{2} + C_{1} } \right] \\ & + Q_{2} \left[ {\frac{{A_{1} }}{(4 - m)(3 - m)(1 - m)(2 - m)} - \frac{{Q_{1} }}{24} + \frac{{C_{1} }}{6} + \frac{{C_{2} }}{2}} \right] - C_{3} \\ & - A_{2} \left[ {\frac{{A_{1} }}{(4 - m)(3 - m)(1 - m)(2 - m)} - \frac{{Q_{1} }}{2(4 - m)(3 - m)} + \frac{{C_{1} }}{(3 - m)(2 - m)} - \frac{{C_{2} }}{(2 - m)(1 - m)}} \right] \\ \end{aligned} $$

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Venkitesh, V., Mallick, A. Thermal analysis of a convective–conductive–radiative annular porous fin with variable thermal parameters and internal heat generation. J Therm Anal Calorim 147, 1519–1533 (2022). https://doi.org/10.1007/s10973-020-10384-9

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