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Application of Collocation Spectral Method for Irregular Convective–Radiative Fins with Temperature-Dependent Internal Heat Generation and Thermal Properties

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Abstract

A collocation spectral method (CSM) is developed to solve the fin heat transfer in triangular, trapezoidal, exponential, concave parabolic, and convex geometries. In the thermal process of fin heat transfer, fin dissipates heat to environment by convection and radiation; internal heat generation, thermal conductivity, heat transfer coefficient, and surface emissivity are functions of temperature; ambient fluid temperature and radiative sink temperature are considered to be nonzero. The temperature in the fin is approximated by Chebyshev polynomials and spectral collocation points. Thus, the differential form of energy equation is transformed into the matrix form of algebraic equation. In order to test efficiency and accuracy of the developed method, five types of convective–radiative fins are examined. Results obtained by the CSM are assessed by comparing available results in references. These comparisons indicate that the CSM can be recommended as a good option to simulate and predict thermal performance of the convective–radiative fins.

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Abbreviations

\(b_j\) :

Coefficient of the integral term weight

\(C_i\) :

Dimensionless constants of the integral term weight

\(\bar{{c}}_j\) :

Coefficient of Lagrange interpolation polynomials defined in Eq. 17

\(c_i\) :

Constants of the integral term weight (\(\mathrm {W}{\cdot }\mathrm {m}^{-3}\))

\(D_{i,j}^{\left( 1\right) }\) :

Entries of the first-order derivative matrix

\(D_{i,j}^{\left( 2\right) }\) :

Entries of the second-order derivative matrix

\(F_{i,j}\) :

Entries of spectral coefficient matrix which are defined in Eq. 20

\(G_i\) :

Entries of spectral coefficient matrix which are defined in Eq. 21

\(H_{i,j}\) :

Entries of spectral coefficient matrix which are defined in Eq. 24

\(h_b\) :

Convective heat transfer coefficient corresponding to the temperature difference \(T_b -T_c\) (\(\mathrm {W}{\cdot } \mathrm {m}^{-2}{\cdot } \mathrm {K}^{-1}\))

\(h_i\) :

Lagrange interpolation polynomials

h :

Convective heat transfer coefficient (\(\mathrm {W}{\cdot } \mathrm {m}^{-2}{\cdot } \mathrm {K}^{-1}\))

L :

Fin tip length (\(\mathrm {m}\))

\(l_i\) :

Adjustment parameter

m :

Power exponent of heat transfer coefficient

\(N_{cc}\) :

Convective–conductive parameter

\(N_{rc}\) :

Radiative–conductive parameter

N :

Total number of collocation points

\(P_i\) :

Entries of spectral coefficient matrix which are defined in Eq. 25

\(\dot{q}\left( T\right) \) :

Internal heat generation (\(\mathrm {W}{\cdot } \mathrm {m}^{-3}\))

\(q_c\) :

Convective heat loss (\(\mathrm {W}{\cdot } \mathrm {m}^{-1}\))

\(q_r\) :

Radiative heat loss (\(\mathrm {W}{\cdot } \mathrm {m}^{-1}\))

\(q_{\mathrm{total}}\) :

Total heat loss (\(\mathrm {W}{\cdot } \mathrm {m}^{-1}\))

\(q_{\mathrm{ideal}}\) :

Ideal heat loss (\(\mathrm {W}{\cdot }\mathrm {m}^{-1}\))

\(R_{\mathrm{Benchmark}}\) :

Benchmark results

\(R_{\mathrm{CSM}}\) :

CSM results

\(s_i\) :

CGL collocation points

\(T_b\) :

Temperature at fin base (K)

\(T_\mathrm{c}\) :

Ambient fluid temperature (K)

\(T_s\) :

Radiation sink temperature (K)

T :

Temperature (K)

\(w_i\) :

Weight of integral term defined in Eq. 27

X :

Dimensionless axial coordinate

x :

Coordinate in x-direction (m)

\(\alpha \) :

Coefficient of variable thermal conductivity

\(\beta \) :

Coefficient of surface emissivity

\(\gamma \) :

Dimensionless exponential power factor

\(\delta \) :

Semi-thickness of the fin (m)

\(\delta _0\) :

Semi-thickness at the fin tip (m)

\(\delta _b\) :

Semi-thickness at the fin base (m)

\(\varepsilon \) :

Surface emissivity

\(\varepsilon _{\mathrm{error}}\) :

Integral averaged relative error

\(\eta \) :

Fin efficiency

\(\Theta ^{*}\) :

The last iterative value of dimensionless temperature

\(\Theta _c\) :

Dimensionless ambient fluid temperature

\(\Theta _s\) :

Dimensionless radiation sink temperature

\(\Theta \) :

Dimensionless temperature

\(\lambda _c\) :

Thermal conductivity at convection sink temperature (\(\mathrm {W}{\cdot } \mathrm {m}^{-1}{\cdot } \mathrm {K}^{-1}\))

\(\lambda \) :

Thermal conductivity (\(\mathrm {W}{\cdot } \mathrm {m}^{-1}{\cdot } \mathrm {K}^{-1}\))

\(\sigma \) :

Stefan–Boltzmann constant (\(\mathrm {W}{\cdot } \mathrm {m}^{-2}{\cdot } \mathrm {K}^{-4}\))

i, j :

Solution node indexes

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Acknowledgments

This work was supported by the Natural Science Foundation of China (Nos. 51206043, 51406014, 51176026), and the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20130036120009), and the Fundamental Research Funds for the Central Universities (Nos. 310822151021, 2015MS42).

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Authors and Affiliations

  1. Beijing Key Laboratory of Multiphase Flow and Heat Transfer for Low Grade Energy, North China Electric Power University, Beijing, 102206, China

    Ya-Song Sun

  2. Key Laboratory of Shaanxi Province for Development and Application of New Transportation Energy, School of Automobile, Chang’an University, Xi’an, 710064, China

    Jing Ma

  3. Institute of Thermal Engineering, School of Energy and Power Engineering, Dalian University of Technology, Dalian, 116024, China

    Ben-Wen Li

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  1. Ya-Song Sun
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Correspondence to Jing Ma or Ben-Wen Li.

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Sun, YS., Ma, J. & Li, BW. Application of Collocation Spectral Method for Irregular Convective–Radiative Fins with Temperature-Dependent Internal Heat Generation and Thermal Properties. Int J Thermophys 36, 3133–3152 (2015). https://doi.org/10.1007/s10765-015-1978-0

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  • DOI: https://doi.org/10.1007/s10765-015-1978-0

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