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A Modified Decomposition Solution of Triangular Moving Fin with Multiple Variable Thermal Properties

  • Research Article - Mechanical Engineering
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Abstract

In this paper, we use the modified decomposition method (MADM), a recent mathematical technique to find closed-form solutions of the singular nonlinear heat transfer equation of a longitudinal triangular moving fin when all the thermal parameters vary with the temperature. The energy balance equation for the triangular moving fin is solved using the MADM. The results are compared with those obtained using the differential transform method and numerical solutions obtained using the spectral quasi-linearization method. The effects of the various thermo-physical parameters, such as thermal conductivity parameter, power exponent of heat transfer coefficient, surface emissivity parameter, convection–conduction parameter, radiation–conduction parameter, convection sink temperature, radiation sink temperature, heat generation parameter, and the Peclet Number on the temperature distribution and, therefore, energy transfer efficiency are analyzed.

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Abbreviations

\(N_\mathrm{r}\) :

Radiation–conduction parameter

C :

Constant which represents the temperature

k :

Temperature-dependent thermal conductivity [W/( mK)]

\(k_\mathrm{a} \) :

Thermal conductivity corresponding to ambient    condition [W/(mK)]

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

Surface emissivity corresponding to radiation sinks temperature, \(T_\mathrm{s} \)

T :

Temperature (K)

P :

Fin perimeter (m)

\(T_\mathrm{b} \) :

Fin’s base temperature (K)

\(T_\mathrm{a} \) :

Sink temperature corresponding to \({k_\mathrm{a}}\) (K)

\(T_\mathrm{s} \) :

Sink temperature for radiation (K)

L :

Length of the fin (m)

x :

Axial coordinate of the entire fin (m)

\(A_x \) :

Cross-sectional area of the elementary section of fin at location x (\(\hbox {m}^{2})\)

X :

Dimensionless spatial coordinate

A :

Thermal conductivity parameters

B :

Surface emissivity parameters

\(\varepsilon _G \) :

Heat generation parameters

G :

Heat generation number

Pe :

Peclet number

U :

Velocity of the fin

\(C_p \) :

Specific heat of the fin material

\(t_\mathrm{b} \) :

Base thickness of the fin

\(\alpha \) :

Slope of the thermal conductivity–temperature curve (\(\hbox {K}^{-1}\))

\(\beta \) :

Slope of the surface emissivity–temperature curve (\(\hbox {K}^{-1}\))

\(\gamma \) :

Slope of the heat generation–temperature curve (\(\hbox {K}^{-1}\))

\(\theta \) :

Dimensionless temperature of the fin

\(\theta _\mathrm{a} \) :

Dimensionless convective sink temperature

\(\theta _\mathrm{s} \) :

Dimensionless radiations sink temperature

\(\sigma \) :

Stefan–Boltzmann constant

\(\varepsilon \) :

Emissivity

\(\rho \) :

Density of the fin materials

References

  1. Kem, D.Q.; Kraus, D.A.: Extended Surface Heat Transfer. McGraw-Hill, New York (1972)

    Google Scholar 

  2. Yigit, A.; Mehmet, P.: New perturbation iteration solutions for Bratu-type equations. Comput. Math. Appl. 59, 2802–2808 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aziz, A.; Huq, S.M.E.: Perturbation solution for convecting fin with variable thermal conductivity. ASME J. Heat Transf. 97, 300–301 (1975)

    Article  Google Scholar 

  4. Aksoy, Y.; Pakdemirli, M.; Abbasbandy, S.; Boyacı, H.: New perturbation-iteration solutions for nonlinear heat transfer equations. Int. J. Numer. Methods Heat Fluid Flow 22(7), 814–828 (2012)

    Article  Google Scholar 

  5. Bergman, T.L.; Lavine, A.S.; Incropera, T.P.: Introduction to Heat Transfer. Wiley, New York (2011)

    Google Scholar 

  6. Cengel, Y.A.: Heat Transfer: A Practical Approach, 2nd edn. McGraw-Hill Science, London (2007)

    Google Scholar 

  7. Cihat, A.: A decomposition method for fin efficiency of convective straight fins with temperature-dependent thermal conductivity. Int. Commun. Heat Mass Transf. 32, 831–841 (2005)

    Article  Google Scholar 

  8. Mallick, A.; Ranjan, R.; Sarkar, P.K.: Effect of heat transfer on thermal stresses in an annular hyperbolic fin: an approximate analytical solution. J. Theor. Appl. Mech. 54(2), 437–448 (2016)

    Article  Google Scholar 

  9. Ching-Huang, C.; Cha’o-Kuang, C.: A decomposition method for solving the convective longitudinal fins with variable thermal conductivity. Int. J. Heat Mass Transf. 45, 2067–2075 (2002)

    Article  MATH  Google Scholar 

  10. Chang, M.H.: A decomposition solution for fins with temperature dependent surface heat flux. Int. J. Heat Mass Transf. 48, 1819–1824 (2005)

    Article  MATH  Google Scholar 

  11. Bhowmik, A.; Singla, R.K.; Roy, P.K.; Prasad, D.K.; Das, R.; Repaka, R.: Predicting geometry of rectangular and hyperbolic fin profiles with temperature-dependent thermal properties using decomposition and evolutionary methods. Energy Convers. Manag. 74, 535–547 (2013)

    Article  Google Scholar 

  12. Kundu, B.; Wongwises, S.: A decomposition analysis on convecting-radiating rectangular plate fins for variable thermal conductivity and heat transfer coefficient. J. Frankl. Inst. 349, 966–84 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Mohsen, T.; Qiaobao, Z.: Analytical solution for evaluating the thermal performance and efficiency of convective radiative straight fins with various profiles and considering all non linearities. Energy Convers. Manag. 66, 199–210 (2013)

    Article  Google Scholar 

  14. Torabi, M.; Yaghoobi, H.; Aziz, A.: Analytical solution for convective-radiative continuously moving fin with thermal conductivity. Int. J. Thermophys. 33, 924–941 (2012)

    Article  Google Scholar 

  15. Dogonchi, A.S.; Ganji, D.D.: Convection–radiation heat transfer study of moving fin with temperature-dependent thermal conductivity, heat transfer coefficient and heat generation. Appl. Therm. Eng. 103, 705–712 (2016)

    Article  Google Scholar 

  16. Singla, R.K.; Das, R.: Application of decomposition method and inverse prediction of parameters in a moving fin. Energy Convers. Manag. 84, 268–2811 (2014)

    Article  Google Scholar 

  17. Aziz, A.; Khani, F.: Convection–radiation from a continuous moving fin of variable thermal conductivity. J. Frankl. Inst. 348, 640–651 (2011)

    Article  MATH  Google Scholar 

  18. Ma, J.; Sun, Y.; Li, B.: Spectral collocation method for transient thermal analysis of coupled conductive, convective and radiative heat transfer in the moving plate with temperature dependent properties and heat generation. Int. J. Heat Mass Transf. 114, 469–482 (2017)

    Article  Google Scholar 

  19. Singh, S.; Kumar, D.; Rai, K.N.: Convective–radiative fin with temperature dependent thermal conductivity, heat transfer coefficient and wavelength dependent surface emissivity. Propuls. Power Res. 3(4), 207–221 (2014)

    Article  Google Scholar 

  20. Motsa, S.S.; Magagula, V.M.; Sibanda, P.: A bivariate Chebyshev spectral collocation quasilinearization method for nonlinear evolution parabolic equations. Sci. World J. 2014, 581987 (2014)

  21. Roy, P.K.; Mondal, H.; Mallick, A.: A decomposition method for convective–radiative fin with heat generation. Ain Shams Eng. J. 6, 307–313 (2015)

    Article  Google Scholar 

  22. Adomian, G.: Solving Frontier Problems in Physics: The Decomposition Method. Kluwer Academic Publishers, Dordrecht (1994)

    Book  MATH  Google Scholar 

  23. Hasan, Y.Q.; Zhu, L.M.: Solving singular boundary value problems of higher order ordinary differential equations by modified Adomian decompositions method. Commun. Non Linear Sci. Numer. Simul. 14, 2592–2596 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Fatoorehchi, H.; Abolghasemi, H.: Investigation of Non linear problems of heat conduction in tapered cooling fins via symbolic programming. Appl. Appl. Math. 7, 717–734 (2012)

    MathSciNet  MATH  Google Scholar 

  25. Ching-Hung, C.; Cha’o-Kuang, C.: Application of Adomian’s decomposition Procedure to the analysis of convective–radiative fins. J. Heat Transf. 125, 312–315 (2013)

    Google Scholar 

  26. Bellman, R.E.; Kalaba, R.E.: Quasilinearization and Nonlinear Boundary Value Problems. Elsevier, New York (1965)

    MATH  Google Scholar 

  27. Motsa S.; Sibanda P.: On extending the quasilinearization method to higher order convergent hybrid schemes using the spectral homotopy analysis method. J. Appl. Math. Article ID 879195, 9 pages (2013) https://doi.org/10.1155/2013/879195

  28. Torabi, M.; Aziz, A.; Zhang, K.: A comparative study of longitudinal fins of rectangular, trapezoidal and concave parabolic profiles with multiple nonlinearities. Energy 51, 243–256 (2013)

    Article  Google Scholar 

  29. Sun, Y.-S.; Ma, J.; Li, B.-W.: 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)

  30. Roy, P.K.; Mallick, A.: Thermal analysis of straight rectangular fin using homotopy perturbation method. Alexzandria Eng. J. 55, 2269–2277 (2016)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical Engineering, Indian Institute of Technology (ISM), Dhanbad, Jharkhand, 826004, India

    Pranab Kanti Roy & Ashis Mallick

  2. School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Scottsville, Pietermaritzburg, 3209, South Africa

    Hiranmoy Mondal & Precious Sibanda

Authors
  1. Pranab Kanti Roy
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  2. Ashis Mallick
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  3. Hiranmoy Mondal
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  4. Precious Sibanda
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Correspondence to Pranab Kanti Roy.

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Roy, P.K., Mallick, A., Mondal, H. et al. A Modified Decomposition Solution of Triangular Moving Fin with Multiple Variable Thermal Properties. Arab J Sci Eng 43, 1485–1497 (2018). https://doi.org/10.1007/s13369-017-2983-3

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