Advertisement

The Analysis of Fin Radiation

  • Madassar Manzoor
Part of the Lecture Notes in Engineering book series (LNENG, volume 5)

Abstract

This paper presents the boundary integral equation (BIE) formulation and numerical solution procedure for two-dimensional problems governed by Laplace’s equation and subject to non-linear boundary conditions. The introduction of non-linear terms constitutes a fundamental extension of the BIE method, as previous applications have been restricted entirely to linear problems. Furthermore, non-linearities necessitate the use of iterative solution techniques which present the conceptual disadvantage that a solution is not guaranteed. However, no difficulties were encountered with the Newton-Raphson iterative method employed in this study. The various features of the non-linear BIE formulation are illustrated by the application to a physical problem of relevance in heat exchanger design.

Keywords

Heat Transfer Heat Exchanger Heat Transfer Rate Boundary Integral Equation Boundary Integral Equation Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M.A. Jaswon and J.T. Symm, Integral Equation Methods in Potential Theory and Electrostatics, Academic Press, London, 1977.Google Scholar
  2. 2.
    G.D. Smith, Numerical Solution of Partial Differential Equations, Oxford University Press, 1972.Google Scholar
  3. 3.
    O.C. Zienkiewicz, The Finite Element Method in Engineering, McGraw-Hill, London, 1971.MATHGoogle Scholar
  4. 4.
    F.J. Rizzo and D.J. Shippy, “A Method of Solution for Certain Problems of Transient Heat Conduction”, AIAA Journal, Vol. 8, pp. 2004–2009, 1970.CrossRefMATHADSGoogle Scholar
  5. 5.
    G.T. Symm, “Treatment of Singularities in the Solution of Laplace’s Equation by an Integral Equation Method”, National Physical Laboratory, Report NAC 31, 1973.Google Scholar
  6. 6.
    M. Maiti and S.K. Chakrabarty, “Integral Equation Solutions for Simply Supported Polygonal Plates”, International Journal of Engineering Science, Vol. 12, pp. 793–806, 1974.CrossRefMATHGoogle Scholar
  7. 7.
    W.A. Bell, W.L. Meyer and B.T. Zinn, “Predicting the Acoustics of Arbitrarily Shaped Bodies Using an Integral Approach”, AIAA Journal, Vol. 15, pp. 813–820, 1977.CrossRefMATHADSGoogle Scholar
  8. 8.
    Y.S. Wu, F.J. Rizzo, D.J. Shippy and J.A. Wagner, “An Advanced Boundary Integral Equation Method for Two-Dimensional Electromagnetic Field Problems”, Electric Machines and Electro-mechanics”, Vol. 1, pp. 301–313, 1977.Google Scholar
  9. 9.
    G. Fairweather, F.J. Rizzo, D.J. Shippy and Y.S. Wu, “On the Numerical Solution of Two-Dimensional Potential Problems by an Improved Boundary Integral Equation Method”, Journal of Computational Physics, Vol. 31, pp. 96–112, 1979.CrossRefMATHADSMathSciNetGoogle Scholar
  10. 10.
    D.B. Ingham, P.J. Heggs and M. Manzoor, “The Numerical Solution of Plane Potential Problems by Improved Boundary Integral Equation Methods”, Submitted to the Journal of Computational Physics, 1980.Google Scholar
  11. 11.
    G.T. Symm, “Integral Equation Methods in Potential Theory II”. Proceedings of the Royal Society, A275, pp. 33–46, 1963.MathSciNetGoogle Scholar
  12. 12.
    R. Butterfield, “The Application of the Integral Equation Methods to Continuum Problems in Soil Mechanics”, Roscoe Memorial Symposium, Cambridge, 1972Google Scholar
  13. 13.
    P.K. Banerjee, “Non-Linear Problems of Potential Flow”, In Developments in Boundary Element Methods, Vol. 1, Edited by P.K. Banerjee and R. Butterfield, Applied Science Publishers, London, 1979.Google Scholar
  14. 14.
    A. Ralston, A First Course in Numerical Analysis, McGraw-Hill, New York, 1965.MATHGoogle Scholar
  15. 15.
    E.M. Sparrow and R.D. Cess, Radiation Heat Transfer, Brooks/Cole, Belmont, 1970.Google Scholar
  16. 16.
    D.Q. Kern and A.D. Kraus, Extend Surface Heat Transfer, McGraw-Hill, New York, 1972.Google Scholar
  17. 17.
    R.C. Donovan and W.M. Rohrer, “Radiative and Convective Conducting Fins on a Plane Wall Including Mutual Irradiation”, Journal of Heat Transfer, Vol. 93, pp. 41–46, 1971.CrossRefGoogle Scholar
  18. 18.
    M.N. Schnurr, “Radiation From an Array of Longitudinal Fins of Triangular Profile”, AIAA Journal, Vol. 13, pp. 691–693, 1975.CrossRefADSGoogle Scholar
  19. 19.
    R.D. Karam and R.J. Eby, “Linearised Solution of Conducting Radiating Fins”, AIAA Journal, Vol. 16, pp 536–538, 1978.CrossRefADSGoogle Scholar
  20. 1.
    K.A. Gardner, “Efficiency of extended surface”, Transactions of the ASME, Vol. 67, pp. 621–631, 1945.Google Scholar
  21. 2.
    S. Guceri and C.J. Maday, “A least weight circular cooling fin”, Journal of Engineering for Industry, Vol. 97, pp. 1190–1193, 1975.CrossRefGoogle Scholar
  22. 3.
    I. Mikk, “Convective fin of minimum mass”, International Journal of Heat and Mass Transfer, Vol. 23, pp. 707–711, 1980.CrossRefMATHGoogle Scholar
  23. 4.
    R.K. Irey, “Errors in one-dimensional fin solution”, Journal of Heat Transfer, Vol. 90, pp. 175–176, 1968.Google Scholar
  24. 5.
    M. Levistsky, “The criterion for validity of the fin approximation”, International Journal of Heat and Mass Transfer, Vol. 15, pp. 1960–1963, 1972.CrossRefGoogle Scholar
  25. 6.
    W. Lau and C.W. Tan, “Errors in one-dimensional heat transfer analysis in straight and annular fins”, Journal of Heat Transfer, Vol. 95, pp. 549–551, 1973.CrossRefGoogle Scholar
  26. 7.
    E.M. Sparrow and L. Lee, “Effects of fin-base temperature depression in a multifin array”, Journal of Heat Transfer, Vol. 97, pp. 463–465, 1975.CrossRefADSGoogle Scholar
  27. 8.
    N.V. Suryanarayana, “Two-dimensional effects on heat transfer from an array of straight fins”, Journal of Heat Transfer, Vol. 99, pp. 129–132, 1977.CrossRefGoogle Scholar
  28. 9.
    P.J. Heggs and P.R. Stones, “The effects of dimensions on the heat flowrate through extended surfaces”, Journal of Heat Transfer, Vol. 102, 180–182, 1980.CrossRefGoogle Scholar
  29. 10.
    R.L. Chambers and E.V. Somers, “Radiation fin efficiency for one-dimensional heat flow in a circular fin”, Journal of Heat Transfer, Vol. 81, pp. 327–329, 1959.Google Scholar
  30. 11.
    J.G. Bartas and W.H. Sellers, “Radiation fin effectiveness”, Journal of Heat Transfer, Vol. 82, pp. 73–75, 1960.Google Scholar
  31. 12.
    M.N. Schnurr and C.A. Cothran, “Radiation from an array of gray circular fins of trapezoidal profile”, AIAA Journal, Vol. 12, 1476–1480, 1974.CrossRefADSGoogle Scholar
  32. 13.
    M.N. Schnurr, “Radiation from an array of longitudinal fins of triangular profile”, AIAA Journal, Vol. 13, pp. 691–693, 1975.CrossRefADSGoogle Scholar
  33. 14.
    R.C. Donovan and W.M. Rohrer, “Radiative and convecting conducting fins on plane wall including mutual irradiation”, Journal of Heat Transfer, Vol. 93, pp. 41–46, 1971.CrossRefGoogle Scholar
  34. 15.
    R.G. Eslinger and B.T.F. Chung, “Periodic heat transfer in radiating and convecting fins or fin arrays”, AIAA Journal, Vol. 17, pp. 1134–1140, 1979.CrossRefADSGoogle Scholar
  35. 16.
    D.B. Ingham, P.J. Heggs and M. Manzoor, “Boundary integral equation solution of nonlinear plane potential problems”, accepted for publication in Institute of Mathematics and Its Applications Journal of Numerical Analysis, 1981.Google Scholar
  36. 17.
    G.D. Smith, Numerical Solution of Partial Differential Equations, Oxford University Press, 1972.Google Scholar
  37. 18.
    H.C. Hottel and A.F. Sarofim, Radiative Transfer, McGraw-Hill, New York, 1967.Google Scholar
  38. 19.
    A. Ralston, A First Course in Numerical Analysis, McGraw-Hill, New York, 1965.MATHGoogle Scholar
  39. 20.
    O.C. Ziekiewicz, The Finite Element Method in Engineering, McGraw-Hill, London, 1971.Google Scholar
  40. 21.
    M.A. Jaswon and G.T. Symm, Integral Equation Methods in Potential Theory and Electrostatics, Academic Press, London, 1977.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • Madassar Manzoor
    • 1
  1. 1.Dept. of Mathematical SciencesUniversity of DurhamDurhamEngland

Personalised recommendations