Heat transfer by new families of straight and pin fins: exact solutions

  • Leonid G. HaninEmail author
  • David E. Brown


Fins are surface extensions used to increase the rate of heat transfer from a heated surface to the surrounding cooler fluid or from a heated fluid to the surface. In this article, we describe new families of straight and pin fins for which the temperature distribution along the fin as well as fin effectiveness and efficiency can be computed in closed form. The profile of the new straight fin is a circular arc centered on the fin’s longitudinal axis, while the profile of the new pin fin is a more complex concave parametric curve. Although the pattern of temperature distribution in these fins is remarkably similar to that in the classic straight and pin fins of constant thickness, the geometry of the new fins is strikingly different; in particular, the maximum length of the new fins is finite. We also discovered that both classic fins of constant thickness and novel fins described in this work have the following equivalence property: if the ratio of the heat transfer coefficient on the fin’s tip to that on the lateral surface is not too high, then the temperature distribution for a fin with a non-adiabatic boundary condition at the tip is identical to that for a longer fin of the same kind with adiabatic tip. Our analysis of heat transfer by fins is only based on standard homogeneity, steady-state, and one-dimensionality assumptions. In particular, we completely dispense with the “length-of-arc” assumption that underlies most of the previous works.


Adiabatic boundary condition Effectiveness Efficiency “Length-of-arc” assumption One-dimensional approximation Pin fin Straight fin Temperature distribution 

Mathematics Subject Classification

34A30 34B05 80A20 



We are grateful to the two anonymous referees whose thoughtful suggestions have helped us to improve the article.


  1. 1.
    Kraus AD, Aziz A, Welty J (2001) Extended surface heat transfer. Wiley, New YorkGoogle Scholar
  2. 2.
    Lienhard JH IV, Lienhard JHV (2018) A heat transfer textbook. Phlogiston Press, Cambridge, MAzbMATHGoogle Scholar
  3. 3.
    Murray WM (1938) Heat transfer through an annular disk or fin of uniform thickness. Trans ASME 60:A78–A80Google Scholar
  4. 4.
    Gardner KA (1945) Efficiency of extended surface. Trans ASME 67:621–631Google Scholar
  5. 5.
    Irey RK (1968) Errors in the one-dimensional fin solution. ASME J Heat Transf 90:175–176CrossRefGoogle Scholar
  6. 6.
    Lau Wah, Tan CW (1973) Errors in one-dimensional heat transfer analysis in straight and annular fins. ASME J Heat Transf 95:549–551CrossRefGoogle Scholar
  7. 7.
    Maday CJ (1974) The minimum weight one-dimensional straight cooling fin. ASME J Eng Ind 96:161–165CrossRefGoogle Scholar
  8. 8.
    Snider AD, Kraus AD (1983) Recent developments in the analysis and design of extended surface. J Heat Transf 105:302–306CrossRefGoogle Scholar
  9. 9.
    Graff S, Snider AD (1996) Mathematical analysis of the length-of-arc assumption. Heat Transf Eng 17(2):67–71CrossRefGoogle Scholar
  10. 10.
    Hanin LG, Campo A (2003) A new minimum volume straight cooling fin taking into account the “length of arc”. Int J Heat Mass Transf 46(26):5145–5152CrossRefGoogle Scholar
  11. 11.
    Hanin LG (2008) A new optimum pin fin beyond the “length-of-arc” assumption. Heat Transf Eng 29(7):608–614CrossRefGoogle Scholar
  12. 12.
    Schmidt E (1926) Die Wärmeübertragung durch Rippen. Z. Ver. Deut. Ingenieure 70:885–889 and 947–951Google Scholar
  13. 13.
    Badescu V (2017) Smooth and non-smooth optimal pin fin profiles beyond the Schmidt optimality assumption and “length-of-arc” approximation. Appl Math Model 47:358–380MathSciNetCrossRefGoogle Scholar
  14. 14.
    Snider AD (1982) Mathematical techniques in extended surface analysis. Math Model 3:191–206CrossRefGoogle Scholar
  15. 15.
    Harper DR, Brown WB (1922) Mathematical equations for heat conduction in the fins of air cooled engines. Natl Advis Comm Aeronaut Rep 158:679–708Google Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsIdaho State UniversityPocatelloUSA
  2. 2.Department of MathematicsBrigham Young University-IdahoRexburgUSA

Personalised recommendations