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Conductive Heat Transport Systems

  • Achintya Kumar PramanickEmail author
Chapter
Part of the Heat and Mass Transfer book series (HMT)

Abstract

In this chapter, we directly apply the law of motive force in place of variational formulation as well as optimal control theory for a class of problems pertaining to conductive heat transport mode in the realm of thermal insulation design. From the physics of the principle it has been deduced that a truly minimum exists for such class of problems. To start with, the optimum distribution of limited amount of insulating material on one side of a plane surface as well as a curved wall is obtained assuming that the amount of insulating material does not affect the imposed temperature gradient.

Keywords

Heat Transfer Wall Temperature Motive Force Plane Wall Optimum Profile 
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.

References

  1. 1.
    Courant, R., Robbins, H.: What is Mathematics? (Stewart, I., Revised), pp. 329–397. Oxford University Press, Oxford (2007)Google Scholar
  2. 2.
    Hancock, H.: The Theory of Maxima and Minima. Dover, New York (1960)zbMATHGoogle Scholar
  3. 3.
    Niven, I., Lance, L.H.: Maxima and Minima Without Calculus. MAA, Washington (1981)zbMATHGoogle Scholar
  4. 4.
    Tikhomirov, V.M.: Stories About Maxima and Minima. Mathematical World-I, pp. 3–8. AMS, Rhode Island (1990)Google Scholar
  5. 5.
    Tonti, E.: A systematic approach to the variational formulation in physics and engineering. In: Autumn Course on Variational Methods in Analysis and Mathematical Physics. ICTP, Trieste, 20 Oct–11 Dec 1981Google Scholar
  6. 6.
    Yourgrau, W., Mandelstam, S.: Variational Principles in Dynamics and Quantum Theory, p. 175. Dover, New York (2007)Google Scholar
  7. 7.
    Bejan, A.: Advanced Engineering Thermodynamics, pp. 26–34. Wiley, New York (2006)Google Scholar
  8. 8.
    Müller, I.: A History of Thermodynamics. Springer, New York (2007)zbMATHGoogle Scholar
  9. 9.
    Truesdell, C.: The Tragicomedy of Classical Thermodynamics. CISM, Udine, Courses and Lectures, No. 70. Springer, New York (1983)Google Scholar
  10. 10.
    Buchdahl, H.A.: A variational principle in classical thermodynamics. Am. J. Phys. 55, 81–83 (1987)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Hancock, H.: The Theory of Maxima and Minima, pp. 150–151. Dover, New York (1960)zbMATHGoogle Scholar
  12. 12.
    Biot, M.A.: Variational Principles in Heat Transfer. Oxford University Press, Oxford (1970)zbMATHGoogle Scholar
  13. 13.
    Donnelly, R.J., Herman, R., Prigogine, I. (eds.): Non-Equilibrium Thermodynamics, Variational Techniques and Stability. University of Chicago Press, Chicago (1966)Google Scholar
  14. 14.
    Finlayson, B.A., Scriven, L.E.: On the search for variational principles. Int. J. Heat Mass Transf. 10, 799–821 (1967)CrossRefzbMATHGoogle Scholar
  15. 15.
    Goldstine, H.H.: A History of the Calculus of Variations from the 17th Through 19th Century. Springer, New York (1980)CrossRefzbMATHGoogle Scholar
  16. 16.
    Sieniutycz, S.: Conservation Laws in Variational Thermo-Hydrodynamics. Springer, New York (1994)CrossRefzbMATHGoogle Scholar
  17. 17.
    Sieniutycz, S.: Progress in variational formulations of macroscopic processes. In: Sieniutycz, S., Farkas, H. (eds.) Variational and Extremum Principles in Macroscopic Systems. Elsevier, London (2004)Google Scholar
  18. 18.
    Todhunter, I.: A History of the Calculus of Variations During the Nineteenth Century. Dover, New York (2005)Google Scholar
  19. 19.
    Yourgrau, W., Mandelstam, S.: Variational Principles in Dynamics and Quantum Theory, pp. 162–180. Dover, New York (2007)Google Scholar
  20. 20.
    Prigogine, I.: Remarks on variational principles. In: Donnelly, R.J., Herman, R., Prigogine, I. (eds.) Non-Equilibrium Thermodynamics, Variational Techniques and Stability. Chicago University Press, Chicago (1966)Google Scholar
  21. 21.
    Kantrovich, L.V., Krylov, V.I.: Approximate Methods of Higher Analysis (trans: Benster, C.D.). Interscience, New York (1964)Google Scholar
  22. 22.
    Schechter, R.S.: Variational principles for continuum systems. In: Donnelly, R.J., Herman, R., Prigogine, I. (eds.) Non-Equilibrium Thermodynamics, Variational Techniques and Stability. Chicago University Press, Chicago (1966)Google Scholar
  23. 23.
    Prigogine, I.: Evolution criteria, variational properties and fluctuations. In: Donnelly, R.J., Herman, R., Prigogine, I. (eds.) Non-Equilibrium Thermodynamics, Variational Techniques and Stability. Chicago University Press, Chicago (1966)Google Scholar
  24. 24.
    Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes (trans: Trirogoff, K.N.). In: Neustadt, L.W. (ed.), pp. 1–73, 75–114, 239–256. Wiley-Interscience, New York (1965)Google Scholar
  25. 25.
    Fel’dbaum, A.A.: On the question of synthesizing optimum automatic control systems. In: Transactions of the Second All Union Conference on Automatic Control Theory-II, USSR Academy of Science (1955) (in Russian)Google Scholar
  26. 26.
    Ten Hoor, M.J.: The variational method—why stop half way? Am. J. Phys. 62, 166–168 (1994)CrossRefGoogle Scholar
  27. 27.
    Bejan, A.: How to distribute a finite amount of insulation on a wall with nonuniform temperature. Int. J. Heat Mass Transf. 36, 49–56 (1993)CrossRefGoogle Scholar
  28. 28.
    Bejan, A.: Second-law analysis in heat transfer and thermal design. Adv. Heat Transf. 15, 1–58 (1982)CrossRefGoogle Scholar
  29. 29.
    Bejan, A.: Entropy generation minimization: the new thermodynamics of finite-size devices and finite-time processes. J. Appl. Phys. 79, 1191–1218 (1996)CrossRefGoogle Scholar
  30. 30.
    Bejan, A.: A general variational principle for thermal insulation system design. Int. J. Heat Mass Transf. 22, 219–228 (1979)CrossRefGoogle Scholar
  31. 31.
    Tonti, E.: The reason for analogies between physical theories. Appl. Math. Modell. 1, 37–50 (1976)CrossRefMathSciNetGoogle Scholar
  32. 32.
    Censor, D.: Fermat’s principle and real space time rays in absorbing media. J. Phys. A Math. Gen. 10, 1781–1790 (1977)CrossRefGoogle Scholar
  33. 33.
    Newcomb, W.A.: Generalized Fermat principle. Am. J. Phys. 51, 338–340 (1983)CrossRefGoogle Scholar
  34. 34.
    Bejan, A.: Advanced Engineering Thermodynamics, pp. 705–841. Wiley, New York (2006)Google Scholar
  35. 35.
    Bejan, A.: Shape and Structure, from Engineering to Nature. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  36. 36.
    Lemons, D.S.: Perfect Form, pp. ix–xi. Princeton University Press, Princeton (1997)Google Scholar
  37. 37.
    Pramanick, A.K., Das, P.K.: Method of synthetic constraint, Fermat’s principle and the constructal law in the fundamental principle of conductive heat transport. Int. J. Heat Mass Transf. 50, 1823–1832 (2007)CrossRefzbMATHGoogle Scholar
  38. 38.
    Feynman, R.: The Character of Physical Law, pp. 149–173. MIT Press, Massachusetts (1967)Google Scholar
  39. 39.
    Leff, H.S.: What if entropy were dimensionless? Am. J. Phys. 67, 1114–1122 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    Schmidt, E.: Die Wärmeübertragung durch Rippen. Z. Ver. Dt. Ing. 70, 885–889, 947–951 (1926) (in German)Google Scholar
  41. 41.
    Duffin, R.J.: A variational problem relating to cooling fins. J. Math. Mech. 8, 47–56 (1959)zbMATHMathSciNetGoogle Scholar
  42. 42.
    Jany, P., Bejan, A.: Ernst Schmidt’s approach to fin optimization: an extension to fins with variable conductivity and the design of ducts for fluid flow. Int. J. Heat Mass Transf. 31, 1635–1644 (1988)CrossRefGoogle Scholar
  43. 43.
    Clausius, R.: On the motive power of heat, and on the laws which can be deduced from it for the theory of heat (trans: Magie, W.F.). In: Mendoza, E. (ed.) Reflections on the Motive Power of Fire. Dover, New York (2005)Google Scholar
  44. 44.
    Bejan, A.: Heat Transfer, pp. 42–44. Wiley, New York (1993)Google Scholar
  45. 45.
    Feynman, R.: The Character of Physical Law, pp. 84–107. MIT Press, Massachusetts (1967)Google Scholar
  46. 46.
    Rosen, J.: Symmetry in Science, pp. 134–154. Springer, New York (1995)CrossRefGoogle Scholar
  47. 47.
    Van Fraassen, B.C.: Laws and Symmetry. Oxford University Press, Oxford (1989)CrossRefGoogle Scholar
  48. 48.
    Todhunter, I.: A History of the Calculus of Variations During the Nineteenth Century, pp. 243–253. Dover, New York (2005)Google Scholar
  49. 49.
    Culverwell, E.P.: On the discrimination of maxima and minima solutions in the calculus of variations. Philos. Trans. R. Soc. Lond. A 178, 95–129 (1887)CrossRefzbMATHGoogle Scholar
  50. 50.
    Fourier, J.: The Analytical Theory of Heat (trans: Freeman, A.), pp. 137–144. Dover, New York (2003)Google Scholar
  51. 51.
    Carslaw, H.S.: Introduction to the Theory of Fourier Series and Integrals, pp. 323–328. Dover, New York (1930)zbMATHGoogle Scholar
  52. 52.
    Carslaw, H.S., Jaeger, J.C.: Conduction of Heat in Solids, pp. 6–8. Oxford University Press, Oxford (1959)Google Scholar
  53. 53.
    Tolstov, G.P.: Fourier Series (trans: Silverman, R.A.), pp. 12, 60. Dover, New York (1976)Google Scholar
  54. 54.
    Whittaker, E.T., Watson, G.N.: A Course on Modern Analysis, pp. 224–225. Cambridge University Press, Cambridge (1996)CrossRefGoogle Scholar
  55. 55.
    Carslaw, H.S.: Introduction to the Theory of Fourier Series and Integrals, pp. 284–288. Dover, New York (1930)zbMATHGoogle Scholar
  56. 56.
    Carslaw, H.S.: Introduction to the Theory of Fourier Series and Integrals, pp. 329–361. Dover, New York (1930)zbMATHGoogle Scholar
  57. 57.
    Bridgman, P.W.: Tolman’s principle of similitude. Phys. Rev. 8, 423–431 (1916)CrossRefGoogle Scholar
  58. 58.
    Bellman, R.: Methods of Nonlinear Analysis-I, pp. 304–330. Academic Press, New York (1970)Google Scholar
  59. 59.
    Courant, R.: Differential and Integral Calculus-I (trans: McShane, E.J.), pp. 178–179. Wiley, New York (1967)Google Scholar
  60. 60.
    Bellman, R.: Introduction to the Mathematical Theory of Control Processes-I, pp. 33–34. Academic Press, New York (1970)Google Scholar
  61. 61.
    Bejan, A.: Heat Transfer, p. 40. Wiley, New York (1993)Google Scholar
  62. 62.
    Nusselt, W.: Die Abhängigkeit der Wärmeübergangszahl von der Rohrlänge. VDI Z. 54, 1154–1158 (1910) (in German)Google Scholar
  63. 63.
    Pramanick, A.K., Das, P.K.: Note on constructal theory of organization in nature. Int. J. Heat Mass Transf. 48, 1974–1981 (2005)CrossRefzbMATHGoogle Scholar
  64. 64.
    Kalyon, M., Sahin, A.Z.: Application of optimal control theory in pipe insulation. Numer. Heat Transf. A- Appl. 41, 391–402 (2002)CrossRefGoogle Scholar
  65. 65.
    Özişik, M.N.: Heat Conduction, pp. 17–20. Wiley, New York (1993)Google Scholar
  66. 66.
    Tan, A., Holland, L.R.: Tangent law of refraction for heat conduction through an interface and underlying variational principle. Am. J. Phys. 58, 988–991 (1990)CrossRefGoogle Scholar
  67. 67.
    Bellman, R.: Dynamic Programming. Dover, New York (2003)zbMATHGoogle Scholar
  68. 68.
    Sieniutycz, S.: Dynamic programming approach to a Fermat type principle for heat flow. Int. J. Heat Mass Transf. 43, 3453–3468 (2000)CrossRefzbMATHGoogle Scholar
  69. 69.
    Pramanick, A.K., Das, P.K.: Heuristics as an alternative to variational calculus for optimization of a class of thermal insulation systems. Int. J. Heat Mass Transf. 48, 1851–1857 (2005)CrossRefzbMATHGoogle Scholar
  70. 70.
    Sedov, L.I.: Similarity and Dimensional Methods in Mechanics (trans: Kisin, V.I.). Mir, Moscow (1982)Google Scholar
  71. 71.
    Bejan, A.: Convection Heat Transfer, pp. 37–42. Wiley, New York (2004)Google Scholar
  72. 72.
    Luikov, A.V.: Conjugated heat transfer problems. Int. J. Heat Mass Transf. 3, 293–303 (1961)CrossRefGoogle Scholar
  73. 73.
    Janyszek, H., Mrugala, R.: Riemannian and Finslerian geometry and fluctuations of thermodynamic systems. In: Sieniutycz, S., Salamon, P. (eds.) Nonequlibrium Theory and Extremum Principles. Taylor & Francis, New York (1990)Google Scholar
  74. 74.
    Bejan, A.: Constructal comment on a Fermat-type principle for heat flow. Int. J. Heat Mass Transf. 46, 1885–1886 (2003)CrossRefzbMATHGoogle Scholar
  75. 75.
    Bejan, A.: Advanced Engineering Thermodynamics, pp. 352–356, 464–466, 569–571, 709–721, 782–788, 816–820. Wiley, New York (2006)Google Scholar
  76. 76.
    Bejan, A.: Shape and Structure, from Engineering to Nature, pp. 53–56, 84–88, 99–108, 151–161, 220–223, 234–242, 287–288. Cambridge University Press, Cambridge (2000)Google Scholar
  77. 77.
    Bejan, A., Tondeur, D.: Equipartition, optimal allocation, and the constructal approach to predicting organization in nature. Rev. Gen. Therm. 37, 165–180 (1998)CrossRefGoogle Scholar
  78. 78.
    De Vos, A., Desoete, B.: Equipartition principle in finite-time thermodynamics. J. Non-Equilib. Thermodyn. 25, 1–13 (2000)CrossRefzbMATHGoogle Scholar
  79. 79.
    Lewins, J.: Bejan’s constructal theory of equal potential distribution. Int. J. Heat Mass Transf. 46, 1541–1543 (2003)CrossRefzbMATHGoogle Scholar
  80. 80.
    Tuckey, C.: Nonstandard Methods in the Calculus of Variations. Pitman Research Notes in Mathematics Series, vol. 297. Longman Scientific & Technical, Essex (1993)Google Scholar
  81. 81.
    Bellman, R.: Selective Computation, p. 38. World Scientific, Philadelphia (1985)CrossRefzbMATHGoogle Scholar
  82. 82.
    Lardner, T.J.: Biot’s variational principle in heat conduction. AIAA J. 1, 196–206 (1963)CrossRefzbMATHGoogle Scholar
  83. 83.
    Courant, R., Hilbert, D.: Methods of Mathematical Physics-I, pp. 291–295. Wiley, Berlin (2008)Google Scholar
  84. 84.
    Hildebrand, F.B.: Methods of Applied Mathematics, pp. 89–92, 145–148. Dover, New York (1992)Google Scholar
  85. 85.
    Morse, P.M., Feshbach, H.: Methods of Theoretical Physics-I, pp. 719–726. McGraw-Hill, New York (1953)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringNational Institute of Technology DurgapurDurgapurIndia

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