Skip to main content
SpringerLink
Log in

Unsteady temperature field in slabs with different kinds of time-dependent boundary conditions

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

A closed solution for one-dimensional heat conduction in a slab with nonhomogenous time-dependent boundary condition at one end and homogenous boundary condition with time-dependent heat transfer coefficient at the other end is proposed. The shifting function method developed by Lee and his colleagues is used to derive the solution of the temperature distribution of the slab. By splitting the Biot function into a constant plus a function and introducing two particularly chosen shifting functions, the system is transformed into a partial differential equation with homogenous boundary conditions only. Consequently, the transformed system can be solved by a series expansion method. Two limiting cases, including time-independent boundary condition and constant heat transfer coefficient, are proved to be identical to those in the literature. Three-term approximation used in numerical examples can always result in an error <1 % in the present study, rendering the proposed methodology efficient and accurate. Finally, the influence of parameters of heat flux function or Biot function on the temperature distribution is presented.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

a, b, c, d :

Constants used to express heat flux and Biot functions

A n , B n :

Auxiliary series coefficients

Bi:

Biot function

f 1, f 2 :

Auxiliary time functions

F :

Equal to Biot function minus a constant

g 1, g 2 :

Shifting functions

h :

Time-dependent heat transfer coefficient (W  m−2 K−1)

H :

Time-dependent heat flux (Km−1)

k :

Thermal conductivity (W  m−1  K−1)

L :

Thickness of slab (m)

N n :

Norm of try functions

q n :

Time-dependent generalized coordinate

Q n :

Auxiliary time-dependent function

s 1, s 2 :

Parameters used to express heat flux and Biot functions

t :

Time variable (s)

T :

Temperature function (K)

T r :

Reference temperature (K)

T 0 :

Initial temperature function (K)

v :

Auxiliary function

v 0 :

Initial value of auxiliary function

x :

Space coordinate (m)

X :

Dimensionless coordinate

α :

Thermal diffusivity (m2s−1)

\({\beta_n, \bar{{\beta}}_n}\) :

Auxiliary series coefficients

δ :

Initial value of Biot function

ϕ n :

n-th eigenfunction

\({\gamma_n, \bar{{\gamma}}_n}\) :

Auxiliary series coefficients

ζ :

Auxiliary integration variable

λ n :

n-th eigenvalue

θ :

Dimensionless temperature

\({\theta_0 ,\,\,\bar{{\theta}}_0}\) :

Dimensionless initial temperature functions

τ :

Dimensionless time variable

ω :

Parameter for heat flux function

ξ n :

Auxiliary function

ψ, ψ 0 :

Dimensionless time-dependent heat flux functions

φ :

Auxiliary integration variable

0, m, n :

Indices

References

  1. Özisik M.N.: Boundary Value Problems of Heat Conduction. International Textbook Company, Pennysylvania (1968)

    Google Scholar 

  2. Johansson B.T., Lesnic D.: A method of fundamental solutions for transient heat conduction. Eng. Anal. Bound. Elem. 32, 697–703 (2008)

    Article  MATH  Google Scholar 

  3. Young D.L., Tsai C.C., Murugesan K., Fan C.M., Chen C.W.: Time-dependent fundamental solutions for homogeneous diffusion problems. Eng. Anal. Bound. Elem. 28, 1463–1473 (2004)

    Article  MATH  Google Scholar 

  4. Zhu S.P., Liu H.W., Lu X.P.: A combination of LTDRM and ATPS in solving diffusion problems. Eng. Anal. Bound. Elem. 21, 285–289 (1998)

    Article  MATH  Google Scholar 

  5. Amado J.M., Tobar M.J., Ramil A., Yáñez A.: Application of the Laplace transform dual reciprocity boundary element method in the modelling of laser heat treatments. Eng. Anal. Bound. Elem. 29, 126–135 (2005)

    Article  MATH  Google Scholar 

  6. Bulgakov V., Sarler B., Kuhn G.: Iterative solution of systems of equations in the dual reciprocity boundary element method for the diffusion equation. Int. J. Numer. Method Eng. 43, 713–732 (1998)

    Article  MATH  Google Scholar 

  7. Sahin A.Z.: Analytical solutions of transient heat conduction in semi-infinite solid with time varying boundary conditions by means of similarity transformation. Int. Commun. Heat Mass Transf. 22, 89–97 (1995)

    Article  Google Scholar 

  8. Barletta A., Zanchini E., Lazzari S., Terenzi A.: Numerical study of heat transfer from an offshore buried pipeline under steady-periodic thermal boundary conditions. Appl. Therm. Eng. 28, 1168–1176 (2008)

    Article  Google Scholar 

  9. Caffagni A., Angeli D., Barozzi G.S., Polidoro S.: A revised approach for one-dimensional time-dependent heat conduction in a slab. ASME J. Heat Transf. 135, 03130-1–1031301-8 (2013)

    Article  Google Scholar 

  10. Lee S.Y., Huang T.W.: Exact solutions for heat conduction in non-uniform mediums with general time-dependent boundary conditions. J. Chin. Soc. Mech. Eng. 34, 475–485 (2013)

    Google Scholar 

  11. Lee, S.Y., Huang, C.C.: Analytic solutions for heat conduction in functionally graded circular hollow cylinders with time-dependent boundary conditions. Math. Probl. Eng, p. 8. Article ID 816385 (2013). doi:10.1155/2013/816385

  12. Ivanov V.V., Salomatov V.V.: On the calculation of the temperature field in solids with variable heat-transfer coefficients. J. Eng. Phys. Thermophys. 9, 83–85 (1965)

    Google Scholar 

  13. Ivanov V.V., Salomatov V.V.: Unsteady temperature field in solid bodies with variable heat-transfer coefficient. J. Eng. Phys. Thermophys. 11, 266–268 (1966)

    Article  Google Scholar 

  14. Postol’nik Yu.S.: One-dimensional convective heating with a time-dependent heat-transfer coefficient. J. Eng. Phys. Thermophys. 18, 316–322 (1970)

    Google Scholar 

  15. Kozlov V.N.: Solution of heat-conduction problem with variable heat exchange coefficient. J. Eng. Phys. Thermophys. 18, 100–104 (1970)

    Article  Google Scholar 

  16. Becker N.M., Bivins R.L., Hsu Y.C., Murphy H.D., White A.B., Wing G.M.: Heat diffusion with time-dependent convective boundary condition. Int. J. Numer. Methods Eng. 19, 1871–1880 (1983)

    Article  MATH  Google Scholar 

  17. Holy Z.J.: Temperature and stresses in reactor fuel elements due to time-dependent heat-transfer coefficients. Nucl. Eng. Des. 18, 145–197 (1972)

    Article  Google Scholar 

  18. Özisik M.N., Murray R.L.: On the solution of linear diffusion problems with variable boundary condition parameters. ASME J. Heat Transf. 96, 48–51 (1974)

    Article  Google Scholar 

  19. Moitsheki, R.J.: Transient heat diffusion with temperature-dependent conductivity and time-dependent heat transfer coefficient. Math. Probl. Eng, p. 9. Article ID 347568 (2008). doi:10.1155/2008/347568

  20. Chen H.T., Sun S.L., Huang H.C., Lee S.Y.: Analytic closed solution for the heat conduction with time dependent heat convection coefficient at one boundary. CMES Comput. Model. Eng. Sci. 59, 107–126 (2010)

    MATH  MathSciNet  Google Scholar 

  21. Lee S.Y., Lin S.M.: Dynamic analysis of non-uniform beams with time-dependent elastic boundary conditions. ASME J. Appl. Mech. 63, 474–478 (1996)

    Article  MATH  Google Scholar 

  22. Lee S.Y., Lin S.M., Lee C.S., Lu S.Y., Liu Y.T.: Exact large deflection of beams with nonlinear boundary conditions. CMES Comput. Model. Eng. Sci. 30, 27–36 (2008)

    Article  Google Scholar 

  23. Yatskiv O.I., Shvets’ R.M., Bobyk B.Ya.: Thermostressed state of a cylinder with thin near-surface layer having time-dependent thermophysical properties. J. Math. Sci. 187, 647–666 (2012)

    Article  Google Scholar 

  24. Tu T.W., Lee S.Y.: A new analytic solution for the heat conduction with time-dependent heat transfer coefficient. World Acad. Sci. Eng. Technol. Int. J. Mech. Aerosp. Ind. Mechatron. Eng. 8, 1372–1377 (2014)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, National Cheng Kung University, Tainan, Taiwan, Republic of China

    Sen Yung Lee

  2. Department of Mechanical Engineering, Air Force Institute of Technology, No. 198, Jieshou W. Rd., Gangshan Township, Kaohsiung County, 820, Taiwan, Republic of China

    Te Wen Tu

Authors
  1. Sen Yung Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Te Wen Tu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Te Wen Tu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lee, S.Y., Tu, T.W. Unsteady temperature field in slabs with different kinds of time-dependent boundary conditions. Acta Mech 226, 3597–3609 (2015). https://doi.org/10.1007/s00707-015-1389-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-015-1389-0

Keywords

Search

Navigation