Skip to main content
SpringerLink
Log in

Numerical nonlinear inverse problem of determining wall heat flux

  • Original
  • Published:
Heat and Mass Transfer Aims and scope Submit manuscript

Abstract

The inverse problem of determining time-variable surface heat flux in a plane wall, with constant or temperature dependent thermal properties, is numerically studied. Different kinds of incident heat flux, including rectangular waveform, are assumed. The solution is numerically solved as a function estimation problem, so that no a priori information for the functional waveforms of the unknown heat flux is needed. In all cases, a solution in the form of a piece-wise function is used to approach the incident flux. Transient temperature measurements at the boundary, from the solution of the direct problem, served as the simulated experimental data needed as input for the inverse analysis. Both direct and inverse heat conduction problems are solved using the network simulation method. The solution is obtained step-by-step by minimising the classical functional that compares the above input data with those obtained from the solution of the inverse problem. A straight line of variable slope and length is used for each one of the stretches of the desired solution. The influence of random error, number of functional terms and the effect of sensor location are studied. In all cases, the results closely agree with the solution.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Abbreviations

a, b:

constant in Eq. 19

c e :

specific heat, J kg−1 C−1

C :

capacitor, F

e :

average relative error in Eq. 24

F :

functional defined in Eq. 18

G :

voltage-control current source

h :

heat transfer coefficient, W m−2 C−1

J :

electric current variable, A

j :

heat flux rate, W m−2

j1, j2:

constants

k :

thermal conductivity, W m−1 C−1

K :

constant in computational procedure

L :

thickness of the slab

n :

first value of temperature within the functional

N :

number of volume elements

r :

regularisation parameter

R :

resistor

t :

time, s

t 1 :

constant

T :

temperature, C

V :

voltage, V

x :

spatial co-ordinate

δ:

convergence criteria

ε:

random error value

σ:

standard deviation

μ:

random variable

ρ:

density, kg m−3

ω:

angular frequency

Δx:

thickness of the control volume

Δt:

time interval temperature measurement

sur:

surrounding medium

C:

referred to the capacitor

d :

number of iterations

f :

referred to particular location at the slab

est:

values estimated of fluxes

exact:

values exacts of fluxes

i :

connected to the volume element i, 1≤ i≤ N; also, centre of the volume element

i±Δ:

right and left ends of the volume element

in:

incident

inv:

refers to the solution of the inverse problem

s :

1, 2, ..., m

m :

natural number (total number of temperature measurements)

n :

refers to the first temperature with the stretch

o:

associated to the initial condition

z :

1, 2,... Z, in Eq. 18

Z :

number of stretches of the piece-wise function in the IHCP

∞:

value infinite

References

  1. Stolz G Jr (1960) Numerical solution to an inverse problem of heat conduction for simple shape. J Heat Transfer T ASME 82:20–26

    Google Scholar 

  2. Burggraf OR (1964) An exact solution of the inverse problem in heat conduction theory and publications. J Heat Transfer T ASME 86C:373–382

    Google Scholar 

  3. Sparrow EM, Haji-Sheikh A, Lundgren TS (1964) The inverse problem in heat conduction. J Appl Mechanics 32(3):369–375

    Google Scholar 

  4. Beck JV (1968) Surface heat flux determination using an integral method. Nucl Eng Des 7:170–178

    Article  Google Scholar 

  5. Beck JV, Blackwell B, St Clair CR Jr (1985) Inverse heat conduction, ill-posed problems. Wiley-Interscience, New York

    Google Scholar 

  6. Alifanov OM, Mikhailov VV (1978) Solution of the nonlinear inverse thermal conductivity problem by the iteration method. Translated Inzhenerno-Fizicheskii Zhurnal 35(6):1123–1129

    Google Scholar 

  7. Hills RG, Hensel EC Jr (1986) One-dimensional nonlinear inverse heat conduction technique. Numer Heat Transfer 10:369–393

    Google Scholar 

  8. Kurpisz K (1991) Numerical solution of one case inverse heat conduction problem. J Heat Transfer 113:280–286

    Google Scholar 

  9. Huang CH, Özisik MN (1992) Inverse problem of determination unknown wall heat flux in laminar flow through a parallel plate duct. Num Heat Transfer A 21:55–70

    Google Scholar 

  10. Park HM, Chung OY (1999) Inverse natural convection problem of estimating wall heat flux using a moving sensor. J Heat Transfer 121:828–836

    Google Scholar 

  11. Alhama F, Zueco J, González-Fernández CF (2003) An inverse determination of unsteady heat fluxes using a network simulation method. J Heat Transfer 125(6):1178–1183

    Article  Google Scholar 

  12. González-Fernández CF, Alhama F, López Sánchez JF, Horno J (1998) Application of the network method to heat conduction processes with polynomial and potential-exponentially varying thermal properties. Numer Heat Transfer A 33:549–559

    Google Scholar 

  13. Alhama F, González-Fernández CF (2002) Transient thermal behaviour of phase-change processes in solid foods with variable thermal properties. J Food Eng 54:331–336

    Article  Google Scholar 

  14. Statistic for Windows 2.0 (1986)

  15. Pspice 6.0. (1994) Microsim Corporation, 20 Fairbanks, Irvine, California 92718

  16. Chantasiriwan S (1999) Comparison of three sequential function specification algorithms for the inverse heat conduction problem. Int Commun Heat Mass Transfer 26(1):115–124

    Article  CAS  Google Scholar 

  17. Zueco J, Alhama F, González-Fernández CF (2003) An inverse problem to estimate temperature dependent heat capacity under convection processes. Heat Mass Transfer 39(7):599–604

    Article  Google Scholar 

  18. Zueco J (2003) Solution of inverse heat conduction problem by means of the network simulation method. PhD Thesis, Technical University of Cartagena (in Spanish)

  19. Huang CH, Yan JY, Chen HT (1995) Function estimation in predicting temperature-dependent thermal conductivity without internal measurements. J Therm Heat Transfer 9(4):667–673

    CAS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Thermal Engineering and Fluids, Technical University of Cartagena, Cartagena, 30202, Spain

    J. Zueco

  2. Department of Applied Physics, Technical University of Cartagena, Cartagena, 30202, Spain

    F. Alhama & C. F. González Fernández

Authors
  1. J. Zueco
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. F. Alhama
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. C. F. González Fernández
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to C. F. González Fernández.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zueco, J., Alhama, F. & González Fernández, C.F. Numerical nonlinear inverse problem of determining wall heat flux. Heat Mass Transfer 41, 411–418 (2005). https://doi.org/10.1007/s00231-004-0553-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00231-004-0553-1

Keywords

Search

Navigation