KSME International Journal

, Volume 16, Issue 10, pp 1320–1326 | Cite as

A direct integration approach for the estimation of time-dependent boundary heat flux



In a one-dimensional heat conduction domain with heated and insulated walls, an integral approach is proposed to estimate time-dependent boundary heat flux without internal measurements. It is assumed that the expression of the heat flux is not known a priori. Hence, the present inverse heat conduction problem is classified as a function estimation problem. The spatial temperature distribution is approximated as a third-order polynomial of position, whose four coefficients are determined from the heat fluxes and the temperatures at both ends at each measurement. After integrating the heat conduction equation over spatial and time domain, respectively, a simple and non-iterative recursive equation to estimate the time-dependent boundary heat flux is derived. Several examples are introduced to show the effectiveness of the present approach.

Key Words

Inverse Heat Conduction Problem Integration Approach Boundary Heat Flux 



Heat capacity per unit volume [j/m3-K.]


Thermal conductivity [W/m-K]


Length [m]


Non-dimensionalized heat flux, q/qo[-]


Boundary heat flux [W/m2]


Reference boundary heat flux [W/m2]


Random number ranging-0.5 ≤ R ≥ 0.5 [-]


Non-dimensionalized temperature [-]


Temperature [K]


Initial temperature [K]


Non-dimensionalized temperature at the left end [-]


Non-dimensionalized temperature at the right end [-]


Non-dimensionalized time,at*/1 2 [-]


Time [s]


Unit step function,u(t) =0 fort>0,u(t)=l fort≥0 [-]


Non-dimensionalized coordinate,

x*l [-]x*

Coordinate [m]



Thermal difTusivity [m2/s]


Error level [-]


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Copyright information

© The Korean Society of Mechanical Engineers (KSME) 2002

Authors and Affiliations

  1. 1.Department of Nuclear and Energy EngineeringCheju National UniversityCheju-doKorea

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