An Inverse Technique to Estimate the Heat Flux of a Slab with Transient Heat Conduction

A generalized solution for a two-dimensional (2D) transient heat conduction problem with a partial-heating boundary condition in rectangular coordinates is developed. The solution accommodates three kinds of boundary conditions: prescribed temperature, prescribed heat flux and convective. Also, the possibility of combining prescribed heat flux and convective heating/cooling on the same boundary is addressed. The means of dealing with these conditions involves adjusting the convection coefficient. Large convective coefficients such as 1010 effectively produce a prescribed-temperature boundary condition and small ones such as 10−10 produce an insulated boundary condition. This paper also presents three different methods to develop the computationally difficult steady-state component of the solution, as separation of variables (SOV) can be less efficient at the heated surface and another method (non-SOV) is more efficient there. Then, the use of the complementary transient part of the solution at early times is presented as a unique way to compute the steady-state solution. The solution method builds upon previous work done in generating analytical solutions in 2D problems with partial heating. But the generalized solution proposed here contains the possibility of hundreds or even thousands of individual solutions. An indexed numbering system is used in order to highlight these individual solutions. Heating along a variable length on the nonhomogeneous boundary is featured as part of the geometry and examples of the solution output are included in the results.

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The Inverse Problem in Transient Heat Conduction

Journal of Applied Mechanics ◽

10.1115/1.3629649 ◽

1964 ◽

Vol 31 (3) ◽

pp. 369-375 ◽

Cited By ~ 109

Author(s):

E. M. Sparrow ◽

A. Haji-Sheikh ◽

T. S. Lundgren

Keyword(s):

Heat Flux ◽

Inverse Problem ◽

Heat Conduction ◽

Numerical Calculation ◽

Initial Temperature ◽

Transient Heat Conduction ◽

Analytical Treatment ◽

Initial Temperature Distribution ◽

Plane Slab ◽

Long Cylinder

A general theory is devised for determining the temperature and heat flux at the surface of a solid when the temperature at an interior location is a prescribed function of time. The theory is able to accommodate an initial temperature distribution which varies arbitrarily with position throughout the solid. Detailed analytical treatment is extended to the sphere, the plane slab, and the long cylinder; and it is additionally shown that the semi-infinite solid is a particular case of the general formulation. The accuracy of the method is demonstrated by a numerical example. In addition, a numerical calculation procedure is devised which appears to provide smooth, nonoscillatory results.

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Novel Two-Dimensional Transient Heat Conduction Calculation in a Cooled Rotor: Ventilation Preheating—Blow-Down Flux

Journal of Heat Transfer ◽

10.1115/1.3122777 ◽

2009 ◽

Vol 131 (8) ◽

Cited By ~ 5

Author(s):

J. P. Solano ◽

G. Paniagua

Keyword(s):

Heat Flux ◽

Heat Conduction ◽

Data Reduction ◽

Transient Heat Conduction ◽

Element Model ◽

Rotor Blades ◽

Wall Heat Flux ◽

Two Dimensional ◽

Blow Down ◽

Cross Sectional

This contribution presents an alternative to classical data reduction techniques to measure the heat transfer using thin-film gauges. A finite-element model of the two-dimensional unsteady heat conduction equation is solved in the cross-sectional area of a metallic airfoil bounded with a polyamide sheet on which thermal sensors are deposited. This novel methodology allows capturing all 2D heat conduction effects that are irremediably neglected with the 1D data reduction technique. The application of this technique in a compression tube facility allows an exact evaluation of the initial wall heat flux into cooled rotor blades. During the spinning-up period, the rotor is spun up to nearly its nominal speed (from 0 rpm to 6200 rpm) resulting in preheating due to drag losses. The long duration of this experiment (∼450 s) and the magnitude of the wall temperature increase result in significant 2D conduction effects that are not accounted for using the 1D approach. In addition, short-duration experiments confirm the existence of 2D effects at smaller time scales (∼0.5 s), as well as the influence of the initial nonuniform temperature distribution in the rotor blade. The resulting flux with such an initial condition appears to be the superposition of the wall heat flux at the end of the spinning up before the test and the flux due to the blow-down itself.

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Semi-derivative integral method to transient heat conduction time-dependent heat flux boundary conditions

Thermal Science ◽

10.2298/tsci21s2303h ◽

2021 ◽

Vol 25 (Spec. issue 2) ◽

pp. 303-308

Author(s):

Jordan Hristov

Keyword(s):

Boundary Condition ◽

Heat Flux ◽

Heat Conduction ◽

Power Law ◽

Integral Method ◽

Transient Heat Conduction ◽

Infinite Medium ◽

Time Dependent ◽

Conduction Time ◽

Flux Boundary Conditions

Transient heat conduction in semi-infinite medium with a time dependent heat flux as boundary condition has been solved by a semi-derivative integral-balance method. Two versions boundary fluxes have been considered: power-law and exponential.

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Unsteady Surface Element Method

Journal of Heat Transfer ◽

10.1115/1.3244538 ◽

1981 ◽

Vol 103 (4) ◽

pp. 759-764 ◽

Cited By ~ 21

Author(s):

N. R. Keltner ◽

J. V. Beck

Keyword(s):

Heat Flux ◽

Heat Conduction ◽

Finite Difference ◽

Transient Heat Conduction ◽

The Other ◽

Surface Element ◽

Contact Conductance ◽

The Given ◽

Duhamel’S Integral ◽

Element Method

A method for the solution of transient heat conduction problems, called the unsteady surface element (USE) method, is developed and applied to several problems. The method is intended for thermally contacting bodies of similar or dissimilar geometries such as occur in contact conductance and intrinsic thermocouple problems. The method utilizes Duhamel’s integral in several ways. Two different procedures are presented, one utilizing temperature-based kernels and the other uses heat flux-based kernels. One of the given applications is to the intrinsic thermocouple problem. Several solutions are given and the results agree very well with two finite difference solutions.

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Transient Heat Conduction in Semi-infinite Solid with Specified Surface Heat Flux

Encyclopedia of Thermal Stresses ◽

10.1007/978-94-007-2739-7_413 ◽

2014 ◽

pp. 6164-6171 ◽

Cited By ~ 1

Author(s):

Tomasz S. Wiśniewski

Keyword(s):

Heat Flux ◽

Heat Conduction ◽

Surface Heat Flux ◽

Transient Heat Conduction ◽

Surface Heat

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Novel 2D Transient Heat Conduction Calculation in a Cooled Rotor: Ventilation Preheating — Blowdown Flux

Volume 4: Heat Transfer, Parts A and B ◽

10.1115/gt2008-51308 ◽

2008 ◽

Cited By ~ 1

Author(s):

J. P. Solano ◽

G. Paniagua ◽

A. de la Loma

Keyword(s):

Heat Flux ◽

Heat Conduction ◽

Short Duration ◽

Data Reduction ◽

Transient Heat Conduction ◽

Element Model ◽

Wall Heat Flux ◽

Transient Temperature ◽

Blow Down ◽

Cross Sectional

An alternative to classical data reduction techniques for thin film gauges in short duration facilities is presented. A finite element model of the two-dimensional unsteady heat conduction equation is solved in the cross-sectional area of a metallic airfoil bounded with a polyamide sheet, on which thermal sensors are deposited. As a result, the transient temperature field in the multilayered substrate and the experimental wall heat flux distribution are derived. The methodology allows for capuring all 2D heat conduction effects that are irremediably neglected with the 1D data reduction technique. The application of this technique in a compression tube facility allows an exact evaluation of the initial wall heat flux into cooled rotor blades. During the spinning up period, the rotor of this type of fully rotating transient facilities is spun up to nearly its nominal speed (from 0 RPM to 6200 RPM) resulting in preheating due to drag losses. The long duration of this experiment (∼450 s) and the magnitude of the wall temperature increase result in significant 2D conduction effects that are not accounted for using the 1D approach. In addition, short duration experiments confirm the existence of 2D effects at smaller time scales (∼0.5 s), as well as the influence of the initial non-uniform temperature distribution in the rotor blade. The resulting flux with such an initial condition appears to be the superposition of the wall heat flux at the end of the spinning up before the test and the flux due to the blow-down itself.

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