A new three-dimensional heat flux-temperature integral relationship for half-space transient diffusion. (English) Zbl 1207.80008
The authors start from the non steady heat equation \(\frac{1}{\alpha }\frac{ \partial T}{\partial t}(x,y,z,t)=(\frac{\partial ^{2}T}{\partial x^{2}}+ \frac{\partial ^{2}T}{\partial y^{2}}+\frac{\partial ^{2}T}{\partial z^{2}} )(x,y,z,t)\) posed in the domain \((0,\infty )\times \{x>0\), \(y,z\in (-\infty ,\infty )\}\). The initial condition \(T(x,y,z,0)=0\) is imposed. Fourier’s law gives the expressions of the components of the heat flux in terms of the gradient of the temperature. The main purpose of the paper is to give new integral relationships between the component of the heat flux in the \(x\) -direction and the temperature. The temperature is first expressed in terms of the values of the heat flux in the \(x\)-direction on the surface \(x=0\). This leads to some singular integral equation. Then the heat flux in the \(x\) -direction can be computed in terms of the temperature and of its time derivative. The computations leading to these relationships are drawn using the properties of the operators which are involved and some Abel-like regularization process.
Reviewer: Alain Brillard (Riedisheim)
MSC:
| 80A20 | Heat and mass transfer, heat flow (MSC2010) |
| 45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |
References:
| [1] | Frankel, J. I., Regularization of inverse heat conduction by combination of rate sensor analysis and analytic continuation, Journal of Engineering Mathematics, 57, 181-198 (2007) · Zbl 1122.80010 |
| [2] | Frankel, J. I.; Keyhani, M.; Arimilli, R. V.; Wu, J., A new multidimensional integral relationship between heat flux and temperature for direct internal assessment of heat flux, Zeitschrift für Angewandte Mathematik und Physik, 59, 5, 869-888 (2008) · Zbl 1162.80001 |
| [3] | Frankel, J. I.; Osborne, G. E.; Taira, K., Stabilization of ill-posed problems through thermal rate sensors, AIAA Journal of Thermophysics and Heat Transfer, 20, 238-246 (2006) |
| [4] | Frankel, J. I.; Arimilli, R. V., Inferring convective and radiative heating loads from transient surface temperature measurements in the half space, Inverse Problems in Sciences and Engineering, 15, 463-488 (2007) · Zbl 1202.80016 |
| [5] | Frankel, J. I., Generalizing the method of Kulish to one-dimensional unsteady heat conducting slabs, Journal of Thermophysics and Heat Transfer, 20, 945-950 (2006) |
| [6] | Frankel, J. I.; Kulish, V. V., A new flux integral relationship for half-space transient heat conduction with arbitrary initial condition, Engineering Analysis with Boundary Elements, 32, 162-167 (2008) · Zbl 1244.80003 |
| [7] | Frankel, J. I.; Keyhani, M.; Arimilli, R. V.; Elkins, B., A new, anisotropic, two-dimensional, transient heat flux-temperature integral relationship for half-space diffusion, Zeitschrift für Angewandte Mathematik und Mechanik, 90, 161-170 (2010) · Zbl 1192.80006 |
| [8] | T.E. Diller, Advances in Heat Flux Measurements, Advances in Heat Transfer, vol. 23, Academic Press, 1993, pp. 279-368.; T.E. Diller, Advances in Heat Flux Measurements, Advances in Heat Transfer, vol. 23, Academic Press, 1993, pp. 279-368. |
| [9] | Cook, W. J.; Felderman, E. J., Reduction of data from thin-film heat transfer gages: a concise numerical technique, AIAA Journal, 4, 561-562 (1966) |
| [10] | Piccini, E.; Guo, S. M.; Jones, T. V., The development of a new direct heat flux gauge for heat transfer facilities, Measurement Sciences and Technology, 11, 342-349 (2000) |
| [11] | Loehle, S.; Battaglia, J. L.; Jullien, P.; Ootegem, B. V.; Couzi, J.; Lasserre, J. P., Improvement of high heat flux measurement using a null point calorimeter, Journal of Spacecraft and Rocket, 45, 76-81 (2008) |
| [12] | Frankel, J. I.; Arimilli, R. V.; Keyhani, M.; Wu, J., Heating rate dT/dt measurements developed from in-situ thermocouples using a voltage-rate interface, International Communications of Heat and Mass Transfer, 35, 885-891 (2008) |
| [13] | Beck, J. V.; Blackwell, B.; Clair, C. R.St., Inverse Heat Conduction (1985), Wiley: Wiley NY · Zbl 0633.73120 |
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