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# Restitution of the Temperature Field Inside a Cylinder of Semitransparent Dense Medium From Directional Intensity Data

[+] Author and Article Information
V. Le Dez, D. Lemonnier, H. Sadat

Laboratoire d’Etudes Thermiques, UMR 6608, CNRS-ENSMA, 86960 Futuroscope Cedex, France

J. Heat Transfer 131(11), 112701 (Aug 25, 2009) (14 pages) doi:10.1115/1.3154622 History: Received June 19, 2008; Revised May 14, 2009; Published August 25, 2009

## Abstract

The purpose of this paper is to obtain the temperature field inside a cylinder filled in with a dense nonscattering semitransparent medium from directional intensity data by solving the inverse radiative transfer equation. This equation is solved in a first approach with the help of a discrete scheme, and the solution is then exactly obtained by separating the physical set on two disjoint domains on which a Laplace transform is applied, followed by the resolution of a first kind Fredholm equation.

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## Figures

Figure 3

(a) Retrieved Planck function for N=50 and (b) retrieved Planck function for N=500

Figure 4

(a) Retrieved Planck function for N=27 and (b) retrieved Planck function for N=27

Figure 2

(a) Intensity for a linear temperature field, nλ=1.5, (b) intensity for a linear temperature field, nλ=4.5, (c) intensity for a sinusoidal temperature field, nλ=1.5, and (d) intensity for a sinusoidal temperature field, nλ=4.5

Figure 1

(a) Internal trajectory of a ray emerging in the tomographic plane and (b) schematic description of the emerging intensities at local position 0≤x≤R

Figure 5

(a) Retrieved Planck function with noisy data for N=100 and (b) retrieved Planck function with noisy data for N=27

Figure 6

(a) Retrieved Planck function when using a regularization: α=10−12, nλ=1.5, and N=500 and (b) retrieved Planck function when using a regularization parameter, nλ=4.5, N=100

Figure 7

(a) Retrieved Planck function when using a regularization parameter, N=100, and (b) retrieved Planck function when using a regularization parameter, N=27

Figure 8

(a) Evolution of h for x≤R, nλ=1.5, (b) retrieved Planck function with Eq. 14 on the range [0,R/nλ] for nλ=1.5, (c) evolution of h for x≤R, nλ=4.5, and (d) retrieved Planck function with Eq. 14 on the range [0,R/nλ] for nλ=4.5

Figure 9

(a) Evolution of g, h, and Ψ on [0,R], (b) evolution of g, h, and Ψ on [0,R], (c) evolution of function Ψ on [R/nλ,R] for various absorption coefficients, nλ=1.5, and (d) evolution of function Ψ on [R/nλ,R] for various absorption coefficients, nλ=4.5

Figure 10

(a) retrieved Planck function when using a regularization parameter α=10−9 and nλ=1.5, and (b) retrieved Planck function when using a regularization parameter, α=10−9 and nλ=4.5

Figure 11

(a) evolution of KD for τ0≤nλΩ/nλ2−1, (b) evolution of KD for τ0≤nλΩ/nλ2−1, (c) evolution of KD for τ0>nλΩ/nλ2−1, and (d) evolution of KD for τ0>nλΩ/nλ2−1

## Errata

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