An Inverse Heat Conduction Problem and Improving Shielded Thermocouple Accuracy

Abstract

A shielded thermocouple is a measurement device used for monitoring the temperature in chemically, or mechanically, hostile environments. The sensitive parts of the thermocouple are protected by a shielding layer. In order to improve the accuracy of the measurement device, we study an inverse heat conduction problem where the temperature on the surface of the shielding layer is sought, given measured temperatures in the interior of the thermocouple. The procedure is well suited for real-time applications where newly collected data is continuously used to compute current estimates of the surface temperature. Mathematically we can formulate the problem as a Cauchy problem for the heat equation, in cylindrical coordinates, where data is given along the line r = r ₁ and the solution is sought at r ₁ < r ≤ r ₂. The problem is ill-posed, in the sense that the solution (if it exists) does not depend continuously on the data. Thus, regularization techniques are needed. The ill–posedness of the problem is analyzed and a numerical method is proposed. Numerical experiments demonstrate that the proposed method works well.

Notes

The region r ₀ ≤ r < r ₁ is the metal cylinder and the region r ₁ ≤ r < r ₂ is the shielding material.