ARTICLE

J. Cent. South Univ. (2019) 26: 2100-2108

DOI: https://doi.org/10.1007/s11771-019-4157-9

Resolving double-sided inverse heat conduction problem using calibration integral equation method

CHEN Hong-chu(陈鸿初)

Department of Mechanical, Aerospace and Biomedical Engineering, University of Tennessee, Knoxville, TN, 37996-2210, USA

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Abstract:In this paper, a novel calibration integral equation is derived for resolving double-sided, two-probe inverse heat conduction problem of surface heat flux estimation. In contrast to the conventional inverse heat conduction techniques, this calibration approach does not require explicit input of the probe locations, thermophysical properties of the host material and temperature sensor parameters related to thermal contact resistance, sensor capacitance and conductive lead losses. All those parameters and properties are inherently contained in the calibration framework in terms of Volterra integral equation of the first kind. The Laplace transform technique is applied and the frequency domain manipulations of the heat equation are performed for deriving the calibration integral equation. Due to the ill-posed nature, regularization is required for the inverse heat conduction problem, a future-time method or singular value decomposition (SVD) can be used for stabilizing the ill-posed Volterra integral equation of the first kind.

Key words:　inverse　heat　conduction　problem;　surface　heat　flux　estimation;　calibration　integral　equation　method

Cite this article as: CHEN Hong-chu. Resolving double-sided inverse heat conduction problem using calibration integral equation method [J]. Journal of Central South University, 2019, 26(8): 2100-2108. DOI: https://doi.org/ 10.1007/s11771-019-4157-9.

1 Introduction

Direct heat conduction problems with given boundary conditions and initial condition are widely discussed in the heat transfer textbooks [1, 2]. In comparison, inverse heat conduction problems involve estimating boundary conditions (heat flux, temperature or convective heat transfer coefficient) in situations where sensors cannot be directly mounted on the active surface. Instead, in-depth sensors (sensors that are embedded at the interior location of solid body) or backside sensors (sensors that are placed on the opposite side of the surface of interest) are adopted to indirectly estimate the surface thermal conditions on the surface of interest (usually called active or front surface). Inverse heat condition problems have numerous applications in aerospace and engineering world, such as combustion studies [3, 4], nuclear reactors, shock tunnels, hypersonic reentry studies [5], brakes [6, 7], quenching studies [8], metal solidification, phase change process [9, 10] and arc jets. Estimating surface thermal conditions via in-depth or backside sensors, or in other words, projecting the temperature data measured by in-depth or backside sensors to surface is ill-posed problem [11]. This means that a small amount of noise in the in-depth or backside measurements will be magnified significantly when project to the front surface. The error magnification effect can be explained by the physics of heat diffusion. From front surface to interior, heat diffusion is a damping process while high-frequency noises are damped as heat conducts through a solid body. However, in the reverse direction, high-frequency noises are magnified when projecting the in-depth measured data to the surface. As is known to all, any kinds of measurements contain noise, thus regularization is necessary for all inverse heat conduction problems (IHCP’s) [11]. In addition, in most of the studies for inverse heat conduction problems, the temperature measured by the in-depth thermocouple is assumed as the same as the positional temperature at the location where the sensor is being placed. Nonetheless, this assumption can cause significant delay and attenuation in the prediction of surface thermal conditions. Temperature sensor models that account thermal contact resistance between sensor and the host material [12-14], conduction lead losses through the lead wires as well as sensor capacitance need to be proposed in order to achieve accurate surface thermal estimation.

Up till now, numerous projection techniques have been proposed and established for resolving inverse heat conduction problems, including Beck’s function specification [15-17], space marching methods [18-23], exact solution approach [24], global time method [25], iteration approach [26], Laplace transform method [27, 28], conjugate gradient approach [29, 30] and boundary-element approach [31, 32]. Notwithstanding, all the methods mentioned above need given thermophysical properties of the host material, exact sensor positions, sensor related parameters (for accurately determining positional temperature at the sensor site). These methods are called as “parameter required” methods. Considering that accurate determination of the properties and parameters is non-trivial and challenging, significant amount of bias can occur in the surface estimation due to the uncertainties contained in the input parameters and properties. In contrast, calibration approaches for inverse heat conduction problems are “parameter free” methods that do not need input of any properties and parameters mentioned above. Alternatively, all the properties and parameters are implicitly determined in the calibration campaign possessing known surface thermal condition on the active side and known in-depth temperatures measured by embedded thermocouples.

Depending on the backside boundary conditions, the calibration integral equation approach can be derived and formulated as one-probe [13, 33] or two-probe [34, 35] settings. While the one-probe formulation has strict limitation for backside boundary condition. That is, backside boundary condition should be either adiabatic or heat convection with constant convective heat transfer coefficient. The two-probe formulation has more general applications by removing the restriction on the backside boundary condition. The linear one-probe [13] and two-probe [34] calibration integral equation methods have been numerically verified and experimentally validated [36, 37]. For some applications of inverse heat conduction problems, both the front side surface heat flux and the backside surface need to be estimated simultaneously, for example, for disc brake applications [6, 7]. In such engineering scenario, a “parameter free” double-sided, two- probe calibration integral equation can be derived for predicting both the front side and the backside surface heat fluxes. The problem statement for the double-sided, two-probe calibration integral equation is demonstrated in Section 2. The detailed derivation of the calibration integral equation based on frequency domain analysis is shown in Section 3. Section 4 covers some concluding remarks and future works.

2 Problem statement

Figure 1 describes the double-sided, two-probe inverse heat conduction problem under consideration for this paper. Figure 1 shows the front surface heat flux and the back surface heat flux, and two in-depth thermocouples located at x=b and x=w. The thickness of the one-dimensional slab is L. The net heat fluxes at the front surface (x=0) and at the back surface (x=L) are denoted by and respectively. The introduction of the second probe at x=w removes the requirement of imposing specific boundary condition at the back surface. These thermocouples are assumed “ideal” that possess an instantaneous response (time constant=0) and have no conductive lead losses. For the double-sided, two-probe inverse heat conduction problem, the front and back surface heat fluxes and are unknown in the real engineering practice, which need to be estimated by using the in-depth measured temperature at x=b and x=w by the two thermocouples. In this paper, the “parameter free” calibration integral equation method is applied for resolving this double-sided, two-probe inverse heat conduction problem and the derivation of the calibration integral equation is shown in the following section.

Figure 1 Schematic of double-sided inverse heat conduction problem

3 Derivation of double-sided, two-probe calibration integral equation

The derivation of double-sided, two-probe calibration integral equation is based on frequency- domain analysis of the heat equation. Through evaluating frequency-domain temperature at thermocouple sites (x=b and x=w), the calibration integral equation can be derived. The one- dimensional linear heat equation [2] is defined as

         (1a)

subject to the boundary conditions:

            (1b)

           (1c)

and the initial condition:

                (1d)

Take the Laplace transform of Eq. (1a) subject to the initial condition given in Eq. (1d) where (for simplicity) to get

    (2a)

Rearrange yields

       (2b)

whose general solution in hyperbolic functions form is

                         (3)

The corresponding heat flux (Fourier’s law) in the transformed domain is given:

          (4a)

or upon substituting Eq. (3) for transformed temperature into Eq. (4a) to get

                        (4b)

Let evaluate the transformed temperature at the probe locations using Eq. (3) to get

  (5a)

 (5b)

and evaluate the transformed heat flux at x=0 and x=L using Eq. (4b) to get

            (6a)

        (6b)

Combining Eq. (6a) and Eq. (6b) and using Cramer’s rule, we obtain

             (7a)

                           (7b)

Substituting Eqs. (7) into Eqs. (5) and rearranging yield

 (8a)

 (8b)

Let

 (9a)

                 (9b)

 (9c)

                (9d)

Therefore, Eqs. (8a) and (8b) can be written as

          (10a)

         (10b)

It is our goal to remove the kernels in the frequency-domain and reconstruct a calibration integral equation in terms of data only. To eliminate we need two “calibration” tests. Let us denote the first “calibration” test as “c1” (as a subscript) and the second “calibration” test as “c2”. Doing so yields

      (11a)

     (11b)

     (11c)

    (11d)

Combining Eqs. (11a) and (11b) and using Cramer’s rule, we get

    (12a)

    (12b)

Combining Eqs. (11c) and (11d) and using Cramer’s rule, we get

   (12c)

   (12d)

Let’s denote the third test (the “reconstruction” test case withand to be determined) as the subscript “r”. Therefore,Eqs. (10a) and (10b) can be written as

       (13a)

      (13b)

Combining Eqs. (13a) and Eq. (13b) and using Cramer’s rule, we get

      (14a)

      (14b)

Substituting Eqs. (12) into Eqs. (14) and rearranging yield

    (15a)

and

    (15b)

A five-function convolution theorem needs to be developed. We have derived the convolution theorem for the product of five Laplace transformed functions, which is given by (See Appendix for details)

            (16a)

or alternatively written as

   (16b)

Inverting Eqs. (15a)-Eq. (16b) yields the linear calibration equations for double sided inverse heat conduction problems to obtain based on the third “reconstruction” test as

  (17a)

Inverting Eq. (15b)-Eq. (16b) yields the linear calibration equations for double sided inverse heat conduction problems to obtain based on the third “reconstruction” test as

   (17b)

or in compact forms

           (18a)

  (18b)

   (18c)

And

           (19a)

   (19b)

4 Conclusions

In this paper, a calibration integral equation is derived to resolve the double-sided inverse heat conduction problem with two in-depth thermocouples. The derivation is based on mathematical manipulation of the heat equation in the frequency domain. The host material’s thermophysical properties as well as the sensor locations and sensor related parameters are inherently contained in the final mathematical formulation as a Volterra integral equation of the first kind. The final calibration integral equation includes multiple partial integrations involving dummy variables but can be written in terms of compact form. Future works for this calibration approach for double-sided, two-probe inverse heat conduction problem involve numerical verification and experimental validation.

Appendix: Derivation of five-function convolution

The five-function convolution relationship is constructed from the basic definition of Laplace transform starting from

          (A1)

Rearranging the last two terms in the RHS yields

          (A2a)

or

    (A2b)

Let and when integrate with respect to t, we can treat p as a constant, so . Therefore, Eq. (A2b) can be written as

   (A3)

Through carefully interchanging the orders of integration, we get

   (A4)

Moving e^-sz inward, we obtain

          (A5)

Let η=λ+z, and when integrate with respect to λ, we can treat z as a constant, so dη=dλ. Therefore, Eq. (A5) can be written as

       (A6)

Through carefully interchanging the orders of integration, we get

       (A7)

Moving e^-sv inward, we obtain

       (A8)

Let ξ=η+v, and when integrate with respect to η, we can treat v as a constant, so dξ=dη. Therefore, Eq. (A8) can be written as

   (A9)

Through carefully interchanging the orders of integration, we get

  (A10)

Moving e^-su inward, we obtain

  (A11)

Let and when integrate with respect to ξ, we can treat u as a constant, so Therefore, Eq. (A11) can be written as

 (A12)

Through carefully interchanging the orders of integration, we get

 (A13)

which allow us to identify based on the definition of the Laplace transform

,         (A14)

Hence, we arrive at (let for visual convenience)

 (A15a)

or alternatively written as

          (A15b)

Let

             (A16a)

Eq. (A16a) can be written as

 (A16b)

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(Edited by YANG Hua)

中文导读

用基于校准积分方程的方法解决双面导热反问题

摘要：本文推导了用双面双传感器导热反问题估计固体表面热流密度的校准积分方程。跟传统的用于解决导热反问题的方法相比，基于校准的方法不需要给定温度传感器的位置、材料的热物性参数以及温度传感器的接触热阻、比热容和接线的导热损失。所有的这些参数都已包含在最后推导出的第一类Volterra积分方程中。拉普拉斯变换以及频域内的数学处理被用于校准积分方程的推导过程中。由于导热反问题在数学上是病态的，所以所有的导热反问题都需要进行正规则化处理，将来时间方法或者奇异值分解方法可以被用于稳定病态的第一类Volterra积分方程。

关键词：导热反问题；估计表面热流密度；校准积分方程

Received date: 2019-06-20; Accepted date: 2019-07-26

Corresponding author: CHEN Hong-chu, PhD; E-mail: hchen28@utk.edu; ORCID: 0000-0002-5654-642X