Analytical solution to transient inverse heat conduction problem using Green’s function

Abstract

A transient inverse heat conduction problem concerning jet impingement heat transfer has been solved analytically in this paper. Experimentally obtained transient temperature history at the non-impinging face, assumed to be the exposed surface in real practice, is the only input data. Aim of this study is to estimate two unknown thermo-physical parameters—overall heat transfer coefficient and adiabatic wall temperature—at the impinging face simultaneously. The approach of Green’s Function to accommodate both the transient convective boundary conditions and transient radiation heat loss is used to derive the forward model, which is purely an analytical method. Levenberg–Marquardt algorithm, a basic approach to optimisation, is used as a solution procedure to the inverse problem. An in-house computer code using MATLAB (version R2014a) is used for analysis. The method is applied for a case of a methane–air flame impinging on one face of a flat 3-mm-thick stainless steel plate, keeping Reynolds number of the  gas mixture 1000 and dimensionless burner tip to impinging plate distance equals to 4, while maintaining the equivalence ratio one. Inclusion of both radiation and convection losses in the Green’s function solution for the forward problem enhances the accuracy in the forward model, thereby increasing the possibility of estimating the parameters with better accuracy. The results are found to be in good agreement with the literature. This methodology is independent of flow and heating conditions, and can be applied even to high-temperature applications.

Appendix: Derivation of Green’s function

In this section, the Green’s function of the problem under consideration is derived using classic separation of variable (SoV) technique. The advantage of Green’s function is that solution of a differential equation (say, a transient heat conduction problem) can be represented in terms of Green’s function. The solution procedure to derive the Green’s function for an one-dimensional transient heat conduction problem with transient convective boundary condition at one face and transient heat flux at another boundary is discussed in following five steps.

Step 1 Consider the homogenous version of the PDEs at Eqs. (3, 3.1, 3.2, 3.3) suitable for SoV technique and redefine as below.

$$\frac{{\partial^{2} \psi }}{{\partial x^{2} }} = \frac{1}{\alpha }\frac{\partial \psi }{\partial t} \text{ in } 0 \, < \, z \, < \, a, \, t \, > \, 0 \,\text{(Not modified and homogenous),}$$

$$- \frac{\partial \psi }{\partial x} + H\psi = 0 {\text{ at }} x \, = \, 0, \, t \, > \, 0\, \text{(Modified and homogenous)}$$

$$\frac{\partial \psi }{\partial x} = 0 {\text{ at }} x \, = \, a, \, t \, > \, 0\, {\text{(Modified and homogenous)}}$$

$$\psi (x,0) = g(x) - T_{\text{amb}} {\text{ in }} 0 \, < \, x \, < \, L$$

Step 2 The solution of the formulations at step 1 can be readily referred in literature [33] (Sect. 2-4, 2-5), and the same is given by

$$\psi (x,t) = \sum\limits_{n = 1}^{\infty } {\frac{{2\left( {\lambda_{\text{n}}^{2} + H^{2} } \right)}}{{a\left( {\lambda_{\text{n}}^{2} + H^{2} } \right) + H}}} *\cos \left( {\lambda_{\text{n}} \left( {a - x} \right)} \right)*e^{{ - \lambda_{\text{n}}^{2} \alpha (t)}} \int\limits_{{x^{/} = 0}}^{L} {\left( {g(x^{/} ) - T_{\text{amb}} } \right)} \cos \left( {\lambda_{\text{n}} \left( {a - x^{/} } \right)} \right){\text{d}}x^{/}$$

where$\lambda_{n} \tan (\lambda_{n} a) = H, \, n \, = \, 1,2,3, \ldots$

The eigenvalues ‘$\lambda_{n}$’ are the roots of the transcendental equation above.

Step 3 The equation at step 2 can be rewritten as,

$$\psi (x,t) = \int\limits_{{x^{/} = 0}}^{a} {\left[ {\sum\limits_{n = 1}^{\infty } {\frac{{2\left( {\lambda_{\text{n}}^{2} + H^{2} } \right)}}{{a\left( {\lambda_{\text{n}}^{2} + H^{2} } \right) + H}}} *\cos \left( {\lambda_{\text{n}} \left( {a - x} \right)} \right)*\cos \left( {\lambda_{\text{n}} \left( {a - x^{/} } \right)} \right)*e^{{ - \lambda_{\text{n}}^{2} \alpha (t)}} } \right]} \left( {g(x^{/} ) - T_{\text{amb}} } \right){\text{d}}x^{/}$$

Step 4 From the fundamentals of Green’s function [33] (Chapter 6), the standard expression for the solution to problem at step 1, having homogenous differential equation and boundary conditions with a non-homogenous initial condition, is generalised as,

$$\psi (x,t) = \int\limits_{{x^{/} = 0}}^{a} {G\left( {x,t\left| {\left. {x^{/} ,\tau } \right)_{\tau = 0} F} \right.} \right.} (x^{/} ){\text{d}}x^{/}$$

where ‘G’ is the Green’s function and ‘F’ is the initial condition.

Step 5 Comparing equation at step 4 and equation at step 3, appropriate Green’s function is given by,

$$G\left( {x,t\left| {\left. {x^{/} ,\tau } \right)} \right.} \right. = \sum\limits_{n = 1}^{\infty } {\frac{{2\left( {\lambda_{\text{n}}^{2} + H^{2} } \right)}}{{a\left( {\lambda_{\text{n}}^{2} + H^{2} } \right) + H}}} *\cos \left( {\lambda_{\text{n}} \left( {a - x} \right)} \right)*\cos \left( {\lambda_{\text{n}} \left( {a - x^{/} } \right)} \right)*e^{{ - \lambda_{\text{n}}^{2} \alpha (\text{t} - \tau )}}$$

In this step, as a standard procedure, $t$ is to be replaced by $t - \tau$.