Influence of Input Parameters on the Solution of Inverse Heat Conduction Problem

1. Introduction

The basic theory of heat and structure of solid body is associated with the internal energy of matter which in the first law of thermodynamics is referred to as the internal energy concerned with the physical state of the material. The first law of thermodynamics defines that the flowing heat energy is conserved in the absence of heat sources and sinks. It is, therefore, important to study the influence of thermocouple lead wires and distortion due to the thermocouple cavity in solution of the inverse heat conduction problem. According to the second law of thermodynamics, the heat will be transferred from one body to another body only when the bodies are at two different temperatures level and the heat will flow from the point of higher to the point of lower temperature.

A direct solution of transient heat conduction equation with prescribed initial and boundary conditions yields temperature distribution inside a slab of finite thickness. The direct solution is mathematically considered as well-posed because the solution exists, unique and continuously depends on input data. The estimation of unknown parameters from the measured temperature history is called as inverse problem of heat conduction. It is mathematically known as an ill-posed problem since the solution now does not continuously depend on the input data. Measurement data error in temperature, thermal lagging, thermocouple’s cavity, signal noise, etc. makes stability problem in the estimation of unknown parameters.

Numerical inversion of the integral solution [1], exact solution [2], numerical techniques [3], least-squares method [4], transform methods [5], different series approach [6], variable time-step size [7] have been applied to solve inverse heat conduction problems. Solutions of the ill-posed inverse heat conduction problem have been presented in detail by Beck et al. [8] and Özisik et al. [9]. Tikhonov regularization method [10] has been described for cross-validation criterion for selecting the regularization parameter to obtain a stable approximation to the solution. Kurpisz et al. [11] have presented series with derivatives with temperature to solve inverse thermal problem. Hensel [12] has described space marching numerical methods to solve inverse heat transfer problem. Various mathematical methods and numerical algorithms for solving inverse heat conduction problems are described and compared by Alifanov [13]. Taler and Duda [14] have presented solutions of direct and inverse heat conduction problems.

Inverse heat conduction analysis provides an efficient tool for estimating the thermophysical properties of materials, the boundary conditions, or the initial conditions. Estimation of surface heat flux has been carried out without [15] and with [16] heat conduction and comparison between them shows discrepancies as high as about 27% [17]. Moving window optimization method [18] has been applied to predict the aerodynamic heating in a free-flight of sounding rocket by comparing numerically calculated and measured temperature history. Howard [19] developed a numerical procedure for estimating the heat flux with variable thermal properties using a single embedded thermocouple. Simultaneous identification of the temperature-dependent thermal conductivity and the asymmetry parameter of the Henyey-Greenstein scattering phase function have been shown by Zmywaczyk and Koniorczyk [20].

The conjugate gradient method with adjoint problem for function estimation iterative technique is used to solve IHCP to estimate heat flux and internal wall temperature of the throat section of the rocket nozzle [21]. Heisler [22] have reported supplementary “short-time” temperature-time charts for the center, mid-location and surface of large plates, long cylinders and spheres for the dimensionless time sub-domain. Convective heat transfer coefficient and combustion temperature in a rocket nozzle is determined using transient-temperature response chart [23].

The solution of transient IHCP can be obtained using analytical or numerical schemes in conjunction with measured temperature-time history. The estimation of the unknown parameters can be carried out by employing gradient or non-gradient methods to predict the unknown parameters in a prescribed tolerance limit. The focus of the present work is to investigate the influence of various parameters on the solution of inverse heat conduction problem.

2. Measurement errors

Experimental difficulties [24] are noticed in implanting thermocouples at the surface for temperature measurements. Temperature response delays have been studied to solve IHCP applied to cooled rocket thrust chamber [25]. The temperature measured inside the slab may delay and damp depending on xm as illustrated in Figure 1. A thermocouple indicates temperature lag behind the actual temperature. The effect of the thermocouple sensor dynamics on prediction of a triangular heat flux history has been analyzed with simulated data in a one-dimensional domain by Woodbury [26].

Chen and Danh [27] have carried out experimental studies to obtain transient temperature distortion and thermal delay in a slab due to presence of thermocouple cavity. The distortion of temperature profiles inside the slab may be influenced by the dissimilar thermophysical properties of thermocouple and surrounding materials and by the diameter and depth of the cavity. The temperature distortion [28] inside a slab is a function of the thermocouple cavity diameter d and location xm .

Standard statistical analysis consists of error in the measurement as an additive of true plus random, in zero mean, in constant variance, uncorrelated, normal, bell shaped probability density function, constant variance known, errors in the dependent variables and no-prior information about the parameters. The error in measurement can be obtain using exact analytical solution [29] as

where ε and δτ refer to error in measurement of thermocouple location and in time recording, respectively. One of the important points that must be mentioned, here, is the use of a starting solution. In the case of a solid rocket motor where boundary conditions are suddenly imposed by a wall, there will be high intensity of heat flux on cold wall, and the heat flux during the first few steps in time may not be very accurate. The numerical solution is initiated using an exact analytical solution instead of starting from the initial constant condition. Such solutions can be obtained from to exact analytical solution [30] of transient heat conduction equation. Heat transfer rates to the calorimetric probe are estimated from measurements of temperature and rate of temperature change using energy conservation considerations [31].

An optimization method based on a direct and systematic search region reduction optimization method [32] can be employed to estimate the unknown convective heat transfer coefficient in a typical rocket nozzle. The most attractive feature of the direct search scheme is the simplicity of computer programming. The pseudo-random algorithm, an effective tool for optimization, does not require computation of derivatives but depends only on function evaluation. It works even when the differentiability requirements cannot be ensured in the feasible domain. For initiating the search only an estimate of the feasible domain is needed. Therefore, another advantage of the method is that the starting condition is not crucial; any reasonable value will do.

3. Heat conduction equation

4. Inverse problem of heat conduction applied to a rocket nozzle

The influence of constant (average) thermal conductivity, temperature-dependent thermal conductivity, computational grid in numerical solver, nonlinear boundary condition, cylindrical coordinate and the estimation of the wall heat flux and convective heat transfer is carried out by employing measured temperature history of a rocket nozzle of a solid motor. Solution of linear heat conduction equation is used to estimate the convective heat transfer coefficient with the measured temperature data of outer wall of a rocket nozzle. The running time of rocket motor is 16 s. The nozzle wall thickness L = 0.0211 m. The thermo-physical properties of the material are: ρ = 7900 kg m⁻³, Cp  = 545 J kg⁻¹ K⁻¹, K (average) = 35 Wm⁻¹ K⁻¹. Initial temperature Ti  = 300 K and combustion gas temperature Tg  = 2946.2 K are used in the solution of the heat conduction equation.

5. Estimation of heat flux with two-nodes in a sounding rocket

A two-node exact solution is used to calculate the back-wall temperature as described in Section 3.4. The iterative method described above has been used for estimating aerodynamic heating for a sounding rocket in free flight test. Here, the wall heat flux is estimated using the measured temperature history in conjunction with the iterative technique [30]. The aerodynamic heating rate is estimated for a typical sounding rocket as depicted in Figure 2. The location of thermocouple is marked in the diagram. The thermophysical properties of Inconel and wall thickness are k = 18 Wm⁻¹ K⁻¹, α = 4.47 × 10⁻⁶ m²/s, L = 0.7874 × 10⁻³ m. Figure 3 depicts the measured temperature time history at different locations measured from the tip of the cone in the free flight of a sounding rocket as delineated in Figure 2. It can be observed from temperature history that the initial time delay in thermal response is 6 s. The unknown qw are estimated using an iterative technique which starts with an initial value of wall heat flux and is repeated until |F(qw )| ≤ 10⁻⁴.

A two-node exact solution is used to calculate the wall temperature distribution. The unknown qw are estimated using an iterative technique which starts with an initial value of wall heat flux and is repeated until |F(qw )| ≤ 10⁻⁴. Figure 4 displays the estimated variation of the wall heat flux as a function of flight time of the sounding rocket.

The wall heat flux variation depends on the sounding rocket speed. The increase and decrease of the aerodynamic heating are a function of flight Mach number.

The estimated wall heat flux is compared with Van Driest’s results [44]. Table 7 depicts the estimated values of wall heat flux as a function of flight time at thermocouple location 29 as shown in Figure 2. It can be observed from the table that highest aerodynamic heating occurs during 7–8 s, another significant peak wall heat flux was found at 22 s.

6. Conclusions

Analytical, transient numerical and two-node methods are used to compute temperature distribution in a finite slab. Numerical solution is carried out with temperature-dependent thermal conductivity. Implicit finite difference scheme with two-time level technique is implemented to solve nonlinear problem of heat conduction. Time delay is studied using finite element method with deforming grid strategy. A boundary shifting numerical scheme is used to solve transient heat conduction in radial coordinate. Evidence of temporal accuracy and dependence on time-step is demonstrated in the numerical solving of IHCP. Influence of thermocouple cavity and measurement errors in location and time are discussed. The IHCP is applied to predict the wall heat flux in a rocket nozzle of a solid motor. Wall heat flux is estimated in a free flight of a sounding rocket using the two-node method.

Nomenclature