Solution of inverse heat conduction problem.
Abstract
A onedimensional transient heat conduction equation is solved using analytical and numerical methods. An iterative technique is employed which estimates unknown boundary conditions from the measured temperature time history. The focus of the present chapter is to investigate effects of input parameters such as time delay, thermocouple cavity, error in the location of thermocouple position and time and temperaturedependent thermophysical properties. Inverse heat conduction problem IHCP is solved with and without material conduction. A twotime level implicit finite difference numerical method is used to solve nonlinear heat conduction problem. Effects of uniform, nonuniform and deforming computational grids on the estimated convective heat transfer are investigated in a nozzle of solid rocket motor. A unified heat transfer analysis is presented to obtain wall heat flux and convective heat transfer coefficient in a rocket nozzle. A twonode exact solution technique is applied to estimate aerodynamic heating in a free flight of a sounding rocket. The stability of the solution of the inverse heat conduction problem is sensitive to the spatial and temporal discretization.
Keywords
 analytical solution
 inverse heat conduction problem
 numerical analysis
 deforming grid
 heat transfer coefficient
 heat flux
 random search method
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 wellposed 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 illposed 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], leastsquares method [4], transform methods [5], different series approach [6], variable timestep size [7] have been applied to solve inverse heat conduction problems. Solutions of the illposed 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 crossvalidation 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 freeflight 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 temperaturedependent thermal conductivity and the asymmetry parameter of the HenyeyGreenstein 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 “shorttime” temperaturetime charts for the center, midlocation and surface of large plates, long cylinders and spheres for the dimensionless time subdomain. Convective heat transfer coefficient and combustion temperature in a rocket nozzle is determined using transienttemperature response chart [23].
The solution of transient IHCP can be obtained using analytical or numerical schemes in conjunction with measured temperaturetime history. The estimation of the unknown parameters can be carried out by employing gradient or nongradient 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
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
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 noprior information about the parameters. The error in measurement can be obtain using exact analytical solution [29] as
where
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 pseudorandom 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
3.1 Analytical solution
The computation of the turbulent convective heat transfer coefficient from combustion gases to the rocket nozzle wall is based on the Bartz’s equation [33] incorporating the effects of compressibility, throat curvature and variation of transport properties in the boundary layer. The transient heat conduction in a onedimensional Cartesian coordinate system having two parallel plane surfaces
with following initial and boundary conditions:
where
We now consider the constant thermal property solution and can be written in terms of eigen function
In the above Eq. (5),
3.2 Inverse algorithm
The IHCP is solved by comparing calculated and measured temperature using an iterative technique [30]. In estimating
where
The inverse method for solving a value of
Now, it is possible to estimate convective heat transfer coefficient and combustion gas temperature in conjunction with measured temperature history [35]. The equation for converting the calculated heat flux to the heat transfer coefficient is
In the foregoing equation,
3.3 Numerical methods
It is not always feasible to obtain analytical solution of temperaturedependent thermal conductivity and radiation boundary condition. The CrankNicolson finite difference method with twotime level implicit numerical scheme [36] has been employed to solve the nonlinear conduction problem with the NewtonRaphson method to consider the radiation boundary condition.
Deforming or moving finite elements method [37] is used to solve linear heat conduction equation. The moving finite element [38] is used to consider the time delay in the measurement of back wall temperature.
3.4 Twonodes system of transient heat conduction equation
For only two nodes the system of [39] equations reduce to the following pair of equations:
where 0 and 1 represent node in a slab of finite thickness. These are the exact solutions to the system of two ordinary differential equations which resulted from a twonode finitedifference approximation to the original problem.
where
Solution of the above simultaneous equation calculates the temperature with a given value of
4. Inverse problem of heat conduction applied to a rocket nozzle
The influence of constant (average) thermal conductivity, temperaturedependent 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
4.1 Average thermal conductivity
Prediction of convective heat transfer coefficient is carried out in conjunction with the calculated and measured temperature history at outer surface of nozzle divergent in a solid rocket motor static test. The constant thermal conductivity solution of the linear transient heat conduction problem [30] is
For estimating unknown boundary condition, the heat conduction equation is and solved with the following boundary and initial conditions.
and
Exact analytical solution of transient heat conduction as written in Eq. (13) is used to estimate convective heat transfer on the inner surface of the rocket nozzle. An iterative scheme is used to solve inverse problem [30]. The iteration is carried out till the absolute difference between calculated and measured temperature is less than or equal to 10^{−4}. Table 1 exhibits the comparison between the estimated values of the convective heat transfer coefficient based on the exact solution of heat conduction equation with the calculated values of Bartz [33]. Bartz’s equation calculates conservative estimates for the convective heat transfer to the wall [40].







6  0.2950  0.0098  0.0096  1821.9  2254.2 
7  0.3109  0.0159  0.0158  1810.0  2254.2 
8  0.2996  0.0212  0.0211  1610.3  2254.2 
9  0.3244  0.0301  0.0302  1690.9  2254.2 
10  0.3340  0.0386  0.0385  1669.7  2254.2 
11  0.3416  0.0473  0.0472  1641.9  2254.2 
12  0.3302  0.0529  0.0529  1497.6  2254.2 
13  0.3312  0.0602  0.0604  1443.1  2254.2 
14  0.3409  0.0677  0.0676  1387.0  2254.2 
15  0.3442  0.0781  0.0782  1413.0  2254.2 
16  0.3475  0.0862  0.0861  1383.7  2254.2 
4.2 Temperaturedependent thermal conductivity
An iteration procedure [41] is employed in conjunction with exact solution to predict convective heat transfer coefficient from the measured temperaturetime data at the outer wall of the nozzle as shown in Table 2. The expression for temperaturedependent conductivity is





Iterative method  Beck method 

θ 
Iterative method  Beck method  
6  0.0883  0.0838  0.0099  0.0098  536.6  581.7 
7  0.1067  0.1075  0.0158  0.0159  600.6  587.0 
8  0.1144  0.1116  0.0220  0.0212  592.6  598.4 
9  0.1367  0.1367  0.0302  0.0302  674.2  685.3 
10  0.1522  0.1545  0.0386  0.0385  712.9  693.2 
11  0.1654  0.1690  0.0472  0.0472  737.4  730.0 
12  0.1686  0.1639  0.0529  0.0529  718.2  721.9 
13  0.1773  0.1777  0.0605  0.0605  723.6  725.8 
14  0.1844  0.1813  0.0677  0.0676  723.0  725.1 
15  0.1944  0.2040  0.0781  0.0782  753.6  765.0 
16  0.2083  0.2174  0.0862  0.0862  758.3  770.0 
4.3 Numerical solution with various computational grids
Deforming or moving finite element is used to consider the time delay in temperature at the outer wall of the slab [37]. Estimated values of wall heat flux and heat transfer coefficient are tabulated in Table 3. It can be observed from the table that the estimated wall quantities are having significant influence on the predicted unknown boundary conditions. This example is extended to consider spatial grid changed and temporal dependence on the numerical solution using moving finite element method [38].


Uniform grid  Nonuniform grid  Moving grid  








6  326  3.715  1964.5  3.846  2044.9  4.517  2412.1 
7  342  2.700  1408.8  2.848  1449.6  2.818  1485.9 
8  356  2.698  1436.9  2.840  1531.6  2.820  1512.8 
9  380  2.704  1463.0  2.589  1569.8  2.842  1552.8 
10  402  2.705  1491.4  2.858  1603.3  2.846  1586.7 
11  425  2.704  1518.9  2.852  1632.6  2.845  1618.5 
12  440  2.691  1539.7  2.805  1636.2  2.812  1630.8 
13  460  2.683  1564.6  2.776  1649.7  2.791  1650.6 
14  479  2.673  1588.1  2.738  1657.1  2.764  1665.9 
15  507  2.094  1226.4  2.015  1190.6  2.091  1235.4 
16  528  2.086  1231.8  1.981  1178.5  2.067  1231.6 
4.4 Nonlinear boundary condition
Numerical analysis of nonlinear heat conduction with a radiation boundary condition [36] is carried out to estimate wall heat flux using temperature history on the back wall of the rocket nozzle. The high temperature variation alters thermophysical properties of the material of mild steel. Table 4 shows comparison between the estimated convective heat transfer coefficients with the Bartz solution [33]. Effects of nonlinear IHCP with radiation boundary condition are investigated and results are presented in Table 4.









6  659.8  326  2.3547  950.0  2254.2  3137  2946.2 
7  801.0  342  2.3899  1019.6  2254.2  3122  2946.2 
8  900.7  356  2.2211  992.4  2254.2  3115  2946.2 
9  996.3  380  2.6489  1237.1  2254.2  3113  2946.2 
10  1050.5  402  2.3670  1135.5  2254.2  3108  2946.2 
11  1066.4  425  1.7100  827.3  2254.2  3104  2946.2 
12  1201.8  440  2.8144  1459.2  2254.2  3099  2946.2 
13  1320.0  460  2.6559  1467.0  2254.2  3098  2946.2 
14  1354.8  479  1.7595  991.7  2254.2  3095  2946.2 
15  1383.4  507  1.3810  791.4  2254.2  3094  2946.2 
16  1414.9  528  1.1684  681.8  2254.2  3094  2946.2 
4.5 Heat conduction in a hollow cylinder
A grid point shift strategy [42] is adapted to solve inverse conduction problem in a radial coordinate of rocket nozzle with inner and outer radius of rocket nozzle. The inner and outer radius of the nozzle is 0.0839 m and 0.0105 m, respectively. The purpose of the present example to investigate the influence of radial coordinate on the estimated values of heat transfer coefficient. Table 5 shows the effect of geometrical parameters on the predicted heat transfer coefficient.






θ_{g}, K  θ_{gc}, K 

6  1260.2  326  3.6805  1789.6  2254.2  3316  2946 
7  1175.9  342  3.3995  1628.0  2254.2  3264  2946 
8  1160.7  356  2.4745  1181.4  2254.2  3255  2946 
9  1165.8  380  2.5385  1194.7  2254.2  3290  2946 
10  1196.0  402  2.5348  1261.1  2254.2  3206  2946 
11  1192.3  425  2.3385  1166.4  2254.2  3197  2946 
12  1205.8  440  2.2094  1114.8  2254.2  3187  2946 
13  1211.0  460  2.1333  1229.5  2254.2  2946  2946 
14  1222.1  479  2.0441  1187.5  2254.2  3943  2946 
15  1237.1  507  2.0626  1206.7  2254.2  2946  2946 
16  1249.1  528  2.0027  1180.9  2254.2  2945  2946 
4.6 Estimation of heat flux and heat transfer coefficient
The calculated convective heat transfer coefficients and inner wall temperature are used to determine the wall heat flux and the combustion temperature using Eq. (8). The iterative scheme is based on relation between wall heat flux and convective heat transfer coefficient [35]. Table 6 shows the predicted values of wall heat flux and convective heat transfer coefficient. The IHCP is extended to determine wall heat flux in conjunction with convective heat transfer coefficient. A similar IHCP but referring to the 122 mm mediumrange missile during correction engine operation has been considered by Zmywaczyk et al. [43].









6  1355.6  326  3.2502  2631.2  2254.2  3351  2946 
7  1287.8  342  3.2950  1805.3  2254.2  3113  2946 
8  1315.6  356  3.2974  1861.5  2254.2  3087  2946 
9  1368.9  380  3.2967  1885.9  2254.2  3117  2946 
10  1414.4  402  3.2837  1962.1  2254.2  3088  2946 
11  1463.6  425  3.2718  2049.5  2254.2  3060  2946 
12  1370.8  440  2.3825  1476.0  2254.2  2985  2946 
13  1360.9  460  2.4140  1502.1  2254.2  2968  2946 
14  1370.3  479  2.3625  1520.6  2254.2  2924  2946 
15  1382.5  507  2.3675  1517.2  2254.2  2943  2946 
16  1399.3  528  2.3645  1540.7  2254.2  2934  2946 
5. Estimation of heat flux with twonodes in a sounding rocket
A twonode exact solution is used to calculate the backwall 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
A twonode exact solution is used to calculate the wall temperature distribution. The unknown
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  313.0  320.0  2.756  8.246 
7  341.3  349.4  16.695  14.82 
8  408.0  412.3  19.387  22.647 
9  469.7  469.3  8.657  18.876 
10  495.2  495.8  8.145  10.117 
11  504.1  513.3  9.302  4.569 
12  502.4  521.2  2.757  1.176 
13  495.2  506.8  −7.695  −0.977 
14  487.4  487.6  −0.040  −2.320 
15  477.4  477.0  −0.037  −3.134 
16  467.4  467.0  −0.028  −3.578 
17  457.4  457.5  −0,034  −3.769 
18  445.8  447.2  −0.905  −3.789 
19  438.6  437.5  −0,005  −1.628 
20  444.1  445.9  5.173  4.135 
21  469.1  467.5  13.518  11.524 
22  533.0  521.2  20.618  19.732 
23  556.3  558.4  9.180  13.122 
24  567.1  573.5  6.514  8.581 
25  584.1  587.2  8.592  5.467 
26  596.9  602.2  7.018  3.314 
6. Conclusions
Analytical, transient numerical and twonode methods are used to compute temperature distribution in a finite slab. Numerical solution is carried out with temperaturedependent thermal conductivity. Implicit finite difference scheme with twotime 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 timestep 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 twonode method.
Nomenclature
Biot number,
specific heat
heat transfer coefficient
thermal conductivity,
reference thermal conductivity at
slab thickness
nondimensional parameter,
wall heat flux
temperature
time
distance from the inner surface
dimensionless coordinate
thermal diffusivity
density
nondimensional temperature, = (
nondimensional time,
Bartz
computed
combustion gas temperature
initial value
measured
outer wall
wall
constant thermal conductivity coefficient
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