Inverse Problem of a Piezothermoelastic Cylinder Subject to Transient Heating

An inverse problem of a radially polarized piezoelectric hollow circular cylinder of crystal class 6 mm is investigated. It is assumed that a voltage induced by the action of a time-varying temperature applied to the inner surface of the cylinder is measured on the outer surface. The inverse problem entails a determination of the heating temperature from knowledge of the measured voltage. First, an exact solution to the problem is found by solving the equations of equilibrium and electrostatics for the cylinder subject to the prescribed boundary conditions. Then a least-squares residual method that incorporates Lagrange multipliers for satisfaction of the boundary equations is employed to derive an approximate analytic solution. Both formulations are utilized in order to calculate the unknown heating temperature, and the corresponding temperature, displacement and stress fields in a cylinder of cadmium selenide. Numerical results based upon the least-squares residual formulation are found to compare favorably with those obtained by the exact analysis.     

Inverse estimation of spatially and temporally varying heating boundary conditions of a two-dimensional object

In many dynamic heat transfer situations, the temperature at the heated boundary is not directly measurable and can be obtained by solving an inverse heat conduction problem (IHCP) based on measured temperature or/and heat flux at the accessible boundary. In this study, IHCP in a two-dimensional rectangular object is solved by using the conjugate gradient method (CGM) with temperature and heat flux measured at the boundary opposite to the heated boundary. The inverse problem is formulated in such a way that the heat flux at heated boundary is chosen as the unknown function to be recovered, and the temperature at the heated boundary is computed as a byproduct of the IHCP solution. The measurement data, i.e., the temperature and heat flux at the opposite boundary, are obtained by numerically solving a direct problem where the heated boundary of the object is subjected to spatially and temporally varying heat flux. The robustness of the formulated IHCP algorithm is tested for different profiles of heat fluxes along with different random errors of the measured heat flux at the opposite boundary. The effects of the uncertainties of the thermophysical properties and back-surface temperature measurement on inverse solutions are also examined.     

Inverse radiation problem of temperature field in three-dimensional rectangular enclosure containing inhomogeneous,anisotropically scattering media

An inverse radiation analysis is presented for determining the three-dimensional temperature field in an inhomogeneous, absorbing, emitting and anisotropically scattering media of known radiative properties from the knowledge of the exit radiative energy received by charge-coupled device (CCD) cameras at boundary surfaces. The forward Monte Carlo method was employed to describe the radiative energy propagation. The inverse problem was formulated as an ill-posed matrix equation and solved by least square QR decomposition (LSQR) method. The measured data were simulated by adding random errors to the exact solution of the direct problem. The effects of measurement errors, combinations of CCD cameras, concentration distributions of particles, and coefficient fluctuating errors on the accuracy of the inverse problem were investigated. The results show that the three-dimensional temperature field can be estimated accurately, even for the noisy data.     

BEM/FDM conjugate heat transfer analysis of a two-dimensional air-cooled turbine blade boundary layer

A coupled boundary element method （BEM） and finite difference method （FDM） are applied to solve conjugate heat transfer problem of a two-dimensional air-cooled turbine blade boundary layer. A loosely coupled strategy is adopted, in which each set of field equations is solved to provide boundary conditions for the other. The Navier-Stokes equations are solved by HIT-NS code. In this code, the FDM is adopted and is used to resolve the convective heat transfer in the fluid region. The BEM code is used to resolve the conduction heat transfer in the solid region. An iterated convergence criterion is the continuity of temperature and heat flux at the fluid-solid interface. The numerical results from the BEM adopted in this paper are in good agreement with the results of analytical solution and the results of commercial code, such as Fluent 6.2. The BEM avoids the complicated mesh needed in other computation method and saves the computation time. The results prove that the BEM adopted in this paper can give the same precision in numerical results with less boundary points. Comparing the conjugate results with the numerical results of an adiabatic wall flow solution, it reveals a significant difference in the distribution of metal temperatures. The results from conjugate heat transfer analysis are more accurate and they are closer to realistic thermal environment of turbines.     

An Inverse Problem Method for Overall Heat Transfer Coefficient Estimation in a Partially Filled Rotating Cylinder

The objective of this article is to study the estimation of an overall heat transfer coefficient in a partially filled rotating cylinder. Herein is an inverse analysis for estimating the overall heat transfer coefficient in an arbitrary cross-section of the aforementioned system from the temperatures measured on the shell. The material employs the finite-volume method to solve the direct problem. The hybrid effective algorithm applied here contains the local optimization algorithm to estimate the unknown parameter by minimizing the objective function. The data measured here are simulated by adding random errors to the exact solution. An investigation is made of the impact of the measurement errors on the accuracy of the inverse analysis. Two-optimization algorithms in determining the overall heat transfer coefficient are used. It is determined that the Conjugate Gradient Method is better than the Levenberg-Marquardt Method because the former produces greater accuracy for the same measurement errors. The resulting observation indicates that good agreement exists between the exact value and estimated result for both algorithms.     

Estimation of two-sided boundary conditions for two-dimensional inverse heat conduction problems

A hybrid numerical algorithm of the Laplace transform technique and finite-difference method with a sequential-in-time concept and the least-squares scheme is proposed to predict the unknown surface temperature of two-sided boundary conditions for two-dimensional inverse heat conduction problems. In the present study, the functional form of the estimated surface temperatures is unknown a priori. The whole time domain is divided into several analysis sub-time intervals and then the unknown surface temperatures in each analysis interval are estimated. To enhance the accuracy and efficiency of the present method, a good comparison between the present estimations and previous results is demonstrated. The results show that good estimations on the surface temperature can be obtained from the transient temperature recordings only at a few selected locations even for the case with measurement errors. It is worth mentioning that the unknown surface temperature can be accurately estimated even though the thermocouples are located far from the estimated surface. Owing to the application of the Laplace transform technique, the unknown surface temperature distribution can be estimated from a specific time.     

Numerical method for hyperbolic inverse heat conduction problems

The Laplace transform technique and control volume method in conjunction with the hyperbolic shape function and least-squares scheme are applied to estimate the unknown surface conditions of one-dimensional hyperbolic inverse heat conduction problems. In the present study, the expression of the unknown surface conditions is not given a priori. To obtain the more accurate estimates, the whole time domain is divided into several analysis sub-time intervals. Afterward, the unknown surface conditions in each analysis interval are estimated. To evidence the accuracy of the present method, a comparison between the present estimations and exact results is made. Results show that good estimations on the unknown surface conditions can be obtained from the transient temperature recordings only at one selected location even for the cases with measurement errors.     

On the validity of the quasi-steady state equation for heat or mass transfer problems with an axially moving boundary

The following two questions related to the quasi-steady state equation for heat or mass transfer problems are answered in this paper: (1) “Under what process (moving velocity, time, and length) and material parameters (thermal or mass diffusivity) the quasi-steady state equation is valid ?” and (2) “How large are the errors associated with the quasi-steady state assumption when solving for heat or mass transfer?” To answer these questions, an exact analytical solution for the unidirectional transient (unsteady) heat or mass transfer problems with an axially moving boundary in semi-infinite domains is derived. Then, a comparison between the transient and quasi-steady state solutions is provided for the temperature (or concentration) distribution, temperature (or concentration) gradient at the moving boundary, and thermal (or solutal) boundary layer thickness. It is also demonstrated that the solution developed in this work is the general solution applicable for both the transient state and the quasi-steady state.     

Estimation of the time-dependent heat flux using the temperature distribution at a point by conjugate gradient method

In this paper, the conjugate gradient method coupled with adjoint problem is used in order to solve the inverse heat conduction problem and estimation of the time-dependent heat flux using the temperature distribution at a point. Also, the effects of noisy data and position of measured temperature on final solution are studied. The numerical solution of the governing equations is obtained by employing a finite-difference technique. For solving this problem the general coordinate method is used. We solve the inverse heat conduction problem of estimating the transient heat flux, applied on part of the boundary of an irregular region. The irregular region in the physical domain (r,z) is transformed into a rectangle in the computational domain (ξ,η). The present formulation is general and can be applied to the solution of boundary inverse heat conduction problems over any region that can be mapped into a rectangle. The obtained results for few selected examples show the good accuracy of the presented method. Also the solutions have good stability even if the input data includes noise and that the results are nearly independent of sensor position.     

A decentralized fuzzy inference method for solving the two-dimensional steady inverse heat conduction problem of estimating boundary condition

This paper addresses a new technique for solving the two-dimensional steady inverse heat conduction problem, which named decentralized fuzzy inference (DFI) method. First of all, a group of decentralized fuzzy inference units are designed, and the fuzzy inference for each fuzzy inference unit is conducted which bases on the difference between the measured and the computed temperature at each measuring location. The computed temperatures are obtained by solving the direct heat conduction problem with the finite difference method. And then, inference results of fuzzy inference units are weighted to yield compensation values of the unknown boundary temperatures. The unknown boundary temperatures are estimated by updating guess temperatures continuously with compensation values. Numerical experiments are carried out with different initial guesses, the number of measuring points and measurement errors. Comparing results of DFI method and Levenberg–Marquardt (L–M) method, we can conclude that DFI method is valid.     

Estimation for inner surface geometry of furnace wall using inverse process combined with grey prediction model

In this work the inner surface geometry of a cylindrical furnace wall is estimated using inverse process method combined with grey prediction model. In estimating process a virtual area extended from the inner surface of furnace wall is used for analysis. The heat conduction equation and the boundary condition are first discretized by finite difference method to form a linear matrix equation; the inverse model is then optimized by linear least-squares error method and the temperatures of virtual boundary are obtained from a few of measured temperatures in furnace wall using the linear inverse model; and finally the temperature distribution of system is got by direct process and the inner surface geometry of furnace wall can be estimated accordingly. The result shows that using inverse process combined with grey prediction model the geometry can be exactly estimated from relatively small number of measured temperatures. Moreover, the effects of measurement error, location, and number of measured points on the estimation for inner surface geometry of furnace wall are discussed in detail.     

An inverse problem of finding the time-dependent thermal conductivity from boundary data

We consider the inverse problem of determining the time-dependent thermal conductivity and the transient temperature satisfying the heat equation with initial data, Dirichlet boundary conditions, and the heat flux as overdetermination condition. This formulation ensures that the inverse problem has a unique solution. However, the problem is still ill-posed since small errors in the input data cause large errors in the output solution. The finite difference method is employed as a direct solver for the inverse problem. The inverse problem is recast as a nonlinear least-squares minimization subject to physical positivity bound on the unknown thermal conductivity. Numerically, this is effectively solved using the lsqnonlin routine from the MATLAB toolbox. We investigate the accuracy and stability of results on a few test numerical examples.     

Application of the error propagation theory in estimates of static formation temperatures in geothermal and petroleum boreholes

We used the error propagation theory to calculate uncertainties in static formation temperature estimates in geothermal and petroleum wells from three widely used methods (line-source or Horner method; spherical and radial heat flow method; and cylindrical heat source method). Although these methods commonly use an ordinary least-squares linear regression model considered in this study, we also evaluated two variants of a weighted least-squares linear regression model for the actual relationship between the bottom-hole temperature and the corresponding time functions. Equations based on the error propagation theory were derived for estimating uncertainties in the time function of each analytical method. These uncertainties in conjunction with those on bottom-hole temperatures were used to estimate individual weighting factors required for applying the two variants of the weighted least-squares regression model. Standard deviations and 95% confidence limits of intercept were calculated for both types of linear regressions. Applications showed that static formation temperatures computed with the spherical and radial heat flow method were generally greater (at the 95% confidence level) than those from the other two methods under study. When typical measurement errors of 0.25 h in time and 5 °C in bottom-hole temperature were assumed for the weighted least-squares model, the uncertainties in the estimated static formation temperatures were greater than those for the ordinary least-squares model. However, if these errors were smaller (about 1% in time and 0.5% in temperature measurements), the weighted least-squares linear regression model would generally provide smaller uncertainties for the estimated temperatures than the ordinary least-squares linear regression model. Therefore, the weighted model would be statistically correct and more appropriate for such applications. We also suggest that at least 30 precise and accurate BHT and time measurements along with the respective errors should be obtained for a reliable application of the proposed regression procedure.     

COMPUTATION OF TURBULENT NATURAL CONVECTION IN A RECTANGULAR CAVITY WITH THE k–ϵ– –f MODEL

A hybrid numerical method involving the Laplace transform technique and finite-difference method in conjunction with the least-squares method and actual experimental temperature data inside the test material is proposed to estimate the unknown surface conditions of inverse heat conduction problems with the temperature-dependent thermal conductivity and heat capacity. The nonlinear terms in the differential equations are linearized using the Taylor series approximation. In this study, the functional form of the surface conditions is unknown a priori and is assumed to be a function of time before performing the inverse calculation. In addition, the whole time domain is divided into several analysis subtime intervals and then the unknown estimates on each subtime interval can be predicted. In order to show the accuracy and validity of the present inverse scheme, a comparison among the present estimates, direct solution, and actual experimental temperature data is made. The effects of the measurement errors, initial guesses, and measurement location on the estimated results are also investigated. The results show that good estimation of the surface conditions can be obtained from the present inverse scheme in conjunction with knowledge of temperature recordings inside the test material.     

采用边界元法确定渗流自由面的一种改进方法

渗流自由面的确定是无压渗流分析中非常重要的问题。相对有限元法而言 ,采用边界元法确定渗流自由面具有明显的优势。本文基于传统的确定渗流自由面的迭代边界元法 ,提出了改进的采用边界元法确定渗流自由面的迭代方法 ,并运用算例说明了改进方法的精度和有效性     

Sensitivity analysis and numerical experiments on transient test of compact heat exchanger surfaces

A single-blow transient testing technique considering the effect of longitudinal heat conduction is suggested for determining the average convection heat transfer coefficient of compact heat exchanger surface. By matching the measured outlet fluid temperature variation with similar theoretical curves, the dimensionless longitudinal conduction parameter ?_l, the time constant of the inlet fluid temperature ?⁺, and the number of heat transfer units N_tu can be determined simultaneously using the Levenberg-Marquardt nonlinear parameter estimation method. Both sensitivity analysis and numerical experiments with simulated measurements containing random errors show that the method in the present investigation provides satisfactory accuracy of the estimated parameter N_tu, which characterizes the heat transfer performance of compact heat exchanger surfaces.     

A novel non-iterative inverse method for estimating boundary condition of the furnace inner wall

A novel non-iterative inverse method with two-step scheme based on boundary element method (BEM) is proposed to estimate the boundary condition of furnace inner wall. Firstly, a matrix equation of heat transfer is formed by BEM with some measurement points. Then, the boundary value of discrete nodes on the furnace inner wall can be solved by the least-square error method. Different shapes of the furnace inner wall are considered with different function expressions, where some factors are discussed to validate the performance of present method such as the measured error, the number of measurement points, the distributive position of measurement points and the mixed boundary conditions. The results of numerical examples show that the present method can obtain the great performance on both the inversing accuracy and the computing efficiency even for the large heat flux.     

Iterative algorithms for solving inverse problems of heat conduction in multiply connected domains

The presented paper displays a method of solving the inverse problems of heat transfer in multi-connected regions, consisting in iterative solving of convergent series of the direct problems. For known temperature and flux values at the outer boundary of the region the temperature and flux values at the inner boundaries are sought (the cauchy problem for the Laplace equation). In case of such a formulation of the problem, the solution does not always exist, one of the conditions is met in the mean-square sense, providing the optimization criterion. The idea of the process consists in solving the direct problem in which the boundary condition is subject to iterative changes so as to attain minimum of the optimization criterion (the square functional). Two algorithms have been formulated. In the first of them the heat flux at the inner boundaries of the region, while in the other the temperature were subject to changes. Convergence of both the algorithms have been compared.The numerical calculation has been made for selected examples, for which an analytical solution is known. The effect of random disturbance of the boundary conditions on the solution obtained with iterative algorithms has been checked. Moreover, a function was defined, serving as convergence measure of the solution of the inverse problem solved with the algorithms proposed in the paper. The properties of the function give evidence that it tends to the value exceeding unity.     

A finite element analysis for inverse heat conduction problems

This paper presents an efficient inverse analysis technique based on a sensitivity coefficient algorithm to estimate the unknown boundary conditions of multidimensional steady and transient heat conduction problems. Sensitivity coefficients were used to represent the temperature response of a system under unit loading conditions. The proposed method, coupled with the sensitivity analysis in the finite element formulation, is capable of estimating both the unknown temperature and heat flux on the surface provided that temperature data are given at discrete points in the interior of a solid body. Inverse heat conduction problems are referred to as ill-posed because minor inaccuracy or error in temperature measurements cause a drastic effect on the predicted surface temperature and heat flux. To verify the accuracy and validity of the new method, two-dimensional steady and transient problems are considered. Their surface temperature and heat flux are evaluated. From a comparison with the exact solution, the effects of measurement accuracy, number and location of measuring points, a time step, and regularization terms are discussed. © 1998 Scripta Technica. Heat Trans Jpn Res, 26(6): 345–359, 1997     

Precise Time-Domain Expanding BEM for Solving Non-Fourier Heat Conduction Problems

The precise time-domain expanding boundary-element method (BEM) is presented for solving non-Fourier heat conduction problems. The recursive boundary integral equation is obtained via the precise time-domain expanding method and solved by the BEM, where the radial integral method is used to transform the domain integral into the boundary integral. Also, a self-adaptive judging criterion is used in the solving process. The transformation matrices of domain integrals need to be computed only once, except those related to the heat source. Finally, numerical results show that the present method can obtain stable and accurate results with different time steps.