Non-iteration estimation of thermal conductivity using finite volume method

The finite volume approach is developed for the inverse estimation of thermal conductivity in one-dimensional domain. The differential governing equation of heat conduction is converted to a system of linear equations in matrix form using the temperature data and heat generation at the discrete grid points as well as surface heat flux. The unknown thermal conductivities are obtained by solving the system equations directly. The features of the present method are that no prior information about the functional form of the thermal conductivity is required and no iterations in the calculation process are needed. The accuracy and robust of the present method are verified by comparing examples of inverse estimation of spatially and temperature-dependent thermal conductivities with the exact solutions.     

Inverse Estimation of the Thermal Conductivity in a One-Dimensional Domain by Taylor Series Approach

The Taylor series approximation is developed for the inverse estimation of thermal conductivity in a one-dimensional domain. The differential governing equation of heat conduction is converted to a discrete system of linear equations in matrix form using the temperature measurement and heat generation at the grid points as well as surface heat flux. The unknown thermal conductivity is estimated by solving the linear algebraic equations directly without iterations. The features of the present method are that no prior information about the functional form of the thermal conductivity is required, nor are any initial guesses or iterations in the calculation process needed. The accuracy and robustness of the present method are verified by comparing the results with the analytical solutions for constant, spatial- and temperature-dependent thermal conductivities. The results show that the inverse solutions are in good agreement with the exact solutions.     

Estimation of spray cooling characteristics on a hot surface using the hybrid inverse scheme

A hybrid technique of the Laplace transform and finite-difference methods in conjunction with the least-squares method and experimental temperature data inside the test material is applied to investigate the spray cooling of a hot surface. In this study, the unknown surface temperature and heat flux will be predicted. Their functional forms are unknown a priori in the present study and are assumed to be the functions of time before performing the inverse calculation. The whole time domain is divided into several analysis sub-time intervals and then these unknown estimates on each analysis interval can be predicted. In order to validate the accuracy of the present inverse method, comparisons between the present estimates and previous estimated results are made. The results show that the present estimates of the unknown temperature at various measurement locations agree with the previous estimated results and experimental temperature data. However, the present estimates of the unknown surface heat flux deviate from the previous estimated results for larger times.     

Inverse alloy solidification problem including the material shrinkage phenomenon solved by using the bee algorithm

In the paper we solve the one-phase inverse problem of alloy solidifying within the casting mould, including the shrinkage of metal which results from the difference between densities of the liquid and solid phases. The process is modeled by means of the solidification in the temperature interval basing on the heat conduction equation with the source element enclosed, whereas the shrinkage of metal is modeled by the proper application of the mass balance equation. The investigated inverse problem consists in reconstruction of the heat transfer coefficient on the boundary of the casting mould on the basis of measurements of temperature read from the sensor placed in the middle of the mould. Functional expressing the error of approximate solution is minimized with the aid of Artificial Bee Colony Optimization algorithm.     

Estimation of heat flux on the surface of an initially hot cylinder cooled by a laminar confined impinging jet

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to estimate the unknown space-and time-dependent heat flux at the surface of an initially hot cylinder cooled by a laminar confined slot impinging jet from the knowledge of temperature measurements taken on the cylinder’s surface. It is assumed that no prior information is available on the functional form of the unknown heat flux; hence the procedure is classified as the function estimation in inverse calculation. The temperature data obtained from the direct problem are used to simulate the temperature measurements, and the effect of the errors in these measurements upon the precision of the estimated results is also considered. The results show that an excellent estimation on the space-and time-dependent heat flux can be obtained even the distributions of thermal properties inside the cylinder is unknown.     

预测水冷壁管道温度分布的新方法

为了得到锅炉水冷壁管道周向温度分布和内壁氧化膜生长的特性,通过对其传热过程的研究,建立起预测具有内壁氧化膜的管道温度分布的数值模型.并根据管状热流设备测得的管道热流密度,针对某电厂的水冷壁管道实际运行情况进行计算,结果表明在圆周角为120°时管道各界面的温度和氧化膜厚度最小,在圆周角为0°时管道各界面的温度和氧化膜厚度...     

Inverse problem of estimating the heat flux at the roller/workpiece interface during a rolling process

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to estimate the unknown space-dependent heat flux at the roller/workpiece interface during rolling process from the knowledge of temperature measurements taken within the roller. It is assumed that no prior information is available on the functional form of the unknown heat flux; hence the procedure is classified as the function estimation in inverse calculation. The temperature data obtained from the direct problem are used to simulate the temperature measurements, and the effect of the errors in these measurements upon the precision of the estimated results is also considered. The results show that an excellent estimation on the space-dependent heat flux can be obtained for the test cases considered in this study.     

Particle Swarm Optimization-based algorithms for solving inverse heat conduction problems of estimating surface heat flux

An inverse analysis of estimating a time-dependent surface heat flux for a three-dimensional heat conduction problem is presented. A global optimization method known as Particle Swarm Optimization (PSO) is employed to estimate the unknown heat flux at the inner surface of a crystal tube from the knowledge of temperature measurements obtained at the external surface. Three modifications of the PSO-based algorithm, PSO with constriction factor, PSO with time-varying acceleration of the cognitive and social coefficients, and PSO with mutation are carried out to implement the optimization process of the inverse analysis. The results show that the PSO with mutation algorithm is significantly better than other PSO-based algorithms because it can overcome the drawback of trapping in the local optimum points and obtain better inverse solutions. The effects of measurement errors, number of dimensionalities, and number of generations on the inverse solutions are also investigated.     

Optimization of Heat Fluxes on the Heater and the Design Surfaces of a Radiating-Conducting Medium

In a radiating-conducting planar medium with a boundary as the heater surface using an inverse analysis, this work deals with the design methodologies to understand the inherent relationship between heater surface temperature/flux, design surface temperature/flux, and medium properties. The heat flux on the heater surface is chosen as the fitness function. Subsequently, to achieve maximum and minimum design surface heat fluxes, an optimization was done to evaluate the zone of operation of the heater. In addition, the effect of medium properties on the temperature-flux relationships on both surfaces has been studied. The distance between the two surfaces is also considered a parameter. The medium properties, the distance between the surfaces, and the heater surface temperature have been found to have a great impact on the design surface heat flux. The inverse mixed boundary problem has been solved using the lattice Boltzmann method (LBM), the finite-volume method (FVM), and the genetic algorithm (GA). Results of the present study provide a guideline towards the efficient design of a heater in which conduction and radiation are the dominant modes of heat transfer.     

Thermal quadrupoles approach for two-dimensional heat flux estimation using infrared and thermocouple measurements on the JET tokamak

This paper presents several 2D heat flux calculation methods applied to temperature data of one of the most exposed plasma facing components of the JET tokamak. We have two temperature diagnostics: a high time resolution infrared system is used to measure the surface temperature and two embedded thermocouples to measure the temperature in the bulk at 1 cm depth from the surface. A direct calculation is possible using the surface temperature but the presence of thin carbon layers disturb the measurements and leads us to make an inverse computation of the heat flux using the thermocouple data.     

Estimation of Heat Flux and Thermal Stresses in Functionally Graded Hollow Circular Cylinders

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to estimate the unknown time-dependent heat flux at the inner surface of a functionally graded hollow circular cylinder from the knowledge of temperature measurements taken within the cylinder. Subsequently, the distributions of temperature and thermal stresses in the cylinder can be determined as well. It is assumed that no prior information is available on the functional form of the unknown heat flux; hence the procedure is classified as the function estimation in inverse calculation. The temperature data obtained from the direct problem are used to simulate the temperature measurements, and the effect of the errors in these measurements upon the precision of the estimated results is also considered. Results show that an excellent estimation on the time-dependent heat flux, temperature distributions, and thermal stresses can be obtained for the test case considered in this study.     

Combined non-gray radiative and conductive heat transfer in solar collector glass cover

A rigorous approach for the radiative heat transfer analysis in solar collector glazing is developed. The model allows a more accurate prediction of thermal performance of a solar collector system. The glass material is analysed as a non-gray plane-parallel medium subjected to solar and thermal irradiations in the one-dimensional case using the Radiation Element Method by Ray Emission Model (REM by REM).This method is used to analyse the combined non-gray convective, conductive and radiative heat transfer in glass medium. The boundary surfaces of the glass are specular. The spectral dependence of the relevant radiation properties of glass (i.e. specular reflectivity, refraction angle and absorption coefficient) are taken into consideration. Both collimated and diffuse incident irradiation are applied at the boundary surfaces using the spectral solar model proposed by Bird and Riordan. The optical constants of a commercial ordinary clear glass material have been used. These optical constants (100 values) of real and imaginary parts of the complex refractive index of the glass material cover the range of interest for calculating the solar and thermal radiative heat transfer through the solar collector glass cover. The model allows the calculation of the steady-state heat flux and temperature distribution within the glass layer. The effect of both conduction and radiation in the heat transfer process is examined. It has been shown that the real and imaginary parts of the complex refractive index have a substantial effect on the layer temperature distribution. The computational time for predicting the combined heat transfer in such a system is very long for the non-gray case with 100 values of n and k. Therefore, a simplified non-gray model with 10 values of n and k and two semi-gray models have been proposed for rapid computations. A comparison of the proposed models with the reference non-gray case is presented. The result shows that 10 bandwidths could be used for rapid computation with a very high level of accuracy.     

Inverse Boundary Design Problem of Turbulent Forced Convection Between Parallel Plates With Surface Radiation Exchange

The inverse methodology is employed to estimate the unknown heat flux distribution over the heater surface of a channel formed by two parallel plates with forced convection and surface radiation exchange, from the knowledge of the desired temperature and heat flux distributions over a given design surface. The energy and radiative transfer equations are solved by the finite-volume method and the net radiation method, respectively. The conjugate gradient method is used for minimization of an objective function, which is expressed by the sum of square residuals between estimated and desired heat fluxes over the design surface. The performance and accuracy of the present method for solving inverse problems are evaluated by some numerical experiments.     

An effective 3-D inverse procedure to retrieve cooling conditions in an aluminium alloy continuous casting problem

The paper discusses a three-dimensional numerical solution of the inverse boundary problem for a continuous casting process of an aluminium alloy. Because verified information of heat flux distribution is crucial for a good mould design, effective cooling system and the whole caster in general, the main goal of the analysis presented within the paper is identifying of the heat fluxes along the external surface of the ingot. To model the solidification process, an enthalpy-porosity technique implemented in a commercial package was used. In this method, the phase change interface was determined based on the liquid fraction approach. In the inverse procedure, a sensitivity analysis was used to estimate the boundary condition retrieval. While the measured temperatures required to solve the problem are always burdened by measurement errors, a comparison of the measured and retrieved values showed a computational high accuracy. The average percentage error of the sensors was considerably lower than the maximum percentage error of the numerically simulated measurements. In addition, the computationally effective method was independent of the mesh size, the starting value of the assumed boundary condition and the maximum error of measurements used for calculations.     

Transient surface heating rates from a nickel film sensor using inverse analysis

Convective surface heat transfer measurements play an important role in many industrial, environmental and aerodynamic problems. In most of the cases, the flow is unsteady which results in temperature variation in the body. The surface heating rates are then predicted from the measured temperature histories by suitable heat transfer modeling. In this paper, the temperature history obtained from a nickel film sensor during a flight test is considered to study the effect of sensor thickness on surface heat flux measurements during the flight measurement. Inverse methods using analytical solutions as well as control volume approximations are used to infer the surface heat flux. The experimental temperature data are discretized using cubic-spline method to obtain the closed form solution which is used for inverse analysis. The results are compared with that of standard bench mark results with thin film gauge analysis based on semi-infinite one dimensional medium. No significant change in surface heat flux is observed between inverse and thin film analysis. However, when the thickness of nickel film is increased by 100 times during numerical simulation of inverse method, it is seen that peak surface heat flux increases by 20%.     

Solution of the inverse heat conduction problem described by the Poisson equation for a cooled gas-turbine blade

The presented paper describes a method of solving the inverse problems of heat conduction, consisting in solving the Poisson equation for a simply connected region instead of the Laplace equation for a multiply connected one, like a gas-turbine blade provided with cooling channels. The considered method consists in determining unknown values of the source (heat sink) power in the cooling channels for a given external heat transfer situation to achieve as close as possible an isothermal outer surface. Afterwards the temperature and heat flux distributions at the cooling channel walls are determined. Since the unknown source power is sought, the problem is an inverse one. Taking into account the sought values the method is reckoned among the class of the fictitious source methods and presents an optimization scheme. Using an exemplary gas turbine blade cooling configuration, the results of the calculation obtained with this method have been compared to the results achieved with an inverse method using the boundary element method for a multiple connected region.The results obtained with both methods within the optimization scheme approximated each other. Nevertheless, the results for the inverse method shown in the present paper gave nearly no oscillations, which is important in case of the blades with other geometric features of the cooling channels.     

Inverse shape design for heat conduction problems via the ball spine algorithm

ABSTRACT

In this article, a novel iterative physical-based method is introduced for solving inverse heat conduction problems. The method extends the ball spine algorithm concept, originally developed for inverse fluid flow problems, to inverse heat conduction problems by employing a subtle physical-sense rule. The inverse problem is described as a heat source embedded within a solid medium with known temperature distribution. The object is to find a body configuration satisfying a prescribed heat flux originated from a heat source along the outer surface. Performance of the proposed method is evaluated by solving many 2-D inverse heat conduction problems in which known heat flux distribution along the unknown surface is directly related to the Biot number and surface temperature distribution arbitrarily determined by the user. Results show that the proposed method has a truly low computational cost accompanied with a high convergence rate.     

High heat flux in glass

Transient conduction and nongray radiation in an absorbing, emitting and isotropically scattering glass slab with reflective boundaries is solved by the application of an implicit finite difference/discrete ordinates method. Solutions are presented for predicting temperatures and heat fluxes in a one dimensional slab geometry being externally heated by a constant high heat flux at one boundary. The boundaries are assumed to be optically smooth with Fresnel reflection and have a uniform relative real refractive index. The window material is intensely heated to levels that exceed the softening/melting point temperature in order to categorized the transient onset of these conditions without phase or volume change. The numerical results reveal the effects of the system parameters on the heat transfer characteristics. Comparisons are made between a gray model and a 16 band, evenly spaced, nongray model in the 1 to 4 μm wavelength region. Precise knowledge of the thermal behavior of the windows is essential to the development of glass and ceramic materials in diathermal and optical applications.     

A data method using the inner temperature difference to improve the stability of the inverse heat conduction problems

Abstract

This article proposes a method to construct two series systems for improving the stability of the inverse heat conduction problems (IHCP) in a finite slab. The transfer function between the surface heat flux or temperature and the inner temperature difference is respectively obtained by Laplace transform technique firstly. Then the series systems which can solve IHCP based on the inner temperature difference are constructed by replacing the unsuitable zero and pole points of the transfer function approximated by è approximation. Finally the effects of the series systems are evaluated by a typical example. The results of the evaluation show that this method can obtain the surface heat flux and temperature by the inner temperature difference, and enhance the response speed of the measurement system at the same time. In addition this method can also improve the signal to noise ratio (SNR) of the inverse solutions by selectively amplifying the high SNR parts of the inner temperature difference. The present work provides an effective method to improve the stability of IHCP.     

Temperature and heat flux estimation from sampled transient sensor measurements

Laplace transform is used to solve the problem of heat conduction over a finite slab. The temperature and heat flux on the two surfaces of a slab are related by the transfer functions. These relationships can be used to calculate the front surface heat input (temperature and heat flux) from the back surface measurements (temperature and/or heat flux) when the front surface measurements are not feasible to obtain. This paper demonstrates that the front surface inputs can be obtained from the sensor data without resorting to inverse Laplace transform. Through Hadamard Factorization Theorem, the transfer functions are represented as infinite products of simple polynomials. Consequently, the relationships between the front and back surfaces are translated to the time-domain without inverse Laplace transforms. These time-domain relationships are used to obtain approximate solutions through iterative procedures. We select a numerical method that can smooth the data to filter out noise and at the same time obtain the time derivatives of the data. The smoothed data and time derivatives are then used to calculate the front surface inputs.