Estimation of parameters in multi-mode heat transfer problems using Bayesian inference – Effect of noise and a priori

Parameter estimation problems and heat source/flux reconstruction problems are some of the most frequently encountered inverse heat transfer problems. These problems find their application in many areas of science and engineering. The primary focus of this paper is on the heat transfer parameter estimation for a two-dimensional unsteady heat conduction problem with (a) convection boundary condition and (b) convection and radiation boundary condition. The paper demonstrates the effect of a priori model on the performance of the algorithm at different noise levels in the measured data. The inverse problem is solved using three different a priori models namely normal, log normal and uniform. The posterior PDF is sampled using the Metropolis–Hastings sampling algorithm. Both single-parameter estimation and multi-parameter estimation problems are addressed and the effects of corresponding a priori models are studied. It was found that the mean and maximum a posteriori estimates for thermal conductivity and the convection heat transfer coefficient were insensitive to the a priori model at all the considered noise levels for the single-parameter estimation problem. At high noise levels in the two-parameter estimation problem, the estimates for thermal conductivity and convection coefficient were sensitive to the a priori model. It was also found that the standard deviation of the samples was correlated to the error in estimation in the single-parameter estimation case. In three parameter estimation case, alternate solutions to the same problem were retrieved due to a strong correlation between the convection coefficient and the emissivity. However, a more informative a priori model could address this issue.     

Estimation of dimensionless parameters of Luikov's system for heat and mass transfer in capillary porous media

This work deals with the solution of inverse problems of parameter estimation involving heat and mass transfer in capillary porous media, as described by the linear one-dimensional Luikov's equations. Our main objective is to use the D-optimum criterion to design the experiment with respect to the magnitude of the applied heat flux, heating and final experimental times, as well as the number and locations of sensors. The present parameter estimation problem is solved with Levenberg–Marquardt's method of minimization of the ordinary least-squares norm, by using simulated temperature data containing random errors. Moisture content measured data is not considered available for the inverse analysis in order to avoid quite involved measurement techniques. We show that accurate estimates can be obtained for Luikov, Kossovitch and Biot numbers by using only temperature measurements in the inverse analysis. Also, the experimental time can be reduced if the body is heated during part of the total experimental time.     

A simplex search method for a conductive–convective fin with variable conductivity

An inverse problem is solved for simultaneously estimating the convection–conduction parameter and the variable thermal conductivity parameter in a conductive–convective fin with temperature dependent thermal conductivity. Initially, the temperature field is obtained from a direct method using an analytical approach based on decomposition scheme and then using a simplex search minimization algorithm an inverse problem is solved for estimating the unknowns. The objective function to be minimized is represented by the sum of square of the error between the measured temperature field and an initially guessed value which is updated in an iterative manner. The estimation accuracy is studied for the effect of measurement errors, initial guess and number of measurement points. It is observed that although very good estimation accuracy is possible with more number of measurement points, reasonably well estimation is obtained even with fewer number of measurement points without measurement error. Subject to selection of a proper initial guess, it is seen that the number of iterations could be significantly reduced. The relative sensitiveness of the estimated parameters is studied and is observed from the present work that the estimated convection–conduction parameter contributes more to the temperature distribution than the variable conductivity parameter.     

Methodology for Estimation of Local Convective Heat Transfer Coefficient for Vapor Condensation

This paper aims to present an effective two-dimensional inverse heat conduction technique and an experimental design for accurately estimating the local convective heat transfer coefficient of vapor condensation over a conical surface, given temperature measurements at some interior locations. The functional form for the heat transfer coefficient is not known a priori. The method uses a sequential procedure together with Beck's function specification approach. Solution accuracy and the effects of experimental errors are examined using simulated temperature data. It is concluded that a good estimation of space-variable heat transfer coefficient can be made from the knowledge of transient temperature recordings using the proposed inverse heat conduction problem method. The method is also used in a series of numerical experiments to provide the optimum experimental design for condensation heat transfer investigation.     

SIMULTANEOUS ESTIMATION OF INLET TEMPERATURE AND WALL HEAT FLUX IN TURBULENT CIRCULAR PIPE FLOW

An inverse heat convection problem is solved for simultaneous estimation of unknown inlet temperature and wall heat flux in a thermally developing, hydrodynamically developed turbulent flow in a circular pipe based on temperature measurements obtained at several different locations in the stream. The direct problem of turbulent forced convection is solved with a finite difference method with appropriate algebraic turbulence modelling. Although we seek for two unknown functions, we formulate the inverse problem as one of parameter estimation through the representation of the unknown inlet temperature profile and the wall heat flux distribution by one-dimensional finite element interpolation. Nodal values of the inlet temperature and the wall heat flux at chosen positions are determined as unknown parameters through the Levenberg–Marquardt algorithm for minimization procedure. Numerical results for several testing cases with different magnitudes of measurement errors are examined by using simulated experimental data. The effects of the number and the locations of the temperature measurement points are discussed.     

Application of optimization techniques and the enthalpy method to solve a 3D-inverse problem during a TIG welding process

Tungsten inert Gas (TIG) welding takes place in an atmosphere of inert gas and uses a tungsten electrode. In this process heat input identification is a complex task and represents an important role in the optimization of the welding process. The technique used to estimate the heat flux is based on solution of an inverse three-dimensional transient heat conduction model with moving heat sources. The thermal fields at any region of the plate or at any instant are determined from the estimation of the heat rate delivered to the workpiece. The direct problem is solved by an implicit finite difference method. The system of linear algebraic equations is solved by Successive Over Relaxation method (SOR) and the inverse problem is solved using the Golden Section technique. The golden section technique minimizes an error square function based on the difference of theoretical and experimental temperature. The temperature measurements are obtained using thermocouples at accessible regions of the workpiece surface while the theoretical temperatures are calculated from the 3D transient thermal model.     

An inverse problem to estimate relaxation parameter and order of fractionality in fractional single-phase-lag heat equation

In this paper, an inverse analysis is performed for simultaneous estimation of relaxation time and order of fractionality in fractional single-phase-lag heat equation. This fractional heat conduction equation is applied on two physical problems. In inverse procedure, solutions of a previously validated linear dual-phase-lag model on the physical problems under study have been used as the measured temperatures. The inverse fractional single-phase-lag heat conduction problem is solved using the nonlinear parameter estimation technique based on the Levenberg–Marquardt method. The results of the present study show that the Levenberg–Marquardt method can be successfully applied on the inverse fractional heat transfer problem. The solution procedures employed in the present study for direct and inverse problems have greatly increased the reliability and success of parameter estimation problem. In the present study, for the first time, relaxation time and fractionality of a non-homogeneous medium (i.e. processed meat) have been determined. Also, the results of this study show that the fractional single-phase-lag model can predict the same temperature distribution as the linear dual-phase-lag model for the problem under study. This latter result enables us to consider further generalization of the dual-phase-lag model to fractional dual-phase-lag models.     

Implementation of Geometrical Domain Decomposition Method for Solution of Axisymmetric Transient Inverse Heat Conduction Problems

The objective of this article is to study the performance of iterative parameter and function estimation techniques to solve simultaneously two unknown functions (quadratic in time, and linear in time and space) using transient inverse heat conduction method in conjunction with a geometrical domain decomposition approach, in cylindrical coordinates. For geometrical decomposition of physical domain, a multi-block method has been used. The numerical scheme for the solution of the governing partial differential equations is the finite element method. The results of the present study for a configuration composed of two joined disks with different heights are compared to those of exact heat source and temperature boundary condition using inverse analysis. Good agreement between the estimated results and exact functions has been observed for parameter estimation techniques in contrast to those of function estimation approach. In summary, the results show that the function estimation technique is sensitive to the location of measurement points, but is useful to estimate unknown functions without a priori knowledge of the functions' spatial and/or temporal distributions. However, the function estimation technique suffers from a drawback: its implementation and data extraction are less straightforward than parameter estimation method. Finally, it is shown that the use of geometrical domain decomposition offers the possibility of developing a robust inverse analysis code for general purpose heat conduction problems.     

燃烧室内三维温度场的辐射反问题

本文提出了一种在介质辐射特性已知的条件下,由壁面入射辐射热流的测量值反演燃烧室内三维温度场的方法。该方法是在辐射传递方程离散坐标近似的基础上,用求目标函数极小值的共轭梯度法进行反演计算。通过对吸收系数、散射不对称因子、反照率、壁面黑度和燃烧室大小尺寸等参数对反演精度影响的分析,结果表明,即使存在随机测量误差,这些参数对温度场反演精度的影响也不大,本文所提出的方法可较精确地反演燃烧室内三维温度场。     

Application of Simulated Annealing for Simultaneous Estimation of Parameters in a Cylindrical Fin

An inverse problem is solved for simultaneous three parameter estimation in a large-sized cylindrical fin involving conduction-convection heat transfer. The temperature field used for the estimation of unknowns is calculated from the finite difference method (FDM). The simulated annealing (SA) is applied in this work for minimizing the required objective function. The accuracy of the estimated parameters has been studied for different measurement points, initial guess, measurement errors, and number of iterations; CPU times are also reported. The FDM-SA is observed to reconstruct the temperature field with an excellent accuracy, but ill-posed behavior is observed.     

An inverse problem of parameter estimation for heat and mass transfer in capillary porous media

This work deals with the solution of an inverse problem of parameter estimation involving heat and mass transfer in capillary porous media, as described by the dimensionless linear Luikov’s equations. The physical problem under picture involves the drying of a moist porous one-dimensional medium. The main objective of this paper is to simultaneously estimate the dimensionless parameters appearing in the formulation of the physical problem by using transient temperature and moisture content measurements taken inside the medium. The inverse problem is solved by using the Levenberg-Marquardt method of minimization of the least-squares norm with simulated measurements.     

反演计算不同炉气温度下非灰气体的当量吸收系数的方法

提出了一种反演计算不同炉气温度下非灰气体的当量吸收系数的方法。此方法以实验数据反演得到的非灰气体的当量吸收系数作为物性参数,由正演算计算不同条件下的炉气温度,并以此为定解条件,反演不同炉气温度下的非灰气体的当量吸收系数,反复迭代,直至收敛。反演计算过程中采用区域法与求目标函数最小值的共轭梯度法相结合的方法。与实验结果比较表明,该方法可以较为准确地反演不同炉气温度下的非灰气体的当量吸收系数。最后,用该方法计算并比较了非灰气体的当量吸收系数随炉气温度的变化。图2表2参9     

二维燃烧系统气体发热量的确定

开发了一种求解二维燃烧系统气体发热量和气体温度分布的辐射热负荷反问题的新方法－蒙特卡罗法、共轭梯度法和最小二乘法相结合的方法。讨论了气体吸收系数、固体表面发射率和对流传热系数对反问题计算结果的影响。结果表明，即使测量误差存在，辐射热负荷反问题也能得到较准确的结果。     

Detection of hot spot through inverse thermal analysis in superconducting RF cavities

An inverse heat conduction problem in a superconducting radio frequency (SRF) cavity is examined. A localized defect is simulated as a point-heating source on the inner surface (RF surface) of the evacuated niobium cavity. Liquid helium acts as a coolant on the outer surface of the cavity. By measuring the outer surface temperature profile of the cavity using relatively few sensors, the temperature and location of a hot spot on the inner surface of the niobium are calculated using an inverse heat conduction technique. The inverse method requires a direct solution of a three-dimensional heat conduction problem through the cavity wall thickness along with temperature measurements from sensors on the outer surface of the cavity, which is immersed in liquid helium. A non-linear parameter estimation program then estimates the unknown location and temperature rise of the hot spot inside the cavity. The validation of the technique has been done through an experiment conducted on a niobium sample at room temperature.     

Influences of Soret and Dufour numbers on mixed convective and chemically reactive Casson fluids flow towards an inclined flat plate

The purpose of this study is to examine the magnetohydrodynamic mixed convection Casson fluid flow over an inclined flat plate along with the heat source/sink. The present flow problem is considered under the assumption of the chemical reaction and thermal radiation impacts along with heat and mass transport. The leading nonlinear partial differential equations of the flow problem were renovated into the nonlinear ordinary differential equations (ODEs) with the assistance of appropriate similarity transformations and then we solved these ODEs with the employment of the bvp4c technique using the computational software MATLAB. The consequences of numerous leading parameters such as thermophoretic parameter, local temperature Grashof number, solutal Grashof number, suction parameter, magnetic field parameter, Prandtl number, chemical reaction parameter, Dufour number, Soret number, angle of inclination, radiation parameter, heat source/sink, and Casson parameter on the fluid velocity, temperature, and concentration profiles are discoursed upon  and presented through different graphs. Some important key findings of the present investigation are that the temperature of the Casson fluid becomes lower for local temperature Grashof number and solutal Grashof number. It is initiated that the Casson fluid parameter increases the velocity of the fluid whereas the opposite effect is noticed in the temperature profile. Higher estimation of Prandtl number and magnetic parameter elevated the Casson fluid concentration. Finally, the skin friction coefficient, Nusselt number, and Sherwood number are calculated and tabulated. It is also examined that the Nusselt number is weakened for both the Dufour number and Soret number but the skin fraction coefficient is greater for both the Dufour number and Soret number.     

Effect of chemical reaction and heat generation or absorption on double-diffusive convection from a vertical truncated cone in porous media with variable viscosity

This work is focused on the study of combined heat and mass transfer on double-diffusive convection near a vertical truncated cone in a fluid-saturated porous medium in the presence of a first-order chemical reaction and heat generation or absorption with variable viscosity. The viscosity of the fluid is assumed to be an inverse linear function of the temperature. A boundary layer analysis is employed to derive the non-dimensional non-similar governing equations. The governing equations for this investigation are formulated and solved numerically using the fourth-order Runge–Kutta integration scheme with Newton–Raphson shooting technique. Comparisons with previously published work on special cases of the problem are performed and found to be in excellent agreement. A parametric study illustrating the influence of chemical reaction parameter, heat generation or absorption parameter, viscosity-variation parameter, buoyancy ratio and Lewis number on the fluid velocity, temperature, concentration as well as Nusselt number and Sherwood number is conducted. The results of this parametric study are shown graphically and the physical aspects of the problem are highlighted and discussed.     

Application of a Particle Swarm Algorithm for Parameter Retrieval in a Transient Conduction-Radiation Problem

This article addresses the application the particle swarm optimization (PSO) algorithm as an optimization tool for retrieval of parameters in a combined mode 1-D transient conduction-radiation heat transfer problem. In the chosen problem, the participating medium is absorbing, emitting, and scattering. The boundaries are taken to be diffuse gray. In both direct and inverse methods, the energy equation is solved using the lattice Boltzmann method (LBM) and the finite volume method (FVM) is used to compute the radiative information. In the inverse method, the objective function is minimized using the PSO algorithm. The objective function considered in the inverse formulation is an error function evaluated with the exact and inverse temperature fields for the simultaneous retrieval of the extinction coefficient and the scattering albedo. The inverse analysis constituted the effect of measurement errors on solution efficacies. In addition, the effect of important PSO parameters such as swarm size, inertia factor and constriction factor on the parameter retrieval is considered. For the chosen problem, it is found that the PSO with 20 discrete particles and 50 iterations is adequate for accurate parameter retrieval. The PSO has been found to provide a better accuracy than the genetic algorithm.     

Inverse radiation problem of temperature field in three-dimensional rectangular furnaces

The discrete ordinates method is used to developed a solution to an inverse radiation problem of temperature field in rectangular furnaces. It is assumed that, with the exception of the inhomogeneous temperature field, all aspects of the radiation transport problem are known. A method is developed to determine the inhomogeneous temperature field from specified incident radiation heat fluxes at the centers of boundary walls. The inverse problem is solved using conjugate gradient method that minimize the error between the incident radiation heat fluxes calculated and the experimental data. The results of temperature estimation show that the temperature field can be estimated accurately, even with noisy data.     

Determination of temperature wall functions for high Rayleigh number flows using asymptotics: A systematic approach

This paper presents a systematic approach to determine temperature wall functions for high Rayleigh number flows using asymptotics. An asymptotic analysis of the flow and heat transfer in the near wall region forms the basis for the development of the wall functions. Appropriate normalization of the variables followed by asymptotic matching of the temperature gradients of the inner and outer layers in the overlap region leads to a logarithmic temperature profile as a wall function that has undetermined constants. A key classification that has been made in the present study is the introduction of (1) The direct problem and (2) The inverse problem. The former means that temperature profiles, either from experiments or Direct Numerical Simulations (DNS), are available and the wall function problem finally reduces to one of determining certain constants in a general wall function formula. More radical and of more interest, is the inverse problem. The idea behind this it is that when a temperature profile can be recast into a Nusselt–Rayleigh correlation, it should be perfectly possible for one to start from a Nusselt–Rayleigh correlation and end up with a wall function for temperature. This approach again will have undetermined constants that can be calibrated from either experimental or DNS data. The main advantage of using the inverse problem is the dispensation of the need to measure temperatures accurately within the boundary layer. For both the direct and inverse problems, a graded treatment to determine the constants is presented. The treatment at its highest level will result in a parameter estimation problem that can be posed as an optimization problem. The optimization problem is then solved by state of the art techniques like Levenberg–Marquardt algorithm and Genetic algorithms (GA) and the solutions are compared. While for the direct problem, the approach is illustrated for the infinite channel problem (a simple flow), for the inverse problem, the approach is elucidated for the Rayleigh–Bénard problem (a complex flow). Finally, a blending procedure to arrive at a universal temperature profile that is valid in the viscous sublayer, buffer and the overlap layers is suggested. The key ideas of (1) using optimization techniques for determining the constants in the wall function and (2) obtaining wall functions from the Nusselt numbers by the inverse approach are expected to be useful for a wide class of problems involving natural convection.