INVERSE FORCED CONVECTION PROBLEM OF SIMULTANEOUS ESTIMATION OF TWO BOUNDARY HEAT FLUXES IN IRREGULARLY SHAPED CHANNELS

This article deals with the use of the conjugate gradient method of function estimation for the simultaneous identification of two unknown boundary heat fluxes in channels with laminar flows. The irregularly shaped channel in the physical domain is transformed into a parallel plate channel in the computational domain by using an elliptic scheme of numerical grid generation. The direct problem, as well as the auxiliary problems and the gradient equations, required for the solution of the inverse problem with the conjugate gradient method are formulated in terms of generalized boundary-fitted coordinates. Therefore, the solution approach presented here can be readily applied to forced convection boundary inverse problems in channels of any shape. Direct and auxiliary problems are solved with finite volumes. The numerical solution for the direct problem is validated by comparing the results obtained here with benchmark solutions for smoothly expanding channels. Simulated temperature measurements containing random errors are used in the inverse analysis for strict cases involving functional forms with discontinuities and sharp corners for the unknown functions. The estimation of three different types of inverse problems are addressed in the paper: (i) time-dependent heat fluxes; (ii) spatially dependent heat fluxes; and (iii) time and spatially dependent heat fluxes.     

Estimation of unknown heat source function in inverse heat conduction problems using quantum-behaved particle swarm optimization

The estimation of temporal dependent heat source in transient heat conduction problem is investigated. A stochastic method known as quantum-behaved particle swarm optimization (QPSO) is used to estimate the heat source without a priori information on its functional form, which is classified as the function estimation by inverse calculation. Because of the ill-posedness of this kind of inverse problems, Tikhonov regularization method is applied in this paper to stabilize the solution. Numerical experiments indicate the validity and stability of the QPSO method. Comparison with the conjugate gradient method (CGM) is also presented in this paper.     

Structured Multiblock Grid Solution of Two-Dimensional Transient Inverse Heat Conduction Problems in Cartesian Coordinate System

ABSTRACT

In this study a structured multiblock grid is used to solve two-dimensional transient inverse heat conduction problems. The multiblock method is implemented for geometric decomposition of the physical domain into regions with blocked interfaces. The finite-element method is employed for direct solution of the transient heat conduction equation in a Cartesian coordinate system. Inverse algorithms used in this research are iterative Levenberg-Marquardt and adjoint conjugate gradient techniques for parameter and function estimations. The measured transient temperature data needed in the inverse solution are given by exact or noisy data. Simultaneous estimation of unknown linear/nonlinear time-varying strengths of two heat sources in two joined surfaces with equal and different heights is obtained for the solution of the inverse problems, and the results of the present study for unknown heat source functions are compared to those of exact functions. This study is an attempt to challenge the goal of combining a multiblock technique with inverse analysis methods. In fact, the structured multiblock grid is capable of providing accurate solutions of inverse heat conduction problems (IHCPs) in industrial configurations, including composite structures. In addition, the multiblock IHCP solver is suitable to estimate unknown parameters and functions in these structures.     

Direct and inverse mixed convections in an enclosure with ventilation ports

The numerical study presented in this work describes the direct and inverse mixed convection problems in a slot-ventilated enclosure subjected to an unknown heat flux on one side. Particularly, the interaction of internal natural convection with the cold ventilated flow leads to various flow fields depending on the Richardson number, Reynolds number, and the functional form of the imposed boundary heat flux. Fluid and heat transport structures across the enclosure are visualized by the streamlines and heatlines, respectively. Subsequently, an iterative conjugate gradient method is applied such that the gradient of the cost function is introduced when the appropriate sensitivity and adjoint problems are defined for a domain of arbitrary geometries. In this approach, no a priori information is needed about the unknown boundary heat fluxes to be determined. The accuracy of the heat flux profile solutions is shown to depend strongly on the values of Reynolds number and flux functional forms. Effects of measurement errors on the accuracy of estimation are also investigated. The present work is significant for the flow control simultaneously involving the natural convection and forced convection.     

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.     

Analysis of Thermal Nonequilibrium Inverse Heat Transfer in a Porous Channel

This article is concerned with the one-dimensional transient inverse heat conduction between two parallel plates filled with a porous medium under a nonthermal equilibrium condition between the solid and the fluid phases. The estimation of transient heat flux using the conjugate gradient method (CGM) along with the differential adjoint equations has been carried out under nonthermal equilibrium conditions between two phases. Derivation of the adjoint differential equations in the case of nonthermal equilibrium and calculation of gradient function from coupled adjoint equations are presented in detail here. The transient wall heat flux imposed on the porous boundary is estimated using the aforementioned method, and results show that sensor locations and existing error in the measured data have important effects on the calculated heat flux. Nonetheless, accurate heat flux estimation is quite achievable.     

Three-Dimensional Transient Optimal Boundary Heating of Functionally Graded Plates

An optimal control procedure for estimating the heat fluxes on the boundaries of functionally graded (FG) thick plates to reach the desired domain temperature distributions in a specified time interval of heating is presented. The conjugate gradient method (CGM) is employed for optimization, and the differential quadrature method as an accurate and numerically efficient method in conjunction with the forward finite-difference method are applied to solve the three-dimensional transient heat transfer, sensitivity, and adjoint problems. The validity of the presented optimal control problem is demonstrated by solving different numerical examples. Results show that excellent estimation on the boundary heat fluxes can be obtained with arbitrary initial guesses of these functions.     

Inverse problem of unsteady conjugated forced convection in parallel plate channels

The heat transfer phenomena of the unsteady laminar forced convection in parallel plate channels with wall conduction effects are still not very well understood. An inverse algorithm based on the conjugate gradient method is proposed to estimate the boundary conditions of these problems, and the minimization of object function is used to reduce the estimated error. The estimation of applied heat flux is found to be highly dependent of temperature sensor location and uncertainty, plate thickness, and heating way. The results show that the predicted boundary conditions by the present inverse method are consistent with the initially specified ones.     

Inverse heat transfer problem of thermal contact conductance estimation in periodically contacting surfaces

The problems involving periodic contacting surfaces have different practical applications. An inverse heat conduction problem for estimating the periodic Thermal Contact Conductance (TCC) between one-dimensional, constant property contacting solids has been investigated with conjugate gradient method (CGM) of function estimation. This method converges very rapidly and is not so sensitive to the measurement errors. The advantage of the present method is that no a priori information is needed on the variation of the unknown quantities, since the solution automatically determines the functional form over the specified domain. A simple, straight forward technique is utilized to solve the direct, sensitivity and adjoint problems, in order to overcome the difficulties associated with numerical methods. Two general classes of results, the results obtained by applying inexact simulated measured data and the results obtained by using data taken from an actual experiment are presented. In addition, extrapolation method is applied to obtain actual results. Generally, the present method effectively improves the exact TCC when exact and inexact simulated measurements input to the analysis. Furthermore, the results obtained with CGM and the extrapolation results are in agreement and the little deviations can be negligible.     

Inverse problem of estimating time-dependent heat generation in a frictional heated strip and foundation

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to estimate the unknown time-dependent frictional heat generation for the tribosystem consisting of a semi-infinite foundation and a plane-parallel strip sliding over its surface, from the knowledge of temperature measurements taken within the foundation. It is assumed that no prior information is available on the functional form of the unknown heat generation; hence the procedure is classified as the function estimation in inverse calculation. Results show that an excellent estimation on the time-dependent heat generation can be obtained for the test case considered in this study. The current methodology can be applied to the prediction of heat generation in engineering problems involving sliding-contact elements.     

Estimation of the Boundary Heat Flux in Grinding via the Conjugate Gradient Method

The unknown boundary surface heat flux in workpieces during grinding is estimated by the application of inverse heat transfer analysis. The conjugate gradient method of function estimation is used for the minimization procedure. Simulated temperature measurements are used in the inverse analysis for typical practical cases, in order to show that results more accurate than those available in the literature are obtained with the present solution approach. Actual experimental data are also used in the computations to estimate the surface heat flux.     

On the inverse heat convection problem of the flow over a cascade of rectangular blades

In this study, an inverse algorithm based on the conjugate gradient method is applied to a steady flow over a cascade of rectangular blades to estimate the inlet flow temperature. The objective is to study the difficulties associated with inverse heat convection problems. Therefore, the measurement quantity has been deliberately placed at five different locations over the domain, each of them covered by a unique flow feature. The computation shows that at very low Reynolds number, the accuracy of the inverse method is not affected by the relative position between the estimated and measurement quantities. At a higher Reynolds number, however, the accuracy of the inverse method strongly depends on the relative position between the two quantities. The inverse method only returns satisfactory estimation for some cases but not others.     

Two-Dimensional Inverse Heat Transfer Analysis of Functionally Graded Materials in Estimating Time-Dependent Surface Heat Flux

Two-dimensional transient inverse heat conduction problem (IHCP) of functionally graded materials (FGMs) is studied herein. A combination of the finite element (FE) and differential quadrature (DQ) methods as a simple, accurate, and efficient numerical method for FGMs transient heat transfer analysis is employed for solving the direct problem. In order to estimate the unknown boundary heat flux in solving the inverse problem, conjugate gradient method (CGM) in conjunction with adjoint problem is used. The results obtained show good accuracy for the estimation of boundary heat fluxes. The effects of measurement errors on the inverse solutions are also discussed.     

Estimation of heat flux and temperature distributions in a composite strip and homogeneous foundation

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 and temperature distributions for the system composed of a multi-layer composite strip and semi-infinite foundation, from the knowledge of temperature measurements taken within the strip. 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. Results show that an excellent estimation on the time-dependent heat flux can be obtained for the test case considered in this study.     

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.     

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.     

Inverse Natural Convection Problem with Radiation in Rectangular Enclosure

Inverse thermal problem is applied to natural convective flow with radiative heat transfer. The bottom wall temperature in the 2-D cavity domain is estimated by using gas temperature measurements in the flow field. The inverse problem is solved through a minimization of an objective function using the conjugate gradient method with adjoint problem. The effects of functional form of bottom wall temperature profile, the number and the position of measurement points, and the measurement errors are investigated and discussed. The conjugate gradient method is found to work well in estimating the bottom wall temperature, even when natural convection with radiation phenomena is involved.     

NUMERICAL SOLUTION OF 2-D NONLINEAR INVERSE HEAT CONDUCTION PROBLEMS USING FINITE-ELEMENT TECHNIQUES

A general method is presented for solving different classes of nonlinear inverse heat conduction problems (IHCP) for two-dimensional, arbitrarily shaped bodies. It is based on the systematic use of a finite-element library. It is shown that, following this approach, the conjugate gradient method can be easily implemented. The method offers a very wide field of practical applications in inverse thermal analysis, while reducing very significantly the amount of work which remains specific for each particular IHPC. Two numerical experiments illustrate the influence of data errors and the iterative regularization principle.     

An inverse problem in simultaneous estimating the Biot numbers of heat and moisture transfer for a porous material

A conjugate gradient method based inverse algorithm is applied in the present study in simultaneous determining the unknown time-dependent Biot numbers of heat and moisture transfer for a porous material based on interior measurements of temperature and moisture.It is assumed that no prior information is available on the functional form of the unknown Biot numbers in the present study, thus, it is classified as the function estimation in inverse calculation.The accuracy of this inverse heat and moisture transfer problem is examined by using the simulated exact and inexact temperature and moisture measurements in the numerical experiments. Results show that the estimation on the time-dependent Biot numbers can be obtained with any arbitrary initial guesses on a Pentium IV 1.4 GHz personal computer.     

An iterative regularization method in estimating the transient heat-transfer rate on the surface of the insulation layer of a double circular pipe

In this study, a conjugate gradient method based inverse algorithm is applied to estimate the unknown space- and time-dependent heat-transfer rate on the surface of the insulation layer of a double circular pipe heat exchanger using temperature measurements. It is assumed that no prior information is available on the functional form of the unknown heat-transfer rate; 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. The accuracy of the inverse analysis is examined by using simulated exact and inexact temperature measurements. Results show that an excellent estimation on the space- and time-dependent heat-transfer rate can be obtained for the test case considered in this study.