Unknown Boundary Heat Flux Research Articles

An Adaptive Matrix Method for the Solution of a Nonlinear Inverse Heat Transfer Problem and Its Experimental Verification

An adaptive matrix inverse (AMI) method is presented to identify the temperature and unknown boundary heat flux in a domain of a regular or irregular shape with temperature-dependent properties. The nonlinear problem is broken down into a number of linear submodels, and for each submodel, the temperature is obtained in measuring points. Next, based on the matching degree between the temperatures measured and calculated by each prediction submodel, the submodels are weighted and combined to create the full model for the solution of an inverse nonlinear heat transfer problem. Comparisons are also made with the existing multiple model adaptive inverse (MMAI) algorithm and method based on the Levenberg–Marquardt algorithm (LMA). The results of the presented numerical tests for undisturbed and disturbed “measuring” data indicate that the heat fluxes identified by the AMI method are close to the exact values. The application of the presented method for bodies with an irregular shape is also demonstrated. The AMI method has been experimentally verified during the thick-walled cylinder cooling process. The proposed method can be applied in online diagnostic systems for thermal state monitoring.

Novel inverse heat transfer methodology for estimation of unknown interfacial heat flux of a continuous casting mould: A complete three-dimensional thermal analysis of an industrial slab mould

Accurate quantification of interfacial heat flux between the continuous casting mould and slab/billet is essential for the design and development of the products as well as operation and quality control of the casting process. Apart from empirical correlations, inverse heat transfer methods are typically employed to predict the heat flux variation by assuming a one-dimensional variation of heat flux along the casting direction. The assumption of one-dimensional heat flux often leads to significant error especially in determining the corner heat flux which affects the shrinkage behaviour and thereby, the quality of the products of casting. Previous studies have recognized this error and assumed this error of heat flux as a constant deviation to account for the longitudinal and transverse variation of heat flux. In this work, a novel methodology has been presented to predict the variable interfacial heat flux of a three-dimensional mould domain by estimating the parameters of a characteristic heat flux profile using an inverse heat transfer technique based on Salp Swarm Algorithm (SSA). This methodology uses limited experimental data or on-site plant recorded data. Representative cases of billet and slab moulds have been considered based on the available plant data. Results indicate that the mould corner heat flux is considerably different from that of the typically assumed value of 67% of the centre heat flux as reported in the literature with deviation reaching as high as 56% near the meniscus for a slab casting mould. The proposed methodology is simple to implement and is recommended for studies aimed towards realistic estimation of unknown boundary heat flux in a continuous casting process.

Inverse heat problem of determining unknown surface heat flux in a molten salt loop

Inverse heat transfer problems typically rely on temperature measurements for estimating unknown boundary heat flux, such as that in the water tubes of steam boilers or central receivers in solar tower power plants. In this work, an experimental facility consisting of a molten salt loop that simulates a tube of a solar tower receiver is presented to obtain the outer tube surface temperatures under solar tower power plant operating conditions. The external surface f the pipe in the test section is heated in a controlled manner with an induction heater, which provides a very high nonuniform heat flux. An inverse thermal method has been applied to obtain the incident heat flux onto the receiver tube from the outer surface temperature measurements. To solve the inverse problem, a transient two-dimensional numerical model of a circular pipe flowing molten nitrate salt and subjected to a nonhomogeneous circumferential heat flux has been developed. The heat flux calculation with the inverse method is in accordance with the heat flux estimation based on the calibration of the induction heater. A good agreement between the experimental and calculated temperatures is observed. Furthermore, the deflection of the tube caused by the nonhomogeneous heat flux is measured and is compared to the deflection calculated from the radial temperature profile from the inverse problem solution, and a good agreement between both results is observed.

A Bayesian inference approach: estimation of heat flux from fin for perturbed temperature data

This paper reports the estimation of the unknown boundary heat flux from a fin using the Bayesian inference method. The setup consists of a rectangular mild steel fin of dimensions 250×150×6 mm3 and an aluminium base plate of dimensions 250×150×8 mm3. The fin is subjected to constant heat flux at the base and the fin setup is modelled using ANSYS14.5. The problem considered is a conjugate heat transfer from the fin, and the Navier–Stokes equation is solved to obtain the flow parameters. Grid independence study is carried out to fix the number of grids for the study considered. To reduce the computational cost, computational fluid dynamics (CFD) is replaced with artificial neural network (ANN) as the forward model. The Markov Chain Monte Carlo (MCMC) powered by Metropolis–Hastings sampling algorithm along with the Bayesian framework is used to explore the estimation space. The sensitivity analysis of the estimated temperature with respect to the unknown parameter is discussed to know the dependency of the temperature with the parameter. This paper signifies the effect of a prior model on the execution of the inverse algorithm at different noise levels. The unknown heat flux is estimated for the surrogated temperature and the estimates are reported as mean, Maximum a Posteriori (MAP) and standard deviation. The effect of a-priori information on the estimated parameter is also addressed. The standard deviation in the estimation process is referred to as the uncertainty associated with the estimated parameters.

Performance analysis and feasibility study of ant colony optimization, particle swarm optimization and cuckoo search algorithms for inverse heat transfer problems

In the present work, three recently developed metaheuristic algorithms (ant colony optimization, cuckoo search and particle swarm optimization) are examined for a class of heat transfer problems. Unknown boundary heat fluxes are estimated for conduction, convection and coupled conduction–radiation problems. Direct problems are solved to determine temperature distribution assuming known boundary heat flux. Inverse method is then used to estimate boundary heat flux with the help of the temperature previously determined from the direct problem. To replicate experimental error, effect of noise on temperature data is introduced to examine the robustness of all the algorithms. Effect of time step size and regularization are studied. It is found that all the algorithms are promising and can be used for this class of inverse heat transfer problems. Performance of all the algorithms is comparable. Efficiency of the three algorithms is compared in terms of CPU time. Ant colony optimization algorithm is found to be most efficient followed by particle swarm optimization and cuckoo search algorithms for all the considered heat transfer problems. All the algorithms are also applied to estimate diffusion coefficient of a food material (mushroom) using experimental data.

Inverse analysis for transient thermal load identification and application to aerodynamic heating on atmospheric reentry capsule

A transient inverse heat conduction analysis enables the identification of the unknown boundary heat flux from finite number of temperature data obtained, e.g., in the high temperature structural tests or in the actual operation. Since the prediction of thermal load (heat flux) is difficult, the inverse analysis is expected to improve structural design of high temperature components. The present study develops a computational method of transient inverse heat conduction analysis. The developed code is applied to problems of a simple two-dimensional plate and of an atmospheric reentry capsule. Sequential Function Specification (SFS) method and Truncated Singular Value Decomposition (TSVD) are employed to improve the stability of the inverse analysis. Effects of these regularization methods are numerically discussed.

Simultaneous Estimation of Boundary Heat Flux and Convective Heat Transfer Coefficient of a Curved Plate Subjected to a Slot Liquid Jet Impingement Cooling

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to simultaneously estimate the unknown boundary heat flux and convective heat transfer coefficient in a curved plate cooled by an impinging slot jet from the knowledge of temperature measurements taken within the plate. Subsequently, the distributions of temperature in the plate can be determined. It is assumed that no prior information is available on the functional forms of the heat flux and convective heat transfer coefficient; 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 and locations in these measurements upon the precision of the estimated results is also considered. Results show that an excellent estimation on the heat flux and convective heat transfer coefficient and temperature distributions can be obtained for the two test cases considered in this study.

Estimation of transient boundary flux for a developing flow in a parallel plate channel

Purpose – The purpose of this paper is to develop a numerical model for estimating the unknown boundary heat flux in a parallel plate channel for the case of a hydrodynamically and thermally developing laminar flow. Design/methodology/approach – The conjugate gradient method (CGM) is used to solve the inverse problem. The momentum equations are solved using an in-house computational fluid dynamics (CFD) source code. The energy equations along with the adjoint and sensitivity equations are solved using the finite volume method. Findings – The effects of number of measurements, distribution of measurements and functional form of unknown flux on the accuracy of estimations are investigated in this work. The prediction of boundary flux by the present algorithm is found to be quite reasonable. Originality/value – It is noticed from the literature review that study of inverse problem with hydrodynamically developing flow has not received sufficient attention despite its practical importance. In the present work, a hydrodynamically and thermally developing flow between two parallel plates is considered and unknown transient boundary heat flux at the upper plate of a parallel plate channel is estimated using CGM.

The Inverse Reconstruction of the Heat Transfer Coefficient for the Free Surface Water Jet

This article presents an application of inverse algorithm for reconstruction of heat transfer coefficient (HTC) for a water jet impinging a flat surface. Such an approach, allows for decoupling complex fluid flow from heat conduction in a solid impinged by jet. The approach starts with parameterization of a functional form of unknown boundary temperature and heat flux occurring at the fluid–solid interface. Later, Newton's law of cooling is used to force temporal invariability of HTC. Unknown coefficients of HTC distribution are determined from a least square fit between measured and computed temperatures. Temperatures entering the objective function are recorded by an infrared camera at the surface opposite to impinged one.

Solution of Inverse Convection Heat Transfer Problem Using an Enhanced Particle Swarm Optimization Algorithm

The main objective of this paper is to solve the inverse convection heat transfer problems with particle swarm optimization method. An enhanced particle swarm optimization (EPSO) algorithm is proposed to overcome the shortcoming of earlier convergence of standard PSO algorithms. The performance of EPSO is tested by some benchmark functions; it is shown that EPSO has a strong antilocal trap capability especially for high dimensional multimodal optimization problems. At last, EPSO is used to identify the unknown boundary heat flux in a channel flow. According to the computational results of four test problems, it is clear that the proposed EPSO algorithm is able to estimate the unknown heat flux accurately even when the input data contain measurement error.

The heat transfer coefficient spatial distribution reconstruction by an inverse technique

This article presents a technique for obtaining the spatial distribution of the heat transfer coefficient (HTC) for an impingement heat exchange directly from transient temperature measurements. The HTC retrieval problem is formulated through the boundary temperature and the heat flux. That formulation makes it possible to utilize the superposition principle as the temperature field depends linearly on both the mentioned boundary conditions (BC). The unknown boundary temperature and heat flux are approximated by the set of trial functions chosen by the user. The key point is to utilize the definition of the convective BC in the objective function formulation. This fact allows us to take advantage of the fact that for the impingement heat exchange the HTC is constant in time. Such definition of the minimized function improves the stability of the solution in the presence of the measurement errors.

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.

Three-dimensional inverse transient heat transfer analysis of thick functionally graded plates

In this paper, a three-dimensional transient inverse heat conduction (IHC) procedure is presented to estimate the unknown boundary heat flux of thick functionally graded (FG) plates. For this purpose, the conjugate gradient method (CGM) in conjunction with adjoint problem is used. A recently developed three-dimensional efficient hybrid method is employed to solve variable-coefficient initial-boundary-value differential equations of direct problem as a part of the inverse solution. The accuracy of the inverse analysis is examined by simulating the exact and noisy data for problems with different types of boundary conditions and material properties. In addition to rectangular domain, skew plates are considered. The results obtained show good accuracy for the estimation of boundary heat fluxes.

Estimation of Unknown Boundary Heat Flux in Laminar Circular Pipe Flow Using Functional Optimization Approach: Effects of Reynolds Number

An inverse forced convection problem was studied in this paper. The unknown space-dependent heat flux at the outer boundary of a circular pipe was identified from the temperature measurements within the flow using the algorithm based on an improved conjugate gradient method, which is a combination of the modified inverse algorithm proposed by Ozisik et al. (Huang and Ozisik, 1992, “Inverse Problem of Determining Unknown Wall Heat Flux in Laminar Flow Through a Parallel Plate Duct,” Numer. Heat Transfer, Part A 21, pp. 2615–2618) and the general inverse algorithm based on the conjugate gradient method. The effects of the convection intensity, the number of thermocouples, the location of the thermocouples, and the measurement error on the performance of the modified inverse algorithm method and the improved inverse algorithm were studied thoroughly through three examples. It is shown that the improved inverse algorithm can greatly improve the solution accuracy in the entire computation domain. The accuracy and stability of both the modified inverse algorithm method and the improved inverse algorithm are strongly influenced by the Reynolds number and the shape of the unknown heat flux. Those functions, which contain more high-frequency components of Fourier series, are more sensitive to the increase in the Reynolds number.

Inverse Hyperbolic Conduction Problem in Estimating Two Unknown Surface Heat Fluxes Simultaneously

An inverse hyperbolic heat conduction problem is solved in the present study by an iterative regularization method, that is, the conjugate gradient method, to estimate simultaneously two unknown boundary heat fluxes based on the interior temperature measurements. The inverse solutions will be justified based on the numerical experiments in which two specific cases in determining the unknown boundary heat flux distributions are examined. Results show that the inverse solutions can always be obtained with any arbitrary initial guesses of the boundary heat fluxes and that the position of each sensor should be as close to each boundary as possible to obtain accurate estimations. Finally, it is concluded that accurate boundary heat fluxes can be estimated in the present study when large measurement errors are considered.

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.

An inverse hyperbolic heat conduction problem in estimating surface heat flux by the conjugate gradient method

In the present study an inverse hyperbolic heat conduction problem is solved by the conjugate gradient method (CGM) in estimating the unknown boundary heat flux based on the boundary temperature measurements. Results obtained in this inverse problem will be justified based on the numerical experiments where three different heat flux distributions are to be determined. Results show that the inverse solutions can always be obtained with any arbitrary initial guesses of the boundary heat flux. Moreover, the drawbacks of the previous study for this similar inverse problem, such as (1) the inverse solution has phase error and (2) the inverse solution is sensitive to measurement error, can be avoided in the present algorithm. Finally, it is concluded that accurate boundary heat flux can be estimated in this study.

An Application of DHW Algorithm for the Solution of Inverse Heat Conduction Problem

A damped heat wave (DHW) algorithm is applied for the temperature distribution calculation in a solution of a linear inverse heat conduction problem (IHCP). A nonlinear least squares algorithm is used for calculation of the unknown boundary heat flux history in a one-dimensional medium. The solution is based on the assumption that the temperature measurements are available, at least, at one point of the medium over the whole time domain. Sample calculations, for a comparison between exact heat sources and estimated ones, are made to confirm the validity of the proposed method. The close agreement between the exact and estimated values calculated for both exact and noisy data shows the potential of the proposed method for finding a relatively accurate heat source distribution in a one-dimensional homogeneous finite medium. The proposed method of solving inverse heat conduction problems is very simple and easy to implement.

Simultaneous inverse estimation for boundary heat and moisture fluxes of a double-layer annular cylinder with interface resistance

Based on the conjugate gradient method, this study presents a means of solving the inverse boundary value problem of coupled heat and moisture transport in a double-layer annular cylinder. While knowing the temperature and moisture history at the measuring positions, the unknown time-dependent boundary heat and moisture fluxes can be simultaneously determined. It is assumed that no prior information is available on the functional form of the unknown heat and moisture fluxes. The accuracy of this inverse heat and moisture transport problem is examined by using the simulated exact and inexact temperature and moisture measurements in the numerical experiments. Results show that excellent estimation on the time-dependent boundary fluxes can be obtained with any arbitrary initial guesses. In addition, the contact resistance of heat and moisture transfer at the interface of the two layers is considered in the analysis.

역해석을 이용한 가열로 내 소재의 1차원 열유속 추정에 관한 연구

This study deals with the use of the conjugate gradient method for the simultaneous estimation of two unknown boundary heat fluxes on the slab in reheating furnace. Temperature measurements by the experiment arc used in the inverse analysis. The heat flux estimations for three different cases of measurement locations in the slab are performed: non-skid, skid, and shift-skid zones. The estimated heat fluxes for three cases indicated the three regions having local peak values of heat fluxes. The estimated temperatures at measurement locations were in good agreements with the measured temperatures within 5% relative error.<br/>