Development of the ball-spine algorithm for the shape optimization of ducts containing nanofluid

ABSTRACT

In this paper a novel shape design method is introduced for the numerical solution of inverse heat convection problems (IHCPs) of nanofluids. The proposed method is a novel extension of the ball-spine algorithm (BSA) inverse method, which is recently adapted for inverse heat transfer problems (IHTPs). Here it is shown how, by a novel physical-sense remedy, the method is applicable to IHCPs as well. In this respect, after validation of the proposed method, two types of IHCPs are introduced and numerically solved. In both types of introduced inverse problems, the objective is shape optimization of a duct containing such steady-state incompressible laminar nanofluid flow that it satisfies a prescribed heat flux distribution along the walls of the geometry. The results show merits and robustness of the BSA application in capturing the target geometry corresponding to a given heat flux distribution in forced heat convection problems with a low computational cost.     

Neural Computation to Estimate Heat Flux in an Atmospheric Plasma Spray Process

Temperature is a key parameter in the thermal spray process and is a consequence of the heat flux experienced by the workpiece. This paper deals with the estimation of the heat flux transmitted to a workpiece from an atmospheric plasma spray torch during the preheating process often implemented in thermal spraying. An inverse heat conduction problem solution using a conjugate gradient method was considered to determine the heat flux starting from a known temperature distribution. Results from the later method were used to train an artificial neural network to discover correlations between selected processing parameters and heat flux.     

A semi-analytical boundary collocation solver for the inverse Cauchy problems in heat conduction under 3D FGMs with heat source

Abstract

In this article, the inverse Cauchy problems in heat conduction under 3D functionally graded materials (FGMs) with heat source are solved by using a semi-analytical boundary collocation solver. In the present semi-analytical solver, the combined boundary particle method and regularization technique is employed to deal with ill-pose inverse Cauchy problems. The domain mapping method and variable transformation are introduced to derive the high-order general solutions satisfying the heat conduction equation of 3D FGMs. Thanks to these derived high-order general solutions, the proposed scheme can only require the boundary discretization to recover the solutions of the heat conduction equations with a heat source. The regularization technique is used to eliminate the effect of the noisy measurement data on the accessible boundary surface of 3D FGMs. The efficiency of the proposed solver for inverse Cauchy problems is verified under several typical benchmark examples related to 3D FGM with specific spatial variations (quadratic, exponential and trigonometric functions).     

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 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.     

Implementation of an analytical two-dimensional inverse heat conduction technique to practical problems

Two improvements to practical implementation of a solution to the two-dimensional inverse heat conduction problem are presented. The first concept is useful for experimental data with strong or irregular fluctuations in time. The second procedure improves the spatial resolution for problems where the source of the surface heat flux distribution is moving along the surface. The method is tested against analytical solutions and data from quench cooling experiments. Both procedures are found to enhance the quality of the inverse solution results.     

A rapid method for solving Laplace equation in a doubly connected region

A new methodology is developed for solving 2-D Laplace equation (heat conduction problem) within a doubly connected region with prescribed temperature and heat flux distribution along outer or inner prescribed boundary. Both direct problem (with specified geometry of the other boundary) and inverse problem (with prescribed temperature or heat flux distribution along the other boundary) can be solved by this method. The computation work needed is very simple and can be programed with a very small computer such as Sharp PC-1500. This method can be extended to a 3-D one also.     

A fitting algorithm for solving inverse problems of heat conduction

The paper presents an algorithm for solving inverse problems of heat transfer. The method is based on iterative solving of direct and adjoint model equations with the aim to minimize a fitting functional. An optimal choice of the step length along the descent direction is proposed. The algorithm has been used for the treatment of a steady-state problem of heat transfer in a region with holes. The temperature and the heat flux density were known on the outer boundary of the region, whereas these values on the boundaries of the holes are to be determined. The idea of the algorithm consist in solving of Neumann problems where the heat flux on the outer boundary is prescribed, whereas the heat flux on the inner boundary is guessed. The guess is being improved iteratively to minimize the mean quadratic deviation of the solution on the outer boundary from the given distribution.The results obtained show that the algorithm provides smooth, non-oscillating, and stable solutions to inverse problems of heat transfer, that is, it avoids disadvantages inherent in other computational methods for such problems. The ill-conditioning of inverse problems in the Hadamard sense is exhibited in that a very quick convergence of the fitting functional to its minimum does not imply a comparable rate of convergence of the recovered temperature on the inner boundary to the true distribution.The considered method can easily be extended to nonlinear problems.Numerical calculation has been carried out with the FE program Felics developed at the Chair of Mathematical Modelling of the Technical University of Munich.     

Estimation of front surface temperature and heat flux of a locally heated plate from distributed sensor data on the back surface

We present a new method of solving the three-dimensional inverse heat conduction (3D IHC) problem with the special geometry of a thin sheet. The 3D heat equation is first simplified to a 1D equation through modal expansions. Through a Laplace transform, algebraic relationships are obtained that express the front surface temperature and heat flux in terms of those same thermal quantities on the back surface. We expand the transfer functions as infinite products of simple polynomials using the Hadamard Factorization Theorem. The straightforward inverse Laplace transforms of these simple polynomials lead to relationships for each mode in the time domain. The time domain operations are implemented through iterative procedures to calculate the front surface quantities from the data on the back surface. The iterative procedures require numerical differentiation of noisy sensor data, which is accomplished by the Savitzky–Golay method. To handle the case when part of the back surface is not accessible to sensors, we used the least squares fit to obtain the modal temperature from the sensor data. The results from the proposed method are compared with an analytical solution and with the numerical solution of a 3D heat conduction problem with a constant net heat flux distribution on the front surface.     

Study on heat conduction of functionally graded plate with the variable gradient parameters under the H(t) heat source

Abstract

A hybrid numerical method for heat conduction of functionally graded plate with the variable gradient parameters under the H(t) heat source was studied. A weighted residual equation of heat conduction was considered under thermal boundary conditions. In order to calculate temperature distribution of functionally graded plate with variable gradient parameters, the Fourier transform and inverse Fourier transform were applied and the temperature field was obtained under the H(t) heat source. Results show that the influences of the gradient parameters on temperature distribution are dramatic. But with the increase of gradient parameters, the influences of parameters on the temperature distribution are gradually reduced. When the gradient parameters reach a certain critical value, the temperature does not change anymore. By comparing the temperature distribution of the upper and lower surfaces, it is seen that the temperature presents a gentle downward trend with the increase of the heat source distance, while the temperature does not change with the time in farther distance from heat source. Also, the results show that the influence of the heat source has only partial and limited influence on the temperature, which is in accordance with St. Venant’s Principle. The law of the temperature distribution of the lower surface varies with the gradient parameters, which is also discussed, an optimal gradient parameter with the thermal insulation effect of the functionally graded plate is obtained.     

Monitoring of transient 3D temperature distribution and thermal stress in pressure elements based on the wall temperature measurement

Abstract

Two methods for monitoring the thermal stresses in pressure components of thermal power plants are presented. In the first method, the transient temperature distribution in the pressure component is determined by measuring the transient wall temperature at several points located on the outer insulated surface of the component. The transient temperature distribution in the pressure component, including the temperature of the inner surface is determined from the solution of the inverse heat conduction problem (IHCP). In the first method, there is no need to know the temperature of the fluid and the heat transfer coefficient. In the second method, thermal stresses in a pressure component with a complicated shape are computed using the finite element method (FEM) based on experimentally estimated fluid temperature and known heat transfer coefficient. A new thermometer with good dynamic properties has been developed and applied in practice, providing a much more accurate measurement of the temperature of the flowing fluid in comparison with standard thermometers. The heat transfer coefficient on the inner surface of a pressure element can be determined from the empirical relationships available in the literature. A numerical-experimental method of determination of the transient heat transfer coefficient based on the solution of the 3D-inverse heat conduction problem has also been proposed. The heat transfer coefficient on the internal surface of a pressure element is determined based on an experimentally determined local transient temperature distribution on the external surface of the element or the basis of wall temperature measurement at six points located near the internal surface if fluid temperature changes are fast. Examples of determining thermal and pressure stresses in the thick-walled horizontal superheater header and the horizontal header of the steam cooler in a power boiler with the use of real measurement data are presented.     

An analytical solution for two-dimensional inverse heat conduction problems using Laplace transform

An analytical method has been developed for two-dimensional inverse heat conduction problems by using the Laplace transform technique. The inverse solutions are obtained under two simple boundary conditions in a finite rectangular body, with one and two unknowns, respectively. The method first approximates the temperature changes measured in the body with a half polynomial power series of time and Fourier series of eigenfunction. The expressions for the surface temperature and heat flux are explicitly obtained in a form of power series of time and Fourier series. The verifications for two representative testing cases have shown that the predicted surface temperature distribution is in good agreement with the prescribed surface condition, as well as the surface heat flux.     

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.     

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.     

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.     

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     

AN INPUT ESTIMATION APPROACH TO ON-LINE TWO-DIMENSIONAL INVERSE HEAT CONDUCTION PROBLEMS

Abstract

An on-line methodology to solve two-dimensional inverse heat conduction problems (IHCP) is presented. A new input estimation approach based on the Kalman filtering technique is developed to estimate the two separate unknown heat flux inputs on the two boundaries in real time. A recursive relation between the observed value of the residual sequence with unknown heat flux and the theoretical residual sequence of the Kalman filter that assumes known heat flux is formulated. A real-time least-squares algorithm is derived that uses the residual innovation sequence to compute the magnitude of heat flux. This recursive approach facilitates practical implementation, and its capabilities are demonstrated in several typical cases with discontinuous and time-varying heat flux inputs.     

膜式壁上稳态热流测量技术的误差分析

作者以求解导热反问题为基础测定固定壁面热流密度的新方法,进一步用数值模拟的方法从理论上讨论了导热系统本身的各种因素对此测试方法误差的影响,并对方法的实际应用提出了指导性意见。     

Recovering A Heat Source and Initial Value by a Lie-Group Differential Algebraic Equations Method

We consider an inverse problem of a nonlinear heat conduction equation for recovering unknown space-dependent heat source and initial condition under Cauchy-type boundary conditions, which is known as a sideways heat equation. With the aid of two extra measurements of temperature and heat flux which are being polluted by noisy disturbances, we can develop a Lie-group differential algebraic equations (LGDAE) method to solve the resulting differential algebraic equations, and to quickly recover the unknown heat source and initial condition simultaneously. Also, we provide a simple LGDAE method, without needing extra measurement of heat flux, to recover the above two unknown functions. The estimated results are quite promising and robust enough against large random noise.     

Local least–squares element differential method for solving heat conduction problems in composite structures

Abstract

In this article, a completely new numerical method called the Local Least-Squares Element Differential Method (LSEDM), is proposed for solving general engineering problems governed by second order partial differential equations. The method is a type of strong-form finite element method. In this method, a set of differential formulations of the isoparametric elements with respect to global coordinates are employed to collocate the governing differential equations and Neumann boundary conditions of the considered problem to generate the system of equations for internal nodes and boundary nodes of the collocation element. For each outer boundary or element interface, one equation is generated using the Neumann boundary condition and thus a number of equations can be generated for each node associated with a number of element interfaces. The least-squares technique is used to cast these interface equations into one equation by optimizing the local physical variable at the least-squares formulation. Thus, the solution system has as many equations as the total number of nodes of the present heat conduction problem. The proposed LSEDM can ultimately guarantee the conservativeness of the heat flux across element surfaces and can effectively improve the solution stability of the element differential method in solving problems with hugely different material properties, which is a challenging issue in meshfree methods. Numerical examples on two- and three-dimensional heat conduction problems are given to demonstrate the stability and efficiency of the proposed method.