A modified genetic algorithm for solving the inverse heat transfer problem of estimating plan heat source

This work considers a new approach for solving the inverse heat conduction problem of estimating unknown plan heat source. It is shown that the physical heat transfer problem can be formulated as an optimization problem with differential equation constraints. A modified genetic algorithm is developed for solving the resulting optimization problem. The proposed algorithm provides a global optimum instead of a local optimum of the inverse heat transfer problem with highly-improved convergence performance. Some numerical results are presented to demonstrate the accuracy and efficiency of the proposed method.     

Convective cooling optimization of a blade for a supercritical steam turbine

This paper discusses the problem of blade cooling system optimization connected with conjugate heat transfer (CHT) analysis for reliable thermal field prediction within a steam cooled component. Since the full CHT solution, which involves the main flow, blade material and the coolant flow domains is computationally expensive from the point of view of optimization process, it was decided to reduce the problem by fixing the boundary conditions at the blade surface and solving the task for the interior only (both solid material and coolant). Such assumption, on one hand, makes the problem computationally feasible, and on the other, provides more reliable thermal field prediction than it used to be with the empirical relationships.The analysis involves shape optimization of internal cooling passages within an airfoil. The cooling passages are modeled with a set of four Bezier splines joined together to compose a closed contour. Each passage is fed with cooling steam of constant parameters at the inlet. In the present study the airfoil profile is taken as aerodynamically optimal. The search problem is solved with evolutionary algorithm and the final configuration is to be found among the Pareto optimal cooling candidates.     

舰船燃气轮机高压涡轮动叶多维热耦合设计方法

针对舰船燃气轮机复杂高效冷却叶片设计,基于压力修正算法建立冷却叶片一维管网设计方法;通过快速求解可压缩边界层微分方程获得叶片外换热边界,基于参数化的叶片网格生成方法,采用全隐式有限体积的固体导热求解方法,构建了冷却叶片的耦合传热模型,开发了耦合传热计算程序。对某高压涡轮动叶进行多维热耦合设计,确定冷却流路及冷气分布,通过三维气热耦合计算验证了设计方案的可行性,通过对比分析验证了多维热耦合设计方法对主要流通单元的流量、压力误差小于5%,具备较高的工程应用价值。     

Recovering both the space-dependent heat source and the initial temperature by using a fast convergent iterative method

In this paper, we solve two types of inverse heat source problems: one recovers an unknown space-dependent heat source without using initial value, and another recovers both the unknown space-dependent heat source and the initial value. Upon inserting the adjoint Trefftz test functions into Green’s second identity, we can retrieve the unknown space-dependent heat source by an expansion method whose expansion coefficients are derived in closed form. We assess the stability of the closed-form expansion coefficients method by using the condition numbers of coefficients matrices. Then, numerical examples are performed, which demonstrates that the closed-form expansion coefficient method is effective and stable even when it imposes a large noise on the final time data. Next, we develop a coupled iterative scheme to recover the unknown heat source and initial value simultaneously, under two over specified temperature data at two different times. A simple regularization technique is derived to overcome the highly ill-posed behavior of the second inverse problem, of which the convergence rate and stability are examined. This results in quite accurate numerical results against large noise.     

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.     

热负荷阶跃变化条件下热回流式换热系统缺陷机理及优化方法研究

针对国内某北方核电厂的设备冷却水系统热回流式换热器,分析了不同热负荷下热回流式换热系统的稳态特性及负荷阶跃变化下热回流式换热系统缺陷机理,提出了热回流式换热器系统优化方法。研究表明:在不同热负荷下热回流式换热器系统切换的关键是与不同热负荷对应的具有特定温度的伴流的形成;热负荷阶跃变化下状态转换瞬态过程中存在系统缺陷,其根本原因在于单纯采取调节热回流率的方法,具有较大的时间滞后性;调节换热器冷介质侧的流体流量,改变换热器传热系数,强化了对状态改变的快速响应;采取热回流叠加换热器旁流方法,可以解决原有系统状态转换瞬态过程中存在的缺陷。     

One-step GPS for the estimation of temperature-dependent thermal conductivity

In this paper we are concerned with the estimation of temperature-dependent thermal conductivity of a one-dimensional inverse heat conduction problem. First, we construct a one-step group-preserving scheme (GPS) for the semi-discretization of quasilinear heat conduction equation, and then derive a quasilinear algebraic equation to determine the unknown thermal conductivity under a given initial temperature and a measured temperature perturbed by noise at time T. The new method does not require any prior information on the functional form of thermal conductivity. Several examples are examined to show that the new approach has high accuracy and efficiency, and the number of iterations spent in solving the quasilinear algebraic equation is smaller than five even in a large temperature range.     

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.     

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.     

内部通道肋片结构对Mark Ⅱ叶片冷却性能影响的降维模型研究

一维冷却通道气热耦合计算是分层涡轮叶片冷却结构设计的重要方法。发展了以管道网络算法为核心的内部冷却特性计算程序,并与三维传热计算进行了耦合。通过与MarkⅡ叶片特定实验工况下的结果进行对比,验证了方法的有效性。此外,进一步将带肋结构流道传热特性相关经验公式集总在一维气热耦合算法中,分析了带肋通道改型的MarkⅡ叶片冷却性能。结果显示,带肋结构相比光滑流道能显著提升换热性能,在中径截面处较原方案温度下降15～30 K。     

Reconstruction of a space and time dependent heat source from finite measurement data

This paper deals with an inverse problem of determining a heat source function in heat conduction equations when the solution is known in a discrete point set. Being different from other ordinary inverse source problems which are often dependent on only one variable, the unknown coefficient in this paper not only depends on the space variable x, but also depends on the time t. On the basis of the optimal control framework, the inverse problem is transformed into an optimization problem. The existence and necessary condition of the minimizer for the cost functional are established. The convergence of the minimizer as the mesh parameters tend to zero is also proved. The conjugate gradient method is applied to the inverse problem and some typical numerical experiments are performed in the paper. The numerical results show that the proposed method is stable and the unknown heat source is recovered very well.     

Application of the method of fundamental solutions and radial basis functions for inverse heat source problem in case of steady-state

The paper deals with the inverse determination of heat sources in steady 2-D heat conduction problem. The problem is described by Poisson equation in which the function of the right hand side is unknown. The identification of the strength of a heat source is given by using the boundary condition and a known value of temperature in chosen points placed inside the domain. For the solution of the inverse problem of identification of the heat source the method of fundamental solution with radial basis functions is proposed. The accurate results have been obtained for five test problems where the analytical solutions were available.     

基于流热耦合的燃气轮机透平叶顶冷却设计

燃气轮机透平叶顶区域存在复杂的流动和换热问题,承受很高的热负荷。为了降低透平动叶叶顶温度,在透平叶顶现有结构的基础上提出气膜冷却和气膜+内冷通道冷却两种叶顶冷却方案,并通过流热耦合计算分析冷却升级前后叶顶区域的换热和流动特性。研究发现：叶顶气膜冷却方案可有效降低叶顶温度,特别是叶顶前缘至中弦区域;而气膜＋内冷通道冷却方案基于外部气膜冷却,结合内部冷却通道设计,可进一步降低叶顶尾缘的温度;与原型叶片相比,气膜+内部冷气通道的复合冷却设计可以使叶顶尾缘最高温度降低24 K。     

Inverse boundary design of square enclosures with natural convection

An optimization technique is applied to design of heat transfer systems in which the natural convection is important. The inverse methodology is employed to estimate the unknown strengths of heaters on the heater surface of a square cavity with free convection from the knowledge of the desired temperature and heat flux distributions over a given design surface. The direct and the sensitivity problems are solved by finite volume method. 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 convection heat transfer problems is evaluated by comparing the results with a benchmark problem and a numerical experiment.     

SOLUTION TO INVERSE HEAT CONDUCTION PROBLEM IN NANOSCALE USING SEQUENTIAL METHOD

An inverse heat conduction problem for nanoscale structure is studied. The conduction phenomenon is modeled using the Boltzmann transport equation. Phonon-mediated heat conduction in one dimension is considered. One boundary is exposed to an unknown temperature and the other boundary, where temperature observation takes place, is subject to a known boundary condition. A sequential scheme with constant function specification is employed for inverse estimation of the unknown temperature. Sample results are presented and discussed.     

Iterative algorithms for solving inverse problems of heat conduction in multiply connected domains

The presented paper displays a method of solving the inverse problems of heat transfer in multi-connected regions, consisting in iterative solving of convergent series of the direct problems. For known temperature and flux values at the outer boundary of the region the temperature and flux values at the inner boundaries are sought (the cauchy problem for the Laplace equation). In case of such a formulation of the problem, the solution does not always exist, one of the conditions is met in the mean-square sense, providing the optimization criterion. The idea of the process consists in solving the direct problem in which the boundary condition is subject to iterative changes so as to attain minimum of the optimization criterion (the square functional). Two algorithms have been formulated. In the first of them the heat flux at the inner boundaries of the region, while in the other the temperature were subject to changes. Convergence of both the algorithms have been compared.The numerical calculation has been made for selected examples, for which an analytical solution is known. The effect of random disturbance of the boundary conditions on the solution obtained with iterative algorithms has been checked. Moreover, a function was defined, serving as convergence measure of the solution of the inverse problem solved with the algorithms proposed in the paper. The properties of the function give evidence that it tends to the value exceeding unity.     

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.     

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.     

A Universal Performance Chart for CPU Cooling Devices

This article deals with performance evaluation of the discrete ordinates method in terms of its capacity to provide accurate results in solving radiation mode problems with different complexities. The problem formulation is done keeping in view the heat transfer process in an evaporator used in a coal-based thermal power station. The hot gas mass in the evaporator is assumed to be a heat source with two different shapes, spherical and conical. The evaporator walls that are covered with a water cooling jacket are modeled with a convective boundary condition. The gas is assumed to be gray and absorbing–emitting. The solution of the radiative transport equation is carried out using the discrete ordinates method. Parametric studies are performed for a wide range of aspect ratio, extinction coefficient, and convective heat transfer coefficient. The code is validated by comparing the result of discrete ordinates method with the exact and the discrete transfer method for nonradiative as well as radiative equilibrium conditions.     

Recovering a general space-time-dependent heat source by the coupled boundary integral equation method

As to recover a time-dependent heat source under an extra temperature measured at an interior point, we can reformulate it to be a three-point boundary value problem. We can develop a coupled boundary integral equation method, wherein by selecting two sets of adjoint test eigenfunctions in two sub domains and using polynomials as the trial functions of unknown heat source, the time-dependent heat source is recovered very well and quickly. Four numerical examples, including a discontinuous one, demonstrate the efficiency for the ill-posed inverse heat source problem in a large time duration and under a large noise up to 10–30%. Then, selecting three sets of adjoint test eigenfunctions in three domains: problem domain and two sub domains, and using the Pascal polynomials as trial functions, the unknown space-time-dependent heat source is recovered very fast and accurately from the solution of three coupled boundary integral equations.