Inverse heat conduction estimation of inner wall temperature fluctuations under turbulent penetration

Turbulent penetration can occur when hot and cold fluids mix in a horizontal T-junction pipe at nuclear plants. Caused by the unstable turbulent penetration, temperature fluctuations with large amplitude and high frequency can lead to time-varying wall thermal stress and even thermal fatigue on the inner wall. Numerous cases, however, exist where inner wall temperatures cannot be measured and only outer wall temperature measurements are feasible. Therefore, it is one of the popular research areas in nuclear science and engineering to estimate temperature fluctuations on the inner wall from measurements of outer wall temperatures without damaging the structure of the pipe. In this study, both the one-dimensional(1D) and the two-dimensional(2D) inverse heat conduction problem(IHCP) were solved to estimate the temperature fluctuations on the inner wall. First, numerical models of both the 1D and the 2D direct heat conduction problem(DHCP) were structured in MATLAB, based on the finite difference method with an implicit scheme. Second, both the 1D IHCP and the 2D IHCP were solved by the steepest descent method(SDM), and the DHCP results of temperatures on the outer wall were used to estimate the temperature fluctuations on the inner wall. Third, we compared the temperature fluctuations on the inner wall estimated by the 1D IHCP with those estimated by the 2D IHCP in four cases:(1) when the maximum disturbance of temperature of fluid inside the pipe was 3℃,(2) when the maximum disturbance of temperature of fluid inside the pipe was 30℃,(3) when the maximum disturbance of temperature of fluid inside the pipe was 160℃, and(4) when the fluid temperatures inside the pipe were random from 50℃ to 210℃.     

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.     

基于共轭梯度法对T型管内壁瞬态温度的识别

构建了一种基于已知T型管道外壁面瞬时温度,反演管道内壁面瞬态温度的导热反问题数学模型和求解方法.利用有限单元法对管道模型进行离散,并利用共轭梯度法求解非稳态导热反问题.利用正问题得到的外壁面瞬态温度的数值结果作为反问题的输入条件,反演得到内壁面瞬态温度,此反演结果与作为边界条件的内壁面瞬态温度值进行了对比分析,对比结果...     

A method of fundamental solutions for the radially symmetric inverse heat conduction problem

We propose and investigate an application of the method of fundamental solutions (MFS) to the radially symmetric inverse heat conduction problem (IHCP). In the radially symmetric IHCP data on an inner fixed boundary is determined from Cauchy data given on an outer boundary. This is an inverse and ill-posed problem, and we employ and generalize the MFS regularization approach of Johansson et al. (2008) for the time-dependent heat equation to obtain a stable and accurate numerical approximation with small computational cost.     

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.     

The Estimation of Fluid Temperatures Through an Inverse Heat Conduction Technique

Transient, internal temperatures within an instrumented probe are considered as part of an inverse heat conduction problem (IHCP) to compute the temperature of the surrounding fluid. A linear scheme is used where the exchange coefficients are treated as known parameters.Input data to the IHCP have been generated numerically. When these are uncorrupted, the inverse algorithm works well without stabilization. However, in practice the algorithm must be stabilized, as it is shown that noise is amplified substantially. It becomes necessary both to parameterize spatial variations in the fluid temperature and to utilize a functional specification method to address the noncausal solution.     

A NONLINEAR INDIRECT MEASUREMENT PROBLEM FOR A MULTITHERMOCOUPLE PROBE IMMERSED IN A LIQUID

The problem described herein concerns the processing of the time-dependent, internal temperatures within a multithermocouple probe. These are used to compute the temperature of the surrounding fluid, as part of an inverse heat conduction problem (IHCP). The novel achievement in this work is that the exchange coefficients do not have to be supplied a priori, but instead are an additional solution output. Consequently the IHCP is nonlinear and requires significant stabilization. Four methods are applied successively, until a satisfactory solution is found: the parameterization of spatial variations in fluid temperatures and exchange coefficients; a functional specification method (using future time data) to address the noncausal nature of the solution; a lower bound on the exchange coefficient; and a maximum number of iterations at each time step (in accordance with the discrepancy principle).     

Measurement of Transient Fluid Temperature in a Pipeline

The aim of this paper is to develop a method of determining the heat transfer coefficients on the inner surface of the pipeline and outer surface of the thermometer used to measure the temperature of a fluid flowing under high pressure. The method is based on the solutions to the inverse heat conduction problems for the thermometer and the pipeline wall. The heat transfer coefficients are determined based on the measurement of the temperature of a cylindrical metal thermometer and the temperature of the wall of a cylindrical pipeline. The temperature sensor is located in the pipeline wall close to the inner surface. The correlations for the Nusselt numbers used to determine heat transfer coefficients on the outer surface of the thermometer and the inner surface of the pipeline contain unknown coefficients which are found using the least squares method. The unknown coefficients are selected so that the sum of the squares of differences between the fluid temperature determined based on the measurement of the temperature of the pipeline wall and the fluid temperature obtained from measurements inside the thermometer, calculated for several dozen set time points, is as small as possible.     

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.     

Inverse estimation of spatially and temporally varying heating boundary conditions of a two-dimensional object

In many dynamic heat transfer situations, the temperature at the heated boundary is not directly measurable and can be obtained by solving an inverse heat conduction problem (IHCP) based on measured temperature or/and heat flux at the accessible boundary. In this study, IHCP in a two-dimensional rectangular object is solved by using the conjugate gradient method (CGM) with temperature and heat flux measured at the boundary opposite to the heated boundary. The inverse problem is formulated in such a way that the heat flux at heated boundary is chosen as the unknown function to be recovered, and the temperature at the heated boundary is computed as a byproduct of the IHCP solution. The measurement data, i.e., the temperature and heat flux at the opposite boundary, are obtained by numerically solving a direct problem where the heated boundary of the object is subjected to spatially and temporally varying heat flux. The robustness of the formulated IHCP algorithm is tested for different profiles of heat fluxes along with different random errors of the measured heat flux at the opposite boundary. The effects of the uncertainties of the thermophysical properties and back-surface temperature measurement on inverse solutions are also examined.     

Laminar forced convection in circular and flat ducts with wall axial conduction and external convection

An analytical solution is obtained for laminar forced convection in circular and flat ducts with the presence of axial duct wall conduction and external convection at the outer surface of the duct wall. The eigenvalues for the problem are determined using the solution for the constant temperature boundary condition. The heat transfer results depend on four nondimensional numbers. The wall and fluid temperatures depend strongly on the wall conductance parameter while the heat flux enhancement due to wall conduction is large at short distances from the duct inlet.     

反演高温腔体内壁面温度波动的辐射边界导热反问题

在工业生产中有很多情况需要获得高温腔体内壁温度波动,但在内壁面安装测温装置进行直接测量非常困难,一般通过测量外壁温度再进行反演计算间接获得。而已有反演计算方法未考虑高温壁面与周围环境之间的辐射传热,给反演计算结果带来一定误差,为此建立了考虑辐射边界条件的反演高温腔体内壁面温度波动的导热反问题数学模型,并构造了两组数值试验对数学模型的效果进行检验。计算结果表明,建立的数学模型能够很好的由高温腔体外壁面温度反演得到内壁面温度波动情况。     

Thermocouple Data in the Inverse Heat Conduction Problem

The presence of thermocouples inside a heat-conducting body will distort the temperature field in the body and may lead to significant bias in the temperature measurement. If temperature histories obtained from thermocouples are used in the inverse heat conduction problem (IHCP), errors are propagated into the IHCP results. The bias in the thermocouple measurements can be removed through use of appropriate detailed thermocouple models to account for the dynamics of the sensor measurement. The results of these models can be used to generate correction kernels to eliminate bias in the thermocouple reading, or can be applied as sensitivity coefficients in the IHCP directly. Three-dimensional and axisymmetric models are compared and contrasted and a simple sensitivity study is conducted to evaluate the significance of thermal property selection on the temperature correction and subsequent heat flux estimation. In this paper, a high-fidelity thermocouple model is used to account for thermocouple bias in an experiment to measure heat fluxes from solidifying aluminum to a sand mold. Correction kernels are obtained that are used to demonstrate the magnitude of the temperature measurement bias created by the thermocouples. The corrected temperatures are used in the IHCP to compute the surface heat flux. A comparison to IHCP results using uncorrected temperatures shows the impact of the bias correction on the computed heat fluxes.     

Simple method for monitoring transient thermal stresses in pipelines

This paper presents a seminumerical method for solving inverse heat conduction problems (IHCP) encountered in the monitoring of thermal stresses in pressurized thick-walled elements of steam boilers. The objective is to give a simple and quick method of determining transient temperature histories in thick-walled components based on temperature measurements on the outer thermally insulated surface. The method is suitable for solving one-dimensional problems. However, it can be extended to multidimensional temperature fields. The IHCP will be solved using the control volume approach. The accuracy of the method is demonstrated by comparing computational and experimental results. Gram orthogonal polynomials are used to smooth the measured time-dependent temperature and for evaluating time derivatives of noisy data with high accuracy. Due to the simplicity of the final formulations, the developed method is very useful for estimating the thermal stresses and controlling the fatigue damage of boiler components.     

Geometry estimation for the inner surface in a furnace wall made of functionally graded materials

The aim of this study is to solve an inverse geometry heat conduction problem (shape identification problem) to estimate the unknown geometry of the inner surface in a furnace wall which is made of functionally graded materials (FGMs). The inner surface geometry is estimated from the temperatures of measured points within the furnace wall. The inverse algorithm used in the study is based on the conjugate gradient method (CGM) and the discrepancy principle. The effect of measurement errors and measurement locations on the estimation accuracy is also investigated. Two different examples are discussed. Results show that the unknown geometry of the inner wall surface can be predicted precisely by using the present approach.     

Two-dimensional non-linear inverse heat conduction problem based on the singular value decomposition

In this paper an efficient sequential method is developed in order to estimate the unknown boundary condition on the surface of a body from transient temperature measurements inside the solid. This numerical approach for solving an inverse heat conduction problem (IHCP) takes into account two-dimensional problems, planar or axisymmetric cylindrical, composite materials with irregular boundaries and temperature-dependent thermal properties. The unknown surface condition is assumed to have abrupt changes at unknown times. The regularization procedure used for the solution of the IHCP is based on the singular value decomposition technique. An overall estimate of error is defined in order to find the optimal estimation in the 2D IHCP (linear and non-linear). The stability and accuracy of the scheme presented is evaluated by comparison with the Function Specification Method. This comparative study has been carried out using numerically simulated data, and the parameters considered include shape of input, noise level of measurement, size of time step and temperature-dependent thermal properties. A good agreement was found between both methods. Beside this, the slight differences on estimations and number of future temperatures are discussed in this paper.     

A SERIAL SOLUTION FOR THE 2-D INVERSE HEAT CONDUCTION PROBLEM FOR ESTIMATING MULTIPLE HEAT FLUX COMPONENTS

A serial algorithm for the inverse heat conduction problem (IHCP) has been developed to estimate the individual flux components, one by one, at the unknown boundary, based on the function specification method. The sensitivity coefficient defined in this algorithm brings out the influence of the heat flux components independent of each other. The objective function minimizes the difference in the measured temperature and the contribution of the individual flux component to the thermal field at the sensor location. The serial algorithm developed here could be used with data from both overspecified and underspecified sensors with respect to the number of flux components. The method was tested for delineating independent heat fluxes at the boundary of a two-dimensional solid for both space- and time-varying heat fluxes. Simulated thermal histories obtained from direct solution were used as inputs for the inverse problem for characterizing the new algorithm.

Three types of analyses were done on the results of the IHCP, focused on (1) the convergence of error in estimated temperatures at the different sensor locations, (2) overall error in estimated temperatures for the whole domain, and (3) the total heat energy transferred across the boundary. It is shown that the optimum configuration of independent unknown fluxes is given by the one with minimum energy estimates across the boundary, for both cases.     

Reconstruction of local heat fluxes in pool boiling experiments along the entire boiling curve from high resolution transient temperature measurements

In this paper, we consider a transient inverse heat conduction problem (IHCP) defined on an irregular three-dimensional (3D) domain in pool boiling experiments. Heat input to a circular copper heater of 35 mm diameter and 7 mm thickness is provided by a resistance heating foil pressed to the bottom of the heater. The heat flux at the inaccessible boiling side is estimated from a number of temperature readings in the heater volume. These temperatures are measured by some high-resolution microthermocouples, which are mounted 3.6 μm below the surface in the test heater. The IHCP is formulated as a mathematical optimization problem and solved by the conjugate gradient (CG) method. The arising partial differential equations (PDEs) are solved using the software package DROPS. A simulation case study is used to validate the performance of the solution approach. Finally, we apply the solution approach to the IHCP in pool boiling experiments. The procedure enables the reconstruction of local instantaneous heat flux distribution on the heater surface at different locations along the boiling curve.     

Heat Transfer Analysis for a Concentric Tube Heat Exchanger Including the Wall Axial Conduction

The effect of wall axial conduction on the heat transfer in a concentric tube heat exchanger is examined for the inner flow laminar flow regime. The procedure used for the current analysis combines the analytical solution for the inner fluid with a numerical approximation for the wall conduction and has the capability of handling the temperature variation for the outer fluid. Both parallel and counterflow cases are evaluated for the analysis, and results are presented in terms of the axial variations of fluids and wall temperatures. Effects of the heat capacity rate ratio of the fluids on the temperature variations and on the mean heat flux are also pointed out. The effect of the exchanger length is included for the analysis. It is concluded that the total heat transfer between the fluids is greatly influenced by the wall axial conduction for the counterflow arrangement and is not ignorable when the heat capacity rate ratio of fluids are smaller than unity.     

Estimation for inner surface geometry of furnace wall using inverse process combined with grey prediction model

In this work the inner surface geometry of a cylindrical furnace wall is estimated using inverse process method combined with grey prediction model. In estimating process a virtual area extended from the inner surface of furnace wall is used for analysis. The heat conduction equation and the boundary condition are first discretized by finite difference method to form a linear matrix equation; the inverse model is then optimized by linear least-squares error method and the temperatures of virtual boundary are obtained from a few of measured temperatures in furnace wall using the linear inverse model; and finally the temperature distribution of system is got by direct process and the inner surface geometry of furnace wall can be estimated accordingly. The result shows that using inverse process combined with grey prediction model the geometry can be exactly estimated from relatively small number of measured temperatures. Moreover, the effects of measurement error, location, and number of measured points on the estimation for inner surface geometry of furnace wall are discussed in detail.