3D Inverse Heat Conduction Problem Research Articles

Experimental study of the heat transfer of single-jet impingement cooling onto a large heated plate near industrial conditions

Jet impingement cooling has been intensively studied in the past by many authors because of its important application in the metallurgical industry; however, most of the experiments in the literature are at laboratory scale and, in some cases, not near industrial conditions. In this study, we performed cooling experiments near industrial conditions in a new experimental apparatus with a large nickel plate as test sample, which was heated until 850 ∘C before being cooled by a single circular water jet. Five experimental results are presented with different jet Reynolds numbers, from 9,800 to 120,000, obtained by varying the water flow rate and the nozzle diameter. The presented results are: temperature evolutions during cooling, dissipated heat flux (estimated by solving a 2D inverse heat conduction problem), transverse heat flux at the heat-exchanging surface, and the characteristics (position and velocity) of the rewetting and maximum cooling rate fronts. The increase in the jet Reynolds number increased slightly the heat flux at the stagnation zone, but increased it substantially for positions farther from the impact location. The transverse heat flux increased with the passage of the rewetting front and then decreased, and its magnitude was practically the same regardless of the jet Reynolds number. The change in the nozzle diameter did not affect significantly the heat transfer nor the rewetting front growth, although the heat dissipation was slightly higher with the smaller nozzle, possibly because of the higher jet impact velocity. Finally, we compared the present results with some found in the literature in similar conditions, showing that laboratory experiments are valuable to provide detailed information, especially near the stagnation zone, but large-scale experiments allow obtaining macroscale data of the cooling process.

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

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.

Thermal stress monitoring in thick walled pressure components of steam boilers

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 can be used for providing a much more accurate temperature of the flowing fluid in comparison with standard thermometers. A numerical-experimental method of determination of the transient heat transfer coefficient based on the solution of the 3D-inverse heat conduction problem was proposed. The heat transfer coefficient on the internal surface of a pressure element can be estimated based on an experimentally determined local transient temperature distribution on the external surface of the element or on 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 steam header with the use of real measurement data are presented.

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°C, (2) when the maximum disturbance of temperature of fluid inside the pipe was 30°C, (3) when the maximum disturbance of temperature of fluid inside the pipe was 160°C, and (4) when the fluid temperatures inside the pipe were random from 50°C to 210°C.

A new method for reducing local heat transfer data in multi-microchannel evaporators

Measuring the strong local variation in heat transfer coefficients in multi-microchannel evaporators is related to the inverse heat conduction problem (IHCP). As the local flow heat transfer coefficients change greatly in magnitude from single-phase liquid at the entrance to a peak in slug flow and then to a minimum at the transition to the onset of annular flow and finally a new substantial rise up to the outlet, a significant heat spreading occurs due to the heat transfer process itself, and this has to be accounted for when processing the data. Until now, IHCP has not been introduced in the experimental study of heat transfer in such evaporators when reducing local experimental data. In this paper, a new method for processing experimental local heat transfer data by solving the 3D IHCP is proposed. This method is then applied and validated using two sets of single- and two-phase flow experimental data obtained with infrared (IR) camera temperature measurements. The 14 400 raw pixel temperatures per image from the IR camera are first pre-processed by a filtering technique to remove the noise and then to smooth the data, where the IR camera has undergone a prior inhouse pixel by pixel insitu temperature calibration. Three filtering techniques (Wiener filter, spline smooth, and polynomial surface fitting) are compared. The polynomial surface fitting technique was shown to be more suitable for the current type of data set. Then the 3D IHCP is solved based on a finite volume method using the TDMA (Tridiagonal Matrix Algorithm) solver with a combination of Newton-Raphson iteration and a local energy balance method. Furthermore, the present 3D TDMA method (named as 3D TDMA) is compared with three other post-processing methods currently used in the literature, among which the present one is found to be more accurate for reducing the local heat transfer data in multi-microchannel evaporators.

Transient Data Processing of Flow Boiling Local Heat Transfer in a Multi-Microchannel Evaporator Under a Heat Flux Disturbance

Multi-microchannel evaporators are often used to cool down electronic devices subjected to continuous heat load variations. However, so far, rare studies have addressed the transient flow boiling local heat transfer data occurring in such applications. The present paper introduces and compares two different data reduction methods for transient flow boiling data in a multi-microchannel evaporator. A transient test of heat disturbance from 20 to 30 W cm−2 was conducted in a multi-microchannel evaporator using R236fa as the test fluid. The test section was 1 × 1 cm2 in size and had 67 channels, each having a cross-sectional area of 100 × 100 μm2. The micro-evaporator backside temperature was obtained with a fine-resolution infrared (IR) camera. The first data reduction method (referred to three-dimensional (3D)-TDMA) consists in solving a transient 3D inverse heat conduction problem by using a tridiagonal matrix algorithm (TDMA), a Newton–Raphson iteration, and a local energy balance method. The second method (referred to two-dimensional (2D)-controlled) considers only 2D conduction in the substrate of the micro-evaporator and solves at each time step the well-posed 2D conduction problem using a semi-implicit solver. It is shown that the first method is more accurate, while the second one reduces significantly the computational time but led to an approximated solution. This is mainly due to the 2D assumption used in the second method without considering heat conduction in the widthwise direction of the micro-evaporator.

Numerical solution of a 2D inverse heat conduction problem

We consider a two-dimensional (2D) inverse heat conduction problem which is severely ill-posed, i.e. the solution (if it exists) does not depend continuously on the data. For obtaining a stable approximate solution, we propose an iterative regularization method from a new point of view. On the one hand, we give and prove some order optimal convergence results in the L2-norm and Hr-norm under both a priori and a posteriori stopping rules, on the other hand, we discuss the numerical aspect of the proposed method. Three numerical examples illustrate the behaviour of the regularization method.

Differential-difference regularization for a 2D inverse heat conduction problem

We study the differential-difference regularization for a two-dimensional inverse heat conduction problem, i.e. the heat equation is semi-discretized by a differential-difference equation, where the time derivative and a spatial second-order derivative have been replaced by the finite differences, while the other spatial second-order derivative is preserved. We analyze the properties of the discretized approximation using Fourier transform techniques. Some error estimates, which give the information about how to choose the step lengths in the discretization, show that the semi-discretized form has a ‘regularized effect’. We also proved the unconditional convergence of a discretization scheme involving spatial marching.

Inverse estimation of surface heating condition in a three-dimensional object using conjugate gradient method

Temperature and heat flux on inaccessible surfaces can be estimated by solving an inverse heat conduction problem (IHCP) based on the measured temperature and/or heat flux on accessible surfaces. In this study, the heat flux and temperature on the front (heated) surface of a three-dimensional (3D) object is recovered using the conjugate gradient method (CGM) with temperature and heat flux measured on back surface (opposite to the heated surface). The thermal properties of the 3D object are considered to be temperature-dependent. The simulated measurement data, i.e., the temperature and heat flux on the back surface, are obtained by numerically solving a direct problem where the front surface of the object is subjected to high intensity periodic laser heat flux with a Gaussian profile. The robustness of the formulated 3D IHCP algorithm is tested for two materials. The effects of the uncertainties in thermophysical properties on the inverse solutions are also examined. Efforts are made to reduce the total number of heat flux sensors on the back surface required to recover the front-surface heating condition.

Local heat transfer estimation in microchannels during convective boiling under microgravity conditions: 3D inverse heat conduction problem using BEM techniques

Two-phase and boiling flow instabilities are complex, due to phase change and the existence of several interfaces. To fully understand the high heat transfer potential of boiling flows in microscale's geometry, it is vital to quantify these transfers. To perform this task, an experimental device has been designed to observe flow patterns. Analysis is made up by using an inverse method which allows us to estimate the local heat transfers while boiling occurs inside a microchannel. In our configuration, the direct measurement would impair the accuracy of the searched heat transfer coefficient because thermocouples implanted on the surface minichannels would disturb the established flow. In this communication, we are solving a 3D IHCP which consists in estimating using experimental data measurements the surface temperature and the surface heat flux in a minichannel during convective boiling under several gravity levels (g, 1g, 1.8g). The considered IHCP is formulated as a mathematical optimization problem and solved using the boundary element method (BEM).

Real time heat sources identification by a branch eigenmodes reduced model

This study presents a method to solve a 3D inverse heat conduction problem in real time. The aim is to identify sequentially the time varying strength of two independent heat sources from non intrusive noisy temperatures measured by an infrared camera. Due to the geometry and the boundary conditions of the experimental setup, the modelling of the system leads to a high dimensional nonlinear model. The size of this detailed model forbids to solve the inverse problem. A reduced model is then built using the branch eigenmodes technique. This study shows that the identification of the heat sources strengths is possible using a very low order reduced model. Moreover, the computation times are very small, allowing the sequential identification in real time.

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.

Solution of the stationary 2D inverse heat conduction problem by Treffetz method

The paper presents analysis of a solution of Laplace equation with the use of FEM harmonic basic functions. The essence of the problem is aimed at presenting an approximate solution based on possibly large finite element. Introduction of harmonic functions allows to reduce the order of numerical integration as compared to a classical Finite Element Method. Numerical calculations conform good efficiency of the use of basic harmonic functions for resolving direct and inverse problems of stationary heat conduction. Further part of the paper shows the use of basic harmonic functions for solving Poisson’s equation and for drawing up a complete system of biharmonic and polyharmonic basic functions

Model reduction for the resolution of multidimensional inverse heat conduction problems

For large linear heat conduction systems, it is proposed here to solve an inverse heat conduction problem (IHCP) that consists in the identification of several time-varying thermal solicitations from simulations of measured temperatures. For this inversion, instead of using a detailed model of large size, this one is first transformed into a reduced model. The latter is built with identified dominant eigenmodes of the system leading to a reduced state representation that links the inputs (unknown solicitations) to the outputs (simulated temperatures). The procedure is sequential and uses future time steps. At first, a numerical 2D IHCP is provided: two time-varying heat flux densities are estimated from various positions of two sensors. A specific study on static and dynamic sensitivities is made. An example of a 3D IHCP is also given. The method is particularly interesting in this last case where, at each time step, the resolution of a system of order 9 (the reduced model) takes the place of a system of order 1331 (the detailed model).

A guide for the use of the function specification method for 2D inverse heat conduction problems

The solution of the multidimensional inverse heat conduction problem (IHCP) requires much practice and several runs to determine the many parameters involved in the solution. This paper describes a step by step guide to identifying all of the parameters involved in the function specification method with spatial regularization: namely the number of future temperatures, the value of the regularization coefficient and the sensor locations. It also intends to show the new difficulties that arise from solving a 2D IHCP compared to a ID IHCP. The issue of finding the appropriate number of sensors and their locations so as to obtain a desired surface heat flux spatial resolution is addressed. An experimental set-up was built for the validation of the methodology. It also shows that the determination of surface temperature and flux that vary in both time and space from remote temperature measurements is possible when the sensors are properly placed.