Fast inverse prediction of phase change banks in high temperature furnaces with a Kalman filter coupled with a recursive least-square estimator

An inverse heat transfer procedure for predicting the time-varying thickness of phase-change banks on the inside surface of the walls of high temperature furnaces is presented. The main feature of the inverse method is its unique capability of making fast predictions so that it can be easily integrated to existing real-time control systems of industrial facilities. The method rests on fast computing state-space models (direct model) that are designed to mimic the response of a full finite-difference model of the phase change problem. A Kalman filter coupled with a recursive least-square estimator (inverse method) is employed to estimate the time-varying phase front position from the data collected by a temperature and/or heat flux sensor located in the furnace wall. The inverse heat transfer procedure is thoroughly tested for typical phase change conditions that prevail inside industrial facilities. The effect of the sensor type (temperature sensor or heat flux sensor), of its location and of the measurement noise on the accuracy and stability of the predicted bank thickness is investigated. It is shown that the proposed inverse heat transfer procedure becomes increasingly reliable and accurate for predicting the bank thickness as it shrinks. This feature is of the utmost interest for preventing the sudden and accidental loss of the protective banks of industrial furnaces filled with molten material. Recommendations are also made concerning the type and location of sensors.     

Prediction of a 2-D Solidification Front in High Temperature Furnaces by an Inverse Analysis

This study presents an inverse heat transfer method for predicting the two-dimensional time-varying shape of the phase change protective bank on the inside surface of a wall of a high temperature furnace filled with molten material. The method rests on the Levenberg-Marquardt least-square minimization technique and it relies on non-intrusive temperature measurements. Results indicate that under typical melting furnace operating conditions, the 2-D inverse methodology provides accurate predictions of the moving boundary. For these real time industrial applications however, a simpler yet accurate quasi-2-D inverse method that is nearly 10 times faster than the full 2-D inverse model is proposed.     

Identification of wall heat flux for turbulent forced convection by inverse analysis

An inverse problem for turbulent forced convection between parallel flat plates is investigated. The space- and time-dependent heat flux at the upper wall is estimated from the temperature measurements taken inside the flow. In the present study, the conjugate gradient method is adopted for the estimation of the unknown wall heat flux. No prior information is needed for the functional form of the wall heat flux in the inverse analysis. The effects of the measurement errors, the functional form of the wall heat flux, and the location of the sensors on the accuracy of the estimation are investigated. The reconstruction of the wall heat flux is satisfactory when simulated exact or noisy data are input to the inverse analysis. The sensitivity coefficients are discussed in this paper. As expected, it is shown that the accuracy of the estimation can be improved when the sensors are located closer to the upper wall.     

Heat transfer and banks formation in a slag bath with embedded heat sources

A study was conducted for the heat transfer and the formation of solid banks inside a bath of molten slag. This study is motivated by the need to predict the formation of a protective thermal barrier for the refractory brick walls inside smelting furnaces. A mathematical model for natural convection dominated solid liquid phase change with embedded heat sources is presented. A scale analysis is conducted yielding algebraic expressions for the steady-state minimal side bank thickness and location, molten volume fraction and average Nusselt number at the surface of the bath in terms of the Rayleigh number. The predictions of the scale analysis are validated with numerical results and their range of application are delineated.     

燃烧室内三维温度场的辐射反问题

本文提出了一种在介质辐射特性已知的条件下,由壁面入射辐射热流的测量值反演燃烧室内三维温度场的方法。该方法是在辐射传递方程离散坐标近似的基础上,用求目标函数极小值的共轭梯度法进行反演计算。通过对吸收系数、散射不对称因子、反照率、壁面黑度和燃烧室大小尺寸等参数对反演精度影响的分析,结果表明,即使存在随机测量误差,这些参数对温度场反演精度的影响也不大,本文所提出的方法可较精确地反演燃烧室内三维温度场。     

Geometry estimation of the furnace inner wall by an inverse approach

The geometry of a furnace inner wall is estimated by an inverse method in this work. Based upon the concept of a virtual area, in the analysis process the heat conduction equation with boundary conditions was first discretized by a finite difference method to form a matrix equation. And then the linear least-squares error method was applied to determine the temperature of virtual boundary by inverse process. Finally, the geometry of the furnace inner wall can be obtained by direct process. Furthermore, the effects of the measurement errors, number of measurements and position of measurements on the deviation of geometry prediction are also discussed.     

What is the most suitable fixed grid solidification method for handling time-varying inverse Stefan problems in high temperature industrial furnaces?

This paper presents a simple inverse heat transfer method for predicting the time-varying bank thickness of the phase change protective layer inside a high temperature furnace. The direct problem is handled with the single phase method which is neglecting the latent heat. The inverse method rests on the Adjoint Problem and the Conjugate Gradient Method. It is shown that the iterative inverse procedure based on the single phase method predicts a virtual incident heat flux in the liquid zone that yields accurate time-varying bank predictions. Results also indicate that the benefits of the virtual iterative approach are twofold: the CPU time required for solving the inverse problem is reduced and the lagging effect inherent to the inversion is diminished for non isothermal phase change processes. For typical melting furnace operating conditions, it is shown that the virtual approach doubles the allowable diagnostic frequency for predicting the time-varying bank thickness.     

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.     

反演计算不同炉气温度下非灰气体的当量吸收系数的方法

提出了一种反演计算不同炉气温度下非灰气体的当量吸收系数的方法。此方法以实验数据反演得到的非灰气体的当量吸收系数作为物性参数,由正演算计算不同条件下的炉气温度,并以此为定解条件,反演不同炉气温度下的非灰气体的当量吸收系数,反复迭代,直至收敛。反演计算过程中采用区域法与求目标函数最小值的共轭梯度法相结合的方法。与实验结果比较表明,该方法可以较为准确地反演不同炉气温度下的非灰气体的当量吸收系数。最后,用该方法计算并比较了非灰气体的当量吸收系数随炉气温度的变化。图2表2参9     

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.     

Practical application of inverse heat conduction for wall condition estimation on a rotating cylinder

The solution of the linear, inverse, transient heat conduction problem (IHCP) in a cylindrical geometry is analysed. The rotating cylinder under investigation is experiencing boiling convection induced by the impingement of a water jet. The initial temperature is known, additional temperature measurements in time are taken with sensors positioned at a constant radius within the solid material, and the estimation of the wall heat flux at the external radius is sought. First, simulated temperature measurements inside the cylinder are processed in order to be used to estimate the wall heat flux. When noise is present in the data, some of the simulated results obtained using the least squares method exhibit oscillatory behavior, but these large oscillations are substantially reduced by the implementation of a regularization technique. Real experimental data are also used for the wall condition estimation and for the subsequent building of local boiling curves are plotted and discussed. The question of the possible effect of a temperature dependent conductivity on the reconstructed wall condition is also considered.     

Estimation of the transient surface temperature and heat flux of a steel slab using an inverse method

In the steel industry it is of great importance to be able to control the surface temperature and heating- or cooling rates during heat treatment processes. An experiment was performed in which a steel slab was heated up to 1250 °C in a fuel fired test furnace. The transient surface temperature and heat flux of a steel slab is calculated using a model for inverse heat conduction. That is, the time dependent local surface temperature and heat flux of a slab is calculated on the basis of temperature measurements in selected points of its interior by using a model of inverse heat conduction. Time- and temperature histories were measured at three points inside a steel slab. Measured temperature histories at the two lower locations of the slab were used as input to calculate the temperature at the position of the third location. A comparison of the experimentally measured and the calculated temperature histories was made to verify the model. The results showed very good agreement and suggest that this model can be applied to similar applications in the Steel industry or in other areas where the target of investigation for some reason is inaccessible to direct measurements.     

Estimation of heat flux on the surface of an initially hot cylinder cooled by a laminar confined impinging jet

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to estimate the unknown space-and time-dependent heat flux at the surface of an initially hot cylinder cooled by a laminar confined slot impinging jet from the knowledge of temperature measurements taken on the cylinder’s surface. It is assumed that no prior information is available on the functional form of the unknown heat flux; 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 in these measurements upon the precision of the estimated results is also considered. The results show that an excellent estimation on the space-and time-dependent heat flux can be obtained even the distributions of thermal properties inside the cylinder is unknown.     

Three-Dimensional Pipe Fouling Layer Estimation by Using Conjugate Gradient Inverse Method

In this study, a conjugate gradient method based inverse algorithm is applied to estimate the unknown three-dimensional fouling-layer profiles on the inner wall of a piping system using simulated temperature measurements taken on the pipe wall. The temperature data obtained from the direct problem are used to simulate the exact temperature measurements. Results show that an excellent estimation on the fouling-layer profiles can be obtained for the two cases investigated in this study. The technique presented here can be used as an alternative to ultrasonic waves fouling detection techniques to provide crucial information for the optimization of cleaning schedule for piping systems.     

SIMULTANEOUS ESTIMATION OF INLET TEMPERATURE AND WALL HEAT FLUX IN TURBULENT CIRCULAR PIPE FLOW

An inverse heat convection problem is solved for simultaneous estimation of unknown inlet temperature and wall heat flux in a thermally developing, hydrodynamically developed turbulent flow in a circular pipe based on temperature measurements obtained at several different locations in the stream. The direct problem of turbulent forced convection is solved with a finite difference method with appropriate algebraic turbulence modelling. Although we seek for two unknown functions, we formulate the inverse problem as one of parameter estimation through the representation of the unknown inlet temperature profile and the wall heat flux distribution by one-dimensional finite element interpolation. Nodal values of the inlet temperature and the wall heat flux at chosen positions are determined as unknown parameters through the Levenberg–Marquardt algorithm for minimization procedure. Numerical results for several testing cases with different magnitudes of measurement errors are examined by using simulated experimental data. The effects of the number and the locations of the temperature measurement points are discussed.     

镁质压渣剂在转炉炼钢过程中的开发和应用

为解决炼钢厂废弃镁碳砖难以在转炉炼钢过程中规模化全量应用的问题,研制了以废弃镁碳砖为材料的转炉压渣剂,能够有效降解转炉冶炼后期炉渣泡沫化的高度,增加炉渣的黏度,起到消泡的同时稠化转炉钢渣,实现快速倒炉倒渣（或者不倒渣）出钢的目的,对于提高炉渣的碱度和黏度,优化转炉后期的溅渣护炉工艺有积极的促进作用。     

Measurements of Some Characteristics of Thermal Radiation in a 400-kW Grate-Fired Furnace Combusting Biomass

A comprehensive experimental investigation relevant to thermal radiation has been performed in a grate-fired test furnace. Thermal radiation is the dominating mode of heat transfer in the grate-fired furnace and yet only a few studies have focused on thermal radiation. No previous works, to the authors' knowledge, have been carried out concerning measurements on radiative heat transfer in grate-fired furnaces. In this work measurements of temperature have been carried out for all boundaries and the flue gases at a large number of locations. The gas species volume fraction, particle mass–size distributions, and wall irradiation have also been measured at a number of spatial locations. These data are useful in a computational framework to describe the radiative heat transfer reaching the boundaries. Comparing modeled wall irradiation to the measured one makes it possible to obtain a deeper insight into the thermal radiative transport inside the grate-fired furnace.     

Experimental study of heat transfer in oscillating flow

This paper describes an experimental study of heat transfer in oscillating flow inside a cylindrical tube. Profiles of temperature are taken inside the wall and in the fluid from an instrumented test rig, in different conditions of oscillating flow. Profiles obtained allow the observation of the wall effect on heat transfer. A method using the inverse heat conduction principle allows the characterization of local heat transfers at the fluid-solid interface. Finally, a comparison between global and local approaches of heat transfer shows the difficulty of defining a dimensionless heat flux density to model local heat transfer in oscillating flow.     

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℃.     

Analytical solution of the inverse unsteady wall heat conduction problem and experimental application

Based on the analytical solution of the unsteady heat conduction differential equation, a solution procedure is presented for the inverse unsteady wall heat conduction problem, i.e. for the calculation of the thermal properties of structural elements of existing buildings under real transient conditions, using on-site temperature measurements. Previous procedures, which were based on the finite-difference method, required a considerable number of temperature measurements in space and time within the wall. The advantage of the present analytical procedure is that it requires only two temperature measurements, apart from some information on the outdoor and indoor temperature variations. The two temperature measurements may be taken on the outdoor and indoor wall surfaces at the same time level, or on one of these surfaces at two different time levels. The proposed analytical procedure provides the values of the thermal conductivity and heat capacity of structural elements, and therefore it may be used in practice for ex post checking of the materials used by the constructor, or for load calculation when heating or cooling systems are to be installed in old buildings of unknown wall properties. Experimental examples are presented which show that the proposed analytical procedure may be applied in practice with very good accuracy.