Solutions of selected inverse heat transfer problems and their experimental verifications

Authors:

• Artur Cebula

Abstract

The aim of this monograph is to present a measuring instrument for determination of time-dependent heat flux values on the instrument outer surface. The material temperatures are measured in internal discrete points. A mathematical model is developed to enable reconstruction of conditions prevailing on the outer surface based on measurements performed inside a solid body. The issue is based on the solution of an inverse heat conduction problem. The computational domain is divided into the direct and the inverse region. The regions are divided further into control volumes and then the heat balance equations are written according to the finite volume method. First, the temperature distribution is found in the direct region. Next, it is determined by means of the space marching method in the region of the inverse problem, moving towards the outer surface. The conditions prevailing on the outer surface are reconstructed, i.e. the sought surface temperature and heat flux values are calculated. The developed method makes it possible to reconstruct the outer surface conditions varying in time. It is simple both in notation and implementation. A computational test is carried out to verify the method capacity for reconstruction of fast changes in the heat flux on the element surface. The test consists in a comparison between results obtained by means of the method and CFD calculations. The CFD computations concern the control rod in the Forsmark Nuclear Power Plant. The rod is washed with cool water, and in the cross-flow it is washed by hot water jets. This causes fluctuations in the rod surface temperature. Temperature histories are generated from the CFD model in several selected internal points. They are then set as input parameters for the developed own model. The comparison of the results proves the method capacity for reconstruction of fast changes in the heat flux and the surface temperature values. Then the real measuring instrument is designed and made. The element takes account of the specificity and limitations of the inverse method, which is very sensitive to random measuring errors and to any changes in the temperature sensors location, which should coincide with the location of the numerical mesh nodes. So that fast changes occurring on the outer surface can be reconstructed, thermocouples have to be installed possibly close to the surface. The developed measuring instrument was commissioned and made in co-operation with an industrial partner. Its target application was to analyse transient heat conduction phenomena after the rod failure. The monograph presents two design solutions differing in the way in which the thermocouples are installed. The developed 149 measuring instrument is mounted on a testing stand. The stand is designed to enable impingement of hot air jets to cool and heat up the measuring instrument with a different frequency. The tests are carried out to verify the correctness of the measuring instrument operation. Designing the measuring insert, special care was taken to obtain an accurate configuration of the thermocouples. The location of the thermocouples is verified by means of computer imaging. The verification confirms they are located accurately, as planned. The impact of random measuring errors is minimized as early as at the computational model stage because the temperature transients and derivatives thereof are smoothed using a 9-point averaging filter. An attempt is presented to replace the finite volume method with the finite volume-finite element method (FVFEM). The FVFEM is more complex compared to the classical FVM. However, it has the advantage of being able to modify the location of the mesh nodes. If differences arise in the location of the thermocouples compared to the designed configuration, the method makes it possible to correct the node location. This cannot be done within the FVM. The proposed method is tested through a comparison with other numerical methods, including the FVM and the FEM. Additional calculations are performed to investigate the impact of random measuring errors on the method accuracy. An analysis is also conducted of the effect of the time step on the result. Moreover, calculations are carried out using measuring data obtained from the laboratory stand. The calculation results confirm that the method is appropriate and useful in the case of elements with complex geometries and enables determination of the local heat flux or heat transfer coefficient values. It can be used successfully to develop the measuring instrument just like the previous procedure based on the FVM. Together with relevant methodology, the developed element makes it possible to determine transient heat flux values. This enables determination of the mean and the local value as well. The instrument is free from limitations typical of calorimetric meters, where the measured heat impulse must be short. The monograph also presents experimental testing of a flowmeter where considerations aiming to find the sought quantity through a solution of the inverse problem are continued. In the problem, the heat flux on the outer surface is known and the outer surface temperatures are measured. Based on these values, the fluid flow through the duct is reconstructed. A comparison was also made within the problem between the analytical model on the one hand and the CFD model and the experimental results on the other. The presented issues are especially useful if there is no access to the surface on which thermal or flow conditions are to be determined. This is the case if extreme physical conditions occur (high pressure or temperature) in the flow of aggressive liquids or if design reasons make it impossible to gain access to the analysed surface.