scholarly journals Influence of the heat source location on the stability of the solution to the Cauchy problem

2021 ◽  
Vol 323 ◽  
pp. 00016
Author(s):  
Magda Joachimiak ◽  
Michał Ciałkowski
Keyword(s):  
Inverse Problem ◽  
Heat Source ◽  
Source Function ◽  
Regularization Parameter ◽  
Cauchy Type ◽  
Heat Sources ◽  
Calculation Domain ◽  
The Cauchy Problem ◽  
The Stability ◽  
Stability Of The Solution

In this paper the solution to the Cauchy-type inverse problem for the Laplace’s equation is presented. A modified Tikhonov regularization was applied here. The regularization parameter was chosen using the Morozov principle. The relation between the location of the heat source (function singularity) and the stability of the solution to the inverse problem was analyzed. Variable thermal loads in the area were simulated by changing the location of heat sources along two boundaries of the rectangle calculation domain.

Cauchy type nonlinear inverse problem in a two-layer area

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Michał Ciałkowski ◽  
Aleksander Olejnik ◽  
Magda Joachimiak ◽  
Krzysztof Grysa ◽  
Andrzej Frąckowiak
Keyword(s):  
Inverse Problem ◽  
Cauchy Problem ◽  
Heat Flow ◽  
Gas Turbine ◽  
Cauchy Type ◽  
Content Type ◽  
Temperature Changes ◽  
The Cauchy Problem ◽  
Changes Over Time ◽  
Over Time

Purpose To reduce the heat load of a gas turbine blade, its surface is covered with an outer layer of ceramics with high thermal resistance. The purpose of this paper is the selection of ceramics with such a low heat conduction coefficient and thickness, so that the permissible metal temperature is not exceeded on the metal-ceramics interface due to the loss ofmechanical properties. Design/methodology/approach Therefore, for given temperature changes over time on the metal-ceramics interface, temperature changes over time on the inner side of the blade and the assumed initial temperature, the temperature change over time on the outer surface of the ceramics should be determined. The problem presented in this way is a Cauchy type problem. When analyzing the problem, it is taken into account that thermophysical properties of metal and ceramics may depend on temperature. Due to the thin layer of ceramics in relation to the wall thickness, the problem is considered in the area in the flat layer. Thus, a one-dimensional non-stationary heat flow is considered. Findings The range of stability of the Cauchy problem as a function of time step, thickness of ceramics and thermophysical properties of metal and ceramics are examined. The numerical computations also involved the influence of disturbances in the temperature on metal-ceramics interface on the solution to the inverse problem. Practical implications The computational model can be used to analyze the heat flow in gas turbine blades with thermal barrier. Originality/value A number of inverse problems of the type considered in the paper are presented in the literature. Inverse problems, especially those Cauchy-type, are ill-conditioned numerically, which means that a small change in the inputs may result in significant errors of the solution. In such a case, regularization of the inverse problem is needed. However, the Cauchy problem presented in the paper does not require regularization.

Problems of Heat Source Modeling in Iso–Exothermic Materials Used as Riser Sleeves in Foundry

2006 ◽  
Vol 514-516 ◽  
pp. 1438-1442 ◽  
Author(s):  
Zenon Ignaszak ◽  
Paweł Popielarski
Keyword(s):  
Heat Source ◽  
Source Function ◽  
Exothermic Reaction ◽  
Solution Procedure ◽  
Problem Solution ◽  
Heat Sources ◽  
Source Modeling ◽  
Reaction Heat ◽  
Measurement Systems ◽  
Materials Used

The modeling of heat transfer in materials containing exothermic components must take into consideration the presence of heat sources in the Fourier–Kirchhoff equation. The aim of this investigation was the identification of real and effective thermophysical parameters of the insulating–exothermic materials used as riser sleeves containing these exothermic heat sources. The experiments of steel pouring into the mould, containing different insulating and exothermic sleeves were carried out, using thermocouples measurement systems (thermal analysis of casting–mould system). Then the thermophysical coefficients of these materials were calculated using inverse problem solution. The worked time–dependent formula of exothermic reaction heat (heating yield in W/m3) was called heat source function. The paper presents the basis and the practical expression of heat source by different functions, its justification and the results of simulations using these functions. The numerical system Calcosoft and its Inverse Solution procedure were applied.

scholarly journals Thermoacoustic Instability in a Rijke Tube with a Distributed Heat Source

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Xiaochuan Yang ◽  
Ali Turan ◽  
Shenghui Lei
Keyword(s):  
Heat Source ◽  
Heat Release ◽  
Method Of Lines ◽  
Nonlinear Behaviour ◽  
Heat Sources ◽  
Rijke Tube ◽  
Thermoacoustic Instability ◽  
Acoustic Perturbation ◽  
The Stability ◽  
The Galerkin Method

A Rijke tube with a distributed heat source is investigated. Driven by the widely existing thermoacoustic instability in lean premixed gas turbine combustors, this work aims to explore the physicochemical underpinning and assist in the elucidation and analysis of this problem. The heat release model consists of a row of distributed heat sources with individual heat release rates. The integrated heat release rate is then coupled with the acoustic perturbation for thermoacoustic analysis. A continuation approach is employed to conduct the bifurcation analysis and capture the nonlinear behaviour inherent in the system. Unlike the conventional approach by the Galerkin method, the acoustic equations are originally discretized using the Method of Lines (MOL) to build up a dynamic system. The model is first validated and shown to yield good predictions with available experimental data. Influences of multiple heat sources, time delay, and heat release distribution are then studied to reveal the extensive nonlinear characteristics involved in the case of a distributed heat source. It is found that distributed heat source plays an important role in determining the stability of a thermoacoustic system.

PLANETARY SELF-ORGANIZATION

2020 ◽  
Vol 42 (3) ◽  
pp. 271-282
Author(s):  
OLEG IVANOV
Keyword(s):  
Dynamical System ◽  
Heat Source ◽  
Mantle Convection ◽  
Plate Tectonics ◽  
Rotation Speed ◽  
Self Organization ◽  
Heat Sources ◽  
Kinematic Parameters ◽  
The Earth ◽  
New Type

The general characteristics of planetary systems are described. Well-known heat sources of evolution are considered. A new type of heat source, variations of kinematic parameters in a dynamical system, is proposed. The inconsistency of the perovskite-post-perovskite heat model is proved. Calculations of inertia moments relative to the D boundary on the Earth are given. The 9 times difference allows us to claim that the sliding of the upper layers at the Earth's rotation speed variations emit heat by viscous friction.This heat is the basis of mantle convection and lithospheric plate tectonics.

scholarly journals A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation

2020 ◽  
Vol 18 (1) ◽  
pp. 1685-1697
Author(s):  
Zhenyu Zhao ◽  
Lei You ◽  
Zehong Meng
Keyword(s):  
Cauchy Problem ◽  
Laplace Equation ◽  
Tikhonov Regularization ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Solution Process ◽  
Tikhonov Regularization Method ◽  
Hermite Expansion ◽  
The Cauchy Problem ◽  
Ill Posed

Abstract In this paper, a Cauchy problem for the Laplace equation is considered. We develop a modified Tikhonov regularization method based on Hermite expansion to deal with the ill posed-ness of the problem. The regularization parameter is determined by a discrepancy principle. For various smoothness conditions, the solution process of the method is uniform and the convergence rate can be obtained self-adaptively. Numerical tests are also carried out to verify the effectiveness of the method.

scholarly journals Analysis and Comparison of Some Low-Temperature Heat Sources for Heat Pumps

Energies ◽  
2019 ◽  
Vol 12 (10) ◽  
pp. 1853 ◽  
Author(s):  
Pavel Neuberger ◽  
Radomír Adamovský
Keyword(s):  
Heat Transfer ◽  
Low Temperature ◽  
Heat Source ◽  
Heat Pump ◽  
Heat Exchangers ◽  
Ambient Air ◽  
Heat Transfer Fluid ◽  
Heat Sources ◽  
Ground Heat Exchangers ◽  
Temperature Heat

The efficiency of a heat pump energy system is significantly influenced by its low-temperature heat source. This paper presents the results of operational monitoring, analysis and comparison of heat transfer fluid temperatures, outputs and extracted energies at the most widely used low temperature heat sources within 218 days of a heating period. The monitoring involved horizontal ground heat exchangers (HGHEs) of linear and Slinky type, vertical ground heat exchangers (VGHEs) with single and double U-tube exchanger as well as the ambient air. The results of the verification indicated that it was not possible to specify clearly the most advantageous low-temperature heat source that meets the requirements of the efficiency of the heat pump operation. The highest average heat transfer fluid temperatures were achieved at linear HGHE (8.13 ± 4.50 °C) and double U-tube VGHE (8.13 ± 3.12 °C). The highest average specific heat output 59.97 ± 41.80 W/m2 and specific energy extracted from the ground mass 2723.40 ± 1785.58 kJ/m2·day were recorded at single U-tube VGHE. The lowest thermal resistance value of 0.07 K·m2/W, specifying the efficiency of the heat transfer process between the ground mass and the heat transfer fluid, was monitored at linear HGHE. The use of ambient air as a low-temperature heat pump source was considered to be the least advantageous in terms of its temperature parameters.

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