Alternative analytical solutions of the diffusion (thermal conductivity) equation for an arbitrary initial concentration (temperature) distribution

2002 ◽  
Vol 28 (11) ◽  
pp. 923-924 ◽  
Author(s):  
R. Sh. Malkovich
Keyword(s):  
Thermal Conductivity ◽  
Temperature Distribution ◽  
Analytical Solutions ◽  
Conductivity Equation ◽  
Initial Concentration ◽  
Thermal Conductivity Equation

scholarly journals MATHEMATICAL MODELS OF HEAT TRANSFER IN ELEMENTS OF TURBO GENERATORS (CONTINUED)

2020 ◽  
Vol 2 (1) ◽  
pp. 21-28
Author(s):  
V. I. Havrysh ◽  
◽  
B. O. Bilinskyi ◽  
O. S. Korol ◽  
R. R. Shkrab ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Thermal Conductivity ◽  
Temperature Distribution ◽  
Mathematical Models ◽  
Half Space ◽  
Analytical Solutions ◽  
Internal Heat ◽  
Normal Operation ◽  
Cylindrical Shape ◽  
Temperature Regimes

Previously developed [8] and presented new mathematical models for the analysis of temperature regimes in individual elements of turbo generators, which are geometrically described by isotropic half-space and space with an internal heat source of cylindrical shape. Cases are also considered for half-space, when the fuel-releasing cylinder is thin, and for space, when it is heat-sensitive. For this purpose, using the theory of generalized functions, the initial differential equations of thermal conductivity with boundary conditions are written in a convenient form. To solve the obtained boundary value problems of thermal conductivity, the integral Hankel transformation was used, and as a result, analytical solutions in the images were obtained. The inverse Hankel integral transformation was applied to these solutions, which made it possible to obtain the final analytical solutions of the initial problems. The obtained analytical solutions are presented in the form of improper convergent integrals. Computational programs have been developed to determine the numerical values ​​of temperature in the above structures, as well as to analyze the heat transfer in the elements of turbo generators due to different temperature regimes due to heating by internal heat sources concentrated in the cylinder volume. Using these programs, graphs are presented that show the behavior of curves constructed using numerical values ​​of the temperature distribution depending on the spatial radial and axial coordinates. The obtained numerical values ​​of temperature indicate the correspondence of the given mathematical models for determining the temperature distribution to the real physical process. The software also allows you to analyze media with internal heating, concentrated in the spatial figures of the correct geometric shape, in terms of their heat resistance. As a result, it becomes possible to increase it, to determine the allowable temperatures of normal operation of turbo generators, to protect them from overheating, which can cause the destruction of not only individual elements but also the entire structure.

Determination of nonstationary thermal field of passenger car wheel at braking

2020 ◽  
pp. 40-44
Author(s):  
Alexander Vyacheslavovich Zyablov ◽  
◽  
Sergey Valeryevich Bespalko ◽  
Keyword(s):  
Thermal Conductivity ◽  
Thermal Field ◽  
Thermal Emission ◽  
Euler Method ◽  
Rolling Stock ◽  
Conductivity Equation ◽  
Passenger Car ◽  
Thermal Conductivity Equation ◽  
Nonstationary Field ◽  
Curved Rod

The paper is devoted to modeling of nonstationary field of a passenger car wheel at braking. The calculation is based on formulation of thermal conductivity equation for the wheel tread as a curved rod with the application of linear approximation of thermal field. At formulation of thermal conductivity equation it is necessary to consider a balance of heat in small volume of tread with the consideration for thermal flow from braking shoe, thermal emission to the environment and thermal conductivity in circular direction. The authors have set for the initial equation of thermal conductivity a functional and have formulated conditions of stationarity that leads after integration to the system of the first order differential equations of time. The authors have applied the Euler method at integration. The developed method has been realized in the C++ program. With the use of this application the authors have conducted a research of the thermal field of the passenger car wheel. The method can be used at designing of new rolling stock and for the analysis of reasons of flaws appearance on the surface of car wheels.

The Simulation of Discrete Vessel Effects in Experimental Hyperthermia

1994 ◽  
Vol 116 (3) ◽  
pp. 256-262 ◽  
Author(s):  
R. J. Rawnsley ◽  
R. B. Roemer ◽  
A. W. Dutton
Keyword(s):  
Heat Transfer ◽  
Thermal Conductivity ◽  
Blood Vessel ◽  
Effective Thermal Conductivity ◽  
Blood Vessels ◽  
Temperature Data ◽  
Experimental Temperature ◽  
Conductivity Equation ◽  
Thermal Models ◽  
Thermal Conductivity Equation

The ability of two simple thermal models to predict experimentally measured in vivo temperature profiles was compared. These comparisons were done both with and without the inclusion of separate, discrete blood vessels. The two tissue models were: 1) Pennes’ Bio-Heat Transfer equation (BHTE), and 2) an effective thermal conductivity equation (ETCE). The experimental temperature data were measured (Moros, 1990; Moros et al., 1993) in the thighs of anesthetized greyhound dogs under hyperthermic conditions generated by scanned focused ultrasound. Blood vessels were added to the thermal models in counter-current pairs transiting the model domain. The blood vessels in both models were assumed to have a constant heat transfer coefficient, and an axially varying mixed mean temperature. The vessel locations were determined a posteriori, via inspection of the experimental temperature data. Least square error fits of the predicted model temperatures to the experimental temperature data were obtained by adjusting both (a) the mass flow rate within and (b) the position of each blood vessel, and (c) the value of either the perfusion parameter (W) in the BHTE or the effective thermal conductivity parameter (Keff) in the ETCE. When small numbers (3-4) of blood vessel pairs were included, both of the models showed significant improvement in their ability to predict the experimental temperatures. Although both models performed well in terms of predicting temperatures near large vessels, the BHTE had a statistically significant better ability to predict the complete set of measured temperatures at all locations.

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