Heat Conduction in an Eccentrically Hollow, Infinitely Long Cylinder With Internal Heat Generation

1961 ◽  
Vol 83 (4) ◽  
pp. 510-512 ◽  
Author(s):  
M. R. El-Saden
Keyword(s):  
Heat Conduction ◽  
Temperature Distribution ◽  
Hollow Cylinder ◽  
Heat Generation ◽  
Internal Heat ◽  
Internal Heat Generation ◽  
Theoretical Solution ◽  
Steady Temperature ◽  
Special Case ◽  
Long Cylinder

This paper discusses the steady temperature distribution in an infinitely long, eccentrically hollow cylinder with uniform rate of internal heat generation. An exact theoretical solution is presented. The result is applied to the special case of no internal heat generation, and the rate of heat conduction is obtained.

Hysteresis Heating of Railroad Bearing Thermoplastic Elastomer Suspension Element

2017 ◽  
Author(s):  
Oscar O. Rodriguez ◽  
Arturo A. Fuentes ◽  
Constantine Tarawneh ◽  
Robert E. Jones
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Steady State ◽  
Temperature Distribution ◽  
Heat Generation ◽  
Internal Heat ◽  
Thermoplastic Elastomer ◽  
Internal Heat Generation ◽  
Element Analysis ◽  
Railroad Bearing

Thermoplastic elastomers (TPE’s) are increasingly being used in rail service in load damping applications. They are superior to traditional elastomers primarily in their ease of fabrication. Like traditional elastomers they offer benefits including reduction in noise emissions and improved wear resistance in metal components that are in contact with such parts in the railcar suspension system. However, viscoelastic materials, such as the railroad bearing thermoplastic elastomer suspension element (or elastomeric pad), are known to develop self-heating (hysteresis) under cyclic loading, which can lead to undesirable consequences. Quantifying the hysteresis heating of the pad during operation is therefore essential to predict its dynamic response and structural integrity, as well as, to predict and understand the heat transfer paths from bearings into the truck assembly and other contacting components. This study investigates the internal heat generation in the suspension pad and its impact on the complete bearing assembly dynamics and thermal profile. Specifically, this paper presents an experimentally validated finite element thermal model of the elastomeric pad and its internal heat generation. The steady-state and transient-state temperature profiles produced by hysteresis heating of the elastomer pad are developed through a series of experiments and finite element analysis. The hysteresis heating is induced by the internal heat generation, which is a function of the loss modulus, strain, and frequency. Based on previous experimental studies, estimations of internally generated heat were obtained. The calculations show that the internal heat generation is impacted by temperature and frequency. At higher frequencies, the internally generated heat is significantly greater compared to lower frequencies, and at higher temperatures, the internally generated heat is significantly less compared to lower temperatures. However, during service operation, exposure of the suspension pad to higher loading frequencies above 10 Hz is less likely to occur. Therefore, internal heat generation values that have a significant impact on the suspension pad steady-state temperature are less likely to be reached. The commercial software package ALGOR 20.3TM is used to conduct the thermal finite element analysis. Different internal heating scenarios are simulated with the purpose of obtaining the bearing suspension element temperature distribution during normal and abnormal conditions. The results presented in this paper can be used in the future to acquire temperature distribution maps of complete bearing assemblies in service conditions and enable a refined model for the evolution of bearing temperature during operation.

Simulation of Nanoscale Multidimensional Transient Heat Conduction Problems Using Ballistic-Diffusive Equations and Phonon Boltzmann Equation

2005 ◽  
Vol 127 (3) ◽  
pp. 298-306 ◽  
Author(s):  
Ronggui Yang ◽  
Gang Chen ◽  
Marine Laroche ◽  
Yuan Taur
Keyword(s):  
Heat Conduction ◽  
Boltzmann Equation ◽  
Boundary Conditions ◽  
Heat Generation ◽  
Internal Heat ◽  
Internal Heat Generation ◽  
Computational Time ◽  
Discrete Ordinates ◽  
Fourier Law ◽  
Phonon Boltzmann Equation

Heat conduction in micro- and nanoscale and in ultrafast processes may deviate from the predictions of the Fourier law, due to boundary and interface scattering, the ballistic nature of the transport, and the finite relaxation time of heat carriers. The transient ballistic-diffusive heat conduction equations (BDE) were developed as an approximation to the phonon Boltzmann equation (BTE) for nanoscale heat conduction problems. In this paper, we further develop BDE for multidimensional heat conduction, including nanoscale heat source term and different boundary conditions, and compare the simulation results with those obtained from the phonon BTE and the Fourier law. The numerical solution strategies for multidimensional nanoscale heat conduction using BDE are presented. Several two-dimensional cases are simulated and compared to the results of the transient phonon BTE and the Fourier heat conduction theory. The transient BTE is solved using the discrete ordinates method with a two Gauss-Legendre quadratures. Special attention has been paid to the boundary conditions. Compared to the cases without internal heat generation, the difference between the BTE and BDE is larger for the case studied with internal heat generation due to the nature of the ballistic-diffusive approximation, but the results from BDE are still significantly better than those from the Fourier law. Thus we conclude that BDE captures the characteristics of the phonon BTE with much shorter computational time.

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