Distribution of steady state temperatures and thermoelastic stresses in a cylindrical shell with internal heat generation and cooled on both sides or only on one side

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.

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Fourier-Bessel Series Model for the Stefan Problem With Internal Heat Generation in Cylindrical Coordinates

Volume 6A: Thermal-Hydraulics and Safety Analyses ◽

10.1115/icone26-81009 ◽

2018 ◽

Author(s):

Lyudmyla Barannyk ◽

John Crepeau ◽

Patrick Paulus ◽

Ali Siahpush

Keyword(s):

Differential Equation ◽

Steady State ◽

Heat Generation ◽

Numerical Solutions ◽

Internal Heat ◽

Internal Heat Generation ◽

Time Dependent ◽

Cylindrical Coordinates ◽

System Approaches ◽

Melting And Solidification

A nonlinear, first-order ordinary differential equation that involves Fourier-Bessel series terms has been derived to model the time-dependent motion of the solid-liquid interface during melting and solidification of a material with constant internal heat generation in cylindrical coordinates. The model is valid for all Stefan numbers. One of the primary applications of this problem is for a nuclear fuel rod during meltdown. The numerical solutions to this differential equation are compared to the solutions of a previously derived model that was based on the quasi-steady approximation, which is valid only for Stefan numbers less than one. The model presented in this paper contains exponentially decaying terms in the form of Fourier-Bessel series for the temperature gradients in both the solid and liquid phases. The agreement between the two models is excellent in the low Stefan number regime. For higher Stefan numbers, where the quasi-steady model is not accurate, the new model differs from the approximate model since it incorporates the time-dependent terms for small times, and as the system approaches steady-state, the curves converge. At higher Stefan numbers, the system approaches steady-state faster than for lower Stefan numbers. During the transient process for both melting and solidification, the temperature profiles become parabolic.

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On the Stefan Problem With Internal Heat Generation and Prescribed Heat Flux Conditions at the Boundary

ASME 2019 Heat Transfer Summer Conference ◽

10.1115/ht2019-3703 ◽

2019 ◽

Author(s):

Lyudmyla Barannyk ◽

Sidney D. V. Williams ◽

Olufolahan Irene Ogidan ◽

John C. Crepeau ◽

Alexey Sakhnov

Keyword(s):

Heat Flux ◽

Steady State ◽

Phase Change ◽

Heat Generation ◽

Separation Of Variables ◽

Outer Boundary ◽

Internal Heat ◽

Internal Heat Generation ◽

The Difference ◽

Solid Liquid Interface

Abstract We study the evolution of the solid-liquid interface during melting and solidification of a material with constant internal heat generation and prescribed heat flux at the boundary for a plane wall and a cylinder. The equations are solved by splitting them into transient and steady-state components and then using separation of variables. This results in an ordinary differential equation for the interface that involves infinite series. The initial value problem is solved numerically, and solutions are compared to the previously published quasi-static solutions. We show that when the internal heat generation and the heat flux at the boundary are close in value to each other, the motion of the phase change front takes longer to reach steady-state than when the values are farther apart. As the difference between the internal heat generation and the heat flux increases, the transient solutions become more dominant and the numerical solution of the phase change front does not reach steady-state before the outer boundary or centerline is reached. The difference between the internal heat generation and the heat flux at the boundary can be used to control the motion and speed of the interface. The problem has applications for a nuclear fuel rod during meltdown.

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Analysis of Creep in Cylinders, Spheres and Thin Discs

Journal of Mechanical Engineering Science ◽

10.1243/jmes_jour_1965_007_013_02 ◽

1965 ◽

Vol 7 (1) ◽

pp. 82-92 ◽

Cited By ~ 2

Author(s):

E. M. Smith

Keyword(s):

Steady State ◽

Heat Generation ◽

Time Variation ◽

Creep Behaviour ◽

Internal Heat ◽

Internal Heat Generation ◽

Steady State Creep ◽

Time Interval ◽

Simultaneous Solution ◽

Short Period

A method of analysing the creep behaviour of cylinders, spheres and thin discs is described, in which the effects of time variation of the environmental conditions of temperature, pressure, rotational speed and internal heat generation may be accommodated. Deformation is considered as a series of steps each consisting of a short period of steady state creep followed by instantaneous stress readjustment to resatisfy the equations of equilibrium and compatibility. The relationships for stress readjustment can be written down once for all, since they are independent of the type of creep law used and of the length of time interval taken. The simultaneous solution of these relationships is presented in a form suitable for incorporation in computer programmes.

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