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

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.

Integral Solutions of Phase Change With Internal Heat Generation

2004 ◽  
Author(s):  
Ali Siahpush ◽  
John Crepeau
Keyword(s):  
Differential Equation ◽  
Phase Change ◽  
Heat Generation ◽  
Numerical Solutions ◽  
Internal Heat ◽  
Melting Rate ◽  
Internal Heat Generation ◽  
Stefan Number ◽  
Temperature Boundary ◽  
Solid Liquid

This paper presents solutions to a one-dimensional solid-liquid phase change problem using the integral method for a semi-infinite material that generates internal heat. The analysis assumed a quadratic temperature profile and a constant temperature boundary condition on the exposed surface. We derived a differential equation for the solidification thickness as a function of the internal heat generation (IHG) and the Stefan number, which includes the temperature of the boundary. Plots of the numerical solutions for various values of the IHG and Stefan number show the time-dependant behavior of both the melting and solidification distances and rates. The IHG of the material opposes solidification and enhances melting. The differential equation shows that in steady-state, the thickness of the solidification band is inversely related to the square root of the IHG. The model also shows that the melting rate initially decreases and reaches a local minimum, then increases to an asymptotic value.

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.

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

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.

Computational Analysis of Laminar Natural Convection in Rectangular Enclosures of Different Aspect Ratios With Different Heating Conditions

2012 ◽  
Author(s):  
Mosfequr Rahman ◽  
Charles Walker ◽  
Gustavo Molina ◽  
Valentin Soloiu
Keyword(s):  
Heat Flux ◽  
Natural Convection ◽  
Rayleigh Number ◽  
Temperature Gradient ◽  
Aspect Ratio ◽  
Heat Generation ◽  
Internal Heat ◽  
Internal Heat Generation ◽  
External Temperature ◽  
Aspect Ratios

Natural convection in rectangular enclosures is found in many real-world engineering applications. Included in these applications are the energy efficient design of buildings, operation and safety of nuclear reactors, solar collector design, passive energy storage, heat transfer across multi-pane windows, thermo-electric refrigeration and heating devices, and the design-for-mitigation of optical distortion in large-scale laser systems. A common industrial application of natural convection is free air cooling without the aid of fans and can happen on small scales such as computer chips to large scale process equipment. The enclosure phenomena can loosely be organized into two large classes: (1) horizontal enclosures heated from below and (2) vertical enclosures heated from the side. In addition to temperature gradient convection strength within the enclosure can vary due to the existence of heat sources with different strength. Numerical simulations are conducted for free convective flow of air with or without internal heat generation in two-dimensional rectangular enclosures of different aspect ratios. The objective of this numerical study is to investigate the effects of external temperature gradient, internal heat generation and aspect ratio (AR) of enclosure (ratio of the length of the isothermal walls to their separation distance), in free convective laminar flow of a fluid. Two-dimensional rectangular enclosures of different aspect ratio (1, 2, 4, 6, 8, and 10) with two adiabatic side walls and isothermal bottom (hot) and top (cold) walls are considered for the first configuration. Whereas for the second configuration, two adiabatic top and bottom walls, isothermal left side (cold) and right side (hot) walls are considered. Two principal parameters considered for the flow of fluid are the external Rayleigh number, RaE, which represents the effect due to the differential heating of the isothermal walls, and the internal Rayleigh number, RaI, which represents the strength of the internal heat generation. The effect of external temperature gradient and aspect ratio on natural convection has been observed by varying the value of external Rayleigh number (RaE) equal to 2×104, 2×105, and 2×106 and keeping the internal Rayleigh number constant (RaI = 2×105). Similarly, the effect of internal heat generation and aspect ratio on natural convection has been observed by varying the value of internal Rayleigh number (RaI) equal to 2×104, 2×105, and 2×106 and keeping the external Rayleigh number constant (RaE = 2×105). Significant changes in flow patterns and isotherms have been observed for all cases. Also the variation of average heat flux ratio (convective heat flux/corresponding conduction heat flux) along the hot and cold walls, and the convection strength have been calculated for all cases. It is found that the aspect ratio has a significant effect in fluid flow and heat transfer in the enclosures. The average heat flux ratio and the strength of convection increase with aspect ratio as the enclosure shape changes square (AR = 1) to shallow (AR > 1).

Numerical Solution for Variable Viscosity and Internal Heat Generation Effects on Boundary Layer Flow Over an Exponentially Stretching Porous Sheet with Constant Heat Flux and Thermal Radiation

2014 ◽  
Vol 30 (4) ◽  
pp. 395-402 ◽  
Author(s):  
A. M. Megahed
Keyword(s):  
Heat Flux ◽  
Heat Generation ◽  
Variable Viscosity ◽  
Internal Heat ◽  
Constant Heat Flux ◽  
Internal Heat Generation ◽  
Constant Heat ◽  
Suction Parameter ◽  
Viscosity Parameter ◽  
Exponentially Stretching

AbstractA numerical study has been carried out to analyze the constant heat flux, internal heat generation, variable viscosity and thermal radiation effects on the flow and heat transfer of a Newtonian fluid over an exponentially stretching porous sheet. Using a similarity transformation, the governing partial differential equations are transformed into coupled, non-linear ordinary differential equations with variable coefficients. Numerical solutions to these equations subject to appropriate boundary conditions are obtained by using an efficient Chebyshev spectral method. The effects of various physical parameters such as viscosity parameter, the suction parameter, the radiation parameter, internal heat generation or absorption parameter and the Prandtl number on velocity and temperature are discussed by using graphical approach. Moreover, numerical results indicate that in the presence of constant heat flux, the skin-friction coefficient as well as Nusselt number is strongly affected by the viscosity parameter, suction parameter, radiation parameter and the internal heat generation parameter.

Heat Transfer From Solids With Variable Thermal Conductivity and Uniform Internal Heat Generation Using Homotopy Perturbation Method

2008 ◽  
Author(s):  
Rahim Gul ◽  
Zafar H. Khan ◽  
Waqar A. Khan
Keyword(s):  
Heat Transfer ◽  
Thermal Conductivity ◽  
Perturbation Method ◽  
Heat Generation ◽  
Homotopy Perturbation Method ◽  
Outer Boundary ◽  
Internal Heat ◽  
Internal Heat Generation ◽  
Dimensionless Temperature ◽  
Homotopy Perturbation

Homotopy perturbation method (HPM) is employed to investigate the effects of temperature dependent thermal conductivity and internal heat generation on the dimensionless temperature distribution and heat transfer from solids of arbitrary shapes (rectangular, cylindrical and spherical). Dirichlet and Robin boundary conditions are applied at the outer boundary of the solids.

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