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.

Effects of Internal Heat Generation on Solidification

2005 ◽  
Author(s):  
John C. Crepeau ◽  
Ali Siahpush
Keyword(s):  
Heat Generation ◽  
Solid Phase ◽  
Freezing Point ◽  
Internal Heat ◽  
Internal Heat Generation ◽  
Square Root ◽  
Stefan Number ◽  
Temperature Boundary ◽  
Geometry Problem ◽  
Solid Liquid

We present solutions for solid-liquid phase change in materials that generate internal heat. This problem is solved for both cylindrical and semi-infinite geometries. The analysis assumes a temperature profile in the solid phase and constant temperature boundary conditions on the exposed surfaces. We derive differential equations governing the solidification thickness for both geometries as functions of the Stefan number and the internal heat generation (IHG). For the cylindrical geometry, the solidification layer obtains a steady-state value which is related to the inverse of the square root of the IHG. The solutions to the semi-infinite geometry problem show that when the surface is cooled to below the freezing point, a solidification layer forms along the edge and begins to grow until it reaches a maximum, then begins remelt.

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.

Finite element thermal analysis of a moving porous fin with temperature-variant thermal conductivity and internal heat generation

2020 ◽  
Vol 1 (1) ◽  
pp. 110
Author(s):  
Gbeminiyi Sobamowo ◽  
Keyword(s):  
Heat Transfer ◽  
Thermal Conductivity ◽  
Finite Element ◽  
Heat Generation ◽  
Peclet Number ◽  
Numerical Solutions ◽  
Internal Heat ◽  
Internal Heat Generation ◽  
Péclet Number ◽  
Porous Fin

This paper focuses on finite element analysis of the thermal behaviour of a moving porous fin with temperature-variant thermal conductivity and internal heat generation. The numerical solutions are used to investigate the effects of Peclet number, Hartmann number, porous and convective parameters on the temperature distribution, heat transfer and efficiency of the moving fin. The results show that when the convective and porous parameters increase, the adimensional fin temperature decreases. However, the value of the fin temperature is amplified as the value Peclet number is enlarged. Also, an increase in the thermal conductivity and the internal heat generation cause the fin temperature to fall and the rate of heat transfer from the fin to decrease. Therefore, the operational parameters of the fin must be carefully selected to avoid thermal instability in the fin.

Analysis of Nonlinear Heat Transfer in Solids of Various Geometries with Variable Thermal Conductivity and Heat Source

2017 ◽  
Vol 377 ◽  
pp. 1-16
Author(s):  
Raseelo Joel Moitsheki ◽  
Oluwole Daniel Makinde
Keyword(s):  
Heat Transfer ◽  
Thermal Conductivity ◽  
Exact Solutions ◽  
Heat Generation ◽  
Numerical Solutions ◽  
Internal Heat ◽  
Variable Thermal Conductivity ◽  
Internal Heat Generation ◽  
Temperature Dependent ◽  
Power Law Function

In this paper we consider heat transfer in a hot body with different geometries. Here, the thermal conductivity and internal heat generation are both temperature-dependent. This assumption rendered the model considered to be nonlinear. We assume that thermal conductivity is given by a power law function. We employ the preliminary group classification to determine the cases of internal heat generation for which the principal Lie algebra extends by one. Exact solutions are constructed for the case when thermal conductivity is a differential consequence of internal heat generation term. We derive the approximate numerical solutions for the cases where exact solutions are difficult to construct or are nonexistent. The effects of parameters appearing in the model on temperature profile are studied.

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.

Numerical Treatment of the MHD Convective Heat and Mass Transfer in an Electrically Conducting Fluid over an Infinite Solid Surface in Presence of the Internal Heat Generation

2003 ◽  
Vol 58 (11) ◽  
pp. 601-611 ◽  
Author(s):  
N. T. Eldabe ◽  
A. G. El-Sakka ◽  
Ashraf Fouad
Keyword(s):  
Mass Transfer ◽  
Heat Generation ◽  
Solid Surface ◽  
Numerical Solutions ◽  
Internal Heat ◽  
Internal Heat Generation ◽  
Conducting Fluid ◽  
Electrically Conducting ◽  
Linear Partial Differential Equations ◽  
Electrically Conducting Fluid

Numerical solutions of a set of non-linear partial differential equations are investigated. We obtained the velocity distribution of a conducting fluid flowing over an infinite solid surface in the presence of an uniform magnetic field and internal heat generation. The temperature and concentration distributions of the fluid are studied as well as the skin-friction, rate of mass transfer and local wall heat flux. The effect of the parameters of the problem on these distributions is illustrated graphically.

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