scholarly journals Application of Heat Balance Integral Methods to One-Dimensional Phase Change Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
S. L. Mitchell ◽  
T. G. Myers
Keyword(s):  
Phase Change ◽  
Least Squares ◽  
Heat Balance ◽  
Integral Method ◽  
New Method ◽  
Error Measure ◽  
One Dimensional ◽  
Integral Methods ◽  
Heat Balance Integral Method ◽  
The Heat Balance

We employ the Heat Balance Integral Method (HBIM) to solve a number of thermal and phase change problems which occur in a multitude of industrial contexts. As part of our analysis, we propose a new error measure for the HBIM that combines the least-squares error with a boundary immobilisation method. We describe how to determine this error for three basic thermal problems and show how it can be used to determine an optimal heat balance formulation. We then show how the HBIM, with the new error measure, may be used to approximate the solution of an aircraft deicing problem. Finally we apply the new method to two industrially important phase change problems.

The Heat Balance Integral in Steady-State Conduction

1976 ◽  
Vol 98 (3) ◽  
pp. 466-470 ◽  
Author(s):  
A. A. Sfeir
Keyword(s):  
Heat Balance ◽  
Integral Method ◽  
Two Dimensional ◽  
One Dimensional ◽  
Heat Balance Integral Method ◽  
Extended Surfaces ◽  
Different Shapes ◽  
The One ◽  
Biot Numbers ◽  
The Heat Balance

The Heat Balance Integral Method is applied to solve for the heat flow and temperature distribution in extended surfaces of different shapes and boundary conditions. In most cases the analysis is found to be identical to the exact two-dimensional solutions at Biot numbers for which the one-dimensional analysis is almost 100 percent off. Other possible extensions of the method are briefly described.

scholarly journals Approximate solutions to one-phase Stefan-like problems with space-dependent latent heat

2020 ◽  
pp. 1-33
Author(s):  
J. BOLLATI ◽  
D. A. TARZIA
Keyword(s):  
Latent Heat ◽  
Heat Balance ◽  
Integral Method ◽  
Approximate Solutions ◽  
Convective Condition ◽  
One Dimensional ◽  
Stefan Number ◽  
Integral Methods ◽  
Approximate Integral ◽  
The Heat Balance

The work in this paper concerns the study of different approximations for one-dimensional one-phase Stefan-like problems with a space-dependent latent heat. It is considered two different problems, which differ from each other in their boundary condition imposed at the fixed face: Dirichlet and Robin conditions. The approximate solutions are obtained by applying the heat balance integral method (HBIM), the modified HBIM and the refined integral method (RIM). Taking advantage of the exact analytical solutions, we compare and test the accuracy of the approximate solutions. The analysis is carried out using the dimensionless generalised Stefan number (Ste) and Biot number (Bi). It is also studied the case when Bi goes to infinity in the problem with a convective condition, recovering the approximate solutions when a temperature condition is imposed at the fixed face. Some numerical simulations are provided in order to assert which of the approximate integral methods turns out to be optimal. Moreover, we pose an approximate technique based on minimising the least-squares error, obtaining also approximate solutions for the classical Stefan problem.

scholarly journals Heat-balance integral to fractional (half-time) heat diffusion sub-model

2010 ◽  
Vol 14 (2) ◽  
pp. 291-316 ◽  
Author(s):  
Jordan Hristov
Keyword(s):  
Diffusion Equation ◽  
Heat Balance ◽  
Integral Method ◽  
Time Derivative ◽  
Heat Diffusion ◽  
Half Time ◽  
Heat Diffusion Equation ◽  
Parabolic Profile ◽  
Heat Balance Integral Method ◽  
The Heat Balance

The fractional (half-time) sub-model of the heat diffusion equation, known as Dirac-like evolution diffusion equation has been solved by the heat-balance integral method and a parabolic profile with unspecified exponent. The fractional heat-balance integral method has been tested with two classic examples: fixed temperature and fixed flux at the boundary. The heat-balance technique allows easily the convolution integral of the fractional half-time derivative to be solved as a convolution of the time-independent approximating function. The fractional sub-model provides an artificial boundary condition at the boundary that closes the set of the equations required to express all parameters of the approximating profile as function of the thermal layer depth. This allows the exponent of the parabolic profile to be defined by a straightforward manner. The elegant solution performed by the fractional heat-balance integral method has been analyzed and the main efforts have been oriented towards the evaluation of fractional (half-time) derivatives by use of approximate profile across the penetration layer.

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