Invariance, Conservation Laws, and Exact Solutions of the Nonlinear Cylindrical Fin Equation

2014 ◽  
Vol 69 (5-6) ◽  
pp. 195-198 ◽  
Author(s):  
Saeed M. Ali ◽  
Ashfaque H. Bokhari ◽  
Fiazuddin D. Zaman ◽  
Abdul H. Kara
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Partial Differential Equation ◽  
Heat Exchange ◽  
Thermal Diffusivity ◽  
Conservation Laws ◽  
Cylindrical Shape ◽  
Method Of Multipliers ◽  
Double Reduction ◽  
Temperature Dependent

Fins are heat exchange surfaces which are used widely in industry. The partial differential equation arising from heat transfer in a fin of cylindrical shape with temperature dependent thermal diffusivity are studied. The method of multipliers and invariance of the differential equations is employed to obtain conservation laws and perform double reduction.

scholarly journals A NOTE ON TRANSIENT HEAT TRANSFER PROBLEMS WITH TEMPERATURE-DEPENDENT THERMAL CONDUCTIVITY AND THERMAL DIFFUSIVITY

2020 ◽  
Vol 19 (1) ◽  
pp. 90
Author(s):  
R. P. S. da Gama ◽  
J. R. Cerqueira ◽  
R. M. S. da Gama
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Thermal Diffusivity ◽  
Nonlinear Term ◽  
Numerical Procedure ◽  
Transient Heat Transfer ◽  
Temperature Dependent ◽  
Kirchhoff Transformation ◽  
Transfer Problems ◽  
Temperature Dependent Thermal Conductivity

In this work it is presented a numerical procedure for solving transient heat transfer problems in which the thermal diffusivity is strongly dependent on the temperature, with the aid of the Kirchhoff transformation associated to an usual finite difference approach. The first step consists of eliminating the nonlinear terms associated to the derivatives with respect to the position, by means of a Kirchhoff transformation, giving rise to a partial differential equation with only one nonlinear term (involving the coefficient of the derivative with respect to the time). The advance in time is carried out assuming the thermal diffusivity evaluated at a known temperature, giving rise to a semi-implicit scheme. Comparisons between this approach and the usual hypothesis are carried out in order to illustrate the effect of the dependence between the temperature and the thermal diffusivity. Some typical results are presented, based on the (6H-SiC) Silicon Carbide properties.

scholarly journals Метод восстановления моделей тепломассопереноса по пространственно-временным распределениям параметров

2021 ◽  
Vol 47 (24) ◽  
pp. 9
Author(s):  
Н.Ю. Быков ◽  
А.А. Хватов ◽  
А.В. Калюжная ◽  
А.В. Бухановский
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Partial Differential Equation ◽  
Phase Transitions ◽  
Laser Radiation ◽  
High Efficiency ◽  
Design Method ◽  
Temperature Dependent ◽  
Partial Differential ◽  
Correct Structure

An algorithm of the generative design method for reconstruction of heat transfer models from the available data is proposed. The method is applied to generate a partial differential equation describing the process of heating and evaporation of a metal, the surface of which is heated by laser radiation. The high efficiency of the method was demonstrated for the purpose of reconstructing the correct structure of the equation, indicating additional processes accompanying heating as phase transitions, and also for determining the values of the temperature-dependent coefficients of the derivatives.

scholarly journals Symmetries, Conservation Laws, and Wave Equation on the Milne Metric

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
Ahmad M. Ahmad ◽  
Ashfaque H. Bokhari ◽  
F. D. Zaman
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Wave Equation ◽  
Conservation Laws ◽  
Mathematical Physics ◽  
Lie Point Symmetries ◽  
Noether Symmetries ◽  
Physical Systems ◽  
Action Integral ◽  
Lorentzian Metric

Noether symmetries provide conservation laws that are admitted by Lagrangians representing physical systems. For partial differential equation possessing Lagrangians these symmetries are obtained by the invariance of the corresponding action integral. In this paper we provide a systematic procedure for determining Noether symmetries and conserved vectors for a Lagrangian constructed from a Lorentzian metric of interest in mathematical physics. For completeness, we give Lie point symmetries and conservation laws admitted by the wave equation on this Lorentzian metric.

scholarly journals A Numerical Well-Balanced Scheme for One-Dimensional Heat Transfer in Longitudinal Triangular Fins

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
I. Rusagara ◽  
C. Harley
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Partial Differential Equation ◽  
Nonlinear Partial Differential Equation ◽  
Transfer Coefficients ◽  
Partial Differential ◽  
One Dimensional ◽  
Heat Transfer Coefficients ◽  
Highly Nonlinear ◽  
Triangular Fin

The temperature profile for fins with temperature-dependent thermal conductivity and heat transfer coefficients will be considered. Assuming such forms for these coefficients leads to a highly nonlinear partial differential equation (PDE) which cannot easily be solved analytically. We establish a numerical balance rule which can assist in getting a well-balanced numerical scheme. When coupled with the zero-flux condition, this scheme can be used to solve this nonlinear partial differential equation (PDE) modelling the temperature distribution in a one-dimensional longitudinal triangular fin without requiring any additional assumptions or simplifications of the fin profile.

scholarly journals Analytic transient solutions of a cylindrical heat equation

Filomat ◽  
2021 ◽  
Vol 35 (8) ◽  
pp. 2617-2628
Author(s):  
K.Y. Kung ◽  
Man-Feng Gong ◽  
H.M. Srivastava ◽  
Shy-Der Lin
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Partial Differential Equation ◽  
Heat Conduction ◽  
Heat Equation ◽  
Heat Conduction Equation ◽  
Separation Of Variables ◽  
Transient Heat Conduction ◽  
Transient Temperature ◽  
Transient Solutions

The principles of superposition and separation of variables are used here in order to investigate the analytical solutions of a certain transient heat conduction equation. The structure of the transient temperature appropriations and the heat-transfer distributions are summed up for a straight mix of the results by means of the Fourier-Bessel arrangement of the exponential type for the investigated partial differential equation.

A Two-Temperature Model of the Regenerative Solid-Vapor Heat Pump

1994 ◽  
Vol 116 (4) ◽  
pp. 297-304 ◽  
Author(s):  
T. A. Fuller ◽  
W. J. Wepfer ◽  
S. V. Shelton ◽  
M. W. Ellis
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Partial Differential Equation ◽  
Heat Pump ◽  
Flow Direction ◽  
Time And Space ◽  
Partial Differential ◽  
Thermodynamic Performance ◽  
Heat Regeneration ◽  
Two Temperature

A thermally driven heat pump using a solid/vapor adsorption/desorption compression process is thermodynamically analyzed. Heat regeneration between the two adsorbent beds is accomplished through the use of a circulating heat transfer (HX) fluid. Effective heat regeneration and system performance requires that steep thermal profiles or waves be established in the beds along the path of the HX-fluid flow direction. Previous studies by Shelton, Wepfer, and Miles have used square and ramp profiles to approximate the temperature profiles in the adsorbent beds, which, in turn, enable the thermodynamic performance of the heat pump to be computed. In this study, an integrated heat transfer and thermodynamic model is described. The beds are modeled using a two-temperature approach. A partial differential equation for the lumped adsorbent bed and tube is developed to represent the bed temperature as a function of time and space (along the flow direction), while a second partial differential equation is developed for the heat transfer fluid to represent the fluid temperature as a function of time and space (along the flow direction). The resulting differential equations are nonlinear due to pressure and temperature-dependent coefficients. Energy and mass balances are made at each time step to compute the bed pressure, mass, adsorption level, and energy changes that occur during the adsorption and desorption process. Using these results, the thermodynamic performance of the heat pump is calculated. Results showing the heat pump’s performance and capacity as a function of the four major dimensionless groups, DR, Pe, Bi, and KAr, are presented.

scholarly journals Modelling heat transfer in heterogeneous media using fractional calculus

2013 ◽  
Vol 371 (1990) ◽  
pp. 20120146 ◽  
Author(s):  
Dominik Sierociuk ◽  
Andrzej Dzieliński ◽  
Grzegorz Sarwas ◽  
Ivo Petras ◽  
Igor Podlubny ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Partial Differential Equation ◽  
Heat Flux ◽  
Transfer Function ◽  
Transfer Process ◽  
Heterogeneous Media ◽  
Heat Transfer Process ◽  
Order Partial Differential Equation ◽  
Partial Differential

This paper presents the results of modelling the heat transfer process in heterogeneous media with the assumption that part of the heat flux is dispersed in the air around the beam. The heat transfer process in a solid material (beam) can be described by an integer order partial differential equation. However, in heterogeneous media, it can be described by a sub- or hyperdiffusion equation which results in a fractional order partial differential equation. Taking into consideration that part of the heat flux is dispersed into the neighbouring environment we additionally modify the main relation between heat flux and the temperature, and we obtain in this case the heat transfer equation in a new form. This leads to the transfer function that describes the dependency between the heat flux at the beginning of the beam and the temperature at a given distance. This article also presents the experimental results of modelling real plant in the frequency domain based on the obtained transfer function.

Effect of Arbitrary Nonsteady Wall Temperature on Incompressible Heat Transfer

1962 ◽  
Vol 84 (4) ◽  
pp. 347-351 ◽  
Author(s):  
T. R. Goodman
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Partial Differential Equation ◽  
Heat Conduction ◽  
Wall Temperature ◽  
Integral Method ◽  
Nonsteady Heat ◽  
Uniform Heat ◽  
Special Cases ◽  
Title Problem

The title problem is solved using an integral method and ignoring viscous dissipation. A partial differential equation is derived which yields as special cases Lighthill’s non-uniform heat-transfer formula and the nonsteady heat conduction in a slab. The differential equation is then specialized to the nonsteady but uniform heat transfer on a flat plate. Comparisons with other solutions are made when available, and it is shown that the integral method produces accuracy of a few per cent in these limiting cases.

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