scholarly journals Transient Formulations for a Heat-Generating Fin with a Temperature-Dependent Heat Transfer Coefficient and Thermal Conductivity

2019 ◽  
Vol 2 (2) ◽  
Keyword(s):  
Heat Transfer ◽  
Boundary Conditions ◽  
Governing Equation ◽  
Neumann Boundary ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Temperature Dependent ◽  
Discrete Equations ◽  
Highly Nonlinear ◽  
Temporal Derivative

Numerical calculations and scalar transport analyses are carried out for transient heat transfer in a heat generating fin with a temperature-dependent heat transfer and conduction coefficients. The highly nonlinear governing equation, satisfies the Dirichlet and Neumann boundary conditions at both ends of the problem domain.Integral representation of the governing equation over the discretized problem domain is achieved via the Green’s second identity together with the so called free- space Green function . This element-driven approach togetherwith the finite difference approximation of the temporal derivative result in discrete equations which are recursive in nature. At the boundaries of any of the adjacent elements, compatibility conditions and/or boundary conditions are enforced to guarantee scalar continuity. After the resulting system of discrete equations are numerically solved and assembled, they yield the transient history of the scalar variables at any particular point in time. Several numerical tests are carried out to ensure the convergence and accuracy of the formulation by comparing numerical results with those found in literature.

scholarly journals Some Exact Solutions of Nonlinear Fin Problem for Steady Heat Transfer in Longitudinal Fin with Different Profiles

2014 ◽  
Vol 2014 ◽  
pp. 1-16 ◽  
Author(s):  
M. D. Mhlongo ◽  
R. J. Moitsheki
Keyword(s):  
Heat Transfer ◽  
Boundary Conditions ◽  
Neumann Boundary ◽  
Dirichlet Boundary Conditions ◽  
Temperature Dependent ◽  
Fin Efficiency ◽  
Highly Nonlinear ◽  
Steady State Heat ◽  
Steady State Heat Transfer ◽  
Steady Heat

One-dimensional steady-state heat transfer in fins of different profiles is studied. The problem considered satisfies the Dirichlet boundary conditions at one end and the Neumann boundary conditions at the other. The thermal conductivity and heat coefficients are assumed to be temperature dependent, which makes the resulting differential equation highly nonlinear. Classical Lie point symmetry methods are employed, and some reductions are performed. Some invariant solutions are constructed. The effects of thermogeometric fin parameter, the exponent on temperature, and the fin efficiency are studied.

Enhanced Passive Thermal Management of Grid-Scale Power Routers Utilizing Ionic Wind

2014 ◽  
Author(s):  
Noris Gallandat ◽  
J. Rhett Mayor
Keyword(s):  
Heat Transfer ◽  
Electric Field ◽  
Mixed Boundary ◽  
Control Volume ◽  
Vertical Channel ◽  
Mixed Boundary Conditions ◽  
Thermal Design ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Ionic Wind

This paper presents a numerical model assessing the potential of ionic wind as a heat transfer enhancement method for the cooling of grid distribution assets. Distribution scale power routers (13–37 kV, 1–10 MW) have stringent requirements regarding lifetime and reliability, so that any cooling technique involving moving parts such as fans or pumps are not viable. Increasing the air flow — and thereby enhancing heat transfer — through Corona discharge could be an attractive solution to the thermal design of such devices. In this work, the geometry of a rectangular, vertical channel with a corona electrode at the entrance is considered. The multiphysics problem is characterized by a set of four differential equations: the Poisson equation for the electric field and conservation equations for electric charges, momentum and energy. The electrodynamics part of the problem is solved using a finite difference approximation (FDA). Solutions for the potential, electric field and free charge density are presented for a rectangular control volume with mixed boundary conditions.

scholarly journals Effective Methods for Solving Band SLEs after Parabolic Nonlinear PDEs

2018 ◽  
Vol 177 ◽  
pp. 07004 ◽  
Author(s):  
Milena Veneva ◽  
Alexander Ayriyan
Keyword(s):  
Heat Transfer ◽  
Order Approximation ◽  
Linear Equations ◽  
Neumann Boundary ◽  
Nonlinear Pdes ◽  
Temperature Dependent ◽  
Piecewise Constant ◽  
Transfer Processes ◽  
Diagonally Dominant ◽  
Symbolic Algorithms

A class of models of heat transfer processes in a multilayer domain is considered. The governing equation is a nonlinear heat-transfer equation with different temperature-dependent densities and thermal coefficients in each layer. Homogeneous Neumann boundary conditions and ideal contact ones are applied. A finite difference scheme on a special uneven mesh with a second-order approximation in the case of a piecewise constant spatial step is built. This discretization leads to a pentadiagonal system of linear equations (SLEs) with a matrix which is neither diagonally dominant, nor positive definite. Two different methods for solving such a SLE are developed - diagonal dominantization and symbolic algorithms.

Cooling of a Hot Torus

2015 ◽  
Vol 138 (4) ◽  
Author(s):  
Rajai S. Alassar ◽  
Mohammed A. Abushoshah
Keyword(s):  
Heat Transfer ◽  
Aspect Ratio ◽  
Boundary Conditions ◽  
Heat Transfer Rate ◽  
Governing Equation ◽  
Transfer Rate ◽  
Radial Direction ◽  
Convective Boundary Conditions ◽  
Convective Boundary ◽  
Toroidal Coordinates

The problem of a hot torus left to cool in a medium of known temperature is studied. We write the governing equation in toroidal coordinates and expand the temperature in terms of a series in the angular direction. The resulting modes in the radial direction are numerically obtained. We consider both isothermal and convective boundary conditions and study the effect of Biot number and aspect ratio on the heat transfer rate.

A Frequency Accurate Temporal Derivative Finite Difference Approximation

1997 ◽  
Vol 05 (04) ◽  
pp. 371-382 ◽  
Author(s):  
Peter A. Orlin ◽  
A. Louise Perkins ◽  
George Heburn
Keyword(s):  
Finite Difference ◽  
Spatial Frequency ◽  
Frequency Domain ◽  
Temporal Frequency ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Linear Matrix ◽  
Matrix Formulation ◽  
Temporal Derivative ◽  
Spatial Frequency Domain

A method is presented for designing temporal derivative finite difference approximations that achieve specified accuracy in the frequency domain. A general average value approximation with undetermined coefficients is fitted in the spatial frequency domain to attain the desired properties of the approximation. A set of constraints to insure that the approximation convergences as the grid spacing approaches zero and satisfies the Lax Equivalence Theorem are imposed on the fitted coefficients. The specification of the underlying partial differential equation is required in order to replace the temporal frequency domain dependence of the approximation with an explicit spatial frequency domain relation based on the dispersion relation of the PDE. A practical design of the approximations is pursued using an heuristic zero placement method which results in a linear matrix formulation.

Closed-form solution for a rectangular stepped fin involving all variable thermal parameters and nonlinear boundary conditions

2016 ◽  
Vol 231 (5) ◽  
pp. 992-1010 ◽  
Author(s):  
Kuljeet Singh ◽  
Ranjan Das ◽  
Rohit K Singla
Keyword(s):  
Heat Transfer ◽  
Boundary Conditions ◽  
Decomposition Method ◽  
Nonlinear Boundary ◽  
Adomian Decomposition Method ◽  
Nonlinear Boundary Conditions ◽  
Thermal Parameters ◽  
Physical Parameters ◽  
Temperature Dependent ◽  
Adomian Decomposition

In this paper, the implementation of the Adomian decomposition method is demonstrated to solve a nonlinear heat transfer problem for a stepped fin involving all temperature-dependent means of heat transfer and nonlinear boundary conditions. Unlike conventional insulated tip assumption, to make the present problem more practical, the fin tip is assumed to disperse heat by convection and radiation. Thermal parameters such as the thermal conductivity, the surface heat transfer coefficient and the surface emissivity are considered to be temperature-dependent. Adomian polynomials are first obtained and then a set of Adomian decomposition method results is validated with pertinent results of the differential transformation method reported in the literature. Effects of different thermo-physical parameters on the temperature distribution and the efficiency have been exemplified. The study reveals that for a given set of conditions, the stepped fin may perform better than the straight fin.

Solving One-Dimensional Partial Differential Equations

2021 ◽  
pp. 191-215
Keyword(s):  
Heat Transfer ◽  
Fluid Flow ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Material Properties ◽  
Final Part ◽  
Temperature Dependent ◽  
Partial Differential ◽  
One Dimensional

This chapter describes the pdepe command, which is used to solve spatially one-dimensional partial differential equations (PDEs). It begins with a description of the standard forms of PDEs and its initial and boundary conditions that the pdepe solver uses. It is shown how various PDEs and boundary conditions can be represented in standard forms. Applications to the mechanics are presented in the final part of the chapter. They illustrate how to solve: heat transfer PDE with temperature dependent material properties, startup velocities of the fluid flow in a pipe, Burger's PDE, and coupled FitzHugh-Nagumo PDE.

Water absorption by swelling seeds. II. Surface condensation boundary condition

Soil Research ◽  
1978 ◽  
Vol 16 (3) ◽  
pp. 291 ◽  
Author(s):  
N Collis-George ◽  
MD Melville
Keyword(s):  
Porous Medium ◽  
Boundary Conditions ◽  
Water Absorption ◽  
Water Vapour ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Vapour Transport ◽  
Supply Source ◽  
Surface Condensation ◽  
Flow Pathway

The previously developed model of water absorption by a swelling seed has been applied to imbibition and germination of wheat seed with the limiting supply condition of purely water vapour transport both with and without the presence of a porous medium. Experimental imbibition and germination data show that with vapour transport, the rate of imbibition and germination of wheat decreases with increasing distance from the supply source and that competition for water vapour occurs between seed. This competition effect was verified by values of the flow pathway shape factor for vapour transport to seed, measured in a three-dimensional electrical analogue. After accounting for heat production and losses with vapour absorption and using appropriate values of the flow pathway shape factors, the equilibrium seed absorption isotherms were used to derive surface condensation boundary conditions for several imbibition situations. These calculations suggest that in the absence of porous media a significant temperature rise of the seed retards the rate of imbibition. By comparison, with a close-packed porous medium around the seed, the large thermal conductivity of the medium should maintain nearly isothermal conditions. Using a finite difference approximation, the swelling seed model and condensation boundary conditions are analysed to give predicted imbibition data which can be compared with experimental data. These results show good agreement between predicted and experimental imbibition in the absence of a porous medium. For seed embedded in a porous medium, the predicted imbibition rate significantly exceeds the experimental rate, but no account could be taken for restriction of seed swelling by the porous medium. The results are discussed in terms of their relevance to the sowing of seed in 'dry' seed bed conditions.

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