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.

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.

Highly Subcooled Flow Boiling: A Model for Estimating Heat Transfer Behind Sliding Bubbles in a Narrow Channel

2012 ◽  
Author(s):  
Arif B. Ozer ◽  
Donald K. Hollingsworth ◽  
Larry. C. Witte
Keyword(s):  
Heat Transfer ◽  
Flow Field ◽  
Heat Generation ◽  
Wall Temperature ◽  
Flow Boiling ◽  
Narrow Channel ◽  
Heated Wall ◽  
Uniform Heat ◽  
Proposed Model ◽  
Sliding Bubbles

A quenching/diffusion analytical model has been developed for predicting the wall temperature and wall heat flux behind bubbles sliding in a confined narrow channel. The model is based on the concept of a well-mixed liquid region that enhances the heat transfer near the heated wall behind the bubble. Heat transfer in the liquid is treated as a one-dimensional transient conduction process until the flow field recovers back to its undisturbed level prior to bubble passage. The model is compared to experimental heat transfer results obtained in a high-aspect-ratio (1.2×23mm) rectangular, horizontal channel with one wide wall forming a uniform-heat-generation boundary and the other designed for optical access to the flow field. The working fluid was Novec™ 649. A thermochromic liquid crystal coating was applied to the outside of the uniform-heat-generation boundary, so that wall temperature variations could be obtained and heat transfer coefficients and Nusselt numbers could be obtained. The experiments were focused on high inlet subcooling, typically 15–50°C. The model is able to capture the elevated heat transfer rates measured in the channel without the need to consider nucleate boiling from the surface or microlayer evaporation from the sliding bubbles. Surface temperatures and wall heat fluxes were estimated for 17 different experimental conditions using the proposed model. Results agreed with the measured values within ±15% accuracy. The insight gathered from comparing the results of the proposed model to experimental results provides the basis for a better understanding of the physics of subcooled bubbly flow in narrow channels.

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.

The numerical solution of the heat conduction equation in one dimension

1964 ◽  
Vol 60 (4) ◽  
pp. 897-907 ◽  
Author(s):  
M. Wadsworth ◽  
A. Wragg
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Heat Conduction ◽  
Boundary Conditions ◽  
Numerical Method ◽  
Heat Conduction Equation ◽  
Linear Equations ◽  
Conduction Equation ◽  
Partial Differential ◽  
Direct Numerical Method

AbstractThe replacement of the second space derivative by finite differences reduces the simplest form of heat conduction equation to a set of first-order ordinary differential equations. These equations can be solved analytically by utilizing the spectral resolution of the matrix formed by their coefficients. For explicit boundary conditions the solution provides a direct numerical method of solving the original partial differential equation and also gives, as limiting forms, analytical solutions which are equivalent to those obtainable by using the Laplace transform. For linear implicit boundary conditions the solution again provides a direct numerical method of solving the original partial differential equation. The procedure can also be used to give an iterative method of solving non-linear equations. Numerical examples of both the direct and iterative methods are given.

On the Approximate Solutions of Heat-Conduction Problems

1963 ◽  
Vol 85 (3) ◽  
pp. 203-207 ◽  
Author(s):  
Fazil Erdogan
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equation ◽  
Heat Conduction ◽  
Approximate Solutions ◽  
Integral Transforms ◽  
Partial Differential ◽  
The Given

Integral transforms are used in the application of the weighted residual methods to the solution of problems in heat conduction. The procedure followed consists in reducing the given partial differential equation to an ordinary differential equation by successive applications of appropriate integral transforms, and finding its solution by using the weighted-residual methods. The undetermined coefficients contained in this solution are functions of transform variables. By inverting these functions the coefficients are obtained as functions of the actual variables.

The integral of geometric Brownian motion

2001 ◽  
Vol 33 (1) ◽  
pp. 223-241 ◽  
Author(s):  
Daniel Dufresne
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Brownian Motion ◽  
Laplace Transform ◽  
Finite Time ◽  
Geometric Brownian Motion ◽  
Time Interval ◽  
Finite Time Interval ◽  
Special Cases ◽  
The Laplace Transform

This paper is about the probability law of the integral of geometric Brownian motion over a finite time interval. A partial differential equation is derived for the Laplace transform of the law of the reciprocal integral, and is shown to yield an expression for the density of the distribution. This expression has some advantages over the ones obtained previously, at least when the normalized drift of the Brownian motion is a non-negative integer. Bougerol's identity and a relationship between Brownian motions with opposite drifts may also be seen to be special cases of these results.

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 Types of flow in Green-Naghdi Thermoelasticity Theory

2020 ◽  
Vol 5 (6) ◽  
Author(s):  
Gilmar Garbugio ◽  
Henrique De Almeida Silva Mascalhusk Bernardo Leite
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Free Energy ◽  
Heat Conduction ◽  
Helmholtz Free Energy ◽  
Hyperbolic Pde ◽  
Thermal Signal ◽  
Parabolic Pde ◽  
Different Types ◽  
Entropy Flux

In this paper we will show a review of the Green-Naghdi thermoelasticity theory.Such model have beautiful foundations which contains since the law of heat conduction proposed byFourier, fundamentals variables such as Helmholtz free energy, entropy, flux of heat and etc. and the constitutive hypothesis. The nature of partial differential equation of parabolic (PDE) type derived from classical thermoelasticity lead us to a mathematical inconsistency well-know as thermal signal speed paradox. We will show how Green-Naghdi model solved those mathematical inconsistency introducing different types of flow and giving us a hyperbolic PDE.

scholarly journals Heat Conduction On Pan Using Portable Parabolic Stove With Addition Of Flat Mirror

2020 ◽  
Vol 6 ◽  
Author(s):  
Benedictus Mardwianta ◽  
Abdul Haris Subarjo ◽  
Wayan Wiardefan
Keyword(s):  
Heat Transfer ◽  
Heat Conduction ◽  
Solar Radiation ◽  
Surface Temperature ◽  
Experimental Method ◽  
Wall Temperature ◽  
Open Space ◽  
The Sun ◽  
Parabolic Mirror ◽  
Flat Mirror

This research aims to develop the parabolic stove with addition of some flat mirrors around the parabolic mirror. It will increase the heat transfer of conduction in the pan. The parabolic itself has around and concave shape, making it suitable for concentrating solar energy. The experimental method was carried out in this research and the test was carried out in an open space with solar radiation intensities with ranging from 169.6 W/m² to 974.4 W/m². The results of heat conduction on a pan without the addition of a flat mirror generate a 105.15 Watt, addition of one flat mirror will generate a 174.82 Watt, addition of two  flat mirrors will generate a 259.24 Watt, addition of three flat mirrors will generate a 342.79 Watt and addition of four flat mirrors will generate a 412.26 Watt. The heat conduction depends on the intensity of the sun caught by the reflector. If the sun intensity decreases, the surface temperature between of the outer pan wall (T1) and the inner wall temperature (T2) will decrease too. Keywords: Heat conduction, sun intensity, parabolic stove

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