Analysis of a Nonlinear Model of Heat Transfer Through a Rectangular Fin: Exact Solutions and Their Multiplicity

2016 ◽  
Vol 12 (2) ◽  
Author(s):  
Mohammad Danish ◽  
Shashi Kumar ◽  
Surendra Kumar
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Thermal Conductivity ◽  
Ordinary Differential Equation ◽  
Exact Solutions ◽  
Reaction Diffusion ◽  
Substitution Method ◽  
Porous Catalyst ◽  
Exact Analytical Solutions ◽  
The Heat Transfer Coefficient

Abstract Exact analytical solutions for the temperature profile and the efficiency of a nonlinear rectangular fin model have been obtained in the forms of well-known algebraic/non-algebraic functions. In the considered nonlinear fin model, the thermal conductivity and the heat transfer coefficient have been assumed to vary as distinct power-law functions of temperature thereby yielding a nonlinear BVP in a 2nd order ODE (ordinary differential equation). These exact solutions have been obtained by employing the derivative substitution method which not only include the solutions of previously studied simplified cases of the same problem but also the solutions of a similar problem of reaction-diffusion process occurring in a porous catalyst slab. These exact solutions have been successfully validated against their numerical counterparts. Besides, effects of various parameters on the obtained solutions have been studied, and the conditions for their existence, uniqueness/multiplicity and stability/instability are analyzed and discussed in detail.

Analysis of Heat Transfer in a Cylindrical Spine Fin with Variable Thermal Properties

2018 ◽  
Vol 387 ◽  
pp. 10-22 ◽  
Author(s):  
Nkejane Fallo ◽  
Raseelo Joel Moitsheki ◽  
Oluwole Daniel Makinde
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Thermal Conductivity ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Numerical Techniques ◽  
Temperature Dependent ◽  
Approximate Analytical Solutions ◽  
Differential Transform ◽  
The Heat Transfer Coefficient

In this paper we analyse the heat transfer in a cylindrical spine fin. Here, both the heat transfer coefficient and thermal conductivity are temperature dependent. The resulting 2+1 dimension partial differential equation (PDE) is rendered nonlinear and difficult to solve exactly, particularly with prescribed initial and boundary conditions. We employ the three dimensional differential transform methods (3D DTM) to contract the approximate analytical solutions. Furthermore we utilize numerical techniques to determine approximate numerical solutions. The effects of parameters, appearing in the boundary value problem (BVP), on temperature profile of the fin are studied.

scholarly journals Asymptotic and Dynamical Analyses of Heat Transfer through a Rectangular Longitudinal Fin

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
C. Harley
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Thermal Conductivity ◽  
Ordinary Differential Equation ◽  
Heat Transfer Coefficient ◽  
Dynamical Analysis ◽  
Temperature Dependent ◽  
Highly Nonlinear ◽  
The Relationship ◽  
Steady Heat

The steady heat transfer through a rectangular longitudinal fin is studied. The thermal conductivity and heat transfer coefficient are assumed to be temperature dependent making the resulting ordinary differential equation (ODE) highly nonlinear. An asymptotic solution is used as a means of understanding the relationship between key parameters. A dynamical analysis is also employed for the same purpose.

Linearization properties, first integrals, nonlocal transformation for heat transfer equation

2016 ◽  
Vol 30 (28n29) ◽  
pp. 1640024 ◽  
Author(s):  
Özlem Orhan ◽  
Teoman Özer
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Thermal Conductivity ◽  
Ordinary Differential Equation ◽  
Heat Transfer Coefficient ◽  
First Integrals ◽  
Heat Transfer Equation ◽  
Order Ordinary Differential Equation ◽  
Linearization Methods ◽  
Nonlocal Transformation

We examine first integrals and linearization methods of the second-order ordinary differential equation which is called fin equation in this study. Fin is heat exchange surfaces which are used widely in industry. We analyze symmetry classification with respect to different choices of thermal conductivity and heat transfer coefficient functions of fin equation. Finally, we apply nonlocal transformation to fin equation and examine the results for different functions.

scholarly journals The Cauchy Problem for the Generalized Hyperbolic Novikov–Veselov Equation via the Moutard Symmetries

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2113
Author(s):  
Alla A. Yurova ◽  
Artyom V. Yurov ◽  
Valerian A. Yurov
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Cauchy Problem ◽  
Exact Solutions ◽  
Airy Function ◽  
The Real ◽  
The Cauchy Problem

We begin by introducing a new procedure for construction of the exact solutions to Cauchy problem of the real-valued (hyperbolic) Novikov–Veselov equation which is based on the Moutard symmetry. The procedure shown therein utilizes the well-known Airy function Ai(ξ) which in turn serves as a solution to the ordinary differential equation d2zdξ2=ξz. In the second part of the article we show that the aforementioned procedure can also work for the n-th order generalizations of the Novikov–Veselov equation, provided that one replaces the Airy function with the appropriate solution of the ordinary differential equation dn−1zdξn−1=ξz.

The Optimum Dimensions of Circular Fins with Variable Thermal Parameters

1980 ◽  
Vol 102 (3) ◽  
pp. 420-425 ◽  
Author(s):  
P. Razelos ◽  
K. Imre
Keyword(s):  
Heat Transfer ◽  
Thermal Conductivity ◽  
Variable Thermal Conductivity ◽  
Linear Variation ◽  
Transfer Coefficients ◽  
Base Thickness ◽  
Optimum Length ◽  
Heat Transfer Coefficients ◽  
Trapezoidal Profile ◽  
The Heat Transfer Coefficient

Optimum dimensions of circular fins of trapezoidal profile with variable thermal conductivity and heat transfer coefficients are obtained. Linear variation of the thermal conductivity is considered of the form k = k0(1 + εT/T0), and the heat transfer coefficient is assumed to vary according to a power law with distance from the bore, expressed as h = K[(r − r0)/(r0 − re)]m. The results for m = 0, 0.8, 2.0, and −0.4 ≤ ε ≤ 0.4, have been expressed by suitable nondimensional parameters which are presented graphically. It is shown that considering the thermal conductivity as constant, the optimum base thickness and volume of the fin are inversely proportional to the thermal conductivity of the material of the fin, while the optimum length and effectiveness are independent of the properties of the material used.

scholarly journals A Comprehensive Review on Theoretical Aspects of Nanofluids: Exact Solutions and Analysis

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 725 ◽  
Author(s):  
Nadeem Ahmad Sheikh ◽  
Dennis Ling Chuan Ching ◽  
Ilyas Khan
Keyword(s):  
Heat Transfer ◽  
Thermal Conductivity ◽  
Thermal Expansion ◽  
Exact Solutions ◽  
Transfer Rate ◽  
Coefficient Of Thermal Expansion ◽  
Literature Survey ◽  
Heat Transfer Characteristics ◽  
Hybrid Nanofluids ◽  
Mass Expansion

In the present era, nanofluids are one of the most important and hot issue for scientists, physicists, and mathematicians. Nanofluids have many important and updated characteristics compared to conventional fluids. The thermal conductivity, thermal expansion, and the heat transfer rate of conventional fluids are not up to the mark for industrial and experimental uses. To overcome these deficiencies, nanoparticles have been dispersed into base fluids to make them more efficient. The heat transfer characteristics through symmetry trapezoidal-corrugated channels can be enhanced using nanofluids. In the present article, a literature survey has been presented for different models of nanofluids and their solutions—particularly, exact solutions. The models for hybrid nanofluids were also mentioned in the present study. Furthermore, some important and most used models for the viscosity, density, coefficient of thermal expansion, coefficient of mass expansion, heat capacitance, electrical conductivity, and thermal conductivity are also presented in tabular form. Moreover, some future suggestions are also provided in this article.

EXPLICIT AND EXACT SOLUTIONS TO THE mKdV-SINE-GORDON EQUATION

2008 ◽  
Vol 22 (15) ◽  
pp. 1471-1485 ◽  
Author(s):  
YUANXI XIE
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Exact Solutions ◽  
Gordon Equation ◽  
Variable Separation ◽  
Sine Gordon Equation ◽  
Simple Manner

By introducing an auxiliary ordinary differential equation and solving it by the method of variable separation, rich types of explicit and exact solutions of the mKdV-sine-Gordon equation are presented in a simple manner.

scholarly journals An investigation into the effects of heat transfer on the motion of a spherical bubble

2004 ◽  
Vol 45 (3) ◽  
pp. 361-371 ◽  
Author(s):  
P. J. Harris ◽  
H. Al-Awadi ◽  
W. K. Soh
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Numerical Results ◽  
Transfer Effects ◽  
Rigid Boundary ◽  
Spherical Bubble ◽  
Heat Transfer Effects ◽  
Surrounding Fluid

AbstractThis paper investigates the effect of heat transfer on the motion of a spherical bubble in the vicinity of a rigid boundary. The effects of heat transfer between the bubble and the surrounding fluid, and the resulting loss of energy from the bubble, can be incorporated into the simple spherical bubble model with the addition of a single extra ordinary differential equation. The numerical results show that for a bubble close to an infiniterigid boundary there are significant differences in both the radius and Kelvin impulse of the bubble when the heat transfer effects are included.

Inverse analysis for the determination of heat transfer coefficient

2013 ◽  
Vol 91 (12) ◽  
pp. 1034-1043 ◽  
Author(s):  
Ali Fguiri ◽  
Naouel Daouas ◽  
M-Sassi Radhouani ◽  
Habib Ben Aissia
Keyword(s):  
Heat Transfer ◽  
Thermal Conductivity ◽  
Heat Transfer Coefficient ◽  
Transfer Coefficient ◽  
Source Strength ◽  
Hot Wire ◽  
Source Power ◽  
Heating Source ◽  
The Heat Transfer Coefficient

The parallel hot wire technique is considered an effective and accurate means of experimental measurement of thermal conductivity. However, the assumptions of infinite medium and ideal infinitely thin and long heat source lead to some restrictions in the applicability of this technique. To make an effective experiment design, a numerical analysis should be carried out a priori, which requires a precise specification of the heating source strength and the heat transfer coefficient on the external surface. In this work, a more accurate physical and mathematical modeling of an experimental setup based on the parallel hot wire method is considered to estimate the two above-mentioned parameters from noisy temperature histories measured inside the material. Based on a sensitivity analysis, the heating source strength is estimated first using early time measurements. With such estimated value, determination of the heat transfer coefficient using temperatures measured at later times is then considered. The Levenberg–Marquardt (LM) method is successfully applied using a single experiment for the inverse solution of the two present parameter estimation problems. Estimates of this gradient-based deterministic method are validated with a stochastic method (Kalman filter). The effects of the measurement location, the heating duration, the measurement time step, and the LM parameter on the estimates and their associated confidence bounds are investigated. Used in the traditional fitting procedure of the parallel hot wire technique, the estimated heating source power provides a reasonable agreement between fitted and exact values of the thermal conductivity and the thermal diffusivity.

Quasi-Steady-State Temperature Distribution in Periodically Contacting Finite Regions

1981 ◽  
Vol 103 (4) ◽  
pp. 739-744 ◽  
Author(s):  
B. Vick ◽  
M. N. O¨zis¸ik
Keyword(s):  
Heat Transfer ◽  
Thermal Conductivity ◽  
Steady State ◽  
Temperature Distribution ◽  
Interface Temperature ◽  
Quasi Steady State ◽  
Exact Analytical Solutions ◽  
One Dimensional ◽  
Steady State Temperature ◽  
Regular Cycle

Heat transfer across two surfaces which make and break contact periodically according to a continuous regular cycle is investigated theoretically and exact analytical solutions are developed for the quasi-steady-state temperature distribution for a two-region, one-dimensional, periodically contacting model. The effects of the Biot number, the thermal conductivity and thermal diffusivity of the materials and the duration of contact and break periods on the interface temperature and the temperature distribution within the solids are illustrated with representative temperature charts.

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