The Transversal Method of Lines (TMOL) Applied to Unsteady Conduction in Large Plates, Long Cylinders and Spheres With Prescribed Surface Heat Flux

2011 ◽  
Author(s):  
Antonio Campo ◽  
Ramin Soujoudi ◽  
Adelina Davis
Keyword(s):  
Heat Flux ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Surface Heat Flux ◽  
Time Derivative ◽  
Method Of Lines ◽  
Surface Heat ◽  
First Order ◽  
Temperature Distributions ◽  
Cylinders And Spheres

The Transversal Method Of Lines (TMOL) or Rothe method is a general technique for solving parabolic partial differential equations that uses a two-point backward finite-difference formulation for the time derivative and differential spatial derivatives. This hybrid approach leads to transformed ordinary differential equations where the spatial coordinate is the independent variable and the time appears as an embedded parameter. The transformed ordinary differential equations may have constant or variable coefficients depending on the coordinate system and are first-order accurate. In this work, TMOL is applied to the 1-D heat equation for large plates, long cylinders and spheres with constant thermophysical properties, uniform initial temperature and prescribed surface heat flux. The analytic solutions of the adjoint heat equations are performed with the symbolic Maple software. It is demonstrated that the approximate semi-analytic TMOL temperature distributions for the three simple bodies are much better than first-order accurate. This signifies that TMOL temperature distributions are not only valid for short times, but they are valid for the entire heating period involving short, moderate and long times.

Transversal Method of Lines for Unsteady Heat Conduction With Uniform Surface Heat Flux

2014 ◽  
Vol 136 (11) ◽  
Author(s):  
Antonio Campo ◽  
José Garza
Keyword(s):  
Heat Flux ◽  
Differential Equations ◽  
Heat Equation ◽  
Ordinary Differential Equations ◽  
Surface Heat Flux ◽  
Method Of Lines ◽  
Surface Heat ◽  
Second Order ◽  
Partial Differential ◽  
Stationary Heat

The transversal method of lines (TMOL) is a general hybrid technique for determining approximate, semi-analytic solutions of parabolic partial differential equations. When applied to a one-dimensional (1D) parabolic partial differential equation, TMOL engenders a sequence of adjoint second-order ordinary differential equations, where in the space coordinate is the independent variable and the time appears as an embedded parameter. Essentially, the adjoint second-order ordinary differential equations that result are of quasi-stationary nature, and depending on the coordinate system may have constant or variable coefficients. In this work, TMOL is applied to the unsteady 1D heat equation in simple bodies (large plate, long cylinder, and sphere) with temperature-invariant thermophysical properties, constant initial temperature and uniform heat flux at the surface. In engineering applications, the surface heat flux is customarily provided by electrical heating or radiative heating. Using the first adjoint quasi-stationary heat equation for each simple body with one time jump, it is demonstrated that approximate, semi-analytic TMOL temperature solutions with good quality are easily obtainable, regardless of time. As a consequence, usage of the more involved second adjoint quasi-stationary heat equation accounting for two consecutive time jumps come to be unnecessary.

scholarly journals Convective Flow Of Micropolar Fluids Along An Inclined Flat Plate With Variable Electric Conductivity And Uniform Surface Heat Flux

1970 ◽  
Vol 6 (1) ◽  
pp. 69-79 ◽  
Author(s):  
Md Jashim Uddin
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Heat Flux ◽  
Differential Equations ◽  
Flat Plate ◽  
Electric Conductivity ◽  
Surface Heat Flux ◽  
Convective Flow ◽  
Surface Heat ◽  
Micropolar Fluids

Magnetohydrodynamic (MHD) twodimensional steady convective flow and heat transfer of micropolar fluids flow along an inclined flat plate with variable electric conductivity and uniform surface heat flux has been analyzed numerically in the presence of heat generation. With appropriate transformations the boundary layer partial differential equations are transformed into nonlinear ordinary differential equations. The local similarity solutions of the transformed dimensionless equations for the velocity flow, microrotation and heat transfer characteristics are assessed using Nachtsheim- Swigert shooting iteration technique along with the sixth order Runge-Kutta-Butcher initial value solver. Numerical results are presented graphically in the form of velocity, microrotation, and temperature profiles within the boundary layer for different parameters entering into the analysis. The effects of the pertinent parameters on the local skin-friction coefficient (viscous drag), plate couple stress and the rate of heat transfer (Nusselt number) are also discussed and displayed graphically. Keywords: Convective flow; Micropolar fluid; Heat transfer; Electric conductivity; Inclined plate; Locally self-similar solution DOI: http://dx.doi.org/10.3329/diujst.v6i1.9336 DIUJST 2011; 6(1): 69-79

scholarly journals Stagnation-Point Flow toward a Vertical, Nonlinearly Stretching Sheet with Prescribed Surface Heat Flux

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Sin Wei Wong ◽  
M. A. Omar Awang ◽  
Anuar Ishak
Keyword(s):  
Heat Flux ◽  
Differential Equations ◽  
Stagnation Point ◽  
Surface Heat Flux ◽  
Velocity Ratio ◽  
Surface Heat ◽  
Stagnation Point Flow ◽  
Buoyant Flows ◽  
Keller Box Method ◽  
Nonlinearly Stretching Sheet

An analysis is carried out to study the steady two-dimensional stagnation-point flow of an incompressible viscous fluid towards a stretching vertical sheet. It is assumed that the sheet is stretched nonlinearly, with prescribed surface heat flux. This problem is governed by three parameters: buoyancy, velocity exponent, and velocity ratio. Both assisting and opposing buoyant flows are considered. The governing partial differential equations are transformed into a system of ordinary differential equations and solved numerically by finite difference Keller-box method. The flow and heat transfer characteristics for different values of the governing parameters are analyzed and discussed. Dual solutions are found in the opposing buoyant flows, while the solution is unique for the assisting buoyant flows.

Heat Transfer Characteristics of Steady Electroosmotic Flows in Two-Dimensional Straight Microchannels

2003 ◽  
Author(s):  
Keisuke Horiuchi ◽  
Prashanta Dutta
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Nusselt Number ◽  
Joule Heating ◽  
Surface Heat Flux ◽  
Surface Heat ◽  
Two Dimensional ◽  
Heat Transfer Characteristics ◽  
Temperature Distributions ◽  
Electroosmotic Flows

Analytical solutions for the temperature distributions, heat transfer coefficients and Nusselt numbers of steady electroosmotic flows are obtained for two-dimensional straight micro-channels. This analysis is based on infinitesimal electric double layer (EDL) in which flow velocity becomes “plug-like” uniform except very close to the wall. Both constant surface temperature and constant surface heat flux conditions are considered in this study. Separation of variables techniques are applied to obtain analytical solutions of temperature distributions from the energy equation in which Joule heating is a significant contributor due to the applied electric field. The thermal analysis considers interaction among inertial, diffusive and joule heating terms in order to obtain the thermally developing behavior of electroosmotic flows. Heat transfer characteristics are presented for low Reynolds number microflows where the viscous and electric field terms are very dominant. For the parameter range studied here (Re ≤ 0.7), the Nusselt number is independent of the thermal Peclet number, except in the thermally developing region. In both isothermal and constant surface heat flux boundary conditions, the Nusselt number becomes constant in the fully developed region for a uniform volumetric heat generation. Analytical results for no Joule heating cases are also compared with the classical heat transfer results, and in the thermally fully developed region an excellent agreement is obtained between them.

scholarly journals New analytical solution for natural convection of Darcian fluid in porous media prescribed surface heat flux

2011 ◽  
Vol 15 (suppl. 2) ◽  
pp. 221-227 ◽  
Author(s):  
Domiry Ganji ◽  
Hasan Sajjadi
Keyword(s):  
Heat Flux ◽  
Porous Media ◽  
Differential Equations ◽  
Iteration Method ◽  
Surface Heat Flux ◽  
Nonlinear Differential Equations ◽  
Variational Iteration Method ◽  
Surface Heat ◽  
Variational Iteration ◽  
He’S Variational Iteration Method

A new analytical method called He's Variational Iteration Method is introduced to be applied to solve nonlinear equations. In this method, general Lagrange multipliers are introduced to construct correction functional for the problems. It is strongly and simply capable of solving a large class of linear or nonlinear differential equations without the tangible restriction of sensitivity to the degree of the nonlinear term and also is very user friend because it reduces the size of calculations besides; its iterations are direct and straightforward. In this paper the powerful method called Variational Iteration Method is used to obtain the solution for a nonlinear Ordinary Differential Equations that often appear in boundary layers problems arising in heat and mass transfer which these kinds of the equations contain infinity boundary condition. The boundary layer approximations of fluid flow and heat transfer of vertical full cone embedded in porous media give us the similarity solution for full cone subjected to surface heat flux boundary conditions. The obtained Variational Iteration Method solution in comparison with the numerical ones represents a remarkable accuracy.

The Numerical Method of Lines facilitates the instruction of unsteady heat conduction in simple solid bodies with convective surfaces

2020 ◽  
pp. 030641902091042
Author(s):  
Antonio Campo
Keyword(s):  
Heat Conduction ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
Method Of Lines ◽  
Adjoint System ◽  
One Dimensional ◽  
First Order ◽  
The Method Of Lines ◽  
Unsteady Heat Conduction

The present study on engineering education addresses the Method of Lines and its variant the Numerical Method of Lines as a reliable avenue for the numerical analysis of one-dimensional unsteady heat conduction in walls, cylinders, and spheres involving surface convection interaction with a nearby fluid. The Method of Lines transforms the one-dimensional unsteady heat conduction equation in the spatial and time variables x, t into an adjoint system of first-order ordinary differential equations in the time variable t. Subsequently, the adjoint system of first-order ordinary differential equations is channeled through the Numerical Method of Lines and the powerful fourth-order Runge–Kutta algorithm. The numerical solution of the adjoint system of first-order ordinary differential equations can be carried out by heat transfer students employing appropriate routines embedded in the computer codes Maple, Mathematica, Matlab, and Polymath. For comparison, the baseline solutions used are the exact, analytical temperature distributions that are available in the heat conduction literature.

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