Simple Algebraic Treatment of Unsteady Heat Conduction in Solid Spheres with Heat Convection Exchange to Fluids Covering the Entire Time Domain

2016 ◽  
Author(s):  
Marcelo Marucho ◽  
Antonio Campo-Garcia
Keyword(s):  
Heat Conduction ◽  
Time Domain ◽  
Heat Convection ◽  
Entire Time ◽  
Algebraic Treatment ◽  
Unsteady Heat Conduction ◽  
Solid Spheres

Accurate analytical/numerical solution of the heat conduction equation

2014 ◽  
Vol 24 (7) ◽  
pp. 1519-1536 ◽  
Author(s):  
Antonio Campo ◽  
Abraham J. Salazar ◽  
Diego J. Celentano ◽  
Marcos Raydan
Keyword(s):  
Differential Equation ◽  
Heat Conduction ◽  
Numerical Solution ◽  
Time Domain ◽  
Heat Conduction Equation ◽  
Separation Of Variables ◽  
Method Of Lines ◽  
Conduction Equation ◽  
Content Type ◽  
Entire Time

Purpose – The purpose of this paper is to address a novel method for solving parabolic partial differential equations (PDEs) in general, wherein the heat conduction equation constitutes an important particular case. The new method, appropriately named the Improved Transversal Method of Lines (ITMOL), is inspired in the Transversal Method of Lines (TMOL), with strong insight from the method of separation of variables. Design/methodology/approach – The essence of ITMOL revolves around an exponential variation of the dependent variable in the parabolic PDE for the evaluation of the time derivative. As will be demonstrated later, this key step is responsible for improving the accuracy of ITMOL over its predecessor TMOL. Throughout the paper, the theoretical properties of ITMOL, such as consistency, stability, convergence and accuracy are analyzed in depth. In addition, ITMOL has proven to be unconditionally stable in the Fourier sense. Findings – In a case study, the 1-D heat conduction equation for a large plate with symmetric Dirichlet boundary conditions is transformed into a nonlinear ordinary differential equation by means of ITMOL. The numerical solution of the resulting differential equation is straightforward and brings forth a nearly zero truncation error over the entire time domain, which is practically nonexistent. Originality/value – Accurate levels of the analytical/numerical solution of the 1-D heat conduction equation by ITMOL are easily established in the entire time domain.

Semi-analytical unsteady heat conduction in large plane walls with heat convection exchange

2019 ◽  
Vol 29 (2) ◽  
pp. 536-552
Author(s):  
Antonio Campo ◽  
Yunesky Masip
Keyword(s):  
Heat Conduction ◽  
Heat Conduction Equation ◽  
Small Time ◽  
Heat Convection ◽  
Temperature Profiles ◽  
Content Type ◽  
One Dimensional ◽  
Space Coordinate ◽  
Unsteady Heat Conduction ◽  
Large Plane

Purpose The purpose of this study is to address one-dimensional, unsteady heat conduction in a large plane wall exchanging heat convection with a nearby fluid under “small time” conditions. Design/methodology/approach The Transversal Method of Lines (TMOL) was used to reformulate the unsteady, one-dimensional heat conduction equation in the space coordinate and time into a transformed “quasi-steady”, one-dimensional heat conduction equation in the space coordinate housing the time as an embedded parameter. The resulting ordinary differential equation of second order with heat convection boundary conditions is solved analytically with the method of undetermined coefficients. Findings Semi-analytical TMOL dimensionless temperature profiles of compact form with/without regressed terms are obtained for the whole spectrum of Biot number (0 < Bi < ∞) in the “small time” sub-domain. In addition, a new “large time” sub-domain is redefined, that is, setting a smaller critical dimensionless time or critical Fourier number τcr = 0.18. Originality/value The computed dimensionless center, surface and mean temperature profiles in the large plane wall accounting for all Biot number (0 < Bi < ∞) in the “small time” sub-domain τ < τcr = 0.18 exhibit excellent quality while carrying reasonable relative errors for engineering applications. The exemplary level of accuracy indicates that the traditional evaluation of the center, surface and mean temperatures with the standard infinite series retaining a large number of terms is no longer necessary.

The Lumped Capacitance Model for Unsteady Heat Conduction in Regular Solid Bodies With Natural Convection to Nearby Fluids Engages the Nonlinear Bernoulli Equation

2018 ◽  
Vol 10 (3) ◽  
Author(s):  
Antonio Campo
Keyword(s):  
Heat Conduction ◽  
Natural Convection ◽  
Heat Equation ◽  
Biot Number ◽  
Bernoulli Equation ◽  
Convective Coefficient ◽  
Unsteady Heat Conduction ◽  
The Mean ◽  
Capacitance Model ◽  
Solid Bodies

For the analysis of unsteady heat conduction in solid bodies comprising heat exchange by forced convection to nearby fluids, the two feasible models are (1) the differential or distributed model and (2) the lumped capacitance model. In the latter model, the suited lumped heat equation is linear, separable, and solvable in exact, analytic form. The linear lumped heat equation is constrained by the lumped Biot number criterion Bil=h¯(V/S)/ks < 0.1, where the mean convective coefficient h¯ is affected by the imposed fluid velocity. Conversely, when the heat exchange happens by natural convection, the pertinent lumped heat equation turns nonlinear because the mean convective coefficient h¯ depends on the instantaneous mean temperature in the solid body. Undoubtedly, the nonlinear lumped heat equation must be solved with a numerical procedure, such as the classical Runge–Kutta method. Also, due to the variable mean convective coefficient h¯ (T), the lumped Biot number criterion Bil=h¯(V/S)/ks < 0.1 needs to be adjusted to Bil,max=h¯max(V/S)/ks < 0.1. Here, h¯max in natural convection cooling stands for the maximum mean convective coefficient at the initial temperature Tin and the initial time t = 0. Fortunately, by way of a temperature transformation, the nonlinear lumped heat equation can be homogenized and later channeled through a nonlinear Bernoulli equation, which admits an exact, analytic solution. This simple route paves the way to an exact, analytic mean temperature distribution T(t) applicable to a class of regular solid bodies: vertical plate, vertical cylinder, horizontal cylinder, and sphere; all solid bodies constricted by the modified lumped Biot number criterion Bil,max<0.1.

Solution to shape identification problem of unsteady heat-conduction fields

2003 ◽  
Vol 32 (3) ◽  
pp. 212-226 ◽  
Author(s):  
Eiji Katamine ◽  
Hideyuki Azegami ◽  
Yasuhiro Matsuura
Keyword(s):  
Heat Conduction ◽  
Identification Problem ◽  
Shape Identification ◽  
Unsteady Heat Conduction
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