Charis Harley

An implicit series solution for a boundary value problem modelling a thermal explosion

An implicit series solution admitted by a boundary value problem modelled by a Lane-Emden equation of the second kind is obtained. The boundary value problem was derived by Frank-Kamenetskii to model the steady temperature in a vessel in which a thermal explosion is taking place. The Lane-Emden equation is reduced to an autonomous second-order ordinary differential equation by means of a coordinate transformation. The autonomous second-order ordinary differential equation is reduced to a first-order Abel equation. A power series solution of the first-order Abel equation is obtained. The power series solution of the Abel equation is transformed into an implicit series solution of the original Lane-Emden equation satisfying the boundary conditions of the original problem. We show that the implicit power series solution is valid for values of the dimensionless Frank-Kamenetskii parameter @d<0.02.

Instability of invariant boundary conditions of a generalized Lane–Emden equation of the second-kind

Abstract A generator of Lie point symmetries admitted by a Lane–Emden equation of the second-kind for arbitrary shape factor k is used to determine invariant boundary conditions admitted by the equation. The generator of Lie point symmetries is then used to reduce the order of the Lane–Emden equation. A phase plane analysis of the reduced equation indicates that the stability of the invariant boundary condition y ′ = 0 on the line x = 0 changes with changing shape factor k. We show that for values of the shape factor k > 1 the boundary condition y ′ = 0 is stable on the line x = 0 while it is unstable for k ⩽ 1 .

Alternate Derivation of the Critical Value of the Frank-Kamenetskii Parameter in Cylindrical Geometry

Abstract Noether’s theorem is used to determine first integrals admitted by a generalised Lane- Emden equation of the second kind modelling a thermal explosion. These first integrals exist for rectangular and cylindrical geometry. For rectangular geometry the first integrals show the symmetry of the temperature gradients at the rectangular walls. For a cylindrical geometry the first integrals show the dependence of the critical parameter on the temperature gradient at the cylinder wall. The well known critical value for the Frank-Kamenetskii parameter, δ = 2, is obtained in a very natural way.

STEADY STATE SOLUTIONS FOR A THERMAL EXPLOSION IN A CYLINDRICAL VESSEL

Steady state solutions of a heat balance equation modeling a thermal explosion in a cylindrical vessel are obtained. The heat balance equation reduces to a Lane–Emden equation of the second-kind when steady state solutions are investigated. Analytical solutions to this Lane–Emden equation of the second-kind are obtained by implementation of the Lie group method. The classical Lie group method is used to obtain the well-known solution of Frank-Kamenetskii for the temperature distribution in a cylindrical vessel. Using an extension of the classical Lie group method a non-local symmetry is obtained and a new solution describing the temperature distribution after blow-up is obtained.

Two Hybrid Methods for Solving Two-Dimensional Linear Time-Fractional Partial Differential Equations

A computationally efficient hybridization of the Laplace transform with two spatial discretization techniques is investigated for numerical solutions of time-fractional linear partial differential equations in two space variables. The Chebyshev collocation method is compared with the standard finite difference spatial discretization and the absolute error is obtained for several test problems. Accurate numerical solutions are achieved in the Chebyshev collocation method subject to both Dirichlet and Neumann boundary conditions. The solution obtained by these hybrid methods allows for the evaluation at any point in time without the need for time-marching to a particular point in time.

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

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.

A comparison of two hybrid methods for applying the time-fractional heat equation to a two dimensional function

The solution of the time-fractional diffusion equation in two-dimensions allows one to apply this fractional partial differential equation to an image. We present a computationally efficient method that generalizes trivially to temporal derivatives of fractional order. Comparisons of the present method with analytical solutions indicate a small error and justify this method as an application tool for the time-fractional diffusion equation to either a two-dimensional function or an image.

FIRST INTEGRALS OF FIN EQUATIONS FOR STRAIGHT FINS

First integrals admitted by second-order nonlinear ordinary differential equations modeling the temperature distribution in a straight fin are obtained. After imposing the boundary conditions these first integrals give a relationship between temperature at the fin tip and the temperature gradient at the base of the fin in terms of the fin parameters. These first integrals are plotted and analyzed. The results obtained show how the temperature at the fin tip can be controlled by the temperature gradient at the base for fixed fin parameters.

A Numerical Well-Balanced Scheme for One-Dimensional Heat Transfer in Longitudinal Triangular Fins

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.

Steady Thermal Analysis of Two-Dimensional Cylindrical Pin Fin with a Nonconstant Base Temperature

Steady heat transfer through a pin fin is studied. Thermal conductivity, heat transfer coefficient, and the source or sink term are assumed to be temperature dependent. In the model considered, the source or sink term is given as an arbitrary function. We employ symmetry techniques to determine forms of the source or sink term for which the extra Lie point symmetries are admitted. Method of separation of variables is used to construct exact solutions when the governing equation is linear. Symmetry reductions result in reduced ordinary differential equations when the problem is nonlinear and some invariant solution for the linear case. Furthermore, we analyze the heat flux, fin efficiency, and the entropy generation.