Approximate Analytical Method to Stefan Problem for Spheres with Wide Temperature Range of Phase Transition

2014 ◽  
Vol 627 ◽  
pp. 145-148 ◽  
Author(s):  
Ryoichi Chiba
Keyword(s):  
Phase Change ◽  
Temperature Profile ◽  
Solid Sphere ◽  
Transient Temperature ◽  
Transform Method ◽  
Differential Transform ◽  
Apparent Specific Heat ◽  
Heat Method ◽  
The One ◽  
Analytical Series

The two-dimensional differential transform method is applied to solve the one-dimensional phase change problem for a solid sphere with time-dependent boundary temperature. The problem assumes that the phase change occurs over a range of temperatures and the initial temperature of the sphere is an arbitrary constant. An approximate analytical (series) solution is derived for the temperature profile in the melting or solidifying sphere. The solution is based on the apparent specific heat method. Numerical results illustrate the effects of the Stefan number, which is the ratio of sensible heat to latent heat, on the transient temperature profile in the sphere.

scholarly journals A Series Solution for Heat Conduction Problem with Phase Change in a Finite Slab

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Ryoichi Chiba
Keyword(s):  
Heat Capacity ◽  
Phase Change ◽  
Temperature Profile ◽  
Series Solution ◽  
Thermal Loading ◽  
Heat Conduction Problem ◽  
Finite Thickness ◽  
Transient Temperature ◽  
Temperature Dependent ◽  
Transform Method

A two-dimensional differential transform method is applied to solve one-dimensional phase change problems in a slab of finite thickness, which is subjected to convective thermal loading at one surface and a constant prescribed temperature at the other. In the problems, the initial temperature of the slab does not necessarily have to be the same as the fusion temperature. A series solution is derived for the temperature profile in the melting or solidifying slab with temperature-dependent thermal conductivity and volumetric heat capacity. The latent heat effect of the phase change is incorporated into the temperature-dependent heat capacity. Numerical results demonstrate the effects of the temperature-dependent parameters on the transient temperature profile of the slab.

Application of a Differential Transform Method to the Transient Natural Convection Problem in a Vertical Tube with Variable Fluid Properties

2016 ◽  
Vol 71 (2) ◽  
pp. 185-193
Author(s):  
Ryoichi Chiba
Keyword(s):  
Natural Convection ◽  
Special Functions ◽  
Vertical Tube ◽  
Differential Transform Method ◽  
Transient Temperature ◽  
Temperature Dependent ◽  
Transform Method ◽  
Recursive Procedure ◽  
Transient Natural Convection ◽  
Differential Transform

AbstractThe transient natural convection of a viscous fluid in a heated vertical tube is studied using the two-dimensional differential transform method (DTM). A time-dependent Dirichlet boundary condition is imposed for tube wall temperature. The partial differential equations for the velocity and temperature fields within the tube are solved by the DTM while considering temperature-dependent viscosity and thermal conductivity of the fluid. As a result, tractable solutions in double-series form are derived for the temperature and flow velocity. The transformed functions included in the solutions are obtained through a simple recursive procedure. Numerical results illustrate the effects of temperature-dependent properties on transient temperature and flow behaviour, including the Nusselt number and volumetric flow rate. The DTM gives accurate series solutions without any special functions for nonlinear transient heat transfer problems which are advantageous in finding the derivative or integral.

The Significance of Fin Profile and Convective-Radiative Fin Tip on Temperature Distribution in a Longitudinal Fin

2019 ◽  
Vol 26 ◽  
pp. 93-105
Author(s):  
Partner Luyanda Ndlovu
Keyword(s):  
Heat Transfer ◽  
Boundary Condition ◽  
Temperature Distribution ◽  
Physical Parameters ◽  
Transform Method ◽  
Wide Range ◽  
Differential Transform ◽  
Nonlinear Transient ◽  
Numerical Solver ◽  
The One

In this article, the one dimensional nonlinear transient heat transfer through fins of rectangular, convex parabolic and concave parabolic is studied using the two dimensional Differential Transform Method (2D DTM). The thermal conductivity and heat transfer coefficient are modeled as linear and power law functions of temperature respectively. The fin tip dissipate heat to the ambient temperature by convection and radiation. A comparison is made between the proposed convectiveradiative fin tip boundary condition and the adiabatic (insulated) fin tip boundary condition which is widely used in literature. It is found that the fin with a convective-radiative tip dissipates heat to the ambient fluid at a faster rate when compared to a fin with an insulated tip. The results further show that the longitudinal fins of parabolic profiles dissipate more heat when compared to the conventional rectangular fin profile. The accuracy of the analytical method is demonstrated by comparing its results with those generated by an inbuilt numerical solver in MATLAB. Furthermore, a wide range of thermo-physical parameters are studied and their impact on the temperature distribution are illustrated and explained.

scholarly journals Differential Transform Method with Complex Transforms to Some Nonlinear Fractional Problems in Mathematical Physics

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Syed Tauseef Mohyud-Din ◽  
Farah Jabeen Awan ◽  
Jamshad Ahmad ◽  
Saleh M. Hassan
Keyword(s):  
Initial Conditions ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Mathematical Tool ◽  
Transform Method ◽  
Alternative Approach ◽  
Differential Transform ◽  
Fractional Differential ◽  
Analytical Series ◽  
Adomian’S Polynomials

This paper witnesses the coupling of an analytical series expansion method which is called reduced differential transform with fractional complex transform. The proposed technique is applied on three mathematical models, namely, fractional Kaup-Kupershmidt equation, generalized fractional Drinfeld-Sokolov equations, and system of coupled fractional Sine-Gordon equations subject to the appropriate initial conditions which arise frequently in mathematical physics. The derivatives are defined in Jumarie’s sense. The accuracy, efficiency, and convergence of the proposed technique are demonstrated through the numerical examples. It is observed that the presented coupling is an alternative approach to overcome the demerit of complex calculation of fractional differential equations. The proposed technique is independent of complexities arising in the calculation of Lagrange multipliers, Adomian’s polynomials, linearization, discretization, perturbation, and unrealistic assumptions and hence gives the solution in the form of convergent power series with elegantly computed components. All the examples show that the proposed combination is a powerful mathematical tool to solve other nonlinear equations also.

scholarly journals Analysis of Transient Thermal Distribution in a Convective–Radiative Moving Rod Using Two-Dimensional Differential Transform Method with Multivariate Pade Approximant

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1793
Author(s):  
Ganeshappa Sowmya ◽  
Ioannis E. Sarris ◽  
Chandra Sen Vishalakshi ◽  
Ravikumar Shashikala Varun Kumar ◽  
Ballajja Chandrappa Prasannakumara
Keyword(s):  
Heat Generation ◽  
Padé Approximant ◽  
Differential Transform Method ◽  
Transient Temperature ◽  
Two Dimensional ◽  
Temperature Dependent ◽  
Transform Method ◽  
Pade Approximant ◽  
Differential Transform ◽  
Transient Thermal

The transient temperature distribution through a convective-radiative moving rod with temperature-dependent internal heat generation and non-linearly varying temperature-dependent thermal conductivity is elaborated in this investigation. Symmetries are intrinsic and fundamental features of the differential equations of mathematical physics. The governing energy equation subjected to corresponding initial and boundary conditions is non-dimensionalized into a non-linear partial differential equation (PDE) with the assistance of relevant non-dimensional terms. Then the resultant non-dimensionalized PDE is solved analytically using the two-dimensional differential transform method (2D DTM) and multivariate Pade approximant. The consequential impact of non-dimensional parameters such as heat generation, radiative, temperature ratio, and conductive parameters on dimensionless transient temperature profiles has been scrutinized through graphical elucidation. Furthermore, these graphs indicate the deviations in transient thermal profile for both finite difference method (FDM) and 2D DTM-multivariate Pade approximant by considering the forced convective and nucleate boiling heat transfer mode. The results reveal that the transient temperature profile of the moving rod upsurges with the change in time, and it improves for heat generation parameter. It enriches for the rise in the magnitude of Peclet number but drops significantly for greater values of the convective-radiative and convective-conductive parameters.

NUMERICAL AND STABILITY ANALYSIS OF THE TRANSMISSION DYNAMICS OF SVIR EPIDEMIC MODEL WITH STANDARD INCIDENCE RATE

2019 ◽  
Vol 4 (2) ◽  
pp. 349 ◽  
Author(s):  
Oluwatayo Michael Ogunmiloro ◽  
Fatima Ohunene Abedo ◽  
Hammed Kareem
Keyword(s):  
Reproduction Number ◽  
Disease Spread ◽  
Asymptotically Stable ◽  
Equilibrium Solutions ◽  
Transform Method ◽  
Globally Asymptotically Stable ◽  
Differential Transform ◽  
Mathematical Techniques ◽  
Fourth Order Method ◽  
Theoretical Results

In this article, a Susceptible – Vaccinated – Infected – Recovered (SVIR) model is formulated and analysed using comprehensive mathematical techniques. The vaccination class is primarily considered as means of controlling the disease spread. The basic reproduction number (Ro) of the model is obtained, where it was shown that if Ro<1, at the model equilibrium solutions when infection is present and absent, the infection- free equilibrium is both locally and globally asymptotically stable. Also, if Ro>1, the endemic equilibrium solution is locally asymptotically stable. Furthermore, the analytical solution of the model was carried out using the Differential Transform Method (DTM) and Runge - Kutta fourth-order method. Numerical simulations were carried out to validate the theoretical results. 

Numerical Approximations to Solution of Biochemical Reaction Model

2011 ◽  
Vol 9 (1) ◽  
Author(s):  
Ahmet Yildirim ◽  
Ahmet Gökdogan ◽  
Mehmet Merdan
Keyword(s):  
Analytical Solution ◽  
Approximate Analytical Solution ◽  
Transformation Method ◽  
Reaction Model ◽  
Biochemical Reaction ◽  
Graphical Form ◽  
Kutta Method ◽  
Transform Method ◽  
Runge Kutta Method ◽  
Differential Transform

In this paper, approximate analytical solution of biochemical reaction model is used by the multi-step differential transform method (MsDTM) based on classical differential transformation method (DTM). Numerical results are compared to those obtained by the fourth-order Runge-Kutta method to illustrate the preciseness and effectiveness of the proposed method. Results are given explicit and graphical form.

Sign in / Sign up

Export Citation Format

Share Document