scholarly journals NUMERICAL APPROACHES FOR A MATHEMATICAL MODEL OF AN FCCU REGENERATOR

2008 ◽  
Vol 7 (1) ◽  
pp. 71
Author(s):  
J. C. Penteado ◽  
C. O. R. Negrao ◽  
L. F. S. Rossi
Keyword(s):  
Mathematical Model ◽  
Finite Difference ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Finite Difference Methods ◽  
Implicit Finite Difference ◽  
Conservation Principles ◽  
Continuously Stirred Tank Reactor ◽  
Time Changes ◽  
Numerical Approaches

This work discusses a mathematical model of an FCCU (Fluid Catalytic Cracking Unit) regenerator. The model assumes that the regenerator is divided into two regions: the freeboard and the dense bed. The latter is composed of a bubble phase and an emulsion phase. Both phases are modeled as a CSTR (Continuously Stirred Tank Reactor) in which ordinary differential equations are employed to represent the conservation of mass, energy and species. In the freeboard, the flow is considered to be onedimensional, and the conservation principles are represented by partial differential equations to describe space and time changes. The main aim ofthis work is to compare two numerical approaches for solving the set of partial and ordinary differential equations, namely, the fourth-order Runge-Kutta and implicit finite-difference methods. Although both methods give very similar results, the implicit finite-difference method can be much faster. Steady-state results were corroborated by experimental data, and the dynamic results were compared with those in the literature (Han and Chung, 2001b). Finally, an analysis of the model’s sensitivity to the boundary conditions was conducted.

scholarly journals Implicit Finite Difference Simulation of Prandtl-Eyring Nanofluid over a Flat Plate with Variable Thermal Conductivity: A Tiwari and Das Model

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3153
Author(s):  
Nidal H. Abu-Hamdeh ◽  
Abdulmalik A. Aljinaidi ◽  
Mohamed A. Eltaher ◽  
Khalid H. Almitani ◽  
Khaled A. Alnefaie ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Thermal Conductivity ◽  
Finite Difference ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Variable Thermal Conductivity ◽  
Solid Particles ◽  
Engine Oil ◽  
Working Fluid ◽  
Implicit Finite Difference

The current article presents the entropy formation and heat transfer of the steady Prandtl-Eyring nanofluids (P-ENF). Heat transfer and flow of P-ENF are analyzed when nanofluid is passed to the hot and slippery surface. The study also investigates the effects of radiative heat flux, variable thermal conductivity, the material’s porosity, and the morphologies of nano-solid particles. Flow equations are defined utilizing partial differential equations (PDEs). Necessary transformations are employed to convert the formulae into ordinary differential equations. The implicit finite difference method (I-FDM) is used to find approximate solutions to ordinary differential equations. Two types of nano-solid particles, aluminium oxide (Al2O3) and copper (Cu), are examined using engine oil (EO) as working fluid. Graphical plots are used to depict the crucial outcomes regarding drag force, entropy measurement, temperature, Nusselt number, and flow. According to the study, there is a solid and aggressive increase in the heat transfer rate of P-ENF Cu-EO than Al2O3-EO. An increment in the size of nanoparticles resulted in enhancing the entropy of the model. The Prandtl-Eyring parameter and modified radiative flow show the same impact on the radiative field.

scholarly journals Comparison between standard and non-standard finite difference methods for solving first and second order ordinary differential equations

2015 ◽  
Vol 4 (2) ◽  
pp. 316
Author(s):  
Abdulrahman Yaghoubi ◽  
Hashem Saberi Najafi
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Exact Solutions ◽  
Ordinary Differential Equations ◽  
Finite Difference Methods ◽  
Second Order ◽  
Difference Methods

<p>In this paper, we solve some first and second order ordinary differential equations by the standard and non-standard finite difference methods and compare results of these methods. Illustrative examples have been provided, and the results of two methods compared with the exact solutions.</p>

scholarly journals A Step Block Algorithm for Solving First Order Initial Value Problems in Ordinary Differential Equations

2017 ◽  
Vol 2 (1) ◽  
Author(s):  
Emmanuel O Adeyefa ◽  
Oluwatosin Fadaka
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problems ◽  
Finite Difference Methods ◽  
Initial Value ◽  
Step Method ◽  
Block Algorithm ◽  
One Step ◽  
Difference Methods

The implementation of the newly formulated polynomials, ADEM-B orthogonal polynomials,  valid in the interval [-1, 1] with respect to weight function is our major focus in this work. The polynomials, which serve as basis function are employed to develop finite difference methods. Varying off-step points are considered for only One-Step method for the solution of the initial value problems of Ordinary Differential Equations (ODEs).  By selection of points for both interpolation and collocation, threeimportant class of block finite difference methods are produced. The methods are analyzed for their basic properties and findings show that they are accurate and convergent.

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