A Mathematical Model for the Analysis of Fluid Flow in a Scroll

1986 ◽  
Vol 108 (1) ◽  
pp. 6-11 ◽  
Author(s):  
Shou-Rue Chen ◽  
Samuel S. Lee ◽  
Yuan Mao Huang
Keyword(s):  
Mathematical Model ◽  
Fluid Flow ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Flow Condition ◽  
Three Dimensional ◽  
Coordinate Transformations ◽  
Governing Equations ◽  
The Finite Difference Method

A three-dimensional mathematical model has been developed to simulate the flow condition in a scroll. Coordinate transformations are used as an effective tool to make the model universal, and the final governing equations are solved by the finite difference method. Three cases of scroll geometry have been investigated and the results are compared with one another to show the effects of scroll geometry on the flow condition at the outlet of the scroll.

Studi Perpindahan Panas Material pada Sistem Koordinat Segitiga

2017 ◽  
Vol 1 ◽  
pp. 114
Author(s):  
Imam Basuki ◽  
C Cari ◽  
A Suparmi
Keyword(s):  
Heat Transfer ◽  
Mathematical Model ◽  
Heat Conduction ◽  
Partial Differential Equations ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Numerical Solution ◽  
Difference Method ◽  
The Finite Difference Method ◽  
The Mathematical Model

<p class="Normal1"><strong><em>Abstract: </em></strong><em>Partial Differential Equations (PDP) Laplace equation can be applied to the heat conduction. Heat conduction is a process that if two materials or two-part temperature material is contacted with another it will pass heat transfer. Conduction of heat in a triangle shaped object has a mathematical model in Cartesian coordinates. However, to facilitate the calculation, the mathematical model of heat conduction is transformed into the coordinates of the triangle. PDP numerical solution of Laplace solved using the finite difference method. Simulations performed on a triangle with some angle values α and β</em></p><p class="Normal1"><strong><em> </em></strong></p><p class="Normal1"><strong><em>Keywords:</em></strong><em>  heat transfer, triangle coordinates system.</em></p><p class="Normal1"><em> </em></p><p class="Normal1"><strong>Abstrak</strong> Persamaan Diferensial Parsial (PDP) Laplace  dapat diaplikasikan pada persamaan konduksi panas. Konduksi panas adalah suatu proses yang jika dua materi atau dua bagian materi temperaturnya disentuhkan dengan yang lainnya maka akan terjadilah perpindahan panas. Konduksi panas pada benda berbentuk segitiga mempunyai model matematika dalam koordinat cartesius. Namun untuk memudahkan perhitungan, model matematika konduksi panas tersebut ditransformasikan ke dalam koordinat segitiga. Penyelesaian numerik dari PDP Laplace diselesaikan menggunakan metode beda hingga. Simulasi dilakukan pada segitiga dengan beberapa nilai sudut  dan  </p><p class="Normal1"><strong> </strong></p><p class="Normal1"><strong>Kata kunci :</strong> perpindahan panas, sistem koordinat segitiga.</p>

scholarly journals Vibration Analysis of Axially Functionally Graded Non-Prismatic Euler-Bernoulli Beams Using the Finite Difference Method

2021 ◽  
Author(s):  
Valentin Fogang
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Difference Method ◽  
Vibration Analysis ◽  
Mass Density ◽  
Functionally Graded ◽  
Governing Equations ◽  
Mass System ◽  
The Finite Difference Method

This paper presents an approach to the vibration analysis of axially functionally graded (AFG) non-prismatic Euler-Bernoulli beams using the finite difference method (FDM). The characteristics (cross-sectional area, moment of inertia, elastic moduli, and mass density) of AFG beams vary along the longitudinal axis. The FDM is an approximate method for solving problems described with differential equations. It does not involve solving differential equations; equations are formulated with values at selected points of the structure. In addition, the boundary conditions and not the governing equations are applied at the beam&rsquo;s ends. In this paper, differential equations were formulated with finite differences, and additional points were introduced at the beam&rsquo;s ends and at positions of discontinuity (supports, hinges, springs, concentrated mass, spring-mass system, etc.). The introduction of additional points allowed us to apply the governing equations at the beam&rsquo;s ends and to satisfy the boundary and continuity conditions. Moreover, grid points with variable spacing were also considered, the grid being uniform within beam segments. Vibration analysis of AFG non-prismatic Euler-Bernoulli beams was conducted with this model, and natural frequencies were determined. Finally, a direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of AFG non-prismatic Euler-Bernoulli beams, considering the damping. The results obtained in this paper showed good agreement with those of other studies, and the accuracy was always increased through a grid refinement.

A Remark on the Courant-Friedrichs-Lewy Condition in Finite Difference Approach to PDE’s

2014 ◽  
Vol 6 (5) ◽  
pp. 693-698 ◽  
Author(s):  
Kosuke Abe ◽  
Nobuyuki Higashimori ◽  
Masayoshi Kubo ◽  
Hiroshi Fujiwara ◽  
Yuusuke Iso
Keyword(s):  
Mathematical Model ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Numerical Experiments ◽  
Numerical Solutions ◽  
Hyperbolic Partial Differential Equations ◽  
View Point ◽  
Rounding Errors ◽  
The Finite Difference Method

AbstractThe Courant-Friedrichs-Lewy condition (The CFL condition) is appeared in the analysis of the finite difference method applied to linear hyperbolic partial differential equations. We give a remark on the CFL condition from a view point of stability, and we give some numerical experiments which show instability of numerical solutions even under the CFL condition. We give a mathematical model for rounding errors in order to explain the instability.

Finite-Difference Method Appliance to the Computation of Ground Additional Stress and Settlement of Storage Tank

2011 ◽  
Vol 243-249 ◽  
pp. 2638-2642
Author(s):  
Xu Dong Cheng ◽  
Wen Shan Peng ◽  
Lei Liu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Storage Tank ◽  
Three Dimensional ◽  
Additional Stress ◽  
The Finite Element Method ◽  
The Finite Difference Method

This paper adopts the Finite-difference method to research the distribution of ground additional stress and distortion in differently isotropic and non-isotropic foundation conditions, and uses the Finite-difference method to compare with the Finite-element method and the three-dimensional settlement method used by the code. Through comparative analysis, the reliability and superiority of Finite-difference method used for calculating ground additional stress and settlement are justified.

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