Attaining exponential convergence for the flux error with second- and fourth-order accurate finite-difference equations. II. Application to systems comprising first-order chemical reactions

2005 ◽  
Vol 26 (6) ◽  
pp. 633-641 ◽  
Author(s):  
Manfred Rudolph
Keyword(s):  
Finite Difference ◽  
Chemical Reactions ◽  
Fourth Order ◽  
Difference Equations ◽  
Exponential Convergence ◽  
First Order ◽  
Finite Difference Equations

A Two-Dimensional Numerical Solution for Elastic Waves in Variously Configured Rods

1971 ◽  
Vol 38 (1) ◽  
pp. 62-70 ◽  
Author(s):  
J. L. Habberstad
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Difference Equations ◽  
Pressure Pulse ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
First Order ◽  
Finite Difference Equations ◽  
Viscosity Term

The exact equations of motion governing elastic, axisymmetric wave propagation in a cylindrical rod are approximated by a first-order finite-difference scheme. This difference scheme is based on a displacement rather than a velocity formulation, thereby making it unnecessary to explicitly introduce an artificial viscosity term into the finite-difference equations. The resulting difference equations are used in conjunction with the boundary and initial conditions 10 study: (a) a pressure pulse applied to the end of a semi-infinite bar, (b) a bar composed of two materials joined together at some point along its length, and (c) a bar containing a discontinuity in cross section. The numerical results so obtained are compared to available experimental data and other analytical-numerical solutions.

Numerical Simulation of Reacting Flows in Radiant Porous Burners

2005 ◽  
Author(s):  
Timothy W. Tong ◽  
Mohsen M. M. Abou-Ellail ◽  
Yuan Li ◽  
Karam R. Beshay
Keyword(s):  
Porous Media ◽  
Finite Difference ◽  
Reaction Zone ◽  
Chemical Reactions ◽  
Difference Equations ◽  
Solid Wall ◽  
Combustion Products ◽  
Fine Grid ◽  
Finite Difference Equations ◽  
Porous Burner

The present paper presents, numerical computations for flow, heat transfer and chemical reactions in an axisymmetric inert porous burner. The porous media re-radiate the heat absorbed from the gaseous combustion products by convection and conduction. In the present work, the porous burner species mass fraction source terms are computed from an ‘extended’ reaction mechanism, controlled by chemical kinetics of elementary reactions. The porous burner has mingled zones of porous/nonporous reacting flow, i.e. the porosity is not uniform over the entire domain. Therefore, it has to be included inside the partial derivatives of the transport governing equations. Finite-difference equations are obtained by formal integration over control volumes surrounding each grid node. Up-wind differencing is used to insure that the influence coefficients are always positive to reflect the real effect of neighboring nodes on a typical central node. Finite-difference equations are solved, iteratively, for U, V, p’ (pressure correction), enthalpy and species mass fractions, utilizing a fine grid of (80×60) nodes. The eighty grid nodes in the axial direction are needed to resolve the detailed structure of the thin reaction zone inside the porous media. The radial grid is extended inside the annular solid wall of the porous burner, to compute the wall temperature. The porous burner uses a premixed CH4-air mixture, while its radiating characteristics are computed numerically, using a four-flux radiation model. Sixteen species are included, namely CH4, CH3, CH2, CH, CH2O, CHO, CO, CO2, O2, O, OH, H2, H, H2O, H2O, H2O2, involving 49 chemical reaction equations. It was found that 1000 iterations are sufficient for complete conversion of the computed results with errors less than 0.1%. The computed temperature profiles of the gas and the solid show that, heat is conducted from downstream to the upstream of the reaction zone. Most stable species, such as H2O, CO2, H2, keep increasing inside the reaction zone staying appreciable in the combustion products. However, unstable products, such as HO2, H2O2 and CH3, first increase in the preheating region of the reaction zone, they are then consumed fast in the post-reaction zone of the porous burner. Therefore, it appears that their important function is only to help the chemical reactions continue to their inevitable completion of the more stable combustion products.

Numerical Computation of Reacting Flow in Porous Burners With an Extended CH4-Air Reaction Mechanism

2004 ◽  
Author(s):  
Timothy Tong ◽  
Mohsen Abou-Ellail ◽  
Yuan Li ◽  
Karam R. Beshay
Keyword(s):  
Porous Media ◽  
Reaction Mechanism ◽  
Finite Difference ◽  
Reaction Zone ◽  
Chemical Reactions ◽  
Difference Equations ◽  
Combustion Products ◽  
Reacting Flow ◽  
Finite Difference Equations ◽  
Porous Burner

The present paper presents, numerical computations for flow, heat transfer and chemical reactions in an axisymmetric inert porous burner. The porous media re-radiate the heat absorbed from the gaseous combustion products by convection and conduction. In the present work, the porous burner species mass fraction source terms are computed from an ‘extended’ reaction mechanism, controlled by chemical kinetics of elementary reactions. The porous burner has mingled zones of porous/nonporous reacting flow, i.e. the porosity is not uniform over the entire domain. Therefore, it has to be included inside the partial derivatives of the transport governing equations. Finite-difference equations are obtained by formal integration over control volumes surrounding each grid node. Up-wind differencing is used to insure that the influence coefficients are always positive to reflect the real effect of neighboring nodes on a typical central node. Finite-difference equations are solved, iteratively, for U, V, p’ (pressure correction), enthalpy and species mass fractions, utilizing a grid of (60×40) nodes. The sixty grid nodes in the axial direction are needed to resolve the detailed structure of the thin reaction zone inside the porous media. The porous burner uses a premixed CH4-air mixture, while its radiating characteristics are computed numerically, using a four-flux radiation model. Sixteen species are included, namely CH4, CH3, CH2, CH, CH2O, CHO, CO, CO2, O2, O, OH, H2, H, H2O, HO2, H2O2, involving 49 chemical reaction equations. It was found that 900 iterations are sufficient for complete conversion of the computed results with errors less than 0.1%. The computed temperature profiles of the gas and the solid show that, heat is conducted from downstream to the upstream of the reaction zone. Most stable species, such as H2O, CO2, H2, keep increasing inside the reaction zone staying appreciable in the combustion products. However, unstable products, such as HO2, H2O2 and CH3, first increase in the preheating region of the reaction zone, they are then consumed fast in the post-reaction zone of the porous burner. Therefore, it appears that their important function is only to help the chemical reactions continue to their inevitable completion of the more stable combustion products.

scholarly journals Finite-difference equations of quasistatic motion of the shallow concrete shells in nonlinear setting

2020 ◽  
Vol 7 (1) ◽  
pp. 48-55 ◽  
Author(s):  
Bolat Duissenbekov ◽  
Abduhalyk Tokmuratov ◽  
Nurlan Zhangabay ◽  
Zhenis Orazbayev ◽  
Baisbay Yerimbetov ◽  
...  
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Difference Equations ◽  
Constant Load ◽  
Time Interval ◽  
Test Case ◽  
Algebraic Equations ◽  
Nonlinear Properties ◽  
Finite Difference Equations ◽  
Concrete Shells

AbstractThe study solves a system of finite difference equations for flexible shallow concrete shells while taking into account the nonlinear deformations. All stiffness properties of the shell are taken as variables, i.e., stiffness surface and through-thickness stiffness. Differential equations under consideration were evaluated in the form of algebraic equations with the finite element method. For a reinforced shell, a system of 98 equations on a 8×8 grid was established, which was next solved with the approximation method from the nonlinear plasticity theory. A test case involved computing a 1×1 shallow shell taking into account the nonlinear properties of concrete. With nonlinear equations for the concrete creep taken as constitutive, equations for the quasi-static shell motion under constant load were derived. The resultant equations were written in a differential form and the problem of solving these differential equations was then reduced to the solving of the Cauchy problem. The numerical solution to this problem allows describing the stress-strain state of the shell at each point of the shell grid within a specified time interval.

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