How to Deal with Several Reservoirs

1991 ◽  
Author(s):  
James C. G. Walker
Keyword(s):  
Steady State ◽  
Euler Method ◽  
Coupled Systems ◽  
Algebraic Equations ◽  
Coupled Equations ◽  
Linear Algebraic Equations ◽  
Linear Differential ◽  
Surface Reservoir

The previous chapter showed how the reverse Euler method can be used to solve numerically an ordinary first-order linear differential equation. Most problems in geochemical dynamics involve systems of coupled equations describing related properties of the environment in a number of different reservoirs. In this chapter I shall show how such coupled systems may be treated. I consider first a steady-state situation that yields a system of coupled linear algebraic equations. Such a system can readily be solved by a method called Gaussian elimination and back substitution. I shall present a subroutine, GAUSS, that implements this method. The more interesting problems tend to be neither steady state nor linear, and the reverse Euler method can be applied to coupled systems of ordinary differential equations. As it happens, the application requires solving a system of linear algebraic equations, and so subroutine GAUSS can be put to work at once to solve a linear system that evolves in time. The solution of nonlinear systems will be taken up in the next chapter. Most simulations of environmental change involve several interacting reservoirs. In this chapter I shall explain how to apply the numerical scheme described in the previous chapter to a system of coupled equations. Figure 3-1, adapted from Broecker and Peng (1982, p. 382), is an example of a coupled system. The figure presents a simple description of the general circulation of the ocean, showing the exchange of water in Sverdrups (1 Sverdrup = 106 m3/sec) among five oceanic reservoirs and also the addition of river water to the surface reservoirs and the removal of an equal volume of water by evaporation. The problem is to calculate the steady-state concentration of dissolved phosphate in the five oceanic reservoirs, assuming that 95 percent of all the phosphate carried into each surface reservoir is consumed by plankton and carried downward in particulate form into the underlying deep reservoir. The remaining 5 percent of the incoming phosphate is carried out of the surface reservoir still in solution.

Ozone mass transfer in water treatment: hydrodynamics and mass transfer modeling of ozone bubble columns

2001 ◽  
Vol 1 (2) ◽  
pp. 123-130 ◽  
Author(s):  
M.G. El-Din ◽  
D.W. Smith
Keyword(s):  
Mass Transfer ◽  
Steady State ◽  
Liquid Phase ◽  
Plug Flow ◽  
Axial Dispersion ◽  
Bubble Columns ◽  
Algebraic Equations ◽  
Mixed Flow ◽  
Back Flow ◽  
Linear Algebraic Equations

Most of the mathematical models that are employed to model the performance of bubble columns are based on the assumption that either plug flow or complete mixing conditions prevail in the liquid phase. Although due to the liquid-phase axial dispersion, the actual flow pattern in bubble columns is usually closer to being mixed flow rather than plug flow, but still not completely mixed flow. Therefore, the back flow cell model (BFCM), that hypothesises both back flow and exchange flow to characterise the liquid-phase axial dispersion, is presented as an alternative approach to describe the hydrodynamics and mass transfer of ozone bubble columns. BFCM is easy to formulate and solve. It is an accurate and reliable design model. Transient BFCM consists of NBFCM ordinary-first-order differential equations in which NBFCM unknowns (Yj) are to be determined. That set of equations was solved numerically as NBFCM linear algebraic equations. Steady-state BFCM consists of 3 × NBFCM non-linear algebraic equations in which 3 × NBFCM unknowns (qG,j, Xj, and Yj) are to be determined. Those non-linear algebraic equations were solved numerically using Newton–Raphson technique. Steady-state BFCM was initially tested using the pilot-scale experimental data of Zhou. BFCM provided excellent predictions of the dissolved ozone profiles under different operating conditions for both counter and co-current flow modes.

scholarly journals Approximate Stochastic Response of Hysteretic System With Fractional Element and Subjected to Combined Stochastic and Periodic Excitation

2021 ◽  
Author(s):  
Fan Kong ◽  
Renjie Han ◽  
Yuanjin Zhang
Keyword(s):  
Fractional Order ◽  
Harmonic Balance Method ◽  
System Response ◽  
Statistical Linearization ◽  
Algebraic Equations ◽  
Stochastic Response ◽  
Hysteretic System ◽  
Linear Algebraic Equations ◽  
Non Linear ◽  
Linear Differential

Abstract A method based on statistical linearization is proposed, for determining response of the single-degree-of-freedom (SDOF) hysteretic system endowed with fractional derivatives and subjected to combined periodic and white/colored excitation. The method is developed by decomposing the system response into a combination of a periodic and of a zero-mean stochastic components. In this regard, first, the equation of motion is cast into two sets of coupled fractional-order non-linear differential equations with unknown deterministic and stochastic response components. Next, the harmonic balance method and the statistical linearization for the fractional-order deterministic and stochastic subsystems are used, to obtain the Fourier coefficients of the deterministic component and the variance of the stochastic component, respectively. This yields two sets of coupled non-linear algebraic equations which can be solved by appropriate standared numerical method. Pertinent numerical examples, including both softening and hardening Bouc-Wen hysteretic system endowed with different fractional-orders, are used to demonstrate the applicability and accuracy of the proposed method.

Diffusion Transport in Electrochemical Systems: A New Approach to Determining of the Membrane Potential at Steady State

2009 ◽  
Vol 283-286 ◽  
pp. 487-493 ◽  
Author(s):  
Robert Filipek ◽  
Krzysztof Szyszkiewicz ◽  
Bogusław Bożek ◽  
Marek Danielewski ◽  
A. Lewenstam
Keyword(s):  
Steady State ◽  
Electrical Potential ◽  
Original System ◽  
Potential Difference ◽  
Constant Field ◽  
Electrical Potential Difference ◽  
Algebraic Equations ◽  
Liquid Junction ◽  
Electrochemical Systems ◽  
Linear Algebraic Equations

Ionic concentrations and electric field space profiles in one dimensional membrane are described using Nernst-Planck-Poisson (NPP) equations. The usual assumptions for the steady state NPP problem requires knowledge of the boundary values of the concentrations and electrical potential difference. In analytical chemistry the potential difference may not be known and its theoretical prediction from the model is desirable. The effective methods of the solution to the NPP equations are presented. The Poisson equation is solved without widely used simplifications such as the constant field or the electroneutrality assumptions. The first method uses a steady state formulation of NPP problem. The original system of ODEs is turned into the system of non-linear algebraic equations with unknowns fluxes of the components and electrical potential difference. The second method uses the time-dependent form of the Nernst-Planck-Poisson equations. Steady-state solution has been obtained by starting from an initial profiles, and letting the numerical system evolve until a stationary solution is reached. The methods have been tested for different electrochemical systems: liquid junction and ion selective electrodes (ISEs). The results for the liquid junction case have been also verified with the approximate solutions leading to a good agreement. Comparison with the experimental results for ISEs has been carried out.

Steady-State Simulation of an Automotive Air Conditioning System Running on CO2

2007 ◽  
Author(s):  
Pedro P. Morais Filho ◽  
Jose´ Alberto R. Parise ◽  
Rui P. Marques da Silva
Keyword(s):  
Steady State ◽  
Cycle Performance ◽  
Algebraic Equations ◽  
Climate Control ◽  
Ambient Temperatures ◽  
Automotive Air Conditioning ◽  
Linear Algebraic Equations ◽  
State Regime ◽  
Suction Line Heat Exchanger ◽  
Steady State Simulation

This work presents a semi-empirical simulation of an automotive climate control system equipped with a transcritical vapor compression cycle running on carbon dioxide. The cycle components (a compressor, a throttling valve, an evaporator, a gas cooler, a suction accumulator and a suction line heat exchanger) were modeled to study the operation of the system, in the steady-state regime, under high ambient temperatures. The model took into account the severe conditions of tropical climates since the temperature at the inlet of the gas cooler is one of the predominant factors in the transcritical cycle performance. To assess the performance of the cycle, the thermodynamic model, reduced to a set of non-linear algebraic equations, was solved by a modified Newton-Raphson method. Reasonable agreement was found when results predicted by the model were compared with experimental data available in the literature.

Bending of normally loaded simply supported rectangular plates in the large-deflection range

1969 ◽  
Vol 4 (3) ◽  
pp. 190-198 ◽  
Author(s):  
A Scholes ◽  
E L Bernstein
Keyword(s):  
Linear Equations ◽  
Rectangular Plates ◽  
Algebraic Equations ◽  
Simply Supported ◽  
Aspect Ratios ◽  
Linear Algebraic Equations ◽  
Non Linear ◽  
Out Of Plane ◽  
Linear Differential ◽  
Good Agreement

Means of solving the non-linear differential equations of plate bending are revieweed and a method based on minimizing the corresponding energy integral is selected as offering most advantages. The energy intergral can be approximated either by using finite-difference approximatons or by assuming a form of displacement variation. Two sets of non-linear algebraic equations (in the in-plane and out-of-plane deflections) are thus formed and, by substitution alternately in each set, the resulting linear equations are solved. Results for simply supported rectangular plates have been worked out in some detail; these are compared with tests made on plates of various aspect ratios. Good agreement on maximum values of stress and deflection was obtained.

The Damping Forced Vibration of Rectangular Stiffened Plates with Elastic Boundary Edges Including Boundary Damping

2014 ◽  
Vol 577 ◽  
pp. 205-208
Author(s):  
An Fu Zhang ◽  
Mei Ping Sheng ◽  
He Ye Xiao
Keyword(s):  
Forced Vibration ◽  
Stiffened Plates ◽  
Stiffened Plate ◽  
Algebraic Equations ◽  
Element Analysis ◽  
Boundary Damping ◽  
Coupled Equations ◽  
Elastic Boundary ◽  
Linear Algebraic Equations ◽  
Plate System

The paper proposes an analytical-numerical method for damping forced vibration problem of the rectangular stiffened plates with elastic boundary restraints. By enforcing applicable continuity conditions between plate and beams, coupled equations of beams-plate system are established. The displacement function is expressed as a modified two-dimensional Fourier series, which converts the differential equations of plate-beam systems into linear algebraic equations. Furthermore, Rayleigh’s proportional damping is introduced into beams-plate system so that damping parameters of stiffened plate can be inserted into vibration equations. It is derived that the present method demonstrates good agreement with standard finite element analysis. In subsequent analysis, it can be found that the boundary damping can reduce vibration energy of stiffened plate to same extent.

scholarly journals Lucas Polynomial Approach for System of High-Order Linear Differential Equations and Residual Error Estimation

2015 ◽  
Vol 2015 ◽  
pp. 1-14 ◽  
Author(s):  
Muhammed Çetin ◽  
Mehmet Sezer ◽  
Coşkun Güler
Keyword(s):  
Differential Equations ◽  
Approximate Solutions ◽  
Variable Coefficients ◽  
High Order ◽  
Linear Differential Equations ◽  
Algebraic Equations ◽  
Order Linear ◽  
Lucas Polynomials ◽  
Linear Algebraic Equations ◽  
Linear Differential

An approximation method based on Lucas polynomials is presented for the solution of the system of high-order linear differential equations with variable coefficients under the mixed conditions. This method transforms the system of ordinary differential equations (ODEs) to the linear algebraic equations system by expanding the approximate solutions in terms of the Lucas polynomials with unknown coefficients and by using the matrix operations and collocation points. In addition, the error analysis based on residual function is developed for present method. To demonstrate the efficiency and accuracy of the method, numerical examples are given with the help of computer programmes written inMapleandMatlab.

Iterative Methods for Solving Convection-diffusion Problem

2002 ◽  
Vol 2 (4) ◽  
pp. 410-444 ◽  
Author(s):  
Petr N. Vabishchevich
Keyword(s):  
Approximate Solution ◽  
Steady State ◽  
Iterative Methods ◽  
Diffusion Problem ◽  
Convective Transport ◽  
Symmetric Matrices ◽  
Algebraic Equations ◽  
Convection Diffusion ◽  
Linear Algebraic Equations ◽  
Diffusion Problems

AbstractTo obtain an approximate solution of the steady-state convectiondiffusion problem, it is necessary to solve the corresponding system of linear algebraic equations. The basic peculiarity of these LA systems is connected with the fact that they have non-symmetric matrices. We discuss the questions of approximate solution of 2D convection-diffusion problems on the basis of two- and three-level iterative methods. The general theory of iterative methods of solving grid equations is used to present the material of the paper. The basic problems of constructing grid approximations for steady-state convection-diffusion problems are considered. We start with the consideration of the Dirichlet problem for the differential equation with a convective term in the divergent, nondivergent, and skew-symmetric forms. Next, the corresponding grid problems are constructed. And, finally, iterative methods are used to solve approximately the above grid problems. Primary consideration is given to the study of the dependence of the number of iteration on the Peclet number, which is the ratio of the convective transport to the diffusive one.

On Improved Approximate Solution of the Fredholm Integral Equation

2006 ◽  
Vol 6 (3) ◽  
pp. 264-268
Author(s):  
G. Berikelashvili ◽  
G. Karkarashvili
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Fredholm Integral Equation ◽  
Algebraic Equations ◽  
Order Convergence ◽  
One Dimensional ◽  
Averaging Operator ◽  
Linear Algebraic Equations ◽  
Convergence Solution

AbstractA method of approximate solution of the linear one-dimensional Fredholm integral equation of the second kind is constructed. With the help of the Steklov averaging operator the integral equation is approximated by a system of linear algebraic equations. On the basis of the approximation used an increased order convergence solution has been obtained.

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