Simultaneous solution algebraic

Ac Che limic of Knudsen screaming Che flux relacions (5.25) determine Che fluxes explicitly in terms of partial pressure gradients, but the general flux relacions (5.4) are implicic in Che fluxes and cheir solution does not have an algebraically simple explicit form for an arbitrary number of components. It is therefore important to identify the few cases in which reasonably compact explicit solutions can be obtained. For a binary mixture, simultaneous solution of the two flux equations (5.4) is straightforward, and the result is important because most experimental work on flow and diffusion in porous media has been confined to pure substances or binary mixtures. The flux vectors are found to be given by... 

Since a stable steady state is sought, the method of false transients could be used for the simultaneous solution of Equations (5.29) and (5.31). However, the ease of solving Equation (5.29) for makes the algebraic approach simpler. Whichever method is used, a value for UAext pQCp is assumed and then a value for Text is found that gives 413 K as the single steady state. Some results are... 

The determination of the steam density, pj, therefore requires the simultaneous solution of two algebraic equations. This represents an IMPLICIT algebraic loop and cannot be solved within a simulation program without the incorporation of a trial and error convergence procedure. 

For other reaction networks, a similar set of equations may be developed, with the kinetics terms adapted to account for each reaction occurring. To determine the conversion and selectivity for a given bed depth, Ljh equations 23.4-11 and -14 are numerically integrated from x = 0 to x = Lfl, with simultaneous solution of the algebraic expressions in 23.4-12, -13, -15, and -16. The following example illustrates the approach for a series network. 

Digital simulation is a powerful tool for solving the equations describing chemical engineering systems. The principal difficulties are two (1) solution of simultaneous nonlinear algebraic equations (usually done by some iterative method), and (2) numerical integration of ordinary differential equations (using discrete finite-difference equations to approximate continuous differential equations). 

One of the most common problems in digital simulation is the solution of simultaneous nonlinear algebraic equations. If these equations contain transcendental functions, analytical solutions are impossible. Therefore, an iterative trial-and-error procedure of some sort must be devised. If there is only one unknown, a value for the solution is guessed. It is plugged into the equation or equations to see if it satisfies them. If not, a new guess is made and the whole process is repeated until the iteration eonverges (we hope) to the right value. 

Tjoa, I.-B., and Biegler, L. T., Simultaneous solution and optimization strategies for parameter estimation of differential-algebraic equations systems, I EC Research, 30, 376 (1991). 

Models can have the characteristic of different types and sizes of equation sets relative to a general set of algebraic equations. Some common example situations include physical property models and models containing differential equations. In posing the mathematical problem to be solved, a completely simultaneous solution approach can be used or a "mixed mode" that combines specialized solution techniques within the overall EO approach. 

There are now four equations and four unknowns. But the solution of these simultaneous nonlinear algebraic equations is difficult. It would be even more difficult if the reactions were not first-order. 

The calculation of temperatures and equilibrium compositions of gas mixtures involves simultaneous solution of linear (material balance) and nonlinear (equilibrium) algebraic equations. Therefore, it is necessary to resort to various approximate procedures classified by Carter and Altman (Cl) as (1) trial and error methods (2) iterative methods (3) graphical methods and use of published tables and (4) punched-card or machine methods. Numerical solutions involve a four-step sequence described by Penner (P4). 

As before, we have to incorporate corrections for the ligand isotopes and mass differences, but we now have 2 equations with two unknowns which can be solved simultaneously. The algebraic manipulations are quite messy and will not be presented here. Many of the terms in the solution have small insignificant coefficients and can be eliminated. The solution simplifies to ... 

Analytical solution methods such as those presented in Chapter 2 are based on solving the governing differential equation together with the boundary conditions. Tliey result in solution functions for the temperature at every point in the medium. Numerical methods, on the other hand, are based on replacing the difi erential equation by a set of n algebraic equations for the unknown temperatures at n selected points in the medium, and the simultaneous solution of these equations results in the temperature values at those discrete points. 

Biegler, L. T. Strategies for Simultaneous Solution and Optimization of Differential-Algebraic Sy.stems, m Proceedings of tie Third International Conference on Foundations of Computer-Aided Process Design, Snowm tss, CO, 1989, p. 155-179 (1990). 

This section begins with the most rigorous and numerical models. These models are potentially the most accurate, but require physical property data and simultaneous solution of differential and algebraic equations. Generally speaking, simpler models are more accessible to engineers and easier to implement. They can be very useful as long as the inherent limitations are understood. 

Stagewise calculations require the simultaneous solution of material and energy balances with equilibrium relationships. It was demonstrated in Example 1.1 that the design of a simple extraction system reduces to the solution of linear algebraic equations if (1) no energy balances are needed and (2) the equilibrium relationship is linear. 

By solving the following simultaneous linear algebraic equations 97g(b V, Cj)/dbi = 0, I = 1,2,..., Nb, the closed-form solution of the most probable coefficient vector b can be... 

Equation (7.18) represents a set of N nonlinear algebraic equations, which must be solved by a trial and error method such as Newton-Raphson or successive substitution for y . j (Appendix A). This is the characteristic difference between the implicit and explicit types of solution the easier explicit method allows sequential solution one at a time, while the implicit method requires simultaneous solutions of sets of equations hence, an iterative solution at a given time... 

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