Simultaneous solution algebraic equations

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... 

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. 

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. 

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... 

As a result of eq. (10.12), the algebraic equations obtained by Laplace transforming DDEs are always transcendental. Sets of DDEs may be transformed into sets of simultaneous (transcendental) algebraic equations. Note that eq. (10.12) also implies that the value of the solution on the initial interval is required in order to perform the transforms. 

It has been proposed to represent the TF with 10 CSTR staged units 0.3 ft in length. Develop solutions to this problem using a finite difference method of solving an ordinary differential equation and a lumped parameter model employing a method of solution of simultaneous linear algebraic equations. 

For the numerical solution of the mass transfer model equations using the finite element method, the system must first be clearly defined. The finite element method is based on the numerical approximation of the dependent variables at a specific nodal location, where a set of simultaneous linear algebraic equations is produced that can be solved either directly or iteratively. 

Numerical Solution of Simultaneous Linear Algebraic Equations... 

In the remainder of this section, we give several examples of systems drawing from chemical engineering applications that yield sets of simultaneous linear algebraic equations. In the following sections of this chapter, we discuss several methods for the numerical solution of such problems and demonstrate the application of these methods on the computer. 

NumerPcal Solution of Simultaneous Linear Algebraic Equations Chapter 2 Table 2.3 Number of operstlone needed by Cramer s rule... 

The most widely used method for solution of simultaneous linear algebraic equations is the Gauss elimination method. This is based on the principle of converting the set of n equations in n unknowns ... 

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