Equations To Be Solved

For the nonisothermal reactors we need to solve the mass- and energy-balance equations 

For a single reaction in a steady-state CSTR the mass-balance equation on reactant A and temperature T give the equations 

These are two coupled algebraic equations, which must be solved simultaneously to determine the solutions Cj(x) and T(t). For multiple reactions the + 1 equations are easily written down, as are the differential equations for the transient situation. However, for these situations the solutions are considerably more difficult to find We will in fact consider theaolutions of the transient CSTR equations in Chapter 6 to describe phase-plane trajectories and the stability of solutions in the nonisothermal CSTR. 

These are first-order ordinary differential equations that have two initial conditions at the inlet to the reactor, 

We consider here two of the most basic equations in computational chemistry the Poisson equation and the Schrodinger equation. The Poisson equation yields the electrostatic potential due to a fixed distribution of 

We begin with a word on the general properties of these equations and how they can be derived. Even though the FD representation is not variational in the sense of bounds on the ground-state energy, the iterative process by which we obtain the solution to the partial differential equations (PDFs) can be viewed variationally, i.e., we minimize some functional (see below) with respect to variations of the desired function until we get to the lowest action or energy. This may seem rather abstract, but it turns out to be practical since it leads directly to the iterative relaxation methods to be discussed below. (See Ref. 98 for a more complete mathematical description of minimization and variational methods in relation to the FE method.) 

For the Poisson problem, we make up an action functional S[ )] that, when minimized, yields the Poisson equation  

A functional yields a number when the function over the whole domain (cj) here) is specified. We then write a pseudodynamical equation (actually a steepest-descent equation) for the updates of c)  

This equation, when iterated, will repeatedly move downhill on the action surface until the minimum is reached. Notice that this equation looks like a diffusion equation with a source term. It can be proved mathematically that, for this case, there is only one extremum, and it is a minimum. At the minimum (which is the point where the potential stops changing). 

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Here all couplings are ignored except the direct couplings between the initial and final states as in a two-level atom. The coupled equations to be solved are... 

In what follows, the subscript M will be omitted to simplify the notations. If the initial point is P po, qa) and we are interested in deriving the value of A(= Am) at a final point Q p, q) then one integral equation to be solved is... 

This method has a simple straightforward logic for even complex systems. Multinested loops are handled like ordinary branched systems, and it can be extended easily to handle dynamic analysis. However, a huge number of equations is involved. The number of unknowns to be solved is roughly equal to six times the number of node points. Therefore, in a simple three-anchor system, the number of equations to be solved in the flexibiUty method is only 12, whereas the number of equations involved in the direct stiffness method can be substantially larger, depending on the actual number of nodes. 

Successive Substitutions Let/(x) = 0 be the nonlinear equation to be solved. If this is rewritten as x = F x), then an iterative scheme can be set up in the form Xi + = F xi). To start the iteration an initial guess must be obtained graphically or otherwise. The convergence or divergence of the procedure depends upon the method of writings = F x), of which there will usually be several forms. However, if 7 is a root of/(x) = 0, and if IF ( 7)I < I, then for any initial approximation sufficiently close to a, the method converges to a. This process is called first order because the error in xi + is proportional to the first power of the error in xi for large k. 

Implicit Methods By using different interpolation formulas involving y, it is possible to cferive imphcit integration methods. Implicit methods result in a nonhnear equation to be solved for y so that iterative methods must be used. The backward Euler method is a first-order method. 

Discretization of the governing equations. In this step, the exact partial differential equations to be solved are replaced by approximate algebraic equations written in terms of the nodal values of the dependent variables. Among the numerous discretization methods, finite difference, finite volume, and finite element methods are the most common. Tlxe finite difference method estimates spatial derivatives in terms of the nodal values and spacing between nodes. The governing equations are then written in terms of... 

Compute a new set of values of the T) tear variables by solving simultaneously the set of N energy-balance equations (13-72), which are nonlinear in the temperatures that determine the enthalpy values. When linearized by a Newton iterative procedure, a tridiagonal-matrix equation that is solved by the Thomas gorithm is obtained. If we set gj equal to Eq. (13-72), i.e., its residual, the hnearized equations to be solved simultaneously are... 

AIChE monograph Senes, AIChE, New York, 81, No. 15 (1985)]. Homotopy methods begin from a known solution of a companion set of equations and follow a path to the desired solution of the set of equations to be solved. In most cases, the path exists and can be followed. In one implementation, the set of equations to be solved, call tf x), and the companion set of equations, call it g x), are connected together by a set of mathematical homotopy equations ... 

For dilute-gas systems one form of the equation to be solved in conjunction with these experimental data is... 

There will be one CPHF equation to be solved for eaeh perturbation. If it is an eleetric or magnetie field, there will in general be three eomponents if it is a... 

The differential equation to be solved for a cylindrical system assuming unidirectional current flow conditions apply is ... 

In this chapter we described Euler s method for solving sets of ordinary differential equations. The method is extremely simple from a conceptual and programming viewpoint. It is computationally inefficient in the sense that a great many arithmetic operations are necessary to produce accurate solutions. More efficient techniques should be used when the same set of equations is to be solved many times, as in optimization studies. One such technique, fourth-order Runge-Kutta, has proved very popular and can be generally recommended for all but very stiff sets of first-order ordinary differential equations. The set of equations to be solved is... 

This equation is coupled to the component balances in Equation (3.9) and with an equation for the pressure e.g., one of Equations (3.14), (3.15), (3.17). There are A +2 equations and some auxiliary algebraic equations to be solved simultaneously. Numerical solution techniques are similar to those used in Section 3.1 for variable-density PFRs. The dependent variables are the component fluxes , the enthalpy H, and the pressure P. A necessary auxiliary equation is the thermodynamic relationship that gives enthalpy as a function of temperature, pressure, and composition. Equation (5.16) with Tref=0 is the simplest example of this relationship and is usually adequate for preliminary calculations. 

Now consider foe external force acting on the system to be composed of a series of instantaneous impacts, each of which can be expressed mathematically by a delta function. The response of the system can then be represented by a function G(f ) The differential equation to be solved then takes on the form... 

Successive Substitutions Let fix) = 0 be the nonlinear equation to be solved. If this is rewritten as x = Fix), then an iterative scheme can be set up in the form x,. Fixf). To start the iteration an initial... 

As the continuity equation, the NS equations, and the transport equations for the turbulent variables are highly nonlinear, any CFD-calculation is essentially iterative. Generally, the convergence rate of simulations depends on the number of grid points and on the number of equations to be solved. 

The number of equations to be solved is, among other things, related to the turbulence model chosen (in comparison with the k-e model, the RSM involves five more differential equations). The number of equations further depends on the character of the simulation whether it is 3-D, 21/2-D, or just 2-D (see below, under The domain and the grid ). In the case of two-phase flow simulations, the use of two-fluid models implies doubling the number of NS equations required for single-phase flow. All this may urge the development of more efficient solution algorithms. Recent developments in computer hardware (faster processors, parallel platforms) make this possible indeed. 

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