Simultaneous ODE

Equations (2.22) and (2.23) become indeterminate if ks = k. Special forms are needed for the analytical solution of a set of consecutive, first-order reactions whenever a rate constant is repeated. The derivation of the solution can be repeated for the special case or L Hospital s rule can be applied to the general solution. As a practical matter, identical rate constants are rare, except for multifunctional molecules where reactions at physically different but chemically similar sites can have the same rate constant. Polymerizations are an important example. Numerical solutions to the governing set of simultaneous ODEs have no difficulty with repeated rate constants, but such solutions can become computationally challenging when the rate constants differ greatly in magnitude. Table 2.1 provides a dramatic example of reactions that lead to stiff equations. A method for finding analytical approximations to stiff equations is described in the next section. 

Solution A program for evaluating the adiabatic reactor is given below. Subroutine Reactor solves the simultaneous ODEs for the concentrations and temperature. The equation for temperature includes contributions from both reactions according to the methods of Section 5.2. 

Equation (8.22) for a(0) is also special because, due to symmetry, there is only one adjacent point, a(l). The overall set may be solved by any desired method. Euler s method is discussed below and is illustrated in Example 8.5. There are a great variety of commercial and freeware packages available for solving simultaneous ODEs. Most of them even work. Packages designed for stiff equations are best. The stiffness arises from the fact that VJJ) becomes very small near the tube waU. There are also software packages that will handle the discretization automatically. 

This introduces the gas-phase residence time VgjQg as a new parameter. It also introduces an ambiguity regarding the term kgAi(a — Ug). There is no resistance to mass transfer within a pure component so kgAj oo and a — Ug 0. Thus, kgAi(a — Ug) is an indeterminate form of the oo x 0 variety. Its value must continue to equal the rate at which oxygen is transferred into the liquid phase. Equation (11.5) remains true and the pair of simultaneous ODEs become... 

Turning to the substrate balance, yeast cells contain about 50% carbon. The cell mass is measured as total dry weight, not just carbon. This gives Yx/s = 2 when S is measured as the carbon equivalent of glucose. A reasonable value for Yxis is 1 so that half the carbon goes into biomass and half meets the associated energy requirements. The maintenance coefficient in carbon-equivalent units is 0.008 h . Using these parameter estimates, the three simultaneous ODEs for 5" > 0, become... 

The infinite set of simultaneous ODEs has been transfonned to a set of moment equations that can be solved sequentially. The first three members of the set are ... 

Equations (14.1)-(14.3) are a set of simultaneous ODEs that govern the performance of an unsteady CSTR. The minimum set is just Equation (14.2), which governs the reaction of a single component with time-varying inlet concentration. The maximum set has separate ODEs for each of the variables... 

POLYMATH. AIChE Cache Corp, P O Box 7939, Austin TX 78713-7939. Polynomial and cubic spline curvefitting, multiple linear regression, simultaneous ODEs, simultaneous linear and nonlinear algebraic equations, matrix manipulations, integration and differentiation of tabular data by way of curve fit of the data. 

CONSTANTINIDES, Applied Numerical Methods with Personal Computers, McGraw-Hill, 1987. Nonlinear regression, partial deferential equations, matrix manipulations, and a mere flexible program for simultaneous ODEs. 

In Example 2.1, Maple was used to solve two simultaneous first order ODEs. The same methodology can be used to solve more than two simultaneous ODEs. Eor example, the material balance equations for the time dependent concentration of each species (A, B, and C) in an isothermal batch reactor with reversible series... 

The false-transient method can be applied to convective diffusion equations in a manner similar to that used for velocity profiles. Finite-difference approximations are written for the spatial derivatives. Second-order approximations can be used for first derivatives since they involve only the same five points needed for the second derivatives. The result is a set of simultaneous ODEs with (false) time as the independent variable. The computational template of Figure 16.3 is unchanged. The next two examples illustrate its application to problems where axial diffusion is negligible. Such problems are also readily solved by the method of lines as described in Chapter 8. Cases with significant axial diffusion are troublesome for the method of lines and require special boundary conditions for the method of false transients. They are treated in Section 16.2.4. 

Coupled Simultaneous ODE 93 operator D = d/dx. Thus, rewrite using operators to see... 

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