Numerical Solutions to Sets of First-Order ODEs

4 NUMERICAL SOLUTIONS TO SETS OF FIRST-ORDER ODEs 

The computation is begun by setting bold = b, and toid = 0. Rates are 

Solution The following is a complete program for performing the calculations. It is written in Basic as an Excel macro. The rather arcane statements needed to display the results on the Excel spreadsheet are shown at the end. They need to be replaced with PRINT statements given a Basic compiler that can write directly to the screen. The programming examples in this text will normally show only the computational algorithm and will leave input and output to the reader. 

The results from this program (with added headers) are shown below  

These results have converged to four decimal places. The output required about 2 s on what will undoubtedly be a slow PC by the time you read this. 

The computation is begun by setting a0id = o, b0u = bo, and t0id = 0. Rates are computed using the old concentrations and the marching equations are used to calculate the new concentrations. Old is then replaced by new and the march takes another step. 

As shown by Equations 2.9 and 2.10, the design equations for batch reactors are sets of first-order ODEs of the type known as initial-value problems. Time t is the independent variable. The dependent variables are the component concentrations a, b,c. and, in subsequent sections, the state variables of temperature and pressure. These ODEs all have the form 

SO fast that this brute-force method of solving and testing for convergence will take only a few seconds for most of the problems in this book. 

The following statements output the results to the Excel spreadsheet 

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Numerical Solutions to Sets of First-Order ODEs 47... 

Example 4.3 represents the simplest possible example of a variable-density CSTR. The reaction is isothermal, first-order, irreversible, and the density is a linear function of reactant concentration. This simplest system is about the most complicated one for which an analytical solution is possible. Realistic variable-density problems, whether in liquid or gas systems, require numerical solutions. These numerical solutions use the method of false transients and involve sets of first-order ODEs with various auxiliary functions. The solution methodology is similar to but simpler than that used for piston flow reactors in Chapter 3. Temperature is known and constant in the reactors described in this chapter. An ODE for temperature wiU be added in Chapter 5. Its addition does not change the basic methodology. 

This is a fairly simple set of first-order ODEs. The set is difficult to solve analytically, but numerical solutions are easy. 

This set of first-order ODEs is easier to solve numerically than the algebraic equations that result from setting all the time derivatives to zero. The initial conditions are Uout = do, bout = bo, at t = 0. The long-time solution to the ODEs will satisfy Equations 4.1 provided that a steady-state solution exits and is accessible from the initial conditions. As discussed in Chapter 5, some CSTRs have multiple steady states and the achieved steady state depends on the initial conditions. There may be no steady state. Recall the chemical oscillators of Chapter 2. Stirred tank reactors can exhibit oscillations or even a semirandom behavior known as chaos. The method of false transients will then fail to achieve a steady state. Another possibility is a metastable steady state. Operation at a metastable steady state requires a control system and cannot be reached by the method of false transients. Metastable steady states arise mainly in nonisothermal systems and are discussed in Chapter 5. 

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. 

The mass balance given by (18-5), (18-6), and (18-7) corresponds to an ordinary differential eqnation that is second-order dne to diffusion and nonlinear when n 0, 1 due to the rate of depletion of reactant A via chemical reaction. Numerical integration is required to generate basic information for 4 a( A), except when n = 0, 1. Second-order ODEs are solved numerically by reducing them to a set of two coupled first-order ODEs, which require two boundary conditions for a unique solution. The procedure is illustrated for porous wafers. If the dimensionless gradient of molar density is defined by d /dr] = then... 

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