Numerical methods false transients

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. 

Solution With Z>, = 0, a reaction wiU never start in a PFR, but a steady-state reaction is possible in a CSTR if the reactor is initially spiked with component B. An anal5dical solution can be found for this problem and is requested in Problem 4.12, but a numerical solution is easier. The design equations in a form suitable for the method of false transients are... 

A simple numerical example sets = 1, bi = 0, and k = 5. Suitable initial conditions for the method of false transients are o = 0 and bo=l. Suppose the residence time for the composite system is t - -t2 =. The question is how this total time should be divided. The following results were obtained ... 

N202, N203, N205, N03, 03, and possibly others. The various reaction coordinates will differ by many orders of magnitude and the numerical solution would be quite difficult even assuming that the various equilibrium constants could be found. The method of false transients would ease the numerical solution but would not help with the problem of estimating the equilibrium constants. 

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. 

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