The Method of False Transients

The method of false transients converts a steady-state problem into a time-dependent problem. Equations (4.1) govern the steady-state performance of a CSTR. How does a reactor reach the steady state There must be a startup transient that eventually evolves into the steady state, and a simulation of 

The ODEs governing the unsteady CSTR are obtained by adding accumulation terms to Equations (4.1). The simulation holds the volume constant, and 

This set of first-order ODEs is easier to solve than the algebraic equations where all the time derivatives are zero. The initial conditions are that a ut = no, bout = bo. at t = 0. The long-time solution to these ODEs will satisfy Equations (4.1) provided that a steady-state solution exists and is accessible from the assumed initial conditions. There may be no steady state. Recall the chemical oscillators of Chapter 2. Stirred tank reactors can also exhibit oscillations or more complex behavior known as chaos. It is also possible that the reactor has multiple steady states, some of which are unstable. Multiple steady states are fairly common in stirred tank reactors when the reaction exotherm is large. The method of false transients will go to a steady state that is stable but may not be desirable. Stirred tank reactors sometimes have one steady state where there is no reaction and another steady state where the reaction runs away. Think of the reaction A B — C. The stable steady states may give all A or all C, and a control system is needed to stabilize operation at a middle steady state that gives reasonable amounts of B. This situation arises mainly in nonisothermal systems and is discussed in Chapter 5. 

Solution Write a version of Equation (4.4) for each component. Divide through by Qin = Qout and substitute the appropriate reaction rates to obtain 

Then use a first-order difference approximation for the time derivatives, e.g., 

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Example 4.2 used the method of false transients to solve a steady-state reactor design problem. The method can also be used to find the equilibrium concentrations resulting from a set of batch chemical reactions. To do this, formulate the ODEs for a batch reactor and integrate until the concentrations stop changing. This is illustrated in Problem 4.6(b). Section 11.1.1 shows how the method of false transients can be used to determine physical or chemical equilibria in multiphase systems. 

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. 

The method of false transients begins with the inlet stream set to its steady-... 

Solution Example 4.5 was a reverse problem, where measured reactor performance was used to determine constants in the rate equation. We now treat the forward problem, where the kinetics are known and the reactor performance is desired. Obviously, the results of Run 1 should be closely duplicated. The solution uses the method of false transients for a variable-density system. The ideal gas law is used as the equation of state. The ODEs are... 

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 ... 

The accumulation term is zero for steady-state processes. The accumulation term is needed for batch reactors and to solve steady-state problems by the method of false transients. 

The time derivative is zero at steady state, but it is included so that the method of false transients can be used. The computational procedure in Section 4.3.2 applies directly when the energy balance is given by Equation (5.28). The same basic procedure can be used for Equation (5.25). The enthalpy rather than the temperature is marched ahead as the dependent variable, and then Tout is calculated from Hout after each time step. 

The method of false transients cannot be used to find a metastable steady state. Instead, it is necessary to solve the algebraic equations that result from setting the derivatives equal to zero in Equations (5.29) and (5.30). This is easy in the current example since Equation (5.29) (with daout/dr = Q) can be solved for Uout- The result is substituted into Equation (5.30) (with dTout/dt = Q) to obtain a single equation in a single unknown. The three solutions are... 

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... 

Equilibrium Compositions for Single Reactions. We turn now to the problem of calculating the equilibrium composition for a single, homogeneous reaction. The most direct way of estimating equilibrium compositions is by simulating the reaction. Set the desired initial conditions and simulate an isothermal, constant-pressure, batch reaction. If the simulation is accurate, a real reaction could follow the same trajectory of composition versus time to approach equilibrium, but an accurate simulation is unnecessary. The solution can use the method of false transients. The rate equation must have a functional form consistent with the functional form of K,i,ermo> e.g., Equation (7.38). The time scale is unimportant and even the functional forms for the forward and reverse reactions have some latitude, as will be illustrated in the following example. 

Example 7.12 Use the method of false transients to determine equilibrium concentrations for the reaction of Example 7.11. Specifically, determine the equilibrium mole fraction of component A at r=550K as a function of pressure, given that the reaction begins with pure A. 

FIGURE 7.6 Equilibrium concentrations calculated by the method of false transients for a non-elementary reaction. 

Solution A rigorous treatment of a reversible reaction with variable physical properties is fairly complicated. The present example involves just two ODEs one for composition and one for enthalpy. Pressure is a dependent variable. If the rate constants are accurate, the solution will give the actual reaction trajectory (temperature, pressure, and composition as a function of time). If ko and Tact are wrong, the long-time solution will still approach equilibrium. The solution is then an application of the method of false transients. 

These four equations are perfectly adequate for equilibrium calculations although they are nonsense with respect to mechanism. Table 7.2 has the data needed to calculate the four equilibrium constants at the standard state of 298.15 K and 1 bar. Table 7.1 has the necessary data to correct for temperature. The composition at equilibrium can be found using the reaction coordinate method or the method of false transients. The four chemical equations are not unique since various members of the set can be combined algebraically without reducing the dimensionality, M=4. Various equivalent sets can be derived, but none can even approximate a plausible mechanism since one of the starting materials, oxygen, has been assumed to be absent at equilibrium. Thermodynamics provides the destination but not the route. 

We have considered thermodynamic equilibrium in homogeneous systems. When two or more phases exist, it is necessary that the requirements for reaction equilibria (i.e., Equations (7.46)) be satisfied simultaneously with the requirements for phase equilibria (i.e., that the component fugacities be equal in each phase). We leave the treatment of chemical equilibria in multiphase systems to the specialized literature, but note that the method of false transients normally works quite well for multiphase systems. The simulation includes reaction—typically confined to one phase—and mass transfer between the phases. The governing equations are given in Chapter 11. 

The time derivatives are dropped for steady-state, continuous flow, although the method of false transients may still be convenient for solving Equations (11.11) and (11.12) (or, for variable Kh, Equations (11.9) and (11.10) together with the appropriate auxiliary equations). The general case is somewhat less complicated than for two-phase batch reactions since system parameters such as V, Vg, Vh and At will have steady-state values. Still, a realistic solution can be quite complicated. 

Solution The reactions are the same as in Example 12.5. The steady-state performance of a CSTR is governed by algebraic equations, but time derivatives can be useful for finding the steady-state solution by the method of false transients. The governing equations are... 

The general material balance of Section 1.1 contains an accumulation term that enables its use for unsteady-state reactors. This term is used to solve steady-state design problems by the method of false transients. We turn now to solving real transients. The great majority of chemical reactors are designed for steady-state operation. However, even steady-state reactors must occasionally start up and shut down. Also, an understanding of process dynamics is necessary to design the control systems needed to handle upsets and to enable operation at steady states that would otherwise be unstable. 

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. 

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