Reactor, batch transient equations

In the analysis of batch reactors, the two flow terms in equation (8.0.1) are omitted. For continuous flow reactors operating at steady state, the accumulation term is omitted. However, for the analysis of continuous flow reactors under transient conditions and for semibatch reactors, it may be necessary to retain all four terms. For ideal well-stirred reactors, the composition and temperature are uniform throughout the reactor and all volume elements are identical. Hence, the material balance may be written over the entire reactor in the analysis of an individual stirred tank. For tubular flow reactors the composition is not independent of position and the balance must be written on a differential element of reactor volume and then integrated over the entire reactor using appropriate flow conditions and concentration and temperature profiles. When non-steady-state conditions are involved, it will be necessary to integrate over time as well as over volume to determine the performance characteristics of the reactor. 

Insertion of these rate laws in mass balances of ideal reactors (batch/plug flow or transient CSTR) leads to systems of semi-linear, first-order, partial differential equations, with a single family of characteristics [Eq. (139)]. 

Equations 11 and 12 are only valid if the volumetric growth rate of particles is the same in both reactors a condition which would not hold true if the conversion were high or if the temperatures differ. Graphs of these size distributions are shown in Figure 3. They are all broader than the distributions one would expect in latex produced by batch reaction. The particle size distributions shown in Figure 3 are based on the assumption that steady-state particle generation can be achieved in the CSTR systems. Consequences of transients or limit-cycle behavior will be discussed later in this paper. 

Note that this is exactly the transient CSTR equation we derived previously, and elimination of the flow terms yields the batch reactor. Keeping aU these terms and allowing Uq, v, V, and Cao to vary with time yields the semibatch reactor. 

Note that setting one of the terms on the left side of the equation equal to zero yields either the batch reactor equation or the steady-state PFTR equation. However, in general we must solve the partial differential equation because the concentration is a function of both position and time in the reactor. We will consider transients in tubular reactors in more detail in Chapter 8 in connection with the effects of axial dispersion in altering the perfect plug-flow approximation. 

Next consider the response of a PFTR with steady flow to a pulse injected at f = 0. Wc could obtain this by solving the transient PFTR equation written earher in this chapter, but we can see the solution simply by following the pulse down the reactor. (This is identical to the transformation we made in transforrning the batch reactor equations to the PFTR equations.) The S(0) pulse moves without broadening because we assumed perfect plug flow, so at position z the pulse passes at time z/u and the pulse exits the reactor at time T = L/u. Thus for a perfect PFTR the RTD is given by... 

In Chapter 3, the analytical method of solving kinetic schemes in a batch system was considered. Generally, industrial realistic schemes are complex and obtaining analytical solutions can be very difficult. Because this is often the case for such systems as isothermal, constant volume batch reactors and semibatch systems, the designer must review an alternative to the analytical technique, namely a numerical method, to obtain a solution. For systems such as the batch, semibatch, and plug flow reactors, sets of simultaneous, first order ordinary differential equations are often necessary to obtain the required solutions. Transient situations often arise in the case of continuous flow stirred tank reactors, and the use of numerical techniques is the most convenient and appropriate method. 

The CSTR operator, Rc, has an identical term to describe accumulation under transient operation. The algebraic sum of the two other terms indicates the difference of in-flow and out-flow of that species. This operator also describes semibatch or semicontinuous operation in cases where the volume can be assumed to be essentially constant. In the more general case of variable volume, V must be included within the differential accumulation term. At steady state, it is a difference equation of the same form as the differential equation for a batch reactor. 

Batch Reactors. One of the classic works in this area is by Gee and Melville (21), based on the PSSA for chain reaction with termination. Realistic mechanisms of termination, disproportionation, and combination, are treated with a variety of initiation kinetics, and analytical solutions are obtained. Liu and Amundson (37) solved the simultaneous differential equations for batch and transient stirred tank reactors by using digital computer without the PSSA. The degree of polymerization was limited to 100 the kinetic constants used were not typical and led to radical lifetimes of hours and to the conclusion that the PSSA is not accurate in the early stages of polymerization. In 1962 Liu and Amundson used the generating function approach and obtained a complex iterated integral which was later termed inconvenient for computation (37). The example treated was monomer termination. 

In an ideal continuous stirred tank reactor, composition and temperature are uniform throughout just as in the ideal batch reactor. But this reactor also has a continuous feed of reactants and a continuous withdrawal of products and unconverted reactants, and the effluent composition and temperature are the same as those in the tank (Fig. 7-fb). A CSTR can be operated under transient conditions (due to variation in feed composition, temperature, cooling rate, etc., with time), or it can be operated under steady-state conditions. In this section we limit the discussion to isothermal conditions. This eliminates the need to consider energy balance equations, and due to the uniform composition the component material balances are simple ordinary differential equations with time as the independent variable ... 

Reactors do not always run at steady state. In fact, many pharmaceuticals are made in a batch mode. Such problems are easily solved using the same techniques presented above because the plug flow reactor equations are identical to the batch reactor equations. Even CSTRs can be run in a transient mode, and it may be necessary to model a time-dependent CSTR to study the stability of steady solutions. When there is more than one solution, one or more of them will be unstable. Thus, this section considers a time-dependent CSTR as described by Eq. (8.51) ... 

Example 4.2 applied the method of false transients to a CSTR to find the steady-state output. A set of algebraic equations was converted to a set of ODEs. Chapter 16 shows how the method can be applied to PDEs by converting them to sets of ODEs. The method of false transients can also be used to find the equilibrium concentrations resulting from a set of batch chemical reactions. Formulate the ODEs for a batch reactor and integrate until the concentrations stop changing. Irreversible reactions go to completion. Reversible reactions reach equilibrium concentrations. 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. 

Multiphase reactors can be batch, fed batch, or continuous. Most of the design equations derived in this chapter are general and apply to any of these operating modes. They will be derived for unsteady operation. The unsteady material balances include the inventories in both phases and mass transfer between the phases so that steady-state solutions fonnd by the method of false transients will be true transients if the initial conditions are correct. Compare Section 10.6. 

Now we can really see why the CSTR operated at steady state is so different from the transient batch reactor. If the inlet feed flow rates and concentrations are fixed and set to be equal in sum to the outlet flow rate, then, because the volume of the reactor is constant, the concentrations at the exit are completely defined for fixed kinetic parameters. Or, in other words, if we need to evaluate kab and kd, we simply need to vary the flow rates and to collect the corresponding concentrations in order to fit the data to these equations to obtain their magnitudes. We do not need to do any integration in order to obtain the result. Significantly, we do not need to have fast analysis of the exit concentrations, even if the kinetics are very fast. We set up the reactor flows, let the system come to steady state, and then take as many measurements as we need of the steady-state concentration. Then we set up a new set of flows and repeat the process. We do this for as many points as necessary in order to obtain a statistically valid set of rate parameters. This is why the steady-state flow reactor is considered to be the best experimental reactor type to be used for gathering chemical kinetics. 

Remarkably, the form of the steady-state PFR equations is identical to the form of the fully transient well-mixed batch reactor with no volume change. The only difference is that instead of the derivative with respect to real time, the PFR equations involve the derivative with respect to reduced time. This is a very significant result. It shows us why the steady-state PFR is also such a useful reactor for kinetic studies—its model equations are quite simple Whichever form of the equations used is simply a matter of preference they all mean the very same thing. 

This result is very nice because it shows us that at the steady state the PFR has the same governing equation as the transient batch reactor, except that instead of the real time the differential is given in terms of the differential holding time. 

We have seen how the kinetics fit into a reactor equation as a constitutive relationship. Elow and reaction come together in these systems to affect the rate of accumulation. Hence when we refer to the "rate" we must be careful to be specific about the reactor—if it is a constant volume batch reactor, then rate means the chemical rate. If the reactor is a transient CSTR or PER, then the rate of change of the concentration at the exit of the reactor is not the chemical rate alone. Mixing effects are important and we have seen how to begin to account for the fluid mechanics in a reactor through the empirical measure of the r.t.d. The r.t.d. does affect the outcome from the reactor, but the sensitivity to the r.t.d. depends upon the kinetics and their fimctional form. 

This last equation is identical to that of a batch, isothermal, consunt-volume, transient batch reactor with a unique residence time I ... 

Transient reactor operation plays an increasingly important role in bioprocessing and has to some extent already been considered (classification, see Fig. 3.31 fed-batch culture, see Fig. 3.37 situation, see Fig. 4.4 guidelines to solution, see Sect. 4.2 and Fig. 4.5 structured cell model concept, see Fig. 4.7 application, see Chap. 6). Both balanced and frozen conditions have also been considered in Fig. 3.34. A biosystem is in balanced condition when the mechanism is fully adapted, as in a quasi-steady-state (if x ). All different equations can be reduced to algebraic equations. A biosystem is in frozen condition of the initial state (if x x ) and the mechanism may be neglected due to the fact that the slowest step is rate determining ( rds concept ). By this procedure, equations are reduced to parameters so that the number of equations is reduced (e.g., the case of dropwise addition of substrate). This is the case of steady state CSTR. 

In the case of transient operation, an accumulation term, that is, a differential term with respect to time, has to be added to equations 79 and 80 for being able to describe the observations. A batch reactor with uniform concentrations throughout the entire reactor but without continuous feed addition and effluent removal, is inherently operating in a transient regime. The corresponding reactor model equation is analogous to that of a plug flow reactor with the derivative taken with respect to time rather than with respect to position ... 

This law can be applied to steady-state or unsteady-state (transient) processes and to batch or continuous reactor systems. A steady-state process is one in which there is no change in conditions (e.g., pressure, temperature, composition) or rates of flow with time at any given point in the system. The accumulation term in Equation (7.2) is then zero. (If there is no chemical or nuclear reaction, the generation term is also zero.) All other processes are unsteady-state. In a batch reactor process, a given quantity of reactants is placed in a container, and by chemical and/or physical means, a change is made to occur. At the end of the process, the container (or adjacent containers to which material may have been transferred) holds the product or products. In a continuous process, reactants are continuously removed from one or more points. A continuous process may or may not be steady-state. A coal-fired power plant, for example, operates continuously. However, because of the wide variation in power demand between peak and slack periods, there is an equally wide variation in the rate at which the coal is fired. For this reason, power plant problems may require the use of average data over long periods of time. However, most industrial operations are assumed to be steady-state and continuous. 

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