CSTR volume element

In the ideal CSTR, the fluid concentration is uniform and the fluid flows in and out of the reactor. Under the steady state condition, the accumulation term in the general material balance, eq. (3.70), is zero. Furthermore, the exit concentration is equal to the concentration in the reactor. For a volume element of fluid (F,), the mass balance for the limiting reactant becomes (Levenspiel, 1972)... 

It can be readily discerned that the reactor equation for the batch reactor (5.12) and the plug-flow reactor (5.13) are identical. In the former, the concentration changes with time, in the latter, with location. In contrast to the situation in the other two ideal reactors, the residence time T in a CSTR is only an average, as every volume element has a different residence time throughout the reactor. 

Ideal continuously stirred reactor (CSTR) for which a volume element entering the CSTR will become uniform dispersed with all the other volume elements in the reactor. The initial outlet will be equal to the ratio of the tracer volume divided by the reactor volume times the initial tracer concentration, and would then exponential decay in time. 

The reactor models considering complete mixing may be subdivided into batch and continuous types. In the continuous stirred tank reactor (CSTR) models, an entering fluid is assumed to be instantaneously mixed with the existing contents of the reactor so that it loses its identity. This type of reactor operates at uniform concentration and temperature levels. For this reason the species mass balances and the temperature equation may be written for the entire reactor volume, not only over a differential volume element. Under steady-state conditions, the species mass and heat balances reduce to algebraic equations. 

The continuous-stirred-tank reactor (CSTR) is also a well-stirred reactor so there are no concentration gradients anywhere in the reactor volume. We again consider the entire reactor contents to be the reactor volume element as in Figure 4.13, and V = Vr, Since the reactor is well stirred, the relations in Equations 4.3 and 4.4 apply to the CSTR also. The difference between the CSTR and batch reactor is the flow streams shown in figure 4,13. We denote the feed... 

In Figures 3.4 and 3.5, the RTDs of ideal reactors are presented together with the RTD of a real reactor. The ideal, continuously operated stirred tank reactor (CSTR) has the broadest RTD between all reactor types. The most probable residence time for an entering volume element is t = 0. After a mean residence time t = t), 37% of the tracer injected at time t = 0 is still present in the reactor. After five mean residence times, a residue of about 1% still remains in the reactor. This means that at least five mean residence times must pass after a change in the inlet conditions before the CSTR effectively reaches its new stationary state. 

In reactors with some degree of backmixing, however, this cannot be achieved, and there is always a distribution of the residence times of the volume elements. The largest possible distribution of residence times is found in CSTRs, in which the residence times of individual volume elements are spread throughout the time frame, from zero to infinity. 

In the second case, let us assume that each individual CSTR is segregated. As a result, the following derivation is valid, provided that the volume elements are broken up and mixed with each other between each step in the series ... 

The heat balance can be written either globally over the whole reactor volume or locally for a differential element dV. The global heat balance is similar to the CSTR ... 

The Fractional Tubularity Model. Piston flow has a2 = 0. A CSTR has a2 = 1. Real reactors can have 0 < a2 < 1, and a model that reflects this possibility consists of a stirred tank in series with a piston flow reactor as indicated in Figure 15.1(a). Other than the mean residence time itself, the model contains only one adjustable parameter. This parameter is called the fractional tubularity, tp, and is the fraction of the system volume that is occupied by the piston flow element. Figure 15.1(b) shows the washout function for the fractional tubularity model. Its equation is... 

The age of a fluid element is defined as the time it has resided within the reactor. The concept of a fluid element being a small volume relative to the size of the reactor yet sufficiently large to exhibit continuous properties such as density and concentration was first put forth by Danckwerts in 1953. Consider the following experiment a tracer (could be a particular chemical or radioactive species) is injected into a reactor, and the outlet stream is monitored as a function of time. The results of these experiments for an ideal PFR and CSTR are illustrated in Figure 8.2.1. If an impulse is injected into a PFR, an impulse will appear in the outlet because there is no fluid mixing. The pulse will appear at a time ti = to + t, where t is the space time (r = V/v). However, with the CSTR, the pulse emerges as an exponential decay in tracer concentration, since there is an exponential distribution in residence times [see Equation (3.3.11)]. For all nonideal reactors, the results must lie between these two limiting cases. 

For non-isothermal or non-linear chemical reactions, the RTD no longer suffices to predict the reactor outlet concentrations. From a Lagrangian perspective, local interactions between fluid elements become important, and thus fluid elements cannot be treated as individual batch reactors. However, an accurate description of fluid-element interactions is strongly dependent on the underlying fluid flow field. For certain types of reactors, one approach for overcoming the lack of a detailed model for the flow field is to input empirical flow correlations into so-called zone models. In these models, the reactor volume is decomposed into a finite collection of well mixed (i.e., CSTR) zones connected at their boundaries by molar fluxes.4 (An example of a zone model for a stirred-tank reactor is shown in Fig. 1.5.) Within each zone, all fluid elements are assumed to be identical (i.e., have the same species concentrations). Physically, this assumption corresponds to assuming that the chemical reactions are slower than the local micromixing time.5... 

The discussion in Section 8.3.2 indicated that dramatic reductions in reactor volume requirements can be obtained by using a number of individual CSTRs connected in a series flow configuration rather than an individual CSTR. The effluent from one reactor in the cascade serves as the feed stream for the next reactor. Although the composition and temperature are uniform throughout any individual reactor, there can be marked changes in temperature, pH, and composition as one follows an element of fluid from the first tank in a cascade to the last. Similar situations arise when chemostats are connected in series. 

The simplest kinetic reactor model is the CSTR (continuous-stirred-tank reactor), in which the contents are assumed to be perfectly mixed. Thus, the composition and the temperature are assumed to be uniform throughout the reactor volume and equal to the composition and temperature of the reactor effluent However, the fluid elements do not all have the same residence time in the reactor. Rather, there is a residence-time distribution. It is not difficult to provide perfect mixing of the fluid contents of a vessel to approximate a CSTR model in a commercial reactor. A perfectly mixed reactor is used often for homogeneous liquid-phase reactions. The CSTR model is adequate for this case, provided that the reaction takes place under adiabatic or isothermal conditions. Although calculations only involve algebraic equations, they may be nonlinear. Accordingly, a possible complication that must be considered is the existence of multiple solutions, two or more of which may be stable, as shown in the next example. 

Establish a control volume for a differential or finite element (e.g., CSTR) of the system to be modeled sketch the element and indicate all inflow-outflow paths. 

For every element of feed that enters a CSTR, the perfect mixing assumption implies that the entering element is instantaneously converted to the concentration of the bulk volume. Conversion of reactants might then be viewed to occur as a result of mixing and dilution rather than from reaction alone. 

In contrast to the ideal CSTR, backmbdng is excluded in an ideal tubular reactor, characterized by a plug flow pattern of the fluid, with uniform radial composition and temperature. The material balance for a small volume system element (AV) shown in Figure 2.9 at the reactor steady state is written as... 

The arrangement that is chosen is largely based on intuition. The parameters in the model are the relative volumes of the CSTR s and PFR s that ms e up the model (minus one), and the circulation flow rates, divided by the reactor feed. The parameters are estimated by fitting of RTD-measurements. Then the conversion of a given chemical reaction in the simulated reactor is calculated. Since all elements are ideal reactors, the calculation methods of Chapter 3 may be applied, and so the entire reactor system can be described by relatively simple calculations. The calculated conversion is compared to the measured conversion, and when there are deviations, the model is adjusted. By trial and error one may arrive at a model that describes the real reactor satisfactorily. 

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