CSTR equation

Solution The limits you can calculate under part (a) correspond to the three apexes in Figure 15.14. The limits are 0.167 for a PFR (Equation (1.47)), 0.358 for a CSTR (Equation (1.52)), and 0.299 for a completely segregated stirred tank. The last limit was obtained by integrating Equation (15.48) in the form... 

It is readily apparent that equation 8.3.21 reduces to the basic design equation (equation 8.3.7) when steady-state conditions prevail. Under the presumptions that CA in undergoes a step change at time zero and that the system is isothermal, equation 8.3.21 has been solved for various reaction rate expressions. In the case of first-order reactions, solutions are available for both multiple identical CSTR s in series and individual CSTR s (12). In the case of a first-order irreversible reaction in a single CSTR, equation 8.3.21 becomes... 

A CSTR is operated in series with a PFR, with a fraction ce of the flow in bypass around the CSTR. Equations are to be found for the Gain and phase angle of a frequency response. 

Three different arrangements of a CSTR and a PFR with bypass are shown on the sketch, t = fraction of total volume occupied by the CSTR battery. /3 = fraction of the total flow that goes directly to the CSTR. Equations for (tr) are to be developed for the three cases. 

We took the 4- sign on the square root term for second-order kinetics because the other root would give a negative concentration, which is physically unreasonable. This is true for any reaction with nth-order kinetics in an isothermal reactor There is only one real root of the isothermal CSTR mass-balance polynomial in the physically reasonable range of compositions. We will later find solutions of similar equations where multiple roots are found in physically possible compositions. These are true multiple steady states that have important consequences, especially for stirred reactors. However, for the nth-order reaction in an isothermal CSTR there is only one physically significant root (0 < Ca < Cao) to the CSTR equation for a given T. ... 

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. 

These are two coupled algebraic equations, which must be solved simultaneously to determine the solutions Cj(x) and T(t). For multiple reactions the + 1 equations are easily written down, as are the differential equations for the transient situation. However, for these situations the solutions are considerably more difficult to find We will in fact consider theaolutions of the transient CSTR equations in Chapter 6 to describe phase-plane trajectories and the stability of solutions in the nonisothermal CSTR. 

The jacketed reactor may be assumed to have mixed flow (either by stirring the jacket or by mixing through natural convection) so the jacket also obeys the CSTR equation. Therefore, for the jacketed reactor with a single reaction we write the three equations... 

For a CSTR we solve the transient CSTR mass-balance equation because we want the time dependence of a pulse injection without reactiou The transient CSTR equation on a species of concentration C is... 

Note that this is simply the transient CSTR equation with the reaction term omitted, which implies either no reaction (r = 0) or that the species in question is an inert in a reacting system so that its stoichiometric coefficient is zero. 

This situation describes an emulsion reactor in which reacting drops (such as oil drops in water or water drops in oil) flow through the CSTR with stirring to make the residence time of each drop obey the CSTR equation. A spray tower (liquid drops in vapor) or bubble column or sparger (vapor bubbles in a continuous liquid phase) are also segregated-flow situations, but these are not always mixed. We wiU consider these and other multiphase reactors in Chapter 12. 

Solving the CSTR equation, we obtain a residence time of... 

Before we proceed, note that these equations look identical in form to the adiabatic CSTR equations of Chapter 6,... 

If the fluid is mixed, the transient CSTR equation becomes dC ... 

Since the O2 concentration remains constant, we do not need to solve the maSS-balance equation in the gas phase. Both phases are mixed, so we must solve the CSTR equation for in the liquid phase,... 

By placing the impeller within a draft tube within the reactor, the fluids are forced to pass through the impeller, where the bubbles are redispersed by impacting on the impeller surfaces. The draft tube is placed in the center of the reactor so the fluids recirculate repeatedly (a recycle reactor) to allow bubbles to be repeatedly redispersed in the draft tube. The overall reactor becomes well mixed and is therefore described by the CSTR equations. The rapid flow of this reactor enhances the mass transfer rate and thus increases the overall reaction rate if it is limited by mass transfer of a reactant from the liquid phase into the bubbles. 

Note that these are the conventional CSTR equations in a and p phases with the mass transfer reactions between phases added. 

Substitution of Equation (11.19) in the general equation for the CSTR [Equation (11.21)] yields ... 

Auxiliary function for adiabatic Non-isothermal CSTR equation. 

Plugging this expression for y into (3.8) gives us the following equivalent nonadiabatic nonisothermal CSTR equation, which depends solely on xa. ... 

The great difference with the CSTR (Equation 8.2) is that here the reaction rate varies within the reactor volume instead of being constant. Hence, the reaction rate is in the integral term. If the initial conversion is not zero, the equation becomes... 

Solution This solution illustrates a possible definition of the delta function as the limit of an ordinary function. Disturb the reactor with a rectangular tracer pulse of duration At and height A/t so that A units of tracer are injected. The input signal is Cm =0,t < 0 Ct = A/At, 0 < t < At Cin = 0, and t > At. The outlet response is found from the dynamic model of a CSTR, Equation (14.2). The result is... 

---