CSTR graphical solution

ILLUSTRATION 8.7 DETERMINATION OF CSTR SIZE REQUIREMENTS FOR CASCADES OF VARIOUS SIZES— GRAPHICAL SOLUTION... 

Figure 14.11 Basis for graphical solution for multistage CSTR (for A +. .. - products)...

The following example illustrates both an analytical and a graphical solution to determine the outlet conversion from a three-stage CSTR. 

Figure 4.10.20a shows the graphical solution of both equations for the example of a conversion of 95%. The area under the function of 1/(1 — Xa) versus Xa represents Da. For the given example, a value of 19 is needed in a CSTR (rectangle with area 20 x 0.95), whereas for the PFR the respective (dashed) area is only 3. For a cascade consisting of four CSTRs of equal size we get a Da value in between these two extremes of 4.4 (for Xa = 95%). 

The reader can also return to Illustrative Example 9.17 for the three CSTRs problem. Figure 11.8 provides the graphical solution for the case of equal volumes (a) and the case where the total volume of three reactors has been minimized (b). 

Figure 4-3 Graphical solution of the design equation for an ideal CSTR for the Monod rate equation and the parameter values given in Example 4-8, Part A. Two different steady-state outlet concentrations are possible, Ca = 0.20 and 10 g/1.

Why didn t we see this bizarre behavior, e.g., washout and multiple steady states, previously The reason is that our previous examples involved normal kinetics. In all of our past work, the rate increased monotonically as the reactant concentration increased. In such a case, the graphical solution to the CSTR design equation looks as shown in Figure 4-5. 

Figure 4-5 Illustration of the graphical solution of the design equation for an ideal CSTR for a second-order rate equation. There is only one possible intersection between the —r curve and the...

Figure 3.11 makes it obvious that the level-set method for equation (3.6) gives much more meaningful numerical results and clearer graphical representations of the multiple steady state solutions of the CSTR problem (3.3). 

To solve equation (4.86) for y and K > 0 we use the graphical level set method that we have introduced in Chapter 3 for the adiabatic and nonadiabatic CSTRs and draw the surface z = F(y, K), as well as the y versus K curve of solutions to equation (4.86) in order to exhibit and study the bifurcation behavior of the underlying system. 

C, is the solution of the system of Equations (65) and (66). Graphically this means that C, is the intercept of the kinetics curve with the straight line of gradient (F/F,) and the A -intercept is the entrance concentration of the / -th CSTR, i.e. the exit concentration of the (i — l)th CSTR. The estimation of the concentrations in the CSTRs one after the other is shown in Figure 5. 

One should compare the analytical solution given by equation (2-15) with the Laplace transform and matrix results for startup behavior of a series of n CSTRs with first-order irreversible chemical reaction. The three solutions are equivalent. An alternative proof of the analytical solution that does not require mathematical rigor is based on graphical comparison of the numerical results in Figure 2-3 with the solution given by equation (2-15). The numerical and analytical solutions are indistinguishable. 

The function f(C) describes a surface in R" which the locus of CSTR concentrations must intersect. From this, solving for CSTR solutions involves searching the n-dimensional space of concentrations that satisfy f(C) = 0. This may be automated by computer. For plots in this procedure may be demonstrated graphically. In Figure 4.19(a), we show the rate field and feed point for the autocatalytic system in Ca-Cb space. 

The reader may now return to the CSTR illustrative examples in Chapter 9 dealing with CSTRs in series and attempting to minimize the volume requirements for these cases. Figure 11.7 provides a graphical representation of the solutions to Illustrative Examples 9.14, 9.15, and 9.16. Figure 11.7(a) provides the solution for the volume of one reactor and for two reactors of equal volumes. 

Solution. From a graphical perspective, the CSTR volume and the TF volume are equal. The details of the calculations follow ... 

A further specific aspect of the reactor heat balance is the multiplicity of solutions to the system of equations. This situation may arise with a CSTR in which an exothermal reaction is performed. The mass balance [Eq. (12)] is coupled with the heat balance [Eq. (13)], which gives a system of equations [Eq. (14)] that is represented graphically in Figure 11.4. 

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