9 ug 7. CRE Tutorial Problem. or a recycle reactor the relationship between the volume and other parameters is given by V R in R r or simple kinetics such as first order reaction (under isothermal conditions), for a given volume, PR will have higher conversion and CSTR will have a lower conversion. t low recycle ratio, the recycle reactor will behave like a PR and at high recycle ratio, it will behave like a CSTR. t intermediate recycle ratio, the conversion in a recycle reactor will be between that of a PR and CSTR. However, for a few type of unusual kinetics, (again, under isothermal conditions), for a given volume, a recycle reactor behavior may not go monotonically from a PR behavior to that of a CSTR when recycle ratio is increased. We will illustrate this with an eample. kc Consider a liquid phase reaction B, where the reaction rate is given by r. The reaction is kc - conducted at isothermal conditions and the inlet concentration of is mol/lit. The rate constants are k. s - and k 3 lit mol. The volumetric flow rate is lit s -. We want to convert 95% of. Determine the volume of the reactor, if the reaction is to be conducted in a (i) PR (ii) CSTR (iii) Recycle reactor with recycle ration R= and R=. 9 ug 7. CRE Tutorial Problem. or a recycle reactor the relationship between the volume and other parameters is given by V R in R r or simple kinetics such as first order reaction (under isothermal conditions), for a given volume, PR will have higher conversion and CSTR will have a lower conversion. t low recycle ratio, the recycle reactor will behave like a PR and at high recycle ratio, it will behave like a CSTR. t intermediate recycle ratio, the conversion in a recycle reactor will be between that of a PR and CSTR. However, for a few type of unusual kinetics, (again, under isothermal conditions), for a given volume, a recycle reactor behavior may not go monotonically from a PR behavior to that of a CSTR when recycle ratio is increased. We will illustrate this with an eample. kc Consider a liquid phase reaction B, where the reaction rate is given by r. The reaction is kc - conducted at isothermal conditions and the inlet concentration of is mol/lit. The rate constants are k. s - and k 3 lit mol. The volumetric flow rate is lit s -. We want to convert 95% of. Determine the volume of the reactor, if the reaction is to be conducted in a (i) PR (ii) CSTR (iii) Recycle reactor with recycle ration R= and R=.
Solution: or a PR, V in r Q = lit/s, C-in = mol/lit. -in = mol/s. =. kc kc.. in r kc k C 3 3 in kc 3 Hence 3 r k C. Volume = in 3 r ln 3 5 99.573 85 353.8 =7958 lit or CSTR, -in + V(r)out = -out. C-out = C-in (-)=.5 mol/lit r out kc..5 out kc out 3.5 4.65 4 -out = Q C-out = *.5 =.5 mol/s V = (-in -out)/(-r-out) =45 lit or recycle reactor, we know the result when R = and R = infinity. Let us get the answer when R =. V R in R r When R =, V 3 ln 3 5.475 795.8 ln.475 3.475 5.475 =896 lit.475
When R = V 3 3.633 3 ln 3 5 3 795.8 ln.633 3.633 5.633 =95 lit.633 Thus, for this reaction, when the recycle ratio is changed from to infinity, the volume vs recycle ratio trend is not monotonic, and it goes through a minimum. The plot of vs is given below r 9 8 /(-r ) / (lit-s/mol) 7 6 5 4 3...3.4.5.6.7.8.9 Conversion table of vs r is given here (Note: Values are approimate) X (no units) r (lit-s/mol) 3.9 553.5.399 969.4.5985 453.6.875 97 5 Infinity Thus, the minimum is between.8 and.9.
There is an optimal recycle ratio which will minimize the volume of the reactor. It can be derived analytically. R in V R in G R r r R R To minimize the volume, set dv G ' R dr To differentiate inside integral, note that if, br ( ) f b a G ' R f R, b f R, a R R R a( R) Here, b(r ) =, a constant a( R) = R / (R+), obviously a function of R, and b( R) G R f R, then a( R) f R, R r in Hence, f R r in b, R a and R ( R) f R, a R in and f R, b r R R R r in Hence, for minimization of volume, G (R ) =, we get R in in r r R ( R ) R o R R We can cancel -in, and one (R+), and then re-arranging, we get r R R R r This can be re-written as the integral. R. r R R R r R R. i.e. The, must chosen such that it is the average value of
Of course, if separation of the product stream is easy and economical, then optimal conditions would change. Similarly, if two reactors are acceptable, then a CSTR + PR is better. In our particular eample, if only one reactor is acceptable, then recycle reactor is the best. Now, what is the optimal value of recycle ratio, which will minimize the volume? It depends on the final conversion desired. Eample: If the final conversion is less than.8, then (/-r) vs is a continuously decreasing function, and CSTR is the best choice. If =, we know that a recycle reactor is the best, and we need to find the optimal R. One can obtain the values of volume vs R and if it is done with enough granularity (i.e. small steps of R) we can identify the best R. Once the R is finalized, the volume can be found. Otherwise, the entire function can be written in terms of R, including the integration and optimization can be done. V R in R r R R kcin V R k C in kcin R k C k R in R kc in kc in V R ln kc in k k R R kc in kc in ln kc in k k V R R R k C in kc in ln R R R k R k Differentiate this w.r.t. R and set it to zero, and solve for R. This will be lengthy, but is doable (or use a software like Mathematica to get analytical solution, or an implementation of numerical method such as fminsearch in Matlab). That recycle ratio is the optimal one. or this problem, the function fminsearch, gives R-best =.735. The corresponding volume is,795 lit.