Chapter 2: Conversion and Reactor Sizing

Reactors in Series: CSTR-PFR-CSTR

Using either the data in Table E2-W.1, calculate the reactor volumes V1, V2, and V3 for the CSTR/PFR/CSTR reactors in series sequence shown in Figure E2-W.1 along with the corresponding conversion.

                 Diagram showing a sequence of CSTR, PFR, and CSTR reactors in series for calculating reactor volumes V1, V2, and V3 along with corresponding conversions. Component A enters the first CSTR, achieving conversion X1 = 0.4 with flow FA1, then flows into the PFR reaching conversion X2 = 0.7 with flow FA2, and finally enters the second CSTR, achieving a final conversion of X3 = 0.8 with flow FA3.

                                     Figure E2-W.1 CSTR/PFR reactor in series.

 

Table E2-W.1 Processed Data

X

0

0.2

0.4

0.6

0.8

\( -r_{A} \left( \frac{\text{mol}}{\text{dm}^{3} \cdot \text{s}} \right) \)

0.010

0.0091

0.008

0.005

0.002

\( \left( \frac{1}{-r_{A}} \right) \left( \frac{\text{dm}^{3} \cdot \text{s}}{\text{mol}} \right) \)

100

110

125

200

500

\( \frac{F_{A0}}{-r_{A}} \left( \text{dm}^{3} \right) \)

200

220

250

400

1000

 

Solution

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Solution

We again use the plot of \( \left( \frac{F_{A0}}{-r_{A}} \right) \text{ vs. } X \)

                Levenspiel plot for a CSTR/PFR/CSTR sequence, showing the relationship between conversion (X) and reciprocal rate F_A0 over -r_A. The y-axis ranges from 0 to 1000, and the x-axis ranges from 0 to X_3, with marked points at X_1 (0.4), X_2 (0.7), and X_3 (0.8). Shaded regions under the curve represent the reactor volumes for each segment in the sequence.

                   Figure E2-W.2 Levenspiel plot for CSTR/PFR/CSTR sequence.


(a)  The CSTR design equation for Reactor 1 is

        \( V_{1} = \left( \frac{F_{A0} X}{-r_{A1}} \right) \)                                                  (E2-W.1)

        at X = X1 = 0.4 the (FA0/-rA1) = 300 dm3

        V1 = (300 dm3) (0.4) = 120 dm3

       The volume of the first CSTR is 120 dm3.



(b)Reactor 2: PFR The differential form of the PFR design is

         \( \frac{dX}{dV} = \frac{-r_{A}}{F_{A0}} \)             (E2-W.2)

      Rearranging and integrating with limits

            when V = 0       X = X1 = 0.4

            when V = V2     X = X2 = 0.7


         \( V = \int_{X_{1}}^{X_{2}} \left( \frac{F_{A0}}{-r_{A}} \right) dX = \int_{0.4}^{0.7} \left( \frac{F_{A0}}{-r_{A}} \right) dX \)                              (E2-W.3)


      Choose three point quadrature formula with \( \Delta X = \frac{X_{2} - X_{1}}{2} = \frac{0.7 - 0.4}{2} = 0.15 \)


          \( V_{2} = \frac{\Delta X}{3} \left[ \frac{F_{A0}}{-r_{A}(0.4)} + \frac{4 F_{A0}}{-r_{A}(0.55)} + \frac{F_{A0}}{-r_{A}(0.7)} \right] \)                         (E2-W.4)


      Interpreting for (FA0/-rA) at X = 0.55 we obtain

                         \( \left( \frac{F_{A0}}{-r_{A}} \right)_{X=0.55} = 370 \, \text{dm}^{3} \)

      The volume of the PFR is          

       

\( V_{2} = 119 \, \text{dm}^{3} \)


(c)  Reactor 3: CSTR

            Balance


          \( \text{in} - \text{out} + \text{generation} \)

          \( F_{A2} - F_{A3} + r_{A3} V_{3} = 0 \)

          (E2-W.5)


            Rearranging

                                                             

\( V_{3} = \frac{F_{A2} - F_{A3}}{-r_{A3}} \)

                                               (E2-W.6)

                                                           

\( F_{A2} = F_{A0} (1 - X_{2}) \)                                              (E2-W.7)

                                                           

\( F_{A3} = F_{A0} (1 - X_{3}) \)                                              (E2-W.8)

                                                         

\( V_{3} = \frac{F_{A0}}{-r_{A3}} (X_{3} - X_{2}) \)                                            (E2-W.9)

                                           

\( V_{3} = 600 \, \text{dm}^{3} \, (0.7 - 0.4) = 180 \, \text{dm}^{3} \)


            The volume of the last CSTR is 180 dm3.


Summary

CSTR

\( X_{1} = 0.4 \)

\( V_{1} = 120 \, \text{dm}^{3} \)

PFR

\( X_{2} = 0.7 \)

\( V_{2} = 119 \, \text{dm}^{3} \)

CSTR

\( X_{3} = 0.8 \)

\( V_{3} = 180 \, \text{dm}^{3} \)

 

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