2. Chemical Reaction Engineering 1 제 2 장 Conversion and Reactor Sizing 반응공학 1

3. Objectives a.To define conversion (X) and space time (  ) b.To rewrite the mole balances in terms of conversion for a batch reactor, CSTR, PFR, and PBR. c.In expressing -r A as a function of conversion (X), a number of reactors and reaction system can be sized or a conversion be calculated from a given reactor size. - To relate the relative rates of reaction of reactants and products.

4. Definition of Conversion Consider the general equation Choose A as our basis of calculation (The basis of calculation is most always the limiting reactant ) Question - How can we quantify how far a reaction has progressed ? - How many moles of C are formed for every mole A consumed ? The convenient way to answer these question is to define conversion.

5. Design Equations The longer a reactant is in the reactor, the more reactant is converted to product until either equilibrium is reached or the reactant is exhausted. consequently, the conversion X is a function of reaction time The number of moles of A that remain in the reactor after a time t Batch system

6. The number of moles of A in the reactor after a conversion X The mole balance on species A for a batch system In term of conversion by differentiating equation The design equation for a batch reactor in differential form is The differential form for a batch reactor

7. For constant-volume batch reactor For a variable volume batch reactor When the volume is varied by For the most common batch reactors where volume is not predetermined, the time necessary to achieve a conversion X is The integral form for a batch reactor

8. If F A0 is the molar flow rate of species A fed to a system at steady state, the molar rate at which species A is reacting within the entire system will be F A0 X. The molar flow rate Rearranging gives Flow systems

9. The entering molar flow rate, F A0 (mol/s) For liquid systems : C A0 is commonly given in term of molarity For gas systems : C A0 can be calculated from the entering T and P using the ideal gas law or some other gas law For an ideal gas (see Appendix B) : C A0 : the entering concentration v 0 : the entering volumetric flow rate

10. -The design equation for a CSTR -conversion of flow system -Combining (2-12) with (2-11) CSTR or Back-mixing Reactor (2-11) (2-12) Equation to determine the CSTR volume necessary to achieve a specified conversion X. Since the exit composition from the reactor is identical to the composition inside the reactor, the rate of reaction is evaluated at the exit condition. F A0 FAFA (2-13) design equation for a CSTR X -rA-rA 1 Area

11. -General mole balance equation -conversion of flow system -The differential form of the design equation -Volume to achieve a specified conversion X Tubular Flow Reactor (PFR) F A0 FAFA (2-14) (2-15) (2-16) X -rA-rA 1 Area

12. -General mole balance equation -conversion of flow system -The differential form of the design equation Packed-Bed Reactor (PBR) F A0 FAFA (2-17) (2-18) -The catalyst weight W to achieve a specified conversion X

13. Design equation for a batch reactor Summary of Design Equation F A0 FAFA Design equation for a CSTR F A0 FAFA Design equation for a PFR F A0 FAFA Design equation for a PBR

14. Summary of Design Equation 반응시간은 - N A0 에 비례 - X 에 비례 - 반응속도 (r A ) 에 반비례 - 반응기 부피에 반비례 F A0 FAFA FAFA FAFA 반응기 부피 ( 촉매의 무게 ) 는 - F A0 에 비례 - X 에 비례 - 반응속도 (r A ) 에 반비례

15. Applications of the design equation for continuous-flow reactor For a first-order reaction : The rate of disappear of A, -r A, is almost always a function of the concentrations of the various species present. When a single reaction is occurring, each of the concentrations can be expressed as a function of the conversion x; consequently, -r A, can be expressed as a function of X. F A0 FAFA V = F A0 kC A 0  0 X dX 1-X = F A0 kC A 0 ln (1-X) -

16. Consider the isothermal gas-phase isomerization A B How to use the raw data of chemical reaction rate? The laboratory measurements give the chemical reaction rate as a function of conversion. (at T=500K, 830kPa(=8.2atm), Reactant=Pure A) X -r A (mol/m 3 -sec) 1/-r A (m 3 -sec/mol) F Ao /-r A (m 3 ) 0 0.1 0.2 0.4 0.6 0.7 0.8 0.45 0.37 0.30 0.195 0.113 0.079 0.050 2.22 2.70 3.33 5.13 8.85 12.7 20.0 0.89 1.08 1.33 2.05 3.54 5.06 8.0

17. Levenspiel Plot Greatest rate Small rate 

18. -For irreversible reactions, the maximum value of X is that for complete conversion, i.e. X=1.0. -For reversible reactions, the maximum value of X is the equilibrium conversion, i.e. X=Xe. How to use the raw data of chemical reaction rate?

19. For vs. X, the volume of a CSTR and the volume of a PFR can be represented as the shaded areas in the Levenspiel plots. Given –r A as a function of conversion. Constructing a Levenspiel plot. Here we plot either or as a function of X. Reactor Size

20. Example 2-2. Sizing a CSTR The reaction, A  B, is carried out in a CSTR. Molar flow rate of A is 0.4 mol/sec. (1) Using data in the previous Table, calculate the reactor volume necessary to achieve 80% conversion in a CSTR (2) Shade a area in Figure 2-2 that would give the CSTR volume necessary to achieve 80% conversion (1) rArA 1 ( ) X=0.8 = 20 m 3 -sec/mol V=(0.4 mol/sec)(0.8)(20 m 3 -sec/mol) =6.4 m 3 =6400 liter (2)

21. Example 2-3. Sizing a PFR The reaction, A  B, is carried out in a PFR. Molar flow rate of A is 0.4 mol/sec. (1) Using data in the previous Table, calculate the reactor volume necessary to achieve 80% conversion in a PFR (2) Shade a area in Figure 2-2 that would give the PFR volume necessary to achieve 80% conversion (3) Make qualitative sketches of conversion (X) and rate of reaction (-r A ) with respect to reactor volume (1) V =  0 0.8 dX -r A =  0 0.8 dX -r A F A0 By applying Appendix A-23 (Five Point Quadrature Formula): X=0.8/4=0.2 3 0.2 ( ) V=[0.89+4(1.33)+2(2.05)+4(3.54)+8] =2.165m 3 =2165 liter

22. Example 2-3. Sizing a PFR (b)

23. Example 2-3. Sizing a PFR (c) By applying Simpson’s rule in Appendix A.4, we can calculate V for X=0.2, 0.4, 0.6, 0.8 (See the text, page 52). The results are as follows. X -r A (mol/m 3 -sec) V (dm3) 0 0.45 0 0.2 0.30 218 0.4 0.195 551 0.6 0.113 1093 0.8 0.05 2165 전환율을 조금 더 높이기 위해 서는 반응기 부 피가 많이 늘어 나야 한다.

24. Example 2-4. Comparing CSTR and PFR Sizes For isothermal reaction of greater than zero order, the PFR will always require a smaller volume than the CSTR to achieve. What if zero order reaction?

25. Reactors in series Define conversion The conversion X defined as the “total number of moles” of A that have reacted up to that point per mole of A fed to the “first” reactor. (assumption : no side stream withdrawn and the feed stream enters only the first reactor in the series)

26. PFR-CSTR-PFR in series The relationships between conversion and molar flow rate V1V1 X=0 F A0 X 1 F A1 V3V3 X 2 F A2 V 2 X 3 F A3 F A1 = F A0 - F A0 X 1 F A2 = F A0 - F A0 X 2 F A3 = F A0 - F A0 X 3 where similar definitions exist for X 1 and X 3

27. V1V1 X=0 F A0 X 1 F A1 V3V3 X 2 F A2 V 2 X 3 F A3 Reactor 1: Reactor 2 : Reactor 3 : F A1 = F A0 - F A0 X 1 F A2 = F A0 - F A0 X 2 F A3 = F A0 - F A0 X 3 -r A2 is evaluated at X 2 for the CSTR In this series arrangement

28. F Ae X 2 =0.8 Different Schemes of Reactors in Series Two CSTRs in series Two PFRs in series a PFR and a CSTR in series F A0 X 1 =0.4 F Ae X 2 =0.8 F Ae X 2 =0.8 F A0 X 1 =0.4 F A0 F Ae X 2 =0.8 X 1 =0.5 F A0 a CSTR and a PFR in series

29. Two CSTRs in Series F A0 X 1 =0.4 F Ae X 2 =0.8 Reactor 1 Reactor 2 = F Ao (X 1 -X o ) -r A1 0

30. Example 2-5: Two CSTRs in Series F A0 X 1 =0.4 F Ae X 2 =0.8 What is the volume of each of Two reactors? X A [F Ao /-r A ] (m 3 ) 0.0 0.1 0.2 0.4 0.6 0.7 0.8 0.89 1.09 1.33 2.05 3.54 5.06 8.0 Reactor 1 [F Ao /-r A ] x=0.4 =2.05 m 3 V 1 =([F Ao /-r A ] x=0.4 )(X 1 -X 0 )=(2.05)(0.4-0)=0.82 m 3 Reactor 2 [F Ao /-r A ] x=0.8 =8.0 m 3 V 1 =([F Ao /-r A ] x=0.8 )(X 2 -X 1 )=(8.0)(0.8-0.4)=3.2 m 3

31. Example 2-5: Two CSTRs in Series Therefore, V 1 +V 2 =0.82+3.2=4.02 m 3 What is the reactor volume to achieve 80% Conversion in a single CSTR? [F Ao /-r A ] x=0.8 =8.0 m 3 V 1 =([F Ao /-r A ] x=0.8 )(X 1 -X 0 ) =(8.0)(0.8-0)=6.4 m 3 The sum of the two CSTR reactor volumes (4.02 m 3 ) in series is less than the volume of one CSTR (6.4 m 3 ) to achieve the same conversion (X=0.8)

32. [F Ao /-r A ] (m 3 ) F A0 X 1 =0.4 F Ae X 2 =0.8 F A0 FAFA X=0.8 V total = 4.02 m 3 V total = 6.4 m 3 Example 2-5: Two CSTRs in Series

33. A PFR by a Large Number of CSTRs in Series Approximating a PFR with a number of small, equal-volume CSTRs of V i in series 54321 12345

34. A PFR by a Large Number of CSTRs in Series 80 60 40 20.35.53.65.74.8 X 12345 54 321 As we make the volume of each CSTR smaller and increase the number of CSTRs, the total volume of the CSTRs and the PFR will become identical. The performance of a PFR is equal to that of a number of (N  ) CSTRs in Series. Can you verify this mathematically?

35. Two PFRs in Series Reactor 1 Reactor 2 F Ae X 2 =0.8 F A0 X 1 =0.4

36. Two PFRs in Series  0 X1X1 dX + -r A F A0 V Total = V 1 + V 2 =  X2X2 -r A F A0 dX = X1X1  X2X2 -r A F A0 0

37. Sizing PFR in Series F Ae X 2 =0.8 F A0 X 1 =0.4 What is the volume of each of Two reactors? Molar flow rate of A is 0.4 mol/sec. X A [F Ao /-r A ] (m 3 ) 0.0 0.1 0.2 0.4 0.6 0.7 0.8 0.89 1.09 1.33 2.05 3.54 5.06 8.0 Reactor 1 By applying Simpson’s rule in Appendix A.4 (Text page 60), 3 0.2 ( ) V1=V1=[0.89+4(1.33)+2.05] =0.551 m 3 =551 liter Reactor 2 By applying Simpson’s rule in Appendix A.4 (Text page 60), 0.2 ( ) V2=V2=[2.05+4(3.54)+8.0] =1.614 m 3 =1614 liter 3 Therefore, V 1 + V 2 =0.551 m 3 + 1.614 m 3 =2.165 m 3 < 4.02 m 3 (Two CSTR in Series)

38. Combination of CSTR and PFR in Series V 3 =  X2X2 dX -r A F A0 = F Ao (X 1 -X o ) -r A1 0 Reactor 1 V1V1 = F Ao (X 2 -X 1 ) -r A2 Reactor 2 V2V2 Reactor 3 X3X3

39. Example 2-7: Liquid-Phase Isomerization n-C 4 H 10 i-C 4 H 10 X 0.0 0.2 0.4 0.6 0.65 -r A (kmol/m 3 -h) 39 53 59 38 25 Calculate the volume of each of the three reactors for an entering molar flow rate n-butene of 50 kmol/h.

40. Example 2-7: Liquid-Phase Isomerization F Ao = 50 kmol/h X 0.0 0.2 0.4 0.6 0.65 -r A (kmol/m 3 -h) 39 53 59 38 25 [F Ao /-r A ](m 3 ) 1.28 0.94 0.85 1.32 2.0 (a) Reactor 1 (X 1 =0.2) = F Ao (X 1 -X o ) -r A1 0 V1V1 = (0.94)(0.2)=0.188 m 3 (b) Reactor 2 (X 2 =0.6) V 2 =  0.2 dX -r A F A0 0.6

41. Example 2-7: Liquid-Phase Isomerization By applying Simpson’s three point formula in Appendix A.4 (Text page 64), 0.2 ( ) V2=V2=[0.94+4(0.85)+1.32] =0.38 m 3 3 (c) Reactor 3 (X 3 =0.65) = F Ao (X 3 -X 2 ) -r A3 V3V3 = (2.0)(0.65-0.6)=0.1 m 3

42. Reactor Sequence F A0 F Ae X 2 =0.8 X 1 =0.5 Reactor 1 Reactor 2 1 0

43. PFR CSTR F A0 F Ae X 2 =0.8 X 1 =0.5 Total volume= V total =V 1 +V 2 = 305 dm 3 Scheme A Reactor Sequence

44. CSTR PFR Total volume= V total =V 1 +V 2 = 262.3 dm 3 F Ae X 2 =0.8 X 1 =0.5 F A0 Reactor Sequence Scheme B

45. F Ae X 2 =0.8 X 1 =0.5 F A0 F Ae X 2 =0.8 X 1 =0.5 Scheme A Scheme B V total =V 1 +V 2 = 262.3 dm 3 V total =V 1 +V 2 = 305 dm 3 Scheme B will give the smaller total volume for an intermediate conversion of 50%. However, the relative sizes of the reactors depend on the intermediate conversion. Reactor Sequence What if zero order reaction?

46. Choosing Reactor Sequence X [F Ao /-r A ] 0 MFR PFR

47. Space Time Space-time : The time necessary to process one reactor volume of fluid based on entrance conditions. Also called the holding time or mean residence time. A space-time of 2 min means that every 2 min one reactor volume of feed at specified condition is being treated by the reactor.

48. Space Time Space-time : The time necessary to process one reactor volume of fluid based on entrance conditions. Also called the holding time or mean residence time. Consider the tubular reactor, which is 20m long and 0.2 m 3 in volume. The dashed line represents 0.2 m 3 of fluid directly upstream of the reactor. The time it takes for this fluid to enter the reactor completely is the space time. 20m

49. A space-velocity of 5 hr -1 means that five reactor volumes of feed at specified condition are being fed into the reactor per hour. Difference in the definitions of SV and - space time : the entering volumetric flow rate is measured at the entrance condition - space velocity : other conditions are often used Space Velocity Definition of Space-velocity

50. LHSV ( liquid hour space velocity) - v 0 is frequently measured as that of a liquid at 60 or 75 0 F, even though the feed to the reactor may be a vapor at some higher temperature. GHSV ( gas hour space velocity) - v 0 is normally measured at standard temperature and pressure LHSV and GHSV

51. It is usually convenient to report –r A as a function of concentration rather than conversion. - PFR design equation : molar flow rate : flow system conversion : For the special case when v = v 0 For reaction rate depending only on the concentration

52. Differentiating yields A typical curve for determining the space time,  For reaction rate depending only on the concentration

53. Homework P2-7 B P2-8 B P2-9 B Due Date: Next Week