Lecture 5: Continuous Stirred Tank Reactors (CSTRs)

10.37 Chemical and Biological Reaction Engineering, Spring 2007 Prof. William H. Green Lecture 5: Continuous Stirred Tank Reactors (CSTRs) This lecture covers: Reactions in a perfectly stirred tank. Steady State CSTR. Continuous Stirred Tank Reactors (CSTRs) Batch Continuous Plug Flow Reactor CSTR Open system Steady State Continuous Close system Well mixed Transient Open system Steady State Well mixed Figure 2. A batch reactor. Figure 1. A plug flow reactor, and continuous stirred tank reactor. Mole Balance on Component A In-Out+Production=Accumulation FAo − FA + rAV = 0 (steady state) V= FAo − FA −rA FA = FAo − X A FAo In terms of conversion, XA? What volume do you need for a certain amount of conversion? V= FAo X A −rA where rA is evaluated at the reactor concentration. This is the same as the exit concentration because the system is well mixed. For a liquid phase with constant P: FAo = C Ao v0 (v0 =volumetric flow rate) FA = C Av0 V C Ao X A = v0 −rA V τ = ← average time a volume element of fluid stays in the reactor v0 Cite as: William Green, Jr., course materials for 10.37 Chemical and Biological Reaction Engineering, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. τ= C Ao X A −rA Consider: 1st Order Reaction Kinetics − rA = kC A concentration or conversion? Æ convert rate law from CA to XA C A = C Ao (1 − X A ) − rA = kC Ao (1 − X A ) τ= C Ao X A XA ⎪⎫ = ⎬ reactor size in terms of conversion and rate constant kC Ao (1 − X A ) k (1 − X A ) ⎪⎭ Æ rearrange to find how much conversion for a given reactor size XA = τ≡ 1 ≡ k τk 1+τ k average reactor residence time average time until reaction for a given molecule We can now define a “Damköhler number” reaction rate − rAoV , rAo is the reaction rate law at the feed conditions Da = = flow FAo For a liquid at constant pressure with 1st order kinetics: Da = kτ ⇒ XA = Da 1 + Da therefore: As DaÆ, XAÆ1 As Da∞, XAÆ0 (molecule probably leaves before it can react) For a liquid at constant pressure with 2nd order kinetics: − rA = kC A2 = kC Ao 2 (1 − X A ) τ= 2 C Ao X A C Ao X A XA = = 2 2 −rA kC Ao (1 − X A ) kC Ao 2 (1 − X A ) solving for conversion: XA = (1 + 2τ kC Ao ) − 1 + 4τ kC Ao 2τ kC Ao 10.37 Chemical and Biological Reaction Engineering, Spring 2007 Prof. William H. Green Lecture #5 Page 2 of 4 Cite as: William Green, Jr., course materials for 10.37 Chemical and Biological Reaction Engineering, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. Da = kC Ao 2 V = τ kC Ao C Ao vo Thus, conversion can be put in terms of Da. XA = (1 + 2 Da ) − 1 + 4 Da 2 Da How long does it take for a CSTR to reach steady state? In-Out+Production=Accumulation FAo − FA + rAV = dN A dt For a liquid at constant density this is: C Ao − C A + rAτ = τ dC A dt Ænon-dimensionalize C Cˆ A = A C Ao t tˆ = τ τ C Ao dCˆ A C Ao − C Ao Cˆ A − kC Ao Cˆ Aτ = dtˆ τ Da dCˆ A ⎛ P ⎞ ˆ + ⎜ 1 + kτ ⎟ C A = 1 ⎟ dtˆ ⎜⎝ ⎠ dCˆ A + (1 + Da ) Cˆ A = 1 ˆ dt with initial conditions: Cˆ A = 0, tˆ = 0 we have the solution: Cˆ A = ( 1 ˆ 1 − e− (1+ Da )t 1 + Da ) In nondimensional terms, it exponentially approaches a new steady state with a characteristic time τ 1 + Da . 10.37 Chemical and Biological Reaction Engineering, Spring 2007 Prof. William H. Green Lecture #5 Page 3 of 4 Cite as: William Green, Jr., course materials for 10.37 Chemical and Biological Reaction Engineering, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. Cˆ A 1 1 + Da 1 tˆ 1 + Da Figure 3. Approach to steady state in a continuous stirred tank reactor (CSTR). The time at which ½ of the steady state concentration of CA is achieved is the half time: ln(2) τ 1 + Da CSTRs in Series (Liquid and at constant pressure) CA0 Da2 Da1 CA2 Figure 4. Two tanks in series. The output of the first tank is the input of the second tank. 1st order reaction kinetics C A1 = C A0 1 + Da1 For the second reactorÆ iterate C A2 = C A0 (1 + Da1 )(1 + Da2 ) If the CSTRs are identical, C An = C A0 (1+ Da ) n Æ many CSTRs in series looks like a plug flow reactor. 10.37 Chemical and Biological Reaction Engineering, Spring 2007 Prof. William H. Green Lecture #5 Page 4 of 4 Cite as: William Green, Jr., course materials for 10.37 Chemical and Biological Reaction Engineering, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].