10.37 Chemical and Biological Reaction Engineering, Spring 2007 Prof. K. Dane Wittrup Lecture 10: Non-ideal Reactor Mixing Patterns This lecture covers residence time distribution (RTD), the tanks in series model, and combinations of ideal reactors. Non-Ideal Mixing PFR CSTR Figure 1. Ideal PFR with pulse input. A pulse input will yield an output profile that is a pulse input. Figure 2. Ideal CSTR with pulse input. A pulse input will yield an output profile that is a sharp peak with a tail. Real mixed tank stagnant bypassing mixing recirculation eddies volumes Figure 3. A real mixed tank. In a real mixed tank there are portions that are not well mixed due to stagnant volumes, recirculation eddies, and mixing bypasses. In a real PFR there is back-mixing and axial dispersion. In a packed bed reactor (PBR) channeling can occur. This is where the fluid channels through the solid medium. Residence Time Distribution A useful diagnostic tool is the residence time distribution (RTD). The residence time is how long a particle stays in the reactor once entering. E (t) dt ≡ Probability that a fluid element entering the vessel at t=0 exits between time t and t+dt. Probability density function for exit time, t, as a random variable. Cite as: K. Dane Wittrup, 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].t E t dt Probability that fluid element exits before time t. ∫ ( ) 0 ∞ E t dt Probability of exiting at time later than t. ∫ ( ) t ∞ mean t =∫tE ( ) t dt =τ 0 ∞ =∫ E ( ) dt = 1normalized t 0 ∞ variance =σ 2 = t −τ 2E t dt (measures the broadness of the distribution) ∫( ) ( ) 0 E after t1 before t1 t1 t Figure 4. E(t) versus t. At a given time point, some material has exited and some material will still exit at a later time. Experimental Determination of E(t) Inflow should be something measurable -Absorbance -Fluorescence -pH -salt-conductivity -radioactivity Use one of two types of input concentration curves: Pulse Cin Step Cin tt Figure 5. Two types of input. A pulse input is a spike of infinite height but zero width, ideally. A step input is a constant concentration over a period of time. 10.37 Chemical and Biological Reaction Engineering, Spring 2007 Lecture 10 Prof. K. Dane Wittrup Page 2 of 7 Cite as: K. Dane Wittrup, 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].A pulse input allows for easy interpretation because all materials enter the reactor at once. t Cin input detector t Cin curve Figure 6. Schematic of a residence-time distribution experiment. The input curve enters the reactor; a detector detects concentration changes in the output stream. out E ( ) t = tC (t ) ∫Cout ( ) t dt 0 PFR (Ideal) t Cin τ t Cin t0 Figure 7. Pulse input in ideal PFR. A pulse input in an ideal PFR becomes a pulse output. E (t)=δ (t −τ ) = 0x ≠ 0 xδ( ) == ∞ x = 0 ∞ ∫δ( ) x dx = 1 −∞ ∞ ∫ f ( ) ( x δ x − a) dx = f ( ) a −∞ CSTR (Ideal) Transient material balance: In-Out+Production=Accumulation 10.37 Chemical and Biological Reaction Engineering, Spring 2007 Lecture 10 Prof. K. Dane Wittrup Page 3 of 7 Cite as: K. Dane Wittrup, 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].Since all the material is added at once, In=0. The tracer used is non-reactive. Therefore there is no production. This gives: 0 −ν C + 0 = V dC 0 dt ( ) 0 −t τ V C t = C e , τ= ν0 C t t τ t = ∫( ) τ E ( ) ∞( ) = e − C t dt 0 CSTR Figure 8. Pulse input in an ideal CSTR. In an ideal CSTR, a pulse input leads to a sharp peak with a tail. ∞−t τ mean residence time =∫ te dt =τ τ0 CSTR (non-ideal mixing) Bypassing: Divide input into 2 streams 0 Figure 9. A bypass is modeled by dividing the input stream into two streams, one of which does not enter the reactor. V 0ν Bν SB ν ν 10.37 Chemical and Biological Reaction Engineering, Spring 2007 Lecture 10 Prof. K. Dane Wittrup Page 4 of 7 Cite as: K. Dane Wittrup, 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].Figure 11. Residence-time distribution for dead volumes. When a dead volume is present, a decreased amount of material is observed in the output stream. measureable V=VSD+ VD Vτ= SD SD <τνideal 0 PFR (Non-ideal) Ebypass portion Emixed tt combineEPerfect mixing τ= V ν0 V Bypass τ= νt SB Figure 10. Residence-time distribution determination for a bypass. Dead volumes: Stagnant regions not getting mixed VD VSD Eideal dead volume t present Channeling channeling bed channel PFR-like Figure 12. Channeling. In channeling, the residence-time distribution will show peaks for each channel as well as the one for the main portion of the reactor. 10.37 Chemical and Biological Reaction Engineering, Spring 2007 Lecture 10 Prof. K. Dane Wittrup Page 5 of 7 Cite as: K. Dane Wittrup, 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].Figure 13. A pulse input can become an axially dispersed pulse output in a non-ideal PFR. There are two common models for dispersion in a tubular reactor: -Tanks in a series -Taylor dispersion model (based on the Peclet number) To model the PFR as several tanks in a series, break the reactor volume, V, into n V CSTRs of volume each. n −τ(t τE t ) t= (ei ,n − τi =1) !τ nin 1 2 4 10 PFR t E(t) Figure 14. n tanks in series. The output of tank 1 is the input to tank 2. The output is sampled at tank n for dispersion. n−1 Axial Dispersion 1 2 3 n Figure 15. E(t) plots for 1, 2, 4, and 10 tanks and a PFR. Notice how the E(t) curve approaches the PFR pulse as more tanks are used. The numbers above represent numbers of CSTRs. Without enough CSTRs, the peak is not a good approximation to the narrow peak for a PFR when there is a pulse input. 10.37 Chemical and Biological Reaction Engineering, Spring 2007 Lecture 10