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- 1. Ch E 441 - Chemical Kinetics and Reaction Engineering<br />Residence Time Distributionsin Chemical Reactors<br />
- 2. Residence Time Distributions<br />The assumption of a “perfectly mixed” reactor often falls short of reality.<br />Residence time distributions are used to model the imperfect mixing behavior of real reactors.<br />Cumulative age, F(t)<br />External age, E(t)<br />Internal age, I(t)<br />
- 3. Residence Time Distributions<br />Gas-liquid CSTR (A(g) + B(l) C(l))<br />Reaction occurs at gas-liquid interface<br />Liquid phase is perfectly mixed<br />Rate is proportional to bubble surface area<br />Residence time of gas bubble in reactor is proportional to bubble volume<br />Larger bubble escape rapidly<br />Smaller bubbles may remain in reactor until consumed<br />Understanding of RTDs is necessary for analysis<br />
- 4. Residence Time Distributions<br />PBR<br />Sections of the catalyst bed may offer less resistance to flow, resulting in a preferred pathway through the bed.<br />Molecules flowing through the “channel” do not spend as much time in the PBR as those taking another path.<br />Consequently, there is a distribution of residence time for the PBR.<br />
- 5. Residence Time Distributions<br />CSTR<br />Short-circuiting may occur (the direct movement of material from inlet to outlet.<br />Dead zones may exist (regions with a minimum of mixing and thus virtually no reaction takes place).<br />
- 6. Residence Time Distributions<br />Concepts that must be addressed in approaching a solution to such problems:<br />distribution of residence times occurs<br />quality of mixing varies with position in reactor<br />a model must used to describe the phenomenon<br />Accounting for nonideality requires<br />knowledge of macromixing (RTD)<br />application of the RTD to a reactor (micromixing) to predict reactor performance.<br />
- 7. RTD Functions<br />In any reactor, the RTD can affect performance<br />Ideal Plug Flow and Batch Reactors<br />Every atom leaving reactor is assumed to have resided in the reactor for exactly the same duration. No axial mixing.<br />Ideal CSTR<br />Some atoms leave almost immediately, others remain almost forever. Many leave after spending a period of time near the mean residence time. Perfect mixing.<br />RTD is characteristic of mixing in a reactor.<br />RTDs are not unique to reactor type. Different reactor types can have the same RTD.<br />
- 8. Measurement of RTD<br />RTD is measured experimentally by use of an inert tracer injected into the reactor at t = 0. Tracer concentration is measured at effluent as a function of time.<br />Tracer must be non-reactive and non-absorbing on reactor walls/internals.<br />Tracer is typically colored or radioactive to allow detection and quantification.<br />Common methods of injection are pulse and step inputs.<br />
- 9. Pulse Input RTD Measurement<br />An amount of tracer No is suddenly (all at once) injected into the feed of a reactor vessel with flow at a steady state.<br />Outlet concentration is measured as a function of time.<br />reactor<br />effluent<br />feed<br />detection<br />injection<br />
- 10. Pulse Input RTD Measurement<br />pulse injection<br />pulse response<br />C<br />C<br />reactor<br />0<br />-<br />+<br />0<br />-<br />+<br />t<br />t<br />effluent<br />feed<br />detection<br />injection<br />
- 11. Pulse Input RTD Measurement<br />Injection pulse in system of single-input and single-output, where only flow (no dispersion) carries tracer material across system boundaries. <br />The amount of tracer materialN leaving the reactor between t and t+t for a volumetric flowrate of iswhere t is sufficiently small that the concentration of tracer C(t) is essentially constant over the time interval.<br />
- 12. Pulse Input RTD Measurement<br />Dividing by total amount of tracer injected, No yields the fraction of material that has a residence time between t and t+t:<br />where E(t) represents the residence-time distribution function.<br />
- 13. Step Input RTD Measurement<br />step injection<br />step response<br />C<br />C<br />reactor<br />t<br />t<br />effluent<br />feed<br />detection<br />injection<br />
- 14. Step Input RTD Measurement<br />In general, the output concentration from a vessel is related to the input function by the convolution integral (Levenspiel):where the inlet concentration takes the form of either a perfect pulse input (Dirac delta function), imperfect pulse injection, or a step input.<br />
- 15. Step Input RTD Measurement<br />Considering a step input in tracer concentration for a system of constant :<br />constant can be brought<br />outside the integral<br />
- 16. Divide by Co<br />F(t) fraction of molecules that have spent a time t or less in reactor (Cumulative age)<br />Differentiate to obtain RTD function E(t)<br />Step Input RTD Measurement<br />
- 17. Step Input RTD Measurement<br />Advantages<br />Easier to carry out experimentally than pulse test<br />Total amount tracer in feed need not be known<br />Disadvantages<br />Often difficult to maintain a constant tracer concentration in feed.<br />differentiation of data, often leads to large error.<br />Requires large amount of tracer, which in some cases can be expensive.<br />
- 18. RTD Characteristics<br />E(t) is sometimes called the exit-age distribution function. <br />If the age of an atom is regarded as the amount of time it spends in the reactor, E(t) is the age distribution of the effluent.<br />E(t) is the most often used distribution function for reactor analysis.<br />
- 19. Fraction of exit stream that has resided in the reactor for a period of time shorter than a given value of t:<br />Fraction of exit stream that has resided in the reactor for a period of time longer than a given value of t:<br />Integral Relationships<br />
- 20. Integral Relationships<br />
- 21. Mean Residence Time<br />The nominal holding time, , is equal to the mean residence time, tm.<br />The mean value of the time is the first moment of the RTD function, E(t).<br />can be used to determine reactor volume<br />
- 22. 1st moment – mean residence time<br />2nd moment – variance (extent of spread of the RTD)<br />3rd moment – skewness (extent RTD is skewed relative to the mean)<br />Other Moments of the RTD<br />
- 23. Normalized RTD Function, E()<br />A normalized RTD is often used to allow comparison of flow profiles inside reactors of different sizes, where<br />for an ideal CSTR<br />
- 24. Internal-Age Distribution, I()<br />Fraction of material inside the reactor that has been inside for a period of time between and +<br />
- 25. RTD in a Batch or PFR<br />Simplest case<br />Spike at t = (or = 1) of infinite height and zero width with an area of one<br />
- 26. Effluent concentration is identical to that of reactor contents.<br />A material balance for t > 0 on inert tracer injected as a pulse at t = 0<br />RTD in a CSTR<br />
- 27. RTD in a CSTR<br />Recall definition of E(t), and substitute:<br />
- 28. Ideal Reactor Response to Pulse<br />Batch/PFR<br />CSTR<br /><br />1<br />E<br />E<br /><br />1<br />t<br /><br />
- 29. Laminar Flow RTD<br />Velocity profile in a pipe (cylindrical coordinates) is parabolic according to:<br />Time for passage of an element of fluid is<br />
- 30. The fraction of total fluid passing between r and r+dr is d/0:<br />Laminar Flow RTD<br />
- 31. Laminar Flow RTD<br />Combining<br />
- 32. Laminar Flow RTD<br />The minimum time the fluid will spend in the reactor is<br />Therefore, the complete RTD function is<br />
- 33. Laminar Flow RTD<br />The RTD appears graphically as<br />1<br />E<br />0.5<br /><br />
- 34. RTD of PFR and CSTR in series<br />CSTR (s) followed by PFR (p)<br />CSTR output will be delayed by a time of p<br />
- 35. RTD of PFR and CSTR in series<br />PFR (p) followed by CSTR (s) <br />PFR output will delayed the introduction of the pulse to the CSTR by a time of p<br />Regardless of the order, the RTD is the same. However, the RTD is not a complete description of structure for a particular reactor or system of reactors (see Example 13-4).<br />

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