Nonideal Flow Characterized by Residence Time Distribution (RTD)Pulse input of tracerF(t)RTD-方差的加和性方差的加和性一次矩的加和性二次矩的加和性s s2 假设界面处无返混 ( 小)有对矮胖反应器：端部的返混贡献很大对矮胖反应器：端部的返混贡献很大对细长反应器：端部返混贡献可忽略对细长反应器：端部返混贡献可忽略E1E2E3 卷积（E(t): 传递函数的概念）Chap9 - Summary1. E(t)dt: fraction of material exiting the reactor that has spent between time t and t+dt in the reactor.2. The mean residence time3. The variance about the mean residence time isis equal to the space time for constant volumetric flow, = 0 4. The cumulative distribution function F(t) gives the fraction of effluent material that has been in the reactor a time t or less:5. The RTD functions for an ideal reactor arePlug flowCSTRLaminar flow6. The dimensionless residence time is7. The internal-age distribution, I(), , gives the fraction of material inside the reactor that has been inside between a time and a time +d（自学，（自学，P633634）*Simple diagnostics and troubleshooting using the RTD for ideal reactors*8. Segregation modelFor multiple reactions9. Maximum mixedness: For multiple reactions*迟混与早混（见网络学堂补充材料）迟混与早混（见网络学堂补充材料）*O. Levenspiel, P358(a)(b)(c, d, e)Chapter 10Models for Nonideal ReactorsOverviewUse the RTD to evaluate parametersModel of reactor flow patternsTanks-in-series modelDispersion model10.1 Some guidelinesRTD data + Kinetics + Model = PredictionGuidelines to develop models for nonideal reactors1. The model must be mathematically tractable.2. The model must realistically describe the characteristics of the nonideal reactors. The phenomena occurring in the nonideal reactor must be reasonably described physically, chemically, and mathematically.3. The model must not have more than two adjustable parameters. 10.1.1 One-parameter modelsNonideal CSTRs include a reactor dead volume VD, no reaction takes placeNonideal CSTRs with a fraction of fluid bypassing the reactor, exiting unreactedTanks-in-series modelDispersion model* This parameter is most always evaluated by analyzing the RTD determined from a tracer test. Examples: 10.1.2 Two-parameter modelsC(t)tVsVD0bs010.2 Tanks-in-series (T-I-S) model123PulseV1 = V2 = Vi = 01 = 2 = iFirst reactor:Second reactor:ODE: Solution:Same to third reactor .n CSTRs:(C2 = 0 at t = 0)RTD for equal-size tanks in series: Vtotal/n: calculated from RTDLevenspiel book (3rd)First order reaction:E()n = 10n = 4n = 2n: non-integer, or integerFirst order reaction:1n =If n = 2.53, you might calculate the conversions for n = 2 and n = 3 to bound the value.Graphical method of evaluating the performance of N tanks in series for any kineticsLevenspiel, p329-rACACA0CA1CA2CA3CA4Parallel lines for same size tanksSlope: For microfluid:一级等温反应一级等温反应二级等温反应二级等温反应Chemical conversion of Macrofluids10.3 Dispersion modelMolar flow rate of tracer (FT) by both convection and dispersionDa: effective dispersion coefficient, m2/sPulse tracer balancedispersion10.4 Flow, reaction, and dispersion10.4.1 Balance equationsSecond-order ODESimilar to A:1st-order reactionDimensionlessDamkhler numberfor first-order reactionDamkhler numberfor first-order reactionPeclet numberl: characteristic length termPer: reactor Peclet number, it uses reactor length, LPef: fluid Peclet number, it uses characteristic length that determines the fluids mechanical behavior Empty tube: Packed bed: bed porosity: pipe diameter10.4.2 Boundary conditionsBoundary conditions for closed vessels and open vessels.Closed-closed vessels: assume that there is no dispersion or radial variation in concentration either upstream (closed) or downstream (closed) of the reaction sectionOpen-open vessels: dispersion occurs both upstream (open) and downstream (open) of the reaction sectionDa=0Da0Da=0z = 0z = LdispersionDa0Da0Da0z = 0z = LdispersionClosed-closed vesselOpen-open vessel10.4.2A Closed-closed vessel boundary conditionEntrance boundary conditionz = 0z = LExit conditionDanckwerts boundary conditionsz = 00-0+FAEntrance: CA0CA(0+)z0-0+z=0Example: CSTR, CA0 CA,exitExit: CA(L-)L-L+CA(L+)10.4.2B Open-open systemOpen-openBoundary conditionz = 0z = L10.4.2C Back to the solution for a closed-closed systemAnalytical solutionOutside the limited case of a first-order reaction, a numerical solution of the equation is required.Three ways to find Da, and the Peclet number1. Laminar flow with radial and axial molecular diffusion theory2. Correlations from the literature for pipes and packed beds3. Experimental tracer data10.4.4 Dispersion in a tubular reactor with laminar flowThe molecules on the center streamline (r = 0) exited the reactor at a time t = /2;The molecules traveling on the streamline at r = 3R/4 exited the reactor at time:Radial diffusion in laminar flowConvective-diffusion equation for tracer transport in both the axial and radial directionLaminar flowAris-Taylor dispersion coefficientAverage axial concentration10.4.5 Correlations for DaPlease refer to the book, p674-676.10.4.6 Experimental determination of DaUnsteady-state tracer balanceMass of tracer injected, MCalculating Per using tm and 2 determined from RTD data for a closed-closed systemOpen-Open vessel boundary conditionsAt the entrance:At the exit:Per 100Calculate :Calculate :Calculate Per:Case 1. The space time is known. Case 2. The space time is unknown. This situation arises when there are dead or stagnant pockets that exist in the reactor along with the dispersion effects. To analyze this situation we first calculate tm and 2 from the data as in case 1. Then, . finding the effective reactor volumeRTD tm, 2 Per 10.4.7 Sloppy tracer inputsOpen-Open system10.5 Tanks-in-series vs. dispersion modelEquivalency between models of tanks-in-series and dispersionorwhere10.6 Numerical solutions to flows with dispersion and reactionSteady state:Analytical solutions to dispersion with reaction can only be obtained for iso-thermal zero- and first-order reactions. Case A. Aris-Taylor analysis for laminar flowClosed-closedBoundary cond.Open-openBoundary cond.Case B: Full numerical solution10.7 Two-parameter models modeling real reactors with combinations of ideal reactors10.7.1 Real CSTR modeled using bypassing and dead spaceDead zonebypassing VsCA0, 0bCA0CASVd12CA0 = b +sBalance at junction :ModelingFor first-order reaction, a mole balance on CSTR (Vs)RTD10.7.1B Using a tracer to determine the model parameters in CSTR-with-dead-space-and-bypass model CT0, 0bCT0CTSVd=(1-)V12CT0 = b +sTracer balance for step inputModelsystemVs=(1-)0Vs=VJunction balance:Evaluating model parameters10.7.2 Real CSTR modeled as two CSTRs with interchange0V1V2, CA20CA111Using a tracer to determine the model parameters in a CSTR with an exchange volumeReactor 1: Reactor 2: where,阅读：p693-694参看Levenspiel: Chemical Reaction Engineering (3rd), Chap12-Compartment Models, p283.Summary1.The models for predicting conversion from RTD data are:a. Zero adjustable parameters(1) Segregation model(2) Maximum mixedness modelb. One adjustable parameter(1) Tanks-in-series model(2) Dispersion modelc. Two adjustable parameters: real reactor modeled as combinations of ideal reactors2. Tanks-in-series model: use RTD data to estimate the number of tanks in seriesFor a first-order reaction3. Dispersion model: for a first-order reaction, use the Danckwerts boundary conditions4. Determine Da(a) For laminar flow the dispersion coefficient is (b) Correlations. (c) Experiment in RTD analysis to find tm and 2Closed-closedOpen-open5. If a real reactor is modeled as a combination of ideal reactors, the model should have at most two parameters.0V1V2, CA20CA111 VsCA0, 0bCA0CASVd12CA0 = b +s6. The RTD is used to extract model parameters.7. Comparison of conversions for a PFR and CSTR with the zero-parameter and two-parameter models. Xseg symbolizes the conversion obtained from the segregation model and Xmm that from the maximum mixedness model for reaction orders greater than one.with