CHAPTER 8: Mixing in Chemical Reactors

1 CHAPTER 8: Mixing in Chemical ReactorsCopyright 2022 by Nob Hill Publishing, LLCThe three main reactor types developed thus far batch, continuous-stirred-tank,and plug-flow Reactors are useful for modeling many complex Chemical to this point we have neglected a careful treatment of the fluid flow patternwithin the this CHAPTER we explore some of the limits of this approach and develop methodsto address and overcome some of the more obvious / 130 Scope of problemThe general topic of Mixing , even in the restricted context of Chemical Reactors , isan impossibly wide one to treat this CHAPTER , we will restrict ourselves to fluid-phase natural approach to describing Mixing is to solve the equations of motion of fluid systems, the type of fluid flow is obviously important, and we shouldconsider both laminar and turbulent flow, and various mechanisms of diffusion(molecular diffusion, eddy diffusion).

2 Using fluid mechanics to describe all cases of interest is a difficult problem, bothfrom the modeling and computational perspectives. Rapid developments incomputational fluid dynamics (CFD), however, make this approach increasinglyattractive [1].2 / 130 Residence-time distributionA second, classical approach to describing Mixing is to use simple tests toexperimentally probe the system of empirical testing approaches do not use any of the structure of the equationsof motion, but they can provide some rough features of the Mixing taking place inthe system under this CHAPTER we first develop this classical approach, and find out what kinds ofunderstanding it can also identify some of the limitations of this approach.

3 Nauman and Buffhamprovide a more in-depth treatment of many of the classical topics covered in thischapter, and provide many further citations to the research literature [15].3 / 130 Convection and diffusionOne might intuitively expect that to enhance Mixing and reduce spatial variation inconcentration, one should seek conditions that maximize the rate of this notion is correct for Mixing on the finest length scales, it is generallymuch more important in macroscopic scale processes to decrease variations on thelarger length in this regime is enhanced primarily by improving the convection, anddiffusion plays only a small simple terms, one does not expect to appreciably decrease the time required tomix the cream in one s coffee by increasing the coffee temperature (enhanceddiffusion); one instead turns a spoon a few times (enhanced convection).

4 On the finest length scales, Mixing is accomplished readily for small molecules by therandom process of molecular diffusion; in fact, the random molecular motions arethe only effective Mixing processes taking place on the finest length / 130 Residence-Time Distribution DefinitionConsider an arbitrary reactor with single feed and effluent streams depicted in thefollowing figureWithout solving for the entire flow field, which might be quite complex, we wouldlike to characterize the flow pattern established in the reactor at steady residence-time distribution of the reactor is one such characterization ormeasure of the flow / 130 Gedanken ExperimentImagine we could slip some inert tracer molecules into the feed stream and couldquery these molecules on their exit from the reactor as to how much time they hadspent in the assume that we can add a small enough amount of tracer in the feed so that wedo not disturb the established flow of the tracer molecules might happen to move in a very direct path to theexit.

5 Some molecules might spend a long time in a poorly mixed zone before finallyfinding their way to the to their random motions as well as convection with the established flow, whichitself might be turbulent, we would start recording a distribution of residence timesand we would create the residence-time probability density or the reactor is at steady state, and after we had collected sufficient residence-timestatistics, we expect the residence-time distribution to also settle down to a / 130 Probability densityLetp( ) represent the probability density or residence-time distribution, andP( ) theintegrated form sop( )d ,probability that a feed molecule spends time to +d in the reactorP( ),probability that a feed molecule spends timezero to in the reactorThe two versions of the probability function obviously contain the same information andare related byP( ) = 0p( )d ,p( ) =dP( )d 7 / 130 Measuring the RTDAs a thought experiment to define the RTD, querying tracer molecules on their exitfrom the reactor is a fine we plan to actually measure the RTD, so we require an implementableexperiment with actual cannot measure the time spent by a particular tracer molecule in the reactor; tous, all tracer molecules are identical.

6 We can measure concentration of tracermolecules in the effluent, however, and that will prove sufficient to measure an experiment in which we measure the concentration of tracer in the feedand effluent streams over some time period, while the reactor maintains a steadyflow the definition of the RTD in the previous section, the effluent tracerconcentration at some timetis established by the combined exit of many tracermolecules with many different residence / 130 Convolution integralThe concentration of molecules that enter the reactor at timet and spend timet t in the reactor before exiting is given bycf(t )p(t t )dt .These molecules are the ones leaving the reactor at timetthat establish effluentconcentrationce(t), so we havece(t) = t cf(t )p(t t )dt ( )The inlet and outlet concentrations are connected through this convolution integralwith the residence-time we conduct the experiment so that the feed tracer concentration is zero before aninitial timet= 0, then the integral reduces toce(t) = t0cf(t )p(t t )dt ,cf(t) = 0,t 0( )9 / 130 Tracer concentrations to RTDN otice we can change the variable of integration in Equation to establish anequivalent representationce(t) = t0cf(t t )p(t )dt ( )

7 Which is sometimes a convenient connection between the inlet and outlet concentrations, and the RTD, allowsus to determine the RTD by measuring only tracer next describe some of the convenient experiments to determine the / 130 Step responseIn the step-response experiment, at time zero we abruptly change the feed tracerconcentration from steady valuec0to steady convenience we assumec0= 0. Because the feed concentration is constant atcfafter time zero, we can take it outside the integral in Equation and obtaince(t) =cf t0p(t )dt =cfP(t)So for a step-response experiment, the effluent concentration versus time providesimmediately the integrated form of the residence-time distributionP( ) =ce( )/cf,step response( )11 / 130 Pulse and impulse responsesAn impulse response is an idealized experiment, but is a useful concept.

8 As we willsee it provides the RTD directly rather than in the integrated motivate the impulse-response experiment, imagine we abruptly change the inlettracer concentration from zero to a large value and return it to zero after a shorttime as sketched in the following (t)a (t)area=a0 Such a test is called a pulse test. The pulse test is no more difficult to implementthan the step test; it is merely two step changes in feed concentration in rapidsuccession. In some ways it is a superior test to the step response, because byreturning the tracer concentration to zero, we use less tracer in the experiment andwe cause less disruption of the normal operation of the / 130 From pulse to impulseThe impulse response is an idealized limit of the pulse response.

9 Consider a family ofpulse tests of shorter and shorter duration t, as sketched in the maintain constant total tracer addition by spiking the feed with higher andhigher concentrations so that the productcf t=ais impulse response is the limit of this experiment as t 0. We call thislimiting feed concentration versus time function the delta function,a (t). It is alsocalled the Dirac delta functionor an impulse, hence the name, impulse / 130 Impulse responseThe constantais the amplitude of the delta main property of the delta function is that, because it is so narrowly focused, itextracts the value of an integrand at a point in the interval of integration, g(t) (t)dt=g(0),allg(t)( ) (t)dt= 1,normalizedSo if we can approximatecf(t) =a (t), then we have from Equation (t) =a t (t )p(t t )dt =ap(t)So for an experiment approximating an impulse, the effluent concentration versustime provides the residence-time distribution directlyp( ) =ce( )/a,impulse response( )14 / 130 Continuous-Stirred-Tank Reactor (CSTR)

10 We next examine again the well-stirred the following step-response experiment: a clear fluid with flowrateQfenters a well-stirred reactor of time zero we start adding a small flow of a tracer to the feed stream and measurethe tracer concentration in the effluent expect to see a continuous change in the concentration of the effluent streamuntil, after a long time, it matches the concentration of the feed / 130 Mass balanceAssuming constant density, the differential equation governing the concentration ofdye in the reactor follows from Equation (cf c),c(0) = 0( )in whichcis the concentration of the dye in the reactor and effluent CHAPTER 4, we named the parameter =VR/Qfthe mean residence time.