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1、化化 学学 反反 应应 工工 程程Chapter 11 Basics of Non-Ideal Flow So far we have treated two flow patterns,plug flow and mixed flow.These can give very different behavior(size of reactor,distribution of products).We like these flow patterns and in most cases we try to design equipment to approach one or the othe
2、r because lone or the other often is optimum no matter what we are designing for.lthese two patterns are simple to treat.化化 学学 反反 应应 工工 程程But real equipment always deviates from these ideals.How to account for this?That is what this and following chapters are about.Overall three somewhat interrelate
3、d factors make up the contacting or flow pattern:1.the RTD or residence time distribution of material which is flowing through the vessel;2.the state of aggregation of the flowing material,its tendency to clump and for a group of molecules to move about together;3.the earliness and lateness of mixin
4、g of material in the vessel.化化 学学 反反 应应 工工 程程Let us discuss these three factors in a qualitative way at first.Then,this and the next few chapters treat these factors and show how they affect reactor behavior.The Residence Time Distribution,RTD Deviation from the two ideal flow patterns can be caused
5、 by channeling of fluid,by recycling of fluid,or by creation of stagnant regions in the vessel.Figure 11.1 shows this behavior.In all types of process equipment,such as heat exchangers,packed columns,and reactors,this type of flow should be avoided since it always lowers the performance of the unit.
6、化化 学学 反反 应应 工工 程程化化 学学 反反 应应 工工 程程Setting aside this goal of complete knowledge about the flow,let us be less ambitious and see what it is that we actually need to know.In many cases we really do not need to know very much,simply how long the individual molecules stay in the vessel,or more precisely
7、,the distribution of residence times of the flowing fluid.This information can be determined easily and directly by a widely used method of inquiry,the stimulus-response experiment.This chapter deals in large part with the residence time distribution(or RTD)approach to nonideal flow.We show when it
8、may legitimately(合理地)be used,how to use it,and when it is not applicable what alternatives to turn to.化化 学学 反反 应应 工工 程程In developing the“language”for this treatment of nonideal flow(see Danckwerts,1953),we will only consider the steady-state flow,without reaction and without density change,of a sing
9、le fluid through a vessel.State of Aggregation of the Flowing Stream Flowing materials is in some particular state of aggregation,depending on its nature.In the extremes these states can be called microfluids and macrofluids,as sketched in Figure.11.2.化化 学学 反反 应应 工工 程程Figure 11.2 Two extremes of agg
10、regation of fluidMicrofluidMacrofluid化化 学学 反反 应应 工工 程程Two-Phase Systems.A stream of solids always behaves as a macrofluid,but for gas reacting with liquid,either phase can be a macro-or microfluid depending on the contacting scheme being used.The sketches of Figure.11.3 show completely opposite beha
11、vior.Single-Phase Systems.These lie somewhere between the extremes of macro-and microfluids.化化 学学 反反 应应 工工 程程Figure 11.3 Examples of macro-and microfluid behavior.化化 学学 反反 应应 工工 程程Earliness of MixingThe fluid elements of a single flowing stream can mix with each other either early or late in their f
12、low through the vessel.For example,see Fig.11.4.Usually this factor has little effect on overall behavior for a single flowing fluid.However,for a system with two entering reactant streams it can be very important.For example,see Fig.11.5.化化 学学 反反 应应 工工 程程Figure 11.4 Examples of early and of late mi
13、xing of fluid.化化 学学 反反 应应 工工 程程Figure 11.5 Early or late mixing affects reactor behavior.化化 学学 反反 应应 工工 程程Role of RTD,State of Aggregation,and Earliness of Mixing in Determining Reactor Behavior In some situations one of these factors can be ignored;in others it can become crucial.Often,much depends
14、 on the time for reaction,the time for mixing ,and the time for stay in the vessel .In many cases has a meaning somewhat like but somewhat broader(更广的).化化 学学 反反 应应 工工 程程11.1E,THEAGEDISTRIBUTIONOFFLUID,THERTDIt is evident that elements of fluid taking different routes through the reactor may take dif
15、ferent lengths of time to pass through the vessel.The distribution of these times for the stream of fluid leaving the vessel is called the exit age distribution E,or the residence time distribution RTD of fluid.E has the units of time-1.化化 学学 反反 应应 工工 程程停留时间分布密度函数停留时间分布密度函数E(t)函数函数对于同时进入反应器入口的对于同时进入
16、反应器入口的N个流体粒子个流体粒子，若在出口处，若在出口处进行检测，则其中进行检测，则其中停留时间介于停留时间介于tt+dt之间之间的流体粒的流体粒子个数子个数dN所占的分率为所占的分率为E(t)dtdN/N我们定义我们定义E(t)为为停留时间分布密度函数停留时间分布密度函数。如：在某时刻进入反应器入口的如：在某时刻进入反应器入口的100个流体粒子，到达出个流体粒子，到达出口时停留时间为口时停留时间为56min的粒子有的粒子有8个，若取个，若取t=5min，dt=t 1min，则此时则此时E(t)dt=dN/N=N/N=8/100=0.08；E(t)=dN/(dt*N)=N/(t*N)=0.0
17、8 min-1当 dt0，则，则E(t)dt是一个瞬时是一个瞬时(如如t=5min时时)的分率的分率化化 学学 反反 应应 工工 程程停留时间分布函数停留时间分布函数F(t)函数函数对于同时进入反应器入口的对于同时进入反应器入口的N个流体粒子个流体粒子，若在出口，若在出口处进行检测，则其中处进行检测，则其中停留时间介于停留时间介于0t之间之间的流体粒的流体粒子所占的分率为子所占的分率为F(t)我们定义我们定义F(t)为为停留时间分停留时间分布函数布函数。如：在某时刻进入反应器入口的如：在某时刻进入反应器入口的100个流体粒子，到达个流体粒子，到达出口时停留时间为出口时停留时间为05min的粒子
18、有的粒子有20个，若取个，若取t=5min，则此时，则此时F(t)=20/100=0.2。F(t)是一个累积（如是一个累积（如t=05min）的分率。）的分率。化化 学学 反反 应应 工工 程程停留时间分布停留时间分布停留时间分布的定量描述停留时间分布的定量描述停留时间分布密度函数停留时间分布密度函数 E(t)E(t)=0,t 0 red fluid and only red fluid in the exit stream is younger than age t.Thus we have=But the first term is simply the F value,while the
19、 second is given by Eq.1.So we have,at time t,(7)化化 学学 反反 应应 工工 程程and on differentiating(8)In graphical form this relationship is shown in Fig.11.13.These relationships show how stimulus-response experiments,using either step or pulse inputs can conveniently give the RTD and mean flow rate of fluid
20、in the vessel.We should remember that these relationship only hold for closed vessels.When this boundary condition is not met,then the Cpluse and E curves differ.化化 学学 反反 应应 工工 程程Figure 11.13 Relationship between the E and F curves.输入曲线输入曲线响应（输出）曲线响应（输出）曲线c0(t)脉冲法测定反应器的脉冲法测定反应器的E分布曲线分布曲线M 为示踪剂为示踪剂的加
21、入量的加入量停留时间分布的统计特征值停留时间分布的统计特征值1.平均停留时间平均停留时间(mean residence time)2.方差方差(variance)对闭式容器对闭式容器(closed vessel),若采用无因次停留时间若采用无因次停留时间 :1.活塞流模型活塞流模型理想反应器的停留时间分布理想反应器的停留时间分布t=0tE(t)01.001.Plug Flow2.Mixed Flow理想反应器的停留时间分布理想反应器的停留时间分布化化 学学 反反 应应 工工 程程化化 学学 反反 应应 工工 程程Figure 11.14 Properties of the E and F cu
22、rves for various flows.Curves are drawn in term of ordinary and dimensionless time units.Relationship between curve is given by Eqs.7 and 8.存在滞流区存在滞流区非理想反应器的停留时间分布非理想反应器的停留时间分布使使 t t 减小减小存在沟流存在沟流存在沟流存在沟流非理想反应器的停留时间分布非理想反应器的停留时间分布存在短路存在短路非理想反应器的停留时间分布非理想反应器的停留时间分布存在短路存在短路层流反应器层流反应器非理想反应器的停留时间分布非理想反应器
23、的停留时间分布化化 学学 反反 应应 工工 程程For closed vessel,at any time these curves are related as follows:Fall dimensionless,E=化化 学学 反反 应应 工工 程程EXAMPLE 11.1 FINDING THE RTD BY EXPERIMENTThe concentration reading in Table E11.1 represent a continuous response to a pulse input into a closed vessel which is to be used
24、 as a chemical reactor.Calculate the mean residence time of fluid in the vessel and tabulate and plot the exit age distribution E.化化 学学 反反 应应 工工 程程Table E11.1Time t,(min)Tracer Output Concentration,Cpulse(gm/liter fluid)0 05 310 515 520 425 230 135 0化化 学学 反反 应应 工工 程程SOLUTIONThe mean residence time,f
25、rom Eq.4,is The area under the concentration-time curve.Area=(3+5+5+4+2+1)5=100 gmmin/liter化化 学学 反反 应应 工工 程程gives the total amount of tracer introduced.To find E,the area under this curve must be unity;hence,the concentration readings must each be divided by the total area,giving Thus we have 0 5 10
26、 15 20 25 30 0 0.03 0.05 0.05 0.04 0.02 0.01,min-1 t,min 化化 学学 反反 应应 工工 程程Figure E11.1 Plot of Et distribution of Example 11.1.化化 学学 反反 应应 工工 程程11.2 CONVERSION IN NON-IDEAL FLOW REACTORS To evaluate reactor behavior in general we have to know four factors:1.the kinetics of the reaction2.the RTD of f
27、luid in the reactor3.the earliness or lateness of fluid mixing in the reactor4.whether the fluid is a micro or macro fluid For microfluids in plug or mixed flow we have developed the equation in the earlier chapters.For intermediate flow we will develop appropriate models in Chapters 12Chapters 12,1
28、313,and 1414.化化 学学 反反 应应 工工 程程To consider the early and late mixing of a microfluid,consider the two flow patterns shown in Fig.11.17 for a reactor processing a second-order reaction:In(a)the reactant starts at high concentration and reacts away rapidly because n 1;In(b)the fluid drops immediately t
29、o a low concentration.Since the rate of reaction drops more rapidly than does the concentration you will end up with a lower conversion.Thus,for microfluids:Late mixing favors reactions where n 1;Early mixing favors reactions where n 1;However,the earliness of mixing has no effect on the reaction wh
30、ere n=1.(12)化化 学学 反反 应应 工工 程程Figure 11.17 This shows the latest and the earliest mixing we can have for a given RTD.化化 学学 反反 应应 工工 程程For macrofluids,imagine little clumps of fluid staying for different lengths of time in the reactor(given by the E function).Each clump reacts away as a little batch r
31、eactor,thus fluid elements will have different compositions.So the mean composition in the exit stream will have to account for these two factors,the kinetics and the RTD.In words,then化化 学学 反反 应应 工工 程程In symbols this becomes(13)化化 学学 反反 应应 工工 程程From Chapter 3 on batch reactor we have(16)These are te
32、rms to be introduced the performance equation,Eq.13.化化 学学 反反 应应 工工 程程Note:For first-order reactions,the reaction result of macrofluid is identical to that of microfluid.On the other hand,when a reaction with any kinetics occurs in a plug flow reactor,it can always get the same result no matter the r
33、eacting fluid is macro or micro.化化 学学 反反 应应 工工 程程The Dirac Delta Functions,.One E curve which may puzzle(使迷惑)us is the one which represents plug flow.We call this the Dirac function,and in symbols we show it as(17)which says that the pulse occurs at t=t0,as seen in Fig.11.18.化化 学学 反反 应应 工工 程程Figure
34、11.18 The E function for pulg flowE,s-1t0Infinitely high zero width,but with area=1,(t-t0)化化 学学 反反 应应 工工 程程The two properties of this function which we need to know are Once we understand what this means we will see that it is easier to integrate with a function than with any other.For example,化化 学学
35、 反反 应应 工工 程程EXAMPLE 11.4 CONVERSION IN REACTORS HAVING NON-IDEAL FLOW The vessel of Example 11.1 is to be used as a reactor for a liquid decomposing with rate Find the fraction of reactant unconverted in the real reactor and compare this with the fraction unconverted in a plug flow reactor of the sa
36、me size.化化 学学 反反 应应 工工 程程SOLUTIONFor the plug flow reactor with negligible density change we have and with from Example 11.1 Thus the fraction of reactant unconverted in a plug flow reactor equals 1.0%.化化 学学 反反 应应 工工 程程For the real reactor the fraction unconverted,given by Eq.13 for macrofluids,is f
37、ound in Table E11.4.Hence the fraction of reactant unconverted in the real reactor Table E11.4 5 0.03 1.53 0.2154 (0.2154)(0.03)(5)=0.0323 10 0.05 3.07 0.0464 .=0.011615 0.05 4.60 0.0100 .=0.002520 0.04 6.14 0.0021 .=0.000425 0.02 7.68 0.005 .=.0.000130 0.01 9.21 0.0001 .=.0化化 学学 反反 应应 工工 程程From the
38、 table we see that the unconverted material comes mostly from the early portion of the E curve.This suggests that channeling and short-circuiting can seriously hinder(阻碍)attempts to achieve high conversion in reactors.Note that since this is a first-order reaction we can treat it as a microfluid,or
39、a macrofluid,whatever we wish.In this problem we solve the plug flow case as a microfluid,and we solved the nonideal case as a macrofluid.化化 学学 反反 应应 工工 程程EXAMLPE 11.5 REACTION OF A MACROFLUID Dispersed noncoalescing droplets()react(A R,)as they pass through a contactor.Find the average concentratio
40、n of A remaining in the droplets leaving the contactor if their RTD is given by the curve in Fig.E11.5.Figure E11.5 13E,min-1t,min化化 学学 反反 应应 工工 程程SOLUTIONEquation 13 is the pertinent(有关的)performance equation.Evaluate terms in this expressing.For The batch equation from Chapter 3Chapter 3 is化化 学学 反反 应应 工工 程程With E=0.5 for 1 t 3,Eq.13 becomes Soor