1、化 學(xué) 反 應(yīng) 工 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,one or the other often is optimum no matter what we are designing for. these two patterns are simple to treat.,化 學(xué) 反 應(yīng) 工 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 interrelated

3、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

4、mixing of material in the vessel.,化 學(xué) 反 應(yīng) 工 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 c

5、aused 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

6、 of the unit.,化 學(xué) 反 應(yīng) 工 程,化 學(xué) 反 應(yīng) 工 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 p

7、recisely, 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. W

8、e show when it may legitimately(合理地) be used, how to use it, and when it is not applicable what alternatives to turn to.,化 學(xué) 反 應(yīng) 工 程,In developing the “l(fā)anguage” for this treatment of nonideal flow (see Danckwerts, 1953), we will only consider the steady-state flow, without reaction and without dens

9、ity change, of a single 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 學(xué) 反 應(yīng) 工 程,Figure 11

10、.2 Two extremes of aggregation of fluid,Microfluid,Macrofluid,化 學(xué) 反 應(yīng) 工 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

11、 completely opposite behavior.,Single-Phase Systems. These lie somewhere between the extremes of macro- and microfluids.,化 學(xué) 反 應(yīng) 工 程,Figure 11.3 Examples of macro- and microfluid behavior.,化 學(xué) 反 應(yīng) 工 程,Earliness of Mixing,The fluid elements of a single flowing stream can mix with each other either ea

12、rly or late in their flow 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 學(xué) 反 應(yīng) 工 程,Figure 11.4 Examp

13、les of early and of late mixing of fluid.,化 學(xué) 反 應(yīng) 工 程,Figure 11.5 Early or late mixing affects reactor behavior.,化 學(xué) 反 應(yīng) 工 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 cruc

14、ial. Often, much depends 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(更廣的).,化 學(xué) 反 應(yīng) 工 程,11.1 E, THE AGE DISTRIBUTION OF FLUID, THE RTD,It is evident that elements of fluid taking different routes

15、through the reactor may take different 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 學(xué) 反 應(yīng) 工 程,停留時(shí)間分布密度函數(shù) E(t)

16、 函數(shù),對(duì)于同時(shí)進(jìn)入反應(yīng)器入口的 N 個(gè)流體粒子，若在出口處進(jìn)行檢測(cè)，則其中停留時(shí)間介于 t t + dt 之間的流體粒子個(gè)數(shù) dN 所占的分率為 E(t) dt dN / N 我們定義 E(t) 為停留時(shí)間分布密度函數(shù)。 如：在某時(shí)刻進(jìn)入反應(yīng)器入口的 100 個(gè)流體粒子，到達(dá)出口時(shí)停留時(shí)間為 5 6 min 的粒子有 8 個(gè)，若取 t = 5 min，dt = t 1 min， 則此時(shí) E(t) dt = dN / N = N / N = 8 /100 = 0.08； E(t) = dN / (dt*N) = N / (t *N) = 0.08 min-1 當(dāng) dt 0，則 E(t) dt

17、是一個(gè)瞬時(shí)( 如t = 5 min 時(shí) )的分率,化 學(xué) 反 應(yīng) 工 程,停留時(shí)間分布函數(shù) F(t) 函數(shù),對(duì)于同時(shí)進(jìn)入反應(yīng)器入口的 N 個(gè)流體粒子，若在出口處進(jìn)行檢測(cè)，則其中停留時(shí)間介于 0 t 之間的流體粒子所占的分率為 F(t) 我們定義 F(t) 為停留時(shí)間分布函數(shù)。 如：在某時(shí)刻進(jìn)入反應(yīng)器入口的 100 個(gè)流體粒子，到達(dá)出口時(shí)停留時(shí)間為 0 5 min 的粒子有 20 個(gè)，若取 t = 5 min，則此時(shí) F(t) = 20 /100 = 0.2。 F(t) 是一個(gè)累積（如 t = 05 min ）的分率。,化 學(xué) 反 應(yīng) 工 程,停留時(shí)間分布,停留時(shí)間分布的定量描述,停留時(shí)間分布密

18、度函數(shù) E (t),E(t) = 0, t 0 E(t) 0, t0,歸一化條件,化 學(xué) 反 應(yīng) 工 程,停留時(shí)間分布,停留時(shí)間分布的定量描述,停留時(shí)間分布函數(shù) F (t),化 學(xué) 反 應(yīng) 工 程,We find it convenient to represent the RTD in such a way that the area under the curve is unity, or,This procedure is called normalizing(歸一化) the distribution, and Fig. 11.6 shows this.,We should note

19、one restriction on the E curve that the fluid only enters and only leaves the vessel one time. This means that there should be no flow or diffusion or upflow eddies(旋渦) at the entrance or at the vessel exit. We call this the closed vessel boundary condition.,化 學(xué) 反 應(yīng) 工 程,With this representation the

20、fraction of exit stream of age between t and t + dt is,E dt,The fraction younger than age t1 is,(1),whereas the fraction of material older than t, shown as the shaded area in Fig. 11.6, is,(2),The E curve is the distribution needed to account for nonideal flow.,化 學(xué) 反 應(yīng) 工 程,Figure 11.6 The exit age d

21、istribution cure E for fluid flowing through a vessel: also called the residence time distribution, or RTD.,化 學(xué) 反 應(yīng) 工 程,Experimental Methods (Nonchemical) for Finding E,The simplest and most direct way of the E curve uses a physical or nonreactive tracer(示蹤物). Fig. 11.7 shows some of these. Because

22、the pulse and the step experiments are easier to interpret, the periodic(周期的) and random(任意的) harder, here we only consider the pulse and the step experiment.,We next discuss these two experimental methods for finding the E curve. We then show how to find reactor behavior knowing the E curve for the

23、 reactor.,化 學(xué) 反 應(yīng) 工 程,Figure 11.7 Various ways of studying the flow pattern in vessels.,化 學(xué) 反 應(yīng) 工 程,The Pulse Experiment,Let us find the E curve for a vessel of volume V m3 through which flows m3/s of fluid. For this instantaneously introduce M units of tracer (kg or moles) into the fluid entering t

24、he vessel, and record the concentration-time of tracer leaving the vessel. This is the Cpulse curve. From the material balance for the vessel we find,(3),化 學(xué) 反 應(yīng) 工 程,Fig. 11.8 The useful information obtainable from the pulse trace experiment.,化 學(xué) 反 應(yīng) 工 程,(4),All this is shown in Fig. 11.8.,To find t

25、he E curve from the Cpulse curve simply change the concentration scale such that the area under the curve is unity. Thus, simply divide the concentration readings by M/, as shown in Fig. 11.9.,(5a),At exit, the tracer fraction of age t t + dt is: Cpulsedt / M, From the definition of E, we get: Edt =

26、 Cpulsedt / M, So, E = Cpulse / (M/),化 學(xué) 反 應(yīng) 工 程,Figure 11.9 Transforming an experimental Cpulse curve into an E curve.,化 學(xué) 反 應(yīng) 工 程,We have another RTD function E . Here time is measured in terms of mean residence time = t / . Thus,(5b),E is a useful measure when dealing with flow models which come

27、up in Chapters 13, 14, and 15. Figure 11.10 shows how to transfer E into E .,One final reminder, the relationship between Cpulse and E curves only holds(適用于) exactly for vessels with closed boundary conditions.,化 學(xué) 反 應(yīng) 工 程,Closed Vessel, constant,化 學(xué) 反 應(yīng) 工 程,Figure 11.10 Transforming an E curve into

28、 an E curve.,化 學(xué) 反 應(yīng) 工 程,The Step Experiment,Consider m3/s of fluid flow through a vessel of volume V. Now at time t = 0 switch from ordinary fluid to fluid with tracer of concentration Cmax= , and measure the outlet tracer concentration Cmax versus t, as shown in Fig. 11.11.,A material balance rela

29、tes the different measured quantities of the output curve of a step input,化 學(xué) 反 應(yīng) 工 程,Figure 11.11 Information obtainable from a step tracer experiment.,化 學(xué) 反 應(yīng) 工 程,and,=,where is the flow rate of tracer in the entering fluid.,化 學(xué) 反 應(yīng) 工 程,The dimensionless form of the Cstep curve is called the F cur

30、ve. It is found by having the tracer concentration rise from zero to unity, as shown in Fig. 11.12.,Figure 11.12 Transforming an experimental Cstep curve to an F curve.,化 學(xué) 反 應(yīng) 工 程,Relationship between the F and E curve,To relate E with F imagine a steady flow of white fluid. Then at time t = 0 swit

31、ch to red and record the rising concentration of red fluid in the exit stream, the F curve. At any time 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 second is given by Eq. 1. So we have, at time t,(7),化

32、學(xué) 反 應(yīng) 工 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 in the vessel. We should remember that these relati

33、onship only hold for closed vessels. When this boundary condition is not met, then the Cpluse and E curves differ.,化 學(xué) 反 應(yīng) 工 程,Figure 11.13 Relationship between the E and F curves.,脈沖法測(cè)定反應(yīng)器的 E 分布曲線,M 為示蹤劑的加入量,停留時(shí)間分布的統(tǒng)計(jì)特征值,1. 平均停留時(shí)間 (mean residence time),2. 方差 (variance),對(duì)閉式容器(closed vessel) ,若采用無因次停

34、留時(shí)間 :,1. 活塞流模型,理想反應(yīng)器的停留時(shí)間分布,1. Plug Flow,2. Mixed Flow,理想反應(yīng)器的停留時(shí)間分布,化 學(xué) 反 應(yīng) 工 程,化 學(xué) 反 應(yīng) 工 程,Figure 11.14 Properties of the E and F curves for various flows. Curves are drawn in term of ordinary and dimensionless time units. Relationship between curve is given by Eqs. 7 and 8.,存在滯流區(qū),非理想反應(yīng)器的停留時(shí)間分布,使 t

35、 減小,存在溝流,存在溝流,非理想反應(yīng)器的停留時(shí)間分布,存在短路,非理想反應(yīng)器的停留時(shí)間分布,存在短路,層流反應(yīng)器,非理想反應(yīng)器的停留時(shí)間分布,化 學(xué) 反 應(yīng) 工 程,For closed vessel, at any time these curves are related as follows :,Fall dimensionless, E =,化 學(xué) 反 應(yīng) 工 程,EXAMPLE 11.1 FINDING THE RTD BY EXPERIMENT,The concentration reading in Table E11.1 represent a continuous resp

36、onse to a pulse input into a closed vessel which is to be used as a chemical reactor.,Calculate the mean residence time of fluid in the vessel and tabulate and plot the exit age distribution E.,化 學(xué) 反 應(yīng) 工 程,Table E11.1,Time t, (min),Tracer Output Concentration, Cpulse,(gm/ liter fluid),0 0 5 3 10 5 1

37、5 5 20 4 25 2 30 1 35 0,化 學(xué) 反 應(yīng) 工 程,SOLUTION,The mean residence time, from Eq.4, is,The area under the concentration-time curve.,Area =,= ( 3+5+5+4+2+1 ) 5 = 100 gmmin / liter,化 學(xué) 反 應(yīng) 工 程,gives the total amount of tracer introduced. To find E, the area under this curve must be unity; hence, the conc

38、entration readings must each be divided by the total area, giving,Thus we have,0 5 10 15 20 25 30,0 0.03 0.05 0.05 0.04 0.02 0.01, min-1,t, min,化 學(xué) 反 應(yīng) 工 程,Figure E11.1 Plot of Et distribution of Example 11.1.,化 學(xué) 反 應(yīng) 工 程,11.2 CONVERSION IN NON-IDEAL FLOW REACTORS,To evaluate reactor behavior in gen

39、eral we have to know four factors:,1. the kinetics of the reaction 2. the RTD of fluid in the reactor 3. the earliness or lateness of fluid mixing in the reactor 4. whether the fluid is a micro or macro fluid,For microfluids in plug or mixed flow we have developed the equation in the earlier chapter

40、s. For intermediate flow we will develop appropriate models in Chapters 12, 13, and 14.,化 學(xué) 反 應(yīng) 工 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 concentrat

41、ion and reacts away rapidly because n 1; In (b) the fluid drops immediately to a low concentration. Since the rate of reaction drops more rapidly than does the concentration you will end up with a lower conversion.,(12),化 學(xué) 反 應(yīng) 工 程,Figure 11.17 This shows the latest and the earliest mixing we can ha

42、ve for a given RTD.,化 學(xué) 反 應(yīng) 工 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 reactor, thus fluid elements will have different compositions. So the mean composition in the exit

43、stream will have to account for these two factors, the kinetics and the RTD. In words, then,化 學(xué) 反 應(yīng) 工 程,In symbols this becomes,(13),化 學(xué) 反 應(yīng) 工 程,From Chapter 3 on batch reactor we have,(16),These are terms to be introduced the performance equation, Eq.13.,化 學(xué) 反 應(yīng) 工 程,Note: For first-order reactions,

44、 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 reacting fluid is macro or micro.,化 學(xué) 反 應(yīng) 工 程,The Dirac Delta Functions, . One E curve which may

45、 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 學(xué) 反 應(yīng) 工 程,Figure 11.18 The E function for pulg flow,E, s-1,t0,Infinitely high zero width, but with area = 1,

46、 ( t - t0 ),化 學(xué) 反 應(yīng) 工 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 學(xué) 反 應(yīng) 工 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 sam