1、化化 學學 反反 應應 工工 程程 The Dispersion Model Choice of ModelsModels are useful for representing flow in real vessels, for scale up, and for diagnosing poor flow. We have different kinds of models depending on whether flow is close to plug, mixed, or somewhere in between. and deal primarily with small devi

2、ation from plug flow. There are two models for this: the dispersion model and the tanks-in-series model. Use the one that is comfortable for you. 化化 學學 反反 應應 工工 程程They are roughly equivalent. These models apply to turbulent flow in pipes, laminar flow in very long tubes, flow in packed beds, shaft k

3、ilns, long channels, screw conveyers, etc.For laminar flow in short tubes or laminar flow of viscous materials these models may not apply, and it may be that the parabolic velocity profile is the main cause of deviation from plug flow. We treat this situation, called the pure convection model, in .I

4、f you are unsure which model to use go to the chart at the beginning of . It will tell you which model should be used to represent your setup.化化 學學 反反 應應 工工 程程13.1 AXIAL DISPERSION Suppose an ideal pulse of tracer is introduced into the fluid entering a vessel. The pulse spreads as it passes through

5、 the vessel, and to characterize the spreading according to this model (see Fig. 13.1), we assume a diffusion-like process superimposed(添加) on plug flow. We call this dispersion or longitudinal (縱向的, 軸向的) dispersion to distinguish it from molecular diffusion. The dispersion coefficient D (m2/s) repr

6、esents this spreading process. Thus large D means rapid spreading of the tracer curve small D means slow spreading D = 0 means no spreading, hence plug flow化化 學學 反反 應應 工工 程程Figure 13.1 The spreading of tracer according to the dispersion model. 小小 結結 軸向分（擴散）模型軸向分（擴散）模型基本假定：基本假定：對于偏離了理想平推流的管式反應器，假設在主對

7、于偏離了理想平推流的管式反應器，假設在主流體的流動方向上疊加了一個反向的軸向分（擴）流體的流動方向上疊加了一個反向的軸向分（擴）散效應，因而帶來了流體粒子的返混。散效應，因而帶來了流體粒子的返混。特點：特點：徑向上，濃度分布均一；徑向上，濃度分布均一；軸向上，流體的流速軸向上，流體的流速u 和分散系數和分散系數D 均為恒定值均為恒定值化化 學學 反反 應應 工工 程程Also uLD is the dimensionless group characterizing the spread in the whole vessel. We evaluate D or D/uL by recordi

8、ng the shape of the tracer curve as it passes the exit of the vessel. In particular, we measure curve theof spread theof measure aor variance,exit by the passes curve eor when th passage, of mean time2t化化 學學 反反 應應 工工 程程iiiiitCtCtCdttCdtt00The variance(方差) is defined as 20020022tCdtCdttCdtCdttt(2) Th

9、ese measures, and and , are directly linked by theory to D and D/uL. The mean time, for continuous or discrete(離散的) data, is defined as t2(1) 化化 學學 反反 應應 工工 程程or in discrete form 2222ttCtCttCtCttiiiiiiiiii(3) The variance represents the square of the spread of the distribution as it passes the vesse

10、l exit and has units of (time)2. It is particularly useful for matching experimental curves to one of a family of theoretical curves. Figure 13.2 illustrates these terms. 化化 學學 反反 應應 工工 程程Figure 13.2 化化 學學 反反 應應 工工 程程Consider plug flow of a fluid, on top of which is superimposed some degree of backm

11、ixing, the magnitude of which is independent of position within the vessel. This condition implies that there exist no stagnant pockets and no gross(明顯的) bypassing or short-circuiting of fluid in the vessel. This is called the dispersed plug flow model, or simply the dispersion model. Figure 13.3 sh

12、ows the conditions visualized(形象化地). Note that with varying intensities of this model should range from plug flow at one extreme to mixed flow at the other. As a result the reactor volume for this model will lie between those calculated for plug and mixed flow. 化化 學學 反反 應應 工工 程程Figure 13.3 Represent

13、ation of the dispersion (dispersed plug flow) model. 化化 學學 反反 應應 工工 程程Since the mixing process involves a shuffling(攪亂) or redistribution of material either by slippage(滑動, 錯動) or eddies(旋渦), and since this is repeated many, many times during the flow of fluid through the vessel we can consider thes

14、e disturbances(干擾, 騷動) to be statistical(統計學的) in nature, somewhat as in molecular diffusion. For molecular diffusion in the x-direction the governing differential equation is given by Ficks law:2 CCtx (4)2D化化 學學 反反 應應 工工 程程where D, the coefficient of molecular diffusion, is a parameter which unique

15、ly(唯一地) characterizes the process. In an analogous(類似的) manner we may consider all the contributions to intermixing of fluid in the x-direction to be described by a similar form of expression, or 2 CCtx2D (5)where the parameter D, which we call the longitudinal or axial dispersion coefficient, uniqu

16、ely characterizes the degree of backmixing during flow. 化化 學學 反反 應應 工工 程程We use the terms longitudinal (縱向的) and axial (軸向的) because we wish to distinguish mixing in the direction of flow from mixing in the lateral (橫向的)or radial(徑向的) direction, which is not our primary concern. These two quantities

17、 may be quite different in magnitude. For example, in streamline flow of fluids through pipes, axial mixing is mainly due to fluid velocity gradients, whereas radial mixing is due to molecular diffusion alone. 軸向分（擴散）模型的數學模型軸向分（擴散）模型的數學模型Cd xLuc0u()CC+dxxuuruA CDrCA(C+d x)xxrCuA (C+dx)xDrCAxruA CD()

18、rCAC+dxxx()rCuA C+dxxDrCAxrCA dxt+ += =+ + +化化 學學 反反 應應 工工 程程In dimensionless form where z = x / L and , the basic differential equation representing this dispersion model becomes Ltutt/22 CCCuLzzD (6)where the dimensionless group , called thevessel dispersion number (分散準數), is the parameter that me

19、asures the extent of axial dispersion. Thus uLD化化 學學 反反 應應 工工 程程flow mixed hence ,dispersion large flow plug hence ,dispersion negligible 0uLuLDDThis model usually represents quite satisfactorily flow that deviates not too greatly from plug flow, thus real packed beds and tubes (long ones if flow is

20、 streamline).化化 學學 反反 應應 工工 程程Fitting the Dispersion Model for Small Extents of Dispersion, D/uL 0.01 If we impose an idealized pulse onto the flowing fluid then dispersion modifies this pulse as shown in Fig.13.1. For small extents of dispersion (if D/uL is small) the spreading tracer curve does no

21、t significantly change in shape as it passes the measuring point (during the time it is being measured). Under these conditions the solution to Eq.6 is not difficult and gives the symmetrical(對稱的) curve of Eq.7 shown in Figs.13.1 and 13.4.This represents family of gaussian curves, also called error

22、or normal curves.化化 學學 反反 應應 工工 程程211exp 4/2/CuLuL(7)DDThe equations representing this family are 223222231exp4/exp44/ or 12 or 2tuLuLLutuLLuVLtvuLtuLuEE1EED4D /EDDDD(8)mean of E curve化化 學學 反反 應應 工工 程程Figure 13.4 Relationship between D/uL and the dimensionless E curve for small extents of dispersion

23、, Eq.7. 化化 學學 反反 應應 工工 程程Note that D/uL is the one parameter of this curve. Figure 13.4 shows a number of ways to evaluate this parameter from an experimental curve: by calculating its variance, by measuring its maximum height or its width at the point of inflection(拐點), or by finding that width whi

24、ch includes 68% of the area.Also note how the tracer spreads as it moves down the vessel. From the variance expression of Eq.8 we find that L 22curve tracerofwidth L or 化化 學學 反反 應應 工工 程程Fortunately, for small extents of dispersion numerous simplifications and approximations in the analysis of tracer

25、 curves are possible. First, the shape of the tracer curve is insensitive to the boundary condition imposed on the vessel, whether closed of open (see above Eq.11.1). So for both closed and open vessels Cpulse = E and Cstep = F . For a series of vessels the and of the individual vessels are additive

26、, thus, referring to Fig.13.5 we have t2(9) bababaoveralluLuLvVvVttt化化 學學 反反 應應 工工 程程Figure 13.5 Illustration of additivity of means and of variances of the E curves of vessels a,b,n.2223322 overallababLLuuDD (10)and 化化 學學 反反 應應 工工 程程The additivity(可加性) of times is expected, but the additivity of va

27、riance is not generally expected. This is a useful property since it allows us to subtract for the distortion of the measured curve caused by input lines, long measuring leads, etc.This additivity property of variance also allows us to treat any one-shot tracer input, no matter what its shape, and t

28、o extract from it the variance of the E curve of the vessel. So, on referring to Fig.13.6, if we write for a one-shot input222outin (11)化化 學學 反反 應應 工工 程程Figure 13.6 Increase in variance is the same in both cases, or . 2222outin 化化 學學 反反 應應 工工 程程Aris (1959) has shown, for small extents of dispersion,

29、 that 222outin22outin2 uLtttD (12)Thus no matter what the shape of the input curve, the D/uL value for the vessel can be found.The goodness of fit for this simple treatment can only be evaluated by comparison with the more exact but much more complex solutions. From such a comparison we find that th

30、e maximum error in estimate of D/uL is given byerror5% when 0.01uLD化化 學學 反反 應應 工工 程程Large Deviation from Plug Flow, 0 0. .0 01 1uLDHere the pulse response is broad and it passes the measurement point slowly enough that it changes shape-it spreads-as it is being measured. This gives a nonsymmetrical

31、E curve.An additional complication(并發癥) enters the picture for large D/uL: What happens right at the entrance and exit of the vessel strongly affects the shape of the tracer curve as well as the relationship between the parameters of the curve and D/uL.化化 學學 反反 應應 工工 程程Let us consider two types of b

32、oundary conditions: either the flow is undisturbed(未受干擾) as it passes the entrance and exit boundaries (we call this the open b.c.), or you have plug flow outside the vessel up to the boundaries (we call this the closed b.c.). This leads to four combinations of boundary conditions, closed-closed, op

33、en-open, and mixed. Figure 13.7 illustrates the closed and open extremes, whose RTD curves are designated as Ecc and Eoo.化化 學學 反反 應應 工工 程程Figure 13.7 Various boundary conditions used with the dispersion model. Plug flow,D = 0Same D,everywhere Closed vesselChange in flow pattern at boundaries Open ve

34、sselUndistrubed flow at boundaries化化 學學 反反 應應 工工 程程Now only one boundary condition gives a tracer curve which is identical to the E function and which fits all the mathematics of , and that is the closed vessel. For all other boundary conditions you do not get a proper RTD. In all cases you can eval

35、uate D/uL from the parameters of the tracer curves: however, each curve has its own mathematics. Let us look at the tracer curves for closed and for the open boundary conditions. 化化 學學 反反 應應 工工 程程Closed Vessel. Here an analytic expression for the E curve is not available. However, we can construct t

36、he curve by numerical methods, see Fig.13.8, or evaluate its mean and variance exactly, as was first done by van der Laan(1958). Thus 222/2V or 1V221uLtt vttvtetuLuLEEEEDDD(13)化化 學學 反反 應應 工工 程程Figure 13.8 Tracer response curves for closed vessels and large deviation from plug flow.化化 學學 反反 應應 工工 程程O

37、pen Vessel. This represents a convenient and commonly used experimental device, a section of long pipe (see Fig. 13.9). It also happens to be the only physical situation (besides small D/uL) where the analytical expression for the E curve is not too complex. The results are given by the response cur

38、ves shown in Fig. 13.10, and by the following equations, first derived by Levenspiel and Smith (1957). 化化 學學 反反 應應 工工 程程Figure 13.9 The openopen vessel boundary condition.化化 學學 反反 應應 工工 程程22,22,2,21(1)exp4(D /)4 (D /)()exp4D4DDD12 or 12DD28tE ooE, ooE oot oooouLuLuLuttttVttuLvuLtuLuL E EE E, oo, oo(

39、14)(15)化化 學學 反反 應應 工工 程程Figure 13.10 Tracer response curves for “open” vessels having large deviations from plug flow.化化 學學 反反 應應 工工 程程CommentFor small D/uL the curve for the different boundary conditions all approach the “small deviation” curve of Eq. 8. At large D/uL the curves differ more and mor

40、e from each other.(b) To evaluate D/uL either match the measured tracer curve or the measured to theory. Match is simplest, through not necessarily best; however, it is often used. But be sure to use the right boundary conditions.22化化 學學 反反 應應 工工 程程(c) If the flow deviates greatly from plug (D/uL la

41、rge) chances are that the real vessel doesnt meet the assumption of the model (a lot of independent random fluctuations). Here it becomes questionable whether the model should even be used. I hesitate when D/uL 1。 (d) You must always ask whether the model should be used. You can always match values,

42、 but if the shape looks wrong, as shown in the accompanying sketches, dont use this model, use some other model.2化化 學學 反反 應應 工工 程程 (e) For large D/uL the literature is profuse and conflicting, primarily because of the unstated and unclear assumptions about what is happening at the vessel boundaries.

43、 The treatment of end additivity of variances is questionable. Because of all this we should be very careful in using the dispersion model where backmixing is large, particularly if the system is not closed.(f) We will not discuss the equations and curves for the open-closed or closed-open boundary

44、conditions. These can be found in Levenspiel (1996).化化 學學 反反 應應 工工 程程 化化 學學 反反 應應 工工 程程EXAMPLE 13.1 D/uL FROM A Cpulse CURVE On the assumption that the closed vessel of Example 11.1, , is well represented by the dispersion model, calculate the vessel dispersion number D/uL. The C versus t tracer res

45、ponse of this vessel is t, min 0 5 10 15 20 25 30 35 Cpulse, gm/liter 0 3 5 5 4 2 1 0 化化 學學 反反 應應 工工 程程SOLUTIONSince the C curve for this vessel is broad and unsymmetrical, see Fig. 11.E1, let us guess that dispersion is too large to allow use of the simplification leading to Fig. 13.4. We thus star

46、t with the variance matching procedure of Eq. 18. The mean and variance of a continuous distribution measured at a finite number of equidistant locations is given by Eqs. 3 and 4 as iiitCtC and 22222iiiiiiiiit Ct Ct CtCCC化化 學學 反反 應應 工工 程程Using the original tracer concentration-time date, we find 223

47、5542 120(5 3)(10 5)(30 1)300 min(25 3)(100 5)(900 1)5450 miniiiiiCtCt C Therefore 222min5 .4720300205450min1520300t and 211. 0)15(5 .472222t化化 學學 反反 應應 工工 程程Now for a closed vessel Eq. 13 relates the variance to D/uL. Thus 22/DD0.211221uLeuLuLDIgnoring the second term on the right, we have as a firs

48、t approximation 0 106D.uL化化 學學 反反 應應 工工 程程Correcting for the term ignored we find by trial and error that D0.120uLOur original guess was correct: This value of D/uL is much beyond the limit where the simple gaussian approximation should be used. 化化 學學 反反 應應 工工 程程EXAMPLE 13.3 D/uL FROM A ONE-SHOT INP

49、UTFind the vessel dispersion number in a fixed-bed reactor packed with 0.625-cm catalyst pellets. For this purpose tracer experiments are run in equipment shown in Fig. E13.3.The catalyst is laid down in a haphazard manner above a screen to a height of 120 cm, and fluid flows downward through this p

50、acking. A sloppy pulse of radioactive tracer is injected directly above the bed, and output signals are recorded by Geiger counters at two levels in the bed 90 cm apart. 化化 學學 反反 應應 工工 程程 Figure E13.3 化化 學學 反反 應應 工工 程程The following data apply to a specific experimental run. Bed voidage = 0.4, superf

51、icial velocity of fluid (based on an empty tube) = 1.2 cm /sec, and variances of output signals are found to be22221239sec64sec ,andFind D/uL.化化 學學 反反 應應 工工 程程SOLUTION Bichoff and Levenspiel (1962) have shown that as the measurements are taken at least two or three particle diameters into the bed, t

52、hen the open vessel boundary conditions hold closely. This is the case here because the measurements are made 15 cm into the bed. As a result this experiment corresponds to a one-shot input to an open vessel for which Eq. 12 holds. Thus 221222sec 253964化化 學學 反反 應應 工工 程程or in dimensionless form 361cm

53、)(0.4) (90cm/sec 2 . 1)sec 25(22222Vvfrom which the dispersion number is 2D10.014272uL化化 學學 反反 應應 工工 程程13.3 CHEMICAL REACTION AND DISPERSIONOur discussion has led to the measure of dispersion by a dimensionless group D/uL. Let us now see how this affects conversion in reactors.Consider a steady-flow

54、 chemical reactor of length L through which fluid is flowing at a constant velocity u, and in which material is mixing axially with a dispersion coefficient D. Let an nth-order reaction be occurring. nAA- products.AkCr By referring to an elementary section of reactor as shown in Fig. 13.18, the basi

55、c material balance for any reaction component化化 學學 反反 應應 工工 程程Figure 13.18 Variables for a closed vessel in which reaction and dispersion are occurring. 化化 學學 反反 應應 工工 程程inputoutputdisappearance by reactionaccumulationbecomes for component A, at steady state. 0onaccumulatireactionby ncedisappearain)

56、(outin)(outdispersion axialflowbulk (17) The individual terms (in model A/time) are as follow:A, flowcross-sectionalmoles Aentering by bulk flowvelocity areavolume , mol/slCuS(4.1)化化 學學 反反 應應 工工 程程 leaving by bulk flow = A, llCuSentering by axial dispersion = AAldNdCSdtdl D Dleaving by axial dispers

57、ion = lldldCSdtdNAAD Ddisappearance by reaction = mol/s , )()(AAlSrVr化化 學學 反反 應應 工工 程程Note that the difference between this material balance and that for the ideal plug flow reactors of is the inclusion of the two dispersion terms, because material enters and leaves the differential section not only

58、 by bulk flow but by dispersion as well. Enter all these terms into Eq. 17 and dividing by gives S lAAA, A, A( )()0lllllldCdCCCdldlurll D D化化 學學 反反 應應 工工 程程Now the basic limiting process of calculus(微積分) states that for any quantity Q which is a smooth continuous function of l 2121021limlimlllQQQdQl

59、lldl So taking limits as 0lwe obtain 2AA2uD0nAdCd CkCdldl(18a) In dimensionless form where z = l/ Land vVuLt/ , this expression becomes化化 學學 反反 應應 工工 程程or in terms of fractional conversion 21AAA0A2D(1)0nnd XdXk CXuL dzdz (18c) This expression shows that the fractional conversion of reaction of react

60、ant A in its passage through the reactor is governed by three dimensionless groups: a reaction rate group , the dispersion group D/uL, and the reaction order n. 1A0nk C2AAA2D0nd CdCk CuL dzdz(18b) 化化 學學 反反 應應 工工 程程First-Order Reaction. Equation 18 has been solved analytically by Wehner and Wihelm (1