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1、化化 学学 反反 应应 工工 程程Chapter 13 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.Chapter 13Chapter 13 and 1414
2、deal primarily with small deviation 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 学学 反反 应应 工工 程程They are roughly equivalent.These models apply to turbulent flow in pipes,laminar flow in very long tubes,flow
3、in packed beds,shaft kilns,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 pro the main cause of deviation from plug flow.We treat this situation,called the pure convection mod
4、el,in Chapter 15Chapter 15.If you are unsure which model to use go to the chart at the beginning of Chapter 15Chapter 15.It will tell you which model should be used to represent your setup.化化 学学 反反 应应 工工 程程13.1 AXIAL DISPERSION Suppose an ideal pulse of tracer is introduced into the fluid entering a
5、 vessel.The pulse spreads as it passes through 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(纵向的,轴向的)dispersion to distinguish it from molecular diffusion.The
6、dispersion coefficient D(m2/s)represents 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 学学 反反 应应 工工 程程Figure 13.1 The spreading of tracer according to the dispersion model.化化 学学 反反 应应 工工 程程Also is th
7、e dimensionless group characterizing the spread in the whole vessel.We evaluate D or D/uL by recording the shape of the tracer curve as it passes the exit of the vessel.In particular,we measure 化化 学学 反反 应应 工工 程程The variance(方差)is defined as(2)These measures,and and ,are directly linked by theory to
8、D and D/uL.The mean time,for continuous or discrete(离散的)data,is defined as(1)化化 学学 反反 应应 工工 程程or in discrete form(3)The variance represents the square of the spread of the distribution as it passes the vessel exit and has units of(time)2.It is particularly useful for matching experimental curves to
9、one of a family of theoretical curves.Figure 13.2 illustrates these terms.化化 学学 反反 应应 工工 程程Consider plug flow of a fluid,on top of which is superimposed some degree of backmixing,the magnitude of which is independent of position within the vessel.This condition implies that there exist no stagnant p
10、ockets and no gross(明显的)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 shows the conditions visualized(形象化地).Note that with varying intensities of this model should range from plug flow at one extreme to mi
11、xed 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 学学 反反 应应 工工 程程Figure 13.3 Representation of the dispersion(dispersed plug flow)model.化化 学学 反反 应应 工工 程程where D,the coefficient of molecular diffusion,is a parameter which u
12、niquely(唯一地)characterizes the process.In an analogous(类似的)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 where the parameter D,which we call the longitudinal or axial dispersion coefficient,uniquely character
13、izes the degree of backmixing during flow.轴向分（扩散）模型的数学模型轴向分（扩散）模型的数学模型Cd xLuc0uuu+=+化化 学学 反反 应应 工工 程程In dimensionless form where z=x/L and ,the basic differential equation representing this dispersion model becomes where the dimensionless group ,called thevessel dispersion number(分散准数),is the parame
14、ter that measures the extent of axial dispersion.Thus 化化 学学 反反 应应 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 sma
15、ll)the spreading tracer curve does not 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(对称的)curve of Eq.7 shown in Figs.13.1 and 13.4.This represents family of g
16、aussian curves,also called error or normal curves.化化 学学 反反 应应 工工 程程The equations representing this family are(8)mean of E curve化化 学学 反反 应应 工工 程程Figure 13.4 Relationship between D/uL and the dimensionless E curve for small extents of dispersion,Eq.7.化化 学学 反反 应应 工工 程程Note that D/uL is the one paramete
17、r 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(拐点),or by finding that width which includes 68%of the area.Also note how the tracer spreads as
18、it moves down the vessel.From the variance expression of Eq.8 we find that L L or 化化 学学 反反 应应 工工 程程Fortunately,for small extents of dispersion numerous simplifications and approximations in the analysis of tracer curves are possible.First,the shape of the tracer curve is insensitive to the boundary
19、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,thus,referring to Fig.13.5 we have 化化 学学 反反 应应 工工 程程The additivity(可加性)of times is expected,but th
20、e additivity of variance 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 it
21、s shape,and to 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 input化化 学学 反反 应应 工工 程程Figure 13.6 Increase in variance is the same in both cases,or .化化 学学 反反 应应 工工 程程Aris(1959)has shown,for small extents of dispersion,that Thus no matte
22、r 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 the maximum error in estimate of D/uL is given by化化
23、 学学 反反 应应 工工 程程Let us consider two types of boundary conditions:either the flow is undisturbed(未受干扰)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
24、 boundary conditions,closed-closed,open-open,and mixed.Figure 13.7 illustrates the closed and open extremes,whose RTD curves are designated as Ecc and Eoo.化化 学学 反反 应应 工工 程程Figure 13.7 Various boundary conditions used with the dispersion model.Plug flow,D=0Same D,everywhere Closed vesselChange in flo
25、w pattern at boundaries Open vesselUndistrubed flow at boundaries化化 学学 反反 应应 工工 程程Closed Vessel.Here an analytic expression for the E curve is not available.However,we can construct the curve by numerical methods,see Fig.13.8,or evaluate its mean and variance exactly,as was first done by van der Laa
26、n(1958).Thus(13)化化 学学 反反 应应 工工 程程Figure 13.8 Tracer response curves for closed vessels and large deviation from plug flow.化化 学学 反反 应应 工工 程程Open 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 situa
27、tion(besides small D/uL)where the analytical expression for the E curve is not too complex.The results are given by the response curves shown in Fig.13.10,and by the following equations,first derived by Levenspiel and Smith(1957).化化 学学 反反 应应 工工 程程Figure 13.9 The openopen vessel boundary condition.化化
28、 学学 反反 应应 工工 程程(14)(15)化化 学学 反反 应应 工工 程程Figure 13.10 Tracer response curves for“open”vessels having large deviations from plug flow.化化 学学 反反 应应 工工 程程Comment(a)For 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
29、 more and more 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.化化 学学 反反 应应 工工 程程(c)If the flow deviates greatly from plug(D/uL
30、 large)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,but
31、if the shape looks wrong,as shown in the accompanying sketches,dont use this model,use some other model.化化 学学 反反 应应 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.The treatmen
32、t 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 conditions.These
33、 can be found in Levenspiel(1996).化化 学学 反反 应应 工工 程程 化化 学学 反反 应应 工工 程程EXAMPLE 13.1 D/uL FROM A Cpulse CURVE On the assumption that the closed vessel of Example 11.1,Chapter 11Chapter 11,is well represented by the dispersion model,calculate the vessel dispersion number D/uL.The C versus t tracer respo
34、nse 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 学学 反反 应应 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 start with th
35、e 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 and 化化 学学 反反 应应 工工 程程Using the original tracer concentration-time date,we find Therefore and 化化 学学 反反 应应 工工 程程Now for a closed ves
36、sel Eq.13 relates the variance to D/uL.Thus Ignoring the second term on the right,we have as a first approximation 化化 学学 反反 应应 工工 程程Correcting for the term ignored we find by trial and error that Our original guess was correct:This value of D/uL is much beyond the limit where the simple gaussian app
37、roximation should be used.化化 学学 反反 应应 工工 程程EXAMPLE 13.3 D/uL FROM A ONE-SHOT INPUTFind 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
38、 above a screen to a height of 120 cm,and fluid flows downward through this packing.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 学学 反反 应应 工工 程程 Figure E13.3 化化 学学 反反 应应 工工 程程The foll
39、owing data apply to a specific experimental run.Bed voidage=0.4,superficial velocity of fluid(based on an empty tube)=1.2 cm/sec,and variances of output signals are found to beFind D/uL.化化 学学 反反 应应 工工 程程SOLUTION Bichoff and Levenspiel(1962)have shown that as the measurements are taken at least two o
40、r three particle diameters into the bed,then 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 化化 学学 反反 应应 工工 程程or in dim
41、ensionless form from which the dispersion number is 化化 学学 反反 应应 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 chemical reactor of length L throu
42、gh 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.By referring to an elementary section of reactor as shown in Fig.13.18,the basic material balance for any reaction component化化 学学 反反 应应 工工
43、 程程Figure 13.18 Variables for a closed vessel in which reaction and dispersion are occurring.化化 学学 反反 应应 工工 程程becomes for component A,at steady state.(17)The individual terms(in model A/time)are as follow:(4.1)化化 学学 反反 应应 工工 程程 leaving by bulk flow=entering by axial dispersion=leaving by axial dispe
44、rsion=disappearance by reaction=化化 学学 反反 应应 工工 程程Note that the difference between this material balance and that for the ideal plug flow reactors of Chapter 5Chapter 5 is the inclusion of the two dispersion terms,because material enters and leaves the differential section not only by bulk flow but b
45、y dispersion as well.Enter all these terms into Eq.17 and dividing by gives 化化 学学 反反 应应 工工 程程Now the basic limiting process of calculus(微积分)states that for any quantity Q which is a smooth continuous function of l So taking limits as we obtain(18a)In dimensionless form where and,this expression beco
46、mes化化 学学 反反 应应 工工 程程or in terms of fractional conversion (18c)This expression shows that the fractional conversion of reaction of reactant 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.(18b)化化
47、学学 反反 应应 工工 程程First-Order Reaction.Equation 18 has been solved analytically by Wehner and Wihelm(1956)for first-order reactions.For vessel with any kind of entrance and exit conditions the solution is(19)化化 学学 反反 应应 工工 程程Figure 13.19 is a graphical representation of these results in useful form,prep
48、ared by combining Eq.19 and Eq.5.17,and allows comparison of reactor sizes for plug and dispersed plug flow.For small deviations from plug flow D/uL becomes small,the E curve approaches gaussian;hence,on expanding the exponentials and dropping higher order terms Eq.19 reduces to 化化 学学 反反 应应 工工 程程Fig
49、ure 13.19 Comparison of real and plug flow reactors for the first-order reaction A products,assuming negligible expansion;from Levenspiel and Bischoff(1959,1961).化化 学学 反反 应应 工工 程程Equation 20 with Eq.5.17 compares the performance of real reactors which are close to plug flow with plug flow reactors.T
50、hus the size ratio needed for identical conversion is given by(22)while the exit concentration ratio for identical reactor size is given by(23)化化 学学 反反 应应 工工 程程It is used in a manner similar to the charts for first-order reactions.To estimate reactor performance for reactions of order different from