化学反应工程Chapter 13.ppt

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 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 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 profile is the main cause of deviation from plug flow.We treat this situation,called the pure convection model,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 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 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.小小 结结 轴向分（扩散）模型轴向分（扩散）模型基本假定：基本假定：对于偏离了理想平推流的管式反应器，假设在主流对于偏离了理想平推流的管式反应器，假设在主流体的流动方向上叠加了一个反向的轴向分（扩）散体的流动方向上叠加了一个反向的轴向分（扩）散效应，因而带来了流体粒子的返混。效应，因而带来了流体粒子的返混。特点：特点：径向上，浓度分布均一；径向上，浓度分布均一；轴向上，流体的流速轴向上，流体的流速u 和分散系数和分散系数D 均为恒定值均为恒定值化化 学学 反反 应应 工工 程程Also is the 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 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 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Figure 13.2 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 pockets 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 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 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Since the mixing process involves a shuffling(搅乱)or redistribution of material either by slippage(滑动,错动)or eddies(旋涡),and since this is repeated many,many times during the flow of fluid through the vessel we can consider these disturbances(干扰,骚动)to be statistical(统计学的)in nature,somewhat as in molecular diffusion.For molecular diffusion in the x-direction the governing differential equation is given by Ficks law:化化 学学 反反 应应 工工 程程where D,the coefficient of molecular diffusion,is a parameter which uniquely(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 characterizes the degree of backmixing during flow.化化 学学 反反 应应 工工 程程We use the terms longitudinal(纵向的)and axial(轴向的)because we wish to distinguish mixing in the direction of flow from mixing in the lateral(横向的)or radial(径向的)direction,which is not our primary concern.These two quantities 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.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 parameter that measures the extent of axial dispersion.Thus 化化 学学 反反 应应 工工 程程This 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 streamline).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 small)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 gaussian 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 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(拐点),or by finding that width which 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 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 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Figure 13.5 Illustration of additivity of means and of variances of the E curves of vessels a,b,n.and 化化 学学 反反 应应 工工 程程The additivity(可加性)of times is expected,but the 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 its 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 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 the maximum error in estimate of D/uL is given by化化 学学 反反 应应 工工 程程Large Deviation from Plug Flow,Here 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 E curve.An additional complication(并发症)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 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 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 flow pattern at boundaries Open vesselUndistrubed flow at boundaries化化 学学 反反 应应 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 Chapter 11Chapter 11,and that is the closed vessel.For all other boundary conditions you do not get a proper RTD.In all cases you can evaluate 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 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 Laan(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 situation(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 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 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 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 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 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 conditions.These 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 response 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 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 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 vessel 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 approximation 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 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 following 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 or 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 dimensionless 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 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.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 程程Figure 13.18 Variables for a closed v