化学反应工程Chapter 13.ppt
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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 profile is the main cause of deviation from plug flow.We treat this situation,called the pure convect
4、ion 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 ent
5、ering 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 diffusi
6、on.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
7、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.W
8、e 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 defin
9、ed 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 t
10、erms.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 sho
11、rt-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 re
12、actor 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 ed
13、dies(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 Fick
14、s 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 p
15、arameter 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 radi
16、al(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
17、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
18、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 f
19、lowing 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
20、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 an
21、d 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
22、 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 approximatio
23、ns 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 ves
24、sels 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 usef
25、ul 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
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