流体力学混合在单螺杆挤出机-

1、毕业设计（论文）外文资料翻译 3 中文3685字 附件一：外文资料翻译译文 流体力学混合在单螺杆挤出机 Ravlndran Chella 和 Julio M. Ottlno* Massachusetts 州Amherst， Massachusetts 大学，化学工程系 01003卷 矩形空腔流 图5为一个序列的一个接口，已进行二维矩形腔流拉伸步骤，在长度增长的 界面，L（t）伴随着条纹厚度减少而减少，它被定义为相邻的接口之间的平均垂 直距离，因此 L（t） ?s（t）：常数，Biggs和Middleman（1974b）使用一个简化的标记 和细胞（MAC ）技术（Harlow和Amsden, 1

2、970）来追踪该接口的位置。然而，他 们只考虑水平接口以及他们认为小拉伸比率的情况。 图5对两个相邻的垂直拉伸的流体层之间的接口在二维矩形腔流的步骤顺序示意图 在一个典型的数值模拟中，变形及连续线拉伸（或表面）是使用有限数量的 粒子模拟。对于幅度的一个或两个数量级的相对伸展的线变形，包含所述线路分 离的单个颗粒，定义并不清晰，对每一个粒子的初始浓度（每单位长度的粒子数 量）会有一段时间在这几乎不可能重建。（如果粒子流混乱，这个问题会急剧变 得严重。）当进行线路中的示踪粒子模拟时，相同的问题会出现在实验工作中， 另一方面，该线路不能过于集中，因为它不是被动接口，如果线路是可溶性示踪 剂模拟，问题

3、将会扩散。一般来说，这似乎很难遵循传统的跟踪方法或实验的102 或相对较高的拉伸比拉伸，数值误差可能会使它不可能实现可逆性预期规则运动 （Khakhar等人,1984）,界面的长度变化的关系可以用有限的材料进行拉伸计算 L(t) l(0)(C：MM?)1/2 |dx (11) 该组包含该接口的差分线元件的初始取向的需要被指定，对于垂直界面（垂 直于移动板块）M?=（O,I）和水平界面（平行移动板块）M? = （1,0），以及所有的 行元素，由于它是在初始配置，所以用公式11计算是相对简单的。这里使用的 方法可以进行计算任意大的拉伸比，为了能够运用公式11，一种光腔流场的数 学描述是有必要的，在

4、这种情况下，参与关于瞬态问题利用稳态速度分布的误差 比较小，例如稳态操作条件下迅速达到正常操作条件（Bigg和Middleman,1974b Erwin和Moktharian，1981），由公式1可以得出这一流程最简单的说明。 亠 145 J* 药ElIWUJllftdEOD- 图6比较简化为矩形空腔流获得使用SFT（-）和公式12得到 W / H = 15 然而，使用公式1和公式11结合以确定L（t）的值，在方向和变形经过由材 料元件移动到其互补的位置变化假设是必要的。但是，计算表明，混合实现假定 取向的变化是非常敏感的方向，因此需要开发一个流场的数学描述，并不需要这 样任意假设。在n -

5、s方程的数值解这个流场（公式I,公式2）是可能的，它似乎 并不需要计算拉伸比率或更高的基于当前的跟踪技术，此外，一个半解析处理允 许对不同参数的影响更易于可理解。因此，在附录中，Kantorovich Galerkin方 21 2 2 2 法（Ka ntorovich and Krylov, 1964） 被用来获得一个近似的解析解的稳态，蠕 变流动腔流方程。根据公式 A. 8，A. 15和A. 22 2 + 5 )/(5|sinh( qx2)sin( o2x2)+B1 5sinh( a1x2)cos(o2x2) + qcosh( qx2 )sin( 5X2) 12a V 2=42X1(1-X1

6、)(1-2X1)A2cosh(c1x2)sin( o2x2)+B1sinh( cfX2)sinh( o2x2)+B2 5 sinh( o,x2)cos(o2x2)(12b) 在公式A.22中A , B1,和B2的作用仅仅被定义为腔的长宽比。 虽然这些方程满足边界条件下速度的平均移动量，但仅在使用它们计算流线 时相对准确，对于复杂的纵横比，与那些得到更准确的数值方法（Pan和Acrivos, 毕业设计（论文）外文资料翻译 1967）;以及坐标的最大和最小坐标重合几乎完全与SFT的相应互补值的位置 （图6），这些方程就不适用了。 通过最初垂直接口，使用公式11和公式12,计算相对拉伸为两个不同方

7、面比率在图7中表示。在特有的循环时间，纵横比对界面的相对拉伸只有很小 的影响。关于单调递增的均值曲线的振荡周期值约等于Vu/H ,振荡周期可以由 图8得出，当拉伸率（=d L（t）/ dt）时，作图的接口特定速率准确显示了相同特 征的振荡，这样的振荡特征需要重新定位（图3b ）。 601 01020 3D 4C 50 图7接口的矩形腔流函数的计算与速度场由公式12得出，最初垂直界面（垂直于移动板） 除以腔成体积相等的通道纵横比的相对拉伸 图8无因次的特定接口的拉伸率在矩形空腔流（W / H = 15，最初垂直界面） 毕业设计（论文）外文资料翻译 9 图9相对拉伸中矩形腔流接口的初始方向的影响（

8、W/H = 15） 对单一的接口长度影响初始方向如图 9所示，该混合程度的初始取向可通 过图9中工件的坐标表现。研究发现，每一种情况下计算出的界面面积的实际值 对初始取向的依赖性非常小，在图8中可以查找原因，一个最初垂直界面区域（垂 直于流线）和一个最初水平界面区域（几乎平行流简化）之间存在巨大差异，极 大实现越来越多的最初垂直界面缩小成为水平对齐。 同时发现混合相对等于甚至 大于位移的初始位置，接口相对的界面区域可以认为是近似关系 图10变化在沿矩形空腔流动的流线行进的差分材料元件的标高（a）和方位（b）所示可以 由公式12计算出速度（F是正常材料平面之间的角度和x1轴） 62 二二 图11

9、比较的界面拉伸矩形空腔流预测了SFT（），预测使用的流场比值12（ ） , （ W/ H =15,最初垂直界面） 也二垫0.625a） L（o）-S（o）L（0） 而在挤出机混合分析中速度计算可以由等式 12得出，这并不包括另外概念上的问题，这与SFT的计算量相比明显增加了，因此，确定流体元件的取向变 化与该流场获得的信息是否可以被纳入使用 SFT结果准确混合计算是有用的， 图10中，表示典型的时间差分线元的取向变化的关系，也表示在图中的上面部 分是元素相应的X2坐标（图10a）,虚线表示最大值和最小值的位置。图表明这 里本身能够快速建立坐标，可以忽略材料元件的初始位置或方向， 因此，当围绕

10、x1轴方向旋转到界面区域时，相关因素旋转接近1800 SFT的研究与假设是边界 旋转近180，通过材料元素混合的预测是否有用，现在得到验证 50 50100150200 2W 1*= Vot/H 图12腔纵横比对拉伸与使用SFT预测矩形方腔流的初始垂直界面的影响 使用SFT计算初始垂直界面的变形与使用图11中12式相比，旋转流体元 素在空间旋转180-两条曲线的数值有较好的一致性，然而，使用SFT得到的 振荡周期是使用公式12得到结果的三倍以上，这与再分配时的值大概一致 （由 Shearer （1973 ）定义，以从腔体的一侧完全置换流体的其它部分所需要的时间 ） 使用SFT计算 27W 8H

11、 (14) 使用这两种不同的流场的初始垂直界面混合预测之间的公式，即使在图11 所示的比较大的拉伸比也适用，这似乎很奇怪，因为SFT预测水平的接口不变 形以及接口的很大一部分是近于水平拉伸比。 然而，对于有限次的界面是从来没 有完全意义上的水平，SFT中预测一个小而有限的拓展与公式 12的结果一致。 预测弱混合的实现使用公式12得到纵横比，采用SFT确认（图12），SFT 中相关要素按180回转，由此可见，通过公式12可以计算出复制的矩形腔流混 合的主要特点，从而，在三维空间中使用挤出流是有利的，因为它相对公式12 简单了。 由于缺少实验数据，实验数据的理论预测比较难，可行性实验数据不完整（例

12、 如Bigg和Middleman,1974b）两者都是因为不确定二维流动是否在实验装置中 实现和并不是大多数据在有利的情况下测得（较大的纵横比）。但是，综合实验 程序正在进行中（Chien, 1984）。 从空腔流得到的结论在挤出机中的应用应谨慎，但应注意的是，流速在整个 挤出机中的横截面的分布可防止确切坐标中的矩形腔和轴向距离沿着所述挤出 机连续时间之间转化，另外，从拓扑的角度来看，如果我们考虑两种流体混合， 说A和B最初在腔流水平层状，然后在侧壁的两条接触线，最后存在于整个运 动，然而，挤出机最初充满，随后 A和B作为参考，相邻的水平层将有明显区 分，没有接触线，当在垂直界面时将会出现类似

13、情况。 图13通过挤出机的流场中的引入相邻的水平层的两种液体混合产生的层状结构的示意图 图14迹线在挤出机通道材料元素 从以上讨论中可以很明显得出，该方法在用于分析三维挤出流量的二维空腔 流混合是可能的没有准确的扩展，但是近似关系的可能性有待继续探讨。 分析单螺杆挤出机的混合 关于在挤出机中混合方法的分析主要与用于所述螺旋环形混合器类似，修改 是必要的，但是，通过公式12算出的速度场，得出一个完全的分析方法是不可 能的，由在流体元件迹线的总数不连续可排除 SFT。 图13是挤出机中通道的两种液体的混合示意图，截面切割和轴向切割显示 由混合作用所产生的层状结构。至于螺旋环形混合器，av和s用作最

14、大混合度 的局部措施，混合参数0,和分部在任一通道截面对应流场的不均匀性中， 并在 进料面上条纹的方向及厚度分布，对于许多应用来说在第一个片刻来描述这些分 布应该是足够了。 力矩轴向配置和横截面的混合参数分布的装置文件可以如下确定（图14所 示）：（1）许多不同材料的平面确定在进料平面，每个对应界面区域中的原料的 位置和方向。选择平面的数量应足够大，从而这个变量计算分布的影响可以忽略 不计；当然，实际数字依赖所取得的结果；在实践中，200-300因素被认为足够 条纹厚度幅度下降三个数量级。注意 RTD被发现对混合参数分布到所选材料的 元素不敏感。（2）公式2用于所述流场的数学描述来计算这些材料

15、每个平面的拉 伸过程。（3）均值和所述混合参数分布的情况由几个轴向位置确定，这种方法是 非常通用的，并且可以被应用到其它混合器中去。 对于连续流动系统的宏观混合效率是由下列关系式确定 （Ottino等人，1981） 1/2 I na v （15） z（D：D ）Z Z 0 V z 在更详细的计算中，检查上混合绑定是很有意义的，通过设置公式15中右侧的 eff（z） = 1获得。通常情况下，定义在上部混合预测值显著高于大多数实际混 合流量（Ottino和Macosko， 1980 ； Ottino， 1983），但考虑到估计模型参数 对混合模型参数的影响，计算绑定上混合模型参数对于SFT特别简单

16、。 22 ：Inav（Z）0 （16） 43（1，）cos 9 L （1）si n（2日）屮 取函数N和含有 ，二的函数以及L/H函数的比例常数的平均值（需要考 虑其上的平均停留时间的影响）。因此，由公式16来看，影响混合的相关参数为 N，二，和L/H。W / H的影响只能间接地通过移动流体单元的垂直坐标变 化。 在此基础上，当上限值增大时，混合的可能性将被增大，然后由公式16得 出，混合参数方程可通过：（1）保持L / H和二不变，增加（2）保持，和不 毕业设计(论文)外文资料翻译 变，增加L/H ； (3)保持L/H和 不变，当九时，增加二；当r Cmin时，减 小二，可由以下方程得出 (

17、17) i (13：2)1/4 21/2 jiji 一 e 6 _ 一 m i n- 4 以及(4)H L/H和二保持不变，减少H。 这些结论与定性实验结果相一致( Maddock，1959； Sheridan, 1975),在 下一节中将使用更完美的分析方法进行测试。 11 毕业设计（论文）外文资料翻译 附件二：外文资料原文 Fluid Mechanics of Mixing in a Single-Screw Extruder Ravindran Chella and Julio M. Ottlno* Departme nt of Chemical Engin eeri ng. Uni v

18、ersity of Massachusetts, Amherst, Massachusetts 0 1003 Rectangular Cavity Flow Figure 5 is a diagram of a seque nee of steps in the stretch ing of an in terface that has bee n subjected to two-dime nsi onal recta ngular cavity flow. The in crease in len gth of the in terface, L(t), is accompanied by

19、 a decrease in the striation thickness, defined as the average perpendicular dista nce betwee n n eighbori ng in terfaces, so that for long times L(t) ?s(t) ： con sta nt. Biggs and Middlema n (1974b) used a simplified Marker-a nd-Cell (MAC) tech ni que (Harlow and Amsde n, 1970) to track the positi

20、on of the in terface. However, they only con sidered the case where the in terface was horiz on tal, and they assumed small stretch ratios. Figure5. Schematic diagram of sequence of steps in the stretching of an interface between two adjacent vertical fluid layers in two dimensional rectangular cavi

21、ty flow. In a typical numerical simulation, the deformation and stretching of continuous lines (or surfaces) is modeled using a finite number of particles. For a relative stretch of one or two orders of magn itude as the line deforms the in dividual particles compris ing the line separate, making th

22、e line less clearly defined. For every initial concentration of particles (number of particles per unit len gth) there will be a time bey ond which it becomesn early impossible to recon struct the line. (This problem is magnified dramatically if the flow is chaotic.) An identical problem arises in e

23、xperimental work when a line is simulated in terms of tracer particles. On the other hand, the line cannot be too concen trated as it would not behave as a passive in terface. If the line is simulated in terms of a soluble tracer, the problem is diffusi on. In gen eral, it appears to be extremely di

24、fficult to follow stretching by conventional tracking techniques and/or experiments for relative stretch ratios 2 of orderlO or higher. Numerical errors might make it impossible to achieve the reversibility expected from regular moti ons (Khakhar et al, 1984). The cha nge in len gth of the in terfac

25、e may be # 毕业设计（论文）外文资料翻译 calculated using the relation for the stretching of a finite material line L（t）二丄（0）（C： MM?）1/2 |dx (11) The set of in itial orie ntati ons of the differe ntial li ne eleme nts compris ing the in terface n eed to be specified. For a vertical in terface (perpe ndicular to th

26、e moving plate) M? = (0,1), and for a horiz on tal in terface (parallel to the movi ng plate) M? = (l,0), for all the line eleme nts. The evaluation of the integral in eq 11 is relatively simple as it is over the initial configuration. The approach used here can be carried out to arbitrarily large s

27、tretch ratios. In order to apply eq 11, a mathematical description of the cavity flow field is needed. The error involved in the use of a steady-state velocity profile for an essentially transient problem is relatively small in this case, as steady-state operati ng con diti ons are rapidly atta ined

28、 un der no rmal operat ing con diti ons (Bigg and Middleman, 1974b; Erwin and Moktharian, 1981). The simplest such description for this flow is that give n by eq 1. Figure6. Comparison of streamlines for rectangular cavity flow obtained using the SFT ()and eq 12 for W/H = 15. However, in using eq 1

29、in conjunction with eq 11 to determine L(t), assumptions are n ecessary regard ing the cha nges in orie ntatio n and deformati on un derg one by a material eleme nt in moving to its complementary location. However, computations indicate that the mixing achieved is extremely sensitive to the assumed

30、change in orientation at the flights. It is therefore desirable to develop a mathematical description of the flow field that does not entail such arbitrary assumptions. While a numerical solution to the Navier-Stokes equations for this flow field (eq A.l, 2)is possible, it does not seem feasible to

31、compute stretch ratios of order or higher based on curre nt track ing tech niq ues. Additi on ally, a semia nalytical treatme nt allows for easier visualizati on of the effect of different parameters. Hence, in the Appendix,the Kantorovich-Galerkin method (Kan torovich and Krylov, 1964) is used to o

32、bta in an approximate an alytical steady-state, creep ing flow solution to the cavity flow equations. From eq A.8, A.15, and A.22 21 2 2 2 2 V 1 Xt (1-XJ A(二2)/；6sin hC1x2)sinG72x2) B12sinhG1x2)cosC2x2) e -1 cosh1x2)si n(二 2x2) 12a V 2 = 42X1-XJ(1-2XJ A2 coshG/lsin(匚2x2) B1 sin h.1x2)sin h(二2x2) B2二

33、2 sin h(Gx2)cos(二2x2)(12b) where A2, Br ,and 耳 are functions only of the cavity aspect ratio, defined in eq A.22. Even though these equati ons satisfy the boun dary con diti on on the velocity at the movingplate only in the mean the streamlines calculated using them are in good agreement, for large

34、aspect ratios, with those obta ined by more accurate nu merical methods (Pa n and Acrivos, 1967); also,the maximum and mi nimum x2 coord in ates of the streamli nes coin cide almost exactly with the locati on of the corresp onding compleme ntary pla nts of the SFT (Figure 6). The relative stretch ex

35、perie need by an in itially vertical in terface, calculated using eq 11 and 12, is shown in Figure 7 for two different aspects ratios. The aspect ratio has only a small in flue nce on the relative stretch of the in terface; the period of oscillatio n of the curves about a monotonically increasing me

36、an value is approximately equal to VU / H , a characteristic recirculation time.The periodic oscillation can be seen more clearly in Figure 8, where the specific rate of stretch ing of the in terface a(= d In L(t)/dt) plotted vs. time shows the same characteristic oscillation. Thus the cavity flow h

37、as weak reorientation (Figure 3b). 0 P Figure7. Relative stretch of interface in rectangular cavity flow as a function of the channel aspect ratio, calculated with the velocity field of eq 12, for an initially vertical interface (perpendicular to the moving plate) Figure8. Nondimensionalized specifi

38、c rate of stretching of interface in rectangular cavity flow (W/H = 15, initially vertical interface). i/LRlCflL 1 H0PIJQNTAL ZEF?叱 E 6二 r 二J H Figure9. Influence of initial orientation on relative stretch of interface in rectangular cavity flow (W/H = 15) The in flue nce of the in itial orie ntati

39、on of the in terface on the no rmalized in terface len gth is shown in Figure 9. The apparent sensitivity of the mixing level to the initial orientation is an artifact of choice of coord in ates in Figure 9. When the actual amount of in terfacial area in each case is calculated, the dependence on th

40、e initial orientation is found to be very small. The reason for this can be see n in Figure 8, where the in itial large differe nces betwee n an in itially vertical in terface (almost perpe ndicular to the flow streamli nes) and an in itially horiz on tal in terface (almost parallel to the flow stre

41、amli nes), n arrows con siderably as more and more of the in itially vertical in terface becomes alig ned horiz on tally. Also, mixi ng is found to be relatively insen sitive even to large displaceme nts in the in itial locati on of the in terface. The relative stretch of the in terface can be appro

42、ximated by the relati on az- FigurelO. Change in elevation (a) and orientation (b) of a differential material element in traveling along a streamline in rectangular cavity flow, calculated with the velocity field of eq 12 (F is the angle between the normal to the material plane and the x1 axis). Fig

43、ure11. Comparison of interface stretching in rectangular cavity flow predicted by the SFT (-) with that predicted using the flow field of eq 12 ( ),(W/H = 15, initially vertical interface). (13) 显二也7 :十0.625百 L(0)ILS(0)L(0) 毕业设计（论文）外文资料翻译 While the use of the velocity field given by eq 12 in the ana

44、lysis of mixing in the extruder in volves no additi onal con ceptual difficulty, the computati onal effort is con siderably in creased compared with the SFT. Hence it is useful to determine whether information obtained with this flow field regarding the change in orientation of the fluid elements ne

45、ar the flights can be incorporated into mixing calculations using the SFT with satisfactory results. Figure 10 shows a typical plot of the cha nge in orie ntati on of a differe ntial li ne eleme nt with time. Also in dicated, in the upper portion of the figure, are the corresp ondingx2 coord in ates

46、 of the eleme nt (Figure 10a), the dashed lines indicating the locations of the maximums and the minimums. The pattern shown here is found to quickly establish itself regardless of the initial location or orientation of the material element. Thus the material elements are rotated through nearly 180

47、while reversing their x1 flow direct ion n ear the boun daries. The usefu In ess of the SFT in making predict ions of mixing with the material elements assumed to rotate through 180 at the boundaries is now exam in ed. Figure12. Influence of cavity aspect ratio on stretching of an initially vertical

48、 interface in rectangular cavity flow as predicted using the SFT. The deformati on of an in itially vertical in terface calculated using the SFT, with the fluid elements rotated through 180at the flights, is compared to that calculated using eq 12, in Figure 11. Numerically the two curves are in goo

49、d agreement; however, the period of the oscillation obtained using the SFT is more than three times that obtained using eq 12 and is approximately in agreement with the value of the redistribution time (defined by Shearer (1973) as the time required to displace fluid completely from one side of the

50、cavity to the other) calculated usi ng the SFT *27 W t redis 二(14) 8 H The agreeme nt betwee n the mixi ng predict ions for an in itially vertical in terface using these two different flow fields, even for the relatively large stretch ratios shown in Figure 11, seems rather surpris ing as the SFT pr

51、edicts no deformati on of a horiz on tal in terface, and a large portion of the in terface is n early horiz on tal at these large stretch ratios. However, for fin ite times the in terface is n ever perfectly horiz on tai, and the SFT predicts a small but fin ite stretch in agreeme nt with the predic

52、t ions of eq 12. The weak depe ndence of the mixing achieved on the cha nnel aspect ratio predicted using eq 12 is con firmed usi ng the SFT (Figure 12). The SFT with the 180 rotatio n of the material eleme nts at the flights is thus see n to duplicate the prin cipal features of mixing in the recta

53、ngular cavity flow as predicted using eq 12, and its use in the an alysis of mixing in the three-dime nsional extruder flow is favored over eq 12 because of its relative simplicity. Comparis on of the theoretical predict ions with experime ntal data is difficult because of a scarcity of experime nta

54、l data. The available experime ntal data are in complete (e.g., Bigg and Middleman, 1974b) both because of the uncertainity about whether two-dimensional flow was achieved in the experimental setup and because not many data were taken under conditions of in terest here (large aspect ratios). However

55、, a comprehe nsive experime ntal program is un derway (Chie n, 1984). The conclusions obtained from the cavity flow should be applied with care to extruders. It should be no ted that the distributi on of velocities across the extruder cross-sect ion preve nts an exact coord in ate tran sformatio n b

56、etwee n successive times in the recta ngular cavity and axial dista nce along the extruder. Also, from a topological point of view, if we consider the mixing of two fluids, say A and B, in itially layered horiz on tally in the cavity flow, the two con tact li nes at the side wall are present through

57、out the entire motion. However, an extruder filled initially with A and subseque ntly fed with A and B as adjace nt horiz on tal layers would have A wett ing and boun daries completely, with no con tact lin es. An an alogous situatio n is obta ined for a vertical in terface. Figure13. Schematic diag

58、ram of lamellar structure generated by the extruder flow field in the mixing of two Figure14. Pathline of material elements in extruder channel. From the above argume nts it is appare nt that no rigorous exte nsion of the approach used to an alyze mixi ng for the two-dime nsio nal cavity flow is pos

59、sible to the three-dime nsi onal extruder flow; however, the possibility of approximate relati ons will be explored. Analysis of Mixing in Single Screw Extruder The approach used to an alyze mixing in the extruder is similar in prin cipal to that used for the helical annular mixer; modifications are

60、 necessary, however, as a completely analytical approach is not possible using the velocity field give n by eq 12 and is precluded for the SFT by the disc on tin uities in the fluid eleme nt pathli nes at the flights. Figure 13 is a diagram of the mixing of two fluids in the extruder cha nn el. Cros