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THE EFFECT OF A SOLID WALL ON THE CLOSURE OF A SPHERICAL CAVITATION POCKET Yu. L. Levkovskii and G. G. Sudakova Inzhenerno-Fizicheskii Zhurnal, Vol. 15, No. 2, pp. 241-247, 1968 UDC 532.528 We examine the effect of a solid wall on the closure of a spherical cavitation cavity or pocket. It is demonstrated that asymmetric flow substantially reduces the induced pressure and rate of closure for the cavity containing the gas, while in the case of a vapor cavity asym- metry of flow leads to conclusions qualitatively different from those which follow from the classical Rayleigt solution for an infinite fluid. Based on the method of reflections, the mathemati- cal apparatus for the description of the phenomena in- volved in the closure of spherical cavities containing a gas near a solid boundary has been developed in detail 1. The radial oscillations at the wall of a spherical cavity containing a large quantity of gas have been studied in 2. We employ the same methods below to examine the closure at the wall of a cavity with a re- latively low gas content, as is charaeteristic for the phenomenon of vapor cavitation. Limiting ourselves to the reflection of a single source and a single dipole, which is equivalent to ne- glecting powers of e = R/2b above the first as small in comparison with unity, we can write the flow po- tential in the form q = R2 dR i 1 x + y)z/2 4 R* db (X2- y 2 dt 3/2 , (x -2b) -k- g.,.lZ/ x - 2b (x -2b) f3/z ?9 The first terms in the braces correspond to the radial and translational motion of a real sphere, while the second terms correspond to the reflection of that motion at the point with the abscissa x = 2b. The kinetic energy of the liquid can be calculated from the values of the potential and its derivatives at the boundary surfaces, these values having been de- termined from the boundary conditions; equally accu- rate is the following value of the kinetic energy: v = 2,p , ( d, / (l + ) + o m : ,b dt : -3- -) () There is no term which takes into consideration the mutual effect of the radial and translational motion on the magnitude of the kinetic energy, since that term has the order O(e2). The second term in (2) represents the kinetic energy for the motion of a sphere in an in- finite fluid, since the influence of the wall introduces a connection into the order, i.e., O(e2). Use of the value for the kinetic energy from (2) by means of the method of generalized coordinates enables us to derive a sys- tern of differential equations for the radial and trans- lational motion of the cavity: 24 2eh- +1=0, (3) (3 ) where t? b dR ( O_ t u2 db 1/2 dt 2 , p ,o arab RoP , 6- Pgo dd2 Pc Pc The system of differential equations (3) has been derived with the assumption that the gas in the pocket is adiabatically compressed, i.e., y = 4/3. With the cavity removed to infinity from the wall we have 0, 0, and 0, and consequently, the left- hand part of the second equation is identically zero, while the first equation changes into the familiar equation for the radial motion of a cavity in an infi- nite fluid + -2- u + 1 = o. (4) The solution of this equation can be derived analyti- cally 3J. The velocity of the pocket boundary is given by the relationship * = -3-1- 48 (1- 1-3)/2. (5) The maximum velocity sic on closure has the value ?9 . f (1 +36)4-25663 J/2 max = 38453 5.1.10 - (1 + 128) 1/2 8 -3/2 . (6) The approximate equality here and below corresponds to low gas contents, i.e., 5 10 -2. The minimum cavity radius is 36 36 mi, 1 + 36-5 /z 1+38 (7) The maximum pressure corresponds to the instant of closure, and when the condition e _ 1 is satisfied we hay e 705 , ,O,ax= Pmx-Po _ Po 2(I + 3)(1 + 36 -6a/) 2 (1 +9g) 271 2716 (s) In the case of a vapor cavity (5 = 0) we can calcu- late 4 the time of its complete closure: . 0.915. (9) In analogy with the method employed in the solution of Eq. (4), excluding the time t and treating the radius ? as an independent variable, we can reduce the order of system (3) to unity: vvl(l+ e)- v - 2e- - 2el ) (lo) 1- z- +1 =0, rl + I$ (3 -I- v ) -i- 6a = O v (10) whe re v=, v= dv , d r Analytically, the systems of differential equations (3) and (10) cannot be integrated. The solution is achieved numerically, with the use of computers; sys- tem (3) was solved by the Runge-Kutta method, and system (10) was solved by the iteration method. The difference in the results of the solutions does not ex- ceed 1%. Simultaneous with the determination of the kinematic characteristics of cavity motion, we calcu- lated the pressure at the wall and at the critical point b, where the pressure is at its maximum. The utiliza- tion of the Lagrange-Cauchy integral and the value of the potential from (1) make it possible for us to derive an expression for the pressure at this point in the form = P-Po Po = 4s (2+ 1) .4e1(5 -, 1)- 16 a . (11) In the case of an infinite fluid a at a great distance from the cavity * = 2 (2+ ). (12) Consequently,. with sufficiently adequate distance between the cavity and the wall-in which case the sec- ond and third terms in (11) can be neglected-the pres- ence of the wall under the condition of equality be- tween the radial velocities and accelerations leads to a doubling of the pressure. The initial conditions for which the solution of the system of equations (3) and (10) was carried out cor- responds to the assumption that the cavity is at rest at the initial instant of time and that the pressure dif- ference between infinity and the inside of the cavity is given.by Ap = P0(1 - 5) when t = 0, ? = 1, f = rio, = = 0, B = 0. The cavity growth phase which follows the compression phase subsequent to the cavity reaching its minimum dimension ?min was not calculated, since the solution-as in the case of an infinite fluid-for min is symmetric. The numerical calculations were carried out for four values of the gas contents: 5 = 0, 10 -4, 10 -3, 10 -2 , and for seven values of the initial distance from the wall: f0 = i.1, 1.2, 1.5, 2.0, 5.0, 10, 100. For the case in which fi0 o, we use an analytical solu- tion. Figure 1 as an example shows the results from the calculation of cavity closure for the case/30= 1.5, 5 = = 10 -4 in the form of curves showing ?, fl, , , = = /3/Bo, o = T/AT*, and as a function of the relative radius 7. Analysis of these results shows that on clo- sure the velocity of the cavity toward the wall is of the same order of magnitude as the radial velocity. The basic translational displacement of the cavity occurs during the final stages of closure, and when 77 = min the velocity of the translational motion reaches its maximum. The picture for the radial motion corre- sponds qualitatively to the closure of a gas-filled cav- ity in an infinite fluid; however, quantitatively speak- ing, the velocity, acceleration, and induced pressure are considerably smaller, while the minimum radius and closure time are greater. Figure 2 shows the re- suits from a calculation of the velocity as a function of both the radius and the initial distance from the wall for a gas content of 5 = 0. Our attention is drawn to the fact that, qualitatively, the effect of a reduction in the initial distance from the wall is analogous to the effect of an increase in the gas content when the cavity in an infinite fluid is closed. The results from the cal- culations of the minimum cavity radius ?min(f0; 5) are shown in Fig. 3. Analysis of the derived results shows that the greater the initial distances, the smaller the gas con- tent and that these distances are affected by the pres- fo 2 /0 +_f -/0 -fo Fig. 1. ,/ J, 2/ Functions 7, B, 7, B, X, , and versus the relative radius for fi0 = 1.5 and 5 = 10-4: 1) ; 2) fl; 3) ; 4) *; 5) 6) a; 7) . 706 /0-3 W-e I W /0 3 to r f Fig. 2. Radial velocity of closing vapor cavity (6 = 0) as a function of the radius for various initial distances from the wall: 1) G0 = 1.1; 2) 1.2; 3) 1.5; 4) 2; 5) 5; 6) 10; 7) 100. ence of the wall. Unanticipated is the substantial in- fluence of the wall on the closure of a cavity with a gas content of 6 _ 10 -4 at a distance as large as/0 = i00. This result indicates the strong effect of even slight asymmetry of flow on the final stage of closure; this is also borne out by the qualitative difference in the be- havior of the vapor cavity (6 = 0) at the wall and in an infinite fluid: despite the absence of gas in the cavity, complete closure of the cavity does not occur. The explanation of the derived result lies in the fact that in the presence of a wall a portion of the potential energy-which the fluid exhibits at the initial instant of time-changes into the kinetic energy of translational motion, so that the presence of the wall leads to a re- duction in the velocity of radial motion fortheboundary of the cavity, while in the case of a vapor cavity it leads to the appearance of conditions under which its complete closure proves to be impossible. This con- clusion is a consequence of the assumption that the cavity retains its spherical shape. Indeed, because of the accelerated translational motion, the diagram showing the pressure distribution over the surface of ! i0-2 fO-: a! 2 3 # 5 8 8 tO 20 30 40 o Fig. 3. Minimum cavity radius min(fl06) and function Xmin(B0): 1) 6 = 10-2; 2) 10-3; 3) 10-4; 4) 0; 5) min(0). the cavity is exceedingly nonuniform, in connectionwith which-in the later stages of closure-the cavity must undergo strong deformation, subsequently collapsing. It can be demonstrated that the distribution of pres- sures over the surface of a moving sphere is described by the relation where ,=I 9cosO -5 + IS cosO , (13) 8 2 0 = arctg fig-. x In connection with the condition that the cavity re- tain its spherical shape, in the over-all balance of forces applied to the boundary of the cavity, the sur- face-averaged pressure plays a role: =_j_14 m $ cos* 0 sin 0 d 0 - . sin 0 0 -t- - cos0 sin0d0 = - (14) 4 0 The last integral in (14) is equal to zero; consequently, the average pressure resulting from the nonsteady na- ture of the motion is equal to zero. Thus, because of the translational motion of the cavity, a pressure is developed on the cavity surface, the average magni- tude of this pressure is determined by the Velocity, and it is a negative value, which is equivalent to the presence in the cavity of a gas with a pressure deter- mined from relationship (14). When the cavity attains its minimum dimension, its velocity of translational motion and, consequently, its tensile stresses are at their maximum. Figure 4 shows the results from the calculation of the maximum pressure at the wall at the instant of cavity closure, referred to the pressure induced by ! f J / / / )/ f / / I 2 3 z 5 8 /0 20 30 0 o / / / Fig. 4. Maximum pressure on wall at the instant of cavity closing as a function of 0: 1) 5 = 10-2; 2) 10-3; 3) 10 -4 . 707 the cavity on its closure in an infinite fluid at the same distance. With a reduction in the initial distance from the wall, the ratio of the induced pressure to the pressure on cavity closure in an infinite fluid initially dimin- ishes, reaching a minimum at fl0 1.35, and then in- creasing slowly, which is explained by the significant influence of the approach of the cavity to the wall dur- ing the closure process when Do 1.35. Accurate to 0.1%, the quantity is found to be independent of gas content 5; the curve of the function Xmin(B0) is shown in Fig. 3. The general trend toward a reduction in the /* ratio with a reduction in G0 is obviously associated with a reduction in the radial acceleration of the cavity at the instant of closure. As we can see from the graph, for low gas contents the inducedpres- sure may be many orders of magnitude smaller than in an infinite fluid and, consequently, the large values for the induced pressure predicted by theory 3, 4 at the instant of closure are in actual fact impossible. The time for the complete closure of the cavity is greater than that calculated by Rayleigh 4; however, it does not differ from that quantity significantly. This investigation permits us to draw the impor- tant conclusion that attempts to determine the kine- matic characteristics of cavity motion during the final stages of closure-involving the use of the assumption of spherical symmetry of flow-cannot yield satisfac- tory results. The asymmetry existing under real con- ditions is caused by the proximity of the boundaries or by other nonuniformities in the flow as, for example, adjacent cavities, and so it must lead to an excessively pronounced quantitative difference between the the- oretical and actual parameters of cavity motion. The experimentally recorded and unexplained pronounced reduction relative to the theoretical value for the ra- dial velocity of the cavity during the final closure stages 5 is in all probability precisely a result of this circumstance. NOTATION p, p, T, and are the pressure, density, kinetic energy, and potential of the fluid flow; P0 = P - Ps is the pressure difference at infinityand of the saturated vapors; is the dimensionless pressure; Pg0 is the initial gas pressure in cavity; 6 is the relative gas content; y is the adiabatic exponent; R and b are the radius of spherical cavity and the distance from its center to the walI as a function of time; R 0 and b 0 are the same at the initial instant; y is the relative change of cavity distancefrom wall; t is the time; z = = t/R0(P0/p) 1/2 is the dimensionless time; AT* is the g time of complete closing of the vapor cavity in the in- finite fluid; x and y are the Cartesian coordinates, ori- gin at sphere center, with the x-axis directed to the wall. REFERENCES 1. R. Cowl, Underwater Explosions Russiantrans- lation, IL, 1950. 2. A. N. Korovkin and Yu. L. Levkovskii, IFZh Journal of Engineering Physics, 12, no. 2, 1967. 3. A. D. Pernik, Problems of Cavitation in Rus- sian, Sudpromgiz, 1966. 4. Rayleigh, Phil. Mag., 34, 94, 1917. 5. R. D. Ivany, F. G. Hammit, and T. M. Mitch- ell, Trans. ASME, ser. D, no. 3, 1966. 17 November 1967 708 包装机械设计课程设计指导书包装机械设计课程设计指导书 一、课程设计的目的课程设计的目的 1. 包装机械课程设计是该课程的延续，通过设计实践，进一步学习掌握包装机械设计的一般方法。 2. 培养学生综合运用所学专业基础课、 专业课理论知识与生产实际进行有效结合的能力。 3. 培养综合运用机械制图、机械设计基础、机械制造基础等相关知识进行工程设计的能力。 4. 培养使用手册、图册、有关资料及设计标准规范的能力。 5. 提高技术总结及编制技术文件的能力。 6提高学生独立分析问题和解决问题的能力。 7. 为毕业设计教学环节的实施奠定基础。 二、设计内容与基本要求设计内容与基本要求 1. 设计内容设计内容 完成题目：间歇双端扭结式裹包机扭结手设计间歇双端扭结式裹包机扭结手设计 2. 基本要求基本要求 (1) 课程设计必须独立完成，每人应完成扭结手部件设计装配图、所有零件图、传动系统简图、包装机的工作循环图、较复杂零件的三维建模图及 AutoCAD 设计图，能够较清楚地表达各部件的空间位置及有关结构。 (2) 根据设计任务书要求，在全面掌握扭结式裹包机的结构、性能、工作原理、传动系统及其它执行机构的组成、运动规律的基础上，掌握扭结手的结构、组成及运动规律，认真分析扭结手运动规律及动力传动系统。合理的确定尺寸、运动及动力等有关参数。 (3) 结构装配图要正确、完整的表达其工作原理、性能要求、零件间的装配关系、零件的主要结构形状及在装配、检验、安装时所需要的尺寸和技术要求。 (4) 正确的运用手册、标准，设计图样必须符合国家标准规定。说明书必须用工程术语，文字通顺简练，字迹工整。 (5) 各组成零件的视图符合图样标准；能够正确地表达出零件的结构形状；能够正确地标注尺寸及相应的公差；准确地给出零件在使用、制造、检验时应达到的一些技术要求。 1(6) 要以主要执行机构为基础， 按包装工艺流程将各执行机构的运动规律表示出来。 三、设计步骤设计步骤 1. 方案确定方案确定 (1) 根据包装对象及方法确定有关尺寸参数、运动参数及动力参数。 (2) 根据所求得的有关运动参数及给定的原始数据，绘制工作循环图。 2. 装配图和零件图设计装配图和零件图设计 (1) 草图设计：设计各零件尺寸，确定视图比例，确定展开图及截面图的总体布局；据各部件的受力条件， 在有关支撑部位画出轮廓。 并检验各传动件运动过程中是否干涉。 (2) 结构图设计：确定各零、部件的固定方式；确定润滑、密封及调整方式；确定部件形状及尺寸，完成展开图及截面图的绘制。 (3) 尺寸、公差配合标注，填写明细表及装配图技术要求。 (4) 零件图设计 3. 编写设计计算说明书编写设计计算说明书 四、设计内容四、设计内容 1间歇双端扭结式糖果包装机的主要技术参数间歇双端扭结式糖果包装机的主要技术参数 (1) 生产能力 200350 块min (2) 糖块规格 圆柱形(直径长度) 1332 长方形(长宽高) 271611 (3) 包装纸规格 商标纸 宽 90 ，内衬纸 宽 30 (4) 电机 理糖电机 0.37kW，主电机 0.75kW (5) 外形尺寸 14506501620 2间歇双端扭结式糖果包装机的组成及工作原理间歇双端扭结式糖果包装机的组成及工作原理 间歇双端扭结式糖果包装机主要由料斗、理糖部件、送纸部件、工序盘以及传动操作系统等组成。可完成单层或双层包装材料的双端扭结裹包。其包装工艺流程图如图 1所示。 包装机工作时,主传送机构带动工序盘 2 作间歇转动，如图 2 所示，随着工序盘 2 的转动，分别完成对糖果的四边裹包及双端扭结。在第 1 工位，工序盘 2 停歇时，送糖杆7、接糖杆 5 将糖果 9 和包装纸 6 一起送入工序盘上的一对糖钳手内，并被夹持形成 U形状。然后，活动折纸板 4 将下部伸出的包装纸(U 形的一边)向上折叠。当工序盘转动到第工位时，固定折纸板 10 已将上部伸出的包装纸 (U 形的另一边)向下折叠成筒状。 2固定折纸板 10 沿圆周方向一直延续到第工位。在第工位，连续回转的两只扭结手夹紧糖果两端的包装纸，并且完成扭结。在第工位，钳手张开，打糖杆 3 将已完成裹包的糖果成品打出，裹包过程全部结束。 图 1 包装工艺流程图 1送糖；2糖钳手张开、送纸；3夹糖；4切纸；5纸、糖送入糖钳手； 6接、送糖杆离开；7下折纸；8上折纸；9扭结；10打糖 图 2 包装扭结工艺路线图 1扭结手；2工序盘；3打糖杆；4