細(xì)顆粒在曲線流道中的運(yùn)動(dòng)

   液體在曲線流道內(nèi)轉(zhuǎn)向時(shí)，產(chǎn)生離心力，在該力作用下固體顆粒擠向流道外壁面因（它們的密度大于載體介質(zhì)密度)。應(yīng)該考慮在水泵流道內(nèi)通常只發(fā)生液體絮流，因此固體顆粒轉(zhuǎn)向時(shí)，獲得脈動(dòng)速度，結(jié)果力求在液流斷面上均勾分布。估計(jì)脈動(dòng)速度和離心力相互對(duì)固體顆粒在液流中分布的影響。假設(shè)在離心力作用下，顆粒轉(zhuǎn)向時(shí)沉降速度增大，因?yàn)槌两邓俣扰c顆粒迎面阻力有關(guān)。在半經(jīng)方向上，在顆粒運(yùn)動(dòng)擬固定狀態(tài)，實(shí)際上等式為

式中d-所研究額粒的直徑:
u-轉(zhuǎn)向時(shí)顆粒圓周速度;
r-轉(zhuǎn)向時(shí)跟液體一起運(yùn)動(dòng)的顆粒軌線曲事半徑；

w.-顆粒沉降速度。于是固體顆粒沉降速度為

在液流中引用圓柱表面， 它與曲線流道些面等距(圖2-2 2 (o]。令固體顆粒在這的一個(gè)表面的一面的體積濃度等于P，在絮流混合時(shí)，單位時(shí)間內(nèi)經(jīng)過(guò)單位表面的固體顆粒的體積為。

由于只指向同一方向的離心力的作用，固體顆體積PW。從小濃度區(qū)移動(dòng)到較高濃度區(qū)。在穩(wěn)定過(guò)程中，固體顆粒濃度分布應(yīng)該這樣，要使顆粒經(jīng)過(guò)所研究表面ab的總移動(dòng)量等于零。于是有


從等式可以看出，脈動(dòng)速度越大和顆粒沉降速度越小，比值△P/P就越小，即固體顆粒在液流中分布就越均勻?？梢哉J(rèn)為脈動(dòng)速度與轉(zhuǎn)向時(shí)速度成正比，即
v'=ku

式中k——比例系數(shù)。
于是在轉(zhuǎn)向時(shí)，固體顆粒在液流中濃度分布用下列比值確定
為了求出這樣的條件，即在這種條件下由于脈動(dòng)速度，固體顆粒在液流中分布均勻，研究AP/P=0.1的個(gè)別情況。

用下列方程估算C1值：采用迎面阻力系數(shù)C=0.5，固體顆粒密度為2650kg/m3。

由于脈動(dòng)速度在轉(zhuǎn)向時(shí)固體顆粒濃度變化為0.1，顆粒直徑與流線轉(zhuǎn)向半徑之比等于

如果轉(zhuǎn)向半徑例如為300mm那么顆粒尺寸大約為0.006mm因此絮流對(duì)固體顆粒分布的影響，只有存在很細(xì)顆粒時(shí)將是明顯的。
  二、大顆粒在曲線流道中的運(yùn)動(dòng)
    下面研究具有足夠大顆粒的固液混合物沿著平面曲線流道的流動(dòng)[圖2-2-2 (b)].同時(shí)，與轉(zhuǎn)向時(shí)發(fā)生的顆粒沉降速度的影響相比，可以忽略絮流混合的均衡影響.
    固體顆粒運(yùn)動(dòng)為擬固定運(yùn)動(dòng)，即在其軌線每一點(diǎn)，離心力與迎面阻力相平衡?？梢哉J(rèn)為，當(dāng)與向心加速度相比時(shí)，顆粒在沿軌線曲率半徑運(yùn)動(dòng)時(shí)徑向加速度很小。此外假定固體顆粒沿著液體圓周速度等于u的流線運(yùn)動(dòng)。
    顆粒在徑向上動(dòng)平衡條件，與細(xì)顆粒時(shí)一樣，具有下列形式

式中 u——轉(zhuǎn)向時(shí)顆粒運(yùn)動(dòng)圓周速度;
w——顆粒沉降速度，由顆粒在點(diǎn)a沿半徑方向的速度表征(轉(zhuǎn)動(dòng)為0角)，

顆粒沿半徑方向的運(yùn)動(dòng)速度
式中常數(shù)（const）值與固體顆粒在流道入口斷面上的初始位置有關(guān)。

懸浮在液流中固體顆粒在轉(zhuǎn)向時(shí)，在徑向擠壓流道外壁面。下面研究在液流轉(zhuǎn)向時(shí)，顆粒在斷面內(nèi)如何分布。在液流轉(zhuǎn)過(guò)角時(shí)，處于在半徑之間入口的顆粒，在壁面上集中，在軌道cb與流道壁面交點(diǎn)區(qū)其增大，與經(jīng)過(guò)入口斷面相同數(shù)量的顆粒，在單位時(shí)間通過(guò)角度 的流道斷面。假定顆粒在流道入口均勻分布，可以估計(jì)在點(diǎn)b落在壁面上顆粒數(shù)量與流道入口斷面上固體顆粒，總數(shù)量之比為

式(2-2-7)中包括r1和r2兩個(gè)未知變量。對(duì)所研究示意圖，從公共中心引出流道壁面半徑，所以在一個(gè)半徑變化和第二個(gè)半徑固定時(shí)，流道寬度將發(fā)生變化。從等式(2-2-7)中可以看出，流道半徑成比例變化，即保持幾何相似，不會(huì)導(dǎo)致固體顆粒沿流道斷面上濃度分布相似，在流道幾何尺寸變化。固體顆粒濃度變化a倍，同時(shí)流道尺寸增大，導(dǎo)致固體顆粒在其斷面上分布不均創(chuàng)度減少

固體顆粒粒度對(duì)固體顆粒在液流斷面上分布不均勾度有顯著影響。實(shí)際上，C-er pad,即混合物中固體顆粒越細(xì)，轉(zhuǎn)向時(shí)流道型面上的固體顆粒濃度越小。對(duì)混合物在轉(zhuǎn)向時(shí)流動(dòng)的定性分析，說(shuō)明了葉輪葉片入口邊不均勻磨損的原因，實(shí)際上，液流轉(zhuǎn)向時(shí)在葉輪葉片入口之前發(fā)生的過(guò)程更為復(fù)雜。但是可以認(rèn)為，在質(zhì)的方面，轉(zhuǎn)向時(shí)固體顆粒在流道內(nèi)重新分布形態(tài)，以及顆粒尺寸和液體轉(zhuǎn)向角對(duì)這種重新分布的影響程度是相同的。
    在實(shí)際條件下，AF泡沫泵液流轉(zhuǎn)向時(shí)，經(jīng)常發(fā)生液流與流道內(nèi)(前)壁面脫流，即流道工作斷面縮小。但是在這種情況下，隨著濃度的增加，在內(nèi)壁而上保持濃度不均勻分布。相當(dāng)大的顆粒甚至在脫流區(qū)后面液流斷面上擠壓外壁面。

Movement of Fine Particles in Curved Channels

The centrifugal force is produced when the liquid is turning in the curved channel. Under this force, the solid particles are squeezed to the outer wall of the channel (their density is greater than that of the carrier medium). It should be considered that only liquid flocculation usually occurs in the flow passage of the pump, so when the solid particles turn, the fluctuating velocity can be obtained, and the results should be uniformly distributed on the flow section. The influence of fluctuating velocity and centrifugal force on the distribution of solid particles in liquid flow is estimated. It is assumed that under the action of centrifugal force, the settling velocity increases when the particles turn, because the settling velocity is related to the particle head-on resistance. In the semimeridian direction, in the quasi-fixed state of particle motion, the equation is actually

D - The diameter of frontal grains studied in this study:

The circumferential velocity of particles in u-turn;

R-radius of curvature of particle trajectory moving with liquid during steering;

W. - settling velocity of particles. So the settling velocity of solid particles is

The cylindrical surface is introduced in the fluid flow, which is equidistant from some surfaces of the curved channel (Fig. 2-22 (o). The volume concentration of solid particles on one side of this surface is equal to P, and the volume of solid particles passing through the unit surface in unit time is equal to P in flocculation mixing.

Because of the centrifugal force in the same direction, the solid particle volume PW. Move from low concentration area to high concentration area. In the process of stabilization, the concentration distribution of solid particles should be so that the total movement of particles through the studied surface AB is equal to zero. Thereupon

It can be seen from the equation that the larger the fluctuating velocity and the smaller the settling velocity of particles, the smaller the ratio of P/P, that is, the more uniform the distribution of solid particles in liquid flow. It can be considered that the pulsating speed is proportional to the turning speed.

V'=ku

K-proportional coefficient in formula.

When turning, the concentration distribution of solid particles in liquid flow is determined by the following ratio.

In order to find out the conditions under which solid particles distribute uniformly in liquid flow due to fluctuating velocity, the individual case of AP/P=0.1 is studied.

The C1 value was estimated by the following equation: the head-on drag coefficient C=0.5 and the solid particle density 2650 kg/m3 were used.

The ratio of particle diameter to streamline turning radius is equal to 0.1 as the fluctuating velocity changes to 0.1 during turning.

If the turning radius is 300 mm, for example, the particle size is about 0.006 mm, so the effect of flocculation on the distribution of solid particles will be obvious only when there are very fine particles.

2. Motion of Large Particles in Curved Channels

Next, we study the flow of solid-liquid mixture with large enough particles along a plane curved channel [Fig. 2-2 (b)]. At the same time, the equilibrium effect of flocculation can be neglected compared with the effect of particle settling velocity when turning.

The motion of solid particles is quasi-fixed, i.e. the centrifugal force is balanced with the head-on resistance at every point of its trajectory. It can be considered that the radial acceleration of particles moving along the radius of curvature of the trajectory is very small when compared with the centripetal acceleration. In addition, it is assumed that solid particles move along streamlines with liquid circumferential velocity equal to u.

The dynamic equilibrium condition of particles in the radial direction is the same as that of fine particles.

U-the circumferential velocity of particle motion in steering;

W - The settling velocity of particles is characterized by the velocity of particles along the radius direction at point a (rotation is 0 angle).

Particle velocity along radius

The const value of the formula is related to the initial position of solid particles on the entrance section of the channel.

When the solid particles suspended in the liquid flow turn, they are on the outer wall of the radial extrusion channel. Next, we study how the particles are distributed in the cross section when the flow is diverted. At the liquid flow angle, the particles at the entrance between radius concentrate on the wall, and increase at the intersection point between the track CB and the wall of the runner. The same number of particles passing through the entrance section pass through the angle section at unit time. Assuming that the particles are evenly distributed at the entrance of the channel, the ratio of the number of particles falling on the wall at point B to the number of solid particles on the entrance section of the channel can be estimated.

Formula (2-2-7) includes two unknown variables R1 and r2. For the schematic diagram, the wall radius of the runner is drawn from the common center, so the width of the runner will change when one radius changes and the second radius is fixed. It can be seen from equation (2-2-7) that the radius of the runner varies proportionally, that is to say, keeping geometric similarity will not cause the concentration distribution of solid particles along the cross section of the runner to be similar and the geometric size of the runner to be changed. When the concentration of solid particles changes a times and the size of the runner increases, the non-uniform distribution of solid particles on its cross-section decreases.

The particle size of solid particles has a significant influence on the uneven distribution of solid particles on the liquid flow section. In fact, the finer the solid particles in the mixture, the smaller the concentration of the solid particles on the channel profile when turning. Qualitative analysis of the mixture flow during steering shows the cause of uneven wear at the blade inlet. In fact, the process of fluid flow before the blade inlet is more complex. However, it can be concluded that the redistribution pattern of solid particles in the runner, as well as the effect of particle size and liquid steering angle on the redistribution are the same in terms of quality.

Under the actual conditions, when the AF foam pump is turning to the liquid flow, the liquid flow and the internal (front) wall defLOW often occur, that is, the working section of the flow path shrinks. However, in this case, with the increase of concentration, the concentration distribution is not uniform on the inner wall. A considerable amount of particles even extrude the outer wall on the flow section behind the stripping zone.