渣浆泵液混合物在旋转流道中的相对运动
   推导固液混合物液流在旋转流道内相对运动的伯努利方程具有实际意义。固体颗粒沿其轨线运动，液体则沿流线运动。固液混合物在旋转流道出口的总能最大于入口的总能量因而，所研究的液流运动，只有从外面将能量引入固液泥合物液流中才有可能。

一、固液混合物在固定流道内的运动

研究固液混合物相对运动，即认为流道是固定的，同时混合物全体以角速度-w (此处o为流道角速度)绕轴线旋转。因为运动是定常的，在研究它时，在每一点上必须引用向心加速度和哥氏加速度。哥氏力方向垂直于固体颗粒轨线和液体流线，所以不做功。与固液混合物单位质量有关的离心力，与其是否作用在固体颗粒或者液体上无关，做功为
    流经叶轮的固液混合物流量用Q表示，固体颗粒和液体的流量分别用Q,和Q。表示，于是， 单位时间内消耗在克服固液混合物流过时流道内压降(n- p) 的功T一Q. (p:-pi)。. 

假定在流道入口固体颗粒和和液体相对速度相问，并等于w.在流道出口它们不同，分别为UT和wo.固体颗粒和液体固体颗粒、液体和混合物的密度分别为P.P和p.在流过人口运动到出口时动能变化为

固液混合物在旋转流道内运动时静水头变化。

从得到的等式中看出，在旋转流道内静水头变化，等于单位重量固液混合物的压降，与相同密度与质液体在旋转流产内运动时的静水头不同之处在于，混合物的固相和液相动能差乘以固体颗粒质量浓度K和附加水力损失，在固体颗粒和液体相对速度差值不大时，与均质液体液流压降相比，固液混合物液流压降的减少，只与水力损失增加有关。
    第二种情况具有实际意义， 就是具有不同粒度固体颗粒的混合物液流绕轴线旋转流逃内的运动。

    二、固液混合物在旋转流道内的运动
    下面研究现象定性方面，即相同粒度固体颗粒在固定流道内的分布。假定固体颗粒相当小并且其速度与载体局部速度相等。假定流体流动为絮流流动，同时在重力作用下，休颗粒以等于其沉降速度W'从液体中下降，这个速度不同于颗粒在静止水中自由下沉的沉降速度W.由于在液流中速度U的脉动作用，固体颗粒将参与絮流混合过程。
    研究两个平行于液流运动方向的平面，它们相互处在混合路径长度的距离上。采用处在单位体积流体内上平面区域内颗粒数量等于n,在下平面区域内为m. +On..固体颗粒从上层运动到下层，是受到重力作用和液流紊流度的双重影响。而固体颗粒从下层运动到上层，只有依靠紊流运动才能实现。
    单位时间内经过单位面积从上层进入下层的固体颗粒数量等于n,u' +n,W'.在从液体下层到上层交换的稳定过程时，应该进人同样数量的颗粒(n, + An,)v'=n,v' +nW'.因而，在下层和上层之间颗粒数量变化，可以根据A1/n, =W'/u'求出，这种变化表明，悬浮颗粒沉降速度越大，它们在液流中分布均匀性越差。脉冲速度v增大，将导致An,减少，因而导致固体颗粒在液流中分布更加均匀。
    当絮流中沉降速度 W变为等于脉中速度。时，比值Sn/n,=1,就是说处在上层所有颗粒下降到下层，即固体颗粒液流大部分不再悬浮，并且它们将不多加紊流混合过程。
    下面研究绕轴线施转的流道，它垂直于纵断面的平面。在相对运动时，液流在流道中速度等于1。固体颗粒与液体起在旋转流道内， 例如在叶轮叶片之间空间内运动，它们处在离心力场中。加速度法向分量(在相对运动时)为
    为了估算这加速度值，给出下列吸入短管直径为125mm挖泥深几何多数和运动参数: "o=8m/s，断面平均半径R= 150mm,w= 150L/s, cosp=0.85, r,= 150mm。在这种情况下1a1 >0也就是说比重力加速度大50多倍。因此悬浮在液体中固体颗

粒的沉降速度大于混合物在固定流道中运动的速度。

在估算混合物在流道中运动所产生的水力损失，具有重要意义的是粒径级配分为两类的可能性:细颗粒，虽然存在很强的加速度场但仍然参加絮流混合过程:较大颗粒，在离心力场影响下，不再参加紊流交换并在液流中沿单独轨线运动。在所有固体颗粒参加絮流交换时，固液混合物可以作为均质液体进行研究，认为在液说中的损失与密度相同的均质液体一样。在存在不参加絮流交换过程的固体颗粒时，液体不能作为均质液体来研究，确定流道中的水力损失，必须考虑非均质液流。
  悬浮颗粒处在加速度场中，其沉降速度由下列等式确定

式中d-一粒径;

C- 颗粒迎面阻力系数。

对于细颗粒，可以认为迎面阻力系数(报据斯托克斯定律)为

式中，v一载体运动黏度。

因此，在旋转流道中运动的固体颗粒沉降速度为

决定已知粒径径固体颗粒在液流中分市特性的因素: 可以这样估价

认为混合物液流在流道中平均相对速度为w，根据与脉冲速度的关系

对于旋转流道中运动相似工况，液流中加速度与n2D成正比（式中，n为流道转速，D为流道特征尺寸），而速度与nD成正比。

  假定dr是对于给定的混合物粒径级配时颗粒界限直径，所有粒径d<dr的颗粒称为细颗粒，而粒径d>dr的颗粒称为粗颗粒。这就意味细颗粒参加液流的紊流混合，而粗颗粒沿着完全确定的单独轨迹运动，不参加紊流交换。
  如果沉降速度小于脉冲速度，那么颗粒就悬浮，如果沉降速度大于脉冲速度，那么颗粒就逐渐下沉。可以假定，对于界限直径的颗粒，比值Wr/v' (式中， Wr为粒径d:颗粒的沉降速度)，即它们可以从给定层均匀地下沉，与处在悬浮状态一样。
  我们估算界限粒径所采用判据之值dr n人，采用pr=2650kg/m3, Po =1000kg/m3，

对渣浆泵混合物在旋转流道中运动特征的分析表明，在相同混合物粒径及配对，旋转流道中的损失不能模拟。如果模型的轴转速等于1450r/min，而实型的转速为600r/min，那么颗粒界限尺寸在模型试验时比实验小1/1.55倍。因此，利用与实际条件相同的粒径级配固体颗粒进行模型泵试验，不能给出有关实型泵在抽关送固液混合物时实际特性的正确概念。

应该考虑砂砾圆液混合物颗粒界限尺寸相当小。例如，如果采用泵转速n=600r/min。即10r/s.那么d,=0.4mm.在砂砾混合物中粒径小于0.4mm颗粒含量一般不大，所以通常不能认为在砂砾固液混合物流动时在旋转流道中的损失与具有相同密度的均质液体的情况一样。
    从所得到的颗粒界限尺寸判据看出，在确定dr时主要参数是流道转速，因为用挖泥泵抽送固液混合物的载体介质黏度一般变化很小。

Relative Motion of Slurry Pump Mixture in Rotating Channel

It is of practical significance to derive Bernoulli equation for the relative motion of liquid-solid mixture in a rotating channel. Solid particles move along their tracks, while liquids move along streamlines. The total energy of solid-liquid mixtures at the outlet of the rotating channel is greater than that at the inlet. Therefore, it is only possible to introduce energy from the outside into the solid-liquid sludge flow.

I. The Movement of Solid-liquid Mixtures in Fixed Channels

The relative motion of solid-liquid mixtures is studied. It is considered that the runner is fixed and the mixtures rotate around the axis at an angular velocity of - w (where o is the angular velocity of the runner). Because motion is steady, centripetal acceleration and Coriolis acceleration must be used at every point in studying it. Coriolis force is perpendicular to solid particle trajectory and liquid streamline, so no work is done. The centrifugal force related to the unit mass of a solid-liquid mixture has nothing to do with whether it acts on solid particles or liquids.

The flow rate of solid-liquid mixture through impeller is expressed by Q, and the flow rate of solid particles and liquid is expressed by Q, and Q, respectively. Thus, work T-Q. (p:-pi) is consumed per unit time to overcome the pressure drop (n-p) in the outdated flow passage of solid-liquid mixtures. .

It is assumed that the relative velocities of solid particles and liquids at the entrance of the runner are interrelated and equal to w. They are different at the exit of the runner. The densities of solid particles and liquid solid particles, liquids and mixtures are P.P and P.

Static Head Change of Solid-liquid Mixture in Rotating Channel

From the equation obtained, it can be seen that the change of static head in the rotating channel is equal to the pressure drop of solid-liquid mixture per unit weight. The difference between the static head of the same density and mass liquid moving in the rotating abortion is that the kinetic energy difference between the solid phase and the liquid phase of the mixture is multiplied by the mass concentration of solid particles K and the additional hydraulic loss in the solid phase. When the relative velocity difference between particles and liquids is small, the pressure drop of solid-liquid mixture decreases only because of the increase of hydraulic loss compared with that of homogeneous liquid.

The second case has practical significance, that is, the liquid flow of mixtures with different particle sizes flows around the axis of rotation to escape.

2. The Movement of Solid-liquid Mixture in the Rotating Channel

The following qualitative aspects of phenomena are studied, i.e. the distribution of solid particles with the same particle size in a fixed channel. It is assumed that the solid particles are fairly small and their velocities are equal to the local velocities of the carrier. It is assumed that the fluid flow is a flocculating flow, and that under the action of gravity, the suspended particles decrease from the liquid at a settling velocity equal to W', which is different from the settling velocity W of particles in stationary water. Because of the pulsation of velocity U in the liquid flow, the solid particles will participate in the flocculating mixing process.

Two planes parallel to the direction of fluid flow are studied. They lie at the distance of the length of the mixing path. The number of particles in the upper plane region of a unit volume fluid is equal to n, and in the lower plane region is m. +On.. The movement of solid particles from the upper layer to the lower layer is influenced by gravity and turbulence. The movement of solid particles from the lower layer to the upper layer can only be achieved by turbulent motion.

The number of solid particles per unit time passing from the upper layer to the lower layer is equal to n, u'+n, W'. In the stable process of exchange from the lower layer of liquid to the upper layer, the same number of particles (n,+An,) v'=n, v'+nW'. Therefore, the change of the number of particles between the lower layer and the upper layer can be calculated according to A1/n,= W'/u'. The results show that the larger the settling velocity of suspended particles, the worse the uniformity of their distribution in liquid flow. The increase of pulse velocity V will lead to A and decrease, which will lead to more uniform distribution of solid particles in liquid flow.

When the settling velocity W in the flocculation becomes equal to the velocity in the pulse. When the ratio of Sn/n is equal to 1, that is to say, all the particles in the upper layer descend to the lower layer, that is, most of the solid particles are no longer suspended, and they will not add much turbulent mixing process.

Next, we study the flow path running around the axis, which is perpendicular to the plane of the longitudinal section. In relative motion, the velocity of liquid flow in the channel is equal to 1. Solid particles and liquids move in a rotating channel, such as in the space between impeller blades, and they are in the centrifugal force field. The normal component of acceleration (in relative motion) is

In order to estimate this acceleration value, the following geometric majority and movement parameters of the dredging depth with the diameter of the suction short pipe 125 mm are given: "o = 8 m/s, the average cross-section radius R = 150 mm, w = 150 L/s, cosp = 0.85, r, = 150 mm". In this case, 1A1 > 0 is more than 50 times the acceleration of gravity. So solid particles suspended in liquids

The settling velocity of particles is faster than that of mixture moving in a fixed channel.

In estimating the hydraulic loss caused by mixtures moving in the channel, it is important to divide the particle size distribution into two categories: fine particles, although there is a strong acceleration field, still participate in the flocculation mixing process: larger particles, under the influence of centrifugal force field, no longer participate in turbulent exchange and along a single trajectory in the fluid flow. Sports. When all solid particles participate in flocculation exchange, solid-liquid mixtures can be used as homogeneous liquids. It is considered that the loss in liquid theory is the same as that in homogeneous liquids with the same density. When there are solid particles which do not participate in the flocculation exchange process, the liquid can not be studied as homogeneous liquid. To determine the hydraulic loss in the channel, the heterogeneous liquid flow must be considered.

The settling velocity of suspended particles in acceleration field is determined by the following equation

D-1 particle size in the formula;

C- particles