Vane pumps and fans of the basic theory

Vane pumps and fans of the basic theory

    To discuss the principle and performance of pumps and fans is to study the fluid flow in the pump and fan to find the relationship between the fluid flow and the geometry of the flow components to determine the appropriate flow path shape in order to obtain compliance Hydraulic (pneumatic) performance. Fluid flow through the pump and fan over the flow of components in the table below.

Vane pump and fan over-current components operating characteristics of the role of motion analysis and research suction chamber fixed immobile fluid to the working impeller is relatively simple and relatively easy impeller rotation to complete the conversion of energy is more complex and difficult extrusion chamber fixed immobile fluid Pressing out the pipeline is relatively simple and easy. It is easy to see from the above table. To carry out the research on the basic theory of vane pump and fan, the main energy should be focused on the research of the fluid flow law in the impeller runner.
§1-1 Flow Analysis of Fluid in Impeller I. Flow Analysis of Fluid in Centrifugal Impeller (1) Projection of Impeller Flow Path and Its Flow Analysis Assumptions
1, the impeller runner projection Figure 1-1 shows a centrifugal impeller runner projection. Left part of the figure (not to look at the connection between the front and rear cover) shows the shape of the impeller front and rear cover; the right part of the figure (first I have not seen point O, â… , â…¡ ...)
Figure 1-1 impeller axial projection, plane projection and axial cross-sectional diagram
1-front cover; 2-rear cover; 3-blades; 4,5-blade inlet and outlet are plane projection images of the impeller obtained by cutting out the front cover, and the plane of the blade surface can be seen Projection. In order to see the blade surface shape, often attached to the axial (also known as meridian plane) projection map.
The axial projection of an impeller is a diagram obtained by projecting a series of points on an impeller blade onto the same axial plane by rotational projection. The approach is to first project a set of lines connecting the axial planes of lines I, II ... with the impeller blades on the right (for ease of description, set the blades infinitely thin) onto the vertical axial plane OO 'using a rotation projection method , And then projected to the left, can be obtained with this set of intersection line shape exactly the same axial projection line (as shown on the left front and rear cover connection between the line), that impeller axial projection .
The axial projection and the plane projection of the impeller can clearly express the geometry of the centrifugal impeller, which has important practical significance and value in the manufacture of the model and the localization of the imported equipment. In order to describe and analyze the convenience, usually only the axial projection and plane projection of the impeller simply painted as shown in Figure 1-2.
2, the flow analysis Assuming that the fluid flow in the impeller is quite complicated, in order to analyze the flow patterns, often make the following assumptions:
(1) The blade in the impeller is infinitely infinitely thin, that is to say, the blade of the impeller is some non-thickness bone line (or profile line). Constrained by the vane lines, the trajectories of the fluid micelles exactly coincide with the vane lines.
(2) the fluid is the ideal fluid, that is, ignoring the viscosity of the fluid. Therefore, the flow loss in the impeller due to the non-uniform velocity field due to the viscosity can be ignored for a moment.
(3) Flow is steady flow, ie the flow does not change with time.
(4) The fluid is incompressible, and this is not very different from the actual situation, because the liquid volume changes very little pressure difference is very small, and the gas pressure drop is very small volume changes are often negligible.
(5) The flow of fluid in the impeller is an axisymmetric flow. That is, on the same radius of the circle, the fluid micelles have the same size of the speed. That is to say, the shape of the streamlines in each laminar flow surface (the flow surface is the surface formed by one revolution of the flowline about the axis of the impeller) is exactly the same, so only one streamline need to be studied for each flow surface.
(B) the movement of the fluid within the impeller and its speed triangle
1, the impeller fluid movement and its speed Triangular impeller rotation, the fluid on the one hand and the impeller for rotation, that implicate the movement, the speed is called the implicated speed, said; at the same time in the impeller flow along the leaves outward flow , That is, relative motion, the speed is called the relative speed, said. Therefore, the fluid movement in the impeller is a compound movement, that is, absolute movement, the speed is called absolute speed, as shown in Figure 1-3. Since the velocity is a vector, the absolute velocity equals the vector sum of the involved velocity and the relative velocity, ie:
 = +
(A) Circumferential motion (b) Relative motion (c) Absolute motion A vector composed of these three velocity vectors is called a velocity triangle or a velocity map, as shown in Figure 1-5 .
The velocity triangle is the basis for studying the energy transformation of the fluid in the impeller and its parameters. The velocity triangles shown in Figure 1-5 can be made at any point in the flow path of the impeller. However, when the research on one-dimensional flow of fluid in the impeller usually adopts one-dimensional flow, it is mainly to know the fluid at the impeller blade inlet and outlet Case. Because the speed triangle from these two places can compare the speed of the fluid before and after the impeller changes, in order to understand the energy obtained after the fluid flows through the impeller. In order to distinguish the two parameters, the subscripts "1, 2" are used to indicate the impeller blade inlet and outlet parameters respectively; and the subscript "ï‚¥" is used to indicate the parameter of the blade with infinite and infinite infinity.
In velocity triangles, define: Absolute velocity  =  (,); Flow angle  =  (, -); Blade mounting angle y =  (blade tangent direction, -). Obviously, when the fluid flows along the profile of the blade, the flow angle is equal to the mounting angle, ie  = y. In addition, in order to facilitate the calculation, often the absolute velocity is decomposed into two mutually perpendicular velocity components: one is the projection in the radial direction, expressed by r, r = sin, called the radial velocity; one is The projection in the circumferential tangential direction, expressed by u, u = cos, is called circumferential partial velocity, as shown in Figure 1-5.
2, the calculation of the speed triangle In the speed triangle, as long as the three known conditions can be made. According to the pump and fan design parameters used, you can easily determine the u,  r and  1,  2 corner to make the speed triangle. Practice as follows:
(1) circumferential speed u is:
U = (1-1)
Where D - impeller diameter (for the import and export of speed triangle, respectively, D1, or D2, into), m;
n-- impeller speed, r / min.
(2) The radial speed of absolute speed r is:
 r = (1-4)
Where qVT - theoretical flow, that is, the flow through the impeller, m3 / s; b - impeller blade width, m;
排 - crowding coefficient, is to consider the blade thickness of the crowd of the degree of crowding coefficient, its value is equal to the actual effective flow area and non-leaf flow area ratio, the pump, import, export crowding coefficient were:  1 = 0.75 ~ 0.88; 2 = 0.85 ~ 0.95.
(3) 2 and 1 angle when the blade infinitely long,  2 =  2y; and  2y in the design can be selected based on experience. The same  1 can also be based on experience, inhalation conditions and design requirements set.
In determining u2, 2,  2r, you can make a proportionate exit speed triangle, also identified in u1, 1,  1r, you can be made in proportion to the import speed triangle.
Fluid Flow Analysis in Axial Flow Impeller (I) Projection Flow and Flow Analysis of Impeller Flow Path Assumptions The axial projection and plane projection of the axial flow impeller are shown in Figure 1-7.
Fluid flow in the axial flow impeller is also a very complex space motion, in the analysis and calculation of fluid flow through the axial flow impeller, often make the following assumptions:
(1) Consider fluid flowing through an axial impeller very similar to an airplane flying in the atmosphere. Therefore, when studying the theory of impeller of axial-flow pump and fan, the theory of wing can be used in addition to the method used in studying centrifugal pump and fan.
(2) The assumption of cylinder-layer independence, that is, the fluid micro-clusters in the impeller flow on the cylindrical surface (called the flow surface) with the axis of the pump and the fan as the axis, and the flow interaction on the adjacent two cylindrical surfaces Irrelevant, that is to say there is no radial fractional velocity of the fluid micelles in the flow area of ​​the impeller.
(3) The fluid is incompressible.
According to assumption (2), a large number of cylindrical flow surfaces can be made in the impeller. The flow conditions on these cylindrical flow surfaces can be different, but the research methods are the same. Thus, as long as you know the flow on one flow surface, the flow on the other flow surface will get a similar solution. For this purpose, a tiny cylinder layer can be cut by two infinitely close cylinder surfaces of r and r + dr, taken out and cut along the generatrix to produce a plan view, and a plane cascade or straight line blade as shown in FIG. 1-8 is obtained The grid (the so-called cascade is the same blade leaves at equal intervals). Since each airfoil in the cascade has the same flow around, it is general to study the flow around an airfoil. Therefore, the flow of axial flow pump and fan impeller fluid flow, simplified along the length of the blade corresponding to several cylindrical surface of the cascade flow around the airfoil, the air flow around the cylindrical surface of several series together , You get the flow of fluid in the axial flow impeller.
The main characteristics of the cascade parameters:
Column line - grid in the airfoil corresponding points of the connection.
Airfoil in the grid - cross-section of the blades cut across each flow surface flowing around the cascade.
Pitch - the distance between airfoils in the grid t, t = 2Ï€r / z, where r is the radius of the cylindrical flow surface and z is the number of blades.
Density - chord l and the pitch t ratio l / t.
Mounting Angle - The chord length of the airfoil and the included angle 列 between the column lines.
(B) the impeller fluid movement and its speed Triangle and centrifugal impeller, the flow surface of any fluid micelles absolute velocity equal to implicate the speed and relative velocity vector sum, namely:

As shown in Figure 1-9 cascade, at the inlet of the cascade, the fluid flows at an absolute velocity of 1 into the impeller; due to the rotation of the impeller, a circumferential velocity u1 is generated; with respect to the impeller, the fluid flows at a relative velocity w1 into the impeller cascade . Imported by the three speed vector  1, u1, w1 speed triangle. Similarly, at the exit of the cascade, the exit velocity triangle is formed by u2, 2 and w2 respectively, as shown in Figure 1-9.
From the flow analysis assumption (2), we can see that the circumferential velocities in the same radius before and after the cascade are equal, that is, u1 = u2 = u, and due to the continuity of the fluid flow and the assumption of incompressibility, the axis of relative speed and absolute velocity The components are also equal, that is, w1a = w2a = wa, 1a = 2a = a. Therefore, the import and export speed triangle drawn together, as shown in Figure 1-10.
Because the theoretical basis of axial flow impellers is the wing theory, and the work of a single wing and cascade is different, so in order to apply the wing theory to the cascade, need to make some special treatment.
The principle difference between a cascade and a single wing is that the flow velocities in the front and back of the cascade differ in the direction in which the cascade changes the direction of the gate flow and the individual wings do not change the direction of the incoming flow. Due to  1a =  2a, the effect of the cascade on the fluid affects only the circumferential component of the velocity. Therefore, during the calculation of the cascade, we take the geometric mean value of the relative velocities w1 and w2 before and after the cascade as the inflow velocity infinitely equivalent to a single airfoil, the magnitude and direction of which are represented by the velocity triangle 1-10).
(1-5)
(1-6)
If used as a graph method, according to the principle of parallelogram, just connect the middle points E and B of the CD line in Figure 1-10, and the connection BE determines the size and direction of w.
In the calculation of the cascade speed triangle, there is a difference with the centrifugal impeller speed triangle, that is, the calculation of the axial component of the absolute speed, which is calculated as follows:
 (m / s)
Where Dh - impeller hub diameter, m.
§1-2 vane pump and fan energy equation

First, the derivation of energy equation (centrifugal impeller, for example)
After the fluid enters the impeller, the blades work on the fluid to increase its energy. Using the momentum moment theorem in fluid mechanics, we can establish the relationship between the blade's work on the fluid and the change of the fluid's movement state, and deduce the energy equation.
1, the prerequisites The assumptions in the previous section, abbreviated as:
Blade is "",  = 0, [ = const.,],  = const., Axisymmetric.
2, The control body and coordinate system Take the space between the front of the impeller, the rear cover and the blade inlet 1-1 section and the blade outlet 2-2 section as the control body, as shown in Figure 1-11. Rotating the impeller as a relative coordinate system, the flow is considered a steady flow.
3, momentum moment theorem and its analysis In a steady flow, the unit of time out of the fluid flowing into the control body and a change of momentum moment of a shaft, equal to the sum of all the external forces acting on the control body fluid on the same axis .
Suppose the flow rate of outflow and inflow control unit per unit time is qVT, the density of the fluid is , and the vertical distance between the absolute velocity vector of the inlet and outlet of the blades and the shaft is respectively l1 = r1cos1 and l2 = r2cos2. Therefore, the unit of time outflow into the control body of fluid on the shaft moment of momentum K were:
K2 = qVT2l2 = qVT2r2cos2, K1 = qVT1l1 = qVT1r1cos1
The role of the external body fluid in the control of the torque on the shaft are:
(1) The moment produced by the mass force. When studying the fluid's absolute motion, the mass force has only gravity, and due to the symmetry, the sum of the moments of gravity on the axis of rotation is zero.
(2) Torque generated by the surface force. It includes the impeller front and rear cover, 1-1 and 2-2 outside the control plane of the fluid and the blade moment of action on the fluid. Surface forces are stressful because they do not consider stickiness. Through the 1-1 and 2-2 control surfaces acting on the fluid in the direction of the pressure along the impeller radial direction, their torque on the shaft is zero; by the front and rear cover plate acting on the fluid pressure is symmetrical, and due to And the shaft parallel, so the torque on the shaft is also zero. The role of the control body fluid surface force on the shaft torque, only the shaft through the blade to the fluid torque.
4, the derivation of the process set all external forces on the shaft torque and M, then according to momentum moment theorem:
M = K2 - K1 = qVT (2r2cos2-1r1cos1) (1-7)
In the above formula, the moment M is the torque transmitted by the prime mover to the impeller. When the impeller rotates at an isometric velocity, the power delivered to the fluid is:
P = M = qVT (2r2cos2-1r1cos1)
Since u2 = r2, u1 = ωr1, 2u = 2cos2, 1u = 1cos1, into the above formula was:
P = qVT (u22u- u11u)
The unit of gravity fluid flow through the impeller when the energy obtained, that is, the infinite number of blades when the theoretical head HT ï‚¥ is:
 (m) (1-8)
Although the centrifugal impeller is used as an example, the axial impeller is also established. Therefore, the formula is the energy equation of the vane pump and the fan. Since Euler was first exported in 1756, Also known as the Euler equation.
For fans, it is customary to use the full pressure to represent the energy obtained by the fluid, ie the energy obtained when the gas per unit volume flows through the impeller. Since pT = gHT, so the energy equation of the fan is:
 (Pa) (1-9)
For axial-flow pumps and fans, since u1 = u2 = u, substituting (1-8), (1-9) for the two equations results in a simplified version of the axial-flow pump and the fan energy equation:
 (m) (1-10)
 (Pa) (1-11)
Second, the energy equation analysis Energy equation to the impeller fluid work done with the fluid parameters of motion linked, so it is calculated based on the impeller. In the process of derivation, the flow of fluid in the inlet and outlet of the impeller is only involved due to avoiding the complicated fluid flow inside the impeller. Therefore, this method has been widely used in blade machinery.
Analysis of the energy equation (1-8) can be considered from the following three aspects:

1, should be clear energy equation:



The applicable conditions and significance of the energy equation have been described (shown in red). From (1-8), the theoretical head HT HT obtained by the fluid is independent of the density of the fluid being delivered. That is to say, if the impellers are the same size, rotate at the same speed and have the same flow rate, the theoretical lift of the same liquid column or column height can be obtained no matter whether the water is delivered, air or even other density fluids. Obviously, the theoretical total pressure in Eqs. (1-9) is different because the total pressure is related to the density.
2, to improve the infinite number of blades theory can head several measures (1) inhalation conditions. In (1-8) u11u reflects the pump and fan suction conditions, reducing u11u can also increase the theoretical head. Therefore, in the design of pumps and fans, it is generally best to make 1 ≈ 90 (ie, the fluid is approximately radial inflow at the inlet, 1u≈0) to obtain a higher energy head.
(2) impeller diameter D2, peripheral speed u2. From (1-8), (1-9) can be seen that the impeller theoretical head and impeller diameter D2, the circumferential velocity u2 is proportional to. Because u2 = D2n / 60, so when the other conditions are the same, increase the impeller diameter D2 and increase the speed n can improve the theoretical head. However, as will be seen later (Section 6), increasing D2 increases the impeller's frictional losses, reducing the efficiency of the pump and fan, while increasing the structural size, weight, and manufacturing cost of the pump and fan. In addition , But also by the material strength, process requirements and other restrictions, it can not be excessive increase D2. Increase the speed, can reduce the impeller diameter, thus reducing the size and weight of the structure, can reduce manufacturing costs, while increasing the speed of the efficiency and other properties will also be improved. Therefore, the use of increased speed to improve the pump and fan theoretical head is currently widely used method. At present, large-scale thermal power plant feed pump speed has reached 7500r / min. However, the increase in rotational speed is also limited by material strength and pump cavitation (Section I, Chapter II) and fan noise (Section V, Section V), so the speed can not be increased indefinitely .
(3) The component of the absolute speed in the circumferential direction 2u. Increasing 2u also increases the theoretical head, while 2u is related to the type of impeller, that is, the outlet mounting angle 2y, as discussed in the third section.
3, the second form of energy equation In order to more clearly understand the physical concept (1-8), from the impeller blade inlet and outlet velocities know triangle:
, Where i = 1 or i = 2 into equation (1-8), have:

(1-12)
The theoretical fluid obtained can be divided into two parts:
The first part, Hst, shows the added value of the static head of the fluid flowing through the impeller. For axial flow pumps and fans, the first term of Hst 等 is equal to zero since u1 = u2 = u, which means that axial flow pumps and fans can have a lower head than centrifugal mode under the same conditions; To increase the static head of axial-flow pumps and fans, we must try to improve the w1 , for which purpose, the area of ​​the blade inlet should be smaller than its outlet area. In practice, the inlet of the axial flow impeller blade is often slightly thickened, making the airfoil section (2y> 1y) one of the methods.
The second part of Hdï‚¥: fluid flow through the impeller when the added value (or simply dynamic pressure head). This kinetic energy head in the guide vane or volute behind the impeller in the partial conversion into static head (or static head). However, from the hydrodynamic point of view, the loss of the static head into the kinetic head is small, whereas the loss of the kinetic head into the static head is greater. Therefore, in the design of pumps and fans, in order to improve the efficiency of pumps and fans, on the one hand should strive to reduce the proportion of kinetic energy head, on the other hand try to make the diversion part of the design is reasonable, so that streamline to reduce the loss.
Finally, it should be pointed out that since the energy equation is based on several basic assumptions of the flow analysis, the fluid supplied by the impeller should be completely and completely captured by the fluid. This is impossible in practice. Because the flow of fluid in the impeller is very complicated, various losses will be generated in the flow and the energy obtained by the fluid will be reduced. Therefore, applying the conclusions reached in this section to engineering practice needs to be revised in the following sections.
§ 1-3 blade outlet installation angle on the theoretical head

First, the three types of centrifugal impeller (Figure P19 Figure 1-12)
I, backward impeller,  2y  <90 , the blade bending direction and impeller rotation opposite;
II, Radial impeller, 2y = 90, the blade outlet is radial;
III, forward impeller,  2y > 90 . The blade bending direction and the same direction of rotation of the impeller.