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Hydraulics and Pneumatics Transmission

Hydraulics and Pneumatics Transmission. FLUID MECHANICS WITH HYDRAULICS. 1 Fluid Properties 2 Mechanics of fluids at rest 3 Mechanics of fluids in motion 4 Energy loss of fluids in motion 5 Flow of fluids in clearances and orifices 6 Hydraulically sticking 7 Hydraulically shocking

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Hydraulics and Pneumatics Transmission

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  1. Hydraulics and Pneumatics Transmission

  2. FLUID MECHANICS WITH HYDRAULICS 1 Fluid Properties 2 Mechanics of fluids at rest 3 Mechanics of fluids in motion 4 Energy loss of fluids in motion 5 Flow of fluids in clearances and orifices 6 Hydraulically sticking 7 Hydraulically shocking 8 Cavitation

  3. 1 Fluid Properties 1.1 Density Fluid density was defined as mass per unit volume hydraulic oil ρ=890-910kg/m3 1.1 Compressibility and expansibility Suppose the volume is the function of pressure and temperature , or V=f(t,p ) , and t↑→V↑,p↑→V↓ The volumetric increment can be approximated by is total differential, thus

  4. where expansibility coefficient compressibility coefficient 1 Fluid Properties Found by experiment: In hydraulic transmission problems, p≤32Mpa, the volume variation caused by pressure variationis Consequently, hydraulic oil may be regarded as uncompressible.

  5. 1.3 Viscosity 1.3.1 cohesion force and adhesion force There are cohesion forces among fluid particles, while there are adhesion forces among fluid particles and solid wall. Adhesion forces are usually greater than cohesion force except mercury. 1.3.2 dynamic viscosity Consider two parallel plates, placed a small distance Y apart, the space between the plates being filled with the fluid.

  6. The lower surface is assumed to be stationary, while upper one is moved parallel to it with a velocity U by the application of a force F, corresponding to some area A of the moving plates The particles of the fluid in contact with each plate will adhere to it. The velocity gradient will be a straight line. The action is much as if the fluid were made up of a series of thin sheets.

  7. (1) Experiment has shown that for a large class of fluids If a constant of proportionality μis now introduced, the shearing stress τbetween any two thin sheet of fluid may be expressed by

  8. 1 Fluid Properties Above equation is called Newtow’s equation of viscosity and in transposed form it serves to define the proportional constant which is called the dynamic viscosity. The viscosity is the inherent property of fluids, but shown only in fluid flow. The viscosity is a measure of its resistance to shear or angle deformation. The viscosity accounts for energy losses associated with the transport of fluids in ducts and pipes. The friction forces in fluid flow result from the cohesion and momentum interchange between molecules in the fluid.

  9. 1 Fluid Properties As temperature increases, the viscosity of all liquids decreases, while the viscosity of all gases increase. This is because the force of cohesion, which diminishes with temperature, predominates with liquids; while with gases the predominating factor in the interchange of molecules between layers of different velocity. 1.3.3 Kinematic viscosity In many problems including viscosity there frequently appear the value of viscosity divided by density. This is defined as kinematic viscosity E.g. the kinematic viscosity of 30# mechanic oil is 30 厘斯

  10. 1 Fluid Properties 1.3.3 Relative viscosity a. Definition : °E= t1 /t2 t1 –-time the measured liquid (200mL, T°C) passes through the viscosity-meter orifice(2.8mm); t2 —time the pure water (200mL, 20°C) passes through the viscosity-meter orifice(diameter=2.8mm). b. Transposed relation The lubrication oil is named according to its kinematic viscosity E.g. For 30#mechanical oil, its kinematic viscosity is 30 厘斯 The gasoline is named according to its octane The diesel oil is named according to its freezing point

  11. 2 Mechanics of fluids at rest Fluid statics is the study of fluids in there is no relative motion between fluid particles. The only stress is normal stress, pressure, so it is the pressure that is of primary interest in fluid statics. 2.1 static pressure property a. The pressure is defined as the force exerted on a unit area b. The pressure is the same in all direction Note geometric relation and neglect higher order term, thus

  12. z A dz dx dy x y 2 Mechanics of fluids at rest 2.2 Basic differential equation Assume the pressure is the function of space coordinates, or p=f(x,y,z) . Consider a infinitesimal element in the figure. Assume that a pressure p exists at the center A of this element, the pressure each of sides can be expressed by using chain rule from calculus with p(x,y,z)

  13. z A dz dx dy x y 2 Mechanics of fluids at rest Newton’s second law is written in vector for constant mass system. ΣF=ma This results in three component equations, Where ax,ay,and az are the components of the acceleration of the element.

  14. (2) 2 Mechanics of fluids at rest Division by the element’s volume dxdydz yields The equation expresses the relation between pressure variation and acceleration.

  15. (3) The quantity is referred to as the piezometric head (4) p0 2 Mechanics of fluids at rest 2.3 Examples included in fluid statics. E1: liquid at rest Solution: Another form is written as Where p—----the pressure at a point; ρgh---the pressure caused by liquid column weight; p0------the pressure caused by external force (either gases or liquid or solid).

  16. Explanation: ① The term is used to convert pressure to a height of liquid. ② Neglect the pressure caused by the fluid weight when studying gases. ③ Neglect the pressure caused by the liquid weight and the pressure caused by atmosphere on hydraulic transmission. ④ The equal-pressure surface is a horizontal plane. ⑤ The free surface is a special equal-pressure surface (p=pa). (4) 2 Mechanics of fluids at rest

  17. pa z x a v When x=z=0,p=pa, thus 2 Mechanics of fluids at rest E2:trolley in a linear acceleration Solution : The liquid is at rest relative to the trolley, so the reference frame is established on the trolley. According to Equation (1) The equal-pressure surface is not a horizontal plane but a slope

  18. z x y pa pa When x=y=z=0,p=pa, thus 2 Mechanics of fluids at rest Solution : The the reference frame is established on the container. According to Equation (1) E3: Rotating container The constant-pressure surface is a paraboloid of revolution

  19. 2 Mechanics of fluids at rest 2.4 Absolute pressure, gage pressure and vacuum If pressure is measured relative to absolute zero, it is called absolute pressure. When measured relative to atmosphere as a base, it is called gage pressure . If the pressure is below that of the atmosphere, it is designated as a vacuum. fuchsin

  20. 2 Mechanics of fluids at rest 2.5 Forces on plane Areas and on curved surface a. Forces on plane Areas b.Forces on curved surfaces Where Ax, Ay and Az are project areas in three directions

  21. 3 fluid kinematics and dynamics 1736 ~ 1813 3.1Description of fluid motion a. Lagrangian description In the study of particle mechanics, attention is focused on individual particles, motion is observed as a function of time, the position, velocity, and acceleration of each particle are listed where x, y and z are transient position coordinates。 This description is easily acceptable but difficult as the number of particles becomes extremely large in a fluid flow.

  22. 3 Mechanics of fluids in motion 1707 ~1783 Convective acceleration Local acceleration b. Eulerian description An alternative to following each fluid particle separately is to identify points in space and the observe the velocity of particles pass each point. The flow properties, such as velocity, are functions of both space and time. where x, y and z are the position coordinates of the flow field

  23. 3 Mechanics of fluids in motion 3.2 Key concepts a. Ideal fluid: A fluid is presumed to have no viscosity b. Incompressible and compressible fluid An incompressible fluid is the one whose density remains relatively constant. Generally speaking, liquids can be considered as incompressible fluids while gases as compressible fluids c. Steady flow: Where the quantities of interest do not depend on time. d. Path line: A path line is the locus of points traversed by a given particle as it travels in the flow field. Note that a path line is a history of the particle’s locations (LaGrange description)

  24. 3 Mechanics of fluids in motion e. Streamline: A streamline is a line possessing following property: the velocity vector of each particle occupying a point on the streamline is tangent to the streamline (Eulerian description) In a steady flow, pathlines and streamlines are all coincident.

  25. 3 Mechanics of fluids in motion f. Stream tube: A stream tube is a tube whose walls are steam line. Note that no fluid can cross the walls of a stream tube since the velocity is tangent to a stream line People often sketch a stream tube with a infinitesimal cross section in the interior of flow for demonstration purposes. g. Flow cross section A plane or curved surface at right to the direction of velocity.

  26. 3 Mechanics of fluids in motion h. Flow rate and mean velocity The quantity of fluid flowing per unit time across any section is called the flow rate. In dealing with incompressible fluids,volume flow rate is commonly used, whereas mass flow rate is more convenient with compressible fluids. The mean value of the velocity in a cross section is called the mean velocity. This indicates that the volume flow rate is equal to the magnitude of the mean velocity multiplied by the flow area at right to the direction of velocity.

  27. u2 2 2 dA2 1 dA1 1 u1 3 Mechanics of fluids in motion 3.3 Equation of continuity Assume a incompressible fluid steadily flows in the infinitesimal stream tube. The following figure represents a short length of a stream tube The fixed volume between the two fixed sections of the stream tube is called the control volume. According to mass conservation law, in the time dt, the mass flowing in the control volume must be equal to the mass flowing out the control volume.

  28. 3 Mechanics of fluids in motion The equation can be simplified, thus The equation can be integrated along flow section The equation indicates the mean velocity is inversely proportional to the flow area.

  29. 3 Mechanics of fluids in motion 3.4 Differential equation of steady flow for ideal fluid Consider steady flow of a ideal fluid. Use a infinitesimal cylindrical element, with length ds and cross-section dA, in the s-direction of the stream. The forces acting on the element are pressure forces and the weight. Summing up the forces in the s-direction, there results The acceleration of the s-direction

  30. 3 Mechanics of fluids in motion Apply Newton’s second law , we have simplify the expression, we have The equation is called Eulerian equation 3.5 Bernoulli equation a. The Bernoulli equation on following assumptions: (1) Ideal fluid; (2) Steady flow; (3) An infinitesimal stream tube; (4) Constant density; (5) Inertial reference frame. Consider geometric relation Thus

  31. 3 Mechanics of fluids in motion The above expression is integrated along the stream line, the result where Bernoulli equation indicates that the total energy of a fluid flowing from 1 cross section to 2 cross section remains constant though one energy form can be converted into another. Bernoulli • Daniel (1700-1782), Swiss mathematician, who showed that as the velocity of a fluid increases, the pressure decreases, a statement known as the Bernoulli principle.

  32. 热水 冷水 温水 3 Mechanics of fluids in motion E1: Manufacture a shower In order to suck hot water into the tube, the pressure inside the tube need be lower than atmospheric pressure. A good idea is to increase kinematic energy, that is to say, to decrease the diameter of the tube. E2: The lift force of an airplane In order to make an airplane lift, the pressure under the wing need be higher than that on the wing. A good idea is to make the wing have different curve surfaces

  33. 3 Mechanics of fluids in motion b. The Bernoulli equation on following assumptions: (1) Real fluid; (2) Steady flow; (3) An infinitesimal stream tube; (4) Constant density; (5) Inertial reference frame. The ideal fluid flow or inviscid flow does not cause energy losses; while a real fluid flow or viscous flow will cause energy losses. If energy losses are considered the Bernoulli equation can be written as following where h’—energy losses caused by friction forces

  34. 3 Mechanics of fluids in motion c. The Bernoulli equation on following assumptions: (1) real fluid; (2) steady flow; (3) a real tube; (4) constant density; (5) inertial reference frame; (6) cross sections of gradually varied flow A real tube can be considered as consisting of countless infinitesimal stream tubes. Consequently, we can integrate the above equation along the cross-section of a real tube Rewrite the integration

  35. Please note: 3 Mechanics of fluids in motion Note that in the cross section of gradually varied flow Hence Let We can obtain where v1 and v2 ----mean velocities; α1 andα2----kinetic energy correction factors, α=1~2. The selected cross sections should ensure that the stream lines across the cross section are approximately parallel (gradually varied flow)

  36. 3 Mechanics of fluids in motion d. Example Venturi meter A Venturi meter consists of one tube with a constricted throat which produces an increased velocity accompanied by a reduction in pressure. The meter is used for measuring the flow rate of both compressible and incompressible. Assuming D1=200mm, D2=100mm, the height of the mercury column h=45mm，Calculate the flow rate of water.

  37. Next, calculating parameter z1=z2=0, let , 3 Mechanics of fluids in motion Solution First, selecting two flow cross section I-I and II-II; Second, select potential energy base line O-O; Then, writing the Bernoulli equation between cross section I-I and II-II; We can obtain

  38. Use continuity equation 3 Mechanics of fluids in motion Inserting this value of v2 in foregoing expression, we obtain

  39. thus 3 Mechanics of fluids in motion According to static pressure equation, select equal pressure planeO1O1， Finally, the flow rate is

  40. 3 Mechanics of fluids in motion Substituting data for these variables, we obtain the ideal throat flow rate As there is some friction losses between cross section 1-1 and 2-2, the true velocity is slightly less than the value given by the expression. Hence, we may introduce a discharge coefficient C, so that the flow rate is given

  41. 3 Mechanics of fluids in motion 3.6 momentum equation a. Momentum theorem and d'Alembert principle The expression of momentum theorem is The d'Alembert principle expression of momentum theorem is

  42. 3 Mechanics of fluids in motion b. The derivation of momentum equation Assumptions: (1) Incompressible fluid; (2) Steady flow; (3) An infinitesimal stream tube; (4) Constant density; (5) Inertial reference frame. Use a infinitesimal stream tube between section 1-1, with a velocity u1 and cross-section dA1 , and 2-2, with a velocity u2 and cross-section dA2, as the control volume. It may be note that the control volume is fixed.

  43. 3 Mechanics of fluids in motion Assume that time Δt lapses，the fluid flows from cross section1-1 and 2-2 to cross section 1’-1’ and 2’-2’. The variation of the fluid momentum is Both sides are divided by Δt, then taking limit

  44. External resultant force Inertial force 3 Mechanics of fluids in motion Momentum change rate caused by position variation Momentum change rate caused by time variation The expression is written into d'Alembert principle equation Steady flow force Transient flow force

  45. 3 Mechanics of fluids in motion The momentum change rate caused by time variation is equal to zero when flowing steadily. The momentum change rate caused by position variation is calculated as following The integration of momentum change rate

  46. 3 Mechanics of fluids in motion Where v1, v2 ---mean velocity on cross section 1-1 and 2-2 respectively； β1, β2 ---momentum correction factors on cross section 1-1 and 2-2 respectively， β=1～4/3. External forces The external forces acting on the fluid inside the control volume can be classified three types: (1)pressure forces on cross sections; (2) weight force of the fluid inside the control volume; (3) Restrictive force of the control volume, that is

  47. 3 Mechanics of fluids in motion c. Momentum equation of incompressible fluid Assumptions: (1) Incompressible fluid; (2) Steady flow; (3) An real tube. Explanation: The resultant force acting on the fluid inside the control volume is equal to that in unit time the momentum fluxing out the control volume is subtracted by the momentum fluxing in the control volume. It may be noted that the equation is vector equation. Solution steps ①Select a control volume；② Express all external forces in a figure；③ Select a reference frame；④ Write component momentum equations；⑤ calculate parameters;⑥Sometimes the Newton’s third law is applied.

  48. E1: Calculate the force acting on the tube, assume followings are known: 3 Mechanics of fluids in motion c. Examples Solution Select the “y” shaped tube as a control volume. Express all external forces as shown in the Figure Select the reference frame as shown in the Figure

  49. Consequently 3 Mechanics of fluids in motion c. List component equations of momentum In x direction: In y direction: According to Newton’s third law, the forces acting on the tube are

  50. E2: Stable analysis of directional control valve Solution Both are stable.

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