Hydraulics and Pneumatics Transmission

1 Fluid Properties2 Mechanics of fluids at rest3 Mechanics of fluids in motion4 Energy loss of fluids in motion5 Flow of fluids in clearances and orifices6 Hydraulically sticking7 Hydraulically shocking8 CavitationFLUID MECHANICS WITH HYDRAULICS

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

In hydraulic transmission problems, p32Mpa, the volume variation caused by pressure variation isConsequently, hydraulic oil may be regarded as uncompressible.----------------------------------------------------------------------------------- Found by experiment:1 Fluid Properties

1.3.1 cohesion force and adhesion force1.3 Viscosity1.3.2 dynamic viscosityThere 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. Consider two parallel plates, placed a small distance Y apart, the space between the plates being filled with the fluid.

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.

Experiment has shown that for a large class of fluids If a constant of proportionality is now introduced, the shear stress between any two thin sheets of fluid may be expressed by

Above equation is called Newtows equation of viscosity and in transposed form it serves to define the proportional constant which is called the dynamic viscosity. The viscosity is the natural property of fluids, but shown only in fluid flow. The viscosity is a measure of 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.1 Fluid Properties

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 viscosityE.g. the kinematic viscosity of 30# mechanic oil is 30 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 Fluid Properties

1.3.3 Relative viscositya. Definition : E= t1 /t2b. Transposed relation t1 -time the measured liquid (200mL, TC) passes through the viscosity-meter orifice(2.8mm);t2 time the pure water (200mL, 20C) passes through the viscosity-meter orifice(diameter=2.8mm).1 Fluid PropertiesThe lubrication oil is named according to its kinematic viscosity E.g. For 30#mechanical oil, its kinematic viscosity is 30 The gasoline is named according to its octaneThe diesel oil is named according to its freezing point

2 Mechanics of fluids at rest Fluid statics is the study of fluids 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 areab. The pressure is the same in all direction Note geometric relation and neglect higher order term, thus

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 ) 2 Mechanics of fluids at rest

Newtons second law is writtenin vector for constant mass system. F=ma This results in three componentequations, Where ax, ay, and az are the components of the acceleration of the element.2 Mechanics of fluids at rest

Division by the elements volume dxdydz yields The equation expresses the relation between pressure variation and acceleration.2 Mechanics of fluids at rest

2.3 Examples included in fluid statics.E1: liquid at restSolutionAnother form is written asWhere p----the pressure at a point; gh---the pressure caused by liquid column weight; p0------the pressure caused by external force (either gas or liquid or solid).2 Mechanics of fluids at rest

2 Mechanics of fluids at rest

E2: Trolley in a linear accelerationSolution 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 surface.2 Mechanics of fluids at rest

E3: Rotating containerSolution The the reference frame is established on the container. According to Equation (1)The constant-pressure surface is a parabaloid of revolution2 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.2 Mechanics of fluids at restfuchsin

2.5 Forces on plane Areas and on curved surfacea. Forces on plane Areasb.Forces on curved surfacesWhere Ax, Ay and Az are project areas in three directions------------------------------------------------------------------------2 Mechanics of fluids at rest

3 fluid kinematics and dynamics3.1 Description of fluid motiona. 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 listedwhere 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.

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 field3 Mechanics of fluids in motion

3.2 Key concepts a. Ideal fluid:b. Incompressible and compressible fluidc. 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 particles locations (Lagrange description)A fluid is presumed to have no viscosity An incompressible fluid is the one whose density remains relatively constant. Generally speaking, liquids can be considered as incompressible fluids while gases as compressible fluids3 Mechanics of fluids in motion

e. Streamline: A streamline is a curved 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, path lines and streamlines are all coincident.3 Mechanics of fluids in motion

f. Stream tube: A stream tube is a tube whose walls are steamlines. 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 (angle) to the direction of velocity. 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. 3 Mechanics of fluids in motion

3.3 Equation of continuity Assume an 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.3 Mechanics of fluids in motion

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

3.4 Differential equation of steady flow for ideal fluid Consider steady flow of an ideal fluid. Use a infinitesimal cylindrical element, with length ds andcross-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-direction3 Mechanics of fluids in motion

Apply Newtons second law , we havea. The Bernoulli equation on following assumptions:The equation is called Eulerian equation3.5 Bernoulli equation Consider geometric relation Thus (1) Ideal fluid; (2) Steady flow; (3) An infinitesimal stream tube; (4) Constant density; (5) Inertial reference frame.3 Mechanics of fluids in motionsimplifying the expression, we have

3 Mechanics of fluids in motionorwhere

The physical significance of the equation: 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 pressuredecreases, a statement known as the Bernoulli principle. 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---------------------------------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 followingwhere henergy losses caused by friction forces3 Mechanics of fluids in motion

c. The Bernoulli equation on following assumptions: (1) real fluid; (2) steady flow; (3) a real pipe; (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 tubeRewrite the integration3 Mechanics of fluids in motion

Note that in the cross section of gradually varied flow HenceLet We can obtainwhere v1 and v2 ----mean velocities; 1 and2----kinetic energy correction factors, =1~2. The selected cross sections should ensure that the stream lines across the cross section are approximately parallel (gradually varied flow)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.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 obtain3 Mechanics of fluids in motion

Inserting this value of v2 in foregoing expression, we obtain3 Mechanics of fluids in motion

According to static pressure equation, select equal pressure planeO1O1Finally, the flow rate is3 Mechanics of fluids in motion

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 Substituting data for these variables, we obtain the ideal throat flow rate3 Mechanics of fluids in motion

a. Momentum theorem and d'Alembert principleThe expression of momentum theorem is3.6 momentum equation The d'Alembert principle expression of momentum theorem is3 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.3 Mechanics of fluids in motion

Assume that time t lapsesthe fluid flows from cross section1-1 and 2-2 to cross section 1-1 and 2-2.Both sides are divided by t, then taking limit The variation of the fluid momentum is3 Mechanics of fluids in motion

Momentum change rate caused by time variationThe expression is written into d'Alembert principle equationTransient flow forceMomentum change rate caused by position variationSteady flow force3 Mechanics of fluids in motion

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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 rate3 Mechanics of fluids in motion

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Where v1, v2 ---mean velocity on cross section 1-1 and 2-2 respectively 1, 2 ---momentum correction factors on cross section 1-1 and 2-2 respectively =14/3.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 is3 Mechanics of fluids in motion

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Solution steps Select a control volume Express all external forces in a figure Select a reference frame Write component momentum equations calculate parameters;Sometimes the Newtons third law is applied.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.c. Momentum equation of incompressible fluid Assumptions: (1) Incompressible fluid; (2) Steady flow; (3) An real tube.3 Mechanics of fluids in motion

c. Examples SolutionSelect the y shaped tube as a control volume.Express all external forces as shown in the FigureSelect the reference frame as shown in the Figure-------------------------------------------------------------------------3 Mechanics of fluids in motion

c. List component equations of momentumIn x direction: According to Newtons third law, the forces acting on the tube are3 Mechanics of fluids in motionIn y direction:-------------------------------------------------------------------------------

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E2: Stablility analysis of directional control valveSolutionBoth are stable.3 Mechanics of fluids in motion

Synthetical application of three equationsTo sum up, solve the equation including one unknown numberso as to exhibit your clear mind of thoughts.

No.equitionpurposeInitial workGeneral sequence1Continuity equitionSolved for velocitySelect cross sectionsfirst2Bernoulli equitionSolved for pressureSelect cross sections and potential energy base line second3Momentum equitionSolved for Restrictive force Select control volumethird

Energy loss is usually called power loss or head loss. Head loss is the measure of the reduction in the total head (sum of elevation head, velocity head and pressure head) of the fluid as it moves through a fluid system. Head loss includes friction loss and local loss As the fluid flows through straight pipes fiction loss occurs. Losses due to the local disturbance of the flow are called local losses . The viscous friction will cause energy loss, the loss is called the loss caused by viscous force. The fluid particles do not move in uniform linear motion but they follow random paths, or exists the loss caused by inertial force4 Energy losses of fluids in motion

When v is small, a red line appearslaminar flow.When v is great, the red line disappearsturbulent flow.4.1 Reynolds regime experiment4 Energy losses of fluids in motion

vchigher critical velocity ; vc----lower critical velocity Actually, velocity is not the only factor that determines whether the flow regime is laminar or turbulent.4.2 Reynolds number vmean velocity; dtube diameter;kinematic viscosity.4 Energy losses of fluids in motion The flow regime depends on three physical parameters:Velocity, geometric scale and viscosity.

To noncircular ductA ratio of the inertial force to the viscous force 4.3 The physical significance of Reynolds number The physical nature of laminar flow is that the viscous force is dominant; while the physic nature of turbulent flow is that the inertial force is dominant.4 Energy losses of fluids in motion

4.4 The friction loss in laminar flow a. Velocity profile in laminar flow Use a small cylindrical element, with length L andcross-section r2. The forces acting on the element are pressure forces and the friction force.4 Energy losses of fluids in motion

Integrating and determining the constants of integration from the fact that u=0,when r=R, we obtainSumming up the forces in the flow direction ( parabolic profile )4 Energy losses of fluids in motion

c. Average velocity d. Kinetic energy correction factor and momentum correction factor e. Friction loss expressionb. Flow rate4 Energy losses of fluids in motion(-----friction factor) --------------------------------------------------------------------------------------

b. Laminar boundary layer4.5 Friction loss in turbulent flowa. The characteristic of turbulent flow A distinguishing characteristic of turbulence is its irregularity and no two particles may have identical even similar motion, so statistical mean of evaluation must be employed There can be no turbulence next to a smooth wall.Therefore, immediately adjacent to a smooth wall there will be a laminar or viscous sublayer. The thickness of the laminar boundary layer iswhere d-----diameter of a tube; ReReynolds number; -----friction factor in turbulent flow.4 Energy losses of fluids in motion

c. Classification of tubes If the roughness of tubes is considered there will be two type of tubes.4 Energy losses of fluids in motion

4 Energy losses of fluids in motionColebrook equation (Re>4000)Where is the turbulent friction factor.

4.5 Local lossesExpression of local pressure lossExpression of local head lossExpression of local pressure loss of standardized elements Loss due to the local disturbance of the flow in conduits such as changes in cross section, projecting gaskets,elbow, valves, and similar items are called local losses4 Energy losses of fluids in motion(v--exit velocity)

4 Energy losses of fluids in motion

5 flow of fluids in orifices and clearances5.1 flow in orificesa. Classification of orificesb. Flow characteristic in thin wall orifices

Selecting cross section 1-1and 2-2. Additionally selecting O-O as potential energy base line. According to Bernoulli equation Inserting these values in Bernoulli equation, we obtain

According to flow rate expression in a tubed. Universal expression in a orificec. Flow characteristic in long orifices5 flow of fluids in orifices and clearances

Flow characteristic: laminar5.3.1 Gap between parallel plates a. flow by the action of pressure difference5.2 flow inside a gap or clearance5 flow of fluids in orifices and clearances

The relationship between p and x is linear, therefore u has noting to do with p and x; while p is independent of u and y. Consequently

The flow rate passing through is Integrating and determining the constants of integration from the fact that u=0 when y=0 and u=0 when y=h, we obtain From the expression we conclude that because the leakage flow rate through a clearance is proportional to the cube of the clearance height the clearances among components require high dimensional accuracy degree and hence the initial cost of hydraulic elements is very high.5 flow of fluids in orifices and clearances

The flow rate isThe total flow rate is Integrating and determining the constants of integration from the fact that u=0 when y=0 and u=0 when y=h, we obtain a. flow by the action of shearing5 flow of fluids in orifices and clearances

5.3.2 annular gap a. Concentric annular gap5 flow of fluids in orifices and clearances

b.Eccentric annular gapIntegrating above expression from 0 to , yield From the expression we conclude that when components are assembled utterly eccentrically the leakage flow rate is 2.5 times that when utterly concentrically. We conclude that the clearances among components require high positional accuracy degree and hence the initial cost of hydraulic elements is very high.-------------------------------------------------------------------------------------5 flow of fluids in orifices and clearances

5.3.3 Gap between nonparallel plates The gap may be considered as a gap between parallel plates due to a infinitesimal length. In this manner, the expression of gaps between parallel plates can still be used, as long as the length L is replaced with dx, and the pressure drop replaced with dp. Using a infinitesimal element, with length dx in the x-direction ofthe stream.5 flow of fluids in orifices and clearances

a. Flow rateb. Pressure distributionIt may be found that p(x) is a nonlinear curve5 flow of fluids in orifices and clearances

c. Nonlinear factor analysisIn order to make analysis convenient, we define5 flow of fluids in orifices and clearances

It may be seen that: On the other hand, The curvature of p(x) is inversely proportional to 4-th power of the gap height. The more narrow the gap is, the larger The curvature of p(x) .5 flow of fluids in orifices and clearances

6. Hydraulically sticking 6.1 Concept Moving a cylindrical body in the barrel frequently need to exert a great force, even not able to move it. This phenomenon is call hydraulic stick.6.2 Cause Shape deviation and position deviation result in nonparallel clearances.6.3 Nonlinear pressure force According to the geometric significance of integration, the magnitude of the force is equal to the area under the cure

6.4 examples6. Hydraulically sticking

6.4 Solution a. Raising the accuracy degree of components( dimension accuracy degree,shape accuracy degree and position accuracy degree) b. Machining balancing grooves or anti-stiction grooves After machining pressure balancing grooves, pressure distribution curves are cut into many segments, lowering unbalance forces to a great degree. 6. Hydraulically sticking

7 hydraulically shocking The phenomenon encountered when the velocity of a liquid is abruptly decreased due to valve movement is called hydraulic shock or water hammer. It is possible to damage pipes, elements and systems, even to injure persons. When studying hydraulic shock the liquid is not incompressible and the piping is not rigid.7.1 Physic model and physic process Consider a single horizontal pipe of length L and diameter d. The upstream end of the pipe is connected to a reservoir and a valve is situated at downstream end.

Let us assume that the valve is closed instantaneously. The lamina of liquid next to the valve will be brought to rest, its kinematic energy is converted into pressure energy.When the lamina is compressed, the wall of the pipe will be stretched. Then , the second lamina will be brought to rest, its kinematic energy is converted into pressure energy. Next , the third lamina will be brought to rest, its kinematic energy is converted into pressure energy.7 hydraulically shocking

Under an excess pressure, some liquid starts to flow back into the reservoir. The pressure energy is converted into kinematic energy. The reverse velocity will produce a pressure drop that will be below the normal pressure. After the total liquid in the pipe be brought to motion, The pressure will reach minimum. If there were not damp, the periodic process would last forever.7 hydraulically shocking

7.2 Calculation of maximum pressure rise Assume that the valve is closed instantaneously, the velocity of liquid will abruptly decrease from v to v. The cross area of the pipe will become (A+A ), the pressure will rise to (p+p), and the pressure wave from m-m to n-n in time t Selecting an element, writing momentum equationWhere7 hydraulically shocking

Neglect higher order quantityThe resultant force isInsert these in the momentum equation, we obtainWhere c---travel velocity of the shock wave7 hydraulically shocking

c. Some measures1 Close valves as slowly as possible2 Use accumulators to absorb shock3 Arrange buffering valves and anti-overload valves4 Use rubber hoses to connect elements 7 hydraulically shocking

8. Cavitations When the local pressure becomes equal to the vapor pressure of the liquid, small vapor bubbles are generated and these bubbles collapse when they enter a high-pressure region. The collapse is accompanies by very large local pressures that last for only a small fraction of a second. These pressure spikes may reach a wall, where they can, after repeated applications result in significant damage. The phenomenon is called cavitations. In order to prevent cavitations from occurring, the system pressure, including all local pressures, should always be ensured to be above the atmospheric pressure.

The most popular/prevalent nomenclature at present is CFDcomputational fluid dynamicsBe perpendicular to/normal to/ be orthogonal to