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 8 Cavitation

1 Fluid Properties 1.1 Density
Fluid density was defined as mass per unit volume hydraulic oil ρ= kg/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

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

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.

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 μ is now introduced, the shear stress τbetween any two thin sheets of fluid may be expressed by (1)

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 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 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

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

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 area b. 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. A x y z dx dy dz 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 )

x y z dx dy dz 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.

Division by the element’s volume dxdydz yields (2) The equation expresses the relation between pressure variation and acceleration.

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

(4) 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).

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) pa x z a v When x=z=0, p=pa, thus The equal-pressure surface is not a horizontal plane but a slope surface.

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

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

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

3 fluid kinematics and dynamics
1736 ~ 1813 3.1 Description 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 The most popular/prevalent nomenclature at present is CFD—computational fluid dynamics 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.

3 Mechanics of fluids in motion
1707 ~1783 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 Convective acceleration Local acceleration

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)

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.

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. Be perpendicular to/normal to/ be orthogonal to g. Flow cross section A plane or curved surface at right (angle) to the direction of velocity.

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.

3.3 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 u2 2 u1 1 dA1 dA2 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.

The equation can be simplified, thus The equation can be integrated along flow cross section, yielding The equation indicates the mean velocity is inversely proportional to the flow area.

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 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

Apply Newton’s second law , we have simplifying 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

or where

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 ( ), Swiss mathematician, who showed that as the velocity of a fluid increases, the pressure decreases, a statement known as the Bernoulli principle.

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. 冷水热水温水 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

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

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 tube Rewrite the integration

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) Please note:

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.

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; Next, calculating parameter z1=z2=0, let , We can obtain

Use continuity equation Inserting this value of v2 in foregoing expression, we obtain

According to static pressure equation, select equal pressure planeO1O1， thus Finally, the flow rate is

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

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

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.

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

Momentum change rate caused by time variation Momentum change rate caused by position variation The expression is written into d'Alembert principle equation Transient flow force Steady flow force External resultant force Inertial force

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

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

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.

c. Examples E1: Calculate the force acting on the tube, assume followings are known: 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

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

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

Synthetical application of three equations
equition purpose Initial work General sequence 1 Continuity equition Solved for velocity Select cross sections first 2 Bernoulli equition Solved for pressure Select cross sections and potential energy base line second 3 Momentum equition Solved for Restrictive force Select control volume third To sum up, solve the equation including one unknown number so as to exhibit your clear mind of thoughts.

4 Energy losses of fluids in motion
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 force

4.1 Reynolds regime experiment When v is small, a red line appears—laminar flow. When v is great, the red line disappears—turbulent flow.

Laminar flow Turbulent flow vc’—higher critical velocity ; vc----lower critical velocity Actually, velocity is not the only factor that determines whether the flow regime is laminar or turbulent. The flow regime depends on three physical parameters: Velocity, geometric scale and viscosity. 4.2 Reynolds number v—mean velocity; d—tube diameter; ν—kinematic viscosity. Corresponding to two critical velocity，the two critical Reynolds numbers are Lower critical Reynolds number: Higher critical Reynolds number:

To noncircular duct v—mean velocity; ν—kinematic viscosity; R—hydraulic radius(R=A/x), x—wetted perimeter; 4.3 The physical significance of Reynolds number A ratio of the inertial force to the viscous force 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.

4.4 The friction loss in laminar flow a. Velocity profile in laminar flow Use a small cylindrical element, with length L and cross-section πr2. The forces acting on the element are pressure forces and the friction force.

Summing up the forces in the flow direction where Let Integrating and determining the constants of integration from the fact that u=0,when r=R, we obtain ( parabolic profile )

b. Flow rate c. Average velocity d. Kinetic energy correction factor and momentum correction factor e. Friction loss expression (λ-----friction factor)

4.5 Friction loss in turbulent flow a. 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 b. Laminar boundary layer 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 is where d-----diameter of a tube; Re—Reynolds number; λ-----friction factor in turbulent flow.

c. Classification of tubes If the roughness Δ of tubes is considered there will be two type of tubes. δ>Δ----Hydraulically smooth δ<Δ时---- Hydraulically rough When Re is very great

Colebrook equation (Re>4000) Where λ is the turbulent friction factor.

4.5 Local losses 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 losses Expression of local pressure loss (v--exit velocity) Expression of local head loss Expression of local pressure loss of standardized elements

Homework: Assuming that a 60 degree curved pipe is located in horizontal plane, (fluxing out to atmosphere), the inlet and outlet diameter are 80 and 40mm respectively, outlet velocity is 8m/s, the local loss coefficient ξ=0.1. Calculating the x and y component of the force acting on the pipe

5 flow of fluids in orifices and clearances
5.1 flow in orifices a. Classification of orifices Thin wall orifice: Thick wall orifice: Long orifice: b. Flow characteristic in thin wall orifices Coefficient of contraction

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

c. Flow characteristic in long orifices According to flow rate expression in a tube d. Universal expression in a orifice Thin wall orifice Long orifice

5.2 flow inside a gap or clearance Geometric characteristic: Flow characteristic: laminar 5.3.1 Gap between parallel plates a. flow by the action of pressure difference Simplifying the foregoing expression and inserting

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

Integrating and determining the constants of integration from the fact that u=0 when y=0 and u=0 when y=h, we obtain The flow rate passing through is 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.

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

5.3.2 annular gap a. Concentric annular gap Letting , using forgoing expression, we obtain

b.Eccentric annular gap Integrating above expression from 0 to π, 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.

5.3.3 Gap between nonparallel plates Using a infinitesimal element, with length dx in the x-direction of the stream. 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.

a. Flow rate b. Pressure distribution It may be found that p(x) is a nonlinear curve

c. Nonlinear factor analysis In order to make analysis convenient, we define

It may be seen that: On one hand, When p(x) is concave. When p(x) is convex. 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) .

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 examples

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.

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. …… After the total liquid will be brought to rest, the total kinematic energy of liquid will be converted into pressure energy. The pressure will reach maximum

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.

7.2 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+ΔA ), the pressure will rise to (p+Δp), and the pressure wave from m-m to n-n in time Δt Selecting an element, writing momentum equation Where

Neglect higher order quantity The resultant force is Since Insert these in the momentum equation, we obtain Where c---travel velocity of the shock wave

c. Some measures 1 Close valves as slowly as possible 2 Use accumulators to absorb shock 3 Arrange buffering valves and anti-overload valves； 4 Use rubber hoses to connect elements

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.

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