Hydraulics and Pneumatics Transmission

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Hydraulics and Pneumatics Transmission
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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
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1 Fluid Properties
1.1 Density Fluid density was defined as mass
per unit volume
hydraulic oil ?890-910kg/m3
1.1 Compressibility and expansibility
Suppose the volume is the function of
pressure and temperature , or Vf(t,p ) , and
t??V?,p??V?
The volumetric increment can be approximated by
is total differential, thus
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1 Fluid Properties
Found by experiment
In hydraulic transmission problems,
p32Mpa, the volume variation caused by pressure
variation is
Consequently, hydraulic oil may be regarded as
uncompressible. ----------------------------------
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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.
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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.
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Experiment has shown that for a large class of
fluids
If a constant of proportionality µ is now
introduced, the shear stress tbetween any two
thin sheets of fluid may be expressed by
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1 Fluid Properties
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.
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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 ??
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1 Fluid Properties
1.3.3 Relative viscosity
a. Definition E t1 /t2
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(diameter2.8mm).
b. Transposed relation
The lubrication oil is named according to its
kinematic viscosity E.g. For 30mechanical oil,
its kinematic viscosity is 30 ??
The gasoline is named according to its octane The
diesel oil is named according to its freezing
point
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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
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2 Mechanics of fluids at rest
2.2 Basic differential equation
Assume the pressure is the function of space
coordinates, or pf( 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 )
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2 Mechanics of fluids at rest
Newtons second law is written in vector
for constant mass system.
SFma
This results in three component equations,
Where ax, ay, and az are the components of
the acceleration of the element.
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2 Mechanics of fluids at rest
Division by the elements volume dxdydz
yields
The equation expresses the relation between
pressure variation and acceleration.
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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
?gh---the pressure caused by liquid column
weight p0------the pressure caused
by external force (either gas or liquid or solid).
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2 Mechanics of fluids at rest
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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 surface.
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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 parabaloid of
revolution
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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
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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 ---------------------------------------
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3 fluid kinematics and dynamics
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
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.
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3 Mechanics of fluids in motion
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
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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 particles locations (Lagrange
description)
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3 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.
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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.
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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.
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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.
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3 Mechanics of fluids in motion
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.
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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 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
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3 Mechanics of fluids in motion
Apply Newtons 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
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3 Mechanics of fluids in motion
or
where
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3 Mechanics of fluids in motion
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 pressure decreases, a
statement known as the Bernoulli principle.
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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 ---------------------------------
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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 henergy losses caused by friction forces
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3 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 tube
Rewrite the integration
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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 a1
anda2----kinetic energy correction factors, a12.
The selected cross sections should ensure that
the stream lines across the cross section are
approximately parallel (gradually varied flow)
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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 D1200mm,
D2100mm, the height of the mercury column
h45mm, Calculate the flow rate of water.
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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
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3 Mechanics of fluids in motion
Inserting this value of v2 in foregoing
expression, we obtain
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3 Mechanics of fluids in motion
According to static pressure equation, select
equal pressure planeO1O1,
Finally, the flow rate is
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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
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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
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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.
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3 Mechanics of fluids in motion
Assume that time ?t 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 ?t, then taking limit
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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
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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
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3 Mechanics of fluids in motion
Where v1, v2 ---mean velocity on cross section
1-1 and 2-2
respectively ß1, ß2 ---momentum
correction factors on cross section
1-1 and 2-2 respectively, ß14/3.
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
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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 ?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.
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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 ---------------------------------------
----------------------------------
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3 Mechanics of fluids in motion
c. List component equations of momentum
In x direction
In y direction
According to Newtons third law, the forces
acting on the tube are
--------------------------------------------------
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3 Mechanics of fluids in motion
E2 Stablility analysis of directional control
valve
Solution
Both are stable.
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Synthetical application of three equations
No. 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.
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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
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4 Energy losses of fluids in motion
4.1 Reynolds regime experiment
When v is small, a red line appearslaminar
flow. When v is great, the red line
disappearsturbulent flow.
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4 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.
The flow regime depends on three physical
parameters Velocity, geometric scale and
viscosity.
4.2 Reynolds number
vmean velocity dtube diameter ?kinematic
viscosity.
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4 Energy losses of fluids in motion
To noncircular duct
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.
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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 and cross-section pr2.
The forces acting on the element are pressure
forces and the friction force.
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4 Energy losses of fluids in motion
Summing up the forces in the flow direction
Integrating and determining the constants of
integration from the fact that u0,when rR, we
obtain
( parabolic profile )
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4 Energy losses of fluids in motion
b. Flow rate
c. Average velocity
d. Kinetic energy correction factor and
momentum correction factor
e. Friction loss expression
(?-----friction factor)
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4 Energy losses of fluids in motion
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
ReReynolds number ?-----friction
factor in turbulent flow.
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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.
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4 Energy losses of fluids in motion
Colebrook equation (Regt4000)
Where ? is the turbulent friction factor.
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4 Energy losses of fluids in motion
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
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4 Energy losses of fluids in motion
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5 flow of fluids in orifices and clearances
5.1 flow in orifices
a. Classification of orifices
b. Flow characteristic in thin wall orifices
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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
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5 flow of fluids in orifices and clearances
c. Flow characteristic in long orifices
According to flow rate expression in a tube
d. Universal expression in a orifice
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5 flow of fluids in orifices and clearances
5.2 flow inside a gap or clearance
Flow characteristic laminar
5.3.1 Gap between parallel plates a. flow by
the action of pressure difference
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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
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5 flow of fluids in orifices and clearances
Integrating and determining the constants of
integration from the fact that u0 when y0 and
u0 when yh, 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.
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5 flow of fluids in orifices and clearances
a. flow by the action of shearing
Integrating and determining the constants of
integration from the fact that u0 when y0 and
u0 when yh, we obtain
The flow rate is
The total flow rate is
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5 flow of fluids in orifices and clearances
5.3.2 annular gap a. Concentric annular gap
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5 flow of fluids in orifices and clearances
b.Eccentric annular gap
Integrating above expression from 0 to p, 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. --------------------------------------------
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5 flow of fluids in orifices and clearances
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.
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5 flow of fluids in orifices and clearances
a. Flow rate
b. Pressure distribution
It may be found that p(x) is a nonlinear curve
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5 flow of fluids in orifices and clearances
c. Nonlinear factor analysis
In order to make analysis convenient, we define
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5 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) .
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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
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6. Hydraulically sticking
6.4 examples
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6. 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.
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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.
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7 hydraulically shocking
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.

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