液压与气压传动(英汉双语)第2版-教学课件-第2章-Chapter

Hydraulic

and

pneumatic

pressure

transmissionChapter

2Fundamental

HydraulicFluid

MechanicsChapter

2

Fundamental

Hydraulic

Fluid

MechanicsPerformances

of

the

Hydraulic

OilHydrostaticsHydrodynamicsCharacteristics

of

Fluid

Flow

in

PipelineFlow

Rate

and

Pressure

Features

ofOrificeHydraulic

Shock

andCavitationChapterlistChapter

2

Fundamental

Hydraulic

Fluid

MechanicsPerformances

of

the

Hydraulic

OilThe

Main

performancesThe

requests

and

choice

of

hydraulic

oilChapter

2

Fundamental

Hydraulic

Fluid

MechanicsThe

Main

performancesDensity

(kg/m3), is

the

bulk

modulus

of2.

Compressibilitythe

coefficient

ofcompressibilityelasticity(2-1)(2-2)is

defined

as

the

ratio

of

the

change

in

pressure

( )torelative

change

in

volume

( )

while

the

temperature

remainsconstant.4Chapter

2

Fundamental

Hydraulic

Fluid

MechanicsFig.2-1 The

sketch

ofviscosityThe

experiments

haveproved

that

friction

forcebetween

the

two

fluidmolecules

can

be

describedas(2-3)Where isviscositycoefficient,

also

kinematicviscosity.3.

ViscosityThe

sketch

of

viscosity

is

illustrated

by

Fig.

2-1.Cohesion

betweentwomolecules……Chapter

2

Fundamental

Hydraulic

Fluid

MechanicsThere

are

three

methods

to

describe

theviscosity:absolute

viscosity,

Kinematic

viscosity

and

relative

viscosity.(1)

Dynamic

viscosity

or

absolute

viscosity

μ(Pa•s)

or

(N

•s/m2)(2)

Kinematic

viscosityν(mm2/s)(2-4)(3)

Relative

viscosity

(conditional

viscosity)The

relative

viscosity which

used

in

China

is

tested

by

theviscometer,

suchas

Fig.2-2.Chapter

2

Fundamental

Hydraulic

Fluid

Mechanics7Fig.2-2

PrincipleofviscometerChapter

2

Fundamental

Hydraulic

Fluid

MechanicsTake

the

note describes

the

viscosity:The

conversion

formula

between

theand

kinematic

viscosity

is(m2/s)(4)

Viscosity-temperature

:For

the

viscosityless

than15 and

thetemperature30

℃～150℃,theviscosity-temperature

formula

is

describe

as

following(Wecan

alsolook

up

fromFig.2-3):(2-5)(2-6)(2-7)(5)

Viscosity-pressure(2-8)(6)

Others

performances:

physical

and

chemical,

such

asanti-inflammability,anti-oxygenation,anti-concreting,

anti-foam

and

anti-corrosion

etc

.Chapter

2

Fundamental

Hydraulic

Fluid

MechanicsFig.

2-3

Theviscosity-temperature

of

homemadeoilsChapter

2

Fundamental

Hydraulic

Fluid

MechanicsisThe

hydraulic

oil

in

a

hydraulic

system

atrecommended

generally.2.1.2

The

requests

and

choice

of

hydraulic

oilRequestThe

oil

plays

two

roles

of

transmission

energy

and

lubrication

on

the

surfaces

ofworking

interaction.The

requests

for

the

hydraulic

fluids

are:

appropriate

viscosity,

the

good

inproperty

of

favorable

viscosity－temperature,

a

good

lubricity,

chemically

andenvironmentally

stabilities,

compatible

with

othersystem

materials

and

so

on.ChoiceThe

hydraulic

oil

should

be

chosen

in

according

to

the

request

of

hydraulicpump.The

hydraulic

oil

viscosity

adapted

for

different

hydraulic

pumps

is

listed

in

Tab.

2-2.10Chapter

2

Fundamental

Hydraulic

Fluid

MechanicsTab.

2-2

The

range

of

viscosity

of

hydraulic

oiladapted

topumpsTypesviscosities

(10-6

m2/s)TypesViscosities(10-6

m2/s)5～40℃①40～80℃①5～40℃①40～80℃①VanePumpsP<

7MPa30～5040～75Gearpumps30～7095～165P≥7MPa50～7050～90Radialpistonpumps30～5065～240Screw

pumps30～5040～80Axialpistonpumps30～7070～150①5～40℃、40～80℃

are

described

the

temperatures

of

hydraulic

system.11Chapter

2

Fundamental

Hydraulic

Fluid

MechanicsCharacteristics

of

HydrostaticsThe

basic

formula

ofhydrostaticsThe

principle

of

Pascal

applicationEffect

of

fluid

pressure

on

curved

surfaces2.2

HydrostaticsChapter

2

Fundamental

Hydraulic

Fluid

MechanicsCharacteristics

of

HydrostaticsThe

hydrostaticsStatic

pressure:

the

action

force

in

normal

on

a

unit

area.

It

isintituled

pressure

in

physics

and

action

force

in

engineeringusually.The

characteristics

of

hydrostaticsIn

any

homogeneous

fluid

system

at

rest,

thepressureincreases

with

the

depth

of

the

fluid.Pressure

at

any

point

in

a

homogeneous

fluid

system

at

restacts

perpendicularly

to

surfaces

in

contact

with

the

fluid.Chapter

2

Fundamental

Hydraulic

Fluid

Mechanics14Fig.

2-4

The

distribution

of

forces

ina

container

with

rest

fluidFormula

(2-9)

divide

by2.2.2 The

basic

formula

ofhydrostatics1.

The

basic

formula

of

hydrostaticsThe

acting

pressures

on

the

fluid

at

rest

,

in

a

container

include

the

weight,forceon

the

fluid

surface,

shown

in

Fig.

2-4a.The

total

balance

force

formula

is(2-9),then(2-10)Chapter

2

Fundamental

Hydraulic

Fluid

MechanicsThe

formula

(2-10)

is

the

basic

equation

for

hydrostatic.

Itstatesthat

the

distribution

status

of

hydrostatics

as

following:The

pressure

on

a

rest

fluid

contained

involves

two

parts

:(2-11)The

pressure

is

increased

with

the

depth

h;Isotonic

pressure

surface,

that

is,

the

pressures

are

all

equal

at

the

surfaceconsisted

by

all

points

at

given

depth

h,

such

as

at

the

line

of

A-A;Conservation

of

energy(2-12)Here,

theas

pressure

energy

at

per

unit

mass

fluid.Chapter

2

Fundamental

Hydraulic

Fluid

Mechanics2.

The

definition

ofpressureAbsolute

pressureRelative

gauge

pressure：The

pressures

measured

by

apressure

gauge

are

all

relative

pressureVacuum

(negative

pressure)The

units

of

pressure

and

relations

between

different

pressures

：1Pa＝1N/m2；1bar＝1×105

Pa＝1×105

N/m2;1at＝1kgf/cm2=9.8×104

N/m2;

1mH2O＝9.8×103

N/m2;1mmHg＝1.33×102

N/m2.The

relationship

of

three

pressures

is

shown

in

Fig.

2-5.Chapter

2

Fundamental

Hydraulic

Fluid

MechanicsFig.

2-5

Absolute,

relative

and

vacuum

pressureChapter

2

Fundamental

Hydraulic

Fluid

MechanicsExample

2-1:

The

oil

is

full

in

a

container.

For

a

given

condition,

thedensityofoil ,

the

action

force

on

this

piston

surface

F＝1000N,the

areaof

pistonＡ=1×10-3(m2),if

the

mass

ofpiston

is

neglected,

trytocalculate

the

static

pressure

p

at

h

=

0.5ｍ,

as

shown

in

Fig.

2-6.Fig.

2-6

Calculation

of

fluid

static

pressureChapter

2

Fundamental

Hydraulic

Fluid

Mechanics192.2.3 The

principle

of

PascalThe

principle

of

Pascal:

pressure

exerted

on

a

confined

liquid

istransmittedundiminished

in

all

directions

and

acts

with

equal

force

on

all

equal

areas.Its

application

is

shown

in

Fig.

2-7.Fig.

2-7

The

example

of

PascalprincipleChapter

2

Fundamental

Hydraulic

Fluid

Mechanics2.2.4 Effect

of

fluid

pressure

on

curved

surfacesWhen

the

wall

is

plane

：F=PAWhen

wall

is

a

curved

surface

：Example

2-2.

Fig.

2-8

shows

a

cylindrical

member

of

inside

radii

r

of

length

.Calculation:

the

effect

force

Fx

on

the

right

segment

of

the

cylinder

at

x

direction.Fig.

2-8

Effect

force

on

the

inner

surface

of

thecylinderChapter

2

Fundamental

Hydraulic

Fluid

Mechanics2.3

HydrodynamicsEquation

of

continuity—conservation

of

massBernoulli

Equation

—conservation

of

energy2.3.3 Equation

of

momentum—conservation

ofmomentumChapter

2

Fundamental

Hydraulic

Fluid

MechanicsFig.

2-9

sketch

of

conservationmassFor

incompressible

flow,,The

equations

of

continuity,

Bernoulli

and

momentum

are

basicmotion

equations

that

describe

the

dynamics

laws

in

flowingfluid2.3.1 Theequation

of

continuity—conservation

of

massaccording

to

the

conservation

ofmass,（2-14）(2-15)Or constant（2-16）Formula

(2-16)

is

the

equation

of

flowcontinuity.Chapter

2

Fundamental

Hydraulic

Fluid

MechanicsThe

assumptions:

no

energy

loss(meansin-viscid

and

incompressible),

according

theequation

of

Bernoulli—Conservation

of

energy.OrFormulas

(2-17)is

the

well-knowBernoulliequation.

It

states

that

ideal

fluidincludepressure

energy,

potential

energy,

and

kineticenergy.

These

three

energies

can

be

transferredbetween

each

other,

but

the

total

energy

isalways

invariable.2.3.2 Bernoulli

Equation

—conservation

of

energyFig.

2-10

Sketch

ofBernoulli

equation(2-17)1.

Ideal

equation

ofBernoulliChapter

2

Fundamental

Hydraulic

Fluid

Mechanics2.

Real

equation

ofBernoulliIn

many

hydraulic

systems,

the

energies

can

be

lost

(the

total

loss

is

described

ashw),

on

the

other

hand,

the

real

velocity

is

a

non-uniform

distribution

and

set

akinetic

correction

factor to

offset

this

lost,

and

the

coefficient

defined

by:(2-18)Here

α=1.1

when

it

is

turbulent

flow,

and

α=2

when

laminar

flow,

but

usually

inpractice

set

the

α=1.After

introducing

the

energy

loss

and

kinetic

correctionfactor ,

the

equation(2-17)

will

be

change

to(2-19)Notes：see

p27,

(1)

across-section

area1

and

2

should

be

selectedalong

the

streamline

direction

of

fluid

flow……Chapter

2

Fundamental

Hydraulic

Fluid

Mechanics3.

Application

example

of

the

equation

ofBernoulliExample

2-3

The

Venturi

meter

shown

reduces

the

pipe

diameter

from0.1m

to

a

minimum

of

0.05m

as

shown

in

Fig.

2-11.

Calculate

the

flow

rate

andthe

mass

flux

assuming

idealconditions.Fig.

2-11

Venture

meterChapter

2

Fundamental

Hydraulic

Fluid

MechanicsExample

2-4.

Try

to

analyse

the

condition

of

a

pump

drawing

into

oilfrom

a

reservoir

by

the

equation

of

Bernoulli

(Fig.

2-12).

Set

the

pressureat2-2

across-section

is

p2,

the

pressureat

1-1

across-section

is

p1,

andp1=pa.

and

the

distance

from

pump

orifice

to

hydraulic

oil

surface

is

h.Fig.

2-12

Setup

of

hydraulic

pumpChapter

2

Fundamental

Hydraulic

Fluid

Mechanics2.3.3Equationof

momentum-conservationof

momentumFig.

2-13

Sketch

of

oil

flow

througha

pipeline

with

a

pressure

vesselFig.

2-14

Sketch

of

oil

flowthrough

apipelineFig.

2-15

Sketch

of

oilthrough

curved

passagesIn

any

system

of

above,

the

rate

of

change

ofmomentum

in

the

system

equals

the

net

appliedexternal

force.The

equation

looks

the

same

as

therelationship(2-20)(2-21)Chapter

2

Fundamental

Hydraulic

Fluid

MechanicsBecause

q=Av,

soAssume

a

frictionless,

incompressible

liquid

in

a

cylindrical

passage

asshown

in

Fig.2-14.(2-22)The

force

balance

is,

from

equation

(2-20):(2-23)(2-24)Chapter

2

Fundamental

Hydraulic

Fluid

MechanicsFig.2-15,

is

a

change

in

momentum

as

defined

in

equation2-20.The

forces

can

be

resolved

into

a

component

Fx

which

is

axial

to

theinlet

direction

and

a

component

Fy

which

is

normal

to

theinletdirection.(2-25)Chapter

2

Fundamental

Hydraulic

Fluid

MechanicsExample

2-5.

Fig.

2-16

shows

a

sketch

of

a

spool

valve.

When

oil

fluid

flowthrough

the

valve,

calculate:

the

axial

effect

force

of

oil

fluid

on

the

spoolsurface.Fig.

2-16

Hydraulic

dynamic

on

the

spool

valveChapter

2

Fundamental

Hydraulic

Fluid

MechanicsExample

2-6.

Fig.

2-17

shows

a

sketch

of

a

poppet

valve,

where

thepoppetcore

is

2 .

When

fluid

rate

flow

q

through

the

valve

under

the

pressure

andthe

fluid

flow

direction

at

both

statuses

of

out-flowing

Fig.

2-17a

and

in-flowing

Fig.

2-17

b,

calculate:

action

force

magnitude

and

direction

on

thispoppet

core.Fig.

2-17

Hydraulic

dynamic

on

the

poppetvalveChapter

2

Fundamental

Hydraulic

Fluid

MechanicsFor

two

cases

above

the

fluid

action

pressures

on

thepoppet

are

all

equal

to

F.

The

action

directions

areshownin

Fig.2-17a

and

Fig.2-17b

respectively.For

the

Fig.

2-17a

the

fluid

dynamic

pressure

makes

thepoppet

orifices

tendto

be

closed,and

forthe

Fig.2-17btend

to

be

opened.

So

we

should

be

considered

according

tothe

detail

status

and

could

not

consider

all

tend

spool

orificeto

be

closed

in

any

conditions.Chapter

2

Fundamental

Hydraulic

Fluid

Mechanics2.4 Characteristics

of

Fluid

Flow

in

PipelineStates

of

fluid

flow

and

ReynoldsnumberLosses

along

circle

parallelpipeMinor

losses

in

pipe

systemChapter

2

Fundamental

Hydraulic

Fluid

MechanicsWhen

a

continuity

viscous

fluid

flows

through

variable

section,fluidwill

lose

parts

of

energy.

This

can

be

presented

by

the

pressure

loss

hw

andkinetic

correction

factor ,

i.e.,

in

the

above

mentioned

realfluidBernoulli’s

equationhere

hw

includes

two

parts:

pressure

losses

along

parallel

pipes

andminor(or

local)losses.2.4.1 States

of

fluid

flow

and

Reynolds

numberthere

are

three

main

states

of

flow,

such

as

laminar,

transition

andturbulent

ina

pipe.Now

take

Fig.2-18forexample.Chapter

2

Fundamental

Hydraulic

Fluid

MechanicsThe

Reynolds

number

wasobserved

to

be

a

ratio

of

theinertial

force

to

the

viscousforce.The

experiment

proved

that,Reynolds

number,

isconsistedof

threeparameters.(2-26)Fig.

2-18.

Setup

of

Reynolds

test1

-Overflow

pipe

2

-Supply

pipe

3,6-Reservoir4,

8

-Check

vale

5-Small

pipe

7

-Large

pipeChapter

2

Fundamental

Hydraulic

Fluid

Mechanics36pipesRecrpipesRecrsmooth

metalpipe2320Smooth

pipe

witheccentric

annularitygap1000hosepipe1600-2000Column

valve

orifice260smooth

pipe

withconcentric

annularitygap1100Poppet

valve

orifice20-100is

a

critical

value

between

laminar

and

turbulence

usuallydetermined

by

experimental

data.

(show

in

Tab.2-3)Tab.

2-3

Familiar

critical

Reynolds

number

based

on

different

pipe

materialFor

flow

in

noncircularducts(2-27)Here

R

is

hydraulic

radius,

defined

by:(2-28)Chapter

2

Fundamental

Hydraulic

Fluid

Mechanics2.4.2 Losses

along

circle

parallelpipeThe

losses

due

to

viscosity

in

equal

diameter

pipe

is

referred

aslosses

in

parallel

pipe,

which

will

change

with

the

different

flowingstates.1.

Losses

in

parallel

pipe

at

laminar

flow(1)

Velocity

profile

in

a

laminar

pipe

flowFig.

2-19

Laminar

flow

in

a

circle

pipeChapter

2

Fundamental

Hydraulic

Fluid

MechanicsAs

show

in

Fig.2-19,

a

force

balance

in

the

x-direction

yields,thus(2-29)SetthenIntegrate it

and

under

the

boundaryof

u=0

at

r=R,

we

obtain(2-30)It

says

that

velocity

profile

in

a

laminar

pipe

flow

along

radii

direction

is

aparabolaprofile

and

the

maximum

velocity

is

at

the

axis

center

r=0

andChapter

2

Fundamental

Hydraulic

Fluid

Mechanics（2-32）Formula

(2-32)

says

that

the

average

velocity

is

1/2

of

themaximumvelocity.(2-31)(3)

Average

velocity

inpipeAccording

to

the

definition

of

average

velocity,(2)

The

flow

rate

in

pipeFrom

formula

(2-30)Integrate

it

we

obtainChapter

2

Fundamental

Hydraulic

Fluid

Mechanics(4)

Losses

along

circle

parallel

pipeFrom

formula

(2-32),

the

loss

isDo

some

change,

The

formula

(2-33)

canbe

written

as(2-33)(2-34)Where is

the

resistance

coefficient

along

a

circle

pipe.

In

theory,,

but

in

a

practicalcase, for

a

metalpipe,for

a hosepipe

because

influence

of

temperature

need

to

beconsidered.Chapter

2

Fundamental

Hydraulic

Fluid

Mechanics2. Losses

in

parallel

pipe

at

turbulenceflowWhen

turbulence

flow

has

happened,

Theexperimenthas

shown

that

resistance

coefficient

isThe

resistance

coefficient

can

be

calculated

by

experimental

formula

as

followsfor

water-power

slippery

pipe,（2-35）（2-36）Here

∆

is

related

with

material

of

pipe,

such

as

steel

tube

0.04mm,copper

pipe0.0015～0.01mm，aluminum

0.0015～0.06

mm

and

hosepipe0.03mm.The

velocity

is

well

distribution

at

turbulence

flow,

the

maximum

velocity

asChapter

2

Fundamental

Hydraulic

Fluid

Mechanics2.4.3 Minor

losses

in

pipe

systemThe

reasons

of

minorlosses:Usually

the

minor

losses can

be

calculated

by(2-37)Then

we

can

calculate

the

flow

rate

except

the

rating

rate

bypressure

loss

formula

,(2-38)Chapter

2

Fundamental

Hydraulic

Fluid

MechanicsThe

total

energy

losses

in

a

whole

hydraulic

systemcan

be

summed

after

calculating

out

several

section’slosses

by(2-39)Chapter

2

Fundamental

Hydraulic

Fluid

MechanicsFlow

Rate

and

Pressure

Features

ofOrificeThin

wall

orificeStubby

orifice

or

slotorificePlate

clearanceCylinder

annular

clearanceChapter

2

Fundamental

Hydraulic

Fluid

Mechanics2.5.1 Thin

wall

orificeThin

wall

orifice

defined

as

the

radio

of

flow

length

L

todiameter

of

orifice

d

is

less

than

0.5

as

shown

in

Fig.

2-20,

usuallythe

orifice

is

sharpedged.Fig.

2-20

Fluid

flow

through

orificeChapter

2

Fundamental

Hydraulic

Fluid

MechanicsHereis

the

speedcoefficient.For

the

orifice

before

and

after

section

1-1

and

2-2,The

Bernoulli

equation

is(2-40)Then

we

can

obtain(2-41)Chapter

2

Fundamental

Hydraulic

Fluid

MechanicsThe

fluid

flow

rate

that

flows

through

this

orifice

asbelow,Where：A0—the

across-section

area

of

this

orifice

；

Cc—thesection

contraction

coefficient

，；

Cd—flow

ratecoefficient

，Cd=CvCc。(2-42)Chapter

2

Fundamental

Hydraulic

Fluid

Mechanics0.10.20.30.40.50.60.7Cd0.6020.6150.6340.6610.6960.7420.804This

is

the

reason

of

low

resistance

losses

when

fluid

flowsalongthe

length

of

the

pipe

in

thin

orifice.

It

has

less

sensitivity

totemperature,

and

thin

orifice

is

thus

usually

used

to

throttle

adjustor.Poppet

and

spool

valve

orifices

are

similar

to

the

thin

orifice,

so

bothare

all

used

to

the

hydraulic

componentorifices.In

the

case

of

completecontraction,

, can

becalculated(2-43)In

the

case

of

Re＞105，

=0.60～0.61in

the

case

of

incomplete

contraction, can

be

selected

by

Tab.

2-4Tab.

2-4

Flow

rate

coefficients

in

incomplete

contractionChapter

2

Fundamental

Hydraulic

Fluid

MechanicsFig.

2-21

Sketch

of

cylinderspool

orificeA

is

a

valve

seatB

is

a

spool

coreThe

flow

rate

coefficient

can

be

obtained

byFig.

2-22,

the

Reynolds

number

can

becalculated

by

following,The

flow

rate

that

flow

through

the

orificeis

calculated

below

by

equation

as

follow(2-44)If

xv>>Cr，neglect

Cr

,the

flow

rate

as(2-45)(2-46)Chapter

2

Fundamental

Hydraulic

Fluid

MechanicsFor

a

hydraulic

valve

whatever

flowing

in

or

out, is

theanglebetween

streamline

and

spool

line

and

is

called

speed

direction

angle,it

isusually

.Fig.

2-22

Flow

coefficienton

the

orifice

of

spoolvalveChapter

2

Fundamental

Hydraulic

Fluid

MechanicsThe

poppet

valve

orifice

is

shown

in

Fig.

2-23,When

poppet

moves

up

adistanceof ,

the

average

diameter

of

, ,

then

theflow

rate

isFig.

2-23

Orifice

shape

ofpoppet

valveFig.2-24

Flow

coefficient

ofpoppet

valve

orifice(2-47)Where

the

flow

rate

coefficient

can

be

obtained

by

Fig.

2-24Chapter

2

Fundamental

Hydraulic

Fluid

Mechanics2.5.2

Stubby

orifice

or

slotorificeThe

stubby

orifice

is

defined

as

，

slot

orificeThe

flow

rate

equation

for

thestubbyorifice

is

the

same

as

formula

(2-42),but

the

flow

ratecoefficient can

beobtained

from

the

curve

in

Fig.

2-25.The

flow

rate

equation

for

slotorifice

obeys

the

formula

(2-31),i.e.Fig.

2-25

Flow

rate

coefficients

inStubby

orificeChapter

2

Fundamental

Hydraulic

Fluid

MechanicsFig.

2-26

Flow

in

parallel

plainclearance（2-50）2.5.3 Plate

clearanceThe

fluid

flows

under

pressuredifferential

andvelocity as

shown

in

Fig.

2-26.The

flow

rate

fluid

flow

through

theplainplate

clearance

is（2-48）The

formula（2-48）has

two

statuses:1）Fluid

flow

at

pressure

differential

:（2-49）2）Fluid

flow

by

viscosity

shear

：Chapter

2

Fundamental

Hydraulic

Fluid

MechanicsFig.

2-27

Sketch

ofconcentric

clearance

flowIf

the

motion

direction

of

cylinder

is

the

same

asthe

direction

of

pressure

differential,

the

symbolin

(2-51)

chooses

“+”,

otherwise

“－”.the

flowrate

is2.5.4 Cylinder

annular

clearance1. The

flow

rate

equation

in

a

concentric

annular

orificeFig.

2-27

shows

a

sketch

of

concentric

clearance

flowLet’s

consider

annular

clearance

expanded

alongthe

length

direction

is

the

same

as

a

plainplateclearance,

so

substituting into

formula(2-48)(2-51)(2-52)Chapter

2

Fundamental

Hydraulic

Fluid

MechanicsFor

very

small

clearances,is

very

small

and,

thenBecause

of

small

clearance

，

，can

be

considered

as

Plates

clearance

flow,

theincremental

flow

iswhereFig.

2-28

Eccentricannularorifice2.

The

flow

rate

equation

in

eccentric

annular

orificeas

shown

in

Fig.2-28,we

can

abtain(2-53)(2-54)(2-55)Chapter

2

Fundamental

Hydraulic

Fluid

MechanicsIf

e=h0，the

flowis

greaterthanit

wouldbe

indicatedbythe

use

ofequation(2-51).Substitute

（2-54）into（2-55）(2-56)(2-57)Integrating:Or(2-58)Chapter

2

Fundamental

Hydraulic

Fluid

Mechanics3.

The

flow

rate

through

a

conical

annularclearanceBecause

of

machining

irregularities,

such

as

piston

or

bore,

valve

core

or

seatcore,

some

degree

of

conic

must

always

be

expected,

as

shown

in

Fig.2-29.Fig.

2-29

Fluid

flow

through

aconical

annular

clearanceConverse

coneSequence

coneWhen it

iscalledinverse

degree

of

conic

asshown

in

Fig.

2-29

a;otherwise

sequence

degreeof

conic

as

shown

in

Fig.2-29bChapter

2

Fundamental

Hydraulic

Fluid

MechanicsFor

the

status

of

Fig.

2-29

a,

substituting intoformula(2-51),Because

h=h1+xtanθ,substitutinginto

formula(2-59)

：Integrating

and

substitutingintoWe

obtain

the

flow

rate

as(2-59)(2-60)(2-61)(2-62)Chapter

2

Fundamental

Hydraulic

Fluid

MechanicsWhen

，flow

rate

isSubstituting

formula

(2-62）andinto

(2-64)，,When

u0=0,we

have(2-63)Integrating

formula

(2-61)

the

pressure

distribution

in

this

clearance

flowing,

andsubstituting

the

boundary

condition

at

h=h1，p=p1,

we

obtain(2-64)(2-65)(2-66)Chapter

2

Fundamental

Hydraulic

Fluid

Mechanics60For

the

status

of

Fig.

2-29b,

the

sequence

degree

of

conic

the

flowrate

formula

is

the

same

as

the

formula

(2-62)

,

butpressuredistribution

when

isor(2-67)(2-68)Chapter

2

Fundamental

Hydraulic

Fluid

Mechanics4.

Hydraulic

lock

andforceIf

there

is

a

eccentricity

“e”

between

spool

core

and

seat

due

to

setting,

as

shownin

Fig.2-30.Eccentric

with

inverseorder

conical

annularEccentric

withinorder

conical

annularSection

figure

at

any

pointSpool

core

notched balance

pressureFig.

2-30

Fluid

flow

through

a

conical

annular

clearance

with

eccentricChapter

2

Fundamental

Hydraulic

Fluid

MechanicsThe

value