3.1 Systems
(
体系
)
versus Control Volumes
(
控制体
)
System
：
an arbitrary quantity of mass of fixed
identity.
Everything external to this system is denoted by
the term
surroundings
, and the system is separated from
its surroundings by it‘s boundaries through which no mass
across. (Lagrange
拉格朗日
)
Chapter 3 Integral Relations
（
积分关系式
）
for a Control Volume in One

dimensional Steady Flows
Control Volume (CV)
:
In
the neighborhood of our
product the fluid forms the environment whose effect on
our product we wish to know. This specific region is called
control volume
, with open boundaries through which mass,
momentum and energy are allowed to across. (Euler
欧拉
)
Fixed CV, moving CV, deforming CV
3.2 Basic Physical Laws of Fluid Mechanics
All the laws of mechanics are written for a system
, which
state what happens when there is an interaction between
the system and it’s surroundings.
If
m
is the mass of the system
Conservation of
mass
(
质量守恒
)
Newton’s
second law
Angular
momentum
First law of
thermodynamic
It is
rare
that we wish to follow the ultimate
path of a specific particle of fluid. Instead it is
likely that the fluid forms the
environment
whose effect on our product we wish to know,
such as how an airplane is affected by the
surrounding air, how a ship is affected by the
surrounding water. This requires that the basic
laws be rewritten to apply to a
specific region
in
the neighbored of our product namely a
control
volume ( CV)
.
The boundary of the CV is called
control
surface(CS)
Basic Laws for system
for CV
3.3 The Reynolds Transport Theorem (RTT)
雷诺输运定理
1122
is CV .
1
*
1
*
2
*
2
*
is
system
which
occupies
the
CV
at
instant
t
.
:
The amount of per unit mass
The total amount of in the CV is
:
t+dt
t+dt
t
t
s
:
any property of fluid
t+dt
t+dt
t
t
s
In the like manner
s
1

D flow
:
is only the function of s
.
For steady
flow
:
t+dt
t+dt
t
t
ds
R T T
If there are several one

D inlets and outlets
:
Steady , 1

D only in inlets and outlets, no matter
how the flow is within the CV .
3.3
Conservation of mass (
质量守恒
)
(Continuity Equation)
f
=m
dm/dm=1
Mass flux (
质量流量
)
For incompressible flow:
体积流量

Leonardo da Vinci in 1500
If only one inlet and one outlet
壶口瀑布是我国著名的第二大瀑布。两百多米宽的黄河河面，突然
紧缩为
50
米左右，跌入
30
多米的壶形峡谷。入壶之水，奔腾咆哮，势如奔马，
浪声震天，声闻十里。
“
黄河之水天上来
”
之惊心动魄的景观。
Example:
A jet engine working at design condition. At the inlet of the nozzle
At the outlet
Please find the mass flux and velocity at the outlet.
Given gas constant
T1
=865K
，
V1=288 m/s
，
A1=0.19
㎡
；
T2
=766K
，
A2=0.1538
㎡
R=287.4 J/kg.K
。
Solution
According to the conservation of mass
Homework: P185 P3.12, P189P3.36
3.4 The Linear Momentum Equation
(
动量方程
)
（
Newton
’
s Second Law
）
Newton’s
second law
:
Net force on the system or CV (
体系或控制体受到的合外力
)
：
Momentum flux (
动量流量
)
1

D in & out
steady RTT
flux
For only one inlet and one outlet
According to continuity
2
－
out, 1
－
in
Example: A fixed
control volume
of a streamtube in steady
flow has a uniform inlet (
1
,A1
,V1
aa楦ore楴i
(
2
,A2
,
V2
) . Find the net force on the control volume.
Solution
:
Neglect the weight of the fluid. Find the force on the
water by the elbow pipe.
Example:
1
2
1
2
Solution:
select coordinate
,
control volume
In the like manner
Find the force to fix the elbow.
Solution:
coordinate,
CV
Net force on the control volume:
Where F
ex
is the force on the CV by pipe,( on elbow)
1
2
F
ex
Surface force: (1)
Forces exposed by cutting though solid bodies
which protrude into the surface.(2)Pressure,viscous stress.
A fixed vane turns a water jet of
area A through an angle
without
changing its velocity magnitude.
The flow is steady, pressure pa is
everywhere, and friction on the
vane is negligible. Find the force F
applied to vane.
A water jet of velocity V
j
impinges
normal to a flat plate which moves
to the right at velocity V
c.
Find the
force required to keep the plate
moving at constant velocity and the
power delivered to the cart if the
jet density is 1000kg/m3
the jet area is 3cm2, and
Vj=20m/s,Vc=15m/s
Neglect the weight of the jet and plate,and
assume steady flow with respect to the moving
plate with the jet splitting into an equal upward
and downward half

jet.
Home work:
P190

p3.46
P191

p3.50
P192

p3.54
P192

p3.58
Derive the thrust(
推力
) equa瑩潮 for 瑨e 橥琠en杩ge
⸠
a楲 dra朠楳 ne杬ect
Solu瑩tn:
:
mass flux of fuel
x
Balance with thrust
Coordinate, CV
Example:
In a ground test of a jet engine,
p
a
=1.0133
×
10
5
N/m
2 ,
Ae=0.1543m2,Pe=1.141
×
105N/m2,
Ve=542m/s
, .
Find the thrust force.
Solution:
F16 R=65.38KN
x
coordinate
A rocket moving straight up. Let
the initial mass be M
0
,and assume a
steady exhaust mass flow and exhaust
velocity v
e
relative to the rocket. If
the flow pattern within the rocket
motor is steady and air drag is neglect.
Derive the differential equation
of vertical rocket motion v(t) and
integrate using the initial condition v=0
at t=0 .
Example:
Solution:
The CV enclose the rocket,cuts through
the exit jet,and accelerates upward at
rocket speed v(t).
coordinate
z
v(t)
Z

momentum equation:
v(t)
z
3.5 The Angular

Momentum Equation
(Angular

Momentum)
：
Net moment(
合力矩
)
Example:
Centrifugal (
离心
)pump
The veloc楴i of 瑨e flu楤 楳
changed from v
1
to v
2
and
its pressure from p
1
to p
2
.
Find (a).an expression for
the torque T
0
which must be
applied those blades to
maintain this flow. (b).the
power supplied to the pump.
blade
w
For incompressible flow
1

D
Continuity
:
Solution: The CV is chosen .
blade
w
Pressure has no contribution
to the torque
are blade rotational speeds
Work on per unit mass
Homework: P192

p3.55; P194

p3.68, p3.78 ; P200

p3.114,p3.116
Brief Review
•
Basic Physical Laws of Fluid Mechanics:
•
The Reynolds Transport Theorem:
•
The Linear Momentum Equation:
•
The Angular

Momentum Theorem:
•
Conservation of Mass:
Review of Fluid Statics
•
Especially :
Question
When fluid
flowing
…
Bernoulli(1700~1782)
What relations are there in
velocity, height and pressure?
Several Tragedies in History:
•
A little railway
station in 19
th
Russia.
•
The ‘Olimpic’ shipwreck in the Pacific
•
The bumping accident of B

52 bomber of
the U.S. air force in 1960s.
3.6 Frictionless Flow:
The Bernoulli Equation
1.Differential Form of Linear Momentum Equation
Elemental fixed streamtube CV of variable area
A(s),and length ds.
Linear momentum relation in the
streamwise direction:
one

D,steady,frictionless flow
For incompressible flow,
=const.
Integral between any points 1 and 2 on the streamline:
A Question:
Is the Bernoulli
equation a
momentum or
energy equation?
Hydraulic and energy grade lines for frictionless flow in a duct.
Example 1:
Find a relation between nozzle discharge
velocity and tank free

surface height h.
Assume steady frictionless flow.
1,2 maximum information is known or desired.
h
1
2
V
2
Solution:
h
1
2
V
2
Continuity:
Bernoulli:
Torricelli 1644
According to the Bernoulli
equation, the velocity of a
fluid flowing through a hole in
the side of an open tank or
reservoir is proportional to
the square root of the depth
of fluid above the hole.
The velocity of a jet of water from an open pop
bottle containing four holes is clearly related to
the depth of water above the hole. The greater
the depth, the higher the velocity.
Review of
Bernoulli equation
The dimensions of above three items
are the same of length!
Example 1:
Find a relation between nozzle discharge velocity
and tank free

surface height h.
Assume steady frictionless flow.
V
2
h
1
2
Example 2:
Find velocity in
the right tube.
h
A
B
In like manner:
V
Example 3:
Find velocity in the
Venturi tube.
1
2
As a fluid flows through a Venturi tube, the
pressure is reduced in accordance with the
continuity and Bernoulli equations.
Example 4:
Estimate required to keep
the plate in a balance state.
(Assume the flow is steady and frictionless.)
Solution:
For plate,
by lineal momentum equation,
by Bernoulli equation,
Example 5:
Fire hose,Q=1.5m
3
/min
Find the force on the bolts.
1
1
2
2
Solution:
By continuity:
By Bernoulli:
1
1
2
2
By momentum :
Example 6:
Find the aero

force on the blade
(cascade).
A
B
D
C
S
S
Solution:
A
B
D
C
S
S
By continuity,
叶片越弯，做功量越大。
A
B
D
C
S
S
By Bernoulli,
Bernoulli Equation for compressible flow
Specific

heat ratio
For isentropic flow:
Gas Weight neglect
For nozzle:
For diffuser:
Extended Bernoulli Equation
For compressor
多变压缩功
For turbine
多变膨胀功
Home work!
•
Page 206:
P3.158, P3.161
•
Page 207:
P3.164, P3.165
•
《
气体动力学
》
第二章习题第一
部分：
Page 20
33
题
Review of examples:
V
1
2
•
Analysis
•
Choose your control volumn
•
Body force and Surface force
•
Solution
1
1
2
2
x
Find the aero

force on the blade
(cascade).
叶片越弯，做功量越大。
A
B
D
C
S
S
By Bernoulli,
3.7 The Energy Equation
•
Conservation of Energy
Various types of energy occur in flowing fluids.
Work must be done on the
device shown to turn it
over because the system
gains potential energy as
the heavy(dark) liquid is
raised above the
light(clear) liquid.
This potential energy is converted into
kinetic energy which is either dissipated due
to friction as the fluid flows down ramp or
is converted into power by the turbine and
dissipated by friction.
The fluid finally becomes stationary again.
The initial work done in turning it over
eventually results in a very slight increase in
the system temperature.
Energy Per Unit Mass
1
1
2
2
e
First Laws of
Thermodynamics
•
Conservation of Energy
1
1
2
2
The energy equation!
Example:
A steady flow machine takes in air
at section 1 and discharged it at section 2 and
3.The properties at each section are as follows:
section
A,
Q,
T,
P, Pa
Z,m
1
0.04
2.8
21
1000
0.3
2
0.09
1.1
38
1440
1.2
3
0.02
1.4
100
?
0.4
CV
(1)
(2)
(3)
110KW
Work is provided to the machine at the rate of 110kw.
Find the pressure (abs) and the heat transfer
.
Assume that air is a perfect gas with R=287, Cp=1005.
Solution:
Mass conservation:
By energy equation:
CV
(1)
(2)
(3)
110K
W
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