For only one inlet and one outlet According to continuity
mout min m
d (mV ) s F m(Vout Vin ) dt
2－out, 1－ in

Fx m(V 2 x V 1x)
y
Fy m(V 2 y V 1 y )
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
1-D in & out steady RTT
mV
dmV V dm
(linear momentum)
momentum perunit mass
flux
d (mV ) s ( i AiV iV i )out ( i AiV iV i)in (mi V i )out (mi V i )in i i dt i i
s
t t+dt
t
t+dt
: any property of fluid (m, mV , H , E)
d dm
:The amount of
per unit mass
is :
The total amount of
in the CV
CV cv d cv dm
d ( CV ) dt
R=287.4 J/kg.K。
gas constant
Solution
According to the conservation of mass
p p1 AV 1 1 AV 45.1 kg / s m AV RT RT1
p1 AV p2 A2V2 1 1 m 1 AV A V RT RT 1 1 2 2 2 1 2 A1 p1 T2 V2 V1 A p T 565.1 m / s 2 2 1
F m(V 2 V 1)
F x m(V 2 x V 1x ) m V 2


2 1
F sx p 2 A2 mV 2
2 2 4 78.5 d 2 F sx p 2 A2 mV 2 3696 N p 2 2 998 d 2 4

y

o
x
In the like manner
V 1) mV 1 F sy p1 A1 m(0 F sy p1 A1 mV 1 -4642N
1-D flow : is only the function of s .
( s)
(d )in ( dm)in ( Ads)in ( AVdt )in
In the like manner
(d )out ( AVdt )out
s
ds
t t+dt t t+dt

m const
or
dm 0 dt
dV d ( mV ) F ma m dt dt dH M H (r V ) m dt
dQ dW dE dt dt dt
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
d s d cv 1 [(d ) out ( d )in] dt dt dt
d cv [( AV )out ( AV )in ] dt
For steady flow :
d cv 0 dt
RTT

ds ( AV ) out ( AV )in dt
Homework: P185 P3.12, P189P3.36
3.4 The Linear Momentum Equation (动量方程) （ Newton’s Second Law ）
ds ( i AiV i )out ( i AiV i )in dt i i
i
(m )
i
i in
( m i )out
i
Mass flux (质量流量 m )
For incompressible flow:
( A V )
i i
i out
( AiV i )in
i
Qi AiV i Volume flux
体积流量
If only one inlet and one outlet
Fz m(V 2 z Vmple: A fixed control volume of a streamtube in steady flow has a uniform inlet (1,A1,V1 )and a uniform exit (2,A2,V2) . Find the net force on the control volume.
3.3 Conservation of mass (质量守恒)
(Continuity Equation)
f=m
dm/dm=1
dms ( i AiV i )out ( i AiV i)in 0 dt i i
( A V
i i i
i out
)
( i AiV i )in
Chapter 3 Integral Relations（积分关系式） for a Control Volume in One-dimensional Steady Flows
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) 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
Newton’s second law
d (mV ) s (m V )out (m V )in i i i i F dt i i
mi V i ：Momentum flux (动量流量)
F
:Net force on the system or CV (体系或控制体受到的合外力)
surface(CS)
Basic Laws for system
3.3 The Reynolds Transport Theorem (RTT) 雷诺输运定理
for CV
1122 is CV . 1*1*2*2* is system which occupies the CV at instant t.
If there are several one-D inlets and outlets :
d s ( i i AV i i ) out ( i i AV i i )in dt i i
Steady , 1-D only in inlets and outlets, no matter how the flow is within the CV .
Solution:
F m(V 2 V 1)
m 1 A1V 1 2 A2V 2
2
V1
V2
F x m(V2 x V1x ) m(V 2 V 1 cos )