# 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.

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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 systemfor 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 t s : any property of fluid

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 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)  =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 ( r 1,A1,V1 )and a uniform exit ( r 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 ) equation for the jet engine. air drag is neglect Solution: : 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 velocity of the fluid is 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  For incompressible flow 1-D Continuity : Solution: The CV is chosen.

blade  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:

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 V2V2

Solution: h 1 2 V2V2 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. V2V2 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,

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: sectionA,Q,T,P, PaZ,m 10.042.82110000.3 20.091.13814401.2 30.021.4100?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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