All problems except the followings

Chapter 2•45, 46, 48-52, 58, 63-65, 71

Chapter 3•After 50

Introduction to Fluid Mechanics

Chapter 4Basic Equations in Integral Form

for a Control Volume

Main Topics1. Basic Laws for a System2. Relation of System Derivatives

to the Control Volume Formulation3. Conservation of Mass4. Momentum Equation for

Inertial Control Volume5. Momentum Equation for Inertial Control

Volume with Rectilinear Acceleration6. The Angular-Momentum Principle7. The First Law of Thermodynamics8. The Second Law of Thermodynamics

Relation of System Derivatives to the Control Volume Formulation

• Extensive and Intensive Properties

Eq 1

Relation of System Derivatives to the Control Volume Formulation

Relation of System Derivatives to the Control Volume Formulation

• Reynolds Transport Theorem

Eq 2Physical Interpretation ??

is the rate of change of the system extensive property N. For example, if N= , we obtain the rate of change of momentum

is the rate of change of the amount of property N in the control volume. The term computes the instantaneous value of N in the control volume ( is the instantaneous mass in the control volume). For example, if then and computesthe instantaneous amount of momentum in the control volume.is the rate at which property N is exiting the surface of the control volume. computes the rate of mass transfer leavingacross control surface area element ; multiplying by η computesthe rate of flux of property N across the element; and integratingtherefore computes the net flux of N out of the control volume. Forexample, if N= , then η = and computes the netflux of momentum out of the control volume.

Relation of System Derivatives to the Control Volume Formulation

• Interpreting the Scalar Product

Conservation of Mass

• Basic Law, and Transport Theorem

Conservation of Mass (continuity equation)

Conservation of Mass

• Incompressible Fluids

Steady, Compressible Flow

Momentum Equation forInertial Control Volume

• Basic Law, and Transport Theorem

Momentum Equation forInertial (homogeneous motion, no

acceleration) Control Volume

• Special Case: Control Volume Moving with Constant Velocity

xyz coordinate system of control volumeVelocity must be measured with respect to controlled volume

Momentum Equation forInertial Control Volume

u,v, w are only the scalar components, no sign involved

Assumptions: 1) Incompressible flow 2) Uniform flow

For the mass equation

For the y momentum

u

v

xy

Basic equation: Continuity, and momentum flux in x direction

Assumptions: 1) Steady flow 2) Incompressible flow CV 3) Uniform flow

For x momentum

Basic Laws for a System• The First Law of Thermodynamics

the rate of heat transfer, Q, is positive when heat is added to the systemfrom the surroundings; the rate of work,W, is positive when work is done by thesystem on its surroundings.

Eq 1

Eq 2

Eq 3

The First Law of Thermodynamics• Basic Law, and Transport Theorem

From eq 1 and Reynold Transport Theorem

Eq 4

Work Involves Shaft Work Work by normal Stresses

at the Control Surface Work by Shear Stresses at

the Control Surface Other Work

Eq 5

Since the work out across the boundaries of the control volume is the negative of the work done on the control volume, the total rate of work out of the control volume due to normal stresses is

Work Done by Normal Stresses at the Control Surface

work done on the control volume

Eq 6

Work Done by Shear Stresses at the Control Surface

Eq 7

Control Volume Equation

From eq 1, 4, 5, 6,7

Rearranging this equation, we obtain

Since ρ=1/, where is specific volume, then

Eq 8

The First Law of Thermodynamics

From and Eq 8

h Example 4.16, 4.17

Assumptions: 1) Adiabatic 2) No work 3) Neglect KE 4) Uniform properties at exit 5) Ideal gas

Continuity eq

First Law of Thermodynamics for a CV

From continuity

From the 1st law

For air

Hence

The Second Law of Thermodynamics

dAAQ

CS TAdV

CSsd

CVs

t

1

.

At time t0

eq1

eq2

eq3

From 1, 2 ,3

The Second Law of Thermodynamics

Rate of change of total entropy with in control volm

Total entropy transferring through the surface area of control volm

Total (local heat transfer per unit area into the control volm through surface/local temperature)

Momentum Equation for Inertial Control Volume

• Special Case: Bernoulli Equation

1. Steady Flow2. No Friction3. Flow Along a Streamline4. Incompressible Flow

A

B

Bernoulli equation and x momentum

Applying Bernoulli between inlet and throat

Applying the horizontal component of momentum

The Angular-Momentum Principle• Basic Law, and Transport Theorem

From Transport Theorem

The Angular-Momentum Principle

all the torques that act on the control volume. rate of change of angular momentum

within the control volume + the net rate of flux of angular momentum from the control volume.