fluid mechanics i me362 - aast

Fluid Mechanics I – ME362*

Arab Academy for Science, Technology

and Maritime Transportation

Dr. Ahmed Khalifa Mehanna

Associate Professor

[email protected]

[email protected]

Room No: 223

Course Assistant Lecturer:

Eng. Omar Mostafa

Lecture 7:

Flow Measurements

Fluid Mechanics I – ME362*

Fluid System and Control Volume

Continuity Equation

Application of Continuity

Example & Assignment

Energy Relationships (Bernoulli’s Equation)

Lecture Outline

Matter cannot be created or destroyed - (it is simply changed into a different form of matter).

This principle is know as the conservation of mass and we use itin the analysis of flowing fluids.

The principle is applied to fixed volumes, known as controlvolumes.

Continuity of Fluid Flow

The continuity equation is based upon the conservation of mass as itapplies to the flow of fluids.

For any control volume the principle of conservation of massis:

Mass entering per unit time = Mass leaving per unit time +Increase of mass in the control volume per unit time

For steady flow there is no increase in the mass within thecontrol volume, so

Mass entering per unit time = Mass leaving per unit time

The Equation of Continuity are used in pipes, tubes and ductswith flowing fluids or gases, rivers, overall processes as powerplants, ...... etc.


When a fluid flows at a constant rate in a pipe or duct, the massflow rate must be the same at all points along the length. Consider aliquid being pumped into a tank as shown below.

The mass flow rate at any section is m = ρ A um

ρ = density (kg/ m3)

um = mean velocity (m/s)

A = cross sectional area (m2)

Conservation of Mass

For the system shown the

mass flow rate at points

(1), (2) and (3) must be

same at each point; So:

ρ1A1u1 = ρ2A2u2 = ρ3A3u3

OR

A1u1 = A2u2 = A3u3 = Q


ρ1 A1 v1 = ρ2 A2 v2

•

•


A1 V1 = A2 V2

A1 1 v1

Mass Flow Rate: M = ρAV (kg/s)

Volume Flow Rate: Q = A V (m3/s)

Continuity: ρA1 v1 = ρA2 v2

i.e., mass flow rate the same everywhere

e.g., Flow of River

Fluid Flow Concepts

A2 2 v2

Assuming the water moving in the

pipe is an ideal fluid, relative to its

speed in the 1” diameter pipe, how fast

is the water going in the 1/2” pipe?

Select the correct answer.

v2

v1

a) 2 v1 b) 4 v1 c) 1/2 v1 d) 1/4 v1

Using the Continuity Equation

11

2

1

2

12

2211

v4v)2/1(

1vv

vv

A

A

AA

Example

Example of the use of the continuity principle is to

determine the velocities in pipes coming from a junction.


Total mass flow into the junction = Total mass flow out

of the junction

1 Q1 = 2 Q2 + 3 Q3

When the flow is incompressible (e.g. if it is water)

1 = 2 = 3


332211

321

vvv

QQQ

AAA

If pipe 1 diameter = 50mm, mean velocity 2m/s, pipe 2

diameter 40mm takes 30% of total discharge and pipe 3

diameter 60mm. What are the values of discharge and

mean velocity in each pipe?

Application of Continuity - Assignment

The total energy of a fluid in motion consists of the following components:

Internal energy : Internal energy is a function of temperature. The internal

energy per unit mass of fluid is denoted by U.

Potential energy: This is the energy that a fluid has by virtue of its

position in the Earth’s field of gravity. This energy equal zg,

Pressure energy: This is the energy or work required to introduce the

fluid into the system without a change of volume. If P is the pressure and

V is the volume while m is the mass of fluid, then (P V/m) is the pressure

energy per unit mass of fluid. Thus the pressure energy per unit mass of

fluid is equal to P/ρ.

Kinetic energy: This is the energy of fluid motion. The kinetic energy of

unit mass of the fluid is v2/2,

Total energy: Summing these components, the total energy E per unit

mass of fluid is given by the below equation:

Energy Relationships and the Bernoulli Equation

2

v2

PzgUE

If we neglect piping losses, and have a system without pumps or

turbines (E1=E2)

Dividing throughout by g, these equations can be written in a

slightly different form

Z is called elevation head in meter (m); (Potential energy per unit weight)

P/γ is called pressure head in meter (m); (Pressure energy per unit weight)

V2/2g is called velocity head in meter (m); (Kinetic energy per unit weight)

The Bernoulli’s Equation (Ideal Flow)

.2

v

2

v 22

2

22

21

1

11 Const

gg

Pz

gg

Pz

.2

v

2

v 22

2

22

21

1

11 Const

Pgz

Pgz

In a real pipe line there are energy losses due to

friction Energy loss due to friction written as a

head and given the symbol hLoss 1-2 .

This is often know as the head loss due to

friction.

2-1 2

2

2

2

21

2

1

1

1

2

v

2

vLosshz

gg

Pz

gg

P

The Bernoulli Equation (Real Flow)

Water flows from the tap on the first floor of the building shown below

with a maximum velocity of 20 ft/s. For steady inviscid flow, determine

the maximum water velocity from the basement tap and from the tap on

the second floor (assume each floor is 12 ft tall).

Example

Solution:

fluid mechanics i me362 - aast

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