Post on 12-Mar-2022
Fluid Mechanics I – ME362*
Arab Academy for Science, Technology
and Maritime Transportation
Dr. Ahmed Khalifa Mehanna
Associate Professor
a.khalifa@aast.edu
ahmed_marines@yahoo.com
Room No: 223
Course Assistant Lecturer:
Eng. Omar Mostafa
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.
Continuity of Fluid Flow
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
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.
Application of Continuity
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
Application of Continuity
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