PHAROS UNIVERSITY
ME 259
FLUID MECHANICS FOR ELECTRICAL STUDENTS
Basic Equations for a
Control Volume
Main Topics
 Flow Classification
 Basic Laws for a System
 Relation of System Derivatives
to the Control Volume Formulation
 Conservation of Mass
 Bernoulli Equation
Flow Classification
Classification of Fluid Dynamics
Laminar
Inviscid
µ=0
Viscous
Turbulent
Compressible
Incompressible
ϱ = constant
Internal
External
Basic Laws for a System
Conservation of Mass
Relation of System Derivatives to
the Control Volume Formulation
Extensive and Intensive Properties
Relation of System Derivatives to
the Control Volume Formulation
Reynolds Transport Theorem
FLUID FLOW
• Continuity Equation
• Bernoulli’s Equation
• Momentum Equation
• Energy Equation
Basic Laws
 Conservation of mass: dM/dt=0 for system
∂/∂t ∫ϱ d𐐏 +∫ ϱ ⊽. d Ᾱ=0 for control vol.
Newton’s 2nd law Σ F = ma
Σ F = ∂/∂t ∫ ⊽ ϱ d𐐏 + ∫ ⊽ ϱ ⊽. d Ᾱ
First Law of Thermo: Q - W = dE/dt
Q-w=∂/∂t∫e ϱ d𐐏+∫(p/ ϱ+0.5V2+gz)ϱ⊽. dᾹ
Relation of System Derivatives to
the Control Volume Formulation
Interpreting the Scalar Product
Conservation of Mass
Basic Law, and Transport Theorem
Conservation of Mass
Conservation of Mass
Basic Law for a System
Conservation of Mass
Incompressible Fluids
Steady, Compressible Flow
Definitions
Volume (Volumetric) Flow Rate
• Q = Cross Sectional Area*Average
Velocity of the fluid
v
Volume
• Q = A*v cms
Q = Volume/Unit time
Weight Flow Rate
• W = g*Q N/s
Mass Flow Rate
• M = r*Q kg/s
Q = Area*Distance/Unit Tim
Flow in non-circular sections
Flow rate is determined by:
• Q = A*v
Where,
A = Net flow area
v = average velocity
Example:
Dlarge, i = 0.5 m
Dsmall, o = 0.25 m
Dsmall, i = 0.2 m
Vsmall, i = 1 m/sec
Vlarge, i = 1 m/sec
Find Qlarge and Qsmall
Continuity Equation
Continuity for any fluid (gas or liquid)
• Mass flow rate In = Mass Flow Rate out
• M1 = M2
M1
M2
• r1*A1*v1 = r2*A2*v2
Continuity for liquids
• Q1 = Q2
• A1*v1 = A2*v2
Equation of continuity
Rv (volume flow rate)  A1v1  A2 v2  constant
Volume flow rate has units m3/s
Mass flow rate has units kg/s
17/27
Units and Conversion Factors
Q: m3/sec
M: kg/sec,
Volume Flow Rate:
• 1 L/min = 0.06 m3/h
• 1 m3/sec = 60,000 L/min
• 1 gal/min = 3.785 L/min
Example #1
If d1 and d2 are 50 mm and 100 mm,
respectively, and water at 70° C is
flowing at 8 m/sec in section 1,
determine: v2, Q, W, M.
v1
d1
1
2
d2
v2
Example # 2
If d1, d2 and d3 are 10 cm, 20 cm, and 50
cm, determine Q and the velocities, v2
and v3 if v1 = 1 m/sec.
V1 = 1 m/sec
d1 = 10 cm
d2 = 20 cm
d3 = 50 cm
© Pritchard
Example # 3
Determine the required size standard
Schedule 40 steel pipe to carry 192
m3/hr with a maximum velocity of 6.0
m/sec.
Example # 4
The tank is being filled with water by two 1-D inlets. Air is
at the top of the tank. The water height is h. (a) Find
an expression for the change in water height dh/dt. (b)
Compute dh/dt if D1 = 1 cm, D2 = 3 cm, V1 = 3 m/s, V2 = 2 m/s
and At = 2 m2.
Tank Area At
ra
h
1
rw
2
Example # 5
 Consider the entrance region of a circular
pipe for laminar flow. What is mean velocity
of the fluid.
Ideal Fluids in Motion:
Continuity & Bernoulli’s equation
Assume:
the flow of fluids is laminar (not turbulent) or steady flow
- the fluid has no viscosity (no friction).
A fluid element traces out a streamline as it moves.
The velocity
vector of the element is tangent to the
14-Mar-16
streamline at every point.
The steady flow of a fluid around an air foil,
as revealed by a 24/27
dye tracer that was injected
© Pritchard
into the fluid upstream of the airfoil
Conservation of Energy
Bernoulli’s Equation
Energy cannot be created or
destroyed, just transformed
Three forms of energy in fluid
system:
• Potential
• Kinetic
• Flow energy
Potential Energy
 Due to the elevation of the fluid element
PE  w  z
Where,
w = weight of fluid element
z = elevation with respect to a reference level
Kinetic Energy
 Due to the velocity of the fluid element
2
v
KE  w
2g
Where,
v = average velocity of the fluid element
Flow Energy
 Flow work or pressure energy
 Amount of energy necessary to move a fluid
element across a certain section against
pressure
PE  w
P
g
Where,
p = pressure on the fluid element
Total Energy and Conservation
of Energy Principle
 E = FE + PE + KE
v2
E  w  w z  w
g
2g
P
 Two points along the same pipe:
E1 = E2
 Bernoulli’s Equation:
wv12 wp2
wv22
 wz1 

 wz2 
g
2g
g
2g
wp1
v12
P2
v22
 z1 

 z2 
g
2g g
2g
P1
Heads
P
 Pr essure_ Head
g
z  Elevation _ Head
v2
2g
P
g
Assumption: No energy is added
or lost
Assumption: Energy level remains
constant
 Velocity _ Head
z 
v2
2g
 Total _ Head
Restrictions on Bernoullis’
Equation
 Valid only for incompressible fluids
 No energy is added or removed by pumps,
brakes, valves, etc.
 No heat transfer from or to liquid
 No energy lost due to friction
Application of Bernoulli’s
Equation
 Write Bernoulli’s equation in the direction
of flow,
 Label diagram
 Simplify equation by canceling terms that
are zero, or equal on both sides of the
equation
 Solve equation and find desired result(s)
Example
 A hose carries water at a flow
rate of 0.01 m3/sec. The hose
has an internal diameter of 12
mm, and the gauge pressure at
faucet is 100 kPa. Determine
the pressure at the end of the
hose
DZ = 10 m
Torricelli’s Theorem
 For a liquid flowing from a tank
or reservoir with constant fluid
elevation, the velocity through
the orifice is given by:
v2 
2 gh
h
where, h is the difference in
elevation between the orifice
and the top of the tank
Example: If h = 3.00 m, compute v2
Take Home Experiment
A reservoir of water has the surface at 310m above the outlet nozzle
of a pipe with diameter 15mm. What is the a) velocity, b) the
discharge out of the nozzle and c) mass flow rate.
Water Velocity = (2gh)0.5
= (2x 9.81 x 310)0.5 = 78 m/s
Individual Experiment
Pipe Flow: Ideal flow Assumption and Energy Equation
 The aim is to study Continuity equation and Bernoulli equation as will as
pressure losses due to viscous ( frictional) effects in fluid flows through pipes
Differential Pressure
Gauge- measure ΔP
H
Flow
meter
Reservoir
•
D
Pipe
L
Valve
Schematic of experimental Apparatus
Pipes with different Diameter and Length will be used later for the
experiments to study Energy Equation and pressure losses