CE 319 F
Daene McKinney
Elementary Mechanics of Fluids
Energy
Equation
Energy
•
First Law of Thermodynamics
–
Where
•
•
•
–
Where
•
•
•
–
E = energy of a system (extensive property)
Q = Heat added to the system
W = Work done by the system
Eu = Internal energy of the system
Ek = Kinetic energy of the system
Ep = Potential energy of the system
Intensive property (energy per unit mass)
dE dQ dW


dt
dt
dt
Energy of a system changes as heat
is added to the system or work is
done on the system
E  Eu  Ek  E p
e  u  ek  e p
Energy
•
Kinetic energy per unit mass
kineticenergy of mass MV 2 / 2 V 2
ek 


mass
M
2
•
Potential energy per unit mass
ep 
potentialenergy of mass z

 gz
mass

Energy Equation
•
Reynolds Transport Theorem
 
d

 bd   bV  dA
dt
dt CV
CS
dBsys
•
b = e ; Bsys = E
 
dE d

 ed   eV  dA
dt dt CV
CS
 
dQ dW d


 ( ek  e p  u ) d   ( ek  e p  u ) V  dA
dt
dt dt CV
CS
Work
•
Rate of Work
•
Shaft work
•
•
Flow work
Viscous work
dW dW
dW
dW



dt
dt shaft dt flow dt viscous
dW
dt
dW
dt
dW
dt
work done by a mechanical device which crosses the CS
shaft
work done by pressure forces on the CS
flow
work done by viscous stresses at the CS
viscous
Flow Work
•
Work occurs at the CS when a force associated with the normal stress of the fluid acts
over a distance. The normal stress equals the negative of the fluid pressure.
 
dW
 p1V1  A1
dt 1
W2  p2 A2 l2  p2 A2V2 t
 
dW
 p2 A2V2  p2 V2  A 2
dt 2
 
dW
p 
  pV  A    V  dA
dt flow CS
CS 
Energy Equation
dQ dW

dt
dt
shaft
 
d
V2
p V2

 gz  u ) V  dA
 (  gz  u ) d   ( 
dt CV 2
2
CS 
Kinetic Energy Correction Factor
 
dQ dW
d
V2
p V2


(

gz

u
)

d


(


gz

u
)

V
 dA


dt
dt shaft dt CV 2

2
CS
 
V2  
  (  gz  u ) V  A  
V  dA
2
CS 
CS
p
We can modify this kinetic
energy term by a dimensionless
factor, , so that the integral is
proportional to the square of the
average velocity
1  
 V  dA
AA
where
V 
and
1 V
     dA
A A V 
For nonuniform flows, this term
requires special attention
V2  
V2
V  dA   m

2
2
CS
m  V A  V1 A1  V2 A2
3
so
Kinetic energy correction factor
V12 p1
V22 p2
q  wshaft  1
  gz1  u1   2

 gz2  u2
2

2

q
1 dQ
;
m dt
wshaft 
1 dW
m dt shaft
Energy Equation for Pipe Flow
V12 p1
V22 p2
q  wshaft  1
  gz1  u1   2

 gz2  u2
2

2

wshaft  wturbine  w pump
V12 p1
V22 p2
1
  gz1  w pump   2

 gz2  wturbine  (u2  u1  q)
2

2

V12 p1
V22 p2
1
  z1  h pump  hturbine  hloss   2

 z2
2g 
2g 
h pump 
w pump
g
w
; hturbine  turbine ;
g
u u q
hloss  2 1
g
Ex (7.35)
•
Given: PA= 10 psi (12 in dia. pipe), PB= 40
psi (6 in dia. pipe), Q=3.92 cfs
Find: Horsepower of pump
Solution:
•
•
V A2 p A
VB2 p B
A

 z A  h pump  hturbine  hloss   B

 zB
2g 
2g 
h pump
VB2

2g

( 4.99)  (6.19)
2

 V A2
pB  p A

2

(10  40) *144
62.4
2g
 75.04 ft
Qh 3.92 * 62.4 * 75.04
P

 33.4 hp
550
550
VA 
VB 
Q

AA
Q
AB

3.92
 (1) 2 / 4
 4.99 ft / s;
3.92
 (0.5) / 4
2
 19.96 ft / s
EGL & HGL for a Pipe System
•
Energy equation
V12 p1
V22 p2
1

 z1  hL   2

 z2
2g 
2g

•
•
All terms are in dimension of length (head, or
energy per unit weight)
HGL – Hydraulic Grade Line
HGL 
•
p

EGL – Energy Grade Line
EGL  
•
•
z
V2 p
V2
  z  HGL  
2g 
2g
EGL=HGL when V=0 (reservoir surface, etc.)
EGL slopes in the direction of flow
EGL & HGL for a Pipe System
•
A pump causes an abrupt rise in EGL (and
HGL) since energy is introduced here
EGL & HGL for a Pipe System
• A turbine causes an
abrupt drop in EGL
(and HGL) as energy
is taken out
• Gradual expansion
increases turbine
efficiency
EGL & HGL for a Pipe System
•
•
When the flow passage changes diameter, the
velocity changes so that the distance between
the EGL and HGL changes
When the pressure becomes 0, the HGL
coincides with the system
EGL & HGL for a Pipe System
•
Abrupt expansion into reservoir causes a
complete loss of kinetic energy there
EGL & HGL for a Pipe System
•
When HGL falls below the pipe the pressure
is below atmospheric pressure
HW (7.8)
HW (7.16)
HW (7.27)
HW (7.27)
HW (7.41)