Thermodynamics and Fluid Mechanics 1
MM1TF1
Steady Flow Energy Equation III
Dr Steve Pickering
Extn 13785
Room: Coates B73
stephen.pickering@nottingham.ac.uk
The University of Nottingham
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Steady Flow Energy Equation SFEE
The steady flow energy equation (SFEE) is used
very widely in both thermodynamics and fluid
mechanics to describe fluid flows and
mechanical and thermal energy flows. Internal combustion
engine
Fan heater
Jet engines
Steam turbine
2
Steady Flow Energy Equation

p2
v 22  
p1
v12 
q  w s  u2 
 gz2    u1   gz1   J/kg
2
2 
1
2

by definition, enthalpy h
 u  pν
p
 u
ρ
v v
q  w s  h2  h1   g z2  z1  
2
2
2
2
1
J/kg
This is the way the SFEE is often presented. It applies to any
fluid and any steady flow process
3
Steady Flow Energy Equation
In terms of rate of energy (power)
Watts
(J/s)


v

v


 h2  h1   g z2  z1  
Q W  m

2 

2
2
2
1
4
Work Done in a Non-Flow Process
p2V2
p2
2
  pdV
p1
2

 pdV
1
1
5
V2
V1
Work Done in Flow Process
2
Vdp

1
p2
p1
p1V1
6
V2
V1
Work Done in Reversible Steady Flow
Processes involving Perfect Gases
pv  RT
Isothermal Processes
2
w   vdp
1
p2
w  RT ln
p1
Isentropic (constant entropy) process
 n 
w  
 p2 v 2  p1v1 
 n 1
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Page 43
Worked Example (8)
Calculate the rate of work input needed for
a reversible air compressor operating with
an inlet pressure of 1 bar (absolute) and
an outlet pressure of 10 bar (absolute).
The mass flow rate is 5 kg/s and the inlet
temperature is 300K. Assume the
polytropic compression with n =1.4 and R
= 287.1 J/kgK.
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Page 43
Worked Example (8)
1
Work
 1
T2  p2  n
 
T1  p1 
2
 n 
w  
 p2 v 2  p1v1 
 n 1
 n 
 RT2  T1 
w  
 n 1 
pv  RT
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Page 43
Worked Example (8)
 p2 
T2  T1  
 p1 
n -1
n
n  1 0.4

 0.2857
1.4
n
For air n = 1.4,
T2  30010
0.2857

 579 K (306 )
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Page 43
Worked Example (8)


n
 RT2  T1 
W  m 
 n 1 
1
.
4



W  5
287.1579  300
 0.4 
W  1.4 M W
 n 
 R  c p , W  m c p T2  T1 
Note 
 n 1 
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Page 44
Using SFEE for friction losses in pipes

p2
v 22  
p1
v12 
q  w s  u2 
 gz2    u1   gz1  
2
2 
1
2

For incompress ible flow ρ1  ρ2  ρ
Divide by g and rearrange :
ws q  p2
v22   p1
v12  u2  u1 
    z2      z1   
g g  g
2 g   g
2g 
g
HT2
HT1
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Page 44
SFEE for incompressible fluids
w s q  p2
v 22   p1
v12  u2  u1 
 
 z2 
 z1 


g g  g
2g   g
2g 
g
Re-arranging so that all the thermal terms (u and q)
are grouped together:
 u2  u1   q 
ws
 HT 2  HT 1  

g
g


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SFEE for incompressible fluids
ws
 u 2  u1   q 
 HT 2  HT 1  

g
g


 u 2  u1   q 
The quantity 
has the units of length and

g


is referred to as the friction head or head loss due
to friction.
In a real fluid viscosity causes friction and some of the
head is lost and is converted to internal energy (heat)
ws  g H T 2  H T 1  H f 
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Extended Bernoulli equation
w s  g HT 2  HT 1  H f 
 If there is no shaft work then we get:
HT 1  HT 2  Hf
 This is very like the Bernoulli equation except that
now an allowance is made for friction and so it can
be applied to real fluids
 It is called the Extended Bernoulli Equation EBE
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Page 44
Extended Bernoulli equation
 Assumptions of the
Bernoulli equation
steady flow
no shaft work
incompressible
inviscid flow
 Assumptions of the
Extended Bernoulli
equation
steady flow
no shaft work
incompressible
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Page 44
Extended Bernouilli equation
HT 1  HT 2  Hf
 The amount of energy (head) lost due to friction
will depend on the fluid involved (), the velocity
of the flow, the number and type of bends,
valves etc and the roughness of the surfaces
where the flow takes place.
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Page 45
Incompressible flow in pipes
 For pipes, an empirical (found from
experiment) relationship, called the Darcy
Equation, is found to apply for long, round,
straight pipes, for laminar and turbulent flow:
4fL v 2
Hf 
D 2g
Beware! – in North America, the
formula is:
L v2
Hf  f
D 2g
f - Darcy Friction Factor
Their f is four times as big as the
English f. Fanning Friction Factor
And in Europe, they use the
symbol  = 4f
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Page 45
Incompressible flow in pipes
4fL v 2
Hf 
D 2g
f is the friction factor
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For laminar flow f 
(independent of pipe
Re
roughness)
For turbulent flow its value depends on
k
relative pipe roughness ( k '  ) and
d
Reynolds number
The Moody chart is used to estimate f
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Moody chart
Page 46
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