Mass flow rate
 m  VndA t  V·ndA t
m
mdA  lim
 Vn dA  V ·ndA
t 0 t
m  Mass flow rate through an area A of the control surface   V ·ndA
A
Useful approximations to get simple algebraic
forms for the natural laws
Across the INLETS AND OUTLETS of the CV:
• APPROX 1. Uniform property approximation:
The variation of system properties (such as
density, specific kinetic/potential/internal/total
energy)
CAN BE NEGLECTED
• Therefore, these properties can be given
representative average values at inlets and outlets.
The 1-D flow approximation
• Approx 2. 1-D flow approximation: The flow
velocity at the inlets and outlets has only a
single component normal to the respective
areas.
• Therefore, mass flow rate can be calculated by
AV
m   V ·ndA  VA 
v
A
Deriving the laws of conservation of
mass and energy for an open system
starting from the laws expressed for
a closed system
Conservation of mass for open systems
Mass of the closed system as CV :
Consider a “small process”
 m
mcv process begins    m = mcv process ends  outlets
inlets
dmCV 
m  m
inlets
outlets
For “rate processes” dividing both sides by t
and letting t0
End
dmCV
  m  m
dt
inlets
outlets
For “large processes” integrating both sides
Begin
mcv   m 
inlets
m
outlets
First Law of Thermodynamics for open
systems
For the closed system [small process “green to red”] :
dE   Q   W
where
dE  Eprocess ends  Eprocess begins
Eprocess ends  ECV process ends 
  m(u  V
2
/ 2  gz )
outlets
End
Note: uniform property
approximation at inlets/outlets
is used here
Eprocess begins  ECV process begins    m(u  V 2 / 2  gz )
inlets
Begin
dE  dECV 

outlets
 m(u  V 2 / 2  gz )    m(u  V 2 / 2  gz )
inlets
Splitting work into flow and non-flow
work
W  WCV  Wflow  WCV  Wflow,inlets  Wflow,outlets
End
Flow work: work done at the inlets/outlets due to fluid
flowing in/out against pressure forces. At outlets, work
is done by the system on the surroundings. At inlets,
work is done by the surroundings on the system.
 WCV
Begin
=Work other than flow work such as shaft work,
electric work, boundary displacement work.
Flow work
Wflow,inlets    pA x
inlets
   pv m
inlets
End
Wflow,outlets 
 pA x
outlets

Begin
Wflow  Wflow,inlets  Wflow,outlets 

pv m
outlets
 pv m   pv m
outlets
inlets
Note:
A x
m 
v
First Law of Thermodynamics for open
systems
For the closed system (going from green to red)
dE   Q   WCV   W flow 
End
Begin


dECV     m(u  v 2 / 2  gz ) 
 inlets



    m(u  v 2 / 2  gz )    Q   WCV   W flow 
 outlets

First Law of Thermodynamics
Inserting expression for flow work
Wflow 

pv m   pv m
outlets
inlets
and regrouping terms
dECV 
  m(u  pv  v
2
/ 2  gz )
inlets
   m(u  pv  v 2 / 2  gz )   Q   WCV
outlets
For rate processes dividing both sides by t and
letting t0
Recall enthalpy
defn.:
h=u+pv
dECV
  m(h  V 2 / 2  gz )   m(h  V 2 / 2  gz )  Qnet ,in  WCV
dt
inlets
outlets
For large processes provided all inlet/outlet conditions are
steady (not changing with time) integrate both sides
ECV   m(h  V 2 / 2  gz) 
inlets

outlets
m(h  V 2 / 2  gz)  Qnet ,in  WCV