Consequence Analysis 1.2
Mechanical Energy Balance
2

Ws
dp
u  g






z

F


   2 gc  gc
m


p  pressure
  fluid density
u  average velocity of fluid
  unitless velocity profile correction factor
= 0.5 (laminar) or 1.0 (plug flow)
g  gravitational acceleration
z  height
F  net friction loss
Ws  shaft work
m  mass flow rate
Flow of Liquid Through a Hole
2

Ws
p
u  g
  z  F  
 
 2 gc  gc

m


u 2  u 2  u12  u 2
p   pg
let
then
p
 p 
2 g 

 F  C 
  C1  

  
  
p
u  C1 
2
1
2 g c pg

 Co
Qm   uA  ACo 2  g c pg
2 g c pg

Discharge Coefficient
• For sharp-edged orifices and for Re greater
than 30000, Co approaches the value of 0.61.
• For a well-rounded nozzle, it approaches unity.
• For a short section of pipe attached to a vessel
(with L/D not less than 3), it is approximately
0.81.
• For cases where it is unknown or uncertain,
use a value of 1.0.
Flow of Liquid Through a Hole in a
Tank
z  hL
p

g
p g
g 
2
2 g

 z  F  C1  
 z   C1 
 hL 
 gc
  gc 
  gc 
p
 g c pg

 g c pg

u  C1  2 
 ghL   Co 2 
 ghL 
 

 

Qm   uA   ACo
 g c pg

2
 ghL 
 

Empty Time
m   At hL  total mass above the leak
 g c pg

dhL
dm
  At
 Qm   ACo 2 
 ghL 
dt
dt
 

hL
Co A t
dhL

dt
hL0 2 g p

0
A
t
c g
 2 ghL

1
g
2 g c pg

1
 2 ghL 
g
2 g c pg

Co A
 2 gh  
t
At
0
L
Empty Time
Co At
hL  h 
At
0
L
2 g c pg

g  Co A 
t
 2 gh  
2  At 
2
0
L
By setting hL  0
2 g c pg
1  At    g c pg
0
 ghL  
te 
   2

Co g  A    


By substituting hL into Qm
2 2

 g c pg
gC

0
oA
t
 ghL  
Qm   Co A 2 
At

 



Liquid Discharge
Discharge of pure non-flashing liquid through an
orifice
1/ 2
 2( po  pa )

GL  Cd A 
 2 gh 



GL  liquid mass emission rate (kg/s)
Cd  discharge coefficient (dimensionless,0.6
0.64)
A  discharge hole area (m 2 )
  liquid density (kg/m3 )
po  liquid storage pressure (N/m 2 absolute)
pa  downstream (ambient) pressure (N/m 2 absolate)
g  acceleration of gravity (9.81 m/s 2 )
h  height of liquid above hole (m)
Flow of Vapor Through Holes
(Isentropic Expansion)
2

dp
u
     2 gc

 g
Ws
  z  F  
 gc
m

 dp 
   F  C    
u  u
Thus,
dp
2
1
2
 dp 
u
C  
0
   2 g c
2
1
Flow of Vapor Through Holes
(Isentropic Expansion)
For ideal gas

pv 
p


 constant
(  1) / 
(  1) / 
2




 pa 
 pa 
po
2 g cCo RT 
2
2 
1   

1   

u  2 g cCo
  1 o   po 
M
  1   po 





1/ 
 pa 
Qm   Au  o  
 po 
Au  Co Apo
2/
(  1) / 


 pa 
2 g c M   pa 
    

RT   1  po 

 po 

Choked Pressure
The choked pressure is the maximum
downstream/upstream pressure ratio resulting in
maximum flow through the hole, i.e.
dQm
0
 pa 
d 
 po 
Critical Pressure Ratio
rcrit
 po 
  1 
  

 pa crit  2 
 /(  1)
po  abslute upstream pressure (N/m 2 )
pa  absolute downstream pressure (N/m 2 )
  gas specific heat ratio (C p /Cv ,dimensionless)
Typical value of specific heat ratio range from 1.1 to 1.67, which gives
the critical values of 1.71 to 2.05. Thus for releases of most diatomic
gases (1.4) to atmosphere, upstream pressures over 1.9 bar abs. will
result in sonic flow.
Critical Pressure Ratio
For pressure ratio larger than r_crit,
1. The velocity of fluid at the throat of the leak
is the velocity of sound;
2. The velocity and mass flow rate cannot be
increased further by reducing the
downstream pressure and/or increasing the
upstream pressure.
3. This type of flow is called choked, critical, or
sonic flow.
Gas Discharge Rate
Apo
Gv  Cd

a0
Gv  gas discharge rate (kg/s)
Cd  discharge coefficient (dimensionless  1.0)
A  hole area (m 2 )
ao  sonic velocity of gas = ( RTo / M )1/ 2
M  gas molecular weight (kg/kg-mol)
po  absolute upstream pressure (N/m 2 )
R  gas constant (8310 J/kg-mol/ K)
To  upstream temperature ( K)
  flow factor, dimensionless
Flow Factor
• For subsonic flows
2/ 
2

 2  pa 
 
 


1
po 



  p ( 1) /   

a
1   

  po 
 

1
2
po
for
 rcrit
pa
• For sonic (choked) flows
 2 
  

  1
(  1) / 2(  1)
po
for
 rcrit
pa
Flashing Liquid
Excess energy in superheated liquid
Q  mCp (To  Tb )
Mass of liquid vaporized
Fraction of liquid vaporized
mC p (To  Tb )
Q
mv 

H v
H v
mv C p (To  Tb )
fv 

m
H v
The implied assumption is constant physical property.
Flashing Liquid
Without the assumption
dm 

m  mv
m
mC p
H v
dT
Tb C p
dm
dT

To H
m
v
f v  1  exp  C p (To  Tb ) / H v 
Flow of Flashing Liquid
• If the flow path length is very short, nonequilibrium
condition exists, the liquid flash external to the hole.
The equations describing incompressible fluid flow
through hole apply.
• If the flow path length is greater than 10 cm,
equilibrium flashing conditions are achieved and the
flow is choked. A good approximation is to assume a
choked pressure equal to the saturation vapor
pressure of the flashing liquid. The result will be
applicable for liquids stored at higher pressure.
Flow of Flashing Liquid Stored at
Higher Pressure
Qm  ACo 2  f g c ( p  p sat )
where
A  the area of release
Co  discharge coefficient
 f  density of the liquid
p  pressure within the tank
p sat  saturation vapor pressure of the flashing
liquid at ambient temperature
Flow of Flashing Liquid Stored at
Saturation Vapor Pressure
gc
Qm  A 
 dv / dp 
v  v fg f v  v f
where
v  specific volume
v f  liquid specific volume
v fg  the difference in specific volume
between vapor and liquid
f v  mass fraction of vapor
Flow of Flashing Liquid Stored at
Saturation Vapor Pressure
df v
dv
 v fg
dp
dp
Cp
df v

dT
H v
dp H v

(Clausius-Clapyron equation)
dT Tv fg
H v A g c
Qm 
v fg
C pT
Two-Phase Discharge (1)
The discharge of subcooled or saturated liquids
G2 P  Cd G
2
sub
G
2
ERM
/N
G2 P  two phase mass flow rate (kg/m2 /s)
The effect of subcooling is accounted for by
Gsub  2( p  pv )  f
p  storage pressure (N/m )
2
pv  vapor pressure at sorage temperature (N/m2 )
 f  liquid density (kg/m )
3
Two-Phase Discharge (2)
For saturated liquids, equilibrium is reached if the
discharge pipe size is greater than 0.1 m (length
greater than 10 diameter). The discharge rate is
GERM 
h fg
v fg (TC p )1/ 2
h fg  latent heat of vaporization (kJ/kg)
v fg  change in specific volume (m /kg)
3
T  storage temperature ( K)
C p  liquid specific heat (kJ/kg/ K)
Two-Phase Discharge (3)
For discharge pipe less than 0.1 m, the flashing flow increases
strongly with decreasing length, approaching all liquid flow as
the pipe length approaches zero.
h2fg
L
N

2
2p  f Cd v fgTC p Lc
for 0  L  Lc
L  pipe length to opening (m)
Lc  0.1 m
The discharge rate of flashing liquids from sharp-edged orifice at
vessels can be estimated as though there were no flashing.