Transcript con1.2
Consequence Analysis 1.2
Mechanical Energy Balance
p
dp
u
2 2
g c
pressure fluid density
g g c F W s m 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
W s m
shaft work mass flow rate
Flow of Liquid Through a Hole
p
u
2 2
u
u
2 2
g c
u
1 2
g g c
u
2
F W s m
let
p
p g C
1 2
p
C
1 2
p g
then
u
C
1
Q m
uA
2
g p c g
AC o
C o
2
g p c g
2
g p c g
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
p
u
C
1
Q m
h L g g c
uA
F
2
g p c g
C
1 2
p
g g c
gh L
C o
C
1 2
p g
2
g p c g
gh L
g g c h L
AC o
2
g p c g
gh L
Empty Time
m
A h t L
total mass above the leak
dm
A t dt dh L dt
Q m
AC o
2
g p c g
gh L
h L h
0
L dh L
2
g p c g
2
gh L
C A o A t
0
t dt
1
g
2
g p c g
2
gh L
1
g
2
g p c g
2
gh L
0
C A o t A t
Empty Time
h L
h L
0
C At o A t
By setting
h L
0 2
g p c g
2
gh L
0
t e
1
C g o A t A
2
g p c g
gh L
0
g
2
C A o A t t
2 2
g p c g
By substituting
h L
into
Q m Q m
C A o
2
g p c g
gh L
0 2
gC A o
2
A t t
Liquid Discharge
Discharge of pure non-flashing liquid through an orifice
G L
G L
C A d
2(
p o
p a
) 2
gh
liquid mass emission rate (kg/s) 1/ 2
C d
discharge coefficient (dimensionless,0.6
0.64)
A
2 discharge hole area (m ) 3 liquid density (kg/m )
p o
2 liquid storage pressure (N/m absolute)
p a g
downstream (amb 2 acceleration of gravity (9.81 m/s ) 2 ient) pressure (N/m absolate)
h
height of liquid above hole (m)
Flow of Vapor Through Holes
dp
(Isentropic Expansion)
u
2 2
g c
g g c F W s m dp
C
1 2
dp
u
Thus,
C
1 2
dp
u
2 2
g c
0
Flow of Vapor Through Holes (Isentropic Expansion)
For ideal gas
pv
p
constant
u
2 2
g C c o
2 1
p o
o Q m
Au
o
1
p a p o
( 1) / 2
p a
1/
p o Au
C Ap o o c M o
1 1
p a p o
( 1) / 2
g M c RT
p a p o
2 /
p a p o
( 1) /
Choked Pressure
The choked pressure is the maximum downstream/upstream pressure ratio resulting in maximum flow through the hole, i.e.
d dQ m
p p a o
0
Critical Pressure Ratio
1)
r crit
p o p a
crit
1 2
p o
2 abslute upstream pressure (N/m )
p a
2 absolute downstream pressure (N/m ) gas specific heat ratio ( /
p v
,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
G v
C d Ap o
a
0
G v
gas discharge rate (kg/s)
C d A
discharge coefficient (dimensionless 1.0)
a o M
sonic velocity of gas = (
RT o
/
M
) 1/ 2 gas molecular weight (kg/kg-mol)
p o R
absolute upstream pressure (N/m 2 gas constant (8310 J/kg-mol/ K) )
T o
upstream temperature ( K) flow factor, dimensionless
Flow Factor
• For subsonic flows 2 2 1
p a p o
2 / 1
p a p o
( 1) / 1 2 for
p o p a
r crit
• For sonic (choked) flows 2 1 ( 1) / 2( 1) for
p o p a
r crit
Flashing Liquid
Excess energy in superheated liquid
Q
p o
T b
) Mass of liquid vaporized
m v
Q
H v
p o
H v
T b
) Fraction of liquid vaporized
f v
m v m
p
o
H v T b
) The implied assumption is constant physical property.
Flashing Liquid
Without the assumption
dm
mC p
H v dT
m
v dm m f v
T b T o
C p
H v dT p o
T b
) /
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
Q m
AC o
2
f g c
(
p
p sat
) where
A
the area of release
C
o f
discharge coefficient 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
g c Q m
A
v
v f fg 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
dv dp
v fg df v dp df dT dp dT v
C p
H v
H v
(Clausius-Clapyron equation)
Tv fg
Q m
H A v v fg g c C T p
Two-Phase Discharge (1)
The discharge of subcooled or saturated liquids
G
2
P G
2
P
C d
2
G sub
2
G ERM
/
N
2 two phase mass flow rate (kg/m /s) The effect of subcooling is accounted for by
p G sub
2(
p
p v
2 storage pressure (N/m ) )
f p v
f
2 vapor pressure at sorage temperature (N/m ) 3 liquid density (kg/m )
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
h fg G ERM
h fg v fg
(
TC p
) 1/ 2 latent heat of vaporization (kJ/kg)
v fg
3 change in specific volume (m /kg)
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.
N
h
2
fg
2
p
f
2
C v TC d fg p
L L c
for 0
L
pipe length to opening (m)
L c
0.1 m
L c
The discharge rate of flashing liquids from sharp-edged orifice vessels can be estimated as though there were no flashing .
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