Source Models
Vapor flow through holes and pipes
Vapor flow though holes & pipes
Vapor flow through holes
Steady flow of vapor through pipes
Example
Liquid versus Vapor flow
Liquids –
Incompressible flow
Vapors –
Compressible flow


Kinetic energy term
is negligible


Physical properties
(density) constant
Energy from
pressure converted
to kinetic energy
Temperature,
pressure, density all
change when going
through a hole or
down a pipe
Vapor flow though holes & pipes
Vapor flow through holes


Throttling release
Free Expansion
 Non choked or subsonic
 Choked, critical or sonic
Steady flow of vapor through pipes
Example
Vapor flow through holes
Throttling flow



Small cracks – large frictional loses
Not much energy due to pressure is
converted to kinetic
Models require detailed information on
physical structure of leak
Throttling flow
A throttling device is a valve or crack or
porous material with high resistance to
flow that results in a large pressure
drop.
Throttling flow
First law of thermodynamics
dE
  Q  Ws   min (h  e p  ek )in   mout (h  e p  ek )out
dt
Assume Steady state
Adiabatic
Negligible potential and Kinetic energy effects
Single inlet and outlet
No shaft work
Throttling flow
Hence the process is isenthalpic
Consider the temperature as a function of
pressure and enthalpy
dT  fT ( P, h)
Throttling flow
Take partial
 T 
 T 
dT  
 dP  
 dh
 P h
 h  P
Definition of Joule-Thomsen coefficient
 T 
  
 P h
Throttling flow
If isenthalpic then
Integrate out
 T 

 0
 h  P
out
Tout  Tin 
 dP
in
Most gases have positive Joule-Thomsen
coefficient so as pressure drops,
temperature drops
Vapor flow though holes & pipes
Vapor flow through holes


Throttling release
Free Expansion
 Non choked or subsonic
 Choked, critical or sonic
Steady flow of vapor through pipes
Example
Vapor flow through holes
Free Expansion
Assume
Negligible potential
(ΔZ=0)
No shaft work
Ws=0
Vapor flow through holes
Mechanical Energy Balance

 u2  g
Ws
 
  Z  F 

m
 2 gc  gc
dP
Friction through “hole” is defined as before
 dP 
F  
C 


  
dP
2
1
Vapor flow through holes
Need to have density as a function of
pressure to solve integral – Assume
isentropic flow

Pv 
P

 

 constant
Cp
Cv
Vapor flow through holes
Substitute all into MEB and integrate
You end up with velocity as function of
several terms
u  f  P, P0 , T0 , properties 
As before, mass flow rate from velocity
Qm   uA
Vapor flow through holes
Design equation for subsonic flow
through holes Eq. 4-38

2 g c M   P    P 
Qm  C0 AP0
 


RgT0   1  P0   P0 

2
 1 




Vapor flow though holes & pipes
Vapor flow through holes


Throttling release
Free Expansion
 Non choked or subsonic
 Choked, critical or sonic
Steady flow of vapor through pipes
Example
Choked flow through holes
As you lower the down stream pressure (or
increase upstream pressure) the velocity
increases until it reaches a critical velocity,
the sonic velocity, or speed of sound.
After that the velocity becomes independent
of pressure. Downstream conditions no
longer have an effect on velocity.
Choked flow through holes
For choked, critical or sonic flow
u  a ( SpeedOfSound )   g c RgT / M
So at choked conditions Eq. 4-40
Q 
m choked
 gc M  2 
 C0 AP0


RgT0    1 
 1
 1
For sharp edged orifice C0=0.61, Worst
case scenario C0=1.0
Choked flow through holes

Pchoked  2   1


P0


1


Gas

Pchoked
Monotonic
~1.67
0.487P0
Diatomic (air)
~1.40
0.528P0
Triatomic
~1.32
0.542P0
Vapor flow though holes & pipes
Vapor flow through holes
Steady flow of vapor through pipes

Adiabatic flow of vapor through pipes
 Non choked flows
 Choked flows

Isothermal flow of vapor through pipes
 Non choked flows
 Choked flows
Example
Vapor flow through pipes
There are two cases which we can derive
(with much work) relationships for flow of
vapors through pipes


Adiabatic – which assumes well insulated walls, no
energy loss to surroundings
Isothermal – which assumes constant wall
temperature (submerged pipe)
Vapor flow though holes & pipes
Vapor flow through holes
Steady flow of vapor through pipes

Adiabatic flow of vapor through pipes
 Non choked flows
 Choked flows

Isothermal flow of vapor through pipes
 Non choked flows
 Choked flows
Example
Adiabatic vapor flow in pipes
For compressible
flow it is best to
work things out in
terms of the Mach
number, Ma.
u
Ma 
a
Adiabatic vapor flow through pipes
The book doesn’t even attempt to go
through the derivations, just gives the
equations.
As before, we need to consider both
nonchoked and choked flow.
Adiabatic vapor flow through pipes
For most problems you know




L – length of pipe
d – diameter of pipe
T1, P1 – upstream temperature, pressure
P2 – downstream pressure
To get mass flow rate Qm (mass/time) from
G, mass flux, (mass/area*time) use Qm=G*A
Adiabatic non choked flows in pipes
1) Find pipe roughness from Table 4-1
2) Determine f from Eq. 4-27
1
d

 4log  3.7 

f

3) Determine T2 from Eq. 4-51 (trial & error)
4) Calculation G from Eq. 4-52
5) Calculate Reynolds number to verify Eq 427 is valid
Adiabatic Choked flows in pipes
1) Find roughness from Table 4-1
2) Determine f from Eq 4-27
3) Determine Ma1 from Eq 4-57 (use 4-46
to get Y1) (usually trial & error)
4) Determine mass flux, Gchoked Eq. 4-56
5) Determine Pchoked from Eq 4-54
6) Double check Reynolds number
Vapor flow though holes & pipes
Vapor flow through holes
Steady flow of vapor through pipes


Adiabatic flow of vapor through pipes
Isothermal flow of vapor through pipes
 Non choked flows
 Choked flows
Example
Isothermal non choked flows
1)
2)
3)
4)
Find roughness from Table 4-1
Determine f from Eq. 4-27
Compute G from Eq. 4-63
Double check Reynolds number
For isothermal non choked flow no need
for trial and error, nice analytical
equations
Isothermal choked flows
1) Find roughness from Table 4-1
2) Find f from Eq. 4-27
3) Determine Ma1 from Eq. 4-71 (trial and
error)
4) Determine G from Eq. 4-70
5) Double check the Reynolds number
Vapor flow though holes & pipes
Vapor flow through holes
Steady flow of vapor through pipes
Example