For a pipeline blowdown with the inlet isolated, you generally see a flow rate like:
The terms in the equations are:
qchoked ==> volume flow rate at standard conditions (if not defined assume 14.7 psia and 60F) while the velocity is limited to sonic velocity (MSCF/day or kSCM/day)
qincompr ==> volume flow rate at standard conditions (if not defined assume 14.7 psia and 60F) after the velocity has slowed enough to allow the flow to be assumed to be incompressible (MSCF/day or kSCM/day
Cd ==> Discharge coefficient (for a square edged hole use 0.8, a ragged hole use 0.72, a ductile failure use 1.0)
Aeff ==> area of the opening (in^2 or m^2)
Pup ==> Upstream pressure in absolute units (psia or kPaa)
Pdwn ==> Downstream pressure in absolute units (typically local atmospheric pressure)(psia or kPaa)
SG ==> specific gravity relative to air = 1.0
Rair ==> specific gas constant for air (53.353 ft*lbf/(R*lbm or 287.1 m^2/(K*s^2))
Zup ==> compressibility at upstream conditions (fraction)
Tup ==> upstream temperature in absolute units (R or K)
k ==> adiabatic constant, ratio of specific heats (fraction)
?actual ==> density at actual conditions (it is important to include compressibility for all upstream pressures greater than 3 bara) (lbm/ft^3 or kg/m^3)
?std ==> density as though the gas were at a predefined standard pressure and temperature (compressibility is often disregarded in this computation)(lbm/ft^3 or kg/m^3)
gc ==> only for fps units 32.2 ft*lbm/(s^2*lbf)
UnitCons ==> for FPS units use 490, for SI use 84600
These equations can be solved in either fps or SI systems, care must be taken to ensure that consistent units are used (e.g., an area in ft2 does not cancel a pressure per in2)
The higher-pressure part of the graph is sonic velocity (choked flow), the curved section is incompressible flow. The straight line in the transonic region is simply connecting the end of one curve with the start of another. I have no idea what that flow profile looks like and from extensive research I'm pretty sure that no one else does either. The equations connect to each other at the critical pressure, but that is a mathematical artifact of the derivations--the velocity does not actually go from Mach 1.0 to Mach 0.3 with a 0.01 psi pressure drop. The equations do not properly reflect the transonic region, and the vertical lines on the graph are a guess at the unknowable.
In choked flow the standing shock wave prevents downstream equipment from communicating back to the source (e.g., friction in downstream pipe is not a factor in the mass flow rate). In subsonic flow, the downstream effects are able to communicate back to the source and downstream effects do change mass flow rate.
This effect is vividly evident in a diverging nozzle:
For incompressible flow (i.e., velocity below about 0.6 M), an increase in cross sectional area causes a decrease in velocity at an increasing pressure.
For compressible flow an increase in cross sectional area causes the shock wave to expand and allow an increase in velocity at a constant pressure.
Mathematically, mass flow rate is volume flow rate times density. Volume flow rate is a function of area and velocity (assuming the discharge coefficient is constant):
Above about Mach 0.6, velocity is a function of temperature, without a pressure component. Density is a function of both temperature and pressure, so at a constant temperature, mass flow rate is a linear function of pressure.
At lower velocities, the velocity is a function of the volume flow rate at actual conditions which has a term that is the difference of the ratio of the upstream pressure to the downstream pressure raised to an exponent minus the same ratio raised to a different exponent. With everything except upstream pressure held constant then the mass flow rate term will approximate a parabola (but will deviate).
The equations on the graph each represent one point in the blow down. As upstream pressure comes down, a new flow rate needs to be calculated. The equations need to be recalculated frequently to achieve reasonable results. The graph is the result of holding flow rate constant for one second, determining how much mass was exhausted during that second to find a new upstream pressure for a new calculation. Less frequent calculations will result in lines that are not as smooth, but will have a minor impact on the elapsed time calculation. Before I developed a program to do 1 second calcs I did 1 minute or 5 minute (depending on the mass of gas I was exhausting, bigger volume, less frequent calcs) iterations that gave me reasonable estimates of blowdown duration.
Thanks to member SparWeb for finding a significant bust in this FAQ and helping me fix it.
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