INTELLIGENT WORK FORUMS FOR ENGINEERING PROFESSIONALS
Log In
Come Join Us!
Are you an Engineering professional? Join EngTips Forums!
 Talk With Other Members
 Be Notified Of Responses
To Your Posts
 Keyword Search
 OneClick Access To Your
Favorite Forums
 Automated Signatures
On Your Posts
 Best Of All, It's Free!
 Students Click Here
*EngTips's functionality depends on members receiving email. By joining you are opting in to receive email.
Posting Guidelines
Promoting, selling, recruiting, coursework and thesis posting is forbidden. Students Click Here

Pipelines, Piping and Fluid Mechanics engineering FAQ
Choked mass flow rate of gases
What does the flow profile look like for a Pipeline Blowdown? by zdas04
Posted: 14 Feb 15 (Edited 26 Feb 15)

For a pipeline blowdown with the inlet isolated, you generally see a flow rate like:
The terms in the equations are:
 q_{choked} ==> 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)
 q_{incompr} ==> 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
 C_{d} ==> Discharge coefficient (for a square edged hole use 0.8, a ragged hole use 0.72, a ductile failure use 1.0)
 A_{eff} ==> area of the opening (in^2 or m^2)
 P_{up} ==> Upstream pressure in absolute units (psia or kPaa)
 P_{dwn} ==> Downstream pressure in absolute units (typically local atmospheric pressure)(psia or kPaa)
 SG ==> specific gravity relative to air = 1.0
 R_{air} ==> specific gas constant for air (53.353 ft*lbf/(R*lbm or 287.1 m^2/(K*s^2))
 Z_{up} ==> compressibility at upstream conditions (fraction)
 T_{up} ==> 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)
 g_{c} ==> 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 ft^{2} does not cancel a pressure per in^{2})
The higherpressure 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 derivationsthe 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. 
Back to Pipelines, Piping and Fluid Mechanics engineering FAQ Index
Back to Pipelines, Piping and Fluid Mechanics engineering Forum 

Resources
Leading manufacturers are aligning their people, processes, and tools from initial product ideation through to field service. They do so by providing access to product and enterprise data in the context of each personâ€™s domain expertise. However, it can be complicated and costly to unite engineering with the factory and supply chain. Download Now
This whitepaper describes the role of thermal transient measurement to characterize semiconductor thermal behavior. It focuses on the value measurement derived structure functions provide through interpretation of the heat flow path inside a package for use in thermal characterization, failure diagnosis, and improving simulation thermal model accuracy. Download Now
By leveraging comprehensive digital twins of both product and production, manufacturers can design and validate assembly processes and entire facilities virtually, improving quality and accelerating the rampup to volume production. Download Now
Medical device product development is a highly integrated and regulated process. Implementation of a requirements tracking solution requires attention to a variety of nuanced topics. When presented with the task of tracking the many concept relationships in a project of this type, the initial software solution of choice tends to be a twodimensional text systems. Download Now

