User:Richard.arnold32/sandbox
I am creating a new article with covers the Kantrowitz Limit, an important concept in gas dynamics and supersonic devices such as jet engines, rocket engines, and the recently proposed Hyperloop. The Kantrowitz limit is an ever relevant concept in gas dynamics, and this subject has currently little to no coverage on Wikipedia, with only a very general description of the concept included on its inventor's biographical page, Arthur Kantrowitz.
This is a user sandbox of Richard.arnold32. You can use it for testing or practicing edits. This is not the sandbox where you should draft your assigned article for a dashboard.wikiedu.org course. To find the right sandbox for your assignment, visit your Dashboard course page and follow the Sandbox Draft link for your assigned article in the My Articles section. |
In gas dynamics, the Kantrowitz limit refers to a theoretical concept describing describing choked flow at supersonic or near-supersonic velocities.[1] When a fluid flow experiences a reduction in area, the flow speeds up in order to maintain the same mass-flow rate, per the continuity equation. If a near supersonic flow experiences an area contraction, the velocity of the flow will increase until it reaches the local speed of sound, and the flow will be choked. This is the principle behind the Kantrowitz limit: it is the maximum amount of contraction a flow can experience before the flow chokes, and the flow rate can no longer be increased above this limit, independent of changes in upstream or downstream pressure.
Derivation of Kantrowitz limit
Assume a fluid enters an internally contracting nozzle at cross-section 0, and passes through a throat of smaller area at cross-section 4. A normal shock is assumed to start at the beginning of the nozzle contraction, and this point in the nozzle is referred to as cross-section 2. Due to conservation of mass within the nozzle, the mass flow rate at each cross section must be equal:
For an ideal compressible gas, the mass flow rate at each cross-section can be written as[2],
- where is the cross-section area at the specified point, is the Isentropic expansion factorof the gas, is the Mach number of the flow at the specified cross-section, is theideal gas constant, is the stagnation pressure, and is the stagnation temperature.
Letting the mass flow rates equal at the inlet and throat, and recognizing that the total temperature, ratio of specific heats, and gas constant are constant, the conservation of mass simplifies to,
Solving for A4/A0,
Three assumptions will be made: the flow from behind the normal shock in the inlet is isentropic, or pt4 = pt2 , the flow at the throat (point 4) is sonic such that M4 = 1, and the pressures between the various point are related through normal shock relations, resulting in the following relation between inlet and throat pressures[1],
And since M4 = 1, shock relations at the throat simplify to[2],
Substituting for and in the area ration expression gives,
This can also be written as,[3]
Applications
The
Hyperloop
The Kantrowitz limit is a fundamental concept in the
In order to break through the speed limit set by the Kantrowitz limit, there are two possible approaches. The first would increase the diameter of the tube in order to provide more bypass area for the air around the pod, preventing the flow from choking. This solution is not very practical in practice however, as the tube would have to be built very large, and logistical costs of such a large tube are impractical. Another possible solution as proposed by Elon Musk in his 2013 Hyperloop Alpha paper would have a
For a pod travelling through a tube, the Kantrowitz limit is given as the ratio of tube area to bypass area both around the outside of the pod and through any air-bypass compressor:[5]
where: | |
= cross-sectional area of bypass region between tube and pod, as well as air bypass provide by a compressor onboard the pod | |
= cross-sectional area of tube | |
= Mach number of flow | |
= = isentropic expansion factor
| |
( and are specific heats of the gas at constant pressure and constant volume respectively), |
References
- ^ a b Kantrowitz, Arthur; duP, Coleman (May 1945). "Preliminary Investigation of Supersonic Diffusers" (PDF). Advance Confidential Report L5D20. National Advisory Committee for Aeronautics.
- ^ a b "Compressible Mass Flow Rate". www.grc.nasa.gov (in eng). Retrieved 2017-04-10.
{{cite web}}
: CS1 maint: unrecognized language (link) - ISBN 9781600864414.
- ^ a b c d Musk, Elon (August 12, 2013). "Hyperloop Alpha" (PDF). SpaceX. pp. 3–4. Retrieved August 14, 2013.
- . Retrieved August 13, 2013.
The ability of the classical Kantrowitz limit to predict the restart contraction ratio was assessed, and it was shown to be applicable for the hard unstart/restart configurations.