There is no Kantrowitz limit

Anyone who’s somewhat interested in mass transportation systems is bound to have looked into the Hyperloop concept at one point or another. One of the things that came up during my research was something called the ‘Kantrowitz limit’.

What is the Kantrowitz limit? I admittedly had never heard of it before, but it seemed like something very important that fortunately had technological solutions. Yet there was something unsatisfying about all the popular resources I could find about it. They all seemed to be very surface level, and when I started looking into it the rabbit hole ended up going a lot deeper than I expected, so I am writing this in the hopes that it might be useful to someone else one day.

Kantrowitz limit: the elevator pitch

The Kantrowitz limit is a theoretical compressible gas dynamics restriction that places a limit on the flow velocity. It was first derived in 1945 by Arthur Kantrowitz, a physicist and aerospace engineer, from whom it received its namesake, but remained relatively obscure—Modern Compressible Flow by John D. Anderson, the quintessential textbook for gas dynamics, for example, has no mention of it—until 2013, when it was reintroduced into the mainstream engineering lexicon when Elon Musk proposed his Hyperloop concept for futuristic, high-speed travel.

Musk is a little vague about the Kantrowitz limit in his Hyperloop white paper, and doesn’t really explain any of the technical details, except that it’s an important consideration when designing the Hyperlood pods. As he explains:

Whenever you have a capsule or pod (I am using the words interchangeably) moving at high speed through a tube containing air, there is a minimum tube to pod area ratio below which you will choke the flow. What this means is that if the walls of the tube and the capsule are too close together, the capsule will behave like a syringe and eventually be forced to push the entire column of air in the system.

While this is possibly useful to laymen, it is a phenomenological explanation: it describes what is happening, but doesn’t elaborate on why. Thankfully, the Wikipedia article has a fairly succinct explanation:

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 speed can no longer be increased above this limit, independent of changes in upstream or downstream pressure.

Seems simple enough. The Kantrowitz limit is related to the gas dynamics at near-sonic flow velocities. For vehicles moving at near-supersonic speeds through a tunnel (i.e. Hyperloop pods), the Kantrowitz limit places a design constraint on the ratio of the diameter of the pod to the diameter of the tube.

What’s the hiccup?

All of this is sensible and seems well and good… except that this is a really, really basic concept that is known by anyone who has studied gas dynamics or aerodynamics at all. I’m not trying to be an elitist here: choked flow is probably taught by the second or third week of any undergraduate level gas dynamics course.

So clearly, there should be more to it than this if someone got their name attached to it! Maybe the Wikipedia explanation was incomplete? What do other online sources on it say? There’s a surprising dearth of information—most of the articles I found only give a high level explanation of the Hyperloop. Those that do dive into the relevant physics, despite otherwise widely ranging in credibility, all seem to just repeat what was said in the Hyperloop whitepaper.

Here are a few examples:

This article by Mapping Ignorance repeats the “syringe effect”, and also cites an equation that seems to have been taken from the Wikipedia article¹.
CNBC’s article doesn’t really mention the concept and just explains that compressor fan will “transfer high air pressure from [the pod’s] front to the rear and sides”.
Brilliant.org has a section on the Physics of Hyperloop where they also repeat the same claims about the Kantrowitz limit and the syringe effect. They directly quote the white paper on this.
An article by Jalopnik specifically about the Kantrowitz limit just uncritically rephrases what Musk wrote, but it does cite a paper on the topic! But this paper assumes that you already know what the limit is, so some more digging is required.
I also found this from a site called HyperloopDesign.net which claims that there is no Kantrowitz limit for Hyperloop pods. However, their explanation of the relevant physics seems very dubious², so I’m not going to consider their argument further.

So what is the Kantrowitz limit, really?

Without a satisfactory explanation, I decided to read the original 1945 paper³ by Kantrowitz and Donaldson. They were studying the design of diffusers for supersonic air streams (i.e. air intakes for supersonic planes). In short, the Kantrowitz limit has to do with the stability of normal shocks in variable-area ducts when decelerating gases from supersonic to subsonic velocities.

Choked flow: an explanation

Let’s do a brief aside and discuss the concept of choked flow, since it is rather critical to understanding the issue at hand. You can skip this section if you are already familiar with the relevant physics.

Consider a gas flowing through a variable-area duct⁴. As the area of the duct shrinks, the gas must either speed up or compress (become denser) to maintain the same mass flow rate. The maximum speed that the gas can reach through a reduction in area is the speed of sound. Once this speed is reached, any further reduction in area cannot increase the speed further (and in fact the gas will always reach Mach 1 at the vena contracta, or point with the smallest cross-sectional area, as long as this area is below the critical area).

Example of choked flow — Flow through a converging duct, right at the point where the flow chokes (throat velocity has reached Mach 1).

At this point, any further reduction of the pressure downstream of the duct will not influence the flow rate; it is considered choked. However, increasing the pressure upstream will increase the mass flow rate by increasing the density of the gas, but the volumetric flow rate will remain roughly the same. So there is a “critical” amount of shrinkage the duct area can undergo before the flow reaches Mach 1. This, of course, depends on how much the gas decides to speed up vs. compress when the area is reduced, and the “speed-up-to-densification” ratio depends on the assumptions you make of the gas.

Most commonly, we assume that the gas is calorically perfect⁵, and that the compression process is isentropic. In this case, the relationship between the choking contraction ratio and the airspeed is known as the area-Mach number relation:

$$ \frac{1}{\text{Contraction ratio}} = \frac{A^*}{A} = M\left[\frac{2}{\gamma+1}\left(1 + \frac{\gamma-1}{2}M^2\right)\right]^{-\frac{\gamma+1}{2(\gamma-1)}} $$

Where $M$ is the Mach number at some point in the duct (usually the entrance), $A$ is the area at the point in the duct where $M$ was taken, $A^*$ is the cross-sectional area at the minimum area of the duct (known as the throat), and $\gamma$ is the heat capacity ratio, which varies depending on the gas but for air is roughly equal to 1.4. The interesting thing about this equation is how the contraction ratio only depends on the Mach number; there are no variables for temperature, pressure, density, etc.

Area-Mach number relation — The area-Mach number relation creates a 'U' shaped curve. Inside the U, the flow will be choked, and outside, the flow will be unchoked.

Something worth noting is that there are actually two solutions of $M$ for a given contraction ratio: one for $M<1.0$ and one for $M>1.0$. This means for any given ratio, there is a subsonic speed below which the flow will not choke and a supersonic speed above which the flow will not choke, surprisingly enough.

Accelerating subsonic flow — Subsonic flow is accelerated in a converging nozzle and chokes at the throat. If the duct diverges after, the flow can accelerate to supersonic speeds.

This is also the principle behind rocket engine nozzles: a convergent-divergent nozzle will accelerate a gas from subsonic to supersonic velocities as long as the pressure drop is sufficiently high.

The problem with supersonic deceleration

Since this process is isentropic and therefore reversible, one would imagine that the opposite scenario should also be possible: a converging-diverging inlet that decelerates supersonic gas to subsonic velocities.

Decelerating supersonic flow — Smooth deceleration of a supersonic stream. In practice, this is not achievable.

In practice, however, this kind of arrangement is not possible to achieve and maintain. There are two problems.

The first is that generally, before reaching a supersonic velocity, the system has to first accelerate through a subsonic regime. And a subsonic flow will also choke the inlet, but at a much lower mass flow rate. As the system speeds up in the subsonic regime, some of the gas must get diverted away from the inlet as it cannot “fit” through the throat. The ratio of air diverted away to air entering the inlet is called the capture ratio.

Choked sonic deceleration — Choked inlet at subsonic speeds. Some of the flow that would otherwise have entered the inlet is deflected away due to the pressure buildup.

As the airspeed increases further and turns supersonic, the gas can no longer turn away in time (since it cannot receive information faster than the speed of sound). A bow shock will form in front of the inlet, and the subsonic gas on the other side can turn away from the inlet, maintaining mass continuity.

Choked supersonic deceleration — Once the system settles at its (supersonic) cruising speed, a bow shock will form in front of the inlet. This leads to massive pressure losses, while the flow inside the duct is subsonic.

The second problem is that even if a smooth supersonic deceleration is achieved, it is inherently unstable. Since the outgoing flow is subsonic, information about disturbances downstream (such as pressure fluctuations) would be able to propagate through it, lowering the mass flow rate, if only temporarily. However, since the entering gas is supersonic, information cannot propagate upstream through it, and its mass flow rate will remain unchanged. As Kantrowitz explains, “it therefore appears that isentropic deceleration through the speed of sound in channels is unstable and unattainable in practice.”

Unstable shock — As the shock moves backwards, it becomes stronger as the flow at the new location is traveling faster, which leads to an even greater pressure drop, causing a feedback loop until the shock is ejected from the inlet.

Implications of shock waves

Shocks are highly irreversible phenomena and will lead to significant amounts of energy dissipation. The total losses across the shock wave are captured by the drop in stagnation pressure. The stagnation pressure drop across a shock wave is given by the following equation: $$ \frac{p_{t2}}{p_{t1}} = \left(\frac{\frac{\gamma+1}{2}M_1^2}{1 + \frac{\gamma-1}{2}M_1^2}\right)^{\frac{\gamma}{\gamma-1}} \left(\frac{1}{\frac{2\gamma}{\gamma+1}M_1^2 - \frac{\gamma-1}{\gamma+1}}\right)^{\frac{1}{\gamma-1}} $$ Where $p_t$ is the total, or stagnation pressure, and the subscripts 1 and 2 denote before and after the shock, respectively. The stagnation pressure, is, in turn, directly proportional to the mass flow rate through the duct, as given by the following equation: $$ \dot m = \frac{p_t A}{\sqrt{T_t}}\sqrt{\frac{\gamma}{R}}\left(\frac{\gamma+1}{2}\right)^{-\frac{\gamma+1}{2(\gamma-1)}} $$ For a given duct area $A$, the mass flow rate depends on the total pressure and the total temperature. Because a shock is adiabatic, the stagnation temperature does not change across a shock, so $\dot m \propto p_t$ only.

Pressure recovery of air across a normal shock, depending on supersonic Mach number

Thus, a shock wave will result in total pressure losses. These losses will reduce the mass flow rate through the inlet by the proportional drop in the pressure. A smaller mass flow rate means a smaller throat area is needed to choke the flow.

So the Kantrowitz limit basically is just the area-Mach number relation multiplied by the stagnation pressure ratio across a normal shock.

Effect of the Kantrowitz “limit”

Writing out the equation, the Kantrowitz limit is the area ratio denoted by: $$ \overbrace{\frac{A_4}{A_0}}^{\text{area ratio}} = \overbrace{M\left[\frac{2}{\gamma+1}\left(1 + \frac{\gamma-1}{2}M^2\right)\right]^{-\frac{\gamma+1}{2(\gamma-1)}} }^{\text{area-Mach number relation}} \underbrace{ \left(\frac{\frac{\gamma+1}{2}M^2}{1 + \frac{\gamma-1}{2}M^2}\right)^{\frac{\gamma}{\gamma-1}} \left(\frac{1}{\frac{2\gamma}{\gamma+1}M^2 - \frac{\gamma-1}{\gamma+1}}\right)^{\frac{1}{\gamma-1}} }_{\text{stagnation pressure ratio across normal shock}} $$ Simplifying and rearranging gives us the same form as the Wikipedia article: $$ \frac{A_4}{A_0} = M_0 \left(\frac{\gamma+1}{2+(\gamma -1)M_0^2}\right)^{-\frac{\gamma+1}{2(\gamma-1)}} \left[\frac{(\gamma+1)M_0^2}{(\gamma-1)M^2_0+2}\right]^\frac{-\gamma}{\gamma-1} \left[\frac{\gamma+1}{2\gamma M^2_0-(\gamma-1)}\right]^\frac{-1}{\gamma-1} $$

Plotting it out compared the basic isentropic compression case, it’s clear that the shock will greatly limit the design contraction ratio.

Kantrowitz limit vs isentropic compression — Comparison of the choking contraction ratio with the Kantrowitz limit and with isentropic flow. The Kantrowitz effect means that the flow chokes much sooner than it otherwise would.

The Kantrowitz limit is thus relevant when a supersonic flow needs to be decelerated to subsonic flow, such as is the case in air intake ducts on supersonic planes. It appears repeatedly in the literature mostly in papers dealing with the design of supersonic intake ducts.

And perhaps instead of “limit”, a more appropriate term would be “effect”, as it’s not really a limit at all. It’s an aerodynamics phenomenon that results in suboptimal mass flow/pressure recovery, but it’s not an outright cap on the performance of a system, like the term “limit” would imply.

To my great surprise, it does seem like everyone has gotten this wrong—no one I found correctly explained this, and usually just parroted the “syringe” explanation from the Musk paper, despite the original paper by Kantrowitz being readily available online (and rather short: the meat and bones of it is only about 5 pages).

Overcoming the limit

The Kantrowitz “effect” is undesirable as the pressure losses across the shock lower the mass flow rate through the duct. The total pressure ratio is also known as the pressure recovery, with a perfect pressure recovery being a value of 1.

But as it isn’t a hard limit, there are a few workarounds that can be employed to mitigate its effect. The general goal is to make the shock weaker, which will improve the pressure recovery. The most common way to do this is to move the shock wave into the inlet, behind the throat. In this position, it is stable, and is much weaker than if it were outside and in front of the inlet. This is often referred to as “starting” the inlet.

Overspeeding

The simplest solution is to temporarily overspeed the system above the design Mach number.

As the shock moves through the converging duct, a positive feedback loop will occur: its upstream Mach number will decrease, which means that the shock will weaken. This results in the throat experiencing an increased mass flow rate, which will reduce the shock backpressure which draws the shock further into the duct. Thus the shock will continue moving until it has passed through the throat.

Stable shock — In this position, the shock is stable. An increase in backpressure will push the shock back, but as it approaches the throat, it will weaken, improving the pressure recovery and causing a negative feedback loop that prevents it from being dislodged.

However, the opposite situation is also true. As previously mentioned, if the shock is placed at exactly the throat, and there is a temporary backpressure fluctuation, then that could be enough to knock the shock wave out of the inlet, leading to massive energy losses.

Metastable shock — The shock is at its weakest in the throat (technically, at M = 1.0, there is no shock). But the slightest disturbance or fluctuation in backpressure can have it knocked off and spat out of the inlet, causing a very sudden and disruptive drop in pressure.

This is only a viable option for systems operating at relatively low Mach numbers. Beyond a design value of ~Mach 1.6 the amount of overspeeding needed becomes impractical from an engineering standpoint, and above Mach 1.9 it becomes flat out impossible.

Overspeed graph — The overspeed required to start inlets designed for specific Mach numbers dramatically grows above design Mach numbers of 1.6. The dashed line is y = x, for reference purposes.

Increasing the throat size

Kantrowitz and Donaldson propose a simple solution: increasing the throat size so that it chokes at a subsonic mass flow rate equal to the expected mass flow rate during supersonic operation at the design Mach number. This inlet design is also known as a Kantrowitz-Donaldson inlet, and it is self-starting, because it will naturally settle at the desirable equilibrium point.

A Kantrowitz-Donaldson inlet has a wider throat which will automatically swallow the shock as soon as supersonic speeds are reached.

The downside is that the pressure recovery is lower than that of a normally-sized inlet since the inlet isn’t designed for the desired mass flow rate. The larger throat means the gas will decelerate less before reaching the shock, resulting in more pressure losses. So the tradeoff is whether you want a self-starting inlet that works at ~85% performance or one that needs to be manually started but works at ~95% performance.

Using variable nozzles

In a similar vein, another solution is to adjust the size of the throat in real time to move the shock past the throat before closing it back up. This approach results in high pressure recovery (and therefore high performance), at the cost of complexity and additional weight.

Oblique shocks

The last solution is to use a geometry that will force the airflow to turn, producing a series of oblique shocks instead. These will lead to lower pressure losses than a single normal shock. These changes in angle are called ramps. This can be done through systems like intake ramps or inlet cones, which are a common sight on supersonic aircraft. Inlets that have these shocks occur outside of the inlet are called external compression inlets, while those with shocks that occur inside the intake are called internal compression inlets.