# Why does compressible flow never choke in a thin-plate orifice?

According to this Wikipedia article on orifice plates:

... the flow of real gases through thin-plate orifices never becomes fully choked.

Further to this, there is a reference given, which states the following:

Cunningham (1951) first drew attention to the fact that choked flow will not occur across a standard, thin, square-edged orifice.[10] The mass flow rate through the orifice continues to increase as the downstream pressure is lowered to a perfect vacuum, though the mass flow rate increases slowly as the downstream pressure is reduced below the critical pressure.

Is this true? If so, how/why is this the case? Surely an orifice provides a minimum flow area, which should impose a limit on the mass flow rate, beyond a certain critical pressure drop?

If anyone thinks that the Wikipedia article or the reference are incorrect, then I would be very interested in a reliable reference that contradicts it.

There are practical considerations involved in sizing an orifice plate and the associated piping. For the piping that is carrying flow to the orifice plate, you want a maximum gas or vapor velocity of approximately 200-300 ft/s, and a maximum liquid velocity of approximately 10 ft/s, because higher velocities lead to too much pressure drop per foot of pipe, meaning that too much energy is lost due to friction effects. For the associated orifice plate, you want a reasonable estimate of the flow rate through the pipe, but you also want to somewhat minimize the pressure drop across the orifice plate, because this also represents lost energy. This means that the hole diameter of the orifice plate is "substantial" when compared to the diameter of the piping that contains it. Thus, to answer the question, there is never choked flow through an orifice plate because the orifice plate and associated piping system are deliberately designed to avoid those flow conditions.

• The way that passage in the Wikipedia article reads to me - and in particular, the reference 12 (en.wikipedia.org/wiki/Orifice_plate#cite_note-12) - it seems to be claiming that a thin-plate orifice would never completely choke, for some physical reason, not merely that it is typically designed to avoid it. It also doesn't mention anything about a size threshold for the central hole, where this would occur. Jul 26 '18 at 13:14
• Of course, it is possible that the article is incorrect or badly-worded. In that case, I would be very interested in a reliable reference that contradicts the claim. I admit I haven't searched for the 'Cunningham 1951' that is referenced for the claim. I will add the specific passages from the article to my question. Jul 26 '18 at 13:16
• @Time4Tea, the article is correct - a thin-plate orifice in a typical application will never choke. Pumps provide the fluid velocity through the pipe and orifice. All equipment is "matched" in size, and designed to somewhat minimize cost of equipment. Nobody is going to specify the very large and expensive pump (expensive due to both capital and operating cost) needed to achieve choked flow through the associated orifice. Any chemical engineer who consistently did so would get fired. Jul 26 '18 at 17:04
• I disagree with your interpretation of what the article is claiming. It says nothing about 'typical applications'. It seems to me to be claiming that it is not physically possible to completely choke a thin-plate orifice, regardless of the flow conditions that are imposed. Jul 26 '18 at 18:38
• @Time4Tea, I'm not so much claiming that the Wikipedia article is wrong. You asked if anyone thinks the reference is incorrect. I maintain that the chances of deliberately designing an orifice for choked flow in a real-world application, is slim to none. Due to this, it will be very difficult to find someone who can refute the Wikipedia article. Jul 26 '18 at 19:34

I think the article is correct, that it is not physically possible to fully choke a thin-plate orifice. The following sketch shows a comparison between a thin-plate orifice and a de-Laval nozzle:

Because of the separation that occurs in the orifice, and the fact that the jet streamlines are never going to be perfectly horizontal, it seems reasonable to conclude that the vena contracta for the orifice plate must occur somewhere downstream of the orifice. Therefore, the vena contracta is 'floating in space' and is not constrained by any solid boundary.

Therefore, even after sonic velocity is reached in the vena contracta, if the downstream pressure ($p_2$) continues to decrease, then the streamlines of the jet will become more horizontal and the vena contracta will increase, hence the flow rate will continue to increase.

This is enabled by the fact that, because the vena contracta is 'floating' in the flow, sonic flow is not 'blocking' the flow passage, which means that (unlike with the nozzle), pressure waves can travel upstream to and past the orifice, so reducing the downstream pressure can continue to affect the flow through the orifice.

For the orifice plate, the vena contracta area ($A_{vc}$) will never reach the same size as the orifice ($A_o$), because of the flow separation from the edge of the orifice. So, in a way, it could be said to be 'less efficient' than the nozzle, in terms of passing flow.

In the case of the de Laval nozzle, because it is designed to prevent separation of the flow, the vena contracta is constrained to always be at the point of minimum flow passage area. Because of this, it can't increase, so once sonic velocity is reached there, it is not possible to increase the mass flow rate any further. Also, because the sonic vena contracta completely blocks the flow passage, pressure waves are prevented from travelling upstream past it, so reducing the downstream pressure cannot have any effect on the flow conditions upstream of the v.c.

Does this seem like a plausible explanation?

• The vena contracta does in fact occur downstream of the orifice plate. For flow orifices, the upstream pressure tap is located a bit upstream of the orifice plate, and ideally, the downstream pressure tap is located at the vena contracta. Various piping geometry constraints affect the tap locations, and a discussion of some of the details are given here: en.wikipedia.org/wiki/Orifice_plate Jul 27 '18 at 19:02

I ended up here from a Google search after reading and being confused by a similar statement on the Wikipedia article on choked flow. However, after looking at your diagrams and then digging some more, I've discovered that what's happening is actually kind of the opposite of what you drew up and explained.

It turns out that in the case of a sharp-edged orifice, the vena contracta moves further away from the orifice as the downstream pressure is lowered toward a critical pressure ratio and then as the downstream pressure further decreases, the vena contracta actually moves back toward the orifice. Until the fluid's critical pressure ratio is reached downstream, there are still shock disturbances upstream.

I found a copy of Cunningham's paper here (see this archived copy if that link becomes broken in the future).

Cunningham wrote:

For a well-formed convergent nozzle, the (experimental) maximum flow ratio is essentially identical with the (theoretical) critical-pressure ratio. Evidently the occurrence of sonic velocity at the throat of the nozzle prevents flow response to changes in the discharge pressure.

Contrary to the behavior of the convergent nozzle, the square-edged orifice does not exhibit a maximum flow ratio. Rather, experiment shows that the flow rate (for constant upstream conditions) continues to increase at all pressure ratios between the critical and zero; this range is defined as the "supercritical" range of ratios.

It seems apparent that the shape of the fluid path, diameter of the vena contracta and its distance from the orifice is formed by the gas and/or fluid essentially reflecting off of the walls in the diverging end of the nozzle and applying pressure on the flow that has exited the orifice. Thus, as the pressure continues to decrease after reaching the supercritical ratio threshold, it would intuitively make sense that not only would the vena contracta move toward the orifice but it should also expand in diameter. This is supported by his summarized description of findings on page 2 of the paper, which states:

1. For supercritical flow, the orifice jet converged to a minimum section and then diverged.
2. The minimum area of the jet was found to be a function of the downstream pressure. As the pressure ratio was decreased, the minimum area approached the orifice and became larger.
3. The location of the critical axial pressure approximately coincided with the minimum section of the jet; and it shifted toward the orifice in the same manner as the minimum stream section as the pressure ratio was lowered.

So, to finally state the answer to the question clearly:

By definition, choked flow only occurs when adjusting the downstream pressure does not affect the flow velocity. With a diverging nozzle where attachment is largely maintained, the fluid passing through and maintaining attachment follows a longer path than the fluid at the center of the flow and can thus act to apply a degree of pressure on the flow as any other gas/fluid might - the fluid creates its own minimum discharge chamber pressure. With a square-edged orifice this can't happen so there is never a point at which a decrease in discharge chamber pressure prevents increasing flow velocity.

If you're familiar with general relativity, space-time geodesics and zero-point energy, it's really the same fundamental concepts.