Converting CUP to PSI

Long ago, I had a lovely Swedish Mauser, and needed some reload data. I had one of Steve Faber's early strain gauge systems, but the only data conveniently available used powders I didn't have and was in CUP. What to do?

I collected a handful of SAAMI maximum pressure specifications for rifles that gave both the CUP and the PSI ratings and ran a regression on the data to get the formula that connects them. That didn't take long. My results appeared in Varmint Hunter not long after.

Since I was only interested in rifle pressures, the data I used was rifle data, and the results only applied to rifles, not to handguns. Those interested in a more detailed explanation of what I did can read this:

https://www.shootingsoftware.com/ftp/psicuparticle2.pdf

A couple of years after the article appeared, I found that Dr. Lloyd Brownell had wondered about converting back and forth between the two systems, and had published a conversion curve decades before.

The conversion has a little curvature in it, but over the range of about 35-65 KPSI, the simple straight line model works quite satisfactorily. I was able to choose a peak KPSI for my Mauser, and sometime after I did that, SAAMI arrived at a recommended peak very close to the one I chose. Nice.

So for the simple model that applies only to centerfire rifles, the formula for ANSI specs is:

PSI = -17,902 + 1.51 * CUP

Today, I have much more data readily available and had a fit of ambition enough to run the model with handgun data included. The 357 Magnum was a major outlier, and was omitted. SAAMI specs have a large number of odd human choices in them, and that's apparently one. There are still a couple of strange examples in the data, but none so great as the 357. That left me with 38 data pairs, which is plenty.

For those who want a model that covers all centerfire firearms, you can use this one:

KPSI = 14.39 + .01716*KCUP^2

The raw data have significant random noise. Since the input data is imperfect, the output is not exact, but it is close enough for most practical purposes. Also, the exact results will vary a bit if you choose cartridges different from my selection.

If you have a large batch of nearly identical cartridges, randomly select 10, measure their peak pressure, and then draw and test another 10, you'll practically always get two different numbers. You shouldn't expect modern test equipment to repeat much closer than about 1,000 PSI. Bear that in mind as you compare the model with actual data.

All other factors equal, when powder charge goes up, CUP goes up in an orderly way.

Also, as powder charge goes up, PSI goes up in an orderly way.

It follows that as CUP goes up (or down), PSI goes up (or down) too. That's correlation, and the formula linking the two is easily found.

Years ago, SAAMI put out the word that they were not convertible from one to the other. They may have been overreacting to the confusion caused because what we now know as CUP was believed to be and was called PSI. Or they may have tried regression on one batch of nearly identical cartridges, which will show only the combined random error of both systems (no correlation), or they may have simply not been careful in their choice of words. In any event, their incorrect statement has sown a large amount of misunderstanding.

If you draw random cartridges out of large batch of nearly identical cartridges and measure their pressure, you have an estimate of the average PSI of the large batch. If you draw and test another sample, you'll practically always get a different answer. The error in that process is comparable to the error in the old linear conversion. In that sense, the PSI formula predicts CUP about as well as a PSI test predicts the PSI of the next sample out of the same batch.

The results are close to being the same whether or not the 357 Magnum is in the mix. The stats say that there was something unusual about the human decision making in assigning the 357's PSI rating. You do get a little tighter result for the typical cartridges by leaving it out, but putting it in does not destroy the model.

The little red triangle represents the 357.

Varminterror said:

As with many empericisms, their applicability is limited to a certain range of values.

The easiest criticism of the regressions - pressure is pressure, and no pressure is no pressure. Having a non-zero value for one unit when the other is zero doesn’t make sense, which indicates the rule only applies within the range of values for which the rule applies...

Click to expand...

Yup.

The formula applies only to 12-65 KPSI. Beyond that is uncharted territory.

mcb said:

They are very different measurements so creating an accurate conversion between the two is both impossible and rather meaningless. In general terms CUP (Copper Units of Pressure) is a integral of the entire pressure vs time curve where the piezo transducer pressure is the actual peak pressure measure in the pressure vs time curve. The piezo method is much better as the CUP method is notorious for failing to fully capture peak pressure when dealing with faster burning powders. The piezo transducer method also gives you much more data in addition to the peak pressure, as you get the entire pressure vs time curve. The CUP method was good in its day but the piezo transducer method is far better, give you more data, and in an industrial setting much faster and cheaper to operate.

Click to expand...

We partly agree, but I offer the following:

As explained in the linked article, CUP closely represents peak pressure, and the off-peak contribution to deformation can be regarded as a minor measurement error. The point of both the CUP and PSI measurement is determining a safe operating pressure. Peak pressure breaks guns. The integral under the curve does not.

If you reject the accuracy of the conversion, you also have to reject the standard SAAMI pressure testing regimen. The random error in both is roughly the same.

higgite said:

Shouldn't that be .01716 * KCUP^2?

(Not to be confused with Keurig K-cups, btw. )

Click to expand...

You are correct. It's KCUP. Good catch.

mcb said:

I agree the peak pressure is the critical factor in most cases and that is one of the CUP methods short falling, its a bad integrator. CUP has been document to miss the peak pressure when the peak occurs very quickly (relatively speaking as this is all happening super fast). This is likely due to inertial effects in the CUP system. You have to accelerate the piston that is crushing the copper slug and this can lead to a bandwidth limitation when the peaks get quick.

i.e. You can have two loads for a cartridge the first one using a moderate or slow powder that measures under max pressure using both CUP and Trans but the second load that uses a fast powder might measure under max using CUP but over max pressure when measure with Trans. due to the peak happening too quickly for the CUP system to capture it.

I probably should have started here but why would someone want to convert from one unit to the other?

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Good comments. Thank you.

The problem with CUP is not so much the speed of the powder. It's with higher pressures. At lower pressures, it does pretty well at capturing peak pressure. At higher pressures, it underestimates. It was a bit of a shock to the community when modern pressure measurement equipment became available, it was found that CUP did a decent job of estimating actual pressure for the 30-30 and .303, but was quite a bit short in estimating pressure for the 30-06 and 270. So SAAMI set about to make it clear that what we had been calling PSI was to be known as CUP, and PSI came from piezo systems and ne'er the twain shall meet.

Why would one want to convert from one to the other? In my case, I wanted to load for my Swedish Mauser, and had PSI measuring equipment. But the only pressure spec I had was in CUP, so I wanted a PSI pressure spec I could work to. SAAMI has since published a PSI spec.

Other than that, in a few decades I'm sure that we will look back on CUP as a quaint historical footnote. Modern equipment is much better.

Meanwhile, on a lighter note:

LOL... has it really been that long? I suppose it has. Now I feel old....

There is a simple definition of correlation: If having the values of one variable allows you to make an estimate of another variable that is better than taking a simple average, then the two variables are correlated. Visually that reduces to looking at the scatterplot, like the one I posted earlier, and seeing if there appears to be an obvious underlying curve. If there is no such curve, then the plot will look more like someone spilled a bowl of marbles. Spilled marbles = no correlation. Points grouped around an underlying curve = correlation.

One thing that confuses people is that there is random noise mixed in the numbers. Part of that comes from the random noise of the piezo system, a lot of it comes from random error in the crusher system, and quite a bit comes from people making inconsistent decisions in setting standards (270 vs 308 vs 30-06 for example...same head geometry, same material, different specifications).

Besides finding a good upper PSI limit for my Swedish Mauser, with only CUP data available, there is at least one other case where the conversion is very useful. P O Ackley did a lot of cartridge measurements using a copper crusher system. As was the custom at the time, he reported his CUP numbers as PSI. If you want to make sense of some of his work, it's very helpful to be able to translate his numbers into modern units.

Just reloading for any of the common cartridges? Probably no need to make the conversion. There is indeed plenty of good data available.