The ANSWER to DeflateGate

[tom.kordis]

(My first post. Mods, please move it to proper location.)

This whole DeflateGate nonsense is absurd.

And these physicists are as useless as tits on a boar. The MIT physicist talking about how great MIT physics professors are ... but then fails to provide any calculations. The Columbia physicist saying "he thinks that the balls were manipulated" ... also without running any numbers. They are both useless.

More important, they are both WRONG.
Which they both would have realized instantly, if they had bothered to run some numbers.

Generalities are useful ... as a starting point. Not the finish.

The only way to understand what happened is to do some calculations to show exactly how much the balls would have deflated under the conditions seen.

THEN what happened becomes obvious: Nobody tampered with those footballs.

Brady set them at 12.5 (at the low end of the spectrum, but completely legal) at ~75°F.
Mother Nature then dropped the pressure by an additional 1.5 psi when they were cooled from 75°F to 50°F.
0.5 psi + 1.5 psi = "2 pounds (sic) under" the NOMINAL pressure of 13.0 psi.
__

How engineers do it...

1) PV = nRT => P/T = nR/V
2) For nR/V = constant, P1/T1 = P2/T2 => P2 = P1 * (T2/T1)
3) For P1 = 12.5 psi, T1 = 75°F, T2 = 50°F
(but these equations REQUIRE absolute temperatures (°R) & pressures (psia).
Putting in °F or gauge pressure (psig) produces the wrong answer.
"Wrong answers" are very bad things.!
"Correct answers" are good things.!)
P2 = (12.5+14.7) psia * (50+460)°R / (75+460)°R = 25.9 psia
Converting back to gauge pressure:
P2 = 25.9 psia - 14.7 psi = 11.2 psig.

Note: P2 = the internal pressure of the balls measured at 50°F)

In three steps, & one calculation, you can see that the pressure is 1.8 psi below the NOMINAL (13.0 psig). Brady took out 0.5 psi (by setting them at 12.5, instead of 13.0 psig. Mother Nature takes out 1.3 psi simply by cooling the balls.

The total drop in pressure is exactly what they found: 2 psi lower than the nominal 13 psi.

The "Mother Nature" portion of the pressure drop happens to EVERY SINGLE BALL ever used in the NFL.
__

And then engineers run experiments.
Because sometimes theory isn't exactly right. As in this case.

When you run the experiment, as I have, you find that the real pressure ends up slightly lower than pure theory predicts. Using ball pressurized to 12.5 psi @ 75°F, and then exposed to cold water & cold air at 50°F, I recorded a pressure of 11.0 psi after 70 minutes.

The theoretical pressure lapse rate is 0.051 psi/°F.
The experimental pressure lapse rate is 0.061 psi/°F.

Experiment trumps theory.

[Note: some weathermen (as on The Weather Channel) have asserted that the pressure lapse rate is 0.02 psi/°F (or the equivalent, 0.2 psi/10°F). You get this WRONG answer by using gage pressure instead of absolute pressure. You'd think that 3 f'king PhDs, all consulting with each other, could get this trivial number correct. But noooooo....]

To those who ask, "why didn't the Colts balls deflate?", the answer is "They did." EVERY single ball ever used in any NFL game deflates by these amounts in these temperatures. No exceptions allowed.

To the question, "why didn't they announce that the Colts balls were deflated, the answer is, "They didn't check the Colts balls. There were never challenged as being under-inflated." If they had checked the Colts balls, they would have found exactly the same pressure DROP, from Luck's preferred starting pressure.

To the question, "Why did the pressure come up when they went back inside to measure them at half-time?", the answer is "it takes many hours for the temp of the air INSIDE the ball to return to room temp." The balls are, in essence, very good thermos bottles that nobody can open up. Leather is a pretty good insulator.

In the last couple of days, a source-less report is saying "only one ball was 2 psi under, the others were just a tad under". How do you explain this? The answer is: the report is bogus, the information is false. There is NO EXCEPTION to the calculations above.

It's not called "The Ideal Gas Suggestion".
It's called "The Ideal Gas Law".!

There is no mystery here.

NOBODY tampered with those balls.
The Patriots organization knows this.

The NFL lawyers will do what lawyers do: delay & rack up billable hours.
If they fired the lawyers & hired a competent engineer, this thing would have been over a day after it started.

It took me 2 minutes to do the calc, & 1 day to run the experiment, all on a $600 budget. ($300 of which I got back when I returned 3 of the 4 NFL game balls I bought to test.!)

 

[tom.kordis]

One correction just occurred to me. All the above ASSUMES (a bad thing!) that nobody adds or removes any air after inspection, during the game.

Specifically, it assumes that the ball boys are not ADDING air during the game. If they were to check the pressure during the game, read it at 11.0 psig, realize that Brady likes them set to 12.5 psi and to pump them up to this pressure, then the report that "10 of the 11 balls they checked were just a tad under" could be true.

Someone would have to ask the ball boys about their procedures.

Regardless of this complication, all of the above stands.

And the (perhaps surprising to non-techies) conclusion is that ... EVERY SINGLE GAME in NFL history that has been played at a temp below about 53°F has been played with under-inflated balls.

And nobody ever noticed.

The only reason for this clusterfork is the simple fact that this is the first time anyone bothered to measure the pressure at half-time. And the results didn't match their (wholly uninformed) expectations.

Their expectations were wrong.

It's just that simple.

 

[BradyManny]

Great post. Though I would object: experiment is the basis for science and I think you are misrepresenting that these physicists do not run experiments! There are experimentalists, and there are theoretical folks - maybe the NYT and Globe have only interviewed the latter.

I would say that all the pop-sci guys have come up with the lazy answer, but anyone who has thought about it for any length of time or run an experiment has come to the conclusion you have. Let's hope Columbia does likewise!

I'd also say - the Columbia fellow you reference has exchanged emails with PBPF and I don't think the quote attributed to him is an accurate or full representation of his opinions on the matter.

 

[BradyManny]

And the (perhaps surprising to non-techies) conclusion is that ... EVERY SINGLE GAME in NFL history that has been played at a temp below about 53°F has been played with under-inflated balls.

And nobody ever noticed.

The only reason for this clusterfork is the simple fact that this is the first time anyone bothered to measure the pressure at half-time. And the results didn't match their (wholly uninformed) expectations.

Their expectations were wrong.

It's just that simple.

This is the very heart of what has transpired. NEVER has a ball been measured at half-time, and the NFL was too dense to know what SHOULD happen. An unprecedented measurement was made with a faulty hypothesis.

 

[Buchanty]

And the (perhaps surprising to non-techies) conclusion is that ... EVERY SINGLE GAME in NFL history that has been played at a temp below about 53°F has been played with under-inflated balls.

And nobody ever noticed.

The only reason for this clusterfork is the simple fact that this is the first time anyone bothered to measure the pressure at half-time. And the results didn't match their (wholly uninformed) expectations.

Their expectations were wrong.

It's just that simple.

This is the main reason I liked Reiss's question at the Commissioner's Presser "Has the NFL ever tested the air pressure of footballs in the middle of a game, and how important is that as a frame of reference in this investigation?" Goodell's reply? “I don’t know the answer to that question.”

We do know that the first and only protocol for measuring footballs resulted in no recordings, which left me to surmise....

... there is no baseline for any ball, (ie its referee-accepted pressure from the first and only protocol) nor is there a baseline for what a ball could or should be at halftime. Furthermore it is most probable that their current calculations have no accommodation for the laws of physics, as the NFL has yet to acknowledge them as a factor. If the laws of physics are taken into account then the incidence of half-time game balls being under the 12.5 PSI threshold is likely to be significant.

Thank you, Tom, for performing the test and sharing the results, which demonstrably confirm what is obvious to cogent thinking. It also means that those QB's who swore they could tell the difference in PSI were very much mistaken. Either that or they never played in temperatures lower than 50F

 

[Calhoun44]

Isn't the simplest solution to the "problem" of low ball pressure, just to have all balls be given to the refs 2 hours before the game and moved outside to allow them to get to or close to the outside temperature and then test them a half hour before kickoff. At that point after the balls are deflated/inflated they should remain fairly constant. Seems simple, but I am sure there is a reason not to do it this way, it just seems too easy.

 

[NormZauchin]

One correction just occurred to me. All the above ASSUMES (a bad thing!) that nobody adds or removes any air after inspection, during the game.

Specifically, it assumes that the ball boys are not ADDING air during the game. If they were to check the pressure during the game, read it at 11.0 psig, realize that Brady likes them set to 12.5 psi and to pump them up to this pressure, then the report that "10 of the 11 balls they checked were just a tad under" could be true.

Someone would have to ask the ball boys about their procedures.

Regardless of this complication, all of the above stands.

And the (perhaps surprising to non-techies) conclusion is that ... EVERY SINGLE GAME in NFL history that has been played at a temp below about 53°F has been played with under-inflated balls.

And nobody ever noticed.

The only reason for this clusterfork is the simple fact that this is the first time anyone bothered to measure the pressure at half-time. And the results didn't match their (wholly uninformed) expectations.

Their expectations were wrong.

It's just that simple.

And I might add that footballs in games played in Miami in September in 95 degree heat under a broiling sun are likely over-inflated by halftime.

 

[tom.kordis]

BB said that they are going to inflate the balls to slightly above 12.5 psi from now on, just so that temp changes don't cause the balls to under-inflate. Sorry, BB, this is not in your power. Mother Nature is a cast iron *****, & she does what SHE wants to do. Not even you can force her to do otherwise.

For example, let's set the pressure to the absolute max possible, 13.5 psi at, say, 72°F. Now we can ask the question, "At what temp does the pressure drop to the legal minimum of 12.5 psi?"

Rearrange the equation to: T2 = T1 * (P2/P1). Which gives
T2 = (72 + 460)°R * (12.5 + 14.7) psia / (13.5 +14.7) psia = 532°R * (27.2 psia / 28.2 psia) = 513°R = (513°R - 460°) = 53°F.

Any game played below this temp MUST have under-inflated balls (by the current wrong definition).

There is an interesting "competing effect" that comes into play. The "compliance" of the ball (i.e., how much it compresses when squeezed) has two components: the pressure inside & the flexibility of the leather. As the temperatures get colder, the air becomes a weaker spring because its pressure is lower, but the leather gets stiffer. My experience with footballs & basketballs is that the leather effect predominates. Cold balls feel rock hard.

The inverse happens in hot weather. The air becomes a stronger spring, but the leather gets more flexible.

 

[tom.kordis]

The solution to this problem for the NFL is simple:
there are two components that contribute to the "Inflation State" of the ball: the pressure inside the ball & the temperature of the air inside the ball.

It is IMPOSSIBLE to determine the Inflation State by just measuring the pressure. You MUST measure both the pressure and the temperature of the air inside the ball.

This brings us to some embarrassing memories of people of my advanced age: "bend over & say "Ahhhh". We're gonna take your temperature."

A simple, modified inflation needle, with a thermistor (a simple, calibrated temperature measuring device) on its end will do the job perfectly.

The referee can take the temp of air inside the ball, then records its pressure, & then look up on a chart of the minimum & maximum allowable pressures AT THAT PARTICULAR TEMPERATURE.

The NFL will have to decide the "baseline" temperature. Say 72°F (or whatever).

Then it's trivial to construct the chart of minimum & max allowable pressures for every degree above & below that temp.

And, from this chart, if it ever gets constructed, one would find that at 50°F, the allowable pressures are 11.0 psi to 12.0 psi.

And the Pat's balls during the Indy game are perfectly legal.

Problem solved.

 

[JTinHB]

First of all, great post. A couple of assumptions that could change the numbers using the Ideal Gas Law to calculate the pressure difference at two temperatures in a football:

1. That air is an Ideal Gas. Typically, yes, but if there is any moisture in the air, it may not behave exactly like an Ideal Gas.
2. The volume of a football is constant. Which is kind of the point of the supposed advantage, better grip on a smaller, more compressible ball. Leather and the rubber bladder expand and contract with pressure and temperature.

I'm not doing the math here, but I'm pretty sure the pressure change with moisture is going to be more than with dry air considering that water vapor has a higher gas constant, especially if there is any condensation. That is why we do experiments, to prove the theory. The pressure drop has been replicated multiple times by multiple sources confirming the theory.

People who want to call the Pats cheaters don't care anyway. It amazes me how some some people who otherwise seem to have the intelligence to get this, refuse to. The real "-gate" here is the amount and timing of misinformation that has been leaked to the press.

 

[PatsWickedPissah]

Tom,
FYI
Several of us engineers (me) & a couple PhDs posted similar calculations here on Patsfans starting WAY back the monday after the Colts game. Somebody, don't remember who reminded me to use absolute pressure not gauge pressure like Neil deGrasse Tyson did publicly beclowning himself. Sadly NdGT is one who believes calculations & models over actual data.

Welcome To The Board

 

[tom.kordis]

PWP, cool. I'd be very interested in the technical discussion. Can you point me to it?

Thanks.

BTW, me too. Mechanical engineer, Cornell U, '74. So I've been at this for awhile.

The most interesting part to me is the fact that, experimentally I got a pressure lapse rate (∂P/∂T) that was about 20% higher than theoretical. (experimental: 0,06 psi/°F vs. theoretical 0.051 psi/°F). Since most variations from the IsoVolumetric assumption will result in a smaller volume & therefore a higher pressure, this is a source of interest.

I think that JBinHT mentioned one cause: water vapor. Since the "n" in PV=nRT refers to all gaseous molecules, any water vapor that condenses as the temp drops will result in a lower pressure. I've not yet run any numbers to see the magnitude of the effect.

I think that a second effect could well be adiabatic heating due to compression (whether one uses a hand pump or a motorized one). The air entering the pump may be room temp (75°F), but the air that exits the nozzle inside the ball can be well over 100°F. Thereby raising the internal air temp slightly.

N dG Tyson blew it the first time by using gauge pressure. Then to his credit, he admitted his mistake & recalculated. To his discredit, he made a 2nd dumb mistake, glossed it over & claimed that the balls STILL had to be tampered with.

He didn't show any work, but I believe that his 2nd mistake was attributing 2.0 psi pressure drop to temperature, when we know that TB12 was responsible for (legal) 0.5 psi, and temp difference was only responsible for about 1.3 psi.

 

[SlowGettingUp]

First of all, great post. A couple of assumptions that could change the numbers using the Ideal Gas Law to calculate the pressure difference at two temperatures in a football:

1. That air is an Ideal Gas. Typically, yes, but if there is any moisture in the air, it may not behave exactly like an Ideal Gas.
2. The volume of a football is constant. Which is kind of the point of the supposed advantage, better grip on a smaller, more compressible ball. Leather and the rubber bladder expand and contract with pressure and temperature.

Wetting the football also apparently reduces the pressure:



The other issue that people have not considered is whether the rain might have had a temperature lower than 50 degrees. There might also have been some evaporational cooling given the wet balls and the wind, although the very high humidity might have prevented this.

 

[memento]

There have been plenty of engineers and experimental physicists coming to the Patriots' defense. I'm especially proud of our Defensive Coordinator, Matt Patricia, putting his Aeronautical Engineering courses to work and representing the Renssellaer Polytechnic Institute well

The experiments have been run and the numbers have been crunched to show that it's reasonable to lose up to 2 psi in the footballs from game conditions. We need to know more about the starting conditions of the footballs and the status of the Colts' footballs, which could act like a control group. But it's questionable if the Colts' footballs were even tested or started under similar conditions.