Need PSI Math help (confirmation) for Football PSI app

Greetings,

Am looking for folks who really get the math to verify the formula we have by solving for X following as a test

Starting Temp 80F
Starting PSI 14.5
Temp at second measure 40 F
estimated PSI at second measure X

Starting Temp 28F
Starting PSI 12.5
Temp at second measure 70 F
estimated PSI at second measure X

I just want the verification of what X is to 2 decimal points. I am 99% sure my math and formula work but better safe than sorry.



Thank you

it would be an absolute ***** to program but putting in rain as a factor would make sense.

Assuming you used 14.7 psi for atmospheric air pressure,
I got 12.34 and 14.84 as my answers

Why not just put it online and tweet it to everyone you can? It doesn't have to be fancy, just make a basic calculator where you enter the variables and push the button.

First: The temperatures (T1 & T2) that matters are NOT the outside air temp, but the temp of the air INSIDE the ball! At the moment, nobody measures this. You only have access to the OUTSIDE temperatures.

Therefore, when using outside air temperatures, the unstated assumption is that the air inside the ball has had time to equilibrate to the outside air.

In dry weather, it takes a long time for the temperature to equilibrate. (3 - 5 hours)

When exposed to water, then the temperature equilibrates much, much faster. (30 - 60 minutes)

Note that, when the Pats balls were taken from the dry locker room to the wet field, they cooled off rapidly.
When the wet balls were taken to the dry locker room, they warmed up very, very slowly. (Likely took about 5 - 8 hours to warm up, because there was a layer of cold water trapped in the leather.)

IF the balls in the locker room had been sprayed with warm water at locker room temp (i.e., 75°F), then they would have warmed up ALMOST as fast as they cooled down.

__

Second: this is a dynamics problem, not a statics one.

When the ball is taken out of the warm locker room to a cool field, even tho the outside temperature changes in an instant (as you walk out the door), the inside-the-ball air temperature changes very slowly. And, as we said, it's the inside-the-ball air temp that matters.

The air inside the ball will drop slowly, in an asymptote, a decay curve which approaches, but never quite reaches, the outside air temp.

The pressure changes nearly instantly with the average inside-the-ball air temp, but the inside-the-ball air temp change slowly, because the heat has to conduct thru about 4 layers of material.
__

OK, here's what you need.

Given the assumption that the inside-the-ball air temperature has had enough time to equilibrate to the outside air temp ...

The Pressure Lapse Rate (PLR). If you use this, then you don't have to worry about doing the gauge pressure to absolute pressure or °F to °R conversions, because you're talking about DIFFERENCES in temp & pressure.

The THEORETICAL PLR is 0.051 psi/°F, which should be very accurate for dry footballs.
The EXPERIMENTAL PLR (that I measured) for wet balls between 72°F & 50°F is 0.06 psi/°F.

Put in the starting Pressure & Temperature (P1 & T1). Put in the final temperature (T2).
Choose dry or wet conditions.

For dry conditions, use PLR = 0.051 psi/°F.
P2 = P1 + (0.051 * (T2 - T1)). (Pressure in psig. Temp in °F)

For wet conditions, choose PLR = 0.060 psi/°F.
P2 = P1 + (0.060 * (T2 - T1)). (Pressure in psig. Temp in °F)

Note, this equation will automatically give you higher pressures as the temperature warms up.

More testing may well adjust the experimental PLR slightly.
The theoretical PLR will does not depend on experiments.

Dr Pain said:

Greetings,

I just want the verification of what X is to 2 decimal points.

Click to expand...

Actually, this is the most common fallacy that many people - even at least one professional physicist quoted in the Times - makes when discussing this. You cannot get a result to "2 decimal points" if the input data is significant to only one decimal point. It just doesn't make sense in this context.

What's going on with deflategate is deeper by the way. When people start throwing out a bunch of round or half-integers like "12" and "11.5", you cannot even be sure these are good to one decimal point. They might just be rounded to the nearest half-integer. But many people take these as exact numerical results and derive all kinds of nonsense from that.

The fact that every single one of these anonymous third-hand reports of PSI from 12 different balls were all either on the integer or the half-integer is why you should assume that someone was rounding to the integer or half-integer.

tom.kordis said:

First: The temperatures (T1 & T2) that matters are NOT the outside air temp, but the temp of the air INSIDE the ball! At the moment, nobody measures this. You only have access to the OUTSIDE temperatures.

Therefore, when using outside air temperatures, the unstated assumption is that the air inside the ball has had time to equilibrate to the outside air.

In dry weather, it takes a long time for the temperature to equilibrate. (3 - 5 hours)

When exposed to water, then the temperature equilibrates much, much faster. (30 - 60 minutes)

Note that, when the Pats balls were taken from the dry locker room to the wet field, they cooled off rapidly.
When the wet balls were taken to the dry locker room, they warmed up very, very slowly. (Likely took about 5 - 8 hours to warm up, because there was a layer of cold water trapped in the leather.)

IF the balls in the locker room had been sprayed with warm water at locker room temp (i.e., 75°F), then they would have warmed up ALMOST as fast as they cooled down.

__

Second: this is a dynamics problem, not a statics one.

When the ball is taken out of the warm locker room to a cool field, even tho the outside temperature changes in an instant (as you walk out the door), the inside-the-ball air temperature changes very slowly. And, as we said, it's the inside-the-ball air temp that matters.

The air inside the ball will drop slowly, in an asymptote, a decay curve which approaches, but never quite reaches, the outside air temp.

The pressure changes nearly instantly with the average inside-the-ball air temp, but the inside-the-ball air temp change slowly, because the heat has to conduct thru about 4 layers of material.
__

OK, here's what you need.

Given the assumption that the inside-the-ball air temperature has had enough time to equilibrate to the outside air temp ...

The Pressure Lapse Rate (PLR). If you use this, then you don't have to worry about doing the gauge pressure to absolute pressure or °F to °R conversions, because you're talking about DIFFERENCES in temp & pressure.

The THEORETICAL PLR is 0.051 psi/°F, which should be very accurate for dry footballs.
The EXPERIMENTAL PLR (that I measured) for wet balls between 72°F & 50°F is 0.06 psi/°F.

Put in the starting Pressure & Temperature (P1 & T1). Put in the final temperature (T2).
Choose dry or wet conditions.

For dry conditions, use PLR = 0.051 psi/°F.
P2 = P1 + (0.051 * (T2 - T1)). (Pressure in psig. Temp in °F)

For wet conditions, choose PLR = 0.060 psi/°F.
P2 = P1 + (0.060 * (T2 - T1)). (Pressure in psig. Temp in °F)

Note, this equation will automatically give you higher pressures as the temperature warms up.

More testing may well adjust the experimental PLR slightly.
The theoretical PLR will does not depend on experiments.

Click to expand...

If this isn't the post of the year, then I don't know what is. My God, we have some smart people around here.

Dr Pain said:

You guys are over thinking this. Not doing a product for engineering calculations but the quick calculations giving estimates based on temp with a disclaimer. Something suitable for football officials, press people who failed science, former players desperate for attention, lawyers conducting investigations and TV Science Personalities who are rusty at middle school level science. Tweek some noses.

Click to expand...

First off, thanks to your dedication for spreading a positive word. I don't know how much it will matter if we get negative news on the situation, but it could really take off for a day or two in the media if the team is exonerated.

Second of all, would you consider tweaking it as proposed, to be able to include the more appropriate formulas and relevant factors, or is that too much of a pain in the butt? It may benefit you and your son immensely if it were taken seriously; otherwise, people would simply point out the differences of what is being overlooked.

Dr Pain said:

You guys are over thinking this. Not doing a product for engineering calculations but the quick calculations giving estimates based on temp with a disclaimer. Something suitable for football officials, press people who failed science, former players desperate for attention, lawyers conducting investigations and TV Science Personalities who are rusty at middle school level science. Tweek some noses.

Click to expand...

The equations that I posted are simple & effective.

First, check a box for wet or dry conditions.
If "wet" checked, set PLR to 0.060
If "dry" checked, set PLR to 0.051

Enter P1 (psig), T1 (°F) & T2 (°F)

Use the same equation: P2 = P1 + (PLR * (T2 - T1))

Truncate your answer to 1 significant digit after the decimal point & add "±0.2 psig". That error will cover the variables.
Add a note on temp equilibration.

e.g. 11.4 ±0.2 psig
Note: Answer valid only after inside-the-ball air temps equilibrates with outside air temps.

It doesn't get any simpler...

tom.kordis said:

The equations that I posted are simple & effective.

First, check a box for wet or dry conditions.
If "wet" checked, set PLR to 0.060
If "dry" checked, set PLR to 0.051

Enter P1 (psig), T1 (°F) & T2 (°F)

Use the same equation: P2 = P1 + (PLR * (T2 - T1))

Truncate your answer to 1 significant digit after the decimal point & add "±0.2 psig". That error will cover the variables.
Add a note on temp equilibration.

e.g. 11.4 ±0.2 psig
Note: Answer valid only after inside-the-ball air temps equilibrates with outside air temps.

It doesn't get any simpler...

Click to expand...

I'm not sure if the 0.2 there is correct. Ideally this is done using what's called "interval arithmetic" but here there are several simpler options, and most scientists themselves don't use interval arithmetic, clearly overkill for this problem. Just think about how it's used.

First, I would consider just making a disclaimer about precision.

Second DO NOT TRUNCATE as Tom Kordis above suggests. This is utterly wrong. Use rounding if anything.

You have to understand why things are what they are you see.

Let's say you want to convert a psi delta of 2 into Fahrenheit. Where is the "2" here from?

Maybe "2" is a mathematical construct, exact, someone is asking a question about physics. In this case, you can use an exact value.

But if "2" comes from a measurement, then maybe "2" is due to rounding of "between 1.5 and 2.5. In this case, the result should be given as a range of temperature between the temperatures corresponding to exact PSI deltas of 1.5 and 2.5.

This is all utterly obvious if you think about it. In fact, if it's not obvious to you, don't implement. If you can't see why "truncating to 1 significant digit" is wrong, for instance, then you're not understanding the issue.

Conclusion:

A. You need to include something about significant digits or the use of the program with uncertain input data.

B. Consider optionally outputting the answer as a range if the input data (say pressure) is also a range.

C. Generally if you're kind of trying to learn about apps, it's best to get the basics right first. First, get the simplest app working. Then, add features. But A is critical, just even as some text in the app. The next time I see some talking head on the TV act like a reported value of "2" is accurate to 9 significant digits, I'm going to get even more irritated than this whole ridiculous episode has made me so far.

Man I'm so confused.

Out. Fraking. Standing.
As an engineer I do have a thing about significant digits.
Forget 2 digits to the right of the decimal point. Folks will criticize your app justifiably if so.
Congrats!

PatsWickedPissah said:

Out. Fraking. Standing.
As an engineer I do have a thing about significant digits.
Forget 2 digits to the right of the decimal point. Folks will criticize your app justifiably if so.
Congrats!

Click to expand...

So you recommend 12.5 over 12.52? Just want to make sure I understand

Dr Pain said:

So you recommend 12.5 over 12.52? Just want to make sure I understand

Click to expand...

Yes clearly. The 0.02 is an artifact of computation.
If you only know any psi # to X significant figures (e.g. 12.5 psi is 3 significant figures assuming the last digit is accurate) you can't create additional precision or information by doing a computation.

PatsWickedPissah said:

Yes clearly. The 0.02 is an artifact of computation.
If you only know any psi # to X significant figures (e.g. 12.5 psi is 3 significant figures assuming the last digit is accurate) you can't create additional precision or information by doing a computation.

Click to expand...

Simple example -- let's say you have a ruler than only has one-foot markings (no other markings). You measure some distance and it's a bit over the 5' mark and does seem to be closer to 5' than 6'. So you write the measurement down as 5'. Now someone asks you what a third of that distance is. You punch 5/3 in your calculator and get 1.666666666666666667.

Can you measure 1.666666666666666667' with your ruler? No. If someone put a mark at 1.666666666666666667' from the wall and asked you to measure it with your ruler, the best you could say is 2'. So when you measured 5' with your ruler and someone asked you what a third of your measurement is, your best answer is 2'. One significant digit, just like your measurement.

(Actually, the best answer is to quote error bars on the original measurement and all numbers calculated from the original measurement. And there's a whole field of how error bars transform with different types of calculations. Back at MIT I had a physics lab (Junior Lab, if any of you are alums) where you had to be rigorous about having error bars on your original measurements and transforming them all through the computations (which might have square roots, logs, trig functions, etc.) to have proper error bars on your final results.)

Ken, I assume that you're a mathematician.

I’m a mechanical engineer, with a fair amount of experience as designing, running & reporting on experiments to both outside companies (Boeing, NASA, the FDA) and much more often, to the engineering department within the companies for which I worked. The majority of those experiments & reports ended up defining, or changing, the direction that various projects were headed. They needed to be correct.

My overall comment is “settle down, cowboy.”
You ain’t the only smart guy in the room.

Do you usually find this aggressive sort of approach to be productive on a first introduction?
Or do you get punched in the face periodically?
__

Ken Canin said:

I'm not sure if the 0.2 there is correct.

Click to expand...

Neither am I. That would take a boatload of additional experimentation.

But it is justifiable based on my SMALL SAMPLING of experimental data. I tested 4 balls, of which I could use 3 complete data sets, because the 4th sprung a leak late in the testing cycle. (But it was patently obvious that the 4th ball was following EXACTLY the same trendiness as the other 3. Just bad form to use data when you don’t know for certain when it started to deviate.)

So, as Dr. Pain says, he doesn’t need something that’ll be traceable to NBS Standards. (Showing my age. Is that NIST standards, these days? ISO 9000?)

Engineers like answers. Not questions.
The ±0.2 psi is my sense of the consistency & reliability of the number that pops out of the app, without reference to the error of the input data.

it is intended to give a FEELING to the person reading it of my sense of the error bands likely in the calculation.

It may well be that ±0.3 or ±0.5 turns out to be the case, if you’re trying to get 3 sigma or 6 sigma confidence interval.

But who cares.?
Read what Dr. Pain asked for.

The number that I suggested is adequate to give someone an idea, a feeling for both the number & the precision of the calculation.
__

Ken Canin said:

Ideally this is done using what's called "interval arithmetic" but here there are several simpler options, and most scientists themselves don't use interval arithmetic, clearly overkill for this problem.

Click to expand...

WTF??
There is no need for “interval arithmetic” in any of this.

The simple empirical equation that I gave him IS a “simpler option”.
__

Ken Canin said:

Just think about how it's used.

Click to expand...

Jeebus. Thanks, Ken. I’ll try...
__

Ken Canin said:

First, I would consider just making a disclaimer about precision.

Click to expand...

No. A disclaimer transmits NO information about error or precision.
Providing an error band gives one a SENSE of the precision of the calculation.
That is all that I’m trying to do.

This is the SENSE that I got from running the experiment & observing the results.
__

Ken Canin said:

Second DO NOT TRUNCATE as Tom Kordis above suggests. This is utterly wrong. Use rounding if anything.

Click to expand...

You got me. Wrong word.
Don’t truncate, round off.
__

Ken Canin said:

You have to understand why things are what they are you see.

Click to expand...

PV = nRT ain’t no big mystery, Ken.

And NOTHING in what Dr. Pain asked for requires this level of detail or complexity.
__

Ken Canin said:

Let's say you want to convert a psi delta of 2 into Fahrenheit. Where is the "2" here from?

Click to expand...

Well, Dr. Pain asked for P2 = f(P1, T1, T2), and that’s what I gave him.

He didn’t ask for T2 = g(T1, P1, P2).
But, if he had, then I would have given him this one:

T2 = T1 + ((P2 - P1)/PLR)

with a suggested error band that I’ll work out, when I don’t have to be somewhere, with much more fun, interesting people.
__

Ken Canin said:

Maybe "2" is a mathematical construct, exact, someone is asking a question about physics. In this case, you can use an exact value.

But if "2" comes from a measurement, then maybe "2" is due to rounding of "between 1.5 and 2.5. In this case, the result should be given as a range of temperature between the temperatures corresponding to exact PSI deltas of 1.5 and 2.5.

Click to expand...

Thanks, guy.
I understand very well error analysis & precision of measurements.

Last time: READ WHAT Dr. Pain ASKED FOR.
__

Ken Canin said:

This is all utterly obvious if you think about it. In fact, if it's not obvious to you, don't implement. If you can't see why "truncating to 1 significant digit" is wrong, for instance, then you're not understanding the issue.

Click to expand...

Please, down boy.
I’ve been doing experimental analysis for every company that I’ve ever worked for for over 40 years.

It IS all “utterly obvious”.

You got me. Round off. Don’t truncate.
By the way, I did NOT say "truncating to 1 significant digit”.

(I should have said) “Round to" (and I did say) "1 digit after the decimal point.”

IN THE RANGE OF PRESSURES that we’re discussing, that give either 2 or 3 significant digits. With an error band.

By the way, I HAVE run these experiments.
Have you?

As a result, I DO have a feeling for these numbers.
Do you?
__

Your Conclusions:
[QUOTE="Ken Canin, post: 4178924, member: 35283”]
A. You need to include something about significant digits or the use of the program with uncertain input data...

B. Consider optionally outputting the answer as a range if the input data (say pressure) is also a range...

C. Generally if you're kind of trying to learn about apps, it's best to get the basics right first...
[/QUOTE]
__

My conclusions:
Sometimes, smart guys can be insufferable, pompous assholes.

Just my opinion, of course…
YMMV.

PatsWickedPissah said:

Out. Fraking. Standing.
As an engineer I do have a thing about significant digits.
Forget 2 digits to the right of the decimal point. Folks will criticize your app justifiably if so.
Congrats!

Click to expand...

Please read what I wrote.
If that isn't clear, look at the example that I gave.

tom.kordis said:

Truncate (Error: As Ken pointed out, this should read:] "Round" your answer to 1 significant digit after the decimal point & add "±0.2 psig". That error will cover the variables.
Add a note on temp equilibration.

e.g. 11.4 ±0.2 psig
Note: Answer valid only after inside-the-ball air temps equilibrates with outside air temps.

Click to expand...

"... 1 significant digit after the decimal point ..."

If someone else wants to suggest 2 significant digits after the decimal point, I'll be the first one to challenge that, because neither the measurements nor the empirical equations supports this level of precision.

If someone else wants to suggest 0 significant digits after the decimal point, I'll challenge that. Because 12 psi, 11 psi, 10 psi, etc. is pretty damn useless.

Between 0 & 2, there exists only one integer.
__

"The error band that I am suggesting is ±0.2 psi."

The ±0.2 came out of the experimentation.
It is not PRECISE. It is not EXACT.
It does give someone using the calculations a rough idea of the error associated with the calculation.
That is something useful.

JMO.
But I'm right...

If someone wants to expand the testing that I did to include 100 balls or so, just so that this error band can be nailed down (possibly increased, possibly decreased), be my guest.

Having started my engineering education with slide rules (because calculators didn't yet exist), I pride myself on having a very good feel for numbers. Like most engineers of my generation, who lament the application of clueless over-precision that resulted from the "calculator generations" & digital gauges.

Having been required to produce a full-blown error analysis with every lab report, or receive an "F" grade, I am very familiar with precision of measurement & error analysis.

Ken Canin said:

"... blah, blah, snarky patronizing blah ..."

Click to expand...

tom.kordis said:

"... blah, blah, snarkier reactionary blah ..."

Click to expand...

Ken, we got off on a bad note.
My apologies for my short-tempered response. (You got me right after my avaricious Uncle Samuel just got thru raiding my piggy bank. Still, no excuse.)

Here's my offer of an olive branch.
This ain't nothing to get cranked up about.

Let's start over.
Let's both rein in the snark.

Whaddaya say?

Well, message board threads have a way of becoming somewhat less circumspect over time. I wouldn't worry about it, as it's a quality inherent in the medium.

It is insidious how distracting and at times dividing deflategate has become. This makes me appreciate even more Brady's ability to maintain team unity and focus throughout the hype of a hype filled season.

The more I think about OP's original request and the unfamiliarity of the issues raised, the more I lean towards recommending just implementing more or less exact computation at first with some rounding at the end, and then including some notes or cautions on error bars or significant digits: the app can be a tool that, like any other, including a calculator, can be misused.

Questions are a dime dozen.
Answers might be valuable.
Correct answers are diamonds.

For those who question my answer to Dr. Pain's request ...

1. Be my guest. Questioning any technical assertion is a very good thing. Especially for learning the issues. I'll never give anyone any sort of bad time for asking (reasonable) questions.
   
2. I'd request that you ask those questions in a reasonably respectful manner. I don't need, I don't want "obsequious". Just respectful. I did respond negatively to what I interpreted as "less than respectful" comments.
   
3. I gave Dr. Pain an answer. A simple answer (2 simple equations), as he requested.
   An answer that is based on both theory & experimentation (between 75°F & 50°F).
   In other words, a correct answer.
   I gave him my judgment of appropriate precision of the answer.
   I gave him my judgment of appropriate tolerances.
   I gave him my judgment of an appropriate cautionary message against the most obvious misuse of the calculation.

So, at the end of the day, you're welcome to question the origins & development of my answer. You're welcome to question my judgments about those other issues.

But, when you're done, don't just talk about theory. Don't just say that "you're not sure you agree with my assumptions or answers".

Let's See Your Assumptions.
Let's See Your Answers.
And your justifications for them.

Only after someone gives their answer can we see the difference between yours & mine.

That is useful...