Cabin Altitude calculation?

[msw]

Here's a question maybe one of you smart, really technical folks can answer...............

Some of the Aircraft Flight Manuals I have contain a "Cabin Altitude for Various Airplane Altitudes" graph in them. Typically, the Aircraft Altitude is on the y-axis, going from 0 to about 50,000 feet; and the Cabin Altitude is on the x-axis. Then there are a series of diagonal lines on the graph which are labeled 0 Diff Press (delta p), 1 PSI Diff Press, 2 PSI Diff Press, on up through about 10 PSI delta p. (Nice pressurization system there!) The 0 delta p line is perfectly linear (of course), and the other delta p lines, below about 5 PSI delta p, are "close to" linear at the lower altitude ranges (below about 25,000 feet). Above this, the lines get progressively more non-linear (curved).

Can anyone tell me what the equation is that determines the values on this chart? i.e., if I did not have the chart handy, and if I knew my current aircraft altitude (in feet), and my current cabin differential pressure (in PSI), and wanted to calculate my predicted cabin altitude (in feet) based on these factors, what equation would I use?

 

[Minuteman]

Just a guess, but it may be because ambient pressure as a function of altitude is non-linear (there's no constant "lapse rate").

​

70 mb ≈ 1 PSI

5 PSID would be a sea level cabin altitude at about 11,000 ft (an 11,000 ft delta). An 8,000 ft cabin at 23,000 ft is also 5 PSID (a 15,000 ft delta). So you can have a higher spread between ambient and cabin altitudes for a given differential pressure at higher ambient altitudes because the rate of slope of the Pamb/Altitude curve is flatter as you go higher.

The increasing "curved-ness" of each larger PSID gradient may be determined by the ratio of the slopes at the two points on the Pamb/Altitude curve. At 0 PSID the points on the ambient pressure-as-a-function-of-altitude-curve are the same and the ratio of the slopes is 1. At 10 PSID (at, say, 35,000 ambient and 2,500 cabin), the slope of the pressure curve against altitude is much smaller at FL350 than at 2,500 ft MSL, so the ratio of the slopes at the two points would not one.

Err, stated another way, a non-zero pressure differential results in disproportionate changes in ambient and cabin altitudes because the pressure in a "column of air" doesn't change linearly with altitude.

(But I've been drinking.)

 

[Minuteman]

Yup, there is it, the Barometric formula.

I made a guess at what your chart looked like using this formula:

(I used Google Docs and now want to claw my eyes out)​

The jump at the tropopause for higher differential pressures seems a little dramatic and unrealistic (it is probably due to the cabin pressure extending into the lower altitudes where the temperature lapse rate is non-zero; I'm lazy).

Checking against the 8 PSID case:

Pamb @ FL350 is 3.46 PSI, plus 8 PSI is 11.46 PSI (6,700 ft cabin)
Pamb @ FL480 is 1.85 PSI, plus 8 PSI is 9.85 PSI (10,600 ft cabin)

So the chart is a little high above FL360.

 

[msw]

Thanks, Minuteman. That all makes sense ..... sort of..... I think (and I have not been drinking). And I looked at the link you supplied for the barometric equation....... has way more variables than what I want to wrap my head around, for my purposes. What I was looking for - and maybe it is not "out there" - is a simple formula (equation) that can be used to obtain Cabin Altitude, given the Aircraft Altitude and the cabin pressure differential (delta p) that is in use. Assumptions such as a standard atmosphere (temp and pressure lapse rates) are fine, and also, the formula could be an approximation........ after all, it is for "Pilot Use", not for engineering something. I know that someone with enough math background could look at the charts I referenced and probably derive such an equation.......... but my old college math abilities have faded from dis-use over the decades, and doing that is beyond me at this point. So I was hoping someone had a simple formula to obtain the info I have described.

 

[fish314]

MSW,

Would this chart help? Basically, if all you are trying to do is figure out what PSID you are at given aircraft altitude and cabin altitude, or to figure out what cabin altitude you are at given aircraft altitude and pressure differential this should do it for you. But it isn't a formula, so you would have to print it out and put it in the back of some book you keep with you when you fly or something.

Anyway here is how you would do the math. Say you know the aircraft altitude and the cabin altitude and you want to figure out the delta p: Read from the aircraft altitude (column 1) to the psia (column 4), and then read from the cabin altitude (column 1) to the psia (column 4) and subtract the two numbers. So if you are flying along at 40,000 the pressure (standard day) is about 2.71 psi. Say you've got a cabin altitude of 8000ft… 10.9 psi. 10.9-2.71 is 8.29… call it 8.3 PSID.

Or say you know the PSID that you want, and the altitude that you are at, and you want to know what cabin altitude that is. Let's say your airplane has a max PSID of 8.6 (like mine). You are at 50,000 feet, so that's a PSI of 1.61. 1.61+8.6= 10.21… which is somewhere between 10.5 psi (9,000' cabin altitude) and 10.1 psi (10,000' cabin altitude). So… if your aircraft is limited to an 8.6 PSID, you would need something like 9500'-9700' cabin altitude to fly at 50,000'.

Hope that helps. Of course this is all standard day, etc., so maybe pad the numbers 5-10 % for off standard conditions, but it should be close enough to get you in the ballpark.

Of course, you were originally trying to figure out how to solve the problem if you didn't have your aircraft's chart with you, so I don't know if this is any better.

 

[fish314]

Hey MSW,

I did a quick quadratic regression in Excel and came up with this:

PSI=1/2 *(Flight Level/100)^2 -5*(FL/100)+14.7. This will get you in the ball park also.

Here's how it works. Say you are at 40,000 ft pressure altitude. That's FL400. FL divided by 100 =4. Or, another way to think of this is every 500 feet equals .05, and every thousand feet equals .1. That makes 10,000 feet= 1, 20000 feet= 2, 30000 feet=3, etc.

So it's (1/2*4^2) -5*4 +14.7 = 8-20+14.7 =2.7… Compare that to a charted value of 2.71.

Here's how the numbers compare to the charted values. First column is flight level divided by 100. Second column is the real standard day pressure at that altitude, and the third column is what the bastardized formula that I came up with above yields at the different altitudes. Fourth column is the percent error:

FL/100 PSIA Charted PSIA Computed % Error
-0.5 17.5 17.325 1.00%
-0.45 17.2 17.05125 0.86%
-0.4 16.9 16.78 0.71%
-0.35 16.6 16.51125 0.53%
-0.3 16.4 16.245 0.95%
-0.25 16.1 15.98125 0.74%
-0.2 15.8 15.72 0.51%
-0.15 15.5 15.46125 0.25%
-0.1 15.2 15.205 -0.03%
-0.05 15 14.95125 0.33%
0 14.7 14.7 0.00%
0.05 14.4 14.45125 -0.36%
0.1 14.2 14.205 -0.04%
0.15 13.9 13.96125 -0.44%
0.2 13.7 13.72 -0.15%
0.25 13.4 13.48125 -0.61%
0.3 13.2 13.245 -0.34%
0.35 12.9 13.01125 -0.86%
0.4 12.7 12.78 -0.63%
0.45 12.5 12.55125 -0.41%
0.5 12.2 12.325 -1.02%
0.6 11.8 11.88 -0.68%
0.7 11.3 11.445 -1.28%
0.8 10.9 11.02 -1.10%
0.9 10.5 10.605 -1.00%
1 10.1 10.2 -0.99%
1.5 8.29 8.325 -0.42%
2 6.75 6.7 0.74%
2.5 5.45 5.325 2.29%
3 4.36 4.2 3.67%
3.5 3.46 3.325 3.90%
4 2.71 2.7 0.37%
4.5 2.1 2.325 -10.71%
5 1.61 2.2 -36.65%

So that's a (relatively) quick way to get from Altitude to PSI with pretty decent accuracy (within about 4% unless you go above FL400)… but unfortunately it doesn't get you from PSI back to altitude easily (unless you like solving quadratic equations involving very obscure decimals in your head). But combined with the last post, this should get you close enough for government work.

So if you know that you are at FL300, and cabin altitude 8000, you could figure out your PSID. 300/100=3 so pressure at FL300= 1/2(3^2)-15*3+14.7=4.2 (versus 4.36 actual). 8000 ft cabin altitude= FL080. 080/100=0.8… so pressure at Cabin Alt 8000 is about 1/2*0.8^2 -5*0.8+14.7, which is .32 -4 +14.7=11.02 (versus 10.9 actual). So the PSID is just 11.02-4.2, or about 6.78. The actual PSID is 10.9-4.36=6.54. 6.78 is within 4% of the REAL differential of 6.54… so pretty close.

Does any of this stuff make sense?

 

[fish314]

Darn it. Silly chart didn't come out in a nice format. .

Well, it's space-delineated. I think you can probably get the point from there.

 

[msw]

Thanks Fish. Though it's gonna take me awhile to wade through that! But one quick question: what does the ^ symbol mean in your equations? I am guessing maybe an exponent, i.e. 3^2 is 3 to the second power ("3 squared")?

Darn it. Silly chart didn't come out in a nice format. .

Well, it's space-delineated. I think you can probably get the point from there.

 

[fish314]

Thanks Fish. Though it's gonna take me awhile to wade through that! But one quick question: what does the ^ symbol mean in your equations? I am guessing maybe an exponent, i.e. 3^2 is 3 to the second power ("3 squared")?

Yep, that's it. Since I can't really do a superscript here to show an exponent, the carrot symbol "^" has to suffice. I wish I could have gotten the chart in the middle of my second post to come out looking like a chart-- that might have helped clear things up. As it is, it's tough to see the columns, but they are separated by spaces. But anyways, on each row the first number is the altitude (expressed as Flight level divided by 100). The second number is the real pressure (standard day) at that altitude, in PSI. The third number is what my little formula will give for the pressure in PSI. And the fourth number is the error between the two pressures, expressed as a percent.

I flew tonight and tried this out to see how close I got. My jet has a sensor that displays both PSID and cabin altitude, so it allowed me to check the formula to see how close I could get.

Here's what I did. I was flying along at FL340… so 340 divided by 100 is 3.4. Now 3.4 is a tough number to work with, so I just bumped it up to 3.5 so that I could do the math in my head. 3.5 squared is 12.25, and 1/2 of that is 6.125, which I rounded to 6.1. 3.5 times 5 is 17.5. So 6.1 - 17.5 + 14.7= 3.3

Next, I knew my cabin altitude was 5000', which is FL050. 050/100= 0.5. 0.5 squared is 0.25, and half of that is 0.125… which I rounded again to 0.1. 0.5 times 5 is 2.5, so the final equation comes to 0.1 -2.5 +14.7 = 12.3.

So now, to find my PSID I just subtract the two: 12.3 -3.3 is 9.0. Now, I knew that the real number should have been just slightly below this, because I did the math for FL350 and a cabin altitude of 5000'… but I was really flying at FL340. I wasn't really sure how much the 1000 foot difference would make, so I just guessed and took off 0.1, for a final PSID guess of 8.9. The actual needle was showing 8.7, so I think 8.9 (or even 9.0) are both pretty good guesses. That's an error of 2.29% for 8.9 and 3.44% for 9.0.



By the way, here is a quick way to square a two digit number that ends in 5 in your head: Take the first digit and multiply that by next number. Then tack a '25' on the end, and move the decimal to the correct spot.

For example

55. First digit is a 5, so multiply 5 by the next number, 6, to get 30. Then tack a 25 on the end. 55^2=3025.
75. 7*8=56. So 75^2=5625.
3.5. 3*4=12, so 35^2=1225. But since we're dealing with 3.5, not 35, we know the answer has to be between 3^2 (which is 9) and 4^2 (16). So it must be 12.25, not 1225!

.85^2=.7225
650^2=422500
150000^2=22500000000
.0045^2=.00002025

et cetera, et cetera. Being able to use this little trick is why I rounded to FL350 instead of 340! That way, I could do the whole problem basically in my head.