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Air pressure below sea level

  1. Oct 1, 2012 #1
    If you were to drill a hole in the ground, what is the expression for the air pressure below ground? Above ground it is exponential with distance (at least to an approximation), so what is it for below earth's surface?

    It has to increase but probably with a lesser absolute slope than above ground. Let's assume the usual "earth is a perfect sphere with uniform density" and that the air is isothermal.
     
    Last edited: Oct 1, 2012
  2. jcsd
  3. Oct 2, 2012 #2

    cjl

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    Unless you start getting so far into the earth that the gravitational force is changing, it will be an exponential, just like it is above sea level. There's nothing special about sea level that would cause its behavior to change.
     
  4. Oct 2, 2012 #3
    No, but you do not need to wait until gravitational force starts changing. The pressure change stops being exponential when air compressibility deviates from ideal gas - meaning, from couple hundred atmospheres.
     
  5. Oct 2, 2012 #4
    The whole point of the problem is to take into account the changing gravitational field.
     
  6. Oct 2, 2012 #5

    jbriggs444

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    Science Advisor

    But if that is the point of the problem then the model does not call for pressure to be exponential above ground either.
     
  7. Oct 2, 2012 #6

    cjl

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    That's a good point actually - earth's atmosphere has a scale height of about 8km (if I remember right), so by 50km below the surface, the pressure would be about 400 atmospheres. Given the behavior of N2 at high pressure, this would definitely be sufficient to notice deviations from the ideal gas behavior. On the other hand, 50km down, the gravitational change would be fairly negligible, especially since the gravitational acceleration inside earth is fairly constant for the first 2000km or so.

    One thing that you have to keep in mind when thinking about this problem: the atmosphere is extremely thin. Gravitational changes across the entire thickness of the atmosphere are pretty much negligible, so they can pretty safely be ignored, even if you were to extend the atmosphere significantly in either direction.
     
  8. Oct 3, 2012 #7
    I wrote a little program to calculate the pressure above and below ground.
    I assumed air is an ideal gas.
    And I calculated gravity like this:

    double newR = earthRadius+altitude;
    if(altitude >= 0) {
    gravity = earthGravity*Math.pow(earthRadius, 2)/Math.pow(newR, 2);
    } else {
    gravity = earthGravity/earthRadius*newR;
    }

    Here are some results.
    alt: 0km, pressure: 101.312kPa
    alt: 1km, pressure: 89.418kPa
    alt: 2km, pressure: 78.924kPa
    alt: 3km, pressure: 69.664kPa
    alt: 4km, pressure: 61.492kPa
    alt: 5km, pressure: 54.282kPa
    alt: 6km, pressure: 47.919kPa
    alt: 7km, pressure: 42.303kPa
    alt: 8km, pressure: 37.347kPa
    alt: 9km, pressure: 32.973kPa
    alt: 10km, pressure: 29.112kPa
    alt: 11km, pressure: 25.704kPa
    alt: 12km, pressure: 22.696kPa
    alt: 13km, pressure: 20.041kPa
    alt: 14km, pressure: 17.697kPa
    alt: 15km, pressure: 15.628kPa
    alt: 16km, pressure: 13.802kPa
    alt: 17km, pressure: 12.189kPa
    alt: 18km, pressure: 10.765kPa
    alt: 19km, pressure: 9.508kPa
    alt: 20km, pressure: 8.398kPa
    alt: 21km, pressure: 7.418kPa
    alt: 22km, pressure: 6.552kPa
    alt: 23km, pressure: 5.788kPa
    alt: 24km, pressure: 5.113kPa
    alt: 25km, pressure: 4.517kPa
    alt: 26km, pressure: 3.991kPa
    alt: 27km, pressure: 3.526kPa
    alt: 28km, pressure: 3.115kPa
    alt: 29km, pressure: 2.752kPa
    alt: 30km, pressure: 2.432kPa
    alt: 31km, pressure: 2.149kPa
    alt: 32km, pressure: 1.899kPa
    alt: 33km, pressure: 1.678kPa
    alt: 34km, pressure: 1.483kPa
    alt: 35km, pressure: 1.311kPa
    alt: 36km, pressure: 1.158kPa
    alt: 37km, pressure: 1.024kPa
    alt: 38km, pressure: 0.905kPa
    alt: 39km, pressure: 0.800kPa
    alt: 40km, pressure: 0.707kPa
    alt: 41km, pressure: 0.625kPa
    alt: 42km, pressure: 0.552kPa
    alt: 43km, pressure: 0.488kPa
    alt: 44km, pressure: 0.432kPa
    alt: 45km, pressure: 0.382kPa
    alt: 46km, pressure: 0.338kPa
    alt: 47km, pressure: 0.298kPa
    alt: 48km, pressure: 0.264kPa
    alt: 49km, pressure: 0.233kPa
    alt: 50km, pressure: 0.206kPa
    alt: 51km, pressure: 0.182kPa
    alt: 52km, pressure: 0.161kPa
    alt: 53km, pressure: 0.143kPa
    alt: 54km, pressure: 0.126kPa
    alt: 55km, pressure: 0.112kPa
    alt: 56km, pressure: 0.099kPa
    alt: 57km, pressure: 0.087kPa
    alt: 58km, pressure: 0.077kPa
    alt: 59km, pressure: 0.068kPa
    alt: 60km, pressure: 0.060kPa
    alt: 61km, pressure: 0.053kPa
    alt: 62km, pressure: 0.047kPa
    alt: 63km, pressure: 0.042kPa
    alt: 64km, pressure: 0.037kPa
    alt: 65km, pressure: 0.033kPa
    alt: 66km, pressure: 0.029kPa
    alt: 67km, pressure: 0.026kPa
    alt: 68km, pressure: 0.023kPa
    alt: 69km, pressure: 0.020kPa
    alt: 70km, pressure: 0.018kPa
    alt: 71km, pressure: 0.016kPa
    alt: 72km, pressure: 0.014kPa
    alt: 73km, pressure: 0.012kPa
    alt: 74km, pressure: 0.011kPa
    alt: 75km, pressure: 0.010kPa
    alt: 76km, pressure: 0.009kPa
    alt: 77km, pressure: 0.008kPa
    alt: 78km, pressure: 0.007kPa
    alt: 79km, pressure: 0.006kPa
    alt: 80km, pressure: 0.005kPa

    below sea level:
    alt: -1km, pressure: 114.802kPa
    alt: -2km, pressure: 130.069kPa
    alt: -3km, pressure: 147.364kPa
    alt: -4km, pressure: 166.955kPa
    alt: -5km, pressure: 189.146kPa
    alt: -6km, pressure: 214.283kPa
    alt: -7km, pressure: 242.756kPa
    alt: -8km, pressure: 275.007kPa
    alt: -9km, pressure: 311.536kPa
    alt: -10km, pressure: 352.911kPa
    alt: -11km, pressure: 399.773kPa
    alt: -12km, pressure: 452.849kPa
    alt: -13km, pressure: 512.961kPa
    alt: -14km, pressure: 581.041kPa
    alt: -15km, pressure: 658.144kPa
    alt: -16km, pressure: 745.463kPa
    alt: -17km, pressure: 844.351kPa
    alt: -18km, pressure: 956.339kPa
    alt: -19km, pressure: 1083.158kPa
    alt: -20km, pressure: 1226.770kPa
    alt: -21km, pressure: 1389.396kPa
    alt: -22km, pressure: 1573.550kPa
    alt: -23km, pressure: 1782.077kPa
    alt: -24km, pressure: 2018.198kPa
    alt: -25km, pressure: 2285.560kPa
    alt: -26km, pressure: 2588.291kPa
    alt: -27km, pressure: 2931.061kPa
    alt: -28km, pressure: 3319.161kPa
    alt: -29km, pressure: 3758.574kPa
    alt: -30km, pressure: 4256.076kPa
    alt: -31km, pressure: 4819.336kPa
    alt: -32km, pressure: 5457.032kPa
    alt: -33km, pressure: 6178.988kPa
    alt: -34km, pressure: 6996.319kPa
    alt: -35km, pressure: 7921.608kPa
    alt: -36km, pressure: 8969.095kPa
    alt: -37km, pressure: 10154.893kPa
    alt: -38km, pressure: 11497.240kPa
    alt: -39km, pressure: 13016.772kPa
    alt: -40km, pressure: 14736.844kPa
    alt: -41km, pressure: 16683.885kPa
    alt: -42km, pressure: 18887.799kPa
    alt: -43km, pressure: 21382.428kPa
    alt: -44km, pressure: 24206.064kPa
    alt: -45km, pressure: 27402.035kPa
    alt: -46km, pressure: 31019.367kPa
    alt: -47km, pressure: 35113.534kPa
    alt: -48km, pressure: 39747.301kPa
    alt: -49km, pressure: 44991.682kPa
    alt: -50km, pressure: 50927.025kPa
    alt: -51km, pressure: 57644.233kPa
    alt: -52km, pressure: 65246.154kPa
    alt: -53km, pressure: 73849.142kPa
    alt: -54km, pressure: 83584.832kPa
    alt: -55km, pressure: 94602.145kPa
    alt: -56km, pressure: 107069.552kPa
    alt: -57km, pressure: 121177.634kPa
    alt: -58km, pressure: 137141.988kPa
    alt: -59km, pressure: 155206.499kPa
    alt: -60km, pressure: 175647.045kPa
    alt: -61km, pressure: 198775.696kPa
    alt: -62km, pressure: 224945.446kPa
    alt: -63km, pressure: 254555.576kPa
    alt: -64km, pressure: 288057.715kPa
    alt: -65km, pressure: 325962.692kPa
    alt: -66km, pressure: 368848.284kPa
    alt: -67km, pressure: 417367.979kPa
    alt: -68km, pressure: 472260.880kPa
    alt: -69km, pressure: 534362.909kPa
    alt: -70km, pressure: 604619.467kPa
    alt: -71km, pressure: 684099.754kPa
    alt: -72km, pressure: 774012.955kPa
    alt: -73km, pressure: 875726.542kPa
    alt: -74km, pressure: 990786.962kPa
    alt: -75km, pressure: 1120943.029kPa
    alt: -76km, pressure: 1268172.365kPa
    alt: -77km, pressure: 1434711.295kPa
    alt: -78km, pressure: 1623088.635kPa
    alt: -79km, pressure: 1836163.896kPa
    alt: -80km, pressure: 2077170.461kPa
    alt: -81km, pressure: 2349764.399kPa
    alt: -82km, pressure: 2658079.637kPa
    alt: -83km, pressure: 3006790.328kPa
    alt: -84km, pressure: 3401181.343kPa
    alt: -85km, pressure: 3847227.949kPa
    alt: -86km, pressure: 4351685.864kPa
    alt: -87km, pressure: 4922193.050kPa
    alt: -88km, pressure: 5567384.754kPa
    alt: -89km, pressure: 6297023.529kPa
    alt: -90km, pressure: 7122146.188kPa
    alt: -91km, pressure: 8055229.876kPa
    alt: -92km, pressure: 9110379.766kPa
    alt: -93km, pressure: 10303541.165kPa
    alt: -94km, pressure: 11652739.223kPa
    alt: -95km, pressure: 13178349.821kPa
    alt: -96km, pressure: 14903405.681kPa
    alt: -97km, pressure: 16853942.285kPa
    alt: -98km, pressure: 19059388.765kPa
    alt: -99km, pressure: 21553009.599kPa
    alt: -100km, pressure: 24372403.713kPa
    ...
    alt: -1000km, pressure: 9700986171002809000000000000000000000000000000000000.000kPa

    Obviously some of those numbers are unrealistic since air is not really an ideal gas.
     
  9. Oct 3, 2012 #8

    jbriggs444

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    Science Advisor

    A quick spot check indicates that your program is suspect.

    The pressure ratio for consecutive kilometers in the reported figures deep under the surface is greater than the pressure ratio for consecutive kilometers at the surface. If modelled gravity were decreasing with depth, one would expect the ratio to decline with depth rather than increasing.

    I cannot check the algorithm above surface as readily. The high altitude numbers are reported without enough significant figures to make a clear determination. It appears that the same error is present, however.
     
  10. Oct 3, 2012 #9

    mfb

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    2016 Award

    Staff: Mentor

    The numbers increase by ~13.3% near the surface, and ~13% at -100km altitude. That looks consistent, just a bit high (but I do not know which temperature you used). For positive altitudes, the ratio seems to go down a bit, that is fine.
     
  11. Oct 3, 2012 #10
    It would deviate from exponential behavior because air is compressible. Even if the air acted like an ideal gas, the density of the air will be changing due to the increased pressure. So the pressure would rise faster than an exponential due to change in density even if the air acted like an ideal gas.
    However, the gravitational acceleration will be decreasing with depth. So that would push the rate of increase in the opposite direction.
     
  12. Oct 3, 2012 #11

    mfb

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    2016 Award

    Staff: Mentor

    Exactly that leads to an exponential shape. Pressure rise is proportional to the density multiplied with the gravitational acceleration - and in an ideal gas (and constant temperature), density is proportional to pressure.
    Deviations from the exponential law come from:
    - air is not an ideal gas (very important >50km below the surface)
    - temperature is not constant (ignored here)
    - gravitational acceleration is not constant (small effect)
     
  13. Oct 4, 2012 #12

    jbriggs444

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    Science Advisor

    Bah, my mistake. You are right.
     
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