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Encyclopedia > Barometric formula

The barometric formula, sometimes called the exponential atmosphere or isothermal atmosphere, is a formula used to model how the pressure (or density) of the air changes with altitude. The exponential function is one of the most important functions in mathematics. ... Layers of Atmosphere (NOAA) Air redirects here. ... An isothermal process is a thermodynamic process in which the temperature of the system stays constant; ΔT = 0. ... View of Jupiters active atmosphere, including the Great Red Spot. ... In mathematics and in the sciences, a formula (plural: formulae, formulæ or formulas) is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relatx E=mc² (see special relativity). ... The use of water pressure - the Captain Cook Memorial Jet in Lake Burley Griffin, Canberra. ... Density (symbol: ρ - Greek: rho) is a measure of mass per volume. ... Altitude is the elevation of an object from a known level or datum. ...

Contents

Pressure Equations

There are two different equations for computing pressure at various height regimes below 86 km (or 232,939 feet). Equation 1 is used when the value of Standard Temperature Lapse rate is not equal to zero and equation 2 is used when Standard Temperature Lapse rate equals zero.


Equation 1:

{P}=P_b cdot left[frac{T_b}{T_b + L_bcdot(h-h_b)}right]^frac{g_o cdot M}{R^* cdot L_b}

Equation 2:

{P}=P_b cdot expleft[frac{-g_o cdot M cdot (h-h_b)}{R^* cdot T_b}right]


Where

P = Static Pressure (inches Hg)
T = Standard Temperature (degrees Kelvin)
L = Standard Temperature Lapse Rate (degrees Celsius per foot)
h = Height above Sea Level (feet)
R * = Universal Gas Constant (converted to English units: 8.9494596 X 104 Ft2/sec2 K)
go = Gravitational Constant (32.17405 feet per second/per second)
M = Molar Mass of Earth's Air (28.9644 grams per mole)

The value of subscript b ranges from 0 to 6 in accordance with each of seven successive layers of the atmosphere shown in the table below. In these equations, go, M and R* are each single-valued constants, while P, L, T and h are multi-valued constants in accordance with the table below. It should be noted that the values used for M, go and R* is in accordance with the US Standard Atmosphere, 1976. The reference value for Pb for b = 0 is the defined sea level value, Po = 101325 pascals or 29.92126 inches Hg. Values of Pb of b = 1 through b = 6 are obtained from the application of the appropriate member of the pair equations 1 and 2 for the case when h = hb + 1.:[1]

Subscript b Height above sea level (feet) Static Pressure (inches Hg) Standard Temperature (degrees Kelvin) Temperature Lapse Rate (degrees Celsius per foot)
0 0 29.92126 288.15 -0.0019812
1 36,089.24 6.683245 216.65 0.0
2 65,616.79 1.616734 216.65 0.0003048
3 104,986.87 0.2563258 228.65 0.00085344
4 154,199.48 0.0327506 270.65 0.0
5 167,322.83 0.01976704 270.65 -0.00085344
6 232,939.63 0.00116833 214.65 -0.0006096

Density Equations

The expressions for calculating density are nearly identical to calculating pressure. the only difference is the exponent in Equation 1.


There are two different equations for computing density at various height regimes below 86 km (or 232,939 feet). Equation 1 is used when the value of Standard Temperature Lapse rate is not equal to zero and equation 2 is used when Standard Temperature Lapse rate equals zero.


Equation 1:

{rho}=rho_b cdot left[frac{T_b}{T_b + L_bcdot(h-h_b)}right]^{left(frac{g_o cdot M}{R^* cdot L_b}right)+1}

Equation 2:

{rho}=rho_b cdot expleft[frac{-g_o cdot M cdot (h-h_b)}{R^* cdot T_b}right]


Where

ρ = Mass Density (slugs/in3)
T = Standard Temperature (degrees Kelvin)
L = Standard Temperature Lapse Rate (degrees Celsius per foot)
h = Height above Sea Level (feet)
R * = Universal Gas Constant (converted to English units: 8.9494596 X 104 Ft2/sec2 K)
go = Gravitational Constant (32.17405 feet per second/per second)
M = Molar Mass of Earth's Air (28.9644 grams per mole)

The value of subscript b ranges from 0 to 6 in accordance with each of seven successive layers of the atmosphere shown in the table below. In these equations, go, M and R* are each single-valued constants, while ρ, L, T and h are multi-valued constants in accordance with the table below. It should be noted that the values used for M, go and R* is in accordance with the US Standard Atmosphere, 1976. The reference value for ρb for b = 0 is the defined sea level value, ρo = 1.2250 kg/m3 or 0.0023768908 slugs/in3. Values of ρb of b = 1 through b = 6 are obtained from the application of the appropriate member of the pair equations 1 and 2 for the case when h = hb + 1.:[1]

Subscript b Height above sea level (feet) Mass Density (slugs/in3) Standard Temperature (degrees Kelvin) Temperature Lapse Rate (degrees Celsius per foot)
0 0 2.3768908 x 10− 3 288.15 -0.0019812
1 36,089.24 7.0611703 x 10− 4 216.65 0.0
2 65,616.79 1.7081572 x 10− 4 216.65 0.0003048
3 104,986.87 2.5660735 x 10− 5 228.65 0.00085344
4 154,199.48 2.7698702 x 10− 6 270.65 0.0
5 167,322.83 1.6717895 x 10− 6 270.65 -0.00085344
6 232,939.63 1.2458989 x 10− 7 214.65 -0.0006096

Derivation

The barometric formula can be derived fairly easily using the ideal gas law: Isotherms of an ideal gas The ideal gas law is the equation of state of an ideal gas. ...

rho = frac{M cdot P}{R^* cdot T}

When density is known:

P = frac{rho cdot {R^*} cdot T}{M}

References

  1. ^ a b U.S. Standard Atmosphere, 1976, U.S. Government Printing Office, Washington, D.C., 1976.

See also

  • NRLMSISE-00

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