Barometric formula

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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.

1 Pressure equations
2 Density equations
3 Derivation
4 References
5 See also

[edit] Pressure equations

See also: Atmospheric pressure

There are two different equations for computing pressure at various height regimes below 86 km (or 278,400 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:

${P}=P_b \cdot \left[\frac{T_b}{T_b + L_b\cdot(h-h_b)}\right]^\frac{g_0 \cdot M}{R^* \cdot L_b}$

Equation 2:

${P}=P_b \cdot \exp \left[\frac{-g_0 \cdot M \cdot (h-h_b)}{R^* \cdot T_b}\right]$

where

P

= Static pressure (pascals)

T

= Standard temperature (kelvins)

L

= Standard temperature lapse rate (kelvins per meter)

h

= Height above sea level (meters)

R *

= Universal gas constant for air: 8.31432 N·m / (mol·K)

g 0

= Gravitational acceleration (9.80665 m/s²)

M

= Molar mass of Earth's air (0.0289644 kg/mol)

Or converted to Imperial units:^[1]

where

P

= Static pressure (inches of mercury)

T

= Standard temperature (kelvins)

L

= Standard temperature lapse rate (kelvins per foot)

h

= Height above sea level (feet)

R *

= Universal gas constant (using feet and kelvins and gram moles: 8.9494596×10⁴ kg·ft²·s^-2·K^-1·kmol^-1)

g 0

= Gravitational acceleration (32.17405 ft/s²)

M

= Molar mass of Earth's air (28.9644 g/mol)

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, g₀, M and R^* are each single-valued constants, while P, L, T, and h are multivalued constants in accordance with the table below. It should be noted that the values used for M, g₀, and $R *$ are in accordance with the U.S. Standard Atmosphere, 1976, and that the value for $R *$ in particular does not agree with standard values for this constant.^[2] The reference value for P_b for b = 0 is the defined sea level value, P₀ = 101325 pascals or 29.92126 inHg. Values of P_b 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 $h = h b + 1$ .:^[2]

Subscript b	Height Above Sea Level		Static Pressure		Standard Temperature (K)	Temperature Lapse Rate
Subscript b	(m)	(ft)	(pascals)	(inHg)	Standard Temperature (K)	(K/m)	(K/ft)
0	0	0	101325	29.92126	288.15	-0.0065	-0.0019812
1	11,000	36,089	22632.1	6.683245	216.65	0.0	0.0
2	20,000	65,617	5474.89	1.616734	216.65	0.001	0.0003048
3	32,000	104,987	868.019	0.2563258	228.65	0.0028	0.00085344
4	47,000	154,199	110.906	0.0327506	270.65	0.0	0.0
5	51,000	167,323	66.9389	0.01976704	270.65	-0.0028	-0.00085344
6	71,000	232,940	3.95642	0.00116833	214.65	-0.002	-0.0006096

[edit] 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 geometric km (84,852 geopotential meters or 278,385.8 geopotential 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:

${\rho}=\rho_b \cdot \left[\frac{T_b}{T_b + L_b\cdot(h-h_b)}\right]^{\left(\frac{g_0 \cdot M}{R^* \cdot L_b}\right)+1}$

Equation 2:

${\rho}=\rho_b \cdot \exp\left[\frac{-g_0 \cdot M \cdot (h-h_b)}{R^* \cdot T_b}\right]$

where

ρ

= Mass density (kg/m³)

T

= Standard temperature (kelvins)

L

= Standard temperature lapse rate (kelvins per meter)

h

= Height above sea level (geopotential meters)

R *

= Universal gas constant for air: 8.31432 N·m/(mol·K)

g 0

= Gravitational acceleration (9.80665 m/s²)

M

= Molar mass of Earth's air (0.0289644 kg/mol)

Or converted to English units:^[1]

Where

ρ

= Mass density (slugs/ft³)

T

= Standard temperature (kelvins)

L

= Standard temperature lapse rate (degrees Celsius per foot)

h

= Height above sea level (geopotential feet)

R *

= Universal gas constant (8.9494596×10⁴ ft²/(s·K)

g 0

= Gravitational acceleration (32.17405 ft/s²)

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. The reference value for $ρ b$ for b = 0 is the defined sea level value, $ρ o$ = 1.2250 kg/m³ or 0.0023768908 slugs/ft³. Values of $ρ b$ 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 $h = h b + 1$ ^[2]

In these equations, g₀, 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, g₀ and R^* are in accordance with the U.S. Standard Atmosphere, 1976, and that the value for R^* in particular does not agree with standard values for this constant.^[2].

Subscript b	Height Above Sea Level (h)		Mass Density ( $ρ$ )		Standard Temperature (T') (K)	Temperature Lapse Rate (L)
Subscript b	(m)	(ft)	(kg/m³)	(slugs/ft³	Standard Temperature (T') (K)	(K/m)	(K/ft)
0	0	0	1.2250	2.3768908 x 10^-3	288.15	-0.0065	-0.0019812
1	11,000	36,089.24	0.36391	7.0611703 x 10^-4	216.65	0.0	0.0
2	20,000	65,616.79	0.08803	1.7081572 x 10^-4	216.65	0.001	0.0003048
3	32,000	104,986.87	0.01322	2.5660735 x 10^-5	228.65	0.0028	0.00085344
4	47,000	154,199.48	0.00143	2.7698702 x 10 $- 6$	270.65	0.0	0.0
5	51,000	167,322.83	0.00086	1.6717895 x 10^-6	270.65	-0.0028	-0.00085344
6	71,000	232,939.63	0.000064	1.2458989 x 10^-7	214.65	-0.002	-0.0006096

[edit] Derivation

The barometric formula can be derived fairly easily using the ideal gas law:

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

When density is known:

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

And assuming that all pressure is hydrostatic:

$dP = - \rho g\,dz\,$

Substituting the first expression into the second we get:

$\frac{dP}{P} = - \frac{M g\,dz}{RT}$

Integrating this expression from the surface to the altitude z we get:

$P = P_0 e^{-\int_{0}^{z}{M g dz/RT}}\,$

Assuming constant temperature, molar mass, and gravitational acceleration, we get the barometric formula:

$P = P_0 e^{-M g z/RT}\,$

In this formulation, R is the gas constant, and the term $R T / M g$ gives the scale height (approximately equal to 7.4 km for the troposphere).

(For exact results, it should be remembered that atmospheres containing water do not behave as an ideal gas. See real gas or perfect gas or gas to further understanding)

[edit] References

^ ^a ^b Mechtly, E. A., 1973: The International System of Units, Physical Constants and Conversion Factors. NASA SP-7012, Second Revision, National Aeronautics and Space Administration, Washington, D.C.
^ ^a ^b ^c ^d U.S. Standard Atmosphere, 1976, U.S. Government Printing Office, Washington, D.C., 1976. (Linked file is very large.)