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Implement lapse rate model for atmosphere

Environment/Atmosphere

The Lapse Rate Model block implements the mathematical representation of the lapse rate atmospheric equations for ambient temperature, pressure, density, and speed of sound for the input geopotential altitude. You can customize this atmospheric model, described below, by specifying atmospheric properties in the block dialog.

The following equations define the troposphere

$$\begin{array}{c}T={T}_{0}-Lh\\ P={P}_{0}{\left(\frac{T}{{T}_{0}}\right)}^{\frac{g}{LR}}\\ \rho ={\rho}_{0}{\left(\frac{T}{{T}_{0}}\right)}^{\frac{g}{LR}-1}\\ a=\sqrt{\gamma RT}\end{array}$$

The following equations define the tropopause (lower stratosphere)

$$\begin{array}{c}T={T}_{0}-Lhts\\ P={P}_{0}{\left(\frac{T}{{T}_{0}}\right)}^{\frac{g}{LR}}{e}^{\frac{g}{RT}}{}^{{}^{(hts-h)}}\\ \rho ={\rho}_{0}{\left(\frac{T}{{T}_{0}}\right)}^{\frac{g}{LR}-1}{e}^{\frac{g}{RT}}{}^{{}^{(hts-h)}}\\ a=\sqrt{\gamma RT}\end{array}$$

where:

T_{0} | Absolute temperature at mean sea level in kelvin (K) |

ρ_{0} | Air density at mean sea level in kg/m^{3} |

P_{0} | Static pressure at mean sea level in N/m^{2} |

h | Altitude in m |

hts | Height of the troposphere in m |

T | Absolute temperature at altitude h in kelvin
(K) |

ρ | Air density at altitude h in kg/m^{3} |

P | Static pressure at altitude h in N/m^{2} |

a | Speed of sound at altitude h in m/s^{2} |

L | Lapse rate in K/m |

R | Characteristic gas constant J/kg-K |

γ | Specific heat ratio |

g | Acceleration due to gravity in m/s^{2} |

The Lapse Rate Model block icon displays the input and output metric units.

**Change atmospheric parameters**When selected, the following atmospheric parameters can be customized to be different from the ISA values.

**Acceleration due to gravity**Specify the acceleration due to gravity (

*g*).**Ratio of specific heats**Specify the ratio of specific heats

*γ*.**Characteristic gas constant**Specify the characteristic gas constant (

*R*).**Lapse rate**Specify the lapse rate of the troposphere (

*L*).**Height of troposphere**Specify the upper altitude of the troposphere, a range of decreasing temperature.

**Height of tropopause**Specify the upper altitude of the tropopause, a range of constant temperature.

**Air density at mean sea level**Specify the air density at sea level (

*ρ*_{0}).**Ambient pressure at mean sea level**Specify the ambient pressure at sea level (

*P*_{0}).**Ambient temperature at mean sea level**Specify the ambient temperature at sea level (

*T*_{0}).**Lowest altitude (m)**Specify the lowest altitude above which temperature and pressure lapse.

**Lowest altitude (m)**must be below**Height of tropopause**. Default value is 0 m.

Input | Dimension Type | Description |
---|---|---|

First | Contains the geopotential height. |

Output | Dimension Type | Description |
---|---|---|

First | Contains the temperature. | |

Second | Contains the speed of sound. | |

Third | Contains the air pressure. | |

Fourth | Contains the air density. |

Below the geopotential altitude of 0 km and above the geopotential altitude of the tropopause, temperature and pressure values are held. Density and speed of sound are calculated using a perfect gas relationship.

[1] U.S. Standard Atmosphere, 1976, U.S. Government Printing Office, Washington, D.C.

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