# Lapse Rate Model

Implement lapse rate model for atmosphere

## Library

Environment/Atmosphere

## Description

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)

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

where:

 T0 Absolute temperature at mean sea level in kelvin (K) ρ0 Air density at mean sea level in kg/m3 P0 Static pressure at mean sea level in N/m2 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/m3 P Static pressure at altitude h in N/m2 a Speed of sound at altitude h in m/s2 L Lapse rate in K/m R Characteristic gas constant J/kg-K γ Specific heat ratio g Acceleration due to gravity in m/s2

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

## Dialog Box

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 (ρ0).

Ambient pressure at mean sea level

Specify the ambient pressure at sea level (P0).

Ambient temperature at mean sea level

Specify the ambient temperature at sea level (T0).

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.

## Inputs and Outputs

InputDimension TypeDescription

First

Contains the geopotential height.

OutputDimension TypeDescription

First

Contains the temperature.

Second

Contains the speed of sound.

Third

Contains the air pressure.

Fourth

Contains the air density.

## Assumptions and Limitations

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.

## Reference

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

COESA Atmosphere Model

ISA Atmosphere Model