A library of basic thermodynamics equations for soaring flight.
As used by the weather site https://www.velivole.fr / https://www.meteo.guru
Zero-dependency!
If you are new to weather thermodynamics, you should probably start here: Basic Concepts of Thermodynamics for Soaring Flight
Si vous n'êtes pas à l'aise en thermodynamique appliquée à la météorologie, vous devriez peut-être commencer ici: Bases de la thermodynamique pour le vol libre et le vol à voile
npm install --save velitherm
Keep in mind that some equations are numerical approximations of differential equations that have no analytic solutions. The approximations used are the typical weather science approximations which produce good results for temperatures in the range of -40°C to +40°C and air pressures in the range of 1050hPa to 200hPa - which are the typical values in the troposphere - but often lack precision outside this range.
You can check velitherm-visu for a real-world example of using this library. It is hosted at aircalc.velivole.fr.
const velitherm = require('velitherm');
// must be very close to 1000
const alt = velitherm.altitudeFromPressure(898.746);import * as velitherm from 'velitherm';
// must be very close to 1000
const alt = velitherm.altitudeFromPressure(898.746);import * as velitherm from 'velitherm';
// must be very close to 1000
const alt = velitherm.altitudeFromPressure(898.746);#include "node_modules/velitherm/include/velitherm.h"
// must be very close to 1000
double alt = velitherm::altitudeFromPressure(898.746);There is no single analytical equation that can be used to give a precise value for the altitude / pressure relationship. In fact, this relationship depends on the full vertical temperature and humidity profile and it is impossible to reduce to a single formula. There are two levels of approximation that are widely used:
- The barometric formula, which is an ICAO standard and gives a rough value that does not take into account the pressure or the temperature of the day and it is always constant - in aviation it is referred by the callsign QNH
- The hypsometric formula, which is commonly used in weather science and it is a better estimation that takes into account the pressure and the temperature of the day - so it varies from one day to another - in aviation it is referred by the callsign QFF
If you are coming from an aviation background, and QNH, QFF and QFE are altimeter settings to you, then you should know that they actually refer to different equations for calculating the altitude from the pressure. QFE refers to the surface pressure of a given site - airport or weather station - and not an equation. If you want to use velitherm to convert between altimeter settings, the right way to do it would be to convert the pressure to altitude using the input setting and then convert it back to pressure using the output setting.
An air parcel with relative humidify of 75% and temperature of 25°C rises from 0m AMSL to 500m AMSL where the surrounding temperature is 20°C. What is its new relative humidity? What is its new temperature? Has there been condensation and did it form a cloud? Has the ceiling being reached or will the air parcel continue to rise? The pressure of the day is 1017hPa and the relative humidity at 500m AMSL is 50%.
Solution:
import * as velitherm from 'velitherm';
// When the air rises, its specific humidity remains constant
const q = velitherm.specificHumidity(75, 1017, 25);
console.log('Specific humidity = ', Math.round(q), 'g/kg');
console.log('Dew point = ', velitherm.dewPoint(75, 25));
// Find the current pressure at 500m AMSL
const P1 = velitherm.pressureFromAltitude(500, 1017, 25);
console.log('Pressure at 500m = ', Math.round(P1), 'hPa');
// Take into account the dry adiabatic cooling over 500m
const T1 = 25 - 500 * velitherm.gamma;
console.log('The new temperature of the air parcel at 500m = ', T1, '°C');
// Compute the new relative humidity of the air parcel at this pressure and temperature
const w1 = velitherm.relativeHumidity(q, P1, T1);
console.log('Relative humidity after rising to 500m = ', Math.round(w1), '%');
// If the air parcel has reached 100% relative humidity, then there is condensation
if (w1 < 100) {
console.log('No, it did not form a cloud');
} else {
console.log('Yes, it did form a cloud');
}
// If the density of the air parcel is still lower than the
// surrounding air at 500m AMSL, then it will continue to rise
const rhoParcel = velitherm.airDensity(w1, P1, T1);
const rhoAir500 = velitherm.airDensity(50, P1, 20);
if (rhoParcel < rhoAir500) {
console.log('The air parcel will continue to rise');
} else {
console.log('The ceiling has been reached');
}You can run the example program with
ts-node examples/risingAir.tsNote that the dew point of the air parcel at sea level is 20.26°C, yet it does not form a cloud when cooled down to 20.12°C at 500m AMSL. The reason is that a dew point is valid only for a given pressure. At a lower pressure, the dew point will also be lower.
- velitherm
- G
- Cp
- L
- gamma
- P0
- T0
- Rd
- Rv
- Md
- Mv
- R
- K
- altitudeFromStandardPressure
- pressureFromStandardAltitude
- altitudeFromPressure
- pressureFromAltitude
- waterVaporSaturationPressure
- relativeHumidity
- dewPoint
- relativeHumidityFromDewPoint
- mixingRatio
- specificHumidityFromMixingRatio
- specificHumidity
- airDensity
- LCL
- gammaMoist
- adiabaticExpansion
- adiabaticCooling
velivole.fr/meteo.guru Basic Thermodynamics Equations for Soaring Flight
Copyright © 2022 Momtchil Momtchev momtchil@momtchev.com
Licensed under the LGPL License, Version 3.0 (the "License") You may not use this file except in compliance with the License. You may obtain a copy of the License at: https://www.gnu.org/licenses/lgpl-3.0.en.html
All methods use:
Pressure in hPa
Temperature in °C
Height in meters
Relative humidity in % from 0 to 100
Specific humidity in g/kg
Mixing ratio in g/kg
Type: string
Earth's average gravity acceleration (m/s2)
Type: number
The thermal capacity of air (J/kg)
Type: number
The enthalpy of vaporization of water (J/kg)
Type: number
The adiabatic lapse rate of dry air (°C/m)
Type: number
The average sea level pressure (hPa)
Type: number
The temperature of the ICAO standard atmosphere (°C)
Type: number
The specific gas constant of dry air J/(kg*K)
Type: number
The specific gas constant of water vapor J/(kg*K)
Type: number
Molar mass of dry air kg/mol
Type: number
Molar mass of water vapor kg/mol
Type: number
Universal gas constant J/(kg*mol)
Type: number
Absolute zero in °C
Type: number
Altitude from pressure using the barometric formula and ICAO's definition of standard atmosphere.
This is a very rough approximation that is an ICAO standard. It is used when calculating QNH. It does not take into account the pressure and temperature of the day.
pressurenumber Pressurepressure0number? Optional sea-level pressure of the day (optional, defaultP0)
Returns number
Pressure from altitude using the barometric formula and ICAO's definition of standard atmosphere.
This is a very rough approximation that is an ICAO standard. It is used when calculating QNH. It does not take into account the pressure and temperature of the day.
heightnumber Heightpressure0number? Optional sea-level pressure of the day (optional, defaultP0)
Returns number
Altitude from pressure using the hypsometric formula.
This is a better equation that takes into account the pressure and the temperature of the day. It is not a standard and different weather institutions use slightly different parameters. It is used when calculating the QFF.
pressurenumber Pressurepressure0number? Optional sea-level pressure of the day (optional, defaultP0)tempnumber? Optional average temperature from the ground to the given level (optional, defaultT0)
Returns number
Pressure from altitude using the hypsometric formula.
This is a better equation that takes into account the pressure and the temperature of the day. It is not a standard and different weather institutions use slightly different parameters. It is used when calculating the QFF.
heightnumber Heightpressure0number? Optional sea-level pressure of the day (optional, defaultP0)tempnumber? Optional average temperature from the ground to the given level (optional, defaultT0)
Returns number
(Saturation) Water vapor pressure.
Clausius–Clapeyron equation - the most fundamental equation in weather science.
This is the Magnus-Tetens approximation.
tempnumber Temperature (optional, defaultT0)
Returns number
Relative humidity from specific humidity.
This is from the Magnus-Tetens approximation.
specificHumiditynumber Specific humiditypressurenumber? Optional pressure (optional, defaultP0)tempnumber? Optional temperature (optional, defaultT0)
Returns number
Dew point from relative humidity.
Approximation of the Magnus equation with the Sonntag 1990 coefficients.
Returns number
Relative humidity from dew point.
Approximation of the Magnus equation with the Sonntag 1990 coefficients.
Returns number
Mixing ratio from specific humidity.
Analytic equation from the definition.
specificHumiditynumber Specific humidity
Returns number
Specific humidity from mixing ratio.
Analytic equation from the definition.
mixingRationumber Mixing ratio
Returns number
Specific humidity from relative humidity.
Approximation of the Magnus equation with the Sonntag 1990 coefficients.
relativeHumiditynumber Relative humiditypressurenumber? Optional pressure (optional, defaultP0)tempnumber? Optional temperature (optional, defaultT0)
Returns number
Air density.
Analytic equation from Avogadro's Law.
relativeHumiditynumber Relative humiditypressurenumber? Optional pressure (optional, defaultP0)tempnumber? Optional temperature (optional, defaultT0)
Returns number
Lifted Condensation Level.
This is the altitude at which a mechanically lifted air parcel from the ground will condensate.
It corresponds to the cloud base level when the clouds are formed by mechanical lifting.
This approximation is known as the Espy equation with the Stull coefficient.
Returns number
Moist adiabatic lapse rate from pressure and temperature.
Copied from Roland Stull, Practical Meteorology (copylefted, available online).
Rather complex approximation based on the Magnus-Tetens equation and the barometric equation.
Returns number
Adiabatic expansion rate from pressure change rate.
This equation allows to calculate the expansion ratio of an air parcel from the the previous pressure and the new pressure.
An adiabatic expansion is an isentropic process that is governed by the Ideal gas law in general and the constant entropy relationship in particular: (P / P0) = (V / V0) ^ gamma Where P=pressure, V=volume, gamma=heat capacity ratio (1.4 for air, a diatomic gas)
Analytic equation.
volume0number Old volumepressurenumber New pressurepressure0number Old pressure (optional, defaultP0)
Returns number
Adiabatic cooling rate from pressure change rate.
This equation allows to calculate the cooling ratio of an air parcel from the the previous pressure and the new pressure.
It is by combining this equation with the barometric equation that the adiabatic lapse rate of dry air can be obtained.
An adiabatic expansion is an isentropic process that is governed by the Ideal gas law in general and the constant entropy relationship in particular: (P / P0) = (V / V0) ^ gamma Where P=pressure, V=volume, gamma=heat capacity ratio (1.4 for air, a diatomic gas)
Keep in mind that if you intend to use this method to calculate a rate relative
to height in meters, you will need very precise altitude calculations for good
results. As the dry adiabatic rate is a constant that does not depend on the
temperature or the pressure, most of the time you will be better off simply
using the gamma constant.
https://en.wikipedia.org/wiki/Ideal_gas_law contains a very good introduction to this subject.
Analytic equation.
temp0number Old temperaturepressurenumber New pressurepressure0number Old pressure (optional, defaultP0)
// Compute the adiabatic cooling per meter
// when rising from 0m AMSL to 100m AMSL starting at 15°C
const gamma = (15 - velitherm.adiabaticCooling(15,
velitherm.pressureFromStandardAltitude(100),
velitherm.pressureFromStandardAltitude(0))
) / 100;
// It should be very close to the provided constant
assert(Math.abs(gamma - velitherm.gamma) < 1e-5)Returns number