Velitherm
A library of basic thermodynamics equations for soaring flight.
As used by the weather site https://www.velivole.fr / https://www.meteo.guru
Zerodependency!
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 peutêtre commencer ici: Bases de la thermodynamique pour le vol libre et le vol à voile
Installation
npm install save velitherm
Usage
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 velithermvisu for a realworld example of using this library. It is hosted at aircalc.velivole.fr.
Common JS
const velitherm = require('velitherm');
// must be very close to 1000
const alt = velitherm.altitudeFromPressure(898.746);
ECMAScript 6
import * as velitherm from 'velitherm';
// must be very close to 1000
const alt = velitherm.altitudeFromPressure(898.746);
TypeScript
import * as velitherm from 'velitherm';
// must be very close to 1000
const alt = velitherm.altitudeFromPressure(898.746);
C/C++
#include "node_modules/velitherm/include/velitherm.h"
// must be very close to 1000
double alt = velitherm::altitudeFromPressure(898.746);
Barometric and hypsometric equations
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.
Example
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.
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');
// 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 adiabatic cooling
const T1 = 25  500 * velitherm.gamma;
console.log('The new temperature of the air parcel at 500m = ', (T125)/2, '°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% humidity, 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 (T1 < 20) {
console.log('The ceiling has been reached');
} else {
console.log('The air parcel will continue to rise');
}
API
Table of Contents
 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
velitherm
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/lgpl3.0.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
G
Earth's average gravity acceleration (m/s2)
Type: number
Cp
The thermal capacity of air (J/kg)
Type: number
L
The enthalpy of vaporization of water (J/kg)
Type: number
gamma
The adiabatic lapse rate of dry air (°C/m)
Type: number
P0
The average sea level pressure (hPa)
Type: number
T0
The temperature of the ICAO standard atmosphere (°C)
Type: number
Rd
The specific gas constant of dry air J/(kg*K)
Type: number
Rv
The specific gas constant of water vapor J/(kg*K)
Type: number
Md
Molar mass of dry air kg/mol
Type: number
Mv
Molar mass of water vapor kg/mol
Type: number
R
Universal gas constant J/(kg*mol)
Type: number
K
Absolute zero in °C
Type: number
altitudeFromStandardPressure
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.
Parameters

pressure
number Pressure 
pressure0
number? Optional sealevel pressure of the day (optional, defaultP0
)
Returns number
pressureFromStandardAltitude
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.
Parameters

height
number Height 
pressure0
number? Optional sealevel pressure of the day (optional, defaultP0
)
Returns number
altitudeFromPressure
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.
Parameters

pressure
number Pressure 
pressure0
number? Optional sealevel pressure of the day (optional, defaultP0
) 
temp
number? Optional average temperature from the ground to the given level (optional, defaultT0
)
Returns number
pressureFromAltitude
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.
Parameters

height
number Height 
pressure0
number? Optional sealevel pressure of the day (optional, defaultP0
) 
temp
number? Optional average temperature from the ground to the given level (optional, defaultT0
)
Returns number
waterVaporSaturationPressure
(Saturation) Water vapor pressure.
Clausius–Clapeyron equation  the most fundamental equation in weather science.
This is the MagnusTetens approximation.
Parameters

temp
number Temperature (optional, defaultT0
)
Returns number
relativeHumidity
Relative humidity from specific humidity.
This is from the MagnusTetens approximation.
Parameters

specificHumidity
number Specific humidity 
pressure
number? Optional pressure (optional, defaultP0
) 
temp
number? Optional temperature (optional, defaultT0
)
Returns number
dewPoint
Dew point from relative humidity.
Approximation of the Magnus equation with the Sonntag 1990 coefficients.
Parameters
Returns number
relativeHumidityFromDewPoint
Relative humidity from dew point.
Approximation of the Magnus equation with the Sonntag 1990 coefficients.
Parameters
Returns number
mixingRatio
Mixing ratio from specific humidity.
Analytic equation from the definition.
Parameters

specificHumidity
number Specific humidity
Returns number
specificHumidityFromMixingRatio
Specific humidity from mixing ratio.
Analytic equation from the definition.
Parameters

mixingRatio
number Mixing ratio
Returns number
specificHumidity
Specific humidity from relative humidity.
Approximation of the Magnus equation with the Sonntag 1990 coefficients.
Parameters

relativeHumidity
number Relative humidity 
pressure
number? Optional pressure (optional, defaultP0
) 
temp
number? Optional temperature (optional, defaultT0
)
Returns number
airDensity
Air density.
Analytic equation from Avogadro's Law.
Parameters

relativeHumidity
number Relative humidity 
pressure
number? Optional pressure (optional, defaultP0
) 
temp
number? Optional temperature (optional, defaultT0
)
Returns number
LCL
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.
Parameters
Returns number
gammaMoist
Moist adiabatic lapse rate from pressure and temperature.
Copied from Roland Stull, Practical Meteorology (copylefted, available online).
Rather complex approximation based on the MagnusTetens equation and the barometric equation.
Parameters
Returns number
adiabaticExpansion
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.
Parameters

volume0
number Old volume 
pressure
number New pressure 
pressure0
number Old pressure (optional, defaultP0
)
Returns number
adiabaticCooling
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.
Parameters

temp0
number Old temperature 
pressure
number New pressure 
pressure0
number Old pressure (optional, defaultP0
)
Examples
// 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) < 1e5)
Returns number