velitherm
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Basic Thermodynamics Equations for Soaring Flight (from velivole.fr/meteo.guru)
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/**
* 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
*/
export var velitherm = 'velitherm';
/**
* Earth's average gravity acceleration (m/s2)
*
* @const
* @type {number}
*/
export var G = 9.81;
/**
* The thermal capacity of air (J/kg)
*
* @const
* @type {number}
*/
export var Cp = 1005;
/**
* The enthalpy of vaporization of water (J/kg)
*
* @const
* @type {number}
*/
export var L = 2.5e6;
/**
* The adiabatic lapse rate of dry air (°C/m)
*
* @const
* @type {number}
*/
export var gamma = 0.00976;
/**
* The average sea level pressure (hPa)
*
* @const
* @type {number}
*/
export var P0 = 1013.25;
/**
* The temperature of the ICAO standard atmosphere (°C)
*
* @const
* @type {number}
*/
export var T0 = 15;
/**
* The specific gas constant of dry air J/(kg*K)
*
* @const
* @type {number}
*/
export var Rd = 287.058;
/**
* The specific gas constant of water vapor J/(kg*K)
*
* @const
* @type {number}
*/
export var Rv = 461.495;
/**
* Molar mass of dry air kg/mol
*
* @const
* @type {number}
*/
export var Md = 0.0289652;
/**
* Molar mass of water vapor kg/mol
*
* @const
* @type {number}
*/
export var Mv = 0.018016;
/**
* Universal gas constant J/(kg*mol)
*
* @const
* @type {number}
*/
export var R = 8.31446;
/**
* Absolute zero in °C
*
* @const
* @type {number}
*/
export var K = -273.15;
/**
* Number of feets in one meter
*
* @const
* @type {number}
*/
export var feetPerMeter = 3.28084;
/**
* 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.
*
* @param {number} pressure Pressure
* @param {number} [pressure0] Optional sea-level pressure of the day
* @returns {number}
*/
export function altitudeFromStandardPressure(pressure, pressure0) {
if (pressure0 === void 0) { pressure0 = P0; }
return 44330.0 * (1.0 - Math.pow(pressure / pressure0, 1 / 5.255));
}
/**
* 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.
*
* @param {number} height Height
* @param {number} [pressure0] Optional sea-level pressure of the day
* @returns {number}
*/
export function pressureFromStandardAltitude(height, pressure0) {
if (pressure0 === void 0) { pressure0 = P0; }
return pressure0 * Math.pow(1 - height / 44330.0, 5.255);
}
/**
* 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.
*
* @param {number} pressure Pressure
* @param {number} [pressure0] Optional sea-level pressure of the day
* @param {number} [temp] Optional average temperature from the ground
* to the given level
* @returns {number}
*/
export function altitudeFromPressure(pressure, pressure0, temp) {
if (pressure0 === void 0) { pressure0 = P0; }
if (temp === void 0) { temp = T0; }
return (Math.pow(pressure0 / pressure, 1.0 / 5.257) - 1) * (temp - K) /
0.0065;
}
/**
* 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.
*
* @param {number} height Height
* @param {number} [pressure0] Optional sea-level pressure of the day
* @param {number} [temp] Optional average temperature from the ground
* to the given level
* @returns {number}
*/
export function pressureFromAltitude(height, pressure0, temp) {
if (pressure0 === void 0) { pressure0 = P0; }
if (temp === void 0) { temp = T0; }
return pressure0 *
Math.pow(1.0 - 0.0065 * height / (temp - K + 0.0065 * height), 5.257);
}
/**
* (Saturation) Water vapor pressure.
*
* Clausius–Clapeyron equation - the most fundamental equation in weather
* science.
*
* This is the Magnus-Tetens approximation.
*
* @param {number} temp Temperature
* @returns {number}
*/
export function waterVaporSaturationPressure(temp) {
if (temp === void 0) { temp = T0; }
return 6.1078 * Math.exp(17.27 * temp / (temp + 237.3));
}
/**
* Relative humidity from specific humidity.
*
* This is from the Magnus-Tetens approximation.
*
* @param {number} specificHumidity Specific humidity
* @param {number} [pressure] Optional pressure
* @param {number} [temp] Optional temperature
* @returns {number}
*/
export function relativeHumidity(specificHumidity, pressure, temp) {
if (pressure === void 0) { pressure = P0; }
if (temp === void 0) { temp = T0; }
return specificHumidity / (6.22 * waterVaporSaturationPressure(temp) /
pressure);
}
var Sonntag_1990_b = 17.62;
var Sonntag_1990_c = 243.12;
/**
* Dew point from relative humidity.
*
* Approximation of the Magnus equation with the Sonntag 1990 coefficients.
*
* @param {number} relativeHumidity Relative humidity
* @param {number} [temp] Optional temperature
* @returns {number}
*/
export function dewPoint(relativeHumidity, temp) {
if (temp === void 0) { temp = T0; }
var gamma = Math.log(relativeHumidity / 100) + Sonntag_1990_b * temp /
(Sonntag_1990_c + temp);
return Sonntag_1990_c * gamma / (Sonntag_1990_b - gamma);
}
/**
* Relative humidity from dew point.
*
* Approximation of the Magnus equation with the Sonntag 1990 coefficients.
*
* @param {number} dewPoint Relative humidity
* @param {number} [temp] Optional temperature
* @returns {number}
*/
export function relativeHumidityFromDewPoint(dewPoint, temp) {
if (temp === void 0) { temp = T0; }
var gamma = dewPoint * Sonntag_1990_b / (dewPoint + Sonntag_1990_c);
return Math.exp(gamma - Sonntag_1990_b * temp /
(Sonntag_1990_c + temp)) * 100;
}
/**
* Mixing ratio from specific humidity.
*
* Analytic equation from the definition.
*
* @param {number} specificHumidity Specific humidity
* @returns {number}
*/
export function mixingRatio(specificHumidity) {
return specificHumidity / (1 - specificHumidity / 1000);
}
/**
* Specific humidity from mixing ratio.
*
* Analytic equation from the definition.
*
* @param {number} mixingRatio Mixing ratio
* @returns {number}
*/
export function specificHumidityFromMixingRatio(mixingRatio) {
return mixingRatio / (1 + mixingRatio / 1000);
}
/**
* Specific humidity from relative humidity.
*
* Approximation of the Magnus equation with the Sonntag 1990 coefficients.
*
* @param {number} relativeHumidity Relative humidity
* @param {number} [pressure] Optional pressure
* @param {number} [temp] Optional temperature
* @returns {number}
*/
export function specificHumidity(relativeHumidity, pressure, temp) {
if (pressure === void 0) { pressure = P0; }
if (temp === void 0) { temp = T0; }
return relativeHumidity / 100 *
(0.622 * waterVaporSaturationPressure(temp) / pressure) * 1000;
}
/**
* Air density.
*
* Analytic equation from Avogadro's Law.
*
* @param {number} relativeHumidity Relative humidity
* @param {number} [pressure] Optional pressure
* @param {number} [temp] Optional temperature
* @returns {number}
*/
export function airDensity(relativeHumidity, pressure, temp) {
if (pressure === void 0) { pressure = P0; }
if (temp === void 0) { temp = T0; }
var Psat = waterVaporSaturationPressure(temp);
var Pv = relativeHumidity / 100 * Psat;
var Pd = pressure - Pv;
return 100 * (Pd * Md + Pv * Mv) / (R * (temp - K));
}
/**
* 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.
*
* @param {number} temp Temperature at 2m
* @param {number} dewPoint Dew point at 2m
* @returns {number}
*/
export function LCL(temp, dewPoint) {
return 126.7 * (temp - dewPoint);
}
/**
* 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.
*
* @param {number} temp Temperature
* @param {number} [pressure] Optional pressure
* @returns {number}
*/
export function gammaMoist(temp, pressure) {
if (pressure === void 0) { pressure = P0; }
var tK = temp - K;
var es = 6.113 * Math.exp(5423 * (-1 / K - 1 / tK));
var rs = 0.622 * es / (pressure - es);
var gamma = G * 1e-3 * (1 + 8711 * rs / tK) / (1 + 1.35e7 * rs / (tK * tK));
return gamma;
}
var HCR = 1.4;
/**
* 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.
*
* @param {number} volume0 Old volume
* @param {number} pressure New pressure
* @param {number} pressure0 Old pressure
* @returns {number}
*/
export function adiabaticExpansion(volume0, pressure, pressure0) {
if (pressure0 === void 0) { pressure0 = P0; }
return volume0 * Math.pow(pressure0 / pressure, 1 / HCR);
}
/**
* 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.
*
* @example
* // 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)
*
* @param {number} temp0 Old temperature
* @param {number} pressure New pressure
* @param {number} pressure0 Old pressure
* @returns {number}
*/
export function adiabaticCooling(temp0, pressure, pressure0) {
if (pressure0 === void 0) { pressure0 = P0; }
return (temp0 - K) * Math.pow(pressure / pressure0, (HCR - 1) / HCR) + K;
}
/**
* Convert a Flight Level to pressure
*
* Flight levels are defined as pressure and not as a fixed
* altitude. This means that a flight level can always be converted
* to pressure without needing any other information in a fully
* deterministic way.
*
* A flight level of 115 means 11500 feet measured by barometer
* using the ICAO standard atmosphere.
*
* The returned pressure can then be converted to altitude using
* any of the above functions, taking into account the MSL pressure and
* temperature variations.
*
* @param {number} FL Flight Level
* @returns {number}
*/
export function pressureFromFL(FL) {
return pressureFromStandardAltitude(FL * 100 / feetPerMeter, P0);
}
/**
* Convert pressure to Flight Level.
*
* Flight levels are defined as pressure and not as a fixed
* altitude. This means that a flight level can always be converted
* to pressure without needing any other information in a fully
* deterministic way.
*
* A flight level of 115 means 11500 feet measured by barometer
* using the ICAO standard atmosphere.
*
* The returned Flight Level can be a fractional number, this should
* be rounded to the closest integer as there are no fractional flight levels.
*
* @param {number} P pressure
* @returns {number}
*/
export function FLFromPressure(P) {
return altitudeFromStandardPressure(P, P0) * feetPerMeter / 100;
}
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