Pressure in atmosphere: Difference between revisions

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Ideal gas law
Ideal gas law
<math>P = \frac{\rho}{M} R T</math>, where pressure <math>P</math> is a function of <math>n, T, V</math>, thus <math>P=P(n,T,V)</math> and the hydrostatic assumption <math>dP = - \rho g dz</math> are needed to derive this.
<math>P = \frac{\rho}{M} R T</math>, where pressure <math>P</math> is a function of <math>n, T, V</math>, thus <math>P=P(n,T,V)</math> and the hydrostatic assumption <math>dP = - \rho g dz</math> are needed to derive this.
Assume <math>\frac {dp}{dh}=-\rho(h) g = - \frac{N(h)m}{V}g</math> where <math>N(h)</math> is the number of particles at altitude <math>h</math>. The ideal gas law states that <math>pV = N R T</math>, and thus <math>N = N(h) = \frac{ p(h) V }{RT} </math>. Finally, we have
<math>
\frac{dp}{dh} = -\frac{N(h)m}{V}g = -\frac{p(h) V}{RT}
</math>
----


Assume <math>\frac {dp}{dh}=-\rho(h) g </math> and <math>p = \rho \frac{R}{M}T</math> which gives <math>\rho = \frac{pM}{RT}</math> and we have
Assume <math>\frac {dp}{dh}=-\rho(h) g </math> and <math>p = \rho \frac{R}{M}T</math> which gives <math>\rho = \frac{pM}{RT}</math> and we have
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\frac {dp}{dh}=- g \frac{pM}{RT}
\frac {dp}{dh}=- g \frac{pM}{RT}
</math>
</math>
Actually, both <math>p</math> and <math>T</math> are dependent on altitude <math>h</math>.




Revision as of 16:49, 30 August 2023

Introduction

ISO2533:1975

The case in Toposhere (<10 km).

  • Lapse rate +6.5 °C/km
  • Base temp 19.0 °C
  • Base atmospheric pressure 108,900 Pa equals 1.075 atm
  • Base atmospheric density 1.2985 kg/m3

International Standard Atmosphere

https://en.wikipedia.org/wiki/International_Standard_Atmosphere

Consider the function in GY91's BMP280-3.3, eg at https://startingelectronics.org/tutorials/arduino/modules/pressure-sensor/

Earth's atmosphere's changes in

  • Pressure
  • Temperature
  • Density
  • Viscosity or

Hydrostatic balance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dP}{dh} = -\rho g } The ideal gas law Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P = \rho R T }

Reference atmospheric model

How the ideal gas properties change (mainly) as a function of altitude (etc).

Static atmospheric model

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho = \frac{MP}{RT} } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dP = -g \rho dh} (see above).

Standard atmosphere

Isothermal-barotropic approximation and scale height

Temperature and molecular weight are constant: density and pressure are exponential functions of altitude.

The US standard atmosphere

More realistic temperature function, consisting of eight data points connected by straight lines, which is---of course---an approximation.

https://en.wikipedia.org/wiki/U.S._Standard_Atmosphere

NASA Global Reference Atmospheric Models GRAM

Barometric formula

Models how the pressure of the air changes with altitude with linear temperature change.

Ideal gas law Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P = \frac{\rho}{M} R T} , where pressure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} is a function of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n, T, V} , thus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P=P(n,T,V)} and the hydrostatic assumption are needed to derive this.

Assume and which gives and we have

Actually, both and are dependent on altitude .


Simplified model from Weather.gov

https://www.weather.gov/media/epz/wxcalc/pressureAltitude.pdf

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