 Thermophysical Data - Maple Help

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ThermophysicalData

Maple 2021 enhances the ThermophysicalData package with reference atmosphere models. These describe how air pressure, temperature, density, and molecular weight typically change with altitude.

 > $\mathrm{restart}:\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{with}\left(\mathrm{ThermophysicalData}\right)$
 $\left[{\mathrm{Atmosphere}}{,}{\mathrm{Chemicals}}{,}{\mathrm{CoolProp}}{,}{\mathrm{PHTChart}}{,}{\mathrm{Property}}{,}{\mathrm{PsychrometricChart}}{,}{\mathrm{TemperatureEntropyChart}}\right]$ (1)

Two models are available—the 1976 US Standard Atmosphere (the default), and the International Standard Atmosphere.Both are similar but are valid between different altitude limits.

Altitude can be specified as geopotential (the default) or geometric. Geopotential altitude adjusts for the variation in gravity with altitude.

 > $\mathrm{Atmosphere}\left(3⟦\mathrm{km}⟧,\mathrm{all},\mathrm{altitudetype}=\mathrm{geopotential}\right)$
 ${268.6500}{}⟦{K}⟧{,}{70108.52215}{}⟦{\mathrm{Pa}}⟧{,}{0.9091219586}{}⟦\frac{{\mathrm{kg}}}{{{m}}^{{3}}}⟧{,}{328.5779005}{}⟦\frac{{m}}{{s}}⟧{,}{0.00001693718731}{}⟦{\mathrm{Pa}}{}{s}⟧$ (2)
 > $\mathrm{Atmosphere}\left(3⟦\mathrm{km}⟧,\mathrm{all},\mathrm{altitudetype}=\mathrm{geometric}\right)$
 ${268.6591984}{}⟦{K}⟧{,}{70121.13970}{}⟦{\mathrm{Pa}}⟧{,}{0.9092544424}{}⟦\frac{{\mathrm{kg}}}{{{m}}^{{3}}}⟧{,}{328.5835256}{}⟦\frac{{m}}{{s}}⟧{,}{0.00001693764616}{}⟦{\mathrm{Pa}}{}{s}⟧$ (3)

Altitude is assumed to be in meters if no units are provided.

 > $\mathrm{Atmosphere}\left(3000,\mathrm{all},\mathrm{altitudetype}=\mathrm{geometric}\right)$
 ${268.6591984}{,}{70121.13970}{,}{0.9092544424}{,}{328.5835256}{,}{0.00001693764616}$ (4)

Here we plot air temperature, viscosity, density, pressure, and speed of sound as functions of altitude.

 > $\mathrm{plotOpts}≔\mathrm{size}=\left[400,400\right],\mathrm{background}="LightGrey",\mathrm{axis}=\left[\mathrm{gridlines}=\left[10,\mathrm{color}=\mathrm{white}\right]\right],\mathrm{titlefont}=\left[\mathrm{Arial},16\right],\mathrm{axesfont}=\left[\mathrm{Arial}\right],\mathrm{labelfont}=\left[\mathrm{Arial},12\right],\mathrm{axes}=\mathrm{boxed},\mathrm{labeldirections}=\left[\mathrm{horizontal},\mathrm{vertical}\right]:$

 > $\mathrm{plot}\left(\left[\mathrm{Atmosphere}\left(x,"temperature"\right),x,x=0..80000\right],\mathrm{labels}=\left["Temperature \left(K\right)","Altitude \left(m\right)"\right],\mathrm{plotOpts}\right)$ > $\mathrm{plot}\left(\left[\mathrm{Atmosphere}\left(x,"viscosity"\right),x,x=0..80000\right],\mathrm{labels}=\left["Viscosity \left(Pa s\right)","Altitude \left(m\right)"\right],\mathrm{plotOpts}\right)$ > $\mathrm{plot}\left(\left[\mathrm{Atmosphere}\left(x,"density"\right),x,x=0..80000\right],\mathrm{labels}=\left["Density \left(kg/m3\right)","Altitude \left(m\right)"\right],\mathrm{plotOpts}\right)$ > $\mathrm{plot}\left(\left[\mathrm{Atmosphere}\left(x,"pressure"\right),x,x=0..80000\right],\mathrm{labels}=\left["Pressure \left(Pa\right)","Altitude \left(m\right)"\right],\mathrm{plotOpts}\right)$ > $\mathrm{plot}\left(\left[\mathrm{Atmosphere}\left(x,"speedofsound"\right),x,x=0..80000\right],\mathrm{labels}=\left["Speed of Sound \left(m/s\right)","Altitude \left(m\right)"\right],\mathrm{plotOpts}\right)$ A skydiver free-falls through the atmosphere. The descent is modeled by this differential equation.

 >

where

 • m is mass
 • g is acceleration due to gravity
 • Cd is the drag coefficient
 • ρ(t) is the air density

Air density ρ varies with altitude.

 > $\mathrm{ρ}≔t\to \mathrm{Atmosphere}:-\mathrm{Property}\left("density",y\left(t\right)\right):$

Define the other parameters:

 > $A≔1:$$m≔85:$$\mathrm{Cd}≔0.58:$$g≔9.8:$

Numerically solve the differential equation.

 >
 ${\mathrm{sol}}{≔}{\mathbf{proc}}\left({\mathrm{x_rkf45_dae}}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end proc}}$ (5)

Plot the altitude vs. time, velocity vs. time, and velocity vs. altitude.

 > $\mathrm{plots}:-\mathrm{odeplot}\left(\mathrm{sol},\left[t,y\left(t\right)\right],t=0..250,\mathrm{labels}=\left["time \left(s\right)","Altitude \left(m\right)"\right],\mathrm{plotOpts}\right)$ > $\mathrm{plots}:-\mathrm{odeplot}\left(\mathrm{sol},\left[t,\mathrm{vel}\left(t\right)\right],t=0..250,\mathrm{plotOpts}\right)$ > $\mathrm{plots}:-\mathrm{odeplot}\left(\mathrm{sol},\left[\mathrm{vel}\left(t\right),y\left(t\right)\right],t=0..250,\mathrm{plotOpts}\right)$ >