1. Lecture 05: Heat Engines in the Atmosphere
ENPh257: Thermodynamics
Lecture 05: Heat Engines in the Atmosphere
© Chris Waltham, UBC Physics & Astronomy, 2018

2. Adiabatic changes For a diatomic gas, 𝛾 > 1 𝑇 𝑉 𝛾−1 =constant
On a PV diagram, adiabatic curves are steeper than isothermal hyperbolae, because 𝛾 > 1.
See Fermi p.27:
Solid lines - isothermal
Dotted lines - adiabatic
© Chris Waltham, UBC Physics & Astronomy, 2018

3. Example: the atmospheric lapse rate
Convection currents in the troposphere continually transport air from sea level to higher altitudes and vice versa. The air is largely heated from below, as it is fairly transparent to solar radiation. Air is a poor conductor, so little heat is transferred between layers, and the transformations as air rises or falls can be considered adiabatic. First, we need to look at some hydrostatics.
© Chris Waltham, UBC Physics & Astronomy, 2018

4. The atmosphere How and why does it change with altitude? Hydrostatics
Consider a horizontal slab of air of thickness 𝑑𝑧, where 𝑧 is the altitude above the Earth’s surface. In a stable air mass the pressure below must be balanced by the weight of the slab and the pressure above. Consider a slab of air, area 𝐴, thickness Δ𝑧, with pressure below 𝑃 0 and pressure above, 𝑃 1 : Δ𝑧
© Chris Waltham, UBC Physics & Astronomy, 2018

5. The atmosphere
The weight of the air, 𝜌ΑΔ𝑧𝑔 must be balanced by the upward buoyant force 𝑃 1 − 𝑃 0 𝐴= 𝐴𝛥𝑃
𝑑𝑃 𝑑𝑧 = −𝜌𝑔
This is the barometric equation. Apply the gas laws:
𝑑𝑃 𝑑𝑧 = − 𝑚𝑔 𝑅𝑇 𝑃
Here 𝑚 is the mean molar mass of air.
© Chris Waltham, UBC Physics & Astronomy, 2018

6. Uniform temperature approximation
If the temperature is constant with altitude (not a crazy approximation in absolute units):
𝑃 𝑧 =𝑃 0 exp⁡(− 𝑚𝑔𝑧 𝑅𝑇 )
For a mean surface temperature of 288 K, this gives a scale height 𝑅𝑇/𝑚𝑔 = 8.4 km.
So at 11 km, typical cruising altitude for an airliner, the pressure would be:
𝑃 𝑧 =𝑃 0 exp⁡(− ) = 0.27 𝑃 0 = 27 kPa (in reality the mean value here is 23 kPa).
Note: ~ 40 kPa (6000 m) provides the absolute minimum partial pressure of oxygen to survive.
Not too bad, but we can do better…
© Chris Waltham, UBC Physics & Astronomy, 2018

7. Dry adiabatic atmosphere
At some critical value of 𝑑𝑇/𝑑𝑧 air becomes buoyant and remains so even as it cools adiabatically into the lower pressure environment aloft.
The relationship between temperature and pressure during an adiabatic change is:
𝑃 1 𝛾 −1 𝑇=constant
Differentiate:
𝑑𝑇 𝑑𝑃 = 𝑇 𝑃 1− 1 𝛾
© Chris Waltham, UBC Physics & Astronomy, 2018

8. Dry adiabatic atmosphere
Applying this to the barometric equation gives the dry adiabatic lapse rate, Γ𝑑:
Γ𝑑 = 𝑑𝑇 𝑑𝑧 = 𝑑𝑇 𝑑𝑃 𝑑𝑃 𝑑𝑧 =−𝜌𝑔 𝑇 𝑃 (1− 1 𝛾 )
We know 𝜌=𝑚𝑛/𝑉 and 𝑛=𝑃𝑉/(𝑅𝑇):
Γ𝑑 = 𝑑𝑇 𝑑𝑃 𝑑𝑃 𝑑𝑧 =− 𝑚𝑔 𝑅 (1− 1 𝛾 )
Here 𝑚 is the mean molar mass of air, ≈ 28.9
The result is about 10 K/km (and constant, so it can only be an approximation). The value is the right order of magnitude, but plainly bigger than reality; the top of Grouse Mountain is not 10 C cooler than the city.
© Chris Waltham, UBC Physics & Astronomy, 2018

9. Moist adiabatic atmosphere
The “International Standard Atmosphere” (shown) has a lapse rate of 6.5 K/km.
Local lapse rates vary with temperature and RH – can occasionally be negative (“Inversion”).
The air temperature reaches dew point: cloud formation, release of latent heat, which lowers the lapse rate.
Big subject, no time:
Climate science
Weather forecasting
Aviation, etc. etc.
and
F. W. Taylor in Elementary Climate Physics (Oxford, 2005) p.58.
© Chris Waltham, UBC Physics & Astronomy, 2018

10. Hurricanes as heat engines
Hurricane Harvey, 2017
G$125 damage
© Chris Waltham, UBC Physics & Astronomy, 2018

11. Hurricanes as heat engines
Initiated by ocean temperatures > 26 C to a depth of 60 m.
Moist air starts to rise at B
Air dragged in from A, latent heat added isothermally.
Coriolis force starts it spinning; heat is concerted to kinetic energy (work).
Air cools adiabatically as it rises.
Radiative cooling C to D (isothermal).
Air falls and cools isothermally.
“Super-Carnot” efficiency as heat source boosted by frictional heating.
Contour colours indicate entropy density.
© Chris Waltham, UBC Physics & Astronomy, 2018
Kerry Emanuel, Hurricanes: Tempests in a Greenhouse, Physics Today, August 2006.

12. Hurricanes as heat engines
A simple calculation:
For 𝑇1 = 200 K and 𝑇2 = 300 K, Carnot efficiency 𝜂 = 1/3.
Saturated vapour pressure of water at 27 C is about 3 kPa, 3% of standard air pressure, about 2% of air by mass.
Latent heat of vaporization of water is 2.4 MJ/kg, so latent heat available in this air is 48 kJ/kg.
With a Carnot efficiency of 1/3, 16 kJ/kg is available for kinetic energy.
Equate to ½ 𝑚𝑣2 gives 𝑣 = 175 m/s. Ignoring surface friction.
Highest recorded is 96 m/s at surface (Hurricane Patricia, 2015)
87.2 kPa for Hurricane Patricia
© Chris Waltham, UBC Physics & Astronomy, 2018