The Rankine-Kirchhoff approximations for moist thermodynamics

Science

Depending on the level of approximations used, thermodynamic equations for moist air (a mixture of dry air and the three phases of water) can range from simple and analytic to complicated and analytically intractable. To guarantee analytic solutions, a typical approach is to approximate the latent enthalpies of evaporation and sublimation as independent of temperature. This effectively assumes that the heat capacities of the three phases of water are constant and equal.

In reality, the heat capacities of the three phases are very different, and they also vary somewhat with temperature. A more accurate approach, which still yields analytic expressions, is to treat the heat capacities of the three phases as different but still independent of temperature. This approach was first used by Gustav Kirchhoff and William Rankine in the mid-1800s to derive a highly accurate analytic expression for the saturation vapor pressure. A lack of awareness of this work has led to it being rediscovered independently many times since. Here, the history of this approach is reviewed and a proposal is made that the trio of underlying approximations (ideal gas, constant heat capacities, and zero-volume condensates) be elevated to a named entity to raise awareness of the universe of accurate and analytic expressions they generate.

Impact

The trio of RK approximations generates accurate and analytic:

• saturation vapor pressure
• equivalent potential temperature
• quantity conserved during adiabatic ascent
• lifting condensation and deposition levels
• dewpoint and frost point.

Summary

A set of accurate and analytic expressions for atmospheric thermodynamics can be obtained using the three approximations first deployed by Rankine and Kirchhoff: ideal gas, constant heat capacities, and zero-volume condensates.