A comparison between Carnot and steam cycles, and their implications for moist convection

Convection acts as heat engine that produces kinetic energy by transporting energy from the Earth's surface to the colder troposphere. Previous studies (Pauluis and Held 2002, Goody 2003) have shown that the work done in a moist atmosphere is significantly lower than the amount of work that would have been produced in an atmosphere without water vapor. This paper further explore how water vapor impacts the atmosphere ability to generate kinetic energy by contrasting two idealized thermodynamic cycles: the Carnot cycle and the steam cycle.

The Carnot cycle is probably the best known example of heat engine. It is composed of four distinct steps: (1) isothermal heating at warm temperature; (2) adiabatic expansion; (3) isothermal cooling at constant temperature; and (4) adiabatic compression. These four transformations combined are associated with a energy transport from a warm source to a cold sink, and result in a net production of mechanical work. The mechanical efficiency of the cycle is defined as the ratio of the work produced to the net heating at the warm source. The efficiency of the Carnot cycle is the highest that can be achieved by a close cycle and is given by temperature difference between the energy source divided by the temperature of the warm source.

The steam cycle is an idealized thermodynamic cycle introduced by Pauluis (2009). It corresponds to a Carnot cycle in which the isothermal heating and cooling are replaced by isothermal moistening and drying. Hence, the energy source in the steam cycle is associated with the latent heating of vaporization of the water vapor. The efficiency of the steam cycle depends on the relative humidity through the cycle, and is always less or equal to the Carnot efficiency. The maximum Carnot efficiency can only be achieved for a saturated cycle.

The reduced mechanical output of the steam cycle is then discussed in terms of both the Gibbs free energy. The Gibbs free energy of water vapor is primarily a function of its relative humidity. In an unsaturated steam cycle, water vapor is injected at low relative humidity and low Gibbs energy, and is removed at higher relative humidity and Gibbs free energy. As a result, the addition and removal of water through the steam cycle corresponds to an export of Gibbs free energy, which reduces the ability of the system to produce mechanical work.

Finally, it is shown that the buoyancy flux in non-precipitating convection can be expressed as the work done by a mixed Carnot-steam cycle. In particular, the work done by shallow convection depends on four key parameters: the total energy transport, the depth of convection measured by the temperature difference between the surface and the top of the convective layer, the Bowen ratio (the ratio of the surface sensible heat flux to the latent heat flux); and the relative humidity through the cycle.

References

Goody, R., 2003: On the mechanical efficiency of deep, tropical convection. J. Atmos. Sci., 59, 2287–2832.

Pauluis, O., 2009: Water vapor and mechanical work: comparison between Carnot and steam cycles. Submitted to J. Atmos. Sci. Manuscript available at http://math.nyu.edu/~pauluis/Olivier_Pauluis_Homepage/Bibliography.html

Pauluis, O. and I. M. Held, 2002a: Entropy budget of an atmosphere in radiative-convective equilibrium. Part I: Maximum work and frictional dissipation. J. Atmos. Sci., 59, 125–139.