This problem is about the thermodynamics fundamentals of

global circulation and natural convection in general. We explore the

fundamentals in two parts, in accordance with Fig. P9.5.

(a) Consider a stream of ideal gas with the flow rate

m_ which flows isothermally and reversibly through the system shown in

Fig. P9.5a. The temperature T is constant throughout the system. The inlet and

outlet pressures are Pin and Pout. Invoke the first and

second laws, the ideal gas model, and the isothermal and reversible model and

show that the heat input rate __ and work output rate __ are equal

and

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This problem is about the thermodynamics fundamentals of

global circulation and natural convection in general. We explore the

fundamentals in two parts, in accordance with Fig. P9.5.

(a) Consider a stream of ideal gas with the flow rate

__ which flows isothermally and reversibly through the system shown in

Fig. P9.5a. The temperature T is constant throughout the system. The inlet and

outlet pressures are Pin and Pout. Invoke the first and

second laws, the ideal gas model, and the isothermal and reversible model and

show that the heat input rate Q_ and work output rate __ are equal

and given by

(b) Next, the circulation of the atmosphere can be modeled

as a heat engine that functions in a cycle of four processes (Fig. P9.5b): 12,

isothermal heating and expansion at TH; 23, isobaric cooling at PL;

34, isothermal cooling and compression at TL; and 41, isobaric

heating at PH. The cycle is executed reversibly: There are no

pressure drops from 2 to 3 and from 4 to 1, and locally, there is no

temperature difference between the 23 and 41 streams. The internal

(regenerative) heat transfer __ i occurs across a zero

temperature difference. The heating and expansion process is a model for how

the air warms up and rises to higher altitudes (lower pressures) over

the equatorial zone (TH). The cooling and

compression represent a model for the sinking of the same airstream over the

polar zones (TL). The counterflow formed by the 41 and 23 streams

is a model for the circulation of the atmosphere in the meridional direction.

Use the results of part (a) to calculate the net power

output of the atmospheric heat engine (__ net = __ H

_ WL) and the energy conversion efficiency __ = __ net___

H. Does your resulting expression for __

look familiar? Why?

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