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Inequality of Clausius

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Equations

(Eq1)     ∮ δQ T   ≤  0

Inequality of Clausius

Nomenclature

Q	Heat
QH	Heat associated with high-temperature reservoir
QL	Heat associated with low-temperature reservoir
T	Temperature
TH	Temperature of high-temperature reservoir
TL	Temperature of low-temperature reservoir
Wrev	Reversible work

Details

The inequality of Clausius should be established before consideration of the property entropy. The inequality of Clausius is:

(Eq1)

∮ δQ T   ≤ 0

The inequality of Clausius is a corollary of the second law of thermodynamics. It will be demonstrated to be valid for all possible cycles, including both reversible and irreversible heat engines and refrigerators. Since any reversible cycle can be represented by a series of Carnot cycles, in this analysis a Carnot cycle that leads to the inequality of Clausius only needs to be considered.

Consider first a reversible (Carnot) heat engine cycle operating between reservoirs at temperatures T_H and T_L, as shown in Figure 1:

Figure 1 Reversible heat engine cycle.

Figure 2 Reversible refrigeration cycle.

For this cycle, the cyclic integral of the heat transfer ∮δQ is greater than zero:

∮	δQ = QH − QL < 0

Since T_H and T_L are constant, from the definition of the absolute temperature scale and from the fact this is a reversible cycle, it follows that:

∮

δQ T

=

QH TH

−

QL TL

=  0

If ∮δQ, the cyclic integral of δQ, approaches zero (by making T_H approach T_L) and the cycle remains reverrsible, the cyclic integral of δQ/T remains zero. Thus, it can be concluded that for all reversible heat engine cycles:

∮

δQ ≥ 0

and:

∮

δQ T

=  0

Now consider an irreversible cyclic heat engine operating between the same T_H and T_L as the reversible engine of Figure 1 and receiving the same quantity of heat Q_H. Comparing the irreversible cycle with the reversible one, it is concluded from the second law that:

W_irr < W_rev

Since Q_H − Q_L = W for both the reversible and irreversible cycles, it is concluded that:

Q_H  − Q_L,irr < Q_H  − Q_L,rev

and therefore:

Q_L,irr > Q_L,rev

Consequently, for the irreversible cyclic engine:

∮	δQ  =  QH − QL,irr > 0

and:

∮

δQ T

=

QH TH

−

QL,irr TL

<  0

Suppose that the engine were to become more and more irreversible, but Q_H, T_H, and T_L were kept fixed. The cyclic integral of δQ then approaches zero, and that for δQ/T becomes a progressively larger negative value. In the limit, as the work output goes to zero:

∮

δQ = 0

and:

∮

δQ T

<  0

Thus, it is concluded that for all irreversible heat engine cycles:

∮

δQ ≥ 0

and:

∮

δQ T

<  0

To complete the demonstration of the inequality of Clausius, similar analyses must be performed for both reversible and irreversible refrigeration cycles. For the reversible refrigeration cycle shown in Figure 2:

∮	δQ = −QH + QL < 0

and:

∮

δQ T

=  −

QH TH

+

QL TL

=  0

As the cyclic integral of δQ approaches zero reversibly (T_H approaches T_L), the cyclic integral of δQ/T remains at zero. In the limit:

∮

δQ = 0

and:

∮

δQ T

=  0

Thus, for all reversible refrigeration cycles:

∮

δQ ≤ 0

and:

∮

δQ T

=  0

Finally, let an irreversible cyclic refrigerator operate between temperatures T_H and T_L and receive the same amount of heat Q_L as the reversible refrigerator of Figure 2. From the second law, it is concluded that the work input will be greater for the irreversible refrigerator, or:

W_irr > W_rev

Since Q_H − Q_L = W for each cycle, it follows that:

Q_H,irr − Q_L > Q_H,rev − Q_L

and therefore:

Q_H,irr > Q_H,rev

That is, the heat rejected by the irreversible refrigerator to the high-temperature reservoir is greater than the heat rejected by the reversible refrigerator. Therefore, for the irreverrsible refrigerator:

∮	δQ  =  −QH,irr + QL < 0

and:

∮

δQ T

=  −

QH,irr TH

+

QL TL

<  0

As the engine is made progressively more irreversible, while keeping Q_L, T_H, and T_L constant, the cyclic integrals of δQ and δQ/T both become larger in the negative direction. Consequently, a limiting case as the cyclic integral of δQ approaches zero does not exist for the irreversible refrigerator. Thus, for all reversible refrigeration cycles:

∮

δQ < 0

and:

∮

δQ T

<  0

Summarizing, it is noted that, in regard to the sign of ∮δQ, all possible reversible cycles have been considered (i.e., ∮δQ > 0 and ∮δQ < 0), and for each of these reversible cycles:

∮

δQ T

=  0

Also, all possible irreversible cycles for the sign of ∮δQ (that is, ∮δQ > 0 and ∮δQ < 0) have been considered, and for all these irreversible cycles:

∮

δQ T

<  0

Thus, for all cycles it can be written:

(Eq1)

∮ δQ T   ≤  0

where the equality holds for reversible cycles and the inequality for irreversible cycles. This relation, Eq1, is known as the inequality of Clausius.