# EXAMPLES OF CYCLES

Figure 8.4: Clockwise reversible Carnot cycle: (a) PV plot, (b) TS plot. The crosshatched area is *Q* ; the shaded + crosshatched areas equal *Q _{+}.*

in Fig. 8.4, accomplished in principle by the steps shown in Fig. 2.14. It is apparent that the TS plot is much simpler that the PV plot, involving only horizontal and vertical paths that make a rectangle for *any* working substance.

Along the Carnot cycle’s isothermal paths *ab* and *cd,* the heating energy gained by the working substance is *dQ = TdS,* and applying this to path *ab, **Qab **= T _{+}f°dS >* 0, which is the area under the horizontal line

*ab*in Fig. 8.4(b). This is the sum of the areas of the shaded and crosshatched rectangles. I’ll adopt the notation

*Q*0. Similarly,

_{+}= Q_{a}b >*Q*= T_

_{c(}i*f'*0, and I’ll define Q_ = Q

^{1}dS <_{c(}/| > 0, the crosshatched area under path

*cd.*Furthermore, for the entire cycle Д

*E =*0, so

*AE = Q*—

_{+}*Q_*— И(

_{)у}= 0, and the work by the working substance per cycle is li’by =

*Q*— Q_ (shaded area of

_{+}*abed).*

Key Point 8.3 *On the TS diagram, the heating input is the area under path ab and the heating output is the crosshatched area under path cd. The shaded areas on the PV and TS diagrams represent the net work done per cycle.*

Reversible Carnot Counterclockwise Cycles. The reversible, clockwise Carnot cycle can be run in the reverse direction, which reverses the path arrows, as in Fig. 8.5(a) and (b). Heating energy is gained *by the system *at the lower temperature T_ and *ejected from the system* at the higher temperature *T _{+}:* i.e., the lower temperature reservoir loses energy while the upper temperature reservoir gains energy.

The reversible counterclockwise (CCW) cycle has the characteristics of a refrigerator, whose working substance (refrigerant) takes in energy from the refrigerator compartment at low temperature and rejects it to the kitchen at a higher temperature. It is similarly characteristic of an air conditioner, which removes energy from the space being cooled and typically ejects energy to the warmer outdoors. In addition, such a cycle models a heat pump that removes energy from cool *outdoor* air and delivers energy to the higher-temperature *indoor* space being heated. These are explored in more detail in Sec. 8.5.

For the CCW Carnot cycle, it might appear that *heat* energy flows *uphill, *from lower to higher temperature. That is not true. The temperature rise of

Figure 8.5: Counterclockwise Reversible Carnot cycle: (a) PV plot, (b) TS plot. The crosshatched area = *Q.* : the shaded + crosshatched areas = *Q _{+}.*

the working substance (refrigerant), comes from the adiabatic compression *work* process *da* in Fig. 8.4. That is a *work* process, NOT a heat process! In the CCW Carnot cycle, the heating energy added *to* the working substance from the T_ reservoir keeps the temperature constant as the gas expands.

## Efficiency measures

Thermal Efficiency of Heat Engines. Heat engines generate work to perform a desired task, using energy from a high temperature reservoir. A common measure of efficiency for a heat engine is the ratio of the work per cycle, li),_{v}, *by* the working substance to the energy input *from* the high temperature reservoir:

The reversible Carnot heat engine is important for several reasons. First, it uses two reservoirs, which simulate high and low temperature regions in steam engines and electricity generating plants. Second, there are two adiabatic segments, so all entropy changes occur along the isothermal segments. Third, as for all cycles, there is zero entropy change of the working substance during each cycle, so all net changes occur in the environment .

For the reversible Carnot heat engine, the entropy changes of the high and low temperature reservoirs are A.S'_{+} = *—Q _{+}/T_{+}* and A.S'_ = Q_/T_ respectively. The entropy change of the universe per cycle is

Thus, *Q-/Q+ =* T_/T_{+} and

Figure 8.6: An arbitrary *reversible* clockwise cycle *abcda.* An infinite number of reservoirs heat the working substance along the upper path *ab.* Along the lower path *cd,* energy is sent to an infinity of lower temperature reservoirs. The latter paths are separated by reversible adiabatic paths *be* and *da.* (Adapted from H. S. Leff, “Reversible & irreversible heat engine & refrigerator cycles,” Am. J. Phys. 86, 344-353 (2018) with permission of the American Association of Physics Teachers.)

If the Carnot cycle is *irreversible,* then Eq. (8.4) is replaced by *AS _{univ}* = Q-/T- —

*Q+/T+*> 0, and Eq. (8.5) is replaced by

The efficiency inequality (8.6) holds also for the large class of *reversible *cycles in Fig. 8.6. The highest and lowest temperatures are *T _{+}* and T_. In the temperature-entropy diagram, Fig. 8.6: (i)

*Q- .*the area of rectangle

*ijhgi,*is less than Q_

_{аг}ь, which is the area of the shape

*dchgd*; (ii)

*Q*the area of the rectangle

_{+}.*efhg*is greater than Q+.

_{ar}i»> the area of shape

*abhga.*Thus,

*Q-*

*<*

*Q-,*arb and

*Q+*

*>*<5+,arb- It follows that

*Q-.arb/Q+,arb*

*>*

*Q-/Q+*

*=*T_/T

_{+}and thus,

Key Point 8.4 *Carnot’s major achievement was finding that the reversible Carnot cycle has the highest possible efficiency,* ?/_{car} = 1 — T_*/Т+ for any heat engine cycle operating between reservoirs with temperatures* (T_,T_{+}).

Coefficient of Performance for Heating and Cooling. Referring to Fig. 8.5, several points should be clarified:

- 1. Counterclockwise cycles can be used for heating, cooling or both. If used for heating the specific desired task is to deliver energy
*Q*to the high temperature region. In contrast, for cooling, the desired task is to remove energy_{+}*Q -*from the lower temperature region. - 2. To accomplish either heating or cooling, the counterclockwise cycle requires positive external work on the working substance each cycle.

This is reflected in Fig. 8.6 by the fact that the work along the counterclockwise path dcba is negative (the working substance does negative work). Another view is that the magnitude of the negative work along path *dab* in Fig. 8.5(a) exceeds the positive work along *bed: *also the magnitude of the negative *Q _{+}* exceeds the positive

*Q-*in Fig. 8.5(b).

3. Recognising that external work is needed to perform these tasks, we define the positive external work

Given the possibility of performing heating or cooling with counterclockwise cycles that require such external work, it is common to define a *heating coefficient of performance* for a “heat pump” by

The heating COP is discussed further in Sec. 8.5. The cooling *coefficient of performance* for a refrigerator or air conditioner is defined by