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₊ = Q_ab > 0. Similarly, Q_c(i = T_ f'¹ dS < 0, and I’ll define Q_ = Q_c(/| > 0, the crosshatched area under path cd. Furthermore, for the entire cycle ДE = 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_ _агь, which is the area of the shape dchgd; (ii) Q₊. the area of the rectangle efhg is greater than Q+._ari»> 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