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## Thermodynamic Power Cycles: A Primer

The theoretical basis for a vast majority of the electricity generated worldwide is provided by the thermodynamic power cycles that govern the conversion of heat into mechanical work by heat engines. These power conversion cycles (PCSs) that are at the heart of fossil and nuclear power plants are also vital for electricity generation from renewable energy sources—geothermal, solar thermal, ocean thermal, and biomass. A brief overview of some of the power cycles of relevance is presented below.

### B.1 Carnot Cycle

The Carnot cycle is the ideal thermodynamic process proposed by the French physicist Sadi Carnot in the early 19th century. He proposed a cyclic process consisting of two isothermal and two adiabatic steps operating between two energy reservoirs at different temperatures, as shown in Figure B.1 through the pressure–volume (Pv) and entropy–temperature (Ts) diagrams. Any substance that can undergo a cyclic process can function as the working fluid for the cycle, and the efficiency of the cycle is independent of the working fluid, as formalized by the Carnot theorem : Figure B.1 Carnot cycle. Two graphs are shown. The cycle has four stages and they are numbered 1, 2, 3, and 4. The transition from one phase to another are given as follows. 1 to 2 - isothermal expansion. 2 to 3 - isentropic (adiabatic) expansion. 3 to 4 - isothermal compression. 4 to 1 - isentropic (Adiabatic) compression. Graph a: p-v diagram. The horizontal axis represents the volume, while the vertical axis represents the pressure. A rhombus-shaped plot with slightly curved edges is drawn. At point 1, the pressure is P1 and volume is V1. Similarly, at points 2, 3, and 4, the pressures are P2, P3, and P4 respectively; the volumes are V2, V3, and V4 respectively. The topmost vertex is 1, where pressure is the highest and volume is the lowest. The next is the vertex 2, where the volume is second highest value and pressure is also second highest value. The next vertex is 4, where both volume and pressure are second lowest. The vertex 3 is where the volume is the highest, but pressure is the lowest. The flow is from 1 to 2, 2 to 3, 3 to 4, and 4 to 1. Graph b: T-s diagram. The horizontal axis represents S and the vertical axis represents T. A rectangle is drawn with vertices marked 1, 2, 3, and 4 - top left, top right, bottom right, and bottom left respectively. The coordinates of these points are as follows. 1 (S1, S4), (T1, T2). 2 (S2, S3), (T1, T2). 3 (S2, S3), (T3, T4). 4 (S1, S4), (T3, T4).

All reversible cyclic heat engines operating between two fixed temperatures have the same efficiency.

The utility of the Carnot cycle, an ideal construct, is that it places an upper bound on the efficiency of a heat engine operating between two temperatures.

The heat input to the cycle occurs in the step 1–2, while heat rejection occurs during step 3–4. The efficiency of the engine shown in the above cycle—the Carnot efficiency—is given by equation B.1:

### B.2 Brayton Cycle

The power generation in the gas turbine power plants is based on the Brayton cycle, named in honor of George Brayton, an American engineer. The cycle, which is also sometimes referred to as the Joule cycle, involves isobaric steps rather than isothermal steps, as in the Carnot cycle. The Pv and Ts diagrams for the Brayton cycle are shown in Figure B.2. Figure B.2 Brayton cycle. Two graphs are shown. The cycle has four stages and they are numbered 1, 2, 3, and 4. The transition from one phase to another are given as follows. 1 to 2: isentropic (Adiabatic) compression. 2 to 3: isobaric heat addition. 3 to 4: isentropic (Adiabatic) expansion. 4 to 1: isobaric heat rejection. Graph a: p-v diagram. The horizontal axis represents the volume, while the vertical axis represents the pressure. A plot comprised of two horizontal lines, one connecting 2 and 3 at the top, while the other connecting 1 and 4 at the bottom is shown. The points 1 and 2 and 3 and 4 are connected using two decreasing curves. Volume is the lowest at 2, second lowest is 1, second highest is at 3, and highest at 4. Pressure is highest at 2 and 3; lowest at 1 and 4. Graph b: T-s diagram. The horizontal axis represents S and the vertical axis represents T. The plot is drawn using two vertical lines connecting the points pairs: 2 and 1, and 3 and 4. 2 and 3 and 1 and 4 are connected using 2 increasing curves. T is highest at 3, second highest is 4, third is 2, and lowest at 1. S is highest at 3 and 4; lowest at 1 and 2.

The Brayton cycle may be used for power generation in solar thermal systems.

### B.3 Rankine Cycle

The Rankine cycle (as well as the Rankine absolute temperature scale) is named in honor of William John Macquorn Rankine, a Scottish engineer and physicist. It is the base cycle for vapor–liquid systems, such as the steam power plant . The basic steps are the same as in the Brayton cycle; however, as the working fluid is a condensing vapor, the representation on the P–v and T–s diagrams is quite different, as shown in Figure B.3. The curve in each figure represents the boundary between phases, with the region under the dome representing a mixture of vapor and liquid. Point 3 in the figure is located in the superheated vapor region. Figure B.3 Rankine cycle. Two graphs are shown. The cycle has four stages and they are numbered 1, 2, 3, and 4. The transition from one phase to another are given as follows. 1 to 2: isentropic (Adiabatic) compression. 2 to 3: isobaric heat addition. 3 to 4: isentropic (Adiabatic) expansion. 4 to 1: isobaric heat rejection. Graph a: p-v diagram. The horizontal axis represents V and the vertical axis represents P. A plot with two horizontal lines and two decreasing curves connecting four points is shown. The points are 1, 2, 3, and 4 - bottom left, top left, top right, and bottom right respectively. The pressure is highest at points 2 and 3 and lowest at 1 and 4. The volume is highest at points 3 and 4 and lowest at 1 and 2. A bell shaped curve is drawn, where the peak point is labeled critical point. Graph b: T-s diagram. The horizontal axis represents S and the vertical axis represents T. The following is plotted on the graph. Point 1 is where S and T are the lowest. (S1, T1). From there a vertical line is drawn leading to point 2. Point 2, S values remains to be the same marked as S2, while there is a rise in temperature marked as T2. From there, an increasing line is drawn, followed by a horizontal line, and again an increasing line leading to point 3. At point 3, both S and T are maximum, marked S3 and T3 respectively. From there a vertical line is drawn downward leading to point 4. At point 4, S still remains the same as S3, marked as S4. T has fallen equal to T1, marked as T4. A bell-shaped curve is drawn. The peak point is labeled critical point.

The Rankine cycle using steam as the working fluid may be used in solar thermal systems, as well as some geothermal systems. The temperatures in fossil thermal plants may be as high as ~550°C. The temperatures available in most geothermal systems and ocean thermal systems are generally too low to use steam as the working fluid, necessitating the use of organic working fluids, as mentioned in Chapter 5. Some of the relevant physical properties of a few working fluids for power cycles are shown in Table B.1.

### B.4 Kalina Cycle

Ammonia is commonly used in OTEC Rankine cycles for power production. However, the low temperatures (25°C or less) translate to poor thermal efficiencies of ~3%. Alexander Kalina, an American scientist and engineer, devised in the late 20th century a cycle that utilizes a binary mixture, i.e., ammonia–water mixture, to minimize energy losses associated with the phase changes . The Ts diagram for a simplified Kalina cycle is shown in Figure B.4 . The steps occurring in the cycle are listed below:

1. The base ammonia solution (point 1) is heated and partially vaporized, reaching point 2, whereupon it separates into an ammonia-rich vapor (point 3) and an ammonia-poor liquid (point 5).

2. The ammonia-rich vapor (point 3) expands through a turbine, generating electricity and condensing partially to point 4.

3. The ammonia-poor liquid (point 5) is passed through a heat recovery unit and then depressurized (points 6 and 7).

4. The two streams (points 4 and 7) are recombined to obtain the base mixture (point 8).

5. The base mixture is then further subjected to heat recovery and cooling to reach point 9.

6. Further heating of the base solution results in reaching the initial starting point 1 of the cycle. Figure B.4 Kalina cycle. In the graph, the plot of T against S is drawn. The function involves a total of 8 phases and the flow between them are depicted. Three saturation curves are drawn, ammonia base solution, ammonia rich solution, and ammonia poor solution. All three are bell-shaped, with values falling in the mentioned order of names.

These changes can result in increasing the efficiency of power generation, possibly to as much as 5%.

The Kalina cycle was modified by Professor Haruo Uehara (known as the father of Japanese OTEC) in 1999 by incorporating two-stage turbines and improved heat recovery steps, increasing the power conversion efficiency by a further 10% [4, 5]. Further refinements of these cycles are under research and development.

### B.5 Summary

As seen from equation B.1, the maximum efficiency of power conversion is limited by the difference between the temperatures of the two reservoirs—heat source and heat sink. The temperatures available in many of the primary renewable sources—geothermal and ocean thermal—are too low for the established, conventional power cycles to operate efficiently, or even make it impossible to implement them. Innovative cycles that seek to maximize the power conversion efficiency through modifying the working fluids and refining heat recovery networks are being developed all over the world. The mathematical formulations of these, as well as conventional power cycles, are available in the literature.

### References

1. Goecken, N. A., and R. G. Reddy, Thermodynamics, Second Edition, Plenum Press, New York, 1996.

2. Huang, F. E., Engineering Thermodynamics: Fundamentals and Applications, Second Edition, Macmillan, New York, 1988.

3. Liu, W., et al., “A Review of Research on the Closed Thermodynamic Cycles of Ocean Thermal Energy Conversion,” Renewable and Sustainable Energy Reviews, Vol. 119, 2020, #109581, 11 pages.

4. Wang, J., et al., “Assessment of Off-Design Performance of a Kalina Cycle Driven by Low-Grade Heat Source,” Energy, Vol. 138, 2017, pp. 459–472.

5. Goto, S., et al., “Construction of Simulation Model for OTEC Plant Using Uehara Cycle,” Electrical Engineering in Japan, Vol. 176, No. 2, 2011, pp. 1–13.

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