Introduction to the Carnot cycle (Khan Academy, 21min). The Carnot cycle is an idealized conceptual process in the sense that it provides the maximum possible fractional conversion of heat into work (aka. thermal efficiency, η_{θ}). Note that Khan uses the absolute value when referring to quantities of heat and work so his equations may look a little different from ours. By systematically adding up the heat and work increments through all stages of the process, we can infer an approximate equation for thermal efficiency (Khan Academy, 14min) The steps are isothermal and reversible expansion, adiabatic and reversible expansion, isothermal and reversible compression, and adiabatic/reversible compression. We know how to compute the heat and work for ideal gases of each step based on Chapter 2. In this presentation by KhanAcademy, an additional proof is required (17min) to show that the volume ratio during expansion is equal to the volume ratio during compression. (Note that the presentation by KhanAcademy uses arbitrary sign conventions for heat and work. They prefer to change the sign to minimize the use of negative numbers but it doesn't always work out.) When we put it all together, the equation we get for "Carnot efficiency" is remarkably simple: η_{θ} = (T_{H} - T_{C})/TH, where T_{H}is the hot temperature and T_{C }is the cold temperature. We can use this formula to quickly estimate the thermal efficiency for many processes. We will show in Chapter 5 that this formula remains the same, even when we use working fluids other than ideal gases (e.g. steam or refrigerants).

Comprehension Questions: 1. Should we express temperature in Kelvins or Celsius when calculating the Carnot efficiency? Explain. 2. What value of TC would be necessary to achieve 100% efficiency, even for this idealized, maximally efficient process? Explain. 3. Why is it impractical to reject heat at the value of T_{C }discussed in Question 2 above? What is a more practical temperature for rejecting heat? (Hint: what geographical feature is very closely located near most nuclear power plants? "Geographical features" might include mountains, desserts, large bodies of water, forests, ...) 4. What value of T_{H}would be necessary to approach 100% efficiency, even for this idealized, maximally efficient process? What are the practical limitations on T_{H}? Explain. 5. How can the formula for Carnot efficiency help us to calculate the "lost" work in the presence of a temperature gradient?

Heat Engine Introduction (LearnChemE.com, 6min) introduction to Carnot heat engine and Rankine cycle. The Carnot cycle is an idealized conceptual process in the sense that it provides the maximum possible fractional conversion of heat into work (aka. thermal efficiency, η_{θ}). But it is impractical for several reasons as discussed in the video. When operating on steam as the working fluid, as is common in nuclear power plants, coal fired power plants, and concentrated solar power plants, the Rankine cycle is much more practical, as explained here. This LearnChemE video is short and sweet, but it applies the property of entropy, which is not introduced until Chapter 4. All you need to know about entropy at this stage is that the change in entropy is zero for an adiabatic and reversible process and the change in entropy is greater than zero when you add heat or cause irreversibility. Since entropy is a state function, we can use the steam tables to facilitate accounting for inefficiencies. Entropy becomes essential when using steam as the working fluid because working out ∫PdV of steam is much more difficult than for an ideal gas. We reiterate this video in Chapter 5, where we discuss calculations for several practical cyclic processes.

Comprehension Questions: 1. Why is the Carnot cycle impractical when it comes to running steam through a turbine? How does the Rankine cycle solve this problem? 2. Why is the Carnot cycle impractical when it comes to running steam through a pump? How does the Rankine cycle solve this problem? 3. It is obvious which temperatures are the "high" and "low" temperatures in the Carnot cycle, but not so much in the Rankine cycle. The "boiler" in a Rankine cycle actually consists of "simple boiling" where the saturated liquid is converted to saturated vapor, and superheating where the saturated vapor is raised to the temperature entering the turbine. When comparing the thermal efficiency of a Rankine cycle to the Carnot efficiency, should we substitute the temperature during "simple" boiling, or the temperature entering the turbine into the formula for the Carnot efficiency? Explain.

## Comments

Elliott replied on Permalink

## Carnot Cycles

Introduction to the Carnot cycle (Khan Academy, 21min). The Carnot cycle is an idealized conceptual process in the sense that it provides the maximum possible fractional conversion of heat into work (aka. thermal efficiency,

η). Note that Khan uses the absolute value when referring to quantities of heat and work so his equations may look a little different from ours. By systematically adding up the heat and work increments through all stages of the process, we can infer an approximate equation for thermal efficiency (Khan Academy, 14min) The steps are isothermal and reversible expansion, adiabatic and reversible expansion, isothermal and reversible compression, and adiabatic/reversible compression. We know how to compute the heat and work for ideal gases of each step based on Chapter 2. In this presentation by KhanAcademy, an additional proof is required (17min) to show that the volume ratio during expansion is equal to the volume ratio during compression. (Note that the presentation by KhanAcademy uses arbitrary sign conventions for heat and work. They prefer to change the sign to minimize the use of negative numbers but it doesn't always work out.) When we put it all together, the equation we get for "Carnot efficiency" is remarkably simple:_{θ}η= (_{θ}T)/_{H}- T_{C}TH, whereTis the hot temperature and_{H}Tis the cold temperature. We can use this formula to quickly estimate the thermal efficiency for many processes. We will show in Chapter 5 that this formula remains the same, even when we use working fluids other than ideal gases (e.g. steam or refrigerants)._{C }Comprehension Questions:

1. Should we express temperature in Kelvins or Celsius when calculating the Carnot efficiency? Explain.

2. What value of TC would be necessary to achieve 100% efficiency, even for this idealized, maximally efficient process? Explain.

3. Why is it impractical to reject heat at the value of

Tdiscussed in Question 2 above? What is a more practical temperature for rejecting heat? (Hint: what geographical feature is very closely located near most nuclear power plants? "Geographical features" might include mountains, desserts, large bodies of water, forests, ...)_{C }4. What value of

Twould be necessary to approach 100% efficiency, even for this idealized, maximally efficient process? What are the practical limitations on_{H}T? Explain._{H}5. How can the formula for Carnot efficiency help us to calculate the "lost" work in the presence of a temperature gradient?

Elliott replied on Permalink

## Carnot vs. Rankine Cycles

Heat Engine Introduction (LearnChemE.com, 6min) introduction to Carnot heat engine and Rankine cycle. The Carnot cycle is an idealized conceptual process in the sense that it provides the maximum possible fractional conversion of heat into work (aka. thermal efficiency,

η). But it is impractical for several reasons as discussed in the video. When operating on steam as the working fluid, as is common in nuclear power plants, coal fired power plants, and concentrated solar power plants, the Rankine cycle is much more practical, as explained here. This LearnChemE video is short and sweet, but it applies the property of entropy, which is not introduced until Chapter 4. All you need to know about entropy at this stage is that the change in entropy is zero for an adiabatic and reversible process and the change in entropy is greater than zero when you add heat or cause irreversibility. Since entropy is a state function, we can use the steam tables to facilitate accounting for inefficiencies. Entropy becomes essential when using steam as the working fluid because working out ∫PdV of steam is much more difficult than for an ideal gas. We reiterate this video in Chapter 5, where we discuss calculations for several practical cyclic processes._{θ}Comprehension Questions:

1. Why is the Carnot cycle impractical when it comes to running steam through a turbine? How does the Rankine cycle solve this problem?

2. Why is the Carnot cycle impractical when it comes to running steam through a pump? How does the Rankine cycle solve this problem?

3. It is obvious which temperatures are the "high" and "low" temperatures in the Carnot cycle, but not so much in the Rankine cycle. The "boiler" in a Rankine cycle actually consists of "simple boiling" where the saturated liquid is converted to saturated vapor, and superheating where the saturated vapor is raised to the temperature entering the turbine. When comparing the thermal efficiency of a Rankine cycle to the Carnot efficiency, should we substitute the temperature during "simple" boiling, or the temperature entering the turbine into the formula for the Carnot efficiency? Explain.