# 03.1 - Heat Engines and Heat Pumps: The Carnot Cycle

### 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, ηθ). 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: ηθ = (TH - TC)/TH, where TH is the hot temperature and 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).

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, ...)
4. What value of TH would be necessary to approach 100% efficiency, even for this idealized, maximally efficient process? What are the practical limitations on TH? Explain.
5. How can the formula for Carnot efficiency help us to calculate the "lost" work in the presence of a temperature gradient?