## Student handout: Basics of heat engines

Contemporary Challenges 2022 (4 years)
This brief lecture covers the basics of heat engines.

Let's apply the relationship of heat, entropy, and temperature to a contemporary challenge!

We'd like to maximize the efficiency of any process that is based on heat flow as an input.

##### Energy flow diagram

The efficiency of the machine is \begin{align} \text{efficiency} &= \frac{W}{Q_{\text{in}}} \\ \textit{e.g.} &=\frac{500\text{ J}}{1000\text{ J}} = 50\% \end{align} For a car engine, $T_H\approx 600\text{ K}$ and $T_C\approx 300\text{ K}$.

Remember that $\Delta S=\frac{Q}{T}$, and $\Delta S_{\text{tot}} \ge 0$.

• face A coarse-grained model for transportation

face Lecture

30 min.

##### A coarse-grained model for transportation
Contemporary Challenges 2022 (4 years)

A short lecture introducing the idea that most of the energy loss when driving is going into the kinetic energy of the air.
• assignment Heat pump

assignment Homework

##### Heat pump
Carnot efficiency Work Entropy Heat pump Thermal and Statistical Physics 2020
1. Show that for a reversible heat pump the energy required per unit of heat delivered inside the building is given by the Carnot efficiency: \begin{align} \frac{W}{Q_H} &= \eta_C = \frac{T_H-T_C}{T_H} \end{align} What happens if the heat pump is not reversible?

2. Assume that the electricity consumed by a reversible heat pump must itself be generated by a Carnot engine operating between the even hotter temperature $T_{HH}$ and the cold (outdoors) temperature $T_C$. What is the ratio $\frac{Q_{HH}}{Q_H}$ of the heat consumed at $T_{HH}$ (i.e. fuel burned) to the heat delivered at $T_H$ (in the house we want to heat)? Give numerical values for $T_{HH}=600\text{K}$; $T_{H}=300\text{K}$; $T_{C}=270\text{K}$.

3. Draw an energy-entropy flow diagram for the combination heat engine-heat pump, similar to Figures 8.1, 8.2 and 8.4 in the text (or the equivalent but sloppier) figures in the course notes. However, in this case we will involve no external work at all, only energy and entropy flows at three temperatures, since the work done is all generated from heat.

• face Work, Heat, and cycles

face Lecture

120 min.

##### Work, Heat, and cycles
Thermal and Statistical Physics 2020

These lecture notes covering week 8 of Thermal and Statistical Physics include a small group activity in which students derive the Carnot efficiency.
• group Grey space capsule

group Small Group Activity

30 min.

##### Grey space capsule
Contemporary Challenges 2022 (4 years)

In this small group activity, students work out the steady state temperature of an object absorbing and emitting blackbody radiation.
• group Heat capacity of N$_2$

group Small Group Activity

30 min.

##### Heat capacity of N2
Contemporary Challenges 2022 (4 years)

Students sketch the temperature-dependent heat capacity of molecular nitrogen. They apply the equipartition theorem and compute the temperatures at which degrees of freedom “freeze out.”
• face Equipartition theorem

face Lecture

30 min.

##### Equipartition theorem
Contemporary Challenges 2022 (4 years)

This lecture introduces the equipartition theorem.
• face Introducing entropy

face Lecture

30 min.

##### Introducing entropy
Contemporary Challenges 2022 (4 years)

This lecture introduces the idea of entropy, including the relationship between entropy and multiplicity as well as the relationship between changes in entropy and heat.
• group Energy radiated from one oscillator

group Small Group Activity

30 min.

##### Energy radiated from one oscillator
Contemporary Challenges 2022 (4 years)

This lecture is one step in motivating the form of the Planck distribution.
• group Black space capsule

group Small Group Activity

30 min.

##### Black space capsule
Contemporary Challenges 2022 (3 years)

In this activity, students apply the Stefan-Boltzmann equation and the principle of energy balance in steady state to find the steady state temperature of a black object in near-Earth orbit.
• group Earthquake waves

group Small Group Activity

30 min.

##### Earthquake waves
Contemporary Challenges 2022 (4 years)

In this activity students use the known speed of earthquake waves to estimate the Young's modulus of the Earth's crust.

Learning Outcomes