## Heat shields

• assignment Surface temperature of the Earth

assignment Homework

##### Surface temperature of the Earth
Temperature Radiation Thermal and Statistical Physics 2020 Calculate the temperature of the surface of the Earth on the assumption that as a black body in thermal equilibrium it reradiates as much thermal radiation as it receives from the Sun. Assume also that the surface of the Earth is a constant temperature over the day-night cycle. Use the sun's surface temperature $T_{\odot}=5800\text{K}$; and the sun's radius $R_{\odot}=7\times 10^{10}\text{cm}$; and the Earth-Sun distance of $1.5\times 10^{13}\text{cm}$.
• face Review of Thermal Physics

face Lecture

30 min.

##### Review of Thermal Physics
Thermal and Statistical Physics 2020

These are notes, essentially the equation sheet, from the final review session for Thermal and Statistical Physics.
• assignment Photon carnot engine

assignment Homework

##### Photon carnot engine
Carnot engine Work Energy Entropy Thermal and Statistical Physics 2020

In our week on radiation, we saw that the Helmholtz free energy of a box of radiation at temperature $T$ is \begin{align} F &= -8\pi \frac{V(kT)^4}{h^3c^3}\frac{\pi^4}{45} \end{align} From this we also found the internal energy and entropy \begin{align} U &= 24\pi \frac{(kT)^4}{h^3c^3}\frac{\pi^4}{45} V \\ S &= 32\pi kV\left(\frac{kT}{hc}\right)^3 \frac{\pi^4}{45} \end{align} Given these results, let us consider a Carnot engine that uses an empty metalic piston (i.e. a photon gas).

1. Given $T_H$ and $T_C$, as well as $V_1$ and $V_2$ (the two volumes at $T_H$), determine $V_3$ and $V_4$ (the two volumes at $T_C$).

2. What is the heat $Q_H$ taken up and the work done by the gas during the first isothermal expansion? Are they equal to each other, as for the ideal gas?

3. Does the work done on the two isentropic stages cancel each other, as for the ideal gas?

4. Calculate the total work done by the gas during one cycle. Compare it with the heat taken up at $T_H$ and show that the energy conversion efficiency is the Carnot efficiency.

• assignment Light bulb in a refrigerator

assignment Homework

##### Light bulb in a refrigerator
Carnot refridgerator Work Entropy Thermal and Statistical Physics 2020 A 100W light bulb is left burning inside a Carnot refridgerator that draws 100W. Can the refridgerator cool below room temperature?
• 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.
• assignment Rubber Sheet

assignment Homework

##### Rubber Sheet
Energy and Entropy 2021 (2 years)

Consider a hanging rectangular rubber sheet. We will consider there to be two ways to get energy into or out of this sheet: you can either stretch it vertically or horizontally. The distance of vertical stretch we will call $y$, and the distance of horizontal stretch we will call $x$.

If I pull the bottom down by a small distance $\Delta y$, with no horizontal force, what is the resulting change in width $\Delta x$? Express your answer in terms of partial derivatives of the potential energy $U(x,y)$.

• face Basics of heat engines

face Lecture

10 min.

##### Basics of heat engines
Contemporary Challenges 2022 (4 years) This brief lecture covers the basics of heat engines.
• assignment Power Plant on a River

assignment Homework

##### Power Plant on a River
efficiency heat engine carnot Energy and Entropy 2021 (2 years)

At a power plant that produces 1 GW ($10^{9} \text{watts}$) of electricity, the steam turbines take in steam at a temperature of $500^{o}C$, and the waste energy is expelled into the environment at $20^{o}C$.

1. What is the maximum possible efficiency of this plant?

2. Suppose you arrange the power plant to expel its waste energy into a chilly mountain river at $15^oC$. Roughly how much money can you make in a year by installing your improved hardware, if you sell the additional electricity for 10 cents per kilowatt-hour?

3. At what rate will the plant expel waste energy into this river?

4. Assume the river's flow rate is 100 m$^{3}/$s. By how much will the temperature of the river increase?

5. To avoid this “thermal pollution” of the river the plant could instead be cooled by evaporation of river water. This is more expensive, but it is environmentally preferable. At what rate must the water evaporate? What fraction of the river must be evaporated?

• assignment Power from the Ocean

assignment Homework

##### Power from the Ocean
heat engine efficiency Energy and Entropy 2021 (2 years)

It has been proposed to use the thermal gradient of the ocean to drive a heat engine. Suppose that at a certain location the water temperature is $22^\circ$C at the ocean surface and $4^{o}$C at the ocean floor.

1. What is the maximum possible efficiency of an engine operating between these two temperatures?

2. If the engine is to produce 1 GW of electrical power, what minimum volume of water must be processed every second? Note that the specific heat capacity of water $c_p = 4.2$ Jg$^{-1}$K$^{-1}$ and the density of water is 1 g cm$^{-3}$, and both are roughly constant over this temperature range.

• Thermal and Statistical Physics 2020 A black (nonreflective) sheet of metal at high temperature $T_h$ is parallel to a cold black sheet of metal at temperature $T_c$. Each sheet has an area $A$ which is much greater than the distance between them. The sheets are in vacuum, so energy can only be transferred by radiation.
1. Solve for the net power transferred between the two sheets.

2. A third black metal sheet is inserted between the other two and is allowed to come to a steady state temperature $T_m$. Find the temperature of the middle sheet, and solve for the new net power transferred between the hot and cold sheets. This is the principle of the heat shield, and is part of how the James Web telescope shield works.
3. Optional: Find the power through an $N$-layer sandwich.