- heat engine efficiency
*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 \(20^{o}C\).

What is the maximum possible efficiency of this plant?

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?

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

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

- 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?

*group*Ice Calorimetry Lab*group*Small Group Activity60 min.

##### Ice Calorimetry Lab

The students will set up a Styrofoam cup with heating element and a thermometer in it. They will measure the temperature as a function of time, and thus the energy transferred from the power supply, from which they compute changes in entropy.*assignment*Ice calorimetry lab questions*assignment*Homework##### Ice calorimetry lab questions

This question is about the lab we did in class: Ice Calorimetry Lab.**Plot your data I**Plot the temperature versus total energy added to the system (which you can call \(Q\)). To do this, you will need to integrate the power. Discuss this curve and any interesting features you notice on it.**Plot your data II**Plot the heat capacity versus temperature. This will be a bit trickier. You can find the heat capacity from the previous plot by looking at the slope. \begin{align} C_p &= \left(\frac{\partial Q}{\partial T}\right)_p \end{align} This is what is called the*heat capacity*, which is the amount of energy needed to change the temperature by a given amount. The \(p\) subscript means that your measurement was made at constant pressure. This heat capacity is actually the total heat capacity of everything you put in the calorimeter, which includes the resistor and thermometer.-
**Specific heat**From your plot of \(C_p(T)\), work out the heat capacity per unit mass of water. You may assume the effect of the resistor and thermometer are negligible. How does your answer compare with the prediction of the Dulong-Petit law? -
**Latent heat of fusion**What did the temperature do while the ice was melting? How much energy was required to melt the ice in your calorimeter? How much energy was required per unit mass? per molecule? **Entropy of fusion**The change in*entropy*is easy to measure for a reversible isothermal process (such as the slow melting of ice), it is just \begin{align} \Delta S &= \frac{Q}{T} \end{align} where \(Q\) is the energy thermally added to the system and \(T\) is the temperature in Kelvin. What is was change in the entropy of the ice you melted? What was the change in entropy*per molecule*? What was the change in entropy per molecule divided by Boltzmann's constant?-
**Entropy for a temperature change**Choose two temperatures that your water reached (after the ice melted), and find the change in the entropy of your water. This change is given by \begin{align} \Delta S &= \int \frac{{\mathit{\unicode{273}}} Q}{T} \\ &= \int_{t_i}^{t_f} \frac{P(t)}{T(t)}dt \end{align} where \(P(t)\) is the heater power as a function of time and \(T(t)\) is the temperature, also as a function of time.

*biotech*Microwave oven Ice Calorimetry Lab*biotech*Experiment60 min.

##### Microwave oven Ice Calorimetry Lab

Energy and Entropy 2021 (2 years)heat entropy water ice thermodynamics

In this remote-friendly activity, students use a microwave oven (and optionally a thermometer) to measure the latent heat of melting for water (and optionally the heat capacity). From these they compute changes in entropy. See also Ice Calorimetry Lab.*face*Energy and heat and entropy*face*Lecture30 min.

##### Energy and heat and entropy

Energy and Entropy 2021 (2 years)latent heat heat capacity internal energy entropy

This short lecture introduces the ideas required for Ice Calorimetry Lab or Microwave oven Ice Calorimetry Lab.*group*A glass of water*group*Small Group Activity30 min.

##### A glass of water

Energy and Entropy 2021 (2 years)thermodynamics intensive extensive temperature volume energy entropy

Students generate a list of properties a glass of water might have. The class then discusses and categorizes those properties.*assignment*Calculation of $\frac{dT}{dp}$ for water*assignment*Homework##### Calculation of \(\frac{dT}{dp}\) for water

Clausius-Clapeyron Thermal and Statistical Physics 2020 Calculate based on the Clausius-Clapeyron equation the value of \(\frac{dT}{dp}\) near \(p=1\text{atm}\) for the liquid-vapor equilibrium of water. The heat of vaporization at \(100^\circ\text{C}\) is \(2260\text{ J g}^{-1}\). Express the result in kelvin/atm.*assignment*Directional Derivative*assignment*Homework##### Directional Derivative

Static Fields 2022 (4 years)You are on a hike. The altitude nearby is described by the function \(f(x, y)= k x^{2}y\), where \(k=20 \mathrm{\frac{m}{km^3}}\) is a constant, \(x\) and \(y\) are east and north coordinates, respectively, with units of kilometers. You're standing at the spot \((3~\mathrm{km},2~\mathrm{km})\) and there is a cottage located at \((1~\mathrm{km}, 2~\mathrm{km})\). You drop your water bottle and the water spills out.

- Plot the function \(f(x, y)\) and also its level curves in your favorite plotting software. Include images of these graphs. Special note: If you use a computer program written by someone else, you must reference that appropriately.
- In which direction in space does the water flow?
- At the spot you're standing, what is the slope of the ground in the direction of the cottage?
- Does your result to part (c) make sense from the graph?

*group*``Squishability'' of Water Vapor (Contour Map)*group*Small Group Activity30 min.

##### “Squishability” of Water Vapor (Contour Map)

Students determine the “squishibility” (an extensive compressibility) by taking \(-\partial V/\partial P\) holding either temperature or entropy fixed.*group*Heat and Temperature of Water Vapor (Remote)*group*Small Group Activity5 min.

##### Heat and Temperature of Water Vapor (Remote)

In this introduction to heat capacity, students determine a derivative that indicates how much the internal energy changes as the temperature changes when volume is held constant.-
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.

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

- 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.