We have already encountered thermal energy in two contexts. The first was infrared radiation (Eq. 1.8; p. 10), and the second was in the definition of the kilocalorie (Sec. 5.5; p. 73). Otherwise, heat has often been treated as a form of “waste” in a chain of energy conversion: friction, air resistance, etc. The insinuation was that heat is an unwanted byproduct of no value. Yet 94% of the energy we use today is thermal in nature ¹: we burn a lot of stuff for energy!1 Sometimes heat is what we’re after, but how can we use it to fly airplanes, propel cars, and light up our screens? This chapter aims to clarify how heat is used, and explore limits to the efficiency at which heat can perform non-thermal work. Like the previous chapter, this topic represents a slight detour from the book’s overall trajectory, which otherwise aims to build a steady narrative of what we can’t expect to continue doing, what options we might use to change course, and finally how to bring about such change. Nonetheless, the way we utilize thermal energy is a key piece of the story, and relates to both current and future pathways to satisfying our energy demands. 6.1 Generating Heat 6.2 Heat Capacity 6.3 Home Heating/Cooling.. 6.4 Heat Engines 84 85 86 . 88 Entropy and Efficiency Limits 90 6.5 Heat Pumps .. Consumer Metrics. .. 95 97 6.6 Upshot on Thermal Energy . 99 99 6.7 Problems..

6.1 Generating Heat

Before diving in to thermal issues, let’s do a quick run-down of the various ways we can generate heat. Example 6.1.1 Ways to Generate Heat: Roughly arranged according to degree of sophistication: A locomotive engine as an example heat engine. Photo credit: South Australian Government Photographer.

1. Rub your hands together (or other forms of friction). 6 Putting Thermal Energy to Work 85
2. Harvest sunlight, possibly concentrating it, for heat; drying clothes outside and letting sunlight warm a room through a window are examples.
3. Access geothermal heat in select locations.
4. Burn wood in a fireplace or stove.
5. Burn a fossil fuel for direct heat; gas is often used in homes for space heating, as well as for heating water and cooking.
6. Run electrical current through a coil of wire that glows orange; seen in toaster ovens, hair dryers, space heaters.
7. Use electricity to run a heat pump (Section 6.5).
8. Allow nuclear material to undergo fission in a controlled chain reaction.
9. Contrive a plasma hot enough to sustain nuclear fusion-as the sun has done for billions of years. Table 6.1: Specific heat capacities of common materials. Substance J/kg/°C steel rock, concrete 490 750-950 glass aluminum 840 870 air 1,005 plastic 1,100-1,700 wood 1,300-2,000 alcohol 2,400 3,500 4,184

6.2 Heat Capacity

First, we’ll connect a basic thermal concept to something we already covered in Sec. 5.5 (p.73) in the context of the calorie. The statement that it takes 1 kcal to heat 1 kilogram of H2O by 1°C is in effect defining the specific heat capacity of water. In SI units, we would say that H2O has a specific heat capacity of 4,184 J/kg/°C.2 Very few substances top water’s specific heat capacity. Most liquids, like alcohols, tend to be in the range of 2,000 J/kg/°C. Most non-metallic solids (and even air) come in around 1,000 J/kg/°C. Metals are in the 130-900 J/kg/°C range-lighter metals at the top, and heavier ones at the lower end.⁷ Table 6.1 provides a sample of specific heat capacities for common substances. Knowing the specific heat capacity of a substance allows us to compute how much energy it will take to raise its temperature. A useful and approximate guideline is to treat water as 4,000 J/kg/°C and all other stuff (air, furniture, walls) as 1,000 J/kg/°C. Mixtures, like food, might be somewhere between, at 2,000-3,500, due to high water content. If in doubt, 1,000 J/kg/°C is never going to be too far off. For estimation purposes, deviate from this only for high water-content1 or for metals.⁸ Example 6.2.1 A 2,000 kg pick-up truck is transporting a one-cubicmeter container of water. How much energy will it take to raise the temperature of the whole ensemble by 5°C? A cubic meter of water (1,000 L) is 1,000 kg and has a heat capacity around 4,000 J/kg/°C; the truck is mostly steel, so we might guess 500 J/kg/°C. Multiply each specific heat capacity by the respective mass and the 5 degree change to get 20 MJ to heat the water and 5 MJ to heat the truck for a total of 25 MJ.6 flesh water

6 Putting Thermal Energy to Work 86 To perform computations using specific heat capacity, try an intuitive approach rather than some algorithmic formula. The following should just make a lot of sense to you, and can guide how to put the pieces together: it takes more energy to heat a larger mass or to raise the temperature by a larger amount. It’s all proportional. The units also offer a hint. To go from specific heat capacity in J/kg/°C to energy in J, we need to multiply by a mass and by a temperature change. Example 6.2.2 To compute the amount of energy it will take to heat a 30 kg piece of furnitures by 8°C, we will multiply the specific heat capacity by the mass-to capture the “more mass” quality—and then multiply by the temperature change-to reflect the “more temperature change” element. In this case, we get 240 kJ.

6.3 Home Heating/Cooling

Our personal experience with thermal energy is usually most connected to heating a living space and heating water or food. Indeed, about two-thirds of the energy used in residential and commercial spaces9 relate to thermal tasks, like heating or cooling the spaces, heating water, refrigeration, drying clothes, and cooking. When it comes to heating (or cooling) a home, we might care about two things: how long will it take to change its temperature by some certain amount; and how much energy it will take to keep it at the desired temperature. The first depends on how much stuff is in the house,⁹ how much AT you want to impart, and how much power is available to create11 the heat. The energy required is mass times AT times the catch-all 1,000 J/kg/°C specific heat capacity. The time it takes is then the energy divided by the available power. Example 6.3.1 How long will it take to heat up the interior of a mobile home from 0°C to 20°C using two 1,500 W space heaters? We’ll assume that we must heat up about 6,000 kg of mass.¹⁰ The first job is to find the energy required and then divide by power to get a time. We’ll use the good-for-most-things specific heat capacity of 1,000 J/kg/°C. Multiplying the specific heat capacity by mass and temperature change results in 120 MJ of energy. At a rate of 3,000 W, it will take 40,000 s to inject this much energy, which is about 11 hours. How much it takes to maintain temperature depends on how heat flows out of (or into) the house through the windows, walls, ceiling, floor,

or to remove, if cooling

6 Putting Thermal Energy to Work 87 and air gaps. But it also depends linearly on AT-the difference between inside and outside temperatures-that is being maintained. A house can be characterized by its heat loss rate in units of Watts per degree Celsius.¹¹ This single number then indicates how much power is needed to maintain a certain AT between inside and outside. Box 6.1 explores an example of how to compute the heat loss rate for a house, and Example 6.3.2 applies the result to practical situations.

6 Putting Thermal Energy to Work will require 75 W/°C times 20°C, or 1,500 W16 to keep it warm, while the decent house needs 3,000 W and the shoddy house needs 6,000 W. Once we understand how much power it takes to maintain a certain temperature (AT) in a house, we can anticipate the behavior of the house’s heater. Heaters are typically either on full-blast or off. Regulation is achieved by turning the heat on and off-usually controlled by a thermostat. Given the rating of a heater,¹² it is then straightforward to anticipate the duty cycle: the percentage of time it has to be on to produce an average output meeting the power requirement for some particular ΔΤ. In a sensible world, heaters are characterized by W (or kW). In the U.S., the measure for many appliances is Btu/hr. Since 1 Btu is 1,055 J and one hour is 3,600 s, one Btu/hr equates to 0.293 W.18 A whole house heater-sometimes in the form of a furnace-might be rated at 30,000 Btu/hr (about 10 kW), in which case the three outcomes in Example 6.3.2 would require the heater to be on about 15%, 30%, or 60% of the time19 to maintain AT = 20°C in the three houses. It is also possible to assess how much AT the foregoing heater could maintain in the three houses. It should stand to reason that a house requiring 100 W/°C and having a 10,000 W heater can support a AT as high as 100°C.20 Thus, the three houses from Example 6.3.2 could support AT values of 133°C, 67°C, and 33°C if equipped with a 10 kW (~30,000 Btu/hr) heater. The snug house does not need such a powerful heater installed. The poorly built house can maintain a AT = 33°C differential at full-blast, which means that if the temperature drops below -13°C (8.6°F) outside, it will not be able to keep the inside as high as 20°C. So a house in a cold climate should either be built to better thermal standards, or will require a bigger heater-costing more to heat the home.¹³ Cooling a home (or refrigerator interior, or whatever) is also a thermal process, but in this case involves removing thermal energy from the cooler environment. Removing heat is harder to do, as witnessed by the length of human history that has utilized heating sources starting with fire-compared to the relatively short amount of time when we have been able to produce cooling on demand.¹⁴ Section 6.5 will get into how this is even possible, in principle. For now, just be aware that the rating on air conditioners uses the same units as heaters: how much thermal energy can be moved (out of the cooler environment) per unit time. In SI units, we know this as the Watt. In the U.S., it’s Btu/hr.

6.4 Heat Engines

Now we get to the part where thermal energy can be used to do something other than just provide direct heat to a home. It may seem odd to always

6 Putting Thermal Energy to Work 89 characterize burning fuel as a purely thermal action, since what transpires within the cylinder of a gasoline-burning internal combustion engine seems like more of a little explosion than just the generation of heat. This is not wrong, but neither is it the whole story. The process still begins as a fundamentally thermal event. When the fuel-air mixture ignites, the temperature in the cylinder increases dramatically. To appreciate what happens as an immediate consequence, we turn to the ideal gas law: PV = NkBT. 23 (6.1) P, V and T are pressure, volume, and temperature (in N/m2, m3, and Kelvin). N is the number of atoms or molecules, and kg = 1.38×$10^{-23}$ J/K kB is the Boltzmann constant, which we will see again in Sec. 13.2 (p. 199). The temperature rise upon ignition is fast enough that the cylinder volume does not have time to change. Eq. 6.1 then tells us that the pressure must also spike when temperature does, all else being held constant. The increase in pressure then pushes the piston away, increasing the cylinder volume and performing work.¹⁵ But it all starts thermally, via a sharp increase in temperature. In the most general terms, thermal energy tries to flow from hot to cold-out of a pot of hot soup; or into a cold drink from the surrounding air; or into your feet from hot sand. Some part of this flow can manifest as physical work, at which point the system can be said to be acting as a heat engine. Definition 6.4.1 A heat engine is loosely defined as any system that turns heat, or thermal energy into mechanical energy: moving stuff. Example 6.4.1 Example heat engines: when heat drives motion.

1. Hot air over a car’s roof rises, gaining both kinetic energy and gravitational potential energy;
2. Wind is very similar, in that air in contact with the sun-heated ground rises and gains kinetic energy on an atmospheric scale; 3. The abrupt temperature increase in an internal combustion cylinder drives a rapid expansion of gas within the cylinder; 4. Steam in a power plant races though the turbine because it is flowing to the cold condenser. The last example deserves its own graphic, as important as this process is in our lives: almost all of our electricity generation-from all the fossil fuels and even from nuclear fission-follows this arrangement. Figure 6.2 illustrates the basic scheme. Table 6.2 indicates that 98% of our electricity involves turning a turbine on a shaft connected to a generator, and 84% involves a thermal process as the motive agent for the turbine-most often in the form of steam. This is the physicist’s version, which looks a little different than the chemist’s PV nRT. For a comparison, see Sec. B.4 (p. 381).

turbine boiler high pressure steam 90 heat source, Th 6 Putting Thermal Energy to Work generator steam pump condenser return pump electricity 90 cold water source, T (river, ocean, or cooling towers) Figure 6.2: Generic power plant scheme, in which some source of heat at Th generates steam that flows toward the condenser-where the steam cools and reverts to liquid water, by virtue of thermal contact to a cool source at Te provided by a body of water or evaporative cooling towers. Along the way, the rushing steam turns a turbine connected to a generator, exporting electricity. This basic arrangement is employed for most power plants using fossil fuels, nuclear, solar thermal, or geothermal sources of heat. 6.4.1 Entropy and Efficiency Limits A deep and powerful piece of physics intervenes to limit how much useful work may be extracted out of a flow of heat from a hot source at temperature Th to a cold source at temperature Te. That piece is entropy. You don’t need to fully grasp the deep and subtle concept of entropy in order to follow the development in this chapter and understand the role entropy plays in limiting heat engine efficiency. All the same, it is a stimulating topic that we’ll dip a toe into for some appreciation. Definition 6.4.2 Entropy is a measure of how many ways a system might be organized at the microscopic level while preserving the same internal energy.¹⁶ This definition may be an obscure disappointment to those expecting entropy to be defined as a measure of disorder.¹⁷ Consider a gas maintained at constant pressure, volume, and temperature-thus fixing the total energy in the gas. The atoms/molecules comprising the gas can arrange into a staggeringly large number of configurations: any number of positions, velocities, rotational speeds and axis orientations, or vibrational states of each molecule, for instance-all while keeping the same overall energy. Example 6.4.2 To illustrate, consider a tiny system containing 3 molecules labeled A, B, and C, having a total energy of 6 units split

6 Putting Thermal Energy to Work never happen. Pieces of ceramic strewn about the floor will never spontaneously assemble into a mug and leap from the floor! Energy is not the barrier, because the total energy in all forms is the same29 before and after. It’s entropy: the more ordered states are less likely to spontaneously emerge. To appreciate how pervasive entropy is, imagine how easy it is to spot a “fake” video run backwards. These two laws of thermodynamics, plus a way to quantify entropy changes that we will see shortly, are all we need to figure out the maximum efficiency a heat engine can achieve in delivering work. If we draw an amount of heat, AQh from a hot bath30 at temperature Th, and allow part of this energy to be “exported” as useful work, AW, then we must have the remainder flow as heat (AQc) into the cold bath at temperature Te. Figure 6.4 offers a schematic of the process. The First Law of Thermodynamics31 requires that AQh = AQc + AW, or that all of the extracted heat from the hot bath is represented in the external work and flow to the cold bath: nothing is lost. 31 large hot reservoir at T 92

6 Putting Thermal Energy to Work 93 entropy may not be negative (it can’t decrease). In equation form (symbol definitions in Table 6.3):32

AStot ASC ASh > 0,

(6.3) where we have subtracted ASh since it was a deduction of entropy, while AS, is an addition. We therefore require that ASC ASH (6.4) Now we are in a position to ask what fraction of AQh can be diverted to useful work (AW) within the constraints of the Second Law. We express this as an efficiency,¹⁸ denoted by the Greek epsilon: ε = AW AQh - AQc AQh =V AQh (6.5) The second step applies conservation of energy: AQh = AQc + AW. Example 6.4.4 Actual Efficiency: If a heat engine is observed to remove 30 J from the hot bath and deposit 20 J into the cold bath, as in Figure 6.4, what is the efficiency of this heat engine in producing useful work? Whether we deduce that AW = 10 J and use the first form in Eq. 6.5 or apply the second form using the given heat flows, the answer is 1/3, or 33%. We can add a step to Eq. 6.5 to express it in terms of entropy changes:

6 Putting Thermal Energy to Work 94 6.8 happens when n = 1. We therefore derive the maximum physically allowable efficiency of a heat engine as Emax Th - Tc Th AT

/ Th (6.9) Temperature must be in Kelvin. Recall that T(K)≈ T(°C) + 273. where we have designated AT = Th - Te as the temperature difference between hot and cold baths. A major takeaway is that efficiency improves as AT gets bigger, and becomes vanishingly small for small values of ΔΤ. Example 6.4.5 If operating between a hot bath at 800 K and ambient temperature around 300 K,35 a heat engine could produce a maximum efficiency of 62.5%. Example 6.4.6 A heat engine operating between boiling and freezing water has Th≈ 373 K and AT 100 K, for a maximum possible efficiency of Emax = 0.268, or 26.8%. Example 6.4.7 A heat engine operating between human skin temperature at 35°C and ambient temperature at 20°C has a maximum efficiency of Emax = 15/308≈ 0.05, or 5%. If the cold bath is fixed,¹⁹ the maximum possible efficiency improves as the temperature of the hot source goes up. Conversely, for a given Th the efficiency improves as the cold temperature decreases and thus AT increases. Box 6.3: At the Extreme Limit… If Tc approaches 0 K37, the maximum efficiency approaches 100%. We can trace this back to the relation AQ = TAS, which implies that when T is very small, it does not take much heat (AQ) to meet the requirement for the amount of entropy added to the cold bath (ASC) to be large enough to satisfy the prohibition on net entropy decrease, so the arrow width in Figure 6.4 for AQ, can be rather thin (small) allowing AW to be about as thick (large) as AQh, meaning that essentially all the energy is available to do work and the efficiency can be very high. In practice, Earth does not provide baths cold enough for this effect to kick in, but discussing it is a means to better understand how Eq. 6.9 works. Real heat engines like power plants (Figure 6.2) or automobile engines tend to only get about halfway to the theoretical efficiency due to myriad practical challenges. A typical efficiency for an electrical power plant is 30-40%, while cars are typically in the 15-25% range. In contrast, combustion temperatures around 700-800°C suggest a maximum theoretical efficiency around 60%.

6 Putting Thermal Energy to Work 95 95 6.5 Heat Pumps We can flip a heat engine around and call it a heat pump. In this case, we apply some external work to drive a heat flow opposite its natural direction-like pushing heat uphill. This is how a refrigerator38 works, for instance. Figure 6.5 sets the stage. large hot reservoir at T h

6 Putting Thermal Energy to Work Example 6.5.1 What is the limit to efficiency of maintaining a freezer at -10°C in a room of 20°C? First, we express the temperatures in Kelvin: T≈ 263 K and AT = 30 K.43 The maximum efficiency, by Eq. 6.10, computes to cool ≤ 8.8 (880%). Example 6.5.2 What is the limit to efficiency of keeping a home interior at 20°C when it is -10°C outside? First, we express the temperatures in Kelvin: Th≈ 293 K and AT = 30 K.44 The maximum efficiency, by Eq. 6.11, computes to Ɛheat ≤9.8 (980%). Box 6.4: Is >100% Really Possible? At first, it seems to be spooky and impossible that efficiencies can be greater than 100%. Example 6.5.1 essentially says that as many as 8.8 J of thermal energy can be moved for a mere 1 J input of work! The situation bears analogy to the martial art of Jiu Jitsu, whereby the opponent’s momentum is used to their detriment, requiring little work to direct its flow. In this case, we convince a bundle of thermal energy sitting in the freezer to move outside where it is hotter (uphill; against natural flow) and in the process use less energy than the amount of thermal energy residing in the bundle. The fact that our “efficiency” metrics come out to be greater than 100% is an illusion: an artifact of how we defined cool and Ɛheat. Conservation of energy is not violated; we’re just putting the small piece (AW) in the denominator to form the efficiency metric.²⁰ In this sense, it’s not the usual sort of efficiency measure, which puts the largest quantity (total budget) in the denominator. In the case of heating, it is worth comparing the output of a heat pump to the application of direct heat. Let’s revisit the scenarios explored in Section 6.3. .46 Example 6.5.3 If a house’s thermal performance is 150 W/°C and we want to maintain 20°C inside while the outside temperature is a frigid -20°C, we would need to supply 6,000 W of energy to the home in the form of burning fuel (natural gas, propane, firewood) or electricity for direct-heating application.²¹ But according to Eq. 6.11, a heat pump could theoretically move 6,000 W of thermal energy by only applying 820 W without violating the Second Law, since Ɛheat ≤ 293/40 = 7.3 and 6,000 J (AQh) divided by 7.3 (to get AW) is 820 J.48 Engineering realities will prevent operating right up to the thermody-

.e.g., four space heaters each expending 1,500 W AQh/Cheat

6 Putting Thermal Energy to Work namic limit, but we might at least expect to be able to accomplish the 6,000 W goal of Example 6.5.3 for under 2,000 W. Thus the heat pump has shaved a factor of three (or more) off the energy required to provide heat inside. Heat pumps are very special. As Eq. 6.10 and Eq. 6.11 imply, heat pumps are most efficient when AT is small. Thus a refrigerator in a hot garage must not only work harder to maintain a large AT, it does so less efficiently-making it a double-whammy. For home heating, heat pumps offer the most gain in milder climates where AT is not so brutal. U.S. Goverment 97 Federal law prohibits removal of this label before consumer purchase. ENERGY GUIDE Cooling and Heating Split System Cooling Efficiency Rating (SEER)* 20.50-21.50 THERMOCORE SYSTEMS Model T321Q-H263-C 6.5.1 Consumer Metrics: COP, EER, HSPF When shopping for heat pumps or air conditioners (or freezers/refrigerators), products are specified by the coefficient of performance (COP) or energy efficiency ratio (EER) or heating seasonal performance factor (HSPF), as in Figure 6.6. How do these relate to our Ɛheat and cool values? The first one is easy. Definition 6.5.3 COP: Heat pumps used for heating are specified by a coefficient of performance (COP), which turns out to be familiar already: COP = Ɛheat. (6.12) Example 6.5.4 COP Example: Using the red numbers in Figure 6.5, we can compute Ɛheat, the COP, and then determine the minimum T theoretically permissible (resulting in maximum possible efficiency) if Th= 300 K.49 We go back to the original definition of Eheat as AQh/AW, which for our numbers works out to 30/10, or 3.0 The COP is then simply 3.0. Setting Ɛheat,max= Th/AT equal to 3.0, we find that AT is 100 K, so that the minimum permissible T = 200 K in this case. The EER is different, and perhaps a little odd. EER is defined as the amount of heat moved (AQc), in Btu, per work input (AW), in watt-hours (Wh). What?! Sometimes the world is just loopy. But we can manage this. If handed an EER (Btu/Wh), we can convert it to our same/same numerator/denominator units by converting both numerator and denominator to the same units. We could convert Btu to Wh in the numerator and be done, or convert Wh to Btu in the denominator and be done, or we could convert both numerator and denominator to Joules50 to get there. For illustrative purposes, we’ll pick the last approach. To get from Btu to Joules, we multiply (the numerator) by 1,055. To get from Wh to Joules, we multiply the denominator (or divide the EER construct) by 3,600.51 The net effect is highlighted in the following definition. 14.0 Least Effent 8.2 Range of Similar Models *Seasonal Energy Efficiency Ratin 30.5 Most Eficient Heating Efficiency Rating (HSPF)’ 10.40 -11.50 Least Efficient This system’s eficiency ratings depend on the coil your contractor installs with this unit. The heating eficiency rating vanes slighty in different geographic regions. Ask your contractor for details. 13.5 Most Efficient Range of Similar Models Heating Seasonal Performance Factor For energy cost info, visit productinfo.energy.gov Figure 6.6: Typical heat pump energy label in the U.S., showing an EER around 21 and a HSPF around 11. From U.S. DOE.

6 Putting Thermal Energy to Work Definition 6.5.4 EER: Heat pumps used for cooling are specified by the energy efficiency ratio (EER), which modifies Eq. 6.10 as follows. Єcool = EER Btu 1055J/Btu Wh 3600J/Wh = EER 0.293, (6.13) or the converse EER

Ecool 0.293 ≈3.41 X Ecool. (6.14) Example 6.5.5 EER Example: Using the red numbers in Figure 6.5, we can compute &cool, the EER, and then determine the maximum Th theoretically permissible (resulting in maximum possible efficiency) given a target Tc of 260 K, as we might find in a freezer. We go back to the original definition of cool as AQc/AW, which for our numbers works out to 20/10, or 2.0. The EER is then 3.41 times this amount, or 6.8. Setting Ecool,max = Te/AT equal to 2.0, we find that AT is 130 K, so that the maximum permissible Th = 390 K in this case. Because the theoretical maximum efficiency depends on AT-according to Eq. 6.10 and Eq. 6.11-and therefore can fluctuate as outdoor temperatures change, a seasonal average is often employed, called the SEER (seasonal EER). In a similar vein, the HSPF measures the same thing as the COP, but in units of EER52 and averaged over the heating season. Definition 6.5.5 HSPF: Heat pumps used for heating are sometimes specified by the heating seasonal performance factor (HSPF), which modifies Eq. 6.11 as follows.

6 Putting Thermal Energy to Work mode.²² Houses equipped with electric heat pumps can typically be run for both cooling and heating applications, making them a versatile and efficient solution for moving thermal energy in or out of a house. Heat pumps leveraging the moderate-temperature ground just below the surface as the external thermal bath are called “geothermal” heat pumps, but have nothing to do with geothermal energy (as a source). Compared to heat pumps accessing more extreme outside air temperatures, geothermal heat pumps benefit from a smaller AT, and thus operate at higher efficiency. 99

1. How many Joules does it take to heat your body up by 1°C if your (water-dominated) mass has a specific heat capacity of 3,500 J/kg/°C?
2. How long will it take a space heater to heat the air58 in an empty room by 10°C if the room has a floor area of 10 m2 and a height of 2.5 m and the space heater is rated at 1,500 W? Air has a density59

6 Putting Thermal Energy to Work 100 of 1.25 kg/m3. Express your answer as an approximate number in minutes. 3. When you put clothes on in the morning in a cool house at 15°C, you warm them up to something intermediate between your skin temperature (35°C) and the ambient environment.²³ If your clothes have a mass of 2 kg, how much energy must be deposited into the clothes? If you are emitting power at 100 W, how long will this take? 4. You score this massive 1 kg burrito but decide to put it in the refrigerator to eat later. It comes out at 5°C, and you want to heat it in the microwave up to 75°C before eating it. If the microwave puts energy into the burrito at a rate of 700 W.61 How long should you run the microwave for a high-water-content burrito having an effective specific heat capacity of 3,000 J/kg/°C? 5. Let’s say you come home from a winter vacation to find your house at 5°C and you want to heat it to 20°C. Let’s say the house contains: 500 kg of air;62 1,000 kg of furniture, books, and other possessions; plus walls and ceiling and floor that amount to 6,000 kg of effective63 mass. Using the catch-all specific heat capacity for all of this stuff, how much energy will it take, and how long to heat it up at a rate of 10 kW? Express in useful, intuitive units, and feel free to round, since it’s an estimate, anyway. 6. In a house achieving a heat loss rate of 200 W/°C equipped only with two 1,500 W space heaters, what is the coldest it can get outside if the house is to maintain an internal temperature of 20°C? 7. In a house achieving a heat loss rate of 200 W/°C equipped a 5,000 W heater, what will the internal temperature be if the outside temperature is -10°C and the heater is running 100% of the time? 8. In a super-tight house achieving 100 W/°C equipped with a 5,000 W heater, what percentage of the time will the heater need to run in order to keep the internal temperature at 20°C if the temperature outside is at the freezing point?64 9. How much will it cost per day to keep a house at 20°C inside when the external temperature is steady at -5°C using direct electric heating if the house is rated at 150 W/°C and electricity costs $0.15/kWh? 65 66 10. Provide at least one example not listed in the text in which heat flows into some other form of energy. In the text, we mentioned hot air over a car, wind, internal combustion, and a steam turbine plant. 11. What is the only form of significant electricity production in the

Note that a microwave oven might be rated for 1,500 W, but not all the energy ends up in the burrito, so we pick 700 W to be realistic.

12. If a can of soda (350 mL; treat as water) cools from 20°C to 0°C, how much energy is extracted, and how much is the entropy (in J/K) in the can reduced using the average temperature and the relation that AQ = TAS?
13. What would the maximum thermodynamic efficiency be of some heat engine operating between your skin temperature and the ambient environment 20°C cooler than your skin?
14. We can think of wind in the atmosphere as a giant heat engine operating between the 288 K surface and the top of the troposphere68 at 230 K. What is the maximum efficiency this heat engine could achieve in converting solar heating into airflow?
15. Since the sun drives energy processes on Earth, we could explore the maximum possible thermodynamic efficiency of a process operating between the surface temperature of the sun (5,800 K) and Earth’s surface temperature (288 K). What is this maximum efficiency?69
16. A heat engine pulls 100 J out of a hot bath at 800 K, and transfers 80 J of heat into the cold bath at 300 K. What efficiency does this heat engine achieve in producing useful work, and how does it compare to the theoretical maximum?
17. Human efficiency70 is in the neighborhood of 25%, meaning that in order to do 100 J of external work, we need to eat 400 J of energy content. To investigate whether human energy is working as a heat engine, figure out what the cold temperature, Te, would have to be to achieve this efficiency, thermodynamically.“1 Do you conclude that our biochemistry operates as a heat engine, or no?72
18. A 350 mL can of soda73 at 20°C is placed into a refrigerator having an EER rating of 10.0. How much energy will you have to spend (AW) to remove the thermal energy from the soda and bring it to a frosty 0°C?
19. If a refrigerator works at half of its theoretical &cool limit, how much more energy does it take to maintain an internal temperature of 0°C in a 40°C garage vs. a 20°C house interior? Two things are going on here: even at the same efficiency, the cooling energy scales as AT, but the efficiency also changes for a double-whammy.
20. Changing from direct electrical heating to a heat pump operating with a COP of 3 means spending one-third the energy for a certain thermal benefit. If a house averages 30 kWh/day in heating cost through the year using direct electrical heating at a cost of $0.15/kWh, how long will it take to recuperate a $5,000 installation cost of a new heat pump?