In a real turbine, the entropy of the steam will increase

PROBLEM 37P A common (but imprecise) way of stating the third law of thermodynamics is “You can’t reach absolute zero.” Discuss how the third law, as stated in Section 3.2, puts limits on how low a temperature can be attained by various refrigeration techniques.

PROBLEM 3P A power plant produces 1 GW of electricity, at an efficiency of 40% (typical of today’s coal-fired plants). (a) At what rate does this plant expel waste heat into its environment? ________________ (b) Assume first that the cold reservoir for this plant is a river whose flow rate is 100 m3/s. By how much will the temperature of the river increase? ________________ (c) To avoid this “thermal pollution” of the river, the plant could instead be cooled by evaporation of river water. (This is more expensive, but in some areas it is environmentally preferable.) At what rate must the water evaporate? What fraction of the river must be evaporated?

PROBLEM 4P 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°C at the ocean surface and 4°C at the ocean floor. (a) What is the maximum possible efficiency of an engine operating between these two temperatures? ________________ (b) If the engine is to produce 1 GW of electrical power, what minimum volume of water must be processed (to suck out the heat) in every second?

PROBLEM 2P At a power plant that produces 1 GW (109 watts) of electricity, the steam turbines take in steam at a temperature of 500°C, and the waste heat is expelled into the environment at 20° C. (a) What is the maximum possible efficiency of this plant? ________________ (b)Suppose you develop a new material for making pipes and turbines, which allows the maximum steam temperature to be raised to 600°C. Roughly how much money can you make in a year by installing your improved hardware, if you sell the additional electricity for 5 cents per kilowatt-hour? (Assume that the amount of fuel consumed at the plant is unchanged.)

PROBLEM 5P Prove directly (by calculating the heat taken in and the heat expelled) that a Carnot engine using an ideal gas as the working substance has an efficiency of 1 ? Tc/Th.

PROBLEM 1P Recall below Problem, which concerned an ideal diatomic gas taken around a rectangular cycle on a PV diagram. Suppose now that this system is used as a heat engine, to convert the heat added into mechanical work. (a) Evaluate the efficiency of this engine for the case V2 = 3V1, P2 = 2P1 ________________ (b) Calculate the efficiency of an “ideal” engine operating between the same temperature extremes. Problem An ideal diatomic gas, in a cylinder with a movable piston, undergoes the rectangular cyclic process shown in below Figure. Assume that the temperature is always such that rotational degrees of freedom are active, but vibrational modes arc “frozen out.” Also assume that the only type of work done on the gas is quasistatic compression-expansion work. (a) For each of the four steps A through D, compute the work done on the gas, the heat added to the gas, and the change in the energy content of the gas. Express all answers in terms of P1, P2, V1. and V2. (Hint: Compute ?U before Q, using the ideal gas law and the equipartition theorem.) ________________ (b) Describe in words what is physically being done during each of the four steps; for example, during step A, heat is added to the gas (from an external flame or something) while the piston is held fixed. ________________ (c) Compute the net work done on the gas, the net heat added to the gas, and the net change in the energy of the gas during the entire cycle. Are the results as yon expected? Explain briefly. Figure: PV diagrams

PROBLEM 7P Why must you put an air conditioner in the window of a building, rather than in the middle of a room?

PROBLEM 6P To get more than an infinitesimal amount of work out of a Carnot engine, we would have to keep the temperature of its working substance below that of the hot reservoir and above that of the cold reservoir by non-infinitesimal amounts. Consider, then, a Carnot cycle in which the working substance is at temperature Thw as it absorbs heat from the hot reservoir, and at temperature Tcw as it expels heat to the cold reservoir. Under most circumstances the rates of heat transfer will be directly proportional to the temperature differences: I’ve assumed here for simplicity that the constants of proportionality (K) are the same for both of these processes. Let us also assume that both processes take the same amount of time, so the ?t’s are the same in both of these equations. (a) Assuming that no new entropy is created during the cycle except during the two heat transfer processes, derive an equation that relates the four temperatures Th, Tc, Thw, and Tcw. ________________ (b) Assuming that the time required for the two adiabatic steps is negligible, write down an expression for the power (work per unit time) output of this engine. Use the first and second laws to write the power entirely in terms of the four temperatures (and the constant K), then eliminate Tcw using the result of part (a). ________________ (c) When the cost of building an engine is much greater than the cost of fuel (as is often the case), it is desirable to optimize the engine for maximum power output, not maximum efficiency. Show that, for fixed Th, and Tc, the expression you found in part (b) has a maximum value at Thw = . (Hint: You’ll have to solve a quadratic equation.) Find the corresponding expression for Tcw. ________________ (d) Show that the efficiency of this engine is . Evaluate this efficiency numerically for a typical coal-fired steam turbine with Th = 600° C and Tc = 25° C, and compare to the ideal Carnot efficiency for this temperature range. Which value is closer to the actual efficiency, about 40%, of a real coal-burning power plant?

PROBLEM 8P Can you cool off your kitchen by leaving the refrigerator door open? Explain.

PROBLEM 9P Estimate the maximum possible COP of a household air conditioner. Use any reasonable values for the reservoir temperatures.

PROBLEM 10P Suppose that heat leaks into your kitchen refrigerator at an average rate of 300 watts. Assuming ideal operation, how much power must it draw from the wall?

PROBLEM 11P What is the maximum possible COP for a cyclic refrigerator operating between a high-temperature reservoir at 1 K and a low-temperature reservoir at 0.01 K?

PROBLEM 12P Explain why an ideal gas taken around a rectangular PV cycle, as considered in below Problem 1 and Problem 2, cannot be used (in reverse) for refrigeration. Problem: 1 An ideal diatomic gas, in a cylinder with a movable piston, undergoes the rectangular cyclic process shown in below Figure. Assume that the temperature is always such that rotational degrees of freedom are active, but vibrational modes arc “frozen out.” Also assume that the only type of work done on the gas is quasistatic compression-expansion work. (a) For each of the four steps A through D, compute the work done on the gas, the heat added Lo the gas, and the change in the energy content of the gas. Express all answers in terms of P1, P2, V1. and V2. (Hint: Compute ?U before Q, using the ideal gas law and the equipartition theorem.) ________________ (b) Describe in words what is physically being done during each of the four steps; for example, during step A, heat is added to the gas (from an external flame or something) while the piston is held fixed. ________________ (c) Compute the net work done on the gas, the net heat added to the gas, and the net change in the energy of the gas during the entire cycle. Are the results as yon expected? Explain briefly. Figure: PV Diagrams Problem: 2 To measure the heat capacity of an object, all you usually have to do is put it in thermal contact with another object whose heat capacity you know. As an example, suppose that a chunk of metal is immersed in boiling water (100°C), then is quickly transferred into a Styrofoam cup containing 250 g of water at 20°C. After a minute or so, the temperature of the contents of the cup is 24°C. Assume that during this time no significant energy is transferred between the contents of the cup and the surroundings. The heat capacity of the cup itselfis negligible. (a) How much heat is gained by the water? ________________ (b) How much heat is lost by the metal? ________________ (c) What is the heat capacity of this chunk of metal? ________________ (d) If the mass of the chunk of metal is 100 g, what is its specific heat capacity?

PROBLEM 13P Under many conditions, the rate at which heat enters an air conditioned building on a hot summer day is proportional to the difference in temperature between inside and outside, Th – Tc. (If the heal, enters entirely by conduction, this statement will certainly be true. Radiation from direct sunlight would be an exception.) Show that, under these conditions, the cost of air conditioning should be roughly proportional to the square of the temperature difference. Discuss the implications, giving a numerical example.

PROBLEM 14P A heat pump is an electrical device that heats a building by pumping heat in from the cold outside. In other words, it’s the same as a refrigerator, but its purpose is to warm the hot reservoir rather than to cool the cold reservoir (even though it does both). Let us define the following standard symbols, all taken to be positive by convention: Th = temperature inside building Tc = temperature outside Qh = heat pumped into building in 1 day Qc = heat taken from outdoors in 1 day W = electrical energy used by heat pump in 1 day (a) Explain why the “coefficient of performance” (COP) for a heat pump should be defined as Qh/W. ________________ (b) What relation among Qh, Qc and W is implied by energy conservation alone? Will energy conservation permit the COP to be greater than 1? ________________ (c) Use the second law of thermodynamics to derive an upper limit on the COP, in terms of the temperatures Th and Tc alone. ________________ (d) Explain why a heat pump is better than an electric furnace, which simply converts electrical work directly into heat. (Include some numerical estimates.)

PROBLEM 16P Prove that if you had a heat engine whose efficiency was better than the ideal value (4.5), you could hook it up to an ordinary Carnot refrigerator to make a refrigerator that requires no work input.

PROBLEM 15P In an absorption refrigerator, the energy driving the process is supplied not as work, but as heat from a gas flame. (Such refrigerators commonly use propane as fuel, and are used in locations where electricity is unavailable.*) Let us define the following symbols, all taken to be positive by definition: Qf = heat input from flame Qc = heat extracted from inside refrigerator Qr = waste heat expelled to room Tf = temperature of flame Tc = temperature inside refrigerator Tr = room temperature (a) Explain why the “coefficient of performance” (COP) for an absorption refrigerator should be defined as Qc/Qf. ________________ (b) What relation among Qf, Qc, and Qr is implied by energy conservation alone? Will energy conservation permit the COP to be greater than 1? ________________ (c) Use the second law of thermodynamics to derive an upper limit on the COP, in terms of the temperatures Tf, Tc, and Tr alone.

PROBLEM 18P Derive below equation for the efficiency of the Otto cycle. Equation:

PROBLEM 19P The amount of work done by each stroke of an automobile engine is controlled by the amount of fuel injected into the cylinder: the more fuel, the higher the temperature and pressure at points 3 and 4 in the cycle. But according to below equation, the efficiency of the cycle depends only on the compression ratio (which is always the same for any particular engine), not on the amount of fuel consumed. Do you think this conclusion still holds when various other effects such as friction are taken into account? Would you expect a real engine to be most efficient when operating at high power or at low power? Explain. Equation:

Derive a formula for the efficiency of the Diesel cycle, in terms of the compression ratio \(V_{1} / V_{2}\) and the cutoff ratio \(V_3/V_2\). Show that for a given compression ratio, the Diesel cycle is less efficient than the Otto cycle. Evaluate the theoretical efficiency of a Diesel engine with a compression ratio of 18 and a cutoff ratio of 2.

PROBLEM 17P Prove that if you had a refrigerator whose COP was better than the ideal value (4.9), you could hook it up to an ordinary Carnot engine to make an engine that produces no waste heat.

PROBLEM 21P The ingenious Stirling engine is a true heat engine that absorbs heat from an external source. The working substance can be air or any other gas. The engine consists of two cylinders with pistons, one in thermal contact with each reservoir (see below Figure). The pistons are connected to a crankshaft in a complicated way that we’ll ignore and let the engineers worry about. Between the two cylinders is a passageway where the gas flows past a regenerator: a temporary heat reservoir, typically made of wire mesh, whose temperature varies. Figure: A Stirling engine, shown during the power stroke when thehot piston is moving outward and the cold piston is at rest. (For simplicity,the linkages between the two pistons are not shown.)

PROBLEM 22P A small-scale steam engine might operate between the temperatures 20°C and 300°C, with a maximum steam pressure of 10 bars. Calculate the efficiency of a Rankine cycle with these parameters.

PROBLEM 23P Use the definition of enthalpy to calculate the change in enthalpy between points 1 and 2 of the Rankine cycle, for the same numerical parameters as used in the text. Recalculate the efficiency using your corrected value of H2, and comment on the accuracy of the approximation H2 ? H1.

Calculate the efficiency of a Rankine cycle that is modified from the parameters used in the text in each of the following three ways (one at a time), and comment briefly on the results: (a) reduce the maximum temperature to \(500^{\circ} \mathrm{C}\); (b) reduce the maximum pressure to 100 bars; (c) reduce the minimum temperature to \(10^{\circ} \mathrm{C}\).

In a real turbine, the entropy of the steam will increase somewhat. How will this affect the percentages of liquid and gas at point 4 in the cycle? How will the efficiency be affected?

PROBLEM 27P In below Table, why does the entropy of water increase with increasing temperature, while the entropy of steam decreases with increasing temperature? TABLE: Properties of saturated water/steam. Pressures are given in bars, where 1 bar = 105 Pa ? 1 atm. All values are for 1 kg of fluid, and are measured relative to liquid water at the triple point (0.01°C and 0.006 bar). Excerpted fromKeenan et al. (1978). T (°C) P (bar) Hwater (kJ) H steam (kJ) S water (kJ/K) S steam (kJ/K) 0 0.006 0 2501 0 9.156 10 0.012 42 2520 0.151 8.901 20 0.023 84 2538 0.297 8.667 30 0.042 126 2556 0.437 8.453 50 0.123 209 2592 0.704 8.076 100 1.013 419 2676 1.307 7.355

A coal-fired power plant, with parameters similar to those used in the text above, is to deliver 1 GW (\(10^9\) watts) of power. Estimate the amount of steam (in kilograms) that must pass through the turbine(s) each second.

PROBLEM 28P Imagine that your dog has eaten the portion of below Table 1 that gives entropy data; only the enthalpy data remains. Explain how you could reconstruct the missing portion of the table. Use your method to explicitly check a few of the entries for consistency. How much of below Table 2 could you reconstruct if it were missing? Explain. Table 1: Properties of saturated water/steam. Pressures are given in bars, where 1 bar = 105 Pa ? 1 atm. All values are for 1 kg of fluid, and are measured relative to liquid water at the triple point (0.01°C and 0.006 bar). Excerpted fromKeenan et al. (1978). T (°C) P (bar) Hwater (kJ) H steam (kJ) S water (kJ/K) S steam (kJ/K) 0 0.006 0 2501 0 9.156 10 0.012 42 2520 0.151 8.901 20 0.023 84 2538 0.297 8.667 30 0.042 126 2556 0.437 8.453 50 0.123 209 2592 0.704 8.076 100 1.013 419 2676 1.307 7.355 Table 2: Properties of superheated steam. All values are for 1 kg of fluid, and are measured relative to liquid water at the triple point. Excerpted from Keenan et al. (1978).

PROBLEM 29P Liquid HFC-134a at its boiling point at 12 bars pressure is throttled to 1 bar pressure. What is the final temperature? What fraction of the liquid vaporizes?

PROBLEM 30P Consider a household refrigerator that uses HFC-134a as the refrigerant, operating between the pressures of 1.0 bar and 10 bars. (a) The compression stage of the cycle begins with saturated vapour at 1 bar and ends at 10 bars. Assuming that the entropy is constant during compression, find the approximate temperature of the vapour after it is compressed. (You’ll have to do an interpolation between the values given in below Table 1) ________________ (b) Determine the enthalpy at each of the points 1, 2, 3, and 4, and calculate the coefficient of performance. Compare to the COP of a Carnot refrigerator operating between the same reservoir temperatures. Does this temperature range seem reasonable for a household refrigerator? Explain briefly. ________________ (c) What fraction of the liquid vaporizes during the throttling step. Table 1: Properties of superheated (gaseous) refrigerant HFC-134a. All values are for 1 kg fluid, and are measured relative to the same reference state as in Table 2. Excerpted from Moran and Shapiro 1995). Table 2: Properties of the refrigerant HFC-134a under saturated conditions (at its boiling point for each pressure). All values are for 1 kg of fluid, and are measured relative to an arbitrarily chosen refrence state, the saturated liquid at –40°C. Excerpted from Moran and Shapiro (1995). P (bar) T (°C) Hliquid (kJ) Hgas (kJ) Sliquid (kJ/K) Sgas (kJ/K) 1.0 ?26.4 16 231 0.068 0.940 1.4 ?18.8 26 236 0.106 0.932 2.0 ?10.1 37 241 0.148 0.925 4.0 8.9 62 252 0.240 0.915 6.0 21.6 79 259 0.300 0.910 8.0 31.3 93 264 0.346 0.907 10.0 39.4 105 268 0.384 0.904 12.0 46.3 116 271 0.416 0.902

PROBLEM 32P Suppose you are told to design a household air conditioner using HFC-134a as its working substance. Over what range of pressures would you have it operate? Explain your reasoning. Calculate the COP for your design, and compare to the COP of an ideal Carnot refrigerator operating between the same reservoir temperatures.

PROBLEM 31P Suppose that the throttling valve in the refrigerator of the previous problem is replaced with a small turbine-generator in which the fluid expands adiabatically, doing work that contributes to powering the compressor. Will this change affect the COP of the refrigerator? If so, by how much? Why do you suppose real refrigerators use a throttle instead of a turbine?

PROBLEM 33P Below Table gives experimental values of the molar enthalpy of nitrogen at 1 bar and 100 bars. Use this data to answer the following questions about a nitrogen throttling process operating between these two pressures. (a) If the initial temperature is 300 K, what is the final temperature? (Hint: You’ll have to do an interpolation between the tabulated values.) ________________ (b) If the initial temperature is 200 K, what is the final temperature? ________________ (c) If the initial temperature is 100 K, what is the final temperature? What fraction of the nitrogen ends up as a liquid in this case? ________________ (d) What is the highest initial temperature at which some liquefaction takes place? ________________ (e) What would happen if the initial temperature were 600 K? Explain. Table: Molar enthalpy of nitrogen (in joules) at 1 bar and 100 bars. Excerpted from Lide (1994) .

PROBLEM 35P The magnetic field created by a dipole has a strength of approximately (?0/4?)(?/r3), where r is the distance from the dipole and ?0 is the “permeability of free space,” equal to exactly 4? × 10–7 in SI units. (In the formula I’m neglecting the variation of field strength with angle, which is at most a factor of 2.) Consider a paramagnetic salt like iron ammonium alum, in which the magnetic moment ? of each dipole is approximately one Bohr magneton (9 × 10–24 J/T), with the dipoles separated by a distance of 1 nm. Assume that the dipoles interact only via ordinary magnetic forces. a) Estimate the strength of the magnetic field at the location of a dipole, due to its neighbouring dipoles. This is the effective field strength even when there is no externally applied field. ________________ b) If a magnetic cooling experiment using this material begins with an external field strength of 1 T, by about what factor will the temperature decrease when the external field is turned off? ________________ c) Estimate the temperature at which the entropy of this material rises most steeply as a function of temperature, in the absence of an externally applied field. ________________ d) If the final temperature in a cooling experiment is significantly less than the temperature you found in part (c), the material ends up in a state where ?S/?T is very small and therefore its heat capacity is very small. Explain why it would be impractical to try to reach such a low temperature with this material.

An apparent limit on the temperature achievable by laser cooling is reached when an atom’s recoil energy from absorbing or emitting a single photon is comparable to its total kinetic energy. Make a rough estimate of this limiting temperature for rubidium atoms that are cooled using laser light with a wavelength of 780 nm.

PROBLEM 35P The magnetic field created by a dipole has a strength of approximately (?0/4?)(?/r3), where r is the distance from the dipole and ?0 is the “permeability of free space,” equal to exactly 4? × 10–7 in SI units. (In the formula I’m neglecting the variation of field strength with angle, which is at most a factor of 2.) Consider a paramagnetic salt like iron ammonium alum, in which the magnetic moment ? of each dipole is approximately one Bohr magneton (9 × 10–24 J/T), with the dipoles separated by a distance of 1 nm. Assume that the dipoles interact only via ordinary magnetic forces. a) Estimate the strength of the magnetic field at the location of a dipole, due to its neighbouring dipoles. This is the effective field strength even when there is no externally applied field. ________________ b) If a magnetic cooling experiment using this material begins with an external field strength of 1 T, by about what factor will the temperature decrease when the external field is turned off? ________________ c) Estimate the temperature at which the entropy of this material rises most steeply as a function of temperature, in the absence of an externally applied field. ________________ d) If the final temperature in a cooling experiment is significantly less than the temperature you found in part (c), the material ends up in a state where ?S/?T is very small and therefore its heat capacity is very small. Explain why it would be impractical to try to reach such a low temperature with this material.