Engine and Refrigerators

A power plant produces$1\mathrm{GW}$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 $100 m^3/s$.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?

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 $22°\mathrm{C}$at the ocean surface and $4°\mathrm{C}$ 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 $1\mathrm{GW}$ of electrical power, what minimum volume of water must be processed (to suck out the heat) in every second?

Recall Problem 1.34, 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 $V_2=3V_1,P_2=2P_1$.

(b) Calculate the efficiency of an "ideal" engine operating between the same temperature extremes.

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 $\frac{1- t_c}{t_h}$

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$T_{hw}$as it absorbs heat from the hot reservoir, and at temperature$T_{cw}$as it expels heat to the cold reservoir. Under most circumstances the rates of heat transfer will be directly proportional to the temperature differences:

$\frac{Q_h}{\Delta t}=K\left(T_h-T_{hw}\right) \text{ and } \frac{Q_c}{\Delta t}=K\left(T_{cw}-T_c\right)$

I've assumed here for simplicity that the constants of proportionality $\left(K\right)$ are the same for both of these processes. Let us also assume that both processes take the

same amount of time, so the$\Delta t^{\text{'}}$'s are the same in both of these equations.*

$\left(a\right)$ 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$T_h,T_c,T_{hw}$ and $T_{cw}$

$\left(b\right)$ 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$T_{cw}$using the result of part $\left(a\right)$.

$\left(c\right)$ 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 $T_h$and $T_c$, the expression you found in part $\left(b\right)$ has a maximum value at $T_{hw}=\frac{1}{2}\left(T_h+\sqrt{T_h{T}_c}\right)$. (Hint: You'll have to solve a quadratic equation.) Find the corresponding$T_{cw}.$

$\left(d\right)$ Show that the efficiency of this engine is $1-\sqrt{T_c/T_h}$ Evaluate this efficiency numerically for a typical coal-fired steam turbine with $T_h=600°\mathrm{C}$and$T_c=25°\mathrm{C}$, 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?

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

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

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

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?

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

Explain why a rectangular P V cycle, as considered in Problems 1.34 and 4.1, cannot be used (in reverse) for refrigeration.

Explain why a rectangular $PV$ cycle, as considered in Problems $1.34$ and $4.1$, cannot be used (in reverse) for refrigeration.

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, $T_h-T_c$. (If the heat 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.

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:
$$\begin{gathered} T_h=\text{ temperature inside building } \\ {T}_c=\text{ temperature outside } \\ {Q}_h=\text{ heat pumped into building in }1\text{ day } \\ {Q}_c=\text{ heat taken from outdoors in }1\text{ day } \\ W=\text{ electrical energy used by heat pump in }1\text{ day } \end{gathered}$$
(a) Explain why the "coefficient of performance" (COP) for a heat pump should be defined as Q_h / W.
(b) What relation among Q_h , Q_c, 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 T_h and T_c 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.)

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:
Q_f= heat input from flame
Q_c= heat extracted from inside refrigerator
Q_r= waste heat expelled to room
T_f= temperature of flame
T_c= temperature inside refrigerator
T_r= room temperature

(a) Explain why the "coefficient of performance" (COP) for an absorption refrigerator should be defined as Q_c / Q_f.
(b) What relation among Q_f, Q_c, and Q_r 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 T_f, T_c, and T_r alone.

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.

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.

Derive a formula for the efficiency of the Diesel cycle, in terms of the compression ratio V₁/ V₂ and the cutoff ratio V₃/ V₂. 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.

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 Figure 4.7). 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

gradually from the hot side to the cold side. The heat capacity of the regenerator is very large, so its temperature is affected very little by the gas flowing past. The four steps of the engine's (idealized) cycle are as follows:
i. Power stroke. While in the hot cylinder at temperature T_h, the gas absorbs heat and expands isothermally, pushing the hot piston outward. The piston in the cold cylinder remains at rest, all the way inward as shown in the figure.
ii. Transfer to the cold cylinder. The hot piston moves in while the cold piston moves out, transferring the gas to the cold cylinder at constant volume. While on its way, the gas flows past the regenerator, giving up heat and cooling to T_c.
iii. Compression stroke. The cold piston moves in, isothermally compressing the gas back to its original volume as the gas gives up heat to the cold reservoir. The hot piston remains at rest, all the way in.
iv. Transfer to hot cylinder. The cold piston moves the rest of the way in while the hot piston moves out, transferring the gas back to the hot cylinder at constant volume. While on its way, the gas flows past the regenerator, absorbing heat until it is again at T_h.

(a) Draw a PV diagram for this idealized Stirling cycle.
(b) Forget about the regenerator for the moment. Then, during step 2, the gas will give up heat to the cold reservoir instead of to the regenerator; during step 4 , the gas will absorb heat from the hot reservoir. Calculate the efficiency of the engine in this case, assuming that the gas is ideal. Express your answer in terms of the temperature ratio T_c / T_h  and the compression ratio (the ratio of the maximum and minimum volumes). Show that the efficiency is less than that of a Carnot engine operating between the same temperatures. Work out a numerical example.
(c) Now put the regenerator back. Argue that, if it works perfectly, the efficiency of a Stirling engine is the same as that of a Carnot engine.
(d) Discuss, in some detail, the various advantages and disadvantages of a Stirling engine, compared to other engines.

Derive equation $4.10$ for the efficiency of the Otto cycle.

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 equation 4.10, 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.

At a power plant that produces 1 GW10⁹ watts) of electricity, the steam turbines take in steam at a temperature of 500^o, and the waste heat is expelled into the environment at 20^o
(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^o. 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.)

At a power plant that produces 1 GW10⁹ watts) of electricity, the steam turbines take in steam at a temperature of 500^o, and the waste heat is expelled into the environment at 20^o
(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^o. 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.)

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^oC; 

(b) reduce the maximum pressure to 100 bars; 

(c) reduce the minimum temperature to 10^oC.

Imagine that your dog has eaten the portion of Table 4.1 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 Table 4.2 could you reconstruct if it were missing? Explain.

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: 

$\left(a\right)$ reduce the maximum temperature to $500°\mathrm{C};$ 

$\left(b\right)$reduce the maximum pressure to $100$ bars;

$\left(c\right)$ reduce the minimum temperature to $10°\mathrm{C}$.

A small scale steam engine might operate between the temperatures $20°\text{C}$ and $300°\text{C}$, with a maximum steam pressure of $10$ bars. Calculate the efficiency of a Rankine cycle with these parameters.

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? 

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

In table 4.1, why does the entropy of water increase with increasing temperature, while the entropy of steam decreases with increasing temperature?

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 ${\text{H}}_2$, and comment on the accuracy of the approximation ${\text{H}}_2{\text{≈H}}_1$.

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 $500°\text{C}$; (b)reduce the maximum pressure to 100 bars; (c)reduce the minimum temperature to $10°\text{C}$.

Imagine that your dog has eaten the portion of Table 4.1 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 Table 4.2 could you reconstruct if it were missing? Explain.

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?

Table 4.3. Properties of the refrigerant HFC-134a under saturated conditions (at its boiling point for each pressure). All values are for $1 \mathrm{kg}$ of fluid, and are measured relative to an arbitrarily chosen reference state, the saturated liquid at $-40°$c. Excerpted from Moran and Shapiro (1995).

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 extreme temperatures.

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?

Consider a household refrigerator that uses HFC-134a as the refrigerant, operating between the pressures of $1.0\mathrm{bar}$ and $10\mathrm{bars}$.

(a) The compression stage of the cycle begins with saturated vapor at 1 bar and ends at 10 bars. Assuming that the entropy is constant during compression, find the approximate temperature of the vapor after it is compressed. (You'll have to do an interpolation between the values given in Table 4.4.)

(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 extreme temperatures. Does this temperature range seem reasonable for a household refrigerator? Explain briefly.

(c) What fraction of the liquid vaporizes during the throttling step?

Consider an ideal Hampson-Linde cycle in which no heat is lost to the environment.

(a) Argue that the combination of the throttling valve and the heat exchanger is a constant-enthalpy device, so that the total enthalpy of the fluid coming out of this combination is the same as the enthalpy of the fluid going in.

(b) Let $x$ be the fraction of the fluid that liquefies on each pass through the cycle. Show that

$x=\frac{H_{\text{out }}-H_{\text{in }}}{H_{\text{out }}-H_{\text{liq }}},$

where $H_{\text{in }}$ is the enthalpy of each mole of compressed gas that goes into the heat exchanger, $H_{\text{out }}$ is the enthalpy of each mole of low-pressure gas that comes out of the heat exchanger, and $H_{\text{liq }}$ is the enthalpy of each mole of liquid produced.

(c) Use the data in Table $4.5$ to calculate the fraction of nitrogen liquefied on each pass through a Hampson-Linde cycle operating between 1 bar and 100 bars, with an input temperature of $300 \mathrm{K}$. Assume that the heat exchanger works perfectly, so the temperature of the low-pressure gas coming out of it is the same as the temperature of the high-pressure gas going in. Repeat the calculation for an input temperature of $200 \mathrm{K}$.

Table 4.5 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 $300 \mathrm{K}$, what is the final temperature? (Hint: You'll have to do an interpolation between the tabulated values.)

(b) If the initial temperature is $200 \mathrm{K}$, what is the final temperature?

(c) If the initial temperature is $100 \mathrm{K}$, 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 $600 \mathrm{K}$? Explain.

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.

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.

The magnetic field created by a dipole has a strength of approximately $\left({\mu }_0/4\pi \right)\left(\mu /r^3\right)$, where r is the distance from the dipole and ${\mu }_0$ is the "permeability of free space," equal to exactly $4\pi \times {10}^{-7}$ 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 $\mu$ of each dipole is approximately one Bohr magneton $\left(9\times {10}^{-24} \mathrm{J}/\mathrm{T}\right)$, with the dipoles separated by a distance of $1 \mathrm{nm}$. 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 neighboring 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 $1 \mathrm{T}$, 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 $\partial S/\partial T$ 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.

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.