About 86 % of the world's electrical energy is produced by...

About $86 \%$ of the world's electrical energy is produced by using steam turbines, a form of heat engine. In his analysis of an ideal heat engine, Sadi Carnot concluded that the maximum possible efficiency is defined by the total work that could be done by the engine, divided by the quantity of heat available to do the work (for example, from hot steam produced by combustion of a fuel such as coal or methane). This efficiency is given by the ratio $\left(T_{\text {high }}-T_{\text {low }}\right) / T_{\text {high }}$, where $T_{\text {high }}$ is the temperature of the heat going into the engine and $T_{\text {low }}$ is that of the heat leaving the engine. (a) What is the maximum possible efficiency of a heat engine operating between an input temperature of $700 \mathrm{~K}$ and an exit temperature of $288 \mathrm{~K} ?$ (b) Why is it important that electrical power plants be located near bodies of relatively cool water? (c) Under what conditions could a heat engine operate at or near $100 \%$ efficiency? (d) It is often said that if the energy of combustion of a fuel such as methane were captured in an electrical fuel cell instead of by burning the fuel in a heat engine, a greater fraction of the energy could be put to useful work. Make a qualitative drawing like that in Figure 5.10 that illustrates the fact that in principle the fuel cell route will produce more useful work than the heat engine route from combustion of methane.

About $86 \%$ of the world's electrical energy is produced by using steam turbines, a form of heat engine. In his analysis of an ideal heat engine, Sadi Carnot concluded that the maximum possible efficiency is defined by the total work that could be done by the engine, divided by the quantity of heat available to do the work (for example, from hot steam produced by combustion of a fuel such as coal or methane). This efficiency is given by the ratio $\left(T_{\text {high }}-T_{\text {low }}\right) / T_{\text {high }}$, where $T_{\text {high }}$ is the temperature of the heat going into the engine and $T_{\text {low }}$ is that of the heat leaving the engine.
(a) What is the maximum possible efficiency of a heat engine operating between an input temperature of $700 \mathrm{~K}$ and an exit temperature of $288 \mathrm{~K} ?$ (b) Why is it important that electrical power plants be located near bodies of relatively cool water? (c) Under what conditions could a heat engine operate at or near $100 \%$ efficiency?
(d) It is often said that if the energy of combustion of a fuel such as methane were captured in an electrical fuel cell instead of by burning the fuel in a heat engine, a greater fraction of the energy could be put to useful work. Make a qualitative drawing like that in Figure 5.10 that illustrates the fact that in principle the fuel cell route will produce more useful work than the heat engine route from combustion of methane.

Key Concepts

Carnot Efficiency

Carnot efficiency defines the maximum theoretical efficiency for a heat engine operating between two thermal reservoirs. It quantifies the highest possible fraction of heat converted into work based solely on the temperatures of the heat source and heat sink, illustrating the fundamental limits imposed by the temperature difference between them.

Temperature Differential and Heat Sink

The performance of any heat engine depends critically on the difference in temperature between the heat input and the heat rejection. A large temperature differential allows for higher efficiency, and an effective heat sink (often provided by cool water bodies) is essential to maintain a low exit temperature, thereby maximizing the engine's efficiency.

Second Law of Thermodynamics

The second law of thermodynamics sets the boundary for energy conversion processes by indicating that no heat engine can convert all the heat taken from a thermal reservoir into work. Some energy will always be lost as waste heat to the surroundings, which explains why 100% efficiency is unattainable under normal conditions.

Fuel Cells and Direct Energy Conversion

Fuel cells operate by directly converting chemical energy into electrical energy through electrochemical reactions, bypassing the intermediate conversion to heat. This direct energy conversion can achieve a higher fraction of useful work compared to traditional heat engines, which are limited by the Carnot efficiency due to inherent thermal losses.

Transcript

00:01 One way that we can look at the efficiency of a heat engine is to take a look at the ratio of the temperature in and the temperature out, or the high temperature minus the low temperature over the high temperature, where the high temperature is the heat going into the process and the low temperature is what's coming out.

00:34 And so if we took a look at a process that started at 700 kelvin and went down to 288 kelvin, what would the efficiency of that process be? well, we can go ahead and take a look at our equation here and plug those values in to see how efficient this process is.

01:00 And when we wind up doing the math, we wind up with an efficiency ratio, of 0 .589, which means that it is 58 .9 % efficient.

01:21 Now, a lot of electrical power plants are found near water.

01:29 And so why would we want something near a cool pool of water? and the main reason for that is the cooler the water is, the cooler the exit temperature is going to be.

01:48 And the cooler the exit temperature, that means the cooler the t low is.

02:01 And if we have a lower t low, that means we're going to have a higher efficiency for our system.

02:17 And so if we were to try to figure out, you know, what condition could an engine operate at such that it would be near, you know, 100 % efficient? well, something that's going to be near 100 % efficient is going to have to have the t high and the t high, okay, essentially have this ratio be 1, which means that the t low is going to have to be as close.

02:47 To zero as possible.

02:50 So we're going to have to have temperatures near zero kelvin for a process to be near 100 % efficient.

03:04 Now, we produce energy in various ways.

03:09 One way is to use fossil fuels...

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