Home  ›  To Increase the Maximum Efficiency of a Heat Engine, What Can You Do? Choose All That Apply.

To Increase the Maximum Efficiency of a Heat Engine, What Can You Do? Choose All That Apply.

Written By Wednesday, February 16, 2022 Add Comment Edit

Learning Objectives

Past the end of this section, yous will exist able to:

  • Identify a Carnot cycle.
  • Calculate maximum theoretical efficiency of a nuclear reactor.
  • Explicate how dissipative processes affect the platonic Carnot engine.

Photograph of a novelty toy known as the drinking bird. It is made up of two glass bulbs connected to each other by a glass tube. The upper bulb is shaped like a bird's head, and the tube looks like its neck. The lower bulb, which may be compared to the abdomen, contains methylene chloride that has been colored red. The bottom of the neck is attached to a pivot, and in front of the bird's head is a glass of water.

Effigy 1. A drinking bird (credit: Arabesk.nl, Wikimedia Commons)

The novelty toy known as the drinking bird (seen in Figure 1) is an instance of Carnot'southward engine. It contains methylene chloride (mixed with a dye) in the belly, which boils at a very low temperature—nigh 100ºF . To operate, one gets the bird's caput wet. Equally the water evaporates, fluid moves upward into the caput, causing the bird to become peak-heavy and dip frontward back into the water. This cools down the methylene chloride in the head, and it moves back into the abdomen, causing the bird to become bottom heavy and tip up. Except for a very small input of free energy—the original head-wetting—the bird becomes a perpetual motion motorcar of sorts.

We know from the second constabulary of thermodynamics that a heat engine cannot be 100% efficient, since there must always be some heat transfer Q c to the environment, which is often called waste product heat. How efficient, then, can a heat engine be? This question was answered at a theoretical level in 1824 past a immature French engineer, Sadi Carnot (1796–1832), in his study of the and then-emerging heat engine technology crucial to the Industrial Revolution. He devised a theoretical cycle, now called the Carnot bicycle, which is the most efficient cyclical process possible. The second law of thermodynamics can be restated in terms of the Carnot cycle, and so what Carnot really discovered was this fundamental police. Any heat engine employing the Carnot bicycle is called a Carnot engine.

What is crucial to the Carnot cycle—and, in fact, defines information technology—is that only reversible processes are used. Irreversible processes involve dissipative factors, such equally friction and turbulence. This increases oestrus transfer Q c to the environment and reduces the efficiency of the engine. Plainly, then, reversible processes are superior.

Carnot Engine

Stated in terms of reversible processes, the 2nd law of thermodynamics has a 3rd form:

A Carnot engine operating betwixt two given temperatures has the greatest possible efficiency of any estrus engine operating between these two temperatures. Furthermore, all engines employing only reversible processes take this same maximum efficiency when operating between the aforementioned given temperatures.

Effigy two shows the PV diagram for a Carnot cycle. The cycle comprises two isothermal and two adiabatic processes. Think that both isothermal and adiabatic processes are, in principle, reversible.

Carnot also determined the efficiency of a perfect estrus engine—that is, a Carnot engine. It is e'er truthful that the efficiency of a cyclical heat engine is given past:

[latex]\displaystyle{Eff}=\frac{Q_{\text{h}}-Q_{\text{c}}}{Q_{\text{h}}}=1-\frac{Q_{\text{c}}}{Q_{\text{h}}}\\[/latex]

What Carnot institute was that for a perfect heat engine, the ratio [latex]\frac{Q_{\text{c}}}{Q_{\text{h}}}\\[/latex] equals the ratio of the absolute temperatures of the rut reservoirs. That is, [latex]\frac{Q_{\text{c}}}{Q_{\text{h}}}=\frac{T_{\text{c}}}{T_{\text{h}}}\\[/latex] for a Carnot engine, and then that the maximum or Carnot efficiencyEff C is given past

[latex]\displaystyle{Eff}_{\text{C}}=1-\frac{T_{\text{c}}}{T_{\text{h}}}\\[/latex]

where T h and T c are in kelvins (or any other accented temperature scale). No real heat engine can practice as well as the Carnot efficiency—an actual efficiency of nearly 0.vii of this maximum is usually the best that can be accomplished. But the ideal Carnot engine, like the drinking bird to a higher place, while a fascinating novelty, has nada power. This makes information technology unrealistic for whatever applications.

Carnot's interesting upshot implies that 100% efficiency would be possible only if T c = 0 K—that is, only if the cold reservoir were at absolute goose egg, a practical and theoretical impossibility. But the physical implication is this—the just way to accept all heat transfer go into doing work is to remove all thermal energy, and this requires a common cold reservoir at absolute zero.

It is likewise apparent that the greatest efficiencies are obtained when the ratio [latex]\frac{T_{\text{c}}}{T_{\text{h}}}\\[/latex] is equally small equally possible. Just as discussed for the Otto cycle in the previous department, this means that efficiency is greatest for the highest possible temperature of the hot reservoir and lowest possible temperature of the cold reservoir. (This setup increases the expanse inside the closed loop on the PV diagram; also, information technology seems reasonable that the greater the temperature difference, the easier it is to divert the heat transfer to work.) The actual reservoir temperatures of a heat engine are usually related to the type of oestrus source and the temperature of the environs into which oestrus transfer occurs. Consider the following case.

Part a of the figure shows a graph of pressure P versus volume V for a Carnot cycle. The pressure P is along the Y axis and the volume V is along the X axis. The graph shows a complete cycle A B C D. The path begins at point A, then it moves smoothly down till point B along the direction of the X axis. This is marked as an isotherm at temperature T sub h. Then the curve drops down further, along a different curve, from point B to point C. This is marked as adiabatic expansion. The curve rises from point C to point D along the direction opposite to that of A B. This is also an isotherm but at temperature T sub c. The last part of the curve rises up from point D back to A along a direction opposite to that of B C. This is marked as adiabatic compression. The path C D is lower than path A B. Heat Q sub h enters the system, as shown by a bold arrow to the curve A B. Heat Q sub c leaves the system as shown by a bold arrow near C D. Part b of the diagram shows an internal combustion engine represented as a circle. The hot reservoir is a rectangular section at the top of the circle shown at temperature T sub h. A cold reservoir is shown as a rectangular section in the bottom part of the circle at temperature T sub c. Heat Q sub h enters the heat engine as shown by a bold arrow; work W is produced as output, shown to leave the system, and the remaining heat Q sub c is returned back to the cold reservoir, as shown by a bold arrow toward it.

Figure 2. PV diagram for a Carnot bike, employing only reversible isothermal and adiabatic processes. Heat transfer Q h occurs into the working substance during the isothermal path AB, which takes place at constant temperature T h. Heat transfer Q c occurs out of the working substance during the isothermal path CD, which takes place at abiding temperature T c. The net piece of work output W equals the area inside the path ABCDA. Also shown is a schematic of a Carnot engine operating between hot and common cold reservoirs at temperatures T h and T c. Any estrus engine using reversible processes and operating between these two temperatures will have the same maximum efficiency equally the Carnot engine.

Example 1. Maximum Theoretical Efficiency for a Nuclear Reactor

A nuclear power reactor has pressurized water at 300ºC. (Higher temperatures are theoretically possible but practically not, due to limitations with materials used in the reactor.) Oestrus transfer from this h2o is a complex process (meet Figure 3). Steam, produced in the steam generator, is used to drive the turbine-generators. Eventually the steam is condensed to water at 27ºC and and so heated again to start the cycle over. Calculate the maximum theoretical efficiency for a heat engine operating between these two temperatures.

Diagram shows a schematic diagram of a pressurized water nuclear reactor and the steam turbines that convert work into electrical energy. There is a pressure vessel in the middle, dome shaped at the ends. This has a nuclear core in it. The core is a small square in the center of the reactor. Control rods are shown as sticks of equal length attached to the core. The pressure vessel has some coolant tubes passing through it and then back to a steam chamber. These coolant tubes contain a coolant liquid that transports the heat from the pressure vessel to the steam chamber. This whole system is enclosed in another dome shaped containment structure of steel. The water supply to steam chamber and the steam outlet are seen to come out of this chamber. This steam is now shown to run two steam turbines, one a high pressure one and another low pressure one. The turbines are nearly triangular and segmented in shape. The steam turbine in turn generates power using a turbine generator, which is attached to the turbine system. The turbines are again housed in another chamber which gets the steam from the steam chamber and return the steam as water back to the steam chamber with pipes. A coolant tower is shown near the turbine system, which is shown to supply cool water in tubes to the turbine system to cool the steam back to water.

Figure 3. Schematic diagram of a pressurized water nuclear reactor and the steam turbines that catechumen piece of work into electrical energy. Heat exchange is used to generate steam, in function to avoid contamination of the generators with radioactive decay. Two turbines are used considering this is less expensive than operating a single generator that produces the same amount of electrical free energy. The steam is condensed to liquid before being returned to the rut exchanger, to keep get out steam pressure low and aid the flow of steam through the turbines (equivalent to using a lower-temperature common cold reservoir). The considerable energy associated with condensation must be dissipated into the local environment; in this example, a cooling tower is used so in that location is no direct heat transfer to an aquatic environs. (Annotation that the water going to the cooling belfry does not come into contact with the steam flowing over the turbines.)

Strategy

Since temperatures are given for the hot and cold reservoirs of this oestrus engine, [latex]{Eff}_{\text{C}}=1-\frac{T_{\text{c}}}{T_{\text{h}}}\\[/latex] can be used to calculate the Carnot (maximum theoretical) efficiency. Those temperatures must first be converted to kelvins.

Solution

The hot and common cold reservoir temperatures are given as 300ºC and 27.0ºC, respectively. In kelvins, then, T h = 573 Thousand and T c = 300 K, and so that the maximum efficiency is [latex]\displaystyle{Eff}_{\text{C}}=1-\frac{T_{\text{c}}}{T_{\text{h}}}\\[/latex].

Thus,

[latex]\begin{array}{lll}{Eff}_{\text{C}}&=&ane-\frac{300\text{ K}}{573\text{ K}}\\\text{ }&=&0.476\text{, or }47.6\%\end{array}\\[/latex]

Discussion

A typical nuclear ability station's actual efficiency is about 35%, a little amend than 0.7 times the maximum possible value, a tribute to superior engineering. Electrical power stations fired past coal, oil, and natural gas have greater actual efficiencies (near 42%), because their boilers can attain higher temperatures and pressures. The cold reservoir temperature in whatsoever of these power stations is express by the local environment. Figure 4 shows (a) the exterior of a nuclear power station and (b) the exterior of a coal-fired power station. Both have cooling towers into which water from the condenser enters the tower near the height and is sprayed downwards, cooled by evaporation.

Part a shows a photograph of an operational nuclear power plant in night view. There are dome shaped structures which house radioactive material and vapors are shown to come from two cooling towers. Part b shows a photograph of a coal fired power plant. Several huge cooling towers are shown.

Figure 4. (a) A nuclear ability station (credit: BlatantWorld.com) and (b) a coal-fired ability station. Both have cooling towers in which water evaporates into the surroundings, representing Q c. The nuclear reactor, which supplies Q h, is housed inside the dome-shaped containment buildings. (credit: Robert & Mihaela Vicol, publicphoto.org)

Since all real processes are irreversible, the actual efficiency of a estrus engine tin never be as great equally that of a Carnot engine, as illustrated in Figure 5a. Even with the best heat engine possible, at that place are always dissipative processes in peripheral equipment, such as electrical transformers or auto transmissions. These further reduce the overall efficiency past converting some of the engine's work output dorsum into heat transfer, as shown in Effigy 5b.

Part a of the diagram shows a combustion engine represented as a circle to compare the efficiency of real and Carnot engines. The hot reservoir is a rectangular section above the circle shown at temperature T sub h. A cold reservoir is shown as a rectangular section below the circle at temperature T sub c. Heat Q sub h enters the heat engine as shown by a bold arrow. For a real engine a small part of it is shown to be expelled as output from the engine shown as a bold arrow leaving the circle and for a Carnot engine larger part of it is shown to leave as work shown by a dashed arrow leaving the circle. The remaining heat is shown to be returned back to the cold reservoir as shown by bold arrow toward it for real engines and comparatively lesser heat is given by the Carnot engine shown by a dashed arrow. Part b of the diagram shows an internal combustion engine represented as a circle to study friction and other dissipative processes in the output mechanisms of a heat engine. The hot reservoir is a rectangular section above the circle shown at temperature T sub h. A cold reservoir is shown as a rectangular section below the circle at temperature T sub c. Heat Q sub h enters the heat engine as shown by a bold arrow, work W is produced as output, shown leaving the system, and the remaining heat Q sub c and Q sub f are returned back to the cold reservoir as shown by bold arrows toward it. Q sub f is heat due to friction. The work done against friction goes as heat Q sub f to the cold reservoir.

Figure five. Existent heat engines are less efficient than Carnot engines. (a) Real engines apply irreversible processes, reducing the rut transfer to piece of work. Solid lines represent the bodily procedure; the dashed lines are what a Carnot engine would do between the same ii reservoirs. (b) Friction and other dissipative processes in the output mechanisms of a estrus engine convert some of its work output into rut transfer to the surround.

Section Summary

  • The Carnot cycle is a theoretical wheel that is the most efficient cyclical procedure possible. Any engine using the Carnot cycle, which uses just reversible processes (adiabatic and isothermal), is known as a Carnot engine.
  • Any engine that uses the Carnot wheel enjoys the maximum theoretical efficiency.
  • While Carnot engines are platonic engines, in reality, no engine achieves Carnot's theoretical maximum efficiency, since dissipative processes, such as friction, play a role. Carnot cycles without heat loss may be possible at absolute zero, merely this has never been seen in nature.

Conceptual Questions

  1. Think about the drinking bird at the start of this section (Figure ane). Although the bird enjoys the theoretical maximum efficiency possible, if left to its own devices over fourth dimension, the bird will end "drinking." What are some of the dissipative processes that might crusade the bird'due south motility to cease?
  2. Can improved engineering and materials be employed in heat engines to reduce heat transfer into the surround? Can they eliminate heat transfer into the surroundings entirely?
  3. Does the second law of thermodynamics modify the conservation of energy principle?

Problems & Exercises

ane. A certain gasoline engine has an efficiency of thirty.0%. What would the hot reservoir temperature be for a Carnot engine having that efficiency, if it operates with a cold reservoir temperature of 200ºC?

ii. A gas-cooled nuclear reactor operates between hot and cold reservoir temperatures of 700ºC and 27.0ºC. (a) What is the maximum efficiency of a heat engine operating between these temperatures? (b) Find the ratio of this efficiency to the Carnot efficiency of a standard nuclear reactor (found in Example ane).

3. (a) What is the hot reservoir temperature of a Carnot engine that has an efficiency of 42.0% and a cold reservoir temperature of 27.0ºC? (b) What must the hot reservoir temperature be for a existent rut engine that achieves 0.700 of the maximum efficiency, merely yet has an efficiency of 42.0% (and a cold reservoir at 27.0ºC)? (c) Does your respond imply applied limits to the efficiency of motorcar gasoline engines?

4. Steam locomotives take an efficiency of 17.0% and operate with a hot steam temperature of 425ºC. (a) What would the cold reservoir temperature be if this were a Carnot engine? (b) What would the maximum efficiency of this steam engine be if its common cold reservoir temperature were 150ºC?

v. Practical steam engines utilize 450ºC steam, which is later exhausted at 270ºC. (a) What is the maximum efficiency that such a rut engine can have? (b) Since 270ºC steam is still quite hot, a 2d steam engine is sometimes operated using the exhaust of the first. What is the maximum efficiency of the second engine if its exhaust has a temperature of 150ºC? (c) What is the overall efficiency of the two engines? (d) Prove that this is the same efficiency equally a single Carnot engine operating between 450ºC and 150ºC.

vi. A coal-fired electric power station has an efficiency of 38%. The temperature of the steam leaving the banality is [latex]\text{550}\text{\textordmasculine }\text{C}[/latex] . What percent of the maximum efficiency does this station obtain? (Assume the temperature of the environment is [latex]\text{20}\text{\textordmasculine }\text{C}[/latex] .)

7. Would you be willing to financially back an inventor who is marketing a device that she claims has 25 kJ of heat transfer at 600 Yard, has heat transfer to the environment at 300 K, and does 12 kJ of work? Explain your respond.

eight. Unreasonable Results(a) Suppose you want to design a steam engine that has heat transfer to the environment at 270ºC and has a Carnot efficiency of 0.800. What temperature of hot steam must you utilize? (b) What is unreasonable about the temperature? (c) Which premise is unreasonable?

9. Unreasonable ResultsCalculate the cold reservoir temperature of a steam engine that uses hot steam at 450ºC and has a Carnot efficiency of 0.700. (b) What is unreasonable almost the temperature? (c) Which premise is unreasonable?

Glossary

Carnot cycle: a cyclical process that uses just reversible processes, the adiabatic and isothermal processes

Carnot engine: a rut engine that uses a Carnot cycle

Carnot efficiency: the maximum theoretical efficiency for a heat engine

Selected Solutions to Issues & Exercises

1. 403ºC

iii. (a) 244ºC; (b) 477ºC; (c)Yes, since automobiles engines cannot get too hot without overheating, their efficiency is limited.

five. (a) [latex]{\mathit{\text{Eff}}}_{\text{1}}=1-\frac{{T}_{\text{c,one}}}{{T}_{\text{h,i}}}=1-\frac{\text{543 K}}{\text{723 K}}=0\text{.}\text{249}\text{ or }\text{24}\text{.}9\%\\[/latex]

(b) [latex]{\mathit{\text{Eff}}}_{2}=ane-\frac{\text{423 1000}}{\text{543 K}}=0\text{.}\text{221}\text{ or }\text{22}\text{.}1\%\\[/latex]

(c) [latex]{\mathit{\text{Eff}}}_{1}=1-\frac{{T}_{\text{c,1}}}{{T}_{\text{h,ane}}}\Rightarrow{T}_{\text{c,i}}={T}_{\text{h,1}}\left(1,-,{\mathit{\text{eff}}}_{1}\right)\text{similarly, }{T}_{\text{c,ii}}={T}_{\text{h,ii}}\left(one-{\mathit{\text{Eff}}}_{ii}\correct)\\[/latex]

using T h,ii = T c,one in higher up equation gives

[latex]\begin{array}{fifty}{T}_{\text{c,two}}={T}_{\text{h,1}}\left(i-{Eff}_{1}\right)\left(1-{Eff}_{2}\right)\equiv{T}_{\text{h,one}}\left(one-{Eff}_{\text{overall}}\right)\\\therefore\left(i-{Eff}_{\text{overall}}\right)=\left(i-{\mathit{\text{Eff}}}_{ane}\right)\left(1-{Eff}_{two}\correct)\\{Eff}_{\text{overall}}=one-\left(1-0.249\right)\left(i-0.221\right)=41.five\%\end{assortment}\\[/latex]

(d) [latex]{\text{Eff}}_{\text{overall}}=1-\frac{\text{423 Yard}}{\text{723 K}}=0\text{.}\text{415}\text{ or }\text{41}\text{.}five\\%\\[/latex]

seven. The rut transfer to the cold reservoir is [latex]{Q}_{\text{c}}={Q}_{\text{h}}-W=\text{25}\text{kJ}-\text{12}\text{kJ}=\text{13}\text{kJ}\\[/latex], and so the efficiency is [latex]\mathit{Eff}=i-\frac{{Q}_{\text{c}}}{{Q}_{\text{h}}}=1-\frac{\text{13}\text{kJ}}{\text{25}\text{kJ}}=0\text{.}\text{48}\\[/latex]. The Carnot efficiency is [latex]{\mathit{\text{Eff}}}_{\text{C}}=one-\frac{{T}_{\text{c}}}{{T}_{\text{h}}}=1-\frac{\text{300}\text{K}}{\text{600}\text{Yard}}=0\text{.}\text{l}\\[/latex]. The actual efficiency is 96% of the Carnot efficiency, which is much higher than the all-time-ever achieved of about 70%, so her scheme is likely to be fraudulent.

ix. (a) -56.3ºC (b) The temperature is also common cold for the output of a steam engine (the local environment). Information technology is beneath the freezing bespeak of water. (c) The assumed efficiency is as well high.

flahertylethe1985.blogspot.com

Source: https://courses.lumenlearning.com/physics/chapter/15-4-carnots-perfect-heat-engine-the-second-law-of-thermodynamics-restated/

0 Response to "To Increase the Maximum Efficiency of a Heat Engine, What Can You Do? Choose All That Apply."

Post a Comment

Subscribe to: Post Comments (Atom)

Iklan Atas Artikel

Iklan Tengah Artikel 1

Iklan Tengah Artikel 2

Iklan Bawah Artikel