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The Nature of Science and Physics

Kinematics

Introduction to One-Dimensional KinematicsDisplacementVectors, Scalars, and Coordinate SystemsTime, Velocity, and SpeedVideo: One-Dimensional KinematicsAccelerationMotion Equations for Constant Acceleration in One DimensionProblem-Solving Basics for One-Dimensional KinematicsFalling ObjectsGraphical Analysis of One-Dimensional Motion

Two-Dimensional Kinematics

Dynamics: Force and Newton's Laws of Motion

Introduction to Dynamics: Newton's Laws of MotionDevelopment of Force ConceptNewton's First Law of Motion: InertiaNewton's Second Law of Motion: Concept of a SystemNewton's Third Law of Motion: Symmetry in ForcesVideo: Newton's LawsNormal, Tension, and Other Examples of ForcesProblem-Solving StrategiesFurther Applications of Newton's Laws of MotionExtended Topic: The Four Basic Forces—An Introduction

Further Applications of Newton's Laws: Friction, Drag, and Elasticity

Uniform Circular Motion and Gravitation

Introduction to Uniform Circular Motion and GravitationRotation Angle and Angular VelocityCentripetal AccelerationCentripetal ForceFictitious Forces and Non-inertial Frames: The Coriolis ForceNewton's Universal Law of GravitationVideo: GravitationSatellites and Kepler's Laws: An Argument for Simplicity

Work, Energy, and Energy Resources

Introduction to Work, Energy, and Energy ResourcesWork: The Scientific DefinitionKinetic Energy and the Work-Energy TheoremGravitational Potential EnergyVideo: Potential and Kinetic EnergyConservative Forces and Potential EnergyNonconservative ForcesConservation of EnergyPowerWork, Energy, and Power in HumansWorld Energy Use

Linear Momentum and Collisions

Rotational Motion and Angular Momentum

Introduction to Rotational Motion and Angular MomentumAngular AccelerationKinematics of Rotational MotionVideo: Rotational MotionDynamics of Rotational Motion: Rotational InertiaRotational Kinetic Energy: Work and Energy RevisitedAngular Momentum and Its ConservationVideo: Angular MomentumCollisions of Extended Bodies in Two DimensionsGyroscopic Effects: Vector Aspects of Angular Momentum

Statics and Torque

Fluid Statics

Introduction to Fluid StaticsWhat Is a Fluid?DensityPressureVariation of Pressure with Depth in a FluidPascal's PrincipleGauge Pressure, Absolute Pressure, and Pressure MeasurementArchimedes' PrincipleVideo: BuoyancyCohesion and Adhesion in Liquids: Surface Tension and Capillary ActionPressures in the Body

Fluid Dynamics and Its Biological and Medical Applications

Introduction to Fluid Dynamics and Biological and Medical ApplicationsFlow Rate and Its Relation to VelocityBernoulli's EquationVideo: Fluid FlowThe Most General Applications of Bernoulli's EquationViscosity and Laminar Flow; Poiseuille's LawThe Onset of TurbulenceMotion of an Object in a Viscous FluidMolecular Transport Phenomena: Diffusion, Osmosis, and Related Processes

Temperature, Kinetic Theory, and the Gas Laws

Heat and Heat Transfer Methods

Thermodynamics

Introduction to ThermodynamicsThe First Law of ThermodynamicsThe First Law of Thermodynamics and Some Simple ProcessesIntroduction to the Second Law of Thermodynamics: Heat Engines and Their EfficiencyCarnot's Perfect Heat Engine: The Second Law of Thermodynamics RestatedApplications of Thermodynamics: Heat Pumps and RefrigeratorsEntropy and the Second Law of Thermodynamics: Disorder and the Unavailability of EnergyStatistical Interpretation of Entropy and the Second Law of Thermodynamics: The Underlying Explanation

Oscillatory Motion and Waves

Introduction to Oscillatory Motion and WavesHooke's Law: Stress and Strain RevisitedPeriod and Frequency in OscillationsSimple Harmonic Motion: A Special Periodic MotionVideo: Harmonic MotionThe Simple PendulumEnergy and the Simple Harmonic OscillatorUniform Circular Motion and Simple Harmonic MotionDamped Harmonic MotionForced Oscillations and ResonanceWavesSuperposition and InterferenceEnergy in Waves: Intensity

Physics of Hearing

- Identify a Carnot cycle.
- Calculate maximum theoretical efficiency of a nuclear reactor.
- Explain how dissipative processes affect the ideal Carnot engine.

The novelty toy known as the drinking bird (seen in Figure 1) is an example of Carnot’s engine. It contains methylene chloride (mixed with a dye) in the abdomen, which boils at a very low temperature—about 100ºF . To operate, one gets the bird’s head wet. As the water evaporates, fluid moves up into the head, causing the bird to become top-heavy and dip forward 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 energy—the original head-wetting—the bird becomes a perpetual motion machine of sorts.

We know from the second law of thermodynamics that a heat engine cannot be 100% efficient, since there must always be some heat transfer

What is crucial to the Carnot cycle—and, in fact, defines it—is that only reversible processes are used. Irreversible processes involve dissipative factors, such as friction and turbulence. This increases heat transfer

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

Carnot also determined the efficiency of a perfect heat engine—that is, a Carnot engine. It is always true that the efficiency of a cyclical heat engine is given by:

$\displaystyle{Eff}=\frac{Q_{\text{h}}-Q_{\text{c}}}{Q_{\text{h}}}=1-\frac{Q_{\text{c}}}{Q_{\text{h}}}\\$

What Carnot found was that for a perfect heat engine, the ratio $\frac{Q_{\text{c}}}{Q_{\text{h}}}\\$

equals the ratio of the absolute temperatures of the heat reservoirs. That is, $\frac{Q_{\text{c}}}{Q_{\text{h}}}=\frac{T_{\text{c}}}{T_{\text{h}}}\\$

for a Carnot engine, so that the maximum or $\displaystyle{Eff}_{\text{C}}=1-\frac{T_{\text{c}}}{T_{\text{h}}}\\$

where Carnot’s interesting result implies that 100% efficiency would be possible only if

It is also apparent that the greatest efficiencies are obtained when the ratio

$\frac{T_{\text{c}}}{T_{\text{h}}}\\$

is as small as possible. Just as discussed for the Otto cycle in the previous section, 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 area inside the closed loop on the ${Eff}_{\text{C}}=1-\frac{T_{\text{c}}}{T_{\text{h}}}\\$

can be used to calculate the Carnot (maximum theoretical) efficiency. Those temperatures must first be converted to kelvins.$\displaystyle{Eff}_{\text{C}}=1-\frac{T_{\text{c}}}{T_{\text{h}}}\\$

.Thus,

$\begin{array}{lll}{Eff}_{\text{C}}&=&1-\frac{300\text{ K}}{573\text{ K}}\\\text{ }&=&0.476\text{, or }47.6\%\end{array}\\$

- The Carnot cycle is a theoretical cycle that is the most efficient cyclical process possible. Any engine using the Carnot cycle, which uses only reversible processes (adiabatic and isothermal), is known as a Carnot engine.
- Any engine that uses the Carnot cycle enjoys the maximum theoretical efficiency.
- While Carnot engines are ideal 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, but this has never been seen in nature.

- Think about the drinking bird at the beginning of this section (Figure 1). Although the bird enjoys the theoretical maximum efficiency possible, if left to its own devices over time, the bird will cease "drinking." What are some of the dissipative processes that might cause the bird’s motion to cease?
- Can improved engineering and materials be employed in heat engines to reduce heat transfer into the environment? Can they eliminate heat transfer into the environment entirely?
- Does the second law of thermodynamics alter the conservation of energy principle?

2. 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 1).

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 real heat engine that achieves 0.700 of the maximum efficiency, but still has an efficiency of 42.0% (and a cold reservoir at 27.0ºC)? (c) Does your answer imply practical limits to the efficiency of car gasoline engines?

4. Steam locomotives have 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 cold reservoir temperature were 150ºC?

5. Practical steam engines utilize 450ºC steam, which is later exhausted at 270ºC. (a) What is the maximum efficiency that such a heat engine can have? (b) Since 270ºC steam is still quite hot, a second 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) Show that this is the same efficiency as a single Carnot engine operating between 450ºC and 150ºC.

6. A coal-fired electrical power station has an efficiency of 38%. The temperature of the steam leaving the boiler is

$550\text{\textdegree }\text{C}$

. What percentage of the maximum efficiency does this station obtain? (Assume the temperature of the environment is $20\text{\textdegree }\text{C}$

.)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 K, has heat transfer to the environment at 300 K, and does 12 kJ of work? Explain your answer.

8.

9.

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

5. (a)

${\mathit{\text{Eff}}}_{\text{1}}=1-\frac{{T}_{\text{c,1}}}{{T}_{\text{h,1}}}=1-\frac{\text{543 K}}{\text{723 K}}=0\text{.}\text{249}\text{ or }\text{24}\text{.}9\%\\$

(b)

${\mathit{\text{Eff}}}_{2}=1-\frac{\text{423 K}}{\text{543 K}}=0\text{.}\text{221}\text{ or }\text{22}\text{.}1\%\\$

(c)

${\mathit{\text{Eff}}}_{1}=1-\frac{{T}_{\text{c,1}}}{{T}_{\text{h,1}}}\Rightarrow{T}_{\text{c,1}}={T}_{\text{h,1}}\left(1,-,{\mathit{\text{eff}}}_{1}\right)\text{similarly, }{T}_{\text{c,2}}={T}_{\text{h,2}}\left(1-{\mathit{\text{Eff}}}_{2}\right)\\$

using

$\begin{array}{l}{T}_{\text{c,2}}={T}_{\text{h,1}}\left(1-{Eff}_{1}\right)\left(1-{Eff}_{2}\right)\equiv{T}_{\text{h,1}}\left(1-{Eff}_{\text{overall}}\right)\\\therefore\left(1-{Eff}_{\text{overall}}\right)=\left(1-{\mathit{\text{Eff}}}_{1}\right)\left(1-{Eff}_{2}\right)\\{Eff}_{\text{overall}}=1-\left(1-0.249\right)\left(1-0.221\right)=41.5\%\end{array}\\$

(d) ${\text{Eff}}_{\text{overall}}=1-\frac{\text{423 K}}{\text{723 K}}=0\text{.}\text{415}\text{ or }\text{41}\text{.}5\\%\\$

7. The heat transfer to the cold reservoir is

${Q}_{\text{c}}={Q}_{\text{h}}-W=\text{25}\text{kJ}-\text{12}\text{kJ}=\text{13}\text{kJ}\\$

, so the efficiency is $\mathit{Eff}=1-\frac{{Q}_{\text{c}}}{{Q}_{\text{h}}}=1-\frac{\text{13}\text{kJ}}{\text{25}\text{kJ}}=0\text{.}\text{48}\\$

. The Carnot efficiency is ${\mathit{\text{Eff}}}_{\text{C}}=1-\frac{{T}_{\text{c}}}{{T}_{\text{h}}}=1-\frac{\text{300}\text{K}}{\text{600}\text{K}}=0\text{.}\text{50}\\$

. The actual efficiency is 96% of the Carnot efficiency, which is much higher than the best-ever achieved of about 70%, so her scheme is likely to be fraudulent.9. (a) -56.3ºC (b) The temperature is too cold for the output of a steam engine (the local environment). It is below the freezing point of water. (c) The assumed efficiency is too high.