The Second Law of Thermodynamics

Goals

1. Understand the flow of heat and work in heat engines and heat pumps
2. Derive the concept of efficiency - the ratio of input to output energy
3. Understand the experimental evidence and theoretical limit to efficiency
4. Develop the concept of Entropy to better understand this limit.

Energy Flow of the Heat Engine I

• A heat engine is a machine that takes heat (from chemical combustion or nuclear fission) and produces useful work.

  • Heat engines date to Ancient Greece, but the Greeks treated them like novelties. No historical or archaelogical evidence supports practical use.
  • The symbol for the US Navy Rate Boiler Technician is Hero’s Engine, the first known heat engine circa 50 CE.

Energy Flow of the Heat Engine II

• Spontaneous thermodynamic processes are irreversible - Per the 0^th Law, they stop once thermal equilibrium is reached.
• To get a continuous flow of work, heat engines must be engineered to be reversible so the cycle can repeat.
• Reversible processes must be close to equilibrium, such that only small changes to temperature are needed to reverse the heat flow.
• Reversible processes area an idealization, so it practice one must make the system approximately reversible.

Disorder and Thermo Dynamic Processes

• Underlying Thermodynamic are the concepts of order and disorder.

• Nature will spontaneously tend to greater disorder over time.

  • Think about the state of your bedroom over time.

• Knowing where and when disorder is increasing is a crucial skill in heat engine or pump design.

Heat Engines

• A heat engine does useful work from heating a Working Fluid.

  • Hydrocarbons combust into hot steam and carbon dioxide in Internal Combustion Engines, which does work on a piston.
  • Chemical Combustion or Nuclear Fission heat water into steam in Steam Engines, which does work on pistons (Steam Trains) or turbines (Power Plants or Ships).

• Heat Engines work on a Cyclic Process, the engine is designed such that the machinery returns to it’s initial position.

  • Piston Engines go between (Intake, Compression) power and exhaust strokes.
  • Turbines use circular motion.

Heat Source and Heat Sink

• The Book (physicists) calls these ‘Hot and Cold Reservoirs’.

• ‘On the Boat’ we called these heat sources and heat sinks.

• The heat source is the process that provides the heat.

• The heat sink is the the process that removes the unused heat.

  • So the cycle can repeat.

• Note that since the Heat Engine is Cyclic, \(\Delta U = 0\):

\[ 0 = Q - W \\ Q = W \]

• This is just mathematically stating that the Heat does the work.

Efficiency

• Total Heat is then separated into two amounts: \(Q_H\) the heat generated via the source, and \(Q_C\) heat removed via the sink.

\[ Q = Q_H + Q_C = |Q_H| - |Q_C| \\ W = Q \\ W = |Q_H| - |Q_C| \]

• Graphically:

• Efficiency (e) is simply the ratio of useful work, \(W\), to input heat, \(Q_H\).

\[ e = \frac{W}{Q_H} = \frac{|Q_H| - |Q_C|}{Q_H} = 1 - \frac{|Q_C|}{Q_H} \]

Question

If a heat engine is 25% efficient and produces 4000 KJ of heat at the heat source, how much energy is exhausted from the heat sink?

Answer

\[ e = 1 - \frac{|Q_C|}{Q_H} \\ \frac{|Q_C|}{Q_H} = 1 - e \\ |Q_C| = Q_H (1-e) \\ |Q_C| = 4000 KJ (1 - 0.25) \\ |Q_C| = 3000 KJ \]

If any of you, like me, have had the bad luck of touching the exhaust pipe of a running vehicle, you know they get VERY hot.

Follow-up

A riding lawn mower produces useful work at a rate of 20 KW. How long would it take to produce the 4000 KJ of heat, above?

Answer

\[ e = \frac{W}{Q_H} \\ W = e*Q_H = 0.25*4000 KJ = 1000 KJ \\ P = \frac{W}{t} \\ t = \frac{W}{P} = \frac{1000 KJ}{20 \frac{KJ}{s}} = 50 s \]

IC Engines

• Internal Combustion Engines are those typically used by motor vehicles.

• The fuel is vaporized then ignited creating a hot expanding mixture of gases.

• In regards to efficiency, the compression ratio, r, is an important measurement.

• r is is ratio of maximum to minimum volume of the cylinder containing the piston:

\[ r = \frac{V_{max}}{V_{min}} \]

The Otto Cycle part 1

• Intake stroke: A fuel air mixture is taken into the cylinder when the crank shaft moves the piston down.
  • Intake Valve is open
  • Exhaust Valve is closed
  • Work is done on the piston by the crank shaft.
• Compression stroke: The fuel-air mixture is then compressed as the crank shaft moves the piston back up.
  • Intake Valve is closed
  • Exhaust Valve is closed
  • Work is done on the piston by the crank shaft.
• In a gasoline engine the spark plug then ignites the fuel air mixture.
  • Intake Valve is closed
  • Exhaust Valve is closed

The Otto Cycle part 2

• Power Stroke: The spark combusts the fuel air mixture. This forces the piston down
  • Intake Valve is closed
  • Exhaust Valve is closed
  • Work is done on the crank shaft by the piston.
• Exhaust Stroke: The crank shaft moves the piston up, the exhaust value is open and the piston forces the exhaust out of the cylinder.
  • Intake Valve is closed
  • Exhaust Valve is open
  • Work is done on the piston by the crank shaft.

Thermodynamic processes

• The power stroke and compression stroke happen fast enough that they can be treated as adiabatic processes.

• The maximum possible efficiency is given by:

\[ e = \frac{1}{r^{\gamma-1}} \]

• This is the maximum possible efficiency because it hasn’t factored in:

1. Air-fuel mixture and exhaust are not ideal gases.
2. Friction
3. Heat loss to ambient, ie the cylinders are not perfectly insulated.
4. Turbulent gas flow.

• Real efficiency tend to be around 10 to 20 percent less than ideal efficiency

Question

What is the maximum possible efficiency for the lawn mower in the previous example? Assume a compression ratio of 6 and a diatomic gas (air).

Answer

\[ e = \frac{1}{6^{1.4-1}} = 0.488 \]

Diesel Engines

• The Diesel cycle differs from the Otto cycle in that the fuel is ignited by the adiabatic temperature increase during the compression stroke.
• No spark plugs needed.
• Compression ratios tend to be higher, r = 15-20.
• This results in greater maximum efficiency.
• Example: VW Jetta w/ gasoline engines get around 35 mpg highway; VW Jetta with the Turbo Diesel (TDI) gets around 50 mpg highway.

Refrigerators

• Whereas Heat Engines the heat goes from high to low temperature with some of the heat downing work, In Refrigerators work is done to move heat from low temperatures to high temperatures.

• The working fluid is a gas with an very low boiling point

  • R-114 has a boiling point of 3.5 Celcius at 1 ATM
• Adiabatic compression is used to raise temperature above that of the environment.
• As heat is lost to the environment, the high pressure gas turns to high pressure liquid.
• Adiabatic expansion is used to reduce temperature below that of the cooling space.
• As heat is gained from the cooling space, the low pressure liquid turns to low pressure gas.

• The heat exchange in the cool space occurs due to a phase change in the working fluid, so the working fluids temperature does not change.

Refrigeration Cycle Graphs

The Second Law of Thermo Dyanamics - I

Kelvin-Plank Statement:

it is impossible to devise a cyclically operating thermal engine, the sole effect of which is to absorb energy in the form of heat from a single thermal reservoir and to deliver an equivalent amount of work.

• In other words, It is impossible for a heat engine to convert all of the heat into work.

• In other other words, It is impossible for a heat engine to be 100% efficient.

• As engineers you will encounter people who do not understand this and will try to get you to help them design an engine that violates this law.

  • Such devices are called Perpetual Motion Machines, and to date each one has been shown not to work or to be a fraud.
  • We’re talking millions of attempts over the centuries with zero success.

The Second Law of Thermo Dyanamics - II

Clausius Statement:

It is impossible for any process have its sole outcome be transfering heat from a low temperature body to a high temperature body.

• In other words, you must do work to reverse the spontaneous flow of heat as described by the zeroth law.

• It can be mathematically shown that these two statements are equivalent.

The Carnot Cycle

• This is a heat engine cycle where the working fluid moves between adiabatic compression and expansion without any other thermodynamic processes.

• A Carnot Engine is the ideal heat engine, no other design will give greater efficiency.

\[ e_{Carnot} = 1 - \frac{T_C}{T_H} \]

• In terms of the 2nd Law of Thermodynamics, to be 100% efficient you need to have:

1. A heat sink with \(T_C = 0K\)
2. A heat source with \(T_H = \infty\)

• Both states are physically impossible so 100% efficiency is impossible.

Question

What is the Carnot Efficiency of a heat engine with a heat source of 550 K and a heat sink of 297 K?

Answer

\[ e_{Carnot} = 1 - \frac{T_C}{T_H} = 1 - \frac{297K}{550K} = 0.46 \]

Entropy

• Entropy ,S, is a measure of randomness in a system.

• Consider a deck of 52 playing cards. Order the following decks from least to greatest randomness

  • A standard Poker Deck of 2 colors (Red and Black), 4 suites (Clubs, Hearts, Diamonds, Spades) each with 13 cards (Ace - King)
  • A deck of 26 aces and 26 twos one color, no suites.
  • A deck of cards numbered 1 to 13 in two colors (eg. 2 Red Aces, 2 Black Aces, etc) but no suites.

• You can see the more variety in possible outcomes, the more disorder in a system.

From the idea of possible states: \[ S = k ln(w) \]

• k is Boltzmann Constant (\(1.38\times 10^{-23}\frac{J}{K}\)), w is the total possible microscopic states a macroscopic system can have

Ranking Task

Rank the following by Entropy, assume the same mass of each .

• Steam at 373 K
• Ice at 273 K
• Water at 300 K

Entropy and the Second Law

In a Closed System, Entropy can only stay the same or increase. Entropy cannot spontaneously decrease.

• To have a cyclic process, Entropy must be reduced.
• The only way to reduce entropy in a system is to open it to the environment.
• This is opening to the environment to reduce Entropy is the Heat Sink in Heat Engines.
• Hence all Heat Engines must exhaust heat to their surroundings.

The Kelvin-Planck and Clausius Versions of the Second Law are consequences of the behavior of Entropy.