# Thermal Reservoirs and Heat Engines

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Thermal Reservoirs and Heat Engines
A conceptual schematic of a heat engine.  Two channels of heat transfer, with thermal reservoirs of two different temperatures TH and TL, are shown, along with one channel of work transfer.. (URL for notes:

• The thermal reservoirs are assumed to be large enough that their temperatures don’t change as heat flows in or out of them.  • The arrows indicate the direction of energy transfer (heat flow or work) in the forward time direction.  • According to the first law of thermodynamics we have Q = W + q.   • The efficiency h of this engine is defined as the ratio of work energy output to the heat energy input, so we have h = W/Q = 1 – q/Q

All heat engines: Receive heat from a high-temperature source (solar energy, blast furnace, ocean, land surface, etc.) Convert part of this heat to work Reject the remaining waste heat to a low-temperature sink (the atmosphere, rivers, etc.) Operate on a cycle Have a working fluid

boiler turbine pump condenser
Qin boiler Compress to boiler pressure turbine pump Wout condenser Qout Wnet = Wout - Win = Qin - Qout Heat Engine Components

• The Rankine cycle is a thermodynamic cycle.
• Like other thermodynamic cycles, the maximum efficiency of the Rankine cycle is given by calculating the maximum efficiency of the Carnot cycle. • It is named after William John Macquorn Rankine, a Scottish scientist.

• Rankine cycles describe the operation of steam heat engines commonly found in power generation plants. • In such power plants, power is generated by alternately vaporizing and condensing a working fluid (in many cases water, although refrigerants such as ammonia may also be used).

• The working fluid in a Rankine cycle follows a closed loop and is re-used constantly.
• Water vapor seen billowing from power plants is evaporating cooling water, not working fluid. (NB: steam is invisible until it comes in contact with cool, saturated air, at which point it condenses and forms the white billowy clouds seen leaving cooling towers).

Understanding Thermal Reservoirs
Bodies that don’t change their temperature even though heat is being added or subtracted. Blast furnce; hot enough that heat removed does not change the temperature of the furnace. Large lake or ocean; large enough that temperature changes only very slowly in spite of heat entering or leaving at the surface. Land beneath the surface; temperature remains constant even though heat energy is transferred from the surface.

The Underground Motels and Church in Coober Pedy, Australia
outside °F, inside constant 70 °F

RUN GEOEXCHANGE MOVIE

or net work/total heat in
Thermal Efficiency Basic Heat Engine LTER= Low Temperature Energy Reservoir HTER= High Temperature Energy Reservoir The thermal efficiency of a cycle (or more precisely a forward heat engine) is defined as the ratio of net work output, W, to the heat supplied at high temperature, Q1, i.e. or net work/total heat in

Calculate the maximum theoretical thermal efficiency of a coal-fired power station that heats steam to 510°C and cools it in a condenser at 30°C. Answer: Maximum efficiency = (THOT – TCOLD)/THOT = [( ) – (30+273)] / ( ) = 480 / 783 = 0.61 or 61%

2. The temperature of the gases in a car engine during combustion is 1800°C. The exhaust is expelled at 80°C. Calculate the maximum theoretical thermal efficiency of the engine. Answer: Maximum theoretical efficiency = (THOT – TCOLD)/THOT = [( ) – (80+273)] / ( ) = 1720/2073 = 0.83 or 83% Of course, in both case, the actual efficiency will be smaller. Students should consider why.

Second Law of Thermodynamics: Kelvin-Planck Statement
It is impossible for any device that operates on a cycle to receive heat from a single reservoir and produce a net amount of work. Thus, a heat engine must exchange heat with a low temperature sink as well as a high temperature source to keep operating.

or No heat engine can have a thermal efficiency of 100%, or for a power plant to operate, the working fluid must exchange heat with the environment as well as the furnce (must have waste heat).

If all the energy transfer processes in a given heat engine are reversible, we can just as well reverse all the arrows, and run the heat engine “backwards in time”.  (A kitchen refrigerator is a common example of a heat engine running in reverse.)

A Real Refrigerator

Second Law of Thermodynamics
The second law of thermodynamics is a general principle which places constraints upon the direction of heat transfer and the attainable efficiencies of heat engines. In so doing, it goes beyond the limitations imposed by the first law of thermodynamics. It's implications may be visualized in terms of the waterfall analogy. The maximum efficiency which can be achieved is the Carnot efficiency.

Waterfall Analogy

Heat Engine Cycle and the Laws of Thermo

The Otto Cycle A schematic version of the four-stroke engine cycle

The Diesel Engine The diesel internal combustion engine differs from the gasoline powered Otto Cycle by using a higher compression of the fuel to ignite the fuel rather than using a spark plug ("compression ignition" rather than "spark ignition").

Some properties of crude oil fractions.
Processing crude oil - refining Name Carbon chain length Boiling range oC Petroleum gases 1-4 <5 Naphtha 5-9 20-180 Gasoline 5-10 20-200 Kerosine 10-16 Gas oil (diesel oil) 14-20 Lubricating oil 20-50 Fuel oil 20-70 330 upwards Residue >70 Non-distillable Some properties of crude oil fractions. How bubble caps work

In the diesel engine, air is compressed adiabatically with a compression ratio typically between 15 and 20. This compression raises the temperature to the ignition temperature of the fuel mixture which is formed by injecting fuel once the air is compressed.

Diesel Engine Theoretical Efficiency
Since the compression and power strokes of this idealized cycle are adiabatic, the efficiency can be calculated from the constant pressure and constant volume processes. The input and output energies and the efficiency can be calculated from the temperatures and specific heats:

Theoretical Diesel Efficiency (what does we assume?)
It is convenient to express this efficiency in terms of the compression ratio rC = V1/V2 and the expansion ratio rE = V1/V3. The efficiency can be written and this can be rearranged to the form Theoretical Diesel Efficiency (what does we assume?)

Nuclear power plant; thermal rods in water
Simulation of temperatures inside a nuclear reactor. From Argonne

A nuclear power station
A nuclear power station. The nuclear reactors are inside the two cylindrical containment buildings in the foreground—behind are the cooling towers (venting water vapor). Steam from cooling towers

Nuclear power generator
U.S. commercial pressurized water reactor (PWR) nuclear power plants Nuclear power generator

The picture above shows the release of steam from geothermal plants in Santa Rosa -- geothermal plants tap the heat within the Earth to produce energy in the form of electricity.The heat trapped within the Earth, which was generated during its formation billions of years ago and through the decay of radioactive elements with rocks, is trying to escape Earth as a heat engine

Convection, also occurs within the Earth as hot, less dense portions of the mantle rise and displace cooler, denser rocks, which then sink into the mantle -- in summary the cooler,  dense rocks sink in the mantle, whereas the warmer rocks within the mantle rise by a process called mantle convection (shown by red arrows in the diagram above).

Andes Mountains in South America, which in this particular locale are composed of sediments that formed on the seafloor -- miles below the sea surface -- millions of years ago.  Now these sediments rest on the top of the mountains,  miles above the sea surface -- how can this be?  How can something as heavy as the surface of the Earth rise to such a high elevation? Mountains of molten rock, or lava, on the big island of Hawaii. Once again, how can molten (liquid) rock at temperatures more than 1000oC  find its way to the surface of the Earth and why should this happen in Hawaii?

Here is a map of the major plates that make up the surface of the Earth.These plates are formed by a strong, rigid surface layer of rocks between 80 and 300 kilometers-thick.

• Harry Hess was the first to come up with an explanation for the mid-ocean ridges -- he suggested that the seafloor was created by volcanism within the rift valley along the axis of the ridge.  • With time the seafloor and underlying crust will spread away from the ridge in opposite directions on either side --  thereby creating a mobile seafloor -- like a conveyor belt -- very interesting idea, which he called seafloor spreading.

• How do we know that the seafloor is spreading at the mid-Atlantic ridge?
• Reversals in the polarityof magnetic materials in the seafloor that formed at geologic periods when the Earth’s magnetic field was reversed. • What does that mean for us on Earth if the magnetic field reverses?

• The time-scale of the magnetic field reversals is shown at the top.
• Regions with orange or yellow patterns  denote time of "normal polarity" or a magnetic direction with the same direction as today's field. • The white regions represent times when the field was in the opposite (or reversed) direction from what it is today.

Polar Reversals

Here you see a map of the mid-ocean ridge system
Here you see a map of the mid-ocean ridge system. The ridge in the Pacific is called the East Pacific Rise, in the Atlantic is is called the Mid-Atlantic Ridge, and in the Indian Ocean it is either the Southwest Indian Ridge, Central Indian Ridge or Southeast Indian Ridge.

Formation of Hurricane
A Giant Heat Engine Waves in the trade winds in the Atlantic Ocean—areas of converging winds that move along the same track as the prevailing wind—create instabilities in the atmosphere that may lead to the formation of hurricanes. Hurricanes form when the energy released by the condensation of moisture in rising air causes a chain reaction. The air heats up, rising further, which leads to more condensation. The air flowing out of the top of this “chimney” drops towards the ground, forming powerful winds

Hurricane Katrina on August 28 at 1:00 pm EDT
Hurricane or tropical cyclone/typhoon Hurricane Katrina on August 28 at 1:00 pm EDT Schematic representation of flow around a low-pressure area in the Northern hemisphere. The pressure gradient force is represented by blue arrows, the Coriolis acceleration (always perpendicular to the velocity) by red arrows Why do these two storms rotate in opposite directions? Image of Cyclone Catarina on March 26, 2004, the first South Atlantic hurricane ever recorded

Sadie Carnot • French engineer (Paris) • Father was involved in the French revolution and was exciled • Sadie Carnot joined the military and became interested in steam engines. • He worked with his brother on steam engines and did experiments similar to those of Joule 20 years before Joule. • He died of Cholera at the age of 36.

Carnot Cycle • The most efficient heat engine cycle is the Carnot cycle, consisting of two isothermal processes and two adiabatic processes. The Carnot cycle can be thought of as the most efficient heat engine cycle allowed by physical laws. • When the second law of thermodynamics states that not all the supplied heat in a heat engine can be used to do work, the Carnot efficiency sets the limiting value on the fraction of the heat which can be so used. • In order to approach the Carnot efficiency, the processes involved in the heat engine cycle must be reversible and involve no change in entropy. • This means that the Carnot cycle is an idealization, since no real engine processes are reversible and all real physical processes involve some increase in entropy.

Engine Cycles • For a constant mass of gas, the operation of a heat engine is a repeating cycle and its PV diagram will be a closed figure. • The idea of an engine cycle is illustrated below for one of the simplest kinds of cycles. • If the cycle is operated clockwise on the diagram, the engine uses heat to do net work. • If operated counterclockwise, it uses work to transport heat and is therefore acting as a refrigerator or a heat pump.

The Clausius Theorem and Inequality

• The equality above represents the Clausius Theorem and applies only to the the ideal or Carnnot cycle. • Since the integral represents the net change in entropy in one complete cycle, it attributes a zero entropy change to the most efficient engine cycle.

• The Clausius Inequality applies to any real engine cycle and implies a negative change in entropy on the cycle. • That is, the entropy given to the environment during the cycle is larger than the entropy transferred to the engine by heat from the hot reservoir.

So, what is entropy?? (we will do it in more detail later)
• The second law of thermodynamics (the entropy law or law of entropy) was formulated in the middle of the last century by Clausius and Thomson following Carnot's earlier observation that, like the fall or flow of a stream that turns a mill wheel, it is the "fall" or flow of heat from higher to lower temperatures that motivates a steam engine.

• The key insight was that the world is inherently active, and that whenever an energy distribution is out of equilibrium a potential or thermodynamic "force" (the gradient of a potential) exists that the world acts to dissipate or minimize.

• All real-world change or dynamics is seen to follow, or be motivated, by this law.
• So whereas the first law expresses that which remains the same, or is time-symmetric, in all real-world processes the second law expresses that which changes and motivates the change, the fundamental time-asymmetry, in all real-world process.

• Clausius coined the term "entropy" to refer to the dissipated potential and the second law, in its most general form, states that the world acts spontaneously to minimize potentials (or equivalently maximize entropy), and with this, active end-directedness or time-asymmetry was, for the first time, given a universal physical basis.

• The balance equation of the second law, expressed as S > 0, says that in all natural processes the entropy of the world always increases, and thus whereas with the first law there is no time, and the past, present, and future are indistinguishable, the second law, with its one-way flow, introduces the basis for telling the difference.

• The active nature of the second law is intuitively easy to grasp and empirically demonstrate. If a glass of hot liquid, for example, as shown in the Fig., is placed in a colder room a potential exists and a flow of heat is spontaneously produced from the cup to the room until it is minimized (or the entropy is maximized) at which point the temperatures are the same and all flows stop.

A glass of liquid at temperature TI is placed in a room at temperature TII such that . The disequilibrium produces a field potential that results in a flow of energy in the form of heat from the glass to the room so as to drain the potential until it is minimized (the entropy is maximized) at which time thermodynamic equilibrium is reached and all flows stop. refers to the conservation of energy in that the flow from the glass equals the flow of heat into the room.

• The active macroscopic nature of the second law posed a direct challenge to the "dead" mechanical world view which Boltzmann tried to meet in the latter part of the 19th century by reducing the second law to a law of probability following from the random collisions of mechanical particles.

• His father, Ludwig Georg Boltzmann was a tax official
• His father, Ludwig Georg Boltzmann was a tax official. His grandfather, who had moved to Vienna from Berlin, was a clock manufacturer, and Boltzmann’s mother, Katharina Pauernfeind, was originally from Salzburg. • He received his primary education from a private tutor at the home of his parents. Boltzmann attended high school in Linz, Upper Austria. At age 15, Boltzmann lost his father. • Boltzmann studied physics at the University of Vienna, starting in Among his teachers were Josef Loschmidt, Joseph Stefan, Andreas von Ettingshausen and Jozef Petzval. • Boltzmann received his PhD degree in 1866 working under the supervision of Stefan; his dissertation was on kinetic theory of gases. In 1867 he became a Privatdozent (lecturer). After obtaining his doctorate degree, Boltzmann worked two more years as Stefan’s assistant. It was Stefan who introduced Boltzmann to Maxwell's work. Ludwig Eduard Boltzmann (Vienna, Austrian Empire, February 20, 1844 Duino near Trieste, September 5, 1906)

• In 1869, at age 25, he was appointed full Professor of Mathematical Physics at the University of Graz. • In 1869 he spent several months in Heidelberg working with Robert Bunsen and Leo Königsberger and then in 1871 he was with Gustav Kirchhoff and Hermann von Helmholtz in Berlin. • In 1873 Boltzmann joined the University of Vienna as Professor of Mathematics and where he stayed till 1876.

• In 1872, long before women were admitted to Austrian universities, he met Henriette von Aigentler, an aspiring teacher of mathematics and physics in Graz. • She was refused permission to unofficially audit lectures, and Boltzmann advised her to appeal; she did, successfully. • On July 17, 1876 Ludwig Boltzmann married Henriette von Aigentler; they had three daughters and two sons. Boltzmann went back to Graz to take up the chair of Experimental Physics. • He was shortly the president of the U. of Graz, but later moved to Munich, then back to Vienna. He then moved to Berlin and eventually back to Graz.

• Following the lead of Maxwell who had modeled gas molecules as colliding billiard balls, Boltzmann argued that the second law was simply a consequence of the fact that since with each collision nonequilibrium distributions would become increasingly disordered leading to a final state of macroscopic uniformity and microscopic disorder.

• Because there are so many more possible disordered states than ordered ones, he concluded, a system will almost always be found either in the state of maximum disorder or moving towards it. • Entropy always increases.

Clausius Statement of the Second Law
It is impossible to construct a device that operates in a cycle and produces no effect other than the transfer of heat from a lower-temperature body to a higher-temperature body. Both versions of the Second Law are negative statements which cannot be proven. They are, however, based on empirical evidence and no experiment has yet been found to contradict this law.

Perpetual Motion Machines
• A machine that violates either the first or second laws of thermodynamics is called a Perpetual Motion Machine. • What does that mean to you? Do you think you could build one? Are there some people that do?

Turns out there are two types:
PMM1 - violates the first PMM2 - violates the second law

boiler turbine pump condenser
Qin W net,out boiler Compress to boiler pressure turbine pump Gen condenser Qout PMM1

• Claims to run on heat supplied by the resistance units run off of the generator.
• Remainder of electrical energy is available for power. • Idea is to start it externally and it would run indefinitely.

Problem** • Initially the resistance heat would balance the heat externally used to start it. • After it is disconnected system stops with no energy. It can’t create energy out of nothing. • Violates the first law.

Qin W net,out boiler turbine pump Qout PMM2 Compress to boiler
pressure turbine pump Qout PMM2

• This inventor wants to use all his waste heat so sends it back to the pump and skips the condenser. • BUT now he has only one thermal reservoir. • Violates the second law (we can’t have a heat engine with 100% efficiency)

• Surprising number of quacks:
J.W. Kelly collected millions between for the “hydrodynamic-pulsating-vacu-engine” which could push a railroad train 3,000 miles on 1 liter of WATER. (After he died investors found a hidden motor.) More recently investors wanted to invest 2.5 million in an “energy augmentor” but their lawyer wanted an expert opinion at which the inventor fled. I remember when I was younger (a long time ago) someone trying to sell a device you put on your carburetor to increase your gas mileage by a factor of 10. It cost \$20.

• Do any of you have a good story to tell
• Do any of you have a good story to tell? There are lots of them around. • Here an understanding of thermodynamics will help you avoid any of these false claims.

Reversible processes:
• One that can be run in reverse without leaving any effect on the surroundings...both are returned to their original state. • Intuition tells you that this doesn’t happen (and it doesn’t). • Example is put a hot cup of fluid in a cold room, it cools and will actually take more to heat it up again than it gave off to the room. • Reversible processes DO NOT occur in nature

• What is one of the biggest sources of irreversibility in nature?
• Is there some way to avoid it completely? • We are back to our perpetual motion machines.

Heat Pump • A heat pump is a device which applies external work to extract an amount of heat QC from a cold reservoir and delivers heat QH to a hot reservoir. • A heat pump is subject to the same limitations from the second law of thermodynamics as any other heat engine and therefore a maximum efficiency can be calculated from the Carnot cycle. • Heat Pumps are usually characterized by a coefficient of performance which is the number of units of energy delivered to the hot reservoir per unit work input.

Air Conditioners and Heat Pumps
• Air conditioners and heat pumps are heat engines like the refrigerator. • They make good use of the high quality and flexibility of electric energy in that they can use one unit of electric energy to transfer more than one unit of energy from a cold area to a hot area.

• For example, an electric resistance heater using one kilowatt-hour of electric energy can transfer only 1 kWh of energy to heat your house at 100% efficiency. • But 1 kWh of energy used in an electric heat pump could "pump" 3 kWh of energy from the cooler outside environment into your house for heating. • The ratio of the energy transferred to the electric energy used in the process is called its coefficient of performance (CP). • A typical CP for a commercial heat pump is between 3 and 4 units transferred per unit of electric energy supplied.

Coefficient of Performance
The coefficient of performance (CP) for a heat pump is the ratio of the energy transferred for heating to the input electric energy used in the process. In reference to the standard heat engine illustration, the coefficient is defined by

There is a theoretical maximum CP, that of the Carnot cycle:
• For a refrigerator, however, the useful quantity is the heat extracted, QC , not the heat exhausted. • Therefore, the coefficient of performance of a refrigerator is expressed asFor consumer refrigerators in the U.S., the coefficient of performance for refrigerators is typically recast into a number called the Energy Efficiency Ratio.

Energy efficiency rating
EER = COPr rating used for refrigerators and AC * usually between 8 and 12 but heat pumps can go to 17

• Why do we care about processes that turn heat into mechanical work?
• A device that performs such a process is called a heat engine.  • According to the first law of thermodynamics, energy is always conserved, so we obviously cannot “produce” energy with a heat engine.

• We put energy into the process in the form of heat, and extract a (generally lesser) amount of energy from the process in the form of mechanical work.  • The reason for doing this is that energy in the form of mechanical work is often more useful than the same amount of energy in the form of heat.  • In a sense, mechanical work is coherent energy, whereas heat energy is incoherent.

William Thomson, 1st Baron Kelvin, (26 June 1824–17 December 1907) was an Irish-Scottish mathematical physicist, engineer, and outstanding leader in the physical sciences of the 19th century. • He did important work in the mathematical analysis of electricity and thermodynamics, and did much to unify the emerging discipline of physics in its modern form.

• He also enjoyed a second career as a telegraph engineer and inventor, a career that propelled him into the public eye and ensured his wealth, fame and honour.

Second Law Broken • One of the most fundamental rules of physics, the second law of thermodynamics, has for the first time been shown not to hold for microscopic systems. • The demonstration, by chemical physicists in Australia, could place a fundamental limit on miniaturisation, because it suggests that the micro-scale devices envisaged by nanotechnologists will not behave like simple scaled-down versions of their larger counterparts - they could sometimes run backwards.

• The second law states that a closed system will remain the same or become more disordered over time, i.e. its entropy will always increase. It is the reason a cup of tea loses heat to its surroundings, rather than being heated by the air around it. • "In a typical room, for example, the air molecules are most likely to be distributed evenly, which is the overall result of their individual random motion", says theoretical physicist Andrew Davies of Glasgow University. ” • But because of this randomness there is always a probability that suddenly all the air will bunch up in one corner." Thankfully this probability is so small it never happens on human timescales.

CARNOT Principles • The efficiency of an irreversible heat engine is always less than the efficiency of of a reversible process operating between the same two heat reservoirs. • Efficiencies of all reversible heat engines operating between the same two reservoirs are the same.

First statement: High Temp Reservoir at Th Qh Qh Wrev reversible
irreversible Wirrev Ql,rev Ql Low Temp Reservoir at Tl

• If we assume that the irreversible process is more efficient (contradicts Carnot’s first principle) it will thus deliver more work. • Thus we have a system that produces net work Wirrev - Wrev with no net heat exchange. • This is a clear violation of the Kelvin-Planck statement of the 2nd law.

Rankine Steam Plant Cycle

• There are four processes in the Rankine cycle, each changing the state of the working fluid.
• These states are identified by number in the diagram above. •Process 4-1: First, the working fluid is pumped (ideally isentropically..no change in entropy..) from low to high pressure by a pump. • Pumping requires a power input (for example mechanical or electrical

•Process 1-2: The high pressure liquid enters a boiler where it is heated at constant pressure by an external heat source to become a superheated vapor. • Common heat sources for power plant systems are coal, natural gas, or nuclear power. • Which do you think is the most economical? • What are the consequences of each?

• Process 2-3: The superheated vapor expands through a turbine to generate power output. Ideally, this expansion is isentropic. • This decreases the temperature and pressure of the vapor. • Process 3-4: The vapor then enters a condenser where it is cooled to become a saturated liquid. • This liquid then re-enters the pump and the cycle repeats.

Real Rankine cycle (non-ideal)
• In a real Rankine cycle, the compression by the pump and the expansion in the turbine are not isentropic. • In other words, these processes are non-reversible and entropy is increased during the two processes (indicated in the figure as ∆S). • This somewhat increases the power required by the pump and decreases the power generated by the turbine. It also makes calculations more involved and difficult.

Rankine cycle with reheat
• In this variation, two turbines work in series. • The first accepts vapor from the boiler at high pressure. After the vapor has passed through the first turbine, it re-enters the boiler and is reheated before passing through a second, lower pressure turbine. • Among other advantages, this prevents the vapor from condensing during its expansion which can seriously damage the turbine blades.

Regenerative Rankine cycle
• The regenerative Rankine cycle is so named because after emerging from the condenser (possibly as a sub-cooled liquid) the working fluid is heated by steam tapped from the hot portion of the cycle. • This increases the average temperature of heat addition which in turn increases the thermodynamic efficiency of the cycle.