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:http://www.colorado.edu/ASEN/asen3113)

• The thermal reservoirs are assumed to be large enough that theirtemperatures 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 η of this engine is defined as the ratio of work energyoutput to the heat energy input, so we have η = W/Q = 1 – q/Q

All heat engines:

1. Receive heat from a high-temperature source (solarenergy, blast furnace, ocean, land surface, etc.)

2. Convert part of this heat to work

3. Reject the remaining waste heat to a low-temperaturesink (the atmosphere, rivers, etc.)

4. Operate on a cycle

5. Have a working fluid

boiler

condenser

pump turbine

Qin

Qout

Compress toboiler pressure Wout

Wnet = Wout - Win = Qin - Qout

Heat Engine Components

• The Rankine cycle is a thermodynamic cycle.• Like other thermodynamic cycles, the maximumefficiency of the Rankine cycle is given bycalculating the maximum efficiency of the Carnotcycle.• It is named after William John Macquorn Rankine, aScottish scientist.

• Rankine cycles describe the operation of steam heatengines commonly found in power generation plants.• In such power plants, power is generated byalternately vaporizing and condensing a working fluid (inmany cases water, although refrigerants such asammonia may also be used).

• The working fluid in a Rankine cycle follows a closedloop and is re-used constantly.

• Water vapor seen billowing from power plants isevaporating cooling water, not working fluid. (NB: steamis invisible until it comes in contact with cool, saturatedair, at which point it condenses and forms the whitebillowy clouds seen leaving cooling towers).

Understanding Thermal Reservoirs

Bodies that don’t change their temperature even thoughheat is being added or subtracted.

1. Blast furnce; hot enough that heat removed does notchange the temperature of the furnace.

2. Large lake or ocean; large enough that temperaturechanges only very slowly in spite of heat entering orleaving at the surface.

3. Land beneath the surface; temperature remainsconstant even though heat energy is transferred fromthe surface.

The Underground Motels and Church in Coober Pedy, Australia

outside 90-120 °F,inside constant 70 °F

RUN GEOEXCHANGE MOVIE

Thermal Efficiency

LTER= Low Temperature Energy ReservoirHTER= High Temperature Energy ReservoirThe thermal efficiency of a cycle (or more precisely a forward heatengine) is defined as the ratio of net work output, W, to the heatsupplied at high temperature, Q1, i.e.

or net work/total heat in

Basic Heat Engine

1. Calculate the maximum theoretical thermal efficiency of a coal-firedpower station that heats steam to 510°C and cools it in a condenser at30°C.

Answer: Maximum efficiency = (THOT – TCOLD)/THOT

= [(510+273) – (30+273)] / (510+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 thermalefficiency of the engine.

Answer: Maximum theoretical efficiency = (THOT – TCOLD)/THOT

= [(1800+273) – (80+273)] / (1800+273)

= 1720/2073 = 0.83 or 83%

Of course, in both case, the actual efficiency will besmaller. Students should consider why.

Second Law of Thermodynamics: Kelvin-Planck Statement

It is impossible for any device that operates on a cycle toreceive heat from a single reservoir and produce a net amountof work.

Thus, a heat engine must exchange heat with a low temperaturesink as well as a high temperature source to keep operating.

or

No heat engine can have a thermal efficiency of 100%, orfor a power plant to operate, the working fluid mustexchange 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, wecan just as well reverse all the arrows, and run the heat engine “backwards intime”.  (A kitchen refrigerator is a common example of a heat engine runningin reverse.)

A Real Refrigerator

Second Law of Thermodynamics

The second law of thermodynamics is a general principle which placesconstraints upon the direction of heat transfer and the attainableefficiencies of heat engines. In so doing, it goes beyond the limitationsimposed by the first law of thermodynamics. It's implications may bevisualized 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 EngineThe diesel internal combustion engine differs from the gasolinepowered Otto Cycle by using a higher compression of the fuel toignite the fuel rather than using a spark plug ("compressionignition" rather than "spark ignition").

How bubble caps work

Processing crude oil - refining

Non-distillable>70Residue

330 upwards20-70Fuel oil

370-60020-50Lubricatingoil

260-34014-20Gas oil(diesel oil)

180-26010-16Kerosine

20-2005-10Gasoline

20-1805-9Naphtha

<51-4Petroleumgases

Boiling rangeoC

Carbon chainlength

Name

Some properties of crude oil fractions.

In the diesel engine, air iscompressed adiabatically with acompression ratio typicallybetween 15 and 20. Thiscompression raises thetemperature to the ignitiontemperature of the fuel mixturewhich is formed by injecting fuelonce 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 volumeprocesses. The input and output energies and the efficiency can be calculatedfrom the temperatures and specific heats:

It is convenient to express this efficiency in terms of the compressionratio rC = V1/V2 and the expansion ratio rE = V1/V3. The efficiency can bewritten

and this can be rearranged to the form

Theoretical Diesel Efficiency (what does we assume?)

Simulation of temperatures inside a nuclear reactor. From Argonne

Nuclear power plant; thermal rods in water

A nuclear power station. The nuclear reactors are inside thetwo cylindrical containment buildings in theforeground—behind are the cooling towers (venting watervapor). Steam from cooling towers

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

The picture above shows the release of steam from geothermalplants in Santa Rosa -- geothermal plants tap the heat within theEarth to produce energy in the form of electricity.The heat trappedwithin the Earth, which was generated during its formation billions ofyears 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 denseportions of the mantle rise and displace cooler, denserrocks, which then sink into the mantle -- in summary thecooler,  dense rocks sink in the mantle, whereas the warmerrocks within the mantle rise by a process called mantleconvection (shown by red arrows in the diagram above).

Andes Mountains in SouthAmerica, which in this particularlocale are composed of sedimentsthat formed on the seafloor -- milesbelow the sea surface -- millions ofyears ago.  Now these sedimentsrest on the top of the mountains, miles above the sea surface -- howcan this be?  How can somethingas heavy as the surface of theEarth rise to such a high elevation?

Mountains of molten rock, orlava, on the big island ofHawaii. Once again, how canmolten (liquid) rock attemperatures more than1000oC  find its way to thesurface of the Earth and whyshould this happen inHawaii?

Here is a map of the major plates that make upthe surface of the Earth.These plates areformed by a strong, rigid surface layer of rocksbetween 80 and 300 kilometers-thick.

• Harry Hess was the first tocome up with an explanation forthe mid-ocean ridges -- hesuggested that the seafloor wascreated by volcanism within therift valley along the axis of theridge. • With time the seafloor andunderlying crust will spreadaway from the ridge in oppositedirections on either side -- thereby creating a mobileseafloor -- like a conveyor belt -- very interesting idea, which hecalled seafloor spreading.   

• How do we know that theseafloor is spreading at themid-Atlantic ridge?

• Reversals in thepolarityof magneticmaterials in the seafloorthat formed at geologicperiods when the Earth’smagnetic field wasreversed.

• What does that mean forus on Earth if the magneticfield reverses?

• The time-scale of the magnetic field reversals is shown at thetop.

• Regions with orange or yellow patterns denote time of"normal polarity" or a magnetic direction with the samedirection as today's field.

• The white regions represent times when the field was in theopposite (or reversed) direction from what it is today.

Polar Reversals

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

Hurricanes form when the energy releasedby the condensation of moisture in rising aircauses a chain reaction. The air heats up,rising further, which leads to morecondensation. The air flowing out of the topof this “chimney” drops towards the ground,forming powerful winds

Waves in the trade winds in theAtlantic Ocean—areas ofconverging winds that move alongthe same track as the prevailingwind—create instabilities in theatmosphere that may lead to theformation of hurricanes.

Formation of Hurricane

A Giant Heat Engine

HurricaneKatrina on

August 28 at1:00 pm EDT

Schematic representation of flowaround a low-pressure area in theNorthern hemisphere. The pressuregradient force is represented byblue arrows, the Coriolisacceleration (always perpendicularto the velocity) by red arrows Image of Cyclone Catarina on March 26, 2004, the

first South Atlantic hurricane ever recorded

Why do these two stormsrotate in oppositedirections?

Hurricane or tropical cyclone/typhoon

Sadie Carnot

• French engineer 1796 - 1832 (Paris)

• Father was involved in the French revolution and was exciled

• Sadie Carnot joined the military and became interested in steamengines.

• He worked with his brother on steam engines and didexperiments 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 twoisothermal processes and two adiabatic processes. The Carnot cyclecan be thought of as the most efficient heat engine cycle allowed by physicallaws.• When the second law of thermodynamics states that not all the suppliedheat in a heat engine can be used to do work, the Carnot efficiency sets thelimiting value on the fraction of the heat which can be so used.• In order to approach the Carnot efficiency, the processes involved in theheat engine cycle must be reversible and involve no change in entropy.• This means that the Carnot cycle is an idealization, since no real engineprocesses are reversible and all real physical processes involve someincrease in entropy.

Engine Cycles• For a constant mass of gas, the operation of a heat engine is a repeatingcycle and its PV diagram will be a closed figure.

• The idea of an engine cycle is illustrated below for one of the simplest kindsof cycles.• If the cycle is operated clockwise on the diagram, the engine uses heat to donet work.• If operated counterclockwise, it uses work to transport heat and is thereforeacting as a refrigerator or a heat pump.

The Clausius Theorem and Inequality

• The equality above represents the Clausius Theorem and applies onlyto the the ideal or Carnnot cycle.• Since the integral represents the net change in entropy in onecomplete cycle, it attributes a zero entropy change to the most efficientengine cycle.

• The Clausius Inequality applies to any real engine cycle and impliesa negative change in entropy on the cycle.• That is, the entropy given to the environment during the cycle islarger than the entropy transferred to the engine by heat from the hotreservoir.

So, what is entropy?? (we will do it in more detail later)

• The second law of thermodynamics (the entropy law or lawof entropy) was formulated in the middle of the last centuryby Clausius and Thomson following Carnot's earlierobservation that, like the fall or flow of a stream that turns amill wheel, it is the "fall" or flow of heat from higher to lowertemperatures that motivates a steam engine.

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

• All real-world change or dynamics is seen to follow, orbe motivated, by this law.• So whereas the first law expresses that which remainsthe same, or is time-symmetric, in all real-worldprocesses the second law expresses that which changesand motivates the change, the fundamental time-asymmetry, in all real-world process.

• Clausius coined the term "entropy" to refer to thedissipated potential and the second law, in its mostgeneral form, states that the world acts spontaneouslyto minimize potentials (or equivalently maximizeentropy), and with this, active end-directedness or time-asymmetry was, for the first time, given a universalphysical basis.

• The balance equation of the second law, expressed asS > 0, says that in all natural processes the entropy ofthe world always increases, and thus whereas with thefirst law there is no time, and the past, present, andfuture 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 tograsp and empirically demonstrate. If a glass of hot liquid, forexample, as shown in the Fig., is placed in a colder room apotential exists and a flow of heat is spontaneously producedfrom the cup to the room until it is minimized (or the entropyis maximized) at which point the temperatures are the sameand all flows stop.

A glass of liquid at temperatureTI is placed in a room attemperature TII such that . Thedisequilibrium produces a fieldpotential that results in a flow ofenergy in the form of heat fromthe glass to the room so as todrain the potential until it isminimized (the entropy ismaximized) at which timethermodynamic equilibrium isreached and all flows stop.refers to the conservation ofenergy in that the flow fromthe glass equals the flow ofheat into the room.

• The active macroscopic nature of the second law posed adirect challenge to the "dead" mechanical world view whichBoltzmann tried to meet in the latter part of the 19th centuryby reducing the second law to a law of probability followingfrom the random collisions of mechanical particles.

Ludwig EduardBoltzmann(Vienna, Austrian Empire,February 20, 1844 Duino near Trieste,September 5, 1906)

• His father, Ludwig Georg Boltzmann was a taxofficial. His grandfather, who had moved to Viennafrom Berlin, was a clock manufacturer, andBoltzmann’s mother, Katharina Pauernfeind, wasoriginally from Salzburg.

• He received his primary education from a privatetutor at the home of his parents. Boltzmann attendedhigh school in Linz, Upper Austria. At age 15,Boltzmann lost his father.

• Boltzmann studied physics at the University ofVienna, starting in 1863. Among his teachers wereJosef Loschmidt, Joseph Stefan, Andreas vonEttingshausen and Jozef Petzval.

• Boltzmann received his PhD degree in 1866 workingunder the supervision of Stefan; his dissertation was onkinetic theory of gases. In 1867 he became aPrivatdozent (lecturer). After obtaining his doctoratedegree, Boltzmann worked two more years as Stefan’sassistant. It was Stefan who introduced Boltzmann toMaxwell's work.

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

• In 1872, long before women were admitted to Austrianuniversities, he met Henriette von Aigentler, an aspiringteacher 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 vonAigentler; they had three daughters and two sons.Boltzmann went back to Graz to take up the chair ofExperimental Physics.• He was shortly the president of the U. of Graz, but later moved toMunich, then back to Vienna. He then moved to Berlin and eventuallyback to Graz.

• Following the lead of Maxwell who had modeled gasmolecules as colliding billiard balls, Boltzmann argued thatthe second law was simply a consequence of the fact thatsince with each collision nonequilibrium distributions wouldbecome increasingly disordered leading to a final state ofmacroscopic uniformity and microscopic disorder.

• Because there are so many more possible disorderedstates than ordered ones, he concluded, a system willalmost always be found either in the state of maximumdisorder or moving towards it.• Entropy always increases.

Clausius Statement of the Second Law

It is impossible to construct a device that operates in a cycleand produces no effect other than the transfer of heat from alower-temperature body to a higher-temperature body.

Both versions of the Second Law are negative statementswhich cannot be proven. They are, however, based onempirical evidence and no experiment has yet beenfound to contradict this law.

Perpetual Motion Machines

• A machine that violates either the first or second laws ofthermodynamics is called a Perpetual Motion Machine.

• What does that mean to you? Do you think you couldbuild one? Are there some people that do?

Turns out there are two types:

a. PMM1 - violates the first

b. PMM2 - violates the second law

boiler

condenser

pump turbine

Qin

Qout

Compress toboiler pressure Gen

W net,out

PMM1

• Claims to run on heat supplied by the resistance units runoff 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 externallyused to start it.

• After it is disconnected system stops with no energy. It can’tcreate energy out of nothing.

• Violates the first law.

boiler

pump turbine

Qin

Qout

Compress toboiler pressure

W net,out

PMM2

• This inventor wants to use all his waste heat so sends itback 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 enginewith 100% efficiency)

• Surprising number of quacks:

1. J.W. Kelly collected millions between 1874-98 for the“hydrodynamic-pulsating-vacu-engine” which could push arailroad train 3,000 miles on 1 liter of WATER. (After hedied investors found a hidden motor.)

2. More recently investors wanted to invest 2.5 million in an“energy augmentor” but their lawyer wanted an expertopinion at which the inventor fled.

3. I remember when I was younger (a long time ago) someonetrying to sell a device you put on your carburetor to increaseyour gas mileage by a factor of 10. It cost $20.

• Do any of you have a good story to tell? There are lots ofthem around.

• Here an understanding of thermodynamics will help youavoid any of these false claims.

Reversible processes:

• One that can be run in reverse without leaving any effecton the surroundings...both are returned to their originalstate.

• Intuition tells you that this doesn’t happen (and itdoesn’t).

• Example is put a hot cup of fluid in a cold room, it coolsand will actually take more to heat it up again than it gaveoff to the room.

• Reversible processes DO NOT occur in nature

• What is one of the biggest sources of irreversibility innature?

• Is there some way to avoid it completely? (No I don’tknow enough about super-conductivity to know if you canget away from friction but I can say that in all conventionalsystems you can’t. The energy needed to achievesuperconductivity will be a lot more than will lost throughfriction.)

• We are back to our perpetual motion machines.

Heat Pump• A heat pump is a device which applies external work to extract an amount ofheat 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 ofthermodynamics as any other heat engine and therefore a maximumefficiency can be calculated from the Carnot cycle.• Heat Pumps are usually characterized by a coefficient of performance whichis the number of units of energy delivered to the hot reservoir per unit workinput.

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

• For example, an electric resistance heater using one kilowatt-hour ofelectric energy can transfer only 1 kWh of energy to heat your house at100% efficiency.• But 1 kWh of energy used in an electric heat pump could "pump" 3kWh of energy from the cooler outside environment into your house forheating.• The ratio of the energy transferred to the electric energy used in theprocess is called its coefficient of performance (CoP).• A typical CoP for a commercial heat pump is between 3 and 4 unitstransferred per unit of electric energy supplied.

Coefficient of PerformanceThe coefficient of performance (CoP) for a heat pump is the ratio of theenergy transferred for heating to the input electric energy used in theprocess. In reference to the standard heat engine illustration, the coefficientis defined by

There is a theoretical maximum CoP, that of the Carnot cycle:

• For a refrigerator, however, the useful quantity (what you want to achieve) isthe heat extracted, QC , not the heat exhausted.• Therefore, the coefficient of performance of a refrigerator is expressed as

For consumer refrigerators in the U.S., the coefficientof performance for refrigerators is typically recast intoa number called the Energy Efficiency Ratio.Energy efficiency rating

EER = 3.412 COP 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 intomechanical work? • A device that performs such a process is called aheat engine. • According to the first law of thermodynamics, energy isalways 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 energyfrom the process in the form of mechanical work. • The reason for doing this is that energy in the form ofmechanical work is often more useful than the sameamount 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 anIrish-Scottish mathematical physicist,engineer, and outstanding leader in thephysical sciences of the 19th century.• He did important work in themathematical analysis of electricity andthermodynamics, and did much to unify theemerging discipline of physics in itsmodern form.

• He also enjoyed a second career as a telegraphengineer and inventor, a career that propelled him intothe public eye and ensured his wealth, fame and honour.

Second Law Broken

• One of the most fundamental rules of physics, the second law ofthermodynamics, has for the first time been shown not to holdfor microscopic systems.

• The demonstration, by chemical physicists in Australia, couldplace a fundamental limit on miniaturisation, because it suggeststhat the micro-scale devices envisaged by nanotechnologists willnot behave like simple scaled-down versions of their largercounterparts - they could sometimes run backwards.

• The second law states that a closed system will remain thesame or become more disordered over time, i.e. its entropy willalways increase. It is the reason a cup of tea loses heat to itssurroundings, rather than being heated by the air around it.

• "In a typical room, for example, the air molecules are mostlikely to be distributed evenly, which is the overall result oftheir individual random motion", says theoretical physicistAndrew Davies of Glasgow University. ”

• But because of this randomness there is always a probabilitythat suddenly all the air will bunch up in one corner."Thankfully this probability is so small it never happens onhuman timescales.

CARNOT Principles

• The efficiency of an irreversible heat engine is always lessthan the efficiency of of a reversible process operatingbetween the same two heat reservoirs.

• Efficiencies of all reversible heat engines operating betweenthe same two reservoirs are the same.

First statement:

High Temp Reservoir at Th

Low Temp Reservoir at Tl

reversibleirreversible

Qh

Ql

Wirrev

Qh

Ql,rev

Wrev

• If we assume that the irreversible process is moreefficient (contradicts Carnot’s first principle) it will thusdeliver 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 statementof the 2nd law.

Rankine Steam Plant Cycle

• There are four processes in the Rankine cycle, eachchanging the state of the working fluid.• These states are identified by number in the diagramabove.

•Process 4-1: First, the working fluid is pumped (ideallyisentropically..no change in entropy..) from low to highpressure by a pump.• Pumping requires a power input (for examplemechanical or electrical...in a power plant you would usesome of the power generated for this pump.)

•Process 1-2: The high pressure liquid enters a boilerwhere it is heated at constant pressure by an externalheat source to become a superheated vapor.• Common heat sources for power plant systems arecoal, oil, 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 aturbine to generate power output. Ideally, this expansionis isentropic. Can this really be isentropic?• This decreases the temperature and pressure of thevapor.

• Process 3-4: The vapor then enters a condenser where itis cooled to become a saturated liquid. What is a saturatedliquid versus non-saturated?• This liquid then re-enters the pump and the cyclerepeats.

Real Rankine cycle (non-ideal)

• In a real Rankine cycle, the compression by the pumpand the expansion in the turbine are not isentropic.• In other words, these processes are non-reversible andentropy is increased during the two processes (indicatedin the figure as ∆S).

• This somewhat increases the power required by thepump and decreases the power generated by the turbine.It also makes calculations more involved and difficult (nolonger a linear process that can be solved analytically).

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

Regenerative Rankine cycle• The regenerative Rankine cycle is so named becauseafter emerging from the condenser (possibly as a sub-cooled liquid) the working fluid is heated by steam tappedfrom the hot portion of the cycle.

• This increases the average temperature of heat additionwhich in turn increases the thermodynamic efficiency of thecycle.