Termodynamik Föreläsning 9 - Karlstad University · Termodynamik Föreläsning 9 Analys av Värmemaskiner, Entropi i Steady Flow Jens Fjelstad 2010 10 01 1 / 18 Innehåll TFS 2:a

Termodynamik – Föreläsning 9Analys av Värmemaskiner, Entropi i Steady Flow

Jens Fjelstad

2010–10–01

1 / 18

Innehåll

• TFS 2:a upplagan (Çengel & Turner)◦ 8.1–8.6◦ 7.10–7.12

• TFS 3:e upplagan (Çengel, Turner & Cimbala)◦ kapitel 23◦ 8.10–8.12

• TD 6:e upplagan (Çengel & Boles)◦ 9.1–9.7◦ 7.10–7.12

2 / 18

Analys av Värmemaskiner

Ottocykeln, Dieselcykeln

3 / 18

Verkliga & Ideala Cykler

• Verklig värmemaskin: verklig, komplicerad cykel◦ ej kvasistatisk◦ involverar friktion: tryckförluster i kanaler◦ oönskad värmetransport i flera delar av maskinen

• Approximera verklig cykel med Ideal cykel◦ internt reversibel (⇒ kvasistatisk)◦ inga tryckförluster◦ ingen oönskad värmetransport

• Ideal cykel ej nödvändigtvis reversibel:

ηth,ideal < ηth,rev

• Effektivitet mäts via jämförelse med Carnotcykeln• kvasistatisk⇒ kan illustrera i tillståndsdiagram

1 ! BASIC CONSIDERATIONS IN THE ANALYSIS OFPOWER CYCLES

Most power-producing devices operate on cycles, and the study of powercycles is an exciting and important part of thermodynamics. The cyclesencountered in actual devices are difficult to analyze because of the pres-ence of complicating effects, such as friction, and the absence of sufficienttime for establishment of the equilibrium conditions during the cycle. Tomake an analytical study of a cycle feasible, we have to keep the complexi-ties at a manageable level and utilize some idealizations (Fig. 1). When theactual cycle is stripped of all the internal irreversibilities and complexities,we end up with a cycle that resembles the actual cycle closely but is madeup totally of internally reversible processes. Such a cycle is called an idealcycle (Fig. 2).

A simple idealized model enables engineers to study the effects of themajor parameters that dominate the cycle without getting bogged down in thedetails. The cycles discussed in this chapter are somewhat idealized, but theystill retain the general characteristics of the actual cycles they represent. Theconclusions reached from the analysis of ideal cycles are also applicable toactual cycles. The thermal efficiency of the Otto cycle, the ideal cycle forspark-ignition automobile engines, for example, increases with the compres-sion ratio. This is also the case for actual automobile engines. The numericalvalues obtained from the analysis of an ideal cycle, however, are not neces-sarily representative of the actual cycles, and care should be exercised in theirinterpretation (Fig. 3). The simplified analysis presented in this chapter forvarious power cycles of practical interest may also serve as the starting pointfor a more in-depth study.

Heat engines are designed for the purpose of converting thermal energy towork, and their performance is expressed in terms of the thermal efficiencyhth, which is the ratio of the net work produced by the engine to the totalheat input:

(1)

Recall that heat engines that operate on a totally reversible cycle, such asthe Carnot cycle, have the highest thermal efficiency of all heat enginesoperating between the same temperature levels. That is, nobody can developa cycle more efficient than the Carnot cycle. Then the following questionarises naturally: If the Carnot cycle is the best possible cycle, why do wenot use it as the model cycle for all the heat engines instead of botheringwith several so-called ideal cycles? The answer to this question is hardware-related. Most cycles encountered in practice differ significantly from theCarnot cycle, which makes it unsuitable as a realistic model. Each idealcycle discussed in this chapter is related to a specific work-producing deviceand is an idealized version of the actual cycle.

The ideal cycles are internally reversible, but, unlike the Carnot cycle,they are not necessarily externally reversible. That is, they may involve irre-versibilities external to the system such as heat transfer through a finite tem-perature difference. Therefore, the thermal efficiency of an ideal cycle, ingeneral, is less than that of a totally reversible cycle operating between the

hth !Wnet

Q in or hth !

wnet

qin

2 | Fundamentals of Thermal-Fluid Sciences

N

OVEN

ACTUAL

IDEAL

175ºC

WATER

Potato

FIGURE 1Modeling is a powerful engineeringtool that provides great insight andsimplicity at the expense of some lossin accuracy.

P

Actual cycle

Ideal cycle

v

FIGURE 2The analysis of many complexprocesses can be reduced to amanageable level by utilizing someidealizations.

FIGURE 3Care should be exercised in the interpre-tation of the results from ideal cycles.© Reprinted with special permission of KingFeatures Syndicate.

4 / 18

Luftstandard Antagandet

• Gaskraftcykler ofta:◦ öppna, gas passerar in i och ut från maskinen i olika tillstånd, dvs

arbetsmediet fullföljer inte en kretsprocess◦ inre förbränning respektive utsläpp av avgaser ger värmetransport

in respektive ut• Förenkling som möjliggör analys (luftstandard antagandet):◦ arbetsmediet är luft, som arbetar i en kretsprocess och beter sig

som en ideal gas◦ alla delprocesser är internt reversibla (⇒ kvasistatiska)◦ förbränningsprocess ersätts med värmetransport in från extern

källa◦ avgasutsläpp ersätts med värmetransport ut som återför

arbetsmediet till ursprungstillståndet• Kalluftstandard antagande: dessutom konstanta

värmekapaciteter vid rumstemperatur (25◦ C)5 / 18

Exempel ≈ 23.21 TFS3

En gas med totala massan m = 0,003 kg fullföljer en cykelbestående av följande delprocesser• 1 7→ 2: isokor värmetransport in från P1 = 95 kPa, T1 = 17◦ C

till P2 = 380 kPa• 2 7→ 3: isentrop expansion till P3 = 95 kPa• 3 7→ 1: isobar värmetransport ut till tillstånd 1Under antagande om kalluftstandard:a) illustrera cykeln i P–v resp. T –s diagramb) bestäm nettoarbetet per cykelc) bestäm den termiska verkningsgraden

6 / 18

Kolvmotorer (Reciprocating Engines)

• Arbetar mellan◦ högsta läge TDC (top dead center) med volym Vmin = VTDC , och◦ lägsta läge BDC (bottom dead center) med volym Vmax = VBDC

(förflyttningsvolym)

• Slaglängd (stroke): höjdskillnad mellan TDC & BDC• Kompressionsförhållande:

r =Vmax

Vmin=

VBDC

VTDC• Medeleffektivt tryck (MEP): hypotetiskt

konstant tryck som ger samma arbetesom den verkliga cykeln

MEP =Wnet

Vmax − Vmin=

wnet

vmax − vmin

room temperature (25°C, or 77°F). When this assumption is utilized, the air-standard assumptions are called the cold-air-standard assumptions.A cycle for which the air-standard assumptions are applicable is frequentlyreferred to as an air-standard cycle.

The air-standard assumptions previously stated provide considerable sim-plification in the analysis without significantly deviating from the actualcycles. This simplified model enables us to study qualitatively the influenceof major parameters on the performance of the actual engines.

4 ! AN OVERVIEW OF RECIPROCATING ENGINESDespite its simplicity, the reciprocating engine (basically a piston–cylinderdevice) is one of the rare inventions that has proved to be very versatile andto have a wide range of applications. It is the powerhouse of the vast major-ity of automobiles, trucks, light aircraft, ships, and electric power genera-tors, as well as many other devices.

The basic components of a reciprocating engine are shown in Fig. 10. Thepiston reciprocates in the cylinder between two fixed positions called thetop dead center (TDC)—the position of the piston when it forms the small-est volume in the cylinder—and the bottom dead center (BDC)—the posi-tion of the piston when it forms the largest volume in the cylinder.The distance between the TDC and the BDC is the largest distance that thepiston can travel in one direction, and it is called the stroke of the engine.The diameter of the piston is called the bore. The air or air–fuel mixture isdrawn into the cylinder through the intake valve, and the combustion prod-ucts are expelled from the cylinder through the exhaust valve.

The minimum volume formed in the cylinder when the piston is at TDCis called the clearance volume (Fig. 11). The volume displaced by the pis-ton as it moves between TDC and BDC is called the displacement volume.The ratio of the maximum volume formed in the cylinder to the minimum(clearance) volume is called the compression ratio r of the engine:

(3)

Notice that the compression ratio is a volume ratio and should not be con-fused with the pressure ratio.

Another term frequently used in conjunction with reciprocating engines isthe mean effective pressure (MEP). It is a fictitious pressure that, if it actedon the piston during the entire power stroke, would produce the same amountof net work as that produced during the actual cycle (Fig. 12). That is,

or

(4)

The mean effective pressure can be used as a parameter to compare theperformances of reciprocating engines of equal size. The engine with a largervalue of MEP delivers more net work per cycle and thus performs better.

MEP !Wnet

Vmax " Vmin!

wnet

vmax " vmin 1kPa 2

Wnet ! MEP # Piston area # Stroke ! MEP # Displacement volume

r !Vmax

Vmin!

VBDC

VTDC

Power and Refrigeration Cycles | 7

N

Intake valve

Exhaustvalve

BoreTDC

BDC

Stroke

FIGURE 10Nomenclature for reciprocatingengines.

TDC

BDC

Displacement volume

(a) Clearance volume

(b)

FIGURE 11Displacement and clearance volumesof a reciprocating engine.

7 / 18

Gnistantändningsmotorer (Bensinmotor)Fyrtaktsmotor

Reciprocating engines are classified as spark-ignition (SI) engines orcompression-ignition (CI) engines, depending on how the combustionprocess in the cylinder is initiated. In SI engines, the combustion of theair–fuel mixture is initiated by a spark plug. In CI engines, the air–fuelmixture is self-ignited as a result of compressing the mixture above its self-ignition temperature. In the next two sections, we discuss the Otto andDiesel cycles, which are the ideal cycles for the SI and CI reciprocatingengines, respectively.

5 ! OTTO CYCLE: THE IDEAL CYCLE FOR SPARK-IGNITION ENGINES

The Otto cycle is the ideal cycle for spark-ignition reciprocating engines. Itis named after Nikolaus A. Otto, who built a successful four-stroke enginein 1876 in Germany using the cycle proposed by Frenchman Beau deRochas in 1862. In most spark-ignition engines, the piston executes fourcomplete strokes (two mechanical cycles) within the cylinder, and thecrankshaft completes two revolutions for each thermodynamic cycle. Theseengines are called four-stroke internal combustion engines. A schematic ofeach stroke as well as a P-v diagram for an actual four-stroke spark-ignitionengine is given in Fig. 13(a).


N

Wnet = MEP(Vmax – Vmin)

Vmin Vmax V

MEP

P

TDC BDC

Wnet

FIGURE 12The net work output of a cycle isequivalent to the product of the meaneffective pressure and thedisplacement volume.

qin

qout

4

3

2

1

Patm

P

P

Compressionstroke

Power (expansion)stroke

Air–fuelmixture

(a) Actual four-stroke spark-ignition engine

(b) Ideal Otto cycle

Isentropiccompression

AIR(2)

(1)

End ofcombustion

Exhaust valveopens

Ignition

TDC BDC

Intake

Exhaust

Intakevalve opens

Expansion

Compression

IsentropicIsentropic

AIR

(4)–(1)

Air–fuelmixture

AIR(2)–(3)

Exhauststroke

Intakestroke

AIR(3)

(4)

Exhaustgases

Isentropicexpansion

v = const.heat addition

v = const.heat rejection

qin qout

v

TDC BDC v

FIGURE 13Actual and ideal cycles in spark-ignition engines and their P-v diagrams.





N


Vmin Vmax V

MEP

P

TDC BDC

Wnet


qin

qout

4

3

2

1

Patm

P

P

Compressionstroke


Air–fuelmixture




AIR(2)

(1)

End ofcombustion

Exhaust valveopens

Ignition

TDC BDC

Intake

Exhaust

Intakevalve opens

Expansion

Compression


AIR

(4)–(1)

Air–fuelmixture

AIR(2)–(3)

Exhauststroke

Intakestroke

AIR(3)

(4)

Exhaustgases

Isentropicexpansion



qin qout

v

TDC BDC v






N


Vmin Vmax V

MEP

P

TDC BDC

Wnet


qin

qout

4

3

2

1

Patm

P

P

Compressionstroke


Air–fuelmixture




AIR(2)

(1)

End ofcombustion

Exhaust valveopens

Ignition

TDC BDC

Intake

Exhaust

Intakevalve opens

Expansion

Compression


AIR

(4)–(1)

Air–fuelmixture

AIR(2)–(3)

Exhauststroke

Intakestroke

AIR(3)

(4)

Exhaustgases

Isentropicexpansion



qin qout

v

TDC BDC v


8 / 18

Ottocykeln – Ideal “Gnistcykel”





N


Vmin Vmax V

MEP

P

TDC BDC

Wnet


qin

qout

4

3

2

1

Patm

P

P

Compressionstroke


Air–fuelmixture




AIR(2)

(1)

End ofcombustion

Exhaust valveopens

Ignition

TDC BDC

Intake

Exhaust

Intakevalve opens

Expansion

Compression


AIR

(4)–(1)

Air–fuelmixture

AIR(2)–(3)

Exhauststroke

Intakestroke

AIR(3)

(4)

Exhaustgases

Isentropicexpansion



qin qout

v

TDC BDC v






N


Vmin Vmax V

MEP

P

TDC BDC

Wnet


qin

qout

4

3

2

1

Patm

P

P

Compressionstroke


Air–fuelmixture




AIR(2)

(1)

End ofcombustion

Exhaust valveopens

Ignition

TDC BDC

Intake

Exhaust

Intakevalve opens

Expansion

Compression


AIR

(4)–(1)

Air–fuelmixture

AIR(2)–(3)

Exhauststroke

Intakestroke

AIR(3)

(4)

Exhaustgases

Isentropicexpansion



qin qout

v

TDC BDC v






N


Vmin Vmax V

MEP

P

TDC BDC

Wnet


qin

qout

4

3

2

1

Patm

P

P

Compressionstroke


Air–fuelmixture




AIR(2)

(1)

End ofcombustion

Exhaust valveopens

Ignition

TDC BDC

Intake

Exhaust

Intakevalve opens

Expansion

Compression


AIR

(4)–(1)

Air–fuelmixture

AIR(2)–(3)

Exhauststroke

Intakestroke

AIR(3)

(4)

Exhaustgases

Isentropicexpansion



qin qout

v

TDC BDC v


Major car companies have research programs underway on two-strokeengines which are expected to make a comeback in the future.

The thermodynamic analysis of the actual four-stroke or two-stroke cyclesdescribed is not a simple task. However, the analysis can be simplified sig-nificantly if the air-standard assumptions are utilized. The resulting cycle,which closely resembles the actual operating conditions, is the ideal Ottocycle. It consists of four internally reversible processes:

1-2 Isentropic compression2-3 Constant-volume heat addition3-4 Isentropic expansion4-1 Constant-volume heat rejection

The execution of the Otto cycle in a piston–cylinder device together witha P-v diagram is illustrated in Fig. 13b. The T-s diagram of the Otto cycle isgiven in Fig. 16.

The Otto cycle is executed in a closed system, and disregarding thechanges in kinetic and potential energies, the energy balance for any of theprocesses is expressed, on a unit-mass basis, as

(5)

No work is involved during the two heat transfer processes since both takeplace at constant volume. Therefore, heat transfer to and from the workingfluid can be expressed as

(6a)and

(6b)

Then the thermal efficiency of the ideal Otto cycle under the cold air stan-dard assumptions becomes

Processes 1-2 and 3-4 are isentropic, and v2 ! v3 and v4 ! v1. Thus,

(7)

Substituting these equations into the thermal efficiency relation and simpli-fying give

(8)

where

(9)

is the compression ratio and k is the specific heat ratio cp /cv.Equation 8 shows that under the cold-air-standard assumptions, the ther-

mal efficiency of an ideal Otto cycle depends on the compression ratio ofthe engine and the specific heat ratio of the working fluid. The thermal effi-ciency of the ideal Otto cycle increases with both the compression ratio and

r !Vmax

Vmin!

V1

V2!

v1

v2

hth,Otto ! 1 "1

r k"1

T1

T2! a v2

v1b k"1

! a v3

v4b k"1

!T4

T3

hth,Otto !wnet

qin! 1 "

qout

qin! 1 "

T4 " T1

T3 " T2! 1 "

T1 1T4>T1 " 1 2T2 1T3>T2 " 1 2

qout ! u4 " u1 ! cv 1T4 " T1 2qin ! u3 " u2 ! cv 1T3 " T2 21qin " qout 2 # 1win " wout 2 ! ¢u 1kJ>kg 2


N

T

s

1

2

3

4v =

const.

v = const. qout

qin

FIGURE 16T-s diagram of the ideal Otto cycle.• 1 7→ 2: (int.rev.+ad.⇒) isentrop kompression

• 2 7→ 3: isokor värmetransport in

• 3 7→ 4: isentrop expansion

• 4 7→ 1: isokor värmetransport ut

9 / 18

Ottocykeln – Kalluftstandardanalys

• TD1: wnet = qin − qout

• qin = u3 − u2 = cv (T3 − T2)

• qout = u4 − u1 = cv (T4 − T1)

• 1 7→ 2 isentrop: T1/T2 = (v2/v1)k−1

• 3 7→ 4 isentrop: T4/T3 = (v3/v4)k−1

• Verkningsgrad:

ηth,Otto =wnet

qin= 1− qout

qin= 1− T4 − T1

T3 − T2= 1− T1(T4/T1 − 1)

T2(T3/T2 − 1)

v1 = v4, v2 = v3 ⇒ T1/T2 = T4/T3 ⇒ T4/T1 = T3/T2

10 / 18



• qin = u3 − u2 = cv (T3 − T2)

• qout = u4 − u1 = cv (T4 − T1)

• 1 7→ 2 isentrop: T1/T2 = (v2/v1)k−1

• 3 7→ 4 isentrop: T4/T3 = (v3/v4)k−1

• Verkningsgrad:

ηth,Otto =wnet

qin= 1− qout

qin= 1− T1

T2

v1 = v4, v2 = v3 ⇒ T1/T2 = T4/T3 ⇒ T4/T1 = T3/T2

r =vmax

vmin=

v1

v2⇒ T1

T2=

1r k−1

10 / 18



• qin = u3 − u2 = cv (T3 − T2)

• qout = u4 − u1 = cv (T4 − T1)

• 1 7→ 2 isentrop: T1/T2 = (v2/v1)k−1

• 3 7→ 4 isentrop: T4/T3 = (v3/v4)k−1

• Verkningsgrad:

ηth,Otto = 1− 1r k−1

10 / 18

Ottocykeln – Begränsningar hos Verklig Cykel• Termisk verkningsgrad hos verklig cykel typiskt ∼ 25%− 30%◦ huvudsakligen irreversibiliteter

• Öka ηth genom att öka kompressionsförhållandet r?◦ begränsas av att flampunkten för bränsle–luft blandningen nås vid

tillräckligt stort r (självantändning – “knackning”)◦ förhindras traditionellt genom blytillsatser (ökat oktantal), men

miljömässigt dåligt• Öka ηth genom att öka k?◦ k ≤ 3/2, minskar med storleken (“komplexiteten”) hos molekylerna◦ bensin: komplex organisk molekyl, litet värde på kthe specific heat ratio. This is also true for actual spark-ignition internal

combustion engines. A plot of thermal efficiency versus the compressionratio is given in Fig. 17 for k ! 1.4, which is the specific heat ratio value ofair at room temperature. For a given compression ratio, the thermal effi-ciency of an actual spark-ignition engine is less than that of an ideal Ottocycle because of the irreversibilities, such as friction, and other factors suchas incomplete combustion.

We can observe from Fig. 17 that the thermal efficiency curve is rathersteep at low compression ratios but flattens out starting with a compressionratio value of about 8. Therefore, the increase in thermal efficiency with thecompression ratio is not as pronounced at high compression ratios. Also,when high compression ratios are used, the temperature of the air–fuel mix-ture rises above the autoignition temperature of the fuel (the temperature atwhich the fuel ignites without the help of a spark) during the combustionprocess, causing an early and rapid burn of the fuel at some point or pointsahead of the flame front, followed by almost instantaneous inflammation ofthe end gas. This premature ignition of the fuel, called autoignition, pro-duces an audible noise, which is called engine knock. Autoignition inspark-ignition engines cannot be tolerated because it hurts performance andcan cause engine damage. The requirement that autoignition not be allowedplaces an upper limit on the compression ratios that can be used in spark-ignition internal combustion engines.

Improvement of the thermal efficiency of gasoline engines by utilizinghigher compression ratios (up to about 12) without facing the autoignitionproblem has been made possible by using gasoline blends that have goodantiknock characteristics, such as gasoline mixed with tetraethyl lead.Tetraethyl lead had been added to gasoline since the 1920s because it is aninexpensive method of raising the octane rating, which is a measure of theengine knock resistance of a fuel. Leaded gasoline, however, has a veryundesirable side effect: it forms compounds during the combustion processthat are hazardous to health and pollute the environment. In an effort tocombat air pollution, the government adopted a policy in the mid-1970s thatresulted in the eventual phase-out of leaded gasoline. Unable to use lead, therefiners developed other techniques to improve the antiknock characteristicsof gasoline. Most cars made since 1975 have been designed to use unleadedgasoline, and the compression ratios had to be lowered to avoid engineknock. The ready availability of high octane fuels made it possible to raisethe compression ratios again in recent years. Also, owing to the improve-ments in other areas (reduction in overall automobile weight, improvedaerodynamic design, etc.), today’s cars have better fuel economy and conse-quently get more miles per gallon of fuel. This is an example of how engi-neering decisions involve compromises, and efficiency is only one of theconsiderations in final design.

The second parameter affecting the thermal efficiency of an ideal Ottocycle is the specific heat ratio k. For a given compression ratio, an idealOtto cycle using a monatomic gas (such as argon or helium, k ! 1.667) asthe working fluid will have the highest thermal efficiency. The specific heatratio k, and thus the thermal efficiency of the ideal Otto cycle, decreases asthe molecules of the working fluid get larger (Fig. 18). At room temperatureit is 1.4 for air, 1.3 for carbon dioxide, and 1.2 for ethane. The working


N

2 4 6 8 10 12 14Compression ratio, r

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Typicalcompressionratios forgasolineengines!

th,O

tto

FIGURE 17Thermal efficiency of the ideal Ottocycle as a function of compressionratio (k ! 1.4).

0.8

0.6

0.4

0.2

2 4 6 8 10 12

k = 1.667

k = 1.4

k = 1.3

Compression ratio, r

! th

,Otto

FIGURE 18The thermal efficiency of the Ottocycle increases with the specific heatratio k of the working fluid.

the specific heat ratio. This is also true for actual spark-ignition internalcombustion engines. A plot of thermal efficiency versus the compressionratio is given in Fig. 17 for k ! 1.4, which is the specific heat ratio value ofair at room temperature. For a given compression ratio, the thermal effi-ciency of an actual spark-ignition engine is less than that of an ideal Ottocycle because of the irreversibilities, such as friction, and other factors suchas incomplete combustion.

We can observe from Fig. 17 that the thermal efficiency curve is rathersteep at low compression ratios but flattens out starting with a compressionratio value of about 8. Therefore, the increase in thermal efficiency with thecompression ratio is not as pronounced at high compression ratios. Also,when high compression ratios are used, the temperature of the air–fuel mix-ture rises above the autoignition temperature of the fuel (the temperature atwhich the fuel ignites without the help of a spark) during the combustionprocess, causing an early and rapid burn of the fuel at some point or pointsahead of the flame front, followed by almost instantaneous inflammation ofthe end gas. This premature ignition of the fuel, called autoignition, pro-duces an audible noise, which is called engine knock. Autoignition inspark-ignition engines cannot be tolerated because it hurts performance andcan cause engine damage. The requirement that autoignition not be allowedplaces an upper limit on the compression ratios that can be used in spark-ignition internal combustion engines.

Improvement of the thermal efficiency of gasoline engines by utilizinghigher compression ratios (up to about 12) without facing the autoignitionproblem has been made possible by using gasoline blends that have goodantiknock characteristics, such as gasoline mixed with tetraethyl lead.Tetraethyl lead had been added to gasoline since the 1920s because it is aninexpensive method of raising the octane rating, which is a measure of theengine knock resistance of a fuel. Leaded gasoline, however, has a veryundesirable side effect: it forms compounds during the combustion processthat are hazardous to health and pollute the environment. In an effort tocombat air pollution, the government adopted a policy in the mid-1970s thatresulted in the eventual phase-out of leaded gasoline. Unable to use lead, therefiners developed other techniques to improve the antiknock characteristicsof gasoline. Most cars made since 1975 have been designed to use unleadedgasoline, and the compression ratios had to be lowered to avoid engineknock. The ready availability of high octane fuels made it possible to raisethe compression ratios again in recent years. Also, owing to the improve-ments in other areas (reduction in overall automobile weight, improvedaerodynamic design, etc.), today’s cars have better fuel economy and conse-quently get more miles per gallon of fuel. This is an example of how engi-neering decisions involve compromises, and efficiency is only one of theconsiderations in final design.

The second parameter affecting the thermal efficiency of an ideal Ottocycle is the specific heat ratio k. For a given compression ratio, an idealOtto cycle using a monatomic gas (such as argon or helium, k ! 1.667) asthe working fluid will have the highest thermal efficiency. The specific heatratio k, and thus the thermal efficiency of the ideal Otto cycle, decreases asthe molecules of the working fluid get larger (Fig. 18). At room temperatureit is 1.4 for air, 1.3 for carbon dioxide, and 1.2 for ethane. The working


N

2 4 6 8 10 12 14Compression ratio, r

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Typicalcompressionratios forgasolineengines!

th,O

tto

FIGURE 17Thermal efficiency of the ideal Ottocycle as a function of compressionratio (k ! 1.4).

0.8

0.6

0.4

0.2

2 4 6 8 10 12

k = 1.667

k = 1.4

k = 1.3


! th

,Otto

FIGURE 18The thermal efficiency of the Ottocycle increases with the specific heatratio k of the working fluid.

11 / 18

Kompressionsantändning (Dieselmotor)

• Endast luft komprimeras, större r möjligt• Bränsle sprutas in i cylindern nära TDC, antänds spontant• Dieselcykeln:◦ 1 7→ 2: isentrop kompression◦ 2 7→ 3: isobar värmetransport in◦ 3 7→ 4 isentrop expansion◦ 4 7→ 1: isokor värmetransport ut

6 ! DIESEL CYCLE: THE IDEAL CYCLE FOR COMPRESSION-IGNITION ENGINES

The Diesel cycle is the ideal cycle for CI reciprocating engines. The CIengine, first proposed by Rudolph Diesel in the 1890s, is very similar to theSI engine discussed in the last section, differing mainly in the method ofinitiating combustion. In spark-ignition engines (also known as gasolineengines), the air–fuel mixture is compressed to a temperature that is belowthe autoignition temperature of the fuel, and the combustion process is initi-ated by firing a spark plug. In CI engines (also known as diesel engines),the air is compressed to a temperature that is above the autoignition temper-ature of the fuel, and combustion starts on contact as the fuel is injected intothis hot air. Therefore, the spark plug and carburetor are replaced by a fuelinjector in diesel engines (Fig. 20).

In gasoline engines, a mixture of air and fuel is compressed during thecompression stroke, and the compression ratios are limited by the onset ofautoignition or engine knock. In diesel engines, only air is compressed dur-ing the compression stroke, eliminating the possibility of autoignition.Therefore, diesel engines can be designed to operate at much higher com-pression ratios, typically between 12 and 24. Not having to deal with theproblem of autoignition has another benefit: many of the stringent require-ments placed on the gasoline can now be removed, and fuels that are lessrefined (thus less expensive) can be used in diesel engines.

The fuel injection process in diesel engines starts when the pistonapproaches TDC and continues during the first part of the power stroke.Therefore, the combustion process in these engines takes place over alonger interval. Because of this longer duration, the combustion process inthe ideal Diesel cycle is approximated as a constant-pressure heat-additionprocess. In fact, this is the only process where the Otto and the Diesel cyclesdiffer. The remaining three processes are the same for both ideal cycles.That is, process 1-2 is isentropic compression, 3-4 is isentropic expansion,and 4-1 is constant-volume heat rejection. The similarity between the twocycles is also apparent from the P-v and T-s diagrams of the Diesel cycle,shown in Fig. 21.

Noting that the Diesel cycle is executed in a piston–cylinder device,which forms a closed system, the amount of heat transferred to the workingfluid at constant pressure and rejected from it at constant volume can beexpressed as

(10a)

and(10b)

Then the thermal efficiency of the ideal Diesel cycle under the cold-air-standard assumptions becomes

hth,Diesel !wnet

qin! 1 "

qout

qin! 1 "

T4 " T1

k 1T3 " T2 2 ! 1 "T1 1T4>T1 " 1 2kT2 1T3>T2 " 1 2

"qout ! u1 " u4S qout ! u4 " u1 ! cv 1T4 " T1 2 ! h3 " h2 ! cp 1T3 " T2 2 qin " wb,out ! u3 " u2S qin ! P2 1v3 " v2 2 # 1u3 " u2 2


N

Gasoline engine Diesel engine

Sparkplug

Fuelinjector

AIR

Air–fuelmixture Fuel spray

Spark

FIGURE 20In diesel engines, the spark plug isreplaced by a fuel injector, and onlyair is compressed during thecompression process.

1

2 3

4

P


s

v

1

2

3

4

T

P = constant

v = constant

(a) P- v diagram

(b) T-s diagram

qin

qout

qout

qin

FIGURE 21T-s and P-v diagrams for the idealDiesel cycle.






(10a)

and(10b)


hth,Diesel !wnet

qin! 1 "

qout

qin! 1 "

T4 " T1

k 1T3 " T2 2 ! 1 "T1 1T4>T1 " 1 2kT2 1T3>T2 " 1 2



N


Sparkplug

Fuelinjector

AIR


Spark


1

2 3

4

P


s

v

1

2

3

4

T

P = constant

v = constant

(a) P- v diagram

(b) T-s diagram

qin

qout

qout

qin







(10a)

and(10b)


hth,Diesel !wnet

qin! 1 "

qout

qin! 1 "

T4 " T1

k 1T3 " T2 2 ! 1 "T1 1T4>T1 " 1 2kT2 1T3>T2 " 1 2



N


Sparkplug

Fuelinjector

AIR


Spark


1

2 3

4

P


s

v

1

2

3

4

T

P = constant

v = constant

(a) P- v diagram

(b) T-s diagram

qin

qout

qout

qin


r = v1/v2, rc = v3/v2

ηth,Diesel = 1− 1r k−1

r kc − 1

k(rc − 1)

ηth,Diesel(r) ≤ ηth,Otto(r)

12 / 18

Övrigt om Otto & Diesel

• Större r möjligt i Dieselmotor, typisk termisk verkningsgrad35%− 40%

• Isobar värmetransport in ej bra approximation (Dieselcykeln),bättre ideal cykel: Duala cykeln◦ 1 7→ 2: isentrop kompression◦ 2 7→ X : isokor värmetransport in◦ X 7→ 3: isobar värmetransport in◦ 3 7→ 4: isentrop expansion◦ 4 7→ 1: isokor värmetransport ut

• Både Dieselcykeln & Ottocykelngränser av Duala cykeln

We now define a new quantity, the cutoff ratio rc, as the ratio of the cylin-der volumes after and before the combustion process:

(11)

Utilizing this definition and the isentropic ideal-gas relations for processes1-2 and 3-4, we see that the thermal efficiency relation reduces to

(12)

where r is the compression ratio defined by Eq. 9. Looking at Eq. 12 care-fully, one would notice that under the cold-air-standard assumptions, the effi-ciency of a Diesel cycle differs from the efficiency of an Otto cycle by thequantity in the brackets. This quantity is always greater than 1. Therefore,

(13)

when both cycles operate on the same compression ratio. Also, as the cutoffratio decreases, the efficiency of the Diesel cycle increases (Fig. 22). For thelimiting case of rc ! 1, the quantity in the brackets becomes unity (can youprove it?), and the efficiencies of the Otto and Diesel cycles become identical.Remember, though, that diesel engines operate at much higher compressionratios and thus are usually more efficient than the spark-ignition (gasoline)engines. The diesel engines also burn the fuel more completely since theyusually operate at lower revolutions per minute and the air–fuel mass ratio ismuch higher than spark-ignition engines. Thermal efficiencies of large dieselengines range from about 35 to 40 percent.

The higher efficiency and lower fuel costs of diesel engines make themattractive in applications requiring relatively large amounts of power, such asin locomotive engines, emergency power generation units, large ships, andheavy trucks. As an example of how large a diesel engine can be, a 12-cylin-der diesel engine built in 1964 by the Fiat Corporation of Italy had a normalpower output of 25,200 hp (18.8 MW) at 122 rpm, a cylinder bore of 90 cm,and a stroke of 91 cm.

Approximating the combustion process in internal combustion engines as aconstant-volume or a constant-pressure heat-addition process is overly simplis-tic and not quite realistic. Probably a better (but slightly more complex)approach would be to model the combustion process in both gasoline anddiesel engines as a combination of two heat-transfer processes, one at constantvolume and the other at constant pressure. The ideal cycle based on this con-cept is called the dual cycle, and a P-v diagram for it is given in Fig. 23. Therelative amounts of heat transferred during each process can be adjusted toapproximate the actual cycle more closely. Note that both the Otto and theDiesel cycles can be obtained as special cases of the dual cycle.

hth,Otto 7 hth,Diesel

hth,Diesel ! 1 "1

r k"1 c r kc " 1

k 1rc " 1 2 drc !

V3

V2!

v3

v2


N

0.7

! th,D

iese

l


0.6

0.5

0.4

0.3

0.2

0.1

2 4 6 8 10 12 14 16 18 20 22 24

Typicalcompression

ratios for dieselengines

rc = 1 (Otto)

234

FIGURE 22Thermal efficiency of the ideal Dieselcycle as a function of compression andcutoff ratios (k ! 1.4).

1

2

3

4

P


X

qin

qout

v

FIGURE 23P-v diagram of an ideal dual cycle.

EXAMPLE 3 The Ideal Diesel Cycle

An ideal Diesel cycle with air as the working fluid has a compression ratio of18 and a cutoff ratio of 2. At the beginning of the compression process, theworking fluid is at 14.7 psia, 80°F, and 117 in3. Utilizing the cold-air-standard assumptions, determine (a) the temperature and pressure of air at

13 / 18

Stirlingmotorn

• Slutet system, “extern förbränning”• Två områden med variabel volym,

Varmt & Kallt, gas skyfflas mellan dessa• Ideal cykel: Stirlingcykeln, Reversibel

ideala verkningsgraden maximal!• Användningsområden (hittills):◦ ubåtar◦ “solkraftverk”◦ kylanläggningar◦ värmepumpar◦ ...

14 / 18

Gasturbinmotorer – Braytoncykeln

• Arbetar oftast i öppen process med intern förbränning, ersättmed sluten cykel för analys

• Ideal cykel: Braytoncykeln◦ 1 7→ 2: isentrop kompression◦ 2 7→ 3: isobar värmetransport in◦ 3 7→ 4: isentrop expansion◦ 4 7→ 1: isobar värmetransport ut

N

an external source, and the exhaust process is replaced by a constant-pressure heat-rejection process to the ambient air. The ideal cycle that theworking fluid undergoes in this closed loop is the Brayton cycle, which ismade up of four internally reversible processes:

1-2 Isentropic compression (in a compressor)2-3 Constant-pressure heat addition3-4 Isentropic expansion (in a turbine)4-1 Constant-pressure heat rejection

The T-s and P-v diagrams of an ideal Brayton cycle are shown in Fig. 27.Notice that all four processes of the Brayton cycle are executed in steady-flow devices; thus, they should be analyzed as steady-flow processes. Whenthe changes in kinetic and potential energies are neglected, the energy bal-ance for a steady-flow process can be expressed, on a unit–mass basis, as

(14)

Therefore, heat transfers to and from the working fluid are(15a)

and(15b)

Then the thermal efficiency of the ideal Brayton cycle under the cold-air-standard assumptions becomes

Processes 1-2 and 3-4 are isentropic, and P2 ! P3 and P4 ! P1. Thus,T2

T1! a P2

P1b 1k"12>k

! a P3

P4b 1k"12>k

!T3

T4

hth,Brayton !wnetqin

! 1 "qoutqin

! 1 "cp 1T4 " T1 2cp 1T3 " T2 2 ! 1 "

T1 1T4>T1 " 1 2T2 1T3>T2 " 1 2

qout ! h4 " h1 ! cp 1T4 " T1 2qin ! h3 " h2 ! cp 1T3 " T2 2

1qin " qout 2 # 1win " wout 2 ! hexit " hinlet


Compressorwnet

Turbine

Combustionchamber

Freshair

Exhaustgases1

23

4

Fuel

FIGURE 25An open-cycle gas-turbine engine.

Compressor Turbine

1

23

4Heatexchanger

Heatexchanger

wnet

qin

qout

FIGURE 26A closed-cycle gas-turbine engine.

P

s = const.

s = const.

2

1 4

3

s

T

2

3

4

1

P = const.

P = const.

(a) T-s diagram

(b) P-v diagram

qout

qin

qout

qin

v

FIGURE 27T-s and P-v diagrams for the idealBrayton cycle.

N




(14)


and(15b)



T1! a P2

P1b 1k"12>k

! a P3

P4b 1k"12>k

!T3

T4


! 1 "qoutqin

! 1 "cp 1T4 " T1 2cp 1T3 " T2 2 ! 1 "

T1 1T4>T1 " 1 2T2 1T3>T2 " 1 2




Compressorwnet

Turbine

Combustionchamber

Freshair

Exhaustgases1

23

4

Fuel


Compressor Turbine

1

23

4Heatexchanger

Heatexchanger

wnet

qin

qout


P

s = const.

s = const.

2

1 4

3

s

T

2

3

4

1

P = const.

P = const.

(a) T-s diagram

(b) P-v diagram

qout

qin

qout

qin

v


N




(14)


and(15b)



T1! a P2

P1b 1k"12>k

! a P3

P4b 1k"12>k

!T3

T4


! 1 "qoutqin

! 1 "cp 1T4 " T1 2cp 1T3 " T2 2 ! 1 "

T1 1T4>T1 " 1 2T2 1T3>T2 " 1 2




Compressorwnet

Turbine

Combustionchamber

Freshair

Exhaustgases1

23

4

Fuel


Compressor Turbine

1

23

4Heatexchanger

Heatexchanger

wnet

qin

qout


P

s = const.

s = const.

2

1 4

3

s

T

2

3

4

1

P = const.

P = const.

(a) T-s diagram

(b) P-v diagram

qout

qin

qout

qin

v


N




(14)


and(15b)



T1! a P2

P1b 1k"12>k

! a P3

P4b 1k"12>k

!T3

T4


! 1 "qoutqin

! 1 "cp 1T4 " T1 2cp 1T3 " T2 2 ! 1 "

T1 1T4>T1 " 1 2T2 1T3>T2 " 1 2




Compressorwnet

Turbine

Combustionchamber

Freshair

Exhaustgases1

23

4

Fuel


Compressor Turbine

1

23

4Heatexchanger

Heatexchanger

wnet

qin

qout


P

s = const.

s = const.

2

1 4

3

s

T

2

3

4

1

P = const.

P = const.

(a) T-s diagram

(b) P-v diagram

qout

qin

qout

qin

v


rP = P2/P1

ηth,Brayton = 1− 1

r (k−1)/kP

15 / 18

Reversibelt Stationärt Flödes–Arbete

16 / 18

Internt Reversibel Process vid Stationärt Flöde

• TD1 vid stationärt flöde (1 ingång, 1 utgång):

q − w = h2 − h1 +V2

2 − V21

2+ g(z2 − z1)

• Differentiell form (infinitesimalt liten komponent):

δq − δw = dh + dke + dpe

• Gibbs 1:Tds = dh − vdP

• Internt reversibel process:

δqrev = Tds

17 / 18


• Differentiell form (infinitesimalt liten komponent):

δq − δw = dh + dke + dpe

• Gibbs 1:Tds = dh − vdP

• Internt reversibel process:

δqrev = Tds

• För (internt) reversibelt stationärt flöde således:

−δwrev = vdP + dke + dpe

wrev = −∫ P2

P1

vdP −∆ke + ∆pe

17 / 18


• För (internt) reversibelt stationärt flöde således:

−δwrev = vdP + dke + dpe

wrev = −∫ P2

P1

vdP −∆ke + ∆pe

• Inkompressibelt ämne (v konstant):

wrev = −v(P2 − P1)−∆ke −∆pe

• Om inget arbete uträttas (ex. munstycke, rör):

v(P2 − P1) +V2

2 − V21

2+ g(z2 − z1) = 0

Bernoullis Ekvation17 / 18

Störst Effektivitet vid Reversibel Process

• Betrakta verklig och (internt) reversibel process med sammain– och ut–tillstånd

δqact − δwact = dh + dke + dpe

δqrev − δwrev = dh + dke + dpe

• Identiska högerled, och δqrev = Tds:

δwrev − δwact

T= ds − δqact

T

• TD2: ds ≥ δq/T , dvs

δwrev ≥ δwact

• OBS! FUngerar både då w positiv och då w negativ18 / 18

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Termodynamik Föreläsning 9 - Karlstad University · Termodynamik Föreläsning 9 Analys av Värmemaskiner, Entropi i Steady Flow Jens Fjelstad 2010 10 01 1 / 18 Innehåll TFS 2:a