3. Thermodynamic Cycles

8/10/2019 3. Thermodynamic Cycles

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Thermodynamic Cycles

Section 3


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Thermodynamic Cycles

Air-standard analysis is used to perform elementary analysesof IC engine cycles.

Simplifications to the real cycle include:1) Fixed amount of air (ideal gas) for working fluid2) Combustion process not considered3) Intake and exhaust processes not considered4) Engine friction and heat losses not considered5) Specific heats independent of temperature

The two types of reciprocating engine cycles analyzed are:1) Spark ignition Otto cycle2) Compression ignition Diesel cycle


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SI Engine Cycle vs Thermodynamic Otto Cycle

AI

R

CombustionProducts

Ignition

IntakeStroke

FUEL

Fuel/AirMixture

Air

TC

BC

CompressionStroke

PowerStroke

ExhaustStroke

Q in Q out

CompressionProcess

Const volumeheat addition

Process

ExpansionProcess

C

onst volumeheat rejection

Process

ActualCycle

OttoCycle


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Actual SI Engine cycle

TC BC

Ignition


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Process 1 2 Isentropic compressionProcess 2 3 Constant volume heat additionProcess 3 4 Isentropic expansionProcess 4 1 Constant volume heat rejection

v 2

TC TCv

1BC BC

Qout

Q in

Air-Standard Otto cycle

3

4

2

1

vv

vv

r

Compression ratio:


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First Law Analysis of Otto Cycle

1 2 Isentropic Compression

)()( 12 mW

mQuu in

2

1

1

2

1

2

vv

T T

P P

AIR

)()( 1212 T T cuumW

vin

2 3 Constant Volume Heat Addition

mW

mQ

uu in

)()( 23

)()( 2323 T T cuumQ

vin

2

3

2

3

T

T

P

P

AIR Q in TC

11

2

1

1

2

k k r vv

T T


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3 4 Isentropic Expansion

AIR)()( 34 mW

mQ

uu out

)()( 4343 T T cuum

W v

out

4

3

3

4

3

4

v

v

T

T

P

P

4 1 Constant Volume Heat Removal

AIR Qout m

W

m

Quu out )()( 41

)()( 1414 T T cuumQ

vout

1

1

4

4

T

P

T

P

BC

1

1

4

3

3

4 1

k

k

r vv

T T


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23

1243

uuuuuu

QW

in

cycleth

23

14

23

1423 1uuuu

uuuuuu

Cycle thermal efficiency:

thin

thincycle

r r

umQ

k r r

V P Q

P imep

V V

W imep

1/

11

1 111121

Indicated mean effective pressure is:

Net cycle work:

1243 uumuumW W W inout cycle

First Law Analysis Parameters

12

1

23

14 111)()(

1 k v

v

r T T

T T cT T c


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Effect of Compression Ratio on Thermal Efficiency

Spark ignition engine compression ratio limited by T 3 (autoignition)and P 3 (material strength), both ~r k

For r = 8 the efficiency is 56% which is twice the actual indicated value

Typical SIengines9 < r < 11

k = 1.4

1

11 k

const cth r V


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Effect of Specific Heat Ratio on Thermal Efficiency

1

11 k

const cth r V

Specific heatratio (k)

Cylinder temperatures vary between 20K and 2000K so 1.2 < k < 1.4k = 1.3 most representative


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The net cycle work of an engine can be increased by either:

i) Increasing the r (1 2)ii) Increase Q in (2 3)

P

V2 V1

Q in W cycle

1

2

3

(i)

4

(ii)

Factors Affecting Work per Cycle

1

4

4

3 th

incycle

r r

V Q

V V

W imep

1121


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Effect of Compression Ratio on Thermal Efficiency and MEP

k in

r r r

V P Q

P imep 1

11111

k = 1.3


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Thermodynamic Cycles for CI engines

In early CI engines the fuel was injected when the piston reached TCand thus combustion lasted well into the expansion stroke.

In modern engines the fuel is injected before TC (about 15 o)

The combustion process in the early CI engines is best approximated bya constant pressure heat addition process Diesel Cycle

The combustion process in the modern CI engines is best approximatedby a combination of constant volume and constant pressure Dual Cycle

Fuel injection startsFuel injection starts

Early CI engine Modern CI engine


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Early CI Engine Cycle and the Thermodynamic Diesel Cycle

AI

R

CombustionProducts

Fuel injected

at TC

IntakeStroke

Air

Air

BC

CompressionStroke

PowerStroke

ExhaustStroke

Q in Q out

CompressionProcess

Const pressureheat addition

Process

ExpansionProcess

Const volumeheat rejection

Process

ActualCycle

DieselCycle


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Process 1 2 Isentropic compressionProcess 2 3 Constant pressure heat additionProcess 3 4 Isentropic expansionProcess 4 1 Constant volume heat rejection

Air-Standard Diesel cycle

Q in

Qout

2

3

vv

r c

Cut-off ratio:

v 2

TC v

1BC TC

BC


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m

V V P

m

Quu in 23223 )()(

AIR2 3 Constant Pressure Heat Addition

)()( 222333 v P uv P umQin

)()( 2323 T T chhmQ

pin

cr vv

T T

v RT

v RT

P 2

3

2

3

3

3

2

2

Q in

First Law Analysis of Diesel Cycle

Equations for processes 1 2, 4 1 are the same as those presentedfor the Otto cycle


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)()( 34m

W

m

Quu out

AIR

3 4 Isentropic Expansion

)()( 4343 T T cuumW

vout

note v 4=v 1 socr

r

v

v

v

v

v

v

v

v

v

v

3

2

2

1

3

2

2

4

3

4

r r

T T

P P

T v P

T v P c

3

4

3

4

3

33

4

44

11

4

3

3

4

k

c

k

r r

vv

T T


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23

1411hh

uu

mQ

mQ

in

out

cycle Diesel

1111

1 1c

k c

k const c Diesel

r

r

k r V

For cold air-standard the above reduces to:

Thermal Efficiency

11

1 k Ottor

recall,

Note the term in the square bracket is always larger than one so for thesame compression ratio, r , the Diesel cycle has a lower thermal efficiencythan the Otto cycle

Note: CI needs higher r compared to SI to ignite fuel


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Typical CI Engines15 < r < 20

When r c (= v 3/v2) 1 the Diesel cycle efficiency approaches theefficiency of the Otto cycle

Thermal Efficiency

Higher efficiency is obtained by adding less heat per cycle, Q in, run engine at higher speed to get the same power.


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k = 1.3

k = 1.3

The cut-off ratio is not a natural choice for the independent variablea more suitable parameter is the heat input, the two are related by:

111

111

k

in

c r V P

Q

k

k r as Q

in 0, r

c 1


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Modern CI Engine Cycle and the Thermodynamic Dual Cycle

A

IR

CombustionProducts

Fuel injectedat 15 o bTC

IntakeStroke

Air

Air

TC

BC

CompressionStroke

PowerStroke

ExhaustStroke

Q in Q out

CompressionProcess

Const pressureheat addition

Process

ExpansionProcess

Const volumeheat rejection

Process

ActualCycle

DieselCycle

Q in

Const volumeheat addition

Process


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Process 1 2 Isentropic compressionProcess 2 2.5 Constant volume heat additionProcess 2.5 3 Constant pressure heat additionProcess 3 4 Isentropic expansionProcess 4 1 Constant volume heat rejection

Dual Cycle

Q in

Q i n

Q o u t1

1

2

2

2.5

2.5

3

3

44

)()()()( 5.2325.25.2325.2 T T cT T chhuu

m

Q pv

in


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Thermal Efficiency

)()(11

5.2325.2

14

hhuu

uu

mQ

mQ

in

out

cycle Dual

1)1(11

1 1 c

k c

k cconst

Dual r k r

r v

11

1 k Otto r 1

1111 1c

k ck

const c Diesel

r r

k r V

Note, the Otto cycle (r c=1) and the Diesel cycle ( =1) are special cases:

2

3

5.2

3 andwhere P P

vvr c


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The use of the Dual cycle requires information about either:i) the fractions of constant volume and constant pressure heat addition

(common assumption is to equally split the heat addition), orii) maximum pressure P 3.

Transformation of r c and into more natural variables yields

11111 111 k r V P

Qk k r k inc

1

31 P P

r k

For the same inlet conditions P 1, V1 and the same compression ratio:

Diesel Dual Otto

For the same inlet conditions P 1, V1 and the same peak pressure P 3 (actual design limitation in engines):

otto Dual Diesel


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For the same inlet conditions P 1, V1 and the same compression ratio P 2 /P 1:

For the same inlet conditions P 1, V1 and the same peak pressure P 3:

32

141

1

Tds

Tds

QQ

in

out th

x 2.5

P max

Tmax

P o

P o

P r e s s u r e ,

P

P r e s s u r e ,

P

T e m p e r a

t u r e ,

T

T e m p e r a

t u r e ,

T

Specific Volume

Specific Volume

Entropy Entropy


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Finite Heat Release Model

In the Otto cycle it is assumed that heat is released instantaneously. A finite heat release model specifies heat release as a function of crankangle.

The cumulative heat release or burn fraction xb is given by:

n

d

sb a x

exp1)(

where = crank angle s = start of heat release d = duration of heat release

n = form factora = efficiency factor Used to fit to experimental data

0 < xb < 1


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Finite Heat Release

A typical heat release curve consists of an initial spark ignition phase,followed by a rapid burning phase and ends with burning completion phase

The curve asymptotically approaches 1 so the end of combustion is definedby an arbitrary limit, such as 90% or 99% complete combustion where xb = 0.90 or 0.99 corresponding values for efficiency factor a are 2.3 and 4.6

The rate of heat release as a function of crank angle is:

11

n

d

sb

d in

bin x

naQ

d

dxQ

d

dQ

bin dxQdQ

.99


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d dQ

V k

d dV

V P

k d dP

d dP

V d dV

P Rc

d dV

P d dQ

VdP PdV Rc

PdV Q

mR PV

d mcdT mcdU

W QdU

d

v

v

vv

1

anglecrankunit per

gasidealassuming

,change,anglecranksmallafor

cylinder in thegasthecontainingsystemclosedthetoLawFirstApplying



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The cylinder volume in terms of crank angle, V( ), is

2122 )sin(cos121)( R RV r V V d d Differentiating wrt

2122 )sin(cos1sin2

RV

d

dV d

where

sl R

r

S BV d

2

rationcompressio

nt volumedisplaceme4

2

For the portion of the compression and expansion strokes with no heatrelease, where < s and > s + d dQ/d = 0



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Finite Heat Release Model Results

Start of heat release:Engine 1 - 20 o bTCEngine 2 - TC

Duration 40 o


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Finite Heat Release Model Results

3. Thermodynamic Cycles

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