4 ideal models of engine cycles

7/25/2019 4 Ideal Models of Engine Cycles

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44TOPICTOPIC

Ideal modelsIdeal modelsof engine cycleof engine cycle

Chapter 5Chapter 5Sections 5.1 – 5.4Sections 5.1 – 5.4



Engine modeling purposeEngine modeling purpose

• Development of more complete understanding of

engine processes

• Identification of key controlling variables todecrease experimental work

• Prediction of engine behavior over wide range of

design and operating parameters prior to

experiments

• Design optimiation



Classification of modelsClassification of models

• Thermodynamic models !based on energyconservation"

# ero dimensional

# phenomenological # $uai%dimensional

• &luid dynamic models !energy conservation '

e$uation of motion"

• Ideal models

# ideal gas standard cycle

# fuel%air cycle



Closed or open system?Closed or open system?

• (ngine is not a closedsystem) working fluid

does not execute a

thermodynamic cycle

• Cycles analyed here are not thermodynamic cycles) theyare se$uences of processes* intake) compression)combustion) expansion) and exhaust



Real engine cycleReal engine cycle

Pressure%volume diagram of +I engine,

r c - .,/) 0122 rpm) pi - 2,/ atm) pe - 3 atm) imep - 4,5 atm !fig 1%3"



Ideal models of engine processesIdeal models of engine processes

Process Assumptions

Compression !3%4" 3, 6diabatic and reversible !isentropic"

Combustion !4%0" 3, 6diabatic

4, Combustion occurs at

!a" Constant volume

!b" Constant pressure!c" Part at constant volume and part at constantpressure !limited pressure"

0, Combustion is complete

(xpansion !0%/" 3, 6diabatic and reversible !isentropic"

(xhaust !/%1%7"

Intake !7%8%3"

3, 6diabatic

4, 9alve events occur at top% and bottom center

0, :o change in cylinder volume as pressure differencesacross open valves drop to ero

/, Inlet and exhaust pressures constant

1, 9elocity effects negligible



Ideal cycles (Ideal cycles (unthrottled)unthrottled)

!a" constant%volume combustion

!b" constant%pressure combustion

!c" limited pressure combustion



Ideal cyclesIdeal cycles (throttled and supercharged)(throttled and supercharged)

!d" Throttled constant%volume cycle) pi <pe

!e" supercharged constant%volume cycle) pi >pe



Thermodynamic relations for engine processesThermodynamic relations for engine processes

Indicated fuel conversion efficiency

Indicated mean effective pressure

Indicated work per cycle

,

,

c i

f i

f LHV

W m Q

η =

,, f LHV f ic i

d d

m QW imep

V V

η = =

,c i C E W W W = +



Constant-volume cycleConstant-volume cycle

Compression

Combustion

(xpansion

(fficiency

11 2

2

cv r s sv = =

( )1 2 1 2C W U U m u u= − = −

3 2 3 2 0v v u u= − =

44 3

3

c

vr s s

v= = ( )3 4 3 4 E W U U m u u= − = −

( ) ( )3 4 2 1

,

f i f LHV

m u u u u

m Qη

− − − =



Constant-pressure cycleConstant-pressure cycle

Compression

Combustion

(xpansion

(fficiency

11 2

2

cv r s sv

= =

( )1 2 1 2C W U U m u u= − = −

3 2 3 2 0 p p h h= − =

43 2 4 3

2

c

v p p r s s

v= = = ( )

( )

3 4 2 3 2

3 4 4 4 2 2

E

W U U p V V

m h h p v p v

= − + −

= − + −

( ) ( )3 4 2 1 4 4 2 2

, f i

f LHV

m h h u u p v p v

m Q

η − − − + − =



imited-pressure cycleimited-pressure cycle

Compression

Combustion

(xpansion

(fficiency

11 2

2

cv r s sv

= =

( )1 2 1 2C W U U m u u= − = −

3 2 3 3

3 2 3 30 0

a b a

a b a

v v p p

u u h h

= =− = − =

43 3 4 3

3

c b a b

a

vr p p s s

v= = = ( )

( )

3 4 3 3 3

3 4 4 4 3 3

E b b a

b a

W U U p V V

m h h p v p v= − + −

= − + −

( ) ( )3 4 2 1 4 4 3 3

,

b a

f i f LHV

m h h u u p v p v

m Qη

− − − + − =



Ideal gas standard cyclesIdeal gas standard cycles

• Ideal engine processes !with all assumptions

taken above"

• ;orking fluid is an ideal gas• c v and c p are constant throughout cycle

!independent of temperature"

• ($uations which describe engine performance

and efficiency can be further simplified



Constant-volume ideal gas standard cycleConstant-volume ideal gas standard cycle

Compression work

(xpansion work

Temperature rise during combustion is related to the heating value by

&or convenience we will define

;here Q* is the specific internal energy decrease) during isothermal

combustion) per unit mass of working fluid

( )1 2 1 2C vW U U mc T T = − = −

( )3 4 3 4 E vW U U mc T T = − = −

( )3 2v f LHV mc T T m Q− =

* f LHV m QQ

m=




Indicated fuel conversion efficiency becomes

+ince 3%4 and 0%/ are isentropic processes between the same volumes) V 1

and V 2

where , <ence and indicated fuel

conversion efficiency can be written as

( ) ( )

( )3 4 2 1 4 1

3 2 3 2

1, f i

T T T T T T

T T T T η

− − − −= = −

− −

11

1 32 1 4

1 2 3 4

c

T T V V r

T V V T

γ γ

γ

−−

− = = = = ÷ ÷

p vc cγ =4 1 3 2T T T T =

1

11

, f i

cr γ η −= −




&uel conversion efficiency as a

function of compression ratio

In cycle analysis the value of γ can be taken*

! "#$! "#$

1

11

, f i

cr γ η −= −




Two more useful relations, Dimensionless numbers r c ) γ ) and Q*

/(c v T 1 ) aresufficient to describe the characteristics of the constant%volume ideal gas

standard cycle) relative to its initial conditions p1, T 1

or relative to the maximum pressure in the cycle) p3

6 high value of imep= 6 high value of imep= p p33 is desirable, (ngine weight will increase withis desirable, (ngine weight will increase with

increasingincreasing p p33 to withstand the increasing stresses in componentsto withstand the increasing stresses in components

11 1

imep 1 11

1 1

*

c

v c c

r Q

p c T r r γ γ −

= −

÷ ÷ ÷ ÷− −

( ) ( )

1

13 1

1 1imep 11 1 1

с c

c c v c

r r p r r c T Q r

γ

γ γ γ

−

− −= ÷− − + *



imited-pressure ideal gas standard cycleimited-pressure ideal gas standard cycle

Denoting

we get for the limited%pressure cycle

3 3

2 3

and b

a

p V

p V α β = =

( )1

1 11

1 1, f i

cr

γ

γ

αβ η

αγ β α − −= −

− + −

( )1 1

imep

1 1

*

,c f i

v c

r Q

p c T r η γ

= ÷− −

( )3 1

imep 1 1

1 1

*

,

c f i

c v c

r Q

p r c T r γ

η

α γ

= ÷ ÷

− −



Constant-pressure ideal gas standard cycleConstant-pressure ideal gas standard cycle

Taking α - 3 we get for the constant%pressure cycle

( )1

1 11

1, f i

cr

γ

γ

β η

γ β −

−= −

−

( )1 1

imep

1 1

*

,

c f i

v c

r Q

p c T r

η

γ

= ÷

− −

( )3 1

imep 1 1

1 1

*

,

c f i

c v c

r Q

p r c T r γ η

γ

= ÷ ÷− −



Cycle comparison# %ome assumptionsCycle comparison# %ome assumptions

"# ! "#$

&# Q* is defined as the internal energy decrease during isothermal

combustion per unit mass of working fluid, <ence

6 simple approximation for !ma /m" is !!r c -1"=r c, > i,e,) fresh air fills the

displaced volume and the residual gas fills the clearance volume ant the

same density, Then ! A is a constant"

'or this value of all cycles ould e urning a

stiochiometric mi*ture ith an appropriate residual gas fraction

* f LHV f a

LHV a

m Q m mQ Q

m m m

= = ÷ ÷

1

1* c

v c

r Q Ac T r

−= ÷

( )*

vQ c T



Cycle comparisonCycle comparison

3

1 1

12 1 3

8 525 67

*

.

.

c

a

v

r

pQ

c T p

γ = =

= =

Constant volume 2,141 37,0 2,34. 34.

?imited pressure ,122 31,1 2,403 78

Constant pressure 2,0.2 33,. 2,/77 41,0

, f iη 1

imep p 3

imep p 1

max p p



Cycle comparisonCycle comparison



'uel-Air Cycle Analysis'uel-Air Cycle Analysis

4 ideal models of engine cycles

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