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Engine Performance
Section 2
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BC
L
TC
ls
a
B
( ) 2/1222 sincos θ θ alas −+=
The cylinder volume at any crank angle is:
)(4
2
sal B
V y AV V ccc −++=+= π
When the piston is at TC (s= l+a) the cylinder
volume equals the clearance volume Vc
Geometric Properties
For most engines B ~ L
(“square engine”)
Connecting rod
VC
y Piston displacement: y = l + a - s
Compression ratio:
c
d c
TC
BC c
V
V V
V
V r
+==
Maximum displacement, or swept, volume:
L B
V d 4
2π =
Wrist pin
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BC
L
TC
l
VC
s
a
B
Average piston velocity:
Average piston speed for standard auto engine is
about 15 m/s. Ultimately limited by material
strength. Therefore engines with large strokes run
at lower speeds
“Over square engines” (B > L) with light pistons
have higher rev limits
Geometric Properties
( )( )
−+=
2/122 sin/
cos1sin
2 θ
θ θ
π
alU
U
p
p
( ) 2/1222 sincos θ θ alas −+=
dt dsU p =Instantaneous piston velocity:
=
s
rev
stroke
m
rev
stroke LN U p 2
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R = l/a
Piston Velocity vs Crank Angle
TC BC
R=3
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Piston Acceleration
Piston displacement is:
For most modern engines (a/l)2 ~ 1/9
Using series expansion approximate (1-ε)1/2 ~ 1-(ε/2) and subst θ = ωt
So
Substituting
yields
differentiating
2/1
2
2
sin1cos
−+= θ θ la
las
−+= t
lalt as ω ω 2
2
sin2
cos
2/)2cos1(sin2 t t ω ω −=
−−+=)2cos1(
4cos
2
t l
alt as ω ω
+= t
l
at a
dt
sd ω ω ω 2coscos2
2
2
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Piston Inertia Force
The inertia force is simply the piston mass multiplied by the acceleration
+−=−= t
l
at am
dt
sd mForce Inertia ω ω ω 2coscos2
2
2
Primary term Secondary term
• The maximum force occurs at TC, θ = ωt = 0 F ~ amω 2
• The primary term varies at the same speed as the crankshaft
and the secondary term varies at twice the crank shaft speed
• For a very long connecting rod (a/l)
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Torque and Power
Torque is measured off the output shaft using a dynamometer.
Load cell
Force FStator
Rotor
b
N
The torque exerted by the engine is T:
W
)341.1k 1()( :units )2( hpW W J s
rev
rev
rad T N T W ==
⋅⋅=⋅= π ω
J NmbF T =⋅= :units
The power delivered by the engine turning at a speed N andabsorbed by the dynamometer is:
Note: ω is the shaft angular velocity in units rad/s
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The term brake power , , is used to specify that the power is
measured at the output shaft, this is the usable power delivered bythe engine to the load.
bW
Brake Power
iW
The brake power is less than the power generated by the gas in
the cylinders due to mechanical friction and parasitic loads (oil
pump, air conditioner compressor, supercharger, etc…).
The power produced in the cylinder is termed the indicated
power , .
Torque is a measure of an engine’s ability to do work and power is
the rate at which work is done
Note torque is independent of crank speed.
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Indicated Work per Cycle
Given the cylinder pressure data over the operating cycle of the engine one
can calculate the work done by the gas on the piston. This data is
typically given as P vs V
∫= PdV W i
Compression
W0Intake
W>0
Exhaust
W
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Indicated Work per Cycle
Given the cylinder pressure data over the operating cycle of the engine one
can calculate the work done by the gas on the piston. This data is
typically given as P vs V
The indicated work per cycle is given by ∫= PdV W i
Compression
W0Intake
W>0
Exhaust
W 0
WB < 0
A
C
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Gross indicated work per cycle – net work delivered to the piston over
the compression and expansion strokes only:
Wi,g = area A + area C (>0)
Work per Cycle
Pump work – net work delivered to the gas over the intake and exhaust
strokes:
Wp = area B + area C (
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Indicated Work at WOT
The pressure at the intake port is just below atmospheric pressure
The pump work (area B+C) is small compared to the gross indicatedwork (area A+C)
Wi,n = Wi,g - Wp = area A - area B
Pintake
PintakePo
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Indicated Work at Part Throttle
The pressure at the intake port is significantly lower than atmospheric pressure
The pump work (area B+C) can be significant compared to gross indicated
work (area A+C)
Wi,n = Wi,g - Wp = area A - area B
Pintake
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Indicated Work with Supercharging
Engines with superchargers or turbochargers have intake pressures
greater than the exhaust pressure, yielding a positive pump work
Wi,n = area A + area B
Supercharge increases the net indicated work but is a parasitic load
since it is driven by the crankshaft
Pintake
Compressor
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Mechanical Efficiency
Some of the power generated in the cylinder is used to overcome engine
friction and to pump gas into and out of the engine.
The term friction power , , is used to describe collectively these power
losses, such that:
gi
f
gi
f gi
gi
bm
W
W
W
W W
W
W
,,
,
,1
−=
−
==η
f W
Friction power can be measured by motoring the engine.
bgi f W W W −= ,
The mechanical efficiency is defined as:
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Mechanical efficiency depends on pumping losses (throttle position) and
frictional losses (engine design and engine speed).
Mechanical Efficiency, cont’d
Typical values for automobile engines at WOT are:
90% @2000 RPM and 75% @ max speed.
Throttling increases pumping power and thus the mechanical efficiencydecreases, at idle the mechanical efficiency approaches zero.
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cyclecycle W T W N W T N W ∝⋅∝⋅∝ so and
1 kW = 1.341 hp
Rated brake power
Power and Torque versus Engine Speed at WOT
There is a maximum in the brake power
versus engine speed called the rated
brake power (RBP).
At higher speeds brake power decreases as
friction power becomes significant compared
to the indicated power f gib W W W
−= ,
Max brake torque There is a maximum in the torque versus
speed called maximum brake torque (MBT).
Brake torque drops off:
• at lower speeds due to heat losses
• at higher speeds it becomes more difficult to
ingest a full charge of air.
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Indicated Mean Effective Pressure (IMEP)
imep is a fictitious constant pressure that would produce the same
work per cycle if it acted on the piston during the power stroke.
R
p p
R
d i
d
Ri
d
i
n
U Aimep
n
N V imepW
N V
nW
V
W imep
⋅
⋅⋅=
⋅⋅=→
⋅⋅
==2
imep is a better parameter than torque to compare engines for design and
output because it is independent of engine size, Vd.
R
d
d
R
d
b
n
V bmepT
V
nT
V
W bmep
⋅⋅
=→⋅⋅
==π
π
2
2
Brake mean effective pressure (bmep) is defined as:
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The maximum bmep of a good engine design is well established:
Four stroke engines:
SI engines: bmep= 850-1050 kPa*
CI engines: bmep= 700 -900 kPa
Turbocharged SI engines: bmep= 1250 -1700 kPa
Turbocharged CI engines: bmep= 1000 - 1200 kPa
Two stroke engines:
Standard CI engines comparable bmep to four stroke
Large slow CI engines: 1600 kPa
*Values are at maximum brake torque and WOT
Note, at the rated (maximum) brake power the bmep is 10 - 15% less
Can use the maximum bmep in design calculations to estimate engine
displacement required to provide a given torque or power at a specified
engine speed.
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Maximum BMEP
The maximum bmep is obtained at WOT at a particular engine speed
Closing the throttle decreases the bmep
For a given displacement, a higher maximum bmep means more torque
For a given torque, a higher maximum bmep means smaller engine
Higher maximum bmep means higher stresses and temperatures in the
engine hence shorter engine life, or bulkier engine.
For the same bmep 2-strokes have almost twice the power of 4-stroke
2
d
R
d
b
V
nT
V
W bmep
⋅⋅== π
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Vehicle Engine
type
Displ.
(L)
Max Power
(hp@rpm)
Max Torque
(lb-ft@rpm)
BMEP at
Max BT(bar)
BMEP at
Rated BP(bar)
Mazda
Protégé LX
I4 1.84 122@6000 117@4000 10.8 9.9
Honda
Accord EX
I4 2.25 150@5700 152@4900 11.4 10.4
MazdaMillenia S
I4Turbo
2.26 210@5300 210@3500 15.9 15.7
BMW
328i
I6 2.80 190@5300 206@3950 12.6 11.5
Ferrari
F355 GTS
V8 3.50 375@8250 268@6000 13.1 11.6
Ferrari
456 GT
V12 5.47 436@6250 398@4500 12.4 11.4
Lamborghini
Diablo VT
V12 5.71 492@7000 427@5200 12.7 11.0
Typical 1998 Passenger Car Engine Characterist ics
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Road-Load Power
A part-load power level used for testing engines (fuel economy, emissions)
is the power required to drive a vehicle on a level road at a steady speed.
vvv Dav Rr S S AC g M C P ⋅+= )2
1( 2 ρ
*Modern midsize aerodynamic cars only need 5-6 kW (7-8 HP)
to cruise at 90 km/hr, hence the attraction of hybrid cars!
where CR = coefficient of rolling resistance (0.012 - 0.015)
Mv = mass of vehicle
g = gravitational acceleration
ρa = ambient air densityCD = drag coefficient (for cars: 0.3 - 0.5) Av = frontal area of the vehicle
Sv = vehicle speed
The road-load power , Pr , is the engine power needed to overcome
rolling resistance and the aerodynamic drag of the vehicle.
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Drag Force Parameters
Auto manufacturers can improve the drag force by reducing:
Vehicle frontal area:
2005 Corvette is 0.57 m2
Most cars around 0.8 m2
2006 Hummer H3 is 1.56 m2
Drag coefficient CD:
2004 Toyota Prius – 0.26
2005 Porsche Boxster – 0.29
1983 Audi 100 – 0.3
2006 Dodge Challanger – 0.332003 Hummer – 0.57
Formula 1 car – 0.7 to 1.1 (DRS)
http://en.wikipedia.org/wiki/File:Hummer-H3.JPGhttp://upload.wikimedia.org/wikipedia/commons/0/08/Chevrolet-Corvette-C6-2.jpghttp://www.bing.com/images/search?q=prius&FORM=IGRE1http://en.wikipedia.org/wiki/File:Raikkonen_Spain_2009.jpg
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Engine power requirements
F=Mvgα
hpkW S g M PvvC
4735)sin( ==⋅= α
For a 1500 kg car moving at 50 km/hr up a 10o incline:
F = Mva
For a 1500 kg car accelerating (constant) from 0-100 km/hr in 8 s:
hpkW S t S M Pvvva
207145)/( ==⋅=
Ferrari 430 Scuderia has 508 hp at 8500 rpm 4.3L V8 engine
1270 kg and 0-100 km/hr in 3.6 s requires 272 kW (389 hp)
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Automobile transmission
Automatic transmission – gears shift automatically based on input
data from the sensors on the engine and the transmission (e.g., enginespeed, vehicle speed, throttle position, brake pedal position)
Engine operates between 600 – 7000 rpm whereas car wheels rotate at
0 -1800 rpm
Manual transmission – operator shifts gears
Transmission produces high torque at low car speeds and also operatesat highway speeds with the engine operating in the same speed range
transmission changes the gear ratio as the car speeds up
Highest torque is obtained in the mid engine speed range while the
greatest torque is often required at the lowest wheel speed
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Differential provides
Further gear ratio (3:1)
Automobile transmission
Torque converter provides fluid dynamic coupling with torque multiplication during idle andacceleration
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Gears
Gears change the speed of rotation and torque transmitted between shafts
Consider a simple gear set consisting of two gears:
RiRo
ωiωo
Vc, Fc
i
o
R
R
GR =
GR R
R
R RV
ii
o
io
ooiic
ω ω ω
ω ω
=
=
==
ii
i
o
o
ooiic
T GRT R
RT
RT RT F
⋅=
=
== //
Gear ratio (GR) is the number of turns of the input shaft required to give one
revolution of the output shaft
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An automobile is more complicated because you need several gear
ratios so the car can accelerate smoothly (shift for power or fuel
economy?)
Automobile Transmission
Gear GR ωo/ωi To/Ti
1 3:1 1/3 3
2 2.5:1 2/5 5/23 1.5:1 2/3 3/2
4 1:1 1 1
5 (OD) 0.75:1 4/3 3/4
Manual transmission typically has five forward gears and a reverse
Automatic transmission uses two sets of planetary gears to give three
or four forward gear ratios and one reverse
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Manual transmission (two gears)
Bearings
Flywheel
Clutch
CRANKSHAFT
Starter
Splined
shaft/collar
Synchronizer
TO WHEELS
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Automatic transmission planetary gear system
A reduction, B overdrive, C reverse
Planet gears
linked by carrier
Input Out Stationary ωi:ωo
A Sun (S)
Planet
Carrier
(C)
Ring (R) 1 + R/S 3.4:1
B Planet
Carrier (C)
Ring (R) Sun (S)1 / (1 +
S/R)
0.71:1
C Sun (S) Ring (R)Planet
Carrier (C)-R/S -2.4:1
ring gear = 72 teeth and sun gear has 30 teeth.
Locking any two locks up the whole device at a 1:1 gear
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0
50
100
150
200
250
300
350
400
450
500
0 1000 2000 3000 4000 5000 6000 7000 8000
Engine speed (RPM)
T o r q u
e
( F t - l b )
Engine
1st gear (GR=3.54)
2nd gear (GR=2.13)
3rd gear (GR=1.36)
4th gear (GR=1.03)
5th gear (GR=0.72)
1999 Neon DOHC
4210 rpm
3610 rpm
st nd GR
GR1
1
2
2 ω ω =
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Specific Fuel Consumption
In engine testing the fuel consumption is measured in terms of the fuel
mass flow rate . f m
b
f
W
mbsfc
=
hr kW
g
W
misfc
i
f
⋅= :units
Clearly a low value for sfc is desirable since for a given power level
the lesser the fuel consumed the better it is.
The specific fuel consumption, sfc, is a measure of how efficiently thefuel supplied to the engine is used to produce power,
For transportation vehicles fuel economy is generally given as mpg, or
L/100 km (in the future gm of CO2/km).
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Brake Specific Fuel Consumption vs Engine Size
Bsfc decreases with engine size due to reduced heat losses from gas to
cylinder wall.
B L B
BL 1
olumecylinder v
areasurfacecylinder
released energy
lossheat2 ∝==
π
π
Note cylinder surface to volume ratio increases with bore diameter.
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Brake Specific Fuel Consumption vs Engine Speed
There is a minimum in the bsfc versus engine speed curve
b
f
W
mbsfc
=
At high speeds the bsfc increases due to increased friction i.e. smaller bW
iW
Bsfc decreases with compression ratio due to higher thermal efficiency
At lower speeds the bsfc increases due to increased time for heat
losses from the gas to the cylinder and piston wall, and thus a smaller
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Performance Maps
Performance map is used to display the bsfc over the engines full load
and speed range. Using a dynamometer to measure the torque and fuel
mass flow rate for different throttle positions you can calculate:
Constant bsfc contours from a
2 litre four cylinder SI engine
bmep@WOT
d
R
V
nT bmep
⋅⋅= π 2
b
f
W
mbsfc
= )2( T N W b ⋅⋅= π
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Engine Effic iencies
The time for combustion in the cylinder is very short, especially at high
speeds, so not all the fuel may be consumed
HV f
in
HV f
inc
Qm
Q
Qm
Q
ut l heat inptheoretica
t input actual hea
⋅=
⋅==
η
where Qin = heat added by combustion per cycle
mf = mass of fuel added to cylinder per cycle
QHV = heating value of the fuel (chemical energy per unit mass)
The combustion efficiency is defined as:
A small fraction of the fuel may not react and exits with the exhaust gas
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Engine Efficiencies (2)
The thermal effic iency is defined as:
HV f cinth
Qm
W
Q
W
⋅⋅===η
η cycle perinputheat
cycle perwork
HV f cinth
QmW
QW
⋅⋅===
η η
inputheatof rateout power
or in terms of rates
Thermal efficiencies can be given in terms of brake or indicated values
Indicated thermal efficiencies are typically 50% to 60% and brake thermalefficiencies are usually about 30%
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Engine Efficiencies (3)
Fuel conversion effic iency is defined as:
HV f HV f f
Qm
W
Qm
W
⋅=
⋅=
η
Note: η f is very similar to η th, difference is η th takes into account actual
fuel combusted.
W
msfc
f
=
HV f
Qsfc ⋅=
)(
1η
Note: η f is very similar to η th, difference is η th takes into account actual
fuel combusted.
Recall:
Therefore, the fuel conversion efficiency can also be obtained from:
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Volumetric Efficiency
Due to the short cycle time at high engine speeds and flow restrictions
through the intake valve less than ideal amount of air enters the cylinder.
N V
mn
V
m
d a
a R
d a
av
⋅⋅
⋅=
⋅
==
ρ ρ
η
air ltheoretica
inducted airactual
where ρa is the density of air at atmospheric conditions Po, To and for anideal gas ρ a =Po / RaTo and Ra = 0.287 kJ/kg-K (at standard conditions
ρa= 1.181 kg/m3)
The effectiveness of an engine to induct air into the cylinders is measured
by the volumetric efficiency:
Typical values for WOT are in the range 75%-90%, and lower when the
throttle is closed
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Air-Fuel Ratio
For combustion to take place the proper relative amounts of air and fuel
must be present in the cylinder.
For gasoline fuel the ideal AF is about 15:1, with combustion possiblein the range of 6 to 19.
For a SI engine the AF is in the range of 12 to 18 depending on the
operating conditions.
For a CI engine, where the mixture is highly non-homogeneous, the
AF is in the range of 18 to 70.
f
a
f
a
m
m
m
m AF
==
The air-fuel ratio is defined as
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Relationships Between Performance Parameters
By combining equations presented in this section the following additional
working equations are obtained:
)/1(
2
)/1(
)/1(
AF Qmep
n
AF QV T
n
AF QV N W
a HV v f
R
a HV d v f
R
a HV d v f
⋅⋅⋅⋅=
⋅
⋅⋅⋅⋅⋅=
⋅⋅⋅⋅⋅⋅=
ρ η η
π
ρ η η
ρ η η
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