CONSTANT-PRESSURE TURBOCHARGER MATCHING

9th December 2009

Michael de Silva

i

TABLE OF CONTENTS

1 INTRODUCTION ................................................................................................ 12 TURBOCHARGING............................................................................................ 22.1 TurbochargerComponents...............................................................................................................72.2 AvailableTechnology..........................................................................................................................82.2.1 VariableGeometryTurbocharger(VGT) ..............................................................................................82.2.2 Turbine‐sideBypass(Wastegate)......................................................................................................... 102.2.3 TwinEntryTurbine..................................................................................................................................... 102.2.4 Water‐CooledTurbineHousing............................................................................................................. 11

3 TURBOCHARGERMATCHING .......................................................................... 113.1 WinklerTurbochargerMatchingMethod .................................................................................123.1.1 FirstQuadrant ............................................................................................................................................... 153.1.2 SecondQuadrant .......................................................................................................................................... 173.1.3 ThirdQuadrant ............................................................................................................................................. 183.1.4 FourthQuadrant........................................................................................................................................... 19

4 TURBOCHARGERMATCHINGPROJECT............................................................ 214.1 Task1 ....................................................................................................................................................214.2 Task2 ....................................................................................................................................................235 APPENDIXA:MATLABSIMULATIONCODE...................................................... 27

1

1 INTRODUCTION

Dr. Nickolaus A. Otto invented the four-stroke engine in 1876; twenty years later,

Rudolf Diesel created the very first supercharger design. Sir. Dugald Clark

discovered in 1901 that an engine was capable of producing more power if the

volume of air entering the cylinders was artificially increased (Allard, 1986).

It was only in 1902 that Frenchman Louis Renault patented a system that utilised a

centrifugal fan to drive air into the carburettor via its inlet. Five years late, American

Lee Chadwick and J. T. Nicholls implemented a centrifugal compressor to place the

carburettor under pressure thereby increasing its volumetric efficiency, whereas the

first successful modern turbocharger to be driven by the exhaust gasses of the engine

was developed in 1905 by Swiss Dr. Alfred J. Buchi (Humphries, 1992).

The First World War greatly advanced supercharger technology, as this enabled

fighter aircraft and bombers to reach much higher altitudes. Naval vessels also

greatly benefitted from turbocharged diesel engines; in 1925, one of the first

successful applications of a turbocharged diesel engine was in a German ship capable

of 2000 hp (Amann, 1992).

Turbochargers were once again in demand in World War II, where they proved

successful in aircraft such as the B-17 Flying Fortress, P-47 Thunderbolt, and P-38

Lightning (Logan and Roy, 2003). Between 1950 and the Second World War,

turbochargers were commonly found fitted to diesel truck engines in a commercial

capacity. Due to the much higher power densities and improved fuel consumption the

application of turbocharging large diesel truck engines, typically with a displacement

size exceeding 6 litres in capacity, is the industry standard in their production.

2

2 TURBO CHARGING

Supercharging an engine is the process by which a mechanically driven “air pump”,

or compressor, is utilised to artificially provide a much larger amount of fuel-air

mixture to the engine cylinders than normally taken in by naturally aspirated (non-

supercharged) engines.

The basic concept of most of the supercharging devices available is an increase in

pressure of the fuel-air mixture delivered to the cylinders results in an increase in the

density and the mass flow rate of air drawn into the cylinders during each intake

stroke (Mezger, 1978). It is therefore possible to burn more fuel during a combustion

cycle, if the same ratio of air-to-fuel is maintained, and thereby generate more power.

Therefore, a supercharged engine is able to significantly generate more power and

torque than a naturally aspirated, yet identical engine in terms of displacement size.

Conversely, a lighter supercharged engine with a smaller displacement size is capable

of producing the same power and torque as a heavier and larger normally aspirated

engine.

Figure 1: Supercharged engine; the compressor is driven

off the crankshaft (BorgWarner Turbo Systems, 2009)

Figure 2: Turbocharged engine; the compressor is driven

by a gas turbine running off the engine’s exhaust gasses

(BorgWarner Turbo Systems, 2009)

When the source of mechanical power is obtained directly from the engine crankshaft,

the system is classified as a supercharger, where a compressor is directly driven off

the crankshaft. A turbocharger (turbo supercharger) has its centrifugal compressor

3

driven by the engines exhaust gasses; the compressor is axially connected to a

centripetal radial-inflow exhaust-gas turbine via a common shaft. The turbocharger

system is ineffective at low engine speeds due to the introduction of an exhaust

backpressure, and as such the energy of the exhaust gasses are insufficient to

efficiently operate the turbocharger (Schorr, 1979). Therefore, superchargers are

more suited for operation at low engine speeds.

Turbochargers offer two main advantages; the power density of the engine is

increased, and particularly in diesel engines, the specific fuel consumption may be

improved. Exhaust emission due to turbocharging is not adversely effected, as studies

have previously shown, and claims have been made to suggest that typical fuel

consumption is reduced by 20-25% (Schruf and Mayer, 1981). Of course, the

operational characteristics of a turbocharged engine greatly depend on its application.

The characteristics include maximum power output to high torque over a wide engine

RPM (Revolutions Per Minute) while maintaining low fuel consumption.

Turbocharged engines are greatly preferred in industrial and military applications due

to their high power densities.

Figure 3: Turbocharger arrangement, with the compressor and turbine sections (Zinner, 1978)

A logical overview of the mechanical arrangement of a Turbocharger is shown above

in Figure 3, as well as both the compressor and turbine sections. As seen in the

4

illustration above, the hot exhaust gases drive the turbine that is connected to a shaft,

common to both the turbine and compressor, and is held in place by a bearing system.

Figure 4: Diesel engine Turbocharger (BorgWarner Turbo Systems, 2009)

The bearing system not only provides the required lubrication for the continued

operation of the shaft but it also serves to isolate both the chambers from each other.

Since the power generated by the gas turbine drives the compressor impeller, they

both rotate as the same radial velocity, and the air drawn in from the ambient air inlet

is compressed for delivery to the cylinders. The control system, as shown in Figure 4,

serves to alter the operational parameters of the Turbocharger based on the engines

performance (MacInnes, 1984).

The centrifugal forces exerted by the compressor impeller accelerate the air to

extremely high velocities thereby reaching a high level of kinetic energy. This

kinetic energy is transformed into an increase in static pressure, in the diffuser section

where its cross-sectional area increases gradually. With the rise in air pressure, the

velocity of the air decreases and the air flowing along the perimeter of the diffuser is

collected by a volute and delivered to the air outlet and flows to the cylinders of the

engine. The process by which the high velocity, low pressure air is converted into a

low velocity, high-pressure stream is called diffusion. The ambient air is drawn into

the compressor at the ambient temperature of the immediate surroundings, and at a

similar atmospheric pressure. Once compressed, this air is delivered to the air outlet

at temperatures nearing 200°C.

While the compression of the air drawn in by the compressor increases the boost

pressure and charge density, it results in an increase in the temperature of the air as

well. As described by Charles’ Law, an increase in the volume of the air takes place

5

and therefore slightly reduces the charge density. A more immediate concern due to

the increase in temperature is the excessive thermal stresses placed on the engine and

its operating parts, which could ultimately results in pre-ignition or detonation. Figure 3,

illustrates the implementation of an intercooler (or aftercooler) between the

compressor and cylinder intake; this cooling is achieved either with air or water. The

reduction in air temperature not only improves the volumetric efficiency, due to the

density of the air increasing as a result of drop in volume – as determined by Charles’

Law, but also ensures the increase in compression ratio without the overheating of the

cylinder heads. The higher compression offers for improved low RPM output (Stone,

1999).

Turbochargers were typically limited to diesel engines due to the high operational

temperatures of petrol engine exhaust gases far exceeding the operational capabilities

of the materials typically used in the construction of Turbochargers. However,

advances in materials derived from aerospace technology has resulted in the

development of the first Turbocharged petrol engine, developed by BorgWarner

Turbo Systems (2009), and is currently featured in the Type 997 Porsche 911 Turbo

(Paultan.org, 2009).

One of the disadvantages of turbocharging has been commonly termed “Turbo lag” or

“Turbocharger lag”. Since the exhaust gases drive the turbine, Turbo lag is the delay

experienced during the time taken for the turbine to ‘spool’ up and match the exhaust

air velocity. During this period, less power output is experienced until the turbine

spools up to reach the necessary radial velocity to supply optimal boost pressure. As

the demand of torque fluctuates based on the need to accelerate and decelerate the

turbine rotor, to and from high radial velocities, the resulting delay may take several

seconds due to the rotational inertia of the turbine rotor affecting its response to

changes in the exhaust air flow.

Another phenomenon affecting turbochargers is commonly known as the “Surge

effect”. The surge limit is shown in plots of typical turbocharger compressor maps,

Figure 5 and Figure 6. If the compression energy given to the gas by the compressor falls

below that needed to prevail over the adverse pressure gradient between the ambient

air inlet and compressor outlet, the flow of air suddenly collapses. The sudden

6

collapse in airflow results in the output pressure of the compressor dropping to a

sufficient level, thereby re-establishing in the flow of air and this process continues to

repeat in a cyclic manner. Both the turbocharger compressor maps in Figure 5 and Figure 6

show the surge limit and compressor efficiency in terms of the pressure ratio and rate

of non-dimensional mass flow.

Figure 5: Typical Turbocharger compressor map (Logan

and Roy, 2003)

Figure 6: “Typical performance map (pressure ratio plotted

against mass flow parameter, where N is the rotational

speed of the rotor and T and P the absolute temperature and

pressure) for a compressor without a vaned diffuser. The

surge limit is shown clearly on the left, and the limit on the

right is that set by choking of the flow. Between the two are

the loops of constant efficiency. The horizontal arcs

represent constant-speed conditions” (Garrett, Newton and

Steeds, 2000)

Turbocharging is achieved one of two ways: constant pressure and pulse

turbocharging. In constant pressure turbocharging, the exhaust gas is essentially

discharged into a large manifold (plenum) so as to reduce the magnitude of its

pulsations before entering the turbine. While energy conversion efficiency within the

turbine is greater than in pulse turbocharging, much of the kinetic energy of the gas is

lost on leaving the exhaust ports. This is due to the exhaust gas, initially travelling at

a near sonic velocity, ultimately mixing turbulently with the relatively larger volume

of gas in the large manifold as it discharges. Further more, potentially significant heat

loses from the exhaust gas may occur if the plenum is not thermally insulated.

7

The primary disadvantage of pulse turbocharging is its inherently lower efficiency in

energy conversion in comparison with constant pressure turbocharging. Appropriate

manifold design will compensate for this disadvantage, and account for the decrease

in turbine efficiency due to pulsation of the turbine flow. In modern automotive

applications pulse turbocharging is the de facto standard, whereas constant pressure

turbocharging is implemented in engines with a relatively large displacement size and

is therefore typically utilised in large industrial engines.

2.1 TURBOCHARGER COMPONENTS

The components of a typical turbocharger are detailed in Figure 7. The compressor

impeller is connected to that of the turbine via a common shaft that is mounted via a

bearings and bearing housing. The bearings not only serve as a means for reducing

contact friction, but also as a means for maintaining the isolation between both the

compressor and turbine housing. The lubrication of this bearing system is achieved

by oil that is fed under pressure from the engine into the bearing housing, which also

serves the purpose of acting as a coolant for the heat generated (Curless, 1985). The

oil inlet and drain are shown in Figure 8.

Figure 7: “Typical turbocharger components: (1) compressor

housing; (2) compressor wheel; (3) thrust bearing; (4)

compressor back-plate; (5) turbine housing; (6) turbine

wheel; (7) bearing; (8) bearing housing” (Logan and Roy,

2003).

Figure 8: Typical turbocharger schematic depicting

directional flow of ambient air and exhaust gases

(Garrett, Newton and Steeds, 2000)

8

The turbine driven by the exhaust gases consists of both the turbine wheel and the

turbine housing, where the energy of the exhaust gases enables the rotation of the

turbine. Once the exhaust gases pass through the blades of the turbine, they are

discharged from the turbine housing through the exhaust outlet, shown in Figure 8.

As the velocity of the exhaust gases is proportional to the speed of the engine, the

rotational velocity of the turbine wheel is thus proportional to the operational speed of

the engine. Turbines are therefore ineffective at lower engine RPMs as the exhaust

gas energy needs to overcome the inertia of the turbine rotor, to set it in motion, and

thereby provide boost pressure; at low RPMs, turbocharged engines therefore tend to

behave like that of a naturally aspirated equivalent.

Similarly, the compressor consists of the compressor wheel (impeller) and compressor

housing. The compressor impeller therefore rotates at same rotational velocity as that

achieved by the turbine wheel and compresses the ambient air drawn in from the air

inlet.

2.2 AVAILABLE TECHNOLOGY

As turbocharges are implemented in a wide variety of applications, the requirements

of these turbocharging systems vary greatly. The following techniques and solutions

are utilised in turbocharged engines so as to obtain the optimum output, depending on

the application of choice. Typical solutions vary in terms of geometrical alterations,

to additional compensatory mechanisms. A few of these techniques are presented in

the following section.

2.2.1 Variable Geometry Turbocharger (VGT)

A Variable Turbine Geometry turbocharger is commonly known as a Variable

Geometry Turbocharger (VGT) as well as a Variable Nozzle Turbine (VNT). These

are a group of turbochargers designed to allow for the effective aspect ratio

(commonly called A/R ratio) to be altered depending on operational conditions.

9

The optimum aspect ratio of a Turbocharged engine at low engine RPMs is

significantly different to that of an engine operating at much higher RPMs. Too large

an aspect ratio will result in the engine being unable to generate sufficient boost

pressure at low RPMs. Conversely, too small an aspect ratio will result in the choking

of the engine and ultimately leading to lower power output (Hishikawa, Okazaki and

Busch, 1988).

Figure 9: Porsche VGT with the turbine guide vanes

“semi-closed” (Paultan.org, 2009).

Figure 10: Exhaust flow through the VGT with the turbine

guide vanes “semi-closed” (Paultan.org, 2009).

Figure 11: Porsche VGT with the turbine guide vanes

“open” (Paultan.org, 2009).

Figure 12: Exhaust flow through the VGT with the turbine

guide vanes “open” (Paultan.org, 2009).

The ability to alter the aspect ratio in Variable Geometry Turbochargers allow them to

maintain the optimum aspect ratio during acceleration of the engine, and are therefore

extremely efficient at higher engine RPMs (Okazaki, Matsuo, Matsudaira and Busch,

1988). As a result, many VGT configurations do not require the implementation of a

wastegate for controlling the turbochargers maximum boost pressure; a higher

turbocharger efficiency is also achieved that than with a wastegate.

10

2.2.2 Turbine-side Bypass (Wastegate)

Turbine-side bypass, more commonly called wastegate, is a valve that diverts a

controlled portion of the exhaust gases away from the turbine wheel. The diversion of

the exhaust gasses allows for maintaining the boost pressure of the turbocharger, the

rotational velocity of the turbine is regulated and as such the rotational velocity of the

compressor impeller is controlled (Lundstrom and Gall, 1986). This control is

typically triggered once the boost pressure has reached a particular maximum

threshold, and is achieved by means of a spring-loaded diaphragm specifically

selected to open the exhaust bypass valve once the compressor output pressure

reaches the predetermined maximum threshold. Such an implementation is shown in

Figure 13.

Figure 13: Typical implementation of a turbine wastegate (Garrett, Newton and Steeds, 2000)

2.2.3 Twin Entry Turbine

As pulse turbocharging is the preferred form of

turbocharging, the turbine wheel is subjected to

variable pressure. The pulsation of the entering gases

are optimised in twin entry turbine implementations

as it allows for much higher pressure ratios to develop Figure 14: Twin Entry Turbine

(BorgWarner Turbo Systems, 2009)

11

in a much shorter time, and therefore results in an increase in efficiency.

The chance of interference is greatly reduced during charge exchange cycles as the

engine cylinders are divided into two groups for the exhaust inlets. As a result, a

more efficient mass flow passes through the turbine resulting in a more desirable level

of torque produced in the engine, especially at lower RPMs.

2.2.4 Water-Cooled Turbine Housing

Turbochargers are not limited to automotive

applications, and one of their first uses was in marine

engines. Most marine engine assemblies are greatly

overcrowded and therefore, water-cooled turbine

housing reduces the chance of injuries to maintenance

personnel.

Such installations in relatively large marine engines require a great amount of cooling,

especially those components in direct contact with exhaust gasses; this heated water

may be further used in secondary applications such as cabin heating.

3 TURBOCHARGER MATCHING

The performance of a turbocharger, as illustrated in Figure 6, is dependant upon the

engine speed while being limited by the surge and choke lines. Delivery of the

appropriate amount of air into the engine cylinders requires the matching of the size

of both the compressor and turbine stages to the swept volume, capacity, and the

power rating of the engine (Weaving, 1990)

Matching turbochargers with engines with a limited or constant speed range has

typically proven to be a rather straightforward task.

However, the process or matching turbochargers with automotive engines that are

typically characterised by their greatly varying speeds are accompanied by complex

matching issues.

Figure 15: Water-cooled Turbine

Housing (BorgWarner Turbo

Systems, 2009)

12

The conditions for matching have been derived with the assumption of constant-

pressure turbocharging, and the effect of pulsation is taken into consideration by the

use of a variable. The following four conditions ensure the matching of a

turbocharger with an engine:

• As the compressor and turbine share the same shaft, the compressor shaft

power equals that of the turbine shaft power,

€

PC = PT

• The rate of non-dimensional mass flow through the turbine is related to the

rate of non-dimensional mass flow of air and rate of non-dimensional mass

flow of fuel through the compressor,

€

˙ m T = ˙ m C + ˙ m f = ˙ m C1+λtotast

λtotast

where

€

ast is the stoichiometric air-fuel ratio and

€

λtot is the relative air ratio.

• The operating point must lie on the flow characteristic of the engine at the

given engine speed.

• The rotational velocity of the compressor impeller should be equal to the

rotational velocity of the turbine wheel, as the pressure ratio and rate of air

flow are fixed by the power balance; Both the compressor and turbine

typically share a common shaft, so that,

€

NC = NT

3.1 WINKLER TURBOCHARGER MATCHING METHOD

The Winkler method, developed by Professor Gustav Winkler and detailed in his

1982 paper titled “Matching turbocharger and engine graphically”, is a means to

allow those without access to simulation tools to easily accomplish the task of

turbocharger matching to engines of choice. The greatest success of this process is its

ability to enable engineers, lacking in-depth knowledge in turbocharging and

thermodynamics, to be able to carry out the task of matching successfully – even

without much knowledge of the inner workings of the turbocharger in question. This

is particularly useful during the product development process where alterations in the

13

design continue to take place; the process of re-matching the engine with the altered

operating conditions is made considerably easier by this method.

A Winkler diagram consists of four quadrants sharing four common axes, where an

axis is dedicated to each one of the four fundamental parameters of a turbocharged

system. These parameters are described below,

x-axis: Turbine pressure ratio,

€

PT /P0

y-axis: Compressor pressure ratio,

€

PC /P0

z-axis: Mass flow rate,

€

˙ m T / APρT cT

w-axis: Mass flow rate relative to inlet parameters,

€

˙ m T / APρT cT PT / P0( ) T0 /TC

The Winkler’s graphical iterative method, as suggested by the name, is one that is

achieved in an iterative fashion across the four quadrants of the Winkler diagram to

determine the pressure ratios and rate of non-dimensional mass flow so as to obtain

the best turbocharging performance being matched to a specific engine (Zinner,

1978).

Figure 16: Winkler’s graphical iterative method. The

example above illustrates how the iterations are

performed, ultimately leading to the solution rectangle

(Winkler, 1982).

Figure 17: Direct solution obtained from known values of

compressor and turbine pressure ratios (Winkler, 1982).

Each quadrant in the Winkler diagram is a graph that indicates the relationship

between the four fundamental conditions of a turbocharged system (an elucidation to

these fundamental conditions was provided earlier) with regards to one another based

14

on the various quadrant characteristics – also known as the “K” values. The

definitions of these quadrant characteristics and their equations are shown below,

Figure 18: Schematic representation of a turbocharged

four-stroke engine (Winkler, 1982)

Turbocharger efficiency parameter

€

K0 = fηTCTCT0

Engine load parameter

€

K1 =1−ηi −η1fλCηi

γ t −1γ t

PiP0

Engine speed parameter

€

K2 = fλCT0TC

cm4c0

Piston loading parameter

€

K3 = K1 ⋅K2

Turbine size parameter

€

K4 =αATAP

T0TC

Winkler’s iterative graphical method is typically started at a relatively low

compressor pressure ratio,

€

PC /P0( ) ≈ 2, and the iterations across the quadrants are

carried out until the final solution rectangle converges to the values representing the

matching conditions of the turbocharger to the specific engine in question. The

iterative procedure is further elucidated in Figure 16.

The following sections detail the derivation of all the Winkler parameters and their

respective equations.

15

3.1.1 First Quadrant

A priori knowledge of the exhaust temperature is required to establish the power

equilibrium of the turbocharger. The First Law of Thermodynamics states that energy

is conserved; hence, considering Einstein’s equation of relativity in the form

€

m =Ec2

posits the fact that mass is conserved. Therefore, the fuel energy that is neither

transformed into useful mechanical work nor lost to the environment via cooling may

be found inextricably in the extremely hot exhaust gasses. This can be written as,

€

TT −TC( )mTcp = 1−ηi −η1( )mFCV (1)

The following parameters are introduced,

€

Pi = ηimFCV

VS (2)

€

mT = fλCρCVS (3)

€

ρC =PCRCTC

(4)

Substituting Equation 2 (in terms of

€

mF ) and Equation 3 (in terms of

€

mT ) into

Equation 1, yields the following relationship between the compressor pressure ratio

and temperature ratio,

€

TT −TC( ) fλCρCVScp = 1−ηi −η1( ) PiVSCVηi

CV (5)

Substituting of Equation 4 into Equation 5 and further simplifying,

€

TT −TC( )PCcpRCTC

VS =

1−ηi −η1( )fλCηi

PiVS (6)

Expansion and simplification yields,

€

TTPCcpRCTC

−TCPCcpRCTC

=1−ηi −η1( )fλCηi

Pi (7)

16

€

TTPCTC

− PC =1−ηi −η1( )fλCηi

PiRCcp

(8)

Substituting

€

RCcp

=γ t −1γ t

into Equation 8, above,

€

PCTTTC

−1

=

1−ηi −η1( )fλCηi

Piγ t −1γ t

(9)

€

PCTTTC

=1+

1−ηi −η1( )fλCηi

Piγ t −1γ t

(10)

€

TTTC

=1+1−ηi −η1( )fλCηi

γ t −1γ t

PiP0P0PC

(11)

So as to be able to isolate the engine characteristic parameters in Equation 11, the

characteristic parameter

€

K1 is introduced. As such, Equation 11 simplifies to 13,

€

K1 =1−ηi −η1fλCηi

γ t −1γ t

PiP0

(12)

€

TTTC

=1+K1P0PC

(13)

Winkler (1982) states, “The power equilibrium of the turbocharger is described by

what is sometimes called the first law of turbocharging,” and an expression describing

the relationship between the pressure ratios of the turbine and compressor may be

obtained in consideration of this law,

€

PCP0

= 1+ fηTCcpTcpC

TTT01− P0

PT

γT −1γT

γCγC−1

(14)

Since,

€

TTT0

=TTTC

TCT0

,

€

PCP0

= 1+ fηTCTCT0

cpTcpC

TTTC

1− P0PT

γT −1γT

γCγC−1

(15)

17

€

K0 is introduced, to isolate the engine characteristics,

€

K0 = fηTCTCT0

(16)

€

TTTC

, defined in Equation 13, and

€

K0 are substituted into Equation 15,

€

PCP0

= 1+K0

cpTcpC

1+K1

P0PC

1−

P0PT

γT −1γT

γCγC−1

(17)

3.1.2 Second Quadrant

The characteristic lines in the second quadrant represent a relationship between the

rate of non-dimensional mass flow and the compressor pressure ratio – the air

breathing characteristics of a four-stroke engine (Winkler, 1982). As such, the non-

dimensional mass flow rate is evaluated by the following equation, which relates

turbine mass flow rate to the compressor mass flow rate,

€

˙ m T = f ˙ m C = fλCρCVSn2

(18)

The following parameters are introduced,

€

VS = APS (19)

€

cm = 2Sn (20)

€

ρC =PCRCTC

(21)

The above equations were substituted into Equation 18, yielding,

€

˙ m T = fλCPC

RCTC

APS n2

(22)

€

˙ m TAP

= fλCPC

RCTC

S cm

2S12

= fλCPC

RCTC

cm

4 (23)

18

Substituting

€

RC = R0 into Equation 21 yields

€

R0 =P0T0ρ0

; substituting this into the

previous equation yields,

€

˙ m TAP

= fλCPC

P0

T0ρ0

TC

cm

4= fλC

T0ρ0PC

P0TC

cm

4 (24)

Further simplification is obtained by the introduction of

€

K2 ,

€

K2 = fλCT0TC

cm4c0

(25)

Hence,

€

˙ m TAPρ0c0

= K2PC

P0

(26)

3.1.3 Third Quadrant

Equation 26 is rearranged, as follows, so as to establish a new relationship between

the mass flow rate and the flow conditions before the turbine stage,

€

˙ m TAP

= K2PC

P0

ρ0c0 (27)

The following parameters are introduced,

€

cT = KRTT (28)

€

c0 = KRT0 (29)

€

R =PTρTTT

(30)

Substituting Equation 29 and Equation 30, in terms of

€

ρ0 , into Equation 27 yields,

€

˙ m TAP

= K2PC

P0

P0

RT0

KRT0 (31)

19

Substituting Equation 30 and replacing

€

KRT0 by

€

cT into Equation 31 results in,

€

˙ m TAP

= K2PC

P0

P0

PT

ρTTT

T0

cTT0

TT

= K2PC

P0

ρTTT P0

PTT0

cTT0

TT

(32)

€

˙ m TAPρT cT

= K2PC

P0

P0

PT

TT

T0

T0

TT

(33)

Since

€

x 1x

= x ,

€

˙ m TAPρT cT

= K2PC

P0

P0

PT

TT

T0

T0

= K2PC

P0

P0

PT

TT

T0

(34)

Considering Equation 13 in the form

€

TTTC

= 1+K1P0PC

,

€

˙ m TAPρT cT

= K2PC

P0

1+ K1

P0

PC

P0

PT

TC

T0

(35)

Introducing

€

K3 = K1 ⋅K2 ,

€

˙ m TAPρT cT

PT

P0

T0

TC

= K2

PC

P0

1+

K3

K2PC

P0

(36)

3.1.4 Fourth Quadrant

The characteristic lines in the fourth quadrant relate the rate of non-dimensional mass

flow through the turbine’s exhaust to its pressure ratio. This rate of non-dimensional

mass flow through an equivalent nozzle area is therefore given as,

€

˙ m T = αATρT 2RTTT ψ (37)

where

€

ψ is the turbine flow function parameter for the value of

€

AT and is defined as,

€

ψ =γG

γG −1P4P3

2γG−

P4P3

γG+1γG

(38)

20

Multiplying the left and right hand sides of Equation 37 by

€

1APρTcT

PTP0

T0TC

gives,

€

˙ m TAPρT cT

PT

P0

T0

TC

= αATρT 2RTTT ψ1

APρT cT

PT

P0

T0

TC

(39)

€

K4 is introduced to isolate engine parameters, and is defined as,

€

K4 =αATAP

T0TC

(40)

Introducing

€

cT = γT RTTT and Equation 40 into Equation 39, yields,

€

˙ m TAPρT cT

PT

P0

T0

TC

= K4ψPT

P0

2γT

(41)

21

4 TURBOCHARGER MATCHING PROJECT

4.1 TASK 1

The Ford Puma 2.2L diesel engine is a turbo diesel engine that has a single

turbocharger interfaced with three of its cylinders, hence having a total of two

turbochargers. Winkler’s iterative graphical method is utilised with the assumption of

constant-pressure turbocharging, thereby greatly simplifying the task of matching.

The parameters of each of the Winkler graph quadrants are calculated below as the

first step in the Winkler turbocharger matching process. The operating conditions of

the engine have been supplied in the project specifications.

€

K0 = fηTCTCT0

=1.048× 0.4527×

410.203298

= 0.65306 (42)

€

K1 =1−ηi −η1fλCηi

γ t −1γ t

PiP0

=

1− 0.3821− 0.21321.048 ⋅1.8478 ⋅0.3821

0.351.35

18.81821

(43)

€

K1 =0.40470.739935

0.351.35

18.8182( ) = 2.66841 (44)

€

cm = 2Sn = 2 94.61000

200060

= 6.3067 m/s (45)

€

K2 = fλCT0TC

cm4c0

=1.048 ⋅1.8478

298410.203

6.30674 ⋅ 340

= 6.5237×10−3 (46)

€

K3 = K1 ⋅K2 =17.385×10−3 (47)

The aspect area of all the pistons is further tripled, as the turbocharger is interfaced

with three cylinders,

€

AP = nπr2( ) = nπ D2

2

n

πD2

4

= 3( ) π (8.6)

2

4

=174.26 cm2 (48)

The flow pulse factor is given as

€

α =1, and therefore,

€

K4 =αATAP

T0TC

=3.7

174.26

298410.203

=18.097×10−3 (49)

22

The four quadrants were plotted, utilising the Winkler graph parameters that were

previously calculated, and Winkler’s iterative graphical method was implemented to

determine the pressure ratios of the compressor and turbine as well as their associated

temperature ratio. Once the iterative process was completed, the solution rectangle

was plotted as Figure 19. A developed MATLAB simulation verified the plotted

results yielding an acceptable difference of ~ 2-5% between them (Appendix A).

The following values were obtained from the graph,

€

TTTC

≈ 2.05 (50)

€

PCP0

≈ 2.6 (37.71 PSI of boost pressure) (51)

€

PTP0

≈ 2.55 (52)

Figure 19: Converged solution of the iterative Winkler’s graphical method

23

4.2 TASK 2

This task requires the same turbocharger to be matched to the Ford Puma engine, as in

the previous task, at a decreased engine speed of 1700 rev/min and a compressor

outlet temperature of 320K. As in the previous task, the Winkler graph parameters

are calculated below,

€

K0 = fηTCTCT0

=1.048× 0.4527×

320298

= 0.50945 (53)

€

K1 =1−ηi −η1fλCηi

γ t −1γ t

PiP0

=

1− 0.3821− 0.21321.048 ⋅1.8478 ⋅0.3821

0.351.35

18.81821

(54)

€

K1 =0.40470.739935

0.351.35

18.8182( ) = 2.66841 (55)

€

cm = 2Sn = 2 94.61000

170060

= 5.3607 m/s (56)

€

K2 = fλCT0TC

cm4c0

=1.048 ⋅1.8478

298320

5.36074 ⋅340

= 7.108×10−3 (57)

€

K3 = K1 ⋅K2 =18.9678×10−3 (58)

The aspect area of all the pistons is further tripled, as the turbocharger is interfaced

with three cylinders,

€

AP = nπr2( ) = nπ D2

2

n

πD2

4

= 3( ) π (8.6)

2

4

=174.26 cm2 (59)

The flow pulse factor is given as

€

α =1 in the specifications, hence,

€

K4 =αATAP

T0TC

=3.7

174.26

298320

= 20.4898×10−3 (60)

To ensure the surge effect does not take place the compressor surge line was plotted

in the second quadrant on the Winkler graph, as per the project specifications

indicated values of non-dimensional mass flow rates and pressure ratios of the

compressor surge line.

24

However, the mass flow rate of the compressor specified in the project specifications

needed to be converted from

€

˙ m C into the same form as the z-axis,

€

˙ m TAPρT cT( )

.

The density of air at a temperature of 20°C and pressure of 1 atm or 101.325 kPa,

defined as Normal Temperature and Pressure (NTP), is 1.204 kg/m3. However, at sea

level the density of air is approximately 1.2 kg/m3, and as such

€

ρ0 was taken as 1.2

kg/m3.

Hence, as per Equation 18,

€

f ˙ m CAPρ0c0

=˙ m T

174.2610×103

1.2( ) 340( )

(61)

Since the calculated value for the aspect area of all the pistons in Equation 56 is in

cm2, care has been taken to convert this to m2 in Equation 58. The values of the

converted form of the non-dimensional mass flow rate of the compressor at their

respective pressure ratios are tabulated in Table 1.

Table 1: Converted values for plotting the compressor surge line in the second

quadrant

€

˙ m TAPρT cT( )

€

PCP0

0.00330 1.317 0.00398 1.5 0.00538 1.717 0.00641 1.983 0.00884 2.317 0.01266 2.733 0.01561 3.25 0.01828 3.75

Due to the avoidance of the “surge effect”, the reduction in the boost pressure of the

turbocharger prevents straightforward matching with the engine at its lower speed. A

solution to this problem is achieved with the implementation of a Variable Geometry

Turbocharger (VGT) and a variable turbine aspect area,

€

AT .

25

Calculating the value for K4 in the fourth quadrant, as performed in Equation 60,

shows

€

AT as an inextricable variable in the formulation. Therefore, the iterative

Winkler graphical method may not be applied in this particular scenario.

It is, however, possible to determine the solution rectangle based on K1, K2, and K3.

Since, it is the value of K4 that determines the curvature of the curve in the fourth

quadrant – multiple plots of varying values for K4 will allow for a manual means for

inspecting the correct curve that intersects the vertex of the solution rectangle in the

fourth quadrant.

Once an appropriate value is obtained for K4, Equation 60 may be re-evaluated to

determine the required turbine aspect area. Utilising the specified value for indicated

mean effective pressure of 18.8182 bars,

€

Pi =qC

λtotastρCεViηi (62)

where the variables above have the following values and definitions,

•

€

qC : the Higher Heating Value (HHV) of diesel, taken as 42.5 MJ/kg

•

€

αst : stoichiometric air-fuel ratio for gasoline, taken as 14.3

•

€

λtot : the fuel equivalence ratio, i.e. the ratio of fuel-air mix to stoichiometric

fuel-air mix, taken as 1.8

•

€

εVi : volumetric efficiency, i.e. how well the cylinder fills, taken as 0.97

Substituting the values above and solving Equation 62 for

€

ρC ,

€

ρC =18.8182×105

42.5×106

1.8×14.3

0.97( ) 0.3821( )

= 3.075 kg/m3 (63)

Substituting the above value for the density, where

€

R = 287 J kg-1 K-1 is the universal

gas constant, and converting from Pascal’s to bars,

€

PC = ρCRTC =3.075( ) 287( ) 320( )100×103

= 2.82 bar or 40.9 PSI boost pressure (64)

€

∴PCP0

= 2.82 (65)

26

The dashed solution rectangle was positioned in Figure 20 with its upper most side

between K1 and K2 with a vertical displacement of

€

PCP0

= 2.82. As such, its vertex in

the fourth quadrant suggests the actual value of K4 is approximately 0.0143. The

following values were obtained from the graph,

€

TTTC

≈ 1.95 (66)

€

PTP0

≈ 4.29 (67)

The effective turbine aspect area from the obtained value for K4 is calculated below,

€

AT =APK4

α

TCT0

=174.26× 0.0143

1

320298

= 2.58226 cm2 (68)

Figure 20: Winkler’s graphical solution based on three fixed positions for K1, K2,

and K3. K4 is required to pass through the solutions vertex in the fourth

quadrant, and is hence determined.

27

5 APPENDIX A: MATLAB SIMULATION CODE

function result = winkler_1

% Engine characteristics and operating parameters

B = 81; % Engine Bore [mm]

S = 94.6; % Engine stroke [mm]

T_0 = 298; % Compressor inlet temperature [K]

P_0 = 1; % Compressor inlet pressure [Bar]

C_0 = 340; % velocity of sound [m/s]

alpha = 1; % Flow pulse factor

f = 1.048; % Mass ration of exhaust gas to air

gamma_t = 1.35; % Gas isentropic exponent

% Specifications for Task 1:

n = 2000; % Engine speed [rev/min]

lambda = 1.8478; % Volumetric efficiency

P_i = 18.8182; % Mean indicated pressure

eta_i = 0.3821; % Indicated efficiency

eta_1 = 0.2132; % Energy loss to coolant relative to fuel energy

T_c = 425; % Compressor temperature [K]

eta_tc = 0.4527; % Turbo charger efficiency

A_t = 3.7; % Effective turbine flow area [cm2]

% Assumption and data collected by author

gamma_c = 1.4; % In the compressor the gas is air

P_t_P_0 = 3; % Turbine pressure ratio

P_c_on_P_0 = 3; % compressor pressure ratio

C_pt = 1107;

C_pc = 1004.5;

% Solution

K_0 = f * eta_tc * (T_c / T_0);

K_1 = ((1 - eta_i - eta_1)/(f * lambda*eta_i)) * ( (gamma_t - 1)/gamma_t ) *

(P_i/P_0);

C_m = 2 * (n * (S/1000))/60;

K_2 = f * lambda * (T_0/T_c) * (C_m / (4 * C_0));

K_3 = K_1 * K_2;

A_p = 3*((pi * (B/10)^2)/4); % The multiplication by 3 is for 3

pistons per stroke

K_4 = (alpha * (A_t/A_p)) * sqrt(T_0/T_c);

% The error variable as an indication of convergence

error = 1;

28

% Calculation of Pc over P0

term_1 = 1;

term_2 = K_0*(C_pt/C_pc);

term_3 = 1 + K_1 * (1/P_c_on_P_0);

term_4 = 1 - (1/P_t_P_0)^( (gamma_t-1)/gamma_t);

P_c_on_P_0 = ( term_1 + term_2 * term_2 * term_3 * term_4) ^ ( gamma_c / (gamma_c

- 1));

while (error > 0.1)

% Calculation of Tt/Tc

T_t_on_T_c = 1+ (K_1 * ( 1 / P_c_on_P_0));

% Calculation of non-dimensionalised mass flow rate

nd_mass = K_2 * P_c_on_P_0;

% Calculation of 3rd quadrant vertical axes

relative_mass_flow = nd_mass * sqrt( 1 + K_3/( nd_mass));

% Calculation for the 4th quadrant

zai = sqrt( ( gamma_t / (gamma_t +1) ) * ( (1/P_t_P_0)^(2/gamma_t) -

(1/P_t_P_0)^((gamma_t + 1)/gamma_t) ));

denominator = K_4* zai * sqrt(2/gamma_t);

P_t_P_0 = relative_mass_flow/denominator;

% All the values at that stage are recorded

answer = [P_c_on_P_0 , P_t_P_0 , nd_mass , relative_mass_flow];

% Calculation of new Pc over P0

term_1 = 1;

term_2 = K_0*(C_pt/C_pc);

term_3 = 1 + K_1 * (1/P_c_on_P_0);

term_4 = 1 - (1/P_t_P_0)^( (gamma_t-1)/gamma_t);

P_c_on_P_0 = ( term_1 + term_2 * term_2 * term_3 * term_4) ^ ( gamma_c / (gamma_c

- 1));

error = abs(answer(1) - P_c_on_P_0);

end

result = answer;

REFERENCES

i Allard, A. (1986) Turbocharging and Supercharging, 2nd edition, Yeovil,

Somerset: Patrick Stephens Ltd.

ii Amann, C.A. (1992) ‘Air-Injection Supercharging—A Page from History’, SAE Paper No. 920843.

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iv Curless, T. (1985) Turbochargers: Theory, Installation, Maintenance, and Repair,

Osceola: Motorbooks International.

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vi Hishikawa, A. Okazaki, Y. and Busch, P. (1988) ‘Developments of Variable

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viii Logan, E. and Roy, R. (2003) Handbook of Turbomachinery, 2nd edition, Boca Raton, FL: CRC Press.

ix Lundstrom, R.R. and Gall, J.M. (1986) ‘A Comparison of Transient Vehicle

Performance Using a Fixed Geometry, Wastegate Turbocharger and a Variable Geometry Turbocharger’, SAE Paper No. 860104.

x MacInnes, H. (1984) Turbochargers, Tucson, AZ: HP Books.

xi Mezger, H. (1978) ‘Turbocharging Engines for Racing and Passenger Cars’, SAE

Paper No. 780718. xii Okazaki, Y., Matsuo, E., Matsudaira, N. and Busch, P. (1988) ‘Development of a

Variable Area Radial Turbine for Small Turbocharger’, ASME Paper No. 88-GT-102.

xiii Paultan.org (2009) How does Variable Turbine Geometry work?,

http://paultan.org/2006/08/16/how-does-variable-turbine-geometry-work/, Date accessed 8 December 2009.

xiv Schorr, M.L. (1979) Turbocharging: The Complete Guide, Osceola: Motorbooks

International.

xv Schruf, G. and Mayer, A. (1981) ‘Fuel Economy for Diesel Cars by Supercharging’, SAE Paper No. 810343.

xvi Stone, R. (1999) Introduction to Internal Combustion Engines, 3rd edition,

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