Senior design report for Eaton Corporation on improving TVS supercharger volumetric efficiency. Specifically looking at the Dwell and Seal times within the supercharger.

1. TVS: Improving Supercharger Volumetric
Efficiency
ME 4790
Spring 2019
Group 15
Group members: Robert Beneteau, Alex Brunk, Devin Singer
Faculty Adviser: Dr. Tianshu Liu
Sponsoring Company: Eaton Corporation

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Disclaimer
This project report was written by students at Western Michigan University to fulfill an
engineering curriculum requirement. Western Michigan University makes no representation that
the material contained in this report is error-free or complete in all respects. Persons or
organizations who choose to use this material do so at their own risk.

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Abstract
Superchargers are generally associated with fast cars used in road racing and drag racing.
With governments around the world pushing for regulations requiring vehicles to have better fuel
efficiency and high gas mileage ratings, different sized superchargers are becoming increasingly
used in vehicles of all types to achieve this. A supercharger is a very complex system that has a
lot of moving parts. The advanced supercharger engineering team at Eaton Corporation has taken
up the task of trying to understand each aspect of the supercharger one by one. They have sought
out a team to begin in understanding the first part of this task, optimal dwell and seal.
To achieve the solution of optimal dwell and seal times for a supercharger system, a
series of steps were created. As each step gets accomplished, the analytical model would become
closer to the result. After discussion the team came up with four steps. The first step was to
determine an area of dwell in relation to the rotation of the rotor. Next, we wanted to determine
how the pressure of the volume between lobes increases with the change in mass flow. Then,
find the pressure of the control volume at the end of seal. Finally, find the amount of pressure
leakage of all sides of the control volume. An equation or system of equations were determined
for each step as outlined in the following report. A MATLAB program was also developed in
order to easily input, change, and plot results as the project progressed.

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Contents
Disclaimer..........................................................................................................................................2
Abstract.............................................................................................................................................3
Table of Figures .................................................................................................................................5
Table of Equations..............................................................................................................................5
Definition of Symbols..........................................................................................................................6
Background........................................................................................................................................7
Introduction .......................................................................................................................................8
Initial Concepts ..................................................................................................................................8
Analytical Process..............................................................................................................................9
Step 1.............................................................................................................................................9
Step 2...........................................................................................................................................10
Step 3...........................................................................................................................................12
Step 4...........................................................................................................................................12
Pistons Cylinder Relation..................................................................................................................14
Results.............................................................................................................................................15
Conclusion.......................................................................................................................................19
Recommendations ............................................................................................................................19
Acknowledgments ............................................................................................................................21
References .......................................................................................................................................22
Appendices ......................................................................................................................................23
Appendix A (MATLAB Code) ......................................................................................................23
Appendix B (Decision Matrix/ Gantt Chart)....................................................................................24
Appendix C (ABET Outcomes) .....................................................................................................25
Appendix D (Robert Beneteau Resume).........................................................................................35
Appendix E (Alex Brunk Resume).................................................................................................36
Appendix F (Devin Singer Resume)...............................................................................................37

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Table of Figures
Figure 1: How a Supercharger Works...................................................................................................7
Figure 2: Depiction of Dwell and Seal..................................................................................................9
Figure 3: Front View of Supercharger Inlet Geometry.........................................................................10
Figure 4: Simplified Inlet Geometry...................................................................................................10
Figure 5: L1 leakage diagram .............................................................................................................13
Figure 6: Basic Piston/Cylinder Model...............................................................................................14
Figure 7: Piston Cylinder Leakage Diagram .......................................................................................15
Figure 8: Chamber Pressure vs Time ..................................................................................................16
Figure 9: Pressure Loss (Left) and Pressure Overshoot (Right)..............................................................17
Figure 10: Pressure Settling Time (Left) and Steady State Pressure (Right)............................................18
Figure 11: Minimum Pressure............................................................................................................18
Figure 12: Peak Pressures..................................................................................................................19
Table of Equations
Equation 1..........................................................................................................................................9
Equation 2..........................................................................................................................................9
Equation 3..........................................................................................................................................9
Equation 4..........................................................................................................................................9
Equation 5: Bernoulli’s Equation .......................................................................................................10
Equation 6: Basic Mass Flow ............................................................................................................11
Equation 7: Combined Bernoulli’s and Mass Flow Equation ...............................................................11
Equation 8: Valve Flow Equation ......................................................................................................11
Equation 9: Stagnation Speed of Sound..............................................................................................11
Equation 10: Leakage Mass Flow ......................................................................................................12
Equation 11: Distance for Leakage to Occur Along Rotor ...................................................................13
Equation 12: Distance for Leakage o Occur Along Inlet ......................................................................13

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Definition of Symbols
 𝐶 𝑑 = Orifice Coefficient
 𝑐0 = Speed of Sound
 𝛾 = Specific Heat Ratio
 𝑃𝑣 = Pressure Inside the Control Volume
 𝑃0 = Atmospheric Pressure
 𝑚̇ = Mass Flow
 dA = Change in area of the inlet
 Atc = Inlet Area While Closing
 Ato = Inlet Area While Opening
 AInlet = Area of the inlet
 t = Time
 dt = Change in time
 RPM = Revolutions per minute
 𝜃0 = Position of Angle (degrees)
 router = Outer radius of simplified geometry
 rinner = Inner radius of simplified geometry
 A(t) = Combination of Opening and Closing Inlet Area Equations
 L = Leakage
 L1 = Leakage across surface number 1
 L2 = Leakage across surface number 2
 L3 = Leakage across surface number 3
 L4 = Leakage across surface number 4

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Background
A supercharger is essentially an air compressor that supplies a higher pressure of air into
the manifold of an internal combustion engine and allows for someone to get higher power out of
their engine by increasing the pressure of the air to fuel mixture. A basic supercharger has
“screws” or “rotors” inside the housing which are spun using a belt driven system. When the
rotors spin, air is drawn into the open volume between the lobes, and then forced into the
manifold of the engine where it is then compressed.
Currently, Eaton designs and manufactures precision Roots-type positive displacement
superchargers for highly specific automotive applications. Although supercharging is most
associated with high-performance and drag racing, Eaton Roots-type superchargers are currently
being used more frequently by manufacturers to improve fuel efficiency because they provide the
option of using smaller, more efficient engines to achieve desired driving characteristics.
As time goes on, governments throughout the world are implementing stricter vehicle
emissions standards, which in turn starts the process of engine downsizing, yet the market strives
for more powerful systems. To meet the demand of increased power tied with high efficiency and
lower emissions, Eaton’s TVS2 superchargers focus on improved thermal efficiency, reduced
weight, a greater range of airflow, and high and low-pressure systems to fit any range of vehicle.
Unlike turbocharging however, superchargers provide instantaneous throttle response across the
entirety of the engine’s power band. With the idea of financial penalties looming over the heads
of automakers, the implementation of Eaton’s supercharger systems will increase engine
performance while knocking down the levels of carbon dioxide produced by the vehicles. If Eaton
can produce a cheaper, more efficient supercharger then they will be able to gain a larger market
share of production automobile superchargers.
Figure 1: How a Supercharger Works

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Introduction
The advanced supercharger engineering team at the Eaton proving grounds wanted a
Senior Design team to help them identify key features of their superchargers that will greatly
affect the overall efficiency and performance of the system. Specifically looking at the Dwell
and Seal times within the supercharger. The Dwell stage is the point in which the inlet to the
volume is open and air can fill the volume, but there is no output. The Seal stage directly follows
the Dwell stage and consists of neither an input of air into the volume, nor an output of air into
the manifold of the vehicle. As the name of the stage states, the air is “sealed” between the lobes
of the rotor and the housing. Finding the optimal times for these two instances will in turn allow
the supercharger to push as much air into the manifold as possible increasing the pressure ratios
which also increases the efficiency.
It was determined that the best way to accomplish this task would be to use an analytical
method solving several equations, and then creating a GT Power and a MATLAB program that
would allow for quick input and plotting of different solutions to determine the optimal Dwell
and Seal times. The thought would then be for the Eaton Engineers to use this program to help
them design future superchargers by understanding how the twist on the lobes of the rotors and
the size of the rotors, which all effect the Dwell and Seal times, will affect the future
supercharger.
Initial Concepts
In analyzing the Dwell and Seal times of the supercharger, the mass flow through the
inlet and the pressure build up in the volume between lobes was key. Due to the complex
geometry of an actual supercharger, it was decided that the best course of action to complete the
task would be to start with simplified geometries and relate them back to a supercharger. A series
of steps were created that would allow the group to have a baseline of requirements to follow to
complete the project. Those steps are outlined in 1-4 below.
1. Determine the area of Dwell with relation to the rotational velocity of rotor.
2. Determine how the pressure of the volume between lobes increases with the
change in mass flow.
3. Find the pressure of the control volume at the end of Seal.
4. Find the amount of pressure leakage of all sides of the control volume.

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Analytical Process
Step 1
During the Dwell stage of a supercharger, the volume between lobes is filled with
incoming air. As simple as this may sound, there are many variables that come into play. With
the rotation of each rotor, the inlet area changes with respect to the position of the lobe. Also,
since air is a compressible fluid, entering at some unknown velocity, by the time the seal stage
occurs, there can be a pressure within the volume higher than atmospheric pressure. From this,
the pressure of air within the manifold can be maximized, allowing for a greater than 100%
volumetric efficiency of the supercharger.
Figure 2: Depiction of Dwell and Seal
With all above, an equation for the area of Dwell time in relation to the rotational
velocity had to be found before anything else could happen. Using the idea of a linear system to
simplify the geometry, the areas of the inlet are simply based off a sector as seen in Figure 3 and
Figure 4. The equations are as follows;
𝐶𝑙𝑜𝑠𝑖𝑛𝑔: 𝐴 𝑡𝑐 = 𝐴 𝑖𝑛𝑙𝑒𝑡 − 𝑑𝐴 ∗ 𝑡
Equation 1
𝑂𝑝𝑒𝑛𝑖𝑛𝑔: 𝐴 𝑡𝑜 = 𝑑𝐴 ∗ 𝑡
Equation 2
Where:
𝐴 𝑖𝑛𝑙𝑒𝑡 = [
1
2
∗ 𝜃0 ∗ ( 𝑟𝑜𝑢𝑡𝑒𝑟
2
− 𝑟𝑖𝑛𝑛𝑒𝑟
2 )]
Equation 3
𝑑𝐴 = [
1
2
∗ ( 𝑟𝑝𝑚 ∗
2𝜋
60
) ∗ ( 𝑟𝑜𝑢𝑡𝑒𝑟
2
− 𝑟𝑖𝑛𝑛𝑒𝑟
2 )] 𝑑𝑡
Equation 4

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These equations show that the opening and closing of the inlet area for the Dwell phase are a
piecewise function in relation to time and that the rate of the change of this area is an indirect
relation to the rpm of the supercharger.
Figure 3: Front View of Supercharger Inlet Geometry
Figure 4: Simplified Inlet Geometry
Step 2
When talking about the mass flow of a fluid and how the pressure relates to it, the
analysis usually revolves around that of an incompressible fluid, like what is used in hydraulics.
Analyzing the air entering the supercharger in an incompressible manner allows a for a base
point in which to build the necessary inlet mass flow equation from. Doing this required a
combination of Bernoulli’s equation, and a generic mass flow equation seen below.
𝑃0 +
𝑣0
2
2
+ 𝑔𝑧0 = 𝑃1 +
𝑣1
2
2
+ 𝑔𝑧1
Equation 5: Bernoulli’s Equation
router
rinner
θ

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𝑚̇ = 𝜌 ∗ 𝐴 ∗ 𝑣
Equation 6: Basic Mass Flow
These two equations were then combined and resulted in one generalized equation that
would solve for pressure inside the control volume shown below. With some of the variables in
Bernoulli’s equation being zero like height and atmospheric velocity, it allows for it to be
properly plugged into the generic mass flow equation. The combination of the equation results in
the equation shown below.
𝑃1 = −[
(
𝑚̇
𝜌 ∗ 𝐴
)2
2
∗ 𝜌] + 𝑃0
Equation 7: Combined Bernoulli’s and Mass Flow Equation
However, using intuition, it was realized that the generalized equation above will not
work for the project. The equation does not consider of the compressibility of air as it is entering
the control volume. Continuing with this equation, assuming the air was incompressible would
just lead to an answer of P1=P0 as everything would simply equal zero or cancel out. However,
moving forward it was determined that a mass flow equation and pressure equation for a
compressible fluid must be used in order to achieve an adequate solution. Using the final
pressure equation above, a rough estimation can be made as to what variables and aspects will be
needed for an equation that considers the compressibility of the air and knowing that the
compressibility of air entering the control volume is key to the overall volumetric efficiency of
the supercharger, an equation for valve flow was used.
𝑚̇ = 𝜌 ∗ 𝐴(𝑡) ∗ 𝑐 ∗ √
2
𝛾 − 1
((
𝑃𝑣
𝑃0
)
2
𝛾 − (
𝑃𝑣
𝑃0
)
𝛾+1
𝛾 )
Equation 8: Valve Flow Equation
Where: 𝑐 ( 𝑆𝑡𝑎𝑔𝑛𝑎𝑡𝑖𝑜𝑛 𝑆𝑝𝑒𝑒𝑑 𝑜𝑓 𝑆𝑜𝑢𝑛𝑑)
𝑐 = √ 𝛾 ∗ 𝑅 ∗ 𝑇0
Equation 9: Stagnation Speed of Sound
Using this mass flow equation, having all other variables known, one could solve for the
pressure in the volume, Pv. However, rearranging the equation to solve for Pv will result in an
overall complex solution. To overcome this issue, it was determined that to get adequate results,
the overall project would be simplified and related to a piston cylinder type event where actions
mimic that of what a supercharger does.

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Step 3
Continuing to look at the main mass flow equation, Equation 8, a large issue arose to
finding the final pressure within the control volume. For the solution to find the pressure in the
supercharger, the mass flow of air at the inlet was needed to calculate it. However, this became
an issue as the mass flow at the inlet was not known and was difficult to give an approximate
value as it is not constant, nor linear throughout the change in rpm of the system.
In an attempt to combat this issue of an unknown mass flow rate, it was determined that
simulation of a Piston-cylinder event must be used to find a generalized mass flow value that
could then be used in a MATLAB program to find the pressure in the control volume. It was
determined that the best program for this task would be GT Power. Other simulation software’s
such as ANSYS Fluent and Simulink were looked at, but it was decided that each software was
not capable of solving for what was needed and that GT Power would be the best course of
action as if was designed for engine simulations.
To ultimately find the final pressure at the end of the time of Seal, based on intuition, the
amount of time spent in the Seal stage will ideally be zero. As previously stated, during the Seal
stage, there is neither an input nor output to the system, yet leakage will still exist, causing a loss
of pressure to the specific control volume.
Step 4
Being that all the parts within the supercharger are moving, inevitably there will be small
gaps between each of the lobes and the housing of the supercharger. There will be leakage within
the system from these gaps. Leakage is the spilling of air from the transfer volume (higher
pressure) into the surrounding areas (lower pressure) and back to ambient. Leakage can be found
on all surfaces however, the leakage on three of the four surfaces will be constant. The leakage
on the fourth surface will depend on the rotation of the lobes and the overall size of the inlet. An
equation for leakage of the control volume was found to be;
𝑚̇ 𝑙 = 𝜌0 ∗ 𝐶 𝑑 ∗ 𝐴 𝑙 ∗ 𝑐0 ∗ √
2
𝛾 − 1
((
𝑃0
𝑃𝑣
)
2
𝛾 − (
𝑃0
𝑃𝑣
)
𝛾+1
𝛾 )
Equation 10: Leakage Mass Flow
Where;
 𝐶 𝑑 = Orifice Coefficient
 𝑐0 = Speed of Sound
 𝛾 = Specific Heat Ratio
 𝑃𝑣 = Pressure Inside the Control Volume
 𝑃0 = Atmospheric Pressure
 𝐴 𝑙 = Leakage Area

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The overall leakage is related the change in Dwell area. The leakage area on L2, L3, and
L4 are constant during Dwell but depend on the overall pressure in the control volume. As the air
gets compressed more, the pressure in the control volume will become higher and more leakage
will occur. When the transfer volume is either fully closed off or fully open to the inlet, then L1
is;
𝐿1 = 2 ∗ 𝑙𝑜𝑏𝑒 𝑟𝑎𝑑𝑖𝑢𝑠 ∗ 𝑔𝑎𝑝 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑐𝑎𝑠𝑒 𝑎𝑛𝑑 𝑙𝑜𝑏𝑒
Equation 11: Distance for Leakage to Occur Along Rotor
During any time in between when the transfer volume is either fully open of fully closed, leakage
on L1 will only exist on one lobe as the other lobe is exposed to the inlet as shown in Figure 7.
Thus, the area equation becomes;
𝐿 = 𝑙𝑜𝑏𝑒 𝑟𝑎𝑑𝑖𝑢𝑠∗ 𝑔𝑎𝑝 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑐𝑎𝑠𝑒 𝑎𝑛𝑑 𝑙𝑜𝑏𝑒
Equation 12: Distance for Leakage o Occur Along Inlet
Figure 5: L1 leakage diagram

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Pistons Cylinder Relation
However, as stated earlier, the compressibility of the gas becomes a serious factor on the
performance of the supercharger. To help analyze the effects of compressible air flow, the
geometry of the supercharger had to be simplified further than just the inlet geometry. To do this
the simplified geometry was to be that of a piston/cylinder. A piston/cylinder model was used to
simulate that of the supercharger because even though the geometry is vastly different, the
concept is very much the same. Air is drawn in to an expanding volume, then pushed out the
same volume as it begins to decrease in size.
Figure 6: Basic Piston/Cylinder Model
Since the supercharger can be thought of as a piston/cylinder setup, the inlet can roughly
be considered that of a valve. Because of this, a predetermined equation can be considered for
inlet mass flow with relation to pressure which can be seen below. This is the same valve flow
equation as used in the earlier section under Step 2.
𝑚̇ = 𝜌 ∗ 𝐴(𝑡) ∗ 𝑐 ∗ √
2
𝛾 − 1
((
𝑃𝑣
𝑃0
)
2
𝛾 − (
𝑃𝑣
𝑃0
)
𝛾+1
𝛾 )
Equation 13: Equation of Valve Mass Flow
Where: 𝑐 ( 𝑆𝑡𝑎𝑔𝑛𝑎𝑡𝑖𝑜𝑛 𝑆𝑝𝑒𝑒𝑑 𝑜𝑓 𝑆𝑜𝑢𝑛𝑑)
𝑐 = √ 𝛾 ∗ 𝑅 ∗ 𝑇0
Equation 14: Stagnation Speed of Sound Equation
Even though that the end goal is to find the pressure within the volume based on a given
mass flow, to solve directly for the final pressure would cause for a complex solution. Thus, to
solve for this, an estimated guess is needed to be made to match that of the mass flow instead.
This was done using GT Power engine simulation software. The results from GT Power were
then plugged into a MALAB program for ease of calculations. Also, to further relate the piston

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cylinder geometry to a supercharger, a set of boundary conditions for leakage must be
determined. These are shown I the figure below.
Figure 7: Piston Cylinder Leakage Diagram
However, the overall leakage inthe systemisdeterminedtobe verysmall andwasneglected in
the analysisandresults.Leakage addedagreatamountof complexitytothe resultsandhada very
minimal effectonthe overall system.
Results
A GT Powerprogramwascreatedand usedto testthe simple pistonmodel.Thispistonmodel
can be seen below inFigure8.To keepthe data recorded at a manageable level the intakewasheldat
constant30mm diameterandwasopenedwhenthe pistonstartedtomove backandthenclosedwhen
the pistonreachedthe back of the cylinder.The distance thatthe pistontraveledwas115mm andwas
alsoheldconstantfor thistest. It alsoshouldbe notedthatleakage wasalsotakenoutof the model to
helpsimplifythe resultsandallow the testtofocusonthe Dwell phase,the intake openandthe piston
movingback,and the endpressure atthe start of the Seal phase,the intake closedandpistonatthe
back of the cylinder.
Figure 8: Test Piston Cylinder Dimensions

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The test wasthenrun at 10 differentvelocitiesrangingfrom10
𝑚
𝑠
to 100
𝑚
𝑠
inintervalsof 10
𝑚
𝑠
.
While the testwasrunningthe chamberpressure,loss,overfill,settlingtime,steadystate value,
maximum,andminimumpressure valueswere recordedwithrespecttothe velocityandtime.The first
graph producedisthe average chamberpressure foreachvelocitywithrespecttotime,thiscanbe seen
inFigure 9. Inthisgraph whenthe pistoninitiallystartstomove backthe pressure drops whichdrawsin
the air thenthe pressure peaksbefore itstartstoequalize afterthe intake closes.Thisgraphshowsthat
the resultingpressure isdependedonthe volume of the chamberandthe velocityof the piston.
Figure 9: Chamber Pressure vs Time
The nextsetof data recordedisthe pressure differencesfromatmosphericpressureof 1 Bar.
The final pressure lossisshownin Figure10 Left.Thiswas recordedasthe difference inBarof the
atmosphericpressure andthe final pressureinthe chamber.The graphon the rightin Figure10, is the
percentovershootwhichrepresentsthe percentage of pressure differential fromatmosphericpressure.
In thiscase,for 90 and 100 m/s of pistonvelocity,the MATLABcode thatwas writtenassumesthat
steadystate existsata chamberpressure of 1 bar. Ascan be seenin Figure9, steadystate for 90 and
100 m/sexistcloserto0.9 bar, thuscreatingthe error inthe percentovershootgraphonthe rightin
Figure 10. Furtherinvestigationof the MATLABcode wouldhave tobe done to solve the incorrect
percentovershootvaluesshown.

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Figure 10: Pressure Loss (Left) and Pressure Overshoot (Right)
Thenthe settlingtime andsteadystate final valueswere recordedforeachvelocityandare
shownin Figure 11 Leftand Rightrespectively. The ChamberPressure SettlingTime plotonthe leftin
Figure 11 showsthatas the pistonvelocityincreases,the amountof airthat entersthe volume
decreases,thusallowingthe pressureinside the volume tostabilize muchmore quicklythan whenmore
air isable to enterthe volume.
In the ChamberPressure SteadyState Final Value isarepresentationof the inletwaitingtoclose
several tensof milliseconds afterthe volumewascompletelyexpanded,whatthe internal pressure
wouldnormalize to. Byanalyzingthisgraphmore closely,itcanbe seenthatas the pistonvelocity
increasesthe steadystate pressure valuedecreases atanexponentialrate.

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Figure 11: Pressure Settling Time (Left) and Steady State Pressure (Right)
The minimumpressure wasalsorecordedandisshownin Figure12 Left andin Figure12 Rightis
the time ittakesto reach the minimumpressure.Thiscorrespondstothe initial dropinpressure asthe
pistonstartsto move backwardsand is expectedasthe firstpart of the Dwell Phase. Fromthese graphs
the fasterthe pistonmovesthe lowerthe pressure dropisdue tothe volume of the cylinderexpanding
withthe rate of the piston. To counteract thispressure dropasthe velocityincreasesthe intake must
be increasedtoallowformore air to enter.
Figure 12: Minimum Pressure

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The peak pressure wasalsorecordedandisshownin Figure 13 Leftand in Figure 13 Rightis the
time ittakesto reach the peakpressure.Tooptimize the Dwellphase,the peakpressure isavery
importantmeasurementasthe more pressure inthe volume increasesthe efficiencyof the
supercharger.
Figure 13: Peak Pressures
Conclusion
From these graphswitha fixedintakeandvolume there isanoptimal velocityof the piston to
maximize the pressurewithinthe cylinder.Thiscanbe usedtocorrespondthe velocityof the pistonto
the rotational velocityof the superchargerrotors.Since the superchargersystemisdirectlydrivenfrom
the engine,givenatargetcruise speedof the vehiclewillallow forselectingarotational velocityof the
superchargerforoptimal peakefficiencytofitthe vehicle’sneeds.Also, usingthe graphshownin Figure
9, the ideal Dwell phaseduration canbe designedtoendatpeakpressure valuesforthe range of
operational speeds.Fromthis optimizationof Dwell duration,the resultwill be avolume filledwith
slightlypressurizedairatthe start of the Seal phase,trappingthe airin the volume allowingthe
superchargertodelivermore airthanit can technicallydeliveratatmosphericpressures.
As the data from Figure 9 suggests,there isan optimal range speedinwhich asuperchargerwill
operate. Engineersmayhave control overthistosome degree asthe geometryof the supercharger
playsa large role.Dependingonthe use of the superchargerunit, propergeometrical sizingwill be
neededtofitthe applicationneeds.Fromthis,the datapresented onlyrepresentsasmall portion of the
possible range fordesign.
Recommendations
From thistest,a relationshipcanbe seenbetweenthe pistonspeed,volume,andintake area.
The nextstepswouldbe torelate thispistoncylindermodeltothe superchargerhousingandscrews
that have similargeometriccharacteristics.The horizontal velocityof the pistoncouldbe

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mathematicallyrelatedbacktothe rotationof the screws usingthe equationof conversionbetween
linearvelocitytoangularvelocity seenbelow.
𝑃𝑖𝑠𝑡𝑜𝑛 𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦
𝑆𝑐𝑟𝑒𝑤 𝑅𝑎𝑑𝑖𝑢𝑠
= 𝑅𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦
Equation 15: Relation of Linear Velocity and Rotational Velocity
Thisthencouldbe ultimatelybe showninrelationtothe rpmof the engine basedfromgearing
ratiosbetweenthe twosystems. Withthe relationof linearvelocitytorotational velocity,the pressure
withincanbe superchargervolume canbe foundmore easily byfindingthe rate atwhichthe volume
betweenthe lobesonthe screw expands.The intake areacanalsobe relatedbackto the supercharger
model byconsideringthe average intakeareaasthe screw rotates or a more in-depthmodel couldbe
createdwithan intake areathat changeswithtime.
Furthermore,toverifythissimplifiedmodel anAnsysmodel canbe createdtotestthe more
complex superchargermodel andthencanbe comparedto the simplifiedmodel withthe mathematical
relationsthatwere found. Thiswouldallow the simplifiedGTPowermodel tobe usedto geta base line
readingwhile designednewsuperchargermodels.Thiswouldbe verybeneficial asthe GT Powermodel
takesa fractionof the time to runwhencomparedto a complex Ansysmodel.

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Acknowledgments
We would like to give a special thanks to everyone who is listed below. Our project would not have
been possible if it weren't for the great help from the engineers at Eaton and the faculty here at Western
Michigan University.
 Nathan Deville (Eaton Corporation)
 Ali Merat (Eaton Corporation)
 Andrew Meyers (Eaton Corporation)
 Mark VanWingerden (Eaton Corporation)
 Dr. Tianshu Liu (Western Michigan University Faculty)
 Dr. Shiva Om Bade Shrestha (Western Michigan University Faculty)

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References
[1] Milburn, S. M. (1994). Introducing a High Efficiency Variable Positive Displacement Automotive
Supercharger. SAE Technical Paper Series. doi:10.4271/940845
[2] TVS2 Technology: Improving Supercharger Efficiency and Capability Ouwenga D. Hopkins J.
Swartzlander M.
[3] Ganesan, V. (2012). Internal combustion engines. New York: McGraw-Hill.
[4] Brynych, P.,Macek,J., Vitek, O.,& Cervenka, L. (2013). 1-D Model of Roots Type Supercharger.
SAE Technical Paper Series. doi:10.4271/2013-01-0927
[5] Pohorelsky, L., Zak, Z., Macek,J., and Vitek, O., "Study of Pressure Wave Supercharger Potential
using a 1-D and a 0-D Approach," SAE Int. J. Engines 4(1):1331-1353, 2011

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Appendices
Appendix A (MATLAB Code)

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Appendix B (DecisionMatrix/ Gantt Chart)

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Appendix C (ABET Outcomes)

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Appendix D (Robert Beneteau Resume)

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Appendix E (Alex Brunk Resume)

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Appendix F (Devin Singer Resume)