TORQUE GENERATION DURING THE UNSTEADY EXPANSION PROCESS IN
CURVED CHANNELS OF A WAVE DISC ENGINE
By
Raul Quispe-Abad

A DISSERTATION
Submitted to
Michigan State University
In partial fulfillment of the requirements
for the degree of
Mechanical Engineering – Doctor of Philosophy
2017

ABSTRACT
TORQUE GENERATION DURING THE UNSTEADY EXPANSION PROCESS IN CURVED
CHANNELS OF A WAVE DISC ENGINE
By
Raul Quispe-Abad
Worldwide demand for power has been growing exponentially, but its production is
causing undeniably negative effects; environmental regulations are changing and becoming
stricter. The desire for economical but environmentally friendly engines is focusing on toward
alternative methods to produce power. “The Wave Disc Engine” (WDE) is a proposed
technology to surpass these requirements. The reduction of mechanical parts in the drive train
compared with an IC engine and the use of CNG or any other renewable fuel gas make this WDE
an attractive technology to generate power. This new engine concept is a radial rotor in which
the typical processes of an Internal Combustion Engine (Compression, Combustion, and
Expansion) are realized. Several prototypes were built between 2011 and 2013. For torque
production, the unsteady expansion process of outflowing combusted gases is harnessed. This is
a new engine concept with incipient research, investigating the mechanism to generate power
under unsteady-state conditions. This research work focuses on determining factors that
contribute to produce torque in radial rotor channels under unsteady-state conditions.
Computational fluid dynamic numerical simulations and analytical method were employed in
this investigation.
The study initially focuses on the influence of channel parameters (width, height and
length); and concludes that channel length and pressure side area all influence torque generation.
Both length and pressure side area combine to raise the efficiency and power generated.

Because of the unsteady expansion of the gas, an alternative approach was used to
evaluate the performance. The Exergetic efficiency produced results for the channel geometry
and conditions tested in the range of 31 to 67%. In addition to that, the approach revealed
between 82 to 89% of the exergy, initially contained in the channel, still has the potential to be
converted into torque in subsequent stages.
In addition, a zero dimensional macroscopic approximate balance equation was derived
based on the first law of thermodynamics to calculate the unsteady generated work from the
unsteady expansion process. Results show prolonging the duration of unsteady expansion
process enhances the isentropic extracted work toward the maximum value. In addition to that,
the gas expands more efficiently at lower pressure ratios.
The impact on the tangential force by the parameters: beta angle, area of influence, and
static pressure on pressure and suction sides of a constant cross-section channel, are investigated.
The first two parameters change inversely but when combined show similar values at each
pressure and suction wall location. Also, most of the generated torque was found in zones near
the channel outlet.
Furthermore, the torque generation composed of the action of two effects: the change of
the angular momentum of the fluid within channel and the outflow rate of the angular
momentum at the channel outlet is investigated. These two components are referred as unsteady
and steady effects respectively based on the mechanism to produce torque. Results show the
torque production benefits when the channel opens quickly. The increase of rotational velocity
approximates the quick opening. Unsteady effects produce a significant part of the generated
torque and the steady effect can be small at high speed.

Copyright by
RAUL QUISPE-ABAD
2017

Dedicated to my wife
Thank you for your support all these years during the PhD program

v

TABLE OF CONTENTS

LIST OF TABLES ..................................................................................................................... viii
LIST OF FIGURES ...................................................................................................................... ix
KEY TO ABBREVIATIONS .................................................................................................... xiv
CHAPTER 1.
INTRODUCTION ............................................................................................ 1
1.1.
Motivation ....................................................................................................................... 1
1.2.
Structure of the Dissertation ............................................................................................ 1
1.3.
Methods ........................................................................................................................... 2
1.4.
The Inspiration for the Wave Disc Engine idea .............................................................. 2
1.5.
Working Principles of the WDE ..................................................................................... 5
1.6.
Previous work on the WDE ........................................................................................... 10
1.6.1. Thermodynamic Cycle of the WDE ............................................................... 10
1.6.2. Mechanism for Torque Generation ................................................................. 10
1.6.3. Unsteady Expansion in channels .................................................................... 12
1.6.4. Re-utilization of exhaust gases ....................................................................... 12
1.6.4.1. Return channel................................................................................. 13
1.6.4.2. Parallel row of turbine blades.......................................................... 13
1.7.
Expansion efficiency ..................................................................................................... 15
CHAPTER 2.
INFLUENCE OF CHANNEL PARAMETERS: WIDTH, HEIGHT AND
LENGTH AT CONSTANT CROSS-SECTION CHANNELS .................................................... 17
CHAPTER 3.
EXERGY ANALYSIS OF THE UNSTEADY EXPANSION PROCESS .... 23
3.1.
The necessity of an alternative method to evaluate the expansion efficiency............... 23
3.2.
Exergy analysis of a single channel .............................................................................. 24
3.3.
Exergy rate analysis at transient state ........................................................................... 27
3.4.
Exergetic efficiency for control volume under unsteady state ...................................... 35
CHAPTER 4.
ENERGY ANALYSIS OF THE UNSTEADY EXPANSION PROCESS .... 40
4.1.
Zero-dimensional Macroscopic Analytical Model of the Unsteady Expansion Process
....................................................................................................................................... 40
4.2.
Analysis of the approximate general equation for extracted work ................................ 52
4.2.1. The flowing energy in the summation term .................................................... 53
4.3.
Unsteady generated work for isentropic process ........................................................... 53
CHAPTER 5.

ROLE OF PRESSURE AND FORCE IN TORQUE GENERATION .......... 61
vi

5.1.

5.2.

5.3.

Unsteady Expansion Process at Instant Opening .......................................................... 61
5.1.1. Pressure distribution on channel walls ............................................................ 61
5.1.2. Force and Torque distribution on channel walls ............................................. 71
Unsteady Expansion Process at gradual opening .......................................................... 79
5.2.1. Pressure distribution on channel walls ............................................................ 79
5.2.2. Force and Torque distribution on channel walls ............................................. 84
Impact on tangential force (torque generation) by changing θ parameter .................... 87

CHAPTER 6.
MOMENT OF MOMENTUM ANALYSIS ................................................... 92
6.1.
Moment of Momentum equation applied to a single channel of the WDE................... 92
6.2.
Analytical Analysis with respect to the Inertial Reference Frame ................................ 94
6.2.1. The Rate of Change of the Angular Momentum in the Channel .................... 95
6.2.1.1. 1st term ............................................................................................. 97
6.2.1.2. 2nd term ............................................................................................ 98
6.2.1.3. 3rd term ............................................................................................ 99
6.2.2. The Net Rate of Flux of Angular Momentum at the Outlet cross section with
Absolute Variables ................................................................................................ 100
6.3.
Numerical analysis of the Integral Form of the Angular Momentum Equation ......... 102
6.3.1. Unsteady Expansion with Constant Cross Section Channel ......................... 103
CHAPTER 7.
CONCLUSIONS AND RECOMMENDATIONS ....................................... 119
7.1.
Conclusions ................................................................................................................. 119
7.2.
Recommendations for future work .............................................................................. 121
APPENDICES ............................................................................................................................ 122
APPENDIX A: Incident expansion wave travel time ............................................................. 123
APPENDIX B: Centrifugal force effects ................................................................................ 125
APPENDIX C: Components of rate of change of angular momentum................................... 126
APPENDIX D: Numerical simulation set-up.......................................................................... 128
REFERENCES ........................................................................................................................... 129

vii

LIST OF TABLES

Table 1-1: General principle of torque generation according to the travelling direction of
expansion wave [7] .................................................................................................... 11
Table 1-2: Results from numerical simulation [7] ........................................................................ 13
Table 1-3: Comparison of a single stage and 2 stage rotors WDE ............................................... 14
Table 2-1: Geometry conditions for each constant cross-section channel .................................... 18
Table 3-1: Comparison of percentage values with respect to the GO@10K case ........................ 32
Table 4-1: Detailed description of the unsteady expansion process at each stage “i” .................. 46
Table 4-2: Development of the ρ� according to Series Taylor expansion for each ti ..................... 49
Table 4-3: Detailed development of �ρ for each stage “i”............................................................. 50
Table 4-4: Results for PR=3, 5 and 7 (“same duration” refers to the isentropic unsteady work
produced according to the duration of the actual work extracted. “maximum
isentropic work” refers to the maximum value at different duration time than the
actual work extracted )............................................................................................... 60

viii

LIST OF FIGURES

Figure 1-1: Steps toward the WDE ................................................................................................. 4
Figure 1-2: Future compact configuration of the WDE connected to two turbochargers .............. 5
Figure 1-3: Overlap period for early Intake with Scavenging period ............................................. 6
Figure 1-4: Intake process and early stage of Compression process .............................................. 6
Figure 1-5: Beginning and end of the Combustion process ............................................................ 7
Figure 1-6: Unsteady expansion process ........................................................................................ 8
Figure 1-7: (a) WDE-II Rotor designed and built by the MSU WDE research group & (b) space
distribution for processes ............................................................................................. 9
Figure 1-8: Geometric time available per cycle for the Unsteady Expansion Process for a 2 cycle
per revolution WDE-II Rotor....................................................................................... 9
Figure 1-9: Parallel and Serial flow configuration of a radial turbine. Source [9] ....................... 15
Figure 2-1: Comparison of channel outlet, pressure side and suction side areas with respect to
reference case-1. ........................................................................................................ 19
Figure 2-2: (a) Constant cross-section channel configurations for cases-1 to 3 and (b) meaning of
θ and βout angles. ........................................................................................................ 19
Figure 2-3: Duration of expansion process for constant cross-section channels varying their
geometric parameters (Var =geometric parameter changed)..................................... 20
Figure 2-4: (a) Expansion efficiency [%] and (b) Expansion wave travel time to reach channel
inlet (EWTT) [s] for constant cross-section channels varying their geometric
parameters. ................................................................................................................. 21
Figure 2-5: (a) Energy generated by one channel, (b) Power generated by engine rotor of 24
channels rotating at 10,000 rpm................................................................................. 22
Figure 3-1: Control volume considered for exergy rate balance .................................................. 26
Figure 3-2: Exergy (maximum work available) for a constant cross-section curved channel at
10,000 rpm at several Pressure ratios (PR) ................................................................ 26
Figure 3-3: Exergy variation along expansion time for a constant cross-section channel at several
rotational velocities and PR=7 ................................................................................... 27

ix

Figure 3-4: Percentage values of the Wcv, Ed and Efe with respect to equal ΔE in several cases
during the expansion process. .................................................................................... 31
Figure 3-5: Plots for |dE/dt|, Ėd, Ẇcv, & Ėfe for several rotational velocities in absolute values.
Blue arrows represent the flow time when the channel outlet is fully open. ............. 34
Figure 3-6: Exergetic Efficiency [%] and Loss/work generated for several rotational velocities 37
Figure 3-7: Comparison of % of exergy destruction (equal exergy for all cases) with mass flow
rate. Arrows represent the time when the outlets fully open. .................................... 38
Figure 4-1: Schematic of the wave disc engine with curved channels (two cycles per revolution).
................................................................................................................................... 40
Figure 4-2: Unsteady expansion process for stage “i” .................................................................. 45
Figure 4-3: Convergence of the Unsteady Isentropic Work as a function of Number of finite
stages at PR=7 with 10,000 rpm. ............................................................................... 56
Figure 4-4: Density (a), magnitude of outlet velocity (b) and dρ/dt (c) with several inflectiontime during the unsteady expansion process (PR=7 and 10,000 rpm) ...................... 57
Figure 4-5: Density (a), magnitude of outlet velocity (b), and dρ/dt (c), with several durationtimes of the unsteady expansion process (PR=7 and 10,000 rpm) ............................ 58
Figure 4-6: Unsteady Isentropic Work vs time at several PR= 3, 5 and 7, discharging to
surrounding environment pressure of 101325 Pa. ..................................................... 59
Figure 5-1: Torque coefficient (cm) history of a constant-cross section curved channel for instant
opening ...................................................................................................................... 62
Figure 5-2: Pressure plot on the suction and pressure surface according to radial coordinates
(left) and static pressure contour (right) for points 1 to 3. ......................................... 63
Figure 5-3: Pressure gradients and pressure on the suction and pressure walls ........................... 64
Figure 5-4: Increasing trend of the net pressure distribution for points 1 to 3. ............................ 65
Figure 5-5: Velocity magnitude vectors for points 1 and 3 .......................................................... 66
Figure 5-6: (a) cm torque oscillation and (b) Non-dimensional net area of differential pressure. 68
Figure 5-7: Pressure plot on the suction and pressure sides at each radial coordinates (left) and
static pressure contour (right) for points 4 to 6.......................................................... 69
Figure 5-8: Pressure plot on the suction and pressure surface according to radial coordinates
(left) and static pressure contour (right) for points 7 to 9. ......................................... 70
Figure 5-9: Nomenclature for forces, angles and channel walls ................................................... 72
x

Figure 5-10: Channel shape and sinβ variation along the radial coordinate of a constant cross
section channel........................................................................................................... 74
Figure 5-11: Δ(force-θ) and Δ(r*force-θ) plots at points 1, 2, 3 & 4 according to radial
coordinates. ................................................................................................................ 75
Figure 5-12: Δ(force-θ) and Δ(r*force-θ) plots at points 5, 6, 7 & 8 according to radial
coordinates. ................................................................................................................ 76
Figure 5-13: Δ(force-θ) and Δ(r*force-θ) plot at point 9, according to radial coordinates. ......... 77
Figure 5-14: Δ(force-θ) at points 1, 2, 3, 4, 5, 6, 7, 8 & 9 along radial direction ........................ 77
Figure 5-15: Δ(r*force-θ) at points 1, 2, 3, 4, 5, 6, 7, 8 & 9 along radial direction...................... 78
Figure 5-16: Torque coefficient (cm) history of a constant-cross section curved channel for
gradual opening ......................................................................................................... 79
Figure 5-17: Pressure plot on the suction and pressure surface according to radial coordinates
(left) and pressure contour (right) for points 1 to 3. .................................................. 81
Figure 5-18: Pressure plot of the suction and pressure surface according to radial coordinates
(left) and pressure contour (right) for points 4 to 6. .................................................. 82
Figure 5-19: Pressure plot of the suction and pressure surface according to radial coordinates
(left) and pressure contour (right) for points 7 to 9. .................................................. 83
Figure 5-20: Pressure plot of the suction and pressure surface according to radial coordinates
(left) and pressure contour (right) for points 10 to 11. .............................................. 84
Figure 5-21: Δ(p*sinβ) at points 1, 2, 3, 4, 5 & 6 based on radial coordinates ............................. 85
Figure 5-22: Δ(p*sinβ) at points 7, 8, 9, 10 & 11 based on radial coordinates. ............................ 86
Figure 5-23: Δ(r*p*sinβ) at points 1, 2, 3, 4, 5 & 6 based on radial direction ............................. 86
Figure 5-24: Δ(r*p*sinβ) at points 7, 8, 9, 10 & 11 based on radial direction ............................. 87
Figure 5-25: Area of influence projected by a single isobaric line-contour on pressure and suction
walls ........................................................................................................................... 89
Figure 5-26: β & area of influence (APS & ASS) on pressure and suction walls created by linecontour when θ angle is changed ............................................................................... 90
Figure 6-1: Control Volume selected for the Angular Momentum Equation Analysis ................ 92
Figure 6-2: Orientation of radial and tangential unit vectors in cylindrical coordinates .............. 94

xi

Figure 6-3: Changes in local tangential velocity after the head of the expansion wave passes (r1,
θ1) position (t2 > t1). ................................................................................................... 98
Figure 6-4: Graphical method to find the time when expansion wave reaches the inlet wall
(GO@30,000 rpm) ................................................................................................... 104
Figure 6-5: Expansion duration and expansion wave time to reach channel inlet...................... 105
Figure 6-6: Triangle velocities for (a) positive tangential velocity and (b) negative tangential
velocity in absolute and relative system at the channel outlet. The βout angle and tip
speed (U1) are same for (a) and (b) velocity sketches. ............................................ 107
Figure 6-7: (a) Absolute tangential velocities at the channel outlet with gradual opening rotating
at 10,000 rpm, (b) Detail “X” .................................................................................. 108
Figure 6-8: (a) Absolute tangential velocities at the channel outlet with gradual opening rotating
at 50,000 rpm, (b) Detail “X” .................................................................................. 109
Figure 6-9: (a) Absolute tangential velocities at the channel outlet with instant opening rotating
at 10,000 rpm, (b) Detail “X” .................................................................................. 110
Figure 6-10: Rate of Change of the Angular Momentum and components. Expansion wave
impinges inlet wall “↓”. ......................................................................................... 112
Figure 6-11: 20 fixed points selected within the channel of the GO20Krpm case ..................... 113
Figure 6-12: Time rate of local tangential velocity and density for Gradual Opening at 20,000
rpm ........................................................................................................................... 114
Figure 6-13: Rate of change of the angular momentum with rotating and stationary channel
conditions at 50,000 rpm ......................................................................................... 115
Figure 6-14: (a) ᚁ and mass-weighted average values of (b) radial velocity & (c) tangential
velocity at the channel outlet for several rotational velocity conditions, (d) Detail of
Tangential Velocity ................................................................................................. 116
Figure 6-15: Area ratios of Rate of change/ Torque & Surface Outlet/Torque [%] for several
rotational velocities (IO=Instant Opening & GO=Gradual Opening) ..................... 117
Figure 6-16: History of torque extracted, time rate of change and the net rate of flux of the
angular momentum during the expansion process (Gradual Opening at (b) 10000
rpm, (c) 20000 rpm, (d) 30000 rpm, (e) 40000 rpm & (f) 50000 rpm). “↓”, time
when inlet is fully open. “↓”, time when Expansion Wave reaches the inlet port. 118
Figure 7-1: Incident expansion wave travel time at channel rotational velocity of 10,000 rpm 123
Figure 7-2: Incident expansion wave travel time at channel rotational velocity of 20,000 rpm 123

xii

Figure 7-3: Incident expansion wave travel time at channel rotational velocity of 40,000 rpm 124
Figure 7-4: Incident expansion wave travel time at channel rotational velocity of 50,000 rpm 124
Figure 7-5: Centrifugal effects for channel rotating at 30,000 rpm ............................................ 125
Figure 7-6: Centrifugal effects for channel rotating at 40,000 rpm ............................................ 125
Figure 7-7: The term rρ(∂Vθ/∂t) plotted for five time steps at every fixed point in the channel at
GO20Krpm case ...................................................................................................... 126
Figure 7-8: The term ρVθVr plotted for five time steps at every fixed point in the channel at
GO20Krpm case. ..................................................................................................... 126
Figure 7-9: The term rVθ(∂ρ/∂t) plotted for five time steps at every fixed point in the channel at
GO20Krpm case. ..................................................................................................... 127
Figure 7-10: Numerical simulation set-up for 2-dimensional cad models ................................. 128
Figure 7-11: Numerical simulation set-up for 3-dimensional cad model ................................... 128

xiii

KEY TO ABBREVIATIONS

A

area

𝐴𝐴𝑟𝑟𝑟𝑟𝑟𝑟

Reference area value

𝐴𝐴𝑜𝑜𝑜𝑜𝑜𝑜

Outlet area of the channel

AoI

Area of influence

cm

Torque coefficient

CNG Compressed natural gas
CV

Control volume

e

specific total energy of the system

𝑒𝑒𝑓𝑓𝑓𝑓

Specific flow exergy at the exit

𝑒𝑒⃗𝜃𝜃

Unit vector in the azimuthal (tangential) direction

𝑒𝑒⃗𝑘𝑘

Unit vector in the k direction

𝑒𝑒⃗𝑟𝑟

Unit vector in the radial direction

E

Exergy

𝐸𝐸𝑐𝑐𝑐𝑐

Exergy of the control volume considered

𝐸𝐸̇𝑖𝑖𝑖𝑖

Total Energy rate that is coming in the system

𝐸𝐸𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 Total Energy contained in the system

𝐸𝐸̇𝑑𝑑

Exergy destruction rate

𝐸𝐸̇𝑜𝑜𝑜𝑜𝑜𝑜

Total Energy rate that is coming out the system

𝐹𝐹⃗𝑝𝑝

Vector force component due to pressure on the wall

𝐹𝐹⃗𝑠𝑠

Vector surface forces over the control volume

EWTT Expansion wave travel-time

xiv

𝐹𝐹⃗𝑣𝑣

𝐹𝐹𝜃𝜃

Vector force component due to viscous effects
Scalar value of tangential component of force

𝑓𝑓𝑓𝑓

Finite value

𝑔𝑔

Gravity

h

Specific enthalpy

h0

Specific enthalpy at the dead state

IR

Inner radius

ke

Specific kinetic energy

KE

Kinetic energy

LHS

Left hand side

𝐿𝐿𝑟𝑟𝑟𝑟𝑟𝑟

Reference length value

msyst

Mass contained in the system

𝑚𝑚̇𝑒𝑒

Mass flow rate at the exit

n

Number of finite stages, normal direction

OR

Outer radius

p

Absolute static pressure

p0

Pressure at the dead state

pe

Specific potential energy

Pin_i

Area-weighted average value of static pressure at inlet for time “i”

Pin_i-1 Area-weighted average value of static pressure at inlet for previous time respect to “i”
PPS

Pressure on pressure side

PSS

Pressure on suction side

PS

Pressure side

xv

�0
P

Mass-weighted average pressure at the beginning of the expansion process

r

Particle’s radial position at r-θ plane

𝑟𝑟𝑜𝑜𝑜𝑜𝑜𝑜

Radius at the channel outlet position

R

Radius of curvature

𝑅𝑅𝑐𝑐𝑐𝑐

Radio of the center of gravity of the channel

𝑅𝑅𝑜𝑜𝑜𝑜𝑜𝑜

Radius at the channel outlet position

PR

Static Pressure ratio

RHS

Right hand side

SLT

Second Law of Thermodynamic

s

Specific entropy

S

Entropy

SS

Suction side

𝑠𝑠0

Specific entropy at the dead state

S0

Entropy at the dead state

,,,
𝑆𝑆𝑔𝑔𝑔𝑔𝑔𝑔

Entropy generation rate per unit volume

𝑡𝑡𝑥𝑥

Time when density’s inflection (undulation) point occurs during the expansion process

𝑡𝑡2𝑠𝑠
T

Torque magnitude

Tj

Temperature on the boundary “j” where the heat transfer occurs

𝑡𝑡

Time

Total duration time of the isentropic expansion process

�⃗𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 Vector torque on the shaft
𝑇𝑇
T0

Temperature at the dead state

�0
T

Mass-weighted average temperature at the beginning of the expansion process

xvi

u

Specific internal energy

U

Internal energy, tip speed

U0

Internal energy at the dead state

V

Volume of the channel or element

V𝑐𝑐𝑐𝑐

Velocity magnitude at the center of gravity of the channel

V

Velocity Magnitude with respect to inertial reference frame

𝑉𝑉𝑟𝑟𝑟𝑟𝑟𝑟

Reference velocity value

𝑉𝑉𝜃𝜃

Tangential velocity component with respect to inertial reference frame

𝑊𝑊̇𝑐𝑐𝑐𝑐

Time rate of energy transfer by work

V0

Volume of the channel at the dead state

Vout

Outlet velocity magnitude with respect to inertial reference frame

𝑉𝑉𝑟𝑟

Radial velocity component with respect to inertial reference frame

𝑊𝑊𝑎𝑎

Actual work

𝑊𝑊𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 Isentropic work

WDE Wave Disc Engine
𝑊𝑊𝑥𝑥𝑥𝑥𝑥𝑥

Relative velocity in xyz coordinates

𝑊𝑊𝜃𝜃

Relative velocity component in 𝜃𝜃- direction

𝑊𝑊𝑟𝑟

Relative velocity component in r- direction

×

Cross product

Greek Symbols
𝛽𝛽𝑜𝑜𝑜𝑜𝑜𝑜
ρ

Channel outlet angle
Density

xvii

𝜌𝜌𝑟𝑟𝑟𝑟𝑟𝑟

Reference density value

𝜔𝜔𝑖𝑖

Rotational velocity of the channel in the stage “i” [rps]

𝜔𝜔

Rotational velocity

𝜃𝜃̇

Rotational velocity

Θ

Specific total energy for a flowing fluid

ÎŚ

Viscous dissipation

Subscripts
𝑖𝑖𝑖𝑖

In

𝑐𝑐𝑐𝑐

Center of gravity

𝑐𝑐𝑐𝑐

Control volume

𝑜𝑜𝑜𝑜𝑜𝑜

Out

𝑖𝑖

“i” represents an element

𝑐𝑐𝑐𝑐

Control surface

r

Radial direction (cylindrical coordinates)

θ

Azimuthal direction (tangential direction)

Superscripts
→
̇

Vector quantity
Partial derivative with respect to time

xviii

CHAPTER 1.
1.1.

INTRODUCTION

Motivation
“The Wave Disc Engine” has been considered an attractive technology to produce power.

Several reasons could point out to make this a very promising technology. The use of this new
Wave Disc Generator Technology was predicted to be as 74 million low-cost WDG hybrid
vehicles on American roads, using CNG or any other renewable fuel gas, and the impact of this
would save 3.5 million barrels of imported oil per day [1]. Another reason was the reduction of
mechanical parts in the drive train of this new technology compared with an IC engine.
Additionally, the supremacy of unsteady over steady flow as a mechanism of mechanical energy
transfer in fluid flows where there is not combustion and body forces [2].
Since this is a new engine concept with little knowledge of the mechanism to generate
power under unsteady-state conditions, based on the knowledge of the author, this research work
focuses on the investigation of the unsteady expansion process to produce torque in radial rotor
channels.

1.2.

Structure of the Dissertation

This dissertation is a compilation of several investigations to determine factors that contribute to
produce torque in radial rotor channels under unsteady state conditions. It starts reviewing
previous work investigating the mechanism to generate power under unsteady conditions. As an
extension of this, some previous work on re-utilization of burned gases is presented. Chapter 2
discusses the influence on torque generation by varying channel geometric parameters. Chapter 3
proposes exergetic approach to evaluate the performance of an unsteady expansion process
generating torque and characterizes losses and remaining useful exergy. Chapter 4 analyzes the

1

unsteady expansion process based on the first law of thermodynamics using an approximate
derived equation. Chapter 5 investigates pressure distribution on channel walls as a result of
instant and gradual opening conditions. In addition, force distribution as the result of geometric
parameter (beta angle distribution at pressure and suction sides) and channel outlet opening
method, is included in the investigation. Also, the combination of line-contour orientation
(expansion wave) with channel length increase (θ angle), is discussed. Chapter 6 discusses the
torque generation composed of two terms: rate of change and outflow rate of the angular
momentum to give understanding of the working mechanisms in torque generation. Chapter 7
provides conclusions and recommendation for future work.

1.3.

Methods

This research was accomplished with data generated from two sources: numerical simulations
and an approximate derived equation. Numerical simulations were performed using the
commercial software, fluent with 2- and 3-dimensional cad models. Chapter 2 results are based
on a 2-dimensional cad model in which boundary layer effects on top and bottom are not taken
into account, only those from pressure and suction sides of the channel. Results in Chapters 3, 5
and 6 are based on a 3-dimensional cad model. Details of numerical simulation set up are given
in Appendix D. The working fluid used throughout the entire dissertation was air that follows the
ideal gas law.

1.4.

The Inspiration for the Wave Disc Engine idea
The idea for the WDE came to be through a series of well-developed modifications. The

axial wave rotor has been used widely in its four-port version: straight channels with the flow
going in and out through the rotor in axial direction; one of the applications was as a topping
2

device for gas turbine cycles (Figure 1-1(a)). This design presented a disadvantage during the
scavenging process due to the flow leaving in axial direction, causing recirculation of one-burned
gases [3]. A substitute configuration was proposed by Jenny and Bulaty, where the rotor had a
conical shape with oblique channels and the flow entering and leaving the channels at the front
and rear, respectively, and the axial effect still highly interferes with the scavenging process [3].
A breakthrough in new configurations, locating the combustion process inside the channels of
the wave rotor, was later proposed to simplify the porting (Figure 1-1 (b)) [3]. According to
literature, NASA’s wave rotor research was focused on the concept of combustion inside of
wave rotor channels [3], [4], [5] . Further modification was proposed in 2004 [3]; this was the
introduction of the Wave Disc Engine (WDE) concept (Figure 1-1 (c)). The WDE is a radialflow rotor concept which addresses the uncorrected imperfection in the second modification of
the wave rotor (Figure 1-1(b)). Thus, this new design will improve both the scavenging process
through centrifugal forces, and the self-aspirating advantage. Initially the idea of radial rotor with
straight channel was suggested, then, curved radial channels were proposed, due to their
advantages over straight channels. One benefit is the greater length which might be allocated
with the same disc diameter. Curved channels allow the adjustment of the inlet and outlet
channel angles to improve the work extraction [3]. The compression process is accomplished
within the channel by compression waves, and so this process can be done without a compressor.
Another idea was the WDE attached to an external turbine, a proposal configured to enhance the
work extraction of the exhaust gases (Figure 1-1 (d)).

3

(a)

(b)

(c)

(d)

Figure 1-1: Steps toward the WDE
Another configuration envisioned is a compact WDE with two turbochargers (Figure
1-2).
This research project at the MSU turbomachinery Lab, to build a prototype, was funded
by ARPA-E from January 2010 to January 2012, and then extended to May 2013 with $2.5 and
$0.5 million, respectively, and since inception investigation have been dedicated to the research
of this proposed engine concept.

4

Figure 1-2: Future compact configuration of the WDE connected to two turbochargers
1.5.

Working Principles of the WDE
This new engine concept is a radial rotor in which the typical processes of an Internal

Combustion Engine (Intake, Compression, Combustion, and Expansion) are executed. The intake
process delivers the fresh air-fuel mixture and fills the channel while creating turbulence to
augment the mixing process before ignition of the mixture. The onset of this process overlaps the
scavenging period and the mass flow rate of the incoming fresh mixture is intensified due to the
suction effect produced by the exhaust gases (Figure 1-3 (a)). The outlet of the channel then
begins to close and at that time the fresh mixture is ready to fully fill the channel (Figure 1-3(b)).
The later stage of this process also overlaps the compression stage when the outlet port of the
channel is fully closed (Figure 1-4(a)). Fresh mixture still enters the channel until it is fully
closed.

5

(a)

(b)

Figure 1-3: Overlap period for early Intake with Scavenging period
The Compression process starts when the outlet port quickly closes. A compression wave
or shock wave is created that travels back toward the inlet port and compresses unsteadily the
fresh mixture entering the channel (Figure 1-4(b)). The degree of compression by the wave
depends on the velocity of the incoming flow into the channel.

(a)

(b)

Figure 1-4: Intake process and early stage of Compression process

6

The next process is combustion. The outlet and inlet ports of the channel are closed, and the
combustion happens in an enclosed channel. Therefore, it is called constant volume combustion
(Figure 1-5 (a) and (b)). The time available for combustion is constrained by the geometry of the
engine and the rotational velocity. Thus, complete combustion of the mixture is limited by this
time allotted.

(a)

(b)

Figure 1-5: Beginning and end of the Combustion process
The unsteady expansion is the final process in the cycle and torque generation begins with
gradual opening of the channel outlet (Figure 1-6(a)). This process is limited as well by the
available geometric time of the engine at a particular rpm. The scavenging overlaps the intake
process and begins; following a short interval, the front of the expansion wave strikes the inlet
wall of the channel (Figure 1-6 (b)).

7

(a)

(b)

Figure 1-6: Unsteady expansion process
Figure 1-8 shows an example of the available geometric time per cycle. The available time was
calculated based on the WDE-II rotor design built by the WDE research group at MSU (Figure
1-7) and is valid only for this particular rotor. It appears the available geometric time for
Combustion and Expansion + scavenging + Inlet (P0) are 68.6% and 24.4%, respectively, of the
total cycle time. These percentages are kept constant as the rpm increases but the absolute time
for each processes decreases.

8

(a)

(b)

Figure 1-7: (a) WDE-II Rotor designed and built by the MSU WDE research group & (b) space
distribution for processes

Figure 1-8: Geometric time available per cycle for the Unsteady Expansion Process for a 2 cycle
per revolution WDE-II Rotor

9

1.6.

Previous work on the WDE
1.6.1. Thermodynamic Cycle of the WDE
A transient thermodynamic model was proposed to explicitly calculate the cycle

efficiency of the WDE. The formula was stated to calculate the work produced by the burned gas
during the expansion process [6]:
𝑡𝑡4

𝑊𝑊𝑒𝑒 = − � 𝑚𝑚̇𝑜𝑜𝑜𝑜𝑜𝑜 ℎ𝑡𝑡𝑡𝑡𝑡𝑡.𝑜𝑜𝑜𝑜𝑜𝑜 𝑑𝑑𝑑𝑑 + 𝐸𝐸𝑡𝑡3 − 𝐸𝐸𝑡𝑡4
𝑡𝑡3

Where 𝑡𝑡3 and 𝑡𝑡4 represent the initial and final state of the pure expansion process. This equation

was used to derive the final expression of the thermal efficiency in terms of mass-average
temperature.
𝜂𝜂 = 1 − 𝛾𝛾

𝑇𝑇�4 − 𝑇𝑇�1
𝑇𝑇�3 − 𝑇𝑇�2

The unsteadiness of the expansion process was considered in this formula but the
expression was changed to have as a function of mass-average temperature values.

1.6.2. Mechanism for Torque Generation
Previous work on studying the mechanism was presented by Sun [7]. It stated the torque
generation principle for a traditional steady turbine is not valid for the WDE operating in
unsteady flow conditions. This principle refers to a fluid particle passing through a blade
passage; centrifugal force creates a pressure gradient toward the center of the curvature; as a
result of this, force is applied to the blade. According to Newton’s Law, the normal direction for
steady state conditions:
(1-1)

𝜕𝜕p 𝜌𝜌. 𝑉𝑉𝜃𝜃 2
=
𝜕𝜕𝜕𝜕
R
10

The principle was tested for steady and unsteady conditions using the concept from
Equation (1-1). Unsteady condition test did not prove the validity in the two channel shapes
tested. Thus, it was concluded the steady state principle does not apply for torque generation
under unsteady state conditions [7] and more factors need to be considered for torque generation
mechanism.
Another topic in this study was the direction of the pressure gradient when the expansion
wave travels inside the channel [7]. A positive pressure gradient is generated when the expansion
wave moves toward the inlet and the pressure decreases from the inlet to the outlet. A negative
pressure gradient generates an opposite effect. Conclusions were summarized in Table 1-1 and
blade angles refers to the inlet or outlet angles of the blade [7].

Table 1-1: General principle of torque generation according to the travelling direction of
expansion wave [7]
Another conclusion from this research [7], was the influence of the outlet opening in
channel geometries. The efficiency of the expansion increased with the extension of the
expansion duration, and these higher values were the result of having a smaller outlet opening.
This was defined as the smallest distance from the trailing upper wall to the lower wall of the
suction surface.

11

1.6.3. Unsteady Expansion in channels
Previous work shows efficiency is reduced for two primary reasons: Kinetic Energy
leaving the channel outlet and incomplete work extraction from the burned gases [7]. Preliminary
research was conducted for a hypothetical variation of the average density in the channel,
considering different time duration of the expansion process (Figure 1-3). In this figure, the
magnitude of the outlet velocity in the channel decreases as the duration time of the expansion
process increases.
A work extraction investigation by Sun [7] concludes greater work is generated in the
WDE during the expansion process. The work generated is negligible in other WDE processes
during the CFD simulation. Therefore, focusing on the unsteady expansion process for work
generation is valuable. For this investigation, the performance of channel geometry (efficiency)
was calculated using the energy heat addition as ideal (maximum) work: the net internal energythe difference between initial and final state of the expansion process-through the patching of
pressure and temperature.
Sun [7], focusing on the CFD investigation of unsteady expansion, concluded that, in a
single channel, the source of losses were primarily from Kinetic Energy leaving the channel, and
under-expansion of exhaust gases. The next finding was that convergent channels show the best
performance, from 2.2% to 3.3%.

1.6.4. Re-utilization of exhaust gases
Options to re-utilize exhaust gases to improve engine efficiency are found in the
literature:

12

1.6.4.1.

Return channel

Re-directing the pressure and kinetic energy of the gases for additional work extraction,
to take advantage of the exhaust gases during the unsteady expansion, to the entrance of a lower
pressure contiguous channel, is the return channel’s purpose.
Several works have been done on this concept. A numerical study by Sun [7] found that
efficiency increases 56.1% by re-using the exhaust energy in a return channel. This study was
performed using a convergent channel, and the comparison with a base-line design configuration
is provided in Table 1-2.

Table 1-2: Results from numerical simulation [7]

1.6.4.2.

Parallel row of turbine blades

A MSU research team began investigating this new concept in 2010, sponsored by
ARPA-E (Advanced Research Projects Agency-Energy) of the US Department of Energy. Using
CFD, one study was accomplished for the WDE with the combination of re-injection passage
and a second row of turbine blades [8]. This configuration delivered the highest efficiency (5.5%
efficiency), but which of them made the greatest contribution was not stated. However, the
exhaust gases did not yet fully expand within the external blade passages, and adding additional
rows of turbine blades was suggested.

13

Another CFD investigation for two stage rotor configuration (first stage: a convergent
channel/ second stage: a turbine blade) with guide vanes between them, found the efficiency
increased from 3.96% to 11.01% for a 2 cycles per rotation engine [7]. See Table 1-3 for
comparison with a single stage rotor (only convergent channel).

Table 1-3: Comparison of a single stage and 2 stage rotors WDE
A multi-stage serial radial turbine, the hybrid wave engine, was investigated by Dyntar
[9]. This proposed configuration was planned for use in MEMS technology (micro engines). The
parallel flow configuration (Figure 1-9(a)) was dismissed because the mass flow rate of exhaust
gases from the gas generator was high and therefore, not suitable for this application. The high
flow rate passing through the blades do not properly transfer the energy into torque. Thus, this
work proposed the serial flow configuration, where the exhaust gases expand in two steps in the
same set of blades (Figure 1-9 (b)). The flow experiences two stages of expansion: at highpressure and low-pressure, and the total maximum efficiency reported was 10% [9].

14

(a) Parallel flow configuration

(b) Serial flow configuration

Figure 1-9: Parallel and Serial flow configuration of a radial turbine. Source [9]
1.7.

Expansion efficiency
The isentropic efficiency of a turbine is one performance factor criteria to evaluate the

performance of an expansion process, and is calculated in terms of the work done by the fluid
passing through the blade passage [10]:
𝜂𝜂𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 =

𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤
𝑊𝑊𝑎𝑎
=
𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 (𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚)𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑊𝑊𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖

(1-2)

Equation (1-2) is used in turbomachines when the expansion process is at steady state
conditions; this means all fluid particles passing through the turbine undergo the same
thermodynamic conditions from inlet to outlet.
Previous work [7] defined a method to evaluate the unsteady expansion process:
𝜂𝜂 =

𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤
𝑊𝑊𝑎𝑎
=
𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎
𝑄𝑄

(1-3)

The heat addition is calculated based on [7]:

(1-4)

𝑄𝑄 = 𝑈𝑈𝑓𝑓 − 𝑈𝑈𝑖𝑖
15

Where 𝑈𝑈𝑖𝑖 is the internal energy in the channel before patching and 𝑈𝑈𝑓𝑓 is after patching.

16

CHAPTER 2.

INFLUENCE OF CHANNEL PARAMETERS: WIDTH,

HEIGHT AND LENGTH AT CONSTANT CROSS-SECTION CHANNELS
This section determines torque generation influence by changing geometric parameters of
cross-section (width and height) and length of the channel.
All channels initiate the unsteady expansion process at the same total energy. Due to the
geometric parameter change, the kinetic energy of the fluid within the channel might vary but
does not affect the results because they are negligible. Thus, the internal energy is the influential
factor to preserve identical initial conditions of the fluid within the channel. The working fluid is
air which follows the ideal gas law, and because of that, the internal energy contained in the
channel depends on two factors: temperature and mass. To meet these two conditions, the
unsteady expansion process in all cases starts at the same temperature and pressure (2100 [K]
and 709275 [Pa], respectively). There is an additional requirement to keep the same mass:
channel volume must be identical; to achieve this geometrical state, the length of the channel was
varied to adjust to the volume-goal value.
Additional parameters such as outlet blade angle (βout), tip speed (U), outer and inner
radiuses (OR and IR) were kept constant (Figure 2-2(b)). Table 2-1 depicts a summary of the
geometry characteristics for each case. All cases had channels rotating at 10,000 rpm, with
channel outlet instantly opening.
Equation (1-3) was used to evaluate the performance of the unsteady expansion process
in all cases.
In Table 2-1, cases 1 to 3 show the variation of height when theta angle is increased with
channel width kept constant; and thus channel length becomes longer. The channel height was

17

reduced to keep the volume-goal value constant. Figure 2-2 (a) schematically portrays the
changes in Table 2-1 with the application of theta and beta angles (Figure 2-2(b)).

Table 2-1: Geometry conditions for each constant cross-section channel
For cases 1 to 3, the expansion duration time was extended as theta increased (Figure 2-3)
and this is linked to the channel length increase. The length increase is compared with respect to
theta=40deg case-1 and is as follows: 9.2% and 18.5%, respectively. The same trend is observed
in travel-time of expansion wave to reach the channel inlet wall (Figure 2-4 (b)), and as theta
value increases, the head of the expansion waves has a greater distance to travel. Additionally,
the combination of length increase with height decrease resulted in 1.6% and 3.6% rise of
pressure side area with respect to case-1, respectively (Figure 2-1). However, suction side area
decreased in 1.4% and 3.0% respectively. The results of increase and decrease of areas did not
produce an important improvement on the net tangential force. Thus, those cases (1 to 3)
produced similar values of expansion efficiency (Figure 2-4(a)). The biggest increase in
efficiency was 0.7%. A gain based on extended duration of expansion process was counteracted
by the reduced channel height to generate torque.

18

Figure 2-1: Comparison of channel outlet, pressure side and suction side areas with respect to
reference case-1.

OR

IR

(a)

(b)

Figure 2-2: (a) Constant cross-section channel configurations for cases-1 to 3 and (b) meaning of
θ and βout angles.
In the following cases, (4 and 5) was considered the extension of the channel length but
with the same channel height as case-1 (reference case). The changing parameter was the
channel width. As a result of this, the areas of the pressure side were increased 11.3% and 23%
with respect to the base line case, respectively. Also, the areas of the suction side were increased
8.6% and 17.3%, respectively. In addition, the expansion duration time and the EWTT were

19

enlarged (Figure 2-3 and Figure 2-4 (b)). Another parameter to point out is expansion efficiency
with an improvement of 1.4% and 3.7% in respect to the base line case-1 (Figure 2-4). The
reduced width did not diminish significantly the case with the highest efficiency (3.7%) due to
the boundary layer affecting more cross-sectional area if case-5 is compared to case-3.

Figure 2-3: Duration of expansion process for constant cross-section channels varying their
geometric parameters (Var =geometric parameter changed)
The last change in geometric dimension of the cross-section pertains to case-6, with the
same theta value as the base line case-1, but with the width reduced to increase the height value
and still preserve the volume-goal value. The modifications produced an increase in pressure and
suction side areas of 13.1% and 13.6%, respectively. Results in Figure 2-4(a) show an
improvement of 0.62%, in expansion efficiency. However, the duration of expansion process and
EWTT carry the same value as in case-1 (Figure 2-4(b)). The contribution of additional pressure
side area was not substantial enough to generate extra torque if the duration of expansion process
is not extended.
The pressure side area increase produces a rise in the net tangential force if this condition
is met: PPS > PSS. The rise of θ angle or channel length results in pressure side area increase even
20

if height is either reduced or kept constant. However, the reduced height produced lower values
of area increase. For example, 3.6% and 23.5% were the results for case-3 and -5 respectively.
The rise of the channel length extends the duration time of the unsteady expansion
process and the head of the expansion wave will propagate longer distances. Thus, the duration
increase of the expansion process contributes to extend the application time of the tangential
force on the pressure and suction sides.
Results show that the highest efficiency happened when the duration time and the
pressure side area, were increased. Therefore, the length of the channel and pressure side area,
influence the expansion efficiency. Both parameters combine to raise efficiency. A method to
improve it is by increasing geometric parameters: θ angle (channel length) and channel height.

(a)

(b)

Figure 2-4: (a) Expansion efficiency [%] and (b) Expansion wave travel time to reach channel
inlet (EWTT) [s] for constant cross-section channels varying their geometric parameters.
Results of energy generated in Figure 2-5(a) refer to the gas expansion of one channel,
and this value is translated into a rotor’s specific number of channels. Thus, these results are
interpreted as the power generated by the engine rotor and values are plotted in Figure 2-5.

21

A note from Figure 2-4(a) is concerned with the expansion efficiency results in which
low values are based on the criterion of the internal energy change. Using a different approach to
evaluate the unsteady expansion process is desirable, and this will be the next chapter topic.

(a)

(b)

Figure 2-5: (a) Energy generated by one channel, (b) Power generated by engine rotor of 24
channels rotating at 10,000 rpm

22

CHAPTER 3.

EXERGY ANALYSIS OF THE UNSTEADY EXPANSION
PROCESS

3.1.

The necessity of an alternative method to evaluate the expansion efficiency
Typically for steady state processes in turbomachines, turbine or compressor efficiency is

calculated based on the idea a unit of mass goes from an initial state at the inlet to final state at
the outlet. All fluid elements experience the same trajectory in the thermodynamic diagram
during a specific process (expansion or compression). Based on these conditions, the actual and
the ideal (maximum) work are determined to calculate the efficiency. In the case of the WDE
configuration for the expansion process, gases only leave the channel during the expansion and
thus each fluid element traverses different paths along the channel. Therefore, every fluid
particle undergoes different thermodynamic conditions. The application of the criterion above
might give misleading results in the evaluation of the performance of the device.
In previous research [7], the performance of a single channel was approached based on
the internal energy increased due to patching; this was considered as a heat addition. Thus, the
work generated was compared to this value.
Up to now, all efficiency concepts were based on the energy approach of the First Law of
thermodynamics. An alternative approach to evaluate the efficiency based on the combination of
the First and Second law of thermodynamics, and this method is supported by Dincer and Rosen
[11] (2013), Rosen and Etele [12] (2004), and many other researchers. Thermodynamic losses
happening inside the system usually are not appropriately identified and evaluated with the
energy analysis approach [11], and especially when the system is so highly unsteady it requires
special attention. Exergy approach assists in overcoming the drawbacks of the energy analysis
[11].

23

3.2.

Exergy analysis of a single channel
Exergy is defined as: “… the maximum theoretical work obtainable from an overall

system consisting of a system and the environment as the system comes into equilibrium with the
environment (passes to the dead state)” [13]. Exergy concept does not specify the method to
generate the work-either steady or unsteady-but only refers to the maximum potential associated
with the initial state respect to a reference state (dead state). The dead state defined through all of
this research is T0=300 [K] and p0= 101325 [Pa]. Exergy (E) is calculated based on Equation
(3-1) [13]:
(3-1)

𝐸𝐸 = (𝑈𝑈 − 𝑈𝑈0 ) − 𝑝𝑝0 (𝑉𝑉 − 𝑉𝑉0 ) − 𝑇𝑇0 (𝑆𝑆 − 𝑆𝑆0 ) + 𝐾𝐾𝐾𝐾 + 𝑃𝑃𝑃𝑃

For the case of a control volume in which fluid properties are not uniform, equation (3-1)
is written in Integral form:
𝐸𝐸 = � � 𝜌𝜌(𝑢𝑢 − 𝑢𝑢𝑟𝑟𝑟𝑟𝑟𝑟 )𝑑𝑑𝑉𝑉 − � 𝜌𝜌(𝑢𝑢0 − 𝑢𝑢𝑟𝑟𝑟𝑟𝑟𝑟 )𝑑𝑑𝑉𝑉 � − 𝑝𝑝0 � � 𝑑𝑑𝑑𝑑 − � 𝑑𝑑𝑑𝑑 �
𝐶𝐶𝐶𝐶

𝐶𝐶𝐶𝐶

𝐶𝐶𝐶𝐶

𝐶𝐶𝐶𝐶0

2

�⃗ �
�𝑉𝑉
− 𝑇𝑇0 � � 𝜌𝜌(𝑠𝑠 − 𝑠𝑠𝑟𝑟𝑟𝑟𝑟𝑟 )𝑑𝑑𝑉𝑉 − � 𝜌𝜌(𝑠𝑠0 − 𝑠𝑠𝑟𝑟𝑟𝑟𝑟𝑟 )𝑑𝑑𝑉𝑉 � + � 𝜌𝜌 �
� 𝑑𝑑𝑉𝑉
2
𝐶𝐶𝐶𝐶

𝐶𝐶𝐶𝐶

+ � 𝜌𝜌(𝑧𝑧 − 𝑧𝑧0 )𝑔𝑔𝑔𝑔𝑉𝑉
𝐶𝐶𝐶𝐶

𝐶𝐶𝐶𝐶

And considering that the control volume does not change and all cell elements in the
channel keep the same z-coordinate, it yields:

24

(3-2)
𝐸𝐸 = � � 𝜌𝜌(𝑢𝑢 − 𝑢𝑢𝑟𝑟𝑟𝑟𝑟𝑟 )𝑑𝑑𝑉𝑉 − � 𝜌𝜌(𝑢𝑢0 − 𝑢𝑢𝑟𝑟𝑟𝑟𝑟𝑟 )𝑑𝑑𝑉𝑉 �
𝐶𝐶𝐶𝐶

𝐶𝐶𝐶𝐶

− 𝑇𝑇0 � � 𝜌𝜌(𝑠𝑠 − 𝑠𝑠𝑟𝑟𝑟𝑟𝑟𝑟 )𝑑𝑑𝑉𝑉 − � 𝜌𝜌(𝑠𝑠0 − 𝑠𝑠𝑟𝑟𝑟𝑟𝑟𝑟 )𝑑𝑑𝑉𝑉 �
𝐶𝐶𝐶𝐶

𝐶𝐶𝐶𝐶

2

�⃗ �
�𝑉𝑉
+ � 𝜌𝜌 �
� 𝑑𝑑𝑉𝑉
2
𝐶𝐶𝐶𝐶

All terms in Equations (3-1) and (3-2) are based on a stationary control volume chosen
according to Figure 3-1. All variables with the subscript “0” are calculated at dead state

conditions. Also, the subscript “ref” means the variable is calculated at reference values of

Temperature (288.16 [K]) and absolute pressure (101325 [Pa]).

As a starting point for this single channel analysis, Exergy was calculated based on
Equation (3-1) and the gas initial conditions in the channel at different pressure ratios and results
are plotted in Figure 3-2. These values should be considered as the upper limit value (maximum
work) in the expansion process based on Exergy definition stated above. Thus, in Figure 3-2, the
blue triangles are the maximum expected work at every specific pressure ratio for this channel.
Then, an expansion case is simulated numerically for the same geometry and gas initial
conditions (10,000 rpm with instant opening at PR=7) and the work extracted is plotted in the
graph as well. If this value is compared to the exergy at the same PR, it would result in 4.81%
efficiency of expansion for this simulated case. The approach to determine the efficiency using
the internal energy patching criteria predicts a lower value of 3.6%. Efficiency based on the
exergy method predicts higher values and can be considered a more realistic value because the
extracted reference work value is the maximum. Afterwards, exergetic efficiency will be defined
to evaluate the performance of unsteady processes.

25

Figure 3-1: Control volume considered for exergy rate balance

Figure 3-2: Exergy (maximum work available) for a constant cross-section curved channel at
10,000 rpm at several Pressure ratios (PR)
The next step is to monitor the exergy in the channel along with the flow time. Each
exergy value was calculated at each time step interval of 1E-08 [s] using Equation (3-2) under
these conditions: Instant Opening at 10,000 rpm (IO@10K); Gradual Opening at 10,000
(GO@10K), 20,000 (GO@20K), 30,000 (GO@30K), 40,000 (GO@40K) and 50,000
(GO@50K) rpm. The results are shown in Figure 3-3, and reveal the increase of initial exergy
value as the tip speed of the channel increases. In the calculation for exergy, the kinetic energy

26

component raises Exergy 7.2% (@50Krpm) compared to GO@10K. Two limits can be
recognized, based on the conditions tested: at the lowest rotational velocity (GO@10K), and
instant opening of the channel outlet (IO@10K). The history of exergy shows that as the rpm
increases the behavior tends to resemble as the instant opening case.

rpm increases

Figure 3-3: Exergy variation along expansion time for a constant cross-section channel at several
rotational velocities and PR=7
3.3.

Exergy rate analysis at transient state
Since the expansion is an unsteady process, tracking exergy variables as the expansion

progresses, and as an overall value, is appropriate to consider the exergy rate analysis. The
control volume is according to Figure 3-1, and the general exergy rate balance equation is [13]:
𝑑𝑑𝐸𝐸𝑐𝑐𝑐𝑐
𝑇𝑇0
𝑑𝑑𝑉𝑉𝑐𝑐𝑐𝑐
= �(1 − ) 𝑄𝑄̇𝑗𝑗 − �𝑊𝑊̇𝑐𝑐𝑐𝑐 + 𝑝𝑝
� + � 𝑚𝑚̇𝑖𝑖 𝑒𝑒𝑓𝑓𝑓𝑓 − � 𝑚𝑚̇𝑒𝑒 𝑒𝑒𝑓𝑓𝑓𝑓 − 𝐸𝐸̇𝑑𝑑
𝑑𝑑𝑑𝑑
𝑇𝑇𝑗𝑗
𝑑𝑑𝑑𝑑
𝑗𝑗

𝑖𝑖

𝑒𝑒

(3-3)

Based on the working features of this engine and simplifications for the purpose of this

analysis, some idealizations are applied:

27

(a) The boundaries of the control volume are considered adiabatic and channel walls are treated
as if there is no heat flux through them during CFD simulations.
(b) The control volume size does not increase and stays stationary throughout the process
(c) There is only one outflow that goes through the channel outlet and no incoming flow.
After these constraints, Equation (3-3) becomes:
𝑑𝑑𝐸𝐸𝑐𝑐𝑐𝑐
= −𝑊𝑊̇𝑐𝑐𝑐𝑐 − 𝑚𝑚̇𝑒𝑒 𝑒𝑒𝑓𝑓𝑓𝑓 − 𝐸𝐸̇𝑑𝑑
𝑑𝑑𝑑𝑑

(3-4)

Equation (3-4) is the standard form that will be used for all calculations in this chapter.
The final term (𝐸𝐸̇𝑑𝑑 ) in this equation is exergy destruction rate and represents the speed at which
exergy is destroyed at a specific flow time, and must be equal or greater than zero but never a
negative value. 𝐸𝐸̇𝑑𝑑 is calculated indirectly when solving for it in Equation (3-4) and the other
terms should be known:

𝐸𝐸̇𝑑𝑑 = −

𝑑𝑑𝐸𝐸𝑐𝑐𝑐𝑐
− 𝑊𝑊̇𝑐𝑐𝑐𝑐 − 𝑚𝑚̇𝑒𝑒 𝑒𝑒𝑓𝑓𝑓𝑓
𝑑𝑑𝑑𝑑

(3-5)

The second term on the right hand side of (3-4) represents the flow exergy rate at exit (e):
𝐸𝐸̇𝑓𝑓𝑓𝑓 = 𝑚𝑚̇𝑒𝑒 𝑒𝑒𝑓𝑓𝑓𝑓

And is calculated based on Equation (3-6):

Where:

̇ = 𝑚𝑚̇𝑒𝑒 [ℎ − ℎ0 − 𝑇𝑇0 (𝑠𝑠 − 𝑠𝑠0 ) +
𝐸𝐸𝑓𝑓𝑓𝑓

𝑒𝑒𝑓𝑓𝑓𝑓 = ℎ − ℎ0 − 𝑇𝑇0 (𝑠𝑠 − 𝑠𝑠0 ) +

𝑉𝑉 2
2

𝑉𝑉 2
+ 𝑔𝑔𝑔𝑔]
2

(3-6)

+ 𝑔𝑔𝑔𝑔 , is called specific flow exergy and all properties are

taken at the channel outlet. All variables with the subscript “0” are calculated at the dead state

conditions.

28

Equations (3-4), (3-5), and (3-6) refer to uniform properties at every instant in time, but
the unsteady expansion process demonstrates that properties are not uniform in the channel; for
this reason two terms in those equations will be written in integral form:
(a) The term 𝐸𝐸𝑐𝑐𝑐𝑐 will be computed based on Equation (3-2)
(b) The term 𝑚𝑚̇𝑒𝑒 𝑒𝑒𝑓𝑓𝑓𝑓 will be computed based on:

̇ = �� 𝜌𝜌𝑉𝑉𝑟𝑟 (ℎ − ℎ𝑟𝑟𝑟𝑟𝑟𝑟 )𝑑𝑑𝐴𝐴 − � 𝜌𝜌𝑉𝑉𝑟𝑟 (ℎ0 − ℎ𝑟𝑟𝑟𝑟𝑟𝑟 )𝑑𝑑𝐴𝐴 �
𝑚𝑚̇𝑒𝑒 𝑒𝑒𝑓𝑓𝑓𝑓 = 𝐸𝐸𝑓𝑓𝑓𝑓
𝐶𝐶𝐶𝐶

𝐶𝐶𝐶𝐶

2

�⃗ �
�𝑉𝑉
− 𝑇𝑇0 �� 𝜌𝜌𝑉𝑉𝑟𝑟 �𝑠𝑠 − 𝑠𝑠𝑟𝑟𝑟𝑟𝑟𝑟 �𝑑𝑑𝐴𝐴 − � 𝜌𝜌𝑉𝑉𝑟𝑟 �𝑠𝑠0 − 𝑠𝑠𝑟𝑟𝑟𝑟𝑟𝑟 �𝑑𝑑𝐴𝐴 � + � 𝜌𝜌𝑉𝑉𝑟𝑟 �
� 𝑑𝑑𝐴𝐴
2
𝐶𝐶𝐶𝐶
𝐶𝐶𝐶𝐶
𝐶𝐶𝐶𝐶

̇ = �� 𝜌𝜌𝑉𝑉𝑟𝑟 (ℎ − ℎ𝑟𝑟𝑟𝑟𝑟𝑟 )𝑑𝑑𝐴𝐴 − � 𝜌𝜌𝑉𝑉𝑟𝑟 (ℎ0 − ℎ𝑟𝑟𝑟𝑟𝑟𝑟 )𝑑𝑑𝐴𝐴 �
𝐸𝐸𝑓𝑓𝑓𝑓
𝐶𝐶𝐶𝐶

𝐶𝐶𝐶𝐶

− 𝑇𝑇0 �� 𝜌𝜌𝑉𝑉𝑟𝑟 �𝑠𝑠 − 𝑠𝑠𝑟𝑟𝑟𝑟𝑟𝑟 �𝑑𝑑𝐴𝐴 − � 𝜌𝜌𝑉𝑉𝑟𝑟 �𝑠𝑠0 − 𝑠𝑠𝑟𝑟𝑟𝑟𝑟𝑟 �𝑑𝑑𝐴𝐴 �
𝐶𝐶𝐶𝐶

(3-7)

𝐶𝐶𝐶𝐶

2

�⃗ �
�𝑉𝑉
+ � 𝜌𝜌𝑉𝑉𝑟𝑟 �
� 𝑑𝑑𝐴𝐴
2
𝐶𝐶𝐶𝐶

Once values are calculated for each time step, a criterion is chosen to determine the end
of the expansion process in order to calculate Ecv, Ed, Efe and Wcv, which account for the entire
process. This criterion will lead to the comparison of several cases and are as follows:
(a) When the absolute pressure at the channel inlet reaches the area-weighted average value of
101325 [Pa], or
(b) When all cases undergo the same net exergy change (ΔE)
The following formulas calculate values for the entire process:
𝛥𝛥𝛥𝛥𝑐𝑐𝑐𝑐 = 𝐸𝐸𝑓𝑓 − 𝐸𝐸𝑖𝑖

29

(3-8)

Where 𝐸𝐸𝑓𝑓 is determined based on criteria chosen above to terminate the expansion

process.

𝑡𝑡

𝑖𝑖

̇ 𝑑𝑑𝑑𝑑 = � 𝐸𝐸𝑓𝑓𝑓𝑓,𝑖𝑖
̇
𝐸𝐸𝑓𝑓𝑓𝑓 = � 𝐸𝐸𝑓𝑓𝑓𝑓
∗ ∆𝑡𝑡
0

𝑡𝑡

0
𝑖𝑖

𝑊𝑊𝑐𝑐𝑐𝑐 = � 𝑊𝑊̇𝑐𝑐𝑐𝑐 𝑑𝑑𝑑𝑑 = � 𝑊𝑊̇𝑐𝑐𝑐𝑐,𝑖𝑖 ∗ ∆𝑡𝑡
0

𝑡𝑡

0

𝑖𝑖

𝐸𝐸𝑑𝑑 = � 𝐸𝐸̇𝑑𝑑 𝑑𝑑𝑑𝑑 = � 𝐸𝐸̇𝑑𝑑,𝑖𝑖 ∗ ∆𝑡𝑡
0

(3-9)

(3-10)

(3-11)

0

The exergy analysis follows to compare cases described by using variables which
represent the overall process. In Figure 3-4, percentages of the overall variables are plotted based
on the two criteria. All percentages are calculated based on the net exergy change for either
criterion case. Both criteria to establish the termination of the expansion process show similar
trends of the overall variables and percentage values as well. Therefore, equal net exergy is
chosen as the criteria used in discussions (Figure 3-4 (b)).
The trend of the overall value percentage for Ed is opposite to Wcv (Figure 3-4). The
increase in the rotational velocity enhances work generation but reduces the exergy destruction
value. One factor associated with this effect is how quickly the outlet opens, and this will be
discussed later.

30

(a) End process based on inlet pressure

(b) Equal ΔE

Figure 3-4: Percentage values of the Wcv, Ed and Efe with respect to equal ΔE in several cases
during the expansion process.
In the graph above, flow exergy has the highest percentages-ranging between 80 to 90%
for this particular channel geometry-showing high potential to produce more power in
subsequent stages. Another note of the graph is the increase in work generation due to gas
expansion in the channel reducing the flow exergy value leaving the channel, indicating the
production of work either in the channel or in further stages. However, results in Table 3-1
suggest converting the maximum possible available exergy into work because some exergy will
be destroyed if not present in the flow exergy. An explanation of this follows.

31

Table 3-1: Comparison of percentage values with respect to the GO@10K case
Column Ai shows produced work at case “i” with respect to GO@10K case (net
difference). In column Bi, the Ai results is added to the % flow exergy of case “i”, respectively,
and is expected to have the same Bi for all cases if exergy is not destroyed (ideal condition). The
justification for this idea is if exergy is not utilized in the channel then it will be used as flow
exergy. According to Table 3-1, however, Bi increases with the rotational velocity. All Bi values
are then compared with respect to GO@10K case (BGO@10K), and results are shown in column Ci.
These results are interpreted as the exergy not destroyed with respect to GO@10K, i.e. the value
of CGO@50K=1.7 is the exergy not destroyed if run at 50,000 rpm. If these results are added to the
respective %Ed cases, then all cases result in the same value and are shown in column Di.
The trend of overall variable values has been discussed. The next figure shows how these
exergy variables change along the expansion process. The rate of each variable is used to
describe the process. The rate of E (|dE/dt|) was calculated by using a second order finite
difference of each value resulting from equation (3-2). Equations (3-5) and (3-7) are for Ėd and
Ėfe respectively. Thus, in Figure 3-5, graphs for several rotational velocities are shown, and the
blue arrow lines define the flow time when the channel outlet is fully open.
According to Figure 3-5, three regions can be approximately defined:
(a) The region immediately after opening (region-I)
(b) During the period of maximum work rate (region-II)
32

(c) When Ėd starts to decrease until process completion (region-III)
In region I, the instant opening case shows that |dE/dt|, Ėd & Ėfe spike in their values due
to the nature of channel outlet opening immediately. In the five other cases, as the rotational
velocity increases, the spike tends to mirror the immediate opening case, i.e. GO@50K case
tends to be as the sudden opening.
In region-II, the maximum work rate increases and shifts toward the left as the rotational
velocity increases. The graph shows this effect is influenced by the time position when the
channel outlet is fully open, allowing the fluid to follow a trajectory parallel to the channel walls.
In region-III, all variable rates tend to converge and decrease to a close zero value.
As presented here, flow exergy at the outlet holds the biggest percentage-ranging
between 80 to 90%-of the available potential energy to produce work, thus showing great
potential to increase the total power in the wave engine if additional stages for expansion are
considered.

33

(a) IO@10K

(b) GO@10K

(c) GO@20K

(d) GO@30K

(e) GO@40K

(f) GO@50K

Figure 3-5: Plots for |dE/dt|, Ėd, Ẇcv, & Ėfe for several rotational velocities in absolute values.
Blue arrows represent the flow time when the channel outlet is fully open.

34

3.4.

Exergetic efficiency for control volume under unsteady state
In this section, the channel geometry needs to be assessed to determine the effectiveness

of the energy conversion of the overall expansion process. The exergy concept used to evaluate
efficiency is called Exergetic Efficiency or Second Law Efficiency (Îľ). Equation (3-4) is the
starting point for this discussion, where the idealization conditions still apply for the expansion
process.
𝑑𝑑𝐸𝐸𝑐𝑐𝑐𝑐
̇ − 𝐸𝐸̇𝑑𝑑
= −𝑊𝑊̇𝑐𝑐𝑐𝑐 − 𝐸𝐸𝑓𝑓𝑓𝑓
𝑑𝑑𝑑𝑑

To be considered for the overall process, the equation is integrated throughout the process
time:
𝑡𝑡

𝑡𝑡
𝑡𝑡
𝑡𝑡
𝑑𝑑𝐸𝐸𝑐𝑐𝑐𝑐
̇
̇
ďż˝
𝑑𝑑𝑑𝑑 = − � 𝑊𝑊𝑐𝑐𝑐𝑐 𝑑𝑑𝑑𝑑 − � 𝐸𝐸𝑓𝑓𝑓𝑓 𝑑𝑑𝑑𝑑 − � 𝐸𝐸̇𝑑𝑑 𝑑𝑑𝑑𝑑
0 𝑑𝑑𝑑𝑑
0
0
0
𝑡𝑡 𝑑𝑑𝐸𝐸𝑐𝑐𝑐𝑐

The first integral term: ∍0

𝑑𝑑𝑑𝑑

𝑑𝑑𝑑𝑑 is equivalent to ∆𝐸𝐸𝑐𝑐𝑣𝑣 = 𝐸𝐸𝑖𝑖 − 𝐸𝐸𝑓𝑓

∆𝐸𝐸𝑐𝑐𝑐𝑐 = −𝑊𝑊𝑐𝑐𝑐𝑐 − 𝐸𝐸𝑓𝑓𝑓𝑓 − 𝐸𝐸𝑑𝑑

(3-12)

In aerospace engines where exhaust gases are ejected to the atmosphere at high
temperatures and velocities to generate thrust, the large flow exergy leaving the system is
accounted for as a loss [12], [11]. In this case exergetic efficiency is:
𝜀𝜀 =

𝑊𝑊𝑐𝑐𝑐𝑐
× 100 [%]
|∆𝐸𝐸𝑐𝑐𝑐𝑐 |

When the outlet port is opened the gases are released to generate work in the WDE, but
work can still be generated in subsequent stages. Therefore, Efe should not be included in the
efficiency calculation. Thus, Equation (3-12) yields:
∆𝐸𝐸𝑐𝑐𝑐𝑐 + 𝐸𝐸𝑓𝑓𝑒𝑒 = −𝑊𝑊𝑐𝑐𝑐𝑐 − 𝐸𝐸𝑑𝑑

35

∆𝐸𝐸𝑐𝑐𝑐𝑐 has a negative value, meaning |∆𝐸𝐸𝑐𝑐𝑐𝑐 | > |∆𝐸𝐸𝑐𝑐𝑐𝑐 + 𝐸𝐸𝑓𝑓𝑓𝑓 | , and so the sum of the two

LHS terms represents the exploited exergy in the channel during the expansion process. If RHS
consists of work extraction and exergy destruction, the exploited exergy becomes reversible
work or isentropic work, and is written as 𝑊𝑊𝑟𝑟𝑟𝑟𝑟𝑟 = 𝑊𝑊𝑐𝑐𝑐𝑐 + 𝐸𝐸𝑑𝑑 or 𝑊𝑊𝑟𝑟𝑟𝑟𝑟𝑟 − 𝑊𝑊𝑐𝑐𝑐𝑐 = 𝐸𝐸𝑑𝑑 .
If 𝐸𝐸𝑑𝑑 = 𝑇𝑇0 𝑆𝑆𝑔𝑔𝑔𝑔𝑔𝑔 , the equation results in the following expression:
𝑊𝑊𝑟𝑟𝑟𝑟𝑟𝑟 − 𝑊𝑊𝑐𝑐𝑐𝑐 = 𝑇𝑇0 𝑆𝑆𝑔𝑔𝑔𝑔𝑔𝑔

(3-13)

This Equation (3-13) is called the Gouy-Stodola theorem [14]. Thus, the Exergetic
Efficiency for the expansion process yields:
𝜀𝜀 =

𝑊𝑊ℎ𝑒𝑒𝑒𝑒𝑒𝑒 𝑊𝑊𝑟𝑟𝑟𝑟𝑟𝑟 = 𝑊𝑊𝑐𝑐𝑐𝑐 + 𝐸𝐸𝑑𝑑

𝑊𝑊𝑐𝑐𝑐𝑐
× 100 [%]
𝑊𝑊𝑟𝑟𝑟𝑟𝑟𝑟

(3-14)

The Exergetic Efficiency is evaluated based on (3-14) and results are presented in Figure
3-6. The ratio of Ed/Wcv (Loss/Work) in percentages is included as well.
Based on Figure 3-6, exergy destruction takes place in three different conditions: instant
opening, gradual opening at low rotational velocity, and gradual opening at high rotational
velocity. The first two conditions show low exergetic efficiency values in contrast to the third
condition, in which the efficiency increases to a quasi-constant value as the rotational velocity
increases. Each of these three conditions produces different patterns of exergy destruction values
based on the port outlet opening.
Bejan [15] performed a two dimensional analysis on a small fluid element dx dy as an
open system by using the second law of thermodynamics. This tiny element was subjected to
mass fluxes, energy transfer, and entropy transfer interactions, all going through the control
surface. As a result an expression was derived to estimate the entropy generation rate per unit
volume [W/m3.K]:
36

̇ ′′′ = −
𝑆𝑆𝑔𝑔𝑔𝑔𝑔𝑔

1
𝜇𝜇
𝑞𝑞⃗. ∇𝑇𝑇 + 𝛷𝛷
2
𝑇𝑇
𝑇𝑇

And considering that 𝑞𝑞⃗ = −𝑘𝑘 ∇𝑇𝑇 , the equation above becomes:
̇ ′′′ =
𝑆𝑆𝑔𝑔𝑔𝑔𝑔𝑔

1
𝜕𝜕𝜕𝜕 2
𝜕𝜕𝜕𝜕 2
𝜕𝜕𝜕𝜕 2
𝜇𝜇
𝑘𝑘
[(
)
+
(
)
+
(
)
]
+
𝛷𝛷
𝑇𝑇 2
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝑇𝑇

(3-15)

Based on (3-15), two actions produce local irreversibility: heat transfer and viscous
effects. The first action is a tendency for a spontaneous energy transfer by conduction in the
direction of decreasing temperature when spatial temperature varies. The second effect is related
to velocity gradient in the fluid. Both effects will be used to interpret the destruction of exergy in
the channel during gas expansion.

(a) Equal P-in

(b) Equal ΔE

Figure 3-6: Exergetic Efficiency [%] and Loss/work generated for several rotational velocities

37

(a)

(b)
Figure 3-7: Comparison of % of exergy destruction (equal exergy for all cases) with mass flow
rate. Arrows represent the time when the outlets fully open.
The rate of exergy destruction and mass flow rate plots are shown in Figure 3-7 for
comparison; and an analogous trend between same case plots is found. Instant opening and high

38

rotational velocity cases destroy exergy at higher rate at the beginning of the process but their
overall exergy destruction values are smaller than the slow outlet opening case (see legend
Figure 3-7(a)). The high 𝑚𝑚̇ is linked with the full or faster outlet opening area. This condition

generates high spatial temperature gradients along with viscous losses which cause a spike in the
exergy destruction rate when the channel outlet opens instantly. A sharp increase in the exergy
destruction rate is also produced in the quick outlet opening cases. The slow opening case -10000
rpm- destroys exergy at very low rates in the beginning, but continues destroying for longer time
which results in the highest overall value of exergy destruction.
Based on results, the operation of the rotor at high velocities reduces losses or destroys
less exergy.

39

CHAPTER 4.

ENERGY ANALYSIS OF THE UNSTEADY EXPANSION
PROCESS

4.1.

Zero-dimensional Macroscopic Analytical Model of the Unsteady Expansion Process
Figure 4-1 depicts all four processes of the wave engine. The unsteady expansion

process, the final process of the WDE cycle, is analyzed.

Figure 4-1: Schematic of the wave disc engine with curved channels (two cycles per revolution).
Energy analysis is performed using the rate form of the general energy balance equation
for a transient-flow process considering the inertial control volume from Figure 6-1: the control
volume is the area surrounded by the two dotted concentric circles. During the expansion process
the control volume only exchanges mass through the outlet port. This process starts when the
outlet port begins to open, either quickly or gradually.
The rate of energy change in the system is equivalent to the rate of net energy transfer
through the control volume [16]:

40

𝑑𝑑𝐸𝐸𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠
= 𝐸𝐸̇𝑖𝑖𝑖𝑖 − 𝐸𝐸̇𝑜𝑜𝑜𝑜𝑜𝑜
𝑑𝑑𝑑𝑑

Where:

(4-1)

Specific total energy of the system:

e = u + ke + pe = u +

The total energy of the system:

V2
+ gz
2

𝐸𝐸𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = m𝑠𝑠𝑠𝑠𝑠𝑠 . e

(4-2)

(4-3)

The specific total energy for a flowing fluid:
Θ = h + ke + pe = h +

V2
+ gz
2

Ė mass,b = ṁ Θ = ṁ (h +

V2
+ gz )
2

Amount of energy transport:

(4-4)

(4-5)

Rate of energy transfer by heat, work and mass:

Where: 𝑏𝑏 = 𝑖𝑖𝑖𝑖 𝑜𝑜𝑜𝑜 𝑜𝑜𝑜𝑜𝑜𝑜

𝐸𝐸̇𝑏𝑏 = 𝑄𝑄̇𝑏𝑏 + 𝑊𝑊̇𝑏𝑏 + Ė mass,b

(4-6)

Replacing (4-2), (4-5) and (4-6) in Equation (4-1):
V𝑠𝑠𝑠𝑠𝑠𝑠 2
𝑑𝑑(m𝑠𝑠𝑠𝑠𝑠𝑠 . [u𝑠𝑠𝑠𝑠𝑠𝑠 + 2 + gz𝑠𝑠𝑠𝑠𝑠𝑠 ])
𝑑𝑑𝑑𝑑

(4-7)

= [𝑄𝑄̇𝑖𝑖𝑖𝑖 + 𝑊𝑊̇𝑖𝑖𝑖𝑖 + ṁ 𝑖𝑖𝑖𝑖 (h𝑖𝑖𝑖𝑖 +

V𝑖𝑖𝑖𝑖 2
+ gz𝑖𝑖𝑖𝑖 ) ] − [𝑄𝑄̇𝑜𝑜𝑜𝑜𝑜𝑜 + 𝑊𝑊̇𝑜𝑜𝑜𝑜𝑜𝑜
2

V𝑖𝑖𝑖𝑖 2
+ ṁ 𝑜𝑜𝑜𝑜𝑜𝑜 (h𝑜𝑜𝑜𝑜𝑜𝑜 +
+ gz𝑜𝑜𝑜𝑜𝑜𝑜 ) ]
2
In Equation (4-7), macroscopic properties of the fluid within the channel or the
incoming/outgoing flow streams are considered to be uniform but properties actually change
with the position. Having representative values for the fluid within the channel or flow streams at

41

the outlet port is necessary to derive a zero-dimensional macroscopic balance equation. Thus,
mass-weighted average or volume-weighted average will be used accordingly.
ďż˝ at the surface boundary [17]:
For the representative property ∅
∅𝜌𝜌|𝑣𝑣⃗𝑖𝑖 . 𝐴𝐴⃗𝑖𝑖 | ∑𝑛𝑛𝑖𝑖=1 ∅𝑖𝑖 𝜌𝜌𝑖𝑖 |𝑣𝑣⃗𝑖𝑖 . 𝐴𝐴⃗𝑖𝑖 |
�=∍
∅
= 𝑛𝑛
∑𝑖𝑖=1 𝜌𝜌𝑖𝑖 |𝑣𝑣⃗𝑖𝑖 . 𝐴𝐴⃗𝑖𝑖 |
∫ 𝜌𝜌|𝑣𝑣⃗𝑖𝑖 . 𝐴𝐴⃗𝑖𝑖 |

(4-8)

ďż˝ of the control volume [17]:
For the representative property ∅
∫ ∅𝜌𝜌 𝑑𝑑𝑉𝑉 ∑𝑛𝑛𝑖𝑖=1 ∅𝑖𝑖 𝜌𝜌𝑖𝑖 |𝑉𝑉𝑖𝑖 |
ďż˝
∅=
= 𝑛𝑛
∑𝑖𝑖=1 𝜌𝜌𝑖𝑖 |𝑉𝑉𝑖𝑖 |
∫ 𝜌𝜌 𝑑𝑑𝑉𝑉

(4-9)

Temperature, internal energy, enthalpy and velocity magnitude will be calculated based
on Equations (4-8) and (4-9) for volume and surface, respectively. Volume-weighted average
will be used to calculate the representative pressure and density values in the channel according
to equation (4-10) [17].
𝑛𝑛

1
1
� = � ∅𝑑𝑑𝑉𝑉 = � ∅𝑖𝑖 |𝑉𝑉𝑖𝑖 |
∅
𝑉𝑉
𝑉𝑉

(4-10)

𝑖𝑖=1

Changing to representative variables in (4-7):

(4-11)

�𝑠𝑠𝑠𝑠𝑠𝑠 2
V
𝑑𝑑(m𝑠𝑠𝑠𝑠𝑠𝑠 . [u� 𝑠𝑠𝑠𝑠𝑠𝑠 + 2 + gz�𝑠𝑠𝑠𝑠𝑠𝑠 ])
𝑑𝑑𝑑𝑑

�𝑖𝑖𝑖𝑖 2
V
= [𝑄𝑄̇𝑖𝑖𝑖𝑖 + 𝑊𝑊̇𝑖𝑖𝑖𝑖 + ṁ 𝑖𝑖𝑖𝑖 (h� 𝑖𝑖𝑖𝑖 +
+ gz� 𝑖𝑖𝑖𝑖 ) ] − [𝑄𝑄̇𝑜𝑜𝑜𝑜𝑜𝑜 + 𝑊𝑊̇𝑜𝑜𝑜𝑜𝑜𝑜
2

�𝑖𝑖𝑖𝑖 2
V
+ ṁ 𝑜𝑜𝑜𝑜𝑜𝑜 (h� 𝑜𝑜𝑜𝑜𝑜𝑜 +
+ gz� 𝑜𝑜𝑜𝑜𝑜𝑜 ) ]
2
Several conditions are applied to Equation (4-11) based on ideal WDE working
conditions of a single channel and fixed control volume chosen.
Condition-1:
(a) The rotor is flat (2-D), then variables z�𝑠𝑠𝑠𝑠𝑠𝑠 , z�𝑖𝑖𝑖𝑖 𝑎𝑎𝑎𝑎𝑎𝑎 z�𝑜𝑜𝑜𝑜𝑜𝑜 = 0
42

(b) There is not work done on the system or work input, 𝑊𝑊̇𝑖𝑖𝑖𝑖 = 0

(c) The control volume chosen does not exchange heat with the surroundings. Thus,
𝑄𝑄̇𝑖𝑖𝑖𝑖 𝑜𝑜𝑜𝑜 𝑄𝑄̇𝑜𝑜𝑜𝑜𝑜𝑜 = 0. The same condition applies as well to channel walls.

(d) There is not incoming flow to the control volume and to the channel.
Based on condition-1, equation (4-11) becomes:
(4-12)

�𝑠𝑠𝑠𝑠𝑠𝑠 2
V
𝑑𝑑(m𝑠𝑠𝑠𝑠𝑠𝑠 . [u� 𝑠𝑠𝑠𝑠𝑠𝑠 + 2 ])
�𝑜𝑜𝑜𝑜𝑜𝑜 2
V
= −[𝑊𝑊̇𝑜𝑜𝑜𝑜𝑜𝑜 + ṁ 𝑜𝑜𝑜𝑜𝑜𝑜 (h� 𝑜𝑜𝑜𝑜𝑜𝑜 +
) ]
𝑑𝑑𝑑𝑑
2

Solving Equation (4-12) for the generated work:

𝑊𝑊̇𝑜𝑜𝑜𝑜𝑜𝑜

�𝑠𝑠𝑠𝑠𝑠𝑠 2
V
𝑑𝑑(m𝑠𝑠𝑠𝑠𝑠𝑠 . [u� 𝑠𝑠𝑠𝑠𝑠𝑠 + 2 ])
�𝑜𝑜𝑜𝑜𝑜𝑜 2
V
ďż˝
=−
− ṁ 𝑜𝑜𝑜𝑜𝑜𝑜 (h𝑜𝑜𝑜𝑜𝑜𝑜 +
)
𝑑𝑑𝑑𝑑
2

Expressing the rate equation for a finite interval of time δt:
δW𝑜𝑜𝑜𝑜𝑜𝑜
𝛿𝛿𝛿𝛿

�𝑠𝑠𝑠𝑠𝑠𝑠 2
V
𝛿𝛿(m𝑠𝑠𝑠𝑠𝑠𝑠 . [u� 𝑠𝑠𝑠𝑠𝑠𝑠 + 2 ]) 𝛿𝛿m𝑜𝑜𝑜𝑜𝑜𝑜
�𝑜𝑜𝑜𝑜𝑜𝑜 2
V
=−
−
(h� 𝑜𝑜𝑜𝑜𝑜𝑜 +
)
𝛿𝛿𝛿𝛿
2
𝛿𝛿𝛿𝛿

Removing δt from the denominator:

(4-13)

�𝑠𝑠𝑠𝑠𝑠𝑠 2
�𝑜𝑜𝑜𝑜𝑜𝑜 2
V
V
ďż˝
δW𝑜𝑜𝑜𝑜𝑜𝑜 = − 𝛿𝛿(m𝑠𝑠𝑠𝑠𝑠𝑠 . [u� 𝑠𝑠𝑠𝑠𝑠𝑠 +
]) − 𝛿𝛿m𝑜𝑜𝑜𝑜𝑜𝑜 (h𝑜𝑜𝑜𝑜𝑜𝑜 +
)
2
2
Considering that:
𝛿𝛿𝐸𝐸𝑠𝑠𝑠𝑠𝑠𝑠 = 𝐸𝐸𝑠𝑠𝑠𝑠𝑠𝑠,𝑗𝑗+1 − 𝐸𝐸𝑠𝑠𝑠𝑠𝑠𝑠,𝑗𝑗 Or
𝛿𝛿 �m𝑠𝑠𝑠𝑠𝑠𝑠 . �u� 𝑠𝑠𝑠𝑠𝑠𝑠 +

� 𝑠𝑠𝑠𝑠𝑠𝑠 2
V
2

�� = 𝑚𝑚sys,j+1 �u� j+1 +

2
ďż˝ j+1
V

2

+ 𝑔𝑔z�j+1 � − 𝑚𝑚sys,j �u� j +

Convention of symbols:
j = beginning of the process (j=1)
j+1= end of the process (j=2)
Then:
43

ďż˝ 2j
V
2

+ 𝑔𝑔z�j �

�𝑠𝑠𝑠𝑠𝑠𝑠 2
�22
�12
V
V
V
𝛿𝛿 �m𝑠𝑠𝑠𝑠𝑠𝑠 . �u� 𝑠𝑠𝑠𝑠𝑠𝑠 +
�� = 𝑚𝑚sys,2 �u� 2 + � − 𝑚𝑚sys,1 �u� 1 + �
2
2
2

(4-14)

Replacing result of (4-14) into Equation (4-13):
δW𝑜𝑜𝑜𝑜𝑜𝑜

�12
�22
�𝑜𝑜𝑜𝑜𝑜𝑜 2
V
V
V
ďż˝
= 𝑚𝑚sys,1 �u� 1 + � − 𝑚𝑚sys,2 �u� 2 + � − 𝛿𝛿m𝑜𝑜𝑜𝑜𝑜𝑜 (h𝑜𝑜𝑜𝑜𝑜𝑜 +
)
2
2
2

(4-15)

Now, the conservation of mass principle is applied to the fixed control volume:
𝑚𝑚̇in − 𝑚𝑚̇𝑜𝑜𝑜𝑜𝑜𝑜 =

yields:

𝑑𝑑𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠
𝑑𝑑𝑑𝑑

With the condition-1 that no incoming flow entering the control volume 𝑚𝑚̇in = 0, it
−𝑚𝑚̇𝑜𝑜𝑜𝑜𝑜𝑜 =

𝑑𝑑𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠
𝑑𝑑𝑑𝑑

Expressing this instantaneous equation for a finite time δt:

Taking into account that:

−

𝛿𝛿𝑚𝑚𝑜𝑜𝑜𝑜𝑜𝑜 𝛿𝛿𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠
=
𝛿𝛿𝛿𝛿
𝛿𝛿𝛿𝛿

(4-16)

𝛿𝛿𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠 = 𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠,𝑗𝑗+1 − 𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠,𝑗𝑗 = 𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠,2 − 𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠,1

Replacing in (4-16):

−𝛿𝛿𝑚𝑚𝑜𝑜𝑜𝑜𝑜𝑜 = 𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠,2 − 𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠,1

(4-17)

Replacing (4-17) into (4-15):
δW𝑜𝑜𝑜𝑜𝑜𝑜

(4-18)
�12
�22
�𝑜𝑜𝑜𝑜𝑜𝑜 2
V
V
V
ďż˝
= 𝑚𝑚sys,1 �u� 1 + � − 𝑚𝑚sys,2 �u� 2 + � + (𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠,2 − 𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠,1 )(h𝑜𝑜𝑜𝑜𝑜𝑜 +
)
2
2
2

Equation (4-18) is the energy balance for a finite time “δt” during the unsteady expansion

process. To describe the progress of the process, the criteria of “Finite Stages” for intervals of Δt
time, is applied by using the concept in Figure 4-2.

44

�isentropic
T
Sďż˝ isentropic

Figure 4-2: Unsteady expansion process for stage “i”

Considering an arbitrary finite stage “i” and dropping the index “sys” to simplify the
expression, equation (4-18) becomes:
δW𝑜𝑜𝑜𝑜𝑜𝑜,𝑖𝑖

2
2
�1,i
�2,i
�𝑜𝑜𝑜𝑜𝑜𝑜,𝑖𝑖 2
V
V
V
ďż˝
= 𝑚𝑚1,i �u� 1,i +
� − 𝑚𝑚2,i �u� 2,i +
� + (𝑚𝑚2,𝑖𝑖 − 𝑚𝑚1,𝑖𝑖 )(h𝑜𝑜𝑜𝑜𝑜𝑜,𝑖𝑖 +
)
2
2
2

45

(4-19)

Describing every time-step of the unsteady expansion process from the beginning to the
end:
stage

Finite work done

“i”
1
2

𝛿𝛿𝛿𝛿𝑜𝑜𝑜𝑜𝑜𝑜,1 = 𝑚𝑚1,1 �u� 1,1 +

2
2
2
�1,1
�2,1
�𝑜𝑜𝑜𝑜𝑜𝑜,1
V
V
V
� − 𝑚𝑚2,1 �u� 2,1 +
� + (𝑚𝑚2,1 − 𝑚𝑚1,1 )(h� 𝑜𝑜𝑜𝑜𝑜𝑜,1 +
)
2
2
2

𝛿𝛿𝛿𝛿𝑜𝑜𝑜𝑜𝑜𝑜,3 = 𝑚𝑚1,3 �u� 1,3 +

2
2
2
�1,3
�2,3
�𝑜𝑜𝑜𝑜𝑜𝑜,3
V
V
V
� − 𝑚𝑚2,3 �u� 2,3 +
� + (𝑚𝑚2,3 − 𝑚𝑚1,3 )(h� 𝑜𝑜𝑜𝑜𝑜𝑜,3 +
)
2
2
2

𝛿𝛿𝛿𝛿𝑜𝑜𝑜𝑜𝑜𝑜,2 = 𝑚𝑚1,2 �u� 1,2 +

3
4

𝛿𝛿𝛿𝛿𝑜𝑜𝑜𝑜𝑜𝑜,4

5
.

𝛿𝛿𝛿𝛿𝑜𝑜𝑜𝑜𝑜𝑜,5

2
2
2
�1,2
�2,2
�𝑜𝑜𝑜𝑜𝑜𝑜,2
V
V
V
� − 𝑚𝑚2,2 �u� 2,2 +
� + (𝑚𝑚2,2 − 𝑚𝑚1,2 )(h� 𝑜𝑜𝑜𝑜𝑜𝑜,2 +
)
2
2
2

2
2
2
�1,4
�2,4
�𝑜𝑜𝑜𝑜𝑜𝑜,4
V
V
V
ďż˝
= 𝑚𝑚1,4 �u� 1,4 +
� − 𝑚𝑚2,4 �u� 2,4 +
� + (𝑚𝑚2,4 − 𝑚𝑚1,4 )(h𝑜𝑜𝑜𝑜𝑜𝑜,4 +
)
2
2
2
2
2
2
�1,5
�2,5
�𝑜𝑜𝑜𝑜𝑜𝑜,5
V
V
V
ďż˝
= 𝑚𝑚1,5 �u� 1,5 +
� − 𝑚𝑚2,5 �u� 2,5 +
� + (𝑚𝑚2,5 − 𝑚𝑚1,5 )(h𝑜𝑜𝑜𝑜𝑜𝑜,5 +
)
2
2
2

.
.
n-1

n

𝛿𝛿𝛿𝛿𝑜𝑜𝑜𝑜𝑜𝑜,𝑛𝑛−1

𝛿𝛿𝛿𝛿𝑜𝑜𝑜𝑜𝑜𝑜,𝑛𝑛

2
2
�1,𝑛𝑛−1
�2,𝑛𝑛−1
V
V
= 𝑚𝑚1,𝑛𝑛−1 �u� 1,𝑛𝑛−1 +
� − 𝑚𝑚2,𝑛𝑛−1 �u� 2,𝑛𝑛−1 +
� + (𝑚𝑚2,𝑛𝑛−1
2
2

2
�𝑜𝑜𝑜𝑜𝑜𝑜,𝑛𝑛−1
V
)
2
2
2
2
�1,n
�2,n
�𝑜𝑜𝑜𝑜𝑜𝑜,𝑛𝑛
V
V
V
ďż˝
= 𝑚𝑚1,n �u� 1,n +
� − 𝑚𝑚2,n �u� 2,n +
� + (𝑚𝑚2,𝑛𝑛 − 𝑚𝑚1,𝑛𝑛 )(h𝑜𝑜𝑜𝑜𝑜𝑜,𝑛𝑛 +
)
2
2
2

− 𝑚𝑚1,𝑛𝑛−1 )(h� 𝑜𝑜𝑜𝑜𝑜𝑜,𝑛𝑛−1 +

Table 4-1: Detailed description of the unsteady expansion process at each stage “i”

Matching parameters between two consecutives stages: the final state of the system
during stages “i” will have the same conditions for the initial state of the system in the following
stage “i+1”. For instance, if stage 1 and 2 are matched, it yields:
𝑚𝑚2,1 �u� 2,1 +

2
2
�2,1
�1,2
V
V
� = 𝑚𝑚1,2 �u� 1,2 +
ďż˝
2
2

46

Because: 𝑚𝑚2,1 = 𝑚𝑚1,2 ;

2
2
�2,1
�1,2
uďż˝ 2,1 = uďż˝ 1,2 ; V
=V
, and the same condition is applied

for process 2 and 3, 3 and 4 … n-1 and n processes.

𝑛𝑛

Then the summation of all the 𝛿𝛿𝛿𝛿𝑜𝑜𝑜𝑜𝑜𝑜,𝑖𝑖 in Table 4-1 yields:

2
2
�1,1
�2,n
V
V
� − 𝑚𝑚2,n �u� 2,n +
ďż˝
2
2
2
2
�𝑜𝑜𝑜𝑜𝑜𝑜,1
�𝑜𝑜𝑜𝑜𝑜𝑜,2
V
V
+ �𝑚𝑚2,1 − 𝑚𝑚1,1 � �h� 𝑜𝑜𝑜𝑜𝑜𝑜,1 +
� + �𝑚𝑚2,2 − 𝑚𝑚1,2 � �h� 𝑜𝑜𝑜𝑜𝑜𝑜,2 +
ďż˝
2
2
2
2
�𝑜𝑜𝑜𝑜𝑜𝑜,3
�𝑜𝑜𝑜𝑜𝑜𝑜,4
V
V
ďż˝
ďż˝
+ �𝑚𝑚2,3 − 𝑚𝑚1,3 � �h𝑜𝑜𝑜𝑜𝑜𝑜,3 +
� + �𝑚𝑚2,4 − 𝑚𝑚1,4 � �h𝑜𝑜𝑜𝑜𝑜𝑜,4 +
ďż˝
2
2
2
�𝑜𝑜𝑜𝑜𝑜𝑜,5
V
ďż˝
+ �𝑚𝑚2,5 − 𝑚𝑚1,5 � �h𝑜𝑜𝑜𝑜𝑜𝑜,5 +
� … … . … + (𝑚𝑚2,𝑛𝑛−1
2
2
2
�𝑜𝑜𝑜𝑜𝑜𝑜,𝑛𝑛−1
�𝑜𝑜𝑜𝑜𝑜𝑜,𝑛𝑛
V
V
− 𝑚𝑚1,𝑛𝑛−1 )(h� 𝑜𝑜𝑜𝑜𝑜𝑜,𝑛𝑛−1 +
) + (𝑚𝑚2,𝑛𝑛 − 𝑚𝑚1,𝑛𝑛 )(h� 𝑜𝑜𝑜𝑜𝑜𝑜,𝑛𝑛 +
)
2
2

�(𝛿𝛿𝛿𝛿𝑜𝑜𝑜𝑜𝑜𝑜,𝑖𝑖 ) = 𝑚𝑚1,1 �u� 1,1 +
𝑖𝑖=1

(4-20)

The mass of the system is expressed in terms of volume-weighted average:
𝑚𝑚1,i = ρ�1,i V

𝑚𝑚2,i = ρ�2,i V

(4-21)

The volume V of the channel is constant at all stages. Replacing (4-21) into (4-20):
𝑛𝑛

𝑊𝑊𝑜𝑜𝑜𝑜𝑜𝑜 = �(𝛿𝛿𝛿𝛿𝑜𝑜𝑜𝑜𝑜𝑜,𝑖𝑖 )
𝑖𝑖=1

= ρ�1,1 V �u� 1,1 +

2
2
2
�1,1
�2,n
�𝑜𝑜𝑜𝑜𝑜𝑜,1
V
V
V
� − ρ�2,n V �u� 2,n +
� + V�ρ�2,1 − ρ�1,1 � �h� 𝑜𝑜𝑜𝑜𝑜𝑜,1 +
ďż˝
2
2
2

+ V�ρ�2,2 − ρ�1,2 � �h� 𝑜𝑜𝑜𝑜𝑜𝑜,2 +
+ V�ρ�2,4 − ρ�1,4 � �h� 𝑜𝑜𝑜𝑜𝑜𝑜,4 +

2
2
�𝑜𝑜𝑜𝑜𝑜𝑜,2
�𝑜𝑜𝑜𝑜𝑜𝑜,3
V
V
ďż˝
� + V�ρ�2,3 − ρ�1,3 � �h𝑜𝑜𝑜𝑜𝑜𝑜,3 +
ďż˝
2
2

2
2
�𝑜𝑜𝑜𝑜𝑜𝑜,5
�𝑜𝑜𝑜𝑜𝑜𝑜,4
V
V
ďż˝
� + V�ρ�2,5 − ρ�1,5 � �h𝑜𝑜𝑜𝑜𝑜𝑜,5 +
�…….…
2
2

+ V(ρ�2,𝑛𝑛−1 − ρ�1,𝑛𝑛−1 )(h� 𝑜𝑜𝑜𝑜𝑜𝑜,𝑛𝑛−1 +

2
2
�𝑜𝑜𝑜𝑜𝑜𝑜,𝑛𝑛−1
�𝑜𝑜𝑜𝑜𝑜𝑜,𝑛𝑛
V
V
) + V(ρ�2,𝑛𝑛 − ρ�1,𝑛𝑛 )(h� 𝑜𝑜𝑜𝑜𝑜𝑜,𝑛𝑛 +
)
2
2

The index notation of the above expression will be translated into the flow time notation
as follows:
47

(a) Index “0” corresponds to the initial state of stage 1 and “1” to the end state of stage 1.
(b) Index “1” correspond to the initial state of stage 2 and “2” to the end state of stage 2
(c) Index “2” correspond to the initial state of stage 3 and “3” to the end state of stage 3, and so
on.
Condition-2:
�𝑜𝑜𝑜𝑜𝑜𝑜,𝑖𝑖 ) are regarded as constant values during
Parameters at the outlet boundary (h� 𝑜𝑜𝑜𝑜𝑜𝑜,𝑖𝑖 , V

each finite stage “i” but varies at each finite stage.
𝑊𝑊𝑜𝑜𝑜𝑜𝑜𝑜

2
�02
�n2
�𝑜𝑜𝑜𝑜𝑜𝑜,1
V
V
V
ďż˝
= V �ρ�0 �u� 0 + � − ρ�n �u� n + � + (ρ�1 − ρ�0 ) �h𝑜𝑜𝑜𝑜𝑜𝑜,1 +
ďż˝
2
2
2

+ (ρ�2 − ρ�1 ) �h� 𝑜𝑜𝑜𝑜𝑜𝑜,2 +

+ (ρ�4 − ρ�3 ) �h� 𝑜𝑜𝑜𝑜𝑜𝑜,4 +

(4-22)

2
2
�𝑜𝑜𝑜𝑜𝑜𝑜,2
�𝑜𝑜𝑜𝑜𝑜𝑜,3
V
V
ďż˝
� + (ρ�3 − ρ�2 ) �h𝑜𝑜𝑜𝑜𝑜𝑜,3 +
ďż˝
2
2

2
2
�𝑜𝑜𝑜𝑜𝑜𝑜,5
�𝑜𝑜𝑜𝑜𝑜𝑜,4
V
V
ďż˝
� + (ρ�5 − ρ�4 ) �h𝑜𝑜𝑜𝑜𝑜𝑜,5 +
�…….…
2
2

+ (ρ�𝑛𝑛−1 − ρ�𝑛𝑛−2 ) �h� 𝑜𝑜𝑜𝑜𝑜𝑜,𝑛𝑛−1 +

+ (ρ�𝑛𝑛 − ρ�𝑛𝑛−1 ) �h� 𝑜𝑜𝑜𝑜𝑜𝑜,𝑛𝑛 +

2
�𝑜𝑜𝑜𝑜𝑜𝑜,𝑛𝑛−1
V
ďż˝
2

2
�𝑜𝑜𝑜𝑜𝑜𝑜,𝑛𝑛
V
��
2

Series Taylor expansion is applied for the density parameter to simplify Equation (4-22):
ďż˝
∂ρ

ďż˝
∂2 ρ

ρ�(t+∆t) = ρ�t + ( ∂t )𝑡𝑡 ∆𝑡𝑡 + ( ∂t2 )𝑡𝑡

(∆𝑡𝑡)2
2

ďż˝
∂3 ρ

+ ( ∂t3 )𝑡𝑡

(∆𝑡𝑡)3
6

ďż˝
∂n ρ

+ … + ( ∂tn )𝑡𝑡

(∆𝑡𝑡)𝑛𝑛
𝑛𝑛!

+⋯

The volume-weighted average density will be considered for a second order-accurate, and

higher order terms will be a truncation error:
ďż˝
∂ρ

ďż˝
∂2 ρ

ρ�(t+∆t) = ρ�t + ( ∂t )𝑡𝑡 ∆𝑡𝑡 + ( ∂t2 )𝑡𝑡

(∆𝑡𝑡)2
2

48

Translating the stage “i” in terms of time step:
Time
step
0

At t0=0

1

At t1 = t0 +Δt= 0 +Δt =1Δt

2

At t2 = t1+ Δt= Δt+Δt=2Δt

3

At t3 = t2+ Δt= 2Δt+Δt= 3Δt

ρ�(t0 =0) = ρ�0

ρ�Δt = ρ�(0+∆t) = ρ�0 + (

∂ρ�
∂2 ρ� (∆𝑡𝑡)2
)0 ∆𝑡𝑡 + ( 2 )0
∂t
∂t
2

ρ�2Δt = ρ�(Δt+∆t) = ρ�Δt + (
ρ�3Δt = ρ�(2Δt+∆t)

∂ρ�
∂2 ρ� (∆𝑡𝑡)2
)Δt ∆𝑡𝑡 + ( 2 )Δt
∂t
∂t
2

= ρ�2Δt + (

4

+(

At t4 = t3+ Δt= 3Δt+Δt= 4Δt

ρ�4Δt = ρ�(3Δt+∆t)

∂2 ρ�
(∆𝑡𝑡)2
)
∂t 2 2Δt 2

= ρ�3Δt + (

…

…..

….

…

…..

….

n

At tn = t(n-1)+ Δt= (n-1)Δt+ Δt=
nΔt

∂ρ�
) ∆𝑡𝑡
∂t 2Δt

∂ρ�
) ∆𝑡𝑡
∂t 3Δt

∂2 ρ�
(∆𝑡𝑡)2
+ ( 2 )3Δt
∂t
2
ρ�((n−1)Δt+∆t) = ρ�(n−1)Δt + (
+(

∂ρ�
)
∆𝑡𝑡
∂t (n−1)Δt

∂2 ρ�
(∆𝑡𝑡)2
)
∂t 2 (n−1)Δt 2

Table 4-2: Development of the ρ� according to Series Taylor expansion for each ti

49

i
1
2
3
4

Expressions from Table 4-2 are replaced into the ρ� variable of equation (4-22):

ďż˝
∂2 ρ

ďż˝
∂ρ

ďż˝
∂2 ρ

2

ďż˝
∂ρ

ďż˝
∂2 ρ

ďż˝
∂ρ

ďż˝
∂2 ρ

ďż˝
∂ρ

) − ρ�0 = ( ∂t )0 ∆𝑡𝑡 + ( ∂t2 )0

(∆𝑡𝑡)2
2

ρ�3 − ρ�2 = ( ρ�2Δt + ( ∂t )2Δt ∆𝑡𝑡 + ( ∂t2 )2Δt
ρ�4 − ρ�3 = ( ρ�3Δt + ( ∂t )3Δt ∆𝑡𝑡 + ( ∂t2 )3Δt

ďż˝
∂ρ

ďż˝
∂ρ

ďż˝
∂2 ρ

(∆𝑡𝑡)2
2

(∆𝑡𝑡)2
2

ďż˝
∂ρ

ďż˝
∂2 ρ

)
∆𝑡𝑡 + ( ∂t2 )(n−1)Δt
∂t (n−1)Δt

ďż˝
∂2 ρ

(∆𝑡𝑡)2
2

ďż˝
∂ρ

ďż˝
∂2 ρ

(∆𝑡𝑡)2
2

Table 4-3: Detailed development of ρ� for each stage “i”

50

(∆𝑡𝑡)2
2

ďż˝
∂2 ρ
ďż˝
∂2 ρ

) − ρ�3Δt = ( ∂t )3Δt ∆𝑡𝑡 + ( ∂t2 )3Δt

ρ�𝑛𝑛 − ρ�𝑛𝑛−1 = ( ρ�(n−1)Δt + ( ∂t )(n−1)Δt ∆𝑡𝑡 + ( ∂t2 )(n−1)Δt
ďż˝
∂ρ

2

) − ρ�2Δt = ( ∂t )2Δt ∆𝑡𝑡 + ( ∂t2 )2Δt

ďż˝
∂2 ρ

ďż˝
∂ρ

ďż˝
∂ρ

(∆𝑡𝑡)2

) − ρ�Δt = ( ∂t )Δt ∆𝑡𝑡 + ( ∂t2 )Δt

ρ�𝑛𝑛−1 − ρ�𝑛𝑛−2 = ( ρ�(n−2)Δt + ( ∂t )(n−2)Δt ∆𝑡𝑡 + ( ∂t2 )(n−2)Δt
( ∂t )(n−2)Δt ∆𝑡𝑡 + ( ∂t2 )(n−2)Δt

n

(∆𝑡𝑡)2

ρ�2 − ρ�1 = ( ρ�Δt + ( ∂t )Δt ∆𝑡𝑡 + ( ∂t2 )Δt

…
n-1

ďż˝
∂2 ρ

ďż˝
∂ρ

ρ�1 − ρ�0 = ( ρ�0 + ( ∂t )0 ∆𝑡𝑡 + ( ∂t2 )0

(∆𝑡𝑡)2
2

(∆𝑡𝑡)2
2

) − ρ�(n−2)Δt =

) − ρ�(n−1)Δt =

(∆𝑡𝑡)2
2

(∆𝑡𝑡)2
2

The next step is to insert results from Table 4-3 into Equation (4-22).
𝑊𝑊𝑜𝑜𝑜𝑜𝑜𝑜

2
�02
�n2
�𝑜𝑜𝑜𝑜𝑜𝑜,1
V
V
∂ρ�
∂2 ρ� (∆𝑡𝑡)2
V
ďż˝
= V �ρ�0 �u� 0 + � − ρ�n �u� n + � + (( )0 ∆𝑡𝑡 + ( 2 )0
) �h𝑜𝑜𝑜𝑜𝑜𝑜,1 +
ďż˝
∂t
2
2
∂t
2
2
2
�𝑜𝑜𝑜𝑜𝑜𝑜,2
∂ρ�
∂2 ρ� (∆𝑡𝑡)2
V
∂ρ�
ďż˝
+ ( ( )Δt ∆𝑡𝑡 + ( 2 )Δt
) �h𝑜𝑜𝑜𝑜𝑜𝑜,2 +
� + ( ( )2Δt ∆𝑡𝑡
∂t
∂t
∂t
2
2
2
�𝑜𝑜𝑜𝑜𝑜𝑜,3
∂2 ρ�
(∆𝑡𝑡)2
V
∂ρ�
ďż˝
+ ( 2 )2Δt
) �h𝑜𝑜𝑜𝑜𝑜𝑜,3 +
� + ( ( )3Δt ∆𝑡𝑡
∂t
2
2
∂t

2
�𝑜𝑜𝑜𝑜𝑜𝑜,4
∂2 ρ�
(∆𝑡𝑡)2
V
∂ρ�
ďż˝
+ ( 2 )3Δt
) �h𝑜𝑜𝑜𝑜𝑜𝑜,4 +
� + ( ( )4Δt ∆𝑡𝑡
∂t
2
2
∂t

2
�𝑜𝑜𝑜𝑜𝑜𝑜,5
V
∂2 ρ�
(∆𝑡𝑡)2
∂ρ�
ďż˝
+ ( 2 )4Δt
) �h𝑜𝑜𝑜𝑜𝑜𝑜,5 +
� … … . … + ( ( )(n−2)Δt ∆𝑡𝑡
2
2
∂t
∂t
2
�𝑜𝑜𝑜𝑜𝑜𝑜,𝑛𝑛−1
V
∂ρ�
∂2 ρ�
(∆𝑡𝑡)2
ďż˝
) �h𝑜𝑜𝑜𝑜𝑜𝑜,𝑛𝑛−1 +
� + ( )(n−1)Δt ∆𝑡𝑡
+ ( 2 )(n−2)Δt
2
2
∂t
∂t
2
�𝑜𝑜𝑜𝑜𝑜𝑜,𝑛𝑛
∂2 ρ�
(∆𝑡𝑡)2
V
ďż˝
+ ( 2 )(n−1)Δt
) �h𝑜𝑜𝑜𝑜𝑜𝑜,𝑛𝑛 +
��
2
2
∂t

Gathering terms as a summation:
𝑊𝑊𝑜𝑜𝑜𝑜𝑜𝑜

�02
�n2
V
V
= Vρ�0 �u� 0 + � − Vρ�n �u� n + �
2
2
𝑛𝑛

+ V ∆𝑡𝑡 { �[(

𝑊𝑊𝑜𝑜𝑜𝑜𝑜𝑜 = m0 �u� 0 +

𝑖𝑖=1

2
�𝑜𝑜𝑜𝑜𝑜𝑜,𝑖𝑖
V
∂ρ�
∂2 ρ�
∆𝑡𝑡
)(𝑖𝑖−1) + ( 2 )(𝑖𝑖−1) ( ) ][h� 𝑜𝑜𝑜𝑜𝑜𝑜,𝑖𝑖 +
]}
2
∂t
2
∂t

�02
�n2
V
V
� − mn �u� n + �
2
2

(4-23)

𝑛𝑛

1
∂ρ�
∂2 ρ�
∆𝑡𝑡
2
�𝑜𝑜𝑜𝑜𝑜𝑜,𝑖𝑖
+ ( ) V ∆𝑡𝑡 { �[( )(𝑖𝑖−1) + ( 2 )(𝑖𝑖−1) ( ) ][2 h� 𝑜𝑜𝑜𝑜𝑜𝑜,𝑖𝑖 + V
]}
2
∂t
∂t
2
𝑖𝑖=1

Equation (4-23) is the approximate general equation to calculate work extraction during
the unsteady expansion process based on conditions -1 and -2.

51

4.2.

Analysis of the approximate general equation for extracted work
Equation (4-23) is an approximate mathematical expression derived from the energy

balance equation with conditions -1 and -2 to calculate the work of the unsteady expansion
process.
The right hand side of Equation (4-23) has three terms: the first and second terms
represent the total energy of the fluid within the control volume at the initial and final state of the
process, respectively. The third term expresses the total fluid energy leaving the channel thru the
outlet, depicted as a summation term dependent of time and therefore, this third term represents
the unsteadiness of the expansion process. Equation (4-23) demonstrates the physics as follows:
ďż˝
of

𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊
𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒
𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒
𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑡𝑡ℎ𝑟𝑟𝑟𝑟
ďż˝=ďż˝
�−�
ďż˝+ďż˝
ďż˝
𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒
𝑎𝑎𝑎𝑎 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠
𝑎𝑎𝑎𝑎 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑠𝑠𝑡𝑡𝑎𝑎𝑎𝑎𝑎𝑎
𝑐𝑐ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜

ďż˝ 20
V
2

(4-24)

When Total energy is mentioned, it primarily implies the uďż˝ value because the magnitude

is small compared to uďż˝ . The work extracted according to Equation (4-24) is mainly

dependent on flowing energy leaving the channel and the final state because the total energy at
initial state is a fixed condition. The uďż˝ value at final state must be the lowest for maximum work
extraction and occurs when the process trajectory follows a constant entropy line obtaining
u� 𝑛𝑛,𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 value for the fluid within the control volume (Figure 4-2).

The flowing energy is negative due to the decreasing rate of density change, diminishing

the net value of the first two terms, and the lower this rate, the greater the generation of torque.
Thus, maximizing work extraction will depend on how this third term is modulated by the
geometry of the channel.

52

4.2.1. The flowing energy in the summation term
The summation in Equation (4-23) contains constant parameters (V and Δt) and the time�𝑜𝑜𝑜𝑜𝑜𝑜,𝑖𝑖 ). The rotational speed (ωi) is
dependent variables (the rate of ρ� change, h� 𝑜𝑜𝑜𝑜𝑜𝑜,𝑖𝑖 and V

considered constant in the analysis as in the actual operation of the engine.
ďż˝
∂ρ

ďż˝
∂2 ρ

The first and second partial derivatives of the density function, � ∂t � and � ∂t2 �, play an

influential role in the value and symbol of the summation term. The first derivative captures the
slope of the curve and the second derivative accounts for the curvature, but the focus will be on
the first derivative because the second derivative is a higher order term.
During the actual unsteady expansion process, the density decreases throughout the
whole process (emptying process). Thus, the rate of density change in the channel has a negative
value ranging from –fv <

ďż˝
∂ρ
∂t

≤ 0. The physical effect refers to the speed of fluid mass being

expelled from the channel. Higher rates produce bigger values of

ďż˝
∂ρ
∂t

, thus higher flowing

energy diminishes the work generated. For this reason, fast discharging process can have a
negative effect on torque generation.

4.3.

Unsteady generated work for isentropic process
The isentropic process produces maximum values of torque, and this section investigates

the influence of the time dependent variables under isentropic conditions.
The unsteady expansion is a complex process in the operation of an actual engine;
inheriting the effects of compression waves, created during the combustion process, propagating
back and forth inside the channel. The expansion process starts with a gradual opening of the
outlet port, expelling the fluid from the channel with a gradual increase in the fluid’s relative

53

velocity. This opening creates an expansion wave that spreads toward the channel inlet. During
this period, the magnitude of the absolute velocity of the gases at the outlet reaches a maximum
value, and then decreases. Subsequently, the inlet port opens and fresh mixture starts to fill the
channel again. An overlapping period happens between the end of the expansion and beginning
of the inlet processes. The inlet process benefits from the effect of gases leaving the channel,
which draw the incoming fluid. The outlet port closes when most of the gases have already left
the channel, and is the end of the Unsteady Expansion Process.
A number of constraints are applied to simplify the complexity of this process: there is no
influence from the combustion process and the relative gas velocity within the channel is zero (at
rest) until the outlet port opens instantly and the fluid accelerates through the channel toward the
outlet. At the final step of the process, the mass-weighted average value of fluid velocity reduces
to the channel velocity, and the volume-weighted average value of pressure within the channel is
equalized with surroundings.
At every finite stage, the flow expands to the same mass-weighted average value of the
�𝑜𝑜𝑜𝑜𝑜𝑜,𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ) at the channel outlet. Thus, it yields:
isentropic temperature (T
Then:

𝑇𝑇�𝑜𝑜𝑜𝑜𝑜𝑜,1 = 𝑇𝑇�𝑜𝑜𝑜𝑜𝑜𝑜,2 = 𝑇𝑇�𝑜𝑜𝑜𝑜𝑜𝑜,3 = 𝑇𝑇�𝑜𝑜𝑜𝑜𝑜𝑜,4 = ⋯ = 𝑇𝑇�𝑜𝑜𝑜𝑜𝑜𝑜,𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖

(4-25)

ℎ�𝑜𝑜𝑜𝑜𝑜𝑜,1 = ℎ�𝑜𝑜𝑜𝑜𝑜𝑜,2 = ℎ�𝑜𝑜𝑜𝑜𝑜𝑜,3 = ℎ�𝑜𝑜𝑜𝑜𝑜𝑜,4 = ⋯ = ℎ�𝑜𝑜𝑜𝑜𝑜𝑜,𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖

(4-26)

To investigate the influence of density on the torque generation some idealizations are
applied. The fluid in this idealization is air and behaves as an ideal gas. A density function is
built to imitate how the volume-weighted average value of density changes within the channel
during the process. This function depicts the channel’s instant opening and the density's
decreasing rate. This rate value starts at zero, falls to a negative value, and continues decreasing.

54

It reaches the lowest negative value (called inflection point at the density curve), and begins
increasing until reaching zero value again when the pressure inside the channel is equalized to
the surrounding pressure and thus, the outlet relative velocity is zero.
The summation term of Equation (4-23) appears as a function of ∆𝑡𝑡 as well. This

parameter depends on the number of finite stages according to Equation (4-27):
∆𝑡𝑡 =

𝑡𝑡2𝑠𝑠
𝑛𝑛

(4-27)

Based on this statement, several calculations were performed using different number of
finite stages in Equation (4-23). Results in Figure 4-3 indicate the unsteady work value
converges at around 1,000 finite stages. Thus, all calculations are considered to have more than
5,000 finite stages.
To determine any impact on the torque generation based on Equation (4-23), two options
are considered in the density function variation:
(a) Same duration of expansion process but the inflection point of the density function increases
(Figure 4-4).
(b) Duration of expansion time increases but the inflection point remains constant and occurs in
the middle of each duration time (Figure 4-5).
For both options of the density function is plotted the density, magnitude of the outlet
velocity and the rate of density change (Figure 4-4and Figure 4-5).

55

Figure 4-3: Convergence of the Unsteady Isentropic Work as a function of Number of finite
stages at PR=7 with 10,000 rpm.
Shifting the position of the inflection point in the density function toward the right causes
both a concentration of higher outlet velocities and rate of density change toward the right
(Figure 4-4 (b) and (c)).
The curves of the rate of density change show the earlier and later inflexion points
produce the highest negative values (Figure 4-4 (c)). These results predict trends of channel
outlet velocity and rate of density change if the volume-weighted average results of density from
numerical simulation produce similar plots to Figure 4-4(a).

56

Inflection point
(tx) moves
toward right

(a)

(b)

(c)
Figure 4-4: Density (a), magnitude of outlet velocity (b) and dρ/dt (c) with several inflectiontime during the unsteady expansion process (PR=7 and 10,000 rpm)
The second option considers the influence of the total duration time of the unsteady
expansion process on work generation (Figure 4-5 and Figure 4-6). The extension of the
expansion process duration causes lower rates of density change and thus, lower channel outlet
velocities. According to Equation (4-23), the unsteady work is the result of net positive
contribution of both first terms, and by the negative contribution of the summation term. This
last term is negative due to the decreasing rate of density change (first partial derivative with
respect to time). With shorter expansion duration, the rate of density change increases, and the
57

summation term also increases and thus, reduces the unsteady isentropic generated work. In
addition to this change, the summation term contains the outlet velocity which increases as well,
having a greater effect of this summation term in the unsteady isentropic generated work. The
increase in the channel outlet velocity can be interpreted as fast outgoing of kinetic energy that
was not exploited in the generation of work.

Duration of
expansion process
increases

(a)

(b)

(c)
Figure 4-5: Density (a), magnitude of outlet velocity (b), and dρ/dt (c), with several durationtimes of the unsteady expansion process (PR=7 and 10,000 rpm)
The initial and final conditions (the first two terms in Equation (4-23)) become fixed in
the isentropic process, then, the work generated will be higher as long as the summation term
58

becomes smaller. Therefore, extending the duration of the unsteady expansion process appears as
a controlling parameter to maximize the unsteady work extraction. This conclusion is consistent
with that in Chapter 2, but channel side areas are not included in these findings because
thermodynamic energy analysis does not provide evidence of the influence on the unsteady work
due to the area of pressure and suction sides.
In Figure 4-6 the unsteady isentropic generated work is plotted at different pressure ratios
(PR). Each point indicates the maximum isentropic work expected (y-coordinate) according to
the process duration, respectively (x- coordinate). The values of t1, t2 and t3 are the threshold
flow times, and following that point the production of unsteady isentropic work starts to
decrease. The graph depicts the increase of expansion duration produces the increase of work
extraction in all plots.
MAX isent Unst Work
Potential
unsteady work
PR=7

Δt3
uW3
Potential
unsteady work
PR=5

Δt2
uW2

Potential
unsteady work
PR=3

Δt1
uW1

tCFD

t1

t2

t3

Figure 4-6: Unsteady Isentropic Work vs time at several PR= 3, 5 and 7, discharging to
surrounding environment pressure of 101325 Pa.

59

Table 4-4 provides a summary of the actual unsteady generated work results from
numerical simulations. The corresponding unsteady isentropic generated work values are
determined in Figure 4-6 as uW1, uW2 and uW3 at PR= 3, 5 and 7, respectively, based on the
duration of the actual extracted work. The graph indicates additional unsteady work can be
extracted if the expansion process duration is extended. In addition, Table 4-4 provides
expansion efficiencies and indicates work generation at lower pressure ratio is more efficient.
The unsteady expansion efficiencies were calculated based on two criteria: same duration time
and maximum unsteady isentropic work.
PR

Actual UnstW

UnstW_isent

(numerical

Duration

Ρexp

Ρexp

(actual process)

(same duration)

(max isent-work)

simulation)
[J]

[J]

[ms]

[%]

[%]

3

0.40

0.90

0.20897

44.6

29.2

5

0.73

2.62

0.21057

27.9

17.4

7

0.97

5.31

0.21717

18.3

12.2

Table 4-4: Results for PR=3, 5 and 7 (“same duration” refers to the isentropic unsteady work
produced according to the duration of the actual work extracted. “maximum isentropic work”
refers to the maximum value at different duration time than the actual work extracted ).
Expansion efficiency based on the energy approach does not distinguish between losses that
generate entropy and potential energy to produce work in subsequent stages. Exergy approach
considers all aspects.

60

CHAPTER 5.

ROLE OF PRESSURE AND FORCE IN TORQUE
GENERATION

This chapter focuses on the role of pressure and force in the generation of torque. A
constant cross-section curved channel is chosen for an unsteady expansion process, at instant and
gradual openings, both with rotational velocity of 10,000 rpm. The torque coefficient through the
flow time is plotted. Cm is calculated based on equation:
𝐶𝐶𝑚𝑚 =

𝑇𝑇

1
2
2 𝜌𝜌𝑟𝑟𝑟𝑟𝑟𝑟 𝑉𝑉𝑟𝑟𝑟𝑟𝑟𝑟 𝐴𝐴𝑟𝑟𝑟𝑟𝑟𝑟 𝐿𝐿𝑟𝑟𝑟𝑟𝑟𝑟

(5-1)

Torque value (T) is calculated based on the cross product summation of pressure and
viscous force vectors together with position vector, with respect to the center of the channel
rotation.
For the purpose of analysis, the focus will be on the effect of pressure forces in torque
generation. Viscous forces are very small compared with pressure forces, and are negligible in
this study. Therefore, results are not affected when torque due to pressure forces is compared
with cm or torque history values provided by CFD simulation.

5.1.

Unsteady Expansion Process at Instant Opening
5.1.1. Pressure distribution on channel walls
A step by step description of the unsteady expansion in the channel during torque

generation is performed. According to Figure 5-1, 9 points at different flow time locations are
selected to analyze this case.
In Figure 5-2, Figure 5-7, and Figure 5-8, the pressure distribution on the pressure and
suction side of the channel are plotted for each point selected from Figure 5-1, the abscissa on

61

the graphs refers to radial coordinate locations. Additionally, the circumferential average of the
velocity magnitude in radial coordinates was included in each graph. On the right hand side of
graphs were plotted line-contours of pressure, and radial coordinates are indicated.
Flow time when
expansion wave
reaches the inlet
wall

Figure 5-1: Torque coefficient (cm) history of a constant-cross section curved channel for instant
opening
Figure 5-1 shows the increasing trend of the torque coefficient (points 1-3) reaching a
maximum value at point 3. The slope of the curve plotted (approximately a line) by the three
initial points has a high steep value due to the instant opening condition imposed during the
simulation. A less steep trend will be observed for a gradual opening situation. The instant
opening forces the fluid to leave the channel in a radial direction. During this initial time (Figure
5-2), the head of the expansion wave at point 1 is aligned approximately with the circumferential
arc at radial coordinate 0.09638 [m] and modulates to relatively reach a perpendicular position
with respect to the pressure and suction sides (Figure 5-2).

62

Figure 5-2: Pressure plot on the suction and pressure surface according to radial coordinates
(left) and static pressure contour (right) for points 1 to 3.

63

During the unsteady expansion process, different pressures are exerted on the pressure
and suction walls corresponding to the same radial coordinate; as a result, a differential pressure
is exerted in, or against, the rotational direction, which is translated into generated torque. These
different pressure values are due to pressure gradients generated by the expansion wave from
head to tail when it propagates toward the channel inlet. The expansion wave produces linecontours of constant pressure from its head to tail, and these line-contours are expected to remain
approximately perpendicular to the channel walls [18]. In some instances, this condition does not
prevail, as seen later on. In general, the orientation of line-contours yields different pressure
values at the pressure and suction sides with same radial coordinate (Equation (5-2) and Figure
5-3). This concept is used to explain results in this section.
𝑃𝑃𝑝𝑝𝑝𝑝1 > 𝑃𝑃𝑝𝑝𝑝𝑝1 𝑎𝑎𝑎𝑎 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑟𝑟1

(5-2)

Figure 5-3: Pressure gradients and pressure on the suction and pressure walls
In the beginning of the process, two effects increase the differential pressure area (linecontour graphs in RHS of Figure 5-2 and Figure 5-4): the orientation change of the head of the
expansion wave (first line-contour) going from point 1 to 3, and the pressure gradient along the
expansion wave from head to tail. The change in orientation of the head of the expansion wave is

64

one of the factors that causes “s” to gradually increase, and in consequence the net area grows
from point 1 to 3 (Figure 5-4).

Figure 5-4: Increasing trend of the net pressure distribution for points 1 to 3.
In Figure 5-5, Velocity vector plots of velocity magnitude show the fluid closer to the
suction side has higher local velocity magnitudes than the fluid closer to the pressure side during
the period from point 1 to 3. Thus, it produces lesser pressure values on the suction side than the
pressure side at the same radial coordinates as the expansion wave propagates from point 1 to 3.
After point 3, cm plot oscillates until point 7, and then decreases through the end of the
expansion (Figure 5-1). Figure 5-7 shows pressure line-contours and pressure distribution on the
65

walls for points from 4 to 6. Point 4 refers to the lowest cm value, which then goes up to point 5,
and falls down again to point 6. As seen in Figure 5-7, the orientation of line-contours at the tail
of the expansion wave changes from being almost perpendicular (in respect to upper and lower
walls), inclines to the right, and returns to perpendicular. In point 4, the line-contours at radial
locations 0.9 to 1.0 keep the vertical orientation, and then become less steep when moved to
point 5 (maximum “cm” value). This effect projects more area on upper and lower walls for
differential pressure. At point 6, the line-contours return to approximately the vertical position
and the differential pressure is again reduced (last picture on the right of Figure 5-7).

Point-1

Point-3

Figure 5-5: Velocity magnitude vectors for points 1 and 3
Next, at point 7, the line-contours of the channel outlet region inclines to the right again
and covers more area of differential pressure, as seen in Figure 5-8. The head of the expansion
wave impinges on the inlet wall of the channel between points 7 and 8 (0.07 [ms]) and the static
pressure on both walls decreases abruptly until the end of the expansion, and an identical
situation occurs with the differential pressure. The head of the expansion wave now becomes a
right running wave that propagates through the incoming incident left running wave. This causes
a distortion in line-contours and reduces the differential pressure to zero when the reflected wave
travels back to the outlet port of the channel.
66

The oscillating behavior (Figure 5-6) of the cm plot from points 3 to 7 is caused by two
effects: the orientation of the line-contours near the channel outlet and the net area increase of
the differential pressure produced by the wave propagation. At point 4, cm decreases when the
orientation of line-contours become perpendicular in the region near channel outlet and also
reduces the differential pressure approximately at radius of 0.1 m and thus, the net area with
respect to point 3 (Figure 5-6 (b)). At point 5, cm value is raised by the orientation of the linecontours in the region near the channel outlet where the slopes become less steep toward the
right, causing an increase in differential pressure net area (Figure 5-6(b)). At point 6, cm
diminishes due to the change in orientation of line-contours (returning to perpendicular again)
but not at the level of point 4, because the net area increased and this positively compensates. At
point 7, cm value does not recover, as in point 5, even when the orientation of line-contours
benefits positively and the net area is similar (Figure 5-6 (a) & (b)). This result indicates that
additional influential factors need to be considered.
Two extra factors are investigated in the next section: the component of the force due to
pressure in the tangential direction and the impact on the torque distribution by the radial
position.
In this section some conclusions are made: during the propagation of the incident head of
the expansion wave around 62% of the total torque is extracted, approximately 80% of the length
of the channel works actively for the generation of torque according to positive values of the
differential pressure shown, and the remaining 20% corresponds to the channel zone length near
the inlet wall.

67

(a)

(b)

Figure 5-6: (a) cm torque oscillation and (b) Non-dimensional net area of differential pressure.

68

Figure 5-7: Pressure plot on the suction and pressure sides at each radial coordinates (left) and
static pressure contour (right) for points 4 to 6.

69

Figure 5-8: Pressure plot on the suction and pressure surface according to radial coordinates
(left) and static pressure contour (right) for points 7 to 9.

70

5.1.2. Force and Torque distribution on channel walls
Force and torque distribution are presented in this section: the following convention is
used. The expression for the Torque in a specific radial position is defined as:
�⃗𝑟𝑟𝑟𝑟 = 𝑟𝑟⃗ × 𝐹𝐹⃗𝑝𝑝 + 𝑟𝑟⃗ × 𝐹𝐹⃗𝑣𝑣
𝑇𝑇

(5-3)

�⃗𝑟𝑟𝑟𝑟 = 𝑟𝑟⃗ × 𝐹𝐹⃗𝑝𝑝
𝑇𝑇

(5-4)

𝐹𝐹⃗𝑝𝑝 = 𝐹𝐹⃗𝑡𝑡𝑡𝑡𝑡𝑡−𝑝𝑝 + 𝐹𝐹⃗𝑟𝑟𝑟𝑟𝑟𝑟−𝑝𝑝

(5-5)

As was stated at the beginning of this chapter, the moment due to viscous effect is
considered negligible, then:

The vector force component due to pressure:

The force associated with the generation of torque:
𝐹𝐹⃗𝑡𝑡𝑡𝑡𝑡𝑡−𝑝𝑝 = (𝐹𝐹𝑝𝑝 )(𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠) 𝑒𝑒⃗𝜃𝜃

𝐹𝐹⃗𝑡𝑡𝑡𝑡𝑡𝑡−𝑝𝑝 = (𝑃𝑃. 𝐴𝐴)(𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠) 𝑒𝑒⃗𝜃𝜃

(5-6)

𝐹𝐹⃗𝑡𝑡𝑡𝑡𝑡𝑡−𝑝𝑝−𝑃𝑃𝑃𝑃 = (𝑝𝑝𝑃𝑃𝑃𝑃 . 𝐴𝐴𝑃𝑃𝑃𝑃 )(𝑠𝑠𝑠𝑠𝑠𝑠𝛽𝛽𝑃𝑃𝑃𝑃 ) 𝑒𝑒⃗𝜃𝜃

(5-7)

𝐹𝐹⃗𝑡𝑡𝑡𝑡𝑡𝑡−𝑝𝑝−𝑆𝑆𝑆𝑆 = (𝑝𝑝𝑆𝑆𝑆𝑆 . 𝐴𝐴𝑆𝑆𝑆𝑆 )(𝑠𝑠𝑠𝑠𝑠𝑠𝛽𝛽𝑆𝑆𝑆𝑆 ) 𝑒𝑒⃗𝜃𝜃

(5-8)

∆𝐹𝐹⃗𝑡𝑡𝑡𝑡𝑡𝑡−𝑝𝑝 = [𝑝𝑝𝑃𝑃𝑃𝑃 𝐴𝐴𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑠𝑠𝛽𝛽𝑃𝑃𝑃𝑃 − 𝑝𝑝𝑆𝑆𝑆𝑆 𝐴𝐴𝑆𝑆𝑆𝑆 𝑠𝑠𝑠𝑠𝑠𝑠𝛽𝛽𝑆𝑆𝑆𝑆 ] 𝑒𝑒⃗𝜃𝜃

(5-9)

Following the convention in Figure 5-9:
Pressure side:

Suction side:

Calculating the differential pressure for both at the same radial coordinate “r”:

If "Force − θ" is the force in the tangential direction:

And Δ(Force − Ɵ) = 𝑝𝑝𝑃𝑃𝑃𝑃 𝐴𝐴𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑠𝑠𝛽𝛽𝑃𝑃𝑃𝑃 − 𝑝𝑝𝑆𝑆𝑆𝑆 𝐴𝐴𝑆𝑆𝑆𝑆 𝑠𝑠𝑠𝑠𝑠𝑠𝛽𝛽𝑆𝑆𝑆𝑆 , then:
∆𝐹𝐹⃗𝑡𝑡𝑡𝑡𝑡𝑡−𝑝𝑝 = [Δ(Force − θ)] 𝑒𝑒⃗𝜃𝜃
71

(5-10)

So, the magnitude of Equation (5-10) when plotted in graphs:
∆𝐹𝐹𝑡𝑡𝑡𝑡𝑡𝑡−𝑝𝑝 = Δ(Force − θ)

(5-11)

Torque plots are now calculated based on Equation (5-4); but considering only the torque
component in the rotation direction:
�⃗𝑟𝑟𝑟𝑟 = 𝑟𝑟⃗ × 𝐹𝐹⃗𝑝𝑝 = 𝑟𝑟𝐹𝐹𝑡𝑡𝑡𝑡𝑡𝑡−𝑝𝑝 𝑒𝑒⃗𝑘𝑘
𝑇𝑇

Replacing results from Equation (5-11), which regards forces on pressure and suction
walls, it yields:
�⃗𝑟𝑟𝑟𝑟 = 𝑟𝑟Δ(Force − θ) 𝑒𝑒⃗𝑘𝑘
𝑇𝑇

𝑇𝑇𝑟𝑟𝑟𝑟 = 𝑟𝑟. ∆𝐹𝐹𝑡𝑡𝑡𝑡𝑡𝑡−𝑝𝑝 = Δ(r ∗ Force − θ)

(5-12)

Then, Equation (5-12) is used to build torque plots.

Figure 5-9: Nomenclature for forces, angles and channel walls
In previous section torque generation (or torque coefficient) based on pressure
distribution on the channel walls was discussed. But tangential force directly contributes to
gaining or losing torque. Thus, to calculate tangential forces, “β” angle must be known. At any

72

specific radial coordinate, it is formed by the unit vector perpendicular to the channel wall area
(f1 or f3) and the unit vector perpendicular to the circumference (f2) (Figure 5-9). “β” angle
favors a greater value of tangential component on the pressure or suction sides (Equations (5-7)
or (5-8)) when approaching 90 degrees. β values closer to zero will result in low values of
tangential force at the pressure or suction walls. Thus, values of β angle near 90° effects torque
generation positively (Equation (5-9)) because of tangential force on the pressure side is
responsible, but not for the tangential force component acting on the suction side, which
counteracts the torque generation. In addition, values of β near 0° on the suction side work for
the benefit of torque generation if the tangential component has this value. The influence of β
angle in the orientation of the line-contours from head to tail of the expansion wave is a topic to
discuss in Section 5.3.
The geometry of the constant cross section channel is plotted in Figure 5-10 (a). At the
trailing edge of the suction side, the constant cross section is unchanged until the red curve ends
and becomes a green straight line-this was added to reduce the adverse effect of a bigger outlet
area during gradual opening. In Figure 5-10 (b), the sine of β changes from leading to trailing
edge in a decreasing trend for this particular channel geometry. The green straight line added at
the trailing edge generates higher sine of β values on the suction side. At 81.7 mm, on the suction
side, higher values of sine of β are changed to lower values until 92.8 mm with respect to
pressure side. Past this location, another change is made.
From Figure 5-11 to Figure 5-13, the net tangential force and torque distribution were
plotted for points 1 to 9, according to Figure 5-1. To discuss figures in this section, Δ(Force − θ)
or Δ(r ∗ Force − θ) will be referred to simply as force or torque. Positive values of force or

73

torque are interpreted as torque production or gain, and negative values have the opposite
meaning.
A crest appears at point 1 and continues moving and spreading to the left as the head of
expansion wave travels toward the inlet wall for the following points in time. At point 1,
negative values for force and torque are due to higher β values on the suction wall until “a1”
position, where β values of pressure and suction sides are equalized (see Figure 5-10). At “a2” in
Figure 5-11(a), force and torque both reach maximum values and begin to decrease due to the
sine of β increase in the suction side. At “a3” location, both force and torque become zero, and
sine of β of green straight (suction side) and blue curve (pressure side) have the same value. The
decreasing trend of the force (or torque) is the result of increasing values of β for the green
straight line (Figure 5-10 (b)). At the location of 96.38 mm (Figure 5-2 and Figure 5-11(a), point
1), force and torque reach the lowest value and begin increasing because of the action of the
expansion wave on both walls.

(a)

(b)

Figure 5-10: Channel shape and sinβ variation along the radial coordinate of a constant cross
section channel

74

(a)

(b)

(c)

(d)

Figure 5-11: Δ(force-θ) and Δ(r*force-θ) plots at points 1, 2, 3 & 4 according to radial
coordinates.
For points 2 to 7, the pressure and suction sides have the same pressure values in the
channel region between the inlet wall and the head of the expansion wave (before it impinges the
inlet wall), and β values cause a bigger component from the applied force on the suction side in
the tangential direction, resulting in negative values of force and torque (Figure 5-11 and Figure
5-12). Based on this channel geometry, around 18% of the channel length does not produce

75

positive force (or torque) because of bigger β angles in the suction side with respect to the
pressure side. This percentage refers to the zone adjacent to the channel inlet.

(a)

(b)

(c)

(d)

Figure 5-12: Δ(force-θ) and Δ(r*force-θ) plots at points 5, 6, 7 & 8 according to radial
coordinates.

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Figure 5-13: Δ(force-θ) and Δ(r*force-θ) plot at point 9, according to radial coordinates.

Figure 5-14: Δ(force-θ) at points 1, 2, 3, 4, 5, 6, 7, 8 & 9 along radial direction
Figure 5-14 and Figure 5-15 display the force and torque, respectively, as the expansion
process propagates from point 1 to 9. The shape of the force distribution appears similar to the

77

torque distribution along the radial coordinate, but with an order of magnitude around 10. The
highest local values of torque (or force) are generated during the initial stage of the unsteady
expansion process: points 1 to 3. As the head of expansion wave propagates toward the inlet
wall, the torque (force) with smaller values is spread over the length of the channel. This fact can
be called dissipation effect of the action of pressure because of βSS > βPS.
If a significant fraction of the torque is generated at the pressure and suction side of the
region near the channel outlet, βPS > βSS becomes a necessary design condition to maximize the
work extraction.

Figure 5-15: Δ(r*force-θ) at points 1, 2, 3, 4, 5, 6, 7, 8 & 9 along radial direction

78

5.2.

Unsteady Expansion Process at gradual opening
5.2.1. Pressure distribution on channel walls
This section investigates the unsteady expansion process when the channel outlet opens

gradually at 10,000 rpm. The torque coefficient history is shown in Figure 5-16, and 11 points
are selected to analyze the process.
Head of the
expansion wave
impinges the inlet
wall

Outlet fully
open

Figure 5-16: Torque coefficient (cm) history of a constant-cross section curved channel for
gradual opening
At point 1, the fractional opening of the channel outlet creates an expansion wave made
of concentric circle-contours of pressure. This configuration produces higher pressure on the
suction side, which is why cm is negative (Figure 5-16 and Figure 5-17). Subsequently, at point
2, the effect of a slightly greater opening of the outlet causes the shape and position of the head
of the expansion wave to become a front line perpendicular to the pressure and suction sides.
Positive torque is not produced in the region very close to the channel outlet until
complete opening is achieved at point 9 (see left plots in Figure 5-17, Figure 5-18, and Figure
5-19). The gradual opening produces adverse effects on the pressure distribution on both

79

pressure and suction sides, a disadvantageous condition with respect to instant opening because a
significant portion of the generated torque comes from this region.
The head of the expansion wave impinges the inlet wall when the channel outlet is not yet
fully opened; this causes the pressure to begin dropping, but not as rapidly as the instant opening
case, slowing presure loss during the early stage. The pressure diferential between channel
pressure and manifold pressure drives the acceleration of the fluid, thus the increase of the
tangential velocity contributes to torque generation, and an explanation is given in Section 6.3.1.
Therefore, cm values continue increasing till point 4 and then keep approximately the same value
to point 9; this location in time marks the beginning of the cm decrease, the channel outlet has
already been completely opened (point 9 in Figure 5-19), and these conditions appear to be
correlated.
In all points in time, there is no positive net pressure area in the interval from r=0.05 [m]
(at inlet wall) to r=0.06 [m] contributing to the generation of torque.
The results indicate instant opening of channel outlet with respect to the gradual opening
creates more favorable conditions in the region near the channel outlet, maximizing torque
generation. This positive condition indicates increasing the speed at which the outlet port opens
are steps toward the optimal situation: instant opening.

80

Figure 5-17: Pressure plot on the suction and pressure surface according to radial coordinates
(left) and pressure contour (right) for points 1 to 3.

81

Figure 5-18: Pressure plot of the suction and pressure surface according to radial coordinates
(left) and pressure contour (right) for points 4 to 6.

82

Figure 5-19: Pressure plot of the suction and pressure surface according to radial coordinates
(left) and pressure contour (right) for points 7 to 9.

83

Figure 5-20: Pressure plot of the suction and pressure surface according to radial coordinates
(left) and pressure contour (right) for points 10 to 11.
5.2.2. Force and Torque distribution on channel walls
In this section the force and torque differential distribution is presented in terms of
pressure and sine of β. The magnitude of the force differential (Equation (5-9)) is proportional to
the equation in terms of pressure and sine of β.

Then:

∆𝐹𝐹𝑡𝑡𝑡𝑡𝑡𝑡−𝑝𝑝 ≈ [𝑝𝑝𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑠𝑠𝛽𝛽𝑃𝑃𝑃𝑃 − 𝑝𝑝𝑆𝑆𝑆𝑆 𝑠𝑠𝑠𝑠𝑠𝑠𝛽𝛽𝑆𝑆𝑆𝑆 ]
∆(𝑝𝑝 ∗ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠) = [𝑝𝑝𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑠𝑠𝛽𝛽𝑃𝑃𝑃𝑃 − 𝑝𝑝𝑆𝑆𝑆𝑆 𝑠𝑠𝑠𝑠𝑠𝑠𝛽𝛽𝑆𝑆𝑆𝑆 ]

The same is done for the torque differential Equation (5-12):

84

(5-13)

𝑇𝑇𝑟𝑟𝑟𝑟 = 𝑟𝑟. ∆𝐹𝐹𝑡𝑡𝑡𝑡𝑡𝑡−𝑝𝑝 ≈ Δ(r ∗ p ∗ sinβ)

∆(𝑟𝑟 ∗ 𝑝𝑝 ∗ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠) = 𝑟𝑟. [𝑝𝑝𝑃𝑃𝑃𝑃 𝑠𝑠𝑠𝑠𝑠𝑠𝛽𝛽𝑃𝑃𝑃𝑃 − 𝑝𝑝𝑆𝑆𝑆𝑆 𝑠𝑠𝑠𝑠𝑠𝑠𝛽𝛽𝑆𝑆𝑆𝑆 ]

(5-14)

Then, Equations (5-13) and (5-14) are used to plot curves of force (Figure 5-21 and
Figure 5-22) and torque (Figure 5-23 and Figure 5-24), respectively.
Force plots present higher order of magnitude values with respect to the torque plots.
Both force and torque graphs exhibit similar trends:
(a) Negative force and torque values near the channel outlet
(b) Positive force and torque values at the central region of the channel
(c) Negative force and torque near the channel inlet
These results show that gradual opening causes undesirable effects near the channel
outlet, limiting torque extraction.

Figure 5-21: Δ(p*sinβ) at points 1, 2, 3, 4, 5 & 6 based on radial coordinates

85

Figure 5-22: Δ(p*sinβ) at points 7, 8, 9, 10 & 11 based on radial coordinates.

Figure 5-23: Δ(r*p*sinβ) at points 1, 2, 3, 4, 5 & 6 based on radial direction

86

Figure 5-24: Δ(r*p*sinβ) at points 7, 8, 9, 10 & 11 based on radial direction
5.3.

Impact on tangential force (torque generation) by changing θ parameter
The pressure and tangential forces (torque) applied to channel walls for a specific θ angle

has been investigated. The θ angle was mentioned in Figure 2-2 (b) for constant cross-section
channels made of two concentric circular arcs. In this section, the degree of impact on the
tangential force (torque) is determined by varying the θ angle. The areas of influence (APS or
ASS) and the β angle (Figure 5-9), both along the pressure and suction walls, change by the
alteration of θ angle.
When the head of the expansion wave propagates through the channel, isobaric linecontours appear from head to tail. The projection of each isobaric line-contour on pressure and
suction walls creates different areas. These two areas are delimited by two concentric lines: r1
and r2 (Figure 5-25), and the projected areas are called “area of influence (AoI)”. The pressure
on pressure side AoI is greater than the pressure on suction side AoI if the expansion wave is a
87

left running wave. Thus, the isobaric line-contour produces a differential pressure between
pressure and suction walls.
The differential force depends on six parameters: pPS , pSS , APS , ASS , sinβPS and sinβSS ,

according to Equation (5-9); the areas of influence and β angle will be the focus in this section.
To study the impact of the isobaric line-contour in the differential force, equation (5-9)
will be utilized; this differential force is not confined to a specific radial location, but to an area
location. Thus, the length-weighted average of sine of β will be considered in that equation
instead of sin (β) only. The reason for this is β angle changes at each radial coordinate on the
area of influence. The modified equation yields:
∆𝐹𝐹⃗𝑡𝑡𝑡𝑡𝑡𝑡−𝑝𝑝 = [𝑝𝑝𝑃𝑃𝑃𝑃 𝐴𝐴𝑃𝑃𝑃𝑃 ���������
𝑠𝑠𝑠𝑠𝑠𝑠𝛽𝛽𝑃𝑃𝑃𝑃 − 𝑝𝑝𝑆𝑆𝑆𝑆 𝐴𝐴𝑆𝑆𝑆𝑆 ��������
𝑠𝑠𝑠𝑠𝑠𝑠𝛽𝛽𝑆𝑆𝑆𝑆 ] 𝑒𝑒⃗𝜃𝜃

(5-15)

The length-weighted average of sin (β) is defined as:
∫ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠
𝐿𝐿𝑎𝑎𝑎𝑎𝑎𝑎

(5-16)

∑𝑛𝑛𝑖𝑖 𝑠𝑠𝑠𝑠𝑠𝑠𝛽𝛽𝑖𝑖 . ∆𝑙𝑙𝑖𝑖
𝐿𝐿𝑎𝑎𝑎𝑎𝑎𝑎

(5-17)

������
𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 =

Expressed as summation:

Where:

������
𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 =

Larc: refers to the total arc length on the pressure or suction side of the area of influence
Δli: small length element
βi: beta angle at the center of the small length element “i”

88

This complies with
PPS > PSS

Figure 5-25: Area of influence projected by a single isobaric line-contour on pressure and suction
walls
Results in Figure 5-26 correspond to channel geometry drawn by two concentric circle
arcs to have a constant cross-section channel. The four graphs depict the variation of θ angle
from 30 to 60 degrees, respectively, and as the θ angle is increased, the channel length increases
also.
In Section 5.1.2, it is stated that beta angles near 90 degrees on the pressure wall or near 0
degrees on the suction wall benefits the generation of torque; this was concluded for a tangential
force when applied at each radial coordinate. Here, the net tangential force is based on area of
influence, and the orientation of line-contours causes this area to grow or shrink.
For the channel geometry described in Figure 5-26, β angle portrays a decreasing trend
from leading to trailing edge. The opposite manner happens to the area of influence (APS or ASS)
of both pressure and suction sides. These results are used to calculate the terms of “𝐴𝐴𝑃𝑃𝑃𝑃 ���������
𝑠𝑠𝑠𝑠𝑠𝑠𝛽𝛽𝑃𝑃𝑃𝑃 ”

and “𝐴𝐴𝑆𝑆𝑆𝑆 ��������
𝑠𝑠𝑠𝑠𝑠𝑠𝛽𝛽𝑆𝑆𝑆𝑆 ” from Equation (5-15), and the results are plotted in Figure 5-26. The

intersection of the upper line-contour with the pressure side defines the radial coordinate used to
plot these two terms. Radial coordinate of 60, 70, 80, 90 and 100 mm were used for each θ angle.

89

Figure 5-26 shows that both 𝐴𝐴𝑃𝑃𝑃𝑃 ���������
𝑠𝑠𝑠𝑠𝑠𝑠𝛽𝛽𝑃𝑃𝑃𝑃 and 𝐴𝐴𝑆𝑆𝑆𝑆 ��������
𝑠𝑠𝑠𝑠𝑠𝑠𝛽𝛽𝑆𝑆𝑆𝑆 resulted in similar values at each

radial coordinate. Then, Equation (5-15) is re-written as:

Or

∆𝐹𝐹⃗𝑡𝑡𝑡𝑡𝑡𝑡−𝑝𝑝 ≈ 𝐴𝐴𝑃𝑃𝑃𝑃 ���������
𝑠𝑠𝑠𝑠𝑠𝑠𝛽𝛽𝑃𝑃𝑃𝑃 [𝑝𝑝𝑃𝑃𝑃𝑃 − 𝑝𝑝𝑆𝑆𝑆𝑆 ] 𝑒𝑒⃗𝜃𝜃

(5-18)

∆𝐹𝐹⃗𝑡𝑡𝑡𝑡𝑡𝑡−𝑝𝑝 ≈ 𝐴𝐴𝑆𝑆𝑆𝑆 ��������
𝑠𝑠𝑠𝑠𝑠𝑠𝛽𝛽𝑆𝑆𝑆𝑆 [𝑝𝑝𝑃𝑃𝑃𝑃 − 𝑝𝑝𝑆𝑆𝑆𝑆 ] 𝑒𝑒⃗𝜃𝜃

(5-19)

(a) θ=30 deg

(a) θ=40 deg

(a) θ=50 deg

(a) θ=60 deg

Figure 5-26: β & area of influence (APS & ASS) on pressure and suction walls created by linecontour when θ angle is changed

90

The amount of net tangential force over an area of influence depends on 𝐴𝐴𝑃𝑃𝑃𝑃 ���������
𝑠𝑠𝑠𝑠𝑠𝑠𝛽𝛽𝑃𝑃𝑃𝑃 (or

������ varies from lower values
𝐴𝐴𝑆𝑆𝑆𝑆 ��������
𝑠𝑠𝑠𝑠𝑠𝑠𝛽𝛽𝑆𝑆𝑆𝑆 ) and pressure difference (𝑝𝑝𝑃𝑃𝑃𝑃 − 𝑝𝑝𝑆𝑆𝑆𝑆 ). The outcome of A𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠

near channel inlet to greater toward outlet. Also, the trend is less steep as θ angle changes from

30 to 60. Thus, the net tangential force becomes greater in regions closer to the channel outlet. In
addition, due to greater radius, greater torque values can be accomplished in that region.

91

CHAPTER 6.
6.1.

MOMENT OF MOMENTUM ANALYSIS

Moment of Momentum equation applied to a single channel of the WDE
The Equation (6-1) is a general formulation of the angular momentum principle for an

inertial and non-deforming control volume [19] at any instant in time.
�⃗𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 =
𝑟𝑟⃗𝑥𝑥𝐹𝐹⃗𝑠𝑠 + � 𝑟𝑟⃗𝑥𝑥𝑔𝑔⃗𝜌𝜌𝜌𝜌𝑉𝑉 + 𝑇𝑇
𝐶𝐶𝐶𝐶

𝜕𝜕
�⃗ 𝜌𝜌𝜌𝜌𝑉𝑉 + � 𝑟𝑟⃗𝑥𝑥𝑉𝑉
�⃗ 𝜌𝜌𝑉𝑉
�⃗ . 𝑑𝑑𝐴𝐴⃗
� 𝑟𝑟⃗𝑥𝑥𝑉𝑉
𝜕𝜕𝜕𝜕 𝐶𝐶𝐶𝐶
𝐶𝐶𝐶𝐶

(6-1)

This investigation is focused on 2-dimensional models because all geometries
investigated reflect that characteristic, and the initial concept of the WDE was to have a rotor
with flat shroud and hub in the channels. Thus, the generation of torque is the result of the
dynamic action of the fluid with the side walls. Figure 6-1 shows the control volume selected for
the application of equation (6-1).

Figure 6-1: Control Volume selected for the Angular Momentum Equation Analysis
The three terms on the LHS of Equation (6-1) are described as follows:
𝑟𝑟⃗𝑥𝑥𝐹𝐹⃗𝑠𝑠 : represents the angular momentum of surface forces over the control volume (they can be
due to pressure or friction). This term might be ignored due to the amount it represents when
compared with the torque in the shaft.

92

∭𝐶𝐶𝐶𝐶 𝑟𝑟⃗𝑥𝑥𝑔𝑔⃗𝜌𝜌𝜌𝜌𝑉𝑉 : refers to the torque generated by body forces. The direction of the torque

generated is perpendicular to the rotation direction and therefore does not contribute to torque
generation. Then, Equation (6-1) becomes:
�⃗𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 =
𝑇𝑇

𝜕𝜕
�⃗ 𝜌𝜌𝜌𝜌𝑉𝑉 + � 𝑟𝑟⃗𝑥𝑥𝑉𝑉
�⃗ 𝜌𝜌𝑉𝑉
�⃗ . 𝑑𝑑𝐴𝐴⃗
� 𝑟𝑟⃗𝑥𝑥𝑉𝑉
𝜕𝜕𝜕𝜕 𝐶𝐶𝐶𝐶
𝐶𝐶𝐶𝐶

During the unsteady expansion process, the fluid crosses the control surface through only
one outlet, and the equation becomes:
�⃗𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 =
𝑇𝑇

(6-2)

𝜕𝜕
�⃗ 𝜌𝜌𝜌𝜌𝑉𝑉 + �
�⃗ 𝜌𝜌𝑉𝑉
�⃗ . 𝑑𝑑𝐴𝐴⃗
� 𝑟𝑟⃗𝑥𝑥𝑉𝑉
𝑟𝑟⃗𝑥𝑥𝑉𝑉
𝜕𝜕𝜕𝜕 𝐶𝐶𝐶𝐶
𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜

In Equation (6-2), the first term on the right hand side represents the rate of change of
fluid’s angular momentum within the control volume. The second term on the right hand side
refers to the outflow rate of the angular momentum through the outlet area of the channel [19],
[20]. The first and second terms hold the Joules units.
In a typical turbomachinery analysis, the first term of the RHS in Equation (6-1) vanishes
due to steady state condition and becomes:
�⃗𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 = � 𝑟𝑟⃗𝑥𝑥𝑉𝑉
�⃗ 𝜌𝜌𝑉𝑉
�⃗ . 𝑑𝑑𝐴𝐴⃗ = �
𝑇𝑇
𝐶𝐶𝐶𝐶

𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜

�⃗ − �
𝑟𝑟𝑉𝑉𝜃𝜃 (𝜌𝜌𝑉𝑉𝑟𝑟 𝑑𝑑𝑑𝑑) 𝑘𝑘

𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖

And considering uniform flow properties at the inlet and outlet:

�⃗
𝑟𝑟𝑉𝑉𝜃𝜃 (𝜌𝜌𝑉𝑉𝑟𝑟 𝑑𝑑𝑑𝑑) 𝑘𝑘

�⃗
�⃗𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 = 𝑚𝑚̇(𝑟𝑟𝑜𝑜𝑜𝑜𝑜𝑜 𝑉𝑉𝜃𝜃−𝑜𝑜𝑜𝑜𝑜𝑜 − 𝑟𝑟𝑖𝑖𝑖𝑖 𝑉𝑉𝜃𝜃−𝑖𝑖𝑖𝑖 )𝑘𝑘
𝑇𝑇

This shows the extracted torque is based on the change of the fluid’s angular momentum
(𝑟𝑟 ∆𝑉𝑉𝜃𝜃 ), and this change is called turbomachinery principle. The fluid changes its angular

momentum when going through a blade passage between inlet and outlet, and this change
converts into torque.

93

For the unsteady expansion of the WDE, both RHS terms are kept and only one border
crossing is considered (fluid outflow) in the second RHS term. Thus, the first RHS term refers to
the turbomachinery principle in terms of the rate of “change of the angular momentum” within
the channel to produce torque. The second RHS term indicates the rate of flow of the angular
momentum through outlet area and suggests it is linked to mass flow rate (see Equation (6-18)
for a 2-D case resulting from Equation (6-2)). Additional explanation of this last term is given in
Section 6.2.2.

6.2.

Analytical Analysis with respect to the Inertial Reference Frame

The present analysis is accomplished in cylindrical coordinates with the following
conventions:
�⃗ = 𝑉𝑉
�⃗𝑟𝑟 + 𝑉𝑉
�⃗𝜃𝜃
𝑉𝑉

(6-3)

�⃗𝑟𝑟 = 𝑉𝑉𝑟𝑟 𝑒𝑒⃗𝑟𝑟
𝑉𝑉

(6-5)

𝑟𝑟⃗ = 𝑟𝑟𝑒𝑒⃗𝑟𝑟

(6-4)

�⃗𝜃𝜃 = 𝑉𝑉𝜃𝜃 𝑒𝑒⃗𝜃𝜃
𝑉𝑉

(6-6)

Orientation of the unit vectors is according to Figure 6-2.

Figure 6-2: Orientation of radial and tangential unit vectors in cylindrical coordinates
�⃗𝑧𝑧 = 0):
The RHS terms of Equation (6-2) will be analyzed separately and in 2-D (𝑉𝑉
94

𝜕𝜕

�⃗ )𝜌𝜌𝜌𝜌𝑉𝑉
(I) = 𝜕𝜕𝜕𝜕 ∭𝐶𝐶𝐶𝐶(𝑟𝑟⃗ × 𝑉𝑉

(6-7)

�⃗ )𝜌𝜌 𝑉𝑉
�⃗ ∙ 𝑑𝑑𝐴𝐴⃗
(II) = ∬𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜(𝑟𝑟⃗ × 𝑉𝑉

(6-8)

6.2.1. The Rate of Change of the Angular Momentum in the Channel
Replacing (6-3) in (6-7):
�⃗𝑟𝑟 + 𝑉𝑉
�⃗𝜃𝜃 ��𝜌𝜌�
𝜕𝜕��𝑟𝑟⃗ × �𝑉𝑉
𝜕𝜕
�⃗𝑟𝑟 + 𝑉𝑉
�⃗𝜃𝜃 ��𝜌𝜌𝜌𝜌𝑉𝑉 = �
� �𝑟𝑟⃗ × �𝑉𝑉
𝑑𝑑∀
𝜕𝜕𝜕𝜕 𝐶𝐶𝐶𝐶
𝜕𝜕𝜕𝜕
𝐶𝐶𝐶𝐶
�⃗𝑟𝑟 + 𝑟𝑟⃗ × 𝑉𝑉
�⃗𝜃𝜃 ]𝜌𝜌)
𝜕𝜕([𝑟𝑟⃗ × 𝑉𝑉
𝑑𝑑𝑉𝑉
𝜕𝜕𝜕𝜕
𝐶𝐶𝐶𝐶

=ďż˝

�⃗𝑟𝑟 = 0 does not produce rotation. Then:
𝑟𝑟⃗ × 𝑉𝑉

�⃗𝜃𝜃 )𝜌𝜌]
𝜕𝜕
𝜕𝜕[(𝑟𝑟⃗ × 𝑉𝑉
�⃗ )𝜌𝜌𝜌𝜌𝑉𝑉 = �
� (𝑟𝑟⃗ × 𝑉𝑉
𝑑𝑑𝑉𝑉
𝜕𝜕𝜕𝜕 𝐶𝐶𝐶𝐶
𝜕𝜕𝜕𝜕
𝐶𝐶𝐶𝐶

(6-9)

Working inside of the integral:

�⃗𝜃𝜃 )𝜌𝜌�
�⃗𝜃𝜃 �
�⃗𝜃𝜃 �
𝜕𝜕�(𝑟𝑟⃗ × 𝑉𝑉
𝜕𝜕�𝑟𝑟⃗𝜌𝜌 × 𝑉𝑉
𝜕𝜕�𝑉𝑉
𝜕𝜕(𝑟𝑟⃗𝜌𝜌)
�⃗𝜃𝜃 ×
=
= (𝑟𝑟⃗𝜌𝜌) × �
� − 𝑉𝑉
=
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
= (𝑟𝑟⃗𝜌𝜌) × �

= �𝑟𝑟⃗ ×

�⃗𝜃𝜃
𝜕𝜕𝑉𝑉
𝜕𝜕𝑟𝑟⃗
𝜕𝜕𝜕𝜕
�⃗𝜃𝜃 × [ 𝜌𝜌 + 𝑟𝑟⃗ ] =
� − 𝑉𝑉
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕

�⃗𝜃𝜃
𝜕𝜕𝑉𝑉
𝜕𝜕𝑟𝑟⃗
𝜕𝜕𝜕𝜕
�⃗𝜃𝜃 × � 𝜌𝜌 − (𝑉𝑉
�⃗𝜃𝜃 × 𝑟𝑟⃗)
� 𝜌𝜌 − �𝑉𝑉
=
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕

(6-10)

In the first term of expression (6-10), the partial derivative of the tangential velocity
vector:

It is known that:

�⃗𝜃𝜃 𝜕𝜕(𝑉𝑉𝜃𝜃 𝑒𝑒⃗𝜃𝜃 )
𝜕𝜕𝑉𝑉
𝜕𝜕𝑒𝑒⃗𝜃𝜃 𝜕𝜕𝑉𝑉𝜃𝜃
=
= 𝑉𝑉𝜃𝜃
+
𝑒𝑒⃗ =
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕 𝜃𝜃
𝜕𝜕𝑒𝑒⃗𝜃𝜃
= −𝜃𝜃̇𝑒𝑒⃗𝑟𝑟
𝜕𝜕𝜕𝜕
95

Then:
�⃗𝜃𝜃
𝜕𝜕𝑉𝑉
𝜕𝜕𝑉𝑉𝜃𝜃
= −𝑉𝑉𝜃𝜃 𝜃𝜃̇𝑒𝑒⃗𝑟𝑟 +
𝑒𝑒⃗
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕 𝜃𝜃

(6-11)

In the third term for expression (6-10), the partial derivative of the radial vector:
𝜕𝜕𝑒𝑒⃗𝑟𝑟 𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝑟𝑟⃗ 𝜕𝜕(𝑟𝑟𝑒𝑒⃗𝑟𝑟 ) 𝜕𝜕𝜕𝜕
=
= 𝑒𝑒⃗𝑟𝑟 + 𝑟𝑟
= 𝑒𝑒⃗𝑟𝑟 + 𝑟𝑟𝜃𝜃̇𝑒𝑒⃗𝜃𝜃 = 𝑒𝑒⃗𝑟𝑟 + 𝑟𝑟𝜔𝜔𝑒𝑒⃗𝜃𝜃
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕

Where 𝜃𝜃̇ = 𝜔𝜔

𝜕𝜕𝑟𝑟⃗ 𝜕𝜕𝜕𝜕
= 𝑒𝑒⃗ + 𝑟𝑟𝜔𝜔𝑒𝑒⃗𝜃𝜃
𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝑟𝑟

(6-12)

Replacing (6-11) and (6-12) in (6-10):
= �𝑟𝑟⃗ × [−𝑉𝑉𝜃𝜃 𝜃𝜃̇𝑒𝑒⃗𝑟𝑟 +

Writing explicitly:
= �(𝑟𝑟𝑒𝑒⃗𝑟𝑟 ) × [−𝑉𝑉𝜃𝜃 𝜃𝜃̇𝑒𝑒⃗𝑟𝑟 +

𝜕𝜕𝑉𝑉𝜃𝜃
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
�⃗𝜃𝜃 × [ 𝑒𝑒⃗𝑟𝑟 + 𝑟𝑟𝜔𝜔𝑒𝑒⃗𝜃𝜃 ]� 𝜌𝜌 − (𝑉𝑉
�⃗𝜃𝜃 × 𝑟𝑟⃗)
𝑒𝑒⃗𝜃𝜃 ]� 𝜌𝜌 − �𝑉𝑉
=
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕

𝜕𝜕𝑉𝑉𝜃𝜃
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝑒𝑒⃗𝜃𝜃 ]� 𝜌𝜌 − �(𝑉𝑉𝜃𝜃 𝑒𝑒⃗𝜃𝜃 ) × [ 𝑒𝑒⃗𝑟𝑟 + 𝑟𝑟𝜔𝜔𝑒𝑒⃗𝜃𝜃 ]� 𝜌𝜌 − ((𝑉𝑉𝜃𝜃 𝑒𝑒⃗𝜃𝜃 ) × (𝑟𝑟𝑒𝑒⃗𝑟𝑟 ))
=
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕

= (0) + 𝑟𝑟𝑟𝑟
= [𝑟𝑟𝑟𝑟

𝜕𝜕𝑉𝑉𝜃𝜃
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝑒𝑒⃗𝑘𝑘 + 𝜌𝜌𝑉𝑉𝜃𝜃 𝑒𝑒⃗𝑘𝑘 − (0) + 𝑟𝑟𝑉𝑉𝜃𝜃
𝑒𝑒⃗ =
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕 𝑘𝑘

𝜕𝜕𝑉𝑉𝜃𝜃
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
+ 𝜌𝜌𝑉𝑉𝜃𝜃
+ 𝑟𝑟𝑉𝑉𝜃𝜃 ]𝑒𝑒⃗𝑘𝑘
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕

(6-13)

Replacing (6-13) in (6-9):

𝜕𝜕
𝜕𝜕𝑉𝑉
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
�⃗ )𝜌𝜌𝜌𝜌𝑉𝑉 = � �𝑟𝑟𝑟𝑟 𝜃𝜃 + 𝜌𝜌𝑉𝑉𝜃𝜃 + 𝑟𝑟𝑉𝑉𝜃𝜃 � 𝑑𝑑𝑉𝑉 𝑒𝑒⃗𝑘𝑘
� (𝑟𝑟⃗ × 𝑉𝑉
𝜕𝜕𝜕𝜕 𝐶𝐶𝐶𝐶
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝐶𝐶𝐶𝐶

(6-14)

Considering:

𝑉𝑉𝑟𝑟 = 𝑟𝑟̇ and 𝑉𝑉𝜃𝜃 = 𝑟𝑟𝜃𝜃̇, into (6-14):

𝜕𝜕
𝜕𝜕𝑉𝑉
𝜕𝜕𝜌𝜌
�⃗ )𝜌𝜌𝜌𝜌𝑉𝑉 = � �𝑟𝑟𝑟𝑟 𝜃𝜃 + 𝜌𝜌𝑉𝑉𝜃𝜃 𝑉𝑉𝑟𝑟 + 𝑟𝑟𝑉𝑉𝜃𝜃 � 𝑑𝑑𝑉𝑉 𝑒𝑒⃗𝑘𝑘
� (𝑟𝑟⃗ × 𝑉𝑉
𝜕𝜕𝜕𝜕 𝐶𝐶𝐶𝐶
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
𝐶𝐶𝐶𝐶

Analyzing every term in (6-15):

96

(6-15)

6.2.1.1.

1st term
𝜕𝜕𝑉𝑉𝜃𝜃
𝜕𝜕𝜕𝜕
[+][+][−/+] = [−/+]
𝑟𝑟

𝜌𝜌

The contribution would be according to the sign of the rate of change of the absolute
tangential velocity and the direction of this velocity (positive if moving in the rotation direction
of the channel and negative otherwise). Then:
If the fluid element moves in the rotation direction:
+

The result is:
If accelerates: +

If decelerates: +

𝜕𝜕𝑉𝑉𝜃𝜃
𝜕𝜕𝜕𝜕

𝜕𝜕𝑉𝑉𝜃𝜃
𝜕𝜕𝜕𝜕

𝜕𝜕𝑉𝑉𝜃𝜃
𝜕𝜕𝜕𝜕

= (+)(+) = (+)

= (+)(−) = (−)

If the fluid element moves against the rotation direction:
−

The result is:
If accelerates: −

If decelerates: −

𝜕𝜕𝑉𝑉𝜃𝜃
𝜕𝜕𝜕𝜕

𝜕𝜕𝑉𝑉𝜃𝜃
𝜕𝜕𝜕𝜕

𝜕𝜕𝑉𝑉𝜃𝜃
𝜕𝜕𝜕𝜕

= (−)(+) = (−)

= (−)(−) = (+)

The torque generation gain is when the local tangential velocity rate decreases and this
velocity is in the same direction of the channel rotation. The second option for gain is when the
local tangential velocity rate increases and this tangential velocity goes against the direction of
the channel rotation.

97

The first statement for torque generation is explained graphically in Figure 6-3, where a
small element in the channel (inertial frame of reference) experiences a local change in tangential
velocity. This change is to a lower value after the head of the expansion wave passes that
location. The gradient of the local tangential velocity at that location decreases but the local
tangential velocity is still in the direction of the channel rotation. This effect produces a gain in
the torque generation as stated above.

Figure 6-3: Changes in local tangential velocity after the head of the expansion wave passes (r1,
θ1) position (t2 > t1).
6.2.1.2.

2nd term
𝜌𝜌 𝑉𝑉𝜃𝜃 𝑉𝑉𝑟𝑟
[+][? ][+] = [? ]

The contribution is according to the sign of the Absolute Tangential Velocity.
If the fluid particle moves in the direction of the channel rotation:
𝜌𝜌 𝑉𝑉𝜃𝜃 𝑉𝑉𝑟𝑟
[+][+][+] = [+]

If the fluid particle moves in the direction opposite the channel rotation:

98

𝜌𝜌 𝑉𝑉𝜃𝜃 𝑉𝑉𝑟𝑟
[+][−][+] = [−]

For the second term, the gain in torque generation happens when the direction of the local
tangential velocity is against the direction of the channel rotation.

6.2.1.3.

3rd term
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
[+][? ][? ] = [? ][? ]
𝑟𝑟 𝑉𝑉𝜃𝜃

The contribution is according to the sign of the Absolute Tangential Velocity and the rate
of change of the density. Two cases are considered:
If the fluid particle moves in the direction of the channel rotation:
If there is an increase in the local density value
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
[+][+][+] = [+]
𝑟𝑟 𝑉𝑉𝜃𝜃

If there is a decrease in the local density value

𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
[+][+][−] = [−]
𝑟𝑟 𝑉𝑉𝜃𝜃

If the fluid particle moves in the direction opposite the channel rotation:
If there is an increase in the local density value
𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
[+][−][+] = [−]
𝑟𝑟 𝑉𝑉𝜃𝜃

If there is a decrease in the local density value

𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
[+][−][−] = [+]
𝑟𝑟 𝑉𝑉𝜃𝜃

99

The gain in the torque generation is favored by two possibilities of the flow behavior. The
first might be when the local tangential velocity is in the same direction of the channel rotation
and the local density value decreases. The other possibility could be when the local tangential
velocity goes against the direction of the channel rotation and the local density value increases.
There are two other parameters not mentioned in the discussion above: r and ρ. Having

the highest values for those two parameters is desirable. In this case, “r” parameter holds highest
values for local radial positions in the channel close to the outlet port. The “ρ” parameter has the
highest values in the region before the head of the expansion wave; however it does not
contribute to torque generation gain. The referred region is not favorable to produce torque due
to the behavior of the fluid, reflected in the parameters of 𝑉𝑉𝜃𝜃 ,

𝜕𝜕 𝑉𝑉𝜃𝜃
𝜕𝜕𝜕𝜕

𝑜𝑜𝑜𝑜

𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕

.

6.2.2. The Net Rate of Flux of Angular Momentum at the Outlet cross section with
Absolute Variables
Replacing (6-3) into (6-8):
�⃗𝑟𝑟 + 𝑉𝑉
�⃗𝜃𝜃 ])𝜌𝜌 �𝑉𝑉
�⃗𝑟𝑟 + 𝑉𝑉
�⃗𝜃𝜃 � ∙ 𝑑𝑑𝐴𝐴⃗
(𝐼𝐼𝐼𝐼) = ∬𝐶𝐶𝐶𝐶 (𝑟𝑟⃗ × [𝑉𝑉

Considering in cylindrical coordinates that:
𝑑𝑑𝐴𝐴⃗ = 𝑑𝑑𝑑𝑑𝑒𝑒⃗𝑟𝑟

And relations in (6-4), (6-5) and (6-6) in cylindrical coordinates:
� ((𝑟𝑟𝑒𝑒⃗𝑟𝑟 ) × [𝑉𝑉𝑟𝑟 𝑒𝑒⃗𝑟𝑟 + 𝑉𝑉𝜃𝜃 𝑒𝑒⃗𝜃𝜃 ])𝜌𝜌 [𝑉𝑉𝑟𝑟 𝑒𝑒⃗𝑟𝑟 + 𝑉𝑉𝜃𝜃 𝑒𝑒⃗𝜃𝜃 ] ∙ (𝑑𝑑𝑑𝑑𝑒𝑒⃗𝑟𝑟 ) =
𝐶𝐶𝐶𝐶

� (0 + 𝑟𝑟𝑟𝑟𝜃𝜃 𝑒𝑒⃗𝑘𝑘 ])𝜌𝜌 [𝑉𝑉𝑟𝑟 𝑑𝑑𝑑𝑑 + 0] = � 𝑟𝑟𝑟𝑟𝜃𝜃 𝜌𝜌 𝑉𝑉𝑟𝑟 𝑑𝑑𝑑𝑑 𝑒𝑒⃗𝑘𝑘
𝐶𝐶𝐶𝐶

𝐶𝐶𝐶𝐶

100

(6-16)

�⃗ )𝜌𝜌 𝑉𝑉
�⃗ ∙ 𝑑𝑑𝐴𝐴⃗ = � 𝑟𝑟𝑟𝑟𝜃𝜃 𝑉𝑉𝑟𝑟 𝜌𝜌𝜌𝜌𝜌𝜌 𝑒𝑒⃗𝑘𝑘
(𝐼𝐼𝐼𝐼) = � (𝑟𝑟⃗ × 𝑉𝑉
𝐶𝐶𝐶𝐶

𝐶𝐶𝐶𝐶

Analyzing the term in (6-16):
𝑟𝑟 𝑉𝑉𝜃𝜃 𝑉𝑉𝑟𝑟 𝜌𝜌
[+][+/−][+][+] = [−/+]

At the outlet, the variable “r” becomes constant for every fluid element crossing the

outlet boundary. The radial local velocity “𝑉𝑉𝑟𝑟 ” keeps a positive value at the outlet while “𝑉𝑉𝜃𝜃 ”

switches to (–) or (+).

For the second term in Equation (6-2), the gain in the torque generation happens when the
direction of the local tangential velocity at the outlet is against the channel rotation direction in
the integral. Likewise, torque benefits when the local radial velocity and the area swept by the
integral retain higher values during the opening process. Mass flow rate is a variable inherently
expressing the behavior of those two parameters (local radial velocity and area swept) as well as
the fluid density.
The outer radius (r) does not change in the analysis; otherwise geometrical condition of
the rotor had to be modified. In general, torque increases as this value becomes greater.
For the purpose of further analysis, this second RHS term is re-written as the second RHS in
Equation (6-18):
𝑅𝑅𝑅𝑅𝑅𝑅2𝑛𝑛𝑛𝑛 = �

𝑐𝑐ℎ−𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜

𝑟𝑟𝑉𝑉𝜃𝜃 𝑉𝑉𝑟𝑟 𝜌𝜌𝜌𝜌𝜌𝜌 = �

𝑐𝑐ℎ−𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜

𝑟𝑟𝑉𝑉𝜃𝜃 𝑑𝑑𝑚𝑚̇

Overall values can be used as an approximation to analyze this second RHS in plots at
every time step through expansion flow time. Then, area-weighted average value for 𝑉𝑉𝜃𝜃 and
integral of mass flow rate (𝑚𝑚̇) can be utilized:

𝑅𝑅𝑅𝑅𝑅𝑅2𝑛𝑛𝑛𝑛 = 𝑚𝑚̇𝑉𝑉𝜃𝜃 𝑟𝑟
101

(6-17)

Thus, the 𝑅𝑅𝑅𝑅𝑅𝑅2𝑛𝑛𝑛𝑛 will be mainly as a function of two parameters
6.3.

Numerical analysis of the Integral Form of the Angular Momentum Equation
The Computational Fluid Dynamic Analysis is accomplished based on Equation (6-2)

with cylindrical coordinate relations of (6-3) to (6-6). Then, it becomes:
�⃗𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 =
𝑇𝑇

𝜕𝜕
�⃗𝑟𝑟 + 𝑉𝑉
�⃗𝜃𝜃 )𝜌𝜌𝜌𝜌𝑉𝑉 + �
�⃗𝑟𝑟 + 𝑉𝑉
�⃗𝜃𝜃 )𝜌𝜌(𝑉𝑉
�⃗𝑟𝑟 + 𝑉𝑉
�⃗𝜃𝜃 ). 𝑑𝑑(𝐴𝐴𝑒𝑒⃗𝑟𝑟 )
� (𝑟𝑟𝑒𝑒⃗𝑟𝑟 )𝑥𝑥(𝑉𝑉
(𝑟𝑟𝑒𝑒⃗𝑟𝑟 )𝑥𝑥(𝑉𝑉
𝜕𝜕𝜕𝜕 𝐶𝐶𝐶𝐶
𝑐𝑐ℎ−𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜

Considering that: 𝑑𝑑(𝐴𝐴𝑒𝑒⃗𝑟𝑟 ) = 𝑑𝑑𝑑𝑑𝑒𝑒⃗𝑟𝑟 , because 𝑒𝑒⃗𝑟𝑟 does not change in this direction
The equation becomes:

=

𝜕𝜕
� ([𝑟𝑟𝑒𝑒⃗𝑟𝑟 𝑥𝑥𝑉𝑉𝑟𝑟 𝑒𝑒⃗𝑟𝑟 ] + [𝑟𝑟𝑒𝑒⃗𝑟𝑟 𝑥𝑥𝑉𝑉𝜃𝜃 𝑒𝑒⃗𝜃𝜃 ]) 𝜌𝜌𝜌𝜌𝑉𝑉
𝜕𝜕𝜕𝜕 𝐶𝐶𝐶𝐶
+ďż˝

𝑐𝑐ℎ−𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜

([𝑟𝑟𝑒𝑒⃗𝑟𝑟 𝑥𝑥𝑉𝑉𝑟𝑟 𝑒𝑒⃗𝑟𝑟 ] + [𝑟𝑟𝑒𝑒⃗𝑟𝑟 𝑥𝑥𝑉𝑉𝜃𝜃 𝑒𝑒⃗𝜃𝜃 ])𝜌𝜌([𝑉𝑉𝑟𝑟 𝑒𝑒⃗𝑟𝑟 . 𝑑𝑑𝑑𝑑𝑒𝑒⃗𝑟𝑟 ] + [𝑉𝑉𝜃𝜃 𝑒𝑒⃗𝜃𝜃 . 𝑑𝑑𝑑𝑑𝑒𝑒⃗𝑟𝑟 ])

Also, taking into account that:
𝑟𝑟𝑒𝑒⃗𝑟𝑟 𝑥𝑥𝑉𝑉𝑟𝑟 𝑒𝑒⃗𝑟𝑟 = 𝑟𝑟𝑉𝑉𝑟𝑟 (𝑒𝑒⃗𝑟𝑟 𝑥𝑥𝑒𝑒⃗𝑟𝑟 ) = 0

𝑟𝑟𝑒𝑒⃗𝑟𝑟 𝑥𝑥𝑉𝑉𝜃𝜃 𝑒𝑒⃗𝜃𝜃 = 𝑟𝑟𝑉𝑉𝜃𝜃 (𝑒𝑒⃗𝑟𝑟 𝑥𝑥𝑒𝑒⃗𝜃𝜃 ) = 𝑟𝑟𝑉𝑉𝜃𝜃 𝑒𝑒⃗𝑘𝑘

𝑉𝑉𝑟𝑟 𝑒𝑒⃗𝑟𝑟 . 𝑑𝑑𝑑𝑑𝑒𝑒⃗𝑟𝑟 = 𝑉𝑉𝑟𝑟 𝑑𝑑𝑑𝑑 cos(0) = 𝑉𝑉𝑟𝑟 𝑑𝑑𝑑𝑑 (1) = 𝑉𝑉𝑟𝑟 𝑑𝑑𝑑𝑑

𝑉𝑉𝜃𝜃 𝑒𝑒⃗𝜃𝜃 . 𝑑𝑑𝑑𝑑𝑒𝑒⃗𝑟𝑟 = 𝑉𝑉𝜃𝜃 𝑑𝑑𝑑𝑑 cos(90) = 𝑉𝑉𝜃𝜃 𝑑𝑑𝑑𝑑(0) = 0
The equation becomes:

𝜕𝜕

= 𝜕𝜕𝜕𝜕 ∭𝐶𝐶𝐶𝐶([0] + [𝑟𝑟𝑉𝑉𝜃𝜃 𝑒𝑒⃗𝑘𝑘 ]) 𝜌𝜌𝜌𝜌𝑉𝑉 + ∬𝑐𝑐ℎ−𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜([0] + [𝑟𝑟𝑉𝑉𝜃𝜃 𝑒𝑒⃗𝑘𝑘 ])𝜌𝜌([𝑉𝑉𝑟𝑟 𝑑𝑑𝑑𝑑 ] + [0]) =
𝜕𝜕

𝜕𝜕

= 𝜕𝜕𝜕𝜕 ∭𝐶𝐶𝐶𝐶 𝑟𝑟𝑉𝑉𝜃𝜃 𝜌𝜌𝜌𝜌𝑉𝑉𝑒𝑒⃗𝑘𝑘 + ∬𝑐𝑐ℎ−𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 𝑟𝑟𝑉𝑉𝜃𝜃 𝜌𝜌𝑉𝑉𝑟𝑟 𝑑𝑑𝑑𝑑 𝑒𝑒⃗𝑘𝑘 = (𝜕𝜕𝜕𝜕 ∭𝐶𝐶𝐶𝐶 𝑟𝑟𝑉𝑉𝜃𝜃 𝜌𝜌𝜌𝜌𝑉𝑉 + ∬𝑐𝑐ℎ−𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 𝑟𝑟𝑉𝑉𝜃𝜃 𝜌𝜌𝑉𝑉𝑟𝑟 𝑑𝑑𝑑𝑑)𝑒𝑒⃗𝑘𝑘
And finally, Equation (6-18) will be used for calculation of torque at each time step as

well with the two terms in the scalar form.

102

�⃗𝑠𝑠ℎ𝑎𝑎𝑎𝑎𝑎𝑎 = (
𝑇𝑇

𝜕𝜕
� 𝑟𝑟𝑉𝑉 𝜌𝜌𝜌𝜌𝑉𝑉 + �
𝑟𝑟𝑉𝑉𝜃𝜃 𝑉𝑉𝑟𝑟 𝜌𝜌𝜌𝜌𝜌𝜌 )𝑒𝑒⃗𝑘𝑘
𝜕𝜕𝜕𝜕 𝐶𝐶𝐶𝐶 𝜃𝜃
𝑐𝑐ℎ−𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜

(6-18)

The sign convention in Equation (6-18) for the generation of torque will be negative and
when any of the RHS terms have this sign they will be counted as a contribution for total torque
production. A positive value would not be desirable, for it consumes the produced torque.

6.3.1. Unsteady Expansion with Constant Cross Section Channel
In Figure 6-16 is shown the torque history and the progress of the two RHS terms of
Equation (6-18) along with the flow time. The rotational velocity is the changing variable while
the channel geometry is kept same for all cases. In the same graph, dashed blue arrow indicates
the time when the outlet port is fully open. This arrow moves to the left as the rotational velocity
increases. This effect reflects the geometric and kinematic conditions of the channel (arc length
and rotational velocity, respectively) and produces the maximum instantaneous torque as
location shifts to the left.
The second dashed green arrow indicates the flow time when the head of the expansion
wave reaches the inlet port wall; this position was found to be where a sudden increase happens
in ∆𝑝𝑝𝑖𝑖𝑖𝑖−𝑖𝑖 plots. These values in plots were quantified by calculating the reduction of the areaweighted average value of static pressure at the wall during the current time with respect to a

previous time step (Equation (6-19)). This is a graphical method in which the eye determines the
position. This becomes an approximated value due to the area-weighted average value at the inlet
wall and the graphical method. In Figure 6-4 is shown an example, the time when the expansion
wave reaches the inlet wall. Plots for other cases can be found in Appendix A.
∆𝑝𝑝𝑖𝑖𝑖𝑖−𝑖𝑖 = 𝑝𝑝𝑖𝑖𝑖𝑖_𝑖𝑖 − 𝑝𝑝𝑖𝑖𝑛𝑛_𝑖𝑖−1
103

(6-19)

In Figure 6-5 was plotted all travel-times for the cases under study. Apparently, all cases
portray the same travel-time for the head of the expansion wave to reach the inlet wall. However,
there is a small deviation with respect to the GO10K case if considered as a reference case. This
deviation increases as the rotational velocity increases, and the maximum deviation is 4.4% at
GO@50K. This implies the head of the expansion wave takes more time to reach the inlet wall
as the rotational velocity increases; this might be associated with centrifugal force effects on the
fluid. Furthermore, in Figure 6-4, the area-weighted average value of static pressure at the inlet
wall (Pin) decreases and ∆𝑝𝑝𝑖𝑖𝑖𝑖−𝑖𝑖 values oscillate periodically before the head of the expansion
wave reaches its location. This effect appears to be linked to centrifugal force effects as well
because this conduct is enlarged as the rotational velocity increases.

Figure 6-4: Graphical method to find the time when expansion wave reaches the inlet wall
(GO@30,000 rpm)
All explanations have hitherto been concerned with time-locations of when channel outlet
is fully open and the head of expansion wave impinges the channel inlet, and the following is the
discussion of generated torque results. Figure 6-16 describes the progress of the torque
generation (blue plot) at each time step. Also, there are two additional plots: the rate of change of
104

the angular momentum (red plot) and the outflow rate of the angular momentum through channel
outlet (green plot). Both graphs were plotted based on the two RHS terms of Equation (6-18),
respectively, and the torque generation (blue plot) results when both plots are summed.
A comment for special recognition in Figure 6-16 is the sign of plots. According to
previous Sections (6.2.1 and 6.2.2), a negative value means a gain in torque or torque generated
during the expansion and a positive value is the opposite meaning, but for easy reading of
graphs, the sign have been inverted.
The rate of outflow of the angular momentum -the green plot- is mainly shaped by 𝑚𝑚̇ and

𝑉𝑉𝜃𝜃 since “rout” coordinate is a constant value for all fluid particles at the channel outlet. All plots
(green curves) start with a growing negative value, reaching a maximum and then decreasing

until reaching a zero value, according to Figure 6-16. This zero value is correlated with zero
tangential velocity (Figure 6-14 (c) and (d)). Thereafter, the rate of outflow of the angular
momentum has positive values, but not fully for GO50Krpm case (Figure 6-16 (f)).

Figure 6-5: Expansion duration and expansion wave time to reach channel inlet

105

The 𝑚𝑚̇ reaches a maximum value at the time the channel outlet is fully open and the full

area of channel outlet favors this maximum, except for instant opening case (Figure 6-14 (a)).

The channel rotational velocity increase causes earlier complete outlet opening, resulting in plots
with its maximum 𝑚𝑚̇ moving to the left.

The instant opening case has its maximum 𝑚𝑚̇ at the beginning as the outlet is fully open.

The starting point of maximum outflow rate of the angular momentum cannot be correlated since
it is mostly influenced by the tangential velocity direction which adds or subtracts to the net of
this outflow rate term. However, the time when 𝑚𝑚̇ decreases is correlated with the decrease of

the rate of outflow of the angular momentum, because the fluid is already following the channel
shape, except for the GO50Krpm case.
The second agent taking an active role in the outflow rate of the angular momentum is the
tangential velocity. The mass-weighted average of 𝑉𝑉𝜃𝜃 provides a glimpse of how it influences the
rate of outflow of angular momentum. This second torque term becomes negative if 𝑉𝑉𝜃𝜃 direction

is same as rotational channel velocity (Figure 6-7(a) and (b)). In all gradual opening cases, at the
beginning of the process, 𝑚𝑚̇ grows gradually as the channel outlet opens in the same manner.
This fact with the quick increase of 𝑉𝑉𝜃𝜃 produces a rise in the negative second torque term.

However, the following increase of 𝑚𝑚̇ does not continue producing a rise in the negative second
torque term because 𝑉𝑉𝜃𝜃 begins to decline until zero value.

In graphs that follow: Figure 6-7, Figure 6-8 and Figure 6-9, 𝑉𝑉𝜃𝜃 distributions at the

channel outlet are plotted. The abscissa coordinates indicate the fractional opening of outlet area,
measured in terms of angular units with respect to the pressure side.
The tangential velocity of the fluid particle at the outlet location is influenced by tip
speed of the channel, pressure gradient in tangential direction, shape of pressure side, and

106

opening speed of channel outlet. These factors act at each location of the outlet differently and
because of that, three regions are considered: near pressure side (NPS), free stream (FS) and far
from pressure side (FPS).
The direction change of absolute tangential velocity from positive to negative is depicted
by the velocity diagram in Figure 6-6, this also shows the relationship with relative velocity and
tip speed. This comparison is done for the same channel geometry and therefore βout angle and
tip speed (U1) are the same values for both (a) and (b) diagrams.

(a) Positive absolute tangential velocity (𝑉𝑉𝜃𝜃 )

(b) Negative absolute tangential velocity (𝑉𝑉𝜃𝜃 )

Figure 6-6: Triangle velocities for (a) positive tangential velocity and (b) negative tangential
velocity in absolute and relative system at the channel outlet. The βout angle and tip speed (U1)
are same for (a) and (b) velocity sketches.

107

Figure 6-7 shows 𝑉𝑉𝜃𝜃 distributions at the channel outlet at each selected point of Figure

5-16. This plot refers to the channel gradual opening case rotating at 10,000 rpm. At point-1, the
fluid flow trajectory in the NPS region is constrained by the pressure side shape and thus, the
fluid develops negative local tangential velocities which overcome the tip speed of the channel.
The FS region appears to be not highly influenced by the pressure side shape and thus, the
negative tangential velocity imposed by the wall shape is not enough to counteract the tip speed
of the channel. On the other side of the opening, or FPS region, as the outlet opens more, the
flow moves with a positive tangential velocity because it is not constrained to follow the channel
shape. The pressure gradient in the tangential direction appears to contribute to this direction
along with the tip speed of the channel.

(a)

(b)

Figure 6-7: (a) Absolute tangential velocities at the channel outlet with gradual opening rotating
at 10,000 rpm, (b) Detail “X”
At point-2, both positive and negative tangential velocities continue increasing as the
fluid flow accelerates. As the channel outlet opens more area, the effect of the pressure side
shape spreads to zones near FS region and causes more portion of fluid to adjust its trajectory as
parallel to the pressure side shape. Therefore, the positive absolute tangential flow velocity
108

decreases and finally results in the negative tangential velocity which contributes to torque
generation. In subsequent points (3 to 8), the fluid continues accelerating due to pressure
gradients and keeps changing direction by the influence of the channel shape. Thus, greater
regions of fluid move with absolute tangential velocity as more outlet area is opened.
At the channel rotational velocity of 50,000 rpm, the tip speed is higher than the previous
case and to produce torque by negative tangential velocities the fluid must counteract this effect.
The plot of 𝑉𝑉𝜃𝜃 distribution at the channel outlet is shown in Figure 6-8. The first four curves are

of the period when the generated torque is negative (Figure 6-16 (f)). The effects of the pressure
side shape and pressure gradient are not enough to produce higher velocities in the opposite
direction of the rotation and overcome the tip speed. In addition, the fast opening effect of
spreading parallel fluid trajectories to other regions do not influence sufficiently for negative
tangential velocities.

(a)

(b)

Figure 6-8: (a) Absolute tangential velocities at the channel outlet with gradual opening rotating
at 50,000 rpm, (b) Detail “X”
The plot in Figure 6-9 displays more regions of negative tangential velocities from the
beginning of the expansion process at instant opening condition. During a short period of time,

109

fluid flow with tangential velocities in the rotation direction occurs, which is why small negative
torque is produced (Figure 6-16(a)). The first six curves belong to that time interval. The fluid
flow develops negative tangential velocities in regions near the pressure and suction side.
The instant opening with respect to gradual opening presents better tangential velocity
distribution at the channel outlet. The instant opening is an ideal condition to produce superior
tangential velocity distributions, and the increase in rotational velocity can reproduce the quick
opening at the outlet in the operation of a real engine. The benefit of fast opening by the
rotational velocity increase is overshadowed by the tip speed effect which the fluid must
overcome to achieve negative tangential velocities.

(a)

(b)

Figure 6-9: (a) Absolute tangential velocities at the channel outlet with instant opening rotating
at 10,000 rpm, (b) Detail “X”
Rate of Change of angular momentum is the other component contributing to the total
generated torque by the unsteady expansion of a gas. This rate of change is composed of three
terms according to Equation (6-15). Each term will be analyzed to determine the predominant
factor in the rate of change. The first term to examine is 𝜌𝜌𝑉𝑉𝜃𝜃 𝑉𝑉𝑟𝑟 .
110

The history of the term 𝜌𝜌𝑉𝑉𝜃𝜃 𝑉𝑉𝑟𝑟 during the unsteady expansion process is plotted in Figure

6-10 and shows a small contribution to the total value of the rate of change. With the increase of
channel tip speed, term 𝜌𝜌𝑉𝑉𝜃𝜃 𝑉𝑉𝑟𝑟 will decrease even more its contribution to torque generation.

This decreasing contribution can be seen in Figure 6-10 (a) and (b) where the negative values
become bigger and the positive values decrease as the rotational velocity speeds up from 20,000
to 40,000 rpm. The comparison with respect to the sum of 𝑟𝑟𝑟𝑟

𝜕𝜕𝑉𝑉𝜃𝜃
𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕

+ 𝑟𝑟𝑉𝑉𝜃𝜃 𝜕𝜕𝜕𝜕 indicates a significant

part of the rate of change comes from the sum of these two terms aforementioned, and so the
focus should be on them.

In Figure 6-11, 20 fixed points within the channel are selected to calculate the time rates
of 𝑉𝑉𝜃𝜃 and 𝜌𝜌 to determine which flow field variable produces the highest time rate. Results in

Figure 6-12 show the time rate of 𝑉𝑉𝜃𝜃 is 1000 times bigger than the time rate of 𝜌𝜌 and because of
that 𝑟𝑟𝑟𝑟

𝜕𝜕𝑉𝑉𝜃𝜃
𝜕𝜕𝜕𝜕

is the main contributor to the rate of change. The plots in Appendix C show the

contribution of each term: 𝑟𝑟𝑟𝑟

𝜕𝜕𝑉𝑉𝜃𝜃
𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕

, 𝜌𝜌𝑉𝑉𝜃𝜃 𝑉𝑉𝑟𝑟 , 𝑟𝑟𝑉𝑉𝜃𝜃 𝜕𝜕𝜕𝜕 and indicates the small contribution of 𝜌𝜌𝑉𝑉𝜃𝜃 𝑉𝑉𝑟𝑟

with respect to the other two. Also, these graphs support the conclusion of the term having
𝜕𝜕𝜕𝜕

𝜕𝜕𝑉𝑉𝜃𝜃
𝜕𝜕𝜕𝜕

with respect to the term having 𝜕𝜕𝜕𝜕 , which contributes significantly to the rate of change. These
results indicate the expansion wave create favorable conditions for the flow field variables
expressed as a time rate of change.
Additionally, torque generation through the rate of change term benefits from the local
𝜕𝜕𝑉𝑉

deceleration or acceleration in the tangential direction, ( 𝜕𝜕𝜕𝜕𝜃𝜃 ). Selected points 8 to 20 show local
tangential velocities in the rotation direction of the channel decelerating. This result contributes
to the gain in torque. Afterward, the fluid, with local tangential velocity opposite the channel
rotation, accelerates at radial location between 97.5 to 100 mm.
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(a) Gradual opening at 20,000 rpm

(b) Gradual opening at 40,000 rpm
Figure 6-10: Rate of Change of the Angular Momentum and components. Expansion wave
impinges inlet wall “↓”.
In addition, the

𝜕𝜕𝑉𝑉𝜃𝜃
𝜕𝜕𝜕𝜕

benefits from the speed at which the outlet port opens. The faster the outlet

opens, the greater is the time rate of local tangential velocity. The quicker opening of the channel
outlet speeds up the process of decelerating and further accelerates the fluid element. Therefore,

112

the rotational velocity increase raises the positive effect on torque generation and this can be
corroborated in Figure 6-16 from (b) to (f), in which the crest of the rate of change moves to the
left side and raises its tip value.
The

𝜕𝜕𝑉𝑉𝜃𝜃
𝜕𝜕𝜕𝜕

is a predominant term among the other terms of the rate of change, contributing

significantly to torque generation. In unsteady expansion processes, the second mechanism
working for the generation of torque is the time rate of the tangential velocity.

Figure 6-11: 20 fixed points selected within the channel of the GO20Krpm case
Centrifugal effect is another factor to be considered when the rotational velocity
increases. In Figure 6-13 is compared the torque history produced by the gas expansion in both a
stationary and rotating channel at 50,000 rpm. The channel outlet opens at the same speed in
both cases. The rotating channel exhibits the history of higher torque values which demonstrates
the effect of centrifugal forces acting on the fluid during the expansion process. Two other
comparison cases are found in Appendix B for the channel rotating at 30,000 and 40,000 rpm.

113

(a)

(b)
Figure 6-12: Time rate of local tangential velocity and density for Gradual Opening at 20,000
rpm

114

Figure 6-13: Rate of change of the angular momentum with rotating and stationary channel
conditions at 50,000 rpm
Figure 6-16 displays all cases tested: Instant opening at 10,000 rpm and Gradual Opening
at 10000, 20000, 30000, 40000 and 50000 rpm. The torque value posted on each graph was
calculated based on the same exergy change in all cases.
The rate of change of angular momentum (RofCH) and the outflow rate of angular
momentum (RofOF) exhibit opposite plot features in all cases. The RofCH works for the
generation of torque from the start of the channel outlet opening to a specific time when RofOF
starts to generate consistent torque and then the RofCH decreases. Both torque terms
complement to continue torque generation until the end of the process. Concerning the
comparison at 10,000 rpm for instant and gradual opening, the first case generates higher torque
value of 52.7% with respect to gradual opening, which makes the instant opening against the

115

gradual opening more desirable. Thus, the increase in rotational velocity was performed to
reproduce the instant opening, but other effects were triggered by the increase.

(a)

(b)

(c)

(d)

Figure 6-14: (a) ᚁ and mass-weighted average values of (b) radial velocity & (c) tangential
velocity at the channel outlet for several rotational velocity conditions, (d) Detail of Tangential
Velocity
No correlation was found between the time when inlet is fully open or the expansion
wave impinges the inlet wall with any maximum in RofCH or RofOF plots, which makes this
unsteady expansion process a complex phenomenon.

116

The percentage of total values for RofCH and RofOF with respect to overall generated
torque is presented in Figure 6-15. These percentages of RofCH and RofOF exhibit the same
opposite characteristics as the plots in Figure 6-16. The rotational velocity increase favors
RofCH within the channel but diminishes the RofOF at the channel outlet. Coincidentally, these
two terms converge to a same overall percentage value at 30,000 rpm and then diverge.
Results in Figure 6-15 indicate the RofCH and RofOF are influenced by the outlet
opening style but inversely proportional. Comparing instant and gradual opening at 10,000 rpm,
the RofOF takes advantage of the quick opening effect to produce higher torque percentage with
respect to the gradual opening case. The RofCH benefits from the gradual opening to generate
higher torque percentage with respect to the instant opening case.

Figure 6-15: Area ratios of Rate of change/ Torque & Surface Outlet/Torque [%] for several
rotational velocities (IO=Instant Opening & GO=Gradual Opening)

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Torque=0.948 J

Torque=0.621 J

(a)

(b)

Torque=1.495 J

Torque=2.117 J

(c)

(d)

Torque=2.221 J

Torque=2.355 J

(e)

(f)

Figure 6-16: History of torque extracted, time rate of change and the net rate of flux of the
angular momentum during the expansion process (Gradual Opening at (b) 10000 rpm, (c) 20000
rpm, (d) 30000 rpm, (e) 40000 rpm & (f) 50000 rpm). “↓”, time when inlet is fully open. “↓”,
time when Expansion Wave reaches the inlet port.

118

CHAPTER 7.
7.1.

CONCLUSIONS AND RECOMMENDATIONS

Conclusions
Factors positively influencing the generation of torque in radial rotor channels during the

unsteady expansion process of the wave disc engine were determined.
In chapter 2, pressure side area and channel length were found to go hand-in-hand to
improve generation of torque. The channel, in which the rise of pressure side area is a result of
increasing height and channel length, showed the largest improvement in expansion efficiency
when compared to channels in which height was reduced and length increased. The area increase
produces more net force applied in the direction of the channel rotation and the increase in
duration of the expansion process contributes to extend the application time of this force.
In chapter 3, exergy approach proves to be an appropriate method that evaluates
objectively the performance of the unsteady expansion process. In exergetic efficiency, work
extracted is compared with the sum of this work and destructed exergy; this term refers to the
exergy no longer available because it was destroyed; flow exergy still can be used in subsequent
expansion stages, however. The expansion efficiency values range between 37 to 67%. These
results become 3 to 5% higher when the efficiency is calculated based on the extracted work over
the internal energy added. In addition, flow exergy appears as potential exergy to use in further
stages, ranging between 81 to 89% of fluid’s exergy change within the channel.
In chapter 4, a zero dimensional macroscopic approximate balance equation was derived
based on the first law of thermodynamics to calculate the unsteady generated work from the
unsteady expansion process. Results show prolonging the duration of unsteady expansion
process enhances the isentropic extracted work toward the maximum value. Longer duration
times produce lower time rates of density change in the channel and channel outlet velocities.

119

Thus, the dynamic action time of the fluid with channel walls is extended. This result is
consistent with the finding in Chapter 2; area is not regarded in this conclusion because the
thermodynamic approach does not provide evidence of it, however. In addition, the gas expands
more efficiently at lower pressure ratios. The pressure differential between the channel pressure
and the surroundings becomes lower causing reduced time rate of density change and channel
outlet velocities.
In chapter 5, the pressure and torque distribution on pressure and suction sides of the
channel are analyzed. Two cases were considered: instant and gradual opening of the channel
outlet. The instant opening causes pressure differentials which improve torque generation at the
early stage of the unsteady expansion, but the gradual opening triggers adverse effects producing
torque opposite the channel rotation direction. Furthermore, both cases indicate approximately
80% of channel length works to produce torque, and a significant fraction in the instant opening
case is produced in the region near the channel outlet. The remaining length does not contribute
actively on torque generation. The instant opening is an ideal case, exhibiting benefits at the
early stage of the process, and the channel rotational velocity increase can reproduce this effect.
Furthermore, torque generation, composed of the action of two effects: the change of the
fluid’s angular momentum within channel and the outflow rate of the angular momentum at the
channel outlet, is investigated. These two components are referred as unsteady and steady
effects, respectively, based on the mechanism to produce torque. Results show torque production
benefits when the channel opens quickly. The increase of rotational velocity approximates the
quick opening. Unsteady effects produce a significant part of the generated torque and the steady
effect can be small at high speed.

120

7.2.

Recommendations for future work
Fluctuations appear in the torque history as a consequence of the fluctuation in plots of

the rate of change of angular momentum. These effects are the result of orientation change on the
line-contours of pressure in the region near the channel outlet. Future work is recommended to
determine the causes, as it influences the torque generation.
The channel outlet’s instant opening is an ideal condition to generate torque, and the
rotational velocity increase aims to reproduce this condition. An adverse effect appears in the
outflow rate of angular momentum, and the investigation into reversing this effect is a
recommended future work, a potential option for enhancing torque generation.
In the exergy section, available remaining exergy in terms of flow exergy is quantified
resulting in high percentages. As a continuation, an investigation into improving torque
generation by adding more rows of radial turbines can be done by applying the exergy approach
to evaluate efficiency and exergy destruction.
In each CFD numerical simulation, the fluid working conditions at the beginning of the
unsteady expansion process had a zero relative velocity with respect to the channel. The reason
for that was to focus the investigation on mechanisms which generate torque without
interference. Future work is to investigate this torque mechanism by adding a real complex
condition inherited from the previous process: combustion.

121

APPENDICES

122

APPENDIX A: Incident expansion wave travel time
Figures present the time when the head of the expansion wave impinges the inlet wall.

Figure 7-1: Incident expansion wave travel time at channel rotational velocity of 10,000 rpm

Figure 7-2: Incident expansion wave travel time at channel rotational velocity of 20,000 rpm

123

Figure 7-3: Incident expansion wave travel time at channel rotational velocity of 40,000 rpm

Figure 7-4: Incident expansion wave travel time at channel rotational velocity of 50,000 rpm

124

APPENDIX B: Centrifugal force effects
Figures depict the influence of centrifugal forces on the rate of change of angular
momentum within the channel

Figure 7-5: Centrifugal effects for channel rotating at 30,000 rpm

Figure 7-6: Centrifugal effects for channel rotating at 40,000 rpm

125

APPENDIX C: Components of rate of change of angular momentum
The following figures show plots of the three terms that constitute the rate of change of
the angular momentum within the channel: 𝑟𝑟𝑟𝑟

𝜕𝜕𝑉𝑉𝜃𝜃
𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕

, 𝜌𝜌𝑉𝑉𝜃𝜃 𝑉𝑉𝑟𝑟 and 𝑟𝑟𝑉𝑉𝜃𝜃 𝜕𝜕𝜕𝜕 .

Figure 7-7: The term rρ(∂Vθ/∂t) plotted for five time steps at every fixed point in the channel at
GO20Krpm case

Figure 7-8: The term ρVθVr plotted for five time steps at every fixed point in the channel at
GO20Krpm case.

126

Figure 7-9: The term rVθ(∂ρ/∂t) plotted for five time steps at every fixed point in the channel at
GO20Krpm case.

127

APPENDIX D: Numerical simulation set-up

Figure 7-10: Numerical simulation set-up for 2-dimensional cad models

Figure 7-11: Numerical simulation set-up for 3-dimensional cad model

128

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129

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