Hyperloop Alpha - Conceptual study

Tiago Manuel Lourenco da Silva Santos Coelho

Thesis to obtain the Master of Science Degree in

Mechanical Engineering

Supervisor: Prof. Jose Maria Campos da Silva Andre

Examination Committee

Chairperson: Prof. Viriato Sergio de Almeida Semiao

Supervisor: Prof. Jose Maria Campos da Silva Andre

Member of the Committee: Prof. Fernando Jose Parracho Lau

June 2016

ii

Acknowledgments

Firstly, I would like to thank Prof. Jose Maria Andre for accepting this thesis and for all the support

and insightful discussions throughout the development of this work.

To my friends, for all the fun and good times, but most importantly for their motivation.

A special thanks to my family for all their support and patience during this endeavour.

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iv

Resumo

Nesta tese vamos analisar um novo conceito de transporte proposto por Elon Musk, o Hyperloop

Alpha. Este sistema consiste numa capsula que atinge velocidades de Mach 0.7 – 0.8, dentro de um

tubo em ambiente rarefeito (100 Pa). A capsula esta equipada com um compressor axial na area frontal,

que recebe o escoamento para reduzir efeitos de estrangulamento entre a capsula e o tubo, o ar rece-

bido e comprimido para ser utilizado na suspensao pneumatica e uma pequena parte e expandida para

propulsao. Tres problemas foram analisados: o tubo em ambiente rarefeito, o escoamento compressıvel

dentro do tubo e o compressor axial na frente da capsula. As principais conclusoes foram: a evacuacao do

tubo pode ser feita por bombas de vacuo industriais; o compressor axial na frente da capsula reduz, ate

um certo limite, o estrangulamento do escoamento em torno da capsula ao viajar no tubo; a suspensao

pneumatica fornece a sustentacao necessaria para a capsula mas os constragimentos dimensionais impos-

tos, o elevado aquecimento por friccao e a instabilidade pneumatica afectam a fiabilidade do sistema; as

principais forcas de arrasto sao a tensao de corte na suspensao (470 N) e a tensao de corte em camada

limite turbulenta ao longo da capsula (40 N); as condicoes adversas do escoamento resultam numa baixa

eficiencia e leve carregamento das pas no compressor axial apresentando tambem problemas em rejeitar

calor eficientemente. O Hyperloop oferece a possibilidade de uma maior eficiencia energetica para um

transporte de alta velocidade mas enfrenta alguns problemas praticos.

Palavras-chave: Hyperloop, Tubo com ambiente rarefeito, Escoamento compressıvel, Suspensao pneumatica,

Compressor axial integrado

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vi

Abstract

The Hyperloop Alpha, a new transportation concept idealized by Elon Musk, is analyzed. This system

consists of a capsule travelling at high subsonic speeds Mach 0.7 – 0.8, inside a low pressure tube (100

Pa). An axial compressor takes air from the front area of the capsule to alleviate choking effects between

the capsule and the tube, the compressed air feeds the gas bearing suspension beneath and a fraction

of this air is expanded at the aft region to produce a small thrust. Three main problems are addressed:

the characteristics of the low pressure tube, the constraints of compressible flow around the capsule

and the on-board axial compressor. The main conclusions are: the air evacuation of the tube can be

achieved with standard industrial vacuum pumps; the on-board compressor reduces choking around the

capsule, but to a limited extent; the air bearing suspension proposed may provide the required lift but the

very tight dimensional constraints, the high skin-friction heating and the risk of pneumatic instability

hinder the suspension feasibility; the air bearing suspension (470 N) and the capsule’s turbulent skin

friction (40 N) are the main drag components; the axial compressor operates in limiting conditions at low

efficiency with poor stage loading resulting in a large number of stages, where heat rejection cannot be

accomplished efficiently. The Hyperloop offers a good energy efficiency but faces many feasibility issues,

such as the tube deflection and tightness, the air bearing suspension reliability and the axial compressor’s

low efficiency and large dimensions.

Keywords: Hyperloop, Low pressure tube, Compressible flow, Air bearing suspension, On-board axial

compressor

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viii

Contents

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Hyperloop Alpha . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2.1 Similar innovative high-speed ground transportation concepts . . . . . . . . . . . . 4

1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Low pressure tube 7

2.1 Vacuum pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Pressure range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.2 Roughing pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Vacuum system modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Energy requirements & cost estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.1 Steady state operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.2 Tube pumpdown costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 What other aspects should be considered? . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Capsule aerodynamics 18

3.1 On-board axial compressor operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1.1 Blockage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Suspension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.1 Wedge pad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2.2 Externally pressurized air bearing . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 Capsule drag estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

ix

3.3.1 Frontbody . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3.2 Main body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3.3 Afterbody . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.4 What other aspects should be considered? . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4 On-board axial compressor 41

4.1 Aerodynamic Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.1.1 Reynolds number effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.1.2 Mach number effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.1.3 Compressor limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.1.4 Very low Reynolds axial compressor . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.1.5 Flow field scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.2 Meanline preliminary design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2.1 Determination of annulus dimensions and rotational speed . . . . . . . . . . . . . . 48

4.2.2 Meanline calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2.3 Compressor characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5 Conclusions and Future Work 56

5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.2 Strengths and weaknesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.2.1 Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.2.2 Energy and Feasibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.3 Possible design changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.3.1 Axial fan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.3.2 Suspension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.4 Suggestions for Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

References 63

Appendix A Compressible Flow A-1

A.1 Quasi-one-dimensional flow model [1][2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-1

A.2 One-dimensional flow model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-4

A.2.1 Fanno flow - adiabatic frictional flow . . . . . . . . . . . . . . . . . . . . . . . . . . A-4

Appendix B Axial compressor extrapolations A-5

Appendix C Heat exchanger convection coefficients A-6

x

List of Figures

1.1 Hyperloop capsule in tube cutaway with attached solar panel arrays. Modified from [3]. . 2

1.2 Hyperloop capsule [3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Hyperloop speed profile. Modified from [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4 Tube cross-section cut. Capsule front. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.5 Suspension pad [3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.6 Hyperloop proposed route. Modified from [3] . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.7 Innovative high-speed ground transport concepts . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 Cost for different pressure ranges. Adapted from [4] . . . . . . . . . . . . . . . . . . . . . 8

2.2 Vacuum pump cost and pumping speed. Dry pumps [Left]: Scroll and Dry rotary lobe;

Oil-sealed [right]: Rotary and roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Typical roughing pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 Vacuum motor Rating and Pumping Speed. Dry pumps [left]: Scroll and Dry rotary lobe;

Oil-sealed [center; right]: Rotary and Roots . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5 Vacuum pump power requirement profile. . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.6 Design 1 - Flush Air-lock chamber (side view). . . . . . . . . . . . . . . . . . . . . . . . . 13

2.7 Design 2 - Docking Air-lock chamber (front view). . . . . . . . . . . . . . . . . . . . . . . 13

2.8 Integrated air-locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1 Capsule (not to scale). Flow is based on the capsule’s reference . . . . . . . . . . . . . . . 18

3.2 Drag divergence Mach. Source: [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3 Converging streamtube into the compressor inlet . . . . . . . . . . . . . . . . . . . . . . . 20

3.4 Straight streamtube into the compressor inlet . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.5 Diffused streamtube into the compressor inlet . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.6 Fanno flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.7 Limit speed for the Hyperloop Alpha specifications. Restrictions are underlined. . . . . . 23

3.8 Compressor choking alleviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.9 Pad views . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.10 Compressibility effects on wedge. Source: [5] . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.11 Cut-view externally pressurized air bearing (Recess volume is purposely exaggerated) . . 28

3.12 Rectangular pad [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

xi

3.13 Flow control locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.14 Air bearing film flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.15 Tube deflection caused by own weight and capsule load. (Not to scale) . . . . . . . . . . . 31

3.16 Hyperloop capsule sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.17 Smooth cowl profile based on NACA 16 series, 4% thickness . . . . . . . . . . . . . . . . . 34

3.18 Mach number in compressor stream tube. Adapted from [2] . . . . . . . . . . . . . . . . . 34

3.19 Momentum balance to the compressor streamtube . . . . . . . . . . . . . . . . . . . . . . 34

3.20 Boattail with symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.21 Momentum balance to the capsule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.23 Drag components for cruise conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.1 Measured pressures on a cascade of C4 airfoils for various Reynolds number. Stagger 36.5o,

camber 30o, solidity 1.0, thickness/chord ratio 0.10. Inlet Mach number < 0.15. Taken

from [7], [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2 Pressure rise across shock wave required to separate boundary layer. Laminar vs Turbulent.

Taken from [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.3 Mach number influence. Bucket-shape regions. Source: [8] . . . . . . . . . . . . . . . . . . 43

4.4 Reynolds number influence. Dashed lines are an approximation on the limits for different

flow behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.5 Qualitative polytropic efficiency trend with assumed uncertainty . . . . . . . . . . . . . . 46

4.6 Expected flow turning trend for maximum efficiency with assumed uncertainty . . . . . . 46

4.7 Stage velocity triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.1 Energy estimates for a journey of 563 km (factor contribution inside bar) . . . . . . . . . 58

5.2 Perceived factor impact on the system feasibility, with associated uncertainty . . . . . . . 59

xii

List of Tables

2.1 Vacuum pump trendlines used in Figures 2.2 and 2.4 . . . . . . . . . . . . . . . . . . . . . 10

2.2 Typical undesired loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Compression processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Pump specifications for capsule handling assuming a polytropic process with n = 1.2 . . . 14

2.5 Estimated undesired gas air loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1 Guideway alternatives. Source: [9] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.1 Compressor characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2 Compressor dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.3 Expected design performance and size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

B.1 Optimistic polytropic efficiency case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-5

B.2 Pessimistic polytropic efficiency case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-5

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xiv

Nomenclature

Acronyms

CFD Computational Fluid Dynamics.

EDS Electrodynamic suspension.

EMS Electromagnetic suspension.

MFR Mass flow ratio.

NPR Nozzle pressure ratio.

Greek symbols

α Pads angle with vertical axis of the capsule.

β Blade angle (Chapter 4).

β Boattail angle (Chapter 3).

Γ Control volume boundary.

γ Specific heat ratio.

∆T0stage Stage stagnation temperature increase.

δ Boattail vertical displacement (Chapter 3).

ηis Isentropic efficiency.

ηm Mechanical efficiency.

ηp Polytropic efficiency.

Λ Bearing speed number (Chapter 3).

λ Air bearing stiffness.

µ Dynamic viscosity.

θ Pad angle with tube surface.

ρ Density.

xv

σ Bearing frequency number.

Ω Control volume.

Roman symbols

A Area.

B Blockage (Chapter 3 - Section 3.1.1).

B Pad width (Chapter 3 - Section 3.2).

Cf Average skin-friction coefficient

C Conductance.

c Speed of sound.

Cd Drag coefficient.

Cf Skin friction coefficient.

Cp Pressure coefficient.

cp Specific heat capacity.

Cz Compressor axial flow velocity.

D Diameter.

D Drag force (Chapter 3 - Section 3.3).

Dh Hydraulic diameter.

di Inside diameter of heat exchanger tubes.

do Outside diameter of heat exchanger tubes.

E Steel Young’s modulus.

F Force.

f Darcy friction factor.

H Cowl smoothness factor.

H Normalized gap height (Chapter 3 - Section 3.2).

h Gap height (Chapter 3).

h Pad-tube gap height.

h0 Absolute enthalpy.

hfg Water enthalpy of vaporization.

xvi

hi Tube side convection coefficient.

ho Shell side convection coefficient.

I Tube moment of area.

k Specific heat ratio.

k Steel tube conduction.

L Length.

l Support spacing. (Chapter 3 - Section 3.2)

m Massflow rate.

M Mach number.

N Rotational speed.

n Polytropic index.

npads Number of pads

W Power

P Normalized pressure.

P Pad perimeter.

p Pressure.

Q Heat transfer rate.

Q Throughput (Chapter 2)

q Volumetric flow rate.

R Gas constant.

r Radius.

rp Pressure ratio.

Re Reynolds number.

S Pump volumetric flow rate.

S Pumping Speed.

T Temperature.

t Time.

Tvap Water vapor temperature.

xvii

u Velocity vector.

U Velocity.

Uclean Overall heat transfer coefficient for clean heat exchanger.

V Volume.

W Flow velocity relative to compressor blade (Chapter 4).

W Load (Chapter 3).

W Work (Chapter 2).

w Normalized load.

x Normalized pad length.

Subscripts

0 Initial conditions (Chapter 1)

0 Section 0 (Chapter 3)

0 Stagnation property (Chapter 4)

1 Final conditions (Chapter 1)

1 Section 1 (Chapter 3)

a Ambient conditions.

c Chord length.

comp Compressor.

cr Critical conditions.

exit Compressor outlet.

f Friction.

L Reference length.

nozzle Nozzle outlet.

r Recess conditions.

ref Reference case.

roots Roots pump.

w Water.

Superscripts

* Critical conditions.

xviii

Chapter 1

Introduction

1.1 Motivation

The Hyperloop is a concept vehicle idealized by Elon Musk, and publicly released in end-August,

2014. The first implementation would be a high speed connection linking the Californian corridor1 from

Los Angeles to San Francisco (563 km) in 35 minutes.

Musk [3] explains the motivation and general technical layout of the concept, labelled the Hyperloop

Alpha2. The paper makes a bold proposition about a new mode of transportation. Faster, cheaper, safer

and energetically self-sustained. Upon public release, the document invites the community to participate

in the development of this project.

The purpose of our work is to study the Hyperloop claims and the technical feasibility of the concept,

with a focus on energy requirements.

1.2 Hyperloop Alpha

The description of the first version of this concept can be found in [3]. The key points are:

Tube: Two (one-way) evacuated tubes of 2.23 m diameter with an average elevation of 6 m and 563 km

length, connect the stations (see Figure 1.1).

Energy: The Hyperloop system is powered by the solar panel arrays that cover the top surface of

the tube (see Figure 1.1).

Speed: The capsule (see Figures 1.1 and 1.2) consists of a streamlined cylindrical body, which is intended

to reach top speeds of Mach 0.91. The mission intended (Figure 1.3) is very ambitious, compressibil-

ity effects are present during nearly the whole journey and a transonic regime during more than half.

1An alternative project for the same route, the California High Speed Rail (CHSR) was recently approved exceeding $68billion of projected budget [10].

2In this study, Hyperloop refers, in short, to the Hyperloop Alpha (the first version of this concept).

1

Figure 1.1: Hyperloop capsule in tube cutaway with attached solar panel arrays. Modified from [3].

However, in this analysis we consider a slightly more conservative Mmax = 0.80 as the upper-bound for

calculations. An efficient aerodynamic design must be achieved to guarantee this speed requirement.

Figure 1.2: Hyperloop capsule [3]

Figure 1.3: Hyperloop speed profile. Modified from [3].

Partial vacuum: To reduce aerodynamic drag and compressor power requirements, the capsule runs

2

inside a low pressure tube (∼ 100 Pa).

Propulsion: Acceleration is achieved by external linear induction motors, separating the rotor inside

the capsule and stator mounted outside at certain locations along the length of the tube (see Figure 1.1).

Air density is low, enabling the capsule to coast for most of the journey, several stators outside the tube

will provide a periodic boost along the tube length. Secondary propulsion is achieved by expanding the

compressed air.

Blockage ratio: In order to reduce capital costs, which are closely related to the tube cross section

Atube, high blockage ratiosAcapsuleAtube

are explored, where Acapsule is the cross section area of the capsule.

See Figure 1.4, where dark gray is the Acapsule and light gray is the Atube.

On-board axial compressor

TubeSuspensionpads

Acapsule

Atube

Figure 1.4: Tube cross-section cut. Capsule front.

Axial compressor: Air is ingested on the front of the capsule alleviating choking effects (see Figure 1.4

and 1.2), providing pressurized air for the suspension (air cushion) and secondary thrust.

Suspension: The capsule has 28 pads along its length that provide a weight-carrying lift through a

wedge effect and an air cushion which uses the pressurized air from the compressor (Figure 1.5). Re-

tractable wheels are considered for low speeds.

Figure 1.5: Suspension pad [3]

Capacity: The capsules carry 28 passengers (see Figure 1.2) with an average departure time of 2 minutes

and a minimum of 30 seconds. This yield an average capacity of 840 passengers/hour, and a maximum

of 3360 passengers/hour.

Route: The Hyperloop route connecting Los Angeles and San Francisco (Figure 1.6) was found consid-

3

ering the following factors [3]: 1. Maintaining the tube as closely as possible to existing rights of way

to minimize purchase costs. (e.g., following the interstate road I-5). 2. Limiting the maximum capsule

speed to Mach 0.91 for aerodynamic considerations. 3. Limiting accelerations on the passengers to 0.5g.

4. Optimizing locations of the linear motor tube sections driving the capsules. 5. Local geographical con-

straints, including location of urban areas, mountain ranges, reservoirs, national parks, roads, railroads,

airports, etc. The route must respect existing structures.

.

Figure 1.6: Hyperloop proposed route. Modified from [3]

1.2.1 Similar innovative high-speed ground transportation concepts

Past high-speed ground transportation concepts, despite having interesting technology features, could

not find financial support.

Aerotrain

In the 70’s, the French engineer, Jean Bertin, created the Aerotrain3 4, a high speed train whose main

features are the air cushion and the high power propulsion (Figure 1.7a). Prototypes and test tracks

were built. The concept was licensed to a company in the US, which built models on different propulsion

systems (Linear induction motor, jet, etc), settling for linear induction propulsion. Unfortunately, the

linear induction technology was still not well understood and the system was very inefficient. The program

in the US was canceled, which likely influenced the French government to opt for a more conventional

high speed train, the TGV (Train a Grande Vitesse).

3Video - http://www.tillierworld.com/index_ENG4Video (francais) - https://www.youtube.com/watch?v=oBFPy-KbrJA

4

(a) Aerotrain (b) Swissmetro

Figure 1.7: Innovative high-speed ground transport concepts

Swissmetro

More recently, in the beginning of 1992, another innovative concept was studied at EPFL (Ecole Poly-

technique Federale de Lausanne), the Swissmetro[11][12]. This high-speed project combines a partially

vacuumed tube (1/10 of atmospheric pressure) with Maglev levitation and linear induction propulsion

(Figure 1.7b). Due to the complicated topography of Switzerland, an underground solution was sought,

which would guarantee the straight path needed for high speeds and easier access to city centers. However,

it was shut-down due to lack of funding 5.

1.3 Objectives

This thesis will address the conceptual feasibility of key aspects in the Hyperloop, focusing on energy

estimates and the flow field inside the tube. The main topics covered are: vacuum generation inside

the tube, capsule high-speed aerodynamics and pneumatic suspension and the on-board axial compressor

operation. The contributions of this thesis include:

Vacuum

1. Vacuum pump selection.

2. Cost of pumping down the entire tube.

Aerodynamics

1. Estimation of the flow field around the capsule.

2. Estimate of the maximum blockage ratio for the desired speed.

3. Analyze the pneumatic suspension.

4. Estimate the capsule drag forces.

5https://swissmetro.wordpress.com [Accessed: 2015-09]

5

Axial Compressor

1. Analysis of the on-board axial compressor operating in a rarefied high speed environment.

2. Efficiency and stage loading of the compressor.

3. Estimate the on-board axial compressor specifications.

1.4 Thesis Outline

Chapter 1 introduced the Hyperloop concept, its main features and challenges. Chapter 2 discusses

the vacuum generation, analyzing pumps, pumpdown time, power consumption and cost of the vacuum

system. Chapter 3 focuses on the flow field around the capsule, namely, the permissible blockage ratio due

to choking effects, the viability of the pneumatic suspension and the capsule aerodynamic forces. Chapter

4 discusses the uncertainty of the axial compressor performance in a low Reynolds, high Mach flow and

provides a preliminary mean-line design of the axial compressor (to access the required power and number

of stages). Chapter 5 presents the main conclusions about the system feasibility and suggestions of future

work to explore the assumptions discussed in the previous chapters.

6

Chapter 2

Low pressure tube

Vacuum conditions provide a low density environment, which results in lower aerodynamic losses.

The drag force (Eq. 2.1) is proportional to the fluid density ρ, the square of the relative velocity U , the

vehicle cross-sectional area A and the drag coefficient Cd.

Fdrag =1

2ρ U2 A Cd(Re,M,Design) (2.1)

However, one must be cautious when drawing conclusions from this simple formula since the dependence

of the Cd with high subsonic speed (high Mach) on a low pressure environment (low Reynolds) is not

trivial. This problem will be discussed in more detail in Chapters 3 and 4.

The fluid density is also directly linked to the power needed to operate the on-board compressor (W =

m∆h0), and by consequence, the size of the batteries and other auxiliary systems.

This chapter investigates the applications of vacuum technology in this project.

2.1 Vacuum pumps

Our goal is a feasible and economical level of vacuum, which relies on high volumetric flow rate pumps

to achieve a low pressure environment and provide a relatively acceptable cost and robust system.

2.1.1 Pressure range

From Figure 2.1, we can see that mechanical pumps achieve a rough-medium vacuum of 10 Pa. The

pressure range of interest is 10 to 105 Pa. Therefore, most vacuum pumps can be mechanical positive-

displacement pumps, also known as roughing pumps (because they are used to start the vacuum process

extracting large amounts of gas from the chamber creating a rough vacuum). The pressure reported in

[3] as a best compromise is around 100 Pa.

7

.

Figure 2.1: Cost for different pressure ranges. Adapted from [4]

2.1.2 Roughing pumps

Oil-sealed and Dry

The two main choices for the primary vacuum stage, are oil-sealed pumps and dry pumps. The option

depends on volumetric flow rate, cost and contamination constraints of the vacuum application. When

high pumping speeds, low cost and low maintenance are desirable, oil-sealed pumps offer a very good

selection of proven technology. Volume flow rate is the main vacuum design requirement of the Hyperloop

due to the large tube volume.

Speed and cost

Available catalogs [13] provide more information on roughing pump costs. Considering the massive

size of the proposed system [3] (a tube with 563 km length and diameter greater than2.2m), high pumping

capacity is of utmost importance. Figure 2.2 shows which pump offers the best return in acquisition cost.

Although roots pumps are well ahead for pumping speeds greater than 200 m3/h, they are very sensible

to overheating1 and must be paired with a more robust rotary pump [4].

For pumping speeds higher than 300 m3/h, a pump combination of rotary vane/piston and roots offers

the best capacity for a given cost. This is a standard choice in the industry and research when large

vacuum vessels need to be evacuated.

Pumping speed and power

The motor power needed to drive the pump is also available in the catalog [13]. Figure 2.4 suggests

that a single-stage rotary vane/piston and roots offers the best operational economy amongst all roughing

1Its an air cooled pump with a specific mechanical clearance, which allows higher speeds but can only start below 4×103

to 8 × 103 Pa, depending on pressure and speed, to avoid excessive thermal expansion which could grip the rotor.

8

0 1000 2000 3000 4000 5000 60000

5000

10D000

15D000

20D000

25D000

30D000

35D000

40D000

45D000

PumpingDSpeed,Dm3/h

Roots

SingleDRotaryDVaneD+DRoots

SingleDRotaryDVane

DualDRotaryDVane

Scroll

DryDRotary

Acq

uisi

tonD

Cos

t,DC

Figure 2.2: Vacuum pump cost and pumping speed. Dry pumps [Left]: Scroll and Dry rotary lobe;Oil-sealed [right]: Rotary and roots

(a) Roots. Source: Pffeifer (b) Rotary vane. Source: KurtJ. Lesker Company.

(c) Rotary piston. Source: Pf-feifer

Figure 2.3: Typical roughing pumps

pumps. We fitted trendlines (Table 2.1) to the original data points (see Figures 2.2 and 2.4). We believe

these lines represent adequately the global trends for selection purposes.

2.2 Vacuum system modeling

Pumpdown time

When dimensioning a vacuum system, after defining the operating pressure, the next major constraint

is pumpdown time. This requirement will determine the size and number of pumps needed and therefore

the overall acquisition cost. A mass balance is applied to a fixed control volume Ω with a boundary Γ

9

Figure 2.4: Vacuum motor Rating and Pumping Speed. Dry pumps [left]: Scroll and Dry rotary lobe;Oil-sealed [center; right]: Rotary and Roots

Table 2.1: Vacuum pump trendlines used in Figures 2.2 and 2.4

Vacuum pumpx = Pumping speed (m3/h)

Cost ($) Power (kW)

Single stage rotary 43.0 x+ 1.37× 103 43.0 x+ 1.37× 103

Single stage rotary + Roots 13.0 xroots + 1.20× 104 43.0 xroots + 1.37× 103

Roots 5.85 xroots + 1.02× 104 43.0 xroots + 1.37× 103

Dual stage rotary 55.6 x+ 7.63× 103 43.0 x+ 1.37× 103

Scroll 143 x+ 5.91× 103 43.0 x+ 1.37× 103

Dry Rotary 250 x+ 4.50× 103 43.0 x+ 1.37× 103

coincident with the walls of the tube,

∫Ω

∂

∂tρ dΩ +

∫Γ

ρ~u · ~n dΓ = 0 (2.2)

where ~u denotes the fluid velocity and ~n represents the outward unit normal.

For consistency with the vacuum technology literature and assuming an ideal gas at constant temperature,

the mass flux will be given in (Pa.m3/s) units, where Q = m RT . Assuming a uniform pressure in V,

Vdp

dt= Qin −Qout (2.3)

Where Qin is the leakage mass flow rate and Qout is the massflow of the pump, where S is the volumetric

flow rate and p is the inlet chamber pressure:

Qout = p · S (2.4)

10

In order to solve the differential equation (2.3),the time-pressure dependence must be known, from the

pump operation curve. However, for an initial estimate, it will be assumed that the volume flow rate is

independent of time. In addition, it is assumed that during pumpdown of the tube, inflow mass rates are

negligible (Qin << Qout).

We can rearrange (2.3) and integrate, ∫dp

p=

∫− SVdt (2.5)

The pressure evolution inside the tube from an initial pressure p0, is thus given by,

p(t) = p0 exp

(− SVt

)(2.6)

and the time required to reach the final pressure p1,

t1 =V

Sln

(p0

p1

)(2.7)

It must be taken into account that the pumping capacity (volumetric flow rate) of the rotary + roots

combination has two distinct stages. As previously explained, the roots pump only operates below a

relatively low pressure of 5 × 103 Pa. Until this pressure level is reached a rotary pumping capacity is

used. In vacuum systems, the system time constant VS , characterizes the operation.

Leakage

Leaks are any undesired flow path into the vacuum chamber. Leakage always exists. Therefore a

tightness criteria must be set according to the system requirements. Most leakage is found on the vessel

joints and ports (welds, flanges, etc).

A vacuum chamber is never perfectly tight, therefore if left isolated from the vacuum pump, the vessel

pressure will continuously rise over time. Most importantly is to ascertain the acceptable leak rate of the

system, making sure the desirable equilibrium pressure is reached and maintained.

An acceptable leakage criteria will be discussed in the last chapter.

According to vacuum textbooks and catalogs [4], [13], typical leak rates are as follows.

Table 2.2: Typical undesired loads

Leakage rate (Pa.m3.s−1)

Min Max

Welding 10−9 10−7

Flanges 10−7 10−6

Pumps 10−6 10−5

Surface loads Negligible at this pressure

11

2.3 Energy requirements & cost estimates

Assuming a uniform pressure and mass density inside the tube, the vacuum pump operates on an

increasing pressure differential, at the same time, density is dropping although the volumetric flow rate

remains constant, resulting in a diminishing mass flow rate. The equations that express these relations

are found in Table 2.3. These two opposite effects dictate the pump power requirement, from Figure 2.5

it can be seen that the power (normalized by the isothermal power) rapidly increases, reaches a maximum

around 4× 104 Pa and then decays. At the start, the pump experiences an increasing specific work and

the mass flow is high, however at a certain pressure level, which corresponds to the system time constant

(V/S), the decreasing mass flow is low enough that it dominates the increasing work needed to compress

air to the atmosphere. Thus the power required to drive the pump is considerably low during most of its

operation and is dominated by friction losses at low pressures.

Table 2.3: Compression processes

Process Specific Work (J/kg) Massflow (kg/s) Pressure (Pa)

Isothermal RT ln(p0p(t)

)ρ0 exp

(−SV t)S p0 exp

(−SV t)

Adiabatic γγ−1RT

[(p0p(t)

) γ−1γ − 1

]

Polytropic nn−1RT

[(p0p(t)

)n−1n − 1

]

Figure 2.5: Vacuum pump power requirement profile.

Friction can be accounted for by assuming a mechanical efficiency, (ηm = 0.80). The specific work also

depends on the compression process – isothermal, adiabatic or polytropic – as seen in Figure 2.5. For this

case, it shall be assumed a polytropic process since it is expected high pumping speed to evacuate the tube.

12

2.3.1 Steady state operation

At nominal conditions, it is considered that, capsule handling (entering/exiting the tube) and leakage

flow rates are the main gas loads.

Capsule handling

In the terminals, the capsule must receive passengers and/or cargo in an efficient and practical way.

An interface between the atmospheric conditions and the vacuum tube must exist. This interface, an

air-lock chamber, must handle a periodic gas load.

Capsule

Atmospheric Pressure

Air-lock chamber

Vacuum

P = 105 PaP = 100 Pa

Figure 2.6: Design 1 - Flush Air-lock chamber (side view).

The capsule is nearly flush with the chamber to ensure a minimal volume load, Figure 2.6. Considering

the capsule length L = 30 m, and diameter D = 1.5 m and an occupation percentage of 95% results in a

volume load of 3 m3. The expected departure interval[3] varies from 30 s to 2 min, a safe margin of 50%

is assumed, 15 s and 1 min. The specifications are given in Table 2.4.

Docking

VacuumP = 100 Pa

Capsule

Atmospheric Pressure

P = 105 Pa

Figure 2.7: Design 2 - Docking Air-lock chamber (front view).

In the second design, Figure 2.7, the capsule is docked to the station for passenger and/or cargo

transfer. Battery replacement and capsule turn-around must also be considered. This design although

cheaper (since it requires nearly no pumpdown of an airlock chamber), could bring some technical or

13

logistic problems if fast change-over is a priority.

Design 1 (Figure 2.6) will be considered since it is faster and simpler, despite possibly less efficient.

Process integration

An alternative method of operation, focusing on the scheduling of the capsules, opens the possibility

to integrate the pressure conversions. This is possible since on arrival, (see Figure 2.8), the pressure must

be raised up to atmospheric conditions to connect with the station. On the other hand, the departure

chamber needs to equalize the tube pressure (100 Pa). By connecting the chambers at specific times, an

equilibrium pressure peq = 50.5 kPa is reached before pumpdown starts, the operating costs are reduced

by a factor of 50%, as seen in Table 2.4. Another option would be to release the gas load in the tube and

use the several pumps across the tube to normalize the pressure level if a given criteria is exceeded.

Arrival

Departure

Intake Outside0venting

PumpdownExhaust Start0journey

Arrive0at0station

m.

Peq0=0500kPa

A1 A2 A3 A4

B2 B3 B4B1

m.

P0=01000Pa

P0=01050Pa Peq0=0500kPa Peq0=0500kPa

Peq0=0500kPaP0=01050Pa

P0=01000Pa

time

Figure 2.8: Integrated air-locking

Table 2.4: Pump specifications for capsule handling assuming a polytropic process with n = 1.2

Pumping Speed Average Power Peak Power Daily cost

(m3.h−1) (kW) (kW) ($)

Min Max Min Max Min Max Min Max

Regular operation 719 | 2149 1433 | 4299 65 130 150 302 135 270

Process integration 551 | 1652 1101 | 3304 33 65 75 151 68 135

Min: 1 min pumpdown; Max: 30s pumpdown

In the pumping speed column, the two values should be read as, rotary | roots

Daily cost ($) = 20h × average power (kW) × electricity pricea ($0.1038/kW.h)

aRetrieved at June 2015 from: http://www.eia.gov/electricity/monthly/epm_table_grapher.cfm?t=epmt_5_6_a

Leakage flow rates. In such a massive system, each section’s tightness must be tested before assembly

to ensure a negligible pressure rise over time. This strongly depends on the welding process and testing

conducted, making it impossible to give an accurate value. Typical leakage flow rates (10−7 Pa.m3/s)

(see Table 2.2) can be used for roughly optimistic estimate purposes.

14

Assuming one connection per pipe, based on a pipe length of 12 m (considering cargo container dimen-

sions2.)

Table 2.5: Estimated undesired gas air loads

Leakage rateNumber of connections

Tube leakage rate

(Pa.m3.s−1) (Pa.m3.s−1)

Welding 10−7 46 667 4.67× 10−3

Flanges 10−7 46 667 4.67× 10−3

Pump 10−6 100 1× 10−4

The total undesired load is estimated to be 9.43× 10−3 Pa.m3/s. For the following design criteria,

p(t) ≤ 1.01 pdesired

pressure shouldn’t rise more than 1% of its nominal value. Neglecting other effects, the time required to

have this pressure rise inside the tube is,

∆t =∆p V

Q=

1× (2.19× 106)

9.43× 10−3>> 1 year

This estimate shows by a safe margin that, the system can be considered tight and analyzed as leakage

independent.

2.3.2 Tube pumpdown costs

As previously suggested, given the size of the vessel and the considerable time needed, the process

should be considered polytropic with an assumed polytropic index of n = 1.2. The energy needed to

bring the tube pressure from atmospheric to 100 Pa, considering a volume (V = 2.19×106 m3) and a

mechanical efficiency (ηm = 0.80) is given by

W = mair ×n

n− 1RT

(p0

p1

)n−1n

− 1

× 1

ηm= 3600 GJ

In [3] the required vacuum pump costs are expected to be around $10 million USD. The assumed peak

power available is 100 MW, below estimated in [3]. A pumpdown time is selected to meet these conditions.

A 2-day pumpdown is achieved with an installed capacity of 195 000 m3/h for rotary pumps and 585

000m3/h for the Roots pumps. The average power is 21 MW. The peak power rounds up to 48 MW. The

electricity price (see Table 2.4) in the Californian region is 0.1038 $/kW.h, which for the energy needed

(10× 105 kW.h) gives an approximate operation cost of $104 000.

Considering the trendlines in Table 2.1, for a single stage rotary pump + roots combination with a

maximum pumping speed of 7000 m3/h, at least 85 vacuum pump systems are needed. The estimated

pump acquisition cost is $8.8 million USD.

2http://www.evergreen-marine.com/tei1/jsp/TEI1_Containers.jsp

15

For the given pumpdown time, the total (acquisition + pumpdown + installation) cost is estimated to

be around the targeted $10 million USD and the peak power doesn’t exceed 50 MW.

Tube cost

Typical values for steel structures (low carbon steel) which include supply, fabrication, transportation

and man-hours can be round up to3 ∼ $4/kg . Considering the tube dimensions, length L = 2×(563×103)

= 1126×103 m, internal diameter = 2.23 m, thickness = 2×10−2 m and ρsteel = 7.9×103 kg/m3, results

on an estimated cost of $5billion. This number is only an order-of-magnitude estimate, it depends widely

on the optimization and scale of the processes involved. However, one can conclude that the tube cost

dominates the project cost. Vacuum pumping costs are less than 1% of the tube cost.

2.4 What other aspects should be considered?

Sectioning the tube. In order to meet with quality and safety standards, the tube should be divided

into segments facilitating testing procedures, maintenance operations and eventual transportation prob-

lems. Although providing proper compartmentalization could reasonably increase the structure cost, it

will ensure a more efficient and safe system.

Maintenance. Its important to monitor in real-time the structural integrity and leakage evolution of

the system, proceeding to the needed repairs, depending on its urgency. Even if no major problems are

detected, a periodic maintenance must be performed. During this period the service can be halved by

scheduling the journeys to run on a single tube.

Thermal expansion Steel expands and contracts ( 12.0 × 10−6m/(m K)) therefore expansion joints

must be installed and be compatible with the low pressure environment. A temperature rise of 10 K on

a 563 km tube, gives an expansion around 68 m. Tolerances in the material, dimensional and surface,

must be carefully specified.

2.5 Summary

The results from this investigation can be summarized as follows:

1. A rotary and roots pump combination exhibit a good performance for the pressure level and size

of this system.

2. Leakage rates are the major operational unknown in the vacuum system. The industry standards

may be overly optimistic and difficult to meet in a vessel of this size. Leakage influences the energy

efficiency (and therefore competitiveness) of the system. Ensuring a reasonable vessel tightness at

a moderate cost is undoubtedly a huge engineering challenge of this project.

3Approximated value for cost estimates of steel structures.

16

3. The pumping power reaches a maximum around half the atmospheric pressure. As the pressure

decreases, the energy required to evacuate the rarefied air is smaller and friction starts to dominate.

4. In steady operation, even assuming a negligible leakage rate, the tube receives periodic amounts

of air from the stations airlocks. The airlock should be nearly flush with the capsule to minimize

pumping power and time for embark/disembark.

5. In steady state, the pump power requirements per tube are estimated to be around 100 kW with a

peak power of 230 kW.

6. A vacuum pumping system to evacuate the tube in 2 days costs around $10 million USD. The tube

is estimated to cost around $5 billion USD. Vacuum pumping costs are negligible (< 1%) compared

to the infrastructure cost.

7. The tube should be sectioned to allow for emergency situations and maintenance.

8. The tube thermal expansion must be accounted for in the design of the system.

17

Chapter 3

Capsule aerodynamics

The capsule achieves high subsonic to transonic speeds inside the low pressure tube. In order to

minimize cost, the tube cross-section area should be minimized, this implies a smaller capsule and a

higher blockage ratio. A schematic of the capsule is presented in Figure 3.1.

Freestream

Frontbody Main body Afterbody

Laminar Transitional TurbulentCowl

Shoulder

Boattail

Pads

Capsule

On-boardCompressor

U

TubePads

α

Pad angle

Pads

Figure 3.1: Capsule (not to scale). Flow is based on the capsule’s reference

In this high speed compressible it is important to minimize drag and guarantee that critical conditions

are avoided. The flow, given the compressibility of gas under high speeds and the viscous effects, is

characterized by two non-dimensional numbers, Mach number (M) and Reynolds number (Re).

M =U

c(3.1) Re =

ρUL

µ(3.2)

Where U denotes the relative velocity, c is the local speed of sound. ρ is the fluid density, L is the

characteristic length and µ represents the dynamic air viscosity.

Compressibility effects

In order to compete with commercial airlines (making a transportation shift valuable to the customer),

speeds similar or higher than aircraft cruise speed should be achieved. For this reason, we shall consider

M = 0.8 − 0.9 as the steady cruise speed. This is a compressible transonic regime, where the effects of

the shock-boundary layer interaction can severely limit the capsule performance (see Figure 3.2).

18

CD

Mdrag

divergence

1 M∞

Figure 3.2: Drag divergence Mach. Source: [1]

Reynolds effects

The flow proprieties for air at pressure 100 Pa and temperature 300 K are given by density ρ =

1.16×10−3 kg/m3, dynamic viscosityµ = 1.85×10−5 Pa.s. The capsule has a length L = 30 m and speed

U = 300 m/s. The Reynolds number based on the capsule length is,

ReL = 5.6× 105

The flow is likely laminar in a significant portion of the capsule, however the Reynolds number is higher

than the critical Reynolds, ReL > Recr. Flow transition will occur making it possible to trip the boundary

layer. Turbulent flow in the afterbody region (see Figure 3.1), improves the capsule performance, attaining

a higher momentum boundary layer which can better resist separation.

3.1 On-board axial compressor operation

The flow in the compressor inlet must be constrained to Mach 0.6, in order to avoid excessive aerody-

namic losses [2] [7] [8] [14]. A useful ratio for high subsonic air breathing vehicles is the mass flow ratio

(MFR) given by

MFR =A1

A2=M2

M1

[1 + γ−1

2 M21

1 + γ−12 M2

2

] γ+12 (γ−1)

(3.3)

The Mach-Area continuity equation provides a practical way to observe the effects of the compressor

operation. The MFR should take into account the following factors:

• Alleviate choking

• Avoid excessive drag

• Provide enough air for the suspension

MFR > 1

The flow is accelerated as it enters the compressor, causing a converging streamtube (Figure 3.3). If the

19

capsule travels at M > 0.6, this violates the previous restriction on the compressor inlet. In addition,

the flow decelerates around the capsule enhancing separation. This regime is not feasible if we expect a

reliable compressor operation.

0 12

M2 > M0

The compressorrecieves flowat excessive speed

Flow accelerates aroundthe capsule

A1 < A0

M1 > M0

Figure 3.3: Converging streamtube into the compressor inlet

MFR = 1

The streamtube forms straight lines that meet with the tip of the cowl (Figure 3.4). The flow is received

at cruise speed, which could hinder the compressor performance at high subsonic speeds (M > 0.6)

A0 = A1

M0 = M1

p0 = p1

M2 M0

0 1 2

The compressorcannot receivehigh cruise speeds

∼

Figure 3.4: Straight streamtube into the compressor inlet

MFR < 1

The flow is decelerated from the freestream conditions to the inlet of the compressor (Figure 3.5). This

MFR enables an isentropic external deceleration, being the only configuration that satisfies the compressor

inlet restriction.

0

12

M2 < M0

The compressorrecieves flowat adequate speed

Flow accelerates aroundthe capsule

A1 > A0

M1 < M0

Figure 3.5: Diffused streamtube into the compressor inlet

The main purpose of the intake of axial compressors, when the vehicle travels at high subsonic

Mach (typically, M ∼ 0.8) is to decelerate the incoming flow and reduce distortion, assuring compressor

efficiency and reliability. The interior of the intake usually acts as a diffuser. However, in this case the

20

high speed laminar flow cannot withstand weak pressure gradients, hence internal diffusion should be

kept to a minimum to prevent stalled flow into the compressor.

3.1.1 Blockage

In order to reduce upfront capital costs, the tube diameter should be minimized. However, the

capsule travels at high subsonic speeds where the flow may reach sonic conditions. This can have serious

detrimental effects on the drag (and stability) of the capsule. Shock waves could form around the capsule

which would cause premature separation and excessive drag. Furthermore, choking effects would increase

the upstream flow density in front of the capsule. The critical blockage ratio (B∗) is

B∗ =AcapsuleA∗tube

(3.4)

Isentropic flow

For a given capsule area (Acapsule), dictated by passenger comfort needs, a minimum tube area (A∗tube)

must be found to avoid choked flow around the capsule. For isentropic flow with constant specific heat,

the ratio only depends on the cruise Mach number,

A

A∗ =Tube cross-section

Bypass cross-section=

A∗tube

A∗tube −Acapsule

(3.5)

If we rearrange the expression, the critical blockage ratio is found,

B∗ = 1−(A

A∗

)−1

(3.6)

Equation 3.6 is only valid for isentropic flow without a compressor, giving a baseline for the critical

blockage. Friction around the mainbody and the on-board compressor must be taken into account

Friction effects around the mainbody

The flow around the capsule is divided into several regimes (see Figure 3.1). Around the mainbody

(see Figure 3.6) the flow accelerates from subsonic to transonic at the shoulder (section 3). This flow has

a constant cross-section, is almost adiabatic and is dominated by friction (Fanno line). The boundary

layer is considered laminar along a considerable extent given the Reynolds number based on the capsule

length is around 5× 105 and the pressure gradient is slightly favourable.

Using the Blasius average friction coefficient (Equation 3.7), taken from [15], the Moody’s friction factor

is

Cf,laminar =1.33√ReL

= 1.75× 10−3 (3.7)

f = 4× Cf,laminar = 7× 10−3

According to [3], the tube and capsule dimensions are,

Dtube = 2.5 m Dcapsule = 1.5 m L = 30 m (approximated)

21

Main body

Adiabatic and frictional

Macr = 1

0 2 3 41

Figure 3.6: Fanno flow

The hydraulic diameter (Dh = 4AP ) is defined by the ratio of the flow cross-sectional area and the wetted

perimeter (capsule cross-sectional perimeter).

f L

Dh= 7.87× 10−2

Assuming Mach 1 at section 3, the upstream condition in section 2 (see Figure 3.7) is,

Mcr,2 < 0.79

On-board axial compressor alleviation

As previously discussed, at high subsonic speeds, MFR < 1. The flow that does not enter the

compressor, accelerates around the capsule. Considering the possibility of having shock-wave/boundary

layer interaction on the cowl (section 1-2 in Figure 3.5), a very smooth profile with a small angle is

necessary. Assuming this condition is satisfied, we neglect friction in the cowl and consider the flow is

isentropic.

The area relation between the freestream (section 0) cross-sectional area A0 (the streamtube that goes

around the capsule) and the cross-sectional area A2 (the same streamtube in section 2) (see Figure 3.7),

A0

A2=

(1 +

1−MFR

(B H − 1)−1

)1−B H

1−B(3.8)

H =

(1− t

Dcapsule

)2

(3.9)

Where t/Dcapsule is a factor for the smoothness of the cowl. Using this expression, where A0 and A2 are

the external streamtube flows and, simultaneously restricting the Mach number at the compressor to 0.6.

22

For MFR = 0.93 and a tube diameter Dtube = 2.5 m, the critical capsule speed is,

Mcr,0 ∼ 0.7 (3.10)

This limiting Mach is below the cruise Mach number considered by Musk [3] (M = 0.91).

The blockage ratio for a given cruise Mach is presented in Figure 3.8. It is interesting to see that for low

blockage ratios adding a compressor does not bring a relevant advantage. However, as speed increases,

a capsule cruising without a compressor would be affected by choking at lower speeds. For a blockage

ratio, B = 0.36, a capsule without compressor could only reach cruise speeds of Mach 0.4, where as using

a compressor it can go up to Mach 0.7. For a cruise Mach 0.8, the tube diameter is 4 m, a 1.6 increase

factor.1

The usefulness of using an on-board axial compressor is limited by aerodynamic losses (M < 0.6), which

is a limiting factor to transonic cruise Mach.

Main body

Adiabatic and frictional

Macr = 1Macr = 0.79

Frontbody

Isentropic

Macr = 0.70

Macomp = 0.60

Freestream

0 2 3 41

A0

A2

Figure 3.7: Limit speed for the Hyperloop Alpha specifications. Restrictions are underlined.

Depending on the capsule design, a high subsonic freestream flow (M > 0.85) on average may produce

sonic conditions locally, due to the curvature of the shoulder. Therefore, this curvature should be smooth.

It should be noted that Mach 1 at section 3 in Figure 3.7, is an undesirable limit because the flow would

be supersonic in the boattail section 3-4.

3.2 Suspension

In order to minimize mechanical friction and maximize the on-board compressor usefulness, the capsule

travels in a pressurized air film, avoiding the high costs of conventional Maglev technology. Musk [3]

proposes 2 alternatives of bearing-like suspension based on viscous flow (fluid film lubrication model).

1Chin [16] presented a similar conclusion about the feasible cruise speed for the given blockage ratio.

23

Cruise Mach

Critica

l bl

ocka

ge r

atio

Compressor blockage

No compressor blockage

Figure 3.8: Compressor choking alleviation

1. Wedge pad, a passive aerodynamic suspension;

2. Air bearing, a compressor pressurized pad;

The capsule and pad specifications [3] can be found below (see Figures 3.1 and 3.9):

Capsule mass estimate: 15× 103 kg Number of pads: 28

Pad dimensions (L × B): 1.5 m ×0.9 m Total pad area: 37.8 m2

Ground clearance (h): ∼ 1× 10−3 m Ambient pressure: 100 Pa

Pad angle (θ): 0.05o Suspension angle (α): 40o

L

hB

(a) Pad dimensions

L

hb1

θ

(b) Wedged pad

Figure 3.9: Pad views

In cases 1 and 2 the lubrication model applies, the gap between the lower surface of the pad and the ground

is 1.0 to 1.5 mm. A reference velocity U = 300 m/s, for air at pressure pa = 100 Pa and temperature

T = 300 K, the air density is ρ = 1.16× 10−3 kg/m3 and the absolute viscosity is µ = 1.85× 10−5 Pa.s .

Hamrock [17] suggests a modified Reynolds number for fluid film lubrication

Re =ρUh2

µL(3.11)

The modified Reynolds number is 1.25× 10−2, thus viscous forces are greater than inertia forces and the

fluid film lubrication model applies.

24

3.2.1 Wedge pad

At low Reynolds number pads with a slight angle behave like a wedge (Figure 3.9), creating an

overpressure between the pad and the ground.

The thin fluid film is described by the Reynolds equation (Eq. 3.12), where h is the gap height, p is the

static pressure, µ is the dynamic viscosity and U is the flow velocity. The equation is obtained from the

Navier-Stokes equations and mass continuity, with following simplifications:

1. Newtonian fluid;

2. Fluid viscous forces dominate over fluid inertia forces;

3. Fluid body forces are negligible;

4. The variation of pressure across the fluid film is negligibly small (i.e. ∂p∂z = 0);

5. The fluid film thickness is much less than the width and length of the pad and thus curvature effects

are negligible. (i.e. h << B and h << L).

The derivation of Equation 3.12 along with the discussion of each term can be found in Hamrock [17].

∇.

(−h3 p

12 µ∇p

)+U

2

∂

∂x(p h) +

∂

∂t(p h) = 0 (3.12)

The expression is non-linear and cannot be analytically solved, a limiting case approach or numerical

tools are needed.

In Equation 3.12, we assume steady state conditions ∂∂t = 0 and an infinite width pad. The equation can

be nondimensionalized to give insight on the flow behavior,

∂

∂x

(PH3 ∂P

∂x

)= Λ

∂

∂x(P H) (3.13)

P =p

pax =

x

LH =

h

b1(3.14)

Λ =6 µ U L

pa h2(3.15)

For high speed flows, the speed number Λ→∞,

1

Λ

∂

∂x

(PH3 ∂P

∂x

)=

∂

∂x(PH) (3.16)

the only way for pressure to remain finite is for

∂

∂x(PH)→ 0

Which implies that

PH = constant

25

and that

pah = constant

This indicates that the pressure gradient is no longer proportional to speed, limiting the amount of lift

generated by the wedge.

1 10 100 10000

0.1

0.2

0.3

0.4

0.5

0.6

Speed Number, Λ

Nor

mal

ized

load

cap

acity

Figure 3.10: Compressibility effects on wedge. Source: [5]

Figure 3.10 was obtained considering an infinite width wedged pad at a ratio of b1/h = 2.2 (similar to

the pad geometry in Figure 3.9) and isothermal flow between the pad and the tube considering its very

small gap height (∼ 1.5 mm), it shows that the wedge effect stabilizes to a given load around a speed

number (Equation 3.15) of Λ = 100, due to compressibility effects.

In our case, the pad speed number is high Λ ∼ 500, according to Figure 3.10 the normalized load(w = W

B L pa

)stabilizes around 0.42. The load carried by the 28 wedge pads is,

W = (0.42× 1.5× 0.9× 100)× 28 = 1588 N

The resultant lift for isothermal high-speed flow is negligible for the assumed conditions.

Wedge pad skin-friction

However, the typical assumption of isothermal flow, presented above given the small air gap, is not

accurate due to the high skin-friction heating (pressure drag is also present but is not considered for the

heating calculations). For the flow conditions, ρ = 1.16 × 10−3 kg/m3, U = 200 m/s, Awedge,in = b1 B

where b1 = 2.2× 10−3 and B = 0.9 m. The massflow that enters the wedge gap inlet is

m ∼ ρ U2Awedge,in (3.17)

m = (1.16× 10−3)300

2(2.2× 10−3 × 0.9) = 3.4× 10−4 kg/s

26

For a dynamic viscosity µ = 2.00× 10−5 Pa/m/s and an average gap height hm = 1.5× 10−3 m, the pad

yields a friction shear stress of

τw = µdu

dy|y=0 (3.18)

τw ∼ µU

hm= 2.00× 10−5 300

1.5× 10−3= 4 Pa

Resulting in a skin-friction power of

Pd = τw Af U (3.19)

Pd = 4 (1.5× 0.9) 300 = 1620 W

The heat transfered to the air and the surrounding walls needs a more detailed numerical study, since

various heat transfer mechanisms are present and the variation of air properties with temperature must

be accounted (e.g. dynamic viscosity). If we assume approximately adiabatic walls (which in practice

would have a heat dissipation capability), the increase of air temperature would be around,

∆T =Pd

m cp,air(3.20)

∆T ∼ 1620

(3.4× 10−4) 1005∼ 4700 K

The air massflow that is absorbing the skin-friction heating is very low resulting in a large temperature

increase. The actual increase in air temperature would be smaller if we consider a proper heat sink in

the area of the pads.

The increase in the air temperature would actually improve the wedge pad weight-carrying capability,

since the increase in the flow velocity results in larger pressure gradients beneath the wedge pad. It is

expected that it could provide a reasonable amount of lift, however an accurate measure of the lift force

could only be obtained through computational fluid dynamics simulations and experimental data.

3.2.2 Externally pressurized air bearing

Musk [3] suggests that the air from the on-board compressor is used to create an overpressure under

the pads. The air bearing needs a constant mass flow rate of pressurized air in order to have a steady and

reliable weight-carrying air cushion. This suspension is also known as externally pressurized bearing, it

uses one or more pressurized air-jets from a supply reservoir at pressure ps to a small recess at pressure

pr which then expands to ambient pressure pa (Figure 3.11). The pressure necessary given the capsule’s

weight, pad dimensions and ambient pressure can be estimated. Assuming pads have a tilt angle α = 40o

(see Figure 3.1) and applying a mild safety factor of 1.15 [6] is,

∆p =1.15 (15× 103)× 9.81

(1.5× 0.9)× 28 cos(40)= 6 kPa (3.21)

The pad area is 1.35 m2, giving a load of 8000 N per pad. The recess pressure is pr = 6 kPa at T = 400

K. In report [3], Musk suggests that the pads are fed by a network of grooves. Rowe [6] presents a similar

27

prpa pa

psṁ

h

dt

Figure 3.11: Cut-view externally pressurized air bearing (Recess volume is purposely exaggerated)

geometry rectangular pad (see Figure 3.12), where the pad area is A = B ×L = 1.35 m2, a/B = 0.2 and

A = 0.62 are empirical geometry factors, giving an effective area Aeff = AA = 0.837 m2.

Figure 3.12: Rectangular pad [6].

This result in a weight-carrying load of

W = (pr − pa)×Aeff = 7784 N (3.22)

This small difference, can easily be corrected by slightly increasing the pad area or raising the recess

pressure by less than 4%.

When designing air bearings choked flow should be avoided because it may give rise to pneumatic hammer

instability [6]. We consider the supply pressure ps = 10kPa, which gives a pressure ratio prps

= 0.6, higher

than the critical pressure ratio p∗

ps= 0.528 for choked flow. The air feed massflow rate m (Figure 3.13) is

usually controlled by an orifice or the cross-sectional area of the throat (circle 1) or inherently controlled

by the recess height hr at the throat annular area (At = πdthr) (circle 2) [6].

prpa pa

ps

1

23 3dt

hrh

ṁ

Figure 3.13: Flow control locations

28

The flow available to the pads is limited since it was imposed M = 0.6 on the compressor face, considering

the compressor inlet cross-section occupies around 85% of the capsule cross-section (Acapsule = 1.77 m2),

flow velocity is U = 204 m/s and no pressure losses occur, the air that enters the compressor is known,

qcomp = 361 m3/s

For an air density of ρ = 1.16× 10−3 kg/m3 the massflow is

mcomp = 0.43 kg/s

Despite the unchoked flow in the recess (Figure 3.13, circle 2), we considered that the air along the gap

of constant height h (from 2 to 3 in Figure 3.13) behaves like a Fanno flow. Given the high pressure ratio

between the recess volume and the ambient conditions prpa

= 60 and the friction along the gap, the flow

would reach critical conditions at the edge of the pad (circle 3).

The maximum mass flow rate per unit area is given by the upstream stagnation conditions, p0 = 6

kPa, T0 = 400 K (assuming cooled gas after compression), according to Equation A.20.

m

A∗ =p0√γ

√R T0

(2

γ + 1

) γ+12 (γ−1)

The flow area of pads is,

A? =[2 (B + L) h

]× 28 = 0.134 m2 (3.23)

For the required overpressure (6 kPa), T0 = 400K, γ = 1.4 and R = 287 J/kg.K,

m = 0.04p0√T0

A? = 1.61 kg/s (3.24)

This value clearly exceeds the compressor capacity, mcomp = 0.43 kg/s, the gap height must be reduced

to meet this constrain, where P = 2 (B + L) = 4.8 m is the pad perimeter and the number of pads

npads = 28, the maximum flow area is

Amax = 0.036 m2 → hmax =AmaxP npads

= 0.27 mm (3.25)

This small gap height can bring feasibility issues to the system.

Externally pressurized suspension skin-friction

The externally pressurized suspension pads shear the flow at high speed while having an applied

pressure gradient in the flow direction. According to Rowe [6], Powell [18] and Hamrock [17], the film

friction for journal bearings can be estimated using Couette flow.

From subsection 3.2.2, the flow velocity U = 300 m/s, the dynamic viscosity µ = 2.29 × 10−5 Pa/m/s,

temperature T = 400 K, recess pressure p0 = 6 kPa. The pad geometry, where h = 0.3× 10−3 m is the

gap height , L = 1.5 m is the pad length and the pad width B = 0.9 m. The friction area of the pad

29

is influenced by the recess volume and the jet arrangement, the friction area per pad is Af = 0.837 m2.

The air bearing pad yields a friction shear stress of

τw = µdu

dy|y=0 (3.26)

τw ∼ µU

h= 20 Pa

The total drag for the 28 pads is,

Dpads = τwAfnpads = 470 N (3.27)

The air film flow is more complex than what we assumed for the drag estimate Equation 3.27. From the

capsule’s reference point, the floor is moving at Ufloor = Ucapsule = 300 m/s and a pressure difference is

present beneath the pad between the recess pressure pr = 6 kPa and ambient pressure pa = 100 Pa. In

Figure 3.14, for the velocity profile 1 it is probable that the shear stress is lower than in velocity profile

2, given the competing effects of the floor velocity and pressure gradient weaken the velocity gradient at

the pad wall. In velocity profile 2, the opposite occurs and the shear stress is expected to be higher. A

numerical study using the fluid film Reynolds equation can give more accurate insight of the flow field

beneath the air bearing pads at such high speeds.

prpa pa

ps

Uwall

1 2

Figure 3.14: Air bearing film flow.

Suspension viability

Air bearing limitations [18] are as follows:

a) Requirements of the pressure and flow of the gas supply (the on-board compressor);

b) Space availability (the capsule and tube dimensions);

c) Manufacturing capability, which dictates the minimum practical clearance (the tube and pad surface

geometric tolerances and the tube maximum deformation wavelength).

In our case, all these factors present challenges for an enclosed capsule inside a tube. However, given

the maximum gap height computed above (hmax = 0.27 mm), manufacturing capability is the current

limiting factor since all the pad variants rely on very small tolerances in the distance between the pads

and the ground. For the capsule traveling at 300 m/s, we consider it may safely transit step irregularities

about half as high as its gap clearance hs = hmax2 = 0.135 mm, where Wcapsule = 1.47 × 105 N is the

30

capsule’s weight. The pads-ground interaction must be very stiff Kd in order to keep the design clearance

(Equation 3.28).

Kd =Wcapsule × 1.15

∆hs∼(1.47× 105

)× 1.15

0.135× 10−3∼ 1× 109 N/m (3.28)

From Rowe [6], the air bearing film stiffness Ka can be defined as

Ka =psAeffh

k × npads =11× 103 × 0.837

0.135× 10−3× 1× 28 = 2× 109 N/m (3.29)

where ps = 10 kPa is the supply pressure, Aeff = 0.837 m2 is the effective pad area, h = 0.135 mm is the

gap height and k = 1 is a non-dimensional factor which usually varies from 0 to 2 [6]. The air bearing

film has the same order of magnitude stiffness that is needed for the pads-ground interaction. This shows

that the air bearing needs a more detailed study to access the suspension stiffness and damping since

Equation 3.29 does not account for high speed effects. In addition, a second suspension, to absorb the

vibrations from the pads is needed.

Deflections around 0.2 mm would compromise significantly the system. This means that the design

constraint of the tube is not the maximum allowed stress but rather the maximum allowed deflection.

A steady state analysis is adequate since the structural wave speed cL for steel (E = 210 × 109 Pa ,

ρsteel = 7850 kg/m3) is cL =√

Eρsteel

= 5000 m/s, which is much higher than the capsule’s nominal speed

U = 300 m/s.

The maximum deflection of the tube ymax (not considering reinforcement stringers) can analyzed as a

cantilevered beam fixed at both ends subjected to an uniform load, where the total load W is the capsule

and the tube’s own weight.

ymax =W l3

384 E I= 1.87× 10−3m (3.30)

Where, according to [3]

Tube moment of area: Ix = 8.95× 10−2 m4 Average support spacing: l = 30 m

Tube self-weight + capsule weight: W = 4.8× 105 N Steel Young’s modulus: E = 210× 109 Pa

l

ymax

hmaxhmin

Figure 3.15: Tube deflection caused by own weight and capsule load. (Not to scale)

The value ymax found in Equation 3.30 shows that tube exhibits an excessive deflection for the dimen-

sional constraints required by the air pads, compliance with the the gap height computed in Equation 3.25

may be difficult to achieve (see Figure 3.15). The misalignment of the pads gap height across the capsule

(hmax and hmin ) has consequences in the pads recess pressure and the compressor flow requirements.

31

Most importantly, there must be a structural design change if the suspension is expected to avoid contact

with the tube. The distance between supports must be shortened or reinforcement ribs used. These

measures come with significant cost penalties, including the inability to use conventional tube construc-

tion. Tight tolerances also imply that even small earthquake deformations could require a costly system

shut-down and realignment of the tube.

Even if the stiffness of the pads’ suspension is high enough to avoid contact, the deflection of the tube

originates a periodic disturbance possibly compromising ride comfort. For a nominal 300 m/s velocity

and 30 m support spacing, the capsule is subjected to a frequency of approximately

freq =300 m/s

30 m= 10 Hz (3.31)

An harmonic tube deformation with a 10 Hz frequency and ymax2 = 9.35×10−3 m amplitude, corresponds

to a vertical movement y of

y(t) = 9.35× 10−3 sin(20πt) (3.32)

ay(t) = −3.7 sin(20πt) (3.33)

The vertical acceleration ay exceeds 0.37g, which may not be tolerated by passengers.

The current suspension and tube structure is therefore, unsuitable for transportation.

Suspension alternatives

The Hyperloop system uses air bearing suspension to reduce infrastructure costs, however its feasibility

could not be proven and the tight tolerances necessary may have significant effect on costs. Conventional

high speed ground suspensions such as Maglev or steel wheels are more matured technologies.

The air bearing cost was obtained from the structural cost of the Transrapid which is the vehicle with

the highest guideway tolerances in [9].

Maglev technologies, electromagnetic suspension (EMS, i.e. magnetic attraction) used in Germany and

electrodynamic (EDS, i.e magnetic repulsion) used in Japan, provide larger gaps making construction

tolerances more permissible, also offering greater operational and safety margins in earthquake-prone

regions.

Despite the uncertainty of these cost estimates, it could cost twice as much as the air bearing. A new

maglev system was developed in the US, the Inductrack2, having a lift/drag ratio higher than 200 and

higher stability efficiency, however more detailed studies are needed to access this technology.

Wheels are stable for speeds up to 90 m/s, but, for the capsule’s nominal speed U = 300 m/s, stability,

overheating and structural lifecycle costs of the tube can bring significant disadvantages for a long-term

operation.

2https://web.archive.org/web/20050309114627/http://www.skytran.net:80/press/sciam01.htm

32

Table 3.1: Guideway alternatives. Source: [9]

Elevated Guideway costsLift/Drag

Gap heightReliability

guideway ($M/km) (mm)

Air bearing 5(1) 150 0.3 to 1

High friction heating. ∆T ∼ 4700 K

High geometric constraints are required

(guideway deflection < 0.01 mm)

Suspension stiffness not proven for high speed rarefied flow

High compression system or high pressure reservoir needed

Maglev 12 > 100 8 to 150

Proven for speeds up to 150 m/s

Lighter structure since mechanical traction is not a factor

Higher gap height and overcome greater grades

Wheels 14(2) 10 000(3) -

Stable for speeds up to 90 m/s

Wheel-track maintenance and heating could cause

problems at higher speeds affecting lifecycle costs

(1) Obtained by subtracting the maglev technology costs from the Transrapid cost breakdown and introducing a factor of 1.25

(2) Cost is increased by the elevated guideway requirement, for on-grade would be $2M/km.

Source: Final Report on the National Maglev Initiative - http://ntl.bts.gov/DOCS/TNM.html(3) Source: [19]

3.3 Capsule drag estimates

Freestream Frontbody Main body Boattail Wake/Jet plume

Isentropic Aprox. Isentropic Adiabatic and frictional

0 2 3 4 51

Figure 3.16: Hyperloop capsule sections

Due to the complexity of the flow, the drag contributions are computed progressing from the front to

the afterbody region.

3.3.1 Frontbody

The flow around the capsule, from sections 1 to 2 (see Figure 3.16), should be subsonic and the

boundary layer over the lateral surface of the capsule, sections 2 to 3, should remain laminar to minimize

drag. To achieve this, the capsule’s frontbody must have a smooth cowl profile (ex.: Figure 3.17) that

doesn’t allow the flow to have local shocks nor leading edge boundary layer transition.

33

Figure 3.17: Smooth cowl profile based on NACA 16 series, 4% thickness

Inlet drag

The flow that enters the compressor is limited to Mach 0.60 (MFR < 1, see Section 3.1.1). Usually in

the aircraft industry, the inlet acts as a diffuser at high subsonic speeds (Figure 3.18). However, in the

Hyperloop since the flow around the pod is bounded and limited by choking effects, the diffuser would

not be of any help since it adds exterior blockage area to diffuse the flow with the added penalty of more

pressure losses in the inner diffuser wall.

A short and thin inlet is recommended, in order to reduce blockage and shock losses, while minimizing

flow distortion into the compressor.

Inlet drag results from the change in momentum from the freestream to the intake plane. A momentum-

v0

A0

M0

p0

v1

A1

M1

p1

v2

A2

M2

p2

0

12

M0

M1

M2

Compressor streamtube

External decelaration

Inlet area

Nacelle

Hub

Compressor

Figure 3.18: Mach number in compressor stream tube. Adapted from [2]

balance to the compressor streamtube (Figure 3.19) is given as

v0A0M0

p0

v1A1M1

p1

0

12

3

p

p

Figure 3.19: Momentum balance to the compressor streamtube

34

m(U1 − U0) = p0A0 +

∫ A1

A0

pdA− p1A1 (3.34)

By subtracting a constant pressure p0 from all the surfaces of the control volume, we can rewrite the

momentum equation as

m(U1 − U0) =

∫ A1

A0

(p− p0)A− ([p1 − p0]A1) (3.35)

The pressure integral on the compressor streamtube gives the inlet drag

Dinlet ≡∫ A1

A0

(p− p0)dA = m(U1 − U0) + ([p1 − p0]A1) (3.36)

The flow properties are, the static pressure and velocity in the freestream (section 0) p0 = 100 Pa and

U0 = 240 m/s, the static pressure and velocity at the inlet (section 1) p1 = 110 Pa and U1 = 205 m/s, the

compressor massflow mcomp = 0.43 kg/s and the inlet area A1 ≈ 1.77 m2. The resulting inlet drag is

Dinlet = 33N

The cowl axial force can be neglected since resultant force acts on a very small cross-section area and the

capsule’s pressure drag force is computed in subsection 3.3.3.

3.3.2 Main body

Capsule friction drag

The capsule travels at high speeds in a low density environment. The flow properties are ρ = 1.16×

10−3 kg/m3, µ = 1.85 × 10−5 Pa.s, L = 30 m, Dcapsule = 1.5 m, U = 300 m/s. The Reynolds number

based on the capsule’s length is,

ReL = 5.6× 105

The Reynolds number is in the range of common transition Reynolds over smooth surfaces, however there

are no adverse pressure gradients along the capsule’s length. On a first approximation, we may assume

the surface is smooth and the freestream turbulence is low enough (this is an important approximation

which may only be validated through experiment) to keep the flow laminar while the pressure gradient

is approximately zero, slightly favourable.

According to [20], for adiabatic subsonic flows, compressibility effects on the friction coefficient are neg-

ligible. Therefore, the friction coefficient can be estimated using the Blasius equation for a flat plate

(repeated here for convenience),

Cf,laminar =1.33√ReL

= 1.75× 10−3

The frontbody can be considered short compared to the capsule’s length, therefore friction effects are

only relevant along the wall of the capsule (sections 1 - 2 in Figure 3.16). According to Equation 2.1, this

35

gives a friction drag of

Df,laminar = 13 N

However, if disturbances are present the boundary layer could transit to turbulent conditions much

earlier. Using Schoernherr correlation [21], we can compute the turbulent skin friction coefficient from

the beginning of the capsule,

1√Cf,turbulent

= 4.06× log(ReL Cf,turbulent)− 0.725 (3.37)

Cf,turbulent = 5.53× 10−3

Which results on a turbulent skin friction drag of,

Df,turbulent = 40 N

The skin friction drag is increased by a factor of 3 if the flow is turbulent from the front of the capsule.

3.3.3 Afterbody

The flow should be laminar and subsonic along the capsule’s length but towards the boattail (Fig-

ure 3.20), the boundary layer must be tripped assuring turbulent conditions to withstand the adverse

pressure gradient.

nsitionalTurbulent

β

δ

U

l

rm

Figure 3.20: Boattail with symmetry

Experimental data about boattail drag

For a subsonic regime, boattail angles around (β ∼ 16o) minimize drag [22]. Making the angle

too small for a given displacement (δ), increases drag, probably due to the increase in skin friction

drag surpassing the pressure drag [23]. However, if transonic flow occurs around the boattail, a smoother

geometry with angles around (β ∼ 8o) and boattail length l and capsule radius rm, l/rm = 3.54 minimizes

separation by attenuating the pressure gradients around the boattail curvature thus reducing viscous and

shock induced separation [24], [25].

36

Payne [22] presents the collected data of several aircraft afterbody drag experiments at high-subsonic

flow regimes but the Reynolds numbers are much higher than the characteristic ones of Hyperloop.

In addition, the Reynolds effects cannot be predicted reliably. The afterbody configuration should be

evaluated as a whole, not simply as the sum of the separate effects. For example, the junction of the

main body with the boattail, creates a suction peak that increases drag and may perturb the boundary

layer.

Available wind-tunnel data and correlations can only provide information about the trend and order-of-

magnitude of drag estimates.

Local separation criterion

In this preliminary study, pressure drag is estimated using a simple separation criterion given by Page

[26], which considers flow separation when a rise in static pressure reaches Cp,crit = 0.38 downstream of

the lowest pressure region.

A BC

D

x

pCpA

pB

pD

Figure 3.21: Momentum balance to the capsule

If we take the dynamic pressure of the lowest pressure point as a reference, the pressure coefficient can

be rewritten as

Cp,crit =psep − pmin

12 γ M

2max pmin

(3.38)

where psep is the pressure at flow separation, pmin is the minimum pressure achieved on the boattail

before the adverse pressure gradient, γ is the specific heat ratio, Mmax is the maximum Mach number

which would correspond to the minimum pressure point (no supersonic conditions considered).

The minimum pressure achieved is 56 Pa, where the maximum Mach number is 0.95, the specific heat

ratio is taken as 1.4, which yields a pressure at separation of 70 Pa.

In the x-axis direction, only boundaries A and C account for the x-component of the resultant pressure

force. This yields an axial force of

Fx = ΣpA = 15N (3.39)

Abeyounis [27] and Presz [28], analyzed several separation criteria for boattails and concluded that none

was sufficiently accurate without experimental data correction. This simple method, also doesn’t take

into account the Reynolds nor the Mach numbers but it is practical for a preliminary estimation, where

the geometry is not entirely known.

37

The pressure drag coefficient is Cd,p = 0.35, this value is uncertain given the flow conditions such as

low Reynolds, blockage effects, disturbances from the capsule pads and transonic conditions, which need

detailed studies to isolate their effect. We believe Page’s criterion assumes an excessive pressure re-

covery since it may not take into account the boundary layer development along the capsule’s length.

A more conservative pressure rise is Cp,crit = 0.15. Repeating the analysis, yields a pressure force of

Fx = ΣpA = 30N , resulting in a pressure drag coefficient of Cd,p = 0.70.

The pressure drag values are low, and even if the unaccounted drag effects result in a higher pressure drag

coefficient, Cd,p = 1, the resultant pressure drag is 45 N. For U = 300 m/s, requires a propulsion power

of only 13.5 kW, which for the whole journey (563 km) represents around 8 kW.h of electric energy.

Nozzle propulsion

According to [3], a fraction of the airflow at pnozzle from the compressor exhaust (mfrac = 0.29 kg/s)

is expanded at the back of the capsule to the tube at ambient pressure p0 (see Figure 3.22) generating

thrust Fg,

Fg = mfrac Unozzle + (pnozzle − p0)Anozzle (3.40)

NPR =pnozzlep0

= 21

The pressure ratio between the compressed air, pnozzle and the ambient outside pressure, p0 is high. Two

different nozzle designs are considered, a convergent and a convergent-divergent (De Laval) nozzle (see

Figure 3.22). A simple convergent nozzle results in a choked underexpanded flow, a convergent-divergent

BoattailConvergent nozzle

Divergent nozzleThroat

Figure 3.22: Boattail/Nozzle shapes. Upper: Convergent, Bottom: Convergent-Divergent

nozzle accelerates the flow to supersonic speed and can be designed to fully expand the flow.

Convergent nozzle - Underexpanded flow

From the report data [3],

mfrac = 0.29 kg/s Vnozzle = 347 m/s P ∗nozzle = 0.53× 2100 = 1113 Pa P0 = 100 Pa

This provides a thrust of

Fg = 166 N

Convergent-divergent nozzle - fully expanded flow

For an isentropic full expansion, pnozzle,end ∼ p0,

38

mfrac = 0.29 kg/s AA∗ = 2.97 T

T0= 0.42 M = 2.63

Two situations are considered, Musk[3] states that the compressor flow is cooled before being expanded,

T0 = 300 K. The temperature after the expansion is T = 126 K, resulting in an incredibly cooled flow

which generates a lower thrust,

Vnozzle = 592 m/s, Fg = 172 N (+4% Fgconv)

On the other hand, if the flow is not cooled, T0 = 857 K, T = 360 K

Vnozzle = 1000 m/s, Fg = 290 N (+75% Fgconv)

A convergent-divergent, non-cooled flow, produces the highest thrust. However, a supersonic jet of 1000

m/s inside a tube must be accounted for, as some problems may arise - structural, wake, noise, etc.

An important side note, as we have seen in the suspension section, the bearings require almost all

of the massflow that enters the compressor to maximize the small gap between the pad and the ground.

Therefore, expanding the compressed flow cannot be achieved with the current suspension mechanism.

3.4 What other aspects should be considered?

Aerodynamic stability

The capsule must have stable pitch, roll and yaw. This could be achieved using compressed air or magnetic

tugging on a central or dual guide rails.

Pressure wave inside tube

An overpressure wave, where air density ρ = 1.16× 10−3 kg/m3, flow speed U = 242 m/s, M = 0.7, and

blockage ratio β = 0.36, has an intensity of

∆p ∼ ρU2

1−M2β

(1 +

3

2β

)= 75 Pa (3.41)

3.5 Summary

The results from this chapter can be summarized as follows:

1. The capsule’s top speed is limited by choking effects in the annular flow between the capsule and

the tube. This speed limit depends on blockage ratio, cruise speed and the capsule friction.

2. The on-board compressor alleviates choking by reducing the pod blockage, allowing higher speeds

in a smaller tube. However, its role is limited by aerodynamic losses inside the compressor since

the flow cannot enter the compressor above Mach 0.6 to prevent severe shock wave losses in the

compressor blades. Therefore only a limited amount of air can enter the intake and blockage caused

by the capsule cannot be entirely eliminated.

39

3. For the tube and capsule dimensions stated by Musk [3], the maximum cruise speed is approximately

Mach 0.7. The capsule’s minimum size is constrained by passenger needs, thus the tube diameter

must increase to achieve higher speeds.

4. It was not possible to ascertain the lift generated by the passive wedge pads given the high heating

effects present.

5. Air bearing pads provide enough lift to support the capsule. The compressor can deliver sufficient

flow and pressure for its operation. Nevertheless the air gap height should be less than 0.37 mm,

since the compressor air supply is limited. For the required overpressure beneath the pads, the flow

in the gap is choked and pneumatic instability can arise. The air pads viability, faces geometric

constraints that may be difficult to implement since the gap height and the deflection of the tube

must be compatible.

6. The low density environment reduces the capsules propulsion requirements, the total drag is ex-

pected to be around 573 N. The two highest components (see Figure 3.23) are the suspension pads

friction (470 N) and the skin friction on the capsule’s body (40 N) in turbulent conditions from the

beginning of the capsule.

7. In this Reynolds number range, the aerodynamic design should enhance a laminar boundary layer

over the majority of the main body. Before reaching the boattail, the boundary layer should

transition to turbulent to withstand a larger adverse pressure gradient, reducing separation losses.

8. Expansion of the compressed air through a nozzle at the rear, can yield a significant amount of

propulsion (170 N); especially, if the gas is not cooled and fully expanded in a convergent/divergent

nozzle (290 N). However, this might be difficult to achieve since the suspension needs nearly all the

compressor massflow to maximize the gap height. In addition, a supersonic jet inside the tube may

be a problem.

Drag force (N)

Friction onmain body

Inlet drag Form dragFriction on thesuspension pads

470

4033

30573

470510543

Figure 3.23: Drag components for cruise conditions

40

Chapter 4

On-board axial compressor

The compressor in front of the capsule is one of the major innovations in this project, setting it apart

from conventional solutions. According to Elon Musk[3], its purpose is threefold. It avoids choking,

allowing a narrower tube thus reducing upfront capital costs. Secondly, it supplies the externally pres-

surized air bearings pads. Lastly, it provides thrust by expanding the compressed air.

This chapter will discuss the proposed solution of an on-board compressor operating in a high-speed, low

pressure environment.

4.1 Aerodynamic Considerations

4.1.1 Reynolds number effect

Conventional compressor blade designs assume that a turbulent boundary layer is present on the

suction side of the blade, allowing for a higher pressure rise while preventing separation or stall [2], [7],

[8]. It is common for the laminar boundary layer formed in the leading edge, to separate when adverse

pressure gradients are encountered (e.g. after the suction peak), creating a separation bubble, which later

reattaches as a turbulent boundary layer [7]. The blade designs account for this effect and, above a certain

Reynolds, laminar separation-reattachment is usually not very detrimental for the blades’ performance

[7]. The problem arises when the Reynolds number is so low that the flow may not be able to reattach.

The blade chord length, c, represents a suitable length scale for the Reynolds number evaluation, as

well as the relative velocity to the blade, W . Many authors [2], [7], [8], [14] point out the dangers of

working with low Reynolds, and establish a minimum chord Reynolds around 2 × 105, to ensure a tur-

bulent boundary layer.

Cascade test data of Rhoden (1956) shown in Figure 4.1(a), suggests that at the lowest Reynolds tested(0.3× 105

), there is large separation with no reattachment. Figure 4.1(b) shows that below Rec ∼ 1×105

the cascade suffers from a rapid increase in total pressure loss and seriously reduced flow deviation, which

41

is probably due to the phenomenon of laminar separation with no reattachment. However, cascade tests

(a) (b)

Figure 4.1: Measured pressures on a cascade of C4 airfoils for various Reynolds number. Stagger 36.5o,camber 30o, solidity 1.0, thickness/chord ratio 0.10. Inlet Mach number < 0.15. Taken from [7], [2]

(steady, low turbulence) do not reproduce compressor test rig results. Since the effects of unsteadiness

and high turbulence inherent to turbomachinery, tend to provide a higher mixing at the boundary layer

and thus enhance its ability to withstand pressure rise [7]. For this reason, for low Reynolds, it is difficult

to predict if a laminar boundary layer will result in a large-scale separation (no reattachment) or an

acceptable laminar separation with turbulent reattachment (separation bubble).

Cumpsty [7] refers studies from Schlichting and Das (1969) on the turbulence influence on laminar transi-

tion indicating that the flow easily transitions to turbulent conditions thus reducing major losses, provided

that the turbulence intensity is higher than 2.5 per cent at a Reynolds number ∼ 1.6× 105.

Koch (1981) also examined the stalling pressure rise capability of axial-flow compressor stages, finding a

much-reduced sensitivity to Reynolds number above 2× 105. The reason is higher turbulence due to the

unsteadiness inherent to a compressor stage [8].

The effect of Reynolds number on shock-boundary layer interaction, may be examined from the work of

Donaldson and Lange (1952), Figure 4.2. For a fixed Reynolds number, the turbulent boundary layer

sustains a pressure rise ∼ 10 times higher than its laminar counterpart.

The previous arguments reinforce the importance of ensuring turbulent flow in low Reynolds operation.

We shall now address compressibility effects in the compressor performance.

4.1.2 Mach number effect

Compressibility effects are critical in high subsonic compressors. Major efforts were made to estimate

and delay the drag-rise Mach (above which losses increase rapidly due to shock-boundary layer interac-

tion), usually recurring to thinner profiles [7] and limiting the Mach number in the compressor face (M

= 0.6), as mentioned in the previous chapter. Operating in high subsonic range raises mass flow rate

efficiency (higher mass flow per unit area) but limits the operating range of the machine (stall/surge

onset), as seen on Figure 4.3.

42

Figure 4.2: Pressure rise across shock wave required to separate boundary layer. Laminar vs Turbulent.Taken from [2]

Figure 4.3: Mach number influence. Bucket-shape regions. Source: [8]

The operating range bucket is thinner as the Mach number approaches unity, meaning that at high

Mach compressibility phenomena are very sensitive to small operating deviations. Excessive incidence

may cause the flow to accelerate locally to sonic conditions creating shock waves that may cause boundary

layer separation.

4.1.3 Compressor limitations

It is therefore expected, that an axial compressor operating on such limiting conditions would experi-

ence considerable losses given its low Reynolds and transonic Mach. The compressibility limits the inlet

Mach number to about (M = 0.6).

Figure 4.4 provides some insight on the constraints associated with the on-board axial compressor.

43

Chord Reynolds Number

Minimum Reynoldsrecommended in compressordesign guidelines

50 Pa 100 Pa 1000 Pa 10 000 Pa

0.2 mBlade chord, m

0.5 m 1 m 5 m0.1 m

at fixed pressure, p = 100 Pa

Tube pressure, Paat fixed chord, c = 0.2 m (assumed)

10 m2 m

Hyperloop design region

Ultra low Reynolds Low Reynolds

Laminar separation Attached flow"rough" surface

Attached flow"smooth" surface

Separation bubble w/turbulent reattachment

If high turbulence,"bypasses" separation

Large separationBubble bursting

Transition Reynolds

Figure 4.4: Reynolds number influence. Dashed lines are an approximation on the limits for differentflow behavior

The chord Reynolds is dangerously low for the actual Hyperloop conditions, this limits the pressure rise

achievable per stage. It also shows that it is difficult to shift the design region to higher Reynold numbers

(∼ 2× 105) while keeping the core characteristics of the project, since for a tube pressure equal to (100

Pa) the blade chord would be too large (10 m). On the other hand, keeping the chord length constant at

a reasonable 0.2 m, the pressure inside the tube would have to rise to 5000 Pa. At this moment, given

the assumptions made, we are left with three options:

1. Adjust the tube working pressure, which given the higher air density, would increase the power

consumption of the whole system by a factor of 50;

2. Explore the possibility of having a laminar axial compressor, since the Reynolds is very low (below

critical), assuming transition to turbulent does not occur;

3. Induce high turbulence and unsteadiness or use a boundary-layer trip mechanism to promote early

transition, reducing the separation losses.

Neither option is an ideal solution but given the framework of current design, the first option is not

considered since it would cause a heavy power requirement penalty in the project. Secondly, designing a

laminar compressor seems impractical. Therefore, the possibility of early transition on very-low Reynolds

will be assumed.

4.1.4 Very low Reynolds axial compressor

It is also important to consider the cost of developing a new compressor for this particular low

Reynolds regime. There isn’t a market for this technology since the efficiency is expected to be poor. A

one-application compressor might bring reliability issues.

44

Viscous effects

Viscosity plays a significant role in this Reynolds range. According to Cumpsty[7], the following

effects are expected:

1. Limited pressure rise. Exceeding this limit may cause flow instability (stall or surge) affecting

compressor reliability and escalating losses;

2. Increased blockage, separated regions reduce the cross-sectional area of the main flow, increasing the

flow velocity. This reduces the amount of work input the blade can pass to the fluid and aggravates

operation at high subsonic Mach numbers;

3. The boundary layer changes the effective blade geometry promoting incidence losses.

Following these effects, a lower efficiency and softer stage loading may be expected. Possible qualitative

trends are presented below.

Efficiency penalty

As previously mentioned, most studies found significant efficiency losses when operating the compres-

sor below Re = 2× 105. This is typically the lowest Reynolds considered in the literature, but it is still

considerably higher than the Reynolds at which Hyperloop would operate (∼ 5×103). Scarce information

can be found on this range. Since compressor blades are not usually designed to operate at such low

Reynolds, significant efficiency penalties are expected.

The lowest Reynolds axial compressor study found is described by Heidelberg [29], Re = 3.6× 104, with

a maximum polytropic efficiency, ηp = 0.72. Roberts [30] among other studies, states that maximum

efficiency (assumed separation is minimized) drops significantly with Reynolds below 1.5× 105.

In lack of lower Reynolds number data, efficiencies were extrapolated using Equation 4.1, based on

compressor literature and reference cases.

1− ηp1− ηp,ref

=

(Re

Reref

)−n

(4.1)

Equation 4.1 suggests that viscous losses follow a power correlation [29] (although deviations are reported).

In this case, separation is likely the primary loss mechanism, therefore typical n coefficients are only

useful as guidelines. The power coefficient n was varied to a give reasonable qualitative trend. Given

the uncertainty present, we extrapolate an optimistic and a pessimistic case, thus assuming that our

design case is in between those limits, the results are in Table B.1 and Table B.2 and can be seen in

Figure 4.5 along 3 possible efficiency scenarios (explained in subsection 4.1.5). A sharp drop is expected

below Re = 1.5× 105, from compressor rig experiments, since the nature of the flow changes. Boundary

layer separation induced by gradual adverse pressure gradient or a shock wave are expected to hinder the

compressor efficiency.

45

Hyperloop design region

Extrapolatedregion

Scenario 1

Scenario 2

Scenario 3

Heidelberg [29]Cumptsy [7]

Figure 4.5: Qualitative polytropic efficiency trend with assumed uncertainty

Reduced stage loading

The stage loading must be reduced to account for thinner velocity profile near the boundary layer,

which makes the flow weaker against adverse pressure gradients, causing premature separation.

In order to establish the pressure rise per stage a separation criteria must be applied. Typically, in

preliminary conditions for turbulent flow, the de Haller criterion or the diffusion factor are used, which

limits the amount of deceleration that a boundary layer can sustain before stall [2], [7], [8]. In the

absence of data on stage loading for this low Reynolds range, values for blade loading are taken from [29].

Contrary to the extrapolation made for the efficiency penalty, a coarse qualitative trend is assumed and

3 possible blade loading scenarios (explained in subsection 4.1.5) are presented in Figure 4.6. A trade-off

between acceptable efficiency and blade loading must be reached by the designer to have an acceptable

number of stages, typically hindering efficiency. Figure 4.6 shows scattered points for the blade loading

but a downward trend is visible for lower Reynolds.

Hyperloop design region

Scenario 1

Scenario 2

Scenario 3

Extrapolatedregion

Constant massflow,variable efficiency [29]

Max efficiency,variable massflow [29]

Figure 4.6: Expected flow turning trend for maximum efficiency with assumed uncertainty

46

Transition and blade design

Blades should be designed to promote early transition. According to Cumpsty [7], turbulence inside

multistage axial compressors is typically high (around 5%) and high flow unsteadiness is present due

to blade passing, enhancing transition. Roberts [30] recommends a sharp-leading edge. Although this

reduces operating range, it achieves better performance for low Reynolds applications due to its desta-

bilizing effect on the boundary layer, which is thought to enhance transition. An interesting approach is

presented in [31], where optimization algorithms are used to design blades. This technique can lead to

unexpected but efficient designs, assuming that the transition models are physically accurate. Although

the tested Reynolds is higher, the study suggests that a fairly steep adverse pressure gradient in the

front of the blade is beneficial to promote transition, avoiding large laminar separation losses. Thin pro-

files, such as DCA (Double Circular Arc) are also preferred at high Mach, minimizing shock wave losses [7].

Regarding numerical simulations of this phenomena, a strong interest in transition prediction exists

nowadays and an accurate model would provide very useful insights on this particular compressor design.

An experimental study is needed to confirm if these effects can indeed cause the intended early transition

and quantify compressor performance. There is the possibility that the Reynolds number is so low, in the

front part of the blade, that these disturbances are smoothed out and a large separation occurs further

downstream.

4.1.5 Flow field scenarios

Given the complexity of the flow field, we shall assume 3 reasonable scenarios (see Figures 4.5 and

4.6) with values for polytropic efficiency ηp, flow turning β and maximum pressure recovery Cpmax ∼

1 −(W2

W1

)2

, where W2 and W1 are the blade relative speeds. Although this formula is derived from

Bernoulli’s equation (lossless, incompressible, steady flow) which despite not being the case we decide to

employ it given its usefulness:

Scenario 1: Attached flow - ηp= 0.60, ∆β = 12o, Cpmax = 0.37;

Transition to turbulent conditions at the leading edge of the blade with minor separation at the trailing

edge.

Scenario 2: Partially attached flow - ηp = 0.50, ∆β = 10o, Cpmax = 0.33;

Flow exhibits a large separation bubble with mild separation at the trailing edge.

Scenario 3: Large separation flow - ηp = 0.40, ∆β = 8o, Cpmax = 0.27;

No transition in the leading edge region with large laminar separation with minor pressure recovery.

In the next section, we will use the values assumed above to perform a first-cut analysis of the axial

compressor through a meanline preliminary design.

47

4.2 Meanline preliminary design

A mean line analysis is a simple and effective tool in this initial axial compressor design. Most design

guidelines in turbomachinery textbooks [2] [7] [8] [14], are mostly valid for turbulent regimes. Extra

attention to viscous effects must be given in this case. According to [3], ambient conditions at the

compressor inlet are in Table 4.1.

Table 4.1: Compressor characteristics

Parameter Symbol Value UnitsInlet temperature T1 298 K

Inlet pressure p1 150 Pa

Inlet Mach number M1 0.6 −Pressure ratio rp 14 −Mass flow rate m 0.43 kg/s

Heat capacity cp 1005 J/kg.K

Specific heat ratio γ 1.4 −Gas constant R 287 J/kg.K

Common design choices and assumptions are:

• No inlet guide vanes (IGV), axial in-flow;

• Constant axial velocity, Cz = 200 m/s;

• Constant mean radius, rm =const.;

• Constant heat capacity of air, cp = 1005 J/kg.K;

• rhub/rtip = 0.4 (could be 0.3);

• Constant mean radius rm;

• ηp depends on the scenario considered (1 to 3).

Other assumptions made will be explicitly noted.

4.2.1 Determination of annulus dimensions and rotational speed

The compressor face flow area is

m = ρ1 A Cz = ρ1 π r2tip

[1−

(rhubrtip

)2]Cz (4.2)

Where the tip radius is

r2tip =

m

ρ1 π

[1−

(rhubrtip

)2]Cz

(4.3)

48

rtip =

√√√√√ 1.00

(1.16× 10−3) π

[1− 0.42

]200

= 0.69 m (4.4)

The hub and mean radius are

rhub = 0.28 m

rmean = 0.49 m (assumed constant)

Usually, compressibility effects become critical before mechanical stresses [8],

Mtip,rel ≤ 1.3 (assumed)

From the inlet velocity triangle,

W 2tip = U2

tip + C2z (4.5)

Utip = 385 m/s

A suitable rotational speed is then evaluated,

N =Utip

2 π rtip(4.6)

N = 88 rev/s = 5300 rpm

Exit conditions

The compressor pressure ratio is rp = 14, resulting in a final stagnation pressure

p0,exit = p1 × rp = 2.1 kPa

From the assumed polytropic efficiencies, the final stagnation temperature can be computed,

T0,exit = T0,1

[p0,exit

p0,1

](n−1)/n

(4.7)

n− 1

n=

1

ηp

γ − 1

γ(4.8)

The outlet conditions are a function of the compressor efficiency. Only one scenario will be carried out

through this preliminary design, scenario 2 (partially attached flow: efficiency = 0.50, Cpmax = 0.33).

The relevant parameters for the other scenarios will be later presented.

Applying the stagnation isentropic relation Equation A.11 and the ideal gas Equation A.9, the local static

conditions at the outlet can be found.

49

Texit = 1316 K

Pexit = 2.1 kPa

ρexit = 5.08× 10−3 kg/m3

Remembering that the axial flow velocity is considered constant, the cross-sectional area of the outlet is

Aexit = 3.5× 10−1 m2

and using the constant mean radius assumption,

rtip,exit = 0.55m

rhub,exit = 0.42m

The outlet Mach number is 0.28.

Table 4.2 summarizes the compressor inlet and outlet dimensions and the rotational speed.

Table 4.2: Compressor dimensions

Parameter Symbol Value UnitsRotational speed N 5301 rpm

Inlet tip speed Utip 385 m/s

Axial velocity Cz 200 m/s

Inlet Mach relative tip speed Mtip,rel 1.3 −Mean radius rm 0.49 m

Inlet

Tip radius rt 0.69 m

Hub radius rh 0.28 m

Outlet

Tip radius rt 0.54 m

Hub radius rh 0.43 m

4.2.2 Meanline calculations

The compressor must provide energy to the fluid to reach the specified pressure at the outlet. How-

ever, only a slight pressure ratio can be achieved per stage.

The energy received by the flow corresponds to an increase in stagnation temperature of

∆T0 = T0,exit − T0,1 = 1316− 298 = 1014 K

A reasonable stagnation temperature rise per stage, T0,stage, is estimated for this compressor.

50

The mean line speed is given by

Um = 2π × rm ×N = 260 m/s

For a purely axial inflow velocity triangle,

tanβ1 =UmCz

; β1 = 53o (4.9)

W1 =Cz

cosβ1= 336 m/s (4.10)

In order to estimate the maximum possible deflection in the rotor (β1 − β2), a criterion is applied to

reduce losses. Usually, in preliminary analysis, a commonly applied criterion, is given by de Haller [8]

which specifies the maximum amount of deceleration to avoid losses. This approach will be followed using

the maximum static pressure rise, Cpmax = 0.33, presented above in the flow scenario assumption. For a

more refined criterion, the diffusion factor [8] [7], which takes into account the stage and blade geometry

utilized.

The maximum deceleration is by given by the ratio of relative velocities between the rotor W1 and

the stator W2, (see Figure 4.7) which can be found from our assumed Cp,max (see subsection 4.1.5),

W2

W1≥√

1− Cp,max ≥ 0.82 (assumed) (4.11)

This deceleration estimate doesn’t take into account the flow compressibility but is sufficient for our

calculation. On this basis the minimum relative velocity is,

W2 = 0.82× 336 = 276 m/s

From the stator velocity triangle (Figure 4.7), The blade angle β2 is

cosβ2 =CzW2

; β2 = 43o

We have the required parameters to estimate temperature rise per stage. At this point, the Reynolds

number should be checked.

Considering a blade chord of 0.2 m, the Reynolds number is given by

Rec =ρ W c

µ(4.12)

Rec =(1.16× 10−3)× 336× 0.2

1.85× 10−5= 4.21× 103

The Reynolds number is indeed much lower than normally encountered in axial compressors, therefore

the precautions are justifiable, in order to achieve a reasonable compressor operation.

51

Figure 4.7: Stage velocity triangles

4.2.3 Compressor characteristics

Number of stages

Using the blade meanline design and neglecting blockage effects, the Euler turbomachinery equation

can be applied to estimate the number of stages required,

∆T0stage =Um Cz (tanβ1 − tanβ2)

cp(4.13)

∆T0stage =260× 200 (tan 53o − tan 43o)

1005= 22 K

The total number of stages nstages assuming a constant aerodynamic loading is

nstages =∆T0

∆T0stage

= 29

The results for the three flow scenarios are given in Table 4.3.

Table 4.3: Expected design performance and size

Scenario ηp Cpmax Flow turning [ o] nstages ∆T0stage [K] Tout [K] Viscosity [Pa.s] Power [kW]

1 0.60 0.37 12 29 26 1030 4.20× 10−5 2622 0.50 0.33 10 47 22 1315 4.85× 10−5 3653 0.40 0.27 8 90 18 1905 6.01× 10−5 577

Blockage effects and other losses are not accounted in this simple analysis, the compressor will most likely

52

need more stages.

Observing the large number of stages and assumed rotor blade chord (0.2 m), the length of the compressor

is problematic possibly exceeding 10 m and weighting more than 1 tonne. This can bring unforeseen size

and weight penalties to the design, as well as efficiency issues to align the stages.

Cooling requirements

An intercooler is needed to cool the hot flow (T > 1000 K) coming from the first compressor. Chin

[16] suggests that the heat exchanger does not have a significant impact in the tube temperature to justify

its subsystems’ volume and weight penalties. The rarefied ambient conditions make the heat exchanger

ineffective and given its size and weight may not be feasible to implement on the capsule.

The requirements of the heat exchanger for a air massflow mair = 0.43 kg/s, an exhaust temperature of

Tout = 1030 K and a specific heat cp = 1.005 kJ/kg.K are

Q = m cp ∆T = 316 kW (4.14)

According to Musk [3] water is the refrigerant fluid. It is worth exploring its latent heat capacity since it

is much higher then its sensible heat capacity. For water at 1 atm, the vapor temperature is Tvap = 373

K and the vaporization enthalpy hfg = 2258 kJ/kg.

Assuming the refrigerant water is allowed to boil but not exceed Tvap, the required water massflow mw is

Q = mw × 4.186× (373− 293) + mw × 2258 = 316 kW (4.15)

mw = 0.122 kg/s

The calculation of the shell and tube convection coefficients are presented in Appendix C. The tube

convection coefficient is high given the favorable phenomena present in phase change. The overall heat

transfer resistance is1

Uclean=

1

ho+dodi

1

hi+do ln(do/di)

2k(4.16)

1

Uclean=

1

328+

0.008

0.005

1

3.7× 104+

0.008 ln(0.008/0.005)

2× 60(4.17)

The resultant overall heat transfer coefficient is

Uclean =1

3.05× 10−3 + 4.32× 10−5 + 3.13× 10−5= 320 W/m2K (4.18)

Water is boiling inside the tubes, robust safety mechanisms must be present to prevent a boiling crisis.

53

The heat exchanger dimensions are,

Shell inside diameter: 0.165 m Number of tubes: 200

Shell thickness: 0.018 m Tube outside diameter: 0.008 m

Shell length: 5 m Tube inside diameter: 0.005 m

Steel density: 7850 kg/m3

The heat exchanger has an approximate mass of 800 kg and a volume of 2.6 m3. The cooling water

has a massflow rate of mw = 0.122 kg/s. Considering a 35 minute journey, we need 256 kg of water

on-board, which given the liquid water specific volume vliq = 1 × 10−3 m3/kg occupies 0.256 m3, but,

as noted by Chin [16], its vapor specific volume is much higher, for water at 1-2 atm vvap = 2-1 m3/kg

resulting in a infeasible storage volume. In order for the vapor storage volume to occupy less than 10%

of the capsule available space, the superheated vapor would have to be stored in pressurized tanks of at

least 10 atm.

Despite the effort to maximize heat transfer using convection boiling, even not accounting for the

second compressor, motor and battery cooling, significant weight and volume penalties exist, which make

the cooling system infeasible for the present case.

4.3 Summary

The investigation on the axial compressor operation resulted in the following considerations:

1. The technology is not well-developed for such a low Reynolds. According to the data collected, no

axial compressor off-the-shelf would match the project’s initial design requirements.

2. A detailed research (CFD simulations and compressor rig experiments) on the compressor design

and performance is of great importance for this project. There is a significant lack of data for this

particular regime and the compressor is crucial in the Hyperloop concept.

3. Compressor polytropic efficiency is probably much lower (ηp ∼ [0.40; 0.60]) than usually assumed

in current state-of-the-art axial compressors (ηp ∼ [0.85; 0.90]).

4. For a turbulent boundary layer, smaller Reynolds yield greater viscous and separation losses and

produce a larger displacement thickness, changing the blades/channel geometry in an adverse way.

This blockage increases the through flow velocity, reducing the work input and aggravating incidence

losses and shock effects.

5. Transition must be promoted as early as possible. This can be accomplished by an adverse pressure

gradient, high free-stream turbulence and/or boundary layer tripping mechanisms such as wires or

roughnesses.

6. A front-loaded (fairly steep adverse pressure gradient) thin blade design that ensures early tran-

sition in the leading edge region, could improve the compressor performance by preventing large

separation/bubble bursting and reduce shock losses.

54

7. The low Reynolds and high Mach flow severely limit the axial compressor operation. Efficiency is

expected to drop along with a reduced stage loading. The assumed efficiency and stage loading

(ηp = 0.6, Cp,max = 0.37) indicate that in a best case scenario the first compressor requires 262

kW.

8. The axial compressor requires around 29 stages to achieve the desired pressure ratio (rp = 14), the

compressor size and weight will be a problem on the feasibility of the whole system. In addition, the

high pressure ratio together with the low efficiency operation results in a high temperature exhaust

flow, that needs to be cooled since it affects other parts of the system.

9. Cooling the compressor exhaust flow presents a major feasibility problem since the ambient con-

ditions are rarefied. The heat exchanger size and weight may not be possible to implement in the

capsule.

10. The tube pressure will likely have to increase to a minimum value deemed acceptable by future

compressor studies.

55

Chapter 5

Conclusions and Future Work

5.1 Conclusions

A preliminary analysis of three key aspects of the Hyperloop proposal was presented in this thesis.

The main findings of the previous chapters are:

Low pressure tube

- Generating a low pressure environment is feasible with industrial vacuum mechanical boosters (roots

and rotary combination). The cost of the pumping system depends heavily on its time constant V/S

(tube volume/pumping volumetric rate). Once the volume of the tube is established (which is a function

of the capsule cruise speed), the desired pumpdown time becomes the determinant factor. The tube

must be projected and tested to guarantee an acceptable tightness level. Leakage rates are the main

unknown of the present vacuum estimates.

- On steady state operation, considering the tube is leakage-independent, the mass of air inside the tube

increases only due to the air introduced during capsule departure and arrival.

- Power requirements reach a peak when the pressure is close to half the atmospheric value. From that

point onward, the decreasing massflow rate reduces the power needed. In the limit, power consumption

of the pump is dominated by the its internal friction.

- The tube pumpdown time is approximately 2 days and the operational cost is $104 000 dollars.

- The vacuum pump acquisition cost is $9 million dollars, which is negligible compared to the estimated

tube cost $5 billion dollars.

Capsule aerodynamics

- The capsule top speed is limited by choking in the annular space around the capsule. This is governed

by the blockage ratio, cruise speed and friction. The on-board compressor allows higher speeds for a

given blockage ratio in the annular space but its role is limited by aerodynamic losses (Mach 0.6 at the

56

inlet). For the given capsule and tube specifications, the maximum cruise speed cannot exceed Mach

0.7. Achieving Mach 0.8 requires an increase of the tube diameter by a factor of 1.6 resulting in a 4 m

diameter tube.

- The aerodynamic wedge effect bearing is governed by the high skin-friction heating effects and its lift

cannot be calculated without CFD simulations or experiments. The externally pressurized air bearing

pads provide the required weight-carrying lift, but the necessary overpressure may lead to choked

flow conditions in the periphery of the pads. This condition can cause pneumatic hammer instability

affecting the reliability of the system. The gap height between the pads and the tube dictates the

massflow requirements from the compressor. The compressor massflow rate is constrained by the

capsule size and the maximum inlet speed (Mach 0.6). For this flow rate the gap height cannot exceed

0.5 mm. This clearance poses a constraint to the pads and tube geometric tolerances and the tube

deflection.

- The major contributions to drag are the suspension pads friction (470 N) and the capsule’s turbulent

skin-friction (40 N). The rear nozzle could partly mitigate these effects but given the mass flow required

for the suspension, no significant mass flow is available for propulsion.

On-board axial compressor

- The low Reynolds and high Mach flow severely limit the axial compressor operation. Efficiency is

expected to drop along with a reduced stage loading. The assumed efficiency and stage loading (ηp =

0.6, Cp,max = 0.37) indicate that in a best case scenario the first compressor needs 262 kW.

- The axial compressor requires around 29 stages to achieve the desired pressure ratio (rp = 14), the

compressor size and weight will be a problem to the feasibility of the whole system. In addition, the

high pressure ratio together with the low efficiency operation results in a high temperature exhaust

flow.

- Cooling the exhaust flow with refrigerant water brings major feasibility issues given the size and weight

of the liquid water and superheated vapor storage and the heat exchanger.

- Promoting early transition is crucial to reduce the possibility of large separation in the blades and

achieving higher performance.

- The lack of comparable experimental or computational data prevents us from making more detailed

statements about the compressor.

5.2 Strengths and weaknesses

5.2.1 Speed

The Hyperloop top speed, for the reasons presented above, is constrained to Mach 0.7-0.8, since

choked flow conditions must be avoided. The airplane reaches Mach 0.8-0.85, however two factors make

57

the Hyperloop faster. Firstly, since it is already in a low pressure environment, the only restrictions for

acceleration are passenger comfort and route straightness. If the route is well planned it can reach its top

speed sooner than the airplane. Secondly, given the thermal loads inside the tube (radiative energy from

the sun and the compressor exhaust air), the speed of sound is higher than in the colder high-altitude

atmosphere where the airplanes travel, allowing higher speeds for the same Mach number.

5.2.2 Energy and Feasibility

The Hyperloop has a reported mass of 15 000 kg and carries 28 passengers on a 563 km journey in

approximately 40 minutes, considering a top-speed of 280 m/s and an average speed of 235 m/s. For this

journey the energy intensity (energy per passenger per kilometer traveled, kJ/PK) of 4 major factors can

be estimated:

1. Given a top-speed of 280 m/s, the kinetic energy needed to accelerate the capsule is 12 m U2 =

5.9× 105 kJ resulting in an energy intensity of 5.9× 105/(28× 563× 103) = 37 kJ/PK;

2. The steady vacuum pumping has an average pumping requirement of 100 kW (Chapter 2), resulting

in an energy intensity of 15 kJ/PK (which considers only the passengers inside one capsule, although

it could also include all the capsules traveling inside the tube at nominal conditions);

3. Considering the aerodynamic drag force 573 N (Chapter 3), the average propulsion power is 573×

235 = 134 kW resulting in 20 kJ/PK;

4. The first compressor would require around 262 kW (best case scenario in Chapter 4) and the second

compressor consumes roughly 190 kW (depends on the temperature of the exhaust flow from the

first compressor), yielding a total 452 kW which results in 70 kJ/PK.

An extra 10 kJ/PK is added for other unaccounted utilities.

The energy estimates are presented in Figure 5.1. Adding up these contributions, the Hyperloop consumes

Energy perpassenger.km

(kJ/PK)

Aerodynamicdrag

On-boardcompressorsoperation

Steady statetube pumping

Capsuleaceleration

Typical short-haulairplane

1000

Hyperloop

Otherutilities

70

20127142

10737

152

70

1510

Figure 5.1: Energy estimates for a journey of 563 km (factor contribution inside bar)

approximately 152 kJ/PK, a factor of 7 compared with typical short-haul aircrafts [32]. In the previous

58

chapters, several assumptions were made to arrive at these estimates. The uncertainty of our calculations

and which factors could have the biggest impact in the Hyperloop are shown in Figure 5.2.

Impact

Uncertainty

Aerodynamicdrag

On-board compressoroperation

Steady statetube pumping

Capsuleacceleration

Air bearingfeasibility

Other utilities

Figure 5.2: Perceived factor impact on the system feasibility, with associated uncertainty

The general conclusion is that the Hyperloop has the potential to achieve lower door-to-door trans-

portation times, more efficiently than its current competitors. However, the capsule’s axial compressor

and the air bearing mechanism (top-right of Figure 5.2) are the critical factors of this system with un-

proven feasibility. The Hyperloop Alpha still presents a high number of risks and practical difficulties

since these are the initial steps of an alternative design. Considering the open-source nature of the project

it is expected that it will evolve into a more matured design faster than conventional projects.

5.3 Possible design changes

5.3.1 Axial fan

The axial compressor could be replaced by a lighter and simpler, 1-2 stage axial fan. This would

be enough to avoid choked flow conditions around the capsule and would reduce the energy, weight and

heating problems of the capsule.

5.3.2 Suspension

The axial fan cannot ensure a high pressure flow to the air bearing suspension (which itself was

not proven entirely feasible) therefore compressed air tanks or electrodynamic suspension (EDS) would

be preferred. Despite the cost escalation, EDS or Inductrack would be solid candidates for the pod

suspension, since it takes advantage of the capsule high speed and provides a higher gap height while

ensuring minimum maintenance across the vacuum tube.

59

5.4 Suggestions for Future Work

- The coupled nature of this project needs a framework for an efficient design exploration. A relevant

effort was put forth by NASA engineers [16], using an Open-Source Multidisciplinary Design Analysis

and Optimization (OpenMDAO) developed in Python, enabling system analysis and optimization by

modeling the different features of the Hyperloop and releasing it to the community.

- Detailed studies on the suspension and axial compressor reliability are of paramount importance since

these constitute crucial parts of the Hyperloop concept.

- After the design is sufficiently defined, CFD simulations could provide a powerful tool to study key

aerodynamic parameters. Essential aspects involve transition prediction, shock-wave detection, drag

analysis, suspension reliability and aerodynamic stability, among several other crucial aspects.

- Upon a more matured design, wind-tunnel experiments (which are considerably expensive for this

conditions) could provide valuable insight and validation concerning the models used.

- A start-up company, Hyperloop Tech, was formed recently and has already received financial support

to study the Hyperloop feasibility. In the last month, they announced an interesting collection of

hardware facilities to test the critical sub-systems (compressor, air bearing and tube test rigs, along

with a dedicated CAD/FEA station).

- In 2015, SpaceX launched a capsule design competition, which will take place in 2016, in order to

accelerate the design exploration and stimulate creativity, challenging student teams and organizations

from all over the world. A special mention to rLoop, which is an online-collaborative international team

that intends to build a capsule and compete in the event.

- Recently (May 16, 2016), Hyperloop One (former Hyperloop Tech) broadcasted an open air propulsion

test on a 2 mile track in Nevada. They received a $80 million financing round and made partnerships

with General Electric (GE) and the French National Railway Company (SNCF).

60

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AIAA SciTech, 2015.

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63

64

Appendix A

Compressible Flow

The study of high-speed transport in a tube is highly dependent on a deeper understanding of gas

dynamics. The laws of thermodynamics allied with the conservation of mass and momentum are indis-

pensable tools to comprehend many physical phenomena in this problem.

A.1 Quasi-one-dimensional flow model [1][2]

The flow inside the tunnel is three dimensional, since we encounter variable area streamtubes. Al-

though we consider that the change in area A = A(x) is gradual, this assumption is reasonable since we

intend to model a streamlined body inside a tunnel, for convenience we can accurately assume changes

in y or z are negligible and flow proprieties only vary with x.

A-1

We apply integral formulation to get the algebraic equations for the quasi-one-dimensional flow.

Continuity

∂

∂t

∫∫∫V

ρdV +

∫∫S

ρ u dS = 0 (A.1)

Assuming steady conditions and integrating over the control volume.

ρ1 u1 A1 = ρ2 u2 A2 (A.2)

X-Momentum

∫∫∫V

∂(ρ u)

∂tdV +

∫∫S

ρ u(~u · ~n) dS =

∫∫S

−p n dS +

∫∫S

tx dS +

∫∫∫V

ρ ~g dV (A.3)

Energy

∫∫∫V

∂

∂t[ρ (e+

u2

2] dV +

∫∫S

ρ (e+u2

2) V dS =

∫∫∫V

ρ q dV −∫∫S

p u dS +

∫∫∫V

ρ (f · u) dV (A.4)

Assuming steady adiabatic flow win no body forces on the control volume:

p1 u1 A1 + ρ u1 A1 (e+u2

1

2) = p2u2A2 + ρ u2 A2 (e+

u22

2) (A.5)

Dividing this by the continuity:

p1

ρ1+ e1 +

u21

2=p2

ρ2+ e2 +

u22

2(A.6)

According to the definition of enthalpy h = e+ pρ , the above equation can be re-written

h1 +u2

1

2= h2 +

u22

2(A.7)

For a steady adiabatic quasi-one-dimensional flow - the total enthalpy is constant:

h0 = const (A.8)

Equation of state - Ideal gas

The gas can be assumed to behave ideally thus thermodynamic equilibrium is achieved respecting the

following expression,

p = ρ R T (A.9)

where ρ is the density, R is the gas constant and temperature T .

Stagnation state

h0 − h =

∫ T0

T

cpdT (A.10)

A-2

p0

p=

(T0

T

)γ/(γ−1)

(A.11)

Mach number

A very convenient variable in compressible flow is the Mach number defined as

M =u

a, (A.12)

where u is the flow velocity and a is the speed of sound at local conditions.

The speed of sound can be deduced from a standing wave control volume analysis. For an ideal gas it is

shown that it depends on temperature T with

a =√γ R T (A.13)

Isentropic flow

Many real processes can be considered ideally as isentropic (adiabatic and reversible). The ratio of

stagnation temperature to static temperature is

T0

T= 1 +

u2

2 cp T(A.14)

T0

T= 1 +

γ − 1

2M2 (A.15)

The ratio of stagnation pressure to static pressure is

p0

p=

(1 +

γ − 1

2M2

)γ/(γ−1)

(A.16)

Applying continuity.

m = ρAu (A.17)

u = M

√γ R T0

1 + γ−12 M2

(A.18)

m

A=

p0√γ

√R T0

M1(

1 + γ−12 M2

) (γ+1)2 (γ−1)

(A.19)

The maximum mass flow rate per unit area is given when the flow is critical M = 1.

m

A∗ =p0√γ

√R T0

(2

γ + 1

) γ+12 (γ−1)

(A.20)

The maximum area reduction for critical conditions at the throat is

A

A∗ =1

M2

[2

γ + 1

(1 +

γ − 1

2M2)] (γ+1)

2 (γ−1)

(A.21)

A-3

The Mach - Area ratio is given by

A1

A2=A1

A∗A∗

A2=M2

M1

[1 + γ−1

2 M12

1 + γ−12 M2

2

] γ+12 (γ−1)

(A.22)

A.2 One-dimensional flow model

Although the one-dimensional model is just a particular case of the above developed (where the area

is constant), some particular flows of interest arise from this analysis.

A.2.1 Fanno flow - adiabatic frictional flow

The region where friction plays a major role. To simplify our analysis, we shall make the following

assumptions:

• Steady (cruise conditions)

• Adiabatic (high speed on a rarefied atmosphere)

• One dimensional (constant area annulus)

• Irreversible (no-slip condition on the walls)

• Ideal gas with constant specific heat

The assumptions are reasonable and enable us to consider this a Fanno flow.

Fanno flow is characterized by the inlet (Ma,P0, T0) and pipe conditions(f LDh

), also since the flow tends

to sonic conditions this serves as a suitable reference state for calculations. The upstream conditions can

be computed fromf L∗

Dh=

1−M2

γ M2+γ + 1

2γln

(γ + 1)M2

2 + (γ − 1)M2(A.23)

A-4

Appendix B

Axial compressor extrapolations

Table B.1: Optimistic polytropic efficiency case

n ReReref

−nRe ηp

(assumed)

0.20 1 5.0× 105 0.91

0.20 1.30 1.6× 105 0.88

0.25 1.90 3.6× 104 0.82

0.25 2.70 1.0× 104 0.75

0.25 3.20 5.0× 103 0.70

0.22 3.9 1.0× 103 0.63

Table B.2: Pessimistic polytropic efficiency case

n ReReref

−nRe ηp

(assumed)

0.1 1 5.0× 105 0.8

0.2 1.3 1.6× 105 0.75

0.2 1.4 1.1× 105 0.73

0.2 1.5 7.1× 104 0.7

0.2 1.6 5.0× 103 0.68

0.25 1.9 1.0× 103 0.61

0.3 3.2 1.0× 103 0.35

0.28 3.6 1.0× 103 0.27

0.25 4.7 1.0× 103 0.05

0.25 4.7 1.0× 103 0.05

A-5

Appendix C

Heat exchanger convection coefficients

The fluid coefficients for the shell and tube side are presented below:

Shell side - Air

Using the Kern method [34] for estimating the shell side convection coefficients,

ho De

k= 0.36

(De Gsµ

)0.55 (cp µ

k

)0.33 (µbµw

)0.14

(C.1)

For a pitch size PT = 0.024 m and an external diameter do = 0.008 m

De =4(P 2

T − π d2o/4)

π do=

4(0.01012 − π 0.0082/4)

π 0.008= 0.00825 m (C.2)

C = PT − do = 0.0101− 0.008 = 0.0021 m (C.3)

As =Ds C B

PT=

0.32× 0.0021× 0.192

0.0101= 0.0128 m2 (C.4)

Gs =m

As=

0.43

0.0128= 33.6 kg/s m2 (C.5)

Re =Gs De

µ=

33.6× 0.00825

3.178× 10−5= 8.722× 103 (C.6)

ho De

k= 0.36

(0.00825 33.6

3.178× 10−5

)0.55(

1005× 3.178× 10−5

0.065

)0.33

(0.99)0.14

= 41.7 (C.7)

ho = 328 W/m2 K

Tube side - Water

For an tube internal diameter of di = 5 × 10−3 m and 4 shell passages (this is the minimum value

chosen to increase the Reynolds number to an acceptable range).

Atp =π di

2

Nt4

=π × (5× 10−3)

4

133

4= 6.53× 10−4 m2 (C.8)

A-6

U =m

ρ Atp=

0.123

1000× (6.53× 10−4)= 0.2 m/s (C.9)

Re =1000× 0.19× (5× 10−3)

9× 10−4= 1056 (C.10)

The convection coefficient of the water flow is difficult to estimate accurately since

∆T = Twall − Tvap,w >> 20 K (C.11)

The wall temperature is much higher than the water vaporization temperature, and therefore two-phase

flow (forced convection boiling) occurs. It is important to ascertain which phenomenon dominates since

nucleate boiling will enhance convection, however if film boiling dominates, the convection coefficient will

drop since a gas film will coat the inner surface of the tube.

Incropera [33] suggests the following correlation for two-phase flow,

hihsp

= 0.6683

(ρlρv

)0.1

X0.16(1− X

)0.64f(Fr) + 1058

(q′′

s

m′′ hfg

)0.7 (1− X

)0.8Gs,f (C.12)

f(Fr) = 1, Gs,f = 1

From Gnielinski[33] correlation we find hsp

hsp = 183 W/m2K (C.13)

Vapor quality is calculated from

X(x) =q′′

s π D x

m hfg=

27.5 π × 0.05 x

0.123× 2258= 0.0155 x (C.14)

where x follows the tube axial direction. The flow has a low vapor quality, meaning that nucleate boiling

dominates and a high convection coefficient is expected.

The resulting convection coefficient can be found

hihsp

= 204

hi = 3.7× 104 W/m2K

Further iterations could slightly improve the heat exchanger performance but for this initial design, we

consider the convection coefficients are reasonable.

A-7

A-8