Hyperloop Alpha - Conceptual study

Tiago M.L.S. Santos Coelho

Department of Mechanical Engineering, Instituto Superior Tecnico, University of Lisbon,Av. Rovisco Pais, 1049-001 Lisbon, Portugal

June 2016

Abstract

The Hyperloop Alpha, a new transportation concept idealized by Elon Musk, is analyzed. Thissystem consists of a capsule travelling at high subsonic speeds Mach 0.7 – 0.8, inside a low pressuretube (100 Pa). An axial compressor takes air from the front area of the capsule to alleviate chokingeffects between the capsule and the tube, the compressed air feeds the gas bearing suspension be-neath and a fraction of this air is expanded at the aft region to produce a small thrust. Three mainproblems are addressed: the characteristics of the low pressure tube, the constraints of compressibleflow around the capsule and the on-board axial compressor. The main conclusions are: the air evacu-ation of the tube can be achieved with standard industrial vacuum pumps; the on-board compressorreduces choking around the capsule, but to a limited extent; the air bearing suspension proposed mayprovide the required lift but the very tight dimensional constraints, the high skin-friction heating andthe 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 compressoroperates in limiting conditions at low efficiency with poor stage loading resulting in a large number ofstages, where heat rejection cannot be accomplished efficiently. The Hyperloop offers a good energyefficiency but faces many feasibility issues, such as the tube deflection and tightness, the air bearingsuspension reliability and the axial compressor’s low efficiency and large dimensions.

Keywords: Hyperloop, Low pressure tube, Compressible flow, Air bearing suspension, On-boardaxial compressor

1. Introduction

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 to link the Californian corridor 1 from LosAngeles to San Francisco (563 km) in 35 minutes. Musk [1] explains the motivation and general technicallayout of the concept, labeled the Hyperloop Alpha. The paper makes a bold proposition about a newmode of transportation. Faster, cheaper, safer and energetically self-sustained. Upon public release, thedocument invites the community to participate in the development of this project. The purpose of ourwork is to study the Hyperloop claims and the technical feasibility of the concept.

The description of the first version of this concept can be found in [1]. The key points are:

Tube: Two (one-way) evacuated tubes of 2.23 m diameter with an average elevation of 6 m and 563 kmlength, connect the stations (see Figure 1).Energy: The Hyperloop system is powered by the solar panel arrays that cover the top surface of thetube (see Figure 1).Speed: The capsule (see Figures 1 and 2) consists of a streamlined cylindrical body, which is intendedto reach top speeds of Mach 0.91. The capsule’s speed profile is very ambitious, compressibility effectsare present during nearly the whole journey and a transonic regime during more than half. However, inthis 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 high speed requirement.Partial vacuum: To reduce aerodynamic drag and compressor power requirements, the capsule runsinside a low pressure tube (∼ 100 Pa).Propulsion: Acceleration is achieved by external linear induction motors, separating the rotor inside

1

Figure 1: Hyperloop capsule in tube cutaway with attached solar panel arrays. Modified from [1].

Figure 2: Hyperloop capsule [1]

the capsule and stator mounted outside at certain locations along the length of the tube (see Figure 1).Air density is low, enabling the capsule to coast for most of the journey, several stators outside the tubewill provide a periodic boost along the tube length. Secondary propulsion is achieved by expanding thecompressed air.Blockage ratio: In order to reduce capital costs, which are closely related to the tube cross sectionAtube, high blockage ratios

AcapsuleAtube

are explored, where Acapsule is the cross section area of the capsule.See Figure 3, where dark gray is the Acapsule and light gray is the Atube.

On-board axial compressor

TubeSuspensionpads

Acapsule

Atube

Figure 3: Tube cross-section cut. Capsule front.

Axial compressor: Air is ingested on the front of the capsule alleviating choking effects (see Figure 3and 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 awedge effect and an air cushion which uses the pressurized air from the compressor. Retractable wheelsare considered for low speeds.Capacity: The capsules carry 28 passengers (see Figure 2) with an average departure time of 2 minutesand a minimum of 30 seconds. This yield an average capacity of 840 passengers/hour, and a maximumof 3360 passengers/hour.Route: The Hyperloop route connecting Los Angeles and San Francisco was found considering thefollowing factors [1]: 1. Maintaining the tube as closely as possible to existing rights of way to minimizepurchase costs. (e.g., following the interstate road I-5). 2. Limiting the maximum capsule speed to Mach0.91 for aerodynamic considerations. 3. Limiting accelerations on the passengers to 0.5g. 4. Optimiz-ing locations of the linear motor tube sections driving the capsules. 5. Local geographical constraints,including location of urban areas, mountain ranges, reservoirs, national parks, roads, railroads, airports,

2

etc. The route must respect existing structures.

2. Low pressure tube

Vacuum conditions provide a low density environment, which results in lower aerodynamic losses. Thedrag force (Eq. 1) is proportional to the fluid density ρ, the square of the relative velocity U , the vehiclecross-sectional area A and the drag coefficient Cd.

Fdrag =1

2ρ U2 A Cd(Re,M,Design) (1)

The design tube pressure, reported in [1], is 100 Pa. At this pressure level, mechanical displacementpumps offer the best economy and pumping speed. The best vacuum pumps for this system are therotary and roots pump combination.

2.1. Pumpdown time

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).

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

p(t) = p0 exp

(− SVt

)(2)

and the time required to reach the final pressure p1,

t1 =V

Sln

(p0p1

)(3)

2.2. Steady state operation

At nominal conditions, it is considered that, capsule handling (entering/exiting the tube) and leakageflow rates are the main gas loads.

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

Capsule

Atmospheric Pressure

Air-lock chamber

Vacuum

P = 105 PaP = 100 Pa

Figure 4: Design 1 - Flush Air-lock chamber (side view).

The capsule is nearly flush with the chamber to ensure a minimal volume load, Figure 4. Consideringthe capsule length L = 30 m, and diameter D = 1.5 m and an occupation percentage of 95% results in avolume load of 3 m3. The expected departure interval[1] 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 1.

2.3. Tube pumpdown

The process is polytropic, given the size of the vessel and the considerable time needed, with an assumedpolytropic 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

[poutp(t)

n−1n − 1

]× 1

ηm= 3600 GJ

3

Table 1: 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 270Process 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

In [1] the required vacuum pump costs are expected to be around $10 million USD. The assumed peakpower available is 100 MW, below estimated in [1]. A pumpdown time is selected to meet these conditions.A 2-day pumpdown is achieved with an installed capacity of 195000m3/h for rotary pumps and 585000m3/hfor the Roots pumps. The average power is 21 MW. The peak power rounds up to 48 MW. The electricityprice (see Table 1) in the Californian region is 0.1038 $/kW.h, which for the energy needed (10×105kW.h)gives an approximate operation cost of $104 000.For a single stage rotary pump + roots combination with a maximum pumping speed of 7000 m3/h, atleast 85 vacuum pump systems are needed. The estimated pump acquisition cost is $8.8 million USD.For the given pumpdown time, the total (acquisition + pumpdown + installation) cost is estimated tobe around the targeted $10 million USD and the peak power doesn’t exceed 50 MW.

2.4. Tube cost

Typical values for steel structures (low carbon steel) which include supply, fabrication, transportation andman-hours can be round up to1 ∼ $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, resultson an estimated cost of $5billion. This number is only an order-of-magnitude estimate, it depends widelyon the optimization and scale of the processes involved. However, one can conclude that the tube costdominates the project cost. Vacuum system costs are less than 1% of the tube cost.

3. Capsule aerodynamics

The capsule (Figure 5) travels at high subsonic speeds in a low density environment. The Mach is 0.7 -0.8 and the Reynolds based on capsule length is 5× 105.

Freestream

Frontbody Main body Afterbody

Laminar Transitional TurbulentCowl

Shoulder

Boattail

Pads

Capsule

On-boardCompressor

U

TubePads

α

Pad angle

Pads

Figure 5: Capsule (not to scale). Flow is based on the capsule’s reference

3.1. Blockage

In order to reduce investment costs the tubes’ cross-section is minimized. At high subsonic speeds, theblockage ratio is alleviated using the compressor on-board. However, the flow in the compressor inletmust be constrained to Mach 0.6, in order to avoid excessive aerodynamic losses [6], [4], [3], [8]. A useful

1Approximated value for cost estimates of steel structures.

4

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)

. (4)

For the capsule’s nominal speed, Mach 0.8 - 0.9, the flow must be decelerated from the freestreamconditions to the inlet of the compressor (Figure 6). MFR < 1 enables an isentropic external deceleration,thus satisfies the compressor inlet restriction. The blockage ratio for a given cruise Mach is presented in

0

12

M2 < M0

The compressorrecieves flowat adequate speed

Flow accelerates aroundthe capsule

A1 > A0

M1 < M0

Figure 6: Diffused streamtube into the compressor inlet

Figure 7. It is interesting to see that for low blockage ratios adding a compressor does not bring a relevantadvantage. However, as the blockage ratio is increased, a capsule cruising without a compressor wouldbe affected by choking at lower speeds. For a blockage ratio, B = 0.36, a capsule without compressorcould only reach cruise speeds of Mach 0.4, where as using a compressor it can go up to Mach 0.7. For acruise Mach 0.8, the tube diameter is 4 m, a 1.6 increase factor. At high subsonic speeds the compressoris limited by aerodynamic losses (M < 0.6), which is a limiting factor to the usefulness of an on-boardaxial compressor.

Cruise Mach

Critica

l bl

ocka

ge r

atio

Compressor blockage

No compressor blockage

Figure 7: Compressor choking alleviation

3.2. Suspension

In order to minimize mechanical friction and maximize the on-board compressor usefulness, the capsuletravels in a pressurized air film, avoiding the high costs of conventional Maglev technology. Musk [1]proposes 2 alternatives of bearing-like suspension based on viscous flow (lubrication model).

1. Wedge pads, a passive aerodynamic suspension;

2. Air bearing, a compressor pressurized pad;

5

In cases 1 and 2 the lubrication model applies, the gap between the lower surface of the pad and the groundis 1.0 to 1.5 mm. A reference velocity U = 300 m/s, for air at pressure pa = 100 Pa and temperatureT = 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 [7] suggests a modified Reynolds number for fluid film lubrication

Re =ρUh2

µL(5)

The modified Reynolds number is 1.25× 10−2, thus viscous forces are greater than inertia forces and thefluid film lubrication model applies.The capsule and pad specifications [1] can be found below (see Figures 5 and 8):

Capsule mass estimate: 15× 103 kg Number of pads: 28Pad dimensions (B × L): 1.5 m ×0.9 m Total pad area: 37.8 m2

Ground clearance (h): ∼ 1× 10−3 m Ambient pressure: 100 PaPad angle (θ): 0.05o Suspension angle (α): 40o

L

hB

(a) Pad dimensions

L

hb1

θ

(b) Wedged pad

Figure 8: Pad views

The thin fluid film is described by the Reynolds equation,

∇(−h3 p12 µ

∇p)

+U

2

∂

∂x(p h) +

∂

∂t(p h) = 0 (6)

Wedge pad: It was not possible to ascertain the increased lift generated by the passive wedge padsgiven the skin-friction high heating effects which could lead to an increase in air temperature around∆T ∼ 4700 K.

Air bearing: The air bearing can provide sufficient lift given a recess pressure of 9.4 kPa and a totalsurface area of 37.8 m2. The compressor delivers 0.4 kg/s, the gap height for this mass flow rate ishmax = 0.27 mm. However, given the large pressure ratio the flow is choked. This can bring pneumatichammer instabilities and affect the suspension performance.

For the maximum gap height computed (hmax = 0.27 mm), manufacturing capability is the currentlimiting factor since all the pad variants rely on very small tolerances in the distance between the padsand the ground. For the capsule traveling at 300 m/s, we consider it may safely transit step irregularitiesabout half as high as its gap clearance hmax

2 = 0.135 mm. The pads-ground interaction must be very stiff(∼ 1 × 109 N/m) in order to keep the design clearance. In addition, a second suspension, to absorb thevibrations from the pads is needed.Deflections around 0.2 mm would compromise significantly the system. This means that the designconstraint of the tube is not the maximum allowed stress but rather the maximum allowed deflec-tion. A steady state analysis is adequate since the structural wave speed 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 acantilevered beam fixed at both ends subjected to an uniform load, where the total load W is the capsuleand the tube’s own weight.

ymax =W L3

384 E I= 1.87× 10−3m (7)

Where, according to [1]Tube moment of area: Ix = 8.95× 10−2 m4 Average support spacing: L = 30 mTube self-weight + capsule weight: W = 4.8× 105 N Steel Young’s modulus: E = 210× 109 Pa

The value ymax found in Equation 7 shows that tube exhibits an excessive deflection for the di-mensional constraints required by the air pads, compliance with the the gap height computed (hmax =

6

l

ymax

hmaxhmin

Figure 9: Tube deflection caused by own weight and capsule load. (Not to scale)

0.27 mm) may be difficult to achieve (see Figure 9). The misalignment of the pads gap height acrossthe capsule (hmax and hmin ) has consequences in the pads recess pressure and the compressor flowrequirements. Most importantly, there must be a structural design change if the suspension is expectedto avoid contact with the tube. The distance between supports must be shortened or reinforcement ribsused. These measures come with significant cost penalties, including the inability to use conventionaltube construction. Tight tolerances also imply that even small earthquake deformations could require acostly 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 tubeoriginates a periodic disturbance possibly compromising ride comfort. For a nominal 300 m/s velocityand 30 m support spacing, the capsule is subjected a vertical acceleration ay that exceeds 0.37g, whichmay not be tolerated by passengers.

3.3. Propulsion

Expansion of the compressed air through a nozzle, would 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 not be possible due to size/weight considerations in the capsule or handling the effects of asupersonic jet inside the tube.However, the mass flow rate from the compressor is almost entirely directed to the suspension, leavingno air to expand for propulsion.

3.4. Drag estimates

The low density environment reduces the capsules propulsion requirements, the total drag is expected tobe around 573 N (see Figure 10). The two highest contributors are the air bearing pads (470 N), and theturbulent friction on the capsule’s main body (40 N).

Drag force (N)

Friction onmain body

Inlet drag Form dragFriction on thesuspension pads

470

4033

30573

470510543

Figure 10: Drag book-keeping for cruise conditions

7

4. On-board axial compressor

The compressor in front of the capsule is one of the major innovations in this project, setting it apartfrom conventional solutions. According to [1], its purpose is threefold. It avoids choking, allowing anarrower tube thus reducing upfront capital costs. Secondly, it supplies the externally pressurized airbearings 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, lowpressure environment.

4.1. Compressor limitations

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 11: Reynolds influence in compressor

Figure 11 provides some insight on the constrains associated with the compressor operation. The chordReynolds is dangerously low for the actual Hyperloop conditions, this limits the pressure rise achievableper stage and stalling effects could easily affect operation reliability. It also shows that it is difficult toshift the design region to higher Reynold numbers (∼ 2×105) while maintaining the core characteristics ofthe project, since if the tube pressure is kept constant (100 Pa) the blade chord would be surrealisticallylarge (10 m). On the other hand, keeping the chord length constant at a reasonable value (0.2 m), thepressure inside the tube would have to rise to 5000 Pa.

4.2. Very-low Reynolds compressor

We explore the possibility of high turbulence, unsteadiness or using a boundary-layer trip is sufficientto promote early transition, reducing large separation losses. Viscosity and compressibility will play asignificant role in the compressor performance. The following effects are expected:

i) Efficiency penalty (Figure 12a)

ii) Reduced stage loading (Figure 12b)

4.2.1 Flow field scenarios

Given the complexity of the flow field, we shall assume 3 reasonable scenarios (see Figures 12a and 12b)with values for polytropic efficiency ηp, flow turning β and maximum pressure recovery Cpmax (bothrelated to blade loading):

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 trailingedge.

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.

8

Hyperloop design region

Extrapolatedregion

Scenario 1

Scenario 2

Scenario 3

Heidelberg [29]Cumptsy [7]

(a) Qualitative polytropic efficiency

Hyperloop design region

Scenario 1

Scenario 2

Scenario 3

Extrapolatedregion

Constant massflow,variable efficiency [29]

Max efficiency,variable massflow [29]

(b) Expected flow turning trend for maximum efficiency

Figure 12: Qualitative trends with assumed uncertainty

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.

The scenarios assumed enable us to perform a first-cut analysis of the axial compressor through a meanlinepreliminary design.

4.3. Mean line preliminary design

The main dimensions are given in Table 2. The results from this preliminary design for the three flowscenarios are given in Table 3.

Table 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

Table 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

The low Reynolds and high Mach flow severely limit the axial compressor operation. Efficiency isexpected 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. The axialcompressor requires around 29 stages to achieve the desired pressure ratio (rp = 14), the compressor sizeand weight will be a problem on the feasibility of the whole system. In addition, the high pressure ratio

9

together with the low efficiency operation results in a high temperature exhaust flow, that needs to becooled since it affects other parts of the system.The cooling water has a massflow rate of mw = 0.122kg/s. Considering a 35 minute journey, we need 256kg of water on-board, which given the liquid water specific volume vliq = 1× 10−3 m3/kg occupies 0.256m3, but, as noted by [5], its vapor specific volume is much higher, for water at 1-2 atm vvap = 2−1m3/kgresulting 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 atleast 10 atm.

5. Conclusions

In transportation systems, a useful metric of the transport efficiency is the energy required per passengerper kilometer traveled, referred here in kJ/PK units. The energy estimates of this work are presented inFigure 13.

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 13: Energy estimates for a journey of 563 km (factor contribution inside bar)

Adding up these contributions and assuming an energy spent by other smaller utilities, the Hyperloopconsumes approximately 152 kJ/PK, an efficiency factor of 7 compared with typical short-haul aircrafts[2].The general conclusion is that the Hyperloop has the potential to achieve lower door-to-door transporta-tion times, more efficiently than its current competitors. However, the capsule’s axial compressor and theair bearing mechanism are the critical factors of this system with unproven feasibility. The HyperloopAlpha still presents a high number of risks and practical difficulties since these are the initial steps of analternative design. Considering the open-source nature of the project it is expected that it will evolveinto a more matured design faster than conventional projects.

References

[1] E. Musk; Hyperloop Alpha. http://www.spacex.com/sites/spacex/files/hyperloop_

alpha-20130812.pdf. [Accessed: 2014-12].

[2] Analysing the options for 757 replacement. Aircraft Commerce, 2005.

[3] H. S. G. R. H. Cohen. Gas Turbine Theory. Prentice Hall, 2001.

[4] N. A. Cumpsty. Compressor Aerodynamics. Longman Scientific & Technical, 1989.

[5] J. C. et al. Open-source conceptual sizing models for the hyperloop passenger pod. AIAA, 2015.

[6] S. Farokhi. Aircraft Propulsion. John Wiley & Sons, 2014.

[7] B. Hamrock. Fundamentals of Fluid Film Lubrication. NASA Reference Publication, 1991. pp. 147 -163.

[8] P. H. C. Peterson. Mechanics and Thermodynamics of Propulsion. Addison-Wesley, 1992.

10