Hyperloop - The Train-In-A-Vacuum-Tube Fantasy (1)

Since 2012 the global media, both transport specialist and general publications, have been intrigued by Hyperloop, a revolutionary concept enabling passenger- and freight-carrying vehicles to move through hermetically-sealed vacuum tubes, at speeds of between 1,100 and 1,200 km/h (around 330 m/s, close to breaking the sound barrier). Hyperloop’s logo is a figure „8“ on its side. The fascination in what Hyperloop has to offer is growing, as more and more countries, politicians, institutions and transport providers come under its spell. It is worthwhile, therefore, for us to undertake a critical examination of some of the technical characteristics of the Hyperloop idea.

Conventional road and older conventional railway systems are built for vehicles to legally travel at speeds of slightly more than 100 km/h. But such speed limits personal temporal-spatial activities. Supposing one has to travel 200 km, out and back, in a day, for a business meeting. That means that around two hours are spent travelling in each direction. But four hours out of a normal eight-hour business day is a sizeable amount of time, and potentially wasted worker productivity. Geographically, it serves as a restraint on the distances apart that business activities relying on frequent interpersonal contacts can be located.

Ambient Resistance

Various factors limit the speed of road and older conventional railway vehicles to a bit more than 100 km/h, and not all these factors are legislative. The main controlling factor is the quadratic dependence of dynamic pressure, and thus also aerodynamic resistance, on the square of speed:

F_a = p_d . S . k_t . C_x = 0.5 . ρ . v² . S . k_t . C_x / 3.6²

F_a … aerodynamic resistance (N),
p_d … dynamic air pressure (Pa),
S … vehicle frontal area (m²),
k_t … tunnel factor,
C_x … coefficient of vehicle longitudinal shape, 
ρ … specific air weight (kg/m³),
v … speed of movement (km/h).

Increases in aerodynamic resistance with speed squared (and hence also increases in the aerodynamic component of energy consumption) together with increases of tractive power needed to overcome aerodynamic resistance which occurs with velocity to the power of three (cubed) have very disadvantageous results for vehicle movement. The specific weight of air at or near ground level is relatively heavy, around 1.25 kg/m³. Moving air at around 100 km/h - a storm force wind - creates a high dynamic pressure of 0.48 kPa. This is the pressure which a vehicle travelling at 100 km/h at ground level must overcome.

Aviation technology has very effectively overcome the challenges of reducing aerodynamic resistance. Following take off, commercial airliners gene-rally ascend to an altitude of about 10,000 m. Here, thanks to reduced static pressure, the air is around four times thinner, with only 0.3 kg/m³ of its specific weight near ground level. The result is that flying at this altitude saves energy, and hence fuel consumption, the aerodynamic resistance being so much lower than nearer the ground at lower altitudes. However, the force applied to the underside of the aircraft’s wings is much lower at high altitudes. So, to keep flying at around 10,000 m without stalling, it is necessary to maintain a reasonably high cruising speed, usually of around 900 km/h. 

To summarise, energy savings accrued by commercial aircraft flying at high altitudes with thinner air involve two essential expenditures of energy. One is the energy required to climb to 10,000 m, the other, that necessary to accelerate to around 900 km/h. The formula for calculating the specific energy involved is:

e_p = E_p / m = m . g . h / m / 3,600 = g . h / 3,600 = 9.81 . 10,000 / 3,600 = 27.3 kWh/t

e_p … specific potential energy (kWh/t),
E_p … potential energy (kWh),
m … weight (t),
g … gravitational acceleration (m/s²),
h … height, or flight altitude (m).
 
To accelerate an aircraft to a speed of 900 km/h, following specific kinetic energy is needed:

e_k = E_k / m = 0.5 . m . v² / m / 3,600 = 0.5 . v² / 3.6² / 3,600 = 0.5 .(900/3.6)² / 3,600 = 8.7 kWh/t

e_k … specific potential energy (kWh/t),
E_k … potential energy (kWh),
m … weight (t),
v … speed (km/h).

To create these conditions for flight in low density air, with low aerodynamic resistance, all the engines of an aircraft must produce specific potential and kinetic energy: 

e = e_p + e_k = 27.3 + 8.7 = 36 kWh/t.

The benefits from high altitude, low aerodynamic resistance flight are most advantageous on long-haul flights cover-ing distances of thousands of kilometres. They are not accrued by aircraft on short-haul flights, of only a few hundred kilometres. This energetic paradox of air transport is accompanied by a time paradox. If we take into account the time spent travelling from one’s origin to the airport, check-in and waiting time, and at the end of the flight baggage reclaim time, and travel time from the airport to one’s destination, all these activities could add up to two hours to the overall journey time. 

Even though one might be flying at around 900 km/h, the average end-to-end journey time means that for journeys of up to 1,000 km the achievable average speed may be no greater than 300 km/h. We could express this by means of the following formula:

v_p = L / (T₀ + T) = L / (T₀ + L / v) = 900 / (2 + 900 / 900) = 300 km/h

v_p … overall end-to-end average transport speed (km/h),
L … distance (km),
T₀ … times before departure and after arrival (h),
T … flight time (h),
v … flight speed (km/h).

Therefore, a typical short-haul flight in Europe (say, of around two hours’ duration), will only result in an average end-to-end journey time of between three and four hours.

Aircraft engines have an efficiency level of only around 30 %. To create economical energetic flight conditions 120 kWh/t (36/0.3) of fuel energy must be expended. How does this compare with rail transport? 
- a conventional 160 km/h express train with a specific electric energy consumption of 0.025 kWh/tkm could cover a distance of 4,800 km with 120 kWh/t of energy. This is the same amount that a plane requires to take it into the high altitude conditions to enable economical flight from an energetic point of view. 
- a high speed train travelling at 320 km/h with a specific electric energy consumption of 0.040 kWh/tkm will be able to cover 3,000 km with 120 kWh/t - the amount that a plane requires to take it into the high altitude conditions to enable economical flight from an energetic point of view.

We can therefore conclude that short-haul commercial airline flights, of the type found in the USA and in Europe, are not only very inefficient both in terms of the overall end-to-end (not terminal-to-terminal) journey time they offer, but also in terms of the energy they consume. 

Around half a century ago there was practically no means of transport available - either train, plane or car - that could provide a fast means of communication between urban centres spaced several hundred kilometres apart. The solution was found in the development of high speed trains, designed to be highly aerodynamic, using purpose-built tracks, at speeds of between 300 and 350 km/h. There were essentially two differing approaches to this: 
- the Shinkansen technology, evolved in Japan. Here the high speed network is mostly separate from the conventional one, usually without the transition of high-speed trains to the conventional network.
- the European version, where high-speed railways are conceived as a complement to the existing railway system, and where networks enable high speed trains to use conventional lines as well. 

In the end, though, the development of high speed rail technology was not quite straightforward. During the second half of the 20th century, apart  from upgrading conventional railways, often from only a 120 km/h to a 160 km/h maximum line speed, there were many unconventional proposals, such as monorail systems with wheeled vehicles or magnetic levitation. Their promoters had great hopes, but there was no massive take-up of these technologies, for many reasons. But in retrospect it is possible to isolate two particular and specific factors:
- so far none of these alternative technologies have not delivered any significant improvements over established railway technology (both conventional and high speed) so there was no sense in investing any further time and money in developing them to the point where they could be commercially implemented,
- during the in-depth analyses of these technologies it became apparent that there had been a lot of problems - not only of detailed aspects, but also of basic principles.

It is also apparent that the creators and promoters of these alternative technologies behaved somewhat irrationally, becoming so enthusiastic over their new ideas that they lost sight of the essentials of technical and economic judgement.

The Magic Of The Vacuum Tube  

Vacuum or pneumatic tubes for the transport of objects are not a new idea. The earliest were developed around 1799 by William Murdoch, a Scottish engineer and inventor. By the Victorian era they were used widely, to send telegrams between buildings, and to send canisters containing messages and money around large department stores. In the 19th century there were also proposals that such tubes could be used for the movement of freight, and even passengers. Isambard Kingdom Brunel’s „Atmospheric Railway“ between Exeter and Newton Abbot (originally planned for extension thence to Plymouth) was a practical example in the early 1840s of carriages being moved above a pneumatic tube. The aim of these efforts was obvious: to reduce the aerodynamic resistance of the environment by use of a vacuum and thus to achieve higher speed and lower energy consumption. And, unlike aircraft, without the need to raise to a high altitude.

Building on the experiences gained during the 19th century, the idea was furthered during the 20th. One pioneer in this respect was Robert Goddard, an American physicist and later the designer of rocket technology. He invented the Vacuum Tube Transport System (Vactrain) in 1906, while involved in research on vacuum-based transport systems at Worcester Polytechnic Institute. In 1914 Boris Weinberg, a Russian professor at Tomsk Polytechnic University, published a book entitled „Motion Without Friction“, regarded as a significant research advancement in this field.

During the 20th century some monumental schemes for vacuum tube transport systems evolved. One significant one was by Robert M. Salter, who proposed a tube linking Los Angeles and New York, enabling speeds of up to 4,800 km/h. In Switzerland in the 1970s Marcel Jufer of the École Polytechnique Fédérale de Lausanne proposed a scheme known as Swissmetro, which would have created a vacuum tube running from Genève via Lausanne, Bern and Zürich to St. Gallen, with a branch from Zürich to Basel, in all 411 km. The economic viability of these projects was considered dubious, and they soon faded from the scene. 

The main obstacles to the realisation of such projects are nether finance, nor the readiness of entrepreneurs to take them on, but the laws of physics. Some of the technical nonsense incorporated in these „amazing“ projects is evident from their brief marketing descriptions:
- a vehicle moves in a vacuum while simultaneously being carried along on a pillow of compressed air at the base of a horizontal tube,
- in a vacuum tube the compressed air gives the vehicle an acceleration rate of 15 m/s2 and a similar deceleration rate using compressed air for braking.
There are many other such statements which could easily be dismissed using basic calculations. Moreover, we are not talking about complicated calculations - only those found in secondary school level mathematics and physics.

The traction power needed to overcome aerodynamic drag increases with the cube of speed:

P_a = F_a . v / 3,6 = p_d . S . k_t . C_x . v / 3,6 = 0,5 . ρ . v³ . S . k_t . C_x / 3,6³

P_a … traction power needed to overcome aerodynamic drag (kW), 
F_a … aerodynamic drag (N),
p_d … dynamic air pressure (Pa),
S … front surface (m²),
k_t … tunnel factor,
C_x … vehicle shape coefficient in longitudinal axis,
ρ … air density (kg/m³),
v … vehicle speed (km/h).

Increasing the speed of travel four-fold from the usual 300 km/h of high speed railway to 1,200 km/h, would lead to an increase in traction power needed to overcome the drag by 64 times (4³ = 64) under unchanged conditions (without the need to reduce the density of the air). In addition, focus-ing on the movement of a short capsule in a pipeline result in an adverse effect on aerodynamics: 
- short vehicles have a higher specific drag than long vehicles because of the ratio of front end area to length (hence long trains and aircraft with slender fuselages are used),
- a vehicle travelling in a tunnel or tube with very limited clearance generates an increase in the airflow velocity in the limited space between the sides and roof of the tunnel and the vehicle - a piston effect, which considerably increases aerodynamic drag.

From the above it is clear that if transport at speeds of up to 1,200 km/h through vacuum tubes is to be really achievable, and if it has to be provided at a level comparable with trains running at around 300 km/h on a conventional high speed railway, a merely small reduction in air pressure will be insufficient. Therefore, a very high vacuum must be used inside the tube to reduce aerodynamic drag - an absolute air pressure of only 0.1 to 1 kPa (air weight of 0.00125 kg/m³ to 0.0125 kg/m³), a thousandth to one hundredth of the normal atmospheric air pressure at the Earth’s surface 100 kPa (1.25 kg/m³).

The graph attached shows that the specific traction power needed to overcome the aerodynamic drag comparable to a conventional high speed rail vehicle (roughly 10 kW per seat at 300 km/h on open track) is, for a vehicle in a vacuum tube with a capacity of 40 passengers, only achieved when the air pressure in that tube is reduced to one 4,000th of the natural atmospheric pressure.

The graph shows the dependence on the absolute air pressure of traction power needed to overcome aerodynamic drag at 1,200 km/h within a tube with substantially reduced atmospheric pressure, assuming a 40-passenger vehicle with a diameter of 2.75 m, a shape factor of 0.2, and a tunnel factor of 3.7.

Similarly, to achieve the specific energy consumption needed to overcome aerodynamic drag comparable to a conventional high speed rail vehicle (about 0.033 kWh/km per seat at 300 km/h on open track) for the 40-passenger capsule in a vacuum tube, it would be necessary to reduce air pressure in the tube to one 15,000th of the normal level of atmospheric pressure.

Vacuum technology is not a new feature on railways. Vacuum, or suction brakes were used widely in the past, and are still in use on some systems, such as the Rhätische Bahn. The reason behind their use was the simple design of steam pumps. However, the air dilution was not high for vacuum brakes compared to proposed vacuum tube transport. The two-chamber automatic rail brake worked with a pressure the equivalent of a 52 cm column of mercury. This corresponds to an absolute air pressure of 32 kPa - the reduction to around a third of the normal atmospheric pressure of 100 kPa.

The article will continue.