What is the Point of a Pulley?

In its simplest implementation, a pulley simply turns a force in one direction into a force in another direction. This might be useful if it’s easier to apply a force in one direction than the other (e.g. to pull downwards rather than push upwards).

A pulley really comes into its own when it is combined with another pulley to create a system known as a block and tackle, allegedly invented by Archimedes in the third century BC.

The simplest block and tackle, the gun tackle, uses two pulleys, which are usually mounted above one another (for clarity, they are shown separated in the diagram below).


gun-tackle

The load (W) is shared between the tension in the rope and the mount that attaches the block and tackle to the ceiling, and thus you only have to pull with a force equal to half the weight in order to lift it. However, you will have to pull the rope twice as far and thus the conservation of energy is not violated (lifting a one hundred newton weight through one metre is the same amount of work as applying a fifty newton force over two metres).

The Luff tackle uses three pulleys in theory, but this is usually accomplished by running a rope over the top pulley twice. In this case the force required to lift a weight is reduced to one-third of its actual value, but again you have to pull further: three times the height required in this case.luff-tackle

A Luff tackle, shown separated.

The double tackle uses four pulleys, but similar to the Luff tackle, this is usually accomplished by running the rope twice over both pulleys.

double-tackle

In theory, this process can be continued indefinitely (one is reminded of Archimedes’ alleged remark: “Give me a place to stand and I will move the Earth“) but it quickly becomes impractical to do so, as the length of rope required, and the distance pulled through become unmanageable.

block-tackle-eg

Source [PDF]

A block and tackle can also be created using pulleys of different sizes.

Why Can’t We Get to Absolute Zero?

The temperature of a substance is a measure of the average kinetic energy of the particles in that substance. As the average kinetic energy of the particles increases (i.e. they move faster), the temperature of that substance increases.

Some of the particles in a very hot substance will be moving slower than some of the particles in a very cold substance, but the average speed of the particles in a hot substance will be faster than the average speed of the particles in a cold substance. The number of particles at each speed in a gas is governed by something called a Maxwell-Boltzmann distribution, and is shown for air in the graph below:

air-maxwell-boltzmann

The average speed for particles of air* at 0°C is around 400 metres per second, for air at 100°C it is about 460 m/s and for air at 1000°C it is about 860 m/s. (Note also that at 1000°C there is far more variation in the speeds of particles than for air at 0°C.) At absolute zero, the coldest possible temperature, particles have a minimum of kinetic energy, and therefore the lowest possible speed. (They cannot have a kinetic energy of zero and actually be stationary because of something called degeneracy pressure.)

So why can’t we cool something all the way to absolute zero?

For something to cool down, it has to lose thermal energy. In order to lose thermal energy, this thermal energy has to go somewhere, and thermal energy only ever moves from hot to cold.** For example: a warm can of drink placed into a cold fridge loses thermal energy to its surroundings until it reaches the same temperature as the fridge’s interior.

Therefore, in order to bring something to absolute zero it would have to be surrounded by something that is colder than absolute zero, and this is impossible: hence you cannot achieve a temperature of absolute zero.

The closest we’ve ever got to absolute zero is less than 100 picokelvin, or 100 trillionths of a degree above absolute zero at the Low Temperature Lab at the University of Aalto in Finland. Interestingly though, it would be possible for something to feel colder than absolute zero due to wind chill.

* Obviously air is made up of different gases travelling at different speeds, so this is an average, weighted by the masses and prevalences of the different gases that make up air.

** It would actually be more accurate to say that the net movement of thermal energy is always from hot to cold. Some energy may go from cold to hot, but more will always go in the opposite “direction”.

Desalination

Desalination is the removal of salt(s) from seawater to create fresh water for drinking and irrigation. This is obviously very important on ships and submarines, but is also important on a national level: Israel produces fifty percent of its water via desalination, and the world’s largest desalination plant, the Jebel Ali Desalination Plant in the United Arab Emirates, produces 636 million litres of drinkable water per day.

jebel-ali-desalination

The Jebel Ali MSF desalination plant

There are two primary methods of desalination: distillation and reverse osmosis.

Distillation is the simplest method of desalination: seawater is boiled, and the water boils away as steam and is then collected and condensed back into liquid water, leaving the salt behind. The most common distillation method is Multi-Stage Flash (MSF) Distillation, which operates by feeding seawater through a series of chambers, each at a lower pressure than the first. The low pressure reduces the water’s boiling point (thereby saving energy) and as water reaches each stage it immediately boils (“flashes”) into steam. This steam is collected by condensers and the heat given off in this process (i.e. the latent heat of vaporisation) is used to pre-heat the seawater entering the chambers. MSF distillation uses around 50-90 megajoules of energy total per cubic metre of water produced.

Other distillation methods of desalination include mechanical vapour-compression, in which steam is mechanically compressed into liquid water, thereby generating heat that can be used to generate more steam; and multi-effect distillation (MED) in which steam produced in in one stage (known as an “effect”) is used to boil water into steam for the next effect, reusing energy that would otherwise be wasted. In MED, like MSF, each effect has a lower temperature and pressure than the previous one, so the inevitable decrease in temperature due to lost energy does not affect the boiling of water into steam.

reverse-osmosis-plant

The interior of a large reverse osmosis desalination plant.

Reverse osmosis is the most commonly used method of desalination. In normal osmosis, water moves from an area of low solute (low-salt) concentration to an area of high solute (high-salt) concentration through a semi-permeable membrane along a concentration gradient, thus equalising the concentration on either side of the membrane. (The membrane is constructed so that it allows water to pass through it, but not the solute.) In reverse osmosis high pressures (in the region of five megapascals) are used to overcome the osmotic pressure, forcing water to go from a high-salt concentration to a low-salt one. This is different from filtration, because filtration operates by size-exclusion (a difference in size), whereas reverse osmosis relies on a difference in concentration. Reverse osmosis desalination uses around 11-20 megajoules of energy per cubic metre of water.

Reverse osmosis is only one of a number of membrane desalination processes. Other membrane processes include electrodialysis reversal, which uses an electric current to push salt ions through ion exchange membranes*; and nanofiltration, which uses nanometre-sized (0.1 nm-1.0 nm) filters to remove salts.

* If electrical current is used to move salt ions, then the movement of salt ions can create an electrical current. This is the basis of reversed electrodialysis, a method of electricity generation using the difference in salt concentration between saltwater and fresh water patented in 1977 by Sidney Loeb at Ben Gurion university in Israel. Loeb was also the inventor of a semi-permeable membrane that made reverse osmosis desalination practical.

The Kármán Line

The Kármán Line, at 100 kilometres above sea level, is commonly taken to be the boundary between Earth’s atmosphere and space. But why?

An aircraft generates lift by moving through air.

L=\frac{1}{2} AC_L \rho v^2

where L is the lift generated, A is the wing area, C_L is the lift coefficient, \rho is the density of air, and v is speed through the air.

density-temperature-elevation

Data from NASA MSIS E-90 atmosphere model. Note that density is plotted on a logarithmic scale and (absolute) temperature on a linear scale.

The density of air changes with elevation (altitude is height above ground, elevation is height above sea level). Therefore, keeping all other factors the same (which would be the case for an aircraft, which cannot change the area of its wings or its lift coefficient), as the air becomes less dense an aircraft must increase its speed to stay airborne. A Boeing 747-400 has a wing area (planform) of 541.2 square metres and a lift coefficient of around 0.5. Assuming that it flies empty, with a mass of 178 800 kilograms (weight 1 754 000 N), the speed it will need to fly to generate the required lift changes rapidly as the air becomes thinner. (At an elevation of 200 km a 747-400 would have to fly at nearly twenty thousand times the speed of sound.)

speed-elevation-boeing-747

Speed required at elevation for a Boeing 747-400.

The cruising speed of a 747-400 is about Mach 0.85 or 290 metres per second, which limits the 747-400 to an elevation of around 16 km. (In reality it flies much lower, around 12 km, for safety and fuel consumption reasons.) The record elevation for sustained horizontal flight by a ground-launched aircraft is 36 240 m by a MiG-25M, which was, at the time, one of the fastest military aircraft with a top speed of Mach 2.83.

The Kármán Line, at an elevation of 100 km, is the point at which the atmosphere is so thin (more than a million times thinner than air at sea level) that an aircraft would have to be moving at a speed faster than it could orbit in order to stay in the air. Thus, it is the greatest altitude at which an aircraft could fly in a straight line rather than following the curvature of the Earth.

orbital-speed-requirement

speed-orbit-flight-elevation

As you can see from the graph above, for our Boeing 747-400, the Kármán Line is around 61 km, rather than the full one hundred kilometres, because the 747-400 is not the most aerodynamic of aircraft, designed for bulk passenger carrying and cost-efficiency rather than performance.

Throats, Nozzles and Shock Diamonds

rocket-engine-shapes

All rocket engines feature a “throat”, a narrowing of the exhaust nozzle. In the photographs above this throat is obvious at the top of the images.

de-laval-nozzle

A de Laval nozzle (also known as a convergent-divergent nozzle) as used in rocket engines.

The question of why rocket engines all feature throats might seem like it has an obvious answer: just try blowing air out of your mouth with your mouth wide open. But the physics behind rocket throats is a little bit more complicated than that.

The nozzle of a rocket engine is designed to accelerate exhaust gases to Mach 1 at the throat, causing a process known as choking or choked flow.

Normally the rate at which a gas can flow out of a pressurised container, such as the combustion chamber of a rocket engine, is limited by the difference between the pressure of the interior of the container and the pressure of the atmosphere surrounding the container: the container tries to push gas out, and the atmosphere tries to push the gas back in. In choked flow, this dependence disappears: the outside pressure has no effect on the rate at which gas is ejected, external pressure cannot force its way past the supersonic shockwave that forms at the throat. Choked flow produces the greatest rate of flow of exhaust gases, and therefore the highest possible thrust: you cannot get the exhaust gas particles to move any faster, but you can push through more of them per second.

As a gas passes through a narrowing in a pipe its pressure decreases and its speed increases; and as the pipe expands the pressure increases and the speed decreases.* This is known as the Venturi effect.

\frac{dv}{v}=\left( \frac{1}{M^2-1} \right) \frac{dA}{A}

Where \frac{dv}{v} is the rate of change of the velocity of the gas, M is the speed of the gas as a fraction of the speed of sound (i.e. the gas’s Mach number) and \frac {dA} {A} is the change in the area that the gas is flowing through.

There is a limit to the Venturi effect, and that is when the fluid reaches the (local) speed of sound, as happens during choked flow. At this point, the Venturi effect is reversed: instead of the gas slowing as the nozzle expands, its velocity increases (because the \frac{1}{M^2-1} term becomes positive rather than negative).

Thus the shape of a de Laval nozzle is designed to first accelerate the gas (by narrowing) to sonic speeds at the throat, and then because the Venturi effect has been reversed, to accelerate the gas further (by expanding) to supersonic speeds. The faster the exiting exhaust gas is, the more thrust that will be produced.

Once the gas has exited the throat, the shape of the diverging (widening) part of the de Laval nozzle is such that the exhaust gases are directed backwards parallel to the body of the rocket (as shown in the middle diagram below), giving the maximum possible thrust in the direction that you want the rocket is to travel in.

nozzle-expansion

An underexpanded nozzle (top) is inefficient because some of the exhaust gas is propelled backwards at an angle to the rocket’s direction of travel, i.e. pushing it right/left and back/forth rather than up. An overexpanded nozzle (bottom) is more efficient than either of the previous two, but the jet of exhaust that it produces is unstable, which could lead to your rocket veering off course.

The degree of expansion of your nozzle depends on the ambient pressure, and so nozzles are often overexpanded at low altitudes and underexpanded at very high altitudes, giving a “sweet spot” during its journey where it operates at the best possible efficiency.

When a nozzle is over- or underexpanded, a complex process can cause shock waves to form in the exhaust flow. Unburnt fuel passing through these shock waves is compressed and burnt, causing bright “shock diamonds” to form.

shock-diamonds-xcor

Shock diamonds in the exhaust of the XCOR Liquid Oxygen-Methane engine.

shock-diamonds-f16

Shock diamonds in the exhaust of an F-16 during takeoff.

* This is an interesting example of the conservation of energy: the energy of the fluid is a combination of the fluid’s kinetic and potential energies, and as the speed (and therefore the kinetic energy) of the fluid increases, the potential energy (i.e. its pressure) must decrease.