{"id":2555,"date":"2019-09-20T12:13:53","date_gmt":"2019-09-20T19:13:53","guid":{"rendered":"https:\/\/c21-wp.phas.ubc.ca\/?post_type=article&p=2555"},"modified":"2019-10-11T17:01:47","modified_gmt":"2019-10-12T00:01:47","slug":"air-transport","status":"publish","type":"article","link":"https:\/\/c21.phas.ubc.ca\/article\/air-transport\/","title":{"rendered":"Air Transport"},"content":{"rendered":"

The central question when considering any form of transport is how much energy $E_C$\u00a0 is required to move a unit mass a unit distance. This will determine the resources and environmental consequences of moving goods and people around the globe. The units of $E_C$ are J\/kg\/m (or MJ\/tonne\/km), which reduces to m\/s2<\/sup>, so clearly a lot depends on g<\/em>.<\/p>\n

A little aerodynamics[note] J.D.Anderson, Introduction to Flight<\/em>, McGraw-Hill, 6th Edition (2008) [\/note]\n<\/strong><\/p>\n

The total drag on an aircraft is equal to sum of the \u201cparasite drag\u201d and \u201cinduced drag\u201d. The \u201cparasite drag\u201d $D_p$ is caused by airflow being slowed as it passes the body. It has the form (Eqn.1):<\/p>\n

$\\begin{equation} D_p = \\dfrac{1}{2} C_{D,0} \\rho S v^2 \\tag{1} \\end{equation}$<\/p>\n

Here $C_{D,0}$ is a dimensionless coefficient of drag that is determined by the overall shape of the aircraft (the \u201czero\u201d means drag at zero lift \u2013 see below), $\\rho$ is the density of air, $S$ the wing plan area (normally this would be the frontal area, but in aerodynamics the convention is different; the change makes $C_D$ much smaller than you might expect), and $v$ the air speed.<\/p>\n

The lift $L$ has the same form, with a coefficient of lift $C_L$ replacing that of drag. You can imagine that the total aerodynamic force on the wing is mostly vertical but angled backwards a little due to air being deflected downwards (to generate the lift). Thus the lift contributes to the total drag force, and this is called induced drag, <\/em>$D_i$, which is a dependent on the effective aspect ratio of the wing (length \/ mean chord), $A$. Expressions for this and total drag $D$ are given in Eqns.2-5:<\/p>\n

$\\begin{equation} L = \\dfrac{1}{2} C_L \\rho S v^2 \\tag{2} \\end{equation}$<\/p>\n

$\\begin{equation} D_i = \\dfrac{1}{2} C_{D,i} \\rho S v^2 \\tag{3} \\end{equation}$<\/p>\n

$\\begin{equation} C_{D,i} = \\dfrac{C^2_L}{\\pi A} \\tag{4} \\end{equation}$<\/p>\n

$\\begin{equation} D = D_p + D_i = \\dfrac{1}{2} (C_{D,0} + \\dfrac{C^2_L}{\\pi A})\\rho S v^2\\tag{5} \\end{equation}$<\/p>\n

In level flight (i.e. most of most journeys), $L=W$, the weight of the aircraft. This constraint means that there is preferred flying speed where the drag is at a minimum; at higher speeds, the parasite drag becomes large; at lowers speeds, the induced drag becomes large.<\/p>\n

\"\"

Fig.1. Drag contributions for a Boeing 747 as a function of velocity. Note kN = MJ\/km. In practice, airliners fly somewhat faster than the minimum drag speed as the shape of the total drag curve means there is a relatively small penalty in fuel cost for the resulting reduction in airtime (crew salaries, meals, customer satisfaction etc.). It is also safer: consider what would happen if you flew on the low side of the drag curve, and you hit some turbulence which briefly reduced your airspeed.<\/p><\/div>\n

At the minimum total drag speed, $v_{MD}$, the specific energy cost of transport is at a minimum. A little algebra gives Eqn.6:<\/p>\n

$\\begin{equation} v^2_{MD} = \\dfrac{W}{\\rho S}\\sqrt{\\dfrac{1}{\\pi A C_{D,0} }}\\tag{6} \\end{equation}$<\/p>\n

Evaluating the energy cost at this speed gives the simple result (Eqn.7):<\/p>\n

$\\begin{equation} E_C = \\dfrac{D}{m} = 2g\\sqrt{\\dfrac{C_{D,0}}{\\pi A}} \\tag{7} \\end{equation}$<\/p>\n

One major factor has been forgotten here: the mechanical efficiency $\\eta$ of the aircraft\u2019s engines. This is important because so far we have only considered the mechanical energy, not the total energy expended. The ratio of the two, $\\eta$ , is generally about 1\/3; i.e. 2\/3 of the chemical energy produced by burning fuel is lost as useless heat. Hence, Eqn.7 is really just the mechanical energy cost of transport; the real thermal<\/em> energy cost of transport is about three times higher and is given by Eqn.8:<\/p>\n

$\\begin{equation} E_C = \\dfrac{D}{m} = \\dfrac{2g}{\\eta} \\sqrt{ \\dfrac{C_{D,0}}{\\pi A}} \\tag{8} \\end{equation}$<\/p>\n

The above result contains all the engineering complexity of the airframe in just two numbers; the drag coefficient $C_{D,0}$ depends on the shape, and is 0.0155[note] Sometimes expressed as the minimum drag coefficient, which is double the zero-lift coefficient.[\/note] for a 747[note] Boeing 747, http:\/\/en.wikipedia.org\/wiki\/Boeing_747-400<\/a> [2019-09-20].[\/note] (slightly lower for more modern airliners); the effective aspect ratio $A$ is 7.4 (slightly higher for more modern airliners). All the thermodynamic complexity of the engines is rolled into the efficiency $\\eta$.<\/p>\n

Putting in the numbers: E<\/em>c<\/sub> = 1.5 J\/kg\/m = 1.5 MJ\/tonne\/km<\/p>\n

Three other factors we have ignored:<\/p>\n