Lesson 1 · 35 min
Reading an Aircraft Like a Designer
A Sopwith Camel and a Boeing 747 look nothing alike, yet a designer can compare them with the same half-dozen numbers: how heavy the aircraft is, how much of that weight is the aircraft itself, how hard the wing and the engines work, and how cleanly it slips through the air. This lesson defines those numbers. The rest of the module follows them through eighty years of aircraft.
Learning objectives
- Define takeoff gross weight \(\Wo\), empty weight \(\We\) and the empty-weight fraction \(\We/\Wo\), in Raymer's notation.
- Compute wing loading \(\WS\), power loading \(W/P\) and thrust-to-weight ratio \(\TW\), and convert them between SI and US units.
- Find the aspect ratio \(A = b^2/S\) and the drag area \(f = \CDz S\) of a wing or an aircraft.
- Derive \(\LDmax = \tfrac12\sqrt{\pi A e/\CDz}\) from the parabolic drag polar and use it to estimate an aircraft's aerodynamic efficiency.
Why designers compare numbers, not pictures
Conceptual design starts long before anyone draws the final shape. A designer asked for a new aircraft first asks what similar aircraft have achieved: how heavy they were for their payload, how big a wing they needed, how much power. Those answers come from historical data, and they are the starting point of the sizing method you will use in the coming weeks. Raymer's book builds its first weight estimate on exactly this kind of data.
This module therefore reads the history of aircraft design through the numbers that a designer would extract from each aircraft. Most of them come from one careful source: Laurence Loftin's Quest for Performance (NASA SP-468, 1985), whose appendix lists the weights, dimensions, engines, speeds and estimated drag of about ninety aircraft from 1914 to 1982. The Design Trends Explorer lets you plot all of them.
Weights: \(\Wo\), \(\We\) and the empty-weight fraction
Raymer divides the takeoff gross weight \(\Wo\), the weight at the start of the design mission, into three parts:
Weight build-up
\[ \Wo = W_{\text{crew}} + W_{\text{payload}} + \Wf + \We \]\(\Wf\) is the fuel weight and \(\We\) the empty weight: structure, engines, landing gear, fixed equipment and avionics, everything that is not crew, payload or fuel.
The empty-weight fraction \(\We/\Wo\) measures how much of the aircraft's weight is the aircraft itself. A low value leaves more of \(\Wo\) for fuel and payload, so it is one of the best single indicators of structural and technological progress. It also depends strongly on the type of aircraft and its size, which is why Raymer correlates it statistically against \(\Wo\) for each class of aircraft (Raymer, Chapter 3; you will use these correlations for the first weight estimate).
Loadings: how hard the wing and the engines work
Wing loading \(\WS\)
Weight per unit wing area, \(\Wo/S\). In steady level flight the wing carries the weight, so \(\WS = \tfrac12\rho V^2 \CL\): a high wing loading means high speeds, especially for takeoff and landing.
Power and thrust loading
For propeller aircraft, the power loading \(W/P\) (kg/kW or lb/hp; lower is more powerful). For jets, the thrust-to-weight ratio \(\TW\), dimensionless (higher is more powerful).
Loadings
\[ \frac{W}{S} = \frac{\Wo\,g}{S}\ \ \text{(N/m}^2\text{)}, \qquad \frac{W}{P} = \frac{\Wo}{P}\ \ \text{(kg/kW)}, \qquad \frac{T}{W} = \frac{T}{\Wo\,g} \]Raymer quotes \(\WS\) in kg/m² in SI (lb/ft² in US units). Multiply by \(g\) for N/m².
Raymer calls \(\TW\) and \(\WS\) the two most important parameters of a conceptual design: together with \(\Wo\) they fix the size of the wing and of the engines. Raymer devotes his Chapter 5 to choosing them. Lesson 4 shows how flaps made possible the jump in wing loading of the 1930s.
Example 1.1 — The loadings of a P-51D Mustang
Loftin lists the P-51D at \(\Wo = 10\,100\ \text{lb}\), \(S = 233\ \text{ft}^2\) and \(1490\ \text{hp}\). Find its wing loading in kg/m², lb/ft² and N/m², and its power loading in kg/kW.
Show solution
Convert first: \(\Wo = 10\,100(0.4536) = 4581\ \text{kg}\), \(S = 233(0.0929) = 21.65\ \text{m}^2\), \(P = 1490(0.7457) = 1111\ \text{kW}\).
\[ \begin{aligned} \frac{W}{S} &= \frac{4581}{21.65} = 211.6\ \text{kg/m}^2 = \frac{10\,100}{233} = 43.3\ \text{lb/ft}^2 \\ \frac{W}{S} &= 211.6(9.81) = 2076\ \text{N/m}^2 \\ \frac{W}{P} &= \frac{4581}{1111} = 4.12\ \text{kg/kW}\quad (6.8\ \text{lb/hp}) \end{aligned} \]Seven times the wing loading of a First World War fighter (about \(30\ \text{kg/m}^2\)), with half its power loading: Lessons 3 and 4 explain how that became possible.
Aerodynamic efficiency: \(A\), \(\CDz\) and \(\LDmax\)
The wing's aspect ratio is its span squared over its area, \(A = b^2/S\): long, slender wings have a high \(A\). For a biplane, Loftin uses the span and the area of one wing, so a biplane with two equal wings has \(A = b^2/(S/2)\).
For most of this course, the drag of a whole aircraft is modeled with the parabolic drag polar: a zero-lift (parasite) part, plus a part that grows with the square of the lift coefficient.
Lift, drag and the drag polar
\[ \colL{L} = \tfrac12\rho V^2 S\,\CL, \qquad \colD{D} = \tfrac12\rho V^2 S\,\CD, \qquad \CD = \CDz + K\CL^2,\quad K = \frac{1}{\pi A e} \]\(e\) is the Oswald efficiency factor, about 0.7 to 0.85 for a complete aircraft. The product \(f = \CDz S\) is the drag area, the area of a flat plate with the same zero-lift drag.
The ratio \(L/D = \CL/(\CDz + K\CL^2)\) is the aircraft's aerodynamic efficiency. To find its maximum, set the derivative with respect to \(\CL\) to zero:
\[ \frac{\dd}{\dd\CL}\!\left(\frac{\CL}{\CDz + K\CL^2}\right) = \frac{\CDz - K\CL^2}{(\CDz + K\CL^2)^2} = 0 \quad\Rightarrow\quad K\CL^2 = \CDz \]At the best \(L/D\) the induced drag equals the zero-lift drag, \(\CD = 2\CDz\), and
Maximum lift-to-drag ratio
\[ \CL^* = \sqrt{\frac{\CDz}{K}}, \qquad \LDmax = \frac{1}{2\sqrt{K\CDz}} = \frac12\sqrt{\frac{\pi A e}{\CDz}} \]Loftin computed every \(\LDmax\) in his tables this way, with \(e\) between 0.70 and 0.75. Two routes lead to a better aircraft: lower \(\CDz\) (cleaner shapes, Lesson 4) or higher \(A\) (longer wings, limited by structural weight).
Example 1.2 — How efficient was the DC-3?
The Douglas DC-3 has \(A = 9.14\) and \(\CDz = 0.0249\). With \(e = 0.75\), estimate \(\LDmax\) and the lift coefficient at which it occurs. Its wing area is \(91.7\ \text{m}^2\); what is its drag area?
Show solution
Loftin's table gives \(\LDmax = 14.7\). The whole aircraft, with two engines, landing gear and a tail, has the zero-lift drag of a flat plate of about \(2.3\ \text{m}^2\), a square 1.5 m on a side.
Eighty years in one chart
Put the numbers of many aircraft on one chart against the year of their first flight, and the history of aircraft design appears as a set of trends. Figure 1.2 shows four of them. Look for the sudden changes: they mark the technical revolutions that the next five lessons describe.
Check your understanding
Key takeaways
- \(\Wo = W_{\text{crew}} + W_{\text{payload}} + \Wf + \We\). The empty-weight fraction \(\We/\Wo\) measures how much of the aircraft is aircraft.
- Wing loading \(\WS\) and power loading \(W/P\) (or \(\TW\) for jets) say how hard the wing and the engines work; with \(\Wo\) they size the aircraft.
- \(A = b^2/S\); the drag polar is \(\CD = \CDz + \CL^2/(\pi A e)\), and \(f = \CDz S\) is the drag area.
- \(\LDmax = \tfrac12\sqrt{\pi A e/\CDz}\), reached when the induced drag equals the zero-lift drag.
- Historical data are the designer's first tool. Next, Lesson 2 goes back to the beginning: how the first designers found the lift equation and the wing that made the Wright Flyer work.