Lesson 2 · 40 min

The Empty-Weight Fraction

Before the aircraft is drawn, nobody can add up the weight of its wing, fuselage and systems. What the designer can do is ask what fraction of their takeoff weight similar aircraft have spent on themselves. The answer is a one-line power law per class of aircraft, fitted to history. Used well, it is surprisingly good; used carelessly, it is the largest error in the first estimate.

Learning objectives

Evaluate the statistical empty-weight fraction \(\We/\Wo = A\,\Wo^{\,C}K_{vs}\) for a class of aircraft.
Convert the coefficient \(A\) between pounds and kilograms, and explain why \(C\) does not change.
Choose the right class, and apply variable-sweep and technology factors.
Judge how far to trust the trend, using the scatter of real aircraft around it.

A power law per class

Within a class of aircraft, the empty-weight fraction falls slowly as aircraft get bigger: some items (avionics, the cockpit, minimum gauges of skin) do not grow in proportion. Each class is fitted with

Statistical empty-weight fraction

\[ \frac{\We}{\Wo} = A\,\Wo^{\,C}\,K_{vs} \]

\(C\) is negative and small. \(K_{vs} = 1.04\) for a variable-sweep wing, 1.00 otherwise. On log–log axes the trend is a straight line of slope \(C\).

Empty-weight trends (source: Raymer, Table 3.1). \(A\) is the coefficient for \(\Wo\) in lb; the kg column is converted for this module. Check the values against the edition you use.
Class	\(A\) (lb)	\(A\) (kg)	\(C\)	\(\We/\Wo\) at a typical \(\Wo\)

Pounds and kilograms

Because \(\We/\Wo\) is a ratio, the trend must give the same fraction whatever the unit of \(\Wo\). With \(\Wo\) in lb equal to \(\Wo\) in kg divided by 0.4536:

Converting the coefficient

\[ A_{\text{kg}} = A_{\text{lb}}\,(0.4536)^{-C} \]

The exponent \(C\) is the same in any unit; only \(A\) changes. Using a pound \(A\) with a weight in kilograms is one of the most common sizing errors.

Example 2.1 — The business jet's trend

The jet-transport trend has \(A = 1.02\), \(C = -0.06\) for \(\Wo\) in lb. (a) Convert \(A\) to kg. (b) Evaluate \(\We/\Wo\) at \(\Wo = 6943\ \text{kg}\) both ways.

Show solution

\[ \text{(a)}\ A_{\text{kg}} = 1.02(0.4536)^{0.06} = 1.02(0.9537) = 0.9727 \] \[ \text{(b)}\ 0.9727(6943)^{-0.06} = 0.5721 \] \[ 1.02\left(\frac{6943}{0.4536}\right)^{-0.06} = 1.02(15\,307)^{-0.06} = 0.5721 \]

The same fraction either way, as it must be. Typed in the lb form with 6943 instead of 15 307, the answer would be 0.6000: a 5% error that the sizing equation would amplify about fourfold (Lesson 7).

Choosing the class, and adjusting for technology

The trend is only as good as the match between the new design and the aircraft behind the statistics. Three questions to ask:

Which class does the mission resemble? A business jet has no class of its own in the table; it is closest to a jet transport. A turboprop trainer is closer to a jet trainer than to a general-aviation single.
Is the weight inside the data? A trend fitted to 50 t transports says little about a 5 t one. Extrapolating far beyond the data is a guess dressed up as a calculation.
Is the technology the same? The trends describe the aircraft they were fitted to. A design with extensive composite structure, or one with unusual features (a variable-sweep wing, a pressurized cabin on a small aircraft, a high-g structure), needs an adjustment.

Technology is handled with a factor on the trend, \(\We/\Wo = A\,\Wo^{\,C}K_{vs}K_{\text{tech}}\). A common first assumption for an extensively composite airframe is a factor around 0.9; a conservative designer might use 1.05 for an unfamiliar configuration. Whatever the choice, write it down: it is an assumption the later weight estimates (Module 10) will test.

How good is the trend?

Module 1's NASA data give an independent check. Figure 2.1 plots the empty-weight fractions of those aircraft against the trend for their class.

Figure 2.1 The trend for a class (blue line) and the aircraft of that class in Module 1's data (points, from Loftin's NASA survey). The readouts give the average ratio of the real fractions to the trend, and the spread. Hover over a point to name the aircraft. For the jet classes the trend sits in the middle of the data; for the Second World War bombers it does not.

For jet transports, jet fighters and light single-engine aircraft, the real aircraft scatter around the trend by about ±15%, with an average within a few percent of it. The Second World War bombers lie about 50% above the military cargo and bomber trend: the trend describes modern jet bombers and cargo aircraft, with different structures, materials and missions. Statistics from the wrong era are statistics of the wrong aircraft.

Check your understanding

Key takeaways

\(\We/\Wo = A\,\Wo^{\,C}K_{vs}\), with \(A\) and \(C\) for the class and \(K_{vs} = 1.04\) for variable sweep.
\(A_{\text{kg}} = A_{\text{lb}}(0.4536)^{-C}\); \(C\) does not change with units.
Choose the class by mission, stay inside the data, and apply a stated technology factor.
Real aircraft scatter about ±15% around the trend, and the sizing equation amplifies that.
Next, Lesson 3 estimates the aerodynamic ingredient of the fuel fraction, \(L/D\).