Lesson 1 · 35 min

Sizing from a Conceptual Sketch

Module 2 ended with a set of requirements and a mission profile. This module turns them into one number: the takeoff gross weight \(\Wo\). It is the first number of a new design and the one everything else scales from: the wing, the engines, the cost. It comes from a rough sketch, a handful of historical statistics and the Breguet equations, in an afternoon.

Learning objectives

Derive the sizing equation from the weight build-up and name its three ingredients.
Compute the crew and payload weight from a requirement.
Explain why the sizing equation must be solved by iteration, and carry out one iteration by hand.
Read the solution as the crossing of two curves: the empty-weight fraction history predicts and the one the design can afford.

The weight build-up

Module 1 split the takeoff weight into four parts:

Takeoff weight build-up

\[ \Wo = W_{\text{crew}} + W_{\text{payload}} + \colW{\Wf} + \colW{\We} \]

The crew and payload are known: the requirements state them. The fuel and empty weights are not, because both depend on how big the aircraft turns out to be. The trick is to write them as fractions of \(\Wo\), which are much easier to estimate than the weights themselves: an airliner's fuel fraction depends mostly on its range, and its empty-weight fraction on its class and size, not on its exact weight. Dividing by \(\Wo\) and solving gives

The sizing equation

\[ \Wo = \frac{W_{\text{crew}} + W_{\text{payload}}}{1 - \colW{\dfrac{\Wf}{\Wo}} - \colW{\dfrac{\We}{\Wo}}} \]

Only the crew and payload are in the numerator. The denominator is the share of the takeoff weight left for them, usually 0.1 to 0.3.

Empty-weight fraction

From statistics of earlier aircraft of the same class: \(\We/\Wo = A\,\Wo^{\,C}\) (Lesson 2).

Fuel fraction

From the mission profile, segment by segment, with \(L/D\) from the sketch (Lesson 3) and \(C\) from the engine type (Lesson 4), combined in Lesson 5.

Crew and payload

From the requirements, at standard masses per person and bag, plus cargo, weapons or equipment.

The business jet of this module

One design runs through all seven lessons, so that each ingredient can be seen in its place. Its requirements:

Requirements of the module's business jet
Payload	8 passengers at \(100\ \text{kg}\) each, with baggage
Crew	2 pilots at \(90\ \text{kg}\) each
Design range	\(4600\ \text{km}\) at \(850\ \text{km/h}\) (about Mach 0.8 at \(12\ \text{km}\))
Reserve	45 minutes of loiter, plus an allowance for trapped fuel
Engines	Two turbofans

A first sketch gives it a wing of aspect ratio 8, a fuselage about \(14\ \text{m}\) long, and two engines on the rear fuselage. By Lesson 6, those numbers and the requirements will have sized it.

Example 1.1 — Crew and payload

Find \(W_{\text{crew}} + W_{\text{payload}}\) for the business jet.

Show solution

\[ W_{\text{crew}} = 2(90) = 180\ \text{kg}, \qquad W_{\text{payload}} = 8(100) = 800\ \text{kg} \] \[ W_{\text{crew}} + W_{\text{payload}} = 980\ \text{kg} \]

Under a tonne. Since the denominator of the sizing equation will be about 0.14, the aircraft will weigh about seven times that.

Why it takes iteration

If both fractions were fixed numbers, the sizing equation would give \(\Wo\) at once. But the empty-weight fraction depends on \(\Wo\): bigger aircraft of a class have slightly smaller empty-weight fractions. So \(\Wo\) appears on both sides, and the equation is solved by iteration:

Guess \(\Wo\).
Evaluate the empty-weight fraction at the guess, \(\We/\Wo = A\,\Wo^{\,C}\).
Compute a new \(\Wo\) from the sizing equation.
Repeat from step 2 with the new value until it stops changing.

The fuel fraction does not depend on \(\Wo\) in this first estimate (Lesson 6 shows when it does), so it is computed once.

Example 1.2 — The first iteration

Lessons 2 to 5 will show that the business jet has \(\Wf/\Wo = 0.2867\) and, as a jet transport, \(\We/\Wo = 0.9727\,\Wo^{-0.06}\) with \(\Wo\) in kg. Starting from a guess of \(10\,000\ \text{kg}\), carry out the first two iterations.

Show solution

\[ \frac{\We}{\Wo} = 0.9727(10\,000)^{-0.06} = 0.5598, \qquad \Wo = \frac{980}{1 - 0.2867 - 0.5598} = 6382\ \text{kg} \] \[ \frac{\We}{\Wo} = 0.9727(6382)^{-0.06} = 0.5750, \qquad \Wo = \frac{980}{1 - 0.2867 - 0.5750} = 7088\ \text{kg} \]

The guesses swing back and forth and settle quickly: Lesson 6 finishes the iteration at \(\Wo = 6943\ \text{kg}\).

Figure 1.1 The sizing equation as two curves. Blue: the empty-weight fraction the statistics predict for an aircraft of each weight. Violet: the fraction the design can afford, \(1 - \Wf/\Wo - (W_{\text{crew}} + W_{\text{payload}})/\Wo\). The design closes where they cross. The numbered points are the iterations from the starting guess. Try a heavier payload or a larger fuel fraction, and see how quickly the crossing runs away.

Check your understanding

Key takeaways

\(\Wo = W_{\text{crew}} + W_{\text{payload}} + \Wf + \We\), and so \(\Wo = (W_{\text{crew}} + W_{\text{payload}})/(1 - \Wf/\Wo - \We/\Wo)\).
The empty-weight fraction comes from statistics, the fuel fraction from the mission, the crew and payload from the requirements.
Because \(\We/\Wo\) depends on \(\Wo\), the equation is solved by iteration; graphically, where the trend and the affordable curve cross.
Next, Lesson 2 looks closely at the empty-weight statistics.