Weight Fractions

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Weight is a big deal.

We start the design of an airplane by estimating the gross weight (takeoff weight, , or “w-naught”) because weight plays such a significant role in almost everything an airplane does. A heavier airplane means:

• More wing area is required to maintain the desired takeoff and landing speeds (which in turn weighs more and adds drag in the form of wetted area).
• We have to burn more fuel to meet intended speed and range requirements; the additional fuel is now more weight that must be carried by even larger wings.
• Performance suffers, particularly:
  • Rate of climb
  • Climb gradient
  • Takeoff performance (e.g. distance to clear a 50′ obstacle)
  • Landing distance (approach speed,V_ref , increases with stall speed, so with more weight, the brakes have to scrub off more kinetic energy, which increases with the square of speed).

In short, it’s often said that weight is the enemy.

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Weight fractions

Let’s start with some definitions:

• is the total, or gross weight (takeoff weight).
• is the crew weight.
• is the payload weight (baggage).
• is the fuel weight.
• is the empty weight of the aircraft.

Putting all of this together:

$$\scriptsize W_0 = W_{crew}+W_{payload}+W_{fuel}+W_{empty}$$

Another way we can think of this it to make each piece a fraction of the gross weight:

$$ \scriptsize \frac{W_{crew}}{W_0} + \frac{W_{payload}}{W_0} + \frac{W_{fuel}}{W_0} + \frac{W_{empty}}{W_0} = 100 \text{%} $$

We could also rewrite the first equation above like this…

$$\scriptsize W_0=W_{crew}+W_{payload}+(\frac{W_{fuel}}{W_0})W_0+(\frac{W_{empty}}{W_0})W_0$$

…and continue to rearrange the terms until we have:

$$\scriptsize W_0-(\frac{W_{fuel}}{W_0})W_0-(\frac{W_{empty}}{W_0})W_0=W_{crew}+W_{payload}$$

$$\scriptsize W_0=\frac{W_{crew}+W_{payload}}{1-(W_{fuel}/W_0)-(W_{empty}/W_0)}$$

Notice now that we already know two of those variables (crew weight and payload weight, since we selected them in our design requirements) and if we can estimate the fuel weight fraction and the empty weight fraction, we can determine .

We’ll take up estimating the fuel fraction and empty weight fraction in the next couple of posts, and then put it all together to determine our estimate of gross weight.