Wing Loading

Wing loading, measured in lb/ft², is basically a measure of how hard the wing has to work to keep the airplane in the air. A lightly-loaded wing has the advantages of a lower stall speed and the ability to achieve a higher turn rate. On the downside, low wing loading can result in a bumpier ride in turbulence, and it also means we’re carrying more area (drag) and weight than we might otherwise need once we’re off the ground.

Conversely, higher wing loading yields a better ride on a turbulent day and minimizes drag and weight, but at the expense of stall speed and turn rate. Since we’re not designing an explicitly aerobatic airplane or a dogfighter, we’re not as concerned about an eye-watering turn rate, and the benefits of a smoother ride and reduced drag are appealing. Still, we want a reasonably low stall speed–so is there a compromise to be had?

The answer, of course, is to make use of flaps so that we can have the best of both worlds: more lift and drag when we want it (the landing regime), less wing area when we don’t (cruise regime). We’ll start with the assumption that we’ll include single-slotted flaps to achieve our desired landing speed, and we’ll keep wing area to a minimum to help with our desired cruise speed.

The lift equation

We might as well bite the bullet and get familiar with the lift equation now.

$$ \scriptsize L=\tiny \frac{1}{2} \scriptsize \rho V^{2} S C_L \>\>\> \text{or} \>\>\> L=qS C_L $$

• is lift in pounds, and for now is assumed to equal the weight of the aircraft.
• (rho) is the density of air, which in the US customary system is measured in slugs. Density varies with altitude, temperature and humidity, so we’ll use sea-level standard conditions (59° F) unless specified otherwise. This is 0.002377 slugs/ft³.
• is velocity in feet/second.
• is sometimes lumped together as (dynamic pressure). We’ll discuss a little more in our next installment.
• or is the wing planform reference area in ft². The reference area is not quite the actual wing area, because we’re including the part of the wing buried in the fuselage and the reference area might be squared off at the wingtips even if the wings have a nicely sculpted rounded shape.
• is the wing’s lift coefficient. The best way to think of this is as a measure of how much lift the wing produces per square foot of area and per psi of dynamic pressure. It’s important to note that this is the whole-wing coefficient (finite span) and that is different from the lift coefficient of an airfoil (2D section, or infinite span). The value you look up for an airfoil in Theory of Wing Sections or Riblett’s GA Airfoils will be the 2D value, which we denote with a small “c” (). The difference is that the wing of finite span has losses at the tips.

Wing Loading

By rearranging the terms of the lift equation we get:

$$ \scriptsize \frac{W}{S}= \tiny \frac{1}{2} \scriptsize \rho V_{stall}^{2} C_{Lmax} $$

Because we’re primarily interested in the minimum size the wing can be at our desired stall speed, we’re using stall speed for (58 mph) and the maximum lift coefficient the wing can generate. Because we intend to use a relatively high aspect ratio and slotted flaps, we will use 2.0. Therefore our wing loading will be:

$$ \scriptsize \frac{W}{S}= \tiny \frac{1}{2} \scriptsize (0.002377 \text{ slugs/ft}^{3} ) \left( 58 \text{ mi/hr} \cdot \frac{5,280 \text{ mi}}{\text{hr}} \cdot \frac{\text{hr}}{3,600 \text{ sec}} \right) ^{2} \cdot 2.0 $$

$$ \scriptsize \frac{W}{S}=17.2 \text{ lb/ft}^{2} $$

Next we’ll examine power loading.