Lift-to-Drag Ratio

Before we can estimate the airplane’s weight, we need to know a little more about its performance. The lift-to-drag ratio, L/D, is an important figure to estimate because it gives us a relative idea of the amount of drag the airplane has to overcome to keep the airplane aloft in cruise configuration. Let’s go through the variables one at a time.

$$ \scriptsize \frac{L}{D}=\frac{1}{\frac{\normalsize q \>C_{do}}{\normalsize W/S}+ \normalsize \left( \frac{W}{S} \right) \left( \frac{K}{q} \right)}$$

Dynamic Pressure

Our old friend Mr. Bernoulli tells us that the sum of static pressure (the force exerted over a given area) and dynamic pressure (akin to the kinetic energy of the air flow) is constant along a streamline at a given height. In our case, dynamic pressure at cruise speed (200 mph) and altitude (8,000 ft) is:

$$ \scriptsize q = \tiny \frac{1}{2} \scriptsize \rho_{cruise} V_{cruise}^{2} $$

$$ \scriptsize q = \tiny \frac{1}{2} \scriptsize \left( 0.001867 \text{ slugs/ft}^{3} \right) \> \left( 200 \text{ mi/hr} \cdot \frac{5,280\text{ ft}}{\text{mi}} \cdot \frac{\text{hr}}{3,600 \text{ sec}} \right)^{2} = 80.32 \text{ lb/ft}^{2}$$

Equivalent Skin Friction Coefficient

This is a figure that represents the aerodynamic cleanliness of the design. For now, we’ll use Raymer’s estimate for a smooth design with a single engine and fixed landing gear as a starting place.

$$ \scriptsize C_{fe} = 0.0065 $$

Wetted Area Ratio

The wetted area of an object is the total external surface area. If you dunked it in water, this is the amount of area that would get wet. The ratio we’re interested in is the wetted area of the whole airplane in proportion to reference area of the wing. We don’t know either of those figures yet, but Raymer provides an estimate to use as a starting point for a single-engine, conventional design. We hope to keep this number to a minimum by necking down the aft portion of the fuselage (like a tadpole) but we’ll stick with his figure for now.

$$ \scriptsize \frac{S_{wet}}{S_{ref}}=3.8 $$

Parasitic Drag Coefficient

This coefficient tells us how much parasite drag (as distinguished from induced drag–the drag due to lift creation) the airplane will have as a function of its wetted area and dynamic pressure (a combination of speed and density altitude).

$$ \scriptsize C_{d0}=C_{fe} \frac{S_{wet}}{S_{ref}} $$

$$ \scriptsize C_{d0}=0.0065 \cdot 3.8 = 0.0247$$

Wing Loading

Earlier, we calculated that the wing loading should be 17.2 lb/ft². It stands to reason that W/S plays a part in L/D because a large wing that doesn’t have to work very hard to do its job keeping the plane aloft is dragging around a lot of extra skin area and producing unwanted drag.

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

Drag-due-to-lift

We’ve figured the parasite drag already; now what about drag due to lift? This drag, also known as induced drag, depends largely on the span loading (weight/span) which is baked into the following equation in the variable (aspect ratio). We’ll arbitrarily select a fairly high value of = 8.5, which will produce a long span and help reduce induced drag, giving us a better rate of climb.

$$ \scriptsize K = \frac{1}{0.75 \pi A}=\frac{1}{0.75 \pi (8.5)} = 0.050 $$

L/D Ratio

Now we have everything we need to get an estimate of the L/D ratio.

$$ \scriptsize \frac{L}{D}=\frac{1}{\frac{\normalsize q \>C_{do}}{\normalsize W/S}+ \normalsize \left( \frac{W}{S} \right) \left( \frac{K}{q} \right)}$$

$$ \scriptsize \frac{L}{D}=\frac{1}{\frac{\normalsize 80.32 \text{ lb/ft}^{2} \cdot \>0.0247}{\normalsize 17.2 \text{ lb/ft}^{2}}+ \left(17.2 \text{ lb/ft}^{2} \right) \left( \frac{\normalsize .050}{\normalsize 80.32 \text{ lb/ft}^{2}} \right)}$$

$$ \scriptsize \frac{L}{D}=7.93 $$

Next we’ll move on to look at the concept of weight fractions.