Last week, we saw how we can approximate an airplane’s lift coefficient at any small angle of attack. Now, we can zoom in on drag, looking at how the drag coefficient depends on the lift coefficient and how this information can further inform aerospace engineers on how to design planes.

It turns out that an airplane’s drag coefficient, C_{D}, has a quadratic relationship with its lift coefficient, C_{L}. More specifically, we can write the following approximate equation:

C_{D} = C_{D, 0} + KC_{L}^{2}.

Here, K is a constant whose value is specific to each individual plane. C_{D, 0}, often called the “zero lift drag coefficient,” is another constant whose value is specific to each plane. This presence of C_{D, 0} tells us that even when an airplane is generating no lift, it still experiences drag. Overall, however, this equation conveys that increasing the lift coefficient comes at the cost of increasing the drag coefficient, and that this cost gets more significant the higher the lift coefficient becomes.

Therefore, there exists an “optimal” lift coefficient to fly with, one that allows a plane to produce enough lift to keep its weight aloft while also not producing too much drag for the thrust from the engines to counteract. We can obtain this optimal lit coefficient if we fly at (L/D)_{max}, the maximum lit-to-drag ratio. Since L = ½ ρ u^{2 }S C_{L} and D = ½ ρ u^{2 }S C_{D}, the ratio L/D can be calculated by dividing C_{L} by C_{D}.

Since different planes have different values of K and C_{D, 0}, it then makes sense that different planes have different values of (L/D)_{max}. In general, engineers have been able to design newer planes to have higher lift-to-drag ratios. The 1903 Wright Flier, which completed the first ever powered flight, had an (L/D)_{max} of 8. In contrast, the Boeing 747, which completed its first flight in 1969, has an (L/D)_{max} of 18. Nature, however, still outperforms human-made designs, with a cruising albatross bird having an (L/D)_{max} of 20.

The concept of a maximum lift-to-drag ratio has numerous applications, one of which is designing a plane to have the largest possible range. When range is maximized, fuel expenditure per unit distance is minimized, and therefore a plane can fly the greatest distance possible. Using numerous approximations, engineers have derived the following equation that estimates the maximum range for a propeller-driven aircraft:

Range = (1/g)* (n/f_{p}) *(C_{L}/C_{D})* log(m_{i} / m_{e})

In this equation, we see that range is linearly proportional to the ratio C_{L}/C_{D}. This precisely means that range is maximized when flying at (L/D)_{max}. To those who are curious, g represents the gravitational field strength on Earth (9.81 m/s/s), n is a quantity that measures propeller efficiency, f_{p} represents the mass of fuel consumed per unit time per unit power, and m_{i} and m_{e} represent the mass of the plane fully loaded with fuel and the mass of the plane without any fuel, respectively.

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