How to calculate the airspeed for maximum specific range?

Question

Given the following parameters for a small business aircraft:

• Wing surface: $S=30m^2$
• Lift Drag Polar in clean configuration: $C_D = 0.022 + 0.047 {C_L}^2$
• Maximum Lift Coefficient $C_{L_{max}}=1.35$
• Static sea-level Thrust: $T_0 = 12KN$
• Thrust variation with altitude: $\frac{T}{T_0}=\frac{\rho}{\rho_0}$
• Aircraft weight: $W = 50 KN$
• Thrust specific fuel consumption (constant): $C_T = 0.7 N/Nh$

I need to calculate:

• the optimum lift coefficient for maximum specific range
• the airspeed for maximum specific range when operating at the specified weight and sea level conditions.

For $C_{L_{opt}}$ I have found a value of 0.395 using the lift drag polar and the fact that $C_{L_{opt}}= \sqrt{\frac{1}{3}C_{D_0}\pi A e}$.

I am then assuming that the airspeed for maximum range is given by $V=\sqrt{\frac{W}{S}\frac{2}{\rho}\frac{1}{C_{L_{opt}}}}$.

In my calculations, I am using $W=50000/9.80665=5098.581 KG$ and $\rho=1.225$, which gives me a value of $V = 26.504 m/s$

However this seems not to be the correct answer. What am I missing?

Answer 1

Check the units. Using $$W = 50000N$$ instead of $W=5099kg$ you get $$V=83m/s$$ (vs. $V=26.5\sqrt{m}$).