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## Q. 14.5

The Flettner rotor-powered ship shown in Figure 14.23A had two rotors, each 3 m diameter and 15 m tall. If ω = 200 rpm and the speed of the wind relative to the rotor is 30 km/h, ﬁnd the force applied to each rotor by the wind.

## Verified Solution

To ﬁnd the force generated by each rotor, we will use the results for a spinning cylinder as shown in Figure 14.22 to determine the lift and drag coefficients. First we calculate the rotational velocity as

Vθ = Rω = (0.0015 km)(200 rpm)(2$\pi$ rad/rev)(60 min/h) = 113 km/h

Dividing this value by the wind speed gives us the spin ratio WD/2U. Thus we have

$\frac{V_θ}{U}=\frac{1133\ km/h}{30\ km/h}=3.75$

From Figure 14.22 we ﬁnd CL = 8.9 and CD = 3.8. Thus the lift and drag forces are

FL = CL $\frac{1}{2}$ρU2DL = (8.9) $\left(\frac{1}{2}\right)$ (1.2 kg/m3)(8.33 m/s)2(3 m)(15 m) = 16.7 kN

FD = CD $\frac{1}{2}$ρU2DL = (3.8) $\left(\frac{1}{2}\right)$ (1.2 kg/m3)(8.33 m/s)2(3 m)(15 m) = 7.1 kN

where we have assumed air at 20°C in calculating the density. The force applied by the wind to each rotor is thus given by Fwind = 7.1 kNi + 16.7 kNj as shown in Figure 14.23B. This force acts at an angle of θ = tan−1(16.7/7.1) = 67° to the left of the relative wind direction.