A ship tows a submerged cylinder, 1.5 m in diameter and 22 m long, at U = 5 m/s in fresh water at 20°C. Estimate the towing power in kW if the cylinder is (a) parallel, and (b) normal to the tow direction.

(a) 120 kW

(b) 800 kW

Explanation:

Given:

diameter: 1.5 m

length = 22 m

U = 5 m/s

temperature = 20°C

For water at 20°C, take ρ = 998 kg/m³ and µ = 0.001 kg/m⋅s

To find:

power in kW

(a) if the cylinder is parallel

Length / Diameter = L / D = 22 / 1.5 = 14.6 = 15

Re (L) = ρ*U*L / µ = 998 * 5 * 22 / 0.001 = 109780000 = 1.1E8

C (D. Frontal) ≈ 1.1

Force = F = 1.1 * ρ/2 * U² * π / 4 * D

= 1.1 (998 / 2) (5) ² (π / 4) (1.5) ²

= 1.1 * 499 * 25 * 0.785 * 2.25

= 24000 N

Power = Force * Displacement / time

= F * U

= 24000 * 5

= 120000

Power = 120 kW

b) if the cylinder is normal to the tow direction.

Re (L) = ρ*U*D / µ = 998 * 5 * 1.5 / 0.001 = 7485000 = 7.5E6

C (D. Frontal) ≈ 0.4

Force = F = 0.4 * ρ/2 * U² * D * L

= 0.4 (998 / 2) (5) ² (1.5) (22)

= 164670 ≈ 165000

Power = Force * Displacement / time

= F * U

= 165000 * 5 = 825000

≈ 800 kW

Power = 800 kW