Vandelay Jet Boat A boat is propelled by a jet of water as s

Vandelay Jet Boat. A boat is propelled by a jet of water, as shown in the figure below. Water at 0.03 m³/s is pumped into a 15-cm diameter pipe at the front of the boat and exits from a 5-cm diameter pipe at the rear of the boat. The frictional force resisting the boat’s motion is given as

F_D = 12v²

where v is the boat’s velocity relative to still surrounding water and the coefficient 12 has the units of N s² m^-2. The center of the inlet and outlet pipes are the same depth below water. The water temperature is 20 °C.

(a)Calculate the steady velocity of the boat relative to still water. (15 points)

(b)If you are attempting to redesign the boat to achieve a faster velocity, name two changes that you could make. (10 points)

Solution

Given data,

Volume discharge = 0.03m³/s,

Inlet diameter, d_i = 15cm = 0.15m,

So, Inlet area = d_i²/4 = x (0.15)²/4 = 0.0177m²

Exit diameter, d_o = 5cm = 0.05m,

So, Exit area = d_o²/4 = x (0.05)²/4 = 1.96x10^-3m²

Calculation for inlet velocity

Inlet velocity, V_i = volume discharge/inlet area = 0.03/0.0177 =1.695m/s

Calculation for exit velocity

Exit velocity, V_o = volume discharge/exit area = 0.03/1.96x10^-3 = 15.31m/s

Calculation for Thrust produced

Thrust produced = change in momentum = mass flow rate x change in velocity

= density x volume discharge (Exit velocity – Inlet velocity)

                        = 998.2 x 0.03 x (15.31-1.695)              density of water at 20^oC = 998.2kg/m³

                        = 407.6N

a)

The boat achieves a steady velocity when the thrust produced is equal to the drag force.

ie. Thrust = Drag force

407.6 = 12V²

V² = 407.6/12 =33.97

V = 5.83 m/s

b)

For the boat to achieve a faster velocity, the following options can be adopted:-