Solution Naval Numerical 3

Numerical 3: A ship 150 m In length, 24 m breadth, displaces 25000 tonne when floating at a draught of 9 m in sea water of density 1025 kg/m3. The ship's propeller has a diameter of 5.8 m, a pitch ratio of 0.9 and a blade area ratio of 0.45. With the propeller operating at 2 rev/sec, the following results were recorded: Apparent slip ratio = 0.06 Thrust power = 3800 kW Propeller efficiency 64% The Taylor wake fraction is given by: Wt= 0.5Cb -0.05.Calculate EACH of the following for the above condition:(a) The ship's speed;(b) The real slip ratio;(c) The thrust per unit area of blade surface;(d) The torque delivered to the propeller.

Solution: Given that: - Length (L) = 150 m 
Breadth (B) = 24 m
Displacement (∆) = 25000 tonne 
Draught (d)  =9 m
Seawater density (ρ)  = 1025 kg/m3. =1.025 T/m3
Diameter of Propeller(D) = 5.8 m
Pitch ratio = 0.9
Blade area ratio (BAR) = 0.45
Rotational speed (N) = 2 rev/s
Apparent slip (Sa) = 0.06
Thrust Power (Tp) = 3800 kW 
Propeller efficiency (η) = 64%
Wt= 0.5Cb -0.05
In this Formula we know Cb is  block coefficient
Cb = Δ/L x B x d x ρ
Cb = 25000/150 x 24 x 9 x 1.025 
Cb = 0.7527
Now, Wt = 0.5 x 0.757 - 0.05
Wt =  0.326

(a) Ship’s speed = V = Vt(1-Sa)
Vt = P x N x 60/1852 knots
Pitch ratio = P/D 
0.9 = P/5.8
P = 5.22
Using the value of P and N to get Vt
Vt = 5.22 x 2/1852 (N is already in rps not required to multiply with 60)
Vt = 20.29
Sa = 0.06 (given)
Now, V = 20.29(1-0.06)
V = 19.073

(b) The real slip ratio; Va = V(1-Wt)
Va = 19.073(1-0.326)
Va = 12.855 knots
Sr = (Vt - Va)/Vt = (20.29-12.855)/20.29  = 0.366

(c) Thrust per unit area; = Thrust(T)/Blade area(Ad)
Thrust =Thrust power(Tp)/ Speed of advance (Va)
Va need to be converted to m/s
Thus 12.885 x 1852 /3600 =6.628 m/s
T = 3800/6.628 = 573.325 kN
BAR=Ad/(πD2/4)
BAR (given)=0.45
D (given) =5.8 m2
Putting the values Ad = 0.45 x π(5.8)2 / 4
Ad = 11.889 m2
Thrust per unit area = 573.325 / 11.889 = 48.223 kN/ m2

(d) The torque delivered to the propeller
We know that delivered power (dp) = torque on shaft x 2πN; (N in Rps)
Thrust power Tp = delivered power (dp) x transmission efficiency (η)
Tp = T x 2πN x η
T = Tp /  2πN x η
T = 3800/ 2 x 3.14 x 0.64
T = 472.7 kN-m