1. A survey vessel has a 10 ft diameter, B 5-90 propeller with a pitch of 10 ft. The propeller speed is 200 rpm, the boat speed is 20 knots, and the thrust reduction factor (t) is 0.12, wake fraction (w) is 0.18, and the relative rotational efficiency ηR is 1.0. The propeller operates as indicated by the Wageningen (Troost) Series B propeller charts. Determine:

a. Thrust b. Shaft torque c. EHP of the boat d. The propeller shaft power (delivered power) PD e. The (Quasi) PC or ηD

The propeller is also tested at zero ship speed (bollard pull) and it is found that the engine limits the torque to 50,000 lbf ft. Determine:

f. the propeller rpm and thrust at this condition a. Prop. Thrust.

p_over_dpd

:= Given variables Vs 20knot:= d 10ft := f:=p 10 t n_rpm 200:=

ρ 1.9905lbsec2

ft4⋅:= ηR 1:= t .12:= w .18:=

VA Vs 1 w−( )⋅:= VA 8.437

ms

= Velocity of Approach

J1VAn d⋅

:= nn_rpm60 sec⋅

:= n 3.3331s

= Advance Ratio J1 0.83=

Use the B 5-90 prop curve to determine KT and KQ KT .12:= KQ .023:=

Thrust KT ρ⋅ n2⋅ d4

⋅:= Thrust 2.654 104

× lb=

b. Shaft Torque

Torque KQ ρ⋅ n2⋅ d5

⋅:= Torque 5.087 104

× lb ft⋅= PD 2 π⋅ n⋅

Torque

550lb

ftsec⋅

hp⋅

⋅:= c. Shaft powerr delivered

PD 1.937 103× hp=

PE Thrust 1 t−( )⋅Vs

550lb

ftsec⋅

hp⋅

⋅:= d. EHP

PE 1.433 103× hp=

e. Quasi Efficiency ηD

PEPD

:= ηD 0.74=

f. Propeller rpm and thrust at 50,000.

Torquemax 50000 lb⋅ ft⋅:= Advance_velocity 0:=

noTorquemax

ρ KQ⋅ d5⋅ ηR⋅

:= no 3.305

1s

= nq no 60⋅ sec⋅:=

nq 198.286=

Thrust q KT ρ⋅ nq2

⋅ d4⋅:= Thrustq 9.391 107

× s2 lb=

2. A propeller is to be selected for a single-screw container ship with the following features: EHP = 80000 HP, ship speed = 25 kts, maximum propeller diameter = 34 ft,

w = 0.249, t = 0.18, ηR = 1.0, centerline depth, h = 25 ft a. Using the maximum prop diameter, determine the optimum B 5-90 design. Use the metrics below to confirm your design. a. P/D b. KT (optimum) c. KQ (optimum) d. ηo (optimum) e. J f. Developed HP g. The (Quasi) PC or ηD h. RPM From the consideration of cavitation, determine: i. The predicted cavitation (%) using the Burrill correlation j. The expanded area ratio (EAR) to provide 5% cavitation for a commercial ship. Assume the operating conditions are similar to the B 5-90 propeller.

Given V2 25 knot⋅:= d2 34 ft⋅:= w2 .249:= t2 .18:= ηR 1:= EHP 80000hp⋅:= h 25:=

First we must combine a couple of equations in order to get all the information we know in terms of KT and J.

R2 550lb

ftsec⋅

hp⋅

⎛⎜⎜⎝

⎞⎟⎟⎠

EHPV2

⋅:= T2R2

1 t2−:= Kt

T2

ρ n2⋅2 d2

4⋅

:=d2n2 n2⋅

J2V2 :=

n2

550lb

ftsec⋅

hp⋅

⎛⎜⎜⎝

⎞⎟⎟⎠

EHP( )⋅

ρ V23

⋅ d22

⋅ 1 t2−( )⋅ 1 w2−( )2⋅0.55=

Kt

J22 →

Now we can plot the function KT = 0.55 * J2 on the B 5-90 curve graph. Drawing a verticle line where the function plot and each KT - P/D intersect will provide a value for KT and ηo. Starting with a logical P/D (.5 for example), step though P/D values, recording KT and ηo. Take note at the peak value for ηo, That will determine optimal values. Using the curves posted on the web, I found: P/D = 1.2 KT=.29 ηo = .6

Kt .29:= a. P/D = 1.2

J2Kt.55

:= Kq .055:= b. KT(opt) = .29

c. KQ(opt) = .055 ηo2 .6:=

Q2 Kq ρ⋅ n2⋅ d5

⋅:= d. ηo = .6

J2 0.726= e. J = 0.726

PC ηo21 t−

1 w−⎛⎜⎝

⎞⎟⎠

⋅ ηR⋅:=

PD2EHPPC

:= PD2 1.242 105× hp= f. HP = 124200 HP

n2 V21 w2−

J2 d2⋅⋅:= g. PC = .644

n2 1.2841s

= PC 0.644= h. RPM = 77.012

N2 n2 60⋅ s⋅:= N2 77.012=

Cavitation Calculations EAR 90:= P_over_D_ans 1.2:= h 25ft:=

AE EARπ d2

2⋅

4⋅:= assume AD ~ AE

AP AE 1.067 0.229 P_over_D_ans( )−[ ]⋅:= VR V2 1 w2−( )⋅⎡⎣ ⎤⎦

2 0.7π n⋅ d2⋅( )2+⎡⎣

⎤⎦

1

2:=

τC

T

AP

12ρ⋅ VR

2⋅

:= τC 5.421 10 10−×

1

A s2⋅

= AE 7.591 103× m2

=

σ0.7R

2026ft4

sec464.4

ft3

sec4h⋅+

VR2 4.836 N2

1s⋅⎛⎜

⎝⎞⎟⎠

2⋅ d2

2⋅+

sec2

ft2⋅:=

σ0.7R 1.095 10 4−×=

CτC A⋅ s2

⋅ 0.3064+ 0.523σ0.7R0.2

⋅−

0.0305σ0.7R0.2

⋅ 0.0174−

:= % cavitation C 17.791−=

Negative cavitation indicates that it is not a problem with at this speed

i. Cavitation = - 17.8%

τCn C .0305 σ0.7R0.2⎛

⎝⎞⎠⋅ . 0174×−⎡

⎣⎤⎦⋅ .3064− .523σ0.7R

0.2⋅+:=

ApnT

.5 ρ⋅ τCn⋅ VR2

⋅⎛⎝

⎞⎠

:= EARn

Apn

1.067 .229 1.2⋅ π⋅342( )4

−⎡⎢⎣

⎤⎥⎦

:=

j. = EAR is much less than one, Changing to meet these requirements would not be necessary. (This will be considered extra credit)

3. List the advantages and disadvantages of the fixed pitch propeller, controllable pitch propeller, and waterjet propulsion systems. List the best applications (or platform(s)) for each propulsor and supporting reasons considering the mission of the platform. (expectation: half a page of concise thought).

For full credit - A brief discussion similar to that in chapter 6 of the text, At least 2 advantages and 2 disadvantages of each and an example of where each has been used sucessfully.