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25-Nav-A2 Hydrodynamics of Ships (I)_ Resistance and Propulsion · December 2019

Question 6 of 9: Cavitation Check by Burrill's Method

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

National Exams — December 2019 — 16-Nav-A2 Hydrodynamics of Ships I: Resistance and Propulsion. Three-hour, closed-book exam; a data sheet, a propeller (Wageningen B4-55) chart and a Burrill cavitation chart are supplied. Format: Questions 1–7 are compulsory (attempt all seven), then one of Questions 8 or 9. All nine are solved below for completeness. Units follow the paper (mixed SI, with the historic Imperial-unit legend that duplicates on the supplied Burrill sheet noted where relevant).

Reference texts: Larsson & Raven, Ship Resistance and Flow (SNAME) — model-scale resistance testing, Froude/Reynolds scaling and the ITTC 1978 performance-prediction method; Lewis (ed.), Principles of Naval Architecture, Vol. II — Resistance, Propulsion and Vibration (SNAME) — propeller geometry, open-water B-series design and cavitation; Carlton, Marine Propellers and Propulsion (Butterworth-Heinemann) — Wageningen B-series charts and the Burrill back-cavitation criterion.

Question 6: Cavitation Check by Burrill's Method (10 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Propeller from Question 5: $D=1.7$ m, $n=6.0$ rps, $P/D=0.90$, $Z=4$, $A_E/A_O=0.55$, $T=68.9$ kN, $V_A=4.691$ m/s. Shaft depth $h_0=2.40$ m; $p_v=10$ kPa; $p_{atm}=101$ kPa; $\rho=1025\ \text{kg/m}^3$ (sea water, 15 °C).

QuantityValue
Propeller $D$, $n$, $P/D$1.7 m, 6.0 rps, 0.90
Blade area ratio $A_E/A_O$ / blades $Z$0.55 / 4
Thrust $T$ (from Q5) / advance speed $V_A$68.9 kN / 4.691 m/s
Shaft depth $h_0$2.40 m
$p_v$ / $p_{atm}$10 kPa / 101 kPa

Find. (i) the approximate extent of back (suction-side) cavitation; (ii) three negative effects of cavitation.

Approach. Compute the local cavitation number $\sigma_{0.7R}$ and thrust-loading coefficient $\tau_c$ at the $0.7R$ blade section from the data-sheet formulae, converting the expanded blade area $A_E$ to the projected area $A_P$ that Burrill's method uses, then locate the point on the Burrill diagram (page 8) against the family of back-cavitation-percentage contours.

σ₀.₇ᵣ (local cavitation number)τ_c0.20.40.60.81.00.10.20.30.40.5approx. band of the chart's"suggested upper limit—merchantpropellers (1943)" curve (page 8)design point: σ₀.₇ᵣ≈0.43, τ_c≈0.24Burrill diagram (page 8): design point vs. limit bandpoint sits above the merchant-ship limit band → back cavitation expected
Figure 4 — Design point on the Burrill diagram, plotted against the approximate location of the "suggested upper limit — merchant ship propellers (1943)" curve read from the supplied chart.
  1. Disk, expanded and projected blade areas. $$A_O=\frac{\pi D^2}{4}=\frac{\pi(1.7)^2}{4}=2.270\ \text{m}^2,\qquad A_E=(A_E/A_O)\,A_O=(0.55)(2.270)=1.248\ \text{m}^2,$$ $$\frac{A_P}{A_E}=1.067-0.229\frac{P}{D}=1.067-0.229(0.90)=0.861 \;\Rightarrow\; A_P=(1.248)(0.861)=\boxed{1.075\ \text{m}^2}.$$
  2. Static pressure at the shaft centreline. $$p_0=p_{ATM}+\rho g h_0=101{,}000+(1025)(9.806)(2.40)=\boxed{125{,}100\ \text{Pa}}.$$
  3. Resultant velocity and dynamic pressure at $0.7R$. $$V_R=\sqrt{V_A^2+(0.7\pi nD)^2}=\sqrt{4.691^2+(0.7\pi(6.0)(1.7))^2}=\sqrt{4.691^2+22.43^2}=22.92\ \text{m/s},$$ $$q_{0.7R}=\tfrac12\rho V_R^2=\tfrac12(1025)(22.92)^2=\boxed{269.1\ \text{kPa}}.$$
  4. Local cavitation number. $$\sigma_{0.7R}=\frac{p_0-p_v}{q_{0.7R}}=\frac{125{,}100-10{,}000}{269{,}100}=\boxed{0.428}.$$
  5. Thrust-loading coefficient. $$\tau_c=\frac{T}{A_P\,q_{0.7R}}=\frac{68{,}900}{(1.075)(269{,}100)}=\boxed{0.238}.$$
  6. Compare with the Burrill diagram. The point $(\sigma_{0.7R},\tau_c)=(0.43,0.24)$ plots above the "suggested upper limit for merchant ship propellers (1943)" curve on the supplied chart, in the region bracketed by the chart's 10%–20% back-cavitation contours. The diameter cap imposed by the hull ($D_{max}=1.7$ m) has produced a somewhat heavily-loaded propeller: expect on the order of 10–20% back (suction-side) cavitation over the blade at this radius — a direct consequence of the diameter constraint in Question 5, trading cavitation margin for the available hull clearance.
QuantityResult
Projected blade area $A_P$1.075 m²
Local cavitation number $\sigma_{0.7R}$0.428
Thrust-loading coefficient $\tau_c$0.238
(i) Expected back cavitation≈ 10–20% of blade area
(ii) Negative effectsthrust/efficiency loss; erosion; noise & vibration (see below)

(ii) Three negative effects of cavitation: (1) thrust and torque breakdown — once a vapour cavity covers an appreciable fraction of the blade, the effective lifting surface is disrupted and delivered thrust (and propulsive efficiency) fall off sharply; (2) material erosion — when vapour bubbles are swept into a higher-pressure region and collapse violently against the blade surface, the resulting micro-jet impacts pit and erode even hardened bronze/stainless propeller alloys over time; (3) noise and vibration — collapsing cavities generate broadband underwater noise and unsteady blade forces that excite hull vibration and are a major contributor to detectable acoustic signature.

Check
The 10–20% figure is read from the Burrill diagram's family of back-cavitation contours as supplied with the exam.