25-Nav-B5 Marine Control Systems · May 2017
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format: National Examination 98-Mar-B5 Fluid Machinery, May 2017 — closed book, three hours, 60 marks. Section A is calculative (Q1–Q5) and Section B is descriptive (Q6–Q8); the rubric asks for four questions of Section A plus two of Section B (six questions, each of equal value, 10 marks). All eight questions are solved in full as a study resource. General constants supplied with the paper: g = 9.81 m/s², patm = 100 kPa, pvapour = 2.34 kPa (20 °C), ρwater = 1000 kg/m³, ρair = 1.21 kg/m³ (15 °C), cp,air = 1.005, cv,air = 0.718 kJ/kg·K.
Reference texts. S. L. Dixon & C. A. Hall, Fluid Mechanics and Thermodynamics of Turbomachinery (7th ed.); R. A. Sabersky, A. J. Acosta, E. G. Hauptmann & E. M. Gates, Fluid Flow: A First Course in Fluid Mechanics (4th ed.); H. Cohen, G. F. C. Rogers & H. I. H. Saravanamuttoo, Gas Turbine Theory; R. W. Fox, A. T. McDonald & P. J. Pritchard, Introduction to Fluid Mechanics.
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.
A liquid boils, at any given temperature, once its local static pressure falls to the vapour pressure pv corresponding to that temperature. In a hydraulic machine, local pressure drops below the bulk/inlet pressure wherever the flow accelerates around a curved surface — the suction (low-pressure) side of an impeller or runner vane, the throat of a draft tube, or the tip-vortex core near a blade tip. If the local pressure reaches pv, dissolved gas and vapour flash out of solution and form small vapour-filled cavities (bubbles). These bubbles are carried by the flow into a region of higher pressure (downstream of the low-pressure zone, e.g. further along the blade or into the diffuser), where the surrounding liquid pressure again exceeds pv and the bubble collapses (implodes).
The damage mechanism is not the boiling itself but the collapse. A cavity collapsing near or on a solid surface does so asymmetrically: liquid rushes in fastest from the side away from the wall, forming a high-speed re-entrant micro-jet (velocities can exceed 100 m/s) that punches through the bubble and strikes the surface, together with a violent local pressure/shock wave (transient pressures reported in the GPa range). A single collapse event does negligible damage, but cavitation occurs at very high frequency (many thousands of collapses per second over an eroding area), so the repeated micro-jet impacts and shock loading fatigue the metal surface, producing the pitted, sponge-like erosion characteristic of cavitation damage. Because the process is fundamentally a fatigue mechanism, even hard, corrosion-resistant alloys (stainless steel, bronze) erode over time if the machine operates persistently in a cavitating condition.
The parts most at risk are those with the lowest local static pressure for a given operating condition: on pumps, the impeller-eye/vane leading-edge suction surface (where the flow first accelerates into the blade passage) and, at part-load or off-design flow, the vane tips; on turbines, the runner blade suction (low-pressure) surface near the outlet/trailing edge and the draft-tube entrance, where the flow has already given up most of its head and pressure is at its lowest before recovery in the draft tube.
"Setting" is the vertical elevation of the pump (impeller centreline) or turbine (runner centreline) relative to the free surface of the water it draws from or discharges to. It matters because it directly controls the absolute pressure available at the machine's lowest-pressure point: too high a setting (pump too far above the supply, or turbine runner too far above tailwater) drops that local pressure toward the vapour pressure and triggers cavitation, exactly as described in Part I.
The governing parameter is the net positive suction head available, $\text{NPSH}_{avail}=\dfrac{p_{atm}-p_{vapour}}{\rho g}-\Delta z-h_L$, which must exceed the machine's own required NPSH (a function of its design, found by test) with a safety margin. The setting is therefore fixed by: local atmospheric pressure (site elevation), water temperature (vapour pressure), suction-side friction losses, and the machine's critical cavitation parameter $\sigma_c=\text{NPSH}/H$ (from manufacturer data, itself a function of specific speed — higher specific-speed machines are inherently more cavitation-prone and need a lower, or even negative/submerged, setting).
The setting convention differs by machine type because the direction of the critical low-pressure point differs. A centrifugal pump is set as low as practicable relative to its supply reservoir (or even flooded/submerged) because it must draw water up to itself against a suction lift — the higher the pump above the supply, the lower the inlet pressure. A reaction turbine (Francis/Kaplan) is set relative to tailwater level: the runner outlet/draft-tube throat is the critical low-pressure point, so raising the runner above tailwater lowers pressure there in the same way, and the allowable setting height (often called the turbine's "draft head") is limited by $\sigma_c$ at the design head. An impulse turbine (Pelton) operates at atmospheric pressure throughout the runner (the jet strikes free-standing buckets in air), so cavitation setting is not a limiting factor in the same sense — only splash clearance above tailwater matters.
Incorrect setting — installing the machine too high for the available NPSH — causes onset of cavitation: falling head and efficiency, rough/noisy operation from bubble collapse, vibration, and progressive pitting erosion of the vanes/runner that shortens the machine's service life and can eventually cause mechanical failure of blade material. Because the erosion is cumulative fatigue damage, a machine can run acceptably for a period after cavitation onset before failure becomes apparent, which makes correct setting at the design stage far cheaper than a field fix.
