Question 7 of 8: Pump and Turbine Cavitation and Setting

Question 7: Pump and Turbine Cavitation and Setting (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.

Part I — formation and collapse. A liquid boils whenever its local static pressure falls to the saturation vapour pressure at the local temperature. In a hydraulic machine that happens not by heating but by acceleration: wherever the flow speeds up — over the suction surface of a blade near its leading edge, through the throat of a nozzle, in the vortex core shed from a runner — Bernoulli's equation converts static pressure into kinetic energy, and if the static pressure reaches $p_v$ (2.34 kPa for water at 20 °C, from page 13) vapour cavities nucleate. Nucleation in practice occurs at microscopic gas pockets trapped in surface crevices or on entrained particles rather than in the pure liquid, so the practical inception pressure is close to, but slightly above, the theoretical vapour pressure, and dissolved air content matters. The governing parameter is the cavitation number $\sigma=\left(p-p_v\right)/\tfrac{1}{2}\rho V^2$: cavitation appears when it falls below a critical value characteristic of the geometry.

The bubbles are swept downstream into a region of higher pressure — further along the blade passage, or into the draft tube — where the vapour condenses. Because vapour condenses essentially instantaneously, the cavity does not deflate gently; the surrounding liquid rushes inward and the cavity implodes. In an unbounded liquid the collapse is spherically symmetric and merely produces a pressure pulse. Adjacent to a solid boundary, which is precisely where the damage occurs, the wall obstructs the inflow on one side, the collapse becomes asymmetric, and the far side of the bubble accelerates inward faster than the near side, folds through the cavity and forms a re-entrant microjet that strikes the wall at speeds of the order of 100–500 m/s. The sequence is sketched in Figure 7.1. The water-hammer pressure of such an impact, $\rho c V_{jet}$, reaches the order of a gigapascal over a spot a few micrometres across — far above the yield strength of any engineering alloy. The remnant torus then collapses in its turn and radiates a spherical shock wave that reinforces the attack.

Damage therefore proceeds by mechanical fatigue rather than by chemical attack. Individual impacts plastically deform the surface; repeated at the several-kilohertz rate at which bubbles collapse, they work-harden it, initiate sub-surface cracks, and eventually detach grains. The characteristic appearance is a spongy, deeply pitted surface, often with an incubation period of tens or hundreds of hours before mass loss becomes measurable. Corrosion accelerates the process where the passivating oxide film is continually stripped, which is why cavitation resistance correlates with hardness, work-hardening capacity and corrosion resistance together: austenitic stainless steels and stellite overlays perform far better than plain carbon steel or cast iron. Beyond the metal loss, cavitation produces a distinctive gravel-rattling noise, broadband vibration, and — when the cavity grows large enough to block a passage — a collapse in head and efficiency.

The vulnerable locations follow the pressure minima. In a pump, damage appears on the suction (back) surface of the impeller vanes near the leading edge, where incidence accelerates the flow most sharply, and at the eye of the impeller and the inlet shroud; at strongly off-design flows the pressure minimum migrates to the pressure surface instead, so damage can appear on either face. Wear rings and, in multistage machines, the first-stage impeller only, are also classic sites. In a reaction turbine the pressure falls continuously through the machine, so the minimum lies at the runner outlet: damage concentrates on the outlet edges of Francis blades near the band, on the discharge side of Kaplan blades and their tip clearances, and on the draft-tube cone where the swirling vortex rope forms at part load. In a Pelton wheel the whole machine runs at atmospheric pressure so classical cavitation is largely absent; the buckets are damaged instead by droplet erosion and by silt abrasion, which is a different mechanism with a similar appearance. Guide vanes, labyrinth seals and the volute tongue are secondary sites in all reaction machines.

Part II — the meaning and importance of setting. The setting is the elevation of the machine relative to the lower water surface — the sump level for a pump, the tailwater level for a turbine. It matters because it is the one design variable that directly controls the absolute pressure at the point in the machine where cavitation begins. For a pump, the net positive suction head available follows the page-16 relation

$$\text{NPSH}_{a}=\frac{p_{atm}-p_{v}}{\rho g}-\Delta z-h_L$$

With the exam constants, the leading term is $(100-2.34)\times 10^{3}/(1000\times 9.81)=9.96$ m — the entire budget available at sea level for a cold-water pump. Every metre the pump is raised above the sump, and every metre of friction and entrance loss in the suction line, is deducted from it. Cavitation is avoided only while $\text{NPSH}_a>\text{NPSH}_r$, the requirement of the particular impeller at the particular flow, with a margin typically of one metre or 10–30 %. The corresponding turbine criterion is Thoma's, $\sigma=\text{NPSH}/H$ with setting $\Delta z=\left(p_{atm}-p_v\right)/\rho g-\sigma_c H$.

The parameters that enter are therefore: atmospheric pressure at the site, which falls roughly 1.2 m of water per 1000 m of altitude and can also drop a further 0.3 m in a deep depression; vapour pressure, which rises steeply with temperature (0.24 m of water at 20 °C but 3.2 m at 70 °C, so hot-water service devours the budget); suction or draft-tube friction and entrance losses at the maximum flow, including strainers and any partly closed valve; the range of lower water level, since the setting must satisfy the worst case — minimum sump level for a pump, minimum tailwater for a turbine; the machine's own requirement, NPSH_r or $\sigma_c$, read from the manufacturer's curve or estimated from suction specific speed, and evaluated at the runout flow rather than the best-efficiency point; and finally transient and off-design conditions such as start-up, runaway and part-load vortex-rope operation.

The setting differs sharply between machine types because $\sigma_c$ and NPSH_r rise steeply with specific speed. A high-head, low-specific-speed Francis runner may need only $\sigma_c\approx 0.05$: on the 160 m head of Question 3 that permits a setting of $9.96-0.05\times 160=+2.0$ m, so the runner may sit above tailwater. A low-head Kaplan of high specific speed may need $\sigma_c\approx 1$: on a 12 m head that gives $9.96-12=-2.0$ m, and the runner must be drowned two metres below tailwater — which is why axial machines are built into deep, expensive concrete substructures. A Pelton wheel, running in air, is instead set just far enough above maximum tailwater that the buckets never run submerged, sacrificing that head entirely. Pumps show the same spread: an ordinary end-suction unit can lift several metres, a high-speed multistage boiler feed pump may require positive submergence and a booster, condensate and cryogenic pumps are set below their vessels as a matter of course, and an inducer is fitted specifically to reduce NPSH_r and buy back setting height.

The consequences of getting it wrong are one-sided. Set the machine too high and the operator finds noise, vibration, a head-capacity curve that breaks down at high flow, progressive pitting of the impeller or runner, accelerated bearing and seal wear from the unsteady radial loads, loss of efficiency and eventually loss of prime or complete flow breakdown, with repair costs and outage far exceeding any saving. Set the machine too low and the penalty is purely economic — deeper excavation, more concrete, permanent dewatering provision, harder access for maintenance and, for a turbine, a longer draft tube. Because the first failure mode is progressive and expensive while the second is a one-off capital cost, prudent practice is to err downward, keep an explicit NPSH margin, and check the setting against minimum tailwater and maximum water temperature rather than against average conditions.