22-Mec-B6 Advanced Fluid Mechanics · December 2018
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format: National Examinations, December 2018 — 16-Mec-B6 Fluid Machinery. Closed book, three hours, 22 pages. Section A is calculative (Questions 1–5) and Section B descriptive (Questions 6–8); candidates answer four questions from Section A and two from Section B — six questions of ten marks each, 60 marks in all. Reference data for individual questions are supplied on pages 11–17, and the nomenclature, general constants and reference equations on pages 18–22. All eight questions are solved below.
Check — angle conventions are taken from the paper's own attachments, and they are not the same across the paper. The compressor velocity diagram on page 11 strikes $\alpha_1$ and $\beta_1$ off the axial component $C_{X1}$, so every blade and vane angle in Questions 1 and 2 is measured from the axial direction. The steam-turbine diagram on page 16 strikes $\theta$, $\phi$, $\gamma$ and $\delta$ off the tangential (blade-motion) direction, and the Francis diagram on page 15 defines $\alpha_1$ between $V_1$ and $u_1$ — also tangential. Questions 4 and 5 therefore use the tangential reference. Every constant below is the paper's own page-19 value: $g = 9.81\ \text{m}\,\text{s}^{-2}$, $\rho_{\text{water}} = 1000\ \text{kg}\,\text{m}^{-3}$, $c_p = 1005\ \text{J}\,\text{kg}^{-1}\text{K}^{-1}$ and $c_v = 718\ \text{J}\,\text{kg}^{-1}\text{K}^{-1}$, from which $R = c_p - c_v = 287\ \text{J}\,\text{kg}^{-1}\text{K}^{-1}$ and $k = c_p/c_v = 1.3997$ — not a textbook 1.40.
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 when its local absolute pressure falls to its vapour pressure at the local temperature, and it makes no difference to the physics whether that condition is reached by raising the temperature at constant pressure or by lowering the pressure at constant temperature. In a hydraulic machine it is the second route that matters. Water at 15 °C has a vapour pressure of only 1.71 kPa absolute (page 19), so wherever the local static pressure in the flow falls to about 1.7 kPa — some 98 kPa below atmospheric — vapour cavities nucleate, growing from microscopic gas nuclei and dissolved air already present in the water. The pressure falls that low in exactly the places where the velocity is highest or the elevation greatest: on the suction surfaces of blades near their leading edges, in the tip clearance of a Kaplan runner, at the runner exit of a Francis machine and in the throat of a draft tube set too high above tailwater.
What makes cavitation destructive is not the formation of the bubbles but the manner of their collapse. A vapour cavity is not a gas bubble that can cushion itself: it contains vapour that condenses almost instantaneously the moment the bubble is carried into a region of higher pressure, so the cavity does not deflate gently but implodes. If the collapse occurs in the free stream it is spherically symmetric and merely noisy. If it occurs close to a solid surface the collapse is asymmetric, because the wall inhibits the inflow of liquid from one side; the far side of the bubble accelerates inwards faster and forms a re-entrant microjet that pierces the bubble and strikes the surface at velocities of the order of hundreds of metres per second. The resulting impact pressures are measured in gigapascals over a very small area, well above the yield strength of any turbine material. Each event does no visible harm; but the events repeat thousands of times a second on the same few square millimetres, and the surface fails by fatigue — work-hardening, then micro-cracking at grain boundaries, then loss of grains — leaving the characteristic deeply pitted, spongy, honeycombed appearance. Where the collapse occurs slightly further out, the associated shock wave produces the same effect less intensely. The damage is accompanied by a distinctive noise like gravel passing through the machine, by vibration, and by a fall in head and efficiency once the vapour blockage becomes large enough to alter the flow pattern.
On a Francis turbine the vulnerable regions are those where the pressure is lowest, which means the outlet end of the machine and not the inlet. Specifically: the suction (back) surfaces of the runner blades near the trailing edge, where the pressure has already fallen through the runner and the local velocity over the curved surface reduces it further; the runner band and crown near the blade exit, and the outer part of the blade where the peripheral velocity is highest; the draft tube inlet cone and the runner hub downstream of it, where the vortex core sits; and, at off-design flows, the leading-edge suction surface, where incorrect incidence produces a local suction peak. Cavitation is governed for a given design by the Thoma parameter of page 21, $\sigma = [(p_{\text{atm}}-p_{\text{vapour}})/\rho g - \Delta z]/H$, in which $\Delta z$ is the height of the runner above tailwater. The parameter says plainly what the remedy is: set the runner lower — even below tailwater level — so that the static pressure at the runner exit is raised. A machine of high specific speed needs a larger $\sigma$ and therefore a deeper setting, which is why high-specific-speed Kaplan units are routinely set below tailwater while a low-specific-speed Francis may be set above it. Beyond the setting, the defences are to avoid prolonged operation far from the design point, to use cavitation-resistant materials such as stainless steel or a stainless overlay at the vulnerable areas, and to admit a small quantity of air into the draft tube to cushion the collapse.
A Francis turbine driving a synchronous generator must run at constant speed, so power is changed by changing the flow, and the flow is changed by the wicket gates (guide vanes) that surround the runner. These are pivoted vanes, all linked to a single regulating ring and driven by the governor's servomotor; rotating the ring changes the throat area between adjacent vanes and simultaneously changes the guide-vane angle $\alpha_1$, and hence the whirl handed to the runner. Since the power is $\rho g Q H \eta$, reducing the flow reduces the power essentially in proportion. Because the vanes also change the direction of the flow, the machine can be regulated over a wide range without gross shock losses, although efficiency does fall away at part gate because the incidence at the runner leading edge is no longer correct.
The consequence in the penstock is water hammer. The column of water in the penstock has momentum, and closing the gates decelerates it. Any deceleration produces a pressure rise at the gate given in the extreme by the Joukowsky relation $\Delta p = \rho a \Delta V$, where $a$ is the celerity of the pressure wave in the pipe — of the order of 1000 to 1400 m/s in a steel penstock, reduced somewhat by the elasticity of the pipe wall. For the Question 3 plant, with a penstock velocity of 4.7 m/s, an instantaneous stoppage would generate a head rise of roughly $a\Delta V/g \approx 1200 \times 4.7/9.81 \approx 575\ \text{m}$ — nearly forty times the 15.67 m operating head, and far beyond what the pipe could contain. The pressure wave travels up the penstock to the reservoir, reflects there as a rarefaction, returns to the gate, and the cycle repeats with a period of $4L/a$, decaying under friction. On the negative half-cycle the pressure can fall to vapour pressure, causing column separation and a far more damaging rejoining impact.
The rate of closure is therefore the whole question. The critical time is $T_c = 2L/a$, the time for a wave to travel to the reservoir and back. If the gates close in a time shorter than $T_c$ the closure is hydraulically instantaneous: no relief can arrive from the reservoir before the gate is shut, and the full Joukowsky pressure develops. If the closure takes longer than $T_c$, successive reflections from the reservoir progressively relieve the gate pressure, and the surge is reduced roughly in the ratio $T_c/T_{\text{closure}}$ (the Allievi or Michaud result). Governors are therefore deliberately set with closure times of several seconds — typically five to ten — against critical times measured in fractions of a second. The counter-pressure is that a slow closure means the machine cannot shed load quickly, so on a full load rejection the runner accelerates towards runaway speed; the design must satisfy both constraints at once. The standard resolutions are to fit a surge tank or standpipe near the top of the penstock, which provides a free surface close to the machine and shortens the effective inertial length; to fit a pressure-relief valve or a Francis-machine bypass that opens as the gates close so that the total flow changes slowly even though the flow through the runner changes quickly; or, on an impulse machine, a deflector that diverts the jet instantly while the needle closes slowly.
A Francis runner has fixed blades, and a generating machine runs at fixed speed, so both $U_2$ and the outlet blade angle $\beta_2$ are constants of the problem. The design condition is arranged, as in Question 4, so that the water leaves the runner with no whirl at all: the absolute exit velocity is purely axial (or radial into the draft tube), all the angular momentum is extracted, and the draft tube receives a clean, straight, swirl-free flow which it can diffuse efficiently to recover the exit kinetic energy.
Reduce the load and the radial (through-flow) velocity $V_{f2}$ falls in proportion to the flow, while $U_2$ and $\beta_2$ do not change. The relative velocity therefore still leaves along the blade, but it is shorter; when it is added to the unchanged blade speed the resultant absolute velocity no longer closes onto the axial direction but retains a substantial tangential component in the direction of runner rotation. The figure shows the two triangles superimposed. The water consequently enters the draft tube spinning, and the energy in that spin is simply lost — it is angular momentum the runner failed to extract, which is one reason part-load efficiency falls.
Worse, the swirl is unstable. Below about 60 per cent of best-efficiency flow the swirling core in the draft-tube cone breaks down into a helical, precessing vortex known as the vortex rope, whose centre is at very low pressure and frequently cavitates, becoming visible as a corkscrew of vapour. The rope precesses about the axis at roughly a quarter to a third of the runner speed, and the rotating low-pressure region generates a strong periodic pressure pulsation that is transmitted through the whole water passage. The consequences are severe and well documented: pressure surges in the draft tube and penstock, heavy vibration of the unit and the powerhouse, noise, fatigue loading of the runner, shaft and draft-tube liner, and power swings on the generator. Where the rope frequency coincides with a natural frequency of the water column, the resonance can be violent enough to be a limit on operation. This is why Francis units are generally not run for long periods in the rough zone between about 40 and 60 per cent of rated output. The standard palliative is to admit atmospheric air, or compressed air, into the draft-tube cone through the runner cone or the tube wall: the air cushions the pressure pulsations and prevents the vapour core from collapsing sharply, at a small cost in efficiency. Fins or a fluted draft-tube cone can also break up the rope. The Kaplan machine avoids the problem altogether by adjusting its runner blade angle with load, which keeps the exit swirl near zero over the whole range — which is precisely the property that recommended it for the plant of Question 3.
