Question 1 of 8: Hydro Turbines

Question 1: Hydro Turbines (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. Two independent machines. Part I is a Kaplan unit under test:

Part II is a high-head Francis machine: P = 483.2 MW, N = 200 rev/min, H = 312.4 m, runner diameter 5.82 m, water at 15 °C (p_vapour = 1.71 kPa), runner centreline at El. 117.3 m and downstream water level at El. 132.8 m.

Find. Part I: the two pipe velocities, the hydraulic power delivered by the water and the overall turbine-generator efficiency. Part II: the power specific speed, the critical Thoma coefficient read from the chart, the maximum permissible runner setting it implies, and how that compares with the setting the station actually has.

Approach. Part I is the steady-flow energy equation of page 20 applied between the two pressure taps, which gives the net head, then P = ρgQH; Part II is the turbine specific speed of page 21 used to enter the cavitation chart, followed by the Thoma definition rearranged for the runner elevation.

1. Part I (a) — convert the flow to velocities at the two measuring sections. Continuity for an incompressible fluid gives V = Q/A, with A = πD²/4: $$A_1=\frac{\pi(6.4)^2}{4}=32.17\ \text{m}^2, \qquad A_2=\frac{\pi(7.0)^2}{4}=38.48\ \text{m}^2$$ so that $$\boxed{V_1=\frac{354}{32.17}=11.00\ \text{m/s},\qquad V_2=\frac{354}{38.48}=9.20\ \text{m/s}}$$ The draft tube is deliberately the larger pipe: it recovers velocity head that would otherwise be thrown away downstream.
2. Part I (b) — assemble the net head from the energy equation. Page 20 gives the pump-and-turbine energy equation; with the turbine work as the only extraction, the head made available to the machine is the drop in total head between the taps, $$H=\frac{p_1-p_2}{\rho g}+(z_1-z_2)+\frac{V_1^2-V_2^2}{2g}$$ Each term is now evaluated. The pressure term uses the gauge readings directly, because both taps are referred to the same atmosphere: $$\frac{p_1}{\rho g}=\frac{226\,000}{1000\times 9.81}=23.04\ \text{m},\qquad \frac{p_2}{\rho g}=-4.50\ \text{m}$$ a difference of 27.54 m. The elevation term is 5.00 m and the velocity term is $$\frac{V_1^2-V_2^2}{2g}=\frac{11.00^2-9.20^2}{2\times 9.81}=\frac{36.49}{19.62}=1.86\ \text{m}$$
3. Part I (b) — net head and hydraulic power. Adding the three contributions, $$H=27.54+5.00+1.86=\boxed{34.40\ \text{m}}$$ and the hydraulic power carried by the water into the machine follows from P = ρgQH (page 21): $$P_{\text{hyd}}=1000\times 9.81\times 354\times 34.40=1.195\times 10^{8}\ \text{W} =\boxed{119.5\ \text{MW}}$$ Note how small the velocity-head contribution is: 1.86 m out of 34.40 m, about five per cent. It is still worth carrying, because at this power level five per cent is 6 MW.
4. Part I (c) — electrical output and combined efficiency. The electrical output is the measured 110 MW, so the turbine-generator efficiency is the ratio of what leaves the terminals to what the water delivers: $$\eta=\frac{P_{\text{el}}}{P_{\text{hyd}}}=\frac{110.0}{119.5} =\boxed{0.921\ \ (92.1\ \text{per cent})}$$ This is a combined hydraulic and electrical efficiency — it contains the runner losses, the draft-tube losses downstream of the tap, the bearing and windage losses and the generator losses together. For a large Kaplan set with its own generator, 92 per cent is a thoroughly normal figure.

Part II moves to a different machine and a different question: not how well the runner converts energy, but how deep it has to be buried so that it does not cavitate.

5. Part II (a) — power specific speed of the Francis machine. Page 21 gives the turbine specific speed in the non-dimensional (SI) form $$N_s=\frac{\omega\sqrt{P}}{\rho^{1/2}\,(gH)^{5/4}}$$ with ω = 2πN/60 = 2π(200)/60 = 20.94 rad/s. Substituting P = 483.2 MW, ρ = 1000 kg/m³ and gH = 9.81(312.4) = 3064.6 m²/s²: $$N_s=\frac{20.94\sqrt{483.2\times 10^{6}}}{\sqrt{1000}\,(3064.6)^{5/4}} =\frac{20.94\times 21\,982}{31.62\times 22\,795}=\boxed{0.638\ \text{rad}}$$ A value of about 0.6 rad is squarely in the Francis band of the page-12 chart, which is what one expects of a 312 m head.
6. Part II (b) — read the critical Thoma coefficient off the chart. Entering the page-12 plot on the horizontal axis at Ω_sp = 0.638 rad and following the Francis line gives $$\boxed{\sigma_c \approx 0.08}$$ The chart is logarithmic on both axes, so the reading should be quoted with its uncertainty; a careful reader would call it 0.08 with a plausible band of roughly 0.07 to 0.11. The consequence of that band is examined in the note below the results table.
7. Part II (c) — convert the coefficient into a permissible setting. The Thoma definition on page 21 is $$\sigma=\frac{1}{H}\left[\frac{p_{\text{atm}}-p_{\text{vapour}}}{\rho g}-\Delta z\right]$$ in which Δz is the runner elevation measured above tailwater. Rearranging for the setting at the critical value, $$\Delta z=\frac{p_{\text{atm}}-p_{\text{vapour}}}{\rho g}-\sigma_c H$$ The barometric term at 15 °C is $$\frac{100\,000-1710}{1000\times 9.81}=10.02\ \text{m}$$ so that $$\Delta z=10.02-0.08(312.4)=10.02-24.99=\boxed{-15.0\ \text{m}}$$ The sign is the whole answer: the runner must be set 15.0 m below the downstream water surface. On a 312 m head the barometric allowance of ten metres is simply not enough on its own.
8. Part II (d) — compare with the setting the station actually has, and comment. The runner centreline is at El. 117.3 m and the surge-chamber water surface at El. 132.8 m, so the actual setting is $$\Delta z_{\text{actual}}=117.3-132.8=\boxed{-15.5\ \text{m}}$$ The station therefore has 0.5 m more submergence than the chart demands. Expressed the other way round, the plant operates at $$\sigma_{\text{plant}}=\frac{10.02-(-15.5)}{312.4}=0.0817$$ against a critical 0.08 — a margin of about two per cent. The comment the question is fishing for is that the design is consistent with the Thoma criterion but only just: there is essentially no allowance for a low barometer, for warmer water than 15 °C, for a depressed tailwater level at low river flow, or for part-load operation, where the swirl leaving a fixed-blade Francis runner raises the local velocities and pushes the true critical coefficient up. In practice a designer would want the runner deeper than the bare criterion, and would confirm the setting by model cavitation test rather than by chart alone.