22-Mec-A6 Fluid Machinery · May 2014
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format. National Examination 07-Mec-A6-1 Fluid Machinery (May 2014) — closed book, three hours, 60 marks. Section A is calculative (Q1–Q5) and Section B is descriptive (Q6–Q8); the rubric asks for four 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. K. Turton, Principles of Turbomachinery; H. Cohen, G. F. C. Rogers & H. I. H. Saravanamuttoo, Gas Turbine Theory (for the axial compressor stage); R. W. Fox, A. T. McDonald & P. J. Pritchard, Introduction to Fluid Mechanics (pump energy equation and affinity laws).
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.
The efficiency-versus-load curves reflect how each machine controls flow and how well its internal velocity triangles stay matched as the discharge is throttled away from the design point. A turbine is most efficient when the water enters the runner with the incidence its blades were shaped for; any departure from design flow introduces shock (incidence) losses at inlet and swirl losses at exit, and the shape of the resulting efficiency curve is essentially a map of how quickly those mismatch losses grow.
(a) Francis best, Pelton worst at high load. At full load the Francis (reaction) runner is working exactly at its design velocity triangle: the whole runner passage is flooded, flow is guided smoothly by the fixed blades and adjustable guide vanes, and friction is spread over a large wetted area at modest velocity — so peak efficiency is high, typically 92–94%. The Pelton (impulse) machine, by contrast, converts the entire head into one high-speed jet and then relies on buckets to reverse it; windage of the spinning runner in air, bucket-edge splash and the residual kinetic energy leaving the buckets are irreducible losses that cap its peak efficiency a few points below the best reaction machines at high load.
(b) Kaplan stays high to low load. The Kaplan is a double-regulated machine: both the guide vanes and the runner blades pitch. As load falls the runner blades re-stagger to keep the inlet incidence near zero, so the velocity triangles stay well matched over a very wide flow range. Its efficiency curve is therefore broad and flat, holding high efficiency down to perhaps 30–40% load where a fixed-blade machine would already be badly mismatched.
(c) Francis drops soon as load decreases. A Francis runner has fixed blades; only the guide vanes move. Closing the guide vanes changes the inlet whirl but the runner cannot follow, so the incidence angle grows rapidly and the discharge leaves with a large residual swirl that spirals in the draft tube (the "vortex rope"). These shock and swirl losses climb quickly, so the Francis efficiency curve falls off comparatively steeply below about 60–70% load.
(d) Pelton high even at very low load. A Pelton is regulated by a spear (needle) valve that narrows the nozzle while keeping the jet velocity essentially constant at $\sqrt{2gH}$. Because the bucket sees the same jet speed and the same optimum speed ratio at any flow, the bucket velocity triangle is unchanged as load drops — only the number of active jets or the jet diameter changes. Its efficiency curve is therefore almost flat, staying high even at very low load, which is exactly why Peltons suit installations with widely varying flow.
(e) Choice of machine. The governing parameter is specific speed, which pairs with the available head. High head with low flow gives a low specific speed suited to a Pelton (large head, small swallowing capacity); medium head and flow suit a Francis; and low head with large flow gives a high specific speed suited to an axial Kaplan. Beyond the head match, the expected load profile matters: a site with strongly varying flow favours the flat curves of a Pelton or Kaplan, whereas a base-loaded Francis is ideal where the plant runs near design most of the time. Cavitation limits (the Thoma setting of Question 2) and cost also enter — Kaplan runners with movable blades are mechanically complex and expensive, justified only where their part-load efficiency pays back.
