25-Nav-B5 Marine Control Systems · December 2016
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format: National Examination 98-Mar-B5 Fluid Machinery, December 2016 — 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 kJ/kg°C, k = 1.4, R = 0.287 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, 2nd ed.; H. Cohen, G. F. C. Rogers & H. I. H. Saravanamuttoo, Gas Turbine Theory, 6th ed.; R. W. Fox, A. T. McDonald & P. J. Pritchard, Introduction to Fluid Mechanics, 8th ed.
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) Why twisted blades are necessary. The blade (peripheral) speed is $U=\omega r$, so it increases linearly from root to tip — on a large blade the tip may run at more than twice the root speed. The axial velocity of the working fluid, however, is nearly uniform across the annulus. Because the velocity triangle is built from the fixed axial velocity and the radius-dependent blade speed, the relative flow angle onto the blade changes continuously along the span. If the blade had a single fixed angle it would only match the flow at one radius; everywhere else the incidence would be wrong, causing separation, shock losses and — on a compressor — local stall. Twisting the blade so that its metal angle follows the local flow angle keeps the incidence small and the loss low over the whole span.
(b) Base vs tip velocity diagrams. At the base (root) the blade speed $U$ is small, so with the same axial velocity the relative velocity is steeply inclined and the blade is highly staggered; the triangle is “tall and narrow.” At the tip $U$ is large, so the relative velocity is much more nearly tangential and the blade is flatter; the triangle is “short and wide.” The nozzle (absolute) angle and the whirl also change: for the common free-vortex design the whirl velocity falls with radius as $C_w r=\text{const}$, so the inlet and outlet angles both open out from root to tip, as sketched above.
(c) Degree of reaction, root to tip. The degree of reaction is the fraction of the stage enthalpy (or pressure) drop that occurs in the moving (rotor) blades rather than in the fixed nozzles: $R=\dfrac{\text{enthalpy drop in rotor}}{\text{total stage enthalpy drop}}$. For a free-vortex twisted blade the reaction is low at the root — often designed close to zero (impulse) so the root does not need a large pressure drop across a short, structurally loaded section — and rises toward the tip, typically approaching 0.5 (fifty-per-cent reaction) or more. Physically, the larger tip speed does more work through a rotor pressure drop, while the root behaves more like an impulse section. This radial variation of reaction is the direct consequence of the twist and the free-vortex whirl distribution.
