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 compressors need many stages. A compressor raises pressure, so its flow moves against an adverse pressure gradient: the fluid is decelerated (diffused) as it is compressed. Boundary layers on the blades and end-walls are prone to separate under a rising pressure, so each stage can only tolerate a modest amount of diffusion before the blades stall. This limits the pressure ratio and temperature rise achievable per stage (typically only about $1.2$–$1.4$:1 and a few tens of kelvin), so many stages are needed to reach an overall ratio of 3 or 5. The limiting factors are the de Haller number / diffusion factor (how much the relative velocity may be reduced before separation), blade-tip Mach number (compressibility and shock losses), and the stall/surge margin of the whole compressor.
(b) How one turbine stage drives seven compressor stages. A turbine expands the gas, so its flow runs with a favourable pressure gradient — the fluid accelerates through the passages, which keeps the boundary layers attached even under a large pressure drop. A single turbine stage can therefore take an enthalpy drop several times larger than a compressor stage can add. In addition the turbine works at much higher temperature, so the same enthalpy drop $c_p\Delta T$ corresponds to a large pressure ratio, and the higher speed of sound allows higher blade speeds. Because work per unit mass is $c_p\Delta T$ and the mass flow is common to both, one high-temperature, highly-loaded turbine stage can supply the work absorbed by the seven lightly-loaded compressor stages it drives.
(c) Why the two shafts run at different speeds. The LP and HP compressors operate in very different conditions: the HP compressor handles hotter, denser air in a smaller-diameter annulus, while the LP compressor handles cold, less-dense air in a larger annulus. Each compressor has an optimum blade speed set by its aerodynamic loading and by the tip Mach-number limit. Mounting them on separate (concentric) shafts — a twin-spool arrangement — lets the HP spool run fast (8500 rev/min) to keep its small-diameter blades adequately loaded, while the LP spool runs slower (6500 rev/min) so its large-diameter tips stay below the Mach limit. Decoupling the spools also greatly improves the surge margin and off-design and starting behaviour, because each spool is free to settle at its own aerodynamically matched speed.
