25-Nav-B5 Marine Control Systems · December 2017
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format. 98-Mar-B5 Fluid Machinery, National Examinations December 2017 — three hours, closed book. Section A is calculative (Questions 1–5) and Section B descriptive (Questions 6–8); the rubric asks for four of Section A and two of Section B, six questions of ten marks each for a sixty-mark paper. Reference data for individual questions are supplied as Attachments (pages 10–17) and a general nomenclature/constants/equations sheet occupies pages 18–22. All eight questions are solved here, because the set is a study resource rather than a timed attempt.
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.; F. M. White, Fluid Mechanics, 8th ed.; R. W. Fox, A. T. McDonald & P. J. Pritchard, Introduction to Fluid Mechanics, 9th ed.; Y. A. Çengel & M. A. Boles, Thermodynamics: An Engineering Approach, 9th ed. Constants are those printed on page 19 of the paper (g = 9.81 m/s², ρwater = 1000 kg/m³, ρair = 1.21 kg/m³ at 15 °C and 1.19 kg/m³ at 20 °C, cp = 1.005 kJ/kg°C, cv = 0.718 kJ/kg°C, patm = 100 kPa, pvapour = 1.71 kPa at 15 °C). Every reference equation quoted below is one of those printed on pages 20–22, and is identified as such where it is first used.
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 asymmetry the cross-section shows — many compressor stages driven by a handful of turbine stages, even though the two must exchange almost exactly the same power — comes from a single fact of fluid mechanics: a fluid can be decelerated only gently, but may be accelerated as violently as one likes. Everything else in the answer is a consequence.
A compressor blade passage is a diffuser. The relative velocity entering the rotor must be slowed down, and the static pressure rises along the passage, so the boundary layer on the blade surface is climbing an adverse pressure gradient. A boundary layer in an adverse gradient loses momentum near the wall and separates once the gradient becomes too steep — and separation in a compressor is not a modest efficiency penalty but a system-level failure: the blade row stalls, the stall cell rotates around the annulus, and if it propagates the whole machine surges, with violent flow reversal. Design practice therefore limits the diffusion in each row, conventionally through the de Haller number W2/W1 ≥ 0.72 or an equivalent diffusion factor. That limit caps the turning each row may impose, and hence caps the whirl change, and hence — through w = UΔCY — caps the work per stage. In practice a subsonic axial stage is limited to a temperature rise of roughly 20 to 30 K, which at these conditions is a pressure ratio of only about 1.25 to 1.4 per stage. Question 5 of this paper is a worked example: the first N1 stage of this very machine gives 24.6 K and a ratio of 1.33.
The consequence is arithmetic. The gas generator must reach an overall pressure ratio of 14.1 (page 16). At about 1.3 per stage that requires
$$n \approx \frac{\ln 14.1}{\ln 1.33}=\frac{2.65}{0.285}\approx 9\ \text{to}\ 11\ \text{stages}$$spread across the two spools — which is what the attachment's cross-section shows for the combined N1 and N2 compressors. That is the minimum the compressor can have: fewer stages would demand more work from each, more turning, more diffusion, and the machine would stall on the running line or lose its surge margin at part speed. Splitting the compression into two spools running at different speeds (6805 and 8395 rev/min at peak load, from the attachment) is the same constraint expressed a second way — it lets each group of stages run near its own optimum incidence over a wide speed range, which a single long spool cannot do.
A turbine blade passage is a nozzle. The flow accelerates, the pressure gradient is favourable, the boundary layer stays firmly attached, and separation is simply not the governing concern. A turbine stage can therefore be turned through 90° or more, and can extract three to four times the work of a compressor stage at the same blade speed. That is why two or three turbine stages can drive nine or eleven compressor stages: the work per stage is far larger, even though the total work exchanged is the same.
But the turbine cannot be reduced to a single stage either, and the reasons are the ones the question calls “limiting conditions”. First, blade stress. The centrifugal stress in a blade root scales as ρN²A — the well-known AN² parameter — and the turbine runs at over 1000°C (1077°C at the gas-generator inlet, page 16), where the creep strength of even a single-crystal superalloy is a small fraction of its cold strength. That places a hard ceiling on the product of annulus area and speed, and therefore on blade speed, and therefore on UΔCY per stage. Second, aerodynamic loading: pushing all the expansion through one stage drives the stage loading coefficient Δh0/U² above about two, at which point the exit swirl and the leaving kinetic energy become large and stage efficiency falls away. Third, the flow would go supersonic in the nozzle and the shock losses would take back what the extra loading gained. Fourth, cooling: the first row must be film-cooled, and every kilogram of cooling air is compressor work thrown away, so a designer prefers to drop the gas temperature quickly across an early stage and let the later rows run uncooled — which needs more than one stage.
In practice the compressor stage count is set from the bottom up: fix the required pressure ratio, divide by the largest stage ratio the diffusion limit allows with an adequate surge margin, and round up. The turbine stage count is set from the top down: fix the work the compressor and the load demand, divide by the largest work per stage that blade stress at the operating temperature and the loading limit permit, and round up. The two answers come out different by a factor of three or four because one calculation is governed by an adverse pressure gradient and the other by metal temperature, and those constraints have nothing to do with each other. In this machine the free power turbine (N3) is a further stage group, sized separately because it runs at the 3000 rev/min the generator requires rather than at the gas-generator speed.
