22-Mec-A6 Fluid Machinery · December 2013
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Reference texts: Fox & McDonald, Introduction to Fluid Mechanics (turbomachinery chapter); S.L. Dixon & C.A. Hall, Fluid Mechanics and Thermodynamics of Turbomachinery; R.K. Turton, Principles of Turbomachinery; Cohen, Rogers & Saravanamuttoo, Gas Turbine Theory. Constants used (exam reference sheet): g = 9.81 m/s², ρwater = 1000 kg/m³, Patm = 100 kPa.
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.
Each blade of an axial compressor is an aerofoil that turns and diffuses the air, and like any aerofoil it stalls when its angle of attack becomes too large. In a compressor the angle of attack is set by the incidence of the relative velocity onto the blade, which depends on the ratio of axial velocity to blade speed. If the mass flow falls (for example when a downstream throttle closes) while the rotor speed is held, the axial velocity drops, the incidence onto the blades rises, and the boundary layer on the suction surface separates — the stage stalls. Stall usually appears first as rotating stall, where one or more cells of separated flow travel around the annulus, and if the disturbance grows it can trigger surge, a violent axial oscillation of the whole machine that can damage blades and flame-out the combustor.
The conditions that provoke stall are therefore low flow at high speed (throttled or accelerating engine), operation above the design pressure ratio, distorted or reduced inlet flow, and off-design speeds where the front and rear stages are mismatched. Because each stage can only tolerate a limited pressure rise before its blades stall, a high overall pressure ratio must be built from many stages, each doing a modest, safe amount of diffusion. Designers add stall margin by using variable inlet guide vanes and variable stators, bleed valves, and sometimes twin spools, so that the compressor stays clear of the surge line across the running range. The stall limit is thus the single most important constraint fixing the number of stages and the complexity of an axial compressor.
(a) Why water horsepower rises to a peak then falls. The hydraulic (water) power delivered is $P_{hyd}=\rho g Q H$. At shut-off ($Q=0$) no water is delivered, so despite the maximum head the product $QH$ is zero. As flow opens up, $Q$ increases faster than $H$ falls, so $QH$ — and the water horsepower — climbs. Beyond the best-efficiency region the head droops steeply toward zero at maximum flow, so $QH$ collapses again. The result is a curve that rises from zero, peaks near the best-efficiency point, and returns to zero at run-out — the same hump as the efficiency curve because $P_{hyd}$ is $\rho gQH$ and efficiency is $P_{hyd}/P_{brake}$.
(b) Why the brake–water power gap narrows then widens. The brake (mechanical) power is what the shaft absorbs; it is non-zero even at shut-off because the impeller still churns water and overcomes bearing, disk-friction and recirculation losses. Near the best-efficiency point these losses are smallest relative to the useful output, so the water power rises to meet the brake power and the gap between them (the total loss) shrinks to a minimum. At high flow the losses climb again — friction rises with the square of velocity and the flow angles no longer match the blades, so hydraulic and shock losses grow — while the useful water power is falling; the gap therefore widens once more, ending larger than it began (at run-out the brake power is still substantial while the water power has returned to zero). The narrowing-then-widening gap is exactly the inverse of the efficiency hump.
