22-Mec-A6 Fluid Machinery · December 2016
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Reference texts: Dixon & Hall, Fluid Mechanics and Thermodynamics of Turbomachinery (7th ed.); Cohen, Rogers & Saravanamuttoo, Gas Turbine Theory (6th ed.); Turton, Principles of Turbomachinery; Çengel & Boles, Thermodynamics (9th ed.); Fox & McDonald, Introduction to Fluid Mechanics (9th ed.). Constants from the paper: $g=9.81\ \text{m/s}^2$, $c_p=1.005\ \text{kJ/kg\,K}$, $k=1.4$, $R=0.287\ \text{kJ/kg\,K}$, $\rho_{water}=1000\ \text{kg/m}^3$, $p_{atm}=100\ \text{kPa}$, $p_{vap}=2.34\ \text{kPa}$.
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.
(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. In a turbine stage the nozzles deliver a strongly tangential absolute velocity, and the free-vortex law $C_w r=\text{const}$ makes that whirl fall with radius while the blade speed $U=\omega r$ rises with it. At the base (root) $U$ is small and $C_{w1}$ is at its largest — in the sketch above $C_{w1}=2U$ — so the relative velocity $W_1$ still carries a large forward whirl, is steeply inclined to the axial direction (about $45^\circ$ here) and the blade section is highly staggered; the triangle is narrow in the $U$ direction and the flow is turned a great deal. At the tip $U$ has grown while $C_{w1}$ has shrunk until the two are equal — in the sketch $C_{w1}=U$, so $W_1$ is purely axial — the relative flow is much more nearly axial, the blade section is flatter and the triangle is wide in the $U$ direction with much less turning. The axial velocity is the same at both radii, so the entire change is in the tangential direction, which is precisely what the twist accommodates.
(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.
