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.
Specific speed is a dimensionless shape number, $N_s=\omega Q^{1/2}/(gH)^{3/4}$, formed from the design speed, flow and head at the best-efficiency point. Because it is independent of size, it groups all geometrically similar pumps onto one number and therefore fixes the type of impeller that will run efficiently at a given duty. A designer computes $N_s$ from the required duty and reads off the impeller family that peaks in efficiency there.
At low specific speed the duty is high head with small flow, and the efficient impeller is a narrow, large-diameter radial (centrifugal) type: water enters axially at the eye and is turned through 90° to leave radially, gaining most of its energy from centrifugal action. As specific speed rises the impeller becomes wider and shorter — a mixed-flow shape in which the water leaves partly radially and partly axially. At high specific speed (large flow, low head) the efficient machine is an axial (propeller) impeller in which flow passes straight through parallel to the shaft, energy coming mainly from the aerofoil lift of the blades rather than from radius change.
Thus specific speed and duty are linked directly: high head and low flow → low $N_s$ → radial; low head and high flow → high $N_s$ → axial. The flow direction through the impeller migrates from radial, through mixed, to axial as $N_s$ increases (see the sketch above).
The same idea classifies hydraulic turbines using the turbine specific speed $N_s=\omega P^{1/2}/[\rho^{1/2}(gH)^{5/4}]$. Low specific speed corresponds to very high head with small flow: the efficient machine is the Pelton wheel, an impulse turbine whose buckets are struck by one or more high-velocity jets — a partial-admission, tangential-flow runner. Moderate specific speed (medium head, moderate flow) suits the Francis turbine, a radial/mixed-flow reaction runner (the type classified in Question 2, $\Omega_{sp}\approx1.4$). High specific speed (low head, very large flow) suits the Kaplan (or propeller) turbine, an axial-flow reaction runner with adjustable blades — exactly the machine of Question 1.
The runner-shape progression mirrors the pump case: Pelton (tangential buckets) → Francis (radial-inward, mixed discharge) → Kaplan (axial propeller). Increasing specific speed always trades head for flow at fixed power, and the runner opens out from a narrow bucket wheel to a wide propeller.
