22-Mec-B6 Advanced Fluid Mechanics · December 2017
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format. 16-Mec-B6 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.
A pump does not suck. Liquid is pushed into the impeller eye by whatever absolute pressure exists at the free surface, less everything the liquid must spend on the way. The elevation limit is therefore the point at which the absolute pressure at the impeller eye falls to the vapour pressure of the liquid at its own temperature; below that, the liquid boils in the eye, the vapour bubbles are swept into the higher-pressure part of the impeller and collapse violently, and the pump cavitates — losing head, making a noise like gravel, and pitting the impeller until it fails.
The accounting is the net positive suction head, which page 22 of the paper gives as
$$\text{NPSH}=\frac{p_{\text{atm}}-p_{\text{vapour}}}{\rho g}-\Delta z-h_L$$and the setting is acceptable only while the NPSH available from that expression exceeds the NPSH required by the pump at its duty flow, with a margin. The factors that must be taken into account are each a term in that balance or a threat to it.
The absolute pressure over the liquid surface. For an open sump this is the local barometric pressure, which falls with altitude — roughly one metre of water head per 900 m of elevation, so a pump installed in the interior of British Columbia has appreciably less to work with than the same pump at sea level. It also falls in a passing low-pressure system. For a closed vessel it is the vessel pressure, and a suction drawn from a vacuum tower or a condenser hot well may have almost nothing available.
The vapour pressure of the liquid, and hence its temperature. This is the term that most often decides the answer. Water at 15 °C has a vapour pressure of 1.71 kPa (0.17 m) and leaves nearly the whole barometric head available; the same water at 80 °C has 47 kPa (4.8 m), and boiler-feed condensate at saturation has none at all — which is precisely why condensate and boiler-feed pumps are always set below their hot wells and deaerators, never above them. Volatile liquids behave the same way at ordinary temperatures.
The static lift Δz. Every metre the pump sits above the surface is a metre subtracted directly. This is the quantity being solved for, and it is the only one the installer fully controls.
The suction-line friction and fitting losses hL. These scale with the square of the flow, so the worst case is the maximum flow the pump will ever run at, not the rated duty. Suction lines are consequently sized one or two nominal sizes larger than the discharge, kept short and straight, and fitted with the fewest possible bends; a partially closed suction valve or a clogged strainer is a common field cause of cavitation in a pump that was correctly set on paper.
The pump's own NPSH required, which grows with flow. The manufacturer's NPSHR curve rises steeply to the right of the best efficiency point, so a pump run out along its curve can cavitate at a setting that is perfectly safe at rated duty. A margin of at least 0.5 to 1.0 m, or a ratio of about 1.3, is normal practice.
Transients and the worst credible level. The governing surface level is the lowest the sump will reach, not the normal one; start-up, sudden valve opening and the drawdown of a wet well all reduce the available head momentarily. Dissolved gas coming out of solution, and the vortex that forms if the submergence over the suction bell is inadequate, are further practical limits that a bare NPSH calculation does not capture.
Expressed in the paper's own notation, the same criterion is the pump Thoma coefficient σc = NPSH/H (page 22), which must exceed the critical value for the pump's specific speed — the direct analogue of the turbine setting calculation in Question 1 Part II.
(a) Why the water power rises from zero to a peak and then declines. Water power is the product of two quantities that the pump's own characteristic drives in opposite directions:
$$P_{\text{water}}=\rho g Q H$$At shut-off (Q = 0) the pump develops its maximum head but delivers nothing, so the product is zero — all of the shaft input is going into churning and heating the trapped liquid. As the valve is opened, Q grows quickly while the head falls only slowly, so the product rises. Further out along the curve the head begins to fall steeply, because the backward-curved blading gives a descending characteristic and because the internal friction and shock losses grow with the square of the flow. At the far end, run-out, the head reaches zero: the pump is passing all the flow the system will take and generating no pressure rise at all, so the product is zero again. A quantity that is zero at both ends of the range and positive in between must have a maximum somewhere between, and that is what the curve shows. The efficiency curve behaves the same way and for the same reason, since it is the water power divided by a brake power that never reaches zero; its peak defines the best efficiency point, the flow at which the incidence onto the blades matches the design angles and the shock loss vanishes.
(b) Why the gap between brake power and water power first narrows and then widens. The vertical distance between the two curves is the total loss — hydraulic, volumetric, mechanical and disc friction — and it behaves as it does because those losses do not all depend on flow in the same way.
At shut-off the water power is zero but the brake power is not: the impeller is still spinning, still shearing the liquid in the casing, still overcoming bearing and seal drag and disc friction. Every watt entering the shaft is a loss, and it appears as heat in the trapped liquid — which is why a pump must never be left running against a closed discharge valve for long. The gap therefore starts at the full shut-off power.
As flow increases, the water power rises much faster than the brake power, because the brake power of a backward-curved impeller rises only gently. The gap therefore narrows, and it reaches its minimum at the best efficiency point, where the flow enters the impeller and the volute at the angles they were designed for, incidence and shock losses are at their smallest, and the largest possible fraction of shaft power is being converted.
Beyond the best efficiency point the trend reverses. The head is now collapsing, so the water power turns over and starts to fall, while the brake power keeps climbing with flow. At the same time the losses themselves grow rapidly: friction losses scale as Q², and the incidence at the impeller inlet and the mismatch between the impeller discharge and the volute throat both grow worse the further the pump is run from its design flow, so shock losses rise steeply. One curve falling while the other rises means the gap widens quickly, and at run-out it exceeds the shut-off value, which is exactly what part (b) describes. This is also the practical reason that a pump run far to the right of its best efficiency point overloads its driver, cavitates (NPSHR is rising there too) and suffers the worst vibration and bearing loads — and the reason that mixed-flow and axial machines, whose brake power rises more steeply still, are usually started against an open valve while a radial pump is started against a closed one.
