22-Mec-B6 Advanced Fluid Mechanics · May 2018

Question 7 of 8: Compressor and Pump Characteristics (10 marks)

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format: National Examinations, May 2018 — 16-Mec-B6 Fluid Machinery. Closed book, three hours, 21 pages. Section A is calculative (Questions 1–5) and Section B descriptive (Questions 6–8); candidates answer four from Section A and two from Section B, six questions of ten marks each for a total of 60. Reference data for individual questions are supplied on pages 11–16 and the nomenclature, constants and reference equations on pages 17–21. All eight questions are solved below.

Check — angle conventions taken from the paper's own attachments. The compressor attachment on page 11 strikes $\alpha_1$ and $\beta_1$ off the axial component $C_{X1}$, so in Questions 1 and 2 all blade and vane angles are measured from the axial direction. The steam-turbine attachment on page 13 and the Francis attachment on page 14 strike $\theta$, $\phi$, $\gamma$, $\delta$, $\alpha$ and $\beta$ off the tangential (blade-motion) direction, so Questions 3 and 5 use the tangential reference. Mixing the two is the single most common way to lose all the marks on a velocity-diagram question. Every constant used below ($g = 9.81\ \text{m}\,\text{s}^{-2}$, $\rho_{\text{water}} = 1000\ \text{kg}\,\text{m}^{-3}$, $\rho_{\text{air}} = 1.21\ \text{kg}\,\text{m}^{-3}$ at 15 °C, $c_p = 1.005$ and $c_v = 0.718\ \text{kJ}\,\text{kg}^{-1}\text{K}^{-1}$) is taken from the paper's page 18 rather than from a textbook.

Reference texts

Dixon, S. L. and Hall, C. A., Fluid Mechanics and Thermodynamics of Turbomachinery, 7th ed. — Ch. 3 (two-dimensional cascades), Ch. 5 (axial compressors), Ch. 4 (axial turbines), Ch. 9 (hydraulic turbines).
Cohen, H., Rogers, G. F. C. and Saravanamuttoo, H. I. H., Gas Turbine Theory, 7th ed. — Ch. 3 (jet propulsion cycles), Ch. 5 (axial flow compressors), Ch. 6 (compressor and turbine matching, surge and stall).
Turton, R. K., Principles of Turbomachinery, 2nd ed. — Ch. 2 (Euler equation), Ch. 4 (pump characteristics), Ch. 8 (fans and their control).
White, F. M., Fluid Mechanics, 8th ed. — Ch. 11 (turbomachinery: specific speed, pump curves, Pelton and Francis turbines).
Fox, R. W. and McDonald, A. T., Introduction to Fluid Mechanics, 10th ed. — Ch. 10 (fluid machinery, similarity rules).
Çengel, Y. A. and Boles, M. A., Thermodynamics: An Engineering Approach, 9th ed. — Ch. 9 (gas power cycles, Brayton and turbojet).

Question 7 — Compressor and Pump Characteristics (10 marks)

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.

Part I — Stalling in an axial flow compressor

What stalling is. Each compressor blade is an aerofoil working in a cascade, and like any aerofoil it develops its pressure rise by turning the flow while keeping the boundary layer attached to its suction surface. The blade is designed for one relative inlet angle. When the actual relative flow arrives at an angle appreciably greater than the blade's metal angle — that is, at high positive incidence — the adverse pressure gradient on the suction surface becomes too steep for the boundary layer to negotiate, the flow separates, and the blade loses most of its ability to deflect the air and to raise the pressure. That is blade stall. It is worth being precise that a compressor blade row is a diffuser: it is trying to decelerate the relative flow, so it is inherently far closer to separation than a turbine row, which accelerates.

Stall rarely appears uniformly around the annulus. Small non-uniformities in incidence mean one or two passages separate first; the blockage they create deflects flow away from themselves and into the passages on one side while increasing incidence on the other, so the stalled region propagates around the annulus at roughly half rotor speed relative to the blades. This is rotating stall. Its practical significance is twofold: the compressor delivers a reduced and unsteady pressure rise, and the blades passing in and out of the stall cell are subjected to an alternating aerodynamic load at a frequency that can coincide with a blade natural frequency, causing high-cycle fatigue failure. If the disturbance grows until the whole compressor can no longer sustain the pressure ratio imposed on it by the downstream system, the flow reverses violently and the machine surges — a global, axial oscillation of the entire flow that can extinguish the combustor and damage the machine.

Conditions under which it occurs. Incidence rises whenever the axial velocity falls relative to the blade speed, so the trigger is any condition that reduces mass flow at a given rotational speed, or raises rotational speed at a given mass flow:

Throttling towards low mass flow at constant speed — the classical approach to the surge line on the compressor map.
Off-design speed, especially starting and low-speed running. At low speed the compressor cannot generate its design pressure ratio, so the density rise through the machine is far below design. The rear stages then see too much volume flow (high axial velocity, negative incidence, tending to choke) while the front stages see too little (low axial velocity, high positive incidence, tending to stall). This front-stage stall at part speed is the single most common stalling condition in a gas turbine.
Rapid acceleration, when the fuel schedule raises turbine work faster than the airflow can respond, momentarily driving the operating point towards the surge line.
Distorted or disturbed inlet flow — crosswinds, intake separation, aircraft manoeuvre — which creates a sector of low axial velocity and hence locally high incidence.
Blade damage, erosion, fouling or excessive tip clearance, all of which lower the stall margin permanently.

Effect on design and on the number of stages. Stall sets a hard ceiling on how much a single stage may be asked to do. The stage pressure ratio depends on the deflection $\Delta V_\theta$ the blades achieve, and the achievable deflection is limited by the diffusion the suction surface can sustain — conventionally expressed as a limit on the de Haller number $W_2/W_1$ (typically not below about 0.72) or on Lieblein's diffusion factor. In practice an axial stage is limited to a pressure ratio of roughly 1.15–1.4 in subsonic designs. A gas turbine needing an overall ratio of 6.8, as in Questions 1 and 2, therefore cannot be built with two or three heavily loaded stages: it needs eight lightly loaded ones, each at 1.271. Higher-pressure-ratio engines need correspondingly more stages, which is precisely why large industrial and aero compressors carry fifteen or more.

Because a multi-stage machine also suffers the front-stage/rear-stage mismatch described above, extra design measures follow directly from the stall problem: interstage bleed (blow-off) valves that dump air from the middle of the compressor at low speed to raise front-stage axial velocity; variable inlet guide vanes and variable stator rows in the front stages, which are closed at part speed to reduce incidence; and multi-spool arrangements, in which the front stages run on their own shaft at their own speed so that each group can stay near its own design incidence. Every one of these adds weight, cost and complexity, and each exists because of stall. Finally, the designer must retain a deliberate surge margin between the running line and the surge line, which means the compressor is never operated at its own peak efficiency point — a permanent efficiency penalty accepted in exchange for operability.

Part II — Centrifugal pump characteristic curves

Figure 7.1 — Characteristic curves for a typical mixed-flow centrifugal pump. The vertical bars mark the gap between brake horsepower and water horsepower — the power lost inside the machine — at shut-off, at the best efficiency point and at runout.

(a) Why the water horsepower rises to a peak and then declines. Water horsepower is the useful hydraulic power actually delivered to the liquid, $P_w = \rho gQH$, or in the customary units of the figure $P_w\,[\text{hp}] = QH/3960$ with $Q$ in gpm and $H$ in feet. It is therefore the product of two quantities that move in opposite directions along the characteristic. At shut-off the pump develops its maximum head, about 80 ft, but the flow is zero, so the product is zero: the pump is churning liquid round the casing and delivering nothing. As the discharge valve is opened the flow rises rapidly while the head falls only slowly — the head curve is almost flat over the first third of the range — so the product climbs steeply. Further out, the head curve steepens and begins to fall faster than the flow rises; the product passes through a maximum and then declines. At the pump's maximum capacity (runout, where the head has fallen to whatever the system can no longer resist, or in the limit to zero) the product returns towards zero.

The peak occurs at about 10 500 gpm on this pump, coinciding with the best efficiency point, and the arithmetic is straightforward: at that duty the head is about 60 ft, so $P_w = 10\,500 \times 60/3960 = 159\ \text{hp}$ (the plotted peak reads about 165 hp); at 15 000 gpm the head has fallen to about 30 ft, giving $15\,000 \times 30/3960 = 114\ \text{hp}$, a clear decline despite the 43 % larger flow, and by the end of the curve near 16 000 gpm the water horsepower is down to about 60 hp. The reason the peak of the water-horsepower curve sits so close to the efficiency peak is that the brake horsepower curve is very nearly a straight line, so $\eta = P_w/P_b$ and $P_w$ reach their maxima at almost the same flow.

(b) Why the gap between brake and water horsepower first narrows and then widens beyond its initial value. The difference $P_b - P_w$ is everything the driver supplies that does not end up in the discharged liquid: it is the total internal loss. Reading it across the curve tells the whole story of how a centrifugal pump behaves off design.

At shut-off the water horsepower is zero, so the entire brake horsepower — about 135 hp on this machine — is loss. With no through-flow the impeller simply recirculates liquid in the passages and the volute, and every watt goes into disc friction on the impeller shrouds, recirculation and churning, mechanical friction in bearings and seals, and ultimately into heating the trapped liquid. (That is why running a centrifugal pump against a closed valve for more than a short time is dangerous: the liquid can boil.)

As flow increases towards the design point, the loss falls, even though the friction losses through the passages are growing with the square of velocity. The dominant loss at low flow is not friction but incidence (shock) loss plus recirculation: away from the design flow the liquid meets the impeller vanes at a large angle of incidence and separates, exactly as described for the compressor in Part I. That loss decreases steeply as the flow approaches the value for which the vane angles were drawn. At the best efficiency point the incidence is nominally zero, recirculation has died away, and only friction and leakage remain; the gap reaches its minimum, here about 40 hp (roughly 160 hp of water horsepower against 200 hp brake), consistent within chart-reading accuracy with the plotted peak efficiency of about 84 %.

Beyond the BEP both mechanisms reverse. The incidence angle now grows in the opposite (negative) sense and separation returns on the other face of the vanes, while the frictional and mixing losses — which scale roughly with the square, and the disc-friction and turbulent-dissipation terms with the cube, of the flow — grow rapidly. Meanwhile the water horsepower has passed its own peak and is falling. The two effects compound: brake horsepower continues to climb almost linearly (this is a mixed-flow pump, whose power curve does not flatten in the way a radial pump's does), so at 15 000 gpm the brake horsepower is already about 238 hp against a water horsepower of 114 hp (a loss of about 124 hp), and at the end of the curve near 16 000 gpm it is about 250 hp against roughly 60 hp — a loss of some 190 hp, well above the 135 hp lost at shut-off. This is the practical reason a mixed-flow or axial-flow pump must not be run far out on its curve without checking the driver rating: the motor can be overloaded at runout even though the pump is delivering less useful power than it did at its best point.

Topic. Aerodynamic stall and surge in axial compressors; the shape of centrifugal-pump characteristic curves and the loss budget behind them.

Key relations. Incidence $i = \beta_{\text{flow}} - \beta_{\text{metal}}$ with $\tan\beta = (U - V_\theta)/V_x$; de Haller criterion $W_2/W_1 \ge 0.72$; overall ratio from $n$ stages $r_c = r_{\text{stage}}^n$; water power $P_w = \rho gQH$ (or $QH/3960$ in gpm and feet); pump efficiency $\eta = P_w/P_b$; loss $= P_b - P_w$.

Why this works. Both halves of this question are the same idea seen in two machines: a blade row works well only near the incidence it was drawn for, and the penalty for departing from that incidence is separation. In the compressor the penalty is stall, rotating stall and eventually surge, and the designer's response is to load each stage lightly and use many of them. In the pump the penalty is the shock loss that dominates the low-flow end of the curve, which is why the loss falls as the duty approaches the BEP even though friction is rising. The pump's power curves then follow from arithmetic: water horsepower is a product of a rising flow and a falling head, so it must vanish at both ends and peak in between, while brake horsepower is that product divided by an efficiency which itself collapses at both ends.

Common pitfall. Confusing rotating stall with surge. Rotating stall is a local, circumferentially travelling cell with the machine still delivering flow; surge is a system-level axial oscillation with flow reversal. Both matter but they are different phenomena with different remedies. On the pump side, the frequent error is to assume the loss is smallest at the smallest flow because friction is smallest there; the loss is in fact largest at shut-off, because the useful output is zero.

Source. Cohen, Rogers & Saravanamuttoo, Gas Turbine Theory, 7th ed., §5.8–5.9 and Ch. 6 (compressor characteristics, surge and stall); Turton, Principles of Turbomachinery, 2nd ed., Ch. 4; White, Fluid Mechanics, 8th ed., §11.3.

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