25-Nav-B5 Marine Control Systems · May 2018

Question 8 of 8: Fan Control (10 marks)

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

Notes on this paper

Paper format: National Examinations, May 2018 — 98-Mar-B5 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, because the set is a study resource rather than a three-hour sitting.

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).

Subject note. Although this paper is listed under “Marine Control Systems”, page 1 of the examination itself reads 98-MAR-B5, FLUID MACHINERY, and every question is a turbomachine question with zero marine-control-systems content. The solutions below answer the paper as printed, citing turbomachinery texts rather than any control-systems reference.

Question 8 — Fan Control (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.

(a) The three constructions on the head–flow diagram

Figure 8.1 — The three control methods on head–flow axes. Dotted blue is the original fan characteristic, dotted grey the original system characteristic, and the grey dot the design operating point; the solid red curve and dot show the changed characteristic and the new operating point in each case.

The operating point of any fan-and-duct combination is the single point where the fan characteristic (the head the fan can generate at each flow) intersects the system characteristic (the head the ducting demands at each flow). A control method can only work by moving one of those two curves. Dampers move the system curve; inlet vanes and speed change move the fan curve. That is the structural answer to part (a), and each diagram shows which curve has moved and where the intersection has gone.

(b) Why and how the flow is reduced in each case

(i) Effect of duct dampers

A duct system's characteristic is essentially $H_{\text{sys}} = KQ^{2}$: all the loss is turbulent friction, entry, bend and fitting loss, each of which scales with the square of velocity and therefore with the square of flow. Closing a damper inserts an additional and increasingly severe local resistance, so the system constant $K$ rises. The parabola therefore rotates upward about the origin and becomes steeper, and its intersection with the (unaltered) fan curve slides back and up along that fan curve. The flow falls and the head rises.

The mechanism is dissipative. The fan is still doing work on the air at a head higher than the duty requires, and the excess head is thrown away as turbulence and heat across the damper blades. The fan itself is also being pushed away from its design point towards lower flow, so its own efficiency falls at the same time. Absorbed power does fall — because the flow has fallen — but far less than proportionally, so damper control is the least efficient of the three methods. It is also the cheapest to install and the simplest to operate, which is why it survives in small and intermittently controlled systems. A practical caution: on a fan with a rising-to-a-peak (unstable) region in its characteristic, closing the damper too far can drive the operating point into that region and provoke fan surge or stall, exactly as described for the compressor in Question 7.

(ii) Effect of inlet vanes

Inlet guide vanes do not add resistance to the flow path in any significant way; they change what the fan is capable of. Set at an angle, they impart pre-whirl — a tangential velocity component $C_{Y1}$ in the same direction as the impeller rotation — before the air reaches the blades. The work the fan can do per unit mass is given by the Euler equation, exactly as in Question 1:

$$w = U\left(C_{Y2} - C_{Y1}\right)$$

Adding positive pre-whirl increases $C_{Y1}$ and so reduces $C_{Y2} - C_{Y1}$ directly. The fan generates less head at every flow, so its whole characteristic falls; intersected with the unchanged system parabola, the new operating point lies at lower flow and lower head. Progressively opening the vanes towards the fully swirling position lowers the characteristic further.

Figure 8.2 — Inlet velocity triangles without and with positive pre-whirl. Swirl in the direction of rotation eats into the whirl change the impeller can produce, lowering the whole fan characteristic.

Because the reduction is achieved by taking work out of the fan rather than by throttling the air, inlet-vane control is markedly more efficient than damper control: the absorbed power falls roughly in step with the reduced duty over the upper part of the range. The vanes also pre-align the flow with the impeller blades, which limits the incidence penalty and holds the fan efficiency up better than throttling does. Its limits are that the achievable turndown is modest (typically to about 60–70 % of design flow before efficiency falls away sharply), and that at large vane angles the swirl itself becomes a loss.

(iii) Effect of fan speed

Reducing the driving-motor speed changes the fan characteristic through the fan (affinity) laws, which are the similarity relations printed on page 20 applied to one machine at two speeds, so the diameter terms drop out:

$$\frac{Q_2}{Q_1} = \frac{N_2}{N_1}, \qquad \frac{H_2}{H_1} = \left(\frac{N_2}{N_1}\right)^{2}, \qquad \frac{P_2}{P_1} = \left(\frac{N_2}{N_1}\right)^{3}$$

The whole characteristic is therefore rescaled: every point moves to a lower flow in proportion to speed and to a lower head in proportion to speed squared. Cutting the speed to 80 % of design gives 80 % of the flow, 64 % of the head and just 51 % of the power. The system curve is untouched, and the new intersection lies down and to the left along it.

Speed control is by a wide margin the most efficient of the three methods, and the reason can be read directly from the geometry of the diagram. A pure duct system has the characteristic $H = KQ^{2}$, which is the same parabolic form as the fan's own similarity relation $H \propto N^{2}$ with $Q \propto N$. The scaled operating point therefore lands on the system curve automatically and, more importantly, on the fan's own similarity parabola, which means the fan is still running at the same velocity-triangle shape and hence at very nearly the same efficiency as at design. Nothing is throttled and no work is added and then discarded — the fan is simply made into a smaller fan. The cubic power law is what delivers the large energy savings that justify variable-frequency drives on large fan and pump installations. The qualifications are that a system with a static-head component (an outdoor pressure difference, or a filter with a fixed minimum resistance) does not have a purely parabolic curve, so the operating point drifts off the similarity parabola and the efficiency benefit is reduced; and that the drive itself has losses, and reduced speed must be checked against motor cooling and against any blade or shaft critical speed the reduced range may cross.

Comparison of the three control methods
Method	Which curve moves	How the flow is reduced	Energy penalty
Duct dampers	System curve steepens	Added throttling resistance; fan rides back up its own curve	Highest — surplus head dissipated as heat
Inlet vanes	Fan curve falls	Pre-whirl reduces $C_{Y2} - C_{Y1}$ and hence the head generated	Moderate — work is never added in the first place
Fan speed	Fan curve rescales	Affinity laws: $Q \propto N$, $H \propto N^{2}$, $P \propto N^{3}$	Lowest — efficiency preserved on a parabolic system

Topic. Fan flow control: system versus fan characteristics, pre-whirl, and the affinity laws.

Key relations. System curve $H = KQ^2$; operating point at the intersection of fan and system curves; Euler work $w = U(C_{Y2} - C_{Y1})$; affinity laws $Q \propto N$, $H \propto N^2$, $P \propto N^3$ at fixed diameter, from the similarity equations on page 20.

Why this works. Every one of the three methods is a statement about which of the two intersecting curves is being altered, and recognising that is most of the answer. Dampers cannot change what the fan can do, so they must change what the duct demands; vanes and speed cannot change the duct, so they must change the fan. The efficiency ranking then follows from where the surplus energy goes. Throttling adds head that is immediately destroyed across the damper. Pre-whirl prevents the head from being generated at all, so the energy is never spent. Speed reduction goes further still, because a purely quadratic system curve coincides with the fan's own similarity parabola, so the machine moves to a geometrically similar operating condition and keeps its efficiency — the cube law on power is then the direct payoff.

Common pitfall. Drawing the damper case with the fan curve moved, or the speed case with the system curve moved. Also common is applying the affinity laws to a system with significant static head as though the operating point stayed on the similarity parabola; it does not, and both the flow reduction and the power saving are then smaller than the cube law suggests. Finally, negative pre-whirl (swirl against rotation) raises the characteristic and increases both head and power — useful as a boost, useless as a control.

Source. Turton, Principles of Turbomachinery, 2nd ed., Ch. 8 (fans and their control); Fox & McDonald, Introduction to Fluid Mechanics, 10th ed., §10.4–10.5 (similarity rules and flow control).

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