22-Mec-B6 Advanced Fluid Mechanics · December 2018

Question 7 of 8: Fan Blade Shape

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

Notes on this paper

Paper format: National Examinations, December 2018 — 16-Mec-B6 Fluid Machinery. Closed book, three hours, 22 pages. Section A is calculative (Questions 1–5) and Section B descriptive (Questions 6–8); candidates answer four questions from Section A and two from Section B — six questions of ten marks each, 60 marks in all. Reference data for individual questions are supplied on pages 11–17, and the nomenclature, general constants and reference equations on pages 18–22. All eight questions are solved below.

Check — angle conventions are taken from the paper's own attachments, and they are not the same across the paper. The compressor velocity diagram on page 11 strikes $\alpha_1$ and $\beta_1$ off the axial component $C_{X1}$, so every blade and vane angle in Questions 1 and 2 is measured from the axial direction. The steam-turbine diagram on page 16 strikes $\theta$, $\phi$, $\gamma$ and $\delta$ off the tangential (blade-motion) direction, and the Francis diagram on page 15 defines $\alpha_1$ between $V_1$ and $u_1$ — also tangential. Questions 4 and 5 therefore use the tangential reference. Every constant below is the paper's own page-19 value: $g = 9.81\ \text{m}\,\text{s}^{-2}$, $\rho_{\text{water}} = 1000\ \text{kg}\,\text{m}^{-3}$, $c_p = 1005\ \text{J}\,\text{kg}^{-1}\text{K}^{-1}$ and $c_v = 718\ \text{J}\,\text{kg}^{-1}\text{K}^{-1}$, from which $R = c_p - c_v = 287\ \text{J}\,\text{kg}^{-1}\text{K}^{-1}$ and $k = c_p/c_v = 1.3997$ — not a textbook 1.40.

Reference texts

Dixon, S. L. and Hall, C. A., Fluid Mechanics and Thermodynamics of Turbomachinery, 7th ed. — 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. 5 (axial flow compressors), Ch. 6 (compressor and turbine matching, surge and stall), Ch. 3 (jet propulsion cycles).
Turton, R. K., Principles of Turbomachinery, 2nd ed. — Ch. 2 (Euler equation), Ch. 8 (fans, blade form and control).
White, F. M., Fluid Mechanics, 8th ed. — Ch. 11 (turbomachinery: specific speed, Francis and Kaplan turbines, cavitation).
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 — Fan Blade Shape (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.

The three outlet velocity diagrams asked for in part (a), drawn for a common impeller speed and a common radial (through-flow) velocity, together with the head–flow characteristics they produce. The only thing that changes between them is the direction of the relative velocity leaving the blade, and everything else follows from it.

(a) Outlet velocity diagrams and the head–flow characteristic

Take a common impeller tip speed $U_2$ and a common radial velocity $V_{f2}$, which is the component that carries the flow, and let $\beta_2$ be the angle of the blade at exit measured from the tangential direction opposite to the motion. The outlet whirl is then $V_{\theta 2} = U_2 - V_{f2}/\tan\beta_2$, and the three blade forms differ only in the sign of the second term. For backward-curved blades $\beta_2$ is acute, the relative velocity leans back against the rotation and the whirl is less than the blade speed — about 68 per cent of it in the figure. For radial blades $\beta_2 = 90^{\circ}$, the relative velocity is purely radial and the whirl is exactly the blade speed. For forward-curved blades $\beta_2$ is obtuse, the relative velocity leans into the rotation and the whirl exceeds the blade speed — about 132 per cent of it. The absolute velocity leaving the impeller therefore grows steadily from the backward to the forward form, and with it the kinetic energy that the volute must convert into pressure.

The Euler head is $H = U_2 V_{\theta 2}/g$, so substituting the whirl expression gives the ideal characteristic directly: $$H = \frac{U_2}{g}\left(U_2 - \frac{Q}{\pi D_2 b_2 \tan\beta_2}\right)$$ Since the radial velocity is proportional to the flow rate, this is a straight line in $Q$ whose slope is set entirely by $\beta_2$. The backward-curved fan gives a line that falls steeply with flow; the radial fan gives a horizontal line; the forward-curved fan gives a line that rises with flow. Real characteristics are the ideal lines less the shock and friction losses, which grow roughly with the square of the departure from the design flow, so all three curves bend over at high flow; but the ordering survives, and the forward-curved fan retains a characteristic that is flat or humped over most of its range while the backward-curved fan falls away steadily and monotonically. The figure above shows the three curves with those shapes on common axes.

(b) Advantages of each form, and which is more common

The forward-curved fan's advantage is size. Because it develops a whirl greater than its own tip speed, it produces more head and more flow than the other two for the same impeller diameter and the same rotational speed, so a given duty can be met by a smaller, slower, quieter and cheaper machine. This is why the multi-vane forward-curved impeller — the squirrel cage — is universal in domestic and light commercial air handling, in furnace blowers and in fan-coil units, where the duty is fixed, the pressure is low and space is the binding constraint.

Against that stand three real disadvantages. First, the forward-curved fan is an overloading machine: its power demand, $P = \rho g Q H$, rises steeply and without limit as the flow increases, because the head does not fall away to check it. If a filter is removed or a damper is opened wide, the fan runs out along its curve and can overload and burn out its motor — so the motor must be sized for the runout condition, not the design condition, which wastes both capital and efficiency. Second, most of the head is produced as kinetic energy rather than as pressure inside the impeller, and it must be recovered by diffusion in the volute; diffusion is lossy, so the peak efficiency of a forward-curved fan is typically 55 to 70 per cent against 75 to 85 per cent for a good backward-curved one. Third, the flat or humped characteristic gives poor stability: a nearly horizontal head–flow curve intersects a system curve at a poorly defined point, and a rising portion can produce surging and hunting when two fans operate in parallel or when the fan works into a large volume.

The backward-curved fan reverses each of these. Its power characteristic reaches a maximum and then falls, so it is non-overloading: a motor sized for the peak of the power curve is safe at every point of the fan's range, whatever the system does. Its steeply falling head–flow curve intersects any system curve at a single, well-defined and stable operating point, and it makes the fan easy to control by damper or by speed. Because more of the head is developed as static pressure within the impeller and less as kinetic energy, less diffusion is needed and the efficiency is markedly higher — which over a duty cycle measured in tens of thousands of hours dominates the whole-life cost. The blades are also less prone to dust build-up than the many small forward-curved vanes, and the aerofoil version of the backward-curved blade is the quietest and most efficient form of all.

Which is more common depends on where one looks, and the honest answer names both markets. By unit count the forward-curved squirrel cage is the more numerous, because it dominates the very large population of small, low-pressure air-handling products. By installed power, and in every application where the fan is large, runs continuously, or must work against a variable system, the backward-curved impeller is the standard choice, and it is the one specified for industrial ventilation, boiler forced- and induced-draught service, mine ventilation and building air handling of any size. Radial-bladed fans occupy a third, narrower niche: they are the least efficient of the three, but their simple, self-cleaning, robust blades tolerate dust, fibres and material-laden gas that would foul either curved form, so they survive in material-handling and exhaust duties where erosion and fouling, not efficiency, set the design.

Topic. The effect of impeller blade form on the outlet velocity triangle, the head–flow characteristic, the power characteristic and the efficiency of a centrifugal fan.

Key relations. Euler head $H = U_2 V_{\theta 2}/g$; outlet whirl $V_{\theta 2} = U_2 - V_{f2}/\tan\beta_2$; continuity at the impeller periphery $Q = \pi D_2 b_2 V_{f2}$; fan power $P = \rho g Q H / \eta$; degree of reaction, which falls as $\beta_2$ increases past $90^{\circ}$.

Why this works. The Euler equation makes the head depend only on the tip speed and the outlet whirl, and the outlet whirl is fixed by where the relative velocity points as it leaves the blade. Because the radial velocity is proportional to the flow, the whirl becomes a linear function of flow whose slope carries the sign of $\cot\beta_2$ — positive for backward, zero for radial, negative for forward. That single sign propagates through everything: it sets whether the head falls, stays flat or rises with flow; whether the power curve turns over or runs away; and how much of the head appears as impeller pressure rather than as kinetic energy needing recovery in the volute, which in turn sets the efficiency. Recognising that the three fans differ in one quantity, and that all of the consequences flow from it, is what turns this from a list of memorised facts into an argument.

Common pitfall. Drawing the three diagrams with different radial velocities or different tip speeds, which hides the very comparison the question is asking for; they must be drawn on a common $U_2$ and a common $V_{f2}$. The second pitfall is answering "backward-curved is more common" or "forward-curved is more common" without qualification — the mark is for the reasoning, and the reasoning divides the market by duty, size and running hours.

Source. Turton, Principles of Turbomachinery, 2nd ed., Ch. 8 (fans, blade form and control); Dixon & Hall, Fluid Mechanics and Thermodynamics of Turbomachinery, 7th ed., §7.2–7.4 (centrifugal machines, slip and characteristics); Fox & McDonald, 10th ed., §10.3–10.4.

← Question 6 Next: Question 8 →