Question 6 of 8: Blade Design

Question 6: Blade Design (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 — vane curvature. The distinguishing feature is the blade outlet angle $\beta_2$, measured between the blade tangent at exit and the direction of rotation. A backward-curved vane has $\beta_2<90^\circ$: the trailing edge sweeps away from the direction of rotation, so the relative velocity leaving the passage has a component opposing the blade motion. A forward-curved vane has $\beta_2>90^\circ$, its trailing edge leaning into the direction of rotation, so the relative velocity adds to the blade motion. A radial vane ($\beta_2=90^\circ$) is the intermediate case. The exit velocity triangles in Figure 6.1 make the consequence immediate. With the blade speed U₂ and the radial (through-flow) velocity V_f2 the same in each case, the whirl component is

$$C_{w2}=U_2-\frac{V_{f2}}{\tan\beta_2}$$

which exceeds U₂ for a forward-curved vane, equals it for a radial vane, and falls short of it for a backward-curved vane. Taking the illustrative values of Figure 6.1 ($U_2=30$ m/s, $V_{f2}=6$ m/s), the forward-curved vane at 150° gives $C_{w2}=40.4$ m/s and the backward-curved vane at 30° gives 19.6 m/s. Euler's equation, $H_{th}=U_2C_{w2}/g$, therefore returns theoretical heads of 123.5 m, 91.7 m and 60.0 m for the forward, radial and backward arrangements at the same size and speed. Forward curvature is by far the more powerful arrangement for a given impeller diameter and rotational speed — which is exactly why small, cheap ventilation fans use it.

The penalty appears in the absolute exit velocity, and therefore in the burden placed on the diffuser. Combining the whirl and radial components, the forward-curved impeller discharges at $C_2=40.8$ m/s against 20.5 m/s for the backward-curved one — almost exactly four times the kinetic energy, since energy goes as the square. A forward-curved machine therefore leaves most of the energy it has imparted in kinetic form at the impeller tip and depends on the volute or vaned diffuser to convert it into static pressure. Diffusion is the hardest thing a fluid machine has to do: the flow decelerates against a rising pressure, boundary layers thicken and separate readily, and diffuser efficiency is a strong function of off-design incidence. The backward-curved machine, by contrast, delivers a lower absolute velocity, achieves a larger part of its pressure rise inside the impeller passage itself (that is, it has a higher degree of reaction), and asks much less of the diffuser. In consequence its overall efficiency is typically five to ten points better and far flatter across the operating range, and its noise output is lower.

The characteristic curves in the lower panel of Figure 6.1 follow directly. Because $C_{w2}=U_2-V_{f2}/\tan\beta_2$ and $V_{f2}\propto Q$, the ideal head varies with flow as

$$H_{th}=\frac{U_2^{2}}{g}-\frac{U_2}{g\,A\tan\beta_2}\,Q$$

The slope is negative for a backward-curved vane, zero for a radial vane and positive for a forward-curved one. A rising head-flow characteristic is inherently unstable: over any part of the curve where head increases with flow, a small disturbance is amplified rather than damped, and the machine can hunt or surge, with two possible operating points on some system curves. Real forward-curved fan characteristics therefore show the familiar hump at part flow, with a dip that must be avoided in operation. The backward-curved characteristic falls monotonically, gives one unambiguous intersection with any system curve, and is self-correcting. Backward curvature also makes the machine “non-overloading”: its power draw peaks at a flow within the operating range and then falls, so the motor cannot be overloaded if the duct resistance drops, whereas a forward-curved fan's power rises continuously with flow.

In practice the choice runs: backward-curved (often aerofoil-bladed) for large process fans, boiler draught fans and virtually all centrifugal pumps, where efficiency, stability and motor protection dominate; forward-curved for small, low-cost, low-speed ventilation and domestic units, where a high head from a compact wheel matters more than efficiency; radial for dust-laden or abrasive service, where the self-cleaning straight blade survives erosion better than either curved form.

Part II — number of vanes. The vane count is a compromise between two opposing effects, and both degrade the pump if pushed too far.

Too few vanes. The Euler head assumes perfect guidance — that the fluid leaves exactly along the blade direction. Real flow does not: within the rotating passage the fluid tends to retain its orientation in the absolute frame, producing a relative eddy that circulates opposite to the impeller rotation. The result is slip: the actual whirl velocity is less than the Euler value by a slip factor $\sigma=C_{w2,actual}/C_{w2,Euler}$, typically 0.8 to 0.9. Slip worsens as the passages widen, because a wide passage guides the flow less firmly; the classical slip correlation puts the slip velocity in proportion to $\pi U_2\sin\beta_2/Z$, so the head deficit scales inversely with the number of vanes Z. With too few vanes the head generated falls well below the design intention, the impeller must be made larger or faster to compensate, and the flow within each over-wide passage becomes markedly non-uniform — a jet-and-wake structure with separation on the suction side. That non-uniformity is itself a loss, and it produces a strongly unsteady, low-frequency pressure field that shows up as noise and as radial thrust on the shaft.

Too many vanes. Every additional vane adds wetted surface, so skin-friction losses rise roughly in proportion to the vane count. More importantly, the vanes have finite thickness and each occupies part of the flow area. At inlet, where the radius is small and the circumference short, the blockage is most severe: the blade-thickness coefficient can easily fall below 0.8, accelerating the flow locally and depressing the static pressure just where the fluid is already closest to its vapour pressure. Cavitation inception therefore moves to a higher suction head, and the pump's NPSH requirement worsens. Narrow passages also increase the hydraulic radius penalty and make the impeller harder and more costly to cast or machine. The head does rise slightly as slip is suppressed, but the efficiency falls, and the gain is asymptotic — beyond a certain count little further head is recovered while friction keeps accumulating.

The optimum accordingly falls in a narrow band. For most centrifugal pumps of moderate specific speed the count is five to eight vanes, chosen from empirical rules such as $Z\approx 6.5\left[(D_2+D_1)/(D_2-D_1)\right]\sin\left[(\beta_1+\beta_2)/2\right]$, which reflects the same physics: long, narrow passages guide well with few vanes, while short, wide passages need more. Low-specific-speed impellers, whose passages are long relative to their width, manage with fewer; high-specific-speed mixed-flow impellers use fewer still. Designers also avoid integer relationships between the impeller vane count and the number of volute tongues or diffuser vanes, since coincident blade passing generates strong pressure pulsations at the blade-passing frequency; an odd vane count with an even diffuser count (or vice versa) is standard practice. Inducers and some sewage pumps break the rules deliberately: a two-vane or single-vane impeller sacrifices head and efficiency to pass solids, accepting the slip and the vibration as the price of non-clogging service.