3.8 Exercises
149
3.8.8 Axial Fan (Idealised Flow): Free Vortex and Non-Free
Vortex
Consider the axial fan from the previous exercise and assume a rotor blade shape
with constant work and constant axial velocity along the radius.
1. Determine the velocity triangles at the hub and the tip.
2. Determine the degree of reaction and the work coefficient at hub and tip.
3. Verify the realisability of the fan blades by calculating the deceleration ratios
w 2 /w 1 and the axial chord solidities, assuming a Zweifel coefficient of unity, at
hub and tip. Assume constant axial velocity in the streamtubes at hub and tip.
4. Perform the calculation once more, but now with the work ΔW proportional to
the radius. Compare with constant work distribution.
A: I: R hub = 0.36; ψ hub = 1.28; R tip = 0.84; ψ tip = 0.32; (w 2 / w 1 ) hub = 0.766; (w 2 / w 1 ) tip = 0.768;
(σ a ) hub = 2.17; (σ a ) tip = 0.46; II: R hub = 0.60; R tip = 0.80.
3.8.9 Inlet Guide Vane with a Centrifugal Fan
Argue that, with a centrifugal fan with backward curved blades (Fig. 3.5), the characteristic is modified by an inlet guide vane ring. Consider pre-swirl generating
a tangential component of the absolute velocity in the rotation sense (closing the
guide vanes) and flow rate reduction so that the direction of the relative velocity at
the inlet remains the same. Observe that rotor work stays approximately the same
due to the changed velocity triangle at the rotor outlet. The incidence at the volute
entrance changes, but whether or not this decreases efficiency depends on the operating point. Conclude that with adjustable inlet guide vanes, in principle, characteristics may be shifted to lower or higher flow rates. In practise, there is some loss of
efficiency due to incidences.
3.8.10 Change of Rotational Speed with Centrifugal and Axial
Fans
Consider a centrifugal fan with backward curved blades (Fig. 3.5). Analyse the effect of the change of rotational speed. Remark that the shape of the velocity triangles is kept, if all velocity components vary proportional to the speed of rotation.
Incidence angles stay the same. Conclude that similarity of the velocity triangles is
obtained for flow rate proportional to rotational speed and energy rise proportional
to rotational speed squared. So, similarity of velocity triangles is realised on parabolas through the origin in a map of characteristic curves. Remark that losses stay
proportional to the rotor work and internal efficiency is thus maintained, provided
that loss coefficients are constant. Observe that volumetric efficiency is also maintained, provided that contraction coefficients are constant. So, global efficiency is
149
3.8.8 Axial Fan (Idealised Flow): Free Vortex and Non-Free
Vortex
Consider the axial fan from the previous exercise and assume a rotor blade shape
with constant work and constant axial velocity along the radius.
1. Determine the velocity triangles at the hub and the tip.
2. Determine the degree of reaction and the work coefficient at hub and tip.
3. Verify the realisability of the fan blades by calculating the deceleration ratios
w 2 /w 1 and the axial chord solidities, assuming a Zweifel coefficient of unity, at
hub and tip. Assume constant axial velocity in the streamtubes at hub and tip.
4. Perform the calculation once more, but now with the work ΔW proportional to
the radius. Compare with constant work distribution.
A: I: R hub = 0.36; ψ hub = 1.28; R tip = 0.84; ψ tip = 0.32; (w 2 / w 1 ) hub = 0.766; (w 2 / w 1 ) tip = 0.768;
(σ a ) hub = 2.17; (σ a ) tip = 0.46; II: R hub = 0.60; R tip = 0.80.
3.8.9 Inlet Guide Vane with a Centrifugal Fan
Argue that, with a centrifugal fan with backward curved blades (Fig. 3.5), the characteristic is modified by an inlet guide vane ring. Consider pre-swirl generating
a tangential component of the absolute velocity in the rotation sense (closing the
guide vanes) and flow rate reduction so that the direction of the relative velocity at
the inlet remains the same. Observe that rotor work stays approximately the same
due to the changed velocity triangle at the rotor outlet. The incidence at the volute
entrance changes, but whether or not this decreases efficiency depends on the operating point. Conclude that with adjustable inlet guide vanes, in principle, characteristics may be shifted to lower or higher flow rates. In practise, there is some loss of
efficiency due to incidences.
3.8.10 Change of Rotational Speed with Centrifugal and Axial
Fans
Consider a centrifugal fan with backward curved blades (Fig. 3.5). Analyse the effect of the change of rotational speed. Remark that the shape of the velocity triangles is kept, if all velocity components vary proportional to the speed of rotation.
Incidence angles stay the same. Conclude that similarity of the velocity triangles is
obtained for flow rate proportional to rotational speed and energy rise proportional
to rotational speed squared. So, similarity of velocity triangles is realised on parabolas through the origin in a map of characteristic curves. Remark that losses stay
proportional to the rotor work and internal efficiency is thus maintained, provided
that loss coefficients are constant. Observe that volumetric efficiency is also maintained, provided that contraction coefficients are constant. So, global efficiency is