7.7 Exercises
279
We take a tongue clearance of about 10 % of the radius: about 22.5 mm. The
opening angle of the scroll is calculated for the first sector of 30° with b 3 = b 2 , because the flow expansion by the width leap cannot be sudden. This way, the tongue
clearance becomes 20 mm. For the rest of the scroll, the full width is used, but the
scroll is made somewhat larger than calculated with the logarithmic spiral.
The width leap to the scroll becomes quite large: 40 mm/150 mm. Common
practice is then to widen the rotor at the outlet, for instance to 60 mm. Then, the
strongest possible deceleration is realised in the rotor, but at rotor outlet the flow is
then separated. We may accept that a velocity reduction of 0.7 is possible between
the inlet flow just after rotor entrance and the outlet flow just before rotor exit,
so before slip. Enlargement of the rotor width at outlet is commonly used with
fans with slightly backward leaning blades or forward curved blades. For forward
curved blades, typically the outlet width is taken equal to the inlet width, because
the diameter ratio is near to unity (see Fig. 3.19). Figures 7.21 and 7.22 show the
geometry. A final calculation of the performance of the fan can now be made (see
Exercise 7.7.6).
7.7 Exercises
7.7.1. Reason that, with a radial fan or pump, the similitude laws with a change
of rotational speed, Q ~ n, ∆p 0 ~ n
2
, stay valid in the presence of incidence at the rotor and the scroll inlets.
7.7.2. Q = 0.20 m
3
/s, H m = 30 m, η global = 0.75 are measured at a pump with
n = 1450 rpm. The fluid is water with ν = 10
−6
m
2
/s. Determine flow rate, head and
efficiency of the pump when doubling the rotational speed for a similar flow. Assume a volumetric efficiency and a mechanical efficiency of respectively 1 and
0.9 and assume that these values do not change. Correct the internal efficiency for
imperfect similarity by means of Pfleiderer’s formula (7.14). Correct head and flow
rate according to the formulae of Casey and co-authors (7.10–7.12). Consider the
method of Casey et al. (7.9) as an alternative for the efficiency correction. Assume
R a = 20 μm for that. Determine an approximation for the rotor diameter and the rotor
width at outlet assuming a work coefficient 0.4 and a flow angle − 70° at the rotor
outlet. Note that the correction on efficiency is small with (7.14) and that it even
becomes smaller with (7.9).
A: Hm = 120.80 m, Q = 0.401 m
3
/s, d 2 ≈ 400 mm, b 2 ≈ 26.5 mm.
7.7.3. Consider the pump of the previous exercise once more. Take as approximations for rotor diameter and rotor width at outlet 400 mm and 25 mm. Determine for n = 1450 rpm the head and the flow rate for operation with a light oil with
ν = 200 10
- 6
m
2
/s in the operating point homologous to Q = 0.20 m
3
/s and H m = 30 m
with water. Correct for imperfect similitude with the formulae of Casey and coauthors (7.10–7.12). Use the formula of Haaland to describe the Moody diagram
(Sect. 2.3.1 in Chap. 2).
A: H m = 28.83 m, Q = 0.196 m
3
/s.
279
We take a tongue clearance of about 10 % of the radius: about 22.5 mm. The
opening angle of the scroll is calculated for the first sector of 30° with b 3 = b 2 , because the flow expansion by the width leap cannot be sudden. This way, the tongue
clearance becomes 20 mm. For the rest of the scroll, the full width is used, but the
scroll is made somewhat larger than calculated with the logarithmic spiral.
The width leap to the scroll becomes quite large: 40 mm/150 mm. Common
practice is then to widen the rotor at the outlet, for instance to 60 mm. Then, the
strongest possible deceleration is realised in the rotor, but at rotor outlet the flow is
then separated. We may accept that a velocity reduction of 0.7 is possible between
the inlet flow just after rotor entrance and the outlet flow just before rotor exit,
so before slip. Enlargement of the rotor width at outlet is commonly used with
fans with slightly backward leaning blades or forward curved blades. For forward
curved blades, typically the outlet width is taken equal to the inlet width, because
the diameter ratio is near to unity (see Fig. 3.19). Figures 7.21 and 7.22 show the
geometry. A final calculation of the performance of the fan can now be made (see
Exercise 7.7.6).
7.7 Exercises
7.7.1. Reason that, with a radial fan or pump, the similitude laws with a change
of rotational speed, Q ~ n, ∆p 0 ~ n
2
, stay valid in the presence of incidence at the rotor and the scroll inlets.
7.7.2. Q = 0.20 m
3
/s, H m = 30 m, η global = 0.75 are measured at a pump with
n = 1450 rpm. The fluid is water with ν = 10
−6
m
2
/s. Determine flow rate, head and
efficiency of the pump when doubling the rotational speed for a similar flow. Assume a volumetric efficiency and a mechanical efficiency of respectively 1 and
0.9 and assume that these values do not change. Correct the internal efficiency for
imperfect similarity by means of Pfleiderer’s formula (7.14). Correct head and flow
rate according to the formulae of Casey and co-authors (7.10–7.12). Consider the
method of Casey et al. (7.9) as an alternative for the efficiency correction. Assume
R a = 20 μm for that. Determine an approximation for the rotor diameter and the rotor
width at outlet assuming a work coefficient 0.4 and a flow angle − 70° at the rotor
outlet. Note that the correction on efficiency is small with (7.14) and that it even
becomes smaller with (7.9).
A: Hm = 120.80 m, Q = 0.401 m
3
/s, d 2 ≈ 400 mm, b 2 ≈ 26.5 mm.
7.7.3. Consider the pump of the previous exercise once more. Take as approximations for rotor diameter and rotor width at outlet 400 mm and 25 mm. Determine for n = 1450 rpm the head and the flow rate for operation with a light oil with
ν = 200 10
- 6
m
2
/s in the operating point homologous to Q = 0.20 m
3
/s and H m = 30 m
with water. Correct for imperfect similitude with the formulae of Casey and coauthors (7.10–7.12). Use the formula of Haaland to describe the Moody diagram
(Sect. 2.3.1 in Chap. 2).
A: H m = 28.83 m, Q = 0.196 m
3
/s.