4 Performance characteristics
4. The diffuser losses are obtained by applying Bernoulli’s Equation (1.7) be- tween impeller and diffuser outlet
4.2 Best efficiency point
100
nq = 250
0.5 1.0 1.5 0
0.5 1.0 1.5
2.0 nq = 100 80 70
60 50
40 30 20
nq = 250 nq = 20
40 60 80 H
Hopt
η ηopt
1.5 1.0
0.5 0
q* q*
1 2 3
P Popt
nq = 250
nq = 20 nq = 50 nq = 100 nq = 150
Fig. 4.11. Influence of specific speed on the shape of the pump characteristics
4.2 Best efficiency point
According to Chap. 3.7 the fluid tends to retain its angular momentum (its
“swirl”) ρ×Q×r2×c2u downstream of the impeller. Experience shows that the losses reach a minimum when the volute casing is designed so that the fluid at the design point flows according to the conservation of angular momentum cu×r = c2u×r2. This means that the volute cross sections, especially the end cross section or “throat”, have to be selected according to Eq. (3.15). If c2u from Eq. (T3.2.8) is substituted in Eq. (3.15), the collector characteristic is obtained as a linear relationship HLe = f(QLe) according to Table 4.1, Eq. (T4.1.1). The intersection of this straight line with the characteristic of the impeller according to Eq. (T3.3.7) or (T3.3.11) supplies the flow rate at which angular momentum conservation is satisfied in the volute throat area. Theoretically, the maximum efficiency should be expected at the resulting flow rate. Equation (T4.1.3) supplies this best efficiency point flow rate (in short the “best efficiency point” or BEP), Fig. 4.12. Equation (T4.1.3) can be derived from Eq. (3.15): Q = c2u×r2×Jsp with c2u from Eq. (T3.2.7) which yields Eq. (T4.1.3).
The volute/diffuser characteristic cannot be measured. It has meaning only near the best efficiency point. The concept is very useful for evaluating the shift of the best efficiency point due to a change of the volute or the diffuser throat area.
QA QB
V2 V1
C
A B
Im p e lle r 2 Im p e lle r 1 H
Ș
Fig. 4.12. Location of the best efficiency point
Table 4.1 Diffuser/volute characteristic and best efficiency point
Dimensional representation Dimensionless representation Eq.
E 2 s 1 s
La Q Q Q Q
Q = + + +
E 3 s
Le Q Q Q
Q = + +
°¿
°¾
½
°¯
°®
−
η
= 2
*m 1 u 1 sp 2 2 2 Le h 2
Le u
d c J r u
Q g
H u °¿
°¾
½
°¯
°®
π ϕ −
η
=
ψ 2
*m 1 u 1 Le sp
2 q h 2
Le u
d c Q Q J
f b 2 2
4.1.1
°¿
°¾
½
°¯
°®
−
β
− τ γ η
= u
d c tan u A f
Q g
H u
2
*m 1 u 1 2B 2 2 q
2 2 La
h 2
°¿
°¾
½
°¯
°®
−
β τ
− ϕ γ η
=
ψ u
d c 2 tan
2
*m 1 u 1 2B
2
h La 4.1.2
La Le sp
q 2 2B 2
2 2 q th v
, opt
Q Q J
f b 2 tan
u A Q f
+ π β τ
γ
= η
La Le sp
q 2 B 2 2
v th q
, opt
Q Q J
f b 2 tan
f + π β τ
γ
= η
ϕ 4.1.3
According to Fig. 4.12, a reduction of the best efficiency point flow rate from QB
to point C (with impeller 2) is achieved by a reduction of the volute throat area, which causes the volute characteristic to change from V1 to V2. Conversely, two different impellers (impellers 1 and 2 in Fig. 4.12) in a given casing would have their best efficiency points QA and QB on the same volute characteristic V1. If such calculations are performed for a given pump, it is recommended to substitute the actual slip factor calculated from the experiment according to Eq. (T3.2.9).
Table 4.1 and Eq. (3.15) apply to both diffusers and volute casings. Diffusers and volutes with rectangular cross sections can be calculated using Eq. (4.13):
¸¸¹
¨¨ ·
©
= §
3 3 3 3
Le
sp d +2e
a + 2 1 ln b z
J (4.13)
Experience shows that the measured best efficiency point is near the theoretical value calculated from Eq. (T4.1.3). For a volute with a circular cross section of radius r3q at the throat the integral is:
4.2 Best efficiency point 163
°¿
°¾
½
°¯
°®
+
− + π
= z
q 3 z
q z 3
Le r
2r r 1
1 r r z sp 2
J (4.13a)
According to the definition η = Pu/P = Pu/(Pu + ΣPv) = 1/(1 + ΣPv/Pu) the effi- ciency reaches its maximum when the ratio of all losses to the useful power ΣPv/Pu becomes minimal. In view of the different types of losses (Chaps. 3.6 and 3.7) and the interaction of the main flow with the flow in the impeller sidewall gaps (which influences disk friction and leakage losses) it is a “coincidence”
rather than a strictly derivable physical law that the best efficiency point can be roughly described by the intersection of the diffuser and the impeller characteris- tics. The finding that the diffuser characteristic describes the conditions quite well is also explained by the fact that the diffuser losses according to Fig. 4.7 have a distinct minimum near the BEP flow while the impeller losses exhibit an entirely different character (also refer to Figs. 4.5 to 4.7 in this regard).
With increasing hydraulic losses the best efficiency point shifts towards smaller flow rates. This finding is not reflected by the concept of the diffuser characteris- tic since the hydraulic efficiency does not enter in Eq. (T4.1.3). Equation (T4.1.3) merely supplies the flow rate where the angular momentum conservation is satis- fied in the throat of the collector at a given circumferential velocity c2u (i.e. c2u at Hopt,th). Equation (T.4.1.3) does not imply any assumptions on hydraulic losses.
With excessive hydraulic losses, such as occur when pumping highly viscous me- dia, the calculation according to Table 4.1 is unable to predict the best efficiency point. Notwithstanding this, the best efficiency points are still approximately situ- ated on the collector characteristic as shown in Chap.13.1.
The position of the efficiency optimum depends on the following parameters:
• With low specific speeds the BEP flow rate is largely determined by the collec- tor throat since the losses in the volute or diffuser dominate in comparison to the impeller losses.
• Above approximately nq > 75 the best efficiency point is increasingly influ- enced by the flow rate at shockless impeller entry since the impeller losses then gain in significance. But the deceleration losses in the diffuser have also a strong impact at q* < 1 as demonstrated by Fig. 4.7.
• Strictly speaking, the calculation applies only to the actual volute up to the throat since the hydraulic losses in the downstream diffuser/discharge nozzle are not taken into account. The greater these losses are, the more the actual best efficiency point shifts to the left of the theoretical value from Table 4.1. The same applies to diffusers: the calculation is valid only up to the diffuser throat.
Hence: if the volute or diffuser is followed by components causing major losses, a slightly over-dimensioned throat area is recommended so that the ki- netic energy at the diffuser inlet and the associated losses are reduced.
• Because of the short distance between the impeller outlet and the diffuser throat, the flow is scarcely able to develop according to the angular momentum conservation. Greater deviations between theory and experiment must therefore
be expected when calculating diffusers rather than volutes. The losses in the re- turn channels contribute to this. It should be remembered in this context that the conservation of angular momentum describes a flow without external forces. Such are however exerted by the presence of vanes and, in general, by any structures causing friction and flow deflection.
• A local expansion of the cross section near the volute throat has only a small effect on the position of the best efficiency point if the remainder of the volute is too narrow. Similarly it is not possible to shift the best efficiency point to- wards a smaller flow rate using a local constriction without adapting the re- mainder of the volute accordingly. (A local constriction would be comparable to a venturi nozzle which has only a minor effect on the throughput of a long pipe.) With low specific speeds, however, it is possible to shift the BEP by modifying the throat area and the subsequent diffuser.
Sometimes (a) different impellers are employed in a given casing or (b) a given impeller is used in modified or different casings. Both cases shall be explained with reference to the experiments depicted in Fig. 4.11. Curves 1 to 3 in Fig. 4.13 show measurements with three different impellers in the same volute casing which was designed for nq = 16. Curve 1 represents the baseline test with the impeller calculated for nq = 16. The impeller tested with Curve 2 was designed for nq = 21, while Curve 3 was measured with a nq = 13 impeller. The best efficiency points of the three different impellers are situated close to the intersection of the three Q-H- curves with the volute characteristic of nq = 16. The resulting best efficiency points are all very close to nq = 16, although the impellers were dimensioned for nq = 13, 16 and 21. These tests confirm that the casing largely determines the flow rate at BEP with small nq and that the impeller losses are of minor influence – even though the best efficiency point of the oversized impeller (nq = 21) is situ- ated a few percent to the right of the intersection with the volute characteristic, while the undersized impeller is found slightly to the left.
Figure 4.14 shows how a given impeller behaves in different casings. Both tests were done with the impeller designed for nq = 21. Curve 2 represents its perform- ance in a casing designed for nq = 16, while curve 4 was tested with the volute throat area increased by 37%. Because of differing hydraulic losses the character- istics 2 and 4 increasingly diverge at high flows – contrary to the idealized depic- tion in Fig. 4.12. This behavior cannot be expressed by the equations in Table 4.1 unless the various combinations are calculated with different hydraulic efficien- cies.
Except for very low partload operation, the power consumption is basically de- termined by the impeller alone. This follows from Euler’s equation. The maxi- mum efficiency obtained increases with the flow rate or the useful power Pu since the secondary losses (disk friction, mechanical and leakage losses) are largely in- dependent of the impeller and casing. Consequently, the ratio of the secondary losses to the power consumption diminishes with increasing Pu.
In order to expand the operation range of a given pump, attempts are some- times made to shift the best efficiency point by modifying the volute or diffuser throat area as shown by Fig. 4.14.