3 Pump hydraulics and physical concepts
3.7 Basic hydraulic calculations of collectors
At the impeller outlet the fluid has the circumferential velocity c2u = c2×cos α2 and the specific kinetic energy Ekin = ½c22. Up to moderate specific speeds, cos α2 is close to 1.0. Equation (T3.3.9) yields c2u/u2 = ½ψth for swirl-free inflow. The theoretical specific work is Yth = ½ψth×u22. Using these expressions we can derive the kinetic energy at the collector inlet Ekin/Yth = ψth/4 as a fraction of the energy transferred by the impeller. An acceptable efficiency can consequently be achieved only if the kinetic energy available at the impeller outlet is effectively decelerated in the diffusing elements. The collector design attempts to convert the greatest possible part of Ekin into static pressure.1
In Table 2.1 the following diffusing elements were introduced: volute casings, vaned or un-vaned diffusers or concentric annular chambers, all of which can be designed by relying on similar principles.
According to Newton's law of inertia, a mass retains its momentum if not pre- vented by external forces to do so. As a consequence, the fluid has the tendency to maintain its angular momentum (its “swirl”) ρ×Q×r2×c2u downstream of the impel- ler unless it is influenced by structures or wall friction. In an essentially vaneless casing the flow thus follows the law cu×r = c2u×r2 = constant (apart from friction effects). The circumferential component of the flow velocity in any element of the casing cross section is therefore calculated from cu = c2u×r2/r. The diffusing ele- ments have to be designed so that the flow field in the collector is compatible with the conservation of angular momentum. If the angular momentum is conserved, rotational symmetry around the impeller is achieved and interaction between the flows in the collector and impeller is minimized.
The diffusing elements are dimensioned for the flow rate QLe = Qopt + Qs3 + QE
where QE is the volumetric flow through a device for axial thrust balancing and Qs3 the leakage of the inter-stage seal.
Volute casing: Consider any cross section of a volute located at the circumferen- tial position ε°, Fig. 3.17. The fluid dQ = cu×b×dr = c2u×r2/r×b×dr flows through an area element dA = b×dr on the radius r in the cross section A. The flow rate through the entire cross section is obtained by integration Q(ε) = c2u×r2׳ b/r×dr.
From the start of the volute (point S in Fig. 3.17) to the cross section A, the impel- ler discharges the flow Q(ε) = Qoptε°/360° in the design point. In order to allow the fluid to flow according to cu×r = c2u×r2 = constant, cross section A must there- fore be designed in such a way that Eq. (3.14) is satisfied:
³ = ° ε
A z r
r 2u 2
opt o r c 360 dr Q r
b (3.14)
1 Exceptions are axial pumps with very high specific speeds, which under certain condi- tions, have to be designed without diffusing elements, Chap. 7.6.6.
3.7 Basic hydraulic calculations of collectors 103
r2 rz
A S
Section A
rz r dr
rA
α3Β
rz
b
Fig. 3.17. Volute casing
Experience confirms that the losses become approximately minimal if a volute casing is dimensioned so that the fluid at the design point is able to flow in accor- dance with the angular momentum conservation. This means that the cross sec- tions of the volute casing – and most importantly the end cross section or “throat area” – have to be selected according to Eq. (3.14). The throat area then follows from Eq. (3.15):
= ³
³ =
= A
z r Ler SP SP
2 2 2u
u 2
Le dr
r z b J with J
r c Q or dr r b c r z
Q (3.15)
zLe is the number of partial volutes forming the casing. Most frequently used are:
single volutes with ε = 360° wrap angle (i.e. zLe = 1) and double volutes with 180°
each and zLe = 2. Occasionally three or four volutes are used to reduce the size of the casing. Double volutes are also designed with wrap angles εsp < 180° where zLe = 360°/εsp must be used accordingly.
The swirl r2×c2u at the volute inlet is obtained from Eqs. (T3.2.7) or (T3.7.8). If a vaned or vaneless diffuser is installed upstream of the volute, its outlet swirl c4u×r4 must be inserted into Eq. (3.14) or (3.15).
The volute cutwater forms a camber angle α3B with the circumferential direc- tion which must be selected in accordance with the approach flow angle. It is ob- tained from the following equations that apply to both volutes and diffusers:
meridional component with blockage (Table 0.1),
for volutes set IJ3 = 1.0 3 3
3 m Le
3 d b
c Q π
= τ
′ (3.16)
circumferential component according to conserva-
tion of angular momentum (dz = d3): 3 u 2 2 u
3 d
c d
c = (3.17)
approach angle with blockage:
3u m 3 c3
tan c ′
′=
α (3.18)
vane inlet angle: α3B=α3′+i3 (3.19)
The incidence is selected in the range of i3 = ±3°.
It is advisable to shape the cutwater as an elliptical profile, which is less sensi- tive to changes in incidence (or pump flow rate). The profile can be applied either symmetrically as in Fig. 3.17 or be made asymmetric to obtain small incidences at partload. Any profiling must be made very short to keep stresses in the cutwater within acceptable levels. Not much profiling is possible with small pumps.
The diffuser which usually follows the actual volute must be designed accord- ing to Chap. 1.6. For hydraulic and design reasons, a distance is required between the impeller outlet and the cutwater. In the space thus created the fluid flows ac- cording to the laws of the vaneless diffuser, but the strong secondary flow created in the volute has an impact on the wall-near flow.
Diffuser: As shown by Fig. 3.18, a diffuser consists of zLe vanes forming triangu- lar inlet sections and closed channels (“diffusing channels” as per Table 0.2), which can be calculated according to Chap. 1.6. The flow through the triangular inlet section can be imagined to behave in a similar way as a partial-volute. The smaller the number of diffuser vanes, the more this is the case. When the number of diffuser vanes is large the flow does not truly follow the relationship cu×r = c2u×r2 = constant. The diffuser throat area at a3 cannot be designed strictly according to Eq. (3.15). Nevertheless Eq. (3.15) forms a physically sound basis to calculate the throat area, if an appropriate correction factor fa3 is applied, Chap.7.
For a given inlet width b3, the diffuser inlet area can thus be determined from Eq. (3.20).
¿¾
½
¯®
−
= 1
c d b z
Q exp 2 2 f d a
u 2 2 3 Le
Le 3 3
a
3 (3.20)
This relationship corresponds to the integral of Eq. (3.15) for a volute with rec- tangular cross section. As for the rest, the diffuser can be calculated according to similar principles as a volute: (1) Blade angle α3B according to Eqs. (3.16) to (3.19) where i3 = ±3°; (2) Diffusing channel according to Chap. 1.6; (3) Profiling as described above (for details see Chap. 7).
Vaneless diffuser: In a vaneless diffuser, depicted in Fig. 3.19, the flow is able to develop according to the angular momentum conservation and the pressure in-
3.7 Basic hydraulic calculations of collectors 105
creases in radial direction following Eq. (1.28). The meridional component is cal- culated from continuity as per Eq. (3.21).
b r
b r c b r 2
cm QLe = 3m 2 3
= π (3.21)
Setting cu = c2u×r2/r and neglecting friction, the flow angle is obtained from the following equation:
b tan b b c
b c c
tan c 3 3
u 2
3 m 3 u
m = = α
=
α (3.22)
Equation Eq. (3.22) shows that the flow angle remains constant in a vaneless dif- fuser of constant width as long as friction is neglected.
In reality, the circumferential component cu drops due to wall friction in com- parison with the angular momentum conservation employed without shear stresses. As a result the flow angle grows with increasing radius. According to [B.1] this follows from the relationship:
t3
d3
r3
a4 a3
α3B
Fig. 3.18. Vaned diffuser
r cm dr cu
c α b4
b3
b3
r2
r4
a) b) c)
Fig. 3.19. Vaneless diffuser. a parallel walls; b conical walls; c velocity vectors
1 rr 3 3
2 f 2
u 2
u 1
tan b
r 1 c r c
r c
2
−
¿¾
½
¯®
¸
¹·
¨©
§ −
+ α
= (3.23)
The flow angle in viscous flow can be deduced from Eq. (3.23) leading to:
¸¸¹
¨¨ ·
©
§ −
+ α
=
α 1
r r b c r tan tan
2 4 3 f 2 3
4 (3.24)
The meridional velocity results from continuity and is therefore not affected by the friction.
The friction losses in a vaneless diffuser of constant width can be determined approximately as follows: In a ring element (as shown in Fig. 3.19c) the velocity is c = cu/cos α = c2u×r2/(r×cos α) and the wall shear stress is τ = ½ρ×cf×c2. Thus the power dPd = c×dF = ½ρ×cf×c3×dA is dissipated, Chap. 1.5. Integrating dPd
from r2 to r4 with ζLR = 2×Pd/(ρ×Q×u22) the coefficient of the friction losses ζLR ac- cording to Eq. (T3.8.23) is obtained which comprises both sides of the vaneless diffuser.
Equation (T3.8.23) shows that the friction losses diminish with growing flow rate since the circumferential component cu drops. At the same time the flow an- gle α increases and the friction path length is shortened. For this reason the best efficiency point shifts towards a higher flow rate if a vaneless diffuser is em- ployed in place of a volute or a diffuser.
There is an optimum flow angle at the impeller outlet that has to be realized in order to place the efficiency optimum at the specified design flow rate. This means that the width b2 or b3 cannot be selected arbitrarily.
Since cu drops with increasing flow, the pressure recovery drops as well. Hence a vaneless diffuser is a stabilizing element – for as long as the flow does not sepa- rate. The smaller the outflow angle α2 from the impeller, the greater the losses be- come in the vaneless diffuser. Consequently, a vaneless diffuser yields poor effi- ciencies with low specific speeds. If the vaneless diffuser is designed so that its width decreases from the inside to the outside (b4 < b3), the flow angle according to Eq. (3.22) grows with increasing radius. Flow path length and losses diminish accordingly.
If the vaneless diffuser width b3 is designed larger than the impeller outlet width b2,shock losses occur due to deceleration of the meridional velocity from c2m to c3m which can be calculated according to Eq. (1.12): 2g×ΔH = (c2m' - c3m)2. The corresponding loss coefficient is shown as Eq. (T3.8.24) in Table 3.8; it in- cludes the effect of the blade blockage at the impeller outlet. This sudden decel- eration creates a minor pressure recovery that can be calculated from Eq. (1.11).
According to the principle of angular momentum conservation the circumferential component c2u is not affected by the expansion from b2 to b3.
Volute casings and diffusers behave like vaneless diffusers with regard to the expansion from b2 to b3.