3 Pump hydraulics and physical concepts
3.1 One-dimensional calculation with velocity triangles
ble). Likewise, turbine operation of a pump (Chap. 12) is calculated in the direc- tion of the flow, but follows the course of diminishing pressure. The numbering of the calculation stations is retained: the turbine calculation sequence therefore is from station 6 to 1. The following shall be determined:
• Circumferential speeds: u = ω×r = π×d×n/60 = π×d×n(s) (n in rpm; n(s) in s)
• Absolute velocities: c
• Relative velocities: w
• Angle in the absolute reference frame: Į
• Angle in the relative reference frame (rotating reference frame): β
• Blade or vane angles on a component: subscript B. Unless otherwise specified this means the camber angle of the impeller blades, the diffuser vanes, the re- turn vanes or the volute cutwater.
• Passage width between the blades or vanes: a (shortest distance, Table 0.2)
• Width in the meridional section: b
• Meridional velocity components: subscript m, where: cm = QLa/(fq×π×d×b); note that wm = cm applies.
• Circumferential velocity components: subscript u, where u = cu + wu
• cu is counted as positive when acting in the direction of u, negative when op- posite to the u-direction
• wu is counted as positive if acting opposite to the direction of u
• Outer, mean and inner streamlines are described by the subscripts a, m, i; for example c1m,m, β1B,a. The subscript m always refers to the streamline which starts or ends on the inlet or outlet diameter calculated as the geometric mean value (d1m and d2m on the impeller and d3m and d4m on the diffuser). The one- dimensional calculation assumes representative conditions on the mean stream- lines defined in this way. At low specific speeds frequently only the outer and inner streamlines are considered while three, five or more streamlines are ad- visable at high specific speeds.
• When the fluid flow enters or leaves a cascade there is a blockage effect due to the finite blade thickness. The flow velocity in the meridional section under- goes (arithmetically) a sudden change at these points. It is, therefore, possible to define velocities with and without blockage at any calculation station; quan- tities with blockage are described by a raised stroke, e.g. c1m'.
• Two velocities must be distinguished at the inlet of a blading: (1) the velocity vector, which is calculated from the velocity triangle (vector diagram); (2) the mean velocity resulting from the continuity equation. This is given the sub- script q to avoid confusion; hence cq = Q/A and wq = Q/A, where A is the local cross section under consideration.
• The flow rate of the impeller QLa consists of the useful flow rate Q plus leakage through the annular seal at the impeller inlet Qsp and the balance flow QE, re- sulting in QLa = Q + Qsp + QE. In addition there may be an intermediate take- off in the case of multistage pumps.
The flow around a blade is considered from the view of an “observer sitting in the blade”. Consequently, the relative velocities are relevant for the impeller,
3.1 One-dimensional calculation with velocity triangles 71
while the absolute velocities are used for the volute or diffuser calculation. The relationship between circumferential impeller speed u, relative velocity w and ab- solute velocity c is obtained from the rules of vector addition which can be illus- trated as velocity triangles (refer also to Chap. 1.1). The equations describing the geometry of triangles therefore apply to the calculation of all velocities, their components in circumferential or meridional direction and the angles α and β. Inlet triangle: Consider the velocity relationships on the impeller inlet shown in Table 3.1. The meridional velocity immediately in front of the impeller blade leading edges is c1m = QLa/A1, where A1 is calculated in accordance with the posi- tion of the leading edge, Eq. (T3.1.2). Immediately after the leading edges the me- ridional velocity is increased to c1m' = τ1×c1m due to the blade blockage. The latter is obtained from τ1 = t1/{t1-e1/(sinβ1BsinλLa)}, Eq. (T3.1.7), using the sketch in Table 3.1 (with the pitch of the blades t1 = ʌd1/zLa).1 If the blades are not perpen- dicular to the front shroud (λLa ≠ 90°), additional fluid is displaced, since the blade blockage obviously becomes larger than with λLa = 90°. This influence is taken into account by λLa as shown by the sketch in Table 0.1. The circumferen- tial components of the absolute or relative velocity are not affected by the block- age, as follows from the conservation of angular momentum.
The fluid flow to the impeller is mostly axial (α1 = 90°). The circumferential component of the absolute inflow velocity is therefore c1u = 0. However, if a de- vice for pre-rotation control is installed or when the inlet casing or the return vanes generate an inflow with α1≠ 90°, the circumferential component is obtained from Eq. (T3.1.3). Figure 3.1 shows the inlet triangles for non-swirling inflow α1 = 90°, pre-rotation α1 < 90° (“pre-swirl”) and counter-rotation α1 > 90°. It can be seen that the approach flow angle β1 of the blades is increased by pre-swirl and reduced by counter-swirl. As a result of the blade blockage c1m grows to c1m'so that the approach flow angle increases from β1 to β1'.
The difference between blade angle β1B and flow angle β1' is known as inci- dence: i'1 = β1B - β1'. If the incidence is zero, the blade has only a displacement ef- fect on the flow; local excess velocities are correspondingly low. They grow with increasing incidence until the flow separates, since incident flow generates a cir- culation around the leading edge at i1≠ 0, Fig. 3.1d. The flow angles without and with blockage are calculated from Eqs. (T3.1.6 and 3.1.8).
With a certain flow rate (i.e. a specific c1m') blade and flow angles are identical (β1B = β1') and the incidence becomes zero. This flow situation, called “shockless entry”, is calculated from the condition tan β1' = tan β1B from Eq. (T3.1.10), where ϕ1,SF = c1m'/u1 is the flow coefficient introduced in Chap. 3.4. If the approach flow angle drops below the blade angle (i1 > 0), the stagnation point is situated on the pressure surface of the blade. If the pump flow rate exceeds the value of the
1 With profiled blades, the effect of the blockage may not be easily defined in certain con- ditions. Generally, e1 must be selected where the cross section is narrowest relative to the next blade. The effect of the angle λ (if any) must be considered iteratively.
No swirl α1 = 90° (c1u = 0)
a) b)
c) d)
Pressure surface
Suction surface
Counter rotation: α1 > 90° (c1u < 0)
Pre-rotation α1 < 90° (c1u > 0) α1
w1u
β1' β1
w1'
i1' c1'
β1B
c1u
-c1u
c1m' c1m
w1' w1
β1B α1
β1' β1
i1'
β1 β1' β1B
w1' w1
α1
c1mc1m' i1'
u1
β1B
i1' β1' w1' u1
u1
α1'
Fig. 3.1. Velocity triangles at the impeller inlet
shockless entry, the incidence is negative and the stagnation point is located on the blade suction surface.
Outlet triangle: The velocity relationships at the impeller outlet are shown in Fig. 3.2a and Table 3.2. The meridional velocity downstream of the impeller is obtained from Eq. (T3.2.2). Blade blockage is still present immediately upstream of the impeller outlet and the velocity is correspondingly greater than downstream of the trailing edge: c2m' = c2m×τ2, Eq. (T3.2.3). Again, the blockage does not af- fect the circumferential component. The absolute velocity c2 and outflow angle α2
are relevant for the design of the diffusing elements, Eqs. (T3.2.10 and 3.2.13).