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Centrifugal Pump Handbook
Butterworth-Heinemann
Copyright © 2010 Elsevier Ltd.
All right reserved. ISBN: 978-0-08-094265-0
Chapter One
Physical Principles
1.1 ENERGY CONVERSION IN CENTRIFUGAL PUMPS
In contrast to displacement pumps, which generate pressure hydrostatically, energy is converted in centrifugal pumps by hydrodynamic means. A one-dimensional representation of the complex flow patterns in the impeller allows the energy transfer in the impeller to be computed from the fluid flow momentum theorem (Euler equation) with the aid of vector diagrams as follows (Fig. 1.1):
The torque acting on the impeller is defined as:
TLA = ρQLA(c2uR2 - couR1) (1)
With u = Rω, the energy transferred to the fluid from the impeller is defined as:
PLA = TLAω = ρQLA(c2uu2 - couu1) (2)
The power transferred per unit mass flow to the fluid being pumped is defined as the specific work YLA done by the impeller. This is derived from equation (2) as:
YLA = PLA/ρQLA = c2uu2 - couu1 (3)
The useful specific work Y delivered by the pump is less than that done by the impeller because of the losses in the intake, impeller and diffuser.
These losses are expressed in terms of hydraulic efficiency ηh:
Y = ηhYLA = ηh(c2uu2 - couu1) (4)
The specific work done thus depends only on the size and shape of the hydraulic components of the pump, the flow rate and the peripheral velocity. It is independent of the medium being pumped and of gravitational acceleration. Therefore any given pump will transfer the same amount of energy to completely different media such as air, water or mercury.
In order to use equation (4) to calculate the specific work done by the pump, the flow deflection characteristics of the impeller and all the flow losses must be known. However, these data can only be determined with sufficient precision by means of tests.
In all the above equations the actual velocities must be substituted.
If it were possible for the flow to follow the impeller vane contours precisely, a larger absolute tangential flow component c2u∞ would be obtained for a given impeller vane exit angle ß2 than with the actual flow c2u, which is not vane-congruent (see Fig. 1.1). The difference between c2u∞ and c2u is known as "slip". However, slip is not a loss that causes any increase in the pump's power consumption.
In order to examine the interrelationship between a pump and the pumping plant in which it is installed (Fig. 1.2), it is necessary to consider the energy equation (Bernoulli equation). In terms of energy per unit mass of pumped fluid it can be written:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
1.2 POWER, LOSSES AND EFFICIENCY
Equations (1) to (4) hold only as long as no part load recirculation occurs. The same assumption is implied below.
The impeller flow QLA generally comprises three components:
• the useful flow rate (at the pump discharge nozzle): Q;
• the leakage flow rate (through the impeller sealing rings): QL;
• the balancing flow rate (for balancing axial thrust): QE.
Taking into account the hydraulic losses in accordance with equation (4), the power transferred to the fluid by the impeller is defined as:
PLA = ρ(Q + QL + QE) Y/ηh (6)
The power input required at the pump drive shaft is larger than PLA because the following losses also have to be taken into account:
• disc friction losses PRR (impeller side discs, seals);
• mechanical losses Pm (bearings, seals);
• frictional losses PER in the balancing device (disc or piston).
The power input required at the pump drive shaft is calculated from:
P = ρ(Q + QL + QE) Y/ηh + PRR + Pm + PER (7)
If the volumetric efficiency ηv is:
ηv = Q/Q + QL + QE (8)
the power input required by the pump can be written as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
Pump efficiency is defined as the ratio of the useful hydraulic power PQ = ρ * Q * Y to the power input P at the pump drive shaft:
η = PQ/P = ρ * Q * Y/P = ρ * Q * g * H/P (10)
Pump efficiency can also be expressed in the form of individual efficiencies:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10a)
If the hydraulic losses and disc friction losses are combined to:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10b)
efficiency may also be expressed as:
η = ηhR * ηv * ηm (10c)
In the above equation hm represents the mechanical efficiency:
ηm = 1 - Pm/P (10d)
The internal efficiency incorporates all losses leading to heating of the pumping medium, such as:
• hydraulic losses;
• disc friction losses;
• leakage losses including balancing flow, if the latter is returned to the pump intake as is normally the case
ηi = ρ * Q * ρ * H/P - Pm (10e)
The power balance of the pump expressed as equation (7) is illustrated in Fig. 1.2a, where:
Pvh = hydraulic power losses = ρ * Q * g * H * (1/ηh - 1) PL = (QL + QE) * ρ * g * H/ηh
PER = frictional losses in the balancing device
PRec = hydraulic losses created by part load recirculation at the impeller inlet and/or outlet.
The pump efficiency depends on various factors such as the type and size of pump, rotational speed, viscosity (Reynolds number), hydraulic layout, surface finish and specific speed. Figure 1.3 shows the range of typical efficiencies. Maximum efficiencies are obtained in the range nq [congruent to] 40 to 60. At higher specific speeds efficiency starts to fall off due to an increasing proportion of hydraulic losses. With reduced specific speed there is above all a considerable increase in the proportion of disc friction losses, leakage and balancing flow losses.
For given values of density, flow rate and head the power input required by the pump can be calculated from:
P = ρ * Q * g * H/η
1.3 TOTAL HEAD OF THE PUMP
In pump design it is normal practice to define the useful mechanical energy transferred to the pumped medium in terms of unit weight under gravitational acceleration, instead of unit mass as in equation (3).
This value, which is expressed in units of length, is known as the head H:
H = Y/g (11)
By solving equation (5) for Y, the head is defined as:
H = Y/g = Pd - Ps/ρ * g + Zd - Zs + c2d - c2s/2g (12)
The head can be determined by measuring the static pressure in the suction and discharge nozzles and by computing the flow velocities in the suction and discharge nozzles.
In summary, the head is a unit of energy and corresponds to the total head between the suction and discharge nozzles as defined by the Bernoulli equation. It is independent of the properties of the pumped medium, and for a given pump depends only on the flow rate and the peripheral speed of the impeller. In contrast to this, the pressure difference created by the pump and power consumption are proportional to fluid density. However, for extremely viscous fluids (such as oils) the pumping characteristics depend to a certain extent on viscosity.
1.4 THE TOTAL HEAD REQUIRED BY A PUMPING PLANT
In order to ensure that the correct pump is selected for a given application, the total head HA required by the plant at design flow must be determined. This is again derived from the Bernoulli equation including all relevant head losses in the circuit (Fig. 1.2). Generally the pump draws from a reservoir whose surface is at a pressure pe and delivers to a second reservoir at a pressure pa (pa and/or pe can be at atmospheric pressure). Using the definitions shown in Fig. 1.2, the plant head requirement is then defined as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)
where:
Hvs = total head loss in the intake line
Hvd = total head loss in the discharge line
The head of the pump H must be selected to match the near HA required by the plant: HA = H in accordance with equations (12) and (13).
Normally it is not advisable to apply large safety factors when determining the head requirement, since this leads to oversizing of the pump. Oversizing not only increases investment cost but is unfavorable because the pump then operates at part load, causing unnecessarily high energy consumption and premature wear.
1.5 CAVITATION AND SUCTION BEHAVIOR 1.5.1 Physical principles
The flow path over the impeller vane leading edge, as with an aerofoil, is subject to local excess velocities and zones in which the static pressure is lower than that in the suction pipe. As soon as the pressure at any point falls below the saturation pressure corresponding to the temperature of the liquid, vapor bubbles are formed. If the liquid contains dissolved gases, these are separated out to an increasing extent with falling pressure. The vapor bubbles are entrained in the flow (although the bubble zone often appears stationary to the observer) and implode abruptly at the point where the local pressure again rises above the saturation pressure. These implosions can cause damage to the material (cavitation erosion), noise and vibrations.
The criteria used for determining the extent of cavitation can best be illustrated by a model test at constant speed. For this purpose the suction head is reduced in steps at constant flow rate while measuring the head and observing the impeller eye with a stroboscope. Figure 1.4 shows such a test, with the suction head (NPSH) represented by a dimensionless cavitation coefficient. Points 1 to 5 correspond to the following situations:
1. Cavitation inception, NPSHi: at this suction head the first bubbles can be observed on the impeller vanes (at higher suction heads no more cavitation occurs).
2. As the suction head is reduced, the bubble zone covers an increasing length of the impeller vane.
3. Start of head drop, NPSH0 (0% head drop): if the suction head is reduced below the value at point 3, the head will start to fall off.
4. 3% head drop, NPSH3, is a widely used cavitation criterion. This point is much easier to measure than the normally gradual onset of fall-off in the head, and is less subject to manufacturing tolerances.
5. Full cavitation, NPSHFC: at a certain suction head, the head falls off very steeply ("choking").
Figure 1.4 also shows the distribution of cavitation bubbles over the impeller vanes and the amount of material removed by cavitation erosion, as a function of cavitation coefficient.
Figure 1.4a shows typical curves for the visual cavitation inception NPSHinc, the NPSH required for 0%, 3% head drop and full cavitation (choking) as a function of flow. At shockless flow – where inlet flow angles match blade angles – NPSHinc shows a minimum. At flow rates above shockless, cavitation occurs on the pressure side of the blades, below shockless on the suction side. NPSHinc and NPSH0% often show a maximum at part load where the effects of inlet recirculation become effective. When the rotation of the liquid upstream of the impeller, induced by the recirculation, is prevented by ribs or similar structures this maximum can be suppressed and the NPSH will continue to increase with reduction in flow. NPSH3% and full cavitation usually do not exhibit a maximum at part load; if they do, this in an indication of an oversized impeller eye which may be required in some applications where operation at flow rates much above BEP is requested.
Figure 1.5 shows the cavitation bubble distribution in the entire impeller eye.
1.5.2 Net positive suction head (NPSH)
The difference between total head (static head plus dynamic head) and the vapor pressure head at the pump inlet is defined by DIN 24260 as the total suction head HH:
HH = pges - pD/ρ * g (14)
(Continues...)
Excerpted from Centrifugal Pump Handbook Copyright © 2010 by Elsevier Ltd. . Excerpted by permission of Butterworth-Heinemann. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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