CIVE240001 Fluid Mechanics

Section 2: Pumps

1. Some General Notes about Fluid Machinery Fluid machines either take energy from a fluid and convert it into mechanical energy or vice versa. Machines which take energy from a fluid are called turbines and machines which give energy to a fluid are called pumps and fans. From a theoretical viewpoint there is no difference between the two, in practice there is also a great deal of similarity. 1.1 Types of pumps and turbines Both pumps and turbines can be split into two distinct groups, positive displacement and rotodynamic 1.1.1 Positive displacement

• reciprocating pumps - often used for temporary site drainage • reciprocating engines - gas, petrol, diesel • gear pumps – two intermeshing gear wheels, often used for the oil pump

in reciprocating engines • rotor/stator pumps – essentially a double helix stator with a single helix

rotor inside it – used for pumping food, concrete, sewage and many other fluids

The behaviour of such pumps is not readily amenable to analytical treatment by the methods of fluid mechanics.

1.1.2. Rotodynamic Centrifugal, mixed flow and axial flow pumps

• centrifugal, mixed flow and axial flow pumps – centrifugal pumps are used for high head, low flow situations and axial flow pumps for low head high flow e.g. land drainage

• radial, mixed flow and axial flow turbines - radial flow turbines, often referred to as Francis turbines are used for high head hydroelectric sites and axial flow turbines for low head sites e.g. tidal barrage schemes

• Pelton wheels - for very high heads, essentially one or more jets and a wheel with buckets on, called an impulse turbine, the equivalent for a (highly inefficient) pump would be a rotating wheel with buckets attached

2

• Banki and Turgo wheels - other less common forms of high head impulse turbines

These notes will concentrate on rotodynamic pumps with the occasional reference to turbines. Before any analysis can be undertaken a relationship has to be developed between torque, power and the normal hydraulic parameters of head and flow. The next section does this for a turbine, by reversing the signs it applies to a pump. 1.1.3 Why should Civil Engineers study pumps? The heading to this sub section is a perfectly reasonable question to ask since normally we are not involved in their manufacture, we are simply users of them. There are several reasons why it is useful for us to have some idea of how pumps work and the different types of pumps available, the more important of these are as follows:-

• matching pumps to pipelines i.e. which is the most suitable size and type of pump for any given pipeline

• to understand the relationship between head and flow in a pump • to determine the power requirements of pumps

2. Flow through a rotating curved passage 2.1 Introduction All rotodynamic machines make use of the effect that occurs when fluid passes through a rotating curved passage. For turbines energy is transferred from the fluid to the rotating curved passage and for pumps the energy is transferred from the passage to the fluid. The moving part of a turbine is called a runner1, this consists of two concentric discs with curved plates between them rather like the ventilated discs used for vehicle brakes except that the plates are not usually flat but close together at the outer periphery and more widely spaced at the centre. In the case of a turbine fluid enters the runner around the outer periphery then flows across the curved blades between the two plates and leaves through a hole at the centre of the runner, this is called the eye. The runner is rotating with an angular velocity Ω, as the fluid flows over the curved blades thus angular momentum is transferred from the fluid to the runner2. Symbols used in analysis

u1 peripheral velocity at outer tip of passage u2 peripheral velocity at inner tip of passage V1 absolute velocity of fluid at inlet

1 For a pump the equivalent name is impeller. 2 In a pump the fluid flows from the eye to the outer periphery and angular momentum is given to the fluid by the impeller.

3

V2 absolute velocity of fluid at outlet Vf1 radial component of V1 Vf2 radial component of V2 Vw1 tangential component of V1 Vw2 tangential component of V2

the subscript f denotes the flow velocity and w the whirl velocity. 2.2 Development of an expression relating power to flow Since the runner is rotating about a shaft then only the force in the circumferential direction performs useful work, hence we need to find the change in angular momentum3 in this direction, changes of momentum in the radial direction have no effect because they do not generate a moment about the axis of rotation of the runner. It can be assumed that the conditions at entry and exit from the runner are uniform in both magnitude and direction, this is tantamount to saying that angular momentum is constant around the periphery of the runner. Figure 1 shows part of a runner, the fluid enters with a velocity V1 then flows through the runner and leaves with a velocity V2, in the process of passing through the runner energy is transferred from the fluid to the runner, most of this performs useful work, a small amount is lost in friction. Any particle of mass δm passing through the runner has a momentum tangential to the shaft of

rwVmδ

3 Angular momentum is defined as the moment of momentum.

4

Figure 1 where

rwV is the radial component of the absolute velocity vector at any radius r hence the angular momentum (moment of momentum) can be written

rVmrwδ

At entry if the velocity and direction of flow are uniform then the angular momentum per unit time is 11

rVQ wρ , similarly at exit the angular momentum per unit time is

22rVQ wρ , therefore the rate of increase of angular momentum of the fluid is

( )1122

rVrVQ ww −ρ

and this equals the torque4, T, exerted on the fluid by the runner. Therefore the torque exerted on the runner by the fluid is

( )21 21 rVrVQT ww −= ρ this is called Euler’s equation. Note that this is a momentum equation therefore is valid regardless of the path taken by the fluid particles. Civil Engineers are much more interested in power than torque; for turbines, we need to know how much power a potential hydro-electric site will provide and for pumps we need to know the power consumption so that we can determine the size of the pump and the operating costs. Power, P, is the rate of doing work = work done/s = TΩ where Ω is the angular velocity of the runner or impeller, we can write

( )

( )2211

2211

uVuVQPurbut

rVrVQTP

ww

ww

−=∴=Ω

Ω−Ω=Ω=

ρ

ρ

Therefore the work done per unit weight of fluid is

( )21 21uVuV

QgQ

ww −ρρ

( )21 21

1 uVuVg ww −= (2.1)

this has the units of head and is the form in which we generally use it. It is a fundamental equation which will be used throughout our analysis. 3. Pumps 3.1. Centrifugal pumps

4 Torque about a given axis is defined as the rate of change of angular momentum about that axis.

5

(a) (b)

Figure 2 Consider Figure 2 which shows a simple centrifugal pump. Figure 2(a) shows a cross section along the line of the drive shaft and Figure 2(b) a cross section through the impeller. The flow path through the pump can be most easily described by considering Figure 2(a), the fluid enters the eye of the impeller with no whirl velocity, in a pure centrifugal pump it is immediately turned through 900 and enters the impeller. As it flows through the impeller it is given energy, after it leaves the impeller and enters the volute (casing of the pump), the velocity (hence kinetic energy) is very high. For the fluid to flow along the pipeline we need a high pressure but do not want a high velocity because this would result in a high head loss, also it would be dangerous since it may give rise to significant water hammer pressures. Hence we need to convert the high kinetic energy to potential energy with a minimum of energy loss, this is achieved in the volute. This flow pattern can be summarized as follows

• fluid enters the impeller through the eye with no whirl (tangential) velocity

• as the fluid flows through the impeller it receives energy and is discharged into the casing of the pump with a high velocity hence high kinetic energy

• the velocity of the fluid is too high for the pipeline therefore the volute has to reduce the velocity and hence convert the kinetic energy into potential energy

• as the fluid flows round the volute and into the pipeline the velocity has to remain constant to minimise energy losses and, incidentally, satisfy the assumptions made in deriving Euler’s equation

This type of pump results in a high head and a low flow. If it is necessary to increase the flow then either the area or the velocity has to be increased. Any significant increase in the velocity of flow would result in an unacceptably high loss of energy hence the design has to be altered to increase the area of flow. This is done by widening the volute and the impeller particularly at the inlet. By doing this the fluid will enter the impeller and be receiving energy before it is flowing entirely in the radial direction i.e. the flow will have a component in the axial direction within the impeller, this is called a mixed flow pump, it delivers a higher flow and a lower head than a purely radial flow pump. If this design is taken to the limit then the flow becomes entirely axial and will produce high flow volumes and a low head, in practice the impeller becomes a propeller within the pipe and the pump is referred to as axial. 3.1.1.Volutes Clearly the design of the volute is important since a poorly designed one would not efficiently convert kinetic to potential energy resulting in a pump of low efficiency. Volutes will be considered in more detail later in this section.

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3.1.2. Variations on a simple centrifugal pump There are many variations on the simple type of centrifugal pump, for example if it is necessary to increase the flow for the same head then it is possible to have double suction machines where the fluid enters from both sides of the eye. The impeller then looks like two single impellers back to back, this doubles the flow at the same head, it also reduces the thrust on the bearings because of its symmetrical design. If it is required to increase the head for the same flow then it is possible to have multi stage machines with several impellers driven on one shaft, the outlet from one becomes the inlet to the next and so on. 4. Simplified analysis of pumps 4.1 Introduction Consider Figure 3 which shows a small part of the impeller complete with vector triangles of the flow pattern at entry and exit. Firstly consider the situation at entry,

V2

Vw2

R2

Vf2

U2

V1 Vt1

R1

U1

=

ß

Q

Figure 3

unlike turbines it is not usual for pumps to have guide vanes at entry therefore the fluid enters the impeller with no whirl velocity hence the absolute velocity V1 is equal to the flow velocity

1fV and the inlet vector triangle under design conditions is as

shown. Consideration of this triangle leads to two important points

• since there is no whirl component at inlet velocity 11 fVV = (which is the

component along the radial, this means that the angle between the vector V1 and the vector u1 (the speed of the impeller tip at inlet) is a right angle

7

• in order to minimise energy loss the fluid should impinge on the blade tangentially, in terms of the vector triangle this means that the angle between u1 and R1 should equal the blade angle, α (defined as the angle between the tangent to the pitch circle and the leading edge of the impeller blade).

Figure 4(a) shows the fluid entering the runner tangentially, this is called shockless entry, and Figure 4(b) shows the fluid entering the impeller non-tangentially i.e. under non design conditions. When this happens the follow occur

• impact losses occur • boundary layer separation takes place • eddies arise which give rise to some back flow into the inlet pipe, this

causes the incoming flow to have some whirl velocity The result of this non tangential entry is a dramatic drop in the efficiency of the pump. 4.2 Analysis of centrifugal pump behaviour Consider the inlet triangle in Figure 3, under design conditions we can immediately write down

111

11

1

11

V and60

,tanBD

QVNDuuV

f ππα ====

where

D1 diameter of the impeller at inlet N speed of rotation in RPM Q flow through the pump B1 width of the impeller at inlet

Therefore we can write

( )

⎟⎟⎠

⎞⎜⎜⎝

⎛=∴

⎟⎟⎠

⎞⎜⎜⎝

⎛=

−

NBDQ

NDBDQ

12

12

1

1

11

60tan

60tan

πα

ππα

This defines the entry angle of the vane if 01=wV .

For the exit triangle three cases have to be considered as follows

(i) Forward facing blades β < π/2 Forward facing blades are ones which face in the same direction as the rotation as shown in Figure 5(a), the symbols have the same meaning as for the inlet triangle with the subscripts changed to 2

8

V2

Vt2

Vw2

U2

R2ß

Figure 5(a)

(ii) Radial blades β = π/2 This case is shown in Figure 5(b), the vector triangle is right angled

hence 22 22 and fw VRuV ==

V2 Vt2

Vw2U2

R2

=

=

= 90°ß

Figure 5(b)

(iii) Backward facing blades β > π/2 This is shown in Figure 5(c), for the time being all that needs be noted

from this triangle is that V2 is smaller than in the other two cases.

9

V2

Vt2

Vw2

U2

R2

ß

For analytical purposes it is easiest to consider the triangle resulting from the forward facing blades however the results obtained will apply to all the cases with no changes in the signs. Using equation 2.1 with

1wV set to zero and recalling that since we are now considering pumps the sign will change then we can say that for an inviscid fluid the head difference across the pump would be

guVw 22

from the triangle in Figure 5(a) we can write

( )βπ −−= cot22 2 fw VuV

βcot22 2 fw VuV +=∴ (4.1)

The head imposed on the fluid5 is the energy given to it g

uVw 22 less any losses, hi, in

travelling through the impeller. As the fluid leaves the impeller and enters the volute a relatively small amount of the total energy is potential (i.e. pressure) energy much of it is kinetic; this has to be converted to potential energy by the volute and diverging delivery pipe. However efficiently the volute converts the kinetic energy to potential there is still a head loss, hv. We can now write the energy conservation equation in the form

5 Remember that head is energy per unit weight of fluid.

10

gv

hhHg

uV pvi

w

2

222 +++=

where vp is the velocity of flow in the outlet pipe. In order to advance this analysis we must be able to evaluate the losses hi and hv, it is not possible to do this analytically therefore the same method will be used as for minor losses in pipes. We assume that the loss in the impeller is proportional to 2

2R since this is the velocity of flow relative to the impeller, similarly the loss in the volute is assumed to be proportional to 2

2V hence we can now write the energy equation as

gv

gV

kg

Rk

g

uVH p

viw

222

222

2222 −−−=

vp is very small compared to the other terms and can safely be ignored. We can write

2222 22 fw VVV +=

Using equation 4.1 we can now write

( )( )

( )[ ]g

VVuukVkVuuH

VV

R

VVuu

VVuuV

VVuV

ffvfif

ff

ff

ff

ff

2coseccot2coseccot22

cosecsin

also

coseccot2

cot1cot2

cot

222

22

222

22

222

222

222

22

222

22

22

222

22

2222

22

22

22

22

ββββ

ββ

ββ

ββ

β

++−−+=∴

==

++=

+++=∴

++=

Tidying this expression up yields

( ) ( ) ( )[ ]

22

22

222

22

2

22

and60

but

2cosec1cot22

BDQVDNu

gkkvkVuku

H

f

ivfvfv

ππ

ββ

==

+−−+−=

aswritten -re bely convenientcan equation thehence

22 QCQNBNAH −+=∴

where A, B and C are constants defined by the properties of the pump.

11

This equation demonstrates that the H – Q relationship for a pump is parabolic, examples of the shape of the curves are given in Figure 6 for different blade angles. It is clear that, unlike most of the equations encountered in fluid mechanics, this one has dimensions.

Figure 6 For water backward facing blades are usually preferred because the absolute velocity is lower therefore the velocity head in the volute is less resulting in a reduced head loss and a higher efficiency. They give a smaller head for a particular size and flow rate, this disadvantage is usually outweighed by the greater efficiency obtained. For liquids the usual objective is to increase the pressure substantially i.e. the volute is required to convert kinetic head to potential head. On the other hand fans are required to move large quantities of air with very little change in pressure hence forward facing blades may be preferred.

H - Q Curves for Pumps

0.00

10.00

20.00

30.00

40.00

50.00

60.00

0 0.02 0.04 0.06 0.08 0.1 0.12

Flow

Hea

d

Backward

Radial

Forward

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4.3 Manometric Efficiency Definition of manometric head – Manometric head is the difference in head that would be recorded on a manometer connected between the suction and delivery flanges of the pump. The ratio of the manometeric head, Hm to the Euler head is called the manometric efficiency, it can be written as

22

uVgH

w

mman =η

It represents the effectiveness of the pump in producing pressure from the energy given to the fluid by the impeller. The heads H and Hm are related by the expression

( ) hgVVHH sd

m +−

+=2

22

where Vd is the velocity of flow in the delivery pipe, Vs is the velocity of flow in the suction pipe and h is the difference in level between the inlet and outlet flanges of the pump 4.4 Design of the Volute The volute is the part of the pump where most of the energy losses occur. No additional torque is given to the fluid once it has left the impeller hence the angular momentum is constant6 hence we can write

constanta=rV w

The radial velocity (velocity of flow) does not vary around the circumference (or at least it should not); the combination of a uniform radial velocity with a free vortex gives a pattern of spiral streamlines which should be matched by the shape of the volute. This situation can only be achieved at design conditions – at all other conditions there are increased shock losses and variations of pressure hence variation of radial velocity around the impeller. This results in a rapid reduction in efficiency. 4.4.1 Types of Volute 4.4.1.1 Variable Velocity Volute The velocity in the volute starts at zero and increases to Vp. This is the cheapest and least efficient design of volute. 6 In practice this is not quite true since some friction loss occurs.

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4.4.1.2 Constant Velocity Volute The flow velocity in the volute is kept constant, the increasing flow is compensated for by linearly increasing the cross sectional area of the volute. The energy losses amount to approximately 25% and can be approximated by the expression

( )g

VVh

vwl 2

22−

=

where Vv is the velocity of flow in the volute 4.4.1.3 Whirlpool Volute The velocity decreases in the radial direction across the volute with aconsequent increase in pressure, the loss of energy is comparatively small. The decrease in velocity creates a free vortex (v = C/r) where C is the vortex constant. The free vortex converts high velocity low pressure fluid to low velocity high pressure fluid. 4.5 Speed to Commence Pumping When there is no flow through the pump the fluid moves as a forced vortex hence we can write

guH2

22=

But 60

22

NDu π= where D2 is the diameter of the impeller

Hence we can write

2

222

2

260

2x3600

DgH

N

gNDH

π

π

=∴

=

Where H is the static head through which the water must be lifted before flow can commence. This speed must be further increased to obtain the required delivery. 4.6 Axial and Mixed Flow Pumps In order to increase the flow in a centrifugal pump the width of the impeller has to be increased, if this was not done the velocities, hence friction losses, would become unacceptably high. By doing this the flow, as it passes through the impeller, is partially in the axial direction and partially in the radial direction hence the description of mixed flow. Mixed flow pumps usually operate at lower heads than pure axial flow pumps. When it is required to increase the flow still further the design is changed radically to give an axial flow pump which can discharge very high flows at low heads. An axial flow pump is rather like a ship’s propeller, it is the converse of

14

an axial flow (Kaplan) turbine and similar in appearance although a pump would not usually have adjustable blades. The analysis applied earlier to centrifugal pumps applies equally well to both mixed and axial flow pumps. In axial flow pumps an element of fluid enters and leaves the pump at the same radius hence u1 = u2 = u Therefore work done/unit weight of fluid is given by

( )12 ww VV

gu

−

There is usually no whirl velocity at inlet therefore 0

1=wV however

2wV varies with radius which could cause difficulties with the analysis , this is overcome by designing the impeller as a free vortex i.e. CrVw =

2. Now ru Ω= therefore we can write the

work done/unit weight of fluid as

gC

rC

gr Ω

=Ω

Hence the expression for work done/unit weight of fluid applies regardless of radius. This is not true close to the hub where the assumption of a free vortex will not be valid. 5. Matching a pump to a pipeline Matching a pump top a pipeline is a mixture of hydraulics, site engineering and economics. By considering the Darcy-Weisbach expression for head loss it is readily seen that a small increase in diameter will cause a significant reduction in head loss hence a reduction in pump size, this reduces both capital and running costs of the pump but may increase the capital and construction costs of the pipeline. A further restriction may be that the retention time of the fluid in the pipe may be limited, for example when pumping sewage, clearly this restricts the diameter of the pipe. When all these factors have been considered a pipe diameter is chosen which enables a friction head to be calculated, the total head is then determined by adding this to the static head, this is the head that the pipe has to overcome. Pumps come in a family of different sizes but all geometrically similar. For each pump in the series the manufacturer will supply the H – Q curve for a specific pump speed or more correctly the part of the curve around the point of maximum efficiency. The point where this intersects the H – Q curve for the pipeline gives the flow. This is shown graphically in Figures 7 and 8. In Figure 7 the operating point does not align itself anywhere near the point of maximum efficiency hence the pump will not operate efficiently and the operating costs will always be higher than they ought to be. In Figure 8 the operating point is reasonably well aligned with the point of maximum efficiency hence the cost of running the pump will be minimised.

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Matching Pump to Pipeline

0.00

10.00

20.00

30.00

40.00

50.00

60.00

0 0.02 0.04 0.06 0.08 0.1 0.12

Flow

Hea

d

0

10

20

30

40

50

60

70

80

90

η

Pump

Pipeline

η

Operating point

Figure 7 – An inefficient system

Matching a pump to a pipeline

0.00

10.00

20.00

30.00

40.00

50.00

60.00

0 0.02 0.04 0.06 0.08 0.1 0.12

Flow

Hea

d

0

10

20

30

40

50

60

70

Pump

Pipeline

Operating point

hstatic

η

η

Figure 8 – An efficient system

From these diagrams it is clear that the part of the H – Q that lies under the point of maximum efficiency is quite short thus the remainder of the curve is of little interest. Pump manufacturers use this property to put the efficient part of the H - Q for all the pumps in an homologous series onto one graph. An example7 of this is shown in Figure 9. The short length of curve is that part of the curve under the highest point of the efficiency curve, this enables the user, once the H – Q for the pipeline is known to read off the model number of the pump most suitable for the job.

7 This diagram is reproduced by kind permission of Weir Pumps Ltd., Glasgow.

16

Figure 9 If one pump cannot produce sufficient head then two or more pumps may be used in series; for the great majority of pipelines this would not be considered a good

17

arrangement, it would be better to specify a larger pump. However such an arrangement is frequently used in deep boreholes. If the pump cannot produce sufficient flow then two or more pumps are used in parallel. This arrangement is very common is water supply and sewerage pumping stations. In fact in these designs it is usual to have several pumps running in parallel, with the pumps cutting in sequentially as the flow increases. When pumps are running in parallel it is essential to fit reflux (non return) valves so that one pump does not drive another pump as a turbine. 6. Dynamic Similarity for Pumps Many pumps are too large to be tested in a laboratory therefore it is essential to be able to test small geometrically similar ones and then scale the results up to the required size. Like any other scaling problem in fluid mechanics this is achieved by scaling up dimensionless variables. 6.1 Geometric Similarity Geometric similarity is a prerequisite of dynamic similarity. Geometric similarity must be preserved for all the hydraulically important part of the pump, for example entrance and discharge passages, impeller and diffuser (if one is fitted). Machines which are geometrically similar in these respects from a homologous series (from the Greek homos – same and logos – ratio). 6.2 Dynamic similarity Dynamic similarity means a fixed ratio of forces, to achieve this kinematic similarity is also required i.e. a fixed ratio of velocities. In practice this means that the inlet and outlet velocity triangles must be geometrically similar. 6.3 Analysis The variables in the analysis are as follows:-

H difference of head across the machine (energy/unit weight. of fluid)

[L]

N rotational speed [T-1]

P power transferred from impeller to fluid [ML2T-3]

g gravitational force [LT-2]

ρ density of the fluid [ML-3]

µ viscosity of the fluid [ML-1T-1]

Q flow through the pump [L3T-1]

18

D diameter of the impeller [L]

k mean height of roughnesses in the machine [L]

Hence we can write

( ) 0,,,,,,,, =kDQgPNHf μρ

g is included because we need a definition of velocity head. Now we need to have parameters to describe the fluid, the size of the machine and the flow pattern. The obvious parameters would be ρ, D and N. However this would have the disadvantage that when they were combined with H then we would have a π group D/H which would not be of any use. A method of avoiding this is to combine g and H into one variable gH (energy per unit mass). Hence we can write

( ) 0,,,,,,, =kDQPNgHf μρ By applying Buckingham’s second π theorem it is clear that there are 5 π groups hence we can write

( ) 0,,,, 54321 =πππππf

Using ρ, D and N as the recurring variables we can write

555

444

3

222

111

5

4

333

2

1

cba

cba

cba

cba

cba

ND

ND

ND

PND

gHND

ρπ

ρπ

μρπ

ρπ

ρπ

=

=

=

=

=

By expressing these π groups as dimensionless equations we can write

LLMTL

TLLMTL

TMLLMTL

TMLLMTL

TLLMTL

ccba

ccba

ccba

ccba

ccba

5555

4444

3333

2222

1111

35

1334

1133

3232

2231

−−

−−−

−−−−

−−−

−−−

=

=

=

=

=

π

π

π

π

π

Considering 1π we can write

19

( )( )( )

( )headSpecificDN

gHacM

bbTcaL

221

1

1

11

11

20

202023

=∴

−=∴=

−=∴=−−=+−

π

Similarly we can write

( )

( )

Dk

NDQ

ND

ND

NDP

=

=

=

=

=

5

34

2

3

23

352

discharge Specific

No. Reynolds of form a

hence inverted is hisNormally t

π

π

ρρπ

ρμπ

ρπ

Consider the specific capacity ND is a typical velocity of the impeller8 and 2DQ is a

typical velocity of the fluid hence if the specific capacities of the model and prototype machines are equal then the ratios of all the velocities are equal therefore kinematic similarity is achieved i.e. the velocity vector triangles is similar. Consider the Reynolds number, clearly for complete similarity this will have to be equal in both model and prototype. If we assume that the density and viscosity of both the prototype and model fluids are the same or, at least similar then

( ) ( )prototype2

model2 NDND = . Since the ratio

model

prototype

DD

may be quite large and the square

of this ratio even larger the speed of rotation of the model has to be very much larger than that of the prototype, in practice this is usually very hard to achieve; this is called scale effect. Fortunately the variation of efficiency and other non-dimensional parameters with the Reynolds number is not great therefore the requirement that the Reynolds numbers be equal is not usually necessary i.e. scale effect can usually be neglected. For practical purposes dynamic similarity is satisfied when

ρ35223 and,NDP

DNgH

NDQ are equal in both model and prototype.

Dk can safely be

ignored because great efforts are made to make the surface of the impeller as smooth as possible. 8 This can readily be seen by recalling that

60NDu π

=

20

6.4 Unit speed, quantity and power A pump which is designed to operate under one set of conditions may be required to operate under a set of different conditions, for example when the speed of rotation of the power supply is changed or when it is being moved from one pipeline to another. The methodology for doing this is to scale the values for N,P and Q to the values for unit head (H = 1 metre). This is a similar process to the simple unitary method taught in primary schools to work out the cost of, for example, apples although in this case it is not linear. First consider the specific capacity

NQNQ

DNQ

NDQ

uu

u

u

=∴

= 33

where Qu is the flow when the head generated is 1 metre and Nu is the corresponding speed of rotation, D is the same since it is the same machine. Similarly for specific head

quantity)unit (

and

speed)(unit

definitionby 1but

2222

HQQ

HNN

HDN

gHDN

gH

u

u

u

u

u

=

=∴

=

=

Finally consider power

23

3

3

5353

H

PP

HNNP

NNPP

DNP

DNP

u

uu

u

u

=∴

⎟⎠⎞

⎜⎝⎛=

⎟⎠⎞

⎜⎝⎛=∴

=ρρ

Since ooQHgP ηηηρ == u clear that isit then , in other words these unitary transformations ensure that the machine continues to run at its maximum overall efficiency oη .

21

In summary we can now say that once an H – Q curve has been obtained for a small machine then the results can not only be scaled to any machine within the homologous series but also, for a particular machine, scaled to a different set of operating conditions. 6.5 Specific Speed For a pump the important design variables are N, H and Q, it would be very useful if all three could be combined into one dimensionless variable which uniquely described the type of pump necessary to satisfy the design condition for a particular application without any regard to the specific size of the pump. In order to see the relevance of this consider a specific pump and recall that it is a member of a homologous series which are likely to vary in size from quite small ones to very large ones. They will be all of the same type, either centrifugal, mixed or axial flow, however the particular values of H and Q at maximum efficiency may be very different. In other words it is not possible to look at the actual physical values of H and Q and know whether we need a centrifugal, mixed or axial flow machine. The concept of specific speed allows this problem to be overcome. We need a dimensionless number which includes N, H and Q but does not include the size (measured by D). This is easily achieved by combining the specific head and the specific discharge; recall that all the π groups are dimensionless therefore we can multiply them together in any way we wish and still retain the dimensionless property. Thus we can write

( ) 23

2

23

22

3

gHQN

DNgHNDQ

=

⎟⎠⎞

⎜⎝⎛

which removes the size of the machine, D. This number is quite large therefore it is usual to take the square root

( ) 43

gH

QN

This is called Addison’s shape number for pumps and when plotted on a graph against efficiency allows a designer to know which type of pump to use for specific situation. Unfortunately engineers usually use the number without the g which gives it dimensions, in this form it is known as the specific speed. This has the serious disadvantage of requiring the user to be careful which units are used to calculate it; N is almost invariably in RPM, H is usually in meters or feet but Q can be in a wide variety of units, for example m3/s, m3/hour, ℓ/s, gallons/minute (gpm), cubic feet/s (ft3/s). Just to create even more confusion the gallons may be either Imperial or US. 7. Cavitation 7.1 Vapour Cavitation

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Vapour cavitation is the process that takes place when the pressure in a liquid falls below the vapour pressure at that temperature with the result that the liquid boils. This causes small bubbles of vapour to form which are carried along in the flow. When the flow enters a zone of higher pressure the bubbles suddenly collapse as the vapour turns back to liquid. The surrounding liquid rushes in from all sides and collides in the middle of the cavity giving rise to very high local pressures. If the bubbles are in contact or near a solid surface then intense pitting of the surface can result which will ultimately lead to failure from fatigue. The sequence of formation and collapse of the bubbles may take place at a very high frequency. Cavitation causes vibration and noise (imagine the sound of gravel flowing through the pump), the flow becomes very disturbed and the efficiency falls rapidly. Look at the axial flow impeller (the one painted a silver colour) in Lecture Theatre A, the surface is pitted by cavitation, some text books contain photographs of ships propellers which are severely damaged by cavitation. 7.2 Air Cavitation Air cavitation is similar to vapour cavitation although not usually as severe, it is simply air coming out of solution when the pressure drops, it has a similar effect of the efficiency of the machine.

I.M. Goodwill, March, 2006.