Hydraulic Machines D.D.

5/25/2018 Hydraulic Machines D.D.

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Prof Dumitru DINU

HYDRAULIC MACHINES


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CONTENTS

1. INTRODUCTORY CONCEPTS2. HYDRAULIC PUMPS AND MOTORS

2.1Volumetric pumps2.1.1 Piston pump2.1.2 Pumps with radial pistons2.1.3 Pumps with blades2.1.4 Pumps with axial pistons2.1.5 Pumps with sprocket wheels2.1.6 Others types of volumetric pumps2.1.7 Characteristics of volumetric pumps

2.2 Hydrodynamic pumps2.2.1 Building and classification2.2.2 Turbo pump theory2.2.3 Turbo pumps in network2.2.4 Computation of centrifugal pumps2.2.5 Parallel and series connection of centrifugal

pumps

2.2.6 Suction of centrifugal pumps2.2.7 Axial pumps

2.3 Ejectors2.4 Volumetric hydraulic motors

2.4.1 Hydraulic cylinders2.4.2 Motors with radial pistons2.4.3 Motors with blades2.4.4 Motors with axial pistons2.4.5 Oscillating rotary motors

2.5 Turbines2.5.1 Peltons turbine2.5.2 Francis turbine2.5.3 Kaplans turbine

3. CONTROL AND AUXILIARY APPARATUS3.1 Control apparatus

3.1.1 Distribution apparatus3.1.2 Flow monitoring apparatus

3.1.3 Pressure monitoring apparatus

3.2 Auxiliary apparatus

3.2.1 Conduits

3.2.2 Filters

3.2.3 Tanks

3.2.4 Accumulators


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4. MEASURING APPARATUS4.1 Apparatus which determine the physical properties of fluids

4.1.2 Density measurement

4.1.3 Viscosity measurement

4.2 Measuring instruments for the level of liquids4.3 Pressure measuring instruments

4.3.1 Devices with liquids

4.3.2 Devices with elastic elements

4.3.3 Devices with transducers

4.4 Velocity measuring instruments

4.4.1 Pitot-Prandlt tube

4.4.2 Mechanical anemometers

4.4.3 Thermic anemometers

4.4.4 Optical measuring instruments

4.5 Flow measurement

4.5.1 Volumetric methods4.5.2 Methods based on throttling the stream

section of the fluids

4.5.3 Methods based on exploring

the velocity field in the flow section

4.5.4 Flowmeters with variable crossing

section

4.5.5 The ultrasound flowmeters

4.5.6 The electromagnetic flowmeter

4.5.7 Diluting methods

BIBLIOGRAPHY


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1. INTRODUCTORY CONCEPTS

Hydropneumatic systems transmit the mechanical energy from a leading

element to a led one by means of fluids.

Depending on the way the energy is transmitted, hydropneumatic systems may

be classified as follows:

- hydropneumatic systems of hydrostatic type;- hydropneumatic systems of hydrodynamic type;- hydropneumatic systems of sonic type.For the hydropneumatic systems of hydrostatic type, potential energy is sent by

means of fluids.

In Fig 1. such a system is schematically shown. The hydraulic generator GH, in

fact a volumetric pump, takes over the mechanical energy transmitted by the electrical

engine ME, turns it into potential hydraulic energy and transmits it by means of pipes

and other control, monitoring and adjusting devices to the hydraulic motor MH, which is

also of volumetric type. This, in its turn, converts the hydraulic energy into mechanical

energy used by the working equipment OL.

Systems of hydrodynamic type use the kinetic energy of the fluid. They are also

called turbo couplings or turbo transmissions. In figure 1.2 the scheme of a turbo

transmission is shown.

Fig.1.1.

The mechanical energy received from shaft 1 is turned into kinetic energy by the

hydrodynamic pump 2. In turbine 3, kinetic energy is turned into mechanical energy,

which is taken over by shaft 4.


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This transmission system has besides a coupling role, the role of variable

regulator. Invented in 1904 by professor Ftinger, turbo transmission was designed to

couple the shaft of a naval Diesel engine with the propeller, thus also accomplishing

substantial rotation decrease.

Systems of hydrodynamic type are high power systems.

Fig.1.2

Systems of sonic type are based on pressure wave propagation supplied by a mono or

threephase sonic generator (a hydraulic cylinder or three hydraulic cylinders at 120 o),

to a mono threephase receiver (motor).

By the alternate movement of the piston, an area of high pressure is generated,

which is sent along conduit 2 to the driving piston 3. (Fig.1.3.). So, as in the above-

mentioned systems, the mechanical energy is converted into hydraulic energy (this time

hydrosonic) and then back into mechanical energy.

Fig.1.3.

The transmission of energy is made under very high pressures 1,000 2,000

daN/ 2

cm . The distance between the two pistons must be a whole number multiple ofwavelength . If we note with the propagation speed of the pressure wave and with n

the rotation in rot/s of the crank, then n.

We must underline that sonication, i. e. energy transmission through conduits by

means of pressure waves, was founded as a science by Gh. Constantinescu, a Romanian

scientist.


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A hydropneumatic system represents an assembly of elements by means of

which we can produce and direct in a controllable manner the hydraulic and pneumatic

energy stored in a fluid with the help of a motor that turns it again into mechanical

energy.

To carry out the generating functions of hydraulic energy, its reconvertion into

mechanical energy, directing of the fluid agent, control and adjustment of theparameters, there are a large variety of hydraulic elements, which we shall study below.

Pumps and compressors represent the generating elements of hydraulic and

pneumatic energy.

Hydraulic and pneumatic motors convert the energy of the fluid into mechanical

energy. Within the control elements we distinguish the directing (distributing) elements,

flow adjusting ones (chokes), pressure regulators (valves).

Hydropneumatic systems contain auxiliary elements that in spite of their name

are of vital importance for the smooth working of the assembly, achieving the fluiddirecting (pipes), its filtering (filters), storing (tanks), sealing, vibration and flow shock

damping.

We mustnt forget the measuring equipment for the working parameters of the

installation.

In table 1.1. there are shown, according to STAS 7145 76, some of the

symbols for the elements the hydropneumatic transmission systems.


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Table 1.1

1. Pumps1.1.One-way discharging adjustable pump1.2.Two-way discharging adjustable pump

1.3.One-way discharging non adjustable pump

1.4.Two-way discharging non adjustable pump

2. Motors and pump-motor units

2.1.Circular irreversible hydrostatic motor withconstant capacity

2.2.Reversible hydrostatic motor with constantcapacity

2.3.Irreversible hydrostatic motor withadjustable capacity

2.4.Reversible hydrostatic motor withadjustable capacity

2.5.Non adjustable pump-motor with reversedirection


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2.6.Adjustable pump-motor with reverse fluiddirection

2.7.Linear motor (cylinder) with simpleoperating piston

2.8.Linear motor (cylinder) with doubleoperating piston with uni and bilateral rod

2.9. Linear motor (cylinder) differential3. Hydrostatic transmissions

3.1.Non adjustable hydrostatic transmissionwith one way rotation

3.2.Adjustable hydrostatic transmission pumpwith one way rotation

4. Hydrostatic distributorsDiscrete

4.1. With two channels and two positions4.2.With two channels and three positions4.3.With four channels and two positions4.4.With four channels and three positions


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Continuous (servo-distributors)

4.5.Mechanical and hydraulic distributors withone active edge

4.6.Electro hydraulic distributors5. Pressure valves

5.1.Normal closed

5.2.Normal open

5.3. With differential control

5.4.Safety valve with external operating control

5.5.Reducing valve6.Hydraulic resistors and flow regulators6.1.Fixed or adjustable hydraulic resistor

6.2.Regulator for constant flow (with fixedresistor) and normal open (two-way) valve

6.3.Fixed or adjustable chok


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6.4.Flow regulator with detour valve6.5.Adjustable resistor with manual control7.Auxiliary devices7.1. Hydraulic accumulator

7.2. Filter

7.3. Cooler

7.4. Manometer7.5.Flow -meter

Compared to mechanical or electrical systems, hydropneumatic systems have

a series of advantages:

- a lower weight and volume, compared to their power;- reliability and silent working;- important possibilities of automation, standardization, normalization,

modulation;

- continuous speed adjustment;- quick at normal working parameters;- stopping within a short time;- possibility to achieve forces and important momentum, as well as high

powers while control and operating can easily be done.


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Hydropneumatic systems have also some disadvantages:

- a high degree of accuracy of its components, which require complexmanufacture technology;

- possibilities to stop up inlets/outlets;- working under pressure with all the dangers implied;

a high price, because of high quality materials required to manufacture the

elements.


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2. HYDRAULIC PUMPS AND MOTORS

Pumps and hydraulic motors, i.e. hydraulic machines, are the basic elements of a

hydraulic system. Hydraulic machines turn the mechanical energy into a hydraulic oneand the other way round, being characterized by mechanical power Nm with its

components: force F, speed v or momentum M and rotation n as well as by hydraulic

power Nhwith its components flow Q and load H.

If we refer to the energetic conversion, we may group hydraulic machines by the

direction of this transformation into hydraulic generators (pumps) that convert

mechanical energy into hydraulic energy, and hydraulic motors, that convert hydraulic

energy into mechanical energy. There is also another category of hydraulic machines,

i.e. hydraulic transformers (couplings or clutches), that convert mechanical energy into

mechanical energy with other parameters, by means of hydraulic energy, or hydraulic

energy into hydraulic energy, by means of mechanical energy.

For generating hydraulic machines (MHG), if referring to their characteristic

power, the following conversion may be written:

Nm(M, n) MHG Nh(Q, H) (2.1-1)

There are generating hydraulic equipment for which the hydraulic power

(secondary) is also obtained from a hydraulic power (primary).

Nh(Qp, Hp)

MHG

Nh(Qs, Hs). (2.1-2)

For hydraulic motors (MHM) we have the transformation:

Nh(Q, H) MHM Nm(M, n). (2.1-3)

Hydraulic transformers are in fact a combination of generating and motive

hydraulic machines. By the manner in which the transformation takes place we can

distinguish between hydraulic equipment in a closed circuit (2.1-4) or in an open circuit

(2.1-5):

Nm(Mp, np) MHG Nh(Q, H)

MHM Nm(Ms, ns) (2.1-4)

Nh(Qp, Hp) MHM

Nm(M, n) MHG

Nh(Qs, Hs) (2.1-5)

We must underline the fact that there is a large variety of reversible hydraulic

equipment which can work both as a pump or as a motor.


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In a hydraulic machine the conversion of position, potential or kinetic energy

takes place. Referring to the type of load that is transformed we may classify hydraulic

equipment into volumetric equipment and turbo equipment.

Volumetric (hydrostatic) machines process potential pressure energy. Turbo

machines (hydrodynamic machines) process potential pressure energy and kinetic

energy. There is also another category of hydraulic machines now very rare, whichconvert the position potential energy, but which were widely spread in the past. They

are the hydraulic elevators (MHG) and water wheels (MHM). There are also motive

hydraulic motors that transform only the kinetic energy (Pelton activated turbines).

Volumetric hydraulic machines can be classified into:

- linear or alternative (with piston, plunger, with piston and membrane);- rotating (with radial or axial pistons, with blades, with sprocket wheels, with

screws).

Turbo equipment achieves the conversion of energy by hydrodynamic

interactivity between the rotor with profiled blades and the fluid. From the point of viewof the rotation they can be classified into pumps with a side channel, centrifugal pumps

and axial pumps. When presenting the hydraulic equipment we shall take into

consideration the two classifying criteria.

2.1. Volumetric pumps

Volumetric pumps convert mechanical energy into hydraulic energy, which is in

the form of potential pressure energy. This is achieved by means of closed spaces

between the fixed and the mobile parts of the pump, this process being a discontinuous

one. Volumetric pumps are, to a great extend, reversible, they can work as a pump or as

a motor, according to the liquid that comes in the body of the unit with under pressure

or over pressure.

The pressure of the volumetric pumps is generally high-250-300 bar, and the

flows extend to a very large scale 1-8,000 l/min. Their power can be up to 3,500 kW. In

the case of rotating volumetric pumps, rotations range from 3,000 to 5,000 rot/min, and

sometimes they can get up to 15,000 - 30,000 rot/min.

2.1.1. Piston pumps

The piston pump is a volumetric hydraulic pump, which achieves the pumping

effect by an alternate rectilinear movement of a piston inside a cylinder (fig.2.1.)


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Fig.2.1.

Piston pumps can be with simple or double effect (fig.2.1.) As it can be noticed

from their simple working principle, for the pumps with simple effect the flow range

has a strong discontinuous character (fig.2.3.), which is improved in the case of double

effect pumps. (fig.2.4.)

Fig.2.2.

We shall calculate the mean and instantaneous flows for a piston pump.

The relation gives the volume of discharged liquid for one stroke of the piston

(cylinders):

V =4

2Dh (2.1-6)

where D is the diameter of the piston, and h = 2 r, its stroke.

Noting with n the rotations in rot /min for the driving shaft, we can calculate the

mean flow:

Qmed =4

2D2 r

60

n. (2.1-7)

To compute the instantaneous flow, we shall first determine the speed of thepiston. Starting from the value of the distance

x = 1 cos + r cos = 1 cos- r cos (2.1-8)

and noticing that


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sin

r=

sin1

(2.1-9)

or else

sin = l

r

sin . (2.1-10)

so

cos = 22

2

sin1l

r , (2.1-11)

which being unfolded in this series and the first two terms retained (the error is

very much decreased because

1

ris sub-unitary) we may write:

cos 22

2

sin12

11

r , (2.1-12)

and we get:

x = lr cos - 22

sin2

1

l

r, (2.1-13)

and

v =

2sin

21sin

rr

dt

dx. (2.1-14)

The instantaneous flows will be:

Q =

2sin

21sin

44

22 rr

Dv

D. (2.1-15)


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Fig. 2.3

We define the pulsation coefficient of the flow as the ratio:

% = 100minmax

med

Q

QQ . (2.1-16)

Since maxQ obtained when =2

, and 0minQ , (fig.2.3), we shall get:

%314100

60

302

4

4%2

2

rD

rD

. (2.1-17)

For pumps with simple effect piston the flow pulsation is high. For this reason

the pumps are equipped with hover containers that are placed in the vicinity of theworking cylinder.

The pumps with double effect piston overflow in the returning area of the piston

with a lower flow. The instantaneous flow for the area 2, will be (fig.2.4):

Qx =

2sin

21sin

4

22r

rdD

. (2.1-18)

Fig.2.4


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Because curves Q and Qx intersect only on the abscissa axis, the pulsation

coefficient of the flow remains approximately the same as for simple effect pumps.

Their advantage, not negligible, is that they also overflow on the return stroke of the

piston.

The classical piston pumps are less and less frequently seen in the hydraulic

installations due mainly to the high pulsation coefficient of the flow.

2.1.2. Pumps with radial pistons

Pumps with radial pistons are rotary volumetric pumps with variable flow. The

pulsation coefficient of the flow is very diminished, thus having beneficial effects on the

extent of hydraulic oscillations introduced in the transmission system.

They may be classified into pumps with external suction and with internalsuction.

Pumps with radial pistons and external suction (fig.2.5) mainly consist of stator

1, rotor 2, pistons 3 coupled by means of piston rods 4 to the eccentrically axle 5 (with

variable eccentricity). The excentricity of the pistons axle gives the possibility that their

movement be different, some being in suction, others in discharge.

Fig.2.5

Pumps with radial pistons and internal suction (fig.2.6) consist of stator 1,

eccentrically rotor 2, piston 3, central axle 4, which contains the suction channels 6. Due

to the eccentricity e of the rotor, the pistons carry on an alternate movement of stroke

2e, being in turns in suction/discharge. The pistons are pressed to the walls of the stator


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by the force of springs or by the centrifugal force only. By modifying the eccentricity

we can adjust the flow of the pump.

Fig.2.6

The cylindricality of the z cylinders of diameter d or the volume of discharged

liquid for one rotation will be:

ezd

V 24

2 . (2.1-19)

For the rotation min/rotn we shall have the mean flow:

ze

dnez

dQmed

460

2

4

22

. (2.1-20)

Fig.2.7


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To calculate the instantaneous flow that ranges between a minimum and the

maximum value, first we establish the speed of contact point A of the piston with the

stator (fig.2.7). The absolute speed v is made up of speed1v in relation to the center of

the rotor1

O and2v the movement speed of the piston inside the cylinder. We note the

variable distance1

AO with .

Then we shall have:

1v , (2.1-21)

dt

dv

2 .

From the triangle AOO 21 we get:

cos2222 eeR (2.1-22)

From which:

2

2

2222sin1coscoscos

R

eReReee (2.1-23)

As 1R

e, we may leave out the second term of the radical. Then:

Re cos . (2.1-24)

The speed of the piston will be:

sin2 edt

dv (2.1-25)

For the interval ,o when increases; the speed2v decreases as the sign

from the relation (2.1-25) shows us.

We shall consider speed in modulus flow of the j pistons that are in discharge,

each being in the position 20i :

i

j

i

i ed

Q

1

2

sin4

. (2.1-26)


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If we note bythe instantaneous position angle of the first piston in discharge

and byz

2 , the angle between two pistons, then the position angle of the piston to

the given point M will be:

1 ii

. (2.1-27)

In the case of an even number of pistons, z = 2k, we shall have k pistons in

discharge and k pistons in suction. We can rewrite the equation (2.1-26) knowing that j

= k:

1sin....2sinsinsin4

2

ked

Qi .(2.1-28)

By transforming the sum between the braces into a product, we shall get:

2

1sin

2sin

2sinsin1

k

k

i

k

i

. (2.1-29)

The maximum value of this sum is obviously obtained when

12

1sin

k or 22

1

k , so

2

12

k . (2.1-30)

The minimum value could be obtained for

02

1sin

k , or .02

1

k But, because 0 ,(2.1-31)

hence

2

12

12

1

kkk . (2.1-32)

So, the minimum value of the argument of function sinus is 2

1

k or else

2

12

1

kk . (2.1-33)


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The minimum value of the sum in the relation (2.1-29) is obtained for 0 .

Going back to the relation (2.1-28) whose sum may be written in the form of

(2.1-29) and bearing in mind the considerations on the instantaneous position angle of

the first discharging piston for the maximum values of the flow, we may write:

2sin

2sin

4

2

max

k

ed

Q , (2.1-34)

2

1sin

2sin

2sin

4

2

min

k

k

ed

Q . (2.1-35)

Now we are able to write the pulsation coefficient of the flow for the pumps

with an even number of radial pistons:

10042

1002

1sin1

2sin

1

2

1002

1sin1

2sin

2sin

2%

ktg

kkk

k

k

k

k

k

(2.1-36)

In the case of pumps with an odd number of radial pistons 2k+1, we may

distinguish between two cases: either k+1 pistons are in discharge, therefore:

2,0

, (2.1-37)

or k pistons discharge, and then:

,

2. (2.1-38)

We shall compute the maximum and minimum flows for both hypotheses

and we shall notice that they are identical.

We shall write expressions maxQ and minQ for the two cases:

1. k+1 discharging pistons


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2sin

21sin

4

2

max

k

ed

Q , (2.1-39)

2

sin

2sin

21sin

4

2

min

kk

ed

Q . (2.1-40)

2. k repressed pistons

2sin

2sin

4

2

max

k

ed

Q , (2.1-41)

2sin

2sin

2sin

4

2

min

k

k

ed

Q . (2.1-42)

But

2

2

212

21

2 zzkkk . (2.1-43)

The angles being supplemental, it results in

2

1sin2

sin

kk , (2.1-43)

therefore the maximum and minimum flows shall be equal for the two situations

we come across with during the working of the pumps with an odd number of radial

pistons.

Taking into consideration the relations (2.1-41) and (2.1-42) as well as (2.1-20)

we can compute the pulsation of the flow for this type of pumps:

.100

124122

10012

sin1

12sin

12sin

12100

2sin1

2sin

2sin

12%

ktg

k

k

k

k

k

k

k

k

k

k

(2.1-45)


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In fig.2.8 the variation of the instantaneous flow for a pump with 9 radial

pistons is shown.

Fig.2.8

On studying table 2.1 it can be noticed that pumps with more pistons have a

lower pulsation coefficient and that pumps with an odd number of pistons are from this

point of view preferred to those with an even number of pistons.

Table 2.1

z odd number z even number

z % z %

3 14,022 2 157

5 4,973 4 32,515

7 2,527 6 14,022

9 1,526 8 7,80711 1,020 10 4,973

12 3,444

The force required to rotate the impeller of the pump is a perpendicular force on

direction 1AO ; we shall note it by F. Force F is decomposed into two directions: 1AO

(component F - the force with which the liquid, having the pressure p, acts upon a

piston of a diameter d) and 2AO (component N which acts upon the bearing of the

pump) (fig.2.9).


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Fig.2.9

The force with which the liquid acts upon the piston is equal and has opposite

direction to the force with which the pistons acts upon the liquid.

pdF4

2 . (2.1-46)

tgFT . (2.1-47)

We notice that:

sinsinR

e . (2.1-48)

Thus:

fR

earctgp

dT

sinsin

4

2

. (2.1-49)

The maximum value of T is obtained for 090 .

The torque corresponding to a piston is:

sinsincos

4

2

R

earctgeRp

dTMr . (2.1-50)

The total torque shall be:

j

i

iirt TM1

. (2.1-51)

where j is the number of discharging pistons.

The relation shall give the power of the pump:

rtMP . (2.1-52)


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2.1.3. Pumps with blades

Pumps with blades are volumetric pumps for which variable spaces are limitedby blades, impeller, stator and front lids.

They can be with external or internal suction (fig.2.10) and (fig.2.11).

Fig.2.10 Fig.2.11

By the number of suction-discharge for one rotation, the pumps with blades can

be with simple action (fig.2.10) and (fig.2.11) or multiple action. In fig.2.12 a double

action pump with blades is shown.

Fig.2.12

Pumps with blades and simple action are pumps with variable flow, their

adjustment being made by modifying eccentricity e. Pumps with multiple action

have a constant flow.

To calculate the flow we use the scheme in fig.2.13, for which we have

done the following denotations:

R, rthe stator radius and the impeller radius respectively; b the breadth

of a blade; - the angle between two consecutive blades; z- the number of blades

[20].

In fig.2.13 it is shown the blade coupling 1-2 in two position: at the beginning of

the discharge 21 , and at the end of discharge '2'1 , .


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Fig. 2.13

To calculate the volume V between the blades (blades of breadth b and

negligible thickness) we shall write first the elementary volume:

ddbdV . (2.1-53)

Knowing that

cos1 eRMO (V.cap.2.1.2)

and

21 ,

we can write

.cossin22cos2sinRe42

2sin2sin2

1

2sinsin2

2

cos2

12

2

1222

12

2

12

22

22

cos 2

1

2

1

erRb

eeRrR

b

dreRb

ddbV

eR

r

(2.1-54)

The maximum value of V is obtained when

.1cos12

cos 1212

and (2.1-55)


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(which means that21 ):

sin

22sinRe4

2

222

max

erR

bV . (2.1-56)

At the end of discharge the relation will calculate the volume among blades:

.cossin22

cos2

sinRe42

'

1

'

2

2'

1

'

222

cos'

'

2

'1

erR

b

ddbVeR

r(2.1-57)

We calculate the extreme of the function '1'2' V

.0

2

cos

2

cos

2

sin

2

sin *

'

1

'

2

'

1

'

2

'

1

'

2

'

eReb

d

dV(2.1-58)

.02

sin'

1

'

2

(2.1-59)

.2

'

1

'

2

(2.1-60)

For 12

''

1

'

2 ,02

d

dVis negative,

and for '1'2

''

1

'

2 ,02

d

dVis positive, so the extreme point when

2

'

1

'

2

represents a minimum.

sin

22sin4

2

222

min

eeRrR

bV . (2.1-61)

* The term in brackets cannot be cancelled since

2cos

2cos

'

1

'

2

eR


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The relation will give the volume discharged by the couple of blades (1,2):

2sin4minmax2,1

eRbVVV . (2.1-62)

The z inter-blade space shall discharge for one rotation the volume:

2sin42,1

zeRbVz . (2.1-63)

For a rotation of n [rot/s] the mean theoretical flow of the pump is obtained with

the help of the relation:

znzeRbnzeRbQmed

sin4

2sin4 ,

becausez

2 . (2.1-64)

When z is big,

.sinzz

Then:

.24 benDz

nzeRbQmed

(2.1-65)

The formula (2.1-65) is used to calculate the flow for pumps with a finite

number of blades. It obviously represents an approximation, higher or lower, according

to a greater or smaller number of blades.

To establish the instantaneous flow of a pump with blades, we shall first

calculate the volume of fluid that exists in the interstice ii 1 between two

blades:

.cossin22

cos2

sin42

1

1

2

122

cos

i

i

ii

ii

eR

r

i

eeRrR

bddbV

(2.1-66)

The instantaneous flow of the couple of blades shall be:


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29

.dt

dVq ii (2.1-67)

By leaving out the term that contains 2e and bearing in mind that

dt

d,

we shall successively obtain:

,2

sin2

sin2 1

iii beRq

(2.1-68)

,

2

sin

2

sin2

ii beRq (2.1-69)

,coscos iii beRq (2.1-70) .coscos 1 iii beRq (2.1-71)

The total instantaneous flow of a pump with blades shall be equal to the sum of

instantaneous flows of the j interstices being in discharge:

j

i

iii beRQ1

1 .coscos (2.1-72)

We shall study the pulsation of the flow first for a pump with an even number of

blades: z=2k. We shall then have j = k interstices being in discharge, for

any :2

,2

.2

sin2

sin2coscos

coscoscoscos

111

11

1

1

kkbeRkbeR

beRbeRQ k

k

i

iii

(2.1-73)

iQ is maximum when

22,

22

,12

sin

11

1

kk

or

k

(2.1-74)


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30

and it is minimum when

.2

02

sin

1

1

k

or

k

(2.1-75)

But

2

1

,2

,2

1

1

or

k

so

(2.1-76)

Under these circumstances:

2sin2max

kbeRQ (2.1-77)

and

*

min2

1sin2

sin2

kkbeRQ (2.1-78)

The relation gives- the pulsation coefficient of the flow for a pump with an even

number of blades2k :

.10042

1002

1sin1

2sin

2100

sin4

21sin1

2sin2

%

ktg

k

kk

k

k

znzbeR

kkbeR

(2.1-79)

* It can be noticed that 2

12

1

kandk

are supplemental angles, so the value of sinus

function remains the same.


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31

For a pump with an odd number of blades2k+1- we have two situations: k+1

interstices under discharge when

0,

21

and k interstices under discharge when

2,01

.

Computing in the same way as for the pump with the even number of interstices

we shall get the relations formaxQ and minQ .

1. k+1 discharged interstices

2

1sin2max

kbeRQ . (2.1-80)

.2

sin2

1sin2min

kkbeRQ (2.1-81)

3. k repressed interstices.

2sin2max

kbeRQ (2.1-82)

.2sin2 2

min

kbeRQ (2.1-83)

The values of maxQ and minQ are equal because the angles 2

1

k and

2

k are supplemental.

Bearing in mind the above shown demonstration there results that the pulsation

of the flow for a pump with an odd number of blades is:


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32

100

124122100

12sin1

12sin

12sin

12

100

sin4

21

2sin2

%

ktg

kkk

kk

k

k

znzbeR

kkbeR

(2.1-84)

By comparing the relations (2.1-79) with (2.1-36) and (2.1-84) with (2.1-45) we

can notice that the pulsation of the flow for pumps with radial pistons is identical to the

one of the flow for pumps with blades (leaving out the term 2e ), that suggests an

analogy between those two types of pumps. The space between two blades behaves like

a radial cylinder with piston during suction and discharge phases.

By equaling the hydraulic power with the power at the shaft of the motor we can

determine the necessary theoretical moment:

.2 pQnMt (2.1-85)

n is expressed in rotations per second.

In (2.1-85) we introduce the value of the mean flow given by (2.1-64):

.sin2

2

sin4

zeRbp

z

n

znzeRbp

Mt

(2.1-86)

Taking into account the mechanical and viscous frictions, the couple developed

by the motor will be:

.sin2

zeRbp

zMM t

(2.1-87)

2.1.4. Pumps with axial pistons

The pumps with axial pistons accomplish the flow of fluid by the alternatemovement of a certain number of pistons inside some cylinders that are placed in an

impeller, which have their axes parallel to the impeller axis of rotation. This manner of

placement gives the pumps a low clearance and equilibrium due to the symmetry of the

masses in rotation. The alternate movement of the piston is achieved by means of a

slanted disk. Its adjustable slanting allows the change of flow of the pumps. For some

pumps slanting the block of cylinders accomplishes the change of flow.


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33

In fig.2.14 the working scheme of a pump with axial pistons and slanting disk is

shown:

1. the block of cylinders (rotor);2. cylinders;3. pistons;4. slanting disk;5. cardan joint;6. connecting rods with spherical joints;7. fixed part of the suction /discharge channels (distribution element).

Fig.2.14.

The electrical driving motor transmits the rotation to the block of cylinders and,

by means of the cardan joint 5, to the slanting disk on which the extremities of the

cylinder rods are propped.

The suction and discharge are accomplished by means of the fixed distribution

element 7, which has channels in the area where the pistons are in suction, or in

discharge.

To calculate the flow of the pump with axial pistons let us consider two systems

of axes (fig.2.14.) xOyzand 111 zOyx that are rotated between them with an angle

around their common axis Oy . The coordinates of a certain M point in the system of

axes that is not rotated can be written with respect to the coordinates of the same point

in the rotated system of axes, (fig.2.15.) as:

sincos 11 zxx (2.1-88)

.sincos 11

1

xzz

yy


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34

Fig.2.15.

In fig.2.16. then are shown the positions of the spherical joint A, joined with the

disk and of the spherical joint B, joined to the piston, that belong to the same connecting

rod, during the rotation with an angle. [20]

Fig.2.16.

With respect to the systems of axes in fig. 2.14. point A has the following

coordinates :

- to 111 zOyx

cos

sin

0

11

11

1

rz

ry

x

A

A

A

(2.1-89)

- to xOyz (see relations 2.1-88)

.coscos

sin

sincos

1

1

1

rz

ry

rx

A

A

A

(2.1-90)


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35

Coordinates y and z of point B with respect to the system xOy are:

,cos

sin

2

2

rz

ry

B

B

(2.1-91)

coordinate Bx is to be determined knowing the constant length l of the connecting rodAB.

We shall then write:

.1 2222 ABABAB zzyyxx (2.1-92)

Relation (2.1-92) represents an equation of 2nddegree with the unknown .Bx

By solving it we get:

sincos1rxB

.coscossin2coscossin 222

222

1

22

1

2

2

2 rrrrl (2.1-93)

It can be noticed that Bx is negative. This is the reason why we chose the sign -

before the root.

The velocity of the piston can be obtained by deriving Bx with respect to time:

.

coscossin2coscossin

coscossin2cossin22coscossin2cossin2

sinsin

22

21

222

1

22

1

2

2

2

21

22

1

2

1

1

.

rrrrrl

rrrr

rxv Bp

(2.1-94)

When the slanting angle of the disk is enough small, we may consider .1cos The velocity of the piston, in modulus, which becomes:

,sinsin1 rvp (2.1-95)

The instantaneous flow of a piston with diameter d will be:

,sinsin4

1

2

rd

qi (2.1-96)

and the instantaneous flow of the j pistons that are under discharge is:


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36

j

i

i

j

i

ii rd

qQ1

1

1

2

.sinsin4

(2.1-95)

The mean flow of the pistons of d diameter and stroke sin2 1rh , inside the

impeller of rotation n will be:

60sin2

4 1

2n

zrd

Qm

. (2.1-96)

To establish the maximum and minimum flow, we have to draw the attention

that the problem is similar to that presented in chapter 2.1.2. This is also the maximum

and minimum of the sums of sinuses

j

li

isin , for the j pistons that are under

discharge, with an even number z = 2k or odd z = 2k + 1 of pistons.

Therefore, we can write the maximum and minimum flows for the pumps withan even number of axial pistons:

,

2sin

2sin

sin4

1

2

max

k

rd

Q (2.1-97)

.2

1sin

2sin

2sin

sin

4

1

2

min

k

k

rd

Q (2.1-98)

In this case the pulsation of the flow, will be:

10042

1002

1sin1

2sin

1

2

1002

1sin1

2sin

2sin

2%

ktg

kkk

k

k

k

k

k

(2.1-99)

For the pump with an odd number of axial pistons we shall have:


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37

2sin

21sin

sin4

1

2

max

k

rd

Q . (2.1-100)

2

sin

2sin

21sinsin4

1

2

min

kk

rd

Q . (2.1-101)

100

124122100

12sin1

12sin

12sin

12

1002

sin1

2sin

2sin

12%

ktg

kk

k

k

k

k

k

k

k

k

(2.1-102)

We can notice that the pulsation of the flow for the pump with axial pistons is

the same with the pulsation of the flow for pumps with radial pistons and pumps with

blades.

To create pressure p, the piston acts upon the liquid with the force:

.4

2

pd

F (2.1-103)

Force F is decomposed into a tangent component T and a normal one N

(fig.2.17).

Fig.2.17

The tangent force T has the value:


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38

.sin4

sin2

pd

FT (2.1-104)

The resistant moment of a piston will be:

.sinsin4sinsin1

2

1

rpd

rFTMr (2.1-105)

z pistons will have a resistant moment:

z

i

irt rpd

M1

1

2

.sinsin4

(2.1-106)

The relation will give the power consumed by the pumps:

,rtMP

(2.1-107)

.

310.97

min/81,9

..620.71

min/81,9

kWrotnNmM

PCrotnNmM

P

rt

rt

(2.1-108)

2.15. Pumps with sprocket wheels

They are volumetric pumps that are widely spread especially due to their simple

building.

As the sprockets come out of gear, a variation of volume in an excessive sense is

created in the suction room. The spaces between the sprocket represent active cups that

carry the fluid. When the sprocket come into gear the volume decreases and a

hydrostatic pressure is created (fig.2.18).

Pumps with sprocket wheels are classified according to several criteria: by the

type of gear (external or internalfig.2.18 a and b), by the level of pressure (low,

medium and high), by the number of rotors (with two or more, fig.2,19), by the profile

of sprockets (evolventric or cycloid), by the sprockets position (straight or slanting).


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39

Fig.2.18 Fig.2.19

The computation of the flow for this type of pumps can be done in a simple

manner; considering the hypothesis that the cross sections of the empty spaces is equal

to that of filled spaces and that the degree of coverage is equal to a unit; a hypotheses

that induces a quite high error.

Thus:

Sg = Sp. (2.1-109)

The cross section of all the cups for the two sprocket wheels that are in gear will

be:

2222

42

1

442 ie

ie

t DDDD

S

. (2.1-110)

By considering the bottom of the sprocket equal to its head maa 21 (the

sprocket modulus) and knowing that the sprocket modulus is

pm , we can write

(fig.2.20):

zmDDDD

S ieiet2

224

2

. (2.1-111)

Let the breadth of the sprocket be mb . The volume transported for one turnwill be:

zmV 32 , (2.1-112)

and the flow:


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40

min/1102 63 nzmQ . (2.1-113)

as m is given in mm, and rotation is considered in rot/min.

For a more accurate computation of the flow we can use two methods: the

geometrical method (more complicated) or the method of equivalence between the

energy transmitted to the liquid and the mechanical work consumed to drive the

sprocket wheels.

By using fig.2.20 we shall further present the second analytical method of flow

computing for the pumps with sprocket wheels. [12,20].

Fig.2.20

The mechanical work consumed to rotate the sprocket wheels with angle d inculcates

the energy pdV upon the liquid:

MdpdV . (2.1-114)

In relation (2.1-114) M is the torque.


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41

Pressure p acts upon the outline of the sprocket wheels. This intricate outline can

be replaced with a simpler one121 BCOAO . On the straight lines of this outline there

act four resultant forces of pressure. This replacement has been made according to the

theorem in mechanics, which states that the resultant of the projection of pressure forces

on a certain surface is equal to the product between pressure and the projection of the

surface on a plane, that is perpendicular on the resultant.

The total torque will be:

22212

2

2

''

2

1'

1

''

1

22

2222

e

ee

rb

p

Fr

FFr

FM

(2.1-115)

We denote the segment PC by x, and notice that rrOO 221 , consequently we

can apply the theorem of the median for the triangle COO 21 :

4

42 22

2

2

12 rrx

. (2.1-116)

Hence:

22

2

2

1 2 rrx . (2.1-117)

Replacing in relation (2.1-115), we shall get:

2222

xrrpb

M re . (2.1-118)

Knowing that dV = Qdt, and dtd and using the relations (2.1-118) and

(2.1-114), we may write:

222 xrrbQ re . (2.1-119)

Magnitude x is variable in time:


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42

.011

rb

rbb

tgtr

tgrrPKCKx

(2.1-120)

In relation (2.1-120) we used the property of the evolvement

1

11 CKCK andthe fact that t 0 (the real driving segment begins in D and ends in

0

1

12 brDKDKE ).

Noting by PKPKl 21 the length of the half of the theoretical driving

segment and by EPDPl 1 the length of the half of the real driving segment, weshall get:

10 llrb . (2.1-121)

So:

11 ltrltrllx bb . (2.1-122)

We can write the instantaneous flow in the form of a time function:

2'122222 2 ltlrtrrrbtQ bbre . (2.1-123)

The time in which the real driving segment is covered, is obtained by using the

properties of the evolvement:

,2 1

trl b (2.1-124)

.2

1

br

lT (2.1-125)

The flow Q(t) has a periodical variation; Tt ,0 .

To compute the pulsation of the flow first we have to establish the mean flow.

The volume discharged by a pair of sprockets during a period T is:


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43

br

l

bre

T

dtltrrrbdttQV

2

0

2122

0

. (2.1-126)

By making the change of the variable

,

,1

dtrdy

ltry

b

b

(2.1-127)

we get:

.33

2 2221

222

1

1

lrr

r

bl

dyyrrr

bV

re

b

l

l

re

b

(2.1-128)

Knowing that the number of sprockets is z and the wheels turn with rotation n,

the mean flow will be:

2'2233

lrrr

blznzVQ re

b

m

. (2.1-129)

The maximum value of the relation (2.1-123) is obtained forbr

lt

1

:

22max re rrbQ . (2.1-130)

For t =0 orbr

lt

12

the flow has the minimum value:

2'22min lrrbQ e . (2.1-131)

We are now able to determine the pulsation of the flow for a pump with sprocketwheels:

.100

3

3%

2'22

1

lrrz

lr

re

b

(2.1-132)


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44

The moment applied to the driving wheel is determined from the relation (2.1-

114):

222 xrrpbpQdt

pdV

d

pdVM re

. (2.1-133)

The moment will be maximum for x=0:

22max re rrpbM . (2.1-134)

By making the same approximations as in relation (2.1-111) we get:

.2max lzpbmM . (2.1-135)

The maximum force applied to the liquid will be:

rr

MF max . (2.1-136)

The power expressed with respect to the moment and to the angular velocity

is written with the known formula:

MP . (2.1-137)

2.1.6. Other types of volumetric pumps

The pumps with diaphragm (fig.2.21) This type of pump is mostly used

when the circulating fluid mustnt come into contact with the parts of the pump or

mustnt be contaminated by the lubricating oil.


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45

Fig.2.21

It consists of one or more metallic diaphragm 1 between two concave disks 2.

The diaphragms move elastically under the action of the piston 8 and liquid of

working 7 (oil).

The volume variation in the working room, that is superior to the diaphragm,

ensures suction (through valve 4) and discharge (through valve 3) of the fluid.

Pump 5 carries out the compensation for the losses of oil due to the non-

tightness of the piston. Valve 6 is a limiting valve for the discharge pressure.

The pump with screw (fig.2.22)

The number of rotors (two or more) can classify pumps with screw, by the shape

of the thread (rectangular, trapezoidal, and cycloid), by the number of starts (one, two ormore).

In fig.2.22 it is presented the scheme

of a pump with screw with two rotors

(screws), of which one is driving. The

driving rotor has a thread right and the

other one left.

Fig. 2.22


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46

By the relative rotation of the two rotors the liquids get into the suction room A,

and fill the clearance between the rotors in the area that is not driven. The liquid will be

transported in the discharge room R, on a straight trajectory, without flow pulsations.

The working of this pump is similar to that of the endless piston.

The pump with cycloid gearings (fig.2.23)

This type of pump consists of two cycloidal

shaped rotors, of which one is driving,that rotate

conversely.

The hachured area represents the section of

suctioned liquid due to the rotation of the

cycloid gearing that is (in the next moment to

that shown in the figure ) to be repressed.

Fig.2.23

The pump with roll (fig.2.24)

The pump with rolls is another type of volumetric

rotary pump with an eccentric rotor. Suction and

discharging are carried out due to the variation of

volume in the space among the rotor, stator androlls. The rolls are made of plastic with a metallic

core. Due to rotation they are pushed on the

walls of the stator by the centrifugal force, thus

separating the variable volumes.

Fig.2.24

Fig.2.24

2.17. Characteristics of volumetric pumps

One of the main characteristics of the volumetric pumps is the characteristic

flow-pressure. The real flow represents a slight decrease with respect to pressure, due to

the increase of volumetric losses. Over a certain pressurelim

p the decrease of the flow


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47

is obvious (fig.2.25). Function pfN , which represents the variation of power, isapproximately linear up to value

limp , after which its increase is even more obvious.

(fig.2.25). After the same valuelimp , the curve of efficiency, pf has a strong

descending carriage.

In fig.2.26 there are shown the characteristics pfQ for a pump withadjustable flow at different eccentricities (or tipping angles in the case of pumps withaxial pistons).

Figure 2.27 shows the mechanical characteristic moment-pressure-rotation.

The slope of these curves,n

M

, shows us the litheness of the mechanical

characteristic.

Fig. 2.25 Fig. 2.26

Fig. 2.27


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48

2.2 Hydrodynamic pumps

2.2.1 Building and classification

Pumps or hydrodynamic generators process the potential energy of pressure and

kinetic energy, by means of an impeller equipped with blades.

The blades of the impeller are usually placed between two parallel disks; one is

fixed on the shaft (the crown) and the other one that contains the inlet of the fluid (the

ring). The fluid passes through the suction pipe, gets into the rotor where a kinetic

energy is inculcated upon it, which afterwards is converted into potential energy in the

spiral room and in the discharge pipe. Some centrifugal pumps are equipped with a

stator with blades that have the role to convert the kinetic load into pressure load and to

direct the fluid. In fig.2.28 it is schematically represented a centrifugal pump with the

following components:

Fig.2.28

1. The suction flange that makes the connection with the suction conduit.2. Ring.3.Network of blades.

4. The crown of the rotor.5. The axis of the pump.6. The tightening system of the axle.7. The spiral room that collects the fluid from the periphery of the statorand contributes to the convention of kinetic pressure into potential

pressure.

8.The stator that has the role to direct the stream and converts the kineticenergy into pressure energy.

9. The diffuser, that also contributes to the conversion of the kinetic loadin pressure load and makes the connection with the discharging conduit.


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49

Hydrodynamic pumps or turbo pumps may be classified by the specific rotation

or dynamic rapidity, that can be considered as the rotation of a pump geometrically

similar with the given one, which absorbs a power of 1 H.P. at a load of 1m:

4/5H

Pnn

HPS (2.2-1)

Specific rotationsn and rotation n measured with the tachometer, obviously

cannot have the same dimension.

In table 2.1 there are shown the classification of turbo pumps and the shape of

the meridian suction of their rotor, with respect to specific rotation.

Table 2.1

Type of

pump

Pump

with

lateral

channel

Centrifugal pump with rotor Axial

pump

slow normal rapid diagona

l

The

shape in

meridia

nsection

of the

rotor

K 0,04

0,2

0,2

0,4

0,4

0,8

0,8

1,55

1,55

2,6

2,6

6,2

Sn 840 4080 80

150

150

300

300

500

500

1200

qn 2,2 - 11 1122 22 - 41 4182 82 -135

135 -380

In order to classify the turbo pumps we can also use their characteristic rotation

or kinematic rapidity:


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50

4/3H

Qnnq (2.2-2)

as well as the characteristic number

4/32gH

QnK . (2.2-3)

Between these values there are the relations:

Knn qSHP 19365,3 . (2.2-4)

2.2.2. Turbo pumps theory

Inside the rotor of the turbo pump, the liquid particles carry out a complex

movement.

Following the outline of the blade, the particle covers a relative trajectory 1-2,

but, at the same time, the rotor turns, the movement of the particle with respect to a

reference system joined to the frame of the pump being '21 - the absolute trajectory.(fig.2-29).

The basic theoretical equations of the turbo pumps applied to the case of

centrifugal pumps are obtained for the following hypotheses:

a) Between two consecutive blades of the rotor of the centrifugal pump, the flow of thefluid is stationary, in the shape of some streamlines that take the curvature of the

blade.

b) Inside the pump we dont have hydrodynamic losses.c) The rotor consists of an infinite number of blades with negligible thickness.

Thus, noting by symbol 1 the inlet in the inter blade channel, and by 2 the outlet,

we shall have (fig.2.29 and fig.2.30):

- the relative inlet and outlet velocities in and from the rotor1

w and 2w tangent in

any point to the stream line that has the shape of blade;- peripheral velocities that are due to the rotation with speed of the rotor on the

circles with radii1

R and2

R , 11 Ru and 22 Ru ;

- absolute velocities1v and

2v that result from the making up of the relative and

peripheral velocities:


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51

.

,

222

111

uwv

uwv

(2.2-5)

Fig.2.29

Fig.2.30

Absolute velocity decomposes into a tangent component ,a load component:

.cos

,cos

22

11

2

1

vv

vv

u

u

(2.2-6)

and a normal component, a flow component:

.sin

,sin

22

11

2

1

vv

vv

m

m

(2.2-7)

The theoretical volumetric flow of liquid at inlet, equal to that at outlet, will be:

,2221 2211 mmv

vbRvbRQt

(2.2-8)

where1b and

2b are the thickness of the blades at inlet and outlet, respectively.


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The fundamental equation of turbo machines, applied in the case of centrifugal

pumps can be obtained in several ways:

a) by applying the theory of variation for the moment of movement quantity (impulse)We shall further consider an ideal centrifugal pump (the impeller with an infinite

number of very thin blades).

The movement quantities at inlet and outlet 1 and 2 are1um

vQ and2um

vQ , and

their moments 11 RvQ um and 22 RvQ um .

The variation of the moment for the movement quantity between these two

points will be:

.1212

12

12

RvRvQ

RvRvQM

uuv

uum

t

(2.2-9)

The power, in the case of rotation with angular velocity , will be given by therelation:

.1212 1212 uvuvQRvRvQMP uuvuuv tt (2.2-10)

The relation expresses the power of an ideal pump with an infinite number of

blades:

Tv HgQP t . (2.2-11)

Equaling the last two relations we get:

g

vuvuH

uu

T

12 12

, (2.2-12)

expression that represents the fundamental equation of ideal centrifugal pumps. Euler

has inferred it for hydraulic wheels long before the invention of centrifugal pumps.

b) by applying Bernoullis equation for the relative movement between the points 1and 2.

In Bernoullis equation for relative movement


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53

fhzp

g

uwz

p

g

uw

2

2

2

2

2

2

1

1

2

1

2

1

22 (2.2-13)

we consider 21 zz .

The pressure load created in the rotor will be:

fhg

uu

g

wwpp

22

2

1

2

2

2

2

2

112

. (2.2-14)

The load TH will be equal to the increase of the water pressure at outlet of the

rotor plus the increase of kinetic energy plus the losses of load:

fT hg

vvppH

2

2

1

2

212

. (2.2-15)

From the relations (2.2-14) and (2.2-15) we get the expression:

g

vv

g

uu

g

wwHT

222

2

1

2

2

2

1

2

2

2

2

2

1

. (2.2-16)

which is the fundamental equation of turbo machines applied to centrifugal pumps, in

velocities.

From the velocity triangle we have:

.cos2

,cos2

222

2

2

2

2

2

2

111

2

1

2

1

2

1

uvuvw

uvuvw

(2.2-17)

By replacing (2.2-17) into (2.2-16) we get the fundamental equation of turbo

machines applied to centrifugal pumps, similar to equation (2.2-12):

g

vuvuvuvu

gH

uu

T

12 12

111222 coscos1

. (2.2-18)

The fundamental equation may also be written in the form:

12 12 uuTT vuvugHY (2.2-19)

where TY is the specific energy, the energy of mass unit.


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2.2.3. Turbo pumps in network

The pump load or the pressure difference between the input and output of the

liquid in a pump is independent from the network in which it works.

The working parameters depend on and are defined by the network that a pump

services.

In fig.2.31 it is schematically shown a simple hydraulic system in which a pump

P sucks liquid from the tank aR , with a pressure ap and whose level of liquid has the

quote az to the reference plane N N and discharges it into the tank rR in which the

pressure is rp and the level of liquid is at the quote rz .

Vacuum gauge V measures the inlet pressure in the pump ip , and manometer M

the outlet pressure from the pump ep . ah and rh are the load losses in the suction,

respectively discharging conduits. The velocities of the fluid on suction and discharge

are av and rv .

Applying Bernoullis equation to the suction route, we get:

iii

iaaa

a Hg

vpzh

g

vpz

22

22

. (2.2-20)


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Fig.2.31

On the discharging route we shall have:

e

ee

er

rr

r Hg

vpzh

g

vpz

22

22

. (2.2-21)

The load of the pump will be:

.2

2

22

22

ar

ar

ra

arar

arie

hg

vvpz

hhg

vvppzzHHH

(2.2-22)

Relation (2.2-22) signifies the pump functions, namely: the liquid lifting on the

height z , the pressure rise from ap to rp , the alteration of the liquid kinetic energy

by increasing its velocity, the overcome of the losses on the suction and discharging

routes.

The losses on the routes are local and linear:

ra

raarg

v

d

lhhh

,

2

2 . (2.2-23)


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The suction and discharging routes, having conduits of diameters ad and rd are

covered by the flow Q:

ra

ragd

Q

d

lh

,42

2

,2

16

. (2.2-24)

and

442

222118

2ar

ar

ddg

Q

g

vv

. (2.2-25)

By replacing (2.2-24) and (2.2-25) into (2.2-22) we get:

2

,4442

11118Q

ddddg

pzH

ra ar

. (2.2-26)

The expression:

ra ar

rddddg

K,

4442

11118

(2.2-27)

is constant for a certain network.

We denote by

,

pzHS

(2.2-28)

the static load.

In this case the load expression becomes:

2QKHH rS . (2.2-29)

Function (2.2-29) stands for the network characteristics and represents, as it canbe noticed, a parabola. Should the flow be reversed (emptying the tank through the

network), the expression would become:

2QKHH rS . (2.2-30)


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Figure 2.32 shows many more characteristics of networks at the same static load,

but which have some alterations for rK (different diameters of conduits, bends, different

taps, etc.).

Fig.2.32

Analytically or experimentally we can determine the function QfH - theinterior characteristics or the machine characteristic.

In the case of a finite number of blades, due to the variation of velocity in the

inter- blade channel, the value of the product22 u

vu is decreased.

Consequently, the conveyed specific energy will be smaller. We may write:

pH

H

gH

gH

Y

Y

T

T

T

T

T

T 1 . (2.2-31)

where p = 0,20,45 according to the model proposed by Pfleiderer.

TH is the theoretical height for a pump with a finite number of blades for the

case when we circulate a liquid without viscosity. The real height may be written in the

form:

rT hHH (2.2-32)

where rh is the dissipation due to viscosity, proportional to the square of the flow,

2

11QKhr (2.2-33)


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and the shock losses2r

h due to the fact that for flows different from the rated flow NQ ,

the inlet angle of the stream of liquid1 differ from the inlet constructive angle of the

blade.

2

2

1

Nr Q

QKh

r

. (2.2-34)

Then:

2

2

2

1 1

N

rQ

QKQKh . (2.2-35)

Returning to the fundamental equation of centrifugal pumps, we notice that TH

is bigger as11 u

vu is smaller and nil when the inlet in the impeller is normal 01 90 :

g

vuH

u

T

22 . (2.2-36)

In fig.2.30 we notice that:

22 22ctgvuv mu . (2.2-37)

But the normal component of the outlet velocity has the value:

222 bD

Qvm

. (2.2-38)

Taking into consideration relations (2.2-36), (2.2-37) and (22.-38) we may write:

2

22

2

2

ctgbD

Qu

g

uHT . (2.2-39)

The theoretical load of a centrifugal pump with an infinite number of blades has

a linear variation with respect to the flow. The bending of the line depends on the angle

2 (fig.2.33).

The theoretic manometer height is maximum when 02 90 , in other words

when the blades of the impeller are curved forward.

Pumps with 02 90 and those with

0

2 90 have a smaller efficiency than

those with 02 90 , due to the high losses of energy at the inlet of the liquid into the

collecting channel (high acceleration inculcated upon the liquid in the inter- blade


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channel). Centrifugal pumps with 02 90 have also an instability of energy. These

disadvantages make us prefer pumps with 02 90 , although their theoretical

manometer height is lower.

Fig.2.33

Considering the relations (2.2-31), (2.2-35) and (2.2-39) we may write the

expression of the real load:

2

2

2

1222

2

2 11

NQ

QKQKctg

bD

Qu

pg

uH

. (2.2-40)

In fig.2.34 it is shown the interior characteristic of the pump that resulted from

the superposition of the linear variation of the theoretical load with the parabolic

variation of the dissipation due to viscosity and shocks.

The working point of a pump in a certain network is found at the crossing

between the network characteristic with the interior characteristic (fig.2.35).

Fig.2.34


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Fig.2.35

The optimal running of a hydraulic system will take place when the duty point is

in the area of maximum efficiency. The curve Q is experimentally obtained, afterdependence QP has been determined.

To improve the pump performances within the hydraulic system, we may change

the position of the duty point, by modifying the network characteristic. This may be

achieved in several ways. A simple way is to modify constant rK by varying the local

strength coefficients and the adjusting parts. We may also change the static load of the

network. Fig.2.33 shows the sliding of the duty point of the pump when the network

characteristics are altered.

2.2.4. Computation of centrifugal pumps

For a real pump, the thickness of blades has an influence on the velocities at

inlet and outlet of the liquid to and from the rotor. In fig.2.36 we noted by s the

thickness of the blade and by t the pitch of the blade. We shall analyze the state of theradial velocities in points O, little before inlet in the impeller, l, at inlet in the impeller,

2, at outlet of the impeller, and 3, immediately after outlet of the impeller. Nothing by

the circle bow corresponding to the thickness of the blade, we shall get:

111 sins . (2.2-41)

From the continuity equation of the flow ( mv - the radial flow component)

between the points O and 1, we get:

,11111 01 btvbtv mm (2.2-42)

wherez

Dt 11

(zthe number of blades)

Further we shall have:


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l

m

mm

v

t

tvv

0

01

11

1

(2.2-43)

where1

11

1t

t

is the decrease coefficient of the section due to the thickness of

blades.

To avoid shocks at the inlet section, the blades are rounded.

Fig.2.36

Similarly, at the outlet from the impeller, we shall have:

32222 32 btvbtv mm (2.2-44)

As the construction of blades at the outlet from the rotor is edged, 02 and

32 mm vv . (2.2-45)

The influence of the outlet angle has been discussed in the previous chapter.

The angle2 has values that range between 14 and

030 , rarely higher.

When computing the impeller dimensions (fig.2.37) we start from thediameter of the driving shaftdcomputed with respect to the torque for a certain

rotation of the driving motor. The power of the driving motor may be computed

with respect to the load H, and flow Q of the pump, and, obviously, with respect to

the efficiency of transmission.


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The diameter of the hub is adopted

ddn 5,12,1 . (2.2-46)

The pump should be computed for a flow 'Q higher than Q , as we have to

take into consideration the volumetric losses:

QQ 15,1......03,1' . (2.2-47)

The velocity of the liquid through the conduit,Sv , is adopted between 2 and

4m/s, the higher value corresponding to a load at lower suction.

From the continuity equation it results that:

2'

4n

S

S d

v

QD

. (2.2-48)

Diameter1D is adopted bigger than SD , so that the inlet edge should be

outside the curvature area of the stream lines:

mmDD S 15.....51 . (2.2-49)

The thickness of the blade at the inlet in the rotor is computed taking into

consideration the radial (flow) component of the velocity, little before the inlet in

the impeller.

1111 sin10 vvv mm . (2.2-50)

Thus:

1111

'

1sin vD

Qb . (2.2-51)

Generally1v can be taken equal to Sv . If

0

1 90 , we may write:

11

1

SvD

Qb . (2.2-52)

Assuming that, in a first approximation 8,01 , we may determine the

velocity triangle at inlet by means of formulae:

60

1

1

nDu

(2.2-53)


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and

1

1

1u

vtg (for 01 90 ). (2.2-54)

The necessary manometric load of the pump H is established beforehand

depending on the necessities of the installation.

For a certain hydraulic efficiencyh we may write:

h

T

HH

(2.2-55)

and according to (2.2-31)

pHH TT 1 . (2.2-56)

For radial pumps the computation relation of coefficient p is:

2

2

11

12

D

Dzp

, (2.2-57)

where is a coefficient experimentally established. For centrifugal pumps with a

stator with blades can be established by means of the relation:

2sin6,065,055,0 . (2.2-58)

For a pump with bladeless stator, its values are a little higher.

From the relation (2.2-39) where 'QQ it results2

u and then 2D :

n

uD

2

2

60 . (2.2-59)

For the case when 12 2DD the pump is well designed, with low friction

losses. When2

D is much higher, we must choose a pump with more serial

impellers, and when2

D is lower, the flow and load characteristics require a pump

with more parallel impellers.


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A pump ensures the water directing with the greatest number of blades, but

which have a detrimental effect in what regards the increase in the friction losses.

When establishing the number of blades we must take into consideration these

aspects. The computation for the number of blades is:

2sin5,6

21

12

12

DD

DD

z . (2.2-60)

To compute the coefficient p we need the number of blades that is established

for the hypothesis that 12 2DD , this is to be checked by mean of the relation (2.2-59).

If the error is too high, the computation must be reconsidered, acting upon some

parameters, within reasonable limits, and if not, we resort to serial or parallel of several

impellers as above stated.

The number of impellers i is established by the relation

,H

Hi

(2.2-61)

Fig. 2.37

where

,22

2 uKDH (2.2-62)

min/, rotmD and

4

105,1.....3,1

K for a stator with blades,

4104,1.......1 K for a stator without blades.


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2.2.5 Parallel and series connection of centrifugal pumps

To increase the flow or the load of a hydraulic system, we use parallel or series

connections of pumps.

a) Parallel connection (fig.2.38)In the case when two or more pumps are connected in parallel it is achieved an

increase of the flow for a constant load. For two pumps well have

21 QQQc , (2.2-63)

expression that is in fact the relation of continuity.

21 HHHc . (2.2-64)

signifies self-equilibrium of the system pump-network.

Fig. 2.38

When two identical pumps are parallel connected (fig.2.39) the interior

characteristic is obtained by doubling the abscissa of the points on the interior

characteristic for one pump.

The duty point of the system cF will be at the crossing between the interior

characteristic and the network characteristic. The efficiency of parallel connected

centrifugal pumps depends on the characteristic of the network.


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It can be noticed that in the case of network R,

the increase in flow Q as compared to a system

with one pump is more important than the

increase 'Q in the case of network R. It is

noticed that in the case of parallel connected

pumps there appears an increase of load, that also

depends on the characteristic of the network.

Fig.2.39

The efficiency of the two identical pumps is 21 .

The efficiency is the ratio between the useful and consumed power:

21 P

HQ

P

HQ cFcF . (2.2-65)

In a coupled regime, each pumps works in duty point F, and cF QQ2

1 .

Thus:

ccHQPP2

121 . (2.2-66)

The efficiency of the coupling will be:

cccc

cccc

CP HQHQ

HQ

PP

HQ

2

1

2

121

. (2.2-67)

In the case of the parallel connected of two or more identical pumps, the

general efficiency will be equal to the efficiency of each pump.

When parallel connecting two pumps with different characteristics, the

problem is much more complex. The characteristic of coupling is obtained in a


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similar way, by summing up the characteristics abscissae of the two pumps at a

constant load, 21 QQQH cc (fig.2.40).

On the diagram of the coupling the critical pointcrP appears, situated at the

quote of critical load crH , corresponding to the crossing between the smaller pump

characteristic and the ordinate.

If the duty point of the system is belowcrP , as in the case of the

characteristic of the network R, then the parallel connecting of the two pumps is

justified. In the case of characteristic 'R , the duty point is above crP , the smaller

pump working on the braking characteristic. In this case the flow of the coupling is

lower than the flow of one pump (the big one), thus the coupling becoming

unjustified.

Fig.2.40.

The efficiency of a coupling of two different pumps will be given by the relation

[8]:

2

2

1

1

2

2

1

1

QQ

Q

HQHQ

HQ c

cc

cc

CP

. (2.2-68)


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b) Seriesconnection (fig.2.41)To increase the load we use the series connection of two or more centrifugal

pumps. The flow that passes through two series connected pumps is the same:

21 QQQc , (2.2-69)

and the load

21 HHHc . (2.2-70)

Fig.2.41

To plot the characteristic of the assembly we sum up the ordinates of the

characteristic point for each pump. Fig.2.42. shows the common characteristic of two

identical seriesconnected pumps.

From fig.2.42. it can be noticed that

in the case of characteristic R we get a

higher increase of the load than in the case

of characteristic 'R . It can be noticed thatin series - connection an increase in flow is

also obtained.

Fig.2.42.

The efficiency of the coupling is equal to the efficiency of each pump taken

separately.


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cccc

cccc

CP HQHQ

HQ

PP

HQ

2

1

2

121

. (2.2-71)

For different pumps seriesconnected, the characteristic of the coupling is also

obtained by summing up the ordinates of the points on the characteristics of the twopumps (fig.2.43).

Fig.2.43

Here there is also a critical point corresponding to the load abscissa O of the

smaller pump. In networks whose characteristics the duty point is belowcrP it is

irrational to use two pumps whose total flow is lower than that of a single pump.

The efficiency of the coupling when series connecting two different pumps

will be [8]:

2

2

1

1

2

2

1

1

HH

H

HQHQ

HQ c

cc

cc

CP

. (2.2-72)

For reason of strength of materials, the peripheral velocities of the impellers

cannot exceed certain values.


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Since the maximum theoretical load depends on the peripheral velocity of the

impeller, thus being limited by it, to increase the load on a single unit, we use pumps

with several seriesconnected impellers (fig. 2.44.)

Fig.2.44.

Also, obtaining higher flows is limited by

rotation and by the outlet diameter from the

impeller, as well as by the circulating velocity of the

liquid. By using double impellers and by parallel

connecting them within a pump (fig.2.45), we

achieve the increase in flow and also the selfequilibrium of the axial thrusting forces.

Fig.2.45

To simultaneously obtain high loads and

flows on a single pump, we can use several

impellers that are axially series and parallel

connected (fig.2.46)

Fig.2.46

2.2.6 Suction of centrifugal pumps

The suction of centrifugal pumps is due to the depression generated in the

impeller; in fact it is due to the difference of pressure between the impeller and the

suction tank. In the case when the pump sucks water from an atmospheric pressure

(barometric) ba pp and the depression in the impeller would attain vacuum, the

theoretical maximum suction height would be:


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mhp

H bbaspt 33,10

. (2.2-73)

Fig. 2.47 shows a centrifugal pump that sucks from a pressure ap . We shall

consider three reference points: athe level of liquid, Othe highest point before inletin the impeller, 1 immediately after inlet in the impeller. We shall consider as

reference points the level of the liquid that is to be sucked and that is under motion with

velocityav * .

* If ba pp then we have the case of acentrifugal pump that sucks from a river.


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We apply Bernoullis relation between the considered points:

rirraasp

raasp

aa

hhH

g

vp

hHg

vp

g

vp

2

22

2

11

2

00

2

. (2.2-74)

where rah are the local and linear losses on the suction itinerary, and rirh the loadloss at the inlet in the channels of the impeller. This loss of load may be written under

the form:

,2

2

1

g

vhrir (2.2-75)

where is the local coefficient of loss at the inlet in the pump.

If suction is being made from a tank 0av , the suction load will be:

raa

asp hg

vppH

21

2

11

. (2.2-76)

The maximum load at sucking would be when 01 p , but it is known that in

real practice the maximum depression in a moving liquid corresponds to the absolute

saturation pressure of the liquid at the respective temperature, the moment when the

cavitation phenomenon appears:

vpp 1 .

Thus:

rava

asp hg

vppH

21

2

1

max

. (2.2-77)

The term g

v

21

2

1 depends on the

design characteristics of the hydraulic

machine, and it can be expressed with respect

to the effective load of the pump H by means

of cavitation coefficient :

Fig. 2.47


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Hg

v

21

2

1 . (2.2-78)

We rewrite expression (2.2-77):

rava

asp hHppH max

. (2.2-79)

The cavitation coefficient is given by the experimental relation [8]:

rotationspecificnna SS 3/4 . (22-80)

Several values are granted for coefficient a in the literature of the subject:4

1029,2 (Thoma); 41020,2 (Stepanoff);

41016,2

(EscherWyss).

It has been established that coefficient a also depends on the specific rotation.

Coefficient may also be written [8]:

HC

Qn 103/4

, (2.2-81)

where C is Rudnevs cavitation coefficient and has the values:

.150.....801000.....800

80.....50800.....600

S

S

nforC

andnforC

Relation (2.2-79) shows us a maximum suction height, which for different

reasons doesnt correspond to the real suction height. Thus, velocity1v of inlet in the

impeller may have a higher value, generating cavity suction conditions.

Thus, to establish the needed suction height we operate on the cavitation

coefficient by considering a:

4,1.....2,1lim . (2.2-82)

or, in a simple manner, by directly operating on the suction load, reducing it, to:

max75,0 aspasp HH . (2.2-83)

According to relation (2.2-81) the suction height will be:

rava

asp hHpp

H lim

. (2.2-84)


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The suction height on the centrifugal pumps is under the influence of a series of

factors.

In the case of suction from an atmospheric pressure ba pp , the suction height

depends on the variation of this pressure with the weather state, latitude and especially

with the height of the place. Should we denote by0

p , the pressure at sea level, the

pressure variation with height mz might be written under the form: zppb 50 104,21 . (2.2-85)

The suction height depends through vp , on the nature of the vehicled fluid and

on its temperature.

We can go as far as that the suction height comes be negative * when:

ra

v hHpp

lim

0

. (2.2-86)

In this case the pressure in the tank of suction should be increased or, in the case

when the tank is open, this should be mounted above the pump, at a corresponding

height.

2.2.7 Axial pumps

According to the classification shown in chapter 2.2.1, axial pumps are at the

extremity of the specter of specific rotation for pumps 1200.....500CPS

n . For this

type of pumps, the specific energy is obtained by a partial conversion of kinetic energy

in the inter- blade channel, the moving of the fluid being performed axially.

*In the case of circulating water at temperatures

higher than C060 . For water at 0105 , mHasp 7.....6 .


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In fig.2.48 an axial pump is schematically shown. It is mainly made up of:

directing device 1, hub with blades 2, that together with axle 7 is the mobile part of the

pump, rectifying device 3, carcass 4, together with elbow 5 and stuffing box 6,

careening of impeller 8.

Generally axial pumps have blades with afixed pitch. For axial pumps of high powers we

can use impellers with variable pitch for different

load situations.

The design of the blade is similar to the

design of the naval propeller, namely a sequence

of hydrodynamic profiles disposed under

different placed angles from hub to periphery.

The directing device ensures a shockless

input of the fluid particles into the impeller, and

the rectifying device, apart from converting a part

of kinetic energy into pressure energy is designedto direct the fluid jet in an axial direction.

Fig. 2.48

In fig.2.49 we have considered a cylindrical section through the pump, at adistance r, section from which we have taken only one element of the directing device,

impeller and rectifying device.

Unlike centrifugal pumps, the peripheral velocity at the input into the impeller

1u is equal to the peripheral velocity at the output from the impeller

2u :


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ruuu 21 . (2.2-87)

The absolute velocity at the input into impeller,1v ,

results from the composition of relative velocity,1

w ,

tangent to the blade, with the peripheral velocity, 1u . Therole of the profile in the directing device is to result an

absolute velocity at output, as near as possible on the

direction of velocity1v . Also, at input in the directing

device velocity av has to have an axial direction,0

90a .

At outlet from impeller, velocity2w , tangent to

the trailing edge, composed with peripheral velocity,2

u

(equal to1

u ), will give the absolute output from impeller

velocity,2

v .

The profiles of the stator will have to direct the

output velocity in point 3, as much as possible to the axial

direction.

The profiles of the blades influence one another.

The problem is that of a network of profiles with pitch t.

We can consider that the profile of the impeller in fig. 2.49

is attacked with velocity w , a mean of velocities 1w and

2w .

Hydraulic Machines D.D.

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Transcript of Hydraulic Machines D.D.