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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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.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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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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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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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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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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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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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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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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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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.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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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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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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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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.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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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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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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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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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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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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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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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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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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.
,
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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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
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 .