notes_pumps_final.pdf

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Department of Civil and Environmental Engineering

CVNG 1001: Mechanics of Fluids

INTRODUCTIONHydrodynamic Machines

A hydromachine is a device used either for extracting energy from a fluid or to addenergy to a fluid. There are many types of hydromachines and Figure 1 below illustratessome common types.

Hydrodynamic Machines

Machines where energy isadded to the fluid, i.e.

work is done on the fluid

Machines where energy isextracted from the fluid, i.e.

work is done by the fluid

RotodynamicMachines

PositiveDisplacement

Machines

Pumps, Fans,Compressors

Pumps,Compressors

Turbines Motors

RotodynamicMachines

PositiveDisplacement

Machines

Cased Uncased Reciprocating Rotary Impulse Reaction

MixedFlow

RadialFlow

(Centrifugal Pumps)flow is in the planeof impeller rotaionor perpendicular to the axis of its

rotation

AxialFlow

flow is perpendicular to the plane of impeller

rotationor along its axis of

rotation

Propellers Pelton Wheel

MixedFlow

RadialFlow

(Francis)

AxialFlow

(Kaplan)

Vane

Gear

Piston

CrankDriven

DirectDriven

SwashPlate

Gear

Vane

Screw

Lobe

Figure 1

Hydrodynamic machines may be classified according to the direction of energy transfer(energy added or extracted) or the type of action (rotodynamic or positive displacementmachines). Rotodynamic machines (momentum transfer machines) have a rotating part(runner, impeller or rotor) that is able to rotate continuously and freely in the fluid. Thismotion allows an uninterrupted flow of fluid which promotes a steadier discharge than

positive displacement machines. Positive displacement machines have a moving boundary (such as a piston or a diaphragm) whereby fluid is drawn or forced into a finitespace. This motion causes an intermittent or fluctuating flow and the flow rate isgoverned by the magnitude of the finite space of the machine and the frequency withwhich it is filled and emptied. The term "pump" is used when the fluid is a liquid,

Academic year: 2006-2007 Page 1 of 16


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however, when the fluid is a gas, terms such as compressors, or fans (or blowers) areused. A compressor is a machine whose primary objective is to increase the pressure ofthe gas, which is accompanied by an increase in the density of the gas. A fan or blower isa machine whose primary objective is to move the gas. Static pressures remain almostunchanged, and therefore the density of the gas is also not changed.

Basic Equations for Rotodynamic MachineryThe flow patterns in rotodynamic machines are three-dimensional, with significantviscous effects and flow separation patterns. However, an analytical analysis requiressimplification of the actual conditions. Therefore, viscosity is neglected and an idealizedtwo-dimensional flow is assumed throughout the rotor region. Figure 2 below defines thecontrol volume that encompasses the impeller region, which contains a series of vanes. Inthe control volume, the flow enters through the inlet control surface and exits through theoutlet surface.

1

2

Outlet

Inlet

b2

b1

Figure 2

A portion of the control volume, at an instant in time, is shown in Figure 3 below. Theidealized velocity vectors are diagrammed at the inlet, location 1, and the outlet, location

2. The parameters used in the diagram are defined below.v = absolute velocity of fluidu = peripheral (or circumferential) velocity of blade ( r u = )R = fluid velocity relative to bladevw = velocity of whirl, i.e., component of absolute velocity of fluid in direction

tangential to the runner circumferencevn = component of absolute velocity of fluid in a direction normal to the runner

circumference



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r = radius from axis of runner = angular velocity of runnerSuffix 1 refers to conditions at inlet to runnerSuffix 2 refers to conditions at outlet from runner.

Axis of Rotation

1

2

r 1

r 2

u1vw1

vn1v1R1 = v 1-u 1

1 1

u2vw2

vn2v2

R2 = v 2-u 2

2

2

x2

x1

Figure 3

The power passed on to the rotor from the fluid (or the power passed to the fluid from therotor) is due to the change in tangential momentum. There may be changes ofmomentum in other directions also, but the corresponding forces have no moments aboutthe axis of rotation of the rotor.

Now, the torque about a given fixed axis is equal to the rate of increase of angularmomentum about the axis. Therefore, the torque on the fluid must be equal to the angularmomentum of the fluid leaving the rotor per unit time minus the angular momentum ofthe fluid entering the rotor per unit time.The angular momentum of fluid entering the control volume per second, I 1, is the productof the mass flow rate ( m&), the tangential velocity (v w1) and the radius (r 1).

111 r vm I w&=



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The angular momentum of fluid leaving the control volume per second, I 2, is the productof the mass flow rate ( m&), the tangential velocity (v w2) and the radius (r 2).

222 r vm I w&= However, the mass flow rate, m&, is the product of the fluid density, , and the flow rate,Q.

Qm =& And the torque, T, is given by ( )1122 r vr vQT ww =

The power, P, is the product of torque, T, and the angular velocity, . Therefore,( ) ( ) 11221122 r vr vQr vr vQT P wwww ===

However, r 1 = u 1 and r 2 =u 2. Therefore,( )1122 uvuvQP ww =

Now, 111cos vv

w=

and 222cos vv

w=

Therefore, ( )111222 coscos uvuvQP =

The power transferred between the fluid and the rotor is given by QgH , where H is the pressure head across the rotodynamic machine.

For an idealized situation, where there are no losses( )111222 coscos uvuvQQgH =

And ( )g uvuv H 111222 coscos = , where H is the theoretical pressure head across the

machine.

In the case of a turbine, H will be negative (as energy is extracted from the fluid) and it iscommon to use the reverse order of terms in the brackets, for the above equation. For

turbines,( )

guvuv

H 222111coscos

= .

Basic Equations for Centrifugal PumpsHowever, for pumps, H retains the form,

( )g

uvuv H 111222

coscos = and can be

applied to centrifugal pumps.

If the whirl or tangential component of the velocity is zero at the inlet ( 1 = 90), then theflow is completely radial and the maximum head is obtained.



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Since 0cos 111 = vu ,( )

g

uv H 222max

cos =

Now, 222

22 cottan

n

n vv

x == and 222 xuvw =

Therefore, 2222 cot nw vuv = and 22222 cotcos nvuv =

( ) ( )2

22

22222

222222

max cotcotcot

g

vu

g

u

g

vuu

g

vuu H nnn =

=

=

By continuity, 22222 2 nn vbr v AQ == , where A 2 is the area perpendicular to v n2.

This implies 222

222

max cot2

br Q

gu

gu

H =

However, u 2 = r 2 and 22

222

222

222

2max cot2

cot2

b

Qgg

r br

Qg

r g

r H ==

And generally, one can write Qaa H 10max = , where a 0 and a 1 are constants for a givenrotodynamic pump. This theoretical head is linearly related to the discharge through the

pump; although the y-intercept, a 0, is constant, the slope of the line (a 1) is dependent onthe blade angle, 2. Figure 4 shows the effect of the blade angle. A forward-curving bladehave 2 >90, radial blades have 2 =90 whilst backward-curving blades have 2 90

2=90

2


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where H p is the actual head across the pump. If the overall pump efficiency, , is definedas the ratio of the output power to the input power, then the input power is given as

pQgH Power Input =

In addition, losses affect the H-Q curve, as shown in Figure 5.

H

2>90

2=90

2


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density of fluid [ML -3] viscosity of fluid [ML -1T-1]P Power transferred between fluid and rotor [ML 2T-3]

However, since g only enters the analysis as a variable associated with H , the term gH is

used, rather than g or H separately, to simplify the analysis. Gravity does not enter the problem in any other way.

Using the Tabular method, the following dimensionless s can be derived forgeometrically similar machines.

31 ND

Q= 222 D N

gH = 23 ND

= and 534 D N

P

=

Alternatively 3 may be re-written as

23

ND= , which resembles the Reynolds

number.

Similarity Laws applied to Centrifugal PumpsThese similarity relationships may be used to obtain the optimum performance when agiven pump is used in a different location or the performance of two different pumps.

These equations are used and based on the fact that most centrifugal pumps either:1. Have a variable speed motor, so that the pump can be changed to obtain the

required head-discharge relationship while retaining the same impeller (constant

D), or2. Have a constant speed motor, so that the pump speed is fixed (constant N) and

consequently different diameter impellers have to be used to vary the requiredhead-discharge relationship.

Case 1 Case 2

1

2

4

The relationships are known as the Affinity Laws .



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Q1.A pump is fitted with a variable speed motor. At 1200rpm it delivers 0.12m 3/s of water.What speed of rotation would be required to increase the discharge to 0.15m 3/s?

= 1500 rpm

Specific speedThe specific speed of a pump may serve as a guide for pump selection. Thedimensionless s may be used, and the dependence on the rotor diameter, D, iseliminated.

c1, c2, c3 and c 4 are constants

Where the dimensionless parameter, N s, isconsidered to be the Specific Speed. However since the acceleration due to gravity, g, isconstant, the specific speed is sometimes givenas the dimensional parameter, N s.



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N s 10 to 70 Centrifugalhigh head, relatively small discharge

N s 70 to 170 Mixed Flowmoderate load, moderate discharge

N s > 110 Axial Flow

low head, large discharge

N s is defined as the speed (rpm) at which the pump operating discharges 1m 3/s against1m head (that is, N=N s when Q=1m 3/s and H=1m).

The specific speed is calculated at the normal operating point of the pump (which may ormay not be at the maximum efficiency), with all other variable having the correspondingvalues. The specific speed is the same for all similar pumps (pumps in a homologousseries), regardless of size.

Performance Characteristics of Centrifugal PumpsPumps usually run at constant speed, and of particular interest are: The variation in head, H, with the discharge, Q The variation of efficiency, , with the discharge, Q The variation of Power, P, with the discharge, Q

These relationships are usually supplied by the pump manufacturer, and are provided as pump characteristic curves. Typical characteristic curves are shown in the figure below.

Head

Efficiency

Power

Discharge

Figure 6 Typical Pump Characteristic Curves

A particular machine may be tested at a fixed head while the load (and speed N) is varied,and for these results to be applicable to other pumps in the same homologous series, thecharacteristic curves is plotted using the dimensionless parameters:

, , and instead of Q, P, H and respectively.



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Selection of Centrifugal PumpsFactors that influence the selection of a pump for a given application are:

The head-discharge relationship (comparison of various H-Q curves) Efficiency Power requirements

Stability of the head-discharge relationships Operational Flexibility (operational conditions may change during use of the

pump)Generally, a pump with a relatively steep head-discharge curve is selected to promotestability during operation of the pump. If a pump has an H-Q curve where it is horizontal(or close to horizontal) over a given discharge range, this indicates that when pumpingagainst this head the discharge could fluctuate significantly in an uncontrolled mannercausing problems with surge and water hammer.

Consider the simplified pump system diagram shown in Figure 7 below.

PUMP

DISCHARGE LEVEL

SUCTION LEVEL

SUCTION PIPE

DELIVERY PIPE

Vs

Vd

hFD

HD

HS

hFS

HT

Figure 7

One can state that: HT = H

S + h

FS + H

D + h

FD H T total head against which the pump must dischargeH S static suction lift; which is the water level in the sump to the datum level of the

pumph FS head loss due to frictional losses and minor losses in the suction pipeH D static delivery lift; which is the head required to pump water to the level of thedischarge pointh FD head loss due to frictional losses and minor losses in the delivery pipe



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The total static head is the sum of H S and H D.

Head losses due to frictional losses may be determined using Darcys equation

where is Darcys friction factor, L is the length of the pipe, v is the mean velocity offlow in the pipe and d is the pipe diameter. Minor losses may include exit or entry losses.

The pump is normally located as close as possible to the sump or suction well to reducethe frictional loss in the suction pipe.

Once the system has been established for the pump, the system curve (or systemcharacteristic) can be determined. The system curve is the relationship between head anddischarge for the required system. The system curve is overlain on the pumpcharacteristic to determine the operating point.

Head

Discharge

System load curve

Pump Charcteristic Curve

Operating Point

Figure 8

Pumps in Series and in ParallelPumps may be used in series or in parallel to achieve the desired Head-Discharge

relationship. Consider two identical pumps in series, shown in Figure 9 below.

PUMP 1 PUMP 2 A B

Figure 9



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Since the discharge through pump 1 must equal the discharge through pump 2, one canwrite Q 1 = Q 2 = Q. However, the head across points A and B is twice the head for asingle pump, at a given discharge. Therefore, the H-Q characteristic curve for twoidentical pumps in series is obtained by doubling the head, at a given discharge, shown inFigure 10 below.

H

Q

H-Qcurve for single pump (Pump 1)

H-Qcurve for two identical pumps in series (Pump 1 + Pump 2)

Figure 10

Now, consider two identical pumps in parallel (at the same horizontal level), shown inFigure 11 below.

PUMP 1

A B

PUMP 2

Figure 11

Pumps 1 and 2 are identical, therefore at a given discharge Q, the head delivered will bethe same for each pump. Since the discharge entering at A (Q T) must equal the dischargeleaving at B (Q T), then by continuity Q T = Q 1 +Q 2, where Q 1 and Q 2 is the dischargethrough pump 1 and pump 2 respectively. For identical pumps Q 1 = Q 2 = Q, and Q T =2Q.However, the head across points A and B is equal to that for a single pump, at the givendischarge, Q. Therefore, the H-Q characteristic curve for two identical pumps in parallelis obtained by doubling the discharge, at a given head, shown in Figure 12 below.



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H

Q

H-Qcurve for single pump (Pump 1)

H-Qcurve for two identical pumps in parallel (Pump 1 + Pump 2)

Figure 12

CavitationCavitation occurs when pressure falls below vapour pressure (at the appropriatetemperature). The liquid boils and bubbles form in large numbers. At locations wherethe pressure increases again, these bubbles implode violently. This implosion results invery high velocities as the liquid rushes in to fill the void. High pressures and hightemperatures are also generated at these locations.

This acting at or near solid surfaces can cause damage including fatigue failure. This phenomenon is accompanied by noise and vibration and is very destructive. Apart from physical damage, cavitation causes a reduction in the efficiency of the machine. Everyeffort must therefore be made to eliminate cavitation and this can be done by ensuringthat the pressure is everywhere greater than the vapour pressure.

Conditions are favourable for cavitation where the velocity is high or the elevation is highand particularly where both conditions occur. For centrifugal pumps, the lowest pressureoccurs near the centre of the impeller where the water enters (Figure 13).

Applying the energy equation between the surface of the liquid in the supply reservoirand the impeller (where the pressure is minimum), for steady conditions:

where v is the velocity of flow where the pressure is minimum.However, the velocity in the supply reservoir, v o, is equal to zero and the equationreduces to:



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datum

Pmin

SUCTION LEVELz= HS

hFS

Po

Zo

Figure 13

Dividing throughout by g, the equation has the form:

and

The pressure in the supply reservoir, p o, is usually atmospheric pressure, but notnecessarily always.

However, for a given pump design, operating under specified conditions, may betaken as a particular portion of the head developed.That is,

This implies

c is related to the specific speed of the pump and is a property of the pump.

Now for cavitation not to occur, p min , must be greater than p v (where p v is the vapour pressure of the liquid at the operating temperature).

To prevent cavitation (or ) where



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The above expression is known as Thoma's Cavitation Parameter.

The value H is known as the Net Positive Suction Head (NPSH), and is given by:

Therefore,

To avoid cavitation one can ensure that the NPSH or is as large as possible. Techniquesused to increase the NPSH include:

Increase in the sump pressure (or the pressure of the supply reservoir, p o) Decrease in the vapour pressure, p v, by reducing the temperature of fluid Decrease the losses in the suction pipe, h FS Decrease the static suction lift, z

At the point where cavitation will occur c = and

If p o, p v are considered to be constant, for a given application, and h FS is considered to benegligible, one can write

Therefore, the maximum value of z, z max , can be obtained from

Since for this given application, z max gives the value of z at which cavitation would occur,

if for this given p o and p v, then cavitation is prevented.



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