Centrifugal Pumps: Elementary Theory

The actual flow patterns in a turbopump are highly threedimensional with significant viscous effects and separation patternst ki l T t t i lifi d th f th di l fltaking place. To construct a simplified theory for the radial-flowpump, it is necessary to neglect viscosity and to assume idealizedtwo-dimensional flow throughout the impeller region.

Consider a control volume thatencompasses the impellerregion. Flow enters through theinlet control surface and exitsthrough the outlet surface. Notegthat a series of vanes existswithin the control volume, andthat they are rotating about the

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y gaxis with an angular speed .

Pump theory

A portion of the controlvolume is shown at ani i i

V2instant in time.

The idealized velocityvectors are diagrammed atthe inlet, location 1, and theoutlet, location 2.

In the velocity diagrams, Vis the absolute fluid velocity,

V1

Vt is the tangentialcomponent of V, and Vn isthe radial, or normal,component of V.

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Pump theory

The peripheral orcircumferential speed of thebl d i h i h Vblade is u =r, where r is theradius of the control surface.

V2

The angle between V and uis . The fluid velocitymeasured relative to thevane is v.

The relative velocity is

V1

assumed to be alwaystangent to the vane; that is,perfect guidance of the fluidp gthroughout the controlvolume takes place.

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Pump theory

The angle between u and vis designated as . V2Since perfect guidancealong the vane is assumed, designates the blade angle aswell.

Equating the torque T actingon the fluid to the flux ofangular momentum (= mass

V1

x tangential velocity xradius) through the controlvolume we get for steadyg yflow

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Pump theory

The power delivered to thefluid is V2

From the velocity vectordiagrams V = Vcos sodiagrams, Vt Vcos , sothe above Eq. can be writtenas

[1]For the idealized situation in

V1

which there are no losses,the delivered power must beequal to QHt, in which Ht isq Q t tthe theoretical pressure headrise across the pump.

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Pump theory

Then from Eq. [1], we get Euler'sturbomachine relation, V2

[2] V[2]Insight on the nature of flow throughan impeller region can be obtainedi hi E F h l f i

V1

using this Eq. From the law of cosineswe can write

These can be substituted into the Eulers relation to provide,

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Pump theory

[3]

Th fi h i h h d id h i i ki iThe first term on the right-hand side represents the gain in kineticenergy as the fluid passes through the impeller; the second termaccounts for the increase in pressure across the impeller. This canbe seen by applying the energy equation across the impeller andsolving for Ht:

[4]

Eliminating Ht, between Eqs. [3] and [4] and neglecting z2 z1, asit is often much smaller than (p2 - p1)/, we get the pressuredifference

This is equivalent to the second term in Eq. [3].20

Pump theory

[2]

Now returning to Eulers Equation, it can be seen that a "bestdesign" for a pump would be one in which the angular momentumentering the impeller is zero, so that maximum pressure rise cantake place. Then 1 = 90o, Vn1 = V1, and Eq. [2] becomes

[5]

From the geometry of the outlet velocity triangle, V2

So, Eq. [5] takes the formq

[6]21

Pump theory

Applying the continuity principle at the outlet region to the controlvolume provides the relation

[7]

In which b is the width of the impeller at location 2 IntroducingIn which b2 is the width of the impeller at location 2. IntroducingEq. [7] into Eq. [6], and recalling that u2= r2, we get

[8][8]

If Q = 0, Eq. [8] gives the shut-off head (i.e. the head at zerodi h ) fdischarge) for a pump as

[9]Th ll d h ff h d f if l i lThe actually measured shut-off head of centrifugal pumps is onlyabout 60 percent of the theoretical value given by Eq. [9].

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Pump theory

For a pump running at constant speed, Eq. [8] takes the form

[10][10]

in which ao and a1 are constants. Equation [10] gives thetheoretical head curve and is seen to be a straight line with agnegative slope.

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QThe effect of the blade angle 2 is shown in the figure. A forwardcurving blade (2 > 90o) can be unstable and cause pump surge,where the pump oscillates in an attempt to establish an operatingpoint. It may cause only rough operation in a liquid pump, but it canbe a major problem in compressor operation.

Backward curving vanes (2 < 90o) are generally preferred.24

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