2 centrifugalpumptheory inviscidfluid singlephase

5/27/2018 2 CentrifugalPumpTheory InviscidFLuid SinglePhase

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Advanced Artificial Lift Methods

Electrical Submersible Pump

Advanced Artificial Lift MethodsPE 571

Chapter 1 - Electrical Submersible Pump

Centrifugal Pump TheoryInviscid FluidsSingle Phase
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ESPs are multi stage centrifugal pumps. The two main components of a

centrifugal pump are the impeller and the diffuser.

The Impeller takes the power from the rotating shaft and accelerates the fluid.

The diffuser transforms the high fluid velocity (kinetic energy) into pressure.

Theoretical Head Developed by an Impeller

Principles of an Centrifugal Pump
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The main components of an ESP including:

Impellers Casing

Diffusers Shaft

Thrust washers Bushing

Impeller

Washer

Diffuser

Geometry of an Centrifugal Pump

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Geometry of an Centrifugal Pump

Impeller

Diffuser

Impeller

Diffuser
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True Velocity Profile of Fluid Inside an Impeller

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Assumptions:

1. Two dimensions: radial and tangential direction.

2. The impeller passages are completely filled with the flowing fluid at all time

(no void spaces)

3. The streamlines have a shape similar to the bladesshape

4. Incompressible, inviscid, and single phase fluid

5. The velocity profile is sysmetric.

The head calculated based on these assumptions is known as the

theoretical head

Assumptions

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Velocities at the intake and outlet of an impeller

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Exit Velocity Triangle

Entrance Velocity Triangle


Velocities at the intake and outlet of an impeller
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Velocity at One Point on the Impellers Blade
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Triangle Fluid Velocity


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Known 3 operational parameters:

1. Angle, b: knowing pump blade geometry

2. Tangential velocity, U: knowing the rotational speed

3. Radial velocity, vr: knowing the flow rate.

Therefore, the velocity triangle is completely determined.

What we need now is to find the pressure increment developed by one impelleras a function of those 3 operational parameters and the fourth one, namely the

fluid density


Conclusion on Triangle Fluid Velocity


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Based on a Free Body Diagram

r R + dr


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Mass Balance

Mass balance equation under steady state conditions in cylindrical coordinate:

Note that the fluid at the outlet of the impeller has two components: v rand vq.

However, the change of vq respect to qis zero.

Hence: constant



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Mass Balance

The flow rate entering the pump intake is given (ri= r):

or

Rotational speed is related to the tangential velocity U by:

Hence, we know three parameters:



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Mass Balance

Three parameters:

Combining with the triangle velocity gives:


Ad d A ifi i l Lif M h d


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Momentum Equation

For S.S; incompressible and single phase fluid; the momentum equations in the

cylindrical coordinates are given:


Ad d A tifi i l Lift M th d


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Total Pressure Losses Along the Streamline

If the fluid is inviscid; No change of velocity in z and q (symmetric velocity)

direction; Neglect the pressure drop due to gravity:

Total derivative of pressure respect to the radius:

Therefore:


Streamline Trajectory



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Streamline Geometric Relationship




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Therefore, the total pressure losses along the streamline can be express as:

From the triangle geometric relationship:

Hence:




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Simplifying this equation gives




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Finally, the pressure difference across a streamline is given:

Integrate this equation gives the pressure increase across one stage:

By definition:

Hence:




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Using the geometrical relationships:

This equation can be expressed as the Euler Equation:

Field unit:




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Pump Head Definition

Definition for the pump head:

Head is an indirect measurement of pressure that does not depend on the fluid

density. That means for low viscous fluids, the pump performance can b uniquely

defined in terms of head.

In other words, the pump performance, in pressure, depends on the density ofthe fluid being pumped, but when this performance is expressed in head, the

pump performance is independent of the fluid being pumped




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Pump Head Definition




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Head Losses

Due to the Leakage and recirculation of fluid inside the impleller.

Hydraulic losses including:

Diffusion loss due to divergence, or convergence

Fluid shock loss at the inlet

Mixing and eddying loss at the impeller discharge

Turning loss due to turning of the absolute velocity vector

Separation losses

Friction losses

Mechanical losses




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Leakage and Recirculation Losses


Recirculation

Leakage



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Theoretical diagram

Diagram with recirculation




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Theoretical head (Euler head)

Leakage/Recirculation losses

Flow rate, Q

Head,

H



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Hydraulic Losses

Pumps are designed trying to achieve a no pre-rotation condition close to the

best efficiency point, since this condition minimize shock-losses. In other words,

shock losses increase as we move away from the BEP.




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Hydraulic Losses

Other losses including friction, mixing, change in direction of fluid, separation,

etc. also contribute significantly to the total losses due to hydraulic.



Hydraulic losses

Flow rate, Q

Head,

H



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Friction Losses

Friction losses increases with increasing flowrate and viscosity.



Friction losses

Flow rate, Q

Head,

H



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Mechanical Losses

These losses include disk friction and frictional losses in bearings. The most

significant loss is the thrust bearing loss. The mechanical losses do not have any

effect on head and capacity of a pump but increase the brake hoursepower.




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Total Losses



Flow rate, Q

Head,

H

Leakage/Recirculation losses

Hydraulic losses

Friction lossesActual Head



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The hydraulic horsepower is the energy transmitted to the fluids by the pump.

The break horsepower is the energy required by the pump shaft to turn. Some of

this energy is dissipated inside the pump.

The ratio between the hydraulic horsepower and the break horsepower is the

pump hydraulic efficiency.


Horsepower



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In practice, a pump is tested by running it at a constant speed and varying the

flow by controlling the choke. During the test, Q, DP, and the break horsepower

are measure at several points. The DP is then converted to head and the overal

efficiency of the pump is calculated. Based on these data, we can develop the

pump performance.

The performance curve of a centrifugal pump can be summarized in only one

curve of head vs. flowrate for all low viscous fluids.


Pump Performance



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Pump Performance



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Manufacturers also provide polynomial equations to describe the catalog pump

performance curves.


Pump Performance



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Do the calculation for these correlations:


Pump Performance
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2 centrifugalpumptheory inviscidfluid singlephase

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