1. Performance of Centrifugal Pumps
9/24/2012
Performance of Centrifugal Pumps
Fluid Machines:
Positive displacement
Energy transfer occurs as a result of the movement of the volume boundaries in which the fluid is enclosed in.
Dynamic
These are devices that use vanes or blades to direct the fluid. The fluid is not confined to a volume
Positive Displacement Gear pump
Dynamic
ME 322 THERMO-FLUIDS LAB-1
11/17/2018
ME 383 Fluid Mechanics

2. Performance of Centrifugal Pumps
9/24/2012
Performance of Centrifugal Pumps
Fluid Machines:
Fluid machines can also be categorized according to the geometry of the fluid path into,
Radial (centrifugal)
Flow direction is mainly in the radial direction
Axial
Flow direction is mainly in the axial direction
ME 322 THERMO-FLUIDS LAB-1
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ME 383 Fluid Mechanics

3. Performance of Centrifugal Pumps
9/24/2012
Performance of Centrifugal Pumps
Fluid Machines: Turboprop Engine
Axial Turbine
Centrifugal Compressor
Axial Compressor
ME 322 THERMO-FLUIDS LAB-1
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4. Performance of Centrifugal Pumps
9/24/2012
Performance of Centrifugal Pumps
Fluid Machines:
Theoretical principle: Angular Momentum Concept
The angular momentum is defined as the moment of momentum
𝐻 = 𝒱 𝑟 × 𝑉 𝜌𝑑𝒱
The conservation of angular momentum applied to a CV
𝜕 𝜕𝑡 𝐶𝑉 𝑟 × 𝑉 𝜌𝑑𝒱 + 𝐶𝑆 𝑟 × 𝑉 𝜌 𝑉 ∙𝑑 𝐴 = 𝑇 𝑆ℎ𝑎𝑓𝑡 + 𝑟 × 𝐹 𝑠 + 𝐶𝑉 𝑟 × 𝑔 𝑑𝑚
This is a general equation applied to any fluid machine. A much simpler equation can be obtained using few simplifying assumptions.
ME 322 THERMO-FLUIDS LAB-1
11/17/2018
ME 383 Fluid Mechanics

5. Performance of Centrifugal Pumps
9/24/2012
Performance of Centrifugal Pumps
Fluid Machines:
Theoretical principle: Angular Momentum Concept
𝜕 𝜕𝑡 𝐶𝑉 𝑟 × 𝑉 𝜌𝑑𝒱 + 𝐶𝑆 𝑟 × 𝑉 𝜌 𝑉 ∙𝑑 𝐴 = 𝑇 𝑆ℎ𝑎𝑓𝑡 + 𝑟 × 𝐹 𝑠 + 𝐶𝑉 𝑟 × 𝑔 𝑑𝑚
Assumptions;
Steady flow
Torque due to body forces and viscous forces are negligible
Torque due to pressure changes is neglected
We get
𝐶𝑆 𝑟 × 𝑉 𝜌 𝑉 ∙𝑑 𝐴 = 𝑇 𝑆ℎ𝑎𝑓𝑡
If we apply this equation to a CV enclosing the fluid machine and further assuming that the flow enters the machine at radius 𝑟 1 and exits at 𝑟 2 .
ME 322 THERMO-FLUIDS LAB-1
11/17/2018
ME 383 Fluid Mechanics

6. Performance of Centrifugal Pumps
9/24/2012
Performance of Centrifugal Pumps
Fluid Machines:
𝐶𝑆 𝑟 × 𝑉 𝜌 𝑉 ∙𝑑 𝐴 = 𝑇 𝑆ℎ𝑎𝑓𝑡
Flow is uniform at inlet and exit.
Axe of rotation is in the z-direction.
Applying the above assumptions we get,
𝑇 𝑆ℎ𝑎𝑓𝑡 = 𝑇 𝑠ℎ𝑎𝑓𝑡 𝑘 = 𝑟 2 𝑉 𝑡 2 − 𝑟 1 𝑉 𝑡 1 𝑚 𝑘
Hence, in scalar form
𝑇 𝑠ℎ𝑎𝑓𝑡 = 𝑟 2 𝑉 𝑡 2 − 𝑟 1 𝑉 𝑡 1 𝑚
This is called the , the Euler Turbomachine Equation
And the rate of work done is given by,
𝑊 = 𝜔 ∙ 𝑇 𝑆ℎ𝑎𝑓𝑡
𝑊 =𝜔 𝑇 𝑠ℎ𝑎𝑓𝑡 =𝜔 𝑟 2 𝑉 𝑡 2 − 𝑟 1 𝑉 𝑡 1 𝑚 OR
𝑊 = 𝜔 𝑟 2 𝑉 𝑡 2 −𝜔 𝑟 1 𝑉 𝑡 1 𝑚 = 𝑈 2 𝑉 𝑡 2 − 𝑈 1 𝑉 𝑡 1 𝑚
ME 322 THERMO-FLUIDS LAB-1
11/17/2018
ME 383 Fluid Mechanics

7. Performance of Centrifugal Pumps
9/24/2012
Performance of Centrifugal Pumps
Fluid Machines:
𝑊 = 𝜔 𝑟 2 𝑉 𝑡 2 −𝜔 𝑟 1 𝑉 𝑡 1 𝑚 = 𝑈 2 𝑉 𝑡 2 − 𝑈 1 𝑉 𝑡 1 𝑚
If we divide by 𝑚 𝑔 we get,
𝐻= 𝑊 𝑚 𝑔 = 1 𝑔 𝑈 2 𝑉 𝑡 2 − 𝑈 1 𝑉 𝑡 1
This is defined as the theoretical pump head (rate of work done per rate of weight flow rate).
For a centrifugal pump assuming the inlet flow is axial, 𝑉 𝑡 1 =0, then
𝐻= 𝑈 2 𝑉 𝑡 2 𝑔
How to evaluate 𝑉 𝑡 2 ?
ME 322 THERMO-FLUIDS LAB-1
11/17/2018
ME 383 Fluid Mechanics

8. Performance of Centrifugal Pumps
9/24/2012
Performance of Centrifugal Pumps
Fluid Machines:
How to evaluate 𝑉 𝑡 2 ?
𝑉 𝑡 2 = 𝑈 2 − 𝑊 2 cos 𝛽 2 = 𝑈 2 − 𝑉 𝑛 2 sin 𝛽 2 cos 𝛽 2 = 𝑈 2 − 𝑉 𝑛 2 cot 𝛽 2
𝐻= 𝑈 2 𝑈 2 − 𝑉 𝑛 2 cot 𝛽 2 𝑔
ME 322 THERMO-FLUIDS LAB-1
11/17/2018
ME 383 Fluid Mechanics

9. Performance of Centrifugal Pumps
9/24/2012
Performance of Centrifugal Pumps
Fluid Machines:
How to evaluate 𝑉 𝑡 2 ?
𝐻= 𝑈 2 𝑈 2 − 𝑉 𝑛 2 cot 𝛽 2 𝑔
For an impeller of width w, the volume flow rate is,
𝑄=𝜋 𝐷 2 𝑤 𝑉 𝑛 2
Hence, we can write the theoretical head as,
𝐻= 𝑈 2 2 𝑔 − 𝑈 2 cot 𝛽 2 𝜋 𝐷 2 𝑤𝑔 𝑄
Then we can write in general,
𝐻= 𝐶 1 − 𝐶 2 𝑄
Where 𝐶 1 and 𝐶 2 are functions of the machine geometry and speed.
𝐶 1 = 𝑈 2 2 𝑔 , 𝐶 2 = 𝑈 2 cot 𝛽 2 𝜋 𝐷 2 𝑤𝑔
ME 322 THERMO-FLUIDS LAB-1
11/17/2018
ME 383 Fluid Mechanics

10. Performance of Centrifugal Pumps
9/24/2012
Performance of Centrifugal Pumps
Fluid Machines:
ME 322 THERMO-FLUIDS LAB-1
11/17/2018
ME 383 Fluid Mechanics

11. Performance of Centrifugal Pumps
9/24/2012
Performance of Centrifugal Pumps
Fluid Machines:
How to compute real pump performance ?
Use the energy equation
𝐻 𝑝 = 𝑝 𝜌𝑔 + 𝑉 2 2𝑔 +𝑧 𝑑 − 𝑝 𝜌𝑔 + 𝑉 2 2𝑔 +𝑧 𝑠
Where the subscript d and s denote the discharge and suction sides, respectively.
𝐻 𝑝 = 𝑝 𝑑 − 𝑝 𝑠 𝜌𝑔 𝑉 𝑑 2 − 𝑉 𝑠 2 2𝑔 + 𝑧 𝑑 − 𝑧 𝑠
𝐻 𝑝 = 𝐻 𝑠𝑡𝑎𝑡𝑖𝑐 + 𝐻 𝑘𝑖𝑛𝑒𝑡𝑖𝑐 + 𝐻 𝑒𝑙𝑒𝑣𝑎𝑡𝑖𝑜𝑛
In the current experiment, the total pressure head, 𝐻 𝑝 , is calculated by measuring 𝐻 𝑠𝑡𝑎𝑡𝑖𝑐 , 𝐻 𝑘𝑖𝑛𝑒𝑡𝑖𝑐 , 𝐻 𝑒𝑙𝑒𝑣𝑎𝑡𝑖𝑜𝑛 at a constant speed.
ME 322 THERMO-FLUIDS LAB-1
11/17/2018
ME 383 Fluid Mechanics

12. Performance of Centrifugal Pumps
9/24/2012
Performance of Centrifugal Pumps
Fluid Machines:
Similarity Rules
For two different pumps to be dynamically similar, assuming geometric similarity, we must have the flow coefficient satisfy the equality,
𝑄 1 𝜔 1 𝐷 1 3 = 𝑄 2 𝜔 2 𝐷 2 3
The dimensionless head and power coefficients are functions of the flow coefficient, such that
ℎ 𝜔 2 𝐷 2 = 𝑓 1 𝑄 𝜔 𝐷 3 𝑎𝑛𝑑 𝒫 𝜌 𝜔 3 𝐷 5 = 𝑓 2 𝑄 𝜔 𝐷 3
𝐻= ℎ 𝑔 . Hence for similar flow coefficient we have,
ℎ 1 𝜔 1 2 𝐷 1 2 = ℎ 2 𝜔 2 2 𝐷 2 2
And
𝒫 1 𝜌𝜔 1 3 𝐷 1 5 = 𝒫 2 𝜌 𝜔 2 3 𝐷 2 5
ME 322 THERMO-FLUIDS LAB-1
11/17/2018
ME 383 Fluid Mechanics