Turbomachinary
^{[1]}This article discusses some of fluid mechanics' biggest applications: Turbomachinaey. Turbomachinery is any device that uses a rotating shaft to supply or extract energy from a system. Turbomachinary can therefore be split into two broad categories:
- Turbines: machines that extract energy from a (fluids) system
- Pumps: machines that add energy to a (fluids) system
Note that not all turbomachines use rotating shafts (such as a hand pump) but they are still considered under turbomachinery.
This article discusses terminology and classifications as well as calculations and methods of matching the correct turbomachinery to the correct system.
Contents |
Textbook Readings
Cengel and Cimbala, Fluid Mechanics: Fundamentals and Applications, (2nd ed, Singapore, McGraw Hill Education, 2010), pp. 761 - 837 (Chapter 14).
Classification and Terminology
^{[2]}As discussed in the introduction, turbomachinery can be split into two broad groups: pumps and turbines. Note that both of these devices add or subtract energy from a system by changing the fluid pressure rather than its velocity. This is easily proven by a pump whose inlet and outlet diameters are the same (and hence, by continuity, the velocities are the same) and whose inlet and outlet have the same elevation. From the general energy equation we can see that the work done by a pump (or extracted by a turbine) is given by the pressure difference.
Further classification of turbomachinery is into:
- Positive displacement machines which are machines that transfer fluid into a closed volume, attempt to compress or expand the fluid and increase or decrease its pressure, thereby sucking in or blowing fluid out. Examples of such machines are the heart or the water meter in households.
- dynamic machines which don't have a closed volume.
Within these machines, common components are Impeller blades/Runner blades/Buckets - these are the blades of the fan/pump/turbine. The first name i used for pumps while the latter two is used for turbines
Pumps
Pump Efficiency
^{[3]}Generally speaking, the useful head that a pump provides can be found from the general energy equation by rearranging for the h_{pump} term. Thats is:
By multiplying the useful head by mass flow rate and gravitational acceleration, we can ccalculate the amont of power provided, known as the water horse power even though the units are not in horsepower and the fluid may not be water). That is,
Where H is the useful head calculated from the general energy equation. The amount of power that the pump has, or can have, is known as the break horse power and is given by:
- bhp = Ẇ_{shaft} = ωT_{shaft}
Where ω is the rotational speed in radians/sec.
The pump efficiency is therefore given by:
Pump Cavitation and Net Positive Suction Head
^{[4]}When pumping fluids, it is often possible for the pressure to drop below the cavitation pressure which causes some of the fluid to effectively boil locally and cause bubbles to form. These bubbles are then transported to the rest of the fluid where they collapse and introduce turbulence and friction, and can eventually damage the pump blades. To limit this we make sure that the pressure within the pump remains above the vapor pressure by defining the net positive suction head (NPSH) where:
i.e. it is the difference between the pump's inlet (for simplicity) stagnation pressure head and the vapour pressure head. Pump manufacturers usually give this value for different flow rates and it is the minimum pressure required to remain above the vapor pressure and eliminate cavitation.
Dynamic Pumps
^{[5]}There are three main types of Dynamic pumps:
- Centrifugal pumps
- Axial pumps
- Mixed pumps
The relevant calculations for each is discussed below.
Centrifugal Pumps
^{[6]}In centrifugal pumps, the fluid enters axially and leaves radially. That is, the fluid enters at the center of the pump, in the same direction as the shaft and then is discharged tangentially along the radius of the blades.
There are three types of centrifugal pump blades:
- Backward inclined blades- most common, yield the highest efficiency and produce intermediate pressure difference (relative to the other two types)
- Straight blades (or radial blades)- produce the largest pressure rise for the most amount of volume flow rates, but this pressure reduces rapidly beyond the point of maximum efficiency
- Forward inclined blades- constant pressure rise, though lower than the other two
The volume flow rate for these pumps is given by:
Where:
- r_{1} = small radius
- r_{2} = large radius
- V_{1,n} = normal velocity at r_{1}
- V_{2,n} = normal velocity at r_{2}
- b_{1} = circumferential cross-sectional area at r_{1}
- b_{2} = circumferential cross-sectional area at r_{2}
The Torque provided by the pump is given by:
Where:
- V_{1,t} = tangential velocity at r_{1}
- V_{2,t} = tangential velocity at r_{2}
Which, when we define the angle α to be the angle between the velocity vector and the normal component, can be rewritten as:
Then, the work provided by the shaft is given by:
And the pump efficiency can be calculated as explained above.
Axial Pumps
^{[7]}Axial pumps differ from centrifugal pumps in that the flow is completely axial. The pumps are actually fans, having blades that act like the wing of a plane and create high flow rates. The blades usually have some twist, with the intention of gaining a lower pitch angle (θ) at the tip than at the root. This is because the tangential speed of the blade increases with radius, that is:
- u_{0} = ωr
At a given radius, the the relative velocity between the fluid and the blades is given by:
- V_{relative} ≈ V_{in} - V_{blade}
Notice here that the velocities are added as vectors. The magnitude of V_{blade} is given by the equation above, i.e. V_{blade} = u_{0} = ωr.
Now we define the angle of attack, α, to be the angle between the blade and the relative velocity of the fluid and for efficiency we keep this angle constant. In fact, the twist in the blades is made for this reason. Since the relative velocity increases towards the tip of the blade, it means that the amount of lift/thrust gained towards the end of the blade is higher than at the root and this is why fan blades usually get larger towards the tip.
There is usually swirl associated with axial pumps which can cause problems, especially when transferring fluids in ducts or pipes. To counter these effects, one could:
- Install a second axial pump in series with the first, but turning in the opposite direction to counter the swirl
- Install guiding stator blades that don't move but guide the flow to counter the swirl
Turbines
^{[8]}Just like pumps, turbines come in two types: positive displacement turbines and dynamic turbines. Positive displacement turbines are the same as positive displacement pumps running in reverse, however they are rarely used for power generation and are rather used for flow measurements.
^{[9]}The calculations for the efficiency of turbines is the same as with pumps except that the terms are inverted. Instead of efficiency being the water horsepower over the break horse power, the efficiency is calculated by taking the break horse power (i.e. the energy generated) over the water horse power (theoretical energy supplied by fluid).
Dynamic Turbines
^{[10]}There are two main types of dynamic turbines:
- Impulse turbines- the fluid is made to go through a nozzle to convert most of its available energy into kinetic energy and then to hit bucket shaped containers on a wheel, making the wheel spin
- Reaction Turbines - the fluid enters tangentially at high pressure and is guided by guide vanes towards blades, which on impact, cause them to spin
References
"Textbook" refers to Cengel and Cimbala, Fluid Mechanics: Fundamentals and Applications, (2nd ed, Singapore, McGraw Hill Education, 2010).