TURBOMACHINES
[ physical interpretation: what are we doing today? ]

Turbomachines are fluid machines that are based on a
spinning rotor

The rotor will typically have blades, deflectors, or buckets on it
to effect interaction with the fluid

We can loosely divide turbomachines into two categories,
pumps and turbines

Pumps add energy to the fluid and turbines remove it
87-351 Fluid Mechanics
TURBOMACHINES
[ physical interpretation: what are we doing today? ]

Who cares?
87-351 Fluid Mechanics
TURBOMACHINES
[ radial, mixed, and axial flow machines ]

Turbomachines can be further subdivided into three other categories
depending on whether they are radial, mixed flow, or axial flow configurations

This is defined by the manner in which the flow moves relative to the
machine rotor
axial flow turbomachine
radial flow turbomachine

In radial flow machines, there exists a significant radial flow component at the
inlet, exit, or both -- in mixed (not pure radial) machines, the flow can have
some radial and axial components through the rotor row
87-351 Fluid Mechanics
TURBOMACHINES
[ centrifugal pumps
]

Centrifugal pumps represent one of the most common
radial flow turbomachines

There are two main components to the machine, a rotating impeller and a
stationary housing or “volute”
a centrifugal pump
87-351 Fluid Mechanics
TURBOMACHINES
[ centrifugal pumps
]




As the impeller rotates, it pulls fluid in through the eye at its
centre and then is thrown radially outward to the walls of the
The
casings are generally shaped to reduce the velocity and
casing
kinetic energy of the flow, converting this to a gain in pressure
energy
a centrifugal pump
Pumps can be single or double suction (double suction reduces
inlet
velocity)
Pumps
can also be single or multistage --- discharge from the
first impeller flows into the eye of the second stage, each stage
augments the pressure
87-351 Fluid Mechanics
TURBOMACHINES
[ centrifugal pumps: theory ]


For analysis, we simplify the three dimensional, unsteady flow in a pump
to a steady (in the mean) one dimensional flow
We consider simple vector triangles to resolve the velocity directions and
magnitudes at pump inlet and outlets
a centrifugal pump
87-351 Fluid Mechanics
TURBOMACHINES
[ centrifugal pumps: theory ]

The absolute velocity, V, of the flow entering or leaving the passage is a
vector sum of the blade velocity, U, and the relative velocity W
V = W + U - [1]
where:
U1 = r1w - [2]
U2 = r2w - [3]

here 1 denotes entrance
conditions and 2 denotes exit
87-351 Fluid Mechanics
TURBOMACHINES
[ centrifugal pumps: theory ]

We know from the moment of momentum equation that the torque
required to rotate the pump impeller is given by
- [4]
- [5]

here the Vq1 and Vq2 are the
tangential components of the
absolute velocities V1 and V2
87-351 Fluid Mechanics
TURBOMACHINES
[ centrifugal pumps: theory ]

Quantification of the power added to the fluid by the pump can be easily had
by examining the following
 We know
- [6]

subbing in our expression
[5] for Tshaft
- [7]

which we can write
(employing U = wr)
- [8]

[8] shows us how power is
transferred to the fluid
87-351 Fluid Mechanics
TURBOMACHINES
[ centrifugal pumps: theory ]

It is also important for us to quantify the head a pump supplies to a fluid,
this can be had via [9]
- [9]

combining [9] with [8] we can write
- [10]

[10] represents the ideal head rise a
fluid experiences in passing through
a pump

We realize that this amount will be
ultimately compromised by the head
losses through the pump components
87-351 Fluid Mechanics
TURBOMACHINES
[ centrifugal pumps: theory: net positive suction head (NPSH) ]

Pressures can become very low on the suction side of a pump

In some situations pressures can drop to below the vapor pressure of the
fluid, at this pressure bubbles will form in the liquid and the liquid will
effectively “boil” at the current temperature

Cavitation can significantly reduce efficiency and cause the
pump structural damage
It is the difference between the total head on the suction side
near the pump impeller inlet, ps/g + V2s/2g, and the liquid vapor
pressure head, pv/g that characterizes the potential for cavitation
This difference is called the NPSH,
- [11]
where



There are two types of NPSH, the NPSH required, NPSHR and
the NPSH available, NPSHA
87-351 Fluid Mechanics
TURBOMACHINES
[ centrifugal pumps: theory: net positive suction head (NPSH) ]



The NPSHR refers to that amount of head that must be maintained or
exceeded to avoid cavitation
The NPSHA refers the head that actually occurs for the entire hydraulic
system, we can determine this value by calculation if the system
parameters are known, otherwise it is determined from experiment
If we apply the energy equation between the liquid free surface and
suction side of the impeller, we get
- [12]

Where ShL represents the losses from the free surface to the impeller inlet

Therefore, the head available at the inlet is
- [13]
87-351 Fluid Mechanics
TURBOMACHINES
[ centrifugal pumps: theory: net positive suction head (NPSH) ]

Then we can say
- [14]

Then we can say that to successfully avoid cavitation
- [15]

We learn from [14] that as the pump elevation, z, or hL increase,
the NPSHA decreases, thus there is always a finite height (above
some datum) at which the pump will not operate
87-351 Fluid Mechanics
TURBOMACHINES
[ centrifugal pumps: theory: net positive suction head (NPSH): example ]
GIVEN: Water is pumped at
0.5 cfs, at this
flowrate the NPSHR
is 15 ft, water
temperature is 80oF
and atmospheric
pressure is 14.7 psi
REQD: Determine the max height above the water free surface, z 1 that
the pump can be situated to avoid cavitation – the only loss to
be considered is the inlet filter that has a loss coefficient K L=20
87-351 Fluid Mechanics
TURBOMACHINES
[ centrifugal pumps: theory: net positive suction head (NPSH): example ]
SOLU: 1. We know the available
NPSH can be computed
- [E1]
2. Our max elevation will occur
when the limiting condition of
NPSHA=NPSHR
- [E2]
3. The only headloss we have to consider is the minor
loss, so let us pick up the velocity
- [E3]
87-351 Fluid Mechanics
TURBOMACHINES
[ centrifugal pumps: theory: net positive suction head (NPSH): example
]
SOLU: 4. The velocity then
- [E4]
5. Now we can solve for hL
- [E5]
6. Now we pick up the vapor pressure and specific
weight at 80oF from tabulated values, and solve for z1max
- [ans]
87-351 Fluid Mechanics
TURBOMACHINES
[ centrifugal pumps: theory: pump parameters and similarity laws ]

We have learned that the principal dependent pump variables are the actual
head rise, ha, the shaft power, Wdotshaft, and of course the pump efficiency, h

It is reasonable to expect that these variables will depend on the
geometrical configuration which we would typically represent by a
characteristic diameter, D, and other relevant lengths, li; other important
variables would include the flowrate, Q, the shaft rotational speed, w, the
fluid viscosity, m, and finally, the fluid density, r

The results of dimensional analysis, show us that three primary parameters
will emerge from the variables mentioned above, CH, CP, and h
87-351 Fluid Mechanics
TURBOMACHINES
[ centrifugal pumps: theory: pump parameters and similarity laws ]

Where
- [16]
- [17]
- [18]

CH is deemed the head rise coefficient, CP is the power
coefficient, and of course, h represents the efficiency
87-351 Fluid Mechanics
TURBOMACHINES
[ centrifugal pumps: theory: pump parameters and similarity laws ]

We observe from the preceding expressions [16-18], that the typically high Reynolds
numbers that are associated with pumped flows will render the last pi term negligible,
and relative roughness pi term will also be neglected on the basis of the highly irregular
shape of the pump casing being the governing geometric factor

This said, we can say that the similarity laws may be expressed as
- [19]
- [20]

Here, we call the
bracketed term, the
flow coefficient, CQ
- [22]
- [21]
87-351 Fluid Mechanics
TURBOMACHINES
[ centrifugal pumps: theory: pump parameters and similarity laws ]

Now we can say that if two pumps are operated at the same flow coefficient
- [23]
then
- [24]

Here, subscripts 1 and 2 refer to two
geometrically similar pumps

The existence of these pump scaling
laws allow us to utilize laboratory
data to predict different operating
conditions
- [25]
- [26]
87-351 Fluid Mechanics
TURBOMACHINES
[ centrifugal pumps: theory: pump parameters and similarity laws
]

Pictured above we have typical performance data for a 12 inch
impeller centrifugal pump (left) and dimensionless characteristic
curves that represent a family of geometrically similar pumps
87-351 Fluid Mechanics
TURBOMACHINES
[ centrifugal pumps: theory: pump parameters and similarity laws: example ]
GIVEN: An 8 inch centrifugal pump operating at 1200 rpm is geometrically
similar to the 12 inch pump depicted in the performance
characteristic curves shown in the figure below. Assume the 12 inch
pump is operating at 1000 rpm and water is the working fluid at 60 oF
REQD:
For peak efficiency predict the discharge, actual head
rise, and shaft horsepower for the smaller pump
87-351 Fluid Mechanics
TURBOMACHINES
[ centrifugal pumps: theory: pump parameters and similarity laws: example
]
SOLU:
1. Now, we have learned, for a geometrically similar family of
pumps, the flow coefficient will have the same value for the same
efficiency
2. So, at peak efficiency, CQ=0.0625, therefore for our 8 inch pump
- [E1]
- [ans]
87-351 Fluid Mechanics
TURBOMACHINES
[ centrifugal pumps: theory: pump parameters and similarity laws: example
]
SOLU:
3. Now, we can pick up the actual head rise and shaft
horsepower following the same methodology, as at peak
h, CH=0.19, and CP=0.014
thus
- [ans]
- [ans]
87-351 Fluid Mechanics
TURBOMACHINES
[ centrifugal pumps: theory: pump parameters and similarity laws: example
]
SOLU:
4. We just solved for shaft horsepower (supplied to the
shaft), now we seek to determine the power actually
acquired by the fluid
- [E2]
5. From this we can compute the pump’s efficiency, h
- [E3]
thus, this corroborates our graphical finding earlier in the
solution !
87-351 Fluid Mechanics
TURBOMACHINES
[ centrifugal pumps: theory: pump parameters and similarity laws: specific speed ]

The diameter, D, can be eliminated when the flow coefficient and head rise
coefficient are combined
- [26]

This dimensionless parameter, Ns, is called the specific speed

It is customary to specify a value of specific speed at the flow coefficient
corresponding to peak efficiency

Pumps with low flows and high heads will have lower specific speeds

Pumps with high flows and low heads will have higher specific speeds

The specific speed, Ns, is used to help determine which pump type is most
appropriate for a given application
87-351 Fluid Mechanics
TURBOMACHINES
[ centrifugal pumps: theory: pump parameters and similarity laws: specific speed ]
- [26]

If the flowrate, head, and speed are specified, Ns can be
computed and appropriate pump type can be selected
87-351 Fluid Mechanics
TURBOMACHINES
[ turbines: a brief look
]

Most turbines are classified as either “impulse” or “reaction” turbines

An impulse turbine is driven by a tangential inflow, the total head
of this flow (elevation, pressure, velocity), is converted to velocity
head at the exit of the flow supply nozzle
There is no pressure drop across the rotor in an impulse turbine
(all of the pressure drop occurs at the nozzle)
A reaction turbine is driven by an axial flow, the rotor in a reaction
turbine experiences a large static pressure drop


impulse
reaction
TURBOMACHINES
[ turbines: a brief look
]

A Pelton Wheel is an example of an impulse turbine, you can find
one in your lawn sprinkler

An automotive turbocharger is an example of a reaction turbine
(gas compressor)
single stage turbo
intercooled, double stage compressor