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Rotodynamic Machines (Massey 13) Introduction Types of machines Turbines
Impulse turbines—The Pelton Wheel Reaction turbines—Francis turbine Basic Equation for rotodynamic machinery (See 13.3.4) Similarity laws and Specific speed for turbines Performance characteristics of turbines
Rotodynamic pumps Centrifugal pumps Basic Equation applied to centrifugal pumps (see 13.4.2) Similarity laws and specific speed for pumps Performance characteristics of pumps
Cavitation Introduction A fluid machine is a device either for converting the energy held by a fluid into mechanical energy or vice versa. Turbines and Pumps A machine in which energy from the fluid is converted directly to the mechanical energy of a rotating member is known as a turbine If the initial mechanical movement is a reciprocating action, then the term engine or motor is used. A machine in which the transfer of energy is from the moving parts of the machine to the fluid takes place is called a pump. The term "pumps" is used when the fluid is a liquid. When the fluid is a gas, terms such as compressors, or fans (or blowers) are used. A compressor is a machine whose primary objective is to increase the pressure of the gas. This is accompanied by an increase in the density of the gas.
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A fan or blower is a machine whose primary objective is to move the gas. Static pressures remain almost unchanged, and therefore the density of the gas is also not changed. The essential principle of a rotodynamic machine is based on the tangential velocities set up at its rotor. In turbines, the initial tangential momentum is reduced by the rotor and thus work is done by the fluid on the rotor that is then converted to useful power. In pumps, energy from the rotor is used to impart (or increase) tangential momentum of the fluid, and the resulting increase in tangential velocity is then converted to pressure energy.
Types of Machines Turbines There are two types of turbines, the impulse and the reaction. In both types the fluid passes through a runner having blades. The momentum of the fluid in the tangential direction is changed and so a tangential force on the runner is produced. The runner therefore rotates and performs useful work, while the fluid leaves it with reduced energy. For any turbine, the energy held by the fluid is initially in the form of pressure.
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• Concerned with the power generated from a given head. • Two types--impulse and reaction 1. Impulse Turbines Conversion of the pressure to kinetic energy in a jet by allowing the fluid to pass through nozzles. The jets of fluid impinge on the moving blades of the runner where all of their kinetic energy is practically lost. The Pelton Wheel Named after Lester A. Pelton (1829-1908), the Pelton wheel is an efficient machine well suited to high heads (>500m). Maximum power output is typically about 80 MW, but it can be as high as 400 MW. Realizes efficiency as high as 80% largely due to improvement in the shape of the vanes upon which the jet impinges. They are capable of working over a wide range of conditions, a desirable characteristic for a turbine since they cannot always work at full load. Physical description: A circular rotor (frequently horizontally mounted) with several spoon-shaped buckets evenly spaced round its periphery. One or more nozzles are mounted so that each directs a jet along a tangent to the circle through the centres of the buckets. Down the centre of each bucket is a splitter ridge which divides the oncoming jet into two equal portions and, after flow round the smooth inner surface of the bucket, the fluid leaves it with a relative velocity almost opposite in direction to the original jet. The deflection of the jet leaving the bucket is limited to about 165°. While 180° is desirable, it will result in the leaving jet from one bucket interfering with the neighbouring bucket. Most of the energy is transmitted, as the absolute velocity of the fluid leaving the bucket is small. Useful power for generation is produced by the forces tangential to the direction of whirl of the rotor, even though there may be other components of forces resulting from the impact of the jet on the buckets. This force is produced by the change in the absolute velocity component in the direction of whirl (called the velocity of whirl). Refer to the diagram below for erivation of the power generated by the velocity of whirl.
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Velocity vector diagram at inlet Velocity vector diagram at outlet The change of the whirl component between the inlet and outlet is given by:
( ){ } ( )θπθπ −+=−−−=∆ coscos 2121 RRRuvvw ( )θcos11 kR +=
Now ρQrateflowmass =
So ( )wvQmomentumofflowofRate ∆= ρ
( )rvQwheelonTorque w∆= ρ
( ) ( )uvQrvQoutputPower ww ∆=∆= ρωρ
The energy at the wheel is in the form of kinetic of the jet and is given by
212
1 vQρ per unit time.
So the wheel efficiency is given by:
( ) ( )212
1
2
21 v
vu
vQ
uvQ www
∆=
∆=
ρ
ρη
Substituting for wv∆ from above and putting uvR −= 11 gives ( )( )
21
1 cos12v
kuvuw
θη −−=
The maximum efficiency is theoretically obtained when 5.0vu
1= . The
actual value is about 0.46 and an efficiency value between 0.85-0.9 results. Loss of efficiency is due to: • Energy to overcome friction in the bearings • Energy to overcome friction between the wheel and the atmosphere
(windage)
v1
u R1
vw2
θ u
v2 R2=kR1
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Note that the frequency of the power generated is linked to the angular velocity of the wheel. Any changes in angular velocity will alter the frequency of the electrical output. The machine is designed to operate at maximum efficiency, and so even when power demand drops, the machine is still required to operate at maximum efficiency. A reduction in required output would therefore have to be followed by a proportional reduction in the input power. The input power can be controlled by altering the initial velocity of whirl. But if this is altered, the optimal ratio of u/v will change. Thus, to maintain efficiency v cannot be altered. Reduction in the input power is achieved by reducing Q, the flow. And since Q is reduced, to maintain the same velocity v, the area of the jet must be reduced. The cross-sectional area of the jet is achieved by means of a spear valve in the nozzle Design Considerations
Ratio of bucket width to jet diameter-- about 4 to 5 Ratio of wheel diameter to jet diameter--minimum of 10
2. Reaction Turbines The Francis turbine was developed by James. B. Francis (1815-1892). It is a radial-flow reaction turbine. Its components are described in the Massey textbook; also see figure below. It is particularly suited for medium heads (between 15m to 300m) and overall efficiencies exceeding 90% have been achieved. Other types of pumps include the axial-flow turbine such as the Kaplan turbine. Net head across a reaction turbine --This is the difference between the head at inlet (gross head of the reservoir, less any losses along the pipeline to the inlet) and the head at outlet from it (see diagram on right).
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H= Total head at inlet to machine - Total head at discharge to tail race Mechanism for generation of power in the Francis turbine: Fluid enters the volute or scroll case and passes through the guide vanes mounted on the runner. The fluid passes the blades of the runner where it is deflected and so its angular momentum is changed. From the centre of the runner the fluid is turned into the axial direction and flows to waste via the draft tube. Note the shape of the volute. The cross-sectional area is decreasing along the fluid path such as to keep the fluid velocity constant in magnitude. The guide vanes direct the fluid on the runner at the angle appropriate to the design. The angle of the vane can be changed to alter the flow rate and hence the power output. The lower end of the draft tube must be submerged below the level of the tailrace to ensure that the turbine is full of water. Basic Equation for Rotodynamic Machinery v = absolute velocity of fluid u = peripheral velocity of blade at point considered R = relative velocity between fluid and blade vw = velocity of whirl, i.e., component of absolute velocity of fluid in
direction tangential to runner circumference. r = radius from axis of runner
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ω = angular velocity of runner Suffix 1 refers to conditions at inlet to runner Suffix 2 refers to conditions at outlet from runner.
The power passed on to the runner from the fluid is due to the change in tangential momentum. There may be changes of momentum in other directions also, but the corresponding forces have no moments about the axis of rotation of the rotor. Now,
axisaboutmomentumangularofincreaseofRateaxisfixedgivenabouttorque = Therefore, the torque on the fluid must be equal to the angular momentum of the fluid leaving the rotor per unit time minus the angular momentum of the fluid entering the rotor per unit time. Consider a small particle of mass δm at the inlet. Its momentum tangential to the rotor is δmvw1, and its angular momentum is δmvw1r1. If the mass flow rate is m& then the rate at which angular momentum passes through a small cross-sectional area having uniform velocity vw1 and radius of curvature r1 is 11w rvm&δ The total rate at which angular momentum enters the rotor is ∫ 11w rvm&δ .
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Similarly, the rate at which angular momentum leaves the rotor is ∫ 22w rvm&δ The rate of increase of angular momentum of the fluid is given by:
mdrvmdrv 11w22w && ∫∫ −
From Newton's Third Law of Motion, the torque exerted on the rotor by the fluid is:
∫∫ −= mdrvmdrvT 22w11w && Above is known as the Euler's equation. Note that the above equation involves initial and final state of the fluid. It applies regardless of the path taken by the fluid between inlet and outlet; also, it is independent of any losses occurring due to friction between the blades and the fluid, changes of temperature. For a rotor, the shaft work done in unit time interval is
∫∫ −= mdrvmdrvT 22w11w && ωωω
∫∫ −= mdvumdvu 2w21w1 && since ru ω= The shaft work done by the fluid per unit mass is obtained by dividing the above equation by the total mass flow rate m& . Thus,
( )∫∫ −= mdvumdvum1fluidofmassunitperdonework 2w21w1 &&&
2w21w1 vuvu −=
if the products uvw are individually constant. The energy available per unit mass of the fluid is gH, where H = the net head. The (theoretical) hydraulic efficiency is:
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gHvuvuefficiencyHydraulic 2w21w1 −
= (if the products uvw are uniform).
Note, this is not the overall efficiency, because a fraction of this energy is lost to overcome (for example) friction in the bearings. Refer to the velocity diagrams above. The ideal condition, the one which would minimise losses due to eddy formation, occurs when the relative velocity at the inlet is in line with the inlet edge of the blade. If there is an appreciable departure, the fluid is forced to change direction suddenly, resulting in violent eddies and dissipation of energy as heat. In rotodynamic machines, there is allowance made for different inlet directions, and the favourable alignment can be achieved by adjusting the direction of the guide vanes. At the outlet, the direction of the relative velocity R2 is determined by the outlet angle of the blade and the geometry of the outlet diagram then determines the magnitude and direction of the absolute velocity v2. For high efficiency, the velocity of the fluid at outlet, and hence the kinetic energy, should be small. The desirable velocity at outlet is one without whirl, that is, one that is perpendicular to the tangential velocity. If such is achieved, then the hydraulic efficiency is given by:
gHvu 1w1
Similarity Laws and Specific Speed The development and utilization of turbomachinery in engineering practice has benefited greatly from the application of dimensional analysis. It has enabled turbine and pump manufacturers to test and develop a relatively small number of turbomachines, and subsequently produce a series of commercial units that cover a broad range of head and flow demands. Geometric similarity is a pre-requisite of dynamic similarity. For fluid machines the geometric similarity must apply to all significant parts of the
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system—the rotor, the entrance and discharge passages and for a turbine, the conditions in the tailrace or sump. Machines that are similar in these respects form a homologous series. Recall that dynamic similarity implies kinematic similarity, that is, corresponding velocities in a constant ratio. The velocities represented in the vector diagrams above in one machine must be similar to the vector diagrams in the other machine. If kinematic similarity (fixed ratio of velocities) and dynamic similarity (fixed ratio of forces) exists, then certain dimensionless parameters representing these ratios are the same for each of the systems being compared. For determining these dimensionless parameters, the following variables may be considered: Dimensional formula D rotor diameter, here chosen as a suitable measure of
the size of the machine [L]
Q volume rate of flow through the machine [L3T-1] N rotational speed [T-1] H difference of head across machine, i.e., energy per
unit weight [L]
g Weight per unit mass [LT-2] ρ density of fluid [ML-3] µ viscosity of fluid [ML-1T-1] P Power transferred between fluid and rotor [ML2T-3] Keeping D, N and ρ as the repeating variables, the following groups have been derived:
534
2
322231 DNPND
DNgH
NDQ
ρΠ
µρΠΠΠ ====
Refer once again to the vector diagram of the velocities, and recall that for kinematic similarity, then the ratio of the velocities must be similar in both machines.
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Consider the ratio of the fluid and the blade velocities, v and u respectively in the diagram. The fluid velocity is given by Q/A and A is proportional to D2. The blade velocity, u, is proportional to ND. Thus,
NDD
Q2
1 =Π
That is, kinematic similarity of two geometrically similar machines is achieved if Q/ND3 (called the discharge number of flow coefficient) is the same for each.
The product of 1Π and 3Π gives :
µ
ρDD
Q2 and if 2D
Q is taken as the
fluid velocity, then the product represents the Reynolds number. An expression for the hydraulic efficiency can be obtained by combining the
power parameter 4Π with 1Π and 2Π as ( )214
ΠΠΠ
× , which is: QgHP
ρ
The relations connecting the variables may be written as:
0,, 532231 =
DN
PDN
gHND
Qρ
φ
or,
0,, 53222 =
DN
PDN
gHρ
ηφ
Explanation of specific speed: For a turbine using a particular fluid, the operating conditions are expressed by values of N, P and H. It is important to know the range of these conditions covered by a particular shape of turbine. With this information, we can decide on which turbine to choose. We therefore need some parameter that can define all the machines
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belonging to a particular homologous series, independent of the size represented by D. That is to say, we need a parameter involving all the operating conditions—N, P and H—but one that does not involve the size, D. The size D can be eliminated by dividing ( ) ( ) 4
52
21
4 ΠΠ by to give:
( ) 452
1
21
gH
NP
ρ
The specific speed for a turbine is defined as the speed at which the turbine should operate to generate 1 KW from a 1 m head. In computing the value it is customary to use values of N, P and H associated with maximum efficiency. For similarity of flow in machines of a homologous series, each Π parameter must be unchanged throughout the series. If the dimensionless parameters Π ’s are unchanged, then the specific speed, derived from the division of two of these dimensionless parameters also much be unchanged. A particular value of this parameter therefore relates all combinations of the N, P and H for which the flow conditions are similar for that homologous series. For a given homologous series, we are interested in the operating conditions at which maximum efficiency occurs. Define a parameter Kn, the value of the specific speed at which this occurs. So, whatever the operating conditions—the values of N, P and H—all machines of a particular homologous series have a particular value of Kn at maximum efficiency. Traditionally, ρ, g are considered invariant—cold water is most times the only fluid used in turbines, and we are concerned with operations on the surface of the earth where g does not vary much from 9.81 m/s2. Thus the specific speed is usually written without the ρ and g quantities and is of the form:
45
21
H
NP
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Performance Characteristics of turbines Under normal conditions, a turbine will be required to work with an almost constant head. It may be desired to operate the machine under conditions other than optimal. Therefore a plot, called the characteristic curve, is developed that shows the manner in which discharge, power output and efficiency changes with speed. It is more useful to develop plots using the dimensionless parameters so that the curves can be used not only for the turbine being tested but also for other machines in the same homologous series. The parameters are:
( ) ( ) ( ) 21
2122
32 gH
ND,gHD
Q,gHD
P
ρ
Basic Equations Applied to Centrifugal Pumps Note the assumptions upon which the equations are based, namely, that flow is steady, uniform velocities at inlet and outlet with respect to magnitude and angle made with the radius. Note that the equation for the energy
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imparted to the fluid by the impeller is similar to that for a turbine (above), except the signs are reversed. It is,
1w12w2 vuvumassunitperfluidondonework −=
This equation can be transformed to the following making use of the expressions for R1 and R2:
( ) ( ) ( ){ }21
22
21
22
21
22 RRuuvv
21massunitperfluidondonework −+−+−=
In the equation for work done on fluid, we may set vw1 to 0 as initially the fluid approaches the impeller without any whirl. Thus the equation simplifies to:
2w2vumassunitperfluidondonework =
This energy increases the piezometric head of the fluid by an amount Hm, called the manometric head. But the total energy supplied to the fluid (per
unit weight) to produce this head is given by gvu 2w2 , which is known as the
Euler head. The ratio of the manometric head to the Euler head is known as the manometric efficiency and is:
2w2
mvu
gH
Similarity Laws and Specific Speed for Pumps While quantities of primary interest for turbines are N, H, P, for pumps we are more concerned with N, H and Q. As had been done above for turbines, we can also derive an expression that is independent of size, D. But just as we obtained an expression in N, H and P for the turbines from the dimensionless parameters, we can also derive
for pumps an expression in N, H and Q from 4
32
211
Π
Π. This gives:
( ) 43
21
sgH
NQN =
The definition for the specific speed is the speed at which the pump operating for discharging 1 m3/s of water against a head of 1 m.
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From the dimensionless parameters given above for turbines, several useful results can be derived for pumps. Consider the case for constant D (the same pump under consideration but under different operating conditions) and for ρ and g fixed, we get the following:
3534
2222
31
NPDN
P
NHDN
gH
NQND
Q
∝⇒=
∝⇒=
∝⇒=
ρΠ
Π
Π
These relations are often known as the affinity laws for pumps and they allow performance characteristics at any one speed to be converted to any other speed. Performance Characteristics of Pumps Pumps normally run at constant speed, with the interest being the manner in which the head H varies with discharge Q and the variation of efficiency and power required with Q. Thus the characteristic curve resembles that shown below. A particular machine may be tested at a fixed head while the load (and speed N) is varied, and for these results to be applicable to other pumps in the same homologous series, the characteristic is plotted using the
dimensionless parameters 53223 DNP,
DNgH,
NDQ
ρin place of Q, H and P
respectively.
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Cavitation Cavitation occurs when pressure falls below vapour pressure (at the appropriate temperature). The liquid boils and bubbles form in large numbers. At locations with higher pressure the bubbles collapse as the vapour condenses. The liquid rushing in to fill the void collides at the centre resulting in very high pressures (~1GPa). This acting at or near solid surfaces can cause damage including fatigue failure. This phenomenon is accompanied by noise and vibration. Apart from physical damage, cavitation causes a reduction in the efficiency of the machine. Every effort must therefore be made to eliminate cavitation and this can be done by ensuring that the pressure is everywhere greater than the vapour pressure. (Air in solution is released as the pressure falls and this leads to air cavitation.) Conditions are favourable for cavitation where the velocity is high or the elevation is high and particularly where both conditions occur. In reaction turbines, the minimum pressure is usually at the outlet end of the runner blade on the leading side. Between the minimum pressure point and the final discharge point the following equation may be written:
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gphz
g2v
gp atm
f
2min
ρρ=−++
zg
pg
phg2
v minatmf
2−−=−
ρρ
For a machine being operated at a net head H across the machine, a parameter cσ can be defined such that
H
zgp
gp minatm
c
−−= ρρσ
For cavitation not to occur, pmin must be greater than the vapour pressure, pv, that is
cσσ > where H
zgp
gp vatm −−
= ρρσ
The above expression is known as Thoma's Cavitation Parameter. The expression can be used to determine the maximum elevation zmax of the turbine above the tail water to avoid cavitation and is:
Hgp
gpz cvatm
max σρρ −−=
Pumps in Series and in Parallel (Discussed in class)