Notice Board


Thursday, July 27, 2017

Fluid Machines Notes

A fluid machine is a device which converts the energy stored by a fluid into mechanical energy or vice versa . The energy stored by a fluid mass appears in the form of potential, kinetic and intermolecular energy. The mechanical energy, on the other hand, is usually transmitted by a rotating shaft. Machines using liquid (mainly water, for almost all practical purposes) are termed as hydraulic machines. In this chapter we shall discuss, in general, the basic fluid mechanical principle governing the energy transfer in a fluid machine and also a brief description of different kinds of hydraulic machines along with their performances. Discussion on machines using air or other gases is beyond the scope of the chapter.
The fluid machines may be classified under different categories as follows:
Classification Based on Direction of Energy Conversion.
The device in which the kinetic, potential or intermolecular energy held by the fluid is converted in the form of mechanical energy of a rotating member is known as a turbine . The machines, on the other hand, where the mechanical energy from moving parts is transferred to a fluid to increase its stored energy by increasing either its pressure or velocity are known as pumps, compressors, fans or blowers .
Classification Based on Principle of Operation
The machines whose functioning depend essentially on the change of volume of a certain amount of fluid within the machine are known as positive displacement machines . The word positive displacement comes from the fact that there is a physical displacement of the boundary of a certain fluid mass as a closed system. This principle is utilized in practice by the reciprocating motion of a piston within a cylinder while entrapping a certain amount of fluid in it. Therefore, the word reciprocating is commonly used with the name of the machines of this kind. The machine producing mechanical energy is known as reciprocating engine while the machine developing energy of the fluid from the mechanical energy is known as reciprocating pump or reciprocating compressor.
The machines, functioning of which depend basically on the principle of fluid dynamics, are known as rotodynamic machines . They are distinguished from positive displacement machines in requiring relative motion between the fluid and the moving part of the machine. The rotating element of the machine usually consisting of a number of vanes or blades, is known as rotor or impeller while the fixed part is known as stator. Impeller is the heart of rotodynamic machines, within which a change of angular momentum of fluid occurs imparting torque to the rotating member.
For turbines, the work is done by the fluid on the rotor, while, in case of pump, compressor, fan or blower, the work is done by the rotor on the fluid element. Depending upon the main direction of fluid path in the rotor, the machine is termed as radial flow or axial flow machine . In radial flow machine, the main direction of flow in the rotor is radial while in axial flow machine, it is axial. For radial flow turbines, the flow is towards the centre of the rotor, while, for pumps and compressors, the flow is away from the centre. Therefore, radial flow turbines are sometimes referred to as radially inward flow machines and radial flow pumps as radially outward flow machines. Examples of such machines are the Francis turbines and the centrifugal pumps or compressors. The examples of axial flow machines are Kaplan turbines and axial flow compressors. If the flow is party radial and partly axial, the term mixed-flow machine is used. Figure 1.1 (a) (b) and (c) are the schematic diagrams of various types of impellers based on the flow direction.

Classification Based on Fluid Used
The fluid machines use either liquid or gas as the working fluid depending upon the purpose. The machine transferring mechanical energy of rotor to the energy of fluid is termed as a pump when it uses liquid, and is termed as a compressor or a fan or a blower, when it uses gas. The compressor is a machine where the main objective is to increase the static pressure of a gas. Therefore, the mechanical energy held by the fluid is mainly in the form of pressure energy. Fans or blowers, on the other hand, mainly cause a high flow of gas, and hence utilize the mechanical energy of the rotor to increase mostly the kinetic energy of the fluid. In these machines, the change in static pressure is quite small.
For all practical purposes, liquid used by the turbines producing power is water, and therefore, they are termed as water turbines or hydraulic turbines . Turbines handling gases in practical fields are usually referred to as steam turbine, gas turbine, and air turbine depending upon whether they use steam, gas (the mixture of air and products of burnt fuel in air) or air.
In this section, we shall discuss the basic principle of rotodynamic machines and the performance of different kinds of those machines. The important element of a rotodynamic machine, in general, is a rotor consisting of a number of vanes or blades. There always exists a relative motion between the rotor vanes and the fluid. The fluid has a component of velocity and hence of momentum in a direction tangential to the rotor. While flowing through the rotor, tangential velocity and hence the momentum changes.
The rate at which this tangential momentum changes corresponds to a tangential force on the rotor. In a turbine, the tangential momentum of the fluid is reduced and therefore work is done by the fluid to the moving rotor. But in case of pumps and compressors there is an increase in the tangential momentum of the fluid and therefore work is absorbed by the fluid from the moving rotor.
Basic Equation of Energy Transfer in Rotodynamic Machines
The basic equation of fluid dynamics relating to energy transfer is same for all rotodynamic machines and is a simple form of " Newton 's Laws of Motion" applied to a fluid element traversing a rotor. Here we shall make use of the momentum theorem as applicable to a fluid element while flowing through fixed and moving vanes. Figure 1.2 represents diagrammatically a rotor of a generalised fluid machine, with 0-0 the axis of rotation and  the angular velocity. Fluid enters the rotor at 1, passes through the rotor by any path and is discharged at 2. The points 1 and 2 are at radii  and  from the centre of the rotor, and the directions of fluid velocities at 1 and 2 may be at any arbitrary angles. For the analysis of energy transfer due to fluid flow in this situation, we assume the following:
(a)  The flow is steady, that is, the mass flow rate is constant across any section (no storage or depletion of fluid mass in the rotor).
(b)  The heat and work interactions between the rotor and its surroundings take place at a constant rate.
(c)  Velocity is uniform over any area normal to the flow. This means that the velocity vector at any point is representative of the total flow over a finite area. This condition also implies that there is no leakage loss and the entire fluid is undergoing the same process.
The velocity at any point may be resolved into three mutually perpendicular components as shown in Fig 1.2. The axial component of velocity  is directed parallel to the axis of rotation , the radial component  is directed radially through the axis to rotation, while the tangential component  is directed at right angles to the radial direction and along the tangent to the rotor at that part.
The change in magnitude of the axial velocity components through the rotor causes a change in the axial momentum. This change gives rise to an axial force, which must be taken by a thrust bearing to the stationary rotor casing. The change in magnitude of radial velocity causes a change in momentum in radial direction
However, for an axisymmetric flow, this does not result in any net radial force on the rotor. In case of a non uniform flow distribution over the periphery of the rotor in practice, a change in momentum in radial direction may result in a net radial force which is carried as a journal load. The tangential component  only has an effect on the angular motion of the rotor. In consideration of the entire fluid body within the rotor as a control volume, we can write from the moment of momentum theorem
where T is the torque exerted by the rotor on the moving fluid, m is the mass flow rate of fluid through the rotor. The subscripts 1 and 2 denote values at inlet and outlet of the rotor respectively. The rate of energy transfer to the fluid is then given by
where  is the angular velocity of the rotor and  which represents the linear velocity of the rotor. Therefore  and  are the linear velocities of the rotor at points 2 (outlet ) and 1 (inlet) respectively (Fig. 1.2). The Eq, (1.2) is known as Euler's equation in relation to fluid machines. The Eq. (1.2) can be written in terms of head gained 'H' by the fluid as
In usual convention relating to fluid machines, the head delivered by the fluid to the rotor is considered to be positive and vice-versa. Therefore, Eq. (1.3) written with a change in the sign of the right hand side in accordance with the sign convention as
Components of Energy Transfer It is worth mentioning in this context that either of the Eqs. (1.2) and (1.4) is applicable regardless of changes in density or components of velocity in other directions. Moreover, the shape of the path taken by the fluid in moving from inlet to outlet is of no consequence. The expression involves only the inlet and outlet conditions. A rotor, the moving part of a fluid machine, usually consists of a number of vanes or blades mounted on a circular disc. Figure 1.3a shows the velocity triangles at the inlet and outlet of a rotor. The inlet and outlet portions of a rotor vane are only shown as a representative of the whole rotor.
Fig 1.3 (a)Velocity triangles for a generalised rotor vane
Fig 1.3 (b)Centrifugal effect in a flow of fluid with rotation
Vector diagrams of velocities at inlet and outlet correspond to two velocity triangles, where  is the velocity of fluid relative to the rotor and  are the angles made by the directions of the absolute velocities at the inlet and outlet respectively with the tangential direction, while  and  are the angles made by the relative velocities with the tangential direction. The angles  and  should match with vane or blade angles at inlet and outlet respectively for a smooth, shockless entry and exit of the fluid to avoid undersirable losses. Now we shall apply a simple geometrical relation as follows:
From the inlet velocity triangle,
Similarly from the outlet velocity triangle.
Invoking the expressions of  and  in Eq. (1.4), we get H (Work head, i.e. energy per unit weight of fluid, transferred between the fluid and the rotor as) as
The Eq (1.7) is an important form of the Euler's equation relating to fluid machines since it gives the three distinct components of energy transfer as shown by the pair of terms in the round brackets. These components throw light on the nature of the energy transfer. The first term of Eq. (1.7) is readily seen to be the change in absolute kinetic energy or dynamic head of the fluid while flowing through the rotor. The second term of Eq. (1.7) represents a change in fluid energy due to the movement of the rotating fluid from one radius of rotation to another.
More About Energy Transfer in Turbomachines
Equation (1.7) can be better explained by demonstrating a steady flow through a container having uniform angular velocity  as shown in Fig.1.3b. The centrifugal force on an infinitesimal body of a fluid of mass dat radius gives rise to a pressure differential dp across the thickness dof the body in a manner that a differential force of dpdA acts on the body radially inward. This force, in fact, is the centripetal force responsible for the rotation of the fluid element and thus becomes equal to the centrifugal force under equilibrium conditions in the radial direction. Therefore, we can write
with d= dA dr ρ where ρ is the density of the fluid, it becomes
For a reversible flow (flow without friction) between two points, say, 1 and 2, the work done per unit mass of the fluid (i.e., the flow work) can be written as
The work is, therefore, done on or by the fluid element due to its displacement from radius  to radius  and hence becomes equal to the energy held or lost by it. Since the centrifugal force field is responsible for this energy transfer, the corresponding head (energy per unit weight)  is termed as centrifugal head. The transfer of energy due to a change in centrifugal head  causes a change in the static head of the fluid.
The third term represents a change in the static head due to a change in fluid velocity relative to the rotor. This is similar to what happens in case of a flow through a fixed duct of variable cross-sectional area. Regarding the effect of flow area on fluid velocity  relative to the rotor, a converging passage in the direction of flow through the rotor increases the relative velocity and hence decreases the static pressure. This usually happens in case of turbines. Similarly, a diverging passage in the direction of flow through the rotor decreases the relative velocity and increases the static pressure as occurs in case of pumps and compressors.
The fact that the second and third terms of Eq. (1.7) correspond to a change in static head can be demonstrated analytically by deriving Bernoulli's equation in the frame of the rotor.
In a rotating frame, the momentum equation for the flow of a fluid, assumed "inviscid" can be written as
where  is the fluid velocity relative to the coordinate frame rotating with an angular velocity .
We assume that the flow is steady in the rotating frame so that We choose a cylindrical coordinate system  with z-axis along the axis of rotation. Then the momentum equation reduces to
where  and  are the unit vectors along and direction respectively. Let  be a unit vector in the direction of and be a coordinate along the stream line. Then we can write

More About Energy Transfer in Turbomachines
Taking scalar product with  it becomes
We have used . With a little rearrangement, we have
Since is the velocity relative to the rotating frame we can replace it by . Further  is the linear velocity of the rotor. Integrating the momentum equation from inlet to outlet along a streamline we have
Therefore, we can say, with the help of Eq. (2.1), that last two terms of Eq. (1.7) represent a change in the static head of fluid.
Energy Transfer in Axial Flow Machines 
For an axial flow machine, the main direction of flow is parallel to the axis of the rotor, and hence the inlet and outlet points of the flow do not vary in their radial locations from the axis of rotation. Therefore,  and the equation of energy transfer Eq. (1.7) can be written, under this situation, as

Hence, change in the static head in the rotor of an axial flow machine is only due to the flow of fluid through the variable area passage in the rotor.
Radially Outward and Inward Flow Machines
For radially outward flow machines, , and hence the fluid gains in static head, while, for a radially inward flow machine,  and the fluid losses its static head. Therefore, in radial flow pumps or compressors the flow is always directed radially outward, and in a radial flow turbine it is directed radially inward.
Impulse and Reaction Machines The relative proportion of energy transfer obtained by the change in static head and by the change in dynamic head is one of the important factors for classifying fluid machines. The machine for which the change in static head in the rotor is zero is known as impulse machine . In these machines, the energy transfer in the rotor takes place only by the change in dynamic head of the fluid. The parameter characterizing the proportions of changes in the dynamic and static head in the rotor of a fluid machine is known as degree of reaction and is defined as the ratio of energy transfer by the change in static head to the total energy transfer in the rotor.
Therefore, the degree of reaction,
Impulse and Reaction Machines
For an impulse machine R = 0 , because there is no change in static pressure in the rotor. It is difficult to obtain a radial flow impulse machine, since the change in centrifugal head is obvious there. Nevertheless, an impulse machine of radial flow type can be conceived by having a change in static head in one direction contributed by the centrifugal effect and an equal change in the other direction contributed by the change in relative velocity. However, this has not been established in practice. Thus for an axial flow impulse machine . For an impulse machine, the rotor can be made open, that is, the velocity V1 can represent an open jet of fluid flowing through the rotor, which needs no casing. A very simple example of an impulse machine is a paddle wheel rotated by the impingement of water from a stationary nozzle as shown in Fig.2.1a.
Fig 2.1(a) Paddle wheel as an example of impulse turbine
 (b) Lawn sprinkler as an example of reaction turbine
A machine with any degree of reaction must have an enclosed rotor so that the fluid cannot expand freely in all direction. A simple example of a reaction machine can be shown by the familiar lawn sprinkler, in which water comes out (Fig. 2.1b) at a high velocity from the rotor in a tangential direction. The essential feature of the rotor is that water enters at high pressure and this pressure energy is transformed into kinetic energy by a nozzle which is a part of the rotor itself.
In the earlier example of impulse machine (Fig. 2.1a), the nozzle is stationary and its function is only to transform pressure energy to kinetic energy and finally this kinetic energy is transferred to the rotor by pure impulse action. The change in momentum of the fluid in the nozzle gives rise to a reaction force but as the nozzle is held stationary, no energy is transferred by it. In the case of lawn sprinkler (Fig. 2.1b), the nozzle, being a part of the rotor, is free to move and, in fact, rotates due to the reaction force caused by the change in momentum of the fluid and hence the word reaction machine follows.
The concept of efficiency of any machine comes from the consideration of energy transfer and is defined, in general, as the ratio of useful energy delivered to the energy supplied. Two efficiencies are usually considered for fluid machines-- the hydraulic efficiency concerning the energy transfer between the fluid and the rotor, and the overall efficiency concerning the energy transfer between the fluid and the shaft. The difference between the two represents the energy absorbed by bearings, glands, couplings, etc. or, in general, by pure mechanical effects which occur between the rotor itself and the point of actual power input or output.

Therefore, for a pump or compressor,
For a turbine,

The ratio of rotor and shaft energy is represented by mechanical efficiency .

Principle of Similarity and Dimensional Analysis
The principle of similarity is a consequence of nature for any physical phenomenon. By making use of this principle, it becomes possible to predict the performance of one machine from the results of tests on a geometrically similar machine, and also to predict the performance of the same machine under conditions different from the test conditions. For fluid machine, geometrical similarity must apply to all significant parts of the system viz., the rotor, the entrance and discharge passages and so on. Machines which are geometrically similar form a homologous series. Therefore, the member of such a series, having a common shape are simply enlargements or reductions of each other. If two machines are kinematically similar, the velocity vector diagrams at inlet and outlet of the rotor of one machine must be similar to those of the other. Geometrical similarity of the inlet and outlet velocity diagrams is, therefore, a necessary condition for dynamic similarity.
Let us now apply dimensional analysis to determine the dimensionless parameters, i.e., the π terms as the criteria of similarity for flows through fluid machines. For a machine of a given shape, and handling compressible fluid, the relevant variables are given in Table 3.1
Table 3.1 Variable Physical Parameters of Fluid Machine
Variable physical parameters
Dimensional formula
= any physical dimension of the machine as a measure of the machine's size, usually the rotor diameter
= volume flow rate through the machine
L3 T -1
= rotational speed (rev/min.)
= difference in head (energy per unit weight) across the machine. This may be either gained or given by the fluid depending upon whether the machine is a pump or a turbine respectively
=density of fluid
 = viscosity of fluid
ML-1 T -1
= coefficient of elasticity of fluid
ML-1 T-2
= acceleration due to gravity
LT -2
P = power transferred between fluid and rotor (the difference between P and H is taken care of by the hydraulic efficiency 
ML2 T-3
In almost all fluid machines flow with a free surface does not occur, and the effect of gravitational force is negligible. Therefore, it is more logical to consider the energy per unit mass gH as the variable rather than alone so that acceleration due to gravity does not appear as a separate variable. Therefore, the number of separate variables becomes eight: D, Q, N, gH, ρ, µ, and . Since the number of fundamental dimensions required to express these variable are three, the number of independent π terms (dimensionless terms), becomes five. Using Buckingham's π theorem with D, N and ρ as the repeating variables, the expression for the terms are obtained as,
We shall now discuss the physical significance and usual terminologies of the different π terms. All lengths of the machine are proportional to , and all areas to D2. Therefore, the average flow velocity at any section in the machine is proportional to . Again, the peripheral velocity of the rotor is proportional to the product ND . The first π  term can be expressed as

Similarity and Dimensional Analysis
Thus,  represents the condition for kinematic similarity, and is known as capacity coefficient or discharge coefficient The second  term  is known as the head coefficient since it expresses the head in dimensionless form. Considering the fact that ND  rotor velocity, the term  becomes , and can be interpreted as the ratio of fluid head to kinetic energy of the rotor, Dividing  by the square of  we get
The term  can be expressed as  and thus represents the Reynolds number with rotor velocity as the characteristic velocity. Again, if we make the product of  and , it becomes  which represents the Reynolds's number based on fluid velocity. Therefore, if  is kept same to obtain kinematic similarity,  becomes proportional to the Reynolds number based on fluid velocity.
The term  expresses the power in dimensionless form and is therefore known as power coefficient . Combination of  and  in the form of  gives . The term 'PQgH' represents the rate of total energy given up by the fluid, in case of turbine, and gained by the fluid in case of pump or compressor. Since P is the power transferred to or from the rotor. Therefore  becomes the hydraulic efficiency for a turbine and  for a pump or a compressor. From the fifth  term, we get
Multiplying , on both sides, we get
Therefore, we find that  represents the well known Mach number , Ma.
For a fluid machine, handling incompressible fluid, the term  can be dropped. The effect of liquid viscosity on the performance of fluid machines is neglected or regarded as secondary, (which is often sufficiently true for certain cases or over a limited range).Therefore the term  can also be dropped.The general relationship between the different dimensionless variables (  terms) can be expressed as

Therefore one set of relationship or curves of the  terms would be sufficient to describe the performance of all the members of one series.

Similarity and Dimensional Analysis
or, with another arrangement of the π terms,
If data obtained from tests on model machine, are plotted so as to show the variation of dimensionless parameters  with one another, then the graphs are applicable to any machine in the same homologous series. The curves for other homologous series would naturally be different.
Specific Speed 
The performance or operating conditions for a turbine handling a particular fluid are usually expressed by the values of and , and for a pump by and . It is important to know the range of these operating parameters covered by a machine of a particular shape (homologous series) at high efficiency. Such information enables us to select the type of machine best suited to a particular application, and thus serves as a starting point in its design. Therefore a parameter independent of the size of the machine is required which will be the characteristic of all the machines of a homologous series. A parameter involving and but not is obtained by dividing  by . Let this parameter be designated by  as
Similarly, a parameter involving and but not is obtained by divining  by  and is represented by  as
Since the dimensionless parameters  and  are found as a combination of basic π terms, they must remain same for complete similarity of flow in machines of a homologous series. Therefore, a particular value of  or relates all the combinations of and or and for which the flow conditions are similar in the machines of that homologous series. Interest naturally centers on the conditions for which the efficiency is a maximum. For turbines, the values of and , and for pumps and compressors, the values of and are usually quoted for which the machines run at maximum efficiency.
The machines of particular homologous series, that is, of a particular shape, correspond to a particular value of  for their maximum efficient operation. Machines of different shapes have, in general, different values of . Thus the parameter  is referred to as the shape factor of the machines. Considering the fluids used by the machines to be incompressible, (for hydraulic turbines and pumps), and since the acceleration due to gravity dose not vary under this situation, the terms and  are taken out from the expressions of  and . The portions left as  and  are termed, for the practical purposes, as the specific speed  for turbines or pumps. Therefore, we can write,
 (specific speed for turbines) = 
 (specific speed for turbines) = 
The name specific speed for these expressions has a little justification. However a meaning can be attributed from the concept of a hypothetical machine. For a turbine,  is the speed of a member of the same homologous series as the actual turbine, so reduced in size as to generate unit power under a unit head of the fluid. Similarly, for a pump,  is speed of a hypothetical pump with reduced size but representing a homologous series so that it delivers unit flow rate at a unit head. The specific speed  is, therefore, not a dimensionless quantity.
The dimension of  can be found from their expressions given by Eqs. (3.5) and (3.6). The dimensional formula and the unit of specific speed are given as follows:
Specific speed
Dimensional formula
Unit (SI)
1/2 T -5/2 L-1/4
kg 1/2/ s5/2 m1/4
3/4 T-3/2
3/4 / s3/2
The dimensionless parameter  is often known as the dimensionless specific speed to distinguish it from .

Figure 26.1 Typical PELTON WHEEL with 21 Buckets
Hydropower is the longest established source for the generation of electric power. In this module we shall discuss the governing principles of various types of hydraulic turbines used in hydro-electric power stations.
Impulse Hydraulic Turbine The Pelton Wheel
The only hydraulic turbine of the impulse type in common use, is named after an American engineer Laster A Pelton, who contributed much to its development around the year 1880. Therefore this machine is known as Pelton turbine or Pelton wheel. It is an efficient machine particularly suited to high heads. The rotor consists of a large circular disc or wheel on which a number (seldom less than 15) of spoon shaped buckets are spaced uniformly round is periphery as shown in Figure 26.1. The wheel is driven by jets of water being discharged at atmospheric pressure from pressure nozzles. The nozzles are mounted so that each directs a jet along a tangent to the circle through the centres of the buckets (Figure 26.2). Down the centre of each bucket, there is a splitter ridge which divides the jet into two equal streams which flow round the smooth inner surface of the bucket and leaves the bucket with a relative velocity almost opposite in direction to the original jet.
Figure 26.2 A Pelton wheel
For maximum change in momentum of the fluid and hence for the maximum driving force on the wheel, the deflection of the water jet should be . In practice, however, the deflection is limited to about  so that the water leaving a bucket may not hit the back of the following bucket. Therefore, the camber angle of the buckets is made as . Figure(26.3a)
The number of jets is not more than two for horizontal shaft turbines and is limited to six for vertical shaft turbines. The flow partly fills the buckets and the fluid remains in contact with the atmosphere. Therefore, once the jet is produced by the nozzle, the static pressure of the fluid remains atmospheric throughout the machine. Because of the symmetry of the buckets, the side thrusts produced by the fluid in each half should balance each other.
Analysis of force on the bucket and power generation Figure 26.3a shows a section through a bucket which is being acted on by a jet. The plane of section is parallel to the axis of the wheel and contains the axis of the jet. The absolute velocity of the jet  with which it strikes the bucket is given by
Figure 26.3(a)Flow along the bucket of a pelton wheel
(b) Inlet velocity triangle
(c)Outlet velocity triangle
where,  is the coefficient of velocity which takes care of the friction in the nozzle. is the head at the entrance to the nozzle which is equal to the total or gross head of water stored at high altitudes minus the head lost due to friction in the long pipeline leading to the nozzle. Let the velocity of the bucket (due to the rotation of the wheel) at its centre where the jet strikes be . Since the jet velocity  is tangential, i.e.  and are collinear, the diagram of velocity vector at inlet (Fig 26.3.b) becomes simply a straight line and the relative velocity is given by
It is assumed that the flow of fluid is uniform and it glides the blade all along including the entrance and exit sections to avoid the unnecessary losses due to shock. Therefore the direction of relative velocity at entrance and exit should match the inlet and outlet angles of the buckets respectively. The velocity triangle at the outlet is shown in Figure 26.3c. The bucket velocity remains the same both at the inlet and outlet. With the direction of being taken as positive, we can write. The tangential component of inlet velocity (Figure 26.3b)
and the tangential component of outlet velocity (Figure 26.3c)

where  and  are the velocities of the jet relative to the bucket at its inlet and outlet and  is the outlet angle of the bucket.
From the Eq. (1.2) (the Euler's equation for hydraulic machines), the energy delivered by the fluid per unit mass to the rotor can be written as
(since, in the present situation, 
The relative velocity  becomes slightly less than  mainly because of the friction in the bucket. Some additional loss is also inevitable as the fluid strikes the splitter ridge, because the ridge cannot have zero thickness. These losses are however kept to a minimum by making the inner surface of the bucket polished and reducing the thickness of the splitter ridge. The relative velocity at outlet  is usually expressed as  where, K is a factor with a value less than 1. However in an ideal case ( in absence of friction between the fluid and blade surface) K=1. Therefore, we can write Eq.(26.1)
If Q is the volume flow rate of the jet, then the power transmitted by the fluid to the wheel can be written as
The power input to the wheel is found from the kinetic energy of the jet arriving at the wheel and is given by . Therefore the wheel efficiency of a pelton turbine can be written as
It is found that the efficiency  depends on  and  For a given design of the bucket, i.e. for constant values of  and K, the efficiency becomes a function of  only, and we can determine the condition given by  at which  becomes maximum.
For  to be maximum,
 is always negative.
Therefore, the maximum wheel efficiency can be written after substituting the relation given by eqn.(26.5) in eqn.(26.4) as

The condition given by Eq. (26.5) states that the efficiency of the wheel in converting the kinetic energy of the jet into mechanical energy of rotation becomes maximum when the wheel speed at the centre of the bucket becomes one half of the incoming velocity of the jet. The overall efficiency  will be less than  because of friction in bearing and windage, i.e. friction between the wheel and the atmosphere in which it rotates. Moreover, as the losses due to bearing friction and windage increase rapidly with speed, the overall efficiency reaches it peak when the ratio  is slightly less than the theoretical value of 0.5. The value usually obtained in practice is about 0.46. The Figure 27.1 shows the variation of wheel efficiency  with blade to jet speed ratio  for assumed values at k=1 and 0.8, and  . An overall efficiency of 85-90 percent may usually be obtained in large machines. To obtain high values of wheel efficiency, the buckets should have smooth surface and be properly designed. The length, width, and depth of the buckets are chosen about 2.5.4 and 0.8 times the jet diameter. The buckets are notched for smooth entry of the jet.
Figure 27.1 Theoretical variation of wheel efficiency for a Pelton turbine with blade speed to jet speed ratio for different values of k
Specific speed and wheel geometry The specific speed of a pelton wheel depends on the ratio of jet diameter and the wheel pitch diameter. (the diameter at the centre of the bucket). If the hydraulic efficiency of a pelton wheel is defined as the ratio of the power delivered to the wheel to the head available at the nozzle entrance, then we can write.
Since [  and 
The specific speed  = 
The optimum value of the overall efficiency of a Pelton turbine depends both on the values of the specific speed and the speed ratio. The Pelton wheels with a single jet operate in the specific speed range of 4-16, and therefore the ratio D/d lies between 6 to 26 as given by the Eq. (15.25b). A large value of D/d reduces the rpm as well as the mechanical efficiency of the wheel. It is possible to increase the specific speed by choosing a lower value of D/d, but the efficiency will decrease because of the close spacing of buckets. The value of D/d is normally kept between 14 and 16 to maintain high efficiency. The number of buckets required to maintain optimum efficiency is usually fixed by the empirical relation.
n(number of buckets) = (27.2)
Govering of Pelton Turbine : First let us discuss what is meant by governing of turbines in general. When a turbine drives an electrical generator or alternator, the primary requirement is that the rotational speed of the shaft and hence that of the turbine rotor has to be kept fixed. Otherwise the frequency of the electrical output will be altered. But when the electrical load changes depending upon the demand, the speed of the turbine changes automatically. This is because the external resisting torque on the shaft is altered while the driving torque due to change of momentum in the flow of fluid through the turbine remains the same. For example, when the load is increased, the speed of the turbine decreases and vice versa . A constancy in speed is therefore maintained by adjusting the rate of energy input to the turbine accordingly. This is usually accomplished by changing the rate of fluid flow through the turbine- the flow in increased when the load is increased and the flow is decreased when the load is decreased. This adjustment of flow with the load is known as the governing of turbines.
In case of a Pelton turbine, an additional requirement for its operation at the condition of maximum efficiency is that the ration of bucket to initial jet velocity  has to be kept at its optimum value of about 0.46. Hence, when U is fixed.  has to be fixed. Therefore the control must be made by a variation of the cross-sectional area, A, of the jet so that the flow rate changes in proportion to the change in the flow area keeping the jet velocity  same. This is usually achieved by a spear valve in the nozzle (Figure 27.2a). Movement of the spear and the axis of the nozzle changes the annular area between the spear and the housing. The shape of the spear is such, that the fluid coalesces into a circular jet and then the effect of the spear movement is to vary the diameter of the jet. Deflectors are often used (Figure 27.2b) along with the spear valve to prevent the serious water hammer problem due to a sudden reduction in the rate of flow. These plates temporarily defect the jet so that the entire flow does not reach the bucket; the spear valve may then be moved slowly to its new position to reduce the rate of flow in the pipe-line gradually. If the bucket width is too small in relation to the jet diameter, the fluid is not smoothly deflected by the buckets and, in consequence, much energy is dissipated in turbulence and the efficiency drops considerably. On the other hand, if the buckets are unduly large, the effect of friction on the surfaces is unnecessarily high. The optimum value of the ratio of bucket width to jet diameter has been found to vary between 4 and 5.
Figure 27.2(a) Spear valve to alter jet area in a Pelton wheel
(b) Jet deflected from bucket
Limitation of a Pelton Turbine: The Pelton wheel is efficient and reliable when operating under large heads. To generate a given output power under a smaller head, the rate of flow through the turbine has to be higher which requires an increase in the jet diameter. The number of jets are usually limited to 4 or 6 per wheel. The increases in jet diameter in turn increases the wheel diameter. Therefore the machine becomes unduly large, bulky and slow-running. In practice, turbines of the reaction type are more suitable for lower heads.
Francis Turbine
Reaction Turbine: The principal feature of a reaction turbine that distinguishes it from an impulse turbine is that only a part of the total head available at the inlet to the turbine is converted to velocity head, before the runner is reached. Also in the reaction turbines the working fluid, instead of engaging only one or two blades, completely fills the passages in the runner. The pressure or static head of the fluid changes gradually as it passes through the runner along with the change in its kinetic energy based on absolute velocity due to the impulse action between the fluid and the runner. Therefore the cross-sectional area of flow through the passages of the fluid. A reaction turbine is usually well suited for low heads. A radial flow hydraulic turbine of reaction type was first developed by an American Engineer, James B. Francis (1815-92) and is named after him as the Francis turbine. The schematic diagram of a Francis turbine is shown in Fig. 28.1
Figure 28.1 A Francis turbine
A Francis turbine comprises mainly the four components:
(i) sprical casing,
(ii) guide on stay vanes,
(iii) runner blades,
(iv) draft-tube as shown in Figure 28.1 .
Spiral Casing : Most of these machines have vertical shafts although some smaller machines of this type have horizontal shaft. The fluid enters from the penstock (pipeline leading to the turbine from the reservoir at high altitude) to a spiral casing which completely surrounds the runner. This casing is known as scroll casing or volute. The cross-sectional area of this casing decreases uniformly along the circumference to keep the fluid velocity constant in magnitude along its path towards the guide vane.
Figure 28.2 Spiral Casing
This is so because the rate of flow along the fluid path in the volute decreases due to continuous entry of the fluid to the runner through the openings of the guide vanes or stay vanes.
Guide or Stay vane:
The basic purpose of the guide vanes or stay vanes is to convert a part of pressure energy of the fluid at its entrance to the kinetic energy and then to direct the fluid on to the runner blades at the angle appropriate to the design. Moreover, the guide vanes are pivoted and can be turned by a suitable governing mechanism to regulate the flow while the load changes. The guide vanes are also known as wicket gates. The guide vanes impart a tangential velocity and hence an angular momentum to the water before its entry to the runner. The flow in the runner of a Francis turbine is not purely radial but a combination of radial and tangential. The flow is inward, i.e. from the periphery towards the centre. The height of the runner depends upon the specific speed. The height increases with the increase in the specific speed. The main direction of flow change as water passes through the runner and is finally turned into the axial direction while entering the draft tube.
Draft tube:
The draft tube is a conduit which connects the runner exit to the tail race where the water is being finally discharged from the turbine. The primary function of the draft tube is to reduce the velocity of the discharged water to minimize the loss of kinetic energy at the outlet. This permits the turbine to be set above the tail water without any appreciable drop of available head. A clear understanding of the function of the draft tube in any reaction turbine, in fact, is very important for the purpose of its design. The purpose of providing a draft tube will be better understood if we carefully study the net available head across a reaction turbine.
Net head across a reaction turbine and the purpose to providing a draft tube . The effective head across any turbine is the difference between the head at inlet to the machine and the head at outlet from it. A reaction turbine always runs completely filled with the working fluid. The tube that connects the end of the runner to the tail race is known as a draft tube and should completely to filled with the working fluid flowing through it. The kinetic energy of the fluid finally discharged into the tail race is wasted. A draft tube is made divergent so as to reduce the velocity at outlet to a minimum. Therefore a draft tube is basically a diffuser and should be designed properly with the angle between the walls of the tube to be limited to about 8 degree so as to prevent the flow separation from the wall and to reduce accordingly the loss of energy in the tube. Figure 28.3 shows a flow diagram from the reservoir via a reaction turbine to the tail race.
The total head  at the entrance to the turbine can be found out by applying the Bernoulli's equation between the free surface of the reservoir and the inlet to the turbine as
where  is the head lost due to friction in the pipeline connecting the reservoir and the turbine. Since the draft tube is a part of the turbine, the net head across the turbine, for the conversion of mechanical work, is the difference of total head at inlet to the machine and the total head at discharge from the draft tube at tail race and is shown as in Figure 28.3
Figure 28.3 Head across a reaction turbine
Therefore, = total head at inlet to machine (1) - total head at discharge (3)
The pressures are defined in terms of their values above the atmospheric pressure. Section 2 and 3 in Figure 28.3 represent the exits from the runner and the draft tube respectively. If the losses in the draft tube are neglected, then the total head at 2 becomes equal to that at 3. Therefore, the net head across the machine is either  or . Applying the Bernoull's equation between 2 and 3 in consideration of flow, without losses, through the draft tube, we can write.
Since , both the terms in the bracket are positive and hence is always negative, which implies that the static pressure at the outlet of the runner is always below the atmospheric pressure. Equation (28.1) also shows that the value of the suction pressure at runner outlet depends on z, the height of the runner above the tail race and , the decrease in kinetic energy of the fluid in the draft tube. The value of this minimum pressure should never fall below the vapour pressure of the liquid at its operating temperature to avoid the problem of cavitation. Therefore, we fine that the incorporation of a draft tube allows the turbine runner to be set above the tail race without any drop of available head by maintaining a vacuum pressure at the outlet of the runner.

Runner of the Francis Turbine
The shape of the blades of a Francis runner is complex. The exact shape depends on its specific speed. It is obvious from the equation of specific speed that higher specific speed means lower head. This requires that the runner should admit a comparatively large quantity of water for a given power output and at the same time the velocity of discharge at runner outlet should be small to avoid cavitation. In a purely radial flow runner, as developed by James B. Francis, the bulk flow is in the radial direction. To be more clear, the flow is tangential and radial at the inlet but is entirely radial with a negligible tangential component at the outlet. The flow, under the situation, has to make a 90o turn after passing through the rotor for its inlet to the draft tube. Since the flow area (area perpendicular to the radial direction) is small, there is a limit to the capacity of this type of runner in keeping a low exit velocity. This leads to the design of a mixed flow runner where water is turned from a radial to an axial direction in the rotor itself. At the outlet of this type of runner, the flow is mostly axial with negligible radial and tangential components. Because of a large discharge area (area perpendicular to the axial direction), this type of runner can pass a large amount of water with a low exit velocity from the runner. The blades for a reaction turbine are always so shaped that the tangential or whirling component of velocity at the outlet becomes zero . This is made to keep the kinetic energy at outlet a minimum.
Figure 29.1 shows the velocity triangles at inlet and outlet of a typical blade of a Francis turbine. Usually the flow velocity (velocity perpendicular to the tangential direction) remains constant throughout, i.e.  and is equal to that at the inlet to the draft tube.
The Euler's equation for turbine [Eq.(1.2)] in this case reduces to
where, is the energy transfer to the rotor per unit mass of the fluid. From the inlet velocity triangle shown in Fig. 29.1
Substituting the values of  and  from Eqs. (29.2a) and (29.2b) respectively into Eq. (29.1), we have
Figure 29.1 Velocity triangle for a Francis runner
The loss of kinetic energy per unit mass becomes equal to . Therefore neglecting friction, the blade efficiency becomes
can be written as
The change in pressure energy of the fluid in the rotor can be found out by subtracting the change in its kinetic energy from the total energy released. Therefore, we can write for the degree of reaction.

Using the expression of from Eq. (29.3), we have
The inlet blade angle  of a Francis runner varies  and the guide vane angle angle  from . The ratio of blade width to the diameter of runner B/D, at blade inlet, depends upon the required specific speed and varies from 1/20 to 2/3.
Expression for specific speed. The dimensional specific speed of a turbine, can be written as
Power generated for a turbine can be expressed in terms of available head and hydraulic efficiency  as
Hence, it becomes
Again, ,
Substituting  from Eq. (29.2b)
Available head equals the head delivered by the turbine plus the head lost at the exit. Thus,
with the help of Eq. (29.3), it becomes
Substituting the values of H and N from Eqs (29.7) and (29.6) respectively into the expression  given by Eq. (29.5), we get,
Flow velocity at inlet  can be substituted from the equation of continuity as
where B is the width of the runner at its inlet
Finally, the expression for  becomes,
For a Francis turbine, the variations of geometrical parameters like  have been described earlier. These variations cover a range of specific speed between 50 and 400. Figure 29.2 shows an overview of a Francis Turbine. The figure is specifically shown in order to convey the size and relative dimensions of a typical Francis Turbine to the readers.
Figure 29.2 Installation of a Francis Turbine

Higher specific speed corresponds to a lower head. This requires that the runner should admit a comparatively large quantity of water. For a runner of given diameter, the maximum flow rate is achieved when the flow is parallel to the axis. Such a machine is known as axial flow reaction turbine. An Australian engineer, Vikton Kaplan first designed such a machine. The machines in this family are called Kaplan Turbines.(Figure 30.1)
Figure 30.1 A typical Kaplan Turbine
Development of Kaplan Runner from the Change in the Shape of Francis Runner with Specific Speed
Figure 30.2 shows in stages the change in the shape of a Francis runner with the variation of specific speed. The first three types [Fig. 30.2 (a), (b) and (c)] have, in order. The Francis runner (radial flow runner) at low, normal and high specific speeds. As the specific speed increases, discharge becomes more and more axial. The fourth type, as shown in Fig.30.2 (d), is a mixed flow runner (radial flow at inlet axial flow at outlet) and is known as Dubs runner which is mainly suited for high specific speeds. Figure 30.2(e) shows a propeller type runner with a less number of blades where the flow is entirely axial (both at inlet and outlet). This type of runner is the most suitable one for very high specific speeds and is known as Kaplan runner or axial flow runner.
From the inlet velocity triangle for each of the five runners, as shown in Figs (30.2a to 30.2e), it is found that an increase in specific speed (or a decreased in head) is accompanied by a reduction in inlet velocity . But the flow velocity  at inlet increases allowing a large amount of fluid to enter the turbine. The most important point to be noted in this context is that the flow at inlet to all the runners, except the Kaplan one, is in radial and tangential directions. Therefore, the inlet velocity triangles of those turbines (Figure 30.2a to 30.2d) are shown in a plane containing the radial ant tangential directions, and hence the flow velocity  represents the radial component of velocity.
In case of a Kaplan runner, the flow at inlet is in axial and tangential directions. Therefore, the inlet velocity triangle in this case (Figure 30.2e) is shown in a place containing the axial and tangential directions, and hence the flow velocity  represents the axial component of velocity  .The tangential component of velocity is almost nil at outlet of all runners. Therefore, the outlet velocity triangle (Figure 30.2f) is identical in shape of all runners. However, the exit velocity  is axial in Kaplan and Dubs runner, while it is the radial one in all other runners.
(a) Francis runner for low specific speeds
(b) Francis runner for normal specific speeds
(c) Francis runner for high specific speeds
(d) Dubs runner
(e) Kalpan runner
(f) For allreaction (Francis as well as Kaplan) runners
Outlet velocity triangle
Fig. 30.2 Evolution of Kaplan runner form Francis one
Figure 30.3 shows a schematic diagram of propeller or Kaplan turbine. The function of the guide vane is same as in case of Francis turbine. Between the guide vanes and the runner, the fluid in a propeller turbine turns through a right-angle into the axial direction and then passes through the runner. The runner usually has four or six blades and closely resembles a ship's propeller. Neglecting the frictional effects, the flow approaching the runner blades can be considered to be a free vortex with whirl velocity being inversely proportional to radius, while on the other hand, the blade velocity is directly proportional to the radius. To take care of this different relationship of the fluid velocity and the blade velocity with the changes in radius, the blades are twisted. The angle with axis is greater at the tip that at the root.
Fig. 30.3 A propeller of Kaplan turbine
Different types of draft tubes incorporated in reaction turbines The draft tube is an integral part of a reaction turbine. Its principle has been explained earlier. The shape of draft tube plays an important role especially for high specific speed turbines, since the efficient recovery of kinetic energy at runner outlet depends mainly on it. Typical draft tubes, employed in practice, are discussed as follows.
Straight divergent tube [Fig. 30.4(a)] The shape of this tube is that of frustum of a cone. It is usually employed for low specific speed, vertical shaft Francis turbine. The cone angle is restricted to 8 0 to avoid the losses due to separation. The tube must discharge sufficiently low under tail water level. The maximum efficiency of this type of draft tube is 90%. This type of draft tube improves speed regulation of falling load.
Simple elbow type (Fig. 30.4b) The vertical length of the draft tube should be made small in order to keep down the cost of excavation, particularly in rock. The exit diameter of draft tube should be as large as possible to recover kinetic energy at runner's outlet. The cone angle of the tube is again fixed from the consideration of losses due to flow separation. Therefore, the draft tube must be bent to keep its definite length. Simple elbow type draft tube will serve such a purpose. Its efficiency is, however, low(about 60%). This type of draft tube turns the water from the vertical to the horizontal direction with a minimum depth of excavation. Sometimes, the transition from a circular section in the vertical portion to a rectangular section in the horizontal part (Fig. 30.4c) is incorporated in the design to have a higher efficiency of the draft tube. The horizontal portion of the draft tube is generally inclined upwards to lead the water gradually to the level of the tail race and to prevent entry of air from the exit end.
Figure 30.4 Different types of draft tubes

Cavitation in reaction turbines
If the pressure of a liquid in course of its flow becomes equal to its vapour pressure at the existing temperature, then the liquid starts boiling and the pockets of vapour are formed which create vapour locks to the flow and the flow is stopped. The phenomenon is known as cavitation. To avoid cavitation, the minimum pressure in the passage of a liquid flow, should always be more than the vapour pressure of the liquid at the working temperature. In a reaction turbine, the point of minimum pressure is usually at the outlet end of the runner blades, i.e at the inlet to the draft tube. For the flow between such a point and the final discharge into the trail race (where the pressure is atmospheric), the Bernoulli's equation can be written, in consideration of the velocity at the discharge from draft tube to be negligibly small, as
where,  and  represent the static pressure and velocity of the liquid at the outlet of the runner (or at the inlet to the draft tube). The larger the value of , the smaller is the value of  and the cavitation is more likely to occur. The term  in Eq. (31.1) represents the loss of head due to friction in the draft tube and z is the height of the turbine runner above the tail water surface. For cavitation not to occur  where  is the vapour pressure of the liquid at the working temperature.
An important parameter in the context of cavitation is the available suction head (inclusive of both static and dynamic heads) at exit from the turbine and is usually referred to as the net positive suction head 'NPSH' which is defined as
with the help of Eq. (31.1) and in consideration of negligible frictional losses in the draft tube , Eq. (31.2) can be written as
A useful design parameter  known as Thoma's Cavitation Parameter (after the German Engineer Dietrich Thoma, who first introduced the concept) is defined as
For a given machine, operating at its design condition, another useful parameter  known as critical cavitaion parameter is define as
Therefore, for cavitaion not to occur  (since, 
If either or is increased,  is reduced. To determine whether cavitation is likely to occur in a particular installation, the value of may be calculated. When the value of  is greater than the value of  for a particular design of turbine cavitation is not expected to occur.
In practice, the value of  is used to determine the maximum elevation of the turbine above tail water surface for cavitation to be avoided. The parameter of increases with an increase in the specific speed of the turbine. Hence, turbines having higher specific speed must be installed closer to the tail water level.

Performance Characteristics of Reaction Turbine
It is not always possible in practice, although desirable, to run a machine at its maximum efficiency due to changes in operating parameters. Therefore, it becomes important to know the performance of the machine under conditions for which the efficiency is less than the maximum. It is more useful to plot the basic dimensionless performance parameters (Fig. 31.1) as derived earlier from the similarity principles of fluid machines. Thus one set of curves, as shown in Fig. 31.1, is applicable not just to the conditions of the test, but to any machine in the same homologous series under any altered conditions.
Figure 31.1 performance characteristics of a reaction turbine (in dimensionless parameters)
Figure 31.2 is one of the typical plots where variation in efficiency of different reaction turbines with the rated power is shown.
Figure 31.2 Variation of efficiency with load
 Comparison of Specific Speeds of Hydraulic Turbines
Specific speeds and their ranges of variation for different types of hydraulic turbines have already been discussed earlier. Figure 32.1 shows the variation of efficiencies with the dimensionless specific speed of different hydraulic turbines. The choice of a hydraulic turbine for a given purpose depends upon the matching of its specific speed corresponding to maximum efficiency with the required specific speed determined from the operating parameters, namely, (rotational speed), (power) and (available head).
Figure 32.1 Variation of efficiency with specific speed for hydraulic turbines
Governing of Reaction Turbines Governing of reaction turbines is usually done by altering the position of the guide vanes and thus controlling the flow rate by changing the gate openings to the runner. The guide blades of a reaction turbine (Figure 32.2) are pivoted and connected by levers and links to the regulating ring. Two long regulating rods, being attached to the regulating ring at their one ends, are connected to a regulating lever at their other ends. The regulating lever is keyed to a regulating shaft which is turned by a servomotor piston of the oil
Figure 32.2 Governing of reaction turbinePumps
Rotodynamic Pumps
A rotodynamic pump is a device where mechanical energy is transferred from the rotor to the fluid by the principle of fluid motion through it. The energy of the fluid can be sensed from the pressur and velocity of the fluid at the delivery end of the pump. Therefore, it is essentially a turbine in reverse. Like turbines, pumps are classified according to the main direction of fluid path through them like (i) radial flow or centrifugal, (ii) axial flow and (iii) mixed flow types.
Centrifugal Pumps
The pumps employing centrifugal effects for increasing fluid pressure have been in use for more than a century.The centrifugal pump, by its principle, is converse of the Francis turbine. The flow is radially outward, and the hence the fluid gains in centrifugal head while flowing through it. Because of certain inherent advantages,such as compactness, smooth and uniform flow, low initial cost and high efficiency even at low heads, centrifugal pumps are used in almost all pumping systems. However, before considering the operation of a pump in detail, a general pumping system is discussed as follows.
General Pumping System and the Net Head Developed by a Pump
The word pumping, referred to a hydraulic system commonly implies to convey liquid from a low to a high reservoir. Such a pumping system, in general, is shown in Fig. 33.1. At any point in the system, the elevation or potential head is measured from a fixed reference datum line. The total head at any point comprises pressure head, velocity head and elevation head. For the lower reservoir, the total head at the free surface is  and is equal to the elevation of the free surface above the datum line since the velocity and static pressure at are zero. Similarly the total head at the free surface in the higher reservoir is ( ) and is equal to the elevation of the free surface of the reservoir above the reference datum.
The variation of total head as the liquid flows through the system is shown in Fig. 33.2. The liquid enters the intake pipe causing a head loss  for which the total energy line drops to point corresponding to a location just after the entrance to intake pipe. The total head at can be written as
As the fluid flows from the intake to the inlet flange of the pump at elevation  the total head drops further to the point C (Figure 33.2) due to pipe friction and other losses equivalent to  . The fluid then enters the pump and gains energy imparted by the moving rotor of the pump. This raises the total head of the fluid to a point D (Figure 33.2) at the pump outlet (Figure 33.1).
In course of flow from the pump outlet to the upper reservoir, friction and other losses account for a total head loss or down to a point . At an exit loss  occurs when the liquid enters the upper reservoir, bringing the total heat at point (Figure 33.2) to that at the free surface of the upper reservoir. If the total heads are measured at the inlet and outlet flanges respectively, as done in a standard pump test, then
Figure 33.1 A general pumping system
Figure 33.2 Change of head in a pumping system
Total inlet head to the pump = 
Total outlet head of the pump = 
where  and  are the velocities in suction and delivery pipes respectively.
Therefore, the total head developed by the pump,
The head developed is termed as manometric head . If the pipes connected to inlet and outlet of the pump are of same diameter,  and therefore the head developed or manometric head is simply the gain in piezometric pressure head across the pump which could have been recorded by a manometer connected between the inlet and outlet flanges of the pump. In practice, ( ) is so small in comparison to  that it is ignored. It is therefore not surprising o find that the static pressure head across the pump is often used to describe the total head developed by the pump. The vertical distance between the two levels in the reservoirs  is known as static head or static lift. Relationship between , the static head and , the head developed can be found out by applying Bernoulli's equation between and and between and (Figure 33.1) as follows:
Between and ,
substituting  from Eq. (33.2) into Eq. (33.3), and then with the help of Eq. (33.1),
we can write
Therefore, we have, the total head developed by the pump = static head + sum of all the losses.
 The simplest from of a centrifugal pump is shown in Figure 33.3. It consists of three important parts: (i) the rotor, usually called as impeller, (ii) the volute casing and (iii) the diffuser ring. The impeller is a rotating solid disc with curved blades standing out vertically from the face of the disc. The impeller may be single sided (Figure 33.4a) or doublesided (Figure 33.4b). A double sided impeller has a relatively small flow capacity.
Figure 33.3 A centrifugal pump
The tips of the blades are sometimes covered by another flat disc to give shrouded blades (Figure 33.4c), otherwise the blade tips are left open and the casing of the pump itself forms the solid outer wall of the blade passages. The advantage of the shrouded blade is that flow is prevented from leaking across the blade tips from one passage to another.
(a) Single sided impeller
(b) Double sided impeller
(c) Shrouded impeller
Figure 33.4 Types of impellers in a centrifugal pump

As the impeller rotates, the fluid is drawn into the blade passage at the impeller eye, the centre of the impeller. The inlet pipe is axial and therefore fluid enters the impeller with very little whirl or tangential component of velocity and flows outwards in the direction of the blades. The fluid receives energy from the impeller while flowing through it and is discharged with increased pressure and velocity into the casing. To convert the kinetic energy or fluid at the impeller outlet gradually into pressure energy, diffuser blades mounted on a diffuser ring are used.
The stationary blade passages so formed have an increasing cross-sectional area which reduces the flow velocity and hence increases the static pressure of the fluid. Finally, the fluid moves from the diffuser blades into the volute casing which is a passage of gradually increasing cross-section and also serves to reduce the velocity of fluid and to convert some of the velocity head into static head. Sometimes pumps have only volute casing without any diffuser.
Figure 34.1 shows an impeller of a centrifugal pump with the velocity triangles drawn at inlet and outlet. The blades are curved between the inlet and outlet radius. A particle of fluid moves along the broken curve shown in Figure 34.1.
Figure 34.1 Velocity triangles for centrifugal pump Impeller
Let  be the angle made by the blade at inlet, with the tangent to the inlet radius, while  is the blade angle with the tangent at outlet.  and  are the absolute velocities of fluid at inlet an outlet respectively, while  and are the relative velocities (with respect to blade velocity) at inlet and outlet respectively. Therefore,
Work done on the fluid per unit weight = 
A centrifugal pump rarely has any sort of guide vanes at inlet. The fluid therefore approaches the impeller without appreciable whirl and so the inlet angle of the blades is designed to produce a right-angled velocity triangle at inlet (as shown in Fig. 34.1). At conditions other than those for which the impeller was designed, the direction of relative velocity  does not coincide with that of a blade. Consequently, the fluid changes direction abruptly on entering the impeller. In addition, the eddies give rise to some back flow into the inlet pipe, thus causing fluid to have some whirl before entering the impeller. However, considering the operation under design conditions, the inlet whirl velocity and accordingly the inlet angular momentum of the fluid entering the impeller is set to zero. Therefore, Eq. (34.1) can be written as
Work done on the fluid per unit weight 
We see from this equation that the work done is independent of the inlet radius. The difference in total head across the pump known as manometric head, is always less than the quantity  because of the energy dissipated in eddies due to friction.
The ratio of manometric head and the work head imparted by the rotor on the fluid  (usually known as Euler head) is termed as manometric efficiency . It represents the effectiveness of the pump in increasing the total energy of the fluid from the energy given to it by the impeller. Therefore, we can write
The overall efficiency  of a pump is defined as
where, is the volume flow rate of the fluid through the pump, and is the shaft power, i.e. the input power to the shaft. The energy required at the shaft exceeds  because of friction in the bearings and other mechanical parts. Thus a mechanical efficiency is defined as
so that

Slip Factor
Under certain circumstances, the angle at which the fluid leaves the impeller may not be the same as the actual blade angle. This is due to a phenomenon known as fluid slip, which finally results in a reduction in  the tangential component of fluid velocity at impeller outlet. One possible explanation for slip is given as follows.
In course of flow through the impeller passage, there occurs a difference in pressure and velocity between the leading and trailing faces of the impeller blades. On the leading face of a blade there is relatively a high pressure and low velocity, while on the trailing face, the pressure is lower and hence the velocity is higher. This results in a circulation around the blade and a non-uniform velocity distribution at any radius. The mean direction of flow at outlet, under this situation, changes from the blade angle at outlet  to a different angle  as shown in Figure 34.2 Therefore the tangential velocity component at outlet  is reduced to  , as shown by the velocity triangles in Figure 34.2, and the difference  is defined as the slip. The slip factor  is defined as
Figure 34.2 Slip and velocity in the impeller blade passage of a centrifugal pump
With the application of slip factor , the work head imparted to the fluid (Euler head) becomes  . The typical values of slip factor lie in the region of 0.9.
Losses in a Centrifugal Pump
•  Mechanical friction power loss due to friction between the fixed and rotating parts in the bearing and stuffing boxes.
•  Disc friction power loss due to friction between the rotating faces of the impeller (or disc) and the liquid.
•  Leakage and recirculation power loss. This is due to loss of liquid from the pump and recirculation of the liquid in the impeller. The pressure difference between impeller tip and eye can cause a recirculation of a small volume of liquid, thus reducing the flow rate at outlet of the impeller as shown in Fig. (34.3).
Figure 34.3 Leakage and recirculation in a centrifugal pump

Characteristics of a Centrifugal Pump
With the assumption of no whirl component of velocity at entry to the impeller of a pump, the work done on the fluid per unit weight by the impeller is given by Equation( 34.2). Considering the fluid to be frictionless, the head developed by the pump will be the same san can be considered as the theoretical head developed. Therefore we can write for theoretical head developed  as
From the outlet velocity triangle figure( 34.1)
where is rate of flow at impeller outlet and is the flow area at the periphery of the impeller. The blade speed at outlet  can be expressed in terms of rotational speed of the impeller as
Using this relation and the relation given by Eq. (35.2), the expression of theoretical head developed can be written from Eq. (35.1) as
where,  and 
For a given impeller running at a constant rotational speed.  and  are constants, and therefore head and discharge bears a linear relationship as shown by Eq. (35.3). This linear variation of  with is plotted as curve Iin Fig. 35.1.
If slip is taken into account, the theoretical head will be reduced to . Moreover the slip will increase with the increase in flow rate . The effect of slip in head-discharge relationship is shown by the curve II in Fig. 35.1. The loss due to slip can occur in both a real and an ideal fluid, but in a real fluid the shock losses at entry to the blades, and the friction losses in the flow passages have to be considered. At the design point the shock losses are zero since the fluid moves tangentially onto the blade, but on either side of the design point the head loss due to shock increases according to the relation
Figure 35.1 Head-discharge characteristics of a centrifugal pump
where  is the off design flow rate and  is a constant. The losses due to friction can usually be expressed as
where,  is a constant.

Equation (35.5) and (35.4) are also shown in Fig. 35.1 (curves III and IV) as the characteristics of losses in a centrifugal pump. By subtracting the sum of the losses from the head in consideration of the slip, at any flow rate (by subtracting the sum of ordinates of the curves III and IV from the ordinate of the curve II at all values of the abscissa), we get the curve V which represents the relationship of the actual head with the flow rate, and is known as head-discharge characteristic curve of the pump.
Effect of blade outlet angle
The head-discharge characteristic of a centrifugal pump depends (among other things) on the outlet angle of the impeller blades which in turn depends on blade settings. Three types of blade settings are possible (i) the forward facing for which the blade curvature is in the direction of rotation and, therefore, (Fig. 35.2a), (ii) radial, when  (Fig. 35.2b), and (iii) backward facing for which the blade curvature is in a direction opposite to that of the impeller rotation and therefore,  (Fig. 35.2c). The outlet velocity triangles for all the cases are also shown in Figs. 35.2a, 35.2b, 35.2c. From the geometry of any triangle, the relationship between  and  can be written as.
which was expressed earlier by Eq. (35.2).
Figure 35.2 Outlet velocity triangles for different blade settings in a centrifugal pump
In case of forward facing blade, and hence cot  is negative and therefore  is more than . In case of radial blade, and In case of backward facing blade, and Therefore the sign of , the constant in the theoretical head-discharge relationship given by the Eq. (35.3), depends accordingly on the type of blade setting as follows:
For forward curved blades 
For radial blades 
For backward curved blades 
With the incorporation of above conditions, the relationship of head and discharge for three cases are shown in Figure 35.3. These curves ultimately revert to their more recognized shapes as the actual head-discharge characteristics respectively after consideration of all the losses as explained earlier Figure 35.4.
For both radial and forward facing blades, the power is rising monotonically as the flow rate is increased. In the case of backward facing blades, the maximum efficiency occurs in the region of maximum power. If, for some reasons, Qincreases beyond  there occurs a decrease in power. Therefore the motor used to drive the pump at part load, but rated at the design point, may be safely used at the maximum power. This is known as self-limiting characteristic. In case of radial and forward-facing blades, if the pump motor is rated for maximum power, then it will be under utilized most of the time, resulting in an increased cost for the extra rating. Whereas, if a smaller motor is employed, rated at the design point, then if increases above  the motor will be overloaded and may fail. It, therefore, becomes more difficult to decide on a choice of motor in these later cases (radial and forward-facing blades).
Figure 35.3 Theoretical head-discharge characteristic curves of a centrifugal pump for different blade settings
Figure 35.4 Actual head-discharge and power-discharge characteristic curves of a centrifugal pump

Flow through Volute Chambers
Apart from frictional effects, no torque is applied to a fluid particle once it has left the impeller. The angular momentum of fluid is therefore constant if friction is neglected. Thus the fluid particles follow the path of a free vortex. In an ideal case, the radial velocity at the impeller outlet remains constant round the circumference. The combination of uniform radial velocity with the free vortex (  =constant) gives a pattern of spiral streamlines which should be matched by the shape of the volute. This is the most important feature of the design of a pump. At maximum efficiency, about 10 percent of the head generated by the impeller is usually lost in the volute.
Vanned Diffuser
A vanned diffuser, as shown in Fig. 36.1, converts the outlet kinetic energy from impeller to pressure energy of the fluid in a shorter length and with a higher efficiency. This is very advantageous where the size of the pump is important. A ring of diffuser vanes surrounds the impeller at the outlet. The fluid leaving the impeller first flows through a vaneless space before entering the diffuser vanes. The divergence angle of the diffuser passage is of the order of 8-10 ° which ensures no boundary layer separation. The optimum number of vanes are fixed by a compromise between the diffusion and the frictional loss. The greater the number of vanes, the better is the diffusion (rise in static pressure by the reduction in flow velocity) but greater is the frictional loss. The number of diffuser vanes should have no common factor with the number of impeller vanes to prevent resonant vibration.
Figure 36.1 A vanned diffuser of a centrifugal pump
Cavitation in centrifugal pumps
Cavitation is likely to occur at the inlet to the pump, since the pressure there is the minimum and is lower than the atmospheric pressure by an amount that equals the vertical height above which the pump is situated from the supply reservoir (known as sump) plus the velocity head and frictional losses in the suction pipe. Applying the Bernoulli's equation between the surface of the liquid in the sump and the entry to the impeller, we have
where,  is the pressure at the impeller inlet and  is the pressure at the liquid surface in the sump which is usually the atmospheric pressure, Z1 is the vertical height of the impeller inlet from the liquid surface in the sump, is the loss of head in the suction pipe. Strainers and non-return valves are commonly fitted to intake pipes. The term must therefore include the losses occurring past these devices, in addition to losses caused by pipe friction and by bends in the pipe.
In the similar way as described in case of a reaction turbine, the net positive suction head 'NPSH' in case of a pump is defined as the available suction head (inclusive of both static and dynamic heads) at pump inlet above the head corresponding to vapor pressure.

Again, with help of Eq. (36.1), we can write
The Thomas cavitation parameter s and critical cavitation parameter  are defined accordingly (as done in case of reaction turbine) as
We can say that for cavitation not to occur,
In order that s should be as large as possible, must be as small as possible. In some installations, it may even be necessary to set the pump below the liquid level at the sump (i.e. with a negative vale of ) to avoid cavitation.