1. Turbomachine principles, Velocity triangles
Module II

2. Equation of continuity
Consider the flow of a fluid with density , through the element of area dA, during the time interval dt.
If c is the stream velocity the elementary mass is dm = .c.dt.dAcos, where  is the angle subtended by the normal of the area element to the stream direction.
The velocity component perpendicular to the area dA is cn = ccos  and so dm = cndAdt.
The elementary rate of mass flow is therefore,

3. The first law of thermodynamics—internal energy
The first law of thermodynamics states that if a system is taken through a complete cycle during which heat is supplied and work is done, then
where dQ represents the heat supplied to the system during the cycle and dW the work done by the system during the cycle.
During a change of state from 1 to 2, there is a change in the property internal energy,
For an infinitesimal change of state

4. The steady flow energy equation
where h is the specific enthalpy, 1/2c2, the kinetic energy per unit mass and gz the potential energy per unit mass.
Defining stagnation enthalpy by h0 = h + 1/2c2 and assuming g(z2- z1) is negligible, then
Most turbomachinery flow processes are adiabatic (or very nearly so) and it is permissible to write Q=0. For work producing machines (turbines) Wx > 0, so that
For work-absorbing machines (compressors) Wx < 0,

6. The momentum equation—Newton’s second law of motion
The momentum equation relates the sum of the external forces acting on a fluid element to its acceleration, or to the rate of change of momentum in the direction of the resultant external force.
Many applications of the momentum equation in the study of turbomachines e.g. the force exerted upon a blade in a compressor or turbine cascade caused by the deflection or acceleration of fluid passing the blades.

7. Euler’s equation of motion
For the steady flow of fluid through an elementary control volume that, in the absence of all shear forces, following relation is obtained,
This is Euler’s equation of motion for one-dimensional flow and is derived from Newton’s second law.
By shear forces being absent we mean there is neither friction nor shaft work.
However, it is not necessary that heat transfer should also be absent.

8. Bernoulli’s equation For an incompressible fluid,  is constant,
where stagnation pressure is p0 = p+1/2c2
If the fluid is a gas or vapour, the change in gravitational potential is generally negligible and therefore,
Control volume in a streaming fluid.
i.e. the stagnation pressure is constant (this is also true for a compressible isentropic process).

9. Control volume for a generalized turbomachine.
Moment of momentum
For a system of mass m, the vector sum of the moments of all external forces acting on the system about some arbitrary axis A–A fixed in space is equal to the time rate of change of angular momentum of the system about that axis, i.e.
where r is distance of the mass centre from the axis of rotation measured along the normal to the axis and c is the velocity component mutually perpendicular to both the axis and radius vector r.
Control volume for a generalized turbomachine.

10. Law of moment of momentum
Swirling fluid enters the control volume at radius r1 with tangential velocity c1 and leaves at radius r2 with tangential velocity c2. For one-dimensional steady flow,
which states that, the sum of the moments of the external forces acting on fluid temporarily occupying the control volume is equal to the net time rate of efflux of angular momentum from the control volume.

11. Euler’s pump and turbine equations
For a pump or compressor rotor running at angular velocity , the rate at which the rotor does work on the fluid is,
where the blade speed U =r.
Thus the work done on the fluid per unit mass or specific work is
This equation is referred to as Euler’s pump equation.
For a turbine the fluid does work on the rotor and the sign for work is then reversed. Thus, the specific work is
This equation is referred to as Euler’s turbine equation.

12. Euler’s pump and turbine equations
In a compressor or pump the specific work done on the fluid equals the rise in stagnation enthalpy. Thus, combining eqns,
and
We, have
This relationship is true for steady, adiabatic and irreversible flow in compressors or in pump impellers.
Euler’s pump or compressor equation.

13. Euler’s pump and turbine equations
Euler’s pump compressor equation.
A change in total enthalpy is equivalent to a change in tangential flow speed and/or tangential engine speed
For engines with little change in mean radius U2 = U1 (e.g. axial turbines, axial compressors, fans) the change in total enthalpy is entirely due to change in tangential flow speed h0 = Uc  blades are bowed.
For engines with large change in mean radius (e.g. radial engines) the change in enthalpy is to a large degree due to the change in radius h0 = Uĉ centrifugal effect, possibility for larger change in enthalpy.

14. Leonhard Euler
Leonhard Euler ( ) was arguably the greatest mathematician of the eighteenth century and one of the most prolific of all time; his publication list of 886 papers and books fill about 90 volumes.
Remarkably, much of this output dates from the last two decades of his life, when he was totally blind.
Though born and educated in Basel, Switzerland, Euler spent most of his career in St. Petersburg and Berlin.
Euler's prolific output caused a tremendous problem of backlog: the St. Petersburg Academy continued publishing his work posthumously for more than 30 years.
Euler married twice and had 13 children, though all but five of them died young.

15. Rothalpy
In a compressor or pump the specific work done on the fluid equals the rise in stagnation enthalpy. Thus, combining eqns,
and
We, have
This relationship is true for steady, adiabatic and irreversible flow in compressors or in pump impellers.
Rearranging and using h0 = h + 1/2c2

16. Rothalpy…
According to the above reasoning a new function I has been defined having the same value at exit from the impeller as at entry.
The function I is widely used as rothalpy, a contraction of rotational stagnation enthalpy, and is a fluid mechanical property of some importance in the study of relative flows in rotating systems.
As the value of rothalpy is apparently unchanged between entry and exit of the impeller it is deduced that it must be constant along the flow lines between these two stations.
Thus, the rothalpy can be written generally as

17. Turbomachine System Inlet Outlet
A turbomachine consists of one or several stages

18. Turbomachine System Stage denotations stator inlet rotor inlet
rotor outlet
The thermodynamic and kinetic properties in a stage are defined by velocities
Velocity triangles are used to design blade shapes

19. Compressor stage
Turbine stage

20. Jet Engine

21. Turbine animation

22. Components of energy transfer in Euler turbomachine equation
The change in magnitude of the axial velocity components through the rotor gives rise to an axial force, which must be taken by a thrust bearing to the stationary rotor casing.
The change in magnitude of the radial velocity component s produces radial force. Neither has any effect on the angular motion of the rotor. This force is taken care by journal bearing
The whirl or tangential components Cw produce the rotational effect.

23. Components of flow velocity in a generalised fluid machine
Components of energy transfer in Euler turbomachine equation
V=Absolute velocity
Va1— Axial velocity in a direction parallel to the axis of the rotating shaft.
Vf1— Radial velocity in the direct ion normal to the axis of the rotating shaft .
Vw1 — whirl or tangential velocity in the direction normal to a radius.
Similarly, exit velocity V2 can be resolved into three components; that is, Va2 , Vf2 , and Vw2
Components of flow velocity
in a generalised fluid machine

24. Turbine blades

25. Velocity triangle
Vector diagrams of velocities at inlet and outlet correspond to two velocity triangles,
where Vr is the velocity of fluid relative to the rotor and
1 and 2 are the and outlet respectively angles made by the directions of the absolute velocities (V) at the inlet with the tangential direction,
1 and 2 are the angles made by the relative velocities (Vr) 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 undesirable losses.

26. Velocity triangle
To calculate torque from the Euler turbine equation, it is necessary to know the velocity components: Vw1 , Vw2 , and the rotor speeds U1 and U2 or the relative velocities Vr1 , Vr 2; radial velocities Vf1 , Vf2 as well as U1 and U2
These quantities can be determined easily by drawing the velocity triangles at the rotor inlet and outlet

27. Components of energy transfer in Euler turbomachine equation
where T is the torque exerted by the rotor on the moving fluid, m is the mass flow rate of fluid through the rotor.
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
Above equation is known as Euler's equation in relation to fluid machines.

28. Components of energy transfer in Euler turbomachine equation
Above equation can be written in terms of head gained 'H' by the fluid as
Note that both of the above equations 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.

29. Components of energy transfer in Euler turbomachine equation
From the inlet velocity triangle,

30. Components of energy transfer in Euler turbomachine equation
Euler's equation in relation to fluid machines,
Becomes,
The first term i.e. (V12 - V22 )/2g represents the energy transfer due to change of absolute kinetic energy or dynamic head or dynamic pressure of the fluid during its passage between the entrance and exit sections.
In a pump or compressor, the discharge kinetic energy from the rotor, may be considerable .
Normally, it is or dynamic head or dynamic pressure that is required as useful energy. Usually the kinetic energy at the rotor outlet is converted into a static pressure head by passing the fluid through a diffuser.
In a turbine, the change in absolute kinetic energy represents the power transmitted from the fluid to the rotor due to an impulse effect .

31. Components of energy transfer in Euler turbomachine equation
The other two terms of are factors that produce pressure rise within the rotor itself, and hence they are called “internal diffusion.”
The centrifugal effect, (U12 - U22 )/2g , is due to the centrifugal forces that are developed as the fluid particles move outwards towards the rim of the machine. This effect is produced if the fluid changes radius as it flows from the entrance to the exit section. This causes a change in static head of the fluid through the rotor.
The third term, (Vr22 - Vr12 )/2g , represents the energy transfer due to the change of the relative kinetic energy of the fluid. If Vr2 > Vr1 the passage acts like a nozzle and if Vr2 < Vr1 , it act s like a diffuser. This causes a change in static head of the fluid across the rotor.

33. 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, U1 = U2 and the equation of energy transfer 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.

34. Radially Outward and Inward Flow Machines
For radially outward flow machines, U2 > U1 and hence the fluid gains in static head, while, for a radially inward flow machine U2 < U1 , 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.
Radial turbomachine

35. 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,
where, H is,

36. Impulse and Reaction Machines
For an impulse machine R = 0, because there is no change in static pressure in the rotor.
Thus for an axial flow impulse machine U1 = U2, Vr1 = Vr2.
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

37. Impulse and Reaction Machines
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 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.

38. 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 Vw2 the tangential component of fluid velocity at impeller outlet.
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.

39. Slip Factor
The mean direction of flow at outlet, under this situation, changes from the blade angle 2 at outlet to a different angle 2’
Therefore the tangential velocity component at outlet Vw2 is reduced to V’w2 , velocity triangles, and the difference is defined as the slip.

40. End of Module II