Turbine Speed Calculations

TF209 Chapter 18 – Rankine Cycle © L.C. Spencer 2004 Page 82 of 100

Chapter 17: Velocity Diagrams for Turbines

Basic Vector DiagramsIn a turbine, steam is directed at high velocity onto moving blades attached to a spinning rotor. The direction of steamflow is roughly parallel to the direction of blade movement, but it has an axial component so the steam can progressthrough the turbine. Blades redirect the steam flow giving it avelocity component in the opposite direction to the blade motion. Ifblade entry and exit velocities are the same (relative to the blades).the absolute steam velocity is reduced as it passes through the blades.Thus the steam’s kinetic energy is reduced as it passes through therotor, and a force is applied on the moving blades, doing work byapplying a torque to the rotating shaft.

To analyse the above processes, we use vector diagrams. In these, weneed to determine both absolute and relative velocities of the steam.By absolute velocity, we mean the velocity a stationary observerwould detect (i.e. steam velocity relative to the frame of the turbine).The relative velocity is the velocity relative to a particular movingobserver (e.g. relative to the moving blades).

The velocity vector diagram shows the magnitude and direction ofthe steam velocities and the rotor blade speed.

ub represents the speed of the rotor blades (perpendicular to theshaft).

c1 represents the absolute velocity of the steam, as it leaves thenozzles or fixed blades, and enters the moving blade passage.

α1 is the angle at which steam leaves the nozzles or fixed bladesbefore entering the rotor blade passages. Note the convention usedgiving all angles relative to the (downstream) turbine shaft.

v1 represents the velocity of steam entering the moving bladepassage relative to the blades, and the angle of entry is β1.

Similarly, c2 and v2 represent the absolute and relative velocities ofthe steam leaving the moving blades, at angles α2. and β2.respectively.

The two velocity vector diagrams can be combined to show acomplete velocity vector diagram across a row of moving blades.

There are two directions of movement to consider in a turbinestage:

(i) the rotation in the whirl direction;

(ii) the movement in the axial direction.

It is possible to separate the absolute velocities of the steam into itswhirl and axial components.


As a result, velocities in the vector diagram can be separated into whirl and axial components, as shown in the diagram.

ΔCw.is the sum of the absolute velocities of the steam in the whirldirection (direction of rotation), and:

ΔCw = C1sinα1 + C2sinα2Also,

ΔCw = ΔVwtherefore:

ΔVw = V1sinβ1 + V2sinβ2

c1a and c2a are the components of the absolute velocities in the axial direction, and:C1a = C1cosα1 and C2a = C2cosα2

Finally, the blade speed can be calculated from the velocity vector diagram by:

UB = C1sinα1 − V1sinβ1

= V2sinβ2 − C2sinαTo find other information on the vector diagram, we use the sine and cosine rules.

Sine Rule: asinA = b

sinB = csinC

Cosine Rule: cosA = b2 +c2 − a22bc

Specific velocity vector diagram can be developed for impulse and reaction turbines.

Velocity Vector Diagrams for Impulse TurbinesIn impulse turbines, there is no pressure drop across the moving blades. Both the fixed and moving blades are symmetrical.Therefore, the blade angles are equal, that is:

β1=β2As there is no pressure drop across the blades, there is noincrease in the velocity relative to the blades, i.e.:

V1=V2In the axial direction, we have:

C1a = C1cosα1 = V1cosβ1

C2a = C2cosα2 = V2cosβ2

therefore axial components of the absolute velocities are the same:

C1a =C2aAlso,

ΔVw = V1sinβ1 + V2sinβ2 = 2V sinβ

Velocity Vector Diagrams for Reaction TurbinesIn reaction turbines, there is a pressure drop as steam flows through the moving blades, so the steam accelerates relative tothe blades as it passes through the rotor.

The fixed blades are the same size as the moving blades, butreversed in curvature.

With a pressure drop in the blades, both fixed and movingblades act as nozzles.

The angles become:α1 = β2 α2 = β1


and therefore: C1 = V2C2 = V1

Also, from symmetry: C1a =C2a

Power Output of a Turbine StageThe power output of a turbine stage P is the rate of work done in a turbine stage ˙ W and can be written as:

P = ˙ W = ˙ m UBΔCw = ˙ m UBΔVw (kW or kJ/s)where: ˙ m is the mass flow rate (kg/s)

UB is the speed of the rotor blades (m/s)ΔCw , ΔVw are the sum of components of absolute and relative velocities in whirl direction.

Utilisation FactorUtilisation Factor (also called the Diagram Efficiency) gives an indication of the efficiency of a turbine stage. It is thefraction of the energy available to a stage that becomes work output.

Therefore, the utilisation factor is defined as:

UF = Rotor Work / Total energy available at inlet to rotor blades

The total energy at inlet is equal to the rotor work plus the rejected energy, that is:

Total energy at inlet = ˙ W + 12 ˙ m C2

2

The utilisation factor becomes: UF =

˙ W ˙ W + 12 ˙ m C2

2

Axial ThrustAxial thrust Ra is the force in the axial direction acting along the rotor. It is important to consider because it may movethe rotor in the axial direction, and cause fouling of the rotor blades with the casing blades.

Axial thrust arises from two causes:(i) components of the absolute velocities in the axial direction;(ii) pressure drop across the rotor blades (as in a reaction turbine), giving a net force exerted by the steam overthe blade area projected in the axial direction.

Therefore, the total axial thrust acting on a rotor blade becomes:

Ra = ˙ m (C1a −C2a) + A(P1−P2 )

=˙ m (C1a −C2a) + AδP

In an impulse turbine:δP = 0 across the blades, andC1a =C2a

and therefore: Ra = 0

In a reaction turbine:C1a =C2a

and therefore: Ra = AδP


Review and Reinforce - Chapter 17

17.1. How do we use steam velocities to limit end thrust in a reaction turbine stage?

17.2. Where would you measure this on a turbine vector diagram?

Turbine Speed Calculations

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