Velocity Triangle for Turbo-machinery BY P M V Subbarao Professor Mechanical Engineering Department I I T Delhi

U Vri Vai Inlet Velocity Triangle Vai Vri U U Vre Vae Exit Velocity Triangle Vre

U Vri Vai Vre Vae bi ai ae be Vai: Inlet Absolute Velocity Vri: Inlet Relative Velocity Vre: Exit Relative Velocity Vae:Exit Absolute Velocity ai: Inlet Nozzle Angle. bi: Inlet Blade Angle. be: Exit Blade Angle. ai: Exit Nozzle Angle.

Vf Vrc Vr Va Vw U

Flow through Blades U Vri Vre Vai Vae Vni

Fluid Dynamics of Blades

U Vri Vai Vre Vae bi ai ae be The stream is delivered to the wheel at an angle ai and velocity Vai. The selection of angle ai is a compromise. An increase in ai, reduces the value of useful component (Absolute circumferential Component). This is also called Inlet Whirl Velocity, Vwi = Vai cos(ai). An increase in ai, increases the value of axial component, also called as flow component. This is responsible for definite mass flow rate between to successive blade. Flow component Vfi = Vai sin(ai) = Vri sin(bi). The absolute inlet velocity can be considered as a resultant of blade velocity and inlet relative velocity. The two points of interest are those at the inlet and exit of the blade.

U Vri Vai Vre Vae bi ai ae be If the stream is to enter and leave the blades without shock or much losses, then relative velocity should be tangential to the blade inlet tip. Vri should enter at an angle bi, the inlet blade angle. Similarly, Vre should leave at be, the exit blade angle. A blade is said to be symmetric if bi = be. The flow velocities between two successive blade at inlet and exit are Vfi & Vfe. The axial (basic useful) components or whirl velocities at inlet and exit are Vwi & Vwe.

Impulse Turbine

U Vri Vai Vre Vae bi ai ae be Newton’s Second Law for an Impulse Blade: The tangential force acting of the jet is: F = mass flow rate X Change of velocity in the tangential direction Tangential relative velocity at blade Inlet : Vri cos(bi). Tangential relative velocity at blade exit : -Vre cos(be). Change in velocity in tangential direction: -Vre cos(be) - Vri cos(bi). -(Vre cos(be) + Vri cos(bi)). Tanential Force,

The reaction to this force provides the driving thrust on the wheel. The driving force on wheel Power Output of the blade, Diagram Efficiency or Blade efficiency:

U Vri Vai Vre Vae bi ai ae be

Define Blade Speed Ratio, f

For a given shape of the blade, the efficiency is a strong function of f. For maximum efficiency:

Impulse-Reaction turbine This utilizes the principle of impulse and reaction. There are a number of rows of moving blades attached to the rotor and equal number of fixed blades attached to the casing. The fixed blades are set in a reversed manner compared to the moving blades, and act as nozzles. The fixed blade channels are of nozzle shape and there is a comparatively small drop in pressure accompanied by an increase in velocity. The fluid then passes over the moving blades and, as in the pure impulse turbine, a force is exerted on the blades by the fluid. There is further drop in pressure as the fluid passes through the moving blades, since moving blade channels are also of nozzle shape. The relative velocity increases in the moving blades.

U Vri Vai Vre Vae bi ai ae be The reaction effect is an addition to impulse effect. The degree of reaction p va vr

First law for fixed blades: 2 1 First law for moving blades:

U Vr1 Va1 Vr2 Va2 b1 a1 a2 b2 If the stream is to enter and leave the blades without shock or much losses, then relative velocity should be tangential to the blade inlet tip. Vr1 should enter at an angle b1, the inlet blade angle. Similarly, Vr2 should leave at b2, the exit blade angle. In an impulse reaction blade, Vr2 > Vr1. The flow velocities between two successive blade at inlet and exit are Vf1 & Vf2. The axial (basic useful) components or whirl velocities at inlet and exit are Vw1 & Vw2.

U Vr1 Va1 Vr2 Va2 b1 a1 a2 b2 Newton’s Second Law for an Impulse-reaction Blade: The tangential force acting of the jet is: F = mass flow rate X Change of velocity in the tangential direction Tangential relative velocity at blade Inlet : Vr1 cos(b2). Tangential relative velocity at blade exit : -Vr2 cos(b2). Change in velocity in tangential direction: -Vr2 cos(b2) – Vr1 cos(b1). -(Vr2 cos(b2) + Vr1 cos(b1)). Tangential Force,

The reaction to this force provides the driving thrust on the wheel. The driving force on wheel Power Output of the blade, Diagram Efficiency or Blade efficiency:

First law for fixed blades: First law for moving blades:

Losses in nozzle, Nozzle blade loss factor, f

For a given shape of the blade, the efficiency is a strong function of U/Va1. For maximum efficiency: