Theory of turbo machineEffect of Blade Configuration on

Characteristics of Centrifugal machines

Unit 2(Potters & Wiggert Sec. 12.2.1, &-607)

Expression relating Q, H, Pdeveloped by Rotary machines

• Rotary Machines include: Centrifugal (or radial),Axial, andMixed types

• In such machines when fluid passes through blade passage static pressure changes.

• Axial flowMixed flow →→

Centrifugal (Unit # 2)• ↓

<— Axial flow (Unit #4)

CENTRIFUGAL MACHINE

12.2.1

A typical radial flow pump.

We already know from Mechanics

1. For a rotary machine• Power = Angular velocity x Torque

= Mass flow rate x Head• Torque = Rate of change of angular

momentum= Mass x [Abs. Circum. velocity x radius (in-out)]

T = [ρ Q] (r2Vt2 – r1Vt1)

Idealized radial-flow impeller (a) impeller; (b) velocity diagrams.

Relative Velocity(Fluid entering periphery)

Power (In terms of flow rate & Blade angle)

• From velocity triangle:Vt= Vncotα = u – Vncotβ

where Vn is radial component of V• From above

P = ρ Q(u2Vt2 – u1Vt1) = ρ Q(u2Vn2 cotα2 – u1Vn1 cotα1 ) (5)

• NOTE1. To minimize entrance loss

Blade angle β is equal to the entry angle of fluid to the blade.2. To minimize exit loss

Fluid entry angle (α) is equal to the angle of the guide vane3. α = Angle between tip and absolute velocity

β = Angle between tip and relative velocity

Symbols to be used• Velocities:

V - Absolute fluid velocityv - Relative fluid velocityu - peripheral speed of blade

• Subscripts:1 - inlet

2 - outletn - normal component

t - tangential component• Geometry:

b - blade widthr - blade radiusα - angle between V and u vectorsβ - angle between v and u vectors

Head• Power, P = Weight flow rate x Head = P = (ρ Qg) H• Head of fluid column,

H = P/(ρ Q .g)] (6)Substituting P from Eq.5 we get

(7)

• For highest head cot α1 = 0; i.e α1 = 90– (8)

• Substituting:Flow rate, Q = Vn.2π r b; Tip velocity u2= wr2 , we can

get– (9)

( )g

VuVug

VuVuH nntt )cotcot( 1112221122 αα −=

−=

( )g

VuugVuH nt )cot( 222222 β−

==

Qgbg

rH 22

22

2cotπ

βωω−=

2

Summary of what we have learnt • From geometry

Vn2 = V2-Vt

2 = v2- (u –Vt )2

u Vt = (V2+ u2 –v2)/2 (12)where u = velocity of blade,

Vt= tangential component of absolute velocity of fluid

• From (4) & (12) (13)

•

• Head = Kinetic energy gain + Pressure rise

gVVuu

gVVH

gVuVVuV

QgPH

rr

rr

2)()(

2

)(

21

22

21

22

21

22

21

21

21

22

22

22

−−−+

−=

−+−−+==

ρ

SUMMARYSUMMARY

• Blade angle (β) is ideally the angle between the relative velocity (Vr) and blade-tip velocity (u) vectors

• To draw the vector diagram note that the blade-tip velocity and relative velocity vector are in the same rotational (clockwise or anticlockwise) direction. Third side of the triangle is the absolute velocity vector which is in opposite direction.

• Power = [blade velocity x tangential component of absolute velocity] inlet – outlet

• Flow ~ Rotor circumference x width x Normal velocity

What we have learnt• Blade angle (β) is ideally the angle between the relative

velocity (Vr) and blade-tip velocity (u) vectors• To draw the vector diagram note that the blade-tip velocity

and relative velocity vector are in the same rotational (clockwise or anticlockwise) direction. The arm of the triangle is the absolute velocity vector which is in opposite direction.

• Power = [blade velocity x tangential component of absolute velocity] inlet – outlet

• Flow ~ Rotor circumference x width x Normal velocity

Blade shapes• Straight (radial) blade wheel

• Forward curve wheel• Backward curve wheel

Vector diagram of a centrifugal pump/fan

FLOW CHARACTERISTICS• Head = Power delivered to fluid

Fluid flow rate (weight) H = Pw /(ρQ g) = (u2Vt2 – u1Vt1 )/g

• For maximum head, Vt1 = 0Η = u2Vt2 /g

• From velocity diagram, Vt2= u2-Vn2cotβ2 • Flow rate discharge, Q = 2 πr2 bVn2• So, H = [u2

2-(Q/ 2 πr2 b) u2cotβ2]/g• = A – B.Q cotβ2

Efficiency

• Ideal Head varies linearly with discharge (Q). • Head (H) increases or decreases with Q

depending on blade angle β2

• With valve shut off . i.e Q = 0

• For pumps/fans: • Efficiency =

where P is the power consumed PQgHρη =

gu

H22=

Ideal H vs Q characteristics

Effect of blade configuration on Performance

• Depending upon the value of exit blade angle the head increases or decreases with increase in flow

• Energy transfer ~ Vt2. From velocity diagram, for a given tip velocity, u forward & radial curve blades transfer more energy

• Backward blades give higher efficiency• Forward and radial are smaller in size for the same

duty, but have lower efficiency• Centrifugal compressor uses radial blades for

better strength against high speed rotation

Characteristics of different types of blades

• Owing to the losses the actual characteristic is different from theoretical linear shape

• Power consumption varies with flow Q

• Efficiency varies with Q with highest value being in the design condition

Home work

1. Show that the manometric head for a pump having a discharge Q and running at a speed N can be expressed by an equation of the form Hm=AN2+BNQ+CQ2, where A,B,C are constants.

Example1. A centrifugal pump impeller is 255 mm diameter, the

water passage 32 mm wide at exit, and the vane angle at exit 30. The effective flow area is reduced by 10% because of vane thickness. The manometric efficiency is 80% when the pump runs at 1000 rpm and delivers 50 litre/s. Calculate the manometer head measured between inlet and outlet flange of the pump assuming 47% of the discharge head is not converted into pressure head. Assume the pump delivers maximum head.