Theory of turbo machine Effect of Blade Configuration on Characteristics of Centrifugal machines Unit (Potters & Wiggert Sec. 1..1, &-607)
Expression relating Q, H, P developed by Rotary machines Rotary Machines include: Centrifugal (or radial), Axial, and Mixed types In such machines when fluid passes through blade passage static pressure changes.
< Axial flow (Unit #4) Axial flow Mixed flow Centrifugal (Unit # )
CENTRIFUGAL MACHINE 1..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] (r V t r 1 V t1 )
(a) impeller; Idealized radial-flow impeller (b) velocity diagrams.
Relative Velocity (Fluid entering periphery)
Power (In terms of flow rate & Blade angle) From velocity triangle: V t = V n cotα = u V n cotβ where V n is radial component of V From above P = ρ Q(u V t u 1 V t1 ) = ρ Q(u V n cotα u 1 V n1 cotα 1 ) (5) NOTE 1. To minimize entrance loss Blade angle β is equal to the entry angle of fluid to the blade.. To minimize exit loss Fluid entry angle (α) is equal to the angle of the guide vane 3. α = Angle between tip and absolute velocity β = Angle between tip and relative velocity
Symbols to be used Velocities: V - Absolute fluid velocity v - Relative fluid velocity u - peripheral speed of blade Geometry: b - blade width r - blade radius α - angle between V and u vectors β - angle between v and u vectors t n Subscripts: 1 -inlet -outlet - normal component - tangential component
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 ( u V u V ) u V cotα u V cot (7) H = t 1 t1 ( n 1 n1 α1) For highest head cot α 1 = 0; i.e α 1 = 90 g uvt u( u Vn cot β) (8) H = = g g Substituting: Flow rate, Q = V n.π r b; Tip velocity u = wr, we can get ω r g (9) = ( ) H = ω cot β πb g Q g
Summary of what we have learnt From geometry V n = V -V t = v - (u V t ) u V t = (V + u v )/ (1) where u = velocity of blade, V t = tangential component of absolute velocity of fluid From (4) & (1) (13) Head = Kinetic energy gain + Pressure rise g V V u u g V V H g V u V V u V Qg P H r r r r ) ( ) ( ) ( 1 1 1 1 1 1 + = + + = = ρ
SUMMARY Blade angle (β) is ideally the angle between the relative velocity (V r ) 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 (V r ) 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 = P w /(ρq g) = (u V t u 1 V t1 )/g For maximum head, V t1 = 0 Η = u V t /g From velocity diagram, V t = u -V n cotβ Flow rate discharge, Q = πr bv n So,H = [u -(Q/ πr b) u cotβ ]/g = A B.Q cotβ
Efficiency Ideal Head varies linearly with discharge (Q). Head (H) increases or decreases with Q depending on blade angle β With valve shut off. i.e Q = 0 For pumps/fans: Efficiency = where P is the power consumed η = H = u g ρqgh P
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 ~ V t. 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 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 of blades
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 H m =AN +BNQ+CQ, where A,B,C are constants.
Example 1. A centrifugal pump impeller is 55 mm diameter, the water passage 3 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.