Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Velocity Triangle for Turbo-machinery BY P M V Subbarao Professor Mechanical Engineering Department I I T Delhi U Vai Vri Vai Vri Inlet Velocity Triangle U U Vae Vre Vre Exit Velocity Triangle ae Vae ai U bi be Vai Vre 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. Vri Vf Vrc Vr Vw Va U Flow through Blades Vre U U Vae Vni Vri U Vai Fluid Dynamics of Blades ae Vae • • • • • • • • • ai U bi be Vai Vre Vri 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. ae Vae ai U bi be Vai Vre Vri • 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 ae Vae ai U bi be Vai Vri Vre 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, FA mVre cos b e Vri cos b i The reaction to this force provides the driving thrust on the wheel. The driving force on wheel FR mVre cos b e Vri cos b i Power Output of the blade, Pb m U Vre cos b e Vri cos b i Diagram Efficiency or Blade efficiency: Power ouput d Kinetic Power of inlet steam m U Vre cos b e Vri cos bi d m Vai2 2U kVri cos b e Vri cos bi d Vai2 2UVri k cos b e cos bi d 2 Vai ae Vae ai U bi be Vai Vre Vai cos a i U Vri cos bi Vri Vai cos a i U Vri cos b i cos b e 2U Vai cos a i U k 1 cos bi d 2 Vai cos b e 2U Vai cos a i U k 1 cos bi d 2 Vai 2 U U cos b d 2 cos a i k 1 Vai Vai cos bi e Define Blade Speed Ratio, f cos b e d 2fcos a i f k 1 cos bi For a given shape of the blade, the efficiency is a strong function of f. For maximum efficiency: d d 0 df cos b e 2cos a i 2f k 1 0 cos bi cos a i cos a i 2f 0 f 2 d ,max cos a i cos b e 2 cos a i cos a i 1 k 2 cos bi d ,max cos b e f cos a i k 1 cos bi 2 2 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. ae Vae ai U Vre bi be Vai Vri The reaction effect is an addition to impulse effect. The degree of reaction The enthalpy drop in the moving blades The enthalpy drop in the stage p va vr First law for fixed blades: h0 h1 V12 V02 2 0 First law for moving blades: h1 h2 V r22 Vr21 2 1 2 a2 Va2 a1 Vr2 U b1 b2 Va1 Vr1 • 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. h1 h2 V r22 Vr21 2 • 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. a2 Va2 a1 Vr2 U b1 b2 Va1 Vr1 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, FA mVr 2 cos b 2 Vr1 cos b1 The reaction to this force provides the driving thrust on the wheel. The driving force on wheel F mV cos b V cos b R r2 2 r1 1 Power Output of the blade, Pb m U Vr 2 cos b 2 Vr1 cos b1 Diagram Efficiency or Blade efficiency: Power ouput d Kinetic Power of inlet steam m U Vr 2 cos b 2 Vr1 cos b1 d m Va21 First law for fixed blades: h0 h1 V12 V02 2 First law for moving blades: h1 h2 h0 h2 V r22 Vr21 2 V12 V02 2 V r22 Vr21 2 2 2 Va21 V02 V r 2 Vr1 h0 h2 2 2 The enthalpy drop in the moving blades The enthalpy drop in the stage h1 h2 h0 h2 V r22 Vr21 h1 h2 2 h0 h2 Va1 V02 V r22 Vr21 Vr 2 2 2 V r1 Va1 V0 1 Vr 2 2 2 Vr1 V V a1 0 1 2 2 2 2U Vr 2 cos b 2 Vr1 cos b1 d Va21 2 2 V r 2 V r21 Va1 V0 1 Va1 cos a1 U Vr1 cos b1 Va1 cos a1 U Vr1 cos b1 Losses in nozzle, Nozzle blade loss factor, f Va1 Actual absolute inlet velo city f Isoentropi c Velocity at nozzle exit Vn,iso 2U Vr 2 cos b 2 Vr1 cos b1 stage 2 Vn,iso 2 2 2 2Uf V r 1 Va1 V0 cos b 2 Vr1 cos b1 1 stage Va21 2 Va1 cos a1 U Vr1 cos b1 2 2Uf stage 2 Va1 cos a1 U 2 Va1 cos a1 U 2 cos b1 Va1 V0 cos b 2 cos b1 cos b1 1 Va21 U 2 2 f Va1 stage U cos a1 Va1 cos b1 U 2 2 f Va1 stage U cos a 1 Va1 cos b1 2 U cos a 2 1 V Va1 1 0 cos b 2 1 Va1 cos b1 cos b 1 Va21 2 U cos a 2 1 V Va1 0 1 cos b 2 1 Va1 cos b1 cos b 1 Va21 For a given shape of the blade, the efficiency is a strong function of U/Va1. For maximum efficiency: d stage U d V a1 0