Three-dimensional losses and correlation in turbomachinery

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Three-dimension losses and correlation in turbomachinery refers to the measurement of flow-fields in three dimensions, where measuring the loss of smoothness of flow, and resulting inefficiencies, becomes difficult, unlike two-dimensional losses where mathematical complexity is substantially less.

Contents

Three-dimensionality takes into account large pressure gradients in every direction, design/curvature of blades, shock waves, heat transfer, cavitation, and viscous effects, which generate secondary flow, vortices, tip leakage vortices, and other effects that interrupt smooth flow and cause loss of efficiency. Viscous effects in turbomachinery block flow by the formation of viscous layers around blade profiles, which affects pressure rise and fall and reduces the effective area of a flow field. Interaction between these effects increases rotor instability and decreases the efficiency of turbomachinery.

In calculating three-dimensional losses, every element affecting a flow path is taken into account—such as axial spacing between vane and blade rows, end-wall curvature, radial distribution of pressure gradient, hup/tip ratio, dihedral, lean, tip clearance, flare, aspect ratio, skew, sweep, platform cooling holes, surface roughness, and off-take bleeds. Associated with blade profiles are parameters such as camber distribution, stagger angle, blade spacing, blade camber, chord, surface roughness, leading- and trailing-edge radii, and maximum thickness.

Two-dimensional losses are easily evaluated using Navier-Stokes equations, but three-dimensional losses are difficult to evaluate; so, correlation is used, which is difficult with so many parameters. So, correlation based on geometric similarity has been developed in many industries, in the form of charts, graphs, data statistics, and performance data.

Types of losses

Three-dimensional losses are generally classified as:

  1. Three-dimensional profile losses
  2. Three-dimensional shock losses
  3. Secondary flow
  4. Endwall losses in axial turbomachinery
  5. Tip leakage flow losses
  6. Blade boundary layer losses

Three-dimensional profile losses

Effect on efficiency by blade profile losses Effect of three dimensional profile losses on efficiency of turbomachinery.png
Effect on efficiency by blade profile losses

The main points to consider are:

Three-dimensional shock losses

Shock losses due to accumulation of flow Three dimensional shock losses.png
Shock losses due to accumulation of flow
Generation of secondary flow due blade profile Secondary flow losses.png
Generation of secondary flow due blade profile

The main points to consider are:

Secondary flow

The main points to consider are:

ζs = (0.0055 + 0.078(δ1/C)1/2)CL2 (cos3α2/cos3αm) (C/h) (C/S)2 ( 1/cos ά1)
where ζs = average secondary flow loss coefficient; α2, αm = flow angles; δ1/C = inlet boundary layer; and C,S,h = blade geometry.

Endwall losses in axial flow in turbomachinery

Endwall losses due to vortex Endwall losses in turbomachinery.pdf
Endwall losses due to vortex

The main points to consider are:

ζ = ζp + ζew      ζ = ζp[ 1 + ( 1 + ( 4ε / ( ρ2V21V1 )1/2 ) ) ( S cos α2 - tTE )/h ]
where ζ=total losses, ζp=blade profile losses, ζew=endwall losses.
η = ή ( 1 - ( δh* + δt*)/h ) / ( 1 - (  Fθh +  Fθt ) / h )
where η=efficiency in absence of endwall boundary layer, where h refers to the hub and t refers to the tip. The values of Fθ and δ* are derived from the graph or chart.

Tip-leakage flow losses

Tip leakage losses due to tip endwall Tip leakage losses in turbomachinery.pdf
Tip leakage losses due to tip endwall

The main points to consider are:

QL = 2 ( ( Pp - Ps ) / ρ )1/2
a/τ = 0.14 ( d/τ  ( CL )1/2 )0.85
ζL ~ ( CL2 * C * τ * cos2β1 ) / ( A * S * S * cos2βm )
ζW ~ ( δS* + δP* / S ) * ( 1 / A ) * ( ( CL )3/2) * ( τ / S )3/2Vm3 / ( V2 * V12 )

See also

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