# Three-dimensional losses and correlation in turbomachinery

Last updated

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

## Three-dimensional profile losses

The main points to consider are:

• Profile losses that occur due to the curvature of blades, which includes span-wise mixing of flow field, in addition to two-dimensional mixing losses (which can be predicted using Navier-Stokes equations).
• Major losses in rotors that are caused by radial pressure gradient from midspan to tip (flow ascending to tip).
• Reduction in high losses between annulus wall and tip clearance region, which includes the trailing edge of a blade profile. This is due to flow mixing and flow redistribution at the inner radius as flow proceeds downstream.
• Between the hub and annulus wall, losses are prominent due to three-dimensionality.
• In single-stage turbomachinery, large radial pressure gradient losses at exit of flow from rotor.
• Platform cooling increases the endwall flow loss and coolant air increases profile loss.
• Navier-Stokes identifies many of the losses when some assumptions are made, such as unseparated flow. Here correlation is no longer justified.

## Three-dimensional shock losses

The main points to consider are:

• Shock losses continuously increase from the hub to tip of the blade in both supersonic and transonic rotors.
• Shock losses are accompanied by shock-boundary-layer interaction losses, boundary-layer losses in profile secondary flow, and tip clearance effects.
• From the Mach number prospective, fluid inside rotor is in supersonic phase except at initial hub entry.
• The Mach number increases gradually from midspan to tip. At the tip, the effect is less than secondary flow, tip clearance effect, and annulus wall boundary-layer effect.
• In a turbofan, shock losses increase overall efficiency by 2% because of the absence of tip clearance effect and secondary flow being present.
• Correlation depends on many parameters and is difficult to calculate.
• Correlation based on geometric similarity is used.

## Secondary flow

The main points to consider are:

• The rotation of a blade row causes non-uniformity in radial velocity, stagnation pressure, stagnation enthalpy, and stagnation temperature. Distribution in both tangential and radial directions generates secondary flow.
• Secondary flow generates two velocity components Vy, Vz, hence introducing three-dimensionality in the flow field.
• The two components of velocity result in flow-turning at the tailing end of the blade profile, which directly affects pressure rise-and-fall in turbomachinery. Hence efficiency decreases.
• Secondary flow generates vibration, noise, and flutter because of unsteady pressure field between blades and rotor–stator interaction.
• Secondary flow introduces vortex cavitation, which diminishes flow rate, decreases performance, and damages the blade profile.
• The temperature in turbomachinery is affected.
• Correlation for secondary flow, given by Dunham (1970), is given by:
`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

The main points to consider are:

• In a turbine, secondary flow forces the wall boundary layer toward the suction side of the rotor, where mixing of blade and wall boundary takes place, resulting in endwall losses.
• The secondary flow carries core losses away from the wall and blade boundary layer, through formation of vortices. So, peak loss occurs away from endwall.
• Endwall losses are high in stator (Francis turbine/Kaplan turbine) and nozzle vane (Pelton turbine), and the loss distribution is different for turbine and compressor, due to flows being opposite to each other.
• Due to the presence of vortices, large flow-turning and secondary flow result to form a complex flow field, and interaction between these effects increases endwall losses.
• In total loss, endwall losses form the fraction of secondary losses given by Gregory-Smith, et al., 1998. Hence secondary flow theory for small flow-turning fails.
• Correlation for endwall losses in an axial-flow turbine is given by:
`ζ = ζp + ζew      ζ = ζp[ 1 + ( 1 + ( 4ε / ( ρ2V2/ρ1V1 )1/2 ) ) ( S cos α2 - tTE )/h ]`
where ζ=total losses, ζp=blade profile losses, ζew=endwall losses.
• The expression for endwall losses in an axial-flow compressor is given by:
`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

The main points to consider are:

• The rotation of a rotor in turbomachinery induces a pressure differences between opposite sides of the blade profile, resulting in tip leakage.
• In a turbomachinery rotor, a gap between the annulus wall and the blade causes leakage, which also occurs in the gap between the rotating hub and stator.
• Direct loss through clearance volume, as no angular momentum is transferred to fluid. So, no work is done.
• Leakage, and its interaction with other losses in the flow field, is complex; and hence, at the tip, it has a more pronounced effect than secondary flow.
• Leakage-flow induced three-dimensionality, like the mixing of leakage flow with vortex formation, entrainment process, diffusion and convection. This results in aerodynamics losses and inefficiency.
• Tip leakage and clearance loss account for 20–40% of total losses.
• The effects of cooling in turbines causes vibration, noise, flutter, and high blade stress.
• Leakage flow causes low static pressure in the core area, increasing the risk of cavitation and blade damage.
• The leakage velocity is given as:
`QL = 2 ( ( Pp - Ps ) / ρ )1/2`
• The leakage flow sheet due to velocity induced by the vortex is given in Rains, 1954:
`a/τ = 0.14 ( d/τ  ( CL )1/2 )0.85`
• Total loss in clearance volume is given by two equations-
`ζL ~ ( CL2 * C * τ * cos2β1 ) / ( A * S * S * cos2βm )`
`ζW ~ ( δS* + δP* / S ) * ( 1 / A ) * ( ( CL )3/2) * ( τ / S )3/2Vm3 / ( V2 * V12 )`

## Related Research Articles

A steam turbine is a device that extracts thermal energy from pressurized steam and uses it to do mechanical work on a rotating output shaft. Its modern manifestation was invented by Charles Parsons in 1884.

The Tesla turbine is a bladeless centripetal flow turbine patented by Nikola Tesla in 1913. It is referred to as a bladeless turbine. The Tesla turbine is also known as the boundary-layer turbine, cohesion-type turbine, and Prandtl-layer turbine because it uses the boundary-layer effect and not a fluid impinging upon the blades as in a conventional turbine. Bioengineering researchers have referred to it as a multiple-disk centrifugal pump. One of Tesla's desires for implementation of this turbine was for geothermal power, which was described in Our Future Motive Power.

The turbofan or fanjet is a type of airbreathing jet engine that is widely used in aircraft propulsion. The word "turbofan" is a portmanteau of "turbine" and "fan": the turbo portion refers to a gas turbine engine which achieves mechanical energy from combustion, and the fan, a ducted fan that uses the mechanical energy from the gas turbine to accelerate air rearwards. Thus, whereas all the air taken in by a turbojet passes through the turbine, in a turbofan some of that air bypasses the turbine. A turbofan thus can be thought of as a turbojet being used to drive a ducted fan, with both of these contributing to the thrust.

A labyrinth seal is a type of mechanical seal that provides a tortuous path to help prevent leakage. An example of such a seal is sometimes found within an axle's bearing to help prevent the leakage of the oil lubricating the bearing.

Centrifugal compressors, sometimes called radial compressors, are a sub-class of dynamic axisymmetric work-absorbing turbomachinery.

The General Electric TF39 was a high-bypass turbofan engine that was developed to power the Lockheed C-5 Galaxy. The TF39 was the first high-power, high-bypass jet engine developed. The TF39 was further developed into the CF6 series of engines, and formed the basis of the General Electric LM2500 marine and industrial gas turbine. On September 7, 2017 the very last active C-5A powered with TF39 engines made its final flight to Davis-Monthan Air Force Base for retirement. The TF39 was effectively retired, and all remaining active C-5 Galaxys are now powered by General Electric F138-GE-100 (CF6) engines.

An axial compressor is a gas compressor that can continuously pressurize gases. It is a rotating, airfoil-based compressor in which the gas or working fluid principally flows parallel to the axis of rotation, or axially. This differs from other rotating compressors such as centrifugal compressor, axi-centrifugal compressors and mixed-flow compressors where the fluid flow will include a "radial component" through the compressor. The energy level of the fluid increases as it flows through the compressor due to the action of the rotor blades which exert a torque on the fluid. The stationary blades slow the fluid, converting the circumferential component of flow into pressure. Compressors are typically driven by an electric motor or a steam or a gas turbine.

Turbomachinery, in mechanical engineering, describes machines that transfer energy between a rotor and a fluid, including both turbines and compressors. While a turbine transfers energy from a fluid to a rotor, a compressor transfers energy from a rotor to a fluid.

A radial turbine is a turbine in which the flow of the working fluid is radial to the shaft. The difference between axial and radial turbines consists in the way the fluid flows through the components. Whereas for an axial turbine the rotor is 'impacted' by the fluid flow, for a radial turbine, the flow is smoothly orientated perpendicular to the rotation axis, and it drives the turbine in the same way water drives a watermill. The result is less mechanical stress which enables a radial turbine to be simpler, more robust, and more efficient when compared to axial turbines. When it comes to high power ranges the radial turbine is no longer competitive and the efficiency becomes similar to that of the axial turbines.

Industrial fans and blowers are machines whose primary function is to provide and accommodate a large flow of air or gas to various parts of a building or other structures. This is achieved by rotating a number of blades, connected to a hub and shaft, and driven by a motor or turbine. The flow rates of these mechanical fans range from approximately 200 cubic feet (5.7 m3) to 2,000,000 cubic feet (57,000 m3) per minute. A blower is another name for a fan that operates where the resistance to the flow is primarily on the downstream side of the fan.

In fluid dynamics, a secondary flow is a relatively minor flow superimposed on the primary flow, where the primary flow usually matches very closely the flow pattern predicted using simple analytical techniques that assume the fluid is inviscid.

The primary application of wind turbines is to generate energy using the wind. Hence, the aerodynamics is a very important aspect of wind turbines. Like most machines, there are many different types of wind turbines, all of them based on different energy extraction concepts.

Dr. Cheng Xu is a Chinese American aerodynamic design engineer and engineering manager. He is a Fellow of the American Society of Mechanical Engineers and a member of the Technical Committee on Energy and Power Systems, IASTED. He also served as a guest editor of International Journal of Rotating Machinery.

In turbomachinery, Degree of reaction or reaction ratio (R) is defined as the ratio of the static pressure drop in the rotor to the static pressure drop in the stage or as the ratio of static enthalpy drop in the rotor to the static enthalpy drop in the stage.

Compressor characteristic is the curve to show the behaviour of fluid, like change in pressure, temperature, entropy, flow rate etc. as it passes through a Dynamic compressor at different compressor speeds. The function of a compressor is to increase the pressure of a fluid passing through it, so that the exit pressure is higher than the inlet pressure. Due to this property, compressors are used in a wide range of machines, such as refrigerators, cars, jet engines and industrial processes. These curves are plotted between various parameters and some are as follows

In turbomachinery, the slip factor is a measure of the fluid slip in the impeller of a compressor or a turbine, mostly a centrifugal machine. Fluid slip is the deviation in the angle at which the fluid leaves the impeller from the impeller's blade/vane angle. Being quite small in axial impellers(inlet and outlet flow in same direction), slip is a very important phenomenon in radial impellers and is useful in determining the accurate estimation of work input or the energy transfer between the impeller and the fluid, rise in pressure and the velocity triangles at the impeller exit.

Francis turbine converts energy at high pressure heads which are not easily available and hence a turbine was required to convert the energy at low pressure heads, given that the quantity of water was large enough. It was easy to convert high pressure heads to power easily but difficult to do so for low pressure heads. Therefore, an evolution took place that converted the Francis turbine to Kaplan turbine, which generated power at even low pressure heads efficiently.

An axial turbine is a turbine in which the flow of the working fluid is parallel to the shaft, as opposed to radial turbines, where the fluid runs around a shaft, as in a watermill. An axial turbine has a similar construction as an axial compressor, but it operates in the reverse, converting flow of the fluid into rotating mechanical energy.

Blade solidity is an important design parameter for the axial flow impeller and is defined as the ratio of blade chord length to pitch.

Joseph Katz is an Israel-born American fluid dynamicist, known for his work on experimental fluid mechanics, cavitation phenomena and multiphase flow, turbulence, turbomachinery flows and oceanography flows, flow-induced vibrations and noise, and development of optical flow diagnostics techniques, including Particle Image Velocimetry (PIV) and Holographic Particle Image Velocimetry (HPIV). As of 2005, he is the William F. Ward Sr. Distinguished Professor at the Department of Mechanical Engineering of the Whiting School of Engineering at the Johns Hopkins University.

## References

• Chapter 4,5,6 In Fluid dynamics and Heat Transfer by Budugur Lakshminarayana
• Fluid dynamics and Heat Transfer by James George Knudsen, Donald La Verne Katz
• Turbomachinery: Design and Theory (Marcell Dekker) by Rama S.R. Gorla
• Handbook of Turbomachinery, 2nd Edition (Mechanical Engineering, No. 158) by Earl Logan, Jr; Ramendra
• Turbines Compressors and Fans by S M Yahya
• Principles of Turbomachinery by R K Turton
• Turbomachinery Flow Physics and Dynamic Performance by Meinhard Schobeiril
• Torsional Vibration of Turbo-Machinery by Duncan Walker
• Turbomachinery Performance Analysis by R. I. Lewis
• Fluid Machinery: Performance, Analysis, and Design by Terry Wright
• Fluid Mechanics and Thermodynamics of Turbomachinery by S L Dixon and C.A Hall
• Turbo-Machinery Dynamics by A. S. Rangwala

## Journals

• K. F. C. Yiu; M. Zangeneh (2000). "Three-Dimensional Automatic Optimization Method for Turbomachinery Blade Design". Journal of Propulsion and Power. 16 (6): 1174–1181. doi:10.2514/2.5694.
• Piotr Lampart. "Tip Leakage Flows in Turbines" (PDF). Task Quarterly. 10: 139–175.
• Horlock J H, Lakshminarayana B (1973). "Secondary Flows: Theory, Experiment, and Application in Turbomachinery Aerodynamics". Annual Review of Fluid Mechanics. 5: 247–280. doi:10.1146/annurev.fl.05.010173.001335.
• D. R. Waigh; R. J. Kind (1998). "Improved Aerodynamic Characterization of Regular Three-Dimensional Roughness". AIAA Journal. 36 (6): 1117–9. doi:10.2514/2.491.
• J. D. Denton; W. N. Dawes (1998). "Computational fluid dynamics for turbomachinery design". Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science. 213 (2): 107–124. doi:10.1243/0954406991522211.