Labyrinth seals are used mainly in high-speed turbo machinery as they reduce gas leakage from high-pressure region to the low pressure regions. The gas flow through the seals as well as its swirling oscillation within the cavities which are formed between the seal teeth can create a net pressure and shear forces that will act on the rotor. These forces will contribute to the destabilization of rotors that are marginally stable, for a reliable design of the turbo machinery, it is important to be able to predict these forces. They can be seen as an accessory to the primary seal. Depending on the size of the gas turbine, they can be used in static and dynamic application.
In the static condition they are used where casting parts are not joined to permit differences during thermal expansion, during this expansion, they minimize leakage.
The dynamic application used in both the turbine and compressor are for inter-stage seals. This is because of the pressure increase during the compression stage of the gas turbine cycle. These seals are needed to prevent backflow of gas from the discharge to the inlet end of the casing. The conditions of the seals will have a direct impact on the performance of the compressor.
There are number of different types of configuration of labyrinth seals that can be used, this section of the report will focus on two of these. To be able to control and correctly predict the leakage proves essential for the efficiency of the turbine.
The gas flow through a labyrinth seal can be described as:
"Swirling gas at high pressure enters through the clearance between the first tooth of the labyrinth seal and the wall opposite to it to first cavity of labyrinth seal, expanding somewhat and altering to its rotational momentum by the first friction of cavity walls which may rotate as it speeds quite different from the inlet swirl.
This rotation is in general non-axis-symmetric and time dependent due to small but nevertheless important vibration of the rotor. Once the gas crosses several such cavities it emerges at the other
end of the labyrinth seal at significantly reduced pressure. A significant assumption
which facilities the semi analytic treatment of this very complex three dimensional
unsteady flow is that the gas pressure in each labyrinth cavity as well as the
circumferential velocity in each cavity are independent of the radial and axial coordinates within the cavity"
Appropriate boundary layers needs to be used in the estimation of the circumferential momentum transfer from gas to wall. For a perfect centric rotation of the rotor, this flow can be assumed to be axis-symmetric and also considered to be in a steady state.
Labyrinth seal geometries 
NT is the tooth number, this varies from 5 to 18
Rs is the shaft radius
Cr is clearance between teeth and rotor surface
Li is labyrinth seal pitch, this is equal of seal height
Bi is seal height
Pi is Pressure
Vi is circumferential velocity
Pin is tooth inlet pressure
Pout is the outlet pressure
Vo is the inlet swirl velocity
i is the leakage at each tooth
m& i is the mass flow rate of the circumference of gap which is created by the clearance
By assuming the seal geometry is axially symmetric i.e. there is no motion of the rotor axis. Pressure and circumferential velocity within each cavity are also assumed to be uniform
The gap created by the clearance can be denoted by ANARsi when the teeth is on the rotor and ANARsi when the teeth is on the stator.
The annular flow areas can be defined by
ANARsi =Ï€ 2Rsi + Cri Cri , when the teeth is on the stator
ANARri = Ï€ 2Rsi + 2Bi + Cri Cri when the teeth is on the rotor
The rotor shear can be defined as:
RSA = 2Ï€RsiLiari
The stator shear area can be defined as
SSA = 2Ï€RsiLiasi
The dimensionless stator shear area is
for teeth on the stator
for teeth on the rotor
Pressure distribution and leakage flow-rate calculation
If the rotor spins with a constant speed, without any eccentricity, the flow is time independent. For this steady state situation a continuity equation says
M(dot) 1 = m(dot) 2 =LNT = m(dot)
The flow rate m(dot) is dependent on the pressure difference of Pin - Pout , the inlet temperature Tin as well as the geometry of seals. For the calculations it is assumed that the gas in each cavity obeys the perfect gas law
Pi = ÏRTi
Where Pi is pressure
Î¡ is density
Ti is the temperature in cavity i
R is the gas constant
By using the Modified Neumann Method calculations for m(dot) can be given as
M(dot)i = Î¼1iANARi
In this equation Vermes' residual kinetic energy carry-over factor which is given by
Î¼1i is the discharge flow coefficient and this can be defined as
Î¼1i = , Si =( ) - 1
The pressure ratio can be defined as
Where Ui = axial gas velocity at tooth (i) = specific heat ratio
The formulas above are only valid for subsonic flow by assuming there is no chocking occurrence in the restriction. The possibility of critical flow about the very last tooth of the seals is always present, critical condition should be checked at the output.
Chocked flow of gas within the last restriction will be present if
â‰¤ () for air this value will be 0.528
If a restriction is chocked, the leakage flow rate equation of the last tooth should be replaced with
M(dot)NT = Î¼NT ANARNT ) / âˆš1- Î±
CIRCUMFERENTIAL VELOCITY DISTRIBUTION
In the i-th labyrinth cavity the bulk circumferential velocity Vi results a viscous shear stress Ï„si and Ï„ri which occurs at the surfaces of the rotor and stator. Both stresses have an influence on momentum balance.
The mass of the gas multiplied by the circumferential velocity in the same cavity is equal to the circumferential momentum of gas.
In a steady state, the circumferential momentum equation is given as
M(dot) iVi - m(dot) iâˆ’1Viâˆ’1 = RSF - SSF
Where RSF is the rotor shear force which is defined as
RSF = Ï„ri (2Ï€RsiariLi)
SSF is the stator shear force, this can be defined as
SSF = Ï„si (2Ï€RsiasiLi)
By substituting these into the circumferential momentum equation we get
M(dot) Vi âˆ’Viâˆ’1 = 2Ï€RsiLi (Ï„riari âˆ’ Ï„siasi)
With the use of this formula, the circumferential velocities in the labyrinth cavities can be calculated, after the shearing stress has been calculated.
Moody produced the following as an approximate representation for the pipe-friction factor, this is given as
f = ai 1/3 where Re =
a1 = 1.375 x 10-3 , b1 = 2 x 104, b2 = 106, e/Dh is relative roughness
This formula should give values that are between 5% of Moody diagram for 4000 â‰¤ Re 107 as well as e/Dh â‰¤ 0.01. If e/Dh is greater than 0.01 it will underestimate f quite significantly
According to Blasius, the shear stress for a turbulent flow within a smooth pipe is
Ï„ = 0.5 fÏV 2
Where Ï„si is for the smooth stator surface and Ï„ri corresponds to the rotor surface, they can be defined with the aid of moody's wall friction-factor model, Ï„si is
Ï„si = ÏiVi2ai
The bulk circumferential flow moves relatively to the rotating rotor surface with a velocity Rsiw - Vi, this makes the shear stress of the rotor surface of the i-th cavity to be
Ï„ri = Ïi(Rsiw -V2i)a1
In this case the Reynolds number Re is defined by
Re = this applies to the teeth on the stator
Re = this applies to the teeth on the rotor
As the Kinematic viscosity v = the Reynolds number of the teeth on the stator can be written as
And the equivalent Reynolds number of the teeth on the rotor will be
Dhi can be defined as the hydraulic diameter given as
Dh = 4
this is equates to
This method can be used to minimize leakage.
Example of previous study using CFD analysis
A thorough understanding of the flow in labyrinth seals at certain engine conditions is key to the development of an improved seal concept as this will enhance and help in predicting the engine performance.
For the success of this mission, there was three technological blocks that were identified at NLR, these includes:
An experimental method facilitated by and advanced seal test rig.
A numerical method which is based on CFD model
A semi-empirical engineering model
The seal test rig used was very advanced; it could simulate flows at extreme engine conditions for example, high rotational speeds, high temperatures as well as high pressures.
From a CFD analysis carried out by B.I. Soemarwoto , J.C. Kok, K.M.J. de Cock et. al, the following method was used to evaluate the performance of gas turbine labyrinth seals.
In their report they focused on the application of computational fluid dynamics to assess the seals. They carried out a comparison between numerical and experimental results with a tested seal model. By the use of three test cases, the first consist of a labyrinth seal configuration with a straight honeycomb land, the second test case uses the same labyrinth seal but the honeycomb land was replaced by a solid smooth land, the third case involves a stepped labyrinth seal.
Fig A.B. Geometry of a stepped labyrinth seal configuration Case 1
Fig A.C. Geometry of a labyrinth seal with a smooth land, Case 2
Fig A.C Geometry of three canted knives with a smooth land case 3
I have attached a copy of their analysis and results within this report
An analysis of the topology and losses that was seen along the streamline was performed, this showed the sealing mechanism of flow through this type of seal is due to losses related with the turbulent mixing as well as the shear which was dominant within the vicinity of the knife edge and this was to a lesser extent about the step wall. For a straight 3 knife edge, the effect of the honeycomb was analysed as a reduction production of losses, these were due to:
A less intense turbulent mixing around the knife-edge
Prevention of a formation of a coherent shear layer
A mitigation of the flow impingement on the step wall
An inhibiting flow separation at each notch
By comparing two different seal configuration i.e. the canted-knife seal and the straight- knife seal, the canted-knife seal has lower performance and can be attributed to the mitigated flow impingement on the step wall. Less intense turbulent mixing and shear within the vicinity of the knife-edge also applies to its lower performance.
Influence of leakage flow on rotor dynamics.
The non uniform circumferential pressure distribution in the labyrinth chambers can exert certain forces on the rotating shaft. To be able to include the fluid induced forces related to rotor dynamics analyses there has been a lot of researches who have found the rotor dynamic coefficients for labyrinth seals. In this section of the report straight-through teeth on rotor and stator, teeth-on-rotor seals as well as inter-locking seals is investigated. Due to the cross-coupled stiffness which proves to the most significant parameter used in rotor dynamic analyses, it is important to be able to destabilize the stiffness coefficient. The concept of a labyrinth seal having the capabilities of exerting a destabilizing force on a rotor was 1st examined by Alford (1965). He did this by the use of a simple one-dimensional model which neglected circumferential flow in the seal.
This can be classified into 3 categories
Single control volume analyses: Underlines that the simplication of the flow can be broken into two parts. (a) a leakage flow (b) circumferential flow.
Multi-control volume analyses
Computational fluid analyses.
The control volume can be defined as the labyrinth chamber. This includes, area of interception between leakage flow and circumferential flow. Jenny 1980 , used a CFD model to construct a simple single control volume model. This had three dimensional flow in the circumferential direction and replaced with a single core flow of the mean circumferential velocity. The most important aspect of this analysis is the introduction of a tangential momentum coefficient. This describes the amount of circumferential momentum that imparted a chamber from the leakage flow.
Iwatsubo (1980) added a time dependency of area change which was due to a change in the rotor position. The resulting equations were solved by the use of a finite difference approach. An analytical separation of variables approach was later used by Iwatsubo et al (1982). From this approach the stiffness and damping coefficients recalculated. Childs and Scharrer (1986) added to this analysis by the inclusion of the variation of area in the circumferential direction due to eccentricity. Wyssmann et al 1984 was first to introduce the two control volume approach. This included calculations for the share stresses between leakage flow and circumferential flow by the aid of two control volumes for the circumferential flow. One of these was used for the bulk of labyrinth chamber and the other used in the area of leakage flow.
Figure A. D Straight- through labyrinth seal
Normann and Weiser (1990), applied the use of a three control volume technique. This was similar to the two control volume method but in this case, control volume within the leakage flow was divided into two regions;
The region above labyrinth chamber
The region above labyrinth fin.
In order to achieve accuracy of a model it is advised to use methods of Childs and Scharrer (1988). To achieve a more accurate model, for flow characteristics within the seal it important to obtain better predictions of dynamic coefficients by the implementation of a tangential momentum parameter.
Dynamic coefficients can be determined by an integration of pressure perturbations around as well as along the shaft. A detailed method can be found in Williams (1992). Due to the complexity of a three dimensional flow in labyrinth seals, it is important to make a set or a number of tractable equations, which can describe the flow and simple assumptions can be made. These include;
The pressure within a labyrinth chamber and circumferential velocity are constant in the axial direction, and as a result, are functions of angular position. This is only applicable in the perturbed case.
Each seal cavity has a constant temperature.
Gas will be assumed to be ideal.
Eccentricity of rotor is relatively small compared to the radical seal clearance.
Within the cavity, pressure variations are assumed to be negligible relative to pressure difference across a seal tooth.
Acoustic resonance frequency of a cavity is higher than the rotation of frequency.
The share stress contribution to dampen coefficients and stiffness is negligible.
To develop equations which describes the flow within the labyrinth seal, principles of conservation of mass and circumferential momentum should be applied to each labyrinth chamber that serves as a control volume. Below is a 2-D view of a cavity control volume
Fig A.E. Cavity control volume 
The resulting circumferential momentum equation is:
A typical resulting equation for rotor shear stresses is:
Where Dhi is the hydraulic diameter, the constants m0 and n0 are given four turbulent flow between smooth annular surfaces as m0 is equal to -0.25 and n0 is equal to 0.079.
To be able to solve these equations, a perturbation analysis should be performed with the eccentricity ratio = This is the perturbation parameter.
Other equations to be derived includes:
The zeroth order circumferential momentum equation
Mass flow rate
Rotor dynamic coefficients
With the above equations determined the only equations unknown are the shear stresses and circumferential velocity. These have been derived previously. After the pressure fluctuations have been derived, the rotor dynamic coefficients can now be calculated with the equation
The X and Y force components can be derived by the integration of pressure surrounding the seal and determining the dynamic coefficient forces. The use of perturbation analysis will lead to grouping of the like terms and yield the final solutions to the stiffness and damping coefficient.
To be able to implement the theory, the mass flow rate through the seals should be determined for the zeroth order. The leakage through a labyrinth seal can be typically modelled as an adiabatic throttling process. There is an occurrence of pressure drop which takes place in each of the annular spaces due to an assumed isentropic expansion. The resulting velocity is completely lost in the chamber for an ideal labyrinth seal. In straight-through labyrinth used in industry, an amount of the kinetic energy connected with each stage is carried over to the next stage.
Effect of mass flow calculation on dynamic coefficients
With the use of a parametric study of the effect of the mass flow rate on dynamic coefficient, the dynamic coefficients can be determined based on the calculated mass flow rate (nominal) and the mass flow rates of 50% to 200% of the nominal value for various operating conditions. The results will give an indication of the sensitivity of dynamic coefficients to mass flow rate. As the rotor dynamic analysis is very important, special attention has to be paid to the cross-coupled stiffness. The conditions for these cases is presented below in table 1.
Table 1. Operating conditions for example case
From a parametric study for cross coupled stiffness shown in figure 2 below shows that calculations based on mass flow rates of 50% and 200% on the nominal mass flow rate lead to a cross-coupled stiffness value that is significantly different from the values obtained with the mass flow rate. When relating to high preswirl and teeth-on-rotor configurations, the differences are less pre-nounced. Figures show that, while a great variance in mass flow rate from nominal value will lead to a large variance in coupled stiffness, the small difference that occur in mass flow rate does not significantly change the calculated mass flow rate values achieved for cross-coupled stiffness. The results shows a small difference of about 10% or less between the actual and calculated mass flow rate do not detract massively from the accuracy of cross-coupled stiffness calculations.
Fig.2 Effect of mass flow rate on cross coupled stiffness
For principal stiffness, graph showed on fig. 2, shows that principal stiffness is quite sensitive for mass flow rate. The level of sensitivity increases with rotor speed, this happens with any differences in the principal speed of roughly 40% at the highest speed for any large variances from the normal mass flow rate. Small differences between the actual and calculated masses flow rates should not have any significant effect on the principal stiffness but significant differences in the mass flow could still affect the stiffness calculations.
This aspect of the report dealt with characteristics of Labyrinth seals, their uses, advantages as well as disadvantages. It also analysed in detail some seal configurations, pressure distribution as so on.
Gas flow through a labyrinth seal was described as well as assumptions needed to be able to carry out a three dimensional unsteady flow, appropriate boundary layers needed to estimate the circumferential velocity.
There are different types of method that can be used to calculate rotor dynamic coefficients for labyrinth seals with an example used above. Calculations for the mass flow rate uses the relation between mass flow rate through a single annular orifice as well as the pressure drop across the orifice. An integration of mass flow rate can then be done in order to match the sum of pressure drops due to orifices with the specified pressure drop across the seal.
A parametric analysis shows a small variation between calculated and experimented mass flow rate should not detract the accuracy of the dynamic coefficient significantly.
Overall the project went well, although finding information needed for this section was quite difficult, by asking my tutor and other Aero colleagues at the University as well as browsing through the University's library , I could gather enough information which help me in the writing of this report. My team members where really helpful as well all worked well as a team. Even though we were short on time, the effectiveness of our team work seem to pay off.
If given the opportunity to do a similar project, I would like to work in a lab and perform some test so I can have my own data for analysis. As some of the data used in this report where quite old, new and up to date data will be ideal. The only problem with this is, the cost of carrying out such a test. Therefore I would create my own Computational fluid analysis model, as this is a cheaper and effective method that can produce accurate results.