Computational Fluid Dynamic Simulation of a Straight
Labyrinth Seal with Emphasis on Thermal Characterization
by
Paul Hiester
An Engineering Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF SCIENCE
Major Subject: Mechanical Engineering
Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Project Adviser
Rensselaer Polytechnic Institute
Hartford, Connecticut
December, 2013
Table of Contents
List of Tables .................................................................................................................... iv
List of Figures .................................................................................................................... v
List of Symbols ................................................................................................................ vii
Acknowledgements........................................................................................................... ix
Abstract .............................................................................................................................. x
1. Introduction and Background ...................................................................................... 1
2. Labyrinth Seal Rotor and Stator Geometry ................................................................. 4
3. CFD Boundary Condition Setup .................................................................................. 7
3.1
Fluent CFD Setup and Solution Convergence ................................................... 9
3.2
CFD Grid Independence Study ........................................................................ 12
4. Seal Clearances Sensitivity Study – Leakage and Viscous Heating.......................... 15
4.1
Results Overview ............................................................................................. 15
4.2
Leakage and Viscous Heating - Methodology ................................................. 18
4.3
Leakage and Viscous Heating - Results and Discussion ................................. 20
5. Seal Convection Heat Transfer Characterization ...................................................... 26
5.1
Heat Transfer with Viscous Heating ................................................................ 26
5.2
Heat Transfer without Viscous Heating ........................................................... 29
5.3
Results Summary and Discussion .................................................................... 30
6. 3D CFD Analysis with Various Honeycomb Seal Land Geometry .......................... 32
6.1
Geometry Definition ........................................................................................ 32
6.2
3D Solution Post-Processing and Discussion .................................................. 35
7. Conclusions................................................................................................................ 39
8. References.................................................................................................................. 40
9. Appendix.................................................................................................................... 41
ii
© Copyright 2013
by
Paul Hiester
All Rights Reserved
iii
List of Tables
Table 1 CFD geometry: Rotor, stator and land dimensions .............................................. 5
Table 2 CFD Geometry: Lab seal detail dimensions ......................................................... 6
Table 3 CFD boundary condition summary ...................................................................... 8
Table 4 Summary of Fluent solver settings and options.................................................... 9
Table 5 CFD grid sizes for grid independence study ...................................................... 13
Table 5 Summary of grid sensitivity results .................................................................... 14
Table 7 Results summary for 5mil clearance case ........................................................... 21
Table 8 Results summary for 10mil clearance case ......................................................... 21
Table 9 Results summary for 15 mil clearance case........................................................ 22
Table 10 Leakage and windage summary for clearances (5-15mil) ................................ 22
Table 11 Summary of heat transfer studies ..................................................................... 26
Table 12 Average rotor/stator heat transfer coefficients for heating and cooling ........... 29
Table 13 Average heat transfer coefficients for heating; No viscous heating ................. 30
Table 14 Absolute value of axial to swirl velocity ratio .................................................. 31
Table 15 Comparison of 2D and 3D analyses for 5 mil radial clearance ....................... 38
iv
List of Figures
Figure 1 Locations where lab seals are typically used gas turbine engines ....................... 1
Figure 2 Key components of a labyrinth type seal; Non-Stepped ..................................... 2
Figure 2 Typical honeycomb used for seal lands [Pratt and Whitney] .............................. 3
Figure 4 CFD geometry: Rotor, stator and land description ............................................. 4
Figure 5 CFD Geometry: Lab seal detail description ....................................................... 5
Figure 6 CFD boundary condition description .................................................................. 7
Figure 7 CFD monitor plane ............................................................................................ 10
Figure 8 Total temperature convergence at XX; 5 mil clearance. ................................... 11
Figure 9 Total temperature convergence at ZZ; 5 mil clearance ..................................... 11
Figure 10 Mass flow rate convergence; 5 mil clearance ................................................. 12
Figure 11 CFD grid sizing regions .................................................................................. 13
Figure 12 Total temperature contour plot for various grid densities ............................... 14
Figure 13 Total temperature, stream function and pressure contours; 5 mil clearance ... 16
Figure 14 Contours of tangential velocity quantities; 5 mil clearance ............................ 17
Figure 15 Control volume for viscous work method 1 calculation ................................. 19
Figure 16 Total temperature contour assuming no viscous work; 5 mil clearance ......... 19
Figure 17 Leakage versus clearance for 2:1 seal pressure ratio ..................................... 23
Figure 18 Windage heat rate versus seal clearance for 2:1 seal pressure ratio ................ 23
Figure 19 RPMF versus seal clearance comparison ........................................................ 24
Figure 20 Total temperature rise versus clearance for 2:1 seal pressure ratio ............... 25
Figure 21 Application of wall temperature boundary conditions .................................... 26
Figure 22 Total temperature contours for heated wall condition .................................... 28
Figure 23 Total temperature contours for cooled wall condition .................................... 28
Figure 24 Total temperature contours for heated wall condition; No viscous heating .... 30
Figure 25 3D Geometry with 1/8" honeycomb land ........................................................ 32
Figure 26 3D geometry with 1/32" honeycomb land ...................................................... 33
Figure 27 Honeycomb schematic .................................................................................... 33
Figure 28 3D mesh with 1/8" honeycomb ....................................................................... 34
Figure 29 3D CFD pressure contours for 1/8" honeycomb ............................................. 35
Figure 30 3D CFD total temperature contours for 1/8" honeycomb .............................. 36
v
Figure 31 3D CFD velocity magnitude contours for 1/8" honeycomb ............................ 37
Figure 32 3D CFD swirl velocity contours for 1/8" honeycomb ................................... 37
Figure 33 3D CFD RPMF contours for 1/8" honeycomb ................................................ 38
Figure 34 3D CFD pressure contours for 1/32" honeycomb ........................................... 41
Figure 35 3D CFD total temperature contours for 1/32" honeycomb ............................. 41
Figure 36 3D CFD velocity magnitude contours for 1/8" honeycomb ............................ 42
Figure 37 3D CFD swirl velocity contours for 1/32" honeycomb .................................. 42
Figure 38 3D CFD RPMF contours for 1/32" honeycomb ............................................. 43
vi
List of Symbols
t – Distance between adjacent knife edges (inches)
h – Knife edge radial height relative to rotor base (inches)
w – Axial width of knife edge base (inches)
cr – Radial clearance between rotor and stator land (mils)
b – Axial width of knife edge tip (mils)
θ- Knife edge angle relative to rotor base (degrees)
f1 – Fillet radius between knife edge base and rotor base (mils)
f2 – Fillet radius between knife edge tip and knife edge base (mils)
r – Outer radius of rotor base (inches)
rL,1 – Inner radius of stator at domain inlet (inches)
rL,2 - Inner radius of stator at domain exit (inches)
tL - Thickness of seal land (inches)
L1 – Axial distance to start of stator land (inches)
Ls – Axial distance to end of stator land (inches)
L3 – Axial length of stator and rotor (inches)
Pup – Upstream static pressure (psia)
Tup – Upstream total temperature (psia)
RPMFup –Upstream non-dimensional swirl velocity
Pdown – Downstream static pressure (psia)
Ω – Angular speed (rad/s)
PR – Pressure Ratio (-)
U – Linear wheel speed in tangential direction (ft/s)
RPMF – Non-dimensional swirl velocity; Normalized by wheel speed (-)
Vrel – Tangential velocity relative to the rotor (ft/s)
Mrel - Mach number relative to the rotor (-)
γ – Ratio of specific heats (-)
R – Gas constant for air (-)
Ts - Static temperature (F)
Tt – Total temperature (F)
vii
Vr – Radial velocity (ft/s)
Vz – Tangential velocity (ft/s)
Ψ – Stream function
̇ – Heating due to viscous forces (BTU/s)
𝑄𝑣𝑖𝑠𝑐
̇
𝑄𝑐𝑜𝑛𝑣𝑒𝑐
– Heat transfer due to convection (BTU/s)
Cp- Specific heat at constant pressure (BTU/lbm-R)
τ – Shear stress (psi)
H – Heat transfer coefficient (BTU/hr-ft2-F)
Tf - Fluid temperature used in evaluating heat transfer coefficient (F)
Tsurf – Surface temperature used in evaluating heat transfer coefficient (F)
A – Surface Area (sqin)
tfoil – Thickness of honeycomb wall (mils)
Cell Size – Size of honeycomb cell (inches)
viii
Acknowledgements
I would like to thank and acknowledge RPI Hartford and in particular my
advisor, Ernesto Gutierrez-Miravete.
Ernesto provided valuable guidance and
encouragement during the course of this project and went out of his way to ensure the
project was successful. Also, I would like to acknowledge my wife and family who
provided needed encouragement positive feedback.
ix
Abstract
Labyrinth seals are commonly used in the gas turbine industry in secondary flow
systems to seal between rotating and static components. The performance of these seals
from a pressure loss and flow rate perspective is extremely important. Seal flow rates
significantly affect overall engine cycle performance and pressure loss characteristics
can significantly affect rotor thrust balance.
Seal thermal behavior is equally as
important. Gas turbine components typically operate at or near material capability
limits. Commonly, limiting temperatures occur at or near rotating seals. Temperatures
are typically increased due to elevated heat transfer coefficients and additional heat loads
due to viscous heat generation (windage).
This paper explores the leakage and thermal characteristics a straight through
type labyrinth seal using CFD at typical gas turbine operating conditions.
x
1. Introduction and Background
Labyrinth (lab) seals are used to seal between high and low pressure regions of
rotor/stator systems. Lab seal are prevalent in industries where durability and robustness
are paramount including industrial gas turbines, steam turbines, and gas turbines for
aircraft. Figure 1 shows the locations lab seals are typically used in a gas turbine type
aircraft engine, and Figure 2 shows the components of typical lab seal.
Lab seals can be found in the low and high pressure compressors on the stator
shrouds. A stator shroud is located at the blade hub and isolates the upstream, lower
pressure stage, from the downstream higher pressure stage. Without effective sealing, a
significant portion of the compressor flow would leak backwards through the
compressor.
These leakages are undesirable because they reduce compressor
efficiencies, and can lead to compressor operability issues like stall and surge.
Similarly, in the high and low pressure turbines, lab seals are used to prevent
flow from leaking between adjacent stages. Like the compressor, turbine lab seals are
typically located near the stator vane hubs. However, low pressure turbines sometimes
use lab seals at the rotating blade tips. This usage requires the blades to have a rotating
shroud. High leakages in the turbine erode turbine efficiencies, and can contribute to hot
gases being ingested into undesirable regions.
Figure 1 Locations where lab seals are typically used gas turbine engines
Figure 2 shows the typical makeup of a lab seal. As mentioned, lab seals are
situated between a rotating (rotor) and static (stator) component. One or more seal teeth,
or “knife edges”, are positioned on the rotor. The design of the seal teeth affect the seal
leakage and thermal characteristics. Seal teeth can be perpendicular to the rotor, or
posited at an angle, as shown in Figure 2. Additionally lab seal teeth are sometimes
“stepped”, meaning downstream teeth are positioned at progressively lower radii.
Seal teeth are designed to be in close proximity with a stator land, which is a
special region of the stator where designers anticipate interaction between the rotor and
stator. These interactions are commonly referred to as “rubs” because the rotor teeth rub
into the stator land.
Figure 2 Key components of a labyrinth type seal; Non-Stepped
Rubs occurs when the rotor component grows radially outward more quickly
than the stator component. In gas turbine operation, centrifugal forces play a role in this
radial growth as do rotor and stator component temperatures.
Rubs are undesirable, as they can damage or destroy the tips of the seal teeth.
However, they are also typically unavoidable during aircraft engine transient operation.
2
In order to mitigate potential damage from rubs, stator lands are often made of a
porous material. This allows the seal teeth to cut into stator land without significant
wear or damage. A porous material of choice in the gas turbine industry is metal
honeycomb.
Honeycomb comes in various cell sizes and can be made of various
materials. Figure 3 shows several varieties of honeycomb used in lab seal stator lands.
The honeycomb shown range in cell size between 1/32” - 3/8”.
Figure 3 Typical honeycomb used for seal lands [Pratt and Whitney]
The ability to accurately model leakage flows of lab seals is extremely important.
Leakages can significantly affect compressor and turbine efficiencies.
Thermal
characteristics are also increasingly important. Velocity gradients are generally high in
in lab seal, which generally result in high heat transfer coefficients and the production of
non-negligible viscous heating, or windage.
3
2. Labyrinth Seal Rotor and Stator Geometry
Due to computational constraints, it is typical to use 2D axi-symmetric models for
rotating component analysis. This section describes the 2D geometry to be used for the
CFD analysis. Figure 4 and Table 1 describe the dimensions of the rotor, stator and the
position of lab seal land. Subsequently, Figure 5 and Table 2 provide a detailed
parameterization of the lab seal teeth and stator land.
As shown, the study geometry consists of a rotor and stator section 5 inches in axial
length with the seal positioned in the middle. The flow rate for this study is from left to
right. The upstream annular gap is 0.4 inches and downstream annular gap is 0.1 inches.
It is worth briefly discussing the tapering of the downstream annular gap. In rotor
stator systems with small axial through flows, strong recirculation vortices can be
generated. If these vortices exist near a flow outlet it is possible for the outlet to have
regions that reverse flow, enter the domain rather than exit. Tapering the downstream
annular gap prevents these large vortices and reduces the possibility of have significant
reverse flow.
Figure 4 CFD geometry: Rotor, stator and land description
4
Table 1 CFD geometry: Rotor, stator and land dimensions
The study lab seal geometry consists of three seal “teeth” and is a “straight
through” type as opposed to a “stepped” type. The seal teeth are canted at an angle with
the tips of the teeth pointed toward the direction of the flow. The dimensions presented
in Table 2 are representative of seals typically used in large high bypass-ratio aircraft gas
turbine engines. However, the seal presented does not represent an actual engine design.
Figure 5 CFD Geometry: Lab seal detail description
5
Table 2 CFD Geometry: Lab seal detail dimensions
6
3. CFD Boundary Condition Setup
The final goal of this study is to determine leakage and thermal characteristics
across a range of seal clearances. In this section, one clearance value and a fixed set of
operating conditions will be considered. This will allow for critical review of the CFD
grid, grid independence, CFD setup, and CFD results.
These details will provide
confidence in the quality of the CFD results, and build a strong foundation for
subsequent studies.
Figure 6 and Table 3 describe the boundary conditions used in this section. This set
of boundary conditions was chosen because it is expected to challenge the analytical
toolset.
The case described has a large pressure ratio across the seal, is rotating at high
angular speeds, and is expected to have high relative Mach numbers in the domain.
Additionally the clearance, cr, is very small and will necessitate a fine mesh in the
clearance gap.
Figure 6 CFD boundary condition description
7
Table 3 CFD boundary condition summary
Reviewing the boundary condition set in more detail, several key observations can
be made. First, according to Equation 1, the seal is operating at a 2 to 1 pressure ratio
(PR=2). This is typically the highest pressure ratio for which this seal type is used, and
the highest for which there is significant research.
Next, using Equations 2-5, the relative Mach number can be calculated. For internal
air systems in aircraft engines, supersonic and transonic flows are avoided. If we assume
negligible swirl velocity in the flow (RPMF=0), the Mach number relative to the rotating
structure is 0.78, approaching the transonic regime.
𝑃𝑅 =
𝑃𝑢𝑝
(1)
𝑃𝑑𝑜𝑤𝑛
𝑈=𝑟𝜔
𝑅𝑃𝑀𝐹 =
(2)
𝑉𝑡𝑎𝑛
(3)
𝑈
𝑉𝑡𝑎𝑛,𝑟𝑒𝑙 = |𝑉𝑡𝑎𝑛 − 𝑈|
8
(4)
𝑀𝑟𝑒𝑙 =
𝑉𝑟𝑒𝑙
(5)
√𝛾𝑅𝑇𝑠
Finally, take note that the radial clearance for this configuration, cr, is 5 mils. 5
mils is the tightest clearance that is practically achievable in current gas turbine design
for large aircraft engines. Because of the large pressure ratio, small radial clearance, and
high rotating speeds, this is a challenging configuration.
3.1 Fluent CFD Setup and Solution Convergence
This section provides details regarding the setup of the CFD analysis, and the
procedure to ensure proper solution convergence. Grid generation details are provided
in the subsequent section.
The CFD grids were generated using ANSYS Workbench version 14.0, and the
CFD solutions were generated using ANSYS Fluent 14.0. Table 4 provides details
regarding the Fluent solver settings and physic modeling options.
Table 4 Summary of Fluent solver settings and options
9
CFD convergence is monitored internal to the Fluent solver by the residuals of
various conservation equations, however it is known that residuals do not always provide
accurate evidence of convergence. CFD analysts generally prefer to monitor physical
quantities of interest in addition to residuals for determination solution convergence.
Because the focus of this paper is on seal leakage and thermal characterization, it
was decided that flow rates and temperatures be monitored for appropriate CFD
convergence. Figure 7 shows planes over which physical quantities will be monitored.
Monitoring will include: the flow rate across plane YY, and the total temperatures at
planes XX and ZZ. Total temperature is calculated according to Equation 6 and will be
reported on a mass weighted average basis. Note that Vrel in Equation 6 is this case is the
velocity relative the inertial reference frame.
𝑇𝑡 = 𝑇𝑠 +
𝑉𝑟𝑒𝑙 2
2𝐶𝑝
(6)
Figure 7 CFD monitor plane
Numerical iterations were subsequently performed on the CFD case.
7000
iterations were executed while monitors tracked the convergence of the solution.
Figures 8-10 show the convergence history of the aforementioned monitors. These
figures show the solution is well converged after 7000 iterations. It is worth noting that
10
the mass flow monitor converges after several hundred iterations, but the total
temperature monitors require significantly more iterations to converge.
Figure 8 Total temperature convergence at XX; 5 mil clearance.
Figure 9 Total temperature convergence at ZZ; 5 mil clearance
11
Figure 10 Mass flow rate convergence; 5 mil clearance
3.2 CFD Grid Independence Study
The quality of any finite element solution is dependent on the quality and length
scale of the constituent elements. CFD solutions are particularly prone to numerical
discretization error due to the higher order equations being considered in the finite
volume formulations.
While these formulations are beyond the scope of this paper, it is important that grid
independence be considered before critical review of CFD results. Grid independence is
performed by running several solutions with grids of increasing quality. The solutions
are compared, and grid impendence is achieved when the solution does not change
appreciably as grid quality is changed.
In order to judge grid independence quantitatively, mass flow rate and total
temperature will be queried after each solution is converged and compared. These
quantities will be queried in the same manner as described in the previous section.
Figure 11 and Table 5 provide details regarding four meshes of varying quality.
Note that “clearance gap” refers to the 5 mil radial clearance.
12
Figure 11 CFD grid sizing regions
Table 5 CFD grid sizes for grid independence study
Figure 12 shows total temperature contour plots for the various mesh densities. All
the solutions show similar temperature distribution and trends. Table 5 summarizes the
computed flows and total temperatures.
Between the “Medium” and “Fine” grid
densities there is less than 2% change in mass flow rate, and less than 1% change in
mass weighted average total temperature at planes XX and ZZ. Because “Medium” and
“Fine” densities produce similar results, and computational resources are limited, the
“Medium” grid density was chosen for subsequent studies.
13
Figure 12 Total temperature contour plot for various grid densities
Table 6 Summary of grid sensitivity results
14
4. Seal Clearances Sensitivity Study – Leakage and Viscous Heating
It has been established that adequate convergence monitors are in place, and a
quality grid is being used. Now, meaningful studies can be conducted with confidence in
the results. The study presented in this section will investigate how various clearances
affect the leakage and heat generation characteristics of the lab seal. Three clearance
values will be considered: 5 mils, 10 mils, and 15mils.
First a general review of each clearance case will be presented and observations
will be made that will enable better interpretation of subsequent material. Next, the
discussion of results will detail the behavior of leakage rates and viscous heat generation
for the various clearances conditions. This section will conclude with a summary of
findings.
4.1 Results Overview
This section will provide an overview of the results for the 5 mil clearance case, and
provide a foundation for more detailed results post-processing.
The pressure, stream function and total temperature are three quantities of interest
for rotor/stator systems. Figure 13 shows the contour plots of theses quantities for the 5
mil clearance case.
From Figure 13 several observations can be made. First the pressure contours
provide confirmation in the expected result of large pressure drops across each labyrinth
seal tooth. There is a 90 psi drop across the first seal tooth, a 70 psi drop across the
second and a 90 psi drop across the third.
Secondly, there is a significant total temperature rise across the seal from plane XX
to ZZ, about 100F. There is also a significant total temperature increase from the
domain inlet to plane XX (upstream of the seal), about 100F.
However there is
negligible total temperature rise from plane ZZ (downstream of the seal) to the domain
outlet. The reason for these observations will be discussed when velocity results are
discussed.
15
Lastly, Figure 13 shows a contour plot of stream function. Stream function is a
measure of recirculation for two dimensional flows is defined by Equation 7, where ψ is
the stream function.
𝒗=∇ × 𝛙
(7)
1 𝛿𝜓
𝑟 𝛿𝑧
1 𝛿𝜓
𝑉𝑧 =
𝑟 𝛿𝑟
𝑣𝑟 = −
The stream function plot show several areas of recirculating flow. Large
recirculation zones exist upstream and downstream of the seal and smaller recirculation
zones exist within between adjacent seal teeth. Recirculation vortices are expected in
rotor stator systems, and the observed recirculation patterns near the seal teeth are
consistent with previous research [reference].
Figure 13 Total temperature, stream function and pressure contours; 5 mil clearance
16
Tangential velocity, or swirl velocity plays a significant role in the physics of rotor
stator systems. Figure 14 shows contour plots of swirl velocity, RPMF (Equation 3), and
percent swirl velocity.
The flow enters the domain with no tangential velocity. As it flows toward the
seal section the swirl velocity increases because the rotor is doing work on it through
viscous forces. At the plane XX upstream of the seal section, the swirl velocity is about
400 ft/s. Given that linear velocity of the rotor is roughly 1250 ft/s by Equation 2, the
non-dimensional swirl velocity is about 0.3.
As the flow traverses the seal section, the swirl velocity increases and at the
downstream plane reaches about 600 ft/s, non-dimensionally about 0.5.
The third contour plot in Figure 14 shows that the swirl velocity is the dominate
velocity component for a majority of the domain.
The exception is at the radial
clearance gap. This is expected as the axial flow contributes significantly to the resultant
velocity as the flow is accelerated through the small gap.
Figure 14 Contours of tangential velocity quantities; 5 mil clearance
17
Now that the velocities have been discussed, a better discussion of the total
temperature contours in Figure 13 can be provided. The equation for total temperature is
provided as Equation 6. Note that the second term includes relative velocity. At the
domain inlet, the swirl velocity is zero. As the flow approaches the seal, the swirl
velocity increases, thus total temperature increased. As the flow goes through the seal
section again, swirl velocity increases along with a corresponding total temperature
increase. At the exit to the seal section however, the swirl velocity is a 600 ft/s (nondimensionally 0.5). From the exit of the seal section to the exit of the domain, swirl
velocity cannot significantly increase. This is because the flow already has a swirl
velocity that is the average of the rotor and stator linear velocities and rotor and stator
have roughly equal surface areas. More concisely, the RPMF is 0.5 and the rotor and
stator viscous forces are balanced.
4.2 Leakage and Viscous Heating - Methodology
This section describes the methodology for determining the leakage flow rate, and
the viscous heat generation in the seal section. The subsequent section will provide a
discussion of the results for the three clearances cases.
The leakage flow rate is easily output by Fluent. The leakage flow rate is quoted
as the mass flow across section YY in Figure 7.
The calculation of windage heating will be performed two ways. The first
method will use Equation 9 calculate the viscous heat generation. Equation 9 describes
that the amount of heat or work transfer to a control volume is a function of the inlet and
outlet total temperatures, the mass flow rate, and the constant pressure specific heat.
Figure 15 shows the control volume being employed.
The quantities required for Equation 9 will be obtained from the Fluent solution.
The mass weighted average total temperatures will be queried at the upstream and
downstream seal planes, XX and ZZ respectively. The mass flow rate will be queried
across plane YY, and the specific heat will be assumed constant at the average static
temperature in the seal section.
18
̇ = 𝑊̇ = 𝑚̇𝐶𝑝 Δ𝑇𝑡
𝑄̇ = 𝑄𝑣𝑖𝑠𝑐
(9)
Figure 15 Control volume for viscous work method 1 calculation
Note there is no mechanical crossing control volume boundary, thus the total
temperature rise in the systems is due purely to viscous work. Figure 16 shows a total
temperature contour of the 5 mil clearance case with the viscous heat option turned off.
As expected there is no total temperature rise in the CFD domain.
Figure 16 Total temperature contour assuming no viscous work; 5 mil clearance
The second method of calculating viscous work involves integrating the shear
stress at the rotor and stator wall. Equation 10 is the full differential form of the energy
conservation equation, and equation 11 is the reduced form for the problem at hand.
19
𝛿
1
̂) = − (𝛁 ∙ (1 𝜌𝑣 2 + 𝜌𝑈
̂) 𝒗) − (𝛁 ∙ 𝐪) − (𝛁 ∙ p𝐯) −
𝜌𝑣 2 + 𝜌𝑈
(
𝛿𝑡 2
2
(𝛁 ∙ [𝜏 ∙ 𝐯 ]) + ρ(𝐯 ∙ 𝐠)
(10)
1
̂) 𝒗) = −(𝛁 ∙ [𝜏 ∙ 𝐯 ])
(𝛁 ∙ (2 𝜌𝑣 2 + 𝜌𝑈
(11)
Integrating Equation 11 yields an equation that relates the change in total
temperature to the total shear stress and velocity.
Δ𝑇𝑡 =
(∮ 𝝉 ∙ 𝒗)
(12)
𝑚̇ 𝐶𝑝
It was previously shown in Figure 14 that the tangential velocity is the dominate
velocity component, so it is appropriate to replace the vectors in Equation 12 with the
scalars 𝜏𝑥𝜑 , and Vtan . The shear stress in the x-theta (axial-circumferential) direction is
easily queried from Fluent for both the rotor and stator walls, and the swirl velocity can
be approximated as the average tangential velocity between upstream and downstream
planes.
4.3 Leakage and Viscous Heating - Results and Discussion
In this section, the viscous heating and leakage flow results will be discussed for
the 5, 10, and 15 mil clearance cases.
The results for the 5 mil clearance case are provided in Table 7. The leakage flow
through the seal was 0.9173 lbm/s. The observed increase in total temperature through
the seal region was 111F. The control volume approach for calculating heat generation
yielded 27.27 Btu/s, and the shear stress method yielded 25.63 Btu/s. These numbers are
in good agreement. Note that the method 2 heat generation value would imply a 105F
seal adiabatic temperature increase.
20
Table 7 Results summary for 5mil clearance case
Two subsequent CFD analyses were run for 10 and 15 mil radial clearances. The
same methodology described for the 5 mil case was used. Tables 8 and 9 provide results
summaries for the 10 and 15 mil clearances cases respectively. Table 10 summarizes the
quantities of interest for the three clearance levels.
Table 8 Results summary for 10mil clearance case
21
Table 9 Results summary for 15 mil clearance case
Table 10 Leakage and windage summary for clearances (5-15mil)
Figure 17 shows a plot of leakage rate versus clearance. As expected the leakage
flow rate increases fairly linearly with clearance. The larger gap allows increased flow
at a given pressure ratio.
22
4
3.5
Leakage (lbm/s)
3
2.5
2
1.5
1
0.5
0
0
5
10
15
20
Clearance (mils)
Figure 17 Leakage versus clearance for 2:1 seal pressure ratio
The windage heat generation versus clearance plot is presented as Figure 18. The
heat generation due to windage increases with increasing seal clearance. This is an
interesting result and requires further explanation.
Windage Heating (Btu/s)
45
40
35
30
25
20
0
5
10
15
20
Clearance (mils)
Figure 18 Windage heat rate versus seal clearance for 2:1 seal pressure ratio
23
The increased viscous heating is caused by lower swirl velocities (RPMFs)
upstream of the labyrinth seal section. More flow is being pulled through the domain as
the seal clearance is increased, consequently the flow tends to preserve its angular
momentum, or lack thereof.
Since the flow enters the domain with zero swirl velocity, it is expected that as the
flow rate increases, the swirl velocity (or RPMF) upstream of the seal section will drop.
Figure 19 shows that this is indeed true. The RPMFs upstream of the seal section drop
as seal clearances are increased.
Figure 19 also shows the downstream RPMF is roughly the same for each clearance
case, about 0.5. So, for the larger clearance cases more work needs to be added to
achieve the same downstream condition.
Figure 19 RPMF versus seal clearance comparison
Figure 20 shows that the adiabatic temperature rise across the seal decreases with
increasing clearance. This is an expected result because the increase in widage heat is
more than offset by the increase in heat capacitance due to increased flow rates.
24
Comparing the 5 mil clearance case to the 10 mil clearance case, the flow rate
increased roughly 100%, and the windage heat increased by about 30%. According to
Adiabatic Total Temperature Rise (F)
Equation 9, this requires the temperature difference to decrease as is true in Figure 20.
120
110
100
90
80
70
60
50
40
0
5
10
15
20
Clearance
Figure 20 Total temperature rise versus clearance for 2:1 seal pressure ratio
25
5. Seal Convection Heat Transfer Characterization
Up to this point heat transfer has been neglected, that is the walls in the CFD
domain were assumed to be adiabatic. In this section wall temperatures will be assigned
in the seal region, and heat transfer coefficient will be calculated. First coefficients will
be determined using formulations that includes viscous heating, then viscous heating
will be turned off and coefficients will be recalculated. A comparison between these two
approaches will be presented. For this section only a 5 mil clearance will be considered.
5.1 Heat Transfer with Viscous Heating
A cooling and heating temperature condition will be considered. Wall temperatures
of 500F (heating condition), and 1500F (cooling condition) will be applied
independently to the rotor and stator walls. Table 11 describes the four cases to be
considered. Note that the previously detailed boundary conditions and Fluent setup
remain same for this study.
Table 11 Summary of heat transfer studies
Figure 21 Application of wall temperature boundary conditions
26
In order to calculate heat transfer coefficients the following approach will be
used. Given the same control volume considered in Figure 15 the energy balance yields
Equation 13, where ΔTt is the difference between the upstream and downstream total
temperatures.
̇ + 𝑄𝑐𝑜𝑛𝑣𝑒𝑐
̇
𝑄̇ = 𝑄𝑣𝑖𝑠𝑐
= 𝑚̇𝐶𝑝 Δ𝑇𝑡
(13)
For this study it will be assumed that Qvisc is the same as calculated in the
previous section.
There is some error in this assumption since it is known that
temperature affects viscosity, and thus viscous heating. However as noted, the heat
transfer coefficient calculations will be performed in subsequent sections without the
inclusion of viscous heating terms.
Heat transfer coefficients are calculated using Equations 14 and 15. When rotor heat
transfer coefficients are calculated the fluid temperature, given in Equation 15, will be
calculated assuming total temperature relative to the rotor (Equation 6). Combining
Equations 13-15 yields an expression for the heat transfer coefficient, Equation 16.
̇
𝑄𝑐𝑜𝑛𝑣𝑒𝑐
= 𝐻𝐴(𝑇𝑠 − 𝑇𝑓 )
𝑇𝑓 =
𝐻=
𝑇𝑡,𝑢𝑝+ 𝑇𝑡,𝑑𝑜𝑤𝑛
2
̇
𝑚̇𝐶𝑝 (𝑇𝑡,𝑑𝑜𝑤𝑛 −𝑇𝑡,𝑢𝑝 )−𝑄𝑣𝑖𝑠𝑐
(𝑇
−𝑇𝑡,𝑢𝑝 )
]
𝐴[𝑇𝑠𝑢𝑟𝑓 − 𝑡,𝑑𝑜𝑤𝑛
2
(14)
(15)
(16)
Figure 22 shows total temperature contours for the heated rotor and heated stator,
cases 1 and 2 from Table 11. The fluid temperature contours are consistent with the
27
applied wall temperatures. Note that wall temperatures applied to the stator affect the
downstream fluid temperature to a greater extent than wall temperatures applied to the
rotor. Figure 23 also shows this trend. This observation leads one to expect that the heat
transfer coefficients on the stator should be significantly higher than on rotor.
Figure 22 Total temperature contours for heated wall condition
Figure 23 Total temperature contours for cooled wall condition
28
Additionally, Figures 22 and 23 show that although the stator has more of an effect
on the downstream temperature, the rotor temperature boundary condition effectively
sets the temperature environment in the pockets between the seal teeth.
Table 12 gives shows the calculated average heat transfer coefficients for each of
the respective runs. Note this table was calculated using Equation 16 with the viscous
heating work, Qvisc, from the previous section. As expected the stator heat transfer
coefficient are indeed significantly higher than the rotor.
Table 12 Average rotor/stator heat transfer coefficients for heating and cooling
5.2 Heat Transfer without Viscous Heating
It was noted that the viscous heat rate may be slightly different with applied wall
temperatures because the fluid temperature is being affected by the convection to the
walls. In order to asses this error and gain confidence in the heat transfer coefficient
values, cases 1 and 2 were re-run without the viscous heat term, Qvisc = 0. These are
presented as cases 1a and 2a in Table 13.
Figure 24 shows the total temperature contours for these analyses. As expected,
the trends from the previous section still hold true. The stator heat transfer is higher,
resulting in a lower downstream temperature condition compared to the rotor. Also the
rotor temperature boundary condition continues to be responsible for setting the
temperature environment inside the seal pockets.
29
Figure 24 Total temperature contours for heated wall condition; No viscous heating
The heat transfer coefficients are presented in Table 13. As expected the stator
heat transfer coefficients are significantly higher than the rotor. Also, the heat transfer
coefficients compare well with those presented in Table 12. Both rotor and stator heat
transfer coefficient are lower by about 10%.
Table 13 Average heat transfer coefficients for heating; No viscous heating
5.3 Results Summary and Discussion
From Tables 12 and 13 several observations can be made. As noted, the average
heat transfer coefficient on the stator is significant higher than on the rotor. This is an
expected result since the solid stator land sees a significant axial or through flow
velocity. The rotor, on the other hand, only sees this velocity contribution at the seal
tips. The other portions of rotor are swirl velocity dominated. This can be seen in
Figure 25 which shows the absolute value of the ratio of axial to swirl velocity.
30
Table 14 Absolute value of axial to swirl velocity ratio
Table 12 also shows the heat transfer coefficients for a heated wall are very
similar to those for a cooled wall. The values for the rotor are about 6% different, and
the values for the stator are less than 1% different.
Finally, Table 13 shows that heat transfer coefficients calculated for analyses run
without viscous heating match fairly well with those calculated using analyses with
viscous heating. Values for both rotor and stator are within about 10%.
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6. 3D CFD Analysis with Various Honeycomb Seal Land Geometry
This section will detail the 3D CFD analysis for two honeycomb land geometries: a
1/8” honeycomb land, and a 1/32” honeycomb land. A 5 mil clearance will be used for
this study. It is expected that honeycomb land geometry will have a significant effect on
seal leakage, and a moderate effect on seal windage heating. Heat transfer calculations
will not be considered in this section.
6.1 Geometry Definition
The rotor geometry is the same as in previous sections, the only change is to the
treatment of the stator land. Figure 25 shows the 3D geometry for the fluid domain. The
domain includes 1/8” honeycomb, 5 mil foil size, and has a seal clearance of 5 mils. As
pictured, two honeycomb cells are included in the circumferential direction and 6 are
included in the axial direction. The geometry is a revolved through a small angle so the
cut faces are rotationally periodic.
Figure 26 shows the geometry for the 1/32”
honeycomb land which has two cells circumferentially 32 axially and has a foil size of 2
mils.
Figure 25 3D Geometry with 1/8" honeycomb land
32
Figure 26 3D geometry with 1/32" honeycomb land
Figure 27 shows a schematic for the honeycomb cells along with pertinent
dimensions.
Figure 27 Honeycomb schematic
33
The 3D geometry was meshed using the same procedure as the 2D geometry. A
fine mesh was applied in the seal section of the domain, and coarser mesh in the
upstream and downstream regions. Figure 28 shows several views of the CFD mesh.
The bottom image shows the honeycomb cells and their proximity to the seal teeth. The
mesh includes 1.1 million nodes and 5.3 million cells.
Figure 28 3D mesh with 1/8" honeycomb
34
6.2 3D Solution Post-Processing and Discussion
This section reviews the 3D CFD solution and presents comparisons to the
equivalent 2D study. The contour plots in the section were taken at the mid plane. For
the 1/8” honeycomb geometry the sector extends from -2 deg to +2 and the mid plane is
at 0 deg.
Figure 29 shows the pressure contours for the 1/8” honeycomb land 3D study. The
contours are very similar to the 2D analysis with 90, 70 and 90 psi pressure drops across
first, second and third knife edges respectively.
Figure 29 3D CFD pressure contours for 1/8" honeycomb
Figure 30 shows contours of total temperature. Here there is significant difference
between the similar 2D analysis (Figure 13). Although the radial clearance between the
knife edge and the seal land is 5 mils for both analyses, the large honeycomb effectively
increases flow area allowing significantly more leakage. The increase in seal leakage
mitigates the total temperature increase according to Equation 9.
This flow area increase effect is shown in Figure 31. At the mid plane, the first knife
edge is positioned directly under a solid portion of the seal land. This is similar to the
35
treatment in the 2D analysis. However, for the second and third knife edges, the knife
edge is offset from the honeycomb solid. The velocity contour shows the flow
accelerating through these gaps.
Figure 30 3D CFD total temperature contours for 1/8" honeycomb
Figures 32 and 33 show the contours for swirl velocity and RPMF. These
contours also differ significantly relative to the 2D analysis. The mass weighted RPMF
at the upstream plane, XX is 0.16 compared to 0.322 for the 2D analysis. However,
given the significant increase in leakage this is understandable as the large leakage flow
preserves more of its angular momentum (RPMF=0.0 at the domain inlet).
If we
compare the 3D results to the 15 mil clearance 2D case (which has a similar leakage
flow), we see the upstream RPMF is in fair agreement.
The downstream mass weighted RPMF at plane YY is also very low, nearly 0.
This is an reasonable result. As was previously discussed, a significant portion of the
flow is directed into the honeycomb cells.
This mitigates the angular momentum
increase imposed by the rotor. Clearly this effect is not represented in the 2D study.
36
Figure 31 3D CFD velocity magnitude contours for 1/8" honeycomb
Figure 32 3D CFD swirl velocity contours for 1/8" honeycomb
37
Figure 33 3D CFD RPMF contours for 1/8" honeycomb
Table 15 provides a comparison between the 2D and 3D 5 mil clearance cases.
Interestingly, the 1/32” honeycomb analysis produces a leakage level significantly less
the solid land. In this case the honeycomb cell size (0.03125”) is close to the size of the
tip of the knife edge (0.010”). This limits the flow’s ability to find a smooth streamline,
and increases the turbulent friction [1]. The honeycomb in this situation discourages
flow.
Note contour plots for the 1/32” honeycomb analysis are provided in the
appendix.
Table 15 Comparison of 2D and 3D analyses for 5 mil radial clearance
38
7. Conclusions
This paper explored several aspects of gas turbine labyrinth seals. The leakage
rate, windage, and adiabatic temperature rise was investigated for various seal
clearances. Next, the adiabatic boundary conditions were removed and heat transfer
characteristics were studied.
A 2D axisymmetric model with a solid seal stator land was used for the above
studies. A grid independence study was conducted and thorough solution convergence
checks were presented.
The results for the 2D analysis were generally as expected. For a given pressure
ratio, an increase in seal clearance resulted in an increase in leakage. The windage heat
also increased with seal clearance due to the drop in upstream RPMF. The adiabatic
total temperature rise across the seal dropped as seal clearance was increased due to
additional heat capacitance provided by more leakage flow.
Heat transfer results for the 2D analysis were also as expected. For a solid seal
land, heat transfer coefficients were higher than the rotor heat transfer coefficients due to
the significant axial velocity component.
Heating versus cooling did not have a
significant effect on heat transfer coefficients on neither the rotor nor the stator.
Removing viscous heating from the solution equations also did not appreciably change
heat transfer coefficients.
Honeycomb geometry was studied using a rotationally periodic 3D CFD sector
model. Two honeycomb geometries were studied: 1/8” honeycomb, and 1/32”
honeycomb. The results of the 1/8” honeycomb study were generally as expected. This
large honeycomb increased leakage levels and reduced total temperature rise.
An
unexpected, but explainable result was the low swirl velocities at the downstream plane.
The 1/32” honeycomb results were surprising in that the leakage level was reduced
relative to the 2D solid land results. However further research determined this to in fact
be an expected result [1].
39
8. References
[1] Choi, Dong-Chun, and David L. Rhode, 2004, “Development of a Two-Dimensional
Computation Fluid Dynamics Approach for Computing Three-Dimensional Honeycomb
Labyrinth Leakage” ASME vol. 126
[2] Yan, X., Feng Z., and Feng, Z. 2010, “Effect of Inlet Preswirl and Cell Diameter and
Depth on Honeycomb Seal Characteristics” ASME Paper 122506-1.
[3] He, K., Li, J., Yan, X., and Feng, Z. “Investigations of the Conjugate heat transfer
and windage effect in stepped labyrinth seals”. International Journal of Heat and Mass
Transfer 55 (2012) 4536-4547
[4] Yan, X., Li, J., Song, L., and Feng, Z. 2009, “Investigations on the Discharge and
Total Temperature Increase Characteristics of the Labyrinth Seals With Honeycomb and
Smooth Lands”. ASME Paper 041009-1
[5] Proctor, Margaret, and Irebert R. Delgado. 2004, “Leakage and Power Loss Test
Results for Competing Turbine Engine Seals”. NASA & US Army Research Laboratory
Report NASA/TM-2004-213049
[6] Ozturk, H.K., Turner, A.B., Childs, P.R.N., et al. “Stator well flows in Axial
Compressors”. International Journal of Heat and Fluid Flow 21 (2000) 710-716
[7] Bird, R. Byron, Warren, E. Stewart, and Lightfoot, N. Edwin. Transport Phenomena
Second Revision. New York: Wiley, 2007.
40
9. Appendix
Figure 34 3D CFD pressure contours for 1/32" honeycomb
Figure 35 3D CFD total temperature contours for 1/32" honeycomb
41
Figure 36 3D CFD velocity magnitude contours for 1/8" honeycomb
Figure 37 3D CFD swirl velocity contours for 1/32" honeycomb
42
Figure 38 3D CFD RPMF contours for 1/32" honeycomb
43