...

Denecke2008-LabyrinthSeals.pdf

by user

on
Category: Documents
6

views

Report

Comments

Transcript

Denecke2008-LabyrinthSeals.pdf
Proceedings of ASME Turbo Expo 2008: Power for Land, Sea and Air
GT2008
June 9-13, 2008, Berlin, Germany
GT2008-51429
INTERDEPENDENCE OF DISCHARGE BEHAVIOR, SWIRL DEVELOPMENT AND
TOTAL TEMPERATURE INCREASE IN ROTATING LABYRINTH SEALS
J. Denecke: J. Farber, K. Dullenkopf and H.-J. Bauer
lnstitut fuer Thermische Stroemungsmaschinen
Universitaet Karlsruhe (TH)
76131 Karlsruhe, Germany
ABSTRACT
swirl is known or can be estimated with reasonable confidence
an analytical approach to determine the swirl development from
chamber to chamber is proposed. Given this swirl development
along the seal axis, the overall total temperature increase can be
calculated. Based on the final dimensionless equation for the total
temperature change the interdependent influences of discharge
behavior, swirl development and the total temperature increase on
each other are discussed.
Leakage flows between stationary and rotating components
are one of the main sources for losses in turbo machines. Therefore, their reduction is a main goal in the design of modern aircraft engines. Contactless seals, mainly labyrinth seals are key
elements either to seal rotating parts or to control the amount of
leakage flow for internal use in the secondary air system. Even
though new seal types like, brush seals, carbon seals etc. will be
seen more often in advanced gas turbines, labyrinth seals will
continue to play an important role in the primary and secondary
air system and thus improved design tools are a necessity for more
efficient and reliable engines.
In the design process but also during the life time of the engine the characterization of contactless seals e.g. their discharge
behavior, the development of the circumferential velocity (swirl)
and the loss induced total temperature increase (windage heating)
are of special interest for designers and operators. Despite of
today's efficient CFD methods, analytical models remain a valuable tool as they provide for reasonably estimates fast with small
computational effort.
Additionally, analytical models are especially suited to improve the understanding of the complex interdependency of the
aforementioned parameters. As one limit of the swirl in rotating
seals, the equilibrium swirl is defined in this paper and a simple
method to determine its value is presented. In this context, the
influences of the rotor-stator area ratio and the stator roughness on
the equilibrium swirl are taken into account. In the case the inlet
INTRODUCTION
The customer's increasing requirements on gas turbines such
as performance, life-time, weight, costs and emissions are fulfilled
partly by raising turbine inlet temperatures and pressure ratios. To
achieve further improvements, the optimization of every single
engine component is of particular importance. Rotating seals
serve many significant purposes throughout the engine, e.g. to
reduce stage losses at shrouds, reduce axial thrust on bearings or
meter the cooling air to prevent hot gas ingress. While matching
the discharge behavior of seals to requirements in a given design
is certainly the most important aspect, additional characteristics
such as heat transfer, total temperature increase due to internal
losses (windage heating) and the swirl development must be taken
into account for an optimized design.
As an example, the increase of the cooling air temperature in
labyrinth seals caused by windage heating, noticeably affects the
lifetime of cooled blades downstream. The outlet temperature
and swirl of labyrinths sealing the direct transfer pre-swirl systems strongly influence the cooling air total temperature in the
*Currently at BASF SE, Ludwigshafen a. R., Germany. Please address all
correspondence to this author. Email: [email protected]
Copyright @ 2008 by ASME
1717
relative frame of reference. The interstage seal exit swirl locally
changes the incidence angle of downstream blades and thereby
influences the stage losses. As a last example, self-excited rotor
dynamic instabilities are largely influenced by the swirl development within the seal. To avoid these rotor dynamic instabilities
in high-pressure compressors swirl brakes are used and in gas
turbines sometimes damping rings or stiffening elements are applied. These examples clearly illustrate why the interdependency
of discharge behavior, swirl development and windage heating in
labyrinth seals has been a recurring subject of research.
A number of publications on this subject are available. For
example Tipton et al. and Stocker et al. [1, 2] reported in their fundamental studies on labyrinth seals that the dominant influences
on the total temperature change across the seal are the mass flow
rate, rotational speed and the number of teeth or rotor surface area,
respectively. McGreehan and Ko [3] presented experimental data
of windage heating in several labyrinth seal configurations. A 'fin
to fin' calculation was derived from shrouded-disk correlations
which illustrates the strong interdependence of swirl and windage.
An easily applicable correlation was provided by Millward and
Edwards [4] of their windage heating data. Scherer et al. [5] found
from numerical simulations, that the seal clearance has an impact
on the temperature increase by changing the discharge characteristic. The authors also described the important influence of the
circumferential velocity at the inlet. Their work was continued
by Denecke et al. [6, 7] who investigated the swirl development
experimentally and by numerical flow simulations.
As an extension this paper aims to to describe the interdependency of the aforementioned parameters by means of simple
analytical approaches. Analytical or semi-empirical formulas allow to obtain fast estimates with small computational effort. Even
though parameters might have to be fitted to experimental or CFD
data, the solid physical basis of these models minimizes scaling
errors and reduces the risks of inter- or even extrapolation of given
data points.
At first non-dimensional numbers for the discharge behavior,
swirl development and windage heating are defined in the following section, as they form the basis for analytical and empirical
models. All data in this paper will be presented by means of these
non-dimensional numbers. Then, the equilibrium swirl is defined
and a simple method to determine its value is presented. The
equilibrium swirl can be thought of as the limiting value which
would be reached in an infinite seal (on a constant radius with an
incompressible fluid). Therefore, it is suited as a first estimate of
the swirl in a given seal geometry, taking into account rotor-stator
area ratio and the stator roughness. For the case that the inlet
swirl is known or can be estimated with reasonable confidence
an analytical approach to determine the swirl development from
chamber to chamber is proposed. The average swirl along the seal
axis is an important input parameter for the calculation of the overall total temperature increase. Based on the final dimensionless
form of the equation the interdependent influences of discharge
behavior, swirl development and total temperature increase are
discussed.
NON-DIMENSIONAL NUMBERS
A major step in the dimensional analysis of seals was the
experimental verification published by Wittig et al. [8], showing
that the discharge coefficient Cv in the non-rotating case is a
function of the seal geometry, pressure ratio, Reynolds number,
ratio of specific heats K, and turbulence level Tu, alone. The
discharge coefficient Cv for subcritical pressure ratios is defined
in Equation 1.
riz
Cv= -.m;d
.
m;d=
Q;d · Ptot,in ·A
vr;;;
(1)
To incorporate the influence of rotation, Waschka et al. [9] found
the velocity ratio vaxju to be well suited to correlate the discharge
coefficient Cv and the Nusselt number. Denecke et al. [6] conducted a dimensional analysis to find suitable non-dimensional
numbers which describe the loss induced total temperature increase (windage heating) and the circumferential velocity development (swirl) in contactless seals. Within that study pairs of
similar geometries were compared, which were scaled from laboratory to typical engine conditions such that the discharge behavior
was kept identical. This allowed a separate investigation of the
influence of mass flow and rotation on windage heating or swirl
development. At a constant circumferential Mach number and
identical radius, deviations of the mass flow between laboratory
and engine operating conditions lay within the numerical uncertainty. In an extended analysis, conditions were compared where
the circumferential Mach number Mtan was kept constant while
the circumferential Reynolds number was doubled. The deviations of windage heating, swirl and discharge behavior in between
the two cases were smaller than 5% in the majority of variations
considered. As a conclusion, the following dimensionless numbers were found to uniquely define the discharge behavior Cv,
the exit-swirl Kout = Ctan,out /U and the windage heating number
a= 2·cp·ATtorju 2 in rotating contactless seals:
i;. }
=
f (1t, Reax, M,an, K;n,
Geometry, Pr, K)
(2)
The tangential Mach number in Equation 2 can be equivalently
replaced by the velocity ratio vaxju as defined by Waschka et
al. [9].
Copyright @ 2008 by ASME
1718
0.60
In the next section the theoretical limit of the exit swirl Kout is
derived, which might also serve as a first estimate for this value.
Reax 10000 - Reax 30000 ------Reax 50000 --------
0.55
0.50
0.45
CONCEPT OF EQUILIBRIUM SWIRL
In a rotating labyrinth seal the fluid is accelerated in circumferential direction by wall friction on rotor and decelerated by
the stator. After a developing length in an ideal infinite seal with
an incompressible fluid a constant circumferential velocity c1an,eq
is reached regardless of the inlet conditions. Downstream of the
developing region, the torques acting on rotor and stator would
be equal (Ms = MR). In a real seal the circumferential velocity
will approach this limiting value asymptotically. For an analytical
calculation of the equilibrium swirl Keq = Ctan,eq/U it is assumed
that wall friction can be described by Equation 3.
A.
4=
't
0"
""""'
0.25
0.15
0.6
l
I
=-
1
- · { R(x) ·dAR/S
Figure 1.
(4)
AR/S }A
Thus, the torques of rotor and stator, which are the integral of the
wall shear stresses in circumferential direction 'ttan,R and 'ttan,s
can be approximated by Equation 5.
L
'ttan,R/S · dAR/S
For the equilibrium condition (Ms
1 = MR
Ms
';;:j
RR/S · 'ttan,R/S ·AR/S
(5)
= MR) Equation 6 follows.
= 'ttan,R · RR ·AR
1.1
1.2
1.3
1.4
= ----:------=-----1 + (&. ~. R 1/4 )4/7
AR 0,3162 etan,S
(9)
If the stator also has a smooth surface and the Blasius friction parameters can be applied then Equation 9 can be further simplified
to Equation 10.
K
)2
eq-
(7)
1
(2- nR)
(&
1 + AR
)4/7
(10)
In Figure 1 the equilibrium swirl Keq is plotted against the area
ratio of rotor to stator surface following Equation 10. Typical
labyrinth seals with teeth on the rotor and a smooth stator wall
will have an area ratio of about As/AR ';;:j 0.7 which leads to an
equilibrium swirl of Keq = 0.55.
In order to incorporate abrasive liners such as honeycomb materials in this approach the friction parameters for a fully rough
Equation 7 can be rewritten for the equilibrium swirl Keq as follows.
As . !!!:..S_. R (nR-ns))
( AR mR etan,S
1.0
(6)
'ttan,s · Rs ·As
-nR . ( roRR- Ctan,eq
As = mR . Retan,R
AR
ms Ret-nss
Ctan,eq
an,
Ctan,eq
roR
0.9
Equilibrium swirl according to Equation 9 versus area ratio.
Keq
Assuming large radii (Rs/RR ';;:j 1) and inserting the relation for
friction (Equation 3) leads to the subsequent function for the area
ratio.
Keq =
0.8
Equation 8 shows that the equilibrium swirl Keq is influenced by
the area ratio As/AR, the circumferential Reynolds number for the
stator and the friction parameters mR/S and nR/S· The influence of
the area ratio will lead to an equilibrium swirl larger than Keq =
0, 5 if the teeth are applied on the rotor and a value smaller than
Keq = 0, 5 if they are applied on the stator. The friction parameters
mR/S and nR/S should be determined experimentally due to the
complex three-dimensional boundary layers and chamber flow
fields ([10] and [11]). However, in case of common parameters for
rotor and stator the influence of the chosen friction parameters on
the swirl are small because discrepancies of rotor and stator torque
partly cancel each other out (compare Equation 8). Therefore,
the Blasius friction parameters for turbulent boundary layers m =
0.0791 and n = 0.25 are used for the rotor in order to simplify
Equation 8 to Equation 9.
I
MR/S = RR/S
0.7
As/AR
As an additional simplification area averaged radii are defined in
Equation 4.
RR/S
l!l6;:--
HC
0.20
I
'i
0,35
0.30
(3)
CJ = P/2 c2 = m Re-n
0.40
(8)
+1
Copyright @ 2008 by ASME
1719
1.6
{
:E
~
E
·c:::l
~::l
<:1'
1.6
convergent, s=0.5mm, z=6, M...=0,775
Ko=O.S
~if
1.4
1.2
-;::::
-~
l.O
E
::l
__ .... --
0.8
__
. --
0.6
11)
·g>
~
1.0
0.8
~
0.6
11)
11)
1.2
·c:
:9
divergent, s=0.5mm, z=6, n=L4
1.4
11)
>
0.4
- - n=u
n=L4
n=2.o
0.2
"i
0.4
~
0.2
- - M,,.=0,465
------- M...=0,620
---- M...=0,775
a:l
0.0
0.0
-1.0
0.0
2.0
1.0
3.0
4.0
-1.0
5.0
l.O
0.0
Figure 2.
2.0
3.0
4.0
5.0
Labyrinth length x/t
Labyrinth length x/t
Figure 3.
Influence of pressure ratio and inlet swirl on swirl development.
Influence of rotational speed and inlet swirl on swirl develop-
ment.
regime according to Nikuradse in Equation 11 can be applied.
_1_ = _ 2 O l
v'A
'
og
(k/Dhyd)
3,7
compared to the pressure ratio. However, the CFD data of the
divergent seal configuration displays a significantly different equilibrium swirl compared to the convergent seal. The fact that the
real equilibrium swirl Keq,CFD is smaller than the one calculated
using Equation 10 has qualitatively been found in all investigated
divergent seals. Besides the differences of the boundary layers
(e.g. different position of stagnation points) another major cause
is the reduced mixing of the circumferential wall jet velocity component in a divergent seal. In contrast, in a convergent seal the
jet is redirected into the chamber at the step of the stator which
enhances the mixing. To take this effect into account Equation 10
was extended by the geometry factor 'If.
(11)
As roughness height k the honeycomb diameter dHc is used. This
has to be judged as a rough approximation as honeycombs show
complex friction characteristics (compare [12]). The resulting
equilibrium swirl lays far below the value for a smooth stator. As
expected the higher friction of the large honeycomb diameter with
1/ 16" shows a smaller equilibrium swirl than the small honeycomb diameter with 1/32". This result is supported by previously
published measurements [7].
For further analysis Figure 2 shows the swirl development
calculated by numerical flow simulations (CFD). The standard
k-E turbulence model as implemented in the commercial software
package Fluent v.6.2.1 was used with logarithmic wall functions
and axisymmetric grids. The model used has been described in
detail in a previous paper [6]. The ordinate in Figure 2 shows the
ratio of the local swirl K above a labyrinth fin over the equilibrium
swirl Keq· The abscissa represents the axial position of the fin. All
curves tend to a common limit which is the equilibrium swirl. The
value Keq,CFD derived from flow simulations of the convergent
labyrinth seal is almost identical to the value Keq,Eq.IO estimated
using Equation 10. This is illustrated by their ratio 'If= 0, 97. If
the pressure ratio 1t is reduced, the equilibrium swirl is already
reached after a shorter seal length, because a smaller mass flow
has to be accelerated or decelerated by the rotor. However, the
equilibrium swirl itself does not depend on the pressure ratio. This
behavior has been observed under all operating conditions and for
all labyrinth geometries studied by the authors [6].
The influence of rotational speed on the relative swirl development in Figure 3 is relatively small compared to that of the
pressure ratio 1t. This can be explained by the small influence
of the rotational speed on the mass flow rate through the seal
Keq='lf
1+
(
)4/7
(12)
&
AR
For the cases presented in Figure 2 and Figure 3 the geometry
factor was determined to be
'l'konv
= 0, 97
'l'div =
for the convergent seal
(13)
0, 88 for the divergent seal.
These values may serve as a first approximation for the influence
of the seal geometry. For a straight through seal a further reduction
of 'If is expected. Unfortunately, no flow simulations of straight
through seals were available for the present study. Additionally, a
weak influence of the axial Reynolds number, the acting centrifugal forces and the chamber geometry is expected. On the basis of
the equilibrium swirl the swirl development along the seal axis
can be calculated, as described in the following section.
Copyright @ 2008 by ASME
1720
l.O
SWIRL DEVELOPMENT
The basis for the discussion of influences on the swirl development is the momentum balance in circumferential direction.
Equation 14 defines the swirl change in a single chamber.
I[
~
0.81--
~
0.6
= MR,i-MS,i
.
2
m·Ri. ro
CFD M,.0.61
--------- CFD Mtan=0.82
------- Equation 17
:E
e
Ki+l -Ki
CFD Mtan=0.31
- - - - CFD Mtan=0.46
-A--
~=0.183
~=0.222
~=0.257
~=0.310
·E!
~
g.
(14)
0.4
II)
!;!
·.:=
~
The swirl development is proportional to the reciprocal value of
the mass flow rate. Secondary influences of the mass flow rate on
rotor and stator torque cannot be easily estimated. From Equation
5 and Equation 3 the proportionality in Equation 15 results. The
rotational speed (roRR,i) dominates the torque values while the
radius Ri in Equation 14 is approximately canceled out for the
benefit of a rotor surface length AR/RR (for large radii). The
average chamber swirl K results from the inlet swirl Ko and the
influences discussed before.
AR p
2
-
2
MR ,I· oc --(roRR
·) (1-K) ·RR ,I· ·AR,I·
4 2
,I
0.0
0.0
+ Ctan,O e-S.tjl,
2.0
l.O
3.0
4.0
Labyrinth length x/t
Figure 4.
Swirl development.
the sense of Equation 16.
K(x)
Ko -l;·xT
- = ( 1-e-l;·xT) +-·e
Keq
(15)
Keq
(17)
Even though not all assumptions by Black et al. can be transferred
to labyrinth seals, the comparison to flow simulation results show
that Equation 17 is able to predict the swirl development from
inlet to outlet realistically. In contrast to Equation 16, Equation 17
is only valid at discrete values along the seal axis, which are the
averaged swirls above the seal fins (x/t = 0, 1,2, ... ). The losses
in the seal chambers are modeled as a roughness in between the
fins by the friction parameter ~.
The friction parameter~ can be approximated by Equation 18
if good estimates for inlet and exit swirl are known. The friction
parameter ~ is actually a function of the dimensionless numbers
in Equation 2, if it was to be used to predict an unknown exit swirl.
The accuracy is improved if the average swirl above the first and
the last fin is used instead of inlet and exit values.
To the authors knowledge no integral or half-empirical correlation for the swirl development has been published so far. However, for the calculation of the total temperature change in the next
section, an estimate for the swirl development is needed. Solving
Equation 14 would result in a Bulk Flow model which can only be
iteratively solved. Therefore, a different much simpler approach
in order to estimate the swirl development is proposed: There
exist a large number of investigations on plain cylindrical seals.
According to Black et al. [13] the swirl development in a plain
cylindrical seal is:
Ctan (x) = 1/2 · roRR ( 1 - e-S.tjl-)
0.2
(16)
~ ~ ~ ·in (
The second term on the right hand side of Equation 16 is the
contribution of the inlet boundary condition Ko to the local swirl
development. Here xjL is the relative axial position and~ can be
interpreted as a friction parameter which is defined by the length
of the developing region l. This length in tum marks the point
where the velocity ratio Ctan,eq / (roR) = 0, 5 is reached. Therefore,
the first term on the right hand side incorporates the limit for an
infinite seal as a boundary condition, which is the equilibrium
swirl Keq· Several authors have shown that surface roughness on
rotor or stator influences the equilibrium swirl in cylindrical seals
such that it deviates from the value of K = 0,5 [14].
The idea in this study is that Equation 16 might also be
applied to the swirl development in a labyrinth seal discretized by
the fins. Hence, for labyrinth seals Equation 17 can be written in
Z
Kin - Keq )
Kout -Keq
(18)
The influence of abrasive liners is inherently accounted for by
a different friction parameter ~ and the appropriate equilibrium
swirl Keq· In Figure 4, exemplary results are shown for the swirl
development in a convergent labyrinth seal with four fins. The
analytical solution is compared to results from numerical flow
simulations which had been validated to experimental results in a
previous publication [7].
It can be concluded, that the swirl development follows a
simple exponential function. Even though good estimates of inlet
and exit swirl have to be available, these results for the axial swirl
development are a fundamental requirement for the calculation of
the total temperature change as described in the following section.
Copyright @ 2008 by ASME
1721
TOTALTEMPERATURECHANGE
1.6
As an illustration of the influences on the total temperature
change in labyrinth seals Figure 5 shows results of numerical flow
simulations. The dimensionless total temperature cr is plotted
versus the labyrinth seal length. Again, the value for !lit was
computed as an average value at the tip of the labyrinth fins. The
influence of rotational speed is smaller than that of the inlet swirl
Ko in this dimensionless representation. If the swirl stays constant
along the seal length (equilibrium swirl), the complete work input
of the rotor is dissipated. In this case the total temperature changes
linearly from chamber to chamber. If the inlet swirl is smaller
than the outlet swirl (which is generally the case), the fluid has
to be accelerated by the rotor and the labyrinth seal acts like a
compressor (with miserable efficiency). The velocity gradients to
the rotor are higher than in the linear case and therefore, the total
temperature shows a steeper increase. The curvature of cr is due
to the exponential swirl development as described in the previous
section. Once the equilibrium swirl is almost reached the further
total temperature change is again linear with x / t. If the inlet swirl
is higher than the equilibrium swirl the fluid is decelerated and
the labyrinth seal acts like a turbine. Depending on the length of
the seal the total temperature can even decrease in this case.
The influence of the total temperature change on the discharge
behavior can be estimated by Equation 19 which was derived from
measurements by [15]. For typical gas turbine total temperature
levels the resulting mass flow changes can often be neglected.
1.4
!lm- o 3 !lit
0
~
~u
'tR,tan =
4~~) · lPm · (IDRR -
'tR,tan
·dx
Ctan (x))
2
__.
E
2
0.6
5
0.4
Ko=0.3
--• Ko=O.S
0.2
0.0
1.0
2.0
3.0
4.0
5.0
6.0
Labyrinth fin number
Figure 5.
Total temperature development.
change, which is proportional to the rotor surface and the reciprocal mass flow rate.
If the average swirl velocity is reduced, e.g. by honeycomb
abrasive liners the total temperature change increases. An increased inlet swirl leads to a strong reduction of the total temperature change. Axial flow generated turbulence and stagnation
points increase the wall shear stress and thereby the the total temperature increase. Their influence or that of flow field changes
due to rotation or axial Reynolds number effects can only be accounted for by adjusting the friction parameters. A comparison
to the experimental data by the authors for smooth walls shows
a mean deviation of 15% with maximum values of 40%. As a
tendency, for convergent stepped seals higher and for divergent
stepped seals lower values than the experimental data were predicted by Equation 23.
Equation 23 can be rewritten in a dimensionless form. The
first term represents the circumferential to axial Reynolds number
ratio weighted by the friction parameter n. The area ratio of the
rotor surface and the product of rotor radius and hydraulic diameter of the seal chamber describe the influence of the number of
teeth and the scale of the labyrinth geometry. The radius RR can
be canceled if instead of the rotor surface AR a representative rotor
length is introduced. The last term of the equation describes the
average swirl level relative to the rotor surface and is weighted by
the friction parameter n. The integral for averaging the swirl development can be approximated by a sum in order to simplify the
application of Equation 23 in a 'chamber to chamber' approach.
Additionally, the following assumptions were made: Walls are
adiabatic. The tangential and axial components of the wall shear
stresses can be calculated independently. The rotor torque can
be approximated by Equation 21, where Equation 22 is averaged
along the axis. To compute the wall shear stress Equation 3 is
assumed to be applicable.
±·i
0.8
Fl
(20)
= RR ·AR ·
~
&
The basis for an analytical calculation of the total temperature
change is the energy balance.
MR
1.0
Ko=O.O
- - M...=0,465
M...=0,620
M...=0,775
~
(19)
---;;---' T;
1.2
convergent, s=0.5mm, z=6, n=1,4
(21)
(22)
Combining Equation 20 through 21, Equation 23 follows. It is
a simple formula for the determination of the total temperature
Copyright @ 2008 by ASME
1722
K
cr~
m
4
(1-n)
· - [(1-K·)2-n
z i=l
Reax
"-v-"'
velocity ratio
area ratio
m
m
weighted average
Mtan =
p
Pr =
R
Raas = 287.2
Retan
Pm·U·dhydjJl
Reax = wrt·R
=
ml
s
t
T
Tu = ~
U = roR
X
z
Greek letters:
L\
K
mls
m2
c
mls
Cp
llkg·K
Cv
dhyd
m
kglm·s
f.l
1t = Ptot,in1Pl1at,ollt
p
cr = 2·cp·!!..1iotluz
ro
~
radls
A.
Subscripts:
1,2, ... ,z
eq
id
in I out
R
rad I tan
s
speed of sound
area
fluid velocity
specific heat capacity
discharge coefficient
hydraulic diameter
stat
kglm3
Nlm 2
't
NOMENCLATURE
y'KRaas I'stat,in
JHpjk
Qideal
SUMMARY
Analytical models are especially suited to improve the understanding of the complex interdependency of the discharge
behavior, the development of the circumferential velocity (swirl)
and the loss induced total temperature increase (windage heating)
in labyrinth seals. Additionally, they provide for reasonably estimates fast with small computational effort. Therefore, several
analytical calculation methods for the aforementioned parameters
are presented in this paper. The uncertainty can be reduced by
fitting the friction parameters to numerical flow simulations or
experimental results.
In this paper, the equilibrium swirl was defined as the theoretical limit of the swirl in rotating seals and a simple method
to determine its value was presented. In this context, the influences of the rotor-stator area ratio and the stator roughness on the
equilibrium swirl were taken into account. In case that the inlet
swirl is known or can be estimated with reasonable confidence
an analytical approach to determine the swirl development from
chamber to chamber was proposed. Once the swirl development
along the seal axis is known, the overall total temperature increase
can be calculated using the formulas given in this paper. Based
on the final dimensionless form of the equations the interdependent influences of discharge behavior, swirl development and total
temperature increase were discussed.
a=
Ula
n
n
The relation in Equation 24 shows that the seal radius only has a
minor influence on the dimensionless total temperature change as
was proposed in a previous paper [6].
The analytical approach in Equation 24 quantifies the dominant influences of the discharge behavior, rotational speed and
inlet swirl on the total temperature change. By fitting the fricton
parameters to a specific seal geometry and range of operating
conditions the accuracy of Equation 24 can be further improved.
swirl ratio
length of dev. region
seallength
friction coefficient
kgls
mass flow rate
circumferential Mach number
s- 1
rotational speed
friction coefficient
Nlm 2 pressure
Prandtl number for air
~ expansion function
m
radius
1lkgK
specific gas constant
circumferential Reynolds no.
axial Reynolds number
m
seal clearance
m
seal pitch
K
temperature
inlet turbulence intensity
mls
rotor circumferential velocity
mm
axial coordinate
number of fins (= number of
constrictions)
m
m
L
(24)
l
swirl
A
= Ctanlu
l
1 i=z
R etan
wlm·K thermal conductivity
k
From these ideas Equation 24 follows.
I ax
I tot I dyn
difference
ratio of specific heats
dynamic viscosity
pressure ratio
density
windage heating coefficient
wall shear stress
angular velocity
swirl friction coefficient
pipe friction coefficient
the i-th fin or the condition of the
fluid exiting the i-th constriction resp.
equilibrium
ideal
inlet I outlet condition
Rotor
radial I tangential I axial direction
Stator
static I total I dynamic Value
Copyright @ 2008 by ASME
1723
,
References
[1] Tipton, D., Scott, T., and Vogel, R., 1986. Labyrinth Seal
Analysis: Volume III - Analytical and Experimental Development of a Design Model for Labyrinth Seals. Tech. Rep.
AFWAL-TR-85-2103 Vol. III, Allison Gas Turbine Division of General Motors Corporation, Indianapolis, Indiana,
January.
[2] Stocker, H., Cox, D., and Holle, G., 1977. "Aerodynamic Performance of Conventional and Advanced Design
Labyrinth Seals with Solid-Smooth, Abradable and Honeycomb Lands". NASA-CR-135307.
[3] McGreehan, W., and Ko, S., 1989. "Power Dissipation in
Smooth and Honeycomb Labyrinth Seals". ASME-Paper
89-GT-220.
[4] Millward, J., and Edwards, M., 1996. "Windage Heating of
Air Passing Through Labyrinth Seals". ASME Journal of
Turbomachinery, 118, pp. 414-419.
[5] Scherer, T., Waschka, W., and Wittig, S., 1992. "Numerical
Predictions of High-Speed Rotating Labyrinth Seal Performance: Influence of Rotation on Power Dissipation and
Temperature Rise". ICHMT 1992 International Symposium
on Heat Transfer, in Turbomachinery, Athens, Greece, 26,
pp. 1514-1522.
[6] Denecke, J., Farber, J., Dullenkopf, K., and Bauer, H.-J.,
2005. "Dimensional Analysis of Rotating Seals". ASMEPaper GT-2005-68676.
[7] Denecke, J., Dullenkopf, K., Wittig, S., and Bauer, H.-J.,
2005. "Experimental Investigation of the Total Temperature Increase and Swirl Development in Rotating Labyrinth
Seals". ASME-Paper GT-2005-68677.
[8] Wittig, S., Dorr, L., and Kim, S., 1983. "Scaling Effects
on Leakage Losses in Labyrinth Seals". ASME Journal of
Engineering for Power, 105, pp. 305-309.
[9] Waschka, W., Wittig, S., and Kim, S., 1992. "Influence
of High Rotational Speeds on the Heat Transfer and Discharge Coefficientsin Labyrinth Seals". ASME Journal of
Turbomachinery, 114(2), pp. 462-468.
[10] Hirs, G., 1973. "A Bulk-Flow Theory for Turbulence in
Lubricant Films". ASME Journal of Lubrication Technology,
95, pp. 137-146.
[11] Childs, D., and Ramsey, C., 1991. "Seal-RotordynamicCoefficient Test Results for a Model SSME ATD-HPFTP
Turbine Interstage Seal With and Without a Swirl Brake".
ASME Journal ofTribology, 113, pp. 198-203.
[12] Ha, T., and Childs, D., 1994. "Annular Honeycomb-Stator
Turbulent Gas Seal Analysis, Using a New Friction Factor Model Based on Flat Plate Tests". ASME Journal of
Tribology, 116, pp. 352-360.
[13] Black, H., Allaire, P., and Barrett, L., 1981. "Inlet Flow
Swirl in Short Turbulent Annular Seal Dynamics". In Papers presented at the 9th International Conference, on Fluid
Sealing, The Netherlands, pp. Paper D4, pp.141-152.
[14] Nakabayashi, K., Yamada, Y., and Kishimoto, T., 1982.
"Viscous Frictional Torque in the Flow Between Two Concentric Rotating Rough Cylinders". Journal of Fluid Mechanics, 119, pp. 409-422.
[15] Komotori, K., and Mori, H., 1971. "Leakage Characteristics
of Labyrinth Seals". 5th International Conference on Fluid
Sealing 1971, 5(4), pp. 45-63.
Copyright @ 2008 by ASME
1724
Fly UP