Download A New Analogy Function for the Naphthalene Sublimation

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A NEW ANALOGY FUNCTION FOR THE NAPHTHALENE SUBLIMATION TECHNIQUE
TO MEASURE HEAT TRANSFER COEFFICIENTS ON TURBINE AIRFOILS
M. Haring
Swiss Federal Institute of Technology
E PF L-LTT
Lausanne, Switzerland°
B. Weigand
ABB Power Generation Ltd.
Gas Turbine Development
Baden, Switzerland
Abstract
The naphthalene sublimation technique is based on the
analogy between mass and heat transfer. This analogy
is only fully valid for incompressible flow and if the
Prandtl and Schmidt number are equal. In the present
investigation the energy- and mass transfer equations
were solved simultaneously to establish an analogy
function which allows the calculation of the Nusselt
number from the Sherwood number in function of the
Mach, the Prandtl and the Schmidt number.
For a laminar flow this new analogy function is based on
similarity solutions of the conservation equations for
high Mach number flows. Also a numerical investigation
was conducted to study the influence of the pressure
gradient and the Soret effect as well as varying fluid
properties. For a turbulent flow, a flat plate solution was
established for Pr=1. Energy and mass transfer
equations were additionally solved for a two dimensional
duct flow to study the influence of the Prandtl number
on the analogy function independently.
The resulting analytical and numerical solutions are
shown for various pressure gradients, Prandtl and Mach
numbers. In addition, approximations for the analogy
function are derived. The influence of the present
theory on heat transfer measurements on a turbine
airfoil is shown. The theory is validated against
experimental results in Haring et. al. (1995) showing a
good agreement between the heat transfer coefficients
calculated with the new analogy function and
measurements of actual heat transfer.
Nomenclature
Symbol Unit
Description
c
cp
cf
C
C,
mass concentration (naphthalene)
specific heat at constant pressure
skin friction coefficient
Chapman Rubesin Parameter
see Eq. (1) and (2)
[kgN/kg]
[J/(kg K)]
[
[
[
- ]
- ]
- ]
dh
D
Ec
f, G
fn,gn
g
[m]
[m2/s]
[
-]
[ -]
_]
[-]
[
h
h
H
K
L
I
[m]
[m]
m
[kg/(m2s)]
[J]
[m]
[J]
1-1
M
Nu
n
Pr
Prt
p
Re
Sc
Sc t
Sh
t
T
[
-]
-]
-]
[-]
[
[
1-1
[Pa]
-]
-]
-]
_]
[ ]
[
[
[
[
T05
[K]
[K]
Tr
u,v
[K]
[m/s]
up
w
[
x
[m/s]
yZ
-]
[m]
_]
x
[
a
[W/m2K]
hydraulic diameter
mass diffusion coefficient
Eckert number
non-dimensional stream function
see eq. (39)
non-dimensional stagnation
enthalpy
enthalpy
duct hight
stagnation enthalpy
see eq. (19)
thermal entrance length
mixing length
mass flux
Mach number
Nusselt number
exponent in the analogy function
(eq. 3)
Prandtl number
turbulent Prandtl number
pressure
Reynolds number
Schmidt number
turbulent Schmidt number
Sherwood number
integration variable (eq. 23)
temperature
total temperatur (free stream)
recovery temperature
velocity components in the x,
directions, respectively
mean velocity in the duct
Sutherland exponent (eq. 18)
coordinate axes
y
x =x/ h/(hRe Pr)
heat transfer coefficient
Presented at the International Gas Turbine and Aeroengine Congress and Exposition
Houston, Texas - June 5-8, 1995
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aT[kg/(ms)] thermal diffusion factor
see eq. (31)
[-1
a1, 3 1
mass transfer coefficient
[m/s]
R
[-]
wedge angle (eq. 27)
boundary layer thickness
[m]
5
eddy diffusivity for momentum
Em
[-]
Illingworth variable (x-coordinate)
[ ]
Illingworth variable (y-coordinate)
rl
[ ]
stream function
yr
[ ]
coefficient of heat conductivity
[W/mK]
[-]
eigenvalues
dynamic viscosity
[Pa s]
kinematic viscosity
[m2/s]
v
p
[kg/m3]
density
PN
8
flow over a flat plate Gnielinski (1975) established an
empirical correlation in function of the Prandtl, the
Schmidt and the Reynolds number, which gives values
for n between n=0.45 (ReL=3.e5) and n=0.74
(ReL=1.e7, Sc=2.5, Pr=0.7). All these studies were
conducted at low Mach numbers (incompressible flow).
The objective of the present investigation is to develop
an analogy function, that is also valid for the
compressible flow range and for different Schmidt and
Prandtl numbers.
Analysis
Laminar flow with zero pressure gradient
To establish an analogy function the continuity-, the
[kg/m3]
partial density naphthalene
momentum-, the energy- and the mass transfer
1-1
non-dimensional enthalpy
equations have to be solved for the boundary layer flow.
Continuity equation
Subscripts:
h
i
N
p
w
S
homogeneous solution
refers to initial conditions
refers to naphthalene
particular solution
refers to wall conditions
refers to free stream conditions
ax (Pu) + ya (pv) = 0(4)
Momentum equation in x direction
p u au + v a u _ dp + a µau
ay) = dx ay ay
ax
Introduction
The naphthalene sublimation technique has been used
for many years to study heat transfer phenomena. The
()
5
Momentum equation in y direction
d
Y
method is based on the analogy between the energy
equation and the mass transfer equation and is only fully
(6)
= 0
Energy equation
valid for the incompressible flow range and if the
Schmidt number is equal to the Prandtl number. In most
aHaH
p u ax + p v ay =
PrPr 1 tay ( ^ u `aYI ) } + Pr Lay \µ ay/J
applications this is not the case. Sublimates such as
Naphthalene, Thymol, Jod, Pardibrombenzol, etc. have
a Schmidt number between Sc=2 and 2.5, whereas the
Prandtl number of air is Pr=0.7. Empirical correlations
are commonly applied to correlate the heat and mass
transfer coefficients when the Schmidt number and the
(7)
Mass transfer equation
Prandtl number are not equal. Often power law relations
are used for the Nusselt and the Sherwood number
Nu = C, Rem Prn
\
Sh = C, Rem Scn
(8)
(2)
For a "similar" laminar boundary layer flow this set of
equations can be reduced to ordinary differential
For the same Reynolds number the analogy function
becomes
Nu =Sh (SOn
J
pl uax + v ay =ay (pia )
(1)
equations by introducing the Illingworth Stewardson
variables
y
(3)
rl (x,Y) _ P S u Sp
^s
where the exponent n is an empirical constant. Chen
0
ps
dy(9)
x
and Goldstein (1992) applied a numerical value of
n=1/3, the same accounts for Berg (1991), Brakel and
=^(x)= f Ps us µs d x Cowel (1986), and Kestin and Wood (1970) whereas
(10)
Presser (1968) gives values of n=0.33 for laminar and
n=0.44 for turbulent flow conditions. For the turbulent
2
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00
which allow a separation of the stream function yr, the
velocity u and the stagnation enthalpy H in a product of
"size" times "shape"
iV(x,Y) = G(4) f(ii)
(11)
u(x,Y) = uS(4) f'(il)
(12)
H(x,y) =H8(^) g(TI)•
(13)
To normalize the boundary conditions the nondimensional variables O = (g - g w )/ (1 - g w ) for the
+K$
t(' (f„)Pr del
00(20)
(f „ ) Pr r (f..) Pr(f,f„),drIdt .1
r (f„)Pr del
0J
and the concentration distribution in the boundary layer
is given by:
00
enthalpy and c = (c-c w )/(cg-c w ) for the concentration
shall be introduced.
c = 1 - (f(f„)Sc dTl / J (f„)Sc del
(21)
Introducing (9) and (10) into (5) to (8) leads to
momentum-, energy and mass transfer equations in the
following form:
Once the enthalpy- and the concentration distribution in
the boundary layer are known, the Nusselt and
(Cf")'+ff"+mo u
d s P5 - f' 2 =0
(14)
K(Cf'f")'+(CO')'+PrfEY=0 (15)
(Cc')'+fScc'=0
(16)
us dp
with the associated boundary conditions
1=0: f=f'=0
f'=1
i
0=0
0=1
c=0
c=1
Sherwood number can be calculated. By introducing
the Illingworth Stewardson variables ri and E the analogy
function becomes:
Nu To s TW e ,
Sh - T r- TW c' 1=0
where O' and c' can be derived from eqs. (20) - (21).
(17)
C is the Chapman Rubesin Parameter, which takes the
variation of the fluid properties in account based on the
Sutherland law (0.5 <w < 1):
0
Pr
Pr r (f „ ) -Pr
O'I
=
f )— 1+K j(f')
(f'f")'dildt
1=00
0J
(f „ ) Pr
f
0
P µ
s _ s\w-1 _(
p µ C
JI
w-1
T' 8-)
c ,
(18)
and the parameter K is defined as
,0=0
(23)
dn
00
= (f„)Sc/ j(f')Scdi(24)
The Prandtl and Schmidt number dependency
functions can be approximated numerically, leading to
K _ Pr - 1) ( U52
l
H I-
the analogy function
(19)
1
Nu T - T w Pr 3
(Sc) (1+K{0.249 0.15644log, o (Pr)])
Sh - T r - T W
The set of eqs. 14-17 can be solved analytically for the
simple case of zero pressure gradient and C=1. The
enthalpy distribution in the boundary layer becomes:
00
r (f „ ) Pr d _
(22)
(25)
K can be expressed in function of the Mach number,
the Prandtl number and the wall to gas temperature
ratio:
0
„ ) Pr ( (f „ )
0=1- ^J +K J^ (f
Pr (f, f')' dry dt
OJ
$ (f,,)Pr dTl
K =
2(Pr-1) I1+0.5(K-1) Mg 2 ] (K -1) M5 2
I
(26)
1 + 0.5(K -1) Mgt - TS^ [ (K -1) M3 2 +2]
The result obtained (eqs. (25)-(26)) shows that the
transfer function for Nu/Sh is reduced to the classical
solution Nu/Sh=(Pr/Sc) 113 for the case of Pr=1 for all
3
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0.76
Mach numbers. For Pr#1 the classical solution is
Nu
obtained for M5- 0 because in this case To is equal T r .
Sh
For the case of very high Mach numbers eq. (26) shows
the interesting result of K=2(Pr-1).
lical solution
0.74
auskas (1985)
0.72
Influence of variable fluid properties
0.70
Nu
Sogin (1961)
M=0.2
Sh
M=0.4
0.65
0.68
0.0
0.4 p 0.6
0.8
1.0
Influence of the Soret effect
Mass diffusion is a function not only of concentration
M = 0.6
gradients but also of other potential gradients acting on
the system. At higher Mach numbers the primary
concern is the coupling effects of mass concentration
and temperature gradients. Although pressure gradients
M = 0.8
M = 1.0
0.50
0.5
0.2
Fig. 2 Influence of the pressure gradient (Pr=1,
Sc=2.5, C=1)
0.60.
0.55
ant solution
0.70
M = 0.05
Eckerl(1942)
0.6
0.7
0.8
0.9
and body forces, among others, can also induce mass
diffusion, they are not generally important in the type of
application considered. If the thermo-diffusion or Soret
effect is taken into account eq. (8) becomes:
1.0
w
Fig. 1 Influence of variable fluid properties (Pr=O. 7,
Sc=2.5, dp/dx=0, T /T8=0.9)
p
To study the influence of variable fluid properties, eqs.
(14) to (17) were solved numerically using a shooting
method (Press et. al (1989)). For a zero pressure
(
u ac
ax
D ^c + a p Dc (1-c) aT ai
ac' ay I
ay) ay (P ay ) ayT
(29)
At room temperature the concentration of naphthalene
at the surface is approximately c W , N = 3.e-4 kg N /kg,
gradient and a Chapman Rubesin Parameter of C=1,
eq. (25) was verified. Fig. 1 shows that the influence of
the exponent w in the Sutherland law (eq. 18) on the
therefor c(1-c)can be simplified to c. This reduces
analogy function can be neglected for the temperature
equation (29) to an ordinary differential equation after
applying the Illingworth Stewardson transformation.
variations presented here (T/T5=0.8-1.2).
With the temperature distribution in the boundary layer
approximated by the Crocco Buseman integral
(Schlichting (1965)) eq. (29) becomes:
Influence of the pressure gradient
The above shown results were based on the assumption
of a zero pressure gradient. In the leading edge region
the influence of pressure gradient has to be taken into
account. Fig. 2 shows the numerical solution of eqs. (14)
to (17) for different pressure gradients, presented by the
c"+fScc'-
u52a11c f'f„1.
2
=0.
(30)
cp (T6 + 2 cp [1 - f' 2 l)
p
parameter:
Eq. (30) was solved numerically and the result is shown
=
ug du5
d^ (27)
in Fig. 3. It can be seen that the influence of the Soret
effect depends strongly on the thermo-diffusion factor
aT. For a binary mixture aT can be calculated in function
The results are compared with measurements
conducted by Zukauskas (1985) and Sogin (1961) on a
of the molecular weight, the thermal conductivity and
the temperature (Hirschfelder (1954)). At room
temperature the thermo-diffusion factor for the
pressure gradient can be additionally taken into account
naphthalene air mixture becomes aT=0.16, which
shows that the Soret effect has to be taken into account
cylinder in cross flow and a simplified solution
established by Eckert (1942). The influence of the
by modifying the analytical solution:
Nu Nu
Sh=oh
S )analyt. (1-0.02[)
for Mach numbers higher than M=1.5.
(28)
ll
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component in y direction becomes zero. Introducing
the turbulent Prandtl number Pr t and the non-
0.7
ShNu
dimensional parameters
0.6
Hi _ Hw ;u=u ;Y =h;x -h;p=P
(33)
0
the energy equation can be written as:
0.5
0
1
2
m
e =
R ax a { Pr
+ E Pr
3M 4
Y
C
Fig. 3 Influence of the Soret effect (Pr=0.7, Sc=2.5,
tJ
C=1, dp/dx=0)
L ((1 - P) + C
+
y
Turbulent flow
When the boundary layer is turbulent the energy and
r
`
m+
aH }
(1 Pr l)u ^U Ec
\
t/
Y
with the boundary conditions
the mass transfer equations can not be solved in the
same way as for laminar flow. The solution proposed by
y=0
H'=0
the authors is based=0
on a simple model for the velocity
H
y= 1
distribution and on the von Karman analogy. Energy
and mass transfer equations were additionally solved for
(34)
x=0 H= 1
and the Eckert number defined as Ec=u 0 2/(c p AT). The
velocity distribution u of the fully developed duct flow,
fully developed channel flow to study the influence of
the Prandtl number.
appearing in eq. (34), was calculated from the
momentum equation using the mixing length model for
the eddy viscosity E m +=1 2 jau/ayj/v. The well known
mixing length distribution given by Nikuradse and
Flat plate solution for Pr=1
Van Driest (1951) solved the momentum equation for a
turbulent boundary layer flow on a flat plate. The integral
method was used to obtain surface friction based on a
modified by a Van Driest damping term near the wall was
applied:
velocity profile derived from the mixing length theory.
To take the variation of fluid properties into account the
=h[0.14-0.08(y/h) 2 -0.06(y/h) 4 ] (1 -e - Y 26
Crocco Buseman temperature distribution was
assumed. The skin friction coefficient is given by an
implicit relationship:
)
(35)
For more detailed information the reader is referred to
Cebeci and Bradshaw (1988).
A solution of eq. (34) can be derived by splitting the
problem into a solution for the homogeneous equation
0.242 arc sinal + arc sin[31
a (cf T W /T5) 1i2
= 0.41 + log (Re x cf) - w log 10 (TS)
1Y
and a particular integral. A particular solution can be
found easily since the second term of the right side of
eq. (34) is only a function of y. By introducing H P =H p (y)
(31)
into eq. (34) the following ordinary differential equation
can be derived:
where a1 and R1 are functions of the Mach number and
the gas to wall temperature ratio (Jischa (1982)). Once
the skin friction coefficient is known an analogy function
1H p
between heat and mass transfer can be derived
applying the von Karman analogy to obtain Nusselt and
aY
= Ec u ua
1 _ (1 + S m + ) Pr
aY
Sherwood numbers:
Nu P1 + 5 '
Sh = Sc
For air the Prandtl number is equal to Pr=0.7 and the
turbulent Prandtl number is approximately Prt=0.9
2 { Sc - 1 + ln[(5 Sc + 1)/6]}
leading to a ratio of Pr/Prt=1. This assumption permits
the solution of eq. (36) analytically and results in the
-
1+5'
(36)
(1 + ER,+ Pr
Pr tJ
particular solution
2 { Pr - 1 + ln[(5 Pr + 1)16]}
2
(32)
H p =(1-Pr) Ec 2 .
(37)
Fully developed channel flow
In order to study the effect of different Prandtl numbers
on the analogy function a solution for fully developed
channel flow was derived. In this case the velocity
5
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It should be noted that eq. (37) satisfies the two
(0.6526+0.07065log, 0 M)
boundary conditions for y=0 and y=1 for H. For values
of the Prandtl number where Pr/Prt^1, eq. (36) has to
be solved numerically.
This analogy function was compared with the flat plate
solution, and the results expressed in form of the
exponent n (eq. 3) are shown in Fig. 4. A good
The homogenous part of eq. (34) is given by
Re aH = a
1
+ E r"—+
agreement between the flat plate and the channel flow
1 a H
solution for Pr=0.7 and Sc=2.5 can be seen.
(38)
Prt ay
J y
ax ayy ^(Vr
0.70
n
which is known as the turbulent Graetz-Problem. A
solution can be found by introducing
0.60
channel flow solution
which reduces eq. (38) to the Sturm-Liouville
Eigenvalue problem. This problem was solved
numerically by using a forth-order Runge-Kutta method.
The resulting eigenvalues and constants were
compared with solutions given by Shah and London
0.55
(1978) and a good agreement was found. The general
0.50
solution of the energy equation (34) is the addition of
the homogenous and the particular solution
0.1
00
cn 9n(y) e n+HP
(40)
n=1
seen that the effect of Prandtl number on the analogy
X
2—
cn 12
7 n
x- 8 (1-Pr)Ec
g (1) e
0.9
function becomes important for higher Prandtl
numbers.
-
Nu(x) =
M
numerically. Fig. 5 shows the exponent n in the analogy
function (3) for different Prandtl numbers and it can be
Co
e n
0.7
valid any longer and equation (36) has to be solved
boundary conditions by using the orthogonality of the
eigenfunctions. Finally the Nusselt and the Sherwood
number are derived:
cn g
0.5
For other fluids than air the simplification Pr/Prt=1 is not
The integration constants c n can be found via the
4
0.3
Fig. 4 Exponent n for the turbulent boundary layer
flow (Pr=O.7, Sc=2.5)
a2
n=1
flat plate solution
0.65
(39)
H = fn (X)'gn (y)
H=Hh+Hp=
(43)
1Sc^
0.7
(41)
n
—
Xn
n=1
0.6
Channel flow solution
cn gn'(1) e
Sh(x) = 4
0.5
X n2 X
Flat plate solution
n 1(42)
2
1
C 2 gn'( 1 ) e
n=1
0.4
0
2
4
6
8
10
Pr
^n
Fig. 5 The exponent n in the analogy function for
different Prandtl numbers (Sc=2.5, M=O. 1)
With the Nusselt and Sherwood number being known
an analogy function was established by integrating over
the thermal entrance region (L=20 dh)
The theory is validated against experimental results
(Haring et. al (1995)) comparing the naphthalene
sublimation (mass transfer data) with the liquid crystal
technique (heat transfer data) in compressible flow.
Both techniques were used on a cylinder, on a flat plate
ti
i
Nu _ Nu(x)dx/ Sh(x)dx)
Sh
0
0
L
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and a turbine airfoil and show a good agreement
between the two methods.
exponent n in the analogy function becomes very
important.
Measurements on a turbine airfoil
a [W/m 2 Kl
1200
The naphthalene sublimation technique was applied to
measure the local heat transfer coefficient on a turbine
airfoil. The measurements were conducted in a linear
test facility consisting of five blades. The test facility and
the measurement equipment are described in detail in
Haring et al (1994). To obtain the Mach number
Measurements
TEXSTAN
800
distribution the airfoil surface pressures were measured
at midspan of the airfoil surfaces (both on suction and
pressure sides, Fig.6, right axis) forming the two center
flow passages of the cascade. On the suction side the
velocity rapidly increases. After the strong acceleration
400
the velocity increases only slightly to reach the
maximum at s/L=0.8, from where a decrease can be
observed. On the pressure side the acceleration is
almost uniform. The sublimation losses were measured
in 595 points (85 on the perimeter, 7 heights).
-1.0
OC [W/m 2 K]
1200
-0.2
— a
nbyEq.25and43
n=0.33
Pressure side
1.2
1000
800
0.6
1.0
Suction side
1.0
Fig. 7 Numerical comparison with TEXSTAN (M2=0. 7
Re2=1 '900'000, Tu=2%)
0.8
The experimental data is compared with the numerical
^'
b
b^
600
0.2
s/L
M
1400
.
-0.6
results obtained with the boundary layer code
TEXSTAN (Crawford (1986)), using the Lam Bremhorst
0.6
PTM k-c model (Schmidt and Patankar (1991)). With the
400
0.4
200
0.2
-1.0
-0.6
-0.2
0.2
0.6
exception of the stagnation point region a good
agreement between predicted and measured values
was found (Fig.7).
Conclusions
The naphthalene sublimation technique to measure
heat transfer coefficients is based on the analogy
between the energy and the mass transfer equation.
This analogy is only fully valid in the incompressible flow
range and it the Prandtl and Schmidt number are equal.
1.0
s/L
Pressure side Suction side
The theory described in this paper allows the
application of the sublimation technique in the
Fig. 6 Heat transfer coefficients on a turbine airfoil for
different exponents n in the analogy function (Eq. 3)
(M2=0.7 Re2=1'900'000) and Mach number
distribution.
compressible flow range and for different Prandtl and
Schmidt numbers.
•
For laminar boundary layer flow the energy and
the mass transfer equations were solved based on
similarity solutions. A numerical investigation has been
Fig. 6 (left axis) shows the measured heat transfer
distribution on a turbine airfoil. The present analogy
function for laminar and turbulent boundary layer flow
was used to evaluate the measured mass transfer data
and the result is compared with a constant exponent
n=0.33 in equation (10). In the laminar flow region at low
conducted to study the influence of the pressure
gradient, variable fluid properties and the Soret effect.
• The influence of the pressure gradient and
variable fluid properties on the analogy function is small,
whereas for Mach numbers higher then M=1.5 the
Soret effect has to be taken into account.
Mach number the differences between the two analogy
•
functions are very small whereas at higher Mach number
For the turbulent boundary layer flow a flat plate
solution (for Pr=1) established by van Driest was used
and in the turbulent flow region the influence of the
to obtain an analogy function. The energy and mass
7
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transfer equations were then solved for a fully
developed channel flow to study the influence of the
Prandtl number. For Pr=0.7 and Sc=2.5 a good
agreement was found between the two solutions.
• The theory is validated against experimental
results (Haring et. at (1995)) comparing the
naphthalene sublimation with the liquid crystal
technique in compressible flow. Both techniques were
used on a cylinder, on a flat plate and a turbine airfoil and
show a good agreement between the two methods.
•
Measurements conducted with the naphthalene
sublimation technique on a typical turbine airfoil
(M2=0.7, Re2=1'900'000, Tu=2%) were compared to
data obtained with a numerical boundary layer code
(TEXSTAN) and good agreement was found except for
the stagnation region.
Acknowledgments
The coating of the airfoil and the thickness
measurements, to obtain the heat transfer
measurements shown in Fig. 7 and 8 were conducted
at the Technical University Darmstadt, Dept. of Flight
Propulsion, Darmstadt, Germany, by Dipl.-Ing. J.Richter.
The authors kindly acknowledge the encouragement
and helpful discussions with Dr. Harasgama, ABB Gas
Turbines Development, concerning the development
of the above theory. Additionally the authors would like
to thank ABB Power Generation Ltd. for permission to
publish this paper.
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