Proceedings of IMECE2005
2005 ASME International Mechanical Engineering Congress and Exposition
November 5-11, 2005, Orlando, Florida USA
IMECE2005-82365
NUMERICAL ANALYSIS OF VAPOR BUBBLE GROWTH AND WALL HEAT TRANSFER DURING
FLOW BOILING OF WATER IN A MICROCHANNEL
Abhijit Mukherjee
[email protected]
Rochester Institute of Technology
76, Lomb Memorial Drive
Rochester, NY 14623
Phone- 585.475.5839, Fax- 585.475.7710
ABSTRACT
The present study is performed to analyze the wall heat
transfer mechanisms during growth of a vapor bubble inside a
microchannel. The microchannel is of 200 µm square cross
section and a vapor bubble begins to grow at one of the walls,
with liquid coming in through the channel inlet. The complete
Navier-Stokes equations along with continuity and energy
equations are solved using the SIMPLER method. The liquid
vapor interface is captured using the level set technique. The
bubble grows rapidly due to heat transfer from the walls and
soon turns into a plug filling the entire channel cross section.
The average wall heat transfer at the channel walls is studied
for different values of wall superheat and incoming liquid mass
flux. The results show that the wall heat transfer increases with
wall superheat but is almost unaffected by the liquid flow rate.
The bubble growth is found to be the primary mechanism of
increasing wall heat transfer as it pushes the liquid against the
walls thereby influencing the thermal boundary layer
development.
INTRODUCTION
Microchannel heat sinks with liquid cooling are
extensively used in various applications such as electronic chip
cooling. At sufficiently high wall superheats, flow boiling
takes place through microchannels and is capable of removing
very high wall heat fluxes. During flow boiling, bubbles
nucleate on the microchannel walls and eventually grow big
enough to fill up the entire channel cross-section. The wall
heat transfer from the channel wall to the liquid is affected by
the bubble nucleation and growth inside the channels.
Kandlikar (2004) observed that at low Reynolds number,
the convective boiling is diminished in microchannels and
nucleate boiling plays a major role with periodic flow of liquid
and vapor slugs in rapid succession. He compared the transient
Satish G. Kandlikar
[email protected]
Rochester Institute of Technology
76, Lomb Memorial Drive
Rochester, NY 14623
Phone- 585.475.6728, Fax- 585.475.7710
conduction under the approaching rewetting liquid slug and the
heat transfer in the evaporating meniscus region of the liquidvapor interfaces in the contact line region to the nucleate
boiling phenomenon.
Jacobi and Thome (2002) argued that the experimental
studies that show heat-flux dependence of the convection
coefficient along with relative independence from quality and
mass flux cannot be ascribed only to the nucleate boiling
mechanism. They developed a hypothesis that microchannel
evaporation is thin-film dominated.
Thome et al. and Dupont et al. (2004) developed threezone flow boiling heat transfer model to describe evaporation
of elongated bubbles in a microchannel and compared the time
averaged local heat transfer coefficient with several
independent experimental studies. Their numerical model
consisted of sequential and cyclic passage of a) a liquid slug, b)
an evaporating elongated bubble with a thin liquid film around
it and c) a vapor slug, through a microchannel. The model had
three adjustable parameters, the initial thickness of the liquid
film, the minimum thickness of the liquid film at dryout, and
the bubble departure frequency. The comparison of the results
with experimental data indicated limited success of the model.
Mukherjee and Kandlikar (2005) numerically studied
growth of a vapor bubble inside a microchannel during flow
boiling. A bubble was placed at the center of the microchannel
surrounded by superheated liquid. The incoming liquid
superheat and the flow rate were varied. The results indicated
that the bubble growth was strongly influenced by the liquid
superheat whereas the effect of the liquid flow rate was
comparatively negligible. The effect of gravity was also found
to be negligible on bubble growth.
The various numerical and experimental studies indicate
that wall heat transfer during flow boiling depends strongly on
the wall heat flux but weakly on the mass flux. However, there
1
Copyright © 2005 by ASME
z
∂
∂z
non-dimensional quantity
vector quantity
NUMERICAL MODEL
Computational Domain
Figure 1 shows the typical computational domain. The
domain is 3.96x0.99x0.99 non-dimensional units in size.
Cartesian coordinates are used with uniform grid.
North Wall
---->
1
--------------> Outlet
0.75
0.5
<--
0.25
Inlet -------------->
<--- Bubble at wall
0
-0.5
-0.25
Z*
2
0
0.25
1
0.5
0
4
X*
--->
NOMENCLATURE
A
wall area
Cp
specific heat at constant pressure
d
grid spacing
g
gravity vector
H
Heaviside function
h
heat transfer coefficient
latent heat of evaporation
hfg
k
thermal conductivity
L
length of bubble
L1
upstream bubble cap location
L2
downstream bubble cap location
length scale
l0
m
mass transfer rate at interface
ms
milliseconds
Nu
Nusselt number
p
pressure
Re
Reynolds number
SH
superheat
T
temperature
∆T
temperature difference, Tw-Tsat
t
time
time scale
t0
u
x direction velocity
velocity scale
u0
v
y direction velocity
w
z direction velocity
x
distance in x direction
y
distance in y direction
z
distance in z direction
βT
coefficient of thermal expansion
κ
interfacial curvature
µ
dynamic viscosity
ν
kinematic viscosity
ρ
density
σ
surface tension
τ
time period
φ
level set function
ϕ
contact angle
Subscripts
evp
evaporation
in
inlet
l
liquid
sat
saturation
v
vapor
w
wall
∂
x
∂x
∂
y
∂y
Superscripts
*
→
Y*
is no general agreement on whether the dominant wall heat
transfer mechanism is nucleate boiling or thin film evaporation.
In the present study, complete numerical simulation of a
growing vapor bubble inside a microchannel during flow
boiling is being carried out. The objective is to analyze and
explain the effect of wall heat flux and mass flux on bubble
growth and the corresponding wall heat transfer.
-- Top Wall
3
South Wall
Fig 1 – Computational domain
The liquid enters the domain at x* = 0 and leaves the
domain at x* = 3.96. To take advantage of symmetry and
reduce computation time, a nucleating cavity is placed
equidistant from the walls in the x-y plane. The two horizontal
walls in the x-z planes are named as South Wall (y* = 0) and
North Wall (y* = 0.99). The vertical wall in the x-y plane is
named the Top Wall (z* = 0.495).
The number of computational cells in the domain is
320x80x40, i.e. 80 grids are used per 0.99l0. This grid size is
chosen from previous work of Mukherjee and Kandlikar (2005)
to minimize numerical error and optimize computation time.
Variable time step is used which varied typically between 1e-4
to 1e-5. Negligible change in the results is observed when
calculations are repeated with half the time step, which ensured
that calculations are time step independent.
Method
The complete incompressible Navier-Stokes equations are
solved using the SIMPLER method [Patankar, 1980], which
stands for Semi-Implicit Method for Pressure-Linked Equations
Revised. The continuity equation is turned into an equation for
the pressure correction. A pressure field is extracted from the
given velocity field. At each iteration, the velocities are
corrected using velocity-correction formulas.
The
computations proceed to convergence via a series of continuity
satisfying velocity fields. The algebraic equations are solved
using the line-by-line technique, which uses TDMA (TriDiagonal Matrix Algorithm) as the basic unit. The speed of
convergence of the line-by-line technique is further increased
by supplementing it with the block-correction procedure
[Patankar, 1981]. The multi-grid technique is employed to
solve the pressure equations.
Sussman et al. (1994) developed a level set approach
where the interface was captured implicitly as the zero level set
of a smooth function. The level set function was typically a
smooth function, denoted as φ . This formulation eliminated the
problems of adding/subtracting points to a moving grid and
automatically took care of merging and breaking of the
2
Copyright © 2005 by ASME
interface. Furthermore, the level set formulation generalized
easily to three dimensions. The present analysis is done using
this level set technique.
The liquid vapor interface is identified as the zero level set
of a smooth distance function φ . The level set function φ is
negative inside the bubble and positive outside the bubble. The
interface is located by solving the level set equation. A fifth
order WENO (Weighted, Essentially Non-Oscillatory) scheme
is used for left sided and right sided discretization of φ [Fedkiw
et al., 1998]. While φ is initially a distance function, it will not
remain so after solving the level set equation. Maintaining φ as
a distance function is essential for providing the interface with
a width fixed in time. This is achieved by reinitialization of φ .
A modification of Godunov's method is used to determine the
upwind directions. The reinitialization equation is solved in
fictitious time after each fully complete time step. With
d
∆τ =
, ten τ steps are taken with a third order TVD (Total
2u0
Variation Diminishing) Runge Kutta method.
Governing Equations
Momentum equation r
r
r
∂u r r
ρ ( + u .∇u ) = −∇p + ρg − ρβT (T − Tsat ) g
∂t
r
r
− σκ∇H + ∇.µ∇u + ∇.µ∇u T
H is the Heaviside function given by H = 1 if φ ≥ + 1.5d
H = 0 if φ ≤ −1.5d
(9)
H = 0.5 + φ /(3d ) + sin[ 2πφ /(3d )] /(2π ) if | φ | ≤ 1.5d
where d is the grid spacing
Since the vapor is assumed to remain at saturation temperature,
the thermal conductivity is given by –
k = k l H −1
(10)
The level set equation is solved as r r
∂φ
+ (u + uevp ).∇φ = 0
∂t
(11)
After every time step the level-set function φ , is reinitialized
as∂φ
= S (φ0 )(1− | ∇φ |)u0
∂t
(12)
φ ( x,0) = φ0 ( x)
S is the sign function which is calculated as (1)
S (φ0 ) =
φ0
(13)
φ0 2 + d 2
Energy equation ∂T r
+ u.∇T ) = ∇.k∇T for φ > 0
∂t
T = Tsat for φ ≤ 0
ρC p (
Continuity equation r
r m
∇.u = 2 .∇ρ
ρ
(2)
(3)
The curvature of the interface -
κ (φ ) = ∇.(
∇φ
)
| ∇φ |
(4)
The mass flux of liquid evaporating at the interface r k ∇T
m= l
h fg
(5)
The vapor velocity at the interface due to evaporation –
r
r
k ∇T
m
(6)
uevp =
= l
ρ v ρ v h fg
To prevent instabilities at the interface, the density and
viscosity are defined as -
ρ = ρv + (ρl − ρv )H
(7)
µ = µ v + ( µl − µ v ) H
(8)
Scaling Factors
The governing equations are made non-dimensional using a
length scale and a time scale. The length scale l0 given by the
channel width/height and is equal to 200 microns. Thus for
water at 100o C, and Re = 100, the velocity scale u0 is
calculated as 0.146 m/s. The corresponding time scale t0 is
1.373 ms.
The non-dimensional temperature is defined as
T − Tsat
*
(14)
T =
Tw − Tsat
The Nusselt number (Nu) is calculated based on the areaaveraged heat transfer coefficient ( h ) at the wall given by,
1A
(15)
h = ∫ hdA
A0
where A is the wall area and h is obtained from
∂T
− kl
| wall
∂y
for horizontal walls
(16a)
h=
Tw − Tsat
∂T
| wall
∂z
and h =
for vertical walls
Tw − Tsat
The wall Nusselt number is defined as,
hl
Nu = 0
kl
− kl
3
(16b)
(17)
Copyright © 2005 by ASME
The bubble is also seen to form vapor patches on the vertical
walls at 0.311 ms.
0.000 ms
1
0.5
Y*
0.75
0.25
0
-0.5
-0.25
Z*
Initial Conditions
The bubble is placed at x* = 0.99, y* = 0 and z* = 0, with
0.1l0 radius in the domain shown in Fig. 1. All velocities in the
internal grid points are set to zero. The liquid inlet temperature
is set to 102o C and the wall temperature is set to the specified
superheat (T* = 1). The vapor inside the bubble is set to
saturation temperature of 100o C (T* = 0). The initial liquid
temperature inside the domain is set equal to the inlet liquid
temperature of 102oC. All physical properties are taken at 100o
C. The contact angle at the walls is specified as 40o, which is
obtained from the experimental data of Balasubramanian and
Kandlikar (2004).
4
0
0.25
0.5
3
2
1
X*
0
0.136 ms
1
Boundary Conditions
The boundary conditions are as following –
0.5
Y*
0.75
0.25
At the inlet (x* = 0) :-
0
(18)
Z*
u = u0; v = w = 0; T = Tin; φ x = 0
-0.5
-0.25
Constant inlet flow velocity has been specified in the
numerical calculations. In parallel microchannel heat
exchangers constant inlet flow velocity is necessary to
maintain stable operating conditions, which can be
achieved using flow restrictions at the inlet, (Kandlikar et
al. 2005.)
• At the outlet (x* = 3.96) :-
•
3
2
1
X*
0
0.271 ms
<-------- ------
-------- ---------
--- --L2 ---------
---------------
-->
1
0.75
0.5
------ ->
<-------- L1
0.25
0
-0.5
-0.25
Z*
ux = vx = wx = Tx = 0; φ x = 0
4
0
0.25
0.5
Y*
•
(19)
4
0
0.25
0.5
3
2
1
X*
0
0.311 ms
At the plane of symmetry (z* = 0) :-
1
•
*
(20)
0.5
0.25
*
At the walls (y = 0, y = 0.99) :-
0
-0.5
-0.25
Z*
u = v = w = 0; T = Tw; φ y = − cos ϕ
(21)
4
0
0.25
0.5
where ϕ is the contact angle
•
Y*
uz = vz = w = Tz = 0; φ z = 0
0.75
3
2
1
X*
0
Fig 2 – Bubble growth, Tw = 8K, Re = 100
*
At the wall (z = 0.495):u = v = w = 0; T = Tw; φ z = − cosϕ
Re - 100
(22)
1
0.9
Bubble Cap Location (mm)
RESULTS AND DISCUSSION
In the present study a bubble is placed on the microchannel
wall during flow boiling of water with constant incoming liquid
superheat of 2K. The wall superheat is varied as 5K, 8K and
10K. The Reynolds no. is varied as 50, 100 and 200, which
correspond to inlet liquid velocities of 0.073 m/s, 0.146 m/s and
0.29 m/s respectively. As the bubble grows inside the
microchannel, the bubble growth rate and the wall heat transfer
are studied for the different values of wall superheat and
Reynolds number.
Figure 2 shows the growth of a vapor bubble inside the
microchannel with Re = 100 and wall superheat of 8K. The
bubble grows predominantly in the direction of flow, due heat
transfer from the wall and the surrounding superheated liquid.
The time in each frame is indicated at the upper right corner.
SH - 5K
SH - 8K
SH - 10K
0.8
0.7
<-------L2
0.6
0.5
0.4
0.3
<--- L1
0.2
0.1
0
0
0.1
0.2
Time(ms)
0.3
0.4
Fig 3 – Effect of wall superheat on bubble growth
4
Copyright © 2005 by ASME
Figure 3 shows the effect of the wall superheat on the
bubble growth inside the microchannel with Re = 100. The
plot shows the bubble cap locations from the microchannel inlet
as a function of time. The upstream and downstream bubble
cap locations, L1 and L2 respectively, are measured from the
channel inlet as indicated in the third frame in Fig. 2. The
upstream cap distance L1 stays almost constant initially and
increases slightly as the bubble fills up the channel length. The
downstream cap distance L2 increases linearly at the beginning
of the bubble growth but afterwards increases exponentially as
the bubble turns into a vapor plug. The results indicate that the
bubble grows faster with increase in wall superheat.
Re - 100
50
SH - 5K
SH - 8K
SH - 10K
Nu North
40
30
20
10
SH - 8K
0
Re - 50
Re - 100
Re - 200
0.8
0.1
0.2
Time(ms)
0.3
SH - 5K
SH - 8K
SH - 10K
40
<-------L1
0.2
0
0.1
0.2
Time(ms)
0.3
0.4
50
0.4
0
0
Re - 100
<-------L2
0.6
Nu South
Bubble Cap Location (mm)
1
0.4
30
20
10
Fig 4 – Effect of Re on bubble growth
0
0
0.1
0.2
Time(ms)
0.3
0.4
Re - 100
50
SH - 5K
SH - 8K
SH - 10K
40
Nu Top
Figure 4 plots the bubble end cap locations as a function of
time for different Reynolds no. with a fixed wall superheat of
8K. It is seen that initially both L1 and L2 increase slightly
with higher Re as the higher mass flux pushes the bubble
forward. However, at the later stages of bubble growth after
0.25 ms, L2 increases faster with decrease in the Re indicating
that bubble growth rate decreases with increase in Re.
Comparing figures 3 and 4, it can also be seen that the effect of
Reynolds number on the bubble growth is insignificant as
compared to the effect of wall superheat.
Figure 5 plots the wall heat transfer at the two horizontal
and one vertical wall (since the bubble growth is symmetrical
about the central vertical plane) of the microchannel as a
function of time. The wall heat transfer in each case is very
high initially as the incoming liquid contacts the heated wall.
The wall heat transfer decreases with time as the thermal
boundary begins to develop. At all the microchannel walls, it
can be seen that the heat transfer improves with increase in the
wall superheat. This is because the bubble grows faster with
increase in the wall superheat, pushing the liquid towards the
wall on the opposite side, thereby thwarting the thermal
boundary layer development.
At the North Wall, the wall heat transfer decreases initially
but can be seen to increase at the later stages of bubble growth,
as the bubble changes to vapor plug and elongates. The vapor
plug elongates and in a sweeping action presses the already
developed thermal boundary layer at its downstream towards
the wall thereby causing increased wall heat transfer.
30
20
10
0
0
0.1
0.2
Time(ms)
0.3
0.4
Fig 5 – Effect of wall superheat on heat transfer
At the South Wall, the wall heat transfer decreases
continuously as the bubble base elongates, thereby diminishing
the liquid contact with the wall. Similarly at the Top Wall, the
wall heat transfer decreases initially due to thickening of the
thermal boundary layer, but decreases more rapidly at the later
stages due to formation of vapor patches as shown in Fig. 2.
5
Copyright © 2005 by ASME
SH - 8K
50
Re - 50
Re - 100
Re - 200
Nu North
40
seen that the rate of change of L2 is initially about 1 m/s which
increases to 2 m/s at later stages of bubble growth, whereas the
specified incoming liquid velocity is only around 0.1 m/s. Thus
the boundary layer development and the wall heat transfer are
primarily influenced by the bubble growth and not by the
incoming liquid mass flux.
T*:
30
SH - 5K
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.5
0.5
Y
0.4
20
X
Z
10
0.5
0
0
0
0.1
0.2
Time(ms)
0.3
1
2
3
4
0.4
SH - 8K
0.3
0.3
SH - 8K
0.2
Y
50
X
Z
Re - 50
Re - 100
Re - 200
40
0.3
0.3
Nu South
0
1
2
3
4
30
SH - 10K
0.3
20
Y
Z
0 .2
0 .2
X
10
0.3
0
0
0.1
0.2
Time(ms)
0.3
0.4
50
Re - 50
Re - 100
Re - 200
Nu Top
40
30
20
10
0
0.1
0.2
Time(ms)
1
2
3
4
Fig 7 – Temperature field at the central vertical plane
at 0.3 ms, Re = 100
SH - 8K
0
0
0.3
0.4
Fig 6 – Effect of Re on heat transfer
Figure 6 shows the effect of Reynolds number on the wall
heat transfer with constant wall superheat of 8K. Compared to
Fig. 5 it can be seen that Reynolds number, or the mass flux has
little effect on the wall heat transfer. This is because the
velocities associated with bubble growth are much higher
compared to the incoming liquid velocity. From Fig 4 it can be
Figure 6 also shows that the heat transfer at the North Wall
increases slightly after 0.3 ms with a decrease in Re. This is
because the bubble growth rate increases with a decrease in Re,
thereby pushing the thermal boundary layer more effectively.
However, as the bubble growth rate increases, the formation of
vapor patch at the walls also takes place more rapidly. This
explains the gradual decrease in wall heat transfer with
decrease in Re at the South and Top Walls.
Figure 7 compares the temperature field around the vapor
bubbles at the central x-y vertical plane inside the
microchannel, for different wall superheats. Non-dimensional
temperature contours are plotted between 0 and 1 with intervals
of 0.1. The plot shows the thermal boundary layers at the
North and South Walls. The wall heat transfer increases with
wall superheat at the North Wall due to two reasons. The faster
growing bubble pushes the thermal boundary layer thinner and
at the same time the bigger bubble affects a larger area along
the bubble length. At the South wall the wall heat transfer
decreases with increase in superheat due to larger bubble base
at the wall.
6
Copyright © 2005 by ASME
Summarizing the above results it can be said that the wall
heat transfer inside the microchannel is affected by the
approaching liquid vapor interface of a growing bubble. With
increase in wall superheat, bubble growth rate increases
pushing the liquid more rapidly against the wall and thereby
increasing the wall heat transfer. This explains the findings of
previous researchers that heat transfer coefficient during flow
boiling through microchannels is dependent on the wall heat
flux. However, increase in bubble growth rate at higher heat
fluxes also accelerates vapor patch formation at the walls which
is detrimental to wall heat transfer and may lead to early CHF
at higher heat fluxes.
The incoming liquid mass flux decreases the bubble
growth rate thereby hindering the nucleate boiling mechanism
and decreasing wall heat transfer. On the other hand increase
in incoming liquid mass flux delays vapor patch formation
thereby increasing the wall heat transfer. Thus as a net effect,
the incoming liquid mass flux does not significantly affect the
wall heat transfer, as seen from Fig. 6.
The liquid velocity generated due to bubble growth is
much higher compared to the incoming liquid velocity and thus
the nucleate boiling plays a major role during flow boiling in
microchannels.
CONCLUSIONS
1. Numerical simulation is carried out for a growing vapor
bubble during flow boiling of water inside a microchannel.
2.
The wall heat transfer is found to increase with wall
superheat and the bubble growth rate.
3.
The wall heat transfer is found to be unaffected by the
changes in the incoming liquid flow rate.
4.
The wall heat transfer is primarily increased due to the
motion of the evaporating liquid-vapor interface as it
grows and pushes against the opposite microchannel walls
and prevents the thermal boundary layer development.
ACKNOWLEDGMENTS
The work was conducted in the Thermal Analysis and
Microfluidics Laboratory at RIT.
REFERENCES
Balasubramanian, P., and Kandlikar, S. G., 2004,
Experimental Study of Flow Patterns, Pressure Drop and Flow
Instabilities in Parallel Rectangular Minichannels, Proc. of 2nd
International Conference on Microchannels and Minichannels
2004, Rochester, pp. 475-481, ICMM2004-2371.
Dupont V., Thome J. R., and Jacobi, A. M., 2004, Heat
Transfer Model for Evaporation in Microchannels. Part II:
Comparison with Database, International Journal of Heat and
Mass Transfer, 47, pp. 3387- 3401.
Fedkiw, R. P., Aslam, T., Merriman, B., and Osher, S.,
1998, A Non-Oscillatory Eulerian Approach to Interfaces in
Multimaterial Flows (The Ghost Fluid Method), Department of
Mathematics, UCLA, CAM Report 98-17, Los Angeles.
Jacobi, A. M. and Thome, J. R., 2002, Heat Transfer
Model for Evaporation of Elongated Bubble Flows in
Microchannels, Journal of Heat Transfer, 124, pp. 1131-1136.
Kandlikar, S. G., 2004, Heat Transfer Mechanisms during
Flow Boiling in Microchannels, Journal of Heat Transfer, 126,
pp. 8-16.
Kandlikar, S. G., Willistein, D. A., and Borrelli, J., 2005,
Experimental Evaluation of Pressure Drop Elements and
Fabricated Nucleation Sites for Stabilizing Flow Boiling in
Minichannels and Microchannels, Proc. of 3rd International
Conference on Microchannels and Minichannels 2005,
Totonto, Canada, ICMM2005-75197.
Mukherjee, A., and Kandlikar, S. G., 2005, Numerical
Study of Growth of a Vapor Bubble during Flow Boiling of
Water in a Microchannel, Journal of Microfluidics and
Nanofluidics, 1, no. 2, pp. 137-145.
Patankar, S. V., 1980, Numerical Heat Transfer and Fluid
Flow, Hemisphere Publishing Company, Washington D.C.
Patankar, S. V., 1981, A Calculation Procedure for TwoDimensional Elliptic Situations, Numerical Heat Transfer, 4,
pp. 409-425.
Sussman, M., Smereka, P., and Osher S., 1994, A Level
Set Approach for Computing Solutions to Incompressible TwoPhase Flow, Journal of Computational Physics, 114, pp. 146159.
Thome J. R., Dupont V., and Jacobi, A. M., 2004, Heat
Transfer Model for Evaporation in Microchannels. Part I:
Presentation of the Model, International Journal of Heat and
Mass Transfer, 47, pp. 3375- 3385.
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