The vapor flow patterns in heat pipes are examined during the start-up transient phase. The vapor core is modelled as a channel flow using a two dimensional compressible flow model. A nonlinear filtering technique is used as a post process to eliminate the non-physical oscillations of the flow variables. For high-input heat flux, multiple shock reflections are observed in the evaporation region. The reflections cause a reverse flow in the evaporation and circulations in the adiabatic region. Furthermore, each shock reflection causes a significant increase in the local pressure and a large pressure drop along the heat pipe

Catton, I.

Ghoniem, N, M.

Issacci, F.

English

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NA SA-C_-2 J_ _6 7
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/'1. _._ /_.
VAPOR FLOW PATTERNS DURING A START-UP
TRANSIENT IN HEAT PIPES
F. Isucd, N. M. Ghonlem, and L Cmtton
Department of Mechanical, Aerospace end Nuclear Engineering
University of California, Los Angeles
Los Angeles, California
ABSTRACT
The vapor flow patterns in heat pipes are
examined during the start-up transient phase. The
vapor core is modelled as a channel flow using a two
dimensional compressible flow model. A nonlinear
filtering technique is used as a post process to
eliminate the non-physical oscillations of the flow
variables. For high-input heat flux. multiple shock
reflections are observed in the evaporation region.
The reflections cause a reverse flow in the
evaporation and circulations in the adiabatic region.
Furthermore. each shock reflection causes a
significant increase in the local pressure and a large
pressure drop along the heat pipe.
NOMENCLATURE
b channel width
Cp specific heat
hls latent heat
k thermal conductivity
L heat pipe total length
L a adiabatic section length
L c condenser length
L e evaporator length
m" mass flux
Nx number of grid points in x-direction
Ny number of grid points in y-direction
p pressure
q" input heat flux
R gas constant
Re Reynolds number
T Temperature
t time
u axial velocity
v vertical velocity
x axial coordinate
y vertical coordinate
,5 liquid layer thickness
viscosity
p density
¢ general dependent variable
O r referenced value of 0
Subscripts
c condenser
eff effective
o initial value
sat saturation
INTRODUCTION
The transient behavior of vapor flow dunng the
start-up phase of heat pipe operation has been
analyzed in order to examine the flow patterns for
high-input heat fluxes. Start-up is a transient
process through which the heat pipe starts its
operation from a static condition and arrives at
steady-state operation. During a start-up transient.
the working fluid of a heat pipe may initially be
very cold or frozen in the wick. Here. the liquid is
assumed to be very cold and the vapor, which isJ
; thermodynamically in equilibrium with the liquid, is
at very low pressure. The vapor core response ,o a
sudden high-input heat flux is the primary, focus of
this work.
41
https://ntrs.nasa.gov/search.jsp?R=19980027096 2020-06-16T00:51:16+00:00Z
The past two-dimensLonal (2D) studies are
limited to incompressible flows (Tien and Rohani.
1974; Faghri. 1986; and Pcterson and Tien, 1987)
and comwessible flow studies ate limited to I D (e.g.,
Chen and Faghri, 1989). At UCLA. the study of
dynamic behavior of heat pipes began by analyzing
the liquid and vapor phases separately (Roche,
1988; Isszcci et at., 1988). The objective was to
develop very detailed computer codes for each
phase and then to combine them in z comprehensive
code. Roche (1988) developed a numerical code for
the liquid phase at the st•n-up _'ansientmode by
assuming the vapor phase at a constant temperature
lind pressure. He used • kinerlctheory approach to
model the phase change at the liquid-vapor
interface. The vapor dynamics of heat pipes was
studied by Issacciet al. (1989) using a 2D transient
model with the SIMPLER method (Patankar, 19g0).
The primary advantage of SIMPLER is the staggered
grid which makes the numerical scheme very stable
and, consequently, the computational time very low.
Although this method yielded interesting results,
the resultingcomputational code was limited to low
compressibility flows. However, in the start-up
mode of a heat pipe, there are regions of high as
well as low compressibility.
A filteringtechnique, based on the concept
introduced by Engquist et _. (1989), was used in a
study of ID compressible vapor dynamics by Issacci
et al. (1990a). It was shown that the centered-
difference scheme (CDS) used with nonlinear
filteringyields a second order, stablesolutionin cell
Reynolds problems and is able to capture a shock
without oscillations. The same method was also used
in a 2D analysis of vapor dynamics in heat pipes by
Issacci et al. (1990b). The shock reflectionin the
evaporation region was observed to affect the flow
pattern and the pressure drop along the pipe. This
study is continued here for high-input heat fluxes.
The corresponding Mach number is about one and
the flow patterns show a strong flow circulation in
the evaporation and adiabatic regions and,
consequently, a large pressure drop along the heat
pipe.
GOVERNING EQUATIONS
Vapor flow in the heat pipe core is modeled as
channel flow, as is shown in Fig. I. The bottom
boundary of the channel is s thin porous medium
which contains the liquid. The input heat flux to the
evaporator and the temperature of the outer surface
of the condens-.r are sp_.cifie:.. The planar side
walls arc assumed adiabatic,
The equations governing the vapor flow are the
continuity, momentum, and energy equations, which
are time dependent, viscous, and compressible. An
equation of state (EOS) is used to relate pressure,
b
Y
0
x VAPOR 1_
x=O x=L
Fig. I. Vapor flow model in a he.at pipe
density, and temperature within the vapor core.
These equations in Cartesian coordinates (x,y) are
Continuity
8(pu) + 8(p,)
+-'-Z- =o (1)
x-Momentum
8(.ou) + 8(pu,) + 8(puv) _ 8
y-Momentum
8
8(9v) + d(PUv) + _(Pvv) °_ + _.I/_ _.1 + -_I/_ --_/
i
Energy
C_O(pT) +O(puT)+O(ovT)] ap + ap +v_ + Oli Sr_T
_ _2
EDS
p = .oRT (5)
Initial and Boundary Condition,_
The working liquid is assumed initially to be at
a very low temperature (close to the freezing point).
Further, there is no input heal and the stagnant
vapor is in thermodynamic equilibrium with the
42
liquid. Theboundariesof the vapor core are shown
in Fig. I. A no-slipcondition for the velocityand an
adiabatic condition for the temperature are assumed
on the side walls, i.e., at x = 0 and x = L,
u=0 , v=0 , a/'l&=0 (6)
On the center line the symmetry condition implies,
aty = b.
,/u/6,=o , v=0 , ar/_=0 (7)
The boundary conditions at the liquid-vapor
interface are the challenging ones. The liquid flow is
assumed to be in a porous medium of thickness, B,
which is much smaller than the vapor core
thickness, b. The axial velocity is assumed zero on
this boundary, i.e., at y = 0,
u = 0 (8)
To assign boundary conditions for the temperature
and vertical velocity, the liquid-vapor interface is
divided into three regions. In the evaporation zone
the input flux. ¢". is a given parameter and the
input flow is approximated at y = O and O < x < Le
by
pv=m"-¢'lh/8(T) , T = Tsar(P) (9)
where hfg is the heat of vaporization and m" is the
input mass flux. Conduction in the liquid layer is
neglected. It should be noted that this boundary
condition ignores the thermal characteristics of the
heat pipe wall and the working fluid-wick layer. The
temperature is assumed to be the saturation
temperature of the liquid corresponding to the
interface pressure, Tsar(p). In the adiabatic zone,
the boundary conditions at y = 0 and Le<x<(l. e +
La) are
v =0 . aT/o_=0 (I0)
In the condensation zone the temperature of the
outer surface, Tc, is given. By equating the heat of
evaporation to the heat conduction in the liquid
layer, the outflow from this region is approximated
at y =Oand (Le + Lc) <z._.L, by
(II)
where keff is the effective conductivity of the liquid
layer and the temperature is assumed to be the
saturation temperature corresponding to the
4,3
pressure at the interface. This neglects transient
conduction in the fluid layer which may be an
important contribution to the process.
SOLUTION METHOD
The five governing equations, (i)-(5), with the
initial and boundary conditions given by equations
(6)-(I1), have been solved numerically for the five
variables: p, pu, pv, T, and p. A finite-difference
method has been used with backward Euler
discretization in time and centered differences in
space. Dealing with a wide range of input heat
fluxes, the method deals effectively with the cell
Reynolds-number problem as well as shock-
capturing complexity. Discretization of the advective
terms in this problem plays a significant role. The
centered-difference scheme (CDS) is ill-behaved for
grid Reynolds numbers larger than two because the
coefficients of the difference equations are not
necessarily positive. As a consequence, solutions to
the difference equations may not be unique,
resulting in nonphysical spatial oscillations. In a
study of the ID forms of the governing equations,
(1)-(5), by Issacci et at. (1990a), standard methods
such as upwinding and power law and a more recent
scheme, CONDIF, were examined. It was shown that
these schemes are well.behaved but that they cause
oscillations and overshoot at high compressibility
and, except for CONDIF, they cause high numerical
diffusion at low compressibility as well. The CONDIF
scheme was used by lssacci et al. (1990c) in a study
of the start-up u'ansient process in heat pipes. It
was shown that for input heat fluxes > I kW/m 2 in
an Na-filled heat pipe, a shock is created in the
evaporation region. Therefore, the CONDIF scheme
was found to be unsuitable for shock capturing.
In the present work the same technique used in
the study by Issacci et at. (1990a,b) has been used
to solve the 2D form of the governing equations for
a compressible vapor flow. The governing equations
are nonlinear and coupled. An iterative method is
used to solve each equation separately, i.e.,
equauons (1)-(5) for p, pu, pv, T, and p, respectively.
The SOR method is used to solve the differenced
equations. Iteration on the equations were stopped
after convergence of all the v_iables. This iteration
method required the least storage and fewest
calculations.
RESULTS
For the results shown in this section, the winking
fluid is liquid sodium at. initially, Po = 10"2 aern and
To = g00 K. The geometry dimensions are b = 5 cm.
Le = La = l.c = b. The caiculational grid for each
computation was chosen by inspection of the
calculated L2-norm error.The L2-norm Js defined as
. F'  -F l'l°
(12)
where Nz and Ny are numbers of grid points in x-
and y-directions,respectively.Here, qtisone of the
dependent variables (p, pu, pv, T, or P) and @r isthe
calculated 1)on the finestmesh of calculations.
Figure 2 shows the flow patterns in the vapor
core for m high-input heat flux. q_' =106W/m 2. The
Reynolds number based on the vapor thickness is
Re = 100 and the computations were done for an
optimized grid of 41 x 121. The transient
development of the vertical mass flux, pv, at
differenttimes is shown in Fig. 2. At t : 0, the vapor
is stagnant. Applying the input heat flux,
evaporation takes place in the evaporator 0 <x/b _l.
Since the input heat flux is high, a shock wave is
created (Fig. 2a). The vapor flow develops above
the evaporator and in the adiabatic region and the
shock travels above the evaporator until it hits the
upper boundary (Fig. 2b) where the vertical mass
flux is blocked, pv = 0. At this point, the vapor is
compressed and the vapor pressure increases which
causes a positive vertical pressure gradient and the
shock reflects back (Fig. 2c). The figure also shows
reverse flow in the top left corner which is caused
by shock reflection. When the reflected shock
reaches the lower boundary (Fig. 2d) and the vapor
is compressed within the evaporation region, the
vertical pressure gradient is reduced and the inflow
mass flux tries again to fill the evaporation region
(Fig. 2e). As the vapor fills the evaporation region,
another shock reflection occurs. Each reflection of
the shock wave causes a significant increase in the
local pressure and, consequently, a large pressure
drop along the heat pipe.
The same multiple shock reflection and refill of
vapor in the evaporator region was also observed
with lower input heat flux, q_' el0 s W/m 2 (lssacci
el. al, 1990b). However, at high-input heat flux (q,"
=106W/m2), the flow pattern in the adiabatic
region, 1 <x/b < 2, is significantly different from
that for lower heat fluxes. Figure 2d shows that
when the shock reflection reaches the lower
boundary of the evaporator, circulation is initiated
in the adiabatic region. This circulation ceases when
the vapor flow refills the evaporator region (Fig. 2e).
However, the second reflection will cause a stronger
circulation in the adiabatic region, as shown in Fig.
2f. Figures 2g through 2t demonstrate the
subsequent sets of reflection and refill of the vapor
flow with circulation strength increasing with time.
When the process reaches steady-state conditions,
circulation occupies a significant portion of the
region. Circulation in the adiabatic region was also
found and reported by Issacci ct al. (1989) in an
analysis of the operational transient mode of heat
pipes.
In order to show the overall picture of flow
pattern development. Fig. 3 was prepared to depict
the vapor flow fields at different times. Figure 3a
shows the flow field at an early stage of the
transient process after the first shock reflection.
Here the flow reversal is shown to be in the left top
corner of the evaporation region. The shock
reflection initiates flow circulation at the entrance to
the adiabatic region shown in Fig. 3b. Vortex
formation in the adiabatic region develops with time
to a stronger flow circulation (Fig. 3c) and another
vortex can be seen in the left corner of the
evaporation region. In the flow field at steady state
(Fig. 3d), it is seen that flow circulation in the
adiabatic region develops to two vortices.
CONCLUSIONS
A nonlinear filtering technique has been used to
analyze the start-up vapor dynamics in heat pipes.
Large variations in vapor compressibility during the
transient phase render it inappropriate to use the
SIMPLER scheme or to use standard schemes for the
discretization of advective terms.
For a high heat flux, the stun-up uransient phase
involves multiple shock reflections from the line of
symmetry in the evaporator region. Each shock
reflection causes a significant increase in the local
pressure and a large pressure drop along the heat
pipe. Furthermore, shock reflections cause flow
reversal in the evaporation region and flow
circulations in the adiabatic region. The vapor
vortex formation in the evaporator is only transient
and is observed to disappear as the dynamics
approach steady state. However the circulation in
the adiabatic region grows with time. and in the
steady-state condition the circulation occupies a
significant portion of the region.
The pressure increase due to shock reflection
may cause the overall pressure of the evaporator to
be greater than the saturated vapor pressure. When
this occurs, condensation takes place by mist
nucleation in the vapor. Therefore, the condensation
rate is expected to decrease.
The time for the vapor core to reach a steady-
state condition depends on the input heat flux. the
heat pipe geometry, the working fluid, and the
condenser conditions. However. the vapor transient
time is on the order of seconds. Depending on the
time constant for the overall system, the vapor
transient time may be very short. Therefore, the
vapor core may be assumed quasi-steady in
transient analysis of a heat pipe operation.
44
Fig. 2. Transient development of the ve.nical mass flux; high-inpu; hea[ flux
45
# _I*,W.OOOOBOOWG.WOO_S.OOBOW_._S°.o.
I ttllltlt010*oo*
I lllllllploooo.oB.._.... .... . ...... .._
t If/lolPlooIJo........°. ...............
f lt/P//ttpsIn..°.. ............... ° .....
t p f.* opt//t... ..........................
I t I! *l///,o. * _
iii""'"'" ...........................
_[[_fff_:. ...........................
(l)
I,,,, i; " .;;'2 ......
I _ P I * l/Ill * .*./
IIIII_I' " ..................................
Co)
, ,, ,,,, .. .:. .::':,. ...... , ,,,
(c)
(d)
Fig. 3. Vapor flow fields a_ different _imes (arrows indicate direction; t_il indicates magnitude)
46
ACKNOWLEDGMENT
This work was supported by NASA Lewis under
Contract No. NAG3-899 and NASA Dryden under
Contract No. NCC2-374 Supp. 2, and by the U.S.
Department of Energy, Office of Fusion Energy,
Great No. DE-FGO3-86ER52126, with UCLA.
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47


Vapor Flow Patterns During a Start-Up Transient in Heat Pipes

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