Investigation of Mixed Flow Turbine Under Inlet Pulsating Flow
OSCILLATORY FLOW IN PULSATING HEAT PIPES...
Transcript of OSCILLATORY FLOW IN PULSATING HEAT PIPES...
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OSCILLATORY FLOW IN PULSATING HEAT PIPES
WITH ARBITRARY NUMBERS OF TURNS
Yuwen Zhang
Senior Member AIAA
Department of Mechanical Engineering
New Mexico State University
Las Cruces, NM 88003
Amir Faghri
Associate Fellow AIAA
Department of Mechanical Engineering
University of Connecticut
Storrs, CT 06269
ABSTRACT
Oscillatory flow in Pulsating Heat Pipes (PHPs) with
arbitrary numbers of turns is investigated numerically.
The PHP is placed vertically with evaporator sections at
the top and the condenser sections at the bottom. The
governing equations, obtained by analyzing
conservation of mass, momentum, and energy of the
liquid and vapor plugs, are nondimensionalized and the
problem is described by eight nondimensional
parameters. The numerical solution is obtained by
employing an implicit scheme. The effects of the
number of turns, length of heating and cooling section,
and charge ratio on the performance of the pulsating
heat pipe were also investigated.
NOMENCLATURE
A dimensionless amplitude of pressure
oscillation
Ac cross sectional area of the tube, m²
B dimensionless amplitude of displacement
C integration constant
cp specific heat at constant pressure, J/kgK
cv specific heat at constant volume, J/kgK
d diameter of the heat pipe, m
g gravitional acceleration, m/s²
h heat transfer coefficient, W/m²K
H dimensionless heat transfer coefficient
hfg latent heat of vaporization, J/kg
L length, m
L* dimensionless length
M dimensionless mass of vapor plugs
mv mass of vapor plugs, kg
n number of turns
P dimensionless vapor pressure
pv vapor pressure, Pa
dimensionless parameter defined by eq. (22)
Rg gas constant, J/kgK
t time, s
T temperature, K
x displacement of liquid slug, m
X dimensionless displacement of liquid slug
Greek Symbols
γ ratio of specific heats
charge ratio
θ dimensionless temperature
Θ dimensionless temperature difference
νe effective viscosity, m²/s
ρ density, kg/m³
τ dimensionless time
phase of oscillation
ω dimensionless angular frequency
ω0 dimensionless inherent angular frequency
Subscripts
c condenser
e evaporator
h heating
i ith liquid slug or vapor plug
L left
p plug
R right
v vapor
INTRODUCTION
Pulsating heat pipes (PHPs) are made from a long
capillary tube bent into many turns with the evaporator
and condenser sections located at these turns [1]. The
unique feature of PHPs, compared with the
conventional heat pipe [2], is that there is no wick
structure to return the condensate to the heating section,
and therefore there is no counter current flow between
the liquid and vapor. PHPs can be applied in a wide
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range of practical problems including electronics
cooling [3]. Gi et al. [4] investigated an "O" shaped
oscillating heat pipe as it applied to cooling a CPU of a
notebook computer. Due to the simplicity of the PHP
structure, its weight will be lower than that of
conventional heat pipe, which makes PHPs ideal
candidates for space applications.
Since the PHP was invented in the early nineties,
limited experimental and analytical/numerical
investigations on PHPs have been reported. The
experiments mainly focused on some preliminary
results for visualization of flow patterns and
measurement of temperature and effective thermal
conductivity. Miyazaki and Akachi [5] presented an
experimental investigation of heat transfer
characteristics of a looped PHP. They found that heat
transfer limitations that usually exist in traditional heat
pipes were not encountered in the PHP. The test results
suggested that pressure oscillation and the oscillatory
flow excite each other. A simple analytical model of
self-excited oscillation was proposed based on the
oscillating feature observed in the experiments.
Miyazaki and Akachi [6] derived the wave equation of
pressure oscillation in a PHP based on self-excited
oscillation, in which reciprocal excitation between
pressure oscillation and void fraction is assumed. They
also obtained a closed form solution of wave
propagation velocity by solving the wave equation.
Miyazaki and Arikawa [7] presented an experimental
investigation on the oscillatory flow in the PHP and
they measured wave velocity, which was fairly agreed
with the prediction of Ref. [3].
Lee et al. [8] reported that the oscillation of bubbles is
caused by nucleate boiling and vapor oscillation, and
the departure of small bubbles are considered to be the
representative flow pattern at the evaporator and
adiabatic section respectively. Hosoda et al. [9]
investigated propagation phenomena of vapor plugs in a
meandering closed loop heat transport device. They
observed a simple flow pattern appearing at high liquid
volume fractions. In such conditions, only two vapor
plugs exist separately in adjacent turns, and one of them
starts to shrink when the other starts to grow. A
simplified numerical solution was also performed with
several major assumptions including neglecting liquid
film which may exist between the tube wall and a vapor
plug. Shafii et al. [10] presented thermal modeling of a
vertically placed unlooped and looped PHP with three
heating sections and two cooling sections. The
dimensional governing equations were solved using an
explicit scheme. They concluded that the number of
vapor plugs is always reduced to the number of heating
sections no matter how many vapor slugs were initially
in the PHP. Zhang et al. [11] numerically investigated
oscillatory flow and heat transfer in a U-shaped
miniature channel. The two sealed ends of the U-shaped
channel were the heating sections and the condenser
section was located in the middle of the U-shaped
channel. The U-shaped channel was placed vertically
with two sealed ends (heating sections) at the top. The
effects of various nondimensional parameters on the
performance of the PHP were also investigated. The
empirical correlations of amplitude and circular
frequency of oscillation were obtained. Zhang and
Faghri [12] proposed heat transfer models in the
evaporator and condenser sections of a PHP with one
open end by analyzing thin film evaporation and
condensation. The heat transfer solutions were applied
to the thermal model of the PHP and a parametric study
was performed. Both Shafii et al. [10] and Zhang and
Faghri [12] found that heat transfer in a PHP was due
mainly to the exchange of sensible heat because over
90% of the heat transferred from the evaporator to the
condenser is due to sensible heat. The role of
evaporation and condensation on the operation of PHPs
was mainly on the oscillation of liquid slugs and the
contribution of latent heat on the overall heat transfer
was not significant.
In the present study, an analysis of oscillatory flow in a
PHP with arbitrary number of turns will be presented.
The governing equations are first nondimensionalized
and the parameters of the system will be reduced to
several dimensionless numbers. The nondimensional
governing equations are then solved numerically and
the effects of various parameters on oscillatory flow in
the PHP will be investigated.
PHYSICAL MODEL
A schematic of the pulsating heat pipe under
investigation is shown in Fig. 1. A tube with diameter d
and length 2nL is bent into n turns with the two ends
sealed. The evaporator sections of the PHP are at the
upper portion and each of them has a length of Lh. The
condenser sections with length Lc are located at the
lower portion of the PHP. The adiabatic sections,
located between evaporation and condenser sections,
have length of La. The wall temperatures at the
evaporator and condenser sections are Te and Tc,
respectively. The liquid slugs with uniform length 2Lp
are located at the bottom of the PHP [7,10]. The
location of each liquid slug can be represented by the
displacement, xi, which is zero when the liquid slug is
exactly in the middle of the turns. When the liquid slug
shifts to the right, the displacement is positive. When
the liquid slug shifts to the left, the displacement is
negative. The operation of the PHP is accomplished by
oscillation of the liquid slugs due to evaporation and
condensation in the vapor plugs.
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Governing equations
The momentum equation for the liquid slug in Fig. 2
can be expressed as
Fig. 1 Pulsating heat pipe
Fig. 2 Heating and cooling sections
ppiccivivi
pc dLxgAAppdt
xdLA 22)(2 1,,2
2
(1)
where Ac=πd²/4 is the cross sectional area of the tube.
Equation (1) can be rearranged as
p
iviv
i
p
iei
L
ppx
L
g
dt
dx
ddt
xd
2
32 1,,
22
2
(2)
where νe is the effective kinetic viscosity of the liquid.
The energy equation of the vapor plugs is obtained by
applying the first law of thermodynamics to each plug
dt
dx
dt
dxdp
dt
dmTc
dt
Tcmdii
v
iv
ivp
ivviv 12
,
,
,,
4
)( (3)
Equation (3) can be rearranged as
dt
dx
dt
dxdp
dt
dmRT
dt
dTcm ii
v
iv
iv
iv
viv
12
,
,
,
,4
(4)
It is assumed that the behavior of vapor plugs in the
evaporators can be modeled using ideal gas law
1,1,
2
11,4
])[( vgvpv TRmdxLLp (5a)
niTRmdxxLLp ivgiviipiv ,...3,2,4
])(2[ ,,
2
1,
(5b)
1,1,
2
1,4
])[( nvgnvnpnv TRmdxLLp (5c)
Substituting eq. (5b) into eq. (4) to eliminate vapor plug
pressure, pv,i yields
])(2[ 1
1
,,
,
,
,
,
iip
ii
ivgiv
iv
iv
iv
vivxxLL
dt
dx
dt
dxTRm
dt
dmRT
dt
dTcm
(6)
i.e.
])2[(
])2[(1
1
11
1
1
,
,
,
iip
iipvi
iv
iv
iv xxLL
xxLLdt
d
dt
dT
Tdt
dm
m
(7)
where γ=cp/cv is the specific heat ratio of the vapor.
Integrating eq. (7), a closed form of the mass of the
vapor plug is obtained
nixxLLTCm iipiviiv ,...3,2,])(2[ 1
1
1
,,
(8)
Similarly, the masses of the first and last vapor plug are
])[( 1
1
1
1,11, xLLTCm pvv (8a)
])[(1
1
1,11, npnvnnv xLLTCm
(8b)
where Ci is the integration constant. Substituting eqs.
(8a, 8, 8b) into eqs. (5a,b,c), the pressures of the vapor
plug are
niTd
RCp iv
gi
iv ,...2,1,4
1
,2,
(9)
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The masses of the vapor plugs increase due to
evaporation and decrease due to condensation
fg
RcLccivc
fg
RhLhiveeiv
h
LLTTdh
h
LLTTdh
dt
dm ))(())(( ,,,,,,,
(10)
where, Lh,L and Lh,R are lengths of evaporator sections
that are in contact with the vapor plug, and Lc,L and Lc,R
are lengths of condenser sections that are in contact
with the vapor plug (Fig. 2).
acip
acipip
Lh LLxL
LLxLxLLL
1
11
, 0
)( (11a)
acip
acipip
Rh LLxL
LLxLxLLL
0
)(,
(11b)
cip
cipipc
Lc LxL
LxLxLLL
1
11
, 0
)( (11c)
cip
cipipc
Rc LxL
LxLxLLL
0
)(,
(11d)
Nondimensional governing equations
In order to nondimensionalize the governing equations,
a reference state of the PHP needs to be specified. At
this reference state, the pressure and temperature of all
of the vapor plugs are p0 and T0, respectively. The
displacement of all of the liquid plugs at the reference
state are xi=x0. According to eq. (9), the constants Ci for
different vapor plugs are the same and can be expressed
as
1
0
2
04
TpR
dCC
g
i
(12)
The masses of the vapor plugs at the reference state are
])[(4
00
0
2
1,0 xLLpTR
dm p
g
v (13a)
nnLLpTR
dm p
g
iv ,...3,2,)(2
0
0
2
,0 (13b)
])[(4
00
0
2
1,0 xLLpTR
dm p
g
nv
(13c)
The average mass of the first and last vapor plugs is
ivp
g
nvvmLLp
TR
dmmm ,00
0
21,01,0
0 )(22
(14)
Substituting eq. (12) and (14) into eqs. (8a, 8, 8b, 9)
)(2
1
1
0
1,
0
1,
p
pvv
LL
xLL
T
T
m
m
(15a)
)(2
)(2 1
1
0
,
0
,
p
iipiviv
LL
xxLL
T
T
m
m
(15b)
)(2
1
0
1,
0
1,
p
npnvnv
LL
xLL
T
T
m
m
(15c)
1
0
,
0
,
T
T
p
p iviv (16)
By defining
L
L
x
xX
m
mM
P
PP
T
T pii
iv
i
iv
i
iv
i 00
,
0
,
0
, (17)
eqs. (15-16) become
)1(22
1 1
1
11
XM (18a)
niXX
M ii
ii ,...3,2,)1(2
1 1
1
(18b)
)1(22
11
11
n
nn
XM (18c)
1,...2,1,1 niP ii
(19)
Introducing the nondimensional variables to eq. (2) and
defining dimensionless time as
2d
te (20)
eq. (2) becomes
niPPXd
dX
d
Xdiii
ii ,...2,1,)(32 1
2
02
2
(21)
where ω0 and are two dimensionless parameters
defined as
2
4
0
2
42
02 ehpep LL
dp
L
gd
(22)
Substituting eq. (17) and (20) into eqs. (10), one obtains
))(())(( *
,
*
,
*
,
*
, ciRcLccieRcLcei LLHLLH
d
dM
(23)
where
000
2
0
0
2
0 44
T
T
T
T
hp
dRThH
hp
dRThH e
ec
c
efg
ee
efg
cc
(24)
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The dimensionless lengths of vapor plug in the heating
and cooling section in eq. (23) are
*
1
*
11,*
,10
1)(1
hi
hiiLh
LhLX
LXX
L
LL
(25a)
*
*,*
,10
1)(1
hi
hiiRh
RhLX
LXX
L
LL
(25b)
*
1
*
11
*,*
,0
)(
ci
ciicLc
LcLX
LXXL
L
LL
(25c)
*
**,*
,0
)(
ci
ciicRc
RcLX
LXXL
L
LL
(25d)
where
L
LL
L
LL c
ch
h ** (26)
The system is described by nine nondimensional
parameters including the number of turns, n, and the
parameters defined in eqs. (22, 24 and 26). If the
reference temperature is chosen to be the average of Te
and Tc, the dimensionless temperature of heating and
cooling sections is
11 ce (27)
where
ce
ce
TT
TT
(28)
At this point, the number of dimensionless parameters
that describe the system are further reduced to eight.
Initial conditions
The reference state of the PHP is chosen to be the initial
state of the system. The initial conditions of the system
are
niXX i ,...2,1,0,0 (29)
niPi ,...2,1,0,1 (30)
nii ,...2,1,0,1 (31)
0,)1(22
1 0
1
XM (32a)
niM i ,...3,2,0,1 (32b)
0,)1(22
1 0
1
XM n
(32c)
NUMERICAL SOLUTION
The oscillatory flow in a pulsating heat pipe is
described by eqs. (18-19), (21) and (23) with initial
conditions specified by eqs. (29-32). It is noted that eq.
(21) is an ordinary differential equation of forced
vibration. If the vapor pressure difference between the
two vapor plugs at two ends of the liquid slug is
iiii APP cos1 (33)
the analytical solution of eq. (21) can be obtained and it
will have the following form [11].
)cos( iiii BX (34)
However, the amplitude and angular frequency are
unknown a priori and the pressure difference between
the two vapor plugs depends on heat transfer in two
vapor plugs. The amplitude and angular frequency of
pressure oscillation must be obtained numerically. The
results of each time step are obtained by solving the
dimensionless governing equations using an implicit
scheme. The numerical procedure for a particular time
step is outlined as follows:
1. Guess the dimensionless temperatures of all vapor
plugs, θi
2. Obtain the dimensionless vapor pressure, Pi, from
eq. (19)
3. Calculate the dimensionless displacement of liquid
slug, Xi, from eq. (21)
4. Calculate the mass of the vapor plugs, Mi, using eq.
(23)
5. Calculate the nondimensional temperature of the
vapor plugs, θi, from eqs. (18a, b, c)
6. Compare θi obtained in step 5 with the guessed
values in step 1. If the differences meet a tolerance
go to the next step; otherwise, steps 2-5 are
repeated until a converged solution is obtained
The time step independent solution of the problem can
be obtained when time step is Δτ=10-5, which is then
used in all numerical simulations in the following
section.
RESULTS AND DISCUSSIONS
Figure 3(a) shows the comparison of the liquid slug
displacement obtained by the present model and the
model of Zhang et al. [11], who studied oscillatory flow
in a U-shaped miniature channel. The present result is
obtained by set the number of turn n=1. It can be seen
that the agreement between the results obtained by the
present model and Ref. [11] is excellent. Figure 3(b)
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shows the comparison of the liquid slug displacements
obtained by the present model and the model of Shafii
et al. [10]. The results of Shafii et al.'s model [10] was
obtained by using the following parameters: Lh=0.1m,
La=0m, Lc=0.1m, Lp=0.2m, d=3.34mm, Te=123.4°C,
Tc=20°C, and he=hc=200W/m²K. The present results
were obtained by using the corresponding
nondimensional parameters: ω0²=1.2×104, =1.2×105,
Θ=0.15, He=Hc=3000, Lh*=0.5, Lc
*=0.5 and n=2. It
can be seen that the results obtained by using the
present model agreed very well with the results
obtained by Shafii et al.'s model [10], which employed
dimensional parameters and was applicable only to
PHPs with two turns. The phase of the oscillation of
two vapor plugs are the same for the first several
periods. Steady oscillation is established after τ=0.09, at
which time the amplitudes of oscillation for the two
liquid slugs are the same. The phase difference for the
oscillation of the two liquid slugs is equal to π, which
means that the oscillation of the liquid slug in the PHP
with two turns is symmetric after steady oscillation is
established. The amplitude and circular frequency for
oscillation in a PHP with two turns are the same as
those for a U-shaped channel. At the parameters
specified above, the amplitude and circular frequency
of oscillation for both n=1 and n=2 are B=0.31489,
ω=597.78, respectively.
(a) Comparison with Zhang et al. [11]
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
Xi
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8Present
Zhang et al. [11]
(b) Comparison with Shafii et al. [10]
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
Xi
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8i=1, Present
i=2, Present
i=1, Shafii et al. [10]
i=2, Shafii et al. [10]
Fig. 3 Comparison of the present results with Zhang et
al. [11] and Shafii et al. [10]
(a) Dimensionless temperature of vapor plugs
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
i
0.8
0.9
1.0
1.1
1.2
1.3
1.4i=1
i=2
i=3
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
Mi
0.2
0.4
0.6
0.8
1.0
1.2
1.4i=1
i=2
i=3
Fig. 4 Dimensionless temperature and mass of the
vapor plugs (n=2)
(a) Dimensionless displacement of liquid slugs
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
Xi
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8i=1
i=2
i=3
(b) Dimensionless temperature of vapor plugs
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
i
0.8
0.9
1.0
1.1
1.2
1.3
1.4i=1
i=2
i=3
i=4
Fig. 5 Displacement of liquid slugs and dimensionless
temperature of the vapor plugs (n=3)
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Figure 4(a) shows the dimensionless temperature of
three vapor plugs. The maximum temperature of the
vapor plug can exceed the heating wall temperature of
PHP due to compression of the vapor plug. The
variations of the dimensionless temperatures of first and
third vapor plug become identical after steady
oscillation is established at τ=0.09. Figure 4(b) shows
the variation of dimensionless mass of the vapor plugs.
The mass of first and third vapor plugs were different at
the beginning but they are identical after steady
oscillation is established. The mass of the second vapor
plug is twice of that of first or third vapor plugs after
steady oscillation is established. The history of vapor
plug temperature and mass further indicated that the
oscillations in the PHP with two turns are symmetric.
The differences between the oscillations in PHP with
one or two turns can be observed before steady
oscillation is established. After steady oscillation is
established, the oscillation in the U-shaped channel is
same as that in the PHP with two turns.
Figure 5 shows oscillatory flow in a PHP with three
turns. The time required to establish steady oscillation
for the PHP with three turn is longer that for the PHP
with two turns. Upon steady oscillation is established,
the dimensionless displacement of the first and third
liquid slugs become identical, which means the
oscillation is symmetric for the PHP with three turns.
Figure 5(b) shows the dimensionless temperature of the
vapor plugs for the PHP with three turns. The
dimensionless temperatures of vapor plugs of odd
number are identical once steady oscillation is
established. The dimensionless temperature of vapor
plugs of even number are also identical upon steady
oscillation is established but their phase difference with
odd numbered vapor plugs is π. The amplitude and
circular frequency of oscillation are same as those of
n=1 and 2. Figure 6 shows oscillatory flow in a PHP
with four turns. The time required to establish steady
oscillation for the PHP with four turns is about τ=0.13,
which is longer than that for PHP with three turns. The
dimensionless displacement of the odd numbered liquid
slugs become identical upon steady oscillation is
established. The dimensionless displacement of the
even numbered liquid slugs are also identical after
τ=0.13. Figure 5(b) shows the dimensionless
temperature of the vapor plugs for the PHP with four
turns. Similar to the case with three turns, the phase
difference between odd and even numbered vapor plugs
is also π. The increase in the number of turns from three
to four did not result in any changes in the amplitude
and circular frequency of oscillation.
(a) Dimensionless displacement of liquid slugs
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
Xi
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8i=1
i=2
i=3
i=4
(b) Dimendionless temperature of vapor plugs
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14i
0.8
0.9
1.0
1.1
1.2
1.3
1.4i=1
i=2
i=3
i=4
i=5
Fig. 6 Displacement of liquid slugs and imensionless
temperature of the vapor plugs (n=4)
Table 1. Amplitude and circular frequency of
oscillatory flow in PHPs
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Numerical solutions are then performed for PHPs with
different numbers of turns. The results show that the
amplitude and frequency of oscillation of PHPs are not
affected by number of turns until n=6. When the
number of turns is increased to 6, the amplitude and
frequency of oscillation for different liquid slugs begin
to differ. The amplitude and circular frequency of
oscillation for different number of turns are shown in
Table 1. The amplitude and circular frequency for
different liquid slugs in the same PHP are slightly
different when the number of turns is greater than five.
(a) Odds number liquid slugs
0.480 0.485 0.490 0.495 0.500
Xi
-0.4
-0.2
0.0
0.2
0.4i=1
i=3
i=5
i=7
i=9
(b) Even number liquid slugs
0.480 0.485 0.490 0.495 0.500
Xi
-0.4
-0.2
0.0
0.2
0.4i=2
i=4
i=6
i=8
i=10
Fig. 7 Displacement of liquid slugs (n=10)
i
0 1 2 3 4 5 6 7 8 9 10 11
Xi
-0.4
-0.2
0.0
0.2
0.4=0.4833
=0.4862
=0.4891
=0.4918
Fig. 8 Distribution of the displacement of liquid slugs
(n=10)
(a) Odds number liquid slugs
0.480 0.485 0.490 0.495 0.500
Xi
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4i=1
i=3
i=5
i=7
i=9
(b) Even number liquid slugs
0.480 0.485 0.490 0.495 0.500X
i
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4i=2
i=4
i=6
i=8
i=10
Fig. 9 Displacement of liquid slugs (Lh*= Lc
*=0.45)
Figure 7 shows the dimensionless displacements of
liquid slugs for a PHP with 10 turns after steady
oscillation is established. The phase of oscillation for
all odd numbered liquid slugs are very close to each
other, and the phase of oscillation for all even number
liquid slugs are also very close to each other. The phase
difference between any two adjacent liquid slugs is
approximately π. Figure 8 shows the overall
displacements of liquid slugs at four different times.
The oscillation of any two adjacent liquid slug is nearly
always in opposing directions. The delay of oscillation
for the ith slug relative to the (i-2)th slug is also clearly
seen from Fig. 8. Figure 9 shows the displacements of
liquid slugs for Lh*=Lc
*=0.45. The overall distribution
of the displacements of liquid slugs at four different
times is shown in Fig. 10. The delay of oscillation for
the ith slug relative to the (i-2)th slug is more significant
when the lengths of heating and cooling sections is
reduced. The amplitude and circular frequency of
oscillation are listed in Table 2. Both amplitude and
circular frequency of oscillation are decreased because
the available heating and cooling section areas are
decreased. Figure 11 shows the displacements of liquid
slugs for the charge ratio of 0.45. The overall
distribution of the displacements of liquid slugs at four
different times is shown in Fig. 12. The oscillation for
the ith slug relative to the (i-2)th slug is more closer to
each other. The amplitude and circular frequency of
oscillation for an increased charge ratio are also listed
American Institute of Aeronautics and Astronautics
9
in Table 2. The amplitude of oscillation is decreased
because the mass of the liquid slug is larger for a large
charge ratio. On the other hand, the circular frequency
of oscillation is increased.
i
0 1 2 3 4 5 6 7 8 9 10 11
Xi
-0.4
-0.2
0.0
0.2
0.4
0.6=0.4844
=0.4913
=0.4941
=0.4971
Fig. 10 Distribution of the displacement of liquid slugs
(Lh*= Lc
*=0.45)
(a) Odds number liquid slugs
0.480 0.485 0.490 0.495 0.500
Xi
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
i=1
i=3
i=5
i=7
i=9
(b) Even number liquid slugs
0.480 0.485 0.490 0.495 0.500
Xi
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4i=2
i=4
i=6
i=8
i=10
Fig. 11 Displacement of liquid slugs (=0.45)
i
0 1 2 3 4 5 6 7 8 9 10 11
Xi
-0.4
-0.2
0.0
0.2
0.4
0.6
Fig. 12 Distribution of the displacement of liquid slugs
(=0.45)
Table 2. Effects of heating/cooling section length and
charge ratio on amplitude and circular frequency
American Institute of Aeronautics and Astronautics
10
CONCLUSIONS
Oscillatory flow in a pulsating heat pipe with arbitrary
number of turns is investigated in the present study. The
governing equations that describe the oscillatory flow
were nondimensionalized and the parameters that
describe the system were reduced to eight
nondimensional parameters. The results show that the
increase of the number of turns has no effect on the
amplitude and circular frequency of oscillation when
the number of turns is less or equal to five. When the
number of turns is increased to more than five, the
amplitude and circular frequency of oscillation for
different liquid slugs are shown. Both amplitude and
circular frequency of oscillation will be decreased when
the lengths of heating and cooling sections are
decreased. When the charge ratio is increased, the
amplitudes of oscillation are decreased and the circular
frequency of oscillation is increased.
Acknowledgments
This work was partially supported by NASA Grant
NAG3-1870 and NSF Grant CTS 9706706.
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