oscillatory flow in pulsating heat pipes with...

American Institute of Aeronautics and Astronautics

1

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

ivivi

p

iei

Lpp

xLg

dtdx

ddtxd

232 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

dtdx

dtdxdp

dtdm

Tcdt

Tcmd iiv

ivivp

ivviv 12

,,

,,

4)( (3)

Equation (3) can be rearranged as

dtdx

dtdxdp

dtdm

RTdt

dTcm ii

viv

iviv

viv1

2,

,,

, 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

iiivgiv

iviv

ivviv xxLL

dtdx

dtdx

TRm

dtdm

RTdt

dTcm (6)

i.e.

])2[(

])2[(11

11

1

1

,

,

,

iip

iipvi

iv

iv

iv xxLL

xxLLdtd

dtdT

Tdtdm

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[ 11

1

,,

(8) Similarly, the masses of the first and last vapor plug are

])[( 11

1

1,11, xLLTCm pvv (8a)

])[(11

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

niTdRC

p ivgi

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

hLLTTdh

hLLTTdh

dtdm ))(())(( ,,,,,,,

(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

acipipLh LLxL

LLxLxLLL

1

11, 0

)( (11a)

acip

acipipRh LLxL

LLxLxLLL

0)(

, (11b)

cip

cipipcLc LxL

LxLxLLL

1

11, 0

)( (11c)

cip

cipipcRc 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

TpRdCC

gi

(12)

The masses of the vapor plugs at the reference state are

])[(4 00

0

2

1,0 xLLpTR

dm pg

v (13a)

nnLLpTR

dm pg

iv ,...3,2,)(2 0

0

2

,0 (13b)

])[(4 00

0

2

1,0 xLLpTR

dm pg

nv

(13c)

The average mass of the first and last vapor plugs is

ivpg

nvv mLLpTR

dmmm ,00

0

21,01,0

0 )(22

(14)

Substituting eq. (12) and (14) into eqs. (8a, 8, 8b, 9)

)(21

1

0

1,

0

1,

p

pvv

LLxLL

TT

mm

(15a)

)(2)(2 1

1

0

,

0

,

p

iipiviv

LLxxLL

TT

mm

(15b)

)(2

1

0

1,

0

1,

p

npnvnv

LLxLL

TT

mm

(15c)

1

0

,

0

,

TT

pp iviv (16)

By defining

LL

xxX

mm

MPP

PTT pi

iiv

iiv

iiv

i 00

,

0

,

0

, (17)

eqs. (15-16) become

)1(221 1

1

11

XM (18a)

niXX

M iiii ,...3,2,

)1(21 1

1

(18b)

)1(22

11

11

nnn

XM (18c)

1,...2,1,1 niP ii

(19) Introducing the nondimensional variables to eq. (2) and defining dimensionless time as

2dte (20)

eq. (2) becomes

niPPXd

dXd

Xdiii

ii ,...2,1,)(32 1202

2

(21)

where ω0 and are two dimensionless parameters defined as

2

40

2

420 2 ehpep LL

dpLgd

(22)

Substituting eq. (17) and (20) into eqs. (10), one obtains

))(())(( *,

*,

*,

*, ciRcLccieRcLce

i LLHLLHd

dM

(23)

where

000

20

0

20 44

TT

TT

hpdRThH

hpdRThH e

ec

cefg

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,*

, 101)(1

hi

hiiLhLh LX

LXXL

LL

(25a)

*

*,*

, 101)(1

hi

hiiRhRh LX

LXXL

LL

(25b)

*1

*11

*,*

, 0)(

ci

ciicLcLc LX

LXXLL

LL

(25c)

*

**,*

, 0)(

ci

ciicRcRc LX

LXXLL

LL

(25d)

where

LLL

LLL 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

TTTT

(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 01

XM (32a)

niM i ,...3,2,0,1 (32b)

0,)1(22

1 01

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)


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


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


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)


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


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


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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.

REFERENCES

1. Akachi, H., Looped Capillary Heat Pipe, Japanese

Patent, No. Hei6-97147, 1994. 2. Faghri, A., Heat Pipe Science and Technology,

Taylor & Francis, Washington, DC, 1995. 3. Akachi, H., Polasek, F., and Stulc, P., Pulsating

Heat Pipes, Proceedings of the 5th International Heat Pipe Symposium, Melbourne, Australia, 1996, pp. 208-217.

4. Gi, K., Maezawa, Kojima, Y., and Yamazaki, N.,

CPU Cooling of Notebook PC by Oscillating Heat Pipe, Proceedings of the 11th International Heat Pipe Conference, Tokyo, Japan, 1999, pp. 166--169.

5. Miyazaki, Y., and Akachi, H., Heat Transfer

Characteristics of Looped Capillary Heat Pipe, Proceedings of the 5th International Heat Pipe Symposium, Melbourne, Australia, 1996, pp. 378-383.

6. Miyazaki, Y., and Akachi, H., Self Excited

Oscillation of Slug Flow in a Micro Channel, Proceedings of the 3rd International Conference on Multiphase Flow, Lyon, France, June 8-12, 1998.

7. Miyazaki, Y., and Arikawa, M., Oscillatory Flow in the Oscillating Heat Pipe, Proceedings of the 11th International Heat Pipe Conference, Tokyo, Japan, 1999, pp. 131-136.

8. Lee, W.H., Jung, H.S., Kim, J.H., and Kim J.S.,

Flow Visualization of Oscillating Capillary Tube Heat Pipe, Proceedings of the 11th International Heat Pipe Conference, Tokyo, Japan, 1999, pp. 131-136.

9. Hosoda, M., Nishio, S., and Shirakashi, R.,

Meandering Closed-Loop Heat-Transport Tube (Propagation Phenomena of Vapor Plug), Proceedings of the 5th ASME/JSME Joint Thermal Engineering Conference, San Diego, CA, March 15-19, 1999.

10. Shafii, M.B., Faghri, A., and Zhang, Y., Thermal

Modeling of Unlooped and Looped Pulsating Heat Pipes, ASME J. Heat Transfer, Vol. 123, No. 6, 2001, pp. 1159-1172.

11. Zhang, Y., and Faghri, A., Analysis of Liquid-

Vapor Pulsating Flow in a U-shaped Miniature Tube, International Journal of Heat and Mass Transfer, Vol. 45, No. 12, 2002, pp. 2501-2508.

12. Zhang, Y., and Faghri, A., Heat Transfer in a

Pulsating Heat Pipe with Open End, Int. J. Heat Mass Transfer, Vol. Vol. 45, No. 4, 2002, pp. 755-764.

oscillatory flow in pulsating heat pipes with...

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