oscillatory flow in pulsating heat pipes...

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


2

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.


3

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)


4

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)


5

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)


6

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)


7

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


8

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


(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


(=0.45)

Table 2. Effects of heating/cooling section length and

charge ratio on amplitude and circular frequency


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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