Theoretical study of oscillatory phenomena in a horizontal closed-loop pulsating heat
pipe with asymmetrical arrayed minichannel
☆
Ching-Ming Chiang
a, Kuo-Hsiang Chien
b, Han-Ming Chen
a, Chi-Chuan Wang
c,⁎
aDepartment of Mechanical Engineering, National Taiwan University, Taipei, Taiwan
bGreen Energy and Environment Research Laboratories, Industrial Technology Research Institute, Hsinchu, Taiwan c
Department of Mechanical Engineering, National Chiao-Tung University, Hsinchu, Taiwan
a b s t r a c t
a r t i c l e i n f o
Available online 2 June 2012 Keywords:
Pulsating heat pipe
Asymmetrical arrayed minichannel Number of turns
Filled liquid ratio Frequency ratio
This study develops an analytical model applicable for predicting the oscillatory motion of a pulsating heat pipe (PHP) with asymmetrical arrayed minichannel configuration. The analytic model takes into account the temperature difference between the average evaporating region and the average condensing region as the thermally driven force for the oscillatory motion. It is found that the dominant parameters affecting the PHP include configurations and dimensions of the flow channel, the number of turns, the filled liquid ratio, the frequency ratio, the operating temperature, and the temperature difference between the average evaporating region and the average condensing region are investigated in this study. The calculated results show that the closed-loop pulsating heat pipe with asymmetrical arrayed minichannel under lower number of turns, lowerfilled liquid ratio, higher operating temperature and higher temperature difference between the average evaporating region and the average condensing region for the frequency ratio of unity could attain a better performance of the oscillatory motions.
© 2012 Elsevier Ltd. All rights reserved.
1. Introduction
With continuous shrinkage of integrated circuits and endless ris-ing of capability/functions of electronic devices, the dissipatris-ing power of internal chip array is also substantially increased according-ly. Hence, with efficient heat removal from the electronic devices, damage or even failure of the whole system is evitable. It is therefore imperative to provide an efficient cooling technique to meet the ris-ing demand for the current electronic industry.
One of the novel designs for effective heat removal is the pulsating heat pipe which was originally proposed by Akachi[1]. The pulsating heat pipe (PHP) or called the oscillating heat pipe (OHP) is made of slender tubes having serpentine configurations. By filling appropriate amount of workingfluid into PHP, the PHPs are able to function prop-erly with liquid slugs and vapor plugs being distributed arbitrarily and unevenly within the whole tube. In an effort to avoid the short-coming of stratified flow, Cai et al.[2] suggested that the interior diameter should not be larger than 3 mm.
In general, the PHP have three types: (1) the closed-loop type, which has two ends interconnected with each other and the working fluid could be circulated in the tube; (2) the closed-loop with check valve type, in whichflow direction of the working fluid could be con-trolled; (3) the closed-end type, which has two ends sealed off and
onlyfluid flow without circulation in the tube. Zhang et al.[3] exper-imentally investigated the closed-loop PHP and showed that a better thermal performance than that of the closed-end. The closed-loop PHP with check valve is not popular because installing the check valve is normally accompanied with much higher expense. As a re-sult, the closed-loop PHP is often regarded as the most probable can-didates as far as PHP is concerned. Note that the closed-loop PHP is a passive device which can be used in the absence of any external or electrical powers. Some of the main advantages of the closed-loop PHP when compared to the conventional heat pipe are the absence of the wick structure and the ability to transport heat to sufficient long distance.
The closed-loop PHP is composed of the evaporating, condensing and adiabatic regions during normal operation. The evaporating and condensing regions are the regimes of the received and removed heat, respectively. The adiabatic region between the evaporating and con-densing region is optional in terms of practical application. There were considerable studies concerning the performance of PHP either experimentally or theoretically. For instance, the effects of hydraulic di-ameter, the number of turns, workingfluids, filling liquid ratio, inclined angles, and heat transfer rate on the performance of PHP were reported by Refs.[2,4–12], Refs.[4,7,13], Refs.[3,4,8–10], Refs.[2–4,6,8,9,11–13], Refs.[5,7], and Refs.[2,3,5,7,10–12], respectively. A detailed review about the relevant researches was summarized by Zhang and Faghri
[14]. Besides, the impinging jets can be used to cooling high heatflux of the electronic devices. Thefluidic flow and heat transfer associated with pulsating impinging jets were numerically analyzed by Refs. ☆ Communicated by W.J. Minkowycz.
⁎ Corresponding author at: EE474, 1001 University Road, Hsinchu, Taiwan 300. E-mail address:[email protected](C-C. Wang).
0735-1933/$– see front matter © 2012 Elsevier Ltd. All rights reserved.
doi:10.1016/j.icheatmasstransfer.2012.05.019
Contents lists available atSciVerse ScienceDirect
International Communications in Heat and Mass Transfer
[15,16]. Yen et al.[17]simulated the effects of the amplitude, frequency, permeability of the porous block, and porous blockage ratio that affect the pulsatingflow in a channel with a porous-block-mounted heat source. The lid-driven cavity has been found in electronic cooling for practical application. Aminossadati and Ghasemi [18]formulated a
two-dimensional square cavity under unsteady laminar mixed convec-tionflow. They showed that the average heat transfer rate of the cavity with an oscillating wall is lower than that of a constant velocity wall for low Richardson number.
Despite some efforts that had been undertaken, there is still a need for detailed physical interpretation of this process subject to in fluenc-ing parameters. This is because theflow oscillatory phenomenon in a closed-loop PHP is rather complex. Some dominant mechanisms for this process includes configurations and dimensions of the flow chan-nel, the number of turns, thefilled liquid ratio, the frequency ratio, the operating temperature, the temperature difference between the average evaporating region and the average condensing region, the combination of workingfluid with channel material, and the dynamic contact angle of advancing with receding. Nevertheless, the compli-cated relationships between these effects have not been completely investigated. It is therefore crucial to conduct parametric study on the oscillatory phenomena in a closed-loop PHP. To attain the better uniformity of temperature and higher capability of heat transport, the closed-loop PHP with asymmetrical arrayed minichannel is adopted. The advantages of the closed-loop PHP with asymmetrical arrayed minichannel can be validated by comparing its performance with that having constant cross-sectional area for creating more unbalancing capillary force to boost the PHP. Hence, the objective of this study is to develop an analytical model for predicting the oscilla-tion mooscilla-tion in a pulsating heat pipe with and without asymmetrical arrayed minichannel pertaining to horizontal arrangement.
2. Theoretical model
Fig. 1(a) denotes a schematic with detailed dimension for the pre-sent modeling. The closed-loop PHP contains an evaporating and a condensing region with an adiabatic region in between. The saturated liquid slugs and vapor plugs are distributed arbitrarily and unevenly in the closed-loop PHP as shown inFig. 1(a). The evaporating region of the closed-loop PHP received external heat input during the oper-ation, and the workingfluid absorbs heat from the channel wall. This basic modeling is similar to that of Ma et al.[8]but with a much more comprehensive revision. The basic assumptions are as follows:
(1) The PHP is a closed-loop type and operates horizontally with little influence of gravity.
(2) The liquid slug is incompressible and the vapor plug is as-sumed to behave as an ideal gas.
(3) Both liquid slug and vapor plugflowing in the channel are con-sidered one-dimensional.
(4) Fully developed laminarflow prevails along the channel. (5) The minor pressure losses including the bends, gradual
expan-sion and contraction sections are negligible.
(6) The heat transfer between the liquid slug, vapor plug and channel wall is not considered.
(7) The influence of liquid film and the shear force along the liquid/ vapor interface is negligible.
(8) The capillary force along theflow path is assumed to be constant. (9) The closed-loop PHP is constituted of rectangular minichannel
with two cross-sectional areas as shown inFig. 1(a).
As evaporating region received the heat from surroundings, the sat-urated temperature and pressure are described by the Clapeyron– Clausius equation: ln Pe Pr sat≅ hfg R 1 Tr− 1 Te sat: ð1Þ
The foregoing relationship is also applicable to the condensing region. This can be done by expanding Eq.(1)using Taylor series and introducing ideal gas equation of state, and neglecting the higher order terms from this operation. Hence, the pressure Nomenclature
A cross-sectional area (m2)
a channel width (m)
B perimeter (m)
c damping coefficient (N s m− 1)
Dh hydraulic diameter (m) F force (N) Fc capillary force (N) Fd driving force (N) Ff frictional force (N) Fi inertial force (N)
Fv elastic restoring force (N)
f friction factor
fn undamped natural frequency (Hz)
G magnitude of driving force (N) hfg latent heat of vaporization (J kg− 1)
k spring stiffness (N m− 1) L length (m) m mass (kg) N number of turns P pressure (N m− 2) R gas constant (J kg− 1K− 1) Re Reynolds number S capillary force (N) T temperature (°C) t time (s) V volume (m3) x coordinate (m) Greek symbols γ frequency ratio
ΔT temperature difference between the average evapo-rating region and the average condensing region (°C)
ζ damping ratio
θa dynamic contact angle of advancing (°)
θr dynamic contact angle of receding (°)
μ dynamic viscosity (kg m− 1s− 1) ρ density (kg m− 3)
σ surface tension (N m− 1) φ filled liquid ratio (%) ω driving frequency (rad s− 1)
ωn undamped natural frequency (rad s− 1)
Subscripts
a adiabatic region
c condensing region
e evaporating region
i smallflow channel
j bigflow channel
l liquid slug
r reference
sat saturated
t total
difference between the evaporating and condensing region can be rearranged as Pe−Pc¼ hfgρv;c Te Te−Tc ð Þ: ð2Þ
However, the heat input into the evaporating region leads to an increase of saturated temperature and consequently the expansion of vapor plug. The heat output from the condensing region leads to a decrease of temperature and the contraction of vapor plug. The evaporation and condensation processes brought about the oscillatory motion in the closed-loop PHP. In addition, it is expected that the evaporating and condensing temperature varies along the evaporator and condenser. With a prescribed driving frequency ofω, the temperature difference between the average
evaporating region and the average condensing region can be re-written as
Te−Tc¼
Te−Tc
2 ð1þ cosωtÞ: ð3Þ
By combining Eqs.(2) and (3), the thermally driven force for the oscillatory motion with regard to asymmetrical arrayed minichannel can be obtained by Fd¼ hfgρv;c 2Te Aiþ Aj Te−Tc 1þ cosωt ð Þ; ð4Þ
where Aiand Ajare the cross-sectional area of differentflow channels.
c)
a)
b)
Fig. 1. Schematic of a closed-loop PHP model. (a) The closed-loop PHP with asymmetrical arrayed minichannel (all dimensions with out of scale are in mm). (b) The capillary force and dynamic contact angle along theflow path. (c) The individual forces.
The oscillatoryflow starting up inside the horizontal channel and the pressure gradient along theflow path can be readily calculated by dP dx¼ fρ 2Dh dx dt 2 : ð5Þ
The frictional force stemming fromfluid flow in the asymmetrical arrayed minichannel can be evaluated by integrating Eq.(5). Thus, Ff¼ NV i 2D2 h;i fl⋅Rel ð Þiϕμlþ fðv⋅RevÞi½ð1−ϕÞμv þNVj 2D2 h;j fl⋅Rel ð Þjϕμlþ fðv⋅RevÞj½ð1−ϕÞμv n o dx dt; ð6Þ
whereφ is the filled liquid ratio, representing the volume of the workingfluid relative to the total volume inside the closed-loop PHP, viz.φ=Vl/Vt.
As stated before, the variation of the pressure difference of vapor plug is linear along theflow path with ideal gas behavior. For this rea-son, the required work subject to expansion and compression pro-cesses of the vapor plug can be determined by
Fv¼ ρv RT 1−ϕ ð Þ A2i NViþ A2j NVj ! " # x: ð7Þ
The interface divided by the liquid slug and vapor plug is called the meniscus as shown inFig. 1(b). From Faghri[19], the dynamic contact angle of advancing and receding areθa= 84° andθr= 33°,
re-spectively, and is also depicted inFig. 1(b). Based on the Laplace equation, the capillary force for asymmetrical arrayed minichannel can be found as
Fc¼ Fc;a−Fc;r¼ σ cosθð a− cosθrÞ Biþ Bj
: ð8Þ
3. Governing equation with parametric designs
Fig. 1(c) illustrates the interaction of the individual forces on the oscillatory motion. By substituting Eqs.(4), (6)–(8)into the Newton's sec-ond law of motion, namely,ΣF=m(d2x/dt2), a governing equation of the oscillatory motion of working fluid with asymmetrical arrayed
minichannel in the x direction yields x″þ 2ζωnx′þ ω 2 nx¼ G mð1þ cosωtÞ þ S m ð9Þ where ζ ¼2mcω n; ð10Þ ωn¼ ffiffiffiffiffi k m r ; ð11Þ m¼ Vt½ϕρlþ 1−ϕð Þρv; ð12Þ
The governing equation, Eq.(9), is an alternative form as a single degree of freedom system subject to harmonic excitation with the variable x(t). The associated boundary and initial conditions are x(0) = 0 and x′(0)=0, respectively.
For all conditions of the oscillatory motion, an optimum design is desirable in this study. Hence in the case of the under-damped, the general solution x(t), can be derived as
x tð Þ ¼ e−ζωnt −mG 1 ω2 n þ ω 2 n−ω 2 ω2 n−ω2 2 þ 2ζωð nωÞ2 " # −mS 1 ω2 n ( ) cosωn ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1−ζ2 q h i t þ G m −γ 2ζωnω ω2 n−ω2 2 þ 2ζωð nωÞ2 −ζ 1 ω2 n þ ω 2 n−ω 2 ω2 n−ω2 2 þ 2ζωð nωÞ2 " # ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1−ζ2 q −S m ζ ω2 n ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1−ζ2 q 8 > > > > < > > > > : 9 > > > > = > > > > ; sinωn ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1−ζ2 q h i t 8 > > > > > > > > > < > > > > > > > > > : 9 > > > > > > > > > = > > > > > > > > > ; þ mG 1 ω2 n þ ω 2 n−ω 2 ω2 n−ω2 2 þ 2ζωð nωÞ2 cosωt þ 2ζωnω ω2 n−ω2 2 þ 2ζωð nωÞ2 sinωt " # þmS 1 ω2 n ( ) ; ð13Þ where γ ¼ωω n: ð14Þ
From the foregoing expression, the parametric influences on the closed-loop PHP are investigated accordingly. The associated parameters are summarized as follows:
(1) Configurations and dimensions of the flow channel: N1, N2 and N3. To avoid the concern of stratified flow, that the interior flow channel should be less than 3 mm is appropriate. The mark N1 and N2 denote the rectangular cross-sectional area of flow channels are Ai
(1 × 1 mm2) and A
j(2 × 1 mm2), respectively. Both N1 and N2 are constant cross-sectional area of the minichannels. The mark N3 denotes
the PHP comprised of two cross-sectional areas (i.e., N1 and N2) as an asymmetrical arrayed minichannel which is shown inFig. 1(a). The height of the threeflow channels is 1 mm.
(3) Operating temperature (T): 40, 50, 60, 70 and 80 °C. This temperature reckon on that of the adiabatic region, which is defined as T¼ Teþ Tc
2 : ð15Þ
(4) Filled liquid ratio (φ): 30%, 40%, 50%, 60% and 70%.
(5) Temperature difference between the average evaporating region and the average condensing region (ΔT): 1, 3, 5, 7 and 9 °C. ΔT is given by
ΔT ¼ Te−Tc: ð16Þ
(6) Frequency ratio (γ): 0.1, 0.5, 1, 2 and 5. 4. Results and discussion
In the present study, the nomenclature N3 = 6 denotes the con fig-uration and dimension of theflow channel (N3) with six (6) number of turns. The oscillatory motions for the case of the under-damped condition are shown inFig. 2. The undamped natural frequency is often called the system's natural frequency, denoted by fn
fn¼ωn
2π: ð17Þ
The abscissa and ordinate are thefilled liquid ratio and system's natural frequency, respectively. It is found that system's natural fre-quency shows a minimum value at afilled liquid ratio around 50% for allflow channel and operating temperature. The result is ascribed to the rising the effective spring stiffness of the vapor when thefilled liquid ratio is increased. It gives rise to a higher system's natural fre-quency when thefilled liquid ratio is higher or lower than the mini-mum value. The larger spring stiffness implies a much larger movement of workingfluid.Fig. 2also reveals that higher system's natural frequency as well as lower damping ratio is a consequence of higher operating temperature. In addition, the closed-loop PHP with asymmetrical arrayed minichannel with lower number of turns is preferable for having a higher system's natural frequency.
Fig. 3shows the displacement and velocity response of the closed-loop PHP with asymmetrical arrayed minichannel (N3), and operat-ing temperature is 80 °C pertainoperat-ing to the number of turns of 3 and 15, respectively.Fig. 3(a) shows the dependence of displacement re-sponse for asymmetrical arrayed minichannel for variedfilled liquid ratio. It is assumed that temperature difference between the average evaporating region and the average condensing region is 5 °C and at a
frequency ratio of 1. It can be seen that, for afilled liquid ratio of 30% and the number of turns of 3, the closed-loop PHP with asymmetrical arrayed minichannel has higher amplitude of the oscillatory motion. This is because a reduction of the frictional force and the elastic re-storing force is encountered when the number of turns andfilled liq-uid ratio are significantly decreased. Furthermore, the higher filled liquid ratio also raises the inertial force and reduces the oscillatory amplitude. Irrespective of the system's natural frequency being kept unchanged for thefilled liquid ratio from 30% to 70%, the lower num-ber of turns also improves rapid periodic movement as shown in
Fig. 3(b). The steady oscillatory motion in terms of the displacement
0.5 1.5 2.5 3.5 20 40 60 80 100
fn
(Hz
)
N3=3, T=80°C N3=3, T=70°C N3=3, T=60°C N3=6, T=80°C N3=3, T=50°C N1=3, T=80°C N2=3, T=80°C N3=6, T=70°C N3=3, T=40°C N3=9, T=80°C N1=3, T=70°C N2=3, T=70°C N3=6, T=60°C N3=9, T=70°C N2=3, T=60°C N3=12, T=80°C N1=6, T=80°C N2=6, T=80°C N3=12, T=70°C N3=15, T=80°C N2=6, T=70°C N2=9, T=80°CFig. 2. Relationship offilled liquid ratio and system's natural frequency.
-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0 30 60 90 120 150
t (s)
x
(
m
)
a
-2.5 -1.5 -0.5 0.5 1.5 2.5 3.5 4.5 0 30 60 90 120 150t (s)
x
' (
m
/s
)
b
Fig. 3. Effect of number of turns on the displacement and velocity response for the asymmetrical arrayed minichannel subject to variousfilled liquid ratio.(a) Displacement response.(b) Velocity response.
and velocity response are established after one period operation for all under-damped cases.
Fig. 4shows the effect of operating temperature on displacement and velocity response. As shown inFig. 2, the asymmetrical arrayed minichannel with a number of turns of 3 has the highest system's nat-ural frequency at an operating temperature of 80 °C. Therefore, it is clear that the oscillatory motion primarily depends on the operating temperature. In fact, the system's natural frequency increases with the rising operating temperature. This is attributed to the dramatic change of vapor density at a constant filled liquid ratio. On the other hand, with the rise of the operating temperature, an appreciable reduction of the period, and a rise of the amplitude with velocity of the oscillatory motion are also observed. The higher operating tem-perature contributes to decrease viscosity and latent heat of vaporiza-tion, resulting in a smaller frictional force. Nevertheless, the higher operating temperature also helps to increase thermally driven force and decreases the inertial force. Besides, higher heat input into the evaporating region also makes way in rising the operating tempera-ture. In these regards, the foregoing interactions manage to enhance the amplitude and velocity of the oscillatory motion. However, the lower surface tension also reduces the capillary force despite the amplitude of the oscillatory motion decreases slightly.
For the effect of the number of turns on the displacement and veloc-ity response at an operating temperature of 80 °C and afilled liquid ratio of 30%, the results are shown inFig. 5. For a larger cross-sectional area, higher amplitude and velocity of oscillatory motion are seen due to higher thermally driven force with lower frictional force in theflow channel. The amplitude of oscillatory motion is affected by the capillary force. As shown inFig. 5, when the number of turns is as large as 9, it has the lower and slower velocity induced by the lowest system's natural frequency for all cases. For the uniform hydraulic diameter con figura-tion for allflow channels, lower frictional force and higher thermally driven force are expected. With the number of turns of 3 along with a filled liquid ratio of 30% at an operating temperature of 80 °C, best per-formance is achieved as far as oscillatory motion is concerned. The con-cept of flow channel applicable to asymmetric staggered array is consistent with that of Holley and Faghri[5].
Fig. 6illustrates the influence of temperature difference between the average evaporating region and the average condensing region on the displacement and velocity response for the relations of asymmetri-cal arrayed minichannel. It can be observed that the amplitude and pe-riod of oscillatory motion are simply proportional to the temperature difference. With the rise of the temperature difference between the
-0.1 0.0 0.1 0.2 0.3 0 30 60 90
t (s)
x
(
m
)
a
-3 -1 1 3 5 0 30 60 90t (s)
x' ( m /s )b
Fig. 4. Effect of operating temperature on the displacement and velocity response for the asymmetrical arrayed minichannel.(a) Displacement response.(b) Velocity response.
-0.1 0.1 0.3 0.5 0.7 0 30 60 90 120 150 180
t (s)
x
(
m
)
a
-2.5 -0.5 1.5 3.5 5.5 0 30 60 90 120 150 180t (s)
x
' (
m
/s
)
b
Fig. 5. Effect of configurations and dimensions on the displacement and velocity response subject to number of turns.(a) Displacement response.(b) Velocity response.
average evaporating region and the average condensing region, the ef-fect of latent heat contribution into the PHP is also increasing, leading to a substantial change of vapor volume. This phenomenon is applicable in both evaporator and condenser. As a result, the oscillatory amplitude is significantly increased.
As mentioned by Shafii et al.[11], a large proportion of the heat transfer in the PHP is contributed by sensible heat. For this purpose, the higher amplitude and the faster velocity of the oscillatory motion between the evaporating and condensing region are essential in im-proving the heat transfer performance. The effect of frequency ratio on the displacement and velocity response is shown inFig. 7with asymmetrical arrayed minichannel configuration. It is evident that the oscillatory motion becomes more pronounced when the frequen-cy ratio is equal to 1. This is actually due to the resonance character-istic. An appreciable reduction in oscillatory magnitude is seen when the frequency is above and below 1.
5. Conclusions
The study provides a theoretical study upon the oscillatory phe-nomena in a horizontal closed-loop pulsating heat pipe with asym-metrical arrayed minichannel. The analytic model takes into account the temperature difference between the average evaporating region
and the average condensing region as the thermally driven force for the oscillatory motion. The dominant parameters affecting the PHP such as mechanisms primarily depend on the configurations and di-mensions of theflow channel, the number of turns, the filled liquid ratio, the frequency ratio, the operating temperature, and the temper-ature difference between the average evaporating region and the av-erage condensing region are investigated in this study. The calculated results show that the closed-loop pulsating heat pipe with asymmet-rical arrayed minichannel with lower number of turns, lowerfilled liquid ratio, higher operating temperature and higher temperature difference between the average evaporating region and the average condensing region for the frequency ratio of unity could achieve a better performance due to larger oscillatory motions.
Acknowledgements
The authors are indebted to the the Energy R&D foundation funding from the Bureau of Energy of the Ministry of Economic Affairs, Taiwan.
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