Modeling of reversal flow and pressure fluctuation in rectangular microchannel

International Journal of Heat and Mass Transfer 102 (2016) 1024–1033

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Modeling of reversal flow and pressure fluctuation in rectangular microchannel Hui He, Peng-fei Li, Run-gang Yan, Liang-ming Pan ⇑ Key Laboratory of Low-grade Energy Utilization Technologies and Systems (Chongqing University), Ministry of Education, Chongqing 400044, China

a r t i c l e

i n f o

Article history: Received 29 November 2015 Accepted 28 June 2016

Keywords: Microchannel Reversal flow Pressure fluctuation

a b s t r a c t Solving the flow instability in microchannel continues to be a topic of current research. In view of this, the current paper presents an analytical model to predict the pressure fluctuation by the analysis of bubble growth in a rectangular microchannel. To facilitate the analysis, bubble growth in the rectangular microchannel is assumed to be composed of three stages, namely, free growth, partially confined growth and fully confined growth. The interfacial velocity of bubble, being used to investigate the relationship between bubble reversal flow and pressure fluctuation, is determined by solving the conservation equations of the momentum of the liquid column coupled with the equations of the force balance at the bubble interface. The model reveals that when the length of fully confined bubble expands to some extent, the tail interface of bubble will reverse resulting in dramatically pressure increase. Additionally, the dependent factors, including Boiling number, nucleation site position, transverse shape and inlet restrictor, of pressure fluctuation are also analyzed based upon the current model, which denote that a smaller aspect ratio corresponds to a premature pressure fluctuation, and the magnitude of pressure fluctuation increases with the increasing Boiling number, decreases as the position of nucleation site moving downstream of the channel. In addition, the magnitude of pressure fluctuation decreases as the increase of pressure drop multiplier parameter, however, accompanied with the penalty of increasing pressure drop. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction With the rapid progress of MEMS and lTAS technology, flow boiling heat transfer in microchannels is widely applied due to their compact sizes and effective heat transfer through its high specific surface area. However, the main disadvantage of this approach to fabricate cooling devices is the flow instability characterized by pressure drop and reversing flow, which can lead to high amplitude temperature oscillations with premature critical heat flux (CHF) and mechanical vibrations. A number of studies provided evidence to the sensitivity of two-phase microchannel systems to flow instabilities. Oscillating pressure fluctuations and visualizations showing cyclical backflow were encountered in many experiments. Qu and Mudawar [1] have investigated hydrodynamic instability and pressure fluctuations in a water cooled two-phase microchannel heat sink containing 21 parallel 231
173 lm microchannels. They have identified two types of two-phase flow instability, namely, severe pressure oscillations and mild parallel channel instability. Wu et al. [2–5] reported

⇑ Corresponding author. Fax: +86 23 65102280. E-mail address: [email protected] (L.-m. Pan). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.06.102 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.

two modes of two-phase flow instability in multi-parallel microchannels having a hydraulic diameter of 186 lm, i.e. the liquid/two-phase alternating flow (LTAF) and the liquid/twophase/vapor alternating flow (LTVAF). Xu et al. [6] measured the dynamic unsteady flow in a compact heat sink which consisting of 26 rectangular microchannels with 300 lm width and 800 lm depth, and three types of oscillations were identified, i.e. large amplitude/long period oscillation superimposed with small amplitude/short period oscillation and small amplitude/short period oscillation. Thermal oscillations were always accompanying the above two oscillations. Wang et al. [7] investigated dynamic instabilities of flow boiling of water in parallel microchannels as well as in a single microchannel, and two types of unstable oscillations were reported, one with long-period oscillations and another with short-period oscillations in temperature and pressure. Bogojevic et al. [8] carried out a series of experiments to investigate pressure and temperature oscillations during the flow boiling instabilities in a microchannel silicon heat sink with 40 parallel rectangular microchannels. Their results revealed that two types of twophase flow instabilities with appreciable pressure and temperature fluctuations were observed, that depended on the heat to mass flux ratio and inlet water temperature. These were high amplitude/low frequency and low amplitude/high frequency instabilities.

H. He et al. / International Journal of Heat and Mass Transfer 102 (2016) 1024–1033

1025

Nomenclature A A+ Bo Ca cl Dh f FE FM Fr g H 00 H hlv Ja L M p p+ Dpþ f Pr qin R1, R2 R+ Re t t+ T u u0 u+

bubble projected area (m2) dimensionless area given by Eq. (27) Boiling number Capillary number specific heat of liquid (J/kg K) equivalent diameter of the channel (m) Fanning friction factor evaporation momentum force (N) liquid inertia force (N) surface tension force (N) gravitational acceleration (m s2) height of microchannel (m) channel height subtracts liquid film thickness (m) latent heat of water (kJ/kg) Jacob number channel length (m) pressure drop multiplier parameter pressure (Pa) pressure nondimensionalized by ql;out u20 frictional pressure drop nondimensionalized by ql;out u20 Prandtl number heat flux (kW/m2) principal radii of bubble (m) dimensionless radius given by Eq. (28) Reynold number time (s) dimensionless time defined as Eq. (35) temperature (°C) velocity (m/s) initial liquid velocity in the channel inlet (m/s) dimensionless velocity defined as Eqs. (15)–(17)

A number of researchers confirmed that the high amplitude/low frequency oscillation is attributed to the presence of flow reversal. Because of which reverse vapor flow in parallel microchannels will cause the flow mal-distribution, as vapor–liquid interface in each channel may temporally extend into different directions, either forward or backward [9]. Hetsroni et al. [10] observed periodic wetting and rewetting boiling in triangular microchannel heat sink, and the pressure oscillations coincided with the reversed flow. Chang and Pan [11] found that the magnitude of pressure oscillations may be used as an index for the appearance of reversed flow. Barber et al. [12,13] emphasized that the confined bubble growth can cause pressure fluctuations due to the reversed flow. Tuo and Hrnjak [14] revealed that reversed flow due to confined bubble longitudinal expansion caused periodic oscillations of the evaporator inlet pressure and the pressure drop based on simultaneous flow visualization and pressure measurements. Hence, preventing the reversed flow was crucial for the application of the flow boiling heat transfer in microchannels. Gedupudi et al. [15] concluded that the reversed flow can occur only if there exists upstream compressibility, i.e. a volume of trapped condensable or non-condensable gas or an expanding component such as a bellows. Generally, inlet header served as a buffer tank, providing significant compressible volume upstream of the heated microchannel tubes. Such volume may be able to intermittently retain and discharge the backflow vapor [16]. The two-phase flow stability is generally influenced by the boundary conditions. Preliminary experiments conducted by Brutin and Tadrist [17] indicated a strong dependence of the boundary conditions on the thermo-hydraulic behavior in the microchannel. Wang et al. [18] showed that the resistance associated with the configuration of the connections between the external circuit and

W W00 We X1 Z

width of microchannel (m) channel width subtracts liquid film thickness (m) Weber number length of liquid column (m) bubble length (m)

Greek symbols a thermal diffusivity (m2/s) c aspect ratio of the channel d liquid film thickness (m) d+ dimensionless liquid film thickness defined as Eq. (31) k thermal conductivity (W/m K) q density (kg/m3) q+ dimensionless density defined as Eqs. (13) and (14) r surface tension (N/m) Subscripts b bubble exp channel expansion f friction i interface, initial in inlet l liquid 1 tail interface of bubble 2 nose interface of bubble out outlet orf orifice s saturation v vapor w wall

the inlet and outlet plena influenced flow reversal as well as flow instability characterized by pressure fluctuation. In pursuit of the solutions to eliminate or mitigate the pressure fluctuation caused by the bubble reversal flow, many references have devoted the researches to this matter so far. Qu and Mudawar [19] placed a throttling valve upstream of the test module and increased the overall pressure drop to eliminate upstream compressible flow instability. Kosar et al. [20] fabricated 20 lm wide 400 lm long restrictors in the inlet of 200 lm wide microchannels to successfully eradicate flow oscillations. Subsequently, Kosar et al. [21] proposed a dimensionless parameter, M, accounting for the pressure drop increase imposed by the inlet restrictors, to correlate the extent of flow instability suppression, and revealed that the onset of unstable boiling in the microchannels asymptotically increased with M and upstream compressible volume instabilities were completely eradicated at sufficiently high M values. Bergles and Kandlikar [22] were perhaps the first researcher to comprehensively discuss apparent flow instabilities in microchannels and suggested various means for suppression. In a latter study, Kandlikar et al. [23] experimentally investigated several methods to reduce the occurrence of flow instabilities in channels having hydraulic diameter of 333 lm by throttling valves and artificially drilled nucleation sites. It should be noted that, as mentioned by Kandlikar [24], the nucleation characteristics in the microchannel were crucial for bubble dynamics, pressure drop and etc. Because the microchannel interior surfaces were generally smooth due to the micro fabrication processes [25], resulting in the increase of nucleation wall superheat due to the absence of cavities of all sizes. Once boiling was triggered, the superheated liquid would readily change phase causing an explosive vapor growth and subsequently reversal flow. Therefore, in addition to placing the inlet restrictors,

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introducing nucleation cavities of radii satisfying nucleation criteria in the microchannels was the another solution to eliminate or mitigate the pressure fluctuation. Kuo and Peles [26] experimentally studied the ability of reentrant cavities to suppress flow boiling oscillations and instabilities in microchannels, found that structured surfaces formed inside channel walls (viz. reentrant cavities) can, to an extent, assist mitigating the rapid bubble growth instability. Xu et al. [27] utilized seed bubbles to trigger boiling heat transfer and control thermal non-equilibrium of liquid and vapor phases in parallel microchannels, which can improve flow and heat transfer performance in microchannels without apparent oscillations of pressure drops. However, although those aforementioned studies confirmed the effectiveness of inlet throttling and artificial nucleation bubble to eliminate flow instabilities in microchannels, no emphasis was placed on the quantitative relationship between bubble dynamic and flow instability in microchannel. Pursuit of the relationship between bubble dynamic and flow instability in microchannel, Kandlikar [28] has ever elucidated that as a bubble nucleated and occupied the entire channel, the stability of the flow depended on the balance of the inertia force of the liquid and the evaporation momentum force at the upstream meniscus between the expanding bubble and the upstream liquid. The ratio of these forces was given by the parameter K1,

K1 ¼

FE FM

ð1Þ

Based on the criteria proposed by Wang et al. [7], and introducing the properties of water at the mean test section pressure employed, which was quite close to 1 bar, the criteria for unstable boiling with long-period oscillations can be expressed as,

K 1 P 1:45
103

ð2Þ

Lee et al. [29] established a generalized instability model for micro-channels. It was similar to the parameter K1, but took the effects of both inlet orifice and channel expansion into account, i.e.,

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi FE R¼ F M þ F exp þ F orf

ð3Þ

However, the instability criteria revealed by these previous models in the microchannel are qualitative, and are irrelevant to the bubble dynamic processes. A fundamental understanding of the relationship between bubble dynamic and flow instability is vital for the improvement of flow and heat transfer performance in microchannels. In view of this, an analytical model in present paper has been proposed to investigate the pressure fluctuation by the analysis of bubble growth in a rectangular microchannel, and the dependent factors (including inlet restrictor, nucleation site position, transverse shape) of pressure fluctuation are also analyzed based upon the current model. 2. Mathematic models 2.1. Bubble growth In the microchannel, the pressure drop is very sensitive to the bubble dynamics. As mentioned by Steinke and Kandlikar [30], the bubble nucleation and subsequent bubble growth will increase the pressure drop in the microchannel. Barber et al. [12,13] visually investigated the bubble growth in a rectangular microchannel by mean of high-speed video, and highlighted that three main stages of bubble growth were observed in the particular microchannel during flow boiling, namely unconfined bubble growth, partial bubble confinement and full bubble confinement. Therefore, based on the visualization phenomena given by Barber et al. [12,13],

Bubble growth in a rectangular microchannel is assumed to be divided into three stages: free growth, partially confined growth, fully confined growth. Usually, the growth period from a small radial bubble to an elongated vapor slug is very quick indeed [31,32], and the reversed flow happens in the last stage due to explosive and confined growth. Consider an isolated bubble nucleation in a horizontal microchannel with a rectangular crosssectional dimension (see Fig. 1) of H
W. The initial conditions for the confined growth model depend on the preceding growth from triggered nucleation on a heated surface to a radius approximately equal to the microchannel height H at the time t1, which as shown in Fig. 1. Lee et al. [31] have ever explored experimentally bubble dynamics in a single trapezoid microchannel, and proved that the bubble at the stage of free growth grew with a spherical shape. Therefore, bubble at the stage of free growth can be estimated by the well-established model for spherical, unconfined bubble growth at constant superheat, i.e.,

R ¼ 2:5

Ja 1=2

Prl

ðal tÞ1=2

ð4Þ

where Ja is the Jakob number being defined as,

Ja ¼

t1 ¼

ql cl ðT w  T s Þ qv hlv

ð5Þ

H002 Prl

ð6Þ

6:25al Ja2

When the channel width is larger than the microchannel height, it will take some time for the bubble to grow approximately to the channel width W as it moves along the channel. This is defined as partially-confined growth, in which the bubble is confined fully along the height, except for the very thin liquid film below and above the growing bubble between the bubble and the heating wall. Partially-confined growth is assumed for the growth from A ¼ p4  H002 at time t = t1, through A ¼ p  R2 to A ¼ p4  W 002 , and the heat balance for bubble growth is given by,

2 00 dA H qv hlv ¼ qin A 3 dt

ð7Þ

Substituting A ¼ p  R2 into Eq. (7),

4 00 dR H qv hlv 0 ¼ qin R 3 dt

ð8Þ

The initial condition at time t0 ¼ t  t 1 ¼ 0 is R ¼ H00 =2,

R¼

H00 3t0 e 4s 2

In which,

ð9Þ 00

s ¼ H qqinv hlv . Fully confined growth is assumed for the

growth from A ¼ p4  W 002 at time t = t2, and then the bubble grows along axial direction that is constrained in the transversal directions. Given that the appearance of reversal flow is at the stage of bubble fully confined growth, and the reversal flow can result in a sharp pressure drop in the microchannel. Therefore, a precise model to predict the fully confined bubble growth is of paramount importance. When bubble is at the stage of fully confined growth in the microchannel, the growth of elongated bubble is mainly governed by the evaporation of thin liquid film around the bubble as asserted by Li et al. [32] who modeled the time rate of elongated bubble length by assuming that heat flux transferred by conduction from the wall was only used to phase change, and found that there was a good agreement between model prediction and experimental data related to the bubble length. Barber et al. [13] predicted the confined bubble area in the microchannel based upon

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H. He et al. / International Journal of Heat and Mass Transfer 102 (2016) 1024–1033

fully confined

H

H''

liquid film

vapor core

partially confined

Wall qin

W''

W Fig. 1. Cross-sectional schematic of confined bubble growth in a microchannel.

energy balance, viz. set the power imposed at the channel wall equal to the bubble growth over time, and found that the bubble area obtained from energy balance was qualitatively in agreement with the observed case at the flow conditions shown. Additionally, Agostini et al. [33] modeled the elongated bubble velocity in the microchannel by assuming that heat flux from the wall yielding the enthalpy was absorbed by the elongated bubble. Their results revealed that the elongated bubble velocity predicted by their model slightly under-predicted the data but the agreement was still good (mean absolute error was MAE = 8.9% and the mean error was ME = 6.8%). Therefore, energy balance, i.e. heat flux transferred by conduction from the wall contributing to the phase change, given by Eq. (10) is plausible for bubble fully confined growth.

p 4

H00 qv hlv

p  dz W 00 þ z 00 ¼ qin dt 4

ð10Þ

The initial condition at time t 00 ¼ t  t2 ¼ 0 is Z ¼ 0, thus,

 0  1 H002 Prl 00 4 t4s ln ðW00 Þ  4t00  p 3 H 625al Ja2 B C ps z ¼ W 00 e ps  1 ¼ W 00 @e  1A 4 4

p

ð11Þ

2.2. Conversation equations Given that the measurements of pressure drop in microchannel are specifically tailored to the pressure difference between the inlet plenum and outlet plenum, and one-dimensional conversation equations are sufficient for the pressure drop characteristics in the microchannel. Zu et al. [34] proved that the axial pressure distributions in the microchannel predicted by the 1-D and 3-D models for the same bubble growth rate are in good agreement. Therefore, for convenience, the analytical model predicting the pressure fluctuation resorts to the one-dimensional conversation equations in this paper. Additionally, as mentioned in Section 2.1, bubble growth in a rectangular microchannel can be divided into three stages, i.e. free growth, partially confined growth, fully confined growth. Barber et al. [12] investigated the relationships between the pressure fluctuation and bubble dynamics in the microchannel by mean of the simultaneous visualization and measurement, and revealed that the pressure drop was almost constant when the bubble was at the stage of free and partially confined growth. Therefore, the present model places the main emphasis on the stage of bubble fully confined growth.

In the liquid region, mass moves toward the interface at a velocity of ul1 with respect to a stationary observer, and the upstream interface is moving at a velocity of ui1 . The control volume in Fig. 2 is so thin that it can be negligible accumulation of mass within it, and the conservation of mass for this control volume defines as follow,

ql ðul1  ui1 Þ ¼ qv ðuv  ui1 Þ

This equation simply states that there is no capacity of mass accumulation at the interface. Hence, phase changes are pure exchanges of mass between two phases. The conservation equation of mass for the control volume can be converted into dimensionless form by using following dimensionless variables:

qþl ¼ ql =ql;out

ð13Þ

qþv ¼ qv =qv ;out

ð14Þ

uþl1 ¼ ul1 =u0

ð15Þ

uþi1 ¼ ui1 =u0

ð16Þ

uþv ¼ uv =u0

ð17Þ

In which, ql;out and qv ;out are liquid and vapor density respectively at the saturation temperature with respect to the channel outlet pressure, u0 is the initial liquid velocity in the channel inlet. Based upon the aforementioned dimensionless variables, the conservation equation of mass becomes,

ðuþl1  uþi1 Þ ¼

qv ;out qþv þ ðu  uþi1 Þ ql;out qþl v

ð18Þ

The momentum balance normal to the interface requires,

X Fi Ai

¼ qv ðuv  ui1 Þuv  ql ðul1  ui1 Þul1

ð19Þ

in which,

X

Fi ¼ FE  FM  Fr

ð20Þ

i¼E;M;r

where Fi is the forces acting on the interface, which includes the backward evaporation momentum force FE, forward liquid inertia force FM and surface tension force Fr, which are defined as follows,

 F E ¼ qv

2.2.1. Conversation equations for bubble interfacial At the bubble interface, the system must satisfy the conservation of mass and momentum. The transport of mass and momentum at the interface is schematically indicated in Fig. 2.

ð12Þ

F M ¼ ql

qin A 2qv hlv Ai

 2 G

ql

Ai

2 Ai

ð21Þ

ð22Þ

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H. He et al. / International Journal of Heat and Mass Transfer 102 (2016) 1024–1033

control volume

liquid phase

p plenum u0

inlet restrictor

Pin

l( ul1-dZ/dt)dAi

v( uv-dZ/dt)dAv

ul1

pb

Z

uv

vapor phase

g

interface

Wall

liquid film

X1

Fig. 2. Mass flux and force-momentum interactions at tail interface of the fully confined bubble along the microchannel.

 F r ¼ 2r

 1 1 Ai þ R1 R2

ð23Þ

R1 and R2 are the principal radii of curvature of the interface surface. Substituting Eqs. (21)–(23) into the Eq. (19) and converting it into dimensionless form, one obtains,

 2   ql;out þ þ 2 1 qþl 1 1 2 2 þ q ðA Þ ðBoÞ  q  þ þ þ v l 4 qþv qv ;out R1 R2 We ¼

qv ;out þ þ q ðu  uþi1 Þuþv 1  qþl ðuþl1  uþi1 Þuþl1 ql;out v v 1

ð24Þ

2

where Bo and We are the Boiling number and Weber number being defined as follows, respectively,

Bo ¼

qin

ql hlv u0

We ¼

Dh q2l;out u20

ql;out r

A Ai

R R ¼ Dh þ

0:67Ca3 2 3

1 þ 3:13Ca þ 0:504Ca0:672 Re0:589  0:352We0:629

ð30Þ

in which, the dimensionless liquid film thickness, dþ , denotes the liquid film thickness nondimensionalized by hydraulic diameter, i.e.,

ð26Þ

dþ ¼

ð27Þ

d Dh

ð31Þ

2.2.2.2. Liquid momentum. The momentum equation with respect to the liquid column at the tail interface side of the bubble under the condition of fully confined growth shows as below,



pin  pb þ r ð28Þ

2.2.2. Conversation equation for liquid column 2.2.2.1. Liquid continuity. In the one-dimensional model, liquid velocity profiles can be assumed uniform over the channel crosssection. Hence, the dimensionless liquid continuity for bubble fully confined growth is given by,

uþl1 14 pH00þ W 00þ ¼ uþi1 Hþ W þ

dþ ¼

ð25Þ

It should be noted that the dimensionless area, Aþ , and dimensionless radius, Rþ , in Eq. (24) are given by,

Aþ ¼

their model, and those correlations advocated by Kenning et al. [35] were based on the liquid film thickness having the constant value along the axial direction of channel. Shikazono and Han [36] investigated the local and instantaneous liquid film thicknesses (thickness of flat film region) in high aspect ratio rectangular quartz tubes with Dh  0.2, 0.6 and 1.0 mm by mean of laser confocal displacement meter, and found that the liquid film thicknesses in high aspect ratio rectangular tubes could be predicted well using the circular tube correlation provided that hydraulic diameter was used for those related characteristic numbers, i.e.

ð29Þ

As can be seen from Eq. (29), the relationship between the interfacial velocity and liquid column velocity is closely linked to the liquid film thickness below and above the growing bubble between the bubble and the heating wall. It should be noted here that the liquid film thickness is assumed constant along the axial direction of channel when bubble is fully confined. So far, much experimental effort has been directed to the study on the liquid film thickness in the microchannel, which was assumed to be constant along the axial direction of channel in their experiments. Kenning et al. [35] summarized the liquid film thickness correlations which were apt for micro tube and discussed the effect of those correlations on

1 1 þ R1 R2



¼ ql X 1

dul1 þ Dpf 1 dt

ð32Þ

The dimensionless form of the above expression by mean of dividing by ql;out u20 gives,

pþin  pþb þ

  duþ 1 1 1 ¼ qþ= X þ1 þl1 þ Dpþf 1 þ þ þ We R1 R2 dt

ð33Þ

þ where X þ 1 and t are dimensionless liquid column length and time denoted respectively by,

X þ1 ¼

X1 L

ð34Þ

tþ ¼

t L=u0

ð35Þ

Generally, the changes in pþ b with time are small enough for bubble temperature relative to reference conditions calculated from the linearized Clausius–Clapeyron equation. The frictional pressure drops, Dpþ f 1 , in the liquid columns downstream of the bubble can be calculated by,

Dpþf 1 ¼ 2f qþl ðuþl1 Þ X þ1 2

L Dh

ð36Þ

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H. He et al. / International Journal of Heat and Mass Transfer 102 (2016) 1024–1033

where f is the Fanning friction factor for steady laminar flow on a channel of cross sectional aspect ratio c with an expression as follow based upon Qu and Mudawar [1].

  f  Re ¼ 24 1  1:3553c þ 1:9467c2  1:7012c3 þ 0:9564c4

ð37Þ

Qu and Mudawar [1] summarized that the low coolant flow rate and small channel size of microchannels yielded mostly laminar flow for the liquid phase. However, it should be noted here that some investigators, such as Lee and Lee [37], Kim et al. [38] etc., considered turbulent flow on the liquid column. Actually, the liquid phase turbulent flow may be encountered in the microchannels when appearance of explosive bubble growth [30]. Therefore, when the Reynolds numbers span the laminar–turbulent transition (the conventional value of Re = 2000 for transition to turbulent flow), the calculation of frictional pressure drops in the liquid columns should use the friction factor for steady turbulent flow. However, Kenning et al. [35] conducted a series of experiments in a capillary tube with internal diameters of 0.8 mm and 0.4 mm respectively to investigate the relationship between pressure drop and bubble confined growth, and revealed that whether transition actually occurred in their experiments or was delayed by the high acceleration was not known. Additionally, Kenning et al. [35] developed two procedures to predict the pressure pulse (pressure drop) in the capillary tube by employing standard correlations for the shear stress in steady laminar flow and turbulent flow respectively, and concluded that the two procedures produced the same prediction of the pressure pulse and the turbulent shear stress caused only small increases in the maximum pressure. Therefore, the current model resorts to the friction factor for steady laminar flow to calculate the liquid phase frictional pressure drops. 2.3. Inlet condition Kandlikar [39], Tadrist [40] and Gedupudi et al. [15] proved that upstream compressibility was necessary for reversal flow. Upstream compressibility can stem from a volume of trapped non-condensable gas, an expanding component such as a bellows and even subcooled boiling in the preheater. Gedupudi et al. [15] derived the upstream compressible volume, Vc, representing many small bubbles near the superheated wall region in a preheater based upon the assumption that the evaporation and condensation occur on the small bubble, i.e.,

dV þc 2=3 ¼ Gþc ðV þc Þ ðpþin  pþin;i Þ dt þ

ð38Þ

2.4. Numerical solution To obtain the relationship between pressure characteristic of the channel and the bubble dynamics, some equations should be dealt with as follows. Substitute the Eqs. (18), (29) and (39) into Eq. (24), and the interfacial velocity can be represented by the inlet pressure as shown in the form of Eq. (42).

uþi1 ¼ f ðpþin Þ

It should be noted here that the right side of Eq. (33) is the differential form of uþ i1 by substituting the Eq. (36) into Eq. (33). Hence, to avoid the differential form of uþ i1 in the Eq. (33), the Eq. (42) should be differentiated with respect to the time and substituted into Eq. (33). Accordingly, the Eq. (33) can be rearranged as the form of Eq. (43).

dpþin ¼ f ðpþin ; X þ1 Þ dt þ

dX þ1 ¼ f ðpþin ; X þ1 Þ dtþ

ð39Þ

In contrast to the effect of upstream compressibility on the reversal flow, inlet restrictor of the microchannel, with the penalty of increased pressure drop and pumping power, can mitigate the reversal flow or eliminate the pressure fluctuation. In view of this, the current model employs a pressure drop multiplier parameter, M proposed by Kosar et al. [21], to take the effect of inlet restrictor on the pressure fluctuation into account, i.e., in the absence of reversal flow,



pþplenum ¼ pþin þ ð1:5 þ MÞ

qþl uþl1

2

2

ð40Þ

and for the case of reversal flow,

pþplenum ¼ pþin  M

qþl ðuþl1 Þ2 2

ð44Þ

The solution of the Eqs. (43), (44) is obtained through the Runge–Kutta method.

3. Model verification To validate correctness of the current model, the experimental results of Barber et al. [12], who investigated the relationships between the pressure fluctuations and bubble dynamics in the microchannel with cross-sectional dimension of 400
4000 lm (hydraulic diameter 727 lm) by employing FC-72 as the experimental working fluid, are used for verification. It should be noted here that the results revealed by Barber et al. [12] are meaningful. In conjunction with obtaining high-speed images, sensitive sensors are used to record the experimental data. Bubble nucleation and confined growth are observed in the microchannel, and both the high-speed images and the recorded data are highly synchronized. Hence, it is persuasive to explain the flow instability characterized by pressure fluctuation based upon the bubble dynamic in the microchannel. Fig. 3 is the quantitative comparisons of the experimental bubble interfacial velocity with the present model under the work condition of heat flux qin = 4.26 kW/m2, mass flow rate

0.4

Interfacial velocity of bubble (m/s)

dV þc dtþ

ð43Þ

Finally, differentiating the X þ 1 with respect to the time gives

þ where V þ c and Gc are the dimensionless compressible volume and compressibility parameter respectively. Based on the liquid continuity, the velocity of liquid column can be expressed as,

uþl1 ¼ uþl1;i þ

ð42Þ

0.3 0.2

ui1 ui2

utail unose

Present model Experimental data

0.1 0.0 -0.1 -0.2 0.10

0.15

0.20

0.25

0.30

0.35

0.40

t (s)

ð41Þ

Fig. 3. Comparison of experimental bubble interfacial velocity with the results of present model.

H. He et al. / International Journal of Heat and Mass Transfer 102 (2016) 1024–1033

0.6 0.5

0.045

ui1 ui2

0.040 0.035

bubble length

0.4 0.3

0.030 0.025

Reversed flow

0.2

0.020

0.1

0.015

0.0

0.010

-0.1

0.005

-0.2 0.15

0.20

0.25

0.30

0.35

bubble length (m)

Interfacial velocity at nose and tail (m/s)

0.7

16000

0.045 Reversed flow

0.040

12000

ui1

10000

Bubble length Pressure

8000 6000 4000 2000 0.15

0.035 0.030 0.025 0.020 0.015

Bubble length (m)

14000 Tail velocity of bubble (m/s)

_ = 1.33
105 kg/s and inlet temperature T = 34 °C at atmom spheric pressure. As elucidated in Fig. 3, the trend of interfacial velocity at both nose and tail developing with time by comparison of experimental data and present model agrees quite well, although the nose velocity of current model is slightly higher than that obtained by the experiment. Continuing this analysis, it is plausible for the comparative result because the friction between liquid and bubble is neglected in present model. However, when the length of the bubble extends to some extent, the evaporation momentum force is dominant in the forces acted on the interface of bubble and the friction between liquid and bubble makes no difference to the nose velocity. Kandlikar [41] has ever quantitatively studied the scale effects on different forces, including inertia, surface tension, shear, gravity and evaporation momentum force, applicable during flow boiling in microchannels, and found that the evaporation momentum force acting on the evaporating interface played a dominate role when bubble grew in micro/mini-channel having diameters ranging from 10 mm down to 10 lm with the working fluid of FC-72 at atmospheric pressure. Therefore, the large evaporation momentum force encountered at the inception of nucleation is responsible for the rapid interface movement and the presence of reversed flow in microchannels. Fig. 4 is the relationship between the bubble length and the interfacial velocity. General characteristics of single bubble growth in the rectangular channel with a high aspect ratio are divided into three stages subsequently as mentioned in Section 2.1, i.e. free growth, partially confined growth, fully confined growth. Once a nucleated bubble is developed, the growth period from the nucleation to the fully confined vapor bubble filling the microchannel cross-section is very short. In practice, as the bubble grows so fast, it is a major impediment to modeling and predicting the growth of bubble as well as the pressure drop in this period. Fortunately, the fully confined growth is highlighted in the present model, for which the reversed flow happens in this stage. Additionally, according to the Barber et al. [12] who investigated the relationships between the pressure fluctuations and bubble dynamics in the microchannel by mean of the simultaneous visualization and measurement, and revealed that the pressure drop was almost constant when the bubble was at the stage of free and partially confined growth, the first two stages of bubble growth can be simplified as elucidated in Section 2.1. When the bubble grows to a certain size, it begins to slide along the channel. At the beginning, the velocity of the bubble is hypothesized as same as the liquid. As the bubble evolves in length, the interface velocities both at the bubble nose and bubble

Inlet pressure (Pa)

1030

0.010 0.005

0.20

0.25

0.30

0.35

0.40

0.000

t (s)

Fig. 5. Relationship between confined bubble growth and pressure fluctuation for FC-72 at atmospheric pressure in channel 0.4
4
80 mm (heat flux _ = 1.33
105 kg/s). qin = 4.26 kW/m2, mass velocity m

tail are different. The evaporation momentum force prevents interface of the bubble tail moving to the channel outlet. For the interface at the nose, the situation is the opposite. Accordingly, when the length of the bubble extends to some extent, as mentioned above, the evaporation momentum force is dominant in forces acted on the interface of bubble and the bubble tail interface starts to recede toward the channel inlet. Therefore, the reversed flow happens, as illustrated in Fig. 4. With respect to the microchannel, the pressure drop is very sensitive to the bubble dynamics. As mentioned by Steinke and Kandlikar [30], the bubble nucleation and subsequent bubble growth will increase the pressure drop in the microchannel. As can be seen from Fig. 5, the bubble dynamic in the microchannel is crucial for the pressure fluctuation. Once a nucleated bubble is developed, the vapor bubble quickly fills up the microchannel cross-section. After fully confined growth of the bubble, it begins to expand along the microchannel and there is a sharp increase in pressure at the inlet channel due to the confined growing bubble which blocks the liquid flowing through the microchannel. When the length of the bubble extends to some extent, the evaporation momentum force can overcome other forces and cause reversed flow of the tail interface. Subsequently, the confined bubble quickly expands as well as pushes the fluid in the microchannel towards the inlet leading to an increase of inlet pressure. This process keeps on until the confined bubble is purged from the microchannel. Once the fresh liquid comes into the channel resulting in the flow of the bubble toward outlet channel, the pressure of inlet channel begins to decrease. To verify the validity of the pressure fluctuation, the experimental results of Barber et al. [12] are used again. Fig. 6 is the comparison of the experimental inlet pressure fluctuation with the present model. Obviously, the experimental data demonstrates that the reversed flow plays a more pronounced function on the pressure fluctuation than that in the present model. The inlet pressure increases steeply when the interfacial velocity of bubble tail is reversed. The inlet pressure is almost the same before the reversed flow happens, as the experimental data shown. However, the inlet pressure is closely related to the bubble dynamic as depicted before, i.e. not only the reversed flow but also the partly and fully confined growth contributes to the inlet pressure increase. Hence, the present model seems to be plausible.

0.000

t (s) Fig. 4. Variations of bubble interfacial velocity and length for FC-72 at atmospheric pressure in channel 0.4
4
80 mm (heat flux qin = 4.26 kW/m2, mass flow rate _ = 1.33
105 kg/s). m

4. Pressure fluctuation dependent factors Given that the main disadvantage of the microchannel to fabricate cooling devices is the flow instability characterized by

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H. He et al. / International Journal of Heat and Mass Transfer 102 (2016) 1024–1033

16000

100000 Present model Experimental data

14000

80000 Pressure drop (Pa)

12000 Inlet pressure (Pa)

Bo=4.5e-4 Bo=4.0e-4 Bo=3.5e-4 Bo=3.0e-4 Bo=2.5e-4

10000 8000

60000

40000

6000

20000 4000 2000 0.10

0.15

0.20

0.25

0.30

0.35

0.40

0 0.000

0.005

0.010

0.020

0.025

0.030

t (s)

t (s) Fig. 6. Comparison of the experimental inlet pressure fluctuation with the present model.

0.015

Fig. 7. Relationship between pressure fluctuation and the Boiling number for water at atmospheric pressure in channel 0.3
1.5
40 mm (nucleation site position Lþ location ¼ 0, no inlet restrictor).

4.2. Nucleation site position pressure drop and reversing flow which can lead to high amplitude temperature oscillations with premature critical heat flux (CHF) and mechanical vibrations, and solving the flow instability in microchannel (or avoiding the pressure fluctuation) continues to be a topic of current research [39]. Therefore, the following sections will discuss the pressure fluctuation dependent factors, including Boiling number, nucleation site position, crosssectional aspect ratio and inlet restrictor of the channel, based upon the current model by employing the water as the working fluid at 101 kPa exit pressure.

4.1. Boiling number Boiling number, representing the ratio of heat added to that which is required to evaporate liquid flow into vapor, is extensively used in empirical treatment of flow boiling since it combines two important flow parameters, qin and G, being defined as,

Bo ¼

qin Ghlv

ð45Þ

As summarized by Kandlikar [30], Boiling number is closely linked to another non-dimensional number, K1, proposed by Kandlikar [28], which represents the ratio of evaporation momentum to inertia forces encountered during boiling at microscale to study nucleation and bubble removal processes.

K1 ¼

FE q ¼ Bo2 l FM qv

ð46Þ

Given that the pressure fluctuation is attributed to the presence of flow reversal, and the large evaporation momentum force (i.e. large Boiling number) encountered at the inception of nucleation is responsible for the rapid interface movement and the presence of reversed flow in microchannels. Fig. 7 demonstrates the relationship between Boiling number and pressure fluctuation. As can be seen from Fig. 7, the amplitude of pressure fluctuation increases with the increasing Boiling number, and a larger Boiling number will result in a premature pressure fluctuation. Based on the criteria proposed by Wang et al. [7], and introducing the properties of water at the mean test section pressure employed, which was quite close to 101 kPa, the criteria for unstable boiling with long-period oscillations in the microchannel can be expressed as Eq. (2). Therefore, based upon Eqs. (2) and (46), a larger Boiling number is reasonable for the premature pressure fluctuation.

Generally, the microchannel interior surfaces are smooth due to the micro fabrication processes [25], and it will result in the increase of nucleation wall superheat due to the absence of cavities of all sizes. Hence, the nucleation characteristics in the microchannel are crucial for bubble dynamics and pressure drop. Once boiling is triggered, the superheated liquid will readily change phase causing an explosive vapor growth. Therefore, to avoid the explosive vapor growth caused by high superheated liquid, some researchers [26] introduce the artificial nucleation cavities of radii satisfying nucleation criteria in the microchannels to reduce the superheat needed by the onset of nucleate boiling. In practice, regardless of whether the origin of the bubble is dissolved gas or liquid phase change, bubbles even can form at specific locations along the channel walls at a few nucleation sites. However, the present model does not involve the onset of nucleate boiling (ONB) in the microchannel which is discussed by Okawa [42] at length. Actually, the initial bubble dynamic in the current model is bubble free growth, and it is estimated by the well-established model, given by Eq. (4), for spherical shape. It should be noted here that bubble at the stage of free growth grows with a spherical shape is proved by Lee et al. [31] who explored experimentally the bubble dynamics in a single trapezoid microchannel. Therefore, irrespective of the onset of nucleate boiling (ONB), the nucleation site position is of paramount importance for the pressure fluctuation in the microchannel. Fig. 8 delineates the relationship between pressure fluctuation and nucleation site position. As can be seen from Fig. 8, the magnitude of the pressure fluctuation decreases as the position of nucleation site moving downstream of the channel. As depicted by Kandlikar [24], the location of the ONB point plays an important role in the stability of the flow boiling process. Because when the nucleation occurs toward downstream of the channel, the flow resistance in the backflow direction is higher than that in the flow direction due to a longer distance from the inlet manifold. In contrast, if the nucleation occurs near the inlet plenum, the flow resistance to backflow is lower. In consideration of the pressure fluctuation being attributed to the presence of flow reversal, the magnitude of the pressure fluctuation, therefore, decreases as the position of nucleation site moving downstream of the channel. 4.3. Cross-sectional aspect ratio From the perspective of effective heat transfer, a higher aspect ratio rectangular has a higher efficiency of heat removal. However,

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H. He et al. / International Journal of Heat and Mass Transfer 102 (2016) 1024–1033

20000

80000 L+location= 0 L+location=1/8

16000

L+location=2/8

Pressure drop (Pa)

Pressure drop (Pa)

64000

L+location=3/8

48000

L+location=4/8

32000

12000

8000

4000

16000

0 0.000

M=0 M=5 M=10 M=20 M=50

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

0 0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

t (s)

t (s) Fig. 8. Dependence of pressure fluctuation on nucleation site position for water at atmospheric pressure in channel 0.3
1.5
40 mm (Boiling number Bo = 4.0e4, no inlet restrictor).

Fig. 10. The effect of pressure drop multiplier parameter on pressure fluctuation for water at atmospheric pressure in channel 0.3
1.5
40 mm (Boiling number Bo = 4.0e4, nucleation site position Lþ location ¼ 0).

based upon the research conducted by Lee et al. [25,43], the crosssection shape and size of the microchannel have notable effect on bubble formation and dynamics, flow-pattern, and etc. Therefore, from the viewpoint of pressure fluctuation dependent factor, the cross-sectional aspect ratio being the subject of study is meaningful. Fig. 9 denotes the effect of aspect ratio on the pressure fluctuation. It should be noted here that the variation of aspect ratio originates solely from the change of channel width W (i.e. the channel height is assumed to be constant in the current model) and the highest aspect ratio channel has a hydraulic diameter of 514 lm which still pertains to the category of microchannel according to Kandlikar et al. [23]. As can be seen from Fig. 9, the incipience of pressure fluctuation is delayed as the increase of aspect ratio. Given that the incipience of pressure fluctuation stems from the presence of bubble reversal flow due to bubble confined growth, therefore, it is anticipated that a smaller aspect ratio corresponds to a premature pressure fluctuation as shown in Fig. 9.

temperature needed by the onset of nucleate boiling which can avoid bubble explosive growth in the microchannel, and the inlet pressure of channel can be elevated with the inlet restrictor in response to the rapid pressure rise as the bubble nucleation begins. However, the present model does not involve the nucleation characteristics in the microchannel, and the initial bubble dynamic in the current model is bubble free growth as discussed in Section 4.2. Therefore, the present model puts emphasis on the effect of inlet restrictor on the pressure fluctuation. Fig. 10 illustrates the relationship between the pressure fluctuation and pressure drop multiplier parameter, M. As can be seen from Fig. 10, the magnitude of pressure fluctuation decreases as the increase of pressure drop multiplier parameter, however, accompanied with the penalty of increasing pressure drop. When the pressure drop multiplier parameter is equal to 50, flow instability still exists, and it is anticipated that the pressure fluctuation will be eliminated as the pressure drop multiplier parameter further increases. In practice, improved micro-fabrication techniques may make it possible to build different flow restrictions at the inlet of channel to mitigate flow instability under different work conditions.

4.4. Inlet restrictor As mentioned in Section 1, two main methods is extensively used to reduce the occurrence of flow instabilities (or eliminate or mitigate the pressure fluctuation caused by the bubble reversal flow), i.e. placing inlet restrictor and introducing nucleation cavities of radii satisfying nucleation criteria in the microchannels. The artificial nucleation cavity can reduce the liquid superheated

Pressure drop (Pa)

80000 = 1/2 = 1/3 = 1/4 = 1/5 = 1/6

60000

40000

20000

0 0.000

0.002

0.004

0.006

0.008 t (s)

0.010

0.012

0.014

0.016

Fig. 9. Pressure fluctuation as a function of aspect ratio of microchannel under the work condition of Boiling number Bo = 4.0e4, nucleation site position Lþ location ¼ 0, and with no inlet restrictor.

5. Conclusions The current paper presents an analytical model to predict the pressure fluctuation by the analysis of bubble growth, which is assumed to be composed of three stages, namely, free growth, partially confined growth and fully confined growth, in a rectangular microchannel. The interfacial velocity of bubble, being used to investigate the relationship between bubble reverasl flow and pressure fluctuation, is determined by solving the conservation equations of the momentum of the liquid column coupled with the equations of the force balance at the bubble interface, and the tail interface of bubble will reverse resulting in dramatically pressure increase when the length of fully confined bubble expands to some extent. The dependent factors, including Boiling number, nucleation site position, transverse shape and inlet restrictor, of pressure fluctuation are analyzed based upon the current model, and reveal that a smaller aspect ratio corresponds to a premature pressure fluctuation, and the magnitude of pressure fluctuation increases with the increasing Boiling number, decreases as the position of nucleation site moving downstream of the channel. In addition, the magnitude of pressure fluctuation decreases as the increase of pressure drop multiplier parameter, however, accompanied with the penalty of increasing pressure drop.

H. He et al. / International Journal of Heat and Mass Transfer 102 (2016) 1024–1033

Acknowledgments The authors are grateful for the support of the Natural Science Foundation of China (Grant No: 51376201) and the Natural Science Foundation Key Project of CQ CSTC (Grant No: cstc2015jcyjBX0130).

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