Younes and Hassan 2012 an Analytical Model of Flow Boiling Heat Transfer for Slug Flow in a Single Circular Horizontal Micro-Channel

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1 Copyright 2012 by ASME

AN ANALYTICAL MODEL OF FLOW BOILING HEAT TRANSFER FOR SLUG FLOWIN A SINGLE CIRCULAR HORIZONTAL MICRO-CHANNEL

Amen M. YounesDepartment of Mechanical and Industrial Engineering,

Montreal, Quebec, [email protected]

Ibrahim HassanDepartment of Mechanical and Industrial Engineering

Montreal, Quebec, [email protected]

ABSTRACTSlug flow is one of the most common flow patterns that

occur during flow boiling in horizontal micro-channels. In thepresent work, an analytical model of flow boiling heat transferis developed for slug flow in a single circular horizontal micro-channel under a uniform heat flux. The heat transfer is affectedmainly by the liquid film thickness confined between the vaporslug and the channel wall. For more physical and reliable flow

boiling heat transfer model, the liquid film thickness variationand pressure gradient effects on the flow boiling heat transfercoefficient are considered. The influence of vapor quality onheat transfer coefficient, vapor velocity and liquid film velocityis studied. The model is constructed based on the conservationequations of the separated two phase flow. The interphasesurface is assumed to be smooth and the flow is a laminar flow.

The obtained model applied for flow boiling of R-134arefrigerant in the slug flow at a narrow vapor qualityrange0 . 0 < < 0 . 1. The heat transfer coefficient showed ahigh increase close to the low vapor quality while decreasesgradually after the peak. Furthermore, the vapor velocityincreases linearly by increasing the vapor quality while, theliquid film velocity decreases.

NOMENCLATURE

channel cross section area, . Boiling number.

Capillary number.

Confinement number. diameter, . bubble frequency, . friction factor. mass flux, . . saturated liquid enthalpy. saturated vapor enthalpy. heat transfer coefficient, . . length, .

mass flow rate,

.

Prandtl number.

inner channel perimeter,. interfacial surface perimeter. pressure. heat flux, . Reynolds number.ui interfacial velocity, .u vapor velocity, .u liquid film velocity, .u liquid slug velocity, . vapor quality.Weber number.Greek symbols

area void fraction. mass flow rate per unit length, /.. liquid film thickness, . dynamic viscosity, ./. density, . surface tension, /. shear stress, .Subscripts bubble. critical.convective boiling dominant. liquid or liquid film.

gas or vapor phase.

hydraulic. inlet. liquid slug.LO liquid only.nucleate boiling dominant.o initial. outlet. saturated. two phase. wall.

Proceedings of the ASME 2012 International Mechanical Engineering Congress & Exposition

IMECE2012

November 9-15, 2012, Houston, Texas, USA

IMECE2012-87468

wnloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 04/29/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use


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INTRODUCTIONFlow boiling in micro-channels is an essential option for

cooling high heat flux micro-devices. Developing thermaldesign guidelines for flow boiling in micro-channels is needed.Flow boiling in micro-channels is distinguished by its uniqueflow patterns. The latter are categorized into three maincommon flow patterns which are bubbly flow, slug flow, andannular flow. Slug flow usually occurs at low and intermediatevapor quality range. This flow pattern plays a significant role inheat transfer enhancement in two-phase flow in micro-channels.

Slug flow is characterized by a travelling train consists ofvarious elongated bubbles separated by liquid slugs. Theelongated bubble is surrounded by a thin liquid film confined

by the channel surface. By browsing the literature, one can seemany experimental and few analytical studies for flow boilingin micro-channels have been performed in the last threedecades. Four major design parameters were conducted in mostof the previous studies, flow boiling heat transfer coefficient,frictional pressure drop, critical heat flux and flow pattern map.

Moriyama and Inoue [1] investigated experimentallyadiabatic flow pattern, pressure drop and heat transfer for two-

phase boiling flow of R-113 in very narrow passages with sizeof35 110 m and proposed a phenomenological model of

boiling in micro-channels. According to their experimentalresults, a sharp rising in heat transfer coefficient was observedin the single-phase region at low quality range [0-0.1], whilethere was not a significant change in heat transfer coefficient inthe two-phase region in terms of vapor quality. Additionally, theorder of two-phase heat transfer coefficient was 2 to 20 timeshigher than that of the liquid single-phase flow. This rangenarrows by decreasing channel size. Meanwhile, they revealedthat the heat transfer coefficient decreases by increasing

Capillary number at low range of values and increasesat high range of Ca values.Moreover, they developed analytical models for the two-phase multiplier of frictional pressure droplvand two-phaseheat transfer coefficient prediction for low quality region; slugflow; and high quality region; film flow. In slug flow model,they neglected the interaction between elongated bubble andliquid slug, but considered the drag force effect. Thus, themodel underestimated the experimental data that were used incomparison. Regard to the liquid-film region, the heat transfermodel was developed based on the liquid-film evaporation andconsidering the significance of surface tension effect. Theaverage liquid film velocity, interfacial velocity, interfacial

shear stress, two-phase multiplier and two-phase pressure dropwere derived and presented in terms of ratio for the filmflow and elongated bubble regions. Their predicted values andthe experimental data were in a good agreement.

Jacobi and Thome [2] proposed a heat transfer model forthe evaporation of the elongated bubble in slug flow regime inmicro channels. The thin-film evaporation was considered as adominant heat transfer mechanism. The model predicts thelocal heat transfer coefficient for the elongated bubble/liquidslug pair in a circular micro channel and it is dependent of two

unknown empirical parameters that are difficult to be estimatedanalytically; the initial liquid film thickness and thenucleation critical radius. The thin liquid film was assumedto be a uniform in the slug pair. The authors used theconduction-limited model of Plesset and Zwick [3] to estimatethe time required for creating two bubble/slug pairs which isused for bubble frequency prediction. In addition, the modewas developed by applying the void fraction equation, massand energy conservation equations on the bubble/liquid slug

pair. The authors neglected the heat transfer to the laminaliquid slug compare to the elongated bubble. Thus, their modelconsidered only the significance of the elongated bubble zonefor heat transfer process. Furthermore, the model wasconfronted to experimental data obtained by Bao et al [4] for R11, with initial liquid film thickness = 12.5 m andT = 28 Kwithin an average relative error less than 10 %Meanwhile, the authors correlated other experimental data

points for R-12 performed by Tran et al. [5] and the modepredicted their data with an average deviation less than 13% fo

T = 23 Kand

= 20 m. In general, their model was able

to predict the effects ofp,q , G and x on the heat transfercoefficient in the elongated bubble zone with a reasonableagreement depending on the estimated values of T,and .

Qu and Mudawar [6] Part I investigated experimentallythe measurement and prediction of saturated flow boiling heattransfer coefficient in rectangular cross-sectional microchannels applied for a water-cooled heat sink consists of 12micro-channels. The cross section size was231 713 m. Adecrease in heat transfer coefficient when the vapor qualityincreases was observed in a low vapor quality rangeFurthermore, a sudden transition to annular flow mode wasobserved also close to the low vapor quality range. Sixcorrelations originally developed for macro-cannels and five

others developed for mini-micro channels were used by theauthors for confronting their experimental data. Most ofcorrelations used in comparison were developed based ondomination of nucleate boiling except the one proposed by Leeand Lee [7] which was developed based on the principle offorced convective boiling domination. Besides, the tested fluidswere different as well. Thus, neither the macro-channecorrelation nor the mini-micro channel correlation captured themeasured heat transfer coefficient trend which showed adecreasing in flow boiling heat transfer coefficient when vaporquality increases. This was attributed due to the influence ofentrained droplets deposition occurs in annular flow modeAbove all, the authors revealed that forced convective boiling is

the dominant heat transfer mechanism for annular flow boilingin micro-channels.Following their work in Part I, Qu and Mudawar [8] Part

II, developed an analytical flow boiling heat transfer model forannular flow in micro-channels. The onset of annular flow inhorizontal tubes was estimated to be in a very low vapor qualityrange of 0.006 0.0064 based on a correlation developed

by Taitel and Duckler [9] that was dependent of Martinellparameter. In the meantime, the model was established basedon applying conservation equations of the mass and momentum



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for two domains, the liquid film and the vapor core includingthe entrained droplets. Both domains were assumed to belaminar and obtained equations were developed and solvednumerically based on a tentative initial value of liquid filmthickness. The principle of heat transfers conduction throughthe thin liquid film was adopted for predicting heat transfercoefficient. Above all, the model was confronted to all saturatedflow boiling experimental data obtained by the authors in part Ifor water-cooled system and the data was captured with 40 %error andMAE of 13.3 %.

Kandlikar and Balasubramanian [10] reviewed a heattransfer correlation developed by Kandlikar [11,12] for flow

boiling in conventional size channels and they extended thecorrelation to be applicable for laminar and transition flow

boiling in mini and micro-channels with a good verification ofthe correlation validity. The extended correlation was presentedas follows,, = 0.6683 .1 .+ 1058.0 .1 . 1, = 1.136 .1 .+ 667.2 .1 . 2Where h,N and h, refer to the two phase heattransfer coefficients for the nucleate boiling dominant andconvective boiling dominant respectively. The fluid surface

parameter is represented byFl. They revealed that the larger ofthe two valuesh,N and h, is the total heat transfercoefficient hfor the boiling in mini-channels 2 0 0 m d


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=0.763 . 6 = 0.00014 . 7Where , is the coalescence vapor quality and , is the

annular vapor quality. The obtained model still includes someparameters that are estimated based on empirical correlationssuch as the maximum bubble frequency

, and the liquid

slug fraction placed to the liquid film. The model wasconfronted to 980 database points for the coalescence bubbleregime for different refrigerants and fluids. The best resultsobtained by the model were for R134a data points, where 93 %of the database points were captured with 30% error band.The obtained trend for the average heat transfer coefficient inthis mode showed a decrease in heat transfer coefficient in theneighborhood close to coalescence vapor quality and thenincreases when the vapor quality increases. It is of importanceto mention to the presented expression of the bubble noselocation with respect to time that was recommended by theauthors for future researches particularly for varying heat fluxapplications.

It is known that the elongated bubble zone in slug flowregime represents the major characteristic of this regime.Further, investigation the bubble velocity and its lengthvariation is a crucial aspect in modeling this flow pattern. Theelongated vapor bubble length effects on its relative velocity forthe flow boiling of R134a in two horizontal micro-channelswith sizes of 509 m and790 m, in a low vapor quality rangeof0.02 to 0.19were investigated experimentally by Agostini etal. [21]. In addition, they proposed a theoretical modeldescribes the relation between the relative velocity of elongated

bubble and its length. They observed that the relative bubblevelocity ( ) increases by increasing the bubble lengthuntil a certain bubble length where there is no a significant

change in the bubble velocity. The bubble relative velocityincreases when the channel size increases. The proposedtheoretical model was stated as follow:

= . 1 + 1 2 2 8

Where = 4 . . 9Co is the confinement number and C is an empirical factoroptimized as C=0.58for R134a.The authors confronted themodel to 484 data points obtained experimentally for R-134aand it captured 90 % of the data within 20 % andMAE of 8.9 %

. The model can be useful for predicting bubble

frequency and transitional vapor quality range of the patternflow map for flow boiling in micro-channels.

Additional experimental investigation on the elongatedbubble hydrodynamic characteristics was performed byRevellin et al. [22]where they collected 440 experimental dataand studied the relation between the elongated bubble velocityand its length for R134-a flowing in a micro-tube with size of = 509 for a wide range of operating conditions. Inaddition they consider the effects of vapor quality, the inlet sub-

cooling, saturation temperature and the micro-evaporator lengthon the elongated bubble velocity. They confronted theirexperimental data to the model proposed by Agostini et al. [21where the model predicted 92 % of the data within =14 %. They conclude that the elongated bubble velocityincreases with increasing the vapor quality and the bubblelength while decreases by increasing the saturation temperatureMeanwhile, the micro-evaporator length did not show anysignificant influence on the bubble velocity.

In this work, a flow boiling heat transfer model for a slugflow regime in a circular micro-channel was developed basedon separated flow model. The obtained differential equationswere solved numerically for the refrigerant R-134a. The heatransfer coefficient, mean liquid film velocity, and vapor corevelocity in terms of vapor quality were presented.

MODEL ANALYSISHeat transfer mechanisms for flow boiling in micro

channels have been studied widely, as in Ref. Kandlikar [23]and Thome and Consolini [24]. In general, there is an

agreement that the dominant heat transfer mechanism for slugflow regime in micro-channels is the convective filmevaporation and the major heat transfer occurs in this zonerather than the liquid slug zone. Thus, in this problem, theelongated bubble zone is modeled using two-phase separatedflow model while the liquid slug zone is considered as a liquidsingle phase. The following assumptions are the current and

previous assumptions that have been adopted for simplifyingthe slug flow problem analysis:1. The flow is a one dimensional steady fully developed flow

boiling and The model domain is discretized into twozones. The elongated bubble zone and the liquid slug zoneas depicted in Fig. (1).

2. The elongated bubble zone is considered as a two-phaseflow and a one dimension separated flow model is appliedfor this zone.

3. The liquid slug moves by a liquid velocity named as uandthis zone is treated as a single-phase flow zone.

4. The bubble moves by a mean vapor velocity named as uand is elongated enough, thus its tail and nose effects areneglected.

5. The liquid film trapped between the elongated bubble andthe channel wall moves by a uniform velocity named as uand its thickness is varied due to influence of inertia force

interfacial stress and pressure gradient.6. The micro-channel is subjected to a uniform and constan

heat flux.7. The liquid slug and liquid film are at saturated status. Thus

the liquid and vapor enthalpies derived in energy balanceare the saturated enthalpies.



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8. The body force, virtual mass, and the interfacial waves areneglected and the interphase surface is assumed to besmooth.

Figure 1: schematic diagram of a slug unit cell.

A control volume element in the elongated bubble zone ischosen and the three conservation equations, mass, momentumand energy equations, for each phase are derived in awaysimilar to that presented by Ghiasiaan [25]. The chosen controlvolume for deriving the mass balance equations for the vaporcore and liquid film is depicted in Fig (2). The fraction of liquidslug flow rate feeding the liquid film is taken into account andthe mass equations of the vapor core, Eq.(10), and liquid film,Eq.(11), can be written as follows: []= 10 [1 ] = + 1 11

Whereml, is the portion of the liquid mass transfer fromthe liquid slug feeding the liquid film, and it can be estimatedas, = 1 ( ) 12

Considering the steady flow state and adding Eq. (10) toEq. (11), the mass balance equation for the whole mixture can

be given as in Eq. (13). [ + 1 (2 )]=0 13The acting forces on the chosen control volume on both

liquid film and vapor core phases are shown in Fig.3. Twoforces were neglected here; the body and virtual mass forces.The body force does not have a significant effect in micro-scalesize, so it was neglected. While the virtual mass force wasneglected for model simplifications. The momentum equationsfor the elongated bubble and liquid film are written as shown inEqs. (14) & (15) respectively.

[

] =

+ 14

1 1 ( ) = 1 + 15The momentum equation of the whole mixture is obtained

by adding Eq. (14) to Eq. (15) and presented as in Eq. (16). [+21 1 ] = 16

Figure 2: a symmetric sketch of the chosen control volume

for applying mass balance.

Figure 3: a symmetric sketch shows the forces effects onthe control volume.

The fraction of liquid mass flows from liquid slug feedingthe liquid film in the elongated bubble zone was neglected inenergy balance equation for simplification. Thus, the energy

balance equation for the whole mixture can be represented byEq. (17) as,

+ 2 + 1 + 2 = 17In the elongated bubble zone, four main parameters have

been chosen as main variable parameters in terms of heatedlength. These parameters are bubble phasic velocity, liquidfilm velocity, pressure variation in elongated bubble zoneand the area void fraction. The model was established basedon the above five equations: The continuity equation of thevapor phase; Eq. (10), the momentum equation of the vapor

phase; Eq. (14), the whole mixture basic equations, continuityequation Eq. (13), momentum Eq. (16), and energy equationEq. (17). The volumetric mass transfer rate at the interphase

was substituted by = in Eq. (14). Further, thederivation of equations was extended and organized in fourmain equations as follows,

+ 21 + ( 2 + ) =0 182 21 +

= [1

]

= [] = ,,

, ,

= 0

2

[1 ]

[] []

[ ]

[] 1 1

= 0

(

)

[1 ]



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+( 2 + ) = 192( ) + ( ) =

20

+ 32 + 1 + 32 + + 2 +

2 = 21The developed system of differential algebraic equations;

equations (18),(19),(20) and (21); are organized andformulated in a mass matrix form. A MATLAB code wasdeveloped for solving the set of equations using a MATLABfunction that implements explicit Runge-Kutta method. Theobtained results are affected by the initial conditioned estimated

based on the used correlation as will be explained in thefollowing section.

INITIAL VALUES AND CONSTITUTIVE PARAMETERS

The parameters, wall shear stress, the interfacial shearstress, the interfacial velocity and the wall liquid frictionfactor are needed to be calculated in order to solve therequired variable parameters , , , and . Thus, the

proper correlations available in the literature are used.Furthermore, the most sensitive initial parameter affects theobtained results is the initial liquid film thickness. Beginningwith the interfacial parameters, the interfacial velocity isestimated to be

= 1 2 ( + ) 22

While the interfacial shear stress can be calculated using thefollowing relation, = 12 ( ) 23

The interfacial friction factor is predicted by employing thecorrelation proposed by Wallis [26]. = 1+300 24Where =0.005 for annular flow in conventional sizechannels and the initial liquid film thickness is estimated usinga modified Taylors law presented in Aussilous and Quere [27].

=(1.34

(1+1.342.5

) ) 25

It is of importance to mention that Capillary number here iscalculated as = ( )based on the superficial vaporvelocity. The initial vapor and liquid film phasic velocitiesare calculated as the following, = ( ) 26 = [1 1 ] 27

Regards to the void fraction, it should be noted that voidfraction estimation for two-phase flow in micro-channels is stilla challenge. Further, most of void fraction databases available

in the literature are based on visual studies and dependent offlow pattern. However, if the bubble nose and tail effects areneglected as a simplification, the initial void fraction for theelongated bubble zone can be calculated as follow, = 2 28

The wall shear stress

w is represented in terms of liquid

film velocity as given in Eq. (29).

= 12 29Where, is the liquid film friction factor and is determined

in terms of liquid film Reynolds number for laminar flow = ( 6 4 ) 30The hydraulic diameterused in Reynolds number can be

defined in terms of void fraction as given in Eq. (31). = ( 1 ) 31Considering the void fraction definition given in Eq. (26)

and the liquid film Reynolds number in terms of annulushydraulic diameter that is given in Eq. (31), the wall shearstress for laminar flow in the elongated bubble zone can be

presented in terms of void fraction as follows, = 34 (1 ) 32RESULTS AND DISCUSSION

The above set of differential equations Eqs. (18), (19), (20)and (21) are solved; numerically using a MATLAB code; forthe saturated refrigerant R-134a flows in a circular micro-channel with size of D = 540 m and operating conditionssimilar to that adopted by Bertsch et al. [28] for T =30andp = 550 kPa,G = 84 kg m. s , q" = 5 2 k W m ; Anaverage bubble length of

= 30 is considered. The

obtained results are depicted and discussed as follows.The flow boiling heat transfer coefficient is predicted bythis model for extended range of vapor quality as is shown inFig. 4. It can be noted that the heat transfer coefficient increase

by increasing the vapor quality in the low vapor quality regionreaching the peak at = 0 . 1and then decreases gradually.

Fig. 4 Heat transfer coefficient for flow boiling of R134-

a at = . , = . in slug flow regimefor average bubble length = . Comparison of thecurrent model with Kandlikar and Balasubramanian [10and Bertsch et al. [28].

4000

5000

6000

7000

8000

9000

10000

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Heattransferc

oefficient,[W/m2.K

]

Vapor quality

Current Model Kandlikar and Balasubramanian [10] Bertsch et al. [28]



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The model was compared with the heat transfer correlationdeveloped by Kandlikar and Balasubramanian [10] and the data

base points obtained by Bertsch et al. [28] for low Reynoldsnumber where the nucleate boiling is dominant. It is ofimportance to remind that the heat transfer coefficient is mainlydependent of the estimated initial value of the liquid filmthickness, which is assumed to be

= 18 here. The

predicted heat transfer coefficient has a trend similar to thatobtained by Bertsch et al [28], particularly at low vapor qualityrange0 . 0 0 . 1. In other words, the model predictedwell the experimental data obtained by Bertsch et al [28] within = 8.03 % in the vapor quality range0 . 0 < < 0 . 5.This can be due to the fact that the model was developed

basically for the low and intermediate vapor quality range.Meanwhile by applying Kandlikar and Balasubramaniancorrelation [10] the Mean Average Error was, = 21.16 %and showed that the heat transfer coefficient decreases fast byincreasing the vapor quality.

Fig. 5 shows the influence of vapor quality on both thevapor mean velocity and liquid film mean velocity obtained by

the current model. It is seen that the mean vapor velocityincreases by increasing the vapor quality, while the liquid filmvelocity decreases. Furthermore, the magnitude value of vaporvelocity is higher than that of the liquid film. This can beattributed to the significant wall shear effects at low mass flux.

Fig. 5 depicts the vapor quality effect on both vapor andliquid film velocities for flow boiling of R134-a, at = . , = . in slug flow regime foraverage bubble length = .

The mean vapor velocity trend predicted by the model wascompared with that obtained by Revellin et al. [22] as shown inFig. 6. The original experimental data points were obtained forR134-a flowing with a high mass flux = 1000 /. in amicro-tube with size of 509 subjected to maximum heatflux equal to = 81.3 / and saturated pressure =770 in the low vapor quality range 0 . 0 < < 0 . 1.

The model accurately predicted bubble velocity trend.However, it overestimated the experimental data with =56.7%. This can be attributed to two main parameters, theestimated fraction of mass feeds the liquid film predicted by

Eq. (12) and the initial liquid film thickness which assumed forthese operating conditions to be = 35 .

Fig. 6 Shows the comparison between vapor velocityobtained by the current model and that one revealed byRevellin et al. [22] for flow boiling of R134-a, at

= . , = . for average bubblelength = and initial liquid film thickness = .CONCLUSIONS

An analytical heat transfer model of flow boiling for slugflow in micro-channels subjected to uniform heat flux wasdeveloped. The influence of vapor quality on heat transfercoefficient, vapor velocity and liquid film velocity wasaddressed. The following conclusions can be drawn.1. The heat transfer coefficient for low Reynolds number

conditions increases by increasing the vapor quality in thelow vapor quality region reaching the peak at

= 0 . 1and

then decreases gradually.2. The model predicted well the experimental heat transfer

data obtained by Bertsch et al [28] at low mass flux = 84 /. within = 8.03 % in the vaporquality range0 . 0 < < 0 . 5 employing the estimatedinitial liquid film thickness = 18 .

3. The vapor velocity increases linearly by increasing thevapor quality while the liquid film velocity decreases. Themodel overestimated the vapor velocity obtained byRevellin et al. [22] at high mass flux = 1000 /. within = 56.7 % in the vapor quality range 0 .0


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Documents

Younes and Hassan 2012 an Analytical Model of Flow Boiling Heat Transfer for Slug Flow in a Single Circular Horizontal Micro-Channel