Radulescu and Robinson 2008 Numerical Study of Marangoni - Thermocapillary Convection Influence During Boiling Heat Transfer in Minichannels

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1 Copyright 2008by ASME

Proceedings of the Sixth International ASME Conference on Nanochannels, Microchannels and MinichannelsICNMM2008

June 23-25, 2008, Darmstadt, Germany

ICNMM2008-62244

NUMERICAL STUDY OF MARANGONI THERMOCAPILLARY CONVECTION INFLUENCEDURING BOILING HEAT TRANSFER IN MINICHANNELS

Cristina RadulescuTrinity College Dublin, Ireland

Mechanical and Manufacturing [email protected]

Anthony J. RobinsonTrinity College Dublin, Ireland

Mechanical and Manufacturing [email protected]

ABSTRACT

Marangoni thermocapillary convection and its contributionto heat transfer during boiling has been the subject of somedebate in the open literature. Currently, for certain conditions,such as microgravity boiling, is being shown that has a

significant contribution to heat transfer [1]. Typically, thisphenomenon is investigated for the idealized case of an isolatedand stationary bubble resting atop a heated solid which isimmersed in a semi-infinite quiescent fluid or within a two-dimensional cavity. However, little information is availablewith regard to Marangoni heat transfer in miniature confinedchannels in the presence of a cross flow. As a result, this paperpresents a numerical study that investigates the influence ofsteady thermal Marangoni convection on the fluid dynamicsand heat transfer around a bubble during laminar flow of water

in a minichannel with the view of developing a refinedunderstanding of boiling heat transfer for such a configuration.This mixed convection problem is investigated for channelReynolds numbers in the range of 0 Re 500 and Marangoninumbers in the range of 0 Ma 17114. The influence of thethermocapillary flow is most pronounced for low Re and highMa numbers showing an average of 40% increase in heattransfer. For low Ma and high Re inertial effects dominate andthe thermocapillary effect is not as noticeable. However, thedisruption of the fully developed flow does tend to enhance theheat transfer at the expense of additional pressure drop.

INTRODUCTIONBased on the experimental results published in 1855 by

Thomson [2], C. G. M. Marangoni [3] later offered a viableexplanation of the effect of surface tension on drops of oneliquid spreading upon another [4]. Subsequent to this severalnumerical and experimental studies [5, 6, 7] established thatthermocapillary Marangoni convection is in fact real physicalphenomenon at gas-liquid and liquid liquid interfacesresulting from gradients in surface tension. The surface tensiongradients can be brought about by variations in the liquidconcentration or temperature. Once the existence of this

phenomenon was confirmed the main focus of the scientific

research work was to quantify the impact on the heat transferenhancement. Starting with the experimental results oMcGrew et. al.[8] and followed by the early analytical work ofLarkin [9], thermal Marangoni convection and its contributionto heat transfer during boiling became the subject of somedebate in the open literature [10]. Recently, it has beenestablished that for certain conditions, such as microgravityboiling, the thermocapillary induced flow has associated with ia significant enhancement of heat transfer due to the liquid flowin the vicinity of the interface [11, 12 and 13]. Despite theresearch conclusions presented in literature there is stilinsufficient information available with regard to Marangoniflow contribution on the heat transfer in miniature confinedchannels [14, 15] especially when it takes place in the presence

of a cross flow [16].Boiling heat transfer in minichannels has become an

increasingly important topic due to its application in thecompact heat exchanger design such as those required forelectronics thermal management. Typically, for electroniccooling applications involving two-phase flow, nucleate boilingis the preferred regime of operation because the small increasein wall superheat is accompanied by a disproportionately largeincrease in the wall heat flux [17, 18]. Apart from the high ratesof heat transfer at relatively low volumetric flow rates theisothermal nature two-phase convective boiling makes this avery attractive technology in contrast with single-phase channecooling.

The objective of this paper is to provide qualitativeinformation regarding the fluid motion and the influence onheat transfer around the centerline of a bubble placed on thebottom heated wall of a rectangular section minichanne(1x1x20 mm) in a cross flow configuration as illustrated inFigure 1. This mixed convection problem is investigated forchannel Reynolds numbers in the range of 0 Re 500 byincreasing the inlet mass flow rate. For a fixed inlet temperaturethe Marangoni number is varied in the range of 0 Ma 17114by increasing the wall temperature of the channel. Furthermore

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the influence of the bubble dimension on the flow pattern andheat transfer is taken into consideration for the followinggeometrical characteristics: Rb/H = 0.1 (B_1), Rb/H = 0.5 (B_2)and Rb/H = 0.75 (B_3).

Figure 1.Physical domain showing a hemispherical bubble near theentrance of the minichannel

NOMENCLATUREB_1 first bubble under considerationB_2 second bubble under considerationB_3 third bubble under considerationCa capillary numberhMa height of Ma recirculation cell [m]

H height of the channel [m]k thermal diffusivity [m

2/s]

lMa length of Ma recirculation cell [m]L length of the minichannel [m]Ma Marangoni number (thermocapillary)n unit normal vector (height of the boundary)

p static pressure [N/m2]

Pr Prandtl numberRb bubble radius (RB_1RB_2RB_3radius of

B_1, B_2, B_3 respectively)[m]Re Reynolds number

R surface tension Re numberl

brefl RV

[16]

Tm avg bulk liquid temperature at inlet [C]Twall heatedwall temperature [C]Vavg liquid inlet average velocity [m/s]Vref reference thermocapillary velocity

l

T T

| [16] [m/s]

u velocity vectorX, y Cartesian coordinatesT temperature difference Twall- Tm [C]

Greek Symbols stagnation point angle dynamic viscosity [Ns/m

2]

liquid surface tension [N/m]T surface tension gradient [N/mK] density [kg/m3] kinematic viscosity [m

2/s]

Laplace divergence operator

Subscriptsavg averageb bubbleg gasl liquidm bulk liquid

Ma Marangoniref referenceT tension

MATHEMATICAL FORMULATIONFigure 1 shows the simplified schematic of the channe

through which water at a mean inlet temperature Tmflows withthe average velocity Vavg. The flow is assumed to behydrodynamically fully developed at the inlet with a parabolicvelocity profile. The heated bottom wall is maintained at aconstant temperature - Twallwhile the top wall is considered tobe insulated. The hemispherical gas bubble is situated on theheated wall creating a cross flow configuration due to the bulkliquid flow directed perpendicular to the bubble axis. This is

located near the entrance of the channel since this is theexpected region of the nucleate boiling flow regime [19] withstratified or slug flow regimes being more likely downstream. Iwas concluded that the bubble nuclei grow slowly to visiblesize in the laminar inlet flow [19]. As a result the flow and heattransfer problem has been simplified by considering steadystate conditions for three different dimensions of the bubblestarting with the incipient stage of growth and at the final stagebefore bubble sliding is anticipated [14]. The bubble shapedeformation is neglected based on the assumption that thecapillary number Ca =

TavgV / is much less that unity.

Lastly, it is well known that the flow around the bubblewithin a small channel is inherently a threedimensiona

problem. However, at the mid-plane of the spherical bubble theflow is approximately twodimensional. In this respect theproblem can be treated qualitatively as a twodimensionaphenomenon as consistent with the work of Bhunia andKamotani [16].

Governing EquationsThe governing continuity, momentum and energy

equations for steady flow are [22, 23 and 24]:The continuity equation is solved in the following form:

u = 0 Eq. 1

Conservation of momentum is described byEq. 2:

upuu

21

+= Eq. 2

The energy equation without internal heat generation is reducedatEq. 3:

TkTu2

= Eq. 3

The governing equations were solved subject to the followingassumptions and boundary conditions: i) the bubble isrepresented as a hemispherical gas-liquid interface, ii) the heaflux zero at the bubble interface, iii) the top wall is insulatediv) the inlet flow is hydro-dynamically fully developed, v) thebottom heated wall is at a constant temperature, vi)gravitational affects are negligible. vii) temperature variations

in physical properties are not considered, viii) the liquid isincompressible and ix) the shear stress in the liquid at the gas-liquid interface of the bubble is balanced by the surface tensionsuch that the boundary condition at the bubble interface is a noslip and a directly applied Marangoni stress given as [24, 27]:

TTn

unu

=

=

0 Eq. 4

Here, nis the unit vector normal to the bubble interface.



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The governing equations were solved numerically with theCFD package Fluent Version 6.3.26 [20] and the physicaldomain and grid was created in the Gambit Version 2.2.30[21].Cartesian coordinates were used with a non-uniform grid of14400 cells and 29420 faces. In order to resolve the flow andtemperature fields accurately, grid clustering near the bubblewas implemented. The accuracy of the resulting simulationswas been confirmed by reproducing the numerical results ofBhunia and Kamotani [16].

RESULTS AND DISSCUSIONThe results presented in this study have been carried out

for water which has a surface tension gradient of d/dT= -0.1477x10

-3 N/mK. The problem is investigated for channel

Reynolds numbers in the range of 0 Re500 by increasing theinlet average velocity between 0.1m/sRe0.5m/s. TheMarangoni number is varied in the range of 50 Ma 17114 byincreasing the difference between the inlet temperature and thewall temperature of the channel (T= Twall-Tm) between 1CTw30C. Consistent with the works of [25, 26 and 27] theMarangoni number has been defined as:

rbmwall PR

H

RTTTMa

=

=

2))(/( Eq. 5

To approximate different stages of bubble growth i.e.nucleation to bubble sliding [14], the bubble size relative to thechannel height has been investigated for Rb/H=0.1 (B_1),Rb/H=0.5 (B_2) and Rb/H=0.75 (B_3).

The primary objective of this study is to provide aqualitative description of the effect of thermocapillaryconvection on the flow field and heat transfer during bubblegrowth in a miniature channel.

The Effect of Marangoni Convection on the Flow FieldFigure 2 presents the streamlines for steady flow around

the bubble for the case Rb/H = 0.1, Re= 100 and Ma = 0, 50,

100, and 300. To provide a baseline case for comparison,steady flow around a bubble with no thermocapillary effect hasbeen simulated for this test case and each test case to follow.This is equivalent to imposing a constant surface tension

(/T=0) such that Ma=0 even though there are temperaturegradients along the interface. Due to these, the surface tensionis highest near the top of the bubble and lowest near the heatedwall. This surface tension variation generates thermocapillaryflow along the bubble surface away from the hot wall towardsthe bulk liquid. Figure 2 illustrates clearly that increasing thedriving potential for thermocapillary flow, which in thissituation is T= Twall-Tm, the influence of the surface tensiondriven flow becomes stronger which is apparent from theincreased deformation of the streamlines as compared with the

baseline Ma=0 case.Upstream (front) side of the bubble: in this region the

thermocapillary action accelerates the liquid flow along thebubble surface. The shear driven flow at interface has the effectof drawing the relatively colder bulk liquid downward towardsthe front corner of the bubble as apparent from the deformationof the near wall streamlines towards the hot front corner of thebubble for the Ma 50 and 100 cases. Downstream side of thebubble: in this region a sizable vortex is formed even at lowMarangoni numbers (Ma=50) when the recirculation cell is

strong enough to cross over the line of symmetry of the bubbletowards the front region. For the highest Marangoni numberobtained for Rb/H = 0.1 (i.e. Ma=300) the high shear rate at thefront bubble interface interacts with the strong rear vortex toform a recirculation cell near front region of the bubble. Withregard to the rear vortex itself increasing Ma has the effect oincreasing the strength of the vortex, as is evident from theincreasing concentration of the streamlines, as well asincreasing the vortex size as it is seen to penetrate deeper intothe bulk liquid and along the heated wall as well as moving

forward along the bubble surface. In order to visualize thevelocity magnitude on x coordinate, Figure 3 shows the case ofMa = 100 together with an information of the geometricadimensions of the recirculation thermocapillary cell.

Figure 2:Streamlines of steady flow for Rb/H = 0.1(B_1) at Re=100

Near the downstream end the surface velocity due to thebulk liquid flow it is in an opposite direction with therecirculation cell created by the thermocapillary flow. The poin

on the bubble surface where the forward and reverse flow meeis known in literature as the stagnation point [16].

As is evident in Figure 2, and which will be discussed laterin more detail, this separation point appears to move closer tothe front region of the bubble with increasing Ma. Figure 3presents the influence of the bubble dimension (B_1 and B_3)on the stagnation point position for the same Re = 100 andtemperature difference = 10C. It is noticed that this point issituated closer to the front region of the bubble (at higherseparation angle ) for the smaller bubble dimension (B_1).



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Figure 3. Magnitude of x-velocity and stagnation point position forB_1, B_3 at Re 100 and T=10C

Figure 4 illustrates the influence of inertial effects of thechannel flow by considering the identical configuration as inFigure 2 but for the higher Reynolds number case of Re=300.


As one would expect the increase in the cross-flowvelocity has an important impact on the flow structure bysuppressing the influence of the Marangoni flow. This is clearconsidering that for Re=300 it takes a Marangoni number of

Ma=300 to roughly reproduce the flow structure that aMarangoni number of Ma=100 was able to produce forRe=100.

With the view of future development of a dynamic bubblegrowth model it seemed instructive to investigate the influenceof the bubble size on the flow and heat transfer within thechannel for this idealized case of steady two-dimensional flowFigures 5 and 6 show the simulated flow patterns aroundbubbles with aspect ratios Rb/H=0.5 and 0.75 respectively for afixed Reynolds number of Re = 100.

Increasing the relative size of the bubble from Rb/H = 0.1to Rb/H = 0.5 has a notable influence on the flow pattern asevident from Figure 5. It must first be noted that the Manumber increases disproportionately compared with theincrease in T and Rb/H since the length scale has been chosenas Rb

2/H for this study to incorporate the influence o

confinement on the flow and heat transfer. Compared with therelatively unconfined case depicted in Figure 2 for Rb/H = 0.1the flow structure within the liquid for the more confined caseof Rb/H = 0.5 indicates that the thermocapillary inducedconvection has a more profound influence on flow for a likedriving temperature differential and a significantly higher Ma.


The presence of the confining upper wall tends to formelongated yet more concentrated recirculation zones at thedownstream end of the bubble compared with the moreunconfined case at identical T. For T=30C, Figure 5 shows



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that confinement effects result in the formation of threerecirculation zones, in particular, an elongated vortex spanninga considerable portion of the top region of the bubble interface.

Figure 6 presents the extreme situation of Rb/H = 0.75where the bubble is at its maximum growth dimension beforesliding in the minichannel due to the inertial forces of the bulkliquid flow would be expected [14]. It is noticed that the flowpattern is altered notably compared with the previous twocases. In this case the confining and adiabatic top wall tends torestrict the rear vortex from encroaching on the frontal region

with increasing T. This pinching of the rear vortex is strongenough that for the highest driving potential of T=30C nosecondary recirculation zones develop since the separationpoint is far enough back on the bubble interface that it does notsignificantly obstruct the frontal shear flow. This is alsoillustrated in Figure 3 where it is shown that the stagnationpoint on the bubble surface, where the forward and reverseflows meet, has the tendency to occur at smaller separationangles, , for the larger bubbles at identical T and Re.

Figure 6.Streamlines for steady flow for Rb/H = 0.75(B_3) at Re=100

The Effect of Marangoni Convection on the Thermal Fieldand Wall Heat TransferThe flow and thermal fields are directly coupled via the non-linear convection term inEq. 3and more indirectly through theMarangoni stress boundary condition given in Eq. 4. As aresult, the interaction of the flow and thermal fields must beunderstood in relation to one another in order to elucidate the

effect on less global parameters such as the stagnation anglebubble surface temperature distribution and resulting wall heatransfer in the vicinity of the bubble.

Figure 7 illustrates the thermal profile for Rb/H = 0.1Re=100 and Ma=0, 50 and 300 case for it is noticed the highervalues of the temperature profile liquid boundary layer formedclose to the heated wall. For Ma=0 the presence of the bubblehas a small influence on the thermal field, acting as a simpleobstruction to the flow. However, for the same waltemperature but with Marangoni convection, a non-symmetric

jet of warm fluid is forced into the bulk of the flow. The size othe warm jet is consistent with the size of the vortices observedin Figure 2 and the penetration depth of the warm jet increaseswith Ma in the same way as the size of the recirculation regionsincreased in Figure 2. Near the front edge of the bubble it isclearly evident that the deformation of the streamlines observedin Figure 2 has associated with it the drawing in of the coolerbulk liquid which will have important implications with regardto the wall heat transfer.

Figure 7:Temperature profile for Rb/H = 0.1 at Re= 100 and Ma=0,

50 and 300.

Figure 8 (a, b) presents the wall heat flux distribution forRb/H = 0.1, Re=100 and 300 with Ma=30, 50, 100 and 200. Thecorresponding heat flux distribution for the situation of noMarangoni flow is also plotted for each temperaturedifferential. It is clear that the thermal and flow fields resultingfrom thermocapillary convection have a direct impact on theheat transfer in the vicinity of the bubble as is evident from thepeaks in the wall heat flux that appear around the bubble.



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(a) B_1at Re=100 and Ma = 30, 50, 100, 200

(b) Heatflux [kW/m2] for B_1at Re 300 and Ma = 30, 50, 100, 200

Figure 8:Wall heatflux for B_1 at Re = 100 and 300.

Upstream (front) side of the bubble: in this region theincrease in the heat flux is stronger due to the combined effectof forced and thermocapillary convection with thethermocapillary component drawing the cooler bulk liquidtowards the heated wall and thinning the thermal boundary

layer as depicted in Figures 2, 4 and 7. For Re=100 a two tothree fold increase in the local heat flux is evident. As expected,the enhancement decreases somewhat with increasing Re,however it is still substantial. Downstream of the bubble theincrease in the wall heat transfer is only evident for the higherMa because it is primarily the warm liquid that was extractedfrom the frontal region that is being recirculated in this area.Here the local heat flux increases by a factor of nearly 1.2 forRe=100 and Ma=200 and tends to improve with increasing Rewith a 1.4 times improvement for Re=300 and Ma=200.

The bubble dimensionless interface temperaturedistributions for Rb/H = 0.1, Re=100 and Ma=0, 30, 50, 100,200 and 300 are plotted in Figure 9. Considering the Ma=0 andMa=50 case which both correspond with T=5C it is clear that

the Marangoni convection tends to diminish the thermalgradients thus reducing the magnitude of T

/over the majority of

the bubble surface. For Ma>0 the general shape of thetemperature profiles are similar. In the front region of thebubble the gradients are steep due to the combined influencesof the forced and thermocapillary convection as cold bulk fluidis drawn towards the surface and accelerated along it. Thesurface temperature decreases to a minimum at the stagnationpoint. Behind the stagnation point the temperature gradientsalong the bubble surface are less steep as the flow transitions

from a combined convection region to a dominantlythermocapillary driven recirculation region where the averageliquid temperature is generally much higher than the bulkliquid. Increasing the Marangoni number has two notableeffects on the bubble surface temperature profile. First thestagnation point moves forward with initial increases in MaHowever, increasing Ma beyond Ma=200 has a minimal effecton the position of the separation point. Secondly, increasing Matends to flatten the temperature profile along the bubbleinterface with large gradients isolated to the front and rear of

the bubble.

Figures 9:Bubble interface temperature B_1 forRe=100 and Ma = 0,30, 50, 100, 200 and 300

Figure 10 shows the effect of the top confining wall on thethermal field around the bubble for which Rb/H = 0.5 for thesame T values in Figure 7.

Figure 10: Temperature distribution for B_2 at Ma =0, 1260, 7600.



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Compared with the unconfined small bubble, the confinementeffects tend to keep the warm recirculation zone near the rear ofthe bubble with strong effects on the front region where theshear flow causes entrainment of the cold bulk fluid towardsthe wall which thins the thermal boundary layer in this regionmore so than for the Rb/H = 0.1 case.

Figure 11 illustrates this point more clearly where the heatflux profiles for Rb/H = 0.1, 0.5 and 0.75 are plotted forRe=100 and T=10C. It is clear that the peak heat flux at thefront of the bubble increases notably with increasing Rb/H. The

peak local heat transfer enhancement at the rear of tends toincrease with increasing Rb/H but to a much lesser extent.

Figure 11: Heat flux profiles for Rb/H = 0.1, 0.5 and 0.75 at Re=100and T=10C.

CONCLUSIONSThis paper presents a twodimensional numerical model

that investigates the influence of steady thermal Marangoniconvection on the fluid dynamics and heat transfer around abubble during laminar flow of water in a rectangularminichannel. This mixed convection problem is investigated forchannel Reynolds numbers in the range of 0 Re 500 andMarangoni numbers in the range of 0 Ma 17114. Thethermocapillary effect has a significant impact on heat transferfor this configuration with an average increase of 35% in theheat flux figures at the downstream of the bubble while thecombination between thermocapillary and forced convectionmechanisms results in an average of 60% increase at the front

of the bubble. Further extension of the present work is expectedbased on a three-dimensional model capable to quantifyprecisely the impact of this flow on heat transfer enhancement.

ACKNOWLEDGEMENTSWe gratefully acknowledge the support from Science

Foundation Ireland that sponsored this research through grantnumber ENMF 249. We acknowledge the help provided withthe CFD software by our colleagues Seamus OShaughnessyand Geoff Bradley.

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http://www.fluent.com/http://www.fluent.com/http://www.fluent.com/software/gambit/index.htmhttp://www.fluent.com/software/gambit/index.htmhttp://www.fluent.com/software/gambit/index.htmhttp://www.fluent.com/http://www.fluent.com/http://www.fluent.com/software/gambit/index.htmhttp://www.fluent.com/

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Radulescu and Robinson 2008 Numerical Study of Marangoni - Thermocapillary Convection Influence During Boiling Heat Transfer in Minichannels