Holographic interferometric study of heat transfer to a sliding vapor bubble

International Journal of Heat and Mass Transfer 55 (2012) 925–940

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International Journal of Heat and Mass Transfer

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Holographic interferometric study of heat transfer to a sliding vapor bubble

Sathish Manickam, Vijay Dhir ⇑Mechanical and Aerospace Engineering Department, Henry Samueli School of Engineering and Applied Science, University of California, Los Angeles, CA 90095, USA

a r t i c l e i n f o

Article history:Received 3 September 2010Received in revised form 20 May 2011Accepted 20 May 2011Available online 13 November 2011

Keywords:Sliding bubblesHolographic interferometryMechanistic models

0017-9310/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.ijheatmasstransfer.2011.10.016

⇑ Corresponding author. Tel.: +1 310 825 9617; faxE-mail address: [email protected] (V. Dhir).

a b s t r a c t

Heat transfer associated with a vapor bubble sliding along a downward-facing inclined heater surfacewas studied experimentally using holographic interferometry. Volume growth rate of the bubbles as wellas the rate of heat transfer along the bubble interface were measured to understand the mechanisms con-tributing to the enhancement of heat transfer during sliding motion. The heater surface was made of pol-ished silicon wafer (length 185 mm and width 49.5 mm). Experiments were conducted with PF-5060 astest liquid, for liquid subcoolings ranging from 0.2 to 1.2 �C and wall superheats from 0.2 to 0.8 �C. Theheater surface had an inclination of 75� to the vertical. Individual vapor bubbles were generated in anartificial cavity at the lower end of the heater surface. High-speed digital photography was used to mea-sure the bubble growth rate. The temperature field around the sliding bubble was measured using holo-graphic interferometry. Heat transfer at the bubble interface was calculated from the measuredtemperature field. Results show that for the range of parameters considered the bubbles continued togrow, with bubble growth rates decreasing with increasing liquid subcooling. Heat transfer measure-ments show that condensation occurs on most of the bubble interface away from the wall. For the param-eters considered condensation accounted for less than 12% of the rate heat transfer from the bubble base.In this study the heater surface showed no drop in temperature as a result of heat transfer enhancementduring bubbles sliding.

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1. Introduction

Boiling is a phase change process associated with very high heattransfer rates. Recent focus on boiling studies has been to break thecomplex process into sub-processes. Nucleation, bubble growth,bubble departure and dynamics and their influence on heat trans-fer from the heater form the basis of such mechanistic models forboiling [1–3]. Bubbles that slide along the heater surface afterdeparture from their original nucleation sites are an integral partof these models. Studies have also shown that the sliding motionof bubbles enhances heat transfer from the heater surfaces,although the cause for this enhancement has been attributed tomany sub-processes including, evaporation of thin microlayerunderneath the bubble, turbulence in the wake of the bubblesand the disruption of the thermal boundary layer due to bubblemotion. Efforts have been made to model the heat transfer fromthe wall due to sliding motion of bubbles [4–7]. However, thereis no clear understanding of the heat transfer to a bubble in thesestudies. In this paper experimental results of the heat transferassociated with a single vapor bubble sliding along a downwardfacing heater surface is presented.

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Early work done on sliding bubbles was mainly focused on theapparent increase in heat transfer rates when sliding bubbles werepresent. Higher rates of heat transfer due to sliding bubbles havebeen reported in tube bundles [8], on single tubes [9], as well asduring flow boiling on vertical flat plates [10]. Studies have beenmade to understand mechanisms that cause the increased ratesof heat transfer. Yan et al. [11] have suggested, based on experi-ments with vapor bubbles sliding under inclined plates and curvedsurfaces, that the evaporation of thin liquid layer under the bub-bles made a significant contribution to heat transfer only in thecase of large bubbles. Addlesee and Cornwell [4] have measuredthe thickness of the liquid film trapped between the sliding airbubble in water using a fiber-optic probe, and found the averagefilm thickness to be about 350 microns. Addlesee and Kew [5] haveestimated the film thickness, by solving the liquid layer momen-tum equation, to be in the range of 50–100 microns.

Kenning et al. [6] have also reported experiments of a water va-por bubble sliding on a downward facing heater surface. Based onthe temperature drop along the wall, they have estimated the li-quid layer thickness to be between 45 and 80 microns. Bayazitet al. [7] have also made a similar attempt, but have modeled theheat transfer to the liquid layer. Based on their experiments withFC-87 as the test liquid, they have estimated that a microlayerthickness of about 50 microns was possible. In a follow-up study,Li et al. [12] measured the liquid microlayer using a reflectance

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Nomenclature

A area (m2)b bubble base length (m)D bubble equivalent diameter (D ¼ ð6V

p Þ1=3)

ds differential distance along the interface (m)h bubble height (m)L width of the heater surface (m)n refractive index of the liquid (m)P point on the interferogram_Q rate of heat transfer (W)q heat flux (W/m2)S fringe orderT temperature (�C)V bubble volume (m3)w bubble width (m)DT temperature difference (�C)

Greek lettersk wavelength (m)H angle of inclination of the heater surface to the verticalh angular location of a point on the bubble interfaceq density (kg/m3)

Subscriptsbase bubble basee bulk liquidi interface or locationnet neto initialsat saturationv vaporw wall

926 S. Manickam, V. Dhir / International Journal of Heat and Mass Transfer 55 (2012) 925–940

based fiber optic probe, and showed that the average microlayerthickness ranges from 2 to 50 microns. Hollingsworth et al. [13] re-ported on the heat transfer enhancement in the wake of the slidingbubbles, based on temperature measurements at the back of theheater surface. Local heat transfer coefficients in the wake showeda sharp increase to about 1000–2500 W/m2K (about 20–50 timesthat of the natural convection value). The maxima of the local heattransfer coefficient occurred at a distance approximately equal tobubble width, from the back of the bubble and at less than onemm of the surface. The enhancement in local heat flux comparedto the natural convection values persisted for short duration overa distance of approximately three times the bubble width.

Qiu and Dhir [14] studied the flow pattern and heat transferassociated with a bubble sliding along a downward facing heatersurface using direct photography and holographic interferometry.Experiments were conducted with PF-5060 as a test fluid, and apolished silicon wafer serving as the heater surface. A thin liquidfilm and a wedge like liquid gap were observed between the bub-ble and the heater surface. Vortices were observed to shed from thebubble in its wake, resulting in significant drop in wall tempera-ture. Manickam and Dhir [15] extended this study and measuredheat transfer through the interface of sliding bubbles for down-ward facing heater surfaces at 75� inclination to vertical. Resultsshowed that condensation occurs on much of the bubble interfaceaway from the wall. Almost all of evaporation for bubble growthoccurred at the bubble interface very close to the wall and alongthe bubble base. Li et al. [16] and Li and Dhir [17] presented a com-panion numerical study for the experimental work described in[14,15]. Complete three-dimensional numerical simulation of thevapor bubble was carried out. Numerical results also showed thatevaporation from the liquid layer underneath the bubble was themajor contributor to the bubble growth. The temperature gradientin the wake of the bubble was significantly altered, possibly lead-ing to enhancement of heat transfer from the heater surface. Lucicet al. [18] have also presented results from experimental measure-ments and numerical simulation of rising water vapor bubbles inpool and flow boiling on a vertical flat plate. Bubble shapes andthe thermal field surrounding the bubble obtained from thenumerical simulations were compared with holographic interfer-ometry data, and appeared to be in general agreement. Sateeshet al. [19] have modeled the wall heat flux, including the effectof sliding motion of a vapor bubble, on a vertical flat plate andon a horizontal cylinder under pool boiling conditions. Estimationsof wall heat flux from their model agreed within ±25% of the exper-imental data available in the literature.

Based on the above review it is clear that sliding bubbles are animportant part of bubble dynamics, especially for flow boiling andpool boiling outside of tubes and boiling on inclined downwardfacing heater surfaces. Although several attempts have been madeto study sliding bubbles in more detail, the current state of under-standing regarding the heat transfer associated with sliding bub-bles is not complete. There are no systematic experimentsconducted to estimate the heat transfer around the bubble inter-face or to assess the effect of various heat transfer parametersaffecting the sliding bubbles. This work presents results from theexperimental study on heat transfer around a vapor bubble slidingalong a downward facing heater surface under pool boiling condi-tions. Direct photography and holographic interferometry are usedto measure volume growth rate and the rate of heat transferaround the sliding bubble, respectively.

2. Description of experiments

2.1. Experimental methodology and test setup

The schematic in Fig. 1 shows a sketch of a bubble sliding alonga heater plate. Measurements are carried out in two steps: heattransfer measurements and volume growth measurements. Forboth the cases, only individual vapor bubbles that slide through awell-defined thermal boundary layer are considered. For volumegrowth measurements, direct high-speed digital photography pro-viding two-orthogonal views of the bubble simultaneously is used.Bubbles are characterized by their width (w), base length (b), andheight (h), as shown in figure. The temperature distribution inthe liquid surrounding the bubble is determined using holographicinterferometry. The vapor inside bubble is assumed to be at satu-ration temperature and the heat flux into the bubble through theinterface (qi) is obtained from the temperature measurements.

The angle of inclination of the downward facing heater surface(H), is fixed at 75� to the vertical, and both wall superheat(DTw = Tw� Tsat) and liquid subcooling (DTsub = Tsat� Te) are variedover a narrow range (where Tw, Te and Tsat are the wall, bulk liquid,and saturation temperature, respectively). A thin liquid layerunderneath the bubble is also shown. No attempt was made tomeasure its thickness. Estimations of possible range of liquid layerthickness will be deduced from the heat transfer measurementsusing an energy balance.

Let qi be the local heat flux measured along the curved interfaceaway from the wall, qbase be the area-averaged heat flux from thebase of the bubble attributed to evaporation from the microlayer,

Fig. 1. Schematic of sliding bubble.

S. Manickam, V. Dhir / International Journal of Heat and Mass Transfer 55 (2012) 925–940 927

_Q net be the net heat transfer rate into the bubble resulting in vol-ume growth, and V be the volume of the bubble at any instant intime. We can write from conservation of energy,

_Q net ¼ _Q base þ _Q i ¼ qbasebwþwZ

qids ¼ qvhfgdVdt

ð1Þ

where ds is the differential distance measured along the curvedinterface of the bubble, b is the bubble base length, w is the bubblewidth, and _Qbase and _Qi are the total heat transfer rates from thebubble base, and bubble interface away from the wall respectively._Qnet and _Qi are determined through volume growth measurements

and interferometry, respectively. Volume growth experiments andthe interferometry measurement, for the same experimentalparameters, were performed separately.

Fig. 2(a) and (b) show a schematic of the test section and theheater surface. The test section consists of a rectangular box madeout of aluminum, the top side of which houses the heater surface,and the other three sides are made of clear glass to permit opticalstudies. The test section is sealed off at the lower end and has avertical open column at the other end. The vertical column servesas a fluid reservoir. The test section also houses a cartridge heaterand a stirrer for heating the bulk liquid, both of which are turned-off when experiments are conducted.

The heater surface in this study is made of polished singlecrystal silicon wafer of length 185 mm and width 49.5 mm. Twodifferent heater surfaces - one with a silicon wafer of 700 lm

thickness, and another with 100 lm thickness, were used in theexperiment. The heater surface with the thinner silicon waferwas used mainly to assess temperature response of the heater sur-face during sliding motion of bubble. At the lower end of the waferassembly, a polished copper plate of 50 � 50 mm2 and 1 mm thick-ness was used. An artificial cavity of 250 lm was drilled into thiscopper plate, which served as the nucleation site for the experi-ments. Kapton foil heaters were affixed on the back side of the wa-fers in three separate groups. The voltage applied across eachgroup of heaters could be controlled independently. Thermocou-ples were also attached to the back of the wafer at several locationsalong the length of the heater.

A set of strain gage heaters was fixed to the copper plate at thebottom of the heated section. By controlling power to these straingage heaters individual vapor bubbles could be generated at thecavity. The entire wafer-assembly, shown in Fig. 2(b), was moldedonto a G-10 (a fiber-glass epoxy laminate material) base using one-part silicone adhesive (high temperature RTV). The setup ismounted on a frame, which could be tilted to the required inclina-tion. Holographic interferometry was used in this study to measurethe temperature field around the sliding vapor bubble. Theory ofholographic interferometry and the method of evaluation of inter-ferograms for heat transfer study of transparent objects are welldiscussed in the literature [20–24]. Fig. 2(c) shows the schematicof the optical layout used in this study. A laser beam from a singlesource is split into two beams - reference beam and an object beamusing a beam splitter. The object beam is made to pass through thetest section. The reference beam meets the object beam on a spe-cially prepared photographic plate. Both the beams also passthrough a shutter arrangement, spatial filters, and a system oflenses as shown in the layout. Note that the center of the objectbeam passes through the test section at a distance of 150 mm fromthe cavity. The interferometry experiments pertain to measure-ments made at this location.

2.2. Experimental procedure

Bubble volume growth measurements were made using directphotography. High-speed digital video images of the bubble intwo orthogonal directions were simultaneously obtained using ahigh-speed digital camera (maximum frame rate per sec-ond = 1240). Due to the limited field of view of the camera, the en-tire length of the heater surface could not be observed at the sametime. Therefore, a set of four experiments, with four different bub-bles obtained at the same experimental conditions, was used to ob-tain the volume growth rate of a bubble along the complete lengthof the heater surface. Based on the shape at a given location alongthe length of the heater surface, the bubbles were approximated tobe either an ellipsoid, or a cylinder, or an oblate spheroid, and thevolume was determined correspondingly. Volume growth mea-surements were obtained over the entire length of the test section.Real-time infinite fringe method was used in the interferometryexperiments. For the light source a He–Ne continuous-wave laser(k = 633 nm) was used. The holographic plates used were 6’’ � 4’’,BB-640 type made by VinTeq, NC. Exposed holographic plates weredeveloped using the development kit JD-3, made by PhotographersFormulary, MT. Test liquid was PF-5060. Refractive index data forliquid PF-5060, was taken from Qiu and Dhir [25].

Typical experimental runs begin first by filling the test sectionwith the working fluid PF-5060 (degassed prior to the experiment).Both the cartridge heater and the stirrer were turned on, and thefluid was heated to attain the desired subcooling. Voltage was ap-plied to the foil heaters to maintain the heater surface at the de-sired temperatures. By controlling the voltage applied to thestrain gage heaters, single bubbles were formed at the nucleatingcavity. With the single bubbles generated, direct photography or

Fig. 2. Schematic of the test section and heater assembly and the optical layout for the holographic interferometry studies.


interferometry studies were performed. Only the very first bubblesliding along the heater surface, in an undisturbed, fully developednatural convection thermal layer was considered in this study.When bubbles were generated sequentially, the time between abubble and the subsequent bubble was such that, an undisturbedfully developed natural convection thermal boundary layer wasonce again present on the heater surface. Details of the experimen-tal procedure adopted in the study and practical aspects ofholographic interferometry experiments can be found in [26].

2.3. Evaluation of the interferograms

The general problem of evaluation of interferograms in threedimensions is really an inverse problem, where the refractive indexdistribution must be determined based on fringe patterns recordedon the interferogram. Consider the case of ideal interferometrywhere the optical system used consists of ideal mirrors and a mono-chromatic light source, and the medium is a two-dimensional phase

object with no end effects. We assume that the change in opticalpath length of the object beam that results in the interference iscaused by the temperature gradients imposed on the phase object.The temperature gradients imposed can be related to change inoptical path length as follows: suppose the z-axis is parallel to theobject beam. The interference pattern recorded in such a case showsat each point P(xi,yi) on the interferogram, the difference in the opti-cal path length (S k) between the object and reference beam. Here,the object beam is parallel to z-axis and Si is the order of the fringeat that location (xi,yi). The optical path length difference is given by:

Sðxi; yiÞ � k ¼ nrðxi; yiÞl� nmðxi; yiÞl ¼ Dnðxi; yiÞl ð2Þ

where nr(xi,yi) is the refractive index at (xi,yi) when there is no tem-perature gradient in the test section and nm(xi,yi) is the refractiveindex with temperature gradient and l is the width of the test sec-tion. This equation is known as the equation for ideal interferome-try [23]. When the infinite fringe method is used, the loci ofconstant phase difference DS are points of constant refractive index


difference in the case of a two-dimensional phase object. If therefractive index changes are only due to changes in temperature,these may also be considered as points of constant temperature dif-ference. Therefore we can write, between any two consecutivefringes located at (xi+1,yi+1) and (xi, yi):

½Sðxiþ1; yiþ1Þ � Sðxi; yiÞ� � k ¼ DS � k ¼ ldndT

DT ð3Þ

This is valid for a two-dimensional phase object in ideal interferom-etry. In the natural convection studies performed during the courseof this work for validation purposes, the two-dimensional phase ob-ject is closely approximated, and the above equation was used with-out any modifications.

Evaluation of interferograms for sliding bubbles is difficult dueto the three-dimensional nature of the problem. Fig. 3(a) shows a

Fig. 3. Path of the measurement beam and the coordinate system for evaluation ofinterferograms.

schematic of the path of the object beam in the measurement pro-cess when sliding bubbles are present. Let z be the direction of theobject beam. The image plane (digital camera recording variationsin x- and y- directions) is placed beyond z = L. As shown in figure,the width of the sliding bubble (w) is smaller than that of the hea-ter surface (L). Assuming the bubble is not wide enough to causeedge effects, it can be seen that the object beam passes throughtwo distinct regions of different temperature fields. In region A,on both sides of the bubble, where n = na, a natural convectionboundary layer exists along the heated surface. In region B, closeto the bubble, where n = nb, the temperature distribution is dueto the interaction between the natural convection as well as thebubble boundary layers. The fringe pattern recorded in the (x,y)plane in this case shows the integrated optical path length differ-ence between z = 0 and z = L. Between each consecutive fringethe path difference still equals one wavelength to satisfy the con-ditions for interference. However, the distribution of the refractiveindex inside the test section that results in this path length differ-ence is unknown. The difficulties associated with evaluation ofinterferograms for the sliding bubbles may, therefore, be attributedto the three-dimensional nature of the problem. Variation of n inthe z-direction, in the case of sliding bubbles, could be due toany or all of the following reasons (see, Fig. 3(a)): (i) the bubblewidth (w) is smaller than the width of the heater section (L); (ii)the bubble interface is curved, i.e., the bubble width varies frombubble base (close to the wall) to the top of the bubble; (iii) thetemperature distributions in regions A and B are not known; and(iv) the length (Dl), in the region B, is not known. In order to eval-uate the interferogram, therefore, na, nb and Dl must be deter-mined. It must be emphasized that both na, and nb will varyalong the path of the object beam, there by making the evaluationof the interferogram an inverse problem, requiring the use ofapproximate methods.

A new methodology, which combines the approach for axi-sym-metric bodies along with additional data from natural convectionand bubble growth experiments, for evaluation of the fringes isdeveloped in this work. Consider the object beam travelingthrough two separate regions of refractive indices na and nb, asshown in Fig. 3(a). Integrating Eq. (2), over the heater width yields,

Sðxi; yiÞ � k ¼Z z¼L

z¼0Dnðxi; yiÞdz ¼ DniL ð4Þ

where Dni is the difference in refractive index (ni) and the refractiveindex (ne) for the bulk evaluated at bulk fluid temperature Te (at agiven (xi,yi)) and can be written as,

Dni ¼ ni � ne ¼1L

Z z¼L

z¼0Dnðxi; yiÞdz ð5Þ

In terms of the three regions Eq. (4) may be written as:

Sðxi; yiÞ � k ¼Z z¼z1

z¼0Dnadzþ

Z z¼z2

z¼z1

DnbdzþZ z¼L

z¼z2

Dnadz ð6Þ

The left hand side of the above equation is constant for a givenworking fluid and light source used in the experiments. Methodol-ogy developed in this work for determining na, nb and Dl can nowbe described in detail.

Consider the path of the object beam shown in Fig. 3(a). On bothsides of the vapor bubble we have region A, where the refractive in-dex is na. We can assume that the thermal (and possibly the hydro-dynamic) influence of the sliding vapor bubble is limited to thebubble boundary layer. Region A, lying outside of the bubble bound-ary layer is part of the natural convection boundary layer on thesolid surface. Therefore at any given point (xi,yi) on the interfero-gram, na has the same value as when only natural convection ispresent on the surface. Provided ideal interferometry conditions


are met during natural convection, determination of na can easily becarried out using the ideal interferometry equation given in Eq. (3).In this work, effort has been made to establish a steady state naturalconvection temperature profile along the heater surface before thevapor bubble begins to slide. The refractive index na in this case(prior to sliding of vapor bubble) is constant in z-direction. This alsomeans that exact placement of z1 and z2 (i.e., the location of regionB) is irrelevant and the first and last integrals in Eq. (6) may becombined, and rewritten as:

Sðxi; yiÞ � k ¼ DnaðL� DlÞ þZ z¼z2

z¼z1

Dnbdz ð7Þ

The above equation implies that the temperature at location (xi, yi)in the na region is the same as the temperature at that locationwhen only natural convection was present (i.e., prior to bubble slid-ing). With na determined from the natural convection profile, theevaluation of the interferogram reduces to that of determining nb

and Dl.The length Dl, may be determined using geometrical approxi-

mations from the interferogram or the volume growth studies. Thiscan be done in two steps. From Fig. 3(a), it is clear that the front ofthe bubble may be treated as an axi-symmetric body (say, sectionof a spheroid or cap shaped bubble). The back of the bubble hasnearly uniform width at any given y location. Therefore, Dl, inthe front of the bubble is evaluated using geometrical consider-ations for axi-symmetric bodies as developed in [24] and [25].Referring to the hologram shown in Fig. 3(b), let y

!be the normal

to the heater surface passing through the center of bubble base.At a given point (xi,yi) on an interference fringe, let ri be the dis-tance of this point in the x-direction from the axisy

!. Let ro be the

perpendicular distance between axis y!

and nearest location onthe fringe where it becomes parallel to the heater surface, signify-ing the end of influence of bubble boundary layer. The effectivelength of the bubble boundary layer Dl, through which the mea-surement beam passes at this point, can be written as,

Dl ¼ 2�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2

o � r2i

qð8Þ

Note that this approximation is valid only for the front portion ofthe bubble, where we can assume the bubble interface as a sectionof a sphere and therefore bubble boundary layer as a spherical ther-mal field. In the back portion, the wake of the bubble affects theregion that is easily seen as slightly more than the width of the bub-ble itself. Therefore for the back portion of bubble Dl is assumed tobe:

Dl ¼ 2� Dltop þw ð9Þ

where Dltop is the effective length, calculated using Eq. (8) at the topof the bubble, and w is the width of the bubble.

In region B, where boundary layer on the bubble and on thesolid surface interact, nb is also a function of z, i.e., nb = n(x,y,z).However, the variation of nb in the z-direction cannot be accountedfor analytically from the interferogram and one must resort toapproximate methods and a heuristic approach. As discussed ear-lier, for axi-symmetric bodies, a polynomial variation of nb hasbeen adopted in several studies. In the current work a simplerapproximation of constant average value for nb is used. Several testcases, involving different variations of nb were used to evaluateboth the bounds of the approximation, as well as to develop aheuristic method for accounting for the actual variation of nb.

Let nb be the average value of nb in the z-direction:

Dnb ¼1Dl

Z z¼z2

z¼z1

Dnbðxi; yiÞdz ð10Þ

Using the above equation, Eq. (7) can be rewritten as:

Sðxi; yiÞ � k ¼Z z¼L

z¼0Dnðxi; yiÞdz ¼ DnbðDlÞ þ DnaðL� DlÞ ð11Þ

Note that irrespective of the variation of na and nb within the testsection, the left hand side of Eq. (11) is constant between any twoconsecutive fringes (to satisfy requirements for interference).Therefore, for any two consecutive fringes represented by (S � 1)at location (xi�1,yi�1) and S at location (xi, yi), we can write:

½ðSÞ � ðS� 1Þ� � k ¼Z z¼L

z¼0½Dnðxi; yiÞ � Dnðxi�1; yi�1Þ�dz ¼ Dni;fringeL

ð12Þ

where Dni;fringe is the difference in average refractive index at loca-tions (xi, yi) and (xi�1,yi�1). Note that Eq. (12) assumes that changein k between these two locations is negligible. Combining Eqs.(12) and (11), we can write for locations (xi, yi) and (xi�1,yi�1):

ðDnb;i � Dnb;i�1Þ ¼1Dl½ðDni;fringeÞL� ðDna;i � Dna;i�1ÞðL� DlÞ� ð13Þ

Note that subscripts i and i � 1 used in referring to Dnb;i and Dnb;i�1

as a shorthand notation for locations (xi, yi) and (xi�1,yi�1) respec-tively. Evaluation of Dnb;i was done using Eq. (13), starting fromthe fringe closest to the undisturbed region representing bulk fluidcondition. Evaluation of nb was carried out assuming no variationsof n within Dl, i.e., simply assuming that nb = nb. The actual variationof nb in the z-direction is evaluated using heuristic methods wherehypothetical experiments performed with several possible varia-tions for nb are considered. Details of the heuristic method em-ployed are provided in the appendix. It must also be noted that nb

is the quantity we are interested in the study, as this provides thelocal variation in temperature in the vicinity of the bubble, therebyenabling us to evaluate the rate of heat transfer into or out of thebubble. The refractive index is converted into temperature datafrom the refractive index property relations given by Qiu and Dhir[25].

Although we need the temperatures at various locations alongthe normal to the bubble interface, it is more convenient to beginby mapping the entire temperature field from the interferogram.Once the entire field is determined, both isotherms as well as heatfluxes around the bubble can be easily determined. The procedureoutlined here requires limited amount of image processing, how-ever, a large set of measurements need to be made on theseimages. In this work an open-source software program, ImageJ, isused for image analysis and measurements from the images. Themethod outlined above was validated with natural convectionstudies on inclined flat plates. Natural convection studies were alsoconducted by placing on the heater a 25 mm long neoprene rubberstrip of 4 mm � 3 mm cross-section. This strip partly covered thewidth of the heater and was used to validate the method developedfor the evaluation of interferograms using an inverse method. Thecross-sectional area and length of the rubber insert similar to theaverage bubble sizes encountered was placed in the middle ofthe heater plate. A simple steady state analysis using the electricalresistance analogy, with known wafer temperature, liquid temper-ature, and steady state heat flux was used to obtain the tempera-ture at the top of the rubber insert. Temperature obtainedthrough the natural convection interferometry studies and ob-tained using the inverse technique as described in the appendixagreed well with the values from the steady state analysis. Detailsof the evaluation procedure, error analysis, and the results from thevalidation studies are available in [26].

2.4. Uncertainty analysis

Uncertainty analysis, on the basis of single sample experimentsfor the current study was performed as per the ASME guidelines.

Table 1Uncertainty in volume growth measurements.

Shape l1 (mm) l2 (mm) l3 (mm) DV/V% Volume (V) mm3

Sphere 1.418 1.418 1.418 43.3 1.49Cylinder 10.635 1.773 1.418 32.2 21.26Oblate spheroid 14.180 2.836 2.836 18.4 78.31


Table 1 presents the uncertainty in volume growth data, for a typ-ical experiment reported in this work. All the length scales and thevolumes of the bubbles shown are typical values encountered in thestudy. The error in each length measurement, Dl, is 0.355 mm forall the bubble shapes considered. Quantifying uncertainty in inter-ferometry measurements using a similar method is not quiteadequate, as only the bulk liquid temperature and length measure-ments are made on the interferograms, which may be measuredwith good accuracy. However, uncertainty in evaluation of interfer-ograms may introduce large uncertainty in determining local tem-perature distribution, and consequently the rate of heat transfer. Asdescribed in the previous section, due to the variation of the shapeof the bubble, and the temperature in the thermal layer adjacent tothe bubble, evaluation of the interferograms for sliding bubbles isan inverse problem, where the actual temperature distribution inthe experiments is back calculated from its observations, whichare the interferograms. The average values of na and nb were calcu-lated as shown in the previous section and the maximum boundsfor the uncertainty in estimating nb was obtained heuristically.

Fig. 4. Bubble shape along the heater surface at varying d

More details of the uncertainty of interferometry experiments areavailable in [26].

3. Results and discussion

3.1. Volume growth rate studies

Fig. 4 presents a collage of images of sliding bubbles at varyingdistances (x) from the cavity, measured along the heater surface forDTw = 0.5 �C and DTsub = 0.3 �C. Note that each image is obtainedthrough a separate experiment, and hence the images are not theimages of the same bubble at different locations, but rather imagesof different bubbles, all produced at the same wall super heat andliquid subcooling. Near the bottom end of the heater x < Lx/4, whereLx is the total length of the test surface, the bubbles are smallest insize, and look like spheres. As the bubbles grow, their shapeschange from spheroids to ellipsoids, to long cylinders and at thefar end of the test surface (x > 0.8Lx) the bubbles appear to be partsegment of spheres or oblate spheroids. The bubbles shapes ob-tained here are in general agreement with the shapes reported inearlier works by Maxworthy [27], Qiu and Dhir [14], and Li et al.[28]. However, it should be noted that in [27], the experimentswere conducted on air bubbles in water, involving no evaporationor heat transfer. Although [14] and [28] are similar to the currentwork, the wall superheat ranges considered in these studies weremuch larger than the current study. Therefore direct comparisonof volume growth rates has not been made. The change in bubbleshapes and the bubble growth occurs quite rapidly. In general

istances from the cavity (DTsub = 0.3 �C, DTw = 0.5 �C).


the time taken for the bubble to slide along the length of the heaterplate is around 2 s for all of the cases reported here. The continuouschanges in the bubble shape observed here indicate a very activeinterface with heat transfer to and from the bubble across theinterface. Volumes of the bubbles were calculated by the approxi-mating the actual shape of the bubbles to the closest geometricalshapes – spheres, cylinders or lamina of circular segments. Lengthmeasurements to determine the volume of these approximate geo-metrical shapes were made from individual frames obtained fromthe high-speed video recordings of the sliding bubble. Knowing theframe rate of the video recordings and the bubble volume at agiven instant, bubble growth rate can be easily determined.

Fig. 5(a) presents the bubble equivalent diameter D, for a set ofliquid subcoolings with wall superheat maintained at 0.5 �C, as afunction of distance from the cavity (same as the sliding distance).Note that the bubble equivalent diameter is defined as the diameterof a sphere of volume equal to that of the sliding bubble. Fig. 5(b)presents increase in bubble equivalent diameter (D � Do) as a func-tion of sliding time, where Do is the bubble equivalent diameter att = 0 s (i.e., at the bottom of the heater section). In both figures, theerror bars represent the uncertainty in bubble equivalent diameter

0 25 50 75 100 125 150 175

1

2

3

4

5

6

7

8

Bubb

le e

quiv

alen

t dia

met

er, D

(mm

)

Distance from the cavity, Lx (mm)

ΔTsub = 0.2 oCΔTsub = 0.4 oCΔTsub = 0.7 oCΔTsub = 1.0 oCΔTsub = 1.2 oC

(a) Bubble equivalent diameter (D , mm) as a function of sliding distance (Lx, mm)

0 300 600 900 1200 1500 1800 2100

0

1

2

3

4

5

6

Incr

ease

in b

ubbl

e eq

uiva

lent

dia

met

er, D

-Do (

mm

)

Time (ms)

ΔTsub = 0.2 oCΔTsub = 0.4 oCΔTsub = 0.7 oCΔTsub = 1.0 oCΔTsub = 1.2 oC

(b) Increase in bubble equivalent diameter

(D-D o, mm), with growth time (t, ms)

Fig. 5. Volume of sliding bubbles – effect of subcooling (DTw = 0.5 �C).

estimated at these experimental conditions. Note that both the sub-cooling as well as wall superheat mentioned in these figures arebased on the average values of the liquid and wall temperaturemeasurements made along the flow direction. Several importantobservations can be made based on the data presented in this fig-ure. First, for the range of subcoolings studied (0–1.2 �C), withDTw = 0.5 �C, all the bubbles continue to grow. Therefore, there isa net energy input into the bubble through the interface underthese conditions, which can be quantified based on the volumetricgrowth rate of the bubbles. Volume growth rate of the bubbles arein the range of 1–2.5 � 10�7 m3/s with lower rates corresponding tohigher subcooling. Note that Qiu and Dhir [14] have also reported avolume growth rate of about 1 � 10�7 m3/s for DTsub = 1.3 �C andDTw = 3.0 �C, for PF-5060. Li et al. [28] have also reported a volumegrowth rate of about 2.5 � 10�6 m3/s, for DTsub = 5.0 �C andDTw = 6.6 �C for the same angle of inclination of the heater surfacewith FC-87. The higher volume growth rates in their study may beattributed to the much larger wall superheat used. This is also sup-ported by Qiu and Dhir’s [14] data for higher wall superheat.

In general, as DTsub increases the volume growth rate decreases.This is due to condensation at the bubble interface, which increaseswith increasing DTsub. For conditions where DTsub < 1 �C, the varia-tions between different experiments in the growth rate are small,and the bubble volumes are within the experimental uncertainty.Several reasons could be attributed to the scatter at low subcoo-lings. Firstly the heater surface could not be maintained at perfectisothermal conditions; in particular the variation in temperaturealong the flow direction observed is about 0.2 �C for the slidingbubble experiments, which is also the difference between two setsof data. The scatter also accounts for the growth rate curves forDTsub = 0.2, 0.4 and 0.7 �C lying close to each other showing no dis-cernible influence of DTsub, on bubble growth rates. As alreadynoted, several experiments had to be performed to observe thebubbles at various distances from the cavity in order to obtain asingle growth rate curve and some amount of scatter therefore isunavoidable.

3.2. Bubble features revealed during interferometry studies

Fig. 6 shows a typical fringe pattern for a bubble as it movesthrough the natural convection field along the heater surface forDTsub = 0.5 �C, and DTw = 0.5 �C. Note that the first frame shows awell-developed natural convection boundary layer. Steady statenatural convection boundary layers similar to these were presentprior to every sliding bubble studied in this work. The fringes arestraight lines parallel to the heater surface. The bubble appearsblack in the images. It is also observed that close to the wall thefront portion of the bubble interface has a curvature, causing awedge-shaped liquid gap close to the wall, as can be seen fromthe frame captured at 30 ms. Back of the bubble appears to besmooth in the third frame shown, although in general for largerbubbles (oblate spheroids) the back of the bubble often appearsragged. The bubble interface is rapidly changing due to forces act-ing on the bubble including bubble and liquid inertia. These fea-tures are more pronounced for the larger bubbles (associatedwith low DTsub, and high DTw). With increased liquid subcoolingthe bubbles are smaller, and appear to have smoother interfaces.

In all cases, the fringe pattern in the wake of the bubble looksenlarged near the wall and distorted in the outer region of theboundary layer. The wake left behind by the bubble is clearly seenin the last frame, where the edge of the boundary layer is com-pletely disturbed. The disturbances introduced in the boundarylayer at any location persist well beyond 2 s, which is about thetime taken for the bubbles to slide the total length of the heatersurface. In general, the features shown here compare well withQiu and Dhir [14]. However, the vortices and bulk mixing observed

Fig. 6. Time sequence of a sliding bubble (DTsub = 0.5 �C, DTw = 0.5 �C).


in the wake of the sliding bubble in their experiments were notpresent in the current study. This could possibly be due to the factthat the size of the bubbles in Qiu and Dhir’s work is much largerdue to the high wall superheat in their experiments. The effect ofthe wake, in the thickening of the boundary layer and the subse-quent increase in the fringe spacing close to edge of the boundarylayer, is also less pronounced in the current study compared totheir work. The general disturbances of the fringes leading to wavyor nonuniform fringes, however, can be observed in both thestudies.

3.2.1. Heat transfer through the top section of the bubble interfaceFig. 7 shows the actual interferogram and the temperature dis-

tribution around the bubble using ideal interferometry (i.e., two-dimensional) analysis for DTw = 0.5 �C and DTsub = 0.7 �C. The ideal

temperature distribution is the same as the interferogram, witheach fringe representing an isotherm and the temperature differ-ence between any two consecutive fringes, given by Eq. (3). Forthe He–Ne laser used in the study, and with PF-5060 as the testfluid, the temperature difference between any two consecutivefringes is 0.025 �C when the bulk fluid is at room temperature.The temperature profile obtained through the ideal interferometrywas corrected by taking into account appropriate Dl, and na as de-scribed in the earlier section. Fig. 8 shows the isotherms of the cor-rected temperature field around the sliding bubble, along with therate of heat transfer through the top portion of the bubble inter-face. The corrected temperature distribution differs from the tem-perature distribution obtain through ideal interferometry inseveral significant ways. Comparing the ideal temperature distri-bution (Fig. 7(b)), and the corrected temperature distribution

(a) Interferogram

0 1 2 3 4 5 6 70.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5Sliding Direction

Dis

tanc

e (m

m)

Distance (mm)

Bubble (Tsat) 57.600 (Tw) 56.971 56.870 56.768 56.667 56.565 56.464 56.438

(b) Ideal interferometry temperature distribution

Fig. 7. Interferogram and the temperature distribution around a sliding bubbleusing ideal interferometry analysis (DTsub = 0.7 �C and DTw = 0.5 �C).

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0


Bubble (Tsat) 57.6 (Tw) 56.971 56.870 56.768 56.667 56.565 56.464 56.438

(a) Corrected temperature distribution

0 1 2 3 4 5 6 7

0 30 60 90 120 150 180-100

-80

-60

-40

-20

0

20

40

Hea

t tra

nsfe

r int

o th

e bu

bble

(mW

/m)

Location on the interface (Degrees)

Q-Ideal Interferometry (mW/m) Q-Corrected (mW/m)

(b) Local heat transfer rate along the bubble interface

Fig. 8. Corrected temperature distribution and heat transfer to a sliding bubble(DTsub = 0.7 �C and DTw = 0.5 �C).


(Fig. 8(b)), firstly, at a given location, the actual temperature ishigher than the values obtained through ideal analysis – we findthat all the isotherms seem to have moved some relative distanceaway from the wall. The amount of displacement of course de-pends on the isotherm and its original position. The net effect ofthis is that the liquid temperatures close to the bubble interfaceare slightly higher than those obtained by the ideal interferometry.Secondly, the effect of the sliding bubble can be seen more dramat-ically in the wake region where the displacement of the fringes ismore pronounced. Rate of heat transfer from the back of the bub-ble, in contact with liquid in the wake region, is lower than thatfrom the front of the bubble. Fig. 8(b) shows heat transfer resultsobtained using both ideal as well as deduced temperature distribu-tion. Results from both the analysis show that evaporation occursat the bubble interface very close to the wall while the rest ofthe bubble interface experiences condensation. The highest valuesof condensation heat transfer occurs close to top most part of theinterface, as the bubble interface at this location (h � 75�) is largelysurrounded by subcooled liquid. The rate of heat transfer obtainedafter correction is similar to that obtained from the ideal interfer-ometric analysis, however the local condensation heat transfer rateat each of the locations is slightly lower than given by idealinterferometry.

Figs. 9 and 10 show the interferograms and the isotherms of thecorrected temperature distribution for a sliding bubble withDTsub = 0.5 �C and 1.0 �C respectively. The wall superheat DTw is0.5 �C for both the cases. The temperature distributions for thetwo cases show features similar to those described earlier. Temper-ature at a given point (x,y) is higher than the values predicted bythe ideal interferometry, and the effect is more pronounced inthe wake region. Heat transfer data obtained from these tempera-ture distributions also show similar features, with higher local con-densation present in each of the locations with increasing DTsub.Notice the higher density of fringes with increased temperaturedifference between the wall and the bulk liquid. With higher sub-cooling, the fringe density close to the wall become higher, and thedifference between consecutive fringes may become indistinguish-able at the spatial resolution used in the study.

Effect of subcooling on heat transfer from the sliding bubbleinterface can be seen in Fig. 11. Both the local heat flux(Fig. 11(a)), as well as the local rate of heat transfer per unit lengthof the bubble (Fig. 11(b)) are presented with the positive valuesdenoting evaporation and negative signs denoting condensationrespectively. Local heat flux was computed from the deduced tem-perature distribution for angular locations (15� 6 h 6 165�) on theinterface (Fig. 3(b)). Heat flux angular locations h < 15� and h > 165�could not be obtained directly from the temperature distributionclose to the wall due to high fringe density. Therefore, the dis-tances between the wall and the normal to the bubble interface

(a) Interferogram

0 1 2 3 4 50.0

0.5

1.0

1.5

2.0


Dis

tanc

e (m

m)

Distance (mm)

Bubble (Tsat) 57.6 (Tw) 57.0125 56.9125 56.8125 56.7125 56.6375

(b) Corrected temperature distribution

Fig. 9. Interferogram and the corrected temperature distribution around a slidingbubble using modified analysis (DTsub = 0.5 �C and DTw = 0.5 �C).

(a) Interferogram

0 1 2 3 4 5 60.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5


Dis

tanc

e (m

m)

Distance (mm)

Bubble (Tsat) 56.1625 56.2375 56.3375 56.4375 56.5375 56.6375 56.7375 57.6 (TW)

(b) Corrected temperature distribution

Fig. 10. Interferogram and the corrected temperature distribution around a slidingbubble using modified analysis (DTsub = 1.0 �C and DTw = 0.5 �C).


at these locations were measured, and the temperature differencebetween interface and wall was used to obtain the temperaturegradient, and thus heat flux, at these locations. Each local heat fluxwas multiplied by an associated chord length on the curved inter-face of the bubble away from the wall to obtain the local rate ofheat transfer. It may be seen from Fig. 11(a) that the local heat fluxfor h 6 5� and 175� 6 h are much higher than the heat flux at otherlocations. It may also be noted from Fig. 11(a) and (b) that theangular locations, h, for maximum condensation heat flux are dif-ferent from the locations for maximum rate of heat transfer perunit length. This is due to the fact that the peak values of chordlengths do not always occur at the same angular location as themaximum heat flux. As mentioned earlier the bubble interface israpidly changing due to the forces acting on it, leading to changesin chord lengths associated with the angular positions. As expectedthe condensation heat flux values increase with increase in subco-oling. Condensation heat transfer rate also shows similar behaviorwithin uncertainty of measurements.

3.2.2. Rate of heat transfer through the bubble base and heat transferpartitioning

Heat transfer through the bubble base was calculated by utiliz-ing Eq. (1). The total energy transfer into or out of the bubblethrough the top interface obtained through interferometry experi-ments was combined with the volumetric growth data for a bubblewith a given wall superheat and liquid subcooling. Fig. 12(a)

presents the results of such an analysis. For the liquid subcoolingand wall superheats considered in this study, the sliding bubblescontinued to grow, although the growth rates become smaller withincreased subcooling. The net rate of heat transfer into the bubble( _Qnet), as shown here, however, remains positive (i.e. evaporationdominates), but decreases in magnitude with increased subcooling.Note that the results presented here are for DTw = 0.5 �C and at adistance of 150 mm from the cavity. As the volume growth datashows, with increased wall superheat, _Qnet will also increase.

As can be inferred from the energy equation Eq. (1), the energyfor net bubble growth _Qnet represents a balance between rate ofheat transfer through the bubble base _Q base and rate heat transferthrough the top curved interface of the bubble away from the wall_Qi. Note that _Q i is the net energy in through the curved interface ofthe bubble, which includes both evaporation (occurring primarilyin h < 15� and h > 165�) and condensation (occurring, in general,15� < h < 165�). However, as Fig. 11 shows for the experimentalconditions investigated, much of the interface away from the wallexperiences condensation. Except for the evaporation that occursalong the curved interface close to the wall, all the energy required

-600

-400

-200

0

200

400

600

800

1000

1200

1400

Hea

tflux

into

the

bubb

le (W

/m2 )

Location on the bubble (Degrees)

ΔTsub = 0.2oCΔTsub = 0.5oCΔTsub = 0.7oCΔTsub = 1.0oC

(a) Local heat flux into the bubble

-15 0 15 30 45 60 75 90 105 120 135 150 165 180 195

-15 0 15 30 45 60 75 90 105 120 135 150 165 180 195

-300

-250

-200

-150

-100

-50

0

50

100

Rat

e of

hea

t tra

nsfe

r int

o th

e bu

bble

(mW

/m)

Location on the bubble (Degrees)

ΔTsub = 0.2oCΔTsub = 0.5oCΔTsub = 0.7oCΔTsub = 1.0oC

(b) Local heat transfer rate into the bubble

Fig. 11. Local heat flux and heat transfer around a sliding bubble – effect ofsubcooling (DTw = 0.5 �C).

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Rat

e of

hea

t tra

nsfe

r int

o th

e bu

bble

(W)

Liquid subcooling ΔTsub (oC)

Qi Qnet Qbase Best fit (Qi) Best fit (Qnet) Best fit (Qbase)

(a) Partitioning of heat transfer into the bubble

0.2 0.4 0.6 0.8 1.0 1.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

0

2

4

6

8

10

12

14

16

Qi/Q

base

(Per

cent

)

Ja [(ρl cpl ΔTsub)/(ρv hfg)]

(b) Ratio of iQ•

with baseQ•

Fig. 12. Partitioning of heat flux into the sliding bubble (DTw = 0.5 �C).


for the bubble growth and to support condensation comes from_Qbase. Both _Q net and _Q base decrease with increasing liquid subcool-

ing, which can be attributed to smaller bubble size and smallerbubble base area. Since _Qbase is calculated by adding _Qnet and _Qi,the variation of _Qbase with liquid subcooling follows the same trendas _Q net .

Net heat loss from the interface _Q i as a percentage of rate ofheat transfer from the base _Q base is shown in Fig. 12(b). Note thatsince only a small fraction of _Qbase is transferred to bulk fluid viacondensation at the interface at low subcoolings, _Q= _Q base is onlyabout 13% for Ja = 1.88. The energy transfer from the bubble baseis by evaporation in the thin microlayer leading to net bubblegrowth.

With the bubble width w, and the bubble base length b(Fig. 1(b)) known and using the _Qbase evaluated as above, the heatflux through the microlayer at the base of the bubble qbase was alsocalculated. Assuming that microlayer has uniform thickness overthe base and one dimensional heat conduction occurs throughthe microlayer we can determine average microlayer thicknessthrough the following relation:

_Qbase ¼ ðw � bÞqbase ¼ ðw � bÞklTw � Tsat

dl

� �ð14Þ

where kl is the liquid thermal conductivity at Tsat and Dl is the aver-age thickness of the microlayer. Average microlayer thicknesses ob-tained using Eq. (14), for the _Qbase shown in Fig. 12(a), are in therange of 4–6 microns. These are lower than the microlayer thick-nesses of 65–100 microns reported by Qiu and Dhir [14] for PF-5060 but at higher subcooling and higher wall superheat. Li et al.[12] reported microlayer thicknesses of about 25 microns for FC-87, also for higher subcooling and higher wall superheat.

It may be noted that prior to sliding motion of bubble naturalconvection is present on the heater surface. The heat flux fromthe wall under natural convection conditions qnc can also be com-puted from the interferograms obtained prior to bubble sliding.From energy balance for heat transfer from the heater surface overan area covered by the bubble base length (b) and width of the hea-ter surface (L) in the direction of the object beam, i.e., z-direction,we can write:

_QLb ¼ qaveLb ¼ _Q nc þ _Qbase ð15Þ_QLb ¼ qncAnc þ qbaseAbase ¼ qncðLb�wbÞ þ qbaseðwbÞ ð16Þ


where _QLb is the rate of heat transfer from the wall in the presenceof sliding bubble, qave is the average heat flux across the heatersurface, and Anc and Abase are the appropriate area for natural con-vection and area of the bubble base respectively.

Results from such calculations are presented in Fig. 13 for alocation Lx at a distance of 150 mm from the cavity. Even with areaaveraging the qave values are about one to two orders of magnitudehigher than the natural convection heat flux qnc. Increase in aver-age heat flux in the presence of sliding bubble is also presentedin Fig. 13(b). The increase in area averaged heat flux qave dropsfrom about 90 times to about 10 times that of natural convectionheat flux, qnc, with increase in liquid subcooling. This decrease inarea-averaged heat flux can be attributed to the smaller bubblesizes and consequent smaller base area (Abase) of the bubble pres-ent at higher subcoolings. Note that this analysis only considersinstantaneous area averaged heat flux in the presence of a slidingbubble.

Note that Qiu and Dhir [14] reported that the time averaged val-ues of local heat flux, obtained from the temperature at the back of

0.01

0.1

1

10

100

1000

10000

100000

Hea

t flu

x fro

m th

e w

all (

W/m

2 )

Liquid subcooling ΔTsub (oC)

qnc qbase qave Best fit (qnc) Best fit (qbase) Best fit (qave)

(a) Heat flux from the wall

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00

10

20

30

40

50

60

70

80

90

100

Incr

ease

in h

eat f

lux

(qav

e-qnc

)/(q nc

)

Ja [(ρl cpl ΔTsub)/(ρv hfg)]

(b) Increase in net heat flux from the wall as a result of bubble motion

Fig. 13. Heat transfer enhancement due to the sliding bubble (DTw = 0.5 �C).

the heater surface in the path of the sliding bubble, for a series ofbubbles, show an increase of about 10% compared to natural con-vection. A similar study based on the temperature drop at the backside of the heater surface, in the wake region of a single slidingbubble was conducted by Hollingsworth et al. [13] under constantheat flux condition. As stated earlier, the maximum local heattransfer coefficients of about 20–50 times that of the natural con-vection was reported, in the wake region of the sliding bubble, al-beit for very short period and over a small section of the heatersurface. In the current work, rate of heat transfer around andunderneath the very first sliding bubble is studied under nearlyconstant wall temperature condition. In the present work, undermuch smaller wall superheat and liquid subcooling no measurableincrease in heat transfer has been observed in the wake region ofthe bubble. No special attempt was made to study the heat transferin the wake region of the sliding bubbles.

4. Conclusions

Heat transfer associated with a single vapor bubble sliding on adownward facing inclined heater surface was studied experimen-tally. The heater was at an inclination of 75� to the vertical andPF-5060 was the test liquid. Measurements were made for DTsub

ranging from 0 to 1.2 �C and DTw ranging from 0.2 to 1.0 �C. Holo-graphic interferometry was used to measure the temperature fieldsurrounding the bubble. Based on the results presented severalimportant conclusions can be drawn:

(a) Schematic of temperature distribution on the heater surface

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

Dis

tanc

e, y

/δ

Temperature, (T-Te)/(Tsat - Te)

Profile - 1 Profile - 2 Profile - 3 Profile - 4 Profile - 5

(b) Test temperature profiles

Fig. 14. Schematic of temperature distribution on the heater surface and testtemperature profiles considered in the study.


(i) Bubble shapes change from spheroids to elongated cylindersand segment of spheres in that order, as the bubble slides onthe heater surface.

(ii) For the range of liquid subcooling and wall superheats studied,the bubbles continue to grow as they slide along the heatersurface. In general, the size of the bubble and the growth rateof the bubble decreases with increase in liquid subcooling.

(iii) Condensation occurs at most of the bubble interface. The mag-nitude of which increases with increasing liquid subcooling.

(a) Ideal interferogram for te

0.0 0.1 0.2 0.3 0.4 0.50.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Imposed (Profile -

Dis

tanc

e (y

/δ)

Temperature (T

(b) Temperature profiles deduc

Fig. 15. Effect of Dl on the ideal interferogram and the te

(iv) Evaporation occurs at locations on the bubble interface closeto the wall, and at the base of the bubble. In all cases evap-oration from the base exceeded condensation at the upperportion of the bubble and resulted in growth in volume ofthe bubble.

(v) Area averaged heat fluxes in the presence of sliding bubbleare about two orders of magnitude higher with the presenceof the vapor bubbles compared to that for naturalconvection.

st profile -4 for various Δ l

0.6 0.7 0.8 0.9 1.0

4)

-Te)/(Tsat-Te)

ImposedΔl = 0.05 LΔl = 0.1 LΔl = 0.2 LΔl = 0.4 LΔl = 0.6 LΔl = 0.8 LΔl = L

ed from the interferograms

mperature profiles deduced from the interferograms.


Acknowledgment

Financial support from NASA, under the Microgravity FluidPhysics Program is gratefully acknowledged. The authors wish tothank Dr. Gopinath Warrier and Dr. Dongming Qiu for their assis-tance and helpful suggestions during the course of this work.

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

ΔTer

r/ΔT im

pose

d

Δl/L

(a) Difference in base temperature

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

(dT/

dx)/(

dT/d

x)im

pose

d

Δl/L

(b) Difference in heat flux

Fig. 16. Maximum error in determining base temperature and heat flux from theinterferograms using ideal interferometry.

Appendix A

A.1. Evaluation of interferograms

Evaluation of interferograms for sliding bubbles is difficult dueto the three dimensional nature of the problem. As shown in theschematic (Fig. 3(a)), while the interferogram shows the variationof n(x,y,z) in the x- and y-direction, it only provides the integratedeffect of n in the z-direction. Therefore evaluation of the variationof n in the thermal layer around the bubble in the z-direction mustbe determined in an inverse manner. The variation of n at a given(xi,yi) in the z-direction could be due to any or all of the followingreasons:

(i) The width of the bubble (w) is smaller than the width of theheater (L).

(ii) Bubble interface is curved.(iii) Temperature distribution on the interface in region B is dif-

ferent from temperature distribution in region A.(iv) The length Dl of the region B is not known.

Note that Eqs. (8) and (9), given earlier can be used to evaluateDl and the bubble width (w) is known from the images obtainedfor volume growth studies. Neglecting any disturbance effects,temperature distribution in Region A is also known from interfero-grams of natural convection obtained prior to bubble sliding. How-ever, the temperature distribution in the region B (present duringthe sliding motion of bubble), and the effect of curvature of thebubble interface can be evaluated only heuristically.

Consider a generic temperature profile T = T(y) imposed over aregion Dl along the width of the heater surface (L) as shown inFig. 14(a). This temperature profile represents the temperature dis-tribution in the region B, where the boundary layer around thesliding vapor bubble exists while the temperature in the fluid lay-ers outside of Dl is uniform. The effect of assuming a particularprofile for T(y) was studied heuristically by evaluating the effectof a set of test temperature profiles.

Fig. 14(b) shows five different test temperature profiles consid-ered in the study. The shape of these profiles and the gradients atthe interface were chosen to represent possible temperature distri-butions in the region B. In addition to that, the effect of Dl was alsostudied heuristically, by considering cases where the given tem-perature profile T(y) was imposed over a range of Dl (i.e.,0.1L 6 Dl 6 L). In obtaining the ideal interferograms, it was as-sumed that the ideal interferometry system (defined in Section 2.4)uses a He–Ne laser and PF-5060 is the working fluid. It was furtherassumed that the difference between the interface temperature(Tsat) and the bulk liquid temperature (Te) is constant(Tsat � Te = 0.5 �C). Note also, that in the following discussion d de-notes the boundary layer thickness and y/d denotes the dimension-less distance normal to the interface.

Ideal interferograms when these profiles (in Fig. 14(b)) are pres-ent on Dl region were drawn based on Eq. (12) beginning withfringe corresponding to Te and computing the location of everysubsequent fringe closer to the interface. Locations of these fringescan be easily determined by computing ðDni;fringe � DlÞ, over smallincrements of y, until Eq. (12) is satisfied for the given test profileand the corresponding Dl. These interferograms were then used to

obtain the temperature profile T(y) using ideal interferometryequation (Eq. (3)). Note that in ideal interferometry we considertwo-dimensional phase objects (i.e., we assume Dl = L). Both theimposed T(y) and the profile obtained from the ideal interfero-grams were compared. In particular, the differences between tem-perature and heat flux at the interface based on the temperatureprofiles from such interferograms to the values from the imposedtemperature profiles were evaluated.

Fig. 15(a) shows a set of ideal interferograms for the profile type4, imposed according to the distribution in Fig. 16 for different Dl.Using the Eq. (3) the temperature profiles were evaluated from theideal interferograms. Fig. 15(b) shows the effect of Dl on tempera-ture profiles deduced from the ideal interferograms for the profiletype 4. Note that with increase in Dl/L, the deduced profile ap-proaches the imposed profile and for Dl = L, these two are identical.Fig. 16(a) shows the maximum difference between the deducedinterface temperature and the imposed interface temperature(DTerr). Fig. 16(b) shows the difference in the temperature gradient


at the interface between the deduced profiles and the imposed pro-file. With the increase in Dl/L, the difference between deduced andimposed temperature and heat flux at the interface decreases. Sim-ilar studies were also performed for the other test profiles shownin Fig. 14(b). Studies were also conducted that considered one typeof profile applied over region B, and a second type of profile appliedover region A (instead of treating T = Te in region A).

Note that in the evaluation of interferograms for sliding bub-bles, Dl was obtained using either Eqs. (8) and (9). Evaluation ofnb was carried out by assuming no variations of n within Dl. There-fore the heuristic analysis, summarized in Fig. 16, provides the lim-its for maximum possible error that may be present in theseevaluations.

Natural convection experiments were conducted to test the in-verse method describe above, as stated in Section 2.4. A rubber in-sert, with a cross-sectional area and length similar to that ofaverage bubble size encountered in the experiments, was placedin the middle of the heater plate. A steady state analysis usingthe heater surface temperature and heat flux was used to obtainthe temperature at the top of the insert. This agreed well withthe temperature at the top of the insert obtained through the nat-ural convection experiments using the inverse method. Further de-tails of this are provided in [25].

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Holographic interferometric study of heat transfer to a sliding vapor bubble

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