International Journal of Heat and Mass Transfer 53 (2010) 3193–3201

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Heat transfer model for gas–liquid slug flows under constant flux

Patrick A. Walsh a,*, Edmond J. Walsh a, Yuri S. Muzychka b

a Stokes Institute, Mechanical & Aeronautical Engineering Dept., University of Limerick, Limerick, Irelandb Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St. John’s, NL, Canada A1B 3X5

a r t i c l e i n f o a b s t r a c t

Article history:Available online 3 April 2010

Keywords:Two-phaseSlug flowNon-boilingThermographyHeat transferMass transferSlug aspect ratio

0017-9310/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.ijheatmasstransfer.2010.03.007

* Corresponding author. Tel.: +353 61213716; fax:E-mail address: Pat.Walsh@ul.ie (P.A. Walsh).

This paper investigates the mechanisms leading to enhanced heat and/or mass transfer rates in two-phase non-boiling slug flows. The problem is analyzed in a minichannel geometry subjected to a constantheat flux boundary. Local Nusselt numbers, obtained using Infrared thermography are analyzed in bothentrance and fully developed flow regions. These novel measurements highlight the physics governingslug-flow heat transfer and results indicate that optimized slug geometries can yield up to an order ofmagnitude heat transfer enhancement. Finally, based on the physics identified, a heat transfer modelis developed which is also applicable to similar mass transfer problems.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

In recent years there has been significant drive in almost allindustries to reduce the size of heat and mass exchange devices.In heat transfer applications, this has been primarily driven by ad-vances in microprocessor performance, as the computing industryis currently curtailed by increased heat flux levels due to reduceddie size and increased numbers of transistors. This problem of ele-vated heat flux has led numerous researchers to investigate manynovel air cooled heat sink designs [1–4] and to examine the effi-ciency of blade geometry in the miniature fans required [5,6]. Also,the advent of microchannel technology has lead to the develop-ment of numerous liquid cooling devices, such as cold plates forprocessor cooling, and compact heat exchangers for industrialapplications. These liquid cooling devices yield significantly en-hanced heat transfer coefficients over their air based counterpartsand hence, allow for further advances in processor performanceand the miniaturization of industrial processes. Various designsof these devices were suggested and prototyped by Mudawar [7]including: spray cooling [8]; jet cooling [9]; various designs of coldpates [10] and microchannels [11] to name but a few.

The latter of these in particular, investigating microchannelheat exchangers, has almost become a research discipline in itself.Numerous designs have been suggested to increase heat transferrates, ranging from different channel layouts to boiling inducedtwo-phase flows. The concept of microchannel cooling is simplybased on increasing the convective surface area to coolant volumeratio. However, despite some early suggestions that an increase in

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Nusselt number above the classical macroscale level may beachieved in microchannels, it has since being proven that heattransfer at the microscale, �250 lm or greater, collapses well withthe long established analytical solutions applicable to macroscalesystems. Also, laminar flows dominate at the microscale and thepressure required to pump fluids significantly increases as channelscales reduce. Hence, existing pumping technologies cannot pro-vide the required pressure drop and flow rate to achieve enhancedheat transfer through turbulent flow in practical environments.The Graetz solution [12,13] in the developed region for laminarflow shows that no gains in fully developed flow Nusselt numberare achievable for increases in flow rate, although the capacity toremove more heat, in line with the energy equation, is possible.Hence, much recent microchannel research has been focused onusing physical structures to induce local turbulent spots or theuse of two-phase flows, both of which enhance the Nusselt numberbeyond its laminar level. The current study focuses on the latter ofthese.

Many different two-phase flow regimes have been studied fromboth pressure drop and heat transfer perspectives. The most com-monly studied two-phase flows are typically encountered in boil-ing heat transfer where the coolant changes phase within theheat exchange device. This mechanism of heat transfer provideshigh local heat transfer coefficients but the process is complex, dif-ficult to accurately control, suffers from dry out issues and requireshigh pressure sealed vessels for containment. Comprehensive re-views on such technologies have been reported by Thome [14]and Kandlikar [15]. Alternatively, two-phase flows without boilinghave rarely been investigated for use in microscale heat exchangedevices. Instead, the bulk of studies analyzing non-boiling two-phase flows are focused on large scale systems such as that studied

Nomenclature

h heat transfer coefficient (W/m2 K)k thermal conductivity (W/m K)q00 heat flux (W/m2)x* inverse Graetz number, x/DRePr (–)w1,2,. . .,n error in measured variables, 1,2,. . .,n (–)y1,2,. . .,n measured variables, 1,2,. . .,n (–)Ca capillary number (–)D diameter of tube (m)F slug passing frequency (1/s)L length of heat transfer tube (m)Ls length of liquid slug (m)L+ dimensionless tube length, L/DPeD (–)L�s dimensionless slug length, Ls/DPeD (–)N number of slugs in tube (–)Nu Nusselt number, hD/kw (�)Pe Peclet number, UD/a (–)Pr Prandtl number, Cpl/k (–)Q volumetric flow rate (m3/s)Re Reynolds number, UD/m (–)U average velocity (kg/s)

WE error in calculated variable, E (%)

Greek symbolsa thermal diffusivity (m2/s)e void fraction, Qg/(Ql + Qg) (�)m kinematic viscosity (m2/s)q density (kg/m3)

SubscriptsDev developed flow regionEnt entrance regiong gaseous phasel liquid phasem bulk meanPois Graetz Poiseuille flowPlug Graetz plug flows slug floww wallx local value

3194 P.A. Walsh et al. / International Journal of Heat and Mass Transfer 53 (2010) 3193–3201

by Kim and Ghajar [16], Hetsroni and Rozenbilt [17]. The flow re-gimes encountered in these large scale systems were characterizedby bubbly, slug, mist, annular, wavy or stratified flows. Addition-ally, many authors have formulated charts in an attempt to identifywhich type of flow regime is most likely under prescribed gas andliquid flow rates [16–18]. Throughout these studies there is littleconsideration given to the local flow field. For example, the lengthof liquid slugs is not typically measured within the slug flow re-gime even though it has been noted to be of importance [19–22].This is especially significant in microscale heat or mass transfer de-vices where capillary forces dominate and the slug flow regime, asit is depicted in Fig. 1, is most often observed. Importantly, the def-inition of slug flow in microscale systems differs somewhat fromthat in macroscale systems. In microscale, liquid slugs and gas bub-bles occupy the entire channel cross-section whereas in macro-scale they may only occupy a percentage of this due to buoyancymismatches between the gas and liquid phases.

The focus of the current study is on this non-boiling two-phaseslug flow regime. Such flows are generated by segmenting a con-tinuous liquid stream with an injected gaseous phase to createwell-ordered trains of segmented liquid slugs and gas bubbles.As seen from Fig. 1, a large number of different configurationsare possible for the same gas and liquid flow rates depending onthe ratio of slug length to channel diameter (Ls/D). In gas–liquidslug flows, the gaseous phase has a negligible contribution to heattransfer since, at the same volumetric flow rate, its thermal capac-ity is <0.1% of the liquid phase when water and air are used. Also ofinterest for comparison with this flow regime, is the theoreticalGraetz solution for single phase flow in the limit of Prandtl number

Fig. 1. Slug flow with varying slug lengths for constant liquid and gas flow rates.

approaching zero. This theoretical result is referred to as the ‘PlugFlow’ limit and has been shown by Bejan through analytical con-siderations [23], to be capable of enhancing the Nusselt numberin the developed region from 4.36 for typical single phase convec-tion to 8 for plug flow under an isoflux boundary. In reality plugflows can only be realized by the use of an inviscid fluid or onewith an infinite thermal conductivity. The slug flow depicted inFig. 1 on the other hand, is more realistic in microscale deviceswhere typical coolants are used since viscous forces dominate atlow Reynolds numbers. Two distinctly different mechanisms whencompared to single phase convection could account for increasedheat transfer rates within this flow regime. These were noted byOliver and Wright [24] and include an increased velocity of the li-quid phase due to the addition of a segmenting gaseous phase andan internal circulation within liquid slugs, as was recently mea-sured by King et al. [25] using Particle Image Velocimetry in micro-channels. Muzychka et al [26] have shown analytically that flowsegmentation, resulting in increased fluid velocity alone, cannotaccount for increased heat transfer rates over single phase or longsolid plug flows. Hence the physical mechanism driving enhancedheat transfer must either result from the internal circulation, or amodified velocity profile within liquid slugs. Fig. 2 illustrates thecirculation within each slug and the difference in slug shapes thatresult from passing the fluid through channels with (a) hydropho-bic and (b) hydrophilic surfaces.

Although most studies do not control the slug length, this hasbeen recognized as a key factor in characterizing the enhanced heattransfer rates offered by segmented flows. A small number ofauthors have experimentally [19–22] and more recently numeri-cally [27–29] investigated the influence of slug length on heat andmass transfer characteristics. Table 1 summarizes the experimentalvariables of these studies and an in-depth analysis of their resultswas recently published by the authors [26]. The focus of the exper-imental studies listed in Table 1 was on bulk tube heat transfer withan isothermal wall boundary condition, although the numericalstudies do show local heat transfer rates. To date, only one experi-mental study has been undertaken to examine local heat transferwith an isoflux boundary condition but this was limited in scope[22]. Hence, a key question is: what Nusselt number enhancement,if any, can be achieved in segmented flows and how does this varywith changes in slug length, void fraction and Reynolds number?

Fig. 2. Schematic of circulation within each slug (a) hydrophobic surface, (b)hydrophilic surface.

P.A. Walsh et al. / International Journal of Heat and Mass Transfer 53 (2010) 3193–3201 3195

The current paper addresses this deficit in the literature byanswering these open questions. An experimental facility is builtto examine the problem in tube flow with a constant wall heat flux.A controlled slug flow regime is produced using water as the liquidphase and air as the segregating gaseous phase. Dimensionlessheat transfer results in terms of Nusselt number are deduced fromthe experimental results and presented for inverse Graetz numbersover three decades, which span both the thermally developing anddeveloped regions. It was found that long slugs showed littleenhancement over single phase flow while short slugs resulted inalmost an order of magnitude enhancement.

2. Analytical models

This section highlights the relevant analytical expressions usedfor characterizing single phase heat transfer in both the entranceand fully developed regions of internal channel flows. The specificproblem under investigation analyzes thermal boundary layerdevelopment in a hydrodynamically developed laminar flow withconstant wall heat flux. Dimensionless heat transfer rates for suchflows are characterized by local Nusselt numbers, as defined by thefollowing equation:

Nux ¼hxD

k¼ q00D

kðTw;x � TmÞð1Þ

In addition, the dimensionless position downstream of the heatedsection entrance is characterized by the inverse Graetz number.

Table 1Parameters of studies available in the literature.

Parameter Oliver and Young [19] Horvath et al. [20]

Ls/D 1.2–74 1.6–20L/D 144 260D 6.35 mm 2.32 mmRe 977–1362 3–220Pr 10 1700Pe 9770–13,620 51,000–374,000L+ 0.00087–0.069 0.0042–0.73L�s 0.000077–0.0061 0.0000025–0.00043� 0.15–0.39 0.5N 1–100 6–80Type Gas–liquid Gas–liquid

This parameter is referred to as x* throughout and is defined bythe following equation:

x� ¼ xD

aUD¼ x

DRePr¼ x

DPeð2Þ

For fully developed laminar tube flow, the velocity profile at anycross-section is known; hence the energy conservation equationcan be solved to provide an exact solution to the temperature field.Such a solution was first reported by Graetz [12,13] and allowed forthe later development of simplified piecewise analytical expres-sions defining the variation of local Nusselt number with dimen-sionless position [30]. These solutions are separated into tworegions where an expression is defined for the entrance regionwhile the Nusselt number in fully developed flow is constant. Muz-ychka and Yovanovich [31] combined such expressions using theaddition of asymptotic limits approach and arrived at the expres-sion of Eq. (3). This defines the variation of local Nusselt numberwith dimensionless position where a constant wall heat flux bound-ary condition is applied and the fluid Prandtl number, Pr, is greaterthan unity

Nux;Pois ¼1:302x�1=3

� �5

þ ð4:36Þ5" #1=5

ð3Þ

Also of interest to the current study is the theoretical solution ob-tained in the limit of Pr approaching zero. This is known as the plugflow limit and can be thought of as either an inviscid fluid stream ora solid rod passing through a channel. Hence, fluid flows as a solidplug with a uniform velocity profile across the channel cross-sec-tion. An analytical expression describing this solution, again interms of asymptotic limits was defined by Muzychka et al. [32],Eq. (4)

Nux;Plug ¼0:886x�1=2

� �2

þ ð7:96Þ2" #1=2

ð4Þ

In order to evaluate local heat transfer rates in liquid–gas slug flowsystems, it is also necessary to introduce a ‘void fraction’ parameter.This is required since the liquid phase is the primary contributor to-ward heat transfer but is only in contact with a percentage of thetotal heat transfer surface, since voids or gas bubbles are present.Hence, heat transfer rates should be normalized using this parame-ter which defines the percentage of the heat transfer channel vol-ume filled with gas at any operating condition. This can becalculated using the expression of Eq. (5), where Qg and Ql are thegas and liquid input volumetric flow rates, respectively

e ¼Q g

Q g þ Q lð5Þ

When analyzing results in subsequent sections, the single phaseheat transfer prediction of Eq. (3) will be used to validate the

Vrentas et al. [21] Narayanan and Lakehal [27]

10.5–61 1.92–6.1128 409.5 mm 1 mm0.7–15.8 1396–21351000, 10,000 5.54210–17,800 7678–11,4723.85–31.2 0.0014–0.00290.00083–0.0093 0.00025–0.000520.016–0.087 0.205–0.4802–12 2–10Solid–liquid Gas–liquid

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experimental set-up by comparing with results from benchmarksingle phase convection tests. Also, the models of Eqs. (3) and (4)along with the void fraction of Eq. (5) will be used to gauge themagnitude of enhancement offered by liquid–gas slug flows, andin developing a model for the heat transfer results obtained.

Fig. 4. Images of two-phase segmented train used to determine the slug length.

3. Experimental facility and data processing

The experimental facility used during this study is shown sche-matic in Fig. 3. This was designed and constructed to: (1) enable acontinuous stream of slugs to be produced, whose length could beaccurately controlled and measured; (2) provide a heated test sec-tion which subjects the internal flow to a constant wall heat fluxboundary condition; (3) obtain high resolution local measure-ments of the heated section surface temperature and (4) developan analysis tool to interpret temperature measurements and aver-age these over the recording period. Each of these aspects of thefacility will be discussed throughout this section along with themethodology employed for extracting results.

Two high precision Harvard PHD 2000 Programmable syringepumps were used to set the gas and liquid volumetric flow ratesdelivered from 100 ml capacity Hamilton glass syringes (Model1100TLL). Glass syringes were used as these were found to providesteadier flow than was attainable using equivalent plastic syringes.The flow from these entered transparent tubing and was combinedin T-junctions where well-ordered segmented slug trains weregenerated. The liquid slugs and gas bubbles were visualized, bothprior to entering and downstream of, the heated section throughtransparent tubing to ensure that consistent slug trains were main-tained and that no coalescence of slugs occurred. Images such asthat shown in Fig. 4 were recorded at the exit to the heated sectionand analyzed using Matlab (Version R2008a) to obtain an accuratemeasure of liquid slug lengths. This resulted in obtaining an aver-age and standard deviation for the slug length from a series of 10images recorded simultaneously with tube surface temperaturesfor each test conducted. A number of different internal bore T-junc-tions were used to generate different length slugs for fixed liquidand gas flow rates. In all cases water was used as the liquid phaseand air as the segmenting gas phase. Finally, it was also noted thatthe generation of well-ordered slug trains was greatly aided whenthe tube cross-section dimension was less than the capillary lengthscale, which for water and air as working fluids is �2 mm.

The heated test section consisted of a 1.5 mm internal borestainless steel tube with wall thickness of only 0.25 mm. Such athin wall was required to minimize both axial conduction lossesand the temperature gradient through the wall so that local walltemperature measurement could be obtained. The length of thestainless steel tube employed was 2 m but only a short section ofthis, 0.5 m length, was heated during experimentation. Before en-try to the heated section, flow was passed through an unheated en-try length of 300 diameters or 0.45 m to ensure hydrodynamicboundary layers became fully developed prior to heating. A con-

Fig. 3. Experimental set-up for measurements o

stant wall heat flux boundary condition was achieved in the heatedsection by Joule heated using a high current DC. power supply.Electrical connections to the tube were made using custom madesharp edged copper ring contacts, ensuring that the thermal en-trance point could be clearly identified. Additionally, using copperas the material for these contacts ensured minimal internal heatgeneration within them, and their sharp edge connection to thetube ensured minimal conductive heat loss. Finally, the stainlesssteel tube was used from new and so the surface was at least ini-tially uncontaminated, hence providing a hydrophilic boundarycondition so that the shape of liquid slugs during testing was sim-ilar to that shown in Fig. 2(b). It was also noted that there was nonoticeable change in this hydrophilic condition, or in heat transfercharacteristics, during the testing period of 3 months. This sug-gests that results are independent of surface contamination byatmospheric air.

External surface temperature measurements of the heated sec-tion were obtained using a high resolution Infrared (IR) thermogra-phy system. This consisted of a FLIR Systems ThermaCam Merlinseries IR camera set-up. Such measurements required the exteriorsurface of the stainless steel test section to be sprayed matt blackto enhance its surface emissivity. The actual emissivity of this sur-face was evaluated prior to testing by simultaneously comparing IRthermography measurements with equivalent measurements fromfour K-type thermocouples. These were calibrated to ±0.1 K andmounted on the tube surface. Calibration was achieved by pump-ing single phase liquid at a minimum of three known temperaturesthrough the tube at a sufficiently high flow rate to ensure that thetemperature drop between thermocouple probes, due to naturalconvection losses, was less than 0.1 K. This procedure resulted indetermining that the test section had a surface emissivity of 0.95,which is typical of a matt black surface. Additionally, during testingboth the IR camera and thermocouples were used to measure thesurface temperature of the tube, hence allowing for a continuouscalibration check of the thermal imaging set-up in-situ.

In attaining local temperature measurements throughoutexperiments, the IR camera was placed at a fixed position withits field-of-view extending over approximately 100 mm length ofthe tube, as is shown in Fig. 5(a). Hence, all local temperature mea-surements were obtained within this region. Also, in order to avoidthermal reflections from surrounding bodies, this section of theapparatus was further enclosed within a matt black box. Recordingof thermal images began immediately prior to supplying electricalcurrent to the test section for heating. Images were recorded for up

f tube surface temperature and slug length.

Fig. 5. (a) Image used for scaling; (b) selecting region of interest for analysis; (c) contour map of transient surface temperature; (d) output profiles of mean, upper and lowersurface temperatures.

P.A. Walsh et al. / International Journal of Heat and Mass Transfer 53 (2010) 3193–3201 3197

to 15 min and at a frequency of 0.2–1 Hz depending on the flowrate. Such extended recording times were necessary to let the heattransfer process reach a quasi steady-state condition where thewall temperature clearly oscillates periodically about a constantmean value. This periodic oscillation is due to the inherently un-steady heat transfer process associated with the periodic passingof liquid slugs and gas bubbles. In general, this quasi steady-statecondition was achieved in approximately 1 min. Also noted is thatthe IR camera recording frequency was typically lower than theslug passing frequency, F, see Table 2. Hence the exact oscillatorynature of the flow could not be captured. However, when record-ings were made over long time periods and averaged, the systemwas capable of accurately capturing average heat transfer rates.

After thermal images were recorded, it was necessary to devel-op an analysis code in Matlab (Version R2008a) for extracting thetransient tube surface temperature profile and averaging this overtime. Fig. 5 illustrates the procedure following by this code. Firstly,a single image without heating, and incorporating a ruler, is usedfor scaling, image (a). In image (b), a region of interest is high-lighted as shown by the dashed lines. The code uses an edge detec-tion algorithm to accurately identify the heated tube edges andsubsequently identify its centerline. Surface temperatures are thenextracted along this line at each pixel location and resulted in onetemperature measurement every 300 lm along the tube length.This latter step was looped for each IR recording over the test

Table 2Range of experimental variables for slug flow experiments.

Parameter Min Max

Re 56.4 1127Ca 6.6 � 10�4 8.0 � 10�3

x* 3.3 � 10�4 1.5 � 10�1

� 0.2 0.8Ls/D 0.45 0.8N 0.7 37.3F [1/s] 1.7 281.5

duration and resulted in obtaining the temperature contour mapshown in (c). The abscissa of this contour plot represents distancealong the tube from the inlet and the ordinate represents each im-age number in recorded sequence, i.e. representative of the timedomain and related through the recording frequency. The colorcontour map represents the tube surface temperature in degreeCelsius with magnitude indicated by the legend shown. The userthen inputs a range of image numbers between which the tubesurface temperature is considered to be in the quasi steady-statecondition discussed in the previous section. An average tempera-ture is then calculated for each axial location. The output of thisaveraging technique is shown in Fig. 5(d) which reports the meansurface temperature profile along the tube length. Also shown arecurves marked ‘upper’ and ‘lower’, these represent three standarddeviations of the periodically fluctuating temperature along thelength of the tube. Importantly, this fluctuation does not representan error, but rather is reflective of the unsteady nature of the heattransfer process when using segmented flows. Finally, the entiresystem was also calibrated using single phase flow theory, wheregood agreement was found for five different flow rates, rangingover three orders of magnitude of inverse Graetz number, x*. Re-sults from these calibration experiments along with those obtainedfrom 40 individual slug flow set-ups are discussed in the followingsection. Table 2 presents a brief summation of the range of exper-imental parameters examined throughout the study.

Before presenting results, an estimate of the error associatedwith the experimental measurements is made using the methodof Kline and McClintock [33], which is based on the expression out-lined in Eq. (6)

WE ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidEdy1

w1

� �2

þ dEdy2

w2

� �2

þ � � � þ dEdyn

wn

� �2s

ð6Þ

Here, ‘W’ represents the error associated with any calculatedparameter ‘E’, while ‘w’ represents the error associated with anymeasured variable ‘y’. The errors associated with most of the

Fig. 6. Dimensioned plot of time averaged wall temperature versus distancehighlighting the influence of the liquid slug length on local surface temperature. Re,void fraction and input heat flux held constant at 112.7, 0.33 and 8.852 W/m2,respectively.

3198 P.A. Walsh et al. / International Journal of Heat and Mass Transfer 53 (2010) 3193–3201

measured parameters were found to be dependent upon the param-eters’ magnitude. However, maximum errors were of order ±50 lmfor tube diameter, ±100 lm for all other length scales, ±0.5% for pre-scribed volumetric flow rates, ±0.5% for supplied heat flux, ±1% fortabulated fluid properties and ±0.1 K for temperature measure-ments. From these intrinsic errors, the total measurement errorfor Nux of Eq. (1), and x* of Eq. (2), were calculated by substitutioninto Eq. (6). This resulted in experimental errors which varied be-tween 8.5% and 6.8% over the range of Nux considered and 4.7% to1.1% over the range of x* values considered. Such errors were at amaximum in the early entrance region and quickly diminished to-wards their lower values as the flow developed. Uncertainty calcu-lations also showed that errors in void fraction measurementsthrough Eq. (5) were negligible due to the precision of the pumpsemployed. However, when measuring void fraction based on inletvolumetric flow rate conditions, errors can result due to an expan-sion of the gas phase between the location of the slug producingmechanism and the heated test section. This can result from varia-tions in both the pressure and temperature of the gas. An estimateof the maximum pressure drop between these locations was madefrom single phase correlations at the highest flow rate consideredand corresponded to �1200 Pa, while the maximum temperaturerise in the test section was 15 �C. Such small variations can beshown through the ideal gas law to result in maximum void fractionerrors of only 4% for the range of parameters considered. It will beseen in the following section that these uncertainty levels are farless than the enhancements observed during experimentation.Hence, all error bars are omitted from the graphs presented so asto enhance their clarity.

4. Results and discussion

This section details the heat transfer results obtained when ana-lyzing the effect of segmenting a continuous Graetz type flow intodiscrete liquid slugs separated by gas bubbles and heated by a con-stant wall flux boundary condition. The discussion of results beginsby first showing the augmentation of heat transfer rates attainablein a dimensioned format and then generalizing these results bypresenting them in dimensionless format. The physics of the floware detailed throughout and correlations based upon the describedflow physics, and the analytical solutions presented earlier, aredetermined. These are put forward to enable accurate predictionsof heat transfer in the entrance and fully developed flow regionsof two-phase non-boiling slug flows. Importantly, these can alsobe applied to similar mass transfer problems as the same physicsgovern this process also.

The first set of results presented in Fig. 6 show a dimensionedplot of the time averaged wall temperature rise versus distancefrom the entrance of the heated section. The theoretical bulk fluidand tube wall temperatures based upon an enthalpy balance andthe continuous Graetz flow solution of Eq. (3) are plotted for refer-ence. In addition to these, three experimental results for seg-mented slug flows are also presented where the slug length todiameter ratio is varied from 1.6 to 14.3. The plot highlights the ef-fect that changing slug length has on the resulting time averagedwall temperature for constant input boundary conditions of: wallheat flux; inlet flow rates of air and water; and hence void fraction.The most striking observation is the obvious change in the walltemperature profile within the entrance region. It is seen that theratio of slug length to tube diameter, Ls/D, has a profound effecton this profile and also on the fully developed temperature differ-ence between the tube wall and the bulk fluid. The general trendsobserved are that: shorter slugs provide augmented heat transferrates throughout the test section; moderate length slugs can resultin a degradation of heat transfer rates within the early entrance re-

gion but augmented in fully developed flow; and long slugs can re-sult in a degradation of heat transfer rates throughout the entiresystem. It is noted that such significant deviations in heat transferrates within the entrance region were not predicted by the numer-ical studies analyzing a constant heat flux boundary [28,29]. Also,none of the experimental studies analyzing isothermal wall bound-aries were able to obtain local heat transfer measurements such asthose obtained herein [19–21].

In order to explain the trends in wall temperature observed, theresult obtained for Ls/D = 5.7 or slug length of 8.6 mm will be fo-cused upon as this result clearly highlights both entrance and fullydeveloped flow phenomena. In the early entrance region, the tubewall temperature increases above that of continuous flow, then at adownstream distance of approximately 9 mm or one slug length,the wall temperature flat lines until approximately 22 mm. Thisphenomenon is due to the internal circulation within moving slugsthat simply results from a combination of viscous dissipation andmass conservation. This causes fluid from tube centerline to beforced towards the tube wall at the slug leading edge and visa ver-sa at the trailing edge. Given that in laminar flow a parabolic veloc-ity profile should develop in the center of such slugs, onecirculation length can be approximated by twice the slug lengthplus a single tube diameter. This is the distance a slug would movedownstream before an element of fluid would complete one circu-lation around the slug and is based upon (1) the no-slip conditionat the wall and (2) a typical centerline velocity for parabolic tubeflow equal to twice the mean slug velocity. Hence, the thermalboundary layer at the slug leading edge will continuously be re-newed with fluid at the inlet temperature until it has reached adownstream distance of one full circulation length, provided ofcourse that Pr� 0. The result is that the wall temperature in-creases to a value where the supplied heat flux can be dissipatedand holds constant until heated fluid from the early entrance re-gion makes its way to the slug leading edge. In Fig. 6, the circula-tion length corresponds to a distance of approximately 20 mmwhich correlates well with the position where the wall tempera-ture begins to rise again. Following this, a similar phenomenon isseen to occur downstream but its amplitude is damped signifi-cantly after each cycle until the flow fully develops and the walltemperature begins to rise linearly with distance.

The most sensible way of generalizing such results is to presentthem in dimensionless form as shown in Fig. 7 where the localNusselt number, Eq. (1), is plotted against dimensionless positionor inverse Graetz parameter, Eq. (2). In this plot, the dimensionless

Fig. 7. Dimensionless plot showing measured and predicted local Nusselt numbersversus inverse Graetz number, x*, for measured single phase, plug and slug flows. Reand e held constant for slug flow tests, equal to 112.7 and 0.33, respectively. Rerange for single phase flow was 56–1127.

Fig. 8. Dimensionless plot showing effect of slug length after normalizing local Nuand Re with the liquid wetting fraction (1 � e), Re and e held constant at 112.7 and0.33, respectively.

Fig. 9. Plot highlighting the effect of slug length to diameter ratio on fullydeveloped Nu magnitude.

P.A. Walsh et al. / International Journal of Heat and Mass Transfer 53 (2010) 3193–3201 3199

position is calculated based on the inlet liquid Reynolds numberonly and hence does not account for the increased fluid velocitydue to segmentation. This allows a direct comparison of heat trans-fer rates achievable when using a constant mass flux of water. Thefirst point to note here is the excellent agreement found betweenthe Graetz solution for single phase flow and experimental valuessince all local Nu measurements fall within 4% of this solution.With regard to slug flows, the plot clearly shows that these can de-grade heat transfer performance within the entrance region butwill typically augment heat transfer in fully developed flows, afinding which is of significant importance in applications wherethe heat exchanger length is of order the thermal developmentlength. In addition, the aforementioned internal circulation effectis clearly evident as the local Nu is seen to undergo a decayingoscillation about its fully developed value.

The first peak of this oscillation corresponds to a downstreamdistance from the thermal entrance of just over one circulationlength which is also equal to the period of oscillation. Furthermore,it is seen that the amplitude of the oscillation varies inversely withslug length, a phenomenon which is believed to be due to smallerslugs having lower thermal mass and so reaching fully developedflow much quicker. It is also worth noting, that in general, the ther-mal entrance length for slug flows is much shorter than for a con-tinuous flow. In fact, it is found that the only parameter whichaffects the entrance length is the liquid slug length since the localNu reaches its steady state value after only traveling one sluglength and oscillates about this value thereafter. Finally, it is alsoclearly evident that the slug length has a significant effect on fullydeveloped Nux as short slugs can greatly enhance this asymptoticvalue over that typical of a single phase flow.

Up to this point, the experimental data has been presented fordirect comparison with a similar continuous flow system but in or-der to gain a comprehensive understanding of the flow physicsgoverning segmented flow heat transfer, the data must be reducedappropriately. To achieve this, both Re and Nux are normalized bythe percentage contact area of the active cooling fluid, i.e. liquidslugs, since the thermal capacity of the gaseous phase is insignifi-cant. This percentage contact area is defined as one minus the voidfraction, (1 � e) see Eq. (5) for its calculation.

The experimental data is plotted in this format in Fig. 8 where thephysics of the flow can now be accurately compared with solutionsfor unsegmented Poiseuille and plug flow. Reporting the data in thisformat highlights some very interesting findings regarding the flowphysics. The first of these relates to the thermal development region

where it becomes clear that their heat transfer characteristics ap-proach that of a continuous Poiseuille type flow for long slugswhereas it approaches the theoretical plug flow limit for short slugs.Furthermore, over the range of experimental testing conducted, itwas confirmed that slugs of length equal to one tube diameter fallupon the plug flow limit and indeed this limit can be exceeded withslugs of even shorter lengths, see Figs. 9 and 10. This finding is incontrast with the numerical studies analyzing local Nu within theentrance region [28,29], as these showed that slug flow characteris-tics in this region always fall on the plug flow limit regardless of sluglength. This experimental finding can be explained by noting thatthe velocity profile in long slugs is identical to that of a continuousPoiseuille flow for the majority of their length, whereas in shortslugs, flatter velocity profiles at the slug leading and trailing endsdominate the heat transfer process. Furthermore, the observationof increased Nu above the plug flow limit is hypothesized to be plau-sible due to an impinging type flow phenomenon existing at the slugleading edge which results from the internal circulation. Thisimpingement point occurs at the leading edge of the slug where itcontacts with the wall. Also noted is that the angle of impinging flowwill be influenced by the wall surface properties, i.e. hydrophobic orhydrophilic, as was illustrated in Fig. 2.

With regard to fully developed Nusselt numbers, NuDev, it isclearly seen that shorter plugs provide far higher values than

Fig. 10. Plot showing the accuracy with which the proposed correlation of Eq. (4)predicts the experimental data in the entrance, transition and fully developedregions.

3200 P.A. Walsh et al. / International Journal of Heat and Mass Transfer 53 (2010) 3193–3201

longer ones. The maximum augmentation shown in Fig. 8 is a 600%enhancement over the fully developed Poiseuille flow limit or a300% enhancement over the theoretical limit of fully developedplug flow. However, for longer slugs it is seen that NuDev reducessignificantly and appears to be approaching the Poiseuille flow lim-it. This enhancement in NuDev results from the aforementionedinternal circulation, as it provides constant mixing of the slug vol-ume. Hence, the shorter the slug, the more isothermal its temper-ature will be. It is believed that the thermal mass of the slug is thedominating factor in this regard.

Now that the physics of the slug flows under analysis have beenexplained, the development of a correlation to accurately predictlocal Nu will be focused upon. This is achieved by deriving separateexpressions for the developing and fully developed asymptoticlimits and later combining these through the blending approachof Churchill and Usagi [34]. Firstly, results within the thermal en-trance region from the wide range of experimental investigationsconducted showed that the local Nu varies between the Poiseuilleand plug flow limits as slug length varies. This behavior was foundto be best correlated using a weighted mean method betweenthese two limits where a slug of length equal to one diameter fallson the plug flow limit, while an infinite length slug falls on thePoiseuille limit. The local Nu then varies linearly between theselimits for slugs of intermediate lengths. Hence, a correlation defin-ing local Nu within the entrance region is defined in the followingequation:

Nuxðs;EntÞ ¼ NuxðPois;EntÞ þ D=LsðNuxðPlug;EntÞ � NuxðPois;EntÞÞ ð7Þ

With respect to the fully developed flow limit, slug length againhas a significant effect on Nusselt number, NuDev. In order to obtainan accurate representation of how NuDev varies with liquid sluglength, an extensive range of tests were conducted with controlparameters ranging between the limits defined in Table 2. Thesetests investigated extensive ranges in Reynolds number, void frac-tion and liquid slug length. Overall, the flow physics observedthroughout these were identical to those already describedthrough Figs. 6–8. In all, 40 slug flow tests were conducted andexamined to determine the effect of slug length on NuDev. In orderto determine the correct relationship, it was first noted that in thelimit of an infinite length slug, NuDev should approach that of aPoiseuille flow. Hence, the difference between the Poiseuille flowlimit and the measured slug flow values are considered for corre-lating, and are presented in Fig. 9. This plot clearly highlights the

effect of liquid slug aspect ratio on the increase in NuDev over thePoiseuille flow limit, where the power law trend identified accu-rately represents the experimental data. Hence, the fully developedasymptotic limit is given by Eq. (8) where NuDev enhancements,over the Poiseuille flow limit, are observed to be inversely propor-tional to the square root of slug length to diameter ratio. Alsoworth noting from Fig. 9, is that there is little scatter in the exper-imental findings from the expression of Eq. (8) when Ls/D is below�3. Above this however, some scatter is observed. It is believedthat this scatter is primarily due to errors resulting from the meth-od used in determining NuDev when the flow had not reached itsfully developed limit within the experimental test section. Anexample of this can be seen from Fig. 8 when Ls/D was equal to14.7. For such cases, the value of NuDev was estimated to equalthe mean of the peaks and troughs observed

Nuxðs;DevÞ ¼ NuxðPois;DevÞ þ 25ðLs=DÞ�1=2 ð8Þ

These limits are now combined in Eq. (9) with a blendingparameter, n, equal to 10 used to ensure an abrupt transition fromentrance to fully developed flow. The accuracy of this correlation isseen in Fig. 10 by comparing predictions to a number of the exper-imental measurements obtained. It is seen that very good agree-ment was found in both the thermal entrance and the fullydeveloped flow regions. However, for longer slugs where Nux oscil-lated within the transition region, the correlation was less accuratein this region. Hence, the proposed correlation can be employedwith a high level of accuracy for values of x/Ls < 1 and x/Ls > 5. How-ever, within the range 1 < x/Ls < 5 the proposed correlation resultsin up a ±20% deviation based on the experimental data. Of course,this correlation does require knowledge of the liquid slug length. Inpractical applications this would have to be either directly mea-sured within the system, or alternatively it could be predicted fromrecently published [35] relationships between pressure drop andslug aspect ratio in gas–liquid slug flows without boiling

NuxðsÞ ¼ ðNuxðs;EntÞÞ1=n þ ðNuxðs;DevÞÞ1=nh in¼10

ð9Þ

Finally, as was noted at the beginning of this section, the de-rived correlation is not limited to modeling heat transfer problemsbut can also be applied to similar mass transfer problems. This isachieved by replacing the Nusselt number throughout with itsanalogous mass transfer equivalent, the Sherwood number, andlikewise changing the Prandtl number used in the definition of in-verse Graetz parameter, Eq. (2), with the Schmidt number. It isnoted that such an analogy should provide an excellent mass trans-fer model within the entrance region, Eq. (7), since this was definedfrom theoretical considerations alone. The model put forward forfully developed flow on the other hand, Eq. (8), incorporates anempirically derived coefficient which may change if applied tomass transfer problems. Further experiments are required to deter-mine if this is the case but the power law relation between thetransfer rate and slug aspect ratio should remain unchanged.

5. Conclusions

The current study reported novel experimental insights into theflow physics associated with segmented slug flows under a con-stant wall heat flux boundary condition. High spatial resolutiontemperature measurements were obtained which allowed for flowcharacteristics within the entrance, transition and fully developedregions to be accurately defined. It was shown that heat transfercharacteristics in the entrance region of slug flows are governedby a combination of conventional single phase and plug flowbehaviors with short slugs approaching the theoretical plug flowlimit and long slugs approaching the single phase flow limit.

P.A. Walsh et al. / International Journal of Heat and Mass Transfer 53 (2010) 3193–3201 3201

Another interesting finding, with regard to the thermal entrancelength, was that slug flows reach their fully developed asymptoticlimit at a distance of one slug length regardless of flow Reynoldsnumber. However, in transitioning from the entrance region tofully developed slug flow, local Nu oscillate about the fully devel-oped limit with a period equal to the length of one internal circu-lation. Finally, in the fully developed flow limit, it was found thatNuDev can be augmented by up to an order of magnitude over thesingle phase limit by utilizing slugs of length to diameter ratioclose to unity. A correlation based on the described flow physicswas proposed to aid in predicting local Nu over the entire rangeof inverse Graetz numbers but it was noted that some error maybe encountered at the transition region due to the observed oscil-lations. Overall, the findings reported herein provide a greaterunderstanding of the physics associated with non-boiling slugflows and correlations are proposed which can be used for predic-tive purposes in either heat or mass transfer applications.

Acknowledgments

The authors acknowledge the financial support of EnterpriseIreland for funding this research.

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