flowboilingheattransfercharacteristicsusingthemodifiedeulerian and wall heat … · 2020. 7....

ORIGINAL

Flow boiling heat transfer characteristics using the modified eulerianand wall heat balance model

C. E. Okon1 & A. Turan1

Received: 28 October 2019 /Accepted: 29 January 2020# The Author(s) 2020

AbstractFlow boiling heat transfer is routinely encountered in nuclear reactors, steam engines and other engineering applications.Although several researchers have carried out different numerical and experimental investigations on flow boiling, the underlyingphysics of the interfacial interaction is still a complex phenomenon to understand in detail. Hence, the numerical simulation andoptimisation regarding the adoption of engineering flow boiling parameters have been conducted in this study using theModifiedEulerian-Eulerian Model (MEEM) and Wall Heat Balance Model (WHBM). To predict interpenetrating flow fields and toprovide detailed relevant information on the flow behaviour, this study considered a uniform axial heating profile for a cylindricalflow channel. The Raynolds Average Navier Stokes (RANS) equation with an appropriate turbulence model are used to predictthe effect of turbulence on the mean flow field, while the MEEMmultiphase sub-models are employed to predict the temperaturedistribution along the wall, the average void fraction, tracking of the single bubble detachment diameter, heat balance at the wall,effect of surface roughness on heat transfer, the effect of aspect ratio, and the critical heat flux. The results obtained from thisstudy are compared with the selected numerical investigation and experimental data presented in the open literature. The presentstudy shows a better approximate prediction (with minimal uncertainties) of both the subcooled boiling heat transfer and thesaturated boiling heat transfer. In summary, this study agrees with extant theories and experimental predictions. Thus, it hasprovided more profound insights into flow boiling heat transfer particularly for flow in a vertical pipe.

NomenclatureAf Cross-sectional area of the liquidcpf The specific heat of the waterdd Bubble departure diameterDdr Droplet diameterDe Equivalent diameterDh Channel Hydraulic diameterfbw Bubble departure frequencyFam The added mass forceFb Buoyancy forceFdu Unsteady forceFqs Quasi-steady forceFs Surface tension forceFsl Linear lift forceG Mass fluxhf Liquid Phase Enthalpyhg Vapor Phase Enthalpy

hfg Latent heat of vapourizationℵ!

wz Interaction drag forceJa Jacob numberk The Boltzmann constantkf Thermal conductivity of the liquidK!

wz Turbulent dispersed forceLd Length for bubble detachmentNw Active nucleation site densityPl The imposed liquid pressurePv The vapor pressure inside the nucleusqm Flow rate of the outside airq! z The heat fluxr Bubble radiusrsvn The spherical vapor nucleusR0 Initial radius of the cavitySq The source termfi The bulk inlet temperatureTl Liquid temperature near the wallTw Wall temperature

GREEK SYMBOLSβ The Inclination angleτ Collision frequencyη Liquid thermal diffusivity

* C. E. [email protected]

1 School of Mechanical, Aerospace and Civil Engineering, Universityof Manchester, Manchester, UK

https://doi.org/10.1007/s00231-020-02854-5

/ Published online: 29 April 2020

Heat and Mass Transfer (2020) 56:2425–2443

http://crossmark.crossref.org/dialog/?doi=10.1007/s00231-020-02854-5&domain=pdfmailto:[email protected]

ℵ!

f The body forceδl,,f Thin film thicknessSUBSCRIPTSb Buoyancyd Departuree uivalentf Liquid Phaseg Vapor Phasel Liquids Surface Tensionv vapourw Wallz z-directionamam Added massdr Dropletdu Unsteadyqs Quasi-Steadysl Linear Liftsvm Spherical Vapor Nucleus

1 Introduction

Flow boiling occurs when a fluid circulates over a heatedsurface by external means such as a pump or due to thenatural buoyancy effect [1]. Several flow patterns characterisesuch a flow field, and the transition between flow regimesdependent upon the channel geometry and thermophysicalproperties of the fluid [2]. A continuous flow of liquid andvapour in a two-phase flow boiling and heat transfer encoun-ters some flow resistance, that causes the fluid pressure to dropat a certain height (i.e., the non-boiling and the boiling height)which is one of the main factors of characterising the flowboiling in a channel. The non-boiling height is the single-phase height in which the subcooled liquid entering the chan-nel receives a quantity of sensible heat. This sensible heatcauses a change in subcooled temperature, resulting insubcooled boiling without any significant effects on the pres-sure drops and fluid density. In the boiling height, the pressuredrop is higher than that in a non-boiling height for the sameflow rate [2]. This higher pressure drop is a result of an in-crease in the flow velocity along the channel and reduction incross-sectional area due to the presence of both liquid and gasphases.

Flow boiling heat transfer is primarily affected by factorssuch as the inertia forces, viscous forces, pressure forces, in-terfacial tension forces, liquid-surface contact angle, exchangeof mass, momentum, and energy between the interacting in-terface between the liquid and vapour phases. These interfacesare bubbles in a continuous liquid flow, a droplet in continu-ous vapour flow, a vapour film in continuous liquid flow and aliquid film in continuous vapour flow [3]. Due to a largenumber of dependencies, it is not possible to obtain a single

heat transfer correlation for the entire flow regimes. The na-ture of the flow regimes depends on the operating conditionssuch as pressure, temperature, mass flux, channel orientation,and fluid properties [2, 4].

The onset of sub-cooled nucleate boiling initiates when thewall temperature at the single-phase forced convection regionreaches to the saturation temperature at a given pressure, andalso the wall superheats necessary to cause nucleation. Theamount of superheat necessary for nucleate boiling to occur isdependent on the inner surface heat flux. Hence, no boilingcan occur when the inner surface temperature is less than thesaturation temperature at a given location along the channel[2]. Therefore, subcooled boiling begins and develops whenthe bulk temperature or the mean enthalpy of the liquid phaseis below the saturation temperature. In the highly subcooledregion, bubbles grow and collapse while attached to the heatedsurface and do not penetrate far into the bulk subcooled flow.Bubbles grow and detach in the low subcooled region becausethe void fraction in this region rises with its length. The sur-face tension and inertia forces are responsible for the bubbleformation in a highly subcooled region, which also holds thebubble to the heated surface. Buoyancy and the frictional(drag) forces are responsible for the bubble detachment froma heated surface in a low subcooled region [2].

2 Background literature

In order to understand the state-of-the-art of flow boiling, acomprehensive review of the background literature has beenconducted. Flow boiling research has seen significant ad-vances in recent years with particular attention given by thenuclear industry. Over the years, numerous numerical andexperimental techniques were employed to understand flow-boiling phenomena. For instance, experimental investigationshave used a combination of high-speed cameras and imageprocessing techniques to visualise the different flow regimesand analyse the different bubble dynamic parameters.Numerical simulations were applied to study and visualisethe bubble dynamics and different flow parameters for differ-ent working fluids and other operating conditions. Analyticalapproaches based on theoretical correlations have been widelyused for studying the interfacial transport phenomenon. Thisapproach is particularly vital in the subcooled boiling regionwhere bubble nucleation at the interfacial boundary is theessential heat transfer mechanism. This method also exploresthe use of interfacial area equation to study particle interactionat the interface and phase change.

2.1 Experimental framework

Prodanovic et al., [5] performed experimental studies using avertical annular test section to observe the effect of parameters

2426 Heat Mass Transfer (2020) 56:2425–2443

such as subcooling, pressure, and flow rate on bubble behav-iour in subcooled flow boiling. High-speed photography atrates between 6-8 kHz revealed at low heat flux, the bubbleshape is spherical and does not change in size with the flowdistance. The final region termed the bubble coalescence re-gion was characterised by large bubbles of different sizes andshapes. Work by Michel et al. [6] experimently investigatedthe effect of gravity on heat transfer of upward flow boiling.This experiment was performed by varying the gravity levelswhile making local temperature measurements and observingthe different flow transitions. The effect of gravity on heattransfer decreases as the mass flux and the heat flux increases.Forced convection was experienced in high mass flux whilefilm evaporation was seen dominant during applied high heatflux. Burak et al. [7], carried out an experimental investigationof flow boiling using a single 1.8 mm× 0.9 mmmicrochannelat low mass velocities to ascertain corresponding effects onthe bubble dynamics. Their experimental observations indi-cated that the rate at which heat transferred from the heatedsurface to the main flow stream is dependent on the heat fluxand vapour quality, and it increases with increasing mass flux.Recent work on flow boiling by Colgan et al., [8] studied theCritical Heat Flux (CFH) under subatomic atmospheric con-ditions. Based on their findings, the mass flux and the inletsubcooling do not have any substantial effect on the criticalheat flux, and also the wall superheat is prolonged at the sub-atmospheric conditions compared to the atmosphericconditions.

Experimental studies of Zhu et al. [9] on naturalcirculation of two-phase flow resistance in a vertical 3 × 3rod bundle showed a rapid occurrence of flow instability un-der natural circulation conditions. This instability results in arapid increase in acceleration pressure drop and frictionalpressure drop under a low mass quality condition. With thegravity, channel pressure drop decreases under the same con-dition. Sira et al., [10] investigated the effect of channel ori-entations on flow boiling behaviours. As expected, theirinvestigation was in agreement with the expected results asachieved by previous researches over the decades. Theyspecifically examined the effect of change in channelorientation on pressure drop, flow pattern, and heat transfercoefficient. Cheng et al. [11] studied the effect of diameter ontwo-phase gas-liquid flow using a small tube of 28.9 mm indiameter and a larger tube of diameter 150 mm. They reportedthat slug flow does appear only in the pipe with small diameterbut did not appear in large diameter. Chen et al. [12] studiedthe two-phase flow of R134 refrigerant in vertical mini pipesand found out that the transition regions of slug-churn andchurn-annular, strongly depend on tube the diameter.However, the transition boundaries for bubbly to churn andbubbly to slug are slightly affected by a change in the magni-tude of diameter. Furukawa et al. [13] studied the effect ofviscosity on transition regions between flow patterns in

vertical tubes with inner diameter of 19.2 mm and height of54 mm. They used water and aqueous glycerol solutions withviscosities of 1 × 10–6 up to 14.7 × 10–6 m2/s and reportedthat an increase in liquid viscosity shifts the bubbly-slug tran-sition boundary to lower superficial air velocities and thefroth-annular transition boundary to regions of higher super-ficial viscosity.

Taylor performed visualisation tests on a vertical annulus toinvestigate the bubble departure frequency [14]. Conclusionsfrom this study established that heat flux, mass flux, degree ofsubcooling, and system pressure are the four major thermal-hydraulic parameters which control the bubble departure fre-quency. Their investigation shows that the bubble departurefrequency increases with an increase in heat flux and pressure,but it decreases with the increase of mass flux and subcooling.Thorncroft et al. [15], measured the waiting time, bubblegrowth, and bubble departure size in an upward verticalsubcooled flow-boiling channel. They correlated the waitingtime against the wall superheat and the growth time with thebulk subcooling.

2.2 Computational framework

A validation and assessment study to predict the critical heatflux in flow boiling inside a 4 × 4 rod bundle was carried outby Harish et al. [16]. They tested the effect of axial powerdistribution by the applied heat flux, which affects the temper-ature distribution that, in turn, affects the coolant phase changeand the critical heat flux. From detailed and systematic vali-dations, mean relative errors of 15% and 7% for tube andannulus cases respectively were recorded. These simulationsidentify the critical heat flux (CHF) location in an annulus anda rod bundle. The wall temperature variations predicted for theconcentric and eccentric annulus as well as a 4 × 4 rod bundlegeometry. Eccentricity found to promote the early occurrenceof CHF in the narrow gap and delay in the broad gap regions.For the 4 × 4 rod bundle considered, the corner rod facing thepressure tube was found to have an early shoot up in the walltemperature. The effect of different turbulence model was alsocarried out and they concluded that RNG k-e accurately pre-dicts both the CHF and the post-dryout wall temperature;while standard k-e and SST k-w models underpredicted andoverpredicted the peak wall temperature, respectively.

In order to understand the nature of bubble behaviour at thereactor’s operating conditions, Eyitayo et al. [17], carried out anumerical study using the Large Eddy Simulation (LES)-mi-cro layer model to investigate the growth rate, the microlayerthickness, and distortion of a single bubble in a highly turbu-lent flow boiling. They investigated the effect of system pres-sure, contact angle, initial bubble size, and bulk velocity onbubble growth. For the analysis of bubble microlayer thick-ness, the results showed that the microlayer thickness in-creases with the increase of both the radius of the micro-

2427Heat Mass Transfer (2020) 56:2425–2443

https://www.sciencedirect.com/topics/engineering/flow-boiling

region and the system pressure. Furthermore, they studied thebubble growth rate for different operating pressures. In partic-ular, a slow growth rate was experienced at system pressuresbetween 1-21 MPa while the growth was rapid at higher pres-sures due to the decrease in liquid-vapour density ratio andsurface tension. This report shows that using Large EddySimulation (LES) to analyse the bubble microlayer andgrowth in subcooled flow boiling ensured a faster computa-tional speed with an excellent accuracy than carrying out a fullDirect Numerical Simulation (DNS) analysis. Overall, thisstudy covers the operating conditions for a boiling water re-actor (BWR) and pressurised water reactor (PWR).

Adrian et al. [18], studied the two-phase flow of air/watermixture using OpenFOAM and compared their findings withthe mechanistic model developed by Petalas et al. [19]. Theirfindings showed a good correlation with the mechanistic ap-proach as all the flow regimes observed were consistent withthe mechanistic prediction. Fangfang et al. [20] studied theinstability mechanisms leading to slug formation by usingDirect Numerical Simulation (DNS) analysis of two-phaseflow in an inclined pipe. They identified different flow re-gimes (i.e., stratified, slug, dispersed, and annular) and thefindings were compared with the theoretical models.Nemitallah et al., [21], carried out an intensive numericalstudy to analyse the current status of two-phase flow boilingcharacteristics using the Eulerian Multiphase flow boilingmodel. In their research, they considered the influence of anon-uniform axial heating profile for a highly pressurised sys-tem of 2 m long with inner and outer diameters of 15.4 mmand 25.4 mm respectively. The mass flow rate and the inlettemperature of the water was 0.161 kg/s and 200oC, respec-tively. Francisco et al. [22], also carried out a numerical inves-tigation to validate the subcooled flow boiling in a verticalchannel using the model developed by Kurul et al. [23].They considered several factors (such as the mesh aspect ratio,heat flux, system pressure, and thermodynamic quantities.)which could enhance the active nucleation along the heatedwall. Their numerical investigation was validated using thebenchmark of the experimental work of Chenturiya [24].Adrian et al. [25], modelled two-phase flow boiling in aBoiling Water Reactor Fuel Assembly using a CFD codeSTAR-CDwhich has been seen to have the capability of track-ing the mass, momentum, and energy of the liquid and vapourphases in each computational cell. Huiying Li et al. [26]modelled the boiling flow within the framework of theEulerian multifluid model, using the RPI model for nucleateboiling and its extension to the non-equilibrium boing andcritical heat flux regime. Mukesh et al. [27], of the BhabhaAtomic Research Centre, Mumbai, India, carried out a CFDsimulation to predict the void fraction and temperature distri-bution inside the rod bundle to identify the growth of any hotspots during the removal of decay heat in an AdvancedHeavy-Water Reactor (AHWR). This boiling water reactor is

a natural circulation based reactor with many passive safetyfeatures and utilizes an isolation condenser system for decayheat removal. It is believed that, during the removal of thedecay heat, there is a possibility of a hot spot occurring insidethe fuel rod cluster. Hence, a boiling Eulerian modelwas developed to obtain the flow field during the decay heatremoval by first considering the effect of different models onthe flow field.

Multiphase flow boiling modelling using CFD is verychallenging due to the inherent complications such as the in-stabilities arising from the turbulent mixing of the differentphases that have different characteristic properties, the wall-fluid interaction encountered during the process, and the in-terfacial heat and mass transfer in the two-phase flow-mixingregion. Despite the numerous research, the two-phase flowboiling phenomenon still remains a challenging case studyin the nuclear industry and other engineering applications.Notwithstanding, the importance of CFD cannot beoveremphasised because it is an indispensable tool used inthe absence of an experimental setup to study and understandseveral complex structures in fluid flows. This present study isaimed at exploring the capabilities of Computational FluidDynamics (CFD) using the modified eulerian and wall heatbalancemodel [28] to optimise the accuracy of some predictedflow boiling parameters. The modified Eulerian two-fluidcoupled model is more robust and can predict flow boilingparameters more accurately in an upward flow through a heat-ed vertical channel. The results in this study can be used as abenchmark for the validation of future study.

3 Modelling description

The pictorial representation, as shown in Fig. 1, explains theconcept established during vertical flow boiling. In practical,the thermodynamic nonequilibrium experienced in two-phaseflow, causes the actual vapour quality to be less than the equi-librium quality. The present study has been implementedusing ANSYS Fluent. Some key information about modellingwith ANSYS Fluent cand be found in the user guide manual[52].

3.1 Turbulence model

The Raynolds Average Navier-Stokes (RANS) method targetsresolving turbulence quantities, and the gradients of theRenolds stress tensor components entering the equation ofmotion represent the total stress tensor in a fluid obtained fromthe averaging operation over Navier-Stokes equations to ac-count for turbulent fluctuations in fluid momentum. This com-ponent is of decisive importance for the correct capturing ofthe flow physics. Equation (1), is a modified standard k − ϵturbulence model with an additional dissipation energy term


(Rϵ) which account for the effective viscosity. This modelprovides information for low Reynolds number effects andimproves the accuracy for rapidly strained flows [1].

∂ ρεð Þ∂t

þ ∂∂xi

ρεuið Þ ¼ ∂∂x j αεμeff∂ε∂x j

� �

þC1ε εk Gk þ C3εGbð Þ−C2ερε2

k−Rε þ Sε

ð1Þ

3.2 Multiphase model

The baseline wall boiling model used for this simulation is theTwo-Phase Eulerian model developed by Kurul et al. [28].This wall heat balance model, as shown in Eq. (2), is used toquantify the wall-to-fluid heat transfer. It does not automati-cally compute the vapour temperature when solving the phaseenergy equation; instead, the model partitioned the total heatflux HB(W) from the wall to the fluid free stream into three

components namely: the liquid phase convective heat flux q̇,

the quenching heat flux q̇ Qð Þ and the evaporation heat fluxq̇ Eð Þ.

HB Wð Þ ¼ HTCl Tw−Tlð Þ 1−NBð Þ

þ Ywt 2klffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiπαl= f bw

p Tw−Tl;yþ� �NB

þ π6

d3bw f bwNwρvHlv ð2Þ

To account for the non-equilibrium boiling and critical heatflux, the value for the vapour temperature, which correspondsto the saturation temperature at a given operating system, thepressure is inputted using the user defined function (UDF).Thus, Eq. (3), shows the modified wall heat-partitioning mod-el which was used in this study.

HB Wð Þ ¼ q̇ Cð Þ þ q̇ Qð Þ þ q̇ Eð Þ þ q̇ Fð Þ� �

f αlð Þ

þ 1− f αlð Þð Þq̇ Vð Þ þ q̇ Gð Þ1− f αlð Þ

¼ f αvð Þ ¼ max 0;min 1; αv−αv;1αv;2−αv;1

� �ð3Þ

where; α_(v,2)=0.95 and α_(v,1)=0.90Table 1 shows different models available in the pub-

lished literature to predict the bubble departure diameter(db) which is one of the most significant dynamic param-eters observed during flow boiling heat transfer. Most ofthe existing bubble departure diameter correlations (suchas Fritz [32], Cole and Rohsenow [33], Tolubinski andKostanchuk [34], Kocamustafaogullari-ishii [35], Klauseret al. [36] and Situ et al. [29]) were formulated based onthe balance of forces acting on the bubble at the depar-ture. Several researchers, such as Huiying Li et al. [26],used the Tolubinski & Kostanchurch model for their re-search. Unal correlation [37] is considered in this presentstudy because it depends solely on the degree of localsubcooling and the wall superheat, and also it shows thebest agreement with the experimental data reported bySitu et al. [29]. The general formulation for bubble depar-ture diameter is as shown in Eq. (4).

db ¼ db;0 Tw−T satð Þm ð4Þ

During the transition from a bubbly to a mist flow, thedrople t d iameter (Dd ) was es t imated us ing theKocamustafaogullari-Ishii correlation provided in Eq. (5) [38].

Dd ¼ Cds σρv j

2vRe−1=6l Re

2=3v

ρvρl

� �−1=3 μvμl

� �2=3ð5Þ

Where Cds is a coefficient, which has a value of about0.028, and jv is the superficial velocity.

The active nucleation site density, defined as the numberof nucleation sites per unit area, has been resolved inthe current study using the Lammet correlation [39].Kocamustafaogullari-Ishii correlation [40] was developed to

Fig. 1 Vapour Quality along thechannel

Table 1 Bubble Departure Diameter Models [29–31]

Model db = db, 0(Tw − Tsat)m

db, 0 m

Fritz0:208θ

ffiffiffiffiffiffiffiffiffiffiffiffiffiσ=gΔρ

p 0Cole

4x10−2ffiffiffiffiffiffiσ

gΔρ

qρlCplρvhfg

1

Cole and RohsenowCfs σgΔρ

� �1=2 ρlCρlT satρvhfg

� �5=4 0

Tolubinsky-Kostanchuk min[dref exp (−ΔTsat/ΔTref), 0.0014] 0Unal

1:21x10−5P0:709ρvhfg

kwρwCpwπb∅

� �1=2 1

Kocamustafaogullari and Ishii2:64x10−5θ Δρρv

� �0:8σ

gΔρ

� �0:5 0

Situ et al.4ffiffiffiffiffi7:3

pb2

π S:ρ f Cpfρ f hfg

� �2 v fffiffiffiffiffiffiffiClur

pPr f

2


account for cases of forced convective boiling at high pres-sure, the Lammert-Chawla model as used in this current re-search gives a more precise, accurate and stable solution ascompared to the Hibiki and Ishii model [41] used by Colomboand Fairweather [42] in their research.

All the published models have provided the nucleation sitedensity (Nw) as a function of the wall superheat, as shown inEq. 6.Where C andm are dimensionless constants with values1.805 and 15,545.54, respectively.

Nw ¼ C Tw−Tsatð Þm ð6Þ

The drag coefficient ( f) was calculated using the UniversalDrag Function [43], computed based on the relative Reynoldsnumber as shown in Eq. (7)

f ¼ CDRe24

ð7Þ

Also, the disperse bubbly, and mist flows were also calcu-lated using Eq. (8);

D!DF

lv ¼ −D!DF

vl ¼Ai8ρcC

Dlv V!

l−V!

v

�� V!l−V!v� �

ð8Þ

In the bubbly flow regime, the lift coefficient (CLlv )was calculated using the formulations of Moraga et al. [26].However, this coefficient is assumed to be zero for dropletflows. The general formulation, as used for the simulation, isshown in Eq. (9),

L!LF

l ¼ − L!LF

l ¼ −ρcCLlvαd V!

l−V!

v

� �� ∇−V!c� �

ð9Þ

The wall lubrication force (F!

wl ) acts as a repulsive forcealong the wall by pushing the bubbles away from the wall.Previous studies as reported in the literature used the Antalet al. model [25] [26] [22] whereas this present study used themodel proposed by Hosokawa et al. [44] and the wall lubri-cation coefficient (Cwl) value of 0.0217 was calculated usingEq. (10). Also, this model employs the Tomiyama formula-tion, which is defined based on the channel hydraulic diame-ter.

F!

wl ¼ Cwlρlαv v!l− v!v� �

II

�� 2 n!w ð10ÞLopez-de-Bertadano model [45] was considered for the

turbulence drift force (see Eq. 11), although several authorsused Burns-et-al Model [46] as recommended in the literature.

D!TDF

lv ¼ D!TDF

lv ¼ ρcCTDkecαd∇ ð11Þ

Where kec is the turbulent kinetic energy.During flow boiling, there is an interfacial heat transfer

(i.e., the interface to liquid transfer and the interface to vapour

transfer). At the onset of the nucleate boiling, the bubble de-parture from the wall transport both heat and mass from tobulk liquid. This phenomenon is known as the interface-liquidtransfer and numerically by Eq. (12);

q̇lit ¼ hlitAi T sat−Tlð Þ ð12Þ

Where hlit is the heat transfer coefficient of the liquid inter-face. This study used two resistance Phase-Interphase heattransfer models switching from Ranz-Marshal to Hughmarkformulation [47], shown in Eq. (13).

hlit ¼ klDd 2þ 0:6Re0:5Pr0:33

� � ð13ÞSimilarly, the interface to vapour heat transfer is calculated

based on the Lavieville et al. correlation [48], which proposedthat vapour retains the saturation temperature by rapid evapo-ration/condensation. Equation (14) is a mathematical expres-sion for vapour phase heat transfer.

q̇vit ¼αvρvCp;v

δtT sat−Tlð Þ ð14Þ

Where δt is the time scale.Also, the wall-vapour mass transfer (evaporation mass

flow, ṁE ) is calculated based on the evaporation heat flux

q̇E defined in Eq. (15);

ṁE ¼ q̇Ehlv þ Cp;l T sat−T lð Þ ð15Þ

Furthermore, the interfacial mass transfer relationship usedin the present study is given by Eq. (16);

ṁlv ¼ ṁlit þ ṁvit ¼ q̇lit þ q̇vitHlv

ð16Þ

3.3 Boundary conditions

Appropriate thermal boundary conditions specified for allflow boundaries and all walls. Setting up the correctboundary conditions is very important due to the rele-vance in the drag and lift exerted on bodies as well asheat and mass transfer across boundaries. Specifically,when considering continuous vapour and liquid phaseswhere evaporation and condensation occur, their bound-ary conditions becomes very complicated as they requirecomprehensive information about the molecular phenom-ena at the interface [49].

The following initial conditions used for our numericalanalysis; At the Inlet; the velocity and the temperature ofthe working fluid as it enters the domain were specified to


be 1.04 m/s and 474.15 K respectively. While at the out-let, a pressure outlet was specified and the value of530.15 K was used as the backflow temperature to ac-count for any reverse flow occurs during part of the iter-ative process. This value can only use it if there is reverseflow, but if there is no reverse flow, the backflow valuewill not have any effect on the solution.

The heat flux of 0.57 MW/m2 was specified at the wallboundary. By defining the heat flux, the CFD algorithm willcalculate what the wall temperature will be in other to achievethat heat flux.

Turbulence settings for near-wall boundary layer are de-pendent upon the turbulence approach used. The boundarylayer is a thin layer of fluid adjacent to the solid surface inwhuch the velocity develops from zero at the wall to the freestream value at soe distance away from the wall. Hence, it isessential to use a very fine mesh to resolve the steep profilewithin the boudary layer because the frictional drag on thesurface and the heat transfer takes place within the boundarylayer. The mesh resolution requirements are normallyexpressed in terms of the dimensional wall distace, whichshould be maintained within a acceptable range for the modelused to achieve accurate results. The enhanced wall treatmentfor the k-epsilon realisable model used because this near-walltreatment will automatically switch between the wall functionand viscous sublayer resolution based on the value of y+. Also,it will blend between the two if the first grid point happens tofall above the buffer region. With the RANS model, no turbu-lence was resolved in the simulation; instead, the mean flowfield was calculated. The turbulence specification methodused for this study is the intensity and hydraulic diameter.Turbulence intensity is a measure of the strength of turbulencefluctuations. The value of 0.05 used in this study means thatthe velocity of the turbulence flunctuations is 5% of the meanvelocity.

The time step size Δt was set to be small enough to resolvetime-dependent behaviour in the flow and was calculatedusing Eq. (17).

Δt ¼ Lg:β:ΔT :Lð Þ1=2

ð17Þ

L=the characteristic length scale,β,ΔT, is the thermal properties of the fluidAlso, the y + value estimated using Eq. (18);

yþ ¼ ρU τyμ

ð18Þ

4 Results and discussion

This section presents the current case study analysis for thenumerical framework and results validated with the most re-liable experimental and numerical data available in the openliterature. Most recent experimental data reported in the openliterature use refrigerant and other materials as the workingfluid, as such, not suitable for the validation of the presentstudy.

4.1 Steady state analysis

Sensitivity analysis has been conducted showing the effect ofmesh resolution and turbulence on the flow field.

Mesh dependence A 2D and 3D Hexahedral mesh presentedin Fig. 2 was created in such a way that the grid points areclose enough together to predict the regions where there arehigh gradients in the flow or temperature field. Also, such thatall the geometric features of interest resolved. These wereperformed for a different number of grid points, as shown inFig. 3. The results show that the number of grid points has anoverriding on the entry length and the void fraction at theoutlet. The results obtained with the refined mesh gave ashorter entry length, (hence longer computational times andlarge memory consumption), and a high vapour volume

Fig. 2 Computational meshused for the current simulation


fraction at the outlet, whereas, low-resolution mesh resultspoint to a longer single-phase entry length, (hence fast com-putation time and low memory consumption), and a lowervapour fraction at the outlet.

Turbulence flow field The choice of turbulence modelplayed a significant role in the accuracy and stabilityof the simulation. The k-epsilon model gave a stablesolution during iterations, and all the residuals met theconvergence criteria, whereas, a little turbulence

fluctuation experienced when using the k-omega forthe simulation. Apart from the continuity, other residualsmet the set convergence criteria. Notwithstanding, fromthe graph of Fig. 3, both k-epsilon and k-omega show astable and smooth transition, and a more accurate resultwas obtained using the k-omega model irrespective ofthe turbulence fluctuations experienced during residualiterations.

N/B: ON-FDSCNB: Onset of the Fully DevelopedSubcooled Nucleate Boiling

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

noitcarFe

muloVruopaV

Axial Length (m)

2D-RUNS 1 (k-epsilon)


3D HEXAMESH

2D-RUNS 2 (k-omega)



Degree of Subcooling = 56.4K System Pressure = 45barMass Flux = 900kg/m2-sInner Heat Flux = 0.5MW/m2

Fig. 3 Mesh Dependence

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noitcarFe

muloVruopaV

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2D-Runs 1 (k-epsilon)

2D-Horizontal (k-epsilon)

2D-Experiment


3D-Hexamesh

2D-Runs 2 (k-omega)




Fig. 4 The output void fractionfor different test cases


Figure 4 and Fig. 5 shows the output void fraction fordifferent test cases and compared with the experiment andthe tracking of the single bubble departure diameter forthree test cases. This study explains the flow dynamicswhen liquid moves along the heated pipe. The fluid nearthe heated wall gains thermal energy, thereby resulting inthe temperature distribution from the heated surface alongthe fluid stream. Though the temperature of the fluid in-creased along the pipe, the primary fluid stream will notboil unless it reaches the saturation temperature, which inthis case is 530.15 K. The length of transition to nucleateboiling is where the fluid gains thermal energy albeit notenough to cause bubble nucleation. This phenomenon hasbeen investigated for different test cases as tabulated inTable 2. Hence, the onset of nucleate boiling begins when

the wall temperature and the fluid near the wall are thesame as the saturation temperature. Bubble growth alsobegins at this stage until detachment occurs at the onsetof fully developed subcooled nucleate boiling (ON-FDSCNB). Spontaneous bubble growth and detachmentcontinues as the vapour nucleus attains a size greater thanthat for unstable equilibrium, as observed in Fig. 5.

Figure (6, 7 and 8) show a comparative analysis of thesubcooled boiling length and the average volume fraction re-ported by Adrian et al. [18], Fransisco el al. [22], Huiying Liet al. [26], compared with the current numerical study andexperiment. The values, as reported in Table 2, show that thewall-boiling model used for the current study accurately pre-dict the boiling phenomena with a smooth transition and flowstability. It also recorded about +0.039 percentage deviation

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)m(rete

maiD

erutrapeD

elbbuB

Axial Length (m)

Runs 4 3D-Hexamesh Runs 1


Fig. 5 The Tracking of SingleBubble Departure Diameter (m)against Axial Length (m)

Table 2 Table showing the gridpoints, the single-phase entrylength, and the percentage of out-put void fraction for different testcases

No. of GridPoints

ONB-FDSCB (m)

VAPOURVOLUMEFRACTION

Percentage Deviation fromExperiment

2D-Runs 1 13,000 1.00 38.90 −0.0612D-Runs 2 30,634 0.70 41.20 −0.0382D-Runs 4 213,689 0.40 48.90 +0.039

3D-Hexamesh 0.60 46.60 +0.011

Adrian et al. STARCD [25]

0.70 40.20 −0.048

Huiyiug Li et al.[26]

0.90 44.90 −0.001

Fransisco et al. [22] 1.19 40.70 −0.043Experiment [24] 0.80 45.00


from the experiment with the non-boiling length of 0.4 m,which is perfect for a stable, steady-state condition. The pre-vious numerical investigation and experimental data used as areference for the current study introduced the wall heat flux atsome distance (0.4 m) away from the channel inlet. This adi-abatic entry length accounts for a single-phase flow regionwhere there is no transfer of heat between the thermodynamicsystem and the environment, neither there is a change in thetemperature of the subcooled liquid flowing along this region.The onset of the single-phase convective heat transfer began atthe point where the uniform wall heat flux is introduced up to

about 1 m from the channel inlet. The set criteria resultsto an increase in the ratio of the entry length to thechannel length, which inturns underpredicted the aver-age void fraction at the channel outlet. Whereas, thecurrent simulation assumed that the entire channellength from the inlet is uniformly heated and the walltemperature at the inlet is assumed to be the same asthe liquid inlet temperature. It is thereby resulting in asmooth transition of the heating effect up to the onsetof incipient boiling and better prediction of the averagevoid fraction.

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3D-Hexamesh (Current Work)


Fig. 6 The output void fractionfor Adrian et al. and Chanturiya

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2D-Huiying Li et al.)

2D-Experiment



Fig. 7 The output void fractionfor Huiying Li et al. andChanturiya


Figure 9 and Fig. 10 show the centreline temperature, andthe degree of wall superheat, respectively. The flow channelshows that the wall temperature increases exponentially alongthe vertical tube. At the channel inlet, the temperature of theprimary phase is 474.15 K. As the liquid moves along theheated pipe, it gains thermal energy from the heated wallthereby resulting in the temperature distribution from the heat-ed surface along the fluid stream starting from the liquid nearthe heated wall. Though the temperature of the fluid increasesalong the pipe, the primary fluid stream will not boil unless itreaches the saturation temperature, which in this case is530.15 K. The entry length where the fluid gains thermal

energy, but it is not enough to cause bubble nucleationknown as the non-boiling height. Hence, the onset ofnucleate boiling begins (if and only if) the wall temper-ature and the fluid near the wall are the same as thesaturation temperature. Spontaneous bubble growth con-tinues when the vapour nucleus attains a size greater thanthat for unstable equilibrium. Previous studies show thatfor a normal operating condition of a water reactor heattransport channel, the wall superheat should not behigher than 10 K [2]. The result obtained in this studyfalls within the benchmark reported in the literature, andit is in good agreement with the experiment.

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2D-Experiment



Fig. 8 The output void fractionfor Francisco et al. andChanturiya

470

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540

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enilretneC

Axial Length (m)

Current Work Experiment BartolomeiDegree of Subcooling = 56.4K System Pressure = 45barMass Flux = 900kg/m2-sInner Heat Flux = 0.5MW/m2

Fig. 9 The CentrelineTemperature (K) against AxialLength (m)


Figures 11 and 12 illustrate the significant effects ofsurface roughness on the total pressure drop and heattransfer coefficient which determines the amount of va-pour volume fraction at the channel outlet. In Fig. 11;P1, P2, W1 and W2 represent the average pressuredrops along the mainstream liquid phase, in the main-stream vapour phase, along the liquid phase pipe wall,and along the vapour phase, respectively. When a fluidis flowing along a heated pipe, the heat transfer andflow behaviour vary with the heat flux and the condi-tions of the fluid. Where the fluid temperature reaches

saturation point, there is an increase in volume and ve-locity of the mixture. The ressure loss encountered dueto the increase in momentum of the mixture as it flowsthrough the channel and vapourises, and increase infrictional pressure drop due to the effective roughnessof the inner channel. The pressure loss is also causedby the buoyancy effect and can lead to an increase ingravitational potential energy as the boiling mixturerises in the channel.

Table 3 gives a comprehensive summary of data pre-dicted for regions of both subcooled nucleate boiling

530

532

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542

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D

Axial Length (m)

Experiment Bartolomei Current WorkDegree of Subcooling = 56.4K System Pressure = 45barMass Flux = 900kg/m2-sInner Heat Flux = 0.5MW/m2

Fig. 10 The Degree of WallSuperheat (K) against the AxialLength (m)

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Fig. 11 Roughness Height (m)against Pressure Drop (Pa)


and saturated nucleate boiling using the initial andboundary conditions stated in section 4.1. The definitionof the initials as used in Table 3

& ONPDSUB (Onset of Partially Developed SubcooledNucleate Boiling)

& ONFDSUB (Onset of Fully Developed SubcooledNucleate Boiling)

& ONPDSAT (Onset of Partially Developed SaturatedNucleate Boiling)

& ONFDSAT (Onset of Fully Developed Saturated NucleateBoiling)

The subcooled nucleate boiling starts from the point(ONPDSUB) where the wall temperature is sufficiently

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ur F

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Fig. 12 The Output Void Fractionagainst Roughness Height (m)

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VVF-ONFDSAT VVF-ONPDSAT VVF-EXITDegree of Subcooling = 56.4K System Pressure = 45barMass Flux = 900kg/m2-sInner Heat Flux = 0.5MW/m2

Fig. 13 Vapor volume fractionalong the flow channel fordifferent heat fluxes


superheated enough to cause nucleation of bubbles of liquidnear the wall. At this point, the onset of saturated boilingbegins, and the value of bulk liquid temperature obtainedwas about 499.43 K. As the wall temperature increased furtherto the point of ONFDSUB, the degree of subcooling, bulkliquid temperature, and wall superheats are 21.29 K,510.64 K, and 11.21 K respectively. The heat transfer untilincipient boiling is a single-phase flow heat transfer of theliquid. Thus, the heat transfer coefficient is almost a constantalong the channel, but when the wall temperature graduallyincreases, there is a corresponding increase in the liquid tem-perature near the wall thereby causing a transition from oneregime to another. In order to control a smooth transition fromcontinuous liquid bubbly flow to a continuous vapour droplet,it was assumed that the interfacial surface topology contains amulti-connected interface and the transition from one flow

regime to another controlled by the vapour volume fraction.There exists a point (PDSAT) where the bulk liquid enthalpyequals the saturation enthalpy, and then boiling is said to oc-cur. After boiling starts, the heat transfer coefficient begins toincrease and then become almost constant after the saturatedboiling initiation. Hence, the interfacial length from the inletto this point is about 0.9 m, while the degree of subcooling, thebulk liquid temperature, and the degree of the wall superheatare approximately, 16.68 K, 531.94 K, and 21.29 K respec-tively. This situation continues until the fully developed satu-rated nucleate boiling by transition from nucleate boiling toforced convection evaporation heat transfer.

In summary, within the bubbly flow region, the vapourphase is dispersed in the continuous liquid in the form ofbubble and the vapour volume fraction is said to

volume fraction is between 0.28 and 0.5. The region where theliquid phase dispersed as droplets in the continuous vapour,the vapour volume fraction in this region is > = 0.5.

Figure 13, shows the vapour volume fractions (for differentheat fluxes) at the onset of the partially developed saturatedboiling (VVF-ONPDSAT), fully developed saturated boiling(VVF-ONFDSAT) and vapour volume fraction at the pipe exit(VVF-EXIT). This scenario can be observed when the bulkliquid enthalpy is higher than the saturation enthalpy. Theamount of heat transfer in the saturated boiling region

controlled by the mass quality or the vapour volume fraction.As reported in Table 3, the vapour volume fraction is about28% at the onset of the saturated boiling, and as the heatflux increased along the pipe, the vapour volume fractionincreases up to 46%. Further increase in the surface heatflux could result to burnout scenario. Burnout ischaracterised by a noticeable increase in surface tempera-ture in response to a change in the wall heat flux. Thischange in temperature can be very drastic, as in the caseof a Pressurized Water Reactor (PWR) or the change can

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Wehtta

ecanalaBtae

H2 )

Axial Length (m)

EVAP. HF Current WorkQUEN. HF Current WorkCONV. HF Current WorkEVAP. HF. Adrian et al. STAR CDQUEN. HF. Adrian et al. STAR CDCONV. HF Adrian et al. STAR CD


Fig. 15 The Heat Balance at theWall

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VVF-1 VVF-2 VVV-3Degree of Subcooling = 56.4K System Pressure = 45barMass Flux = 900kg/m2-sInner Heat Flux = 0.5MW/m2

Fig. 16 Vapour Volume Fractionalong the axial length for differentaspect ratios


be very gradual as for a Boiling Water Reactor (BWR).The condition for critical heat flux approached for a givensystem by varying any of the independent variables. Thefour independent variables that influence critical heat flux,as investigated in this study are; the mass velocity, thechannel inner diameter, the channel length, and the degreeof inlet subcooling. For uniformly heated channels wherethe flow is assumed to be stable, the onset of critical heatflux condition occurs at the exit end of the channel. At ahigh mass velocity, the critical heat flux increases as thesteam mass fraction increases along the flow channel,whereas, at a low mass flux, the steam mass fraction ap-proaches a critical point due to a drastic decrease in theheat transfer coefficient, thereby causing a high localisedheating effect at the channel exit. For predicting the crit-ical heat flux, it is assumed that the fluid fed to the chan-nel is subcooled and contains no entrained vapour, theflow is stableinand oscillations do not occur.

Figure 14 shows the plot of the wall superheats at differentaspect ratios for the onset of partially developedsubcooled nucleate boiling and the onset of fully devel-oped subcooled nucleate boiling, and the pressure drop(DP) against enthalpy rise for different heat flux andsystem pressure (45 bar) and mass flux (900 kg/m2 s)respectively. This chat is a lookup chat (with minimalerror) for design modelling, and the results agree withany experimental data.

Figure 15 shows how the wall heat flux partitioned alongthe heated channel, as explained earlier in section 3.2. Thetotal heat flux balance can be given by splitting the heat fluxinto three different wall regions; the evaporation heat flux,quenching heat flux, and convective heat flux. All the heat

balance accurately predicted and compared with the work ofAdrian et al., as tabulated in Table 4.

Figure 15 shows a smooth spontaneous exponential decaypattern of the convective heat flux up to a channel length of1 m where the bubble nucleation is dominant. In addition,both the quenching and evaporation heat flux values show abetter trend, which is a reflection of an accurate prediction ascompared to the work of Adrian et al.

Most importantly, the results reported by Adrian is contra-dictory with the present work. Here, their results do not reflectthe theory of the physics of the flow. According to Collier [2],the effect of both evaporation heat flux and the quenching heatflux can only be experienced when the bulk liquid gainenough thermal energy from the heated wall to cause bubblenucleation. In essence, both the quenching and evaporationheat flux values should be zero at the channel inlet. The reportof Adrian et al., contradict this norm as it records the values of63.48 W/m2 and 1745.755 W/m2 for hese two types of heatfluxes, respectively.

Figure 16 shows the effect of aspect ratio on the single-phase entry length and the maximum void fraction record-ed along the axial length. The ratio of the length of thechannel to the channel diameter (L/D) investigated forthree different cases using a 2D hexahedral refined meshwith grid points of 13,000. The first case which representsthe diameter (D: 0.00385 m) with an aspect ratio of519.48 has the maximum void fraction (VVF1 = 0.83)and the single-phase entry length of about 0.5 m. Thesecond case (D: 0.0077 m, L/D: 259.74) and third case(D:0.0154 m, L/D: 129.87), recorded the maximum va-pour volume fraction of VVF-2 = 0.61 and VVF-3 = 0.39respectively, also with single-phase entry lengths of 0.9 m

Table 4 Summary of Heat Balance data obtained

CONVECTIVE HEAT FLUX QUENCHING HEAT FLUX EVAPORATION HEAT FLUX VAPOURFRACTION

MINIMUMVALUE (W/m2)

MAXIMUMVALUE (W/m2)

MINIMUMVALUE (W/m2)

MAXIMUMVALUE (W/m2)

MINIMUMVALUE (W/m2)

MAXIMUMVALUE (W/m2)

Adrian et al.STAR CD

33,137.6 1,199,937 63.48199 506,046.7 1745.755 1,000,032 0.00846

Current Study 84,254 570,000 0 251,559 0 234,187 0.07125

Fig. 17 Vapour Volume Fractionalong the axial length for differentaspect ratios


and 1.1 m respectively. In summary, a high aspect ratioresults to a shorter single-phase entry length and increasesthe heat transfer coefficient along the wall while a lowaspect ratio results to a longer single-phase entry lengthwith a reduced heat transfer coefficient along the wall[50].

4.2 Transient Analysis

Figure 17 and Fig. 18 show the transient behaviour of bubblegrowth and detachment for flow boiling along the vertical pipeusing the Volume of Fluid (VOF) multiphase submodels. Asshown in Fig. 17 (Step 2), the fluid near the wall had gainedenough thermal energy to initiate bubble nucleation at the wallcavities. Hence, forces responsible for bubble growth andbubble detachment plays a vital role in this stage. The forma-tion, growth, and detachment of the vapour bubble depend onthe influence of the forces acting on the bubble [51]. Thesurface tension and inertia forces are responsible for the bub-ble formation in the highly subcooled region and also holdsthe bubble to the heated surface, while the buoyancy and thefrictional (drag) force are responsible for the bubble detach-ment from the heated surface in the low subcooled region [2].For a spherical bubble to remain in the bulk fluid after detach-ment from the heated surface, the pressure in it must be higherthan the pressure of the surrounding fluid, and its size must belarger than the critical size of a bubble. This phenomenon ispractically observed by the bubble contours in the current casestudy, as seen in Fig. 17 and Fig. 18.

5 Conclusions

This study considers the 2D and 3D numericalapproachs for simulating a boiling phenomenon for up-ward vertical flow boiling in a pipe using the ModifiedEulerian Model and Wall Heat Balance Model(WHBM). The model and sub-models used in this studywere able to provide more accurate results than the fewreported in the open literature. Parameters investigatedwere the axial distribution of bulk liquid temperature,

axial distribution of the wall adjacent temperature, axialbulk average vapour volume fraction, superheat alongthe wall surface, the bulk liquid subcooling with axiallength, tracking of the single bubble departure diameter,heat balance at the wall, the effect of aspect ratio, theeffect of inlet subcooling and the critical heat flux. Allthe results obtained are in good agreement with theexperimental data used for the validation and agree withthe physics of the theory.

The two critical features established in this studywere the effect of grid points and turbulence on thenon-boiling length and the output void fraction. Theresults obtained for different test cases show how thenumber of grid points and the choice of turbulencemodel could influence the accuracy in resolving the dif-ferent flow fields. Runs1-Runs4, as shown in Table 2and Fig. 3 have different numbers of grid points, andthe results obtained show reasonable percentage devia-tion of the entry length and void fraction, using eitherthe k-epsilon and k-omega models. Longer entry lengthis experienced for low mesh cells, whereas, high meshcells results in a shorter entry length, thereby enhancingeffective/efficient heat transfer out of the channel.

Other key observations deduced from this study can besummarised as below;

& The Modified Eulerian Two-phase Coupled Model ismore robust and can predict more accurately flow boilingparameters in an upward flow through a heated verticalchannel.

& The amount of superheat necessary for nucleate boiling tooccur is dependent on the inner surface heat flux.Therefore, no boiling can occur when the inner surfacetemperature is less than the saturation temperature at agiven location along the channel.

& The adjacent wall temperature close to the inlet is the sameas the fluid inlet temperature because the heating processof the wall undergoes a gradual and sustained thermalconduction effect. As such, one should not expect anabrupt increase in heat flux distribution along the wall.The Onset of Sub-cooled Nucleate Boiling initiates when

Fig. 18 Contour Showing theBubble Growth and Detachmentfor Upward-Vertical Flow(Current Work)


the temperature of the wall at the single-phase regionequals the sum of the saturation temperature at the givensystem pressure, and the wall superheats necessary tocause nucleation.

& The onset of the critical heat flux condition occurs at theexit end of the channel

In conclusion, this current study employed alternative nu-merical phase coupling models as compared to ones used byprevious researchers. The results obtained in this study is com-prehensive, explicit, and more accurate than those reported inthe literature. It is thus, recommended as a validation baselinefor future study.

Acknowledgements I am grateful to the Almighty God for his perfectlove and grace throughout this study. I extend profound gratitude to thePetroleumTechnology Development Fund (PTDF) for funding this study.Also, special thanks go to Dr. Andrea Cioncolini (University ofManchester-UK), Professor Ita O. Akpan, Professor E. J. Uwah,Professor S. O. Udoh and Professor Anthony Akpan (University ofCalabar). Heartfelt gratitude goes to Chukwuma Ogbonnaya (Universityof Manchester-UK), for his assiduity and absolute professional input,which significantly improved the quality of this work.

Open Access This article is licensed under a Creative CommonsAttribution 4.0 International License, which permits use, sharing, adap-tation, distribution and reproduction in any medium or format, as long asyou give appropriate credit to the original author(s) and the source, pro-vide a link to the Creative Commons licence, and indicate if changes weremade. The images or other third party material in this article are includedin the article's Creative Commons licence, unless indicated otherwise in acredit line to the material. If material is not included in the article'sCreative Commons licence and your intended use is not permitted bystatutory regulation or exceeds the permitted use, you will need to obtainpermission directly from the copyright holder. To view a copy of thislicence, visit http://creativecommons.org/licenses/by/4.0/.

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Flow boiling heat transfer characteristics using the modified eulerian and wall heat balance modelAbstractIntroductionBackground literatureExperimental frameworkComputational framework

Modelling descriptionTurbulence modelMultiphase modelBoundary conditions

Results and discussionSteady state analysisTransient Analysis

ConclusionsReferences

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