VolumeofFluid(VOF)ModelingofLiquidFilmEvaporationin

Research ArticleVolume of Fluid (VOF) Modeling of Liquid Film Evaporation inMixed Convection Flow through a Vertical Channel

Hayat El Baamrani Lahcen Bammou Ahmed Aharoune and Abdallah Boukhris

ermodynamics and Energetics Laboratory Faculty of Sciences Ibn Zohr University BP8106 Agadir 80006 Morocco

Correspondence should be addressed to Hayat El Baamrani hayatelbaamrani92gmailcom

Received 20 March 2021 Revised 21 April 2021 Accepted 11 May 2021 Published 24 May 2021

Academic Editor Babak Shotorban

Copyright copy 2021 Hayat El Baamrani et al ampis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

In this paper the volume of fluid (VOF) method in the OpenFOAM open-source computational fluid dynamics (CFD) package isused to investigate the coupled heat and mass transfer by mixed convection during the evaporation of water-thin film ampe liquidfilm is falling down on the left wall of a vertical channel and is subjected to a uniform heat flux density whereas the right wall isassumed to be insulated and dryampe gas mixture consists of air and water vaporampe governing equations in the liquid and in thegas areas with the boundary conditions are solved by using the finite volume method ampe results which include temperaturevelocity and vapor mass fraction are presented ampe effect of heat flux density liquid inlet temperature and mass flow rate on theheat and mass transfer are also analyzed Better liquid film evaporation is noted for the system with a higher heat flux density andinlet liquid temperature or a lower mass flow rate amperefore the VOF method describes well the thermal and dynamic behaviorduring the evaporation of the liquid film

1 Introduction

Falling liquid film evaporation has been widely studied inrecent decades due to its interest in improving coupled heatand mass transfer for various engineering applications suchas seawater desalination distillation drying of agriculturalproducts cooling of electronic components and air con-ditioning Many studies linked to the heat and mass transfercharacteristics are available in the literature Classical worksincluding evaporation of pure liquid film have been largelystudied [1ndash7]

A mixed convection heat and mass transfer during theliquid film evaporation has been the subject of a largenumber of numerical or experimental studies For numericalstudy Van and Lin [8] studied the effects of natural con-vection all along a finite liquid film evaporation in a verticalchannel ampey showed that the assumption of an extremelythin film is only useful for a system with a small liquid massflow rate Yan et al [9 10] presented a numerical study of theeffect of the wall heat flux and inlet liquid mass flow rateduring finite liquid film evaporation of laminar mixed

convection in a vertical channel ampe results indicate that theheat and mass transfer is dominated by latent mode LaterFeddaoui et al [11] presented a numerical study for liquidfilm evaporation inside an insulated vertical channel ampeyreported that the evaporation is higher for low liquid flowrate or high gas flow Reynolds number ampey also treated thecase of liquid film evaporation with turbulent flows [12]Mezaache and Daguenet [13] presented a numerical study ofthe effect of inlet conditions on the evaporation processalong the gas-liquid interface over an inclined plate ampeirresults showed that the effect of inclination is highly de-pendent on the liquid mass flow rate and gas velocity and anincrease in the inclination improves the evaporation byincreasing the vapor mass flow rate Laaroussi et al [14]investigated the mixed convection in a vertical channel ofthin film evaporation on wetted walls ampey compared theBoussinesq approximation with the density change for air-water vapor and air-hexane vapor mixtures Mass diffusioninduces an important flow acceleration in the boundarylayers and the recirculation of flow at the center of thechannel for large mass evaporation rates

HindawiMathematical Problems in EngineeringVolume 2021 Article ID 9934593 12 pageshttpsdoiorg10115520219934593

Cherif et al [15] presented an experimental study of theevaporation by natural and forced convection in a verticalchannel and a thin liquid film falls on two parallel platesampey showed that the thermal yield of evaporation increaseswith the down air velocity and the heat flux density in orderto improve heat and mass transfer during liquid filmevaporation

Cherif et al [16] investigated experimentally the influ-ence of film evaporation on mixed convective heat and masstransfer in vertical parallel channel wetted by a water filmampe results showed that the evaporative cooling increaseswith small heat flux and large air velocities Alla et al [17]studied the evaporation of glycol liquid film in mixedconvection ampeir results showed that the propylene glycolevaporates more than glycol and the heat transfer by thelatent mode is more important than the heat transfer bysensible mode Nasr and Alghamdi [18] investigated nu-merically the evaporation and condensation under mixedconvection of binary liquid filmampey showed that the watercondensation increases by reducing inlet temperature andincreasing the amount of liquid film and an increase of inletliquid concentration of ethylene glycol enhances the eth-ylene glycol evaporation and water vapor condensationZhang et al [19] studied the evaporation by laminar mixedconvection in a vertical three-dimensional rectangularchannel with nitrogen and water vapor ampe results indicatethat the wall temperature and the inlet temperature improvethe heat and mass transfer Alla et al [20] have investigatednumerically two configurations In the first case a uniformheat flow is applied to the wet wall while in the second caseand the same heat flux was applied to preheat the liquid atthe inlet ampe authors indicate that the evaporation rateincreases with the second case Fang et al [21] studied theheat transfer coefficient correlation of the multiphasetransformation inside and outside the tube ampey showedthat the heat transfer coefficient increases with the feed flowrate and ethanol content in the evaporation tube ampe heattransfer coefficient decreases when the heat transfer tem-perature difference between the inside and outside of thetube increases

We also find in the literature some other studies thathave focus on heat and mass transfer with the evaporation ofa liquid film flowing in a porous layer Leu et al [22] nu-merically studied the heat and mass improvement of liquidfilm evaporation with a porous layerampe results showed thatthe heat transfer performance of the liquid film was im-proved by using a porous layer and the average Nusselt andSherwood numbers are increased inversely with porosityporous layer thickness and relative humidity Nasr and Al-Ghamdi [23] analyzed the influence of porous layer thick-ness and porosity on the coupled heat and mass transferperformance during the evaporation of flowing liquid filmampe results indicate that the presence of the porous layerimproves the heat and mass transfer performance at theliquid-gas interface Terzi et al [24] numerically studied theperformance of liquid film evaporation by introducing aliquid saturated porous layer along an inclined channel amperesults indicated that the use of the porous layer improvesthe heat and mass transfer

ampe VOFmethod has been used in the literature in manyproblems involving two-phase flows Piscaglia et al [25]presented a development and validation of a two-phasedynamic VOF solver for the simulation of internal nozzleflows ampe phase change at the interface was modeled usingthe Schnerr and Saver model Large Eddy simulations (LES)are used to model turbulence effect ampe approach can beconsidered to be reliable for investigations of the internalnozzle flows during transient operations Pham and Choi[26] performed a numerical simulation for the waterhammering using interFoam solver ampe pressure-implicitmethod is used for continuity momentum and energyequations of a volume of fluid model discretization ImplicitEuler and central difference schemes were used for temporaland spatial discretization ampe solver is validated with a one-dimensional Stefan problemrsquos benchmark case Amidu et al[27] presented a hybrid VOF- Eulerian multifluid model forsolving a hot low and high void fraction subcooled flowboiling ampey showed that the best estimation of the radialvelocity profile is achieved by using the two-phase roughwall function instead of the classical single-phase wallfunction

In our knowledge all the previous studies deal with theheat and mass transfer during the liquid film evaporationwith the finite volume method by considering boundaryconditions at gas-liquid interface ampe main objective of thepresent work is the use of volume of fluid (VOF) method toinvestigate numerically the mixed convection heat and masstransfer with liquid film evaporation To this end anOpenFoam solver was developed and successfully validatedwith previous numerical results taken from the literatureampe effect of parameters such as the inlet liquid temperaturemass flow rate and wall heat flux density on the heat andmass transfer during water film evaporation in a verticalchannel was investigated

2 Problem Formulation

21 Definition of Geometry ampe present work deals with anumerical study of liquid film evaporation by mixed con-vection in a vertical channel as illustrated in Figure 1 ampegeometry under consideration is two vertical and parallelplates with height H and width b ampe first plate is main-tained at the uniform heat flux qe and wetted by a thin liquidfilm with inlet mass flow B0 and temperature TL0 while thesecond one is dry and insulated ampe gas enters in thechannel with inlet temperature T0 a water-vapor concen-tration W0 and velocity V0

22 Assumptions ampe following simplifying assumptionsare taken into account

Flows and transfers in the two phases are laminar andincompressibleampe effect of the superficial tension is negligibleampe liquid and gas flows are two dimensional andtransientampe channel walls are supposed very thin

2 Mathematical Problems in Engineering

23GoverningEquations In the present work the volume ofthe fluid VOF method [28] is used to track the liquid-gasinterface In this method a single momentum equation isshared by the fluids and the volume fraction of each of thefluids in each computational cell is tracked throughout thedomain [29 30] ampe volume fraction αL indicating the ratioof the liquid volume in a cell to the volume of the cell is equalto one if the cell is completely filled with liquid and is equalto zero if the cell is totally filled with gas

αL(x t) VLiquid

V

0 inside gas phase

0lt αL lt 1 inside transitional region

1 inside liquid phase

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(1)

ampe fluid density and viscosity of a single mixture phasein a computational cell are defined as follows

ρ αLρL + 1 minus αL( 1113857ρv

μ αLμL + 1 minus αL( 1113857μv(2)

ampe continuity equation is

nabla middot (ρUrarr

) 0 (3)

Taking into account the effect of gravity the momentumequation is defined as

z

zt(ρU

rarr) + nabla middot (ρU

rarrUrarr

) minusnablap + nabla middot μ nablaUrarr

+ nablaUrarrT

1113874 11138751113876 1113877 + ρ grarr

(4)

ampe energy equation is

z

zt(ρE) + nabla middot (U

rarr(ρE + p)) nabla middot(knablaT)] + Sh (5)

Leersquos model [31] is adopted in the VOF method and it iswidely used to simulate mass transfer during evaporationand condensation processes In this model the mass transferdepends mainly on the saturate and shared temperaturesampe mass transfer can be described as follows

If TL gtTsat _mLV coeff lowast αLρL

TL minus Tsat( 1113857

Tsat

If TV ltTsat _mVL coeff lowast αVρV

Tsat minus TV( 1113857

Tsat

(6)

24 Boundary Conditions ampe inlet conditions (x 0) are asfollows

In the liquid phase

TL TL0

B B0(7)

In gas phase

TG T0

VG V0

WG W0

(8)

At the wet wall (y 0)

VL 0

minusλzTL

zy qe

(9)

At the dry wall (y b)

VG 0

zTG

zy 0

zWG

zy 0

(10)

3 Solution Method

ampe system of equations with boundary conditions aresolved numerically using the open-source library Open-FOAM [32] which is based on the finite volume methodampe icoReactingMultiphaseInterFoam solver is used in thisstudy based on the VOF method ampe pressure-velocitycoupling is handled by solving the momentum equationsand a pressure equation derived from a combination of thecontinuity and the momentum equations ampis coupling hasbeen ensured by the PIMPLE algorithm (pressure-implicitmethod for pressure-linked equations) Figure 2 shows thePIMPLE flowchart which is a combination of the pressureimplicit with splitting of operatorsrsquo (PISO) algorithm andthe Semi-implicit method for pressure-linked equationsrsquo(SIMPLE) algorithm Although this method is formulatedfor the pseudo-transient simulations with a large courantnumber it permits an under-relaxation in the equation inorder to assist in stabilizing the numerical convergencewhilst maintaining numerical accuracy ampe spatial schemesused to approximate the convective and diffusive fluxes areboth of the second-order accuracy ampe gradient the di-vergence and the Laplacian terms are approximated by theGauss Linear the Gauss linearUpwind and the Gauss linearorthogonal schemes respectively ampese second-orderschemes give very good precision in the results despite theirdifficulty in achieving convergence ampe Euler scheme isemployed for the time derivative discretization Adjustabletime step is used to ensure the courant number limit to 1ampe convergence criteria are satisfied for all calculatedvariables when (Errle 10minus 7)

ampe courant number is given by

Co V middot ΔtΔx

(11)

Mathematical Problems in Engineering 3

31Mesh Independence Validation ampe mesh quality affectsthe stability and the calculation accuracy In order to im-prove the mesh quality the verification of the grid inde-pendence is necessary All domains are meshed using thestructured mesh and finer grids are used in the regions nearthe inlet and the wall as shown in Figure 3 ampe average celllength in the region of the liquid film is 01mm

To examine the independence of mesh size three dif-ferent grid sizes are used which are 118959 263940 and598410 respectively Figure 4 shows the wall temperaturevariation for three different gridsampe result shows that thereis a little difference between the simulation with 263940 and598410 grids Hence the size of grid is 263940 in this study

32NumericalValidation In the present study to check thevalidation and the adequacy of the used numerical schemeswe compared the results obtained by the code with resultsavailable in the literature ampe numerical solver we devel-oped was validated with the results of Yan [9] study con-cerning the evaporation of the pure water film in a verticalchannel which is uniformly heated on the left wall ampecomparison has been done for T0 20degC W0 50TL0 20degC Re 2000 and qe 1000Wm2 ampe comparisonillustrates the evolution of the interfacial temperature(Figure 5(a)) the axial velocity profile (Figure 5(b)) and theinterfacial vapor mass fraction (Figure 5(c)) A goodagreement between our results and those obtained by Yan[9] is observed amperefore the present numerical method

and the used grid layout are considered to obtain accurateresults for practical purpose

4 Results and Discussion

In this study we present the effects of flow conditions onheat and mass transfer during the evaporation of the purewater film in a vertical channel whose left wall is uniformlyheated ampe ranges of physical parameters are listed in Ta-ble 1 ampe relative humidity of the air is assigned to 50 forthe airwater vapor mixture along a vertical channel

Figure 6 illustrates the temperature distributions alongthe y-axis for various x positions Figure 6(a) reveals that thetemperature decreases monotonically with Y at specific axiallocations which means that the direction of heat transfer isfrom the liquid film through to the gas stream We alsoobserve that for Y between 0 and 05 the temperature risesas the flow progresses while for Ygt 05 the temperatureprofile is not affected throughout the flow Figure 6(b) showsthat an increase of the heat flux density applied to the humidwall qe induces an increase of the temperature in the liquidfilm and the region in the gas flow where the temperatureprofile is unaffected and increases by starting from Y 04Figure 6(c) shows the impact of the rise in mass flow rate B0on the temperature profile It is clearly observed that anincrease of themass flow rate B0 decreases the temperature ata given X position In this case we observe also that theregion in the gas flow where the temperature profile is notaffected increases and starts at Y 03

qe

g

b

X

X = HIn

sula

ted

wal

l

δx

0

TL0 B0 T0 V0 W0

Y

Velocity

Figure 1 Schematic diagram of geometry


Figure 7 shows the effects of the applied heat flux on thetemperature profile development along the gas-liquid in-terface It is observed that the temperature increases fromthe inlet to the outlet which is due to the heat transferredfrom the wall to the liquid film As expected the interfacialtemperature is raised for higher applied heat flux qe

In order to investigate the importance of the inlet liquidtemperature on the heat and mass transfer we present inFigure 8 the evolution of the temperature along the gas-liquid interface for different inlet liquid temperature amperesults disclose that the temperature decreases for a higherinlet liquid temperature ampis is due to the evaporationprocess because the evaporation of the liquid film is greateras the liquid temperature rises Consequently a largeamount of energy is taken from the liquid film to ensurelatent heat of evaporation at the interface

Figure 9 shows the distribution of the liquid volumefraction within the channel From this distribution we cansee that the liquid-gas interface is almost smooth and has nowaves all along the channel It is also observed from thisdistribution that the liquid film thickness in the lower part ofthe channel is thinner than that in the upper part which issimply due to the evaporation of the liquid film

Figure 10 presents the evolution of the axial velocityprofiles under various conditions ampe velocity profile isalmost uniform near the inlet As the downstream flow thevelocity profile becomes parabolic ampe maximum velocity

value takes place in the outlet and displaces to the humidwall ampe change of the velocity peak can be explained by thefilm evaporation which is in accord with the results providedby Yan [9] Comparing cases in Figures 10(a)ndash10(d) a verysmall variation of velocity is observed by the change in theinlet liquid temperature mass flow rate and heat fluxdensity We also observe from these figures that the inter-facial velocity always remains constant along the channel

Figure 11 presents the interfacial vapor mass fractionprofiles along the gas-liquid interface It is easy to observethat the interfacial vapor mass fraction follows the sametrend as the interfacial temperature due to the thermody-namic equilibrium at the liquid-gas interface A carefulinspection in Figure 11 discloses that as flow movesdownstream the vapor mass fraction increases from theinlet to the outlet of the channel due to the evaporation of theliquid film as a result of the increase in liquid film tem-perature It is also observed that the increase in the mass flowrate of the liquid film causes a decrease in the interfacialvapor mass fraction

Figures 12(a) and 12(b) show respectively the influenceof the heat flux density and the inlet liquid film temperatureon the interfacial vapor mass fraction along the channel Itcan be noted that a higher heat flux leads to a higher latentheat transferampis results in more water vaporization On thecontrary the interfacial vapor mass fraction increases as theinlet temperature of the liquid film increases

Pimplecorrect end

Yes

Update the velocity field based on thecorrected pressure

Solve the produced linear system ofpressure Solve the produced linear system of

pressure with the new intitial guess

CorrectorConstruct and discrete the pressure

equation based on the updated velocityfield

PredictorPredict the pressure field p momentumequation is solved using pressure fieldand solve the linear system of velocity

Converged orpimpleloop end

No

Yes

No

Start

t = t + Δt

Figure 2 PIMPLE flowchart


Figure 3 Mesh structure of the physical model

20

25

30

35

40

45

50

X000 004 008 012 020016

T W (deg

C)

118959 grids263940 grids598410 grids

Figure 4 Variation of wall temperature in three different grids

0

5

10

15

20

25

30

Study of Yan [9]Present study

B 0 = 002 kgms

B0 = 004 kgms

X000 004 008 012 020016

T i ndash

T0 (

degC)

(a)

00

04

08

12

16

20

U

X = 00078


Y00 02 04 06 1008

B0 = 004 kgms

(b)

Figure 5 Continued


0

10

20

30

40

50

T (deg

C)

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

(a)

0

10

20

30

40

50

60

T (deg

C)

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

(b)

Figure 6 Continued

Table 1 ampe ranges of physical parametersTL0 (degC) 20T0 (degC) 20B0 (kgms) 001 002 004V0 (ms) 21qe (Wm2) 1000 1500 2000H (m) 075b (m) 0015Re 2000

000

001

002

003

004

005

006

B 0 = 002 kgms

B0 = 004 kgms


X000 004 008 012 020016

Wi ndash

W0

(c)

Figure 5 Comparison of the present study with that of Yan [9] Validation of calculated (a) interfacial temperature (b) axial velocity profileand (c) interfacial vapor mass fraction


0

10

20

30

40

50

T (deg

C)

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

(c)

Figure 6 Evolution of axial temperature profiles for (a) B0 002 kgms qe 1000Wm2 TL0 20degC and Re 2000 (b) qe 1500Wm2and (c) B0 004 kgm

0

10

20

30

40

50

X000 004 008 012 020016

T i ndash

T0

(degC)

qe = 1000 Wm2

qe = 1500 Wm2

qe = 2000 Wm2

Figure 7 Effects of wall heat flux on the interfacial temperature along the channel for B0 002 kgms TL0 20degC and Re 2000


07499

07498

07497

07496

07495

07494

07493

07492

07491

0749

07489

06524

06526

06522

0652

06518

06516

06514

06512

0651

06508

06506

02626

02624

02622

0262

02618

02616

02614

02612

0261

02608

02606

00007

00006

00005

00004

00003

00002

00001

00011

00001

00009

00008

Alp

ha li

quid

08

06

04

02

10e + 00

00e + 00

Figure 9 Liquid volume fraction distribution along the channel for B0 002 kgms qe 1000Wm2 and Re 2000

0

5

10

15

20

25

30

TL0 = 20degCTL0 = 30degCTL0 = 40degC

T i ndash

T0

(degC)

X000 004 008 012 020016

Figure 8 Effects of inlet liquid temperature on the interfacial temperature along the channel for B0 002 kgms qe 1000Wm2 andRe 2000

00

05

10

15

20

25

30

35

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

v (m

s)

(a)

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

00

05

10

15

20

25

30

35

v (m

s)

(b)

Figure 10 Continued


00

05

10

15

20

25

30

35

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

v (m

s)

(c)

00

05

10

15

20

25

30

35

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

v (m

s)

(d)

Figure 10 Evolution of axial velocity profiles for (a) B0 002 kgms qe 1000Wm2 TL0 20degC and Re 2000 (b) TL0 40degC(c) B0 001 kgms and (d) qe 1500Wm2

000

001

002

003

004

005

006

X000 004 008 012 020016

Wi ndash

W0

B 0 = 002 kgms

B0 = 004 kgms

Figure 11 Development the interfacial vapor mass fraction along the channel for qe 1000Wm2 TL0 20degC and Re 2000

000

002

004

006

008

X000 004 008 012 020016

qe = 2000 Wm2

qe = 1500 Wm2

qe = 1000 Wm2

Wi ndash

W0

(a)

000

002

004

006

008

Wi ndash

W0

X000 004 008 012 020016

TL0 = 30degCTL0 = 20degC

TL0 = 40degC

(b)

Figure 12 Effects of heat flux density and inlet liquid temperature on the interfacial vapor mass fraction profiles


5 Conclusion

In this work the heat and mass transfer during the evap-oration of thin water film in a mixed convection flow along avertical channel are investigated using the volume of fluid(VOF) method within OpenFOAM CFD package ampe ef-fects of the inlet liquid mass flow rate the heat flux densityapplied to the wet wall and the inlet liquid temperature onthe heat and mass transfers are analyzed and discussed indetail ampe obtained results for water film evaporation showgood agreement compared to the results previously pub-lished in the literature which indicates that the interfacetracking by the VOF method respects well the thermody-namic equilibrium at the liquid-gas interface

ampe summary of the major results are as follows

(1) ampe interfacial temperature is higher for systems withgreater heat flux density qe and also with a lowermass flow rate B0

(2) ampe velocity profile through the channel is slightlyaffected by the change in the inlet liquid temperaturemass flow rate and heat flux density

(3) ampe increase of imposed heat flux density qe and inletliquid temperature TL0 leads to an increase of theevaporation phenomenon

(4) A decrease of mass flow rate B0 improves the waterevaporation

All these results show that the VOF method describeswell the thermal and dynamic behavior during the evapo-ration of the liquid film without imposing equilibriumconditions at the liquid-gas interface ampis insists us todevelop this method in this configuration to benefit from itsinterface tracking advantages

Nomenclature

b Channel width (m)B Mass flow rate of feed water (kgmminus1 sminus1)Co Courant numberg Gravity acceleration (m sminus2)H Channel length (m)k ampermal conductivity (Wmminus1 Kminus1)p Pressure (Pa)qe Heat flux imposed (Wmminus2)Re Reynolds numberT Temperature (degC)t Time (s)U Dimensionless axial velocityv Axial velocity (msminus1)

V Domain volumeW WallW Vapor mass fractionx Coordinate in the axial direction (m)X Dimensionless axial coordinatey Coordinate in the transverse direction (m)Y Dimensionless transversal coordinateGreek symbolsμ Dynamic viscosity (kgmminus1 sminus1)

α Volume fractionδx Local liquid film thickness at x coordinate (m)ρ Density (kgmminus3)Subscripts0 Inlet conditionG Gas phasei InterfaceL Liquid phaseSat Saturationv Vapor

Data Availability

No data were used to support this study

Conflicts of Interest

ampe authors declare that they have no conflicts of interest

Acknowledgments

ampis work made use of the High-Performance ComputingResource in HPC-MARWAN (httpwwwmarwanmahpc) provided by the National Center for Scientific andTechnical Research (CNRST) Rabat Morocco

References

[1] L C Chow and J N Chung ldquoEvaporation of water into alaminar stream of air and superheated steamrdquo InternationalJournal of Heat and Mass Transfer vol 26 no 3 pp 373ndash3801983

[2] J Schroppel and F ampiele ldquoOn the calculation of momentumheat and mass transfer in laminar and turbulent boundarylayer flows along a vaporizing liquid filmrdquo Numerical HeatTransfer vol 6 no 4 pp 475ndash496 1983

[3] C J Chang T F Lin and W M Yan ldquoNatural convectionflows in a vertical open tube resulting from combinedbuoyancy effects of thermal andmass diffusionrdquo InternationalJournal of Heat and Mass Transfer vol 29 no 10pp 1543ndash1552 1986

[4] T R Shembharkar and B R Pai ldquoPrediction of film coolingwith a liquid coolantrdquo International Journal of Heat and MassTransfer vol 29 no 6 pp 899ndash908 1986

[5] O V Trevisan and A Bejan ldquoCombined heat and masstransfer by natural convection in a vertical enclosurerdquo Journalof Heat Transfer vol 109 no 1 pp 104ndash112 1987

[6] D J Nelson and B DWood ldquoFully developed combined heatand mass transfer natural convection between parallel plateswith asymmetric boundary conditionsrdquo International Journalof Heat and Mass Transfer vol 32 no 9 pp 1789ndash1792 1989

[7] H K Wee R B Keey and M J Cunningham ldquoHeat andmoisture transfer by natural convection in a rectangularcavityrdquo International Journal of Heat and Mass Transfervol 32 no 9 pp 1765ndash1778 1989

[8] W M Van and T F Lin ldquoCombined heat and mass transferin natural convection between vertical parallel plates with filmevaporationrdquo International Journal of Heat and MassTransfer vol 33 no 3 pp 529ndash541 1990

[9] W-M Yan ldquoEffects of film evaporation on laminar mixedconvection heat and mass transfer in a vertical channelrdquo


International Journal of Heat and Mass Transfer vol 35no 12 pp 3419ndash3429 1992

[10] W-M Yan ldquoConvective heat and mass transfer from a fallingfilm to a laminar gas streamrdquo Warme- und Stoffubertragungvol 29 no 2 pp 79ndash87 1993

[11] M Feddaoui A Mir and E Belahmidi ldquoNumerical simu-lation of mixed convection heat and mass transfer with liquidfilm cooling along an insulated vertical channelrdquo Heat andMass Transfer vol 39 no 5 pp 445ndash453 2003

[12] M Feddaoui A Mir and E Belahmidi ldquoCocurrent turbulentmixed convection heat and mass transfer in falling film ofwater inside a vertical heated tuberdquo International Journal ofHeat and Mass Transfer vol 46 no 18 pp 3497ndash3509 2003

[13] E Mezaache and M Daguenet ldquoEffects of inlet conditions onfilm evaporation along an inclined platerdquo Solar Energyvol 78 no 4 pp 535ndash542 2005

[14] N Laaroussi G Lauriat and G Desrayaud ldquoEffects of var-iable density for film evaporation on laminar mixed con-vection in a vertical channelrdquo International Journal of Heatand Mass Transfer vol 52 no 1-2 pp 151ndash164 2009

[15] A S Cherif S Ben Jabrallah J P Corriou and A BelghithldquoIntensification of the liquid film evaporation in a verticalchannelrdquo Desalination vol 250 no 1 pp 433ndash437 2010

[16] A S Cherif M A Kassim B Benhamou S HarmandJ P Corriou and S Ben Jabrallah ldquoExperimental and nu-merical study of mixed convection heat and mass transfer in avertical channel with film evaporationrdquo International Journalof ermal Sciences vol 50 no 6 pp 942ndash953 2011

[17] A N Alla M b Feddaoui and H Meftah ldquoNumericalsimulation of heat and mass transfer in the evaporation ofglycols liquid film along an insulated vertical channelrdquoAmerican Journal of Heat and Mass Transfer vol 2 no 2pp 59ndash75 2015

[18] A Nasr and B Al-Ghamdi ldquoSimultaneous heat and masstransfer during evaporation and condensation of a binaryliquid filmrdquo ermal Science vol 23 no 2A pp 661ndash6722017

[19] L Zhang Y Huang J Wu Z Liu and G Yang ldquoEvaporationof water film in a three-dimensional vertical rectangularchannel by laminar mixed convectionrdquo Applied ermalEngineering vol 130 pp 242ndash253 2018

[20] A N Alla M Feddaoui and H Meftah ldquoInternationaljournal of thermal sciences comparison of two configurationsto improve heat and mass transfer in evaporating two-component liquid film flowrdquo International Journal ofermalSciences vol 126 pp 194ndash204 2017

[21] J Fang K Li and M Diao ldquoEstablishment of the falling filmevaporation model and correlation of the overall heat transfercoefficientrdquo Royal Society Open Science vol 6 no 5 ArticleID 190135 2019

[22] J-S Leu J-Y Jang and Y Chou ldquoHeat and mass transfer forliquid film evaporation along a vertical plate covered with athin porous layerrdquo International Journal of Heat and MassTransfer vol 49 no 11-12 pp 1937ndash1945 2006

[23] A Nasr and A S Al-Ghamdi ldquoNumerical study of evapo-ration of falling liquid film on one of two vertical platescovered with a thin porous layer by free convectionrdquo Inter-national Journal of ermal Sciences vol 112 pp 335ndash3442017

[24] A Terzi W Foudhil S Harmand and S B Jabrallah ldquoLiquidfilm evaporation inside an inclined channel effect of thepresence of a porous layerrdquo International Journal of ermalSciences vol 109 pp 136ndash147 2016

[25] F Piscaglia F Giussani A Montorfano J Helie andS M Aithal ldquoA multiphase dynamic-VoF solver to modelprimary jet atomization and cavitation inside high-pressurefuel injectors in OpenFOAMrdquo Acta Astronautica vol 158pp 375ndash387 2019

[26] T Q D Pham and S Choi ldquoNumerical analysis of directcontact condensation-induced water hammering effect usingOpenFOAM in realistic steam pipesrdquo International Journal ofHeat and Mass Transfer vol 171 Article ID 121099 2021

[27] M A Amidu Y Addad and M K Riahi ldquoA hybrid mul-tiphase flow model for the prediction of both low and highvoid fraction nucleate boiling regimesrdquo Applied ermalEngineering vol 178 Article ID 115625 2020

[28] B D N Hirt and W Cyril ldquoVolume of fluid (VOF) methodfor the dynamics of free boundaries Iranrdquo Journal of Scienceand Technology Transactions of Electrical Engineering vol 42no 3 pp 357ndash366 2018

[29] SW JWelch and JWilson ldquoA volume of fluid basedmethodfor fluid flows with phase changerdquo Journal of ComputationalPhysics vol 160 no 2 pp 662ndash682 2000

[30] D L Youngs Time-Dependent Multi-Material Flow withLarge Fluid Distortion Academic Press Cambridge MAUSA 1982

[31] W H Lee A Pressure Iteration Scheme for Two-PhaseModeling Los Alamos Science Laboratory Los Alamos NMUSA 1979

[32] H G Weller G Tabor H Jasak and C Fureby ldquoA tensorialapproach to computational continuum mechanics usingobject-oriented techniquesrdquo Computers in Physics vol 12no 6 p 620 1998


Cherif et al [15] presented an experimental study of theevaporation by natural and forced convection in a verticalchannel and a thin liquid film falls on two parallel platesampey showed that the thermal yield of evaporation increaseswith the down air velocity and the heat flux density in orderto improve heat and mass transfer during liquid filmevaporation

Cherif et al [16] investigated experimentally the influ-ence of film evaporation on mixed convective heat and masstransfer in vertical parallel channel wetted by a water filmampe results showed that the evaporative cooling increaseswith small heat flux and large air velocities Alla et al [17]studied the evaporation of glycol liquid film in mixedconvection ampeir results showed that the propylene glycolevaporates more than glycol and the heat transfer by thelatent mode is more important than the heat transfer bysensible mode Nasr and Alghamdi [18] investigated nu-merically the evaporation and condensation under mixedconvection of binary liquid filmampey showed that the watercondensation increases by reducing inlet temperature andincreasing the amount of liquid film and an increase of inletliquid concentration of ethylene glycol enhances the eth-ylene glycol evaporation and water vapor condensationZhang et al [19] studied the evaporation by laminar mixedconvection in a vertical three-dimensional rectangularchannel with nitrogen and water vapor ampe results indicatethat the wall temperature and the inlet temperature improvethe heat and mass transfer Alla et al [20] have investigatednumerically two configurations In the first case a uniformheat flow is applied to the wet wall while in the second caseand the same heat flux was applied to preheat the liquid atthe inlet ampe authors indicate that the evaporation rateincreases with the second case Fang et al [21] studied theheat transfer coefficient correlation of the multiphasetransformation inside and outside the tube ampey showedthat the heat transfer coefficient increases with the feed flowrate and ethanol content in the evaporation tube ampe heattransfer coefficient decreases when the heat transfer tem-perature difference between the inside and outside of thetube increases

We also find in the literature some other studies thathave focus on heat and mass transfer with the evaporation ofa liquid film flowing in a porous layer Leu et al [22] nu-merically studied the heat and mass improvement of liquidfilm evaporation with a porous layerampe results showed thatthe heat transfer performance of the liquid film was im-proved by using a porous layer and the average Nusselt andSherwood numbers are increased inversely with porosityporous layer thickness and relative humidity Nasr and Al-Ghamdi [23] analyzed the influence of porous layer thick-ness and porosity on the coupled heat and mass transferperformance during the evaporation of flowing liquid filmampe results indicate that the presence of the porous layerimproves the heat and mass transfer performance at theliquid-gas interface Terzi et al [24] numerically studied theperformance of liquid film evaporation by introducing aliquid saturated porous layer along an inclined channel amperesults indicated that the use of the porous layer improvesthe heat and mass transfer

ampe VOFmethod has been used in the literature in manyproblems involving two-phase flows Piscaglia et al [25]presented a development and validation of a two-phasedynamic VOF solver for the simulation of internal nozzleflows ampe phase change at the interface was modeled usingthe Schnerr and Saver model Large Eddy simulations (LES)are used to model turbulence effect ampe approach can beconsidered to be reliable for investigations of the internalnozzle flows during transient operations Pham and Choi[26] performed a numerical simulation for the waterhammering using interFoam solver ampe pressure-implicitmethod is used for continuity momentum and energyequations of a volume of fluid model discretization ImplicitEuler and central difference schemes were used for temporaland spatial discretization ampe solver is validated with a one-dimensional Stefan problemrsquos benchmark case Amidu et al[27] presented a hybrid VOF- Eulerian multifluid model forsolving a hot low and high void fraction subcooled flowboiling ampey showed that the best estimation of the radialvelocity profile is achieved by using the two-phase roughwall function instead of the classical single-phase wallfunction

In our knowledge all the previous studies deal with theheat and mass transfer during the liquid film evaporationwith the finite volume method by considering boundaryconditions at gas-liquid interface ampe main objective of thepresent work is the use of volume of fluid (VOF) method toinvestigate numerically the mixed convection heat and masstransfer with liquid film evaporation To this end anOpenFoam solver was developed and successfully validatedwith previous numerical results taken from the literatureampe effect of parameters such as the inlet liquid temperaturemass flow rate and wall heat flux density on the heat andmass transfer during water film evaporation in a verticalchannel was investigated

2 Problem Formulation

21 Definition of Geometry ampe present work deals with anumerical study of liquid film evaporation by mixed con-vection in a vertical channel as illustrated in Figure 1 ampegeometry under consideration is two vertical and parallelplates with height H and width b ampe first plate is main-tained at the uniform heat flux qe and wetted by a thin liquidfilm with inlet mass flow B0 and temperature TL0 while thesecond one is dry and insulated ampe gas enters in thechannel with inlet temperature T0 a water-vapor concen-tration W0 and velocity V0

22 Assumptions ampe following simplifying assumptionsare taken into account

Flows and transfers in the two phases are laminar andincompressibleampe effect of the superficial tension is negligibleampe liquid and gas flows are two dimensional andtransientampe channel walls are supposed very thin



αL(x t) VLiquid

V

0 inside gas phase



⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(1)






) 0 (3)


z

zt(ρU


rarrUrarr


+ nablaUrarrT

1113874 11138751113876 1113877 + ρ grarr

(4)


z






Tsat



Tsat

(6)


In the liquid phase

TL TL0

B B0(7)

In gas phase

TG T0

VG V0

WG W0

(8)


VL 0

minusλzTL

zy qe

(9)


VG 0

zTG

zy 0

zWG

zy 0

(10)

3 Solution Method



Co V middot ΔtΔx

(11)









qe

g

b

X

X = HIn

sula

ted

wal

l

δx

0

TL0 B0 T0 V0 W0

Y

Velocity










Pimplecorrect end

Yes








No

Yes

No

Start

t = t + Δt




20

25

30

35

40

45

50

X000 004 008 012 020016

T W (deg

C)



0

5

10

15

20

25

30


B 0 = 002 kgms

B0 = 004 kgms

X000 004 008 012 020016

T i ndash

T0 (

degC)

(a)

00

04

08

12

16

20

U

X = 00078


Y00 02 04 06 1008

B0 = 004 kgms

(b)

Figure 5 Continued


0

10

20

30

40

50

T (deg

C)

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

(a)

0

10

20

30

40

50

60

T (deg

C)

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

(b)

Figure 6 Continued


000

001

002

003

004

005

006

B 0 = 002 kgms

B0 = 004 kgms


X000 004 008 012 020016

Wi ndash

W0

(c)



0

10

20

30

40

50

T (deg

C)

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

(c)


0

10

20

30

40

50

X000 004 008 012 020016

T i ndash

T0

(degC)

qe = 1000 Wm2

qe = 1500 Wm2

qe = 2000 Wm2



07499

07498

07497

07496

07495

07494

07493

07492

07491

0749

07489

06524

06526

06522

0652

06518

06516

06514

06512

0651

06508

06506

02626

02624

02622

0262

02618

02616

02614

02612

0261

02608

02606

00007

00006

00005

00004

00003

00002

00001

00011

00001

00009

00008

Alp

ha li

quid

08

06

04

02

10e + 00

00e + 00


0

5

10

15

20

25

30


T i ndash

T0

(degC)

X000 004 008 012 020016


00

05

10

15

20

25

30

35

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

v (m

s)

(a)

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

00

05

10

15

20

25

30

35

v (m

s)

(b)

Figure 10 Continued


00

05

10

15

20

25

30

35

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

v (m

s)

(c)

00

05

10

15

20

25

30

35

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

v (m

s)

(d)


000

001

002

003

004

005

006

X000 004 008 012 020016

Wi ndash

W0

B 0 = 002 kgms

B0 = 004 kgms


000

002

004

006

008

X000 004 008 012 020016

qe = 2000 Wm2

qe = 1500 Wm2

qe = 1000 Wm2

Wi ndash

W0

(a)

000

002

004

006

008

Wi ndash

W0

X000 004 008 012 020016


TL0 = 40degC

(b)



5 Conclusion








Nomenclature




Data Availability




Acknowledgments


References





































αL(x t) VLiquid

V

0 inside gas phase



⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(1)






) 0 (3)


z

zt(ρU


rarrUrarr


+ nablaUrarrT

1113874 11138751113876 1113877 + ρ grarr

(4)


z






Tsat



Tsat

(6)


In the liquid phase

TL TL0

B B0(7)

In gas phase

TG T0

VG V0

WG W0

(8)


VL 0

minusλzTL

zy qe

(9)


VG 0

zTG

zy 0

zWG

zy 0

(10)

3 Solution Method



Co V middot ΔtΔx

(11)









qe

g

b

X

X = HIn

sula

ted

wal

l

δx

0

TL0 B0 T0 V0 W0

Y

Velocity










Pimplecorrect end

Yes








No

Yes

No

Start

t = t + Δt




20

25

30

35

40

45

50

X000 004 008 012 020016

T W (deg

C)



0

5

10

15

20

25

30


B 0 = 002 kgms

B0 = 004 kgms

X000 004 008 012 020016

T i ndash

T0 (

degC)

(a)

00

04

08

12

16

20

U

X = 00078


Y00 02 04 06 1008

B0 = 004 kgms

(b)

Figure 5 Continued


0

10

20

30

40

50

T (deg

C)

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

(a)

0

10

20

30

40

50

60

T (deg

C)

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

(b)

Figure 6 Continued


000

001

002

003

004

005

006

B 0 = 002 kgms

B0 = 004 kgms


X000 004 008 012 020016

Wi ndash

W0

(c)



0

10

20

30

40

50

T (deg

C)

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

(c)


0

10

20

30

40

50

X000 004 008 012 020016

T i ndash

T0

(degC)

qe = 1000 Wm2

qe = 1500 Wm2

qe = 2000 Wm2



07499

07498

07497

07496

07495

07494

07493

07492

07491

0749

07489

06524

06526

06522

0652

06518

06516

06514

06512

0651

06508

06506

02626

02624

02622

0262

02618

02616

02614

02612

0261

02608

02606

00007

00006

00005

00004

00003

00002

00001

00011

00001

00009

00008

Alp

ha li

quid

08

06

04

02

10e + 00

00e + 00


0

5

10

15

20

25

30


T i ndash

T0

(degC)

X000 004 008 012 020016


00

05

10

15

20

25

30

35

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

v (m

s)

(a)

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

00

05

10

15

20

25

30

35

v (m

s)

(b)

Figure 10 Continued


00

05

10

15

20

25

30

35

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

v (m

s)

(c)

00

05

10

15

20

25

30

35

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

v (m

s)

(d)


000

001

002

003

004

005

006

X000 004 008 012 020016

Wi ndash

W0

B 0 = 002 kgms

B0 = 004 kgms


000

002

004

006

008

X000 004 008 012 020016

qe = 2000 Wm2

qe = 1500 Wm2

qe = 1000 Wm2

Wi ndash

W0

(a)

000

002

004

006

008

Wi ndash

W0

X000 004 008 012 020016


TL0 = 40degC

(b)



5 Conclusion








Nomenclature




Data Availability




Acknowledgments


References











































qe

g

b

X

X = HIn

sula

ted

wal

l

δx

0

TL0 B0 T0 V0 W0

Y

Velocity










Pimplecorrect end

Yes








No

Yes

No

Start

t = t + Δt




20

25

30

35

40

45

50

X000 004 008 012 020016

T W (deg

C)



0

5

10

15

20

25

30


B 0 = 002 kgms

B0 = 004 kgms

X000 004 008 012 020016

T i ndash

T0 (

degC)

(a)

00

04

08

12

16

20

U

X = 00078


Y00 02 04 06 1008

B0 = 004 kgms

(b)

Figure 5 Continued


0

10

20

30

40

50

T (deg

C)

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

(a)

0

10

20

30

40

50

60

T (deg

C)

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

(b)

Figure 6 Continued


000

001

002

003

004

005

006

B 0 = 002 kgms

B0 = 004 kgms


X000 004 008 012 020016

Wi ndash

W0

(c)



0

10

20

30

40

50

T (deg

C)

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

(c)


0

10

20

30

40

50

X000 004 008 012 020016

T i ndash

T0

(degC)

qe = 1000 Wm2

qe = 1500 Wm2

qe = 2000 Wm2



07499

07498

07497

07496

07495

07494

07493

07492

07491

0749

07489

06524

06526

06522

0652

06518

06516

06514

06512

0651

06508

06506

02626

02624

02622

0262

02618

02616

02614

02612

0261

02608

02606

00007

00006

00005

00004

00003

00002

00001

00011

00001

00009

00008

Alp

ha li

quid

08

06

04

02

10e + 00

00e + 00


0

5

10

15

20

25

30


T i ndash

T0

(degC)

X000 004 008 012 020016


00

05

10

15

20

25

30

35

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

v (m

s)

(a)

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

00

05

10

15

20

25

30

35

v (m

s)

(b)

Figure 10 Continued


00

05

10

15

20

25

30

35

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

v (m

s)

(c)

00

05

10

15

20

25

30

35

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

v (m

s)

(d)


000

001

002

003

004

005

006

X000 004 008 012 020016

Wi ndash

W0

B 0 = 002 kgms

B0 = 004 kgms


000

002

004

006

008

X000 004 008 012 020016

qe = 2000 Wm2

qe = 1500 Wm2

qe = 1000 Wm2

Wi ndash

W0

(a)

000

002

004

006

008

Wi ndash

W0

X000 004 008 012 020016


TL0 = 40degC

(b)



5 Conclusion








Nomenclature




Data Availability




Acknowledgments


References











































Pimplecorrect end

Yes








No

Yes

No

Start

t = t + Δt




20

25

30

35

40

45

50

X000 004 008 012 020016

T W (deg

C)



0

5

10

15

20

25

30


B 0 = 002 kgms

B0 = 004 kgms

X000 004 008 012 020016

T i ndash

T0 (

degC)

(a)

00

04

08

12

16

20

U

X = 00078


Y00 02 04 06 1008

B0 = 004 kgms

(b)

Figure 5 Continued


0

10

20

30

40

50

T (deg

C)

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

(a)

0

10

20

30

40

50

60

T (deg

C)

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

(b)

Figure 6 Continued


000

001

002

003

004

005

006

B 0 = 002 kgms

B0 = 004 kgms


X000 004 008 012 020016

Wi ndash

W0

(c)



0

10

20

30

40

50

T (deg

C)

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

(c)


0

10

20

30

40

50

X000 004 008 012 020016

T i ndash

T0

(degC)

qe = 1000 Wm2

qe = 1500 Wm2

qe = 2000 Wm2



07499

07498

07497

07496

07495

07494

07493

07492

07491

0749

07489

06524

06526

06522

0652

06518

06516

06514

06512

0651

06508

06506

02626

02624

02622

0262

02618

02616

02614

02612

0261

02608

02606

00007

00006

00005

00004

00003

00002

00001

00011

00001

00009

00008

Alp

ha li

quid

08

06

04

02

10e + 00

00e + 00


0

5

10

15

20

25

30


T i ndash

T0

(degC)

X000 004 008 012 020016


00

05

10

15

20

25

30

35

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

v (m

s)

(a)

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

00

05

10

15

20

25

30

35

v (m

s)

(b)

Figure 10 Continued


00

05

10

15

20

25

30

35

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

v (m

s)

(c)

00

05

10

15

20

25

30

35

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

v (m

s)

(d)


000

001

002

003

004

005

006

X000 004 008 012 020016

Wi ndash

W0

B 0 = 002 kgms

B0 = 004 kgms


000

002

004

006

008

X000 004 008 012 020016

qe = 2000 Wm2

qe = 1500 Wm2

qe = 1000 Wm2

Wi ndash

W0

(a)

000

002

004

006

008

Wi ndash

W0

X000 004 008 012 020016


TL0 = 40degC

(b)



5 Conclusion








Nomenclature




Data Availability




Acknowledgments


References





































20

25

30

35

40

45

50

X000 004 008 012 020016

T W (deg

C)



0

5

10

15

20

25

30


B 0 = 002 kgms

B0 = 004 kgms

X000 004 008 012 020016

T i ndash

T0 (

degC)

(a)

00

04

08

12

16

20

U

X = 00078


Y00 02 04 06 1008

B0 = 004 kgms

(b)

Figure 5 Continued


0

10

20

30

40

50

T (deg

C)

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

(a)

0

10

20

30

40

50

60

T (deg

C)

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

(b)

Figure 6 Continued


000

001

002

003

004

005

006

B 0 = 002 kgms

B0 = 004 kgms


X000 004 008 012 020016

Wi ndash

W0

(c)



0

10

20

30

40

50

T (deg

C)

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

(c)


0

10

20

30

40

50

X000 004 008 012 020016

T i ndash

T0

(degC)

qe = 1000 Wm2

qe = 1500 Wm2

qe = 2000 Wm2



07499

07498

07497

07496

07495

07494

07493

07492

07491

0749

07489

06524

06526

06522

0652

06518

06516

06514

06512

0651

06508

06506

02626

02624

02622

0262

02618

02616

02614

02612

0261

02608

02606

00007

00006

00005

00004

00003

00002

00001

00011

00001

00009

00008

Alp

ha li

quid

08

06

04

02

10e + 00

00e + 00


0

5

10

15

20

25

30


T i ndash

T0

(degC)

X000 004 008 012 020016


00

05

10

15

20

25

30

35

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

v (m

s)

(a)

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

00

05

10

15

20

25

30

35

v (m

s)

(b)

Figure 10 Continued


00

05

10

15

20

25

30

35

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

v (m

s)

(c)

00

05

10

15

20

25

30

35

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

v (m

s)

(d)


000

001

002

003

004

005

006

X000 004 008 012 020016

Wi ndash

W0

B 0 = 002 kgms

B0 = 004 kgms


000

002

004

006

008

X000 004 008 012 020016

qe = 2000 Wm2

qe = 1500 Wm2

qe = 1000 Wm2

Wi ndash

W0

(a)

000

002

004

006

008

Wi ndash

W0

X000 004 008 012 020016


TL0 = 40degC

(b)



5 Conclusion








Nomenclature




Data Availability




Acknowledgments


References




































0

10

20

30

40

50

T (deg

C)

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

(a)

0

10

20

30

40

50

60

T (deg

C)

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

(b)

Figure 6 Continued


000

001

002

003

004

005

006

B 0 = 002 kgms

B0 = 004 kgms


X000 004 008 012 020016

Wi ndash

W0

(c)



0

10

20

30

40

50

T (deg

C)

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

(c)


0

10

20

30

40

50

X000 004 008 012 020016

T i ndash

T0

(degC)

qe = 1000 Wm2

qe = 1500 Wm2

qe = 2000 Wm2



07499

07498

07497

07496

07495

07494

07493

07492

07491

0749

07489

06524

06526

06522

0652

06518

06516

06514

06512

0651

06508

06506

02626

02624

02622

0262

02618

02616

02614

02612

0261

02608

02606

00007

00006

00005

00004

00003

00002

00001

00011

00001

00009

00008

Alp

ha li

quid

08

06

04

02

10e + 00

00e + 00


0

5

10

15

20

25

30


T i ndash

T0

(degC)

X000 004 008 012 020016


00

05

10

15

20

25

30

35

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

v (m

s)

(a)

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

00

05

10

15

20

25

30

35

v (m

s)

(b)

Figure 10 Continued


00

05

10

15

20

25

30

35

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

v (m

s)

(c)

00

05

10

15

20

25

30

35

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

v (m

s)

(d)


000

001

002

003

004

005

006

X000 004 008 012 020016

Wi ndash

W0

B 0 = 002 kgms

B0 = 004 kgms


000

002

004

006

008

X000 004 008 012 020016

qe = 2000 Wm2

qe = 1500 Wm2

qe = 1000 Wm2

Wi ndash

W0

(a)

000

002

004

006

008

Wi ndash

W0

X000 004 008 012 020016


TL0 = 40degC

(b)



5 Conclusion








Nomenclature




Data Availability




Acknowledgments


References




































0

10

20

30

40

50

T (deg

C)

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

(c)


0

10

20

30

40

50

X000 004 008 012 020016

T i ndash

T0

(degC)

qe = 1000 Wm2

qe = 1500 Wm2

qe = 2000 Wm2



07499

07498

07497

07496

07495

07494

07493

07492

07491

0749

07489

06524

06526

06522

0652

06518

06516

06514

06512

0651

06508

06506

02626

02624

02622

0262

02618

02616

02614

02612

0261

02608

02606

00007

00006

00005

00004

00003

00002

00001

00011

00001

00009

00008

Alp

ha li

quid

08

06

04

02

10e + 00

00e + 00


0

5

10

15

20

25

30


T i ndash

T0

(degC)

X000 004 008 012 020016


00

05

10

15

20

25

30

35

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

v (m

s)

(a)

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

00

05

10

15

20

25

30

35

v (m

s)

(b)

Figure 10 Continued


00

05

10

15

20

25

30

35

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

v (m

s)

(c)

00

05

10

15

20

25

30

35

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

v (m

s)

(d)


000

001

002

003

004

005

006

X000 004 008 012 020016

Wi ndash

W0

B 0 = 002 kgms

B0 = 004 kgms


000

002

004

006

008

X000 004 008 012 020016

qe = 2000 Wm2

qe = 1500 Wm2

qe = 1000 Wm2

Wi ndash

W0

(a)

000

002

004

006

008

Wi ndash

W0

X000 004 008 012 020016


TL0 = 40degC

(b)



5 Conclusion








Nomenclature




Data Availability




Acknowledgments


References




































07499

07498

07497

07496

07495

07494

07493

07492

07491

0749

07489

06524

06526

06522

0652

06518

06516

06514

06512

0651

06508

06506

02626

02624

02622

0262

02618

02616

02614

02612

0261

02608

02606

00007

00006

00005

00004

00003

00002

00001

00011

00001

00009

00008

Alp

ha li

quid

08

06

04

02

10e + 00

00e + 00


0

5

10

15

20

25

30


T i ndash

T0

(degC)

X000 004 008 012 020016


00

05

10

15

20

25

30

35

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

v (m

s)

(a)

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

00

05

10

15

20

25

30

35

v (m

s)

(b)

Figure 10 Continued


00

05

10

15

20

25

30

35

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

v (m

s)

(c)

00

05

10

15

20

25

30

35

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

v (m

s)

(d)


000

001

002

003

004

005

006

X000 004 008 012 020016

Wi ndash

W0

B 0 = 002 kgms

B0 = 004 kgms


000

002

004

006

008

X000 004 008 012 020016

qe = 2000 Wm2

qe = 1500 Wm2

qe = 1000 Wm2

Wi ndash

W0

(a)

000

002

004

006

008

Wi ndash

W0

X000 004 008 012 020016


TL0 = 40degC

(b)



5 Conclusion








Nomenclature




Data Availability




Acknowledgments


References




































00

05

10

15

20

25

30

35

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

v (m

s)

(c)

00

05

10

15

20

25

30

35

X = 005X = 01

X = 015X = 02

Y00 02 04 06 1008

v (m

s)

(d)


000

001

002

003

004

005

006

X000 004 008 012 020016

Wi ndash

W0

B 0 = 002 kgms

B0 = 004 kgms


000

002

004

006

008

X000 004 008 012 020016

qe = 2000 Wm2

qe = 1500 Wm2

qe = 1000 Wm2

Wi ndash

W0

(a)

000

002

004

006

008

Wi ndash

W0

X000 004 008 012 020016


TL0 = 40degC

(b)



5 Conclusion








Nomenclature




Data Availability




Acknowledgments


References




































5 Conclusion








Nomenclature




Data Availability




Acknowledgments


References





























































Documents

VolumeofFluid(VOF)ModelingofLiquidFilmEvaporationin