Simulation of transient Rayleigh-Bénard-Marangoni convection induced by evaporation

HAL Id: hal-00691272

https://hal.archives-ouvertes.fr/hal-00691272

Submitted on 25 Apr 2012

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires

Simulation of transient Rayleigh-Bénard-Marangoni convection induced by evaporation

O. Touazi, Eric Chénier, F. Doumenc, B. Guerrier

To cite this version:

O. Touazi, Eric Chénier, F. Doumenc, B. Guerrier. Simulation of transient Rayleigh-Bénard-

Marangoni convection induced by evaporation. International Journal of Heat and Mass Transfer,

Elsevier, 2010, 53 (4), pp.656–664. �10.1016/j.ijheatmasstransfer.2009.10.029�. �hal-00691272�

Rayleigh-Bénard-Marangoni onvetion indued

by evaporation

O. Touazi

a,b

, E. Chénier

a

, F. Doumen

b, ∗

b

a

Université Paris-Est,laboratoire Modélisationet SimulationMulti Ehelle,

MSME FRE3160 CNRS,5 bd Desartes,77454 Marne-la-Vallee, Frane

b

Univ Pierre etMarie Curie-Paris6, Univ Paris-Sud,CNRS, F-91405,

Lab FAST,Bat 502, CampusUniv, Orsay, F-91405.

Abstrat

Numerial simulation of thermal onvetion indued by solvent evaporation in an

initiallyisothermaluidisonsidered. Boththermoapillarityandbuoyanydriving

fores are taken into aount, and a riterium based on the Pelet number is used

to analyzethe stabilityof this transient problem. Critial Marangoni and Rayleigh

numbers are obtained for a large range of Biot and Prandtl numbers. Results of

the non linear simulations are ompared with a previous linear transient stability

analysisbasedon anon normalapproah andwithvisualizations performed during

PIB/Toluene solutions drying experiments. A saling analysis is developed for the

Marangoni problem and orrelations arederived to predit the order of magnitude

oftemperature and veloityasafuntion of

Bi

M a

P r

Key words: freeonvetion, evaporation, stability, heattransfer

1 Introdution

Evaporationof avolatiluidordrying of asolutioninduesa derease ofthe

temperatureatthe freesurfae duetothe vaporizationlatent heat.This situ-

ationan generate onvetive motionduetobothbuoyany and thermoapil-

larydrivingfores[16℄.Severalpointsmaybepointedoutwhenstudyingsuh

∗

doumenfast.u-psud.fr(F. Doumen), guerrierfast.u-psud.fr(B.

Guerrier).

plingbetweenthesystemandthesurroundings.Theevaporatinguxandthen

the temperaturegradientinthe uid depends onthe heat and mass transfers

withthe ambientair.Several authorshavedeveloped numerialortheoretial

studies taking into aount this oupling, for example Merkt and Bestehorn

[7℄, Colinetand o-authors [8℄, Ozen and Narayanan [9℄ and Moussy and o-

authors[10℄.Bothsurfaetension-andbuoyany-drivenonvetionmayour

as studied for example by Medale and Cerisier in several geometries but in

a nonvolatile uid [11℄. Another point that is more speially onsidered in

this paperisthe transient natureof manyexperiments.Indeed, startingfrom

an initial state where the uid is isotherm, evaporation indues a derease

of the surfae temperature. The basi temperature eld orresponding tothe

purediusiveproblem(nullveloityeld)istimedependentsothatalassial

stability analysis based onthe perturbation of a steady ondutive regime is

questionable. Then the predition of ritial onditions for the onset of on-

vetion is a omplex problem. A theoretial linear analysis based on a non

normal approah has been reently developed to take into aount the time

dependent basi state for this same problem(f. [12℄ and referenes herein).

In this paper we present a 2D non linear numerial simulation to study the

development and evolution of onvetive patterns in the transient regime,

when both buoyany and thermoapillary eets an be put forward. A large

rangeofparametersharaterizingtheproblem(Biot,Marangoni,Prandtland

Rayleigh numbers) are investigated. This numerial approah follows an ex-

perimentalstudy wherenumerousexperimentsofdryingofapolymer/solvent

solution have been performed with dierent thiknesses and visosities. Sim-

plifying assumptions follow from experimental results and are presented in

setion 2, as well as the model equations and numerial method. Setion 3

and 4 are devoted to 2D non linear numerial simulations and to the deter-

mination of onvetion thresholds for this transient problem. Comparisons

are made with experimental results and with previous results dedued from

the linear stability analysis. At last, a saling analysis is developed for the

Marangoniproblemin setion5.

2 Problem formulation

2.1 Thermal model

Dryingexperimentsthatunderliethesimulationspresentedinthispaperhave

been performed on the system Polyisobutylene (PIB)/Toluene. Experiments

aredesribedindetailin[13℄andweonlyreallthemainpointsusedtoestab-

lishthesimpliedmodelonsideredhere.Thesolutioninitiallyattheambient

trol parameters used in the experiments are the initial thikness (

0.3mm ≤ e ≤ 14.3mm

0 ≤ ω P ≤ 15%

mer solutionvisosity isstrongly sensitive topolymer onentration[14℄.The

visosity

µ

5.5 × 10 ^{− 4} P a.s

2.4P a.s

ω _P 0%

15%

servations. They are valid only at the beginning of the drying whih is the

time under onsideration inthis paper. When evaporation begins, onvetive

patterns have been observed at the very beginning of the experiment(quasi-

instantaneous or less than 100 s after pouring the solution). They disappear

well before the end of the drying. The Lewis number (ratio of the thermal

diusivitytomassdiusivity)being very large(about

10 ³

onvetive patterns observed in the rst minutes are mainly driven by ther-

maleets.Twoexperimentalobservationsdetailedin[13℄supportthisthesis.

First,a fewexperimentswere onduted withdeuterated solvent, whoseden-

sityishigherthanthepolymerdensity.Inthatase,thedensityofthesolution

dereaseswhenthe polymer onentrationinreases,leadingtoastable situa-

tionif thesolutalRayleigh-Bénardproblemisonsidered.Sinenodierenes

were found with the experiments onduted with the standard solvent, we

an exlude solutal buoyany as adominant mehanism.Seond, freesurfae

temperature elds measured by infrared amera showed that the end of free

onvetionwasrelatedtothedurationofthetransientthermalregime.Solutal

onvetion is then not taken intoaount in this paper.

Moreover, thelayer thikness isassumed to remainonstant.This hypothesis

an beadopted if

P e int ≪ 1

P e int ≡ (ev ev )/α

e

v ev

evaporation and

α

P e ≪ 1

displaement

v ev δ dif f

e

during the problemharateristitime i.e. the diusion time

δ dif f ≡ e ² /κ

the experiments,the Pelet number issmaller than

0.1

rateofhangeofthepolymermassfrationissmallandthephysialproperties

of the solution are assumed onstant.

The free surfae is assumed at. Surfae deformationan be negleted if the

rispationnumber

Cr ≡ (ρνα)/(γe) ≪ 1

Ga ≡ (ge ³ )/(να) ≫ 1

ρ

ν

γ

are shown to be satised for the experiments onsidered here. Then surfae

deformability an be disregarded.

Forsakeofsimpliitythenumerialanalysisisrestrited toatwo-dimensional

geometry sine the study fouses on the onset of onvetion and not on the

desription of onvetive patterns morphology. Comparison of experimental

riori. The solution layer is ontained in a retangular domain of aspet ratio

A = L/e = 20

L

e

solution layer. The vertial and bottom solid walls are supposed to be adia-

bati and non permeable. Evaporation ours at the free surfae only. Heat

andmass transfersbetweenthe freesurfae andthe ambient airare desribed

by global transfer oeients.

Withthe Boussinesq approximation, the Navier-Stokesand energy equations

are:

∇ ~ .~v = 0,

∂

∂t ~v + (~v.~ ∇ )~v = − 1

ρ ∇ ~ p + ν∆~v + gβ T (T − T 0 ) e ~ y ,

∂

∂t T + (~v.~ ∇ )T = α∆T,

where

~v(~x, t)

p(~x, t)

T (~x, t)

temperature eld,

T ₀

e ~ _y

tor in the vertial diretion. In this approximation, the density

ρ

be the density at

T = T 0

β T

ν = µ/ρ

α = k/ρc

oeient, the kinemativisosity and the thermaldiusivity with

µ

nami visosity,

k

c

The polymer mass fration being small, we have used the physial proper-

ties of the toluene exept for the visosity that is very sensitive to poly-

mer onentration. The following values have been used:

ρ = 865kg.m ^{− 3}

k = 0.142W.m ^{− 1} .K ^{− 1}

α = 0.97 × 10 ^{− 7} m ² .s ^{− 1}

β T = 1.07 × 10 ^{− 3} K ^{− 1}

experiment the visosity is also assumed onstant but depends on the initial

polymer onentration, aording to an empirial law (Fig.1) dedued from

visosity measurements performed with a Low Shear 30 rheometer (oaxial

ylinders and imposed deformation)[16℄.

2.2 Initial and boundary onditions

At

t = 0

T (~x, 0) = T ₀

the stability of the system, a perturbation is imposed onthe veloity eld at

t = 0

r

is implemented in the following way: numerial resolution of the problem is

ahievedusinganitevolumesheme.Foreahspatialnode,thevalueallotted

totheveloityat

t = 0

− r/2

+r/2

3.2and [12,17℄).

The solidwallsare adiabatiwithazero veloityboundaryondition.On the

free surfae loated at

y = e

− k ∂T

∂y

!

y=e

= h th (T (x, e, t) − T 0 ) + LΦ ev

The rst term of the r.h.s. is the onvetive heat transfer between the free

surfae and the ambient air where

h _th

seond term,

L Φ ev

L

heat of vaporizationand

Φ ev

Φ _ev = h _m (ρ ^g _Sint − ρ ^g _S ∞ )

where

h m

ρ ^g _Sint

ρ ^g _S ∞

entrationinthe gas phasejustabove the interfaeand farfromthe interfae

respetively. The last one is zero, due to the important air ow rate in the

extratorhood.With the ideal gas law, we get:

Φ ev = h m

M S P V S0 (T (x, e, t))

RT (x, e, t) a(T (x, e, t), ϕ S (x, e, t))

where

M S

a

P V S0

saturatedsolventvaporpressure,

ϕ S

phase at the interfae and

R

the ativity is lose toone for solvent volume fration greaterthan about 0.4

[18℄, so that

Φ ev

a ≃ 1

Moreover, giventhesmallamplitudeoftemperaturevariationsobservedinthe

experiments (a fewdegrees), it ispossibleto use a rst orderdevelopmentof

Φ ev

Φ ev (T (x, e, t)) ≃ Φ ev (T 0 ) + ^∂Φ _∂T

| ^T

(T (x, e, t) − T 0 )

Thethermalboundaryonditionatthefreesurfaeanthen beapproximated

by the following expression

− k ∂T

∂y

!

y=e

= H th (T (x, e, t) − T 0 ) + L Φ ev (T 0 )

with

H th = h th + L ^∂Φ _∂T

| ^T

Ashear stressboundary onditionisimposedatthe freesurfae,givenbythe

µ ∂v x

∂y

!

y=e

= dγ dT

∂T

∂x

!

y=e

(6)

where

γ

Theotherboundaryonditionattheinterfaeonernsthevertialomponent

of the veloity,

v y

variationsof the evaporationux are muhsmaller thanthe ux itself,itan

beshown [12℄ that, inthe limitof

P e int ≪ 1

to:

v y = 0

y = e

Aording toexperimentalresults,thefollowingvalueshavebeen used:

H _th = 28W.m ^{− 2} .K ^{− 1}

L = 396kJ.kg ^{− 1}

Φ ev (T 0 ) = 3.37 × 10 ^{− 4} kg.m ^{− 2} .s ^{− 1}

dγ/dT =

− 0.119 × 10 ^{− 3} N.m ^{− 1} .K ^{− 1}

2.3 Non-dimensional equations

The non-dimensional form of the equations results from saling the lengths

by the thikness of the uid

e

~v

α/e

and time

t

p

e ² /α

ρα ² /e ²

θ = ^T _∆T ^{− T}

∆T

T 0

∆T = ^LΦ _H

^(T

⁾

∆T = 4.8K

stabilityanalysiswhere the basistate isharaterizedby aonstant temper-

ature gradient inthe uid.

The dimensionlessform of equations (1a-1)is:

∇ ~ .~v = 0

∂~v

∂t + (~v.~ ∇ )~v = − ∇ ~ p + RaP rθ~e y + P r∆~v

∂θ

∂t + (~v.~ ∇ )θ = ∆θ

where

P r = ν/α

Ra = β _T g ∆T e ³ /(να)

number.

ary onditions atthe free surfae are :

− ∂θ

∂y

!

y=1

= Bi (θ(x, y = 1, t) + 1) with Bi = H th e

k

∂v _x

∂y

!

y=1

= − Ma ∂θ

∂x

!

y=1

with Ma = − e ∆T µα

∂γ

∂T

!

(10)

v y = 0

y = e

with

Ma

Bi

into aountthe evaporativeux.

The ow and thermal behaviour in the solution are governed by four non-

dimensionalparameters(

Bi

Ma

P r

Ra

parameters used inthe experiments,the initialthikness and visosity [13℄.

2.4 Numerial method

Computations are arried out with a olloated nite volume sheme with a

seond order spae and time disretization [19℄. The disrete sheme is fully

oupled.Thesetofthedisretebalaneequationsissolvedbyanunder-relaxed

Newton'smethodwith theiterativelinear solverBICGSTAB, preonditioned

by an inomplete LU fatorization. The hosen mesh is a standard regular

mesh

L ∗ e = (800 ∗ 40)

∆t = 10 ^{− 3}

obtained with ner grids or time steps did not show any notieable hanges

inthe results.

3 Transient onvetion

3.1 Typialtemperature and veloity elds

Fig.2 gives a typial example of the time evolution of the non dimensional

temperature dierene between the bottom and the free surfae of the solu-

tion for a ongurationwhere onvetion is observed (parameters of the test

ase are given in table 1, initial veloity disturbane amplitude

r = 4

a omparison, the temperature evolution obtained for a pure diusive prob-

lem is also drawn (dashed lineobtained with

Ma = 0

Ra = 0

domains an be observed. At the beginning, heat transfer is dominated by

the diusion and the twourves are superposed (domainI). Then onvetion

atureeld (domainII).This regimeisfollowed by aquasi-steadyregime with

aslowderease ofthetemperaturedierenebetweenthe bottomand thefree

surfae (domain III). At lastfor large times the two urves orresponding to

purediusiveoronvetive regimesgotozero.Indeed the temperatureinthe

uidisuniformatthe endof thethermaltransientregime(notrepresented in

Fig.2).

The stream lines and the temperature on the free surfae are given in Fig.3

for three dimensionless times t=0.5, t=0.6 and t=0.65. We an observe the

vanishing of the entral ell between t= 0.5 and t=0.65 that orresponds to

the seond peak that an be observed on the temperatureevolutionin Fig.2.

It is followed by the quasi-steady regime,with a onstantnumberof ells.

The objetive of this paper is to analyze the three regimes, as a funtion

of the four non dimensional parameters that haraterize the problem,

Bi

Ma

P r

Ra

the onset of onvetion is analyzed in setion 4. Setion 5 is dediated to

the haraterization of the quasi-steady regime, using saling laws to get an

estimation of the temperatureand veloity in this regime.

Toharaterizethepreseneofobservableonvetion,ariteriumbasedonthe

thermal Pelet number was hosen. The thermal Pelet number

P e = e v/α

ompares the relative importane of advetion and diusion. The veloity

v

veloity norm. Then onvetion will be onsidered signiant if, when the

systemissubmittedtoaninitialveloity perturbation,there isatime

t

the perturbation is signiantly amplied, i.e. suh as

dP e(t)/dt > 0

P e(t) > 1

3.2 Eet of the initial disturbane amplitude

As the problem is sensitive to initial onditions, we have rst performed a

preliminarystudy to analyze the inuene of the amplitude of the initialdis-

turbaneonthe timeevolutionofthePeletnumber.The parametersusedfor

this test are given inTab.1.

Fig.4a shows the temporal evolution of the Pelet number in log sale. At

the beginninga linearregime an be observed witha dereasing of the initial

perturbation followed by its ampliation. But at large

t

regime,

P e

P e > 1

i.e.onvetionisobserved;onlythetimedelaydependsontheamplitudeofthe

disturbane. The orresponding time is about

t = 0.1

r = 4

t = 0.03

for

r = 400

sothat the transitionbetween the linearand nonlinear regimean belearly

observed.

Sine the general trends observed in the three regimes and the ourrene of

onvetiondonotdependstronglyontheperturbationamplitude(atleastina

largedomain),weuse

r = 4

ysis of the inuene of the initialperturbation struture has been performed

[17℄ andgivesthe same onlusions:hangingthe initialperturbationindues

only small hanges in the pattern wavelength, in the temperature or in the

Peletvariations.Thethresholdvalueorrespondingtotheonsetofonvetion

isthereforelittlesensitivetothe initialperturbation.This assumptionison-

rmedin setion4.3 by the goodagreementbetween the thresholds obtained

with the non-linear simulationsand the linear stability analysis. Howeverthe

thresholdmustnotbeunderstoodasapreisedelimitationbetweenstableand

unstable domains but rather as a transition region (a fator of about two is

obtained for dierent initialperturbations). This "blurring" eet is inherent

tothe transientharater of the problemunder study [12℄.

4 Stability

4.1 Inuene of the dynami visosity

The visosity of solutions understudy being very sensitive tothe initialpoly-

mer onentration, we rst study the inuene of the dynami visosity

µ

on the Pelet transient behavior. Both buoyany and surfae tension driving

foresare taken intoaount(Rayleigh-Bénard-Marangonionguration)and

resultsare given inFig.5for

e = 1

visosity

µ

For the largest visosities, the Pelet number is a monotone dereasing fun-

tions and the initialperturbation dies down. For smaller visosities, there is

an ampliation of the initial perturbation after some times. However this

ampliation may be very weak and for visosity larger than

5mP a.s

Pelet number is always smaller than one, so that this onguration is "sta-

ble" aording to the riterium dened in setion 3.1. The ritial visosity

orrespondingtothe onsetofonvetionliesbetween

4mP a.s

5mP a.s

this thikness. In the next setion, the same study was performed for other

thiknesses inorder to ompare simulationsand experimentaldata.