A numerical investigation of the evaporation process of a liquid droplet impinging onto a hot substrate

A numerical investigation of the evaporation process of a liquid droplet impinging onto a hot substrate
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  A numerical investigation of the evaporation processof a liquid droplet impinging onto a hot substrate N. Nikolopoulos a , A. Theodorakakos b , G. Bergeles a,* a Department Mechanical Engineering, National Technical University of Athens, 5 Heroon Polytechniou, 15710 Athens, Greece b Fluid Research, Co, Greece Received 20 December 2005Available online 22 August 2006 Abstract A numerical investigation of the evaporation process of   n -heptane and water liquid droplets impinging onto a hot substrate is pre-sented. Three different temperatures are investigated, covering flow regimes below and above Leidenfrost temperature. The Navier– Stokes equations expressing the flow distribution of the liquid and gas phases, coupled with the Volume of Fluid Method (VOF) fortracking the liquid–gas interface, are solved numerically using the finite volume methodology. Both two-dimensional axisymmetricand fully three-dimensional domains are utilized. An evaporation model coupled with the VOF methodology predicts the vapor blanketheight between the evaporating droplet and the substrate, for cases with substrate temperature above the Leidenfrost point, and the for-mation of vapor bubbles in the region of nucleate boiling regime. The results are compared with available experimental data indicatingthe outcome of the impingement and the droplet shape during the impingement process, while additional information for the dropletevaporation rate and the temperature and vapor concentration fields is provided by the computational model.   2006 Elsevier Ltd. All rights reserved. Keywords:  Droplet evaporation; Volume of Fluid Method; Kinetic theory; Leidenfrost temperature 1. Introduction The liquid–vapor phase change process, plays a signifi-cant role in a number of technological applications in com-bustion engines, cooling systems or refrigeration cycles. Inall the aforementioned applications, the dynamic behaviorof the impinging droplets and the heat transfer between theliquid droplets and the hot surfaces are important factors,which affect the mass transfer associated with liquid–vaporphase change.The mechanism of the droplet spreading and theaccompanying heat transfer is governed not only by non-dimensional parameters as the droplet Weber ( We ), theReynolds ( Re ) number, Eckert ( E  c ) number, and Bond( Bo ) number, but also by the temperature of the surface.As the droplet impacts upon the hot solid surface, heat istransferred from the solid to the liquid phase. This energytransfer to the droplet increases its mean temperature,while liquid vaporizes from the bottom of the droplet. If the heat transfer rate is large enough during the impact,liquid vaporized from the droplet forms a vapor layerbetween the solid and the liquid phase, which repels thedroplet from the solid surface. In this case the heat transferreaches a local minimum and the evaporation lifetime of the droplet becomes maximum. This phenomenon was firstobserved by Leidenfrost [1] in 1756 and hence the behavioris known as the Leidenfrost phenomenon. Based on theevaporation lifetime of a droplet, mainly four differentevaporation regimes can be identified depending on thewall temperature; film evaporation, nucleate boiling, tran-sition boiling and film boiling. This work contributes tothe study of transition and film boiling impact regimesonly. 0017-9310/$ - see front matter    2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijheatmasstransfer.2006.06.012 * Corresponding author. Tel.: +30 2107721058; fax: +30 2107723616. E-mail addresses: (N. Nikolopoulos), (A. Theodorakakos), (G. Bergeles). International Journal of Heat and Mass Transfer 50 (2007) 303–319  The collision dynamics of a liquid droplet impinging ona hot surface has been investigated mainly experimentally.Researchers have presented a sequence of photographsshowing the deformation process of liquid droplets impact-ing on a hot surface. Wachters and Westerling [2] wereamong the first to investigate the impact of a saturatedwater droplet of about 2 mm in diameter impinging on apolished gold surface heated to 400   C, while Akao et al.[3] inspected the deformation behavior of various liquiddroplets of 2 mm diameter on a chromium-plate coppersurface heated to the same temperature. Xiong and Yuen[4] measured the time history of a  n -heptane dropletimpinging on a stainless-steel surface heated to tempera-tures between 63   C and 605   C. Chandra and Avedisian[5] performed the same experiment with a temperaturerange from 24   C to 205   C keeping a constant Webernumber  We  = 43 while the same authors in [6] have pre-sented results for the deformation process of a dropletimpinging onto a porous ceramic surface. Naber and Far-rell [7] examined the deformation process of liquid dropletsof 0.1–0.3 mm in diameter impinging on a hot stainless-steel surface, while at the same time Anders et al. [8] inves-tigated the rebounding phenomenon of ethanol dropletsimpacting obliquely on a smooth chromium-plated coppersurface at 500   C.Ko and Chumg [9] investigated experimentally the effectof wall temperature on the break-up process of   n -decanefuel, in the Leidenfrost temperature range of 220–330   C,and demonstrated that wall temperature variation showsa peculiar nonlinear behavior in the droplet break-up prob-ability, especially near 250   C, which corresponds to thetemperature of local maximum droplet lifetime. Manzelloand Yang [10] examined the effect of an additive in a waterdroplet on its collision dynamics on a stainless-steel surfacewith the wall temperature varying from film evaporation tofilm boiling regime for three  Weber  number impacts.Bernardin et al. [11,12] realizing that the impact param-eters can alter the collision outcome, conducted a thoroughseries of experiments, concerning water droplets impingingon a polished aluminium surface, with the main controllingparameters of the phenomenon being droplet velocity,resulting in  We  number from 20 to 220 and surface temper-ature from 100   C to 280   C. They constructed dropletimpact regime maps, which distinguish between the variousboiling regimes for each of the three experimental  We  num-bers investigated. Moreover, the heat flux from the surfacewas measured, for different  We  numbers, drop impact fre-quency and surface temperature, determining the two veryimportant points in the regime map, the Leidenfrost point(LFP) and the critical heat flux point (CHF). The first Nomenclature Bo  Bond number,  ð¼ q liq  gD 2o = r Þ C   p  non-dimensional pressure,  ¼  D  P  = 12 q liq U  2o   c  p  heat capacity (J/kg K) D AB   diffusivity of gas  A  to gas  B  , (= l /( Sc    q )) E  c  Eckert number,  ð¼  U  2o = ð c  p  ð T  liq    T  w ÞÞÞ D o  initial diameter of droplet E  sur  surface energy E  kin  kinetic energy k   thermal conductivity (W/mK), (= c  p    l / Pr ) MB   molecular weight (kg/kmol) ~ n  vector normal to interface of the two phases Oh  Ohnesorge number, (= l liq /( rq liq D o ) 0.5 ) P   pressure Pr  Prandtl number, (= l c  p / k  )  R  universal gas constant (J/kmol K) R  computational radius R o  radius of initial droplet Re  Reynolds number (= q liq D o U  o / l liq ) Sc  Schmidt number (= l /( q D AB  )) SYG   vapor concentration (mass of vapor (kg)/massof gas phase (kg))T temperaturet time ~ T   stress tensor ~ u  velocity U  o  initial velocity of droplet U  l  velocity of an equivalent droplet of the ring V   volume X X  -axis of computational field Y Y  -axis of computational field Z Z  -axis of computational field Z  h  height of spreading droplet We  Weber number,  ð¼  q liq  D o U  2o = r Þ Greek symbols a  volume of fluid (also noted as indicator func-tion) d  vapor height j  curvature (m  1 ) l  dynamic viscosity q  density r  surface tension  r  thermal accommodation coefficient Subscripts gas gas phaseliq liquid phaseb basevap vaporcell computational cellsat saturation pointw substrate or wall 304  N. Nikolopoulos et al./International Journal of Heat and Mass Transfer 50 (2007) 303–319  corresponds to the minimum heat flux point and the secondto the lower temperature boundary of the transitional boil-ing regime.Apart from the above-mentioned controlling parametersfor the description of such a phenomenon, secondaryparameters such as surface roughness, control the evolu-tion of this phenomenon. Most of the researchers ignoredthe effects of surface roughness on droplet heat transfer.Cumo et al. [13], Baumeister et al. [14] and Nishio and Hir- ata [15] observed that rough surfaces require a thickervapor layer between the droplet and the surface to sustainfilm boiling and, therefore, possess a higher LFP tempera-ture. Avedisian and Koplik [16] found that the LFP forwater droplets on porous ceramic surfaces increases withincreasing porosity. Engel [17] observed that surface rough-ness promotes droplet break-up, and Ganic and Rohsenow[18] reported surface roughness enhances liquid–solid con-tact in dispersed droplet flow and hence increases film boil-ing heat transfer. Fujimoto and Hatta [19] and Hatta et al.[20] confirmed that the critical  We  number, above whichwhether or not the droplet is disintegrated during deforma-tion, depends on the kind of surface material. Wachtersand Westerling [2] observed experimentally that the critical We  number, above which disintegration of a dropletimpinging on a hot wall once the droplet is transformedinto an expanding torus is around 80.Bernardin et al. [11,12] used three different surface finishesand reported that although the temperature correspondingto the critical heat flux (CHF) was fairly independent of sur-face roughness, the Leidenfrost point (LFP) temperature wasespecially sensitive to surface finish. They produced regimemaps illustrating not only the well-known boiling curveregimes of liquid film, transition and nucleate boiling, butalso the complex liquid–solid interactions which occur dur-ing the lifetime of the impacting droplet.One more important secondary controlling parameter,essential not only for the description of physics of this phe-nomenon, but also for its numerical simulation, is the valueof contact angles. Bernardin et al. [11,12] using the sessiledrop technique measured the variation of contact anglesfor an aluminum surface, as a function of surface temper-ature, while Chandra et al. [21] studied the effect of contactangles on droplet evaporation, adding varying amounts of a surfactant to water.A few studies have examined the effect of reduced grav-ity. Siegel [22] has reviewed much of the work done on thistopic. The principal findings were that gravity has littleeffect on the nucleate pool boiling heat transfer coefficients.For low wall heat flux, vapor bubble diameters increase atlow gravity. Furthermore, the critical heat flux decreases inthe absence of buoyancy forces while stable film boiling canbe maintained at low gravity, but heat transfer is reduced.Qiao and Chandra [23] performed a series of experiments,using water and  n -heptane, intending to isolate the effect of buoyancy forces on droplet impact and boiling. Theirobjective was to study the effect of gravity and liquid prop-erties on transition from nucleate to film boiling.Due to the highly complex nature of these processes,development of methods to predict the associated heatand mass transfer has often proved to be a difficult task.Nevertheless, research efforts over several decades haveprovided an understanding of many aspects of vaporiza-tion or condensation. Important and interesting numericalsimulations of droplet collisions with a variety of methodshave also been published. The MAC-type solution methodto solve a finite-differencing approximation of the Navier– Stokes equations governing an axisymmetric incompress-ible fluid flow was used by Fujimoto and Hatta [19] andHatta et al. [20]. The simulation of the flow field insidethe liquid droplet has been performed assuming a simplethermal distribution such that temperature becomes lower(higher) on the upper (lower) side of the droplet and higherwith time. The unsteady thermal distribution inside thedroplet is not calculated, assuming the temperature of thedroplet’s bottom to be at the saturation temperature andthat a vapor layer exists between the droplet and solidsurface.A number of analytical studies by Gottfried et al. [24],Wachters et al. [25], Nguyen and Avedisian [26], and Zhang and Gogos [27] are dealing with the Leidenfrost phenome-non and the steady-state droplet film boiling. Indispensablecondition for these studies is that the droplet has a nearlysteady spherical shape, so that the heat transfer rates anddroplet evaporation times can be predicted successfully.Pasandideh et al. [28] used a complete numerical solu-tion of the Navier–Stokes and energy equations, basedon a modified SOLA-VOF method, to model dropletdeformation and solidification, including heat transfer inthe substrate. The heat transfer coefficient at the droplet-substrate interface was estimated by matching numericalpredictions of the variation of substrate temperature withmeasurements. Heat transfer in the droplet was modeledby solving the energy equation, neglecting viscous dissipa-tion, whilst the effect of substrate’s cooling on the droplet’sevaporation was taken into account [29]. Following that,Pasandideh et al. [30], extended the model developed byBussmann et al. [31] and combined a fixed-grid control vol-ume discretization scheme of the flow and energy equationswith a volume tracking algorithm to track the droplet freesurface. Surface tension effects were also taken intoaccount. The energy equation both in the liquid and solidportion of the droplet were solved using the Enthalpymethod in the case of solidification. More recent three-dimensional codes have been used to model complex flowssuch as impact on inclined surfaces resulting in dropletbreak-up, as shown by Zheng and Zhang [32] and splash-ing, according to Ghafouri-Azar et al. [33]. Zheng andZhang [32] developed an adaptive level set method formoving boundary problems in the case of droplet spread-ing and solidification.Zhao and Poulikakos [34,35] studied numerically thefluid dynamics and heat transfer phenomena both in drop-let and the substrate, based on the Lagrangian formulationand utilizing the finite element method in a deforming N. Nikolopoulos et al./International Journal of Heat and Mass Transfer 50 (2007) 303–319  305  mesh. The temperature fields developing in both the liquiddroplet and the substrate during the impingement processwere also determined. Waldvogel and Poulikakos [36]followed the Langrangian formulation including surfacetension and heat transfer with solidification. They investi-gated the effect of initial droplet temperature, impactvelocity, thermal contact resistance and initial substratetemperature on droplet spreading, on final deposit shapesand on the times to initiate and complete freezing. Buttyet al. [37] solved the energy equation in both the dropletand substrate domain, implementing a time and space aver-aged thermal contact resistance between the two thermaldomains. During calculations a regeneration of mesh tech-nique is used, in order to enhance accuracy. Harvie andFletcher [38–40] coupled VOF methodology with a sepa-rate one-dimensional algorithm to model not only thehydrodynamic gross deformation of the droplet, impactingonto a hot wall surface, but also the fluid flow within theviscous vapor layer existing between the droplet and thesolid surface. The height of the vapor layer was assumedto be several orders of magnitude smaller than the dimen-sions of the droplet, resulting in a Knudsen numberapproaching values of the order of 0.1. It is important tonote that the height of the vapor layer does not result fromthe solution of the Navier–Stokes equations, but it wasassumed to be known. Furthermore, they used a kinetictheory treatment in order to calculate conditions existingat the non-equilibrium interface of the vapor layer, solvingthe heat transfer within the solid, liquid and vapor phases.This model was validated for a number of droplet impactconditions, covering a wide range of   We  numbers and ini-tial droplet and surface temperatures.The present investigation studies numerically theimpingement of   n -heptane and water droplets on a hot sub-strate under various temperatures, covering regimes aboveand below the Leidenfrost temperature. Viscous dissipationand surface tension effects are taken into account; the equa-tions are solved numerically with the finite volume method-ology, whilst the Volume of Fluid methodology of Hirt andNichols [41] is used for the tracking of the liquid–gas inter-face. The methodology is coupled with an adaptive localgrid refinement technique, both in 2-D axisymmetric andfully three-dimensional cases, allowing the prediction of details of droplet’s levitation, above the Leidenfrost tem-perature, without any ‘a priori’ assumption for the vaporlayer height. Moreover in contrast to other methodologies,in the case of impact below the Leidenfrost point, theentrapment of vapor between the liquid droplet and thewall is predicted, The evaporation model coupled withVOF methodology is used in an in-house developed CFDcode, predicting not only the deformation of the liquiddroplet and the height of vapor blanket in the case of theabove Leidenfrost temperature, but also the correspondingtemperature and vapor fields. The used model is validatedfor a number of droplet impacts both for low and high  We numbers and substrate temperatures. The heat transferinside the substrate is not solved, as the substrate tempera-ture is considered to be constant, the liquid–gas interface isassumed to be at saturation conditions, whilst the effect of substrate roughness on the droplet spreading is not takeninto account. 2. The numerical solution procedure  2.1. Fluid flow The flow induced by the impact of a droplet on a hotsurface, is considered as two-dimensional axisymmetricfor cases A, B, and C ( n -heptane) and for case D (water)as three-dimensional; the details of the test conditionsinvestigated are summarized in Table 1. The volume frac-tion, denoted by  a , is introduced following the Volume of Fluid Method (VOF) of Hirt and Nichols [41] in order todistinguish between the gas and the liquid phases. This isdefined as: a  ¼  Volume of liquid phaseTotal volume of the control volume  ð 1 Þ where the  a -function is equal to: a ð  x ; t  Þ ¼ 1 ;  forapoint ð  x ; t  Þ insideliquidphase0 ;  forapoint ð  x ; t  Þ insidegasphase0 < a < 1 ;  forapoint ð  x ; t  Þ insidethetransitionalareabetweenthetwophases 8>>><>>>:  ð 2 Þ For a single droplet splashing onto a wall film, the VOFmethodology has been successfully applied and the methodis described in more detail in Nikolopoulos et al. [42].The momentum equation is written in the form: o ð q ~ u Þ o t   þ r  ð q ~ u   ~ u   ~ T  Þ ¼  q ~  g   þ ~  f  r  ð 3 Þ where ~ T   is the stress tensor, ~ u  is the velocity,  q  is the densityof the mixture and  f  r  is the volumetric force due to surfacetension. The value of   f  r  is equal to  f  r  =  r    j    ( $ a ), where  r is the numerical value of the surface tension (for immisciblefluids the value is always positive) and  j  is the curvature of the interface region.The flow field is solved numerically on two or three-dimensional unstructured grids, using a recently developedadaptive local grid refinement technique, following thefinite volume approximation, coupled with the VOF meth-odology; a detailed discussion of the fluid flow model ispresented by Nikolopoulos et al. [42], while the adaptivelocal grid refinement technique is used in order to enhanceaccuracy of the predictions in the areas of interest (i.e. theliquid–gas interface), with minimum computational cost, asshown by Theodorakakos and Bergeles [43]. To accountfor the high flow gradients near the free surface, the cellsare locally subdivided to successive resolution levels, onboth sides of the free surface. As a result, the interface is 306  N. Nikolopoulos et al./International Journal of Heat and Mass Transfer 50 (2007) 303–319  always enclosed by the densest grid region. A new locallyrefined mesh is created every 20 time steps for the cases thatwill be presented afterwards. The numerical cell at whichsubdivision is performed, is locally refined by a factor of 3 for case D or 4 for cases A, B, and C (i.e. in two dimen-sions an initial cell is split into four cells). In that way a newgrid with 1 level of local refinement is created. Obviously,computations are more time efficient on the dynamicallyadaptive grid, than on the equivalent fine resolution uni-form grid.The high-resolution differencing scheme CICSAM, pro-posed by Ubbink and Issa [44] in the transport equation for a  (VOF-variable) is used. The discretization of the convec-tion terms of the velocity components is based on a highresolution convection-diffusion differencing scheme (HRscheme) proposed by Jasak [45]. The time derivative wasdiscretized using a second-order differencing scheme(Crank–Nicolson). Quadrangular (2D) or hexahedron(3D) computational cells are used. Finally, the contactangles at the advancing and receding contact lines areassigned as boundary conditions.  2.2. Heat transfer Heat transfer in the droplet was predicted by solvingthe energy equation, calculating all physical propertiesas a function of the corresponding properties of the liquidand gas (air and vapor) phase. Such properties are den-sity, viscosity, heat capacity and Prandtl number. Allproperties were assumed to vary with temperature andpressure, including the diffusivity of vapor in air ( D AB  ).The surface tension coefficient is assumed to vary alsowith temperature.Heat transfer within the liquid phase is described by thefollowing thermal energy transport equation (enthalpyequation) for incompressible fluids: q D h 0 D t   ¼ rð k    r T  Þ þ  D  P  D t   þ  _ Q ;  ð 4 Þ In this equation,  _ Q  is a source term due to evaporation andis equal to the amount of heat released, when liquid passesthrough the liquid–vapor interface and evaporates: _ Q  ¼  d m = d t V    cell     L ;  L  ¼ ð C   p  ; liq    C   p  ; vap Þ   T  ;  ð 5 Þ where  L  is the latent heat of vaporization of liquid andd m /d t  the evaporation rate of the liquid phase.The value of   ~ Q  term is proportional to the mass flux of liquid molecules which evaporate. Following Langmuir’s[46] approach, whereby the liquid and vapor phases areassumed to be separated by a discrete molecular layer,but including the Schrage’s correction [47] to account formolecular flow towards or away from the liquid surface,the evaporated mass flux is equal tod m = d t   ¼  2     r 2     r     MB vap 2    p    R   1 = 2   P  sat ; liq T  1 = 2liq   P  sat ; vap T  1 = 2vap  ! >  0 ð 6 Þ where   r  is the thermal accommodation coefficient.It seems that thermal accommodation coefficient has notbeen measured with any real confidence yet, but its value isin the range of 0–1. Here, a value of 0.5 has been chosenboth for  n -heptane and water.Apart from the energy equation, an additional trans-port Eq. (7) for the concentration of vapor in the gas issolved ð 1   a Þ   q air D C  D t   ¼ r½ð 1    a Þ  q air    D  AB   r C   þ  d m = d t V    cell   ð 7 Þ where  C   is the concentration of the vapor phase in the gasphase (kg vapor /kg gas ). For the mixed phase of liquid andgas, physical and thermodynamic properties are calculated Table 1Test cases examinedCase A B C DLiquid  n -Heptane  n -Heptane  n -Heptane Water R o  0.00075 0.00075 0.00075 0.0015 U  o  0.8 0.8 0.8 2.34 We  34.52 34.52 34.52 222.10 Re  2156.09 2156.09 2156.09 7638.78 Oh  0.00273 0.00273 0.00273 0.00195 Bo  0 0 0 1.19 E  c  0.034 0.031 0.028 0.060 T  w  (  C) 178 190 210 180 T  liq  (  C) 25 25 25 27Computational domain( X  tot , Y  tot , Z  tot )13.33 R o    6.67 R o  13.33 R o    6.67 R o  13.33 R o    6.67 R o  10 R o    10 R o    6.67 R o Base grid 60    30 (4 levels localrefinement)60    30 (4 levels localrefinement)60    30 (4 levels localrefinement)45    45    30 (3 levels localrefinement)Maximum number of grid nodes 11314 16353 13183 499132 N. Nikolopoulos et al./International Journal of Heat and Mass Transfer 50 (2007) 303–319  307
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