A theoretical framework for comprehensive modeling of steadily fed evaporating droplets and the validity of common assumptions
CComprehensive modeling of steadily fed evaporating droplets
Akkus et al.
A theoretical framework for comprehensive modeling of steadily fedevaporating droplets and the validity of common assumptions
Yigit Akkus a , Barbaros C¸ etin b , Zafer Dursunkaya c ∗ a ASELSAN Inc., 06200 Yenimahalle, Ankara, Turkey b Mechanical Engineering Department, ˙I.D. Bilkent University, 06800 C¸ ankaya, Ankara, Turkey c Department of Mechanical Engineering, Middle East Technical University, 06800 C¸ ankaya, Ankara, Turkey
Abstract
A theoretical framework is established to model the evaporation from continuously feddroplets, promising tools in the thermal management of high heat flux electronics. Usingthe framework, a comprehensive model is developed for a hemispherical water dropletresting on a heated flat substrate incorporating all of the relevant transport mechanisms:buoyant and thermocapillary convection inside the droplet and diffusive and convectivetransport of vapor in the gas domain. At the interface, mass, momentum, and thermalcoupling of the phases are also made accounting for all pertinent physical aspects includ-ing several rarely considered interfacial phenomena such as Stefan flow of gas and theradiative heat transfer from interface to the surroundings. The model developed utilizestemperature dependent properties in both phases including the density and accounts forall relevant physics including Marangoni flow, which makes the model unprecedented.Moreover, utilizing this comprehensive model, a nonmonotonic interfacial temperaturedistribution with double temperature dips is discovered for a hemispherical droplet havinginternal convection due to buoyancy in the case of high substrate temperature. Proposedframework is also employed to construct several simplified models adopting common as-sumptions of droplet evaporation and the computational performance of these models,thereby the validity of commonly applied simplifying assumptions, are assessed. Bench-mark simulations reveal that omission of gas flow, i.e. neglecting convective transportin gas phase, results in the underestimation of evaporation rates by 23–54%. When gasflow is considered but the effect of buoyancy is modeled using Boussinesq approximationinstead of assigning temperature dependent density throughout the gas domain, evapo-ration rate can be underestimated by up to 16%. Deviation of simplified models tendsto increase with increasing substrate temperature. Moreover, presence of Marangoni flowleads to larger errors in the evaporation rate prediction of simplified models.
Keywords: droplet evaporation, steadily fed droplet, thermocapillarity, buoyancy, gasconvection, Stefan flow ∗ e-mail: [email protected] a r X i v : . [ phy s i c s . a pp - ph ] A p r omprehensive modeling of steadily fed evaporating droplets Akkus et al.
Droplet evaporation is a ubiquitous phenomenon observed in various natural processes such ashuman perspiration and industrial applications such as DNA mapping, inkjet printing, biosens-ing, and surface coating [1–4]. In recent years, utilization of droplet evaporation in coolingapplications has been of interest due to its high heat removal capability associated with thelatent heat of vaporization during phase change. While some studies propose the use of dryingdroplets such as spray cooling [5, 6], others suggest the utilization of continuously fed, constantshape droplets [7, 8] similar to sweat-droplets on mammals’ skin. In the absence of feeding,droplet evaporation is a transient process because of the deforming droplet surface. However,in the case of steadily fed droplets, liquid-gas interface preserves its shape and the problembecomes that of a steady state configuration.Regardless of being transient or steady state, droplet evaporation is a complex problembecause of the presence of various energy transport mechanisms in two different phases to-gether with their coupling at the liquid-gas interface where heat and mass transfer take placesimultaneously. Inside the droplet, energy is transferred from the substrate to the interface via conduction and convection. The latter was rarely considered before 2000’s [9–11] but sub-sequent studies [12–26] focused on the convective heat transport inside the droplets, which istriggered by two mechanisms; buoyancy and thermocapillarity [12]. Thermocapillary convec-tion was reported to dominate the buoyancy and dictate the flow pattern inside an evaporatingdroplet [20, 23]. Outside the droplet, energy is transferred from the droplet surface to the am-bient via diffusion of vapor, convection of gas, and conduction in gas phase. In the absence of aforced flow of surrounding gas, natural convection is responsible for the convective heat trans-fer in the gas phase. While the majority of modeling attempts did not account for the effectof buoyancy in gas phase, several experiments [27–29] demonstrated that diffusion-controlledevaporation models considerably underestimate the evaporation rate. In addition, this issuewas also confirmed by empirical [30] and numerical [31, 32] models.The proper coupling of the phases at the interface is essential in building a successfulcomputational model; therefore, mass, force, and energy balances should be properly imposed atthe liquid-vapor interface. The mass balance results in the discontinuity of normal velocities ofliquid and vapor due to the density difference of the phases. Moreover, interfacial gas velocity isdictated by the Stefan flow of air to counteract the diffusion of air towards the interface, throughwhich air cannot penetrate due to its insolubility in liquid. Thermocapillarity produces asurface force in the tangential direction and the curved surface creates a Laplace pressure jump;therefore, tangential and normal force balances need to be constructed accordingly. Energytransferred to the liquid interface is conveyed to the gas phase/surroundings by evaporative heattransfer, conduction to the gas phase and radiation to the surroundings. While evaporation isalways responsible for the majority of the heat transfer, the contribution of the others increase2 omprehensive modeling of steadily fed evaporating droplets
Akkus et al. when the temperatures of the interface and substrate increase.An accurate estimation of evaporative mass flux is vital since it is the major energy transfermechanism at the interface. Although both diffusive and convective components contribute tothe total evaporative flux, majority of the previous studies [16, 18, 19, 21–25, 33] neglectedthe effect of convection. Many of them [18, 19, 21, 22, 25] implemented the semi-empiricalcorrelation of Hu and Larson [33], which was built on the well-known studies of Deegan etal. [34, 35]. In the derivation of this correlation, the sole transport mechanism considered inthe gas phase was the diffusion of vapor. Moreover, the droplet surface was assumed to beisothermal and the correlation was obtained for contact angles less than 90 ◦ . Another vapordiffusion based correlation was suggested by Popov [36], which accounted for the nonuniformityof evaporative mass flux at the droplet surface and valid for all contact angles. This correlationwas also used in subsequent studies [24, 30]. Estimation of convective mass transfer, on theother hand, requires the solution of flow in the gas phase. Among the studies carrying outthe solution of gas flow [23, 31, 32, 37] in droplet evaporation problems, only a few [32, 37]considered the convective transport in the calculation of evaporation rate. In addition, severalstudies [12, 13, 17, 26, 38, 39] implemented correlations based on analogies between heat andmass transfer instead of calculating a mass diffusion based solution of concentration field inthe gas phase. Lastly, the kinetic theory of gases was also adopted in the prediction of theevaporation flux in the literature [20, 40].The present study aims to develop a comprehensive model in order to estimate the evapora-tion rate from steadily fed evaporating droplets. In our previous models, we only considered theliquid flow and energy transfer inside the droplet without [39] and with the thermocapillarityeffect [26], and the evaporative mass flux at the interface was estimated based on an analogybetween heat and mass transfer proposed by [12]. The current study takes a further step andincludes the solution of mass, species, momentum and energy transfer in the gas mixture. Theresultant concentration field of the vapor and flow field of the gas are utilized to estimate theevaporation rate from a droplet. The computational model developed is tested by consideringthe evaporation from a spherical (with 90 ◦ contact angle), sessile and continuously fed waterdroplet—by liquid injection—placed on a heated flat substrate, which was the same configura-tion in our previous work [26] and a previous experimental work [41] thereby enables a directcomparison with previous results.The ultimate objective of the current work is to develop a general theoretical framework forthe modeling steadily fed evaporating droplets by incorporating all relevant physical phenomenainside and outside the droplet as well as with the ones at the interface. Since the steadilyfed droplets are proposed in the thermal management of high heat flux dissipating electroniccomponents, the substrate temperature is expected to reach high values. Although Boussinesqapproximation [23, 30–32] is an alternative approach to solving full compressible Navier-Stokes3 omprehensive modeling of steadily fed evaporating droplets Akkus et al. equations to simplify nonlinearity and improve the numerical convergence, strictly speaking,it is not suitable for air when the temperature difference between the heated surface and thefar field is higher than 15 ◦ C [42, p. 14-15]. The computational model presented utilizes fullcompressible Navier-Stokes equations with temperature dependent thermophysical properties.Therefore, buoyancy effects in both phases together with the varying surface tension alongthe droplet surface (thermocapillarity) are simulated without any approximation, which makesthe present work unprecedented. Moreover, in order to assess the validity of these commonassumptions, the results of the proposed model is compared with several simplified models andcorrelations applying widely used assumptions in the literature .The article is organized as follows: in Section 2, modeling strategy is presented by providingdetailed information about the governing equations and associated boundary conditions in bothliquid and gas domains. In Section 3, the iterative computational scheme is explained step bystep. The resultant flow, temperature, and concentration fields for a certain test configurationare reported and elucidated with and without the presence of Marangoni convection for twodifferent substrate temperatures in Section 4. Moreover, the framework proposed is utilizedto make a benchmark test for the models utilizing simplifying assumptions in droplet evapo-ration modeling to question the validity of these common assumptions. Finally, summary andconclusions are presented in Section 5.
Steady evaporation from a steadily fed liquid droplet resting on a heated substrate to ambientair is modeled. As long as the capillary forces dominate the gravitational ones, the Bondnumber is smaller than unity and the droplet surface assumes a spherical shape. In our model, ahemispherical droplet is considered using a 2-D axisymmetric model as shown in Fig. 1. A largeair volume in the shape of a cylinder encloses the droplet and the flat substrate. Since steadilyfed droplets are considered in the thermal management of electronic components, material ofthe substrate (heat sink) is likely to be a thermally highly conductive metal, which causes anearly constant substrate temperature. Therefore, constant wall temperature is assigned to thesubstrate surface. Moreover, feeding liquid is assumed to be in thermal equilibrium with thesubstrate. 4 omprehensive modeling of steadily fed evaporating droplets
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Figure 1: Problem domain
In the liquid phase, normal component of the interfacial velocity determines the evaporatingmass flux ( ˙ m (cid:48)(cid:48) ev ): u (cid:96) · n = ˙ m (cid:48)(cid:48) ev ρ , (2.1)where n and ρ are unit normal vector and density of the liquid at the interface, respectively,and the subscript (cid:96) designates the liquid phase. When the value of evaporating mass flux isprovided, Eq. (2.1) can be used to determine the normal velocity of the liquid phase at theinterface. However, the same equation cannot be used to evaluate the interfacial gas velocitydue to the presence of additional air flow from the interface to the surrounding gas. This airflow, also known as Stefan flow, is determined by equating the diffusive air transport towardsthe interface and the convective air transport from the interface: D ( ∇ · n ) c air = ( u g · n ) c air , (2.2)where c air is the molar concentration of air and D is the binary diffusion coefficient; the subscript g designates the gas phase. Equation (2.2) is used to estimate the normal component of thegas velocity, i.e. u n = ( D/c air )( ∂c air /∂n ), in the current study. Tangential gas velocity, onthe other hand, is equal to the tangential liquid velocity [30, 43], which is determined fromthe solution of the governing equations in liquid domain. The primary factor determining the5 omprehensive modeling of steadily fed evaporating droplets Akkus et al. tangential liquid velocity is the tangential force balance at the interface.
At the interface between the liquid and gas phases, normal and tangential stress balances aregiven, respectively, as follows: n · ¯¯ τ g · n − n · ¯¯ τ (cid:96) · t = γ ∇ · n , (2.3a) n · ¯¯ τ g · n − n · ¯¯ τ (cid:96) · t = ∇ γ · t , (2.3b)where ¯¯ τ is the deviatoric stress tensor defined as µ ( ∂u i /∂x j + ∂u j /∂x i ), γ is the surface tension,and t is the unit vector in tangential direction. The normal stress balance expresses the interplayof normal forces associated with the pressures (including the Laplace pressure due to curvedinterface) and normal velocities in the two phases. Normal stress balance, however, is notimplemented in the model proposed by the current study for two reasons: the normal velocitiesin both phases are determined from the mass balance relations, and accurate pressure transitionbetween phases is not critical due to the fact that liquid and gas domains are solved separately.Application of tangential stress balance, on the other hand, is essential since the non-uniformdistribution of the interfacial temperature initiates thermocapillary (or Marangoni) convection.The shear stress induced by the gas phase is assumed to be much smaller than that of the liquidphase. The validity of this assumption is confirmed by an a posteriori analysis of the results.Therefore, the effect of gas shear on the interface force balance is neglected. Energy transfer from liquid interface to the ambient occurs via three mechanisms: (i) evapora-tive heat (mass) transfer, (ii) conduction to the gas phase, and (iii) radiation to the surround-ings. The resultant energy balance at the interface is provided below: n · ( − k l ∇ l ) = ˙ m (cid:48)(cid:48) ev h fg − n · ( − k g ∇ T g ) + σ(cid:15) ( T s − T surr ) , (2.4)where k , h fg , σ , and (cid:15) are thermal conductivity, latent heat of vaporization, Stefan-Boltzmannconstant, and emissivity of the liquid surface; T s and T surr are the temperatures of the interfaceand surroundings, respectively. Evaporative heat transfer dominates the others due to the highlatent of vaporization. Therefore, estimation of evaporative mass flux ( ˙ m (cid:48)(cid:48) ev ) has paramounteffect on the temperature and flow fields in both domains. Evaporative mass flux is determinedby the summation of diffusive and convective transfer of the vapor from the interface:˙ m (cid:48)(cid:48) ev = − D ( ∇ · n ) c v + ( u g · n ) c v , (2.5)6 omprehensive modeling of steadily fed evaporating droplets Akkus et al. where c v is the molar concentration of vapor. Therefore, estimation of evaporation flux requiresthe solution of the concentration field of vapor in the gas domain. Steady forms of the conservation equations for mass, linear momentum, and energy in bothliquid and gas phases together with species conservation equation for vapor transport in thegas domain are solved. Governing equations are summarized below: ∇ · ( ρ u ) = 0 (2.6a) ρ ( u · ∇ ) u = −∇ p + ∇ · ¯¯ τ + ρ g (2.6b) ρc p u · ∇ T = ∇ · ( k ∇ T ) + ¯¯ τ : ∇ u (2.6c) u · ∇ c v = ∇ · ( D ∇ c v ) (2.6d)where g is the gravitational acceleration; µ and c p are dynamic viscosity and specific heat,respectively. All fluid properties, including the density, are defined as temperature dependentand the values are taken from the material library of COMSOL Multi-physics software.Symmetry boundary condition is applied along the center line of the droplet and the sur-rounding air volume. No slip and constant temperature boundary conditions are used on thesubstrate. The velocity of the feeding liquid ( ¯u in ) is assumed to be uniform and the tem-perature of the feeding liquid is equal to wall temperature ( T w ). At the liquid-gas interfacemass, tangential force, and energy balances are secured as explained in Section 2.1. Boundaryconditions for the liquid domain are summarized below: ∂ φ u = 0; ∂ φ T = 0 at φ = 0 (2.7a) u = 0; T = T w at φ = π/ u = ¯u in ; T = T w at r = r i (2.7c) u · n = ˙ m (cid:48)(cid:48) ev /ρ, − n · ¯¯ τ · t = ∇ γ · t ; n · ( − k l ∇ T ) = ˙ m (cid:48)(cid:48) ev h fg − n · ( − k g ∇ T g ) + σ(cid:15) ( T − T surr ) at r = r o (2.7d)Evaporative mass flux ( ˙ m (cid:48)(cid:48) ev ) requires the solution of concentration and flow fields in the gasdomain. Therefore, evaporative mass flux together with the temperature distribution in the gasphase ( T g ) are unknown a priori . They are evaluated in an iterative algorithm whose detailsare given in Section 3. Temperature of the surroundings ( T surr ) is assumed to be equal to theambient gas temperature. 7 omprehensive modeling of steadily fed evaporating droplets Akkus et al.
In the gas phase, thermal and hydrodynamic boundary conditions identical to those ofthe liquid phase are utilized at the center line and substrate surface. In addition, no vaporpenetration condition is used at these boundaries. At the outer boundaries, gas temperature,gas pressure, and vapor concentration are assumed to reach their ambient values. At theliquid-gas interface, the phases are considered to be at thermal equilibrium; therefore, surfacetemperature estimated from the solution of liquid domain ( T s ) is assigned to the gas domain. Agas velocity distribution ( u s ) is assigned to the interface by combining its normal and tangentialcomponents. While normal component is calculated based on the Stefan flow (see Eq. (2.2)),tangential one is taken from the solution of the velocity field in the liquid domain. Boundaryconditions for the gas domain are summarized below: ∂ φ u = 0; ∂ φ T = 0; ∂ φ c v = 0 at φ = 0 (2.8a) u = 0; T = T w ; ∂ φ c v = 0 at φ = π/ p = p ∞ ; T = T ∞ ; c v = φ RH c v,sat at the outer boundaries (2.8c) u = u s ; T = T s ; c v = c v,sat at r = r o (2.8d)where φ RH is the relative humidity in the far field and c v,sat is the saturation concentration ofvapor at the corresponding temperature. Primary challenge in the modeling of droplet evaporation is the coupling of a condensed phasewith a gas phase. Transition between the phases should be delicately treated since the resultantflow, temperature, and concentration fields are shaped based on the dynamics at the interface.One way to mitigate the coupling problem can be the utilization of a proper two phase method,which deals with the two phases simultaneously, but this approach, undoubtedly, brings ahigher computational cost. An interface tracking method such as ALE (arbitrary Lagrangian-Eulerian) method can be beneficial in the modeling of a drying droplet, whose surface is incontinuous deformation [32]. However, in the problem of interest, droplets are steadily fedand preserve their interface shape without requiring a special treatment to track the interface.Therefore, the current study offers a modeling strategy, which handles liquid and gas domainsseparately but couples them at the interface properly.The proposed model utilizes an iterative computational scheme (Fig. 2), which enablesthe simultaneous solution of the steady forms of the mass, momentum, energy, and speciesconservation equations with temperature dependent thermo-physical properties. Governingequations are solved using the Finite Element Method (FEM) based solver of COMSOL. Linear8 omprehensive modeling of steadily fed evaporating droplets
Akkus et al. shape functions are used in the discretization of all variables during FEM formulation. Theiterative scheme is implemented using the interface, Livelink TM for MATLAB. Each step of theiterative process is explained in the following sections. Initial estimates of the boundary conditions at feeding surface ( r = r i ) and liquid-vapor interface( r = r o ) are assigned in the initialization step. A uniform velocity distribution is assumed atthe inlet based on the initial guess of total evaporation rate ( ˙ m totev ) from the droplet:¯ u oin = ( ˙ m totev ) o A in ρ , (3.1)where A in is the surface area of the inlet surface and the liquid density ( ρ ) is calculated at thesubstrate temperature ( T w ). The velocity at the inlet is assumed to be perpendicular to theinlet surface and it is confirmed that direction of the inlet velocity has no observable effect onthe results. Distribution of liquid velocity at the liquid-gas interface is assumed uniform in onlythe first iteration as the initial estimate and calculated based on the inlet-outlet area ratio asfollows: ¯ u os = ¯ u in A in A out , (3.2)where A out is the area of the droplet surface. Distribution of the interfacial liquid velocity isiterated on within the computational scheme in subsequent iterations. In a similar manner,a homogeneous heat flux is assigned to the interface based on the initial estimate of totalevaporation rate: ( ˙ q (cid:48)(cid:48) ev ) o = ( ˙ m totev ) o h fg A out . (3.3)This initial estimate is replaced in the following iterations by updating the distribution ofevaporative mass flux ( ˙ m (cid:48)(cid:48) ev ), which is calculated based on Eq. (2.5) after the solution of the gasdomain. Continuity, momentum, and energy equations, Eqs. (2.6a) to (2.6c), with the associated bound-ary conditions, Eqs. (2.7a) to (2.7d), are solved simultaneously for the liquid domain usingCOMSOL which enables the utilization of temperature dependent thermo-physical properties.In this step, solver is applied iteratively to implement both hydrodynamic boundary conditions:normal component of the liquid velocity and tangential stress balance (see Eq. (2.7d)). Sincethe solution of the gas domain is not available in Step-1 of the first iteration, temperature field9 omprehensive modeling of steadily fed evaporating droplets
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Figure 2: Flowchart of the computational schemeof gas near the interface is unknown. Therefore, interfacial heat conduction to the gas phase isomitted in the first iteration, but included in subsequent iterations.
Continuity, momentum, energy, and species transport equations, Eqs. (2.6a) to (2.6d), withthe associated boundary conditions, Eqs. (2.8a) to (2.8d), are solved simultaneously for thegas domain with temperature dependent thermo-physical properties. As stated previously,distributions of interfacial gas velocity and temperature are assigned based on the solutionof liquid domain and the concept of Stefan flow. Interfacial vapor concentration distributioncorresponds to the saturation concentration of vapor at the interfacial temperature, which isimplemented by setting the relative humidity to unity in the solver interface.10 omprehensive modeling of steadily fed evaporating droplets
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In Step-3, evaporative mass flux ( ˙ m (cid:48)(cid:48) ev ( φ )) is calculated based on the interfacial vapor concen-tration ( c s ( φ )) and its gradient in normal direction ( dc s ( φ ) /dn ) as described in Eq. (2.5). Usingthe evaluated evaporative mass flux and Eq. (2.1), normal component of the interfacial liquidvelocity is calculated and stored to be used in the next iterative step. In addition, velocity ofthe feeding liquid is also updated based on the calculated evaporative mass flux. These updatedliquid velocities together with the updated evaporative mass flux are introduced to Step-1 andthis iterative procedure is continued until the convergence of total evaporation rate from thedroplet.Embedded grid generator of COMSOL is utilized to mesh the computational domain. Theboundary between the liquid and gas domain, i.e. the interface, is divided to equal length arcsbefore meshing. Grid generation starts at the interface on these arcs and continues towards theliquid and gas domains with a certain growth rate. Resolution of the solution is controlled byadjusting the number of boundary elements ( i.e. arcs), which directly determines the densityof the resultant mesh at the interface. Mesh independence test is carried out for all casessimulated by increasing the number of boundary elements, thereby increasing the number ofmesh elements. Grid-independence is decided to satisfy when the change in evaporation rate isless than 0.1%. The proposed computational framework is tested to model the evaporation from a hemispheri-cal, sessile, and continuously fed water droplet placed on a heated substrate as shown in Fig. 1.Two cases with different wall temperatures are simulated. Superheat values—i.e. differencebetween wall and ambient temperatures—are selected as 9 ◦ C (Case-1) and 44 ◦ C (Case-2) todemonstrate the effect of increasing substrate temperature. Moreover, both values are suffi-ciently high such that utilization of Boussinesq approximation is rendered questionable in themodeling of the buoyancy in the flow solution. The configurations in the simulated two casesmatch the previous theoretical [26] and experimental [41] studies enabling a direct compari-son. The values of the geometric parameters together with far field conditions are provided inTable 1. The Bond number is calculated as 0.28 confirming the spherical shape of the dropletsurface. Emissivity of the water surface is taken as 0.97 [44]. Binary diffusion coefficient ofwater vapor in air is estimated based on the temperature dependent formulation suggested in[45]. For all other thermophysical properties, temperature dependent values are assigned usingthe material library of COMSOL.The model proposed is also employed to make a benchmark test for the models utilizingsimplifying assumptions in droplet evaporation modeling. The comprehensive model (full-11 omprehensive modeling of steadily fed evaporating droplets
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Table 1: Geometrical parameters and far field conditionsFeeding opening radius (mm) r i r o W H ∞ φ RH ◦ C) T ∞ ◦ C) T w ◦ C) T w FM-1 . Widelyutilized Boussinesq approximation (with temperature dependent properties excluding density)is simulated only in gas phase by reflecting the effect of density change on the body forceterm solely instead of temperature dependent density setting set throughout the domain. Thismodel is called as
FM-2 . A common approach adopted in the literature is undoubtedly theone accounting for only diffusion of heat and vapor in gas phase because of the its relativelylow computational cost; however, it inevitably lacks from the omission of convection transport.To demonstrate the applicability of this approach, simulations are carried out for the diffusionbased model ( DM hereafter). Moreover, the widely used correlation of Hu and Larson [33] isemployed to calculate the evaporation rates to exhibit its validity. The last model tested is theone based on a natural convection correlation for a sphere hanging in air environment, whichwere previously adopted in [12, 17, 26, 39]. Simulations with these models and correlations areperformed and evaporation rate predictions of them are summarized in Table 2.Results demonstrate that full-models ( FM-1 and
FM-2 ) solving the gas flow predict higherevaporation rates compared to those of others. Predictions of
FM-2 are close to those of
FM-1 but start to deviate with increasing substrate temperature due to the incapability of Boussinesqapproximation in capturing the gas flow field in elevated temperature differentials. AlthoughStefan flow, thereby the convective mass flux at the interface, is considered, DM severelyunderestimates evaporation rates due to the absence of convective transport in the gas phase.Correlation proposed by [33] is based on the diffusion of vapor and its prediction is expectedto be similar to that of DM . However, predictions of them are similar only in the case ofMarangoni flow driven droplet. This result is not surprising since the correlation given in [33]employs constant interface temperature equal to the substrate temperature and temperatureof the droplet surface is closer to the substrate temperature in the presence of Marangoniflow [20, 26]. On the other hand, natural convection correlation based model substantiallyunderestimates the evaporation rate. The poor performance of this model is primarily due toits failure to capture the flow field in the gas. In the problem of interest, buoyancy of gas is12 omprehensive modeling of steadily fed evaporating droplets Akkus et al.
Table 2: Evaporation rate (in µ g/s) estimation of different models w/o Marangoni w/ Marangoni Case-1 Case-2 Case-1 Case-2Full-model (
FM-1 ) 19.4 135.4 24.5 238.4Full-model with Boussinesq appr. in gas (
FM-2 ) 19.4 116.2 22.9 200.0Diffusion (in gas) based model ( DM ) 15.0 87.2 16.8 108.6Diffusion (in gas) based correlation of Hu&Larson [33] 17.2 113.3 17.2 113.3Natural conv. (in gas) correlation based model 12.1 67.8 13.4 81.3triggered by the heated flat substrate, whereas the natural convection correlation is built on aflow field triggered by a sphere hanging in air. Therefore, this model is not applicable in anaccurate modeling of the evaporation of a hemispheric droplet since it refers to a geometricallydifferent configuration.Although simulations are performed for two cases with different superheats, resultant flow,temperature and concentration fields are reported for only one case (Case-2) in the rest of thisstudy, since the patterns are similar. Figures 3 and 4 show the temperature and flow fieldsof FM-1 . Figure 5 exhibits the same fields obtained using DM . Whilst Fig. 6 and 7 showthe the distribution of heat flux and temperature along the droplet surface for all models,respectively, Fig. 8 focuses on the temperature patterns near the contact line arising in theabsence of Marangoni flow.A typical natural convection pattern is obtained in the gas domain as a result of FM-1 asshown in Fig. 3a, where air enters the domain from the periphery and moves upwards alongthe axis of the domain due to being heated by the hot wall. This pattern is also quite similarto the results of
FM-2 . However, not only liquid domain but also gas region near the dropletsurface has substantially different resultant patterns based on the presence of thermocapillaryflow. When Marangoni effect is included in the model, a strong surface flow in the direction ofmonotonically decreasing temperature ( i.e. from the substrate to the apex) results in a maincounter clockwise (CCW) vortex (see Fig. 3b) pattern, which is in agreement with previousstudies reporting a similar pattern for water drops with contact angles close to 90 ◦ [12, 21, 23].Although the values of Marangoni numbers are much higher than the threshold value [46] for thecases considered in the current study, the results of the model without Marangoni effect are stillreported to demonstrate the underlying physics in the absence of thermocapillary flow, whosepresence has been contentious in the literature [15, 47] for water droplets and films in air. InFig. 3c, as opposed to the case with Marangoni flow, a clockwise (CW) vortex pattern appearsas a result of the buoyancy of liquid similar to the findings of previous studies [20, 26, 37]. It13 omprehensive modeling of steadily fed evaporating droplets Akkus et al.
T(ºC) a) b) + c) T(ºC)706560555045403530
Figure 3: Temperature field and streamlines by
FM-1 a) in the entire computational domain, b) in droplet and gas region near the droplet surface with thermocapillarity, and c) withoutthermocapillarity.should be noted that buoyancy flow of liquid is dominated by Marangoni flow (Fig. 3b) whenthermocapillarity effect is accounted for in accordance with the previous studies [20, 23, 26].Temperature field also substantially differs based on the presence of thermocapillarity. Whenpresent, the temperature inside the liquid as well as the gas temperature near the surfaceincreases suggesting a higher convective transport of the energy, which can be understoodbetter by examining the strength of the internal liquid flow. a) b)
0 0.004 0.008 0.012u(m/s)0 0.02 0.04 0.06 0.08 0.10u(m/s) c v (mol/m )
2 4 6 8 10 12 c v (mol/m )
2 4 6 8 10 12 droplet (liquid domain) droplet(liquid domain)
Figure 4: Vapor concentration field (left image) and velocity magnitude field with streamlines(right image) by
FM-1 a) with and b) without thermocapillarity. Note that scale bars in a) and b) are different for both vapor concentration and velocity magnitude fields.Figure 4 shows the distribution of the magnitude of the flow velocity in both liquid and gasdomains with and without Marangoni flow together with the distribution of vapor concentrationin gas domain, which is presented as the mirror image at left sides of the figures. Results14 omprehensive modeling of steadily fed evaporating droplets Akkus et al. clearly exhibit that Marangoni flow is much stronger than buoyancy flow explaining the elevatedamount of heat transport (convective) from the substrate to the interface. Gas flow near theinterface is also affected by the interface velocity. When thermocapillary flow is effective, gasflow is in the same direction with the interface velocity, which results in the upward accelerationof gas along the interface (Fig. 4a). On the contrary, when thermocapillary flow is absent,buoyancy driven surface flow (from apex to the wall) is against the direction of gas flow, whichdecelerates the gas near the interface leading to the bending of the streamlines as shown inFig. 4b. Moreover, in the case of buoyancy driven internal liquid flow, Stefan flow of the gasmixture originating from the drop surface is apparent. This flow intensifies near the contactline due to the substantially increased rate of evaporation by manifesting a velocity jet seenin Fig. 4b similar to the finding of [37]. Distributions of vapor concentration in Fig. 4 areconsistent with the flow fields. In the case of Marangoni flow, tangential upward movement ofthe gas flow carries the vapor to the apex region, where concentration isolines are distorted inthe upward direction due to the accumulation of vapor. This accumulation is less apparent inthe absence of thermocapillary flow since Stefan flow aids transportation of vapor in the radialdirection. In addition, the elevated evaporation rate results in the denser vapor zone near thecontact line as seen in Fig. 4b.Although gas flow has a dominant effect on evaporation rate, diffusion based models arefrequently applied for the estimation of evaporation rate. When the gas flow is not taken intoaccount, conduction heat transfer becomes effective in energy transport resulting in a stratifiedtemperature distribution in the gas domain as seen in Fig. 5a. Due to the decrease in evapora-tion rate, evaporative cooling is also reduced, which leads to warmer drops (Fig. 5b and c) thanthe ones obtained in the presence of gas flow (Fig. 3b and c). Although the flow patterns ofliquid inside the droplet are unaffected, distribution of vapor concentration inevitably differs inthe absence gas flow. This difference is especially apparent in the case with Marangoni convec-tion. Instead of elongated isolines, stratified distribution of vapor concentration shows the lackof convective vapor transport in Fig. 5b. Likewise, in the case of buoyancy driven liquid flow,less distorted concentration isolines form as seen in Fig. 5c. However, non-uniformity of vaporconcentration near the interface is still present, which can be linked to the non-uniformity ofevaporation distribution along the interface. A better understanding requires the examinationof heat flux distribution along the interface.Regardless of the presence of thermocapillarity, substrate temperature or the model utilized,interfacial heat flux reaches a maximium at the contact line as seen in Fig. 6. However, theincrease of heat flux towards the contact line is monotonic in the presence of Marangoni flow(Fig. 6a), whereas it is nonmonotanic in the case of buoyancy driven internal liquid flow (seeFig. 6b). Although both
FM-1 and
FM-2 exhibit a near linear variation (except ∼ ◦ portionadjacent the contact line, where flux values increase rapidly) in the case of Marangoni flow, FM- omprehensive modeling of steadily fed evaporating droplets Akkus et al.
69 70 71 72 73 74T(ºC)c v (mol/m )
2 4 6 8 10 12 droplet(liquid domain) a) b) droplet(liquid domain) c) + T(ºC)7065 Figure 5: Results of DM . a) Temperature field in the complete domain. Vapor concentrationfield (left image) and temperature field with streamlines (right image) b) with and c) withoutthermocapillarity. predicts higher heat flux along the interface than FM-2 . DM predicts near uniform (exceptfor ∼ ◦ portions adjacent to the apex and contact line) interfacial flux and it underestimatesthe heat flux along the interface (except ∼ ◦ portion adjacent to the apex) with respect to FM-1 when Marangoni flow is present. This underestimation increases towards the contactline, which is in accordance with the literature [31]. While Case-1 yields lower interfacial heatflux values as expected (the inset of Fig. 6a), both cases result in similar distribution patternsalong the interface.In the case of buoyancy driven liquid flow, all models have a slightly decreasing heat fluxdistribution starting from the apex. However, substrate temperature greatly affects the trend ofthe distributions approaching the contact line. Case-1 yields a minimum flux point around theangular position of 70 ◦ in all models. While DM underestimates the flux values throughout thesurface with respect to full-models ( FM-1 and
FM-2 ), resultant flux distribution of full modelsare nearly identical. In Case-2, DM yields a similar flux pattern to the one in Case-1. However,full models exhibit two dips and one peak in between the dips before the final rise at the contactline, a behavior requiring a close examination. Another conspicuous result is the prediction ofnegative flux values by full-models. Negative flux means heat transfer from ambient to thedroplet. Although perplexing, this result is understandable since the surrounding bulk gas flow16 omprehensive modeling of steadily fed evaporating droplets Akkus et al. a) b)
Angular position (º) Angular position (º) I n t e r f a c i a l hea t f l u x ( W / c m ) -0.50.00.5 Case-1,
FM-1
Case-1,
FM-2
Case-1, DM Case-2,
FM-1
Case-2,
FM-2
Case-2, DM Figure 6: Interfacial heat flux distribution predictions of the models a) with and b) withoutthermocapillarity.is warmer than the ambient due to the natural convection pattern shown in Fig. 3a, whichin turn, leads to a net conduction heat transfer from the gas to the interface. Wheneverthis conduction heat transfer becomes larger than the total of evaporative and radiative heattransfer, net interfacial flux changes its sign.In the presence of Marangoni flow, temperature distribution predictions of all models forboth cases are similar: monotonically increasing temperature towards the contact line as shownin Fig. 7. Since the resultant evaporation rate of FM-1 is higher than that of the others, this a) Angular position (º) Angular position (º) I n t e r f a c i a l t e m pe r a t u r e ( º C ) b) Case-1,
FM-1
Case-1,
FM-2
Case-1, DM Case-2,
FM-1
Case-2,
FM-2
Case-2, DM Figure 7: Interfacial temperature distribution predictions of the models a) with and b) withoutthermocapillarity.model predicts the lowest apex temperature. In the case of buoyancy driven liquid flow, atemperature dip appears in both substrate temperatures but it is more apparent for highersubstrate temperature. Presence of temperature dip was previously reported in experimental[41] and numerical [26] studies. However, in the current study, FM-1 predicts two temperature17 omprehensive modeling of steadily fed evaporating droplets
Akkus et al. dips, which, to the best of the authors’ knowledge, has never been reported in the literature.Reason of the formation of two dips is due to the formation of a local temperature peak in theregion of minimum temperature and this local peak can be attributed to the heating effect dueto conduction heat transfer from the gas. However, local temperature peak does not appear asa result of t
FM-2 . In order to elucidate this, the local liquid temperature distribution togetherwith the local velocity magnitude distribution in the gas phase are plotted in Fig. 8.In the buoyancy driven liquid flow, internal convection pattern drives the warm liquid fromsubstrate to the apex, then it moves towards the contact line along the interface, during whichthe liquid cools down due to the evaporative heat loss. However, conduction heat transfer fromhot substrate starts to heat the liquid near the contact line, which yields an intermediate coolerzone above the substrate (these temperature dips are made more apparent by adjusting thecolor ranges in Fig. 8). However, as shown in Fig. 8a,
FM-1 exhibits a secondary temperaturedip between the primary one and the substrate due to the local heating, which arises because ofthe suppression of evaporation. White arrow in Fig. 8a indicates the region with substantiallylow velocities. This region appears due to the presence of a local vortex which forms under theeffect of strong Stefan flow originating from the contact line region. Local vortex suppresses theconvective transport of vapor from the interface, which reduces the evaporation rate, and con-sequently, the evaporative cooling. Therefore, the liquid is subjected to an immediate heating.After passing the vortex zone, strong Stefan flow suddenly enhances the evaporation resultingin the secondary temperature dip. Following the dip, conduction from the hot substrate raisesthe interfacial temperature. Secondary dip is not shown in the results of
FM-2 (see Fig. 8b)due to the location of vortex region, which is closer to the contact line. Despite the elevatedevaporative cooling beyond the vortex zone, heat transfer from the substrate heats up the liquidnear the contact line. Consequently, a secondary dip cannot form. Different vortex locationpredictions of full-models are linked to the different flow fields in the gas phase. While
FM-2 predicts a gas flow mainly in longitudinal direction,
FM-1 is able to capture the rise of the gasimmediately, which pushes the vortex zone above compared to
FM-2 . DM , on the other hand,always predicts a single temperature well when the internal liquid flow is buoyancy driven. A theoretical framework is introduced to model the steady droplet evaporation with all relevantphysics and a test case—evaporation from a continuously fed constant shape hemisphericalwater droplet resting on a heated flat substrate—is solved. Computational performance of theproposed model, which utilizes temperature dependent thermophysical properties, is comparedwith several simplified models adopting widely used assumptions in the literature. Simulationsdemonstrate that the comprehensive model introduced in the current study (
FM-1 ) yields18 omprehensive modeling of steadily fed evaporating droplets
Akkus et al.
T(ºC)64.0 64.2 64.4 64.6 64.8 65.0 u(m/s)
T(ºC)65.0 65.2 65.4 65.6 65.8 66.0 u(m/s) a) b)
T(ºC)66.0 66.2 66.4 66.6 66.8 67.0 c) Figure 8: Temperature distribution in liquid and velocity magnitude distribution in gas phasenear the contact line with superimposed streamlines as the result of a) FM-1 , b) FM-2 , and c) DM . White arrows show the zone, where a local vortex leads to small velocity magnitudes.Note that there is no gas flow field in the last plot since DM omits the solution of gas flow.the highest evaporation rates compared to other simplified models. When the effect of gasconvection is excluded by employing a diffusion based model in the gas phase, evaporationrates are underestimated by 23–54% with respect to FM-1 . Implementation of Boussinesqapproximation in the gas phase yields the underestimation of evaporation by 14–16% for thehigh substrate temperature cases (superheat value of 44 K), while underestimation is less than6% in low substrate temperature cases (superheat value of 9 K). Deviations of simplified modelsincrease not only with increasing substrate temperature but also with the presence of Marangoniflow (up to 50% increase with respect to the case with buoyancy driven liquid flow for the highsubstrate temperature). In the absence of Marangoni flow, all models predict a temperaturewell zone at the interface but the comprehensive model identifies a secondary dip in this zone,which is reported for the first time in the literature. Based on these findings conclusions of thepresent study are: • Boussinesq approximation fails to capture the physics in the gas phase for the caseswith elevated substrate temperatures, which manifests the need of using temperaturedependent thermophysical properties in such cases. • Diffusion limited approaches in the modeling of gas domain are inadequate in simulatingthe physics since they lead to substantial discrepancies in the prediction of evaporationrates. • Utilization of natural convection or diffusion based correlations in the estimation theinterfacial evaporation flux without solving the gas phase is not appropriate since theevaporation flux is highly sensitive to the physics of the gas flow near the interface, whichis specific to the instantaneous configuration of the problem such as contact angle of thedroplet, substrate temperature or the concentration of vapor in ambient air.19 omprehensive modeling of steadily fed evaporating droplets
Akkus et al. • Physics of the fluid in one phase inevitably affects the other phase as long as propercoupling is established at the interface. For instance, convection mechanism inside thedroplet (buoyancy or Marangoni driven) regulates the interfacial velocities, which deter-mines the flow field of gas near the interface. Likewise, natural convection pattern of gashas paramount effect on the evaporation rate of the droplet. Therefore, omission of aphysical phenomenon in one of the two phases cannot be justified while being consideredin the other phase.Theoretical framework established in this study is demonstrated to be effective in creatingmodels of steady droplet evaporation, which are able to capture unreported physics of dropletevaporation. Using this framework, our future work will focus on studying the evaporationfrom droplets with various contact angles and sizes in different environmental and substrateconfigurations with the ultimate objective of maximizing cooling rates in droplet based thermalmanagement applications.
Acknowledgments
B.C¸ . would like to acknowledge fundings from the Turkish Academy of Sciences through Out-standing Young Scientist Program (T ¨UBA-GEB˙IP) and The Science Academy, Turkey throughTurkey Distinguished Young Scientist Award (BAGEP).
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