Thermal transients during the evaporation of a spherical liquid drop

International Journal of Heat and Mass Transfer 97 (2016) 803–817

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Thermal transients during the evaporation of a spherical liquid drop P. Talbot, B. Sobac ⇑, A. Rednikov, P. Colinet, B. Haut Université libre de Bruxelles, TIPs (Transfers, Interfaces and Processes), C.P. 165/67, av. F.D. Roosevelt 50, 1050 Brussels, Belgium

a r t i c l e

i n f o

Article history: Received 27 July 2015 Received in revised form 13 November 2015 Accepted 4 December 2015 Available online 12 March 2016 Keywords: Drop Evaporation Thermal transients Relaxation Homogenization

a b s t r a c t This work is centered upon the thermal transients taking place during the evaporation of a spherical drop of a pure liquid suspended in a gaseous environment. Based on mass and energy conservation equations, a so-called complete model is developed considering quasi-steady diffusive and Stefan convective transports in the non-isothermal gas phase, and unsteady conduction in the liquid drop. A simplified version of the complete model, the so-called quasi-homogeneous model, is developed using an asymptotic analysis in the limit of small thermal homogenization time in the drop compared to the total drop evaporation time. The models enable highlighting the role of two dimensionless numbers, R and H, characterizing the two thermal transients of the problem: the thermal relaxation transient of the drop interfacial temperature and the thermal homogenization transient of the drop. The values of these two dimensionless numbers are provided for several liquids and their dependence on the evaporation conditions is discussed. It is shown that, when an accurate evaluation of the drop evaporation time is required by the considered application, the use of a fully quasi-steady model should be restricted to systems presenting small values of R compared to one (at least an order of magnitude smaller) and H < 1. For other systems, it appears necessary to use the complete model or the quasi-homogeneous model. A simple formula is proposed to evaluate the relative difference between the drop evaporation times predicted by the complete model and by the fully quasi-steady model. When an accurate evaluation of the time evolution of the drop temperature field is required by the considered application, it appears to be necessary to use the complete model, whatever the system considered in this work. Indeed, the thermal transients can generally take an important part of the drop evaporation time and large temperature gradients can develop in the drop. The use of the complete model reveals that three different types of dynamics can be observed when a drop evaporates, depending on the relative values of three temperatures: the initial drop temperature, the dew point temperature of the gas far from the drop and the established interfacial temperature (i.e. the drop interfacial temperature calculated using a fully quasi-steady model). Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Maxwell, in 1877 [1], proposed the first mathematical model describing the evaporation of a spherical liquid drop suspended in a gaseous environment. The problem stated by Maxwell concerns a simplified system composed of isothermal liquid and gas phases, at fully quasi-steady state, and with a purely diffusive evaporation flux. Subsequent works published in various research fields (combustion, sprays, powders, crystallization. . .) tried to describe more realistic systems by relaxing Maxwell’s hypotheses. The aerosol physicist Fuchs [2] introduced, in 1959, a convective evaporation flux due to Stefan flow in the Maxwell’s model. At the same time, models including thermal transients in the drop were developed in the aerospace and combustion sectors [3–9]. ⇑ Corresponding author. E-mail addresses: [email protected] (B. Sobac), [email protected] (B. Haut). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.12.075 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.

In 1975, Hubbard et al. [10] proposed a model taking into account thermal gradients in the liquid and gas phases. Such a model posed the question of considering homogeneous liquid and gas physicochemical properties. Subsequent other works analyzed the validity of Maxwell’s assumptions for the modeling of drop evaporation [11–14]. However, in all of the works mentioned above, the results are presented for specific practical studied cases, where the components of the system, the gas composition and temperature and/or the pressure are set. In contrast, in one of our previous works [15] and in the present paper, a comprehensive analysis of the evaporation of a suspended spherical drop of a pure liquid in a gas phase is proposed, considering various liquids and a wide range of gas compositions and temperatures. In our aforementioned previous work [15], based on mass and energy balance equations, a fully quasi-steady model (with all relevant time scales being much smaller than the evaporation time) has been developed, including diffusive and convective transports,

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Notations Roman symbols Bi Biot number, – c molar density or concentration, mol/m3 C p heat capacity, J/(mol K) reduced heat capacity, – Cp Dv a diffusion coefficient of vapor in air, m2/s H thermal homogenization number, –  j local molar evaporation flux, mol/(m2 s) L reduced latent heat of vaporization, – L latent heat of vaporization, J/mol Le Lewis number, – M reduced molar mass, – b M molar mass, kg/mol P pressure, Pa r radial coordinate, m R drop radius, m R thermal relaxation number, – b R universal gas constant, J/(mol K) St Stefan number, – t time, s T temperature, K u mass-averaged velocity of the gas mixture, m/s molar-averaged velocity of the gas mixture, m/s u V volume, m3 X mole fraction of vapor in the gas phase, – Greek symbols a thermal diffusivity, m2/s b accommodation coefficient, –

and considering the non-isothermia of the gas phase. The main original feature of this simple model lied in the consideration of the local dependence on the temperature of the physicochemical properties of the gas. It appeared that this inhomogeneity of the gas physicochemical properties can have a significant influence on the evaporation rate, especially at high ambient temperatures. Simplified versions of the model were developed in order to highlight the key mechanisms governing the evaporation process. Concerning heat transfer, the model highlighted that a gas isothermia assumption never appears to be valid for the considered conditions, even at room temperature, due to the large temperature gradient that develops in the gas phase. The assumptions of the model were rigorously discussed and general criteria were established, independently of the liquid–gas couple considered. It appeared that considering drop isothermia with an established drop temperature (i.e. the drop temperature calculated using a fully quasisteady model) remained the most dubious assumption. Therefore, in the continuity of this previous work, the present paper focuses on the thermal transients within a drop during its evaporation. When evaporating into a gas, a drop is initially subject to two simultaneous thermal transients. One thermal transient is the thermal relaxation transient of the drop interfacial temperature which corresponds to the change of the drop interfacial temperature from its initial value to its established value, i.e. the drop interfacial temperature calculated using a fully quasi-steady model. The duration of this thermal relaxation transient is called the thermal relaxation time of the drop. The other initial thermal transient corresponds to the thermal homogenization of the drop. This transient is hereafter referred to as the thermal homogenization transient and its duration is called the thermal homogenization time of the drop. Regarding these two transients, different degrees of simplification are encountered in the models of drop evaporation proposed

D



k

l q s

difference, – relative difference, – thermal conductivity, W/(m K) dynamic viscosity, Pa s density, kg/m3 dimensionless time, –

Subscripts 0 initial conditions a air b boiling crit critical dp dew point ev evaporation g gas phase hom thermal homogenization he heat exchange i drop interface ‘ liquid m mass rel thermal relaxation qs quasi-steady t time v vapor 1 far from the drop Superscripts max maximum

in the literature. The simplest models are fully quasi-steady models, considering a drop of constant and uniform temperature [1,2]. Some authors looked at the temperature distribution inside a drop of constant interfacial temperature [5]. Other authors took into account the thermal relaxation transient but consider a homogeneous temperature inside the drop [3,7,13] to simplify their model. Finally, the most complete models consider both thermal transients [6,8–11]. According to different authors, the thermal transients occurring during the evaporation of a drop are damped in less than 10% [7,9] or 20% [8] of the drop lifetime. Results presented in these papers also showed that the evaporation time can be underestimated when the thermal transients are not considered. For instance, considering the evaporation of a homogeneous drop of hydrocarbons, Williams et al. [3] showed that neglecting the thermal relaxation transient can lead to an underestimation of the evaporation time by up to 20%. In contrast, Law et al. [9] concluded that the evaporation time of an octane drop is only slightly underestimated when the thermal transients are not considered in a model of its evaporation. No author mentioned that the evaporation time could be overestimated by neglecting the thermal transients in a model. Moreover, in these different works, no criteria was proposed to enable predicting easily whether or not the thermal transients should be considered in a model of an evaporating drop, depending on the considered case. Therefore, a wider point of view about the thermal transients occurring during the evaporation of a liquid drop is needed, in order to fully analyze their influence on the evaporation process of the drop. Such a study implies a comprehensive analysis of the involved physical phenomena and of their mathematical formulation. In this work, we propose a model of a spherical drop evaporation taking into account both thermal transients in the drop.

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A well-formulated dimensionless model is presented. Simplified solutions are developed for limiting cases of the problem, leading to the establishment of characteristic times. Criteria coming out of these models allow predicting whether or not the thermal transients should be considered in the modeling of the evaporation of a liquid drop, depending on the situation considered.

2. Statement of the problem 2.1. System description and assumptions We consider a suspended spherical pure-liquid drop (species A) evaporating into the surrounding gas phase of infinite extent. By ‘‘suspended”, we mean that there is no global relative motion between the drop and the gas. The gas phase is a mixture of air (assumed not to be soluble in the liquid) and of the species A. The species A in the gas phase is hereafter referred to as the vapor. The system is schematically represented in Fig. 1. Unless otherwise specified, the validity of the assumptions to be introduced here is discussed in Sobac et al. [15]. They are checked to be satisfied for the results presented in the present paper. The gas phase is assumed to be an ideal mixture of ideal gases; the total gas pressure (Pg ) is the sum of the vapor partial pressure (P v ) and the air partial pressure (Pa ). The gas molar density (cg ) is the sum of the vapor molar concentration (cv ) and the air molar b g (n ¼ g; v or a), with T g concentration (ca ). We have P n ¼ cn RT

b the universal gas constant. Diffuthe temperature of the gas and R sive and Stefan convective heat and mass transports are taken into account in the gas phase and considered to be quasi-steady therein (large diffusivities). The gas physicochemical properties are taken to be time-independent and homogeneous and are just evaluated as functions of the gas composition and temperature at r ¼ 1. For concrete numerical results presented in this paper, the gas physicochemical properties are evaluated using data and equations given in [15–17]. The liquid phase is present in the form of a spherical pure-liquid drop of radius R. Unsteady conductive heat transfer is considered

within the liquid drop. Hence, the two thermal transients mentioned earlier, the thermal relaxation transient and the thermal homogenization transient, are both accounted for in the developed model. The liquid physicochemical properties are considered timeindependent and homogeneous. They are evaluated at the drop initial temperature. These assumptions are discussed later. The liquid–gas interface is considered at local equilibrium, i.e. the gas phase in contact with this interface is saturated with vapor. The latent heat of evaporation is taken constant and evaluated at the initial drop temperature. Radiative heat transfer is neglected. Moreover, the Kelvin’s effect (i.e. vapor pressure dependence on the interface curvature) is neglected. Finally, the total gas pressure, Pg , is considered to be homogeneous and constant, and equal to the ambient atmospheric pressure (P g ¼ 1 atm). 2.2. Equations The problem is written using a molar formulation. The local 



molar flux of evaporation at the interface is written j ¼ j r, where r is the unit radial vector. Vapor is transported into air by diffusion and Stefan convection. Hence, the use of the Fick’s law of binary diffusion [18] allows writing the following equation at the liquid–gas interface: 

j ¼ cv ðuv  ui Þ ¼ cg Dv a rX þ cv ðu  ui Þ

ð1Þ

_ is the receding velocity where uv ¼ uv r is the vapor velocity, ui ¼ Rr of the interface due to evaporation (R being the drop radius), Dv a is the diffusion coefficient of the air–vapor mixture and X ¼ cv =cg is the vapor mole fraction in the gas phase. u ¼ u r is the molaraveraged velocity of the gas, defined as follows:

u ¼ ua ð1  XÞ þ uv X

ð2Þ

where ua ¼ ua r is the air velocity. The steady state mass balance equation writes:

r:ðcg u Þ ¼

1 dðr2 cg u Þ ¼0 r2 dr

ð3Þ

Therefore, u ¼ n=ðcg r2 Þ, where n is an unknown coefficient. At the liquid–gas interface (r ¼ R), the following mass balance equations can be written: 

j ¼ c‘ ui ¼ cg ðu  ui Þ

ð4Þ

where c‘ is the liquid molar density. As cg  c‘ , it implies that 



j ’ cg u at r ¼ R, yielding n ¼ j R2 and

u ¼



j R2 cg r 2

ð5Þ

The steady state mass balance equation for the vapor in the gas phase writes: r:ðcv u  cg Dv a rXÞ ¼ 0. Using Eq. (5), this equation can be rewritten as:

   j R2 dX 1 d 2 dX ¼ c D r g v a dr r 2 dr r 2 dr

ð6Þ

The boundary conditions regarding mass transport in the gas phase are:

Fig. 1. Schematic representation of a suspended drop in an air–vapor mixture. T ‘ and T g are the drop and the gas temperature, respectively. T i ; T 1 and X i ; X 1 are the temperature and the mole fraction of vapor in the gas phase at the drop interface and at r ¼ 1, respectively. The time variable is grayed out where it appears just as a parameter on quasi-steady profiles.

Xðr ! 1Þ ¼ X 1

ð7Þ

Xðr ¼ RÞ ¼ X i

ð8Þ

 cg Dv a dX   j ¼ 1  X i dr r¼R

ð9Þ

Eq. (9) is obtained by the combination of Eqs. (1) and (4). Under the assumption of local liquid–gas equilibrium at the drop interface, the vapor mole fraction in the gas phase at the

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interface is here expressed as a function of the interfacial temperature (T i ) using the Clapeyron’s equation:

Using Eq. (21) in the drop mass balance equation (Eq. (18)) gives a differential equation for the drop radius time evolution:

ð10Þ

ð22Þ

   L 1 1 X i ¼ exp   b Ti Tb R

where T b is the boiling temperature at the atmospheric pressure P g and L is the molar latent heat of vaporization of the liquid. The steady state energy balance equation in the gas phase (r P R), including heat diffusion and Stefan convection, writes:

cg C p g u

  dT g 1 d dT g kg r2 ¼ 2 r dr dr dr

ð11Þ

where C p g and kg are the molar heat capacity and the thermal conductivity of the gas, respectively. In this equation, heat is advected by the mass averaged velocity of the gas u. For the present radial Stefan flow, this velocity is directly related to u as follows: b v =M b g Þu (see [15]), where M b g ¼ ð1  X 1 Þ M b a þ X1 M b v is the u ¼ ðM

b a the molar mass of the air and M bv molar mass of the gas (with M the molar mass of the vapor) evaluated at X ¼ X 1 just like other gas properties (see Section 2.1). The transient energy balance equation inside the drop (0 6 r 6 R), considering heat conduction as the only mechanism of heat transfer (indeed, the velocity of the liquid is zero), is expressed, in spherical coordinates, as:

  @T ‘ 1 @ @T ‘ c‘ C p ‘ k‘ r2 ¼ 2 r @r @t @r

ð12Þ

where T ‘ ; C p ‘ and k‘ are the temperature, the molar heat capacity and the thermal conductivity of the liquid, respectively, and t is the time. The heat transfer boundary conditions are:

T g ðr ! 1Þ ¼ T 1

ð13Þ

T g ðr ¼ RÞ ¼ T ‘ ðr ¼ RÞ ¼ T i  @T ‘  ¼0 @r r¼0   @T ‘  @T g   ¼ k  j L k‘ g @r r¼R @r r¼R

ð14Þ ð15Þ ð16Þ

Eq. (16) is the energy balance equation at the interface. A global mass balance on the drop writes:

  dR 1 cg Dv a 1  Xi ln ¼ dt R c‘ 1  X1

in which X i (dependent on T i through Eq. (10)) is still unknown, however. The thermal conductivity of the gas being considered homogeneous, it can be taken out of the derivative in the energy balance b v =M b g Þu equation in the gas phase (Eq. (11)). Then, using u ¼ ð M and Eq. (5), the gas phase energy balance equation becomes:

b g cg ag d  dT g  dT g M r2 ¼ b v j R2 dr dr dr M

ð23Þ

where ag ¼ kg =ðcg C p g Þ is the thermal diffusivity of the gas. The solution of Eq. (23) satisfying the boundary conditions specified in Eqs. (13) and (14) is:

T1  Ti  T g ðrÞ ¼ T 1  bv 1  exp  M M bg



C p g j R

 1  exp 

kg

b v C p g j R2 1 M bg r kg M

!!

ð24Þ Introducing Eq. (24) in the energy balance equation at the interface (Eq. (16)) gives:

k‘

 @T ‘  ¼ @r r¼R

bv   M T T   1   i C p g j  j L Cp g j R b M b M 1 g exp v kg M bg

ð25Þ

As the liquid thermal conductivity is assumed constant, the energy balance equation within the drop (Eq. (12)) can be rewritten as:

  @T ‘ a‘ @ @T ‘ r2 ¼ 2 @t r @r @r

ð26Þ

where a‘ ¼ k‘ =ðc‘ C p ‘ Þ is the thermal diffusivity of the liquid. A dimensionless form of the equations of the problem is obtained introducing a dimensionless time, s ¼ cg Dv a =ðc‘ R20 Þ t, ~ ¼ R=R0 , temperatures, radial coordinate, ~r ¼ r=R, drop radius, R ~ ~ T ‘ ¼ T ‘ =T b and T g ¼ T g =T b , and molar evaporation flux,

ð18Þ

~j ¼ j R0 =ðcg Dv a Þ. Note that the radial coordinate, r, is converted to its dimensionless form by way of the drop radius, R. As a consequence, the liquid drop radius ranges from ~r ¼ 0 to ~r ¼ 1, at any time. It can be seen in Eq. (21) that cg Dv a =R is an order of magnitude of the molar evaporation flux (as mentioned in [15], the logarithm in Eq. (21) is generally close to unity, except at very high gas temperature or at small X i  X 1 ). Therefore, using Eq. (22), it

Rðt ¼ 0Þ ¼ R0

ð19Þ

can be seen that c‘ R20 =ðcg Dv a Þ is an order of magnitude of the drop evaporation time. The dimensionless forms of Eqs. (10), (21), (22) and (24) are:

T ‘ ð0 6 r 6 R; t ¼ 0Þ ¼ T ‘;0

ð20Þ

dV j 4pR ¼ c‘ dt 

2

ð17Þ

where V is the drop volume. Developing Eq. (17) with V ¼ 43 pR3 gives: 

dR j ¼ dt c‘ The initial conditions of the problem are:

3. Solutions of the problem 3.1. Complete model As all physicochemical properties of the gas are considered homogeneous, the gas molar density and the diffusion coefficient can be taken out of the derivative in Eq. (6). Then, solving Eq. (6) with the boundary conditions stated in Eqs. (7)–(9) leads to: 

j ¼



cg Dv a 1  Xi ln R 1  X1



ð21Þ

   1 X i ¼ exp L 1 T~ i

ð27Þ

  ~j ¼  1 ln 1  X i ~ 1  X1 R

ð28Þ

  ~ 1 dR 1  Xi ¼ ln ~ ds R 1  X1

ð29Þ

T~ g ð~r Þ ¼ T~ 1 

 ~ T~ 1  T~ i M ~j R   1  exp   ~ M~ ~ Le r 1  exp  Le j R

!! ð30Þ

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b b Þ; T~ i ¼ T i =T b ; T~ 1 ¼ T 1 =T b ; M ¼ M b v =M b g and where where L ¼ L =ð RT Le ¼ ag =Dv a is the Lewis number of the system. The partial derivative with respect to time in Eq. (26) can be expressed as:

~ @T ‘ @T ‘ cg Dv a @T ‘ ~r dR  ¼ 2 ~ ~ @t @ s c ‘ R0 @ r R dt

ð31Þ

Therefore, the energy balance equation within the drop (Eq. (26)) can be written, in its dimensionless form, as:

~ @ T~ ‘ @ T~ ‘ ~r dR H  ~ ds @~r @s R

!

1 1 @ @ T~ ~r 2 ‘ ¼ 2 ~ 2 ~r @~r @~r R

!

ð32Þ

where H ¼ cg Dv a =ðc‘ a‘ Þ. Using Eq. (29), Eq. (32) becomes:

!   ~r @ T~ ‘ 1  X i @ T~ ‘ 1 1 @ @ T~ ~r2 ‘ ¼ ln þ ~2 ~ 2~r 2 @~r @s R 1  X 1 @~r H R @~r

ð33Þ

The dimensionless forms of the boundary conditions regarding heat transfer in the system (Eqs. (13)–(15) and (25)) are:

T~ g ð~r ¼ 1Þ ¼ T~ 1 T~ g ð~r ¼ 1Þ ¼ T~ ‘ ð~r ¼ 1Þ ¼ T~ i  @ T~ ‘   ¼0 @~r 

ð34Þ ð35Þ ð36Þ

~r ¼0

 @ T~ ‘   @~r 

~r ¼1

¼

  H 1  Xi ln St‘ 1  X1   T~ 1  T~ i 1  Xi     M Cp H ln 1  X 1 exp  M ln 1X i 1 Le

T~ ‘ ¼ T~ ð0Þ þ H T~ ð1Þ

ð40Þ

Introducing this expansion in the energy balance equation within the drop (Eq. (32)) gives:

!! ~ @ T~ ð0Þ @ T~ ð0Þ @ T~ ð1Þ ~r dR @ T~ ð1Þ H þH  þH ~ ds @s @s @~r @~r R ! !! 1 1 @ @ T~ ð0Þ @ @ T~ ð1Þ 2 2 ~r ~r H þ ¼ ~ 2 ~r 2 @~r @~r @~r @~r R

ð41Þ

Gathering the terms independent of H in Eq. (41) gives:

@ @ T~ ð0Þ ~r 2 @~r @~r

!

¼0

ð42Þ

Note that no time derivative appears at the leading order (Eq. (42)), hence the approximation used implies a quasi-steadiness, which must ensue following a fast initial transient at the times s ¼ OðHÞ. As the boundary conditions Eqs. (35) and (36) should be satisfied for H ! 0 (and hence by T~ ð0Þ ), the solution of Eq. (42) is the homogeneous temperature field:

T~ ð0Þ ¼ T~ i

ð43Þ

Then, gathering the terms proportional to H in Eq. (41) gives:

~ ð0Þ

~ @ T~ ð0Þ ~r dR @T 1 1 @ @ T~ ð1Þ ~r 2  ¼ 2 ~ 2 ~r @~r ~ ds @~r @s @~r R R

!

ð44Þ

Solving Eq. (44) using Eqs. (35), (36) and (43) yields the correction:

ð37Þ

1X 1

with St‘ ¼ C p ‘ T b =L , the Stefan-like number of the liquid phase and

Cp ¼ C p g =C p ‘ .

The dimensionless initial conditions of the system write:

~0 ¼ 1 ~ s ¼ 0Þ ¼ R Rð

ð38Þ

T ‘;0 T~ ‘ ð0 6 ~r 6 1; s ¼ 0Þ ¼ T~ ‘;0 ¼ Tb

ð39Þ

The combination of Eqs. (29) and (33) with their initial conditions (Eqs. (38) and (39)), their boundary conditions (Eqs. (35)–(37)), and on account of Eq. (27), constitutes a general dimensionless model of the evaporation of a drop suspended in an air–vapor mixture, where the thermal transients are taken into account. These equations can be solved numerically using a spatial discretization of Eq. (33) and a Runge–Kutta method to obtain the ~ and T~ ‘ ) during the evaporation. time evolution of each unknown (R

The time evolution of ~j ; T~ g and X i may then be calculated afterwards using Eqs. (27), (28) and (30). This so-called complete model involves 9 dimensionless numbers: Cp ; L, Le, M; H, St‘ ; T~ ‘;0 ; T~ 1 and

X1. It is worth noting that such a complete model was first developed by Hubbard et al. [10] and later on studied by numerous authors [8,11,19–21]. This model is also known as the conduction limit model [8,20], the diffusion limit model [19] or the effective thermal conductivity model [11]. 3.2. Quasi-homogeneous model (H  1) For small values of H, an approximate solution of the complete model can be obtained by using an asymptotic analysis. For this purpose, the drop temperature field is Taylor-expanded up to the first order in H:

  dT~ i ~ 2 ~r 2 1 T~ ð1Þ ¼ R  ds 6 6

ð45Þ

Up to first order in H, the temperature inside the drop can thus be expressed as:

  dT~ i ~ 2 ~r 2 1 T~ ‘ ¼ T~ i þ H R  ds 6 6

ð46Þ

which in particular gives:

 @ T~ ‘   @~r 

~r ¼1

¼H

~2 dT~ i R ds 3

ð47Þ

Introducing Eq. (47) in Eq. (37) finally gives:

  ~2 dT~ i R 1 1  Xi ln ¼ St‘ ds 3 1  X1   T~ 1  T~ i 1  Xi     M Cp ln M 1  X 1 exp  ln 1X i 1 Le

ð48Þ

1X 1

Eqs. (29) and (48) are two ordinary differential equations that ~ enable, on account of Eq. (27), calculating the time evolution of R and T~ i . Thus, with this asymptotic analysis, the complete model containing a partial differential equation is simplified into a model of two ordinary differential equations. This so-called quasihomogeneous model involves 8 dimensionless numbers: Cp ; L, Le, M, St‘ ; T~ 1 ; X 1 and T~ i;0 ¼ T~ ‘;0 . Eq. (46) then allows calculating the ~ temperature field within the drop from the time evolution of R

and T~ i . This model can be compared to the well-known infinite thermal conductivity model (also called rapid mixing model) examined by Williams et al. [3], Law and Sirignano [7,8], Sazhin et al. [11], Sirignano et al. [19] and Aggarwal et al. [20] and which considers a drop of uniform temperature. Indeed, both models consist in a simplified version of the complete model (two ordinary differential equa~ and T~ i . However, the tions) and yield the time evolution of R

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infinite thermal conductivity model is derived from a global energy balance on the drop. On the contrary, our quasi-homogeneous model is based on an expansion at first order of the temperature field inside the drop and thus, through Eq. (46), provides additional information about the time evolution of the temperature field inside the drop. 3.3. Fully quasi-steady model A fully quasi-steady model can be derived from the quasihomogeneous model (already partly quasi-steady itself). In this fully quasi-steady model, we consider a quasi-steady version of Eq. (48). In other words, the left-hand side of Eq. (48) is neglected during the most part of the evaporation process, which on account of Eq. (28) leads to:

T~ i ¼ T~ 1 

1 M Stg

    M ~ ~ exp j R 1 Le

ð49Þ

where Stg ¼ C p g T b =L is the Stefan number of the gas phase. The combination of Eqs. (27)–(29) and (49) constitutes a system ~ in which six of four equations with four unknowns (~j ; T~ i ; X i and R), dimensionless numbers appear: L, Le, M, Stg ; T~ 1 and X 1 . This model is hereafter referred to as the fully quasi-steady model of drop evaporation and has been analyzed in detail in [15]. Note that Eqs. (27), (28) and (49) can be solved numerically using a Newton– Raphson method leading to time-independent values of X i ; T~ i and

~j R. ~ As a consequence of this time independence, Eq. (29) can be straightforwardly integrated, giving:

  ~ 2 ¼ 1  2 ln 1  X 1 s R 1  Xi

ð50Þ

It is worth noting that this fully quasi-steady model leads to the well known R2 -law model [22,23], i.e. the square of the drop radius decreases linearly over time. 4. Characteristic times of the system Two times characterizing the thermal transients in the drop can be distinguished in the studied problem:  the thermal homogenization time of the drop (thom;‘ ), during which the drop homogenizes thermally;  the thermal relaxation time of the drop (trel;‘ ), during which the drop interfacial temperature changes from its initial value to its established value. An order of magnitude of the thermal homogenization time of the drop is given by:

t hom;‘ ’

R20

a‘

ð51Þ

The dimensionless heat transport equation in the drop (Eq. (32)) evidences a dimensionless number:

H¼

c g Dv a c‘ a‘

ð52Þ

As the unit time used to make the equation dimensionless is the typical drop evaporation time scale R20 c‘ =ðDv a cg Þ, it is straightforward to see that H is actually an order of magnitude of the dimensionless thermal homogenization time. As mentioned previously, the gas phase is assumed to be ideal and its physicochemical properties are evaluated as functions of the gas composition and temperature at r ¼ 1. Therefore, if the

b g Þ, we kinetic theory of gases is used (Dv a / T g1:5 ), as cg ¼ Pg =ð RT pffiffiffiffiffiffiffi have H / T 1 . If the correlation developed by Fuller, Schettler and Geddings (Dv a / T 1:75 , see [15]) is used instead, H / T 0:75 g 1 . H

is inversely proportional to c‘ a‘ ¼ k‘ =C p ‘ . For many liquids, the

ratio k‘ =C p ‘ experiences slight variations (less than 10%) when

the temperature varies between the room and the boiling temperature of the liquid. In conclusion, H is mainly influenced by T 1 (it increases when T 1 increases), slightly influenced by T ‘;0 and independent of X 1 . In Appendix A, the value of H is provided for various liquids, at T ‘;0 ¼ 50  C and T 1 ¼ 200  C. These liquids present values of H around 0.2–0.7 at T 1 ¼ 200  C. Therefore, according to the kinetic theory of gases, they have values of H around 0.15–0.55 at T 1 ¼ 20  C and around 0.30–1.15 at T 1 ¼ 1000  C. These values show that, regarding the evaporation process of a drop of these liquids into air, the thermal homogenization transient might occur during a significant part of the evaporation process. For small enough values of H, the quasi-homogeneous model can be used to get further insight into the dynamics of the thermal relaxation transient. A first order Taylor expansion of the logarithmic and exponential terms appearing in Eq. (48), with respect to X i and around X 1 , allows simplifying this equation into:

  ~2 dT~ i R 1 X1  Xi þ Le Cp ðT~ 1  T~ i Þ ¼ St‘ 1  X 1 ds 3

ð53Þ

A Taylor expansion of Eq. (27), with respect to T~ i and around T~ i ¼ 1, gives:

  X i ¼ 1 þ L T~ i  1

ð54Þ

Introducing Eq. (54) in Eq. (53), and rearranging the resulting equation, allows obtaining the following equation regarding the time evolution of the interfacial temperature T~ i :

  ~ 2 dT~ i R L 1 ¼  Le Cp þ T~ i þ Le Cp T~ 1 St‘ ð1  X 1 Þ 3 ds   1 L  1 St‘ 1  X1

ð55Þ

~ still close to 1, For small values of s, and hence for values of R the analysis of Eq. (55) shows that the time evolution of the interfacial temperature is a first order process with characteristic dimensionless time:

R¼

1 Le Cp þ StL‘

1 ð1X 1 Þ

ð56Þ

Therefore, R is an order of magnitude of the dimensionless thermal relaxation time of the drop. Two characteristic times can be seen in the expression of R: Rhe ¼ 1=ðLe Cp Þ and Rev ¼ ð1  X 1 ÞSt‘ =L. Rhe can be rewritten as follows: Rhe ¼

R20 c‘ C p ‘ Dv a cg , kg R20 c‘

with R20 c‘ =ðDv a cg Þ the unit time. It

can be easily calculated that ðR20 c‘ C p ‘ Þ=kg is an order of magnitude of the thermal relaxation time of a non evaporating liquid drop (with a heat capacity C p ‘ and a molar density c‘ ) of radius R0 , immobilized in an infinite gaseous environment (with a thermal conductivity kg ), when there is initially a temperature difference between the drop and the gas. Therefore, Rhe characterizes the dimensionless thermal relaxation time due to the heat exchange between the gas and the drop without evaporation, while Rev characterizes the dimensionless thermal relaxation time due to the evaporation. For many liquids, Rhe is an order of magnitude larger than Rev . Indeed, the Lewis number Le is usually close to one, while

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uids, Rhe ¼ 1=ðLe Cp Þ P 1. On the other hand, Rev can be rewritten b  T 2 =L2 . Using available data, it can as follows: Rev ¼ ð1  X 1 Þ RC p‘

b

be evaluated that, for many liquids, Rev < 0:2. As a consequence, R can often be approximated as follows: R ’ ð1  X 1 ÞSt‘ =L. Rev is mainly influenced by X 1 , slightly influenced by T ‘;0 and independent of T 1 . Indeed, Rev linearly decreases when X 1 increases while, for many liquids, the ratio C p ‘ =L2 experiences slight variations (less than 10%) when the temperature varies between the room and the boiling temperature of the liquid. Hence, as R ’ Rev ; R is mainly influenced by X 1 and slightly influenced by T ‘;0 and by T 1 via Rhe . R decreases when X 1 increases, until reaching 0 when X 1 ¼ 1. In this situation, the drop is in contact with its pure vapor and, therefore, the interface is instantaneously at the boiling temperature, i.e. the thermal relaxation time of the drop is 0. In Appendix A, the value of R is provided for various liquids, at T ‘;0 ¼ 50  C; T 1 ¼ 200  C and X 1 ¼ 0. These liquids present values of R around 0.05–0.2. As these values are provided for X 1 ¼ 0, they should be close to the maximum values of R for these various liquids.

5. Results and discussion In this section, the different models described above are analyzed and compared, in order to give insight into the thermal transients during the evaporation of a drop, quantitatively and qualitatively. The complete model is considered as being the reference, and the ability of the quasi-homogeneous and fully quasisteady models to predict the results obtained by the complete model is discussed. It should be recalled that, to develop the different models presented in this paper, it has been assumed that the gas physicochemical properties are homogeneous (and evaluated at the gas composition and temperature at r ¼ 1). It has been proved that, for a drop of water, this assumption leads to an overestimation of the evaporation flux (when compared to the evaporation flux calculated using a model taking into account the heterogeneity of the gas physicochemical properties), especially for high values of T 1 [15]. Consequently, whereas results presented for low values of T 1 can be interpreted as quantitive results, the results presented for high values of T 1 should be interpreted as qualitative results. Note also that the numerical results presented in this section were

radius and the time is retrieved (R2 -law). These results are coherent with the transient behaviors reported in the literature when unsteady models comparable to the complete model are used [8,10,11,19,20].

5.1. Different types of drop dynamics Depending on the drop initial temperature, T ‘;0 , and the temperature and the vapor mole fraction far from the drop, T 1 and X 1 , 6 different types of dynamics can be observed when a drop is suspended in an air–vapor mixture. These dynamics are presented in Fig. 3 for the case of a water liquid drop. The type of dynamics observed depends on the relative values of three temperatures:

(c)

(b)

τ

T (◦ C)

˜2 R

Ti (◦ C)

(a)

obtained using Antoine’s equations to express the saturation pressure of the considered liquids. Such an equation is indeed accurate in a wider range of temperatures than the Clapeyron’s equation. Finally, the gas pressure is considered at Pg ¼ 1 atm. In Fig. 2, an example of the solution of the complete model is presented. This figure allows highlighting the main thermal mechanisms taking place inside a drop during its evaporation. The results presented in Fig. 2 have been obtained for a drop of water, with an initial temperature T ‘;0 ¼ 20  C, suspended in dry air (X 1 ¼ 0) at T 1 ¼ 200  C. These are classical conditions for a spray drying process. In these conditions, the drop dimensionless evaporation time is 5.45; the values of H and R are 0.18 and 0.04, respectively. The thermal relaxation transient is clearly observed in Fig. 2 (a), where the time evolution of the interfacial temperature is presented. This transient appears to be over after approximately one unit of dimensionless time. In Fig. 2(b), the time evolution of the temperature field inside the drop is presented. The thermal homogenization transient is clearly observed, as a temperature gradient first develops inside the drop, and then eventually vanishes after approximately one unit of dimensionless time. To get more insight into this thermal homogenization transient, the temperature difference between the interface and the center of the drop is presented in Fig. 2(c). It can be observed that this difference increases sharply at the very beginning of the evaporation, until reaching a maximum value, and then disappears after approximately one unit of dimensionless time. Finally, an inset in Fig. 2 (a) presents the time evolution of the square of the dimensionless drop radius. During the thermal transients, it can be seen that this time evolution is not linear. However, after the thermal transients, the classical linear relationship between the square of the drop

ΔT (◦ C)

Cp ¼ C p g =C p ‘ is usually smaller than one. Therefore, for many liq-

τ

τ

r˜

τ

Fig. 2. Solution of the complete model for a water drop with T ‘;0 ¼ 20  C, T 1 ¼ 200  C and X 1 ¼ 0. (a) Time evolution of the interfacial drop temperature. The inset presents the time evolution of the square of the dimensionless drop radius. (b) Time evolution of the temperature field within the drop. The arrow indicates the direction of the time evolution and the dimensionless time step between two successive curves is 0.044. (c) Time evolution of the temperature difference between the interface and the center of the drop.

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Fig. 3. Different types of dynamics when a water drop (with an initial temperature T ‘;0 ) is suspended in an air–vapor mixture. The time evolutions of the drop interfacial ~ 2 , are presented. The plots are restricted in time to s  1 in order to highlight the transients behavior. The temperature, T i , and of the square of the dimensionless radius, R complete model (–) is compared with the fully quasi-steady model (--). (a) Evaporation with a slower initial transient stage [T ‘;0 ¼ 20  C, T i;qs ¼ 47  C and T dp ¼ 18  C]. (b) Evaporation with a faster initial transient stage [T ‘;0 ¼ 40  C, T i;qs ¼ 26  C and T dp ¼ 18  C]. (c) Evaporation with an initial condensation transient stage [T ‘;0 ¼ 20  C, T i;qs ¼ 66  C and T dp ¼ 60  C]. (d) Condensation with a slower initial transient stage [T ‘;0 ¼ 45  C, T i;qs ¼ 42  C and T dp ¼ 46  C]. (e) Condensation with a faster initial transient stage [T ‘;0 ¼ 40  C, T i;qs ¼ 58  C and T dp ¼ 60  C]. (f) Condensation with an initial evaporation transient stage [T ‘;0 ¼ 100  C, T i;qs ¼ 57  C and T dp ¼ 60  C].

 the initial drop temperature, T ‘;0 ;  the dew point temperature of the gas far from the drop, T dp ðX 1 Þ (that can be calculated, using the Clapeyron’s equation (Eq. (10)), as follows: 1 ¼ 1  bR lnðX Þ, or using Antoine’s equaT dp

Tb

L

1

tion; for X 1 ¼ 0 formally T dp ¼ 0 K);  the established interfacial temperature, T i;qs ðT 1 ; X 1 Þ (i.e. the drop interfacial temperature calculated using the fully quasisteady model). If T i;qs ðT 1 ; X 1 Þ < T dp ðX 1 Þ, which is equivalent to the simpler criterion T 1 < T dp ðX 1 Þ, the drop eventually grows by condensation ~ ! 1 for s ! 1, see cases (d–f) in Fig. 3), regardless of the drop (R initial temperature, and even if an initial evaporation stage might be observed (if T ‘;0 > T dp ðX 1 Þ, see case (f) in Fig. 3). When T i;qs ðT 1 ; X 1 Þ > T dp ðX 1 Þ, which is equivalent to T 1 > T dp ðX 1 Þ (cases (a–c) in Fig. 3), the drop eventually completely evaporates. In cases (a) and (b), T ‘;0 > T dp ðX 1 Þ. In these cases, the decrease of the drop radius is monotonic. When the initial drop temperature is below the interfacial established temperature, i.e. T ‘;0 < T i;qs ðT 1 ; X 1 Þ (case (a)), it can be observed in Fig. 3 that the fully quasi-steady model underestimates the drop evaporation time. Indeed, during the thermal relaxation transient, the actual evaporation rate is smaller than the one predicted by the fully quasi-steady model (as, during this transient, T i is smaller than T i;qs ðT 1 ; X 1 Þ and some part of the heat provided by the surroundings is used to heat up the whole drop with less heat for evaporation). When the initial drop temperature

is larger than the interfacial established temperature, i.e. T ‘;0 > T i;qs ðT 1 ; X 1 Þ (case (b)), it can be observed in Fig. 3 that the fully quasi-steady model overestimates the drop evaporation time. Indeed, during the thermal relaxation transient, the actual evaporation rate is larger than the one predicted by the fully quasisteady model (as, during this transient, T i is larger than T i;qs ðT 1 ; X 1 Þ and the heat initially stored into the drop is readily used for evaporation). In case (c), the initial drop temperature is lower than the dew point temperature of the gas far from the drop, i.e. T ‘;0 < T dp ðX 1 Þ. In this situation, the process starts by a condensation of the vapor ~ increases), along with an increase of the interfacial tempera(i.e. R ture. As T i;qs ðT 1 ; X 1 Þ > T dp ðX 1 Þ, the interfacial temperature eventually becomes equal to the dew point temperature of the gas far from the drop. At this critical point, the evaporation rate is null,  j ¼ 0 and the gas composition is homogeneous, X i ¼ X 1 . Then, as the interfacial temperature continues to increase, the evaporation ~ decreases). Due to this initial condensation of the drop starts (i.e. R stage, the fully quasi-steady model underestimates the drop evaporation time. As far as we know, such a clear-cut classification of the dynamics of a drop suspended in an air–vapor mixture was never presented. Most of the authors reported results with the evaporation dynamics of Fig. 3(a) [3,8,10,19,20]. A few authors observed the initial condensation transient stage during evaporation, i.e. the dynamics of Fig. 3(c) [11,24–26]. However, to the best of our knowledge, no study reported cases where the fully quasi-steady model overestimates the drop evaporation time compared to unsteady models (Fig. 3(b)).

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It can also be seen in Fig. 4(a–b) that, for the considered conditions, the highest absolute value of tev is approximately 9%. Thus, regarding a water drop, the results presented in Fig. 4(a–b) show that the drop evaporation time calculated with the fully quasisteady model appears to be a fair approximation of the drop evaporation time predicted by the complete model, for a wide range of evaporation conditions. This fair approximation can be explained by the fact that, for water, the dimensionless thermal relaxation time R (see Section 4) is small compared to one (an order of magnitude smaller). Indeed, for water, R ’ 0:05 at X 1 ¼ 0 (see Appendix A). Hence, during a large part of the evaporation process, the interfacial temperature (and thus the evaporation rate, see Eqs. (27) and (28)) should be the one predicted by the fully quasisteady model. Given the definition of R, this explanation is valid only because H ’ 0:18 is also smaller than one for water at X 1 ¼ 0. Similar conclusions should be reached for liquids such as methanol or ethanol, presenting small values of R compared to one (and H < 1) at X 1 ¼ 0 (see Appendix A). For liquids with H < 1 but presenting larger values of R at X 1 ¼ 0, it could be needed to use the complete model to predict accurately the evaporation time. To illustrate this, calculations have been made for a drop of n-hexane. For this liquid, R ’ 0:2 and H ’ 0:72 at X 1 ¼ 0 (see Appendix A); it is the largest values of R and H presented in Appendix A. In Fig. 4(c–d), tev is presented, for this drop, as a function of the evaporation conditions, T 1 and X 1 , for two values of the initial drop temperature: T ‘;0 ¼ 20  C and T ‘;0 ¼ 55  C. When comparing Fig. 4(a–b) and Fig. 4(c–d), it can be clearly seen that the ability of the fully quasi-steady model to predict the evaporation

5.2. Comparison between the complete, the fully quasi-steady and the quasi-homogeneous models 5.2.1. Ability of the fully quasi-steady model to predict the drop evaporation time The dimensionless drop evaporation time predicted by the fully quasi-steady model can be easily expressed using Eq. (50):

sev;qs ¼ tev;qs

cg Dv a c‘ R20

¼

2 ln

1 

1X 1 1X i;qs



ð57Þ

with X i;qs obtained solving the system of three equations (Eqs. (27), (28) and (49)) presented in Sections 3.1 and 3.3. In Fig. 4(a–b), the relative difference between the drop evaporation times predicted by the complete model (tev ) and by the fully quasi-steady model (tev;qs ), tev ¼ ðtev  t ev;qs Þ=tev;qs , is presented for water as a function of the evaporation conditions, T 1 and X 1 , for two values of the initial drop temperature: T ‘;0 ¼ 20  C and T ‘;0 ¼ 80  C. The analyzed conditions are restricted to those leading to a complete evaporation of the drop (cases (a–c) in Fig. 3). This is why some curves in Fig. 4 are restricted to a range of X 1 values with an upper limit smaller than one. As mentioned in Section 5.1 and observed in Fig. 4(a–b), the fully quasi-steady model can overestimate (tev < 0, when T ‘;0 > T i;qs ðT 1 ; X 1 Þ) or underestimate (tev > 0, when T ‘;0 < T i;qs ðT 1 ; X 1 Þ) the drop evaporation time. At a fixed value of T 1 ; T i;qs increases with X 1 (until reaching T b for X 1 ¼ 1). This actually mainly explains why tev increases with X 1 at a fixed value of T 1 , in the plots presented in Fig. 4(a–b).

Water 0

= 20 C

(c)

n-Hexane T

0

= 20◦ C

tev (%)

T

◦

tev (%)

(a)

T∞ (◦ C) :

X∞ 0

= 80◦ C

(d)

X∞ T

0

= 55◦ C

tev (%)

T

tev (%)

(b)

X∞

X∞

Fig. 4. Relative difference between the drop evaporation times estimated by the complete and the fully quasi-steady models, tev ¼ ðtev  tev;qs Þ=tev;qs , as a function of T 1 and X 1 , for two liquids and two initial drop temperatures. (a–b) Drop of water, with T ‘;0 ¼ 20  C and T ‘;0 ¼ 80  C. (c–d) Drop of n-hexane, with T ‘;0 ¼ 20  C and T ‘;0 ¼ 55  C.

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time of a drop of n-hexane into air is not as good as for a water drop. It is important to note that even if the fully quasi-steady model appears to provide a good estimation of the drop evaporation time for liquids presenting small values of R and H compared to one, it does not, by definition, give information about the time evolution of the temperature field inside the drop, whose evaluation might be important for some applications. An analytical estimation of the absolute difference between the drop evaporation times predicted by the complete model and by the fully quasi-steady model is provided in Eq. (58) below. It is estimated by the time that it would take to change the drop temperature from T ‘;0 to T i;qs by means of the established evaporation heat flux:

t ev  t ev;qs ’

4 3



pR30 c‘ C p ‘ T i;qs  T ‘;0 L



1 4p

 R20 jqs

ð58Þ



where jqs is the established evaporation rate, i.e. the one calculated using the fully quasi-steady model. In the case of an initial drop temperature lower than the established drop temperature (T i;qs  T ‘;0 > 0), the fully quasi-steady model underestimates the drop evaporation time (t ev  t ev;qs > 0). Similarly, the heat stored in the drop when it is initially warmer than at its established state (T i;qs  T ‘;0 < 0) leads to a corresponding overestimation (t ev  t ev;qs < 0). Using Eqs. (21) and (57) in Eq. (58) provides an analytical estimation of the relative difference between the drop evaporation times predicted by the complete model and by the fully quasisteady model:

  t ev  t ev;qs 2 C p ‘ T i;qs  T ‘;0 ¼ tev ’ bev L 3 tev;qs

ð59Þ

where an accommodation coefficient bev has been introduced (bev ¼ 1 if Eq. (58) was exactly valid). As shown in Fig. 5, the actual values of bev are found to range between 0.56 and 1 with a mean value of 0.83 for a water drop. The analytical formula proposed in Eq. (59) is then seen to capture the effect of the thermal transients on the evaporation rate fairly well. Results obtained for a n-hexane drop are similar, with values of bev ranging between 0.58 and 0.98 (not shown in Fig. 5 for clarity). Thus, the formula of Eq. (59), with bev ranging from 0.56 to 1, is valid for a wide range of realistic values of the dimensionless numbers appearing in the model.

Fig. 5. Comparison between the actual relative difference between the drop evaporation times predicted by the complete model and by the fully quasi-steady model, tev , and its analytical estimation for a water drop at initial temperatures T ‘;0 ¼ 20  C and T ‘;0 ¼ 80  C and for various T 1 .

Note that these results concerning the drop evaporation time constitute an original approach compared to the results proposed in the literature. Previous studies comparing the results of the drop evaporation time obtained with models equivalent to the complete model and to the fully quasi-steady model are restricted to a small number of liquids and to specific evaporating conditions, always leading to an underestimation of the drop evaporation time by the fully quasi-steady model [3,8,19,20]. Our approach allows to characterize the ability of the fully quasi-steady model to assess the drop evaporation time through the evaluation of two dimensionless numbers, and proposes an analytical formula that evaluates fairly well the drop evaporation time for a wide range of liquids and evaporating conditions.

5.2.2. Comparison between the quasi-homogeneous and the complete models A comparison between results obtained with the quasihomogeneous model and with the complete model is presented in Fig. 6. These results have been generated considering a drop of water, for evaporation conditions leading to different types of dynamics (evaporation with a slower initial transient stage: Fig. 6(a–d), evaporation with a faster initial transient stage: Fig. 6 (b) and evaporation with an initial condensation transient stage: Fig. 6(c)). It can be observed in Fig. 6(a–d) that, for the considered evaporation conditions, the quasi-homogeneous model yields a fair approximation of the time evolution of the interfacial temperature. The quality of this approximation decreases with an increase of T 1 . The most significant discrepancy between the two models is observed for T 1 ¼ 1000  C (Fig. 6(d)). This is explained by the fact that the quasi-homogeneous model is obtained assuming H  1. Therefore, the discrepancy between the quasi-homogeneous and the complete models is expected to increase with an increase of H (and hence with an increase of T 1 , see Section 4). For a drop of water, H is close to 0.15 at T 1 ¼ 50  C, close to 0.2 at T 1 ¼ 200  C, and close to 0.3 at T 1 ¼ 1000  C. It can be observed in Fig. 6(a–d) that, for the considered evaporation conditions, the quasi-homogeneous model allows an accurate estimation of the time evolution of the drop radius, even when an initial condensation stage is present. Consequently, the quasi-homogeneous model should be able to predict accurately the drop evaporation time. Actually, it can be calculated that, for the evaporation conditions considered in Fig. 4(a–d), the relative difference between the water drop evaporation times predicted by the complete model and by the quasi-homogeneous model is less than 1%. When analyzing the results presented in Fig. 6(e–f), it can also be observed that the quasi-homogeneous model provides a fair approximation of the time evolution of the temperature field inside the drop for T 1 ¼ 50  C (except at s ¼ 0, but this is to be expected as Eq. (46) is not coherent with the initial condition T ‘ ðr; 0Þ ¼ T ‘;0 ). At T 1 ¼ 200  C, the quality of this approximation is a bit decreased, as expected given the increase of H. Values of H for other liquids are presented in Appendix A; they all are between 0.34 and 0.72 at T 1 ¼ 200  C. Hence, for these other liquids, a limited ability of the quasi-homogeneous model to provide a fair approximation of the time evolutions of the interfacial temperature and of the temperature field inside the drop might be expected. However, regarding the drop evaporation time, it appears that, for all the liquids mentioned in Appendix A, the quasi-homogeneous model provides an accurate prediction. For instance, it can be calculated that, for a n-hexane drop, the relative difference between the drop evaporation times predicted by the complete model and by the quasi-homogeneous model is less than 4.5% (for the evaporation conditions considered in Fig. 4(c–d)).

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Fig. 6. Comparison between the quasi-homogeneous (- -) and the complete (–) models for a water drop. (a–d) Time evolution of the interfacial temperature for different evaporation conditions. The inset presents the time evolution of the square of the dimensionless drop radius. The plots are restricted in time to s < 1 in order to highlight the transients behavior. The drop dimensionless evaporation times are [5.61; 34.44; 6.87; 1.04], respectively. (e–f) Time evolution of the temperature field within the drop, for different evaporation conditions. The arrow indicates the direction of the time evolution and the fields are extracted at s ¼ ½0; 0:06; 0:12; 0:19; 0:25; 0:31 for (e) and s ¼ ½0; 0:05; 0:09; 0:14; 0:18; 0:23 for (f). The black bold line represents the established drop temperature field.

The results obtained in this section are similar to the results presented by Aggarwal et al. [20] when comparing their infinite thermal conductivity model to the conduction limit model (equivalent to our complete model). These authors show the same tendency for the time evolution of the interfacial temperature with the crossing of the curves obtained with the two models and with an increase of the deviation between the models when T 1 increases. They also obtain a good approximation of the drop evaporation time using the infinite thermal conductivity model. 5.3. Influence of the evaporation conditions on the thermal transients In this section, we discuss the influence of the evaporation conditions (T ‘;0 ; T 1 and X 1 ) on some characteristics of the thermal transients taking place during the evaporation of a drop. The analyzed conditions are restricted to those leading to a complete evaporation of the drop (cases (a–c) in Fig. 3). This is why, in some of the presented curves of a thermal transient characteristics as a function of X 1 and at a given value of T 1 , the curve is restricted to a range of values of X 1 with an upper limit smaller than 1. In order to highlight the influence of H and R on the thermal transients, results are presented for two liquids showing a large difference between their R values and between their H values: water and n-hexane (see Appendix A). 5.3.1. Thermal relaxation transient The thermal relaxation time of the drop interfacial temperature (t rel;‘ ) is defined such as the time at which the difference between the interfacial temperature and the established interfacial temperature is equal to 1% of the difference between the initial drop tem-

perature and the   T i ðtrel;‘ ÞT i;qs   T ‘;0 T i;qs  ¼ 0:01.

established

interfacial

temperature,

i.e.

In Fig. 7(a–b), the influence of T 1 and X 1 on trel;‘ ¼ t rel;‘ =t ev is presented for a water drop, for two values of the initial drop temperature: T ‘;0 ¼ 20  C and T ‘;0 ¼ 80  C. These results have been generated using the complete model. The results obtained for the two initial drop temperatures are similar. On the one hand, trel;‘ decreases with an increase of X 1 (and is equal to zero at the maximum values of X 1 available for evaporation for any given T 1 T b and at X 1 = 1 for T 1 > T b ). This can be explained by the fact that, at constant T 1 and increasing X 1 ; R, an order of magnitude of the dimensionless thermal relaxation transient, is decreasing (see Section 4), while the driving force for the evaporation is decreasing (and hence the evaporation time is increasing). On the other hand, trel;‘ increases with an increase of T 1 . Indeed, as mentioned previously, R is almost independent of T 1 , while the dimensionless evaporation time (see Eq. 57) obviously decreases with an increase of T 1 . It is worth noting that, for a drop of water, whatever T ‘;0 and X 1 , the thermal relaxation time is relatively short compared to the evaporation time for low values of T 1 . More specifically, when T 1 K 100  C, the maximum of trel;‘ is about 7%. In that situation, the thermal relaxation transient can be negligible. However, for higher values of T 1 ; trel;‘ can reach up to 20–25%. In Fig. 8(a–b), the influence of T 1 and X 1 on trel;‘ is presented for a n-hexane drop at two initial temperatures, T ‘;0 ¼ 20  C and T ‘;0 ¼ 55  C. These results have been generated using the complete model. It can be observed in Fig. 8(a–b) that, qualitatively, the influence of T 1 and X 1 on trel;‘ is similar to that observed for a

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(a)

T

(c)

= 20◦ C

0

T

(e)

= 20◦ C

0

T

= 20◦ C

0

80 40

20

5

(◦ C)

thom

20

max

(%)

10

40

ΔT

(%)

30

trel

60

15

20

10

T∞ (◦ C) :

0 0.0

0.2

0.4

0.6

0.8

1.0

X∞

(b)

T

0

0 0.0

0.2

0.6

0.8

1.0

X∞

(d)

= 80◦ C

0.4

T

0

0 0.0

0.2

0.6

0.8

1.0

0.8

1.0

X∞

(f )

= 80◦ C

0.4

T

0

= 80◦ C

20 25

(◦ C)

10

15

20

(%)

(%)

15

trel

20

max

10

5

5 0 0.0

10

ΔT

thom

15

5

0.2

0.4

0.6

0.8

0 0.0

1.0

0.2

0.4

0.6

X∞

0.8

0 0.0

1.0

0.2

0.4

0.6

X∞

X∞

Fig. 7. Influence of the evaporation conditions (T 1 and X 1 ) on the thermal relaxation transient (a–b), the thermal homogenization transient (c–d) and the amplitude of the thermal heterogeneity (e–f) during the evaporation of a water drop. Two values of the initial drop temperatures are considered: T ‘;0 ¼ 20  C (a,c,e) and T ‘;0 ¼ 80  C (b,d,f).

(a)

T

(c)

= 20◦ C

0

T

(e)

= 20◦ C

0

T

= 20◦ C

0

100

0.2

(b)

0.4

0.6

0.8

T

0

0 0.0

1.0

max

ΔT

20

0.2

0.4

0.6

0.8

1.0

X∞ T

0

0 0.0

T∞ (◦ C) :

0.2

0.4

0.6

0.8

1.0

0.8

1.0

X∞

(f )

= 55◦ C

100

T

0

= 55◦ C

15

20

10

max

60 40

ΔT

(%)

40

(◦ C)

80

60

thom

(%)

30

10

(d)

= 55◦ C

80

trel

40 20

X∞

0 0.0

60

thom

trel 20

(◦ C)

(%)

(%) 40

0 0.0

40

80

60

5

20

0.2

0.4

0.6

X∞

0.8

1.0

0 0.0

0.2

0.4

0.6

X∞

0.8

1.0

0 0.0

0.2

0.4

0.6

X∞

Fig. 8. Influence of the evaporation conditions (T 1 and X 1 ) on the thermal relaxation transient (a–b), the thermal homogenization transient (c–d) and the amplitude of the thermal heterogeneity (e–f) during the evaporation of a n-hexane drop. Two values of the initial drop temperatures are considered: T ‘;0 ¼ 20  C (a,c,e) and T ‘;0 ¼ 55  C (b,d,f).

P. Talbot et al. / International Journal of Heat and Mass Transfer 97 (2016) 803–817

water drop. However, when compared to a water drop, the thermal relaxation transient takes place during a larger part of the evaporation process, especially when X 1 is small. It can be explained by the fact that, at same values of T ‘;0 ; X 1 and T 1 , this n-hexane drop presents a value of R significantly larger than the one of a water drop (the largest value of R presented in Appendix A corresponds to n-hexane, while the smallest corresponds to water). The maximum calculated value of trel;‘ is 85%. It is obtained when considering the evaporation of a n-hexane drop into pure air (X 1 ¼ 0), with T ‘;0 ¼ 55  C and T 1 ¼ 2000  C. For this n-hexane drop and for X 1 ¼ 0; trel;‘ J 20%, whatever T 1 and T ‘;0 considered. The two cases presented in this section show that the thermal relaxation transient takes place during a limited fraction of the water drop evaporation time, but can take place during a significant fraction of the evaporation time of a n-hexane drop. This seems coherent with the definition of R and its values provided in Appendix A.

5.3.2. Thermal homogenization transient The thermal homogenization time of the drop (thom;‘ ) is defined such as the time at which the relative difference between the temperature at the drop center and the temperature at the drop inter  T ðr¼0;thom;‘ ÞT i ðthom;‘ Þ face is equal to 1%, i.e.  ‘  ¼ 0:01, after the T i ðt Þ hom;‘

maximum of this relative difference has been reached (as this difference is obviously equal to zero at time t ¼ 0). In Fig. 7(c–d), the influence of T 1 and X 1 on thom;‘ ¼ t hom;‘ =t ev is presented for a water drop, for two values of the initial drop temperature: T ‘;0 ¼ 20  C and T ‘;0 ¼ 80  C. These results have been generated using the complete model. The results presented in Fig. 7(c– d) reveal that under some specific evaporation conditions, the drop can stay thermally homogeneous during the entire evaporation process (thom;‘ ¼ 0). This situation occurs when T ‘;0 ’ T i;qs ðT 1 ; X 1 Þ. When this condition is not fulfilled, a temperature gradient develops within the drop. For most of the considered evaporation conditions, thom;‘ increases with an increase of T 1 . This can be explained by the fact that t hom;‘ is more or less independent of the temperature far from the drop (t hom;‘ ’ R20 =a‘ ), while the evaporation time decreases with an increase of T 1 . The influence of X 1 on thom;‘ appears to be more complex. This is discussed in the following section of the paper. It is worth noting that, for a drop of water, whatever T ‘;0 and X 1 , the thermal homogenization time is relatively short compared to the evaporation time for low values of T 1 . More specifically, when T 1 K 100  C, the maximum calculated value of thom;‘ is about 5%. For the highest value of T 1 considered here (2000 °C), the calculated value of thom;‘ reaches 45% at X 1 ¼ 0. In Fig. 8(c–d), the influence of T 1 and X 1 on thom;‘ is presented for a n-hexane drop at two initial drop temperatures, T ‘;0 ¼ 20  C and T ‘;0 ¼ 55  C. These results have been generated using the complete model. It can be observed in Fig. 8(c–d) that, qualitatively, the influence of T 1 and X 1 on thom;‘ is similar to the one exhibited by a water drop. However, when compared to a water drop, the thermal homogenization transient takes place during a larger part of the evaporation process. It can be explained by the fact that, at same values of T ‘;0 ; X 1 and T 1 , a n-hexane drop presents a value of H significantly larger than the one of a water drop (the largest H value presented in Appendix A corresponds to n-hexane, while the smallest corresponds to water). For high values of T 1 , it can be observed that the drop is thermally inhomogeneous during almost the entire evaporation process. The two cases presented in this section show that the thermal homogenization transient often takes place during a limited fraction of a water drop evaporation time, but can take place during

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a significant fraction of a n-hexane drop evaporation time. This seems coherent with the definition of H and its values provided in Appendix A.

5.3.3. Amplitude of the thermal heterogeneity of the drop In Fig. 7(e–f), the influence of T 1 and X 1 on the absolute value of the maximum temperature difference that develops inside the ) is presented, for two drop during the evaporation process (DT max ‘ values of the initial drop temperature: T ‘;0 ¼ 20  C and T ‘;0 ¼ 80  C. These results have been generated using the complete model, considering a drop of water. obtained for X 1 = 1 At a given value of T ‘;0 , the value of DT max ‘ (when T 1 P 100  C) appears to be independent of T 1 . It can be explained by the fact that, when X 1 = 1, the drop is in contact with its pure vapor at atmospheric pressure and, therefore, its interface is instantaneously at a temperature T i ¼ T b ¼ 100  C, the maximum temperature that can be reached at the drop interface. It ¼ T b  T ‘;0 ¼ 80  C when explains why, when X 1 ¼ 1; DT max ‘  T ‘;0 ¼ 20  C and DT max ¼ T  T ¼ 20 C when T ‘;0 ¼ 80  C. ‘;0 b ‘  For T ‘;0 ¼ 80 C, the influence of X 1 on DT max appears to be com‘ plex. For T 1 6 1000  C, DT max first decreases with an increase of ‘ X 1 , reaches zero for a value of X 1 depending on the value of T 1 , ¼ 20  C. For T 1 P 1250  C, and then increases to reach DT max ‘ DT max is a monotonically increasing function of X 1 . This can be ‘ explained by the fact that, when T 1 6 1000  C, there is a value of X 1 , depending on T 1 and T ‘;0 and written X 1;crit ðT 1 ; T ‘;0 Þ, such that T ‘;0 ’ T i;qs ðT 1 ; X 1;crit ðT 1 ; T ‘;0 ÞÞ. As mentioned previously, for this particular value of X 1 , the drop is initially at its steady state temperature and no temperature gradient develops in the drop (hence, DT max ¼ 0). For X 1 < X 1;crit ðT 1 ; T ‘;0 Þ or X 1 > X 1;crit ðT 1 ; T ‘;0 Þ, a tem‘ perature gradient develops in the drop, leading, during its thermal homogenization transient, either to an increased temperature at the drop interface compared to the drop center (when T ‘;0 < T i;qs ) or to a decreased temperature at the drop interface compared to the drop center (when T ‘;0 > T i;qs ). On the other hand, when T 1 P 1250  C, T ‘;0 is smaller than T i;qs , whatever the value of X 1 . When T ‘;0 ¼ 20  C, such a behavior is only observed for T 1 6 40  C. For higher T 1 ; T ‘;0 is always smaller than T i;qs . As presented in Fig. 8(c–d), similar results are obtained when considering the evaporation of a n-hexane drop. Using the complete model, it can also be demonstrated that, for the evaporation conditions leading to the results presented in Fig. 7, the maximum temperature difference that develops inside the drop during the evaporation process is reached in less than 5% of the drop evaporation time. Some behaviors highlighted by the results showed in Figs. 7 and 8 were mentioned in previous studies. When studying drop evaporation in typical combustion conditions (high T 1 ), Wise et al. showed that the thermal homogenization transient takes a smaller part of the drop evaporation time when X 1 increases [5]. In the same kind of evaporation conditions, Williams found that the relaxation transient can take a non negligible part of the drop evaporation time [3]. Law showed that an octane drop evaporating at low T 1 and X 1 takes around 10–20% of the drop evaporation time to reach its quasi-steady interfacial temperature [8]. This author also concluded that the complete model should always be used when details on the temperature field inside the drop are needed. Therefore, the present study appears to be coherent with previous investigations. However, whereas the previous conclusions of the literature are only valid for specific evaporation conditions, the present paper proposes a more general picture of the transients during drop evaporation based on two dimensionless numbers.

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6. Conclusion This work is centered upon the thermal transients taking place during the evaporation of a drop of a pure liquid suspended in an air–vapor mixture at atmospheric pressure. Based on mass and energy conservation equations, a so-called complete model is developed considering quasi-steady diffusive and Stefan convective transports in the gas phase, and unsteady heat conduction in the liquid phase. A simplified version of the complete model, the so-called quasi-homogeneous model, is developed. It is obtained by a Taylor expansion (limited to first order) of the drop temperature field with respect to H. A fully quasi-steady model is also presented. The complete model highlights three different types of dynamics when a drop evaporates, depending on the relative values of three temperatures: the initial drop temperature, the dew point temperature of the gas phase far from the drop and the established interfacial temperature (i.e. the drop interfacial temperature calculated using the fully quasi-steady model). The evaporation can occur with a faster initial transient stage, with a slower initial transient stage or even with an initial condensation transient stage. Two dimensionless numbers are highlighted by the developed models: H and R characterize the thermal homogenization transient of the drop and the relaxation transient of the drop interfacial temperature, respectively. H is an order of magnitude of the dimensionless thermal homogenization time of the drop, while R is an order of magnitude of the dimensionless thermal relaxation time of the drop interfacial temperature. Results are presented for two liquids showing a large difference between their R values and between their H values: water and n-hexane (see Appendix A). The complete model is considered as being the reference, and the ability of the quasi-homogeneous and fully quasi-steady models to predict the results obtained by the complete model was discussed. It was shown that the fully quasi-steady model is able to provide a good estimation of the drop evaporation time only for liquids with H < 1 and presenting small values of R compared to one (at least an order of magnitude smaller). A simple formula is proposed to evaluate the relative difference between the drop evaporation times predicted by the complete model and by the fully quasi-steady model. Thus, the use of this formula with the fully quasi-steady model enables estimating the drop evaporation time given by the complete model for all the liquids and evaporation conditions considered in the present work. Moreover, the quasi-homogeneous model, although developed for H  1, also appears to provide a very good estimation of the drop evaporation time, for all the liquids considered in this work. By definition, the fully quasi-steady model does not provide any information about the thermal transients. On the other hand, it has been shown that the quasi-homogeneous model yields a fair approximation of the time evolutions of the interfacial temperature and of the temperature field inside the drop for liquids and evaporating conditions leading to values of H K 0:3. However, among the liquids considered in this work, this condition is only fulfilled for water. Therefore, it appears that the use of the complete model is often needed when an accurate estimation of the evolution of the temperature within the drop is required by the considered application. Using the complete model, the influence of the evaporation conditions on the thermal transients has been analyzed. The ratio of the thermal relaxation time to the drop evaporation time increases with an increase of T 1 and with a decrease of X 1 . It has been shown that, in the same evaporation conditions, liquids

with larger values of R present a thermal relaxation transient that takes place during a larger part of the drop evaporation process. For instance, the thermal relaxation time of a water drop can reach up to 25% of its evaporation time, while the thermal relaxation time of a n-hexane drop can reach up to 85% of its evaporation time. The ratio of the thermal homogenization time to the drop evaporation time can reach zero (i.e. the drop stays thermally homogeneous), for values of T 1 and X 1 leading to an established interfacial temperature equal to the initial drop temperature. When deviating from these critical conditions, a thermal gradient develops in the drop. It has been shown that, in the same evaporation conditions, liquids with larger values of H present a thermal homogenization transient that takes place during a larger part of the drop evaporation process. For instance, the thermal homogenization time of a water drop can reach up to 45% of its evaporation time, while a n-hexane drop can stay thermally inhomogeneous during almost its entire evaporation process. The maximum temperature difference that develops inside the drop (DT max ) is usually reached very soon during the evaporation ‘ is often far from being negligible (it can reach sevprocess. DT max ‘ eral tens of degrees). For a given liquid and a given value of T 1 , this maximum temperature difference is reached instantaneously is simply equal to T b  T ‘;0 . when X 1 ¼ 1. In this situation, DT max ‘ In view of the modeling of spray drying processes, it should be interesting to consider, in a future work, the evaporation of a drop made of a mixture of a volatile solvent and a non volatile component (either soluble or not). On the other hand, it is worth noting that the detailed analysis of drop thermal transients carried out in this paper is of interest for predicting a possible temporary onset of Marangoni instability in a suspended evaporating drop [27]. Moreover, as there is a relative motion between the drop and its environment in many applications, it should certainly be of strong interest to include an external flow around the drop in a future modeling, following for instance an approach similar to that used by Abramzin and Sirigano [28]. Acknowledgements This work was financially supported by the Fonds de la Recherche Scientifique – F.N.R.S. (Research Fellowship of PT, Senior Research Associate Position of PC, F.R.F.C. – ODILE contract nr 24623), by the Interuniversity Attraction Poles Programme (IAP 7/38 MicroMAST) initiated by the Belgian Science Policy Office (BELSPO), and by the European Space Agency (ESA) through the BELSPO Prodex ‘‘Heat Transfer” project. This work was also conducted under the umbrella of COST MP1106 Action. Appendix A. Estimation of the dimensionless numbers

Table 1 Values of the various characteristic dimensionless numbers of the problem for several liquids evaporating into air. These values are obtained for T ‘;0 ¼ 50  C; T 1 ¼ 200  C and X 1 ¼ 0. Liquids

H

R

Acetone Benzene Ethanol n-Hexane Methanol Water

0.61 0.44 0.45 0.72 0.34 0.18

0.11 0.09 0.07 0.20 0.06 0.05

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