2.1. Constitutive relations

Both liquid and gas phases are considered ideal mixtures with heat capacity and enthalpy calculated as $c_{p_{\beta }}=\sum ^{N_{\beta }} Y_{\beta ,i}c_{p_{\beta ,i}}$ and $h_{\beta }=\sum ^{N_{\beta }} Y_{\beta ,i}h_{\beta ,i}$, in terms of the heat capacity $c_{p_{\beta ,i}}$ and thermal enthalpy $h_{\beta ,i}$ of species $i$ in phase $\beta$.

The thermodynamic properties $c_{p_{\beta ,i}}$ and $h_{\beta ,i}$ of pure species are obtained using the NASA polynomials (McBride Reference McBride1993), where the coefficients are obtained whenever possible from the San Diego mechanism database (UCSD 2016). Data for those species not available in the San Diego database were taken from Burcat's database (Goos, Burcat & Ruscic Reference Goos, Burcat and Ruscic2010).

The gas-phase molecular transport coefficients $D_{{g},ij}$ and $k_{{g},i}$ are obtained using the expression derived directly from the kinetic theory (Hirschfelder et al. Reference Hirschfelder, Curtiss and Bird1964) using the transport database of the San Diego mechanism (UCSD 2016), while $k_{{g}}$ is obtained using the standard mixture average formula (Mathur, Tondon & Saxena Reference Mathur, Tondon and Saxena1967):

(2.12)\begin{equation} k_{{g}} = \frac{1}{2}\left( \sum_{i=1}^{N_{g}}X_{{g},i}k_{{g},i} + \frac{1}{\displaystyle\sum_{i=1}^{N_{g}}X_{{g},i}/k_{{g},i}}\right). \end{equation}

For the liquid phase, the mixture thermal conductivity $k_{\ell }$ was obtained from a generalization of Filippov's equation (Filippov Reference Filippov1955):

(2.13)\begin{equation} k_\ell = \sum_{i=1}^{N_\ell} Y_{\ell,i} \left( k_{\ell,i} - \sum_{j=i+1}^{N_\ell}K_{i,j}Y_{\ell,j}\left| k_{\ell,i}-k_{\ell,j} \right| \right), \end{equation}

where Filippov's constant is $K_{i,j}=0.72$. The conductivity of the pure species is computed using the correlation (Svehla Reference Svehla1995)

(2.14)\begin{equation} \log { k_{\ell,i} } = A_{k,i} \log(T) + \frac{B_{k,i}}{T} + \frac{C_{k,i}}{T^{2}} + D_{k,i} + E_{k,i} T + F_{k,i} T^{2}, \end{equation}

in which the coefficients are obtained by fitting with the experimental data published in Kadlec et al. (Reference Kadlec, Henke and Bubnik2010), Engineering ToolBox (2013), Assael et al. (Reference Assael, Charitidou, de Castro and Wakeham1987), Burgdorf et al. (Reference Burgdorf, Zocholl, Arlt and Knapp1999), Tanaka et al. (Reference Tanaka, Itani, Kubota and Makita1988) and Dortmund Data Bank (2019). The viscosity of the liquid mixture is evaluated using the Grunberg and Nissan equation: (Grunberg & Nissan Reference Grunberg and Nissan1949)

(2.15)\begin{equation} \mu_{\ell}=\exp\left( \sum_{i=1}^{N_\ell}X_{\ell,i} \ln\mu_{\ell,i} \right), \end{equation}

where the viscosity of the pure species is obtained using an expression analogous to (2.14) for the experimental results of Kadlec et al. (Reference Kadlec, Henke and Bubnik2010), Engineering ToolBox (2013), Sagdeev et al. (Reference Sagdeev, Fomina, Mukhamedzyanov and Abdulagatov2013), Michailidou et al. (Reference Michailidou, Assael, Huber, Abdulagatov and Perkins2014), Caudwell et al. (Reference Caudwell, Trusler, Vesovic and Wakeham2004), Koller et al. (Reference Koller, Klein, Giraudet, Chen, Kalantar, van der Laan, Rausch and Froba2017) and Wohlfarth (Reference Wohlfarth2008). Even though the momentum equation is not integrated and no viscous terms are included in the formulation, the liquid-phase viscosity is required in (2.9) to obtain the effective diffusivity of the species in the liquid phase. The numerical values of the coefficients $A, B, C, D, E$ and $F$ for density, conductivity and viscosity of the liquid-phase species are given in the Appendix.

2.2. Boundary conditions

Boundary conditions are required at the centre of the droplet and in the far field:

(2.16)\begin{gather} r=0: \quad \frac{\partial T_{\ell}}{\partial r}=\frac{\partial Y_{\ell,i}}{\partial r}=u_{\ell}=0, \end{gather}

(2.17)\begin{gather}r \rightarrow \infty: \quad T_{{g}}-T_{\infty}= Y_{{g},i}-Y_{\infty,i}=0, \end{gather}

while the boundary conditions at the liquid–gas interface are obtained by imposing the continuity of the temperature field and the conservation of species mass and energy in a control volume extending from $r = a(t) - \delta$ to $r = a(t) + \delta$ in the limit $\delta \rightarrow 0$, yielding

(2.18)\begin{align} -{\dot m}''(Y_{{g},i}-Y_{\ell,i})_{r=a}&=-(J_{{g},i}-J_{\ell,i})_{r=a},\quad i=2,\ldots,N_{\ell}, \end{align}

(2.19)\begin{align} -{\dot m}''(Y_{{g},i})_{r=a} &=-(J_{{g},i})_{r=a},\quad i=N_{\ell}+2,\ldots,N_{{g}}, \end{align}

(2.20)\begin{align} \left( T_{\ell} \right)_{r=a} &= \left( T_{{g}} \right)_{r=a} = T_s, \end{align}

(2.21)\begin{align} -\dot m'' \sum_{i=1}^{N_l} \left( Y_{\ell,i} {L}_i(T_s) \right)_{r=a} &= \left( k_{{g}} \frac{\partial T}{\partial r} - k_{\ell} \frac{\partial T}{\partial r} \right)_{r=a} \nonumber\\ &\quad +\alpha_{{eff}}\sigma\left(T_\infty^{4}-T_s^{4}\right) - \sum_{i=1}^{N_l} \left( J_{\ell,i} {L}_{i}(T_s) \right)_{r=a}, \end{align}

where $a(t)$ is the instantaneous time-dependent radius of the droplet at a generic time $t$, determined by the surface mass balance

(2.22)\begin{equation} {{ -{\dot m}''=\rho_{\ell}(u_{\ell}-{\dot a})_{r=a}=\rho_{{g}}(u_{{g}}-{\dot a})_{r=a}, }}\end{equation}

where ${{-\dot m''}}$ is the mass vaporization rate per unit of surface area, $T_s$ is the droplet surface temperature and ${\dot a}={\textrm {d}}a/{\textrm {d}}t$. The vaporization heat of each species is calculated as ${L}_i(T_s) = h_{{g},i}(T_s)-h_{\ell ,i}(T_s) + {L}_{i}^{{ref}}$, with ${L}_i^{{ref}}$ representing the vaporization heat at the reference temperature. Notice that an optically thin radiation model is included in our model through (2.21), where $\sigma$ is the Stefan–Boltzmann constant and $\alpha _{{eff}}=0.93$ is the effective absorption coefficient as found by Yang & Wong (Reference Yang and Wong2001) and Tseng & Viskanta (Reference Tseng and Viskanta2005).

Additionally, imposing the conservation of the chemical potential at the interface we obtain the Clausius equation (Criado-Sancho & Casas-Vázquez Reference Criado-Sancho and Casas-Vázquez1997):

(2.23)\begin{equation} (Y_{{g},i})_{r=a}=\left(Y_{\ell,i}\frac{W_{\ell}}{W_{{g}}}\right)_{r=a} \frac{p_{{atm}}}{{{p(r=a)}}} \gamma_i \exp \left( { \int_{T_{{b},i}}^{T_s}\frac{{L}_i(T)}{R_{{g},i}T^{2}}\mathrm{d}T } \right),\quad i=1,\ldots, N_{\ell}, \end{equation}

where $T_{{b},i}$ is the boiling temperature at atmospheric pressure ($p_{{atm}}=101\,325$ Pa) and $R_{{g},i} = \mathcal {R}/W_i$ and $\gamma _i$ are, respectively, the specific gas constant and the activity coefficient for species $i$, obtained using the UNIFAC method (Fredenslund, Gmehling & Rasmussen Reference Fredenslund, Gmehling and Rasmussen1977). The pressure of the fluid at the droplet surface is equal to the ambient pressure $p(r=a)=p_\infty =101\,325$ Pa in the low-Mach-number approximation used in the paper.

2.3. Initial conditions

We consider here the simplest case of a droplet with uniform composition and temperature that is placed at $t=0$ in an infinitely large homogeneous gaseous ambient. We assume that at $t=0$ the droplet, of uniform temperature and composition, is suddenly placed unperturbed in the gaseous ambient:

(2.24)\begin{equation} t=0: u_\ell = Y_{\ell,i} - Y_{\ell,i,0} = T_{\ell} - T_{d_0} =0;\quad i=1,\ldots, N_{\ell}, \end{equation}

Notice that, because of the quasi-steady assumption for the gas phase, it is not necessary to smooth the initial condition in the gas phase near the droplet interface, as was done in a previous work (Millán-Merino et al. Reference Millán-Merino, Fernández-Tarrazo, Sánchez-Sanz and Williams2020) to avoid numerical instabilities. The comparison between both approaches in this work provides further validation for the approximation used there.

2.4. Numerical method

The set of equations (2.1)–(2.6) together with the boundary conditions (2.16)–(2.21) and initial conditions (2.24) are discretized in a spherical domain using a second-order, finite-volume discretization for the spatial derivatives and a first-order backward Euler discretization for the temporal derivatives. The resulting set of equations is solved using a modified Newton–Raphson method that minimizes an error function $f$ formed subtracting the left- and right-hand-side terms of the equations. A non-uniform grid with typically 80 points is used to discretize a fluid domain that spans $100 a_0$, with a maximum clustering of points at the gas–liquid interface where the maximum gradients are located. The minimum and maximum grid steps are ${\rm \Delta} r/a_0=0.05$ and ${\rm \Delta} r/a_0=28$, respectively.

Even though for this one-dimensional problem simpler alternatives may exist, we chose, in order to track the position of the gas–liquid interface, to implement a moving mesh method (Diddens Reference Diddens2017) that uses a two-step, predictor–corrector strategy that can be easily generalized to two- and three-dimensional geometries. Once the new value of the variables is known, we computed the new position of the interface ${\dot a}={\dot m}''/\rho _{\ell }+(u_{\ell })_{r=a}$ and the location of the grid points that conformed the new mesh using the recession velocity $v_s=r{\dot a}/a(t)$. The values of the variables are recalculated in the new grid and the procedure continues until the normalized difference between the interface position calculated in two consecutive iterations falls below $10^{-5}$. A detailed description of the numerical procedure can be found in Millán-Merino (Reference Millán-Merino2020).

3.1. A first estimate for the time scales

In droplet vaporization problems, it is particularly interesting to find the adequate scale $t^{*}_{{C}}$ to measure time by estimating the droplet lifetime. To do so, we consider here a small spherical droplet with uniform density and temperature vaporizing in a hot, radiating atmosphere, as sketched in figure 1. Leaving aside droplet dilation, the rigorous evaluation of the evaporation time implies the integration of the mass conservation equation for the liquid phase:

(3.1)\begin{equation} {{\dot m = \rho_\ell 4 {\rm \pi}a^{2} \frac{{\textrm{d}}a}{{\textrm{d}}t}={\dot m}'' 4 {\rm \pi}a^{2},}} \end{equation}

to be solved with initial conditions $a(t=0)=a_0$ and where ${{-{\dot m}''=\rho _{g} (u_{g}-{\dot a})}}$ is the evaporation rate per unit surface. A first estimate of the droplet lifetime (Crespo & Liñán Reference Crespo and Liñán1975; Liñán & Williams Reference Liñán and Williams1993) can be obtained comparing the two terms in (3.1) to give

(3.2)\begin{equation} t_{{C}}^{*} \sim \frac{\left(a_0^{3} \rho_{\ell}\right)}{\left(-a_0^{2} \dot m''\right) }\sim \frac{a_0^{2}}{D_{{th}, {g}}} \frac{\rho_{\ell}}{\rho_{{g}}} \frac{{L}_{b}}{c_{p_{g}}(T_{\infty}-T_{b})}, \end{equation}

where ${{-}}\dot m''\sim \rho _{{g}} D_{{th}, {g}} c_{p_{g}}(T_\infty -T_{b})/({L}_{b} a_0)$ was estimated assuming heat conduction is the dominant mechanism driving vaporization in (2.21). In the expression given above, the liquid density $\rho _\ell$ and the enthalpy of vaporization ${L}_{b}$ are evaluated at the boiling temperature $T_{b}$, the gas heat capacity is obtained as $c_{p_{g}} = (c_{p,\infty } + c_{p,{b}})/2$ and the thermal diffusivity $D_{{th},{g}}=k_{{g}}/\rho _{g} c_{p_{{g}}}$, where $k_{{g}} = (k_{{g},\infty } + k_{{g},{b}})/2$. Typical values of the physical properties for common liquid fuels are summarized in table 1.

Table 1. Physical properties of common liquid fuels. Note that $\beta _{b} = L_{b} W/\mathcal {R} T_{{b}}$ is the non-dimensional latent heat of vaporization. The Lewis number $L_e$ of a gaseous species in air is obtained from Smoke & Giovangigli (Reference Smoke and Giovangigli1991) when possible. For the remaining species, it was obtained from mixture average transport model (Kee et al. Reference Kee, Warnatz and Miller1983).

This expression is similar to that developed analytically by Spalding (Reference Spalding1959), who gave the evaporation time in terms of the transfer number $B_{T_{b}}=c_{p_{g}}(T_\infty -T_{b})/{L}_{b}$ yielding

(3.3)\begin{equation} \tilde{t}_{{C}} \sim \frac{a_0^{2}}{{{D_{{th},{g}}}}} \frac{\rho_{\ell}}{\rho_{{g}}} \frac{1}{2\log(1+B_{T_{b}})}. \end{equation}

Except for a constant of order unity, this expression reduces to (3.2) in the limit $B_{T_{b}}\ll 1$. To quantify the relative importance of radiation, we define here the radiative evaporation time as the characteristic time to evaporate a droplet of radius $a_0$ when the radiation coming from the surrounding ambient at temperature $T_\infty$ is the dominant mechanism heating the droplet in (2.21), yielding

(3.4)\begin{equation} {{{\tilde t_{{R}}} \sim \frac{\left( a_0^{3} \rho_{\ell}\right)}{\left( -a_0^{2} \dot m''\right) }\sim \frac{\rho_{\ell} a_0 {L}_{b}}{\alpha_{eff} \sigma (T^{4}_{\infty}-T_{b}^{4})}. }} \end{equation}

Traditionally, the role of radiation in the evaporation of liquid fuel droplets has been neglected, a limit that implicitly considers $\tilde t_{{R}} \gg \tilde {t}_{{C}}$. This hypothesis, as we show below, fails for sufficiently large droplets or high ambient temperatures when $\tilde t_{{R}}/ \tilde {t}_{{C}} = \textit {O}(1)$. Examples of this effect are the numerical simulations carried out by Abramzon & Sazhin (Reference Abramzon and Sazhin2006), Dombrovsky et al. (Reference Dombrovsky, Sazhin, Sazhina, Feng, Heikal, Bardsley and Mikhalovsky2001), Lage & Rangel (Reference Lage and Rangel1993) and Tseng & Viskanta (Reference Tseng and Viskanta2006). They showed that the temporal evolution of the square of the droplet radius clearly deviates from the $d^{2}$-law in droplets of initial radius larger than $a_0 \sim 0.5$ mm and $T_\infty =1000$ K but did not make any comment about it in their papers.

3.2. Radiation effect: an analytical solution

The characteristic evaporation time $\tilde {t}_{{C}}$ given above in (3.3) provides a valid approximation for the evaporation time when $T_{\infty }-T_{b}$ is sufficiently large. If the ambient temperature $T_{\infty }$ is near or below $T_{b}$, this estimate does not take into account that vaporization occurs even if the droplet surface temperature is far below $T_{b}$. In these cases, we can develop a better estimation of the vaporization time using the quasi-steady approximation for both the gas and the liquid phases, as was proposed by Spalding (Reference Spalding1959), but retaining the effect of thermal radiation at the interface. To do so, we consider a spherical droplet of pure fuel vaporizing in an inert nitrogen atmosphere with temperature $T_\infty$, assuming, for these refined estimates, constant properties in the gas phase and homogeneous mass fractions and temperature in the liquid phase, with the latter remaining equal to the surface temperature $T_s$.

Under these conditions, the droplet behaves as a zero-heat-capacity liquid, coupled to the gas only through its instantaneous radius $a(t)$. The quasi-steady state of the gas is then computed without reference to a time variable to determine $T_s$ and $-{\dot m}$. As a result, the time dependence only comes in through (3.1) when the time rate of change of the radius $a$ is linked to the evaporation rate $-{\dot m}$. The surface temperature $T_s$ is then fully determined by the diffusive/radiative state of the gas and needs no initialization. The problem then reduces to integrating mass, species and energy conservation equations in the gas phase, yielding

(3.5)\begin{gather} \frac{\mathrm{d} }{\mathrm{d} r} \left( \rho_{g} u_{g} r^{2} \right) =0, \end{gather}

(3.6)\begin{gather}\frac{\mathrm{d} }{\mathrm{d} r} \left( \rho_{g} u_{g} r^{2} Y \right) = \frac{\mathrm{d} }{\mathrm{d} r} \left( \rho_{g} r^{2} D_{{g}} \frac{\mathrm{d} Y }{\mathrm{d} r} \right), \end{gather}

(3.7)\begin{gather}\frac{\mathrm{d} }{\mathrm{d} r} \left( \rho_{g} u_{g} r^{2} c_{p_{g}}T \right) =\frac{\mathrm{d} }{\mathrm{d} r} \left(r^{2} k_{g} \frac{\mathrm{d} T }{\mathrm{d} r} \right), \end{gather}

with the boundary conditions

(3.8)\begin{gather} r \to \infty:\quad Y=T-T_\infty=0, \end{gather}

(3.9)\begin{gather}r = a:\quad \rho_{{g}} u_{{g}} = -\dot m'' = - \frac{\dot m}{{{4 {\rm \pi}}} a^{2}}, \end{gather}

(3.10)\begin{gather}T-T_s=Y-Y_s=0, \end{gather}

(3.11)\begin{gather}-\frac{{\dot m}}{{{4 {\rm \pi}}} a^{2}} (Y_s-1) =\left(\rho_g D_{g} \frac{\mathrm{d} Y}{\mathrm{d} r}\right)_{r=a}, \end{gather}

(3.12)\begin{gather}-\frac{{\dot m}}{{{4 {\rm \pi}}} a^{2}} {L} =\left(k_{g} \frac{\mathrm{d} T}{\mathrm{d} r}\right)_{r=a}+ \alpha_{eff} \sigma (T^{4}_\infty -T^{4}_s), \end{gather}

(3.13)\begin{gather}\frac{Y_s}{W+Y_s(1-W) } = \exp{ \left( \int_{T_{b}}^{T_s} \frac{{L}(T)}{R_{{g},F}T^{2}} {\textrm{d}}T \right)}, \end{gather}

where $W=W_F/W_{N_2}$ is the fuel-to-nitrogen molecular mass ratio and $R_{g,F} = \mathcal {R}/W_F$ the specific fuel gas constant. Notice that (3.11), (3.12) and (3.13) correspond to the more general equations (2.18), (2.21) and (2.23), respectively, taking into account the simplifications introduced in this section. The system (3.5)–(3.13) is complemented with the mass conservation equation for the liquid phase (3.1) given above.

The gas-phase velocity $u_{{g}}$, species mass fraction $Y$ and temperature $T$ can be obtained in terms of the yet unknown evaporation rate $\dot m$, droplet surface temperature $T_s$ and mass fraction $Y_s$. To do so, we first integrate equation (3.5) with the boundary condition (3.9), to obtain $\rho _{g} u_{g} r^{2} = -{\dot m}/{{4 {\rm \pi}}} = -a^{2} {\dot m}''$. Temperature and mass fraction are then obtained integrating (3.6) and (3.7) with conditions (3.8) and (3.10), yielding

(3.14a,b)\begin{equation} {{\frac{T-T_\infty}{T_s-T_\infty} = \frac{1-\exp(-\varLambda a/r)}{1-\exp(-\varLambda)} \quad \mathrm{and} \quad \frac{Y}{Y_s} = \frac{1-\exp(-\varLambda L_e a/r)}{1-\exp(-\varLambda L_e)},}}\end{equation}

with ${{{\varLambda =-({\dot m c_{p {g}}}/{{4 {\rm \pi}} a k_{{g}}})}}}$ and ${L_e= {k_{{g}} }/{\rho _{{g}} D_{F, {g}}c_{p{g}} } }$ being the Lewis number of the fuel. The expressions (3.14a,b) are later substituted in (3.11)–(3.13) to form an implicit nonlinear equation system for $T_s$, $Y_s$ that can be reduced to

(3.15)\begin{gather} \exp{ \left( -\int_{T_{b}}^{T_s} \frac{\mathrm{L}(T)}{R_{{g},F}T^{2}} {\textrm{d}}T \right) } = 1+ \frac{W}{ \exp(\varLambda L_e) -1}, \end{gather}

(3.16)\begin{gather}\varLambda = \log \left[1 + \frac{B_{T}}{1 - \dfrac{2 \log(1+B_{T_{{b}}})}{\varLambda} \dfrac{\tilde{t}_{C}}{\tilde t_{R}} \dfrac{a}{a_0} } \right], \end{gather}

where $\tilde t_{{C}}$ is given by (3.3) and $B_T= c_{p_g} (T_\infty -T_s)/{L}(T_s)$. In deriving (3.16), we made the approximation $({L}_{{b}}/{L}) [(T_\infty ^{4} - T_s^{4})/ (T_\infty ^{4} - T_{{b}}^{4}) ] \approx 1$.

Finally, once $T_s$ and $Y_s$ are known, the temporal evolution of the droplet radius is obtained using the global mass conservation equation (3.1):

(3.17)\begin{equation} {{\frac{\mathrm{d}a^{2}}{\mathrm{d} t} = -{2 \varLambda}{D_{{th}, {g}}} \frac{\rho_{{g}}}{\rho_{\ell}}. }} \end{equation}

The relative importance of radiation is measured in (3.16) through the parameter

(3.18)\begin{equation} \varepsilon= \frac{\tilde t_{{C}}}{\tilde t_{{R}}}= \frac{a_0 \alpha_{{eff}} \sigma}{{{D_{{th}, {g}}}}{L}_{b} \rho_{{g}} } \frac{T_{\infty}^{4}-T_{{b}}^{4}}{2\log(1+B_{T_{b}})}. \end{equation}

The classical $d^{2}$-law developed by Spalding (Reference Spalding1959) predicts a linear decay of the square of the droplet radius with time $(a/a_0)^{2}=1-C_{C} t$, with $C_{C}$ a known constant. As can be checked in figure 2, where we plot the evolution with time of the surface temperature $T_s$, the square of the droplet radius $a^{2}$ and the vaporization rate obtained by integrating (3.15)–(3.17), the evolution of the droplet clearly deviates from the $d^{2}$-law when the ratio $\tilde {t}_{{C}}/{\tilde t_{{R}}}$ is increased. The curves shown in figure 2 are actually a family of parallel curves plotted at different scales $a_0$. From (3.15) and (3.16) we can see that both $T_s$ and $\varLambda$ can be computed once both $T_\infty$ and the instantaneous droplet radius $a(t)$ are known, independently of the initial droplet size $a_0$. Consequently, the right-hand side of (3.17) becomes identical for two droplets with different initial diameter once they reach the same instantaneous radius, anticipating the same temporal evolution thenceforward.

Figure 2. Ethanol droplet vaporization in a hot nitrogen atmosphere at ambient temperature and pressure of $T_\infty =800$ K and $p_\infty = 1$ atm, respectively. (a) The normalized droplet surface $(a/a_0)^{2}$ versus dimensionless time ${t/t_{C}}$. (b) The normalized droplet surface $(a/a_0)^{2}$ versus surface temperature $T_s$. (c) Dimensionless gasification rate $-\mathrm {d}(a/a_0)^{2}/\mathrm {d} (t/t_{C})$ as a function of the dimensionless time $t/t_{C}$. Different line styles are chosen for each value of $\varepsilon =\tilde t_{C}/\tilde t_{R}$, as shown in the figure legend. Thick colour lines represent the solution of (3.15)–(3.16), thin grey lines depict the asymptotic prediction $\varepsilon \ll 1$ given in (3.19) and thin black lines represent the asymptotic prediction $\varepsilon \gg 1$ defined by (3.28).

3.2.1. Case of small radiation effects $\varepsilon =\tilde {t}_{{C}}/{\tilde t_{{R}}} \ll 1$

As shown in figure 2, in the limiting case in which radiation heating is negligible $\varepsilon =\tilde {t}_{{C}}/{\tilde t_{{R}}} \ll 1$, the evolution of the droplet diameters follows the classical $d^{2}$-law. This can be seen easily from the above system of equations by introducing the expansion

(3.19)\begin{equation} f=f_1 + \varepsilon f_2 + O(\varepsilon^{2}) \end{equation}

in (3.15)–(3.17), with $f=(\varLambda , a^{2}, T_s)$. In the first order we get $\varLambda _1=\log (1+B_{T_1})$, $B_{T_{1}}=c_{p_{g}}(T_\infty -T_{s,1})/{L}_{b}$ and $T_{s_1}$ computed by solving the implicit equation

(3.20)\begin{equation} \exp \left(\frac{{L} (T_{b} - T_{s_1})}{R_{{g},F} T_{b} T_{s_1}} \right) = 1+ \frac{W}{(1+B_{T_1})^{L_e} -1} \end{equation}

derived from (3.15) assuming constant vaporization heat ${L}$. As is clear from this expression, to a first approximation the surface droplet temperature $T_{s_1}$ remains constant during the whole evaporation period and is independent of the initial droplet radius $a_0$. Once $T_{s_1}$ is known we can easily determine the vaporization rate using (3.17) to recover the $d^{2}$-law

(3.21)\begin{equation} \left(\frac{a_1}{a_0} \right)^{2} = 1- \frac{\log(1+B_{T_1})}{\log(1+B_{T_{b}})} \frac{t}{\tilde{t}_{{C}}} \end{equation}

depicted in figure 2 after solving numerically the system of (3.15)–(3.16) for $\varepsilon \ll 1$. The droplet evaporation time $t_{{C}}$ is then easily obtained to give, to a first approximation,

(3.22)\begin{equation} t_{{C}} = \frac{a_0^{2}}{{{D_{{th}, {g}}}}} \frac{\rho_{\ell}}{\rho_{{g}}}\frac{1}{2 \log{\left( 1 + B_{T_1}\right)} }. \end{equation}

The explicit procedure indicated here is a major difference with respect to previous models (Abramzon & Sirignano Reference Abramzon and Sirignano1989; Sazhin Reference Sazhin2006), which required the resolution of a coupled system of equations at each time step. Higher orders of the solution can be computed to give first-order corrections due to the presence of radiation, yielding $T_{s_2}= C_1 (a_1/a_0)$, $\varLambda _2= C_2 (a_1/a_0)$ and $(a_2/a_0)^{2}=C_3 [(a_1/a_0)^{3}-1]$, with

(3.23)\begin{gather} C_0 = \frac{L_e W T_{s_1}}{T_\infty-T_{s_1}} \frac{\textrm{e}^{L_e \varLambda_1}}{(\textrm{e}^{L_e \varLambda_1}-1)^{2}}+ \frac{B_{T_1}+1}{B_{T_1}}\frac{{L}}{R_{{g},F}T_{s_1}} \exp{\left(\frac{{L}(T_{b}-T_{s_1})}{R_{{g},F}T_{b}T_{s_1}}\right)}, \end{gather}

(3.24)\begin{gather}C_1 = \frac{2}{C_0} L_e W \frac{\log(1+B_{T_{b}})}{\log(1+B_{T_1})} \frac{\textrm{e}^{L_e \varLambda_1}}{(\textrm{e}^{L_e \varLambda_1}-1)^{2}} T_{s_1}, \end{gather}

(3.25)\begin{gather}C_2 = \frac{2}{C_0} \frac{{L}}{R_{{g},F}T_{s_1}} \frac{\log(1+B_{T_{b}})}{\log(1+B_{T_1})} \exp{\left(\frac{{L}(T_{b}-T_{s_1})}{R_{{g},F}T_{b}T_{s_1}}\right)}, \end{gather}

(3.26)\begin{gather}C_3 =\frac{2}{3}\frac{C_2}{\log(1+B_{T_1})}. \end{gather}

The accuracy of the asymptotic predictions (3.19) and the radiation-induced departures from the $d^{2}$-law are clearly depicted in figure 2 for $\varepsilon =( 0.1, 0.25)$. As expected, the asymptotic expansion fails when the droplet radius $a/a_0 = O(\varepsilon )$.

3.2.2. Case of dominant radiation $\varepsilon = \tilde t_{{C}}/{\tilde t_{{R}}} = 1/\delta \gg 1$

In this limiting case, radiation dominates the evaporation process and (3.16) provides the scale for the parameter $\varLambda = \textit {O}(1/\delta ) \gg 1$. From (3.15), and taking into account that $\varLambda > 0$, we obtained $T_s = T_b$ to a first approximation, in excellent agreement with the numerical results shown in figure 2 for $\delta = 1/\varepsilon = 1/6$. Therefore, $B_T \simeq B_{T_{{b}}}$ and taking the limit $\delta \to 0$ in (3.16), we obtain

(3.27)\begin{equation} \varLambda \delta = 2 \log (1 + B_{T_{{b}}} ) \frac{a}{a_0}, \end{equation}

which allows the integration of (3.17) to afford a linear droplet diameter time evolution,

(3.28)\begin{equation} \frac{a}{a_0} = 1 - \frac{t}{t_{{R}}}, \end{equation}

which yields the same evaporation time $t_{{R}} = \tilde t_{{R}}$ estimated above in (3.4).

3.2.3. Case $\varepsilon = \tilde t_{{C}}/{\tilde t_{{R}}} = \textit {O} (1)$

In the more general case $\tilde {t}_{{C}}/{\tilde t_{{R}}} = \textit {O} (1)$, (3.16) clearly indicates that the decay of the square of the droplet radius $(a/a_0)^{2}$ is not linear with time and the $d^{2}$-law is not satisfied even when the problem is quasi-steady. The nonlinear evolution of $a^{2}$ becomes evident in figure 2 for $\tilde {t}_{{C}}/{\tilde t_{{R}}} >0.25$, when the evaporation rate and the surface temperature changed substantially with the droplet diameter. Notice that, even in the case $\tilde {t}_{{C}}/{\tilde t_{{R}}} = \textit {O} (1)$, the $d^{2}$-law is recovered in the last stages of the vaporization when $a/a_0 \ll 0$ with the slope of the curves abruptly changing to adopt a linear evolution and the droplet temperature becoming independent of the initial droplet radius $a_0$, as shown in figure 2 and anticipated by (3.20).

Even relatively small radiation effects produce a deviation from the $d^{2}$-law. To take into account this effect, hereafter we measure time in units of the droplet vaporization time,

(3.29)\begin{equation} 1/t_{V} = 1/ t_{{C}} + 1/t_{{R}} , \end{equation}

developed integrating the conservation equations to improve the estimation of the characteristic times scales given above in (3.2) and (3.3). Equation (3.29) corresponds to an interpolation between $t_{{C}}$ and $t_{{R}}$ with proper asymptotic behaviour for large values of either $t_{{C}}$ or $t_{{R}}$. To test the new scaling, we introduce $\tau =t/t_{V}$ in figures 3, 4 and 5, where we depict the temporal evolution of the droplet diameter at different initial ambient temperature. As can be seen in these figures, all curves nearly collapse into a single curve for $n$-heptane and ethanol droplets. A summary of the different time scales is presented in table 2.

Figure 3. Normalized droplet surface as a function of time for $n$-heptane droplets at atmospheric pressure $p_\infty =1$ bar and initial droplet temperature $T_{d_0}=300$ K. (a) Plots of $(a/a_0)^{2}$ versus the normalized time, $t/a_0^{2}$. (b) Plots of $(a/a_0)^{2}$ versus the dimensionless time, $\tau =t/t_{V}$, with $t_{V}$ given by (3.29). The colours denote the parameters $(T_\infty , a_0, t_{C}/t_{R})$: $\textrm {red}=(471\ \textrm {K},\ 0.35\ \textrm {mm},\ 0.10)$, $\textrm {blue}=(555\ \textrm {K},\ 0.35\ \textrm {mm},\ 0.16)$, $\textrm {green}=(741\ \textrm {K},\ 0.35\ \textrm {mm},\ 0.34)$, $\textrm {purple}= (1050\ \textrm {K},\ 0.12\ \textrm {mm},\ 0.30)$. Solid lines: quasi-steady gas-phase simulations. Dashed lines in (b): full transient simulations using the formulation described by Millán-Merino (Reference Millán-Merino2020). Dash-dotted lines in (a): numerical results by Yang & Wong (Reference Yang and Wong2001). Circles: experiments by Nomura et al. (Reference Nomura, Ujiie, Rath, Sato and Kono1996). Triangles: experiments by Lee & Law (Reference Lee and Law1992).

Figure 4. Ethanol droplet vaporization in nitrogen atmosphere at pressure $p_\infty =1$ bar and initial droplet temperature $T_{d_0}=300$ K. (a) Normalized droplet surface $(a/a_0)^{2}$ as a function of the non-dimensional time $\tau =t/t_{V}$, with $t_{V}$ given by (3.29). (b) Dimensionless gasification rate $-\mathrm {d}(a/a_0)^{2}/\mathrm {d}\tau$ as a function of the dimensionless time $\tau$. The squares represent experimental results of Hallett & Beauchamp-Kiss (Reference Hallett and Beauchamp-Kiss2010) and solid lines represent our numerical results. The colours denote the parameters $(T_\infty , a_0, t_{C}/t_{R})$: $\textrm {green}=(703\ \textrm {K},\ 0.8\ \textrm {mm},\ 0.46)$, $\textrm {orange}=(893\ \textrm {K},\ 0.8\ \textrm {mm},\ 0.81)$, $\textrm {purple}=(1050\ \textrm {K},\ 0.7\ \textrm {mm},\ 0.99)$.

Figure 5. Ethanol droplet vaporization in a hot nitrogen atmosphere at ambient temperature and pressure of $T_\infty =800$ K and $p_\infty =1$ bar, respectively, and initial droplet temperature $T_{d_0}=300$ K. (a) Normalized droplet surface $(a/a_0)^{2}$ versus dimensionless time $\tau = t/t_V$, with $t_{{V}}$ given by (3.29). (b) Normalized droplet surface $(a/a_0)^{2}$ versus surface temperature $T_s$. (c) Dimensionless gasification rate $-\mathrm {d}(a/a_0)^{2}/\mathrm {d}\tau$ as a function of the dimensionless time $\tau$. Lines represent different initial radius $a_0$ as shown in the figure legend. The numbers indicate the time at which the radial profiles of temperature and mass fraction are shown in figure 6.

Table 2. Characteristic time definitions.

$^{a}$In the definition of $t_R$, $T_{s_1}$ substitutes $T_{b}$ for ambient temperatures $T_\infty \lesssim T_{b}$.

3.3. Time scales comparison

The classical asymptotic theory (Liñán & Williams Reference Liñán and Williams1993) considers the limit of large heat of vaporization ${L}_{b}/R_{g} T_{b}$ in which the evaporation time $t_{{C}}$ is much longer than the heat diffusive time in both the gas $t_{{{D_{{th},{g}}}}} \sim a_0^{2} / {{D_{{th},{g}}}}$ and the liquid $t_{{{D_{{th},\ell }}}} \sim a_0^{2} / {{D_{{th}, \ell }}}$ phases. In this limit it is assumed that $t_{{R}} \gg t_{{C}} \gg t_{{{D_{{th},\ell }}}} \gg t_{{{D_{{th},{g}}}}}$, so that the quasi-steady approximation is used in both phases and vaporization takes place with the liquid at the boiling temperature $T_{b}$.

As shown in table 3, this hypothesis fails for the liquid phase for moderately large ambient temperatures ($T_\infty$ above 600 K) when the liquid fuel droplets evaporate in a nitrogen atmosphere. The realistic estimations given in this table clearly show that the ambient gas remains in quasi-steady state for the whole range of ambient temperatures considered. This is not the case in the liquid phase as the thermal conduction-to-evaporation characteristic time ratio is of order unity $t_{{{D_{{th},\ell }}}}/t_{C} = \textit {O}(1)$ for $T_\infty >600$ K. Finally, as indicated by (3.18), radiation is negligible only at low temperatures (close to ambient temperature) or small initial droplet diameters. According to table 3, a more realistic ordering of the different time scales for ethanol droplets would be $t_{D_{\ell }} \gg t_{{R}} \sim t_{{{D_{{th},\ell }}}} \sim t_{{C}} \gg t_{{{D_{{th},{g}}}}} \sim t_{D_{{g}}}$, where $t_{D_{\beta }}=a_0^{2}/D_{\beta }$ is the mass diffusion time in phase $\beta$.

Table 3. Characteristic time relations for different ambient temperatures for ethanol, $n$-heptane, dodecane and hexadecane droplets, of initial radius $a_0=0.5$ mm, in nitrogen atmosphere. Liquid mass diffusion time for ethanol is evaluated for ethanol–water mixtures while for $n$-heptane the self-diffusion coefficient is used. Physical properties are evaluated at $T_s$, as obtained from (3.15).

Motivated by the order-of-magnitude analysis shown above, in the present work, we assume that the gas phase is quasi-steady but we consider the full transient problem for the liquid phase, as was already stated for the formulation in § 2. This approach will be compared later with the solution obtained using the full transient formulation for both gas and liquid phases described in Millán-Merino et al. (Reference Millán-Merino, Fernández-Tarrazo, Sánchez-Sanz and Williams2020).

4.1. Vaporization of anhydrous ethanol droplets

Liquid ethanol is a polar substance with a strong hydrophilic character. The –OH group forms hydrogen bonds to neighbouring polar molecules that make ethanol infinitely soluble in water. Ethanol droplets can be easily contaminated with ambient moisture condensing on the droplet surface, making droplet vaporization experiments technically very challenging. Alternatively, to keep the conditions of the experiment under control, numerical analysis emerges as a convenient alternative for analysing the particularities of ethanol vaporization. Keeping in mind the above-mentioned technical limitation, we validated our simulations by comparing in figure 4 our simulations with the experiments by Hallett & Beauchamp-Kiss (Reference Hallett and Beauchamp-Kiss2010) carried out in a nitrogen atmosphere. The agreement for droplets with initial radius $a_0=(0.7, 0.8, 0.8)$ mm and ambient temperature $T_\infty =(1023, 893, 703)$ K is notable with all curves collapsing into one once the scaling $\tau =t/t_V$ is introduced. Also, the droplet vaporization rate shown in figure 4(b) does not depict the quasi-steady vaporization rate that characterizes the $d^{2}$-law, confirming the departures from the ideal theory anticipated above in § 3.

To further investigate these results we plot in figure 5 the temporal evolution of the droplet radius, surface temperature and vaporization rate of pure ethanol droplets with radii $a_0=(0.05, 0.1, 0.25, 0.5, 1.0, 2.5)$ mm in a hot nitrogen atmosphere at $T_\infty =800$ K. Unlike the ratio

(4.1)\begin{equation} \frac{t_{{{D_{{th},\ell}}}}}{t_{{C}}}=O\left[\frac{\rho_{g} {{D_{{th},{g}}}}}{ \rho_\ell {{D_{{th},\ell}}}} \log(1+B_{T_1}) \right]=O(1), \end{equation}

which remains constant with the droplet size, the conduction-to-radiation characteristic time ratio increases proportionally to the droplet radius, yielding

(4.2)\begin{equation} \frac{t_{{C}}}{t_{{R}}} \sim \frac{\alpha_{eff}\sigma a_0}{2 \rho_g {{D_{{th},{g}}}} {L}_{b}} \frac{T^{4}_\infty - T^{4}_{{b}}}{\log\left(1+ B_{T_1} \right)}= \left\{\begin{array}{ll} 0.04 & a_0=0.05\ \mathrm{mm} \\ 0.74 & a_0=1\ \mathrm{mm}. \end{array}\right. \end{equation}

In accordance with this result, the numerical results depicted in figure 5 clearly show significant departures from the $d^{2}$-law for droplets with sufficiently large initial radius, $a_0>0.25$ mm. Both the surface temperature and the droplet vaporization rate do not achieve the steady state predicted by the $d^{2}$-law over the droplet lifetime. Leaving aside the initial heating period of the liquid phase, these numerical results are qualitatively identical to the theoretical predictions plotted in figure 2 and clearly illustrate the relevance of radiation in sufficiently large droplets.

Also, in figure 5 we numerically explore the effect of radiation by excluding the radiation term in (2.21) during the vaporization of a droplet with initial radius $a_0=2.5$ mm. The red dashed line in figure 5 represents the artificial case $\alpha _{{eff}}=0$ and clearly indicates the importance of radiation for the accuracy of the calculations. Its absence in the energy balance would lead to mistakenly predicting significantly lower droplet surface temperatures and a linear evolution of the square of the droplet diameter that do not match with the experimental measurements.

The radial profiles of the temperature within the liquid ethanol are plotted in figure 6 at the time instants indicated in figure 5 for droplet radii $a_0=0.05$ and 1 mm. In small droplets, the liquid temperature becomes uniform early in the droplet lifetime $\tau <0.2$, foreseeing a rapid evolution towards the linear temporal evolution that announces the $d^{2}$-law. Larger droplets with initial radius $a_0=1$ mm need a slightly longer time to achieve uniform temperature $\tau \gtrsim 0.45$ as shown in figures 5 and 6, but the temperature and vaporization rate are significantly higher than in smaller droplets as a consequence of the higher relative importance of radiation. For even larger droplets $a_0=2.5$ mm, uniform temperature in the liquid phase is not achieved until the droplet is almost completely vaporized, with the droplet surface temperature $T_s$ slightly below the temperature at the centre of the droplet, an effect also observed in figure 6 for $a_0=1$ mm. Notably, the droplet surface temperature becomes higher in larger droplets, approaching the boiling temperature $T_{b}=351.44$ K as the droplet radius $a_0$ is increased.

Figure 6. Ethanol droplet temperature profiles as a function of normalized radius $r/a_0$ for instants marked in figure 5. (a) Temperature profiles for initial droplet radius of $a_0=1$ mm at time instants 1 to 8 indicated in figure 5. (b) Temperature profiles for initial droplet radius of $a_0=0.05$ mm at time instants i to vi indicated in figure 5. The dot-dashed line represents the evolution of the surface temperature $T_s$.

5.1. Characteristic evaporation time

In the case of bicomponent droplets, the evaporation time $t_{{V}}$ given by (3.29) is no longer a good estimate of the droplet lifetime. A better estimate of the droplet lifetime can be obtained by considering that, within a droplet of initial radius $a_0$, the liquid components of the droplet are initially segregated. Assuming we only have two components, the less volatile would occupy a sphere of radius $a_1 < a_0$ in the innermost part of the droplet, while the most volatile component would fill the outermost spherical space between the radii $a_1$ and $a_0$. Additionally, for estimation purposes, we consider that the droplet components evaporate sequentially, with the most volatile component evaporating first in a time

(5.2)\begin{equation} \frac{1}{t_{{V}_1}} {{\approx}} \frac{(\mathrm{d} a^{2}/\mathrm{d}t)_{{C}_1}}{a_0^{2}- a_1^{2}}+ \frac{(\mathrm{d} a/\mathrm{d}t)_{{R}_1}}{a_0- a_1}, \end{equation}

and then, after complete evaporation of the outermost liquid, the less volatile component would evaporate in a time

(5.3)\begin{equation} \frac{1}{t_{{V}_2}} {{\approx}}\frac{(\mathrm{d}a^{2}/\mathrm{d}t)_{{C}_2}}{a_1^{2}}+ \frac{(\mathrm{d}a/\mathrm{d}t)_{{R}_2}}{a_1}, \end{equation}

with $(\mathrm {d} a^{2}/\mathrm {d}t)_{{C}_i}$ and $(\mathrm {d}a/\mathrm {d}t)_{{R}_i}$ computed using (3.21) and (3.28), respectively. The overall evaporation time is then obtained by adding the evaporation time of each fluid, yielding

(5.4)\begin{equation} t_{{V}} = t_{{V}_1} + t_{{V}_2}.\end{equation}

Using this scale for the characteristic vaporization time, the curves for different initial compositions collapse, as shown in figures 7 and 11 for ethanol–water and $n$-dodecane–$n$-hexadecane droplet vaporization, respectively.

Figure 7. Effect of initial water content in the droplet $\mathcal {V}$ and ambient relative humidity $\mathcal {H}$ on the evolution of the normalized droplet surface (a,b) and on the evolution of the time derivative of the normalized droplet surface (c,d). Solid lines: numerical simulations; symbols: experiments (Gregson et al. Reference Gregson, Ordoubadi, Miles, Haddrell, Barona, Lewis, Church, Vehring and Reid2019). All results correspond to the vaporization of a droplet of initial radius and temperature $a_0=24\ \mathrm {\mu }$m and $T_{d_0}=293$ K, respectively, at ambient temperature and pressure $T_{\infty }=293$ K and $p_\infty =1$ atm, respectively.

5.2. Moisture and water content effects on ethanol droplet vaporization

We next investigate the effect of water both in the initial droplet composition and in the ambient. This analysis is relevant for technological reasons. Since ethanol is hydrophilic, water from the ambient can easily be absorbed into a fuel tank. Moreover, in combustion applications, the recirculation of exhaust gases is a common practice to reduce flame temperature and emissions of nitric oxides. This gas contains a high content in water vapour that, in addition to ambient humidity, will affect droplet vaporization, as we show below. These changes can lead to variations in the combustion properties (Millán-Merino Reference Millán-Merino2020; Millán-Merino et al. Reference Millán-Merino, Fernández-Tarrazo, Sánchez-Sanz and Williams2020)

In figure 7 we show the effect of both the initial volumetric water content $\mathcal {V}= V_{{\textrm {H}_2\textrm {O}}}/V_{{droplet}}$ and ambient relative humidity $\mathcal {H}$, defined above in (5.1), on the evolution of a droplet evaporating in an inert nitrogen atmosphere at ambient temperature $T_{\infty }=293$ K. According to the order-of-magnitude analysis presented in (4.2), the small value of the parameter $t_{C}/t_{R}=0.002 \ll 1$ anticipates a negligible influence of radiation.

Figure 7(a,c) compares the experimental results with our computations considering a dry atmosphere $\mathcal {H}=0$ and different initial water contents $\mathcal {V}=(0, 0.24, 0.44)$. In all three cases, both the experimental and numerical results follow the classical $d^{2}$-law until the liquid ethanol is vaporized completely. At that instant of time, the slope of the vaporization curve swiftly changes giving way to a new, but also constant, vaporization rate.

The accuracy of the model at $T_\infty =293$ K in the presence of ambient humidity is tested in figure 7(b,d) against the experiments carried out by Gregson et al. (Reference Gregson, Ordoubadi, Miles, Haddrell, Barona, Lewis, Church, Vehring and Reid2019) considering, also, a non-negligible water content inside the droplet with $\mathcal {V}=0.44$. The match between the numerical and experimental results is notable, with the model correctly predicting the droplet lifetime and accurately describing the formation of a liquid water boundary layer after the condensing of ambient moisture that slowly diffuses towards the droplet centre, creating the concentration gradient depicted in figure 9.

Figure 8 analyses, separately, the effect of the droplet water content $\mathcal {V}$ and ambient relative humidity $\mathcal {H}$ on the vaporization of a droplet with initial radius $a_0=0.5$ mm placed in an atmosphere at initial temperature $T_\infty =800$ K. This figure shows the opposite effect that moisture and droplet water content have on the vaporization rates.

Figure 8. Effect of initial water content in the droplet $\mathcal {V}$ and ambient relative humidity $\mathcal {H}$ on the evolution of the normalized droplet surface (a), dimensionless gasification rate (c), net vaporization rate of ethanol (b) and net vaporization rate of water (d). All results correspond to full one-dimensional simulations for evaporation of a droplet of initial radius and temperature $a_0=1$ mm and $T_{d_0}=300$ K, respectively, at ambient temperature $T_{\infty }=800$ K and pressure $p_\infty =1$ bar. Green lines: $\mathcal {V}=0, \mathcal {H}=0$; blue lines: $\mathcal {V}=0.1, \mathcal {H}=0$; red lines: $\mathcal {V}=0.2, \mathcal {H}=0$; violet lines: $\mathcal {V}=0, \mathcal {H}=0.1$; orange lines: $\mathcal {V}=0, \mathcal {H}=0.2$.

In practical applications, liquid ethanol is commonly found with large quantities of dissolved water due to its hygroscopic character (Brown, Keates & Brown Reference Brown, Keates and Brown2011). To account for this effect, we analysed the evaporation of water–ethanol droplets with $\mathcal {V}=(0.1, 0.2)$. In all cases computed, the evaporation rate decreases, resulting in longer lifetimes as we scale up the parameter $\mathcal {V}$. The gasification rate of both water and ethanol follows a similar evolution, as is shown in figure 8(b,d), with a slow monotonic decay after peaking at the beginning of the vaporization process. The profiles of ethanol mass fraction and temperature within the droplet are depicted in figure 9(b,d) for the case $\mathcal {V}=0.2, \mathcal {H}=0$. The large value of the Lewis number of liquid ethanol in water $L_e={{D_{{th},\ell }}}/D_l \simeq 47.0$ at 333 K and the ratios of characteristic times

(5.5a,b)\begin{equation} \frac{t_{{{D_{{th},\ell}}}}}{t_{{C}}} \simeq 0.557 \quad \mathrm{and}\quad \frac{t_{D_\ell}}{t_{C} } \simeq 26.2 \end{equation}

predict the slow migration of the volatile ethanol to the surface of the droplet and the fast temperature homogenization illustrated in figure 9(b), when the temperature profile becomes uniform during the first quarter of the droplet lifetime. Both water and fuel are vaporized simultaneously keeping the mass fraction almost uniform until the end of the droplet life (figure 9).

Figure 9. Structure of the solution for ethanol–water droplets vaporizing in a humid atmosphere at instants marked in figure 8(a). (a,b) Ethanol mass fraction profiles; (c,d) temperature profiles. The orange lines in (a,c) correspond to the case $\mathcal {V}=0, \mathcal {H}=0.2$ and red lines in (b,d) correspond to the case $\mathcal {V}=0.2, \mathcal {H}=0$. Dashed lines show ethanol mass fraction (a,b) and temperature (c,d) at the droplet surface $r=a(t)$.

The effect of ambient relative humidity $\mathcal {H}$ is opposite to that of the water content. During the first stages of the vaporization, moisture condenses on the droplet surface (figure 8) initially increasing the vaporization rate of ethanol and the droplet surface temperature with respect to the case $\mathcal {H}=0$. The condensed water slowly builds a water-rich boundary layer that forms a concentration gradient inside of the droplet (figure 9a) that progressively reduces the gasification rate of ethanol until it falls below the gasification rate computed for a dry ambient ($\mathcal {H}=0$). As before, the temperature within the droplet becomes rapidly uniform, even after the sudden rise in the surface temperature observed when the droplet runs out of ethanol (figure 9c).

The relative importance of the heat release during the condensing of the water vapour on the droplet can be estimated using the coupling condition (2.21). Comparing the first and last terms on the right-hand side of (2.21), and making use of (2.7) and (2.23), yields

(5.6)\begin{equation} \frac{k_{g} \dfrac{\partial T}{\partial r}}{J_{g,\textrm{H}_2\textrm{O}}{L_{\textrm{H}_2\textrm{O}}}} \sim \frac{L_{e_{\textrm{H}_2\textrm{O}}} c_{p_{g}} (T_{\infty}-T_s)}{Y_{{\textrm{H}_2\textrm{O}},\infty} {L_{\textrm{H}_2\textrm{O}}}}\sim O(1). \end{equation}

Using the data collected in table 1 at $T_\infty =800$ K, it is easy to check that this ratio becomes of order unity only during the first stages of the evaporation process. This extra heat rapidly increases the droplet surface temperature, that suddenly jumps from the initial 300 K to reach 335 K almost instantaneously (figure 9), contributing to the accelerated vaporization rate observed in the computations.

In the cases shown in figures 8 and 9, the contribution of thermal radiation is relevant, $t_{C}/t_{R}=0.37$. To isolate the effect of water on the vaporization rate, we plot in figure 10 the temporal evolution of the normalized droplet surface and vaporization rate of a droplet with initial diameter $d_0=0.2$ mm, in which the ratio $t_{C}/t_{{R}} \simeq 0.07$ is sufficiently small to be considered negligible. In this case, we compute a non-constant vaporization rate that clearly deviates from the $d^{2}$-law when ambient humidity is non-zero, while the linear evolution is maintained even at relatively large droplet water contents $\mathcal {V} > 0$. This nonlinearity is introduced by the heat released due to water condensation when $\mathcal {H} \ne 0$, a similar effect to that of radiation discussed in § 3.

Figure 10. Effect of initial water content in the droplet $\mathcal {V}$ and ambient relative humidity $\mathcal {H}$ on the evolution of the normalized droplet surface (a) and dimensionless gasification rate (b). All results correspond to full one-dimensional simulations for evaporation of a droplet of initial diameter and temperature $d_0=0.2$ mm and $T_{d_0}=300$ K, respectively, at ambient temperature $T_{\infty }=800$ K and pressure $p_\infty =1$ bar. Green lines: $\mathcal {V}=0, \mathcal {H}=0$; blue lines: $\mathcal {V}=0.1, \mathcal {H}=0$; red lines: $\mathcal {V}=0.2, \mathcal {H}=0$; violet lines: $\mathcal {V}=0, \mathcal {H}=0.1$; orange lines: $\mathcal {V}=0, \mathcal {H}=0.2$.

5.3. Vaporization of bicomponent n-dodecane–n-hexadecane droplets

In this section we consider droplets of perfectly mixed $n$-dodecane (C$_{12}$H$_{26}$) and $n$-hexadecane (C$_{16}$H$_{34}$), liquid fuels with very different boiling temperatures ($T_{{b},\textrm {C}_{12}\textrm {H}_{26}}=489$ K and $T_{{b},\textrm {C}_{16}\textrm {H}_{34}}=560$ K) and very small vapour pressure under atmospheric conditions ($p_{{v},{\textrm {C}_{12}\textrm {H}_{26}}} \simeq 18$ Pa and $p_{{v},{\textrm {C}_{16}\textrm {H}_{34}}} \simeq 10$ Pa). In figure 11 we compare our numerical results with the experimental measurements of Han et al. (Reference Han, Zhao, Fu, Zhang, Pang and Li2015), performed at ambient temperatures $T_{\infty }=443$ K, sufficiently below the boiling temperatures of both fuels to preclude internal micro-bubble formation and reduce the uncertainties associated with radiation.

Figure 11. Droplet vaporization of $n$-dodecane and $n$-hexadecane mixtures in a hot nitrogen atmosphere at ambient temperature and pressure of $T_\infty =443$ K and $p_\infty =1$ bar, respectively, and initial droplet conditions of $a_0=1.2$ mm and $T_{d_0}=315$ K. (a) The time evolution of droplet normalized surface $(a/a_0)^{2}$. (b) The temporal evolution of the dimensionless gasification rate $\mathrm {d}(a/a_0)^{2}/\mathrm {d}\tau$. Stars: experimental results of Han et al. (Reference Han, Zhao, Fu, Zhang, Pang and Li2015); solid lines: numerical simulations. The colours indicate the droplet composition. Green: pure $n$-hexadecane; blue: 70 % $n$-hexadecane and 30 % $n$-dodecane (by volume); red: pure $n$-dodecane.

In figures 11(a) and 11(b) we plot the temporal evolution of the normalized droplet surface and the vaporization rate, respectively. Notice that numerical solutions show good agreement with experiments. In this case, the contribution of thermal radiation is relatively small $t_{C}/t_{R}=0.16$ and the $d^{2}$-law is approximately followed by both pure fuel curves and by the bicomponent droplet in the last stage of the vaporization, once the volatile component has vaporized.

The radial distribution of hexadecane mass fraction is plotted in figure 12(a) at the instants of time depicted with circles in figure 11(a). By comparing the characteristic time of species diffusion with the droplet lifetime at the initial temperature we obtain $t_{D_\ell }/t_{{C}}=1.7$, a value that anticipates an almost constant droplet composition throughout the droplet lifetime. According to figure 12, this is clearly not the case. This apparent contradiction can be easily understood if we check the droplet temperature distribution included in figure 12(b). Since the heat conduction time is much shorter than the droplet lifetime $t_{{{D_{{th},\ell }}}}/t_{{C}} \simeq 0.05 \ll 1$, the droplet temperature rapidly increases to achieve a uniform value, close to the boiling temperature. That induces a tenfold increase in the mass diffusivity $D_{\ell }$, markedly reducing the ratio $t_{D_\ell }/t_{{C}}$ down to a value of 0.21. This explains the concentration gradient observed during the droplet lifetime depicted in figure 12(a), in which the droplet composition is essentially frozen until the temperature is high enough to bring the volatile component C$_{12}$H$_{26}$ to the droplet surface and start vaporizing. As shown in the case of 70 % C$_{16}$H$_{34}$ depicted in figure 11(a), when the volatile component is depleted at $\tau _6 \sim 0.5$, the time evolution of the normalized droplet surface $(a/a_0)^{2}$ changes its slope and transitions from the curve of pure C$_{12}$H$_{26}$ to that of pure C$_{16}$H$_{34}$ droplets.

Figure 12. Droplet profiles as a function of normalized radius $r/a_0$ for instants marked from 1 to 6 in figure 11. (a) The $n$-hexadecane mass fraction profiles. (b) The temperature profiles. Solid light lines show $n$-hexadecane mass fraction (a) and temperature (b) at the droplet surface $r=a(t)$.