2. Experimental set-ups and methods

In our experiments, we employ the suspended fibre technique (Chauveau et al. Reference Chauveau, Birouk and Gökalp2011, Reference Chauveau, Birouk, Halter and Gökalp2019) to conduct droplet vaporisation experiments. Two sets of experiments are performed: droplet evaporation and droplet combustion. The former tests whether the D ²-law can still hold in the presence of a fibre, while the latter investigates the potential emergence of a non-square diameter law due to buoyant convection. To access the influence of support fibre effects, both experiments are conducted using thinner fibres than those used by Chen et al. (Reference Chen, Yang, Yang and Wei2024). Additionally, to evaluate whether the observed power laws depend on fuel chemistry, we also select several representative liquid fuels, both low and high boiling hydrocarbons, for these experiments.

Figure 2 displays schematic diagrams of our experimental set-ups for droplet evaporation and combustion. The chamber consists of four vertical aluminium framing posts, with glass panels covering the front and rear sides, while the top and bottom remain open. To hold the droplet, we use a ceramic fibre with low thermal conductivity to minimise heat transfer from the fibre to the droplet (Lai & Pan Reference Lai and Pan2023). The fibre has a thermal conductivity of 0.12 W mK^–1 and an average diameter of 35 µm. A fibre segment of 10–15 cm in length is suspended from the top of the chamber, with one end secured to sticky tape across the diagonal extrusions. To prevent the droplet from sliding down along the fibre, a small knot (of ∼140 µm in size) is tied at the fibre’s end.

Figure 2.

Experimental set-ups for (a) droplet evaporation and (b) droplet combustion.

For droplet evaporation, experiments are conducted at an elevated temperature using two ceramic heater plates (XH-RP505020, 95W) positioned on the lateral sides of the chamber. Prior to each experiment, the chamber is preheated to 250 °C using a power supply (HJS-480-0-60, 480W) at 12 V. A steady-state condition is ensured by monitoring the temperature with a K-type thermocouple at a distance of 20 mm from the knot end of the fibre. A droplet of the desired liquid is then placed onto the knot using a syringe needle (inner diameter: 0.2 mm), with the initial diameter D ₀ $=$ 0.7 – 1.7 mm. As for droplet combustion, the droplet is produced in the same manner but is ignited using either an ordinary open-flame igniter or a continuous flame supplied by a burner (lighter). In the latter case, the burner is placed 2 cm below the droplet. The ambient temperature for droplet evaporation is maintained at 70 °C using a heating plate placed on the side of the chamber, while the ambient temperature for droplet combustion is maintained at 25 °C in an air conditioned environment. For both experiments, the relative humidity is maintained at 50 %. The number of experimental realisations for each tested fuel is 10 for evaporation and 50 for combustion.

The starting droplet diameter for all tested fuels ranges from 1.34 to 2.11 mm. Since a droplet typically undergoes thermal expansion during the early stage of vaporisation before burning begins, the initial diameter D ₀ is defined at a re-established starting point t $=$ 0 after this initial heat-up period. The droplet shrinking process is observed from the front side of the chamber and recorded using a high speed CCD camera (Phantom v7.3). The recording speeds of the CCD for droplet evaporation and droplet combustion are 100 and 1000 frames s^–1, respectively. Exposure time is 1 ms. The recording is taken in a backlighting manner with a LED light source (LEDD1B, 14.4W) placed behind the rear side of the chamber. After the recording, the video is stored and converted into a sequence of images of 800 $\times$ 600 pixels in resolution. The number of images captured varies depending on droplet lifetimes, ranging from 1600 to 63 000 for droplet evaporation and 500–3000 for droplet combustion. These images are then post processed using MATLAB to generate a series of snapshots for subsequent data analysis.

Due to the presence of the fibre, the droplet deforms into a raindrop-like shape. However, as long as its length and width remain comparable and do not differ by an order of magnitude – except near the final stages of vaporisation, where the droplet may become more slender – a unique characteristic length scale can still be defined to represent the droplet size. This length scale is expressed as the effective diameter D, determined using the volume-equivalent-sphere approach, in which D is taken to be the diameter of a sphere having the same volume as the deformed droplet (Nomura et al. Reference Nomura, Ujiie, Rath, Sato and Kono1996; Han et al. Reference Han, Yang, Zhao, Fu, Ma and Song2016). In this approach, the droplet profile is assumed to be axisymmetric, with the interface position r(z) measured from the symmetry axis z. By performing a volume integral for the body of revolution with respect to z and subtracting the volume of the knot (of diameter d _knot ) at the end of the fibre, we can determine the effective diameter D according to

(2.1) \begin{equation} \pi \int r^{2}{\rm d}z-\frac{\pi }{6}d_{\textit{knot}}^{3}=\frac{\pi }{6}D^{3}. \end{equation}

It is important to note that the knot’s volume has to be subtracted off. This ensures that D reduces to zero when the droplet vanishes at the end of its vaporisation.

To highlight the importance of using this equivalent volume approach, we also measure D using the projection area approach (Shang et al. Reference Shang, Yang, Xuan, He and Cao2020; Wang et al. Reference Wang, Huang, Qiao, Ju and Sun2020). This approach is based on the D ²-law (1.1), where D ² is interpreted as the droplet area. Figure 3 shows that the equivalent volume approach yields n $=$ 2.03 pretty close to the ideal value n $=$ 2, whereas the projection area approach gives n $=$ 1.70, which is significantly lower than n $=$ 2. The reason why the equivalent volume approach gives a correct result is that it is the droplet volume (or mass) that responds to vaporisation and the associated energy change. That is, the droplet size has to be determined in a manner that is directly linked to the cause of its size change; otherwise, the correct shrinkage power law will not be obtainable. From this perspective, it is inappropriate to determine D using the projection method based on the D ²-law. In fact, treating D ² in the D ²-law (1.1) as the droplet area is actually a misinterpretation since it comes from the result by time integration of the droplet volume ∝D ³ in response to the received conductive heat flow ∝D (see the detailed heat balance in (3.1)). Furthermore, the way to measure D should be objective and not rely on the underlying shrinkage power law, particularly in cases where the shrinkage power law is priorly unknown and needs to be determined.

Figure 3.

Plot of (1−t $ / $ t _life ) against $D/D_0$ in log–log scale for determining the shrinkage exponent n for an evaporating octane droplet. The correct value of n, found to be close to the ideal value 2 of the D ²-law, can only be obtained using the volume-equivalent-sphere approach for determine the effective diameter D of the droplet. In contrast, using the projection area approach leads the value of n to be significantly below 2, indicating its inadequacy.

Figure 4.

(a) Conventional D ²-t plots for droplet evaporation of several representative liquid fuels, showing approximately linear profiles. (b) Corresponding log–log plots of (1−t $ / $ t _life ) versus $D/D_0$ . The measured values of the shrinkage exponent n (mean ± standard error) extracted from the slopes are pretty close to the ideal value 2 according to the D ²-law. In each case, the slope is measured for every realisation (up to $D/D_0= 0.5$ ), and the mean value (in red) and standard error are computed over 10 realisations. Experiments are performed using the suspended fibre technique with an average fibre diameter of 35 μm.

3. Distinct shrinkage kinetics for various droplet vaporisation processes

3.2. Droplet evaporation under forced convection

Bochkareva et al. (Reference Bochkareva, Miskiv, Nazarov, Terekhov and Terekhov2018) experimentally investigated the evaporation of fibre-supported droplets under imposed flow conditions. They examined the vaporisation kinetics of pure water ethanol–water mixtures, and presented their results using the $D^{3/2} $ -t plot, which exhibit a linearly decreasing trend, suggesting that the shrinkage kinetics follow the $D^{3/2} $ -law due to forced convection (Terekhov et al. Reference Terekhov, Terekhov, Shishkin and Bib2010; Starinskaya et al. Reference Starinskaya, Miskiv, Nazarov, Terekhov, Terekhov, Rybdylova and Sazhin2021). However, similar to the reasoning presented in figure 1, this alone does not provide definitive proof, as the data also appear approximately linear and slightly concave when plotted in the D ²-t plot, as shown in figure 5(a). Similar experiments were conducted by Daïf et al. (Reference Daïf, Bouaziz, Chesneau and Chérif1999) who focused on multicomponent droplet vaporisation under forced condition and observed a concave trend in the D ²-t plot, implying the influence of convection. Since vaporisation of blended fuels in these studies may introduce additional complexities, it is not clear whether the observed behaviour reflects purely convective effects or is also affected by compositional variations within the droplets. Therefore, to avoid such complications and to provide a more direct and consistent basis for comparison with our model of pure fuel vaporisation, we use the pure water data from Bochkareva et al. (Reference Bochkareva, Miskiv, Nazarov, Terekhov and Terekhov2018) (shown in figures 4 and 5) to test the $D^{3/2} $ -law.

Figure 5.

(a) The D ²-t plot for an evaporating droplet under forced convection, based on the experimental data from Bochkareva et al. (Reference Bochkareva, Miskiv, Nazarov, Terekhov and Terekhov2018). The concave profile clearly indicates a shrinkage exponent n < 2. (b) Corresponding log–log plot of (1 − t $ / $ t _life ) versus $D/D_0$ for extracting the exponent n from the slope, showing a trend that closely matches the theoretical prediction of 1.5.

Along the above line, we first estimate the values of the droplet lifetime t _life by extrapolating the data to D $=$ 0 using the $D^{3/2} $ -t plot under the assumption that the $D^{3/2} $ -law holds. We then re-analyse the data using the dynamic slope approach. As shown in figure 5(b), the resulting values of the exponent n, indicated by the slopes, closely match the theoretical value 1.5. This not only re-affirms the validity of the $D^{3/2} $ -law but also demonstrates the robustness and reliability of this dynamic slope approach. Moreover, since the droplets are supported by fibres, the consistency of the $D^{3/2} $ -law further confirms that the presence of fibres has no influence on the droplet shrinkage kinetics.

While the $D^{3/2} $ -law has been previously derived (Terekhov et al. Reference Terekhov, Terekhov, Shishkin and Bib2010; Starinskaya et al. Reference Starinskaya, Miskiv, Nazarov, Terekhov, Terekhov, Rybdylova and Sazhin2021), it is worthwhile to re-visit why the exponent n for this forced convection scenario is smaller than that in pure droplet evaporation.

We begin with the energy balance over a vaporising fuel droplet of mass ρ _L V $=$ ρ _L (4π $ / $ 3)R ³ (with $V$ being the droplet volume), expressed in terms of its instantaneous radius R and the liquid fuel’s density ρ _L . This balance states that the rate of energy loss due to vaporisation, ρ _L $\dot{V}$ Δ $H_{v\textit{ap}}$ , associated with the rate of fuel mass reduction ρ _L $\dot{V} =$ 4πR ² $\dot{R}$ ρ _L is supplied by heat transfer from the surrounding gas to the droplet surface (of area 4πR ²)

(3.1) \begin{equation} \rho _{L}4\pi R^{2}\frac{\textrm{d}R}{\textrm{d}t}{\unicode[Arial]{x0394}} H_{v\textit{ap}}=-h{\unicode[Arial]{x0394}} T(4\pi R^{2}). \end{equation}

Here, Δ $H_{v\textit{ap}}$ is the heat of vaporisation, ΔT ≡ T _∞ – T _s is the driving temperature difference between the higher temperature T _∞ of the surrounding gas and the surface temperature T _s of the droplet and h is the heat transfer coefficient of the gas phase. Writing h in terms of the Nusselt number $\textit{Nu}$ $=$ hR $ / $ k, with k being the thermal conductivity of the gas phase, (3.1) can be re-written as

(3.2) \begin{equation} \frac{{\rm d}R^{2}}{{\rm d}t}=-2\alpha \left(\rho /\rho _{L}\right)\left(C_{p}{\unicode[Arial]{x0394}} T/{\unicode[Arial]{x0394}} H_{v\textit{ap}}\right)\textit{Nu}, \end{equation}

where α $=$ k $ / $ ρC _p is the thermal diffusivity of the gas phase (with gas density ρ and heat capacity C _p ) and C _p ΔT $ / $ Δ $H_{v\textit{ap}}$ is the well-known Spalding heat transfer number (Abramzon & Sirignano Reference Abramzon and Sirignano1989).

In the absence of convection, when the fluid velocity U $=$ 0, heat transfer is dominated by conduction and the Nusselt reduces to $\textit{Nu}$ $=$ 1. Under this condition, (3.2) is simplified to

(3.3a) \begin{equation} \frac{{\rm d}R^{2}}{{\rm d}t}=-\left(1/4\right)K_{D2}, \end{equation}

which recovers the classical D ²-law given in (1.1) with the corresponding vaporisation rate constant

(3.3b) \begin{equation} K_{D2}=8\alpha \left(\rho /\rho _{L}\right) \left(C_{p}{\unicode[Arial]{x0394}} T/{\unicode[Arial]{x0394}} H_{v\textit{ap}}\right)\!. \end{equation}

For forced convection due to an imposed flow at velocity U, if convection is strong when the Péclet number Pe $=$ UR $ / $ α is large, $\textit{Nu}$ in (3.2) can be evaluated according to (Deen Reference Deen1998)

(3.4) \begin{equation} \textit{Nu}={\textit{cPe}}^{1/2}={\textit{cRe}}^{1/2}{\textit{Pr}}^{1/2}, \end{equation}

where Re $=$ UR $ / $ ν is the Reynold number (with ν being the kinematic viscosity of the gas phase), Pr $=$ ν $ / $ α is the Prandtl number and c is a dimensionless numerical factor. The exponent 1/2 in (3.4) arises from the slipping thermal boundary layer due to a possible fuel vapour film formed on the droplet surface (Sirignano Reference Sirignano2014).

To understand the derivation of (3.4), we begin with the definition of the Nusselt number, $\textit{Nu}$ $=$ hR $ / $ k, which relates the conductive flux –k ∂T $ / $ ∂r in the gas phase at the droplet surface (r $=$ R) to the ambient heat flux hΔT. Thus, $\textit{Nu}$ can be interpreted as a dimensionless conductive flux at the droplet surface, leading this flux to scale inversely with the thermal boundary layer thickness δ _T according to (Deen Reference Deen1998)

(3.5) \begin{equation} \textit{Nu}=\frac{hR}{k}=\frac{\left.-k\dfrac{\partial T}{\partial r}\right| _{r=R}}{{\unicode[Arial]{x0394}} T}\sim R/\delta _{T}. \end{equation}

The value ofδ _T in (3.5) can be estimated from the energy balance equation $\boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{\nabla }T=\alpha {\nabla} ^{2}T$ in the gas phase, where $\boldsymbol{v}$ is the fluid velocity field. Because of the slipping surface due to the presence of a vapour film, the gas velocity near the droplet surface can be approximated as $\boldsymbol{v}$ ∼ U. Consequently, the convection term $\boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{\nabla }T$ scales as UΔT $ / $ R. Balancing this to the conduction term $\alpha {\nabla} ^{2}T$ ∼ $\alpha$ ΔT $ / $ $\delta$ _T ² within the boundary layer yields δ _T /R ∼ Pe ^–1/2, thereby leading to (3.4). A correlation similar to (3.4) has been proposed for high Re regimes (Ranz & Marshall Reference Ranz and Marshall1952; Abramzon & Sirignano Reference Abramzon and Sirignano1989) in which the exponent of Pr is 1 $ / $ 3 instead of 1 $ / $ 2. This alternative scaling assumes a no-slip boundary condition at the droplet surface, which may result from surface contamination. Nevertheless, in both cases, the dependence of $\textit{Nu}$ on U and R remains the same: $\textit{Nu}$ ∝ U ^1/2 R ^1/2.

With $\textit{Nu}$ ∝ R ^1/2 in (3.4), the power of R on the left-hand side of (3.2) will be reduced by 1/2, yielding

(3.6) \begin{equation} \frac{{\rm d}}{{\rm d}t}R^{3/2}=-\left(3/16\right)c K_{D2}\left(U/\alpha\right)^{1/2}, \end{equation}

which results in the $D^{3/2} $ -law for droplet evaporation under forced convection (Terekhov et al. Reference Terekhov, Terekhov, Shishkin and Bib2010; Starinskaya et al. Reference Starinskaya, Miskiv, Nazarov, Terekhov, Terekhov, Rybdylova and Sazhin2021), as seen in figure 5(b). Compared with (3.3) for the pure conduction case described by the D ²-law, the vaporisation rate constant on the right-hand side of (3.6) includes an additional factor (U $ / $ α)^1/2 , which carries units of square root of inverse length. In other words, there exists an inherent length scale that characterises this forced-convection-driven vaporisation process

(3.7) \begin{equation} \mathrm{\ell }_{\textit{forced}}=\alpha /U, \end{equation}

which modifies the vaporisation rate constant from its D ²-law’s value K _D2 to

(3.8) \begin{equation} K=\left(3\sqrt{2}/8\right)c K_{D2}/\mathrm{\ell }_{\textit{forced}}^{1/2} \,.\end{equation}

Equations (3.7) and (3.8) highlight that, when droplet vaporisation is dominated by convective effects, dimensional considerations necessitate the introduction of an additional inherent length scale into the description of the shrinkage kinetics. This length scale fundamentally alters the form of the vaporisation rate, rendering it markedly different from that in the classical D ²-law. As will also be shown later in § 4, this is not an isolated case but a general principle – similar behaviour also emerges in droplet combustion, where convection likewise modifies the rate law through an associated length scale.

3.3. Departure from the D²-law in self-sustaining droplet combustion

Having experimentally re-confirmed the D ²-law in the presence of a fibre, we proceed to conduct experiments on self-sustaining droplet combustion. For a given fuel, we repeat the experiment 50 times. Typical sequential images are shown in figure 6. In terms of measuring the value of n using the dynamic slope method according to (1.3), to ensure more accurate measurements reflecting effects of combustion, we exclude the data during the heat-up period at the early stage of a droplet combustion process. Figure 7(a) displays the D ²-t plots for the representative fuels, showing slightly convex profiles implying n > 2 according to figure 1. The actual values of n can be determined from the slopes of the corresponding dynamic slope plots (by fitting the data up to $D/D_0$ > 0.5), giving n $=$ 2.56 ± 0.20–2.65 ± 0.17 (mean ± standard error based on 50 experimental realisations), as shown in figure 7(b). The range again overlaps with n $=$ 2.53 ± 0.30–2.62 ± 0.25 obtained by using thicker fibres of diameter ∼100 μm (Chen et al. Reference Chen, Yang, Yang and Wei2024). This suggests that the observed departure from the D ²- law is genuine and insensitive to the presence of support fibres. Such departure is attributed to buoyant convection driven by the flame (Chen et al. Reference Chen, Yang, Yang and Wei2024), which is distinct from droplet evaporation in natural convection under which n < 2 (Misyura Reference Misyura2018).

Figure 6.

(a) Sequential images of a shrinking ethanol droplet during its combustion process. (b) Snapshots of the droplet and the surrounding flame. The flame also diminishes over time. The droplet is suspended at the end of a thin fibre with a diameter of 35 μm.

Figure 7.

(a) The D ²-t plots for droplet combustion of representative liquid fuels. While the curves look approximately linear, they exhibit greater convexity compared with those shown in figure 4 for droplet evaporation, implying n > 2. (b) Corresponding log–log plots of (1 − t $ / $ t _life ) versus $D/D_0$ yield the values of n (mean ± standard error) extracted from the slopes, which are significantly greater than 2, signifying a positive departure from the D ²-law. In each case, n is determined by the slope of individual realisations with the mean value (marked in red) computed over 50 realisations along with its standard error. These results are obtained using the suspended fibre technique with a thin fibre of 35 μm in diameter.

3.4. Droplet combustion in a continuous flame

To further illuminate that it is the flame responsible for the positive departure from the D ²-law in droplet combustion, we place a fuel droplet in a continuous flame supplied by an additional burner to observe its shrinking behaviour. We aim to investigate this droplet combustion process as frequently occurs in real-world fuel combustion applications driven by burners. We begin with low-volatility dodecanol (C₁₂H₂₅OH). Figure 8(a) is the D ²-t plot, displaying convex profiles with the characteristic n > 2. Figure 8(b) presents the result using the dynamic slope method, yielding n $=$ 3.00 ± 0.15 (mean ± standard error based on 50 experimental realisations). This value is not only significantly higher than 2.67 ± 0.17 observed in the self-sustaining case, but also approaches 3, corresponding to the scenario where the droplet experiences constant heating, as described by the second term on the right-hand side of (4.1) later. Similar trends are also observed for high-volatility ethanol, with n $=$ 2.56 ± 0.20 in the self-sustaining case and a markedly higher n $=$ 3.02 ± 0.16 under the continuous flame condition. For ethyl acetate, which is a low-volatility fuel, although n $=$ 2.82 ± 0.20 in a continuous flame is slightly lower than the values above, it remains significantly higher than n $=$ 2.58 ± 0.14 observed in the self-sustaining case.

Figure 8.

(a) A D ²-t plot for combustion of a dodecanol (C₁₂H₂₅OH) droplet in a continuous flame supplied by a burner. The curves exhibit pronounced convexity, clearly indicating n > 2. (b) Corresponding log–log plot of (1 − t $ / $ t _life ) versus $D/D_0$ , showing that the measured value of n from the slope is close to 3 and greater than that of the self-sustained case shown in second panel of figure 7(b).

Figure 9 summarises our findings by plotting measured n values against the boiling points of various fuels for both self-sustaining and burner-assisted combustion experiments, clearly showing that the results consistently exceed those observed in droplet evaporation governed by the D ²-law. Moreover, droplets burning in a continuous flame show even more pronounced deviations from the D ²-law. These apparent departures from the D ²-law in droplet combustion are caused by flame-driven buoyant convection, which will be explained in more detail next.

Figure 9.

Plot of the measured shrinkage exponent n versus boil point for selected pure liquid fuels. For droplet evaporation, the values of n are around the ideal value 2, re-confirming the D ²-law. In contrast, for droplet combustion, the measured n values are around 2.6 for self-sustained droplet combustion and approach 3 for droplet combustion in a continuous flame. These results are well captured by 2.33 < n < 3 according to (4.28b ), signifying a substantial departure from the D ²-law.

4. Unified theory for departures from the D ²-law in flame-driven and non-flame-driven droplet vaporisation processes

Figure 9 reveals that the measured values of n in droplet combustion are significantly greater than 2 in droplet evaporation, especially in the additional burner situation where n approaches 3. Chen et al. (Reference Chen, Yang, Yang and Wei2024) have recently developed a theory for a fuel droplet whose burning is set up by its own flame and shown n $=$ 8 $ / $ 3 ≈ 2.67 due to flame-driven buoyant convection. Here, we develop a more general theory for droplet combustion to explain the origin of the positive departures from the D ²-law and to account for the more pronounced departure observed in the burner-assisted case. This framework also captures the negative departures in purely convection-driven droplet vaporisation without flame, thereby highlighting the unique features that distinguish droplet combustion from non-reactive vaporisation processes.

Figure 10.

Schematic diagram of a vaporising fuel droplet resulting from combustion . The burning can be sustained by the droplet itself or by an additional burner.

For droplet combustion, aside from the heat flow from the ambient gas to the droplet, the energy balance over the droplet may include an extra contribution from the burner. Since a typical burner produces a continuous pre-mixed flame (figure 10), such flame may act as an additional heating source, releasing heat at a rate $\dot{S}$ (>0) toward the droplet. Therefore, we add this additional heating on the right-hand side of the energy balance (3.1), yielding

(4.1) \begin{equation} \rho _{L}{\unicode[Arial]{x0394}} H_{v\textit{ap}}\frac{{\rm d}R^{2}}{{\rm d}t}=-2k \textit{Nu}{\unicode[Arial]{x0394}} T-\frac{\dot{S}}{2\pi R}. \end{equation}

If the heating from the burner is sufficiently intense, the droplet shrinkage is primarily governed by the $\dot{S}$ term. This situation may arise when the burner’s flame temperature is much higher than that of the droplet flame, as in the case of burning of a high-carbon fuel droplet whose sooty red flame burns within a blue flame of the butane lighter used as the burner. Since the buoyant flow induced by the burner’s flame is also much stronger than that generated by the droplet flame, the burner becomes the primary oxygen supply sustaining droplet combustion. In particular, because the heat release originates from combustion within the burner flame – maintained by a continuous oxygen supply driven by the steady buoyant flow set up by the burner – $\dot{S}$ can be regarded as approximately constant, despite inevitable heat losses. Therefore, (4.1) leads to

(4.2) \begin{equation} \frac{{\rm d}R^{3}}{{\rm d}t}=-\frac{3\dot{S}}{4\pi \rho _{L}{\unicode[Arial]{x0394}} H_{v\textit{ap}}}, \end{equation}

which results in the D ³-law. It is more convenient to use the departure exponent m to represent the degree of deviation from the D ²-law where the actual shrinkage exponent n $=$ 2 + m. So (4.2) has

(4.3) \begin{equation} m=1. \end{equation}

If burning is mainly sustained by the droplet itself, the droplet’s vaporisation is supported by the heat flow from the flame generated from the droplet, which is governed by (3.2):

(4.4) \begin{equation} \frac{{\rm d}R^{2}}{{\rm d}t}=-2\alpha \left(\rho /\rho _{L}\right)\left(C_{p}/{\unicode[Arial]{x0394}} H_{v\textit{ap}}\right)\textit{Nu}{\unicode[Arial]{x0394}} T\!. \end{equation}

Here, we emphasise that the Nusselt number $\textit{Nu}$ ΔT has to involve the unknown temperature difference ΔT due to the driven buoyant convection set up by the flame. The velocity scale of such flame-induced convection can be estimated as (Deen Reference Deen1998)

(4.5) \begin{equation} U\sim \left(\mathit{g}\beta {\unicode[Arial]{x0394}} TW\right)^{1/2}. \end{equation}

Rather than the droplet size R, it is the flame width W determining this velocity scale, since the flow is set up at the lower envelope of the flame where combustion is most intense. Balancing the buoyancy force ρβΔTgW ³ to the inertial force ρU ² W ² thus yields (4.5), with g denoting gravitational acceleration and β the volume expansion coefficient of the air. This velocity scale can also be more straightforwardly obtained by balancing the terms in the Navier–Stokes equation for the flow field $\boldsymbol{v}$ around a burning droplet. That is, it is achieved by balancing the buoyancy term Δρg to the inertial term $\rho\boldsymbol{v}$ $\boldsymbol{\cdot}$ ∇ $\boldsymbol{v}$ ∼ρ _∞ U ² $ / $ W in the equation using the Boussinesq approximation $\Delta\rho = \rho_\infty\beta \Delta T$ for the fluid density change in the buoyancy term while keeping the fluid density in the inertial term as the ambient value ρ _∞ (Deen Reference Deen1998). It is worth noting that the Boussinesq approximation U ∼ (g(Δρ $ / $ ρ _∞) W)^1/2 adopted here is formally valid only when the density difference Δρ $ / $ ρ _∞ between the environment and the buoyant fluid is small, and is therefore, strictly speaking, inapplicable to combustion where the density difference is large. Nevertheless, this approximation has been widely employed in the classical theory for convective buoyant thermals and plumes (Batchelor Reference Batchelor1954; Morton et al. Reference Morton, Taylor and Turner1956; Turner Reference Turner1969), where the actual density differences are not small, and it has proven highly successful in capturing the essential physics. More recently, Skvortsov et al. (Reference Skvortsov, Dubois, Jamriska and Kocan2021) demonstrated that the relevant scalings differ little between Boussinesq and non-Boussinesq cases, suggesting that the present use of the Boussinesq form provides a reasonable and physically consistent approximation even when the density contrast is significant. Its adequacy has also been confirmed in our recent theory for droplet combustion, which successfully accounts for the non-square shrinkage kinetic laws observed in experiments (Chen et al. Reference Chen, Yang, Yang and Wei2024).

Such flame-induced buoyant convection described by (4.5) is typically strong, meaning that the Péclet number Pe $=$ UR $ / $ α is generally large. Under this condition, $\textit{Nu}$ in (4.4) can be evaluated as (Deen Reference Deen1998)

(4.6) \begin{equation} {\textit{Nu}}=c\left({\textit{UR}}/\alpha \right)^{\phi }, \end{equation}

where the value of the exponent φ depends on the surface condition of the droplet. Specifically, φ $=$ 1 $ / $ 3 when the droplet surface becomes rigid due to contamination by impurities, and φ $=$ 1 $ / $ 2 when the droplet surface exhibits slip behaviour due to the presence of a vapour film. These values of φ have also been reported in early studies (Clift et al. Reference Clift, Grace and Weber1978; Ranz & Marshall Reference Ranz and Marshall1952).

It is worth noticing that the droplet merely passively receives the convective heat transfer due to the above buoyant flow set up by the flame. That is, from the droplet’s standpoint, since the natural convection here is not generated directly by the droplet but indirectly by the flame, the heat transfer to the droplet is of forced convection type. This is very distinct from self-driven natural convection heat transfer around a hot body (Yuge Reference Yuge1960; Potter & Riley Reference Potter and Riley1980). It is also very different from natural convection-driven droplet evaporation by a heated surface (Misyura Reference Misyura2018).

Since both ΔT and W in (4.5) are also part of the results waiting to be determined, how U varies with R for determining the shrinkage kinetic power law from (4.4) and (4.6) requires the knowledge of how ΔT and W vary with R. In other words, if we can determine how $\textit{Nu}$ varies with R in (4.6) by knowing the relationships of how ΔT and W depend on R, the true kinetic law can be readily determined by substituting the resulting Nu–R relationship into the energy balance equation (4.4). Chen et al. (Reference Chen, Yang, Yang and Wei2024) has adopted this idea to derive the $ D^{8/3} $ -law for a self-sustaining burning droplet, applying specific conditions to derive these relationships. Instead of relying on case specific conditions, we offer a more general approach by considering how ΔT and W vary with R, assuming their relationships with R follow power-law forms. Rather than directly determining the exponents in these relationships, we provide constraints for them, which will enable us to define the range of the shrinkage exponent n in (1.2). As will be demonstrated, this approach can effectively explain why n > 2 due to flame-driven buoyant convection, in contrast to n < 2 in convection-driven droplet evaporation without flame.

We first examine how W responds to changes in R. Experimental observations indicate that the flame gradually diminishes as the droplet shrinks during combustion (see figure 6). This suggests that W scales positively with R in a power-law form

(4.7) \begin{equation} W/\mathrm{\ell }=c_{1}\left(R/\mathrm{\ell }\right)^{\omega }, \end{equation}

where $\mathrm{\ell }$ is a characteristic length scale, c ₁ is a dimensionless factor and ω > 0 is the scaling exponent. Equation (4.7) is physically intuitive since a larger droplet supplies more fuel, resulting in a correspondingly wider flame. It also ensures that the flame is completely extinguished when the droplet vanishes at the end of combustion.

It is important to note that in (4.7), the characteristic length $\mathrm{\ell }$ is not taken to be the initial droplet radius R ₀ to scale lengths. This is because the vaporisation rate constant K in (1.2) represents an intrinsic rate of droplet consumption, derived from the kinetics of the underlying chemical or physical processes. Consequently, K must be composed solely of kinetic parameters or fluid properties, independent of system-specific dimensions. Initial conditions – such as R ₀ – should not influence this intrinsic kinetic description and, by extension, should not appear in the definition of K. If, as assumed here, the burning kinetics are primarily governed by transport phenomena, then K must be determined exclusively by fluid properties. Since K inherently involves the characteristic length $\mathrm{\ell }$ , it follows that $\mathrm{\ell }$ must likewise be defined purely in terms of fluid properties. As it will be shown shortly, this length plays a central role in determining the droplet’s shrinkage kinetics.

How to determine $\mathrm{\ell }$ can be guided by the Péclet number Pe $=$ UR $ / $ α that characterises the relative importance of convection to conduction. Using the buoyancy velocity scale U given by (4.5), Pe can be re-written as

(4.8) \begin{equation} \textit{Pe}=\frac{\textit{UR}}{\alpha }=G^{1/2}\left(\frac{W}{R}\right)^{1/2}, \end{equation}

where G is a new dimensionless number defined as

(4.9) \begin{equation} G=\frac{g\beta {\unicode[Arial]{x0394}} TR^{3}}{\alpha ^{2}}. \end{equation}

This dimensionless group is analogous to the Grashof number in a thermal context, with the kinematic viscosity ν in the denominator replaced by the thermal diffusivity α (≈0.2 cm² s^–1). Because both the temperature difference ΔT and the droplet radius R vary during combustion, G is not constant. Therefore, it is more convenient to relate it to a reference, fluid-property-dependent dimensionless parameter G ₀ by expressing ΔT and R with respect to their characteristic values T ^* and $\mathrm{\ell }$ that depend only on fluid properties. That is, we aim to express G in the form

(4.10a) \begin{equation} G=G_{0}\left(\frac{{\unicode[Arial]{x0394}} T}{T^{*}}\right)\left(\frac{R}{\mathrm{\ell }}\right)^{3}, \end{equation}

where

(4.10b) \begin{equation} G_{0}=\frac{g\beta T^{*}\mathrm{\ell }^{3}}{\alpha ^{2}}. \end{equation}

The characteristic temperature T ^* can be chosen as the reaction temperature T _rxn reflecting the fuel’s heat of reaction ΔH _rxn :

(4.11) \begin{equation} T_{\textit{rxn}}=|{\unicode[Arial]{x0394}} H_{\textit{rxn}}|/C_{p}. \end{equation}

Hence, G ₀ in (4.10) becomes

(4.12) \begin{equation} G_{0}=\frac{g\beta T_{\textit{rxn}}\mathrm{\ell }^{3}}{\alpha ^{2}}, \end{equation}

which is also a dimensionless quantity involving only fluid properties. Since we wish to express G in terms of the dimensionless temperature ΔT $ / $ T ^* and the dimensionless radius R $ / $ $\mathrm{\ell }$ in (4.10a ), it is convenient to choose the characteristic length $\mathrm{\ell }$ such that G ₀ $=$ 1. This choice affects only the numerical prefactor, directly yielding the inherent buoyancy length of the fuel

(4.13) \begin{equation} \mathrm{\ell }=\left(\frac{g\beta T_{\textit{rxn}}}{\alpha ^{2}}\right)^{-1/3} .\end{equation}

As this length characterises the ability of the fuel to drive natural convection through heat release from the flame, the shorter the value of $\mathrm{\ell }$ , the more inclined the droplet is to receive a natural convective heat flow generated from the flame. For commonly used C6–C12 alkanes having |ΔH _rxn | ∼ 50 KJ g^–1 with C _p ≈ 10^–3 (KJg⁻¹) K^–1, T _rxn ∼ 5 $\times$ 10⁴ K according to (4.11). The corresponding buoyancy length is $\mathrm{\ell }$ ∼ 60 μm from (4.13).

Having re-expressed Pe in (4.8) in terms of G $=$ (ΔT $ / $ T _rxn )(R $ / $ $\mathrm{\ell }$ )³ from (4.10a ) with $\mathrm{\ell }$ defined by (4.13), the Nusselt number expression (4.6) can be re-written as

(4.14) \begin{equation} {\textit{Nu}}=c\left(\frac{{\unicode[Arial]{x0394}} T}{T_{\textit{rxn}}}\right)^{\phi /2}\left(\frac{R}{\mathrm{\ell }}\right)^{\phi }\left(\frac{W}{\mathrm{\ell }}\right)^{\phi /2}. \end{equation}

In the above, we have established how W varies with R through (4.7). To further determine how $\textit{Nu}$ depends on R, we also need to establish the relationship between ΔT and R. This will ultimately allow us to derive the shrinkage kinetic law and determine the value of n from (4.4).

Establishing the relationship between ΔT and R is not straightforward. However, since the buoyancy-induced velocity U can be related to ΔT and W via (4.5), understanding how W responds to U allows us to link ΔT to W. With this relationship, we can then determine how ΔT varies with R using (4.7) for W.

Suppose the flame width W is primarily controlled by the bottom envelope of the flame, which can be treated as non-premixed with a narrow reaction zone. In this case, the fuel vapour released from the droplet will be consumed upon encountering the ample oxygen supplied by the buoyancy-induced upward airflow. As a result, a higher U leads to greater fuel vapour consumption at the flame front, causing W to become narrower. This suggests that W varies inversely with U according to

(4.15) \begin{equation} W\infty\, U^{-\gamma }, \end{equation}

with γ > 0.

Alternatively, the response of W to U is somewhat analogous to flow over a blunt-nosed body, where the shape is determined by the superposition of a point source and an imposed uniform flow (Milne-Thomson Reference Milne-Thomson1960). In this analogy, the vaporising droplet acts like a point source, while the upward buoyant convection corresponds to the imposed flow. Drawing from this comparison – where the width of a blunt-nosed body decreases with the flow speed V as $ V^{-1/2} $ (Milne-Thomson Reference Milne-Thomson1960) – the flame width W is similarly expected to be suppressed by the upward buoyant convection described by (4.5).

Substituting (4.5) for U into (4.15) yields

(4.16) \begin{equation} {\unicode[Arial]{x0394}} T/T_{\textit{rxn}}={c_{2}^{\prime}}\left(W/\mathrm{\ell }\right)^{-\lambda}, \end{equation}

where $c_2^{\prime}$ is again a fuel-property-dependent dimensionless factor and the exponent λ is given by

(4.17) \begin{equation} \lambda =1+2/\gamma \gt 1. \end{equation}

As such, (4.16) can also be interpreted as a result of the lateral flame suppression effect induced by the flow in (4.5). This effect is quite similar to that occurring to opposed flow flames (Wickman Reference Wichman1992), with the key distinction being that, in this case, the flow here is not externally imposed but rather self-generated by the flame itself. This contrasts sharply with the microgravity situation where buoyant convection is absent and, consequently, no such flame suppression effect occurs. Additionally, the flames of these two scenarios differ in nature. In the present situation under normal gravity, the flame is not entirely non-premixed. The bottom flame envelope is diffusive and somewhat non-premixed, as the outward vapour fuel diffusion from the droplet is opposed by the upward airflow convection, leading to the formation of a stagnation point – similar to a counterflow diffusion flame (Tsuji Reference Tsuji1982). This is also the region where the most intense burning occurs. Meanwhile, the upper part of the flame is more or less advective and premixed, as the vaporised fuel from the droplet surface is diluted by the upward convection.

In microgravity, because of the absence of natural convection, fuel vapour can only diffuse radially in a spherically symmetric manner, as does the flame. The flame is of non-premixed type, as it separates the vaporised fuel from the droplet from the surrounding oxidiser. Despite this difference, the flame in gravity or microgravity is actually partially pre-mixed, at least in the initial stage of combustion. This is because a droplet typically undergoes a short period of preheating prior to ignition. During this heat-up period, the vaporising fuel from the droplet mixes with the surrounding oxidiser, resulting in a partially pre-mixed flame upon ignition (Millán Merino et al. Reference Millán-Merino, Fernández-Tarrazo and Sánchez-Sanz2021a ).

Having established the relationship between ΔT and W from (4.16), along with the W−R relationship given in (4.7), how ΔT varies with R can be obtained as

(4.18) \begin{equation} {\unicode[Arial]{x0394}} T/T_{\textit{rxn}}=c_{2}\left(R/\mathrm{\ell }\right)^{-\lambda \omega }, \end{equation}

where c ₂ $=$ c ₁ $c_2^{\prime}$ and λω has to satisfy the following constraint that directly follows (4.17):

(4.19) \begin{equation} \lambda \omega \gt \omega . \end{equation}

Equation (4.18) indicates that ΔT increases as R decreases. This follows directly from the lateral flame suppression effect discussed earlier, as described by (4.16) and (4.7). Shaw & Wei (Reference Shaw and Wei2007) observed in their droplet combustion experiments under gravity that the measured gas-phase temperature continues to rise gradually after an initial sharp increase upon ignition. During this temperature climbing period – which persists until a rapid drop in temperature near the end of combustion – the droplet steadily shrinks over time. Similar behaviour has been reported in droplet combustion under various systems or conditions (Won, Baek & Kim Reference Won, Baek and Kim2018; Meng et al. Reference Meng, Lai, Zhang, Willmott and Zhang2025). These experimental observations support the prediction of (4.18). In a microgravity environment, on the contrary, the gas-phase temperature exhibits a relatively stable plateau following its initial rapid rise after ignition, as noted in the same experiments. A comparable temperature evolution has also been observed in droplet evaporation under forced convection (Daïf et al. Reference Daïf, Bouaziz, Chesneau and Chérif1999). The distinct temperature responses observed between gravity and microgravity conditions strongly suggest that buoyant convection plays a crucial role in driving these differences, thereby providing further support for the concept underlying (4.18).

Now, incorporating (4.18) and (4.7) to describe the variations of ΔT and W with R, (4.14) transforms into

(4.20) \begin{equation} {\textit{Nu}}=c'\left(\frac{R}{\mathrm{\ell }}\right)^{\left(1-\lambda \omega /2+\omega /2\right)\phi }, \end{equation}

with $c^{\prime}$ being a lumped dimensionless factor. Because $\textit{Nu}$ must be much greater than unity owing to Pe $=$ UR $ / $ α > >1 in (4.6) and also because R is typically greater than $\mathrm{\ell }$ , the exponent of the R term in (4.20) must be positive. This imposes an additional constrain between λω and ω, regardless of the value of φ

(4.21) \begin{equation} \frac{\omega }{2}+1\gt \frac{1}{2}\lambda \omega .\end{equation}

Moreover, since the majority of the heat released from the flame must supply for the heat flow (k $ / $ R) $\textit{Nu}$ ΔT 4πR ² toward the droplet, this heat flow must exceed the heat loss ρC _p UΔT πW ² caused by the upward buoyant convection out of the flame; otherwise, the droplet vaporisation would not be possible. This requirement leads to

(4.22) \begin{equation} \textit{Nu}\gt \left(1/4\right)\left(UR/\alpha \right)\left(W/R\right)^{2}. \end{equation}

Writing (4.22) in terms of (R $ / $ $\mathrm{\ell }$ ) using (4.5), (4.6), (4.7), and (4.18), in order to satisfy (4.22), the combined exponent of (R $ / $ $\mathrm{\ell }$ ) must satisfy

(4.23) \begin{equation} \left(1-\phi \right)\frac{\lambda \omega }{2}-\left(5-\phi \right)\frac{\omega }{2}+\phi +1\gt 0. \end{equation}

Figure 11.

Blue shadowed region indicating constraints between the exponent λω in (4.18) for ΔT and the exponent ω in (4.7) for W: λω > ω from (4.19), λω < ω + 2 from (4.21) and (1 − φ)λω > (5 − φ) ω − 2(φ + 1) from (4.23). While this region is bounded within the ranges 0 < λω < 3 and 0 < ω < 1, slightly narrower ranges 0 < ω < (1 + φ) $ / $ 2 and (1 + φ) $ / $ 2 < λω < 2 given respectively by (4.24a ) and (4.24b ) can be selected as the sufficient conditions that guarantee satisfying these constraints, as indicated by red shadowed area. Applying these more strict conditions for λω and ω to (4.25b ), we can show that the range of the deviation exponent m satisfies (1 − φ) $ / $ 2 < m < 2. Combining m < 1 from the burner case, we arrive at (1 − φ) $ / $ 2 < m < 1 as described by (4.27).

As a result, we have three constraints between λω and ω : (4.19), (4.21) and (4.23). Figure 11 presents a λω–ω diagram showing the region within which these constraints are satisfied. While the region is bounded within the ranges 0 < λω < 3 and 0 < ω < 1, slightly narrower ranges can be selected below as the sufficient conditions that guarantee satisfying these constraints

(4.24a) \begin{align}&\,\, 0\lt \omega \lt \left(1+\phi \right)/2, \end{align}

(4.24b) \begin{align}& \left(1+\phi \right)/2\lt \lambda \omega \lt 2. \end{align}

The upper bound (1 + φ) $ / $ 2 of ω in (4.24a ) is determined from (4.19) and (4.23). The range (4.24b ) is selected to satisfy (4.19) and (4.21).

Combining (4.20) and (4.18) to account for the overall impacts of $\textit{Nu}$ ΔT in (4.4) and writing T _rxn in terms of |ΔH _rxn | using (4.11), we deduce (4.4) to

(4.25a) \begin{equation} \frac{\textrm{d}}{\textrm{d}t}\left(\frac{R}{\mathrm{\ell }}\right)^{2+m}=-2\left(2+m\right)c'^{\prime}\left(\frac{\alpha }{\mathrm{\ell }^{2}}\right)\left(\frac{\rho }{\rho _{L}}\right)\left(\frac{\left| {\unicode[Arial]{x0394}} H_{\textit{rxn}}\right| }{{\unicode[Arial]{x0394}} H_{v\textit{ap}}}\right), \end{equation}

where $c^{\prime\prime}$ $=$ $c^{\prime}$ c ₂ and the departure exponent m is

(4.25b) \begin{equation} m=\left(1+\frac{\phi }{2}\right)\lambda \omega -\left(1+\frac{\omega }{2}\right)\phi . \end{equation}

After integrating (4.25a ) and writing the result in terms of D as given in (1.2), the corresponding vaporisation constant K is given by

(4.25c) \begin{equation} K={c^{\prime\prime\prime}}\mathrm{\ell }^{2+m}\left(\frac{\alpha }{\mathrm{\ell }^{2}}\right)\left(\frac{\rho }{\rho _{L}}\right)\left(\frac{\left| {\unicode[Arial]{x0394}} H_{\textit{rxn}}\right| }{{\unicode[Arial]{x0394}} H_{v\textit{ap}}}\right)\!, \end{equation}

where $c^{\prime\prime\prime}$ is a lumped dimensionless factor. Equation (4.25c ) indicates that such a K possesses units of mm^2+m s⁻¹, which directly reflects the deviation from the D ²-law by a power m. Additionally, according to (4.13), $\mathrm{\ell }$ ∝ |ΔH _rxn |^−1/3. Consequently, it follows from (4.25c ) that K ∝ $\mathrm{\ell }$ ^m |ΔH _rxn | $ / $ Δ $H_{v\textit{ap}}$ ∝ |ΔH _rxn |^1−m $ ^{/} $ ³ $ / $ Δ $H_{v\textit{ap}}$ . Since m is typically less than unity, this indicates that K increases with ΔH _rxn in a nearly linear manner, but this increase is offset by a comparable decrease in K due to Δ $H_{v\textit{ap}}$ . Such compensation explains why low- and high-carbon fuels exhibit similar lifetimes in droplet combustion, in contrast to large differences in droplet evaporation where K varies solely as $\Delta H_{v\textit{ap}}^{-1}$ as described by 3.3b ). For instance, in droplet combustion, both hexane (C₆H₁₄) and tetradecane (C₁₄H₃₀) exhibit similar lifetimes of approximately 1 s. In contrast, during droplet evaporation, hexane has a much shorter lifetime of approximately 50 s, compared with roughly 750 s for tetradecane.

Next, we derive the range of m using the constraints (4.24a ) and (4.24b ). Using λω > (1 + φ) $ / $ 2 in (4.24b ), we arrive at m > 1 $ / $ 2 − φ $ / $ 4 + φ ² $ / $ 4 − φω $ / $ 2 from (4.25b ) and then m > (1 − φ) $ / $ 2 because of ω < (1 + φ) $ / $ 2 from (4.24a ). With λω < 2 in (4.24b ), (4.25b ) gives m < 2 − φω $ / $ 2. As a result, for the self-sustaining case, m has to satisfy

(4.26) \begin{equation} \left(1-\phi \right)/2\lt m\lt 2-\phi \omega /2. \end{equation}

Note that the upper bound (2 − φω $ / $ 2) in the above is likely greater than unity. However, m cannot exceed 1 since m $=$ 1 corresponds to the extreme n $=$ 3 case where droplet vaporisation is driven by a constant heating source like the burner case.

With (4.26) for the self-sustaining case and m $=$ 1 from (4.3) for the burner case, the actual value of m should range between these two cases, giving

(4.27) \begin{equation} \left(1-\phi \right)/2\lt m\lt 1. \end{equation}

Since φ can be either 1 $ / $ 2 (slipping surface) or 1 $ / $ 3 (no-slip surface), we can have two different ranges for m from (4.27) depending on the surface condition of a droplet:

(4.28a) \begin{align}& 1/4\lt m\lt 1\left(\text{slipping surface}\right)\!, \end{align}

(4.28b) \begin{align}& 1/3\lt m\lt 1\left(\text{no-slip surface}\right)\!. \\[9pt] \nonumber \end{align}

Either 2.25 < n < 3 from (4.28a ) or 2.33 < n < 3 from (4.28b ) adequately covers the measured values n $=$ 2.56 ± 0.20 − 2.67 ± 0.24 for the self-sustaining case shown in figure 7(b). Equation (4.28b ) appears to better fit the experimental data, possibly due to surface immobilisation by impurities. The case n $=$ 3 predicted for the burner-assisted case from (4.2) also explains well the measured value n $=$ 3.00 ± 0.15 in a continuous flame shown in figure 9. Therefore, our theory can capture the experimental values n $=$ 2.56 ± 0.20 − 3.00 ± 0.15 for both the self-sustaining and burner-assisted combustions summarised in figure 9.

It is worth noting that, for droplet evaporation driven by natural convection in the absence of flame, under a fixed elevated temperature ΔT, as studied by Misyura (Reference Misyura2018), the induced flow velocity scales as U ∼ (gβ $\Delta$ TR) $ ^{1/2} $ and the corresponding Nusselt number in the strong convection limit behaves as $\textit{Nu}$ $=$ c(UR $ / $ α) $^{1/2} $ $=$ c(gβ $\Delta$ TR ³ $ / $ α ²) $ ^{1/4} $ , following the analysis by Yuge (Reference Yuge1960). Substituting this scaling into (4.4) yields n $=$ 1.25 < 2, indicating a negative departure from the D ²-law, as reflected in the resulting equation

(4.29a) \begin{equation} \frac{\textrm{d}R^{2}}{\textrm{d}t}=-\left(1/4\right)cK_{D2}\left(\frac{R}{\mathrm{\ell }_{\textit{natural}}}\right)^{3/4}. \end{equation}

Similar to $\mathrm{\ell }$ introduced in (4.15) for droplet combustion, $\mathrm{\ell }_{\textit{natural}}$ in (4.29a ) is the inherent length scale associated with natural convection driven by a fixed ΔT:

(4.29b) \begin{equation} \mathrm{\ell }_{\textit{natural}}=\left(\frac{g\beta {\unicode[Arial]{x0394}} T}{\alpha ^{2}}\right)^{-1/3}\end{equation}

This length scale also modifies the vaporisation rate constant to be

(4.29c) \begin{equation} K=\left(5\boldsymbol{\cdot }2^{1/4}/16\right)c K_{D2}/\mathrm{\ell }_{\textit{natural}}^{3/4}, \end{equation}

which is no longer simply proportional to ΔT, but instead varies more sensitively as ΔT $ ^{5/4} $ , since K _D2 ∝ΔT and $\mathrm{\ell }_{\textit{natural}}$ ∝ ΔT $^{-1/3} $ .

Comparison with the above results highlights the key reason why droplet combustion exhibits n > 2: it originates from the inverse dependence of ΔT on R as described by (4.18). In fact, the negative departure with m $=$ −3 $ / $ 4 in natural convection without flame can be recovered using the same theoretical framework developed for droplet combustion by treating this case as a special limit. This is achieved by setting ΔT ∝ R ⁰ (i.e. independent of R) with λ $=$ 0, W $=$ R with ω $=$ 1, and φ $=$ 1 $ / $ 2 in the Nusselt number expression given by (4.20). This particular set of exponents yields the corresponding value of m via (4.25b ). Similarly, the case of forced convection, which leads to m $=$ −1 $ / $ 2, can be obtained by choosing the parameter set (λ, ω, φ) $=$ (0, 0, 1 $ / $ 2). These results further reinforce the distinctions between flame-driven and non-flame driven vaporisation processes, emphasising the unique influence of combustion-driven convection.

Having table 2, that highlights the features of all these different droplet vaporisation mechanisms, we conclude that the departure of shrinkage kinetic laws from the classical D ²-law is fundamentally attributable to the presence of inherent length scales, denoted by $\mathrm{\ell }_{\textit{inherent}}$ , which arises from the underlying convective mechanisms driving vaporisation processes. This length scale necessitates the impact of the supplied convective heat flux – typically dependent on the droplet radius R – to be expressed in terms of the ratio R $ / $ $\mathrm{\ell }_{\textit{inherent}}$ . This in turn modifies the governing equation, wherein the vaporisation rate K _D2 in the D ²-law is further modulated with a power-law dependence on that ratio, characterised by a deviation exponent −m:

Table 2.

Summary of values of the shrinkage exponent n, inherent lengths and expressions of the vaporisation rate constant K for different droplet vaporisation mechanisms. Here, K _D2 $=$ 8α (ρ $ / $ ρ _L ) (C _p ΔT $ / $ Δ $H_{v\textit{ap}}$ ) is the vaporisation rate constant of the D ²-law.

(4.30) \begin{equation} \frac{\textrm{d}R^{2}}{\textrm{d}t}=-\left(1/4\right)c K_{D2}\left(\frac{R}{\mathrm{\ell }_{\textit{inherent}}}\right)^{-m}. \end{equation}

As such, our theory not only unifies various deviations from the D ²-law under different vaporisation scenarios, but also reveals the central role of inherent convective length scales in shaping the true shrinkage kinetics of droplets.

5. Impacts due to non-idealities

In actual droplet vaporisation experiments, non-idealities inevitably contribute to variations in the measured data. The most common of these are the effects of support fibre and thermal radiation, which can significantly influence droplet evaporation and droplet combustion, respectively. Regardless of these specific non-idealities, the measured vaporisation rate constant, which characterises the vaporisation process, is often found to depend on the initial droplet diameter, contrary to the usual assumption that it does not. This section discusses the issues arising from these factors and their impacts on experimental results.

5.1. Possible impacts from support fibres

Support fibres may or may not affect the shrinkage kinetics of a vaporising droplet. The results we report are largely based on the suspended fibre technique in which a droplet is held at the end knot of a suspended fibre. A droplet can also be held using the cross-fibre technique (Chauveau et al. Reference Chauveau, Birouk, Halter and Gökalp2019; Wang et al. Reference Wang, Huang, Qiao, Ju and Sun2020). In this technique, a droplet is held at the cross-point of two fibres. Compared with the suspended fibre technique, this technique creates more contacts with the fibres, which may render fibre heating effects more pronounced in affecting the shrinkage kinetics of a vaporising droplet.

To test if the cross-fibre technique can alter droplet shrinkage kinetics, we perform both droplet evaporation and combustion experiments for ethanol (whose n value is the lowest used here). In order to demonstrate the difference between these two fibre supporting techniques, we use thicker fibres of diameter ∼100 μm in these experiments. For the evaporation case, we find n $=$ 2.5 ± 0.24 for the cross-fibre technique whereas the suspended fibre technique gives n $=$ 1.96 ± 0.08 close to the ideal value of 2. The more pronounced departure from the D ²-law in the former implies that crossed fibres have a greater impact on heat transfer to the droplet, as they provide additional heating. Since such fibre heating is through conduction across the fibre radius, it supplies a constant heat flow to the droplet, which tends to cause the droplet’s volume to decrease linearly with time, thereby increasing n towards 3.

For the combustion case, we find n $=$ 2.57 ± 0.19 for the cross-fibre technique. It does not have much difference from n $=$ 2.53 ± 0.30 for the suspended fibre technique. But unlike the evaporation case, the measured value of n does not change with fibre supporting methods. Therefore, this positive departure from the D ²-law in droplet combustion cannot be attributed to the fibres.

To explain why fibre heating has a strong impact on droplet evaporation but not on droplet combustion, we compare the fibre heat flux q _f to the ambient heat flux q by the surrounding gas; q _f grows roughly linearly with the driving gas-droplet temperature difference ΔT, as it is mainly by conduction with a constant heat transfer coefficient. In contrast, q is driven by natural convection set up by the flame. Since the corresponding heat transfer coefficient h $=$ q $ / $ ΔT is typically enhanced by this convection with speed increasing with ΔT, h is no longer independent of ΔT, but instead grows as some positive power η of ΔT, causing q $=$ hΔT to grow as ΔT ^{η + 1}. Thus, fibre heating relative to ambient heating will decrease with ΔT according to

(5.1) \begin{equation} q_{f}/q\varpropto {\unicode[Arial]{x0394}} T^{-\eta }. \end{equation}

Since ΔT here is typically high (of ∼10³ K), fibre heating may be less significant than heating by the ambient gas. Additionally, considering that the heat transfer area for fibre heating is a factor (d _f $ / $ D)² smaller than that for the ambient gas (with d _f being the fibre diameter), the heat flow from the fibre could be even weaker compared with that of the latter. Therefore, not only do fibre heating effects not affect the shrinkage kinetic law, but also the ways to support the droplet will not matter, which is exactly what we observe in experiments. For droplet evaporation, since heat transfer is mainly by conduction, both fluxes increase linearly with ΔT, making q _f $ / $ q independent of ΔT. Consequently, q _f $ / $ q in droplet evaporation does not decline with ΔT, unlike in droplet combustion. This explains why droplet evaporation is more sensitive to the presence of fibres. It may account for the sharp decline near the end of evaporation in the D ²-t plot shown in figure 4(a), displaying a significant deviation from the D ²-law.

Asrardel et al. (Reference Asrardel, Muelas, Poonawala and Ballester2024) recently conducted droplet evaporation experiments under various conditions and theoretically analysed potential artefacts caused by non-ideal effects, including fibre heating. They found that the measured evaporation rates are slightly greater than those without a fibre, with the departure being proportional to the ratio of the fibre diameter to the initial droplet diameter, d _f $ / $ D ₀, when d _f $ / $ D ₀ < 0.2. This suggests that using a thinner fibre could help reduce fibre heating effects. This is exactly what we observe using a thinner fibre of d _f ∼ 35 μm in both droplet evaporation and combustion experiments in which the measured values of n do not differ much compared with those using a thicker fibre of d _f ∼ 100 μm. This again supports our claim that the presence of the fibre does not alter the shrinkage kinetic laws, provided that the contacts between a droplet and the fibre are minimised.

5.2. Impacts of radiation in droplet combustion

Because the flame temperature ∼10³K is typically high, thermal radiation may play an important role in droplet combustion (Elperin & Krasovitov Reference Elperin and Krasovitov1995; Marchese & Dryer Reference Marchese and Dryer1997; Millán Merino et al. Reference Millán-Merino, Fernández-Tarrazo and Sánchez-Sanz2021b ), especially for large droplets. However, most of the analyses are largely based on the spherical flame model for the convection-free situation, such as in a microgravity environment where conduction dominates heat transfer. When convection is present, it has been shown that the difference in vaporisation rates with and without radiation is only approximately 10 % or even less for droplet evaporation at elevated temperatures (Abramzon & Sazhin Reference Abramzon and Sazhin2006). This can be attributed to the fact that convection enhances heat transfer to the droplet, making the contribution from radiation less significant. A similar situation may occur during droplet combustion under gravity, where buoyant convection prevails. As shown below, the heat transfer to the droplet in this case is primarily governed by the prevailing buoyant convection, with radiation playing a minor role.

To quantify the impact of radiation on droplet combustion in the presence of buoyant convection, we follow the approach of Marchese & Dryer (Reference Marchese and Dryer1997) to estimate the radiative heat to the droplet (of radius R ∼ 1 mm and surface area A $=$ 4πR ²) received from the flame. Since the frame area is typically much larger than the droplet surface area, this radiative heat can be estimated according to (Incropera & De Witt Reference Incropera and De witt1990)

(5.2) \begin{equation} \dot{Q}_{\textit{rad}} = \varepsilon \sigma _{B}\big({T_{f}}^{4}-{T_{s}}^{4}\big)A, \end{equation}

where $\varepsilon$ ∼ 1 is the emissivity of the droplet, σ _B $=$ 5.67 $\times$ 10⁻⁸ W m⁻² K⁻⁴ is the Stefan–Boltzmann constant and T _f ∼ 10³ K is the flame temperature, which is typically much higher than the droplet’s surface temperature T _s . Using the above information above, we find $\dot{Q}_{\textit{rad}}$ ∼ $\varepsilon$ σ _B T _f ⁴ A ∼ 0.7 W.

The heat flow to the droplet from the ambient gas heating can be evaluated according to

(5.3) \begin{equation} \dot{Q}_{\textit{droplet}}=h{\unicode[Arial]{x0394}} TA. \end{equation}

Here, ΔT, the difference between the flame temperature and the droplet surface temperature, is roughly the flame temperature T _f ∼ 10³ K. The heat transfer coefficient h can be evaluated as (k $ / $ R) $\textit{Nu}$ , where k ∼ 2.62 $\times$ 10⁻² W m⁻¹ K⁻¹ is the thermal conductivity of the gas phase and $\textit{Nu}$ is the Nusselt number. The latter can be estimated as $\textit{Nu}$ ∼ (UR $ / $ α) $ ^{1/2} $ in terms of the flow speed U ∼ 1 m s⁻¹ and the thermal diffusivity α ∼ 0.2 cm² s⁻¹ for the gas phase. With these values in (5.3), we find $\dot{Q}_{\textit{droplet}}$ ∼ 2 W. Therefore, $\dot{Q}_{\textit{rad}}$ is indeed smaller than $\dot{Q}_{\textit{droplet}}$ . This indicates that thermal radiation only plays a minor role compared with the dominant heat transfer from buoyant convection. This contrasts sharply with droplet combustion in microgravity.

5.3. Impacts of initial droplet diameter

The impacts of the initial droplet diameter D ₀ can arise from two sources. The first source comes from the application of an improper kinetic law to describe droplet vaporisation processes. This is in particular relevant in cases where the shrinkage kinetic law deviates from the D ²-law but is nonetheless forced into that framework. To illustrate this, consider a scenario where the shrinkage kinetic law follows a non-square law, as given by (1.3)

(5.4) \begin{equation} \left(D/D_{0}\right)^{n}=1-\left(K/{D_{0}}^{n}\right)t. \end{equation}

If re-writing (5.4) to a D ²-t form for short times

(5.5) \begin{equation} \left(D/D_{0}\right)^{2}=\left(1-\left(K/{D_{0}}^{n}\right)t\right)^{2/n}\approx 1-\left(K^{obs}/{D_{0}}^{2}\right)t, \end{equation}

the observed vaporisation rate constant will vary with D ₀ according to

(5.6) \begin{equation} K^{obs}= \left(2/n\right)\left(K/{D_{0}}^{n-2}\right)\!. \end{equation}

For droplet combustion under gravity with n > 2 shown in figure 7, (5.6) indicates that, if the vaporisation rate constant is measured using the D ²-law (5.5), it will decrease weakly with D ₀, which may partially explain experimental observations (Shang et al. Reference Shang, Yang, Xuan, He and Cao2020).

The second source arises from non-ideal effects that are not accounted for in the original model used to derive the shrinkage kinetic law. Fibre heating and thermal radiation effects, as discussed earlier, fall into this category. In principle, these effects can modify the shrinkage kinetic law by introducing additional heat fluxes to the overall heat balance of the droplet. The relative contributions of these heat fluxes, compared with the heat flux from the ambient gas, generally depend on relevant length scale ratios. For example, the impact of fibre heating is typically quantified by the ratio d _f $ / $ D ₀ and has been observed to increase with it (Asrardel et al. Reference Asrardel, Muelas, Poonawala and Ballester2024). This explains the scatter observed in the plots of the measured time course (1−t $ / $ t _life ) against the normalised droplet size $D/D_0$ , as shown in figure 7. It may also contribute to the variations in the measured values of n seen in figure 9. However, since the average values of n remain approximately the same across different fuels and the observed variations are relatively small (around 10 %), these non-ideal effects are unlikely to significantly alter the shrinkage kinetic laws. Nevertheless, they do influence the vaporisation constant K, causing it to exhibit a slight dependence on D ₀.