Evaporation of Suspended Nanofluid (SiO 2 /Water) Droplets: Experimental Results and Modelling

The results of experimental studies and modelling of the evaporation of suspended water droplets containing silicon dioxide SiO 2 nanoparticles at mass fractions 0.02 and 0.07 are presented. The experimental results are analysed using the previously developed model for multicomponent droplet heating and evaporation. In this model droplets are assumed to be spherical and the analytical solutions to the heat transfer and species diffusion equations are incorporated into the numerical code. They are used at each timestep of the calculations. Silicon dioxide nanoparticles are considered to be a non-evaporating component. It is demonstrated that both experimental and predicted values of droplet diameters to the power 1.5 decrease almost linearly with time, except at the beginning and the final stages of the evaporation process, and are only weakly affected by the presence of nanoparticles. At the final point in this process, the effect of nanoparticles becomes dominant when their mass fraction at the droplet surface reaches about 40 % and a cenosphere-like structure is formed. Both predicted and observed droplet surface temperatures rapidly decrease during the initial stage of droplet evaporation. After about t = 100 s the predicted surface temperature remains almost constant whilst its experimentally observed values increase with time. This might be related to a decrease in the temperature of ambient air in the vicinity of droplets, not taken into account in the model. Both observed and predicted values of the mass fraction of silicon dioxide at the droplet surfaces are shown to increase with time until they reach about 0.4.


Introduction
Experimental studies and modelling of nanofluids have become a new and rapidly growing direction of research in the last two decades [1][2][3][4][5][6]. These nanofluids are now considered to be promising coolants in heat exchangers, heat pipes, solar collectors, and many other energy devices. The most powerful effects of nanoparticles can be exerted in two-phase (gas-liquid) systems and especially in the presence of phase changes-boiling and evaporation. In this case, nanoparticles at the interfacial boundaries can have an important impact on the surface forces, as a result of which the magnitude of the critical heat fluxes during the boiling of the nanofluids can change greatly compared to those of pure liquids. This can affect the rate of droplet evaporation [4,[7][8][9].
Note that so far research into nanofluids has been focussed mainly on convective heat exchange in the single-phase mode rather than on their phase transformation. Evaporation of both suspended and sessile nanofluid droplets, for which the processes at the interfacial boundary can play a significant role, were investigated by Zhong et al. [10] and Sefiane and Bennacer [11]. Evaporation of these droplets can be affected by several parameters, including the composition of the base fluid, size, shape, material, and concentration of nanoparticles, and the surface tension, speed and temperature of the surrounding flow [12]. Despite the existence of a number of available theoretical and experimental publications on the evaporation of nanofluids, this field of science is still being developed.
Chen et al. [13,14] were amongst the first to present the results of measurements of the rate of evaporation and surface tension of droplets of nanofluids suspended from a capillary. Laponite (mass fraction 0.5 %), Ag (silver) and Fe 2 O 3 (mass fractions 0.05 %) nanoparticles were added to the base fluid (water). To stabilise the nanofluid with silver (diameters in the range 10-30 nm) and iron oxide (diameters in the range 30-40 nm) nanoparticles, polyvidone (PVP) was used as a surfactant (mass fraction 1 %). The presence of a surfactant stimulated an approximate doubling of the droplet evaporation rate. The weak effect of the addition of nanoparticles on the droplet surface tension, and the rate of their initial evaporation was demonstrated. When a certain critical size was reached, the rate of droplet evaporation decreased, and this effect was most visible for solutions with silver nanoparticles. For laponite, there was no change in the droplet evaporation rate; the nanofluid with Fe 2 O 3 nanoparticles occupied an intermediate position between them.
The impact of nanoparticles on the evaporation rate of silicone oil and deionised water was investigated experimentally by Zhang et al. [15]. Nanoparticles of silicon dioxide (SiO 2 ), calcium titanate (CaTiO 3 ) and two modifications of titanium dioxide (TiO 2 ) (anatase and rutile) were used. It was shown that the presence of nanoparticles could suppress or intensify the evaporation of a water-based nanofluid with TiO 2 anatase nanoparticles, depending on their concentration. The process of flotation of nanoparticles on the surface of the liquid was considered in more detail by Bochkarev and Polyakova [16].
Kumar and Sathian [17] used non-equilibrium molecular dynamics methods to investigate the evaporation of liquid argon droplets with platinum nanoparticles.
The authors analysed in detail the contribution of each input parameter and showed that an increase in the interaction force at the nanoparticle-base liquid boundary leads to a decrease in the evaporation intensity. A phenomenological model based on the theory of convective nanofluid drying was developed by Wei et al. [18]. Handscomb et al. [19] developed a drying model of a droplet of colloidal liquid when a porous structure was formed during the final stage of droplet evaporation.
The evaporation of nanofluids with nanoparticles of diamond, oxides of silicon, aluminium, and zirconium was investigated experimentally by Minakov et al. [20]. Terekhov and Shishkin [21] showed experimentally that the presence of carbon nanotubes in low concentration does not affect the droplet evaporation rate.
As follows from the experimental results presented by Moghiman and Aslani [22], the addition of small amounts of nanoparticles to a base liquid can lead to an intensification of the evaporation process or its suppression, or have no influence on it whatsoever. As follows from the results presented in [23], the rate of droplet evaporation depends on the specific material of the nanoparticles, the presence of stabilising additives and a number of other factors.
There were several studies of evaporation and combustion of fuel droplets with nanoparticles from high-energy materials including aluminium with surface protection [23][24][25], cerium [26] and its oxide [27], and graphite [28]. The contribution of nanoparticles to the transfer of thermal radiation in nanofluids was investigated by Gan and Qiao [29], Said et al. [30], and Jing and Song [31].
Numerical modelling of nanofluid droplets has been discussed in a number of papers including [25]. Wei et al. [18] and Fisenko and Khodyko [32] drew attention to the fact that accumulation of nanoparticles in the vicinity of the droplet surface leads to a reduction in the effective area of droplet evaporation which leads to a reduction in the rate of evaporation. Note that most attention has been focussed on sessile droplets [6,33,34], the analysis of which is beyond the scope of this paper.
As follows from the brief review of previous research into the heating and evaporation of nanofluid droplets presented above, this problem is far from well understood. Even at a qualitative level, it is difficult to predict the nature of the change in the rate of evaporation of droplets containing nanoparticles of various materials, sizes, and shapes. The aim of this paper is to make a further contribution to this study and clarify some features of nanofluid droplet evaporation which have not been considered so far.
The paper focusses on the experimental study and modelling of the evaporation of suspended water droplets, containing silicon dioxide SiO 2 nanoparticles with mass fractions 0.02 and 0.07, in an ambient temperature of 28.4 • C (room temperature), airflow velocity U = 0.2 ± 0.02 m·s −1 and relative humidity 4.3 %.
The choice of mass fractions of nanoparticles was motivated by our intention to investigate the effect of a high concentration of nanoparticles on the nanofluid droplet evaporation process, including the formation of the cenosphere-like structure during their evaporation. This study is complementary to the previous investigation where the effect of low mass fraction (0.1 %) of these particles was investigated [35]. The initial mass fractions of nanoparticles 0.02 and 0.07 were chosen based on the observation that for these values their agglomeration was not observed until the final stage of the drying process when the cenosphere-like structure was formed. The relative humidity used in our experiments was the lowest that we could effectively control. The chosen air velocity was the lowest for which the effect of ambient conditions outside the chamber where the experiments were performed on the droplet evaporation process could be ignored.
This study was primarily motivated by the fact that silicon oxide nanopowders are widely used in various fields of industry and science. They are actively used as additives in paint products, anti-corrosion, antifriction and hydrophobic coatings, and rubber products. SiO 2 nanoparticles are used as additives for concretes, dry building mixtures, heat-resistant and heat-insulating materials [36].

Experimental Setup and Procedure
A scheme of the experimental setup is presented in Fig. 1. A liquid droplet was fixed on a crosshair of threads with diameters 105 m, and placed above a nozzle. The thread material (horsehair) had a fairly low thermal conductivity (about 0.15 W·(m·K)) and did not absorb water [37]. Heat supplied through these threads was assumed to be small [38]. This type of support allowed us to preserve the spherical shape of the droplet during most of the evaporation process. Also, it prevented droplet movement along the thread and reduced its vibration in the airflow [39].
The gas flow came from a compressed air cylinder into the heating section through a cylindrical channel with a diameter of 150 mm. Then it passed through a narrowing channel with a diameter of 60 mm. The inlet flow rate was controlled by a reducer. To reduce disturbances in the flow, the gas passed through a system of two honeycombs with grid steps 0.5 mm and 0.2 mm. These honeycombs were Fig. 1 The scheme of the setup used in the experiments located at a distance of 380 mm from each other. A profiled converging channel (confuser) was installed at the exit from the working area. This confuser had an outlet diameter of 12 mm and the degree of flow compression was 17. It ensured the formation of a homogeneous flow.
Airflow speed was measured with a KIMO instruments VT110 anemometer at the outlet from the confuser. The anemometer and a rotameter allowed us to perform measurements of velocity in the range 0.1-3 m·s −1 with uncertainty ±3 %. The values of relative humidity at the location of the droplet were measured with a hygrometer probe (AZ Instrument model 872) with measurement uncertainty ±4 %. A closed box shown in Fig. 1 was used in the experiments to ensure that remained constant (cf. [40]).
The time evolution of droplet radii was measured with a Baumer vcxg-04m high-speed video camera. The obtained data were processed using the ImageJ program. Approximating the droplet shapes by that of a spheroid their diameters (d) were estimated as: where a and b are the axes of the spheroid. The measurement uncertainty did not exceed ±0.07 mm.
The sources of errors of measurements of d include systematic errors (e.g. those related to non-sphericity of droplets), and random errors. These were estimated as 0.05 mm and 0.02 mm, respectively, in the range of droplet diameters under consideration, and were considered to be additive, leading to the abovementioned measurement uncertainty of ±0.07 mm. The dependence of these errors on droplet sizes was weak and not considered in our analysis. This led to errors of (d∕d 0 ) 1.5 equal to 1.5(0.07∕2) = 1.5 × 0.035 ≈ 0.05 at the beginning of the evaporation process. These errors increased to about 0.1 at the end of this process.
The droplet diameters were measured simultaneously with the measurements of temperatures in the vicinity of droplet surfaces. These temperatures were measured with an NEC TH7102IR thermal imaging camera at wavelengths = 8−14 m using a TH 71-377 macro lens. These temperature measurements refer to droplet surfaces with a thickness of not more than 9 m. The value of this thickness was determined by the depth of the IR penetration into the droplet at various wavelengths [41].
The thermal imaging camera was installed with a focal length of 6 cm [41][42][43]. Individual temperature measurements were averaged over the area near the central part of the thermal image of the droplet. The data were processed by ThermoTracer software. The systematic and random errors of these measurements were estimated to be 0.2 K and 0.45 K, respectively, leading to a total error of 0.65 K. See [41][42][43] for a more detailed analysis of methods of droplet temperature measurement using thermal imaging.
Typical images of droplets of pure water and nanofluids obtained using this camera are shown in Fig. 2. As follows from this figure, in all cases the droplet surface temperature first drops and then gradually increases during the evaporation process. The formation of the cenosphere-like structure can be clearly seen for the nanofluid droplet with a mass fraction of nanoparticles equal to 0.07. The average mass fractions of nanoparticles were estimated as where Y l0 is the initial mass fraction (0.02 or 0.07) of nanoparticles, d 0 is the initial droplet diameter. The errors of measurements of Y l0 did not exceed 0.5 % for Y l0 = 0.02 and 0.1 % for Y l0 = 0.07 . The error of measurement of d 0 ∕d 3 was 3(0.07∕2) = 3 × 0.035 ≈ 0.1 at the beginning of the evaporation process and increased to about 0.2 at the final stages of this process. This means that the latter errors control the overall errors of measurement of Y l0 and are equal to approximately 10 % (beginning of the evaporation process) or 20 % (end of the evaporation process) for both Y l0 . Nanofluids used in the experiments were prepared based on commercially available spherical nanoparticles of silicon dioxide (SiO 2 ) with median diameters 12 nm supplied by the Evonik company. This size was the result of a compromise between availability and the requirement to minimise particle sizes. Distilled water was used as the base liquid. To prepare nanofluids with certain mass fractions of components the latter were weighed on HR-250AZG electronic scales with measurement error ±0.1 mg. Then the nanoparticles and base liquid were mixed manually. An ultrasonic treatment was used to obtain a stable homogeneous mixture with a minimum degree of particle agglomeration [23]. Note that all nanofluids were prepared without the addition of surfactants.

Description of Models and Approximations
The experimental results obtained using the setup and procedures described in Sect. 2 were analysed using the model of droplet drying developed by the authors of [44] and the model describing the effect of support presented in [45]. In the model described in [44], nanoparticles inside water in droplets are considered to be a non-evaporating liquid.
The spherically symmetric processes inside the droplet were described based upon the transient heat transfer and species diffusion equations [46]: are the temperature and mass fractions of water ( i=0 ) and nanoparticles ( i=1 ) inside the droplets, R and t are the distance from the droplet centre and time, respectively, and eff and D eff are effective thermal and nanoparticle diffusivities, respectively. These diffusivities consider the effects of recirculation inside droplets due to their motion relative to the gas, using the Effective Thermal Conductivity (ETC) and Effective Diffusivity (ED) models [46]. P ≡ P(R, t) accounts for the effects of external heating inside droplets (e.g. thermal radiation).
Equations (1) and (2) were solved analytically assuming that T(R, t) and Y li (R, t) are twice continuously differentiable functions at 0 ≤ R ≤ R d ( R d is the droplet radius), using the standard initial conditions and the following conditions at the droplet surface: h and ṁ d are the convection heat transfer coefficient and the rate of droplet evaporation, respectively, k eff and D eff are the effective thermal conductivity and effective diffusivity, respectively, and l0 and L are water density and specific heat of water evaporation, respectively. The value of ṁ d predicted by the Abramzon and Sirignano model [47] was used (see [46] for the details). The explicit expressions for T and Y li , predicted by the analytical solutions, are given in [46]. They were implemented into the numerical code and used at each timestep for calculations of droplet heating and evaporation [44,46]. This allowed us to take into account the dependence of all input parameters on time and temperature. The advantages of this approach compared with the one based on a purely numerical solution to Eqs. 1 and 2 are described in [46].
The effect of thermal swelling or contraction was modelled as in [46]. In this approach, the droplet sizes were calculated assuming that the liquid density is the same throughout the droplet; this density was estimated for the average droplet temperature. The contribution of the supporting thread was accounted for using the approach described in [45]. In this approach, the analytical solution to the transient heat transfer equation inside a semi-transparent droplet in the presence of thermal radiation was used to find heat transferred to the droplet via the supporting thread [45,46]. It was assumed that this heat is instantaneously and homogeneously distributed throughout the whole volume of the droplet. This assumption can be used when the contribution of heat transferred to the droplet through the thread is much smaller than that transferred to it by convection from the ambient gas. Thus, the analytical solution to the transient heat transfer equation with internal heating Eq. 1 was used to study droplet heating via the supporting thread. In this model, the effect of the thread is considered by using the following expression for P in Eq. 1 [45]: where T sup and T c are the temperatures of the thread and at the droplet centre, S c is the contact area between the droplet and the thread, subscript l (liquid) refers to a mixture of water and nanoparticles.
This approach allowed us to avoid complex 3D calculations of the effects of the support on droplet heating when these effects are small (see [46] for further details).
It was assumed, following [45], that the contact area of the droplet with the thread can be approximated as where d t is the thread diameter, R d the droplet radius. The low thermal conductivity of the supporting thread allows us to assume, following [48], that T sup = T s , where T s is the droplet surface temperature. This approach to considering the effect of the supporting thread is more accurate than the one used in [49] where T sup was identified with the ambient gas temperature.
To estimate the effect of SiO 2 nanoparticles on droplet evaporation, two areas at the surface of the droplet are introduced. These are the areas covered by water ( S w ) and particles ( S p ). Note that where Y l1s is the mass fraction of nanoparticles at the droplet surface, and p and w are the densities of nanoparticles and water, respectively.
Rearranging (8) and (9) we obtain This allows us to present the evaporation rate of a droplet with nanoparticles as: where ṁ d(AS) is the evaporation rate of water without nanoparticles predicted by the classical Abramzon and Sirignano model [46]. In our analysis we assumed that s = 1 (nanoparticles are hydrophilic and covered with water at the droplet surface). As follows from our experiments, droplet diameters stopped reducing when Y l1s reached about 0.4-0.6. At this value of Y l1s nanoparticles form a porous structure through which water can penetrate from the interior of the droplet to its surface. Unfortunately, this process cannot be investigated using the model developed in [44] and it will not be considered in our analysis. Note that in the original version of the model described in [44] it was assumed that solid residue forms a solid ball. In our approach we considered a more general case assuming that the solid substance forms a porous ball after the completion of the drying process.
The maximal observed value of Y l1s (0.6) is less than the maximal random packing density of spherical solids predicted by Radovskii [50]. In the latter paper, the maximal volume fraction of randomly distributed spheres was predicted to be 0.61 (note that the density of nanoparticles is higher than that of water). Note that the model of droplet drying described in this section was applied here for the first time to the investigation of nanofluid droplet evaporation.

Transport and Thermodynamic Properties
The transport and thermodynamic properties of distilled water and their temperature dependencies used in our analysis are described in [49]. The following parameters for SiO 2 nanoparticles were used: density p = 2330 kg·m −3 , thermal conductivity k p = 1.4 W·(m·K) −1 , and specific heat capacity c p = 783 J·(kg·K) −1 . All these properties apply at room temperature. Their temperature dependence in the conditions of the experiments was negligible.
The density and specific heat capacity of the mixture of water ( i = 0 ) and nanoparticles ( i = 1 ) was estimated as [44]: where is the volume fraction of nanoparticles Following [51], liquid viscosity is estimated as: where l0 is the dynamic viscosity of water ( i=0 ).
Formula (15) is not expected to be very accurate but this is not important for our application as the droplet heating and evaporation rates are weak functions of l . They do not depend on l for stationary droplets.
It is assumed that dissolved non-evaporating substances can be treated similarly to non-dissolved substances with masses of particles equal to molecular masses. Thus, the liquid diffusion coefficient can be estimated based on the Wilke-Chang formula [52,53]: where M v is the average molar mass defined as i=0 refers to water, whilst i=1 refers to SiO 2 nanoparticles, v is the Lennard-Jones length (in Å) [46,54]: M v is the average molar mass (in kg·kmol −1 ), inferred from (17).
One of the key parameters of the model is the thermal conductivity of the mixture of water and SiO 2 nanoparticles. Several approximations of this parameter have been suggested. For our application, this approximation should satisfy three key conditions. Firstly, in the absence of SiO 2 nanoparticles it should reduce to the thermal conductivity of water. Secondly, it should be reasonably close to the experimentally observed values of this parameter. Thirdly, the values of the thermal conductivity predicted by this approximation should be reasonably close to the values predicted by well tested approximations for volume fractions of SiO 2 nanoparticles up to about 0.4-0.6 (when the porous structure is expected to begin to form). Following [55], we use the following expression, which satisfies all three conditions: As in the previous formulae, subscript 1 ( 0 ) refers to nanoparticles (water).
In the calculations, it was assumed that the evaporation heat of the nanofluid droplet is based on that of the evaporating component (water). Approximations for air properties were taken from [56]. The water saturation pressure was obtained following the Ambrose-Walton corresponding states method. The vapour diffusion coefficient was calculated using the Wilke-Lee method [57].
Note that in our approach the effect of nanoparticles was taken into account via the modification of thermodynamic and transport properties of nanofluids. The detailed analysis of the processes at the nanoparticle/base liquid interface is beyond the scope of our investigation.

Modelling Versus Experimental Data
As in [49], our focus is on droplet surface temperatures ( T s ) and their diameters (d).
Since all experiments were performed for droplets with initial temperatures equal to the ambient gas temperatures ( T g ), which remained almost constant during the experiments, we will present the results for T g − T s rather than for T s . This approach will allow us to show more clearly the conditions in which droplet evaporation takes place. Following [49,58], we focus on the time evolution of (d∕d 0 ) 1.5 , where d 0 is the initial droplet diameter, which is expected to be an almost linear function of time.
The experiments were performed for the following initial mass fractions of SiO 2 nanoparticles: 0 %, 2 %, and 7 %. As mentioned earlier, the ambient gas temperature was T g = 28.4 • C , relative humidity was 4.3 %, and air velocity U = 0.2 m·s −1 . The droplets were supported by two threads, both 0.105 mm in diameter. The initial droplet diameters were 2.20 mm (pure water), 2.19 mm (water with 2 % nanoparticles), and 2.24 mm (water with 7 % nanoparticles). The experiments with pure water were repeated twice, whilst the experiments with nanofluids were repeated three times for each initial concentration of SiO 2 nanoparticles. The results of the experiments were very close; they differed slightly, due to small differences (up to 4 %) in the initial droplet radii. The results of only one experiment from each series of the experiments will be presented in this section.
Note that for this value of velocity, initial droplet diameter 2.2 mm, and air kinematic viscosity 15.89 × 10 −6 m 2 ·s −1 (at temperature 300 K) we obtain Re = 27.7 .
For thermal diffusivity of air equal to 22.5 × 10 −6 m 2 ·s −1 and mass diffusivity of water vapour in air 26 × 10 −6 m 2 ·s −1 (both at room temperature) [57] we obtain the following values of the Prandtl and Schmidt numbers: Pr=0.7 and Sc=0.6. Using the simplest formulae for the estimation of the Nusselt and Sherwood numbers, assuming that both Spalding heat and mass transfer numbers are small [46]: we obtain Nu≈ 4.8 and Sc≈ 4.7 . Both these numbers are expected to decrease with time due to decreasing droplet diameters during the evaporation process. For finite Spalding heat and mass transfer numbers these values are expected to be slightly smaller. Note that a relatively large value of Re used in our experiments justifies the 1.5 power law for the time evolution of droplet diameters except at the initial stage of droplet evaporation (see [49] for the details). The value of ambient air velocity in the experiments was selected to ensure that we had control over the influence of ambient air humidity on the evaporation process. Investigation of the effect of the ambient air velocity on droplet evaporation was beyond the scope of this paper.
Images of evaporating droplets with initial mass fraction of SiO 2 7 % in the time range 0-1228 s are shown in Fig. 3. At times close to t = 921 s the surface mass fraction of nanoparticles reached about 0.4, droplet size stopped reducing and a porous cenosphere-like structure was formed. As follows from this figure, the droplet shape is reasonably close to spherical until about t = 600 s (10 min), but at the later stage of its evaporation the deviations of the droplet shape from spherical are clearly seen. At this point the model used in our analysis becomes less reliable.
The normalised droplet diameters (d∕d 0 ) 1.5 versus time (t) are shown in Fig. 4. Calculations were performed until the mass fractions of nanoparticles at the droplet surface reached 0.4 when the formation of the cenosphere-like structure at the surface of the droplet was observed. As expected, both experimentally observed and predicted (d∕d 0 ) 1.5 are almost linear functions of time except at the very initial stage (heat-up period) and the final stage of evaporation of droplets for an initial mass fraction of nanoparticles of 7 % (we refer to [49] for a detailed discussion of the linear behaviour of this curve).
As can be seen in Fig. 4, the agreements between observed and predicted values of droplet diameters are close before the cenosphere-like structure at the surface of the droplet is formed. This happened when the surface mass fraction of nanoparticles Nu = 2 + 0.6 Re 1∕2 Pr 1∕3 , Sh = 2 + 0.6 Re 1∕2 Sc 1∕3 , Fig. 3 Images of evaporating droplets with an initial mass fraction of SiO 2 7 % reached about 0.4 (calculations stopped). The deviation between the modelling and experimental results can be attributed to a number of factors which were not taken into account in the model, the most important of which is our assumption about the sphericity of the droplet (see [59] for the details). Once the cenosphere-like structure at the surface of the droplet begins to form, the model can no longer be used. Also, it can be seen in Fig. 4 that there is almost no dependence of the evaporation rate of a droplet on the initial mass fraction of nanoparticles, observed experimentally and predicted by the model, until about t = 300 s. At later times, a slight increase in the evaporation rate with a decreasing initial mass fraction of nanoparticles is observed and predicted by the model. As mentioned earlier, at these times the predictions of the model become less reliable as the model does not consider the effect of nonsphericity of droplets.
The plots of the observed and predicted values of T g − T s versus time for the same parameters as in Fig. 4 are shown in Fig. 5. The symbols show the experimental data. The observed difference T g − T s initially increases and then slowly decreases with time. This result is compatible with those shown in Fig. 2. Note that the modelling results presented are for the time up until the surface mass fraction of nanoparticles reached the threshold value of 0.4 when the porous structure at the droplet surface is expected to develop (see Sect. 3.1 for the details).
As follows from Fig. 5, the predicted and observed initial increase in T g − T s are very close. The observed maximal values of T g − T s are slightly (several K) lower than the predicted ones. After about t = 100 s the predicted values of T g − T s remain almost constant whilst the experimentally observed values of this Fig. 4 Plots of (d∕d 0 ) 1.5 versus time for water and nanofluids. Symbols refer to experimental data, circles-water, down (up) triangles-2 % (7 %) SiO 2 nanofluid. Curves refer to the predictions of the model. Solid-water, dashed thick (thin) curve-2 % (7 %) SiO 2 nanofluid, (a) the initial 50 s (note big error bars), (b) the entire period of observation parameter decrease with time. This might be related to a decrease in the temperature of ambient air in the vicinity of droplets with time, which is not taken into account in the model ( T g in the model is assumed to be constant).
Note that in the experiments, the results of which have been described, the droplet temperature reduces compared with its initial temperature which leads to droplet contraction. Unfortunately, we cannot separate the effects of droplet evaporation and the above-mentioned contraction (both of which are described by the model) based on our experimental data.
Plots of predicted average and surface mass fractions of SiO 2 and observed average mass fractions of SiO 2 ( Y 1 ) versus time for the nanofluids with initial mass fractions of 2 % and 7 % of SiO 2 are presented in Fig. 6. As follows from this figure, the predicted values of average and surface mass fractions of SiO 2 are rather close which indicates that the diffusion of nanoparticles in droplets is quick enough to maintain almost homogeneous distribution of these nanoparticles at all time instants. The agreement between the observed and predicted average values of Y 1 looks almost ideal until the cenosphere-like structure begins to form (when the initial mass fraction of nanoparticles is equal to 0.07) or the observations stopped (when the initial mass fraction of nanoparticles is equal to 0.02). This supports the applicability of our modelling approach to the investigation of the experimental data presented in the paper.
In the case of 7 % of SiO 2 the value of Y 1 = 0.4 was reached after about 1030 s. This is consistent with the results shown in Figs. 2, 3 and 4. Note that the limiting mass fraction of nanoparticles of about 0.4 (when the porous structure at the droplet surface developed) is less than the maximal mass fraction of spherical particles corresponding to their maximal volume fraction of 0.61 predicted in [50] (see our discussion in Sect. 3.1). This is related to the nonsphericity and complex shapes of the nanoparticles used in our experiments. A typical SEM image of an SiO 2 nanoparticle is shown in Fig. 7. Note that in other experiments this limiting mass fraction of nanoparticles at the droplet surface can be different from 0.4, but in all cases it was in the range 0.4-0.6.
Similar limiting mass fractions of nanoparticles can be expected for droplets of nanofluid with lower initial mass fractions of nanoparticles including those with mass fractions 1 % and 2 %. The relevant images of the residues of droplets after the completion of the drying process (including the image for the initial mass fraction 7 %) are shown in Fig. 8. As can be seen in this figure, although these residues are easily recognisable, the limiting mass fractions of nanoparticles cannot be easily measured for the cases of initial mass fractions 1 % and 2 % due to the complexity of the shapes of these residues.

Conclusion
The results of experimental studies and modelling of the evaporation of suspended water droplets containing silicon dioxide SiO 2 nanoparticles at initial mass fractions 0.02 and 0.07 are presented for the first time, to the best of the authors' knowledge. The initial droplet temperatures were the same as ambient temperatures (about 25 • C). The experiments were performed at atmospheric pressure in the presence of air moving at a velocity of 0.2 m·s −1 . The droplets were fixed on a crosshair of threads with diameters 105 m. The time evolution of droplet diameters was measured with a Baumer vcxg-04m high-speed video. The surface temperatures were measured in the layer of thickness of not more than 9 m with an NEC TH7102IR thermal imaging camera at wavelengths in the range 8 to 14 m.
The experimental results were interpreted in terms of the previously developed model for multicomponent droplet heating and evaporation. In this model, droplets are assumed to be spherical and the analytical solutions to the heat transfer and species diffusion equations were incorporated into the numerical code and used at each time step of the calculations. Silicon dioxide nanoparticles were considered to be a non-evaporating component. The effect of supporting threads was taken into account assuming that the heat supplied through them would be instantaneously and homogeneously distributed throughout the whole droplet volume.
It was demonstrated that both observed and predicted values of droplet diameters to the power 1.5 decrease almost linearly with time almost throughout droplet evaporation, and are only weakly affected by the presence of nanoparticles. This law was not followed at the very initial stage (the heat-up period) of droplet evaporation and in the final stages when the mass fraction of nanoparticles at the droplet surface reached about 0.4 and a cenosphere-like structure was formed. In this case, the effect of nanoparticles becomes dominant. The limiting average mass fraction of nanoparticles of about 0.4 is less than the maximal mass fraction of spherical particles corresponding to their maximal volume fraction of 0.61. This is related to the non-sphericity and complex shapes of the nanoparticles used in our experiments. The formation of this structure was not considered by the model used in the analysis.
Both predicted and observed droplet surface temperatures rapidly decreased during the initial stage of droplet evaporation. After about t = 100 s the predicted T g − T s remained almost constant whilst the experimentally observed values Fig. 8 Images of the residues of nanoparticles after the completion of the drying process: (a)-(c) refer to the cases in which the initial mass fractions of nanoparticles were 0.01, 0.02 and 0.07, respectivel of this parameter slowly decreased with time. This might be related to a slight decrease in the temperature of ambient air in the vicinity of droplets which was not considered in the model.
Both observed and predicted values of the average mass fraction of silicon dioxide were shown to increase with time until this mass fraction reached a value close to 0.4.