Heat exchange of an evaporating water droplet in a high-temperature environment

International Journal of Thermal Sciences 150 (2020) 106227

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Heat exchange of an evaporating water droplet in a high-temperature environment G.V. Kuznetsov, Professor, P.A. Strizhak, Professor, R.S. Volkov, PhD * National Research Tomsk Polytechnic University, 30 Lenin Avenue, Tomsk, 634050, Russia

A R T I C L E I N F O

A B S T R A C T

Keywords: Water droplets High-temperature heating Convective heat exchange Heating rates Evaporation rates Droplet temperature field

In this research, we present the results of experiments capturing the characteristics of high-temperature heating and evaporation of water droplets. The experimental parameters are as follows: initial droplet radius 1–2.5 mm, temperature 20–1100 � C, air flow velocity 0–5 m/s. We use two schemes of water droplet fixation in the heated air flow. In the experiments, we record the heating and evaporation of free-falling droplets so that the holder does not interfere with the experimental results. Typical water droplet heating rates range from 0.4 to 92.4 � C/s and evaporation rates from 0.003 to 0.178 kg/(m2s). Criterial processing of our experimental data is based on the classical equations formulated by Ranz and Marshall and equations proposed by a set of research groups and scientists. We determine the variation ranges of the Reynolds numbers, in which one can only use a limited set of Nu(Re,Pr) correlations. Adjustment coefficients are proposed to take into account the droplet surface tempera­ tures and evaporation rates as functions of the gas medium temperature. Finally, we hypothesize as to how modern mathematical models can be modified to bring the simulation results closer to the experimental data in droplet heating and evaporation rates during fast heat supply.

1. Introduction The intensity and typical rates of liquid droplet evaporation are defined by the vapor concentration gradient in the thin layer composed of the ambient gas and vapors of the evaporating liquid [1–6]. It is nearly impossible to determine these concentrations in the regions that are typically much smaller than 1 mm [7–10]. The evaporation of liquid drops is the main process behind many advanced technologies of various industries, such as chemical, petrochemical, oil-refining, and energy sectors [11–15]. Thus, in order to choose the right process conditions, we primarily need to know the liquid evaporation rates (We) and droplet heating rates (Wh). The latter is basically the rate, at which an almost quasi-steady temperature field is generated. It is usually difficult, if at all possible, to experimentally measure the Wh and We in the actual production process [12–20]. Therefore, when calculating the droplet heating and evaporation rates in a gas [21], the researchers often assume that all the energy supplied to the liquid droplet surface is spent on the phase transition. The heat flux to the droplet surface can be calculated using criterial equations obtained for a solid spherical particle. When using this approach, it is hard to consider

the actual physical properties of the processes occurring in a vapor-gas mixture near the droplet surface. With the recent boom in technology, it has become necessary to reliably describe liquid evaporation from the heated droplet surface. The heating temperatures in this case range from 500 � C to 1500 � C, some­ times even reaching 2000 � C, for instance, in flame liquid treatment. Empirical equations for evaporation rates were obtained for a limited temperature range (under 500 � C) in an assumption that water vapor concentration may be linked to the Reynolds and Prandtl numbers ob­ tained in the experimental data processing [22,23]. In other words, the heated gas flow around a droplet entrains vapor molecules. Their con­ centration in the near-wall layer only depends on the inertia and vis­ cosity in this layer. Under such conditions, it is extremely difficult to predict the phase transition rates using models that only consider convective heat exchange. Thus, traditional assumptions and models should be improved to reproduce the evaporation conditions and char­ acteristics more adequately. Conventional approaches to the experimental research of evapora­ tion assume that the temperature distributions in a liquid droplet are homogeneous and all the energy supplied to a droplet is spent on the phase transition. Many renowned scientists including O. Knake, I.N.

* Corresponding author. E-mail addresses: [email protected] (G.V. Kuznetsov), [email protected] (P.A. Strizhak), [email protected] (R.S. Volkov). URL: http://hmtslab.tpu.ru (R.S. Volkov). https://doi.org/10.1016/j.ijthermalsci.2019.106227 Received 29 April 2019; Received in revised form 9 December 2019; Accepted 9 December 2019 Available online 26 December 2019 1290-0729/© 2019 Elsevier Masson SAS. All rights reserved.

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International Journal of Thermal Sciences 150 (2020) 106227

Nomenclature and units B C d G Δm Q Qe R S t Δt T ΔT U V We Wh x, y Nu Pr Re

ν ρ

absolute accuracy emissivity factor Stefan-Boltzmann constant, W/(m2K4) thermal conductivity coefficient, W/(m � C) kinematic viscosity, m2/s density, kg/m3

Subscripts a b conv d e h m rad s sum t w

air boiling convective droplet evaporation heating surface of tubular heater in muffle furnace radiative droplet surface total value tube water

Δ

ε σ

mass-transfer number heat capacity, J/(kg � C) diameter, mm mass flow, l/s decrease in mass over Δt, kg specific heat flux, kW/m2 latent heat of vaporization, kJ/kg radius, mm surface area, m2 time, s time interval, s temperature, � C temperature increment, � C velocity, m/s volume, μl average evaporation rate, kg/(m2s) average droplet heating rate, � C/s coordinates, mm Nusselt number Prandtl number Reynolds number

λ

Superscripts 0 initial value i inner o outer * value at the moment

Greek symbols heat transfer coefficient, W/(m2� C) β fluorophore luminous intensity, cps

α

Stranskiy, N. Fuchs, D. Spalding, M. Yen, L. Chen and M. Renksizbulut, S.S. Kutateladze, D.V. Labuntsov, V.E. Nakoryakov, V.I. Terekhov, A.A. Avdeev, and S.Ya. Misyura developed models to simulate these processes. How­ ever, both the modeling and experimental approaches are limited to the temperature range of 300–500 � C, in which the evaporation character­ istics are in satisfactory agreement with the experimental data (with deviations of less than 10–15%). For gas temperatures of over 500 � C, there are no adequate models so far that would predict phase transition rates with deviations of no more than 10–20% [24]. The main reason of this state of things is the lack of reliable experimental data. Experimental and simulated results by Chen [25], Yuen and Renksizbulut [26,27], building on the ideas by Fuchs [28], Ranz and Marshall [29], are widely known. For many years, there have been discussions over the feasibility of the approach considering the so-called thermal balance at the droplet surface. It is based on the equality of heat spent on the liquid evapora­ tion and the energy supplied to the droplet. Criterial processing under such conditions is reduced to the modification of classical correlations like Nu ¼ f(Re,Pr) using additional factors and effects, for instance, Refs. [14,30–32]. In their monograph [33], Terekhov and Pakhomov give a number of the most widespread equations of the form Nu ¼ f(Re,Pr). According to the research findings from Refs. [33,34], it is important which procedure is used to record the main characteristics of high-temperature heating and evaporation of droplets. There are plenty of high-potential applications for high temperatures in the liquid droplet – gas medium systems [22–24]: direct-contact heat exchangers, fuel ignition in combustion chambers, heat carriers based on flue gases, water vapor and droplets. At the same time, solving the performance problems of both current and emerging heat-power equipment is a key to the further development of the whole global heat and power industry. Therefore, high-temperature heating and evaporation of droplets is attracting more and more interest each year. In the mathematical modeling of the heat exchange between a gas medium and a liquid droplet, a major challenge is adequately describing the temperature fields of the latter. The optical methods of measuring droplet temperature fields (e.g., Refs. [7–10,35–39]) show their high

inhomogeneity. Two stages are distinguished in the process of heating. At the first one, the temperature in all the droplet sections grows rapidly. At the second stage, this growth slows down, and the temperature field becomes quasi-steady. Under rapid phase transformations, the duration of these stages changes considerably. Significant temperature gradients are recorded from the droplet surface to its deep layers almost over the whole droplet lifetime. Such processes greatly affect the temperature distribution in aerosol flows and jets [35,38]. The aim of this research is to experimentally study the hightemperature heating and evaporation of free-falling water droplets or those fixed on holders using well-known recording procedures. We employ typical Nu ¼ f(Re,Pr) correlations to determine the approaches that would allow us to use the latter to reliably predict the values of key heating and evaporation parameters of various liquids. 2. Experimental setup and procedures 2.1. Schemes of droplet generation and placement Fig. 1 shows the schemes of experimental setups. We used three approaches. The first one (Fig. 1a) involves heating and evaporation of a droplet fixed in a hot air flow (Schemes 1–3). The second approach (Fig. 1b) implies the heating and evaporation of a droplet fixed in a horizontal rotary muffle furnace (Scheme 4). The air velocities in the furnace do not exceed 0.1 m/s and only depend on the mechanism of thermogravitational convection. It is impossible to exclude the impact of the heated air moving in an open muffle furnace. Therefore, we esti­ mated the velocities using fine soot particles sized 50–70 μm and highspeed video recording. With this approach, the accuracy of velocity estimation was 15–20% but it was good enough for our purpose. We found that when the temperature in a muffle furnace increased from 20 to 1100 � C, the rate of thermogravitational convection in the air went up to 0.07–0.1 m/s. Therefore, when we further calculated the convective heat flux to the droplet, we took the velocity of the incoming gas me­ dium as equal to 0.1 m/s. It is safe to say that we considered the 2

G.V. Kuznetsov et al.

International Journal of Thermal Sciences 150 (2020) 106227

Fig. 1. Schemes of experimental setups: a – a droplet is fixed on a holder in a high-temperature gas flow (convective heating); b – a droplet is fixed on a holder in a high-temperature gas area (radiative heating) c – a droplet falls under gravity in a high-temperature medium; d – illustration of the droplet heating schemes under study.

maximum possible convective heat flux. The third approach (Fig. 1c) implies heating and evaporation of a free-falling droplet in a hightemperature gas (in a vertically positioned rotary muffle furnace) (Scheme 5). We chose several schemes mostly because we wanted to consider various mechanism of droplet heating and evaporation (convective and radiative heating, with or without a holder, etc.). Further we give a detailed description of each scheme and approach as well as the experimental parameters. In the first scheme (Fig. 1a), a water droplet is placed in a heated air flow (as in Ref. [24]) moving inside a heat-proof quartz cylinder, which can withstand temperatures up to 1800 � C. The cylinder is 96 mm in inner diameter and 100 mm in outer diameter. In order to position a droplet in the flow, illuminate it, and record the ongoing processes, there are four 10-mm orifices in the cylinder. The heated air flow is generated using a Leister CH 6060 compressor with a flow rate of 0–5 m/s and a Leister LE 5000 HT heater with a temperature range of 100–600 � C. The lower limit of the range is chosen because the target applications of the research findings are based on liquid temperatures above 100 � C. The droplet is introduced into the flow by a motorized manipulator. In the experiments, its mobile part moves at 0.05 m/s. The second approach (Fig. 1b) involves placing a droplet inside a rotary muffle furnace LOIP LF 50/500–1200 with a 600-mm heater tube and an operating temperature ranging from 100 to 1200 � C. The droplet is also placed in the flow with the help of a motorized manipulator. According to the third approach (Fig. 1c), we consider the heating and evaporation of a free-falling droplet in a high-temperature gas medium. A vertically positioned LOIP LF 50/500–1200 muffle furnace serves as the heater. Below you will see a detailed description of the five schemes we used (Fig. 1).

diameter. A water droplet was generated using a single-channel electronic dispenser Finnpipette Novus with a volume increment of 0.1 μl. In the experiments, the droplet volume Vd ranged from 5 to 30 μl, which allowed us to vary the initial size (average radius) of the droplet: Rd ¼ 1–2 mm. The newly generated droplets were intro­ duced into a quartz channel at its symmetry axis. The droplets covered the distance from the inlet of the cylinder to its central part in 1 s. Once the droplet reaches the symmetry axis of the quartz channel, we started the video recording of droplet heating and evaporation. � Scheme 2 (Fig. 1a) implied the use of a thin metal tube to generate droplets. We used two types of tubes. The first one was 1 mm in outer diameter (dot ) and 0.7 mm in inner diameter (dit) to generate droplets with a radius Rd ranging from 0.6 to 1.2 mm. The second one was 2.3 mm in outer diameter (dot ) and 1.7 mm in inner diameter (dit) to generate droplets with Rd ¼ 1.1–1.9 mm. An automatic syringe pump was connected to the tubes through a flexible hose. It squeezed a droplet of the required size from the tube. Further actions are similar to those described in Scheme 1. � Scheme 3 (Fig. 1a) has a lot in common with Scheme 2. The exper­ imental procedure was similar to that in Refs. [25,26]. Using a sy­ ringe pump connected to a tube, we generated a water droplet of the necessary size. The water droplet was introduced into the quartz channel. After that, we used a syringe pump to measure the mass flow rate of water (Gw) without changing the droplet size (permis­ sible deviation 0.015 mm). The whole process was captured on camera. Here we assumed (just as in the experiments in Refs. [25, 26]) that the mass flow rate of water (Gw) corresponded to the mass rate of water evaporation from the droplet surface (We). In these experiments, we did not use a perforated sphere as in Refs. [25,26], since a droplet was fixed steadily on a holder after being squeezed from the tube. Moreover, this approach eliminates the impact of the perforated sphere on droplet heating.

� Scheme 1 (Fig. 1a) implied fixing a droplet on a hollow metal holder (as in Ref. [24]) 0.4 mm in inner diameter and 0.6 mm in outer 3

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International Journal of Thermal Sciences 150 (2020) 106227

� Scheme 4 (Fig. 1b) is similar to Scheme 1. The method of droplet generation and placement in a high-temperature air medium is also similar to Scheme 1. What distinguishes this scheme from others is the use of a muffle furnace as the heating medium (radiative heat­ ing). As PLIF cannot be used for temperature measurement inside a tubular heater, we used two thermocouples (type K, temperature range 0–1200 � C, accuracy �1 � C, response time 0.1 s). � Scheme 5 (Fig. 1c) employed droplet discharge through the heated air. The droplet generation system (Fig. 1c) featured a tank with the liquid under study, a supply duct, a roller clamp controlling the droplet supply frequency, as well as a set of tubular metal lugs. We used three types of lugs. The first one was 2 mm in outer diameter (dot ) and 1.6 mm in inner diameter (dit) to generate droplets with Rd � 2.2 mm. The second one was 0.8 mm in outer diameter (dot ) and 0.6 mm in inner diameter (dit) to generate droplets with Rd � 1.5 mm. The third one was 0.4 mm in outer diameter (dot ) and 0.2 mm in inner diameter (dit) to generate droplets with Rd � 1.2 mm. The average droplet velocity Ud approximated 1.7 m/s on entering the hightemperature gas region and 3.9 m/s when leaving. The initial droplet temperature (T0d) was measured with thermocouple (type L, temperature range 0–200 � C, systematic error � 1.5 � C) and varied from 25 to 70 � C. To heat the liquid above 25 � C, we placed a spiral heater in a tank.

the features of droplet temperature field measurement specific to this technique. It is based on the natural fluorescence of fluorophore mole­ cules induced by laser light. Thus, we added the Rhodamine B dye in a mass concentration of 1000 μg/l before the experiments. The intensity of the light irradiated by the fluorophore (β) went down with an increase in water temperature. To illuminate liquid droplets in the experiments, we used a twin solid-state Nd:YAG laser Quantel EverGreen 70 (wavelength 532 nm, pulse frequency 4 Hz, maximum pulse energy 36 mJ). To generate the light sheet, we used cylindrical lenses with an opening angle of 45� . The light sheet was positioned using an optical mirror. Water droplet images were captured with an ImperX IGV-B2020 M camera: frame resolution 2048 � 2048 pix, frame rate 4 fps, 16 bit depth. We used a Nikon 200 mm f/4 AF-D Macro lens with an inter­ ference filter of 600–10 nm. The optical axis of the camera was posi­ tioned perpendicular to the laser sheet plane. We used the Actual Flow software to process the data and produce droplet temperature fields [24]. The experiments had two stages. At the first one, we plotted the calibration curve reflecting the link between droplet luminosity and its temperature. We introduced thermocouple (type K, temperature range of 0–1200 � C, systematic error of �1 � C, response time 0.1 s) into a droplet, so that its junction was in its center (for Schemes 1–3 in Fig. 1). For temperature recording, we used a National Instruments 9213 mod­ ule with a frame rate of 2 Hz. The droplet with a small thermocouple inside was introduced into the high-temperature gas. The droplet tem­ perature field Td(Rd, t) was adjusted by varying the air temperature Ta. The video camera captured the evaporating water droplets and then noise was excluded from each image obtained. The luminosity of the droplet (β) was compared against the corresponding temperature values taken from the thermocouple trend. These were used to plot a calibra­ tion curve T ¼ f(β) (Fig. 2). To provide higher computation accuracy, a droplet section area no larger than 0.5 � 0.5 mm was selected close to the thermocouple junction. For Scheme 5, the above procedure is similar apart from the T0d values taken as the water temperature. The calibration was based on droplet snapshots at the immediate outlet of the batcher. At the second stage, we captured the temperature field in the given section of the water droplet for schemes No. 1,2,3,5. The equipment settings were the same as at the first stage when plotting a calibration curve. At least five experiments were performed for each initial

It seems reasonable to describe in more detail how the temperature is recorded in a tubular muffle heater (Fig. 1b,c). When conducting the experiments according to scheme No. 4 (horizontally placed muffle furnace – Fig. 1b), we measured the temperature in four points around the droplet right before each experiment. For that we used type K thermocouples (temperature range 0–1200 � C, accuracy �1 � C, response time 0.1 s). The dispersion between the thermocouple readings and experimental settings was as follows: �5 � C for Ta � 100 � C; �14 � C for Ta � 300 � C; �29 � C for Ta � 600 � C. The deviations are rather signif­ icant, because the tubular heater was always open in the experiments. We had to remove its side covers to introduce the droplets and to record the process. When conducting the experiments according to scheme No. 5 (vertically placed muffle furnace – Fig. 1c), we first measured the temperature in different sections of the heating chamber (also using type K thermocouples) along the height of the tubular heater. At the same time, the maximum temperatures were observed in the central part of the heater and the minimum ones, in its lower part. The temperature deviations from the settings were as follows: �85 � C for Ta � 600 � C and �135 � C for Ta � 1100 � C. It is also noteworthy that the temperature setting was chosen so that the arithmetic mean of the temperature along the height of the heater matched the temperature requirement (for instance, 600 � C or 1100 � C). Table 1 gives the experimental parameters for the five schemes above. The purpose of this section’s experiments was to determine the droplet heating rates Wh and water evaporation rates We. 2.2. Droplet temperature recording We measured the temperature of water droplets for Schemes 1–3, 5 in the course of their heating and evaporation (as in Refs. [24,34]) using Planar Laser Induced Fluorescence (PLIF). Volkov et al. [39] described

Fig. 2. Typical calibration curve for the PLIF technique.

Table 1 Parameters for five schemes used in the experiments (Fig. 1). Scheme

No. 1

No. 2

No. 3

No. 4

No. 5

δ

T0d (� C) R0d (mm) U0d (m/s) Ta (� C) Ua (m/s) Re

25 1–2 – 100–600 3–5 60–960

25 0.6–1.9 – 100–600 3–5 40–910

25 0.6–1.9 – 100–600 3–5 40–910

25 1–2 – 100–600 0.1 2–20

25–70 1.2, 1.5, 2.2 1.7–3.9 600–1100 0.1 190–940

�1.5 �0.015 �0.06 �3.5 �0.08 –

4

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International Journal of Thermal Sciences 150 (2020) 106227

parameter group (T0d, R0d, U0d, Ta, Ua) and droplet images were captured. Then we used a calibration curve (Fig. 2) to convert the luminosity of each pixel in the image into temperature using the corresponding soft­ ware. This way we obtained a two-dimensional temperature field. The accuracy of PLIF temperature measurement was �2 � C. We used thermocouples to measure the temperature when the droplet was placed according to scheme No. 4 (Fig. 1b), because PLIF measurement was impossible inside the tubular heater of the muffle furnace. For that we used two type K thermocouples (see Section 2.1). The thermocouples were first fixed near the holder so that the sensor of one thermocouple would be in the center of the droplet and that of the second one near its surface. We used a National Instruments NI9213 analog input module for thermocouple sampling and recording their signals. The sampling frequency was 0.5 s. Then, using the information obtained, we calculated the droplet heating and evaporation parame­ ters. The systematic error of thermocouple measurements was �1.2 � C.

calculations because a droplet cannot heat up to a homogeneous tem­ perature field this fast (for less than 0.25 s) in scheme No. 5. Thus, if we take the current temperature in the droplet center as Td* in our calcu­ lations, the calculated values of Wh are 2–3 times lower than the actual ones. For schemes 1–4, the latter approach (using temperature in the droplet center) gives a result that is almost similar to the previous approach. The difference in the Wh values does not exceed 1–1.5 � C. Fig. 4a gives an example of the average temperature variation within the droplet being heated and evaporating (PLIF data), as well as the corresponding values of instantaneous heating rate (calculated using equation (1)) for scheme No. 1. The time increment in the figures is 0.25 s. Clearly (Fig. 4a, on the left), the average temperature fluctuates when the droplet is being heated. This is related to the convective fluxes within the droplet (constant mixing of liquid layers) and the PLIF error. Therefore, when using equation (1) to calculate the instantaneous heating rates, we often get both negative values of Wh (excluded from Fig. 4a) and significantly overstated ones. That is, even with the ho­ mogeneous temperature field, the Wh values obtained using this approach are mostly non-zero (Fig. 4a, on the right). This makes it sensible to use averaged values of Wh for the whole droplet heating time, when analyzing the results. Before discussing the findings, we need to outline a few important points related to Fig. 4b. No Wh values are specified for schemes No. 1–3 in Fig. 4b at Ta > 600 � C because of the experimental equipment limi­ tations: the system used for the air flow generation can only vary its temperature in the range up to 600 � C. For scheme 4 (muffle furnace), the lack of data for high temperatures is conditioned by too high droplet heating and evaporation rates: at temperatures above 500–600 � C, the droplet heating time until the homogeneous temperature field does not exceed 2–4 s. Moreover, the motion of the droplet fixed on the posi­ tioning mechanism from the entrance of the muffle furnace to a com­ plete stop takes about 1.5 s. When moving and right after stopping, the droplet sways preventing us from recording its exact radius. For this reason, the values of Wh at Ta > 600 � C for scheme No. 4 are not there either. There are also limitations of the experimental setup with a freefalling droplet (scheme 5) as to the heating channel length. A droplet only heats up when falling through such a channel if the heating tem­ perature is very high (Ta > 600 � C). However, if we use lower temper­ atures in the muffle furnace, the droplet temperature does not change much relative to its initial value—from 0.5 � C to 3.5 � C with a temper­ ature variation in the range of Ta ¼ 100–500 � C. The temperature var­ iations obtained are comparable with the systematic error of the PLIF technique (2 � C). Thus, the calculation of Wh for scheme 5 at Ta < 500 � C does not seem possible with high accuracy (the error exceeds 50%). Based on experimental and theoretical analysis results of [16, 40–42], we drew a conclusion about the diffusion and thermal Mar­ angoni flow in the experimental conditions under study. There are three factors that decisively affect the fields of rate and temperature under the conditions of incoming air flow: thermal (heating temperature), dy­ namic (incoming flow velocity), and geometric (droplet size). R.S. Vol­ kov et al. [43] and D.V. Antonov et al. [44] show that in the case of droplets with a homogeneous composition (e.g., water), the dynamic factor (air flow velocity) dominates. If a droplet contains several liquid components, the thermal factor (heating temperature) has the greatest effect. Since the research conditions of this paper are close to those of the experiments from Ref. [43], we can draw a conclusion that the findings from Ref. [43] are also true for this study. R.S. Volkov et al. [43] com­ mented on the role of a group of factors, in particular, (1) heat con­ duction; (2) buoyancy (the Rayleigh number influence); (3) surface tension (the Marangoni number influence); (4) evaporation rate and heat of phase transition; (5) velocity of the incoming air flow; (6) change in the diameter of the droplet exposed to evaporation; (7) change in the friction and heat-transfer laws for the system of droplet/air flow, etc. The differences in terms of evaporation rate between fixed and fall­ ing droplets have three main reasons. Let us consider the parameter values based on a droplet with R0d � 1.5 mm. The first one is the

3. Results and discussion 3.1. Temperature fields of droplets and heating rates When processing the experimental results, we captured important patterns of water droplet temperature field generation. These became common for the three recording schemes (i.e., when studying the evaporation of free-falling droplets or those fixed on a holder). Fig. 3 shows the features of temperature field variation. We did not plot temperature fields for the experiments performed according to scheme No. 4, because PLIF measurement was impossible inside the tubular muffle furnace. Kuznetsov et al. [34] measured the temperature using the PLIF technique for a similar scheme under radiative heating. The dif­ ference was in the use of a muffle furnace with a constant volume combustion chamber. To illustrate the temperature distributions in a droplet being heated and evaporating according to scheme No. 4, Fig. 3d gives temperature fields for a similar scheme (using the data from Ref. [34]). Using the data in Table 2, we can calculate droplet heating rates as functions of time and temperature. The results of a series of experiments for various heating rates are presented in Fig. 4. For the numerical es­ timate of the droplet heating rate in the experiments, we used the following equation: � �� Wh ¼ ΔTd Δt ¼ T *d –T 0d Δth : (1) To explain the results obtained (heating and evaporation rates vs. temperature Ta), we calculated the convective (qconv) and radiative (qrad) heat fluxes supplied to the droplet at the stage of its heating and evaporation (equation (2) and (3)): qconv ¼ α⋅ðTa

(2)

Ts Þ;

qrad ¼ σ ⋅ εm ⋅ T 4a

� 4

Ts ;

(3)

where: α ¼ λa∙Nu/(2∙Rd), σ ¼ 5.67 � 10 8 W/(m2� C4); εm ¼ 0.8. Overall, it is clear (Table 2) that the evaporation rates depend on the total heat flux (qsum) supplied to the droplet. In particular, the calcula­ tions confirm that the greatest heat flux provides the highest heating rate (Scheme 5). When calculating the values of Wh (see Table 2) using equation (1), we took the following values as Δth: for schemes No. 1–4 – times (th) of droplet heating until homogeneous temperature field (when the droplet temperature virtually stops growing, and the value of temperature levels out throughout the droplet volume); for scheme No. 5 – the time (th) when the droplet remained inside the tubular heater of the muffle furnace (time of its motion through it). 25 � C was taken as T0d and the average temperature throughout the droplet volume was used as Td* in the calculations. The average temperature value was chosen for Wh 5

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International Journal of Thermal Sciences 150 (2020) 106227

Fig. 3. Temperature fields of water droplets obtained using the PLIF technique: a – Scheme 1 (T0d � 25 � C; R0d � 1.5 mm; Ta � 500 � C; Ua � 3 m/s); b – Scheme 2 (T0d � 25 � C; R0d � 1.5 mm; Ta � 300 � C; Ua � 3 m/s); c – Scheme 3 (T0d � 25 � C; R0d � 1.5 mm; Ta � 400 � C; Ua � 3 m/s); d – Scheme 4 (data from Ref. [34], T0d � 25 � C; R0d � 1.5 mm; Ta � 500 � C); e – Scheme 5 (T0d � 25 � C; R0d � 2.2 mm; Ta � 1100 � C).

6

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International Journal of Thermal Sciences 150 (2020) 106227

rather little. Using various fixation schemes, we managed to determine the impact of this factor by comparing the temperature fields of a sphere and an ellipsoid of the same volumes. The average heating temperatures differed within the measurement accuracy. Therefore, this factor is negligible at a first approximation. Third, the temperature variation rates of the tentative side, end, and back surfaces of the droplets may only vary noticeably at low heating temperatures (e.g., 100 � C) and high air flow velocities (over 4 m/s). Under other heating conditions, the temperature variation rates are negligible. With free-falling droplets, this factor did not manifest itself because of the continuous rotation and, hence, active internal ther­ mogravitational convection. Fourth, a very important role is commonly assigned to heat sink and heat input to the droplet when using holders made of various materials. As a rule, these are nichrome, steel, aluminum, ceramics, phosphorus, etc. Further, in the criterial processing (section 3.3), we will consider these factors in more detail. The general trend, however, is as follows: in all the experiments, the temperature of the free droplet surface was higher than in any of the droplet sections. This shows that the droplet is heated starting from the free surface. Here, the higher the flow tem­ perature and the lower the velocity, the more even the droplet heating and the corresponding temperature fields. Fifth, the key difference of the falling droplets from those fixed on a holder was in the continuous deformation of the moving droplet. This process increased the scattering of experimental data on the liquid droplet dimensions. However, the trend of a decrease in the average size due to evaporation remained unchanged. Overall, the schemes with or without holders can be used to adequately measure the liquid droplet evaporation rates. The results can be adjusted using the data on the impact of holders and fixation schemes given in this study.

Table 2 Droplet temperatures as well as heating times and rates for the five experimental schemes (Fig. 1): T0d � 25 � C; R0d � 1.5 mm; Ua � 3 m/s; U0d � 2 m/s. Values for schemes No. 1–4 correspond to the time when the droplet temperature field becomes virtually quasi-steady; the parameter values for scheme No. 5 are given for the time of droplet travel through the muffle furnace. The droplet heating mechanism is given in bracket. Scheme

Ta � 100 � C Td (� C) Ts (� C) th (s) Wh (� C/s) qconv (kW/ m2) qrad (kW/m2) qsum ¼ qconv þ qrad (kW/m2) Ta � 600 � C Td (� C) Ts (� C) th (s) Wh (� C/s) qconv (kW/ m2) qrad (kW/m2) qsum ¼ qconv þ qrad (kW/m2) Ta � 1100 � C Td (� C) Ts (� C) th (s) Wh (� C/s) qconv (kW/ m2) qrad (kW/m2) qsum ¼ qconv þ qrad (kW/m2)

No. 1 (Conv.)

No. 2 (Conv.)

No. 3 (Conv.)

No. 4 (Conv.þ Rad.)

No. 5 (Conv.þ Rad.)

37 41 23 0.6 15.8

54 58 47 0.6 11.2

49 60 56 0.4 10.6

36 42 29 0.4 0.7

– – – – –

– 15.8

– 11.2

– 10.6

0.4 1.1

– –

74 83 4 12.3 189.5

79 85 4.2 12.9 186.4

72 89 4.5 13.4 184.9

80 84 5.5 10 10.2

28 67 0.225 35.6 195.2

– 189.5

– 186.4

– 184.9

25.7 35.9

25.8 221

– – – – –

– – – – –

– – – – –

– – – – –

42 87 0.238 92.4 526.8

– –

– –

– –

– –

160.7 687.5

3.2. Evaporation rates Fig. 5 shows instantaneous (Fig. 5a, on the right) and average (Fig. 5b,c) values of water droplet evaporation rates obtained experi­ mentally for the five schemes considered in this study. It also gives the generalized experimental results [24] in comparison with several known models [25]. To determine We (Fig. 5), we used the following equation [24]: �� We ¼ ρd R0d –R*d Δtd : (4)

difference in the evaporation and heating surface area: Sd � 26.2 mm2 for schemes 1, 4; Sd � 21.7 mm2 for schemes 2, 3; and Sd � 28.3 mm2 for scheme. The second one is that the calculation time of the average evaporation rate differs a lot: t ¼ 13–140 s for schemes 1–4; t < 0.25 s for scheme 5. Hence, the heating rates were also significantly different (see Fig. 4b): Wh ¼ 1–90 � C/s. Under identical conditions, the measurements of free-falling droplets are hard to perform, since the droplets need to remain in a gaseous medium for several seconds. Such conditions are possible if droplets fall through very long channels. The third one is the droplet shape. When falling (scheme No. 5), it is more spherical than when suspended on a holder (schemes No. 1, 4) or when squeezed from a tube (schemes No. 2, 3). Here is an overview of the key experimental results as well as the corresponding effects. First, the temperature fields of liquid droplets are highly heterogeneous at the heating stage (Fig. 3). Depending on the gas temperature, this stage may last from several seconds to several dozens of seconds. Temperature variations between the central part and the near-surface layers of a droplet may reach several dozens of degrees. The larger the droplet and the greater the heat flux supplied to it, the more noticeable the effect. At the same time, the heating mechanism has a significant impact, in particular the heat supply direction. Only after the thermogravitational convection in the droplet is enhanced does the temperature equalize in different droplet sections. In this case, the dif­ ferences may be several degrees, which is within the temperature measurement accuracy. Second, the impact of the droplets being non-spherical (for instance, ellipsoids, flat disks, dumbbells, parachutes, etc.) on their heating rate is

Scheme 5 suggests that a droplet is discharged into a heated medium, which means that it moves down under gravity rather than rests on the holder, like in Schemes 1–4. The results of calculating the heat exchange coefficient show that a droplet moving in the flow is heated more rapidly than the one that stands still. The higher the heating temperature, the more significant the difference in heating and evaporation rates in these cases. Therefore, at 600 � C, the evaporation rates of water in Scheme 5 are severalfold higher than those in other schemes. For Schemes 1–4, the time of complete droplet evaporation was taken as the Δtd interval. For Scheme 5, it was the time of droplet travel through the inner part of the tubular heater (Fig. 1c). For the satisfactory agreement between the theoretical and experimental results, when calculating the evaporation rates using equation (4), the droplet needs to be spherical. In this case, a 2D image of a droplet may be used to calculate We from its size variation over time. Fig. 5a gives an example of a variation in the size of an evaporating droplet (on the left), as well as the corresponding instantaneous mass evaporation rates (calculated using equation (4)) for scheme No. 1 (on the right). When calculating We as Δtd, we took the time interval of 1 s. We could observe significant oscillations of We over time. That is why further we are going to discuss the mass evaporation rates of droplets average for the whole evapora­ tion period (Fig. 5b,c). Fig. 5c shows the mass evaporation rates corresponding to the ex­ periments from Ref. [25]. Unfortunately, Yuen et al. [25] do not give 7

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International Journal of Thermal Sciences 150 (2020) 106227

Fig. 4. a – Example of variation in the average temperature inside the droplet while being heated and evaporating, as well as the corresponding instantaneous heating rates (calculated using equation (1)) for scheme No. 1 where Ua � 3 m/s; b – Average droplet heating rates (calculated using equation (1)) for the five schemes considered (T0d � 25 � C; R0d � 1.5 mm): No. 1–3 – Ua � 3 m/s; No. 4 – Ua � 0.1 m/s; No. 5 – U0d � 2 m/s.

their primary experimental results (temperature fields, heating rates, etc.). Thus, we recalculated the values of We corresponding to their experiments [25]. For that we used the Nu numbers from Ref. [25] and an equation for the thermal balance at the surface of an evaporating liquid droplet according to equation (5): We ¼ Nu ⋅ λa ⋅ðTa

numbers when Re > 200 and their equality when Re ¼ 0–200. Moreover, the We values calculated using equation (4) (the method used by Yuen and Renksizbulut) become even higher. This confirms the hypothesis about the importance of considering the actual We values. In Refs. [25, 26], the authors used bronze holders in the form of porous spheres for water droplet generation. In this case, some of the energy supplied to the droplet is spent on heating the holder (tube) and the liquid therein. The authors of [25,26] did not emphasize this factor when making their calculations as they assumed that its role is negligible, as mentioned in Ref. [25]. Fig. 5c shows that when a droplet is small and so is the bronze tubular holder, the energy spent on heating them is low. This is confirmed by the We values in Fig. 5c (the data from Refs. [24,25] is almost equal). Thus, when taking this factor into account, the actual adjusted values of evaporation rates will be higher, which will appar­ ently lead to an acceptable agreement of the results from Refs. [24,25]. Strizhak et al. [45] showed that the holder - liquid droplet contact area has quite a significant impact on the liquid evaporation rates (as illustrated by n-heptane). The impact does not only depend on the holder design but also on the thickness of the rod or wires. In Ref. [45], the authors analyzed a large group of papers to show that variations in the droplet support fiber diameter in the range from 14 μm to 225 μm significantly affects the n-heptane evaporation rates. The main findings are that the cross-fiber technique, which uses 14 μm fiber diameters, induces no

(5)

Ts Þ = ð2 ⋅ Rd ⋅ Qe Þ:

Yuen et al. [25] describe the We calculation procedure taking into account the loss of mass, changes in droplet surface area and droplet-holder contact area (i.e., they considered the heat removal from the droplet to the holder). With this in mind, we also compared the We values using the following equation (6): � � (6) We ¼ Δmd S*d ⋅ Δtd ; where: � Δmd ¼ ρd ⋅4 3⋅π ⋅ R*3 dðnÞ *2 S*d ¼ 2⋅π⋅ R*2 dðnÞ þ Rdðn

R*3 dðn � 1Þ

1Þ

� ;

π⋅ R*dðnÞ þ R*dðn

� 1Þ

⋅doh

Clearly (Fig. 5c), with small droplet dimensions, the We values differ negligibly. However, with large droplets, the We differences in Refs. [24, 25] reach 40–45% with rising temperature (curves 1 and 2 in Fig. 5c). Apparently, this is what causes a significant difference in the Nu 8

G.V. Kuznetsov et al.

International Journal of Thermal Sciences 150 (2020) 106227

Fig. 5. Curves of mass rates of water evaporation We vs. air flow temperature Ta. a is an example of a variation in the current size of an evaporating droplet, as well as the corresponding instantaneous mass evaporation rate (calculated using equation (4)) for scheme No. 1 with Ua � 3 m/s; b shows experimental data from this research (calculated using equation (4)) for the five schemes considered (T0d � 25 � C; R0d � 1.5 mm): No. 1–3 – Ua � 3 m/s; No. 4 – U0d � 0.1 m/s; No. 5 – U0d � 2 m/s; c – data [24, 25]: 1 – values from Ref. [24] calculated using equation (4); 2 – values recalcu­ lated from equation (5) for data from Ref. [26]; 3 – values from Ref. [24] calculated using equation (6);.

noticeable heat transfer to the droplet and consequently does not interfere with the evaporation process. In contrast, the classical fiber technique, which uses relatively larger fibers, greatly enhances the droplet evaporation rate as a consequence of increased conduction heat transfer through the fiber. A correlation is proposed [45] to quantify the level of this increase as a function of ambient temperature and the fiber cross-sectional area. Chauveau et al. [46] made a similar conclusion from a comparison of modeling and experimental results on a setup with a water droplet evaporating on a holder. The model presented in Ref. [46] can determine the contribution of the holder to the droplet heating dynamics.

The curves in Fig. 5 can help us make several key conclusions, important for explaining the patterns of droplet shrinkage and trans­ formation of their surface, as well as modeling of these processes. First, during the experiments, the evaporation rate is changing at all times. This stems from a shift in the heat balance towards heating through thermal conductivity or endothermic phase transitions at the surface of an evaporating and shrinking droplet. Thus, it is possible to record instantaneous minimum, maximum, and average phase transi­ tion rates. This experimental information can be effectively used for modeling. The values may vary several-fold even at the same heating temperature. The main impact comes from two processes: (i) a droplet is 9

G.V. Kuznetsov et al.

International Journal of Thermal Sciences 150 (2020) 106227

heated with a transition from a highly heterogeneous temperature field, with maximum temperatures at the surface and minimum in the depth of the droplet, to a quasi-steady one; (ii) a droplet shrinks due to evapo­ ration and a buffer vapor zone is formed between the gas flow and the droplet surface, which reduces heat flux reaching the droplet. The smaller the droplet, the more noticeable the second factor, whereas the first one may play a major role for larger droplets. Secondly, droplet evaporation nonlinearly accelerates in the course of heating and, after a certain period, the rate starts fluctuating within a limited range (usually no more than 5–10% relative to the mean value). Over this period, a quasi-steady temperature field is generated. The greater heat flux is supplied to the droplet surface, the faster the evap­ oration rate reaches this limited variation range. For free-falling drop­ lets, this range of experimental data scattering is somewhat wider due to their surface transformation and the relative velocity and size of droplets changing in the course of their movement and evaporation. An impor­ tant factor is that the greater the relative droplet velocity in a gas, the more vapors are entrained in the trace of the droplet. It means that the buffer layer around the droplet has an uneven thickness. The latter differs significantly in the trace and in the front of the droplet in the course of its accelerated movement. This may be the reason why in the experiments with high relative droplet velocities, the droplets were heated to the quasi-steady temperature field throughout their volume faster than in a motionless gas. The higher the droplet surface temper­ ature, the higher the evaporation rates. The latter increases the energy spent on the endothermic phase transition. This reduces the heat flux reaching the deep layers of the droplet. As a result, its temperature field becomes virtually quasi-steady. Third, the evaporation rates of free-falling droplets can be adequately predicted using the results of experiments, in which droplets were fixed on holders. The results given in this paper show that adjustment coefficients are necessary, which can be derived from curves in Fig. 5. It is important to classify all the factors into primary and secondary ones. At a first approximation, the contribution of the primary factors can be factored in the adjustment coefficients. Fourth, the most adequate correlation between the simulated and experimental characteristics of evaporating liquid droplets can be ob­ tained using a detailed model. This model should take into account the highly nonlinear (exponential) relationship of the vaporization rate vs. droplet surface and gas temperature, buffer vapor layer around the droplet, as well as the convective, conductive, and radiative energy supply to the droplet. When analyzing the evaporation rates obtained in this study and comparing them with the data from Ref. [24], we can see the acceptable agreement only with the same comprehensive model. Therefore, the experimental results can be regarded as an additional indicator that the hypothesis formulated in Ref. [24] are valid. The values of We obtained experimentally in this research are compared further in Table 3 with the data of the known models [25–28,42] shown in Ref. [24]. At the same time, it is clear from Table 3 that the We values obtained experimentally in this research are in the best agreement with the calculated data obtained using the model described by Vysoko­ mornaya et al. [47] (evaporation model including parameters: conduc­ tive, convective and radiation heat transfer, vapor heating, nonlinear

temperature dependence of droplet evaporation rate). Fifth, the heating scheme has a certain impact on the evaporation rate. For instance, Fig. 5b shows that the We values for Schemes 1–4 are almost equal in the temperature range of 100–300 � C. However, in the 300–600 � C range, the We deviation reaches 25%. The maximum evaporation rates are observed in Scheme 1, and the minimum ones within this temperature range, in Scheme 4. Moreover, if a droplet is squeezed through a narrow tube (Schemes 2, 3), We is lower than under the same conditions for Scheme 1. With similar heating rates (Fig. 4), this result primarily stems from the differences in the droplet heating time th (Table 2). So, the th values for Schemes 2 and 3 are higher than for Scheme 1: more energy supplied to the droplet is spent on heating the metal tube and the liquid therein. While in the experiments with Schemes 2 and 3 the droplet is still being heated, Scheme 1 provides its rapid evaporation (decrease in Rd*). 3.3. Criterial processing and generalization Using the experimental results from Ref. [24], we derived the pa­ rameters of liquid droplet heating and evaporation in the form of empirical correlations Nu ¼ f(Re, Pr) from the experimental evaporation rates (We) of water droplets in a high-temperature gas. Then we compared the parameters obtained with the data based on the long-term hypotheses and ideas by Fuchs, Ranz, Marshall, Yuen, and Renksizbulut [25–29] on the physics of droplet evaporation. We plotted several curves of Nusselt numbers vs. Reynolds and Prandtl numbers in the form of Nu ¼ f(Pr,Re). Such equations are commonly used for analytical and nu­ merical studies of heat and mass transfer and endothermic phase tran­ sitions at the liquid – gas interface. We show that a group of equations for calculating the Nu numbers should be used to describe the heat and mass transfer during high-temperature evaporation of liquid droplets for various ranges of Re numbers. Here it is important to take into account the mass rate of liquid evaporation. Its values depend on both the gas temperatures and features of the experimental methods and setup. Below are the main equations in line with the classical concepts from Refs. [25–32]: Nu ¼ 0:37⋅Re0:6 ;

(7)

Nu ¼ 2 þ 0:6⋅Re1=2 ⋅Pr 1=3 ;

(8)

Nu ¼ 2 þ 0:69⋅Re1=2 ⋅Pr 1=3 ;

(9)

�. Nu ¼ 2 þ 0:6⋅Re1=2 ⋅Pr 1=3 ð1 þ BÞ1:4 ;

(10)

where: (7) (8) (9) (10)

is McAdams’ ratio [30]; is Ranz and Marshall’s ratio [29]; is the adjusted equation from Ref. [31]; is the adjusted equation proposed by Renksizbulut and Yuen [25–27].

Table 3 Values of We obtained experimentally in this research compared with the data of known models [25–28,47]. Data Diffusion model of evaporation [28] Phase transformation model [25–27] Model [47] Experiments of this work Scheme No. 1 Scheme No. 2 Scheme No. 3 Scheme No. 4 Scheme No. 5

Ta (� C) 300

400

500

600

800

1100

0.0105 0.0111 0.0182 0.0151 0.00142 0.0139 0.0134 –

0.0141 0.0194 0.0263 0.0242 0.0229 0.0218 0.0206 –

0.0153 0.0249 0.0384 0.0359 0.0343 0.0294 0.0296 –

0.0158 0.0298 0.0577 0.0545 0.0478 0.0445 0.0401 0.1228

0.0173 0.0385 0.2193 – – – – 0.1422

– – – – – – – 0.1807

10

G.V. Kuznetsov et al.

International Journal of Thermal Sciences 150 (2020) 106227

The Re and Pr numbers are traditionally calculated using the following equations [32]: Re ¼ 2⋅Ua ⋅Rd =νa ;

(11)

Pr ¼ νa ⋅ρa ⋅Ca =λa :

(12)

temperatures Ta. In most of the experiments, a droplet either rests in a moving air flow, or, on the contrary, falls under gravity. This leads to high concentrations of water vapors being only in the droplet trace [48]. The known modeling results [47] show that the vapor concentration field around an evaporating droplet in motion is highly inhomogeneous. This parameter ranges from 0.1 to 0.5 at Ud � 0.5 m/s. Thus, a rather narrow gas-vapor trace (its width equaling the droplet diameter) will be observed at a relative droplet velocity in the flow about 3 m/s [48]. The vapor concentration in the droplet trace at a 1-mm distance from its surface will change in the range of 0.4–0.5, whereas on the side facing the incoming air flow and on the lateral sides of the droplet, the vapor concentration will be less than 0.1. Thus, the calculated average con­ centration of vapors around the evaporating droplet will be just over 0.1. In order to take into account the presence of vapor around the droplet when calculating Re and Nu, the authors proceeded from the assumption that the gas-vapor mixture consisted of 90% of air and 10% of water vapors. The Pr numbers were determined considering that the properties of substances depended on temperature. Ts and We were derived from the experimental results from Ref. [24]. The variable parameters were altered in the following range: Rd ¼ 0.5–2 mm; Ua ¼ 1–3 m/s; Ta ¼ 100–600 � C. It is clear from Fig. 6 that curve 2 (Ranz and Marshall’s ratio [29]) is in good agreement with the Nu numbers (curve 6) derived from the experimental values of We when Re ¼ 0–100. The values for curve 3 are calculated using an equation proposed in Ref. [31]. The curve is in acceptable agreement with the Nu numbers on curve 6 when Re ¼ 0–200. When Re > 200, the experimental values of the Nu numbers (curve 6) exceed those obtained using the classical concepts [25–31]. The adjustment coefficient (1 þ B)1.4 proposed by Yuen and Renksizbulut [25–29] does not bring the Nu numbers closer to the experimental ones, since it is in the denominator and understates the corresponding Nu values (Fig. 6). Overall, the differences of Nu numbers as functions of Re between curves 1–4 (Fig. 6) do not exceed 15%. Therefore, for further analysis we will use the Nu numbers obtained using equation (10) proposed by Yuen and Renksizbulut [25–29]. Unlike equations (7)–(9), this approach indirectly considers the main parameters of an evaporating droplet by means of the B values. Apparently, the difference between the Nu numbers calculated using various approaches stems from the accepted assumptions and limitations of the experimental methods and setups used in Refs. [24–26]. However, Yuen and Renksizbulut’s ratio [25] can be used to obtain reliable Nu ¼ f(Re) functions corresponding to the Nu numbers derived from the mass rate of water evaporation (We) [24]. For

When calculating the values of qconv (Table 2), we used equation (9) to determine Nu in schemes No. 1–3, 5 and equation (7) for scheme 4. We made this choice when analyzing the main conclusions from Refs. [25–31]. The correlation obtained by McAdams [30] provides a good description of the process under natural convection [25,26], which corresponds to scheme No. 4. For the forced convection conditions (in line with schemes No. 1–3, 5), Fuchs [28], Ranz and Marshall [29], Rowe, Claxton and Lewis [31] recommend equation (8). According to Refs. [25–27], equations (7)–(10) are good for describing vapor mass transfer as well as heat and mass transfer at the droplet surface when Re < 2000. The coefficients and powers of the Re and Pr numbers are determined empirically and based on the mass rate of liquid evaporation We. For instance, Yuen et al. [25] emphasize that equation (8) can be used for the We values under 0.01 kg/(m2s), whereas equation (9) is good for high We (over 0.1 kg/(m2s)). Moreover, there are several criterial equations linking the Nu, Re, and Pr numbers for a single droplet [33] and differing in both their form and the values of the corresponding coefficients and powers. Renksizbulut and Yuen [25–27] showed that the Nu numbers calcu­ lated using equation (8) are overstated vs. the experimental values ob­ tained by these authors. Therefore, the authors of [25–27] justify the necessity to introduce an adjustment coefficient (1 þ B)1.4. This makes it possible to narrow down the range of Nusselt numbers using equation (10): Nu ¼ f(Pr,Re,B). The values of B (mass-transfer number) are determined empirically. Table 4 gives the values of the mass-transfer numbers. On the other hand, the Nu number may be determined using the equation for the balance of heat at the surface of an evaporating liquid droplet under convective and radiative heating in a gas [33], with neglect of droplet heating: � Qe ⋅ We ¼ α ⋅ ðTa Ts Þ þ σ ⋅ ε⋅ T 4a T 4s ; (13) where: �

�

�

α ¼ λa ⋅Nu ð2⋅Rd Þ; ​ σ ¼ 5:67⋅10 8 W m2 ​ K4 ; ​ ε ¼ 0:8: For convective heating (data from Ref. [24]), the summand σ∙ε∙ (T4a -T4s ) was taken as equal to zero. However, in the experiments with droplet heating in a muffle furnace, the contribution of this summand is much more than the convective one due to the radiation of the ceramic tube. Similar conclusions can be drawn from the analysis of droplet heating conditions, for instance, in the flame or heated combustion products flow (flue gases). Under the conditions of convective heating, we obtain the equation linking the Nu number with the mass rate of water evaporation We for [24]: Nu ¼ 2 ⋅ Rd ⋅ Qe ⋅We = ðλa ⋅ ðTa

(14)

Ts ÞÞ:

Fig. 6 gives a group of Nu ¼ f(Re) curves obtained using equations (7)–(10), (14). When calculating the Nu and Re numbers, we chose the values of νa, ρa, Ca, and λa on the basis of the air and water vapor flow Table 4 Values of coefficient numbers) for water.

B

(mass-transfer

Ta (� C)

Data [25]

100 200 400 600

0.028 0.064 0.14 0.23

Fig. 6. Nu ¼ f(Re) curves obtained using equations (7)–(10), (14): 1 – calcu­ lated using equation (7); 2 – calculated using equation (8); 3 – calculated using equation (9); 4 – calculated using equation (10), values of coefficient B were taken from Ref. [25]; 5 – derived from We using equation (14). 11

International Journal of Thermal Sciences 150 (2020) 106227

G.V. Kuznetsov et al.

that, just as in the equations given in Ref. [33], we can determine the corresponding powers for the Re number in equation (10). Thus, we established that the following equation should be used: for curve 1 (Fig. 7): �. Nu ¼ 2 þ 0:6⋅Re0:55 ⋅Pr 1=3 ð1 þ BÞ1:4 : (15)

temperature variation near the droplet surface, the more intense its heating is. Thus, we obtained an equation for an adjustment coefficient (1-Ts/Ta)Ts/Tb that brings the analytical estimates much closer to the experimental data. Fig. 8 gives the Nu ¼ f(Re) curves for three cases: experimental re­ sults [24]; calculations based on Ranz and Marshall’s ratio [29]; and values derived from the adjusted Ranz and Marshall’s ratio – equation (14). Ratio (16) gives a good agreement of the Nu numbers with the experimental values from Ref. [24]. Thus, equation (16) factors in both the gas medium parameters and the thermophysical characteristics of the evaporating liquid. It can be used for modeling the processes of heat and mass exchange as a criterial heat and mass transfer equation for a single evaporating droplet. To estimate the validity range of the equation obtained (16), we plotted the Nu ¼ f(Re) curves for Re < 20. This range corresponds to the radiative heating of a droplet. To calculate the Nu numbers, we used the data from this study obtained for Scheme 4. The Nu numbers were calculated using equation (13): �� � Nu ¼ 2 ⋅ Rd ⋅ Qe ⋅ We ​ σ ⋅ ε ⋅ T 4a T 4s ðλa ⋅ ðTa Ts ÞÞ: (17)

Fig. 7 gives the adjusted Nu ¼ f(Re) curves. The Nu values obtained are in quite a good agreement with each other. However, it is clear that using such an approach does not provide the absolute agreement throughout the Re number range considered. Apparently, this would require a comprehensive recalculation of all the coefficients and powers in equation (10). This approach is sophisticated and has no link to the physics of the processes involved. Equation (15) do allow Yuen and Renksizbulut’s [22–26] formula adjustment but still do not fully consider the physics of the process. The choice of powers and coefficients does not explain the physical nature of processes taking place in the near-surface gas-vapor and liquid layers but only gives a satisfactory correlation of the Nu and Re numbers for specific experiments (in this case [24]). Overall, the approach proposed by Yuen and Renksizbulut [25–29] usually provokes discussion and should be supplemented by a reasonable choice of values for empirical coefficients and powers in equation (10). Most parameters in the for­ mulas for Nu, Re, and Pr numbers depend on the ambient temperature either directly or indirectly. The values of mass evaporation rate, in turn, depend mostly on the liquid boiling temperatures and the evaporation surface temperature. Thus, to adjust Ranz and Marshall’s ratio [29] (equation (8)), we can use the following equation: �. s=Tb Nu ¼ 2 þ 0:6⋅Re1=2 ⋅Pr 1=3 ð1 Ts =Ta ÞT : (16) The physical significance of the adjustment coefficients in the equation for the Nusselt number (16) is that at various points of time during droplet heating, its surface temperature Ts differs greatly. At the first stage, the values of Ts increase significantly and then this increase slows down and the values of Ts become closer to asymptotes. In this research, we have established a correlation between the asymptotic values of the droplet surface temperature Ts and the heating temperature Ta. According to the mass transfer theory, energy accumulation due to the heat capacity and evaporation plays a major role at the droplet surface. That is why, it is important to account for Tb when analyzing the correlation between Ts and Ta. The maximum droplet surface tempera­ ture Ts equals the boiling temperature Tb. After processing the experi­ mental data, we established that the calculation of the heat transfer coefficient requires the current values of Ts, since the higher the

The use of equation (16) gives a good data correlation in the range of 2 < Re < 5 (Fig. 9). At the same time, when Re ¼ 5–18, the Nu numbers obtained using equations (16) and (17) differ by up to 20%. This result stems from a significant contribution of the radiative component to droplet heating and evaporation. Equation (16) does not factor this in, since it is only formulated for convective heat exchange. Adding the corresponding summand to (17) significantly reduces the difference between the theoretical and experimental values. In the future, it is advisable to experiment with liquids of significantly different thermo­ physical properties, boiling temperature, and vaporization heat to esti­ mate the validity conditions of the approach proposed. Such large-scale experiments have not been conducted yet, because there are no methods to reliably determine liquid droplet temperatures in experiments. Specialized dyes (fluorophores) can help make such measurements in a wide range of the heating temperatures using the PLIF technique. There are dye compositions for water and several combustible liquids (the main ones are discussed in Refs. [7–10,35,49]). At the same time, a number of dyes dissolve in some liquids and laminate when heated and cooled [39]. It is important to consider the corresponding patterns when planning the experiments. The reliable estimate of a droplet surface temperature largely affects the accuracy of its heating and evaporation rate calculation and there is almost no experimental data for the tem­ peratures over 100 � C. Therefore, this research area is one of the most appealing to the world scientific community in the coming years.

Fig. 7. Nu ¼ f(Re) curves obtained using equations (14) and (15): 1 – calcu­ lated using equation (15), values of coefficient B were taken from Ref. [25]; 2 – derived from experimental values of We using equation (14).

Fig. 8. Nu ¼ f(Re) curves from convective heating: 1 – classical Ranz and Marshall’s ratio [29] (calculated using equation (8)); 2 – adjusted Ranz and Marshall’s ratio [29] (determined using equation (16)); 3 – derived from We using equation (14) (experimental data from Ref. [24]). 12

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International Journal of Thermal Sciences 150 (2020) 106227

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Fig. 9. Nu ¼ f(Re) curves when radiative heating dominates: 1 – classical Ranz and Marshall’s ratio [29] (calculated using equation (8)); 2 – adjusted Ranz and Marshall’s ratio [29] (determined using equation (16)); 3 – derived from We using equation (17) (experimental data from this research for Scheme 4).

The results obtained from the criterial processing of the experimental research findings are of great interest for actual practical applications [34]. They are also useful for the development of comprehensive ap­ proaches to the criterial processing of experimental and modeling results using empirical equations, for instance Ref. [49]. Moreover, they can be used for developing models of high-temperature liquid evaporation [46–52] for the most promising fuel and heat and mass transfer tech­ nologies. In particular, due to the generalized findings for the five different heating schemes and varying the values of heat fluxes in wide ranges, it is possible to predict the high-temperature heating and evaporation rates of different liquid droplets. This study shows that the corresponding estimates can be performed adequately under the pre­ vailing convective and radiative heat fluxes (hence, when Reynolds numbers vary more than 100-fold and the heating temperatures, several-fold). 4. Conclusion (i) We established the liquid heating and full evaporation time as well as temperatures of heated water droplets (Td) as functions of the gas medium temperature (Ta). After varying a set of the main experimental parameters, we calculated the functions of water heating and evaporation rates. Under high-temperature heating, the values of these parameters may be 1.3–1.5 times as high as the theoretical and experimental data by other authors. This happens because a reliable calculation of the mass evaporation rates takes into account the energy spent on the heating of the holder and the liquid therein, if hollow tubes are used. (ii) When Re ¼ 0–200, classical equations can be used for a single droplet reflecting the Nu ¼ f(Re,Pr) correlation [22–24,26–28]. For Re > 200, when describing the heat and mass transfer pro­ cesses, it is necessary to consider the actual mass rates of liquid evaporation. (iii) To adjust the values of the Nu numbers derived from the classical Ranz and Marshall’s ratio [26], we proposed an adjustment co­ efficient (1-Ts/Ta)Ts/Tb to factor in the gas medium and evapo­ rating droplet parameters. The equation obtained can be used as a criterial heat and mass transfer equation for an evaporating droplet when predicting the high-temperature heating and evaporation rates. Acknowledgments The research was supported by the Russian Science Foundation 13

G.V. Kuznetsov et al.

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