A quasi-discrete model for heating and evaporation of complex multicomponent hydrocarbon fuel droplets

A quasi-discrete model for heating and evaporation of complex multicomponent hydrocarbon fuel droplets is suggested and tested in Diesel engine-like conditions. The model is based on the assumption that properties of components are weak functions of the number of carbon atoms in the components (n). The components with relatively close n are replaced by the quasi-components with properties calculated as average properties of the a priori defined groups of actual components. Thus the analysis of heating and evaporation of droplets consisting of many components is replaced by the analysis of heating and evaporation of droplets consisting of relatively few quasi-components. In contrast to previously suggested approaches to modelling the heating and evaporation of droplets consisting of many components, the effects of temperature gradient and quasi-component diffusion inside droplets are taken into account. The model is applied to Diesel fuel droplets, approximated as a mixture of 21 components C_nH_2n+2 for 5 ? n ? 25, which correspond to a maximum of 20 quasi-components with average properties for n = n_j and n = n_j+1, where j varies from 5 to 24. It is pointed out that droplet surface temperatures and radii, predicted by a rigorous model taking into account the effect of all 20 quasi-components, are very close to those predicted by the model, using just five quasi-components. Errors due to the assumptions that the droplet thermal conductivity and species diffusivities are infinitely large cannot be ignored in the general case.     

An experimental investigation of sessile droplets evaporation on hydrophilic and hydrophobic heating surface with constant heat flux

Droplet evaporation widely exists in the daily life and industrial production. In most of previous experimental studies, the evaporation of sessile droplets was conducted under a constant substrate temperature condition. However, drops often evaporating on a heating surface under a constant heat flux condition in many practical applications. In this paper, we have carried out an experiment on sessile 3 μl DI water droplets evaporated on hydrophilic and hydrophobic heating surfaces under constant heat flux in the range from 1153 W/m² to 6919 W/m². A high-speed camera was used to record the changing shapes of two sessile droplets on a hydrophilic and a hydrophobic heating surface placed side by side. The droplet height, dynamic contact angle, droplet contact diameter, evaporation mode and evaporation rate are presented.     

Effects of ambient pressure,gas temperature and combustion reaction on droplet evaporation

The effects of ambient pressure, initial gas temperature and combustion reaction on the evaporation of a single fuel droplet and multiple fuel droplets are investigated by means of three-dimensional numerical simulation. The ambient pressure, initial gas temperature and droplets’ mass loading ratio, ML, are varied in the ranges of 0.1–2.0 MPa, 1000–2000 K and 0.027–0.36, respectively, under the condition with or without combustion reaction. The results show that both for the conditions with and without combustion reaction, droplet lifetime increases with increasing the ambient pressure at low initial gas temperature of 1000 K, but decreases at high initial gas temperatures of 1500 K and 2000 K, although the droplet lifetime becomes shorter due to combustion reaction. The increase of ML and the inhomogeneity of droplet distribution due to turbulence generally make the droplet lifetime longer, since the high droplets’ mass loading ratio at local locations causes the decrease of gas temperature and the increase of the evaporated fuel mass fraction towards the vapor surface mass fraction.     

Model for predicting evaporation characteristics of vegetable oils droplets based on their fatty acid composition

In this work, a model for predicting evaporation characteristics (constant of evaporation and evaporation time) of cottonseed oil and diesel fuel has been developed and validated experimentally in the temperature range of 684–917 K under atmospheric pressure.The experimental study is based on the fibre-suspended droplet evaporation technique. The theoretical model for predicting evaporation characteristics is based on the determination of transport properties and thermodynamic properties of different phases of cottonseed oil using the properties of its predominant fatty acids (linoleic, oleic and palmitic). Results show that taking into account convection in the quasi-steady model by the correlation of Ranz and Marshall is enough to give a good prediction of the constant of evaporation of diesel fuel in the studied temperature range. For cottonseed oil, the quasi-steady model gives a good prediction for temperatures from 684 K to 773 K while for temperatures from 773 K to 917 K, it is necessary to take into account the convection and the influence of the heating period of the droplet for a good prediction of the constant of evaporation. For the duration of heating and evaporation time, the model gives a rather good prediction for cottonseed oil for the temperature range from 840 K to 917 K.     

Evaporation characteristics of kerosene droplets with dilute concentrations of ligand-protected aluminum nanoparticles at elevated temperatures

The evaporation characteristics of kerosene droplets containing dilute concentrations (0.1%, 0.5%, and 1.0% by weight) of ligand-protected aluminum (Al) nanoparticles (NPs) suspended on silicon carbide fiber were studied experimentally at different ambient temperatures (400–800 °C) under normal gravity. The evaporation behavior of pure and stabilized kerosene droplets was also examined for comparison. The results show that at relatively low temperatures (400–600 °C), the evaporation behavior of suspended kerosene droplets containing dilute concentrations of Al NPs was similar to that of pure kerosene droplets and exhibited two-stage evaporation following the classical d²-law. However, at relatively high temperatures (700–800 °C), bubble formation and micro-explosions were observed, which were not detected in pure or stabilized kerosene droplets. For all Al NP suspensions, regardless of the concentration, the evaporation rate remained higher than that of pure and stabilized kerosene droplets in the range 400–800 °C. At relatively low temperatures, the evaporation rate increased slightly. However, at relatively high temperatures (700–800 °C), the melting of Al NPs led to substantial enhancement of evaporation. The maximum increase in the evaporation rate (56.7%) was observed for the 0.5% Al NP suspension at 800 °C.     

Experimental study on spreading and evaporation of inkjet printed pico-liter droplet on a heated substrate

In this work, the spreading and evaporation of 2–70 pL droplet (17–50 μm diameter) of water and ethylene glycol jetted by drop-on-demand piezo-driven jetting head on the heated substrate are studied. According to the experimental results, the interfacial oscillation phenomena of water droplet whose Ohnesorge number (Oh) is about 10^?2 is similar to that in inviscid impact driven region, while that of ethylene glycol droplet (Oh ≈10^?1) is similar to that in highly viscous impact driven region followed by capillary driven extra spreading. In addition, various time scales used for nano/micro-liter droplets agree well with the times for interfacial oscillation, viscous damping, extra wetting, and evaporation in pico-liter droplets. In the case of water droplet, the spreading processes end before the evaporation becomes significant. However, in the case of highly viscous ethylene glycol droplet, the extra wetting overlaps the evaporation at high temperature.     

A simplified model for bi-component droplet heating and evaporation

A simplified model for bi-component droplet heating and evaporation is developed and applied for the analysis of the observed average droplet temperatures in a monodisperse spray. The model takes into account all key processes, which take place during this heating and evaporation, including the distribution of temperature and diffusion of liquid species inside the droplet and the effects of the non-unity activity coefficient (ideal and non-ideal models). The effects of recirculation in the moving droplets on heat and mass diffusion within them are taken into account using the effective thermal conductivity and the effective diffusivity models. The previously obtained analytical solution of the transient heat conduction equation inside droplets is incorporated in the numerical code alongside the original analytical solution of the species diffusion equation inside droplets. The predicted time evolution of the average temperatures is shown to be reasonably close to the measured one, especially in the case of pure acetone and acetone-rich mixture droplets. It is shown that the temperatures predicted by the simplified model and the earlier reported vortex model are reasonably close. Also, the temperatures predicted by the ideal and non-ideal models differ by not more than several degrees. This can justify the application of the simplified model with the activity coefficient equal to 1 for the interpretation of the time evolution of temperatures measured with errors more than several degrees.     

New solutions to the species diffusion equation inside droplets in the presence of the moving boundary

Two new solutions to the equation, describing the diffusion of species during multi-component droplet evaporation, are suggested. The first solution is the explicit analytical solution to this equation, while the second one reduces the solution of the differential transient species equation to the solution of the Volterra integral equation of the second kind. Both solutions take into account the effect of the reduction of the droplet radius due to evaporation, assuming that this radius is a linear function of time. The analytical solution has been incorporated into a zero dimensional CFD code and applied to the analysis of a bi-component droplet evaporation. The case of an initial 50% ethanol–50% acetone mixture and droplets with initial diameter equal to 142.7 μm moving in air at atmospheric pressure has been considered. To separate the effect of the moving boundary on the species diffusion equation from a similar effect on the heat conduction equation inside droplets, described earlier, a rather artificial assumption that the droplet temperature is homogeneous and fixed has been made. It has been pointed out that the effect of the moving boundary slows down the increase in the mass fraction of ethanol (the less volatile substance in the mixture) and leads to the acceleration of droplet evaporation.     

Modelling and experiments on vaporization of saline water at low temperatures and reduced pressures

A vapor diffusion model, which takes into account the reduction of droplet temperature during the evaporation process, was used to determine the achievable targets for desalination of seawater at temperatures between 26 °C and 32 °C when the saline water was injected as fine droplets in a low-pressure vaporizer. The temperatures between 26 °C and 32 °C correspond to the warm temperatures of the ocean surface in the tropics. The predictions from the model were verified by a large number of experiments at vacuum pressures between 10 mm and 18 mm mercury. The upper bound of the rate of flow of the saline water in the experiments was 1000 l/h. Typical evaporation time of the droplets was a few hundred milliseconds and this was less than the residence time of the spray provided for in the vaporizer. The yield of fresh water measured in the experiments was between 3% and 4% and matched well with the predictions. Small values of water injection pressures of about 0.1 MPa were found to be adequate when a swirl nozzle, used for garden sprays, was employed. Changes in the height of water injection in the vaporizer did not significantly influence the yield of fresh water.     

Modeling of thermo-physical processes in liquid ceramic precursor droplets injected into a plasma jet

Modeling of liquid ceramic precursor droplets axially injected into a plasma is presented. Droplets undergo heating and solvent vaporization leading to high solute concentration near droplet surface. At a critical solute super-saturation concentration, precipitation is postulated to occur forming a precipitate shell around liquid core. Internal pressurization and rupture of shell occur subsequently. Droplet size, shell porosity and thickness effects were studied. Timescales of internal pressurization and precipitate formation are of the order of microsecond and millisecond, respectively. Small droplets (d ⩽ 5 μm) tend to form thick shells and are less likely to undergo shell fracture compared to larger droplets.     

Experimental and numerical investigation of droplet evaporation under diesel engine conditions

Evaporation of mono-disperse fuel droplets under high temperature and high pressure conditions is investigated. The time-dependent growth of the boundary layer of the droplets and the influence of neighboring droplets are examined analytically. A transient Nusselt number is calculated from numerical data and compared to the quasi-steady correlations available in literature. The analogy between heat and mass transfer is tested considering transient and quasi-steady calculations for the gas phase up to the critical point for a single droplet. The droplet evaporation in a droplet chain is examined numerically. Experimental investigations are performed to examine the influence of neighboring droplets on the drag coefficients. The results are compared with drag coefficient models for single droplets in a temperature range from T = 293–550 K and gas pressure p = 0.1–2 MPa. The experimental data provide basis for model validation in computational fluid dynamics.     

Numerical investigation of the cooling effectiveness of a droplet impinging on a heated surface

Computational fluid dynamics numerical simulations for 2.0 mm water droplets impinging normal onto a flat heated surface under atmospheric conditions are presented and validated against experimental data. The coupled problem of liquid and air flow, heat transfer with the solid wall together with the liquid vaporization process from the droplet’s free surface is predicted using a VOF-based methodology accounting for phase-change. The cooling of the solid wall surface, initially at 120 °C, is predicted by solving simultaneously with the fluid flow and evaporation processes, the heat conduction equation within the solid wall. The range of impact velocities examined was between 1.3 and 3.0 m/s while focus is given to the process during the transitional period of the initial stages of impact prior to liquid deposition. The droplet’s evaporation rate is predicted using a model based on Fick’s law and considers variable physical properties which are a function of the local temperature and composition. Additionally, a kinetic theory model was used to evaluate the importance of thermal non-equilibrium conditions at the liquid–gas interface and which have been found to be negligible for the test cases investigated. The numerical results are compared against experimental data, showing satisfactory agreement. Model predictions for the droplet shape, temperature, flow distribution and vaporised liquid distribution reveal the detailed flow mechanisms that cannot be easily obtained from the experimental observations.     

Interfacial temperature measurements,high-speed visualization and finite-element simulations of droplet impact and evaporation on a solid surface

The objective of this work is to investigate the coupling of fluid dynamics, heat transfer and mass transfer during the impact and evaporation of droplets on a heated solid substrate. A laser-based thermoreflectance method is used to measure the temperature at the solid–liquid interface, with a time and space resolution of 100 μs and 20 μm, respectively. Isopropanol droplets with micro- and nanoliter volumes are considered. A finite-element model is used to simulate the transient fluid dynamics and heat transfer during the droplet deposition process, considering the dynamics of wetting as well as Laplace and Marangoni stresses on the liquid–gas boundary. For cases involving evaporation, the diffusion of vapor in the atmosphere is solved numerically, providing an exact boundary condition for the evaporative flux at the droplet–air interface. High-speed visualizations are performed to provide matching parameters for the wetting model used in the simulations. Numerical and experimental results are compared for the transient heat transfer and the fluid dynamics involved during the droplet deposition. Our results describe and explain temperature oscillations at the drop–substrate interface during the early stages of impact. For the first time, a full simulation of the impact and subsequent evaporation of a drop on a heated surface is performed, and excellent agreement is found with the experimental results. Our results also shed light on the influence of wetting on the heat transfer during evaporation.     

Numerical investigation on the evaporation of droplets depositing on heated surfaces at low Weber numbers

The evaporation of water droplets, impinging with low Weber number and gently depositing on heated surfaces of stainless steel is studied numerically using a combination of fluid flow and heat transfer models. The coupled problem of heat transfer between the surrounding air, the droplet and the wall together with the liquid vaporisation from the droplet’s free surface is predicted using a modified VOF methodology accounting for phase-change and variable liquid properties. The surface cooling during droplet’s evaporation is predicted by solving simultaneously with the fluid flow and heat transfer equations, the heat conduction equation within the solid wall. The droplet’s evaporation rate is predicted using a model from the kinetic theory of gases coupled with the Spalding mass transfer model, for different initial contact angles and substrate’s temperatures, which have been varied between 20–90° and 60–100 °C, respectively. Additionally, results from a simplified and computationally less demanding simulation methodology, accounting only for the heat transfer and vaporisation processes using a time-dependent but pre-described droplet shape while neglecting fluid flow are compared with those from the full solution. The numerical results are compared against experiments for the droplet volume regression, life time and droplet shape change, showing a good agreement.     

Evaporation of droplets into a background gas: Kinetic modelling

A new kinetic model for droplet evaporation into a high pressure background gas, approximated by air, is described. Two regions above the surface of the evaporating droplet are considered. These are the kinetic region, where the analysis is based on the Boltzmann equation, and the hydrodynamic region. It is assumed that the mass fluxes leaving the kinetic region and the corresponding diffusion fluxes in the hydrodynamic region are matched. A modified version of the previously developed method of direct numerical solution of the Boltzmann equation is used. It is assumed that the mass flux leaving the droplet’s surface is the maximal one (evaporation coefficient is equal to 1). The model and numerical algorithm allowed us to calculate the value of the net evaporation coefficient, defined as the ratio of the actual mass flux leaving the kinetic region and the maximal possible mass flux. The values of this coefficient for diesel fuel (approximated by n-dodecane) were shown to be much less than 1 for droplet surface temperatures less than 650 K. For these droplets, the kinetic effects predicted by the new model turned out to be negligible when the contribution of air in the kinetic region was ignored. These effects, however, appear to be noticeable, and larger than those predicted by the approximate analysis, if the contribution of air in the kinetic region is taken into account. It is recommended that the kinetic effects are taken into account when accurate analysis of diesel fuel droplet evaporation is essential.     

Towards a predictive evaporation model for multi-component hydrocarbon droplets at all pressure conditions

In this paper, a new evaporation model for multi-component hydrocarbon droplets is proposed. Compared to previously published models, it has two new features. First, an expression of the Stefan velocity is proposed which ensures gas mass conservation. In addition, the evaporation rate of each species is obtained by the integration of the exact equation of species mass fraction. Second, the heat flux due to species diffusion is taken into account in addition to the classical conduction heat flux between the gas and the liquid droplets. The comprehensive multi-component droplets vaporization model including the above two features is presented for high and low pressure conditions, for which a real and a perfect fluid equation of state (EOS) has been used, respectively. Free convection is also taken into account using the Grashof number in the Kulmala–Vesala correlations [1] for the Sherwood and Nusselt numbers. The model is compared with very accurate experimental data which were recently obtained by Chauveau et al. (2008) [2] at atmospheric pressure and temperature ranges of 473–973 K for n-heptane and 548–623 K for n-decane droplets of 400 μm initial size. A very good agreement with the experimental data including micro-gravity conditions has been obtained. Indeed, the results have confirmed that the free convection process plays a significant role in the evaporation rate of liquid droplets under earth gravity and quiescent conditions. This shows the relevance of the new features of the model. The numerical results have also shown that real fluid EOS is not necessary at atmospheric pressure for the temperature range given above. In addition, the numerical results of the new model are also compared with the experimental data of Birouk (1996) [3] for two-component droplets of n-heptane and n-decane with different compositions of the liquid mixture. Finally, the non-ideality of the mixture is shown to become significant at high ambient pressures and especially at low ambient temperature conditions where a real-gas EOS is needed.     

New approaches to numerical modelling of droplet transient heating and evaporation

New approaches to numerical modelling of droplet heating and evaporation by convection and radiation from the surrounding hot gas are suggested. The finite thermal conductivity of droplets and recirculation in them are taken into account. These approaches are based on the incorporation of new analytical solutions of the heat conduction equation inside the droplets (constant or almost constant h) or replacement of the numerical solution of this equation by the numerical solution of the integral equation (arbitrary h). It is shown that the solution based on the assumption of constant convective heat transfer coefficient is the most computer efficient for implementation into numerical codes. This solution is applied to the first time step, using the initial distribution of temperature inside the droplet. The results of the analytical solution over this time step are used as the initial condition for the second time step etc. This approach is applied to the numerical modelling of fuel droplet heating and evaporation in conditions relevant to diesel engines, but without taking into account the effects of droplet break-up. It is shown to be more effective than the approach based on the numerical solution of the discretised heat conduction equation inside the droplet, and more accurate than the solution based on the parabolic temperature profile model. The relatively small contribution of thermal radiation to droplet heating and evaporation allows us to take it into account using a simplified model, which does not consider the variation of radiation absorption inside droplets.     

Advanced models of fuel droplet heating and evaporation

Recent developments in modelling the heating and evaporation of fuel droplets are reviewed, and unsolved problems are identified. It is noted that modelling transient droplet heating using steady-state correlations for the convective heat transfer coefficient can be misleading. At the initial stage of heating stationary droplets, the well known steady-state result Nu=2 leads to under prediction of the rate of heating, while at the final stage the same result leads to over prediction. The numerical analysis of droplet heating using the effective thermal conductivity model can be based on the analytical solution of the heat conduction equation inside the droplet. This approach was shown to have clear advantages compared with the approach based on the numerical solution of the same equation both from the point of view of accuracy and computer efficiency. When highly accurate calculations are not required, but CPU time economy is essential then the effect of finite thermal conductivity and re-circulation in droplets can be taken into account using the so called parabolic model. For practical applications in computation fluid dynamics (CFD) codes the simplified model for radiative heating, describing the average droplet absorption efficiency factor, appears to be the most useful both from the point of view of accuracy and CPU efficiency. Models describing the effects of multi-component droplets need to be considered when modelling realistic fuel droplet heating and evaporation. However, most of these models are still rather complicated, which limits their wide application in CFD codes. The Distillation Curve Model for multi-component droplets seems to be a reasonable compromise between accuracy and CPU efficiency. The systems of equations describing droplet heating and evaporation and autoignition of fuel vapour/air mixture in individual computational cells are stiff. Establishing hierarchy between these equations, and separate analysis of the equations for fast and slow variables may be a constructive way forward in analysing these systems.     

Water droplet evaporation on Cu-based hydrophobic surfaces with nano- and micro-structures

The characteristics of water droplet evaporation on three different hydrophobic surfaces, PCu (Plain Copper, θ = 115°), MSCu (Micro-Structured Copper, θ = 126°) and NSCuO (Nano-Structured Copper Oxide, θ = 159°) with coating of the same SAM (Self-Assembled Monolayer) material, were experimentally investigated. For industrial heat transfer applications, copper material was used as the substrate, and the simple and cost-effective fabrication technique to prepare the superhydrophobic surface, NSCuO, was introduced. Based on the observations, the behavior of droplet evaporation was divided into three stages: Stage I (constant contact area stage), Stage II (constant contact angle stage) and Stage III (mixed stage). When studying the PCu surface, the Stages I, II, and III were observed, consistent with previous reports. For the MSCu surface, Stages I and III appeared without Stage II, and the pinning period of contact line was the longest among the test samples due to the formation of Wenzel state droplet. In the case of the superhydrophobic NSCuO surface, only Stage III occurred, and the contact line moved freely during the entire evaporation time because of the formation of Cassie state droplet. The total evaporation time of the NSCuO was the longest out of all the samples tested. At the last stage of evaporation, the edge of the droplet shrank at a much faster rate in all surfaces. On the other hand, the shrinking velocity of the droplet height drastically increased only on the NSCuO, which was considered as the unique behavior of superhydrophobic surface. In this experiment, it was found that the surface structure determines the motion of the contact line on the surface, which, in turn, strongly influences the characteristics of the droplet evaporation.     

Experimental study on evaporation characteristics of single and multiple fuel droplets

In this work, evaporation experiments of multiple droplets are carried out in a stagnant hot atmospheric environment (573, 673 and 773 K) using high-speed backlit image technique. Three fuel droplets with nearly same initial diameter are suspended at intersections of two 0.1 mm quartz fibers. The normalized droplet spacing (s/d₀) of three droplets is 2.25. The results show that the evaporation process of single, edge and central fuel droplet containing three stage: initial heating, unsteady evaporation and quasi-steady evaporation stage. Classical d² law is still suitable for edge and central droplet at quasi-steady evaporation stage. The third stage of edge and central droplet accounts for more than 60% of droplet lifetime at low temperatures and about 50% at high temperatures. The evaporation rate constant of edge and central droplet increases and droplet lifetime decreases with increasing ambient temperature. The evaporation time of edge and central droplet at first and third stage is higher than single droplet, but lower than single droplet in the second stage. More importantly, the evaporation interactions between droplets is significant at low temperature. Compared with single droplet, the lifetime of central droplet is increased by 31.8%, 18.6% and 25.9%, respectively.