Diffusion-limited Model Research Articles

Predicting the weathering of fuel and oil spills: A diffusion-limited evaporation model

The majority of the evaporation models currently available in the literature for the prediction of oil spill weathering do not take into account diffusion-limited mass transport and the formation of a concentration gradient in the oil phase. The altered surface concentration of the spill caused by diffusion-limited transport leads to a slower evaporation rate compared to the predictions of diffusion-agnostic evaporation models. The model presented in this study incorporates a diffusive layer in the oil phase and predicts the diffusion-limited evaporation rate. The information required is the composition of the fluid from gas chromatography or alternatively the distillation data. If the density or a single viscosity measurement is available the accuracy of the predictions is higher. Environmental conditions such as water temperature, air pressure and wind velocity are taken into account. The model was tested with synthetic mixtures, petroleum fuels and crude oils with initial viscosities ranging from 2 to 13,000 cSt. The tested temperatures varied from 0 °C to 23.4 °C and wind velocities from 0.3 to 3.8 m/s. The average absolute deviation (AAD) of the diffusion-limited model ranged between 1.62% and 24.87%. In comparison, the AAD of a diffusion-agnostic model ranged between 2.34% and 136.62% against the same tested fluids.

Estimation of Tritium Release from LLCB TBM and Its Ancillary Systems and Tritium Management in Different Locations of ITER

The Indian Lead Lithium Ceramic Breeder (LLCB) Test Blanket Module (TBM) is to be installed in one half of equatorial port #2 for testing in ITER machine. Liquid Pb-Li and solid Li2TiO3 are the tritium breeder materials in LLCB TBM. Tritium permeates through structural materials in particular at higher temperatures, which is a major operational and safety concern. Therefore, tritium flows in different locations of ITER Tokamak complex have been estimated.Tritium transport from LLCB TBM and its ancillary systems into process rooms has been studied and analyzed in this work. A steady state diffusion limited permeation model neglecting surface effects has been used for the analysis. Tritium permeation to the Vacuum Vessel, Pipe Forest Area, Port Cell, Pipe Chase Area, Tokamak Cooling Water System Vault Annex (TCWS-VA) and Tritium Process Room in L-2 level has been estimated.The requirement to be fulfilled in each equatorial port cell is that the tritium concentration in the port cell during maintenance operations should be below the admissible limit for human access (regulatory maximum allowable value < 1 DAC = 3.4 × 105 Bq/m3, Derived Air concentration). The presence of the Detritiation System (DS) in the Port cell has to be taken into account.This admissible limit for human access has to be reached in a sufficiently short time (target = 12 h) after plasma shutdown. Additional release during maintenance and radiological zoning with recommended <10 µSv/h need to be considered. Management of concentration of permeated tritium in different locations considering above requirement has also been discussed in this paper.

Incorporation of engineering knowledge into the modeling process: a local approach

Process modelling is the foundation of developing process controllers for monitoring and improving process/system health. Modelling process behaviours using a pure empirical approach might not be feasible due to limitation in collecting large amount of data. Engineering models provide valuable information about processes’ general behaviours but they might not capture distinct characteristics in the particular process studied. Many recent publications presented various ideas of using limited experimental data to adjust engineering models for making them suitable for certain applications. However, the focuses there are global adjustments, where modification of engineering models impacts the entire model-application region. In practice, some engineering models are only valid in a part of experimental data domain. Moreover, many discrepancies between engineering models and experimental data are in local regions. For example, in a chemical vapour deposition process, at high temperatures a process may be described by a diffusion limited model, while at low temperatures the process may be described by a reaction limited model. To address these problems, this article proposes two approaches for integrating engineering and data models: local model calibration and local model averaging. Through the local model calibration, the discrepancies between engineering’s first-principle models and experimental data are resolved locally based on experts’ feedbacks. To combine models adjusted locally in some regions and also models required little adjustments in other regions, a model averaging procedure based on local kernel weights is proposed. The effectiveness of the proposed method is demonstrated on simulated examples, and compared against a well-known existing global-adjustment method.

Beyond the Model Limit: Parameter Inference Across Scales

In multiscale studies of emergent phenomena, a common approach is to define a microscopic scale generative model and subsequently, by passing to a diffusion limit, to derive a mesoscopic or macroscopic model via a homogenization argument. Microscopic models are often inherently stochastic, while the diffusion limit model may represent distributions or mean field quantities. A canonical example of this type of limiting process is the classical diffusion equation as a limit of Brownian random motion. Being typically based on ordinary or partial differential equations formalism, large scale models are more amenable to inverse problems, where the model parameters are estimated from indirect observations. Due to the intrinsic ill-posed nature of inverse problems, the estimation of the parameters of large scale models may be itself a significant challenge. In some multiscale investigations, the quantities of primary interest are not those characterizing the large scale model, but rather the parameters of the associated microscopic model. The latter are sometimes related to our understanding of the processes that are described by the generative models. In many cases, there is no direct connection between the small scale parameters and the large scale parameters describing the aggregate effects of the underlying processes. Moreover, estimating the unknown parameters of a microscopic, stochastic model from the large scale observations poses a number of challenges, and requires the development of new methodological approaches. In the Bayesian statistical framework, stochastic forward models often lead to intractable likelihood densities, making standard statistical computations a challenge. Such models are addressed by approximate Bayesian computing (ABC). In this paper we propose an alternative approach combining Bayesian statistics and particle techniques for inference beyond the limits of model scales. The viability of the approach is demonstrated on three computed examples describing diffusion, chemical kinetics, and cell culture assay, and a connection with existing ABC methods is discussed.

Modeling the evaporation of sessile multi-component droplets

We extended a mathematical model for the drying of sessile droplets, based on the lubrication approximation, to binary mixture droplets. This extension is relevant for e.g. inkjet printing applications, where ink consisting of several components are used. The extension involves the generalization of an established vapor diffusion-limited evaporation model to multi-component mixtures. The different volatilities of the liquid components generate a composition gradient at the liquid-air interface. The model takes the composition-dependence of the mass density, viscosity, surface tension, mutual diffusion coefficient and thermodynamic activities into account. This leads to a variety of effects ranging from solutal Marangoni flow over deviations from the typical spherical cap shape to an entrapped residual amount of the more volatile component at later stages of the drying. These aspects are discussed in detail on the basis of the numerical results for water-glycerol and water-ethanol droplets. The results show good agreement with experimental findings. Finally, the accuracy of the lubrication approximation is assessed by comparison with a finite element method.

From material design to mechanism study: Nanoscale Ni exsolution on a highly active A-site deficient anode material for solid oxide fuel cells

Nonstoichiometric perovskites with active metal nanoparticles exsolved on the surface have shown promising potential in energy and environmental applications ranging from catalysis to power generation. In this work, a novel Sc-based A-site deficient perovskite material La0.4Sr0.4Sc0.9Ni0.1O3−δ (LSSN) is reported as a highly active anode for intermediate-temperature solid oxide fuel cells. The material is designed using both thermodynamical analysis and ab-initio simulations. The drop in Gibbs free energy for Ni in H2 is substantial in comparison to the other elements, and density functional theory simulations indicate that the segregation of Ni towards the surface is energetically favored. Spherical Ni nanoparticles with well-defined boundaries are exsolved on the surface of LSSN after reduction in hydrogen, and the reduced samples show a high electrochemical catalytic activity in symmetric-cell measurements with an area specific resistance as low as 0.055Ωcm2 at 800°C in humid H2. Insights into the exsolution mechanism are also derived from both experiments and analytical modeling. Experimental observations show a patterned particle distribution, which is consistent with heterogeneous nucleation. The particle size evolution is investigated using the strain, reactant, and diffusion limited analytical models. The mechanistic insights gained here can be broadly applied to design more efficient materials capable of exsolution and to control the nanoparticle growth and coverage.

Modeling the effect of carbon-dioxide gas on cavitation

One of the controlling parameters of the physical and chemical effects produced by acoustic cavitation is the use of dissolved gas as it affects the temperature and pressure obtained at cavity collapse and, the reactions happening in a bubble. It also enhances the nucleation rates by decreasing the threshold required for cavitation by providing dissolved gas nuclei. The present study looks into the effect of carbon dioxide gas on cavitation using a diffusion limited model. The model couples the dynamics of a single bubble with 11 chemical reactions involving 8 reactive species. The effect of mass transport (diffusion of water vapor and radical species) and heat transport (by conduction) is included in the model. Simulations were carried out for different initial compositions of an Ar-CO2− bubble and the results were compared with an experimental study reported in the earlier literature. The results have indicated that intensity of collapse decreases with an increase in CO2 composition in the bubble thereby decreasing the yield of the oxidizing radicals like OH. This is due to the lower polytropic coefficient and higher specific heat of CO2 compared to that of argon. Also, the bubbles grows to a larger extent with an increase in the dissolved CO2 concentration thereby accommodating higher amounts of water vapor and ultimately decreasing the temperature obtained at collapse. Simulations were done for a bubble containing a mole fraction of 95% Ar and 5% CO2 at different values of driving frequencies (213, 355, 647 and 1000kHz) and driving pressure amplitudes (3.22, 5, 7.5 and 10bar). Higher production rate of OH radicals was predicted at a lower driving frequency, for a given driving pressure amplitude and it increased with an increase in the driving pressure amplitude. At a given driving pressure amplitude, the yield of OH radicals decreased with an increase in the CO2 concentration in the bubble for all the driving frequencies used in the simulations.

Diffusion-driven growth of nanowires by low-temperature molecular beam epitaxy

With ZnTe as an example, we use two different methods to unravel the characteristics of the growth of nanowires (NWs) by gold-catalyzed molecular beam epitaxy at low temperature. In the first approach, CdTe insertions have been used as markers, and the nanowires have been characterized by scanning transmission electron microscopy, including geometrical phase analysis and energy dispersive electron spectrometry; the second approach uses scanning electron microscopy and the statistics of the relationship between the length of the tapered nanowires and their base diameter. Axial and radial growth are quantified using a diffusion-limited model adapted to the growth conditions; analytical expressions describe well the relationship between the NW length and the total molecular flux (taking into account the orientation of the effusion cells), and the catalyst-nanowire contact area. A long incubation time is observed. This analysis allows us to assess the evolution of the diffusion lengths on the substrate and along the nanowire sidewalls, as a function of temperature and deviation from stoichiometric flux.

Adsorption of gemini surfactants at the air-water surface

We have studied the behavior of a gemini surfactant, the double chained dodecyl ammonium bromide or 12-2-12, at the surface of aqueous solutions and at room temperature. We have measured the kinetics of adsorption and the equilibrium surface tension. The surfactant has been obtained through a new synthesis method, which has the advantages of being simpler than previous ones and resulting in a product with good purity after the first crystallization. In the absence of salt, 12-2-12 behaves like other charged surfactants. The adsorption is initially diffusion limited, but as adsorption proceeds, and the surface concentration increases, the surface potential reaches large values (of around 200 mV) slowing down considerably the adsorption process. The observations are in line with existing models. The sensitivity of the gemini surfactants to salt is greater than that of single-chain surfactants, owing to their peculiar molecular structure. For single-chain nonionic surfactants or ionic surfactants, adsorption can be described by diffusion limited models. The behavior of 12-2-12 in salt solutions is very different. The very early stages of adsorption are diffusion limited, but very quickly, the behavior changes and strong deviations are observed. Such a behavior could be due to a desorption barrier, the origin of which remains to be elucidated.

Asymmetric collapse by dissolution or melting in a uniform flow.

An advection-diffusion-limited dissolution model of an object being eroded by a two-dimensional potential flow is presented. By taking advantage of the conformal invariance of the model, a numerical method is introduced that tracks the evolution of the object boundary in terms of a time-dependent Laurent series. Simulations of a variety of dissolving objects are shown, which shrink and collapse to a single point in finite time. The simulations reveal a surprising exact relationship, whereby the collapse point is the root of a non-analytic function given in terms of the flow velocity and the Laurent series coefficients describing the initial shape. This result is subsequently derived using residue calculus. The structure of the non-analytic function is examined for three different test cases, and a practical approach to determine the collapse point using a generalized Newton-Raphson root-finding algorithm is outlined. These examples also illustrate the possibility that the model breaks down in finite time prior to complete collapse, due to a topological singularity, as the dissolving boundary overlaps itself rather than breaking up into multiple domains (analogous to droplet pinch-off in fluid mechanics). The model raises fundamental mathematical questions about broken symmetries in finite-time singularities of both continuous and stochastic dynamical systems.

Dissolution process at solid/liquid interface in nanometric metallic multilayers: Molecular dynamics simulations versus diffusion modeling

The dissolution process of Ni into liquid Al in a layered Ni–Al–Ni system is here studied by means of molecular dynamics simulations, using an embedded-atom method type potential. Molecular dynamics simulations allow us to describe the underlying microscopic mechanisms associated with the dissolution process and prove that the number of Ni dissolved is the key variable in following the progress of the reaction occurring in NiAl nanofoils. A diffusion-limited dissolution model has also been developed. This model is tractable enough to use simulation data to extract kinetic and thermodynamic parameters such as solubility, melting temperature and diffusion coefficients. The comparison with experimental data is satisfactory. The present work assesses the predominant role played by diffusion in the dissolution process at the nanoscale in layered systems.

(Keynote) Imaging Electrochemical Growth Using Liquid Cell Transmission Electron Microscopy

Transmission electron microscopy is very helpful in measuring structure and composition after electrochemical growth has finished. However, it is also possible to image structures while they are still growing, by carrying out electrodeposition inside the electron microscope. The trick is to confine the electrolyte in a layer that is thin enough to be transparent to the imaging electrons – of the order of a few hundred nanometers – and to include electrodes with a geometry that allows the reaction to be controlled and visualized. To do this, we use microfabrication techniques to build liquid cells with thin windows of silicon nitride. Once filled with electrolyte, a liquid cell is placed in the microscope and controlled by a potentiostat during imaging. We can then relate changes in morphology to electrochemical parameters, providing a detailed probe of growth mechanisms with good temporal and spatial resolution. In this presentation, we first describe practical aspects of liquid cell electron microscopy, particularly the competing requirements of resolution and diffusion – in other words, the extent to which a liquid layer that is thin enough to provide high resolution images can also replicate a process of interest. We then describe measurements of nucleation and growth during the deposition of metal films, and quantify the onset of growth instabilities. We show movies of the interface evolution of galvanostatically deposited Cu from an acidic electrolyte: initially, the kinetics are consistent with kinetic roughening theory, and later, with diffusion limited growth models. Macroscale models appear to predict the behavior over long length and time scales, but short scale behavior differs from spatially averaged measures, affecting the onset of the instability. We demonstrate some control of interface evolution through pulse plating and inorganic additives. We finally discuss liquid cell microscopy of electroless deposition and of corrosion processes in metal films, and combine metal dissolution with reduction by the electron beam to produce nanostructures. Developments in equipment and modelling are making liquid cell microscopy increasingly quantitative, suggesting exciting possibilities for detailed interrogation of a wider range of electrochemical growth physics.

Reaction rate in an evanescent random walkers system

Diffusion mediated reaction models are particularly ubiquitous in the description of physical, chemical or biological processes. The random walk schema is a useful tool for formulating these models. Recently, evanescent random walk models have received attention in order to include finite lifetime processes. For instance, activated chemical reactions, such as laser photolysis, exhibit a different asymptotic limit when compared with immortal walker models. A diffusion limited reaction model based on a one dimensional continuous time random walk on a lattice with evanescent walkers is presented here. The absorption probability density and the reaction rate are analytically calculated in the Laplace domain. A finite absorption rate is considered, a model usually referred to as imperfect trapping. Short and long time behaviors are analyzed.Received: 20 Novembre 2014, Accepted: 20 March 2015; Edited by: C. A. Condat, G. J. Sibona; DOI: http://dx.doi.org/10.4279/PIP.070003Cite as: M A Re, N C Bustos, Papers in Physics 7, 070003 (2015)This paper, by Miguel A. Re, Natalia C. Bustos , is licensed under the Creative Commons Attribution License 3.0.

Hydrogen kinetics in a-Si:H and a-SiC:H thin films investigated by real-time ERD

Hydrogen effusion from hydrogenated amorphous silicon (a-Si:H) and amorphous silicon carbide (a-Si1−xCx:H) thin films during a temperature ramp between RT and 600°C was studied by in situ real-time elastic recoil detection analysis. Point to point contour maps show the hydrogen depth profile and its evolution with the ramped temperature. This paper proposes a diffusion limited evolution model to study H kinetic properties from total retained H contents recorded in a single ramp. In a compact a-Si:H layer where H predominantly effuses at high temperatures between 500 and 600°C, an activation energy value of ∼1.50eV and a diffusion pre-factor of 0.41×10−4cm2/s were obtained. Applied to an non-stoichiometric a-Si1−xCx:H film in the same range of temperature, the model led to reduced values of activation energy and diffusion prefactor of ∼0.33eV and 0.59×10−11cm2/s, respectively.

Kinetics of hydrogen reduction of titanium-doped molybdenum dioxide

Ti-doped MoO2 was synthesized to broaden the oxygen-to-carbon ratio operating range of MoO2 for partial oxidation of long-chain hydrocarbons by increasing the redox stability. The structure modification causes the hydrogen reduction mechanism to change from three-dimensional nuclei growth with an activation energy of 61.3 kJ mol−1 to a three-dimensional hydrogen diffusion limited model with an activation energy of 317.9 kJ mol−1. Because of the enhanced redox stability, Ti-doped MoO2 has potential as an alternative anode in direct liquid-fed solid oxide fuel cells.

Raman and thermal desorption spectroscopy analyses of amorphous graphite-like carbon films with incorporated xenon

Structural and thermodynamic properties of xenon incorporated in amorphous carbon films deposited by means of a sputtering system free of vacuum pumping during the deposition were investigated by visible Raman (Vis–Raman) scattering and thermal desorption spectroscopy (TDS), respectively. Vis–Raman measurements, carried out before and after the xenon desorption, revealed a polycrystalline material rich in C-sp2 sites that form randomly dispersed nanosized graphite clusters (nanocrystals) of approximately 1 nm. After xenon desorption, a compensating mechanism, activated by the thermal heating, promotes a more ordered C-sp2 network as revealed by the linewidths of both D and G bands, as well as by the ID/IG ratio evolution. The TDS thermograms show that the xenon onset effusing temperature is approximately 120 °C. Besides, they also revealed two different regimes, at low and at high temperatures, associated with desorption of xenon atoms trapped either in an interconnecting void network or within the graphite nanocrystals. From the latter regime, the xenon diffusion free energy (activation energy) was determined to be 1.2 eV (115.7 kJ/mol) on the basis of the diffusion-limited desorption standard model.

Cononsolvency of Water/Methanol Mixtures for PNIPAM and PS-b-PNIPAM: Pathway of Aggregate Formation Investigated Using Time-Resolved SANS

We investigate the cononsolvency effect of poly(N-isopropylacrylamide) (PNIPAM) in mixtures of water and methanol. Two systems are studied: micellar solutions of polystyrene-b-poly(N-isopropylacrylamide) (PS-b-PNIPAM) diblock copolymers and, as a reference, solutions of PNIPAM homopolymers, both at a concentration of 20 mg/mL in D2O. Using a stopped-flow instrument, fully deuterated methanol was rapidly added to these solutions at volume fractions between 10 and 20%. Time-resolved turbidimetry revealed aggregate formation within 10–100 s. The structural changes on mesoscopic length scales were followed by time-resolved small-angle neutron scattering (TR-SANS) with a time resolution of 0.1 s. In both systems, the pathway of the aggregation depends on the content of deuterated methanol; however, it is fundamentally different for homopolymer and diblock copolymer solutions: In the former, very large aggregates (>150 nm) are formed within the dead time of the setup, and a concentration gradient appears at their surface in the late stages. In contrast, the growth of the aggregates in the latter system features different regimes, and the final aggregate size is ∼50 nm, thus much smaller than for the homopolymer. For the diblock copolymer, the time dependence of the aggregate radius can be described by two models: In the initial stage, the diffusion-limited coalescence model describes the data well; however, the resulting coalescence time is unreasonably high. In the late stage, a logarithmic coalescence model based on an energy barrier which is proportional to the aggregate radius is successfully applied.

About precipitates in boron doped Si investigated by dynamical X-ray diffraction

Thickness-dependent Pendellösung oscillations as described in the dynamical theory of diffraction are highly sensitive to strain fields from defects in a host crystal. Based on this, we present a novel approach to determine the precipitation kinetics of oxygen in silicon (Si) at the early stages of clustering at high temperatures. We present in-situ measurements up to 11000C performed with the characteristic Kα1-line at 59.31 keV. The extracted static Debye-Waller factors are evaluated as a function of annealing time within a diffusion limited model of growing spherical precipitates. We investigated moderately p- ([B] ≍ 1015 1/cm3) and highly p+ ([B] ≍ 1018 1/cm3) boron doped Czochralski Si crystals at different nucleation and growth temperatures to determine the nucleation and precipitation kinetics as well as the long time precipitation behavior. At 6500C the diffusion constant found is enhanced compared to the extrapolated value for normal diffusion [1], and it is one order of magnitude lower compared to SIMS data [2]. However, it is close to the value obtained from dislocation unlocking experiments [3]. Moreover, the nucleation rates in p+ material are enhanced at 4500C and 7800C compared to the p- samples. The acceleration at 4500C can be explained with boron enhanced oxygen dimer diffusion, whereas the nucleation rate at 7800C is much too high to be accounted for by the enhanced oxygen dimer diffusivity alone. An analysis of the misfit strain yields a platelet morphology of the precipitates with a higher aspect ratio in the p- than in the p+ case. The long time precipitation behavior at 9000C shows a second growth regime of comparable amplitude in both materials. This can be interpreted as Ostwald ripening and gives access to the surface energy of the precipitates.

Phase Field Modeling of SEI layer morphology in Li-Ion Batteries

Lithium-ion batteries (LIBs) suffer severe performance degradation during their lifetime application because of the undesired chemical reactions, ageing, corrosion, structural integrity compromise, and the threat of thermal runaway. Among those degradation mechanisms, much attention has been focused on LIBs materials decomposition, new components formation and growth due to undesired side reactions. The electrode is covered by a passivation layer named as the solid electrolyte interface (SEI), which is one of dominant LIBs degradation mechanisms. The SEI layer affect the LIBs cyclability, life time, power and rate capability, and even their safety. Growth of the SEI reduces the lifetime of LIBs and increases cell internal resistance.Many researchers have investigated the SEI layer in term of structure, formation and compositions, thickness growth prediction and measurement. The SEI is believed to have multilayered structure, a thin layer of inorganic components close to the anode, and followed by a thick porous organic or polymeric layer close to the electrolyte phase. The composition of SEI layer can be distinguished to include Li2CO3, LiOH, LiF, Li2O, ROCO2Li and RCOLi (1). The SEI on the cross-section is dominated by Li and F and on the basal plane is dominated by organic materials. Heavy organic fragments were found mainly at the surface of the basal SEI. The thickness of the SEI on the basal plane was found to be three to five times smaller than that on the cross-section (2). Compare to the experiment, the simulation work will be more flexible. Yan et al phenomenologically modeled the formation and evolution of the SEI on the graphite electrode. Within the framework of classical nucleation theory (CNT), they qualitatively explained the origin of the two-layer structure of SEI films (3). Broussely et al. proposed their diffusion-limited SEI growth model, then tried to explain the mechanism of lithium loss for LIBs during storage and gave us the rate of lithium loss is proportional to the SEI electronic conductance(4). Kim et al. simulated the effect of electrolytes on the structure and evolution of the SEI (5).Although much work has already been done with the SEI study, the morphology of SEI layer is still not fully understood, especially in term of SEI structure prediction. In this work, we will predict the morphologies of the SEI using the phase field (6-8). The phase field model has been applied in many electrochemical studies to simulation the electrochemical reactions and to model the phenomena as roughness formation and electronically mediated reactions between interfaces in electronic contact. Also the phase field model is more accurate in regions with sharp composition gradients and can treat phase boundaries without the need to explicitly track interfaces. Phase field model can be used as a potentially powerful tool for modeling the structure of the SEI in LIBs.1. F. M. Wang, M. H. Yu, Y. J. Hsiao, Y. Tsai, B. J. Hwang, Y. Y. Wang and C. C. Wan, International Journal of Electrochemical Science, 6, 1014 (2011).2. E. Peled, D. Bar Tow, A. Merson, A. Gladkich, L. Burstein and D. Golodnitsky, Journal of Power Sources, 97–98, 52 (2001).3. J. Yan, B.-J. Xia, Y.-C. Su, X.-Z. Zhou, J. Zhang and X.-G. Zhang, Electrochimica Acta, 53, 7069 (2008).4. M. Broussely, S. Herreyre, P. Biensan, P. Kasztejna, K. Nechev and R. J. Staniewicz, Journal of Power Sources, 97–98, 13 (2001).5. S.-P. Kim, A. C. T. v. Duin and V. B. Shenoy, Journal of Power Sources, 196, 8590 (2011).6. B. C. Han, A. Van der Ven, D. Morgan and G. Ceder, Electrochimica Acta, 49, 4691 (2004).7. J. E. Guyer, W. J. Boettinger, J. A. Warren and G. B. McFadden, Physical Review E, 69, 12 (2004).8. J. E. Guyer, W. J. Boettinger, J. A. Warren and G. B. McFadden, Physical Review E, 69, 13 (2004).

Single and Multicomponent Droplet Models for Spray Applications

An unsteady, spherically symmetric, single component, diffusion controlled gas phase droplet combustion model was developed first, by solving numerically the time dependent equations of energy and species. Results indicated that flame to droplet diameter ratio (flame standoff ratio) increased throughout the droplet burning period, its value being much smaller than that of the quasi-steady case, where it assumes a large constant value. Effects of fuels on important combustion characteristics suggested that combustion parameters were influenced primarily by the fuel boiling point. Droplet mass burning rate variation was smallest for ethanol in comparison with methyl linoleate (biodiesel) and n-heptane. Also, effects of fuels on CO, NO, CO2, and H2O concentrations were determined from the point of view of getting a qualitative trend.For multicomponent spherical combustion of a heptane-dodecane droplet, it was observed that the mass fraction of heptane decreased abruptly to a minimum value as the droplet surface was approached. For a 200μm hexane-decane droplet (at its boiling point), vaporising in conditions of 1 atm and 1000K with Le_l = 10, it was observed that mixing of air and fuel vapour resulted in a higher concentration of hexane at the droplet surface at the end of droplet lifetime thereby altering the vaporisation behaviour. Other conditions remaining same, an increase in Lewis number resulted in a higher mass fraction of hexane being present at the droplet surface. A detailed multicomponent (MC) droplet vaporisation model (diffusion limit model with convection and no internal liquid circulation) was also evolved by numerically solving the transient-diffusive equations of species and energy for a 280 μm (heptane-dodecane) droplet vaporising at 1 atm and 1000 K with Re_σ=100, and Le_l= 10. The present MC model was compared with other existing models and was found to be simpler and quite accurate. The submodels developed in the present work can be implemented in spray analysis.